e e 2x2u ds e s)dws: E e e E[e e 0 sff(xs) + jcsjmgds ; c 2 A:
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we can see (x3t )2 (zt )2 by the comparison theorem [6]. Hence, = 1. By the monotonicity of u0 (x) and '(x), the uniqueness of (28) holds. Thus, we conclude that (28) admits a unique strong solution 3 (xt ). Now, we apply Ito’s formula for convex functions [7, p. 219] to obtain
e0 t u(xt3 ) = u(x) + 3 0
+cs u +
t
+ 0
t
e0 s
1
2 x2 u00
0
2
0 u + Axu0 x=x
ds
e0 s xs3u0 (xs3 ) dws :
Robust Stability of Quasi-Polynomials and the Finite Inclusions Theorem S. Mondié, J. Santos, and V. L. Kharitonov Abstract—A semianalytic tool for the study of the robust stability of quasi-polynomials of retarded type is presented. The approach is based on graphical methods. The finite Nyquist theorem for polynomials is extended to the case of quasi-polynomials with the help of a novel approach. The finite inclusions theorem for quasi-polynomials is derived. It allows to conclude on the stability of polytopic families of quasi-polynomials from a finite number of testing values. Index Terms—Graphical test, quasi-polynomials, robust stability, timedelay systems.
I. INTRODUCTION
By virtue of (5)
E e0 (t^ ) u(xt3^
)
=
u( x) 0 E
t^ 0
e0 s ff (xt3 ) + jc3t jm g dt
where f n g is a sequence of localizing stopping times for the local martingale. Letting n ! 1 and then t ! 1, we get J (c3 ) u(x) and c3 = (ct3 ) 2 A. Moreover, taking into account (18) with = m, we can see by the same calculation as before that
E [e0 t u(xt )] u(x)
0E
t 0
e0 s ff (xs ) + jcs jm g ds ; c 2 A:
E [ 01 e0 t jxt jm dt] < 1, which 0 t liminf t!1 E [e jxt jm ] = 0. Therefore, we deduce u(x) for all c 2 A. The proof is complete. By (4), we have
implies
J (c)
REFERENCES [1] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Boston, MA: Birkhäuser, 1997. [2] A. Bensoussan, Stochastic Control by Functional Analysis Methods. Amsterdam, The Netherlands: North Holland, 1982. [3] M. G. Crandall, H. Ishii, and P. L. Lions, “User’s guide to viscosity solutions of second order partial differential equations,” Bull. Amer. Math. Soc., vol. 27, pp. 1–67, 1992. [4] G. Da Prato, “Direct solution of a Riccati equation arising in stochastic control theory,” Appl. Math. Optim., vol. 11, pp. 191–208, 1984. [5] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions. New York: Springer Verlag, 1993. [6] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processess. Amsterdam, The Netherlands: North Holland, 1981. [7] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus. New York: Springer Verlag, 1988. [8] S. Koike and M. Morimoto, “On variational inequalities for leavable bounded-velocity control,” Appl. Math. Optim., vol. 48, pp. 1–20, 2003. [9] O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations. New York: Academic, 1968. [10] J. L. Menaldi and M. Robin, “On some cheap control problems for diffusion processes,” Trans. Amer. Math. Soc., vol. 278, pp. 771–802, 1983. [11] M. Nisio, Stochastic Control Theory, ISI Lecture Notes 9. New York: MacMillan, 1981.
Over the last decade, the problem of robust stability has received a sustained attention of the control community. An interesting approach to this problem is the following: Given a family, find a testing subfamily such that the stability of the subfamily implies that of the original family. Reducing the number of elements of the testing subfamily is the main issue of this approach. Many results on the topic are available for the case of polynomials which are central in the study of linear systems and some of them were extended to the case of quasi-polynomials. A strong motivation for our study of the robust stability of quasi-polynomial families comes from their connection with the stability of uncertain linear time delay systems [1], [8], [4]. Within this approach one can distinguish two main lines of research. The first one is based on the concept of convex directions [7], and on the edge theorem [3], and the second one is graphical, it is based on the frequency response plots and on the zero exclusion principle, see [3] (for an historical review on robustness of quasi-polynomials see [7] and the references therein). In this note, we study the robust stability of a polytopic family of retarded quasi-polynomials within the graphical framework. The main goal is to reduce the computational complexity associated with the evaluation of the value set of the family along the imaginary axis. A substantial reduction is achieved based on the following two ideas: The first one is the well known fact that the convexity of the polytopic family is inherited by its value set, which allows to restrict the search to the set of extreme quasi-polynomials of the family. The second one, the main contribution of this note, is an extension of finite inclusion/exclusion theorems obtained for polynomials independently in [9] and [2] to the case of quasi-polynomials. Indeed, this allows to perform the stability analysis of the quasi-polynomial family based on the location on the complex plane of the value set of the family computed for a finite number of frequencies only. We would like to emphasize the fact that a pessimistic opinion on a possibility of extension of the finite inclusion/exclusion type results to the case of nonpolynomial families has been expressed in [5].
Manuscript received December 19, 2003; revised April 22, 2005. Recommended by Associate Editor S.-I. Niculescu. This work was supported by CONACyT, México, under Project 41276-Y and Grant 130255. S. Mondié and V. L. Kharitonov are with the Department of Automatic Control, CINVESTAV-IPN, 07360 Mexico City, Mexico (e-mail: smondie@ ctrl.cinvestav.mx). J. Santos is with the Faculty of Mechanical-Electrical Engineering, Veracruzana University, 94740 Mendoza, Mexico. Digital Object Identifier 10.1109/TAC.2005.858649
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In this note, retarded type quasi-polynomials are considered. Without any loss of generality, they can be written in the form
f (s) =
m l=0
where 0 < 1 < 1 1 1 < m = are real polynomials of the form
pl (s)e s
(1)
0 are real numbers and where pk (s)
pl (s)= 0l + 1l s + 1 1 1 + n01;l sn01 ;
l =0; 1; . . . ; m 0 1
with pm (s) = 0m + a1m s + 1 1 1 + sn . The quasi-polynomial f (s) has a finite number of roots in any right half complex plane. In this note, f (s) is assumed to have no roots on the imaginary axis. The note is organized as follows: In Section II, it is shown that one can deduce the number of roots inside a Nyquist contour from the knowledge of the net change of the argument of the quasi-polynomial along a finite segment of the imaginary axis. In Section III, we generalize the finite Nyquist theorem, proven in [2] for polynomials, to the case of retarded type quasi-polynomials. More precisely, we prove that the stability of a retarded type quasi-polynomial can be deduced from the knowledge of the values of the argument of the quasi-polynomial at a finite set of frequencies. In Section IV, the finite inclusions theorem for the robustness analysis of polytopic families of quasi-polynomials is proven and an illustrative example is given in Section V. The note ends with some concluding remarks. II. PRELIMINARY RESULTS In this section, we derive the proof of the Nyquist theorem for quasi-polynomials because it plays a central role in this note. First, we obtain a bound on the contribution of [jR; +j 1) to the argument. Proposition 1: Let " be given and let " be the smallest positive root of the equation
C0 n + C1 n01 1 1 1 + Cn01 = sin "
2
(2)
where
0 Ck =
m l=0
j kl j ;
k = 0; 1; . . . ; n 0 1:
Then, the net change of the argument of f (j! ) for !
