VIINIMAX STUDIES y W. A. Glusser, K. D. Grdihmz, und C. A. Huwey ;repared by ;ONEYWELL, INC. t. Paul, Minn. w George C. Marshall Space Flight Center ATIONAL AERONAUTICS AND SPACE ADMINISTRATION l WASHINGTON, D. C. . AUGUST 1966 https://ntrs.nasa.gov/search.jsp?R=19660022818 2020-03-16T21:14:35+00:00Z - -*- TECH LIBRARY KAFB. NM 0079420 NASA CR- 547 MINIMAX STUDIES By W. A. Glasser, K. D. Graham, and C. A. Harvey Distribution of this report is provided in the interest of information exchange. Responsibility for the contents resides in the author or organization that prepared it. Prepared under Contract No. NAS 8-11206 by HONEYWELL, INC. St. Paul, Minn. for George C. Marshall Space Flight Center NATIONAL AERONAUTICS AND SPACE ADMINISTRATION For sole by the Cleoringhouse for Federal Scientific ond Technical Information Springfield, Virginia 22151 - Price $4.00 ABSTRACT Two brief studies of the effect of non-zero initial conditions on the per- formance according to the minimax criterion and on the selection of minimax controllers from a given set of controllers are reported. The results of two studies of extremal bounded amplitude, bounded rate inputs to linear systems are also reported. The first study of the effect of non-zero initial conditions considers one flight condition for a vehicle of the Saturn V Type with first order gimbal dynamics. The control configuration has pitch rate, lagged pitch attitude and normal acceleration feedbacks. Each of the optimal controllers had one positive pole, one negative pole and a stable complex pair of poles. The positive pole is small and its magnitude decreases with increasing magnitude of initial conditions. The second study of the effect of non-zero initial conditions considers two flight conditions for “Model Vehicle Number 2 for Advanced Control Studies” with no gimbal dynamics. The control configuration has pitch attitude, pitch rate and lateral velocity feedbacks. The optimal gains are found to be monotone functions of the magnitude of initial conditions. Further, the stability of the optimal system tends to increase with increasing magnitude of initial conditions. The first study of extremal inputs is restricted to an oscillator. The theoretical development indicates the relation between several sets of necessary conditions and one sufficient condition. One set of necessary conditions is shown to be sufficient and from these conditions general explicit formulas for extremal inputs are derived. The last study pertains to the development of computational algorithms for extremal inputs for general linear stationary systems. Two algorithms are presented, and an example of computer results obtained from one algorithm is given. ii FOREWORD This document partially comprises the final report prepared by Honeywell, Incorporated for George C. Marshall Space Flight Center, Huntsville, Alabama, 35812 under Contract NAS 8-11206. The application of optimal (minimax) control theory to a piecewise constant approximation of a large launch booster for the first 84 seconds of flight is presented in NASA CR-546. A linear piecewise constant controller is determined which minimizes the maximum of several cost items. The work on this contract was supervised by Mr. C. R. Stone and Dr. E. R. Rang. Section 2 was prepared by Mr. W. A. Glasser. Section 3 was prepared by Mr. K. D. Graham. Sections 4 and 5 were prepared by Dr. C. A. Harvey. Dr. J. Y. S. Luh contributed to the results of Section 4. The linear programming formulation presented in Section 5 was developed by Dr. P. Treuenfels. iii TABLE OF CONTENTS Page SECTION 1 INTRODUCTION 1 Non-Zero Initial Conditions 1 Saturn V Study 1 Model Vehicle Number 2 Study 2 Theore tical Developments 5 Computational Algorithms 5 SECTION 2 MINIMUM WIND EFFECT CONTROL OF A SATURN 6 V LAUNCH VEHICLE WITH NON-ZERO INITIAL CONDITIONS System Representation 7 Analog Computation 14 Digital Computation 17 Simulation Results 18 Conclusions 33 References 34 SECTION 3 THE EFFECT OF NON-ZERO INITIAL CONDITIONS ON SELECTION OF MINIMAX CONTROLLERS Problem Summary Choice of Intervals Choice of Initial Conditions (I. C. ‘s) and Gains Results of Minimax Computations Identification of Controllers Costs, Gain Changes, and Closed Loop Pole Interval I3 Interval I5 Magnitude of Initial Conditions Conclusions SECTION 4 EXTREMAL BOUNDED-AMPLITUDE, BOUNDED- RATE INPUTS FOR A HARMONIC OSCILLATOR 55 Applicable Theory 57 Definition 1 60 Definition 2 61 Definition 3 61 Definition 4 61 Theorem 1 61 Theorem 2 61 Theorem 3 62 Theorem 4 62 35 35 35 36 37 40 40 40 46 51 52 V TABLE OF CONTENTS (continued) Page Derivation of Extremals 71 Explicit Representation of Extremals 76 Conclusions 85 References 87 SECTION 5 EXTREMAL BOUNDED-AMPLITUDE BOUNDED RATE INPUTS 88 Necessary and Sufficient Conditions 90 Formulation of a Computational Algorithm 95 Computer Results 98 Summary 100 vi - ILLUSTRATIONS Figure 4 5 6 7 8 9 10 11 12 Analog Computation of the Performance Index C(U) for Nonze ro Initial Conditions Change in Gains with a Change in Initial Conditions Worst Disturbance Responses of Optimally Controlled System with Zero Initial Conditions Worst Disturbance Responses of Optimally Controlled System with @ = 1 and 2 degrees Worst Disturbance Responses of Optimally Controlled System with Q = 3 and 4 degrees Worst Disturbance Responses of Optimally Controlled System with @ = 0. 5” /set and 1.0” /set Worst Disturbance Response of Optimally Controlled System with @ = 1. 5” /set and @ = 2” /set Worst Disturbance Responses of Optimally Controlled System with k = 1 m/set and 2 m/set Worst Disturbance Responses. of Optimally Controlled System with i = 3 m/set and z = 4 m/set Worst Disturbance R.esponses of Optimally Controlled System with fi = 0. 5 and 1 degree Worst Di.sturbance Responses of Optimally Controlled System with fl= 1.5 and 2 degrees Extremal Input and Approximations Obtained Page 16 23 24 25 26 27 28 29 30 31 32 38 vii - . . .._ SECTION 1 INTRODUCTION The need to design controllers for large launch boosters provides the moti- vation for the minimax studies. Four areas of investigation are discussed. The first two studies are concerned with the effect of non-zero initial condi- tions on the selection of minimax controllers. The remaining studies are aimed at theoretical developments which are necessary for the inclusion of a bound on the time rate of change of the disturbance in the minimax prob- lem statement. The inclusion of such a constraint would yield a closer approximation to disturbances which are encountered in practice. NON- ZERO INITIAL CONDITIONS The purpose for studies in Sections 2 and 3 was to examine the effect of non-zero initial conditions on control cost (performance index) and on selection of minimax controllers for large launch boosters. In both sec- tions, rigid vehicles with linear controllers and bounded amplitude winds are assumed. Saturn V Study The vehicle in Section 2 is a typical Saturn V booster with first order gimbal dynamics for a ten-second flight condition characterized by maxi- mum dynamic pressure and Mach number of about 1. 7. Gains for a good controller for this vehicle with zero initial conditions were known from work on NASA Contract NASw-563 (Honeywell MPG Report 1541-TR 14). This controller had pitch rate, lagged pitch attitude and normal accelera- tion feedbacks. Four values of initial conditions were chosen on each state variable (pitch attitude, pitch rate, lateral velocity, and gimbal angle). The gain grid chosen represented 54 different controllers and contained the controller for zero-initial conditions. The wind velocity had a magnitude of 75 meters per second. It was found that a set of only four controllers minimized the control cost for all of the 16 initial conditions. In particular, the controller for zero initial conditions was also best for small values of initial conditions on pitch attitude, pitch rate, gimbal angle, and all values of lateral velocity consi- dered. The four best controllers all had one positive and one negative real pole, and a stable complex pair of poles. The positive pole was small and its magnitude decreased with increasing magnitude of initial conditions. The closed loop natural frequency and the damping ratio of the complex pair decreased with increasing amplitude of initial conditions. The ranges of values of the positive pole, natural frequency, and damping ratio of the four best controllers are as follows: 0.001738 < real pole < 0.005408 - - 0.780 cps C f < 0. 