IV. ANILLUSTRATIVEEXAMPLE ANDCONCLUSION We consider inner–outer factorization for
G(s) = s(s 0 2)s(s 0 2) (s + 1)1 3: (40)
This example is from [12]. A minimal realization with identity observ-ability gramian is found forG(s) (with four decimal points)
A = 00:1948 01:12640:7476 00:1841 2:15350:9221 00:7242 0:4672 02:6210 B = 0:5458 00:3521 0:6732 CT = 00:4414 00:441400:4291 00:4291 1:6190 1:6190 : Following Step 4 of the algorithm, an orthogonal matrixU can be easily obtained which is listed as follows, together with the results of Step 3 of the algorithm: U = 0:3764 00:7196 0:5835 0:9105 0:1711 00:3764 0:1711 0:6729 0:7196 V = 0:70710:7071 F2 = 1:5119:
SinceF2 6= 0 and m = 1, we need compute a stabilizing solution to
(32). However, due to the existence of zeros at the origin, the stabilizing solution to (32) does not exist. Thus, we used the same method as in [12] to compute a semistabilizing solution, andF1, which are given by
Z = 00:200 720:042 12 00:200 720:956 51 F1= [ 0:014 66 00:069 86 ] :
The state–space realization for the inner factor is obtained as ^
A = 00:091 320:400 54 01:908 680:435 18 B =^ 01:875 000:812 50 ^
C = 00:290 25 1:383 1200:290 25 1:383 12 D = V =^ 0:70710:7071 : Clearly, the mode corresponding to zero eigenvalue of ^A is not ob-servable. Indeed, by transforming the state–space realization into the transfer matrix, we find that
Gi(s) = 11 0:7071s
20 1:4146s
s2+ 2s =
1
1 p2(s + 2)s 0 2 identical to the result in [12].
To conclude this note, we would like to point out that for the case det(F2) = 0, the matrix pencil method as in [1], [4], and [5] can also
be used to compute the solutionF1. However, it is not easy to program, in terms of the accuracy of the stabilizing solution to (29). The method from [11] seems to be more effective in this regard. It should also be pointed out that for the casedet(F2) = 0, the inner factor Gi(s), as in
(37), represents a singular system. Interested readers are referred to [9] for a reduction of singular systems to regular systems using the singular perturbation method.
ACKNOWLEDGMENT
The author would like to thank the reviewer for bringing to his at-tention references [8] and [12].
REFERENCES
[1] W. F. Arnold and A. J. Laub, “Generalized eigen-problem algorithms and software for algebraic Riccati equations,” Proc. IEEE, vol. 72, pp. 1746–1754, 1984.
[2] T. Chen and B. Francis, “Spectral and inner–outer factorizations of ra-tional matrices,” SIAM J. Matrix Anal. Appl., vol. 10, pp. 1–17, 1989. [3] C. C. Chu and J. C. Doyle, “On inner–outer and spectral factorization,”
Proc. IEEE Conf. Decision and Control, 1984.
[4] D. J. Clements and K. Glover, “Spectral factorization via Hermitian pencil,” Linear Alg. Appl., pp. 797–846, 1989.
[5] P. Van Dooren, “A generalized eigenvalue approach for solving Riccati equations,” SIAM J. Sci. Stat. Comput., vol. 2, pp. 121–135, 1981. [6] B. A. Francis, A Course inH Control Theory. Berlin, Germany:
Springer-Verlag, 1987, vol. 88, Lecture Notes in Control and Informa-tion Sciences.
[7] K. Glover, “All optimal Hankel-norm approximation of linear multivari-able systems and theirL -error bounds,” Int. J. Control, vol. 39, pp. 1115–1193, 1984.
[8] S. Hara and T. Sugie, “Inner–outer factorization for strictly proper func-tions withj! -axis zeros,” Syst. Control Lett., vol. 16, pp. 179–185, 1991.
[9] P. Kokotovic, H. K. Khalil, and J. O’Reilly, Singular Perturbation
Methods in Control. Philadelphia, PA: SIAM, 1999.
[10] M. Vidyasagar, Control System Synthesis: A Factorization
Ap-proach. Cambridge, MA: MIT Press, 1985.
[11] Y.-Y. Wang, S.-J. Shi, and Z.-J. Zhang, “A descriptor-system approach to singular perturbation of linear regulators,” IEEE Trans. Automat. Contr., vol. 33, pp. 370–373, Apr. 1988.
[12] X. Xin and T. Mita, “Inner–outer factorization for nonsquare proper functions with infinite and finitej!-zeros,” Int. J. Control, vol. 71, pp. 145–161, 1998.
Control for Nonlinear Descriptor Systems
He-Sheng Wang, Chee-Fai Yung, and Fan-Ren Chang
Abstract—In this note, we study the control problem for nonlinear descriptor systems governed by a set of differential-algebraic equations
(DAEs) of the form _ = ( ), = ( ), =
( ), where is, in general, a singular matrix. Necessary and sufficient conditions are derived for the existence of a controller solving the problem. We first give various sufficient conditions for the solvability of control problem for DAEs. Both state-feedback and output-feedback cases are considered. Then, necessary conditions for the output feedback control problem to be solvable are obtained in terms of two Hamilton–Ja-cobi inequalities plus a weak coupling condition. Moreover, a parameteri-zation of a family of output feedback controllers solving the problem is also provided.
Index Terms—Descriptor Systems, differential games, differential-alge-braic equations (DAEs), dissipation inequalities.
I. INTRODUCTION
For the purpose of control, nonlinear descriptor systems are fre-quently described by a set of differential-algebraic equations (DAEs)
Manuscript received September 1, 1998; revised July 10, 2000 and July 16, 2001. Recommended by Associate Editor H. Huijberts.
