ON THE TIME DOMAIN DESIGN OF LINEAR SAMPLED—DATA SYSTEMS Thesis for the Degree of PhD. MICHIGAN STATE UNIVERSITY Kensie Boss Johnson 1964 THESIS This is to certify that the thesis entitled ON THE TIME DOMAIN DESIGN OF LINEAR SAMPLED-DATA SYSTEMS presented by Kensie Ross Johnson has been accepted towards fulfillment of the requirements for Ph. D. degree in Electrical Engineering £91k / K %e7 Major professor Date May 15’ 1964 0—169 LIBRARY Michigan State University ABSTRACT ON THE TIME DOMAIN DESIGN OF LINEAR SAMPLED-DATA' SYSTEMS by Kensie Ross Johnson The design problems associated with‘linear, sampled-data control systems have been extensively investigated. in the past using, primarily, z-transform methods. Although the transform techniques have not been outmoded for linear control systems, it is felt that a time domain method, using discrete-state models, provides a common basis for the study of sampled-data control systems‘in general. The discrete-state system model used in this thesis is easily ob- tained from the models of the subassemblies and appears as a set of first-order difference equations. The solution of this. system of equa- tions is accomplished by functions of matrices, and the response of systems to particular driving functions is discussed. Procedures are developed for designing single input-single output, sampled-data systems with minirnal, non-minimal and asymptotic .time response. Considerable attentionis devoted to the problem of plant saturation. An extended concept of plant controllability is also dis- cussed inwconjunctionwith asymptotic response. -1- Abstract -2- Kensie Ross Johnson A preliminary investigation is made of the problems associated with the design of state-controllable, multiport plants. Techniques for obtaining desired control functions are developed. It is felt that the area of multiport design is a most fruitful one for further research. ON THE TIME DOMAIN DESIGN OF LINEAR SAMPLED-‘DATA SYSTEMS by Kensie Ross Johnson A THESIS Submitted to Michigan State University in-partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering 1964 ACKNOWLEDGE MENTS The author wishes to thank Dr. H. E. Koenig for his assistance and encouragement during the preparation of this thesis and for his guidance in the research effort which preceded it. The author also wishes to thank the National Science Foundation who contributed to the author's support during the period of research. -ii- CHAPTER I II III IV VI TABLE OF CONTENTS INT RODUCTION .................. DISCRETE-STATE MODELS OF SAMPLED- DATA FEEDBACK CONTROL SYSTEMS ..... A. Mathematical Models for Systems with Negligible Controller Delay . . . ...... Mathematical Models for Systems with Controller Delay ............... SOLUTION AND STABILITY OF DISCRETE- STATE MODELS ................. A. B. C. D. Direct Computation of the State Vector Solution of the Discrete -State Model by Functions of Matrices ............ Solution by the Augmented Matrix Method . . System Stability ................ THE DESIGN OF DIGITALLY CONTROLLED SAMPLED-DATA SYSTEMS IN TERMS OF DISCRETE-STATE MODELS ........... A. B. C. The Design for Minimal-Time Response Design for Non—Minimal-Time Response Asymptotic Time Response .......... SYSTEM WITH MULTIPLE INPUTS AND OUTPUTS SUMMARY .................... -iii- Page 13 17 17 19 23 34 39 41 52 66 77 92 LIST OF FIGURES FIGURE Page I Block Diagram of Sampled-Data System ...... 3 11 Plant Input Signal .................. 4 III Cascaded Sub-Assembly .............. 7 IV Description of a Driver as a Function of n . . . . 28 V System with Multiple Inputs and Outputs ..... 77 VI Multiport Control System .............. 78 LIST OF APPENDICES APPENDIX Page A Discussion of Cont rollability ........... 95 —v- I INTRODUCTION In recent years, the field of automatic control systems has wit- nessed considerable interest in the representation of such control systems by mathematical models which are derived on the basis of the "state" of the system. In most general terms, the state of a 1” dynamic system may be defined as: the minimum collection of numbers which must be specified at time t = tk, in order to predict the entire system behavior at time t 2 tk. In particular, the state of a system may appear as the solution to the set of first-order differential or difference equations which is characteristic of the system under consideration. Prior to the investigations of Kalmanl: 2: 4, Bertramza 3’ 4, Sarachik3 and others5 , automatic control systems have been ana- lyzed and designed using the Laplace or z-transform representation of the system equations. Although the transform techniques have not been outmoded for linear control systems, the system-state model provides a common basis for investigation of control systems in general; that is, continuous systems as well as discrete time systems, nonlinear as well as linear. The shift in interest from Laplace-transform and z-transform representation of control systems to the system-state model is due in part to 1) the desire of a few investigators to arrive at a general time- domain formulation technique for the mathematical modeling of control systems; 2) recently translated technical journals and textbooks, espe- -2... cially from the Russian, by such authors as Lyapunovs, Pontriagin7 and Hahn8 to name several. These contributions to the areas of optimal control processes and the stability of systems have featured systems modeled as a set of first—order differential or difference equations; that is, in state-model form. As a result, renewed interest has been fostered inthe representation Of systems by state-models in order to take advan- tage of the deveIOpments in these areas. A sub-class of automatic control systems is that class of feedback control systems of the linear, time-stationary, digitally-controlled, sampled-data type. The analysis and design of these systems in the past has been carried out primarily by the use of a z-transform model and have been treated extensively in the literatureg. The z-transform methods, however, do not extend to non-linear systems and also become quite un- wieldy in certain linear applications. The use of a discrete-state model; that is, a system modeled as a set of first-order linear difference equa- tions at least provides an alternate to the z-transform technique. . This thesis presents methods for the analysis and design of linear digitally controlled, sampled-data systems by means of the discrete- state model. Some attention is also devoted to the formulation of the system model from the discrete-state model of the sub-assemblies and methods of solution of the discrete-state equations. II DISCRETE-STATE MODELS OF SAMPLED-DATA FEEDBACK CONTROL SYSTEMS A. Mathematical Model For Systems With Negligible Computer Delay Consider the system shown below which is made up of a digital controller D, a zero-order hold H and a continuous plant G which is to be controlled. The sampling Operation is shown schematically by means of a switch. The digital controller is assumed to be one which receives a sequence of numbers (in time) at the input and operates arithmetically on the sequence to produce a number sequence at the output. The samplers in the system under consideration serve to indicate that the input and output of the controller are sequences of numbers. The sampling Operation is assumed to be synchronous and any time delay in the computer for computation is neglected. flMn‘ A s(n) A Asn) _ lct)__ D H G 4: FIG. I If the plant G can be modeled as a system of first-order linear differential equations; that is, as a continous state model of the form IX: Q’X + 6 e(t) then the linear discrete-state model for the plant is of the formlo IXZW) = 02 X2(n-1)+ 623(n-1) where n is understood to signify nT, for n = 0, l, 2, . . . , and xzm) represents the value of the plant state vector at t=nT. The scalar s(n-l) is the input to the plant from the zero-order hold and is of the form shown in Fig. 11. SW A r——1 l l‘"——l ___J | l——_l : . : 'u——- I . I 1 l I __,__I___l t T 2T 3T 4T 5T 5 FIG. 11 The discrete-state approximation to the continuous output c(t) may or may not be an element of the state vector. It is, however, assumed to be a linear combination of the state variables as indi- cated by the algebraic equation -5- c(n) = 08’ 2 X 2m) + f2s(n) where 0&7 2 is a row vector and f2 is a scalar. If c(n) is the first element of the state vector X 2(n), then .fl"2 reduces to [1, O, . . . , (fl . In general, the discrete-state model of a linear, time-invariant plant is, therefore, taken as ’sz) = a2 %2(n-1)+ stm-l) C(n) = 90‘2 X 2(n) +fzs(n) (1) When f2 35 O, the above model describes a plant having direct transmis- sion. If f2 = 0, the plant is said to have no direct transmission. The input-output characteristic of the digital controller D is con- 9 sidered to be linear and given by the recursion formula, c(nT) = a0r(nT) + alr L(n-1)T:l + . , . + amr [(n-mYIj ' (2) -b1c l: (n-1)TJ -b2c I: (n-2)TJ - . . . - - bkc I: (n-k)'Ij where c(n) represents the output at a discrete time t = nT, r(n) repre- sents the input at t = nT and the a1 and b1 are constants. Note that in general the output at t = nT is a linear combination of the k previous out- ~6- puts and (m + 1) previous inputs. It can be shown10 that (2) is easily transformed to the following state model for the controller. -x1 (nf1 _-b1 -b2 -b3 . . . ~b1'; F'xfin-I)‘ " 1 7 1:2 (n) 1 0 O . . . 0 x2(n- 1) 0 X3 (11) 0 I 0 . . . 0 . 