Journal of Control

European Journal of Conlrol (2000)6:358-367

© 2000 EUCA

European

Journal of Control

A Glob al O ptim ization Approach to Sca la r H d H

⁰⁰

Contro l

C.

Guarino Lo Bianco and A. Piazzi*

Univ. di Parma. Dip. di Ingegneria dcll"lnformazione. Parco Area delle Scienzc. 181/A - 43100 Parma. Ilaly

Various appro{lche.\· to the mixed Hd Hoo cOlI/rol problem {Ire propQ.\"ed in the literatllre. The .wlution is often obtained by means of classical techniques (i.e.

soil'illg a Riuati problem) or by means of COli I'ex opti- mization procedures. With the aim to sYlllhesize (/

jixed-stfllclure cOli/roller for anullcerwill plalll. a glo- bal optimizmioll approach to a single-inplil single- oulpul (SISO) H2IHoo problem is proposell. /11 particular, robust stability is guaranteed alld a 1I0lllillai quadrmic cost illdex is millimized by solvillg Oil equi- valent nonCOIII'ex semi-illjinite opljmizmioll problem.

The resulting desigll method that relies 011 a recemly devised genetic/illterral algorilhm is tested by desigllillg IOIl'-order COlltrollers for tll'O example p/(III/s.

Keywords! Genetic algorithms; Hoo control; Interval algorithms; Optimization; Quadratic cost indices

I. Introduction

The mixed fl2/ Hoo approach to finite-dimensional linear time-invariant control systems design has been variously investigated in the recent literature [1,4]. As can be evinced also rrom the K wakernaak's tutorial paper [15], the standard Hoo robust control leaves degrees or rreedom in the controller design. For this reason the Hoo problem is orten coupled to the mini- mization oran H2 (quadratic) nominal cost index.

A procedure ror discrete-time scalar systems is pro- posed by Sznaier [23] by means ora sequence orconvex

·Tel.: +390521905733: Fax: +390521905723;

Email: piazzi@:ce.unipr.it

Corr(!spolll/(!IlCe (Ind offprint requesis 10: A. Piazzi. Univ. di Parma.

Dip. di Ingegncria dclrlnformazionc, Parco Area delle ScicnlC, 181/A - 43100 Parma. haly.

optimization problems which finally yields a subopti- mal finite-dimensional controller. Convex optimiza- tion is also used in [22], which addresses a more general multiobjective Hd Hoo design problem ror multi variable plants. In recent times, the H2/Hoo ap- proach has attracted the attention or researchers ror solving a classical problem: the PID tuning ror an un- certain plant. The reason or such interest lies in poten- tial industrial applications or the new methodologies.

The papers [3] and [24] belong to this current.

In this paper, in the context or continuous-time, sca- lar (single-input single-output) systems, a solution to an H2I Hoo fixed-structure controller design problem is proposed via a global optimization approach. Preli- minary results were presented in [8] ror a plant with multiplicative uncertainties and comparisons with the PID controller design method or Chen et al. [3]

were made. In this work a more general approach is considered by adopting the reed back generalized plant scheme or [23]. Specifically, an arbitrary fixed- structure controller is designed to globally minimize a nominal H2 COSt, subject to robust closed-loop stabi- lity secured through an Hoo constraint and a rerormu- lation or the Lienard-Chipard criterion. A salient reature or this approach lies in the use or a hybrid two-level (genetic/interval) algorithm. This approach permits finding an estimated global minimizer that is feasible with certainty, i.e. the synthesized controller definitely guarantees the rulfillment orthe robust sta- bility and, at the same time, provides a good estimate orthe H2 global minimum. Focusing on multiplicative uncertainty plant models, an alternative approach to scalar H2I 1-/00 fixed-struct ure controller design is pro- vided by Krohling [14] using a homogeneous two-level (genetic/gcnctic) algorithm. This approach, due to the

Rcct:il"f(/ 2 JIIIIC 1999. Acccplc(( ill rCI'isct/ form 6 SCfJlrmber 2()()(). Recu""'U'II(/"d by M. S('iJ"k (IIu/ M. GH,'rs

Scalar Hd Hoo Comrol

ract that a stochastic technique (the genetic routine) cannot guarantee the complete rulfillment or a semi- infinite constraint (the Hoo one ror the case at hand), though quite effective on many instances, cannot ensure with certainty that the estimated optimal con- troller provides closed-loop stability ror all the mem- bers or the uncertain plant rami[y. On the contrary, the approach proposed in this paper using a deter- ministic g[obal technique (an interval procedure) makes sure the ru[fillment orthe Hoo constraint [[9].

In Section 2, we pose and solve the H2/ Hoo problem.

In particular this section shows how to convert lhe Hoo constraint into a semi-infinite inequality over a real bounded interval. Then, Ihe resulting semi-infinite optimization problem is reduced to a finite bound- constrained problem whose estimated global solution can be obtained by using a recently devised genetic/

interval algorithm [7,9]. Section 3 proposes two con- tro[ problems, one associated with a second-order plant and the other with a two-mass-spring plant (in [23], the latter was already examined using the H2/Hoo rramework). For both examples it is shown how to obtain the corresponding generalized plant (that is not the "'actual" plant) in order to set up the

fl d

floo problem. Computational results, simulations, and comparisons are included. Final comments are reported in Section 4.

