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Robust stabilization of distributed systems
Automatica, Vol. 22, No. 1, pp. 77-84, 1986 Printed in Great Britain.
0005 1098/86 $3.00 + 0.00 Pergamon Press Ltd. © 1986 International Federation of Automatic Control
Robust Stabilization of Distributed Systems* PRAMOD P. K H A R G O N E K A R t and K. POOLLA:~
Interpolation theory solves the robust stabilization problem Jbr families of distributed plants given by gain variations and multiplicative uncertainty. Key Words--Robust control; time lag systems; distributed parameter systems; control system synthesis; frequency domain.
solution is known. However, for certain types of families of plants, it is possible to give a concrete constructive solution to this problem. Recently, robust stabilization problems for certain families of finite-dimensional LTI plants have been solved by Kimura (1984), Tannenbaum (1980, 1981, 1982), Khargonekar and Tannenbaum (1985), Khargonekar, Poolla and Tannenbaum (1985), and the references cited there. It is not difficult to see that the techniques of these authors generalize easily to distributed plants which have only finitely many poles and zeros in the closed right half plane including infinity. In this paper, we extend some of these results to a fairly general class of distributed systems defined by Callier and Desoer (1978). These systems contain transmission delays, have possibly infinitely many zeros in the closed right half plane, but have only finitely many poles in the closed right half plane. As expected, our procedures are not easy to carry out in this generality, but jbr the class of systems having a pure transmission delay and.finitely many poles and zeros in the closed right half plane, our procedures are completely constructive and yield concrete bounds j or robust stabilizability. We refer the interested reader to Kimura (1984), Khargonekar and Tannenbaum (1985), Vidyasagar (1985), and the references therein for further literature on the robust stabilization problems. Also, the recent papers by Ackermann (1980), Ghosh and Byrnes (1983), Saeks and Murrey (1982), and Vidyasagar and Viswanadham (1982), on the simultaneous stabilization problem and parameter space design represent important contributions to the robust stabilization problem.
Almtraet--This paper is devoted to the problem of robust stabilization for a class of distributed plants. We consider the gain margin optimization problem and the robust stabilization problem for multiplicative perturbations. Using techniques from interpolation theory and complex variables, we obtain explicit necessary and sufficient conditions for robust stabilizability. A few examples are included to illustrate the results. NOTATION C =/complex numbers} H = open right half plane = {s in C: Re(s) > 0} H = closed right half plane = {s in C: Re(s) >I 0} O, = open disc of radius r = {s in C: Jsj < r} Dr = closed disc of radius r = {s in C: tsJ ~< r} 1. INTRODUCTION
CONSIDER the feedback system shown in Fig. 1. In general terms, the robust stabilization problem may be formulated as follows: let Pk(S) be a family of linear time-invariant SISO plants, where the index k takes values in some set K. Then the robust stabilization problem is to find a (realizable) controller C(s) such that the above closed loop system is internally asymptotically stable for each k in K. If this problem is solvable, then C(s) is a robust controller in the sense that as long as the plant belongs to the set {Pk(S): k in K}, the closed loop system is internally asymptotically stable. Roughly speaking, realizable means that C(s) represents a casual dynamic system, and internally stable means that there are no unstable pole-zero cancellations between C(s) and Pk(S). These notions are defined precisely in section 2. The robust stabilization problem stated above, in its complete generality, is very hard and no general * Received 12 November 1984; revised 6 May 1985. 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 J. Ackermann under the direction of Editor H. Kwakernaak. This work was supported in part by the National Science Foundation under grant No. ECS-8400832. t Department of Electrical Engineering, University of Minnesota, Minneapolis, MN 55455, U.S.A. Department of Electrical Engineering, University of Illinois, Urbana, IL 61801, U.S.A. Previously with the Department of Electrical Engineering, McGill University, Montreal, Canada.
FEEDBACK
SYSTEM
FIG. 1. Feedback system. 77
78
P . P . KHARGONEKAR and K. POOLLA
As in Kimura (1984), Tannenbaum (1980, 1981, 1982), Zames ( 1981 ), it turns out that interpolation theory plays an important role in robust stabilization problems. We end this section with a result from classical interpolation theory (see Helton, 1982: Khargonekar and Tannenbaum, 1985, for details). Let w~, w2. . . . . w~ be a set of distinct points in the open right half plane H and wt + ~, w~+ 2. . . . . Wq be a (possibly repeated) set of points on the boundary of H, i.e., on the imaginary axis including the point at infinity. Suppose we are given a set of points b~, b2 . . . . . hi, bt + ~..... bq in C. Define l x l matrices
A =
i,j = 1,2 ..... I
D~bi ] i , j = B = LW~ + ~'J_]
l,2 . . . . . I.
Let ~-max denote the largest eigenvalue of A - ~B. (It turns out that 2 .... >~ 0.) Set c~~" = m a x ( v /'~....... Jbt+~J. . . . . Ibql). max-
(1)
If the right-hand side turns out to be zero, then we define C~m~= 7;.) We now state a result from the classical interpolation theory which will be useful in the next sections. Here/7: = Is in C: Re(s) >~ 0~ and H : H u t ~j ~. Theorem 1. Let wi in H and bi in C, i = 1, 2 . . . . . q be as above. Then there exists aJunetion f analytic in H, continuous at infinity such that f:Iq --* D~:f(wi) = bi,
i = 1,2 . . . . . q
if and only i[ r
1 <
0(max.
A proof of this result may be found in (Khargonekar and T a n n e n b a u m (1985, Theorem 1.5). This shows that the image of any analytic function which sends w~ to b~ cannot lie in a disc of radius smaller than ~m2~; and given any disc of radius greater than C~m,~,we can find an interpolating function f whose image lies in that disc. Remark 1. For simplicity, in Theorem 1 we have assumed that w i, .j = 1,2 . . . . . l are distinct. It is straightforward (and standard) to generalize the result in case wjs are repeated. In this case, one has an interpolation problem with multiplicities. For such an interpolation problem, the corresponding matrices A, B are more complicated. See Helton (1982) and the references there for the results on interpolation problems with multiplicities.