H.-S. Wang is with the Department of Guidance and Communications Technology, National Taiwan Ocean University, Keelung 202, Taiwan (e-mail: hswang@mail.ntou.edu.tw).
C.-F. Yung is with the Department of Electrical Engineering, National Taiwan Ocean University, Keelung 202, Taiwan (e-mail: yung@mail.ntou.edu.tw).
F.-R. Chang is with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan (e-mail: frchang@ac.ee.ntu.edu.tw).
Digital Object Identifier 10.1109/TAC.2002.804465 0018-9286/02$17.00 © 2002 IEEE
of the form
_xxx1=f(xxx1; xxx2; u) (1)
0 =g(xxx1; xxx2; u) (2)
or in a more compact formE _xxx = F (xxx; u); where E = I 0
0 0 and
xxx = [xxxT
1 xxxT2]T 4= col(x1; . . . ; xn) are local coordinates for an
n-di-mensional state–space manifoldX . In the state–space X , dynamic state variablesxxx1and instantaneous state variablesxxx2are distinguished. The dynamics of the statesxxx1is directly defined by (1), while the dynamics ofxxx2is such that the system satisfies the constraint (2). In many cases, the algebraic constraint (2) of the full DAEs can be eliminated (usu-ally due to the consistence of initial conditions). As a consequence, the DAEs reduce to a well-known state-variable system. Nevertheless, in some cases this kind of elimination is not possible (often due to in-consistent initial conditions), since it may result in loss of accuracy or loss of necessary information. A large class of physical systems can be modeled by this kind of DAEs. The paper by Newcomb and Dziurla [6] gives many practical examples, including circuit and system design, robotics, neural network, etc., and presents an excellent review on non-linear DAEs. Many other applications of DAEs, as well as numerical treatment, can be found in [2].
In this note, we investigate the contractive property of DAEs, namely theH1control problem. Our note is mainly divided into two parts. The first part concerns various sufficient conditions for the solvability of the H1control problem. Both state feedback and output feedback cases are considered. We seek sufficient conditions under which a given DAE has anL2gain no greater than a prescribed positive number with in-ternal stability, and in the mean time, eliminates possible impulse dy-namics and other singularity-induced nonlinearity of the system. We will derive a family of output feedback controllers solving theH1 control problem. The underlying ideas are differential games and dissi-pation inequalities. These ideas were also used by Isidori and Kang [3] and Yung et al. [11], in which they have given the central controller and a family of controllers, respectively, solving theH1output feedback control problem for general nonlinear systems in usual state-variable form, i.e., theE matrix is nonsingular.
The second part of this note is devoted to a converse result, namely the derivation of necessary conditions for solutions of local disturbance attenuation to exist. We obtain necessary conditions given in terms of the existence of nonnegative solutions to two Hamilton–Jacobi inequal-ities, together with a weak coupling condition. A similar result has pre-viously been published in [1] for (W -)input affine nonlinear systems with nonsingularE matrix. In a recent monograph [7], among many other important contributions, van der Schaft addressed a number of issues related to necessary conditions for solutions of local disturbance attenuation to exist (see also [3] for some related work). Our results can be thought of as a parallel extension of the results of [1] and [7] to the DAEs case. As a matter of fact, the results in this note reduce to the ones given in [1] for state–space systems.
This note is organized as follows. In Section II, we will review some notions of DAEs together with some preliminary results for the theory of DAEs, including stability theory and dissipativity. Our main results will be summarized in Sections III and IV. We will concentrate on the output feedback case. However, for the sake of completeness, we will first investigate the state feedback control. We also give a parameteri-zaton of a family of output feedback controllers. In Section IV, we will give a necessary condition for theH1output feedback control problem to be solvable.
II. ELEMENTS FORNONLINEARDESCRIPTORSYSTEMS Consider the following DAE:
E _xxx(t) = F (xxx; u); u 2 U IRm (3)
wherexxx4= col(x1; . . . ; xn) are local coordinates for an n-dimensional
state-space manifoldX . E is a constant matrix and F (0; 0) = 0. The constant matrixE 2 IRn2nis, in general, a singular matrix with rank E = r < n. Without loss of generality, we can assume that
E = I0 0r 0 : The following definition will be used (see [2]).
Definition 1: [2] The DAE (3) is said to be of (uniform) index one if the constant coefficient system
E _w(t) 0 Fxxx(^xxx; 0)w(t) = g(t)
is impulse free for all^xxx in a neighborhood of the graph of the solution, whereFxxxdenotes the Jacobian matrix@F =@xxx.
The index of a DAE can be thought of as the generalization of the nilpotent index [2] of a linear time-invariant descriptor system. The notion of index provides an easy way to guarantee the solvability of a given DAE. Rewrite the DAE (3) in the following form:
Ir 0 0 0 _xxx1 _xxx2 = F1(xxx1; xxx2; u) F2(xxx1; xxx2; u) :
Suppose that the aforementioned DAE is of index one. Then, from Definition 1, it is necessary that(@=@xxx2)F2(xxx1; xxx2; 0) is nonsingular
around the equilibrium pointxxx = 0. Consequently, by the implicit function theorem, there exists a functionh() so that the DAE reduces to an ODE
_xxx1= F1(xxx1; h(xxx1); u)
which is always solvable provided thatF1is smooth enough. This im-plies that DAE (3) is solvable.
In [9], some stability definitions and Lyapunov stability theorems for nonlinear descriptor systems have been given. For the sake of brevity, we do not reproduce those results here. Instead, we will derive an im-proved version of the Lyapunov stability theorem for DAE (3).
Theorem 2: Consider DAE (3) withu = 0. Let Exxx(0) = Exxx0
be given. Suppose that there exists aC3 functionV : IRn 0! IR vanishing at the points whereExxx = 0 and positive elsewhere which satisfies the following properties:
i) (@=@xxx)V = ~VT(xxx)E for some C2function ~V : IRn0! IRn; ii) ~VT(xxx)F (xxx; 0) < 0 for all xxx 6= 0;
iii) ETV~xxx= ~VxxxTE 0, where ~Vxxxdenote the Jacobian of ~V . Then, the equilibrium pointxxx = 0 is locally asymptotically stable and the DAE is of index one.
