Proceedings of the 1996 IEEE International Symposium

Dearborn, MI September 15-18,1996 on Computer-Aided Control System Design MPOI 4:20

3-12/3-1, Guaranteed Cost PID Design for Uncertain Systems: A Multiobjective Approach

Ricardo H. C. Takahashi’ Pedro L. D. Peres Paul0 A. V. Ferreira

Dep. of Electrical Eng. - UFMG Dep. of Telematics - FEEC-UNICAMP AV. AntBnio Carlos 6627, Belo Horizonte - Brazil C.P. 6101 - 13.081-970 - Campinas - Brazil

Email: takaQdt.fee.unicamp.br E m d . {peres}{valente}@dt.fee.unicamp.br

Abstract

This paper presents a methodology for sub-optimal design of PID compensators for systems subject to parametric un- certainties of polytope type and to disturbance signals. The adopted optimality criteria are the Hz and X, norms of the transfer matrices from the disturbance inputs and from the reference input to the controlled output error. The PID pa- rameters selection combines the different optimization cri- teria through a multiobjective technique, here adapted in order to allow the uncertainties to be handled in a minimax sense. True guaranteed cost values for optimization criteria are calculated.

Keywords: multiobjective programming, PID control, 3 t z /a, control, uncertain systems control, guaranteed cost control.

1. Introduction

In recently published papers [I, 21 the traditional prob- lem of PID controllers tunning is being revisited. This work is of practical relevance, since the majority of in- dustrial plants in operation is still based on that clas- sical structure. In this way, some recent advances in control theory may be quickly incorporated into the existing plants, so raising up its currently accepted per- formance levels. In control theory, some recently obtained results refer to controller synthesis under model uncertainty. The explicit accounting for model uncertainties allows the systematic design of controllers which behave in the “better way” in a set of situations, so reducing the need for heuristic adjustments. The uncertainties model here employed is based on the concept of the disturbed model parameters (real plant) pertinence to a known set of possible parameters values, this set defined as a convex polytope. This kind of modelling allows the determination of properties that are common to all el- ements in the set from the verification of the vertices elements only. A discussion on this kind of uncertain- ties representation may be found in [3]. Optimal control techniques have achieved a major rel-

On leave at UNICAMP.

evance in the contemporary control development, ei- ther as an instrument for copping with the increasing problems complexity or as a methodological tool for design techniques systematization. A class of optimiza- tion problems which is receiving concentrated research efforts is the minimization of system ‘H2 and ‘H, norms through a feedback controller. The class of techniques designated by 3 t 2 / ’ H , control possess interesting prop- erties of robust performance under perturbations; for a detailed discussion see [4]. The employment of these op- timality criteria in the specific case of this work has an additional suitability argument, as 3cz and X w norm characterization for polytope type model families can be achieved from the verification of the vertices only

In view of the diversity of objectives which must be simultaneously achieved in the PID controller design, objectives which are in general conflicting, such as dis- turbances rejection and time response to reference input specifications, a multiobjective programming approach is here employed [SI. Such approach aims at finding the so called efficient solutions, or Pareto optimal so- lutions, which are characterized as the solutions set in which any increase in a performance criterion necessar- ily implies in a degradation in some other criterion per- formance index. Minimax-type efficient solutions are here employed, in order to characterize solutions in the presence of uncertainties. In this way, the resulting PID design falls in the class of gzcaranteed cost controllers.

131.