2. The optimal

H 2/ Hoo

controller design

The given generalized plant E and the output-to-input dynamic reedback C(Sjx) are shown in Fig. 1. Two inputs (woo and Wl) are exogenous scalar disturb- ances while the third one is the system control input u.

In the same way E has two generalized controlled outputs «00 and (2) and one sensed (regulated) output y. The fixed-structure, rational, proper controller C(s; x) is parametrized by the vec- tor x := [Xl Xl'" xtlT EX =[x

1, x ii

^{x [X2}

,xiI

^{x ... x}

[XI

,xtI

^{<; ]RI. The transrer functions} between Woo and (00 and between W2 and (2 are denoted as T(x.w"" (s; x)

w,

(,

w_ _u .. _~ ^C _y

C(S; x )

fig. I. The generalized plant ,I"d the fixed-strUClure con- Iroller C(s; x).

359

and T(lWJ (s; x). respectively. The optimal Hd Hoo con- troller design problem aims to find an x' such that the reedback controller C(s; x') internally stabilizes the plant while minimizing the H2 norm

I I

T(lWJ (s; x')lb sub- ject to the Hoc; constraint IIT(",w",(s;x')lIoo:5 l. The rationale behind this problem rormulation is to find an optimal fixed-structure controller that minimizes a nominal H2 cost, subject to the robust closed-loop sta- bilityconstraint. The transrerrunctions T("w",(s; x) and

T(lW, (s; x) orthe generalized plant depend, respective[y, on the unstructured uncertainty associated with the actll<ll plant and on the chosen quadratic index. e.g.

the integral or the squared regulated error in response to a unit-step rererence (cr. Section 3.1).

The nominal characteristic polynomial associated with the internal closed-loop stability be denoted by

~(s; x) = L~=o ~;(x)i and the associated ith-order Hurwitz determinant by H;(x). To avoid degenerate system configurations the rollowing assumptions are introduced.

Assumption I. T< ... w",(s; x) and T(lWJ(S; x) are strictly proper fixed-order rational runctions ror any x E X

o

Assumption 2. The leading coefficient ~,,(x) or the nominal characteristic polynomial is always positive

ror any x E X 0

The ro1!owing rerormulation or the Lienard- Chipard criterion [6, p. 221] can be rruitrully used to ensure the internal stability or the reed back system or

Fig. I.

Property I. The fixed-structure controller C(s; x) internally stabilizes the closed-loop systems ir and only if(1/ 2: 3):

(o(x) > 0; (,(x) > 0, ,,(,) > 0, ... , (,(,) > 0;

H,,_l (x) > 0, H,,_3(X) > 0, ... , Hq(x) > 0, with 1':= 1/- I, q:= 3 ir II is even and 1':= /I - 2,

q := 2 ir II is odd. 0

The numerator and denominator polynomials of the w-rational runction

I

Tc",w".,(jw;

x )1

² =

Tc",w""Uw; x)T(oow_J -jw; x) be denoted by (3(w; x) and a(w; x), respectively.

Property 2. There exists a finite positive w(x) E IR+, depending on x E X, ror which the Hoc; constraint

(I) is equivalent to

~(w; x) - a(w; x) 5 0 Vw E [O,w(x)J. (2)

360

Proof By virtue of the Hoo norm delinition, (I) is clearly equivalent to the inequality

Vw E [0, +00).

and also to

1

^{(j .}

') 1 '

/l(w; x)

T( '" w,X =-(--)" 1

'" '" 0' Wj X Vw E [0, +00). (3)

Inequality (3) can be arranged as

/l(w; x) - o(w; x) ,,0 Vw E [0, +00). (4) Assumption I and the definition of the polynomials a(w; x) and f3(w; x) permit writing:

(a) deg(o(w;x» > dcg(/l(w;x»;

(b) the leading coefficient of a(w;x) is positive;

(c) both a(w; x) and (J(w; x) have only even powers of w.

By virtue of the above statements (a) and (b) the following equation is satisfied:

lim [!3(w; x) - a{w; x)] = -00,

~oo

so that {3(w; x) - a(w; x) reaches a finite maximum over (0, +00). Therefore, the semi-infinite inequality (4) is equivalent \0

/l(w'(x);x) - o(w'(x);x)" 0 (5)

with w'(x) E IR defined as

w'(x) := argmax{!3{w; x) - a(w; x): w E [0, +oo)}.

Inequality (5) is clearly equivalent to (2) if w(x) is an upper bound of w'(x). Todeterminew(x) wecan safely supposethatw'(x) > O. Indeed, ifw'(x) = Othenany positive value of w(x) is an upper bound of w'(x). Let the polynomial (J(w; x) - a(w; x) be described by

( ) "" ( )" .2,.,-2

P2m X W-" + 1'2",-2 X w

+ .

+

p2(X)W²

+

po(x)

with m:= deg(a(w; x»)/2 and P2I,,(x) < 0 "Ix E X.

The maximizer w'(x) must be a root of the derivative polynomial

Ow (Mw;

a

x) - o(w; x))

= 2mp2,.,(x)Jm-1

+

(2m - 2)Pb,,_2(X)Jm-3

+ ... +

2p2(X)W

or, excluding the zero root, of the polynomial mp2I,,(x)w^2m-2

+

(m _ I)P2I,,_2(x)w2m-4

+ ... +

p,(x).

C. Guarino Lo Biallco lim/ A. Pill::i

I'!cnee [w'(x)f is a root of the 1J-polynomial mp2m(x)rr- 1

+

(m - I)P2m_2(X)1}",-2

+ .

