2032 IEEE lntemational Symposium CO Computer Aided Control System Design Pmceetilngs September 18.20, POM*Glaylow, Scotland, U.K.

Linear Controller Design by Goal Programming

A. D. S. Lordelo and P. A. V. Ferreira University of Campinas

School of Electrical & Computer Engineering 13084-970 Campinas, SP Brazil

valente@dt.fee.unicamp.br

Abstract

A computational optimization-based approach for designing fixed-order controllers is presented. The approach focuses classical pole placement design techniques for both certain and uncertain (interval) linear time-invariant plants within a goal propnuning framework. The design objective is the minimization of the overall deviation from the desired per- formance for the closed-loop system, as specified by apoly- tope of characteristic polynomials in the general case. Goal programming, a traditional technique in many different engi- neering design contexts, reveals to be an adequate hasis for integrating a number of existing results in linear controller design by pole placement. Examples illustrate the main char- acteristics of the proposed approach.

Keywords - Controller design, pole placement, uncertain systems, robust control, goal programming.

1 Introduction

THE PROBLEM of designing feedback controllers for lin- ear time-invariant systems in order to guarantee satisfactory closed-loop transient and steady-state responses has a prac- tical solution through pole placement. The basic assumption is that stability and other performance specifications can he met using dynamic output feedback to place the poles of the closed-loop system in appropriate locations of the complex plane. It is well-known that the pole placement design prob- lem can be reduced to the solution of a system of linear al- gebraic equations, as comprehensively discussed in [2] and [61, among others. An extension of thepoinhvise pole place- ment problem, in which the closed-loop poles are placed in specific locations, is the regional pole placement prob- lem, where the objective is to place the closed-loop poles in a suitable region of the complex plane. The regional pole placement problem is usually treated in connection with the substantially more general mbust pole placement problem: the controller must place the closed-loop poles in a specified region in the face of uncertainty with respect to the matbe- matical model of the plant. In many real-world situations, the Inode1 uncertainty reflects on the parameters of the plant,

which has motivated extensive research efforts in parametric robust control theory ([l], [3], [SI).

In this paper the objective of assigning closed-loop poles is replaced by the equivalent problem of assigning charac- teristic polynomials, either in specific polynomial instances (pointwise assignment) or in polytopic families of polyno- mials (regional assignment) adequately defined. Uncertain plants are represented by proper transfer functions with co- efficients varying on a polytopic region. The proposed ap- proach for the robust characteristic polynomial assignment - RCPA - problem is similar to the ones presented in [12], and [9]. In [12], after explicitly characterizing all RCPA con- trollers as a convex set, the distance of a nominal controller from this set is minimized using a nonlinear programming algorithm. Some important developments of the last fifteen years in the area of robust control are employed in [9] to ha- dle the RCPA problem in a linear programming framework.

In RCPA problems, the notion of goal is always present: the design objective is to obtain a controller of the lowest pos- sible order that assigns a goal characteristic polynomial or robustly assigns characteristic polynomials in a goal poly- topic family of polynomials. Since goals can or cannot be attained by a robust controller of fixed order, a relevant in- formation for the designer would be what goal is attainable by a contmller of a given order. In fact, goal adjustments are an important aspect of decision-making processes based on the concept of goal.

Goal programming is a traditional methodology in many different engineering design contexts ([71) and incorporates a number of modeling and solution techniques panicularly suitable for the RCPA problem under consideration. Certain aspects of the characteristic polynomial assignment problem (pintwise and regional) can be interpreted using goal pro- gramming results. The use of interval goal programming with interval coefficients ([XI) enables a substantially sim- pler solution for the RCPA problem than those presented in [I21 and [91 and, as in 191. all controller design problems can be reduced to linear programming problems. The methd- ology is presented for single input-single output continuous time systems, but applies equally well for discrete time sys- tems.

0-7803-7388-W02/$17.00 02002 IEEE . 145

2 Pule placement by goal programming where

In this section, the pole placement design technique is revis- ited in a goal programming framework. For comprehensive treatments of pole placement and goal programming as inde- pendent techniques, see for instance [61 and [13].