Proof: We first show that the DAE has index one. Set ~V (xxx) = ~
V1(xxx1; xxx2)
~
V2(xxx1; xxx2) : It is easy to show that
@2
@xxx2V~T(xxx)F (xxx; 0) jxxx=0= ~VxxxT(0)Fxxx(0; 0)
+FT xx
x(0; 0) ~Vxxx(0) < 0: (4)
Condition iii) implies that(@=@xxx2) ~V1 = 0; this, in turn, implies that(@=@xxx2)F2(0; 0) is nonsingular. Consequently, by the continuity
ofF , the DAE (3) is of index one. Now, consider the constant pair {E, Fxxx(0; 0)}. From inequality (4) and condition iii), we can conclude
that ~Vxxx(0) is a solution satisfying the generalized Lyapunov inequality
~ VT
xxx(0)Fxxx(0; 0) + FxxxT(0; 0) ~Vxxx(0) <0
ETV~
xxx(0) = ~VxxxT(0)E 0:
Hence, by the standard result on Lyapunov stability of linear de-scriptor systems [8], the pair {E, Fxxx(0; 0)} is admissible (i.e., regular,
asymptotically stable and impulse free). This, in turn, implies that the DAE (3) withu = 0 is asymptotically stable by noting that the pair
{E, Fxxx(0; 0)} is a linearization of the DAE around the equilibrium
pointxxx = 0. Q.E.D.
Remark: Because the intrinsic property of descriptor systems, the initial values must be given in the formExxx0. Its rationale can be better understood by investigating linear descriptor systems. Consider the fol-lowing linear differential equation:
E _xxx = Axxx(t) + Bu(t):
Taking Laplace transform of the previous equation yields X(s) = (sE 0 A)01[Exxx(0) + BU(s)]:
We assume the invertibility of the pencil (sE 0 A) so that unique so-lutions of the above equation are obtained for allExxx(0) and U(s). In particular, we point out that the initial conditions must be given in the formExxx(0). As a matter of fact, given u(t) for t 0, the knowledge ofExxx(0) is necessary and sufficient to completely determine xxx(t) for t 0. It is part of the reason that the candidate Lyapunov function V (xxx) should be vanishing at the points where Exxx = 0 rather than xxx = 0. On the other hand, for an index one descriptor system, (2) is simply an algebraic constraint. Therefore, only the part thatEx 6= 0 contributes to the energy function (see also [4], [9], and [10] for more details).
Next, we give an extension of the LaSalle invariance principle. Theorem 3: Consider the DAE
_x =f1(x; y) (5a)
0 =f2(x; y) (5b)
wheref1, f2 are continuously differentiable functions. Suppose the DAE is of index one. Let(x; y) = (0; 0) be an equilibrium point for the DAE (5a), (5b). LetV (x; y) : D 0! IR+= [0; 1) be a smooth positive–definite function on a neighborhoodD of (x; y) = (0; 0), such that _V (x; y) 0. Let S = f(x; y) 2 D j _V = 0g, and suppose that no solution can stay forever inS, other than the trivial solution. Then, the origin is locally asymptotically stable.
Proof: The proof is straightforward. Since the DAE is of index one, as far as the (5b) is concerned, there exists a unique solution y = g(x) such that f2(x; g(x)) = 0, with g(0) = 0, provided by
the implicit function theorem. In this case, iflimt!1x(t) = 0, then limt!1y(t) = 0. The condition limt!1x(t) = 0 is a direct
conse-quence of the usual LaSalle invariance principle. Q.E.D. Remark: In the aforementioned theorem, _V denotes differentiation with respect tot along the solution trajectory of (5a) and (5b). Because the descriptor system is of index one, it possesses a solution which is impulse free.
For the remainder of this section, we will investigate dissipative property of a given DAE. Consider the following DAE:
E _xxx =F (xxx; u); u 2 U IRm; F (0; 0) = 0
y =H(xxx; u); y 2 Y IRp; H(0; 0) = 0 (6)
wherexxx 2 X , together with a function s : U 2 Y 0! IR
called the supply rate. It is well known [7] that a usual state–space system (i.e.,E I, the identity matrix) has L2 gain if it is dissipative with respect to the supply rate
s(u; y)4= 2kuk20 kyk2; > 0:
This result can also be applied to DAE (6). In fact, we have the fol-lowing very important result.
Theorem 4: Consider DAE (6) withExxx(0) = Exxx0given. Suppose that the matrixDTD 0 2I is negative definite and {E, A, G} is impulse observable (the triple {E, A, G} is called impulse observable
if there exists a constant matrixL such that {E, A + LG} is impulse free), where D = @u@ H (xxx;u)=(0;0) A = @xxx@ F (xxx;u)=(0;0) G = @xxx@ H (xxx;u)=(0;0):
Suppose that any bounded trajectory xxx(t) of the system E _xxx = F (xxx(t); 0) satisfying H(xxx(t); 0) = 0 for all t 0 is such thatlimt!1xxx(t) = 0. Suppose also that there exists a C3
functionV : IRn0! IR+vanishing at the points whereExxx = 0 and positive elsewhere which satisfies the following properties:
i) (@=@xxx)V = ~VT(xxx)E for some C2function ~V : IRn0! IRn; ii) Y0= ~4VT(xxx)F (xxx; u) + kyk20 2kuk2 0, for all u 2 U;
iii) ETV~xxx = ~VxxxTE.
Then, the DAE has anL2gain less than or equal to and the equilib-rium pointxxx = 0 is locally asymptotically stable. Moreover, the DAE is of index one.