+

p,(x).

By using the classical Cauchy bound we infer

[W'(X)]2 < I

+

max

{ e m

-j)!P2(,., :11(X)!}.

19$,.,-1 m!p2n,(x)1 Therefore, we finally define w(x) as

w(x) ,~ I

+

max { .

(m - J)lP"m-J)(x)l}

Ig;Sm-1 mlp2I,,(x)1 (6)

o

Remark I. The previous proof does not only demon- strate the equivalence between (I) and (2) but is also fully constructive because it gives a definite expression for a proper w(x).

As previously anticipated, the cost index to be mini- mized is given by the Hz norm IIT(l"'Z(S;x)lb denoted in the following by lex) and computed, for example by means of the determinantal method of Katz [13}

(cf. the Appendix).

By considering Properties I and 2, the addressed

H z!

Hoo fixed-structure controller design can be posed as the following global semi-infinite optimiza- tion problem:

min lex)

.' x

subject to

(o(x) > 0; (,(x) > 0, 6(x) > 0, ... ,

(7)

(,.(x) > 0;

(8)

., Hq(x) > 0; (9)

/l(w; x) - ,,(w; x) ,, 0 Vw E 10,w(x)J. ( 10) If the fixed-structure controller C(s; x) that has to be pertinently chosen by the control engineer admits robust closed-loop stability, then problem (7)-(10) has a solution and evidently the constraints (8) and (9) cannot be active at the global minimum. Hence, there exists a sufficiently small positive ^€ E IR.+ such that problem (7)- (10) is equivalent to the following one:

minl(x)

~€,y (II )

subject to

{o(x) 2": €; {I(X) 2": ^€, 6(x) 2": €; ... , {.(x) 2": €;

( 12)

Scalar fhlll"" CmJlrol

~(w; x) - a(w; x)

:s

0 Vw E [O,w(x)l. ( 14)

It is worth noting that again the constraints (12) and (13) arc not active on the optimal solution (typi- cally the active constraint is (14), cf. the example in Section 3.2) and on the practical viewpoint choosing a correct ~ is not an issue: just select the smallest posi- tive real compatible with the implemented precision of computations.

The optimization problem (11)-(14), which is of semi-infinite type due to the presence of the semi- infinite constraint (14), generally involves nonlinear and nonconvex functions: the J{x) as well as the constraint functions. Hence, conventional algorithms such as those proposed in [12,20] and could be easily trapped in local minima or even end up with ~optimal

solutions" that are not feasible, i.e. ~solutions" not satisfying the semi-infinite constraint(s). With the aim of determining an estimated global minimizer x' to problem (11)-(14), an effective alternative is the global optimization algorithm proposed by the authors [7,9] for standard semi-infinite problems. It is an hybrid algorithm based on a genetic algorilhm at the upper level and on an interval procedure at the lower level to handle the semi-infinite constraints.

By using a penalty method, problem (11)-(14) is transformed into the following (finite) minimization problem:

{

"+' }

~if J(x)

+ 8

^{4J(O"i(X» .}

The penalty function is defined by

( 15)

if 0" ::::; 0,

if 0< (1 < T,

if 0" ::::: T

and its arguments are defined accordingly 10

O";(x)

{

' -(O(x)

E - 61-3 (x)

. - E - H_2;_n_I(X)

max~lo'::ix)dfi(w; x) - o(w; x)}

if;= I,

if;= 2,3, ... ,j, if i=)+ 1, .. ^,,11, ifi= II+ I,

where) = I

+

11/2 if II is even while) = (11

+

I )/2 if /I is odd. The positive parameters M and T used by the penalty function 1:>«(1) must be sufficiently large and small respectively, depending on the chosen numerical accuracy.

]61

The panially elitistic genetic algorithm orrl7] can be used to solve the unconstrained problem (15). While computing the penalty terms 4«(1;(x)), i = I, ... , /I IS

straightforward, the term <1>(0"11+1 (x» needs a special interval procedure, i.e. a deterministic algorithm which uses concepts of interval analysis [18] and con- verges with certainty within a prespecified tolerance, Essentially, it is based on a branch-and-bound techni- que used for exhaustive search on [0, w(x)] and on the use of the so-called inelusion filllCliolls of interval ana- lysis Ill]. This procedure can be regarded as a variant of the interval positivity test (IPT) for multivari- ate functions exposed in [19]. Further details can be found in [7,9].

It should be made clear that the genetic/interval algorithm proposed to solve (11)-(14) does not pro- vide a guaranteed global minimizer, but provides an estimate of the global minimizer that is feasible with certainty. As shown in [9], which also reports extensive results on test problems, this estimate can indeed be an excellent solution of the semi-infinite problem. More- over, the deterministic and global nature of the inter- val procedure makes sure that the estimate satisfies with certainty the semi-infinite constraint(s). This feature of the hybrid algorithm renders the approach interesting to tackle other control design problems, e.g. the optimal worst-case output feedback [10].

3.

Examples

Two examples are reported in the following subsec- tions. For both of them two different fixed-structure controllers have been designed and their performances have been analyzed. The first example considers a second-order fractionary uncertain plant. It is known (see {26, pp. 227-228]) that such models can be used to appropriately describe systems with a changing number of right half-plane poles. The second example is derived from a well-known benchmark problem: the two-mass-spring plant proposed at the 1990 American Control Conference [25].

