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On the optimal tuning of a robust controller for parabolic distributed parameter systems
0005-1098/87$3.00+ 0.00 PergamonJournalsLtd. © 1987InternationalFederationof AutomaticControl
Automatica, Vol. 23, No. 6, pp. 719-728, 1987
Printedin GreatBritain.
On the Optimal Tuning of a Robust Controller for Parabolic Distributed Parameter Systems* SEPPO POHJOLAINEN~:
An asymptotically new optimal selection of control parameters for a multivariable I-controller for stable parabolic distributed parameter systems, derived using slow and fast subsystem decomposition, provides improved performance over previous approaches. Key Words--Control theoi'y; distributed parameter systems; linear optimal regulator; multivariable systems; PID-control; robustness; state-space methods.
behind the original selection kI=(CA-1B) * (Davison, 1976), was to guarantee stability of the closed loop system. Later, when the theory was generalized to distributed parameter systems (Pohjolainen, 1980, 1982), a(CA-1BKI)cC + was found to be a sufficient condition for stability, when the scalar gain e is small. Thus there is an important question, how to fix the control matrix/(1, so as to guarantee both closed loop stability and a good dynamical behaviour. A standard technique to consRler the closed loop system is to augment the state of the original system, so that the dynamics of the controller will be included. Then, tuning of the controller has been transformed to be an optimal feedback control problem. There are several good reasons for using a quadratic criterion to define optimality. This functional has both a clear physical interpretation in the time domain, and an appealing number of theoretical results to find the optimal control does already exist, at least for finite-dimensional systems. In general, there are two well-known methods to find an optimal feedback controller, when using a quadratic criterion. If the systems state can be measured, then standard quadratic regulator theory may be used to tune a multivariable PI-controller (Johnson, 1968). Some further work in this direction has been reported, e.g. by Wong and Seborg (1985), to improve the dynamic behaviour of the system under load and set-point changes, and by Mahalanabis and Pal (1985) to assign the closed loop spectrum. Unfortunately, the requirement that the state should be directly accessible is far too strong for distributed parameter systems. If outputs only are available for control, then a standard method is to assume the initial state
Abstract--Optimal tuning of robust multivariable controllers for stable parabolic distributed parameter systems is discussed. In order to evaluate the behaviour of the closed loop system, a quadratic criterion is selected. Then it is proved that the system may be decomposed into slow and fast subsystems, and that the value of the criterion will mainly depend on the slow subsystem. Tuning of the controller, for this subsystem, turns out to be a standard finite-dimensional quadratic regulator problem. Two examples are given to show the improvement due to the new selection of the control parameters. 1. INTRODUCTION
THE PURPOSE Of a controller (Davison, 1976) is to guarantee output regulation and closed loop stability for linear, stable and unknown systems
Yc(t) = Ax(t) + Bu(t) + w(t) y(t) = Cx(t),
(1.1)
in spite of a class of perturbation signals w(t) and variations in the systems parameters. The structure of the controller depends on the spectrum of the perturbation and reference signals. In the simplest case, which is rather important from the practical point of view, these signals are constant, and the controller is of the following form
u(t) = e. KI
(y(t) - Yref)dt
(1.2)
i.e. a multivariable/-controller. The tuning of the controller (1.2) includes a selection of the control matrix K1. The rationale * Received 1 April 1986; revised 20 March 1987. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor A. K. Mahalanabis under the direction of Editor H. Kwakernaak. * Denotes the right pseudoinverse. :~Tampere University of Technology, Department of Mathematics, P.O. Box 527, SFo33101 Tampere 10, Finland. 719
720
S. POHJOLAINEN
to be a random variable with zero mean and known covariance. As a result, a quadratic trace functional should be minimized (Levine and Athans, 1970; Davison and Ferguson, 1981). This method has also some drawbacks. As far as the author knows, only necessary conditions for optimality are known, see e.g. Hutcheson (1980) and the references therein. In addition, the trace functional is statistical by nature, and so for some initial states the quality of the optimal control may be poor. A new and different solution for the optimal tuning of a multivariable /-controller will be given in this paper. The main idea is to use an ordinary quadratic criterion of the form r 2
operator is an infinitesimal generator of a holomorphic stable semigroup in a Hilbert space. This includes, in particular, finitedimensional systems and parabolic distributed parameter systems.
1.1. Basic notation and C are the fields of real and complex numbers, respectively. E + is the open interval (0, 2) and C ÷ the open positive half plane. R p and C p are p-dimensional Euclidean spaces, and C p×p denotes the class of p × p matrices, with (possibly) complex elements. The domain of an operator A is denoted by D(A), R(A) being the range, and N(A) the nullspace. The resolvent R(A;A) is defined by R()q A) = (A - M) -I. o(A) denotes the spectrum of A and p(A) the resolvent set. By L(X, Y) we denote the space of linear and bounded operators from X to Y.
The augmented closed loop system may be decomposed into slow and fast subsystems, for small positive values of the scalar gain e. As e---~ O, it will be proved that the slow subsystem will dominate the criterion, and at e = O, the initial optimal output feedback control problem turns out to be an ordinary finite-dimensional quadratic regulator problem. The optimal solution may readily be calculated as
2. THE SYSTEM Let X be a complex Hilbert space and U, Y finite-dimensional Euclidean spaces, dim ( U ) = m -> dim (Y) = p. Consider the following system:
K o = I (cA_IB),[(CA_,B)(CA_,B),]_,/2. r
where the system operator A is assumed to be an infinitesimal generator of a stable holomorphic semigroup (Kato, 1976) TA(t), i.e.
(1.4)
A pleasing feature of the solution is that the optimal control matrix may be computed, once the matrix CA-1B is known. Fortunately, this matrix may easily be measured from the open loop step responses. The selections K1 = ap/r[CA-1B] t, where ap is the smallest singular value of the matrix CA-1B, and KI = K°h will then be compared. The result is that the maximal improvement, due to the optimal selection, is large, if the singular values of the matrix CA-1B are widely dispersed. It should be noticed that the solution (1.4) is optimal at e = 0, but because the cost functional is continuous in e, the selection should be good for small positive values of e. Two simulation examples will be presented to strengthen this argument. The first one is a finite-dimensional system, where the singular values of the matrix CA-1B are of very different sizes. The second is more a practical one: the outlet temperature of a fluid, moving in a tube, will be controlled. Finally, it should be stressed that the mathematical analysis in the paper is more general, as compared with the above-mentioned other author's results. The theory applies to general linear systems, where the system
it(t) = Ax(t) + Bu(t) + w
(2.1)
y(t) = Cx(t),
(2.2)
x(O) = x o e X ,
IITA(t)II<--M.e-%
M>-I,
co>0.
The control and measurement operators are linear and bounded, i.e.
BeL(U,X),
CeL(X,Y).
The perturbation w e X is constant. The purpose of a robust/-controller (Davison, 1976; Pohjolainen, 1982)
i?(t) = Cx(t) - Yref
(2.3)
u(t) = eK;q(t),
(2.4)
where Yrefe Y is a given constant reference signal, Kt ~ L(Y, U), ~ e (0, ~), is to guarantee closed loop stability and output regulation, i.e. limy(t)=Yref, for all yrefeY, in spite of the t---.~oo
perturbation w e X, and some system parameter variations. The closed loop system may be written as
sK,]ix [;] = [C
0
w
JLt/] + [--yree] '
(2.5)
in the extended state-space ) ? = X ~ Y. The space ){ is a Hilbert space with the inner product
(x, y) = (x, Y)x + (~, ~)r,
(2.6)
721
Optimal tuning of a robust controller where X=
K~ • K, which satisfies
ix]
,
y=
Jl(0, K °, Xo)= min lim J~(e, KI, Xo)
.
KICK e--~O
= ((rio - CA-~xo), P(rlo- CA-lxo))r,
The extended system operator
(3.2)
where
= r[(CA-~B)(CA-~B)*] -1/2, is an infinitesimal generator of a holomorphic semigroup, and hence the extended system is well-posed, and stability of the spectrum a ( ~ ( e ) ) implies stability of the system. Because stability of the system (2.5) implies regulation, in spite of the perturbation w and some system parameter variations (Pohjolainen, 1982), we shall consider only stability, in the following. A necessary and sufficient condition for the existence of a stabilizing controller was found to be (Davison, 1976; Pohjolainen, 1982) rank [CA -~B] = p;
and o
1
K, = r (CA-'B)*[(CA-1B)(CA-~B)*I-V2" (3.4) Because the criterion J~(e, Kz, Xo) depends continuously on e, as will be proved later, the original cost functional (2.11) may, for small values of e, be written as
J(e, K,, Xo) = ~ Ja(O, 1"(1,Xo).
