0PTlMAL APERIODIO SAMPLING FOR SIGNAL REPRESENTATION AND SYSTEM CONTROL Dfissettafion for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY RONALD. LEE VAN WIEREN 1975 This is to certify that the thesis entitled flflf/Imc fl/E/E'xomc 5/4 MFA/M, Foe Jay/u flgrflgeA/Mflol/ Ail/D 57375;“ GOA/7760b presented by Kay Am 455 1/sz 5/195 5” has been accepted towards fulfillment of the requirements for PA: D degree in 537544 SCIENC E Major professor ABSTRACT OPTIMAL APERIODIC SAMPLING FOR SIGNAL REPRESENTATION AND SYSTEM CONTROL BY Ronald Lee Van Wieren There were two investigations conducted in this thesis. First, signal representation through use of non-periodic sampling was investigated in an effort to reduce the sampling errors inherent in periodically sampled piecewise constant approximations of con- tinuous time signals. Second, non-periodic sampling was applied to the feedback control problem by sampling of the continuous time feed- back control error signal. A performance index for the signal representation problem was proposed which measured the errors caused by passing a continuous- time signal through a sample and hold mechanism. A cost for im- plementation was developed and included to form a system performance measure. The cost of implementation measured the costs for computer utilization, software and data storage, and data communication for each of the sampling methods (periodic, aperiodic, and adaptive) used. The Optimal aperiodic sampling criterion was determined by first Obtaining a derived performance index as a function of the sampling interval sequence, (i. e. a sequence of sampling interval lengths) and Ronald Lee Van Wieren the number of sampling intervals being used. This derived performance index was then minimized with respect to the sampling intervals for a particular number of sampling intervals. The Optimal sampling in- te rval sequence was then determined to be the optimal number of samples and the resulting optimal sampling interval sequence which minimized the derived system performance index. A remote display problem was presented as an example for com- paring periodic, adaptive, and sub—optimal aperiodic sampling techniques used to obtain piecewise constant representations of continuous time signal records. The continuous time signal record used in the display problem was generated by a signal model, then stored, and later sam- pled for transmission via a specified communications network to the remote display. Optimal aperiodic sampling was extended from the signal repre- sentation problem and applied to a control implementation problem. The sampling process for measuring the outputs and actuating the control inputs was to be designed. Two system configurations employing control signal sampling we re considered. The first sampling configuration sam- ples both the input and feedback signals while the second samples only the feedback with the input signal being continuously applied to the system plant. The control law was assumed to be designed previously using either classical or modern techniques. Thus, the only design parameters are the number of samples and the sampling interval lengths. A quadratic control performance index is assumed which measures Ronald Lee Van Wieren control energy, tracking error and terminal error. A cost for im- plementation is developed and included to form a system performance index. This cost of implementation measured the computer utilization, data and program storage, and data communication costs for each method of sampling used. The optimal aperiodic sampling criterion was again computed by determining a derived system performance index which was a function of the number Of samples and the length of each sampling interval. The derived system performance index was minimized with respect to the lengths of sampling intervals for a particular number of sampling in- te rvals. The optimal aperiodic sampling criterion was then determined to be the optimal number of samples and the resulting optimal sampling interval sequence which minimized the system performance index. The results of the investigations conducted on optimal sampling for signal representation of continuous time signals resulted in the following findings. The derivedperformance costs for sub-optimal aperiodic sampling in all cases considered yielded lower performance costs than comparable periodic or adaptive sampling methods applied to the same continuous time signal record with an equal number of sampling intervals. The costs for implementation of sub-optimal aperiodic sampling were in general higher than periodic or adaptive sampling as the number of sampling intervals. This increase was due to the increased computational requirements of aperiodic sampling. Investigations of non-pe riodic sampling for system control indicated Ronald Lee Van Wieren that through use of sub-Optimal aperiodic sampling control performance costs could be reduced below periodic sampling costs using an equal number of sampling intervals in each case considered. In several cases control performance costs using aperiodic sampling of slow responding control Systems were reduced below continuous time performance costs over a fixed control interval. Again Obtaining the optimal sampling interval sequence proved costly with respect to computer processing time except if the number of sampling intervals was small. OPTIMAL APERIODIC SAMPLING FOR SIGNAL REPRESENTATION AND SYSTEM CONTROL BY Ronald Lee Van Wieren A DISSERTATION Submitted to Michigan State University in partial fulfillment Of the requirements for the degree of DOCTOR OF PHILOSOPHY Department Of Electrical Engineering and Systems Science 1975 ACKNOWLEDGEMENTS I am deeply grateful to my advisor, Dr. Robert A. Schlueter, for his friendship, constant encouragement, and professional criticisms made during the preparation of this dissertation. For their interest and advice, I also thank the other members Of my committee: Drs. G. L. Park, J. 5. Frame, J. Preminger, and K. Y. Lee. ii TABLE OF CONTENTS Chapter I. INTRODUCTION. . . ....................... II. SAMPLING FOR SIGNAL REPRESENTATION ...... Signal Representation Problem Statement ..... 1. 2. The Derived Signal Representation Per— formance Index ...................... 2. 2. III. OPTIMAL APERIODIC SAMPLING FOR CONTROL . . 3. l. Aperiodic Sampling for Control Problem Statement .......................... 3. 2. The Derived Control Implementation Per- formance Index ...................... IV. EVALUATION OF SAMPLING METHODS ......... 4. 1 Sampling for Signal Representation ......... 4. 2 Signal Representation Performance Results . . . 4. 3 Sampling for Control Implementation ........ 4. 4 Control Implementation Performance Results . . 4. 5 Control Performance Investigation by System Type ............................. 4. 6. Analysis by System Type ................ V. CONCLUSIONS ........................... 5.1. Summary.... ................... 5.2. Future Research ..................... LIST OF REFERENCES. ............... APPENDICES A . The Sample and Hold Operation ........... B. Periodic, Adaptive, and Aperiodic Sampling. . . C. Numerical Data for Figures found in Chapter IV. iii Page 13 22 25 29 34 39 39 45 66 72 85 105 117 117 121 123 128 131 134 LIST OF FIGURES Figure Page 3. 1 Control System Configuration Using a Sampled Feedback Error Signal ............. . ...... 31 3. 2 Control System Configuration Using a Continuous Input Signal and Sampled Feedback ............ 31 4. 1 Block Diagram of the Remote Display System . . . . 43 4. 2 Computer Program Flowchart for Signal Repre- sentation Investigation .................... 44 4. 3 Signal Representation Costs Using a=—l and Step Input ................................ 51 4. 4 Signal Representation Costs Using a=0 and Step Input. ........................ . ...... 52 4. 5 Signal Representation Costs Using a=l and Step Input ................................ 53 4. 6 Representative Sampling Interval Lengths for Figures 4. 3(a) to 4. 5(a) using N25. . . . . . . ..... 54 4. 7 Signal Representation Costs Using a=0 and Ramp Input. 0 O O O O O O O O O O O O O O O O O O O O ..... O O O O O 59 4. 8 Signal Representation Costs Using a=0 and Para- bolic Input ............................ 6O 4. 9 Signal Representation Costs Using azO and Noise Input. O O O 0 O O O O O O O O O ........ O O O O O O O O O O 61 4. 10 Representative Sampling Interval Lengths for Figures 4. 7(a) to 4. 9(a) using N=5. . . . . . . ..... 62 4. 11 Computer Program Flowchart for Control Per- formance Evaluation ............. . ....... 71 4. 12 Performance Costs for a Type 2 FAST Response Time Control System Using a Step Input . . . . . . . . 77 iv Figure 4.13 Performance Costs for a Type 2 MEDIUM Re- sponse Time Control System Using a Step Input. . . Performance Costs for a Type 2 SLOW ReSponse Time Control System Using a Step Input ........ Representative Sampling Interval Lengths for Figures 4.12(a) to 4.14(a) using N:5 . ......... Performance Costs for a Type 2 MEDIUM Response Time Control System Using aRamp Input, Performance Costs for a Type 2 MEDIUM Re- sponse Time Control System Using a Noise Input. . Performance Costs for a Type 1 MEDIUM Re- aponse Time Control System Using a Step Input. . . Performance Costs for a Type 1 MEDIUM Re— sponse Time Control System Using a Ramp Input. . Performance Costs for a Type 1 MEDIUM Re- aponse Time Control System Using a Noise Input. . Performance Costs for a Type 0 MEDIUM Re- sponse Time Control System Using a Step Input. . . Performance Costs for a Type 0 MEDIUM Re- sponse Time Control System Using a Ramp Input. . Performance Costs for a Type 0 MEDIUM Re- sponse Time Control System Using a NOise Input. . Open Loop Plant Responses of the Type 2, l, and 0 Plants for a Unit Step Input ........... . . . . . Sub-Optimal Sampling Interval Sequences, N=5, for Type 2, 1, and 0 Control Systems .......... Type 2, l, and 0 Feedback Control System Re- sponse to a Unity Step Input using Periodic and Sub-Optimal Aperiodic Sampling ..... . . . ..... Optimal Performance Costs by System Type and Input Signal ......................... Page 78 79 82 88 89 94 95 96 100 101 102 106 107 109 113 Figure A-l A-2 Page Signal Sampling using Impulses ............. 128 The Impulse Holding Operation ............. 129 Periodic Sampling ..................... 131 Adaptive Sampling ..................... 132 Aperiodic Sampling ..................... 133 vi LIST OF TABLES 4. 4. .3(a) to 4. 5(a) ...... .6(a) to 4. 8(a) ...... 12(a) to 4.14(a) . . . . 13(a), 4.16m, and 4.17(a) .............................. Table 4. 1 Data Summary for Figures 4. 2 Data Summary for Figures 4. 3 Data Summary for Figures 4. 4 Data Summary for Figures 4. 5 Data Summary for Figures 4. 6 Data Summary for Figures 4. 7 Numerical Data for Figure vii 4o 4. 18(a) to 4.20(a) . . . . 21(a) to 4.23(a) . . . . .25 ............. Page 50 58 75 87 93 99 112 I. INTRODUCTION The subject Of this thesis deals with the development Of a theory for optimal aperiodic sampling1 for both representation of signals and system control. The design Of sampling criteria has always been based on control performance, signal sampling efficiency, cost of implementation, and ease Of design and analysis. Periodic sampling criteria have Often been used because periodic sampled-data single-input single-output control systems can be easily designed using transform techniques. For periodic sampling the sampling rate is generally constrained to satisfy the sampling theorem. As a result, the sampling rate becomes greater than twice the system bandwidth in order to minimize aliasing and thus provide reasonable control performance. The actual choice of a sampling rate is chosen by a tradeoff between control performance and the economic cost for instrumentation and communications hard- ware. The continuous-time controller is implemented if the cost for communicating data is low and computer processing is not required to implement the system design. The design of sampling criteria for multiple -input multiple-output systems is more involved since the bandwidths are generally quite different for each element Of the transfer function. Because Of this, 1See Appendix A for a definition of sampling as used in the context of this thesis. See Appendix B for discussion of periodic, adaptive, and aperiodic sampling. 2 multi-rate and asynchronous sampling criteria [28] have been intro- duced. Adaptive sampling criteria [4-11] have also been developed which vary sampling rate with respect to changes in output or error signals. If the signals used in an adaptive or aperiodic sampling cri- terion accurately represent the system dynamics, where system dy- namics depend on the system state equation, control inputs, disturbance noise, and initial conditions, the sampling criterion can be considered tuned to the system. Tuning the sampling rule to the system dynamics can improve both control performance and sampling efficiency. Most adaptive sampling criteria have been derived using either an integral-absolute-error performance index or an integral-squared- error performance index which measures the error caused by the sample and hold mechanism over one interval. This performance index can be shown to explicitly depend on the system state equations, control inputs, and initial conditions. To insure a non-trivial sampling rule the integral performance index is augmented by a functionwhich is in- versely pr0portional to the length of the sampling intervals. This add- itional term is implied to represent the costs associated with the sam- ple and hold mechanism on a cost per sample per unit time basis. The performance index is frequently simplified by approximating continuous time signals by a Taylor series expansion. The number of terms used to approximate the continuous signals will determine how well the performance index is tuned to the particular signal. Retaining only the first two terms of the Taylor series approximation will de- grade the representation of the continuous signals in the sampling criteria. [24] As a result the cost function no longer explicitly depends on the System state equation, disturbance noise, and control inputs, but instead depends on the signal and its derivatives. Through use of these approximations, integration of the performance index can be performed analytically yielding a performance measure dependent on the length of the sampling interval, the signal, and the derivatives of the signal. Having the analytical solution Of the performance index, the adaptive sampling rule can be obtained by setting the derivative of the performance index with respect to the sampling interval equal to zero and solving for the sampling interval length. As an alternative to minimizing an integral performance measure, Tomovic and Bekey [6] and Tait [11] proposed that sampling occur only when the sampling error exceeds a predetermined level. This approach did not require signal derivatives and insured an error bound [14] . In a paper by Smith [1] , an evaluation Of adaptive sampling com- pared to periodic sampling was undertaken. The adaptive techniques were shown to have a tendency to sample at the sampling interval ex- tremes if constraints on the minimum and maximum sampling interval were imposed. In general, the result of his study indicated that if the input was known a priori, any adaptive sampling rule could out perform periodic sampling. If the input was not known _a_ priori, periodic sampling was generally more desirable. Aperiodic sampling for signal representation is proposed as an alternative to adaptive sampling. In aperiodic sampling the number Of sampling intervals and the lengths of each sampling interval are chosen to minimize the performance index defined over the entire time interval of interest. The choice Of any particular interval length is dependent on the fixed final time and the length of the other sampling intervals. The continuous time system dynamics and associated im- plementation costs1 are incorporated into the performance index thereby influencing the number of sampling intervals used and the length of each interval. A poor choice of sampling intervals will in- crease the performance cost while a good choice of interval lengths will yield a lower performance cost for a given number of samples. Therefore the aperiodic sampling performance index is derived as a function of the sampling interval sequence. This derived performance index can then be minimized with consideration being given to the system dynamics and costs for implementation to yield the optimal sampling interval sequence. Optimal aperiodic sampling can also provide better control performance than a similar non-optimal sampling criterion. In this case, the performance index used must reflect the performance objec- tives for the control system as well as indicate control implemen- tation costs with respect to computer utilization and data communi- cations. Hsia [24] and Smith [ 1] treated the adaptive sampling problem for control as a special case Of the signal representation problem. Therefore a signal representation type performance index was used for system control without regard to system control objec- tives such as: tracking and final state errors and control energy ex- penditures found in most control type performance indices. The assumption made is that if the sampling process minimizes the errors between the signal and its sampled and hold approximation, good con- trol performance will result. This assumption is clearly invalid and lImplementation costs will be discussed in more detail in Chapter II. therefore the performance index used for the control problem should reflect specific control objectives and the relative implementation costs required to achieve the control Objectives. The approximations made to obtain simple analytic expressions for the adaptive sampling performance indices are not used in the evalua— tion of the Optimal aperiodic sampling for control problem. If these ' approximations were made, the performance Of the resulting sampling rule would depend on the accuracy of the approximation to a particular signal and the sampling rule would thus become dependent on the signal and its derivative and independent of the a priori information available on the system state equation, inputs, disturbances, initial conditions, and control law. The first Optimal sampling problem for control [45, 46] was formulated to obtain an optimal sampled data control where both the level of control over a sampling interval and the length of the sampling interval were chosen together to minimize the performance index. The optimal control law was thus specified by an optimal control sequence- optimal sampling interval sequence combination. Necessary conditions were derived, but were not used to obtain an efficient computational algorithm for the optimal control. An algorithm was developed [47] by discretizing the cost functional and state equations and adjoining constraints on the sampling intervals. The resulting nonlinear pro- gramming problem was solved using a sequential unconstrained mini- mization technique. The optimal sampled-data control problem can be solved directly if the Optimal control sequence can be determined as a unique function of the sampling interval sequence chosen. In this case, the performance index can also be derived as a unique function of the sampling interval sequence. The resulting derived performance index can be minimized using a search algorithm to Obtain the optimal sampling interval sequence. The optimal control sequence can then be determined by specifying the optimal sampling interval sequence. This algorithm solves the Optimal sampling problem directly and is more efficient than