Consider the unit feedback (SISO, time-invariant) control system represented in Figure 1. The transfer functions of a given n-order plant P(s) to be controlled and a series T-

order controller C(s) to he designed are represented as

A :=

-

-

x E R4, z:= [ 21 x2- "' xq ] T , and

T . where t := [ ti t z ... t , ] , t ER"'.

n p ( s ) := alsn + azsn-'+. . . +a,+,, ~ d p ( s ) . := a,+Zs" + a,+3sn--l + ... + azn+z, .

:= x l d . + x2sp-1 + . . . +z,+l,

Matrix A E Rmxe in (3) is the Sylvester &trir associated with the plant. Since any plant, controller or characteristic polynomial-is uniquely specified by A, x or t , they will be

-~ . . ,

. . .. n c ( s ) -

~ - ~ . d c ( s ) :=. x,+zsr + x,+3sp:1 +. .: + xZr+2. sometimes kferred as plant A , controller x and character- istic polynomial t , respectively. According to the controller design technique proposed in this paper;any desired charac-

_ .

- teristic polynomial to is viewed as a goal that can or cannot ..' ~. ~ 1 I

- - c(s) H '('1 h- , he attained by some conwller of a given order T . Let ~

- .

q i= [ ql- Ili

- P : = [ P 1 - P 2 ,.: Pm ] , .

t o := [ ty t; .. . t: ~ ] T ~ ,

... qm ]Ti,. . T . - 7

- - . i~ .

. Fig. 1: Unity-feedback SISO system. . he defined as deviation variables from the goal ~nitially the plant coetfcients (a1, az, . . ~, az,+z) are sup- posed to be precisely known. The design parameters of the controller ( x i , 2 2 , . . . , X ~ ~ + Z ) must he.selected in order to meet performance specifications translated int,j pole loca, tions of the closedLloop system. Classical pole-placement apalysis shqws that there exists a solution for the compen-

'satorequation . . ~

~ - '~ d ~ ( s ) d c ( s ) + np(s)nc(s) = ~ F ( s ) (2)- ProblemGP ~~

for all possible d F ( s ) , where

~

in the sense that AX + q - P = to, with q; t 0, pi 2 0 and Tipi.= 0 for all i -= 1 , 2 , . . . , m. The pole placement design then resbited as the Classical goalprogrammingproblem (1131).

:

.. .~

~.

X,%P d F b ) ~ :=, (s - P l ) b - Pz) . .,.(s - Pn+,),

.. = tis"+'.+ tzs"+'-' + '. . + t,+,+1,

.~

subject to Ax + q - p.= t 0 , Q20, P 2 0 ,

if &#only if T 2 n - 1 'and np(s ) and dp(s)~are coprime, . . ~

If T < n .- 1 or if the controller is constrained to assume a ' ~ particular suucture (for example, PD), as usually require

in practical control system implementations, it is not pas- . sible to guarantee that (2) will have a solution for any de- . . sued set of closed=loop poles pl,pz,. . . ;p,+, or, equiGa-

' t l , t z , . . . r,tn+r+l. For convenience, define m := n + T + 1 and that (2) be re,+.&en as a linear eipation of the form

'

. . . . .

! . . ~

, 'where ai > 0, z = 1 , 2 , . . . , m are given weights associated with deviations andp 2 1, usually p = 1, p = 2 o r p = W.

.~ lenfly, for anydesired p~yno,,g by A necessary and sufficient condition for the existence of a controller 2 of order T assigning the characteristic polyno- mid at t = to will be established on the hasis of the follow- ing Lemma, whose validity extends to all goal progra"ing formulations introduced in this paper. Its pqoof can be found

. .

: ~. := 2T + 2: 1ris

A x = t (3) in(131. ~

- 146

Lemma 1 Lef (x*,q*,p*) be any opfimalsolution ofpmb- lem GP. Then nlpt = 0 for all i = 1,2 , . . . ,m.

Theorem 1- Let up($') be the optimal value ofpmblem GP and x' the corresponding contmller. Then x* assigns the characteristic polinomial of the closed loop sysfem at t = to ifandonly i fup( to) = 0.

Proof: Let (x*, f , p') be an optimal solution of problem GP and define t' := Ai'. If vp(to) = 0, then q' = p* = 0 and one obtains t' = to. If ",(to) > 0, at least one devia- tion variable is positive and by Lemma 1, the controller x* assigns t: below or (exclusive or) above t f , depending on

0

Theorem 1 can be interpreted in terms of classical pole place- ment analysis. Let

which kind of deviation variable is positive.

