0PTIMAL SINGULAR CONTROL THEORY WITH APPLICATION TO VEHICULAR BRAKING Thesis for the Degree of Ph. D. MICHIGAN-STATE UNIVERSITY MICHAEL BISHOP SCHERBA 1959 THESIS _ jWWt-z.’ mug}. ; LIBR A R y Michigan State \‘ UanCrSity n This is to certifg that the thesis entitled OPTIMAL SINGULAR CONTROL THEORY WITH APPLICATION TO VEHICULAR BRAKING presented by Michael Bishop Scherba has been accepted towards fulfillment of the requirements for PhD degree in Electrical Engineering and Systems Science M Wm Major professor I Date WZQL/qé 7 0-169 I I L'HRARV BINDERS . xi‘ new: RY HICIIIEII y amomo av 7' "DAB & SENY 300K BINDERY INC. ABSTRACT OPTIMAL SINGULAR CONTROL THEORY WITH APPLICATION TO VEHICULAR BRAKING by Michael Bishop Scherba General results from both the Maximum Principle and Green's Theorem are specialized to a class of singular control problems encountered in vehicular braking processes. These problems are in the class of nonlin- ear problems in which the control appears linearly. These are of the form i(t) - f(x,t) + B(x,t)u where the n dimensional vector x(t) is the state of the system at time t and the r dimensional vector u is the control vector. The object is to find a control vector which takes the system from some initial state x0 at time to to state xT at time T and minimize the functional T Jlu] - J fo(t,x,u)dt to In the vehicular braking problems, the functional J[u] corresponds to stop- ping distance. This problem is shown to be equivalent to the time optimal problem for the class of functions encountered. Necessary conditions along singular arcs are established using both the Maximum Principle and Green's Theorem. Algorithms for determining op- timal trajectories along both singular and nonsingular arcs are developed using the concept of reachable and controllable sets. The optimal control as a function of the state variables — the closed loop problem - is solved. Application to vehicular braking processes is shown by means of both rate and amplitude limited controls. The result- ing systems are singular "pang-pang" and singular "bang-bang" systems. The inability of a single mathematical performance index to encom— pass all the qualities desired in the vehicular braking system resulted in the development of suboptimal control systems. Favorable comparison with optimal control systems is shown by means of digital and analog techniques. Simulation including real hardware shows application of the theory. Several frequency domain criteria are included to provide in- sight regarding the effect of time delays in suboptimal vehicular braking systems. OPTIMAL SINGULAR CONTROL THEORY WITH APPLICATION TO VEHICULAR BRAKING By Michael Bishop Scherba A THESIS 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 1969 ACKNOWLEDGEMENT I wish to express my gratitude to the members of the Guidance Committee - Dr. Gerald L. Park, Dr. Herman E. Koenig, Dr. John B.Kreer, Dr. Robert O. Barr, and Dr. E. A. Nordhaus for their comments, encour- gement, and continued interest during the course of this research. Thanks is also due to Dr. Richard C. Dubes for assistance during the initial phase of my program. Appreciation is also expressed to my wife and sons for their aid, patience and support. ii CHAPTER I CHAPTER II 2.1 2.2 2.3.1 2.3.2 2.4 CHAPTER III 3.1.1 3.1.2 3.2.1 3.2.2 3.2.3 3.3 CHAPTER IV 4.1 4.2 4.3.1 4.3.2 4.3.3 4.3.4 4.4.1 TABLE OF CONTENTS INTRODUCTION SINGULAR CONTROL THEORY Introduction The Maximum Principle Approach The Green's Theorem Approach Determination of Reachable Region The Second Variation Approach TIME OPTIMAL SINGULAR CONTROL Maximum Principle Approach Closed Loop System Control Green's Theorem Approach Singular Pang-Pang Time Optimal Control N-Dimensional Singular Control Problems The Second Variational Approach OPTIMAL CONTROL OF THE VEHICULAR BRAKING PROCESS Introduction Deve10pment of the System Equation Optimization of the One-Wheel Vehicular Braking Control Problem Analytical Verification of the Optimal Control Equivalence of Minimum Time and Minimum Stopping Distance Criteria An Optimal Digital Control System The One-Wheel Vehicular Braking Control Model with Rate and Amplitude Limited Control iii Page 15 22 27 34 34 38 41 43 46 49 S3 53 56 S8 63 66 68 73 4.4.2 CHAPTER V 5.1 5.2 5.3 5.4 CHAPTER V1 6.1 6.2 6.3 CHAPTER VII APPENDIX I APPENDIX II APPENDIX III APPENDIX IV BIBLIOGRAPHY Solution of the Pang-Pang Singular Control Problem by the Green's Theorem Approach SUBOPTIMAL VEHICULAR BRAKING CONTROL Introduction Block Diagram of Suboptimal Control System Analog Computer Studies Digital Computer Studies NONLINEAR PHENOMENON IN VEHICULAR BRAKING PROCESSES Introduction Time Delay Criteria by Describing Function Technique Effect of Friction-Slip Nonlinearity CONCLUSION Suboptimal One-Wheel Model Control System Program used in Optimal Digital Control System Based on Green's Theorem Approach Differential Forms Alternate Digital Program iv Page 76 79 79 81 84 94 118 118 118 124 131 134 136 141 148 152 Figure 2.1 2.2 2.3 2.4 2.5a 2.6a 3.1 3.2 3.3 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 LIST OF FIGURES Comparison of Trajectories Determination of Optimal Trajectory Permissible Domain of Operation Optimal Trajectory Not Along an w=0 Arc First Special Control Variation Second Control Variation Closed Loop System on Singular Arc Portion of Closed Loop on Nonsingular Arc Decision Element Model of One-Wheel System Force and Torque Diagram Model of Friction-Slip Characteristic Reachable and Controllable Regions Optimal Trajectory Optimal Control for Vehicular Braking System. on Control Signal Optimal Control for Vehicular Braking System. Constraint on Control Signal Optimal Control and State Vectors versus Time Constraint No Digital Optimal Control System Based on Green's Theorem Approach Wave Forms of Digital Control System Diagram for Small Signal Sinusoidal Analysis Reachable Region for 3-Dimensional Problem Optimal Trajectory for 3-Dimensional Problem Page 16 18 19 20 31 31 39 40 41 53 53 55 60 61 62 63 65 69 7O 71 76 78 Figure 5.1 5.2a 5.2b 5.3 5.4 5.5a 5.5b 5.6a 5.6b 5.7a 5.7b 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 Basic Suboptimal Control Basic Block Diagram Suboptimal Control System with Time Delays Velocity Response-Real Actuator X2 Ideal Wheel Velocity Transducer Effect of Wheel Velocity Transducer Effect of Initial Delay in Actuator X1 Velocity Response Effect of Initial Delay in Actuator X1 Solenoid, Control Pressure, and Friction Response Effect of Initial Delay in Actuator X1 Initial Velocity Response Effect of Initial Delay in Actuator X1 Initial Response of Solenoid, Control Pressure, and Friction Response without Compensation Response without Compensation Program I-CSMP Listing of One-Wheel Model Output of Program I Histogram of Wheel Velocity Histogram of Friction Coefficient Histogram of Control Pressure Histogram of Wheel Velocity Histogram of Friction Coefficient Histogram of Control Pressure Velocity Response Suboptimal System Employing a Single Control Pressure Signal Friction and Control Pressure vi Page 80 82 83 86 87 88 89 9O 91 92 93 98 99 100 101 102 103 104 105 106 107 Figure Page 5.18 Wheel Velocity Response 108 Suboptimal System Employing a Single Control Pressure Signal 5.19 Wheel Velocity Response 109 Suboptimal System Employing a Single Control Pressure Signal 5.20 Control Pressure Response 110 Suboptimal System Employing a Single Control Pressure Signal 5.21 Friction Characteristic 111 Suboptimal System Employing a Single Control Pressure Signal 5.22 Wheel Velocity Response-Unbalanced Friction 112 Suboptimal System Employing a Single Control Pressure Signal 5.23 Friction Coefficient and Control Pressure Response 113 SubOptimal System Employing a Single Control Pressure Signal 5.24 Wheel Velocity Response 114 Reference Suboptimal Control System 5.25 Friction Coefficient Response 115 Reference Suboptimal Control System 5.26 Control Pressure Response 116 Reference Suboptimal Control System 5.27 Phase Plane Plot 117 Reference Suboptimal Control System 6.1 S-plane Model of One-Wheel Vehicular Control System 119 6.2 Friction-Slip Curve Linearized about Operating Point 119 6.3 Simplified Incremental Block Diagram 120 One-Wheel Vehicular Control System 6.4 Nonlinear System used to Develop Time Delay Criterion 121 6.5 Typical Nyquist Plot 122 6.6 Wheel Velocity Response 125 vii Figure Page 6.7 Control Pressure Response 126 6.8 Friction Coefficient Response 127 6.9 Output Response of Asymmetrical Relay 128 6.10 Diagram used to Evaluate Variational Frequency and 129 Amplitude of Variables A2.1 3-Simp1ex with Orientation ‘ 145 viii CHAPTER I INTRODUCTION Writers have called attention to the fact that a gap exists be- tween contemporary control theory and control practice [G7]. The gap can be attributed to the fact that theoreticians and designers do not study and solve the same problems in the same order and manner. In the present study - Optimal Control of a Vehicle During Braking — there ap— pear many difficulties. Those of an essentially mathematical nature are of interest to the theoretician. These difficulties are not usually the same as those which concern the designer. The theoretician often finds interesting and worthy of study a simplified version of the de— signer's problem. On the other hand, the designer often will change the design to bypass a difficulty which he has insufficient time to analyze. This thesis is an attempt to decrease the communication gap due to the divergent interests of theoreticians and designers in the area of optimal vehicular braking. Nonlinear systems in which the control appears linearly may be singular control problems [H3]. Chapter 11 presents a unification of singular control theory results found in the literature. The Maximum Principle approach to singular control problems due to Johnson and Gibson [J1], [J2] is presented and extended. The Green's Theorem approach first presented by Miele [M2] and generalized by Haynes [H1] is quite useful in low order systems. In the first approximation model of the vehicular control system, this method 1 is applied. Studies by Snow [S3] concerning reachable and attainable sets supplement the development of the Green's Theorem approach. Geo- metric rules for determining optimal trajectories which contain both singular and non-singular arcs are developed and presented. To establish necessary conditions for minimality of singular arcs, the second variation approach developed by Robbins [R1] and also by Kopp, Kelley, and Moyer [K2], [K3] is presented. In Chapter III the general theory of singular control as presented in the previous sections is applied to low order time optimal control systems. This specialization is directed toward the vehicular braking control problem. The first section employs the necessary conditions obtained from extensions of the Maximum Principle to derive the optimal control law. The mathematical model of the friction-slip characteristic used in the vehicular control system is of an exponential nature. This permits further specialization and simplification of the control law. The result of this approach is a closed loop system, in which the control is a function of state variables, operating as a second order system in the singular mode and as a fourth order system in the non-singular mode. The singular "bang-bang" cases concerned are designated Problem 3.1 and Problem 3.2. Using the Green's Theorem approach, no additional information is obtained for these problems. The necessary conditions for singular arcs to exist are compared and tabulated. It is shown that the necessary conditions of the Maximum Principle approach imply the necessary condition of the Green Theorem approach. The previous problems dealt with bounded control variables. Problem 3.3 is the singular "pang-pang" time optimal control problem and models a vehicular braking control system when the control pressure is rate limited. It is shown that the control law for both the singular "bang- bang" case and the singular "pang-pang" case are identical on the singu- lar portion of the trajectory. The terminology singular ”bang-bang" and singular "pang-pang" is defined in Chapter III. The "pang-pang" problem increased the order of the system to three, since the control u became a state variable and a became the new control signal. The extension of Green's Theorem from two to three dimensions is the traditional Stoke's Theorem. The generalization to higher dimensions will be designated as the n-dimensional Green's Theorem. ,This generali- zation is considered and a procedure for applying Green's Theorem to n-dimensional problems is presented. The concluding section of Chapter III applies the second variation approach to Problem 3.1, the singular "bang- bang" time optimal control problem. The chapters to this point have stressed the theoretical aspects re- lated to the problem of interest. Chapter IV is concerned with the ap- plication of the previous material to the vehicular braking control problem. As such, it is of interest to both the theoreticians and the de- signers. A mathematical model of the one-wheel vehicular braking control model is optimized using the Green's Theorem approach. Reachable and at- tainable sets are obtained and a realizable control algorithm determined. A computer program to automate the procedure is discussed and included in Appendix II. By eliminating the constraint on the control signal and using an impulse function in the control, a simple analytical solution is obtained. This provides a design tool for this particular problem and serves to check the digital computer solution. Implementation of the optimal control based on the Green's Theorem approach shows the feasability of this method. However, the optimal con- trol system model used shows that unavoidable time delays in the IBM 360-65 computer are responsible for the control signal oscillating or chattering about the theoretical value. By means of small signal analy- sis, relations between the variables are derived so that the magnitudes of the oscillations can be predicted. These are primarily of theoretical in- terest, since they would have negligible effect on a hardware system. They do indicate that time delays, such as encountered in various trans- ducers, will affect the practical system. In Chapter IV, the vehicular braking control problem was formulated as a time optimal control problem. By means of Green's Theorem, it is shown that minimizing time to stop the vehicle is equivalent to minimizing Vehicle. stopping distance. A criterion is also derived which shows the relationship necessary for equivalence. The one-wheel vehicular braking control model with rate and amplitude limited control is solved by using the n-dimensional Green's Theorem deve- lopment. The three dimensional trajectory is described. The problems considered to this point, have been aimed at directly assisting in the develOpment of a vehicular control system. Subject to well defined constraints, the optimal control problem has been solved by both the Maximum Principle and the Green's Theorem approach. The de- signer, however, is faced with constraints which are not well defined. Diverse facts such as cost, reliability, variability, and noise sen- sitivity must be considered. The complexity forces the designer - at this stage - to optimize, in some undefined sense, a system which will be called the suboptimal vehicular braking control system. This is done in Chapter V. The basic supoptimal control system developed first is almost indistinguishable from the optimal control system. Since time delays, due to transducers are significant, the final suboptimal system developed takes these into account. Studies using both the analog and the digital computer are conduct- ed, using the optimal control system as a reference system. It is shown that it is possible to compensate for transducer time delays which are no more than approximately 20 milliseconds. Also, if the time delays are in this range, they may be treated linearly. Representative plots and their comparison with respect to the optimal control system are included. In order to predict the effect of various system parameters more easily, it is necessary to develop models which reduce the complexity of the system. Several models are developed in Chapter VI, which are useful under various operating conditions. The first model, which is valid for a properly compensated system having time delays, establishes a criterion relating time delay to ripple frequency present in the system response. Another model developed is appropriate for systems having physically realizable time delays but operating without compensation. This model considers the effects of two nonlinearities, the friction-slip characteristic and the relay characteristic. The use of these models in predicting the effect of system parameters is demonstrated and comparison with computer studies is included. In summary, this thesis presents a unification of singular control theory and specializes the general results to a class of problems en— countered in vehicular braking processes. It develops necessary conditions along singular and nonsingular tra- jectories, which are used to develop algorithms necessary to mechanize time optimal models based on both the Maximum Principle and Green's Theorem. The equivalence of minimum time and minimum stopping distance criteria is proved. Parameter studies of suboptimal systems are made using both analog and digital technique. The suboptimal performance of simulators using real hardware is shown to compare favorably with the optimal control system. Several frequency criteria are developed to permit evaluation of suboptimal control systems. CHAPTER II SINGULAR CONTROL THEORY 2.1 lgtrodggtion This chapter is a survey of some of the currently known mathematical techniques applicable to the study of systems described by nonlinear differential equations in which the control appears linearly, i.e., r i1(c) - fi(x,t) + E bij (x,t)uj 1 . 