20 Years of Passivity Based Control (PBC): Theory and Applications

Transcription

1 20 Years of Passivity Based Control (PBC): Theory and Applications David Hill/Jun Zhao (ANU), Robert Gregg (UTDallas) and Romeo Ortega (LSS) Contents: Preliminaries on passivity (DH). PBC: History, main principles and recent developments (RO). Interconnection and Damping Assignment PBC (RO). PBC in bipedal locomotion (RG). Passivity of switched systems (JZ). CDC Workshop, Shanghai, PRC, 15/12/2009 p. 1/85

2 What is PBC? Why is passivity important? For physical systems it is a restatement of energy conservation. Is a natural generalization (to NL dynamical systems) of positivity of matrices and phase-shift of LTI systems sign preserving property. Term PBC introduced in R. Ortega and M. Spong, Adaptive Motion Control Of Rigid Robots: A Tutorial, Automatica, Vol. 25, No. 6, 1989, pp , to define a controller methodology whose aim is to render the closed loop passive. It was done in the context of adaptive control of robot manipulators. Natural, because mechanical systems and parameter estimators define passive maps. The paper has been cited more than 600 times and is the 13th most highly cited paper out of 4250 published in Automatica since PBC has 2500 hits in Google scholar. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 2/85

3 Passivity as a Design Tool: Foundational Results (Moylan and Anderson, TAC 73): Optimal systems define passive maps. Nonlinear extension of Kalman s inverse optimal control result. (Fradkov, Aut and Rem Control 76): Necessary and sufficient conditions for passivation of LTI systems via state feedback. (Takegaki and Arimoto, ASME JDSM&C 81), (Jonckheere, European Conf Circ. Th. and Design 81): Potential energy shaping and damping injection as design tools for mechanical and electromechanical systems new energy function as Lyapunov function. (Kokotovic and Sussmann, S&CL 89): Stabilization of a NL system in cascade with an integrator using positive realness. (Ortega, Automatica 91): Extension to cascade of two NL systems using Hill/Moylan theorem. (Byrnes, Isidori and Willems, TAC 91): Complete geometric characterization of passifiable systems via minimum phase and relative degree conditions. Backstepping and forwarding are passivation recursive designs that overcome the obstacles of relative degree and minimum phase for systems with special structures. See e.g., (Astolfi, Ortega and Sepulchre, EJC 02). CDC Workshop, Shanghai, PRC, 15/12/2009 p. 3/85

4 PBC Provides a Paradigm Shift for Controller Design Classical formulation: Signal processing viewpoint u + e y 1 1 G y 2 G 2 e u 2 System model and controller are signal processors: G 1 : e 1 y 1, G 2 : e 2 y 2. Control specifications in terms of signals: tracking, disturbance attenuation, etc. Presumes the presence of actuators and sensors not always (physically) true or (conceptually) convenient Fine for isolated" systems but cannot easily accommodate interactions" e.g., the Σ paradigm to model uncertainty. At a more philosophical level: there re no inputs and outputs in nature! CDC Workshop, Shanghai, PRC, 15/12/2009 p. 4/85

5 Passivity Based Control: An Energy Processing Viewpoint View plant as energy transformation multiport device Physical systems satisfy (generalized) energy conservation: Stored energy = Supplied energy + Dissipation Control objective in PBC: preserve the energy conservation property but with desired energy and dissipation functions In other words Desired stored energy = New supplied energy + Desired dissipation PBC = Energy Shaping + Damping Assignment For general systems achieve a passivation objective Three possible formulations. State feedback either to passivize or to change the energy function (and dissipation). Control by Interconnection (CbI) plant and controller are energy transformation devices, whose energy is added up. Decompose the system into passive (or passifiable) sub-blocks and design PBCs for each one of them. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 5/85

6 Advantages of PBC Advantages of energy shaping (over nonlinearity cancellation and high gain) a Handle on performance, not just stability Respect, and effectively exploit, the structure of the system to incorporate physical knowledge, provide physical interpretations to the control action Energy serves as a lingua franca to communicate with practitioners There s an elegant geometrical characterization of power conserving interconnections (via Dirac structures) and passifiable NL systems (in terms of stable invertibility and relative degree) a Euphemistically called nonlinearity domination. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 6/85

7 Applications of PBC Mechanical systems: walking robots, bilateral teleoperators, pendular systems. Chemical processes: mass balance systems, inventory control, reactors. Electrical systems: power systems, power converters. Electromechanical systems: motors, magnetic levitation systems. Transportation systems: underwater vehicles, surface vessels, (air)spacecrafts. Control over networks: formation control, synchronization, consensus problems. Hybrid systems: switched systems, hybrid passivity.. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 7/85

8 Key Property: Cyclo Passivity Definition We say that the m port system with state x R n, and power port variables u, y R m ẋ = f(x) + g(x)u Σ : y = h(x) is cyclo passive if there exists storage (energy) function H : R n R such that H[x(t)] H[x(0)] }{{} stored energy t u (s)h(x(s))ds 0 }{{} supplied energy If H(x) 0 then the system is passive with port variables (u, y) and storage function H(x). Remark For passive systems we have t 0 u (s)y(s)ds H[x(0)] < amount of energy that can be extracted from a passive system is bounded. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 8/85

9 Stabilization via Energy Shaping and Damping Injection With u(t) 0, we have H[x(t)] H[x(0)] Trajectories tend to converge towards points of minimum energy If the minima are strict H(x) qualifies as a Lyapunov function for them To operate the system around some desired equilibrium point, say x, PBC shapes the energy to assign a strict minimum at this point. Furthermore, if we terminate the port with u = K di y, K di = K di > 0 we get Ḣ y K di y 0. Hence, x(t) 0 if h(x) is detectable (for the closed loop system). That is, if h(x(t)) 0 x(t) 0. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 9/85

