Dissipativity M. Sami Fadali EBME Dept., UNR 1 Outline Differential storage functions. QSR Dissipativity. Algebraic conditions for dissipativity. Stability of dissipative systems. Feedback Interconnections Control design. 2 Motivation Passivity requires a positive energy function Generalize the energy supplied. Use state space realizations. Show the connections between inputoutput stability and Lyapunov stability analysis. Dissipative Systems = space of input functions = space of output functions =state space 3 4
Supply Rate Function of the input output and the Dissipative Dynamical System Definition 9.1: A dynamical system is dissipative w.r.t. the supply rate if a storage function that and, satisfies the dissipation inequality Locally absolutely integrable function storage function = energy stored by the system at time 5 = energy externally supplied to in 6 Implications of Definition 9.1 A dissipative system does not create energy. For a motion along a closed trajectory Nonnegative energy required to complete a closed trajectory. 7 Passivity A system is passive if a positive semidefinite storage function s.t. for all admissible inputs solutions and Identical to the definition of Chapter 8 with, supply rate 8
Differentiable Storage Function Take limit as Dissipativity Restated Definition 9.2: A dynamical system is dissipative w.r.t. the supply rate if a continuously differentiable positive definite storage function that, and, satisfies the dissipation inequality ) Differential dissipation inequality 9 Pos. Definite: class 10 ISS and Dissipativity Lemma 9.1: A system is input to state stable (ISS) if and only if it is dissipative w.r.t. the supply rate = functions of class Proof Definition 9.2: A system is input to state stable (ISS) iff it is dissipative w.r.t. the supply rate, Dissipative i.e. Theorem 7.6: ISS Lyapunov function is positive definite and satisfies the condition 11 Theorem 7.3 (p.191) A system is ISS iff an ISS Lyapunov function=. 12
Supply Rate Definition 9.3: Given constant matrices the supply rate is 13 Dissipativity Definition 9.4: A system is dissipative if a storage function s.t. and, Note: This is a special case of the storage function of Definition 9.1 with no statespace model. It is an input output property 14 Dissipativity: Passive Passive= is dissipative with Dissipativity: Strictly Passive Striclty Passive= is dissipative with =stored energy at 15 16
Dissipativity: Finite gain Stable dissipative with Strictly Output Passive dissipative with L L 17 18 Very Strictly Passive Lemma 9.2 dissipative with If is strictly output passive then it has a finitel gain. 19 20
Example: Spring Mass Damper Spring Mass Damper (cont.) continuously differentiable, positive definite (supply rate) 21 Spring mass damper is QSR dissipative and strictly output passive, with With no damping, system is passive., and the 22 Available Storage Theorem 9.1 Maximum amount of energy that can be extracted from a dissipative system at a given time starting from the initial state.. A dynamical system is dissipative if and only if for the available storage is finite. Moreover, for a dissipative system we have So is a possible storage function. 23 24
Proof: Sufficiency Assume. is zero for positive otherwise Let be an arbitrary input that takes the system from to. Show that is a storage function satisfying 25 Proof: Sufficiency (cont.) Available Storage: starting at.. (dissipative) 26 Proof: Necessity Assume is dissipative, then s.t. Finite. 27 Algebraic Condition for Dissipativity Available storage is not a good way to check dissipativity. Under certain assumptions, we can check dissipativity using the state space realization. Check leads to the KYP lemma for LTI passive systems. 28
Assumptions 1. Assume that the state space realization is affine in the input 2. The state space of the system is reachable from, i.e. and and an input s.t. 3. If is dissipative, the available storage is a differentiable function of 29 Theorem 9.2 The affine nonlinear system is dissipative if a differentiable function and functions and satisfying 30 Theorem 9.2 Conditions Proof: Sufficiency 31 32
Sufficiency (cont.) Dissipative. 33 Corollary 9.1 If the affine nonlinear system is dissipative, then a differentiable function satisfying Proved in the sufficiency proof of Theorem 9.2 34 Special Case: Passive Passive= is dissipative with Special Case: Passive LTI LTI Model Guided by LTI, choose Can consider this as a nonlinear version of KYP Lemma. To show this consider LTI case. 35 36
Strictly Output Passive dissipative with Strictly Passive Requires i.e. no real exists for the strictly passive case. The conditions of Theorem 9.2 cannot be satisfied for the affine system with 37 38 Stability of Dissipative Systems Assume a continuously differentiable storage function satisfying Assume that is an equilibrium of the unforced system 39 Theorem 9.3 Let be a dissipative dynamical system w.r.t. the continuously differentiable storage function satisfying and assume 1. The equilibrium is a strictly local minimum for 2. The supply rate satisfies Then is a stable equilibrium of the unforced system 40