(2.8) Hence the selection (3.4) should be good for small values of e, too.
further, if the control matrix KI satisfies
a(CA-1BK,) c C +,
(2.9)
then the system will be stable, for small positive values of e (Pohjolainen, 1982). In order to choose a good control matrix KI, which satisfies (2.9), we shall consider the free system (2.5) i = M(e)x,
x(0) = Xo• A',
(2.10)
under the quadratic criterion
Proof. The spectrum of the operator M(0), a(M(O)) = a(A)t.J {0} is separated in two parts so that the spectrum decomposition (Kato, 1976) is valid. Let F be a circle, centered at the origin, enclosing the origin, and excluding a ( A ) c C in its exterior. Then the spectrum decomposition holds for the perturbed operator M(e), if e is so small that E
- . sup IIR(Z; ~t(0))ll IIBII ~ k < 1,
J(e, Kz, Xo) =
r
(x(t), Qx(t))
r2
Q->0,
Z~F
(3.5)
where IIKII = 1/r, and
]
+-~(u(t), u(t))v dt (2.11) where Q • L(X'), following form
(3.3)
K = [0, K,],
B = [B],
(3.6)
and it is of the and R(A; M(0)) = (M(0) - / ~ . i ) - 1 is the resolvent. The projector P(e) = ~ /
r e(O,~),
and e is small. The control u(t) satisfies equation (2.4). 3. THE OPTIMAL SOLUTION
The next theorem gives the main result of the paper. The structure of the cost functional will be investigated as e ~ 0, and an optimal solution will be found at e = 0.
Theorem 3.1. Let rank [CA-1B] =p, and let
R(~.;M(e))d~.
(3.7)
decomposes the space X = X + ( e ) ~ X - ( e ) according to the decomposition of the spectrum. In particular, the space X+(e) is finite-dimensional, and contains the modes associated with the eigenvalues near the origin. The projector (3.7) decomposes the semigroup generated by M(e) into two parts
T~(~)(t) = Ta+(o(t)P(e ) + T~-(,)(t)(I - P(e)),
(3.1)
where M+(e), M-(e) are the parts of M(e) on X+(e), X-(e), respectively (Curtain and Prit-
Define a modified cost functional Jl(e, KI, Xo) = el(e, Kt, Xo). Then there is a unique optimal
chard, 1978). Because u = el(x, the cost functional (2.11)
[~ = {K, • L(Y, U) [ a(CA-1BKI) = C+}.
722
S. POHJOLAINEN
may be written as
fo
J(e, K,, Xo) =
(T~c(E)(t)Xo,
[ Q + r2K*K]T ac,)(t)xo) dt.
(3.8)
The integral is well-defined for small values of e, because T~(,)(t)Xo is continuous, and by assumption, stable. Substituting the decomposition of the semigroup into (3.8) results in
fo
J(e, K,. ~o) =
separated in two parts. Let I" c C- be a closed, simple, rectifiable curve enclosing o(-CA-1BKt) in its interior. Then, for sufficiently small values of e, the spectrum of the operator B(e) is likewise separated, and the projection
P(e) = ~ i £ R(fl; B(e)) dfl may be defined. Further,
-lf R(fl. e; eB(e))d(efl)
P(e)=~izi
(r~,+.)(t)P(e)~,
[Q + r~g*glr~,+.)(t)P(e)~) dt (3.9-a)
+
fo (Ta+(~)(t)P(e)xo,[Q+r2K*Kl
=~z/ where
x T~,-¢~)(t)(I - P(e))Xo) dt (3.9-b)
F~ = {), = eft I fle F} c C-.
Because
M(e)P(e) = eB(e) = M(e) on X+(e), we have
-lf,
/5(e) = ~z/
+ f f (T~-(~)(t)(l - P(e))Xo, [Q + ~2K*K 1
R()~; eB(e))d),,
R(;t; M(e)P(e))P(e) d;t e
x T~,+¢~)(t)P(e)Xo) dt (3.9-c) +
fo (T~,-(~)(t)or
- P(e))Xo. [Q + r~K*KI
=2-zli£ R(,~;M(e))d,~=P(e).
x T~,-.)(t)(I - P(e))Xo) dt. (3.9-d) In Appendix A, it is proved that
IlT~-¢~)(t)(I- P(e))ll -
(3.10)
M ~ - 1 , W l > 0 where 3/1 and w~ are independent of e. Thus the integral (3.9-d) remains bounded as e ~ 0. The operator M+(e) may be written as M+(e)P(e) = M(e)P(e)= ~
Jr ZR(Z; M(e))dA
i=1
= Bo + eB(e),
(3.11)
T~(~)(t)P(e) = ~z/
e~tR(Z; M(e))P(e) dit e
2~ri
£e~¢'R(fl; B(e)) dfl. (3.14)
Hence
• sup IIR(/~; B ( e ) ) l t , /3~r
where
-lf B,=~izi
The last equality follows from the fact that ['~ encloses all the eigenvalues of M+(e). Proceeding the same manner, we obtain
IIT~(~)(t)P(e)II -< earn" d(F) 2z
= ~ eiB, = Bo + e ~ #-~Bi, i=0
_
)~R().; M(0))
× [-BKR().; M(0))]' dZ e L(X).
where d(I~) is the length of 1", and
The series is uniformly convergent for e satisfying (3.5). The first two components may directly be computed:
fi* = sup Re fl~ C-,
-1£
Bo = ~z/
(3.13)
and e is so small that e .su_p liB(e) - B~II IIe(fl; B011-
~,R(~,;M(0)) d~, = 0,
#eF
Finally, we have, since B, = ~z/
~.R().; M(0))BKR().; M(0)) dZ 0
T~..)(t)P(e) = T~,+.)(t)P(e). IIT~,+(.)(t)P(e)ll = IIT~,¢~)(t)P(e)ll
0
=[CA-1BK, CA -~ - C A - BKt]"
(3.12)
Because K~ e K, o(-CA-~BK~) c C - , the spectrum of B~, o ( B 1 ) = a(-CA-~BKz) U {0} is
- ME" e -~'`.
(3.15)
Using the estimates (3.15) and (3.10), it follows that the components (3.9-b) and (3.9-c) of the
Optimal tuning of a robust controller cost functional are bounded uniformly in e, for small values of e. Next, let us consider the first component (3.9-a). From equation (3.14), we have
723
where t3 is the unique positive definite solution of the associated algebraic Riccati equation 1
O. # + P . O - # P(CA-1B)
T ~(,)(t)P(e) = Tn(~)(et)P(e), × (CA-1B)*P +
and thus
It is easy to see that the solution is
fo(T~(~)(t)P(e)xo, [Q + r2K*K]
P = r[(CA-1B)(CA-1B)*] -1/2,
x r~(o(t)P(e)Xo) dt 1
I = O. (3.21)
(rn(~)(r)P(e)Xo, [Q + r2K*KI
(3.22)
which gives (3.4), too. If we note that
E
x rmo(r)e(e)xo)dr,
(3.16)
where r = et. Because Tm~)(r)P(e)--~ Tmo)(~)P(0 ) uniformly in 1: on compact intervals, as e - * 0 , and [[Tmo(r)P(e)ll-
0"(CA-1BK°)= 0"(~ [(CA-1B)(CA-1B)*]u2) c C +, by (2,8), K~ze / ( , and the proof is complete.
Note 3.1. Let the singular value decomposition for CA-1B be CA-1B = UAV*,
e---~O
=
[Q +
(Tmo)(~')P(0)~ ,
r2K*K]TB(o)()P(O)xo) dr.
(3.17)
where V e C re×m, U e Cp×p, U and V are unitary, and A e R p×m is of the following form
By noting that B(0)= B1, a direct computation
A=
shows Tn(o)(r)P(0)Xo
r
o
0
17ol,
= L - - e - c A - BK'TCA--1 e-CA-1BKtT j k 1'/oJ
where the expressions (3.12) and (A7), from Appendix A, have been used. Taking into account equations (2.12) and (3.6), we obtain •/1(0, Kz, Xo)=
(3.23)
I
0
...... il
0"2 . . . . . .
0
''"
0"p
(3.24)
'''
where 01 -> 02 ~> "0"p>"0 are the singular values (Dahlquist and Bjrrck, 1974). The condition 0"p > 0 is equivalent to rank [CA-1B] =p. It easily follows that the optimal control matrix may be written as "
"
KO =-1 VAT(AAT)_I/2u, '
(e-CA-'nr'*~(0),
(3.25)
r
and
[I + r2KTKde-CA-'BK'Oy d r =
1 IIK°II = - ,
[(~(r), ~(r))v + r2(v(l:), v(r))v] dl:, (3.18)
where ~(0)
=
-CA-1xo +r/o and
r
since V, U* are unitary. The optimum cost (3.22) has a simple form
,
~(r) = 0. ~(r) - CA-1Bv(r)
(3.19)
v(r) = KI~(r),
(3.20)
~(0) 6 Y.
D = r . U(AAT)-I/2u*
(3.26)
and
IIPll = - -r,
Thus, finding an optimal feedback matrix KI, for the criterion Jl(0,/(1, Xo), turns out to be a standard finite-dimensional quadratic optimum control problem! This problem has a unique solution since the system (3.19) is controllable, because rank [CA-1B] = p , and (I, 0) is observable. The unique optimal feedback matrix is
where the spectral norm is used. It is even possible to estimate the improvement due to the optimal selection, compared with Kl=ap/r[CA-1B] *. Assume that the control matrix is given as
K ° = ~ (CA-1B)*P,
KI = 1 VAk U*,
o,,
r kUTO 23:6-C
(3.27)
724
S. POHJOLAINEN
ks
0
""
0
The corresponding formulas for proportional deterioration are given as
0
k2
"'"
0
0 -< II(e(g,) - P(K°))P(K°)-lll
A k E ~mxp,
where
and
B
= Ak=
0
......
1
'
If
( 1 -- ki) 2
max i=1.....p- l
2k i
(3.32)
Kj = 0-p [CA-1B] *, r
0
......