10 Ex. 1 State feedback PBC: PI Control of Power Converters A large class of power converters are modeled by ( ) ( m ẋ = J 0 + J i u i R H(x) + G 0 + i=1 ) m G i u i E (SW ) i=1 where x R n is the converter state (typically containing inductor fluxes and capacitor charges), u R m denotes the duty ratio of the switches, the total energy stored in inductors and capacitors is H(x) = 1 2 x Qx, Q = Q > 0 = x, J i = Ji i m := {0,..., m} are the interconnection matrices, R = R 0 represents the dissipation matrix, and the vector G i R n contains the (possibly switched) external voltage and current sources. The control objective is to stabilize and equilibrium x R n. It is desirable to propose simple, robust controllers. (Hernandez, et al., IEEE TCST 09) CDC Workshop, Shanghai, PRC, 15/12/2009 p. 10/85

11 A Three-phase Rectifier acements E v s i s C r c I r L L CDC Workshop, Shanghai, PRC, 15/12/2009 p. 11/85

12 An Incremental Passivity Property Let x R n be an admissible equilibrium point, that is, x satisfies ( ) ( ) m m 0 = J 0 + J i u i R H(x ) + G 0 + G i u i E, i=1 i=1 for some u R m. The incremental model of the system for the output y = Cx, where C := E G 1 (x ) QJ 1. Q Rm n, E G m (x ) QJ m is passive. More precisely, the system verifies the dissipation inequality V ỹ ũ, where y = Cx and the (positive definite) storage function is given by V (x) := 1 2 (x x ) Q(x x ). CDC Workshop, Shanghai, PRC, 15/12/2009 p. 12/85

13 Corollary: Global Asymptotic Stabilization with a PI Consider a switched power converter described by (SW) in closed loop with the PI controller ż = ỹ u = K p ỹ + K i z, where K p, K i R m m are symmetric positive definite matrices, ỹ = C x. For all initial conditions (x(0), z(0)) R n+m the trajectories of the closed loop system are bounded and such that Moreover, lim C x(t) = 0. t lim t x(t) = x, if ỹ is detectable, that is, if for any solution x(t) of the system the following implication is true: Cx(t) Cx lim t x(t) = x. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 13/85

14 Ex. 2 of Control by Interconnection: A Flexible Pendulum δ( q * ) K 1 m=1 R c q c K 2 q p2 q p1 D Plant energy: H(q p, p p ) = 1 2 p p D 1 (q p )p p + V (q p ) Controller energy: H c (q c, p c, q p2 ) = 1 2 p c (q c q p2 ) K 2 (q c q p2 ) (q c δ) K 1 (q c δ) Controller Rayleigh dissipation function: 1 2 q c R c q c (Ailon/Ortega, SCL 93) CDC Workshop, Shanghai, PRC, 15/12/2009 p. 14/85

15 Ex. 3 PBC via Passive Subsystems Decomposition Electromechanical Systems u i g λ y m λ is flux, θ position, u voltage CDC Workshop, Shanghai, PRC, 15/12/2009 p. 15/85

16 Dynamic Behavior Model (assuming linear magnetics, i.e., λ = L(θ)i) Total energy: H = Rate of change of energies 1 λ 2 2 L(θ) }{{} electrical, H e (λ,θ) λ + Ri = u m θ = F mg F = 1 L 2 θ (θ)i2 + m θ 2 + mgθ 2 }{{} mechanical, H m (θ, θ) Ḣ e = λ L(θ) ( Ri + u) 1 }{{} 2 λ λ 2 L 2 (θ) }{{} i 2 L θ θ = Ri 2 + iu F θ, Ḣ m = θf Adding up: Ḣ = Ri 2 + iu is cyclo passive. However, H is not bounded from below, hence not passive! CDC Workshop, Shanghai, PRC, 15/12/2009 p. 16/85

17 Passive Sub Systems Feedback Decomposition The maps Σ e : (u, θ) (i, F ) and Σ m : (F mg) θ are passive (with storage function H e 0, m 2 θ 2 0, resp.) u Σ e λ y Σ m F -mg Designing a PBC that views" Σ m as a passive disturbance, suggests a nested-loop control configuration CDC Workshop, Shanghai, PRC, 15/12/2009 p. 17/85

18 Classical Nested Loop Controller Often employed in applications, where the inner loop is designed neglecting" the mechanical part. This is rationalized via time scale separation arguments. y * F d C ol C il Σ e F Σ m u λ y -mg Passivity provides a rigorous formalization of this approach, without this assumption, see (Ortega et al s Book, 98). Adopting this perspective allows to prove that the industry standard field oriented control of induction motors is a particular case of PBC. Hence, rigourously prove its global stability. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 18/85

19 State feedback PBC Consider the system Σ : ẋ = f(x) + g(x)u y = h(x), Definition [The set PBC] The state-feedback u SF : R n R m is said to be a PBC (u SF PBC) if and only if there exist functions H d : R n R and h d : R n R m such that u = u SF + v renders the closed loop system Σ d : ẋ = f d (x) + g(x)v, f d (x) := f(x) + g(x)u SF (x) y d = h d (x) cyclo-passive with storage function H d (x). That is, if it verifies Ḣ d y d v (DP E) From Hill-Moylan s Theorem, the new power balance becomes Ḣd = y d v d d, where y d = h d = g H d, d d (x) = H d (x)(f(x) + g(x)u SF(x)) CDC Workshop, Shanghai, PRC, 15/12/2009 p. 19/85

20 Algebraic Characterization of PBCs Assumption Σ is cyclo-passive. That is, there exists H : R n R, d : R n R + such that Ḣ = y u d(x). Proposition u SF P BC if and only if there exist functions H a : R n R and d a : R n R, with d a (x) d(x), such that h (x)u SF (x) = H a (x)(f(x) + g(x)u SF(x)) d a (x). Remark Besides the energy and the dissipation, the output has also been modified is a natural way to satisfy the vector relative degree requirement and to overcome the minimal phase restriction on the plant corresponds to the addition of a current source h h d. (Castanos and Ortega, SCL 09) CDC Workshop, Shanghai, PRC, 15/12/2009 p. 20/85

21 Electrical Circuit Analog of PBC y y d Σ d + frag replacements h h d v Σ + u u SF CDC Workshop, Shanghai, PRC, 15/12/2009 p. 21/85

