H ∞ -control of a distributed parameter system with non-minimum phase

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H∞-control of a distributed parameter system with non-minimum phaseJ. P. GAUTHIER a & C. Z. XU aa Laboratoire d'Automatique et de Génie des Procédés, URA CNRS D1328,43 Boulevard du 11Novembre 1918, Villeurbanne Cedex, 69622, FrancePublished online: 17 Apr 2007.

To cite this article: J. P. GAUTHIER & C. Z. XU (1991): H∞-control of a distributed parameter system with non-minimumphase, International Journal of Control, 53:1, 45-79

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INT. J. CONTROL, 1991, VOL. 53, No.1, 45-79

H"'-control of a distributed parameter system with non-minimumphase

1. P. GAUTHIERt and C. Z. XUt

In this paper weexamine the ability of H oo-control theory to perform actually linearcontrol of distributed parameter systems in non-trivial cases. The process to becontrolled is originally an alumina furnace, which is modelled by trivializing combustion phenomena as a rather special heat exchanger. This process has an intrinsicnon-minimum phase, due to its particular geometry. It is modelled as a non-linearpartial differential system, which is linearized and controlled using HOOtechniques. Algorithms are developed to accomplish this task, in particular tofactorize the (transcendental) transfer function into its inner and outer parts.Simulation and comparisons with classical controllers are performed.

I. IntroductionSince the pioneering work of Zames and Francis (1983), H"'-control theory has

been developed for the study of robustness and sensitivity of finite dimensional lineartime-invariant systems (Francis and Zames 1984, Francis 1987). Recently the theoryhas been extented by Curtain (1988, 1989)to a large class of linear infinite dimensionaltime-invariant systems (the so-called Salmon and Callier-Desoer class). It is also amatter of Factthat comparatively few papers deal with the applicability of H"'-controlto concrete systems.

In this paper, H"'-control is applied following the theoretical results of Zamesand Francis (1983) to a distributed parameter system. Our aim in this paper is toprovide some practical methodology in this area and to test for a rather complicatedproblem the applicability of H"'-control.

The controlled process was an alumina furnace. The alumina is expected to bebaked while moving along, with the product of heating moving in a contraflow. Inthe middle of the furnace combustion occurs and the gaseous combustion products,which are blown at the opposite side of the alumina are pre-heated by the bakedalumina. This results in a non-minimum phase, as will be explained in § 2.

Based on the consideration of this process we construct an academic process withthe same qualitative input-output behaviour: we do not model the combustionphenomena, so we assume that the heat flow rate resulting from combustion isbrought by the combustible. We neglect the thermal effect due to chemical phenomenain alumina baking-it is actually minor; however, we link the flow rates of thecombustible and the gaseous combustion products by a ratio' (0 < IX< 1) to simulatethe combustion. This results in a special contraflow heat exchanger, for which wemake standard assumptions: no thermal losses, constant exchange cofficients, no axial

.diffusion, etc. This exchanger is described by a set of partial differential equationsand limit conditions, and reproduces the non-minimum phase.

Received 29 April 1989. Revised 15 November 1989.t Laboratoire d'Automatique et de Genie des Precedes, URA CNRS D1328,43 Boulevard

du II Novembre 1918,69622 Villeurbanne Cedex, France.

002Q.7119j91 5).00 <0 1991 Taylor&: Frucis Ltd

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46 J. P. Gauthier and C. Z. Xu

The paper is organized as follows. Section 2 gives the modelling of the non-linearcontraflow heat exchanger system by linear partial differential equations. Then it islinearized around an equilibrium point. In § 3 the transfer function representation ofthe linearized system is given. In § 4 we characterize an open set of numerical valuesof physical constants of the process, for which we can actually prove that the transferfunctions are in H'" (right half-plane) and that a non-minimum phase occurs. Thenwe present some H"'-factorization results relative to the transfer functions occurringin the plant. Together with this H"'-factorization we develop a method to computethe unstable zeros of the transcendental plant transfer function P(s). Then, followingthe framework of Francis and Zames (1984), the H"'-controller is designed for thelinearized system. § 5 is devoted to a short description of the numerical simulationalgorithms and the sensitivity performance ofthe designed H"'-controller is comparedwith that of a PID controller. The paper ends with § 6 in which some open problemsare presented.

Let us point out that our aim in this paper is not to improve on H'" theory. Onthe contrary, we decided to deal with the first elementary problem treated by H'"techniques, namely sensitivity minimization which is the real physical problem forthe heat exchanger system under consideration. We tried to evaluate what kind ofpractical computations are needed to achieve H"'-control for distributed parametersystems. Hence this paper may be thought of as a tentative application of H'" techniques to a non-trivial problem.

2. Modelling the heat exchangerThe contraflow heat-exchanger is modelled by a series of first-order partial

differential equations, neglecting the diffusion phenomena. Despite the non-linearityof the controlled system, an explicit characterization of the equilibrium points canbe given. As consequence, the linearized system around an equilibrium point is obtained as a linear infinite dimensional time-invariant system. Based on this linearizedsystem, an H"'-control design is made. The contraflow heat exchanger works withthree input fluids denoted by FL

"FL2 and FL3 , as illustrated in Fig. 1.

"/,

Figure I. Heat exchanger with three input fluids and its geometry. 5, = It(rf - d); 52 = ltd;1= 2ltr2'

The fluid FL , enters the exchanger at x = I" where it is mixed with FL2 (thegaseous combustion products) coming from its right. FL, is very hot, since it carriesthe heat flow rate simulating the combustion.

The purpose of the process is to heat FL3 by mixing FL, with FL 2 . The physicalproperties of these fluids are specified in Table I as well as the notation used for thegeometric description of the heat exchanger. We suppose that FL, and FL 2 have the

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Hr-controt of distributed parameter system

Fluids

Specification FL, FL2 FL3

Specific weight p, p, P2

Specific heat Cp , Cp 'Cp2

Cross-section of path s, 5, 52

Contact circumference of exchanger 1 1 ITemperature

0", x <I, Tj"(x, t) Tj"(x, t) Ti"(x, t)

I, '" x '" 12 T~(x, t) n(x, t)Exchange coefficient k k kMass flow rate <xF ,(t) (l-<X)F,(t) F2

Input temperature To, T0 2 T0 3

Table I.

47

same physical properties and that the mass flow rate FL3 is constant and equal toF 2 •

According to the heat exchange balance, we obtain the following differentialequations describing, respectively, the two parts of the heat exchanger in Fig. I.

(2.1)

The energy balance at the mixing point x = I, gives us

T~(lb t)=(1 :i1.)Ti(I" t)+i1.ToI }

T 2 (/ " t)=T2(1" t)(2.3)

The entering temperatures Ti(12 , t) and Ti(O, t) of the fluids FL2 and FL3 arefixed as

Ti(1 2 , t) = T0 2 }

Ti(O, t) = T0 3 + d(t)(2.4)

(2.5)

where Ti(O, t) is the disturbance to be contended with in practice. In (2.3) and (2.4),TOl> T0 2 and To3 are positive constants. From now on we assume that d(') and itsderivative ci( •) are in L2(0, + co).

