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Application of geometric programming to PID controller tuning with state constraints Carver, Leonard James 1976

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APPLICATION OF GEOMETRIC PROGRAMMING TO PID CONTROLLER TUNING WITH STATE CONSTRAINTS by L. James Carver B.Sc. (E.E.), University of Alberta, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of El e c t r i c a l Engineering We accept this thesis as conforming to the required standard Research Supervisor Members of the Committee The University of British Columbia July, 1976 L. James Carver, 1976 In presenting th i s thesis in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th is thes is for f inanc ia l gain sha l l not be allowed without my wr i t ten permission. Department of L>-g/££>a/frg, ~ST~uZPl£LS The Univers i ty of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date *=? Q-ULy/ n\-4> ABSTRACT In the thesis, geometric programming i s considered as a numeri-cal optimization technique. The problem of minimizing the integral square error of a system characterized by a second order plant with pro-portional-integral-derivative (PID) controller i s investigated. Con-straints are imposed upon the state of the system In order to obtain feasible solutions and conditions that are amenable to the geometric programming technique. The application of geometric programming requires the use of approximation procedures to eliminate untenable conditions i n the object-ive and constraint functions. The techniques u t i l i z e d render solutions that are easily obtainable, usually amounting to solving a set of linear equations and requiring no differentiation of terms. In addition, there i s rapid convergence to an optimum. The accuracy of the results i s dependent upon the validity of the approximations. i i TABLE OF CONTENTS Page ABSTRACT . . . . . . . . . . i i TABLE OF CONTENTS i i i SYMBOL TABLE . . . . . v LIST OF ILLUSTRATIONS v i i i ACKNOWLEDGEMENT ix I. INTRODUCTION 1 1.1 Basic Concept of Optimization 1 1.2 Fundamentals of Geometric Programming . . 1 1.3 Some Properties of Geometric Programming . 7 1.4 Introductory Examples '. 9 II. APPROXIMATION TECHNIQUES . . . 16 2.1 Standard Approximation Procedures 16 2.2 Approximation Using a Non-derivative Technique . . . . . 17 III. GEOMETRIC PROGRAMMING APPLICATION TO A PID CONTROL SYSTEM . . 20 3.1 Introduction . . . 20 3.2 PID Controller and System Definitions . . . . . . . . . 20 3.3 Constraint Derivation . . . . . . . . . . . 23 3.3.1 Approximation of the Constraint by a Straight Line 25 3.3.2 Approximation of the Constraint by a Curved and Straight Line . . . . . . 35 3.3.3 Approximation of the Constraint by the Method of Duffin . . . 45 IV. GEOMETRIC PROGRAMMING APPLICATION TO A PID CONTROL SYSTEM WITH TIME DELAY 54 i i i Page 4.1 System D e f i n i t i o n s and Co n s t r a i n t D e r i v a t i o n . . . . . . 54 4.2 Examination of the Performance Index . 60 4 . 3 Optimization Procedures 66 V. SUMMARY AND CONCLUSIONS . . . . . . . . . . . . . . 80 LIST OF REFERENCES 82 i v SYMBOLS Some common mathematical symbols used i n c l u d e : ^ approximately equal to > greater than > equal to or greater than < l e s s than < equal to or l e s s than A defined as n I 1 summation of terms w i t h index i ranging from m to n n II product of terms w i t h index i ranging from m to n i=m L[ ] Laplace transform j ds i n t e g r a l over the same dimension as the ve c t o r The p r i n c i p a l symbols used are defined below: a. v a r i a b l e associated w i t h e r r o r t r a n s f e r f u n c t i o n 1 exponent of v a r i a b l e i n p r i m a l program v a r i a b l e a s s o c i a t e d w i t h e r r o r t r a n s f e r f u n c t i o n °^  i n i t i a l value of p r i m a l v a r i a b l e a* optimum value of p r i m a l v a r i a b l e b^ c o e f f i c i e n t of c o n s t r a i n t equation v a r i a b l e a s s o c i a t e d w i t h e r r o r t r a n s f e r f u n c t i o n f o r the time delay system c. v a r i a b l e associated w i t h e r r o r t r a n s f e r f u n c t i o n f o r time l delay system v C j c o e f f i c i e n t of a posynomial term C(s) Laplace transform of system output response c ( t ) system output response i n time domain 6^ v a r i a b l e v e c t o r i n dual program 6| optimum value of dual v a r i a b l e exponent of dual v a r i a b l e E(s) Laplace transform of e r r o r t r a n s f e r f u n c t i o n e ( t ) system e r r o r t r a n s f e r f u n c t i o n dual exponent G(s) system t r a n s f e r f u n c t i o n G (s) PID c o n t r o l l e r t r a n s f e r f u n c t i o n c Gp(s) p l a n t t r a n s f e r f u n c t i o n g^(t) k = l . . . n , posynomial c o n s t r a i n t s i n p r i m a l program g Q ( t ) posynomial, p r i m a l o b j e c t i v e f u n c t i o n y p l a n t parameter ISE system o b j e c t i v e f u n c t i o n : I n t e g r a l Square E r r o r K £ PID c o n t r o l l e r gain X Lagrange m u l t i p l i e r X^(6) dual v a r i a b l e a s s o c i a t e d w i t h a forced c o n s t r a i n t i n the p r i m a l program m number of p r i m a l v a r i a b l e s Np number of terms i n the p r i m a l program to^  p l a n t parameter V (t) a posynomial term ^(6) dual o b j e c t i v e f u n c t i o n R(s) Laplace transform of system input r ( t ) system input i n time domain v i T system time delay t ^ v a r i a b l e v e c t o r i n p r i m a l program T s e t t l i n g time s T, PID d e r i v a t i v e time constants d PID i n t e g r a l time constants <j> cost or o b j e c t i v e f u n c t i o n <f> augmented cost f u n c t i o n 3. p r i m a l v a r i a b l e 5 p l a n t parameter v i i V LIST OF ILLUSTRATIONS Figure Page 1 System Configuration 21 2 Constraint Equation 26 3 Linear Approximation of Constraint .28 4 Output Response - Linear Constraint . . . . 34 5 Partial Approximation of Constraint . 38 6 Two Line Approximation of Constraint 39 7 Output Response - Two Line Approx. of Constraint . . . . . 46 8 Duffin Approximation of Constraint . . . . . . . . . . . . 51 9 Output Response - Duffin Approx. of Constraint . . . . . . 53 10 PID Controller with Time Delay System 54 11 Performance Indexes for Time Delay System 62 12 Approximations of Performance Index 67 13 Curve Shift Due to Optimization Procedure . . . . . . . . 72 14 Output Response - Time Delay System [Non-optimum Case] . . 74 15 Non-optimum Case - Comparison of Exact and Approximate ISE 75 16 Output Response - Time Delay System [Optimum Case] . . . . 78 17 Optimum Case - Comparison of Exact and Approximate ISE . . 79 v i i i ACKNOWLEDGEMENT I am indebted to many f o r t h e i r help and encouragement i n the pre p a r a t i o n of t h i s t h e s i s , and i n p a r t i c u l a r , s i n c e r e g r a t i t u d e i s expressed to Dr. E.V. Bohn f o r h i s general guidance and numerous h e l p f u l suggestions. S p e c i a l thanks are due to the author's w i f e , B e t t y , f o r her many years of patience and understanding and to Miss Sa n n i f e r Louie f o r her outstanding job of typ i n g the manuscript. i x 1 I . INTRODUCTION 1.1 B a s i c Concept of O p t i m i z a t i o n The optimal design and c o n t r o l of systems and i n d u s t r i a l pro-cesses has long been of concern to the a p p l i e d s c i e n t i s t and engineer, and, indeed might be taken as the d e f i n i t i o n of the f u n c t i o n and goal of engineering. U s u a l l y i n design problems a system i s c h a r a c t e r i z e d i n mathematical terms by d e f i n i n g a d e s i r e d o b j e c t i v e f u n c t i o n or per-formance index which c o n s i s t s of f i x e d parameters and d e c i s i o n v a r i a b l e s . The o b j e c t i v e of an optimal design i s to choose those v a r i a b l e s which y i e l d an optimum ( u s u a l l y maximum or minimum) value of the performance c r i t e r i o n . T r a d i t i o n a l methods of a c h i e v i n g t h i s are by d i f f e r e n t i a l c a l c u l u s and/or some k i n d of a search over the design v a r i a b l e s , which i s o f t e n slow and exhausting. A r e c e n t l y developed theory c a l l e d geo-met r i c programming o f f e r s a more e f f i c i e n t method of s o l u t i o n i n as much as i t i s computationally more convenient and i n a d d i t i o n r a p i d l y approaches the optimum of the o b j e c t i v e f u n c t i o n . 1.2 Fundamentals of Geometric Programming Geometric programming was f i r s t formulated by Zener (1961) as a method f o r o p t i m i z i n g engineering design. He considered the problem of minimizing the sum of component costs of a u n i t , when each cost depends on products of the design v a r i a b l e s , each v a r i a b l e r a i s e d to an a r b i t r a r y but known power. In a d d i t i o n , i t was r e q u i r e d t h a t the number of components i n the u n i t exceed the number of design v a r i a b l e s by one. Zener showed that the s o l u t i o n of such a problem can always be obtained simply by s o l v i n g a square system of l i n e a r a l g e b r a i c equations. Zener's method was subsequently extended ( D u f f i n , 1962a; 2 Duffin, 1962b; Duffin and Peterson, 1964) to optimization problems of a more general nature, including those with inequality constraints. The most comprehensive reference on geometric programming is the book of Duffin, Peterson and Zener (1967) where the mathematical derivations and some engineering applications are described in detail. The mathe-matical treatment relies on a generalization of the inequality and equality relationships between arithmetic and geometric means, as well as other geometric concepts, such as orthogonality of vectors. Geometric programming derives i t s name from i t s intimate connection with these geometric concepts. The underlying premise is based on the fact that the minimum of the arithmetic mean is equal to the maximum of the geo-metric mean. The basic mathematical concepts of geometric programming deal with real valued functions called posynomials. A posynominal i s given g(t) = I P (t) (1.1) j=l 2 where m p.(t) = c. n t , a i j (1.2) 3 3 i=i Each coefficient c.. i s positive, while the exponents a„ are arbitrary real numbers and the variables t^ are restricted to take on positive values, i.e. the domain of the posynomial i s the positive orthant. Geometric programming deals with minimizing posynomials subject to a certain type of posynomial constraint. This minimization problem is called the primal problem of geometric programming or simply, the  primal program. In the primal program, the objective is to find a vector t* 3 that minimizes the f u n c t i o n g Q ( t ) s u b j e c t to the c o n s t r a i n t s t1 > 0, t 2 > 0 ..., t > 0 (1.3) and g l ( t ) . < 1, g 2 ( t ) * 1, g ( t ) < 1 (1.4) N k Here S k ( t ) = I P . ( t ) , k = 0,1,...,p (1.5) J=M k m where P.(t) = c. n t . i j (1.6) J 3 i = l 1 and M = 1, Mj^  = N k_ 1 + 1, k = l , . . . , p (1.7) The exponents a „ are r e a l numbers and the c o e f f i c i e n t s c are p o s i t i v e . Thus the f u n c t i o n s g^Ct) a r e po'synomials. An important feature of geometric programming i s the c e n t r a l r o l e played by the terms P i n the posynomials g^. Instead of f o c u s i n g on determining the optimal t * , the approach i s to concentrate on evaluat-i n g the minimum of g Q ( t ) and the r e l a t i v e c o n t r i b u t i o n of the terms P_.