0
then the bound is
where kt -< k 2 ~ " 1. The selection kp, = 1 has been used to scale matrices so that IIKill=l/r. Both of the controllers are of the form (3.27). If k~ = 1, •
i=l,...,p,
• ~
thenKz=K~t;ifki=0-p.a71,
e(1/r)CA-'BKI • ~ Ue(l/r)AAkrU*"
The quadratic cost matrix (3.18) is given by
f0"
(3.33)
i=
1. . . . . p, then KI = op/r[CA-1B] *. Because CA-1BKI = ( 1 / r ) U A A k U * , and U is unitary, we have
P(K1) =
__ (1- °4Ol,
Ue-O/r)AA*'( I + ATAk) × e--(1/r)AAk*u * dr:
= r U ( A A k ) _ I ( I + ATAk)U,, (3.28) 2 where the fact that ATAk and AAk are diagonal in R p×p has been used. A straightforward calculation results in
4. E X A M P L E S
In this section two examples will be presented. The first one is a finite-dimensional system, where the singular values are widely dispersed, so that the quadratic cost may be considerably reduced. The second example is a more practical one. The purpose is to control the outlet temperature of a fluid, which is moving in a tube. This process can be modelled by a parabolic partial differential equation. Example 4.1. Consider the system 2 = A x + Bu,
(4.1)
y = Cx,
where
P ( K , ) - P ( K °)
= 2 U[ (AAk)-I(I +
A~Ak)-
2(AAT)-I/ZlU*,
A= I-1 0 0
- 22 1 i l , 0 -
(3.29)
B= Ii
1
and thus the following estimate is obtained
il
(4.2)
•
A simple calculation shows that
r (1 - - ki) 2 2 i=1 ..... p oiki
0 = - min - - - <
l I P ( K , ) - P(K°)II
r (1 - - ki) 2 -<- max 2 i=1 ..... p - 1 o i k i
CA-XB =
(3.30)
The maximal improvement is large, if the singular values 0-i and ki are small. In particular, let 1(1 = % [ C A - X B ] *. r
Then <-f--r [ 1 - ( % ] 1 2 , IlP(gz) - P(/Q~) I1- 2 o p \aX/.]
(3.31)
and much better time responses may be obtained provided that op is small, and 01 is large. It should be noticed that the estimates (3.30) and (3.31) only give bounds for maximum improvement. Because the minimal bound in (3.30) is zero, for some initial states, the optimal selection would not help matters at all.
E-10.5
-11.0 -,] "
The singular values of this matrix are 01 = 16.77, Oz = 0.15. The optimal control matrix (3.4), for r = 1, is given as K~/= [-0.9457 k 0.3251
-0.3251] -0.9457_1"
(4.4)
Another selection for the control matrix could be 1(1 = ap[CA-1B]-I = [-0.6559 I_ 0.6261
0.2982] (4.5) -0.29821"
In this case the maximal proportional difference between these two selections may directly be computed from equation (3.33) as 0-1(1 20"2
--
O'2/2
a-7/ " 100% = 5490%,
Optimal tuning of a robust controller o
725
Y2 Y,
°3I 02
0
Heoters 0
0
0
)
0
Of
0
I
o
I
0.5
I
i
1.5
o Uz
o 04 F
-0021
Ui
F[o. 3. Fluid is moving with velocity v t h r o u g h a tube. Using two heaters the m e a s u r e m e n t s at z = 0.7 and z = 1.45 should be regulated to given reference values.
,
,
I O0
0
, ~-~
200
300
!,~-.,.
400
500
Time
FIG. 1. The measurements yl(t), yz(t), and controls u~(t), uz(t) as functions of time corresponding to the optimal selection of the control matrix (4.4). and thus a considerable reduction in the cost may be expected. Numerical simulations, with e = 0.05, x(0) = 0, and Yref= [0.1, 0.2] T, clearly demonstrate the superiority of the selection (4.4). In Fig. 1, regulation happens almost immediately, when compared with the sluggish behaviour of the outputs, seen in Fig. 2. The effect of the smallest singular value is dearly seen in the time responses of the control signals in both of the figures. Example 4.2. Consider the physical system described in Fig. 3. A simplified mathematical model for the system is given as
where cr is the diffusion coefficient, v the velocity, h the heat transfer coefficient, and To the temperature of the neighbourhood. The boundary conditions are
f ax x(t, O)= 0 Let X = L2[0, 1.5]. differential operator
the
spatial
with the domain D ( m ) = { f e S l f , f ' a.c., fleX, f ( 0 ) = 0 , f ' ( 1 . 5 ) = 0 } , as a SturmLiouville operator, is an infinitesimal generator of a holomorphic stable semigroup. The control operator B = [bl, b2], where bl(z)={~
z~[0.5,0.6] z • [0, 0.5) U (0.6, 1.51
z [1.0, 1.1] z e [ 0 , 1)U (1.1, 1.5]. It is easy to see that measurement operator
- h(x(t, z) - To) + bl(z)u,(t) + b2(z)u2(t)
03
Then
d2 d a = cr-~z2- V - ~ z - h
ax(t, z) a2x(t, z) ax(t, z) ~t 2 =or. Oz 2 - v az
(4.6)
x(O, z) = Xo(Z),
(4.7)
~z(t, 1.5 / =0.
Y~
o~
c
L,x,Y,
B eL(U,X).
c
The
rc11 LC2J
is defined as 1 f0.7+an
02
Clf=-~
0 I
0.7-6/2
f ( z l dz
_!
c2f - a ,1.45- /2 I ( z ) dz,
0 o U2 Dr
0 04 F
0
0010
0
2
~
lO0 '
200 ,
~
n
4~
500
Time
FIo. 2. T h e m e a s u r e m e n t s yl(t), y2(t), and controls ul(t), uz(t), as functions of time, corresponding to the traditional selection of the control matrix (4.5).
where 6 > 0 is small. For simulation, the values of the parameters were fixed as a~=0.05, v = l , h = 0 . 3 , and To = 0. The transfer matrix C A - I B was measured from the open loop process using step responses. The method is described in Davison (1976) and Pohjolainen (1982). The result was (see Fig. 4) CA_IB = [-0.093 L-0.075
0.000] -0.087J"
(4.8)
726
S. POHJOLAINEN o Y~
' 15
Y~
Y'
o,o~
}/-<_
f---~--+
0 05
/
\
o o
U2 U,
U2 C 06 ~--
.-~7
-
-
O5
15
0
0
o
I 15
I0
20
Time
L-0.075
0.000 -0.087]"
The singular values of this matrix are: 01 = 0.1351, a2 = 0.0599. The optimal control matrix, when r = 1, is given as KO=_ [
0.9231 -0.3846
0.3846] 0.92311'
(4.9)
and another, traditional selection, would be
KI=o2[CA_lB]_l=[-0.6439 0.5551
0.000 ] -0.6883 " (4.10)
The maximal proportional loss, due to (4.10) is O'1(1_O'212
20"2
~/
• 100% = 35%,
o Y2
YI__ -o--- -
~
~-o--~----o
0 005
0 oU 2
o os f
U,
~__---o---~
off
-
o o F/ O 0
2
L 200
L 300
; 4 O0
500
FIG. 6. The measurements yl(t), y2(t) and controls ul(t ), u2(t) as functions of time, corresponding to the traditional selection of the control matrix (4.10). At the time t = 250, the flow rate was changed from u = 1 to v = 1.2.
+CA-1B[Ul,u2] = CA_IB = [ - 0 . 0 9 3
0 0 I O ~
i I O0
Time
FIG. 4. Measurement of the matrix CA-1B from the open loop system (4.6). Let I11 = [1, 0] T, and u2 = [0, 1]T. Then the corresponding outputs ya(t)---~[0.093, 0.075] T, y2(t)---* [0.000, 0.087] T, as t---, 0o. Hence we have --[Yl(~), Y2(~)] =
oV
~
Time
FIG. 5. The measurements yl(t), y2(t) and controls ul(t ), u2(t) as functions of time, corresponding to the optimal selection of control matrix (4.9). At the time t = 250, the flow rate was changed from v = 1 to v = 1.2.
and so a much smaller improvement than in the previous example may be expected. The reference signal was selected to be Yref= [0.005, 0.010] T, and the value for the scalar tuning parameter e = 0.2. In order to compare the behaviour of the closed loop systems against load perturbations, the value of the velocity was changed at t = 250 from 1 to 1.2. The results of the simulations are given in Figs 5 and 6. Both regulation and disturbance rejection are clearly better when the optimal controller (Fig. 5) is used. The difference, however, is smaller than in Example 4.1, as was expected. 5. C O N C L U S I O N S
An asymptotically optimal selection for a multivariable robust /-controller has been presented. As the value of the scalar gain E approaches zero, the selection of an optimal output feedback matrix turns out to be a standard finite-dimensional quadratic optimization problem. As is well known, this problem has a unique, simple solution, which in our case can be expressed by using only measurable system parameters. For a class of suitable feedback matrices, the maximal improvement caused by the optimal selection may directly be calculated. Because the quadratic cost functional is continuous in e, the asymptotically optimal feedback matrix should be a good selection for small values of s. There is not much to be said about the systems behaviour, if e is not small. It could be possible, in principle, to improve the tuning by increasing the value of e, as far as the quadratic cost is decreasing. Because the cost functional is also a Lyapunov-function, this would guarantee stability as well. However, in some situations it is natural to keep ~ small: the
Optimal tuning of a robust controller control might be expensive, or the state of a nonlinear system should be kept near the operating point in order to be able to use a linear model, etc. It should also be noticed that the optimal regulation problem is, how to drive a system from one steady state, due one selection of reference and perturbation signals, to another steady state, determined by another set of reference and perturbation signals. This problem can be formulated as the one that has been solved above, but then the initial states x(0), T/(0) do depend on e and Kv This dependence has not been taken into account by our method. This deficiency is perhaps not so important, because additional perturbations, variations in the system parameters, may change the initial state, or the initial state may also be due to feedforward controller or linearization. Extensions to more general classes of perturbation and reference signals, and to a more general class of distributed parameter systems, may be developed, using the ideas of Ukai and Iwazumi (1985), and Pohjolainen (1985).
