genetic algorithm for robust h infinity

8/3/2019 Genetic Algorithm for Robust h Infinity

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IEEE TRANSACTIONS ON INDUSTRIAL E LECTRON ICS, VOL. 43, NO. 5 , OCTOBER 1996

Structured Genetic Algorithm forRobust H'>OControl Systems DesignK. S . Tang , K . F. M a n , Member, ZEEE, and D.-W. Gu

Abstruct- A structured genetic algorithm (SGA) approach isdeveloped for robust controller design based on the concept ofan H" loop-shaping technique and the method of inequalities.Such an SG A is capable of simultaneously searching the ordersand coefficients of the precompensator and postcompensator forthe weigh ted plant. It is, therefore, not necessary to pre define theorder of compensators as in usual practice. A multiple objectiveranking approach is also incorporated so that the design criteriaof extreme plants can be easily achieved. The effectiveness of sucha technique is illustrated by a high-purity distillation columndesign example.I. INTRODUCTION

T IS well known that control system design problems can beI aturally formulated as constrained optimization problems,the solutions of which will characterize acceptable designs.The numerical optimization approach to controller design candirectly tackle design specifications in both the frequency-domain and the time-domain. The optimization problems de-rived, however, are usually very complicated with manyunknowns, many nonlinearities, many constraints, and in mostcases, they are multi-objective with several conflicting designaims which need to be simultaneously achieved. It is alsoknown that a direct parameterization of the controller willincrease the complexity of the optimization problem.

On the other hand, during the last decade or so , H" opti-mization has emerged as a useful tool for robust control systemdesign. It has a sound theoretical background for handlingmodel uncertainties. Based on H" optimization, a variety ofdesign methods have been developed. The H" loop shapingdesign procedure (LSDP) is one of these which has provento be efficient in practical industrial design. The approachinvolves the robust stabilization to additive perturbations ofnormalized coprime factors of a weighted plant. Prior to robuststabilization the open-loop singular values are shaped usingweighting functions to give a desired open-loop shape whichcorresponds to good closed-loop performance. A successfuldesign using LSDP depends on the appropriate choice ofweighting functions, which in turn relies on a designer'sexperience and familiarity with the design approach.In [ I I] , it is proposed to enhance the LSDP by combiningit with numerical optimization techniques. In order to moreeffectively search for optimal solutions to the derived con-

Manuscript received September 20, 1995; revised November 12, 1995.K. S. Tang and K. F. Man are with the Department of Electronic Engineer-D.-W. Gu is with the Department of Engineering, Control Sys tems Re-Publisher Item Identifier S 0278-0046(96)03301-1.

ing, City University of Hong Kong, Hong Kong.search, University of Leicester, Leicester LE1 7R H, U.K.

575

Fig. 1. Robust stabilization with respect to coprime factor uncertainty

strained optimization problems, a multiple objective geneticalgorithm is suggested in [12]. In this mixed optimizationapproach, the structures of the weighting functions are pre-defined by the designer. It is not possible to systematicallysearch for an optimal design among various structured weights.In this paper, a structured genetic algorithm (SGA) is devel-oped. The hierarchical structure of the chromosome makes itpossible to optimize simultaneously over the structures andcoefficients of the weighting functions.

The paper is organized as follows. Section I1 gives a briefintroduction to LSDP and the mixed optimization approach.SGA is explained in detail in Section 111. Such a technique isthen applied to a design exercise of a high-purity distillationcolumn. Several designs, with discussions, are presented inSection IV. Concluding remarks are given in Section V.11. A MIXEDOPTIMIZATIONESIGNAPPROACH

LSDP is based on the configuration as depicted in Fig. 1 ,where (fi,U ) RH", the space of stable transfer functionmatrices, is a normalized left coprime factorization of thenominal plant G. That is, G = l k ' f i , and there existsV , U E RH " such that A h + f i U = I , and*,* +NN*I , where for a real rational function of s , X * denotes X T (- s ) .For a minimal realization of G( s )