0 it + £1 E, x (n) 0 ...1 0 x(n-1) O Lk ...I L. .l .. k ...1 .. .. "x'l (n)-1 C(n) = [aOs 8-1: 8'20 0 ° - 9 am: 0: o o s 0 J ‘ x12 (n) xm (n) where k > m (3b) Iik (n) — or ,xl (n) = 01 X1 (n-l) + B 1 r(n) (4) c (n) = '0’ 1 ’X 1(n) To establish a discrete-state model for the system shown in Fig. I, consider first the cascaded sub-assembly shown in Fig. 111, (i. e. , the system of Fig. I without feedback). Ian—4:57- D Jest" H FIG. III Rewriting (1) and (4), there results 12(11): 02 IX 2(n-1)+ 6 2s(n-l) c(n)= 00“2 ’x2(n)+fzs(n) ,Xlin)= 01 I)(1(n-1)+ Blew) s(n) = «)3 961m) Substituting (8) into (5) and combining this result with (7), the cascade system model is established as s(n) '36:)"— (5) (6) (7) (8) 2(n) 02 62 00’1 ’sz-l) 0 9(1 o 01 9(1m-1) 61 (9) C(11) = [6'2 0 ] + f2 S (n) The presence of the zero in the 2, 1 position of the coefficient matrix is characteristic of an Open-loop system of this type. It will be shown later that the digital computer in the system of Fig. 111 cannot stabilize a plant for which the magnitude of the eigenvalues of the transition ma- trix are greater than unity. When the feedback connection is made, there results e(n) = r(n) - C(n) (10) . When the plant output equation (6) is premultiplied by - 8 1, there results - 510(n)=- 81 00'2 IX2(n)- Blrzsm) or " 61cm)“ 51 “0.2 “291)“ Bifz ‘6'1 “1”” (11) rsm. -9- Substituting (10) and (11) into the state equations yields -" r- -‘-r r— ’LL 0 fight) a2 1&2 “L“ 31 fzaa'I) 1‘“) 0 ._ L. ..J L... a a? _ %2(n- 1) 0 31"“ DJ 61 Assuming that the inverse of the coefficient matrix exists*, then %2 (n: Lyl (n) --) whe re 91191? I012 BM? 21922 0 a1 L... r. 96 2(n- 1) lem— 1) .1. ..J 81 '1) ..J 0 (71+ 61 f2.&1>'161.a-2 <fl+61 £2193)“ r (n) (12) *It can be shown that the inverse exists for all cases except fZaO == -1. -10- Equation (12) can also be written as __ r __ _, 952 (‘1) £11 “12 W 2““) 0 31‘“) Q21fl22 751““) (71+ 6N2 6'1): whe re gs 0.2 52 «5% £21 a (71+ EIIZJI)’1€1J~2Q2 <’L(+ 61f2&1>'1610fl262.§.1 + (in alfzévlrlal 3B 29 Equation (13) along with c(n) = 06—2 2:2(n) + f2 s(n) represent the model for the entire feedback control system. -11.. For notation purposes, let 1 12(5)- ——@11 Q1; 0 (n) = , Q = . ((5 film) LRZIIQZEJ (0+ 6112 ag'lY'l and ”IL-”20] The discrete—state model of the system of Fig. I with negligible computer delay. is then ’Jlm) = O ’Xm-I) + firm) c(n) = J’Xm) + £2 s(n) A (14) where the vector x (n) is referred to as the state vector of the system, the matrix Q is the system state transition matrix and fl is a vector associated with the sampled driving function r (n). It is of interest to note that the term inside the braces of (12) is the same as (9) (the Open- loop system), and the effect of the feedback loop shows explicitly as a premultiplication by a non-singular triangular matrix. The special case of no direct transmission in the plant implies that f2 5 0 in (6). Equation (12) then reduces to I7“, (fl —’)(2(n)1 ZZZ 232.031 A’sz-lfl _o‘ ' = + r(n) 81 46’2 u 11“” 0 a 1 x1“!- 1) 61 L. .. L. _ L. _ L _ L. .. I’szfl I- 71. (7‘ Tag 82 .29:1 Wyn-13) 37 = + r(n) 31W 1'81 “9'2 ”é 1? a1. 2‘1““: £31. (15) _ 02 52401 1 — 2(n-I)1 -0-) = + r(n) ' 610269 -61‘&2 82°b'1+a1 1(n'1) 61 L. ._ L. -1 L. _' (16) Equation (16) together with the relation c(n) = 062 sz) represent a discrete—state model of the entire feedback system providing the plant has no direct transmission. This model is characteristic of plants which do not reSpond instantaneously to changes in the input. -13- B. The Mathematical Model of a System With Computer Delay The problem of computation delay in the digital computer can be thought of as an output delay; that is, the sampled input at t = kT is not available for output until t = (k + 1) T. To incorporate a single unit of delay into the system model, note that it is only necessary to set a0 = 0 in (2). Equation (2) then becomes c(nT) = alr [(n-1)T] + azr [(n-2)T:] + + amr E(n-m)T] -blc [(n-1)T] - bzc [(n-2)T] - . . . - bkc [(n-MT] It is easily seen that the output is not current. The input information is lagging one sample behind. The extension to higher order delays is trivial. For the purposes of design, it is necessary to know the properties of the transition matrix of the system Q as they relate to the properties of the transition matrices for the components. To establish these relations explicitly, let there be q variables in the plant state vector, and let the linear recursion formula for the digital computer as given by (2) contain (m+k) terms. For the sake of simplicity, con- sider the case of no direct transmission inthe plant and no computer delay. The sub-matrices of the system transition matrix corresponding to the components (see Eq. (16)) are "'01 -14- 5‘22 q2 -bl -bz qxq -b k kxk qxl -15.. The detailed form of E as obtained by performing the indicated ma- trix multiplications is: 3.11 3.12 ... alq : bllao bllal ... bllaTn 0 . .. O , : b21a0 b2131 ... b21am 0 ... 0 . l I I o l aql aqq I bqlaO bqlal bqlam 0 0 Q = ------------------- 4 ---------------------------------------- d'11 d'12 'Iq : p11 P12 PIm "bm+1 ”bk 0 O 0 l 1 0 O 0 O 0 O O : O 1 . O 0 . 0 : I . l o o o I o o . o o . 1 o 1.. ' J (q+k) (17) q m where d'h - E 1 C11 331 and pij - -bj aj+1 E 1 Cli bil for jflgm J: 1: -15- The following important points are to be noted: 1. The discrete-state transition matrix is related to the transition matrices of the components by a simple algorithm. 2. The column entries to the left of the partition lines are a function only of the entries in Q2, 81, and .02. ‘ Since [3 1 contains only one non-zero entry, these ,- columns are functions only of the plant parameters. * 3. The column entries to the right of the partition lines are functions of the entries in both the plant and com- puter transition matrices. 4. The entries in the submatrix in the 1, 2 position-are _ functions only of the controller ai; the entries in the 2, 2 submatrix are linear combinations of the controller parametersai and bi. The transition matricesfor systems with computer delay and direct transmission in the plant are formed in exactly the same manner as in the previous example. To avoid repetition, these forms are not included here. 1 III SOLUTION AND STABILITY OF DISCRETE ~STATE MODELS The primary performance characteristics of concern in the de- sign of linear feedback control systems are: 1. System response time 2 . System stability To evaluate the system response the solution of the discrete- state model (14) must, of course, be obtained. There are several methods which can be used to solve (14). These are: 1. Direct computational method 2. Solution by functions of matrices 3. The augmented matrix method These methods will be considered separately in the following discussion. A. Direct Computation of the State Vector Let (14) be written recursively as -17- .. ._ -“'f‘q -. ‘VI‘Iw-lw 4 .r‘ ‘~.‘_ ‘1" -13- Q 1(0) + (61(1) @2 ’XW" $15141) + 15112) «(1) 7((2) (18) ’)(+ firm) The response of the system is, of course, determined entirely by the behavior of the state vector. This follows directly from the defi- nition of the system."state. " The evaluation of (X (n) as given in (18) can be obtained by direct computation; that is, by calculating the indi- cated products of powers of the transition matrix. One measure of the behavior of the state vector as n increases without bound can be obtained by considering the norm of the state vector, H X" , where such a norm may be taken as any of the following: 1. IIXH=max IXI i 2. "IX“: |X|1+ lxl 2+'°'+ IX] 11 i 1 .. 2 2 2 '2' 3. - x + x + . , .. + x ||’)(J||:II1||2 lln] The third norm is the well-known Euclidian norm or length of a vector in. n-space. It is stated here, without formal definition, that in order for the system to respond in a stable mode it is necessary that -19- the state vector x (n) converge to some vector X 0. The following well-known theorem11 establishes conditions for such a convergence: For a sequence of vectors x (1), x (2), . . . , ’Xm), . . to converge to a vector yo it is both necessary and sufficient that "X(k) - x0" —> O as k -->- oo . A test of the above norm, or equivalently, a test of "’X (k)" . as k—ar DO , is sufficient to determine stability and rate of conver- gence. Although analysis by the direct method is possible, it involves excessive computation time for all but the most simple cases, parti- cularly. if the state vector does not converge rapidly. This difficulty is partially avoided by the use of the solution by functions of matrices. Since this approach is relatively new, considerable attention is devoted to it. B. Solution‘of the Discrete-State Model by Functions of Matrices Let (14) be written as (Km-+1) = Qn+1 9((0) + anfl r(O) + and (6 r(l) + + firm) When the eigenvalues of a are distinct, the above equation can be expanded by functions of matrices12 to read -20- ’X(n+1)= 11>‘111+1 + 312%?“ +... 4' 31k)?” 11(0) 4- E511x31+312>\_3+...+ 31k>fi116fim + [311 Ali-1 4- 312))2-1 +... + whirl] firm) L (19) f + [311X1 + 512))2'+ "'+ 311(ka firm“) firm) where the 3, 11, i = 1, 2, 3, . . . , k, are the constituent matrices, and the + )tj j. = 1, 2, 3, . . . , k are the k distinct roots of the polynomial equation det' [a " MA] = 0. For certain classes of input signals (19) can be simplified somewhat as the following example indicates. Example: Step-Function Input Let the driving function r(n) be a unit step function, i. e. , 1(0) = r(1)= r(2) = = r(n) = 1 For convenience let Qn-tl ’X(O) = yo. Equation (19) then becomes -21... + ’X(n+1)=(yo+fl) m1(k?+k?-1+... +A§+ A1) + 1/;2< xg+x§'1+...+xg+).2> (20) + n-l 2 UIkmfiaka +...+,\k+).k) where 1/12: 911‘6 ,i=1,2,...,k. LettingSi=(A11q— l) = A?+>\?-l+,,,+)\i, Aigél A1— the general formula for the system response to a unit step forcing function is X= + 1&1 K [En-1) k1 + (Ia-mi + + mag-3] n-2 -1 +’U:2K[(n-1)A2+(n-2)>\§+... +2}. 