Wong, K. P. and D. E. Seborg (1985). Optimal proportional plus integral control for regulator and tracking problems. Opt. Control Applic. Meth., 6, 335-350. APPENDIX A: PROOF OF THEOREM 3.1 In this Appendix, we shall prove that under the assumptions of Theorem 3.1, for small positive values of e, we have IIT~-~o(t)(l - P(e))ll -< MI" e - ° % (gl) where M1 - 1 and to~ > 0 are independent of e. Proof. Let A • p ( ~ ( e ) ) N p(~t(0)). R(g; d(e)) satisfies
Then, the resolvent
R(A; M(e)) = R(A; M(0)) - eR(;~;.ff(O))BKR(,~; M(e)). (A2) If condition (3.5) is fulfilled, the projector (3.7) may be written as P(e) = P(0) + ePO)(e),
(A3)
where
sg(0))BKR(A;M(e)) d~., and P(e), P°)(e) are holomorphic functions of e. Equations (A2) and (A3) imply that R(g; M ( e ) ) ( l - P(e)) - R()~; M(0))(I - P(0)) = - eR(,~; M(0))[P(1)(e) + BKR(A; M(e))(l - P(e))] = -,R(X;
+ P(0)BKR(;~; M(e))(l - P(e))
Acknowledgement--The author wants to thank Mrs Riitta Saukkonen for her careful typing.
REFERENCES Curtain, R. F. and A. J. Pritchard (1978). Infinite Dimensional Linear Systems Theory. Springer, Berlin. Dahlquist, G. and /~. Bj6rck (1974). Numerical Methods. Prentice-Hall, Englewood Cliffs, New Jersey. Davison, E. J. (1976). Multivariate tuning regulators: the feedforward and robust control of a general servomechanism problem. I E E E Tram. Aut. Control, AC-21, 35-47. Davison, E. J. and I. J. Ferguson (1981). The design of controllers for the multivariable robust servomechanism problem using parameter optimization methods. IEEE Tram. Aut. Control, AC-26, 93-109. Hutcheson, W. J. (1980). The design of feedback controllers for multivariable systems using optimization methods. Ph.D. Thesis, Imperial College, London. Johnson, C. D. (1968). Optimal control of the linear regulator with constant disturbances. I E E E Tram. Aut. Control., AC-13, 416-421. Kato, T. (1976). Perturbation Theory for Linear Operators. Springer, Heidelberg. Levine, W. S. and M. Athans (1970). On the determination of the optimal constant output feedback gains for linear multivariable systems. IEEE Trans. Aut. Control, AC-15, 44-48. Mahalanabis, A. K. and J. K. Pal (1985). Optimal regulator design of linear multivariable systems with prescribed pole locations in the presence of system disturbances, lEE Proc.. 132, 231-236. Pohjolainen, S. (1980). Robust multivariable controller for distributed parameter systems. Ph.D. Thesis, Tampere University of Technology, publication No. 9, Tampere, Finland. Pohjolainen, S. (1982). Robust multivariable PI-controller for infinite dimensional systems. 1EEE Tram. Aut. Control, AC-27, 17-30. Pohjolainen, S. (1985). Robust controller for systems with exponentially stable strongly continuous semigroups. J. Math. Analysis Applic., 111, 622-636. Ukai, H. and T. Iwazumi (1985). General servomechanism problem for distributed parameter systems. Int. J. Control, 42, 1195-1212.
727
+ (I - P(O))P°)(e)
+ (I - P(0))BKR().; sg(e))(I - P(e))] = e_ /( [p(o)p(,)(e) + P(0)BKR()~; M ( e ) ) ( l - P(e))] -
M(0))(I - P(0))[PO)(e)
eR(;[;
+ BKR().; M(e))(l - P(e))].
(A4)
Next, we note that
P(0)p(1)(e) = ~ /
P(0)BKR(A; M(e)) dA
- l f ~l P(0)SKR(~;
= 2~--i
~(e))(I-
P(~)) ,iX
fl (A5)
= -P(O)BK[A-(e)]-I(I - P(e)).
Since, in the first integral, R()~;~l(e))(1 - P(e)) = R(/t; ,~/-(e))(l- P(e)) is holomorphic inside and on F, the value of the second integral is zero, because R(;t;sg(e))P(e)=R(;t;M+(e))P(e) is holomorphic in the exterior of F. Because the integrand is analytic, the curve F may be deformed to be a circle FR, with radius R, and center at the origin. Thus, we have 1
___~_1~ 1 IIR(~;~+(e))llx+
1 IIBII"IIP(0)II" IIP(e)ll • f r ~ -<--" 2:tr R
IdZl
1 1 --<-'r IIBII" IIP(0)II" IIP(e)ll" M(c) .~--}0, as R---~oo. The estimate IIR(Z;~t+(e))llx+(e)<-M(e)/IZl follows easily, since zg+(e) is finite-dimensional and bounded. Now we may continue from equation (A4), by substituting
728
S. POHJOLAINEN
equation (A5) into it:
have IIR(X; ~ t - ( 0 ) ) ( / - P(0))xll 2
R(X; M ( e ) ) ( l - P(e)) - R(X; s~(0))(1 - P(0))
<_ M2(1 ; IICAf'II2) ( ixl I2x + IICA-IxlIZr)
= ~ ( - P ( 0 ) B K [ I M - ( e ) I -1 - R(X; ,ff(e))](l - P(e)))
---~
- eR(X; ~/(0))(1 - P(O))IPO}(e) + BKR(X; ~¢(e))(l - P(t))] - e R ( X ; M ( O ) ) ( I - P(O))[P°)(e)
+ BKR(X; M(e))(1 - P(e))].
max II[M-(e)]-l(1 - P(e))ll-
The last expression is obviously valid for X e p ( M - ( 0 ) ) N p ( M - ( e ) ) . By rearranging the terms, we have
M(e))(l
-
[w----~-ff~os 6 " I l l - P(0)tl + IIP(0)II
IIR(A; ~¢(e))(l - P(e))ll
(A6)
<
' 0,]
,.,
1 (1 + e- ItP
1-k
___ & the following estimate is directly available:
zl~/1=
l[I
--1
X~to
2
2
tlL(x.v))IIXlIx.
(A8)
larg(X-to)l
<~+6,
J
(A10)
sup 114 (1 + e tlPC~)(e)II) cos 6 + 1 ~E[0,~'l 1 -- k cos dt " II1 - P(0)II,
and t ' > 0 is sufficiently small. It is well known (Curtain and Pritehard, 1978) that M - ( e ) is an infinitesimal generator of a strongly continuous semigroup. In addition, equation (A10) imples that
The operator A is, by assumption, an infinitesimal generator of a stable holomorphic semigroup. Thus the resolvent set p ( A ) contains the sector S(w, 6 ) = { X ~ C ,
I I I - P(0)II
where
[ R(X;A)x IIR(X; M - ( 0 ) ) ( I - P(0))xl[ 2 = I_C A - 1 R ( X ; A ) x J [I <-IIR(X;A)It~(x)(1 + ItCA
-< <
Then, from (A6), we obtain
P(e))
= R(X; M - ( O ) ) ( I - P ( 0 ) ) [ I - eP°)(e)]. Because
(A9)
Now, let Z e S(to/2, ~5), and select e to be so small that e.
{I + e[R(X; M-(0))(I - P(0))BK - P(0)BK[M-(e)]-1(I - P(e))]} • R(X;
I1(1 - P(0))xll e,
for Z e S(to, 6). Because [ M - ( e ) ] - a ( l - P(e)) depends continuously on e, and [ M - ( e ) . ] - l ( I - P ( e ) ) ~ [ M - ( 0 ) ] - 1 ( I - P(0)), as e ~ 0, there is an M, and ~ > 0, such that
~l(e))(1 - P(e))}
= e{P(O)BK[~t-(e)]-1R(X;
M2
0<6<~,
}
to
and for X • S(w, 6), the following estimate is valid: M IIR(X; A)ll-< IX - wl" If we substitute this inequality into equation (A8), we finally
IIR(X; M-(e))llx-(.) -< IX - to/21 ' for X ~ S ( w / 2 , 6 ) . Thus the operator M - ( e ) is an infinitesimal generator of a holomorphic stable semigroup, which satisfies IIT ~-(,)(t)ltx-(,) <- MI " e -°'r2. Further,
IIT . - ( ~ ) ( t ) ( l - P(e))xll <- M, . e -~',2 I1(I - P(e))xll - M1 • e -'°'r2 Ilxll
where
M 1= sup M I l I I - P ( e ) I I , t~(0, t") and e " > 0 is sufficiently small. Because both M~ and w~ = w / 2 are independent of e, the proof is complete.