22 Energy-Balancing PBC is Output-Dissipation Preserving The most natural desired storage function candidate is the difference between the stored and the supplied energies: H d (x(t)) = H(x(t)) t 0 h (x(s))u SF (s)ds. Definition [Energy-Balancing] A PBC for the cyclo-passive system Σ is said to be EB (i.e., u SF PBC EB) if and only if y u SF = Ḣa. Proposition u SF PBC EB if and only if, the output and the dissipation remain invariant. That is, if and only if (DPE) holds with y d = y, d d = d. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 22/85

23 The Dissipation Obstacle The storage function is typically used as a Lyapunov function, so it is required that x = arg min H d. Since H d = 0 is a necessary condition it is clear that y d = 0 and d d = 0. EB PBCs, which preserve output and dissipation, impose to the open-loop system that d = ( H ) f = 0, y = (g ) H = 0. This is the so-called dissipation obstacle. Extracted power should be zero at equilibrium EB PBC applicable only for systems without pervasive damping. OK in regulation of mechanical syst. where power = F q, but very restrictive for electrical or electromechanical syst.: power = v i. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 23/85

24 Role of Dissipation Non pervasive 1 L L ϕ L Pervasive 1 L L ϕ L u C 1 R 2 1 C q C 1 u C 1 1 C 1 q C Only the dissipation has changed. i L 0 nonzero power R 2 Equilibria: (i L, v C ) = (0, ) zero extracted power! t lim v C (s)i L (s)ds = t 0 for any stabilizing controller (run down the battery!) CDC Workshop, Shanghai, PRC, 15/12/2009 p. 24/85

25 Application of EB PBC to Mechanical Systems EB PBC are widely popular for potential energy shaping of mechanical systems. In this case x = q p, H(q, p) = 1 2 p M 1 (q)p+v (q), F = 0 I I R, g(q) = 0 G(q), where (q, p) are the generalized coordinates and momenta, M = M > 0 is the inertia matrix, R = R 0 is the dissipation due to friction, G is the input matrix and V is the open loop potential energy. The added energy is H a (q) = V d (q) V (q), V d is the desired potential energy. (EB-PDE) g F H a = 0 becomes g G ( V d V ) = 0, known as the potential energy matching equation. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 25/85

26 CDC Workshop, Shanghai, PRC, 15/12/2009 p. 26/85 EB PBC: An Alternative Viewpoint Given ẋ = f(x) + g(x)u, y = h(x). If we can find a vector function û : R n R m such that the PDE ( ) Ha x (x) [f(x) + g(x)û(x)] = h (x)û(x), can be solved for H a : R n R, and H d (x) = H(x) + H a (x), is such that x = arg min H d (x), then, the state feedback u = û(x) is an EBC, i.e., x is stable with Lyapunov function H d (x) = H(x) t 0 û (x(s))h(x(s))ds. Remark: A necessary condition for the solvability of the PDE is f( x) + g( x)û( x) = 0 h ( x)û( x) }{{} power = 0, i.e. extracted power at the equilibrium x should be zero.

27 CDC Workshop, Shanghai, PRC, 15/12/2009 p. 27/85 Mechanical Systems: Physical Interpretation Passivity Ḣ ( H p ) G(q)u Full actuation m = n, G(q) = I we can assign any function of q with û(q) = H a q (q). Asymptotic stability with v = K di q, K di > 0 q * δ( ) Underactuated case cannot solve k p G(q)û(q) = H a q (q) equivalent to the matching equation. Need total energy shaping. k d q= q * mg In (Ailon/Ortega, 93) done with dynamic extension to inject damping without measuring speeds.

28 Flexible Pendulum δ( q * ) K 1 m=1 R c q c K 2 q p2 q p1 D CDC Workshop, Shanghai, PRC, 15/12/2009 p. 28/85

29 Overcoming the Dissipation Obstacle Proposition [Interconnection and Damping Assignment (IDA PBC)] Fix d d (x) = H d (x)r d(x) H d (x) with R d : R n R n n, R d = R d 0. (i) u SF PBC if and only if g(x)u SF (x) = f(x) R d (x) H d (x) + α(x) for some function α : R n R n such that α H d is identically zero, α = 0 (ii) For any J d : R n R n n, J d = J d, the function α(x) = J d(x) H d (x), satisfies both restrictions: α = 0 and α H d = 0. Furthermore, the closed-loop system, Σ d, takes the port-hamiltonian (PH) form Σ d : ẋ = F d (x) H d (x) + g(x)v, F d (x) := J d (x) R d (x) y d = g (x) H d (x). CDC Workshop, Shanghai, PRC, 15/12/2009 p. 29/85

30 Control by Interconnection Control by interconnection viewpoint, y c y Σ c + - u c Σ I + u Σ - Subsystems: Σ c (control) and Σ (plant) are PH systems Σ I (interconnection). Principle: Select Σ I such that we can add" the energies of Σ and Σ c. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 30/85

31 Controllers by Interconnection are as Old as Control Itself CDC Workshop, Shanghai, PRC, 15/12/2009 p. 31/85

32 They re Pervasive and Efficient CDC Workshop, Shanghai, PRC, 15/12/2009 p. 32/85

33 Adding Energies Definition: The interconnection is power preserving if y u + y c u c = 0. Simplest example: Classical feedback interconnection u = u c Proposition: y y c Σ I power preserving, Σ, Σ c cyclo passive with states x R n, ζ R n c, and energy functions H(x), H c (ζ), resp. Then, interconnection cyclo passive with new energy function H(x) + H c (ζ). Problem: Although H c (ζ) is free, not clear how to affect x? CDC Workshop, Shanghai, PRC, 15/12/2009 p. 33/85

34 Invariant Function Method ξ Principle: Restrict the motion to a subspace of (x, ζ) Say Ω κ {(x, ζ) ζ = F (x) + κ}, (κ determined by the controllers ICs). Then, in Ω 0, H d (x) H(x) + H c [F (x)] It can be shaped selecting H c (ζ). x (o) x 2 Problem Let C(x, ζ) F (x) ζ. Finding F ( ) that renders Ω invariant x 1 x (t) d dt C C=0 0, (P DE) Stymied by pervasive damping for solution of (PDE) but can be overcome selecting new port variables. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 34/85