898 cps - - 0.201 5 5 C 0.063 - Model Vehicle Number 2 Study Data for the vehicle in Section 3 is taken from the data package “Model Vehicle Number 2 for Advanced Control Studies” and perfect gimbal dynamics were assumed. A cost item corresponding to bending moment was included in this study. Two flight conditions were considered: (1) one was sixteen seconds long near Mach 0. 55 with dynamic pressure about one-third of maximum; and (2) the second was eight seconds long at Mach 1 with about eight-tenths maximum dynamic pressure. The controllers had pitch attitude, pitch rate, and lateral velocity feedback gains with the wind disturbance intro- duced in such a manner that the gains could easily be converted to equiva- lent ones for controllers with pitch attitude, pitch rate, and either normal acceleration or attack angle feedback signals. Good controllers for zero initial conditions were known for both flight condi- tions from Honeywell Report 12003-FTRl. Each had relatively high gains and all real poles with one of them positive. The positive pole was small for the first flight condition (real pole at 0.00055) and large for the second one (real pole at 0. 30809). These particular flight conditions were selected because it was expected that the influence of initial conditions would be comparatively large with higher controller gains, and particularly so with the controller having the large positive pole. Three iterations of cost computations were performed. A total of 125 con- trollers was included in each gain grid. Three values of non-zero initial conditions were chosen for pitch attitude, pitch rate, and lateral velocity, with a range of four to one between the minimum and maximum values in each case. The wind velocity was 59 meters per second in the first flight condition and 75 meters per second in the second one. The results for the first flight condition are generally summarized as follows: 0 All gains are monotone non-decreasing/non-increasing with the magnitude of any initial conditions 0 All minimax controllers have a negative real and a stable com- plex pair of closed loop poles 0 The real pole is much closer to the origin than the complex pair and its distance from the origin decreases with increasing magni- tude of initial conditions 3 0 The damping ratio and natural frequency of the complex pair increase slightly with increasing magnitude of initial conditions. 0 The range of values of the real pole Z. is -0.00115 z. < -0.00018. - 0 The range of values of c’ ando, of the complex poles is 0. 64 < c< 0. 87 - - and 0.071< tin cps C 0.094. - - 0 It is possible to select one fixed gain controller which gives good performance for each initial condition. 0 One initial condition, the maximum value of the initial condition on lateral velocity considered, must be excepted for several of the above cone lusi ons. However, this initial condition appears to be larger than need be considered, so its exception is not serious. The conclusions for the second flight condition are similar: 0 All gains are monotone non-decreasing/non-increasing with the magnitude of initial conditions on the state variables. 0 All minimax controllers have real poles and one of them is posi- tive. 0 The distance of the positive poles from the origin decreases with increasing magnitude of initial conditions (0.041 < z. <_ 0. 3291). - 0 It is possible to select one fixed gain controller which gives good performance for each initial condition. 0 One initial condition, the maximum of the initial condition on pitch attitude considered, must be excepted for the third and fourth COP- clus ions. THEORETICAL DEVELOPMENTS The purpose of section 4 is to present theoretical developments applicable to the minimax control problem with a bound on the time derivative of the disturbance. The extremal inputs may be thought of as worst disturbances to the system and the desired result of this study is a means of character- izing such inputs. The discussion is restricted to an oscillator so that explicit results are achieved. The oscillator is general, however, in the sense that extremal inputs may have an arbitrary number of segments on which the input is at its extreme. The theoretical development presented for the oscillator can be generalized. The results of such a generalization are presented in section 5. The discussion of the theory associated with the oscillator indicates the relationship between the necessary conditions for extremal inputs obtained by Gamkrelidze, Bryson, Denham and Dreyfus, and Russell and Schmaedeke, and the sufficient conditions obtained by Russell. The necessary conditions of Russell and Schmaedeke are shown to be sufficient. These conditions are used to determine general explicit formulas for extremal inputs. COMPUTATIONAL ALGORITHMS The purpose of section 5 is to develop computational algorithms which may be used to determine extremal inputs for general linear stationary systems. Necessary and sufficient conditions for extremal inputs are presented. A computational algorithm is formulated based on these conditions. Also a linear programming formulation of an approximation to the problem is given, Results of a computer program developed from this last formulation are presented for an example. 5 SECTION 2 MINIMUM WIND EFFECT CONTROL OF A SATURN V LAUNCH VEHICLE WITH NON-ZERO INITIAL CONDITIONS A natural criterion for launch booster control is the minimum wind effect criterion. Small errors in the system response do not degrade performance. Hence, there is no reason for saying that control performance is not optimum if the controller permits small errors. The major concern is the maximum value that error components attain over the entire launch trajectory. Hence, a desirable control criterion is one that rates controllers (in terms of a performance index) according to their capabilities for holding the maximum normalized error component to a minimum over the launch interval. The synthesis of such a controller presents a formidable task. First, it must be assumed that the launch vehicle can be adequately described over the portion of the launch trajectory of interest by a set of linear, constant- coefficient differential equations. A second and less restrictive assumption is that the controller is linear fixed-gain. Further, it is assumed that the wind disturbance is bounded by a known maximum speed. Under these assumptions, a minimum wind effect controller is synthesized for a Saturn V launch vehicle with non-zero initial conditions. Given the launch vehicle data for the maximum dynamic pressure flight condition, a controller is synthesized which minimizes maximum weighted error components over a fixed time interval with worst disturbances within a given class of bounded amplitude disturbances and a specified vehicle initial condition. The resulting controller is a linear, fixed-gain feedback controller whose optimal gains are a function of vehicle initial conditions. The linear representation of the longitudinal rigid-body body equations of motion of a Saturn V launch booster is chosen to illustrate the synthesis technique of a minimum wind-effect controller for a linear stationary system with non-zero initial conditions and amplitude bounded disturbances. The vehicle data is that for the maximum dynamic pressure flight condition. The synthesis procedure selects the controller gains such that a specified performance index will be minimized for a given disturbance and vehicle initial condition. This results in the need to integrate a system of first- order, piecewise linear, autonomous, ordinary differential equations. The computation may be readily accomplished with the use of either an analog or digital computer. The numerical results indicate that the optimal gains (which minimize the performance index) are a function of the vehicle initial conditions. Furthermore, the vehicle has an unstable closed-loop pole for certain optimal gains. Having determined the optimal gains for a given vehicle initial condition, a linear fixed-gain controller which minimizes the per- formance index for the specified initial condition is determined. SYSTEM REPRESENTATION To illustrate the synthesis of a minimum wind effect controller for a linear stationary system with non-zero initial conditions and amplitude bounded disturbances, a linear representation of the longitudinal rigid body equations of motion of a Saturn V launch vehicle is considered (reference 1). The assumed equations of motion are: ‘$ = -Cl” - 9 . . Z = YlQ+ Yz@+Y3B (1) Q = @ + (VW - g,/v The control equation is: 4 +P = K3 {8 + K29 +[KlYl - Cl7 - ClKl (CM - C&a + +[K1y3 - c-27 - cp1 (Cn/r - c,)l PI Introducing xl = @ (attitude angle); x2 = i (attitude rate); x3 = * z (displacement rate of center of gravity); and x4 = p (gimbal motor defection angle) yields the following set of closed loop equations: . x1 . x2 . x3 . x4 L where: 0 1 0 0 -c1 0 Cl/V -c2 y1 -I- Y2 0 -YJv Y3 kl k2 k3 k4 + 0 -cl/ YJV -k3 V kl = {Kg/~) 1K2 + Klyl - Cl [T+ Kl (CM - c,)]] k2 = K3/7 k3 = (-K3/7v) iKIYl - Cl CT + Kl (CM - c,)l] k4 = {K3 6KlY3 - c27 - C2KJ(CM - c,)] - d/T gw (2) (3) - .