MSc in Computer Engineering, Cybersecurity and Artificial - PowerPoint PPT Presentation

MSc in Computer Engineering, Cybersecurity and Artificial Intelligence Course FDE , a.a. 2019/2020, Lecture 6 Stability analysis of linear dynamical systems Prof. Mauro Franceschelli Dept. of Electrical and Electronic Engineering University of Cagliari, Italy Wednsday, 1st April 2020 1 / 43

Outline Introduction Equilibrium points Lyapunov stability of linear continuous-time dynamical systems Lyapunov stability of linear discrete-time dynamical systems Steady state for constant inputs 2 / 43

Introduction Lyapunov stability and autonomous systems Lyapunov theory studies the stability properties of the state of dynamical systems represented by represented by state variables formal models. It includes several methods which apply to both linear and non-linear systems. A generic nonlinear dynamical system is represented by : ① ( t ) = ❢ ( ① ( t ) , ✉ ( t ) , t ) ˙ In particular:  x 1 ( t ) ˙ = f 1 ( x 1 ( t ) , . . . , x n ( t ) , u 1 ( t ) , . . . , u r ( t ) , t )   . . . . . .  x n ( t ) ˙ = f n ( x 1 ( t ) , . . . , x n ( t ) , u 1 ( t ) , . . . , u r ( t ) , t )  We study autonomous dynamical system, which means: no input is applied to the system, the model is stationary (time-invariant). 3 / 43

Introduction State trajectory An autonomous dynamical system takes the form: ① ( t ) = ❢ ( ① ( t )) ˙ in particular:  x 1 ( t ) ˙ = f 1 ( x 1 ( t ) , . . . , x n ( t ))  . .  . . . .  x n ( t ) ˙ = f n ( x 1 ( t ) , . . . , x n ( t ))  The solution ① ( t ) with initial state ① ( t 0 ) is also called state trajectory. For a linear system it holds ① ( t ) = e ❆ t ① ( t 0 ). For a nonlinear system there may not exist an explicit form for its state trajectory. It is usually computed numerically. 4 / 43

Introduction Example Consider the autonomous nonlinear system with ① ( t 0 ) = [ x 1 , 0 x 2 , 0 ] T : � ˙ x 1 ( t ) = 1 − x 2 1 ( t ) x 2 ( t ) = − x 2 ( t ) ˙ If | x 1 , 0 | < 1 and a = atanh( x 1 , 0 ) the state trajectory is x 1 ( t ) = e ( t + a ) − e − ( t + a )   e ( t + a ) + e − ( t + a ) x 2 ( t ) = x 2 , 0 e − t  If | x 1 , 0 | = 1 the state trajectory is � x 1 ( t ) = x 1 , 0 x 2 ( t ) = x 2 , 0 e − t If | x 1 , 0 | > 1 we can compute the trajectory numerically. 5 / 43

Introduction Example The state trajectory starting from the initial condition ① 0 = [ − 0 . 5 − 5] T : x 2 1 x 1 t = ∞ 0 t 3 = 2 −1 t 2 = 1 −2 t 1 = 0.5 −3 −4 −5 t 0 = 0 −6 −1 −0.5 0 0.5 1 1.5 6 / 43

Introduction Example Examples of state trajectories for different initial states: 5 x 2 4 3 2 1 x 2,e x 1,e 0 −1 −2 −3 −4 −5 −3 −2 −1 0 1 2 3 x 1 7 / 43

Outline Introduction Equilibrium points Lyapunov stability of linear continuous-time dynamical systems Lyapunov stability of linear discrete-time dynamical systems Steady state for constant inputs 8 / 43

Equilibrium point/state: definition Equilibrium point Definition A state ① e is an equilibrium point if each trajectory which starts from ① e in a generic instant of time τ remains in ① e in each following instant of time, i.e., ① ( τ ) = ① e ⇒ ( ∀ t ≥ τ ) ① ( t ) = ① e . The state ① e is an equilibrium point if and only if ❢ ( ① e ) = 0 , The state trajectory is constant only if the variation in time of the state variables is zero. 9 / 43

Equilibrium point/state: definition Example The nonlinear system � ˙ x 1 ( t ) = 1 − x 2 1 ( t ) x 2 ( t ) = − x 2 ( t ) ˙ has two equilibrium points: ① 1 , e = [1 0] T and ① 2 , e = [ − 1 0] T . 10 / 43

Equilibrium point/state: definition Lyapunov stability of an equilibrium point Definition An equilibrium point ① e is said stable if for each ε > 0 there exists a δ ( ε ) > 0 such that || ① (0) − ① e || ≤ δ ( ε ), then || ① − ① e || < ε for all t ≥ 0. Otherwise ① e is an unstable equilibrium point. x (t) δ ε x e x (0) (a) Stability in the sense of Lyapunov implies that if an equilibrium point is stable, then the state trajectory remains arbitrarily close to such point, as long as the initial conditions of the system are sufficiently close to the equilibrium point. 11 / 43

