ME/ECE859-Spring 2014 Homework 3, Due date: 2/10/2014
Monday
0. Re-do the proof of the Lyapunovs theorem in Lecture 8.
1. (Gradient systems) A dynamical system x = F (x), where F (x) = gradV (x) =
(minus gradient of a smooth function V ), is called a gra
ME/ECE859-Spring 2008 Homework 6, Due date: 3/10/08 Monday
1. Re-derive the result of Example 7.1 of the textbook by adding more details. For instance, (a) show the L2 gain of the system (or H norm of G(j) is G
= sup max [G(j)] = sup G(j) 2 .
R R
ME/ECE859-Spring 2008 Homework 4, Due date: 2/11/08 Monday
1. Exercise 4.17 of the textbook. 2. Exercise 4.23 of the textbook. When Q = C T C, the control u(t) = -R-1 B T P x(t) can be considered as the stationary Linear Quadratic (LQ) optimal state
ME/ECE859-Spring 2008 Homework 8 Due date: 3/24/08 Monday
1. Consider the system x1 = x1 + x2 , 1 + x2 1 x2 = -x2 + x3 , x3 = x1 x2 + u
(a) Use Theorem 13.2 to prove that the system is transformable into the controller form. (b) Find the change o
ME/ECE859-Spring 2008 Homework 7, Due date: 3/17/08 Monday
1. Exercise 5.5 of the textbook. 2. Exercise 5.11-(1), (2) of the textbook. 3. Exercise 5.16 of the textbook (Hint: Use V (x) = 4. Exercise 5.18 of the textbook.
x1 0
(y)dy + 1 (x2 + x2 ).
Nonlinear Systems and Control Lecture # 30 Stabilization Control Lyapunov Functions
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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
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 Co
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 st
ME/ECE859-Spring 2008 Homework 9 Due date: 4/14/08 Monday
1. Exercise 14.33 of the textbook. 2. Exercise 14.46 of the textbook. (Hint: Use the Cascade Connection method in Lecture 29, or Theorem 14.5 in the textbook.) 3. Consider the system x1 = x2 ,
ME/ECE859-Spring 2008 Homework 2, Due date: 1/30/08 Wed
The spread of infective diseases can be modeled as a nonlinear system. We introduce here the SIRS model. The population consists of three disjoint groups. The population of susceptible individua
Nonlinear Systems and Control Lecture # 37 Observers Linearization and Extended Kalman Filter (EKF)
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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 +
Nonlinear Systems and Control Lecture # 35 Tracking Feedback Linearization & Sliding Mode Control
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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
Norma
Nonlinear Systems and Control Lecture # 26 Stabilization Feedback Lineaization
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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), defined for all x D Rn , that tran
Nonlinear Systems and Control Lecture # 6 Bifurcation
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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 poi
ME/ECE859-Spring 2014 Homework 2, Due date: 2/3/2014
Monday
1. The spread of infective diseases can be modeled as a nonlinear system. We introduce here the
SIRS model. The population consists of three disjoint groups. The population of susceptible
individ
ME/ECE859-Spring 2014 Homework 5, Due date: 3/17/2010
Monday
1. Exercise 6.10 of the textbook.
2. Exercise 6.14 of the textbook.
3. Exercise 6.17 of the textbook.
1
ME/ECE859-Spring 2014 Homework 4, Due date: 2/19/2014
Wednesday
1. Exercise 4.17 of the textbook.
2. Exercise 4.23 of the textbook. When Q = C T C , the control u(t) = R1 B T P x(t) can be
considered as the stationary Linear Quadratic (LQ) optimal state f
ME/ECE859-Spring 2014 Homework 1
Due date: 1/20/2014 Monday
1. Consider the system
x1 = x1 + 2x2 + x1 x2 + x2
2
x2 = x1 x2 x1 x2
1
(a) Is the right-hand-side function locally Lipschitz? Is it globally Lipschitz?
(b) Find all equilibrium points and determi
ME/ECE 859-Spring 2014 Nonlinear Systems and Control
Instructor: Jongeun Choi; Room 2459 EB; Tel 517-432-3164; E-mail jchoi@egr.msu.edu;
Web http:/www.egr.msu.edu/jchoi/
Class schedule: M W F 9:10-10:00 am, Room 2320 Engineering Building.
Oce Hours: M
EECS 221 A
A Notation B Algebraic Aspects C Normed Vector Spaces D Inner Product Spaces E The Projection Theorem
Vector Spaces
1
A.
! R C Q
Notation
there exists there exists a unique for all eld of real numbers eld of complex numbers eld of rational num
Nonlinear Systems and Control Lecture # 41 Integral Control
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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 # 40 Observers High-Gain Observers Stabilization
p. 1/1
x z y
= = = =
Ax + B(x, z, u) (x, z, u) Cx q (x, z )
u Rp , y Rm , Rs , x R , z R A, B, C are block diagonal matrices 0 1 0 0 1 0 0 0 0 . . . . Ai = . , Bi = .
Nonlinear Systems and Control Lecture # 38 Observers Exact Observers
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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 # 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 z
z1 (t) = r0 cos(t + 0 ), r0 =
2 z1 (0)
Nonlinear Systems and Control Lecture # 2 Examples of Nonlinear Systems
p.1/17
Pendulum Equation
l
mg
ml = -mg sin - kl x1 = , x2 =
p.2/17
x1 = x2 x2 = - g l sin x1 - k m x2
Equilibrium Points:
0 = x2 0 = - g l sin x1 - k m x2
(n,
Nonlinear Systems and Control Lecture # 22 Normal Form
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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] =