(t) to t h i s minimum. Only a f t e r they are determined i s the optimal t * found. Another important feature of geometric programming i s the concept of a d u a l i t y theory which a s s o c i a t e s w i t h each p r i m a l program a " d u a l " programming problem which i s u s u a l l y e a s i e r to s o l v e than the p r i m a l program. The o b j e c t i v e of the dual program i s to f i n d a v e c t o r 6* that maximizes the product f u n c t i o n N *(6) = [ n ( c . / 6 , ) 6 j ] n X ( 6 ) X k ( 6 ) (1.8) j = l 3 2 k=l fc N k where A (6) = £ 6., k = 1 p (1.9) j = \ 2 4 subject to the constraints fil > °» 62 > °» •••> % p > 0 (1.10) N o I 6. = 1 (1.11) N o and I a..6. = 0, i = 1 m (1.12) j=l 1 J 3 Here a „ , c_. , M^ , are the same as in the primal program. The dual program is obtained from the primal program. The constants c_. appearing in the dual function I|J(6) a r e t n e coefficients of the posynominals g appearing in the primal objective function and constraints. The variable <S . i s associated with the j t n term, P., of 3 3 g^» k = 0,1,...,p, so that each 6^ i s associated with one and only one posynomial term P.. . Moreover, each ^.(6) i s associated with the k t n primal constraint ^ ( t ) < 1. The X^, therefore, are similar to Lagrange multipliers. The normality condition (1.11) i s imposed on the dual variables associated with the primal objective function only, while the orthogonality conditions (1.12) apply to a l l dual variables. It should be noted, also, that the i t n constraint in (1.12) i s associated with the i primal variable t^ through the exponents a „ . A primal or dual program is consistent i f there exists at least one vector that satisfies i t s constraints. A primal geometric A, program is superconsistent i f there exists at least one vector t > 0 such that g^t) < 1, k = 1 p (1.13) A vector t i s called primal feasible i f i t satisfies the primal con-straints (1.3) and (1.4) while a vector 6 i s called dual feasible i f i t 5 satisfies the dual constraints (1.10), (1.11), and (1.12). A primal'program i s said to be solvable i f there i s a feasible vector, t, such that g Q(t) < SQ^1-) ^ o r a ^ feasible t. Similarly, a dual program is said to be solvable i f there exists a feasible vector 6 such that \1>(E) > I|J(<5) for a l l feasible 5. In terms of the preceding concepts, the following duality theorem of geometric programming can be stated: Theorem 1. If a primal geometric program is superconsistent and solvable, then: (i) The corresponding dual program i s solvable, ( i i ) The constrainted minimum of the primal program is equal to the constrained maximum of the dual program, i.e. g Q(t*) = i|i(6*) (1.14) ( i i i ) The relations between optimal primal and dual variables are given by: P (t*)/g ( t * ) , j = 1 N 6.* = J 2 X k ( 6 * ) P j ( t * ) , 3 = ^ , . . . ^ k = l p (1.15) and \(«*) [1 - g k(t*)] = 0 k = l,...,p (.1.16) Thus, i f a given primal geometric program is superconsistent and solvable then Theorem 1 implies that instead of solving the primal geometric program, the corresponding dual program may be solved and by (1.14) the maximum of the dual is equal to the minimum of the primal. In addi-tion, the upper and lower bounds on the solution of the primal and dual programs can be obtained by evaluating g Q(t) and for feasible vectors t and 6, respectively. For such t and 6, then g Q(t) > g 0(t*) = K<5*) > iJ>(S). The duality theorem enables the minimum value of the primal 6 o b j e c t i v e f u n c t i o n to be found without a c t u a l l y s o l v i n g the p r i m a l p ro-gram, but r e l a t i o n (1.15) a l s o gives a method to f i n d the minimi z i n g v e c t o r t * from the knowledge of a maximizing v e c t o r 6*. From (1.15) i t f o l l o w s t h a t : m c. n t * a i j = 6 * i K 5 * ) , 3 = 1.....N (1.17) J 1=1 3 ° and f o r k = 1 p, such t h a t X^(6*) > 0 m c. H t * a i j = <5*/X, ( 6 * ) , j = M, ,...,N, (1.18) 3 1=1 1 3 K K K Taking the l o g a r i t h m of both sides of each equation i n (1.17) and (1.18) and rearranging them y i e l d s m J a±. l o g t * = l Q g ( 6 * * ( 6 * ) / c j ) , j = 1 N q (1.19) and m I a ± j l o g t * = l o g [ y ( c . X k ( 6 * ) ] , j = Mk,..., N K (1.20) The optimal p r i m a l v a r i a b l e s t * are thus found by s o l v i n g the above system of l i n e a r equations i n the v a r i a b l e s l o g t * . Knowing th a t each term i n the p r i m a l o b j e c t i v e f u n c t i o n g Q i s a s s o c i a t e d w i t h one of the dual v a r i a b l e s , i t can be seen from (1.15) that each optimal dual v a r i a b l e j = 1 N Q , represents the weight or r e l a t i v e c o n t r i b u t i o n of the term P. to the minimum of g ( t ) . Thus 3 ° by s o l v i n g the dual program, one obtains f i r s t the minimum of the p r i -mal o b j e c t i v e f u n c t i o n and then the r e l a t i v e c o n t r i b u t i o n of each term Pj to the optimal s o l u t i o n . Since the f e a s i b l e t are r e s t r i c t e d to be p o s i t i v e , t * i s als o p o s i t i v e ; i t f o l l o w s then that each P^(t*) i n the o b j e c t i v e f u n c t i o n i s p o s i t i v e . By (1.15), t h e r e f o r e , those dual v a r i a b l e s which correspond to terms in the objective function are positive, i.e. 8* > 0 for j = 1,...,Nq. The remaining 6.., i.e. those 8^ corresponding to terms i n the constraints, are zero or positive, according as the particular con-straint i s loose (g k(t*) < 1) or tight (g f c(t*) = 1) at primal optimum. More precisely relations ( 1 . 9 ) , (1.10) and (1.16) imply: (i) whenever g^(t*) < 1, the optimal dual variables 8*, j = M j^.-.N vanish; ( i i ) i f 8* > 0, for some j = M^,...,^, then ^ ( t * ) = 1. Moreover, by the second equation of (1.15) and the positivity of t*, i t can be concluded that A (6*) > 0 for some k, implies 8* > 0 for K. j j = MK,...,NK. 1.3 Some Properties of Geometric Programming The dual problem of geometric programming consists of maximiz-ing a given function subject to linear equations and nonnegativity con-straints on the variables. A unique and easily obtained solution for the linear equation arises when the number of equations in the dual con-straint is the same as the number of dual variables. This case occurs when the total number of terms, N^, in the primal objective function and constraints exceeds the number of primal variables, m, by one, i.e. N = m + 1 (1.21) P If the number of variables in the dual exceeds the number of equations by one (i.e. the number of terms in the primal program is two greater than the number of variables) then the dual constraints may be solved expl i c i t l y in terms of only one variable, thus reducing the pro-blem to a maximization over a single variable. The next case, in which the number of primal terms exceeds the number of variables by three, V 8 would lead to a dual program in which a two-variable maximization needs to be carried out. Each case is succeedingly more d i f f i c u l t than the previous; the 'degree of d i f f i c u l t y ' of a geometric program may be de-fined by Degree of d i f f i c u l t y = N - m - 1 (1.22) Generally, well-formulated geometric programs can have arbitrary non-negative degrees of d i f f i c u l t y . The linear dual constraints (1.10) - (1.12) have the important property that they are independent of the primal coefficients c_.. Hence the dual optimal solution for a problem with zero degree of d i f f i c u l t y is invariant in the sense that no matter what the numerical values of the coefficients c_. are, the optimal dual variables are the same, since they are uniquely determined by solving the dual constraints. When the primal objective function represents a cost to be minimized, the optimal dual variables measure the relative contribution of the various cost items to the minimum cost. In the case of zero degrees .of d i f f i c u l t y each term in the primal objective function at the optimum has an invari-ant weight represented by the unique solution of the linear dual con-straints, thus providing insight into the engineering or economic structure of the problem. An analysis to find the relative importance of the terms in the primal objective function can be made, in the zero degree of d i f f i c u l t y case, without prior knowledge of the numerical values of the coefficients. From a computational point of view, evaluat-ion of the optimal primal variables from (1.19) and (1.20) for positive values of c_. can be achieved without resolving the programming problem. Thus the optimal primal variables are easily adjusted for any change in 9 the coefficients (a condition that could reflect altered design para-meters or market fluctuations). Complete solution of a geometric programming problem requires either the minimization of a posynomial, subject to posynomial inequality constraints (primal program) or the maximization of a nonlinear product function, subject to linear inequality and equality constraints (dual program). If the degree of d i f f i c u l t y is low i.e . zero or one, solving the dual is preferable since the optimization procedure i s f a i r l y easy. For higher degrees of d i f f i c u l t y , the question is not as clearly decid-able although the dual offers some advantages as has been noted by Duffin, Peterson and Zener (1967). 1.4 Introductory Examples As a preliminary, the general idea of geometric programming w i l l be illustrated by some examples that are indicative of the kind of reasoning and results that pertain to geometric programming. Consider the cost of producing a product to be made up of several factors: the cost of raw materials i s lOOOn" $/year, the cost of operating one machine in the production process is ^ -9J19_ $/y ear, while n r , Z the cost of operation of a second machine is lOOOr $/year. The object-ive is to find values for n and r which minimize the total annual cost, g i V 6 n b y -R 4000 4 = lOOOn*0 + + lOOOr (1.23) nr * In this unconstrained example, the optimal value of the parameters n* and r* may be found by setting the f i r s t partial derivatives equal to zero: A = (.8) (1000) _ J f 0 m l ^ ( ( . 8 ) ( 1 0 0 0 n * - 8 - = 0 (1.24) d n n• . z . . z n* . .. z n n* r* n*r* 10 = .40001,21 + 1 0 0 Q m !L (_ 4000(^1 + 1 Q 0 0 r A ) = Q ( 1 > 2 5 ) oT . ,1.Z . .» • Z n*r* n*r* The values of n* and r* can be found by solving the nonlinear equations (1.24) and (1.25). However, the method of geometric programming does not require the solution of nonlinear equations. Since the minimum cost <(>* is made up of three cost terms, there i s a unique distribution among the terms that contribute to the cost. Let S\, 62, and 63 be respectively, the fractions of the total minimum cost represented by raw materials, operation of machine one and operation of machine two. Then 6 l = i p o p ^ ( 1 . 2 6 ) 6 2 - - * 2 0 0 ( 1 > 2 7 ) <j>*n*r* 6 3 = ^ ^ d.28) Since the weights must sum to unity 61 + 6 2 + <53 = 1 (1.29) Substituting (1.26), (1.27), arid (1.28) into (1.24) and (1.25) gives .861 - 6 2 = 0 (1.30) -.262 + 6 3 = 0 (1.31) Equations (1.29), (1.30) and (1.31) represent a system of three linear equations in three unknowns which has the unique solution 61 = 25/49, 6 2 = 20/49, 63 = 4/49 (1.32) Hence in the optimal design, raw materials contribute 25/49 of the total cost, machine one contributes 20/49, while machine two contributes the remainder or 4/49. Note that to find this optimal cost distribution, the optimal values of the variables n and r did not have to be found. 