Equilibrium point/state: definition Example 5 x 2 4 3 2 � ˙ 1 x 2,e x 1,e x 1 ( t ) = 1 − x 2 1 ( t ) 0 x 2 ( t ) = − x 2 ( t ) ˙ −1 −2 −3 −4 −5 −3 −2 −1 0 1 2 3 x 1 The equilibrium point ① 1 , e = [1 0] T is stable; the equilibrium point ① 2 , e = [ − 1 0] T is unstable. 12 / 43

Equilibrium point/state: definition Asymptotically stable equilibrium points Definition An equilibrium state ① e is said asymptotically stable if it is stable and if t →∞ || ① − ① e || = 0 . lim x (t) δ δ ε ε x e x e x (t) x (0) x (0) (a) (b) Asymptotic stability not only requires that each trajectory of the perturbed system remains in the neighborhood of the equilibrium point [fig. (a)], but also that for t → ∞ such a trajectory converges to the equilibrium point [fig. (b)]. 13 / 43

Equilibrium point/state: definition Example 1 The equilibrium point ① 1 , e = [1 0] T of the system � ˙ x 1 ( t ) = 1 − x 2 1 ( t ) x 2 ( t ) = − x 2 ( t ) ˙ is asymptotically stable. 5 x 2 4 3 2 1 x 2,e x 1,e 0 −1 −2 −3 −4 −5 −3 −2 −1 0 1 2 3 x 1 14 / 43

Equilibrium point/state: definition Example 2 The system � ˙ x 1 ( t ) = − 4 x 2 ( t ) x 2 ( t ) = x 1 ( t ) ˙ has the origin as unique equilibrium point. Such point is stable but not asymptotically. x 2 ε x 1 δ ( ε ) ε /2 0 15 / 43

Equilibrium point/state: definition Domain of attraction Local stability: The property of stability according to Lyapunov is referred to the neighborhood of an equilibrium point. Domain of attraction: The set of possible initial conditions in which the asymptotic stability property holds with respect to the equilibrium point (difficult to be determined exactly in general). An equilibrium point ① e is said globally asymptotically stable if it is asymptotically stable for any initial state. 16 / 43

Equilibrium point/state: definition Domain of attraction Local stability: The property of stability according to Lyapunov is referred to the neighborhood of an equilibrium point. Domain of attraction: The set of possible initial conditions in which the asymptotic stability property holds with respect to the equilibrium point (difficult to be determined exactly in general). An equilibrium point ① e is said globally asymptotically stable if it is asymptotically stable for any initial state. 16 / 43

Equilibrium point/state: definition Domain of attraction Local stability: The property of stability according to Lyapunov is referred to the neighborhood of an equilibrium point. Domain of attraction: The set of possible initial conditions in which the asymptotic stability property holds with respect to the equilibrium point (difficult to be determined exactly in general). An equilibrium point ① e is said globally asymptotically stable if it is asymptotically stable for any initial state. 16 / 43

Equilibrium point/state: definition Domain of attraction Note that if a system has a globally asymptotically stable equilibrium point, then such point is the only equilibrium for the system. 5 x 2 4 Example: 3 2 � ˙ 1 x 1 ( t ) = 1 − x 2 x 2,e x 1,e 1 ( t ) 0 x 2 ( t ) = − x 2 ( t ) ˙ −1 −2 −3 The domain of attraction of ① 1 , e is the semi −4 plane to the right of the line x 1 = − 1 −5 −3 −2 −1 0 1 2 3 x 1 17 / 43

Outline Introduction Equilibrium points Lyapunov stability of linear continuous-time dynamical systems Lyapunov stability of linear discrete-time dynamical systems Steady state for constant inputs 18 / 43

Lyapunov stability of linear continuous-time dynamical systems Possible equilibrium points in linear systems The state equation reduces to  x 1 ( t ) ˙ = a 1 , 1 x 1 ( t ) + a 1 , 2 x 2 ( t ) + . . . + a 1 , n x n ( t )   x 2 ( t ) ˙ = a 2 , 1 x 1 ( t ) + a 2 , 2 x 2 ( t ) + . . . + a 2 , n x n ( t )   ˙ ① = ❆① ⇐ ⇒ . . . . . .    x n ( t ) ˙ = a n , 1 x 1 ( t ) + a n , 2 x 2 ( t ) + . . . + a n , n x n ( t )  Equilibrium point The state ① e is an equilibrium point if and only if it is a solution of the linear homogeneous system of equations: ❆① e = 0 . ❆ non-singular: the only equilibrium point is the origin ① e = 0 . ❆ singular: the equilibrium points form a linear space (null space of ❆ ). A linear autonomous system can either have a single isolated equilibrium point or it has infinite non-isolated equilibrium points. 19 / 43

Lyapunov stability of linear continuous-time dynamical systems Examples � 0 � − 4 Case 1: ❆ = . 1 0 Such a matrix is not singular and the origin is the only equilibrium point. Case 2: x 2 � 2 15 − 2 � ❆ = . 10 3 − 3 5 x 1 0 Such a matrix is singular and the system has an infinite num- −5 ber of equilibrium points which −10 satisfy x 1 = x 2 . −15 −15 −10 −5 0 5 10 15 20 / 43