Selected Topics in Engineering Science Model Predictive Control
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].
a. Describe (in words) the geometry of the viable subset of X.
b. Write a Matlab function that receives A and B and plots the boundaries of the viable set.
c. Write a Matlab function that receives A and B and a horizon N and plots the boundaries of the feasible set XN 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 Qf = 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 Gru¨ne and Pannek, finding a justification for each step taken for the case λ = 0. Then repeat the proof for arbitrary λ ≥ 0 assuming that asymptotic controllability holds with the small control property. Be prepared to discuss your reasoning during class on 11/01.
4 Solve Prob. 3 of Chapter 3 in Gru¨ne and Pannek.
5 Solve Prob. 4 of Chapter 4 in Gru¨ne and Pannek.