2. M A I N R E S U L T S
Let us begin by describing the robust stabilization problems we shall consider in this paper. (2.l) Gain margin problem Let us consider the parametrized family of plants
Pk(s) = kPo(s),k in [a,b].O < a < 1 < h:
(2)
here Po(s) is the nominal plant model (k = 1). If the robust stabilization problem is solvable for this family of plants then, by definition, the controller C(s) guarantees a gain margin of at least (20 log b - 20log a) dB for the nominal closed loop. Further, as in Khargonekar and Tannenbaum (1985), by choosing different regions in the complex plane as the sets where k takes its values, one can similarly formulate gain phase margin, and complex parameter variation problems. (2.2) Muhiplicatice perturbation problem Let Po(s) be the nominal plant model. Suppose we are given a weighting function W(s) which is bounded and analytic in/7 such that W - ~(s) is also bounded and analytic in /4. Now given 6 > 0, consider the family of all plants P(s) such that
P(s) = (1 + W(s)A(s))Po(s)
(3)
where A(s) is a meromorphic function with no poles on the imaginary axis and IA(jo))L~O,
-~
< ~ , ) < m,
and P(s) has the same number of right half plane poles as Po(s). Of course, since A is meromorphic, there may be unstable pole-zero cancellations between Po and 1 + WA. Thus, in general, P and Po may not have the same unstable poles, even though the number of unstable poles of P and Po is the same. This family of plants has been considered in Doyle and Stein (1981), Vidyasagar (1985), and others. Kimura (1984) also considered a related family given by additive perturbations. Using a homotopy argument on the Nyquist plot, Doyle and Stein (1981) and Chen and Desoer (1982) have shown that a controller C(s) stabilizes this family of plants if and only if C(s) internally stabilizes the nominal plant Po(s) and
S
FPo(j(o)Clje))W(juJ)] 1 upi ........... < '" k 1 + Po(flo)c(o)) J 3"
(4)
Out" resuhs will gice the largest ~5jbr given W, Po .[br which one ean find a robust controller. We now describe the class of nominal plant models we consider in this paper. In this we
Robust stabilization essentially follow the convolution algebra approach of Callier and Desoer (1978). (See also Vidyasagar, 1985.) So let A denote the algebra (under pointwise addition and convolution multiplication) of all distributions f : 0, .l(t) =
~
.L(t) +
t<0
79
where the numerator n(s) admits a factorization: n(s) =
e ~ hSnl(s)n2(s)n3(s),
where n~(s) is the possibly infinite Blaschke product of zeros ofn in the open right half plane, i.e. if q l, q2 . . . . . q . . . . . . are zeros of n(st in H, then
.t~6(t - ti),t >10. i=0
j= ~
where J, belongs to L, and {Jil belongs to Ii. Further, let A denote the algebra (under pointwise addition and multiplication) of Laplace transforms of all distributions in A. Now define the subalgebra A = {./in A: e~'.l(t) belongs to A for some a > 0], and let A~ denote the subalgebra of ~4 of Laplace transforms of all elements of A_. Finally, let A2 be the subset of A~ consisting of analytic functions which are bounded away from zero at infinity, i.e. A 2 = {]in A~ : there exists p > 0 such that inf {](s): s i n / 7 , Isl > pl > 01.
Callier and Desoer quotient algebra
(1978) introduced
A ] ' A, = ~'n(s). ( d ( s ) n in A 1, d
in
the
1.
Clearly, n, d have no common zeros. Further, Callier and Desoer (1978) showed that the denominator d factors as d = d~d2
(5)
where d~ is a proper stable rational function having only a finite number of zeros in/7 and d 2 is a unit in A 2. Thus, the class of distributed plants we consider here have only finitely many poles in/q the closed right half plane. For simplicity, we shall assume that the roots of d L in the open right half plane H are distinct. We can now define the transfer functions Po(s) which we shall consider as nominal plant models. Consider the transfer function Pa(s) in A 2 1A ~which has a coprime factorization, i.e. there exist n, u, v in A, and d in A2 such that Po(s) = n(s)d
](s),un + vd = 1,
where a > 0 is chosen such that a # q i for all j; n2(s) is a proper stable rational function having zeros only on the imaginary axis including ~ , and n3(s) is a unit in A> (For details of Blaschke products, see Duren, 1970). It can be shown that n l(s) converges and is analytic everywhere in/7.) Thus, our plants can have pure transmission delays, infinitely many open right half plane zeros, and a finite number of zeros on the imaginary axis including infinity. The simplest example of this class of plants is: Po(s) = e-h*n(s)d
l(s)
where n,d are coprime proper stable rational functions. Also, using certain arguments of Callier and Desoer (1978) one can show that any Po(s) of the form
A2}
as the set of transfer functions of a class of distributed systems. A key result of Callier and Desoer (19781 is that every transfer function in A£ ~A~ has a coprime (or Bezout)factorization, i.e. for every P(s) in A ,5 ~A ~, we can find n, u, v in A ~ and d in A2 such that P=nd-*,un+~'d=
~s + qJl llq/ 2 - all
(6)
Po(s) = e-%2(s)d- ~(s),
where n in A1 and d in A 2 a r e coprime, and sin(s) belongs to Az for some i/> 0, admits a factorization as indicated above. Following Tannenbaum (1980), Youla, Bongiorno and Lu (1974) and Zames (1981), we first show that the problem of internal stabilization is equivalent to an interpolation problem. Recall that !f P = nd-1 and C = uv I are coprime.lactorizations .tor the plant and the controller, then the .teedback system in Fig. 1 is internally stable (t and only if dv + nu is a unit in A2. (See Desoer et al., 1980, for an abstract algebraic theory of feedback systems.) We now have Lemma 1. Consider the plant transjer junction Po(s) =
e-h~nl(s)n2(s)n3(s) dl (s)d2(s)
(7)
as above. Let the zeros old1 in H be pl, P2 . . . . . Pn and the zeros of n2 be zl, z2..... Zm (on the imaginary axis including ~ ) . Then:
(i) A compensator C(s) in A ; ~AI internally stabilizes P0(s) if the complementary sensitivity junction T(s):= PoC(1 + PoC)-l(s)
80
P . P . KHARGONEKAR and K. POOLLA
can be written as T(s) = e-h~n~ (s)T~ (s)
(8)
d~ is a proper stable rational function, we can find T3, a bounded analytic function i n / 7 such that (1 - T) = d~T3.
where TI is a bounded analytic Jhnction in H and T(zi) = 0,
i = 1,2 ..... m,
(9)
T(pj) = 1, j = 1,2 ..... n, and
(10)
T is continuous at ~ and T ( ~ ) # 1.
(11)
Further (11) implies that T3 belongs to A2. It follows that C = TPo~(1 - T) -~ = dzT2(n3T3) - t is in A~ ~A~. Finally, internal stability follows from:
(ii) I f TI (s) is a bounded analytic function satisfying (9)-(11) above, then C(s) = T P o ~(1 - T)- ~(s)
d2Tze-hsnan2n3 + n3T3dld2 = di T2e-hsntn2n3 + n3d2(1 - T) = n3d2
(12) which is a unit in A2.
belongs to A ; I A~ and internally stabilizes Po(s). Proof. For simplicity, we will assume that the poles P; and the zeros z~ are distinct. The general case of repeated poles and zeros can be easily handled by considering (9) and (10) as interpolation conditions with multiplicity (see Remark 1). (i) Let C(s) = uv- ~(s) be a coprime factorization of C in A£ ~A ~. Then, since C internally stabilizes Po, ffl = dldzV + e-h~nlnzn3u is a unit in A2. We can now write the complementary sensitivity function as: T = e-hsnlnzn3~ - 1.
Remark 2. Condition (11) is essentially the wellposedness condition. If n2 is strictly proper, i.e. n2 has at least one zero at oo, then (9) implies (11 ) since one of the z~s is infinity. We thus see that internal stabilization amounts to finding a complementary sensitivity function of the form (8) satisfying (9)-(11). Let us now consider the multiplicative perturbation problem, section 2.2. Suppose we are given the nominal model P0, a weighting function W, and > 0. Then, as we discussed above, the robust stabilization problem for the family (3) is equivalent to finding a complementary sensitivity function satisfying (9)-(11) such that 1 Sup IW(jo)T(jo)l < ~.
Define 7 1 : = n2n3~t-1. Clearly, 7-1 is a bounded analytic function in/7. Further
co
Since T(s) must be of the form (8), we require that T(zi)=O,
i = 1 , 2 ..... m.
(9) Sup In l (jog)e-hJ'°W(J'tn)Tl (j~o}[ <
We can also write T as: T = 1 - dld2v~9-1.