Both problems have been solved by using the genetic/interval algorithm presented in {7,9] which has been coded and compiled with GCC for MS Windows on a PC Pentium III 500 MHz .

3,1, Second-order plant

Let the plant be described by the fractionary uncer- tainty model P given by

p,~ {P(.,), P(.,) ,~ P(s)/[l

+

^t>W,(s)I,

11t>1I00< I} (16)

362

with P(s) the transrer runction or the nominal plant, W2(S) the fixed stable weighting runction, and .6. the sealing ractor or W2(s). As usual, it is assumed that unstable poles or P(s) cannot be cancelled by 1/[1

+

.6.W2(s)J. A unity-reed back control scheme is adopted. The fixed-structure, proper controller C(s; x), with the design parameter vector x belonging to the multi-interval X ~ RI, must guarantee (a) the robust stability orthe closed-loop system, (b) the mini- mization over X or

.r;;

^e2(t; x) dt where e(t; x) is the output tracking error to a unit-step rererence signal evaluated ror the nominal plant P(s). Both require- ments can be rulfilled by matching the problem with the general rramework proposed in the previous sec- tion. Due to the uncertainty model (16), robust stabi- lity is obtained by imposing T(""",,,,,(s; x) := W2{s)/

(I

+

P(s)C(s;x)) [5, see p. 55]. Denoting by £(s;x) the Laplace transrorm or e(r; x), we can set

T,,~(s;x) ,~ £(s;x) ~ I/{s[l

+

P(s)C(s;x)]}. By';"

tue or the above definitions, the generalized con- trolled system can be represented as in Fig. 2.

The actual uncertain plant P is defined by

The uncertain pole or model (17) can belong both to the lert or to the right haIr-plane. Model (17) can be arranged in the general rorm (16) by imposing

W,(s),~ -( -2 )' s+ 1 Two cases are considered ror the fixed-structure controller:

l. PI COl/{ro!

( )

XI

+

^X2S

CI s;x := , S

x E X,~ [0, 100[ x [-500,500[.

Fig. 2. Block diagram of the generalized controlled plant.

C. GUlIrillO Lo BillllCO lind A Pill==i

2. PID cOlI/ro!

C ( . )._ . ^s2 +2X2X3S+X~

2S,X ·- ·\1 ( ) , s s+ a

x E X,~ [0,5000[ x [0, 10[ x [0, 10[,

where (/ = 50 is a "'Tast" pole introduced to obtain a proper controller.

PI con/roller. The genetic/interval algorithm has converged to the estimated global minimizer x' = [19.52 500.01^T with optimal cost index

.r;;

^e2(r; x') dl = 0.105 and with an average computa- tion time or 2 s. 11 is worth noting that x' is on the boundary or X. This corresponds to an almost pole- zero cancellation in the origin or the complex plane ror the optimal controller CI (s; x'). The unit-step re- sponse ror the nominal plant shown in Fig. 3 has an unacceptable oscillatory behaviour.

Marginal improvements on the cost index are ob- tained by enlarging X but with a drawback or larger overshoots in the un ii-step response. Thus, rrom a con- trol engineering viewpoint, a PI controller is inconve- nient ror this kind or uncertain plant so that the next PID controller design is proposed.

PlD cOlltroller. The evaluated optimal solution is x' = [5000 0.5683 l.9151T with nominal perrormance

.r;;

^{e2(t; x') dt = 0.01456 and an a}verage computation time equal to 7 s. Again, the minimizer x' is on the boundary or X and corresponds to the selection or the maximal allowable velocity constant. This solution provides a controller with two complex zeroes (- 1.088 ±j 1.576). The nominal perrormance is defini- tively better than that obtained with the PI controller.

The exhibited unit-step response, shown by the solid line in Fig. 3, is better 100; it has a sharply damped oscillatory behaviour with a considerably smaller overshoot Mp = 25.5%.

1.8 1.6 1.01 1.2

0.8 0.6 0.4 0.2 0 0

1\ •

1\

I I I I (\

I I 1\

I I I \

I I (,

0.2 0.01 0.6 0.8 I 1.2 1.01 1.6 1.8 2

Fig. 3. Unit-step responses for the nominal plant. obtained with C 1 (s: x') (dashed line) and C2(s; x") (solid line).

1.2

IV

0.8 0.6

0"

0.2

o o

0.2 0.<1 0.6 0.8 I 1.2 1.4 1.6 1.8 2

Fig. 4. Unit-step responses for i'l(s), P2(S) and PJ(.r) obtained with C2(J;.'\(·).

Figure 4 shows the step responses associated with the following three plants related to the uncertain model P:

( 18)

( 19)

(20)

PhillIS

P I

(s) and P3(.~) belong to the boundary of p, while

1 \

(.s) has the uncertain pole located at the origin of Ihe complex plane. In spite of the diversity of the three plants. quite similar step responses have been obtained.

The original vcrsion of this benchmark problem was first proposed in the 1990 at the American Control Conference (25) to verify the capability of controllers to reject disturb.mccs in presence of plant uncertain- ties. In the following, we adopt the design specifica- tions proposed by Sznaier (23). The plant is the classical undamped two-mass-spring system shown in Fig. 5. Two unit masses (11/1 = I, 11/2 = I) are coupled by a spring whose elastic constant k is uncer- tain: k E (0.5.2). The controller has to reject a white noise disturbance W2: acting on the second mass by means of a force II working on the first mass. Differ- ently from Szainer. we have adopted a continuous- time fixed-structure controller C(s; x) parametrized

36J

y

~ w,

( ) ( )

on,

J

( Fig. 5. The two-mass-spring plant.

