Introduction to Control of port-hamiltonian systems - Stabilization of PHS

Transcription

1 Introduction to Control of port-hamiltonian systems - Stabilization of PHS - Doctoral course, Université Franche-Comté, Besançon, France Héctor Ramírez and Yann Le Gorrec AS2M, FEMTO-ST UMR CNRS 6174, Université de Franche-Comté, École Nationale Supérieure de Mécanique et des Microtechniques, 26 chemin de l épitaphe, F-253 Besançon, France March, 215 Doctoral course UFC-ST 1 / 14

2 References Relevant references: 1. Brogliato, B., Lozano, R., Maschke, B., and Egeland, O. (27). Dissipative Systems Analysis and Control. Communications and Control Engineering Series. Springer Verlag, London, 2nd edition edition. 2. A.J. van der Schaft, L2-Gain and Passivity Techniques in Nonlinear Control, Lect. Notes in Control and Information Sciences, Vol. 218, Springer-Verlag, Berlin, 1996, p. 168, 2nd revised and enlarged edition, Springer-Verlag, London, 2 (Springer Communications and Control Engineering series), p. xvi Jeltsema. D, van Der Schaft, A. Memristive port-hamiltonian systems, Mathematical and Computer Modelling of Dynamical Systems - MATH COMPUT MODEL DYNAM SYST 1/21; 16(2). DOI:1.18/ van der Schaft, A.J and Jeltsema, D. Port-Hamiltonian Systems: from Geometric Network Modeling to Control, Module M13, HYCON-EECI Graduate School on Control, April 7 1, 29. Doctoral course UFC-ST 2 / 14

3 Outline 1. Stability: definitions 2. Stabilization of PHS 3. Control by interconnection 4. Interconnection and Damping Assignment Passivity Based Control 5. Final remarks and future work Doctoral course UFC-ST 3 / 14

4 Some basic notions on stability We are considering the following class of state space model ẋ(t) = f (x(t)), x() = x, (1) with t R and x R n. Furthermore f : R n R m R n. It is assumed that f (x(t)) satisfies the standard assumptions for existence and uniqueness of solutions, i.e., that f (x(t)) is Lipschitz continuous with respect to x, uniformly in t, and piecewise continuous in t. We want to analyse the dynamics of the system Do the solutions of (1) remain bounded in time? If yes, do they in addition converge to some equilibrium point? If yes, can we say anything of the speed of convergence? Can we modify the solution with some external control input to impose a desired closed-loop behaviour? Doctoral course UFC-ST 4 / 14

5 Some basic notions on stability We are considering the following class of state space model ẋ(t) = f (x(t)), x() = x, (2) with t R and x R n. Furthermore f : R n R m R n. It is assumed that f (x(t)) satisfies the standard assumptions for existence and uniqueness of solutions, i.e., that f (x(t)) is Lipschitz continuous with respect to x, uniformly in t, and piecewise continuous in t. Equilibrium position An equilibrium point x is a solution to ẋ(x ) =, i.e., f (x ) = Without loss of generality we will assume that x =. Doctoral course UFC-ST 5 / 14

6 Some basic notions on stability An equilibrium position x = of system (1) is 1. Stable if for any ɛ > and t, there exists a δ(ɛ, t ) > such that x < δ implies x(t, x ) < ɛ for all t t, 2. Uniformly Stable if δ does not depend on t, 3. Asymptotically Stable if it is stable and for any t there exists a (t ) > such that every solution x(t, x ) of system (1 for which x < satisfies the relation lim x(t, x ) (3) t 4. Uniformly Asymptotically Stable if it is uniformly stable, does not depend on t, and relation (3) holds uniformly with respect to t and x in the domain t, x <, Doctoral course UFC-ST 6 / 14

7 Some notions on stability An equilibrium position x = of system (1) is 1. Globally Asymptotically Stable if it is stable and relation (3) holds for any t and x, 2. Uniformly Globally Asymptotically Stable if it is uniformly stable and relation (3) holds for any t and x uniformly relative to t and x in the domain t, x K, where K is arbitrary compact in the x-space, 3. Exponentially Asymptotically Stable if there exist positive constants, M, and α such that every solution x(t, x ) of system (1), for which x <, satisfies the relation x(t, x) < M x e α(t t ) (4) for all t t, and 4. Globally Exponentially Asymptotically Stable if there exist positive constants M and α such that relation (4) holds for t t and arbitrary t and x. Doctoral course UFC-ST 7 / 14

8 Some notions on stability Figures taken from: Doctoral course UFC-ST 8 / 14

9 Some notions on stability How do we analyse stability? Lyapunov stability theory Lyapunov s direct method (second method) non-linear systems Lyapunov s indirect method (first method) linear systems Lyapunov s direct method allows to determine the stability of a system without explicitly integrating the differential equations. The method is a generalization of the idea that if there is some measure of energy in a system, then we can study the rate of change of the energy of the system to ascertain stability. Doctoral course UFC-ST 9 / 14

10 Some notions on stability Let B ɛ be a ball of size ɛ around the origin, B ɛ = x R n : x < ɛ. Positive definite function A continuous function V : R n R is a locally positive definite function if V () = and for x B ɛ, x V (x) >. If B ɛ is the whole state space, then V (x) is globally positive definite. A positive definite function is like an energy function. Doctoral course UFC-ST 1 / 14

11 Lyapunov s direct method Theorem: Lyapunov stability Let V (x) be a non-negative function with continuous partial derivatives such that V (x) is positive definite on B ɛ, and V locally in x and for all t, then the origin of the system is locally stable (in the sense of Lyapunov). If in addition V (x) when x, then the system is globally stable. Theorem: Asymptotic stability Let V (x) be a non-negative function with continuous partial derivatives such that V (x) is positive definite on B ɛ, and V <, x B ɛ/{} and V () = locally in x and for all t, and then the origin of the system is locally asymptotically stable. If in addition V (x) when x, then the system is globally asymptotically stable. Doctoral course UFC-ST 11 / 14

12 Lyapunov s direct method Figure : taken from: For physical systems: relate the physical energy with Lyapunov functions Doctoral course UFC-ST 12 / 14

13 Stabilization of passive systems Questions: Are passive system asymptotically stable? Can we increase the convergence rate to the origin? Can we shift the stable equilibrium? Doctoral course UFC-ST 13 / 14