35 Port Hamiltonian Systems PH model of a physical system ẋ = [J (x) R(x)] H + g(x)u Σ (u,y) : y = g (x) H u y has units of power (voltage current, speed force, angle torque, etc.) J = J is the interconnection matrix, specifies the internal power conserving structure (oscillation between potential and kinetic energies, Kirchhoff s laws, transformers, etc.) R = R 0 damping matrix (friction, resistors, etc.) g is input matrix. PH systems are cyclo passive Ḣ = H R H + u y. Invariance of PH structure Power preserving interconnection of PH systems is PH. Nice geometric structure formalized with notion of Dirac structures. Most nonlinear cyclo passive systems can be written as PH systems. Actually, in (network) modeling is the other way around! CDC Workshop, Shanghai, PRC, 15/12/2009 p. 35/85

36 Basic CbI for PH Systems Given a PH system, Σ (u,y) ẋ = F (x) H(x) + g(x)u y = g (x) H(x), Ḣ u y where we defined F (x) := J (x) R(x), J = J, R = R 0. PH controller (nonlinear integrators), ζ R m Σ c : ζ = u c y c = ζ H c (ζ), Ḣc = u c y c Standard negative feedback interconnection Σ I : u u c = y + v y c 0 Ḣ + Ḣc v y For ease of presentation, and with loss of generality, we have taken ζ R m. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 36/85

37 Conditions for CbI Proposition Assume there exists a vector function C : R n R m such that F g C = g (CbI P DE) 0 Then, for all functions Φ : R m R, the following cyclo passivity inequality is satisfied Ẇ v y, where the shaped storage function W : R n R m R is defined as W (x, ζ) = H(x) + H c (ζ) + Φ(C(x) ζ). CDC Workshop, Shanghai, PRC, 15/12/2009 p. 37/85

38 Comparison of CbI and State Feedback PBC: Applicability (CbI) (CbI SM ) F g g F g C = g 0 C = 0 (EBC) (Basic IDA) g F g H a = 0 (Basic CbI PS ) F C = g (CbI PS ) F d C = g plus (PO PDE) (F H = F d H PS ) (Basic CbI SM PS ) g F C = 0 (CbI SM PS ) g F d C = 0 g F H a = 0 (PS) g F d H a = 0 plus (PO PDE) (IDA) g F d H a = g (F F d ) H plus (PO-PDE). CDC Workshop, Shanghai, PRC, 15/12/2009 p. 38/85

39 ntsfinal Implication Diagram CbI Basic CbI ps CbI ps CbI sm Basic CbI sm ps CbI sm ps EB Basic IDA PS IDA (Ortega, et al., TAC 08) CDC Workshop, Shanghai, PRC, 15/12/2009 p. 39/85

40 State Feedback PBC and CbI: Connections CbI Dynamic feedback control u = y c + v = ζ H c (ζ) + v, ζ controllers state with energy H c (ζ) free, Generate Casimir functions, C, that make Ω = {(x, ζ) ζ = C(x)} invariant For arbitrary Φ State feedback PBC Facts Ḣ(x) + Ḣ c (ζ) + Φ(C(x) ζ) v y Solve some PDE on H a and define a static state feedback, û(x), that ensures Ḣ + Ḣa v y State feedback PBC is the projection of CbI into the invariant manifold. There is no advantage of dynamic extension from minimum assignment viewpoint. (Astolfi and Ortega, SCL 09) Simpler controllers with CbI. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 40/85

41 Open Problems Is there a CbI version of IDA? What is the modification that is needed to add this degree of freedom? We have fixed the order of the dynamic extension to be m. There are some advantages for increasing their number. Also, we have taken simple nonlinear integrators. CbI does not help for minimum assignment, but certainly has an impact on performance and simplicity. Can CbI be used as an alternative to the current perturbation" framework to formulate the problem of control over networks? Key question: Impact of the network topology on the ability to shape the energy, i.e., to generate the Casimir functions. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 41/85

42 IDA PBC: A Matching Perspective Consider ẋ = f(x) + g(x)u. Assume there are matrices g (x), J d (x) = J d (x), R d(x) = R d (x) 0, where g (x)g(x) = 0, g (x) full rank, and a function H d (x), with x = arg min H d (x), that verify the matching equation g (x)f(x) = g (x)[j d (x) R d (x)] H d (ME) Then, the closed loop system with u = û(x), where û(x) = [g (x)g(x)] 1 g (x){[j d (x) R d (x)] H d f(x)}, takes the port controlled Hamiltonian (PCH) form ẋ = [J d (x) R d (x)] H d, with x a stable equilibrium. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 42/85

43 Universal Stabilizing Property of IDA PBC Lemma If x is asymptotically stable for ẋ = f(x), f(x) C 1 then H d (x) C 1, positive definite, and C 0 functions J d (x) = J d (x), R d(x) = R d (x) 0 such that f(x) = [J d (x) R d (x)] H d x Corollary If û(x) C 1 that asymptotically stabilizes the PCH system, then J d (x), R d (x) C 0 and H d (x) C 1 which satisfy the conditions of the IDA PBC theorem. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 43/85

44 IDA PBC as a State Modulated Interconnection Controller as an (infinite energy) source Σ c : with energy function H c (ζ) = ζ. ζ = u c y c = H c ζ (ζ) State modulated (power preserving) interconnection Σ I u(s) u c (s) = 0 û(x) û(x) 0 y(s) y c (s) Overall interconnected PCH system (with total energy H(x) + H c (ζ)) ẋ ζ = J(x) R(x) g(x)û(x) û (x)g (x) 0 H x (x) H c ζ (ζ) CDC Workshop, Shanghai, PRC, 15/12/2009 p. 44/85

45 When is IDA an Energy Balancing PBC? Plant is a PCH system ẋ = [J(x) R(x)] H x + g(x)u, y = g (x) H x The PDE becomes g (x)[j d (x) R d (x)] H a x = g (x) [J d (x) J(x) R d (x) + R(x)] H x to be solved for H a (x) and form H d (x) = H(x) + H a (x). If R d (x) = R(x), and satisfies R(x) H a x (x) = 0, that is, no shaping of coordinates with damping, then Ḣ a = u y IDA PBC is energy balancing. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 45/85

