RHODES UNIVERSITY
DEPARTMENT OF MATHEMATICS (Pure & Applied)
EXAMINATION : JUNE 2009
APPLIED MATHEMATICS III
Examiners : Dr C.C. Remsing
Dr F.A.M. Frescura
AVAILABLE MARKS : 110
FULL MARKS : 100
DURATION : 3 HOURS
AM3.2 (LINEAR CONTROL)
NB : All questions may be attempted. All steps must be clearly motivated.
Marks will not be awarded if this is not done.
Question 1. [18 marks]
Let A ∈ Rn×n .
(a) Define the transpose matrix A⊤ and then show that the linear map τ : Rn×n → Rn×n , A 7→ A⊤ satisfies the condition
τ (AB) = τ (B) τ (A). Hence deduce that if the matrix A is invertible, then so is its transpose A⊤ , and
−1
⊤
A⊤
= A−1 .
(b) Let φ denote the quadratic form x ∈ Rn 7→ x⊤ Ax. Explain what
is meant by saying that φ is positive definite, and then give two
sets of necessary and sufficient conditions for positive definiteness.
For what values of α is φ(x) = x21 + 2x22 + 3x23 + 2αx1 x3 positive
definite ?
(c) Show that A can be written as A1 + A2 , where A1 is symmetric
and A2 is skew-symmetric. Hence deduce that
φ(x) = x⊤ A1 x
for all x ∈ Rn .
[5,7,6]
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Appl. Maths III
Linear Control
June 2009
Question 2. [22 marks]
(a) Given A ∈ Rm×m , B ∈ Rm×ℓ and x0 ∈ Rm , prove that the
initialized linear control system
ẋ = Ax + Bu
x(0) = x0
has an unique solution curve x(·) : R → Rm .
(b) Consider the linear control system
ẋ1 = ω x2
ẋ2 = −ω x1 + u,
ω ∈ R.
Find the state transition matrix Φ(·, 0).
(c) Determine the solution curve x(·) starting from the origin when
u(·) = 1.
[8,14]
Question 3. [22 marks]
Consider a linear control system (with outputs) Σ given by
ẋ = Ax + Bu
and y = Cx.
(a) Define the terms complete controllability and complete observability (of Σ).
(b) State and prove the Kalman Controllability Criterion.
(c) Consider the control system (with outputs)
1 2
ẋ =
x + b u and y = c x.
0 3
Find b ∈ R2×1 and c ∈ R1×2 such that the system is
i. not completely controllable;
ii. completely observable.
When c = 1 1 , determine x(0) if y(t) = et − 2e3t .
[4,10,8]
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Appl. Maths III
Linear Control
June 2009
Question 4. [16 marks]
(a) Consider a single-input linear control system Σ described by
ẋ = Ax + bu
A ∈ Rm×m , b ∈ Rm×1 .
Given an arbitrary set Λ = {θ1 , . . . , θm } of complex numbers
(appearing in conjugate pairs), prove that if Σ is completely
controllable, then there exists a feedback matrix K ∈ R1×m such
that the eigenvalues of A + bK are the set Λ.
(b) For the control system
ẋ1 = x2
ẋ2 = x3 + u
ẋ3 = x1 − x2 − 2x3
find a linear feedback control which makes all the eigenvalues (of
the system) equal to −1.
[10,6]
Question 5. [20 marks]
Consider a linear dynamical system
ẋ = Ax,
x ∈ Rm .
(a) Give necessary and sufficient conditions for
i. asymptotic stability
ii. neutral stability
iii. instability
of the system (at the origin).
(b) Investigate the stability of the dynamical system
ẋ1 = x2
ẋ2 = −bx1 − ax2 ,
a, b > 0.
(c) Define the term Lyapunov function and then state the First Lyapunov Theorem.
(d) State and prove the Lyapunov Linearization Theorem.
[3,7,3,7]
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Appl. Maths III
Linear Control
June 2009
Question 6. [12 marks]
Consider the following optimal control problem :
Minimize the cost functional
J =
Z
T
0
u2 (t) + x2 (t) dt
subject to
ẋ = −x + u,
x ∈ R, u ∈ R
x(0) = 1
x(T ) = 2.
The final time T > 0 is fixed. Find the optimal control u∗ (·) : [0, T ] → R as
well as (an explicit expression for) the optimal trajectory x∗ (·) : [0, T ] → R.
[12]
END OF THE EXAMINATION PAPER
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