14 Stabilization of passive systems Let us consider system arising from some physical energy model. We then usually have H(t) = H(t ) + u(τ)y(τ)dτ S(x(τ))dτ. }{{}}{{} supplied energy dissipated energy So if H(x) qualifies as a Lyapunov function and S(x) vanishes at x = (and only in x = ), then the system is asymptotically stable! So why do we need the control then? Doctoral course UFC-ST 14 / 14

15 Stabilization of passive systems What if we want to increase the rate of convergence?: damping injection, What if we want to stabilize at some different equilibrium point, x = x, x : Energy shaping, What if S(x) vanishes for some x or S(x) =?: damping injection + Energy shaping Doctoral course UFC-ST 15 / 14

16 PHS are dissipative and passive systems Consider the system ẋ(t) = f (x(t), u(t)), y(t) = h(x(t), u(t)), (5) with t R, x R n, u R m and y R p. Furthermore f : R n R m R n and h : R n R m R p. Let us addition define the supply rate w(t) = w(u(t), y(t)), w(u(τ), y(τ))dτ < Dissipative systems The system (5) is said to be dissipative if there exists a so-called storage function V (x) such that the following dissipation inequality holds: V (x(t)) V (x()) + w(u(τ), y(τ))dτ along all possible trajectories of (5) starting at x(), for all x(), t. Doctoral course UFC-ST 16 / 14

17 Passive systems A particular case of dissipative systems are passive systems: Passive systems Suppose that the system (5) is dissipative with supply rate w(u, y) = u T y and storage function V (x(t)) with V () = ; i.e. for all t we have that Then the system is passive. V (x(t)) V (x()) + u(τ) y(τ)dτ, Passive systems are a subclass of dissipative systems with the specific properties The supply rate is defined by the product between inputs and outputs w(u, y) = u T y, The minimum of the storage function is at the origin V () =. Doctoral course UFC-ST 17 / 14

18 Passive systems The dissipation of a passive system may also be explicitly taken into account: Strictly passive systems A system (5) is said to be strictly state passive if it is dissipative with supply rate w = u y and the storage function V (x(t)) with V () =, and there exists a positive definite function S(x) such that for all t : V (x(t)) V (x()) + u(τ) y(τ)dτ S(x(τ))dτ, If the equality holds in the above equation and S(x) =, then the system is said to be lossless (conservative). The function S(x) is called the the dissipation rate. Doctoral course UFC-ST 18 / 14

19 Passive systems Why are we interested in passive systems? Many physical systems are passive with respect to the storage function defined by their physical energy function and with respect to their natural supply rate (given by the physical inputs and outputs), Its a non-linear approach (does not require any assumption of linearity), The physical energy may be used as a candidate Lyapunov function to analyse stability. A well defined interconnection of passive system is again a passive system. Doctoral course UFC-ST 19 / 14

20 Stability of passive systems Recall the passivity inequality V (x(t)) V (x()) + u(τ) y(τ)dτ, if the storage function V is strictly increasing and x = is an isolated global minimum, then the output feedback u = stabilizes the system at x = (LaSalle s inv. Principle). Damping injection V For many physical systems: Energy = storage function = Lyapunov function Doctoral course UFC-ST 2 / 14

21 Port-Hamiltonian control systems Energy-dissipation is included in the framework of PHS by terminating some of the ports by resistive elements. Dissipative PHS ẋ = ( J(x) R(x) ) H (x) + g(x)u(t), x y = g (x) H x (x). Two geometric structures: J(x) and R(x) = R(x), the damping matrix. The energy-balance is Ḣ = u (t)y(x(t)) H (x(t))r(x(t)) H x x (x(t)), Ḣ u (t)y(t), Dissipative PHS are strictly-passive if the Hamiltonian H is bounded from below. Doctoral course UFC-ST 21 / 14

22 Stability of strictly passive systems Recall the strict passivity inequality V (x(t)) V (x()) + u(τ) y(τ)dτ S(x(τ))dτ, if the storage function V is strictly increasing, x = is an isolated global minimum and S(x(t)) only vanishes at x =, then the output feedback u = asymptomatically stabilizes the system at x = (LaSalle s inv. Principle). Damping injection V S(x(t)) For many physical systems: physical dissipation = dissipation rate Doctoral course UFC-ST 22 / 14

23 Examples: RLC circuit and MSD system Show that the RLC and the MSD are strictly passive and give the corresponding PHS models. Doctoral course UFC-ST 23 / 14

24 Example: RLC circuit Let us consider a simple linear RLC circuit: Constitutive relations u s = V in u r = RI r φ = LI L Q = Cu C Dynamic relations u L = dφ dt, or in integral form φ(t) = φ(t ) + u L (τ)dτ I C = dq dt, or in integral form Q(t) = Q(t ) + I C (τ)dτ Doctoral course UFC-ST 24 / 14

25 Example: RLC circuit Interconnection relations (Kirchkoff s laws): u = voltage law, i = current law Using the interconnection relations together with the constitutive and dynamical relations we obtain the state space model dq = φ dt L dφ = Q dt C R φ L + V in with state variables x = [Q, φ] and input V in. If the initial conditions Q(t ) and φ(t ) are known, together with the profile V in, then the time evolution of the system is fully determined for all t > t. Doctoral course UFC-ST 25 / 14

26 Example: RLC circuit What about the energy of the systems? Energy = Energy stored in the capacitor + Energy stored in the inductor H(x(t)) = 1 2 The time variation of the energy is given by dh(x(t)) dt = H dx x dt = V in ( φ L ( ) ( Q φ = C L ( φ L ) R ) 2 φ Q L 2 C ( Q C ) 2 = V ini L RI 2 L ) ( ) ( ) φ φ + V in R L L ( ) φ 2 L Hence, the balance equation characterizing the time variation of energy can be written as H(t) = H(t ) + V in (τ)i L (τ)dτ RI L (τ) 2 dτ Doctoral course UFC-ST 26 / 14

27 Example: mass-spring-damper system Let us consider a simple linear translational MSD system: Constitutive relations F s = F in F B = Bv B p = Mv M q = K 1 F K Dynamic relations F M = dp dt, or in integral form p(t) = p(t ) + F M (τ)dτ v K = dq dt, or in integral form q(t) = q(t ) + v K (τ)dτ Doctoral course UFC-ST 27 / 14

