Unformatted text preview: Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS
by A. Megretski Problem Set 71
Problem 7.1
A stable linear system with a relay feedback excitation is modeled by
x(t) = Ax(t) + B sgn(Cx(t)),
˙ (7.1) where A is a Hurwitz matrix, B is a column matrix, C is a row matrix, and sgn(y ) denotes
the sign nonlinearity
�
y > 0, 1,
0,
y = 0,
sgn(y ) =
�
−1, y < 0.
For T > 0, a 2T periodic solution x = x(t) of (7.1) is called a regular unimodal limit cycle
if C x(t) = −Cx(t + T ) > 0 for all t → (0, T ), and C Ax(0) > CB . (a) Derive a necessary and suﬃcient condition of exponential local stability of the reg
ular unimodal limit cycle (assuming it exists and A, B , C, T are given).
(b) Use the result from (a) to ﬁnd an example of system (7.1) with a Hurwitz matrix
A and an unstable regular unimodal limit cycle.
Problem 7.2
A linear system controlled by modulation of its coeﬃcients is modeled by
x(t) = (A + B u(t))x(t),
˙
where A, B are ﬁxed nbyn matrices, and u(t) → R is a scalar control.
1 Posted November 5, 2003. Due date November 12, 2003 (7.2) 2
(a) Is it possible for the system to be controllable over the set of all nonzero vectors
x → Rn , x = 0, when n � 3? In other words, is it possible to ﬁnd matrices A, B with
¯
¯ ∞ n > 2 such that for every nonzero x0 , x there exist T > 0 and a bounded function
¯ ¯1
u : [0, T ] ≥� R such that the solution of (7.2) with x(0) = x0 satisﬁes x(T ) = x1 ?
¯
¯
(b) Is it possible for the system to be full state feedback linearizable in a neigborhood
of some point x0 → Rn for some n > 2?
¯
Problem 7.3
A nonlinear ODE control model with control input u and controlled output y is deﬁned
by equations
x1
˙
x2
˙
x3
˙
y =
=
=
= x2 + x2 ,
3
(1 − 2x3 )u + a sin(x1 ) − x2 + x3 − x2 ,
3
u,
x1 , where a is a real parameter.
(a) Output feedback linearize the system over a largest subset X0 of R3 .
(b) Design a (dynamical) feedback controller with inputs x(t), r(t), where r = r(t) is
the reference input, such that for every bounded r = r(t) the system state x(t) stays
bounded as t � �, and y (t) � r(t) as t � � whenever r = r(t) is constant.
(c) Find all values of a → R for which the open loop system is full state feedback
linearizable.
(d) Try to design a dynamical feedback controller with inputs y (t), r(t) which achieves
the objectives from (b). Test your design by a computer simulation. ...
View
Full
Document
 Spring '09
 AlexandreMegretski
 Dynamics, Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, DYNAMICS OF NONLINEAR SYSTEMS, unimodal limit cycle

Click to edit the document details