Cleveland State University

Department of Electrical and Computer Engineering

EEC 641 / 741

Multivariable Control

Homework Assignments

Chapter 1

1.      For the inverted pendulum system, we designed a closed loop controller under the assumption that the cart mass was 1 kg. Find out the largest percentage by which the cart mass can increase before the closed loop system becomes unstable (answer to the nearest 1%).

2.      For the inverted pendulum system, we designed a closed loop controller under the assumption that the pendulum mass was 0.2 kg. Suppose that the pendulum track extends from y = -5 to +5 m, so if |y| > 5 then our system fails. Find out the largest percentage by which the pendulum mass can increase before the system fails to meet the performance requirements (answer to the nearest 1%).

3.      The hovering rocket overshoots the target altitude by about 0.7 m. Find out by what percentage the thrust coefficient can increase before the rocket overshoots the target altitude by 5 m (answer to the nearest 0.1%).

4.      HW01.doc

5.      A measure of “how controllable” a system is can be given by the condition number of the controllability matrix. The smaller the condition number, the more controllable the system is. What is the condition number of the controllability matrix for the inverted pendulum problem? What is the condition number if the pendulum is only half as long?

6.      Sometimes when a simulation “blows up” it just means that the simulation step size is too large. Find the largest value of the simulation step size that gives good results for the inverted pendulum simulation (answer to the nearest 0.01 s).

Chapter 2

1.      A series RC circuit can be modeled as u(t) = v(t) + RC v’(t), where u(t) is the input (the applied voltage), v(t) is the output (the voltage across the capacitor), and v’(t) is the derivative of v(t). Assume that RC = 1.

a.       Derive the transfer function of the system.

b.      Find analytical expressions for the gain and phase of the system.

c.       Generate the bode plot (you can use Matlab’s “bode” function). What is the gain and phase of the frequency response at 1 rad/s?

d.      Generate the response of the system to the input u(t) = sin(t) (you can use Matlab’s “lsim” function). Plot the input and output on the same graph. Verify graphically that the output magnitude is equal to the input magnitude multiplied by the gain, and that the output phase is equal to the input phase added to the phase of the frequency response.

2.      You are given a system with transfer function G(s) = 10 / s / (1 + 0.2s) / (1 + 0.02s) controlled with a single DOF controller K = 1.

a.       Use Matlab’s “margin” function to compute the gain margin and phase margin.

b.      Use Matlab’s “nyquist” function to generate the Nyquist plot. Use the Nyquist plot to graphically determine the gain and phase margins.

c.       Compute the gain margin and phase margin bounds given in Section 2.4.3 and compare them to the true gain margin and phase margin. (You can use Matlab’s “norm” function to compute the bounds. Note that you may need to cancel out the common poles and zeros of a transfer function, using Matlab’s “minreal” function, before you can compute its norm.)

d.      Compare omega_b and omega_c, as defined in Section 2.4.5.

3.      Consider the linearized inverted pendulum: G(s) = (s^2 – 9.81) / s / (s – 3.42) / (s + 3.44) / (s + 0.08) with negative unity feedback

a.       What is the gain margin and phase margin of the closed loop system according to Matlab’s “margin” command? Why is this not reliable?

b.      Generate the Nyquist plot for this system. How can you tell from the Nyquist plot that the closed loop system is unstable?

4.      Consider the transfer function G(s) = (s + 1) / (s + 10)^2.

a.       Analytically (using pencil and paper) compute the infinity norm of G(s).

b.      Use Matlab’s “norm” function to compute the infinity norm of G(s).

5.      Consider the linearized hovering rocket model: G(s) = 0.0313 (s + 0.0016) / s / (s - 0.00079) / (s + 0.0024).

a.       Use Matlab’s “mixsyn” function to minimize the weighted infinity norm with M = 1.5, wb = 1, A = 1e-4, wT = 0, and wu = 1. Plot the control and the output response (for 25 seconds) to a step input.

b.      Repeat for wu = 2. How does the control and output compare to part (a)? Give an intuitive explanation for the differences that you observe.