FiG. 6. The dosed-loop system.

by x E X. The resulting closed-loop system is shown in Fig. 6. where

(21 )

(22)

SZll<lier proposed in 123) to model the spring constant as k := ko

+

^{6., where ko = 1.25 and}

1 .6.1

< lh :=

0.75. Disturbance rejection is achieved by minimizing, for the nominal plant (i.e. k := ko), the RMS value of the control signal II in response to a white noise W2 act- ing on 1112 or, equivalently, by minimizing the lh-noTm orthc transfer function between W:2 and II,

T. ( . ) _ C(s; X)(IIII~

+

ko)

(l'"'l S, X - s2[IIIJIII2s2

+

ko(ml

+

11/2)]-koC(s; x)' (23)

The small-gaill Ihl'orem has been applied to ensure the robust stability of the closed loop system for any

1I.6. 1I

_1X1 <

1 /1.

^{where t::. is a stable} unstructured uncer-

tainlY. The closed-loop system can be arranged in the usual standard form shown in Fig. 7, where the uncertainty 6. has been pulled oul. According to the small-gain theorem for any stable 6. such that

116. 1I

1X1 < I

h. "( >

0, the closed-loop system is robustly stable if M(s;x) is stable and

II M(s; x )111XI

S; "( or, equivalently. defining T< ... ,,(s;x):= M(s;x)h, if

11 7 , .... "' ...

(s;x)lIlXI:5 I. For the two-mass-spring system

364

Fig. 7. The standard form of the uncertain closed-loop system.

I [kJ<w

'-:rfl-~

_{~ 1/1.,s}

,

M ~

II -

1

u

L= - ---

w, (, ..

y

~----II

^C(,;x)

~ [---"

"'ig. 8. The generalized controlled plant for the two-mass- spring system.

it holds Ihal

T. ( .. )_ C(s;x) _s2(1111 +1112) I

(",w,- s,x - s2[m]1Il2S2 + kO(1II1 +1112)] koC(s;x) ,'

(24)

The resulting generalized controlled plant is shown in Fig.8.

[n the following, Iwo fixed-structure controllers have been designed: a second-order controller is com- pared with a third-order controller. Bo\h controllers

afC bipropcr.

Second-orderCOl/lrol/er. The following second-order structure has been chosen:

C ( . . )"_ . (S+X2)(S+X3)

t S,X .- .\1

2 ^~,

s

+

^2X4X5S

+

x5 ⁽²⁵⁾

where x:= [Xl X2 x) X4

x sl

^T E X:= [0.01, IOjx [-2,2[ x [-2,2[ x [0.6, 1J x [0.01,5[. The gen",c/

interval algorithm has converged, with an average computation time of 7 min 51 s, to the estimated glo- bal minimizer x' = (4.7068 6.1646· 10~3 - 0.13685 0.64392 1.964I]T with nominal cost index l(x') =

1.18380. The amplitude Bode plot of the transfer func- tion Tc .. r..."o.-(s;x·) shows that the semi-infinite con- straint is active al the optimal solution (see Fig. 9).

20 dB 10

C. Guarino lA, Bianco {/lid A. Pia~~i

o .... -... .

-10 -20 -30 -40 -50 -60

-70 Rad/s

-80 L..._....,-_....,-_"-,--_-,-_--,----'

lU·.j J(TJ J(T2 WI lff 10¹ 102

Fig. 9. Amplitude Bode plot of the transfer function

T(~w ... (s; x').

2 ~---. dB

-I -2 -3

-4 L_~

Fig. 10. A detail of IT(."w"Uw;x·)I.

In Fig. 10, a detail of the same plot reveals that the constraint limit value is reached at two differenl frequencies.

The closed-loop system has a dominant pole placed at - 0.006671. Its position is practically independent of the uncertain k. The root loci for the nominal plant and for the two extreme plants (k = 0.5 and k = 2) arc reported in Fig. I I.

In Fig. 12 the response of the nominal closed-loop system to a unit impulse is shown. The response peak vallie (close to I) is much smaller than the correspond- ing one found by Sznaier (close to - 15). This is in accordance with the cost index (equal to 1.18380) of the proposed second-order controller compared with the cost index found in [23] (equal to 22.6493).

Third-order cOl1lrofle,.. To further reduce the nom- inal cost index, the following third-order controller

Scalar Hd Hoc COlllrol

,

2

I

^{... _ -..,;;}

",,"-::;! ...

^~

"-

~~"'''' ...

... ~

..

----"""~--

... - ... .

""--~

.,_.'"

o ____________

"'_--e <:r---;-.

-I -2

,~"'--or--::: =-.-,;;...;--~",.

- -

~:.-::.: .. ... ~' ,~ •.••.••. - . .f

i r -- -"",

-3 L--_~ _ ___'__'_L___ _ _ .,;_ _ ___,_l

-1.5 -I -0.5 0 0.5

Fig. 11. Root loci for C1{s; x'): squares indicate the position of the dosed-loop poles.

