Estimation and enlargement of domains of attraction Alexandre Sanfelice Bazanella On leave from Universidade Federal do Rio Grande do Sul Electrical Engineering Department Porto Alegre - RS, BRAZIL Louvain-la-Neuve 1 c INMA - FSA - UCL � Alexandre Sanfelice Bazanella
My research topics • Design of controllers with restricted structure – for multivariable processes – automation of design procedures – optimization of the experimental procedures • Modeling and identification for control – applications for electric power systems - load models • Domains of attraction – enlargement of domains of attraction by control ∗ includes estimation – passivity-based controllers – dynamic extension controllers: robust regulation and enhanced properties – applications of nonlinear control and identification, particularly to electric power systems Louvain-la-Neuve 2 c INMA - FSA - UCL � Alexandre Sanfelice Bazanella
This seminar - domains of attraction • The problem setting: objectives, definitions and Lyapunov’s methods • Estimation of domains of attraction with Lyapunov functions: – how-to and why – some history – the issues • Numerical determination of Lyapunov functions with LMI’s • Passivity-based control for enlargement of the domain of attraction: L g V and IDA Examples/applications: • pendulum • electric power systems Louvain-la-Neuve 3 c INMA - FSA - UCL � Alexandre Sanfelice Bazanella
Problem setting • Consider autonomous nonlinear systems described by x ∈ ℜ n x = f ( x ) (1) ˙ • The (unique) solutions of (1) with initial condition x ( 0 ) = x 0 are x ( x 0 , t ) • The equilibria of (1) are the points x e : f ( x e ) = 0 • System stability concerns: – existence of (desired) equilibrium - operating point – asymptotic stability (a.s.) of the operating point – sufficiently large domain of attraction (DOA) of the operating point Louvain-la-Neuve 4 c INMA - FSA - UCL � Alexandre Sanfelice Bazanella
Related control problem • "Autonomous" control-affine nonlinear systems x ∈ ℜ n u ∈ ℜ m x = f ( x )+ g ( x ) u (2) ˙ • Design u = φ ( x ) such that: – the operating point is kept: g ( x e ) φ ( x e ) = 0 – the operating point is a.s. – the DOA of the operating point is as large as possible Louvain-la-Neuve 5 c INMA - FSA - UCL � Alexandre Sanfelice Bazanella
Lyapunov’s indirect method • Define the Jacobian J ( x ) = ∂ f ( x ) ∂ x – the equilibrium is a.s. if σ ( J ( x e )) ⊂ C − – the equilibrium is unstable if � C + � = / σ ( J ( x e )) 0 – if σ ( J ( x e )) ⊂ C − & σ ( J ( x e )) ∂ C − � = / 0 � then we don´t know Louvain-la-Neuve 6 c INMA - FSA - UCL � Alexandre Sanfelice Bazanella
Lyapunov’s direct method Lyapunov’s direct method: the equilibrium is a.s. if there exists a (Lyapunov) function V ( · ) such that ∂ V ( x ) ∂ x | x = x e = 0 & ∂ 2 V ( x ) ∀ x ∈ R + ⊃ x e ( x e is a minimum of V ( x ) ) | x = x e > 0 ∂ x 2 V ( x ) = ∂ V ( x ) ∂ x f ( x ) < 0 ∀ x ∈ R − ⊃ x e ˙ V ( x ) ≤ 0 The (LaSalle’s) invariance principle: allows to prove a.s. when ˙ Converse Lyapunov theorems: if the equilibrium is a.s. then there exists a Lyapunov function Louvain-la-Neuve 7 c INMA - FSA - UCL � Alexandre Sanfelice Bazanella
Two assumptions for this presentation V ( x ) < 0 – A1 - Whenever we prove a.s. with a Lyapunov function, ˙ ∗ for simplicity only, all results that follow apply mutatis mutandis when the invariance principle is used – A2 - The Jacobians at all equilibria have no eigenvalues with zero real part; not all results that follow apply otherwise Louvain-la-Neuve 8 c INMA - FSA - UCL � Alexandre Sanfelice Bazanella
The domain of attraction (DOA) The domain of attraction of an a.s. equilibrium is the set of all initial conditions which converge to it: D ( x e ) = { x 0 : lim t → ∞ x ( x 0 , t ) = x e } • The DOA is an open, connected, invariant set • The boundary of the DOA is an invariant set • The boundary of the DOA contains critical elements (other equilibria, limit- cycles, etc) • Lyapunov’s direct method allows to obtain analytic estimates of the DOA Louvain-la-Neuve 9 c INMA - FSA - UCL � Alexandre Sanfelice Bazanella
Estimation of DOA • By topological characterization • By invariant set theory • By Lyapunov’s direct method, which we prefer because it provides: – analytical expressions for the DOA – control design tools Louvain-la-Neuve 10 c INMA - FSA - UCL � Alexandre Sanfelice Bazanella