Proof: The proof of the dissipative property is standard, hence omitted. We now prove asymptotical stability and index one property. We first show that the DAE (6) is of index one. Settingu 0 in Y0
and taking the second-order partial derivative ofY0with respect toxxx at(xxx; u) = (0; 0) yields
ATV~
xxx(0) + ~VxxxT(0)A + GTG 0: (7)
Since {E, A, G} is impulse observable, inequality (7) along with con-dition iii) implies that DAE (6) is of index one. To prove asymptotical stability, observe that along any trajectoryxxx() of the DAE with u 0 is such that
dV (xxx(t))
dt 0kyk2 0:
This shows that the equilibrium pointxxx = 0 of the DAE (6) is stable. In addition, observe that any trajectoryxxx() such that _V (xxx(t)) = 0 for allt 0 is necessarily a trajectory of E _xxx = f(xxx; 0) such that xxx(t) is bounded andH(xxx(t); 0) = 0 for all t 0. Hence, by hypothesis, it is concluded thatlimt!1xxx(t) = 0 by using Theorem 3. Q.E.D.
III. THEH1CONTROLPROBLEM
Let6 be a nonlinear system described by the following DAE: E _xxx =F (xxx; w; u); w 2 W IRl; u 2 U IRm
z =Z(xxx; w; u); z 2 Z IRs
y =Y (xxx; w; u); y 2 Y IRp (8)
wherexxx 2 X . Here u stands for the vector of control inputs, w is the exogenous input (disturbances to-be-rejected or signals to-be-tracked), y is the measured output, and finally z denotes the to-be-controlled outputs (tracking errors, cost variables). It is assumed throughout that F (0; 0; 0) = 0, Z(0; 0; 0) = 0 and Y (0; 0; 0) = 0. The standard H1control problem consists of finding, if possible, a controller0 such that the resulting closed-loop system has a locally asymptotically stable equilibrium point at the origin, is of index one, and hasL2gain (fromw toz) less than or equal to . In the state feedback H1control problem we assume thaty = xxx in (8), i.e., that the whole state is available for measurement. We suppose the following.
A1) The matrixD12has rankm and the matrix D11TD110 2I is
negative definite, whereD12 = (@Z=@u)(xxx;w;u)=(0;0;0)andD11 =
A2) Any bounded trajectory xxx(t) of the system E _xxx(t) =
F (xxx(t); 0; u(t)) satisfying Z(xxx(t); 0; u(t)) = 0 for all t 0 is such thatlimt!1xxx(t) = 0.
A3) The matrix pencil A 0 j!E B2
C1 D12 has full column rank
for all! 2 IR [ f1g, where A = (@F =@xxx)(xxx;w;u)=(0;0;0),B2 =
(@F =@u)(xxx;w;u)=(0;0;0), andC1= (@Z=@xxx)(xxx;w;u)=(0;0;0).
Two preliminary lemmas will be needed in the sequel.
Lemma 5: Consider the DAE (8). Assume that assumptions A1)–A3) are satisfied. Suppose the following hypothesis also holds.
H1: There exists a smooth real-valued functionV (xxx), locally de-fined on a neighborhood of the equilibrium pointxxx = 0 in X , which is vanishing at the points whereExxx = 0 and positive elsewhere such that the function
Y1(xxx) = H(xxx; ~V (xxx); 1(xxx); 2(xxx))
is negative semidefinite nearxxx = 0, where the function H : IRn2 IRn2IRl2IRm0! IR is defined on a neighborhood of (xxx; p; w; u) =
(0; 0; 0; 0) as
H(xxx; p; w; u) = pTF (xxx; w; u) + kZ(xxx; w; u)k20 2kwk2 (9)
(@V =@xxx) = ~VT(xxx)E defined as shown in Theorem 4, 1(xxx) =
w3(xxx; ~V (xxx)) and
2(xxx) = u3(xxx; ~V (xxx)), and w3(xxx; p) and u3(xxx; p)
are defined on a neighborhood of(xxx; p) = (0; 0) satisfying @H
@w(xxx; p; w3(xxx; p); u3(xxx; p)) =0 @H
@u(xxx; p; w3(xxx; p); u3(xxx; p)) =0 withw3(0; 0) = 0 and u3(0; 0) = 0.
Then, the feedback lawu = 2(xxx) solves the H1state feedback
control problem. 5 5 5
Proof: The result of the lemma is a direct consequence of The-orem 4. The details are thus omitted.
Next, consider the case in which the statexxx of the DAE (8) is not available for direct measurement. Motivated by the work of Isidori and Kang [3] and Yung et al. [11], we consider a dynamic controller of the form
^
E _ =F (; 1(); 2()) + G()(y 0 Y (; 1(); 2()))
u = 2() (10)
where = col(1; . . . ; n) are local coordinates for the state-space manifoldXc of the controller0. The matrix G(), called the output injection gain, is to be determined. Substitute the controller (10) in (8) to obtain the corresponding closed-loop system as
Ee_xxxe= Fe(xxxe; w) z = Ze(xxxe; w) = Z(xxx; w; ()) (11)
whereEe= E 0
0 E^ , shown in the expressions at the bottom of the page. Again, we try to render the closed-loop system locally dissipative with respect to the supply rate 2kwk2 0 kzk2. Clearly, it suffices to show that there exists a smooth nonnegative functionU(xxxe) with (@U=@xxxe) = ~UTEeandEeTU~
xx
x = ~UxxxTEesuch that
~
UTFe(xxxe; w) + kZ(xxxe; w)k20 2kwk2 0; for allw (12)
and such that the closed-loop system is locally asymptotically stable and is of index one. To state the main result of this section, a further assumption is needed.
A4) The matrixD21= (@Y =@w)(xxx;w;u)=(0;0;0)has rankp.
