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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 81
Problem 8.1
Autonomous system equations have the form
� � � y (t )
y (t)
y (t) =
¨
Q
,
y t)
˙(
y t)
˙( (8.1) where y is the scalar output, and Q = Q� is a given symmetric 2by2 matrix with real
coeﬃcients.
(a) Find all Q for which there exists a C � bijection � : R2 ≥� R2 , matrices A, C , and
a C � function � : R ≥� R2 such that z = � (y, y satisﬁes the ODE
˙)
z t) = Az (t) + �(y (t)), y (t) = C z (t) ˙(
whenever y (·) satisﬁes (8.1). (b) For those Q found in (a), construct C � functions F = FQ : R2 × R ≥� R2
and H = HQ : R2 ≥� R such that HQ (� (t)) − y t) � 0 as t � → whenever
˙(
y : [0, →) ≥� R is a solution of (8.1), and � t) = FQ (� (t), y (t)). ˙(
1 Posted November 19, 2003. Due date November 26, 2003 2
Problem 8.2
A linear constrol system
� x1 (t) =
˙
x2 (t) + w1 (t),
x2 (t) = −x1 (t) − x2 (t) + u + w2 (t)
˙ is equipped with the nonlinear sensor
y (t) = x1 (t) + sin(x2 (t)) + w3 (t),
where wi (·) represent plant disturbances and sensor noise satisfying a uniform bound
wi (t) � d. Design an observer of the form
� t) = F (� (t), y (t), u(t))
˙(
and constants d0 > 0 and C > 0 such that
� (t) − x(t) � C d � t ∀ 0
whenever � (0) = x(0) and d < d0 . (Try to make d0 as large as possible, and C as small
as possible.)
Problem 8.3
Is it true or false that the set � = �F = {P } of positive deﬁnite quadratic forms VP (x) =
¯
�
�
x P x, where P = P > 0, which are valid control Lyapunov function for a given ODE
¯¯
model
x(t) = F (x(t), u(t)),
˙
in the sense that
x¯
inf x� P F (¯, u) � −x2 � x ≤ Rn ,
¯
¯
uR
¯
is linearly connected for all continuously diﬀerentiable functions F : Rn × R ≥� Rn ?
(Remember that a set � of matrices is called linearly connected if for every two matrices
P0 , P1 ≤ � there exists a continuous function p : [0, 1] ≥� � such that p(0) = P0 and
p(1) = P1 . In particular, the empty set is linearly connected.) ...
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 Spring '09
 AlexandreMegretski
 Dynamics

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