28 Example: mass-spring-damper system Using the interconnection relations (Kirchkoff s laws) together with the constitutive and dynamical relations we obtain the state space model with state variables x = [q, p] and input F in. dq = p dt M dp = q dt K 1 B p M + F in Doctoral course UFC-ST 28 / 14

29 Example: MSD system What about the energy of the systems? Energy = Energy stored in the mass + Energy stored in the spring The time variation of the energy is given by dh(x(t)) dt = H dx x dt = F in ( p M H(x(t)) = 1 p 2 1 q M 2 K 1 ( q = ) B ) ( q ) ( p ) ( p ) ( p ) 2 K 1 + F in D M M M ) ( p K 1 M ( p ) 2 = Fin v M BvM 2 M The balance equation characterizing the time variation of energy can be written as H(t) = H(t ) + F in (τ)v M (τ)dτ Bv M (τ) 2 dτ Doctoral course UFC-ST 29 / 14

30 Port-modelling of physical systems Let us look closer to the energy balance H(t) = H(t ) + u in (τ)y(τ)dτ R(x)y(τ) 2 dτ }{{}}{{} supplied energy dissipated energy The balance equations expresses conservation of some physical quantity: Energy, mass, volume, etc... The existence of balance equations is the base for dissipative and passive system theory. Doctoral course UFC-ST 3 / 14

31 Example: the RLC circuit Let us first consider a lossless LC circuit. The energy is H(x(t)) = Q φ C 2 L The interconnection structure just characterize the exchange of energy between the inductor and the capacitor: [ ] 1 J =. 1 The internal dynamics of the system is then given by ] ẋ = J H x = J [ Q C φ L = [ φ L Q C ] Doctoral course UFC-ST 31 / 14

32 Example: the RLC circuit Let us consider the complete RLC circuit, with dissipation and input port. The energy remains the same H(x(t)) = 1 2 Q φ 2 C 2 L The interconnection structure just characterize the exchange of energy between the inductor and the capacitor, but in this case we have to add an additional structure matrix that characterizes the dissipation of the system and an input vector field J = [ ] 1 1, R = [ ] R, gu = [ 1] u The complete dynamics of the system is now given by [ ] [ ] ẋ = (J R) H Q φ + gu = (J R) C φ + gu = L x Q L C R φ L + V in Doctoral course UFC-ST 32 / 14

33 Example: the MSD system Let us first consider a lossless MS system. The energy is H(x(t)) = 1 q 2 1 p K 1 2 M The interconnection structure just characterize the exchange of energy between the mass and the spring: [ ] 1 J =. 1 The internal dynamics of the system is then given by ẋ = J H x = J [ q ] K 1 p = M [ p ] M q K 1 Doctoral course UFC-ST 33 / 14

34 Example: the MSD system Let us consider the complete MSD system, with dissipation and input port. The energy remains the same H(x(t)) = 1 q 2 1 p K 1 2 M The interconnection structure remains the same, but in this case we have to add an additional structure matrix that characterizes the dissipation of the system and an input vector field [ ] [ ] [ 1 J =, R =, gu = u 1 B 1] The complete dynamics of the system is now given by ẋ = (J R) H x [ q ] + gu = (J R) K 1 p + gu = M [ p ] M q K B p 1 M + F in Doctoral course UFC-ST 34 / 14

35 Examples: RLC circuit and MSD system The RLC circuit Energy = Energy stored in the capacitor + Energy stored in the inductor H(x(t)) = 1 2 The time variation of the energy is given by dh(x(t)) dt = H dx x dt = V in ( φ L ( ) ( Q φ = C L ( φ L ) R The time variation of the energy is ) 2 φ Q L 2 C ( Q C ) 2 = V ini L RI 2 L ) ( ) ( ) φ φ + V in R L L H(t) = H(t ) + V in (τ)i L (τ)dτ RI L (τ) 2 dτ ( ) φ 2 L Doctoral course UFC-ST 35 / 14

36 Examples: RLC circuit and MSD system H(x(t)) = φ Q, H() =. L 2 C Hence H qualifies as a potential storage function. Now, H(t) = H(t ) + V in (τ)i L (τ)dτ RI L (τ) 2 dτ. The system is passive if we choose u = V in and y = I L : H(t) H(t ) + V in (τ)i L (τ)dτ. Furthermore, if the we choose the dissipation rate as S(x) = RI L (τ) 2, then the system is strictly passive H(t) = H(t ) + u(τ)y(τ)dτ S(x(τ))dτ. }{{}}{{} supplied energy dissipated energy Doctoral course UFC-ST 36 / 14

37 Examples: RLC circuit and MSD system MSD system Energy = Energy stored in the mass + Energy stored in the spring The time variation of the energy is given by dh(x(t)) dt = H dx x dt = F in ( p M H(x(t)) = 1 p 2 1 q M 2 K 1 ( q = ) B ) ( q ) ( p ) ( p ) ( p ) 2 K 1 + F in D M M M ) ( p K 1 M ( p ) 2 = Fin v M RvM 2 M The balance equation characterizing the time variation of energy can be written as H(t) = H(t ) + F in (τ)v M (τ)dτ Bv M (τ) 2 dτ Doctoral course UFC-ST 37 / 14

38 Examples: RLC circuit and MSD system H(x(t)) = 1 q 2 1 p 2 +, H() =. 2 K 1 2 M Hence H qualifies as a potential storage function. Now, H(t) = H(t ) + F in (τ)v M (τ)dτ Bv M (τ) 2 dτ. The system is passive if we choose u = F in and y = v M : H(t) H(t ) + F in (τ)v M (τ)dτ. Furthermore, if the we choose the dissipation rate as S(x) = Bv M (τ) 2, then the system is strictly passive H(t) = H(t ) + u(τ)y(τ)dτ S(x(τ))dτ. }{{}}{{} supplied energy dissipated energy Doctoral course UFC-ST 38 / 14

39 Examples: RLC circuit and MSD system Some remarks The chosen inputs and outputs correspond to the physical input and outputs of the system: input voltage and input force / current in the inductor and velocity of the mass If we eliminate the resistive components, resistor (R) and damper (B), the supply rate is zero and the system is a lossless (conservative) passive system. Indeed, i.e., the energy is conserved. H(t) = H(t ) + u(τ)y(τ)dτ }{{} supplied energy The product u y has the units of power, i.e., it defines a power product. This has strong implications for modelling: if the input and outputs define power products the power preserving interconnection of physical (passive) systems defines again a physical (passive) system. Doctoral course UFC-ST 39 / 14