2. System Model

Consider a linear time-invariant strictly proper system:

x = A x + Bu + Ew (1)

y = c x

in which A E PXn, B E WX1, E E Snxn and C E %Ixn. Vector z( t ) represents the system states, u(t) its control input, w ( t ) an unknown disturbances vector, and y ( t ) the controlled output. The triple (A, B, C) is not precisely known, but belongs to a polytope P given

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by: In closed loop comes:

u = K Z + H r

where v is the number of polytope vertices. The goal of the controller design is to control the output y(t) in such a way that the error in relation to a reference input r ( t ) is minimized, simultaneously minimizing the disturbance induced from signal w(t) in the output. De- fine this output error as: e(t) = r ( t ) - y(t). In order to attain that aim, the PID compensator structure is em- ployed, which calculates the control action ~ ( t ) through the relation:

de u = Kpe+Ki e d t + K d - J dt (3)

The PID compensator design problem consists in se- lecting the parameters K p , Ki and &. This problem is more conveniently formulated in terms of an augmented states equation. Define as two new state variables:

f = / e d t +c = -arc + a-lr (4)

in which a > 0 and a >> max(lX(A)I), with A(.) denot- ing the eigenvalues of the matrix argument. The vari- able r,(t) is necessary in order to make causal the com- pensator derivative term, and the output error must be redefined as: e(t> = re(t) - y(t). Note that re(t) can arbitrarily approach r ( t ) , just taking the parameter a arbitrarily big. Define the matrices:

A = [ t --a 0 0 0 1 B = [ ~ !

G = [ -0“ ; ;] D = [ 0 0 1 -c 1 0

F = [ O a-l 0 3 ’ E = [ E ’ 0 0 1 ’

-CA -a -CB

H = [ O 0 a-’]’ Ce=[-c 1 0 1

C = [ C 0 O ] K = [ K p K; K d ]

(5) and the variables:

Z = [ z re f ] ’ E = [ e f 6 1 ’ (6) From these comes the augmented states equation:

i = AZ + Bu + Fr + Ew E = GZ + Du + Hr

y = C Z

(7)

The PID compensator design is transformed, in this new state space, into an static output feedback con- troller design: -

U = KC (8)

3. Performance Criteria

In order to account for the control effort in the opti- mization criteria, the above system description is rede- fined, incorporating a new fictitious (‘output’’ propor- tional to the control input:

@ e = [ c;. ] E = [ O- ] (11) CYK a H

In this definition, the parameter CY makes the relative weighting between the output error cost and the control action cost. Define two classes of performance criteria for the closed loop system (9), the first class relative to disturbance re- jection, the second one relative to the reference tracking response. The disturbance rejection criteria are associ- ated to the transfer matrix between the perturbation vector w(t) and the output vector [ e(t) au(t) 1’:

Hwue = Ce(sI - A)-% (12) The reference tracking criteria are associated to the transfer function between the reference signal r ( t ) and the output vector [ e ( t ) au( t ) 1’:

(13) H,,, = Ce(SI - A)-lB + ij

1 1 ~w u e 1 1; = n ( E ’ ~ o E)

llHr+ieII; = q 3 ~ o B ) (16)

The optimization criteria to be employed in this work are the 7 i 2 and 7i- norms of these transfer functions, defined as:

(14) IIffwuell, = SUP {bmaz [ H w u e ( j ~ ) ] ) (15)

WE%+

and matrix Lo is the solution of

(18) 2 L o + LOA + E;Ee = 0 Equation (16) actually does not define the 7 i z norm of the transfer function (13), but the same norm for the transfer function:

(19) H,,, = Ce(sI - A)-% In (19) the portion of the control action ‘(due” to the in- put reference is not accounted for in the augmented out- put, and only the portion ‘(due” to the states appears.

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Such simplification is necessary because the transfer function (13) is not strictly proper, what would make its 3 t 2 norm infinite. The adopted weighting is, how- ever, sufficient for the purpose of bounding the control action in the designed sub-optimal controller.