- Equivalently (2) may be written as: . x = AQX + CR g(t) The open loop set of equations is: where : . x1 . x2 . x3 . x4 = . 0 -c1 Yl +Y2 0 1 0 0 0 q/v -5 0 -YJV Y3 0 0 0 x1 x2 x3 x4 + Ll+ 0 -cl/v Yl/V 0 g(t) (4) u = klxl + k2x2 + k3x3 + k4x4 - kg g(t) The four real parameters kl, k2, kg, and k4 may be thought of as pseudo- gains. However, these parameters must be constrained so that the solutions (of the defining equations for kl, k2, kg, k4) for the gains, time constant and accelerometer location are physically realizable. In order for the time con- stant 7 to be real it is necessary that: [(k4Cl + vk3C2) (CM - Cc) - k4Y1 - vk3Y312 - 4k2 (C1y3 - C2Y1) q P+ - c(g - Y113 0 (5) 9 Denoting by b3 and bg the minimum and maximum values respectively of cM - CG such that CM represents an accelerometer location on the vehicle, it is possible to express the constraint on the k’s in the form: E(k) fl [bg, b9 3 # 4 where : E(k) = iA: [(k4Cl + vk3C2)X - k4y1 - vk3y3 I2 ’ 4k2 (ClY3 - - C2Y4 (C,X - Y,)3 This constraint is just a mathematical way of stating that the k’s must be chosen so that there is some accelerometer location on the vehicle for which the corresponding value of 7 is real. For a set of acceptable gains, the con- trol law is given by: u = kl@ + k2i + k3; + k4/3 - kg g(t) (6) The data used represents that for a typical Saturn class launch booster. Units for the data are meters, radians, and seconds. T = 10, Cl = -0.2165, c2 = 1.1381, Yl + y2 = 27.66, yl/v = 0.0133, y 3 = 17. 65, v = 507. The control criterion (performance index) is defined as: c (u) = max Ci(u) o-=i u> g) I Wit< T gEG -- 10 for each u in the class U, with x(t, u, g) denoting the solution of (2) and di representing a non-zero constant weighting vector for i = 1, 2,. . . . , S where S is a positive integer. A controller is said to be optimal in case it is an element of U which minimizes C(u). For the present problem, it is possible to write the performance index (reference 2) as: Ci(U) = max C Ixi(t) I + C(i(t) 1 (7) tct-0, Tl The functions Xi(t) and Y(t) may be obtained as solutions of sets of piece- wise linear autonomous differential equations. The term k(t) may be expressed as: Xi(t) = di . e *Qt x” (8) . which can be obtained from the solution of the linear system x = AQx with x(0) = x”, For the example being considered, this results in the system of equations given by: . x1 . x2 . x3 . x4 = 0 1 0 0 -c1 0 CJV -5 Y1+Y2 0 -YJV Y3 kl k2 k3 k4 x1 x2 x3 x4 ,x” = @ 0 i0 . Z 0 PO . (9) 11 Forming the inner product. d1 . e AQt x0 yields: (10) If the di 2.x-e chosen such that: dt = ? Oifj # i d;= * di if j = i Then the system of equations given by (10) reduces to: (11) z = x3/ d3 If @o=+o=io= po = 0, the solution of ( 9 ) is identically zero. Consequently, the ix,1 Is indicate the contribution to the cost created when the initial condi- tions for the vehicle are non-zero with no disturbance present. 12 - The second function Y(t) may be expressed as: t Pi = S di . e AQct- 7) CR Yi(T)d’ 0 (12) with Yi(t) given by: YiCt) = (Vw)max sgn [di. e AQ( t- 7) 1 (13) which represents a worst disturbance condition. Substitution of expression (13) into (12) yields: t Pi = (Vw)max s Idi - e AQT c&T 0 (14) To sim bl lify, notation (Vw)max will be replaced by VW in all that follows. The term e QT CR may be determined by solving the set of differential equations: . z1 z2 Z 3 z4 = 0 1 0 0 -c1 0 CJV -57 Yl+Y2 0 -YJV Y3 kl k2 k3 k4 z1 z2 z3 z4 > z(0) = 0 -cJv Q/V -k3 (15) The p*s may be thought of as representing the costs induced by the disturbance for zero vehicle initial conditions. 13 ANALOG COMPUTATION The computation necessary to determine the performance index, C(u), is readily accomplished using an analog computer. For purposes of scaling, the system of equations given by (4) was rewritten as: . Yl . y2 . y3 . y4 = 0 10 0 -c1 0 c1 -5 (Yl+Y$ o -Y1 Y3 -- V V V kl k2 k3v k4 Yl y2 y3 y4 + 0 -c1 YIIV -k3v g(t) V (16) where : Yl . . = @, y2 = @, y3 = z/v, and y4 = P. This was necessitated by the small numerical value of kg (approximately 0.00049) which was optimal. The product k3v is approximately equal to -0. 249 to which the analog potentiometers may be readily adjusted. Accordingly, equations ( 9 ) and (15) are modified. Since the disturbance is normalized with respect to the vehicle velocity, v, the expression for pi now becomes: (17) The expressions e AQt x” and e AQt CR are each evaluated using four integra- tors and the necessary summing and inverting circuits. The inner products 14 are easily formed since the following set of weighting vectors, d’, for this example: d1 = l/2. 3: 0 0 0 ; d2 = . - 0 1 0 0 II ; d3 = ; d4 = 0 0 l/523 0 - 0 0 0 l/ 1. 33 The weighting vectors were selected by determining the maximum value of is used the transient response of each of the parameters for a similar launch vehicle to a disturbance input and then normalizing such that dlQmax = d2imax = d3zmax = d4p max’ The absolute values were formed using two diodes and two summing ampli- fiers. A complete wiring diagram for the analog computer is shown in Figure 1. The computer is scaled such that ten volts equals one degree or one degree per second. Since the performance index as determined from the analog computation is in degrees as opposed to radians for the digital computation, one must convert degrees to radians or vice versa for comparison purposes. The primary use of the analog computer was that of observing the system response for a given initial condition and corresponding optimal gain set. Furthermore, it gave a convenient check of the digital computation results. 15 Ir Figure 1. Analog Computation of the Performance Index C(U) for Nonzero Initial Conditions DIGITAL COMPUTATION A program was written for the Honeywell H- 1800 digital computer to evaluate C(u) by solving the systems of equations described by ( 9 ), (12), and (15). Initially it was assumed that the set of minimizing gains for non-zero initial conditions would be close to the optimal gains for zero initial conditions. Consequently, the refined grid of Example 3 of Reference 1 was chosen to minimize C(u) for non-zero initial conditions. This gain grid consisted of the following set of gains (Gain Grid II): kl = 0.401765, 0.602647, 0. 802647 k2 = 0. 471405, 0.942809, 1.414214 kg = -0. 00049 105, -0. 00036828 k4 = -0.353553, -0. 707107, -1.060066 To more closely observe the dependence of the gains on the vehicle initial conditions, another gain grid refinement was made (Gain Grid III): kl = 0.401765, 0. 502207, 0. 602647 k2 = 1. 178511, 1.414214, 1. 649916 kg = -0.00049105, -0.00042977 k4 = -0. 530330, -0. 707107, -0. 883586 All possible combinations of gains were taken resulting in a total of 54 gain sets. These were conveniently numbered 1 through 54 and consequently any reference to a particular gain set number is only significant with respect to the manner in which the combinations were ordered. 17 The performance index was minimized for the following set of non-zero initial conditions: @O = 1, 2, 3, and 4 deg i. = 0. 5, 1.0, 1. 5 and 2.0 deg/sec . Z 0 = 1, 2, 3, and 4 m/set PO = 0. 5, 1.0, 1. 5, and 2. 