ECE 509 Analysis of Linear Systems Project 1 Due: April 12, 2015 by 11:55 p.m. The following three problems are to be used for practice, and are not to be turned in. While you can use MATLAB to verify your results, be sure to work out all the control design details by hand. Solutions to these problems are available on Isidore for your reference. 1. Problem 8.4 2. Problems 8.5, 8.6 The following problems constitute the actual project. You are expected to turn in a typewritten report with your results. The report should be concise and to the point. 1. Problem 8.7, but instead of the system given there use this one:

= 1 0 12 4 31 5 5 + 111 = 2 0 0 First, verify whether this system is controllable and observable. Here, set the reference

r to be a train of steps alternating between 4 and 4 with a period of your choice. Make sure the period is chosen long enough that steady-state can take place. What happens as you decrease the period of your reference? Simulate the system with your feedback controller, and plot the reference and the output in the same plot to demonstrate tracking. You may use the MATLAB command place to come up with the state feedback gain.

2. Consider the inverted pendulum on a cart. You will design a state-feedback controller for

this system that regulates all states to zero.

a. Open-loop NONLINEAR simulation In order to help you, I have uploaded MATLAB code to Isidore, which uses the original nonlinear equations for the inverted pendulum on a cart. Use this code to simulate the pendulum on a cart in open-loop (that is, without control - note that the actuating force is set to F=0), with the given initial conditions. Plot the four states versus time. You do not need to include a movie in your report this is just there for your benefit.

b. Control of the pendulum Using the linearized equations of the system, design a state-feedback controller that balances the pendulum and regulates the carts position to zero. You may want to try the command place in MATLAB to speed your design. Make sure you use the numerical parameters in the file pend_ol.m to derive the linearized equations.

Apply your controller to the original nonlinear equations (not the linearized ones!) and simulate it for the initial conditions given in the code. Plot the four states versus time. Note that the control signal is F.

Assume that the control hardware only allows up to 4 N of force to the cart. This is the variable Fmax in the code. Use the provided function sat to saturate your control signal F, in such a manner that it does not exceed 4N of force. Make sure you select your closed loop poles carefully, in order for your control signal not to go beyond the limit long enough that instability will ensue. Hints: 1. First design your control system without the saturation, to make sure you are

able to balance the pendulum. 2. Then apply the saturation function, and modify your pole selection as

necessary to maintain closed-loop stability. 3. The code provided shows how to obtain the control signal and plot it in the

main file. 4. If you cannot make your controller work with the saturation, you will still get

points for it, minus a 10% deduction. Report guidelines:

In your report, include your code only for Problem 2, part b. But be sure to upload all your project code to Isidore.

A short and concise report is best four or five pages at most, not counting code. All you need to do is show me your design steps, and prove that your controller works by including the requested plots. No movies are necessary.

Handwritten equations are not allowed. Use Latex or Word to produce your report. All figures must be carefully labeled. The x and y axes should be labeled, a title used,

and a legend should be included when more than one signal is plotted in the same figure.

Presentation, English, and professional appearance count 5% of the total grade. A poorly presented report will automatically be deducted 5%.

Submit your report in electronic form via Isidore! No hard copies will be accepted. All reports must be in PDF format. Any other format will not be accepted.