Define r11(xxx) = 12 @ 2H(xxx; ~VT(xxx); w; u) @w2 w= (xxx);u= (xxx) r12(xxx) = 12 @ 2H(xxx; ~VT(xxx); w; u) @u@w w= (xxx);u= (xxx) r21(xxx) = 12 @ 2H(xxx; ~VT(xxx); w; u) @w@u w= (xxx);u= (xxx) r22(xxx) = 12 @ 2H(xxx; ~VT(xxx); w; u) @u2 w= (xxx);u= (xxx) and set R(xxx) = (1 0 r 1)r11(xxx) r12(xxx) 21(xxx) (1 + 2)r22(xxx)
where1 and 2 are any real numbers satisfying0 < 1 < 1 and
2 > 0, respectively. The following theorem is readily obtained.
Theorem 6: Consider (11). Suppose assumptions A1)–A4) are satis-fied. Suppose hypothesis H1 of Lemma 5 holds. Suppose the following hypothesis also holds.
H2: There exists a smooth real-valued functionQ(xxx), locally de-fined on a neighborhood ofxxx = 0, which is vanishing at the points whereExxx = 0 and positive elsewhere such that the function
Y2(xxx) = K(xxx; ~Q(xxx); ^w(xxx; ~Q(xxx); ^y(xxx; ~Q(xxx))); ^y(xxx; ~Q(xxx)))
is negative definite nearxxx = 0, and its Hessian matrix is nonsingular atxxx = 0. Here ~Q : IRn ! IRn is a smooth function defined by (@Q=@xxx) = ~QTE with ETQ~
xx
x = ~QxxxE, the function K : IRn2
IRn2 IRl2 IRp! IR is defined on a neighborhood of the origin as
K(xxx; p; w; y) = pTF (xxx; w + 1(xxx); 0) 0 yTY (xxx; w + 1(xxx); 0) + 0 w 2(xxx) T R(xxx) 0 w 2(xxx)
and the functionw(xxx; p; y), respectively ^y(xxx; p), defined on a neigh-^ borhood of (0,0,0), respectively (0,0), satisfies
@K(xxx; p; w; y) @w w= ^w(x;p;y)= 0 ^w(0; 0; 0) = 0 respectively @K(xxx; p; ^w(xxx; p; y); y) @y y=^y(x;p)= 0 ^y(0; 0) = 0: Then, if ~ Q(xxx)G(xxx) = ^yT(xxx; ~Q(xxx)) (13) has a smooth solutionG(xxx) near xxx = 0, the nonlinear H1 output feedback control problem is solved by the output feedback controller
(10) with ^E = E. 5 5 5
Proof: Since the result of the theorem is a special case of that given in Theorem 8, we omit the proof here for brevity.
A. Parameterization of Output Feedback Controllers
Recently, Yung et al. [11] have derived a set of parameterized so-lutions to theH1 control problem for general nonlinear systems in state-variable form. They have considered both output feedback and
xxxe= xxx
and
Fe(xxxe; w) = F (xxx; w; 2())
state feedback cases. Indeed, we can extend the technique developed in [11] to give a family ofH1controllers for nonlinear differential-al-gebraic systems.
Motivated by the work of [11], we consider the family of controllers described by the following DAEs:
^ E _ =F (; 1(); 2() + c()) + G()(y 0 Y (; 1(); 2() + c())) + ^g1()c() + ^g2()d() EQ_ =a(; y 0 Y (; 1(); 2() + c())) u = 2() + c() (14)
where and are defined on some neighborhoods of the origins in Xc andIRq, respectively.G() satisfies (13). a(; ) and c() are smooth functions witha(0; 0) = 0 and c(0) = 0. ^g1(), ^g2() and d() are Ck functions (k 1). EQis a constant matrix, and, in general, is singular. The functionsa(; ), c(), ^g1(), ^g2(), d(), and the matrix EQare to-be-determined variables such that the closed-loop system (8)–(14) is dissipative with respect to the supply rate 2kwk20 kzk2, and is locally asymptotically stable with index one.
Observe first that the DAEs describing the closed-loop system (8)–(14) can be put in the form
Ea_xxxa=Fa(xxxa; w) z =Z(xxx; w; 2() + c()) where xxxa 4= col(xxx; ; ), Ea 4= E 0 0 0 E^ 0 0 0 EQ
, and the equa-tion at the bottom of the page holds. In that equaequa-tion, ~F (; ) =4 F (; 1(xxx); 2() + c()) 0 G()Y (; 1(); 2() + c()):
Consider a Hamiltonian functionJ : IR2n+q2 IR2n+q2 IRr ! IR defined as follows: J(xxxa; pa; w) =pTaFa(xxxa; w) + w 0 2(xxx) 2() + c() 0 2(xxx) T 2 R(xxx) w 0 2(xxx) 2() + c() 0 2(xxx) : (15)
It is easy to check that @2J(xxx
a; pa; w)
@w2
(xxx ;p ;w)=(0;0;0)
= 2(1 0 1)(DT11D110 2I)
which is negative definite by A1). Then, by the implicit function the-orem, there exists a unique smooth functionw(xxx~ a; pa), defined on a
neighborhood of the origin, satisfying @J(xxxa; pa; w)
@w w= ~w(xxx ;p )= 0 ~w(0; 0) = 0:
Lemma 7: Consider (8) and (14). Suppose assumptions A1)–A4) are satisfied. Suppose hypotheses H1 of Lemma 5 and H2 of Theorem 6 hold. Furthermore, suppose that the following hypothesis also holds. H3: There exists a smooth real-valued function M(xxxa), locally
defined on a neighborhood of the origin in IR2n+q, which van-ishes at the points where xxxa = col(Exxx; Exxx; 0) = col(0; 0; 0),
is positive elsewhere, satisfies (@M(xxxa)=@xxxa) = M~T(xxxa)Ea
with ~MxxxT (xxxa)Ea = EaTM~xxx (xxxa), and is such that the function
Y3(xxxa) = J(xxxa; ~M(xxxa); ~w(xxxa; ~M(xxxa))) vanishes at the points
wherexxxa= col(Exxx; Exxx; 0) = col(0; 0; 0) and is negative elsewhere.