0 deg SIMULATION RESULTS The numerical results presented herein will be those obtained using Gain Grid III and a disturbance magnitude, VW of 75 m/set. Table 1 identifies the gains which minimized C(u) for the set of initial conditions used. Table 2 lists the initial conditions and the corresponding optimal gains. Table 3 shows the location of the controlled vehicle poles. Approximately 16 minutes of digital computer time was required to determine the optimal gains for the set of initial conditions considered. For small values of Go, i,, /3 and all . values of zoconsidered, the optimal gain set is equal to the optimal gain set for zero initial conditions. The change in gains with a change of initial conditions is shown in Figure 2 which indicates that the gains are functions of the initial conditions. If the initial conditions are sufficiently small, then ~1. >> I Xi 1 and the 1 optimal gain set will be equal to the optimal gains for zero initial conditions. Also if VW is sufficiently large, the optimal gains will be independent of initial conditions. Initially the k’s were chosen such that lki I ( Mi. For all the optimal gain sets, the gain kg is at its maximum value. Consequently a smaller value of C(u) may have been obtained had the bounds on kg been increased in magnitude. 18 Table 1. Identification of Optimal Gains kl -~ -. 7 0.401765 I Gain 14 0.401765 k2 k3 k4 1.414214 - 0.00049 105 -0.707107 1.649916 I - 0.00049 105 -0.707107 Gain 31 ( 0.502207 / Gain 49 0.602647 Gain 8 Gain 14 Gain 31 Gain 49 C(u) in Radians for x” = 0 0. 121686 0. 124523 0.134310 0. 142547 vW = 75 m/set and T = 10 set 19 111.11. ,.I I I I .1..1.111--.. I --. ------- Table 2. Optimal Gain and Minimum Cost for Given Set of Initial Conditions xi 1. 0" 2.0" 3.0" 4. 0" Minimizing Cost C(u) xi = ilo xi = p Gain in Radians 8 0.164166 49 0.194999 49 0.221071 49 0.250300 0.5" /set 8 0.133654 1.0" /set 8 0.145514 1. 5" /set 31 0.150867 2.0" /set 31 0.158254 1 m/set 8 0.137765 2 m/set 8 0.158381 3 m/set 8 0.169858 4 m/set 8 0.185952 0. 5" 8 0.137249 1. 0" 14 0. 152562 1. 5” 31 0.164756 2.0" 31 0.174840 Min C(u) = 0.121686 for x” = 0 and VW ‘= 75 m/set Corresponding Gain Set is Gain 8. 20 Table 3. Location of Controlled Vehicle Poles Root ,-_ .-. -. Gain 8 1 2 3 4 -___-- Gain 14 1 2 3 4 -. Gain 31 1 2 3 4 Gain 49 1 2 3 4 Real Part .- 0.005408 -0.273208 -0.226303 -0.226303 0.005308 -0.229022 -0.248347 -0.248347 0.002385 -0.315827 -0.115094 -0.115094 0.001738 -0.380567 -0.080662 -0.080662 Imaginary Part 0.000000 0.000000 1.113617 1 f = 0.898 cps -1.113617 c= 0.201 0.000000 0.000000 1.227910 1 f = 0.815 cps -1.227910 c = 0.159 0.000000 0.000000 1.261882 f = 0.791cps -1.261882 c = 0.092 0.000000 0.000000 1.268127 f = 0.780 cps -1.268127 c = 0.063 21 -0.5303 K4 J- @-CD-@- t I J CD-@- -0.7071 63 -CD-CD meA--m-v-------- ---_-- ------ -0.00049 ~-@-a+-@- -CD+-CD-e- K3 1.6499 K2 1.4142 0.6026 K1 0.4018 -0.5303 K4 -0.7071 ----------------- --------------- CD-@+--- -a+- 0-B 1 e--cr3-e3 I” -----------------------C-----m J e3-a3-a+ J a-a- y--a3 e-@--CD I I I I I I I I do IN DEG b. IN DEG/SEC -0.000490-cB-cB-@-cB- ~-cI3-+3-cr3-@- K3 _---__-v---------m. --- ---_ ----0 1.6499 -- CB-eB-+B--- K2 <3-e-cD-cB-G3- 1.4142 -- <3-@ 1 ---------------------------- 0.6026 -- K1 ~3-d+-e+-tB-cr3- O-@--e-J a+-ee- 0.4018 I I I I 1 I 2 IN M/SEC k’ IN DEG Figure 2. Change in Gains with a Change in Initial Conditions 22 The responses of the optimally controlled system to a worst disturbance of maximum amplitude of 75 m/set with the initial conditions previously given are shown in Figures 3 through 11. Observation of the analog traces indicates that the vehicle parameter z is the major contributor to the increase in the performance index for non-zero initial conditions over zero initial condi- tions. In fact, without exception C(u) = max Ci(u) = C3 lO An extremal input is an admissible input defined on the interval LO, T 1 that maximizes [y(T) cos 8 + y(T) sin 81 cos @+ v(T) sin 4 for some T, 8 and # where T > 0, 0 2 8 C 277 and 1~1 < ;. Extremal inputs may be characterized as follows. The response of the system (1) to an input v(t) forms a trajectory in a three dimensional Euclidean space with coordinates of a point on the trajectory given by y(t), y(t) and v(t). Let the set of attainability at t = T be the set of all endpoints (points with coordinates y(T), y(T), v(T)) of tra- jectories corresponding to admissible inputs on the interval 0 C t < T. Such - - a set is closed and bounded. An extremal input on the interval 0 _< t _< T is an admissible input to which corresponds a trajectory with an endpoint that is a boundary point of the set of attainability. Hence, an equivalent definition of an extremal input is an input that is a time optimal regulation input with the time reversed. 55 The problem can easily be formulated as one with a phase constraint. Let u(t) = v(t) and introduce the vector x(t) with components xl(t) = y(t), x,(t) = y(t), x3(t) = v(t). Then equation (11, with initial conditions specified and admissible inputs, may be represented by j, = Ax + bu, x(O) = 0, lu I 5 k/IT, where: (2 The phase constraint is then Ix3 I 2 1. A discussion of the development of necessary and sufficient conditions for extremal inputs in problems of this type is given in reference 1, pp. l-2. For this particular problem these conditions say, essentially, that extremal inputs are given by: u(t) = (k/r)sgn [$(t)bl (3) where sgn(0) = 0 and G(t) is a piecewise continuous solution of an adjoint equation with a piecewise continuous right-hand side. The discontinuitie s are allowed only at values of t which are endpoints of maximal intervals in which the corresponding response has Ix3 I = 1. The points of discontinuity of & can be further restricted to occur only at right-hand endpoints of such intervals. A more detailed statement of these conditions will be given in the section on APPLICABLE THEORY. Also it will be shown that allowing at most one discontinuity in $J at t = T, the extremal input with respect to e(T) x(T) is given by (3). Thus, since the response x(T) depends only on the input on the open interval (0, T), extremal inputs correspond to continuous solutions of piecewise continuous adjoint equations. This problem is chosen for two reasons. The first is that it presents a case in which the number of segments or arcs of extremal responses which lie on the phase constraint can be made arbitrarily large. This makes it possible to determine that such segments are interrelated. The second is that an explicit solution can be obtained. 56 In the DERIVATION OF EXT.REMALS the theory will be used to treat a particular case. Ranges of values of k and 8 will be chosen and the extremal inputs will be derived for all values of the parameters T and $J. Formulas for extremal inputs are given in EXPLICIT REPRESENTATION OF EXTREMALS for the parameters k, T, 8 and $ with ranges k > 2.5, T >O, 0 < 8 cnand I@I < ;. - The results are then summarized in the CONCLUSIONS. APPLICABLE THEORY The necessary conditions for the present problem will be based on general necessary conditions given by Gamkrelidze, references 2 and 3, and improved by Bryson, Benham and Dreyfus, reference 4. Then the necessary conditions given by Russell and Schmaedeke, reference 5 will be cited. Comparison of these necessary conditions with sufficient conditions obtained by Russell, reference 1, show that the necessary conditions are also sufficient. The notation in reference 3 will be followed with the exception that the components x0 and Go will not be included in the vectors x and #. Thus f(x, u) = Ax + bu. The phase constraint is represented by requiring x to lie in the region G represented by: G = fx:g(x) = (x.,)~ - 1 _< 0 3. Then p(x., u) = 2x3u and 2 = (0, 0, 2~). H(#, x, u) = #f(x, u) and m($, x) = $Ax, M($, x) =u”E$ H(#, x, u), where U = {u: lu 1 c k/n]. (4) Theorem 25 of reference 2 may be stated as follows (taking note of theorems 1, 22 and 24 of reference 2): 57 Suppose that x(t) is an optimal trajectory of equation (2), corresponding to the optimal control u(t), and that x(t) lies entirely in G for 0 _< t _< T and contains a finite number of junction points. Also suppose that each of its sections which lies on the boundary of G is regular. Let 0 C 71 2 r2 < * * * < 72q- 1 5 Tsq - C= T denote the junction points. Then there exists a piecewise continuous vector #J(t) = ($J,(t’). #J (t), 2 #J (t)) and a piecewise continuous, 3 piecewise smooth scalar-valued function x(t) such that: h= aH(k