R(A) := {t E R"' : t = Ax)

he the range space of all possible characteristic polynomials attainable by a controller of order T . According to Theorem I , up(to) = 0 if to E R(A), which always happens when n p ( s ) and d p ( s ) are coprime and T 2 n - 1 because matrix A bas full column rank under such conditions. If up(t") > 0, then to 6 R(A) . In summary, the controller x* assigns the characteristic polynomial at

t* = to + q* - p*,

and t' = to if to E R(A). Problem GP is always feasible and reduces to a linear program if p = 1 or p = CO, and to a quadratic program if p = 2. As in any goal programming formulation, the objective is to minimize the overall devia- tion from the prescribed goal to, which in turn must reflect the desired performance for the closed-loop system.

Example 1 The second order plant

P(s )= s+ l sa - 2.2s - 2.4

is considered in [12]. It is required to place the closed-loop poles at -1 f j0 .7 and -10. The corresponding goal char- acteristic polynomial is

to = (1.000,12.00,21.49,14.90)

Since the solution of the compensator equation is unique when T = n - 1, the optimal value of GP is up(to) = 0 for all p 2 1 and a > 0. The goal to is actually attained and the lirst order controller is

7.688s + 30.53 C ( S ) =

s+6.513 ' the same presented in [12]. It is interesting to investi- gate if the dominant closed-loop poles -1.0 f j0 .7 (to = (1.00,2.00,1.49)) can be placed using a proportional con- troller (T = 0). Simple root-locus analysis shows that no

proportional gain is capable to accomplish this task, as illus- trated in Figure 2.

Fig. 2: Proportional control - Example 1.

The numerical data ielated to the solution of problem GP

are summarized in Table 1. The damping factor and natu- ral frequency of the desired closed-loop poles -1.0 f j0.7 are 0.82 and 1.22 rads, respectively. The goal programming solutions provide 0.72 and 1.28 rads 0, = l), 0.71 and 1.26

0

(a; = 1, i = 1 , 2 , 3 ) f o t p = 1 , p = 2 m d p = 00

radls (p = 2), and 0.68 and 1.20 rads (p = CO).

Table 1: Goal programming solutions. U U" t' Gain

Y

1 0.301 (1.000,1.850,1.650) 4.05 2 0.217 (1.030,1.848,1.642) 4.00 w 0.141 (1.141,1.859,1.631) 3.83

In principle, nothing can be said a-priori about the perfor- mance or even the stability of the closed-loop system when up(to) > 0. In order to associate performance and stabil- ity with the solutions of problem GP, consider the family of polynomials with coefficients lying in the weighted 1, ball of radius p > 0 ( [ 5 ] ) :

where 13, > 0, i = 1,2, . . . , m are given weights and p 1 1. The set B,(to,fi) represents a ball with radius p in 1 s ace when PI = pZ = . . . = 0,. An 1, ball of polynomals B p ( t o , p ) is Hurwitz if and only if Bp(to ,p) contains only Hunvitz polynomials (all roots in the open left half com- plex plane). The 1, stability margin of a given polynomial to - the largest radius j~ for which Bp(to, p) is Hurwitz - can he computed for instance through the Tsypkin-Polyak locus

p 4

1 47

( [5 ] ) . Several other procedures have been proposed in the literature ([3], [5]). A connection between stability margins and the optimal solutions of problem GP is established in the following theorem.

Theorem 2 Assume that ai = pi, a = 1,2,. . . , m, and let -p* be the maximal radius for which 5,(to, p) is Hurwitz. rfu,(t') 5 p*, then t* E B,(t',p*) and the associated closed-loop system is stable.

Pm08 The proof relies on the fact that

q. + p, = It, - tPI for all i = ~ 2 , . . . , m, ( 5 )

( t , is the a - t h component o f t = A x ) and, as a consequence, that vp(to) is the minimal weighted 1, norm between to and R(A). Hence, if vp(t') 5 p* then t* E 5,(to,p*) and the resulting closed loop system is stable. 0

The fundamental aspect of Theorem 2 is that vp(to) is a mea- s%e of proximity-between t* and t'. It is also known that, assuming invariant degree, the mots of a polynomial vary con$nuously with respect to its coefficients ([31). Hence, for sufficiently small vp(to) the closed-loop system associated with~t' will he stable and its performance will be similar to that specified't&ough to. -.