1,2,..., n (2.1) j 1 It will be shown that when the control appears linearly, a class of solutions which are called singular may appear. In matrix notation, the above equation may be written as x(t) - f(x,t) + B(x,t)u (2.2) In the above equation, t is the independent variable (t = time in the practical cases considered). The n dimensional vector x(t) is the state of the system at time t, and the r dimensional vector u is the control vector. The problems of primary interest are those in which control vectors, i.e., a set of control functions ua(t), are to be found which will take the system from state x at time t to state x at time T. O o T By requiring that the control vectors Optimize some performance criterion, we have an optimal control problem. This criterion is usually a functional which may depend on time, the state of the system, and the control vector. When expressed in integral form it appears as follows: T J[u] - I f (t,x,u)dt (2.3) t o O In much of the study that follows the scalar f will be 1. This 0 is the classical time optimal control problem. The control vector will be selected from a class of functions U depending on the problem. Bounds on the control and its derivative such as Iul_<_l a=1,2,...,r a and Ida] :_1 are considered. The system equations may always be written so that the magnitudes of the control components are normalized, i.e., Iu I < l. a ._ For an optimal control problem, a trajectory is said to be singular, if along the trajectory, the necessary conditions for optimality such as provided by the Pontryagin Maximum Principle are satisfied in a trivial manner. Application of the usual necessary conditions here produces no useful information. A definition due to Hermes [H5] states that a control vector is totally singular when the Maximum Principle yields no informa- tion in time optimal problem for any components of the optimal control. If a trajectory is nonsingular, it is called normal. The trajectories considered in this study have subarea which may be normal and other sub- area which may be singular. 2.2 The Maximum Principle Approach In this section application of the Maximum Principle will show that the usual necessary conditions provide no information regarding the sin- gular control. Hence, other necessary conditions will be developed to provide additional information regarding the control. The practical cases considered later will be time invariant with a single control variable. Consider, Problem 2.1 i,(c) - fi[x(t)] + bi[x(t)Ju(t) 1 = 1,2,..., n (2.4) or in vector notation x(t) - ftx1 + b[x(t)Iu(t) (2.5) where the state vector is the n dimensional vector x and u(t) is a sectionally continuous scalar control function. Assume that u(t) is constrained in magnitude by the relation [u(t)] 5 1 for all t ts[O,T] (2.6) The problem is to drive the initial state x(0) = a to x(t) = b while minimizing the functional T JluJ - L {fo[x(t)1 + b0[x(t)hi(t)}dt (2.7) which can be represented as 10 - fo[x(t)] + bo[x(t)]u(t) x (0) = o (2.8) The Hamiltonian is defined as n n u(x.u.t.p) - Z £i[x(c)1p1(t) +‘.(t) Z bi[x(t)1p1(t) (2.9) i=0 i=0 The vector p is the costate or adjoint vector and is given by ' _.__2§___ t I p1( ) 3x1(t) i = 1.2.000,“ (2.10) or in vector form as 5(t) . _ .22. (2.11) 3x 10 From equation (2.9) and (2.10), we have . ( ) r21 ( ) £j[x(t)] n )ab [x(t)] (2.12) p t " - t '- t The following theorem is one form of the Maximum Principle which gives necessary conditions for Problem 2.1 [A1]. Theorem 2.1 If u*(t) is an Optimal control and if x*(t) is the corresponding optimal trajectory, then there exists a nonzero absolutely continuous * vector valued function p*(t) and a constant po 3 0 such that e * * * * 1) x. (c) - filx (t)] + bilx (c)1u (t) (213) * u 3f [x t)] u 3b [x*(t)] é,*(t) - -2 p *(t) 1 - u*(t)) pj*(t) 3 3=1 1 3x: (t) j=i axju) (2.14) i k x (O) = a x (T) 8 b i=1,2,...,n (2.15) ii) For t s [0,T] and all u(t) satisfying the constraint Iu(t)|g 1, the following relation holds 11 n * a a e u (t) 2 b [x*(c)1p (c) :_u J . In fact, J ICF IDF IGF trajectory, hence is the maximum arc. Similarly JIDF < JICF if m is nega- tive over the entire domain. If m is zero over the entire domain, then is greater than any other admissible the integrals are independent of path, and JIDF = JICF' Now, considering the general case where w may change sign within 18 the admissible domain. It is possible to have several subdomains in which w is positive and several in which w is negative. In order to find the trajectory from I to F, corresponding to the maximum or mini- mum value J, proceed as follows referring to Figure 2.2: X: X: Figure 2.2 Determination of Optimal Trajectory Starting at I, compare IAB versus 18. Since the domain encircled has w > 0. and going CCW,J Compare AB vs. ACB. The domain en- >I IAB IB circled has m > 0, hence < J . Likewise I Z J AB ACB IAD IQD similarly, E— é—fi 19 Therefore I is the maximum integral and IADHMF is the corresponding IADHMF trajectory. This procedure may be summarized as follows: To determine the tra- jectory which makes the integral given by equation (2.44) a maximum, start at the initial point I and proceed so that the subdomains w > 0 are on the left and subdomains where m < 0 are on the right. This means the trajectories will be either on the boundary of the domain or on the arc w I 0. The procedure for minimizing the integral is just the opposite. Hence, the minimizing trajectory is IDAMHF. The singular arcs are those where m'I 0. The nonsingular arcs are those where e(x1,x2) I 0. When comparing trajectories using the above procedure, it is assumed that the admissible control functions are able to generate the trajecto— ries, including the singular arc w I 0. When constructing the domain and its boundary s(x1,x2) I 0, it may not be obvious that the singular arc w I 0 is not admissible in certain cases. XF f(xhxl) . O Figure 2.3 Permissible Domain of Operation 20 The domain of operation is constructed by determining the inter- section of two sets, 81 and $2. The set 31 is the set of points attainable by admissible controls starting from the central point x0. 82 is the set attainable by admissible controls starting from the final point xf with time reversed, or equivalently, the set of points from which it is possible to derive xf using admissible controls. In this domain are found all admissible trajectories. If, arbitrarily, a trajectory m = 0 is drawn, it is not obvious that this trajectory can be generated by an admissible control u(t). If it can, then it is a possible candidate for a singular arc. For example, consider the domain shown in Figure 2.4. This domain is associated with the system *1 . X1 + XZU (2.49) 222 - X2 + Xlu (2.50) IUI 5 1 (2.51) ’2 “III Figure 2.4 Optimal Trajectory Not Along an m = 0 Are 21 The fundamental function w(xl,x2) associated with various perform- ance criteria can have the form shown. An example is given in [H2]. As is evident from this system, the permissible trajectories emanating from any point in or on the boundary of the domain are confined to angles be- tween -45° and +45°. Hence, when point A is reached, the trajectory con- tinues downward instead of going along w I 0. If the slope of the line w I 0 is l, we have the interesting case where the trajectory is now a- long w I 0 and the problem is singular and still "bang-bang". This is because u I +1 along w I 0. 22 2.3.2 Determination of Reachable Regions The Green Theorem Approach necessitates determining regions over which control is possible. Two regions are of interest. The first is the set of states that can be reached, given a class of functions U and initial state x(to) I x0. The total set will be called the Reachable set. Related to this set is the set of states that can be reached at time T by use of admissible controls. This set was called the T-Reachable set by Snow [53]. It should be noted that both the Reachable set and the T-Reachable set are independent of any performance criteria. The second region of interest is the set of states for which there is an admissible control in U that drives the state to a given final state. This set of states will be called the Controllable Set. Related to this set is the set of states for which there is an admissible con- trol U that drives the state to a given final state in time T. This set is defined as T-Controllable by Snow. The name controllable is related to the concept of controllability, which states that a system is control- lable if, given any two states, there is a control which will drive the system from one state to the other in finite time. In the application of the Green's Theorem Approach, only the chara- cter of the Reachable Region and the Controllable Region need be known. The intersection of the Reachable Set and the Controllable Set contain all trajectories from the initial point to the final point. A method of obtaining the Reachable Set was developed by Snow [83]. His method is based on the solution of three Hamilton-Jacobi partial 23 differential equations. The equations are solved by the method of characteristics. The Reachable region is the region bounded by the surfaces S(x,t) I S(x0,t0) where S(x,t) represents the solutions of the Hamilton-Jacobi equations. The method used here will be based on several theorems developed by Hermes and Haynes [H3]. The theorems are directly applicable to Problem 3.2 which will be considered later. The system to be considered is two dimensional with the control function appearing linearly. x1 I f1(xl,x2) + bl(x1,x2)u x1(0) x10 (2.52) x I f2(xl,x2) + b2(x1,x2)u x2(0) 2 x20 (2.53) The control functional u is’a scalar and is in the set of admissible control functions, U. u : {u:|u(t)l 5 1 , c e [0,m1} (2.54) In the development, the solution of equations (2.52) and (2.53) < when a constant control u(t) = a. -1 E o — 1, is applied is designated a ase. It is assumed that fl, f2, b1, and b2 are once continously differ- entiable in an open, simply connected set DC R2. The initial point x0 and the final point xf are always considered to be in D. The Reachable Set (set of points which can be attained from x0) is defined as 2 R(x0) E {x c R : x I ¢u(t,x0), u s U} (2.55) 24 The Controllable Set, (set of points from which xf can be attained in a finite time), is defined as u(xf) s {x s R2 : x 3 (pt: Waxy: u E U} (2.56) The relationship between the T-Controllable Region for a given system and the T-Reachable Region for the system with time reversed is deve- loped by Snow [S3], who shows that if the system is described by n first order equations, the T-Reachable Region for the forward time equation is precisely the same as the (T-to) Controllable Region for the reversed time system. This is not true in general for a system described by a single nth—order differential equation. If a solution to the Optimal Control problem exists the trajectory connecting x0 to xf must lie in R(x0){)R(xf). The following lemmas due to Hermes and Haynes [H3] are the basis for the theorem giving sufficient conditions so that the trajectories ¢1 (', x0), ¢'1(',x°), ¢1(',xf), and ¢'1 (',xf) determine and bound R‘XO)I]R(Xf). The following definitions are used in the theorems and lemmas. My) 5 -b2 (y) f1 (y) + b1 (5') f2()’) y e D (2.57) 6(a,y) is the angle traced out by the ray 5 (o,y) as o varies continuously from -1 to a. The vector 6 is defined as, f1(Y) + bl(Y)a f2(Y) . b2(y)a (2.58) 5(09Y) E The possible directions which solution trajectories can assume at a point y in the two-dimensional Space are given by the vector s(o,y). 25 Lemma 2.1 If A (y) I 0, the set { s (d,y) : Iolfij} of possible directions is bounded by c(-1,y) and e(l,y) with O<|6(1,y)I0} F(xf) {¢'1(-t,xf): c s T(1,xf)}LI{¢'1(-t,xf): c e T(—l,xf), c>0} and there exist t1, t2, t3, t“ > 0 such that 1) ¢1(t1.x0) - ¢”1(-t2.xf) 11) ¢-1(t1,xo) - ¢1(-c,.xf) iii) The trajectory arcs ¢1(t,x0), O :_t :_t1; ¢'I(t,xo), 0 _<_ t i t3; ¢1(-t,xf), 0 i t i t“; ¢'1(-t.xf). 0 1 c i t , all lie in 1) 2 iv) A(x) I 0 in the set F(x0) or F(xf) which properly separates D Theorem 2.1 If a problem satisfies Condition 2.1 and A(y) f 0 for y s S, then S I R(x0)flR(xf) Summarizing, Theorem 2.1, provides a rigorous basis for the de- termination of the region containing all the admissible trajectories. It also shows that this region is bounded by trajectories resulting from application of "bang-bang" control signals. 27 2.4 The Second Variation Approach This section will present equations for minimality of singular arcs over a finite time interval. Special control variations are used to ob- tain a second variation test for singular arcs. The approach is based on the work of Kopp,Ke11y, and Moyer [K2], [K3]. Consider the system of differential equations and boundary conditions £1 I f1(x1,...,xn,u,,...,ur,t) i I 1,2,...,n (2.59) x1(t0) I xio i I l,...,n (2.60) xi(T) I xif i I l,...,m(m §_n) (2.61) The cost functional to be minimized will be formulated in the Mayer form of the calculus of variation, i.e., minimize J(xm+l(T),...xn(T), T) (2.62) Although the minimization is also subject to constraints on the controls, i.e., Iu[§_1, these constraints will present no difficulty since the control corresponding to a singular arc is usually interior to the bound- ary of U. Therefore, in the development that follows, u will be considered to be in the interior of the class of admissible controls U. The Hamiltonian is defined as n 1-1 1 1 Introduce the auxiliary vector pi(t). This turns out to be the ad- joint or costate vector and is defined by the following differential equations and boundary conditions, . n 3f 3H - - - - -—- 1 -- 1 .4 p1 321111 31 3X1 3 an (2 6) BJ p1(T) - axif 1 = m+1,...,n (2-65) Necessary conditions for P to be a minimum are that the Hamiltonian be a minimum.for all admissible controls, * t * * H(u1 + Au1,...,ur + Ant) 1 H(ul ,...,ur ) (2.66) Asterix denotes the optimal controls. Singular subarcs occur when the matrix Huu’ With typical element 3H Bu 8 1 u , is singular over a finite interval of time. Emphasis will be on the case in which a single variable appears linearly in the system equation as in the vehicle braking problem. The total variation in the cost functional J[Lfl due to a variation in the vector u is n n n 32J(xf+eAxf) AJ'I 2 -§l- Ax + 8 Z 2 Ax Ax 8x if i=m+1 Im+l 3x 3x if 1b iIm+1 if 3 if if (2.67) where 0 :_6 §_l. Consider the case where the end points are fixed. Then, due to a a change Au away from the optimal u , * * * Aii I fi(x*+Ax,u +Au,t) - fi(x ,u ,t) i=1,...,n (2.68) I g ’00., 2069 Ax1(t0) O i 1 n ( ) I 1.1,eee’n (2.70) Ax1(tf) 0 29 Consider the equation n . n t s * * 121 piAx1 - 121 p1[f1(x +Ax,u +Au,t) - fi(x ,u ,t)] (2.71) Using the Hamiltonian in equation (2.71), n , * e * * Z 1:14):1 - u(x +4x.u +2....) - no: .u ,t) (2.72) 1.1 Now consider d n E n -—- Ax I p Ax + p Ax (2.73) d, 121 P1 1 1-1 1 1 1:1 1 1 Multiplying through by dt and integrating, n tf tf n . n X piAx1 I J [ Z piAxi + X piAxil dt (2.74) iIl t0 t0 iIl i=1 Due to the boundary conditions n t n * a * * 1-m+1 3‘11 t 1'1 1 1 1 n as * * - z __ (X g“ ,t)AX } C“: (2.75) . 