G( s )= D + C ( s 1 - A ) - l BE [

0278-0046/96$05,00 0 996 IEEE


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576 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 43 , NO . 5, OCTOBER 1996

a normalized coprime factorization of G can be given by [7][N f i ] = + H C 1 B + H D H ] (2 ) ]TwTHTky

R-1/2C R - l / 2 D R-112where H = - ( B D + ZC)R- R = I + DD and thematrix 2 2 is the unique stabilizing solution to the algebraicRiccati equation (ARE)

( A- BS-IDC)Z + Z ( A- BS- lD/C) / Fig. 2. Th e 2-DOF scheme.- ZCR-ICZ + BS-IB = 0 ( 3 )where S = I + D I D .A perturbed model G , of G is defined as

G, = (A? + A,)-l(N + A,) (4)where A U , AN E RHm.To maximize the class of perturbedmodels such that the closed-loop system in Fig. 1is stabilizedby a controller K( s ) , he K ( s ) must stabilize the nominalplant G and minimize y [5] where

From the small gain theorem, the closed-loop system willremain stable ifIIPN a M l l l , < 7 - l . ( 6 )

The minimum value of 7,yo), for all stabilizing controllers

system G are modified to give a desired loop shape.The nominal system and weighting functions W1 andW2 are combined to form the shaped system, G,, whereG, = W2GW1. It is assumed that Wl and W2 are suchthat G, contains no hidden unstable modes.

2 ) A feedback controller, K,, is synthesized w hich robustlystabilizes the normalized left coprime factorization ofG,, with a stability margin t.3) The final feedback controller, K , s then constructed bycombining the H controller K, , with the weightingfunction Wl an d W2 such that

K = W 1 K s W 2 . (10)Fo r a tracking problem, the reference signal is generally fedbetween K , an d W l,so that the closed loop transfer functionbetween the reference T and the plant output y becomes

where A represents the maximum eigenvalue, and X 20 i s the unique stabilizing solution to the following ARE:( A - BS-lDC)X + X ( A- L ? S - l D C )- X B S - l B X + C R - l C = 0 . (8)

A controller which achieves a y > yo is given in [7]byK =

y2(Q/)-1zc/D 1+ B F + y2(Q)-1ZC(C+D F )B I X(9 )

where F = - S - l ( D C + B X ) and Q = (1 - y 2 ) I+ X Z .optimal controller such that the minimum value yo is achieved.

In practical designs, the plant needs to be weighted tomeet closed-loop performance requirements. A design m ethod,known as the loop shaping design procedure (LSD P), has beendeveloped [7],8] to choose the weights by studying the open-loop singular values of the plant, and augmenting the plantwith weights so that the weighted plant has an open-loop shapewhich will give good closed-loop performance.This loop shaping can be done by thle following designprocedure.1) Using a pre-compensator, W,, and/or a post-compensator, W;, the singular values of the nominal

A descriptor system approach may be used to synthesize an

where the reference T is connected through a gain K , 0)W2 (0 )where

K,(O)WZ(O)= lim K,(s)W2(s) (12)s i 0to ensure unity steady-state gain.The above design procedure can be developed further intoa two-degree-of-freedom (2-DOF) scheme as shown in Fig. 2.The philosophy of the 2-DOF scheme is to use the feedbackcontroller K, (s ) o meet the requirements of internal stability,disturbance rejection, measurement noise attenuation, and sen-sitivity minimization. T he precompensator Kp s then appliedto the reference signal, which optimizes the response of theoverall system to the command input. The precompensator Kpdepends on design objectives and can be synthesized togetherwith the feedback controller in a single step via the H LSDP

In LSDP the designer has the freedom to choose theweighting functions. Controllers are synthesized directly. Theappropriate weighting functions will generate adequate optimalyo and will produce a closed-loop system with good robustnessand satisfactory and nonconservative performance. The selec-tion of weighting functions is usually done by a trial-and-errormethod and is based on the designers experience. In [ l l ] , tis proposed to incorporate the method of inequalities (MOI)[141 with LSD P such that it is possible to search for optimalweighting functions automatically and to meet more explicit

~41.