Moreover, initial conditions are given as follows:

T 1(x , 0) = fl(X)} o<x < I,Ti(x,O)=fi(x)

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Ti(X,O)=ft(X)}II <x<12

Ti(x, 0) =fi(x)(2.6)

Hence, the dynamics of the heat exchanger of Fig. I is completely governed by thedifferential equations (2.1) and (2.2) with the boundary conditions (2.3) and (2.4) andinitial conditions (2.5) and (2.6). This results in a system to be controlled, with

(a) FI(t) the control input

(b) Ti(O, i) the disturbance input

(e) Ti(/2, t) the output

Note that the system to be controlled is non-linear with respect to the control,but linear with respect to the disturbance. To reproduce the behaviour of the aluminabaking process, we also set

T0 2 < T0 3 < aToI}

0'5«a<1(2.7)

Expression (2.7) means that the fluid FL 3 is heated in the first part (0 < x < II) andheats the fluid FL 2 in the second part (II < x < 12) , Moreover, the result of 0'5« a isthat FL) will be heated at the output.

The condition (2.7)makes the non-minimum phase phenomenon appear naturallyin the system. Let F I (t) = F0 be increased by a positive unitary step, then the fluidFL 2 cools more the fluid FL 3 contained in the second part of the exchanger, and atthe same time the mixing of FL I with FL 2 heats further the fluid FL 3 contained inthe first part. Since FL 3 moves counter-currently to FL I and FL2 , it is observed atthe output that T i (/2' t) first decreases, that is, before the heated FL 3 contained inthe first part reaches the output. Later Ti(/2, t) increases (from the moment whenthe heated part of FL 3 reaches the output)-see Fig. 2.

In further simulation, the length of time over which rto; t) decreases can be

2S.20.IS.10.S.33 .+,..,....,..,.......,,..,..,....,..,..,..,....-T;rrr...,...~..,....~,..,.....TTT~TT""'~,...,....T'TT.......

0.

Figure 2. Response of the heatexchanger system to a positive stepcontrol (/2-/1 = 1(0)andp, = P'S2/F2 = 0-05.

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BOO-control of distributed parameter system 49

verified to be approximately equal to the travelling time of FL3 over 12 -II to reachthe output (see Fig. 2).

On the other hand, a pure time delay exists in the disturbance-outputapplication-this will clearly appear later in the transfer function W(s).

Given a set of parameters {Tol • T0 2 , T0 3 . a}, verifying (2.7) and a fixed controlinput FI(r) = F0 and no disturbance input d(t)= O. the unique equilibrium point isdescribed by

+ ( PI )?I ( al ) -T 2 (x) = al Co exp I -a - PI x + CI---PII-a

(2.8)

(2.9)

where the constants Co, CI• CO and CI are uniquely determined by the boundaryconditions (2.3) and (2.4), or

al

exp (-lal- PI)II -(I -a)a l -a

---PII-a

PI exp(-f--PI)II -1~_P -a1-a I

o o

o 0

al

1-a exp(~_ PI)12~_PI I-aI-a

o(2.10)

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The above constants are related to the physical parameters of the exchanger by

(2.11 )

Remark 1

The matrix in (2.10) is invertible as soon as IX I ;h PI, IX d(l - IX) ;h PI' IX I > 0, PI> 0,12 > II > 0, and 0 < IX < I. However, IX I = PI or IXd(1 -IX) = PI correspond to the degenerate case. It is assumed that IX I ;h PI and IXd(1 -1X);h Pl' Set

Tj±(x, t) = Thx) + Rhx, .t) }

FI(t) =Fo + .... FI(t), I = I, 2(2.12)

By introducing (2.12) into (2.1)-(2.6), we obtain two series of partial differentialequations relative to the variations of the temperature around an equilibrium point,i.e.

aRi(x, t) aRi(x, t) _ _at = ml(t) ax -m3(R I [x, t) - R2(x, t»

+ _1_dTI(x) .... FI(t)PISI dx

aRi(x, t) aRi(x. t) _ _at = - m2 ax + m4(R I (x, t) - R2(x, t»

aRi(x, t) _ ) aRi(x. t) + +at - (I -1X)ml(t ax - miRI (x, t) - R2 (x, t)

I-lXdTi(x). ()+ -----t.>FI tPISI dx

aRi(x, t) aRi(x, t) + +at = - m2 ax + m4(R I (x, t) - R2(x, t»

(2.13)

II<x<1 2 (2.14)

with the boundary conditions (and corresponding initial conditions)

Ri(ll, t)=(I-IX)Ri(lI' t)

Ri(ll, t)=Ri(lI' t)

Ri(O, t) = d(t)

Ri(/2 , t) = 0

(2.15)

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where

ROO -control of distributed parameter system 51

F 2m2=-

P2S2

(2.16)

It is important to note that the linearized system around an equilibrium point isgoverned by the same equations (2.13), (2.14) and (2.15) but with ml(t) replaced byml = Fo/p,s,. Hence, if we use the same notation RNx, r), i= 1,2, for the statevariables of the linearized system, we obtain the linearized system equations from(2.13), (2.14) and (2.15)simply by replacing ml(t) by mi' Later we refer to (2.13)-(2.15)for the linearized system.

The explicit characterization of the equilibrium points has been given and thelinearized system around an equilibrium point has been described by linear partialdifferential equations with constant coefficients. It is easy to show that the linearizedsystem is well-posed and has a unique solution. It is on the basis of the transferfunctions of this linearized system that the ROO-optimal controller design will be madethroughout the paper.

3. Transfer function representation of the linearized systemIn this section, from the linearized system governed by (2.13)-(2.15), we compute

explicitly the plant and disturbance transfer functions, i.e. P(s) and W(s).By taking the Laplace transform of both sides in (2.13)-(2.15) with zero initial

conditions, we obtain

-(XI ] [R1(X, S)]-(PI + P2S) R2(x , s)

(3.1)

(X, ][_+ ]--1- RI (x, s)

-(PI : ~S) R;(x, s)

(3.2)

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52

where

J. P. Gauthier and C. Z. Xu

(3.3)

The boundary conditions being

[~I (l I ' S)] = [I -IX

Ri(ll, s) 0

[Ri (/2, S)] [0]Ri (0, s) = d(s)

(3.4)

(3.5)

(3.6)

(3.7)

The resolution of the linear differential equations (3.1) and (3.2) with (3.4) leads to

["+ ] [I ]RI (/2, s) -- 0" = exp [A 2(s) (/2 -II)] I-IX

Ri(l2' s) 0 I

{ [Rl (O, S)] I"x exp [AI(s) II]" + exp [AI(s) (II - x)]Ri(O, s) 0

[

I dTl(X)] }x r; 0 dx dxtiFI(s)

