ELEC4631 midterm 21
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ELEC4631 midterm 21
Problem 1 (30 marks in total)
This problem must be done by hand without the aid of a computational device like a calculator or computer software like MATLAB
(a) (10 marks) Determine whether the 3 x 3 matrix
┌ ┐
' '
can be diagonalised or not.
(b) (20 marks) Consider the quadratic form f : R3 → R given by
f (x) = 2x1(2) - 8x1 x2 + 2x2(2) + 6x2 x3 + 2x3(2) ,
where x = (x1 , x2 , x3 )T . Determine an orthogonal matrix V and real constants d1 , d2 , d3 such that f (Vz) = d1 z 1(2) + d2 z2(2) + d3 z3(2), where z = (z1 , z2 , z3 )T .
Problem 2 (40 marks in total)
(a) (15 marks) Consider a system described by the ODE:
x˙ = -x - 3y5
y˙ = 2x3 - y .
Using the candidate Lyapunov function
V (x, y) = x4 + y6 ,
show that the origin is globally asymptotically stable.
(b) Consider a control system described by the nonlinear controlled ODE:
x˙ = -y(x - y) + u
y˙ = -y(sin x + x - y) + u,
where u = η(x, y) is a scalar state-feedback control signal. Answer the following questions:
(i) (20 marks) Using the candidate Lyapunov function V (x, y) = 1 - cos x + (x - y)2 ,
show that you can design u such that it renders the origin asymptotically stable and V˙ (x) = -κ sin2 x for some constant κ > 0.
(ii) (5 marks) Using the same candidate Lyapunov function as in (i), is it possible to establish that the same control law u renders the origin globally asymptotically stable? Explain your answer.
Problem 3 (30 marks in total)
An autonomous linear time-invariant system (with no driving input) is described by the differential equation
y¨ + 3y˙ + 2y = 0,
with initial conditions y(0) = 1 and y˙(0) = -2. Using the observer canonical state- space representation for the system (with zero input), solve the state-space equation to find the solution of the differential equation satisfying the given initial conditions for all t 2 0.
2022-07-07