4. Mono-objective Optimization Problem Formulation

In this section, the PID compensator design problem is formulated, as a mathematical programming problem with the objective functions defined in the former sec- tion. The optimization of each performance criterion separately is firstly approached. Two constraints are convenient for the PID de- sign mathematical programming problem formulation. These constraints are: (i) Maximal response time to the reference signal constraint. Such constraint is need- ful because the optimal controllers (particularly in the case of E, ones) for the reference signal loop may lead to extremely slow responses. Let 1/S be equal to the maximal admissible time constant for the closed loop system. This constraint is introduced as a matrix A eigenvalues location restriction:

K E { K : max ( ~ e [x (“ (K) ) ] ) 5 -6) (20)

and (ii) Constraint on the maximum admissible abso- lute value for parameters K p , K; and Kd:

E KM A { K lKpl 5 k p i lKil 5 ki; l K d l 5 k,} (21)

Let P be an arbitrary element of polytope P of pos- sible system models. Four optimization problems are formulated as:

min max Jh(K,P) h = { l , 2 , 3 , 4 } (22)

K P

s.t.: { K E n KM; P E P } with:

In order to cope with uncertainty, the optimization problems are defined in a minimax sense, so generat- ing the controller which minimizes the cost function J h maximum for any plant contained in the uncertainty polytope. Some computational regards must be made at this moment: (i) There is not, till the moment, any available tool for performing the global optimal calcu- lation of functionals J1 to J4 in the case of systems with output feedback of order less than the plant or- der (this is the case here). Therefore, the algorithms to be employed here will guarantee only local optimal- ity for the resulting controller. In view of this solution intrinsic sub-optimality, it is here suggested the employ- ment of optimization algorithms baed on primal meth- ods which make the search through the feasible solu- tions regions, so furnishing feasible sub-optimal results. (ii) Most of constrained optimization methods admits

searching through infeasible regions in some steps of the trajectory towards the optimal solution. In the case of problems defined by (22), however, the optimization algorithm must restrict itself to walk over trajectories entirely contained in the stabilizing gains set. Such req- uisite is due to the existence of singularities in objective functions J1 to J4 precisely in the frontier between the stabilizing and the non-stabilizing gains regions. (iii) At each optimization algorithm step, an objective func- tion evaluation (3-12 or E, norm) must be performed. There are available computational techniques for calcu- lating upper bounds for ‘H2 and XW norms for uncer- tain systems contained in uncertainty polytopes. Such techniques, however, are computationally expensive, in- volving the solution of an entire optimization problem for each evaluation to be performed. The difficulty pointed out in consideration (iii) suggests the reformulation of optimization problem (22). A new problem is so defined:

in which Pv is the set of the vertices of P. This new problem involves only the norms evaluation at the poly- tope vertices, so being less burdensome. However, once the controller has been found with this new problem formulation, another problem still remains: the “guar- anteed cost” associated to the solution, in this case, is still undetermined. This issue is discussed in section 6 of this work. The initial stabilizing solution, whose necessity is pointed out in remark (ii), may be found through the following auxiliary problem:

2 min K [Re (A,(A)) - 261

i

s.t.: m.ax Re A.(A) < 0 I L - 3

5. Multiobjective Problem Formulation

The optimization problems defined in (22) or (24) have in general competing solutions, what means that their individual solutions minimize only one among the ob- jective functions, letting the other objective functions in non-minimizing points. It is reasonable expecting that, in most situations, the better solution will not necessar- ily coincide with any of that individual optimals, being rather in general a compromise solution among the di- verse performance criteria. The adequate conceptual framework for defining what would be the solutions for this “compromise” problem is furnished by the multa- objective programming theory [5]. The matter is therefore to minimize, in some sense, the objectives vector J E g4, whose components are de- fined in (23). The central concept in multiobjective programming is the efficient solutaon or Pareto-optimal sohtaon. The set X * of efficient solutions z* may be

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characterized by: e* E X * e { ,B 3: # e* / J(e) 5 J(z*);

