In this note we give necessary and sufficient conditions for a linear Hamiltonian system to be input-output decouplable by Hamiltonian feedback, i.e. feedback that preserves the Hamiltonian structure. In a second paper we treat the same problem for nonlinear Hamiltonian systems

Nijmeijer, H.

English

Nijmeijer, Henk

University of Twente Research Information

Systems C Control Letters 6 (1985) 53-58 North-Holland June 1985 Input-output decoupling of Hamiltonian systems: The linear case H. NIJMEIJER and A.J. van der SCHAFT Dept. of Applied Mothemotics, Tweote University of Technology, P.O. Box 217, 7500 A E Enschede. The Netherlands Received 5 June 1984 Abstract: In this note we give necessary and sufficient condi- tions for a linear Hamiltonian system to be input-output decouplable by Hamiltonian feedback, i.e. feedback that pre- serves the Hamiltonian structure. In a second paper we treat the same problem for nonlinear Hamiltonian systems. AMS Subject Clossi/icotion: 93CO5, 93B10, 58FOS. Keywords: Noninteracting control, Linear Hamiltonian sys- tems, Output feedback, Symplectic structure. 1. Introduction A physically well-motivated subclass of the set of all linear systems i=Ax+Bu, y= cx, (1.1) consists of the Hamiltonian systems. Such a Ham- iltonian system is defined as follows. The state space X is even dimensional, say X= R2” and is endowed with a symplectic form J= Furthermore the matrices A, B and C satisfy ATJ + JA = 0 (A is Hamiltonian), Bj’= C. 0.2) It follows that necessarily the number of inputs equals the number of outputs, say U, y E R”‘. Without loss of generality we may assume that B is injective, or equivalently, C is surjective. From an input-output point of view a Hamiltonian sys- tem is fully characterized by the condition G(t) = -GT(-t), or H(s)=HT(-s), where G resp. H is the impulse response resp. transfer matrix of (l.l), see e.g. [1,7,8]. Many physical systems are of this Hamiltonian form, for example linear con- servative mechanical systems. In a second paper [ll] we shall consider nonlinear Hamiltonian sys- tems (cf. [8]), but we mention already that the linearization of a nonlinear Hamiltonian system (e.g. a robot manipulator) also exhibits this linear Hamiltonian structure (1.2). In this paper we wish to tackle the input-out- put decoupling problem for linear Hamiltonian systems; that is we seek for a feedback u = Fx + Go for the Hamiltonian systems (l.l), such that in the modified system 5~ = (A + BF)x + BGu, y=cx, 0.3) with u = (u ,,..., u,,,)~, y = (yl ,..., Y,,,)~, the input ui does not affect the output yi for j # i. Moreover we require the input ui to ‘control’ the output y,., i=l ,***, m. Necessary and sufficient conditions for the solvability of this problem are well known, cf. [2,5]. However in this paper we shall add the extra requirement that the system (1.3) after feed- back is again Hamiltonian. This restricts the al- lowed feedback in the following way. Proposition 1.1. Suppose (1.1) defines a Hamilto- nian system. The system (1.3) is again Hamiltonian if and only if (a> G = I,,,, (b) F = HC for some (m, m)-matrix H with H=HT. Proof (see [7,8]). The triple (A + BF, BG, C) has to satisfy (1.2), i.e. (A+BF)~J+J(A+BF)=~, (~G)~J=C. From the last identity we obtain G = I,,,. The first equation yields FTBTJ+JBF=FTC-CTF=O. 0167-6911/85/$3.30 0 1985, Elsevier Science Publishers B.V. (North-Holland) 53 Volume 6, Number 1 SYSTEMS & CONTROL LETTERS June 1985 Therefore