ME 597: Model Predictive Control Homework 2 Due October 6, 2023 (d) To gain more insight into this problem, add a slack variable, s, to the output constraint such that y(k) 0.5 + s. Constrain the slack variable to be non-negative and add a large weighting on s in the cost function to heavily penalize non-zero values. Play with different values of N to see how the output constraint violation and overall closed-loop behavior changes as a function of the prediction horizon. Provide the same three-subplot figure from the steps above showing the state, output, and input trajectories for a value of N of your choosing. How does increasing the horizon affect the extent of constraint violations (as softened by the slack variable)? How does increasing the horizon affect the rate of convergence to the origin of the closed-loop system? 2. In this problem, you will continue exploring the stability of closed-loop MPC but with a particu- lar focus on determining the set of initial conditions for which the constrained MPC formulation admits a feasible solution, also known as the region of attraction. The goal is to use MPC to regulate a system to the origin with discrete-time state-space matrices 2 A = B = " The states and inputs are constrained such that xk X and uk EU where X = {x = R | [10]x 5}, U = {u R | 1 u 1}. Note that 1 = R denotes a vector of ones. The controller prediction horizon is N = 3 steps and the quadratic cost function penalizes the states and inputs with matrices Q = al and R = I, where a is a scalar used to adjust the relative weighting of the two matrices. Assume there is no terminal cost (P = 0). (a) First, implement the unconstrained case (no state or input constraints) for a few different values of a. Use an initial condition of x(0) = [1 0.5] and a 10 step simulation. Determine the effect of a on the closed-loop system and find a value of a such that the closed-loop system is unstable. Provide example figures of a stable closed-loop trajectory and an unstable closed-loop trajectory along with their corresponding values of a. For each figure, include two subplots where the top subplot shows the state trajectories over time and the bottom subplot shows the input trajectories over time. Be sure to include legends to label the state and input trajectories. (b) Using the unstable value of a, now add the state and input constraints. Apply the input constraint to every input in the prediction horizon. Similar to the previous problem, apply the state constraint starting at the second step of the horizon (i.e., excluding the initial state) up to and including the terminal state. Does the resulting closed-loop system appear to be stable or unstable? Also note the feasibility of the optimization problem at each time step. (c) Implement the constrained MPC formulation with the additional terminal constraint of xN = 0. Provide a figure of the closed-loop state and input trajectories. (d) Finally, find the region of attraction for the constrained MPC formulation from the previous step using both the batch and recursive algorithms discussed in class. You can use MPT's Page 2 of 3