R 1=" is bounded by ".
(3)
2 [R; 1) with
Proof: The quasi-polynomial f (s) can be written as
f (s ) = s n 1 +
m n01 l=0 k=0
kl sk0n e s
(4)
and the argument of f (s) at any point s = jR, R > 0, on the imaginary axis is
arg(f (jR)) = n arg(jR) + arg(Z ) n01 k=0
k0n ej R . When
where Z = 1 + z and z = (jR) R tends to 1, arg(f (jR)) tends to n(=2) because arg(Z ) tends to 0. We observe that for R 1 the complex number Z belongs to the disk of radius = sin ("=2) centered at 1, depicted on Fig. 1, and we observe that m l=0 kl
jzj = sin 2" :
(5)
Fig. 1. Complex number Z .
The module of z satisfies
jz j =
n01 m k=0 l=0 n01 m k=0 l=0
kl (jR)k0n ej R
j kl j (jR)k0n ej R :
By using the facts that
ej R = 1; l = 0; 1; . . . ; m (jR)k0n = Rk0n ; k = 1; 2; . . . ; n and defining a new variable module of z satisfies
> 0 as = 1=R, we conclude that the
jzj C0 n + C1n01 + 1 1 1 + Cn01 = sin 2"
(6)
0" N 0 n 2 0 8[0;jR] "
(7)
where Ck , k = 0; 1; . . . ; n 0 1, are defined in (3). Finally, by solving (6), we find R = 1=" such that the contribution of [R; 1) to the change of the argument is bounded by ". We are now ready to show that the number of unstable roots of a quasi-polynomial of retarded type can be deduced from the knowledge of the change of the argument on the segment [0; jR] of the imaginary axis. Since we restrict our attention to the case of quasi-polynomials with real coefficients, the net change of the argument of f (s) on [0jR; 0] and [0; jR] are the same, so we can reduce the analysis to the set of positive frequencies. Theorem 2: Let " 2 [0; =2) be given and let 8[0;jR] denote the net change of argument of f (s) described by (1) along the finite segment [0; jR], where R is given in Proposition 1. Then, the unique integer N 0 that satisfies
is equal to the number of unstable roots of f (s). Proof: Consider the Nyquist contour which consists of a semicircle of radius R in the right half complex plane and the segment [jR; 0jR] of the imaginary axis. It follows from (4) that the argument of f (s) on the semicircle is
arg(f (Rej ))= n +arg 1+
m n01 l=0 k=0
kl (Rej )k0n e Re
:
Clearly, when R tends to infinity, the second summand tends to zero and we can conclude that when varies from 0 to =2 the net change of the argument of f (s) on the semicircle of the infinite radius is n(=2). The contribution of the positive imaginary semi-axis to the change of the argument can be decomposed into two parts, the first one corresponds to the contribution of the infinite segment [jR; j 1) where, according
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to Proposition 1, R is selected so that the contribution is bounded by " < =2. The second one, denoted by 8[0;jR] , corresponds to the segment [0; jR]. Then, the net change 8 of the argument accumulated on the upper part of the contour is such that
n
2 0 8[0;jR] 0 " 8 n 2 0 8[0;jR] + ":
n
2 0 8[0;jR] ":
n
202
< 8[0;jR] < n
2 + 2:
(9)
Proof: The result follows straighforwardly by substituting N = 0 in (7) and from the bound " < =2. Remark 4: Observe that the fact that m = 0 is crucial for obtaining a finite change of the argument on the semicircle of an infinite radius in Theorem 2. Nevertheless, the study of the stability of quasi-polynomials with m 6= 0 can be easily handled too, because multiplication of f (s) by a factor e0 s does not modify the set of roots. Remark 5: It is possible to analyze the existence of roots of the quasi-polynomial f (s) in a half plane at the right of a vertical line Re (s) = in the complex plane, by studying the stability of f (s 0 ) with the help of Theorem 2. This property is useful when a given stability margin < 0 is desired.
(11) (12)
2 0N 0 " 8[0;jR] n 2 0N + " 0 " < 2 : Since N is a nonnegative integer, 8[0;jR] is such that 8[0;jR] < n 2 + 2 :
(13)
In the following, we prove that n(=2) 0 =2 < 8[0;jR] . Observe first that without any loss of generality we may assume that all differences j +1 0 j have the same sign. If it is not the case and there exists j < r 0 2, such that j +2 0 j +1 and j +1 0 j have opposite sign then either
0 j+2 0 j+1 <
and 0 < j +1 0 j
0
or
0 j+2 0 j+1 < 0
and
0 < j+1 0 j :
In both cases
0 < j+2 0 j < holds. Moreover
r 0 1 =
=
(14)
r01 i=1 j 01 i=1
(i+1 0 i )
(i+1 0 i ) + j+2 0 j+1
+ j+1 0 j +
III. FINITE NYQUIST THEOREM FOR QUASI-POLYNOMIALS In the previous section, it was shown that the stability of a quasipolynomial f (s) can be deduced from the knowledge of the net change of the argument on a finite segment of the Nyquist contour. However, computation of the argument at an infinite number of frequencies on the imaginary axis is still needed. The question is whether or not the knowledge of the argument at a finite number of frequencies is sufficient to conclude on stability of f (s). The answer to this query is the extension of the Finite Nyquist Theorem for polynomials, see [2], to the case of quasi-polynomials. This extension is not straighforward because the proofs presented in [9] as well as in [2] are based on the fact that a polynomial of degree n may have at most n roots in the left half plane of the complex plane and, as a consequence, the total variation of the argument of a polynomial of degree n on the imaginary axis cannot exceed n. Certainly, this property cannot be exploited in the case of quasi-polynomials because in general they have an infinite number of roots in the left half complex plane. On the other hand, a retarded type quasi-polynomial can only have a finite number of roots in the right half complex plane, therefore one may provide for the quasi-polynomials a proof based on the analysis of the roots in this half plane that circumvents the difficulty, and which may be applied to polynomials as well.
