Mathematics for Control Theory Outline of Dissipativity and Passivity Hanz Richter Mechanical Engineering Department Cleveland State University
Reading materials Only as a reference: Charles A. Desoer and Mathukumalli Vidyasagar (2009) [1975], Feedback Systems: Input-Output Properties, Classics in Applied Mathematics, SIAM, ISBN-13: 978-0898716702 (chapter on passivity) Romeo Ortega, Antonio Loria, Per Johan Nicklasson and Hebertt Sira-Ramirez, (1998), Passivity-Based Control of Euler-Lagrange Systems, Springer, ISBN 978-1-4471-3603-3. 2 / 17
Energy conservation u ẋ = Ax+Bu y = Cx y Define u = F, y = ẋ (we know power=uy). Then in a time interval [t 1,t 2 ] we expect: energy stored in m, k = energy delivered by F - energy dissipated in b 3 / 17
Dissipation Inequality: Physical Systems Since the dissipated energy is nonnegative, we can also say: Energy delivered energy stored In this system, E = 1 2 mẋ2 + 1 2 kx2 and t2 Since energy is nonnegative: t 1 u(τ)y(τ)dτ E(t 2 ) E(t 1 ) t2 t 1 u(τ)y(τ)dτ E(t 1 ) = β Passivity is defined using just a constant β on the right-hand side (no need to involve a storage function). 4 / 17
Dissipation Inequality: Other Systems Given an arbitrary LTI system u ẋ = Ax+Bu y = Cx y Is there an energy-like function E(x) so that t2 for any [t 1,t 2 ] and any input u(t)? t 1 u(τ)y(τ)dτ E(x(t 2 )) E(x(t 1 )) 5 / 17
Notation Input and output signals are assumed to take values in R m and R p for some m,p 1. These signals are assumed to be in the L 2e or L 2 classes, as appropriate. Notations like L m 2e could be used, but the superscript will be omitted when the dimension of the value space is clear from the context. 6 / 17
Definition of Passivity System: Σ : { ẋ = f(x,u), x(0) = x0 R n y = h(x,u) with u and y are L 2e signals taking values in R m and R m, respectively. System Σ is passive if there exists a storage function H : R n R 0+ s.t. u y T T 0 u T (t)y(t)dt H(x(T)) H(x(0)) for all u L 2e, all T 0 and all x 0 R n. 7 / 17
LTI Systems: Kalman-Yakubovich-Popov Lemma System: Σ : { ẋ = Ax+Bu, x(0) = x0 R n y = Cx The system is passive with storage function V(x) = 1 2 xt Px iff P = P T 0 and Q = Q T 0 s.t. A T P +PA = Q C = B T P This result (KY Lemma) was a breakthrough in systems theory in the 1960 s. It was cast in the slidework of controls by Kalman. Yakubovich and Popov had worked on the problem previously. 8 / 17
Properties of Passive LTI Systems Passivity in LTI systems implies stability. Also, passivity implies and is implied by the positive-real property: with no poles of G(s) in Re(s) 0 Geometrically: G(jw)+G( jw) 0 w R The Nyquist plot of G(s) is contained in the right-half plane, that is, the phase of G(s) lies within [ 90, 90 ]. Also, the relative degree of G must be 1. These properties are fundamental to establish stability robustness (infinite gain margin, good phase margin) and stability with nonlinear feedback interconnections. 9 / 17
Example: Using the KY Lemmma If a LTI system is defined by A,B and it s up to us to select outputs for feedback, (C matrix), we can make the system passive through the KY lemma. Example: lindemo.mdl, lin_pass_demo.m Set up random but stable 3rd order system A,B. Compute C to render the system passive via KY Check PR property with a Nyquist plot. Verify dissipation inequality Check robustness against any positive gain uncertainties, some phase shift via delay. 10 / 17
Passivity in Nonlinear Systems Besides linear systems born from the KY lemma, which others are passive? There are nonlinear versions of the KYP lemma (Moylan, TAC 74; Savkin, IFAC 93). Euler-Lagrange systems viewed as control systems are passive: d dt L q L q + F q = Mu+Qζ q R n is the vector of generalized coordinates L is the Lagrangian F is a dissipation function (think b q 2 /2) u is the control, ζ are disturbances and M and Q are constant matrices. In mechanical systems, the Lagrangian can be chosen as L(q, q) T (q, q) V(q) (difference between kinetic and potential energies). 11 / 17
Passivity Implies Stability in Nonlinear Systems Definitions: 1. Output strict passivity (OSP): u y T δ o y 2 T +β o for some δ o > 0,β o 2. Zero-state detectability (ZSD): A state-space system is ZSD if the following implication is true: y(t) = 0 t 0 = lim t x(t) = 0 12 / 17
Passivity Implies Lyapunov Stability... Proposition (Hill and Moylan, 1976; van der Schaft, 1996): Consider the system: Σ : { ẋ = f(x)+g(x,u), x(0) = x0 R n y = h(x,u) with u and y taking values in R m. If Σ is OSP with storage function H > 0 and it is ZSD, then x = 0 is a local asymptotically stable equilibrium point. Further, if H is radially unbounded, the stability is global. H is radially unbounded if lim x H(x) = 13 / 17
Passivity theorem A feedback interconnection of passive systems maintains stability under some conditions. u 1 + e 1 H 1 H 2 e 2 + + u 2 1. Input strict passivity (ISP): u y T δ i u 2 T +β i for some δ i > 0,β i This feedback interconnection can be used to capture many control systems. Additional definitions: 2. Finite gain (we studied something equivalent with the SGT). y T γ u T +β for some γ > 0,β Properties must hold for all inputs u L 2e and all T 0. 14 / 17
A version of the passivity theorem (Desoer and Vidyasagar) Suppose H 1 and H 2 map L 2e into itself. Assume that for any u 1 and u 2 in L 2, there are solutions e 1 and e 2 in L 2e. Suppose that: 1. H 2 satisfies a passivity-like inequality relative to the output: H 2 e 2 e 2 T δ o H 2 e 2 2 T +β o for some δ o,β o 2. H 1 satisfies a passivity-like inequality relative to the input: e 1 H 1 e 1 T δ i e 1 2 T +β i for some δ i,β i 3. H 1 has finite gain. Then the feedback system is internally L 2 -stable if δ i +δ o > 0 This means that activity (lack of passivity) of one system can be compensated by passivity of the other. 15 / 17
Passivity of interconnections We only mention these as facts, without proofs (see references). 1. A parallel interconnection (feedforward) of two passive systems remains passive. 2. A negative feedback interconnection of two passive systems remains passive. 3. In the feedback configuration of the previous slides with u 2 = 0, if one of the systems is passive and the other strictly input passive, the feedback combination is L 2 -stable. 16 / 17
Examples and notes on loop transformations With causal weights, techniques similar to those used in the SGT can be applied. Non-causal weights require the more advanced multiplier theory (Desoer and Vidyasagar). Linear invertible shifts can be used as in the SGT. A very general analysis of system interconnections involving active and passive systems has been developed recently by prof. Antsaklis and his group. See for instance: Zhu, Xia and Antsaklis, Passivity analysis and passivation of feedback systems using passivity indices, 2014 American Control Conference. 17 / 17