46 Some Practical Considerations: Integral Action Proposition IDA PBC with integral action u = u es + u di + v where v = K I g H d x with K I = K I Proof Let > 0, preserves stability. W (x, v) = H d v K 1 I v The closed loop ẋ v = J d R d gk I K I g 0 W x W v is clearly PCH. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 46/85

47 Damping Injection with "Dirty Derivatives" If To inject additional damping, instead of the passive output, we can feed back its integral, preserving stability. Of particular interest to obviate velocity measurements in mechanical systems where the passive output is q. Proposition u = u es + K di g H d x is asymptotically stable, then u = u es + u di, where u di = 1 τ u di K di τ with τ > 0, also ensures convergence of x. g H d x CDC Workshop, Shanghai, PRC, 15/12/2009 p. 47/85

48 Proof With the energy W (x, u di ) = H d + τ 2K di u 2 di we have ẋ u di = J d K di τ g K di τ g K di τ 2 W x W u di yielding Ẇ = u T di K 1 di u di CDC Workshop, Shanghai, PRC, 15/12/2009 p. 48/85

49 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 g (x), J d (x) and R d (x). Parameterized IDA Restrict the desired energy function to a certain class, for instance, for mechanical systems (ME) becomes a PDE in M d (q), V d (q), H d (q, p) = 1 2 p M 1 d (q)p + V d(q), imposes some constraints on J d (x), R d (x). Application of Poincare s Lemma (applicable for systems NL in u): H d (x) = F 1 d (x)f(x, u). CDC Workshop, Shanghai, PRC, 15/12/2009 p. 49/85

50 Application to a MEMS Actuator Energy function H(q, p, Q) = 1 2 k(q q ) m p2 + q 2Aɛ Q2. u + - R b m k q q Dynamical model q = p m ṗ = k(q q ) Q2 2Aɛ b m p Q = qq RAɛ + 1 R u Desired equilibrium (q, 0, 0) p is not measurable. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 50/85

51 Algebraic IDA Fix quadratic energy function (ME) solvable with γ 1 = km and H d (q, p, Q) = γ 1 2 (q q ) m p2 + γ 2 2 Q2. J d (Q) = 0 1 m 0 1 m 0 Q 0 2Aɛγ 2 Q 0 2Aɛγ 2, R d = b m r 3 and γ 2, r 3 > 0 free parameters. Note J 23 = Control law where γ i > 0 are arbitrary. Q 2Aɛγ 2 0. û(q, p, Q) = γ 1 pq γ 2 Q + 1 Aɛ qq CDC Workshop, Shanghai, PRC, 15/12/2009 p. 51/85

52 Parameterized IDA PBC Fix H d = 1 2m p2 + ϕ(q, Q) and J d R d = Solution of the PDE: b r 3. ϕ(q, Q) = 1 2 k(q q ) Aɛ qq2 + ψ(q), with ψ(q) free for the equilibrium assignment. Output feedback control û(q, Q) = (r 3 R 1) 1 Aɛ qq r 3Rψ Contains, as a particular case with r 3 = 1 R charge feedback controller. and ψ(q) quadratic, the well known linear CDC Workshop, Shanghai, PRC, 15/12/2009 p. 52/85

53 Poincare s Lemma: Boost Converter Procedure There exists H d : R n R such that H d (x) = F 1 d (x)f(x, u) if and only if [ ] [ F 1 d (x)f(x, û(x)) = ( F 1 d ]) (x)f(x, û(x)). Model (under fast switching), x(0) R 2 >0 ẋ = 0 u 1 u R }{{} J(u) R H x (x) + E 0, H(x) = 1 2L x C x2 2. Control objective: Regulate 1 C x 2 to a desired constant value V > E, verifying C.1 Only x 2 measurable. C.2 u (0, 1). C.3 x R 2 >0. C.4 R is unknown. Main contribution: Stabilization via IDA PBC with a simple static nonlinear output feedback. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 53/85

54 Proposition For all R > 0 the IDA PBC yields ( ) α x2 u = u, 0 < α < 1 x 2 (i) x = ( L RE V 2, CV ) is asymptotically stable with Lyapunov function H d (x) = 1 2L x C x2 2 + κ 1 x 2(1 α) 2 (κ 2 + κ 3 x 1 )x 1 α 2 (ii) Domain of attraction: Ξ α = {x x R 2 >0 and H d (x) H d (0, x 2 )} is such that x(0) Ξ α x(t) Ξ α and lim t x(t) = x (iii) Saturation: x, α (0, 1) s.t. x(0) Ξ α 0 < u(t) < 1 CDC Workshop, Shanghai, PRC, 15/12/2009 p. 54/85

55 Proof IDA Select R a = diag{r a, 1 R } R d = diag{r a, 0}. Integrability Key PDE a K = H a x (x) = 1 β(x 2 ) 1 RC x 2 1 L R ax 1 E + R a RC x 2 β(x 2 ) PDE solvable K 2 x 1 (x) = K 1 x 2 (x) dβ dx 2 (x 2 ) = α x 2 β(x 2 ) where α = 1 R arc L. Thus, u = c 1 x α 2 a Assuming J(β(x)) R d is invertible. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 55/85

56 Proof cont Equilibrium assignment: c 1 such that H d x (x ) = H x (x ) + H a x (x ) = 0 This yields c 1 = u x α. 2 Hessian condition 2 H d x 2 (x ) = 1 L R a u L R a u L 1 C + (R ax 1 +EL)α u Lx 2 + (1 2α)R a u 2 RC Positive definite 1 < α < 1. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 56/85

57 On the Role of J d (x) Propagates the damping via interconnection of strictly passive and passive subsystems ẋ = 0 α(x) α(x) r x Σ 1 is passive and Σ 2 strictly passive (if r > 0) Property is preserved for concatenated structures", for instance ẋ = 0 β(x) 0 β(x) 0 α(x) 0 α(x) r x CDC Workshop, Shanghai, PRC, 15/12/2009 p. 57/85