the symplectic form on R2”’ is zero restricted to the subspace Im( ‘,), and this implies that dimIm g <m. ( 1 Since rank C = m  it follows F = HC for some (m,m)-matrix H. Now from FTC - CFT = 0 this yields CTHTC - CTHC = 0 or H = HT. 0 Such a feedback u = HCx + u = Hy + u, with H = HT, which leaves the Hamiltonian structure invariant, will be called a Hamiltonian feedback. From a physical viewpoint such a feedback adds a ‘potential energy’ iyTHy to the internal energy of the system, while leaving the input channels in- variant (i.e. no precompensation factor G). Now the Hamiltonian input-output decoupling problem can be formulated as follows: Consider the Ham- iltonian system (1.1). Find, if possible, an output feedback u = Hy, with H = HT, such that in k=(A+BHC)x+Bu, y= cx, the transfer functions from ui to vj are zero for j # i and nonzero for i =j, i, j = 1,. . . , m . 2. Geometric and Hamiltonian preliminaries Recall the following notions from the geometric approach to linear system theory, cf. [9,10]. A subspace I/c X is called (A, B)-invariant for the - not necessarily Hamiltonian - system (1.1) if AVc V+ Im B, or equivalently, if there exists a matrix F  such that (A + BF)Vc V. The maximal (A, B)-invariant subspace in Ker C (which exists) is denoted by V*(A, B, C). A subspace R c X is called a con- trollability subspace of (1.1) if R is (A, B)-in- variant and the eigenvalues of A + BF 1 R can arbitrarily be assigned by using an appropriate feedback F which leaves R invariant: (A + BF)R c R. The maximal controllability subspace con- tained in Ker C (which also exists) is denoted by R*(A, B, C). Furthermore a subspace S c X is called (C, A)-invariant if A(SnKerC)cS, or equivalently, if there exists a matrix K such that (A + KC)S c S. The m inimal (C, A)-in- variant subspace containing Im B (which exists) will be denoted as S*(A, B, C). We will now derive some elementary facts on (A, B)- and (C, A)-invariant subspaces of a Hamiltonian sys- tem, see also [7]. Therefore assume that the state space X equals R2” and is endowed with a sym- plectic form J. For a linear subspace VC X the orthogonal complement of V with respect to J, denoted as VI, is given by VI= {XElR2’~~XTJU=0,vUE v}. Proposition 2.1. Let (A, B, C) be a Hamiltonian system. Then (a) ( A, B) is controllable = (C, A) is ob- servable. (b) Vc Iw2” is (A, B)-invariant if and only if V1 is (C, A) inuariant. (c) v*( A, B, C) = (S*( A, B, C)) * . (d) R*(A, B, C) = V*(A, B, C) n (WA B, C))*. Proof. (a) Check the rank conditions for controlla- bility and observability by using (1.2). (b) Suppose V c R 2” satisfies AVc V+ Im. B. Then VI n(Im B)lc (AV)‘. Now XE(AV)* - xTJAu=O,Vu~ V - xTATJv=O VUEV , * AxEV’. Hence (AV)‘= A-‘(VI) (here A-‘W:= {x E R2” ] Ax E W} for a subspace W). Furthermore, since BTJ = C we have (Im B) 1 = Ker C. So V*nKerCcA-‘(VI) or A(V’nKer C)c V*. (c) Follows directly from (b). (In fact it is easily seen that the subspaces V’ and S’ in the v* and S* algorithm, cf. [9,10], are dual in the sense that vi = (S’) A .) 54 Volume 6, Number 1 SYSTEMS & CONTROL LETTERS June 1985 (d) Use the fact that R*(A, B, C)= V*(A, B, C)nS*(A, B, C), cf. [4] and (c). 