3x 1 i 1 1 Using Taylor's expansion, and substituting the Hamiltonian, n t n * a H * SJ Ax I f [H(p,x ,u +Au,t) + Z 2—-(p,x ,u*+Au,t)Ax ax if i 1-m+1 if to 1-1 311 n n 32H * * + k 2 Z S;—;;— (p,x ,+6Ax,u +Au,t) Axiij iIl jIl i j a a 3 an e a - H(p,x ,0 ,t) ' X SIT-(P931 su at)Axi] dt i=1 i (2.76) 30 Substituting (2.76) into (2.67) the total variation due to a variation in control vector u is as follows: ' a * a a t [H(x ,u +Au,t) - H(x ,u ,t)] dt AJ - I f to t * * * * +J f [g2 (Psx :11 + “,t) " 21(1)." a“ at)]Ax dt to 1 3x1 1 n n 2 a H a e + h I Z Z --—- (p,x +6Ax,u +Au,t)AxiAx dt 1-1 1-1 311313 J n n 32J(x +6Ax ) + 8 Z X f (f Ax Ax 3-1 1-1 3X1£3XJ£ 11 if (2.77) At this point, assume that the control U appears linearly in the system equation (2.59). The control will then also appear linearly in the Hamiltonian. When the first integral of (2.77) is expanded using the 2 Taylor expansion, the §_§ term will vanish because the Hamiltonian is auz linear in u. Hence, minimality cannot be established. Therefore, to obtain ad- ditional necessary conditions, further inspection of the second order terms is required. The classical derivation of the Legendre necessary condition was Obtained by employing a special derivation in conjunction with the second variation [GI]. The second variation for the present case is obtained from Kelley. Letting Au I K6x, the second variation is 31 t 2. h a * AJ2 I K2! 1 a (x ,u ,t) 6x16u dt + t0 3X13“ t KZIfEn 311 (1*)Ax4xd+ —-- x ,u ,t t 2 101-1 j-1 311313 1 1 2 n n * _IZEI Z X 3J(X ,t) OX 6x J-1 1-1 axif 3x1, if if (2.78) The first and second control variations used by Kelly, Kopp, and Mayer are shown in Figure 2.5. dL‘ ta (0-) e e 1:1"! :1 (H U 1w Figure 2.5 a) First special control variation b) Second control variation The first special control variation is designated as ¢0'(t,r). The time tIO is the center of an interval 21, and may occur at any interior point of the singular subarc. The parameter T will approach zero in the limit. 32 In minimizing J2, the constraint equations are n 3f1(x,u,t) 3f (x,u,t) 6x - Z ax 6xj + 1 a On 3-1 1 “ (2.79) 6x1(t0) - o i=1,...,n (2.80) 5x1(tf) I 0 iIl,...,m (2.81) Letting A I 3:2; s 32H 1 1 au 3p au 2 1 (2.82) n 3f A1 2 ' 1 ‘SE1 A 1 ‘ A1 1 (2'83) ’ 3-1 1 1’ ’ 01' n 2 a a “1.2 11.1 3.1—1'31; 11.1 ' A1.1 12'8“) the necessary condition obtained by Kelly, Kopp, and Moyer for the sin- gular arc to be minimizing is n n 2 n n 2 ’ 32H 3 H 3 H 1.31:) ___—11.2.... egg—A... t . l l- 1_1 auaxi 1 2 1.1 3u8x1 1:2 131 j=1 axiaxj 1’ 3’ (2.85) An equivalent and more compact form is due to Robbins [R1] 3 d2 3H 3.115;: Tu ) :0 (2.86) The equality part of the sign in the conditions (2.85) and (2.86) means that the conditions are met marginally and the nature of the extremal is still undetermined. Hence, it is necessary to proceed to the second special variation, and so on. Using the Robbins form, the second special variation leads to 33 The general form of the necessary condition is W) Z O (2.88) where k is a positive integer. CHAPTER III TIME OPTIMAL SINGULAR CONTROL 3.1 Maximum Principle Approach The problem to be considered is a time optimal problem where it is desired to drive xo to O in minimum time. Consider the system equations of Problem 3.1. Singular BanngangTTime Optimal Control Problem i1(t) - f1[x1(t), x2(t)] + blu x1(0) = x10 i2(c) - f2[x1(t), x2(T)] + bzu x2(0) - x20 b1 and b2 are constant and u is a scalar This is the Mayer form BIG], if we minimize x0(T) where io(t) - 1 x0(0) - o The control will be constrained to 0 g_u i 1 Again dropping the arguments for notational convenience, the Hamiltonian is H - pa + plfl + plblu + pi2 + p2b2u The costate equations are Po ' 0 _ I .._.. + p1 p1 3x1 p2 BXI ‘ afl sz "Pz ' P2 "'"’+ 92 _"' 8x1 8x2 Mtnbmizing H with respect to u yields an '53-". plbl + p2b2 " O (NCl) 34 (3.1) (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) (3.9) 35 Again, since we are interested in singular arcs, u is assumed to be in the interior of its allowed region and 3H/3u is assumed to exist. Since H - 0, and p0 - 1, (This is permissible since pO functions only as a scale factor). 1 + plfl + p2f2 - o (ncz) (3.10) Differentiating (3.9) élbl + $282 - o (NC3) (3.11) b ( 'afl + afz) + b ( af‘ + afz) o (3 12) 1 p1 3X1 P2 3X1 2 P1 3X2 P2 3X2 . Differentiating (3.10) 3fl . + 3f1 . + , + 8f2 . + sz . + f . 0 P1(‘3;T X1 5;; x2) flpl P2(;;: x1 5;; X2) 292 . . (3.13) Replacing x and p, at, afl ail afz —- -—- - -- + -— + PIIBXI (f1 + b10) + 3x2 (f2 + bZU)] f1(Pl 3x1 P2 axl) 3f f b + afz b f afl + 3f2) 0 leaxl ( 1 1“) 3x2 (f2 2u)] 2(P1 3x1 P2 3x2 (3.14) From (3.11) the coefficient of u is zero while the remaining terms cancel out. Hence (3.11) is the third necessary condition. pl(bl -—— + b2 -——) + p2 (bl -- + b2 ———) I 0 (N04) (3.15) 3x1 3x2 8x1 3x2 This necessary condition‘appears as row 3 of equation(2.39), i.e., pla11 + pza12 - 0 (310 - O) (3.16) where 3f ail . b __l + b ... (3.17) a. ll 1 31! 2 3X2 36 afz 3f2 3X1 3X2 Differentiation of this equation will yield additional information. (3.21) (3.22) p15 + p a + p a + p a = O (3.19) 11 1 11 2 12 2 12 azfl 32f1. 32f1 . azfl . all I b1( x1 + x2 + b2( x1 + x2) 3x12 Bxlaxz 3x28x1 3x22 (3.20) , azfl 32f] 32f azf1 a I b -——— + b -——-—— f + b + b . + b -—— f + b 11 ( 13x12 23x23x1)( 1 10) ( laxlaxz 23X12)( 2 Zn) . 82f2 32f2 azfz 32f2 8112 - (bl + b2 )(fl + blu) + (bl + b2 )(f2 + bzu) 3x 2 3x 3x 3x 8x 3x 2 1 2 1 1 1 2 Substituting in (3.19) the following reSult is obtained, 2 2 2 3f 3f 82f 32f i Z ( — bjpi -E ——$ + b fk-——1 + (.bek -—-—-) . o 1-0 3:0 kxo 3x3 Bxk 3 xkaxj xkax (3.23) In the‘general case the upper summation limit would be n. It is possible to solve this equation for the control u. To be more specific consider the determination of the singular are for, Problem 3.2 Singular Bang-Bang Time Optimal Control x1 I f1 X1 (0) - X10 (3.24) i2 - f2 + b2“ x2(0) I x20 b2I constant (3.25) i0 - 1 x3(0) I O (3.26) Since b1 I 0, N01 implies p2 I 0 on the singular. nca implies p2 - o NC2 implies plf1 I -1 Sfl NC4 implies pl 5;; 3 0 This implies that p1 I 0 or 3—— I 0 x 2 The condition pII 0 would contradict NC2 which requires that H I 0 3f 1 on the Optimal trajectory. Hence NCA implies that «3;; I 0. substituting necessary conditions in (3.19) rather than the more formid— able (3.23), azfl ._ azfl f1 3x 3x ( 2 2U) 3x22 0 ( ) on the singular arc. The optimal control is synthesized as azfl azfl f -———-— + f -——2 laxzaxl 23x u ' ' (3.28) For a practical case to be considered later, it will be convenient to consider 3x1 X1 3x2 and 3f x 3f __3 . - .3 __3 (3.30) 8x1 x1 6x2 This modified version of Problem 3.2 shall be designated as Problem 3.2M. Differentiating with respect to x2 and taking advantage of the neces- 3f sary condition -—- I O on the singular arc, 8x 2 2 2 8 fl - x2 8 fl 2 axlaxz x1 3x2 (3.31) 38 Substituting this in (3.28) gives the optimal control function for this case as x2 u I ( - f1 - f2)/b2 (3.32) 1‘1 This control function can be shown to be a constant. Differentiate (3-32) with respect to time. d“ X2 afl sz f 1 b2- I -— [--f1 + -—--(f2 + bzu)] + -; [x1(f2 + bzu) I x2f1] - dt x1 3x1 ‘ 3x2 x1 3f2 3f2 [_fl + __(fz . .2.” (333) 8x1 8x2 3f From (3»29) and the necessary condition -— I O, and assuming x , f 0, 8x 2 du f x 8f 3f x l 2 2 2 dt x12 x1 3x1 3x2 xl Finally using (3.30) du _ dt Therefore the singular control for this problem is a constant. 3.1.2 Closed Loop System Control The problem of determining the Optimal control as a function of the state of the system is called the closed loop problem. On the singular arc,(3.32) provides this information for the modified problem 3.2M. Equation (3.28) would be required for Problem 3.2. A block diagram for Problem 3.2M will be shown. As (3-28) indicates, three additional func— tion generators would be required to implement Problem 3.2. 39 Closed Loop System On Singular Arc Figure 3.1 When operating on a nonsingular arc, the control as given by (2.19) and applied to Problem 3.2M is, The costate equations for Problem 3.1M are u*(t) - - sgn b2p2(t) P .I 0 ail ‘91'P1_+P2 fi 2 32!] 3f 8x2 2 afz 3x 3f NH 3x2 (3.36) (3.37) (3.38) (3.39) 40 The costate equations require four function generators for Problem 3.1. In Problem 3.1M, since, . Bfl x2 sz x2 91 . -— (p1 -—> + —- (p2 —) (3.40) 3x2 xl 3x2 x1 , Bfl 8f2 3x2 3x2 The block diagram which applies for Problem 3.1M when operating on the nonsingular arc is then given by Figure 3.2. :55 XI 3 XI Fanciwm A NhAt1lur Pu Mu“) "NV PI +- r+ -P -P P. o 1 [hthmbrP—‘t’ «L + , 4 f . Pa )Mnther y. Pi Conctuh t1) x" Generator 3}. Mu. "If N“ {i K Figure 3.2 Portion of Closed Loop System on Nonsingular Arc 41 The complete closed loop system which functions on the complete trajectory requires a decision element to switch u to either the singu- lar control u or the nonsingular control u . The decision element must operate according to the logic demanded by the necessary conditions. If the system is on a nonsingular arc, then it shold transfer to the singular arc. u I us if 92 I 0 p2 I 0 (3.42) u I unS otherwise (3.43) 3.2.1 Green's Theorem Approach Determination of singular and nonsingular arcs associated with the time optimal problem is considered. The system constraints are: X1 - f1(xlsx2) + bl“ (3.44) X2 - f2(xlsx2) + b2“ (3.45) 0 5 u(t) 5.1 0 :.t :_T (3.46) Since the functional to be minimized is, T J[u] I J dt (3.47) it will be necessarg to put this in the form of equation (2.44). Since the procedure basically eliminates u(t) from the equations, it is only necessary to find a vector orthogonal to the column vector T b I [b1 ble. In this case, we may use [—b2 b1] as the orthogonal col- umn VBCCOI'. 42 Multiplying through by [---b2 b1] and solving for dt, equation (3.47) becomes x1 :x2 bldxz ' bzdxl [“1 Jx x ‘B3f2 - 82$, 10’ 20 The fundamental function w(x1,x2) is 3 b2 3 b1 - .__. — — ___ .4 “(xl’xz’ 3x2 ( blfz - bzfl ) 3x1(b1f2 - b2f1 ’ (3 9) afz 8f] 8f2 Bfl 152(1)];— - b2 _) + b1(b1 3— - b2 7) X2 X2 X1 X1 w(xl:x2) ‘ (3.50) 2 (ble ’ bzfl) Since w(x1,x2) I 0 is of primary interest, we obtain the condition on the singular arc as, 3f2 2 3f2 2 3f2 3f (3.51) (be2 - bzfl) # o In Problem 3.2, b1 I 0, therefore a necessary condition for singular arc is afl _§;;_ - o (3.52) This same result was also obtained by the Maximum Principle approach. The necessary conditions involving the costate variables will not appear, since they are not present in the Green Theorem approach. A comparison of the necessary conditions for Problem 3.1 using both approaches is shown as follows: 43 Necessary condition for singular arcs for Problem 3.1 Maximum Principle Approach NC]. Plbl + p2b2 - 0 NCZ l + plf1 + pzf2 I 0 NC3 plbl + pzbz I 0 afl Bfl 8f2 8f2 NC4 p1(b1 -—- + b2 -——) + p2(bl -——-+ b2 -——) I 0 3x1 3x2 8x1 3x2 Necessary condition for singular arcs for Problem 3.1 Green's Theorem Approach afz 3f 3f 3f 1 l 1 2 2 1 1 2 8x 8x 3x 3x 2 2 1 1 It is evident that NCl of the Green's Theorem Approach is obtainable by eliminating the costate variables from NCl and NC4 for the Maximum Principle Approach. Hence the necessary conditions of the Maximum Principle Approach imply the necessary condition of the Green Theorem Approach. 3.2.2 Problem 3.3 Singglar'PangIPang_Time Optimal Control The previous problems involved controls that were bounded. In the next case the control signal will be rate limited in addition to being magnitude limited. Consider the following system equations, i,(t) I f1(x1,x2) (3.53) O x2(t) I f2(x1,x2) + gzu, 0 §_Iu| __ (3.54) A H x3(t) - 8 - v {1| 1 1 (3.55) 44 By letting u I x3, the order of the state equations has been in- creased to three. Also x3 may now be a bounded state variable. These complications are compensated for by having a control signal which is only magnitude limited. In a practical case to be considered later there is only rate limiting. This will be designated in Problem 3.3M and will be considered using both the Maximum Principle Approach and the Green Theorem Approach. Since the general expression is long, the necessary conditions for singular control will be obtained by going to the basic equations directly. The process will be as before; to repeatedly differentiate until no further information is obtainable. For the time optimal program we again introduce x0 I l x0(0) I O (3.56) The Hamiltonian is, again letting p0 I l, H I 1 + plf1 + pzfz + ngZx3 + p3v (3.57) For the nonsingular trajectories, v I -sgn p3 minimizes H. For the sin- gular case, we obtain the first necessary condition p3 I O (NCl) (3.58) If we write the costate equations, 60 - 0 p0 - 1 (3.59) ail sz -8 - p ___ + p -—— (3.60) 1 1 3x1 2 3x1 . afl at -p I p .__... + p _Z (3.61) 2 1 3x 2 3x 2 2 -p . 928 (3.62) 45 Necessary condition, p3 I O on the singular arc, implies p3 I O, or equivalently, p2 - 0, 32 # 0 (N02) (3.63) This in turn implies, p2 I 0. Therefore, from equation (3.61), an 9 —- I 0 (N03) (3.64) 1 3x 2 Using equation (3.57) p2 I O and p3 I O, we have on the singular arc, H I 1 + plf1 (3.65) Since H I O on the optimal trajectory, P1 . _ .l. (NC4) (3.66) fl Assuming p f O, necessary condition 3 becomes ail — - 0 (NC3) (3.67) 3x2 This was expected since the projection of the singular arc in the xl-x2 plane for this problem should be the same as Problem 3.2. The value of the control function u on the singular arc is to be determined next. Differentiating equation (3.65). with respect to time, ail . afl. . p (—x +-—x)+pf-0 (3.68) 1 3X 1 3X 2 ll 1 2 Substituting for £1, and i2 and p1, ail afl afl ail 3f2 p (-—-f + --f + -—-g u) - (f p -- + f p .__) I O (3.69) 1 3x 8x 2 8x 2 1 18x 1 28x 1 2 2 1 1 Using equation (3.71) and.(3.63), OIO. 46 Thus, no new information was obtained by that approach. Differentiate equation (3.67) with respect to time. 82£1 azfl i + x2 - 0 (3.70) 1 axlaxz 3x22 Substituting for £1 and i2, 82£1 azfl f + (f + gzu) = 0 (3.71) Bxlaxz 1 3x22 2 This equation may be solved for u, 32f1 32f} f + f 3x 3x 1 3x 2 2 l 2 2 ug— 32f1 (3.72) 23X2 2 The control signal u (t) for both the singular "bang-bang" case and the singular "pang-pang" control are identical on the singular arc. Assume that u(t) I x3(t) is in the interior of its allowed region, U. Hence, there is no magnitude limiting of u(t). Equivalently, the state vari: able x3 is not bounded. 3.2.3 N-Dimensional Singular Control Problems In the case of Problems where the dimension of the state space is greater than two, the Green's Theorem Approach must be extended. The ex— tension of Green's Theorem from two to three dimensions is the traditional Stoke's Theorem. The generalization to higher dimensions is designated ,as both the generalized Green's Theorem [H1] and the generalized Stoke's Theorem. -The developments are via exterior calculus and differential 47 forms. A brief treatment of differential forms is given in Appendix III. Consider the differential constraint equation, . 