40 1. Stability: definitions 2. Stabilization of PHS 3. Control by interconnection 4. Interconnection and Damping Assignment Passivity Based Control 5. Final remarks and future work Doctoral course UFC-ST 4 / 14

41 Stabilization of passive systems What if we want to increase the rate of convergence?: damping injection, What if we want to stabilize at some different equilibrium point, x = x, x : Energy shaping, What if S(x) vanishes for some x or S(x) =?: damping injection + Energy shaping Doctoral course UFC-ST 41 / 14

42 Stabilization of PHS: Damping injection Consider the energy balance equation of a passive system: H(t) = H(t ) + u(τ)y(τ)dτ S(x(τ))dτ. }{{}}{{} supplied energy dissipated energy And assume that H(x) qualifies as a Lyapunov function candidate. If we select the input u = Ky, with K a positive definite constant matrix, then the energy balance equation becomes: H(t) = H(t ) K y 2 (τ)dτ S(x(τ))dτ, } {{ } } {{ } controller dissipated energy ( ) H(t) = H(t ) Ky 2 (τ)dτ + S(x(τ)) dτ. } {{ } dissipated energy Doctoral course UFC-ST 42 / 14

43 Example: mass-spring-damper system Let us consider a simple linear translational MSD system: Constitutive relations F s = F in F B = Bv B p = Mv M q = K 1 F K Dynamic relations F M = dp dt, or in integral form p(t) = p(t ) + F M (τ)dτ v K = dq dt, or in integral form q(t) = q(t ) + v K (τ)dτ Doctoral course UFC-ST 43 / 14

44 Examples: the MSD system H(x(t)) = 1 q 2 1 p 2 +, H() =. 2 K 1 2 M Hence H qualifies as a storage function and as a candidate Lyapunov function. Now, H(t) = H(t ) + F in (τ)v M (τ)dτ Bv M (τ) 2 dτ. The system is passive if we choose u = F in and y = v M, and furthermore, if we select u = Ky, (F in = Kv M ), then H(t) = H(t ) = H(t ) (K + B) }{{} B (Kv 2 M (τ) + Bv 2 M (τ) ) dτ. 2 vm (τ)dτ We have changed (increased) the system s natural damping. Doctoral course UFC-ST 44 / 14

45 Stabilization of PHS: Energy Shaping Consider the energy balance equation of a passive system: H(t) H(t ) = u(τ)y(τ)dτ }{{} stored energy }{{} supplied energy S(x(τ))dτ. }{{} dissipated energy Assume that we want to change the closed-loop equilibrium to some forced (controlled) equilibrium x = x. In that case H(x ), hence H(x) can no longer be used as Lyapunov function! We need to consider a new Lyapunov function candidate Doctoral course UFC-ST 45 / 14

46 Stabilization of PHS: Energy shaping Let us consider the energy balance equation and assume we have no dissipation H(t) H(t ) = u(τ)y(τ)dτ } {{ } controller The idea is to construct a new energy function, by using the (state) feedback u = β(x) H d (x, x ) = H(x) β(x(τ))y(τ)dτ such that H d, with H d (x ) = qualifies as a Lyapunov function for the closed-loop system. Doctoral course UFC-ST 46 / 14

47 Stabilization of PHS: Energy shaping If this function exist (...why should it exist?) it will be a state function such that H d (x) = H(x) + H a(x), hence H a(x, x ) = β(x(τ))y(τ)dτ The time derivative of H d (x) along the trajectories of the system is given by Ha Ḣ d = Ḣ + Ḣa = Ḣ + ẋ x Ha ẋ = β(x)y x Hence, for dynamical systems of the form ẋ = f (x, u), y = h(x), in order for the function H a to exist, the following PDE should be satisfied H a f (x, β(x)) = β(x)h(x) x Doctoral course UFC-ST 47 / 14

48 Stabilization of PHS: Energy shaping Some remarks Energy shaping requires the solution of a PDE: the matching equation. Not an easy task for general non-linear systems H a(x, x ) = β(x(τ))y(τ)dτ The existence of solutions for the PDE is strongly related with the existence of physical invariants. In the case of port-hamiltonian systems: Casimir functions. For systems arising from physical applications the energy shaping technique has been proven to be a powerful stabilization method. Doctoral course UFC-ST 48 / 14

49 Example: RLC circuit Let us consider a simple linear RLC circuit: Constitutive relations u s = V in u r = RI r φ = LI L Q = Cu C Dynamic relations u L = dφ dt, or in integral form φ(t) = φ(t ) + u L (τ)dτ I C = dq dt, or in integral form Q(t) = Q(t ) + I C (τ)dτ Doctoral course UFC-ST 49 / 14

50 Example: RLC circuit The state space model dq = φ dt L dφ = Q dt C R φ L + V in with state variables x = [Q, φ], output y = φ L = x 2 L and input V in. The energy of the system is given by H(x) = 1 x x C 2 L If V in =, the natural equilibrium is x = (, ). If on other hand V in = V, the forced equilibrium point is x = (x 1, ), with x 1 = CV. Doctoral course UFC-ST 5 / 14

51 Example: RLC circuit The matching equation becomes H a f (x, β(x)) = β(x)h(x) x ( H a x 2 x 1 L Ha x1 x 2 C R x 2 L β(x) ) = x 2 L β(x) Notice that the forced equilibrium corresponding to the x 2 coordinate already is a minimum of the physical energy H(x), hence we only need to shape the closed-loop energy in the x 1 coordinate. Hence and the matching equation becomes H a = H a(x 1 ) H a x 1 x 2 L = x 2 L β(x) Hence, the function H a exists if the feedback is chosen as β(x) = Ha x 1 Doctoral course UFC-ST 51 / 14

52 Example: RLC circuit Beautiful! The matching equation (PDE) is automatically solved for any H a = H a(x 1 ) provided that the state feedback is of the form β(x) = Ha x 1. It only remains to select H a(x 1 ) such that H d = H + H a has a minimum at x = (x 1, ). Recall that the open-loop energy function is H(x) = 1 2 x 1 C x 2 2 L Hence if we chose H a(x 1 ) = 1 ( 1 x1 2 2C + 1 ) x 1 x1 a C a C The closed-loop energy function H d = H + H a H d (x) = 1 (x 1 x1 ) 2 (C + C a) x 2 2 L Doctoral course UFC-ST 52 / 14