11 Since the sum of the weights is equal to unity then the mini-mum cost can be written as 4* = + * < f i l + 62+ 63) = ^ 1 . ^ 2 . ^ 3 ( 1 . 3 3 ) Substituting (1.26), (1.27) and (1.28) into (1.33) gives = ( m ^ ^ H ^ ^ ^ 2 ( i o ^ i i ^ 3 ( 1 . 3 4 ) Rearranging, = ( j y M )«l (^^ )«2 ( I P 0 0 )«3 n * C-8«l " « 2 ) r*<-- 2 62 + 63) ( 1 > 3 5 ) 6j 6 2 63 But by (1.30) and (1.31) the exponents of n* and r* vanish so that ^ = (1000 6 ! 40M « 2 1000 63 (1.36) 01 o 2 03 1 i ,1000 ,25/49,4000 .20/49,1000,4/49 . „ Q ,. ° r * (25749) (20749) (4749 ) =4359.61 (1.37) i.e. the minimum cost is $4359.61. The optimum design variables can now be found. From (1.28) _4 = lOOOr* 49 4359.61 or r* = 0.36 (1.38) Similarly from (1.27) 20 = 4000 49 (4359.61)n*(.36)«2 ° r n* =2.74 (1.39) These are the identical values to be obtained from a solution of the nonlinear equations (1.24) and (1.25). It i s clear that the geometric programming approach was advantageous in that i t required only minor computation of linear equations to obtain a solution. 12 Consider a second example, similar to one appearing in Eveleigh, 1967. For a system that is described by the transfer function G ( s ) " i w * "T7 TT 77 S + X 2S + X}S + 1 the objective is to find values for the system parameters, X j , and x 2 , which w i l l minimize the Integral Square Error (ISE) for the system. This is given by: 2 2 2 Xn "t" Xi Xn Xi X. X-ISE = ~2= J 2 L. = _L + _ 2 (1.41) 2 ( X l x 2 - 1) 2 2 ( X l x 2 - 1) As before, the optimal variables x* and x* may be obtained by setting the f i r s t p a r t i a l derivatives equal to zero USE = 1 _ 2 X 2 3 _ 0 ( 1 , 2 s 3x2 2 ( 2 x l X 2 - 2)2 " 0 USE = 2 x 2 2 X 1 X ? 2 = o (1 43) 3x2 ( 2 X l x 2 - 2) ( 2 X l x 2 - 2 ) z U U . f J ; and solving the nonlinear equations. Alternately, let X q < 2x 2x 2 - 2 (1.44) and substitute into (1.41) to obtain 2 x x ISE = + (1.45) o Thus a related problem is to solve the posynomial objective function (1.45), subject to the constraint -T2— + < 1 (1.46) 2 x 2 X 2 2 x j x 2 -To show the concepts and development of geometric programming as they apply to this problem, consider the introduction of the Lagrangian 13 m u l t i p l i e r X i to form the augmented equation. x 2 x 9 (j, = _ L + + X J C — ° — + (1.47) y a 2 X Q l v 2 x ! X 2 2xxx2 Now l e t &i and 62 be r e s p e c t i v e l y the f r a c t i o n s of the t o t a l ISE r e -presented by the f i r s t and second terms i n the ISE and l e t 63 and 64 be the r e s p e c t i v e f r a c t i o n s of the r e s t r a i n i n g equation. 6 2 = ^  (1.48) * 2 - f y c i - 4 9 ) o r a X, x 63 = 1 0 (1.50) 2 xl x 2<t> a. 2X 2 x 2 X 2 ( ) > a The optimum of the augmented equation, (1.47), can be found by the t r a d i t i o n a l method of t a k i n g p a r t i a l d e r i v a t i v e s and equating to zero. Thus, 9d> .. X x 2X . x X x 2X _ J L = I _ 1 0 1 = i _ f _ L _ 1 0 _ _ i ) = 0 (1 52) 9 X 1 2 2 X l 2 x 2 2 X l 2 X 2 X 1 2 2 X 1 X 2 3<|> 2x X x 2X, 1 2x 2 X x 2Xn = _ 2 _ _ l o _ _ J _ ^ = l _ ( _ ^ . . ^ o = 0 (1.53) 3xo x „ 2 2 X p v x „ «, • ' z o 2xi x 2 2xjx 2 * o 2x xx 2 2xjx 2 3(f) x 2 X .. x 2 X x ^ = _ _ ^ . + _ i _ = i ^ ( _ ^ - + _ J _ o ) = 0 (1.54) 3x 2 2x,x, x x 2 x i x / o x 1 z o o 1 o S u b s t i t u t i n g (1.48) - (1.51) i n t o (1.52), (1.53) and (1.54) gives 61 - 63 - 6 U = 0 (1.55) 26 2 - 63 - 8h = 0 (1.56) - 6 2 + 5 3 = 0 (1.57) 14 Equations (1.55), (1.56) and (1.57) are termed the orthogona-l i t y c o n d i t i o n s . The no r m a l i t y c o n d i t i o n , a p p l i c a b l e only to the p r i m a l o b j e c t i v e f u n c t i o n i s formed by n o t i n g that the f r a c t i o n s , 6 j and 6 2 , of (1.45) must sum to u n i t y . Therefore, 61 + 6 2 = 1 (1.58) Equations (1.55) through (1.58) represent a system of four l i n e a r equat-ions i n four unknowns which has the unique s o l u t i o n Si* = 2/3, 6 2* = 1/3, 6 3 * = 1/3, Sk* = 1/3 (1.59) The general r e l a t i o n s h i p of the a r i t h m e t i c mean to the geometric mean f o r a s e r i e s of weighted terms s t a t e s that the weighted a r i t h m e t i c mean i s equal to or greater than the corresponding weighted geometric mean. From (1.47) t h i s r e l a t i o n s h i p can be expressed as: x x 2 A x 2A r a 1 261 ^ o 2 2 x 1 X 2 6 3 H 2 x i x 2 6 i / x ~ x 2 ~ A x r 2A r. Equation (1.60) i n d i c a t e s the t r a n s i t i o n from the p r i m a l problem ( m i n i -mizing the a r i t h m e t i c s e r i e s ) to the as s o c i a t e d dual problem (maximizing the geometric s e r i e s ) . At the extremes there i s e q u a l i t y between the expressions. Hence d, * = ( ^ ) 6 l ( I _ ) 6 2 / J L ) 6 3 / L - ) 6 ' t X l * ^ l - 6 3 - S u *26 2-6 3-6 l t *-6 2+6 3 , 6 3 + 6 4 9 a ^ 2 6 i ; \ 5 2 ; ^26 3 ; ^ 6 t / 1 2 o 1 (1.61) But by (1.55), (1.56) and (1.57) the exponents of x j * , x 2 * , and X q * vanish so tticit - <2^Sl<t/2<4>63<t;>5' ,Xi63+6" ( 1 - 6 2 ) and from (1.50), (1.54) and (1.55) 15 ^ ( S Q + S ^ ) = 1 or «3+Sif = * i (1.63) Thus, or V " ( 2 ( 2 / 3 ) ) ( ( l / 3 ) ) ( 2 ( l / 3 ) } ( ( l / 3 ) } l 1 ' 3 * 1 ' 3 ) ~ 3 ' 2 (1.65) The theory of geometric programming s t a t e s t h a t the maximum of the dual problem i s equal t o the minimum of the p r i m a l problem. Hence the m i n i -mum of the ISE i s 3/2. I t should be noted that t h i s was obtained w i t h -out f i r s t o b t a i n i n g the optimum of the design v a r i a b l e s , x^ and x 2 . The r e l a t i o n s h i p between the optimal p r i m a l and dual v a r i a b l e s i s given by (1.17) and (1.18). S u b s t i t u t i n g i n values obtained from (1.59) and (1.65) y i e l d s x * 6!*«|>a(6*) = 1 (1.66) x * 2 = S2*<i>a(6*) = i (1.67) o X o * 6 3 * 2 X ! * x 2 * = 6 3* + &u* = * (1.68) i - r = T - ^ r - r = i (1.69) x i * x 2 * 63* + 614.* S o l u t i o n of these equations y i e l d s the optimal parameters x i * = 2, x 2 * = 1, which may be v e r i f i e d by s u b s t i t u t i n g i n t o Equations (1.46) and (1.4.3). 16 II. APPROXIMATION TECHNIQUES 2.1 Standard Approximation Procedures The direct application of geometric programming as an optimiza-tion technique i s restricted by two main limitations - for problems with high 'degree of d i f f i c u l t y ' , the solution could require a multi-variable maximization procedure to be employed in addition to the geometric pro-gramming procedures; and the requirement that each term in the primal objective and constraining equations be positive. Recently several techniques (Avriel, Dembo and Passy, 1975; Avriel and Williams, 1970; Avriel and Williams, 1971; Duffin, 1970) have been developed to over-come the limitations of nonpositive terms. In general, these methods require the introduction of new variables in any objective function which contains negative coefficients, or a rearrangement of the inequa-l i t y constraints i f these contain negative terms. Unfortunately, the resultant functions may s t i l l contain many complicated and inconvenient terms. The method of Duffin, Peterson and Zener, 1967, in their exposi-tory book does not require the introduction of additional terms and yet can s t i l l handle some nonposynomial functions. Hence i t may be potent-i a l l y more useful in the solution of many engineering problems. In this method, a function f (t^ , t 2 , . j t ^ ) that i s not a posynomial might be approximated by a posynomial. There is no unique method of doing this but the following i s typical. Suppose that f ( t 1 , t 2 , . . . , t m ) = g ( t l 5 t 2 t m) + h ( t 1 } t 2 , . . . ,t m) (2.1) where g is a posynomial and h i s not. To approximate h by a single term posynomial a rough estimate of the range of va r i a b i l i t y of each variable t i s made. Let t., be the geometric mean of this range. Then 17 (t p t 2 , . . . ) i s termed the operating p o i n t . Now, i f u(t) i s a s i n g l e -term posynomial such t h a t u ( t ) = c t l a i t 2 a 2 . . . t (2.2) then u ( t ) = u ( t ) A ) a i ( 3 a 2 . . . ( 3 a m (2.3) t T t o t„ m and A a., j = 1,2,...,m (2.4) thus, h i s approximated as t t t h ( t ) % h(t)(-J-)aH^-)a2...(-r)am (2.5) t i t o t 1 ^ m where ai = ^ li~ ]t=F » j = 1' 2 m ( 2 - 6 ) This approximation i s equivalent to expanding l o g h i n a power s e r i e s i n terms of the v a r i a b l e s z_^  = log(t_./t_.) and n e g l e c t i n g a l l but the l i n e a r terms. I f h ( t ) i s p o s i t i v e , f i s approximated by a posynomial The approximation i s such t h a t f and the posynomial have the same value and the same f i r s t p a r t i a l d e r i v a t i v e s at the operating p o i n t t . 2.2 Approximation Using a Non-derivative Technique I t i s i n t e r e s t i n g to note that s i m i l a r approximation r e s u l t s can be obtained without the use of d i f f e r e n t i a t i o n . This r e s u l t i s derived from the b a s i c a r i t h m e t i c - geometric r e l a t i o n s h i p s and the r e s u l t a n t e q u a l i t y at the extremes. Consider the expression a. . ^ a c.t-I -I g(t) = I . t l a i l . . . t / i n i (2.7) 18 where can be e i t h e r p o s i t i v e or negative. At the o p e r a t i n g p o i n t tj,t2»...,t » (2.7) takes on the value g(t) and each term c o n t r i b u t e s a f r a c t i o n of t h i s t o t a l equal to i t s value d i v i d e d by the t o t a l . That i s c.t a i l . . . t a i m A. = 1 1 _ m (2.8) 1 g (t) Now the a r i t h m e t i c - geometric r e l a t i o n s h i p i s given by: c t a i l t a i m X- c . t ."11 . . . t a i * * n ( r ± ± ) A i (2.9) l i i and at the extremes, e q u a l i t y h o l d s . Hence c . t . a i l . . . t a l m g(t) n ( - ^ - i - 2_— . g ( t ) ) A i (2.10) c . t ^ i l . . . t 3 i m i m Since the sum of the weights, A^, must add to u n i t y , then t a H . . . t a i m g ( t ) ^ I(t) n ( _ 1 g T " m ) A i (2.11) t a i l t i m l m where g(t) i s now a posynomial. An a d d i t i o n a l and important b e n e f i t to be r e a l i z e d by t h i s approximation and the one of Sect i o n 2.1, i s t h a t although the o r i g i n a l expression may contain s e v e r a l terms, the approximation can condense them i n t o a s i n g l e term. Hence, techniques of t h i s type may be used as a means of reducing the degree of d i f f i c u l t y of the o r i g i n a l problem s i n c e they can reduce the d i f f e r e n c e between the number of terms and the number of v a r i a b l e s i n the p r i m a l problem. With i n i t i a l l y appearing to be u n i v e r s a l l y a p p l i c a b l e , these methods must be thoroughly i n v e s t i g a t e d before being a p p l i e d to engineer-i n g problems. However, the techniques of approximation, together w i t h the features of geometric programming when j u d i c i o u s l y a p p l i e d , y i e l d considerable b e n e f i t s i n the a n a l y s i s of o p t i m i z a t i o n problems. 