EE 5321 Optimal Control Systems

EE 5321 Optimal Control Systems

Homework for Spring 2009

Updated: Wednesday, March 25, 2009

• Some homework assignments refer to Lewis and Syrmos, Optimal Control,Wiley, 1995.
• For full credit, show all work.
• Some problems require hand calculations. In those cases, do not use MATLAB except to check your answers.

It is OK to talk about the homework beforehand.

BUT, once you start writing the answers, MAKE SURE YOU WORK ALONE.

The purpose of the Homework is to evaluate you individually, not to evaluate a team.

Cheating on the homework will be severely punished.

EE 5321 Homework 1

Static Optimization

Do these problems from Lewis & Syrmos, Optimal Control:

1.2-1

1.2-2

1.2-6

1.2-7, Kalman Gain

EE 5321 Homework 2

DT Optimal Control and LQR

Do these problems from Lewis & Syrmos, Optimal Control:

2.1-1

2.2-1

2.2-2

2.2-3

EE 5321 Homework 3

DT Optimal Control and LQR

1. DT Optimal LQR for Newton’s System.

Newton’s law is given by

and state with y(t) the position and u(t)=F/m= force per unit mass input.

Discretize the system using MATLAB function c2d. Use sample period of T=0.5 sec. Select weighting matrices Q= I, R= 1, SN= 100*I, with I= identity matrix.

a. solve the DT Riccati difference equation to find the optimal Riccati solution. Use N= 10 samples, so that final time is 5 sec. Plot the diagonal elements s11 and s22 vs. time

b. compute the optimal Kalman gain sequence Kk.

c. Simulate the DT system using MATLAB using a ‘for’ loop. Use initial conditions y(0)=10, v(0)=10. Plot the states and control input vs. time. You should get a plot of points defining the three graphs.

d. Simulate the continuous-time dynamics with the DT optimal control LQR, which updates the control every sample period. Use initial conditions y(0)=10, v(0)=10. You will need to use ode23 between the sample points to integrate the CT dynamics, then update the control uk, then use ode23 again. Plot the states and control input vs. time. You will get a figure like Fig. 2.3-11 in the text on p. 78.

2. DT Steady-State LQR for Newton’s System.

a. For the previous situation, find the steady-state DT ARE solution by iterating the Riccati difference equation until convergence.

b. find the steady-state DT ARE solution by using MATLAB function dare.

c. Find the SS Kalman gain.

d. Simulate the continuous-time dynamics with the DT steady-state LQR gain, which updates the control every sample period. Use initial conditions y(0)=10, v(0)=10. Plot the states and control input vs. time. Compare to Problem 1d.

3. Inverted Pendulum Controller Design

The inverted pendulum is given by

with state and

0. Find the open-loop poles in s-plane.

Discretize the system using MATLAB function c2d. Use sample period of T=0.1 sec.

a. Find the open-loop DT poles. Find damping ratio and natural frequencies. Use MATLAB function ddamp.

b. Select weighting matrices R= 1, Q= diag{1, 1, 50, 50}. Find SS DT ARE solution and Kalman gain using MATLAB dare.

c. Find closed-loop poles of DT system. Find closed-loop damping ratio and natural frequencies.

d. Simulate the continuous-time dynamics with the DT steady-state LQR gain, which updates the control every sample period. Simulate for 15 sec with initial condition . Plot the states vs. time on one plot, and the control vs time on another.

3. Inverted Pendulum Controller Design with Fast Response

Repeat the previous problem with R= 0.01. All else the same.

EE 5321 Homework 4

CT Optimal Control Basics

Do these problems from Lewis & Syrmos, Optimal Control:

3.2-1

3.2-2

3.2-3

3.2-5

EE 5321 Homework 5

Dynamic Programming

Do these problems from Lewis & Syrmos, Optimal Control:

6.2-1

6.2-4

6.2-5