(10)
Proof/Sufficiency: Observe first that it follows from condition (7) of Theorem 2 that
n
Finally, the fact that " < =2, implies that there exists only one number multiple of which satisfies (7). Next, an important particular case of Theorem 2 for the stability analysis of quasi-polynomials is derived. Corollary 3: Let 8[0;jR] be the contribution of the finite segment [0; jR] to the net change of argument of the quasi-polynomial f (s) described by (1). Here, R = 1==2 is given in Proposition 1. Then f (s) is stable if and only if
n 2 0 2 < r 0 1 < 2 + 2 8 1 i < r 0 1 : ji+1 0 i j 8 1 i r : f (j!i) 6= 0 8 1 i r : arg f (j!i) i (mod 2):
n
(8)
By the Argument Principle, we know that 8 is equal to N , where N is the number of roots of f (s) inside the open left half complex plane. It follows that
0" N 0
Theorem 6: Quasi-polynomial (1) is stable if and only if there exist an integer r 1, angles i 2 for 1 i r , and real numbers 0 !1 ; < !2 < 1 1 1 < !r such that
j 01
r01 i=j +2
(i+1 0 i )
= (i+1 0 i )+ j+2 0 j + i=1
r01
(i+1 0 i ):
i=j +2
(15)
So, we can exclude j +1 and !j +1 from our consideration. In this way, in a finite number of steps, we arrive at the situation when all the differences i+1 0 i , have the same sign. Now, it follows from (10) that r01
n
2 0 2 < r 0 1 = i=1 (i+1 0 i )
hence, the sum is necessarily a sum of positive differences and we have that
n
r01
(16) 2 0 2 < i=1 ji+1 0 i j : Now, the fact that the differences i+1 0 i , i = 1; . . . ; r 0 1 are
positive, implies that there exists ki r01 i=1
= 1; . . . ; r such that
ji+1 + 2ki+1 0 i 0 2ki j = 8[0;jR] :
(17)
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Combining (16) and (17) along with the fact that the inequality jX j jX + 2kj holds for all integer k and all X such that jX j , yields n 0 < 8 ;jR : 2 2 [0
]
Finally, we can conclude from the last inequality and (13) that
n 2 0 2 < 8[0;jR] < 2 + 2 and Corollary 3 implies the stability of f (s). n
Necessity: Now, we assume that the quasi-polynomial f (s) is stable. It follows from (1) and [12, Lemma 2.14] that f (0) is a positive real number, therefore, we can chose !1 = 0 and 0 = 0. Next, according to Corollary 3, n(=2) 0 =2 < 8[0;jR] < n(=2) + =2 and we can define
i = (i 0 1) ; i = 2; . . . ; n 0 1 2 : = (n 0 1) + < 8
2
n
[0;jR]
Observe that (10) and (11) are satisfied. Finally, let !i , i = 1; . . . ; n be the smallest value of the frequency such that arg f (j!i ) = i with !i > !i01 , i 2. Then, the result follows. Corollary 7: The minimum number of frequencies that are required in order to establish the stability of a given quasi-polynomial (1) is equal to m if n = 2m is even, and it is equal to m + 1 if n = 2m + 1 is odd. Proof: The result follows straighforwardly from (16). Remark 8: The previous gives an alternative proof of the finite Nyquist theorem for polynomials. Moreover, a consequence of the novel approach we use is that the finite Nyquist theorem may be proven for other classes of entire functions. Next, we give a corollary of Theorem 6 which is particularly well suited for robustness analysis purposes. Corollary 9: Quasi-polynomial 1 is stable if there exist an integer r 1, intervals (ai ; bi ) for 1 i r and real numbers 0 !1 < !2 < 1 1 1 < !r such that
n max b1 + n (18) 2 0 ar br 0 a1 0 2 2 8 1 i < r 0 1 : maxfbi 0 ai+1 ; bi+1 0 ai g (19) j 8 1 i r f (j!i) 2 Si = e j > 0; ai 0; ai < i < bi :
(24) (25)
Proof: The conditions (25) imply that fl (j!i ), l = 1; . . . ; T belong to the sectors Si satisfying (23) and (24). The convexity of the value sets VF (!i ), i = 1; . . . ; r which are polytopic families in the
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Fig. 2. Value set of
G at some frequencies.
complex plane with vertexes fl (j!i ), l = 1; . . . ; T implies that the value set T T V F (! i ) = l fl (j!i )jl 0; l = 1 l=1 l=1 belong also to the sectors Si and satisfy Corollary 9 and are stable, hence, the family F is robustly stable. Remark 14: All the previous results are easily extended to the case of quasi-polynomials with complex coefficients. This case is of interest in a number of theoretical applications as, for example, the robust analysis of strictly positive real transfer functions with delay and the control design of time delay systems subject to unstructured multiplicative uncertainty [10]. The main difference is that the analysis cannot be reduced to positive frequencies because the roots of quasi-polynomials with complex coefficients are not symmetric with respect to the real axis. Remark 15: The finite exclusion theorem for polynomials was proved independently in [9]. The novel approach of the proof of Theorem 6 allows to extend it to the case of quasi-polynomials with real or complex coefficients [10]. This result provides necessary and sufficient conditions for a very general type of uncertainty but its practical application is limited. V. ILLUSTRATIVE EXAMPLE Example 16: Consider the four vertex quasi-polynomials studied in [12]
f1 (s) = s3 + 102 s + 3 e0s 5
f 2 (s ) = s 3 + f 3 (s ) = s 3 +
15 2 35 4
8 5
2 s + 3 e0s 8
5 2 s + 3 e0s 2
15 f4 (s) = s3 + 10 2 s + 3 e0s : 4
The value-set of the quasi-polynomial family
G=
4 l=1
l fl ( s ) j l
0;
4 l=1
l
=1
TABLE I SECTORS DESCRIPTION
at the frequencies !0 = 0, !1 = 1:8 and !2 = 10:8, along with the Mikhailov diagram of the central quasi-polynomial (l = 1=4, l = 1; . . . ; 4) are depicted on Fig. 2. The three sectors are characterized precisely in Table I with the real numbers ai , bi and i = maxfbi 0 ai+1 ; bi+1 0 ai g. These sectors satisfy the conditions of Theorem 13, hence the quasi-polynomial family G is stable. VI. CONCLUDING REMARKS In this contribution, the finite Nyquist theorem for quasi-polynomials is proved by using a novel idea where no argument based on the total number of roots is employed. This result leads straightforwardly to the finite inclusions theorem that allows to reduce the stability analysis of a polytopic family of quasi-polynomials to the evaluation of a finite number of quasi-polynomials at a finite number of frequencies. Our current interests concern the possible of extensions of the finite Nyquist theorem/finite inclusions theorem to other classes of entire functions that our novel approach makes possible, the improvement of the practical selection of frequencies in the search procedure, and finally, the exploitation of these results for the design of robust controllers for time delay systems. REFERENCES [1] R. Bellman and K. L. Cooke, Differential-Difference Equations. New York: Academic, 1963. [2] T. Djaferis, Robust Control Design, a Polynomial Approach. Amherst, MA: Kluwer, 1995. [3] M. Fu, A. W. Olbrot, and M. P. Polis, “Robust stability for time-delay systems: The edge theorem and graphical tests,” IEEE Trans. Autom. Control, vol. 34, no. 8, pp. 813–819, Aug. 1989. [4] K. Gu, V. Kharitonov, and J. Chen, Stability of Time-Delay Systems. Boston, MA: Birkhäuser, 2003.