Automatica, Vol. 23, No. 6, pp. 719-728, 1987
Printedin GreatBritain.
On the Optimal Tuning of a Robust Controller for Parabolic Distributed Parameter Systems* SEPPO POHJOLAINEN~:
An asymptotically new optimal selection of control parameters for a multivariable I-controller for stable parabolic distributed parameter systems, derived using slow and fast subsystem decomposition, provides improved performance over previous approaches. Key Words--Control theoi'y; distributed parameter systems; linear optimal regulator; multivariable systems; PID-control; robustness; state-space methods.
behind the original selection kI=(CA-1B) * (Davison, 1976), was to guarantee stability of the closed loop system. Later, when the theory was generalized to distributed parameter systems (Pohjolainen, 1980, 1982), a(CA-1BKI)cC + was found to be a sufficient condition for stability, when the scalar gain e is small. Thus there is an important question, how to fix the control matrix/(1, so as to guarantee both closed loop stability and a good dynamical behaviour. A standard technique to consRler the closed loop system is to augment the state of the original system, so that the dynamics of the controller will be included. Then, tuning of the controller has been transformed to be an optimal feedback control problem. There are several good reasons for using a quadratic criterion to define optimality. This functional has both a clear physical interpretation in the time domain, and an appealing number of theoretical results to find the optimal control does already exist, at least for finite-dimensional systems. In general, there are two well-known methods to find an optimal feedback controller, when using a quadratic criterion. If the systems state can be measured, then standard quadratic regulator theory may be used to tune a multivariable PI-controller (Johnson, 1968). Some further work in this direction has been reported, e.g. by Wong and Seborg (1985), to improve the dynamic behaviour of the system under load and set-point changes, and by Mahalanabis and Pal (1985) to assign the closed loop spectrum. Unfortunately, the requirement that the state should be directly accessible is far too strong for distributed parameter systems. If outputs only are available for control, then a standard method is to assume the initial state
Abstract--Optimal tuning of robust multivariable controllers for stable parabolic distributed parameter systems is discussed. In order to evaluate the behaviour of the closed loop system, a quadratic criterion is selected. Then it is proved that the system may be decomposed into slow and fast subsystems, and that the value of the criterion will mainly depend on the slow subsystem. Tuning of the controller, for this subsystem, turns out to be a standard finite-dimensional quadratic regulator problem. Two examples are given to show the improvement due to the new selection of the control parameters. 1. INTRODUCTION
THE PURPOSE Of a controller (Davison, 1976) is to guarantee output regulation and closed loop stability for linear, stable and unknown systems
Yc(t) = Ax(t) + Bu(t) + w(t) y(t) = Cx(t),
(1.1)
in spite of a class of perturbation signals w(t) and variations in the systems parameters. The structure of the controller depends on the spectrum of the perturbation and reference signals. In the simplest case, which is rather important from the practical point of view, these signals are constant, and the controller is of the following form
u(t) = e. KI
(y(t) - Yref)dt
(1.2)
i.e. a multivariable/-controller. The tuning of the controller (1.2) includes a selection of the control matrix K1. The rationale * Received 1 April 1986; revised 20 March 1987. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor A. K. Mahalanabis under the direction of Editor H. Kwakernaak. * Denotes the right pseudoinverse. :~Tampere University of Technology, Department of Mathematics, P.O. Box 527, SFo33101 Tampere 10, Finland. 719
720
S. POHJOLAINEN
to be a random variable with zero mean and known covariance. As a result, a quadratic trace functional should be minimized (Levine and Athans, 1970; Davison and Ferguson, 1981). This method has also some drawbacks. As far as the author knows, only necessary conditions for optimality are known, see e.g. Hutcheson (1980) and the references therein. In addition, the trace functional is statistical by nature, and so for some initial states the quality of the optimal control may be poor. A new and different solution for the optimal tuning of a multivariable /-controller will be given in this paper. The main idea is to use an ordinary quadratic criterion of the form r 2
operator is an infinitesimal generator of a holomorphic stable semigroup in a Hilbert space. This includes, in particular, finitedimensional systems and parabolic distributed parameter systems.
1.1. Basic notation and C are the fields of real and complex numbers, respectively. E + is the open interval (0, 2) and C ÷ the open positive half plane. R p and C p are p-dimensional Euclidean spaces, and C p×p denotes the class of p × p matrices, with (possibly) complex elements. The domain of an operator A is denoted by D(A), R(A) being the range, and N(A) the nullspace. The resolvent R(A;A) is defined by R()q A) = (A - M) -I. o(A) denotes the spectrum of A and p(A) the resolvent set. By L(X, Y) we denote the space of linear and bounded operators from X to Y.
The augmented closed loop system may be decomposed into slow and fast subsystems, for small positive values of the scalar gain e. As e---~ O, it will be proved that the slow subsystem will dominate the criterion, and at e = O, the initial optimal output feedback control problem turns out to be an ordinary finite-dimensional quadratic regulator problem. The optimal solution may readily be calculated as
2. THE SYSTEM Let X be a complex Hilbert space and U, Y finite-dimensional Euclidean spaces, dim ( U ) = m -> dim (Y) = p. Consider the following system:
K o = I (cA_IB),[(CA_,B)(CA_,B),]_,/2. r
where the system operator A is assumed to be an infinitesimal generator of a stable holomorphic semigroup (Kato, 1976) TA(t), i.e.
(1.4)
A pleasing feature of the solution is that the optimal control matrix may be computed, once the matrix CA-1B is known. Fortunately, this matrix may easily be measured from the open loop step responses. The selections K1 = ap/r[CA-1B] t, where ap is the smallest singular value of the matrix CA-1B, and KI = K°h will then be compared. The result is that the maximal improvement, due to the optimal selection, is large, if the singular values of the matrix CA-1B are widely dispersed. It should be noticed that the solution (1.4) is optimal at e = 0, but because the cost functional is continuous in e, the selection should be good for small positive values of e. Two simulation examples will be presented to strengthen this argument. The first one is a finite-dimensional system, where the singular values of the matrix CA-1B are of very different sizes. The second is more a practical one: the outlet temperature of a fluid, moving in a tube, will be controlled. Finally, it should be stressed that the mathematical analysis in the paper is more general, as compared with the above-mentioned other author's results. The theory applies to general linear systems, where the system
it(t) = Ax(t) + Bu(t) + w
(2.1)
y(t) = Cx(t),
(2.2)
x(O) = x o e X ,
IITA(t)II<--M.e-%
M>-I,
co>0.
The control and measurement operators are linear and bounded, i.e.
BeL(U,X),
CeL(X,Y).
The perturbation w e X is constant. The purpose of a robust/-controller (Davison, 1976; Pohjolainen, 1982)
i?(t) = Cx(t) - Yref
(2.3)
u(t) = eK;q(t),
(2.4)
where Yrefe Y is a given constant reference signal, Kt ~ L(Y, U), ~ e (0, ~), is to guarantee closed loop stability and output regulation, i.e. limy(t)=Yref, for all yrefeY, in spite of the t---.~oo
perturbation w e X, and some system parameter variations. The closed loop system may be written as
sK,]ix [;] = [C
0
w
JLt/] + [--yree] '
(2.5)
in the extended state-space ) ? = X ~ Y. The space ){ is a Hilbert space with the inner product
(x, y) = (x, Y)x + (~, ~)r,
(2.6)
721
Optimal tuning of a robust controller where X=
K~ • K, which satisfies
ix]
,
y=
Jl(0, K °, Xo)= min lim J~(e, KI, Xo)
.
KICK e--~O
= ((rio - CA-~xo), P(rlo- CA-lxo))r,
The extended system operator
(3.2)
where
= r[(CA-~B)(CA-~B)*] -1/2, is an infinitesimal generator of a holomorphic semigroup, and hence the extended system is well-posed, and stability of the spectrum a ( ~ ( e ) ) implies stability of the system. Because stability of the system (2.5) implies regulation, in spite of the perturbation w and some system parameter variations (Pohjolainen, 1982), we shall consider only stability, in the following. A necessary and sufficient condition for the existence of a stabilizing controller was found to be (Davison, 1976; Pohjolainen, 1982) rank [CA -~B] = p;
and o
1
K, = r (CA-'B)*[(CA-1B)(CA-~B)*I-V2" (3.4) Because the criterion J~(e, Kz, Xo) depends continuously on e, as will be proved later, the original cost functional (2.11) may, for small values of e, be written as
J(e, K,, Xo) = ~ Ja(O, 1"(1,Xo).
(2.8) Hence the selection (3.4) should be good for small values of e, too.
further, if the control matrix KI satisfies
a(CA-1BK,) c C +,
(2.9)
then the system will be stable, for small positive values of e (Pohjolainen, 1982). In order to choose a good control matrix KI, which satisfies (2.9), we shall consider the free system (2.5) i = M(e)x,
x(0) = Xo• A',
(2.10)
under the quadratic criterion
Proof. The spectrum of the operator M(0), a(M(O)) = a(A)t.J {0} is separated in two parts so that the spectrum decomposition (Kato, 1976) is valid. Let F be a circle, centered at the origin, enclosing the origin, and excluding a ( A ) c C in its exterior. Then the spectrum decomposition holds for the perturbed operator M(e), if e is so small that E
- . sup IIR(Z; ~t(0))ll IIBII ~ k < 1,
J(e, Kz, Xo) =
r
(x(t), Qx(t))
r2
Q->0,
Z~F
(3.5)
where IIKII = 1/r, and
]
+-~(u(t), u(t))v dt (2.11) where Q • L(X'), following form
(3.3)
K = [0, K,],
B = [B],
(3.6)
and it is of the and R(A; M(0)) = (M(0) - / ~ . i ) - 1 is the resolvent. The projector P(e) = ~ /
r e(O,~),
and e is small. The control u(t) satisfies equation (2.4). 3. THE OPTIMAL SOLUTION
The next theorem gives the main result of the paper. The structure of the cost functional will be investigated as e ~ 0, and an optimal solution will be found at e = 0.