(26) J ( c ) f: J ( z * ) ; e E D}

in which J E SR4 and D is the optimization problem domain. In the case of non-convex problems (as in this case) the locally efSicient solzltions becomes rele- vant, that is, the solutions which are efficient in an open neighborhood of e* (and not necessarily in the whole admissible function domain). As the difference between these definitions is a matter of choice in the function domain V , there will be no effort here in dis- tinguishing between them. However, note that the ef- ficient solutions that will be here determined will be, in general, only locally efficient solutions. For brevity, the statement that an efficient solution must belong to the feasible solutions set has been omitted in the above expression. This assumption will be implicit in the re- mainder of this section. The multiobjective programming, as a methodology, is proper for giving to the designer the tools for attaining the efficient solutions set. The designer must decide, among these solutions, which is the most suitable for effective implementation. The technique here employed in the determination of ef- ficient solutions is based on a relaxation method, which approachs the efficient solutions set following a relax- ation direction, through an scalar relaxation parameter minimization (this is a variation of the goal attainment technique). Define the vector Jo as the utopzan soh- tion, or the vector whose components are the individual optimals of each objective function. Define also the re- laxation direction vector W:

J: = min Jh(K) K

LJ! 1' 1 w4 1 The multiobjective problem is then formulated as a mono-objective optimization problem with the form:

r = min c

s.t.: J ( K ) 5 J o + u.W The original optimization parameters are given in vec- tor K . An additional relaxation scalar parameter U has been defined. The designer may arbitrarily define the relaxation direction vector W, letting a smaller rela- tive relaxation to the more important objectives, and vice-versa. Each different choice will lead to a different solution in the set of efficient solutions. The process of choicing among the efficient solutions is performed in an interactive way, with the designer altering the relax- ation direction and verifying the obtained performance until reaching a suitable solution. The resulting solu- tion Kr from (28) is such that:

Kr E Kr L? { K : J ( K ) 6 Jo + r.W} (29)

It can be demonstrated that problem (28) generates all the efficient solutions of a multiobjective program- ming problem, even in the case of non-convex objective functions [5 ] . This would not be the case of the weight- ing method', for instance. However, program (28) may in some cases generate non-efficient solutions in non- convex problems. In these cases, the set K p contains more than just one element. In order to assure that the final solution of the proposed method belongs to the efficient solutions set, one more step is necessary in the optimization procedure. Define the following complementary problem:

Ki = arg min J,-(K) 3.t. : J(K) 5 J(Kg-1)

The resulting solution K$ from (30) is certainly an ef- ficient solution for the multiobjective problem. The se- quence of objective functions optimization in (30) is . ar- bitrary, so a priority order might be followed, the most important objectives solved in the first place.

6. 'Hz/'H, Guaranteed Costs Calculation

As discussed in section 4, the search algorithm here em- ployed in the determination of efficient solutions for the controller design problem verifies the performance cri- teria only over the uncertainty polytope vertices. Once the controller has been chosen, on the basis of this algo- rithm, the next problem to be solved is the determina- tion of criteria values which are "guaranteed costs" for the uncertainty polytope, which means they are values not less than the maximalvalues assumed by the objec- tive functions in every point inside the polytope. Note that if problem (22) were employed in place of problem (24) the present question would had been already im- plicitly solved. But this procedure would had been to a large extent more computationally expensive. Consider matrices d, B , E and C, of the closed loop system, as defined in equations (5) and (10). Define matrices Ai, and cei as the corresponding closed- loop system matrices for each vertex i of the uncertain- ties polytope. Matrix ,?? does not depend on uncertain parameters. Consider firstly the ?io0 norm guaranteed cost compu- tation for the transfer matrix between the disturbances input and the output error. Define, for each vertex i of the uncertainties polytope:

E'

'The weighting method for multiobjective optimization con- sists of defining a mono-objective problem with objective func- tion taken as a weighted s u m of the multiple individual objective functions.

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In the case of the transfer matrix between the reference signal and the output error, substitute matrix E by matrix B in this expression. Now define the set F6 as:

(32) F6 = {(w, 6 ) E RnX" x 8 /

w = w t > o ; r i 2 0 ~ i = i , . . + I in which v is the number of polytope vertices. The problem of X, norm computation can be defined as the convex mathematical programming problem:

6* = min 6 (33)

llH(s)ll, = J6T (34)

S.t . : (w,6) E F6 and:

This procedure is a direct extension for the uncertain systems case of the one presented in [6]. Consider now the 'I& norm guaranteed cost computa- tion. Define for each vertex i of the uncertainties poly- tope the following matrix, associated to the transfer matrix between the disturbances input and the output error:

. . . . . . . . . . . .