2 +Xn21 + . 2 n-2 n-1 +Vle[(n-1))\k+(n-2)kk+...+2 k+kk] where 1’? ' 1 81 = 1 : >‘1# 1 )‘i ' 1 or k ,X(n+1).=(yro.+ngn)-PK 2; Wu [1131' >‘ijk Si] ~23- Example: Constant Acceleration Input Similarly, constant acceleration response, when a has distinct eigenvalues, is k IX(n+1) = ( 30 + @193) + g 141an Si - 2n)(i S; + )xi%)7i()isifl This method of obtaining a closed forin of the solution to (14) can be extended to include higher order-inputs and repeated eigenvalues. Since the algebra is quite involved, the details are omitted here. C. Solution by the Apgmented Matrix Method An alternate and sometimes more useful solution procedure results when the nonhomogeneous equations are first reduced to homogeneous form. Consider 'X(n+1) = OmeH- Ban) where ’Xm) =X0 (21) Suppose (21) can be reduced to the homogeneous form . rim-r1)— —a‘_ 6*— F% (n; F-¢(O)flr W0“) L—fi(n+1) ) = L. 0 3?; _/€(n)-J , {€(OL L’QQ (22) The obvious benefit derived from this transformation, if generally applicable, lies in the simplicity of the solution; namely, if (22) is written as ymu) A! ?(n) ?(o) = yo then the solution is ? (n+1) #n-i-l % (0) whe re Cl. 6* —%(n+1) 10 fi- 0 (Z; amt-1) - ‘Lfl(n+1)u and y0) - a) This result is simple compared to the solution to the nonhomogeneous case which is n ’X(n+1)= an“ I)((0)+ Z Of'j'l (8 r(j-l) (23) 1‘1 -25.. It is immediately apparent that the benefit derived from reducing (23) to the form of (22) is the elimination of the summation term in (23). This benefit is realized, however, at the expense of increasing the order of the transition matrix. The question remaining is: under'what condi- tions can (23) be reduced to the form of (22)? Several examples using input functions of particular interest will aid in determining these conditions. Example 1: If the driving function in (21) is taken as r(n) =KnP, where p is an integer and K a constant, then r(n+l) = K(n’~1-1)p = K(nP+pnP-1 + ... + pn+1) This sequence represents the solution to the difference equations. and the nonhomogeneous form in (21) can be represented by the homo- geneous form "mun-I Kp p Fr(n) _ 3(0) " rp(n+1) (p- 1) (p- l) rp(n) rp(o) r2(n+ 1) O 1 r2(n) r.2(o) Lz-‘-1(n+ l) i 0 0 :lmL 3(0)‘ -26- _ ’X (n+1)T Cl 6 K o o 0 [X (n) ’X (o) rp(n+ l) 0 l p . . . p 1 rp(n) rp(o) r1 (n+1) L0 O 0 O 1 r1(n) r1(o) l.— —J — l—. — h— — Example 2: If the driver in (21) is of the form r(n) = cos knT = cos Kn then ,an -an e + r(n) — 2 e = r1(n) + 91m) and it follows that jK(n+1) . . r1(n+l) = L2— = r1(n)eJK and r1(n+l) = r1(n)e-JK The equivalent homogeneous state model is — - I— 1 1 _ “(— .... '- Ilbs-+1) (l '5 "5 X (n) PX (o) r1(n+l) = 0 ejK 0 r1(n) r1(o) = 91am) 0 o e’J'K am) 91(0) L. _ 1— J _ ._ _. .. -27,— Example 3: If the driver is of the form r(n) = ne then define Kn r1(n) = e The resultant homogeneous form is ._ ... ._ _. _ ... IX(n+1) a B 0 X (n) r(n+l) . = 0 eK eK r(n) r1(n+1) L 0 O eK r1(n) The above examples cover the driving functions frequently used as a basis of design. There remains, however, the general problem of what is to be done in the case of an. input function which cannot be ex- pressed analytically; that is, a driver whose values are known only as a function of n as shown in Fig. IV. OQO) r(o) r1(o)J -28- r(n) A r(Z) - -— — — .— ' r( 1) -— —' : l l f I r(O) I | I I I I l ' ‘ 4 ‘ ‘ > t 0 T 2T 3T 4T 5T 6T FIG. IV In general if the ( n+1) values of a function are known in an interval 0 S n S nT, then a polynomial of at most degree n can be con- structed to pass through the (n+1) known data points. For example, if the polynomial passing through three data points is r(n) ‘4 Kin2 + Kzn 4- K3 and rim) . n2 r1(n) I n r0(n) = 1 then the nonhomogeneous equivalent of ( 21) is F’X (n+1? r2(n+1) r1(n+ 1) r (n+1) L_._0 In general, using the above technique, and any polynomial input to a -29- Pa 5K1 5 K2 0 1 2 o o 1 o o o 8 K; PX (n7 1 r2(n) 1 r1(n) 1 r (n) .1 _ 0 _ “9((0) 6—10 1‘2“» O r1(0) 0 0 1 id )_ 1. .1 system (providing the input is zero after n samples), the nonhomogeneous equation can be reduced to the augmented homogeneous form 1(n+1)fl rk( n+ 1) r (n+1) k-l r0(n+ 1) b _ where the Ki are the coefficients of the polynomial driver r(n) and the non-zero entries in the 1th nomial (n-I- l)i . I1 6K1 8K2 o 1 k 0 o 1 E) E) 0 o o o 6 Kk-I (3 K1: k l k-l 1 1 l 0 1 = Klnk + Kznk’1-+... + Kk-1n -I- Kk We}? 56(0) row are the coefficients of the poly- # . 7:35 "III. . r: ...;—,.,:.=.,. ‘m V ,7- . ,_ -~LL—~ L- A’TW -30.. From the preceding examples, it is evident that if the driving functions can be written in the form k r(n) = 2 Kj I‘j (n) J'=1 and the rj can be regarded as the solution to a set of first order linear difference equations, then the nonhomogeneous system can be trans- formed to an equivalent homogeneous system. To show this, let r(n) = K1r1(n) + K2r2(n) + . . . + Kkrk(n) Since rj(n) represents the solution to the difference equations — ---I — _" _—7 rk(n+1) a1 1 312 .. . 31k 1 rk(n) .rk(-II+1) 0 a22 . . . a2k rk_nl) r1(n+l) 0 . . . akk r1(n) ._ _ L __ _ _. or R (n+1) 2V IQ (n), it follows that the nonhomogeneous equation ~ .12» I‘m -31.- %(n+1) = a’lm) + 8 r(n) can be written as ~ k ’X.(n+1) = a I/K(n) + ,6 E Kj rj(n) j=i or _r1(n)— %(n+1) = a‘hn) + [BKL BKZHH BKIEI r2(n) a. rk( n)..I The homogeneous equivalent is 71(1)“;- —O~ : 6K1 8K2 3K3 619.7, __%(n)_ 13351) HT ’ " " 7" ’ ' " """ 11(1)“ r2(n+l) = o : [V r2(n) . . l x4 I 3km“: ..6 I .1 LIE-km)— and is the same form as (22). For the particular driving functions con- sidered inthe above examples, the matrix A is upper triangular. -32- Equation (22) can be reduced further, or ”split,” by a simple change of state variables. Let (22) be written as “'1 I'- '- - M— CI 6* - 01’2" 1 74(1)) /6 (n+ 1) o (2" /€ (n) (24) whe re aim-I1) = “(n+1) - ”Kin-t1) is a new set of state variables and W is a matrix undefined as yet. Equation (24) can also be written as Pym“) _ 0. 5*-W+QOF Ian) - WKUII £(n+1) 0 9 /€(n) or' y(n+1) rd 6* - W + a. W F (n) Km“) 0 2" Km) (25) L. ... __ J _. .. The special form of (25) leads to the following: ..." 3..._-:;:.,:—....__.__: «:fl.". Salt?“ a _. -33- Theorem I If the nonhomogeneous set of difference equations X(n+1) = QXm) + B r(n) can be written in the equivalent form gum) a 6* - MD. CM! (n) o (4‘ (n) L. J _ .1 then there exists a matrix M such that 5* - luv/TI (MW where w) is given by 501 2;?3-1. 6H? WW.) (n+1) and the §jk are the constituent matrices for a corres- ponding to the eigenvalue A j' M: After Frame”. Note that such a selection of the matrix W leads to two separate homogeneous systems of equations, and that the original state variables x (n+1) are related to the new state variables y(n+ 1) by the equation 4-34- X(n+1) = 731(n+1)+ ctr/Q (n+1) D . System Stability The problem of system stability must be considered in the design of any linear feedback control system. Instability in a system repre- sented by a discrete-state model is characterized by the fact that one or more of the state variables increases without bound when the system is displaced from an equilibrium state. The following stability theorem was prOposed by‘Kalman with the statement that the proof can be car- ried out by reducing the transition matrix to its Jordan canonic form. An alternate proof is givenhere. Theorem II A stationary linear system is stable if and only if the n zeros of the polynomial equation det [@‘ALG = 0 satisfy I )3' S 1 for i = l, 2, . . . ,n. The (All that satisfy (Ail -= 1 must be of multiplicity one . Proof: Let the system discrete-state model be written as 7((n) = (I) ’K (n-I) + 8 r(n) (26) fififll _ -35- Since the system is assumed to be linear, the stability is independent of the driving functions; therefore, let r(n) = 0, then X(n) = Q xm-l) = @n 7((0) for X(0) arbitrary. The solution is of the form ’X(n) = i311 >\111+ 312 n Ail-1+. ..+ 31k n(n-l) (n-k) Ap-k-I-l n n-l n-p+l + 321A2+322n)2 +...+ 32pn(n-1)...(n-p)/\2 . n 11-1 _ _ n-q+1 + gml Xk+ 5m2n 1k +...+; mqnm 1)...(n q)/\k }l(0) or -1 -k1 ’Xm): 751W)” Vlanf +...+ Vikn(n-1)...(n-k))\n + 1 ,+ UBIAE-I- ’U‘zz 12nd +...+ UBPn(n-1)(n-p)A;-p+1 (26) , 1'1“]. n-q+1 + U‘m1>\11:1+ wnznxm +...+ U’mqn(n-1)...(n-q))\ m Since the vectors Did: 1,2,...m, j = 1,2,...k, p,-.... q. are constants (i.e. all entries are constants), consider the coefficients 0f the y ij. In general, these coefficients are of the form n(n-1)(n-2) . .. (n-i) Af-nkfl -36- where nk is the multiplicity of the kth eigenvalue A k' It is necessary, therefore, to prove the following: Lemma If I z I < 1, then n(n-1)(n-2). . .(n-k) zn'(k+1) tends to zero as n+0o , where k is a fixed positive number and n > k+l. Proof: Since n(n-1)(n-2)...(n-.k) z Mk“) < nk zn’u‘“) and Ink 2 n-(k+l) I ginkl ‘2 n-(k+1)| it follows upon setting I z) = f (the modulus of 2) that I nk z n-(k+1) l = nk 1° n-—(k+1) = nk f, n e-(kJ-l) Since p-(k‘n) is a constant, consider only the behavior of lim [nk (0 n] = lim p}: n—’ on n—voo (o-n Applying L'Hopitals rule k-l times, there results -37- .lim k! - ~1im k! Izln = 0 .n—aoo (-1)k(1n (a )k e-n 1n n—709 (-1)k(1n’a)k Applying the above Lemma to the termsin (26) it follows that the entries in the state vector tendtoward zero if I NI < 1 for all i = 1, 2, . . . ,n. If some I Akl = 1 and are of multiplicity one, then ”km = 0 for m > 1. Consequently, (26) is stable for unit eigenvalues of multiplicity one. To show that these conditions are also necessary, suppose any one of the A1 , i = 1, 2, . . . , n, exceed unity in absolute value. Obviously the entries in the vector 7):]- A ?, associated with the eigenvalue I >‘i l > 1, increase without bound as 11—» co . If any eigenvalue I kjl , - 1, j = 1, 2... .,n, and is of multiplicityp > 1, then (26) con- tains a vector sum 11 n-l .. +1 U31 Aj+ ’ngan ‘l" ... + 'V‘ipn(n-1)...