[

I dTi(X)]+ f' exp [A 2(s) (/2 - x)] - Fo~ dxtiFI(s) (3.8)

(3.9)

Since Ri(l2' s), Ri(O, s) and tiFI(s) are, respectively, the output, disturbance andcontrol for the linearized system, we can easily compute from (3.5)and (3.8) the plantand disturbance transfer functions as follows:

Ri(l2,S) _IP(s)= tiFI(s) = T 22 (s) (NI(s) + N 2(s»

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ROO-control of distributed parameter system

U'( ) _ Ri(l2' s) _ T- 1( )H'S - A - 22 SR 2 (0, S)

where T22(S), N I(S) and N 2(S) are entire functions given by

53

(3.10)

Tn{s)=(O I)exp [-A ,(s) 1,] [I ~IX ~J exp [-Ais) (12-1,)] [~J (3.11)

I', [I -IX oJN 2(s)=(0 I) exp [-A ,(s) 1,]I, ° I

[

dTi(Z)]x exp [A 2(s) (1, - x)] F~x dx (3.13)

For the purpose of the factorization and study of stability later on, we need tocompute the explicit expressions of the transfer functions P(s) and W(s). Straightforward computations give

with

[

A , ., ,(X, s) Al.liX,S)Jexp [-A ,(s) x] =

A ,.2 , (X, s) A ,.n (x, s)

all + a,2sA,. ll(X, s)= 2C,(s) [exp [-).,.,(S) x] -exp [-).,.2(5) x]]

+ Hexp [-).,.,(S) x] + exp [-).,.is)x]]

1X ,A ,. 12(X, s)= C,(s) [exp [-).1.2(S) x] -exp [-).1.1(5) x]]

A ,.21(X, s)= ~(S) [exp [-).,.,(S) x] - exp [-).,.2(S) x]]

all + a12sA , .n{x , s)= 2C,(s) [exp [-).,.2(S) x] -exp [-).",(S) x]]

+Hexp [-).",(S) x] +exp [-)."is) x]]

[A2.ll(X, s) A2.,2(X,S)J

exp [-A2(s) x] =A2,21(X, s) A2.22(X, s)

(3.14)

(3.15)

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54

where

J. P. Gauthier and C. Z. Xu

a2' +a22sA2., ,(x , s) = 2C

2(s)[exp [-).2,,(S) x] - exp [- ).2,2(s) x]]

+ Hexp [-).2,,(S) x] +exp [-).2,2(s) x]]

IX,I-IX

A2,' 2(X, s)= C2(s)

[exp [-).2,is) x] -exp [-).2,,(S) x]]

A2,2' (X, s) = ~(S) [exp [-).2,,(S) x] - exp [-).2,2(S) x]]

a2' + a22sA2,d x, s) = 2Cis) [exp [-).2,2(S) x] -e- exp [-).2,,(S) x]]

+ Hexp [-).2,,(S) x] -i-exp [-).2,2(S) x]]

C,(s) = [(IX, - Pd2+ 2al1a'2s + a'2s2]1/2

).",(s) =HIX, - P, + (1X2 - P2)S + C,(s)]

).,,2(S) =HIX, - P, + (1X2 - P2)S - C,(s)]

(3.16)

(3,17)

(3,18)

(3,19)

(3.20)

(3.21 )

(3.22)

(3,23)

In the above expressions and in the rest of the paper, the complex square root isdefined by

(X + iYjI/2 = [(X + (X 2 + y2)'/2)/2] '/2 + sign (Yj[(- X + (X 2+ y2)'/2)/2J'/2(3.24)

for any reals X and YIt is now a straightforward manipulation to obtain the following expressions of

T22(s), N ,(s) and N 2(S):

W,(s)T22 = 4C,(s) C

2(s)exp [-)."is) /, - ).2,is) (/2-II)] (3.25)

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H'<control of distributed parameter system 55

where

WItS) = 4IX i/J1[exp [- C l(S) I,] - 1] [1 - exp [- C2(S)(l2 - Ill]]

+ [a 11 +a'2S+C,(S)+(Cl(s)-all-a'2s)exp [-C,(s) I,]]

x [a2' + a22s + C2(s) + (Cis) - a2' - a22s) exp [- C2(s) (/2-Ill]] (3.28)

W2(s) = 4IX,P, CO {exp [(IX, - Pili,] - exp [A,,2(S) I,]

Fo IX, - P, - A'.2(S)

_ exp [(IX, - P, - C,(s»I,] - exp [A.'.2(S) I,]}IX, - P, - AI,I(S)

X C2(s) exp [A.2,2(S) (12-Ill] (3,29)

2IX,P,Co exp [(~-p),JW3(s) = - (1 - IX)F0

x{II - .)(~p [- C,I') I,] -I)

x[Ion +on' + C,I'»)

exp [C ~ IX - p, - C2(S»)(12 -I,)J - exp [Ads) (/2-I,)]x ---'=-.>....--------'------~"----------

IX,1_ IX - p, - A2,,(S)D

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56 J. P. Gauthier and C. z. Xu

Up to now we have found explicit expressions for the transfer functions P(s) andW(s), through those of W,(s), Wz(s) and W3(s). In the frequency domain, the linearizedsystem may be represented by Fig. 3.

Figure 3. Open loop for the linearized system.

Since the heat exchanger is a non-minimum phase system, we need to computeall of the unstable zeros of P(s) to achieve the HOO-controller design. Let us proceedwith the HOO-factorization of P(s) and W(s).

4. H"-controUer designIn this section we first characterize an open set of numerical values of physical

constants of the process, for which we can actually prove that the transfer functionsare in HOO (right half-plane) and that a non-minimum phase occurs. Secondly, wedevelop a method to compute the unstable zeros of the plant transfer function inorder to make its HOO-factorization. Thereafter, the theory developed by Francis andZames (1984) is directly applied to obtain the Hoo-optimal controller.

Let us begin with the stability of the linearized system.

4.1. Stability of the linearized system

Physical intuition clearly suggests that any exchanger system is stable in the sensethat their transfer functions are in HOO (right half-plane). In fact, it has been provedthat a classical contraftow exchanger system (i.e. IX = 0 in our case) is stable(Friedly 1972). It is thus reasonable to think that our heat-exchanger system is alsostable with respect to the control as well as the disturbance. In this section wegive a necessary and sufficient condition for P(s) and W(s) E HOO (right half-plane).Based on this condition, we exhibit an open set of physical parameters for our heatexchanger, such that it is stable. We are not able to prove that it is stable in general,however we think that it is.

First, we state some properties of C,(s) and Cz(s) defined in (3.18) and (3.21).