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Fig. 3 . Chromosome structure.design specifications in both the frequency domain and thetime domain.In this mixed optimization approach, the weighting func-tion s W1 and W2 are the design parameters. Control systemdesign specifications are given in a set of inequality con-straints. That is, for a given plant G(s), to find (W 1, Wz)such that

yo(G,WI, W z) = [1+ Lax(ZX)]1'2Iy (13)& ( G , W1, W2) S E , . i = 1. 2 , . . . , n (14)

and

where 4,'s are performance indices, which are algebraic orfunctional inequalities representing rise time, overshoot, etc.,and and E , are real numbers representing the desired boundson yo and &, respectively. Numerical search algorithms m aythen be applied to find solutions to the above optimizationproblems.111. STRUCTUREDENETICALGORITHM

The constrained optimization problems derived by themixed optimization design approach described in the previoussection are usually nonconvex, nonsmooth, and multiobjectivewith several conflicting design aims which need to besimultaneously achieved. In [121, a multiobjective geneticalgorithm [3] is employed to find solutions to such optimiza-tion problems. Successful designs have been achieved. It hasbeen, however, found that due to the order and structureof weighting functions being pre-fixed by the designer, theapproach lacks flexibility in choosing weights of differentforms, which in turn affects the optimality of the design. AnSG A [lo] is thus considered in this paper to optimize overboth the orders and coefficients of the weights, W1 and W2,used in the design.A. Chromosome Coding

SGA differs from standard GA in its hierarchical structurein that each chromosome consists of a multilevel of genes.Fig. 3shows the chromosome representation within SGA fora transfer function where the fundamental structure is

There are two types of genes, known as control genesand coefficient genes, in the chromosome. The control genesin the form of bits decide the activation or deactivation ofthe blocks. The coefficient genes define the value of thecoefficients in each block. The following example explainssuch a hierarchical structure.Example:Control genes: [1,0 , 0; 0, 0, 11Coefficient genes: [0.1, #, #; #, #, #, 0.7, 0.8, 1.51'

These represent a transfer function ofs + 0.1

s2 + 0.7s + 0.8G(s )= 1.5 xThe fundamental structures of Wl and W2 in the desig nexample in Section IV are given as(s + w d ( s + Ulg) (SZ + Ul7S + U l g )S (S + u l l ) ( S + 7U2)(S2 +W3.9 + 204)( S + ?Ug)(S + Wl*)(S2 + W l l S + 2012)

w1=(17)

(s f w13)(s+W14)(s2 + w15S +wl6)wz =(18)

In general, Wl and W, can be diagonal matrices withThe chromosome is a binary string describing the controldifferent diagonal elements.and coefficient genes gc and g r , wheregc E BIZ

where B = [0, 11 and R I , R2 defining the search domainfor the parameters, which usually represents an admissibleregion, e.g., ensuring that the weighting functions are stableand minimum phase.B. Objective Function and Fitness Function

An objective function f is defined, for a chromosome if (13)is satisfied, as the number of violated inequalities in (14). Theprocedure of objective function evaluation is listed as follows.1) For a chromosome I = (W I ,W2) in hierarchy coded2) Calculate G, = WzGWl.3) Find the solutions Z,, X , to (3) and (8).4) Calculate yo(W l ,Wz) by (13).

a) synthesize K , by (9);b) calculate 4,(G, W I , W2) or the present chromo-c) calculate f by

form, generate the corresponding W1 and Wz.