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[5] D. Kaminsky and T. Djaferis, “The finite inclusions theorem,” in Proc. 32nd Conf. Decision and Control, San Antonio, TX, 1993, pp. 508–517. [6] , “The finite inclusions theorem,” IEEE Trans. Autom. Control, vol. 40, no. 3, pp. 549–551, Mar. 1995. [7] V. Kharitonov and A. Zhabko, “Robust stability for time delay systems,” IEEE Trans. Autom. Control, vol. 39, no. 12, pp. 2388–2397, Dec. 1994. [8] S. I. Niculescu, Delays Effects on Stability, A Robust Control Approach. Heidelberg, Germany: Springer-Verlag, 2001. [9] A. Rantzer, “A finite zero exclusion principle, robust control of linear systems and nonlinear control,” in Proc. I. S. MTNS-89, vol. II, 1989, pp. 239–245. [10] J. Santos, S. Mondié, and V. Kharitonov, “The finite inclusions theorem/ finite zero exclusion principle for complex quasi-polynomials,” in Proc. 4th IFAC Workshop on Time Delay Systems, Rocquencourt, France, Sep. 8–10, 2003. [11] J. Santos, “Método para el análisis de la estabilidad robusta de sistemas con retardo,” Ph. D., Dept. Control Automatico, CINVESTAV-IPN, Mexico City, Mexico, 2004. [12] G. Stépan, Retarded Dynamical Systems: Stability and Characteristic Function, ser. Research Notes in Math. London, U.K.: Longman, 1989, vol. 210.
An Output Feedback Precompensator for Nonlinear DAE Systems With Control-Dependent State–Space Marie-Nathalie Contou-Carrere and Prodromos Daoutidis Abstract—This note considers singular systems of nonlinear differential and algebraic equations (DAEs) whose constrained state space depends on the control inputs. A state–space realization of such systems cannot be derived independently of the controller design. An output feedback precompensator is derived, which results in a modified DAE system whose state–space is invariant under any feedback control law and can be used for output feedback controller synthesis. Its application is illustrated by a nonlinear electrical circuit example. Index Terms—Differential and algebraic equations (DAEs), nonlinear control, nonlinear electrical circuit, singular systems.
I. INTRODUCTION Differential-algebraic-equation (DAE) systems of high-index (also referred to as semistate, descriptor, or singular systems) arise naturally as dynamic models of a wide range of engineering applications [1]–[3]. It is by now well-established that such systems behave fundamentally differently from systems of ordinary differential equations (ODEs) (see, e.g., [4]). In order to provide a measure of the difference between DAE and ODE systems, the notion of differential index is commonly used, which corresponds to the minimum number of differentiations of the algebraic equations required to obtain an equivalent ODE system [5]. The original algebraic equations, and those obtained after the differentiations, restrict the solution of such systems to a lower dimensional state–space. As a consequence, arbitrary initial conditions may lead to solutions with impulsive behavior. Moreover, these underlying constraints may depend on forcing inputs and their Manuscript received April 24, 2003; revised June 2, 2004 and April 26, 2005. Recommended by Associate Editor J. M. A. Scherpen. This work was supported in part by Grants NSF/CTS-0234440 and ACS/PRF-38114-AC9. The authors are with the Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2005.858650
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derivatives, thus, requiring them to be sufficiently smooth to guarantee the existence of smooth solutions. For nonlinear DAEs, the underlying constraints may also depend singularly on the forcing inputs, i.e., the system may fail to have a well-defined index for all smooth forcing functions (see, e.g., [3]). Control of linear DAEs has been studied extensively, either assuming arbitrary initial conditions with the objective of removing the impulsive behavior through feedback, or within the more conventional framework of consistent initial conditions yielding smooth solutions (see, e.g., the texts [6]–[8] and the more recent [9]–[11]). Control of nonlinear DAEs has mainly been addressed within the framework of smooth solutions (e.g., [12]–[16]); the key idea to the control of such systems is the derivation of an equivalent state–space realization that can be used for controller synthesis by application of well-known ODE control methods. Such an approach is suitable for nonlinear DAEs whose state–space is control-independent, i.e., for which the underlying constraints do not depend on the control inputs. However, a large class of applications (e.g., circuits with operational amplifiers [6], electrical circuit networks [17], chemical engineering examples [3]) are modeled by high-index DAEs that have a control-dependent state–space due to the presence of control inputs in the underlying constraints. A natural approach to control such systems [3] consists of using a feedback precompensator in order to obtain a system for which the underlying state–space is independent of the new control inputs. The design of such a state feedback precompensator of minimal order has been addressed in [3]. An alternative approach based on dynamic extension which however generally leads to high order compensators was proposed in [18]. In this note, we consider high-index nonlinear DAEs with controldependent state–space which are not controllable at infinity (i.e., no smooth feedback yields an index one closed-loop system). We address the derivation of a minimal order output feedback precompensator to obtain a new DAE system whose state–space does not depend on the new control inputs. We identify sufficient and necessary conditions for the solvability of this problem and construct a class of dynamic compensators that achieve this objective. We illustrate this approach with a nonlinear electrical circuit example. A preliminary version of this paper with a different example was presented at the IFAC World Congress in 2002 [19]. II. PRELIMINARIES We consider nonlinear DAE systems that have the following semiexplicit description [4]: x _
=
f (x)
+ b(x)z + g (x)u
0=
k (x )
+ l (x )z + c(x )u
=
h (x )
y
(1)
2 X IRn is the vector of differential variables, i.e., the
where x vector of variables for which differential equations are available, z 2 Z IRp is the vector of algebraic variables, i.e., the vector of variables that are determined by the algebraic equations (X ; Z are open connected sets), u 2 IRm is the vector of control inputs, f (x); k(x); h(x) are sufficiently smooth vector fields of dimensions n; p; m, respectively, and b(x); g (x); l(x); c(x) are smooth matrices of appropriate dimensions. Note that, in this description, the manipulated inputs u and the algebraic variables z appear in a linear fashion, which is typically the case in practical applications. Systems that are nonlinear in u and/or z can be recast in a linear form using standard dynamic extension techniques (see [20] and [3]). We assume that the system is high-index (i.e., rank [l(x)] = p1 < p, which implies that the algebraic equations are
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we can see (x3t )2 (zt )2 by the comparison theorem [6]. Hence, = 1. By the monotonicity of u0 (x) and '(x), the uniqueness of (28) holds. Thus, we conclude that (28) admits a unique strong solution 3 (xt ). Now, we apply Ito’s formula for convex functions [7, p. 219] to obtain
e0 t u(xt3 ) = u(x) + 3 0
+cs u +
t
+ 0
t
e0 s
1
2 x2 u00
0
2
0 u + Axu0 x=x
ds
e0 s xs3u0 (xs3 ) dws :
Robust Stability of Quasi-Polynomials and the Finite Inclusions Theorem S. Mondié, J. Santos, and V. L. Kharitonov Abstract—A semianalytic tool for the study of the robust stability of quasi-polynomials of retarded type is presented. The approach is based on graphical methods. The finite Nyquist theorem for polynomials is extended to the case of quasi-polynomials with the help of a novel approach. The finite inclusions theorem for quasi-polynomials is derived. It allows to conclude on the stability of polytopic families of quasi-polynomials from a finite number of testing values. Index Terms—Graphical test, quasi-polynomials, robust stability, timedelay systems.
