Ball, J.A.

Schaft, A.J. van der

English

It is shown how the method for inner-outer factorization of stable nonlinear state space systems may be extended to the non-invertible case by replacing a Hamilton-Jacobi equation by a dissipation inequality. The construction of the outer factor is based on the factorization of this inequality

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   University of GroningenNonlinear inner-outer factorizationSchaft, A.J. van der; Ball, J.A.Published in:Proceedings of the 33rd IEEE Conference on Decision and Control, 1994IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1994Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Schaft, A. J. V. D., & Ball, J. A. (1994). Nonlinear inner-outer factorization. In Proceedings of the 33rd IEEEConference on Decision and Control, 1994 (pp. 2549-2552). University of Groningen, Johann BernoulliInstitute for Mathematics and Computer Science.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 08-06-2022proceedings of the 33d confenrnce on Dedsion and control ~ a k e  BuenaVista. FL- December 1994 Nonlinear inner-outer factorization A.J. van der Schaft * J.A. Ball t Abstract It is shown how the method for inner-outer fac- torization of stable nonlinear state space systems as put forward in [11] may be extended to the non-invertible case by replacing a Hamilton-Jacobi equation by a dissipation inequality. The construc- tion of the outer factor is based on the factorization of this inequality. In linear control theory inner-outer factorization (or more generally J-inner-outer factorization) of rational matrices has played an important role e.g. in the theory of 'H, op- timal control. In linear as well as in nonlinear theory [lo], (161, [6] it has been argued that the control design of non- minimum phase stable systems can be based upon the in- verse of the minimum phase (outer) factor, with the inner factor remaining as a limiting element in the closed-loop system. In a series of papers, see e.g. [2], [3], [I], Ball and Helton have investigated inner-outer factorization of non- linear input-output operators and of nonlinear state space systems in discrete time. In the present note we will study inner-outer factorizationof nonlinear state space systems in continuous time, using a quite different approach. Indeed our method will be a kind of "nonlinear spectral factor- ization" and concentrates on finding first the outer factor (instead of starting with the inner factor). The present paper is a continuation of [12] where the invertible case has been studied, and concentrates on the non-invertible case. More details will appear in [4]. Consider a (smooth) nonlinear system i: = a(z)+b(z)u,  U E  R" y = c(z)+d(z)u,  y E RP E : (  (1) where z = ( ~ 1 , .   . ,z,) are local coordinates for the state space manifold M ,  with globally asymptotically stable equi- librium 0 (thus a(0)  = 0). Without loss of generality we assume c(0)  = 0. The problem of inner-outer factorization consists in finding a lossless nonlinear system 0 (the inner factor) and an asymptotically stable and minimum phase nonlinear system R (the outer factor), such that symboli- cally C = O . R .  (2) By this we mean that for every initial condition of C there exist initial conditions of 0 and R such that the input- output behavior of C equals the input-output behavior of 'Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, Fax 31-53-340733, e-mail twarjanbmath.utwente.nl 'Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, U.S.A. 0-7803-1 968-0/94$4.0W1994 IEEE 2549 TP-16 3140 the series interconnection of R followed by 8. Let us recall [14] that a nonlinear system (1) is called lossless with respect to  the supply rate f 11 U IIz -f )I y if there exists a function V ( z )  2 0 (the storage function) such that V ( z ( t l ) ) - V ( z ( t o ) )  = ;&'(I1 u ( t )  1 1 2  - II ~ ( t )  l12)dt(3) for all to,tl and U(.), or equivalently, if V is C', V z ( z )  [a(.) + b(z)ul= ( 4 )  I T  TU U - f [c(z) + d(z)uIT [c(z) + d(z)u] for all r ,  U. (Vz(z) denotes the row vector of partial deriva- tives of V(z).) Taking t o  = 0 and tl = 00 in (3), it follows that (1) is L2-norm preserving Furthermore, a nonlinear (1) is called minimum phase if 0 is a Lyapunov stable equi- librium of its zero-dynamics [8]. Our approach for constructing the outer factor R runs as follows. First we consider the Hamiltonian eztension of - x ( z ) u a  BTc - u T 9 ( z ) u a ,  U. E RP ( 5 )  y = 4.) +d(z)u, Yo = bT(z)p + fl(z)ua, y a  E R" which is