1' xi(t) = £1(x) + g j_1bij(X)uj(t) i = l,2,...,n _ (3.73) Ls. A __n-1 or, i - f(x) + B(X)u(t) (3.74) In order to transform equation (3.73) into the proper form for application of Green's Theorem, it is necessary to eliminate the uj(t). Since j §_n-1, an n-dimensional vector, 9(x), orthogonal to the columns of B can be found. Hence the inner product (W(x).i(t)) I (W(x).f(x)) (3.75) since (f(X).B(X)) I 0 (3.76) Equation (3.75) permits the determination of dt in the functional J[u] , where t J[u] - I f fodt (3.77) t0 Hence, - f (W(x),dx) J[u] I: f0 ( (x),f x (3.78) 0 In the notation and nomenclature forms, equation (3.77) may be written as J[u] I f n (3.79) P where, n is the pfaffian or one-form n X a (X)dxi (3.80) “-f M. 0 > 1 48 The generalized Green's Theorem is (see Appendix III) I dfl I J n (3.81) s r As in the two dimensional form of Green's Theorem, P is a curve from x0 and xf and s is a surface containing the points x0 and xf. The term dn is called the exterior derivative of n and is the differential two-form defined as, n 301 d 5—: d Adx .82 n §-1 x xj 1 (3 ) i. The exterior multiplication sign A is often omitted. An alternate useful form may be obtained by using the rules from differential form theory. dx1 A dxj I -dxj A dxi (3.83) and dx1 A dx1 I 0 (3.84) then dn I wijdxidxj iI1,...,n-l j=(i+1),...,n (3.85) where Ba Ba (1311' —-1 - -—-3- (3.86) 3xJ 3x1 In a 3-dimensional case, the exterior derivative given by (3.80) would be written as dw I mlzdxldxz + mlsdxldx3 + w23dx2dx3 (3.87) and 301 30.2 332 3X1 301 803 ”13 I -—- - -— (3.89) 3x 3x 3 l 49 3&2 303 “’23 " 5;" ‘ :3: (3.90) 3 2 These will be useful in the 3-dimensional case to be considered later. The procedure for applying Green's Theorem to n-dimensional may be summarized as follows: 1) Convert the functional to be minimized to the line integral form by equation (3.79). 2) Use equation (3.86) and wij I 0 to determine singular hypersur- faces. There are no more than (n—l) independent hypersurfaces. The intersection.of hypersurfaces, if it exists, is a singular'arc. 3) Compare trajectories by using the generalized Green's Theorem as given by equation (3.81). The possibilities of singular arcs must be investigated. 3.3 The Second Variation Approach The necessary condition of Kelly, Kopp, and Meyer will be applied in Problem 3.1, the singular "bang-bang" time optimal control problem. Using equations (2.82), (2.83) and (2.84) at1 Am " 5';- ' b1 (3.91) at, “2’1 " 5';- " b2 (3.92) A “I. . “I. A ) 192 - — 191 "— 2:1 191 (3.93 3x1 3X2 8 . 2’2 3x1 1’1 3x2 2’1 2’1 ° ) Simplifying equations (3.92) and (3.93) 50 Bfl sz A1,2 I b1-——-+ b2——- (3.95) 3x1 3x2 3f2 at, A2’2 ' b1"" ' b2“' (3.96) 3x 8x 1 The necessary condition for minimality is, using equation (2.85) 32H 32H 32H .____ 2 ______. ____ 2 a 2A1,l + (A1,1A1,2 + A2,1A1,l) + 2A2,1 3_0 (3.97) x1 3x13:2 8x2 Substituting equations (3.95) and (3.96) 282H 32H 82H _. 2 b1 2 + 2blb2 + b2 2 3_0 (3.98) 3x1 axlaxz 8x2 The Hamiltonian for Problem 3.1 was H I l + plf1 + plblu + p2f2 + p2b2u (3.99) Substituting partial derivatives of H into equation (3.98) 82:1 azf2 azfl 82f2 2 b1 (P1 2 + 132-7) + 2b1b2(P1 + P2 ) + 8x1 3x1 3x13x2 axlax2 azfl azfz b22(91 + pz-—-) :_0 3x22 8x22 (3.100) This is one form of the necessary condition for minimality along the singular arc. Using the alternate form, 3 d an _. __ < 0 3.101 3u(dtz Bu) -' ( ) since a I 5%. plbl + p262 (3.102) The first necessary condition is obtained from equation (3.102) and 51 plb1 + pzb2 I O (3.103) and ' 62p 62 p d2 an 1 2 -—- -) I b -——— + b— (3.104 dt2 3“ 1 8:2 2* dt2 ) then substituting expressions for p1 and pz, 2 3f 3f2 d 3f1 3f2 d 3H 1 -—-( ) I - b1-(pl-—- + pZ-——) - b 2-—(p1——— + p,-——) (3.105) dtz Bu dt 3X1 3X1 2(11: 3X 2 8x2 we obtain a d2 an a 82f1 a2f1 -— --( ) I - - b1p1[ -——--(f1 + blu) + (f2 + bzu)] Bu dt2 3n Bu 3x12 axlax2 32f2 82f2 bp[ (f +bu)+ (f +bu)]- l 2 3X12 1 l 3x13x2 2 2 a2f1 a2f1 bp[ (f +bu)+ (p +bu)] 2 1 axlax2 1 1 3x2 2 2 a2f2 32f2 b2p2[ (fl + blu) 4» (f2 + bzu)] 8x13x2 8x22 (3.106) After taking the partial derivative with respect to u, and using equation 2 8H (3.101), equation (3.100) is obtained. Although the form -(— d - au 8:2 an is quite compact it nevertheless involves quite a few manipulations. If as indicated previously, the equality applies, it is then necessary to proceed further using equation (2.25). Using the somewhat simpler Problem 3.2, the necessary condition for minimality along the singular arc is obtained by substituting b1 = O in 52 equation (3.99) and (3.103), then p2 I 0 NCl (3.107) 82f1 p O NC2 (3.108) 13x2 2 Using H I 0, equation (26) gives p I -—- NC3 (3.109) Hence, NCZ can be written as a2fl 5.0 NC2' (3.110) 2 f1 3x2 CHAPTER 1V OPTIMAL CONTROL OF THE VEHICULAR BRAKING PROCESS 4.1 Introduction The vehicular braking process to be considered first will consist of a single wheel carrying a body on a flat horizontal surface. Figure 4.1 and 4.2 show the model used and the pertinent parameters. Figure 4.1 Model of One-Wheel System Figure 4.2 Force and Torque Diagrams S3 54 The symbols in the figures are defined as follows: Fb braking force developed at the tire-surface interface Fe external force on the vehicle Fr reaction force between the body and axle g acceleration due to gravity J Polar mass moment of inertia of the wheel and associated rotating members M mass of the vehicle N normal force at the tire—surface interface R rolling radius of the wheel Tb torque exerted on the wheel by the brake Tr rolling resistance and bearing friction torque v vehicle velocity u coefficient of friction n slip Q. angular velocity of wheel As the vehicle moves with velocity v, the wheel runs under slip as it transmits driving, braking, or cornering forces to the surface. Slip is defined as the ratio of effective slip velocity in a specified direction to the forward ground speed of the vehicle. Since braking will be the main concern, slip during braking is defined as ".2239. and05n<1 (4.1) The so-called "panic stop" usually results in a slip of 1.0, corre- sponding to zero wheel velocity. The braking force Fb developed at the tire surface interface is due to the friction coefficient u and is 55 defined as Fb I uN (4.2) where N is the normal force at the tire-surface interface. Investigators in the area of tire friction [K13] [N6], have found that the friction characteristic depends on factors such as vehicle velocity, normal load, tire tread pattern, tire inflation pressure, tire temperature, and surface composition. The friction-slip model used in this study is shown in figure 4.3 "5 SUP 41 ‘-° Figure 4.3 Model of Friction-Slip Characteristic The selection of this model is based on investigtions [F6], [K14], which show that regardless of surface composition, the friction coeffi- cient u, usually has a peak value and this peak value occurs when the slip is in a range about the 0.15 point. 56 The optimal control for the one wheel model will use minimum time as the optimum criterion. As will be shown in section 4.3.3 this control will also minimize stopping distance. Optimal control theory will show that the control should bring the state of the system to the peak of friction-slip curve and then keep it there. It should be noted that due to the low value of slip, the wheel velocity will be an appreciable fraction of the vehicle velocity. This will also benefit more complex models such as two wheel and four wheel models which are concerned with lateral stability. When one tire Of the vehicle is subjected to a differ- ent friction characteristic than the opposite tire, a torque tending to rotate or spin the vehicle is developed. When the wheels of the vehicle are rotating.the tendency to spin is reduced and the vehicle has more lateral stability. This problem will be considered in more detail later. 4.2 ‘Qevelopment of the System Equations The differential constraint equations for the Optimal control problem are obtained by referring to Figure 4.2. The normal force N is obtained from the summation of the vertical forces, N I Mg (4.3) The horizontal forces are summed, obtaining, F + M dv 4 The external force Fe is neglible with reSpect to the braking force. Then using the relation (4.2), equation (4.4) becomes. dv Mdt- -uMg (4.5) 57 or 6 - -ug (4.6) Now considering the torques associated with the wheel, d6 rbn —Tb-Tr = J'gf' (4.7) Here the rolling resistance and bearing friction torque are assumed negligible with respect to the brake torque. To make the units of wheel velocity the same as the units of vehicle velocity equation (4.7) is written as d6 2 R-EE-I (uMgR -RKPb)/J (4.8) In the above equation Fb is replaced by uMg and brake torque is assumed to be linearly related to brake pressure Pb, i.e., Tb I KPb (4.9) Typical values for an equivalent one wheel model are R I 1.1 ft. M8 I 5000 lbs. J I 5.0 ft-lbs/sec.2 K I 6.0 ft-lbs/p.s.i. Letting x1 I v (4.10) x2 I R0 (4.11) Here the first state vector corresponds to vehicle velocities and the second state vector corresponds to wheel velocity. The dif- ferential constraint equation then becomes i1 - ~32u (4.12) 22 - 1210“ - 1.329 (4.13) b 58 Constraining the brake pressure to 0 f Pb 5 1200 psi and intro- ducing the control u, 0 f u f 1 equation (4.12) becomes i2 I lZlOn - 1584u (4.14) The friction coefficient n is a function of x1 and x2 which will be designated state variables. I: I u(x1.x2) (4.15) As indicated previously, the friction-slip characteristic has the general shape as shown in Figure 4.3. This shape will be generated by -a(l-x2/x1) _ e-b(l-x2/x1)] u(x1,x2) I no [a (4.16) Recall that the slip is given by n I 1 - x2/x1 , (4.17) The factor no will take into account the surface-tire interface. For example, no I 1 will correspond an interface having the highest friction coefficient such as concrete, while a low friction surface such as ice may have a value of no I .06. This function will have a peak of approximately 0.947 no at a slip of 0.2 when a I 0.225 and b I 23.5. These are values that are used in most of the later computations. 4.3.1 Optimization Of the One-Wheel Vehicular Braking Control Model The problem to be considered is the time optimal regular problem, i.e., take the vehicle from an initial state to the origin in minimum time, subject to the constraints 12,- -32u (x1362) x1(0) - 60. (4.18) {:2- 1210u (x1,x2)-1584u x2(0) - 60. (4.19) 0 5 u f l (4.20) 59 The function n(xl,x2) is given by equation (4.15) with no + l, a I .225, and b I 23.5. This problem when stated in real physical terms is: Find the control pressure P(t), constrained to be between zero and 1200 psi, which will stop a given vehicle initially travelling at 60 ft/Sec (approximately 40 MPH) in minimum time. It should be noted that there is no limitation on pressure rate in this case. This problem will be label- ed Problem 4.1 and will be called the singular "bang-bang" control problem. This problem fits the format of Problem 3.2, where flI I32n (x1,x2) (4.21) f,- 1210u (x1,x2) (4.22) 4.23 H _ €-a(l-x2/x1)-E-b(l-x2/xl) ( > Any of the methods in Chapter III provide the necessary condition for a singular are (4.24) Q) I: I O Q) X N or, Ia(l-X /x ) -b(l-x /x ) x1 x1 Solution of this equation gives an equation relating x1 and x2 on the singular arc. For the values of the constants used, a linear relation exists, x2 I kx1 k is approximately 0.8 (4.26) This problem is most readily solved using the Green Theorem Approach. The regions which are reachable and controllable are found and their in- tersection provides the region containing all permissible trajectories from the initial point xo to the final point xf. In order to find the 60 region containing all the trajectories, Theorem 2.1 is used. If the conditions Of Theorem 2.1 are satisfied then the region containing all trajectories from x to xf are bounded by the four trajectories 0 obtained by solving (4.19) and (4.18) as follows: 1 a) starting at x0, use uIl and obtain 0 . b) starting at x0, use qu+ and obtain 000+. c) starting xf, integrate backwards (reverse time) use uIl, obtain ¢fl° d) starting at xf, integrate backwards use u>0, Obtain ¢f°+. To satisfy the conditions of Theorem 2.1, it was necessary to use u>0 in (b) since uI0 leads to xlIO and 22-0 and the A condition of Theorem 2.1 is not satisfied. Also it is convenient in this study to use 0 5 u S 1 instead of Iulfll. Theorem 2.1 is applicable in either case, since it is only necessary to change (4.19) to izI 1210 u(x1.x2) - 752 - 752a Iulgl (4.27) The results appear in Figure 4.4 Y: Y1 Figure 4.4 Reachable and Controllable Regions 61 The singular arc, wIO, Obtained from the necessary condition is also shown in Figure 4.1. We are now ready to apply the Green Theorem Approach. In this problem, w is positive in the region above the curve mIO, which is a straight line in this case. The global Optimal trajectory is obtained as follows: a) Start at x0, keeping the region w<0 on the right, proceed to the point where the trajectory intersects the wIO line. b) Now proceed along the wIO line until xfis reached. Note that along the wIO, the region w>0 is on the right and the region m<0 is on the left. Also, if the final point xfwas not on the curve w=0 the procedure would be the same except that a boundary of the region containing the admissible trajectories would be reached before xf is reached. Traversing the boundary keeping w>0 on the right or w<0 on the left would then ultimately terminate the trajectory at x Figure f. 4.5 shows a case where xfI(a,0). The Optimal trajectory is xO-a-b—xf. Y1 MIL \AIo Figure 4.5 Optimal Trajectory 62 Having found the optimal trajectory, it is now possible to des- cribe the Optimal Control. The boundaries establish the "bang-bang" values of u while the value of control on the singular arc is deter- mined from (3.3.) and (3.34). The approximate value of u=.735. Figure 4.6 shows the Optimum Control function as a function of time. u(t) -—1(.O t. Figure 4.6 Optimal Control for Vehicular Braking System Constraint on Control Signal A program P2 was written to automate the above procedures. This program does the following: 1) 2) 3) 4) scans the w region of all permissible trajectories finds the proper boundary finds the singular arc finds the Optimal control on all parts of the trajectory including the singular arc The listing of P2 and a typical output is shown in Appendix II. 63 4.3.2 Analytical Verificgtion of the Optimél Control In the previous section, the Green's Theorem Approach showed that the Optimal control is piecewise constant. As shown in Figure 4.3, maxi- mim control of uIl is applied for time t1; then, reduced control is applied until the state vectors both reach zero. Recall that the two state vectors correspond to vehicle velocity and wheel velocity. In this case, a very simple solution is possible if the constraint on u(t) is removed. The Optimal control will consist of an impulse at tIO and then a value Of less than 1.0 for the remaining time. See Figure 4.7 4) lm yo“. U“) .735 Figure 4.7 Optimal Control for Vehicular Braking System No Constraint on Control Signal The impulse Of Figure 4.7 drives the system to the peak Of the friction curve in zero time while the pulse of Figure 4.6 drives the system to the peak of the n-curve in minimum time t1. As will be shown t1 is much less than tf, the time to drive the system to zero. Hence, the impulse method leads to negligible error. From (4.23), the maximum value of the friction coefficient n is 64 0.947, corresponding to x2/x1I0.8. From (4.18), if an impulse is applied at tIO, no change in xl takes place. Applying an impulse of strength 6 and using x2=0.8x1, (4.18) becomes, 48 0+ I dx2 I -1584 J 6dt (4.28) o 60 The strength of the impulse is 6 I .00757 (4.29) If it is assumed that the area of the pulse of Figure 4.3 is equal to the strength of the impulse, then time t1 would be equal to .00757 seconds. This also assumes that u is limited to 1.0. Although this is not accurate, it is sufficient to show that t1 is much less than t . f From (4.18) and using nI0.947, 0 cf I dxl I -32 x .947 I dt (4.30) 60 0 Solving for tf, tf I 1.98 seconds (4.31) Corresponding to this minimum stopping time, the minimum stopping distance is 60 feet. The Optimal control signal during this interval may be found by eliminating n from (4.19) and (4.18). Integrating the resulting equation, 0 1.98 o I dxz I -37.81 I dxl - 1584 I uzdt (4.32) 48 60 o 65 Solving, u - 0.735 (4.33) 2 The impulse solution will differ only from the actual solution because of t1. By assuming n I t/t1 during interval t1, equation (4.19) may be solved to yield, t1 I .0123 seconds (4.34) This agrees with the digital computer solution shown in Appendix II. The results of the analytical approach are summarized in Figure 4.8 u(t) q)- 0072, \‘991 $06. Vdmde Vs loo. )1, Lbh¢e\ Vdomhj 1 .0115 1-992 50‘ - Figure 4.8 Optimal Control and State Vectors vs Time 66 4.3.3 Equivalence of Minimum Time and Minimum StoppingiDistance Criteria In this section it will be shown that minimizing the time to go from xoto xf is equivalent to minimizing the vehicle stopping distance. It has been shown that the intersection of the reachable region R(xo) and the controllable region R(xf) contains the set of all possible trajectories from xo to xf. Also, the construction of this set is in- dependent of the performance criteria imposed by the functional J[u]. Using the Green Theorem Approach, it has been shown that the Optimal trajectory is contained in the boundary of this set unless singular arcs exist. Then the Optimal trajectory will contain portions of the boundary and portions of the singular arc. Hence, the solution is no longer "bang-bang". Two criteria will be equivalent if the singular arcs gene- rated by these criteria are the same. Consider t f t o t f J2[u] - I xldt (4 36) t 0 Since x1 corresponds to vehicle velocity in problcm(4.l), J2[u] is the stopping distance while J1[u] corresponds to the time required to stop. 67 To show that these criteria are equivalent, it will be shown that the critical function, w - 0, which determines the singular arc is the same for both criteria. From equation (4.18), determine dt and substitute in I 1 {xi dxl (4 3 ) J u - . 