Then, the family of controllers (14) with ^E = E solves the H1
output feedback control problem.
Proof: Set U(xxxa) = V (xxx) + M(xxxa). It follows that
(@U=@xxxa) = ~UT(xxxa)EawithEaTU~xxx (xxxa) = ~UxxxT (xxxa)Ea, where
~ Uxxx = ~ Vxxx 0 0 0 0 0 0 0 0 + ~Mxxx :
With the equationsY1(xxx) and Y3(xxx) in hand, we have the following
Hamiltonian equation by using Taylor series expansion: dU dt+kZ(xxx; w; 2() + c())k20 2kwk2 =Y1(xxx) + Y3(xxxa) + w 0 2(xxx) 2() + c() 0 2(xxx) T 1 1r110(xxx) 0 0 2r22(xxx) w 0 2(xxx) 2() + c() 0 2(xxx) + kw 0 ~w(xxxa; ~MT(xxxa))k2R(xx~ x ) + o w 0 1(xxx) 2() + c() 0 2(xxx) 3 + o kw 0 ~w(xxxa; ~MT(xxxa))k3 (16) where ~ R(xxxa)4= 12 @ 2J(xxx a; ~MT(xxxa); w) @w2 w= ~w(xxx ; ~M (xxx ))
and the notationkvk2R~stands forvTRv. It is easy to verify that ~~ R(0) = (1 0 1)(DT11D110 2I). Since Y1(xxx) and Y3(xxxa) are nonpositive,
(16) implies that dU
dt + kZ(xxx; w; 2() + c())k20 2kwk2 0 (17) which, in turn, implies that the closed-loop system has anL2-gain less than or equal to . Set w = 0, rearrange terms, and use (16) to get
dU dt = 0 kZ(xxx; 0; 2() + c())k2+ Y2(xxx) + Y3(xxxa) + 0 2(xxx) 2()+c()0 2(xxx) T 1r11(xxx) 0 0 02r22(xxx) 1 0 2(xxx) 2() + c() 0 2(xxx) + k ~w(xxxa; ~M T(xxx a))kR(xx2~ x ) + o 0 1(xxx) 2() + c() 0 2(xxx) 3 + o k ~w(xxxa; ~MT(xxxa))k3
which is negative semidefinite nearxxxa= 0 by hypothesis. This shows
that the closed-loop system is stable locally around the equilibrium point. We claim that the DAE (14) has index one. To see this, observe that any trajectory satisfying(dU=dt)(xxx(t); (t); (t)) = 0 for all t 0 is necessarily a trajectory of
E _xxx(t) = F (xxx; 0; 2() + c()) (18) such thatxxx(t) is bounded and Z(xxx; 0; 2() + c()) = 0 for all
t 0. This shows that the previous DAE has index one. Moreover,
Fa4= F (xxx; w; 2() + c()) ~ F (; ) + G()Y (xxx; w; 2() + c()) + ^g1()c() + ^g2()d() a(; Y (xxx; w; 2() + c()) 0 Y (; 1(); 2() + c())) :
hypotheses H1 and H3 along with assumption A1) imply that the tra-jectory satisfying(dU=dt)(xxx(t); (t); (t)) = 0 for all t 0 is necessarily a trajectory such thatxxx(t) = (t) and (t) = 0 for all t 0. Setting xxx(t) = (t) 0 and w(t) = 0 in (15), we have ( ~M)Ta(; Y (0; 0; c())0Y (0; 0; c())) < 0; for all 6= 0, where
~ MT(xxx
a) = [( ~Mxxx)T ( ~M)T ( ~M)T]. This shows that the DAE
EQ_ = a(; Y (0; 0; c()) 0 Y (0; 0; c())) (19) has index one and is asymptotically stable. Hence, by hypothesis H2 and the fact that DAE (18) and (19) have index one, we can conclude that the closed-loop system (8)–(14) has index one. Asymptotical sta-bility then easily follows by Theorem 4. Q.E.D. The previous lemma gives a general form of the output feedback con-trollers. However, it does not explicitly specify how we can choose the free system parametersEQ,a(; ) and c() in order to meet the hy-pothesis in Lemma 7. In the sequel, we give a way to meet the condition in Lemma 7, and in the mean time, to reduce the number of indepen-dent variables. Consider the following DAE:
EQ_ = a(; ): (20)
If there exists a smooth functionL(), locally defined on a neighbor-hood of = 0, which vanishes at the points where EQ = 0, is
posi-tive elsewhere, satisfies(@L()=@) = ~L()TEQwithEQT~L() = ~L()TEQ, and is such that ~LT()a(; ) < 0, then we can conclude from Theorem 2 that DAE (20) is locally asymptotically stable and has index one. Henceforth, if some further hypotheses are imposed in the above inequality, the condition in Lemma 7 can be met. This is sum-marized in the following theorem.
Theorem 8: Consider (8) and (14). Suppose assumptions A1)–A4) are satisfied. Suppose hypotheses H1 of Lemma 5 and H2 of Theorem 6 hold. Suppose also that the following hypothesis holds.
H4: There exists a smooth functionL(), defined as previously shown, such that the function
Y4(; w) = ~LT()a(; Y (0; w; 0)) + c()w T
R(0) c()w atw = w+(), viewed as a function of , is negative definite near = 0, and its Hessian matrix is nonsingular at = 0. The func-tionw+() is defined on a neighborhood of = 0, which satisfies (@Y4(; w)=@w)w=w ()= 0 with w+(0) = 0 (This function exists,
forR(0) is nonsingular).