53 Example: RLC circuit H d (x, x ) has a minimum at x = (x1, ) if and only if Ca > C The resulting controller is u = β(x) = 1 ( 1 x1 2 2C + 1 ) x 1 x1 a C a C Doctoral course UFC-ST 53 / 14

54 Final remarks We have revised some concepts from passivity based control techniques: Damping injection and Energy Shaping We have exploited the natural passivity of the system to design stabilizing controllers Works well in many applications, but... we did not see the use of physical invariants, the dissipation obstacle,... What remains for tomorrow Control by interconnection IDA-PBC Irreversible port-hamiltonian systems... Doctoral course UFC-ST 54 / 14

55 1. Stability: definitions 2. Stabilization of PHS 3. Control by interconnection 4. Interconnection and Damping Assignment Passivity Based Control 5. Final remarks and future work Doctoral course UFC-ST 55 / 14

56 Port-Hamiltonian control systems Let us recall the state space model of a port-hamiltonian control system ẋ = ( J(x) R(x) ) H (x) + g(x)u, x y = g (x) H x (x), where where x R n is the state vector, u R m, m < n, is the control action, H : R n R is the total stored energy, J(x) = J(x) is the n n natural interconnection matrix, R(x) = R(x) is the n n damping matrix, g(x), is the n m input map and u, y R m, are conjugated variables whose product has units of power. Ḣ = u y H R H x x, Ḣ u y, Doctoral course UFC-ST 56 / 14

57 Control by interconnection A controlled system may be viewed as a plant system interconnected with a control system exchanging energy The interconnection is power continuous if u (t)y(t) + uc (t)yc(t) =, t Doctoral course UFC-ST 57 / 14

58 Control by interconnection For instance: a negative feedback The negative feedback defines the following relation u = y c y = u c u y + u c yc = y c yc + y y c = Doctoral course UFC-ST 58 / 14

59 Control by interconnection Let us consider the feedback Now u = v y c and u c = y + v c. Let the plant and controller have state variables x and ξ and energy functions H(x) and H(ξ). If the maps u y and u c y c are passive, Then the map (v, v c) (y, y c) is passive with energy function H d (x, ξ) = H(x) + H(ξ). Doctoral course UFC-ST 59 / 14

60 Control by interconnection Assume that the plant and the controller are PHS ẋ =[J(x) R(x)] H (x) + g(x)u Σ : x y =g (x) H x (x) Σ c : ξ =[J c(ξ) R c(ξ)] Hc (ξ) + gc(ξ)uc ξ y c =g c (ξ) Hc ξ (ξ) x X ξ X c Booth are passive systems, so a power preserving interconnection, u = y c, y = u c, yields a passive closed-loop system. Doctoral course UFC-ST 6 / 14

61 Control by interconnection The closed-loop systems looks [ẋ ] [ J(x) g(x)g = c ξ (ξ) ] g c(ξ)g (x) J c(ξ) }{{} [ y yc] [ g(x) = g c(ξ) } {{ } g cl With total energy function J cl (x,ξ) ] [ Hd x (x) H d ξ (ξ) ] H d (x, ξ) = H(x) + H c(ξ) We may equivalently write the closed-loop system as [ R(x) R c(ξ) } {{ } R cl (x,ξ) ] [ Hd x (x) H d ξ (ξ) ] with w = [x ξ]. ẇ = (J cl R cl ) H d w, y cl = g cl H d w Doctoral course UFC-ST 61 / 14

62 Control by interconnection So, what now? We would like to get an energy function in terms of x only: H d = H d (x), so that we can assign the minimum at a desired point and characterize it in terms of the plant system. In order achieve this, we must restrict the dynamics to a submanifold of the (x, ξ) space parametrized by x. This means that we are looking for a submanifold which is dynamically invariant, i.e., ( Ω C = (x, ξ) : ξ = F (x) C F i x ẋ ξ i ) ξ=f i (x) C = Doctoral course UFC-ST 62 / 14

63 Control by interconnection Casimir functions Let us look for structural invariants that relates each state of the controller with the states of the plant: C i (x, ξ i ) = F i (x) ξ i In order to relate all the states of the controller with the state of the plant we define F (x) = [F 1 (x), F 2 (x),..., F nc (x)], and define the following Casimir function C = n (F i (x) ξ i ) = i=1 C is an invariant of the system, hence Ċ = C w ẇ = C w n C i (x, ξ i ) i=1 ( ) H J cl cl = w But furthermore, C is a structural invariants, so it should be invariant with respect to the structure of the system: C J cl = w Doctoral course UFC-ST 63 / 14

64 Control by interconnection Casimir functions Let us look for structural invariants that relates each state of the controller with the states of the plant: n n C = C i (x, ξ i ) = (F i (x) ξ i ) i=1 we obtain the following matching condition [ ] [ F J(x) R(x) g(x)g x (x) I C (ξ) ] [ Hd g C (ξ)g x (x) ] H (x) J C (ξ) R C (ξ) d }{{} ξ (ξ) = Matching condition i=1 Only the term in blue is considered in the matching condition because we want the Casimir functions to be structural invariants of the system: not depend on H d (x, ξ). Doctoral course UFC-ST 64 / 14

65 Control by interconnection The condition for existence of Casimir functions for the closed loop system [ F x may be written out as Matching equations ] [ J(x) R(x) g(x)g (x) I C (ξ) ] g C (ξ)g = (x) J C (ξ) R C (ξ) F (x)j(x) F (x) = Jc(ξ) x x R(x) F x (x) = Dissipation obstacle! R c(ξ) = F (x)j(x) = g c(ξ)g (x) x Doctoral course UFC-ST 65 / 14

66 Control by interconnection The closed-loop dynamic then takes the form [ ] H ẋ = J(x) R(x) x (x) g(x)g Hc C (ξ) ξ (ξ) Using the second and fourth M.C. we get [ ẋ = J(x) R(x) ] ( H x ) F Hc (x)+ (x) x ξ (ξ) Since ξ = F (x) C, we use the chain-rule for differentiation to establish F x (x) Hc ξ Hc (ξ) = (F (x) C) x Doctoral course UFC-ST 66 / 14

67 Control by interconnection Hence we obtain: [ ẋ = J(x) R(x) ] ( H x ) Hc (x)+ (F (x) C) x Or equivalently [ ] Hd ẋ = J(x) R(x) x (x) With closed-loop energy H d (x) = H(x) + H c(f (x) C). Doctoral course UFC-ST 67 / 14