Hamiltonian system living on T'M (with co- ordinates ( z , p ) ) ,  having inputs (u,ua) and outputs ( y ,  y.). Imposing the interconnection U, = y to ( 5 )  leads to the Hamiltonian system x = % ( z , p , u )  C'C: p = - E ( z , p , u )  BH Ya = Z ( ~ ~ P , U )  with Hamiltonian function qz, P ,  U) = PT [a(.) + b(Z)UI + (7) f [c(z) + d(z)uIT [c(z) + d(z)u] Note that for a linear system (1) C'C reduces to the series interconnection of C and its adjoint linear system C', hav- ing transfer matrix GT(-s)G(s) (G(s) being the transfer matrix of E). In [12] we have shown how to obtain the outer factor R by "spectral factorization" of the Hamilto- nian system C'C, assuming the invertibility condition E ( z )  := dT(z)d(z) is invertible for all I (8) In fact, if (8) is satisfied then we may directly compute the inverse system (C*C)-'. The outer factor R is now obtained by computing the stable invariant manifold of the inverse system via the Hamilton-Jacobi equation in P ( z )  Pz(.)[a(z) - b( z)E-'(  x)fl(~)c(~)]+ gc"( z) [ I p  - d(z)E-l(  x)dT(z)]c( x)- ( 9 )  g P z ( I ) b ( I ) E - ' ( I ) b T ( x ) P ~ ( x )  = 0, P(0)  = 0, with Pz(z) = (E(z);..,E(z)). Factorizing E ( z )  = &"(z)d(z) as d T ( z ) d ( z )  = Z(z)d(z) (10) for some m x m matrix d ( r )  (this may be always done; however for d to depend smoothly on x we need to invoke Morse's Lemma), the outer factor is now given as j. = U(.) + b ( x ) ~  j j  = ?(x) + ( S ( Z ) U  ,U E R" , y E R" R :  { (11) ?(z) = J(z )E- ' (x )  [dT(z)c(x) + bT(z)P,T(x)] In this paper we will concentrate on the non-invertible case, i.e. if (8) is not satisfied. First of all, we note that (9) is the Hamilton-Jacobi-Bellman equation corresponding to the cost-functional (with 10 the initial condition) for E, and that H as given in ( 7 )  is the pseudo-Hamiltonian of the Maximum Principle. If (8) is not satisfied then this optimal control problem is singular. Our approach will be heavily motivated by the work of Hill and Moylan [7], and the work of Willems [15] and Schumacher [13] on singular LQ-control where it is shown that the Riccati-equation for the regular LQ optimal control problem may be replaced by a natriz inequality in the singular case. We define the optimal cost for any LO as P'(z0) = inf(J(z0,u) I U admissible, z(co) = 0}(13) Assumption 1 P+(zo) exists for every TO, and P+ is a smooth function on M .  We now consider the dissipation inequality correspond- ing to the pseudo-Hamiltonian (7) e - ( x ) I a ( z )  + b(Z)UI+ ;[.(I) + d(z)u]T(c(x) + d(z)u]  2 0, P ( 0 )  = 0, (14) which should hold for every z and U. It immediately follows from (13) that P+ satisfies (14). Furthermore (compare ~51,1131) Proof Let P be any solution to (14). Consider any in- put function U on the time-interval [0, TI, and integrate (14) from t = 0 to t = T for this particular U to obtain P(,(T)) - P ( x ( 0 ) )  + f I," II y 112 dt  2 0 (15) Now let U be defined on ( 0 , ~ )  such that i-03 lim x ( t )  = 0. Then it follows that and thus by definitionof P+ weobtain P+(z(O)) 2 P ( x ( 0 ) )  0 Thus P+ is completely characterized as the maximal solution to (14), and, in principle, may be computed this way. Now consider the following smooth function of x and U for all ~ ( 0 )  E M .  K + ( I , u )  := P,'(z) [a(.) + b(z)u]+ (17) [c(z) + d(x)uIT [c(z) + d(x)u] Clearly, IC+(O,O) = 0 and K+(r ,u )  2 0. Our next main assumption is Assumption 3 There exists a smooth mapping G : M x R"' -+ R' for some p E N, such that (18) 1 -  K+(z ,  U )  = -G"(z,u)G(z, U)  2 Note that without the smoothness assumption Assump- tion 3 is trivially satisfied since we may take = 1 and Sufficient conditions for the local existence of a smooth G satisfying (18) are provided by the following generalization of Morse's Lemma: L" 4 Suppose the Hessian matrix of I {+ ,  i.e., G ( x ,  U) = JM. has constant rank, say j, on a neighborhood of ( x , ~ )  = (0,O). Then locally near (0,O) there exists a C" mapping G : M x Rm -t RI' such that (18) is satisfied. Proof Can be based on [9]. 0 Now let us define the new system E, defined as j . = a ( z ) + b ( z ) u  , u E R ' "  , z E M  (20) Proposition 2 Let P satisfy (14), then P ( z )  5 P+( I ) ,  for all x. It can be readily checked that in the invertible case (i.e. E ( z )  = fl(~)d(z) being invertible) c coincides with R given in (11). We claim that also in the non-invertible case 2 is the outer factor of E. In order to prove this we first consider the dissipation in equality (14) for c, i.e., 2550 PJz) [a(.) + b(z)u] + ;Q(z,u)G(z,u) 2 0, (21) P ( 0 )  = 0 L e m m a  5 The maximal solution P+ to (21) is P+ = 0. Proof Clearly P = O satisfies (21). Let now P 2 O satisfy (21). Then by adding (21) and (14) for P = P + ,  and using (17), (18) we conclude that P+ + I‘ is a solution to (14). 