I. INTRODUCTION
By virtue of (5)
E e0 (t^ ) u(xt3^
)
=
u( x) 0 E
t^ 0
e0 s ff (xt3 ) + jc3t jm g dt
where f n g is a sequence of localizing stopping times for the local martingale. Letting n ! 1 and then t ! 1, we get J (c3 ) u(x) and c3 = (ct3 ) 2 A. Moreover, taking into account (18) with = m, we can see by the same calculation as before that
E [e0 t u(xt )] u(x)
0E
t 0
e0 s ff (xs ) + jcs jm g ds ; c 2 A:
E [ 01 e0 t jxt jm dt] < 1, which 0 t liminf t!1 E [e jxt jm ] = 0. Therefore, we deduce u(x) for all c 2 A. The proof is complete. By (4), we have
implies
J (c)
REFERENCES [1] M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Boston, MA: Birkhäuser, 1997. [2] A. Bensoussan, Stochastic Control by Functional Analysis Methods. Amsterdam, The Netherlands: North Holland, 1982. [3] M. G. Crandall, H. Ishii, and P. L. Lions, “User’s guide to viscosity solutions of second order partial differential equations,” Bull. Amer. Math. Soc., vol. 27, pp. 1–67, 1992. [4] G. Da Prato, “Direct solution of a Riccati equation arising in stochastic control theory,” Appl. Math. Optim., vol. 11, pp. 191–208, 1984. [5] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions. New York: Springer Verlag, 1993. [6] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processess. Amsterdam, The Netherlands: North Holland, 1981. [7] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus. New York: Springer Verlag, 1988. [8] S. Koike and M. Morimoto, “On variational inequalities for leavable bounded-velocity control,” Appl. Math. Optim., vol. 48, pp. 1–20, 2003. [9] O. A. Ladyzhenskaya and N. N. Ural’tseva, Linear and Quasilinear Elliptic Equations. New York: Academic, 1968. [10] J. L. Menaldi and M. Robin, “On some cheap control problems for diffusion processes,” Trans. Amer. Math. Soc., vol. 278, pp. 771–802, 1983. [11] M. Nisio, Stochastic Control Theory, ISI Lecture Notes 9. New York: MacMillan, 1981.
Over the last decade, the problem of robust stability has received a sustained attention of the control community. An interesting approach to this problem is the following: Given a family, find a testing subfamily such that the stability of the subfamily implies that of the original family. Reducing the number of elements of the testing subfamily is the main issue of this approach. Many results on the topic are available for the case of polynomials which are central in the study of linear systems and some of them were extended to the case of quasi-polynomials. A strong motivation for our study of the robust stability of quasi-polynomial families comes from their connection with the stability of uncertain linear time delay systems [1], [8], [4]. Within this approach one can distinguish two main lines of research. The first one is based on the concept of convex directions [7], and on the edge theorem [3], and the second one is graphical, it is based on the frequency response plots and on the zero exclusion principle, see [3] (for an historical review on robustness of quasi-polynomials see [7] and the references therein). In this note, we study the robust stability of a polytopic family of retarded quasi-polynomials within the graphical framework. The main goal is to reduce the computational complexity associated with the evaluation of the value set of the family along the imaginary axis. A substantial reduction is achieved based on the following two ideas: The first one is the well known fact that the convexity of the polytopic family is inherited by its value set, which allows to restrict the search to the set of extreme quasi-polynomials of the family. The second one, the main contribution of this note, is an extension of finite inclusion/exclusion theorems obtained for polynomials independently in [9] and [2] to the case of quasi-polynomials. Indeed, this allows to perform the stability analysis of the quasi-polynomial family based on the location on the complex plane of the value set of the family computed for a finite number of frequencies only. We would like to emphasize the fact that a pessimistic opinion on a possibility of extension of the finite inclusion/exclusion type results to the case of nonpolynomial families has been expressed in [5].
Manuscript received December 19, 2003; revised April 22, 2005. Recommended by Associate Editor S.-I. Niculescu. This work was supported by CONACyT, México, under Project 41276-Y and Grant 130255. S. Mondié and V. L. Kharitonov are with the Department of Automatic Control, CINVESTAV-IPN, 07360 Mexico City, Mexico (e-mail: smondie@ ctrl.cinvestav.mx). J. Santos is with the Faculty of Mechanical-Electrical Engineering, Veracruzana University, 94740 Mendoza, Mexico. Digital Object Identifier 10.1109/TAC.2005.858649
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In this note, retarded type quasi-polynomials are considered. Without any loss of generality, they can be written in the form
f (s) =
m l=0
where 0 < 1 < 1 1 1 < m = are real polynomials of the form
pl (s)e s
(1)
0 are real numbers and where pk (s)
pl (s)= 0l + 1l s + 1 1 1 + n01;l sn01 ;
l =0; 1; . . . ; m 0 1
with pm (s) = 0m + a1m s + 1 1 1 + sn . The quasi-polynomial f (s) has a finite number of roots in any right half complex plane. In this note, f (s) is assumed to have no roots on the imaginary axis. The note is organized as follows: In Section II, it is shown that one can deduce the number of roots inside a Nyquist contour from the knowledge of the net change of the argument of the quasi-polynomial along a finite segment of the imaginary axis. In Section III, we generalize the finite Nyquist theorem, proven in [2] for polynomials, to the case of retarded type quasi-polynomials. More precisely, we prove that the stability of a retarded type quasi-polynomial can be deduced from the knowledge of the values of the argument of the quasi-polynomial at a finite set of frequencies. In Section IV, the finite inclusions theorem for the robustness analysis of polytopic families of quasi-polynomials is proven and an illustrative example is given in Section V. The note ends with some concluding remarks. II. PRELIMINARY RESULTS In this section, we derive the proof of the Nyquist theorem for quasi-polynomials because it plays a central role in this note. First, we obtain a bound on the contribution of [jR; +j 1) to the argument. Proposition 1: Let " be given and let " be the smallest positive root of the equation
C0 n + C1 n01 1 1 1 + Cn01 = sin "
2
(2)
where
0 Ck =
m l=0
j kl j ;
k = 0; 1; . . . ; n 0 1:
Then, the net change of the argument of f (j! ) for !