0.8 0.6

f\

OA 0.2

o

-0.' V

V

-0.4

-0. 60

!---;':---;'-O-::---:8:---;'1 0;;--:;12;--cI:;'----;,~6

--:,8;:-;!'O'

Fig. 12. Nominal unit impulse response of the dosed·loop system.

has been introduced:

( _ .. )._ . (s

+

.\"2)(.V

+

.\"3)(S

+

^.\"4)

C2 s, x ._ X I ( ' , _

5

+

.\"5)(.\·-

+

^2.\·6X75

+

^{X'7) (26)}

where x:= [XI -'"2 -'"3 -'"4 -'"5 -'"6 -'"7JT EX:= [0.01, 151 x [-2,2[ x [-2,21 x [-20,201 x [0_0001, 201 x [0.5,

iJ

x [0.1,51. Despite the relatively high dimensionality of the search box X, the computation time for this design problem is equal, on average, to a moderate 15 min 42 s. The genetic/interval algorithm has found the following minimizer: x' = [9.0874 - 1.2738· 10-² 2.3415.10-³ 7.9450 19.258 0.55621 1.8308]T with nominal index l(x') = 1.03679. Also, for the third- order controller the Hoo constraint is active at two different frequencies. The closcd-Ioop system has three poles that are practically independent of ^k.

Two of them afC dominating poles (-0.009163 and - 0.004197) while the third one is a fast pole (- 19.26). The positions of the other four poles

3r----~~---1\\

: \

"-

,

^{,/ \}

"'"---

-=L __ J , ______

^-1<

\

• ^{, -'}

\ \.. .

of---_1-1:~-c_~.~~---1

/ \Ioi~

,

-I \

-0::.;--1

~-^---<

'\ /

,...---

- ,

',/

' ,

l---~--~'"'""C----:----"

-3 -1.5 -I -0.5 0 0.5

365

Fig. 13. Root loci for Cl(S;X'): squ;Hes indicatc the positions of the closed-loop polcs.

depending on k are shown in Fig. 13 (k = 0.5, k = 1.25, and k = 2).

4. Co nclusions

The problem of synthesizing a fixed+structure robust controller with mixed H2/ Hoc specifications for an uncertain SISO plant has been faced with the aid of a global optimization technique. Specifically, the H2/ Hoo synlhesis problem can be posed as a semi- infinite optimization problem solvable by mcans of a genetic/interval algorithm developed by the authors [7,9].

The examples of Section 3 highlight the nexibility and the effectiveness of the proposed approach. In par- ticular, an intrinsic advantage of this approach lies in synthesizing easily implementable low-order control- lers that do not require post-design processing for order reduction. A possible limitation of the method could occur in addressing large-scale problems, i.e.

problems where the dimension of the design parameter space is high (for example, 12:20). However, this should not exclude the possibility of successfully addressing more general problems, such as the multi- objective

H d

Hoc control of multivariable plants.

References

1. Bernstein 135. Haddad WM. LQG control with an Hoc performance bound: a Riccati equation approach. IEEE Trans Automatic Control 1989; 34(3): 293-305

2. Boyd SP. Barratt CH. Linear Controller Design: limits of performance. Prentice Hall. Englewoood Cliffs.

New Jersey, 1991

3. Chcn 13-5, Cheng Y -M, Lcc C-H. A genetic approach to mixed Hd Hoo optim'll PID control. IEEE Control Systems Magazine 1995; 15(5): 51-60

366

4. Doylc J. ZhOll K. Glover K. Bodenheimer B. Mixed H2 and flw performance objectives II: optimal control.

IEEE Tr:ms Automatic Control 1994: 39(8): 1575- 1587

5. Doylc

l e .

^{Francis SA.} Tannenbaum AR. Fccdb..1Ck Control Theory. Macmillan Publishing COmplinY.

New York. 1992

6. Gantmachcr FR. The Theory of Matrices. Vol 2.

Chclscli. New York. 1959

7. Guarino Lo Billnco C. Pia7.zi A. A hybrid genetic!

interval algorithm for semi-infinite optimization. In:

Proceedings of the 35th IEEE Conference on Decision and Control. Kobe. Japan. December 1996; 2136-2138 8. Guarino Lo BillnCO C, Piazza A. Mixed Hd H"", fixed-

structure control via semi-infinite optimization. In:

Proceedings of the 7th lFAC Symposium on Computer Aided Control Systems Design. Genl. Belgium. April 1997: 329-334

9. Guarino Lo Bianco C, Piazzi A. A hybrid algorithm for infinitely constrained optimization. International Journal of Systems Science 200 I; 32( I): 91- I 02 10. Guarino Lo Bianco C. Piazzi A. Worst-case output

feedback for uncertain systems. Technical Report TSC02-00. Dipartimento di Ingegneria deWlnforma- zione. University of Parma. Italy, April 2000

II. H:mscn E. Glob,al Optirnizalion Using Interval Analy- sis. Marcel Dekker. New York. 1992

12. Hettich R. Kortanek KO. Semi-infinite programming:

theory. mcthods. and applications. SIAM Review 1993;

35(3): 380-429

13. Katz AK. On the question of calculating the quadratic criterion for regulation. Prikl Mat Mck11952; 16: 362- 364 (in Russian)

14. Krohling RA. Genetic algorithms for synthesis of mixed Ihlll<1O fixed-structures controllers. In: Proceed- ings of the 1998 IEEE ISICjCIRA/ISAS Joint Con- fcrence, Gaithersburg, MD. September 1998. 30-35 15. Kwukernank H. Robust controlund H",,-optimization-

tutorial paper. Autom,Hica 1993; 29(2): 255-273 16. Marshall JE, Gorcchi, Walton K. Korytowski. Time-