0 By Proposition 2 this implies P = 0. This lemma is instrumental in proving the main result: T h e o r e m  6 The zero-dynamics of E is not exponentially unstable. For the proof, based on Lemma 5 and a linearization idea (making use of the linear results described in [13], [15]) we refer to [4]. It follows that if the zero-dynamics of E does not have imaginary eigenvalues, then it will be actually (lo- cally) asymptotically stable, and thus E is an outer factor of C! The inner factor 0 of C is now easily obtained, at  least in the following “right factorization” format: i = a ( z ) +  b(z)u (22) g = G(z,u)  (with driving variables U). Indeed, by considering (14) for P = P+ and (18), we obtain implying that 0 is lossless (from y to y ) ,  with storage func- tion P + .  An explicit input-output representation of 0, however, may not be easily obtainable, due to non-invertibility of E(z )  = A useful property of the inner factor E is that E and C have the same static gains, in the following sense. Consider the set of all controlled equilibria for C ,  i.e., f l ( x ) d ( x ) .  E, = {(z,~) E M x Rmlo(z) + b(z)u = 0) L e m m a  7 Consider C and E. For every (z,~) E E, (24) II 4.1 + d b b  11=11 G ( z , u )  I I  (25) (or equivalently 11 y 11=(1 y 11). Proof Consider the equality PrYzN4z) + b(z)uI+ f[c(z) + d(z)uIT[c(z) + d(z)u] = ;G=(z,u)G(z,u) (26) on E,. 0 Thus, if we compare the step responses of C and for every constant input U, then the static gains of C and (assuming that the corresponding controlled equilibrium ( z , ~ )  of i = a(.) + b(z)u is (globally) asymptotically sta- ble) are equal. Thus for output set-point control of C one may also consider its outer factor 2, which is asymptoti- cally equivalent to C. The control of C thus can be based on 2, and since 2 is minimum phase, inversion techniques can be applied. This idea, which generalizes an old idea in linear control theory (see e.g. [lo]), is discussed in [IS], 161. _____ ~ - _ _  References J.A. Ball, Lossless dynamical systems and nonlinear Livsic-Brodskii nodes, preprint 1992. J.A. Ball, J.W. Helton, Factorization and general properties of nonlinear Toeplitz operators, Operator Theory: Advances and Applications, Vol. 41 (1989), Birkhiiuser, Basel, 25-41. J.A. Ball, J.W. Helton, Inner-outer factorization of nonlinear operators, J. Funct. Anal., 104 (1992), 363- 413. J.A. Ball, A.J. van der Schaft, “Inner-outer factoriza- tion of nonlinear systems: the non-invertible case”, in preparation. P.E. Crouch, A.J. van der Schaft, Variational and HamiZtonian Control Systems, LNCIS 101, Springer, Berlin (1987). F.J. Doyle, F. AllgGwer, M. Morari, A normal form ap- proach to approximate input-output linearization for maximum phase nonlinear systems, preprint. D. Hill, P. Moylan, The stability of nonlinear dissi- pative systems, IEEE Trans. Aut. Contr., AC-21, pp. A. Isidori, Nonlinear Control Systems (2nd edition), Springer, New York (1989). H.Th. Jongen, P. Jonker, F. Twilt, Nonlinear Opti- mization in R”, Peter Lang, Frankfurt a. M. (1983). 708-711, 1976. [IO] M. Morari, E. Zafiriou,- Robust Process Control, Prentice-Hall, Englewood Cliffs, NJ, 1989. [ l l]  A.J. van der Schaft, Lz-gain analysis of nonlinear sys- tems and nonlinear state feedback ‘H, control, IEEE Trans. Aut. Contr., 37 (1992), 770-784. [12] A.J. van der Schaft, J.A. Ball, Inner-outer factoriza- tion of nonlinear state space systems, pp. 529-532 in Systems and Networks: Mathematical Theory and Applications, Volume I1 (eds. U. Helmke, R. Men- nicken, J. Saurer), Akademie Verlag, Berlin, 1994. [13] J.M. Schumacher, The role of the dissipation matrix in singular optimal control, Systems & Control Letters, [14] J.C. Willems, Dissipative Dynamical Systems, Part I: General Theory, Arch. Rat. Mech. Anal., 45 (1972), (151 J.C. Willems, Least squares stationary optimal control and the algebraic Riccati equation, IEEE Trans. Aut. Contr. 16, pp. 621-634, 1971. .- -___ 2, pp. 262-266, 1983. 321-351. 255 1 [16] R.A. Wright, C. Kravaris, Nonminimum-phase com- pensation for nonlinear processes, AIChEj, 38, pp. 26- 40, 1992. 2552 

Nonlinear inner-outer factorization

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Nonlinear inner-outer factorization

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