(n-p))\?p which obviously increases without bound as nave . It is. of interest to note that the property of stability ina linear system represented by a discrete-state model implies that the entries .in the transition matrix @ n tend toward zero uniformly in n. as" _.l -38- Conceptually the stability and reSponse problems are solved once the eigenvalues and the constituent matrices of the transition matrix are obtained. The problem of obtaining the constituent matrices and the eigenvalues will not be considered here. The reader is referred to the appropriate references in the bibliographyll’ 12. IV THE DESIGN OF DIGITALLY CONTROLLED SAMPLED-DATA SYSTEMS IN TERMS OF DISCRETE-STATE MODELS The basic problem encounteredin the design of systems such as that shown in Fig. I is the realization of alinear recursion formula for the digital controller in the form of equation.(2). Whether or not this can be accomplished depends entirely upon the characteristics of the plant and the desired mode of response. Ragazzini and Frankling, and others5' 14 have established design procedures for systems of this type using z-transform techniques. In these procedures, it is standard practice to take the initial conditions aszero and base the design on the reaponse characteristics of standard input signals such as step- functions, ramp functions etc. In a design procedure based on state models, however, it is more convenient to consider the response of the system to an arbitrary set of initial conditions; i. e. , an arbitrary , state vector. In general, problems of inter-sampling ripple, and plant sensitivity are more tractable in the time domain than in the z-domain. For this and other reasons, a design based on state models is believed to give new insight to the design problem. The control system designproblem is primarily that of establishing a control sequence for the digital controller-in the form (2) such that a system insome arbitrary initial state can be brought to equilibrium in +39- -40- the shortest possible time without exceeding acceptable signal levels anywhere in the system. Consider first the conditions under which anarbitrary system initial state vector can be reduced to the null vector. It is Shown in the‘appendix that a ”follow-up" system is reducible to this equivalent; i. e. , the [problem of reducing an initial state vector to a constant is equivalent to reducing an initial state vector to. zero. The design may be based on forcing an'initial state vector to equilibrium along essentially three different modes: 1. Minimal-time response, in which the initial state vector isreduced to zero in the smallest number of sampling intervals N. 2. Non-minimal-time response, in which the initial state vector is reduced to. zero in a finite number of sampling intervals n > N. 3. Asymptotic-time response, in which equilibrium is attained in the limit as n -’ 0° A design procedure based on each of these'three modes of response are considered in the order given. -41.. A. The Design for Minimal-Time Response The minimal-time response design has been considered by several 1, 4, 5. Two casesare of interest: investigators 1. All of the system state variables are measurable. 2. Some of the state variables are not measurable. Kalrnan and Bertram4, and others5, have giventa solution to the problem for case 1. The more general problem stated as case 2 is considered here, i. e. , given a plant in a feedback control configuration asin Fig. I, determine the conditions under which a digital controller recursion formula can be obtained which will force a plant in any, ar- bitrary initial state to equilibrium in the minimum possible number of sampling intervals. It is assumed in the deveIOpment which follows that none of the plant state variables is directly measurable. Let the plant model be written as 12m) CL, 'X 2(n-1) + G 2 s(n) C(n). = 6'2 X 2(n) (27) where.s(n) is the control signalefrom the computer‘as derived from the sampled error, x2(n) isthe plant state vector, c(n) is the plant out- put, and sz) is an arbitrary initial state vector. Recursive application of equation (27) gives -42- 712(1)= Q2 12(0).» 62am) 12(2) 0.: 952(0) + Q2 62 s(1) + 6 2 s(2) (28) 12(3) Q3 012(0) + 03 628(1) + 622 62 s(2)+ [32 3(3) if)” 0-: %2(0) +0.21 62 8(1) + . .. + [32 s(k) Assuming that the control signals s(1), s(2), s(3), . . . s(k) are known for the present, premultiply both sides of the equations in (28) by &’2 to obtain c(1)= 0620.2 962(0) + 062 [3’2 8(1) 0(2) - ”2a: x2(0)+ 042 02 623(1)+p&2 52 3(2) (29) c(k) :- 0030,: 12(0) + fl Zakglfiz 5(1) +... +J232 s(k) The outputs c(i), i = 1, 2, . . . , k, and the inputs s(i), i = 1, 2,. .., k are assumed to be measurable. The k equations in (29) are linear in the k unknowns x01, x02, . . . , ka and can be written as pro III§I§|§IIIIUI|rJQn= ti. . ' '1 I" 1 1' " — -* r- "1 C(3) f31 f32 f33 ... f3k X03 g31 . . . g3k 3(3) = + C(k) Lfkl sz fk3 . . . fkk XOk gkl . . . gkk S(k) I... .. ..J .— _ b _1 1.. .) (30) A unique solution for the unknown initial state variables can be obtained if and only if the coefficient matrix [fij] kk in (30) is nonsingular. Any system which is described by a set of equations such as (30), in which the coefficient matrix [fij] is nonsingular, is said to be observable 1 . Assuming the system to be observable, let (30) be written as C (k) = '5‘ «2(0) + J J (k) (31) The solution for % 2(0) is of the form 12(0) = $1" c(k) - 744.1 (k) (32) and the system state vector 220i) is 752(k) = a}; {2(0) + (2‘51 .62 5(1) + + 62,2 62 e.(t~I--1)+,(5’2 s(k) -44- Substituting (32) into the above equation yields qut) = Q1375'ICUQ- 0-1; 73149.! (k) + [6113162, . . . a2 52,63,1(10 ...1... 0.1; ”+44%? - a; 7-1.4: 621 [12162, a}: 52,..., a282»62] =Q. .... %2(k)= flaunt- [Q1+Q2]J. (k) 56¢” Qal (k) —c( If rs( 1)“ = Q c(2) + Q s(2) @ka _Es(k) __ (33) where y and Q are matrices of order k. -45- After k sampling periods, all components of the state vector at t = nT, n 2 k are known, that is, _c(n-k) 9(2)“) .. y em) c(n-k-I- 1) _I + Q S(n) h — (34) If the system is controllable, then from the discussion on page 101 of the appendix - -( xn1 xn2 S(n+1) = [3.11, 3.12, . . . , 31k] xnk L. -1 where the am, j = l, 2, . . .k, are real constants. Substituting (34) into (35) gives _c(n-k) .- c(n-k-I-l) s(n+1) = Ola) + a Q c(n) = a (Ki n) :(n-k) 1 s(n—k-I— 1) S(n) (35) - ’«u' ..—r 4- :- ‘vmwvm‘s"! - J00. -45- or —c(n-k) ' r'em-k) c(n-k+ 1) s(n-k-I- 1) s(n+1)= U . + a)” . (36) em) an) _. _I 1.. .- where V: Of , and W: QQ are row vectors. If the matrix multiplicationindicated in (36) is carried out, there results s(n+ 1) + w1 s(n) + “’2 s(n-l) + . .. + wk+1 s(n-k) (37) = v1 c(n) -I- v2 c(n-l) + . . . -I- Vk-I-l c(n-k) where the vi, i= 1, 2, . . . , k+1, and the wJ', j = 1, 2, . . ., k+1, are real constants. Since (37) is of the form (2), the derivation is complete. Equation (37) shows that, for any observable and controllable plant, the control sequence can be obtained as a linear combination of the measurable plant outputs c(i), i = (n-k), . . . , n. This control sequence will drive the system from any arbitrary initial state to equilibrium in a minimum number of sampling intervals. To show that this. is actually a minimum, it is only necessary to note that, after the system state has been identi- -47- fied in k samples, it takes k intervals to reduce the system to equili- brium. It follows then, if none of the state variables is directly measurable, that a maximum of 2k sampling intervalsare required to force the system to rest. Suppose now that the matrix 2; in (30) is singularand of rank r < k. In this case, (30) can be rearranged, if necessary, and written as _. _. _ _ r _ _ _ Cr 111 l 712 01140) E11... g1k r88) l s -—-—-— = -----I-—--— ----- + . . I . . I . . a C k-r ?21 : yzz 1191(0) gkl gkkj s(n) m ..J h— .— h — ‘ where 7’11 is a nonsingular r x r submatrix. If the initial state variables ’1, k-r(0) can be measured directly, or approximated, then a solution for the remaining r variables in X50) can be obtained. The system state can then be identified in r samples, where r < k. If the system is controllable, a recursion formula can again be obtained asaabove. Such a system may be described as being semi-observable. If the variables xk—rm) cannot be measured directly or approxi- mated, then the system is said to be unobservable and cannot be con- trolled. It follows then that a system must be controllable and at least semi—observable if a minimal-time response design is to be achieved. -48... Realization of a linearrecursion formula such.as.(37) does not, however, guarantee that the controller-will be‘a practical one. The derived control signals may overdrive the plant or'they may be beyond the capabilities of the computer. If.suchis the case, one must turn to a different type of design such as that discussed in the next sections. The following examples illustrate the design procedure in realizing a minimal-time response design. Example 1: The plant G shown in Fig. I is to be controlled bya digital con- troller D and a zero -order'hold H. Design a controller‘which will force the system from an arbitrary initial state X(0) to equilibrium in the minimum numberof sampling intervals if the plant transfer function is given as 1 (3(5) 3 s(s+1) The discrete-state model of the plant as obtained from the transfer function10 is X1(k+1) 1 1' e-T X1011) T-1+ e‘T II + s(k) x2(k+ 1) 0 e‘T x2(k) 1- e‘T -49- If the sampling interval T = 1.0 second, the plant state model is x1(k+l) l 0. 