Theorem 3.1. Let rank [CA-1B] =p, and let
R(~.;M(e))d~.
(3.7)
decomposes the space X = X + ( e ) ~ X - ( e ) according to the decomposition of the spectrum. In particular, the space X+(e) is finite-dimensional, and contains the modes associated with the eigenvalues near the origin. The projector (3.7) decomposes the semigroup generated by M(e) into two parts
T~(~)(t) = Ta+(o(t)P(e ) + T~-(,)(t)(I - P(e)),
(3.1)
where M+(e), M-(e) are the parts of M(e) on X+(e), X-(e), respectively (Curtain and Prit-
Define a modified cost functional Jl(e, KI, Xo) = el(e, Kt, Xo). Then there is a unique optimal
chard, 1978). Because u = el(x, the cost functional (2.11)
[~ = {K, • L(Y, U) [ a(CA-1BKI) = C+}.
722
S. POHJOLAINEN
may be written as
fo
J(e, K,, Xo) =
(T~c(E)(t)Xo,
[ Q + r2K*K]T ac,)(t)xo) dt.
(3.8)
The integral is well-defined for small values of e, because T~(,)(t)Xo is continuous, and by assumption, stable. Substituting the decomposition of the semigroup into (3.8) results in
fo
J(e, K,. ~o) =
separated in two parts. Let I" c C- be a closed, simple, rectifiable curve enclosing o(-CA-1BKt) in its interior. Then, for sufficiently small values of e, the spectrum of the operator B(e) is likewise separated, and the projection
P(e) = ~ i £ R(fl; B(e)) dfl may be defined. Further,
-lf R(fl. e; eB(e))d(efl)
P(e)=~izi
(r~,+.)(t)P(e)~,
[Q + r~g*glr~,+.)(t)P(e)~) dt (3.9-a)
+
fo (Ta+(~)(t)P(e)xo,[Q+r2K*Kl
=~z/ where
x T~,-¢~)(t)(I - P(e))Xo) dt (3.9-b)
F~ = {), = eft I fle F} c C-.
Because
M(e)P(e) = eB(e) = M(e) on X+(e), we have
-lf,
/5(e) = ~z/
+ f f (T~-(~)(t)(l - P(e))Xo, [Q + ~2K*K 1
R()~; eB(e))d),,
R(;t; M(e)P(e))P(e) d;t e
x T~,+¢~)(t)P(e)Xo) dt (3.9-c) +
fo (T~,-(~)(t)or
- P(e))Xo. [Q + r~K*KI
=2-zli£ R(,~;M(e))d,~=P(e).
x T~,-.)(t)(I - P(e))Xo) dt. (3.9-d) In Appendix A, it is proved that
IlT~-¢~)(t)(I- P(e))ll -
(3.10)
M ~ - 1 , W l > 0 where 3/1 and w~ are independent of e. Thus the integral (3.9-d) remains bounded as e ~ 0. The operator M+(e) may be written as M+(e)P(e) = M(e)P(e)= ~
Jr ZR(Z; M(e))dA
i=1
= Bo + eB(e),
(3.11)
T~(~)(t)P(e) = ~z/
e~tR(Z; M(e))P(e) dit e
2~ri
£e~¢'R(fl; B(e)) dfl. (3.14)
Hence
• sup IIR(/~; B ( e ) ) l t , /3~r
where
-lf B,=~izi
The last equality follows from the fact that ['~ encloses all the eigenvalues of M+(e). Proceeding the same manner, we obtain
IIT~(~)(t)P(e)II -< earn" d(F) 2z
= ~ eiB, = Bo + e ~ #-~Bi, i=0
_
)~R().; M(0))
× [-BKR().; M(0))]' dZ e L(X).
where d(I~) is the length of 1", and
The series is uniformly convergent for e satisfying (3.5). The first two components may directly be computed:
fi* = sup Re fl~ C-,
-1£
Bo = ~z/
(3.13)
and e is so small that e .su_p liB(e) - B~II IIe(fl; B011-
~,R(~,;M(0)) d~, = 0,
#eF
Finally, we have, since B, = ~z/
~.R().; M(0))BKR().; M(0)) dZ 0
T~..)(t)P(e) = T~,+.)(t)P(e). IIT~,+(.)(t)P(e)ll = IIT~,¢~)(t)P(e)ll
0
=[CA-1BK, CA -~ - C A - BKt]"
(3.12)
Because K~ e K, o(-CA-~BK~) c C - , the spectrum of B~, o ( B 1 ) = a(-CA-~BKz) U {0} is
- ME" e -~'`.
(3.15)
Using the estimates (3.15) and (3.10), it follows that the components (3.9-b) and (3.9-c) of the
Optimal tuning of a robust controller cost functional are bounded uniformly in e, for small values of e. Next, let us consider the first component (3.9-a). From equation (3.14), we have
723
where t3 is the unique positive definite solution of the associated algebraic Riccati equation 1
O. # + P . O - # P(CA-1B)
T ~(,)(t)P(e) = Tn(~)(et)P(e), × (CA-1B)*P +
and thus
It is easy to see that the solution is
fo(T~(~)(t)P(e)xo, [Q + r2K*K]
P = r[(CA-1B)(CA-1B)*] -1/2,
x r~(o(t)P(e)Xo) dt 1
I = O. (3.21)
(rn(~)(r)P(e)Xo, [Q + r2K*KI
(3.22)
which gives (3.4), too. If we note that
E
x rmo(r)e(e)xo)dr,
(3.16)
where r = et. Because Tm~)(r)P(e)--~ Tmo)(~)P(0 ) uniformly in 1: on compact intervals, as e - * 0 , and [[Tmo(r)P(e)ll-
0"(CA-1BK°)= 0"(~ [(CA-1B)(CA-1B)*]u2) c C +, by (2,8), K~ze / ( , and the proof is complete.
Note 3.1. Let the singular value decomposition for CA-1B be CA-1B = UAV*,
e---~O
=
[Q +
(Tmo)(~')P(0)~ ,
r2K*K]TB(o)()P(O)xo) dr.
(3.17)
where V e C re×m, U e Cp×p, U and V are unitary, and A e R p×m is of the following form
By noting that B(0)= B1, a direct computation
A=
shows Tn(o)(r)P(0)Xo
r
o
0
17ol,
= L - - e - c A - BK'TCA--1 e-CA-1BKtT j k 1'/oJ
where the expressions (3.12) and (A7), from Appendix A, have been used. Taking into account equations (2.12) and (3.6), we obtain •/1(0, Kz, Xo)=
(3.23)
I
0
...... il
0"2 . . . . . .
0
''"
0"p
(3.24)
'''
where 01 -> 02 ~> "0"p>"0 are the singular values (Dahlquist and Bjrrck, 1974). The condition 0"p > 0 is equivalent to rank [CA-1B] =p. It easily follows that the optimal control matrix may be written as "
"
KO =-1 VAT(AAT)_I/2u, '
(e-CA-'nr'*~(0),
(3.25)
r
and
[I + r2KTKde-CA-'BK'Oy d r =
1 IIK°II = - ,
[(~(r), ~(r))v + r2(v(l:), v(r))v] dl:, (3.18)
where ~(0)
=
-CA-1xo +r/o and
r
since V, U* are unitary. The optimum cost (3.22) has a simple form
,
~(r) = 0. ~(r) - CA-1Bv(r)
(3.19)
v(r) = KI~(r),
(3.20)
~(0) 6 Y.
D = r . U(AAT)-I/2u*
(3.26)
and
IIPll = - -r,
Thus, finding an optimal feedback matrix KI, for the criterion Jl(0,/(1, Xo), turns out to be a standard finite-dimensional quadratic optimum control problem! This problem has a unique solution since the system (3.19) is controllable, because rank [CA-1B] = p , and (I, 0) is observable. The unique optimal feedback matrix is
where the spectral norm is used. It is even possible to estimate the improvement due to the optimal selection, compared with Kl=ap/r[CA-1B] *. Assume that the control matrix is given as
K ° = ~ (CA-1B)*P,
KI = 1 VAk U*,
o,,
r kUTO 23:6-C
(3.27)
724
S. POHJOLAINEN
ks
0
""
0
The corresponding formulas for proportional deterioration are given as
0
k2
"'"
0
0 -< II(e(g,) - P(K°))P(K°)-lll
A k E ~mxp,
where
and
B
= Ak=
0
......
1
'
If
( 1 -- ki) 2
max i=1.....p- l
2k i
(3.32)
Kj = 0-p [CA-1B] *, r
0
......