Xi U) iii- = f(x, u) = Ax + bu w (5) d# _ aHCdJ, X, U) -- - + xct, A(t) ap(x’ u, = - $ A + X(t)X(t) (0, Q,,2u) (6) dt ax ax H(#J(t), x(t), u(t)) = M(#(t), x(t)) [l - X(t) 1 + X(t)m($(t), x(t)) (7) where x(t) is equal to zero when g(x(t)) CO and is equal to one when g(x(t)) = 0, A(t) = a $(t)b sgn [x,(t) 1. The vector $(t) is zero nowhere on CO, ~1. On ’ 2i- 1 -- - C t < 72i, g(x(t)) = 0 and $+( 72i- l) is tangent to the boundary g(x) = 0 at x( T2i- 1) and dA(t)/dt < 0 for 7 2i- 1 et< 7 2i’ i = 1, 2, . . . , q. Furthermore, - at junction points the following jump condition is satisfied: either $J+ (Ti) = #J- (Ti) + Pi grad g(x(‘i’), (8) or $- (pi) + Pigrad g(x(Ti)) = 0, Pi # 0, (9 where pi is a real number. In reference 4 is is shown that the jump condition (8) and (9) may be replaced by: 58 #I- (Tag) = $+ (Eli) + Pigrad g(X(‘2i)) iUS 46 can be defined so that it is continuous at 7 2i- 1’ i- 1, 2, . . . , q. summary of the se results for the problem being considered is that if x(t) Id u(t) are optimum then: (Introducing the notation To = 0 and T2q+l = T, h CT). quation (5) is satisfied (dx/ dt = Ax + bu) and there exists a piecewise contin- :jus G(t) such that: 2 = - $A + $ X(t) #,(t)sgn[x,(t) l(0, 0, 2u) dt here x(t) = 0 if T2i t g(x) yields an equivalent formulation, reference 6. If the formulation indicated by (15) were used then #J(t) would satisfy: d#/dt = - $A + $grad g(x) = - #A + xc(O, 0, 2x3) where c = i$J2sgnx3. Also u(t) would satisfy: u(t) = (k/ IT)sgn[$3(t) I, o < t < T where sgn(0) = 0. The necessary conditions given in reference 5 are also applicable to this problem. They give further information regarding extremal inputs. These results may be summarized in the following definitions and theorems from reference 5 (stated for the present problem in terms of notation given above). Definition 1 (15 (i6: (17: The input u. is extremal if there exists a non-trivial (continuous) solution J T # of ( 13) such that G3(t) u,(t) dt = m;xjT $3(t)u(t)dt, The maximization is 0 Ot taken over all u with lu(t) 1 _< k /a and I / u(7)d71 _< 1 for 0 1 and a sequence 71 2 T2 < T3 < _ -‘_< ‘2N - < T such that [T 2i-1, T2il is an interval of type B for i = 1, 2, . . . , N. c72i, T2i+l 1 is an interval of type Pl for i=l, 2, . . . , N-l and if T2N CT then [‘zN, T 1 is an interval of type Pl. In this case defining p(t) to be equal to 0 on intervals of type B and to be equal to G,(t) - #,(Vk) on each interval of type Pl, where t*E(O, T) is an endpoint of the interval of type Pl yields a continuous function p(t) such that u(t) = (k/@sgn[p(t) I. Note that from Theorem 3 if [T2i, T2i+1]C(0, T) 62 I :hen ~3(‘2i) = ~3(‘2i+i ). Also note that the vector #J(t) with components $1(t), G,(t), p(t) satisfies equation (iG). The se necessary conditions can 3e summarized as follows: if u(t) is an extremal control there exists a zontinuous solution #J(t) of equation (16) such that u(t) is given by equation :17) and on each interval where Iv(t) I = 1 the inequality (20) is satisfied. Extremal inputs defined by Definition 1 are interpreted geometrically in the statement and proof of Lemma 1 of reference 5. It is shown that if a non-zero vector G(t) is chosen to satisfy equation (13) over 0 _ WAX, where x(t) is any response corresponding to an admissible input on CO, T 1. This property can be maintained when Q(t) is taken as a solution of equation (16) if a discontinuity in #3(t) is allowed at t = T and G(t) is continuous on CO, T 1. This is accomplished by setting $3(T) equal to the value of p(T). Furthermore if the corresponding v(T) is less than one in magnitude no discontinuity is required. If the corresponding IV(T) I equals one a discontinuity may appear but the following inequality must be satisfied: v(T) [#3(Tb Q3-(T)IZ0. A sufficient condition for an input to be extremal which is applicable to this problem is given in reference 1. If IV(T) I < 1 and the time scale is reversed Theorem 1 of reference 1 is applicable. Consider, then, the system ;r = Cy + do, 101 _< k/n, g(y) 5 0, where C = -A, d = -b, y(t) = x(T-t), w(t) = u(T-t) and in particular y(T) = 0 and g[y(O) 1 CO. Theorem 1 of reference 1 states: (21) (22) Let w (t) be defined on LO, T 1 and assume cr; (t) transfers y from y(0) to 0 in CO, T 1 with g(y) < 0, and let q(t) be a covariant vector function defined - and continuous on [O, T 1 with the possible exception of points tl, t2, . . . , tr- 1 where r is an odd integer and 0 = to < tl _< t2 < . . C t,r = T. If k is odd, ” - 63 g(&t)) < 1, and if k is even g(&t)) = 1, for tcIk = [t,-1, tk 1 where t(t) is the response to u(t). Let Vl, V2, . . . , Vr- 1 be non-negative real numbers and c(t) be a function defined on CO, Tl, non-negative on each Iak. Also let 1 ItiT) 1 I # 0 and where X(t) = Let H(w, t) = rl = - vc + x(t) at) ;Ig[4(t)l, (v= grad) 0, if tQ2k+i, and X(t) = 1, if tEIZk, for k = 1, 2, q+tt,) - v-ttk) = vkJ&(\) I. rldwforall IUI 0 on each 12k which follows from (20). (2; (26 (27 In the case when iv(T) [ = 1 the theorem of Russell could be modified by requiring (24.) to hold only for k = 2, . . . , r-l since in this case tl = to = 0. This establishes that, also in this case, the inputs determined from the necessary condition are extremal. 64 I’he se results are summarized as follows: a necessary and sufficient condition ‘or u(t) to be an extremal input is that there exist a vector Q(t), continuous IIn (0, T), that satisfies equation (IS), such that u(t) is given by equation (IT), and on each interval in which Iv(t) 1 = 1, the inequality (20) is satisfied. Wurthermore, #-(T) is an external normal to the set of attainability at the point x(T), and if $(T) is an external normal to the set of attainability at x(T), then J&T) = #-(T) when Iv(T) I< 1, and inequality (31) is satisfied, if Iv(T) 1 = 1. Now let 6(T) = (cos 8 cos 8, sin 8 cos @J, sin@ j. Then it is easily shown that $,(t) = cos @ sin (8+T-t) and 92(t)sec @= tan o, cos e-cos( 8+T- t)+tan 8 + &tan Go-tan d), 0, 0, cos(e+T-T2N-1)-cos(B+T-t), cos 8-cos(@+T-t)+tanW 6(tan (PO-tan $9 0, for t = T, for ‘2N - 0. - Thus, u(t) is an extremal input corresponding:: to $J(T) if and only if +This means #J(T) is an external normal to the set of attainability at x(T) where x(T) is the response to u(t). 65 u(t) = (k/ n)sgn[$3(t)sec $1, 0 < t C T (29) where d,(t) sec.@ satisfies (28) on 0 5 t _< T, and Iv(t) 1 < 1, 72i-2 0, T2i- 1 _< t _< 72i; i= 1, 2, . . . , N, - (31) Iv(T~~-~) I = 1, i=l, 2, . . . ,N, (32) COS( 6+T- 7 2j,-2) = cos(8 + T-T 2i-1); i=2, 3,. . . ,N-1, (33) cos &cos( 6+T- T2N) + tan #+ G(tan Go - tan@) = 0, if T2N- 1 < T, (34a) cos 8 - cos(e+T-T 2N-2) + tan ti + 6 (tan Go - tan @) = 0, if T2N-1 = T, (34‘ tT-T2N-l ) [cos(e+T- T2N-2) - cOS( 6+T- T2N- 1) 1 = 0, 6 v(T) (tan Ca - tan go) > 0. - (35) (36) Equations (32) through (35) are 2N equations in the unknown parameters, N, ‘i, i= 1, 2, . . . , 2N, ?. and 6. The relations (23) through (31) are constraints which a solution must satisfy. The simplicity of this problem permits explicit solution of these constrained equations. The nature of T1 is determined in the next paragraph. Consider equation (32 ) with i= 1, namely, I’ Iv< TV) 1 = 1. Equations 03 ) and (20) yield v(T1) = (k/r) 1 sgn[cos(6+T-T1)-cos(e+T-t) ldt. The zeros of cos(e+T-T1)-cos(e+T-t)‘occur at t = 8 +T f (6 + T - T1 + 2mn) for man integer. Since there is at most one zero in the interval (T1 - 27r, T1), and 66 L is greater than two, T1 must be less than 2a. Hence, in the interval (O,Tl), l(t) can change sign at most once. Therefore, either T1 = n/k, or there exists in integer m. such that 0 C2(8+T+mon)-71 *<(a/k) C r1 and T1-2 [2(6+T+mo+T1 1 n/k. That is, if T1 # w/k then there exists an integer m. such that T1 = ? ( &T+molT) + (R/3k) 3 (3% where m satisfies 0 n/2k < 8 + T + mo7T <2n/k. (38) Furthermore, if T1 = ?r/k there can be no zero in (0, i). Hence, there exists an integer m. such that 7r 2(e + T + m-m) - 7, 2 0 < - = 7, 5 2(e + T + m-r) + 2a - 7,. ” I k L ” I This is equivalent to (n/k) - 71 5 8 + T + mo71 5 (r/2k). (39) For i=2, 3, . . . , N-l, T2i-2 and T2i-1 may be determined as follows. general solution of equation (33) for T2i-2 is The ‘2i-2 = e+Tf(e+T - T2i-1 +2m@ where m is an integer. Equations (28) and (29) yield IV(T~~-~) I = 1 and u(t) = (k/n)sgn[cos(8 + T - T2i-1) - cos [e + T - tl for T2i-2 C t < 72i-1. There