632 x1(k) 0. 368 + s(k) x2(k+1) o 0. 368 x2(k) o. 632 _ (38) Obviously the plant is controllable since the transition matrix is non- Singular. The first variable x1(k+ 1) in the state vector isdirectly measurable since. it represents the output. The variable x2(k+1) is not, however, directly measurable. The first sampling period gives x1( 1) _ 1 0.632 x1(0) 0.368 II 4. 5(0) x2(1) o 0.368 x2(0) 0.632 and from a solution to the first equation x2(0) = 1. 58 [ x1(1) - x1(0) - 0.368 s(Ofl Since the state of the system is known after the first sampling, K (k) isdetermined for k 2 1. The requirement is that x (k+2) = 0 for -g_k a 1. From(27) %(k+2)=0= a2 X(k)+ 0.8 s(k)-I- 6 s(k+1) k a 1 -50... 80 0 1 0.863 x1(k) 0.768 0.368 = + s(k)-I- s(k+1) 0 0 0.135 x2(k) 0.231 0.632 Solving for s(k) -s(k) = 1. 58 x1 (k) + l. 24 x2 (k) (39) The variable x2(k) is not measurable and may be eliminated from (39) by using (38). Solving for x2(k) in (38) yields x2(k) = 0. 583 [x1(k) - xfik-I-lZ' . + 0.418 s(k-l) Substituting this into (39), there results -s(k) - 0. 519 s(k-l) = 2. 303 x1(k) — 0. 732 x1(k~1) (40) Equation (40) is the desired recursion formula of the digital controller. This controller will force the plant G in an arbitrary initial state IX (0) to equilibrium in 3 sampling intervals. It is of interest to consider the design based on the requirement that the controller will force the system ' :to follow a unit step input with zero error-after three sampling intervals. :- 1:.»- .. -51- Since x1(k) is the output variable and must follow the step input, the state vector X(k+2), for k 2: 1, must be x(k+2) = = a2 ’X(k) + @6306 + 6s(k+1) O or from (38) 1 1 0.863 x1(k) 0.768 0. 368 = + s(k) + s(k+1) 0 0 0.136 x2(k) 0.231 0.632 Solving the above, there results -s(k) - 0.52 s(k-l) = 2.303 x1(k) - O. 723 x1(k-1) - 1. 58 (41) From the feedback connection x1(k) = 1 - e(k) x1(k-1) = 1 - e(k-l) (42) Substituting (42) into (41) establishes the computer recursion formula s(k) + O. 52 s(k- l). = - 2. 303 e(k) + 0.723 e(k-l) Note that the results of a design based on the response to a step function and the state vector are identical for the above example. -52- The procedure given'above for realizing a minimal-time response design with no constraints on the driving functions can be summarized as follows: 1. Test the system to be controlled in order to determine if it is both controllable and observable. 2. Having satisfied condition 1, allow enough sampling l intervals to determine the initial state variables which I. are not directly measurable. This essentially allows " the unknown forcing functionsto be expressed as a linear combination of the measurable output or the sampled error. 3. From the relationships established in step 2, solve for the required linear recursion formula. If step. 1 in the above procedure shows that the plant is not control- lable and/or not observable inany sense, then the minimal-time response design cannot be realized by this procedure. If for any reason the minimal design cannot be realized (e. g. the plant is not controllable), an asymptotic time response design may be used as dis- cussed in Section C. B. Desigp for Non-Minimal-Time Response In non-minimal-time response systems, the state vector is forced to equilibrium in a finite number of sampling intervals n, where n is -53- greater than the minimum as determined by the procedures of the pre- vious section. To establish under what conditions this type of design can be implemented, let the plant equations be written as 12m) = Q, 'X 2(n—I) + B 2 s(n-I) (43) 01" 3(0) 5(1) )(H- an’)((0)+[:n'15 a“ 8:] 8(2) 2m ‘ 2 2 2 2' 2 82"” 2 ' s(n- 1y I.— (44) Assume that the system state vector has k elements, k < n, and that the system is both controllable and observable*. This amounts to taking more recursive steps in (43) than is necessary for minimal-time response. Partition (44) as “2"“): 0312(0))“ [#11 , 7”£12 ’1': (45) *It can be shown that the following deveIOpment holdsalso for semi- observable systems. -54- whe re 3‘ I am. an: e. 0.: 5,] 12 - [Ia/R2162 a}: 82’ 62] in and r— _ s(O) r“s(n-k) _ s(1) s(n-k+1) J—a = ‘ aim '= s(n-k— l) s(n- 1) _ .1 _. .1 If it is required that xzm) = O (equilibrium), then from (45) 0" a; 762“)” #11 Ja+ W12 Jm Since the system is assumed to be controllable, # 12 is nonsingular and it follows that .1..=- I: a; 94.33- 741; 74.1.2. <46) Equation (46) is a system of k linear equations in n unknowns, where n > k. Given any y, 2(O), this system has a unique solution for Jm if the first (n-k) forcing functions ,1 are specified. Obviously, there ' a exists an infinity of non-trivial solutions for (46) since .4 -a may be -55-. specified arbitrarily. This. additional freedom can be exploitedinthe design by selecting J a and J m in some Optimal sense. One very practical optimizing scheme that can be used to specify J a is to re- quire that the squared norm of J (n- 1) (i. e. J "J ) shall be a minimum subject to the constraint that the plant output c(n) assumes the desired value cd(n). This can be accomplished by first premulti- plying (45) by 06/2 to give c(n) = 0&2 212‘“) = 11036139520” 0&2 #11461“ &2 #121111 or c(n) = 00361121 12“” I fizz/J Let the desired output at time t = nT be specified as cd(n) and define the miss distance as M=ed(n)- 06220.3 712(0)- adj/J The variational method of LaGrange15 can be applied to find the input sequence s(i),. i = 0,1, 2, . . ., (n- 1), such that the squared norm of the driving vectoris minimized subject to the constraint that the miss distance is zero. This variational problem can be formulated as follows . -55- Let en = Cd(n) - (0‘2 Q2 sz) and u = [(JIJH )‘(en- 00'2W’Jil where the A is the LaGrange multiplier. The function u must now be minimized with respect to the elements of J . minu = mir'1J'[(J'.A)+ A (en- $271613] Taking the partial derivatives with respect to the 11 variables in there results the n equations ’0 d 551 "' 2 S1 ' >\ P11 an - " Zs - 532 2 X P12 3' u .. asn - 2 Sn - A pln where the pli’ i=1, 2, . . . , n, are constants. The) above n equations can be written as -57... $5; = 24-19) (47) subject to the constraint 06214.2 0 (48) an 3.4 If is set equal to zero, then (47) gives ,4 = +6) (49) Substituting (49) into (48) gives en " 03‘ 2 fl 22— y Assuming the scalar 00"2 w y is not identically zero >\.=2en.( ”“2344? )-1 (50) ll 0 Substituting (50) into (49), the vector Of driving functions resulting from thisparticular Optimizing process is ’4’... en(DJ-'2#@)-1 Example 1: Consider the plant represented by the discrete-state model x1(n) X2(n) -58.. 1 0. 5 x1(n- 1) 0 0. 5 x2(n-1) 0. 693 + 0.5 Given that the initial state vector at t = 0 is 760. ‘1‘” X2( 0) 10 s(n- 1) findthe sequence of control functions that will take this second-order system to equilibrium in four sampling intervals in such a way that the square of the Euclidean norm of the input vector is minimized. Let u be defined as whe re d eN— u = S'S + >‘T (61(31- HNS) I—S(0)_ X(N) - 03I 9((0) and s = s(1) s(2) ..S‘s’.) It is necessary to evaluate min 11 = min {.S'S+ )\T(e%- S HNS)} (51) -59- subject to the constraint that d eN-HNS=O (52) Differentiating (51) with respect to S and setting equal to zero gives 0 I For N = 4, (52) and (53) give )\ = (H4 Hip-1 2 e2 Substituting A into (53) gives S ~= H;(H4 H19"1 e2 where for four's'ampling periods 1.13 1.068 0.943 0.0625 0.125 0.25 and - —§—S— 8'8 4- AT(edN - HNS)} (53) (54) 0.693 0.5 The) above procedure assures that the norm of the driving vector will ~ 1.13 1.068 0.943 0.693 b 0.0625 0. 125 O. 25 0.5 d -60.. 0.52 ~1.23 -1.23 5.9 10 O 75.03 4.01 1.32 -2.45 -! _I be a minimum for the sampling interval chosen; and certainly helps to keep the input signals from overdriving the plant. . It does not, however, assure that any single entry in J (n-l) will not overdrive the plant. The individual entries in )4— (n- 1) can be maintained below a safe, predetermined level using the above _te_chnique by using a multi- norm constraint onthe driving vector. .A discussion Of this procedure is deferred until Chapter V. If the input levels on the plant are firmly established, the alternate procedure given next can be used asean aid in selecting the driving functions such that none of the s(i), i=0, 1, . . . , n- 1, will overdrive the plant . whe re Let (.46) be written as .1... 3+ #4 y= - 31;; 6L2 95.0 (55) W1 . a it: C t -51- and #='KL;: #11 The detailed form of (55) is .... ... ,— ._ .... ... __ .. 8(n'k) yl hll hlz . . . h1(n'k) 8(0) s(n-k+1) y2 h21 h22 . . . h2(n-k) s(1) . = . + . . (56) 5(n-1) ' i. h (n-k) Aux-1.4) L. ...] ...ykJ L—kl k .. _. ..J where the yi, i=1, 2, . . .k, and the kij' i=1, 2, . . . k, j=l, 2, . . .,n-k), are real constants. If the saturation requirements on the plant dictate that all s(i), i=0, 1, . . .(n- 1) shall not exceed 1- k, where k is a real constant, then the following set of inequalities must be satisfied. I Y1 + 1111 6(0) + + h1(n-k) s(n-k-l)l 5 k t8(0)l S k, |s(n-k)| |s(1)| 5 k, |s(n-k+1)| I yz + 1121 s(O) + + h2(n-—k) s(n-k-1)| s k (57) ls(n-k-1)| i k, [s(n- l)| } yk + hkl 3(0) +.. . .- + hk(n-k) s(n-k- 1). f k q -62- A sufficient condition for the existence of a solution to the inequali- ties in (57) can be obtained as follows. Let lyil 5 k; this condition assures that the origin is included in the requirement space. If the equalitiesare taken in (57), the system of equations defines a set of hyperplanes bounding all possible solutions of(57). Consequently, when lyi] é k, then there exists the Baiti- cular solution 3(0) = 3(1) = 3(2) = = s(n-k-l) S and (57) becomes “k ‘5y1+h118+h128+...+h1(n"k)S 5 k h-k 5 y2+h218+h223+...