58 Choosing J d (x) From Physical Considerations Isotropic (smooth rotor) synchronous motors are more efficient that indented rotor motors Virtual behavior in closed loop Open loop J(x) = 0 0 x (x 1 + Φ) x 2 x 1 + Φ 0 J d (x) = 0 L 0 x 3 0 L 0 x 3 0 Φ 0 Φ 0 Φ is the dq back emf constant. L 0 free parameter, represents stator inductance (L d = L q ). CDC Workshop, Shanghai, PRC, 15/12/2009 p. 58/85

59 cont d Enforce a coupling between electrical and mechanical coordinates in magnetic levitation systems, coordinates x = [λ, y, mẏ] Open-loop interconnection g u i J = λ y Desired interconnection m J d = 0 0 α α 1 0 with α chosen by the designer. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 59/85

60 Other Degrees of Freedom: Change of Coordinates Define x = φ(z), where φ is a diffeomorphism, and apply the construction. Modified (ME) [ g (z)( φ) 1 f(φ(z)) = g (z) Jd (z) R ] d (z) H d where J d (z) = ( φ) 1 J d (φ(z))( φ) = J d (z), R d (z) = ( φ) 1 R d (φ(z))( φ) = R d (z) 0, g(z) = ( φ) 1 g(φ(z)) The choice of φ is a new degree of freedom. New control laws for power systems. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 60/85

61 Adding Dynamics to IDA PBC Proposition Consider the system ẋ = f(x) + g(x)u. Define g : R n R (n m) n to be a full rank left annihilator of g(x), i.e., g (x)g(x) = 0, and rank {g (x)} = n m for all x. Let x R n be an assignable equilibrium. Assume there exists a matrix F : R n R n n and a function H : R n R such that the following holds. (A1) g (x)f(x) = g (x)f (x) H(x) The system in closed loop with the static state feedback u = [g (x)g(x)] 1 g (x){f (x) H(x) f(x)}, can be written in port Hamiltonian (PH) form ẋ = F (x) H(x). Furthermore, if (A2) F (x) + F (x) 0; (A3) ( H) = 0 and ( 2 H) > 0; then x is a (locally) stable equilibrium with Lyapunov function H(x). CDC Workshop, Shanghai, PRC, 15/12/2009 p. 61/85

62 IDA PBC with Dynamic Extension Proposition Assume there exists F 1 : R n+p R n n, F 2 : R n+p R n p and H : R n+p R such that: (B1) g (x)f(x) = g (x)[ F 1 (x, ζ) x H(x, ζ) + F2 (x, ζ) ζ H(x, ζ)] Consider the p dimensional dynamic state feedback controller ζ = F 3 (x, ζ) x H(x, ζ) + F4 (x, ζ) ζ H(x, ζ) u = [g (x)g(x)] 1 g (x){ F 1 (x, ζ) x H(x, ζ) + F2 (x, ζ) ζ H(x, ζ) f(x)}, with arbitrary matrices F 3, F 4. The closed loop system is PH ẋ ζ = F (x, ζ) x H(x, ζ) ζ H(x, ζ) Furthermore, if: (B2) F (x, ζ) + F (x, ζ) 0; (B3) ( H) = 0 and ( 2 H) > 0, for some ζ R p ; then (x, ζ ) is a (locally) stable equilibrium with Lyapunov function H(x, ζ). CDC Workshop, Shanghai, PRC, 15/12/2009 p. 62/85

63 Dynamic Extension is Unnecessary for Stabilization Observation It is clear that we have to compare the sets of solutions of the PDEs (A1), (B1) subject to the constraints, (A2), (A3) and (B2), (B3), respectively. Proposition The following statements are equivalent: (S1) There exists a matrix F : R n R n n and a function H : R n R such that (A1) (A3) hold. (S2) There exists a positive integer p, matrices F 1, F 2 and a function H such that (B1) (B3) hold. Consequently, the equilibrium x is stabilizable via static state feedback IDA PBC if and only if the equilibrium (x, ζ ) is stabilizable via dynamic (state feedback) IDA PBC. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 63/85

64 Proof ((S1) (S2))Trivial. ((S2) (S1)) Assume conditions (B1) (B3) hold. Now, since both PDEs have the same left hand side g (x)[f (x) H(x) F 1 (x, ζ) x H(x, ζ) F2 (x, ζ) ζ H(x, ζ)] = 0. (KE) We now construct functions F (x) and H(x) that satisfy (KE) hence (A1) and conditions (A2), (A3). First, notice that, in view of (B3), one has [ ζ H(x, ζ)] = 0, det{[ ζζ H(x, ζ)] } > 0. Therefore, application of the Implicit Function Theorem to the function ζ H(x, ζ) proves the existence of a function γ : R n R p such that [ ζ H(x, ζ)] ζ=γ(x) = 0 in some open neighborhood of (x, ζ ). Notice, also, that ζ = γ. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 64/85

65 cont d Replacing in (KE) yields g (x)[f (x) H(x) F 1 (x, γ(x))w (x)] = 0, where, W (x) := x H(x, γ(x)),. Now, select F (x) = F1 (x, γ(x)) that, in view of (B2), necessarily satisfies (A2). Replacing F (x) above yields g (x)f (x)[ H(x) W (x)] = 0. A function H(x) that satisfies the equation above is given as H(x) := H(x, γ(x)). To complete the proof it only remains to show that the function H(x) verifies condition (A3). For, note that ( H) = W = 0, ( 2 H) = ( W ) > 0 where (B3), the definition of W (x) and the fact that W (x) = xx [ H(x, γ(x))], have been used. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 65/85

66 Remarks An extension of the (static state feedback) IDA PBC technique that incorporates an (arbitrary) dynamic extension has been presented. It has been shown that, for the purposes of (local) Lyapunov stabilization, no advantage is gained with this extension. The proof of the equivalence relies on the Implicit Function Theorem, from which the local nature of our result is inherited. Hence, the domain of stability of the equilibrium for static feedback may be smaller that the one obtained using dynamic feedback. The version of IDA PBC considered here does not presume any particular structure for the desired energy function. It is known that for some classes of systems, for instance, mechanical, it is convenient to parameterize" the solutions. Studying the effect of a dynamic extension in that case leads to a problem different from the one studied here. Considering dynamic extensions the possibility to improve the transient performance and to remove the need to measure the full state is opened up. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 66/85