Delay Systcms: stability and performance critcria with applications. Ellis Horwood. Chichester, Englund. 1992 17. Menozzi R, Pi;tzzi A, Contini F. Small-signal modeling for microwave FET linear circuits based on a genetic algorithm. IEEE Trans on Circuits and Systems. Part I:

Fundamental Theory and Applications 1996; 43(10):

839-847

18. Moore RE. Methods and Applic;ltions of Interval Analysis. SIAM Press, Philadelphia. PA. 1979 19. Piaui A. Marro G. Robust stability using interval

analysis. International Journal of Systems Science 1996; 27(12): 1381- 1390

20. Polak E. On the mathematical foundation of non- diITerentiable optimiz:ltion in engineering design.

SIAM Review 1987; 29(1): 21-89

21. Popov EP. Thc Dynamics of Automatic Control System. Addison-WeSley, Reading, Mass .. 1962 22. Scherer CW. Multiobjective IId H<1O control. IEEE

Trans Autom:llic Control 1995; 40(6): 1054- 1062 23. Sznaicr M. An exact solution to general SISO mixed

H z!

fi".., problems via convex optimization. IEEE Trans Automatic Control 1994; 39(12): 2511-2517

24. Takahashi RI-IC, Peres PLD, Ferreira PAY. Multi- objl:ctive Ih/l-loo guaranteed cost PID design. IEEE Control Systems Magazine 1997; 17(5): 37-47

C. Gaarinl) l.fl BillllCQ alUl A. Pill::i

25. Wie B. Bernstein DS. A benchmark problem for robust control design. In: Proceedings of the 1990 American Control Conference, San Diego. CA. May 1990: 961- 962

26. Zhou K. Robust and Optimal Control. Prentice I-Iall.

Upper S'lddlc River. NJ. 1996

Appendix

For thc reader's convenience, this appendix exposes succinctly the determinantal method to compute the H2 norm

I I

T(s)lb of a given strictly stable rational function T(s). This method is due to Katz [13] and was subsequently reported in [21J (see also [16, pp.

8- 111). Defining J:=

II T (s)1I2

for ease of notation, it is well known that J =

(Jo

r(/)2d/)t/2, where ret) is the impulse response of the system whose transfer function is T(s) (cf. 12, p. 96]).

Define Tes) according to

8(s) b]s"-l

+ 625', - 2 + .. +

_bn

T(s) = A(s) = aoS'

+

atS'-t

+ ... +

lin

and assume "0

oF

^{0 and A(s) (Hurwit}z) stable for which J is well defined. Introduce the polynomial F(,) ,~ (- 1)"-'

(J,"'-' +12"'-' + ... +

f,,), /; E JR, i = I, ... , II satisfying the following polynomial identity:

8(,)8(-s) ~ F(-s)A(-s)

+

F(s)A(s).

This relation implies that /; must satisfy a system of linear equations for which, under the current assump- tions, a unique solution always exists; then, by virtue of Parscval's theorem, it follows that

Using Cramer's rule, thc following closed-form expression of

11

can be obtained:

g, ao g, a, dot gs a,

/,

~ g" ⁰ a, ao as ^{/2 de< as a, 0 0

0 0 a, ao as a, a,

0 0 a, ao as t12

a,

o o

o o

all

(27)

where

1 2;- 1

g" ='2 L { - I/ - l

b2 ,._jbj {bj= Oifj>II}.

j=1

From (27) we identify the denominator as the II-order Hurwitz delerminant associated with A(s). Then,

367

denoting the numerator of (27) by N and taking into account that HI! = (I"H,,_I (where H,,_I is the Hurwitz determinant of order ^{II -} I) we finally obtain

J~

(

N ) ./2

(10(1" /-1"_1 . (28)

European Journal of Control (2000}6:J68-371

( , 2000 EUCA

European

Journal of Control

Discussion on: 'A Global Optimization Appr oach to Sca lar H j Hoc Control' by C. Guarino Lo Bianco and A. Piazzi

I. Discussion by R.H.C. Takahashi^l

and P.L.D. Pcres

²

1.1. Introduction

In first place, we would like to remark that the authors have tackled an interesting problem. The design of low-order optimal controllers with an Hoo constraint is indubitably of practical significance and probably will be object of continued research effort in the ^COIl- Irol community for the next years.

Our discussion on the paper will be divided in two parts: general comments about the contribution and some spccilic poi IllS 10 be addressed by the authors.

in order to highlight some rclcviml issues in the proce- dure proposed.

1.2. Comments

The general problem of optimal fixed-order H2 con- trollers with an Hoo constraint. as stated in the paper, can be cast as a stHTldard constrained optimization problem th,1I is then transformed into an uncon- strained optimization problcm by means of a penalty method. The algorithm. thercfore, must perform the following calculations:

I. Given a vector of controller parameters. verify the stability of the nominal closed-loop. Assign a -penalty" value to the objcctive function if the stability docs not hold.

2, For the same vector of controller parameters. and for the same nominal closed-loop system, verify

Ilkparlmcnl of Electrical Engineering. Uni\'ersidade Federal de Minas Gerais. Av. Antonio Carlos 6627. 31270·010. Belo HorizonlC. MG. ijrazil; Em .. il: laka(fr epdee.ufmg.br

2School of Ek-clrical ,Ind Compuler Engineering. Universi1y of C.lmpin .. s. Ct' 6101. 13081·970. Campinas. sr, Br .. 7.il:

Email: pcres(" d1.fl'C.llllicamp.br

the feasibility of the Hoo constraint. If the 1-100 constraint docs not hold, add a "penalty" value to the objective function.