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H"'-control of distributed parameter system

Lemma 1

Let a l i' PI and ad(l- a) i' PI; we have for s = x + iy with x ~ 0

57

(4.1 )

(4.2)

(4.3)

(4.4)

The proof of Lemma 1 is a straightforward calculation using (3.24); it is omitted.From Lemma 1 and the definition of A,I.Z(S) and A,2.2(S)-(3.20) and (3.23)-one

has immediately for Re (s) ~ 0

(4.5)

(4.6)

which imply that

- Pllz';;; Re (A,1,2(s) II + A,2.Z(S) (/2 -Itl + P212S)

,;;; ~[(al - PI -Ial - PiI)11 + (1 ~ a - PI -11~ a - PII)(/2 -Itl] (4.7)

Thus

is an H"'-unit (i.e. an H'" element with its inverse in H"'). On the other hand, onecan see easily that I/CI(s), I/C2(s), WI(s) and W2(s) are all analytic in Re (s)~ O.

Now the transfer functions P(s) and W(s) may be written

P(s)= WII(s) (W2(s) + W3(s)) (4.8)

4C I (s) C2(S)W(s)= WI(S) exp [A,1,2(s) II + A,2.2(S)(l2 -Itl + P212s] exp [-pzl2s] (4.9)

in which WI(s), W2(s) and W3(s) are, respectively, defined in (3.28), (3.29) and (3.30).The necessary and sufficient condition for P(s) and W(s) E H'" (right half-plane)

may be stated as follows.

Lemma 2(i) W(s) E H'" (right half-plane) implies that P(s) E H'" (right half-plane)

(ii) W(s) E H'" (right half-plane) iff I/WI(s) is analytic in Re (s)~ 0

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Proof

(i) From (4.9), W(s) E H OO (right half-plane) implies that I/W,(s) is analytic inRe (s)~ O. Then, from (4.8) and what is said above, P(s) is clearly analytic in Re (s) ~ O.Let us set (4.8) in the following form:

First, note that every exponential term in W,(s), W2(s) and W3(s) has exponents withnegative real parts. Moreover the polynomial parts have the same order in the numerator as well as in the denominator for both C,(s)Cis)/W,(s) and (W2(s) + W3(s)). It istherefore clear that C,(s)C2(s)/W,(s), (W2(s) + W3(s)) and I/C,(s)C2(s) are all boundedin Re (s) ~ O. Their product P(s) is thus bounded. By definition, P(s) E H OO (right halfplane).

(ii) W(s) E HOO (right half-plane) => I/W,(s) is analytic in Re (s) ~ 0 is evident by(4.9). Conversely, I/W,(s) is analytic in Re (s) ~ 0

4C,(s) Cis) .=> W,(s) E H OO (right half-plane)

Since

is an Hoo-unit and exp (- /3212s) is an inner function of H OO (right half-plane), theirproduct W(s) E H OO (right half-plane). 0

Note that in the case of the classical simple exchanger, i.e. a = 0, it is easy to verify(by some manipulation of(3.28) with a = 0) that I/W,(s) is analytic in Re (s) ~ O. Belowwe give a sufficient condition for I/W,(s) to be analytic in Re (s) ~ 0 for our case.

Theorem ISuppose that a, "'/3, and al!(I-a)"'/3,. I/W,(s) is analytic in Re(s)~O if

4/3,~(I-a) I-a

(a, +/3, +la,-/3d)C ~a +/3, +11 ~a -/3,1)

[I + exp (-la, - /3d ldJ[1+ exp ( -II ~ a - /3,1(12 -I,))] .

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Hr'-contro! of distributed parameter system

Proof

Rewrite W,(s) in (3.28) as

W,(s) = .4,(s) .4z(s) - (1 - <x)B,(s) Bz(s)

where by identification in (3.28)

.4,(s) = all +a IZs+C ,(s)+(C1(s)-all -a,zs) exp [-CI(s)I,]

.4z(s)= aZI + azzs + Cz(s) + (Cz(s) - a21 - azzs) exp [- Cz(s) (/z -I,)]

8,(s) = 2/31[1 - exp (- C I(S) Id]

I I 1

.4,(s) = all +a1Zs+C l(s) x 1+ C1(s)-all-a 1Zs exp [-C,(s) I,]a,l + a,zs + C 1(s)

It is easily seen from Lemma I that

Ic ,(s) - all - a, zs I

C ()exp [-C,(s) I,] < I

all +al2s+ IS

59

Thus 1/.41(s) E H OO (right half-plane). In the same way, 1/.4z(s) E H OO (right halfplane). We have

I .• () • ( ) E H OO (nght half-plane)

A, s Az s

Moreover

I 1 1

W,(s) = .41(s).4Z(s) x 1 _ (1 _ <x) ~l(S) ~z(s)A,(s) Az(s)

and it is straightforward from the above that

B I (s) Bz(s) .• () • ( ) E H 00 (nght half-plane)A, s Az s

Condition (4.10) implies that

[1 - exp (- C,(s) Id] [1 - exp (- Cz(s) (/z - 'Id]

Hence I/W,(s) E HOO (right half-plane). In particular, I/Wl(s) is analytic in Re (s)~ O.

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Remark

We can derive from (4.18) a stronger but simpler sufficient condition:

(I -IX)[I + exp (-IIX I - 1I11 1dJ[1+ exp ( -I :11X

- 1111(/2 -Id)J---------;:::--=----,-----.,-:--'---------,----'----------;---::-= < I (4.11 )

[I-exp (-IIXI-IIllld{l-exp ( -II ~IX -111[(/2 -ld)J

Note that there exists always IX (0 < IX < I) which verifies this condition for given lXI'Ill> II and 12, since

. (I-IX)[I +eXp (-I1X1-IIdld{1 +exp ( -II ~IX -1111(/2-ld)J

bm =0

.-1- [I-exp (-I1X1-II11Id{l-exp-11 ~IX -1I11(/2-11))J

It is important to observe that the set of physical parameters verifying (4.10) is openand the condition (4.10) is inherent to the appearance of the non-minimum phasephenomenon.

By Lemma 2, the sufficient condition (4.10) implies that P(s) and W(s) E H'" (righthalf-plane), that is to say that the linearized heat exchanger system is stable in thesense defined at the beginning of this section.

Hence conditions (2.7) and (4.10) provide a set of stable linearized systems withnon-minimum phase. It is for this class of heat exchanger systems that the H "'_optimal controller design will be performed below.

4.2. H"'jactorization of P(s) and W(s)The disturbance transfer function W(s) may be written as the product of an H "'_

unit Wo(s) and an inner function, i.e.

(4.12)

where4C I (s) C2(s)

~o(s) = ( ) exp (AI 2(S) II + ,1,2 2(S) (/2 -II) + 11212s) (4.13)WI s . ,

is in H'" (right half-plane) as well as its inverse, that is it is an H "'-unit by (4.9), andexp (- 11212,1,) is the pure time delay of the disturbance output transfer function.

To factorize P(s), we have to know its unstable zeros. According to the previousanalysis, the only unstable zeros of P(s) come from those of the part W2(s) + W3(s).