Lyapunov estimation of DOA Let : • x e = 0 and V ( x e ) = 0 , for simplicity • L a = { x : V ( x ) ≤ a } be the level sets of the Lyapunov function • R − = { x : ˙ V ( x ) < 0 } • L a and R − be connected, so we don’t get overwhelmed by the notation Louvain-la-Neuve 11 c INMA - FSA - UCL � Alexandre Sanfelice Bazanella
Lyapunov estimation of DOA V ( x ) = 0 ∀ x ∈ ∂ R − • Note that ˙ • The level sets contained in R − are invariant sets • Trajectories starting at a bounded level set converge to the equilibrium • Then ˆ D = sup L a ⊆ R − L a , with L a bounded, is the best estimate that can be obtained with a given Lyapunov function • The perfect Lyapunov function: ˆ D = D • A good Lyapunov function: ˆ D ≈ D Louvain-la-Neuve 12 c INMA - FSA - UCL � Alexandre Sanfelice Bazanella
Lyapunov estimation of DOA • How to obtain Lyapunov functions? – Quadratic: V ( x ) = x T P x ∗ can always be obtained solving Lyapunov equation: J T ( x e ) P + J T ( x e ) P = − Q P , Q > 0 – Krasovskii: V ( x ) = f ( x ) T Pf ( x ) – Lur’e-Postnikov – etc • The methods are not always successful in finding a Lyapunov function • Even when they succeed in finding a Lyapunov function, it can be a bad one Louvain-la-Neuve 13 c INMA - FSA - UCL � Alexandre Sanfelice Bazanella
The issues • There is no general systematic way to derive a good Lyapunov function • Estimates of DOA’s can be very poor • "Bad" Lyapunov functions have little value for control design: improving the estimate does not imply improving the real DOA Louvain-la-Neuve 14 c INMA - FSA - UCL � Alexandre Sanfelice Bazanella
� Example - the pendulum T L M x 1 x 2 = ˙ x 2 − a 1 sin ( x 1 ) − a 2 x 2 + b = ˙ where x 1 = δ is the angle, x 2 is the speed, b is the torque, a i > 0 . There are infinite equilibria given by x e = [ x 1 e x 2 e ] T = [ arcsin b a 1 0 ] T Let a 1 = 2 , a 2 = 0 . 1 and b = 0 and study the stability of x o e = [ 0 0 ] T Louvain-la-Neuve 15 c INMA - FSA - UCL � Alexandre Sanfelice Bazanella
Example - the pendulum Consider a quadratic Lyapunov function V Q ( x ) = x T P x By solving the Lyapunov equation for Q = I we get � � 15 . 025 0 . 25 P = 0 . 25 7 . 5 The Lyapunov derivative is V Q ( x ) = 2 x T Pf ( x ) = − x 1 sinx 1 − x 2 2 + 15 ( x 1 x 2 − sinx 1 x 2 ) ˙ Louvain-la-Neuve 16 c INMA - FSA - UCL � Alexandre Sanfelice Bazanella
Example - the pendulum 4 3 2 1 x2 0 −1 −2 −3 −4 −4 −3 −2 −1 0 1 2 3 4 x1 Figure 1: Level curves of the quadratic Lyapunov function and the boundary of R − Louvain-la-Neuve 17 c INMA - FSA - UCL � Alexandre Sanfelice Bazanella
4 3 2 1 x2 0 −1 −2 −3 −4 −4 −3 −2 −1 0 1 2 3 4 x1 Figure 2: Real DOA and its estimate with the quadratic Lyapunov function Louvain-la-Neuve 18 c INMA - FSA - UCL � Alexandre Sanfelice Bazanella
Example - the pendulum Now consider the Lure-Postnikov (energy-like) Lyapunov function � x 1 e V ( x ) = 1 ( b − a 1 sin ( ς )) d ς = 1 2 x 2 2 x 2 2 + b ( x 1 e − x 1 )+ a 1 ( cos x 1 e − cos x 1 ) 2 + x 1 6 4 2 x2 0 −2 −4 −6 −6 −4 −2 0 2 4 6 x1 Figure 3: Level curves of the Lur’e-Postnikov Lyapunov function Louvain-la-Neuve 19 c INMA - FSA - UCL � Alexandre Sanfelice Bazanella
V ( x ) = − a 2 x 2 Its derivative is ˙ 2 ≤ 0 and a.s. is established by the invariance principle. Then R − = ℜ 2 and ˆ D is the largest bounded level curve 4 3 2 1 x2 0 −1 −2 −3 −4 −4 −3 −2 −1 0 1 2 3 4 x1 Figure 4: Real and estimated DOA’s for the pendulum Louvain-la-Neuve 20 c INMA - FSA - UCL � Alexandre Sanfelice Bazanella
Parenthesis - how did I plot the real DOA? • The stable manifold of an equilibrium M s ( x e ) = { x 0 : lim t → + ∞ x ( x 0 , t ) = x e } • The unstable manifold of an equilibrium M u ( x e ) = { x 0 : lim t →− ∞ x ( x 0 , t ) = x e } • The stable and unstable manifolds of an equilibrium are, by definition, invariant sets. Let the jacobian J ( x e ) = ∂ f ( x ) | x = x e have r s eigenvalues in the LHP Theorem 1. x and r u in the RHP . Then dim M s ( x e ) = r s and dim M u ( x e ) = r u . Moreover, the subspace generated by the "stable" (unstable) eigenvectors of J ( x e ) is tangent to M s ( x e ) ( M u ( x e ) ). Louvain-la-Neuve 21 c INMA - FSA - UCL � Alexandre Sanfelice Bazanella
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