Proof Significance of Theorem 9.3 Define the function is continuously differentiable and positive definite in a neighborhood of Hence, is a stable equilibrium of the unforced system 41 Shows that can provide means of constructing a Lyapunov function. Ties dissipativity to stability i. s. Lyapunov. dissipativity is a special case of the results of the theorem but, because it is an input output property, it provides important links between dissipativity and stability i. s. Lyapunov. 42 Corollary 9.2 Let be a dissipative dynamical system w.r.t. the continuously differentiable storage function satisfying Proof From Theorem 9.3 and assume is a strictly local minimum for 2. The supply rate satisfies is the only solution for which =0 Then is an asymptotically stable equilibrium of the unforced system 43 only at the equilibrium The Corollary follows from La Salle s Theorem. 44
Zero state Detectable Definition 9.6: A state space realization is zero state detectable if for any trajectory s.t. for, we have i.e. Some books (Haddad) call this zero state observable. Theorem 9.4 If the system is dissipative and zero state detectable, then the free system is Lyapunov stable if Asymptotically stable if 45 46 Proof Using Theorem 9.2 and corollary 9.1, if is dissipative then Unforced system The stability results follow from Lyapunov stability theorems. 47 Corollary 9.3 Given a zero state detectable affine state space realization, then the unforced system is Lyapunov stable if is passive. Asymptotically stable if (i) finite gain stable, or (ii) strictly output passive, or (iii) very strictly passive. Proof: Follows from Theorem 9.4 with the appropriate choices of the matrices and of the supply rate. 48
Feedback Interconnections Assume that the systems are affine, zero state detectable and completely reachable 49 Theorem 9.5 The feedback interconnection of two dissipative, zero state detectable, completely reachable systems and is stable (asymptotically stable) if for some matrix is negative semidefinite (negative definite) with the supply rate of given by the 50 Proof of Theorem 9.5 Lyapunov function candidate (pos. definite) Substitute zero state detectable:, Result follows from Lyapunov stability theory 51 Corollary 9.4 Under the conditions of Theorem 9.5 If both and are passive, then the feedback system is Lyapunov stable The feedback system is asymptotically stable if one of the following is satisfied 1. Either or is very strictly passive and the other is passive. 2. Both and are strictly passive 3. Both and are strictly output passive 52
Proof Corollary 9.4 Under the conditions of Theorem 9.5 passive stable very strictly passive Proof Corollary 9.4 (Cont.) strictly passive strictly output passive Asymptotically stable 53 Asymptotically stable 54 Corollary 9.5 Under the conditions of Theorem 9.5, if and are finite gain stable with gains and respectively, then the feedback system is stable if (asymptotically stable if ) Proof Corollary 9.5 For finite gain stable and dissipative 55 for stability for asymptotic stability 56
Nonlinear L Gain Inequality Recall: System is finite gain stable with gain is dissipative with supply rate Assume a differentiable storage function if it 57 58 Hamilton Jacobi Inequality Difficult Problem Hamilton Jacobi (H J) Inequality 59 has finite L gain less than or equal to if the Hamilton Jacobi inequality is satisfied. Find a storage function that satisfies the Hamilton Jacobi inequality with the maximum gain Easier: estimate an upper bound on that satisfies the inequality Procedure: (i) Guess a storage function, (ii) Find an approximate upper bound for the gain subject to the H J inequality 60
Example 9.1 Lyapunov function candidate Example 9.1: More Terms 61 62 Hamilton Jacobi Inequality LTI Systems Provided that 63 64
Riccati Equation Strictly Output Passive has finite L gain less than or equal to R i.e. if the Riccati Equation: R has a solution if 65 dissipative with Lemma 9.2: If a differentiable storage function s. t. then is strictly output passive. denotes the Lie derivative. 66 Apply H J Condition Control Design Hamilton Jacobi inequality with Using the control input and the measurement vector, find a controller that stabilized the plant and reduces the effect of disturbances on the output Minimize gain of mapping L gain: minimize the norm of the mapping. L gain: minimize the L gain of the mapping. Choose 67 68
State Feedback Nonlinear L Gain Control Plant Assume State Feedback : selected to optimize the nonlinear L gain of the mapping Full information feedback: the entire state is available. 69 Very difficult to solve the nonlinear L gain problem. Solve a suboptimal problem. Given a desirable exogenous signal attenuation level, find a control such that the L gain of the mapping is less than or equal. Repeat for another controller and iterate till the controller approaches the optimal solution. 70 Theorem 9.6 The closed loop system has a finite L gain if and only if the Hamiltonian inequality has a solution by. The control law is given 71 Proof of Sufficiency Assume satisfies the inequality, and substitute 72