solving the discrete time problem using the sequential unconstrained minimization technique. Moreover, the errors incurred by a priori discretization are eliminated. This later algorithm was used to determine optimal aperiodic [l9] , state dependent [20] , and adaptive [24] control laws for the regulator problem. These problem formulations neglected system disturbances and costs for implemen- tation and used a fixed number Of sampling intervals for all investiga- tions. Another approach to obtaining Optimal sampling processes for control systems was proposed [3, 40] for the case of periodic sam- pling. This control problem included an additional performance cost proportional to the number of uniformly spaced samples used during the particular control interval. An integer programming algorithm was then used to compute the optimal number of sampling intervals and thus the Optimal periodic sampling rate. It should be noted that in order to optimize the performance index for aperiodic sampling, it is desirable that the system input be known for the time in which the system is to be controlled. This restriction is not as severe as it would seem since in general, applications such as numerical control [41] or linear regulator problems [19-21] the input to the system is knowngm or is constant. Optimal aperiodic sampling of continuous time feedback error signals for system control is proposed. A predetermined continuous time optimal or sub—Optimal feedback control law is used tO generate the feedback error signals. The performance Of the control system is determined by a quadratic performance index which is a function of the tracking errors, control energy, and thus implicitly depends on the disturbance noise, system state equation, and initial conditions. The performance index is augmented by a cost for implementation through sampling which approximates computational costs, data stor- age requirements, and communications costs. The performance index is reformulated as a derived function of the sampling interval sequence and then minimized with respect to the number of sampling intervals and the length of each sampling interval. The Optimal sampling interval sequence Obtained is then used in the sampling of the continuous time feedback error signals in an effort to improve overall system performance while yielding acceptable computational and communications costs. In summary, the hypothesis of this research is that signal rep- resentation performance indices are apprOpriate for data acquisition and signal sampling where control using the sampled signal is not an immediate objective. A control type performance index should be used for designing sampling processes for system control. This is why there are two separate problem formulations found in this work. In addition, each problem formulation will use a cost for implementation model to determine the feasibility of system implementation. Heuristic implementation models have always been used by System designers to determine implementation requirements for various system designs. This research will formulate a generalized cost for implementation model based on computer and communications system utilization. Thus in Chapter II a theory for Optimal stochastic aperiodic sampling for signal representation is formulated. The cost functional is derived and used in conjunction with a cost for implementation based on system computational and communication requirements. In Chapter III the optimal stochastic aperiodic sampled-data tracking problem is formu- lated and derived for various feedback configurations. In Chapter IV various tests are performed and comparisons made in order to evalu- ate the performance of several sampling rules for signal representation and system control. Chapter V summarizes the results, reviews the thesis, and suggests further research. II. SAMPLING FOR SIGNAL REPRESENTATION Communications and control very often use sample and hold mechanisms1 in system design. These systems are usually designed around a minicomputer or microprocessor thus enabling the sample and hold devices to be under direct computer control. Therefore with the availability of the computer, the sample and hold devices can be utilized more efficiently through optimal sampling techniques. The function of a sample and hold device is to approximate a continuous signal over discrete intervals. The accuracy of the signal approximation for periodic sampling is dependent on the sampling rate while for adaptive and aperiodic sampling the accuracy is dependent on the length of each sample interval and the number of intervals. In- creasing the number of samples taken over any given time interval for periodic, aperiodic, or adaptive sampling will generally decrease the error in the signal approximation but will increase computational and data communications expense. The signal representation performance index should measure errors in signal approximations and indicate the relative costs for implementation for each method of sampling under consideration. This structure for the design of optimal sampling criteria also provides a framework for understanding the analytic design of non- optimal adaptive sampling criteria [24] . A large class of adaptive 1See Appendix A. 10 sampling criteria can be derived by minimizing a Taylor series approximation of the cost functionals t. + T k -a2Ti 1 1 i 2 liZ—k— S; [[§_(t) —_s_(ti)[] dt+ale (2.1) Ti 1 1 with respect to the sampling intervals Ti for i: 0, 1, . . . , N-l where _s_(t) is any continuous time output or error signal. The first term in each cost functional is a weighted integral of a function of the error _s_(t)-3&1) caused by the sample and hold operation over (ti’ ti +Ti)' The second term represents the so called “cost for sampling. ” The parameters k k 1' 2’ al’ a 2 are permitted to take on the following value 5 k1€ {-1. 0.1}:k2€{1. Z};al. a2e(0.00) This functional ii penalizes sampling interval lengths with respect to the mean of the integral if k = l, the time-weighed integral if k1 : —l, l or the unweighted integral if k1 = O. The signal s(t) can be approximated by the truncated Taylor series over each sampling interval s(t) 2’ s(ti) + s (ti)(t-ti) t2 t1 and the "cost for sampling" can be approximated by either -a2Ti ~ a1[1- aZTi] aie ~ 2 a1[l -aZTi+(a2Ti) /2] The sampling rules can then be determined by minimizing the resulting approximations to the cost functionals ii with respect to Ti for i=0,1,...,N-1. The adaptive sampling rules obtained by Hsia determine the length of only one sampling interval for a particular set of signal 11 measurements. This one step interval determination was not dependent on system dynamics, control laws, or the statistical description of the initial states and system disturbances. The adaptive sampling rules depended only on the values of the signal and its derivative and provid- ed an analytic rule which minimized an approximation to the initial performance index as indicated in (Z. l). The derived adaptive sampling rules were shown to outperform periodic sampling if the parameters of the sampling rules were adjusted appropriately and the number Of samples to be taken along with the system input were known a prim. The criterion used tO compare sampling rule performance was an integral-absolute-error and thus all sampling rules derived from the generalized performance index (2. 1) could be compared [1] . In contrast to adaptive sampling, Optimal periodic sampling with fixed costs for sampling and a variable number Of samples was inves- tigated [3] . The results indicate that optimization with respect to the number of samples was indeed possible and yielded improved perform- ance for the various system dynamics considered. These previous developments in signal sampling have been con— sidered in the formulation of the Optimal aperiodic signal representation problem. The performance index used in adaptive sampling measured sampling errors under the assumption that good signal representation of the feedback error signals implied good system control performance. This assumption was obviously incorrect because the adaptive sampling performance index measured sampling errors rather than Optimizing system performance Objectives (e. g. minimization of final state errors, tracking errors, etc. ). Therefore two problems are formulated in this 12 thesis; the sampling for control problem found in Chapter III and the signal representation problem found in this chapter. A performance index for optimal aperiodic sampling for signal representation will be developed as a function Of system dynamics, sys- tem inputs, disturbance noise, and costs for implementation. The sys- tem dynamics are obtained from the continuous time signal model used to generate the signal record which is to be sampled. The system dynamics will not be discretized _a priori thus reducing system model- ing errors. In addition the length of each sampling interval will not be optimized on an individual basis but instead the entire sampling interval sequence will be optimized with respect to sampling interval lengths for a fixed number of intervals to obtain a sub-optimal aperiodic sampling criterion. If the performance is then optimized over the number of sampling intervals an optimal sampling criterion for signal representa- tion is Obtained. The computational costs required to obtain the optimal aperiodic signal representation as well as communications costs resulting in the transmission of the data which represents the sampled signal will be considered in a cost for implementation model. This model represents a new approach to implementation cost modeling not considered in any of the previous references. Therefore, the signal representation per- formance index will measure the errors in signal representation and indicate the relative costs for implementation for each method Of sampling under consideration. The work which is to follow in this chapter states the signal repre- sentation problem and defines periodic, optimal periodic, aperiodic, sub-Optimal aperiodic, and Optimal aperiodic sampling critera. A 13 general cost function is formulated which can be used to evaluate each sampling criteria with respect to signal representation quality and costs for implementation. 2. 1. Signal Representation Problem Statement Consider the linear time invariant system, z’c_(t) = gm + _1_3_x_1€(t) + gt) (2.2) £(t) = _C_3§(t) + _D_z(t) (2. 3) where £(t) - n dimensional state vector of the system producing y(t); Beat) - r dimensional error signal vector; gt) - r dimensional input signal vector; gt) - m dimensional signal vector; and _w(t) - n dimensional disturbance vector. Matrices A, _B_, and E, _Q are constant and compatible with the above vectors. The initial time to is known, with t > tO being the f fixed terminal time. The initial state §(to) is assumed to be a Gaussian random vector with mean and variance E {35%)} = r_no, COv {x(to), _x(t0)} : 1&0) (2.4) The plant noise process _w_(t) is assumed to be non-zero mean Gaus sian white noise E{y_(t)} 2E0, t6 [tot tf] (2’5) Cov {_vflt), 3(7)} = 311(t)6(t-'r), t, 76 [to’ tf] (2-6) where 6(t - '1') is a delta function. In addition, _x(t0) and fit) are assumed independent. This model was chosen to generate a general signal vector which could be produced by either a communications System or a control 14 system. The actual model dynamics are either known or identified offline from previous output and control histories. With the system specified a signal vector generated by this system is to be sampled at the sampling times ti; such that, tinitial = t0 < t1 < t2 < . . . < tN-l < tN = tfinal (2. 7) with a sample interval being defined as Ti where Tizti+1'ti (2.8) fori=0, 1,...,N-land 1:3[T0’ T1, 0 o o , TN-l] (209) An optimal periodic sampling (OPS) criterion can be specified by selecting the number of sample intervals, N, to satisfy N . 5 NS N (2.10) min max and imposing the equality constraint knowing tf and t0 ‘1: -t g(I_)= {_f_N__o -Ti=0, i=0, 1, . . . , N-lforanyN (2.11) where Nmax is determined by sampler performance specifications and min by Signal representation quality. Optimal periodic sampling is Obtained through the minimization of a signal representation perform- ance index with respect to N satisfying (2. 10) and (2. 11) thus yielding the optimal number of samples N*. Periodic sampling (PS) is specified by a fixed N such that N . =N=N (2.12) min max and requiring (2. 11) be satisfied for the N chosen. An optimal aperiodic sampling (OAS) criterion can be specified by sampling interval constraints of the form 0 < Tmin 3 Ti 5 Tmax (2.13) 15 and given t and t0 f N-l ng) = 2 Ti — [1;f - to] = o (2.14) 1:0 where Tmin is determined by the sampler performance specifications and Tmax by System stability or worst case signal representation quality. The number of sample intervals, N, is chosen to satisfy (2.10). Optimal aperiodic sampling is obtained through minimization of a signal representation performance index with respect to both N and l satisfy- ing (2.10), (2.13), and (2.14)tO yield the Optimal number of sample intervals, N*, and the optimal sample interval sequence, 2* . Sub- optimal aperiodic sampling (SAS) Specifies a fixed N satisfying (2. 12) with the sample interval sequence free but satisfying (2. 13) and (2. l4). Sub-Optimal aperiodic sampling criterion is Obtained through minimization of the Signal representation performance index with respect to '_I‘_ with N satisfying (2. 12). Finally, aperiodic sampling (AS) is specified when N and :1; are chosen to satisfy (2. 12), (2.13), and (2. 14) with no Optimization on N or _'_I'_. Five sampling criteria have been specified. Each of the Optimal criteria require the minimization of a signal representation perfor- mance index with respect to certain variables. The form of the per- formance index is based on various periodic and adaptive sampling cost functionals Similar to (2. 1). A quadratic rather than an absolute error function is chosen since large sampling errors, either positive or negative, should be penalized more than small errors. The measure of the errors introduced by a sample and hold mech- anism does not provide an adequate basis for determining the particular sampling method to be implemented for a particular data acquisition 16 system. The costs for implementation; including computer utilization time, computer program and data storage, and communications Of Out- put data, must also be considered. Therefore, the performance index for a signal representation problem must measure both the errors caused by the sample and hold Operation and the costs for implementa- tion. Thus, the signal representation performance index becomes J = J + J (2.15) O f where N'1 1 t1+1 J =1: {2 — [ (gm-g |>4> £(t) - £(ti) (2. 24) into (2. 22) and then into (2. 21) yields JO:E{:OT —— :a:~[‘+ z(t)-GCx(t)) §H( £(t)-GCx(t))dt} (2.25) 1 s(t) can be any desired signal vector, either generated by a sig- nal model or supplied from an external source. 23 Expanding the integrand in (2. 25) and taking the expectation of each of the three resulting terms using the relationships that E {x' (t)Tx(t )} = _rfi' (Hip/1m +tr {135m} where X(t) is found in (2.4) and tr{ -} is the trace operation, T: C'G'SGS with E{§(t)} -E{3c_(ti)} :E{3<_(t)-§(ti)} : E{§(t)} =28“). (2.26) Therefore the original cost functional found in (2. 25) becomes JO=::——5‘t:+1[§(ct)-é())'§(flé(t)-_G_§_r_g(t) + tr (ll/JUN dt (2.27) Now computing the variance as required in (2. 27) t QM — x(t)-_(t L (t, t) — le (t) +5; 2 (t.7)[_1_3_9_e(t,)+3'(7)1d7 i (2. 28) where _x(ti) is an n x 1 state vector at time ti; 38(t) is an r x 1 error input to the system; \_)v_ ('r) is noise as in (2. 2), a n x 1 vector; (t, t.) is an n x n state transition matrix at time (t ~ti), 1 The mean Of £(t) becomes A “t £1“) = £(tl-m(ti): [513(t. ti)-_I_Jm(ti)+[ <1> (t, 7)[Bue( (t)+wo ]d7- ‘t. 1 (2.29) where \ m(ti) is the mean Of the process at t., and 1 we 18 the mean of the noise process, n x 1 vector. Subtracting (2. 29) from (2. 28): [fin-fin] 'l ,._., IX 3 l >4 7? I )3 E + 3 1.: CL [2(t,t1)-_I_ ][x( t. )-m main t. +S‘H1513(t,7)[_I§ue(t()+W('r)-._1§11,3(t)"27_o]d'T t. 1 (2.30) using (2. 30), and taking the expectation, E{(x(t) m(t))(x(t)- gn’fx'm} = [513 (t,ti) :1] [93(3) -r_n_(ti)] [3(ti) -_Ip_(ti)]'[2 (mi) 41' +£:§t: (t, 'r) ’T-) _V_Vo][W(o.)_ - Worg (t,o.)' deG (2. 31) yields 1: +52 (t, 7 )\Ir('r)<15' (t,'r)d'r (2.32) ti Therefore (2.32) is the variance equation required in (2. 27). The derived cost as found in (2. 27) will be added to the cost for im- plementation, to be given in detail in Chapter IV, to yield the total de- rived performance index as found in (2.15) for the signal representation problem. III. OPTIMAL APERIODIC SAMPLING FOR CONTROL Periodic sampling criteria have often been used in sampled-data control systems to simplify design, analysis, and implementation. Aperiodic and adaptive sampling criteria have become quite practical due to the introduction of digital computers into system design and con- trol implementation. As a result,various aperiodic and adaptive sam- pling criteria have been formulated to improve system performance and reduce sampling costs with respect to periodic sampling. A general framework for the analytical design of non-periodic sam- pling criteria was first proposed by Hsia [24] . A class of adaptive sampling rules were derived from a continuous time integral perfor- mance index which measured the squared error introduced by sampling the error signals of a feedback control system. The performance index was augmented by a ”cost for sampling, " as introduced by Hsia, which was inversely proportional to the sampling interval length in order to insure that the sampling intervals were greater than zero. The perfor- mance index was defined over just one sampling interval and thus the sampling rules obtained determined the length of only one sampling interval for each set of measurements taken of the signal being sampled. The sampling rules were derived by approximating the error signal and the cost for sampling by a Taylor series. These approximations were made in order to derive an analytic expression for the sampling rule. The resulting sampling rules were explicitly dependent on the signal and 25 26 its derivatives and we re not explicitly dependent on the System dynamics, system inputs, system disturbances, and costs of implementation. The dependence of the sampling rule on the a p_ri_o_r_i_ knowledge of the system was destroyed by the approximations made to the cost functional. Each sampling rule was tested by using it to adaptively sample a continuous time feedback error signal. The results of the tests re- vealed that each of the adaptive sampling rules had a tendency to sam- ple at the sampling extremes if constraints on the minimum and maximum sampling intervals were imposed. The underlying assumption of this work was that good signal representation by adaptive sampling of the continuous time error feedback signals implied good system performance over the desired control interval. In general, the performance of a feedback control system imple- mented through use of periodic sampling is dependent on the sampling rate. Periodic sampling requires the sampling rate to be greater than twice the system bandwidth in order to minimize aliasing and thus pro- vide reasonable control performance. The actual sampling rate used is a compromise between control performance and the costs for computa- tion, data storage, data communications, and sampling hardware. Optimal periodic sampling of discrete time control systems was first formulated by Kushner [40] and later extended by Aoki [3] in order to obtain an analytic formulation for the design