+h2(n-k)s s k -k s yk+hk1 s+hk2 s.+... +hk(n-k) s s k or n-k “k “yl 5 S E hll s k ' y]. i=1 .. _ -k 'k "3'2 5 S i h21 5 k "' y2 (58) i=1 n—k ' -k-yk 5s hki :3 k-yk -63- The set of normals to the hype rplanes bounding the solutions to (57) are k - yi 2 2 N'H (59) k + yi 2 2 [hil + hi2 + ... + hi(n-k):l 1 i=1,2,...,(n-k) Let ‘ k - y1 k + Yi "hm" r=min . (n-k-l) 1.1“ llhijll l .. ,2,...,(n-k) J ’ (60) Equation (60) describes a hypersphere R, of radius r > 0, centered at the origin in the requirement space, which contains solutions to (57). It follows that there exists a set of s(i), i=0, 1, 2, . . ., n- 1, which do not exceed the saturation level of the plant if only IYil < k. Obviously, this procedure does not isolate all solutions to (57) but serves to determine if there existsany. inside R. If the requirement is only-to obtaina set of signals which do not overdrive the plant, any set of values inside of R will do. i= 1,2,...,(n-k-1) -54- It remains to inquire whether or not the requirement that lin < k is a realistic one. The vector y is given by y: ' #i; a: “2“” The matrix a: has entries which tendtoward zero monotonically* as n increases (see Theorem 11) while the entries in - Hi; and XZW) are constants; therefore, the entries in 2% decrease if n is allowed to take on larger and larger values. The number of sampling intervals n can be selected arbitrarily by the designer which assures that for some n, the condition I Yi |< k can be met. . If finite-response time is given. in the specification of the design, then assuming the plant can be controlled, the response time can be made minimal or non-minimal asthe designer so chooses. If minimal- response time is selected asa basis of design, there can be no adjust- ment of the input signals, to conform to saturation requirements since the minimal design is unique. Only in non~minimal-response design is it possible to control the levels of the signals. The number of sampling intervals used to establish equilibrium is strictly a function of the parti- cular control problem at hand. The following example illustrates how *Providing a2 has no unit eigenvalues. p -55- this procedure may be applied. Example 2: Consider the plant represented by the discrete-state model s(n-l) II + x2(n) O 0. 367 x2(n- 1) 0.433 Given that the initial state vector is ’)((0) = find the sequence of control functions that will take this system to equilibrium in such a manner that none of the control signals exceed + 3. Equation (55) for N = 4 is s(2) ' 1.9 -1.07 -1. 17 8(0) II 4. 8(3) 0.7 0.34 0.29 s(1) Since | yi I 4 3, s(O) and s(1) may be selected using the equation in (59). Obviously, the minimum value of s will be generated by the form -66- 3 - 1.9 J(1.o7)2.+ (1.17)2 0.69 Letting s(O) = 3(1) = s = 0. 69, the solution for 5(2) and s(3) is s(2) 1.9 -1.07 -l.17 0.69 = + s(3) 0.7 0.34 0.29 0.69 0.35 1. 13 The above solution shows that none of the control functions exceed 1- 3 as required. C. .Asymptotic-Time Response Fora number of reasons, a finite response may be impossible to obtain. Among these reasons are: 1. The plant may not be controllable. 2. The plant may not be observable in any sense. In order to treat those systems which fall under the above classifi- cation, an extended concept of plant controllability is discussed which does not. depend on the existence of the inverse of the matrix # as -67... outlinedinthe previous section. A design realized on. the basis of the extended concept is not, however, a unique design. The Stability Theorem'II assures that the system to be controlled will respond to any bounded driving function in‘a stable mode as long as the eigenvalues of the transition matrix are less than unity in absolute value. . Further, it is noted that the response is. asymptotic if the eigen- values are positive. and oscillatory if the eigenvalues are negative real or complex. The mode of response in either case is asymptotic; that is, the equilibrium point inthe state—space is attained in the limit as t -9- (>0 . To «achieve a particular type of response, the eigenvalues of the system transition matrix must be Specified. There appears to be no optimal way of selecting these eigenvalues other than by ex- perience. .Since this is the normal state of affairsin s-domain or z— domaindesign, nothing-is gained or lost on this point by using state models asa basis of design. Let it be assumed that the desired eigenvalues of the system tran- sition. matrix are given and that the discrete-state transition matrix of the plant is known. The problem is to select the digital controller con- stants so that the system transition matrix has the desired eigenvalues and. constituent matrices. The general form of the system transition matrix Q .as deve10ped in Chapter 'II is -68- ®= Pa. 52-4“. L: 810(9’204 "61””2826'1‘” all for-which the characteristic polynomial is detEQ 4.1.9: det 02 - 8 2 ‘01 - I.- 61‘0'2042 ' 81A'262’6'13' affix (61) Elementary row operations11 on the determinant in (61.) yields a 2 .m (32 ,0», detEQ-ku = det (62) - A 81003 al -A u __ .4 which in detail is all-A 8.12 8.13 . . . 31k : b1]. b1 13.1 bllak : I . I ak1 akz ak3 akin) : b1:190 bklal bklak det . I "Adm - N112 ...... -)\d1k I 401—); -b2 “bk I 0 0 0 0 I 1 - )\ 0 . . . | 0 1 0 o . l o o o o i o 1 -’>\ L— .1163) -59- To relate the eigenvalues of the system to the computer parameters, expand (62) by Laplace's expansion in terms of the submatrix [CL-ml For a sixth order system, the expansion is z‘7‘11’A 5‘12 313 ’bl') 'b2 'b3 D0.) = .121 .122». .123 .(-1)Sij 1 - ,\ 0 8L31 332 2133-). 0 1 -}\ a‘11")‘ a12 b11 ' )‘as “'02 'b3 4" 3.21 fizz-A b21 . a0(-1)Sij 0 - A 0 .ch31 2132 1’31 0 1 -/\ I a11')‘ 8L12 b11 ' ML3 “b1'/\ 'b3 _ _ S.. + a21 a22 A b21 .a1( 1). 1] O 1 O 331 8L32 b31 0 0 7‘ a11')‘ 313 b11 ' >\“112 401‘?) 4% + . . . -+ 321 3.23 b21 .a2('1)slj 0 1 ' A 2131 333') P31 0 0 1 (64) Since the submatrix in the 1, 2 position in (62) is of rank one, any combi- nation of one of the first three columns with any two of the last three columns yields a minor of zero value and D( A ) reduces to -70- mm = lo..- who.- m|+ IE. x2+K2A+K.] c... I}. 13..., was] + [K4 A2+K5A+K51 012 [a0 A3-+a1/\2+32’\:I . [K7 12mm +ng C11 [a0 Aha. was] um = Iaz'WIIOr W] + E0 ,\3+a1 )‘2+a2)«:| I}; )1»ng «5] where the K1, K2, . . . , K9, K1, Kz', K3 are constants determined by the plant transition and output matrices. Expanding the product of the first two terms, (65) .+ [a0 A3+a1,\2+a2)\] I: K1'/\2.+K2')\ +K3:| Equation (65) is the sum of two polynomials whose coefficients are functions of the computer parameters a1 and bi. The first polynomial is of degree six and has coefficients which are linear combinations of the bi. The second polynomial is of degree five and has coefficients which are linear combinations of the a1. The sum of the two poly- nomials is monic and has coefficients which are linear combinations of the a1 and bi. -71- Expanding the characteristic polynomial gives D(>\) = A6 + (P1+b1+Kiao) A5 + (P2+P1b1 + b2+Kia1 ‘I' Kz'ao) A 4 + (p3+p2b1+p1b2+b3+Kiaz + Kz'al + K580) A 3 (66) + (p3b1+p2b2+p1b3+K2'a.2. + K§a1)/\2 + (p3b2+p2b3+K§a2))\ “" P3b3 Equation (66) is' the desired form of the solution for the characteristic polynomial. Note that _e_a_c11_ coefficient can be controlled by the magni- tude of the 3i and bi in the computer transition matrix. This implies that, given sufficient freedom in the magnitude of the computer con- stants, it is possible to apply asymptotic time control to a plant which, for example, is not controllable in the sense of Kalmanl. This is then an extended concept of controllability where the origin of the state-space, or the equilibrium point, is attainedin the limit as t +00 . Note that nothing has been said about the stability of the uncontrolled plant. In fact, there is no reason why an unstable eigenvalue in the plant transi- tion matrix cannot be digitally compensated. This property, however, is well known from z-transform control methods9, but the attendant z- domain sensitivity problem associated with the cancellation of transfer function poles and zeros is. avoided. The cancellation of transfer func- tion poles and zeroes in frequency domain design procedures. is usually ,-72- considered asa fundamental problemg. The sensitivity problem, how- ever, does not have a counterpart in discrete-state system design in terms of state models, i. e. , concern over sensitivity. in z-domain design has no foundation in fact*. Suppose a set of eigenvalues have been specified for the sixth order H system considered above. Let the desired characteristic polynomial I; for the system be written as D()\) = k6+c1 A5+c2 >\4+C3 A3+c4 A2+c5)\ '+C6 (67) Equating (66) and (67), there results. six independent equations in the six unknowns a0, a1, a2, b1, b2, b3, namely "—1 o 0 K1' 0 o _ I‘bl' "pf _c1_ pl 1 0 K2' K1‘ 0 b2 p2 CZ P2 P1 1 K3' K2' K1' ‘03 + 133 = C3 (68) P3 . P2 P1 0 K3' K2' a0 0 C4 0 p3 p2 o 0 K3' a1 0 C5 _0 0 p3 O O 0 _ _a2d _0_I c6 *See reference 2 for a brief discussion on this point. -73- Equation (68) has a solution if the inverse of the coefficient matrix exists. Unfortunately, there is no convenient way to establish the cor- respondence between the entries in the plant transition matrix and the entries in (60); so very little can be said about the existence of the inverse. This approach may be used in general for any order of system once the response has been specified. There is no way, however, to predict the magnitude of a1 and bi resulting from a solution of (68). Conse- quently, there is no assurance that the resulting a1 and bi will define a practical control scheme as discussed on page 48. . Omitting the tedious and somewhat lengthy matrix operations, the general form of the Laplace expansion of the system transition matrix for k elements in the plant state vector and k elements in the computer transition matrix is of the form D(>\) = [Xk+p1 )(k'1+. . .