ME559(EE587) Nonlinear Control and Stability Chapter 01: Introduction to Nonlinear Dynamical Systems The topic of this course is analysis and control of nonlinear dynamical systems. The term “nonlinear” is interpreted in primarily two ways, namely, “not linear” and “not necessarily linear.” The latter meaning is intended here. There are several important differences between linear systems and nonlinear systems. The main difference is that a linear system is governed by the principle of superposition, i.e., L(cx + y)= cL(x)+ L(y), where L is a linear operator; the scalar c belongs to field of the vector space V ; and x, y ∈ V ; there are other differences that accrue from inapplicability of the principle of superposition. For example, a closed-form solution always exists for a set of finite-dimensional linear time-invariant (FDLTI) differential equations; however, this may not be true for a set of nonliear ordinary differential equations. Therefore, it is desirable to obtain a model that yields an approximate solution of the governing equations of a nonlinear dynamical system for the purpose of stability analysis and control. 1 Rudimentary Concepts The general structure for a nonlinear dynamical system in the continuous-time setting is represented as: ˙ x(t)= f [t, x(t),u(t)] ∀t ≥ 0 (1) where t ∈ R + [0, ∞) denotes time; x(t) ∈ R n , where n ∈ N {1, 2, 3, ···}, denotes the state at time t; the input at time t is denoted by u(t) ∈ R m with m ∈ N and m ≤ n. Thus, f : R + × R n × R m → R n . The discrete-time representation of Eq. (1) is a nonlinear map: x k+1 = ϕ k [x k ,u k ] ∀k (2) where k ∈ N 0 {0, 1, 2, 3, ···} denotes instants of discrete time; x k ∈ R n denotes the state at instant k; the input u k ∈ R m , where m ∈ N and m ≤ n, is the input at instant k. Thus ϕ k : R n × R m → R n for all k ∈ N 0 . Next we introduce a few definitions for commonly used terms. Definition 1.1. A dynamical system in Eq. (1) (resp. Eq. (2)) is called forced if the input u(t) serves as the forcing function. The system is called unforced if u(t)=0 ∀t ≥ 0 (resp. u k =0 ∀k ≥ 0). 1
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ME559(EE587) Nonlinear Control and Stability
Chapter 01: Introduction to Nonlinear Dynamical Systems
The topic of this course is analysis and control of nonlinear dynamical systems.
The term “nonlinear” is interpreted in primarily two ways, namely, “not linear” and
“not necessarily linear.” The latter meaning is intended here. There are several
important differences between linear systems and nonlinear systems. The main
difference is that a linear system is governed by the principle of superposition, i.e.,
L(cx + y) = cL(x) + L(y), where L is a linear operator; the scalar c belongs to
field of the vector space V ; and x, y ∈ V ; there are other differences that accrue
from inapplicability of the principle of superposition. For example, a closed-form
solution always exists for a set of finite-dimensional linear time-invariant (FDLTI)
differential equations; however, this may not be true for a set of nonliear ordinary
differential equations. Therefore, it is desirable to obtain a model that yields an
approximate solution of the governing equations of a nonlinear dynamical system
for the purpose of stability analysis and control.
1 Rudimentary Concepts
The general structure for a nonlinear dynamical system in the continuous-time
setting is represented as:
x(t) = f [t, x(t), u(t)] ∀t ≥ 0 (1)
where t ∈ R+ , [0,∞) denotes time; x(t) ∈ Rn, where n ∈ N , 1, 2, 3, · · · ,
denotes the state at time t; the input at time t is denoted by u(t) ∈ Rm with m ∈ N
and m ≤ n. Thus, f : R+ × Rn × R
m → Rn.
The discrete-time representation of Eq. (1) is a nonlinear map:
xk+1 = ϕk[xk, uk] ∀k (2)
where k ∈ N0 , 0, 1, 2, 3, · · · denotes instants of discrete time; xk ∈ Rn denotes
the state at instant k; the input uk ∈ Rm, where m ∈ N and m ≤ n, is the input
at instant k. Thus ϕk : Rn × Rm → R
n for all k ∈ N0. Next we introduce a few
definitions for commonly used terms.
Definition 1.1. A dynamical system in Eq. (1) (resp. Eq. (2)) is called forced
if the input u(t) serves as the forcing function. The system is called unforced if
u(t) = 0 ∀t ≥ 0 (resp. uk = 0 ∀k ≥ 0).
1
Definition 1.2. A dynamical system in Eq. (1) (resp. Eq. (2)) is called autonomous
if the function f does not explicitly depend on the time parameter t (resp. the
function ϕ does not explicitly depend on the time index k); otherwise the dynamical
system is called non-autonomous.
Definition 1.3. In an unforced dynamical system in Eq. (1) with u(t) = 0 ∀t ≥ 0
(resp. Eq. (2) with uk = 0 ∀k ≥ 0), a state xe at a time τ ≥ 0 (resp. xe at an