of periodic sam- pling criteria. A discrete -time performance index was used which con- sisted of the terminal errors squared, a summation of the squared piecewise constant closed loop controls, and a "cost for sampling" term proportional to the number of samples taken over the control interval. The system state equation and control law were also modeled in discrete 27 time with explicit dependence on the sampling rate by substitution of the system state equation and control law into the performance index. The derived performance index was then minimized with respect to the sampling rate through use of an integer programming algorithm. The result was an optimal periodically sampled discrete time feedback con- trol system with consideration being given to sampling costs. These papers [3, 40] were the first attempts at using a control performance index and ”costs for sampling" to obtain optimal sampling for control. A general optimal aperiodic sampled-data control problem was first formulated by Jordan and Polak [46] . A continuous -time cost functional was used with the control being a piecewise constant vector function over each sampling interval. Both the control level over each interval and the length of each sampling interval were selected to mini- mize the performance index. As a result, the optimal control consisted of an optimal piecewise constant vector function and an optimal sam- pling interval sequence. The necessary conditions were established for the control problem but a computational algorithm was not devel- oped to obtain the optimal control. The performance index which was considered was not a function of the number of sampling intervals and did not have provisions for incorporating implementation costs or system disturbances. Tabak and Kuo [47] solved the optimal control problem introduced by Jordan and Palak [46] by discretizing the per— formance index and state equations _a_ priori thus enabling them to solve the non-linear programming problem through use of a sequential unconstrained minimization technique. The optimal aperiodic sampled-data regulator problem was inves- tigated for state dependent sampling by Schlueter and Levis [21] and 28 for constrained sampling times by Schlueter [ l9] . Schlueter developed an efficient computational algorithm for the problem formulated by Jordan and Polak [46] for the special case where the Optimal control could be determined uniquely as a function of the sampling interval se- quence. This optimal control sequence was determined by solving the Kuhn Tucker conditions. The control sequence was then substituted into the performance index to obtain a derived cost functional which was dependent on the sampling interval sequence. The optimal sam- pling interval sequence was then determined by minimizing the derived cost functional subject to the sampling constraints. The optimal sam- pling interval sequence specified the optimal aperiodic sampled-data control law. The optimal control—optimal sampling interval combina— tion was shown to outperform an optimal control law with any periodic or arbitrary aperiodic sampling criteria. All of the previous references were considered in the formulation of the optimal aperiodic sampling for control problem to be presented in this chapter. A continuous time performance index and system state equation were chosen similar to those found in Schlueter [19] and Jordan and Polak [45] and unlike the performance indices and system equations found in Aoki [3] , Kushner [40] , and Tabuk and Kuo [48] . The performance index is augmented by a cost for implementation term proportional to computer computational time requirements, data and software storage requirements, and output data communication costs. Implementation costs were implied by the "cost for sampling” terms as found in Aoki [3] and Hsia [24] but neglected in the other references. The performance index will consider system noise disturbances, which were neglected in earlier references and will be dependent on the Z9 sampling interval sequences as in Jordan and Polak [46] , Tabak and Kuo [47] , and Schlueter [l9] . The number of aperiodic sampling in- tervals used during the control interval will be free and selected opti- mally as in Aoki [3] for the special case of periodic sampling. All other references fix the number of samples used during the control interval. The control law will be specified a m as found in Aoki [3] and Hsia [24] . The design of optimal aperiodic sampling for control implementation, with a specified control law is proposed. The performance index will be formulated with dependence on the system state equation, control inputs, tracking errors, initial conditions, disturbance noise, and augmented by costs for implementation as discussed in Chapter II. The performance index will then be derived as a function-of the sampling interval sequence and implicitly as a function of the number of sampling intervals. Using non-linear programming the derived performance index will be minimized with respect to the sampling interval sequence for a fixed number of sampling intervals to yield the sub-optimal aperiodic sampling criteria. By selecting the number of samples, and through use of repeated opti- mizations the optimal aperiodic sampling criteria for control is obtained. 3.1. Aperiodic Sampling for Control Problem Statement Consider the linear time invariant System, yt) = Ag“) + ggeu) + _vflt) (3.1) x(t) = 23551:) (3. 2) 30 where gt) - n dimensional state vector of the system producing 1(t); 3e“) - r dimensional error signal vector; _z_(t) - r dimensional system input signal vector; y(t) - m dimensional output signal vecotr; and 1v(t) - n dimensional disturbance noise vector. Matrices A, _B_, and g are constant and compatible with the above vectors. The initial time to is known, with tf > tO being the fixed terminal time. The initial state yto) is assumed to be a Gaussian random vector with mean and variance E {£(to)} = _nr_1(to ). COV{§(tO). §(to)} = _Y_(to) (3. 3) The plant noise process w(t) is assumed to be non-zero mean Gaus sian white noise E{:‘_’(t)} =1". 0. 136 [‘10. ff] (3.4) Cov {_vgcc), 3(7)} = _\_Il_(t)6 (t-T), t, 7 e [to, tf] (3.5) where 6 (t - 'r) is a delta function. In addition, _x_(to) and _w_(t) are assumed independent. The performance of the system found in equations (3. l) and (3. 2) will be investigated for two cases of the feedback control error signal vector, Ee(t)' Case 1 will investigate system control by using gem 36(ti) = fiyti) - _gyti) (3. 6) where E and g are r x r and r x m dimensional constant input and feed- back gain matrices respectively and fit) is an arbitary continuous time input signal vector. Figure 3. 1 shows the location of the sample and hold device used in the sampled-data control system being considered 31 in Case I. This is the only case which will be investigated in Chapter IV since it is the most gene rally implemented. wt) “E Figure 3. l - Control System Configuration Using a Sampled Feedback Error Signal _l_i and _C_}_ are constant r x r and r x m dimensional matrices respectively with the other system matrices being defined in equations (3. l) and (3. 2). Case 2 will consider system control using gem = 53m - 9m.) (3. 7) where _I_-I_, _q, and £(t) are as defined earlier. Figure 3. 2 shows the location of the sample and hold device used in the sampled-data control system being considered in Case 2 gem gilt) §(t) (t) - me; If; \ '2: Figure 3. 2. - Control System Configuration Using a Con- tinuous Input Signal and Sampled Feedback 32 where the above sampled-data control system is defined in equations (3.1) and (3. 2) with 1:1, _q, and _g being r x r, m x n, and r x m dimen- sional constant matrices. With the system specified, the feedback control signal generated during the control interval is to be sampled at the sampling times ti; such that, tinitial = to < 1:l < 1:2 < . . . < tN-l < tN = tfinal (3.8) with a sampling interval being defined as T1 where T. = t. - t. (3. 9) 1:[TO,T1,...,T (3.10) N-l ] An optimal periodic sampling criterion for control can be specified by having the number of sampling intervals, N, satisfy N . 5 N5 N (3.11) m1n max and the equality constraint knowing tf and t0 5(3) =[tf111t0 _ Ti = 0 i=0, 1,. . . , N-l, for any N (3. 12) where Nmax is determined by sampler performance specifications and min by system stability. Optimal periodic sampling is obtained through minimization of the derived control implementation performance index with respect to N satisfying (3.11) and (3. 12) thus obtaining the optimal number of samples N]: Periodic sampling for control is specified by a fixed N such that N . =N=N (3.13) m1n max and having (3. 12) satisfied for the N chosen. An optimal aperiodic sampling criterion for control can be speci- fied by having the number of sampling intervals, N, satisfy (3.11). The 33 sampling intervals satisfy constraints of the form 0 5 [(fl’C) - 5(t))'9_(x(t)-§(t)) +g;(t)§ge(t)]dt} :—J 1: 1:0 i (3.18) Matrices X, g, and I): are assumed to be positive semi-definite sym- metric m, m, and r dimensional square matrices respectively and E{ . } is the expectation operator taken over the random variable _x(t). The Jf portion of (3. 17) represents the implementation costs such as, computer computation time, computer program and data storage, and costs for output data communications if necessary as discussed in Chapter 11. Thus, the optimal aperiodic sampling for control problem using the desired sampling criteria and appropriate sampling constraints can be stated as follows: Given the linear dynamical system (3. 1) and (3. 2) with arbitrary system input _z_(t), determine the optimal sampling interval sequence, _'_I‘_*, and the Optimal number of sampling intervals, N*, such that the performance index (3. 17) is minimized for the particular control system specified in (3. 3) through (3. 7). 3.2. The Derived Control Implementation Performance Index The feedback control configurations shown in Figures 3. 1 and 3. 2 will now be used to obtain the derived control performance index. In each case sampling will be involved in the determination of the feedback control signal Be(t)° Sampling of the feedback error signal, He“), will occur at the sampling times ti for i : 0, l, . . . , N-l as determined by 35 the derived control performance index and the optimization routine being used. The performance of the Case 1 feedback control system found in Figure 3. 1 will be determined through use of the performance index as found in (3. 17) with Be“) being defined in (3. 6). The performance index found in (3. 18) plus the implementation costs as discussed in Chapter II yields the total control performance index. The control performance indices for Case 1 and Case 2 will now be derived in terms of the sampling interval sequence, 1, defined in (3.10), and the number of sampling intervals, N, satisfying (3. 11) or (3. l3). Derived Performance Index for Case 1 Substitution of equations (3. 2) and (3. 6) with E = I_, where _I_ is the identity matrix, into (3.18) yields J z E{(§_§_(tN) - _Z_(tN))'X(9_ZE(tN) ‘ -z—(tN)) O 1‘51 t1+1 +§ S [(gyt) - _Z_(t))'_Q(§_3<_(t) -g(t)) :44 t 1:0 1 + (fifti) ‘ §E§(ti))' B<§ ‘S [(§m(t)-§(t))'9(£_rp_(t)-§(t))+tr[§_'QEX(t)l . ’ t. 120 1 + (5(ti) -_G__C_3_rp_(ti))' R (zui) -§_£_Ip_(ti)) +tr[_(_:_'_g'_fg_g_ V(ti)]] dt (3.22) where _T_=[T0,T ,T.,...,TN_1]andTi=t. -t 1, O O O 1 1+1 i. The variance term X(t) found in (3. 22) is obtained from the following equations. The system state is represented by t 50:) = 5130:. ti)_>g(ti) +S _(t. 'r) _B_[§(ti) "99—3151” +3”) (17 t. 1 (3.23) where _A_(t-t.) é(t-'T) (t,t)=e 1, (t,7):e both being state transition matrices of the System found in (3. 1). Taking the expectation of (3. 23) yields t m(t) = (t, t,)m(t.) +5 <1>(t, 7) B[z(t.)—GCm(t.)] +w dt — — 1 — 1 t - — 1 —--— 1 —o 1 (3. 24) Subtracting (3. 24) from (3. 23) yields [s(t)-2%)] = gt. t1) [gap-main t +5. 2 (tn) (-§._<_3.9_L>s gain + 3(7) -310) a. i (3.25) 01‘ [§(t)-m(t)] 2 g(t, ti)[§(t.)-m(t.)] — t 1 -- 1 +3 2(t, 'r)[w('r) -w ]d7' (3. 26) t _ _O i 37 where _e_ tO is uncorrelated with the state §(to), the final expression for the variance becomes 10:) = E{(g:_(t) -_IT_1(t))(§_(t) -_1;n_(t))'} = g (t, ti) E{[§(ti) - _n_3(ti)] [£(ti) - _r_n_(ti)] } '2'“. ti) t t t‘ o o + it 5,; 91“” ) “UV—(""2201 [3(a)-301'}2'(t, a)d7da i i (3. 29) where E{ [35(ti) - {_n_(ti)] [§(ti) - _1_'_n_(ti)] } = _\_’_(ti) and E{ [31(7) - _vzo] [_vy_(o.) - we] '} = $(7)6('r - o), and the integra- tion with respect to 0. yields t X(t)=_€1(t. ti)y_' x (ggtN) - _z_(th + tr [9' 3.911(th N—l'l ti+1 +2 [ (§__rr_1(t)—g(t))'9_(_C__r_n_(t)-§(t)) +tr[g'9__q_\:(t)l t. 120 1 + (at) -§_C__rp_(ti))'_B_(§(t) -§_C_3_1n_(ti)) +tr[g'GRGCV(ti)] dt (3.31) wherel:[T0,T ,T, T andTizt -t 1"" i""N-l] i+l i' The variance term _Y_(t) is derived in equations (3. 23) through (3. 29). IV. EVALUATION OF SAMPLING METHODS In any signal representation or control performance evaluation pro- gram, suitable simulation models have to be developed from which the proposed methods of sampling can be investigated under various signal and system conditions. In particular, the performance of periodic, adaptive, and aperiodic methods of sampling for signal representation and system control will be investigated. The signal which is to be sam- pled will be generated by a known signal model for the signal represen- tation problem and by a known feedback control system for the sampling for system control implementation problem. The initial conditions, system matrices, weighting coefficients, and final time will be speci- fied for each investigation. A step, ramp, parabola, and noise will be used as system inputs with all optimizations being performed using the non-linear programming optimization subroutine ZXPOWL as found of the CDC 6500 computer system. [See reference [53] for a detailed description of the ZXPOWL subroutine. ] 4.1. Samfling for Signal Representation The sampling for signal representation problem is restated from Chapter II: Dete rminc the optimal sampling interval sequence (N*, sz) that SPBCifiCS the optimal sample and hold approximation A _s_(t) = §_(ti), t-[ti , ti“); 1: o, 1, , N—l that minimizes a signal representation performance index (2. 15) subject to the sampling constraints (2. 10), (2.13), and (2.14). 39 40 The signal representation performance index (2.15) not only mea- sures the error S(t) " QUL), t€[tl’ t i: 0919 0") N'l 1+1]; caused by the sample and hold mechanism but also measures the costs for implementation which includes a cost for computer utilization, pro- gram and data storage requirements, and data communications. The following signal representation problem is presented to illus- trate the theory and provide insight into the performance of optimal aperiodic sampling for signal representation. Objective: Investigate the tradeoffs in periodic, adaptive and aperiodic sampling of a data signal record (to be supplied from auxiliary storage upon request) for communications to a remote signal level display panel. General Specifications1 1. The data signal record will consist of 101 data points spanning one unit of time. 2. A minimum of 2 and a maximum of 8 piecewise constant approximations will be allowed for the representation of each signal data record. 3. Each data point inputted from auxiliary storage consists of 3 ASCII‘2 characters. This will also be true for the sampling time and piecewise constant signal level out- put data as obtained through use of periodic, adaptive, or aperiodic sampling. 1Details not specifically stated in ”General Specifications"are left to the discretion of the System Designer. 2ASCII is the commonly used abbreviation for American Standard Code for Information Interchange. 41 4. Output data (time and level information) will be trans- mitted to the remote display via serial data transmission techniques. 5. A start, stOp, and parity bit will be added to each ASCII output character before transmission. 6. Any parity errors occuring during data transmission and detected at the remote display will be indicated by a ”parity error" light. The error indicator will be auto- matically reset after the next signal is received for dis- play. No other means of error detection or correction are necessary. Figure 4. 1 presents the Remote Display System in block diagram form. The most important block shown is the “Main Signal Record Processor and P/S Converter Controller" block. This block represents the computer portion of the display system. The computer will be re- quired to perform the following tasks: a) b) d) request continuous time signal records from the auxiliary storage facility load the signal record into the proper portion of its main memory for purposes of signal representation through sampling execute the predetermined optimization program stored in main memory with respect to the minimization of the de- rived signal representation performance index control the transmission of the resultant signal level and sampling time information to the remote display. This entails the control of the parallel to serial converter via the P/S control line as indicated. (The S/P converter on the display end of the data transmission line is assumed a passive device.) The computer is assumed to be entirely committed to either sig- nal processing through sampling or data transmission via the P/S con- verter and communications line. The Signal Record Processor is 42 assumed too small with respect to memory size and too slow with re- spect to computational speed to allow signal multiplexing and therefore optimal aperiodic sampling will be performed for data compression. The rest of the blocks shown in Figure 4. 1 are explained in the figure or are self-explanatory. A simplified flowchart of the computer program used in the evalu- ation of the three methods of sampling, viz. periodic, adaptive, and sub-optimal aperiodic, for the remote display problem is shown in Figure 4. 2. 43 Auxiliary Storage of Signal Records to be Processed. [1 data request line parallel data 1 Main Signal Record P/S Processor Converter and _ P P/S Converter1 300 Baud Controller Data Trans- P/S mission Line control line S/ P _ Converter2 |‘ , — 5 i l I I 1] and , '- I, Display Parity Checker Parity Error Indicator Figure 4. l - Block Diagram of the Remote Display System lP/S Converter converts data from parallel data inputted on several information lines to serial data which is transmitted to the display using only one transmission line. 2Reverse of P/S Converter. 44 gm...) 1 Obtain the signal record: 1. request signal record from aux. storage or 2. generate the signal record through use of the signal model. l - Initialize the number of sampling intervals :I N = N . min A] '1 Periodically sample the signal record and eval- uate the derived signal representation performance index (JO) and determine the costs for implemen- tation (Jf). Adaptively sample the signal record through use of the sampling rule corresponding to the value of "a" chosen and evaluate the derived signal representation performance index (Jo) and deter- mine the costs for implementation (Jf). Obtain the optimal sampling interval sequence through use of the ZXPOWL optimization sub- routine which minimizes the derived signal representation performance index (JO). Deter- mine the resulting costs for implementation (Jf) necessary for sub-optimal aperiodic sampling of the signal record. No N:N+l Yes CSTOP) Figure 4. 2 - Computer Program Flowchart for Signal R epre s entation Inve stigation 45 4.2. _S_ignal Representation Performance Results The sampling evaluation computer program as found in the simpli- fied flowchart form in Figure 4. 