When Gradient Systems and Hamiltonian Systems Meet

Transcription

1 When Gradient Systems and Hamiltonian Systems Meet Arjan van der Schaft Johann Bernoulli Institute for Mathematics and Computer Science University of Groningen, the Netherlands December 11, 2011 on the occasion of the 60-th birthday of Peter Crouch Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems Meet Pre-CDC Workshop 1 / 32

2 Outline 1 Introduction 2 Linear case 3 On the nonlinear case 4 Conclusions and outlook Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems Meet Pre-CDC Workshop 2 / 32

3 Outline Introduction 1 Introduction 2 Linear case 3 On the nonlinear case 4 Conclusions and outlook Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems Meet Pre-CDC Workshop 3 / 32

4 Introduction When did we meet? Probably the first time I met Peter was in September 1981 during a Warwick-Bremen-Groningen meeting in Edzell, Scotland. Peter published in 1981 the very nice paper Geometric structures in systems theory 1, dealing (among other things) with nonlinear gradient systems, and I published a paper on Hamiltonian systems, continuing on the 1977 paper by Roger Brockett. Next time was probably a nonlinear control conference in Michigan (Copper Peninsula), summer Since then we met at various occasions (conferences), and started our common work on, in particular, realization theory of nonlinear Hamiltonian input-state-output systems 2. My family and I spent Spring 1991 in Tempe (and enjoyed Arizona!). 1 IEE Proceedings, 128, pp , P.E. Crouch, A.J. van der Schaft, Variational and Hamiltonian control systems, Lect. Notes in Control and Information Sciences, Vol. 101, Springer-Verlag, 1987 Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems Meet Pre-CDC Workshop 4 / 32

5 Introduction Gradient versus Hamiltonian systems Where do gradient and Hamiltonian systems agree? Common belief: gradient systems are opposite to Hamiltonian systems. This is certainly true if one considers the dynamical behavior of a standard gradient vector field G(x)ẋ = V x (x) and that of a standard Hamiltonian vector field q = H p (q,p) ṗ = H q (q,p) Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems Meet Pre-CDC Workshop 5 / 32

6 Introduction However, we will consider gradient and Hamiltonian systems that are of the following generalized type: The inner product G(x) of the gradient system may be indefinite. The Hamiltonian system may also include resistive elements (port-hamiltonian systems) Main result : There is an important intersection of the set of gradient and port-hamiltonian systems, which includes the systems studied by Brayton & Moser. Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems Meet Pre-CDC Workshop 6 / 32

7 Outline Linear case 1 Introduction 2 Linear case 3 On the nonlinear case 4 Conclusions and outlook Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems Meet Pre-CDC Workshop 7 / 32

8 Linear case Linear passive systems are port-hamiltonian Consider a minimal linear system Σ(A,B,C) ẋ = Ax +Bu y = Cx and having transfer matrix, x R n,u R m,y R m, H(s) := C(Is A) 1 B The system Σ(A,B,C) is passive if there exists Q = Q T > 0 such that A T Q +QA 0 B T Q = C or, equivalently (by the Kalman-Yakubovich-Popov lemma), the transfer matrix H(s) is positive real. Such a matrix Q is called a storage matrix, in the sense that for all x,u d 1 dt 2 xt Qx u T y Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems Meet Pre-CDC Workshop 8 / 32

9 Linear case In general there are many Q! Fixing a storage matrix Q, and defining J to be the skew-symmetric part of the matrix AQ 1 and R its symmetric part, it follows that any passive system can be written as the linear port-hamiltonian system where ẋ = [J R]Qx +Bu y = B T Qx J = J T, R = R T 0 Note: thus in general there are many port-hamiltonian formulations of passive systems. Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems Meet Pre-CDC Workshop 9 / 32