Now n~(s) is a Blaschke product continuous on the jo-axis and hence
Hence Tis continuous at oo and T ( ~ ) # 1 since d~, dz, v, if- 1 belong to A2. Also T(pj) = 1, j = 1,2 ..... n.
(10)
(ii) Suppose we can find T~, a bounded analytic function i n / 7 such that
Inl(jo)e-hJ~ I = 1,
1
which
for all co.
Thus Sup ]n~ (jtn)e-Jh°~ W ( j o ) T l (jog)l to
= Sup lW(jo)Tl(jog)[. T(s) = e-hsnl Tl(s) satisfies (9)-(11). Then we can find T2, a bounded analytic function i n / 7 such that T(s) = e-hsnlnz T2(s).
(13)
co
Further, (7)-(9) can be written as constraints on T~ in the following form: Tl(zi)=O,
i = 1,2 ..... m,
T~ (p~) = ehp'n ;- ~(pj). Also (10) implies that (1 - T) is divisible by d~, since zeros of d~ are contained in the zeros of(1 - T)I As
is
(Since n~,d are coprime, n~-l(pj) # vc.)
Robust stabilization We thus need to solve the following interpolation problem: find a bounded analytic function
81
junction) WT~ satisfying (16)-(20). Now (14)-(15) and Lemma (1-ii) together imply that with T:=
satisfying WT,(zi) = 0,
i = 1,2 ..... m,
(14)
WT~(p~) = W(pj)ehp~n[ ~(p~), j = 1, 2..... n,
(15)
T=W-le-mnlTl is continuous at oc, and T(oe) ¢ 1. Further, we need to find the largest 6 for which this interpolation problem is solvable. Now let
e-hsnl T1,
the controller C: = TP o 1(1 - T)- 1 = d2 T2(n3 T3)- 1 is in A] ~A, and internally stabilizes the family (3). Conversely, suppose the robust stabilization problem for the family (3) is solvable. Then, by Lemma (1), (13) and (4), it follows that we can find a bounded analytic function WTI :tq --* DI/~ satisfying (15). Then Theorem (1) implies that 6 < ~max(W, Po)-
wj = pj,
bj = W(pj)enP'n[ l(pj), j = 1,2 ..... n.
(We note that some of the wjs may be on the imaginary axis. In this case, we rearrange the order of wjs to fit the notation of Section 1.) Define ~max(W,po) to be the 0~rnax given by (1) for this interpolation data. We can now state one of our main results giving a solution to the multiplicative perturbation robust stabilization problem. Theorem 2. Let W, Po, 6 be as above. Then the robust stabilization problem Jor the family (3) is solvable if and only if < 0~max(W,Po).
(16)
Proof. Suppose t~<~max(W,Po). Consider the interpolation problem of finding a bounded analytic function WTI :tq ~ D 1/~
(17)
satisfying WT~(pj) = W(Pj)ehe'n[ ~P~), j = 1,2 ..... n;
(18)
WT~(zj) = 0, j = 1,2 ..... m, and
(19)
WTl(C~') = 0.
(20)
Now by definition of ~max in (1), emax for this interpolation problem is the same as ~max(W,Po) since zjs and ~ are on the imaginary axis and 0 is certainly less than or equal to ~max(W,Po). Since < 0~max(W~P0),
Theorem (1) implies that we can find a bounded analytic function (in jact, a proper stable rational AUT 22:1-F
Remark 3. Since ~max(W~Po) does not depend upon the zeros of n2 (on the imaginary axis), it follows that robust stabilizability is not affected by the zeros of the plant on the imaginary axis including infinity. However, the transmission delay e -h~ which has an essential zero at ~ has a very significant effect on robust stabilizability, and also on the maximal obtainable gain margin. In other words, even though a zero at ~ of finite multiplicity does not affect bounds on robust stabilizability, the timedelay e-he which has a zero of"infinite multiplicity" at ~ in the right half plane has a significant effect on these bounds. (Note that e -h~ is a singular inner junction. See Duren (1970) for details of singular inner functions.) Remark 4. Note that if Po(s) has only a finite number of zeros in the open right half plane, then n~ (s) is a finite Blaschke product. In this case, the complex numbers bj can be readily computed, and hence, ~max(W,Po) can be found by solving a matrix eigenvalue problem (Section 1 ). On the other hand if n l(s) is an infinite Blaschke product, then one can get good approximations to the bjs and obtain good approximations to ~max(W,Po). In the next section we give a few interesting examples illustrating Theorem 2. Let us now turn to the gain margin problem, section 2.1. The next result gives a solvability criterion for the robust stabilization problem corresponding to the family (2). Theorem 3. Consider the Jamily of plants (2) where P0(s) is given by (7). The robust stabilization problem jor the jamily (2) is solvable if and only if b a<
[~ + ~max(1,Po)] 2 - - ~ 1 "
(21)
82
P . P . KHARGONEKAR and K. POOLLA
Note that ~max(1,Po) is the value of ~max(W,Po) defined above with the weighting function W = 1.
Since O(z) = zO(z), we have that
T(s) = T(s)O(T(s)l. Proof. Let us first formulate the robust stabilization problem in terms of the c o m p l e m e n t a r y sensitivity function. A controller C ( s ) i n A2~A1 internally stabilizes the family (2) if and only if C(s) internally stabilizes Po(s) and
N o w (9) implies that
"['(s) = e-h~nl (s)TL (s)~(T(s)). Further
1 + kPo(s)C(sl # 0, for all k in [a, b] and s in/7.
T(zi) = O, i = 1,2 ..... re, and
(22)
T(pj)=l,
We can rewrite (22) as
N o w the p r o o f of T h e o r e m 2 (with W = 1, 6 = 1/rl implies that
T(s) = PoC(1 + PoC)-l(s) :# (1 - k) 1 for all k in [a, b] and s in ft.
"1 X/''~-- X/6/ ._<~ .... (1,P0). v/ b + x// a r
Thus, the robust stabilization p r o b l e m is solvable for the family (2) if and only if we can find a c o m p l e m e n t a r y sensitivity function T ( s )
T:H ~ C { I - c r ~ , ( l - b ) - l ] w [ ( 1 - a )
j = l , 2 . . . . . n.
Elementary algebra then gives (19). Conversely, suppose (21) holds. Define
1,,~c~]I = : G
+
,/.r
'
satisfying the conditions (9) (12) of L e m m a 1. Now, as in K h a r g o n e k a r and T a n n e n b a u m (1985), consider the conformal equivalence
N o w T h e o r e m 3 implies that we can find an analytic function
O:G~D~:z~O(z)
T:H ~ Dr
, = iv'b + ,/ii)t,£~ - ,/i))-',
such that for some b o u n d e d analytic function TI (s) in/4,
where
and
0(z)=
r(b - a)z II-II-btz)[l+[(1-11-a)z/(l
__
(1
__
b). ~
o)]
(It is straightforward to verify that 0 is one-to-one and onto from G to Dr, see T a n n e n b a u m , 1980.) Clearly, we can write
12
]
2"
'T= e-hSnl{s)'FL(s), i = l, 2 . . . . . m,
{24j
O(z) = zOtz)
T(PS)= I, j = 1,2 . . . . . n,
i251
for some b o u n d e d analytic function 0 on G. Also, one can easily check that
7~ is continuous at 3c and 7~(m) = 0. N o w consider the inverse l of the conformal equivalence 0. A straightforward calculation gives
0(0)=0,
7~(zi) = 0,
(23)
0(l)=l. d = 0 I:D r--* G:w--,Z(w)
Now suppose that the robust stabilization problem is solvable. Then there exists an analytic function
4rw Z(W) = (b - 1)(r - w) 2 - (a - 1)(r + w) 2
T:tq ~ G satisfying (9)-(12). Define the analytic function
T:H
~
O r : s ---,
where
O(T(s)).