3. Compute the f12 norm of the nominal closed-loop system for the same vector of parameters and add it to the objective function.

Due to the non-convexity of the problem, standard optimi7.ation procedures cannot be used efficiently.

In the paper. this objective function is optimized by means of .. gcnetic algorithm that leads to an estimate of its global optimum.

There arc other paJX:rs using evolutive computation methods for global optimization purposes in the litera- ture. The feature that distinguishes the paper from pre- vious contributions is the way it addresses the 1-100 constraint. Instead of directly computing the 1-100 norm. an exhaustive evaluation of a polynomial is performed inside an interval, using a brallch alld bOl/lld technique. The result of such evaluation answers the question if t he controller under consideration is feasi- ble (if it leads to an 1-100 norm less than I) and, if not, it furnishes lin indcx that in some sense means a

"distancc" to the feasible set.

1.3. Spccific 'Joints

In order to highlight the contributions and to justify some choices made in the paper. we would like to ask some questions that in our opinion should be addressed by the authors.

1.3.1. Norm COmfJllf(lfioll

Since the 1-12 and the 1100 norms are well defined only for stable systems, how docs the algorithm handle the unstable transfer functions? For instance, in (he 1100 computation. what would be the meaning of a polynomial maximum for an unstable system?

Discussion on: 'A Global Oplimi=lllioll Approad! 10 S('{IllIr 1f~/lfoo COlllrol' 369

Note that, in general, the penalty value of an un- stable system is defined as a function of a "slability measure" only, in such a way that the norms are com- puted uniquely for stable dosed-loop systems.

1.3.2. COllfillllOIiS Pcnalty Fwu:/iol/

In the paper, a penalty function is used to convert the constrained optimization problem into an uncon- strained problem which is suitable for genetic algo- rithms.

In the context of classical optimization methods, the penalty functions are usually built as continuous func- tions. This is necessary in order to avoid singularities in the gradient computations performed in these methods.

However, there is no need of a continuous penalty function in genetic algorithms, since the gradient com- putation is no longer necessary. Is there a specific rea- son for the use of a continuous penalty function in the algorithm?

1.3.3. Hoo COl/straint

In the paper, there is a claim concerning the Hoo constraint that can be misunderstood by the readers, that is:

.. a stochastic techniqlle (the genetic rOllfine) call not gllarantee the complete jil/fillment of a semi-infi- nile COlis/rail/{ (the Hoo olle for the cW'e at h(lIId) . .. ".

Actually, a genetic algorithm can deal with the pro- blem of finding feasible solutions for IIG(x)lloo < ^"y if the 1100 norm is simply computed for each value x of the vector parameter. It seems that the need ofa deter- ministic global technique as the one proposed in the paper has occurred only because the Hoo constraint is formulated as a semi-infinite constraint. This point should be clearly stated by the authors.

Another point concerning the Hoo constraint is:

why do the authors use a semi-infinite constraint to the Hoo norm? Is there a technical reason for this particular choice?

1.3.4. Othcr Poillls

The following points deserve at least a few words from the authors:

• The procedure proposed assumes some a priori bounds for the controller parameters. How are these bounds defined?

• The examples presented in Section 3 are rather small ones (maximum of I = 7 parameters, taking

around 16 minutes to be solved), but the authors claim that a possible limitation of the method would occur only for I?: 20. Do the authors have numerical experiments to corroborate this affirmation?

• In most of cases, a pole location constraint can provide a good dosed-loop behavior regarding performance specifications that are not explicitly expressed in terms of the H2 and Hoo norms, such as overshoots, oscillatory behavior, etc. In the paper, the authors change the controller structure to cope with this kind of closed-loop behavior specifications. It seems that the method would be able to incorporate other constrains, for instance, to assure a prespecified pole-location inside a given bounded region. Some words about these possibi- lities would be welcome.

• Finally, handling only low order controllers is an advantage or a limitation intrinsic of the proposed method? Some comments about this issue can help to envisage the place and role of genetic algorithms in controller design.

2. Final Comments

by

the Autho rs

C.

Guarin o Lo Bianco

³

and A. Piazzi

⁴

The authors would like to thank Professors R.H.C.

Takahashi and P.L.D. Peres for their enlivening remarks.

Regarding to their general comments we point out:

• Our paper shows how the design of a fixed- structure controller for scalar

H d

Hoo control can be exactly (i.e. without any conservativeness) cast into a semi-infinite (constrained) optimization problem.

• This problem that happens to be generally non- convex is solved by means of an hybrid algorithm combining a global stochastic technique (a genetic algorithm) with a global deterministic technique (an interval procedure); this algorithm is organized about a penalty method that transforms the original problem into a finite unconstrained optimization problem. A detailed description of this hybrid approach is reported in [2] with further improvements inserted in [I].

JUniv. di Parma. Dip. di Ingegneria dcll"lnformazione. Parco Area delle Sden:re. ISI/A. 43100 Parma. haly .

·Univ. di Parma. Dip. di J ngcgncria dclJ·lnformazionc. Parco Area delle Scien:re. 181/A. 43100 Parma. Italy; Email: piazzi@.cc.unipr.it