Then, if^g1() and ^g2() satisfy
~ Q(xxx)^g1(xxx) = 2 T(xxx; 0; 0)r12(xxx) 0 2(1 + 2) T2(xxx)r22(xxx) and ~ Q(xxx)^g2(xxx) = aT(0; Y (xxx; 1(xxx) + (xxx; 0; 0); 0)) respectively, where (xxx; ; )=~w(xxxa;[ ~Q(xxx 0 ) 0 ~Q(xxx 0 )~L() ]),
the family of controllers (14) withd()= ~L() solves the H4 1output
feedback control problem. 5 5 5
Proof: It is straightforward to verify thatM(xxxa)= Q(xxx 0 ) +4
L() satisfies the hypothesis of Lemma 7.
IV. CONVERSERESULT—A NECESSARYCONDITION Suppose that theH1control problem is solved by the output feed-back controller0 which has the following representation:
^
E _ =8(; y)
u =2() (21)
and letU be a smooth function satisfying W (xxx; ; w) = ~Uxxx U~ 8(; Y (xxx; w; 2())F (xxx; w; 2())
+kZ(xxx; w; 2())k20 2kwk2 0 (22)
for all (xxx, , w) in a neighborhood of (0, 0, 0). Consider the case that 6= 0 and ~U(xxx; ) 6= 0. Since 6= 0, we have 8 6= 0. Hence,
from (22), we have inf
8; 2maxw W (xxx; ; w) = 01, because inequality
(22) contains a term linearly in8. Next, consider the case that 6= 0 but ~U(xxx; ) = 0. Suppose that ~Uis nonsingular for every (xxx, ) satisfying ~U(xxx; ) = 0: Then, by the implicit function theorem, the
previous identity has a differentiable solution = `(xxx) with `(0) = 0. The previous statement is needed in the subsequent proof. We take it as a standing assumption.
A5) ~U(xxx; ) = 0 if and only if = `(xxx) for some smooth function
` with `(0) = 0. Furthermore, ~U(xxx; )j=`(xxx)is nonsingular. Setting = `(xxx) in (12) yields
~
VTF (xxx; w; 2(`(xxx))
+kZ(xxx; w; 2(`(xxx))k20 2kwk2 0; 8w: (23) This shows that inf
0; 2()maxw W (xxx; ; w) = Y1(xxx); where = `(xxx).
Hence, the state feedback lawu = 2(`(xxx)) solves the state feedback H1control problem for6. This shows that V is a solution of Y1. A further necessary condition is obtained by restricting to the class of controller0 which produces zero control input u. Consider the Hamil-tonian functionK : IRn2 IRn2 IRl2 IRp! IR defined as K (xxx; p; w; y) = pTF (xxx; w; 0) 0 yTY (xxx; w; 0)
+kZ(xxx; w; 0)k20 2kwk2: (24)
It is easy to verify that @2K (xxx; p; w; y) @w2 (xxx;p;w;y)=(0;0;0;0) = 2(DT 11D110 2I):
This shows that there exists a smooth functionw(xxx; p; y) defined in a^ neighborhood of (0,0,0) such that
@K (xxx; p; w; y)
@w w= ^w(xxx;p;y)= 0 ^w(0; 0; 0) = 0: Furthermore, it is also easy to check that
@2K (xxx; p; ^w(xxx; p; y); y) @y2 (xxx;p;y)=(0;0;0) = 1 2( 2I 0 D11TD11)01D21DT21: (25) Thus, there exists a smooth functiony3(x; p) defined in a neighbor-hood of (0,0) such that
@K (x; p; ^w(xxx; p; y); y)
@y y=y (xxx;p)= 0 y3(0; 0) = 0: Setw3(xxx; p) = ^w(xxx; p; y3(xxx; p)). Then, we have
K (xxx; p; w; y) K (xxx; p; ^w(xxx; p; y); y) (26) for all (xxx, p, w, y) in a neighborhood of the origin and
K (xxx; p; ^w(xxx; p; y); y)
K (x; p; w3(xxx; p); y3(xxx; p)); y3(xxx; p)) (27)
for all (xxx, p, y) in a neighborhood of the origin. We will show that it is necessary
K (xxx; ~P (xxx); w3(x; ~P (xxx)); y3(xxx; ~P (xxx))) 0 (28) for some storage functionP (xxx) with (@P =@xxx) = ~PTE. This is sum-marized in the following statement.