5 ) If yo < E T :some;

nf = C rn,

z = 1where

0, if q5 5 ~i1, else.i =6) if yo 2 E ~ , = n + 1 +yo.'# 'slands for Don't Care Value


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578 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS. VO L 43, NO 5, OCTOBER 1996

To convert the objective function ( f ) o the fitness value, a TABLE 1PA RA M E T E RETTINGF SG Alinear ranking approach [I31 is applied.Population Size 40Generation Gap' 0.2C. Genetic Operations Control Gene

Resolutionrossover and mutation on binary siring 121 are applied 1 bitindependently on different levels of a (chromosome as in a Range B 10. 11standard CA. Crossover 1-point Crossover Crossover Rate = 0.7Mutation Bit Mutation Mutation Rate = 0.05

Coeficient GeneD. Optimization Procedure Resolution 10 bits1) Define the plant G and define the functions &.2) Define the values of E, and cy .3) Define the fundame ntal form of the weighting functionsW, an d W,, and the search domain of R I .Rz.4) Define the parameters for SGA.5) Randomly generate the first population.6) Calculate the objective value and assign the fitness valueto each chromosome.7) Start the SGA cycle:

a) select parents by the Stochastic Universal Samplingmethod [ I ] ;b) generate new chromosomes via crossover and mu-tation;c) calculate the objective values of the new chromo-somes;d) reinsert the new chromosomes into the populationand discard same number of old, low-ranked chro-mosomes.

8) Terminate if (13) and (14) are satisfied; otherwise, repeatth e SGA cycle.IV . EXAMPLE:HE DISTILLATIONOLUMN ESIGN

The proposed algorithm is used to design a feedback controlsystem for the high-purity distillation column described in [6].The column is considered in its LV configuration [9] (LV

Range RI ( 0 , 2)R2 (0, 500)Crossove r 3-point Crossover Crossover Rate = 0.8

Mutation Bit mutation Mutation Rate = 0.12Number of N ew Chromosomes Generated = Generation Gap xPopulation Size

4) The output response to a step demand h( t ) y ] satisfies-0.1 5 y l ( t ) 5 0. 5 for all t , -0.1 5 y 2 ( t ) 5 1.1 foral l t and g2(t) 2 0.9 for all t > 30.

5) The frequency response of the closed-loop transfer func-tion between demand input and plant input is gainlimited to 50 dB and the unity gain crossover frequencyof its largest singular value should be less than 150radlmin.Simulation A : Optimization of Nominal Plant SpecGCations

with Time Delay r1 = r2 = 0.5: The proposed algorithmis used to satisfy the performance design specification for thenominal plant G, using the configuration of Fig. 2.The designcriteria are derived from (13) and (14)YO(Gn, Wl; WZ) FE-, (22)(67(G,,,W1, WZ)I E ~ ,or i = 1, 2, . . . , 16. (23)

For stability robustness, the value of E~ should not be toolarge, and is here taken asE?. = 5.0. (24)indicates that the inputs used are reflux (L)-and boilup (V ) ) ,for which the following model is relevant: The performance functionals 4%(G,, W1, W2) are definedin the Appendix, and the respective prescribed bounds aredecided from the design specifications- and are show n in thesecond column of Table 111. The parameters of the SGA used

in the simulation are tabulated in Table I.It takes about 135 generations to obtain the optimal com-pensators. The weighting functions obtained are

1[(Iyi8 -0.8640 IC,e-r2s

kl, 21 71 1 7-z) = 1 ,082 -1.096] (21)where 0.8 5 k1 , IC2 5 1.2 and 0 5 7-1, 7-2 5 1, and alltime units are in minu tes. The time-delay and actuator-gainvalues used in the nominal model G, are IC1 = IC2 = 1 andr1 = r2 = 0.5. The time-delay element is; approximated by afirst-order Pad6 approximation for nominal plant. The designspecifications are to design a controller which guarantees forall 0.8 5 k l , kz 5 1.2 and 0 5 TI , 2 5 1:

1) Closed-loop stability.2) The output response to a step demand h( t ) [A] satisfies-0.1 5 y l ( t ) 5 1.1 for all t , y l ( t ) 2 .9 for all t > 30and -0.1 5 y ~ ( t ) 0.5 for all t ;3) The output response to a step demand h ( t ) satisfiesy l ( t ) 5 0.5 for all t , yl(t) 2 .35 for all t > 30 and:y2(t) 5 0.7 for all t , and g2(t) 2 0.55 for all t > 30.