Clearly, one can write
Aw) = wO[w),
Robust stabilization for some analytic function ~, on D,. Now let
83
analytic function f in H, Jlfll~ = Sup If(jco)l. Then to
T:zoT=
"P0(T):/7 --* G.
First note that since X(1) = 1, Z(0) = 0, (24)-(25) and T ( ~ ) = 0 imply that T satisfies (10)-(12). Further as t7 w {~ } is compact, continuity of T at implies that the image of T is also a compact subset of D, and hence is contained in a disc of radius < r. Now we can find M sufficiently large such that lO(z)l < M
for all z in C,
IzJ ~< ?.
the proof of Theorem 2 shows that p-
1 .... (W, Po)"
3. Examples In this section we present a few examples to illustrate the concrete nature of our bounds. Consider the plant Po(s) =
e-nS(s - z)l~l(s) ( s - p)(s + l )
, p, z > o,
Notice that T(s) = Tts)O(T(s)) = e-h~nl (s)Tl (s)O(T(s)).
where /61(s) is a unit in A 2. In this case, the interpolation problem (14)-(15) becomes:find
Since 7~t(s) is a bounded analytic function and since I~b(T(s))[ < M for all s in /7, it follows that Tl(s) = TI (s)0(7~h (s)) is also a bounded analytic function on t7. Thus 7-also satisfies (7). Since the image of 7lies in G, the controller
In this case,
C = TPol(1 -- T) -I
~max(W, Po)
solves the robust stabilization problem for the family of plants (2). Remark 4. Results analogous to Theorems (2) and (3) can be obtained for robust stabilization problems which arise from additive perturbations as in Kimura (1984), gain-phase margin as in Khargonekar and Tannenbaum (1985), complex parameter variations as in Doyle, Wall and Stein (1982). We would like to remark that as m Khargonekar and Tannenbaum (1985), the same invariant ~max(W,Po) appears as the crucial quantity in checking solvability of apparently unrelated problems such as the gain margin problem, multiplicative perturbations, complex parameter variations, etc. Thus, the main conclusions of (Khargonekar and Tannenbaum, 1985) on robust stabilization problems/H~-optimization problems are quite valid for a fairly general class of distributed plants. The important contribution of this paper is to show that for plants with a pure delay and finitely many right half plane zeros and poles we can compute the main invariant 7max quite explicitly. Note that Theorem (2) also solves the problem of weighted complementary sensitivity minimization in an H~-optimization setting (Zames, 1981). More specifically, define /~: = inf {J114/TII~, : C internally stabilizes Po}, where Wis) is a weighting function and T = PoC(1 + PoC) 1. (Recall that for a bounded
WTI :/7 --* DI/~ W T l ( ~ ) = O,
=
WTI(p) = W(p)ehP(z + p)(z -- p ) - I
JW
-
l(p)e- hv(z -- p)(Z + p) - 11.
From this formula we can draw the following intuitive conclusion. As the distance between z, p decreases, 0~max reduces. This should be expected since for z = p, we have unstable pole-zero cancellation. The time-delay h reduces the ~-maxby (3 jactor ore- hp. If we consider the robust stabilization problem for the multiplicative perturbation family, section 3, then the robust stabilization problem is solvable if and only if 6 < e-hvlW ltp)(2 --p)(2 + p)-l].
As an example, take z = 3, p = 2, h = 1, W(2) = 1. Then 0~max 0.2e-2 = 0.027. And the bound on 6 for robust stabilizability is =
6 < 0.027. On the other hand, for this plant the maximum obtainable gain margin is 401ogl] + ~m"x(l' P°)]
~max(1,Po)]
+P
e-hPlz
Pl "
For h = 1, z = 3, p = 2, the maximal obtainable gain margin is 1 + 0.2e - 2 ] 40log i - - ~ J = 0.938dB!
84
P.P. KHARGONEKAR and K. POOLLA W e note that one can o b t a i n a general f o r m u l a for
~max(W, P0) for the case of one right half plane at p: ~max(1/1/~Po}
~--
e - hi' I W-
1 (p)n i- 1 (P)I
where n~ (s) is the Blaschke product of the open right half plane zeros of Po(s). We now consider an example of a system with no open right half plane zeros, two right half plane poles, and a pure transmission delay: Po(s) -
e-h~P1(s ) (s-- 1)(s--2)
where/6 (s) is invertible in Az. In this case a routine calculation gives ~max(1, P0) = 0.5e-2h(w/9eZh + 9 -- 14e h -- 3(e h -- 1)). F o r h = 1, ..... (1, Po) = 0.065.
Hence, in this case, the robust stabilization problem for multiplicative perturbation family, section 3 is solvable if and only if 6 < 0.065. And the maximal obtainable gain margin for this plant is 2.26 dB! The effect of a right half plane zero would be to reduce ~m~x even further.
4. CONCLUDING REMARKS
In this p a p e r we have given a solution to certain robust stabilization problems for a large class of distributed plants. In particular, our results give concrete necessary and sufficient conditions for robust stabilizability for plants with pure delay and a finite number of right half plane poles and zeros. Our results provide techniques for the assessment of the impact of time-delays on robust stabilizability. Many open problems remain to be solved in this problem area. For example, so far no explicit results have been obtained for the H~-weighted sensitivity optimization problem of Zames (1981). Also, the multivariable versions of the problems treated in this paper remain essentially unsolved. Finally,
these results need to be e x t e n d e d to systems governed by partial differential equations. REFERENCES Ackermann, J. (1980). Parameter space design of robust control systems. IEEE Trans. Aut. Control, AC-25, 1058. Callier, F. M. and C. A. Desoer (1978). An algebra of transfer functions for distributed time-invariant systems. IEEE Trans. Circuits Svst., CAS-25, 651. Chen, M. J. and C. A. Desoer (1982). Necessary and sufficient conditions for robust stability of linear distributed feedback systems. Int. J. Control, 35, 255. Desoer, C. A., R. W. Liu, J. Murray and R. Saeks (1980). Feedback system design: the fractional representation approach to analysis and synthesis. IEEE Trans. Aut. Control, AC-25, 399. Doyle, J.C.. J. Wall and G. Stein (1982). Performance and robustness analysis for structured uncertainty. Proc 21st IEEE Con[i Dec. Control, pp. 629 636. Doyle, J. C. and G. Stein (1981). Multivariable feedback design: concepts for a classical/modern synthesis. IEEE Trans. Aut. Control, AC-26, 4. Duren, P. L. (1970). Theory of HJ'-spaces. Academic Press, New York. Ghosh, B. K. and C. 1. Byrnes (1983). Simultaneous stabilization and simultaneous pole-placement by non-switching compensation. IEEE Trans. Aut. Cmurol, AC-28, 735. Helton. J. W. (1982). Non-Euclidean functional analysis and electronics. Bull. AMS. 7, I. Khargonekar, P. P., K. Poolla and A. Tannenbaum (1985). Robust control of linear time-invariant plants using periodic compensation. IEEE Trans. Aut. Control, AC-30, 1088. Khargonekar, P. P. and A. Tannenbaum (1985). Non-Euclidean metrics and robust stabilization of systems with parameter uncertainty. IEEE Trans. Aut. Control. AC-30, 1005. Kimura, H. (1984). Robust stabilizability for a class of transler functions. IEEE Trans. Aut. Control, AC-29, 788. Lehtomaki, N. (1983). Practical robustness measures in multivariable control system analysis. Ph.D. dissertation, MIT, Cambridge, Mass. Saeks, R. and J. Murray (1982). Fractional representation, algebraic geometry, and the simultaneous stabilization problem. IEEE Trans. Aut. Control, AC-27, 895. Tannenbaum, A. (1980). Feedback stabilization of plants with uncertainty in the gain factor. Int. J. Control, 32, 1. Tannenbaum, A. (1981). lnvariance and System Theory: Algebraic and Geometric Aspects. Springer-Verlag, Berlin. Tannenbaum, A. (1982). Modified Nevanlinna Pick interpolation and feedback stabilization of linear plants with uncertainty in the gain factor. Int. J. Control, 36, 331. Vidyasagar, M. (1985). Control System Synthesis: A Factorization Approach. MIT Press, Cambridge, Mass. Vidyasagar, M. and N. Viswanadham (1982). Algebraic design techniques for reliable stabilization. IEEE Trans. Aut. Control, AC-27, 1085. Youla, D. C., J. Bongiorno and Y. Lu (1974). Single loop feedback stabilization of linear multivariable dynamic plants. Automatica, 10, 159. Zames, G. (1981). Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses. IEEE Trans. 4ut. Control, AC-16. 301.
