Description
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 Gr¨une 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 Gr¨une and Pannek.
5 Solve Prob. 4 of Chapter 4 in Gr¨une and Pannek.
