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 *fl**2/ *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 (q*uadratic) 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/(!IlC**e (Ind of**fprint 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 *JIII**IC **1999. **Acccplc(( *ill *rCI'isct/ fo**rm *6 *S**C**fJlrmber *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 inp*ut 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 tra}

^{nsrer fu}

^{nctio}

^{ns }between Woo and (00 and between

*W2*and (2 are denoted as

*T(x.w"" (s;*x)

*w, *

(,
*w_ * _{u } .. _{~ } ^{C } _{y }

_{u }

_{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)W*

^{2 }### +

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 **fh**l**ll"" **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 *W**2**(s). *As usual, it is assumed that
unstable poles or *P(s) cannot be * cancelled by
1/[1

### +

*.6.W*

*2*

*(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;;

*x) dt where*

^{e2(}^{t; }*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;;

^{e}

^{2}*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*

^{(r; }*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;;

^{e}

^{2}

^{(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 C*2**(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 2Fig. 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 system364

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 *W**I * lff 10^{1 } 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 C**1{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* *. *

^{Fra}

^{ncis }

^{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 *Ih**l**ll<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 }

^{a}

^{nd }

^{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*

*=*

*Oif*

*j>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 *H**I! *= *(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