0005 1098/86 $3.00 + 0.00 Pergamon Press Ltd. © 1986 International Federation of Automatic Control
Robust Stabilization of Distributed Systems* PRAMOD P. K H A R G O N E K A R t and K. POOLLA:~
Interpolation theory solves the robust stabilization problem Jbr families of distributed plants given by gain variations and multiplicative uncertainty. Key Words--Robust control; time lag systems; distributed parameter systems; control system synthesis; frequency domain.
solution is known. However, for certain types of families of plants, it is possible to give a concrete constructive solution to this problem. Recently, robust stabilization problems for certain families of finite-dimensional LTI plants have been solved by Kimura (1984), Tannenbaum (1980, 1981, 1982), Khargonekar and Tannenbaum (1985), Khargonekar, Poolla and Tannenbaum (1985), and the references cited there. It is not difficult to see that the techniques of these authors generalize easily to distributed plants which have only finitely many poles and zeros in the closed right half plane including infinity. In this paper, we extend some of these results to a fairly general class of distributed systems defined by Callier and Desoer (1978). These systems contain transmission delays, have possibly infinitely many zeros in the closed right half plane, but have only finitely many poles in the closed right half plane. As expected, our procedures are not easy to carry out in this generality, but jbr the class of systems having a pure transmission delay and.finitely many poles and zeros in the closed right half plane, our procedures are completely constructive and yield concrete bounds j or robust stabilizability. We refer the interested reader to Kimura (1984), Khargonekar and Tannenbaum (1985), Vidyasagar (1985), and the references therein for further literature on the robust stabilization problems. Also, the recent papers by Ackermann (1980), Ghosh and Byrnes (1983), Saeks and Murrey (1982), and Vidyasagar and Viswanadham (1982), on the simultaneous stabilization problem and parameter space design represent important contributions to the robust stabilization problem.
Almtraet--This paper is devoted to the problem of robust stabilization for a class of distributed plants. We consider the gain margin optimization problem and the robust stabilization problem for multiplicative perturbations. Using techniques from interpolation theory and complex variables, we obtain explicit necessary and sufficient conditions for robust stabilizability. A few examples are included to illustrate the results. NOTATION C =/complex numbers} H = open right half plane = {s in C: Re(s) > 0} H = closed right half plane = {s in C: Re(s) >I 0} O, = open disc of radius r = {s in C: Jsj < r} Dr = closed disc of radius r = {s in C: tsJ ~< r} 1. INTRODUCTION
CONSIDER the feedback system shown in Fig. 1. In general terms, the robust stabilization problem may be formulated as follows: let Pk(S) be a family of linear time-invariant SISO plants, where the index k takes values in some set K. Then the robust stabilization problem is to find a (realizable) controller C(s) such that the above closed loop system is internally asymptotically stable for each k in K. If this problem is solvable, then C(s) is a robust controller in the sense that as long as the plant belongs to the set {Pk(S): k in K}, the closed loop system is internally asymptotically stable. Roughly speaking, realizable means that C(s) represents a casual dynamic system, and internally stable means that there are no unstable pole-zero cancellations between C(s) and Pk(S). These notions are defined precisely in section 2. The robust stabilization problem stated above, in its complete generality, is very hard and no general * Received 12 November 1984; revised 6 May 1985. 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 J. Ackermann under the direction of Editor H. Kwakernaak. This work was supported in part by the National Science Foundation under grant No. ECS-8400832. t Department of Electrical Engineering, University of Minnesota, Minneapolis, MN 55455, U.S.A. Department of Electrical Engineering, University of Illinois, Urbana, IL 61801, U.S.A. Previously with the Department of Electrical Engineering, McGill University, Montreal, Canada.
FEEDBACK
SYSTEM
FIG. 1. Feedback system. 77
78
P . P . KHARGONEKAR and K. POOLLA
As in Kimura (1984), Tannenbaum (1980, 1981, 1982), Zames ( 1981 ), it turns out that interpolation theory plays an important role in robust stabilization problems. We end this section with a result from classical interpolation theory (see Helton, 1982: Khargonekar and Tannenbaum, 1985, for details). Let w~, w2. . . . . w~ be a set of distinct points in the open right half plane H and wt + ~, w~+ 2. . . . . Wq be a (possibly repeated) set of points on the boundary of H, i.e., on the imaginary axis including the point at infinity. Suppose we are given a set of points b~, b2 . . . . . hi, bt + ~..... bq in C. Define l x l matrices
A =
i,j = 1,2 ..... I
D~bi ] i , j = B = LW~ + ~'J_]
l,2 . . . . . I.
Let ~-max denote the largest eigenvalue of A - ~B. (It turns out that 2 .... >~ 0.) Set c~~" = m a x ( v /'~....... Jbt+~J. . . . . Ibql). max-
(1)
If the right-hand side turns out to be zero, then we define C~m~= 7;.) We now state a result from the classical interpolation theory which will be useful in the next sections. Here/7: = Is in C: Re(s) >~ 0~ and H : H u t ~j ~. Theorem 1. Let wi in H and bi in C, i = 1, 2 . . . . . q be as above. Then there exists aJunetion f analytic in H, continuous at infinity such that f:Iq --* D~:f(wi) = bi,
i = 1,2 . . . . . q
if and only i[ r
1 <
0(max.
A proof of this result may be found in (Khargonekar and T a n n e n b a u m (1985, Theorem 1.5). This shows that the image of any analytic function which sends w~ to b~ cannot lie in a disc of radius smaller than ~m2~; and given any disc of radius greater than C~m,~,we can find an interpolating function f whose image lies in that disc. Remark 1. For simplicity, in Theorem 1 we have assumed that w i, .j = 1,2 . . . . . l are distinct. It is straightforward (and standard) to generalize the result in case wjs are repeated. In this case, one has an interpolation problem with multiplicities. For such an interpolation problem, the corresponding matrices A, B are more complicated. See Helton (1982) and the references there for the results on interpolation problems with multiplicities.