is at most one sign change of u(t) in T2i- 1 - 2W < t < T2i- 1. The maximum length that an interval of constant sign for u(t) can have is 2a/k which is less than n when k is greater than 2. Thus ‘2i- 1 - ‘2i-2 must be less than 27r and is greater than zero by definition. Therefore, 67 ‘2i- 2 = 2(9 + T) - T2i- 1 + 2miV, (40) where mi is such that 72i-l - 271 < 2(e + T) - T2i-l + 2min < ~~~-1, i. e., ‘2i-1 - a<8+T+mi8<7 2i-1’ bl ) Thus, u(t) is of constant sign in (72i- 2, T2i- 1) and hence from (34) , 72i- 1 = 72i-2 + (2V/k). This result, together with equation (41), yields the following: ‘2i-2 = 8 + T + rni7r - (V/k) (42) ‘2i- 1 = B+T +min+(n/k). (43) The integers mi can be readily determined from the relations T2 2 T1 > T2-n and m i+l = mi + 1 for i = 2,3, . . . , N- 1. In case T1 # n/k it is easily shown that m2 = m. + 1. If ~1 = n/k then m2 = m. + 2 if (n/k)-r 5 8 + T + mow < (2Vr/k)-7rr, and m2 q mO + 1 if (2V/k) - n < 8 + T + mo77 5 (77/2k). Note that if T2N- 1 < T, then equation (3.5) implies that equations (42) and (4.3) hold also for i = N. Now consider equations (34a) and (34b). Suppose d = 1, i. e. Iv(T) 1 = 1. If ‘2N = T>T2N-1, go = 0. If T = ~~~-1 then tan go = cos (6 + T - ?2N) - cos 8, Go can also be determined from limiting cases when 6 = 0. Note that when G = $ the value of 6 is immaterial. Suppose now that d = 0. Equation (34) can be written as cos(8+T - 7 2N) = P, 68 (44) where P = p(G, @) = coS6 -t- tan@. (45) It is clear from equation (45) that p must be restricted by the relation IPI _< 1. Thus T2N = 8 + T f [cos -’ p+ 2mnl where 0 5 cos -1 p 5 7r. Since kt’) 1 < 1, 72N CT. It is readily shown that T - T2N C 28. Hence, .T- T2Nisequalto2n- efcos -1 por -8+cos-l p. The three cases: 8 > cos-l p, 6 = cos-‘p and 8 cos-’ p, T - T2N = 2n - 8 f cos-‘0. Suppose = T + 8 - 277 - cos ‘2N- 1 -1 p. Then u(t) changes sign at T + 6 - 217 + cos P and sgn u( T2N+) = sgn[- sin( 2a + cos-’ p) 1 = - 1. Therefore v( T2N) = +l and u(t) = -(k/r) ‘2N -1 cos-1pandT2N=T+e- 2Tr+ cos -IP . In the final case, 8 C cosslp, T2N = T + 8 - cos -1 p or T + 8 - 2n + cos -1 p. If ‘2N = T + 6 - cos-‘p, u(t) = -k/s for TzN 72N 71 - (n/k) and the interval (49) is of positive length if 8 > n - (3a/k). Hence the nature of extremals will depend on the relation of 8 to the values 2W - (37/k), n - (r/k) and 71 - (3r/k). Other such break points in 8 arise from considering the special cases when N < 1. The distribution of the break points in 8 relative to the interval CO, ~1 and the ordering of all break points depend on the value of k. The values 2, 2. 5, and 3 are critical values of k which determine the number and ordering of break points in 8. For example, if k > 3 the break points in 8 for 7 are 7T - (3a/k) and W - (n/k), whereasif k < 3 they are 71 - (r/k) and25 - (371/k). The other break points in 8 are 71 - (r/2k) and m7T - (5a/2k) where m = 2 if 2 2.5. Now consider the case of k > 3 and W - (37/k) _< 8 < 7~ - (5n/2k). In this case as in all cases, if T _< r/k the constraint Iv(t) I _< 1, 0 _< t 5 T, is always satisfied and N and 6 are both zero. 71 The possible values for T2N when T > a/k can be determined as follows. The inequalities r - (5n/2k) 3. Thus, with 8 restricted to [n - (3s/k), n - (5r/2k) 1, the inequality 8 > cos -1 e)/3, i.e., -. p implies cos-lp <2n - 8 - (2n/k) and n/k <(2a - neither (46) nor (4?) is ever satisfied. Hence 7 2N = t when 8 >_ cos-‘p (or equivalently @ z 0) and o. = 0. When @ CO and IV(T) I < 1: ‘2N = T + 8 - cos-‘p if v(T2N) = +l ‘2N = T + 8 - 2n + cos-lo if v(T~~) = -1, subject, of course, to T2N >_ T2N-l. The limiting cases when L(T) I = 1 are T2N = T and T2N - (2n/k), which corresponds to cos.~~ = 8 + (2a/k) in (50), = T - 4(n-8)/3, which corresponds to cos p = (2R + 8)/3 in (5;). When r/k CT C 17 - - 8 + (n/2k) (38) indicates 71 = r/k and it is easily deduced that if T1 = n/k, v(T1) = +l. It is also easy to see that N is at most 1 if iv(T) I C 1. Thus, if N = 1, T2N = T2 = T + 8 - cos-‘p from (50) when IV(T) 1 C 1. Furthermore, as v(T) ranges from +l to -1, T2 ranges from T to T - (2g/k). Hence, if T > 3rr/k, T2 z T1. But if T <3r/k, T - - (2n/k) C a/k so that T2 = T1 for some value of @ As # is decreased from this particular value N = 0 until v(T) = -1 at which point N = 1 and T1 = T2 = T. Thus, the following results are obtained. For n/k @> G.. implies N = 1, 6 = 0, T1 = n/k, T2 = T + 8 - cos -‘p (50) (51) 72 Q:‘#> G4 implies N = 0 G44’ a implies N = 6 = 1, @‘. = $, T1 = T2 = T, where $ and G4 satisfy: tan $ = -COS 8 + COS(T -t 8 - a/k) tan G4 = -COS 8 + COS(e + T/2 i- 7/2k) For 37/k < T _< n - 8 + (n/2k): implies N = 6 = 1, Q. = 0, T1 = a/k, T2 = T implies N = 1, 6 = 0, T1 = r/k, T2 = T + 8 - cos-‘P implies N = 2, 6= 1, go = G8’ 71 = r/k, T2 = T - (an/k), T3 4=T = 7 where tan @ = 8 -CO& + COS( 8 -t 2n/k). Whenr- e+(r/2k) CT _ 4(8+T-@/3+(n/3k) is equivalent to T _<4(~ - 6) - (7r/k). Thus the following result is obtained for 7~ - 8 + (r/2k) CT _< 4(@) - (7lT/k): $‘> 0: N = 6 = 1, o. = 0, T1 = [(n/k) + 4(8 -t T - @l/3, T2 = T 0 > G> @8: N= 1, 6= 0, T1= [(a/k) +4(8+T - n)r>l/3, T2= e+T - COS-1p 73 f18 > h N = 2, 6 = 1, Q’, = a8, 71 = [s/k + 4( e + T - or) ]/3, T2 = T - (2n/k), T3 = T4=T. If 4(a - 8) - (7r/k) CT < B - 6 + 2r/k the above result for @ z 0 holds. For 0 > @ > G2 where tan g2 = -CO& + COS(h + (V/k) + 27 + 81/3); - 6+T - cos -‘p > [(n/k) + 4(e + T - n) l/3. Hence for this range of #; N= 1, 6=0, T1= [(n/k)+4(8+T- n)]/3andT2= e+T-cdp. Then for 8, > @ > P, for some @, N = 0 and u(t) is given by: u(t) = -tk/ II), for0 ct <8+T - 2n+c0&lp tk/ r), for 8 + T - 2V+cos-‘pet C8+T -1 - cos p -tk/ 7T), for 8 + T - cos -lp @l when 4(77 - 6) - (7n/k) CT C [4(n - @/31 + (n/k), and @I >_ G6if [4(7’r- @/31 + r/k 5 T 0: N = 6 = 1, Q. = 0, T1 = [(n/k) + 4( 6 + T - 7~) ]/3, T2 = T cl>@> 8,: N = 1, 6 = 0, T1 = [(n/k) + 4(8 + T - n) l/3, - T2 = 8 + T - COS-lp @,>qO qt6: N=O a,:!% N=6=1, G = $ 0 6’ ‘1 = T2 = T. In the case, [4(7r - 8)/31 + (r/k) _< T CR - 8 + (2n/k) and ‘$1 >_ @, T1 = R/k andv(T1) = -1. Thus T2N= 8+T - 277++0~ - ‘p as long as IV(T) I C 1 and the limiting case occurs if 6 + T - 2n + cos-‘p = T + [4(e - 7r)/3 I. Thus for this range of T the following result is obtained: 74 8 >_ 0: N = 6 = 1, e. = 0, 71 = [(a/k) + 4( 8 f T - 1~) l/3, T2 = T 0 > @>, G2: N= 1, 6=0, T1= [(n/k)+4(8+T-g)1/3, TV= 8+T- cos -‘p ti2 >#>@l: N= 0 4, >_ @J> 8,: N = 1, 6 = 0, T1 = R/k, T2 = 8 + T - 277 + cos-‘jJ 0, >_ ik N = 2, 6 = 1, Go = g5, T1 = s/k, T2 = T + [4( 8 - n)/3 1, T3 = T4 = T. where tanQ5 = -case + cosC(2S + 8)/3 1. Now consider T in the interval ma - 8 + (2s/k) _< T _< (m+l)n - 8 + (8/ 2k), where m is any integer greater than zero. Since (33) is not satisfied, 5 = r/k whenever T2 > (n/k). It is seen that 8 + T - rn71 - (n/k) >_ n/k and that 8+T-n + (r/k)CI’ 0, N = m+l, 6 = 1, Q. = 0, T1 = n/k, T2N = T and - the intermediate Ti are as given above. For 0 > @ >_ g3 the only changes . are that 6 = 0 and T2N = 8 + T - COS-‘p, where tanti3 = -co& + cos(77 - r/k). For g3 > @ > g5, N is reduced from m+l to m with no change in 7i if i < 2m and T2m = e+T-2n+cos-‘p. ForG51 $, 6= 1, go= G5, N = m+l. T2m = T + 4( e - lr)/3, T2m+l = T2m+2 = T and the remaining Ti are unchanged. Consider the interval m?r - 8 + (a/2k) CT 1. In this case 8 + T - (m-1)77 - (R/k) 3 [4(@ + T - m@+(a/k)7/3. Hence the results are similar to those just obtained except that T1 = [4(0 + T - m@+ (r/k) l/3, i.e., $J >_ 0: N = m, 6 = I, go = 0, T1 = [4(e + T - m@+(n/k) 1/3, T2i = 8 + T - (m-i)n - (B/k) ‘2i+1”2i + (2n/k), i=l, 2, . . . , N - 1, T2N = T. 75 0 > Sl 8,: Only changes from ,G 2 0 are 6 = 0, T2N = 8 + T - cos-‘p tij3 > tJ > 8,: Onlychangesfrom GL Oare 6= 0, N= m-l, T2N= 8+T - 27l+ cos -1 p 8, E @: N = m, 6 = 1, go = @,, T2m-2 = T + [4(6 - n)/31, T2m-1=t2m=TandT1 through T2m- 3 are the same as when # > 0. This completes the derivation of the extremals for all T L 0, k > 3, I$1 0, k 1 2. 5, I@ I < 7~/ 2 and 0 C 6 < 7~. To simplify the table the extremals will not be given for the break points in T and 6. For any particular case of interest where T and 6 are break points the extremal could be readily determined from consideration of the neighboring intervals. The following functions of k, 6, @ and T are introduced to simplify future expressions and notation of the dependence on the parameters will be suppressed. hl = r/k h2 = (27r + 8)/3 h3 = (4~+ e)/3 h4 = B+T- cos-1(tan@+cos6) h5 = [4(6 + T - @+0/k) 1/3 76 h6 = 8 + T - 2~ + cos-‘(tan@ + case) h7 = T + [4(6 - r)/31 h8 = 8 + T - TT - (R/k) Let g(k) = 0 if 2. 