67 IDA PBC of Mechanical Systems Model q ṗ = 0 I n I n 0 H q H p + 0 G(q) u where H(q, p) = 1 2 p M 1 (q)p + V (q), rank(g) = m < n. Desired energy is parameterized H d (q, p) = 1 2 p M 1 d (q)p + V d(q), M d (q) = M d (q) > 0 q = arg min V d (q). Desired interconnection and damping matrices J d (q, p) = 0 M 1 (q)m d (q) M d (q)m 1 (q) J 2 (q, p) = J d (q, p) R d (q) = G(q)K v G (q) 0, K v > 0 CDC Workshop, Shanghai, PRC, 15/12/2009 p. 67/85

68 Proposition Assume there is M d (q) = M d (q) Rn n and a function V d (q) that satisfy the PDEs { } G q (p M 1 p) M d M 1 q (p M 1 d p) + 2J 2M 1 d p = 0 G { V M d M 1 V d } = 0, for some J 2 (q, p) = J 2 (q, p) Rn n and a full rank left annihilator G (q) R (n m) n of G, i.e., G G = 0 and rank(g ) = n m. Then, the system in closed loop with u = (G G) 1 G ( q H M d M 1 q H d + J 2 M 1 d p) K vg p H d, takes the desired Hamiltonian form. Further, if M d > 0 (in a neighborhood of q ) and q = arg min V d (q), then (q, 0) is a stable equilibrium point with Lyapunov function H d. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 68/85

69 Proof G G ṗ = G G ( q H + Gu) = G G ( 1 2 q(p M 1 p) V + Gu) G G ( M d M 1 q H d + (J 2 GK v G ) p H d ) = G G ( M d M 1 [ 1 2 q(p M 1 d p) + V d] + (J 2 GK v G )M 1 d p. To prove stability: H d is positive definite and Ḣ d λ min {K v } G M 1 d p 2 0. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 69/85

70 Connection with Controlled Lagrangians PDE s (with J 2 (q, p) = 1 2 n k=1 U k(q)p k, U k = U k ) G { q 1 (M(,k) ) M 1 dm q 1 (Md ) (,k) + U k M 1 d } = 0 G { V q M dm 1 V d q } = 0 { } If J 2 (q, p) = M d M 1 [ q (MM 1 d p)] q (MM 1 d p) M 1 M d, we recover the controlled Lagrangian method All matrices that preserve mechanical structure" (arbitrary Q(q)) [ ] J 2 (q, p) = J 2 above + M d M 1 [ q Q] q Q M 1 M d Gyroscopic (intrinsic) terms are added to the Lagrangian L c (q, q) = 1 2 q M(q)M 1 d (q)m(q) q + q Q(q) V d (q) CDC Workshop, Shanghai, PRC, 15/12/2009 p. 70/85

71 Constructive Solution For Underactuation Degree One Syste Identification of a class of underactuation degree one mechanical systems for which the PDEs are explicitly solved. The KE PDE becomes an algebraic equation and we give a set of solutions. Assume that the inertia matrix and the force induced by the potential energy (on the unactuated coordinate) are independent of the unactuated coordinate. One condition for stability an algebraic inequality that measures our ability to influence, through the modification of the inertia matrix, the unactuated component of the force induced by potential energy. Suitable parametrization of assignable energy functions via two free functions and a gain matrix to address transient performance and robustness issues. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 71/85

72 Parametrization of the Kinetic Energy PDE Then, kinetic energy PDE becomes Assumption A.1 Underactuation degree one: m = n 1. Assumption A.2 G q (p M 1 p) = 0 n i=1 γ i (q) M d q i = [J (q)a (q) + A(q)J (q)], where J (q) is free, γ = M 1 M d (G ), A = [W 1 (G ), W2 (G ),..., Wno (G ) ] with n 0 = n 2 (n 1) and W i = W i, e.g., for n = 3 W 1 = , W 2 = , W 3 = CDC Workshop, Shanghai, PRC, 15/12/2009 p. 72/85

73 Solving the Kinetic Energy PDE The expression above characterizes all solutions to the KE-PDE Assumption A.3 G is function of a single element of q, say q r, r {1,..., n} A.3 satisfied (for partially linearized systems) if the column of M corresponding to the unactuated coordinate depends only on q r. A subset, for which KE PDE becomes algebraic and we can find explicit solutions, is γ r dm d dq r = 2AJ dm d dq r e i Im A Furthermore, G A = 0 A Im G, suggesting dm d dq r e i Im G Proposition For all desired (locally) positive definite inertia matrices M d (q r ) = qr q r G(µ)Ψ(µ)G (µ)dµ + M 0 d where Ψ = Ψ and M 0 d = (M 0 d ) > 0, may be arbitrarily chosen, there exists J 2 such that the kinetic energy PDE holds. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 73/85

74 Solving the Potential Energy PDE Recalling PE PDE: G { V M d M 1 V d } = 0 Can be written as γ (q) V d = s(q), s = G V. Remarks concerning s: For all admissible equilibria q, we have s( q) = 0. G V are forces that cannot be (directly) affected by the control. Since G, M d depends on q r it is reasonable: Assumption A.4 γ, s are functions of q r only. ( M = M(q r ) A generic condition is needed to ensure that the PDE admits a well defined solution: Assumption A.5 γ r (qr ) 0 CDC Workshop, Shanghai, PRC, 15/12/2009 p. 74/85

75 Proposition Under Assumptions A.1 A.5 and M d (q r ) = q r qr the PE PDE are given by G(µ)Ψ(µ)G (µ)dµ + M 0 d all solutions of V d (q) = qr 0 s(µ) dµ + Φ(z(q)), γ r (µ) with z(q) q q r γ(µ) 0 dµ the characteristic of the PE PDE, and Φ an arbitrary γ r (µ) differentiable function. Remarks Identify a set of assignable energy functions parameterized by {Ψ, M 0 d, Φ}. γ r is the element of the coupling term", G M 1 M d, through which we can modify the (unactuated coordinates of the) open loop potential energy. For stability, since Φ(z) is arbitrary, restrictions will only be imposed on s. Namely, γ r that its second derivative, evaluated at qr, is positive. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 75/85