(s+ 1.2800)(s+ 1.5005)w - ( s + 0.8215)(s+ 1.4868)(s + 1.7873)(s2+ 0.5620s+ 1.9844)( s+ 1.7385)(s2+ 1.4946s+ 1.8517)z =

O I36.0976[ 0 36.5854

with yo = 3.6147 which successfully satisfy (22) and (23).The convergence of the objective value is plotted in Fig. 4.Extreme plants GI, G2, Gs, Gg with system parametersshown in Table I1 were used for testing the system's robust-ness. These extreme plant models were judged to be the mostdifficult to obtain simultaneously good performance and it wasfound that the final system was not very robust.


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1

Objective ValueMI

50 . .. .,. . . . . _ . _ . - . I _ . . . .h i I i 1 . . . .. . . . . - 3 . . . .. . . . .- + -

- . -

. . .I I

generationsObjective value versus generations.ig. 4.

TABLE I1EXTREME PLANTS G , FOR j = 1, 2 , 3 , 4

GI 0 0 0.8 0.8Gz 1 1 0.8 1. 2Gs 1 1 1. 2 0.8Gq 1 1 1.2 1.2

Simulation B: Optimization of Plant SpecGcations with TimeDelay 71 = 7 2 = 1: The setting is the same as that forsimulation A, except that the design criteria, (23), are modified.From simulation A, it is realized that the design criteria forTI = 7 2 = 1 are more difficult to achieve. Hence, the designcriteria is modified to

Yo(Gn, Wl,W2) E y (25)q5%(Gm, 1, W 2)I,, for i = I , 2 , . . . , 16 (26)

where G, is the plant with k-1 = k2 = 1 and 71 = 7 2 = 1using a fifth-order Pad6 approximation.It takes about 800 generations to obtain Wl and W2 fo rrobust feedback control. The parameters of Wl an d W2 are(s + 0.8878)(s2 + 0.3161s + 1.1044)s ( s+ 1.1278)(s2+ 1.1356s + 0.1444),

O I

O I

13.1707[ 0 13.1707(s + 0.1678)(s2+ 0.3727s + 0.7161)

- s + 1.5727)(s2+ 1.0049s+ i.8946)w -50.2439[ 0 52.1951

where yo = 3.3047. The closed-loop performances are tabu-lated i n Table I11 and depicted in Fig. 5. All the design criteria

I0 20 40 60 80 100time - min

-0.21 '

I0 20 40 60 60 1Wtime. min

-0.2' ' ' '

51 9

0.70.60.50.40.30.20.10

-0.10 20 40 60 80 100lime - min(b)

(lain . B

lo-* 1 102frequency - rad/min(4

Fig. 5 . System performance for simulation B. (a) 41 - &,. (b) 4 6 - 9 .(c) O I O - d14. (d) dl5 - 416TABLE I11FINAL YSTEM ERFORMANCEOR SIMULATION B

