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

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Equilibrium pair Linearization

Equilibrium pair

Consider a dynamical system ˙ x = f(x, u),

• x+ = f(x, u)
• ,

(1) and the equilibrium pair (xe, ue), if it exists, is such that 0 = f(xe, ue), [xe = f(xe, ue)] . (2) This implies that the solution of the differential [difference] equation ϕ(t, t0, x0, u(·)) [ϕ(k, k0, x0, u(·))] is such that ϕ(t, t0, xe, ue) = xe, [ϕ(k, k0, xe, ue) = xe] (3) for all t ≥ t0 [k ≥ k0].

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Equilibrium pair Linearization

Equilibria

How do you find the pair (xe, ue) for LTI systems? 0 = Axe + Bue, [xe = Axe + Bue]. (4) In continuous time, if A is invertible than xe = A−1Bue and in discrete time the matrix (I − A) is invertible then xe = (I − A)−1Bue,

• therwise...

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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 − xe, ˜ u = u − ue, and the dynamics can be rewritten as ˙ ˜ x [˜ x+] = ∂f(x, u) ∂x

• (x,u)=(xe,ue)

˜ x + ∂f(x, u) ∂u

• (x,u)=(xe,ue)

˜ u + R(x, u), (5) both in continuous and discrete time, then ˙ ˜ x [˜ x+] ≅ A˜ x + B˜ u, (6) A ∂f(x, u) ∂x

• (x,u)=(xe,ue)

, B ∂f(x, u) ∂u

• (x,u)=(xe,ue)

. (7)

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Equilibrium pair Linearization

Topics discussed at the blackboard

Finding equilibrium points xe of continuous and discrete time nonlinear systems Properties of norms Stability and attractivity definition.

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