Since IW2(s) + W3(s) I tends to different values as lsi -+ + CX) in Re (s) ;;.0, it is verydifficult to find explicitly its unstable zeros. Herein, we give a constructive algorithmto compute the unstable zeros of W2(s) + W3(s). It is clear that determining the unstable zeros of P(s) is equivalent to determining those of the following function:

Fo~2(S) = -p-(W2(s) + W3(s)) exp (-AI 2(S) II - ,1,2 is) (12 -II) - 11212s) (4.14)4IX I 1 ..

By considering numerical physical examples of heat exchangers, we have observedby simulation that P(s) has only one unstable zero situated on the real axis. Basedon this practical observation, an algorithm is developed here devoted to the case by

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H""-control of distributed parameter system 61

case proof of the fact that P(s) has only a finite number of unstable zeros and to theircomputation. We are interested in an asymptotic behaviour function .1. 1(s) for .1.z(s),i.e.

lim 1.1. I(s) - .1.2(s)1 = 01·1-+""Re(,,)~O

with .1. 1(s) of the form

5

.1. 1(s) = CI 6 + L CII exp (-ejs)i=1

where

C - C- (ototll l PI) (ot2+ (1 - ot)P2)14 - 0 exp 1 - I 2- ot ot2

(4.15)

(4.16)

(4.17)

(4.18)

(4.19)

(4.20)

(4.21 )

(4.22)

(4.23)

One can observe that .1. 1(s) is an entire function and that ej> 0, i = 1,2, ... ,5, with

emin =min {ei> i = 1, 2, ... , 5} = el (4.24)

The relation (4.15) has an interesting consequence: if 1.1. I(s)1 ;;,: s > 0 outside acompact set in Re (s) ;;:. 0, then there exists necessarily a half-disc with radius suchthat outside it in Re (s) ;;,: 0

1.1.Z(s)1 ;;:. e' > 0

In the following, a criterion is given to decide when it is true that

1.1. I(s)1 > 0, outside of some compact set

(4.25)

(4.26)

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As soon as it is true, an algorithm is given afterwards for finding the half-disc whichcontains all of the possible unstable zeros of ~2(S). Then the Cauchy integral is simplyperformed repeatedly in order to compute all isolated zeros included in the interiorof this half-disc. Of course, if it happens that (4.26) is not satisfied, nothing can besaid about the unstable zeros of ~2(S). Let us begin with some lemmas.

Lemma 3

Whenever Re (s) ;;"0 and lsi;;" Rmin , with

where M is a constant which can be computed explicitly.

Remark 2

The proof of Lemma 3 is tedious but elementary. In Xu (1989), the explicit expression of M is given. We do not reproduce it here.

Lemma 4

Assume that Co # 0 and Co # 0; ~I(S) has no zero in the half-plane Re (s) > Xmax '

with Xmax a finite real number.

ProofSince 81> 0, V i = I, 2, ... , 5, we have

3

1~I(s)l;;" IC161- L ICIiI exp (-8min X)i= 1

where 8min is defined in (4.32) and x = Re (s). It is sufficient for this to take

Xmu = In {1ICIiI/ICI61} (4.28)

o

Remark 3

If Xmu < 0, ~I(S) has no zero in Re (s)> 0, and obviously 1/~I(s) E H'" (right halfplane); moreover

(4.29)

If Xmu > 0, the possible unstable zeros of ~I(S) must be located in the infinite band:o:::; Re (s) :::; X m..

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ROO-control of distributed parameter system 63

(4.30)

In order to determine if ~l(S) has any zero in this infinite band, the followinglemma is used assuming that B;, i = 1,2, ... ,5, are rational numbers.

Lemma 5Assuming that B" i = 1,2, ... ,5, are rational, ~l(S) has no zero in 0:;;;Re (s) :;;; Xm..,

if and only if

,( l\~(s) ds = 0Jr ~l(S)

where ~~(s) = d~l(s)/ds and r is a rectangle covering at least one period of theperiodic function ~l(X + iy) with respect to y (see Fig. 4).

m

_,-~L

0 "-R.

Figure4. Application of the Cauchy integral.

ProofBj , i = 1,2, ... ,5, are rational numbers implies that ~l(X + iy) is a periodic function

of y for any fixed x E [0, xm••]. Thus, ~l(X + iy) has zero in 0:;;;Re (s):;;; Xm.., if andonly if it has zero in the rectangle if Fig. 4. The Cauchy integral (4.30) gives just thenumber of the zeros of ~l(S) in the region bordered by r. 0

If (4.30) is satisfied, under the assumption of Lemma 5 one has again, 1/~l(s) E

Roo (right half-plane) and

111/~l(s)IIHoo = Pmax = sup IA (I. )1 (4.31)c:oe(([)o.wo+LJ Ll1 fW

The application of Lemma 3 to both cases previously considered results in

(4.32)

where M and Pmax are defined above.Since l~l(s)1 > 0 in Re (s) ~ 0 in these cases, it is clear that ~2(S)> 0 for lsi ~ Rm.. =

max {Rm;n' MPm.. } in Re (s) ~ O. We have thus found a half-disc with radius Rm..outside of which l~l(s)1 > O. Hence, all the unstable zeros of ~2(S) are situated in thishalf-disc.

Now, one can apply the Cauchy integral (4.30) on the boundary of the rectangleABCDA (Fig. 5) in order to compute the total number of unstable zeros for ~2(S).

After this step, the rectangle ABCDA is divided into smaller ones. The releated useof the Cauchy integral for each of them gives the number of zeros in the borderedregion. By iterating this procedure, we collect all the isolated zeros of ~2(S) includedin the half-disc.

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1m

Rm..-----;;-~___J----R.

Figure5. Location of the unstable zeros of P(s).

Note that as soon as a rather precise integration method is applied, 'almostinteger' numbers have to be obtained at any step, so that we are sure of the result.

Recall that it would be possible that the Cauchy integral (4.30) is not zero, thatis to say that A,(s) has a countable number of zeros in 0 ~ Re (s) ~ Xm... Nothing canbe done in that case. However, our feeling is that it is never the case for our heatexchanger system.

Let us summarize the different steps for finding the unstable zeros of P(s):

Step I

Compute explicitly A2(s) and dA2(s)lds from (4.14).

Step 2Compute Xm.. from (4.28); if Xm.. < 0, compute Pm.. from (4.29) and jump directly

to Step 4; if X m.. > 0, go to Step 3.

Step 3

Calculate the period L of the periodic function A,(x + iy) with respect to y. Infact this period is equal to the least multiple of {l/el = Qi/Ni> i = 1,2, ... ,5, withcoprime integers QI and Njl (up to 21t). Compute the Cauchy integral (4.30). If it iszero, compute Pmex-from (4.3I)-and jump to Step 4. If it is not zero, nothing canbe done.

Step 4Compute Rm.. = max (Rmln> MPm..) which is exactly the radius of the half-disc

containing all the unstable zeros for A2(s).

Step 5Iterated division in rectangles of ABCDA and repeated use of the Cauchy integral

along each rectangle allow us to find all the isolated unstable zeros for P(s).