R 1=" is bounded by ".
(3)
2 [R; 1) with
Proof: The quasi-polynomial f (s) can be written as
f (s ) = s n 1 +
m n01 l=0 k=0
kl sk0n e s
(4)
and the argument of f (s) at any point s = jR, R > 0, on the imaginary axis is
arg(f (jR)) = n arg(jR) + arg(Z ) n01 k=0
k0n ej R . When
where Z = 1 + z and z = (jR) R tends to 1, arg(f (jR)) tends to n(=2) because arg(Z ) tends to 0. We observe that for R 1 the complex number Z belongs to the disk of radius = sin ("=2) centered at 1, depicted on Fig. 1, and we observe that m l=0 kl
jzj = sin 2" :
(5)
Fig. 1. Complex number Z .
The module of z satisfies
jz j =
n01 m k=0 l=0 n01 m k=0 l=0
kl (jR)k0n ej R
j kl j (jR)k0n ej R :
By using the facts that
ej R = 1; l = 0; 1; . . . ; m (jR)k0n = Rk0n ; k = 1; 2; . . . ; n and defining a new variable module of z satisfies
> 0 as = 1=R, we conclude that the
jzj C0 n + C1n01 + 1 1 1 + Cn01 = sin 2"
(6)
0" N 0 n 2 0 8[0;jR] "
(7)
where Ck , k = 0; 1; . . . ; n 0 1, are defined in (3). Finally, by solving (6), we find R = 1=" such that the contribution of [R; 1) to the change of the argument is bounded by ". We are now ready to show that the number of unstable roots of a quasi-polynomial of retarded type can be deduced from the knowledge of the change of the argument on the segment [0; jR] of the imaginary axis. Since we restrict our attention to the case of quasi-polynomials with real coefficients, the net change of the argument of f (s) on [0jR; 0] and [0; jR] are the same, so we can reduce the analysis to the set of positive frequencies. Theorem 2: Let " 2 [0; =2) be given and let 8[0;jR] denote the net change of argument of f (s) described by (1) along the finite segment [0; jR], where R is given in Proposition 1. Then, the unique integer N 0 that satisfies
is equal to the number of unstable roots of f (s). Proof: Consider the Nyquist contour which consists of a semicircle of radius R in the right half complex plane and the segment [jR; 0jR] of the imaginary axis. It follows from (4) that the argument of f (s) on the semicircle is
arg(f (Rej ))= n +arg 1+
m n01 l=0 k=0
kl (Rej )k0n e Re
:
Clearly, when R tends to infinity, the second summand tends to zero and we can conclude that when varies from 0 to =2 the net change of the argument of f (s) on the semicircle of the infinite radius is n(=2). The contribution of the positive imaginary semi-axis to the change of the argument can be decomposed into two parts, the first one corresponds to the contribution of the infinite segment [jR; j 1) where, according
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to Proposition 1, R is selected so that the contribution is bounded by " < =2. The second one, denoted by 8[0;jR] , corresponds to the segment [0; jR]. Then, the net change 8 of the argument accumulated on the upper part of the contour is such that
n
2 0 8[0;jR] 0 " 8 n 2 0 8[0;jR] + ":
n
2 0 8[0;jR] ":
n
202
< 8[0;jR] < n
2 + 2:
(9)
Proof: The result follows straighforwardly by substituting N = 0 in (7) and from the bound " < =2. Remark 4: Observe that the fact that m = 0 is crucial for obtaining a finite change of the argument on the semicircle of an infinite radius in Theorem 2. Nevertheless, the study of the stability of quasi-polynomials with m 6= 0 can be easily handled too, because multiplication of f (s) by a factor e0 s does not modify the set of roots. Remark 5: It is possible to analyze the existence of roots of the quasi-polynomial f (s) in a half plane at the right of a vertical line Re (s) = in the complex plane, by studying the stability of f (s 0 ) with the help of Theorem 2. This property is useful when a given stability margin < 0 is desired.
(11) (12)
2 0N 0 " 8[0;jR] n 2 0N + " 0 " < 2 : Since N is a nonnegative integer, 8[0;jR] is such that 8[0;jR] < n 2 + 2 :
(13)
In the following, we prove that n(=2) 0 =2 < 8[0;jR] . Observe first that without any loss of generality we may assume that all differences j +1 0 j have the same sign. If it is not the case and there exists j < r 0 2, such that j +2 0 j +1 and j +1 0 j have opposite sign then either
0 j+2 0 j+1 <
and 0 < j +1 0 j
0
or
0 j+2 0 j+1 < 0
and
0 < j+1 0 j :
In both cases
0 < j+2 0 j < holds. Moreover
r 0 1 =
=
(14)
r01 i=1 j 01 i=1
(i+1 0 i )
(i+1 0 i ) + j+2 0 j+1
+ j+1 0 j +
III. FINITE NYQUIST THEOREM FOR QUASI-POLYNOMIALS In the previous section, it was shown that the stability of a quasipolynomial f (s) can be deduced from the knowledge of the net change of the argument on a finite segment of the Nyquist contour. However, computation of the argument at an infinite number of frequencies on the imaginary axis is still needed. The question is whether or not the knowledge of the argument at a finite number of frequencies is sufficient to conclude on stability of f (s). The answer to this query is the extension of the Finite Nyquist Theorem for polynomials, see [2], to the case of quasi-polynomials. This extension is not straighforward because the proofs presented in [9] as well as in [2] are based on the fact that a polynomial of degree n may have at most n roots in the left half plane of the complex plane and, as a consequence, the total variation of the argument of a polynomial of degree n on the imaginary axis cannot exceed n. Certainly, this property cannot be exploited in the case of quasi-polynomials because in general they have an infinite number of roots in the left half complex plane. On the other hand, a retarded type quasi-polynomial can only have a finite number of roots in the right half complex plane, therefore one may provide for the quasi-polynomials a proof based on the analysis of the roots in this half plane that circumvents the difficulty, and which may be applied to polynomials as well.