10 Linear case Linear gradient systems Consider a minimal linear system Σ(A,B,C) with state vector z ż = Az +Bu y = Cz, z R n,u R m,y R m, with transfer matrix H(s). The symmetry (or, reciprocity) condition 3 H(s) = H T (s) is equivalent to the existence of a unique matrix G = G T satisfying A T G = GA B T G = C 3 This should be generalized to Σ eh(s) = H T (s)σ e with Σ e a signature matrix. Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems MeetPre-CDC Workshop 10 / 32

11 Linear case Denoting P := GA it follows that P = P T, and the system Σ(A,B,C) can be rewritten as the linear gradient system Gż = Pz +C T u, G = G T,P = P T y = Cz with inner product G and potential function 1 2 zt Pz. Note that in general G is indefinite. Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems MeetPre-CDC Workshop 11 / 32

12 Linear case Passivity and reciprocity Now consider a system, which is both passive and reciprocal: a passive gradient system. In Willems (1972) it has been shown that there always exists a storage matrix Q satisfying additionally the compatibility condition Q = GQ 1 G Lemma Let Q = Q T > 0 satisfy the compatibility condition. Then there exists a basis in which [ ] [ ] Q1 0 Q1 0 Q =, G = 0 Q 2 0 Q 2 for some k k matrix Q 1 = Q1 T > 0 and (n k) (n k) matrix Q 2 = Q2 T > 0, where 0 k n. Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems MeetPre-CDC Workshop 12 / 32

13 Linear case Proposition Consider a passive gradient system and choose a storage matrix Q satisfying the compatibility condition. In coordinates as above the gradient system takes the form [ Q1 0 0 Q 2 ] ż = Pz +[ C T 1 0 y = [ C 1 0 ] z ] u with the potential function 1 2 zt Pz satisfying, Q 1 > 0,Q 2 > 0 [ ] P1 P P = c Pc T, P P 1 = P1 T 0,P 2 = P2 T 0 2 Conversely, any gradient system with P 1 0 and P 2 0 is passive with storage matrix Q. Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems MeetPre-CDC Workshop 13 / 32

14 Linear case Port-Hamiltonian formulation Multiply the second part of the differential equations (corresponding to z 2 ) by a minus sign, to obtain [ ] [ ] ] Q1 0 P1 P ż = c C T 0 Q 2 Pc T z +[ 1 u P 2 0 y = [ C 1 0 ] z Proposition Transform the co-energy variables z to the energy variables x defined by x 1 = Q 1 z 1, x 2 = Q 2 z 2 In these new variables the passive gradient system takes the following port-hamiltonian form Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems MeetPre-CDC Workshop 14 / 32

15 Linear case Proposition ẋ = ([ ] [ ])[ 0 Pc P1 0 Q 1 Pc T 0 0 P 2 y = [ C 1 0 ][ Q1 1 ] 0 0 Q2 1 x Q 1 2 which is a port-hamiltonian system with [ ] [ ] 0 Pc P1 0 J = Pc T, R = 0, B = 0 0 P 2 and Hamiltonian 1 2 xt 1 Q1 1 x xt 2 Q 1 Call this a reciprocal port-hamiltonian system. 2 x 2 ] x +[ C T 1 0 [ C T 1 0 ], ] u Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems MeetPre-CDC Workshop 15 / 32

16 Linear case Remarks The matrix P c defines the power-conserving interconnection matrix J of the port-hamiltonian system, and thus corresponds to a lossless coupling between the two types of energy storage 1 2 xt 1 Q 1 1 x 1 and 1 2 xt 2 Q 1 1 x 2. The symmetric matrices P 1 and P 2 define the dissipation matrix R, and thus correspond to energy dissipation. The variables x 1,x 2 are called the energy variables, since the energy is 1 2 xt 1 Q1 1 x xt 2 Q 1 while the co-energy is given as 2 x zt 1 Q 1z zt 2 Q 2z 2 with equal values (in this linear case!) In case Q = G or Q = G (corresponding to k = n, respectively k = 0) there is only one type of energy-storage, and the matrix P c describing the coupling between the two energy-domains is void. Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems MeetPre-CDC Workshop 16 / 32