2. M A I N R E S U L T S
Let us begin by describing the robust stabilization problems we shall consider in this paper. (2.l) Gain margin problem Let us consider the parametrized family of plants
Pk(s) = kPo(s),k in [a,b].O < a < 1 < h:
(2)
here Po(s) is the nominal plant model (k = 1). If the robust stabilization problem is solvable for this family of plants then, by definition, the controller C(s) guarantees a gain margin of at least (20 log b - 20log a) dB for the nominal closed loop. Further, as in Khargonekar and Tannenbaum (1985), by choosing different regions in the complex plane as the sets where k takes its values, one can similarly formulate gain phase margin, and complex parameter variation problems. (2.2) Muhiplicatice perturbation problem Let Po(s) be the nominal plant model. Suppose we are given a weighting function W(s) which is bounded and analytic in/7 such that W - ~(s) is also bounded and analytic in /4. Now given 6 > 0, consider the family of all plants P(s) such that
P(s) = (1 + W(s)A(s))Po(s)
(3)
where A(s) is a meromorphic function with no poles on the imaginary axis and IA(jo))L~O,
-~
< ~ , ) < m,
and P(s) has the same number of right half plane poles as Po(s). Of course, since A is meromorphic, there may be unstable pole-zero cancellations between Po and 1 + WA. Thus, in general, P and Po may not have the same unstable poles, even though the number of unstable poles of P and Po is the same. This family of plants has been considered in Doyle and Stein (1981), Vidyasagar (1985), and others. Kimura (1984) also considered a related family given by additive perturbations. Using a homotopy argument on the Nyquist plot, Doyle and Stein (1981) and Chen and Desoer (1982) have shown that a controller C(s) stabilizes this family of plants if and only if C(s) internally stabilizes the nominal plant Po(s) and
S
FPo(j(o)Clje))W(juJ)] 1 upi ........... < '" k 1 + Po(flo)c(o)) J 3"
(4)
Out" resuhs will gice the largest ~5jbr given W, Po .[br which one ean find a robust controller. We now describe the class of nominal plant models we consider in this paper. In this we
Robust stabilization essentially follow the convolution algebra approach of Callier and Desoer (1978). (See also Vidyasagar, 1985.) So let A denote the algebra (under pointwise addition and convolution multiplication) of all distributions f : 0, .l(t) =
~
.L(t) +
t<0
79
where the numerator n(s) admits a factorization: n(s) =
e ~ hSnl(s)n2(s)n3(s),
where n~(s) is the possibly infinite Blaschke product of zeros ofn in the open right half plane, i.e. if q l, q2 . . . . . q . . . . . . are zeros of n(st in H, then
.t~6(t - ti),t >10. i=0
j= ~
where J, belongs to L, and {Jil belongs to Ii. Further, let A denote the algebra (under pointwise addition and multiplication) of Laplace transforms of all distributions in A. Now define the subalgebra A = {./in A: e~'.l(t) belongs to A for some a > 0], and let A~ denote the subalgebra of ~4 of Laplace transforms of all elements of A_. Finally, let A2 be the subset of A~ consisting of analytic functions which are bounded away from zero at infinity, i.e. A 2 = {]in A~ : there exists p > 0 such that inf {](s): s i n / 7 , Isl > pl > 01.
Callier and Desoer quotient algebra
(1978) introduced
A ] ' A, = ~'n(s). ( d ( s ) n in A 1, d
in
the
1.
Clearly, n, d have no common zeros. Further, Callier and Desoer (1978) showed that the denominator d factors as d = d~d2
(5)
where d~ is a proper stable rational function having only a finite number of zeros in/7 and d 2 is a unit in A 2. Thus, the class of distributed plants we consider here have only finitely many poles in/q the closed right half plane. For simplicity, we shall assume that the roots of d L in the open right half plane H are distinct. We can now define the transfer functions Po(s) which we shall consider as nominal plant models. Consider the transfer function Pa(s) in A 2 1A ~which has a coprime factorization, i.e. there exist n, u, v in A, and d in A2 such that Po(s) = n(s)d
](s),un + vd = 1,
where a > 0 is chosen such that a # q i for all j; n2(s) is a proper stable rational function having zeros only on the imaginary axis including ~ , and n3(s) is a unit in A> (For details of Blaschke products, see Duren, 1970). It can be shown that n l(s) converges and is analytic everywhere in/7.) Thus, our plants can have pure transmission delays, infinitely many open right half plane zeros, and a finite number of zeros on the imaginary axis including infinity. The simplest example of this class of plants is: Po(s) = e-h*n(s)d
l(s)
where n,d are coprime proper stable rational functions. Also, using certain arguments of Callier and Desoer (1978) one can show that any Po(s) of the form
A2}
as the set of transfer functions of a class of distributed systems. A key result of Callier and Desoer (19781 is that every transfer function in A£ ~A~ has a coprime (or Bezout)factorization, i.e. for every P(s) in A ,5 ~A ~, we can find n, u, v in A ~ and d in A2 such that P=nd-*,un+~'d=
~s + qJl llq/ 2 - all
(6)
Po(s) = e-%2(s)d- ~(s),
where n in A1 and d in A 2 a r e coprime, and sin(s) belongs to Az for some i/> 0, admits a factorization as indicated above. Following Tannenbaum (1980), Youla, Bongiorno and Lu (1974) and Zames (1981), we first show that the problem of internal stabilization is equivalent to an interpolation problem. Recall that !f P = nd-1 and C = uv I are coprime.lactorizations .tor the plant and the controller, then the .teedback system in Fig. 1 is internally stable (t and only if dv + nu is a unit in A2. (See Desoer et al., 1980, for an abstract algebraic theory of feedback systems.) We now have Lemma 1. Consider the plant transjer junction Po(s) =
e-h~nl(s)n2(s)n3(s) dl (s)d2(s)
(7)
as above. Let the zeros old1 in H be pl, P2 . . . . . Pn and the zeros of n2 be zl, z2..... Zm (on the imaginary axis including ~ ) . Then:
(i) A compensator C(s) in A ; ~AI internally stabilizes P0(s) if the complementary sensitivity junction T(s):= PoC(1 + PoC)-l(s)
80
P . P . KHARGONEKAR and K. POOLLA
can be written as T(s) = e-h~n~ (s)T~ (s)
(8)
d~ is a proper stable rational function, we can find T3, a bounded analytic function i n / 7 such that (1 - T) = d~T3.
where TI is a bounded analytic Jhnction in H and T(zi) = 0,
i = 1,2 ..... m,
(9)
T(pj) = 1, j = 1,2 ..... n, and
(10)
T is continuous at ~ and T ( ~ ) # 1.