20 I I I . GEOMETRIC PROGRAMMING APPLICATION TO A PID CONTROL SYSTEM 3.1 I n t r o d u c t i o n Many i n d u s t r i a l processes can be adequately d e f i n e d by a r e -l a t i v e l y simple mathematical model c o n s i s t i n g of a second order l a g plus dead time. The design of a c o n t r o l system f o r t h i s type of process r e q u i r e s the choice of a c o n t r o l l e r and adjustment of the system para-meters to produce an "optimum" r e s u l t , i . e . u s u a l l y o b t a i n i n g a minimum or maximum value of a performance index or cost f u n c t i o n . Geometric programming can be a p p l i e d to t h i s problem w i t h optimum or near optimum s o l u t i o n s obtained w i t h a minimum of computa-t i o n a l i n t r i c a c i e s . That i s , from an a r b i t r a r y s t a r t i n g c o n d i t i o n , the optimum i s approached i n a s i n g l e computational step by b a s i c a l l y s o l v -i n g only l i n e a r equations. 3.2 PID C o n t r o l l e r and System D e f i n i t i o n s A type of c o n t r o l l e r that i s commonly used i n i n d u s t r i a l pro-cesses i s the p r o p o r t i o n a l - i n t e g r a l - d e r i v a t i v e (PID) c o n t r o l l e r . The t r a n s f e r f u n c t i o n f o r t h i s i s G (s) = K (1 + — + T . S ) (3.1) C " ' C " TjS d" and the c o n t r o l problem r e q u i r e s s e l e c t i o n of the parameters K^, x^, and so as t o optimize a s e l e c t e d performance index such as the i n t e -g r a l square e r r o r f o r the c l o s e d loop system to which the c o n t r o l l e r i s a p p l i e d . I n i t i a l l y l e t the time delay be zero and consider a p l a n t w i t h the open-loop t r a n s f e r f u n c t i o n 2 -P 2 , n y. . 2 s + 2?o) + co n n 21 L e t t i n g n a2 = ^ n (3.3) (3.4) y i e l d s a 3 " V Y G P ( S ) " -s + a^s + a 2 (3.6) R(s) E(s) G p(s) C(s) _ i G (s) A p l a n t = — P 2 ? 0) Y n s + 2r,co s + co n n G (s) A PID c o n t r o l l e r = K (1 + + T ,s) c - C T^S d F i g . 1 System C o n f i g u r a t i o n For a closed-loop system of the form shown i n Figure 1, w i t h the cascade compensation network, G c, given by (3.1) then E(s) _ R(s) 1 + G (s)G (s) c p (3.7) S u b s t i t u t i n g (3.1) and (3.6) i n t o (3.7) gives s(s 2+a 1s+a 2) M s i = n r _ . . R ( s ) s ( s 2 + a l S + a 2 ) + - J ^ ( l + T i s + T i T d s 2 ) 22 s ( s 2 + a 1 s + a 2 ) s 3 + s 2 ( a!+a3K c T d ) + s ( a 2+a 3K c) + a 3 K c / T ± (3.8) L e t t i n g a a = 3 rc O T . 1 (3.9) a x = a 2 + a 3 K c (3.10) aZ ~ a l + a 3 K c T a (3.11) then, E(s) s ( s 2 + a l s + a 2 ) = _ (3.12) I f the i n p u t to the system i s a u n i t s t e p , then R(s) = 1/s, and consequently s 2 + a s + a E(s) = — — (3.13) s +aos +ai s+a o Equation (3.13) i s the e r r o r t r a n s f e r f u n c t i o n f o r the system of F i g . 1. I t has been shown (Newton, Gould, K a i s e r , 1964) that the i n t e g r a l square e r r o r , ISE, can be expressed i n terms of fun c t i o n s l i k e (3.13) by means of Parseval's theorem i n the form I S E 1 • / + J " d B c ( B ) e ( - 8 ) 2 l T3 _j„ d(s)d(-s) From tab u l a t e d values of the i n t e g r a l , the closed-loop system o b j e c t i v e f u n c t i o n i s given by: a a, + (a,2-2a„)a + a 2 a T O T , o 1 1 2 ° 2 2 ... ISE = • = — ^ (3.14) 2 a o a l a 2 ( l — ) ° 1 A a i a 2 The o p t i m i z a t i o n problem i s to determine values of a Q , a\, and a 2 that minimize Equation (3.14). 23 3.3 C o n s t r a i n t D e r i v a t i o n As many aspects of design theory apply to mathematical models, the usefulness of the theory i s determined by how c l o s e l y the model agrees w i t h the p h y s i c a l problem under c o n s i d e r a t i o n . For example, o p t i m i z a t i o n could y i e l d a mathematically r e a l i z a b l e system i n theory but one that p h y s i c a l l y r e s u l t s i n u n r e a l i s t i c responses. That i s , the act of o p t i m i z a t i o n could d r i v e s i g n a l s i n parts of the model correspond-i n g to the f i x e d elements of the c o n t r o l system to such hi g h peak values t h a t the model i s no longer a v a l i d approximation. Thus the design theory i s r e s t r i c t e d i n usefulness. One method of avoi d i n g t h i s condi-t i o n i s to impose c o n s t r a i n t s upon the system to in s u r e that the r e s u l t r ant optimum parameter choice y i e l d s a system that i s p r a c t i c a l . For the PID c o n t r o l l e r problem, a c o n s t r a i n t c o n d i t i o n that could be imposed upon the system i s to force the output response and consequently the e r r o r , e ( t ) , to be a s p e c i f i e d value at some s p e c i f i e d time, t . In t h i s manner, the response can be c o n t r o l l e d so as to avoid a c o n d i t i o n of over-driven s i g n a l s w i t h i n the system. From (3.13) s 2 + + a E(s) = 3 z • • C 3 - 1 5 ) s +o t2S +axs+a Q T h e r e f o r e sHl+aJs+aJs*) E ( s ) = — : : 1 r (3.16) s (l+o^/s+cq/s +a Q/s ) D i v i d i n g the numerator by the denominator and d i s r e g a r d i n g terms greater than s1* gives a -cu • a -a - a ; a +a 2 2a a +a 2 - a a -a a -a 3^ a . E(s) % s ^ l + 2 1 1 2 2 + 1 2 2 1 1 2 2 2 . °)(3.17) S 2 o s s which y i e l d s i n the time domain, 24 :(t) = 1 + (a!-a 2)t + (a 2-a 1-a 1a2+a 2 2)|— + (2a 1a 2+a 2 2a 1-a 1a 1-a 2a 2-a 2 3-a )t 3/6 (3.18) By rearranging terms,(3.18) can be written as a 1(t 2/2+a 1t 3/6)+ a 2(t+a 1t 2/2+a 2t 3/6) + « 2 3t 3/6 - 2(t 3/6)a 1a 2 -(ait 3/6+t 2/2)a 2 2 = 1 - e(t) + ait + a 2t 2/2 - a Qt 3/6 (3.19) Now, c(t) = r(t) - e(t) (3.20) and for a unit step input, r(t) = 1 (3.21) Therefore c(t) = 1 - e(t) (3.22) and c(t) + axt + a 2t 2/2 - a Qt 3/6 = a 1(t 2/2+a 1t 3/6) + a 2(t+a!t 2/2+a 2t 3/6) + a 2 3 t 3 / 6 - 2 ( t 3 / 6 ) a i a 2 - ( t 2 / 2 + a i t 3 / 6 ) a 2 2 (3.23) In general a Q << 1 and the term a Qt 3/6 can be disregarded. Thus by specifying a value of c(t) at a time t, the L.H.S. of Equation (3.23) is a constant, say Y. Therefore, the constraint may be written as Y = b^ai + b 2 a 2 + b 3 d 2 3 - 2b3040^ - b j a 2 2 (3.24) where Y A c ( t ) + a :t + a 2t 2/2 (3.25) b x A t 2/2 + a ] l t 3 / 6 (3.26) b 2 A t + a!t 2/2 + a 2t 3/6 (3.27) b 3 A t 3/6 (3.28) The optimization problem is now to minimize a a + (a 2-2a 9)a + a 2 a 9 ISE = 1 n 2 . ° . 2 2 (3.29) 2a o a i a 2 ( l - a 0 / a i a 2 ) subject to the constraint Y > b\a\ + b20i2 + b 3 a 2 3 ~ 2b3a^a2 - b i < x 2 2 (3.30) Inspection of (3.29) and (3.30) indicate that the equations contain both positive and negative terms, and as such, are not solvable by ordinary geometric programming procedures. In addition, the number of terms compared to the number of variables, indicates a high degree of d i f f i c u l t y . To apply geometric programming to the optimization problem then requires some degree of approximation of the two functions in order to reduce the degree of d i f f i c u l t y and eliminate the negative terms. The following Sections contain various ways of approximation and a comparison of the results achieved by the different methods. 3.3.1 Approximation of the Constraint by a Straight Line Consider the plant of Fig. 1 to have the following parameters a x = 2 ? c o n = 0.25 (3.31) a 2 = o)n2 = 0.01 (3.32) a 3 = u 2y = 0.03 (3.33) 3 n ' In addition, i t i s desired to make the output c(t) have the value 0.5 at time T = 0.25 seconds. Scaling the time axis by a factor of 10, and substituting the above values into the constraint equation, (3.30), gives: Y = 1.15625 = 3.776a! + 3.307a2 + 2.604a 2 3 - 5.2080^2 - 3.776a22 (3.34) A plot of this equation is shown in Fig. 2. Since a large portion of the curve i s nearly linear a simple method of approximating the con-straint i s by a straight line of the form c i a i + c 2 a 2 = 1 (3.35) Figure 2 Constraint Equation 27 To determine the values of the parameters Cj and c 2, a technique that matches ( 3 . 3 4 ) and ( 3 . 3 5 ) at one point plus equating their slopes i s used. (1) at the point a 2 = 0 a i = — = . 3 0 6 2 ( 3 . 3 6 ) 1 c 2 Therefore, c 2 = = 3.2669 (3.37) (2) at the point a2= 0, a1 • - 0.3062 the slope of (3.34) i s d ^ " " 3.307 - ?'.208(.3062) = - 2 ' 2 0 5 2 ( 3 * 3 8 ) and from (3.35) do^ c-^  dcti c 2 (3.39) From (3.37), (3.38), and (3.39) c 2 = 1.4815 (3.40) Therefore the original constraint equation has been approximated by the straight line, 3.2269cti + 1.4815a2 = 1 (3.41) A plot of this function and Equation (3.34) are shown in Fig. 3. The optimization problem is now to minimize (3.29) subject to the constraint of (3.41). Inspection of the equations indicate that only the denominator of (3.29) contains a negative coefficient (since, from (3.31) and (3.32), a j 2 > 2a 2) and, as such, the expression is not a posynomial. However, for small, a -1 a ~ O s ri, , , O (1 — ) % 1 + — — (r> A O N a j a 2 a j a 2 (3.42; and ct a n +(a-,2-2a9)a + a 9 2 a 9 a I S E % S L I — ; - ^ - a + —2-) 2 a Q a i a 2 a i a 2 ' ( 3 . 4 3 ) 29 Both (3.43) and (3.41) are posynomials, containing a total of eight terms. Thus, N = 8 (3.44) P and, since there are three primal variables, m = 3 (3.45) From (1.22) degree of d i f f i c u l t y = N - m- l = 4 (3.46) which indicates that a unique solution would be d i f f i c u l t to obtain by geometric programming. To overcome this, the number of terms in Equat- ' ion (3.43) can be reduced to ultimately yield a zero degree of d i f f i c u l t y problem. Since (3.41) contains two terms, this requires (3.43) to be condensed to a total of two terms also. Take as a nominal starting point O ! = .1531, a 2 = .3375 (3.47) which l i e s at the approximate mid-point of (3.41). The i n i t i a l value a Q, a Q, is chosen as that value which w i l l minimize (3.43). That i s , | I E . ( 1 + A . , ( . ^ l , + J J ^ < i + ^ ! ± i + V ( 3 . 4 8 ) 3a b a i a 2 2 a 2 „ a i a 2 2a 2 2 a i a 2 2 a 0 a f o 1 Solving (3.48) gives , , a 2 a 2 a J a-i + ajz - 2a 2 Substituting, (3.31), (3.32), and (3.47) into (3.49) yields a Q = 0.0031 (3.50) Condensation of the ISE into two terms i s done by the method outlined in Section 2.1 . 30 L e t t i n g , then f o r cj>l A 1 + 04012 aQ = i.0031, a i = .1531, 02 = .3375 (3.51) (3.52) and ch = 1 + — 3 - - 1.609 040:2 3 + n 8 a, = 19.3531 a i ctja2 (3.53) (3.54) 3<fr. 3ai ,3919 cq a j ^ a 2 (3.55) Now 3cj> 3a2 b = 04012 = -.1778 _ _ -1 o 3a 1 o ,0566 (3.56) (3.57) _ 9<f>! _ -1 b l = 0 1 1 804 ' * J = " - 0 5 6 6 (3.58) _ 3*1 _ -1 t>2 = ot2 "g^J/ * = -'0566 (3.59) and, a Q a ! a 2 (3.60) S u b s t i t u t i n g (3.52), (3.53), and (3.57) through (3.59) i n t o (3.60) gives as an approximation to (3.51), the s i n g l e term expression <h Jfe 1.2440(-^-)- 0 5 6 6 Y 1 v aja2 (3.61) 31 Similarily, letting a a, + (a,2-2a„)a + a 2a„ i . O 1 1 2 o 2 2 4>2 A n (3.62) 2a oa 1a 2 then, by rearranging terms n (a 2-2a ) a 2 • 2 = T^- (1 + ~ ) + ( 3- 6 3> ^ 2a 2 aj 2a Qai If a i 2 - 2 a o k A 1 + — (3.64) « 1 then _ a 1 2 k-2a 2 k = 1 + —=z = 1.2776 (3.65) and o „ a, 2-2a 3k 3ai 1 " 2 = -1.8131 (3.66) - 3k r r _ 1 bo " 0 1 1 37^" * k l = -.2173 (3.67) Hence, " l b k Rj k ( 7 - - ) o = .8497a!"- 2 1 7 3 (3.68) <*1 Now from (3.63), <f>2 = .4249ai"" 2 1 7 3a 2 - 1 + . 00005a o ~ 1 ar 1 (3.69) and since, ISE = * i - *2. (3.70) therefore, ISE % . 5 2 8 6 a o - 0 5 6 6 a I - - 2 7 3 9 a 2 - 1 . 0 5 6 6 +... . 0 0 0 0 6 2 a o _ ' ^ V " 1 * 0 5 6 6 a 2 ~ ' 0 5 6 6 (3.71) Eq. (3.71) i s an approximation >6f (3.43) in which the number of terms has been reduced to two. The optimization problem i s now: Minimize ISE & . 5 2 8 6 a o - 0 5 6 6 a r - 2 7 3 9 a 2 - 1 - 0 5 6 6 + .000062aQ- 9 4 3 t a i - l . 0 566 0 2-. 