Solution to problem 2.1-8(e)
MATLAB program:
%ECE457 Homework #1 MATLAB Problem 2.1-8 (e)
%This program generates and plots the signal y(t)=10*sin(5*t).*cos(10*t)
%and also evaluates its signal power.
clear all;
clc;
%y(t) can be written into the sum of t
Nonlinear Systems and Control Lecture # 31 Stabilization Output Feedback
p. 1/1
In general, output feedback stabilization requires the use of observers. In this lecture we deal with three simple cases where an observer is not needed Minimum Phase Relativ
Nonlinear Systems and Control Lecture # 29 Stabilization Passivity-Based Control
p. 1/?
x = f (x, u),
y = h(x)
f (0, 0) = 0 x Theorem 14.4: If the system is
(1) (2)
u yV =
T
V
f (x, u)
passive with a radially unbounded positive definite storage functio
Nonlinear Systems and Control Lecture # 28 Stabilization Backstepping
p. 1/?
= f () + g() = u, Rn , , u R
Stabilize the origin using state feedback View as "virtual" control input to
= f () + g()
Suppose there is = () that stabilizes the origin of
=
Nonlinear Systems and Control Lecture # 27 Stabilization Partial Feedback Linearization
p. 1/1
Consider the nonlinear system
x = f (x) + G(x)u [f (0) = 0]
Suppose there is a change of variables
z= = T (x) = T1 (x) T2 (x)
defined for all x D Rn , that tra
Nonlinear Systems and Control Lecture # 25 Stabilization Feedback Lineaization
p. 1/?
Consider the nonlinear system
x = f (x) + G(x)u f (0) = 0, x Rn , u Rm
Suppose there is a change of variables z = T (x), dened for all x D Rn , that transforms the syst
Nonlinear Systems and Control Lecture # 25 Stabilization Basic Concepts & Linearization
p. 1/?
We want to stabilize the system
x = f (x, u)
at the equilibrium point x = xss Steady-State Problem: Find steady-state control uss s.t.
0 = f (xss , uss ) x =
Nonlinear Systems and Control Lecture # 21 L2 Gain & The Small-Gain theorem
p. 1/1
Theorem 5.4: Consider the linear time-invariant system
x = Ax + Bu, y = Cx + Du
where A is Hurwitz. Let G(s) = C(sI A)1 B + D . Then, the L2 gain of the system is supR G(j
Nonlinear Systems and Control Lecture # 24 Observer, Output Feedback & Strict Feedback Forms
p. 1/1
Definition: A nonlinear system is in the observer form if
x = Ax + (y, u), y = Cx
where (A, C) is observable Observer:
x = A^ + (y, u) + H(y - C x) ^ x ^
Nonlinear Systems and Control Lecture # 32 Robust Stabilization Sliding Mode Control
p. 1/1
Example
x1 = x2 x2 = h(x) + g(x)u, g(x) g0 > 0
Sliding Manifold (Surface):
s = a1 x1 + x2 = 0 s(t) 0 x1 = -a1 x1 a1 > 0
t
lim x1 (t) = 0
How can we bring the tra
Nonlinear Systems and Control Lecture # 34 Robust Stabilization Lyapunov Redesign & Backstepping
p. 1/?