2 will be used to investigate periodic, adaptive, and aperiodic sampling for signal. representation as formu- lated in Chapter II. The following system model and derived signal representation performance index will be used 0 1 10 -1 gm = xjt) + u (t) , 35(t ) = (4.1) 0 0 100 e O 0 y(t) = [10] gt) + [ O] z(t) (4.2) with 8(t) = ue(t) = Z(t) - y(t) (4. 3) where _z_(t) can be one of the following input signals a) 1. 0 (Step) b) t (Ramp) c) t2 (Parabola) for te [to, tf] where t0 = O. O and tf = 1. O. The control signal ue(t) will be sampled for signal representation at the sampling times ti such that, :1: <1: <1: <...< tinitial o 1 2 < t tN-l N : tfinal with a sampling interval being defined as Ti where Ti : t1+1 ' ti The derived signal representation performance index is N-l t. 1 1+1 2 . . 1 1:0 1 The cost for implementation, Jf, for periodic sampling is W2 x K x W J = w x M x U + (4. 5) f l R 46 where M = 800 words of memory wl = . 0001 (word-second)-1 W : 30 bits/output word (record) to be transmitted K = (N+4) words (records)l R = 300 bits/second wZ . 75 seconds-1 The cost for implementation for adaptive and aperiodic sampling is w2 x K x W szwlxMxU+ R (4.6) where M = 1000 words of memory W1 2 . 0001 (word-second)"1 W : 30 bits/output word (record) to be transmitted K : 2 x (N+1) words (records)2 R = 300 bits/second w2 = . 75 seconds-1 1(N+4) is obtained from the fact that specifying periodic data requires 1. 1 data word to indicate the initial time. 2. 1 data word to indicate the time between data points. 3. N + 1 data points. 4. 1 data word to indicate the number of data points sent. Thus the communications costs for periodic sampling will be based on (N + 4) data words to be sent for each sampled signal record which is processed. 22 (N +1) is obtained from the fact that specifying aperiodic or adaptive sampling requires 1. (N + 1) data words to specify the N + 1 signal magnitudes. 2. (N + 1) data words to indicate the times of each signal magnitude change. Thus the communications costs for aperiodic/adaptive sampling depends on 2 (N +1) data words to be sent for each sampled signal record which is processed. 47 In the computer utilization portion of (4. 5) or (4. 6) the computer memory requirements (M) are determined by the number of words re- quired to store the sampling criteria software in the main memory of the computer being used to obtain the signal representation data. The value U, which depends on the number of sampling intervals; N, the complexity of the system software, (i. e. , cost functional subroutine, optimization routines, etc.) and the length of the signal record, is found by determined the computer processing time required, in this case through use of the CDC 6500 computer, to obtain the optimal sampling interval sequence of the input signal record using any particular value of N. The weighting coefficient w is the cost per word-second for the l particular computer being used. In the communications cost portion of (4. 5) and (4. 6) the values of R and W are determined respectively by specifying the minimum com- munications channel capacity to be 300 Baud and that each signal level measurement and sampling time be transmitted by using 3 ASCII char- acters with start, stop, and parity bits being added to each character. The number of words (records) (K) to be transmitted via the communi- cations net work depends on the number of sampling intervals and the type of sampling being used. The weighting coefficient w2 is the cost per second for use of the communications channel for data transmission. In general both weighting factors (i. e. , w and W2) are highly problem 1 specific and will change with system specifications or advances in . technology. A comparison of optimal ape riodic sampling criteria determined for three different performance indices will be made first. The par- ameter "a" will take on the values of -l, 0, and 1 resulting in a time- 48 weighted, unweighted, or average error type performance indices. The sub-optimal aperiodic sampling criterion for each N will be compared with periodic and the appropriate adaptive sampling rule for the value of "a" being used. These adaptive sampling rules were derived in Hsia [24] using the three values of ”a" as indicated. Adaptive Rule 1 (ARI) for a = 0: T. = c / [i1 (t.)] , where c is adjusted to yield the 1 1 e 1 . l . . de81red number of sampl1ng 1n- tervals. (4. 7) Adaptive Rule 2 (ARZ) for a = -1: Ti : CZ/ (he (ti)2> 1/3, where c is adjusted to yield the desired number of sam- pling intervals. (4. 8) Adaptive Rule 3 (AR3) for a = 1: T. = T / ( (11’1 (t.) + 1. 0), where a is adjusted to 1 max e 1 . . yield the deS1red number of sampling intervals. max I Tfinal (4° 9) Therefore given the System, performance index, and cost for im- plementation function, the figures which follow will graphically illus- trate performance cost changes, JO, as a function of the number of sampling intervals used to represent the signal being sampled. In addition the total or System performance costs, J, will also be shown in each figure, (i. c. J 2 Jo + Jf). Both performance costs will be in- vestigated using the three ”a" values for the periodic, optimal aperiodic and the appropriate adaptive sampling criteria. The performance in- dices will be evaluated. for the case of a step input, (z(t) = 1) where §(to)’= [ -l 0] is specified, and the disturbance noise is zero. The performance costs for sub-optimal aperiodic sampling represent the minimized performance cost obtained through use of the ZXPOWL 49 optimization subroutine for the specified number of sampling intervals. (The exact numerical value of each performance cost is listed by figure number in Appendix C. ) The performance of optimal aperiodic sampling for signal repre— sentation will be compared with adaptive and periodic sampling. The control performance costs for aperiodic, adaptive, and periodic sam- pling will be designated as J1, J2, J3, resPectively in this chapter. The optimal performance costs for each sampling criterion, denoted as Ji*, i = l, 2, 3, are defined as the minimum value of Ji over the set of N satisfying the constraint 2 4 N 9'- 8, (2.10). The optimal system performance ratio between adaptive and aperiodic sampling, denoted by (Jz/J1)*, is defined as the maximum ratio of JZ/Jl over the set (2. ll) of feasible N. The optimal system performance ratio between periodic and aperiodic sampling criteria, denoted by (J3/Jl)*, is the maximum ratio J3/J1 over the set of feasible N, (2.11). Sub-optirnal aperiodic sampling reduces signal representation errors compared to periodic or adaptive sampling for each value of N for the three different cases a = -l, a = 0, and a = l as shown in Figures 4. 3(a), 4. 4(a), and 4. 5(a) respectively. The optimal aperiodic (0A) COSt, J], is significantly smaller than the optimal adaptive sampling (OAD) cost, J; or optimal periodic sampling (OPS) cost, J; for each case (a = -l, 0, 1) shown in Table 4. 1. Moreover, the maximum ratio of the performance costs between adaptive and sub-optimal aperiodic sampling (Jz/J1)* and the maximum ratio of performance costs between 50 periodic and sub-optimal aperiodic sampling (J3/Jl)* are quite large. These large performance ratios indicate that selecting the sequence of sampling intervals together rather than individually as is done in adap- tive sampling or using equal length sampling intervals as in periodic sampling, can significantly reduce the errors caused by the sample and hold ope ration. It is also apparent from Table 4. 1 that the optimal performance J: for optimal aperiodic (i=1), Optimal adaptive (i=2), and optimal periodic (i=3) increases as ”a” increases. These increases can be explained by noting that the performance index is prOportional to (l/Ti)a which increases with "a" since in all cases Ti is less than one (tf is equal to one). oA OAD ops AD/OA PS/OA ,. a: J :1: J :1: : .~ * 31" p : "a" J] N] J2 NZ J3 N3 —2 N2 —3 N_3_ -1 .003 8 .036 8 .036 8 17.9 2 15.6 2 0 .040 6 .073 5 .276 8 12.8 5 6.7 5 1 .522 6 .638 6 2.12 8 5.9 5 1.6 4 Table 4.1. Data summary for Figures 4. 3(a) to 4. 5(a). 51 / 1. Z \ \ '—'—'—— Periodic Sampling ‘ —""‘— Adaptive Sampling . 9 \ ----- Aperiodic Sampling 0‘ o U») JLILAIAInlllnlnlnlnLnlnlnlu . 0 p 2 3 4 5 6 7 8 N (Number of Sampling Intervals) (a) 2. 4 1 Periodic Sampling I”. j — - —— Adaptive Sampling 2’ a ’ ’ j ““““ Aperiodic Sampling //’ I 4 ,// l. 6 _1 ’I” r - ’- .: /c/ -: ‘— —*—-——— '/ II -/ 0. 8 — V ‘ J 0' O . I I 1 T I —3 2 3 4 5 6 7 8 N (Number of Sampling Intervals) (b) Figure 4. 3 - Signal Representation Costs Using a = —1 and Step Input 52 Periodic Sampling __..- — Adaptive Sampling ______ Aperiodic sampling N (Number of Sampling Intervals) (a) 3 __ Pe ri odic Sampling —— - — Adaptive Sampling ...... Ape ri odic Sampling 0 v I l 1 1 r 2 3 4 5 6 7 N (Number of Sampling Intervals) (b) 00.1 Figure 4. 4 - Signal Representation Costs Using a = 0 and Step Input 53 6. 0 Periodic Sampling ——-— Adaptive Sampling 5. 0 ------ Aperiodic Sampling 4. 0 3. O 2. 0 l. 0 0- 0 T l I 1 g T “1 2 3 4 5 7 8 N (Number of Sampling Intervals) (a) 6' 0 Periodic Sampling —- —— Adaptive Sampling 5. 0 -- - -- -- Aperiodic Sampling 5‘ o .N 9“ O O llllllllllLlllllllll I, V \ \ I I I I \ \ \ \ I I f I I ‘ l I I I I 1 2 3 4 5 6 7 8 N (Number of Sampling Intervals) (b) Figure 4. 5 - Signal Representation Costs Using a = 1 and Step Input 54 The maximum performance improvement ratios between adaptive or periodic sampling and sub-optimal aperiodic sampling, (Jz/Jl)* and (J3/Jl)* respectively, decrease with increases in ”a". Therefore, the tradeoffs possible in selecting the N sampling intervals together appar- ently provide greater performance improvement in the case when ”a” is small. The length of the sampling intervals for both adaptive and sub- optimal aperiodic sampling criteria are shown in Figure 4. 6 for a = -l, a = 0, and a = l. The length of the sampling intervals becomes less periodic as ”a" increases for both adaptive and sub-optimal aperiodic sampling criteria. Thus, the sampling rate is better tuned to the rate of change of the signal when ”a” increases. (a) Step Input, Adaptive Sampling, 5 sampling intervals (N=5) l 1 1 l — f. *1 t3 {4 a--1 1 L l 1 3:0 1 1 l 1 a=l (b) Step Input, Sub-Optimal Aperiodic Sampling, N=5 1 1 1 1 f, 5 £3 £4. a—-l n n 1 1 3:0 1 ll J azl to f t {'55 Figure 4. 6. Representative Sampling Interval Lengths for Figures 4. 3(a) to 4. 5(a) using N=5. 55 Thus the selection of the proper performance index depends on the particular application and the smallest sampling interval length interval allowable. For this particular problem, a = 1 or a = 0 would be pre- ferred since the sampling criterion would be better tuned to the changes in the signal and since no lower bound exists for the length of the sam- pling interval. Clearly as the number of sampling intervals are increased the sampling errors are reduced. It should be noted that through use of as few as two sampling intervals performance improvements of as much as 15. 6 :1 can be realized through the use of optimal length sampling intervals. The remote di8play problem indicates that through use of optimal aperiodic sampling, reductions in the amount of data necessary to represent the sampled input signal record while maintaining a de- sired signal representation performance index is possible. The System performance costs are graphically presented in Figures 4. 3(b), 4.4(b), and 4. 5(b) as a function of the number of sam- pling intervals, N, used to represent the input signal record. As can be seen, an increase in the number of sampling intervals results in increased costs for implementation in most cases. At points where System performance costs for sub-optimal aperiodic sampling exceeds that for either periodic or adaptive sampling, the computer utilization/ processing costs necessary to obtain the optimal sampling interval sequence, _T_*, and the communications costs necessary to transmit the resultant signal level and sampling time data exceed any perfor- mance improvements resulting from aperiodically sampling the de— sired signal record. These implementation costs become apparent at the 4th sampling interval in Figure 4. 3(b) and at the 6th sampling 56 interval in Figure 4. 4(b). Figure 4. 5(b) shows that sub~optimal aperio- dic sampling costs exceed those of adaptive sampling after 4 sampling 1 and W2 were re- duced, or if a faster computer, more efficient Optimization algorithm, intervals. If the values of the weighting factors w or higher data transmission rate communications channel were used, the costs for implementation could be reduced proportionally. These reductions would make optimal aperiodic sampling more desirable at higher values of N. In Figures 4. 3 and 4. 4 good performance improve- ments can be realized with respect to adaptive and periodic sampling by using a = -l or a =0 with N being either 2 or 3. The ultimate choice of the sampling criterion and the value of N will depend on the perfor- mance level required for Jo. The performance of the Optimal aperiodic sampling for signal representation will now be investigated with an unweighted integral performance index (a = 0) for the following four inputs: a) step, z(t) = 1.0 b) ramp, z(t) : t c) parabolic, z(t) = t2 with initial state covariance and noise covariance being -1 0 0 0 0 E{§(to)} = . V (t )= .30): _ O O 0 0 0 The derived signal representation performance index found in equation (4. 4) and the costs for implementation shown in either equation (4. 5) or (4. 6) will be used. The fourth input is d) noise, z(t) = 0 where the noise and randomly distributed initial states are assumed gaussian with 1 0 1336200)} =9. 1x80): (4.10) 0 1 and l 0 E{1v_} = 9. gm = (4.11) 0 1 The derived performance index for the case of noise disturbances with z(t) = 0 becomes N-l t. 1 1+1 2 J = 2 ~32 St {z(t) - z(ti) -[ l 0] I_n_(t) +[ l 0] _rp_(ti) i=0 Ti 1 1 0 +tr V (t) dt + J (4°12) L 0 1 1} f where the costs for implementation, Jf, are found in (4. 5) or (4. 6). The values of the performance index for periodic, adaptive, and sub-optimal aperiodic sampling criteria using a = 0 are plotted in Figures 4. 4(a), 4. 7(a), 4. 8(a), 4. 9(a) as a function of the number of sampling intervals for the step, ramp, parabolic, and noise inputs respectively. The values of the performance index, Jik' for the Optimal aperiodic (i = 1), Optimal adaptive (i = 2), and Optimal periodic criteria (i = 3) are shown in Table 4. 2 for each of the four input types. The maximum performance improvement ratios, (JZ/J1)* and (J3/Jl)*’ for sub-optimal aperiodic criteria over adaptive and periodic criteria respectively are also given in Table 4. 2. The results indicate that sub-optimal aperiodic sampling produces smaller signal sampling errors as compared to periodic or adaptive sampling for each input and value of N shown in Figures 4. 4(a), 4. 7(a), 4. 8(a), and 4. 9(a). The optimal aperiodic (OA) cost, J*, is lower than either the optimal adaptive (OAD) cost, J: or the optimal periodic >:< sampling (OPS) cost, .1 , for each input considered. The maximum 58 ratio of the performance costs between adaptive and sub-optimal ape ri- odic sampling (JZ/Jl)* and the maximum ratio between periodic and sub-optimal aperiodic sampling (J3/Jl)* are large for a step, ramp, and parabolic input. These performance ratios indicate that optimal selection of the sampling interval sequence is preferable to individual sampling interval selection as found in adaptive or use of equal length sampling intervals as in periodic sampling. Thus using the optimal sampling interval sequence, errors caused by the sample and hold operation can be significantly reduced for these input types. >1: :1: OA OAD OPS (AD/A) (PS/A) >1: 9': 55‘ INPUT * * >3 12 N2 J N _ J J N J _ _ 3 (a-O) 1 2 2 3 J1 1 11 3- STEP .040 .073 5 .276 12.8 5 6.7 5 RAMP .008 .013 6 .062 7.6 2 11.0 6 PARABOLi. 007 .011 8 .069 7. 4 2 11.6 4 NOISE 1.028 8 - - . 029 8 - - 1. 3 5 Table 4. 2. Data Summary for Figures 4. 6(a) to 4. 8(a). The optimal aperiodic cost J): for a noise input is only slightly lower :1: than the optimal periodic cost J 3. In addition the maximum perfor- :1: mance improvement ratio (J3/Jl) is considerably lower than the cor- responding performance ratios for the other inputs. This indicates that periodic sampling is in fact near Optimal if noisy signals are to be sampled. 59 Periodic Sampling —-_ Adaptive Sampling _-.. — -- Aperiodic Sampling 2 3 4 5 6 7 N (Number of Sampling Intervals) (a) 2. 4 Periodic Sampling —--—- Adaptive Sampling 1‘ ------ Aperiodic Sampling ,’ \\ ° 0 r r l I I F 1 2 3 4 5 6 7 8 N (Number of Sampling Intervals) (b) Figure 4. 7 - Signal Representation Costs Using a = O and Ramp Input 60 Periodic Sampling __ .. -— Adaptive Sampling --- - -- Ape riodic Sampling N (Number Of Sampling Intervals) (a) 3 '1 .( ., Periodic Sampling - —- -— Adaptive Sampling - --- - "'" A eriod'c S l' p 1 amp ing A‘ z —-I ,] ~‘~ d / ‘~ / d l/ ’/ q I” / ? m I T I “I 2 3 4 5 6 7 8 N (Number of Sampling Intervals) (b) Figure 4. 8 - Signal Representation Costs Using a = 0 and Parabolic Input 61 Periodic Sampling . ----- Aperiodic Sampling -( ~.‘-~-~_. ‘ ‘--- -- J.- _-_: 0° ° 1 l 1 l l 1 2 3 4 5 6 7 8 N (Number of Sampling Intervals) ‘] (a) ‘,.-’-" ”’ 1.5 -< ’-’ r " / l’.‘~ ” ’ / ‘\‘ l’ - [’r v d / II 1.0 -‘ d / JV .5 "" -‘ Periodic Sampling " """" Aperi odic Sampling 1 0° 0 l l T l T 1 2 3 4 5 6 7 8 N (Number of Sampling Intervals) (b) Figure 4. 9 - Signal Representation Costs Using a = 0 and Noise Input 62 The sampling interval sequences for both adaptive and sub-optimal aperiodic sampling using a = 0 and N = 5 are shown in Figure 4. 10 for each of the four input types. (See Appendix C for the exact numerical length of each sampling interval.) In general, the sampling interval lengths for step, ramp, and parabolic inputs are longer for sub-optimal ape riodi c s ampling. The sampling interval sequence for the noise input using sub-optimal aperiodic sampling is very close to periodic as was indicated by the performance cost for the noise input found in Table 4. 2. (a) Adaptive Sampling (a=0, N25) 1 j 1 1 Step {1 f2 {3 (4 l L l L t, t, t, {'4 Ram? l J l 1 b 1 in t; t, t4 Para 0 a 1 L A 1 £- t t f to I 1 3 ‘f 1“: tf (b) Sub-Optimal Aperiodic Sampling (a=0, N=5) A l L L St f, t,‘ f3 t4 ep 1 j l L ‘ L 4 ‘ Parabola 1, t1 t3 t4 L L l L , t) t2 t3 {4 N01se to t5: ff Figure 4. 10 Representative Sampling Interval Lengths for Figures 4. 7(a) to 4. 9(a) using N=5. 63 As can be seen in Figure 4. 7(b) the costs for implementation added to the performance costs increased the system performance costs for sub-optimal aperiodic sampling above those for periodic and adaptive sampling after 3 sampling intervals. The shape of the cost curve for a ramp input, found in Figure 4. 7(b) for sub-optimal aperiodic sampling, is far from a ”smooth" curve as compared to the curves for periodic and adaptive sampling. The shape of the aperiodic performance curve is due to the fact that an increase in the number of sampling intervals does not necessarily imply a corresponding increase in the computation- al costs (qu) required to obtain the optimal sampling interval sequence _T_*. As can be seen in Figure 4. 7(b) the shift in the system performance for sub-optimal aperiodic sampling is due mainly to the increased costs for implementation. In Figure 4. 8(b) the system performance cost for periodic sampling remains relatively constant while the costs for adaptive and sub-Optimal aperiodic sampling increase above the periodic sampling costs after 4 sampling intervals. The increases in system performance costs for adaptive and aperiodic sampling using a parabolic input are due pri- marily to the increased computational expenses required to obtain the optimal sampling interval sequences 1* as indicated in Figure 4. 