Theorem 9: Consider system (8) and suppose assumptions A1)–A5) hold. Suppose that theH1 control problem is solved by the output feedback controller (21). Suppose that there exists a smooth real-valued functionU(xxx; ), which vanishes at the points where Eexxxe = 0 and is positive elsewhere withEeTU~xxx = ~UxxxTEe, and satisfies (21) for
all (xxx, , w) in a neighborhood of (0, 0, 0). Then, the Hamilton–Jacobi inequalities
Y1(xxx) 0 and K (xxx; ~P (xxx); w3(x; ~P (xxx)); y3(xxx; ~P (xxx))) 0
have solutions V (xxx) and, respectively, P (xxx) (with (@P =@xxx) = ~
PTE) given by V (xxx) = U(xxx; `(xxx)) 0 and, respectively,
P (xxx) = U(xxx; 0) 0. Furthermore, Q(xxx)4= P (xxx) 0 V (xxx) 0. 5 5 5 Proof: It is obvious thatV (xxx) is a solution satisfying Y1(xxx) 0
from our previous observation. It is claimed thatP (xxx) is a solution of inequality (28). To see this, setting = 0 in (12) yields
~
PTF (xxx; w; 0) + ~U(xxx; 0)8(0; Y (x; w; 0))
+kZ(xxx; w; 0)k20 2kwk2 0: (29)
Let ~U(xxx; 0)8(xxx; y) = 5T(xxx; y)y, where 5(xxx; y) is a vector of smooth functions. This can always be done because the func-tion 8(0; y) vanishes at y = 0. Use 5T(xxx; y)y and choose w = ^w(xxx; ~P ; y) in (12) to obtain
K (xxx; ~P ; ^w(xxx; ~P ; y); 5(xxx; Y (xxx; ^w(xxx; ~P ; y)))) 0: (30) Observe that the Hessian matrix ofy05(xxx; Y (xxx; ^w(xxx; ~P ; y))) is non-singular [from assumption A4) and (25)]. Hence, by the implicit func-tion theorem, there exists a unique solufunc-tion, denoted by^y(xxx), satisfying ^y(xxx) 0 5(xxx; Y (xxx; ^w(xxx; ~P ; ^y(xxx))) = 0, ^y(0) = 0. Set y = ^y(xxx) in (30) to obtain
K (xxx; ~P (xxx); ^w(xxx; ~P (xxx); ^y(xxx)); ^y(xxx)) 0:
This shows that
K(xxx; ~P (xxx); w3(xxx; ~P (xxx)); y3(xxx; ~P (xxx))) 0
from (27). In order to complete the proof, we have to show thatQ(xxx)=4 P (xxx) 0 V (xxx) 0. Note that the function U(xxx; ) has the following Taylor series expansion:
U(xxx; ) =U(0; 0) + @U@xxx(0; 0)xxx + @U@(0; 0) + xxx T 1 2@ U@xxx (0; 0) 12@xx@ Ux@(0; 0) 1 2@xx@ Ux@ T (0; 0) 1 2@ U@ (0; 0) xxx + h:o:t: (31)
where “h.o.t.” means high order terms. Let = `(xxx), then we have the following two Taylor series expansions:
@2U @xxx2(xxx; `(xxx)) = @ 2U @xxx2(0; 0) + @ 2U @xxx@(0; 0)`xxx(xxx) + (h:o:t:)xxxxxxj=`(xxx) (32) @2U @@xxx(xxx; `(xxx)) = @ 2UT @xxx@(0; 0) + @ 2U @2(0; 0)`xxx(xxx) + (h:o:t:)xxxj=`(xxx): (33) Note that(@2U=@xxx2)(xxx; `(xxx)) = (@2V =@xxx2)(xxx), and (@2U=@@xxx) (xxx; `(xxx)) = 0. Set xxx = 0 in (32) and (33), respectively, to get
@2U @xxx2(0; `(0)) = @ 2V @xxx2(0) = @ 2U @xxx2(0; 0) + @ 2U @xxx@(0; 0)`xxx(0) and 0 = @@@xxx2U (0; 0) = @@xxx@2UT(0; 0) + @@2U2 (0; 0)`xxx(0)
respectively. Next, observe that @2Q @xxx2(0) = @ 2P @xxx2(0) 0 @ 2V @xxx2(0) = @@xxx2U2(0; 0) 0 @@xxx2U2(0; 0) + @@xxx2U(0; 0)`xxx(0) = 0 @@xxx2U(0; 0)`xxx(0) = `xxxT(0)@ 2U @2 (0; 0)`xxx(0) 0: (34) The last inequality holds by assumption A5). This concludes that Q(xxx) 0 by noting that Q(xxx) has the following Taylor series expansion:
Q(xxx) = Q(0) + @Q@xxx(0)xxx + 12xxxT@2Q
@xxx2(0)xxx + h:o:t: 0:
It is nonnegative around the origin because it vanishes at the origin together with its first-order derivative, and its second-order derivative is positive by (34). This completes the proof. Q.E.D.
REFERENCES
[1] J. A. Ball, J. W. Helton, and M. L. Walker, “ control for nonlinear systems with output feedback,” IEEE Trans. Automat. Contr., vol. 38, pp. 546–559, Apr. 1993.
[2] K. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical
Solution of Initial-Value Problems in Differential-Algebraic Equa-tions. Philadelphia, PA: SIAM, 1996.
[3] A. Isidori and W. Kang, “ control via measurement feedback for general nonlinear systems,” IEEE Trans. Automat. Contr., vol. 40, pp. 466–472, Mar. 1995.
[4] E. Jonckheere, “Variational calculus for descriptor problems,” IEEE
Trans. Automat. Contr., vol. 28, pp. 491–495, May 1988.
[5] W. M. Lu and J. C. Doyle, “ control of nonlinear system via output feedback: Controllers parameterization,” IEEE Trans. Automat. Contr., vol. 39, pp. 2517–2521, Dec. 1994.
[6] R. W. Newcomb and B. Dziurla, “Some circuits and systems applications of semistate theory,” Circuits, Syst., Signal Processing, vol. 8, no. 3, pp. 235–260, 1989.
[7] A. J. van der Schaft, -Gain and Passivity Techniques in Nonlinear Control. Berlin, Germany: Springer-Verlag, 1996, vol. 218, Lecture Notes in Control and Information Sciences.
[8] H. S. Wang, C. F. Yung, and F. R. Chang, “Bounded real Lemma and control for descriptor systems,” in Proc. Inst. Elect. Eng., vol. 145, 1998, pp. 316–322.
[9] H. Wu and K. Mizukami, “Stability and robust stabilization of nonlinear descriptor systems with uncertainties,” in Proc. 33rd Conf. Decision and
Control, Lake Buena Vista, FL, 1994, pp. 2772–2777.
[10] H. Xu and K. Mizukami, “Hamilton-Jacobi equation for descriptor sys-tems,” Syst. Control Lett., vol. 21, pp. 321–327, 1993.
[11] C. F. Yung, J. L. Wu, and T. T. Lee, “ control for more general non-linear systems,” IEEE Trans. Automat. Contr., vol. 43, pp. 1724–1727, Dec. 1998.