(10)
Proof/Sufficiency: Observe first that it follows from condition (7) of Theorem 2 that
n
Finally, the fact that " < =2, implies that there exists only one number multiple of which satisfies (7). Next, an important particular case of Theorem 2 for the stability analysis of quasi-polynomials is derived. Corollary 3: Let 8[0;jR] be the contribution of the finite segment [0; jR] to the net change of argument of the quasi-polynomial f (s) described by (1). Here, R = 1==2 is given in Proposition 1. Then f (s) is stable if and only if
n 2 0 2 < r 0 1 < 2 + 2 8 1 i < r 0 1 : ji+1 0 i j 8 1 i r : f (j!i) 6= 0 8 1 i r : arg f (j!i) i (mod 2):
n
(8)
By the Argument Principle, we know that 8 is equal to N , where N is the number of roots of f (s) inside the open left half complex plane. It follows that
0" N 0
Theorem 6: Quasi-polynomial (1) is stable if and only if there exist an integer r 1, angles i 2 for 1 i r , and real numbers 0 !1 ; < !2 < 1 1 1 < !r such that
j 01
r01 i=j +2
(i+1 0 i )
= (i+1 0 i )+ j+2 0 j + i=1
r01
(i+1 0 i ):
i=j +2
(15)
So, we can exclude j +1 and !j +1 from our consideration. In this way, in a finite number of steps, we arrive at the situation when all the differences i+1 0 i , have the same sign. Now, it follows from (10) that r01
n
2 0 2 < r 0 1 = i=1 (i+1 0 i )
hence, the sum is necessarily a sum of positive differences and we have that
n
r01
(16) 2 0 2 < i=1 ji+1 0 i j : Now, the fact that the differences i+1 0 i , i = 1; . . . ; r 0 1 are
positive, implies that there exists ki r01 i=1
= 1; . . . ; r such that
ji+1 + 2ki+1 0 i 0 2ki j = 8[0;jR] :
(17)
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Combining (16) and (17) along with the fact that the inequality jX j jX + 2kj holds for all integer k and all X such that jX j , yields n 0 < 8 ;jR : 2 2 [0
]
Finally, we can conclude from the last inequality and (13) that
n 2 0 2 < 8[0;jR] < 2 + 2 and Corollary 3 implies the stability of f (s). n
Necessity: Now, we assume that the quasi-polynomial f (s) is stable. It follows from (1) and [12, Lemma 2.14] that f (0) is a positive real number, therefore, we can chose !1 = 0 and 0 = 0. Next, according to Corollary 3, n(=2) 0 =2 < 8[0;jR] < n(=2) + =2 and we can define
i = (i 0 1) ; i = 2; . . . ; n 0 1 2 : = (n 0 1) + < 8
2
n
[0;jR]
Observe that (10) and (11) are satisfied. Finally, let !i , i = 1; . . . ; n be the smallest value of the frequency such that arg f (j!i ) = i with !i > !i01 , i 2. Then, the result follows. Corollary 7: The minimum number of frequencies that are required in order to establish the stability of a given quasi-polynomial (1) is equal to m if n = 2m is even, and it is equal to m + 1 if n = 2m + 1 is odd. Proof: The result follows straighforwardly from (16). Remark 8: The previous gives an alternative proof of the finite Nyquist theorem for polynomials. Moreover, a consequence of the novel approach we use is that the finite Nyquist theorem may be proven for other classes of entire functions. Next, we give a corollary of Theorem 6 which is particularly well suited for robustness analysis purposes. Corollary 9: Quasi-polynomial 1 is stable if there exist an integer r 1, intervals (ai ; bi ) for 1 i r and real numbers 0 !1 < !2 < 1 1 1 < !r such that
n max b1 + n (18) 2 0 ar br 0 a1 0 2 2 8 1 i < r 0 1 : maxfbi 0 ai+1 ; bi+1 0 ai g (19) j 8 1 i r f (j!i) 2 Si = e j > 0; ai 0; ai < i < bi :
(24) (25)
Proof: The conditions (25) imply that fl (j!i ), l = 1; . . . ; T belong to the sectors Si satisfying (23) and (24). The convexity of the value sets VF (!i ), i = 1; . . . ; r which are polytopic families in the
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Fig. 2. Value set of
G at some frequencies.
complex plane with vertexes fl (j!i ), l = 1; . . . ; T implies that the value set T T V F (! i ) = l fl (j!i )jl 0; l = 1 l=1 l=1 belong also to the sectors Si and satisfy Corollary 9 and are stable, hence, the family F is robustly stable. Remark 14: All the previous results are easily extended to the case of quasi-polynomials with complex coefficients. This case is of interest in a number of theoretical applications as, for example, the robust analysis of strictly positive real transfer functions with delay and the control design of time delay systems subject to unstructured multiplicative uncertainty [10]. The main difference is that the analysis cannot be reduced to positive frequencies because the roots of quasi-polynomials with complex coefficients are not symmetric with respect to the real axis. Remark 15: The finite exclusion theorem for polynomials was proved independently in [9]. The novel approach of the proof of Theorem 6 allows to extend it to the case of quasi-polynomials with real or complex coefficients [10]. This result provides necessary and sufficient conditions for a very general type of uncertainty but its practical application is limited. V. ILLUSTRATIVE EXAMPLE Example 16: Consider the four vertex quasi-polynomials studied in [12]
f1 (s) = s3 + 102 s + 3 e0s 5
f 2 (s ) = s 3 + f 3 (s ) = s 3 +
15 2 35 4
8 5
2 s + 3 e0s 8
5 2 s + 3 e0s 2
15 f4 (s) = s3 + 10 2 s + 3 e0s : 4
The value-set of the quasi-polynomial family
G=
4 l=1
l fl ( s ) j l
0;
4 l=1
l
=1
TABLE I SECTORS DESCRIPTION
at the frequencies !0 = 0, !1 = 1:8 and !2 = 10:8, along with the Mikhailov diagram of the central quasi-polynomial (l = 1=4, l = 1; . . . ; 4) are depicted on Fig. 2. The three sectors are characterized precisely in Table I with the real numbers ai , bi and i = maxfbi 0 ai+1 ; bi+1 0 ai g. These sectors satisfy the conditions of Theorem 13, hence the quasi-polynomial family G is stable. VI. CONCLUDING REMARKS In this contribution, the finite Nyquist theorem for quasi-polynomials is proved by using a novel idea where no argument based on the total number of roots is employed. This result leads straightforwardly to the finite inclusions theorem that allows to reduce the stability analysis of a polytopic family of quasi-polynomials to the evaluation of a finite number of quasi-polynomials at a finite number of frequencies. Our current interests concern the possible of extensions of the finite Nyquist theorem/finite inclusions theorem to other classes of entire functions that our novel approach makes possible, the improvement of the practical selection of frequencies in the search procedure, and finally, the exploitation of these results for the design of robust controllers for time delay systems. REFERENCES [1] R. Bellman and K. L. Cooke, Differential-Difference Equations. New York: Academic, 1963. [2] T. Djaferis, Robust Control Design, a Polynomial Approach. Amherst, MA: Kluwer, 1995. [3] M. Fu, A. W. Olbrot, and M. P. Polis, “Robust stability for time-delay systems: The edge theorem and graphical tests,” IEEE Trans. Autom. Control, vol. 34, no. 8, pp. 813–819, Aug. 1989. [4] K. Gu, V. Kharitonov, and J. Chen, Stability of Time-Delay Systems. Boston, MA: Birkhäuser, 2003.