10 for N = 5. In‘ Figure 4. 9(b) the costs for implementation required for sub- optimal aperiodic sampling were so high that periodic sampling had the lowest system costs for each N considered. This implies in con- junction with Figure 4. 10 that if ”noisy" signals are to be sampled for signal representation, based on signal variances as in (2. 32), periodic sampling will provide good signal representation with lower implemen- tation costs than optimal aperiodic sampling for each N considered. 64 In summary, aperiodic sampling as specified in Chapter II yielded improved signal representation performances with respect to periodic or adaptive sampling for all values of ”a" and input signals considered. If signal representation performance costs are to be considered alone in the choice of the sampling method to be used, clearly all results in- dicate that sub-optimal aperiodic sampling should be chosen. In any design problem, in particular the one shown in Figure 4. 1, signal representation performance costs can not be used as the only basis for choosing the method of sampling to implement. AS can be seen in the figures which have been presented, the costs for implemen- tation when considered jointly with the performance costs can outweigh any reductions in sampling errors obtained through use of more sophis- ticated sampling techniques such as optimal aperiodic sampling. Thus if the costs for implementation as introduced in Chapter II and discussed in detail for the remote display problem are low, aperiodic sampling should be employed; if the costs for implementation are high, periodic or possibly adaptive sampling should be employed. The greater the knowledge the system designer has about the problem and the constraints he is required to work within the better the System design will become. The procedure for selecting the particular sampling criterion is to first determine the maximum value of the signal representation costs permitted and then determine the minimum number of sampling inter- vals required to obtain this value Of performance using each sampling criteria. Then selecting the sampling criterion with the lowest total costs assuming that each criterion is only considered in the region above its minimum number of sampling intervals. Another approach would be to select the sampling criteria with the lowest implementation costs over the same set of acceptable number of sampling intervals. 65 EXAMPLE DESIGN PROBLEM Consider the Signal representation performance index cost curves for a ramp input signal using a = 0 as shown in Figure 4. 7. Two pos- sible methods of selecting the best sampling criterion given the system design specifications will be presented. METHOD 1. (See Figure 4. 7(a)) STEP 1. Determine the maximum allowable signal representa- tion performance cost, JO. For this example, the perfor- mance cost will be arbitrarily chosen, thus let Jo = . 04. STEP 2. Select the feasible values of N, where 2 S N S 8 as seen in Figure 4. 7(a), such that STEP 1 is satisfied. Sampling Type Feasible N Minimum N (N?) Periodic Sampling none none Adaptive Sampling 4, 6, 7, 8 4 Sub-Optimal Aperiodic Sampling 4, 5, 6, 7, 8 4 STEP 3. Choose the method Of sampling with the lowest system performance costs. (See Figure 4. 7(b)) For this example, adaptive sampling using 4 sampling intervals yields the lowest total performance cost, J. (The exact numerical values of the performance costs are found in Appendix C.) Thus in this example the use of METHOD 1 indicates that adaptive sampling should be chosen for implementation based on the performance results indicated in Figure 4. 7. 66 METHOD 2. (See Figure 4. 7(a)) STEP 1 and STEP 2 are identical to those in METHOD 1. STEP 3. Choose the method of sampling with the lowest imple- mentation cost. Implementation costs Jf are determined by subtracting the performance cost J0 from the system perfor- mance cost J. Using this procedure, adaptive sampling with N = 4 has the lowest implementation costs, Jf. Thus using this method of sampling criterion selection, adaptive sampling would again be chosen for implementation. AS can be seen in Figure 4. 7, a different choice of the value of JO would lead to different sampling criterion selection. The two methods of sampling type selection are Similar for STEP 1 and STEP 2 but different in STEP 3. STEP 3 causes METHOD 1 to em- phasize total or system performance costs whereas METHOD 2 empha- sizes implementation costs alone. The method used in the selection of the sampling criteria for implementation depends on the System design objectives and allowable system performance. 4.3. Samjliig for Control Implementation The performance of the optimal sampling for control implementa- tion problem will now be investigated. Performance will be investiga- ted as a function of System response times (bandwidth), the system type (the number of poles at the origin in the System plant), and the inputs forcing the system during the control interval. The performance index will be augmented by costs for implementation as discussed in Chapter II. The cost for data communications will be essentially 67 neglected since the controlling computer is assumed to be very near the system under control. The sampling for control problem, as restated from Chapter III, is: Determine the optimal sampling interval sequence (N33, _'I_‘_>:< ) that specifies the optimal sample and hold approximations Gem = ue(ti) , t£[ti , tiH] ; i = 0, 1, , N-l that minimize the control implementation performance index (3. 17). subject to the sampling constraints (3.11), (3.12), and (3.13). The control implementation performance index (3. 17) measures the terminal errors, the system tracking errors, the control energy expenditures and implementation costs. The implementation costs in- clude computer utilization, software and data storage costs for com- puting the optimal sampling intervals, and data communications costs for transmitting the control information from the computer to the system under control. The primary objective of investigating this problem is to evaluate the trade-offs between performance and implementation costs for im- plementating a digital control system through use of a periodic or an optimal aperiodic sampling process. A comparison of performance and costs for implementation will be made for the following cases: [1] a Type 2 (two poles at the origin) system with FAST, MEDIUM, and SLOW speeds of response to a step input. [2] a Type 2 MEDIUM response time system for a step, ramp, and a random noise input. [3] a Type 1 (one pole at the origin) MEDIUM response time System for a step, ramp, and a random noise input. 68 [4] a Type 0 (no poles at the origin) MEDIUM response time System for a step, ramp, and a random noise input. A comparison of the performance and implementation costs for these two sampling criteria will also be compared based on the System type for a particular input signal. The control performance of the feedback control system imple- mented with a continuous time (analog) control is evaluated to provide a basis for evaluating the control performance of the two sampled- data Systems. It is assumed that the implementation costs for the continuous time (analog) control system is prohibitive. The basic assumption made in evaluating and comparing the per- formances of the control laws implemented with the periodic and opti- mal aperiodic sampling criterion are: [l] a second order System will be used in all cases: 0 1 b1 :(t) = _s_(t) + ue(t) (4.13) -al -a2 b2 y(t) = [:1 0] _x_(t) (4.14) [2] the control law is specified by ue(t) = z(t) - y(t) (4.15) over a control interval te[0, l] where z(t) is the programmed control. [3] measurements of the control law are transmitted at the sam- pling times {ti} :51 where 1:0 :tinitial : 0 < t1< < tN—1 < thtfinal :1 where N is restricted to satisfy (4.16) 25N58 (4-17) 69 based on computer software constraints. [4] the control error is generated only at the sampling times and held over each sampling interval such that ue(t) = e(ti) , t [ti , tiH] andi = 0,1, . . ., N-l. [5] the performance objectives for this System are assumed to be met by selecting a performance index which penalizes terminal errors, tracking errors, and control energy expenditures as found in (3. 17). The computational costs are the off-line costs for computing the optimal sampling criterion and the commu- nications costs are the on-line costs for transmitting data between the computer and the System being controlled. The sum of the computational and communications costs result in the costs for control system implementation as discussed in Chapter II. The derived control performance index used is :r = E {(y(tN) - z(th' 5 (y(tN) - 2(th + N-l t 2 £1+1[(y(t) - z(t))' .l (y(t) - z(t)) + . 02ue(t)2] dt} + Jf i=0 i (4. 18) The cost for implementation, Jf, for periodic sampling is w2 x K x W Jf=wlxMxU+ R (4.19) where M = 1700 words of memory wl = .0001 (word-second)-1 W = 30 bits/output word (record) 70 (N+4) words (records)1 K : R = 300 bits/second wZ = .00005 seconds-1 The cost for implementation for aperiodic sampling is wZ x K x W Jf=W1xMXU + R (4.20) where M = 1700 words of memory wl = . 0001 (word-secondfl W = 30 bits/output word (record) K = 2 x (N+l) words (records)2 R = 300 bits/second wz = .00005 seconds-1 In the computer utilization portion of (4. 19) or (4. 20) the computer memory requirements (M) are determined by the number of words re— quired to store the control system software in the main memory of the computer being used for System control. The value of U, which depends on the number of sampling intervals; N, the complexity of the control System software, (i. e. cost functional subroutine, optimization routines, etc. ) and the control time interval, is found by determining the compu- ter processing time required, in this case through use of the CDC 6500 computer, to obtain the optimal sampling interval sequence, _T_*, using any particular value of N. The weighting coefficient w is the cost per 1 word-second for the particular computer being used. 1 See Page 46 , Section 4. 2. 2See Page 46 , Section 4. 2. 71 C1) Data Read-In and Initialization Phase 1 Simulate the continuous time feedback control system and evaluate the derived control per- formance index (Jo). Initialize the number of sampling intervals. N : Nmin r J. r Simulate the feedback control system using per- eriodic sampling of period T = (tf-tO)/N and compute the derived control performance index JO(T, N) and Jf. 1 Obtain the optimal aperiodic sampling interval sequence 3* through use of the ZXPOWL opt- ( irnization subroutine and the derived control per- (( formance index (3. 25). Use «Tj< in the simula- tion of the feedback control system to obtain JO(T_*, N) and Jf for optimal aperiodic sampling. ‘ 8 No IS N=N+l N-‘Nma 7 — Yes Q 1 Figure 4. 11. Computer Program Flowchart for Control Pe r- formance Evaluation. 72 In the communications cost portion of (4. 19) or (4. 20) the values of R and W are determined respectively by specifying the minimum communications channel capacity to be 300 Baud and that each Signal level measurement and sampling time be transmitted by using 3 ASCII characters with start, stop, and parity bits being added to each char- acter. The number of words (records) (K) to be transmitted via the communications network depends on the number of sampling intervals and the type of sampling being used. The weighting coefficient w2 is the cost per second for use of the communications channel for data transmission. In general both weighting factors (1. e. W1 and wz) are highly problem specific and will change with system specifications or advances in technology. Section 4. 4 will investigate three Type 2 control system models having FAST, MEDIUM, and SLOW reSponse times. Performance costs for each System and input being investigated will be found using the computer program shown in simplified flowhcart form in Figure 4.11. 4.4. Control Implementation Performance Results The trade-offs in performance and implementation costs for a sampled-data control implementated with periodic and optimal aperi- odic sampling will be investigated for a Type 2 control system which has been used extensively in research [1, 19, 20, 24, 52] on compari- son and evaluation of adaptive sampling criteria. This system will be investigated for gain settings which result in FAST, MEDIUM, and SLOW response times. The three system models have the form 73 a) FAST response time system : 0 l 10 350.) 33(1) 1 u,,(t) 0 0 100 y(t): [1 0150) b) MEDIUM response time system : 0 l 5 35(t) = 350) + ue(t) 0 0 25 y(t) = [1 01330) c) SLOW response time system : 0 1 2.5 z(t) = z(t) + ue(t) 0 0 6.25 y(t) = [1 0 150) These three systems will be investigated for a z(t)=l, t€[0,l] with initial conditions - l 0 and noise cova rianc e “130) =9 (4.21) (4.22) (4.23) (4.24) step input 74 Therefore given the three response time systems, the per- formance index, and the cost for implementation function, the figures which follow Show the control performance costs and system per- formance costs. Each value of the performance index for the sub- optimal aperiodic sampling represents the minimized performance cost obtained through use of the ZXPOWL optimization subroutine for the specified number of sampling intervals. (The exact numerical values are listed by figure number in Appendix C.) The value of the continuous time feedback control performance costs are obtained by applying the feedback control Signal continuously as the system input control Signal. The performance of this continuous time (analog) control will be denoted J4 and the performance of the periodic and aperiodic sampled data control will be denoted as J3 and J1 respectively. The optimal performance costs for periodic and aperiodic, denoted by J: for i = 3 and i = 1 respectively, are defined as the minimum value of Ji over the set of feasible N (3.11). The optimal system per- formance ratio between periodic and aperiodic sampling (J3/Jl)* is defined as the maximum value of J3/J1 over the set of feasible N (3. 11). The optimal system performance ratio between continuous and aperiodic sampled-data control, (J4/J1)*, is defined as the maximum ratio J4 /J1 over the set of feasible N (3.11). Figures 4.12(a), 4.13(a), and 4. 14(a) graphically represent the values of the derived control performance index using the indicated number of sampling intervals for control of the three Type 2 unity 75 feedback control Systems presented. In each figure sub-optimal aperiodic sampling performance costs will be compared to both periodic sampling and continuous time control. As can be seen in Table 4. 3, continuous time feedback control performance costs (C) increase as the control response time of the system decreases. The main reason for the increase in performance costs is due to the fact that continuous time control is unable to significantly reduce the terminal errors as system response times decrease and is thus penalized by the control per- formance index. The optimal aperiodic (OAS) cost J: is lower than the optimal periodic sampling (OPS) costs J: for each system considered but outperforms continuous time control performance only for the c OAS ops (C/OA)* (PS/OA)* c 1 J J * ontro :1: >1: :1: >1: 4 _3 Response J4 J1 N1 J3 N3 "' * N1 J1 N3 Jl .1. FAST 2. 2 4. 9 8 8. 4 8 .44 8 104 2 MEDIUM 5. 2 2. 3 8 5. 3 5 2. 3 8 104 2 SLOW 11.6 4.5 6 6.8 2 2.6 6 2.3 7 Table 4. 3. Data Summary for Figures 4. 12(a) to 4. 14(a). MEDIUM and SLOW systems. The maximum ratio of the per- formance costs between continuous and sub-optimal aperiodic sam- . >1: , plmg (J4/Jl) increases as system control response decreases. This implies that Optimal aperiodic sampling can be more effective than continuous time feedback control in reducing final state errors for a 76 fixed length control interval for slower responding systems. The max- imum ratio of performance costs between periodic sampling and sub- optimal aperiodic sampling (J3/J1)::< is large for the FAST and MEDIUM control response Systems indicating that the periodic sampled—data system is unstable for N = 2. This conclusion is verified from Figures 4. 12 and 4. 13 respectively. It should be noted that system stability improved and performance cost reductions were made through use of periodic sampling for the SLOW response time system. In the figures which have been presented, performance costs of a continuous time control system, having feedback error signals sam- pled either periodically or ape riodically and applied as a system O 77 70- \ \ 65.) \ ‘ \ 60" \ 5~ 1 ‘ 5 \ Continuous \ 50 - ‘ -—- -- Periodic Sampling 45+ 1 “““ Ap ’od'c S pl' g \ en 1 am in 40-4 ‘ \ 357 \ 30« l ‘ 25 .. “ \ 20‘ 1 \ 1 15" 1 ______ .- \ 10‘ ----‘-..'-—--o—'-'-""‘--'.~~~~ 5‘ ‘*~ 0 r I I 1 1 I 2 3 4 5 6 7 8 N (Number of Sampling Intervals) (a) 75 "1 1 70- ‘, \ 65‘ ‘, \ 60" ‘\ ——--- Periodic Sampling 55" \ ------ Aperiodic Sampling ' 50 - “ \ 45‘ 1 \ 40" “ \ 354 l, ( \ 25 " 1 \ 20+ ‘ \ L-----—---__ 15 --~--o--—-“’~~-‘~ \ 101 «5r 5d 0 r l r I I m 2 3 4 5 6 7 8 N (Number of Sampling Intervals) (b) Figure 4. 12 - Performance Costs for a Type 2 FAST Response Time Control System Using a Step Input 78 Continuous 15 _ \ \ —- Periodic Sampling ., \ \ ------- Aperiodic Sampling .. \ \ \ -————Signal Rep. 3* used for system control 5 — \ ‘( \‘ II \\‘ %—-”—‘-~---§- _____ _. O r I 1 I l j 2 3 4 5 6 7 8 N (Number of Sampling Intervals) (a) 15-1 \‘ \ \ ---- -- Periodic Sampling " \ ‘ ------- Ape riodic Sam ling - \‘ \ ——._—_ Signal Rep. 1 used for « \ \ system control .1 \// .) O I r f I T ‘1 2 3 4 5 6 7 8 N (Number of Sampling Intervals) (b) Figure 4. 13. Performance Costs for a Type 2 MEDIUM Re- sponse Time Control System Using a Step Input. 79 12 -1 a." d /—-r""” n ,a/ 8 /" —-4 / «I a- / / qh\ ‘\\ f” \\ I, .l ‘- ----- Q— ————— .9..-_-.‘..---—-'” 4— ‘ Continuous .. """'"""" Periodic Sampling ““““ Aperiodic Sampling 0 I I I I I j 2 3 4 5 6 7 8 N (Number of Sampling Intervals) (a) ’ 15 «l —- — Periodic Sampling I, 1 ------ Aperiodic Sampling l/l .. l l / . I/’ I .—s--""""'——. - # ‘~~ / a’j” -__...:-<:_’-..——--" 5 _ q , o l t I T T fl N (Number of Sampling Intervals) (b) Figure 4. 14 - Performance Costs for a Type 2 SLOW Response Time Control System Using a Step Input 80 control input, nearly equal or in the case of sub-Optimal aperiodic sam- pling out-perform continuous time feedback control. These per- formance results are explained by the fact that through use of sam- pling, either periodic or sub-optimal aperiodic, delays are intro- duced into the feedback error signal. These delays introduce a mis— representation of the actual input error control signal by indicating to the system under control that the feedback error is greater (smaller) than it actually is. As a result, the system is made to respond faster (slower) than it normally would through use of a continuous time feed- back error signal which has no delays. For example, consider the control error signal ue(t) found in (4. 15). At the initial time, to, ue(to) will have an initial value dependent on z(to) and x1(to). Since the systems being investigated are assumed stable, ue(t) will eventually be reduced depending on the system type and the input signal. If ue(t) is sampled and held until time t1, such that t1 > to, any changes in ue(t) between times t0 and t1 will not become apparent until time tl when ue(t) is again sampled and held. This "delay" in the error feedback signal if used properly (e. g. adjusted through use of sub-optimal aperiodic sampling) will improve system response as indicated in the figures being presented. If the "delay" in the error signal is not adjusted severe system instability can result as shown in Fig.4.12 for periodic sampling. It should be noted that in all cases the performance costs for both periodic and sub-Optimal aperiodic sampling approach the value 81 of performance for the continuous time control as the number of sam- pling intervals increase. This result stems from the fact that using more sampling intervals the feedback control signal becomes a better approximation to the continuous time control. Figure 4. 