0566 (3.72) 32 subject to 3.2669ai + 1.4815a2 < 1 (3.73) Applying geometric programming to the above problem results in the formulation of the dual problem Maximize 62 03 04 subject to the normality arid orthogonality conditions given by (1.11) and (1.12) Therefore &l + <52= 1 (3.75) .05666! - .943462 = 0 (3.76) -.27396! - 1.056662 + 63 = 0 (3.77) -1.05666i - .056662 + 64 = 0 (3.78) Solution of Equation (3.75) through (3.78) results in 61* = 0.9434 (3.79) 6 2* = 0.0566 (3.80) 63* = 0.3174 (3.81) 64* = 1 (3.82) and substitution of these values into/ (3.74). gives .' I|J(6*) = 1.7471 (3.83) From (1.14) the maximum of the dual function i s equal to the minimum of the primal objective function Hence min ISE = max ty(S*) = 1.7471 (3.84) By Equations (1.17) and (1.18) the primal variables that give this 3 3 r e s u l t are determined from ( 1 ) 0 . 5 2 8 6 a o - 0 5 6 6 a r - 2 7 3 9 a 2 - 1 - 0 5 6 6 = 61*^(5*) = 1 . 6 4 8 3 ( 3 . 8 5 ) ( 2 ) 0 . 0 0 0 0 6 2 a 0 _ - 9 l + 3 I + c x r 1 - 0 5 6 6 a 2 " - 0 5 6 6 = 6 2 * * ( S * ) = 0 . 0 9 8 9 ( 3 . 8 6 ) ( 3 ) 3 . 2 6 6 9 a i = 6 3 * / ( S 3 * + 6^*) = 0 . 2 4 0 9 ( 3 . 8 7 ) ( 4 ) 1 . 4 8 1 5 a 2 = 6 ^ / ( 6 3 * + 6 ^ * ) = 0 . 7 5 9 1 ( 3 . 8 8 ) The values of a , a i , and a ? t h a t simultaneously s a t i s f y ( 3 . 8 5 ) through o ( 3 . 8 8 ) are a * = 0 . 0 0 7 8 ( 3 . 8 9 ) o a j * = 0 . 0 7 3 7 ( 3 . 9 0 ) a 2 * = 0 . 5 1 2 4 ( 3 . 9 1 ) From Equations ( 3 . 9 ) , ( 3 . 1 0 ) , and ( 3 . 1 1 ) , the system parameters are given as . o i j * - a 2 K = 2 . 1 2 ( 3 . 9 2 ) a K x i = = 8 . 1 7 ( 3 . 9 3 ) o a * - a d a 3 K c A p l o t of the output response of the system f o r a .unit step displacement input i s shown i n F i g . 4 . From the p l o t i t can be seen that the c o n s t r a i n i n g c o n d i t i o n has been met s i n c e the output has an approximate value of 0 . 5 at a s c a l e d time of t = 2 . 5 seconds. A d d i t i o n -a l l y , a measure of the l a r g e s t e r r o r between input and output during the t r a n s i e n t s t a t e i s c a l l e d overshoot and can be defined by _. ^ , Maximum overshoot , „ A / o n c \ Per cent overshoot = — — - — - , — : z ; — x 1 0 0 ( 3 . 9 5 ) F i n a l d e s i r e d value 1.6 + 0.0 4 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 F i g u r e 4 O u t p u t R e s p o n s e - L i n e a r C o n s t r a i n t 4.5 5.0 t - t i m e 35 From Fig. 4 Per cent overshoot = x 100 = 15% (3.96) The settling time can be defined as the time required for the response to decrease to and stay within a specified percentage, say 5 per cent of i t s f i n a l value. From Fig. 4 T A settling time = 2.7 seconds (3.97) s - ft Finally a measure of the degree of conformity of the approximations of (3.72) and (3.73) to the actual value of the integral square error and constraint can be obtained by comparing a computer calculated value of ISE = 2.0479 with the value of 1.7471 calculated using geometric pro-gramming. From the above, i t can be seen that the methods and proced-ures ut i l i z e d can produce valid results with a minimum of computational complexity. 3.3.2 Approximation of the Constraint by a Curved and Straight Line Although the optimized value of the ISE obtained by making the approximations of Section 3.3.1 is close to the actual value calcu-lated with no approximations, i t is apparent that the agreement between the two can be improved. Furthermore i t is noted, from Fig. 3, that while the straight line given by (3.41) generally follows the original constraint, there are regions of large deviation. An enhancement in this approximation can be obtained i f Equation (3.41) is replaced by two curves - a straight line for that portion of the curve that appears to be f a i r l y linear (04 > 0.16) and a curved line for the non-linear portion of the constraint (04 < 0.16). With values previously given, from Equations (3.31), (3.32), and (3.33) the constraint equation is 1.15625 = 3.776a! + 3.307a2 + 2.604a23 - 5.208aia2 - 3.776a22 (3.98) For the region ai < 0.16, (3.98) can be approximated by an equation of the form c i O ! + c 2 a 2 1 + c r = 1 (3.99) A method that matches (3.98) and (3.99) at two different points, plus making their slopes equal at one of the points can be ut i l i z e d to deter-mine values for C i , c 2, and o". (1) Substituting ai = 0 into (3.98) and solving for a 2 yields a 2 = .5800 (3.100) (2) Similarly at the point ai = 0.16 a 2 = .3998 (3.101) (3) The slope of (3.98) is obtained by taking the derivative of ag with respect to a ^ 9a 2 5.208a - 3.776 Therefore - r — = : — (3.102) 1 3.307 + 7.81a22 - 5.208a.! - 7.552a2 At ai = 0.16, a 2 = 0.3998, 3a 0 - 2.4090 (3.103) Now the slope of (3.99) is . 3a„ . c 1 (3.104) 8an /i . \ ' o 1 c 2 ( l + a)a 2 and substitution of the points (0.0, 0.58) and (0.16, 0.3998) into (3.99) and equating (3.103) to (3.104) gives the following c 2 ( . 5 8 ) 1 + a = 1 (3.105) ci(.16) + c 2(.3998) 1 + a = 1 (3.106) = -2.4090 (3.107) c 2 ( l + a)(.3998) a 37 which can be solved to y i e l d c x = 5.0785 (3.108) c 2 = 11.6031 (3.109) a = 3.5 (3.110) Thus the o r i g i n a l c o n s t r a i n t can be p a r t i a l l y approximated by the equation 5.0785a! + 11.63102^-5 = 1 (3.111) A p l o t of (3.111) and (3.98) i s shown i n F i g . 5. From t h i s , i t can be seen t h a t there i s good agreement between the two equations f o r the region a! < .16 but f o r a\ > .16, the approximation i s no longer v a l i d . However, i n t h i s r e g i o n , the o r i g i n a l c o n s t r a i n t appears to be almost l i n e a r , and i t can be approximated by a s t r a i g h t . l i n e of the form C 3 a ! + ci+a.2 = 1 (3.112) Choosing two poi n t s on the o r i g i n a l c o n s t r a i n t , say (0.16, 0.3998) and (0.306, 0) and s u b s t i t u t i n g i n t o (3.112) gives c 3 = 3.2680 (3.113) ck = 1.1934 (3.114) Thus, the a d d i t i o n a l approximation of the c o n s t r a i n t curve i s 3.2680ai + 1.1934a2 = 1 (3.115) f o r the region a^ > 0.16. A p l o t of both approximations plus the o r i -g i n a l c o n s t r a i n t curve i s shown i n F i g . 6. I t i s apparent t h a t the agreement i s good. The o p t i m i z a t i o n problem i s now given by: a a, + (a 2-2a )a + a 2 a Minimize ISE = —± 2_| (3.116) 2a a i a 2 ( l -o 1 ^ a i a 2 actual constraint approx. of constraint .2 .3 .4 .5 \ 6 Figure 5 Partial Approximation of Constraint — - — actual constraint approx. of constrain 3.2680a 1+ 1.1934a 2 = 1 .3 .4 .5 Y Two Line Approximation of Constraint Subject to 5.0785a! + 11.6031O211* 5 <1 (3.117) a n d 3.2680ai. + 1.1934a2 < 1 (3.118) To f a c i l i t a t e a solution to this problem i t i s necessary to make the same type approximations and condensation of (3.116) as used i n Section 3.3.1. In addition, since there are two constraint equations involved, computational complexity can be avoided i f the problem i s solved i n a piecewise fashion. That i s , i f each constraint i s considered individ-ually i t i s possible to determine i f only one i s vali d and at the same time, the degree of d i f f i c u l t y associated with the problem w i l l remain low. Fir s t consider (3.118) to be the effective constraint and take as i n i t i a l conditions, the point ai = 0.235, a 2 = 0.2. In a manner similar to that of Section 3.3.1, (3.116) can be approximated by the two term expression ISE * .5535a£)- 0 3 6 1 t a i - - 1 8 9 6 a 2 - 1 - 0 3 6 t * + . 000059a o-' 9 6 ^ a j " 1 • 0 3 6 t * a 2 " - 0 3 6 4 (3.119) Now consider a bound on , that is > a. Thus, the optimization pro-blem i s to minimize (3.119) subject to (3.118) and a o - 1 < 1 (3.120) Applying the concept of duality to the problem results i n the formation of the alternate problem of maximizing the dual objective function subject to the constraints of normality and orthogonality. This is given as M a x . K 6 ) . ^ j f i ^ J ^ ^ S ^ ^ 6i 6 2 63 04 05 (3.121) 41 subject to 6i + <S2 = 1 (3.122) .03646! - .963662 = 0 (3.123) -.18966! - 1.036462+63 - 65 = 0 (3.124) -1.036461 - .036462 + 64 = 0 (3.125) Since (3.122) through (3.125) represent a set of four linear equations in five unknowns, there i s no unique solution. However, solving in terms of the variable 63 yields 61* = .9636 (3.126) 6 2* = .0364 (3.127) 63* = 6 3* (3.128) 64* =1 (3.129) 65* = 63* - .2204 (3.130) Substituting into (3.121) now gives 53*-.2204 (3.131) Hence = . 5 4 1 5 ( 1 ^ ) 6 3 a 6 3 * - J 2 2 0 \ 6 3 * + l ) 6 3 * + l (3.132) 63* It has been shown (Duffin, Peterson, Zener, 1967) that the functions ip(6) and itn IJJ(6) have the same maximizing points. Consequently the maximum value of (3.132) can be obtained by taking the natural logari-thm of both sides of the equation, taking the derivative with respect to 63* and then equating to zero. This gives 42 Jin 1K6*) = -.6134+6 3*[An 3.2680-J>n6 3*] + (6 3*-. 2204) Jin a + (6 3*+l) 4n(6 3*+l) (3.133) and 6 3* - 5 3 * + 1 = *n 3* 2 6 8 0 - &n63* - «F + *n a + £ n ( 6 3 * + D + (fipqa) - o (3.134) Therefore Jin 3.2680a = Jin[6 3*/(6 3*+l) ] (3.135) and 6 g* 3.2680a = ^ (3.136) Note 32£ml> 1 , 1 .1 _ , . . _ = _ "F~T + J " " x *v< < 0 for 6 3* > 0 36 3*2 5 3* 63*+l <53*(S3*+1) * Equation (3.136) indicates the conditions for a maximum value of the dual objective function. In addition, from Equation (1.18) the follow-ing holds 5 * 3.2680ai = (3.137) and _i aa1 = 1 (3.138) Substituting a = a-^ into (3.137) yields 3 ' 2 6 8 0 a = 6 l l + T ( 3' 1 3 9 ) which is identical to that condition necessary for a maximum of the dual function as shown in Equation (3.136). This implies that the optimum solution l i e s on the line ai = a. Consequently, the region for 04 > a can be disregarded in the search for a global optimum. That i s , i f a = 0.16, then 0 4 * has a value of 0.16 or less and the constraint given by 3.2680a! + 1.1934a2 = 1 43 should no longer be considered a p p l i c a b l e s i n c e the optimum l i e s i n the region 04 < 0.16. The problem now c o n s i s t s of minimizing (3.116) s u b j e c t to the s i n g l e c o n s t r a i n t given by (3.117). Taking a = 0.16 and s u b s t i t u t i n g i n t o (3.136) y i e l d s 6 3 * = 1.0960 (3.140) Therefore, from (3.132) ty(6*) = .5415(^||||) 1- 0 9 6 0(.16)- 8 7 5 6(2.0960) 2- 0 9 6 0 = 1.6997 (3.141) Equation (3.141) gives a sub-optimal value to the problem t h a t i s a t t a i n a b l e w i t h the c o n s t r a i n t s imposed by (3.117) and (3.120). Under these c o n d i t i o n s the f o l l o w i n g also holds true: .5535ao*- 0 3 6 4 a 1 * " ' 1 8 9 6 a 2 * " 1 , b 3 6 ' t = 6i*iK<5*) = 1.6375 (3.142) .000059a o* _- 9 6 3 6a 1* _ 1- 0 3 6 i t a 2 * - - 0 3 6 l f = S 2 *K<5*) = .0619 (3.143) 3.2680a!* = 6 3 * / ( < 5 3 * + 6 \ * ) = .5229 (3.144) 1.1934a2* = 6 4 * / ( 6 3 * + 64 * ) =.4771 (3.145) .16a!*"1 = 1 (3.146) A s o l u t i o n to the above set of equations gives a * = 0.0053 (3.147) o a i * =0.16 (3.148) a 2* = 0.3998 (3.149) With these values,(3.116) i s re-condensed i n a manner s i m i l a r to before to give ISE % . 5 6 4 9 a ; 0 7 6 8 a 1 - - 2 8 6 7 a 2 - 1 ' 0 7 6 8 + . 000066a Q- 9 2 3 2 af1 * 0 7 6 8 a 2 " ' 0 768 (3.150) which i s now to be minimized subject to the c o n s t r a i n t 5.0785ai + 11.6031a2l+-5 < 1 (3.151) 44 Applying the concept of duality results in maximizing ,.5649N61 ,.000066,69 /5.0785 x6q /11.6031 N6 U /, ,63+61, „ ^(5) = ( — 5 - - ) ( ^ — ) M — ^ — ) d ( ^ — ) 4 ( 6 3 + 6 i t ) d (3.152) subject to Si + 6 2 = 1 (3.153) .07686! - .923262 = 0 (3.154) -.28676i " l-076862 + 63 = 0 (3.155) -1.07686! - .076862 + 4';56g = 0 (3.156) Equations (3.153) through (3.156) can be solved uniquely to give 61' = 0.9232 (3.157) 6 2 ' = 0.0768 (3.158) 6 3 * = 0.3474 (3.159) 6^' = 0.2222 (3.160) Therefore, *<6*> = ( ^ ^ ) ' 9 2 3 2 ( - ^ § ^ = 1.6397 (3.161) which i s the optimum value of the integral square error. From (1.17) and (1.18) the value of the system parameters that give this optimum are determined from . 