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[5] D. Kaminsky and T. Djaferis, “The finite inclusions theorem,” in Proc. 32nd Conf. Decision and Control, San Antonio, TX, 1993, pp. 508–517. [6] , “The finite inclusions theorem,” IEEE Trans. Autom. Control, vol. 40, no. 3, pp. 549–551, Mar. 1995. [7] V. Kharitonov and A. Zhabko, “Robust stability for time delay systems,” IEEE Trans. Autom. Control, vol. 39, no. 12, pp. 2388–2397, Dec. 1994. [8] S. I. Niculescu, Delays Effects on Stability, A Robust Control Approach. Heidelberg, Germany: Springer-Verlag, 2001. [9] A. Rantzer, “A finite zero exclusion principle, robust control of linear systems and nonlinear control,” in Proc. I. S. MTNS-89, vol. II, 1989, pp. 239–245. [10] J. Santos, S. Mondié, and V. Kharitonov, “The finite inclusions theorem/ finite zero exclusion principle for complex quasi-polynomials,” in Proc. 4th IFAC Workshop on Time Delay Systems, Rocquencourt, France, Sep. 8–10, 2003. [11] J. Santos, “Método para el análisis de la estabilidad robusta de sistemas con retardo,” Ph. D., Dept. Control Automatico, CINVESTAV-IPN, Mexico City, Mexico, 2004. [12] G. Stépan, Retarded Dynamical Systems: Stability and Characteristic Function, ser. Research Notes in Math. London, U.K.: Longman, 1989, vol. 210.
An Output Feedback Precompensator for Nonlinear DAE Systems With Control-Dependent State–Space Marie-Nathalie Contou-Carrere and Prodromos Daoutidis Abstract—This note considers singular systems of nonlinear differential and algebraic equations (DAEs) whose constrained state space depends on the control inputs. A state–space realization of such systems cannot be derived independently of the controller design. An output feedback precompensator is derived, which results in a modified DAE system whose state–space is invariant under any feedback control law and can be used for output feedback controller synthesis. Its application is illustrated by a nonlinear electrical circuit example. Index Terms—Differential and algebraic equations (DAEs), nonlinear control, nonlinear electrical circuit, singular systems.
I. INTRODUCTION Differential-algebraic-equation (DAE) systems of high-index (also referred to as semistate, descriptor, or singular systems) arise naturally as dynamic models of a wide range of engineering applications [1]–[3]. It is by now well-established that such systems behave fundamentally differently from systems of ordinary differential equations (ODEs) (see, e.g., [4]). In order to provide a measure of the difference between DAE and ODE systems, the notion of differential index is commonly used, which corresponds to the minimum number of differentiations of the algebraic equations required to obtain an equivalent ODE system [5]. The original algebraic equations, and those obtained after the differentiations, restrict the solution of such systems to a lower dimensional state–space. As a consequence, arbitrary initial conditions may lead to solutions with impulsive behavior. Moreover, these underlying constraints may depend on forcing inputs and their Manuscript received April 24, 2003; revised June 2, 2004 and April 26, 2005. Recommended by Associate Editor J. M. A. Scherpen. This work was supported in part by Grants NSF/CTS-0234440 and ACS/PRF-38114-AC9. The authors are with the Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2005.858650
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derivatives, thus, requiring them to be sufficiently smooth to guarantee the existence of smooth solutions. For nonlinear DAEs, the underlying constraints may also depend singularly on the forcing inputs, i.e., the system may fail to have a well-defined index for all smooth forcing functions (see, e.g., [3]). Control of linear DAEs has been studied extensively, either assuming arbitrary initial conditions with the objective of removing the impulsive behavior through feedback, or within the more conventional framework of consistent initial conditions yielding smooth solutions (see, e.g., the texts [6]–[8] and the more recent [9]–[11]). Control of nonlinear DAEs has mainly been addressed within the framework of smooth solutions (e.g., [12]–[16]); the key idea to the control of such systems is the derivation of an equivalent state–space realization that can be used for controller synthesis by application of well-known ODE control methods. Such an approach is suitable for nonlinear DAEs whose state–space is control-independent, i.e., for which the underlying constraints do not depend on the control inputs. However, a large class of applications (e.g., circuits with operational amplifiers [6], electrical circuit networks [17], chemical engineering examples [3]) are modeled by high-index DAEs that have a control-dependent state–space due to the presence of control inputs in the underlying constraints. A natural approach to control such systems [3] consists of using a feedback precompensator in order to obtain a system for which the underlying state–space is independent of the new control inputs. The design of such a state feedback precompensator of minimal order has been addressed in [3]. An alternative approach based on dynamic extension which however generally leads to high order compensators was proposed in [18]. In this note, we consider high-index nonlinear DAEs with controldependent state–space which are not controllable at infinity (i.e., no smooth feedback yields an index one closed-loop system). We address the derivation of a minimal order output feedback precompensator to obtain a new DAE system whose state–space does not depend on the new control inputs. We identify sufficient and necessary conditions for the solvability of this problem and construct a class of dynamic compensators that achieve this objective. We illustrate this approach with a nonlinear electrical circuit example. A preliminary version of this paper with a different example was presented at the IFAC World Congress in 2002 [19]. II. PRELIMINARIES We consider nonlinear DAE systems that have the following semiexplicit description [4]: x _
=
f (x)
+ b(x)z + g (x)u
0=
k (x )
+ l (x )z + c(x )u
=
h (x )
y
(1)
2 X IRn is the vector of differential variables, i.e., the
where x vector of variables for which differential equations are available, z 2 Z IRp is the vector of algebraic variables, i.e., the vector of variables that are determined by the algebraic equations (X ; Z are open connected sets), u 2 IRm is the vector of control inputs, f (x); k(x); h(x) are sufficiently smooth vector fields of dimensions n; p; m, respectively, and b(x); g (x); l(x); c(x) are smooth matrices of appropriate dimensions. Note that, in this description, the manipulated inputs u and the algebraic variables z appear in a linear fashion, which is typically the case in practical applications. Systems that are nonlinear in u and/or z can be recast in a linear form using standard dynamic extension techniques (see [20] and [3]). We assume that the system is high-index (i.e., rank [l(x)] = p1 < p, which implies that the algebraic equations are
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