17 Linear case Example (Brayton-Moser formulation of RLC circuits) The resistive content function (or current potential) corresponding to the current-controlled resistors in the circuit is of the form 1 2 IT RI for some symmetric matrix R 0 (the resistance matrix), where I denotes the vector of currents through the inductors of the network. Furthermore, the conductive co-content function (or voltage potential) corresponding to the voltage-controlled resistors is of the form 1 2 VT GV for some symmetric matrix G 0 (the conductance matrix), where V is the vector of voltages over the capacitors. Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems MeetPre-CDC Workshop 17 / 32

18 Linear case Example Associated to the circuit-interconnection of the inductors and capacitors (interpreted as a bank of ideal unity transformers), we define the coupling potential I T ΛV where the matrix Λ, only consisting of elements 1, 1,0 can be interpreted as a turn-ratio matrix. The total potential function is then 1 2 IT RI 1 2 VT GV +I T ΛV Finally, the magnetic co-energy 1 2 IT LI (with L the diagonal matrix of inductances) can be identified with 1 2 zt 1 Q 1z 1, and the electric co-energy 1 2 VT CV (with C the diagonal matrix of capacitances) with 1 2 zt 2 Q 2z 2. Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems MeetPre-CDC Workshop 18 / 32

19 Linear case Example The resulting dynamics of the gradient system for u = 0 given as [ ] [ ] L 0 I ż = Pz, z = 0 C V are commonly called the Brayton-Moser equations of an RLC-circuit. The co-energy variables z 1 and z 2 in this case are given as the vector of currents I through the inductors, respectively the vector of voltages V across the capacitors. The corresponding energy variables x 1 and x 2 are the vector of inductor fluxes, and the vector of capacitor charges. Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems MeetPre-CDC Workshop 19 / 32

20 Linear case Example (Consensus algorithm) The standard consensus algorithm for an undirected graph is ẋ = Lx, where L = L T 0 is the (weighted) Laplacian matrix of the graph. This dynamics can be interpreted in two ways: 1. The dynamics is port-hamiltonian with Hamiltonian function H(x) = 1 2 x 2 and resistive matrix R = L. 2. It is a gradient system in z = H x = x, with inner product G = I, and potential function 1 2 zt Lz (sometimes called the Laplacian potential or disagreement function). Formulation can be extended to leader-follower multi-agent systems. Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems MeetPre-CDC Workshop 20 / 32

21 Outline On the nonlinear case 1 Introduction 2 Linear case 3 On the nonlinear case 4 Conclusions and outlook Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems MeetPre-CDC Workshop 21 / 32

22 On the nonlinear case Nonlinear gradient systems Nonlinear gradient systems are defined in local coordinates z for some n-dimensional state space manifold X as systems of the form; see Varaiya, Brockett (1977), Crouch (1978, 1981) G(z)ż = V z (z,u) y = V z (z,u) where G(z) defines a pseudo-riemannian metric on X. An important subclass is obtained by considering V(z,u) = P(z) u T C(z) leading to G(z)ż = P z (z)+ T C z (z)u y = C(z) These systems were finally externally characterized (elegantly!) in the SICON 2005 paper by Cortes, vds and Crouch, after several, nearly successful, attempts by the latter two at the end of the 1980s. Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems MeetPre-CDC Workshop 22 / 32

23 On the nonlinear case The main bottleneck for extending the results for linear passive gradient systems to the nonlinear case resides in the generalization of the compatibility condition to the nonlinear case. This seems to involve the restriction of pseudo- Riemannian metrics to Hessian pseudo-riemannian metrics. Definition A pseudo-riemannian metric defined by the non- singular symmetric matrix G(z) is called Hessian if there exists a function K(z) such that the (i,j)-th element g ij (z) of the matrix G(z) is given as g ij (z) = 2 K z i z j (z) A necessary and sufficient condition for the (local) existence of such a function K(z) is the integrability condition g jk z i (z) = g ik z j (z), i,j,k = 1,,n Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems MeetPre-CDC Workshop 23 / 32