+p1;[ Dk-l-bl >\k-1+. . .+bg (69) k 2 I k-l l k-2 l + 133.0% +...+ak_2)( +ak_1);| B1 A +K2}\ +“'+Kk-1’\ 4.19:] The first term in the first product is the characteristic equation of the plant, the second term is the characteristic equation of the controller, -74- and the second product will be called the control product*. , The following example illustrates how anasymptotic time response design is achieved. Example 1: {T Consider a plant which has the following discrete-state model 1 x1(k+1) 1 0. 5 x1(k) 0. 693 = + s(k) x2(k+1) O 0. 5 x2(k) 0. 5 *It will be noted that the stability of the open-loop system of Fig. HI cannot be controlled by the digital computer. This is apparent when determinant Q - XII ” is considered. For no feedback, the transi- tion matrix has the form ,_ 2 62 0(9’1 @' [0 a1 Thendet [CD -)\’IL] O=det [OZ-MA] det [aw-Mi] The characteristic equation can be written as det [(21- ).u] det [Ch-Ml] = Ek-A1)()\-)\3>...<>.-Ak) (k-A2)<)-A4>...(A-)\mfl which shows that if the plant has any eigenvaluesin its transition matrix of modulus greater than unity it cannot be compensated by the controller and the system will respond in an unstable mode. This is not true, how- ever, for the feedback system considered above. -75- It is desired to control this plant in a feedback arrangement as shown in Fig. I with the asymptotic response given by the system eigen- values >\1=0.2, >\2‘=0.3. A3=o.4, A4=o.5 The transition matrix for the system is F1 -) o. 5 0.693a 0.693a o o 5 -)\ 0 5a 0 5a - A 0 -b1 -)\ -b2 0 o 1 - A The Laplace expansion for the above determinant gives A4 +(b1' 1.5 + 0,693a0) A3 + (b2 - 1. 5b1 + o. 5 - 0,097.10 + 0.693a1))\2 + (0.5101 - 1.5102) - 0.097a1) >\. + 0.5132 From the specified eigenvalues, the desired characteristic polynomial is )(4—1.4 A3.+o.71 )‘2-0.154>\ +0.012 Equating coefficients in the two polynomials the re results -75- o.5 o -o.097 _ r 131—1 —-0.118_ -1.5 -0.097 0.693 a0 = 0.186 —1 0.693 0 .- -_ 31.1 _0.1 L. For-which the solution is Imbl.7 ——0.47_ a0 = 0. 70 La1_ b 0. 20“ The recursion formula employing the above constants will force the system to respond asymptotically and, consequently, is an-infinite- settling-time design. This designis analogous to the z-domain design using a so-called.”staleness" factorg. V SYSTEMS WITH MULTIPLE INPUTS AND OUTPUTS Consider a sampled-data control system having more than one input and output such as the system shown below. _ __ Output FIG. V Let the state model of the plant be given as ’Xm) = OXXm-n + [3 [Q (n-l) C(n) «$9600 where x (n) is an nth order state vector, IQ (n- 1) is an mth order (70) vector of driving functions and C (n) is a kth order vector of measura- ble outputs, n 2. m, k. The output vector expressed as an explicit function of the m inputs is C (n) = 0(9’Q7Un-1) + 0016/3014) (71) -77- .- FIG. VI It is required to design a controller in the multivariable feedback system of Fig. VI that will reduce a given initial state of the plant to zero in a finite time if possible. Two cases are of interest. If all the state variables are measura- ble, the problem has already been considered by Bertram and Sarachik3. If all states are not measurable, they must either be approximated in some manner or sufficient control time must be provided to identify the unknown state variables. Suppose for example, that only the k outputs can be measured. From these outputs the n unknown initial state varia- bles can be computed under certain conditions. To show this, let the first equation in (70) be written recursively as, im- a'an (5)/QM 7((2) - 0.231(0) + dam» + 616(1) ’lin) = anfihm + 623146060) + 6LP'26/61>+...+ gem-1) -79- Premultiplying the above equations by 08’ gives ( ik ) linear scalar equations in the 11 initial state variables, x10, x20, X30,.+ . , an' (3(1) $0}wa (fig/60) 6(2) 1&0} z(0)+ erfi/fiwn (£19361) C(i) 0(9’a} X+ &ai'16/€(0)+...+ diflfiu-n Given IQ(0), fl (1), . . . [e (i-l) and assuming the required inverse exists, the first 11 of these scalar equations can establish the initial state variables. At most i = [n/k] * sampling intervals are required. As in the case of the single variable, this system is called observable. Assuming then that the entries in the initial state vector can be established, the design is realized by writing (70) as 76(0) - , [61) 90k) = CLkQUOH [0546. CF26 .....d€.8] 3(2) (72) KZk-l) *read In/k] as, "the largest integer in n/k. ” 7 v. ""-'”_v-; —‘ (5m . -30- For convenience, let [CU-1 80.192 [5W £6.61 - #N nx( km) In general, #N is rectangular and as such does not have an inverse. Suppose N represents the minimum value of k for which the rank of #N is maximum; .i. e. , there exists n independent vectors among the Nm columns of #N' Then by rearranging columns if necessary, # N may be partitioned as I #N ‘ [fl N1 5 N’ N2 where 1F N 2 is a non-singular nxn submatrix of n N' If this opera- tion cannot be carried out, then the system is not controllable; that is, 14 N must be of maximum rank for the system to be controllable... When (72) is written as /Q N. «(NF QN 7:0” [#Nl» #N2 (73) nx(Nm) flNZ (me 1) the solution for ’QNZ when 1(8) = 0 is -31- IQNzh #152 [QN 00°” Wm fl N1 (74) where the set of inputs K N1 are arbitrary. In the special case where n (the order of the state vector) isan in- tegral multiple of m; i. e. , n = qm, q a positive integer, the matrix I #N is square. If it is also nonsingular, (73) reduces to I); (N = ' #1111 CLN 76(0): n'=qm (75) Therefore, whenthe number of inputs m is an integral multiple of the number of state variables, the set of forcing functions which will take any arbitrary state vector to zero is uniquely specified. This is also the minimal time solution if all the initial state variables are measura- ble. If the solution indicated in (74) exists, the system is said to be state controllable which, incidentally, implies output controllability; the converse, however, is not generally true. Returning tothe more general problem, let the initial state X (0) at time t = to be given. Find the sequence of driving functions which will carry the system through some desired sequence of states % (1), 95(2). 71(3). 96m). For N Z n the plant model can be written as -32- %(N) = 01:: 90°) + #N ’Qm)’ (76) where N’N is asdefined after (72). If xd(N) is the desired state afterN intervals, there results for (73) 7‘)! :I We..." N2 -..-- (77) 16...] If the system is state-controllable, then #N2 is nonsingular and (77) xdm) = QN 76(0) + [Wm give 8 ’QNZ = #172 DOWN) ' aawa) " flNl em] If g N1) is selected arbitrarily, then the driving functions /€ N2 are determined*. The question remaining is how to select the values of KN.- *If ‘# is a square matrix (the number of state variables b ing an integral multiple of the number of inputs) then, of course, 1% N is unique providing the system is controllable. -83- Three mathematically tractable methods of selecting K N which are of practical interest are: 1. selection of IQN such that the norm of the control vector is minimized subject to the constraint that the distance between the actual state and the desired state is zero. This may be accomplished using LaGrange multipliers to minimize the scalar function ’énin $141?qu + XIX 7Cd(N) - all 900) - F/N/(N) (78) N 2. selection of e N such that the distance between the desired state and the actual state is minimized subject to the constraint that the norm of the control vector is equal to some predetermined constant. Again using LaGrange multipliers the problem reduces to minimizing the expression min {( 901m)- aN ’Xm-ZIN £1.55,“ )(d(N)- aNx(0)-#N/€N) (N + x [IQ 13‘ 41. ,QN-K] (79) -84- 3. selection of [Q N such that the distance between the de- sired state and the actual state is minimized subject to a multi—norm constraint on the control vector. Using LaGrange multipliers, the function to minimized is ...1..ng-#N/QN)T%(EN-#NIQN)+A1@IUQNHAZJZUQN) KN +...+) k’ék(/€N)} (80) whe re Ji(flNI=(£§§i/€N'ki) and g i is a diagonal matrix of order meNm and rank r f. Nm, and k1 is the constant associated with the ith constraint. Design procedures based on each of the above constraints will be considered in the order given. Letting 5 N = g: d(N) - QN 75 (0), (78) may be written as 2:“ we: {PM/9+ WNW/8.)} The partial derivative with respect to R-N is -35- Bu - _ T BRN - ZEN WNX where #N is defined after (72). Setting 8u/ a [Q N - 0, it follows that a minimum occurs when gN =§Z¢r§k and EN' 7417 IQN" 0 or CN' "é #17 fig A If ( #N fl qu)-1 is nonsingular, then 7" 2 EN(#Nfl§)-l and the sequence of driving functions is EN = #Nr €N< KIN #EI-l " #3} Wm) - QN%(0))(#Nfl§,-1 ~86- and the state model of the computer has been determined. The norm of the plant driving functions is a minimum, ‘subject to the constraint that the state vector achieves the predetermined desired value. The process depends on the existence of ( #N 3‘ E104. When # N has real entries and is of maximum rank, the product ( ‘3‘ N H g) is positive definite and, therefore, nonsingular*. Note also that since the predeter- mined vector %d(N) is achieved, the system is automatically stable. This procedure assures that the norm of the control vector is a mini- mum but does not assure that any individual entry in the control vector would not exceed some permissible safe upper limit. In order to provide some control overthe magnitude of the plant inputs, the second procedure outlined above may be used. The con- straints on the individual plant inputs may be stated as IrozI 5 K2 IrONmI “<- KNm * if is of rank n, and order anm, so #N #N T is an nxn matrix of rank 11, therefore positive definite. -87- where the Ki, i=0, 1, 2. . . , Nm, are the constants describing the safe upper limits for the respective inputs. The sum of these terms clearly represents the squared norm of [Q N- When such constraints are placed on the inputs, one can only minimize the distance between the desired state and the actual state. Again using LaGrange multipliers, the problem reduces to minimizing the expression in (79) . If (79) is written as zinu =rjeir1:I {(EN-#N’QNITu(EN-WNEN)+’\(R"1I\I/€N)} N Then upon differentiating with respect to [6 one has N’ 91. 