5 6 4 9 a o - 0 7 6 8 a i - - 2 8 6 7 a 2 - 1 - 0 7 6 8 = = 1-5138 (3.162) .000066a o-- 9 2 3 2ai- 1- 0 7 6 8a 2--° 7 6 8 = 6 2*i|)(6') = .1259 (3.163) 5.0785cq = 6 3 ' / ( 6 3 ' + 61/) = .6099 (3.164) l l . e O S l a ^ - 5 = 6i t ,/(6 3' + 6 ^ ' ) = .3901 (3.165) 45 which yield a * = .0035 (3.166) o oV = .1201 (3.167) a 2 ' = .4705 (3.168) Substitution of (3.166), (3.167) and (3.168) into (3.9), (3.10) and (3.11) gives the following optimum controller parameters K = 3.67 (3.169) c T ± = 31.46 (3.170) T . = 2.00 (3.171) a Fig. 7 i s a plot of the output response of the system with the above controller parameters. From the plot i t is seen that 06 Per cent overshoot = ^ * QQ X 100 = 6% and T =1.15 seconds s It i s also evident that the constraint condition (e(t) = .5 at t = 2.5) is met. The computer calculated value of the integral square error i s 1.6612 which compares favorably with the value of 1.6397 determined by geometric programming. This indicates that the method of representing the constraint equation by two less complex expressions and condensing the objective function into a form convenient for geometric programming can yield very good results. 3.3.3 Approximation of the Constraint by the Method of Puffin,  Peterson and Zener The Duffin, et a l , approximation procedure of Section 2.1 is 1.6 very powerful as i t allows the problems of non-posynomial terms and high degrees of d i f f i c u l t y to be overcome. The method does have a drawback as the approximation i s often a very poor representation of a function throughout i t s entire domain. However, i f the condensation can be made in a region that i s close to the optimum, then i t i s pos-sible to obtain valid results. By re-arranging (3.18) an expression for the error can be written as e(t) = 1 + ait + (a 2+a 2 2)t 2/2 + (2a 2a 2 +a 1a 2 2)t 3/6 -[ a 2 t + (a 1+a 1a 2)t 2/2 + (a Q+a 1a 1+a 2a 2+a 2 3)t 3/6 (3.172) Since c(t) = 1 - e(t) (3.173) Therefore -c(t) = ait + (a 2+a 2 2)t 2/2 + (2a xa 2 +aia 2 2)t 3/6 -[ct2t + (ai+a!a 2)t 2/2 + (a o+aiai+a 2a 2+a 2 3) 13/6 (3.174) Let Si £ ot Qt 3/6 + cti(t 2/2+ait 3/6) + ct 2(t+ait 2/2+a 2t 3/6) + a 2 3 t 3 / 6 (3.175) and g 2 A c(t) + ajt + a 2t 2/2 + 2 a i a 2 t 3 / 6 + a 2 2(t 2/2+ait 3/6) (3.176) For the same plant parameters and constraint condition, i.e. e(t) = .5 at t = 2.5 as in Section 3.3.1, Equations (3.175) and (3.176) give: gl = 2.604aQ + 3.776ai + 3.307a2 + 2.604a22 (3.177) g 2 = 1.156 + 5.208040:2 + 3.776a22 (3.178) and for the constraint condition gl = g 2 (3.179) 48 then 2.604aQ + 3.776 a i + 3.307a2 + 2.604a 2 2 = 1.156 + 5.208a].a2 + 3.776a 2 5 > 2 (3.180) To determine a f e a s i b l e p o i n t around which to approximate the c o n s t r a i n t c o n d i t i o n consider the f o l l o w i n g . From (3.174), the slope of the out-put response i s " f t = 3 1 + ( a 2 + « 2 2 ) t + (2aia2 + a x a 2 2 ) t 2 / 2 -[a 2 + ( a 1+a 1a 2)t + ( a r i+a 1 a 1+a 2 a 2 + a 2 3)t 2 / 2 ] (3.181) at t = 0. - ^  = ai - a 2 (3.182) Since the de s i r e d output response has a slope of 0.2, i . e . Ac/At = .5/2.5, f o r s m a l l values of t , then s u b s t i t u t i n g t h i s i n t o Equation (3.182) gives a 2 = .2 + a1 = .2 + .25 = .45 (3.183) Now assume that a Q = 0 and since = g 2 , 3.776a! + 3.307(.45) + 2.604(.45) 3 = 1.156 + 5.208(.45) a i + 3.776(.45) 2 (3.184) from which a i = 0.137 (3.185) I t has been shown by Equation (3.44) t h a t a nominal a Q can be obtained once values of a^  and a 2 are known. From (3.183) and (3.185) then a = / : ° i 9 y «i + a r -? 2 ' a 2^a^a! = .00393 (3.186) 2a 2 S u b s t i t u t i n g (3.186) and (3.183) i n t o (3.184) now gives a! = .1297 (3.187) 49 Using ( 3 . 1 8 3 ) , (3.186) and (3 .187) a condensation of the ISE, Equation ( 3 . 1 4 ) , can be made i n a manner s i m i l a r to that p r e v i o u s l y c a l c u l a t e d . This g i v e s : ISE & . 5 0 7 6 a o - 0 6 3 3 a r - 3 1 0 8 a 2 - 1 - 0 6 3 3 + . 0 0 0 0 6 3 a o " - 9 3 6 7al~1 '• 0 6 3 3 a 2 " - 0 6 3 3 From (3 .177) l e t g = K j + 2 . 6 0 4 a 2 3 where K x A 2 .604a + 3 . 7 7 6 a ! + 3 . 3 0 7 a 2 (3.188) (3.189) (3.190) Now condense Kj i n t o a s i n g l e posynomial term around a Q , a j , a 2 by the method of D u f f i n : K x = 2 . 6 0 4 a Q + 3 . 7 7 6 a : + 3 . 3 0 7 a 2 = 1.9881 and 3K, 3a - = 2.604 a ± (3.191) (3.192) 3a i - = 3.776 (3.193) 3 a 2 _ = 3.307 From which _ 8 K i _ b - a v — _ • K i " 1 = .0051 o o 3a a-j. 1 (3.194) (3.195) 3R\ b i = _ • K i " 1 = .2463 (3.196) .3K, and (3.197) K l ^ f l ( ^ 2 L ) b o ( ^ l ) b i ( ^ 2 ) b 2 = 6 > 1 4 8 4 A O , 0 0 5 1 A I . 2 4 6 3 A 2 . 7 4 8 5 a Q a x a 2 ° (3 .198) 50 Therefore g 1 % 6.1484ao' 0 0 5 1 a 1 - 2 1 + 6 3 a 2 , 7 l f 8 5 + 2.604a23 (3.199) In a similar fashion g 2 can be condensed to give g 2 £ 5 . 6 7 8 3 a 1 ' 1 3 6 5 a 2 ; 8 2 8 9 (3.200) Substituting (3.199) and (3.200) into (3.179), 6.1484a • 0 0 5 1 a 1 - 2 I t 6 3 a 2 - 7 4 8 5 +-2.604a23= 5.6783ai' 1 3 6 6 a 2 ' 8 2 8 9 ° (3.202) which can be re-written as 6.1484an-°051ai.2463^.7^85 + 2.604a 2 3 = 1 (3.202) 5.6783a 1' 1 3 6 6a 2' 8 2 8 9 Therefore 1.0828a o- 0 0 5 1a 1- 1 0 9 7 a 2 " - 0 7 5 1 t + . 4 5 8 6 a i ~ - 1 3 6 5 a 2 2 - t 7 6 1 = 1 (3.203) To satisfy the requirements of geometric programming this i s written as an inequality constraint and the problem in an approximated form i s : Minimize ISE = .5076a o- 0 6 3 8 a 1 - - 3 2 0 8 a 2 - 1 - 0 6 3 3 + .0000.63a • 9 3 6 7 a r 1 • 0 6 3 3 a 2 " - 0 6 3 3 (3.204) subject to 1.0828ao- 0 0 5 1 a 1 - 1 0 9 7 a 2 - 0 7 5 1 t + .4586a! -- 1 3 6 6a 2 2' 1 7 6 1 < 1 (3.205) A plot of the approximate form of the constraint equation i s shown in Fig. 8. It can be seen that over a small region the approxima tion holds to the original constraint but diverges rapidly beyond the region of interest. Solving (3.204) and (3.205) in the standard geometric pro-gramming manner gives Max 4>(6*) = min ISE = 1.66226 (3.206) 3 1 1 ( 1 a_* = 0.0033 (3.207) actual constraint approx. of constraint a i * = 0.1129 (3.208) a 2 * = 0.4793 (3.209) from which, by Equations (3.9), (3.10) and (3.11) K = 3.43 (3.210) c T. X - 30.89 (3.211) T, = 2.23 d (3.212) A plot of the output response i s shown in Fig. 9. It can be seen that the constraint condition has been met and In addition, the response settles right into the 5 per cent limit. The computer calculated value of the integral square error i s 1.6707 which compares favorably with the approximate value of 1.6623. It is interest-ing to note that although the response of Fig. 9 appears better than that of Section 3.3.2 the value of the performance index i s higher (1.66226 versus 1.66106). This i s due to the fact that the response of Fig. 9 rises slower than that of Fig. 7 resulting in a slightly larger error. Per cent overshoot = .04 1,00 x 100 = 4% (3.213) 1.6+ 1.4 + ^2-Lunit step input overshoot 1.0 0.8 0.6 0.4 0.2+ 0.0 0.5 —H 3.5 -+-4.0 4.5 1-0 1.5 2.0 2.5 3.0 Figure 9 Output Response- Duffin Approx. of Constraint 5.0 t-time IV. GEOMETRIC PROGRAMMING APPLICATION TO A PID CONTROL SYSTEM WITH TIME DELAY 4.1 System D e f i n i t i o n s arid C o n s t r a i n t D e r i v a t i o n A l a r g e number of c o n t r o l systems are c h a r a c t e r i z e d by the f a c t that the output responds to a t r a n s i e n t input only a f t e r a given time i n t e r v a l . Due to the time delay e f f e c t , the t r a n s f e r f u n c t i o n of these systems are no longer quotients of polynomials, but u s u a l l y con--Ts t a i n the term e where T denotes the time delay or t r a n s p o r t a t i o n l a g . Figure 10 d e p i c t s the PID c o n t r o l l e r system w i t h time delay. R(B)+ /TX E(S) C ( S ) G (s) A p l a n t = n ' s z + 2?io s + to ' n n G (s) A c o n t r o l l e r = K (1 + + T ,s) c - C T.S d -Ts e A time delay F i g . 10 PID C o n t r o l l e r w i t h Time Delay The o v e r - a l l system t r a n s f e r f u n c t i o n i s Cfs) G c ( s ) G p ( s ) e - T s R< S> 1 + G c(s)G ( s ) e - T S c p (4.1) As a r e s u l t of the numerator e x p o n e n t i a l , there i s a d i r e c t l a g of T seconds between in p u t and output. In a d d i t i o n , the c l o s e d loop 55 performance of the system i s a f f e c t e d by the delay because of the f a c t o r -Ts e i n the denominator. For example, the s t a b i l i t y of the system i s modified by the presence of t h i s f a c t o r . In any a n a l y t i c a l a n a l y s i s , the transcendental t r a n s f e r func-t i o n has c l a s s i c a l l y been considered by approximating the ex p o n e n t i a l by a r a t i o n a l a l g e b r a i c f u n c t i o n , such as the f i r s t few terms of the Macl a u r i n s e r i e s . Thus e " T s 1 - TS•+ ~ X 3 - f ~ + ( 4 > 2 ) Considering only s m a l l values of T, t h i s can be f u r t h e r s i m p l i f i e d to e " T s "JS 1 - Ts (4.3) Let the system of F i g . 10 be defined by the f o l l o w i n g s t a t e equations. x j = c ( t ) (4.4) x 2 = x j (4.5) x 3 = /edt (4.6) Therefore, from F i g . 10 oo 2yU X 1 ( S ) = 1 ~ 2 (4'7) s + 2t,to s + co n n or (4.8) s 2X! + 2?oonsXi + % X i = a>NZYU which can be w r i t t e n as x i + 2?to x i + co 2 x i = co 2 Y U (4.9) j y n 1 n 1 n Since e ( t ) = r ( t ) - x 2 ( t ) (4.10) and f o r r ( t ) = 1 (4.11) then x i = 1 - e (4.12) 56 i l = 6(t) - e (4.13) where 6(t) i s the impulse f u n c t i o n and can be disregarded f o r t > 0. Therefore x i = -e and Equation (4.9) can be r e - w r i t t e n as (4.14) -e = -to 2 ( 1 - e) + 2£(D e + GO 2YU (4.15) n n n ' N o w -Ts U(s) = E J E l S ifc E , ( l - Ts) (4.16) d d or, u ( t ) - e. - T e j (4.17) d d But, from F i g . 10 K E ( S ) e.(s) = K E(s) + — + K T , E ( S ) S (4.18) d C X^S c u which becomes i n the time domain K e,(t ) = K e + — fedt + K x.e (4.19) d x ' c T. J c d X and „ K e e, = K e + -2- + K r.e (4.20) d c T . c d X Therefore, from (4.17) K e u(t) = e J - T (K e + — + K x.e) (4.21) d c c d S u b s t i t u t i n g (4.18) and (4.21) i n t o (4.15) gives K TK e -e = -to 2 ( l - e ) + 2Cco e + GO 2 y [ K e+ — fedt + K T,e-TK e - TK x.e] n n n ' c T . J C d c T. c d x x (4.22) Rearranging, e(l-co 2yK T.T) + e(2?u) -a 2yK T+GO 2yK x.) + n ' c d n n c n ' c d cc 2yK T GO 2yK e(u 2+co 2yK -'——•——) + — - fedt = (3 2 (4.23) n n ' c x . ' x. ' -a. x x 57 As bef o r e , l e t ai = 2ccon (4.24) a 2 = co 2 (4.25) n a? = o) 2v (4.26) n and d e f i n e C ! A ^ n 2YK c (4.27) c 2 A co 2yK x , (4.28) — n c a co 2YK c 3 A - — - ^ (4.29) i S u b s t i t u t i n g i n t o (4.23) gives e ( l - c 2 T ) + e(a 1 + c 2 - c 1 T ) + e ( a 2 + C ! - c 3 T ) + c 3 / e d t = co n 2 (4.30) When t a k i n g Laplace Transforms of the e r r o r the f o l l o w i n g i n i t i a l condi-t i o n s h o l d L[e] = sE(s) - e(0 +) = sE(s) - 1 (4.31) L[e] = s 2 E ( s ) - se(0 +) - e(0 +) = s 2 E ( s ) - s - e ( 0 + ) (4.32) where e(0 +) i s the i n i t i a l r a t e of the e r r o r f o r the system. However, from (4.13) -e = k\ = x 2 (4.33) and from (4.9) x l = x 2 = co n 2 y u - 2?conX! - to n 2x 1 (4.34) I n t e g r a t i n g gives -e = x 2 = co n 2y/udt - lliu^X} - co n 2/xjdt (4.35) Due to the time delay i n the system, there i s no output r e s -ponse to any input f o r the time p e r i o d 0 < t < T+0+. Consequently, x^ = 0 and (4.35) can be w r i t t e n as 58 -e = x 2 = 0) n 2Y/udt (4.36) A d d i t i o n a l l y , since there i s no feedback o c c u r r i n g w i t h i n the system f o r t h i s time p e r i o d K e = K r(t) + — / r ( t ) d t + K x.r(t) (4.37) d c T . J c d l For a u n i t step displacement input r ( t ) = 1, r ( t ) = 6(t) (4.38) and s i n c e T. >> T l e, £ K + K T,6(t) (4.39) d c c d Now u(t) = e.