Lyapunov Redesign (Min-max control) x = f (x) + G(x)[u + (t, x, u)], x Rn , u Rp
Nominal Model:
x = f (x) + G(x)u u = (x)
Stabilizing Control:
V x
[f
Errata List For
Nonlinear Systems Third Edition
Updated on August 12, 2014
Please e-mail error reports to [email protected]
Preface
1. Page xiv, Line 5: Change books to book
Chapter 1
1. Page 10, Line 3: Change Coulombs plus to Coulomb plus
2. Page 24, Secon
Nonlinear Systems and Control Lecture # 36 Tracking Equilibrium-to-Equilibrium Transition
p. 1/1
= f0 (, ) i = i+1 , 1 i - 1 -1 = L h(x) + Lg L h(x) u
f f fb (,) gb (,)
y = 1
Equilibrium point:
0 = f0 ( ) , 0 = i+1 , 1 i - 1 0 = fb ( ) + gb ( ) , , u y
Nonlinear Systems and Control Lecture # 30 Stabilization Control Lyapunov Functions
p. 1/1
x = f (x) + g(x)u,
f (0) = 0, x Rn , u R
Suppose there is a continuous stabilizing state feedback control u = (x) such that the origin of
x = f (x) + g(x)(x)
is
Nonlinear Systems and Control Lecture # 38 Observers Exact Observers
p. 1/1
Observer with Linear Error Dynamics
Observer Form:
x = Ax + (y, u), y = Cx
where (A, C ) is observable, x Rn , u Rm , y Rp From Lecture # 24: An n-dimensional SO system
x = f (x)
Nonlinear Systems and Control Lecture # 41 Integral Control
p. 1/1
x = f (x, u, w ) y = h(x, w ) ym = hm (x, w ) x Rn state, u Rp control input y Rp controlled output, ym Rm measured output w Rl unknown constant parameters and disturbances
Goal:
y (t) r
Nonlinear Systems and Control Lecture # 37 Observers Linearization and Extended Kalman Filter (EKF)
p. 1/1
Linear Observer via Linearization x = f (x, u), 0 = f (xss , uss ), y = h(x) yss = h(xss )
Linearize about the equilibrium point:
x = Ax + Bu , x =
Nonlinear Systems and Control Lecture # 35 Tracking Feedback Linearization & Sliding Mode Control
p. 1/1
SISO relative-degree system:
x = f (x) + g(x)u, f (0) = 0, y = h(x)
h(0) = 0 Lg Lf
-1
Lg Li-1 h(x) = 0, for 1 i - 1, f
h(x) = 0
Normal form:
= f0 (,
Nonlinear Systems and Control Lecture # 23 Controller Form
p. 1/1
Definition: A nonlinear system is in the controller form if
x = Ax + B(x)[u - (x)]
where (A, B) is controllable and (x) is a nonsingular
u = (x) + -1 (x)v x = Ax + Bv
The n-dimensional s
Nonlinear Systems and Control Lecture # 22 Normal Form
p. 1/1
Relative Degree
x = f (x) + g(x)u,
y = h(x)
where f , g , and h are sufficiently smooth in a domain D f : D Rn and g : D Rn are called vector fields on D
y= h x [f (x) + g(x)u] = Lf h(x) + Lg
Nonlinear Systems and Control Lecture # 10 The Invariance Principle
p. 1/1
Example: Pendulum equation with friction
x1 = x2 x2 = a sin x1 bx2 V (x) = a(1 cos x1 ) + 1 2 x2 2
V (x) = ax1 sin x1 + x2 x2 = bx2 2 The origin is stable. V (x) is not negative
Nonlinear Systems and Control
Lecture # 5
Limit Cycles
p. 1/?
Oscillation: A system oscillates when it has a nontrivial
periodic solution
x(t + T ) = x(t), t 0
Linear (Harmonic) Oscillator:
z=
0
0
z1 (t) = r0 cos(t + 0 ),
r0 =
2
z1 (0)
+
2
z2 (0),
z
z2
Nonlinear Systems and Control Lecture # 8 Lyapunov Stability
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Let V (x) be a continuously differentiable function dened in a domain D Rn ; 0 D . The derivative of V along the trajectories of x = f (x) is
n
V (x) =
i=1
V xi
n
xi =
i=1
V xi
fi (x)
Nonlinear Systems and Control Lecture # 9 Lyapunov Stability
p. 1/1
Quadratic Forms
n n
V (x) = xT P x =
i=1 j=1 2
pij xi xj ,
P = PT
2
min (P ) x
xT P x max (P ) x
P 0 (Positive semidenite) if and only if i (P ) 0 i P > 0 (Positive denite) if and only
Nonlinear Systems and Control
Lecture # 6
Bifurcation
p. 1/?
Bifurcation is a change in the equilibrium points or periodic
orbits, or in their stability properties, as a parameter is
varied
Example
x1 = x2
1
x2 = x2
Find the equilibrium points and their
Nonlinear Systems and Control Lecture # 4 Qualitative Behavior Near Equilibrium Points & Multiple Equilibria
p. 1/?
The qualitative behavior of a nonlinear system near an equilibrium point can take one of the patterns we have seen with linear systems. Co
Nonlinear Systems and Control Lecture # 3 Second-Order Systems
p. 1/?
x1 = f1 (x1 , x2 ) = f1 (x) x2 = f2 (x1 , x2 ) = f2 (x)
Let x(t) = (x1 (t), x2 (t) be a solution that starts at initial state x0 = (x10 , x20 ). The locus in the x1 x2 plane of the so
Nonlinear Systems and Control Lecture # 2 Examples of Nonlinear Systems
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Pendulum Equation
l
mg
ml = -mg sin - kl x1 = , x2 =
p. 2/1
x1 = x2 x2 = - g l sin x1 - k m x2
Equilibrium Points:
0 = x2 0 = - g l sin x1 - k m x2
(n, 0) for n = 0, 1, 2,