68 Control by interconnection Let us interpret the control in terms of energy balancing. Since R c =, the energy balance equation of the controller is dh c dt = u c y c Hence, along any invariant submanifold Ω C, we have dh d dt = dh dt + dhc dt and integrating (up to some constant) we obtain = dh dt u y (u c yc = u y) H d (t) = H(t) u (τ)y(τ)dτ } {{ } H c We obtain the general M.E!: H c(t) = u (τ)y(τ)dτ. Doctoral course UFC-ST 68 / 14

69 Example: RLC circuit Let us consider a simple linear RLC circuit: Constitutive relations u s = V in u r = RI r φ = LI L Q = Cu C Dynamic relations u L = dφ dt, or in integral form φ(t) = φ(t ) + u L (τ)dτ I C = dq dt, or in integral form Q(t) = Q(t ) + I C (τ)dτ Doctoral course UFC-ST 69 / 14

70 Example: the RLC circuit Let us consider a RLC circuit, with dissipation and input port. The energy is H(x(t)) = Q φ C 2 L The interconnection and dissipation matrix and input vector field are [ ] [ ] [ 1 J =, R =, gu = u 1 R 1] and the dynamic is ẋ = (J R) H x + gu = (J R) [ ] [ ] Q φ C φ + gu = L Q L C R φ L + V in with state variables x = [Q, φ], output y = φ L = x 2 L and input u = V in If V in =, the natural equilibrium is x = (, ). If on other hand V in = V, the forced equilibrium point is x = (x1, ), with x 1 = CV. Doctoral course UFC-ST 7 / 14

71 Example: RLC circuit Let us synthesis the controller using the M.C.s. From physical considerations we know that we only need to shape the x 1 coordinate: F is only one scalar function. From M.E.2 we obtain that F x 2 =. Then, from M.E.1. we obtain that J c =, and from M.E.3 that R c =. Finally from M.E.4 we have that F x 1 = g c(ξ) and that ξ R. We would like to have ( ) H c(x 1 ) = 1 x 2C 2 a x C a C 1 x1, such that H d (x) = 1 2 (x 1 x 1 ) (C + C a) This is achieved if we select F (x 1 ) = x 1 and C = on the invariant submanifold such that ξ = x 1. The control system is then given by (using condition F = g x 1 c(ξ)) ξ = u c y c = Hc ξ x 2 2 L Doctoral course UFC-ST 71 / 14

72 The pendulum without damping The dynamic equations q = p m ṗ = mg sin(q) + u with state variables x = [p, q], with q the configuration and p the momentum. Propose a PH model and design a stabilizing controller using the Casimir method and a control system given by: ξ = u c y c = Hc ξ Doctoral course UFC-ST 72 / 14

73 Example: pendulum (model) Consider the pendulum without damping q = p m ṗ = mg sin q + u with state variables x = [q, p] T with q the configuration and p the momentum. The port Hamiltonian model is: ( d q dt p ) = y = ( 1) ( 1 ) } 1 {{ } J ) ( H q H p ( H q H p = p m with Hamiltonian :H (q, p) = mg(1 cos q) + 1 2m p2 ) ( ) + u 1 } {{ } g Doctoral course UFC-ST 73 / 14

74 pendulum (extended Casimir functions) Recall: we look for Casimir functions such as: C (q, p, x c) = F (q, p) x c that is functions F (q, p) satisfying: ( 1. input vector field is Hamiltonian: g = 1 ) ( F = J (x) F x = p F q 2. the gradient of F is transversal to g: L gf (x) = = F t x g (x) = F p hence a generating function is: F (q, p) = q ) Doctoral course UFC-ST 74 / 14

75 pendulum (control) For the controller design we choose a function H C (x c) such that H d (x) = H (x) + H c F has a minimum at the desired equilibrium x = (x 1, ). The simplest choice is given by The control is finally obtained is: H C (x c) = cos x c (xc + x 1 )2 u = H C x c (x c) xc = q = sin q (q x 1 ) which is the well-known proportional plus gravity compensation control Doctoral course UFC-ST 75 / 14

76 Control by interconnection Remarks The port-hamiltonian structure provide important information for finding the solutions of the control system, The control has physical interpretation in terms of interconnection and energy balancing The Casimir method can be used to analyse new stability profiles of interconnected systems Doctoral course UFC-ST 76 / 14

77 1. Stability: definitions 2. Stabilization of PHS 3. Control by interconnection 4. Interconnection and Damping Assignment Passivity Based Control 5. Final remarks and future work Doctoral course UFC-ST 77 / 14

78 Interconnection and Damping Assignment Passivity Based Control Consider the following parallel RLC circuit with x = [q C, φ L ], the charge in the capacitor and the flux in the inductance The PH model is ] [ ] H [ ] [ qc 1 1 = R φ L 1 y = [ 1 ] [ H q C H φ L q C H φ L ] [ + u 1] with H(q C, φ L ) = q2 C 2C + φ2 L 2L, the total electromagnetic energy. Doctoral course UFC-ST 78 / 14

79 Interconnection and Damping Assignment Passivity Based Control Recall the dissipation obstacle For the parallel RLCS circuit it becomes: R(x) F (x) =. x F q C (q C, φ L ) = F = F (φ L ). It s only possible to shape the energy in the direction of one coordinate (φ L ). This is known as the dissipation obstacle. Physical interpretation: admissible equilibria are of the form q C = CV, φ L = L R V for any constant V. The power delivered by the source, V φ L L, is nonzero at any equilibrium except for the trivial one. Hence, the source has to provide an infinite amount of energy to keep any nontrivial equilibrium point (Impossible). Notice that pure mechanical systems are free of this problem: any equilibrium point has velocities equal to zero. Doctoral course UFC-ST 79 / 14

80 Interconnection and Damping Assignment Passivity Based Control Overcome the dissipation obstacle using a State modulated control by interconnection (state modulated source) IDA-PBC objective Find a static state-feedback control u(x) = β(x) such that the closed-loop dynamics is a PH system with interconnection and dissipation of the form ẋ = (J d R d ) H d x, H d (x), has a strict local minimum at x, J d (x, u) = J d (x, u) T, the desired interconnection matrix, R d (x, u) = R d (x, u) T, the desired dissipation matrix, Doctoral course UFC-ST 8 / 14