0 3. Main theorem Let (A, B, C) be a Hamiltonian system on (R 2’1, J). A subspace VC Rzrr is called symplectic if V’ n V= 0. We denote by bi the i-th column of the matrix B and by ci the i-th row of the matrix C, i= l,..., m . Furthermore hi is the (2n, m  - l)- matrix derived from B by deleting the i-th column b;, i= l,..., m . Similarly Ci is the (m - 1, 2n)- matrix resulting from C by deleting ci, i = 1,. . . , m . Our main theorem is now as follows. Theorem 3.1, Let (A, B, C) be a m inimal Ham- iltonian system with V*(A, B, C) = 0. Then the Hamiltonian input-output decoupling problem can be solved if and only tf the following two conditions are satisfied: (a) b, E V*(A, B, Ci), i = 1,. . . , m , (b) V*( A, B, ci) = (V*( A, B, Ci)) 1 , i = 1 m . ,...I Proof. For simplicity we will write y = V*(A, B, Ci) and q = V*(A, B, ci). (If) Since bi E V* it follows that y = V*( A, B, Ci) = v”( A, &, Ci) and, since @+ = Cjziv*, that p = V*(A, B, ci) = V*(A, bi, ci). Now the triple (A, bi, ci) is again a Hamiltonian system (with scalar input and output) and there- fore we can apply Proposition 2.1 to conclude that R*(A, b,, ci) = @  n(p)l. Now suppose that ky := R*( A, b,, ci) # 0. Then we have bi E RF and hence A@c@+bi=& and also by definition 87,~ Ker ci. Furthermore @c(@y= y and so RT c Ker Ci. Therefore @  is A-invariant and is contained in Ker C. Hence if @ ’ # 0 we obtain a contradiction with the observability of (A, B, C). It follows that p n (@) I = 0 and hence p is a symplectic sub- space of (lR*“, J) for all i=l,...,m. Since v= (I$+) ’ this also yields that y is symplectic for all i=l ,...,m . Moreover O=l+nn(~)“=~n~= Cvnn j+i implies that the subspaces v;ll,. . . , c are indepen- dent (cf. [lo]). This yields that there exists an (m, 2n)-matrix F  such that (A+BF)vc v, i=l,...,m (this also follows from the ‘noninteracting condi- tion’ bi E I$+). Moreover ImBc iv, A~~&+ i-l i 1 i-l i-l and hence by controllability R *’ = Vi+ @  + - . @  V$ Furthermore because v = (@+) ’ and @  is (A, B)-invariant we obtain by Proposition 2.1 that v is also (C, A)-invariant, i = 1,. . . , m . There- fore there exists an (m, m)-matrix H such that (A+BHC)vcy, i=l,..., m . The last thing we have to show is that we may take H to be a symmetric matrix. Since v is symplectic there exists a symplectic basis (qi, pi) for v, i.e. a basis (qi, Pi)‘(qjI***,q~‘> PjS***>Pf’) with dim v = 2ki, such that J restricted to y equals 0 -Ik, i I ‘ki 0 in this basis. Then because (y)’ = Cj, iT it follows that (ql,. . . , q,,, ql,. . . , p,,) is a symplectic basis for IW’“, with 0 -4, J=L o. i 1 ,I Next we will use the relations (A+BHC)vc v, i=l,..., m . 55 Volume 6, Number 1 SYSTEMS & CONTROL LETTERS June 1985 We have (A+BHC)V;Cc V;, V;CcKerC,, (A+BHC)ec q, QcKerc,. Hence (JA+JBHC)c=(-Q-CTHC)v cJv;z=vy and (-Q-CTHC)fi=fi, where Q = -JA is symmetric. Writing out in the above basis we get where we have symbolically written I*\ 101 In the same way the matrix C can be expressed in this basis as c=; cc: ( 1 1 i c” 22 e; c”. 24 1 Furthermore the matrices Q and H will be parti- tioned according to this basis and we obtain that Q and CTHC have the following form respec- tively: Qn Qn Q13 Q14 Q,, Q, Q23 Q24 Q31 Q32 Q33 Q34 Q41 Q42 Q43 Qw C2wll CT1 h 12c22 C;rh,,c,, Ci!JW22 &h,c,, c:3 h nG2 G$,,c,, Cif&22C22 C,T,h21c,3 c&H22c24 c;h,,cn ~T3hnG.4 ’ and so the matrix Q + CTHC leaves the subspaces I*\ /o\ invariant. This yields chC2, + Q,, = 0, C,Th,,c,, + Q2, = 0, C,Th,,c,, + Q23 = 0, &h,,G, + Q32 = 0, c:lh,2C24 + Q14 = O, C$wll+ Q41= 0, c?