Since the number of unstable zeros in a compact set is necessarily finite for P(s),we invoke the associated Blaschke product

B.(s)= Ii (s -~j) (4.33)j=' s+aj

with at> i = 1, 2, ... , n, being unstable zeros of P(s).It is clear that [Wz(s) + Wis)]B.-'(s) is an H"'-unit. The plant transfer function

P(s) can be factorized by

P(s) = B.(s) WiltS) [Wz(s) + W3(s)]B; '(s) (4.34)

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H"'-control of distributed parameter system 65

Set Po(s) = Wt 1(S)[W2(s) + W3(s)]Bz-

1(s). According to the results of § 4.2, Po(·) eH'" (right half-plane) and its inverse Po I(S) is analytic in Re (s)~ O. Hence

P(s) = Bz(s) Po(s) (4.35)

The factorization results (4.12)and (4.35) will be extensively used in the followingsections.

4.3. H "<controlier

In this section the H "'-control theory developed by Francis and Zames (1984) isapplied directly. Since these techniques are well known, the details are omitted.

P( .) being in H'" (right half-plane), it is clear that all of the stabilizing controllersfor the exchanger system are given by

C(s)= Q(s) (1- P(s) Q(s)) with Q(.) e H'" (right half-plane) (4.36)

The sensitivity function of the closed system is

X(s) = W(s) [I - P(s) Q(s)] (4.37)

In this paper we suppose that the disturbance belongs to the following subset ofL2(0, + oo):

£!2p = {d; d(s) = Wx(s)!(s), V!e H 2 (right half-plane)) (4.38)

where Wx(s) is taken as a strictly proper fraction function in H'" (right half-plane)and W- I(S) is analytic in Re (s) ~ 0 (this is related to the assumption on the natureof the disturbance made at the beginning of the paper).

Following the general theory of Francis and Zames (1984), minimizing the worstL2-effect or £!2p at the output is equivalent to finding Q*(') e H'" (right half-plane)such that

. inf II WxW(l- PQ)IIH~ = IIWxW(l- PQ*)IIH~ (4.39)QeHCO (nght half-plane)

by abuse. To be perfectly correct, the infimum above may be obtained by Q*(s) onlyanalytic in Re (s) ~ O.

If the number of unstable zeros of P(s) is finite and all of these unstable zeros areknown, the solution of (4.39) is immediate (Francis and Zames 1984 propose anexplicit formula). Using the factorization result and the standard H'" theory, one isled to the following problem:

inf IIWxW(l-PQ)IIH~= inf IIV-BzRIIH~ (4.40)Q(s)analytic ReHOIOinRc(s)~ 0

where

V(s) = Wx(s) Wo(s) }

R(s) = Wx(s) Wo(s) Po(s) Q(s)

the unique solution for (4.40) is given by

m (S-b)V(s) - Bz(s) R*(s) = D n ------,} = X o(s)1=1 s+Oj

with m~ n - I (Zames and Francis 1983).

(4.41)

(4.52)

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In (4.42), the Blaschke product and the constant D can be computed explicitlyusing Francis and Zames's algorithm (Francis and Zames 1984). Therefore the H"'controller is obtained for a heat exchanger with

* _ R*(s)Q (s) - WxW(s) P(s)

C(s)= Q*(s)Wx(s) Wo(s)Xo(s)

The weighted sensitivity function is, by (4.37)

Xw(s) = Wx(s) Wo(s) [1- P(s) Q*(s)] exp (-P2/2S)

(4.43)

(4.44)

If P(s) has only one unstable zero, X o(s) will be constant and the H'" (right halfplane) controller in (4.51) will be stable. This is what happens with the numericalexample of the next section. Note that since Wx(s) is assumed to be strictly proper,the improper H'" (right half-plane) controller Q*(s) can be approximated by thesequence Q. (Zames and Francis 1983) so that

(4.45)

(5.1)

5. Numerical simulationsIn this section, we first present the steady-state and dynamical evolution of the

linearized system for a given set of physical parameters. The evolution is illustratedin three-dimensional space to give a physical feeling of the behaviour of the linearizedand the non-linear systems. Afterwards, the theoretical results developed earlier inthe paper are applied here for the H "'-controller design. Finally, the sensitivity performance of the designed H "'-controller is compared with that of a PIO controller.

The method of finite differences is applied to solve numerically the partialdifferential equations of the non-linear and the linearized systems. After discretizationwith respect to the space coordinate, the non-linear system is naturally approximatedby a linear time-varying one of finite dimension, i.e.

dR(t) _ _ _}+----;Jt = ~(t) R(t) + BM1(t) + Dd(t)

R2 (/2, r) = CR(t)

where R(O) is determined by initial conditions and matrices A(t), E, 15 and Care givenexplicitly in the Appendix.

In the same way, the linearized system is approximated by a linear time-invariantsystem.

\ }dR t - _ _+ d~) = ~R(t) + BAF1(t)+ Dd(t)

R 2 (/2' t) = CR(t)

It appears through numerical simulation that when IAF 1(t)1 and Id(t)1enough, the solutions of (5.1) and (5.2) coincide.

(5.2)

are small

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H"'-control of distributed parameter system

Note that the resulting simulation transfer functions are given by

~(s) = ~(SJ _ ~)-l~}W(s) = C(sJ - A)-ID

67

(5.3)

which approximate, respectively, the transcendental transfer functions P(s) and W(s).It is observed by simulation that (5.1) and (5.2) are always stable.

The set of values chosen for physical parameters is given in Table 2.

0·02

Pt0·01

ex2

0·1

P20·05 100 200

Table 2.

ex

0·65

Tot

80

102

15 20 30

Fig.6 illustrates the dynamical evolution corresponding to different excitations.We mention that the initial conditions are taken in the domain of the infinitesimalgenerator so that the solution is smooth. In addition, the effect of the pure time delaycan be observed in Fig. 6(d). The equilibrium solution is shown in Fig. 7.

In the case of the given physical parameters (Table 2), we find that X max < 0 andthat P(s) has only one zero which is real. The use of Cauchy integrals provides thefollowing estimations:

a, E (0'10246086,0'10246086 + 0·32 x 10-9) (5.4)

This fact can be verified on the Nyquist diagram of ~2(S) which encircles the originonce as s moves along a closed half-circle (with radius > Rm.., see Fig. 8). The weightfunction is chosen as

(5.5)

That is, the disturbance is concentrated at low frequencies. One obtains the corresponding optimal sensitivity function

X(s) = 0·2912 exp (-P212s)

Therefore, the H "'-controller is given by

(5.6)

(5.7)Q*(s)= [Wx(s) Wo(s) - 0'2912]

Wx(s) Wo(s) P(s)

Q*(s) being improper, its approximating sequence is taken:

Qn(s) = Q*(s) (_n_)S (5.8)s+n

Theoretically, Q.(s) should be in H'" (right half-plane). However, the zeros of P(s)being computed numerically, the simplification of Q;s singular points with its zerosis not perfect so that in practice, Qn ¢ H'" (right half-plane). However, the unstablezeros of P(s) being well surrounded, no practical problem appears. The numericalsimulation is based on the control diagram Fig. 9, in order to allow the possibilityof comparing H"'-controller with the PID controller. All of the blocks in Fig. 9 aresimulated in the time domain (in particular, Qn(s)'s impulse response shown in Fig. 10is calculated by a fast Fourier transform algorithm).