81 Interconnection and Damping Assignment Passivity Based Control The procedure consists in the matching of the open and (desired) closed-loop vector fields: ( ) H J(x) R(x) x (x) + g(x)β(x) = ( J d (x) R d (x) ) H d x (x) with u = β(x) the state modulated source. Define J a= J d J, J a = J a R a= R d R, R a = R a H a= H d H Then the matching condition becomes ( Ja(x) R a(x) ) H x (x) + g(x)β(x) = ( (J(x) + Ja(x) ) ( R(x) + R a(x) )) H a x (x) with design parameters J a, R a and H a. Doctoral course UFC-ST 81 / 14

82 IDA-PBC: Proposition Suppose that the following PDE is verified g (x) ( J a(x) R ) H ( (J(x) a(x) x (x) = g (x) ) + Ja(x) ( R(x) + R )) H a a(x) x (x) where g (x)g(x) = and H d (x) is such that then the control H d x (x ) =, u = β(x) = (g g) 1 g [ (J d R d ) H d x is such that the closed-loop system takes the PH form 2 H d x 2 (x ) >, (J R) H ] x And closed-loop energy ẋ = (J d R d ) H d x, Ḣ d = H d x R H d d x <, x x and Ḣ d (x ) =. Doctoral course UFC-ST 82 / 14

83 Interconnection and Damping Assignment Passivity Based Control Notice that for systems linear in the control may be interpreted as the optimal solution to the linear least square problem [ (Jd g(x)β(x) = (x) R d (x) ) H d x (x) ( J(x) R(x) ) H ] x (x) }{{} b with where β(x) = g b, subject to the ME (I g g)b = g = (g g) 1 g : Moore-Penrose pseudo-inverse of g and g g projector into rang(g) (I g g) : family of anihilators of g and projector into ker(g) Doctoral course UFC-ST 83 / 14

84 Remarks Degrees of freedom in the design J d and R d are free up to the constraint of skew symmetry and positive semidefiniteness, respectively. H d may be totally, or partially fixed, provided we can ensure H d x (x ) =, 2 H d x (x ) and probably a properness condition. 2 there is an additional degree of freedom in g (x) which is not uniquely defined by g(x). Attention: Requires the solution of a quasilinear PDE the method of characteristics... Doctoral course UFC-ST 84 / 14

85 Interconnection and Damping Assignment Passivity Based Control Notice that the method can be equally applied to general input affine non-linear systems with f (x) a non-linear vector field. f (x) + g(x)β(x) = ( J d (x) R d (x) ) H d x (x) Then the matching condition becomes g (x)f (x) = g (x) ( J d (x) R d (x) ) H d x (x) and the control [ u = β(x) = (g g) 1 g (J d R d ) H ] d x f (x) Doctoral course UFC-ST 85 / 14

86 Integral Action IDA PBC with integral action u = β(x) + v with K I = K I Indeed, define >, preserves stability. v = K I g(x) H d x (x) W (x, v) = H d v K 1 v I The closed-loop system is given by a power preserving interconnection which is again a PHS. [ẋ v ] = [ Jd R d gk I KI g ] [ ] H d x W v Doctoral course UFC-ST 86 / 14

87 Existing Approaches to Solve the Matching Equation Non Parameterized IDA One extreme case fix J d (x), R d (x) and g (x), (ME) becomes a PDE for H d (x), among the family of solutions select one that assigns minimum. Algebraic IDA At the other extreme: fix H d (x), (ME) becomes an algebraic equation in J d (x), R d (x) and g (x), Parameterized IDA Restrict the desired energy function to a certain class, for instance, for mechanical systems H d (q, p) = 1 2 p M 1 (q)p + 1 d 2 V d (q), (ME) becomes a PDE in M d (q) and V d (q), imposes some constraints on J d (x) and R d (x). Application of Poincare s Lemma (applicable for systems NL in u): H(x) = F 1 (x)f (x, u). d Doctoral course UFC-ST 87 / 14

88 Example: parallel RLCS circuit Consider the following parallel RLC circuit with x = [q C, φ L ], the charge in the capacitor and the flux in the inductance The PH model is ] [ ] H [ ] [ qc 1 1 = R φ L 1 y = [ 1 ] [ H q C H φ L q C H φ L ] [ + u 1] with H(q C, φ L ) = q2 C 2C + φ2 L 2L, the total electromagnetic energy. Doctoral course UFC-ST 88 / 14

89 Parallel RLCS circuit: matching equation Choice of structure matrices and associated added Hamiltonian H a (x): 1. choose added structure matrices ( ) ( ) J a(x) = and R a(x) = R R a > R a 2. solve a PDE in H a(x) using the left annihilator g (x) = ( 1 ) of ( ) g (x) =, the matching equation : 1 g (x) (J a R a) H x (x) = g (x) [(J (x) + J a (x)) (R (x) + R a (x))] Ha x (x) becomes: = ( 1 R 1 ) ( H a q c H a φ L ) = 1 H a R q c + Ha φ L Doctoral course UFC-ST 89 / 14

90 Parallel RLCS circuit: closed-loop equilibria The solutions of the matching equation are: H a (q c, φ L ) = Φ (R q c + φ L ) Φ C (R) hence the closed-loop Hamiltonian is H d (q C, φ L ) = H (q C, φ L ) + Φ (R q c + φ L ) = q2 C 2C + φ2 L 2L + Φ (R qc + φ L) The equilibrium in closed-loop is given by: R φ L L q C C = and φ L L + Φ ([ ] ) 1 + R2 C φ L = ξ L Doctoral course UFC-ST 9 / 14

91 Parallel RLCS circuit: two possible H 1. Case 1: Φ (ξ) = k ξ2 2 then: and Φ (ξ) is concave! 2. Case 2: Φ (ξ) = k ξ4 4 then: φ R and k = ( L + R 2 C ) L 2 < φ 1 = ± ( k) ( L + R 2 C ) and k = R and Φ (ξ) is again concave! Doctoral course UFC-ST 91 / 14