24 On the nonlinear case Nonlinear reciprocal port-hamiltonian system The restriction to Hessian pseudo-riemannian metrics is evidenced by the Brayton-Moser formulation of nonlinear RLC-circuits, as well as by the treatment of port-hamiltonian systems in co-energy variables. Consider a nonlinear input-state-output port-hamiltonian system ẋ = [J(x) R(x)] H x (x)+g(x)u y = g T (x) H x (x) Define the co-energy variables z = H x (x) together with the inverse x = H z (z) with H the Legendre transform of H. Using ẋ = 2 H z 2 (z)ż we obtain 2 H z 2 (z)ż = [J(x) R(x)]z +g(x)u y = g T (x)z Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems MeetPre-CDC Workshop 24 / 32

25 On the nonlinear case Now assume there exist coordinates x 1,x 2 such that [ ] 0 Pc J(x) = Pc T 0 [ ] g1 g(x) = 0 [ ] R1 (x) 0 R(x) = 0 R 2 (x) H(x 1,x 2 ) = H 1 (x 1 )+H 2 (x 2 ) Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems MeetPre-CDC Workshop 25 / 32

26 On the nonlinear case Then the system in co-energy variables takes the form 2 H1 (z z1 2 1 ) 0 ] [ R1 (x) P [ż1 = c 0 (z 2 ) ż 2 Pc T R 2 (x) 2 H 1 z 2 2 y = g T 1 (x)z 1 Assuming furthermore that P 1 (z 1 ),P 2 (z 2 ) such that ][ z1 R 1 (x)z 1 = P 1 z 1 (z 1 ), R 2 (x)z 2 = P 2 z 2 (z 2 ) Then define the potential function P(z 1,z 2 ) = P 1 (z 1 )+P 2 (z 2 )+z T 1 P c z 2 z 2 ] + [ ] g1 u 0 Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems MeetPre-CDC Workshop 26 / 32

27 On the nonlinear case Then the system is equivalent to the nonlinear gradient system 2 H1 ] [ (z z1 2 1 ) 0 P ] [ ] [ż1 z = 1 (z) g1 + u 0 2 H1 P (z z2 2 2 ) ż 2 z 2 (z) 0 y = g T 1 (x)z 1 Note 4 that the indefinite Riemannian metric is a Hessian matrix; in fact, with respect to the Legendre transform of the Hamiltonian function. 4 Not bad to have so many great mathematicians in one sentence! Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems MeetPre-CDC Workshop 27 / 32

28 On the nonlinear case Feedback interconnection of gradient systems Consider two gradient systems G i (z i )ż i = P i z i (z i )+ T C i z i (z i )u i y i = C i (z i ), i = 1,2 and interconnect them via the standard negative feedback interconnection u 1 = y 2, u 2 = y 1 Then the interconnected system is again a gradient system with indefinite Riemannian metric G 1 G 2 and mixed potential function P ( z 1 ) P 2 (z 2 )+C T 1 (z 1)C 2 (z 2 ) Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems MeetPre-CDC Workshop 28 / 32

29 Outline Conclusions and outlook 1 Introduction 2 Linear case 3 On the nonlinear case 4 Conclusions and outlook Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems MeetPre-CDC Workshop 29 / 32

30 Conclusions and outlook Conclusions and outlook The intersection of port-hamiltonian and gradient systems is a common class in applications, covering Brayton-Moser systems. For electrical circuits the class of systems corresponds to RLCT-circuits (no gyrators); see the work of Anderson and Willems. Provides new ideas for stability analysis and stabilization. Precise conditions generalizing the compatibility condition to the nonlinear case need to be worked out. Nice geometric theories directly motivated by engineering and physics. See arjan for further info. Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems MeetPre-CDC Workshop 30 / 32

31 Happy Birthday Peter! Arjan van der Schaft (Univ. Groningen) When Gradient Systems and Hamiltonian Systems MeetPre-CDC Workshop 31 / 32