76 Main Stabilization Result Assumption A.6 γ r (q r ) ds dq r (q r ) > 0 ensures (q, 0) is a locally stable equilibrium with Lyapunov function H d (q, p). Assumption A.7 G M 1 e r (q r ) 0, makes it asymptotically stable. Furthermore, if we select Φ(z(q)) = 1 2 [z(q) z(q )] P [z(q) z(q )] with P = P > 0, the control law is of the form u = A 1 (q)p S(q q ) + p A 2 (q r )p. p A n (q r )p + A n+1(q r ) K v A n+2 (q r )p where K v = Kv > 0 is free, S R(n 1) n is obtained removing the r th row from the n dimensional identity matrix. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 76/85

77 Pendulum on a Cart Model ml 2 q 1 + ml cos q 1 q 2 mgl sin q 1 = 0 (M + m) q 2 + ml cos q 1 q 1 ml sin q 1 q 2 1 = v. Can be transformed into q = p ṗ = a sin q 1 e 1 + b cos q 1 1 u Notice that G (q 1 ) = [1, b cos q 1 ]. It can be shown that ψ(q 1 ) cannot be a constant. Propose ψ = k sin q 1. Independently of {Ψ, M 0 d, Φ} assumptions cannot be satisfied outside ( π 2, π 2 ). CDC Workshop, Shanghai, PRC, 15/12/2009 p. 77/85

78 Stability Result The IDA PBC u = A 1 (q 1 )P (q 2 q 2 ) + p A 2 (q 1 )p K v A 3 (q 1 )p + A 4 (q 1 ) ensures asymptotic stability of the desired equilibrium (0, q 2, 0, 0) with a domain of attraction containing the set ( π 2, π 2 ) R3 and Lyapunov function H d (q, p) where M d (q 1 ) = kb2 3 cos3 q 1 kb 2 cos2 q 1 kb 2 cos2 q 1 k(cos q 1 1) + m 0 22, V d (q) = 3a kb 2 cos 2 q 1 + P 2 [ q 2 q b ln(sec q 1 + tan q 1 ) + 6m0 22 kb tan q 1 ] 2, (which is radially unbounded on the set ( π 2, π 2 ).) CDC Workshop, Shanghai, PRC, 15/12/2009 p. 78/85

79 Simulations Trajectories with [q(0), p(0)] = [π/2 0.2, 0.1, 0.1, 0] pendulum starting near the horizontal Positions (q) Control Time (sec) q 1 q Time (sec) 60 Energy Velocities (q) Time (sec) Time (sec) 0.5 q 2 (m) q 1 (rad/sec) q (rad) q (rad) 1 CDC Workshop, Shanghai, PRC, 15/12/2009 p. 79/85

80 Strongly Coupled VTOL Aircraft Model (ɛ 0, possibly large) y ε v 2 ẍ = sin θv 1 + ɛ cos θv 2 v 1 ÿ = cos θv 1 + ɛ sin θv 2 g θ θ = v 2 COG g x Can be transformed into q = p 1 0 ṗ = ɛ cos θ 1 ɛ sin θ u + g ɛ sin θe 3 Objective: Characterize assignable energy functions with (x, y, 0, 0, 0, 0) asymptotically stable. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 80/85

81 Proposition A set of assignable energy functions is characterized by M d (q 3 ) = k 1 ɛ cos 2 q 3 + k 3 k 1 ɛ cos q 3 sin q 3 k 1 cos q 3 k 1 ɛ cos q 3 sin q 3 k 1 ɛ cos 2 q 3 + k 3 k 1 sin q 3 k 1 cos q 3 k 1 sin q 3 k 2 with k 1 > 0 and and the potential energy function V d (q) = g k 1 k 2 ɛ cos q k 3 > 5k 1 ɛ, k 1 ɛ > k 2 > k 1 2ɛ q 1 q 1 k 3 k 1 k 2 ɛ sin q 3 q 2 q 2 + k 3 k 1 ɛ k 1 k 2 ɛ (cos q 3 1) P. Moreover, the IDA PBC law ensures almost global asymptotic stability of the desired equilibrium (q 1, q 2, 0, 0, 0, 0). CDC Workshop, Shanghai, PRC, 15/12/2009 p. 81/85

82 Simulations Effect of tuning (matrix P ) y(m) 2 y(m) 2 3 g 3 g x(m) x(m) CDC Workshop, Shanghai, PRC, 15/12/2009 p. 82/85

83 cont d Upside down simulation g 2 y(m) x(m) CDC Workshop, Shanghai, PRC, 15/12/2009 p. 83/85

84 Some Results at the Frontier General input output theory (Rantzer, SCL 06) Connection between S procedure and KYP Lemma. (Sontag, SCL 08) Secant condition. (Janson and Petersen, TAC 08) Passivity F q. (Iwasaki and Hara, TAC 08) Finite frequency response positive realness. (Willems at al., EJC 08) Behavioral framework. (Ito and Jiang, TAC 09) IiSS and small gain theorem. Various applications (Arcak and Kokotovic, Automatica 01) Observer design. (Pavlov and Marconi, SCL 08) Convergent systems and output regulation. (Alonso et al., SCL 09) Reaction systems. Ships, UAV s, reaction systems, fuel cells,... Port Hamiltonian systems: Chen, van der Schaft, Stramigioli, Fujimoto, Ortega... CDC Workshop, Shanghai, PRC, 15/12/2009 p. 84/85

85 cont d Synchronization and networks (Arcak and Wen, INFOCOM 03) Network flow control. (Pogromsky and Nijmeijer, TCS 06) Notion of almost passivity. (Chopra and Spong, CDC 06) Output synchronization of passive systems. (Arcak/Sontag, CDC 09) Incrementally passive systems. (Nunio, et al., ACC 10) Euler Lagrange systems. CDC Workshop, Shanghai, PRC, 15/12/2009 p. 85/85