3R subject to the requirement, that ’6; [Q N = K. (82) = -fl;(€N-yN/QN) + )(EN = o (81) Solving (81) for jQN gives - )QN=(/\W+ Hgfi/erfllrgEN (83) Substituting this result into the constraint equation (82) gives a system of nonlinear algebraic equations in )k . -33- 6M. E W333.“ T I: A“+#§#N>'J#{.CN=K If A = X 0 is the solution (if it exists), then from (83) (N: DIou+ #5 #N ]-1 #NCSN and the control function has been determined which minimizes the miss distance subject to the one constraint in the driving function. Additional and more restrictive constraints on the inputs can be introduced by considering a multi-norm constraint on the inputs as shown in the expression (80). Each of the constraints are of the form 2 . .2 _ ] r0k I + Iroq ' - Kp (84) Here Kp ,is selected suchthat rOK and qu- do not exceed some pre- assigned safe upper limit. Letting EN: %d(N) - QN 1(0), (80) can be written as 761“ u "' I?“ {‘ EN‘ Kin/€316“ (CN'NN/QNH )‘ 1K§5VeN * /\2K11\13 2KN+"'+ >RKIEJRK’N} Differentiating with respect to the entries of I? N gives or a. as. which for convenience is written as T(eN-#NRN)+2 ARI“: Equation (85) must now be solved subject to the constraints 3RN Q) s: I Q) ..I’ Q.) (:1 ° cu n? 0110' ’1 C :3 K” rIi‘eN' flN’QN: 74‘ 12(981'5/ N KN) n1r1'(eNWN’QI\I) b an =-2(eN- -39- " '2(eN'?/N’QN)TH1+2 A 1’1 -2(eN-flNEN)T#2+2 Alrz +2 #N’QN)T#nm+2 rl (85) r’ ‘. VJ‘M iH-m-W'“ v- =—' J | -_-.-..- - . -90... -. 1 a leg 31 1‘8. N 7K1 KN j. 2 K- N K2 I - Z (86) ,._. 2 _) L. 2...) Solving (85) for R N and substituting (86) into the result gives E‘fld'fllgflNI—IHN en] T 3n Efl+#§#N)'1#§ebfl = K1 T - T T - EA+Z¢NWN) 17“N 6N] 32 Efl+flN#N) 11"; 813 =K2 ' T -1 T T - T E—fl- + flNZ/N’ flN eN] 35%.“..EA” #NflN) IXI’N em] = K}; The required minimization is realized as a solution to a system of non- linear equations. Let the solution (if it exists) be 3 1, A 2, . . . SNm . 2 The controller which minimizes the miss distance with pair-wise constraints on the drivers is then KN=(fl-+ #'§#N)-l #33 3N (87) -91- The above discussion, which is certainly preliminary in nature, suggests several methods of approach which may be used to aid in the design of state control for a multiport plant. Unfortunately, the con- troller characteristic can no longer be modeled as a simple recursion formula. An effort to find a general mathematical model for the input-output characteristic of a controller having multiple inputs and outputs met with little success. Until such a model for the controller can be determined, the multiport system design problem cannot be handled with the same facility as the single input-single output system. The control functions may still be determined with the aid of the above formulas, but the method for determining them from the sampled state variables is unknown. VI SUMMARY The problemsassociated with the analysis and design of linear, sampled-data control systems have received much attention in the past decade. Until recently such analysis and design has been carried out using the z -transform as the mathematical tool. This thesis considers the design of linear, sampled—data control using the discrete—state model. A state model of the system has been derived from the state models of the system components and a simple algorithm established relating the transition matrix of the system to the transition matrices of the components. This. algorithm forms the basis for the design and can be conside red as the counterpart of the z-domain- algorithm D(z) G(z) H(z) = l + D(z) G(z) where D(z) and 6(2) represent respectively the z-transform of the con- troller and plant characteristicsand H(z) is the system transfer function. Functions of matrices were used as a basis for studying the charac— -92- -93- teristics of the system model. It is believed that this is the first appli- cation of functions of a matrix in the design of sampled-data systems. Several general forms for the system-state response to particular drivers have been derived from both the homogeneous and nonhomo- geneous forms (of the system; models and’the.fundamenta1 stability theorem established in terms of functions of matrices. The major contribution of this thesis, however, is believed to be in the area of time domain design of sampled-data systems. A design procedure for minimal-time response was deve10ped for plants which are both observable and controllable in some sense. Such designs are shown to be unique and, consequently, provide no latitude for the control of the level of the plant input signal. A design based on non-minimal response is shown to be more flexible. If the plant is observable and controllable, a design based on non-minimal response always exists. Several methods for determining the "best” control scheme, subject to certain constraints on the magni- tude of the control sequence, are presented. An extended concept of plant controllability has been developed in which the plant achieves equilibrium in the limit as t —9- co . It was found that unstable eigenvalues in the plant transition matrix may be -94- compensated without creating a problem in sensitivity, thereby showing that the so-called sensitivity problem in z-domain design has no founda- tion in fact. A preliminary study of the design of systems with multiple inputs and outputs has been made, and the design procedure is shown to lead to nonlinear algebraic equations._ It is felt that this area represents a fruitful area for further study. APPENDIX A Discussion of Controllability. Let the plant state transition equation for k variables in the state vector be written as X(n+1) = CUM) + 6 r(n+1) The state of the system at successive sampling instants is given by ’Xm- a'Xm) + Bra) 7((2) - @2900) + a 6 r(l) + 6:42) (Kat) - ak %(0)+ CE“ 8 r(1)+... + a6r(k-1)+ Brut) Assuming a is non-singular, that x (0) is given and that it? is .re- quired to establish r(l), r(2), . . . r(k) such that x (k) = 0, it follows from the last expressionthat . 700) = —r(1) a-l [3 M2) @328 _r(3) Q96 _. H_r(k)a-k6 OI‘ X(O)=r(1) C5 1+r(2) CK 2+r(3) Cg3+...+r(k) CK k -95- -96- whe re C31: ' 0:18 The above matrix equationis a system of k scalar equations in the k unknowns r( 1), r( 2), . . . , r(k). This system has a solution for the ri for any X(0) if and only if the k vectors %1, g2, (g3, ... % k are linearly. independent; i. e. , if the vectors 6, a6. a26,..., axis are linearly independent and span the k-dimensional vector space of x . Such a system of vectorsis said to define a system which is "con- trollable. "5- Let the solutionfor the r(i) be writtenas ,- _ -- ..- .. _ r( 1) d 11 . . . a 1k x01 r(2) . x02 ' = i I (A. 1) r(k) O M . .. O( kk ka L. .. 8‘ _I I I...— .— For any given x (0) the solution for the ri is unique; therefore, a unique set of control functions exist that will take any arbitrary state ’XW) to the origin in exactly k sampling intervals providing the system -97- is controllable. The requirement that Q be non-singular is too restrictive for general controllability. Suppose that a is singular. Then as before, for k variables in the state vector, %(k) . ak )00) + 03“ ,6 r(l) + + Q6r(k-1) + 6 r(k) Let XI“) = OfornZ kthen -a“7€= 0346 r<1)+...+ Q8r(k-1)+ 13m.) 01‘ rr( 1)- —a-*?)(= [ark-18. ak-zs . ads] .... r(k) I... [Elk-113 , ak-z 43 dag] .. @ Ifthekvectors 0.1016 , ak'zfi , ...,a6,8 are linearly independent then ¢ '1 exists and -98... "- —' r“- "I r(1) x1(0) r(2) X2(0) . = . $-1Q1f . Eat) ;. (o) ... .. L. k ._ The system represented by O. . is therefore controllable if ak-l B : . - ., Cl 6 , 8 area linearly independent set of vectors. A rather interesting situation arises if i is singular; that is, the vectors ak'l ’3 , ..., a 6 , )6 are linearly dependent. Assume @ has rank r ‘5- R. Then it follows that ‘ r(1) - ak X(0) a Q Q where Q' r(2) .( k) Since - ak %(0) is a known vector, let - a“ ”7((0) - 1140) and partition %(0) and i to give -99_ — 7.” 91“” =- § 11 Q 12 Q1 (A. 2) [92(0)— “Q 21 622; Q2 In; Let 0 be an elementary transformation on the rows of @ which reduces it to the form j? pé. P1 I—‘ 2 where I1 1 is a non- 0 0 singular matrix of order r. — _ Setting 0’ 3(0) = ?'(0) = -,@ CU" 1(0), (A. 2) becomes I r— —‘ "$1” I". I12 I = (A. 3) ff”)... 0 0 'Q 2 .— A .— .— The first equation in (A. 3) is a system of r scalar equations in the k unknowns Q1, Q 2. This system has a unique solution for any arbitrarily specified set of k-r variables satisfying the second equation. There are two possibilities: either the initial conditions are such that f“ ak) annihilates the last k-r initial state variables or it does not. If it does not, which probably is the case, then the system repre- sented by ® is not controllable in the sense of the above definition. Assuming the system is controllable, then from (A. 1) the solution for the control functions is -100- r(1) = 0(11x01+ 0(12x02+...+ dlnXOn “2) = O(21"01’“C9<22"02‘*°“+0(2n"0n r(n) = 0(n1x01+ O