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35

39

25

35

39

25

(e) (f)

Figure 6. (a) Evolution of T,(x, t) from an absolutely continuous initial condition. (b) Evolution of T,(x, t) from an absolutely continuous initial condition. (c) Response T,(x, t)to a disturbance in L2(O, + 00). (tl) Response T,(x, t) to a disturbance in LI(O, + 00). (e)Response T,(x, t) to a step control input. (f) Response T,(x, t) to a step control input.

0 (c) (d)

60

50 35

4039

38

25

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60

so

T2CXI

eo

30

20

10

50 1SO

Figure7. Steady-state solutions.

1m

Figure8. Nyquist diagram of ll2(iw).

YCS)

Figure9. Closed loop for simulation. k =0, PID control; k = I, H""-control.

It is interesting to see in the impulse response, that the time interval between twopeaks is approximately the travelling time of the fluid FL 3 over the distance (/2 -11)'The impulse response qn(t) in Fig. 10 clearly shows that a performant integrationmethod had to be used for the convolution calculation (e.g. Simpson's method). Infact, Fig. 10 also suggests other methods to realize the controller, but this point willnot be developed herein.

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REAL PART

,,/

(a)

(c)

Figure 10. (a) Real and imaginary parts of QN(iw). (b) Impulse response q,.(t) by the inverseFourier transform. (c) Impulse response q,.(t) by the inverse Fourier transform.

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(5.11 )

The PID controller is chosen as

M\(S)=-(Kp+~I+KDS)Y(S) (5.10)

with K; = 2·0; K1 = 0'001; KD = 0'001, which were obtained by manual trial and error.The differences existing between the theoretical optimal sensitivity functions and

the simulated ones are simultaneously investigated here. Indeed, for the H "'-controller, the theoretical and numerical optimal weighted sensitivity functions are given by

: w(s) = Wx(s) ~(s) (1 - ~(s) Q.(S»}

X w(s) = Wx(s) W(s) (1 - P(s) Q.(s»

where W(s) and fi(s) are defined in (5.3). Similarly, for the PID controller we have

X_p(s) = Wx(s) 1i'I_(s)(1 - P_(s) Qp(S»}(5.12)

X p(s)= Wx(s) W(s) (1 - P(s) Qp(s))

.1

weighted sensitivity functioos

1.21..8.6.4.2e '+"'-""""'T"~,-r"TTT"TT"'-rTT~TT""'"',.,.,....':;:;:;:;:;:;;:~:;::;:;:;=MT~;;:;;:;:""~

e.(a)

weighted gensiU vity funct:lons

f""""""",

.5 1. 1.5

(b)

Figure II. Sensitivity functions: (a) h = 12'5; (b) h = 6·25.

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where Qp(s) can be obtained from (5.10) and (4.36). It can be observed that thedifference between X(s) and g p(s) is smaller than that between X W<s) and g W<s) (seeFig. II).

A disturbance fIt) = sin wt with w = 0'12 is applied to the linearized system;the responses obtained for the H "'-controller and the PID controller are given inFig. 12. The range of the responses in the steady state fit well with the sensitivityfunctions in Fig. II. Since we have put a more important weight on the low frequencies, it is natural to observe that the H "'-controller is better than the PID controllerfor the rejection of disturbances concentrated at low frequencies. Finally, a particulardisturbance is applied to the linearized system and to the non-linear system with theH "'-controller and PID controller.

In Fig. 13 we observe that the H"'-controller equally works well on the non-linearheat exchanger system. The same applies for the PID controller (see Fig. 14), whichis better than the H "'-controller for the rejection of this particular disturbance (owingto high frequencies). If the disturbance energy is located around low frequencies, the _

(a)

(b)

Figure 12. Response of the linearized system to a sinusoidal perturbation with (al the H"'·controller; (b) the PID controller.

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-2.

(a)

60. 80. lee. 1213.

-10.

-3.

(b)

t ....

40.

Figure 13. Simulation of Hal-controller on (a) the linearizedsystem;(b) the non-linear system.

implementation of the resulting H "'-controller is always possible from Q*(s) thanksto the existence of the strictly proper approximating sequence. On the contrary,nothing can easily be done for high-frequency located disturbances.

6. ConclusionIn this paper H"'-control of a distributed parameter system with non-minimum

phase is discussed. This system is a rather special heat exchanger (expected to simulatea furnace in a simplified way), which is modelled by a partial differential equation. Itis essentially non-linear with respect to the control input and linear with respect tothe disturbance to be attenuated. It was therefore linearized with respect to thecontrol input.

The stability of the linearized system describing the plant is proven, in the sensethat the transfer functions are in H'" (right half-plane). The problem of sensitivityminimization is then dealt with following the general methodology developed byFrancis and Zames (1984). Hence, it is necessary to determine the unstable zeros of

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T

40. 60.

-l.

-c.

-3.

(a)

(b)

Figure 14. Simulation of PID controlleron (a) the linearized system; (b) the non-linearsystem.

the transfer function of the plant. Based on the idea of approximating the asymptoticbehaviour of the transfer function, numerical algorithms were developed in order to

(a) verify that the number of unstable zeros is finite

(b) localize these zeros

(e) compute these zeros

The H "'-optimal controller minimizing the sensitivity function was then computed, simulated and compared to a standard PID controller. It appears, at firstglance, that there is not a big difference between performances of a PID controllerand the rather complicated H"'-controller. Moreover, in the treated case, the H"'controller is worst for high frequencies. This is partially due to the fact that thecontroller was designed in order to annihilate low-frequency disturbances, but itseems to be also inherent to the method.

A closer look at the results shows that the H"'-controller is really better for lowfrequencies, and it is a matter of evidence that it is very interesting to be able to realizecontrollers that are efficient for the compensation oflow-frequency disturbances. Other-

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H'<controt of distributed parameter system 75

wise, standard methods can be a posteriori applied to increase the performance of thecontroller at high frequencies. Anyway, we consider that the results are positive.

Another positive point is the 'insight' we obtained of the behaviour of this kindof process. In our opinion, the fact that in the impulse response of the H OO-controllerthe same parts repeat (being exponentially attenuated) at instants that are multiplesof the delay should be more precisely examined. There is certainly some generalprinciple behind this observation. Moreover, it also suggests a credible way to realizethe optimal controller.