92 Parallel RLCS circuit: local positivity of H d Consider the Hessian of H d at the equilibrium: 2 1 H d q c, φ 2 = C + ( R2 2 Φ ξ Rq 2 c + φ 2 ) ( L R 2 Φ ξ Rq 2 c + φ 2 ) L ( L R 2 Φ ξ Rq 2 c + φ 2 ) 1 L L + ( 2 Φ ξ Rq 2 c + φ 2 ) L is definite positive iff: 1 1. either: C + ( R2 2 Φ ξ Rq 2 c + φ 2 ) L > or: 1 2. and det 2 H d q c,φ 2 L > i.e. : 1 LC which reduces to: 2 Φ ξ 2 ( Rq c + φ 2 L L + ( 2 Φ ξ Rq 2 c + φ 2 L ( 1 + [ L + R 2 C ] 2 ( Φ ξ Rq 2 c + φ 2 L ) > 1 (R 2 C+L) ) > ) ) > Doctoral course UFC-ST 92 / 14

93 Parallel RLCS circuit: two possible Φ ( Check for the two examples the condition : 2 Φ ξ Rq 2 c + φ 2 ) L > 1 (R 2 C+L) 1. Case 1: Φ (ξ) = k ξ2 2 which is wrong! 2. Case 2: Φ (ξ) = k ξ4 4 then: which leads to a solution! the condition reduces to: ( ) 2 L + R 2 C < L 2 1 ( k) < ( L + R 2 C ) ( φ ) 2 L L Doctoral course UFC-ST 93 / 14

94 Parallel RLCS circuit: IDA-PBC control The control law is given by : β (x) = [ g t (x) g (x) ] 1 g t (x) { [(J (x) + J a (x)) (R (x) + R a (x))] Ha x (x) + (Ja Ra) H x (x) } which becomes : or : β (q C, φ L ) = 1 ( 1 ) [ ( 1 1 R 1 R a ( ) ( qc )] CφL R a L ) ( R Φ ξ (R q C, +φ L ) Φ ξ (R q C, +φ L ) β (q C, φ L ) = (R R a) Φ ξ (R q φ C, +φ L ) R L a L ) Doctoral course UFC-ST 94 / 14

95 The levitating iron ball in a magnetic field The mathematical model φ 1 = Ri + u ẏ = v m v = F m + mg with u input voltage, R electric resistance, mthe mass and g the gravitational constant, F m= Wc (i, y) y electro-mechanical coupling W c= 1 2 L(y)i2 non-linear inductance and the non-linear inductance L(y) = L + k (a + y), with L, a, k > and φ = L(y)i. Doctoral course UFC-ST 95 / 14

96 The levitating iron ball: the PH model The port-hamiltonian model set as state variables: dx dt = x 1 = φ x 2 = y x 3 = mv = p total magnetic flux displacement of the ball kinetic momentum 1 R H 1 x (x) + 1 u }{{}}{{}}{{} J R g y = [ 1 ] H H (x) = = i x x 1 with total energy: H(x) = 1 2 L(x 2 ) x m x 3 2 mgx 2 H x (x) = 1 dl (x 2 dx 2 ) 2 x 1 L(x 2 ) x 3m ( x1 L(x 2 ) ) 2 m g current through the coil, i velocity of the ball electro-motive+gravity force Doctoral course UFC-ST 96 / 14

97 The levitating ball: coupling through energy The coupling between the mechanical and magnetic domain does not occur through the structure matrices J R = R magnetic domain 1 mechanical potential domain 1 mechanical kinetic domain The structure matrix is block-diagonal. The coupling occurs through the Hamiltonian which is not separated H (x) = 1 2 L (x 2 ) x m x 3 2 mgx 2 Doctoral course UFC-ST 97 / 14

98 The levitating ball: open-loop equilibria The equilibria are given by (x, u ) such that: R x 1 (J R) H L(x x (x ) + g u ) + u 2 x = 3 m 1 ( ) ( ) dl 2 dx x x L(x 2 ) m g = and may be parametrized by the current: y = H (x x ) = i : 1 x L y + 1 x2 2mg = a + k y x3 2mg and u = R y they are unstable (see linearised system). Doctoral course UFC-ST 98 / 14

99 The levitating ball: matching equation Choice of structure matrices and associated added Hamiltonian H a (x): 1. choose added structure matrices: J a(x) = and R a(x) = 3 α α What s the physical interpretation? ( 1 2. Use the left annihilator g (x) = 1 ( α ) H x 1 H x 2 H x 3 ( = ), establish the PDE in H a : 1 α 1 ) H a x 1 H a x 2 H a x 3 Doctoral course UFC-ST 99 / 14

100 The levitating ball: added Hamiltonian From : Ha = the added potential is a function H x a (x 1, x 2 ) which does not depend on 3 the velocity and satisfies : with solution: α H x 1 x1 H a (x 1, x 2 ) = = α Ha x 1 Ha x 2 ( L χ x 2 (χ x 1) α For instance, if L (x 2 ) = k ( (x 2 +a) then ) H a (x 1, x 2 ) = 1 x1 3 2k 3α x 1 2 (x 2 + a) + Φ ( x 2 x ) 1 α ( ) dχ + Φ x 2 x 1 α ) Doctoral course UFC-ST 1 / 14

101 The levitating ball: IDA-PBC control The control law is given by : or : β (x) = 1 ( 1 ) }{{} =[g t g] 1 g t α 1 α 1 Ha x (x) + H x (x) α α β (x) = R Ha x 1 (x) + α Ha x 3 (x) + α H x 3 (x) Doctoral course UFC-ST 11 / 14

102 The levitating iron ball: the closed-loop system dx dt = α 1 R α 1 }{{}}{{} J R H d x (x) Doctoral course UFC-ST 12 / 14

103 Remarks Hamiltonian methods for the control of physical systems: 1. use the interconnection of Port Hamiltonian systems and Casimir functions 2. assign closed-loop Hamiltonian function and structure matrices. A control synthesis based on insight of desired physical behaviour in closed-loop: 1. design directly interconnection of the system with environment and indirectly the controller 2. design the closed-loop port Hamiltonian behaviour and deduce the controller Open questions: 1. choice of desired behaviour in closed-loop and relation with performance and robustness 2. parametrization of the matching equations and solution Doctoral course UFC-ST 13 / 14

104 Concluding remarks Energy based modelling: based on the universal concept of energy transfer. Provides physical interpretation to the models and the solutions. Passivity is naturally encountered when working with problems arising from physical applications. Port-Hamiltonian control systems defines a class of non-linear passive systems which encompasses a large class of physical applications. A modelling and control approach which is transversal to different (or combination of) physical domains. Doctoral course UFC-ST 14 / 14