“h,,G, + Q34 = 0, C,T,h,,c,, + Q43 = 0. (3.1) Now the matrix Q is symmetric, so q,2 = q$, qz3 =qT2, q,4= q& and q34= qz. It follows that we may take h,, = hy2 in (3.1). Having fixed h,, and h,, = h& we proceed by considering &+ and @  (or e \ T/;r). Continuing up to c and e in this way we conclude that we may take H satisfy- ing (A+BHC)vcv, i=l,..., m, to be symmetric. Therefore it follows from bi E v*, i=l ,--*, m , that u = Hy + IJ is a Hamiltonian feedback which decouples the system. Clearly the subsystems (A +BHCIv, bi, q), i=l,..., m, are m inimal Hamiltonian systems. (on& if) It is well known that the input-output decoupling by general state feedback is possible if and only if ImB= iIrnBnv, i-l see [5]. Since the precompensation matrix G equals I,,,, it follows that bi E v, i = 1,. . . , m . Because V*(A, B, C) = 0 we must have (A+BHC)v*c, i=l,..., m , for a certain symmetric H. Consider the subspace V= vn p for an iE{l,...,m}. Clearly VC KerC,so(A+BHC)V=AVcVandthusbythe observability of the system I/= 0. Hence for i = 56 Volume 6, Number 1 SYSTEMS & CONTROL LETTERS June 1985 Moreover by controllability we have C;: ,y = R 2”. We will now use Proposition 2.1 for showing that condition (b) is satisfied. Since (A+BHC)vc y* the subspace F* is (C, A)-invariant and so (y*) 1 is (A, B)-invariant. Because (v*)*c(Imbi)*=KerCi we obtain (y+)lc p*, i=l ,..., m . However since and RZfr= JP@ )T? , , 3 i=l 3...,mr wemayconclude(v)‘=@ forall i=l,...,m. cl Remark. Notice that if (a) and (b) are satisfied v is not only the maximal (A, @-invariant subspace contained in Ker Ci, but also equal to the m inimal (C, A)-invariant subspace containing Im b,, i = 1 ,. . ., m . Furthermore it follows that v equals R*(A, B, ci). We conclude from the above theorem that the Hamiltonian input-output decoupling problem is solvable if and only if the Hamiltonian system i=Ax+Bu, y=Cx with VU(A, B, C)=O can be partitioned into m  independent Hamiltonian subsystems after applying Hamiltonian feedback. Furthermore the diagonal elements of the output feedback u = HJJ remain to be freely chosen. This freedom can be used to adjust the behaviour of these subsystems. We leave the case Ir*(A, B, C) # 0 for further research (see also [ll]). In a similar way as in Theorem 3.1 we can deal with the Hamiltonian block-decoupling problem, that there is given a partitioning of the outputs Y= (Yw..dnJT= (_yl,...Qk)T into k disjoint groups (and thus also of the inputs u=(u ,,...I unJT= (_u ,,..., _U,)T) and the question is whether or not we can find a Hamiltonian feedback u = Hx + u, H = HT, such that gi does not affect _vi, j # i, and Q,. controls _vi, i= l,..., k. Writing, as before, Bi for ,Jhe input matrix corresponding to the inputs ui, Bi for the input matrix B with deleting Bi and similarly Ci and Ci we obtain the following sufficient condi- tions. Corollary 3.2. Let (A, B, C) be a m inimal Ham- iltonian system with V*(A, B, C) = 0. Then the Hamiltonian block-decoupling problem is solvable if the following three conditions hold: (a) Im Bi E F(A, B, ei), i = 1,. . . , k, (b) v*( A, B, Ci) = (V*( A, B, ci)) ’ , i = 1,. . . , k, (c) R*(A, Bi,Ci)=O, i=l,..., k. Proof. Let v := V*(A, B, CT) and @  = Tr*(A, B, Ci), i=l , . . . , k. Since Im Bi E ,y we have v = v*( A, &,., Ci) and because v = Cj, il$+ we ob- tain @+ = VY(A, Bi, Ci). The triple (A, Bi, Ci) being Hamiltonian allows us to apply Proposition 2.1 to conclude O=R*(A, Bi,Ci)=pr?(@)‘, j=l , . . . , k. So again, the p’s are symplectic sub- spaces and o=pn(@)‘= CTnn j#i implies that they are independent. The rest of the proof is now as in Theorem 3.1. 