0
then the bound is
where kt -< k 2 ~ " 1. The selection kp, = 1 has been used to scale matrices so that IIKill=l/r. Both of the controllers are of the form (3.27). If k~ = 1, •
i=l,...,p,
• ~
thenKz=K~t;ifki=0-p.a71,
e(1/r)CA-'BKI • ~ Ue(l/r)AAkrU*"
The quadratic cost matrix (3.18) is given by
f0"
(3.33)
i=
1. . . . . p, then KI = op/r[CA-1B] *. Because CA-1BKI = ( 1 / r ) U A A k U * , and U is unitary, we have
P(K1) =
__ (1- °4Ol,
Ue-O/r)AA*'( I + ATAk) × e--(1/r)AAk*u * dr:
= r U ( A A k ) _ I ( I + ATAk)U,, (3.28) 2 where the fact that ATAk and AAk are diagonal in R p×p has been used. A straightforward calculation results in
4. E X A M P L E S
In this section two examples will be presented. The first one is a finite-dimensional system, where the singular values are widely dispersed, so that the quadratic cost may be considerably reduced. The second example is a more practical one. The purpose is to control the outlet temperature of a fluid, which is moving in a tube. This process can be modelled by a parabolic partial differential equation. Example 4.1. Consider the system 2 = A x + Bu,
(4.1)
y = Cx,
where
P ( K , ) - P ( K °)
= 2 U[ (AAk)-I(I +
A~Ak)-
2(AAT)-I/ZlU*,
A= I-1 0 0
- 22 1 i l , 0 -
(3.29)
B= Ii
1
and thus the following estimate is obtained
il
(4.2)
•
A simple calculation shows that
r (1 - - ki) 2 2 i=1 ..... p oiki
0 = - min - - - <
l I P ( K , ) - P(K°)II
r (1 - - ki) 2 -<- max 2 i=1 ..... p - 1 o i k i
CA-XB =
(3.30)
The maximal improvement is large, if the singular values 0-i and ki are small. In particular, let 1(1 = % [ C A - X B ] *. r
Then <-f--r [ 1 - ( % ] 1 2 , IlP(gz) - P(/Q~) I1- 2 o p \aX/.]
(3.31)
and much better time responses may be obtained provided that op is small, and 01 is large. It should be noticed that the estimates (3.30) and (3.31) only give bounds for maximum improvement. Because the minimal bound in (3.30) is zero, for some initial states, the optimal selection would not help matters at all.
E-10.5
-11.0 -,] "
The singular values of this matrix are 01 = 16.77, Oz = 0.15. The optimal control matrix (3.4), for r = 1, is given as K~/= [-0.9457 k 0.3251
-0.3251] -0.9457_1"
(4.4)
Another selection for the control matrix could be 1(1 = ap[CA-1B]-I = [-0.6559 I_ 0.6261
0.2982] (4.5) -0.29821"
In this case the maximal proportional difference between these two selections may directly be computed from equation (3.33) as 0-1(1 20"2
--
O'2/2
a-7/ " 100% = 5490%,
Optimal tuning of a robust controller o
725
Y2 Y,
°3I 02
0
Heoters 0
0
0
)
0
Of
0
I
o
I
0.5
I
i
1.5
o Uz
o 04 F
-0021
Ui
F[o. 3. Fluid is moving with velocity v t h r o u g h a tube. Using two heaters the m e a s u r e m e n t s at z = 0.7 and z = 1.45 should be regulated to given reference values.
,
,
I O0
0
, ~-~
200
300
!,~-.,.
400
500
Time
FIG. 1. The measurements yl(t), yz(t), and controls u~(t), uz(t) as functions of time corresponding to the optimal selection of the control matrix (4.4). and thus a considerable reduction in the cost may be expected. Numerical simulations, with e = 0.05, x(0) = 0, and Yref= [0.1, 0.2] T, clearly demonstrate the superiority of the selection (4.4). In Fig. 1, regulation happens almost immediately, when compared with the sluggish behaviour of the outputs, seen in Fig. 2. The effect of the smallest singular value is dearly seen in the time responses of the control signals in both of the figures. Example 4.2. Consider the physical system described in Fig. 3. A simplified mathematical model for the system is given as
where cr is the diffusion coefficient, v the velocity, h the heat transfer coefficient, and To the temperature of the neighbourhood. The boundary conditions are
f ax x(t, O)= 0 Let X = L2[0, 1.5]. differential operator
the
spatial
with the domain D ( m ) = { f e S l f , f ' a.c., fleX, f ( 0 ) = 0 , f ' ( 1 . 5 ) = 0 } , as a SturmLiouville operator, is an infinitesimal generator of a holomorphic stable semigroup. The control operator B = [bl, b2], where bl(z)={~
z~[0.5,0.6] z • [0, 0.5) U (0.6, 1.51
z [1.0, 1.1] z e [ 0 , 1)U (1.1, 1.5]. It is easy to see that measurement operator
- h(x(t, z) - To) + bl(z)u,(t) + b2(z)u2(t)
03
Then
d2 d a = cr-~z2- V - ~ z - h
ax(t, z) a2x(t, z) ax(t, z) ~t 2 =or. Oz 2 - v az
(4.6)
x(O, z) = Xo(Z),
(4.7)
~z(t, 1.5 / =0.
Y~
o~
c
L,x,Y,
B eL(U,X).
c
The
rc11 LC2J
is defined as 1 f0.7+an
02
Clf=-~
0 I
0.7-6/2
f ( z l dz
_!
c2f - a ,1.45- /2 I ( z ) dz,
0 o U2 Dr
0 04 F
0
0010
0
2
~
lO0 '
200 ,
~
n
4~
500
Time
FIo. 2. T h e m e a s u r e m e n t s yl(t), y2(t), and controls ul(t), uz(t), as functions of time, corresponding to the traditional selection of the control matrix (4.5).
where 6 > 0 is small. For simulation, the values of the parameters were fixed as a~=0.05, v = l , h = 0 . 3 , and To = 0. The transfer matrix C A - I B was measured from the open loop process using step responses. The method is described in Davison (1976) and Pohjolainen (1982). The result was (see Fig. 4) CA_IB = [-0.093 L-0.075
0.000] -0.087J"
(4.8)
726
S. POHJOLAINEN o Y~
' 15
Y~
Y'
o,o~
}/-<_
f---~--+
0 05
/
\
o o
U2 U,
U2 C 06 ~--
.-~7
-
-
O5
15
0
0
o
I 15
I0
20
Time
L-0.075
0.000 -0.087]"
The singular values of this matrix are: 01 = 0.1351, a2 = 0.0599. The optimal control matrix, when r = 1, is given as KO=_ [
0.9231 -0.3846
0.3846] 0.92311'
(4.9)
and another, traditional selection, would be
KI=o2[CA_lB]_l=[-0.6439 0.5551
0.000 ] -0.6883 " (4.10)
The maximal proportional loss, due to (4.10) is O'1(1_O'212
20"2
~/
• 100% = 35%,
o Y2
YI__ -o--- -
~
~-o--~----o
0 005
0 oU 2
o os f
U,
~__---o---~
off
-
o o F/ O 0
2
L 200
L 300
; 4 O0
500
FIG. 6. The measurements yl(t), y2(t) and controls ul(t ), u2(t) as functions of time, corresponding to the traditional selection of the control matrix (4.10). At the time t = 250, the flow rate was changed from u = 1 to v = 1.2.
+CA-1B[Ul,u2] = CA_IB = [ - 0 . 0 9 3
0 0 I O ~
i I O0
Time
FIG. 4. Measurement of the matrix CA-1B from the open loop system (4.6). Let I11 = [1, 0] T, and u2 = [0, 1]T. Then the corresponding outputs ya(t)---~[0.093, 0.075] T, y2(t)---* [0.000, 0.087] T, as t---, 0o. Hence we have --[Yl(~), Y2(~)] =
oV
~
Time
FIG. 5. The measurements yl(t), y2(t) and controls ul(t ), u2(t) as functions of time, corresponding to the optimal selection of control matrix (4.9). At the time t = 250, the flow rate was changed from v = 1 to v = 1.2.
and so a much smaller improvement than in the previous example may be expected. The reference signal was selected to be Yref= [0.005, 0.010] T, and the value for the scalar tuning parameter e = 0.2. In order to compare the behaviour of the closed loop systems against load perturbations, the value of the velocity was changed at t = 250 from 1 to 1.2. The results of the simulations are given in Figs 5 and 6. Both regulation and disturbance rejection are clearly better when the optimal controller (Fig. 5) is used. The difference, however, is smaller than in Example 4.1, as was expected. 5. C O N C L U S I O N S
An asymptotically optimal selection for a multivariable robust /-controller has been presented. As the value of the scalar gain E approaches zero, the selection of an optimal output feedback matrix turns out to be a standard finite-dimensional quadratic optimization problem. As is well known, this problem has a unique, simple solution, which in our case can be expressed by using only measurable system parameters. For a class of suitable feedback matrices, the maximal improvement caused by the optimal selection may directly be calculated. Because the quadratic cost functional is continuous in e, the asymptotically optimal feedback matrix should be a good selection for small values of s. There is not much to be said about the systems behaviour, if e is not small. It could be possible, in principle, to improve the tuning by increasing the value of e, as far as the quadratic cost is decreasing. Because the cost functional is also a Lyapunov-function, this would guarantee stability as well. However, in some situations it is natural to keep ~ small: the
Optimal tuning of a robust controller control might be expensive, or the state of a nonlinear system should be kept near the operating point in order to be able to use a linear model, etc. It should also be noticed that the optimal regulation problem is, how to drive a system from one steady state, due one selection of reference and perturbation signals, to another steady state, determined by another set of reference and perturbation signals. This problem can be formulated as the one that has been solved above, but then the initial states x(0), T/(0) do depend on e and Kv This dependence has not been taken into account by our method. This deficiency is perhaps not so important, because additional perturbations, variations in the system parameters, may change the initial state, or the initial state may also be due to feedforward controller or linearization. Extensions to more general classes of perturbation and reference signals, and to a more general class of distributed parameter systems, may be developed, using the ideas of Ukai and Iwazumi (1985), and Pohjolainen (1985).