(t-T) % K T,6(t-T) (4.40) d c d si n c e r ( t - T ) = 0 f o r 0 < t < T+0 Hence Equation (4.36) becomes -e = x ?(T+0 +) = oo 2yK T , (4.41) n c d and so (4.32) can be w r i t t e n as L[e] = s 2 E ( s ) - s + oo 2yK T, (4.42) n c d Taking the Laplace transform of (4.30) w i t h i n i t i a l c o n d i t i o n s of (4.31) and (4.42) y i e l d s E(s) [ ( l - c 2 T ) s 2 + (a T+ c 2 - C ! T ) s + a 2 + c ^ l - T / T ± ) + c 3 / s ] = ( l - c 2 T ) s + al + c 2 2T - C l T + a 2/s (4.43) or s 2 + 3,s + 3 E(s) : — 2 — - (4.44) s 3 + a 9 s 2 + a i s + a z 1 o where «o • r ^ i <4-45) 59 a + c - c T a. + c_ - c.T »2 • ^ T ^ T ^ < 4- 4 7 ) - 2 J a„ o 1 - c,T (4.48) a + c 2T - c.T 3X ! - C 2 T (4-49) In the a n a l y s i s of Sectio n 3, i t was convenient to impose a c o n s t r a i n t upon the output response of the system. For the system w i t h time delay, i t i s advantageous to consider an a l t e r n a t e c o n s t r a i n t - th a t of making the i n i t i a l r a t e of response of the output to be equal to or l e s s than a f i x e d value. That i s , x o ( 0 + ) = Co = co 2 Y K T , < constant ( 4 . 5 0 ) ^ n c d — From ( 4 . 4 7 ) « 2 + 1 , - 2 and from ( 4 . 4 6 ) C 1 T a i + C 2 a 2 . T = 1 T ( 4 ' 5 1 ) 1 - C o T 1 - C o T c 3T a 2 c, S u b s t i t u t i n g (4.45) i n t o (4.52) gives which when s u b s t i t u t e d i n t o (4.51) gives a1 + c 2,+ a 2T a 2 + a]T + a Q T 2 = i - c T = c o n s t a n t (4.54) From t h i s , i t can be seen t h a t the system v a r i a b l e s a Q , , a 2 are constrained. 60 4.2 Examination of the Performance Index The i n t e g r a l square e r r o r f o r the system of F i g . 10 and Equation (4.44) can be determined from t a b l e s (Newton, Gould and K a i s e r , 1964) t o be (g 2-26 )a + 6 z % ISE = - - 2 - i l i o_2 ( 4 < 5 5 ) 2a a,a, - 2a 2 o 1 2 o In the above equation, a Q, a^, a 2 and 8]_ are f u n c t i o n of the system v a r i a b l e s K , x . , and x , arid are thus v a r i a b l e s themselves, w h i l e 6 i s c l d o a constant. However, from Equation (4.49) a l + c 2 2 t " C 1 T a l " C 1 T c 2 2 t e i = - r^ T z T i T ^ f + T T ^ <4-56> and from Equation (4.47) a i " C 1 T C2 1 - c 2T 0 1 2 1 - c 2T Therefore, c 2 ( - l . + c 2T) &1 = a2 + = a 2 - c 2 (4.58) (1 - c 2T) In a d d i t i o n , i t has been shown i n Se c t i o n 3.3.1, that by d i f f e r e n t i a t i n g the expression f o r the i n t e g r a l square e r r o r w i t h respect to aQ and equating to zero, then a Q can be expressed as a f u n c t i o n of the other v a r i a b l e s i n the equation. That i s / H 1 a„ = a 2 6„ / : (4.59) ° 2 ° v / a i + 3 ! 2 - 26 Q Thus i t can be seen that the number of independent v a r i a b l e s i n the performance index can be reduced to two. When applying the approximation procedures of Se c t i o n 2 to an exp r e s s i o n , i t i s convenient to f i r s t p l o t that expression i n order to (4.57) 61 determine a feasible region over which the approximation can be made. Fig. 11 i s a plot of Equation (4.55) showing a family of constant cost (ISE) curves. It is obvious, that there i s a region of discontinuity of the function and that minimization i s possible only for those values of 012 beyond the discontinuity. Consider now a condensation of Equation (4.55) around the operating point a^ , a 2 into a single term posynomial by the method of Section 2.2. From (4.59), and (4.58), let <(>! = a1 + 9>\2 - 2B Q = ax + a 2 2 - 2a 2c 2 + c 2 2 - 26 q (4.60) which can be approximated by a, . a 2 . a„ . • i % • i ^ A l t r - ) A 2 t r ) A 3 (4.6i) ax a 2 2 a2 where <f>X = a\ + o i 2 2 - 2 a 2 C 2 + C 2 2 - 2g Q (4.62) A : = ^  (4.63) A 2 = 3 — (4.64) * l 2 a 2c 2 A 3 = - — (4.65) Thus, (4.59) can be expressed as • . (l/ 2-Al/ 2 ) (I -A2 -A3/2) f. aQ % $ l a l a 2 (4 . 0 0 ) where r T - ' - A l 2A 2-A 3,-1/2 Q (L $! = [<f>iai 1 a 2 J 3 Q (4 . 0 / ) Now from (4.55) let <|>2 = 2ct ajo^ - 2 a Q 2 (4.68) Substituting (4.66) into (4.68) yields , 3/2-AJ/2 2-A2-A3/2 2 2-2A 2-A 3 „ , Q v <))2 = 2$iaj 1 ct2 2$i^a! 2 a 2 (4.69) which can be approximated by *2 * * 2 ( ( ^ ) 3 / 2 - A L / 2 ( ^ ) 2 - A 2 - A 3 / 2 ) A L T ( A ) 1 - ^ ^ ) 2 - 2 ^ 3 ) ^ ^ . ^ 04 a 2 ctj ct2 where T2 - 2 $ 1 a 1 3 / 2 - A l / 2 a 2 2 - A 2 - A 3 / 2 - 2 $ 1 2 a 1 1 - A l a 2 2 - 2 A 2 " A 3 (4.71) , 2 . 1 a 1 3 / 2 ' 4 l / 2 5 ; 2 " M 3 / 2 A u — ^3—= (4.72) <l>2 2-J--A1- 2-2A 2-A 3 2$,za, o^u ^ 0 A 5 = = r - £ (4.73) • ' <j>2 • Similarly, let <j>3 = aQa!+ (a 2 2-2a 2c 2 +C2 2-23 c )) a Q + 3 Q 2 a 2 (4.74) which becomes, •3 = ( a 1 + a 2 2 - 2 a 2 C 2 + C 2 2 - 2 6 o ) $ 1 a 1 1 / 2 " A l / 2 a 2 1 " A 2 " A 3 / 2 + 3 Q 2 a 2 (4.75) Equation (4.75) can be approximated by ^ r i 3 ( (^ )3/2-A 1/2 (!2 )l-A2-A 3/2 )A 6 ( (^ )l/2-A 1/2 (^ )3-A2-A3/2 )A 7 04 c<2 ai 02 • ( ( ^ ) 1 /2"A1/2 ( ^ ) 2"A2"A 3/2} A 8 ( ^ 1/2-A !/2 ) I-A2-A 3/2} A 9 ( ^ 2 } A ! 0 04 012 011 a2 a2 (4.76) where * 3 = ( a 1 + a 2 2 - 2 a 2 C 2 + c 2 2 - 2 3 o ) $ i a 1 1 / 2 " A 3 / 2 a 2 1 " A 2 " A 3 / 2 + 3 Q 2 a 2 (4.77) 64 - 3 / 2 - A ! / 2 - l-A 2-A 3 / 2 A6 = — — • * ! ( 4 . 7 8 ) $3 - 1 / 2-AT / 2 - 3-Ao-A 3/2 A 7 = — = ^ . 'Sj ( 4 . 7 9 ) $3 - l/2-Ai/2- 2-A2-A3/2 A 8 = — • $!(-2c2) (4.80) <f>3 - 1 / 2 - A 1 / 2 - l-A 2-A 3 / 2 A 9 = — : • ^ ( c ^ B ) • ( 4 . 8 1 ) 7 1 o <P3 A 1 0 = ( 4- 8 2) ^3 and f i n a l l y the integral square error can be approximated by : *3 ISE % ( 4 . 8 3 ) To examine this method and the resultant posynomial expression consider several examples. From Fig. 1 1 , take the point ai = . 1 7 0 , a 2 = . 3 6 which l i e s on the curve ISE = 1 . 6 2 . In addition, assume T = time delay = . 1 or 1 scaled ( 4 . 8 4 ) c 2 = x ? ( 0 + ) = i n i t i a l rate = u 2 Y K x, = 0 . 2 ( 4 . 8 5 ) . ^ n c d From ( 4 . 4 8 ) a 3_ = 2 0.0125 (4.86) '0 1 - c 2 Substituting the above values into the derived expressions yields +l = .1706 (4.87) A x = .9965 (4.88) A 2 = .7597 (4.89) 65 A 3 = -.8441 (4.90) *! = .0089 (4.91) Therefore a Q % .0089a 1- 0 0 1 8a 2- l + 2 5 6 (4.92) S i m i l a r l y , i 2 = -0005 (4.93) A 4 = 1.1026 (4.94) A 5 = -.1026 (4.95) Therefore a. 1.0018 a„ 1.6642 1. 1026 a .0035 ct 1.3247 -.1026 * 2 * [(TI70> ^ ] [ (370> <73§> ] (4.96) = .O204a 1 1' l o l t 2a 2 1- 1 9 7 0 (4.97) and _ <f>3 = .0008 (4.98) A 6 = .9283 (4.99) A 7 = .7068 (4.100) A 8 = -.7855 (4.101) A 9 = .0823 (4.102) A J O = .0678 (4.103) r , " l v1.0018, a 2 s .6624 .9283,, a l .0018, a2 2.6624 .7068 *3 * '00^ <-my <736> > ((7l70> (T36> > . ( (-^L.) • 0018 (J! 2 ) 1. 6624)--7855 ( (-^ 1-) • 00 1 8 • 6624) • 082 3(_^2) • 06 78 . 170 .36 .170 .36 .36 (4.104) = .0164ai - 9 3 0 0 c t 2 1 , 3 1 3 2 (4.105) 66 Equation (4.83) gives + , 10164a/ 9 3 0 ° a i - 3 1 3 2 ISE % —— 4 2 . 8021a,- 1 7 t t 2a 2- 3 8 3 8 (4.106) * 2 .0204a 1 1-10t2 a 21.6970 Now take the p o i n t a^ = .152, a 2 = .375 which a l s o l i e s on the curve ISE = 1.62. With the same time delay and i n i t i a l r a t e of response as given i n Equations (4.84) and (4.85), the ISE can be approximated i n a s i m i l a r manner to give ISE = .8939a 1"- 1 9 5 7a 2"- 2 3 4 8 (4.107) The approximations to the ISE given by Equations (4.107) and (4.106) are compared to the a c t u a l ISE Equation (4.55) i n F i g . 12. I t can be seen that the choice of an operating p o i n t or p o i n t of approxi-mation has a great i n f l u e n c e on the shape of the condensed curve. That i s , f o r a comparatively s m a l l s h i f t of the operating p o i n t , the r e s u l t a n t shape and slope of the condensed curve are very d i f f e r e n t . This i s due to the unique shape of Equation (4.55), making the approximation v a l i d only over a l o c a l r e g i o n , and i n d i c a t e s t h a t whenever approximations are to be made, i t i s important t o thoroughly examine the f u n c t i o n when choosing an operating p o i n t i n order to o b t a i n ? f e a s i b l e r e s u l t s . 4.3 O p t i m i z a t i o n Procedure In g e n e r a l , i t has been shown i n Sectio n 4.2 that the cost (ISE) equation f o r the time delay system can be condensed i n t o a s i n g l e term posynomial of the form ISE % $ a ! n i a 2 n 2 (4.108) where <j>-, _ $ = ^  a r n i a 2 _ n 2 (4.109) <J>2 actual ISE approx. of ISE Figu re 12 Approximations of Performance Index 68 m = ( 3 / 2 ^ ^ 2 ) ^ - ^ ) + ( l / 2 - A 1 / 2 ) ( A 7 + A 8 + A 9 - 2 A 5 ) ( 4 . 1 1 0 ) n 2 = ( 1 - A 2 - A 3 / 2 ) ( A 6 + A 9 - 2 A 5 ) + ( 2 - A 2 - A 3 / 2 ) ( A ^ ) + ( 3 - A 2 - A 3 / 2 ) A T + A 1 0 ( 4 . 1 1 1 ) In addition, Section 4.1 derived a constraint for the system in the form of Equation ( 4 . 5 4 ) . For small a 0, this can be approximated by the linear equation 04T + a 2 < K2 ( 4 . 1 1 2 ) where a x + c 2 + a 2T Kn A constant = ^ = ( 4 . 1 1 3 ) -1 — 1 — C o l or l/K 1(a 1T+a 2) < 1 (4.114) Equations (4.108) and (4.114) are both posynomials in which the total number of terms (three) i s one greater than the number of variables (04 and a 2 ) . Hence, the degree of d i f f i c u l t y i s zero and a solution by geometric programming i s readily accomplished. First the dual i s formed. * < « ) - ( f ^ ) 6 l ( ^ 2 ^ ) 6 2 ( ^ 1 ) 6 3 ( 5 2 + 6 3 ) 6 2 + 6 3 (4.115) Equation (4.115) is to be maximized subject to the normality and ortho-gonality conditions of (1.11) and (1.12), which for this system are: 6l » 1 (4.116) n l 6 l + 6 2 = 0 (4.117) r i 2 6 1 + 63 = 0 (4.118) These have the unique solution 6 l * = 1 (4.119) <52* = -m (4.120) 69 6 3 * = - r i2 (4.121) which when substituted into (4.115) yields the maximum value of iK6), and simultaneously the minimum value of ISE. At an optimum, the follow-ing conditions from Equations (1.17) and (1.18) are satisfied: $ a 1 n l a 2 n 2 = 6 1*if . ( 6 * ) (4.122) (T/K^cx! = 6 2 * / ( 6 2 * + 6 3 * ) (4.123) (1/K!)a 2 = 6 3 * / ( 6 2 * + 6 3 * ) (4.124) The optimum value of the parameters 04 and a 2 are then found from simultaneously satisfying (4.122), (4.123) and (4.124). In addition, from (4.59) a * = cu*B (4.125) ° 2 ° J a 1 * + B 1 2 - 2 3 0 Substitution of a Q * , 0 4 * , and a 2 * into Equations (4.45), (4.46) and (4.47) w i l l yield solutions for the parameter c^, c 2, and C3 which then can be used in Equations (4.27), (4.28) and (4.29) to find values of the controller parameters, K c» x^, and x^ which yield the minimum ISE. It is interesting to note that from Equation (1.10) i t is required that a l l 6^ be equal to or greater than zero for a feasible solution to the optimization problem. From Equations (4.120) and (4,121) this i s possible only i f -Hi = (3/2-A 1/2)(A l t-A 6)+(l/2-A 1/2)(2A 5-A 7-A 8-A 9) > 0 (4.126) -n 2 = ( l - A 2 - A 3 / 2 ) ( 2 A 5 - A 6 - A 9 ) + (2 - A 2 - A 3/2 ) ( A l t - A 8 ) - A 7(3 - A 2 - A 3/2) - A 1 0 > 0 (4.127) Equation (4.126) and (4.127) can be used as a means of determining i f 70 an approximation around a particular point w i l l yield a feasible solution. Consider an example of the optimization procedure. Previously i t has been shown that an approximation of Equation (4.55) around the point cti = .152, ct2 = .375 gives ISE % . 