0 Remark. Condition (c) of Corollary 3.2 implies that the subsystems (A, B, Ci) are left-invertible, cf. [lo], and therefore also right-invertible. In this way the ‘geometric’ decoupling problem with the prerequisite of output controllability coincides with the transfer function approach of [3]. We will conclude this paper with a very simple example. 57 Volume 6, Number 1 SYSTEMS & CONTROL LETTERS June 1985 Example 3.3. Time-reversible Hamiltonian systems have the following additional structure (cf. [6]). There exists a symplectic basis of lRZfi such that A=( -“e ;), B=(;,), c=(c, o), with P = PT, Q = QT and By = C,. Considering only the matrix A we know that if P > 0 there is a symplectic basis for R “’ such that A=(-; ;) with D a diagonal matrix. Setting B, = I,,, = C, (m = n), we clearly have a decoupled system con- sisting of n independent oscillators. However, given B, = C, = I,,, input-output decoupling is possible if and only if P is already diagonal. (Q may be arbitrary since Q + I,, HI, can be freely assigned by the choice of H.) Cl References [l] R.W. Brockett and A. Rabimi, Lie algebras and linear differential equations, in L. Weiss, Ed., Ordinury Dijjeren- tial Equations (Academic Press, New York, 1972). [2] P.L. Falb and W.A. Wolovich, Decoupling in the design of multivariable control systems, IEEE Truns. Automat. Con- trol 12 (1967) 651-659. [3] M.L.J. Hautus and M. Heymann, Linear feedback decou- pling - transfer function analysis, IEEE Tram. Automat. Control 28 (1983) 823-832. [4] AS. Morse, Structural invariants of linear multivariable systems, SIAM J. Control 11 (1973) 446-465. [5] AS. Morse and W.M. Wonham, Status of noninteracting control, IEEE Trans. Automat. Control 16 (1971) 568-580. [6] A.J. van der Schaft, Time-reversible Hamiltonian systems, Systems Control Lett. 1 (1982) 295-300. [7] A.J. van der Schaft, Disturbance decoupling by observa- tion feedback for Hamiltonian systems, System Control Len. 2 (1983) 286-291. (81 A.J. van der Schaft, System theoretic descriptions of physi- cal systems, Doctoral Dissertation, Rijksuniversiteit Groningen (1983); also CWI Tracts, No. 3, Amsterdam (1984). [9] J.M. Schumacher, Dynamic feedback in finite- and in- finite-dimensional linear systems, Mathematical Centre Tract 143 (1981). [lo] W.M. Wonham, Linear Multivariable Control: A  Geometric Approach (Springer, Berlin, 1979). [ll] H. Nijmeijer and A.J. van der Schaft, Input-output de- coupling of Hamiltonian systems: The nonlinear case, in preparation. 58 

Input-output decoupling of Hamiltonian systems:The linear case

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Input-output decoupling of Hamiltonian systems: The linear case

Decoupling in the design of multivariable control systems,

Disturbance decoupling by observation feedback for Hamiltonian systems,

Dynamic feedback in finite- and infinite-dimensional linear systems,

Input-output decoupling of Hamiltonian systems: The nonlinear case, in preparation.

Lie algebras and linear differential equations,

Linear feedback decoupling - transfer function analysis,

Linear Multivariable Control: A Geometric Approach (Springer,

Status of noninteracting control,

Structural invariants of linear multivariable systems,

Time-reversible Hamiltonian systems,

https://research.utwente.nl/en/publications/bd67e259-8d50-4dde-b825-15bf5f68a1fc

Input-output decoupling of Hamiltonian systems: The linear case

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