13 also indicates the performance costs resulting from use of the optimal signal representation sampling interval sequence being applied to the control implementation problem. The results, as seen in Figure 4. 13(a), Show that control performance can be improved in three cases over periodic sampling and in two cases over con- tinuous feedback control. The control performance using sub-optimal aperiodic sampling determined by the control performance index, out-performed the optimal Signal representation sampling interval sequence in all cases as was expected. These results indicate that use of the Optimal sampling interval sequence for signal representation for system control does not necessarily yield good system control performance as was implied in earlier references [1, 24] . The results as seen in Figure 4. 12(b) indicates that for 3 to 7 sampling intervals sub-optimal aperiodic sampling would have to be employed since the control system is extremely unstable using periodic sampling for fewer than 8 sampling intervals. In Figure 4. 13(b) sub- optimal aperiodic sampling would again have to be used if 4 or less sampling intervals were required. If more than 4 sampling intervals could be used either periodic or sub-optimal aperiodic sampling could be used. Figure 4. 14(b) indicates that the total performance costs for 82 periodic and sub-optimal aperiodic sampling are approximately the same until 7 sampling intervals and than the costs for implementation for the aperiodic sampling criterion, in particular the processing costs, become large. The lengths of the sampling intervals which yield the per- formance costs indicated at 5 sampling intervals in Figures 4. 12 through 4. 14 are shown in Figure 4. 15. The initial time tO is 0. 0 with the final time tf being 1. 0 for each sampling interval sequence. Each sampling time ti, for i = 0, 1, 2, 3, 4, 5, is indicated with the length of a sampling interval being ti+ - ti' The exact length of 1 each sampling interval is found in Appendix C. Step Input, Type 2 System, N=5 L m l l FAST 1 l l l MEDIUM 1 1 l L SLOW 1‘6 t, t1 t3 ‘4 f; ‘ t;- Figure 4. 15. Representative Sampling Interval Lengths for Figures 4.12(a) to 4.14(a) using N=5. In Figure 4. 15 the shift in the sampling intervals as the band- width of the Type 2 control system decreases can be attributed to the fact that since the same performance index is used for each system, sampling has to yield large piecewise control signals and delay times to compensate for the "slowness" in the control system reSponse. It 83 —t) should be noted that during the first sampling interval (T0 = t1 0 the control input error signal will reflect the initial conditions of the control system. In this case using the initial conditions of -l. O for the "position" state variable and using a step input having an amplitude of l. 0 at time tO results in an initial input error control signal of 2. 0 (e. g. ue(to) = z(to) - x1(to) or ue(to) = l.0-(-l.0) = 2. 0) during the first sampling interval, T1. The longer sampling interval T1 and the corresponding error feedback delay becomes, the longer the control system is subjected to the value of 2. 0 as the input control error signal. Thus for the SLOW reSponding system the first sampling interval becomes large with respect to the first sampling interval for the FAST responding system in order to better use the delay with the large initial error to help decrease the terminal error which is penalized severely in the performance index. Controllingiof error feedback delay was used so effectively that for MEDIUM and SLOW response time systems the use of optimal aperiodic sampling and in some cases optimal periodic sampling were able to out-perform con- tinuous time control. The conclusions which can be drawn from Figures 4. 12 through 4. 14 are that the slower the system dynamics the more the control performance of the system can be improved through use of periodic or sub—optimal aperiodic sampling. For either of the two "slower" systems considered, periodic and sub-optimal aperiodic sampling yield comparable performance plus costs for implementation. Thus 84 the sub-optimal aperiodic criterion would be implemented since the control performance is significantly lower than the control per- formance for periodic sampling. The number of sampling intervals used in the implementation process would depend on the maximum value of the control per- formance which is acceptable. The number of sampling intervals selected for any particular control implementation would generally be the minimum number such that the control performance remains below this maximum. 85 4. 5 Control Performance Investigation by System Type In the previous section, control performance of a Type 2 feed- back control system having various response times, namely SLOW, MEDIUM, and FAST, were investigated for a step input. In this section the control performance of Type 2, Type 1, and Type 0 feed- back control systems will be investigated using a step, ramp, and noise input signals. Each of the three systems will have a bandwidth of approximately 7 radians per second which corresponds to that of the MEDIUM reSponse time system in the previous section. The per- formance of the Type 2 MEDIUM response time system found in (4. 13), having been previously investigated for a step input will now be investigated using a ramp and noise input. The ramp input will have the form z(t)=t for Oététf with the initial system state and noise statistics being -1 0 0 0 O E13001} = , 31.0.) = , 213“) = 0 0 0 0 0 The resultant control signal ue(t) found in (4. 15) will be sampled to sat- isfy (4. 16) and (4. 17). The derived control performance index and cost for implementation model are identical to those found in (4. 18), (4. l9), and (4. 20) for the step input investigation. The noise input will have the form z(t)=0 for Oététf 86 where the noise and randomly distributed initial states are assumed gaus sian with 1 0 12646)} = 2 , 11.8.) = 0 1 and 1 0 E{x(to)}= _g , “330) = 0 1 The derived performance index for the case of system noise disturb- ances has the form .01 0 J z [i 1 O LIEUN) - z(tNH' .01 [l 1 0 an.(tN) - z(tNll + tr[ 0 0 t1+1 N—l 2 .1 0 31(th + Z [.1 [[ 1 013m - z(t)] + tr[ i=0 0 0 t. 1 .02 0 y(t)] + .02 [ z(ti) - [ 1 0 ]_rn_(ti) ]2 + tr[ y(tin] dt + Jf 0 0 (4.25) with Jf being defined in (4. l9) and (4. 20). The added trace terms are a result of the Specified system statistics. Therefore given the Type 2 system, performance index, and cost for implementation function, the control performance levels for per- iodic and sub-optimal aperiodic sampling criteria are plotted in Figures 4.10, 4.12, and 4.18 for the step, ramp, and noise inputs respectively. As can be seen in Table 4. 4, continuous time control (C), Optimal periodic sampling (OPS), and optimal aperiodic sampling (OA) performance costs decrease using a ramp input rather than a step input and increase Sharply with a noise input. The decrease in 87 performance costs is primarily due to differences in the terminal errors caused by the step and ramp inputs. The large performance cost for the disturbance noise input is attributable to the fact that the per- formance index penalizes large variances in the terminal error on the same level as it penalizes large tracking errors. (see per- fo rmance indexiin (4. 21) Thus it is apparent that with the Specified initial state and noise covariances for the Type 2 system being con- sidered state variances increase significantly over the control interval. As can be seen in Table 4. 4, optimal aperiodic (OA) sampling JT out-performs either continuous time control (C) or Optimal per- iodic sampling (OPS) for each input type considered. The maximum ratio of performance costs between continuous and sub-optimal aperiodic sampling (C/OA)* remains the same for the step and ramp inputs. The maximum ratio of performance costs between periodic and sub-Optimal aperiodic sampling (PS/A)* shows vast improvements 31‘ * c OA ops (C/OA) (PS/0A) 1'5 >1< T 2 J 1* N* * * 34 l3 Ype 4 1 1 J3 N3 1 N4 Jl N3 INPUT T — 1 STEP 5.2 2.3 8 5.3 5 2.3 8 104 2 RAMP .931 .39 8 1.0 4 2.4 8 103 2 NOISE 11.0 8.7 2 8.9 2 1.3 2 1.03 2 Table 4. 4. Data Summary for Figures 4.13(a), 4.16(a), and 4.17(a). 88 \ Continuous ‘ \ \ —- — Periodic Sampling . \ l ------ Ape riodic Sampling 4 ‘\ \ 5 \ 2 _fi \\ 1 I \ \ ,/_——— ‘\ “—0 . \\ ( l/ . \ \ / cl \\ ‘ / 1 _ \\ V \ II \“ d ‘\‘.--~-.~~ q ~‘~*_____~ 0 r I I l I 1 2 3 4 5 6 7 8 N (Number of Sampling Intervals) (a) 12— 1 \ ——---—- Periodic Sampling 1 ______ . . . 1 \ Aperiodic Sampling 101 9. \ 8;: \ ,a’” X’A\ /------” -4 \\ I \\ // 5 \\\ ’l/ \ \\V” 4-1 \x ‘ 3" \ if ..__....,...._-——-——-—-——--—~ 0 I T I I f 1 2 3 4 5 6 7 8 N (Number of Sampling Intervals) (b) Figure 4. 16 - Performance Costs for a Type 2 MEDIUM Response Time Control System Using a Ramp Input 89 10 '- -( - *- ’— -( /” - 4 _. ’- I / 9 ~/ .4 __,_____... _____ ._ ----- "‘ 8: Continuous = 11. 011 _‘ -—--—- Periodic Sampling q ----- Aperiodic Sampling T 7 I I T l I 7 2 3 4 5 6 7 8 N (Number of Sampling Intervals) (a) — - —— Periodic Sampling 30-: ------ Aperiodic Sampling ”-v‘” a”‘. d I" ””’ I -( I/ 20‘ ,4” .. ’I” ". d ’I’ ACII 1 0 I l I I I fl 2 3 4 5 6 7 8 N (Number of Sampling Intervals) (b) Figure 4. 17... Performance Costs for a Type 2 MEDIUM Response Time Control System Using a Noise Input 90 because the periodic sampled-data system was unstable using only two sampling intervals. The maximum performance ratios between per- iodic and sub-optimal ape riodic sampling (PS/OA)* for noise inputs indicate periodic sampling is near Optimal when the system is to be controlled for noise disturbances only. Optimal aperiodic sampling yielded lower performance costs, as seen in Figures 4.13, 4.16, and 4.17, than either periodic or continuous time control for each input and number of sampling in- tervals considered. These performance results are again attributed to the effective use of feedback error signal delay to improve control performance (See page 80 for additional discussion on delay. ) As the number of sampling intervals increase, the delays are reduced yielding a sampled feedback error signal which appears more con- tinuous to the system under control. As is indicated in Figure 4. 12(a) for 8 sampling intervals, the performance costs for periodic sampling would decrease and approach continuous time control as a limit as the number of sampling intervals became large. [52] Likewise for sub- optimal aperiodic sampling as the number of sampling intervals were increased the values of the derived performance index-would approach that of the continuous time control. [52] In Figure 4. 16(b) the control performance costs plus costs for implementation are clearly lower for periodic sampling using 4 or more sampling intervals. The high costs for sub—optimal aperiodic sampling are attributed to the processing involved in obtaining the 91 optimal sampling interval sequence. Using less than 4 sampling in- tervals, periodic sampling results in system instability and thus sub-optimal aperiodic sampling would have to be used. Figure 4. 17(b) indicates that periodic sampling yields almost constant performance costs compared to sub-optimal aperiodic sampling. The increases in costs for aperiodic sampling are due in all cases to increased computational expenses resulting from Optimization of the sampling interval sequence. Since sub-optimal aperiodic and periodic sampling yield almost the same performance and since Optimal aperiodic yields almost periodic samples, periodic sampling should be used for noise inputs. The performance of a Type 1 MEDIUM response time unity feedback control system will now be investigated for a step, ramp, and noise input. The following control system model will be used: 0 l 0 350) = gm + new 0 -5 25 y(t) = [1 0133(t) with the control error Signal for the above system being ue(t) '3 z(t) - Y“) where z(t) is a specified input signal for 0f: t f: tf. The first two input signals being considered have the form : z(t) 1 (step input) and Z(t) t (ramp input) 92 with the initial system state and noise statistics being -1 0 0 O 0 s {1%)}: , yxuo) = . ‘30) = 0 0 0 0 0 The resultant control signal ue(t) found above, will be sampled to sat- isfy (4. l6) and (4. 17). The derived control performance index and cost for implementation model are identical to those found in (4. 18) and (4. l9), and (4. 20) for the step input investigation. The third input, the noise, will have the following form: z(t) = 0 where the noise and randomly distributed initial states are assumed gaussian with l 0 E {14%)} z 2 ' .Yx(to) and E {y(t)} = 9 . ‘Ijm = O 1 The derived control performance index and cost for implementation model are identical to those found in (4. 25), (4.18), and (4.19). It should be noted that for a ramp input a Type 1 unity feedback control system has a steady-state error. Therefore, errors in the final state values using a ramp input will always be present with con- tinuous time feedback control. As can be seen in Table 4. 5, optimal aperiodic (OA) sampling >'.< 1 outperforms either continuous time control (C) or optimal periodic J 93 sampling (OPS) for each input considered. The maximum ratio of performance costs between continuous and sub-optimal aperiodic sampling (C/OA)* occurs for a ramp input. The maximum ratio of performance costs between periodic and sub—optimal aperiodic sam- pling (PS/OA)* also occurs for a ramp input. These optimal im- provement ratios can be attributed to the fact that Optimal aperiodic sampling is able to reduce final state errors to a greater extent for a ramp input than a step input compared to continuous time or periodically sampled control. The maximum performance ratio be- tween periodic and sub-optimal aperiodic sampling (PS/OA)* for noise inputs indicate periodic sampling is near optimal when the system is to be controlled for noise disturbances only. a): C OA ops (C/OA) (Ps/oA)* T l * ' * I E J * .ype >5 . INPUT 1 1 1 STEP 5. 7 3. 8 8 22. 6 8 1. 5 8 8. 5 2 RAMP 1. 0 .29 8 3. 9 8 3.4 8 24.0 4 NOISE .45 .37 2 .38 2 1. 2 2 1.02 3 Table 4. 5. Data Summary for Figures 4. 18(a) to 4. 20(a). Figure 4. 18(a) and 4. 19(a) indicate that delay caused by periodic sampling degraded control performance for the number of sampling in- tervals considered. Sub-optimal aperiodic sampling with its ”controlled delay" was able to outperform continuous time control for N greater than six sampling intervals for a step input and three sampling intervals 94 83 Continuous 75 \ —- -— Periodic Sampling 7O \ ------ Aperiodic Sampling 15‘ 10‘] -~~~-——_____‘_-__—u~ 5‘ ‘D——_“----- 0 1 I I I I I 2 3 4 5 6 7 8 N (Number Of Sampling Intervals) (a) a .. \ . . . q —- -— Perlodlc Sampling 4 \\ ------ Aperiodic Sampling \ N w 4.x U1 0“( \1 00.1 N (Number of Sampling Intervals) (b) Figure 4.18. Performance Costs for a Type 1 MEDIUM Re- sponse Time Control System using a Step Input. 95 7 12 _ f\\ Continuous 4 \ —- — Periodic Sampling / \ ------ Aperiodic Sampling .. / \\ 8 — \ ’ \ C/ \ \ d \ \ q \‘ \‘ _( \ l -( qt.- 0 T I I T I I ' Z 3 4 5 6 7 8 N (Number of Sampling Intervals) (a) 12 - N _. __ Periodic Sampling / \ —————— Ape riodic Sampling \ -( \\ -..—- . \f"" ’I’ \ 4 d IA- ’IA ------ I” ~ .. ’Il ~‘~'V”’ 'I 0 j I I I I *1 2 3 4 5 6 7 8 N (Number of Sampling Intervals) (b) Figure 4.19. Performance Costs for a Type 1 MEDIUM Re- sponse Time Control System using a Ramp Input. 96 .401 /- .39 1 1‘ \ / 38 - / I Continuous = .450 ’ / —" P 'd' S 1' -/ / erlo 1c amp ing 4 ,l’ """" Aperiodic Sampling 4 /’ I -( ” I .37 _k W I T I I I 2 3 4 5 6 7 8 N (Number of Sampling Intervals) (a) —- -— Periodic Sampling ------ Aperiodic Sampling ” (-_-—_* mam-h—*--—~_-_ I I I I fl 2 3 4 5 6 7 8 N (Number of Sampling Intervals) Figure 4. 20. Performance Costs for a Type 1 MEDIUM Re- sponse Time Control System using a Noise Input. 97 for a ramp input since it was more effective in reducing final state errors. In the case of sub-optimal aperiodic sampling where the sampling delays can be "tuned to the system" through variation of the sampling interval lengths, the performance costs have been re— duced. Periodic sampling cannot adjust sampling interval lengths, and thus the delays caused by sampling, as a result performance costs are increased. As was pointed out previously, and as can be seen in Figures 4.18(a) and 4.19(a), the performance cost in periodic and Optimal aperiodic sampling approach the continuous time per- formance cost as N increases. Figure 4. 19(a) indicates the performance costs for the Type 1 control system using the Specified noise input. As can be seen the performance costs for periodic and aperiodic sampling almost identical. This indicates that periodic sampling is near optimal when used to control for noise disturbances. Figure 4. 18(b) indicates sub-Optimal aperiodic sampling to be the only means of implementation using the number of sampling in- tervals investigated. Figure 4. 19(b) indicates sub-optimal aperiodic sampling yields lower performance index plus costs for implementation than periodic sampling for up to 6 sampling intervals. Since the processing costs are large for sub-optimal aperiodic sampling, periodic sampling yields lower overall costs for 7 and 8 sampling intervals even though a performance improvement of 13. 5 was realized through the use of 98 sub-optimal aperiodic sampling at 8 sampling intervals. Figure 4. 20(b) indicate periodic sampling yields almost constant performance costs compared to sub—optimal aperiodic sampling. The increases in costs for aperiodic sampling are due in all cases to in— creased computational expenses resulting from optimization of the sampling interval sequence. The performance of a Type O MEDIUM response time unity feedback control system will now be investigated for a step, ramp, and noise input. The following control system model will be used: 0 l 0 em = §(t) + new -24 —1O 19 y(t) = [1 0 ]§(t) with the control error signal for the above system being new = z(t) - y(t) where z(t) is a specified input signal for 0 1.- t 5 tf. The resulting control signal ue(t), will be sampled to satisfy (4. l6) and (4. 17). The three input signals being considered for this system, viz. step, ramp, and noise, have been Specified in the investigation of the Type 1 system just completed. The derived performance index and cost for im— plementation model to be used for the noise input are found in (4.25), (4.18), and (4.19). It should be noted that for a step input to a Type 0 unity feed- back control system there occurs a steady state error in the output signal with reSpect to the input signal which can not be eliminated by 99 conventional feedback techniques. A ramp used as an input to a Type 0 unity feedback control system yields errors between the input and output signals which increase with time. As can be seen in Table 4. 