L

... 0 (35)

1 ...

E -(A,.W Et + w.& I J [ I

In the case of transfer matrix between the_ reference in- p-ut and the output error, replace matrix E with matrix Bd in the above expression. Define now the set LA as:

(W,A) E RnX" x Rmxm / La= [ W = W ' > O ; A = A ' > O } (36)

Ad( W, A) 2 0 V i = 1, .... v The 'H2 norm guaranteed cost computation problem is defined as the convex mathematical program:

6* =min n ( A ) S.t.: (w, A) E LA

The upper bound for the 'HZ norm is given by relation:

(37)

llH(~>1122 I f5* (38)

7. Numerical Implementation

The design methodology here presented has been im- plemented through the following algorithm: (1) Solve the auxiliary problem (25). (2) Solve problem (24). (3) Solve problems (28) and (30) for a given relaxation direction vector. If the resulting solution does not meet the design requirements, choose another relaxation vec- tor, and repeat this step until a satisfactory solution is found. (4) With the sub-optimal compensator pa- rameters, determine the guaranteed costs for the closed loop uncertain system, through problems (33) and (37) resolution. Steps 1-3 may employ the same optimiza- tion engine, which may be any primal space constrained optimization algorithm. A quasi-Newton method has

been here employed. The routine for step 4 compu- tation has been implemented based on the MATLAB package LMILAB [7]. Example 1: Consider an uncertain system with open loop poles and zeros of the control input-to-output transfer function shown in figure 1. These parame- ters have been randomly generated around a "nomi- nal plant". The execution of design algorithm on these data until step 6 has lead to the results summarized in table 1. The analysis of table 1 reveals a sharp ad-

................................... .... ...... ....... . . . . . . . . . . . . . . . . . . . i o 1 ; : :

-7 d -5 -4 -3 -2 -1 0

Figure 1: Open loop poles (x ) and zeros (0); above: vertex 1, center: vertex 2; bellow: vertex 3

I (1) I (2) I (3) I (4) I (5) (a) I 18.57 I 3.530 I 74.62 I 3.860 I 3.969

Table 1: Controller synthesis results summary. The columns refer to: (1) ' H z optimal disturbance-to- output transfer matrix norm design; (2) 'HZ optimal reference-to-output t.m. norm design; (3) 'H, optimal disturbance-to-output t. m. norm design; (4) H, opti- mal reference-to-output t.m. norm design; (5 ) multiob- jective design. The rows refer to: (a) 'HZ norm of the reference-to-output t.m.; (b) 'H, norm of the reference- to-output t.m.; (c) H2 norm of the disturbancesto- output t.m.; (d) 'H, norm of the disturbances-to-output t.m.. The values are the worst performance index over the set of the polytope vertices.

vantage of the multiobjective solution, when compared with each individual optimization solution. The mul- tiobjective controller achieves in each criterion a per- formance which is always near the best individual op- timization controller [the best controller for each crite- rion is of course the one which individually minimizes that specific criterion). The individual optimization controllers, in turn, are very good in their specific cri- teria, but present a poor performance in at least one of

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GC: WV:

Table 2: Guaranteed costs and worst-vertex costs of the ‘K2 and ‘h!, norms, for the multiobjective-optimal PID closed-loop system. The columns refer to: (I) ‘Hz norm of the disturbances-to-output transfer matrix; (11) ‘ H 2

norm of the reference-to-output transfer matrix; (111) H, norm of the disturbances-to-output transfer matrix; (IV) ‘H, norm of the reference-to-output transfer ma- trix. The rows are: (GC) the guaranteed costs calculated through the developed procedures; (WV) the worst case norm over the set of polytope vertices.

