[Solved] ESC794-Homework 4

1 Consider a discrete-time linear system with two states and one control:

x⁺ = Ax + Bu

Take X to be the unit box centered at the origin and U = [1,1].

1. Describe (in words) the geometry of the viable subset of X.
2. Write a Matlab function that receives A and B and plots the boundaries of the viable set.
3. Write a Matlab function that receives A and B and a horizon N and plots the boundaries of the feasible set X_N corresponding to the terminal set X₀ given by a circle centered at the origin with radius 0.5.

2

Consider the system

x⁺ = Ax + Bu

with A=[1 0 1;0 0 -1;1 2 1];B=[2 0;-1 0;0 1]. For Q = I₃, R = I₂ and Q_f = 10I₃, solve the finite-horizon LQR problem using the Riccati backward recursion. Write code to simulate the control system for any desired horizon. Choose a convenient horizon and plot the resulting trajectories as a function of time and in a 3D phase plot.

3

Work out every step of the proof of Theorem 4.3 in the textbook by Grune and Pannek, finding a justification for each step taken for the case = 0. Then repeat the proof for arbitrary 0 assuming that asymptotic controllability holds with the small control property. Be prepared to discuss your reasoning during class on 11/01.

• Solve Prob. 3 of Chapter 3 in Grune and Pannek.
• Solve Prob. 4 of Chapter 4 in Grune and Pannek.