5 _< k C 3 and g(k) = 1 if k > 3 and define - e. = 0 e,(k) = 71 - [5 + g(k) l(h1/2) 6,(k) = 7~ - 12 + 3g(k) l(h1/2) e3(k) = vr - [ 1 + g(k) l(hl/ 2) e,(k) = [2 - g(k) I?T - [S - 5g(k) l(h1/2) e5 = 77 Then let j( 8, k) be defined implicitly by 6 j(e k)(k) < 6 < 6 dependence on 6 and k will be suppressed fkom now on. j(e, k)+l(k) and the Also introduce the functions s,(j) = 6.. for i, j = 0, 1, 2, 3, 4 where 6.. = 0 if i # j and 6ii= 1. The 11 dependence on j w?ll be suppressed in the future. Further, introduce the functions 4: (T, 6, k), defined implicitly by 1 pi 1 i’= 1,2,..., 11. The pi are defined as < n/ 2 and tanGi + cos.6 = pi, for follows: PI = CO&r + 8 - hl) F2 = cosCh2 + (T + h1)/31 P3 = cos(fl - hl) 77 P4 = cost-e + (T + hl)/21 P5 = codha) p6 = cash - (t - hl)/41 P7 = -Pfj P8 = cost8 + 2hl) 5-J = 9 costh3) 50 = cos[h3 + (T + hl)/31 PII = -P3 The break points in T, i. e., endpoints of intervals in T in which the form of -She extremals remains unchanged, are given below: 3hl(qo + g ql) + [4(?7 - O)-hll (l-g)ql/3 + [7 - 8 + (hl/2) 1 (q2 + gq3) + thl + [4(7r - @/3]~[q3(l-g)+q41 CT - 6 + (hl/2) I&, + 91) + [4(n - @-hl ][(l-g)q2 + gq3 1 + thl + [4(0)/313gq2 + 3hl[Wg)q3 + q41 C4(n - e) - 7hl lC(l-g)q, + gq11 + (I7 - 8 + 2hl)g(qo + q2) + thl+r4(R-0)/3 l&l-g)ql + 3hl [(l-g)q2 + gq3 1 + [2v - Q + (hl/2) lb&q3 + q41 78 T4 = T5 = T2n = {hl + ‘&r-8)/313 [(l-g)qo + gql] + C27r - 8 + (hl/2)lcg(qo + q3,)+q21 +(r-8+2hl)(l-g)ql + Or-e+-2hl)[(l-g)q3 + gq41 + C4(2m - B)-7hl I (l-&i4 (T- 8+2hl) [(l-g)qo+gql 1+(2n- 0+2hl) [g(qo+q3+q4)+q2 1 + C2n-B+(hl/2) 1(1-g)(ql+q3) + n(l-g)q3 + Ehl+[4(28-@/31 3 t l-a4 nn-8+(hl/2)+(3hl/2) [q4+(l-g)(ql+q3) I-~[ql+(l-g)(qo+q4) 1, n> 3 T2n+l = nr-e+2hl-(3hl/2) cq4+(1-g)(ql+q3) I +rcg(q,-q,)+(l-g)(q3-9,) I, n> 3 The significant break points in @ depend on T, 8, and k in a complicated way. But throughout any given 8 interval and T interval they are either 0 or one of the Qi, i= 1, 2, . . . , 11, defined above. &(T, 8, k), i=l, 2. 3,‘4, be defined by: Thus, for Tmdl CT 7 [~,,wg,+@8gl - Let @” = nf 2 and G5 = - 1r/2 and introduce the functions r., i=9, 1, 2, 3, 4, where r. = 1 if (i+ < @ < &+l Extremals flor Ct = 8 will not be 1 and ri = 0 otherwise. defined but they can be easily determined by considering the neighboring intervals. Now the expressions for N, 6, Go and Ti for i= 1, 2, . . . , 2N can be given. ForTCToitisclearthat6=OandN=0. WhenT>To, 6=ro+r4 and @ = r Q1+r4Q4. T, eJo@ anOd k. In this case N and Ti are functions of all the parameters The dependence on T will be shown implicitly by expressing the dependence on m(T) where TrntTJ- 1 < T C TmtTj. All other dependence will be suppressed, e. g. N = N(m). Furthermore N(m) dominates Ti(m), i.e., Ti(m) given by the formulas for i > N(m) are meaningless. 81 IV(l) = 1 - r3 T~( 1) = (1 - r3 - r4) Ehl + T - hl) [q4 + (1-g)q3 13 + Tr4 ~~(1) = r2(h4-T) [l - q4 - (l-g)q31 + T (1-r3) N(2) = 1 + r4 - (r3 +r4) [qltl-g) + q2 + q3gJ - r2b3(l-g) + q41 r,(2) = hl k10+ql+(r3+r4) [q4+(q3-41)(1-g) 13 + h5(l-r3-r4)(q2+q3g) + T b, [ql+q2+(q3-ql)g 1 -I- (ro+rl) [q3(l-g)+q4 13 T2(2) = T(l-r2-r3) + (Tr2 + h r - 2hlr4) (qo+qlg) + h4r2[ql(1-g)+ 43 q2+q3d + [hlr3 + (hl-T)r4 lh3(1-g)+q4 1 ~~(2) = T4(2) = Tr4ho + qlg + clg(l-g) + q4] N(3) = l+r4-(r3+r4.)ql(1-g)-r2(q2+q3g) + (1-r3-r4) [q3(l-g)+q4 1 T1(3) = h5[qo-q1(r3+r4)(1-g)+q2(ro+rl)g$911+ Tb4q1(1-g)+(ro+r1) h2(l-g)+q3glj + hl hr3+r4) h2(l-g)+qggl + q3(l-g)+q4~ T2(3) = [T(l-r3)+h4r3-2hlr41(qo+qlg)+ [T(l-r2 -r3)+h4r21ql(1-g)+ (h4-T)rlq2g + [(T-2hlI (l-r3 -r4)+h6r3+h7r41(1-qo-ql) + 2hl (l-r3 - r4) (q2+q3g) T3(3) = T4(3) = Tr4[l-ql(l-g) 1 + T(l-r3 -r4) [q3(1-g)+q41 N(4) = 2 - r3 - Cl - (l+r3+r4)glqo - [l + (r2 - r3 - r4) (l-g)141 T1(4) = h5+(T-h5)r4[qo(1-g)+qlgl + (hl-h5) [(r2+r3)q1(l-g)+q2+ (qo+q3)gl-(W3 h3Wg)+q4 1 (l-r3 -r,)lg +(h6r3+h~r4)(1-q~-qlg)+(T-2hl)(1-r3-r4)(l-qo-ql-q2g) ~~(4) = Tr4[ql(l-g)+q2g~+(h8+2hl)[qo+q2(l-r3 -r,)lg+T(l-r,)(l-q,-ql-q2g) ~~(4)=r3(4) (1-g)+[T(l-r3)+(h4r3-2hlr4)qo+(h4-T)r2q21g ~~(4) = 76(4) = Tr4q,g N(5)=2-r3-(l+r2 -r3 -r4) [(q,+q4)(1-g)+q1gl+(r3+'4)qog+q3(1-g)+qqg sl(5) = [h5(ro+rl)+hl(r3+r4)~[q~(l-g)+qlg]+h~[(ql+q3)(l-g)+q4g]+ [h5-(4n/3) 1Cq2+(q,+q3)g1+ ET(ro+rl)+[h5-(4~/3)](r3+r4)lq4(l-g) ~~(5) = ~Tr~+h4rl1~q,(1-g~+qlgl+~h6r3+h~r4~[:1-q3+(q3-q~-q4)gl+ T(ro+rl)q4g +h8{(1-r3 -r4)[ql(1-g)+q2g1+q,g3+(h8-n)[q3(1-g)+ q4gl+(T-2hl)(l-r3 -r4)[q2(l-g)+q3gl T3(5)=T+(h8+2hl-T) El-r3 -r4)[ql(1-g)+q2gl+qog]+(h8-r+2hl-T)[q3(1-g)+ T4(5)=T+(h4-T) [r2q1(1-g)+(r39,+r2q2)g ]-2hlr4q,g -[2hl(l-r3 -r,)+(T-h,)r,+(T-h,)r41[q3(l-g)+q4gl T5(5)= T6(5)=Tr4q,g+T( l-r31 [q3(l-g)+q4g 1 N(6)= 3-r3-qo(l-g)-ql+(r3+r4)q,g+(r0-r2+r3+r4-2)q4(l-g) ~~(6) =hl~~q,+q2+(ro+rl)q4~(l-g)+(l-q4)g~+h5ql(l-g)+~h5-~4~/3)~ ECq3+(r3+r4)q41(l-g)+q4gI-(4n/3)[q3(l-g)+q4gl 83 T2(6)=h8 {(l-r3 -r4) Cqo(l-g)+ql l+q2tr3+r4)g j+(h6r3+h7r4) Ctqo+q4)t1-g)+ q1 I+(h8-V) Eq2(1-g)+q3+[qo+q2(l-r3 -r4)+q4]gj+t[h7-(4R/3) lro+ (h4-2drl h4(l-g) T3(6)=(h8+2hl) [(l-r3 -r4) [qo(l-g)+ql 1+q2+q3+(qo+q4)g~-nI:[qo+q2(l-r3 -r4)+ q41g+q3+q2(1-g) 3 +T E[r4qo+(ro+r4)q41(l-g)+r4ql 1 ?4(6)= [T(l-r2-r3)+h4r2 lcqo(l-g)+ql l+(T-2hl)(l-r3-r4) [q2(l-g)+ y3 q4gl+(h6r3+h7r4)(92+q3+94g)+Ttro+r4)q4(l-g)+h8 ho+q2t1-r3-r4) ]g T5(6)=T(l-r3) [q2(l-g)+q3+q4gl+Tr4q2g+(h8+2hl) [qo+q2(l-r3-r4) ]g ~~(6)=T(l-r~) [q2+q3+q4gl+(h4-T)r2q2g+[T+(h4-T)r3-2hlr4]qog T7( 6) = T8( 6) = Tr4qog General expressions can be written for the parameters N and Ti for m > 7. - It is convenient to introduce the integer M(m) defined as follows: M is equal to N, if T2N- 1 < T, and otherwise M is equal to N- 1. Explicitly, M = N-r4~qo+q1+q4~l-g)+q2g~-~1-r3)~q2~1-g)+q3+q4g~-(ro+rl)q4(l-g), and ‘2+1 = ‘2N = T if M = N- 1. With M defined in this way, the range of the index i in the following expressions is 1 < i < M- 1. - - For n > 3; - N(2n+l)=n-r3-qo[(l-g)-( r3+r4)gl+Cq3-(l+r2-r3-r4)q41(l-g)-(ql-q4)g 71(2n+l)= [h5-(n-2)(48/3) lCq,(l-g)+qlgl+:[h5-(n-1)(4n/3) l[q,+(q,+ q3)g!+ hl h1+q3)t1-g)+q4 1 84 T2i(2n+l)=h8+(i-n+l) n++a[(qo-q3)( l-g)+(ql-q4)g I, T2i+1(2n+l)= T2i(2n+1)+2hl N(2n+2)=N(2n+l)+qo+q2+(q.l+q3)g T1t2n+2)=hl[qo~q2+(91+93)8kChg-(n-1)(4T/3) ](ql+q4)(1-g)+~h5-n(4~/3)] Cs,( l-f$+q4g I T2i(2n+2)=T2i(2n+l)-lr[qo+q2+(q1+q3)gl T2i+1(2n+2)= T2i(2n+2)+2hl Also, for n > 7 - T2dn) = [T( ro+r~)+h4r21~qo+q~+(q2-qo)g~+~~(1-r3)+h4r3-2hlr4~qog -. + (h6r3+h7r4)(1-qog)+(T-2h1)(1-r3-r4)[q2+q3+(q4-q2)gl + E[h7-(4n/3) l(ro+rl)+(h4-2r))rZ 3 q4( l-g) The above formulas give a complete explicit representation of the extremal inputs for the problem considered. CONCLUSIONS The necessary conditions for an extremal input given in references 2, 3, and 4 for problems with bounded phase coordinates were discussed with respect to the problem at hand. These conditions give the result that an extremal input is proportional to a Signum function in which the argument of the Signum function is an adjoint solution. The se conditions allow discontinuities in the adjoint solution at certain junction points. The necessary conditions of reference 5, interpreted for this problem show that the se discontinuities are not required. One discontinuity is required 85 if it is desired that the adjoint represent any external normal to the set of attainability. This discontinuity only occurs when the external normal is not unique. It is shown that when the necessary conditions of reference 5 are satisfied the sufficient conditions of reference 1 are satisfied. These conditions imply that u(t) is an extremal input if and only if it can be represented by equations (28) and (29) subject to the constraints (30) through (35). In this representation the junction points and the discontinuity are introduced as parameters. The constraints (30) through (35) are determining equations and inequalities for these parameters. An example shows how these constraints are used to determine these parameters. The results are presented in tabular form for extremal inputs giving explicit formulas for the necessary parameters. Responses having an arbitrarily large number of arcs which lie on the phase constraint boundary can be obtained from extremal inputs if the time interval [o, T 1 is sufficiently long. In such a case the input tends to be in resonance with the oscillator over an intermediate segment of the time interval. During an initial segment the input seeks to get into the proper phase relationship with the oscillator. A final segment of the interval is spent attaining the proper terminal value for the input. 