6, optimal aperiodic (OA) sampling J? outperforms either continuous time control (C) or optimal periodic sampling (OPS) for each input considered. The maximum ratio of performance costs between continuous and optimal aperiodic sampling (C/OA)* occurs for a ramp input. The maximum ratio of performance costs between periodic and sub—optimal aperiodic sam- pling (135/ 0A)* also occurs for a ramp input. These optimal im- provement ratios can be attributed to the fact that optimal aperiodic sampling is able to reduce final state errors to a greater extent for a ramp input than a step input compared to continuous time or per- iodically sampled control. The maximum performance ratio be- tween periodic and sub-optimal aperiodic sampling (PS/ 0A)* for noise inputs indicate periodic sampling is near optimal when the * (3 0A OPS (C/OA) (PS/my“ ~'< * J * * T * | Type 0 J4 Jj‘ NT 33 N3 34 N3 3‘3 N3 INPUT 1 l 1 1 STEP 4.3 1.6 8 12.2 8 2.7 8 11.5 2 RAMP 2.7 .22 8 5.4 8 12.2 8 33.0 3 NOISE .23 .19 2 .19 2 1.2 2 1.02 2 Table 4. 6. Data Summary for Figures 4. 21(a) to 4.23(a). 100 45.. \ . \ Continuous .. \ -—-—— Periodic Sampling 4 ‘\ —————— Aperiodic Sampling 15"" \~ ‘ \ 0 i T T I r I *1 3 4 5 6 7 8 N (Number of Sampling Intervals) (a) 45 j \ —--— Periodic Sampling \ ----- -' Ape riodic Sampling " \ 30 -» \ .. \\ .1 \\ 15 "" ‘\~ ‘ \ din _____ ‘--- ‘_--”—"‘~--~-~«”’ q - O r 1 T l I 1 2 3 4 5 6 7 8 N (Number of Sampling Intervals) (b) Figure 4. 21. Performance Costs for a Type O MEDIUM Re- aponse Time Control System using a Step Input. 101 12 -' / \ Continuous q I \ —-- Periodic Sampling ‘/ \ \ ------ Ape riodic Sampling ----—“——--- ‘—-m-- 0 V 1 ; ----- _s— '43“ '- 2 3 4 5 6 7 8 N (Number of Sampling Intervals) (a) 12 _1 / .1 ’/ \\ --- —- Periodic Sampling / ------ Aperiodic Sampling .. \\\ 8 T \\ .1 \~ 4 \C- \ 4 ,x-m' " “ I — v”’ 4 ”/-"" d ’,—-"”’ 4'- ----- q” 0 1 I f I 1 j 2 3 4 S 6 7 8 N (Number of Sampling Intervals) (b) Figure 4. 22. Performance Costs for a Type O MEDIUM Re- sponse Time Control System using a Ramp Input. 102 ‘ Continuous : . 228 (not shown) __——— Periodic Sampling 21" _..— -— —- Aperiodic Sampling Jo " l,_.._-- “fir” _. m:—--‘"‘"‘ 20 ) “S’w’ 1 /<”/ z ” II t x’ . 19‘; 1 T I l I 1 2 3 4 5 6 7 8 N (Number of Sampling Intervals) (a) 1. 2" . . . ’- -—-- — Periodic Sampling I” “ ------ Aperiodic Sampling /’ . //’ / I/ ’l' 8 "l I,’ qk‘ ”/’ J \ ’I’ .. \\ ,” -( \\ ’l’ \ I 4“ V’ 0 1 I r T T I 2 3 4 5 6 7 8 N (Number of Sampling Intervals) (b) Figure 4. 23. Performance Costs for a Type 0 MEDIUM Re- sponse Time Control System using a Noise Input. 103 the system is to be controlled for noise disturbances only. Performance improvements in sub-optimal aperiodic sampling are again attributed to the ability of aperiodic sampling to "tune sampling delays" to the system and thus minimize the derived per- formance index through use of the optimal sampling interval sequence. Using the ramp input this example was able to show major im- provements in control performance through use of sub-optimal aperiodic sampling. Again as in the past explanations, tuning of the sampling intervals, and thus the feedback control error delays, through use of the sub-Optimal aperiodic sampling techniques as pre- sented in Chapter III, vastly improve control response for, in this example, MEDIUM speed Type O unity feedback control system with ramp inputs. Figure 4. 23(a) indicates the performance costs for the Type 0 control system using the Specified noise input. AS can be seen the performance costs for periodic and sub-optimal aperiodic sampling yielded almost identical performance costs. This indicates that periodic sampling is near optimal when used to control for noise disturbances. Figure 4. 21(b) indicates the total sub-optimal aperiodic sam- pling costs to be lower than periodic sampling for all the sampling intervals considered. As can be seen in the sub-optimal aperiodic sampling case, the precesssing costs increase with an increasing number of sampling intervals and could easily result in the total sub-optimal aperiodic sampling costs exceeding those for periodic 104 sampling using 9 or 10 sampling intervals. Figure 4. 22(b) indicates improved performance plus costs for implementation for sub-optimal aperiodic sampling for all sampling intervals. As can be seen for 7 and 8 sampling intervals, processing costs offset improvements in control performance Shown in Figure 4. 22(a). Figure 4. 23(b) indicates periodic sampling yields almost con- stant performance costs compared to sub-optimal aperiodic sampling. The increases in costs for aperiodic sampling are due in all cases to increased computational expenses resulting from optimization of the sampling interval sequence. 105 4. 6 Analysis bLSystem Type. The lengths of the sampling intervals using N=5 for the Type 2, Type 1, and Type 0 feedback control Systems are Shown in Figure 4. 25(a), 4. 25(b), and 4. 25(c) for a step, ramp, and noise inputs respectively. As can be seen the lengths of the sampling intervals decrease as system type number decreases for both step and ramp inputs. The length of the sampling intervals increase as the type number decreases for the noise input. The changes in the sampling interval lengths with respect to system type are a result of the differences in the Open loop plants being considered for each system type. Consider the open loop reSponses of the three plants to a unit step input for t €[ 0 , 1 ] as shown in Figure 4. 24(a) with the plant transfer function for a step input being shown in Figure 4. 24(b). As can be seen, the Type 2 System reSponds quickly to the step input, having a terminal value of 17. 5 at t : 1. The response of the Type 0 and Type 1 systems are very similar, each being nearly linear over the time interval being considered with the Type 0 system having approximately twice the Slope of the Type 1 system. (See time domain reSponses in Figure 4. 24(b)) AS a result of the sample and hold operation used in system control, piecewise constant signals are applied as inputs to the various open loop plants. (See Chapter III) The responses of each open loop plant, as seen in Figure 4. 24, will proceed as shown 106 Open Loop Reaponse R(3) (3(3) C(8) C(8) = G(s)R(s) R(s) = 1/8 18 17 System C(S) C(t) Type 2 5(S + 5) . 1 l6 ——--—- — 32 5 5t + 12. 5t2 15 1 25 . _1__ l4 s(s+5) S -1/25+t+_l_e'25t 25 0 l9 . _L 13 (s+3)(s+2) s 3.16+6.3e-3t- 2, 9. Se' 1 C(t) 12" 11‘ 1°? 9 ~ 2 8 cl 7 cl 6 .1 5 d 4 II 3 q 2 q 0 l-l 44 I A l 0 I ' I I I I I I I I I 0 .l .2 .3 .4 .5 .6 7 8 .9 l 0 T (seconds) Figure 4. 24. Open Loop Plant Responses of the Type 2, l, and 0 Plants for a Unit Step Input. vy-v-vv rm 107 (a) Step Input, MEDIUM Response Time Systems, N=5. if, :1 ‘3’ (‘4 , Type 2 ti: :1 :3 ‘14 Type 1 ll, :1 I.) ‘tlf t - tTYPe O I: to f .5 (b) Ramp Input, MEDIUM Response Time Systems, N=5. i, 3, t, .3. Type 2 ’1’ It: {13 fl, Type 1 I L ‘ ‘ 1 T e O t., i, t, t, (q 12;. f5 yp (c) Noise Input, MEDIUM Reaponse Time Systems, N =5. 1“. *1: 7’; :4 Type 2 t3 t; t: :4 Type 1 to f, 112 :13 a q” Zype 0 Figure 4. 25. Sub-Optimal Sampling Interval Sequences, N25, for Type 2, l, and 0 Control Systems. 108 until a different piecewise constant input is applied. The length of time each piecewise constant signal is held is used as a control variable for the system. Therefore using the sub-optimal sampling interval sequences as control variables, the resulting System responses are shown in Figure 4. 26 for a step input. The closed 100p step responses as seen in Figure 4. 26 indicate an underdamped response to a step input for the Type 2 system compared to the rather linear overdamped responses of the Type 1 and Type 0 systems. Longer sampling intervals for the Type 2 control system are due to the fact that the control performance index severely penalizes terminal errors making it necessary to reduce the levels of the piecewise constant input error signals as quickly as possible thereby reducing changes in the system state due to non-zero input control levels during the final sampling inte rval. The reduction in the sampling interval lengths for the Type 1 and Type 0 systems are attributable to the fact that each of these systems reSpond slowly to the step input, thus large feedback control errors have to be used effectively to reduce terminal errors. Effective use of the feedback control errors to improve system response is accomplished by sampling earlier in the control interval and thus 109 Sub-Optimal Aperiodic Sampling Re3ponse _____ _. Periodic Sampling Response C(t) Time (Seconds) Type 1 1 cm 0 -1 Time (Seconds) Figure 4. 26. Type 2, l, and 0 Feedback Control System Response to a Unity Step Input .using Periodic and Sub-Optimal Aperiodic Sampling. 110 taking advantage Of the large initial errors between the System state and the step input signal. A Similar explanation is used to explain the differences in the sampling interval lengths, Shown in Figure 4. 25(b), between the Type 2 and Type 0 and 1 systems for a ramp input signal. It should be noted that the terminal errors are larger, as seen in Figure 4. 26, for the Type 1 System than either the Type 2 or Type 0 systems. These large terminal errors for the Type 1 system are due to the relatively Slow response of the Type 1 system and the short control interval which does not allow significant changes to occur to reduce terminal errors. (See Figure 4. 24(b)) The third input considered, the noise input, yields a sampling interval sequences which are Short and periodically spaced at the beginning of the control interval for the Type 2 system compared to the periodically spaced sampling intervals Of the Type 0 and Type 1 systems. (See Figure 4. 25(c)) The noise which occurs early in the control interval has a more severe effect on the terminal errors as the number Of integrators in the plant increase. (See Figure 4. 26(b) for the plant transfer functions.) Thus, the sampling intervals must be chosen smaller as the type number increases to offset the effect Of the increase in noise during the initial part of the control interval. Table 4. 7 summarizes optimal control performance data found in Chapter IV for the Type 2, 1, and 0 unity feedback control systems with respect to step, ramp, and noise inputs for various methods of control. The Optimal performance results present in Table 4. 7 will 111 be compared by method Of sampling versus sytem Type for each input considered. Figure 4. 27 is presented to Show trends and generalities not readily seen in tabular data presentations. Ramp Input (Figure 4. 27(a)) In the unity feedback control systems being considered, a decrease in System type increases the steady state terminal errors. [50] For a ramp input the Type 2 System has no inherent steady state terminal errors, while the Type 1 system has constant steady state terminal errors, and the Type 0 System has terminal errors which increase with time. AS can be seen in Figure 4. 27(a), the continuous time control (C) and Optimal periodic sampling (OPS) control per- formance costs increase with a decrease in system type. These increases in performance costs are attributed to the increased terminal errors which result from changes in System type. Since the control performance index heavily penalizes terminal errors, increases in performance costs with decreases in system type were expected. It should be noted that through use of Optimal aperiodic sampling control performance costs were reduced compared to the other methods Of control being investigated. This reduction in performance cost indicates that controlled feedback error delays, aS discussed in previous sections, are effective in reducing the inherent terminal errors present in the Type 0 and Type 1 control systems having a ramp input. 112 c OA OPS (C/OA)* (PS/OA)* =1: 5 £13: * >1: >1: * a: J Type 2 J4 J1 N1 J3 N3 3.4 Ni J1 N2 INPUT 1 l 1 STEP 5. 2 2. 3 8 5. 3 5 2. 3 8 104 2 RAMP .921 .39 8 1.0 4 2.4 8 103 2 NOISE 11.0 8.7 2 8.9 2 1.3 2 1.03 2 * * c OA OPS (c/OA) (PS/0A) * * * * * Type 1 J4 J’i‘ N1 J: N: l4 Ni _J_3 N3 INPUT J1 1 J1 '1' STEP 5.7 3.8 8 22.6 8 1.5 8 8.5 2 RAMP 1. 0 .29 8 3. 9 8 3.4 8 24. o 4 NOISE .45 .37 2 .38 2 u1.2 2 1.02 3 * * c OA OPS (C/OA) (PS/0A) >1: * Q * * Type 0 J J* N* J N* :14 N4 23 N 4 1 1 3 3 J —- J 3 INPUT 1 1 1 1 STEP 4.3 1. 6 8 12.2 8 2. 7 8 11.5 2 RAMP 2. 7 . 22 8 5. 4 8 12. 2 8 33. o 3 NOISE .23 .19 2 .19 2 1.2 2 1.02 2 Table 4. 7. Numerical Data for Figure 4. 25. /A\ 20 ’ \\ ’ \ I’ \ Type of Sampling used: 15 [I \\ --——— Continuous ’/ \\ -— -—-- Optimal Aperiodic I \ -—---- Optimal Periodic I 5 10 I “ ’,’ STEP INPUT 5 Li \ (a) an" \‘I. OI" \~ .,, O f l I l 2 1 0 System Type 20 . Type of Sampling used: Continuous 15 _- —— Optimal Ape riodic ...... Optimal Periodic 10 RAMP INPUT (b) 5 fl“‘-¢‘ I’)“—" O T I -——“A_T_-_-q— 2 l 0 System Type Type of Sampling used: 7- 0 Continuous _... — Optimal Ape riodic .. .. ......... Optimal Periodic NOISE INPUT (C) .—a o 1111111111111111111114 System Type Figure 4. 27. Optimal Performance Costs by System Type and Input Signal. 114 Stgglnput (Figure 4. 27(b)) For a step input the Type 1 and Type 2 control system have no steady state errors compared to an inherent constant steady state error for the Type 0 System. Thus based on the performance cost results for the ramp input it was expected that performance costs would increase with a decrease in system type for the step input. Instead performance costs for the Type 1 control system exceeded those for the Type 2 and Type 0 Systems for each method of sampling being considered. Therefore since the steady state errors for the Type 1 system are zero, the plant reSponse time must be considered in explaining the large performance costs for the Type 1 system. Clearly from Figure 4. 24 the Type 1 system reSponded slower over the chosen control interval to a step input than did the Type 2 and Type 0 systems. As can be seen in Figures 4. 26 the response of the Type 1 system yields large terminal errors. Thus with the terminal errors being penalized heavily by the control performance index, costs increased as seen in Figure 4. 27(b). The Type 0 system out-performed the Type 1 system even with its inherent steady state errors for a step input. The control performance cost reductions for the Type 0 system are due to the fact that the larger feedback control errors coupled with a faster system response over the control interval resulted in a reduction in terminal errors. For Optimal aperiodic sampling the faster system reSponse time for the Type 0 system along with optimal adjustment of the feedback error delays improved control performance for each input considered. 115 Noise Input (Figure 4. 27(c)) The performance costs with respect to system disturbance noise decreases as system type decreases for each method Of sampling investigated. This implies that control performance costs with reSpect to noise are dependent on System structure, and thus system type. (See the control performance index in equation 4. 21)) Apparently the integration of noise drastically effects the value of the control performance index since performance costs decrease rapidly as the system type decreases. This is consistent with the analysis of sampling times for the different system types since it was found that sampling must be performed faster initially in the control interval for systems with integrators in order to control for the integration effect of the noise that occurs early in the control interval. In summary, this section considered the Open loop response times of the Type 2, l, and 0 system plants subject to a unit.. step input signal. The lengths of the sampling intervals for N=5 were justified using the Open loop plant reSponse times for a step input in an effort to better explain system behaviour using the various methods of sampling under consideration. A summary of the optimal performance costs for each system type, as shown in Figure 4. 27, were used in comparing optimal performance costs for various type systems for a given input. In conclusion this chapter revealed that Optimal aperiodic sam- pling, as Specified in Chapter III, yielded improved control performance 116 with reSpect to Optimal periodic sampling and in several cases of con- tinuous time control. If control performance costs alone are to be considered, eSpecially for the slow responding Type 0 control systems with inherent terminal errors, Optimal aperiodic sampling appears very promising. As in the case Of Signal representation, performance costs alone cannot be considered in the choice of the sampling method to employ. Therefore, the greater the knowledge the system designer has about the control problem at hand and the constraints he is required to work within the better the control system design will become. V. CONCLUSIONS This chapter summarizes the main results of this thesis and sug— gests areas Of future research. 5. 1. Summary This thesis has developed a theory of optimal aperiodic sampling for signal representation and system control. The theory encompasses previous work in adaptive sampling, optimal periodic sampling, and aperiodic sampling in addition to presenting a cost for implementation model. The theory of optimal aperiodic sampling considers the signal rep- resentation and system control problems to be different and are thus considered separately. Past studies have considered the signal repre- sentation and system control problems together under the assumption that good signal representation through sampling leads to good system control. This theory also develops a cost for implementation model which indicates the relative cost required to implement a particular signal representation or system control problem via the selected meth- od Of sampling. Previous studies of sampling methods have neglected or misrepresented implementation costs and thus have reduced the practical significance of the method of sampling being considered. The theory which has been deveIOped Optimizes both the sampling interval sequence and the number Of sampling intervals. Three methods Of sampling were considered in the development Of the Optimal aperiodic sampling theory. Each sampling method studied optimized either the l 17 118 sampling interval lengths or the number of sampling intervals used but never considered joint optimization. Adaptive sampling was studied since it was the first non-periodic method Of sampling which yielded improved performance Over periodic sampling. Adaptive sampling techniques optimized the length of each sampling interval on an indivi- dual basis through use Of a sampling rule Obtained by using various ap— proximations to the original integral performance index. Optimal peri- odic sampling was next studied since it optimized the number of periodi- cally spaced sampling intervals. Finally work in aperiodic sampling which optimized the sampling interval sequence rather than Optimizing the lengths of the individual sampling intervals (as was done in adaptive sampling) was investigated. Concepts from each of these methods of sampling were used in the development Of the theory of optimal ape rio- dic sampling for signal representation and system control. Therefore the theory of optimal aperiodic sampling provides a general framework for investigating various methods of sampling ap- plied to either the signal representation or system control problem. Through use of the proper constraints on the number of sampling inter- vals or the length of each sampling interval, each method Of sampling, viz. adaptive, optimal periodic, and aperiodic, can be investigated using the theory which has been developed for Optimal aperiodic sam- pling. In addition a cost for implementation model is presented which indicates the relative implementation costs associated with the method of sampling being investigated. For the signal representation problem the approach taken was to sample a given continuous time signal record non-periodically adjusting the length of each sampling interval to minimize a derived signal repre- sentation performance index. The derived signal representation perfor- 119 mance index was made dependent on the signal model, initial conditions, input driving signals, and model disturbances. Adjustment Of the sam- pling intervals we re made via the ZXPOWL non-linear programming optimization subroutine found on the CDC 6500 computer System. For each adjustment made in the length Of a sampling interval(s), a re-eval- uation of the derived performance index was made until the minimization of the derived performance index was accomplished. The resulting sampling interval sequence yielded minimum Sampling errors with re- spect to the performance index using the number of sampling intervals allowed and thus was used with the corresponding signal magnitudes to yield a piecewise constant representation of the original continuous time signal. The piecewise constant signal representation required approximately 10% of the storage needed for the continuous time signal record with sampling errors being reduced as much as 95% with respect to an equal number Of periodically spaced sampling intervals. A simplified remote display problem was proposed. The display problem investigation was used as a basis for the introduction of the processing and communications costs as they apply to implementation of various methods of sampling. These costs for implementation were added to the performance costs obtained through minimization of the derived performance index to yield a total performance level reflecting improvements in signal representation through aperiodic sampling and at the same time indicating the relative costs for these improvements in the quality of the signal representation. Two main conclusions can be made as a result of the sampling for signal representation investigations conducted in Chapter IV, Section 4. 