7 1 X0 ’32p(x1,X2) Xf x dx J2[u] ' (4.38) x0 -32u(x1.x2) Since, the general form for the cost function is . . 19. - i2. J [del + de2 [[(3x1 8X2) dxldx2 (4.39) and P1 - - 1 Q1 - o (4.40) 32u(x1,x2) P - x Q - o (4 41) 2 32u(x1,x2) 2 ° Solving for the partial derivatives ax2 3x2 u 3? x 3 (x1 x ) 1 -3 - 1 “ ' 2 2 (4.43) 3x2 3x2 32lJ We are interested in the condition w - 0, where 3Q 3? w - ( - --) (4.44) 3x1 3x2 68 If xlfo and ufO, then referring back to equation (4.16) 3U(xlsxz) ml - (.02 - 3x2 (4045) Typically, as indicated in Figure 4.1, the x #0 and ufO con- ditions are satisfied except at the origin. The origin presents no problem since u(0,0) - 0. Therefore minimizing stopping time is equivalent to minimizing stopping distance in Problem 4.1. f(xl) “(X1 9X2) since the partial derivative is taken with respect to x . Any criteria which results in P - would be equivalent 4.3.4 An Optimal Digital Control System In principle, the implementation of the optimal control is straight forward. Based on the Green's Theorem Approach the steps are as follows: 1) Apply maximum permissible control. 2) Continually solve for w. The condition w=0 indicates that the singular arc has been reached. 3) Reduce the control signal u(t) in order to hold the condition, w-O. Due to time delays, the desired control u(t) is not obtained in zero time. As a result the control oscillates about the predicted value of 0.735. The program shown in Appendix II shows this variation in u(t). The state vectors corresponding to vehicle and wheel velocity are essentially ideal in the digital computer system. 69 The Digital Optimal Control System in block form appears in Figure 4.9 _ 2 i‘ JD$QI+¢i — x‘ A Egan’s w , 3 lini-qrulbr BX: } a $ _ , ‘ i l ("new { F I Punch.» xI ; d: X2. I _ u(t) - Into, «.6» 7 "564 and ‘ I Lumtcr Figure 4.9 Digital Optimal Control System Based on Green's Theorem Approach 70 The significant blocks are the blocks which determine the control signal u(t). If m is positive the digital integration increases u unless u is at its limiting value of 1.0. When u is zero, u would remain con- stant except for the fact that time delays cause w to overshoot. In the digital solution, u changes by 0.01 per integration time interval of 0.000002. The results of this simulation can be summarized by Figure 4.10. The velocity signals which are the state variables x1 and x2 are close to the ideal values. There is a ripple frequency of approximately 300cps. This is a function of the digital integration gain. The peak wheel velocity ripple is approximately 0.020. The m signal has a peak value of approximately 0.012, while the peak value of theAu signal is 0.25. '- L. I" .1. W u(t) .5 oz “ tfi-o *4” (0.) ‘ not (b) ‘0. vehicle Vol salty Vchci Ve\oc\h, *' 4c) t' Figure 4.10 Wave Forms of Digital Control System (a) Control Signal (b) Singular Function (c) Wheel and Vehicle Velocity 71 A sinusoidal analysis based on small signal follows. Consider simplified diagram, w AU($) 1 Au I AX1(’) “1 no (5) _ L AUG) 4h- Figure 4.11 Diagram for Small Signal Sinusoidal Analysis The velocities x and x I tend toward zero slowly when measured 2 on the ripple frequency time scale. Hence, at a given point (x1,x2) the following equations may be considered to apply. x1 ' ‘10 (4.46) x2 . x20 + x Asin wt (4.47) also, x20 8 0.8xlo (4.48) Since w is given by 4.25, and letting k = sz/x10 w - ._1_ (EC-8(1-.8-k sin wt)-b€-b(l".8-k sin wt)) (4.49) Assuming that k<<.2, and using 5x :1 + x, m - _$_ (as-°za(l + ak sin mt) - be-°2b(l + bk sin wt) (4 50) x10 ' -028 '02b Since as - be 72 w I (ak-bk) sin wt/xlo (4.51) w I -5.25 k sin wt/xlo (4.52) 2 m I -5.25 sz sin wt/x10 (4.53) Using (4.19), a relation between sz and Au can be found. The p term has a constant term which cancels the steady state term of u. The sinusoidal variation in u is small relative to the cosinusoidal variation of x2. Therefore, u = u0 + Au cos mt (4.54) and 2anx2 cos 2nft = -1584 Au cos 2nft (4.55) This yields the relationship, 2 2nfx10 /5.25 I1584Au (4.56) or 2 w I 1320 Au/xlo f (4.57) The use of the above equation in conjunction with Figure 4.11, permits determination of the small signal variations in the system. 73 4.4.1 The One-Wheel Vehicular Braking;Control Model with Rate and Amplitude Limited Control This problem is an extension of Problem 4.1 and fits the format of Problem 3.3. It will be designated Problem 4.2. The differential constraint equations are £1 - -32u(x1.x2) x1 (0) - 60 (4.58) i, = 1210u(x1,x2) - 1584..3 x2 (0) = 60 (4.59) i, I v x3 (0) = 0 (4.60) I v I _<_1 (4.61) x3 :_0 (4.62) The function u(x1,x2) is given in Problem 4.1. The control u(t) of Problem 4.1 has been made a state variable with a constraint. It is desired to minimize the time necessary to drive the state from x0 to x The row vector x is [0 0 x3f is not specified. f' f This problem when stated in real physical terms is: Find the control pressure as a function of time which will stop a given vehicle initially traveling at 60 ft./sec. in minimum time. The pressure is rate limited to 12000 psi/sec. In the physical problem under consideration the pressure would normal- ly be amplitude limited (e.g.0§P(t)§_1200psi). This would result in a 74 singular problem which has a bounded state variable (x3). In this problem the upper and lower bounds are not penetrated by the Optimal control and hence the additional complication due to bounded state variables is not en- countered. The approach necessary when bounded state variables occur will not be considered at this time. The development of section 3.2.3 will be applied to Problem 4.2 of section 4.4.4. As may be seen from equations (4.58) to (4.60), the state vector is 3-dimensional while the control in 1-dimensiona1. Hence niil If; F01 i2 - £2 + 0 V (4.63) Lia 101 Ll. also t f J]u] I I dt (4.64) tO T Since there are two independent vectors orthogonal to b I [0 0 1] , dt may be expressed as dt - dxl/fl ' dx1/-32U (4.65) or dt - dxz/fz - dxz/(1210u - 1584x3 (4,66) Applying.equations (3.80) and (3.88) - (3.90) to (4.65) a1 - -1/32u (4.67) 75 612 - a(-l/32u)/3x2 (4.68) “12 I 0 implies a singular surface exists and is given by 8u(x1,x2)/3x2 I 0 (4-69) This surface will contain part of the optimal trajectory as shown in Figure 4.13. It corresponds tO the arc B-C. Its projections in the xl-x2 plane is D-0 and is the same as in Problem 4.1. Applying (3.80) and (3.88) - (3.90) to (4.66), 1 a I 2 lZlOu-1584X3 (4.70) Now 3 l . - .._ 4.71 ml? ax1 (12100-1534x3) ( ) (1)13 - 0 ' (4'72) 3 1 . ___ ( ) (4.73) m 23 3x3 1210U-1584x3 Only “12' 0 implies a singular surface. (4.74) If x1 + 0, this produces the same singular condition given by (4.69). The net result is that two equivalent forms are available to evaluate J[u]. xf dxl. J[ul I J ‘ x0 I32u(31:X2) (4,75) and Xf dXZ - . 6 J[“] [:0 1210u(x1,x2)-1584X3 (4 7 ) 76 4.4.2 Solution of the Pang:Pang Singular Control Problem by the Green Theorem Approach Normally problems of the type formulated in the previous section re- sult in the bang-bang behavior of u(t). This is referred to pang-pang operation with respect to u(t). Several modifications with respect to the procedure used in Problem 4.2 will be necessary. First, no theorem is available for the general case which determined the reachable region in terms of special trajecto- ries. For example, in a 3-dimensional problem with a 2-dimensional control, the reachable region would intuitively appear as shown in Figure 4.12. ¢kn #5-! 4"" Figure 4.12 Reachable Region for 3—dimensional Problem 191 19-1 In the figure, four edges determined by trajectories 0 , ¢ , -l,-l -1,1 1,1 0 , ¢ might be expected. The trajectory 0 is defined as the solution x(t) with control‘ulIl, ule, x(0)Ixo. The other trajectories would be defined in a similar manner. 77 In the case under consideration only one control signal is available. This reduces the reachable region to a surface. Hence, two dimensional theory as given by Theorem 2.1 may be applied. In Figure 4.5, xo-A-B-D-xo shows part of the reachable region. The curve xO-A is part of the tra- jectory due to the application of v(t)-l. The curve xO-D is in the xl-x2 plane and is due to the constraint which requires that x310. This curve cannot be generated by any control v(t) but can be approached as close as desired by using a sufficiently small v(t). The trajectory of the pang-pang problem is shown in Figure 4.13. The trajectory starts at x0, which lies in the xl-x2 plane. The surface w=0 is a plane shown passing through points C, O, and D. Physically the system acts as follows: The pressure is increased at its maximum permissible rate until point A is reached. Then, the pressure is decreased at its maximum per- missible rate until the singular surface is reached. The trajectory then continues from point B to the final point C. Along this trajectory, which is singular, the pressure is maintained at a constant value. The value of pressure required is determined by the singular control law. 78 X. )‘1 Figure 4.13 Optimal Trajectory for 3-Dimensional Problem CHAPTER V SUBOPTIMAL VEHICULAR BRAKING CONTROL 5.1 Introduction Theory indicates that the optimal control drives the state vector to the peak of the friction-slip curve as fast as possible and then holds it there. From apractical point of view the Maximum Principle is not feasible. Figures 3.1 and 3.2 show the block diagram necessary to mechanize the Maximum Principle solution. Since the initial costate vector must be determined on—line, subject to various initial conditions on the state vector, a rather complex and fast system would be needed to satisfy this requirement. This in itself would present a formidable Optimal control problem. The approach using Green's Theorem is more feasible from a practical point of view. Figure 4.6 shows the block dia- gram using this approach. The difficulties associated with this method are due to the block which computes the m function which determines the singular condition. Again this component must Operate with fast response on-line. As indicated in section 4.3.4, the digital solution using the IBM 360-65 produced the wave forms shown in Figure 4.10. The results with respect to optimal stopping time were essentially optimal. The velocity waveforms were also essentially optimal. A slight ripple frequency appeared on the wheel velocity output. This ripple had a peak amplitude of 0.020 fps while the vehicle and wheel velocities were as high as 60fps. The control waveform deviated considerably from the ideal waveform. The ideal control signal appears in Figure 4.8. It should be noted that the digital solution oscillated at 300cps about the ideal value of 0.735. Hence from a practical point of view, this approach can be mechanized. However, in the interest of simplicity 79 80 and economy the following suboptimal control was designed. The basic idea is illustrated in Figure 5.1. (on a. 54 um Basic SubOptimal Control Figure 5.1 The basic premise is that the peak point on the friction—slip curve occurs at essentially the same relative slip regardless of vehicle velocity. Reference to Figure 4.3 shows that this point corresponds to a relative slip of 0.15, i.e., l - xz/x1 I 0.15. This implies that x2 which corresponds to wheel velocity is 85 percent of the vehicle velocity x1 at the peak point. In addition, it is to be noted that the friction coefficient decreases slowly as the relative slip increases. At a slip Of 1.0, the friction coefficient is typically 0.8 of its maximum value. This is the value achieved in the so-called "panic" stop. Therefore, based on this curve the stopping distance can be reduced to 0.8 of the "panic" value. Minimizing stopping distance is however only one factor in making a safe stop. A factor that is equally important is maintaining lateral stability. Due to small variations such as road surface unbalance, 81 brake torque, and wind gusts, yawing may and usually does occur. These effects may be minimized if the slip is kept small. The net result implies that operation should be close to the peak of the friction-slip curve. The most effective operation will be shown to occur at slip values slightly greater than 0.15. 5.2 Block Diagram of Suboptimal Control System The real system must contain transducers which are not ideal. The principal characteristic can be approximated by a time delay. For example, the wheel signal transducer generates an alternating voltage with frequency being proportional to wheel velocity. The electronic processing generates a voltage proportional to wheel velocity. This processing results in a delayed wheel velocity signal. Similar delays are encountered from the other transducers that must be used in the system. Hence. a realistic model of the vehicular braking control system appears as shown in Figure 5.2b. This system was programmed on both analog and digital computers. The analog computer studies permitted interconnecting real components with simu- lated components. The diagram shown in Figure 5.2a shows the four basic elements, the road characteristics, vehicle dynamics, electronic control module, and the braking pressure actuator. All four elements were simulated on both analog and digital computers. In the analog setup, the simulated electronic pressure actuator could be readily replaced by real components. This feature permitted evaluation of various module and actuator designs. R0 AD C HAKM‘I’ tRKT ’1. VELHCLE DVNAMtts 82 Velocutg sands TRANS 006m and B‘SKING COBTROL ELmaomc tomcat M o DOLE. <6n+rol stands masons Asmara: Figure 5.2a Basic Block Diagram 83 17 E. The! (004qu TI ON Figure 5.2b suboptimal Control System With Time Delays 84 5.3 Analog Computer Studies The analog equivalent of the system shown in Figure 5.2 was stud- ied extensively. In these studies, real components such as pressure actuators and electronic control modules would be compared with their simulated models. The primary object was to determine factors which ad- versely affected performance. The optimal control system was used as the reference system. Figure 5.3 shows response characteristics of a sub- optimal system in which all components are simulated except the actuator which is designated X2. As may be observed from the wheel velocity trace, this response is, from a practical stand point, essentially optimal. This is a very low friction case, having a panic coefficient of nominally 0.09. The percentage error due to potentiometer settings is greater at the lower values of friction. To remove this source of error, stopping distances are compared with the skid control on and off. The panic stopping dis- tance, for an initial vehicle velocity of 60 fps, assuming that the vehicle is a point mass, is given by the following equation, Panic Stopping Distance I 56/u (5 1) In this case, the nominal value of 0.09 would predict a value of 622 feet instead of 583 feet. This implies that u is actually 0.096 instead of 0.09. The significant fact is the reduced value of 507 feet. It should be noted that an ideal wheel velocity transducer is used. The er- ratic operation that occurred was due to vacuum pressure going below the design minimum. In summary, this test shows nearly optimal response even with a real actuator in the system. Figure 5.4 shows the effect on the same system by using a transducer 85 that can be approximated by a time delay of 0.020 seconds. Stopping distance is increased slightly, although considerably better than 186 feet, the value corresponding to the panic value of u I 0.3. Actuator X1 has an initial pressure rise characteristic which has adverse effects. This causes the wheel velocity to initially drop to lower than desired values. This can cause lock-up under some road fric- tion conditions. Figures 5.5a and 5.5b show these characteristics. The initial part of this response is magnified in Figures 5.6a and 5.6b. This actuator was one which had been considered satisfactory for systems which did not employ anti-skid controls. An interesting nonlinear phe- nomenon occurs in this set of traces. The Operating point is unstable. The right wheel velocity approaches the vehicle velocity or free wheels, while the left wheel velocity goes to the lock-up condition. This con- dition will be considered in Chapter VI. Figures 5.7a and 5.7b show a typical response characteristic of the system without compensation. As may be noted the wheels alternately free wheel and lock-up. These conditions are also clearly seen on the friction curve. The panic value in this case is 0.8. Since the stopping distance is the same whether the control is on or off, the average coefficient of friction is 0.8. Also, it is to be noted that the pulsing frequency is considerably lower. In this case, it is approximately 3 cps. With com- pensation, pulsing frequencies as high as 20 cps have been encountered. 