AppendixDiscretization of the heat exchanger system

The following notation is used in the paper:

Rtj(t) = R,±(x, t)lx=jh, i= I, 2,j=0, I, ... , n

The intervals [0, II] and [/1,1 2] are divided into nl and n2 parts, respectively, withsampling period h. At the points inside [0, II] and [/1,12] , the partial derivatives withrespect to x are approximated by

iJR,±(x, t)I ~ Rl:j+ I - R'~j_1ox x=jh 2h

For the extreme points on the left and on the right the derivatives are taken as:

OR/±(X,t)! R±'+I-R,±."" I,J ,J

oX x=jh = h

oR±(x t)1 R±.±-R±·_II , "" I,J I,)

= hox x~jh

R(t) = (RI(t))R2(t)

withRI(t) = [Ri.o(t) Ri.l(t) ... Ri.n, -I(t) R;:. I(t) R;:.2(t) R;:.n,(t)]T

R2(t) = [Ri,n,(t) Ri.n,+I(t) Ri.n,+n,-I(t) Ri.n,dt) Ri,n,dt)

Ri.n, +n,(t)]T

Developing (2.13) and (2.15) gives

dR(t) " "---;It = A(t) R(t) + BIiF I (t) + Dd(t)

fit) = CR(t)

where R(O) is obtained with the initial condition,

Al BI CI 0

A(t) =GI £1 HI 0

0 Bz C2 D2

0 £2 H 2 M2

(A I)

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BEl

B=BEz

(A 2)BE3

BE 4

DP1

15=DPz

(A 3)DP3

DP4

c=(0, 0, 0, ... , 0, 1) (A 4)

The matrices in (A l)-{A 4) are:

-(2x I + YI) 2x l 0 0 0 0 0

-Xl -YI Xl 0 0 0 0

0 -Xl -YI XI 0 0 0A I =

0 0 0 0 -Xl -YI XI

0 0 0 0 0 -Xl -YI III X"1

0 0 0 0 0 0

YI 0 0 0 0 0

0 YI 0 0 0 0B I =

0 0 0 YI 0 0

0 0 0 0 YI 0111 XIII

0 0 0 0 0 0

0 0 0 0 0 0

CI =0 0 0 0 0 0

(l-OC)XI 0 0 0 0 0 "1 XIIZ

0 Yz 0 0 0 0 0

0 0 yz 0 0 0 0

GI =0 0 0 0 0 Y2 0

0 0 0 0 0 0 Yz

0 0 0 0 0 0 0 III X rJl

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-Y2 -X2 0 0 0 0

X2 -Y2 -X2 0 0 0

£1=

0 0 0 X2 -Y2 -X2

0 0 0 0 2X2 -(Y2 + 2X2) "1 Xlll

0 0 0 0 0

0 0 0 0 0

H 1 =0 0 0 0 0

(1 - a)Y2 0 0 0 0 "1 xn)

0 0 0 0 YI

0 0 0 0 0

B2=0 0 0 0 0

0 0 0 0 0 "1 X"l

-(2x~ + YI) 2x'l 0 0 0 0 0

-X'i -YI X'I 0 0 0 0

0 -X'i -YI X'I 0 0 0C2=

0 0 0 0 -X'i -YI X'I

0 0 0 0 0 -X'i -YI "2 xnz

with X~ = (1 - a)xl'

0 0 0 0 0 0

Yl 0 0 0 0 0

0 YI 0 0 0 0D2=

0 0 0 Yl 0 0

0 0 0 0 YI 0"2 XII)

0 0 0 0 X2

0 0 0 0 0

£2=

0 0 0 0 0

0 0 0 0 0 "2 XIII

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0 Y2 0 0 0 0 0

0 0 Y2 0 0 0 0

H2=0 0 0 0 0 Y2 0

0 0 0 0 0 0 Y2

0 0 0 0 0 0 0 "2 )( "2

-Y2 -X2 0 0 0 0

X2 -Y2 -X2 0 0 0

M 2=

0 0 0 X2 -Y2 -X2

0 0 0 0 2X2 -(Y2 + 2X2) "2X"z

I dTi(x)IPIS, dx .=0

I dTi(x) IBE,= PIS, dx .=h , BE2 = (0)0' x i

I dTi(x) IPIS,. dx '~(Ol -')h 0, x,

I - IXdTi(x) IPIS, dx .=o,h

I - IXdTi(x) IPIS, dx '=(0, + ')h , BE4 = (0)0'x ,

I - IXdTi(x) IPIS. dx x=(nl+nz-1)h "2x1

Y, X2 0 0

0 0 0 0DP,= , DP2= , DP3= , DP4=

0 "1)( 1 0 ") x 1 0 111 )( 1 0 "2X 1

Their elements are defined by:

(Fo+~F,(t)) F2x, = 2p,s,h ' X2 = -2p-2-s2-h

kl

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Hr'-control of distributed parameter system 79

With the same method of discretization for the linearized system, we obtain a lineartime-invariant system:

dR(t) _ A A

----;It = AR(t) + B6.F l(t) + Dd(t)

y(t) = CR(t)

where R(O), B, J5 and Care the above and A is obtained from A(t) by replacing the

elements Xl and Yl by

REFERENCES

CURTAIN, R. F., 1988, Equivalence of input-output stability and exponential stability forinfinite dimensional systems. Mathematical Systems Theory, 21, 19-48; 1989 aEquivalence of input-output stability and exponential stability. Systems and ControlLetters, 12,235-239; 1989 b, Time and frequency domain methods for infinite dimensional HOO-control. Fifth Symposium on Control of Distributed Parameter Systems,Perpignon, France.

FRANCIS, B. A., 1987, A Course in Hr'-Comrol Theory (Berlin: Springer-Verlag).FRANCIS, B. A., and ZAMES, G., 1984, On HOO-control sensitivity theory for SISO feedback

systems. I.E.E.E. Transactions on Automatic Control, 29,9-16.FRIEDLY, J. c., 1972, Dynamic Behaviour of Processes (Englewood Cliffs, NJ: Prentice Hall).JACOBSON, C. A., and NETT, C. N., 1988, Linear state space systems in infinite dimensional

spaces: the role and characterization of joint stabilizability/detectability. I.E.E.E.Transactions on Automatic Control, 33, 541-549.

KANTOROVICH, L. V., and AKILOV, G. P., 1964, Functional Analysis in Normed Spaces (Oxford:Pergamon).

MOHILLA, B., and FERENCZ, B., 1982, Chemical Process Dynamics (New York: Elsevier).PAZY, A., 1983, Semi-group ofLinear Operators and Application to Partial Differential Equations

(New York: Springer-Verlag).SCHWARTZ, L., 1979, Methodes Mathematiques pour les Sciences Physiques, second edition

(Paris: Hermann), in French.Xu, C. Z., 1989, Quelques resultats concrets sur la commande lineaire par I'approche HOO.

Ph.D. thesis, Universite Claude-Bernard, Lyon, France (in French).ZAMES, G., and FRANCIS, B. A., 1983, Feedback, minimax sensitivity and optimal robustness.

I.E.E.E. Transactions on Automatic Control, 28,585-600.

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H ∞ -control of a distributed parameter system with non-minimum phase