(11)
Further (11) implies that T3 belongs to A2. It follows that C = TPo~(1 - T) -~ = dzT2(n3T3) - t is in A~ ~A~. Finally, internal stability follows from:
(ii) I f TI (s) is a bounded analytic function satisfying (9)-(11) above, then C(s) = T P o ~(1 - T)- ~(s)
d2Tze-hsnan2n3 + n3T3dld2 = di T2e-hsntn2n3 + n3d2(1 - T) = n3d2
(12) which is a unit in A2.
belongs to A ; I A~ and internally stabilizes Po(s). Proof. For simplicity, we will assume that the poles P; and the zeros z~ are distinct. The general case of repeated poles and zeros can be easily handled by considering (9) and (10) as interpolation conditions with multiplicity (see Remark 1). (i) Let C(s) = uv- ~(s) be a coprime factorization of C in A£ ~A ~. Then, since C internally stabilizes Po, ffl = dldzV + e-h~nlnzn3u is a unit in A2. We can now write the complementary sensitivity function as: T = e-hsnlnzn3~ - 1.
Remark 2. Condition (11) is essentially the wellposedness condition. If n2 is strictly proper, i.e. n2 has at least one zero at oo, then (9) implies (11 ) since one of the z~s is infinity. We thus see that internal stabilization amounts to finding a complementary sensitivity function of the form (8) satisfying (9)-(11). Let us now consider the multiplicative perturbation problem, section 2.2. Suppose we are given the nominal model P0, a weighting function W, and > 0. Then, as we discussed above, the robust stabilization problem for the family (3) is equivalent to finding a complementary sensitivity function satisfying (9)-(11) such that 1 Sup IW(jo)T(jo)l < ~.
Define 7 1 : = n2n3~t-1. Clearly, 7-1 is a bounded analytic function in/7. Further
co
Since T(s) must be of the form (8), we require that T(zi)=O,
i = 1 , 2 ..... m.
(9) Sup In l (jog)e-hJ'°W(J'tn)Tl (j~o}[ <
We can also write T as: T = 1 - dld2v~9-1.
Now n~(s) is a Blaschke product continuous on the jo-axis and hence
Hence Tis continuous at oo and T ( ~ ) # 1 since d~, dz, v, if- 1 belong to A2. Also T(pj) = 1, j = 1,2 ..... n.
(10)
(ii) Suppose we can find T~, a bounded analytic function i n / 7 such that
Inl(jo)e-hJ~ I = 1,
1
which
for all co.
Thus Sup ]n~ (jtn)e-Jh°~ W ( j o ) T l (jog)l to
= Sup lW(jo)Tl(jog)[. T(s) = e-hsnl Tl(s) satisfies (9)-(11). Then we can find T2, a bounded analytic function i n / 7 such that T(s) = e-hsnlnz T2(s).
(13)
co
Further, (7)-(9) can be written as constraints on T~ in the following form: Tl(zi)=O,
i = 1,2 ..... m,
T~ (p~) = ehp'n ;- ~(pj). Also (10) implies that (1 - T) is divisible by d~, since zeros of d~ are contained in the zeros of(1 - T)I As
is
(Since n~,d are coprime, n~-l(pj) # vc.)
Robust stabilization We thus need to solve the following interpolation problem: find a bounded analytic function
81
junction) WT~ satisfying (16)-(20). Now (14)-(15) and Lemma (1-ii) together imply that with T:=
satisfying WT,(zi) = 0,
i = 1,2 ..... m,
(14)
WT~(p~) = W(pj)ehp~n[ ~(p~), j = 1, 2..... n,
(15)
T=W-le-mnlTl is continuous at oc, and T(oe) ¢ 1. Further, we need to find the largest 6 for which this interpolation problem is solvable. Now let
e-hsnl T1,
the controller C: = TP o 1(1 - T)- 1 = d2 T2(n3 T3)- 1 is in A] ~A, and internally stabilizes the family (3). Conversely, suppose the robust stabilization problem for the family (3) is solvable. Then, by Lemma (1), (13) and (4), it follows that we can find a bounded analytic function WTI :tq --* DI/~ satisfying (15). Then Theorem (1) implies that 6 < ~max(W, Po)-
wj = pj,
bj = W(pj)enP'n[ l(pj), j = 1,2 ..... n.
(We note that some of the wjs may be on the imaginary axis. In this case, we rearrange the order of wjs to fit the notation of Section 1.) Define ~max(W,po) to be the 0~rnax given by (1) for this interpolation data. We can now state one of our main results giving a solution to the multiplicative perturbation robust stabilization problem. Theorem 2. Let W, Po, 6 be as above. Then the robust stabilization problem Jor the family (3) is solvable if and only if < 0~max(W,Po).
(16)
Proof. Suppose t~<~max(W,Po). Consider the interpolation problem of finding a bounded analytic function WTI :tq ~ D 1/~
(17)
satisfying WT~(pj) = W(Pj)ehe'n[ ~P~), j = 1,2 ..... n;
(18)
WT~(zj) = 0, j = 1,2 ..... m, and
(19)
WTl(C~') = 0.
(20)
Now by definition of ~max in (1), emax for this interpolation problem is the same as ~max(W,Po) since zjs and ~ are on the imaginary axis and 0 is certainly less than or equal to ~max(W,Po). Since < 0~max(W~P0),
Theorem (1) implies that we can find a bounded analytic function (in jact, a proper stable rational AUT 22:1-F
Remark 3. Since ~max(W~Po) does not depend upon the zeros of n2 (on the imaginary axis), it follows that robust stabilizability is not affected by the zeros of the plant on the imaginary axis including infinity. However, the transmission delay e -h~ which has an essential zero at ~ has a very significant effect on robust stabilizability, and also on the maximal obtainable gain margin. In other words, even though a zero at ~ of finite multiplicity does not affect bounds on robust stabilizability, the timedelay e-he which has a zero of"infinite multiplicity" at ~ in the right half plane has a significant effect on these bounds. (Note that e -h~ is a singular inner junction. See Duren (1970) for details of singular inner functions.) Remark 4. Note that if Po(s) has only a finite number of zeros in the open right half plane, then n~ (s) is a finite Blaschke product. In this case, the complex numbers bj can be readily computed, and hence, ~max(W,Po) can be found by solving a matrix eigenvalue problem (Section 1 ). On the other hand if n l(s) is an infinite Blaschke product, then one can get good approximations to the bjs and obtain good approximations to ~max(W,Po). In the next section we give a few interesting examples illustrating Theorem 2. Let us now turn to the gain margin problem, section 2.1. The next result gives a solvability criterion for the robust stabilization problem corresponding to the family (2). Theorem 3. Consider the Jamily of plants (2) where P0(s) is given by (7). The robust stabilization problem jor the jamily (2) is solvable if and only if b a<
[~ + ~max(1,Po)] 2 - - ~ 1 "
(21)
82
P . P . KHARGONEKAR and K. POOLLA
Note that ~max(1,Po) is the value of ~max(W,Po) defined above with the weighting function W = 1.
Since O(z) = zO(z), we have that
T(s) = T(s)O(T(s)l. Proof. Let us first formulate the robust stabilization problem in terms of the c o m p l e m e n t a r y sensitivity function. A controller C ( s ) i n A2~A1 internally stabilizes the family (2) if and only if C(s) internally stabilizes Po(s) and
N o w (9) implies that
"['(s) = e-h~nl (s)TL (s)~(T(s)). Further
1 + kPo(s)C(sl # 0, for all k in [a, b] and s in/7.
T(zi) = O, i = 1,2 ..... re, and
(22)
T(pj)=l,
We can rewrite (22) as
N o w the p r o o f of T h e o r e m 2 (with W = 1, 6 = 1/rl implies that
T(s) = PoC(1 + PoC)-l(s) :# (1 - k) 1 for all k in [a, b] and s in ft.