86 ACTUATOR PRESSURE -200 PSI _AmflMmmWMWWWMNWMMHIIIIIIIHINNIHHNHmm“H“I“J1ALLL .ril 60 FPS 16 CPS VACUUM BELOW MINIMUM /'/ WHEEL VELOCITY \k“‘\\‘\\___ 0 FPS ~| !~ I SECOND MU=.09 STOPPING DISTANCE 507 FEET CONTROL ON 583 FEET CONTROL OFF -'60 FPS VEHICLE VELOCITY --n l- 1 SECOND - 0 FPS Figure 5.3 Velocity Response-Real Actuator X2 Ideal Wheel Velocity Transducer 87 60 FPS WHEEL VELOCITY IDEAL TRANSDUCER MU=.3 J OFPS STOPPING DISTANCE 151 FEET ‘—’60 FPS WHEEL VELOCITY TRANSDUCER wz f‘ 0 FPS STOPPING DISTANCE 157 FEET Figure 5.4 Effect of Wheel Velocity Transducer 88 VEHICLE VELOCITY 60 FPS _. 0 FPS RIGHT REAR WHEEL VELOCITY 60 FPS -— LEFT REAR WHEEL VELOCITY 60 FPS_ WM 0 FPS 1 SECOND ~l [— Figure 5.5a Effect of Initial Delay in Actuator X1 Velocity Response 89 V- mm" " cw; ___J FRICTION COEFFICIENT SOLENOID SIGNAL CONTROL PRESSURE 1000 PSI 1 | 1111')!” 111‘ J11111m 11 1 1 111 111111111111111111111.11111 1 SECOND Ann. 1""— 11111111111111 "'1 “'1“ 11 I1 1 '1 1M IIII 111 1. 11 1111 MW. 100% .' "'1‘." “’1‘1'11'11' ,3 111.11111111.11111111.111111*\0 PS I Figure 5.5b Effect of Initial Delay in Actuator X1 Solenoid, Control Pressure,and Friction Response 90 a————_________________‘____~____—03 FPS ““3 VEHICLE VELOCITY RIGHT REAR WHEEL VELOCITY 60 FPS.. LEFT REAR WHEEL VELOCITY 60 FPS ~« .1 SECOND «1 1— Figure 5.6a Initial Velocity Response Effect Of Initial Delay in Actuator X1 91 100% FRICTION COEFFICIENT SOLENOID SIGNAL F PrrrrrrrrrrrrrP—I-"I CONTROL PRESSURE 1000 P81 .1 SECOND —.1 h— Figure 5.6b Initial Response Effect of Initial Delay in Actuator X1 Solenoid, Control Pressure, and Friction 92 \ RIGHT REAR WHEEL VELOCITY 60 FPS LEFT REAR WHEEL VELOCITY \_ O FPS ' WWW FRICTION COEFFICIENT |—-I SECOND A... A A..- A... __vv ‘“ v“ Figure 5.7a Response Without Compensation Stopping Distance Control On 72 Feet Control Off 72 Feet 93 SOLENOID SIGNAL CONTROL PRESSURE 1000 PSI firm/(m . 60 FPS —\ VEHICLE VELOCITY \ 441.4) WHEEL ACCELERATION "Lu_, .1 SECOND-.4 l.- Figure 5.7b Response Without Compensation 94 5.4 Digital Computer Studies The Continuous Systems Modeling Program (CSMP) on the IBM 360-365 was used to supplement the studies made on the analog computer. The listing of the simple one-wheel model is shown in Figure 5.8. This is designated PROGRAM I. Listings of other models are shown in Appendix IV. Sample outputs and histograms of PROGRAM I are shown in Figures 5.9 to 5.15. The model described by PROGRAM I is almost ideal with respect to transducers. There is no delay in the pressure transducer and a neg- ligible delay (I millisecond) in the wheel velocity transducer. NO compensation is used. In Figure 5.10, the first portion of the wheel velocity Of subopti- mal vehicular control system is shown. In this suboptimal system, the wheel velocity has a peak ripple velocity of approximately 7 feet per second, whereas, the optimal control would have no ripple. More signi— ficant is the friction coefficient, shown in Figure 5.11. After it passes the peak value of 1.0, it varies from between 0.9 and 1.0. Hence, the average value is approximately 0.95 for suboptimal system and just under 1.0 for the optimal system. It should be noted that this is not a realistic suboptimal control system, since the transducers are essen- tially ideal. Figures 5.13 and 5.14 show similar results when the friction coef- ficient is 0.3. The supoptimal U averages 92 percent of the Optimal value of 0.3. The low u case is,with respect to ripple magnitudes, more nearly optimal than the high H case. Stopping distances show that 95 the suboptimal system requires 59 feet at a u of 1.0 and 202 feet at a u of 0.3. This compares with 56 feet and 186 feet for the optimal control system. These values are for an initial velocity of 60 feet per second or approximately 40 miles per hour. Of interest is the ripple frequency, since this may be used as a measure of optimality. The suboptimal system with essentially zero de- lay has a ripple frequency of approximately 11 cps at u I 1.0 and 18 cps when U I 0.3. When transducers, which have time delays are used, these frequencies go down. To compensate for the delays, compensation net- works are employed. In the suboptimal control system, the effectiveness of the compensation can be judged by the ripple frequency generated. The higher ripple frequencies imply that the system is more nearly optimal. A criterion, based on ripple frequency, for estimatihg time delays associated with transducers is established in Chapter VI. The last two figures in this set, Figures 5.12 and 5.15 show the control pressure for the U I 1 and U I 0.3 cases. As generated on the digital computer, the rise and fall rates are 15000 and 45000 psi/second for this case. In the digital studies, use was also made Of the CSMP plotting fea- tures. Typical plots are shown in Figures 5.16 to 5.23. The effect of having unbalanced time delays was studied. Figure 5.16 and 5.17 show the effect of having 20 milliseconds delay in one of the wheel velocity transducers and no delay in the other. Vehicle velocity and pressure transducers also had no delay. The reference velocity in 96 the suboptimal control system was set for 0.5 of the vehicle velocity and the system used the average of the wheel velocity signals to establish the error signal. These results show that, by using proper compensation, this amount of delay may be tolerated. The principal disadvantage is that the wheels lock up at a vehicle velocity of approximately 5 feet per second. In other respects, the supoptimal control operates as desired. This is clearly shown in Figure 5.17. The control pressure quickly brings u to its maximum value. Then, due to the slip reference setting of 0.5, keeps u at a relatively high value for almost the entire stop- ping period. Earlier lockup is very clearly shown in this figure. The value of u is seen to drop to the panic value of 0.8 while the control pressure rises rapidly to its maximun of 1200 psi. The stopping dis- tance for this suboptimal system is 60.21 feet as compared to 56 feet for the optimal control system. The listing of this program, P 252, is shown in Appendix IV. The next set, Figures 5.18 to 5.21, show the effects of having 20 milliseconds delay in both wheel velocity signals. There are no other changes from the previous system. The ripple frequency variation is now quite prominent. The relatively high frequency of 12 cps indicates satisfactory operation at this value of u. The stopping distance has increased slightly to 60.76 feet. The effect of subjecting one side of the vehicle to a peak u 0.95 and the other side to a peak u of 1.0 is shown in Figures 5.22 and 5.23. All other components of the system are as in Program 252. The side of the vehicle which is subjected to a u 0.95 and has a 20 millisecond 97 delay in the transducer locks up early as seen by the WV2D trace in Figure 5.22. Stopping distance was 63.4 feet as compared to the opti- mal value of 57.2 feet. This is a two wheel model which used the average of the two wheel velocity signal to produce a one-dimensional control signal. This example points out the disadvantage of the one dimensional control versus the two dimensional control system. Figure 5.23 clearly indicates the changes in the u characteristic and control pressure that occur after early lock-up. The final figures of this section show the response of the system which is designated as the reference suboptimal control system. As may be noted from Figures 5.24 and 5.25, the wheel velocity response and the friction coefficient response are almost undistinguishable from those determined from the theoretical optimal control system. In Figure 5.23, trajectory O-A-B is nonsingular, while B-C is the singular trajectory. The control pressure response, in Figure 5.26, does not peak as expected in the optimal response. The variation in the flat portion of the con- trol pressure is due to the anticipatory nature of the compensation network. Stopping distance for this system is 57.75 feet. These figures are representative of the digital studies which show that the suboptimal control system considered performs satisfactorily and compares favorably with the reference optimal control system which, it should be noted, does not contain time delays. 98 I‘I‘CONTINUDUS SYSTEM MODELING PROGRAMIIII *‘IPROBLEM INPUT STATEMENTS*** IITLE PARAM FUNCT METHOD PROCED ENDPRO TIMER FINISH LABEL PRINT PRTPLT END PARAH who .& wr- ~10 ASKC B=l..C=l..D=l. MUETIIII.15.-l.1.10..0.O).(.075..6).(.15.1.1.(.57 RKSFX HU1=B*AFGENIHUET19ETAII VVDOT='16.*(HUI+DI VV‘INTGRLI60.09VVDUTI VVD‘DELAY1510015'VV1 HV100T=1210..HU1*0.79*T1 SDIST=INTGRL(0.09VV1 HV1=INTGRL460.09HVIDOTI HVID=DELAYI59.00IVHV117 ETA1A=VVDIHVID lelsbé‘Pll ETA1=1.OINV11/VV HVII‘LIHITIO.96O.OQHV17 ETAIR=0.5‘VVD Y1=DERIVI0.09ETA1A1 ERRIN1=ETA1AIETA1R ERROTI=INSHIERRIN19 1.0.I3.0) EI=ISOOO.*ERRUT1 PI=INTGRL(0.09X11 X1=EI*X11 Xll=DUIIP10E11 lFIEI'levZ IF‘P1130494 XIIIO.O GO TO 7 XII=I.O GO TO 7 IFIPI'IZOO..59596 X11I1.O GO TO 7 X1180.0 CONTINUE P11=DELAYI150.00PII FINTIM=fiooDELT‘.OOO4g VVI1.O ANSKC‘JGG*SCHERBA ETAIR.ETAIA,MUI.ERRIN1,ERROTl,Pll, ETAIROETAIAOHUIsERRINIQERRUTIopll9 B‘OOBQC'OO390303 RESET LABEL LABEL END .STOP ANSKC ("08.3) PRDEL=.CZ VV’PIOSDI VVDPlsSDI Figure.5.8 PROGRAM I CSMP Listing of One-Wheel Model 99 m.¢l N0.m _dd.o N L o.o ,mo.m o~.o _ o.o .¢~.m no.. 0m.~ 00.m m0.o I owed N flow 0.4- uOoh n 0m.~ ~0.m _om.¢n >0.m om.— oo.m 00.~ 00.0 0.0 u (mowhzn ax >> ~32 ax >> ~01 ax >> ~31 ax >> #3! ax >> ‘0: «x >> ~Dt ax >> #0: ax >> #0! xumza ‘0 No #0 ~0 m0 #0 ac m0 do do n0 ‘0 do N0 00 do N0 00 #0 N0 00 wdomo.~ w00h~.o w0-0.m weah~.m m0m0~.~ wmhmm.~ wmmm0.¢ womomoa m¢0m~.~ m¢000.¢ w00-.~ w0~¢0.~ w~0m~.m whom~.0 wmmwh.c wm~0m.m m00m~.m who¢o.m wmhm0.m w000~.~ wom¢m.~ o. o. 0. COO ad): add <~3 dam <~3 gum <~ 0#>3 t01#z#z #0 #0 #0 #0 #0 #0 #0 #0 #0 #0 #0 #0 #0 #0 #0 #0 #0 #0 #0 #0 #0 #0 #0 #0 #0 #0 #0 #0 #0 #0 #0 #0 0 mN¢oo.N w#0#~.m womoo.~ w00#0.# w00~0.N moomm.m m000~.m wowom.~ wh#mm.~ wh¢o#.m mom0¢.m w¢¢00.n m000#.N wmm~0.N w0¢hm.n w00¢¢.n wohom.~ m¢mmo.~ mmmom.m wmmwh.n m¢~0#.m mm00¢.~ w0~0m.m th#0.m wN#mm.n m#omo.~ mo#h#.m w0~m0.¢ m¢000.¢ m#0n~.m mm#om.m wmbm0.m 0.0 0#>3 #0lw000¢.0 #00w000N.0 #01m0000.0 #0Iw0000.m .#00w0000.m #0Iw000¢.m #0Iw000~.m #01w0000.m #01w0000.¢ #00w0000.¢ #00m000¢.¢ #OIw000~.¢ #01m0000.¢ #09m0000.m #0Iw0000.m #0lw000¢.m #0Uw000N.m #04w0000om #0Im0000.~ #0lw0000.~ #01w000¢.N #01m000~.~ #0Iw0000.~ #0lw0000.# #0Im0000.# #01m000¢.# #00m000~.# #01m0000.# N0Iw0000.0 ~00w0000.o N0|w0000.¢ ~01m0000.~ 0.0 wt#h Histogram of Wheel Velocity Figure 5.10 101 \.-----U----- 1-501 -H # #0Imh000.o tDt#x #3! #01w0moo.0 #0Im#Nm0.0 #0lm0#mm.0 #0Iw¢0¢0.0 #01w00N¢.o #0lw0mhh.0 #OIwhm0h.0 #00wmom~.0 #0|w000#.0 #01w#N00.0 #01w0h¢h.0 #OIwMNOm.0 #0Iwom00.o #0lmmm0n.0 #01w0m00.0 #ouwhomo.0 #OUwNmNN.0 #OIwh~0#.0 #0Iw00hm.o #0Iw00ho.o #01m0-¢.o #01m#mF0.0 #00wN¢0¢.0 #0Im#mN0.0 #01w#N0m.o #0IwNN##.o #0lm#00~.o #00w00m0.0 #olwmmm0.0 #0IwMM#o.h #0Iw0#ho.m #0Iw0000.# 0.0 #0! m0|w00m~.¢1 zax#z#z #01w000¢.o #0Iw000~.o #01m0000.0 #01w0000.m #01m0000.m #0lm000¢.m #0lw000~.m #0Iw0000om #0lw0000JV #0lw0000.¢ #01w000¢.¢ #0Iw000~.¢ #0Im0000.¢ #0Iw0000.m #07w0000.m #olw000¢.m #0lm000N.n #0lm0000.m #0Iw0000.N #01w0000.N #0Iw000¢.N #0Iw000N.~ #01w0000.~ #01w0000.# #0Im0000.# #0Iw000¢.# #03w000~.# #0lw0000.# ~0Iw0000.0 _NOImoooo.o ~0Iw0000.¢ N0|m0000.~ 0.0 wt—# Histogram of Friction Coefficient Figure 5.11 102 m0 m0n0~.# I:t#x #& 00 w0000 13!#z#x no No N0 N0 m0 m0 N0 No m0 m0 N0 N0 N0 m0 m0 N0 N0 00 no No N0 N0 no No No N0 m0 m0 m0 N0 N0 N0 001 wmm0#.# wmhmo.0 mahm0.m w00m#.0 meON.# w00N0.# m~mo~.# wmmONo¢ w000N.# wm#h#.# m¢0#h.o mom#h.m mcmh0.h wo#0~.# w00#0.# w000#.h w000#.¢ wONON.# m00m#.# m#oom.0 wNOOm.m mm0fi0.0 m0~0~.# w¢o#¢.o wm0#¢.o w005#.0 w0m0N.# w0n0N.# w00h#.# whom~.0 wocmh.m w000~.~ 0.0 #& #01w000¢.o #01w000~.0 #0Iw0000.0 #0Im0000.m moumoooo.m. #01w000¢.m #00w000~.m #01w0000.m #0Im0000.¢ #0Iw0000.¢ #00w000¢.¢ #0lw000~.¢ #01m0000.¢ #0Iw0000.n #01w0000.m #01w000¢.n #01w000N.m #01w0000.n #0Iw0000.~ #01w0000.~ #0lm000¢.~ #0lw000~.~ #01w0000.~ #01w0000.# #01m0000.# #0Iw000¢.# #0l0000~.# #00w0000.# ~01w0000.0 N00w0000.o ~01w0000.¢ ~0lw0000.~ 0.0 mt#h Histogram of Control Pressure Figure 5.12 103 # #0 w0000.o th#xz + # 0.0 IDI#Z#Z wwooo.~ m0m>0.m wooao.~ moNNO.N w#¢m0.m w¢oam.~ wMN00.m wmo~0.~ whowh.N wh0~#.m m#m0h.~ mm~m0.m w¢#0#.m m#0~o.N w¢00#.m w0¢h0.N w00~0.~ wONNN.m wm000.~ w~m00.m w000#.m wmp¢p.~ w#0mN.m mh0~0.m wN#00.~ wohh~.n w0000.N w#0m#.N w¢¢m~.~ wcomoom wh~¢#.m w#0o~.m 0.0 0#>x #0lw000¢.o #00w000~.o #00w0000.o #00m0000.m #01m0000.m .#01w000¢.m #01w000~.m #00w0000.m #01w0000.¢ #0Iw0000.¢ #0Iw000¢.¢ #00m000~.¢ #0lw0000.¢ #01w0000.m #0Iw0000Jm #00m000¢.m #0lw000N.n #00w0000.m #00w0000.~ #00w0000.~ #0Iw000¢.~ #0lm000~.~ #0Iw0000.~ #00w0000.# #00w0000.# #0Iw000¢.# #0Iw000~.# #00w0000.# ~0Iw0000.0 ~0Iw0000.o ~00w0000.¢ N0Iw0000.~ 0.0 mx#h Histogram of Wheel Velocity Figure 5.13 104 #Olmmooa.N th#x¢t 'T.‘ 11 11111 1------H I ' I "T'---""-""- T. I -""|--"..-"'- " -T-..‘"--'.--|-"" --"""'- " --"" 'H'H' -' 1 III I, "'I' I 1 "'---"'---"-"-" 1 I :‘-""|-' -- - ---". -- -g'--- "--"---'T"..-'-..--.'-'--'-".. ..'.'.".'..".'.|.‘..'.‘:‘.."...'."....'. --'--- 1| l -‘l'. 1'1 ‘l--"'. '.|.'! ¢I -'-------"." Iltlllltllllltltl+ # mt#h m3m¢m> #3! #OImaom~.N #0Iwmo0h.~ #OIwcomh.N #otwh0hh.~ #0Im#00h.~ #0Iwmhwh.~ #OIwmomh.~ #01w0m~h.~ #0Immm¢h.~ #0Iw~m0h.N #01wc#¢h.~ #01whm-.~ #OImmN05.N #00m00Nh.N #0IwN00h.~ #0|m¢00h.~ #01w#h¢h.N #OImm¢Ob.N #0Iw0¢¢h.~ #0Iw#nhh.N #01m0¢0h.N #0lmo0wh.N #01mh00~.N #0Iwom0h.~ #01m0nmh.N #00mm#0h.~ #olw#hMN.N #othFmo.N #0Iwommo.N #0Iw0#om.N #01wm#n0.~ ~0|wo#~0.o 0.0 #3! #00wum¢0.¢l 83t#z#x #01w000¢.0 #01m000~.0 #00w0000.o #OImooomom #OImooooom #0lm000¢.m #00w000Nom #01w0000.m #01w0000.¢ #0lm0000.¢ #0lw000¢.¢ #0lw000~.¢ #08w0000.¢ #01w0000.m #0lw0000.m #01w000¢.n #01w000~.m #0Im0000.n #0lm0000.~ #01w0000.~ #01w000¢.~ #01w000~.~ #0Im0000.~ #01m0000.# #0Iw0000.# #01m000¢.# #01w000~.# #0iw0000.# N00m0000.0 N01w0000oo N0lm0000.¢ ~01m0000.~ 0.0 mt#h Histogram of Friction Coefficient Figure 5.14 105 an:-------..' 0'1--.". -- T- :-':--'-'-'-"' # n0 w0¢¢0.