86 REFERENCES 1. Luh, J. Y. S. , Russell, D. L., Lee, E. B., Gollwitzer, H. E., and Fairchild, D. D., Bounded Phase Coordinate Control, Honeywell Document No. 12006-FRl, Honeywell, Inc., July 1965. 2. Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V. and Mischenko, E . F. , The Mathematical Theory of Optimal Processes, Interscience, 1962. 3. Gamkrelidze, R . V . , Optimal Control Processes with Restricted Phase Coordinates, Izvest. Akad. Nauk S. S. S. R., Ser. Mat., Vol. 24, pp. 315-356. 4. Bryson, A. E., Denham, W. F., and Dreyfus, S. E., Optimal Pro- gramming Problems with Inequality Constraints I: Necessary Con- ditions for Extremal Solutions, AIAA Journal, Vol. 1, No. 11, Nov. 1963, pp. 2544-2550. 5. Schmaedeke, W. W., and Russell, D. L., Time Optimal Control with Amplitude and Rate Limited Controls, J. SIAM Control, Ser. A, Vol. 2, No. 3, 1965,pp. 373-395. 6. Paiewonsky, B., McIntyre, J., Woodrow, P., Terkelsen, F., and Williamson, G., O#n Optimal Control with Bounded State Variables, ARAP Report No. 60, Aeronautical Research Associates of Princeton, Inc., New Jersey, July 1964. 87 SECTION 5 EXTREMAL BOUNDED-AMPLITUDE BOUNDED RATE INPUTS This section considers the problem of determining extremal bounded-amplitude bounded-rate inputs to linear stationary systems. The discussion will be re- stricted to the case of a scalar input. Consider a system represented by the vector differential equation: x(t) = Ax(t) + bw(t) (1) Here it is assumed that w(t) is a scalar input which is a continuous function of time with a piece-wise continuous derivative k(t). It is also assumed that Iw( t) 1 and I&(t) 1 satisfy the constraints (2) and (3). (w(t) 1 c k - (2) (3) The vector, x(t), is an n-vector representing the state (or response) of the system and A and b are constant n x n and n x 1 matrices, respectively. The system is assumed to be initially at rest, i. e. x (0) = w (0) = 0. The input, w(t), can be adjoined to the state of the system by introducing xn + I(t) = w(t) and for convenience set w(t) = v(t). Then the system is represented by: i = k :]x + [Y]v, X(O) = o where x represents the n + l-vector and the constraints are: Ix n + 1 (01 5 k Iv (t) I ( 1 (4) (5) (6) 88 ‘or any T > 0 the set of attainability, K(T), is defined as the set of all x(T) ! hich correspond to solutions of (4) subject to (5) and (6) for 0 < t < T. - - his set is a closed, bounded, convex set in Euclidean n+l dimensional space. nputs which give rise to boundary points of K(T) are defined to be extremal nputs. Since K(T) is convex, it has the property that at any boundary point, ay x. (T), there is a hyperplane, l (T),Bi. e. 9 containing the boundary point which supports There is a vector $ normal to Vsuch that $a [,XB (T) - x(T) 1 20 or any x(T) in K(T). Such a vector, $J, will be called an external normal to c(T) at x.~ (T). ‘bus, it is possible to interpret extremal inputs in the following way. On a .ime interval, LO, T 1 an extremal input gives rise to a response x(T) that is :uch that the projection of x(T) on some vector $ is a maximum. Hence, if .v(t) in (1) is considered as a disturbance and if the significant effects of the disturbance can be described as linear combinations of the state of the system ..hen the worst disturbances are extremal inputs. The results obtained for the problem of determining extremal inputs provide a means for evaluating the performance index for a controller in the minimax problem with bounded - ampli- tude, bounded-rate disturbances. A brief description of such a problem is as follows : Consider a system described by . ,. A ,. X =Ax+bu+cw, x(0) = 0, w (0) =o with u, the scalar control, a linear combination of the components of x and w where w is a scalar disturbance that satisfies /w(t) 1 < k and 1 w(t) ( 5 1. The closed loop system may be written inthe form of equation (4). For a fixed time, T, the performance index of a controller u is given byCi(u) =max Ci(u) 15 ifs with C i(u) = max Id(i) x(T) 1 h w ere the maximum is taken with respect to all allowable v’s and a given set of row vectors, d(i), i’l, 2, . . . , s. 89 With this problem in mind it is desired to develop an efficient computational method to determine the maximum of the scalar product of a given vector,d, and x(T) subject to the constraints (5) and (6). In the following discussion the necessary and sufficient conditions for extremal inputs will be given. Based on these conditions two computational algorithms will be formulated. Then the results of a computer program derived from one of these formulations will be discussed. NECESSARY AND SUFFICIENT CONDITIONS The necessary and sufficient conditions state that extremal inputs are Signum functions of appropriate adjoint solutions. The adjoint solutions are continuous on the open interval (0, T) with a possible discontinuity at T. Determining equations are derived for the junction times (times when the corresponding extremal response enter or leave the phase constraint) and the discontinuity if it occurs. A discussion of the development of these conditions is given in the section on extremal inputs for the harmonic oscillator. The results presented there are written in terms of that specific problem. The generalization of those results for problems of the type considered here is as follows: J T ‘Q(T) x (T) = rl n+l (t) v(t) dt wh ere Q(t) is a row vector satisfying 0 &z-r] A b c 1 0 0 90 (7) hus a precise definition of an extremal input, vo, is that there exists a non- -ivial (continuous ) solution 77 (t) of equation (7) such that fT 77 tt) v,(t) dt = ‘0 .n+l lax J r, ,n+ 1 (t) v(t)dt wh ere the maximization is taken over all v(t) which 0 atisfy (5) and (6) for 0 < t CT with .x n+l (0) = 0. A necessary and sufficient ondition for v(t) to be an extremal input is that: i) There exist a set of junction points 0 < 71 <_ T2 < . . . CT CT and setting 7 = 0 such that g [x(t) 1 CO for 7 2q-F2q - g [x(t) 1 =O gr 72i+l I t I 72i, 2i _ 0 0 (10) 91 .- Furthermore, $J(T-) is an external normal to the set of attainability at the point x(T), and if $ ( T) is an external normal to the set of attainability at x(T) then each of the first n components of $(T) is equal to the corresponding component of # (T-) and and + n+ltT) =‘n+l (T-1 if g Lx(t) 1 0 if g [x(T) 1 = 0 (12) Now suppose 77(T) is a given vector and v(T), 0 7 sq-l 1. t < 72q if Tzq-l <‘I’ (19-I 92 1c/ n+l (0 = 0 * t = -b if kc= T I tc/ n+l ft) = qn+l (t) + 6 [Gn+l (T-1 - qn+l (T)], +T 2q st 0. (24) NOW for determining equations there are the obvious equations lw(T 2i-1) I =k, i = 1, 2, . . . . 9. From the continuity of 4 (t) and equations (15) and (16) it follows that: ‘?(T2i) = ‘?(T 2i + 1) J i = 1~2, . . . . q-2. (25) Also from the continuity of #(t) and equations (16) and (17) one can obtain: (26) 77 n+l(T2q-2) + 6 bn+l(T-) - vn+l(T) I=o if T2q =T (27) If 7 2’4 T2qm1, then 62: 1”::: 9 n+l (T-) =O so that again Vn+l(T2q+ 6 [$n+l(T-) - qn+l(T) 1 =O. Hence one obtains the equation: (T - 7 2q-1) .t’?n+l (T2q) +6 L#n+l (T-1 - Vn+l(T) 1 -I = 0. (28) If 7 2q-1 O 1 if T - T2N = 0 (30) Equations (25) through (30) are determining equations for, q, T., 6, and 1 G n+l(T-) - rl,+p subjecttothe.constraints(22) through (24). FORMULATION OF A COMPUTATIONAL ALGORITHM The first computational algorithm is formulated as a nonlinear programn;ing problem. The second algorithm is based on a finite sum approximation of the integrals involved in the first formulation. Equation (7) can be solved explicitly by determining the eigenvalues and eigen- vectors of the matrix A. Hence 17n+l (t), 0 < t < T can be assumed known - explicitly as a function of t. A value for q can be chosen and then the integral from 0 to T of the product rln+l (t) v(t) may be obtained as a function of TV, i=l, 2, ,,,2q, 6, and+ n+ltT-) - ‘n-l-1 (T) by making use of equations (8) and (13) through (21). That is, (31) The parameters, 7 ., 6, and #Jr,+1 (T-) - rl, + 1 (T) 1 are to be constrained by the inequalities 7 = 0 < T1 2 T2 < . . .

Minimax studies

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server