(1) (11) (111) (IV) 0.490 7.380 0.282 1.353 0.308 6.425 0.189 0.898

the other criteria. Table 2 shows a comparison between the guaranteed cost norms, computed for the above multiobjective- optimal PID, and the corresponding worst-casevertex computed norms. The guaranteed costs, although its conservativeness, are not far from the worst case ver- tex norms. It must be noted that the norms shown in table 2 have been calculated for transfer matrices without any weighting on the control inputs. Remem- ber that these weightings have been introduced for the purpose of preventing the controller gains of becoming unbounded, and may be removed in the costs compu- tation. Therefore, the worst-vertex costs of table 2 are smaller than the corresponding ones of table 1.

8. Conclusion

In this paper, a methodology for the systematic design of PID compensators has been developed, in order to take into account plant model parametric uncertainties and noise disturbances inputs. The presented method- ology combines four objective functions (two related to the tracking response and two related to the distur- bances rejection) through a multiobjective framework, so leading to a uniform performance in those criteria. As an additional feature, the design methodology here presented allows the computation of X2 and X, guar- anteed costs for the resulting controller, so giving a min- imal theoretically assured minimal performance.

References

[l] A. A. Voda and I. D. Landau, “A method for the auto-calibration of PID controllers,” Automatica, vol. 31, no. 1, pp. 41-53, 1995. [2] H. E. Musch and M. Steiner, “Robust PID control for an industrial distillation column,” IEEE Contr. Sys. Magaz., vol. 15, no. 4, pp. 46-55, 1995. [3] P. L. D. Peres, J. C. Geromel, and J. Bernussou, “Quadratic stabilizability of linear uncertain systems in convex-bounded domains,” Automatzca, vol. 29, no. 2,

[4] J. C. Doyle, K. Glover, P. P. Khargonekar, and B. Francis, “State space solutions to the standard 7-12

pp. 491-493, 1993.

and ‘Hw control problems,” IEEE Trans. Aut. Contr., vol. 34, no. 8, pp. 831-847, 1989. [5] V. Chankong and T. K. C. Peng, Multiobjective deczszon making - Theory and methodology. New York: Elsevier, 1983. [6] R. M. Palhares and P. L. D. Peres, “‘Ifm norm calculation for a transfer matrix: a convex approach (in portuguese),” in Proceedings of the XVIII National Conference of Applied and Computational Mathemat- acs, (Curitiba, Brasil), pp. 205-209, 1995. [7] P. Gahinet and A. Nemirovskii, “General purpose LMI solver with benchmarks,” in Proc. 32th Conf. Dec. Contr., (San Antonio, USA), pp. 3162-3165, 1993.

Appendix I 7 - i ~ Guaranteed Costs Derivation

In this appendix it is shown the derivation of expres- sions (35) to (38). Starting with the 7 - i ~ norm definition for a transfer matrix:

H ( s ) = C(s1- A)-’B

llH(s)ll; = t4CWC’) (39)

W : {AW + WA‘ + BB’ = 0 ) one has:

CWC’ 5 A +- tr{CWC‘} 5 t r {A)

This relation is obtained by Schur’s complement. With this:

Considering also the positive definiteness of W and the negativity of matrix A eigenvalues, one has from (39): AW + WA’ + BB’ 5 0 +- IIH(s)lli 5 tr{CWC‘} (42)

Also with Schur’s complement: B‘ ] 2 0 AW+WA’fBB’ I O e

B -(AW+ WAt) (43)

[ ’ Taking together equations (41), (42) and (43) the de- sired result is found:

CN L I‘ n

* l l ~ ( ~ > I l ; 5 trial (44)

Finally, the trace of A is minimized in P and A, while keeping valid the above relation (remember P is the uncertainties set).

Acknowledgements

This work has been supported by CAPES, CNPQ, FAPESP and FAPEMIG.

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