# [Dz—x4! wt#h m0m1m> #n ~0 No 00 No No No ~0 #0 N0 No #01 N0 N0 N0 N0 #0 N0 No 00 N0 N0 N0 00 w0000.0! tDt#z#t w¢0n0.m wommh.~ m#~¢#.0I wco¢o.¢ wh0¢o.# wh0N¢.N w00~o.m w~00~.0 m00#m.m m00#m.~ wo~m0.#I wooo¢o¢ w#00¢.# m#0b0.N m#OF¢.m wo#Oh.¢ w~0¢m.m wm0¢m.~ m0000.0| w¢00¢.¢ w¢00¢.# m¢o~0.m wmomn.m wmm00.n woocm.m whoon.N wm~m0.m0 mm~m0.nl whoam.o whomh.0 wwomh.m w000~.~ 0.0 #m l ,aonmooo¢.o #00w000N.0 #0lw0000.o #01w0000.m #0Iw0000.m #01m000¢.m #0lm000Nom #01m0000.m .#00m0000.¢ #01w0000.¢ #01w000¢.¢ #0Iw000~.¢ #0lm0000.¢ #0Iw0000.n #. 0 I w0.0.0..~ Mm. #01m000¢.n #00w000N.m #0Im0000.m #0lw0000.~ #00w0000.N #00w000¢.~ #0Iw000~.N #0Iw0000.N #01w0000.# #01m0000.# #0Iw000¢.# #0Iw000N.# #01m0000.# N01m0000.0 N01w0000.o ~00w0000.¢ N00m0000.~ 0.0 m:#h Histogram of Control Pressure Figure 5.15 VV 106 Time Delays of Transducers g 8 8 Wheel Velocity WVl 0 ms 6 6 ~ Wheel Velocity WV2 20 ms b 0‘1 8 Vehicle Velocity VV 0 ms \ Control Pressure 0 ms L 1 3 3. 3. ’1 3 S" S‘H I 11 1 1 ~ D O o \\ I I. I a qu E11 .3 3’ 3' 1 8. 2. 8. 3 av 3‘ C3 (3 N .—- 3 § 3 3. 3. . O4» Oy- nu «V (v O O O I: I. :31 ‘ 61b 6le 1 1 1 8. 8.J. 8. A L 4 I: 0) T ' ‘ °o.oo 1.60 2.40 0. u TIME Figure 5.16 Velocity Response Suboptimal System Employing a Single Control Pressure Signal 107 R“ 120.00 T {- c: 9 up 041 r-I-n D s- c: Passsums 3 9 a" 81r- B 1 Time Delays of Transducers a ”8 :1.0 a Wheel Velocity W1 0 ms 0 3 Wheel Velocity wvz 20 ms Vehicle Velocity VV 0 ms ,. Control Pressure 0 ms 23 «I! 4 S 0..° s: =3 i No a. a 4. s: :r 1 l J Compensation 0.15 Derivative Feedback 0 0 Reference Velocity I 0.5 I I. 5.1. a 7 I» 8 1 0 . r: . eoé .,1 4; e as 11 ' °o. oo 0. so 1. oo 1.50 2. TIME Figure 5.17 Friction and Control Pressure Response Suboptimal System Employing a Single Control Pressure Signal HVID 108 S 5% Time Delays of Transducers Wheel Velocity WVl 20 ms Wheel Velocity WV2 20 ms Vehicle Velocity VV 0 ms 8 Control Pressure 0 ms Compensation 0.15 Derivative Feedback Reference Velocity I 0.5 8 6 N 3. 53" a . . 4 1 4 ‘11.!» 0.00 1.20 1.60 0.00 TIME Figure 5.18 . Wheel Velocity Response Suboptimal System Employing a Single Control Pressure Signal NVZD 109 Time Delays of Transducers Wheel Velocity WVl 20 ms s: '2 Wheel Velocity WV2 20 ms 3 Vehicle Velocity VV 0 ms Control Pressure 0 ms Compensation 0.15 Derivative Feedback 53 Reference Velocity - 0.5 3. as 1’ 8. 3. as on =5 r '- 8. J l a H J '‘000 0.00 0.00 1.01.00 as: 'TIIflE Figure 5.19 Wheel Velocity Response Suboptimal System Employing a Single Control Pressure Signal Pl 1 120.00 0001 00.00 110 Time Delays of Transducers Wheel Velocity WVl 20 ms Wheel velocity WV2 20 ms Vehicle Velocity VV 0 ms Control Pressure 0 ms 200.00 j 1 Compensation 0.15 Derivative Feedback Reference Velocity I 0.5 100.” «1:27' 1L 0 0.00 TIE 400.00 F ‘1- ‘r 4? mm 0.00 1.20 1.50 LN Figure 5.20 Control Pressure Response Suboptimal System Employing a Single Control Pressure Signal W1 0. 00 1.00 111 Time Delays of Transducers II Wheel Velocity WVl 20 ms “Ed. Wheel Velocity WV2 20 ms ‘3 Vehicle Velocity VV 0 ms Control Pressure 0 ms 3.41 as I E: : 3 i : 4 '0.00 0.00 0.00 1.00 1.00 0.00 TIME Figure 5.21 Friction Characteristic Suboptimal System Employing a Single Control Pressure Signal 60.00 .1 1 VV L 0 I. (r 112 Time Delays of Transducers O :0 Si Wheel Velocity WVl 0 ms \ Wheel Velocity WV2 20 ms \ Vehicle Velocity VV 0 ms \\ Control Pressure 0 ms 53 \ - \ 3'” \ Compensation 0.15 Derivative Feedback \ Reference Velocity I 0.5 \ \ c: \ I. \ E3“ \ \ \ \ \ 8 \ ° \ I 31 . \ Ea \VS/’VV > wsz SIS; \ , vvvwo \ ¢=+- 7' \ N \ \ \ \ 8. \ °1b ~ 0'. c11.00 0.50 1.00 TIME Figure 5.22 Wheel Velocity Response - Unbalanced Friction Suboptimal System Employing a Single Control Pressure Signal 0u00 M01 0.20 113 ° ‘~ F1ucru»4 g1 ’ /\ I ~~\ - \ r— I \ / \ t I / \\ 8 ' ' ‘ ’ \ n , \— =;$ J»,fo»\./ I>I£5$UEE 8 fié‘ "‘ l 0‘ CD «at: ><‘! 84 cud 0.53 Time Delays of Transducers :51 Wheel Velocity WVl 0 ms 3’ Wheel Velocity wvz 20 ms Vehicle Velocity VV 0 ms Control Pressure 0 ms a! Compensation 0.15 Derivative Feedback is Reference Velocity - 0.5 8 l 1 I _.j °'._00 0.50 1.00 1.50 2.00 A TIME Figure 5.23 Friction Coefficient and Control Pressure Response suboptimal System Employing a Single Control Pressure Signal 30.00 \NVLD 10.00 114 Time Delays of Transducers Wheel Velocity WVl 0 ms Wheel Velocity WV2 0 ms Vehicle Velocity VV 0 ms Control Pressure Pl 0 ms Compensation 0.05 Derivative Feedback Reference Velocity - 0.85 0:00 1.20 1.50 2.00 TIME di- 4% A- Y 3. . °0.00 0.00 Figure 5.24 . Wheel Velocity Response Reference Suboptimal Control System HUI 115 .3? r ‘8 o P 53.. D P 5 :5" Time Delays of Transducers Wheel Velocity WVl 0 ms Wheel Velocity WV2 0 ms :: Vehicle Velocity VV 0 ms N Control Pressure P1 0 ms 6‘ Compensation 0.05 Derivative Feedback Reference Velocity - 0.85 :5... 1 c5 5 J : 1 *a‘ '0.00 0.00 0.00 1.20 1.50 2.0 TIME Figure 5.25 Friction Coefficient Response Reference Suboptimal Control System (X101 ) 60.00 P1 00.00 20.00 190.00 T 116 Time Delays of Transducers Wheel Velocity WVl 0 ms Wheel Velocity WV2 0 ms Vehicle Velocity VV 0 ms Control Pressure P1 0 ms Compensation 0.05 Derivative Feedback Reference Velocity - 0.85 I$11.00 .00 0: I10 0:00 1:20 1:50 2100 TIME I Figure 5.26 Control Pressure Response Reference Suboptimal Control System 117 8 3. 8» 3T 3. 8" C) s: Time Delays of Transducers :2 8 Wheel Velocity W1 0 ms 34:- Wheel Velocity WV2 0 ms Vehicle Velocity VV 0 ms Control Pressure Pl 0 ms 8 e5.. Compensation 0.05 Derivative Feedback " Reference Velocity - 0.85 c: D. l J A J °0.00 20.00 110.00 50.00 00.00 W Figure 5.27 Phase Plane Plot Reference Suboptimal Control System CHAPTER VI NONLINEAR PHENOMENON IN VEHICULAR BRAKING PROCESSES 6.1 Introduction As the complexity of systems increases, it becomes more difficult to predict the effects of the various system parameters. Also, in the vehicular system considered, the wheel-velocities are tightly coupled. This results in the generation of frequencies vastly different than those obtained in the loosely coupled case. The presence of nonlinearities is responsible for the generation of additional frequencies. In this chap- ter, attempts to improve the intuitive feeling for some aspects of the system will be made. 6.2 Time Delay Criteria by Describing;Function Technique The suboptimal control system generates prominent variations in wheel velocity, friction coefficient, and control pressure. These would not be present in the optimal control system. These ripple frequencies are primarily due to the time delay constraints present in the subopti- mal control system with minimal time delays. It will be shown that the ripple frequencies may be used as a measure indicating the degree of op- timality achieved. Describing function techniques will be used to develop criteria. The one-wheel model using s-plane representations is shown in Figure 6.1. The friction-slip curve, shown in Figure 6.2, will be line- arized about a typical operating point. The gain characteristic will' normally be negative; but, as evident from the friction characteristic, may be positive. This gain term will be defined as, It” - Ali/An (6.1) 118 D j 119 vukr ‘ Figure 6.1 S-plane Model of One-wheel Vehicular Control System "I Figure 6.2 Friction-slip Curve Linearized about Operating Point 120 Since the vehicle velocity VV changes with respect to the wheel velocity very slowly, it will be assumed constant. The effect of a changing vehicle velocity will be discussed later. On an incremental basis, the circuit may be represented as shown in Figure The notation is the same as previously defined in Chapter IV. Figure 6.3 Simplified Incremental Block Diagram Onedwheel Vehicular Control System 121 The block involving the u characteristic has a transfer function given by T. F. - 3 " 1‘1 (6.2) where, 2 k1 - kuM R / 8 1 W (6.3) For application of the Describing Function Technique, the final form shown in Figure 6.4 is desirable. _ A _ -,:F . - K 6"" (H155) , S(s-kd Figure 6.4 Nonlinear System used to Develop Time Delay Criterion I T2 is the time constant associated with the lead network. T1 is the total time delay present in the system. 122 Study of this block diagram reveals several significant character- istics of the vehicular control system. First, if both time delays are neglected, the system stability depends on the sign of k1, which in turn depends on the friction-slip gain ku. Since the operating point is usu- ally on the negative slope portion of the friction-slip curve, the system is inherently unstable. This instability, however, has no adverse affect on the optimality of the system. Some of the unusual phenomenon observed is however due to this characteristic. The second characteristic of interest is the chattering of the wheel velocity as it is driven to zero. The chattering is readily ex- plained by means of the Describing Function Technique. Designating the nonlinear element as N(e, w) and the linear element as C(w), the oscil- latory of chattering condition is C(w) I - 1/N(e, w) (6.4) Here e is the amplitude of the input to the relay element. The input signal is assumed to be sinusoidal. A sketch of a typical Nyquist Plot appears as shown in Figure 6.5 In GHQ ‘fifk Re GUM u inseam, Figure 6.5 Typical NYQuist Plot 123 For typical values of T1 I .010 seconds and T2 I .05, the critical point is reached when w is approximately 142 radians per second and N I 2860. A computer study was conducted to establish the accuracy that could be expected. Using a symmetrical relay characteristic which switches between -5000 and +5000, the input was approximately 3. Since the fundamental component of the square wave is 4/0 - 5000, the gain N of the relay element is 2120. This, with the loop gain of 1.32 gives a total gain of 2800. The frequency was 22 cps or 138 radians per second. These values are very close even though the generated waveforms in the system are square, triangular and finally, approximately sinusoidal at the input of the relay element. A study of the NyQuist Plots shows that a simple criterion may be established to evaluate time delays in this system. By neglecting the effect of the gain factor k1, the angle criterion at the critical point - l/N is satisfied by the following condition: Angle of e (1+jmT2) I 0 (6.4) This is equivalent to the condition, Tan w'l‘l I sz, le i "/2 (6.5) Solution of this transcendental equation gives the chatter fre- quency in terms of the two time constants.’ Observation of various solutions shows that le is almost “/2 radians for all cases of interest. Thus, the simplified form below may be used, T1 I 2% seconds (6.6) 124 Thus if m I 142 radians per second, the time delay in the system is approximately .011 seconds. This simplified criterion is quite useful in establishing the time delay in the system. Concerning the effect of the neglected term k1, it is readily shown that at the frequency of chatter, this term is insignificant except at very low velocities where it tends to reduce the chatter frequency. Due to the asymmetric character of the relay element, a Dual Input Describing Function Technique was also investigated. For suboptimal op- eration, due to the small variations which are essentially sinusoidal, no significant additional information was obtained by this method. 6.3 Effect of Friction-slip Nonlinearity In the previous section, the Describing Function Technique was shown to be useful in establishing a criterion for estimating time delays of the suboptimal control system. If compensation is not used, the system performance is adversely affected. The criterion established in the pre- vious section is no longer valid and it is necessary to include the effect of the friction-slip nonlinearity shown in Figure 6.2. For time delays which are in the realizable range - 10 milliseconds to 20 milliseconds - operation will be on the positive slope portion of the friction-slip curve. On this portion of the curve, the gain factor k1 is significant and to a first approximation the following criterion will establish the dominant frequency of the variation: -ij1 Angle of 5 [(k1 + Jm) I -H/2 radians (6.7) 125 Referring to Figure 6.4, it is to be noted that the loss of the lead term due to the compensation network causes the system to make up this phase change by finding a suitable gain factor k1 which reduces the phase of the (s-+ k1) term in the denominator of the transfer function. Several cases where no compensation was used were investigated. The results of a system having no compensation and 10 ms delay is shown in Figures 6.6 to 6.9. Detailed study of the waveforms in these figures indicates that the criterion given by (6.7) accurately predicts system performance. The block diagram shown in Figure 6.10 will be used to illustrate the pro- cedure. Except for the time delays, the diagram is based on the program listing shown in Figure 5.8. Figure 6.10 Diagram Used to Evaluate Variational Frequency and Amplitudes of Variables The procedure is as follows for the system having 10 milliseconds delay. Since operation is on the positive slope portion of the friction- 126 slip curve, k1 is estimated as 40. The transcendental angle criterion (6.7) is then solved for m. The result is approximately 60 radians per second, comparing favorably with the measured frequency of 9.75 cps. The friction-slip coefficient response should lag the pressure re- sponse by 34.5 degrees. Detailed analysis of the response curves in Figure 6.7 and 6.8 shows this to be the case. The pressure amplitude is found by finding the fundamental component of the asymmetrical relay output and dividing by w. The peak to peak fundamental is Jr;— - .78 - 60,000/60 or 995 psi. This is essentially the same as the measured value. From the transfer function l/(s + k1) which relates pressure and wheel velocity, the wheel velocity amplitude is calculated as 13.8 fps peak to peak. This is higher than the measured value which is approxi- mately 10 fps peak to peak. The magnitude of the friction-slip coefficient variation is found from 12100 I WV ° k1 (6.8) This results in a predicted value of 0.46, which is condiderably lower than the measured value of approximately 0.9. Considering the large amplitudes, the results are not unsatisfac- tory. The Dual Input Describing Function Technique was not used here, but would probably improve the accuracy. The presence of other frequen- cies is clearly evident from the pressure response in Figure 6.7. 127 Time Delays of Transducers Wheel Velocity WVl 10 ms Wheel Velocity WV2 10 ms Vehicle Velocity VV 10 ms Control Pressure P1 0 ms No Compensation Reference Velocity = 0.85 ._1 l 1 I i :m 2150 .m “an ‘0” I.“ 2. THE Figure 6.6 Wheel Velocity Response 128 Time Delays of Transducers Wheel Velocity WVl 10 ms Wheel Velocity WV2 10 ms Vehicle Velocity VV 10 ms Control Pressure P1 0 ms No Compensation Reference Velocity I 0.85 1430.00 0‘101 1 50.” —== -==- P1 110.0 -0.m 0.111 0150 1.111 1150 in £50 THE Figure 6.7 Control Pressure Response 1.00 “F 0; Ml 4L 129 a Time Delays of Transducers Wheel Velocity WVl 10 ms Wheel Velocity WV2 10 ms. Vehicle Velocity VV 10 ms Control Pressure Pl 0 ms No Compensation _ Reference Velocity I 0.85 Km £50 an 2.50 TIDE Figure 6.8 Friction Coefficient Response 130 s-r arr I 8. . dei'P I: E {5 as U Time Delays of Transducers Wheel Velocity WVl 10 ms Wheel Velocity WV2 10 ms Vehicle Velocity VV 10 ms Control Pressure Pl 0 ms No Compensation Reference Velocity I 0.85 42.00 .00 1511 11m ~ 1.50 2.1:: 2.50 ”If Figure 6.9 Output Response of Asymmetrical Relay CHAPTER VII CONCLUSION The development of the system equation for the vehicular braking control system shows that the control signal appears linearly. This implies, since the system is nonlinear, the possibility of singular controls. From the unified singular control theory presented, necessary conditions which the time optimal control must satisfy are developed. The class of functions encountered in the vehicular braking control sys- tem are such that the minimum stopping time problem is equivalent to the minimum stopping distance problem. Based on the necessary conditions developed, the closed loop pro- blem is solved and a block diagram showing the mechanization using the Maximum Principle approach is presented. Since the initial costate vector must be determined on-line, subject to various initial conditions on the state vector, any cost functional which takes into account factors such as, cost and simplicity would eliminate this method as a possible candidate. A more practical approach is the mechanization developed by applying the Green Theorem approach. The critical component in this method is the w function block which determined the singular condition. This method is quite possible in applications which are relatively slow. An algorithm for determining the optimal control is presented. For the vehicular brak- ing control system, where significant changes occur in milliseconds, the method becomes costly. At this stage of design, the gap between theory and practice:is apparent. 