~i $i ( Gm) di(G1) $i(Gz) d i (G3) di(G4)1 1.1 1.0212 1.0410 1.0377 1.0306 1.00302 -0.9 -0.9732 -0.9655 -0.9898 -0.9870 -0.96873 0.5 0.3556 0.3625 0.2844 0.4290 0.34714 0.1 0 0 0.0001 0.0002 05 0.1 0.0159 0.0305 0.0232 0.0477 0.00546 '0.5 0.4177 0.4198 0.4272 0.4274 0.43087 -0.35 -0.3981 -0.3966 -0.3851 -0.3873 -0.38438 0.7 0.6012 0.6023 0.6149 0.6162 0.61919101112131416

r16

-0.550.51.1-0.90.10.1

50.0150.0

-0.59720.38371.0124-0.98320.0171

048.44799.0773

-0.59670.39121.0242-0.97860.0330

049.705010.7159

-0.58380.43001.0299-0.97480.04170.000351.04279.8627

-0.58730.33591.0372

-0.98410.02700.0002

51.016710.000

-0.57770.37461.0138-0.96600.0058

047.769.8627

are satisfied except that the 50-dB gain limit is marginallyexceeded by +15(GZ)and +15(G3).Simulation C: Optimization of Overall Plants Specificationswith Extreme Conditions: Since it may not be easy to obtaina controller satisfying the performance specifications for thoseextreme plant models by optimization of the nominal plant ora typical plant, an alternative will simultaneously optimize all


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580 lEEE TRANSACTIONS ON lNDUSTRIAL ELECTRONICS, VOL. 43, NO. 5, OCTOBER 1996

the extreme plants. The design criteria are now redefined as AYO(W1: W2) < E T ,

q5i(G',, W I; W2) < E ; , f o r i = 1, 2 , . . . , 16 ; j = 1; 2 , 3, 4.

A multiple objective SGA (MO-SGA) has been applied here.multiple objective problems. A m ultiple objective ranking [3]MO-SGA is a modified version of SGA for dealing 0 100 200 300 400approach is used. The chromosome I is ranked as GeneraRon

r u n k ( 1 ) = 1 + p (29) Aif I is dominated by other p chromosomes in the population.Dejinition: For an n,-objective optimization problem, I L isdominated by U if

v i = 1 , " ' , n , f i ( U ) 2 i ( U )and

3j I, . . , n, s.t. f , (u) > f , (,u). (30) ' GenerotionFrom (28), 64 objectives need to be achieved. Such hugeamounts of objectives demand a large number of comparisonoperations. Hence, it is simplified into 4 objectives to indicatethe fitness for each extreme plant as before. Define mz1 orextreme plant i = 1. 2 , . . . , 16 and j = 1, 2, 3 , 4 as

The objective f J or extreme plant J is16

fJ= mL3 fo r j = 1, 2 , 3 , 4. (32 )a = 1

After 448 generations, W1 and W2 are obtained and express edas follows: ( s + 0.8956)(s2+ 0.7161s + 1.4888)- (s + 1 .7249 ) ( s2+ 1.9122s + 0.1444)-

76.0976[ 0 47.3171( s + 1.4537)(s2+ 0.1990s + 0.5444)( s + 1 .4498 ) ( s2+ 1.2741s + 1.9551), =

1 .0732[ 0 17.5610with yo = 3.1778. Fig. 6 demonstrates the multiple objectiveoptimization process of the proposed MO-SGA. Trade-offsbetween different objective values can be noticed.The closed loop system responses for the extreme plants aretabulated in Table IV and depicted in Fig. 7.

V. CONCLUSIONThe use of structured genetic algorithm to select weightingfunctions in a mixed optimization approach appears to benovel. The proposed SGA enables the simultaneous searchingon the rtructures and on the coefficients of the weightingfunctions. SGA is particularly suitable for such design ap-proaches. Compared with other numerical search methods,several advantages have been shown in the paper. First, SGA

GeneranonI ,

. . . . . ' . . . . . . ~

i o1k.! . . ! -, . .OO 1w 200 300 4wGeneroHon

Fig. 6 . Multiple objective values versus generations.TABLE IVF I N A L YSTEMERFORMANCEOR SIMULATION

Ei h (G1) ~ ( G z ) h ( G 3 ) h(G4)1 1.12 -0.93 0.54 0.15 0.16 0.57 -0.358 0. 79 -0.5510 0.511 1.112 -0.913 0.114 0.115 50.016 150.0

