Induction Motor Control Design
Nonlinear and Adaptive Control Design for Induction Motors is a unified exposition of the most important steps and concerns in the design of estimation and control algorithms for induction motors. A single notation and modern nonlinear control terminology
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Lyapunov Stability
Consider the ordinary differential equation x˙ = f (x,t) ,
x(t0 ) = x0 , x ∈ Rn
(A.1)
with f ∈ C0 (Rn × R+ , Rn ). It is called autonomous when f does not depend on t, i.e. x˙ = f (x) ,
x(0) = x0
(A.2)
and nonautonomous otherwise. It is called linear if f (x,t) = A(t)x, with A(t) an n×n time-varying matrix, and nonlinear otherwise. A solution of (A.1) over an interval [t0 ,t0 + T ] is denoted by ϕ (t0 + t,t0 , x0 ) ∈ C1 (R+ × R+ × Rn , Rn ), and satisfies (i) ϕ (t0 ,t0 , x0 ) = x0 , dϕ (t0 + t,t0 , x0 ) (ii) = f (ϕ (t0 + t,t0 , x0 ),t), ∀t ∈ [0, T ]. dt A solution of (A.2) over an interval [0, T ] is denoted by ϕ (t, x0 ) with x0 = x(0). Theorem A.1. (Local Existence and Uniqueness) Suppose that there exist positive reals τ , ατ , and βτ such that (i) f (x1 ,t) − f (x2 ,t) ≤ ατ x1 − x2 , ∀x1 , x2 ∈ Br , ∀t ∈ [t0 , τ ], (ii) f (x0 ,t) ≤ βτ , ∀t ∈ [t0 , τ ], with Br (x0 ) = {x ∈ Rn : x − x0 ≤ r}; then (A.1) has only one solution over [t0 ,t0 + T ] for a sufficiently small positive real T such that 0 < T ≤ τ . Theorem A.2. (Global Existence and Uniqueness) Suppose that for each τ ∈ [t0 , ∞) there exist positive reals ατ and βτ such that: (i) f (x1 ,t) − f (x2 ,t) ≤ ατ x1 − x2 , ∀x1 , x2 ∈ Rn , ∀t ∈ [t0 , τ ], (ii) f (x0 ,t) ≤ βτ , ∀t ∈ [t0 , τ ]; then (A.1) has only one solution over [t0 , ∞).
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A Lyapunov Stability
A point xe is called an equilibrium point of (A.1) if f (xe ,t) = 0, ∀t ≥ 0. An equilibrium point is said to be stable if for any ε > 0 and any t0 ∈ R+ there exists δ (t0 , ε ) > 0 such that ϕ (t + t0 ,t0 , x0 ) − xe < ε , ∀t ≥ 0, for every x0 satisfying
x0 − xe < δ (t0 , ε ). If δ can be chosen independently of t0 the equilibrium point xe is said to be uniformly stable. An equilibrium point xe which is not stable is said to be unstable. An equilibrium point xe is said to be attractive if there exists γ (t0 ) > 0 such that lim ϕ (t + t0 ,t0 , x0 ) − xe = 0
(A.3)
t→∞
for every x0 satisfying x0 − xe < γ (t0 ). If γ can be chosen independently of t0 and (A.3) holds uniformly in t0 and x0 , the equilibrium point is said to be uniformly attractive. Let xe be a uniformly attractive equilibrium point: its domain of attraction D(xe ) is defined as D(xe ) = {x ∈ Rn : lim ϕ (t, 0, x) − xe = 0} . t→∞
An equilibrium point is said to be (uniformly) asymptotically stable if it is both (uniformly) stable and (uniformly) attractive. The equilibrium point is said to be exponentially stable if there exist positive constants c, α and r such that
ϕ (t + t0 ,t0 , x0 ) − xe ≤ c x0 − xe e−α (t−t0 ) ,
∀t ≥ t0
(A.4)
for every x0 such that x0 − xe < r; it is called globally exponentially stable if (A.4) holds for every x0 ∈ Rn . The constant α is called the rate of convergence while 1/α is called the time constant. Exponential stability implies uniform asymptotic stability. In the case of linear systems the two properties are equivalent. Theorem A.3. If the origin is a uniformly asymptotically stable equilibrium point for the linear time-varying system x˙ = A(t)x ,
x ∈ Rn ,
then
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