Wong, K. P. and D. E. Seborg (1985). Optimal proportional plus integral control for regulator and tracking problems. Opt. Control Applic. Meth., 6, 335-350. APPENDIX A: PROOF OF THEOREM 3.1 In this Appendix, we shall prove that under the assumptions of Theorem 3.1, for small positive values of e, we have IIT~-~o(t)(l - P(e))ll -< MI" e - ° % (gl) where M1 - 1 and to~ > 0 are independent of e. Proof. Let A • p ( ~ ( e ) ) N p(~t(0)). R(g; d(e)) satisfies
Then, the resolvent
R(A; M(e)) = R(A; M(0)) - eR(;~;.ff(O))BKR(,~; M(e)). (A2) If condition (3.5) is fulfilled, the projector (3.7) may be written as P(e) = P(0) + ePO)(e),
(A3)
where
sg(0))BKR(A;M(e)) d~., and P(e), P°)(e) are holomorphic functions of e. Equations (A2) and (A3) imply that R(g; M ( e ) ) ( l - P(e)) - R()~; M(0))(I - P(0)) = - eR(,~; M(0))[P(1)(e) + BKR(A; M(e))(l - P(e))] = -,R(X;
+ P(0)BKR(;~; M(e))(l - P(e))
Acknowledgement--The author wants to thank Mrs Riitta Saukkonen for her careful typing.
REFERENCES Curtain, R. F. and A. J. Pritchard (1978). Infinite Dimensional Linear Systems Theory. Springer, Berlin. Dahlquist, G. and /~. Bj6rck (1974). Numerical Methods. Prentice-Hall, Englewood Cliffs, New Jersey. Davison, E. J. (1976). Multivariate tuning regulators: the feedforward and robust control of a general servomechanism problem. I E E E Tram. Aut. Control, AC-21, 35-47. Davison, E. J. and I. J. Ferguson (1981). The design of controllers for the multivariable robust servomechanism problem using parameter optimization methods. IEEE Tram. Aut. Control, AC-26, 93-109. Hutcheson, W. J. (1980). The design of feedback controllers for multivariable systems using optimization methods. Ph.D. Thesis, Imperial College, London. Johnson, C. D. (1968). Optimal control of the linear regulator with constant disturbances. I E E E Tram. Aut. Control., AC-13, 416-421. Kato, T. (1976). Perturbation Theory for Linear Operators. Springer, Heidelberg. Levine, W. S. and M. Athans (1970). On the determination of the optimal constant output feedback gains for linear multivariable systems. IEEE Trans. Aut. Control, AC-15, 44-48. Mahalanabis, A. K. and J. K. Pal (1985). Optimal regulator design of linear multivariable systems with prescribed pole locations in the presence of system disturbances, lEE Proc.. 132, 231-236. Pohjolainen, S. (1980). Robust multivariable controller for distributed parameter systems. Ph.D. Thesis, Tampere University of Technology, publication No. 9, Tampere, Finland. Pohjolainen, S. (1982). Robust multivariable PI-controller for infinite dimensional systems. 1EEE Tram. Aut. Control, AC-27, 17-30. Pohjolainen, S. (1985). Robust controller for systems with exponentially stable strongly continuous semigroups. J. Math. Analysis Applic., 111, 622-636. Ukai, H. and T. Iwazumi (1985). General servomechanism problem for distributed parameter systems. Int. J. Control, 42, 1195-1212.
727
+ (I - P(O))P°)(e)
+ (I - P(0))BKR().; sg(e))(I - P(e))] = e_ /( [p(o)p(,)(e) + P(0)BKR()~; M ( e ) ) ( l - P(e))] -
M(0))(I - P(0))[PO)(e)
eR(;[;
+ BKR().; M(e))(l - P(e))].
(A4)
Next, we note that
P(0)p(1)(e) = ~ /
P(0)BKR(A; M(e)) dA
- l f ~l P(0)SKR(~;
= 2~--i
~(e))(I-
P(~)) ,iX
fl (A5)
= -P(O)BK[A-(e)]-I(I - P(e)).
Since, in the first integral, R()~;~l(e))(1 - P(e)) = R(/t; ,~/-(e))(l- P(e)) is holomorphic inside and on F, the value of the second integral is zero, because R(;t;sg(e))P(e)=R(;t;M+(e))P(e) is holomorphic in the exterior of F. Because the integrand is analytic, the curve F may be deformed to be a circle FR, with radius R, and center at the origin. Thus, we have 1
___~_1~ 1 IIR(~;~+(e))llx+
1 IIBII"IIP(0)II" IIP(e)ll • f r ~ -<--" 2:tr R
IdZl
1 1 --<-'r IIBII" IIP(0)II" IIP(e)ll" M(c) .~--}0, as R---~oo. The estimate IIR(Z;~t+(e))llx+(e)<-M(e)/IZl follows easily, since zg+(e) is finite-dimensional and bounded. Now we may continue from equation (A4), by substituting
728
S. POHJOLAINEN
equation (A5) into it:
have IIR(X; ~ t - ( 0 ) ) ( / - P(0))xll 2
R(X; M ( e ) ) ( l - P(e)) - R(X; s~(0))(1 - P(0))
<_ M2(1 ; IICAf'II2) ( ixl I2x + IICA-IxlIZr)
= ~ ( - P ( 0 ) B K [ I M - ( e ) I -1 - R(X; ,ff(e))](l - P(e)))
---~
- eR(X; ~/(0))(1 - P(O))IPO}(e) + BKR(X; ~¢(e))(l - P(t))] - e R ( X ; M ( O ) ) ( I - P(O))[P°)(e)
+ BKR(X; M(e))(1 - P(e))].
max II[M-(e)]-l(1 - P(e))ll-
The last expression is obviously valid for X e p ( M - ( 0 ) ) N p ( M - ( e ) ) . By rearranging the terms, we have
M(e))(l
-
[w----~-ff~os 6 " I l l - P(0)tl + IIP(0)II
IIR(A; ~¢(e))(l - P(e))ll
(A6)
<
' 0,]
,.,
1 (1 + e- ItP
1-k
___ & the following estimate is directly available:
zl~/1=
l[I
--1
X~to
2
2
tlL(x.v))IIXlIx.
(A8)
larg(X-to)l
<~+6,
J
(A10)
sup 114 (1 + e tlPC~)(e)II) cos 6 + 1 ~E[0,~'l 1 -- k cos dt " II1 - P(0)II,
and t ' > 0 is sufficiently small. It is well known (Curtain and Pritehard, 1978) that M - ( e ) is an infinitesimal generator of a strongly continuous semigroup. In addition, equation (A10) imples that
The operator A is, by assumption, an infinitesimal generator of a stable holomorphic semigroup. Thus the resolvent set p ( A ) contains the sector S(w, 6 ) = { X ~ C ,
I I I - P(0)II
where
[ R(X;A)x IIR(X; M - ( 0 ) ) ( I - P(0))xl[ 2 = I_C A - 1 R ( X ; A ) x J [I <-IIR(X;A)It~(x)(1 + ItCA
-< <
Then, from (A6), we obtain
P(e))
= R(X; M - ( O ) ) ( I - P ( 0 ) ) [ I - eP°)(e)]. Because
(A9)
Now, let Z e S(to/2, ~5), and select e to be so small that e.
{I + e[R(X; M-(0))(I - P(0))BK - P(0)BK[M-(e)]-1(I - P(e))]} • R(X;
I1(1 - P(0))xll e,
for Z e S(to, 6). Because [ M - ( e ) ] - a ( l - P(e)) depends continuously on e, and [ M - ( e ) . ] - l ( I - P ( e ) ) ~ [ M - ( 0 ) ] - 1 ( I - P(0)), as e ~ 0, there is an M, and ~ > 0, such that
~l(e))(1 - P(e))}
= e{P(O)BK[~t-(e)]-1R(X;
M2
0<6<~,
}
to
and for X • S(w, 6), the following estimate is valid: M IIR(X; A)ll-< IX - wl" If we substitute this inequality into equation (A8), we finally
IIR(X; M-(e))llx-(.) -< IX - to/21 ' for X ~ S ( w / 2 , 6 ) . Thus the operator M - ( e ) is an infinitesimal generator of a holomorphic stable semigroup, which satisfies IIT ~-(,)(t)ltx-(,) <- MI " e -°'r2. Further,
IIT . - ( ~ ) ( t ) ( l - P(e))xll <- M, . e -~',2 I1(I - P(e))xll - M1 • e -'°'r2 Ilxll
where
M 1= sup M I l I I - P ( e ) I I , t~(0, t") and e " > 0 is sufficiently small. Because both M~ and w~ = w / 2 are independent of e, the proof is complete.