1.0477-0.91590.3843

00.03630.4266-0.39600.6030-0.59380.40651.0292-0.94780.0384

048.89 7816.9133

1.0142-0.97910.32040.00010.01680.4242-0.39670.6023-0.59150.46041.0128-0.94090.01390.0002

49.557513.1862

1.0401-0.91250.41930.00010.03920.4251-0.39630.6034-0.59760.35691.0315

-0.99210.03280.000149.775211.6430

1.0024-0.95250.35780.00010.00180.4218-0.39980.6003-0.59690.37861.0015-0.97090.00190.0002

45.831011.9696

can easily handle the constraints to ensure the stability of theweight functions. Secondly, a multiple objective approach canbe adopted to address the conflicting control design specifica-tions. Finally, the structures of the weighting functions need nolonger be pre-fixed; only a fundamental structure is required,which provides the optimality for the solution over severaldifferent types of weights.In the design example, the performance was evaluated for aselection of extreme plant models chosen by the designer. The


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I0 20 40 60 60 100time - min

-0.2

0.8I-710 20 40 60 80 100

time - min-0.2

O . r - - I I

0 20 40 60 80 100time - min

(b)gain - d 8

-50 1o-z 1 o 1o2frequency. radimin

Fig. 7.(c ) 010- 614. (d ) $15 - 16

System performance for simulation C. (a) $1 - 5 . (b) &, - 49.

problem of efficiently determining the worst-case performanceover the range of plants still remains. Since the proposedalgorithm follows the formulation of an MO 1 which requiresthe choice of several plants only, it is necessary to choosethe most representative plants out of all possible plant models.The design may be further improved if a 2-DOF scheme isused.

APPENDIXCLOSED-LOOPERFORMANCEUNCTIONALSA set of closed-loop performance functionals,{ & ( G D ,W1, W2),i= 1 , 2 , . . . 16}, are defined based on thedesign specifications given in Section IV. Functionals 41-&4are measures of the step-response specifications. Functionals41, $6, $8 and are measures of the overshoo t; 44, 5, 413and 414 are measures of the undershoot; 4 2 , 4 7 , 4 9 and 412are measures of the rise time; and 43 and 410 are measuresof the cross-coupling. Denoting the output response ofthe closed-loop system with a plant GD at a time t to areference step demand h ( t ) b y y % ([h l h ~ ] ) ,= 1, 2,the step-response functionals are

48 = mpxy2([0 .4 0.6], t ) (40)

41 0 = mpxy1([0 lI,t) (42)411 = mpxy2([0 11% (43)$13 - r$nYl ( [o l17t) (45)$14 - mjny2([0 1] , t ) . (46)

49 = - min y2([0.4 0.6], ) (41)t>30

4 1 2 = - y2([0 1 , t ) (44)

The steady-state specifications are satisfied automatically bythe use of integral action. From the gain requirement in thedesign specifications, 415 is the HW -no rm (in dB) of theclosed-loop transfer function between the reference and theplant input415 = S UP 5 { [ I- K(.jw)GD(jw)]-lW1 , / w ) K s ( O ) W 2 ( 0 ) } .

W (47)From the bandwidth requirement in the design specification,416 is defined (in radlmin) as

41 6 = m a x {U }such that

5 { [ l -K(jw)G~(jw)]- l1(jw)Ks(O)W2(0)} 2 .(48)