8 9 3 9 a ! - - 1 9 5 7 a 2 _ - 2 3 4 8 (4.128) For a\, a 2, a 3 as given in Section 3 and T = 1 then a 2 + c + a T Ki 1 _ C g T = .575 (4.129) and from Equation (4.114) the constraint on the system becomes 1.7391 (ax + a 2) < 1 (4.130) Substituting values from (4.128) and (4.130) into (4.115) gives *(«) = (^)6l(i43^)62(i43^)63(62+63)52+53 (4.131) 6i &2 6 3 from which the normality and orthogonality conditions are Sl = 1 (4.132) -.19576! + 6 2 = 0 (4.133) -.23486i + 6 3 = 0 (4.134) These have the unique solution 6X* = 1, 6 2* = .1957, 63* = .2348 (4.135) Substitution of (4.135) into (4.131) gives the maximum of the dual (equal to the minimum of the primal objective function) as: ^(6*) = ( - ^ ^ ) l ( ^ ^ ) . 1 9 5 7 ( ! ^ = l i 5 2 6 1 (4.136) 71 At an optimum Equation (4.122), (4.123) and (4.124) h o l d . Therefore: . 8 9 3 9 a r ' 1 9 5 7 a 2 " - 2 3 I t 8 = <5i*^(6*) = 1.5261 (4.137) 1.7391a! = 6 2 * / ( c 5 2 * + 6 3*) = .4546 (4.138) 1.7391a 2 = 6 3 * / ( 6 2 * + 6 3*) = .5454 (4.139) from which a i * = .2614 (4.140) a 2 * - .3136 (4.141) and from Equation (4.125) a 0 * = 0.0040 (4.142) Equations (4.45), (4.46) and (4.47) can be solved to give c i = .2017 (4.143) c 2 = .202 (4.144) c 3 = .0032 (4.145) The c o n t r o l l e r parameters are Kc = c l / a 3 = 6 - 7 3 (4.146) T ± = c j / c 3 = 63.03... (4.147) T d = c 2 / c i = 1.00 (4.148) A p i c t o r i a l r e p r e s e n t a t i o n of the procedure i s shown i n F i g . 13. The approximate performance index i s shown i n r e l a t i o n s h i p to the c o n s t r a i n t equation and i t i s seen t h a t the p o i n t of condensation a i = .152, a 2 = .375 does not i n i t i a l l y s a t i s f y the c o n s t r a i n t condi-t i o n . At an optimum, the parameters a i and a 2 have been s h i f t e d to l i e on the c o n s t r a i n t curve and there i s tangency w i t h the performance 73 index curve. A p l o t of the output response f o r the system w i t h c o n t r o l l e r parameters given by Equations (4.146), arid (4.147) and (4.148) i s shown i n F i g . 14. From t h i s 45 Per cent overshoot = x 100 = 45% (4.149) and the s e t t l i n g time i s T s = 3.0 seconds (4.150) These values i n d i c a t e a somewhat o s c i l l a t o r y nature f o r the system that could be undesirable. To determine why an " o p t i m i z a t i o n " procedure y i e l d e d these r e s u l t s r e q u i r e s an examination of the approximate and a c t u a l forms of the performance index. As seen from F i g . 13, the o p t i m i z a t i o n procedure has r e s u l t e d i n a s h i f t of the curve of Equation (4.128) to a new p o s i t i o n tangent w i t h the c o n s t r a i n t equation at the p o i n t 04* = .2614, a 2* = .3136. At t h i s new p o s i t i o n , the value of (4.128) i s found from (4.136) to be 1.5261. The system response, F i g . 14, has been determined from con-t r o l l e r parameter values t h a t s a t i s f y (4.136). However, when these values are s u b s t i t u t e d i n the a c t u a l form of the performance index i t s value i s a *a * + (B *2-2B )a * + 3 2 a * ISE = - ° - i 1 ° — ° S - J-i-1.6772 (4.151) 2 2a *a-i*ao* - 2a * o 1 z o This i s higher than the value (ISE = 1.6200) at the i n i t i a l c o n d i t i o n s , (a^ = .152, a 2 = .375) and sin c e the response of F i g . 14 i s derived from equations that contain no approximations, e x p l a i n s why the output r e s -ponse i s non-optimum. In a d d i t i o n from F i g . 15 i t i s evident that at an "optimum" ( i . e . a^* = .2614, 0 2 * = .3136), the agreement between the Figure 15 Non-optimum Case- Comparison of Exact and Approximate ISE Ul 76 approximate and a c t u a l forms of the i n t e g r a l square e r r o r i s not good. This i s evident by the d i s s i m i l a r shapes of the curves and the f a c t that only the p o i n t 04*, a 2* i s common t o both. As p r e v i o u s l y seen, an approximation of (4.55) at the po i n t 04 = .170, ct2 = .36 gives ISE ^ .8021 a i"- 1 7 1 t 2a2~* 3 8 3 8 (4.152) M i n i m i z a t i o n of t h i s equation, s u b j e c t to the c o n s t r a i n t c o n d i t i o n of Equation (4.130) i s as f o l l o w s : K 6 ) = ( ^ 5 1 ( 1 ^ ^ 2 ( 1 ^ 3 9 1 / 3 ( 5 ^ )«2+«3 ( 4 . 1 5 3 ) °1 °2 °3 <$! = 1 (4.154) -V1742S! + 6 2 = 0 (4.155) -.38386! + 6 3 = 0 (4.156) Equations (4.154), (4.155) and (4.156) have the unique s o l u t i o n 6 X* = 1, 6 2* = .1742, 6 3* = .3838 (4.157) S u b s t i t u t i o n of (4.157) i n t o (4.153) gives the maximum of the dual and minimum of (4.152) to be 1K6*) = ( - L g Q g l ) ( 1 > 7 j ^ ) • 1 7 k 2 ( 1 ' 7 3 f 3 1 8 ) • 3838(.5580) -5-580 = L5445 (4.158) In a d d i t i o n .8021ar# 1 7 0 2 a 2 " * 3 8 3 8 = 6 T*^(6*) = 1.5445 (4.159) 1.7391a! = 6 2 * / ( 6 2 * + 63*) = .3122 (4.160) 1.7391ct2 = 6 3 * / ( 6 2 * + 63*) = .6878 (4.161) from which aj* = .1795 (4.162) 77 a 2* = .3955 (4.163) and from Equation (4.125) a 0* = .0048 (4.164) Using these values the controller parameters are found to be K c = 4.58 (4.165) x = 36.16 (4.166) x d = 1.46 (4.167) Substitution of (4.162), (4.163) and (4.164) into Equation (4.55) gives the actual value of the performance index to be a *a * + (6,*2-26 )a *+ 6 2 a * ISE = L 2 _ i _ = L5530 (4.167) 2a *a *a * - 2a * 2 o 1 2 o The agreement between Eqn. (4.158) and (4.167) is good, indicating that the output response with parameter values from (4.165) through (4.167) should be better than that of Fig. 14. Fig. 16 confirms that this i s so. It is seen that 27 Per cent overshoot = j 1 ^ - x 100 = 27% (4.168) and T =2.0 seconds (4.169) s In addition Fig. 17 shows a plot of Equations (4.158) and (4.167). The shape of the curves confirms the close agreement between the two expressions at the point of interest. 80 V. SUMMARY AND CONCLUSIONS A problem expressed as minimize g ( t ) o subject to g k ( t ) <_ Gfc k = 1, ... (5.1) where the g's are posynomials, can be expressed as a geometric program-ming problem. I t i s p o s s i b l e to s o l v e (5.1) by a d i r e c t search i n the t v a r i a b l e s . However, the presence of n o n l i n e a r c o n s t r a i n t s poses par-t i c u l a r d i f f i c u l t i e s , and i t has been demonstrated that i t i s more fea s -i b l e to maximinze a dual f u n c t i o n ^(6) where the c o n s t r a i n t s that 6 must s a t i s f y are l i n e a r . A knowledge of the maximizing v e c t o r 6 provides a complete s o l u t i o n to the problem. G e n e r a l l y , engineering problems l i k e (5.1) c o n t a i n more terms than v a r i a b l e s . This c o n s t i t u t e s a high degree of d i f f i c u l t y and an; e x p l i c i t s o l u t i o n i s not r e a d i l y a v a i l a b l e unless access to extensive computer programs (Zener, 1971; A v r i e l , Dembo, and Passy, 1975) i s pos-s i b l e . In a d d i t i o n , not a l l terms i n the equation are l i k e l y to be pos-i t i v e . These d i f f i c u l t i e s can be overcome by using techniques of approx-imat i o n , i n which non-posynomial equations c o n t a i n i n g many terms can be condensed to p o s i t i v e expressions that c o n t a i n fewer terms. S u c c e s s i v e l y applying t h i s technique can e v e n t u a l l y le a d to a zero degree of d i f f i -c u l t y problem. An approximation technique was developed from the a r i t h -metic-geometric r e l a t i o n s h i p s that r e q u i r e d no d i f f e r e n t i a t i o n of terms making i t p a r t i c u l a r l y easy to apply to complicated expressions. Although the techniques of approximation can be u t i l i z e d to o b t a i n problems that are r e a d i l y s o l v a b l e the method must be a p p l i e d j u d i c i o u s l y . An examination of the intended expression should be made 81 i n order that a f e a s i b l e operating p o i n t be chosen around which to con-r dense the equation. F a i l u r e to do t h i s could r e s u l t i n an o p t i m i z a t i o n of the approximate equation but a f i n a l value of the a c t u a l expression that i s greater than that f o r the o r i g i n a l operating p o i n t . Formulation of a c o n t r o l system problem r e q u i r e s the choice of an o b j e c t i v e f u n c t i o n , such as the i n t e g r a l square e r r o r , plus the i m p o s i t i o n of c o n s t r a i n t s upon"the system performance. For a second order system w i t h p r o p o r t i o n a l - i n t e g r a l - d e r i v a t i v e c o n t r o l l e r , the con-s t r a i n t s can be developed to not only y i e l d a p h y s i c a l l y p l a u s i b l e sys-tem, but a l s o i n a manner to make a s o l u t i o n by geometric programming more v i a b l e . That i s f o r a given o b j e c t i v e f u n c t i o n , a choice of con-s t r a i n t s i s made that mathematically r e s t r i c t s the system v a r i a b l e s to f e a s i b l e v a l u e s , and at the same time lends i t s e l f to the geometric programming concepts. Once a problem has been formulated and expressed i n a s u i t a b l e f a s h i o n , a s o l u t i o n by geometric programming i s e a s i l y accomplished, u s u a l l y amounting to s o l v i n g a set of l i n e a r equations. The minimum value of the o b j e c t i v e f u n c t i o n i s obtained without f i r s t determining parameter values f o r a c h i e v i n g t h i s minimum. However, the power of the method becomes apparent when i t i s seen that the optimum i s approached i n a s i n g l e computational step. T h i s , together w i t h the ease of the mathematical operations i n v o l v e d , make geometric programming an a t t r a c -t i v e o p t i m i z a t i o n technique. 82 REFERENCES A v r i e l , M., Dembo, R. and Passy, U., 1975. S o l u t i o n of Generalized Geometric Programs. I n t e r n a t i o n a l J o u r n a l For Numerical Methods i n Engineering, V o l . 9, 149-168. A v r i e l , M. and W i l l i a m s , A.C, 1970. Complementary Geometric Program-ming. SIAM J o u r n a l of A p p l i e d Mathematics. V o l . 19, No. 1, 125-141. A v r i e l , M. and W i l l i a m s , A .C, 1971. An Extension of Geometric Program-ming w i t h A p p l i c a t i o n s i n Engineering O p t i m i z a t i o n . J o u r n a l of Engine-e r i n g Mathematics. V o l . 5, No. 3, 187-194. D u f f i n , R.J., 1962a. Dual Programs and Minimum Cost. J . Soc. Ind. Appl. Math., 10:1. D u f f i n , R.J., 1962b. Cost M i n i m i z a t i o n Problems Treated by Geometric Means. Opns. Res., 10:5. D u f f i n , R.J., 1970. L i n e a r i z i n g Geometric Programs. SIAM Review V o l . 12, 211-237. D u f f i n , R.J. and Peterson, E.L., 1964. Constrained Minima Treated by Geometric Means. Westinghouse S c i e n t i f i c Paper 64-158-129-P3. D u f f i n , R.J., Peterson, E.L. and Zener, C , 1967. Geometric Programming, Wiley, New York. E v e l e i g h , V.W., 1967. Adaptive C o n t r o l and O p t i m i z a t i o n Techniques. McGraw-Hill. Newton, G.C.; Gould, L.A., K a i s e r , J.F., 1954. A n a l y t i c a l Design of L i n e a r Feedback C o n t r o l s . Wiley, New York. Zener, C , 1971. Engineering Design by Geometric Programming. Wiley-I n t e r s c i e n c e , New York. 

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