If vp(to). is large enough to invalidate the use of the corre- sponding controller, an alternative is to consider the regional Characteristic polynomial assignment problem. For simpiic- ity, all remaining goal programming formulations will be presented in terms of p = 1 A d ai = 1, i = 1,2,. . . , m.

3 Assignment in a polytope

Suppse that the problem of assigning a goal characteristic polynomid is Fplaced by the problem of assigning a char- acteristic polynomial in a polytopic region. The problem is

. to find t L and tu ( tL 5 tu) such that all the roots of the in- terval'polynomid-family t' := [ tL , t u ] lie in the desired re-

.- . gion of closed-loop poles. In [12], the determination of t'is

- -

Problem I G P

m

subject to A x + qL - pL = t", A x + q U - pu = tu , L

U 1) 2 0 , PL t 0,

2 0, PU 2 0. 1)

Theorem 3 Le?v(t') be the optimal value ofpmblem~IGP and x* the corresponding controller. Then x* assigns the

-characteristic polinomial of the closed loop system in t' :=~ [tL,tu] ifandonlyifv(t') = 0. - ~~

Pmofi Let ( x * , q L * , p L * , q u * , p u * ) be an optimal solution .of problem IGP and t ' = As'. If b(t')-= 0, then qL' = pu* = 0 and one obtains t* E t'. If v(t') > 0, then at leastoneofthevduesqp, p?, i = 1,2,.:;,mispositive and the controller x* assigns ti* below to tc or.almve t Y , depending on which kind of deviation variable is positive. 0

Problem IGP exhibits the same fundamental properties of problem GP @ = 1): it is a simple linear programming prob- lem and always has a feasible solution. It is readily seen that the controller x* assigns the characteristic polynomial at

.t* = t L - $' . .

= t" - qU' + p"'.

When v(t') > 0, meaning that t' 6 t', the corresponding controller assigns a characteristic polynomiaLwith minimal deviation (in a sense of the 1? n o d from tJ.

Example 2 - Consider the plant transfer function of Example. 1 and the problem of placing.closed-loop poles in a desired region using a zero order controller. The desired region is illustrated in Figure 3 and corresponds to the root space of. the interval t' = [tL, tu] defined by

.~ - ~ accomplished using a sensitivity method- Anothei possibil- ity involves the concept of mot space of a polpopic family. The Edge Theorem ([41) states that the root space of the hy- perretangle t' is overbounded by the roots of its edges. The

- t L .= (1:00,1:40,1.04), tu = (1.00,2.60,1.94).'

- ideia of selecting t' (that is, edges) in order to confine its root space in a suitable region of the complex plane is proposed

The optimal solution of problem IGP,

in [9] and adopted in this paper. -v( t ' ) = 0, t' = (1.00,1.82,1.62),

A necessw and sufficient condition for the existence of a .conuoller of order T assigning a characteristic polynomial indicates that characteristic &lynomial assignment in t' is in$ is established in terms of the following interval goal possible. The resulting proportional controller C(s) =~ 4.02 pmgramming pmblem ([7]): places the closed-loop poles at -0.910 fj0.890. . 0

14%

' - 1 I

I -2 5 4 -1 5 4 5

Red-

Fig. 3: Root space oft' - Example 2

4 Robust polynomial assignment

Suppose that the plant coefficients a l , a2, . . . , am aie no longer fixed quantities, being known to vary in closed in- tervals defined as

af := [a,",.?], a," 5 ay , i = 1,2,. . . , m.

An interval description for uncertainty in this form gives rise toan intervalSylveslermatrir A' := [ A L , A"], whoselower ( A L ) and upper ( A U ) hounds are obtained when the coeffi- cients of A are replaced by their lower (aL) and upper (au) values, respectively. Given an interval plant A' = [ A L , Au] and an interval polynomial specification t* = [ tL, t"] , the robust characteristic polynomial assignment - RCPA - proh- lem consists in finding a controller x assigning characteristic polynomials in t' for all possible Sylvester matrices in A'. In this paper, the set of all RCPA controllers is defined as

X := {z : A x E t' for all A E A'}.