2. First, minimization of the derived signal representation perfor- mance index, as found in (2. 24), through adjustment of the length of 120 each sampling interval can substantially reduce the sampling errors present in the piecewise constant representation of the continuous time signal record with respect to periodic or adaptive sampling. Second, ob- taining of the Optimal sampling interval lengths, and hence the best piece- wise constant representation using the specified number Of sampling in- tervals, results in increased computational expense and complexity com- 51-» pared to adaptive or periodic sampling. Thus in each situation where continuous time signal records are to be sampled, the system designer _“ '. '1" should consider each sampling alternative at his disposal. ’2’ .‘a The control implementation problem was based on the premise that periodic sampling need not be used to implement feedback control systems designed through use of continuous time techniques. As a result, sampling for signal representation was extended to the sampling of the continuous time feedback error signal generated by the system control law. Again as in the case of signal representation, a performance index was proposed (see (3. 16)). The control implementation performance index resembled conven- tional optimal control performance indices in that it penalized for control energy expenditures, tracking deviations, and final state errors. However, the control implementation index also included a cost of implementation term. The performance index was then derived in terms of the sampling interval sequence, '_I‘_ and the number Of sampling intervals, N. The derived performance index was minimized through use of the ZXPOWL non-linear programming optimization subroutine found on the CDC 6500 computer system. Each attempt at minimization required evaluation Of the derived 121 control. implementation performance index through simulation of the control system using a specified sampling interval sequence. Thus given the required system information as indicated in Chapter IV, Section 4. 3, control of various feedback Systems were investigated with re- spect to sampling of the feedback control error signal. The results of Chapter IV, Section 4. 3, indicate that sub-optimal aperiodic sampling of the continuous time feedback control error signal can improve control performance, based on the derived control imple- mentation performance index, compared to periodic sampling and some cases of continuous time control. Control performance improvements were more apparent as the bandwidth of the closed loop unity feedback System decreased and the system type numbers were reduced. Com- putational cost requirements necessary to Obtain improved control performance through use of sub-optimal aperiodic sampling increased with the number of sampling intervals being optimized such that sam- pling interval Optimizations for a large number of intervals become computationally prOhibitive. Using as few sampling intervals as pos- sible yielded totally unacceptable periodic sampling performance while sub-Optimal aperiodic sampling was able to maintain reasonable system stability through adjustment of sampling delays as discussed in detail in Chapter IV, Section 4. 3. 5. 2. Future Research In both the signal representation and control implementation pro- blems the signal which is to be sampled needs to be known as in the case of signal representation or generated through control system sim- ulation for the control implementation problem. For on-line applica- tions where the continuous time signal record or feedback control error 122 signal are not known a priori further research into on-line signal pre- diction, estimation, and system modelling is necessary. As indicated throughout this thesis optimization of the lengths of each sampling interval is necessary to maximize signal representation or system control performance through use of aperiodic sampling. As was shown in Chapter IV, aperiodic sampling improves performance F but the computational expenditures necessary for optimization made I actual implementation unlikely. Therefore investigation into more computationally efficient Optimization routines, or possible adaptive/ aperiodic hybrid sampling could be further studied. LIST OF REFERENCES LIST OF REFERENCES [ l] M. L. Smith, Jr. , ”An Evaluation of Adaptive Sampling, 11 IEEE Trans. on Automatic Control, Vol. AC-l6, pp. 282-284, June 1971. [ Z] M. Athans, W. S. Levine, and A. H. Levis, "A System for the Optimal and Suboptimal Position and Velocity Control for a String of High-Speed Vehicles, ” 5th International Analogue Computation meetings, 1966. [ 3] M. Aoki and M. T. Li, "Optimal Discrete-Time Control System with Cost for Observation, ” IEEE Trans. on Automatic Control, Vol. AC-l4, pp. 165-175, April 1969. [ 4] R. A. Volz and L. F. 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Dissertation, Department of Electrical Engineering and Systems Science, East Lansing, Mich., 1973. 127 [52] K. T. Ma, "On the Optimal Sample-Data Tracking Problem, " Ph. D. Dissertation, Dept. of Electrical Engineering and Systems Science, East Lansing, MI, 1975. [53] International Mathematical and Statistical, Inc. , ”The IMSL Library 3 Reference Manual, " Houston, Texas, First Edition, May, 1972. APPENDICES APPENDIX A THE SAMPLE AND HOLD OPERATION The operation of a sample and hold mechanism involves two phases. The first phase is the measurement of the continuous time signal magni- tude at the desired sampling instant. The figure below shows a continu- ous signal e(t), with values indicated at various times, ti. ’1“ r‘ e(t) - e*(t) ’1 L t. 1. t, t, t. ' if t. 1:. t, t, t. t; (a) (b) Figure A—l. Signal Sampling using Impulses. The sequence of signal values at the sampling times as shown in Figure A-l (a) may be represented by a series of impulses as shown in Figure A-l (b) with the strengths of the impulses being equal to the magnitude Of e(t) at the corresponding sampling time. Thus the series of impulses as shown in Figure A-l discretize the original signal e(t). N-l e(t) z e“‘(t) = Z e(ti) 6 (t-ti) Os ts tfina1 , N < oo i=0 The term e*(t) is called the sampled signal equaling the continuous time 128 129 Signal at instants t :ti’ Whether e*(t) represents values of the continu- ous signal e(t) at each sampling instant, or if e*(t) is actually a series of impulses at ti - ti length intervals, the impulse representation + 1 may be thought of as a switch which closes instantaneously at the beginning of each interval. If an infinite number of samples were taken the continuous time signal could be represented exactly at any given point by the magnitude of the appropriate impulse function. In all prac- tical applications this is clearly impossible and thus a finite number of samples are used. The magnitude of each impulse can then be quantized, stored, and later used for various calculations as required. After each sample (impulse) is obtained the second, or holding phase Of the sample and hold operation occurs. The ”hold" operation maintains a constant signal level equal to e(ti) until t =ti + 1 at which time sampling again occurs. Figure A-Z (b) indicates the piecewise constant signal used to approximate the continuous time signal e(t) found in Figure A-l (a). The holding operation as depicted in Figure A-2(b) is * ’l '11- “‘ g: I’ l ‘\ e (t) I’q \‘~-’”’p"11 ehlll I \ ~-”.”‘-1 , I I, I I I t. t. t. t; 1. t; t. t. t. I. t: if (a) (b) Figure A-Z. The Impulse Holding Operation. more formally called a zero-order-hold as Opposed to a nth order hold which approximates the signal using a polynomial curve fit between the data points where the degree of the polynomial is proportional to n. 130 Thus the terms "sampling" or "sample and hold" will indicate the representation of signal, as in Figure A-l(a), by piecewise constant levels as indicated in Figure A-2(b). APPENDIX B PERIODIC, ADAPTIVE, AND APERIODIC SAMPLING The ”sampling" of a continuous time signal, as described in Appen- dix A, using equally spaced sampling times is known as periodic sam- w. mm : I.” a pling as shown in Figure B-l. {7 e(t) eh“) 1. t. t. t, +, t, t, 1, +. 1. Figure B-l. Periodic Sampling The sampling interval is defined for Figure B-l as T-t -t i=0,1,2,3,4,5,6,7,8 i ‘ 1+1 i where for periodic sampling T0=T1=T2=T3=T4=T5=T62T 2T8. Periodic sampling as shown in Figure B-l is entirely independent of Sig— nal changes within a sampling interval. The piecewise constant signal eh(t) is determined only by the magnitude Of the original continuous time signal at the sampling instants, ti. The ”sampling" of a signal on a non-periodic basis can be a result of using adaptive or aperiodic sampling rules. Adaptive sampling is a method of sampling where the length of the sampling interval changes as 131 132 a result of the information received about the signal (e. g. , signal mag- nitude, signal derivative) at the end of the previous sampling interval. eh“) /F‘\ e(t) L— t. 1. 1. e. '4 Mt. h *9 Figure B-Z. Adaptive Sampling Consider a typical adaptive sampling rule, Ti:l.0/]é(ti)] i=0, 1, 2, 3,4,5, 6, 7, 8. As can be seen in Figure B-Z the lengths Of the sampling intervals will vary according to the inverse of the absolute value of the continuous time derivative at the sampling time, ti . For the example in Figure B-Z no minimum or maximum sampling interval length constraints were im- posed, thus with a signal derivative approaching zero the sampling in- terval becomes very large as seen in Figure B-Z. During this interval the signal variation (i. e. , during interval ”a”) was entirely neglected. Imposing sampling interval constraints with Tmax being very small would reduce sampling errors of the kind found during interval ”a”, but could drastically increase the number of samples required during the time interval of interest. Aperiodic sampling, in general, also yields non-periodic sampling intervals as does adaptive sampling. In contrast to adaptive sampling where each sampling interval is determined on an individual basis, aperiodic sampling considers a specified number of variable length sampling intervals having a combined length equal tO tfinal-tinitial' 133 Therefore given an initial sampling interval sequence, aperiodic sampling varies the length of each sampling interval to minimize a predetermined sampling error performance index while maintaining the sum of the sampling interval lengths equal to tfinal - tinitial' Adjusting each sampling interval based on minimization of sampling errors rather than signal derivatives will hopefully decrease the possibility Of large sampling errors from going undetected as seen in Figure B-Z. Figure B—3 shows a possible sampling interval sequence which would detect the signal variation during the interval "a". This sampling interval sequence could be Obtained through use of aperiodic sampling assuming that the sampling interval sequence shown in Figure B-3 yielded lower sampling errors than the sampling intervals in Figure B-Z. eh”) “-1 e(t) *0 in '5; t, t. ‘t‘ t‘ *1 +3 8, Figure B-3. Aperiodic Sampling APPENDIX C NUMERICAL DATA FOR FIGURES FOUND IN CHAPTER IV The data presented in this Appendix accurately presents the various performance cost values as graphically presented in Chapter IV. The data found in this Appendix is arranged by Figure number. Figure 4. 3(a) N Periodic Adaptive Aperiodic 2 1.158 1.330 .074 3 . 525 . 696 . 042 4 . 264 . 294 . 041 5 .136 .158 . 013 6 .082 .082 .008 7 .045 .056 .004 8 . 036 . 036 . 003 Figure 4. 3(b) N Periodic Adaptive Aperiodic 2 1.623 1.797 .828 3 1.066 1.313 1.040 4 .881 1.061 1.392 5 .828 1.075 1. 521 6 .850 1.150 1.739 7 .887 1.273 2.100 8 .953 1.403 2.280 134 ‘l. 135 Figure 4. 4(a) N Periodic Adaptive Aperiodic 2 2.315 1.610 .240 3 1. 590 .326 .239 4 1.055 .134 .092 5 .681 .073 .053 6 .485 .088 .040 7 .323 .236 .060 8 .276 .182 .056 Figure 4. 4(b) N Periodic Adaptive Aperiodic 2 2. 780 2.079 .993 3 2.131 .942 1.613 4 1.671 .902 1.037 5 1.372 1.016 1.201 6 1.251 1.156 1.360 7 1.164 1.489 1.943 8 1.193 1.590 2.735 Figure 4. 5(a) N Periodic Adaptive Ape riodic 2 4. 631 4. 822 4. 246 3 4.818 1.909 1.272 4 4.220 1.258 . 764 5 3. 403 . 788 . 580 6 2. 852 . 638 . 522 7 2. 304 . 814 . 538 8 2.124 . 767 . 522 136 Figure 4.5(b) N Periodic Adaptive Aperiodic 2 5.097 5.289 5.131 3 5.359 2. 526 2.349 4 4.837 2.025 2.178 5 4.095 1.705 2.590 6 3.619 1.702 2.542 7 3.146 2.030 3.057 8 3.040 2.134 2.810 Figure 4. 6 (a) Step Input, Sub-Optimal Aperiodic, a=0 [ .0346, .0530, .0617, .3161, .5345 ] a=-l [ .0830, .1037, .1888, .0776, .5469 ] a=l [ .0400, .0428, .0251, .8283, .0635 ] (b) Step Input, Adaptive, a:0 [ .0500, .0500, .0600, .1200, .7200] az—l [ .2100, .2100, .2100, .2100, .1600] a=l [ .0400, .0500, .0700, .2500, .5900] Figure 4. 7(a) N Periodic Adaptive Ape riodic 2 . 563 . 463 . 061 3 . 376 . 081 . 061 4 . 243 . 029 . 024 5 .155 . 060 . 022 6 .110 . 013 . 010 7 . 073 . 022 . 012 8 . 062 . 017 . 008 137 Figure 4. 7(b) N Pe riodic Adaptive Ape riodic 2 1.029 .930 .801 3 .917 .698 1.050 4 .859 .796 .989 5 .846 .977 1. 589 6 .876 1.111 1.347 7 .915 1.323 2.161 8 .979 1.384 1.700 Figure 4. 8(a) N Pe riodic Adaptive Ape riodic 2 . 563 .430 .058 3 . 388 .089 .058 4 .255 .033 .022 5 .167 . 028 . 015 6 .119 . 069 . 033 7 . 079 . 029 . 012 8 . 068 . 011 . 007 Figure 4. 8(b) N Periodic Adaptive Aperiodic 2 1.028 .897 . 737 3 .929 . 706 1.063 4 .876 .801 .972 5 .859 .973 l. 116 6 .886 1.137 1.458 7 . 921 1.283 2.107 8 .985 1.380 1.810 Figure 4. 9(a) 138 N Pe riodic .426 .207 .118 .077 .053 .042 .030 ooxlomubww Figure 4.9(b) N Periodic 2 . 892 3 . 748‘ 4 . 735 5 . 768 6 . 820 7 . 885 8 . 948 Figure 4. 10 (a) Ramp Input, a:0, Adaptive Sub -Opt. Ape r . [.0500, [.0648, (b) Parabola Input, a:0, Adaptive Sub-Opt. Aper. (c) Noise Input, arO, Periodic Sub-Opt. Aper. [.0500, [.0296, [.2000, [.3048, .0500, . 0796, . 0600, . 0383, . 2000, . 2000, Ape riodic .367 .172 .099 .061 .046 .035 .028 Ape riodic .061 .225 .335 .220 .442 .566 . 819 D—‘h-lD-‘Ht—‘r-ll-d .0600, .2539, . 0900, .0661, .2000, .1828, .1200, .1029, . 3300, . 3702, . 2000, .1696, .7200 ] .4986] .4700 ] .4959] .2000] .1428] Figure 4.12(a) N CDKIO‘UWu-PUJN Pe riodic 1.3x107 4.5x107 1.6x107 2.0x105 1082.0 55.4 8.4 Figure 4. 12(b) N CDKJO‘U‘II-FUJN Periodic 1.3x107 4.5x107 1.6x107 2.0x105 1082.20 55.50 8.51 Figure 4. 13(a) N ooxlcmeP-wm Periodic 45779. 223. 0 . 59 . 30 . 01 . 70 . 80 «144mm 139 Ape riodic 140.4 13 11 10 7. 7. . 54 .93 .72 10 89 4.95 Ape riodic 143 16. 15. 14. 12 13 8. Ape riodic 2 NNNNtbxlt-d .43 51 08 97 .72 .53 14 .87 .29 .19 .50 .74 .51 .34 Sig. Rep. 1352. 8 301.1 16.32 4.11 5.31 10.36 6.47 I‘ll 140 Figure 4.13(b) N Periodic Aperiodic Sig. Rep. 2 49779.4 23. 57 1353.2 3 223.2 9.70 301.3 4 5.68 7.44 16.41 5 5. 39 6. 68 4. 21 6 7.11 5.40 5.41 7 7. 80 7. 89 10.46 8 7.97 7.38 6.64 Figure 4.14(a) N Periodic Aperiodic 2 6. 78 6. 59 3 7. 20 4. 68 4 8. 55 4. 55 5 9. 35 4. 54 6 9. 84 4. 50 7 10.17 4. 52 8 10.40 5. 74 Figure 4. 14(b) N Periodic Aperiodic 2 6. 86 7. 90 3 7. 28 6. 52 4 8. 64 7.14 5 9.44 8. 02 6 9.94 9.28 7 10.27 12. 53 8 10. 50 15. 32 Figure 4.15 141 System Response Time, Step Input, Tyep 2 Systems: FAST ME DIUM SLOW [ .0852, .0080, .5010, .0184, .3881] [ .1992, .1139, .2962, .1469, .2437 ] [ .4192, .0845, .0516, .0704, .3742 ] Figure 4.16(a) N CDQO‘U'lI-bUJN Periodic Aperiodic 8938.5 3.91 24.00 1. 50 .990 . 764 1.58 .519 1.83 .510 1.85 .397 1.79 .395 Figure 4.16(b) N 004001pr Periodic Aperiodic 8938. 7 5. 72 24. 09 3. 95 1. 08 6. 53 l. 68 4. 31 l. 93 6. 42 l. 94 6. 58 l. 89 7. 80 Figure 4. 17(a) N mums-1430431 Periodic Ape riodic 8. 92 8. 70 9.15 8. 71 9. 3O 8. 71 9. 39 8. 73 9.47 8. 73 9. 52 8. 73 9. 56 8. 76 Figure 4.17(b) N ooxiome-wa Periodic \O\O\D\O\0\O\O Figure 4.18(a) N 00401114sz Periodic 113. 112. 77. 51. 36. 27. 22. Figure 4. 18(b) N CDKJO‘UTI-FUJN Pe riodic 113. 112. 77. 51. 36. 28. 22. .20 .43 .58 .68 .75 .81 .87 7O 66 24 50 65 99 62 79 76 34 60 75 08 72 142 Ape riodic 14. 69 17.89 19.12 24.46 26. 64 28.21 29.36 Ape riodic 13. 45 7.69 6.83 7.39 5.13 4.10 3. 79 Ape riodic 14. 92 9. 50 9.10 10. 71 9. O4 8. 45 6.48 143 Figure 4. 19(a) Continuous = 1. 051 N Periodic Ape riodic 2 5. 88 1.18 3 12.10 . 613 4 10. 70 .441 5 8. 00 . 656 6 6.04 .412 7 4. 77 . 288 8 3. 94 . 292 Figure 4. 19(b) N Periodic Aperiodic 2 5.97 2.37 3 12.20 1. 55 4 10.79 3.02 5 8.09 3.99 6 6.14 3.89 7 4.87 5.31 8 4.04 5. 68 Figure 4. 20(a) N Periodic Aperiodic 2 . 3760 . 3696 3 . 3837 . 3752 4 . 3888 . 3853 5 . 3922 . 3906 6 . 3947 . 3938 7 . 3965 . 3961 8 . 3978 . 3974 Figure 4. 20(b) N Pe riodic m~10\mh¥>WN Figure 4. 21(a) N Pe riodic .8031 .8373 .8271 .8429 .8467 .8548 .8628 144 Ape riodic 7.268 10.978 6.186 7. 906 9. 544 11.329 13. 565 Continuous = 4. 261 mKJO‘U'IbFUON 52.08 47.87 33.47 23.65 17.91 14.46 12.24 Figure 4. 2 1 (b) N mKJOU'kaUON Periodic 52.17 47.96 33.57 23.75 18.02 14.56 12.35 Ape riodic 4. 57 3.86 2.13 2. 57 2.08 1. 75 1. 64 Ape riodic 6. 05 5. 80 4. 57 5. 84 6. 30 3. 89 7.13 Figure 4. 22(a) N ooximmupww Pe riodic 10. 37 12.69 10.63 8.51 7.06 6.10 5.45 Figure 4. 22(b) N CDKJO\U1I#WN Periodic 10.46 12. 78 10. 73 8.60 7.16 6.21 5. 56 Figure 4. 23(a) N OOQO‘UI$UON Pe rio dic .1944 .1971 .1991 .2003 .2012 .2019 .2023 145 Continuous = 2. 755 Ape riodic . 379 . 384 .300 .280 .253 .232 .225 Ape riodic 2.118 2.271 2.864 2.711 4.358 5.205 5.767 Ape riodic .1903 .1951 .1986 .2001 .2010 .2018 .2023 [‘1— 146 Figure 4. 23(b) N Periodic Aperiodic 2 . 5801 7. 093 3 . 5914 4.126 4 . 5981 5. 776 5 . 6006 6. 901 6 . 6164 8. 079 7 . 6330 10. 372 8 . 6355 11. 927 Figure 4. 25 MEDIUM Re3ponse Time System, Step Input. Type 2 [.1992, .1139, .2962, .1469, .2437] Type1 [.0841, .0345, .0863, .2293, .5658] Type 0 [.0896, .0338, .0818, .1951, .5995] MEDIUM Reaponse Time System, Ramp Input. Type 2 [.2339, .1046, .2763, .1409, .2443] Type1 [.0884, .0439, .0950, .2430, .5268] Type 0 [.0806, .0327, .0828, .1573, .6466] MEDIUM Response Time System, Noise Input. Type 2 [. 0542, . 0709, . 0819, . 0891, . 7038] Type1 [.1823, .1879, .1975, .2025, .2297] Type 0 [.1924, .1947, .1995, .2056, .2076] Inn... “mix; l .!|]\i|1]lli|l}}l|1|ii 178 5912 “I “I N H II" iv] 1 293 0 MI 3 1 1 {Hm