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
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,
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.
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 n-by-n 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 non-zero 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 non-zero x0 , x there exist T > 0 and a bounded function
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?
A nonlinear ODE control model with control input u and controlled output y is deﬁned
= x2 + x2 ,
(1 − 2x3 )u + a sin(x1 ) − x2 + x3 − x2 ,
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
(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. ...
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- Dynamics, Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, DYNAMICS OF NONLINEAR SYSTEMS, unimodal limit cycle