Linearization and system equilibrium Daniele Carnevale Dipartimento - PowerPoint PPT Presentation

Equilibrium pair Linearization Linearization and system equilibrium Daniele Carnevale Dipartimento di Ing. Civile ed Ing. Informatica (DICII), University of Rome “Tor Vergata” Fondamenti di Automatica e Controlli Automatici A.A. 2014-2015 1 / 4

Equilibrium pair Linearization Equilibrium pair Consider a dynamical system x + = f ( x, u ) � � x = f ( x, u ) , ˙ , (1) and the equilibrium pair ( x e , u e ) , if it exists, is such that 0 = f ( x e , u e ) , [ x e = f ( x e , u e )] . (2) This implies that the solution of the differential [difference] equation ϕ ( t, t 0 , x 0 , u ( · )) [ ϕ ( k, k 0 , x 0 , u ( · ))] is such that ϕ ( t, t 0 , x e , u e ) = x e , [ ϕ ( k, k 0 , x e , u e ) = x e ] (3) for all t ≥ t 0 [ k ≥ k 0 ] . 2 / 4

Equilibrium pair Linearization Equilibria How do you find the pair ( x e , u e ) for LTI systems? 0 = Ax e + Bu e , [ x e = Ax e + Bu e ] . (4) In continuous time, if A is invertible than x e = A − 1 Bu e and in discrete time the matrix ( I − A ) is invertible then x e = ( I − A ) − 1 Bu e , otherwise... 3 / 4

Equilibrium pair Linearization Linearization It is possible to linearize the non linear differential [difference] equation defining the state and input variations such as x = x − x e , ˜ ˜ u = u − u e , and the dynamics can be rewritten as x + ] = ∂f ( x, u ) � + ∂f ( x, u ) � ˙ � � x [˜ ˜ x ˜ u ˜ + R ( x, u ) , (5) � � ∂x ∂u � � ( x,u )=( x e ,u e ) ( x,u )=( x e ,u e ) both in continuous and discrete time, then ˙ x + ] ≅ A ˜ x [˜ ˜ x + B ˜ (6) u, A � ∂f ( x, u ) � B � ∂f ( x, u ) � � � , . (7) � � ∂x ∂u � � ( x,u )=( x e ,u e ) ( x,u )=( x e ,u e ) 4 / 4

Equilibrium pair Linearization Topics discussed at the blackboard Finding equilibrium points x e of continuous and discrete time nonlinear systems Properties of norms Stability and attractivity definition. 5 / 4