Instructions

1 due in Moodle, Monday Nov. 25, 4pm,

2 Signed School Cover Sheet attached

3 TYPED only - not handwritten.

4 Follow the course homework guide.

5 Computer output: no discussion ⇒ no marks.

6 Analytical results: no working ⇒ no marks.

7 ♦ icon ⇒ you can use Matlab; else not.

8 Consult only lectures & books−Slotine,Khalil; No Discussion except with instructor

9 No use of Large Language Models

Q1 (17) Optimal Control & NCF .

The following system is a simplified model of a chem-ical reaction where x is a deviation of a resultant concentration from an operating point and the con-trol signal u is a deviation of a reagent concentration from an operating point.

x˙ = ax2 + bu.

Also a > 0, b > 0. Consider optimal control of this system with the cost function

L(x, u) = x 2 + Ru2

(a) Find the steady state optimal control law

(b) Find the closed loop system.

(c) Investigate the closed loop stability.

(d) Also carry out a NCF design. And then calculate the steady state value function

Vc(z) = lim T→∞ Z 0 T L(x(t), u(t))dt with x(0) = z.

(e) ♦Compare the value functions of the optimal control versus the NCF control; both analyti-cally and in plots (you may set a = b = R = 1 for this part of the question).

Q2 (16) Recursive Lyapunov Design .

Consider the NL system

                     x˙ 1 = x2 + a 3 + (x1 − a) 3

x˙ 2 = x1 + u

where a is a known constant.

(a) Design a RLD based control law and prove closed loop stability.

(b) Find the steady state response of the closed loop system to a unit step change in the reference signal.

Q3 (17) Gain Scheduling .

A simplified model of low frequency ship motion is

τ (v)θ ¨+ (1 +  |θ ˙ |)θ ˙ = b(v)u

where θ is the heading angle of the ship; τ (v) is a time constant; b(v) is the open loop gain; u is the control signal which is the rudder angle;  |θ ˙ | is a fluid resistance term. And  << 1 is a known coefficient. Also τ, b depend on the ship speed v according to

τ = τ (v) = α0v and b = b(v) = β0v

where α0, β0 are known constants.

(i) Express the equations in state space form.

(ii) Using linearization investigate the stability of the system at an operating point.

(iii) Using linearization develop a state feedback integral controller, valid near the operating point, to track a reference heading r(t),

(iv) Extend the design to a gain scheduled design by showing how the control gains vary with the speed. Comment on any limitations of the gain-scheduled control law.