It is worth noting that X can he interpreted as the inner so- lutions of the interval linear equation A'x = t' ([IO], [I l l) . When viewed in this context, the set of all RCPA controllers admits a surprisingly simple characterization.

Theorem 4 ([lo]) x E X ifandonly i f s = x 1 - x z , where x l , x2 is a solution to the system of linear inequalities

A L x l - A U x 2 2 tL , A'x' - ALx2 5 tu, x 1 2 0, x2 2 0.

The above characterization in terms of 2m inequalities and 2q non-negative variables is considerably simpler than that proposed in [9], where the number of inequalities involves a combinatorial operation with the vertices of A'. In [12], the lack of an explicit characterization of X prevented the use

of more efficient optimization algorithms. The extension of the goal programming approach to uncertain (interval) plants is made easy in view of Theorem 4. The RCPA problem is restated as the an interval goal pmgramming problem with interval coeficients. See [8] for a detailed discussion about the subject.

Problem I IGP

subject to CLz + qL - pL = tL , c"z + 'I" - p" = tu, t 0, PL t 0,

'I 20, P" t o , L

U 'I

2 t 0, where

CL := [A" - A U ] and

Theorem5 Let w(A',t') be the optimal value of pmblem IIGP and z* = (x '* ,x2*) the corresponding optimal value o f z = ( x 1 , x 2 ) . Then x* = x l + - z2* E X ifand only if w(A', t') = 0.

Pmof: Let (z*,qL*,pL',q"*,p"*) he an optimal solution of problem IIGP. If w(A', t') = 0 then qL' = p"' = 0 and one obtains CLz* 2 tL and C'z' 5 tu, implying that

C" := [A" - A L 1

ALx" - A"x2* 2 tL , A"x" - ALx2* 5 tu, 21- 2 0, 2' 2 0,

and by Theorem 4, x* E X . If w(A',t') > 0 then at least one component i of CLz* or CUz* lies below t" or above

Example 3 - Consider the uncertain version of the second order plant of Examples 1 and 2:

tu, implying that x* g x .

s + a3 s2 - 2.2s + a6

P(s ) =

where a3 E [0.5,1.5] and a6 = [-2.6, -2.21. Assume T = 1 and t ' z [ tL , tU] aspropsedin[12]:

tL = (1.000,9.600,12.81,5.120), tu = (1.000,14.40,30.17,24.68).

The root space of t1 is illustrated in Figure 4. The optimal value of problem IIGP, w(A', t') = 0, indicates that the cor- responding first order controller

149

robustly assigns characteristic polynomials in t l . It is nat- ural to ask if the proposed RCPA problem could be solved by a stable minimum phase controller. To investigate such possibility, the constraint [ 12 -I2 ] z 2 0 imposing non- negative controller coefficients is added to problem IIGP.

.

- The resulting controller,

.~

11.61s + 14.24 cz(s)= s+O.428 '

isalsoasolutionfortheRCPAproblem(u(A', t ') = 0). The root space of the characteristic polynomial dp(s )dc , ( s ) + np(s)nc,(s), a E [aL, a'], illustrated in Figure 5, is en-

0 tirely contained in the mot space of t'.

for characteristic polynomial assignment in term of simple linear programming problems were introduced. In particu- lar, the use of interval analysis results enabled a substantially simpler solution for the RCPA problem when compared to its competing methodologies presented in 1121 and [9]. Pos- sible extensions of the proposed approach in similar lines developed in 191 include the consideration of interval param- eters appearing multiliieasly in the plant transfer function, the introduction of constraints (tolerances) to avoid fragile controllers and the treament of model matching problems, specially those related to the design of PID controllers. The above mentioned extensions and a detailed study of the gee metric properties of RCPA controllers are objects of cupent investigation by the authors.

. .

~- . .

-. . . .

-1 I: . . "..

References .

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Barmish, B. R. (1994). New-Tools for Robusmess of ~

V 1 - 141 Bardett. A. C., C. V. Hollot and H. Lin (1988). Root . .

..

. . I I

Fig. 5 : Root spaceof dp(s )dc , ( s ) + n p ( s ) n c 2 ( s )

- 5 Conclusions ~.

Pointyi@,~regional .&d robust characteristic polynomial as: signment problems-were focused in this paper in a goal pro- 5 m . g framework. Necessary and suflicient conditions

- . .~

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