"1 X/''~-- X/6/ ._<~ .... (1,P0). v/ b + x// a r
Thus, the robust stabilization p r o b l e m is solvable for the family (2) if and only if we can find a c o m p l e m e n t a r y sensitivity function T ( s )
T:H ~ C { I - c r ~ , ( l - b ) - l ] w [ ( 1 - a )
j = l , 2 . . . . . n.
Elementary algebra then gives (19). Conversely, suppose (21) holds. Define
1,,~c~]I = : G
+
,/.r
'
satisfying the conditions (9) (12) of L e m m a 1. Now, as in K h a r g o n e k a r and T a n n e n b a u m (1985), consider the conformal equivalence
N o w T h e o r e m 3 implies that we can find an analytic function
O:G~D~:z~O(z)
T:H ~ Dr
, = iv'b + ,/ii)t,£~ - ,/i))-',
such that for some b o u n d e d analytic function TI (s) in/4,
where
and
0(z)=
r(b - a)z II-II-btz)[l+[(1-11-a)z/(l
__
(1
__
b). ~
o)]
(It is straightforward to verify that 0 is one-to-one and onto from G to Dr, see T a n n e n b a u m , 1980.) Clearly, we can write
12
]
2"
'T= e-hSnl{s)'FL(s), i = l, 2 . . . . . m,
{24j
O(z) = zOtz)
T(PS)= I, j = 1,2 . . . . . n,
i251
for some b o u n d e d analytic function 0 on G. Also, one can easily check that
7~ is continuous at 3c and 7~(m) = 0. N o w consider the inverse l of the conformal equivalence 0. A straightforward calculation gives
0(0)=0,
7~(zi) = 0,
(23)
0(l)=l. d = 0 I:D r--* G:w--,Z(w)
Now suppose that the robust stabilization problem is solvable. Then there exists an analytic function
4rw Z(W) = (b - 1)(r - w) 2 - (a - 1)(r + w) 2
T:tq ~ G satisfying (9)-(12). Define the analytic function
T:H
~
O r : s ---,
where
O(T(s)).
Clearly, one can write
Aw) = wO[w),
Robust stabilization for some analytic function ~, on D,. Now let
83
analytic function f in H, Jlfll~ = Sup If(jco)l. Then to
T:zoT=
"P0(T):/7 --* G.
First note that since X(1) = 1, Z(0) = 0, (24)-(25) and T ( ~ ) = 0 imply that T satisfies (10)-(12). Further as t7 w {~ } is compact, continuity of T at implies that the image of T is also a compact subset of D, and hence is contained in a disc of radius < r. Now we can find M sufficiently large such that lO(z)l < M
for all z in C,
IzJ ~< ?.
the proof of Theorem 2 shows that p-
1 .... (W, Po)"
3. Examples In this section we present a few examples to illustrate the concrete nature of our bounds. Consider the plant Po(s) =
e-nS(s - z)l~l(s) ( s - p)(s + l )
, p, z > o,
Notice that T(s) = Tts)O(T(s)) = e-h~nl (s)Tl (s)O(T(s)).
where /61(s) is a unit in A 2. In this case, the interpolation problem (14)-(15) becomes:find
Since 7~t(s) is a bounded analytic function and since I~b(T(s))[ < M for all s in /7, it follows that Tl(s) = TI (s)0(7~h (s)) is also a bounded analytic function on t7. Thus 7-also satisfies (7). Since the image of 7lies in G, the controller
In this case,
C = TPol(1 -- T) -I
~max(W, Po)
solves the robust stabilization problem for the family of plants (2). Remark 4. Results analogous to Theorems (2) and (3) can be obtained for robust stabilization problems which arise from additive perturbations as in Kimura (1984), gain-phase margin as in Khargonekar and Tannenbaum (1985), complex parameter variations as in Doyle, Wall and Stein (1982). We would like to remark that as m Khargonekar and Tannenbaum (1985), the same invariant ~max(W,Po) appears as the crucial quantity in checking solvability of apparently unrelated problems such as the gain margin problem, multiplicative perturbations, complex parameter variations, etc. Thus, the main conclusions of (Khargonekar and Tannenbaum, 1985) on robust stabilization problems/H~-optimization problems are quite valid for a fairly general class of distributed plants. The important contribution of this paper is to show that for plants with a pure delay and finitely many right half plane zeros and poles we can compute the main invariant 7max quite explicitly. Note that Theorem (2) also solves the problem of weighted complementary sensitivity minimization in an H~-optimization setting (Zames, 1981). More specifically, define /~: = inf {J114/TII~, : C internally stabilizes Po}, where Wis) is a weighting function and T = PoC(1 + PoC) 1. (Recall that for a bounded
WTI :/7 --* DI/~ W T l ( ~ ) = O,
=
WTI(p) = W(p)ehP(z + p)(z -- p ) - I
JW
-
l(p)e- hv(z -- p)(Z + p) - 11.
From this formula we can draw the following intuitive conclusion. As the distance between z, p decreases, 0~max reduces. This should be expected since for z = p, we have unstable pole-zero cancellation. The time-delay h reduces the ~-maxby (3 jactor ore- hp. If we consider the robust stabilization problem for the multiplicative perturbation family, section 3, then the robust stabilization problem is solvable if and only if 6 < e-hvlW ltp)(2 --p)(2 + p)-l].
As an example, take z = 3, p = 2, h = 1, W(2) = 1. Then 0~max 0.2e-2 = 0.027. And the bound on 6 for robust stabilizability is =
6 < 0.027. On the other hand, for this plant the maximum obtainable gain margin is 401ogl] + ~m"x(l' P°)]
~max(1,Po)]
+P
e-hPlz
Pl "
For h = 1, z = 3, p = 2, the maximal obtainable gain margin is 1 + 0.2e - 2 ] 40log i - - ~ J = 0.938dB!
84
P.P. KHARGONEKAR and K. POOLLA W e note that one can o b t a i n a general f o r m u l a for
~max(W, P0) for the case of one right half plane at p: ~max(1/1/~Po}
~--
e - hi' I W-
1 (p)n i- 1 (P)I
where n~ (s) is the Blaschke product of the open right half plane zeros of Po(s). We now consider an example of a system with no open right half plane zeros, two right half plane poles, and a pure transmission delay: Po(s) -
e-h~P1(s ) (s-- 1)(s--2)
where/6 (s) is invertible in Az. In this case a routine calculation gives ~max(1, P0) = 0.5e-2h(w/9eZh + 9 -- 14e h -- 3(e h -- 1)). F o r h = 1, ..... (1, Po) = 0.065.
Hence, in this case, the robust stabilization problem for multiplicative perturbation family, section 3 is solvable if and only if 6 < 0.065. And the maximal obtainable gain margin for this plant is 2.26 dB! The effect of a right half plane zero would be to reduce ~m~x even further.
4. CONCLUDING REMARKS
In this p a p e r we have given a solution to certain robust stabilization problems for a large class of distributed plants. In particular, our results give concrete necessary and sufficient conditions for robust stabilizability for plants with pure delay and a finite number of right half plane poles and zeros. Our results provide techniques for the assessment of the impact of time-delays on robust stabilizability. Many open problems remain to be solved in this problem area. For example, so far no explicit results have been obtained for the H~-weighted sensitivity optimization problem of Zames (1981). Also, the multivariable versions of the problems treated in this paper remain essentially unsolved. Finally,
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