131 132 The mathematical models which have been developed are not sufficiently sophisticated to include noise, variability, cost, reliability and other realistic factors. Inclusion of these factors would subject the models to further constraints and adversely affect the performance. The mathematically expedient models function as reference models, indicating the ultimate that can be expected, and also giving clues as to how the optimal control should function. As a result, a system called the suboptimal vehicular braking control system was developed. This system is optimal in the sense that it heuristically considers cost and simplicity and is suboptimal since minimum stopping distance is slightly greater than the optimal control system subject to a simple cost criterion. The ad- vantages gained far outweigh the effect of slightly greater stopping dis- tances. In zero time delay case, the stopping distance for the suboptimal control system was 57.27 feet as against 57.2 feet for the optimal control system. For systems with time delays, it would be desirable to have opti- mal control models which include time delays. However, by employing pro- per compensation, the suboptimal control systems with realistic time delays compare favorably with optimal control systems having no time delays. Whereas, most of the effort was devoted to one-wheel models, studies of two-wheel models indicate that coupling effects will introduce several new problems. This is especially true if the system is constrained to use one control signal to control two wheels under different friction conditions. Several criteria were developed to assist in the understanding of the non- linear phenomena which take place. The criterion which evaluates time delays present in the system is particularly useful. 133 From the viewpoint of the builder of vehicular braking systems-in particular, the automobile manufacturer - cost is a heavily weighted factor in the performance functional. Elimination of a costly transducer is de- sirable. At the present time, the vehicle velocity transducer is in this category; This leads to a very significant vehicular braking control pro— blem - the optimal control with inaccessible state variables. Based on analog and digital studies already conducted, suboptimal con- trol systems with inaccessible state variables compare favorably with the optimal control system having accessible state variables. Hence, the so- lution to the inaccessible state variable problem is of interest. With the addition of time delays, these significant problems are left for future development. MWHWDKI SUBOPTIMAL ONE-WHEEL MODEL CONTROL SYSTEM ****CONTINUOUS SYSTEM MODELING PROGRAM¥*** " _.. I‘vv-OUQ. Was. :- .ernh- . . 'INPUT"STITEHENTS‘1WV‘""m ‘ IIIPRUBLEM 55 2 la 0,. o o D . (to. O S . 9 9 H: 0 H ’1’ cl. 0.. C . , P 1.1L _. , . m 0 _ 0. 0 H U, . v .. 55 . w r . . V . 11 m _ . W .9 . o . 3. H M . a T 5.0 m (In H w A .. U H 2 . . ' . . ..O I v 0:, . U T! H . 66 l , . 0 _ a. a m 1 Nu ah . V. 0 H D , ., 7.7 . . _ flu V.“ 9 00 _ a . .. .3 u c 7!. I. C. 0 w .. U 0 — l 3 I. I (T . . . RR. “ 0.0 19 . c 123 . 0.R 1T m l . AA. . L.R 00 . 1 a 5 T19 E,E ! a \lfl‘l . alTIT 0’ all! 1 o EEO . E v. 0 00 128 .Tss ll 22 t ... . D V 11 l 9 9 AA .71... . 0990V1 V2 In 1.01.01. 0 VE NU ._ .. TTAT ID770WV WV 0 12 2 9L [MW 00 EE+0 W1..21H 1H ZIYYo 0 DC Ron (I. 9. 220 V00. 0 9 0 O VAt. t. .1 II 0 1...! RI. 1 9 a. IZUV .9“ . _ .0 00V “ISBN 1! 2 I00 VH ED. . )I TTHV 112902V02V +Allllr III 1. NE 19 cc EE+0 oOUU 60V60/ . DchRTY 112 ,= 9VMIE II UUIO oMM. o 9 o/ 9 .7. . IE++R0 9 YP1 F. . UH . .. "MU. Cit oOIoIZ DVOAAERO 99Y M! MI W 9 s. (INC 6 o o OIIOIV VHOIZ‘R o 1.1 v 13...... 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E = = m _ I .100 A21 TVV EHH . . rin nXUO FEE 1%03 BnXu 40;. 9oz. 0.0 213 .= .. = 1 6.0000EIUZ' 4.74365 00 ""*2.9200E”OI ”“"” 1.00006 00 1: 8.00005302” EEE 506 905 800 809 a a a 2.15 = = = .1 laOOOOE‘Ol 11:1 000 EEE 030 601 067 o O O 1.44 .z = a 1.00 21 En!“ .100 000 EEE 601 805 505 800 a a s 22;. .sga . l Rr+u . R1. R0 EES 1.2000EIUI 1.11 000 EEE 221.. 957 204 413 0 o 0 .144 100 A21 TVV EHH 1.4000E-01 1:51 000 EEE 1.6000E-01'” 01:1 nXUO EEE 491 182 310 279 a s e w!44 . IAUD EHH 1.8000E-01 .101. O O O 23.1 ._= a = . 1 RT... rQKI TRD 2.0000E30T"'“ .111 000 EEC... 0.14 752 471 ,o o o .134 2.2000E;Ol 2.40005&01‘" 01:1 000 EFE. .137 348 136 a a o 744 ... .. z 100 A21 EH“ 2.6000E-01 2000 DIMENSION xv1100001. vv1100001. BUFFERIIOOO) AHHmmEXII PROGRAM USED IN OPTIMAL DIGITAL CONTROL SYSTEM BNMH)ON(HEENW3TEEWMQIAPPMMKH I-v—w—wuvuv- CALL PLUTSIBUFFERIII9 4000) CALL PLUT(0909‘12909 3) CALL PLCT(2.09-II.592) EXTERNAL EVAL9 CUT DIMENSION P(519AUXIB. 21,v121.07121 [=1 CUMNUN K9U9F19H COMMON/AREAI/XYIYY91 U308 KLOCP=1 K=99 N=2 P(l)=0. P(3)=.000002 pIZI=20 PI4I=10 Y11I=600 Y121=600 0Y11131a 0" 21:30.0 HRITE(259ZOCOI FURNATI6X9'T'12X'VV' lZX'hV'IZX'U'IZX' FI' lZX‘W'lZX'IB CALL RKGS (P9Y90Y9N9 IBIS EVAL9QUI9 AUXI IFIYIII. LE. 0.11I90 I CC T0 5 KLOCP= KLO0P+1 IFIKLDOP-ISI 29595 CALL SCALE(YY95.O9IOOOO9I910.O) CALL SCALEIXY95909100009191090) CALL AXISIQ.O9O.09'Y'91969Q190909YYIIOQOII9YYIIOOOZI CALL AXISIO. 090.09'X'9 I96.090.09XY(1000119XY110002) CALL LINEIXY9YY9IOOOO9I909OI CALL PLCTC09090.09999) STOP END 136 137 a—.—-— -- w- A 11 SUBROUTINE EVAL 1t.v.ov1 DIMENSION YIZI9DYI2I9PISI COMMON K.U,F1!H u=.225*ExP1.225t1v121/Y1I1-1.11-23.5*Ex§123.5*1v121/v F1=EXP(-.225)‘EXP(.225#YIZIIYIIII-EXPI-23.51*EXP¢23,5 v=u oYflI=3323i?I"*ww DYIZI=1210.#Fl-1584.*V RETURN, END -1.‘ -.—..¢-~.‘ oo—m”. ‘--. 0.. 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A more complete treatment may be found in Flanders [F7]. The objects that occur under integral signs are called exterior differential forms. For example, the line integral, surface integral, and volume integral lead to the following differential forms in 3- dimensional Euclidean space: w - A dx + B dy + C dz (one-form) (A2.1) u = P dydz + Q dzdx + R dxdy (two-form) (A2.2) A = H dxdydz (three-form) (A2.3) In the n-dimensional space, the quantities are called r-forms in n variables. A2.2 Exterior Algebra In the algebra of differential forms the operations of addition and multiplication obey the usual associative and distributive laws. Multiplication, however, is not commutative but anticommutative, i.e., s - A2. dxi A dxj clxj Adxi ( 4) The exterior product is sometimes called the wedge product. Often the product symbol A is omitted. Hence, 8 - 2 9 4 141 142 In the two-form given (A2.2), use of (A2.4) eliminates terms like dzdy. The exterior product has the following properties: (1) (w+C)NH * (wAn) + (CAn) (A2.5) (2) (cw)AC = C(wAC) (A2.6) (3)1 Q A w = (-l)rsw A C , (A2.7) if m has degree r and g has degree 3 (4) (CAw)An = CA(wAn) (A2.8) A2.3 The Exterior Derivative The exterior derivative of a p-form w is a (p+l)-form dm obtained by applying an operator d to transform w to dw. For example if m is a three—form in four variables w = X wijk dxi A dxj A dxk (A2.9) i Where wijk is a function of x1, x2, x3, and x4 and is assumed to be differentiable. This definition is readily generalized. The exterior differential has the following properties: (1) d(w+n) = du1+ dn (A2.12) (2) d(umn) = dm A n + (-1)r w A dn, (A2.13) if m is an r-form and nis an s-form . r}: l—‘_ 1v 143 (3) d(dw) = 0 (1.12.111) w and n are assumed to be differentiable. Property 3 is called the Poincare' lemma. It implies the equality of mixed second partial derivatives. The general case is proved by induction. For simplicity only the O-form is 3 variables is considered, w 8 f(x) (A2.15) Then there results the l-form dm=-8-f-dx+-§-£dy+-3—f-dz (2.16) 3X By 32 Then d d = 8f + 3f + 3f 1 ( w) d(‘a"i')AdX d('57-)A dy d(§-5-)A (12 (A2. 7) Carrying out the differentiation, and using the properties of exterior multiplication, d(dm) = 0 (A2.18) In 3-space, the Poincare' lemma d(dw) = 0 interprets as curl (grad f) = O (A2.19) div (curl V) = 0 (A2.20) A2.4 Integration of Forms The primary purpose of this section is to present the n-dimensional Green's Theorem, also called the n-dimensional Stoke's Theorem. What the classical theorems state for curves and surfaces, these theorems state for the higher-dimensional analogs called manifolds. An n—dimensional manifold consists of a space M and a collection of local coordinates neighborhoods N1, N ... such that each point of 2’ M lies in at least one of the neighborhoods. Whereas, an n-dimensional 144 manifold may not be a Euclidean space, it appears to be Euclidean to a short-sighted observer in the manifold. The proof of the n—dimensional Green's Theorem is simplified if the concepts of chains and Euclidean simplices are introduced. This is done to eliminate the need to chop up manifolds into small pieces. Instead of working with manifolds where things are more difficult, Euclidean spaces may be used where things are relatively more simple. Euclidean simplices are defined as follows: A O-Simplex is a single point (p0). A l-simplex is a directed closed segment on a straight line. It is completely determined by its ordered pair of vertices (P0, Pl)' A 2-simplex is a closed triangle with vertices taken in some definite order. It is determined by the ordered triple (P0, P1, P2). A 3-simplex is similarly the ordered quadruple (P0, P1, Pl, P3). In general, an n-simplex is the closed convex hull (P0, ..., Pn) of (n+1) independent points taken in a definite order. Independent points means that the n vectors (Pl-PO), (PZ-PO), ... (Pn_P0) are linearly in- dependent. The convexity condition implies that the n-simplex is the set of points. P - toPo + ... + tnPn c130. 1 t1 = 1 (A2.21) The boundary 38 of a simplex S is a formal sum of one lower dimension with integer coefficients defined as follows: n a (P0, P1, ..., Pn) - 2 (-l)i(PU, ...y _ ., Pn) (A2.22) i=0 145 For example, the 3-simplex is bounded by four faces, i.e. 3(P09 P1, P2: P39) ' (P19 P29 P3)'(P09 P29 P3)+(P09 P]: P3)- (P0, P1, P2) (A2.23) The terms having positive signs correspond to orientations wnich may be associated with an outward normal if the points are traversed in a counter clockwise direction. See Figure A2-l. Figure A 2.1 3—Simplex with Orientation An n-chain is a formal sum 0 = Z 3181 (A2.24) where a1 are constants and Si are n-simplices. The boundary of the chain is defined as ac - X a1 as1 (A2.25) As a result, the boundary of each chain has zero boundary. 3(30) - 0 (A2.26) For example consider the boundary of the 2-simplexS where S = (PU. P1. P2) (A2.27) 146 Then BS = B(PO, P1, P2) = (P1’ P2)-(PO, P2)+(PO, Pl) (A2.28) and 3(38) = (P2-P1)-(P2-PO)+(Pl—PO) (A2.29) Hence 3(38) - 0 (A2.30) and 3(ac) = 0 (A2.31) It is convenient to have standard models of the simplices. The standard n-simplex is defined as -n s A (R , R ) (A2.32) 0’ " n The points R , ..., R in n-dimensional space are taken as 0 n R0 = (0 ... 0) R1 = (10 ... 0) R2 = (010 ... O) Rn = (00 ... 01) Integration of a n-form defined on a domain N of En which includes n S is written as I _n w =Js‘“ A(x1, ..., xn) dxldxz...dxn (A2.33) S The right side is standard ordinary n—fold integration over the standard n—simplex. Since we wish to integrate a n-form on a manifold M, it is necessary to relate the standard n-simplex to the n-simplex in M (denoted by on). 147 Hence 91.9 o s (112.34) where ¢ is a smooth mapping of the neighborhood N of s"n into M. It can be shown that I m = I d... (A2.35) 30 0 Also since C = Z aidin’ J m ' I dw so 0 (A2.36) This is Stoke's Theorem in its most general form. Recall that C is a chain and 3C is its boundary. APPENDIX IV ALTERNATE DIGITAL PROGRAM ¥**#§0N71Nuous system MODELING PROGRAM9999 #**PROBLEM INPUT STAIEMENTS*“ TITLE FUNCT FUNCT nstwoc PARAH TIMER FINISH PRINT ASKC MUET1=(-9159-1919(C9909039(9075996)9(915919)9(957 MUET2=Cf9159-19’9(09909019(9075996191915919’919571 RKSFX 531098310 HUI=A*AFGEN(MUEIIQETA1’ HU2=B¥AFCEN1HUET29ETA21 VVDOI=-89*(HU1*MUZ*A+B, VV‘INTGRL‘60901VVDUT’ VVD=VV SDISTSINTGRL(0909VV’ HV1=INTGRL(60909HVIDUT) HVICOT=12109‘MU1'C979*11 HVZDOT312100*MUZ'0079*12 HVZ=INTGRL(6O909HVZDOT’ HV11=LIHIT(09960909HV1’ HVIC=DELAY(1990109hV11. 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"M PRTPLT 5115(59969‘OTIH595RRIN1 PRTPLT 5115‘59969’9TIH59WV20 ' PRINT ETAIR9HU1vERRINloERROTl9HVIDDVV09VV9919SDlST END 150 999PROBLEH INPUT. ,S.TATE"ENT$*!$ TITLE BRAKI_NG DYNAMIC§ FUNCT HUETI'I-9159‘I919IO9909019I907599619 (915919 19I95799919 METHOD RKSFX ”3 , “ pARA" A18199A2=99n9A3=999A4:19W PARAM HA8156.9A35318§g331Vfi13g693553991HR§1,3159H131250.,H2= HUI 3A1*.A.FGENI_H.UET1.9.5111.L. HUZ‘AZ‘AFGENIHUETI9ETAZ1 1W3 T” A 39AE§§ALEU§LLI 5.1.5.31..- .... .. HU43A4‘AFGENINUETI9ETA§1 UDOT‘V‘W‘I 1.0/HAL?! 31*321’3313’11” _ USINTGRLI6099UDOT1 VDOTz-UtR-IILKNAJEIL131.- 251311;.“ V8INTGRLIO99VDUT1 RDOT= 31.11v191393339131-3_1;1+121+9.131 3 32-33113 R3INTGRLI099RDUT1 ALPH1=IV+AtR1[UJ3 ALPHZ’IV‘A‘R1/U2 ALPH3=IV-8*R1/U3 ALPH43IV'B‘R1/U4 U13U7H*R U23U+HfiR U3=U+H¥R U43U'H*R SDISTalNTGRLIO. 0,31 HVIDUT=BI/IHR- -O979*TI HVZDOT=BZIIHB?.O.919*TZA . .-M--. --. 3...... - HV3DOT=B3IIHR- -O9 79‘T3 HV4DOT=B4IIHR~O979*T4 HVI‘INTGRLI6O99HVIDOT1 HVZslNTGRL16091Hy200T1 HV3=INTGRLI6099HV3DUT1 WW»: I NTGRLIégul‘XfiDOT 1. HVII=LIHITIO996099WVI1 HV228L1M1T10996039HV21 HV33=LIHITIO996099WV31 HVID’DELAYII990209 HVII1 HVZDsoELAY(1990201HV721 HV3D8DELAYII990209HV331 HV4D=DELAYII99QZO9 HV441 FTAItl9-HVII/Ul £112=1.-wv22/uz3 ETA3'l9-HV33/U3 6 IA 1,. 1 ...- av 44.1.1115...‘ TI’I966‘PII T231966*PZZ -.--v-w-9r,w-.9.n—n —-.-.. .. ..--— ‘9. I...” .---.... .1 _ .9,‘ .— ‘.. ~ , - ~~~ - —- --———-o-v «......9 .- - ----—..~—.—...---H— _4» . .— FINISH PRINT 'aéinuztuatc0577.§3tALpH3b 151 T381.66*P33 74:1.66§P§5#-- . VLWM*_MJ“JW_V Bl=HUl*Hl*CUS(1.85*ALPHlD ez=nu2¢w2*cos¢7.35tALPH27 842MU4‘H4*QDST7.85*AgPflfjfiu L18HU1*H1*SIN(1.85*ALPH1) L28HU2*HZ*S!N(7.85*ALPH2T L3=HU3*H3#SIN(7. aStALPHQT“ LA: MU4*H4}§1N17. 85*ALPH4T _m~m__m“M*m ETA1R=O. Stu ETAlAsu-.zsttuv10+uvzo+uv30+HVAob v1=0€RtVt0..ETA1Al ERR!N1=(ETA1A+.15*YlT-ETA1R E1=1500023§BBQIJ P18!N76RL¢0..X1) P2891 p3-Pl P4=P1 x1=x11#61 x11=IOR(Ylll 7112! Y111=ANDIY11.P1T Y112=AND(EloY12) v11=N07(51)’" Y12=NOT(Pl-IZOO.T 911=DELAVCI..020.PID PZZsPll P33=P11 944:911 HFADsINTGRLIO4oR) YDOT=U*COS(HEADI-V*SIN(HEADD x007=u¢SINQNEADI+v¢c05(HSADT Y=INTGRL(0..Y007) x=INTGRL(OJ;iUfiTT TIMER FINth-A..OEL7=.oooz.DELMINa.000000001.PRDEL U310 .._- . v ....a-v.-....--o —. . . -_.. , .. .....n 9 ~ -_-~‘.' W- -... .... . _ .. .HJ7. . o . - --u..-L-~—— ---—”*7. >7. ‘— .4..~-_-_. ....-. ._ _ "‘~I.u.-~ .... ~.—‘ ---. —'.-a.- MULQHVLoHVZvHV3yHV4gU9V9RQPI'YQXQHEAD PRTPLT MUlgHVlqHVZoHV3oHV4vUoVoR951gMUZ.MU3,MU4 END PARAM RESET LABEL END PARAM RESET LABEL END STOP .‘mu-- ..M-... A1=1..A2=.7,A33;7.Aéhrf“ LABEL 70 PERCENT MU AglogA2'059A3305QA4310 LABEL --.-a-;-—---..—.. 50 PERCENT MU ON ONE S [DE A1 B1 82 B3 B4 BS B6 B7 BB B9 BIBLIOGRAPHY Athans, M., and P. L. Falb: thimal Control: An Introduction to the Theory and Its Applications, McGraw-Hill, New York, 1966. Babuska, 1., M. Prager, and E. Vitaselk: Numerical Processes in Differential Equations, Interscience Publishers, Prague, 1966. Bellman, R. E., and R. E. Kalaba: Modern Analytic and Computational Methods in Science and Mathematics, American Elsevier Publishing Company, Inc., New York, 1965. Bellman, R. E., and H. H. Kagivada, and R. E. 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