REFERENCESJ . E. Baker, Reducing bias and inefficiency in the selection algorithms,in Genetic Algorithms: Proc. 2n d Int. Con6 Hillsdale, NJ: LawrenceErlbaum Associates, 1987, pp. 14-21.L. Davis, Handbook of Generic Algorithms. New York: Van NostrandReinhold, 1991.C. M. Fonseca and P. J. Fleming, Genetic algorithms for multiobiectiveoptimization: Formulation, discussion, and generalization, in GeneticAlgorithms: Proc. Fifth Int. C o n f , S. Forrest, Ed. San Mateo, CA :Morgan Kaufmann, 1993, pp. 41 64 23 .D. J. Hoyle, R. A. Hyde, and D. J. N. imebeer, An H approach totwo degree of freedom design, in Proc. 30th IEEE Con$ Dec. Contr.,Brighton, England, pp. 1581-1585, 1991.K. Glover and D. McFarlane, Robust stabilization of normalizedcoprime factor plant descriptions with H -bounded uncertainty, IEEETrans. Automat. Contr., vol. 34, pp. 821-830, 1989.D . J. N . Limebeer, The specification and purpose of a controller designcase study, in Proc. 30th IEEE Conj: Dec. Contr., Brighton, England,1991, pp. 1579-1580.D. McFarlane and K. Glover, Robust controller design using nor-malized coprime factor plant descriptions, Lecture Notes Control &Information Sciences. Berlin, Germany: Springer-Verlag, 1990, vol.138.D. C. McFarlane and K. Glover, A loop shap ing design procedure usingH m synthesis, IEEE Trans. Automat. C ontr. , vol. 37, pp. 749-769,1992.S. Skogestad, M. Morari, and J. C. Doyle, Robust control of ill-conditioned plants: High-purity distillation, l EEE Trans. Automat.Contr., vol. 33, pp. 1092-1105, 1988.K. S. Tang, C. Y . Chan, K. F. Man, and S . Kwong, Genetic structurefor NN topology and weights optimization, in First IEE/IEEE lnt.Conf Genetic Algorithms in Engineering Systems: Innovations andApplications, Sheffield, U.K., 1995, pp. 250-255.J. F. Whidbome, I. Postlethwaite, and D. W. Gu, Robust controllerdesign using H loop-shaping and the method of inequalities, IEEETrans. Contr. Syst. Technol., vol. 2, pp. 455-461, Dec. 1994.J. F.Whidbome, D. W. Gu , and I. Postlethwaite, Algorithms for solvingthe method of inequalities-A comparative study, in Proc. AmericanControl Con$, Seattle, WA, June 1995.


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582 IEEE TRANSACTIONS O N INDUSTRIAL ELECTRONICS, VO L 43, NO . 5, OCTOBER 1996

[I31 D Whitley, The GENITO R algorithm and selection pressure Why D.-W. Gu was born in Shanghai. China He grad-ranking based allocation of reproductive trials is beyt, in Genetic , uated from the Department of Mathematics, Fu-A l go r i t hm Proc 3r d Int Conf San Mateo, CA Morgan Kaufmann, dan University, Shanghai, China, in 1979, and re-1989, pp 116-121 ceived the M Sc degree in applied mathematics[14] V Zakian and U Al-Naib, Design of dynamical and control systems from Shanghai Jiao Tong university, China, inby the method of inequalit~es, roc Inst Elect En g , vo l 120, no 11 , 1981, and the Ph D degree in control system theorypp 1421-1427, 1973 from the Department of Electrical Engineering, Im-perial College of Science and Technology, London,U K , in 1985From 1981 to 1982, he was a Lecturer at Shang-hai Jiao Tong university He was a postdoctoralresearch assistant in Department of Engineering Science, Oxford University,U K , f ro m 1985 to 1989 In 1989, he was appointed to a UniversityLectureship in Department of Engineering at Leicester University, U K Hi scurrent research interests include robust control, optimal control, optimizationalgorithms, and control system computer-aided design. He is particularlyinterested in combining sophisticated numerical optimization techniques,including semi-intinitive optimization, neural network techniques, and geneticalgorithms in robust control system design. . Man (M91), for a photograph and biography, see this issue, p 518

K. S. Tang, fo r a photograph and biography, see this issue, p. 534.

genetic algorithm for robust h infinity

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