Chapter 3 - due October 5

1.      Derive Equation 3.1.

2.      Consider a system with A = [-1 2; 0 -3], B = [1 0; 0 4], C = [1 1; 1 -1], D = [0 0; 0 0].

a.       Plot the principal gains in dB with a log scale for the frequency axis.

b.      Use the plot to find the principal gains at 10 rad/s.

c.       Use SVD to find the principal gains at 10 rad/s.

d.      Simulate the system with the input u(t) = [2 sin(10t) ; sin(10t)]. Plot the input and output and compute the gain from the plot. Confirm that the gain is between the principal gains that you computed above.

e.       Simulate the system with an input that minimizes the gain. Plot the input and output and compute the gain from the plot. Confirm that the gain is equal to the smallest principal gain.

f.        Simulate the system with an input that maximizes the gain. Plot the input and output and compute the gain from the plot. Confirm that the gain is equal to the largest principal gain.

3.      Consider the mass/spring/damper system that we discussed in class, with two inputs (the two forces on the masses) and one output (the displacement of the top mass).
A = [ 0 0 1 0 ; 0 0 0 1 ;
-k1/m1  k1/m1  -b1/m1  b1/m1 ; k1/m2  -(k1+k2)/m2  b1/m2  -(b1+b2)/m2 ]
B = [ 0 0 ; 0  0 ; 1/m1  0 ; 0 1/m2 ],  C = [ 1 0 0 0 ],  D = [ 0 0 ]
Use the following values for the constants:
k1 = 10, k2 = 1, b1 = 10, b2 = 1, m1 = 1, m2 = 10.

a.       Use SVD to compute the minimum and maximum gains at 1 rad/s.

b.      What is the input at 1 rad/s that will result in a zero output?

c.       Simulate the system for 150 s with the input equal to the minimizing input, plot the output, and use the plot to estimate the minimum gain.

d.      Simulate the system for 150 s with the input equal to the maximizing input, plot the output, and use the plot to estimate the maximum gain.

Chapter 4 - due October 12

Consider the linearized inverted pendulum with two inputs (cart force and angular pendulum acceleration) and two outputs (cart position and pendulum angle).
A = [ 0  1  0  0 ; 0
-F/M  -m*g/M  0 ; 0  0  0  1 ; 0  F/M/L  (M+m)*g/M/L  0 ];

B = [0 0 ; 1/M 0 ; 0 0 ; -1/M/L 1 ], C = [ 1 0 0 0 ; 0 0 1 0 ], D = [ 0 0 ; 0 0 ]

Use the constants given in http://academic.csuohio.edu/simond/courses/eec641/PendCtrl.m.

1.      Find the right and left eigenvectors of the system matrix.

2.      Use the eigenvectors of A to compute the input and output pole directions of the system.

3.      Use SVD to compute the input and output pole directions of the system.

4.      Based on the input and output pole directions, what do you conclude about the controllability and observability of the system?

5.      What is the McMillan degree of the system?

6.      Is the system stabilizable? Is it detectable?

7.      Find the transfer function matrix. Write it out for me as a single matrix.

8.      Find the zeros and poles of the system.

Hand in any Matlab code that you used to obtain your answers.

Chapter 8 (part 1) - due November 9

1.      Compute the P matrix for multiplicative output uncertainty, inverse additive uncertainty, inverse multiplicative input uncertainty, and inverse multiplicative output uncertainty.

2.      In class we found a first order approximation to |sigma| = 0.001 for omega < 2, |sigma| = 0.1 for omega > 30. Find a second order approximation. Use Matlab to graph the two approximations on the same plot.

3.      Use Matlab’s “sysic” function to find the P matrix that we computed in class: G = 1 / (s+p1) / (s+p2), p1 = 1 +/- 0.1, p2 = 4 +/- 1. Verify that your results match the answer that we got in class. Hand in your code and the Matlab results.

4.      Solve Exercise 3.27 in the text.

5.      Solve Exercise 3.29 in the text.

Chapter 8 (part 2) - due November 16

Consider a system with the transfer function G(s) = [ s-a^2 , a(s+1) ; -a(s+1) , s-a^2 ] / (s^2+a^2), where a is a parameter of the system. (This is the transfer function of a symmetric body that is spinning around one axis. The inputs are the two torques applied to the other axes, and the outputs are a linear combination of the angular velocities about those axes.)  The feedback control input u = -y.

1.      Show analytically that the system is internally stable for all values of a.

2.      Suppose that the system has an uncertainty such that the true transfer function Gp = (I+E)G, where E is a uncertainty whose infinity-norm is bounded by c. Find the largest value of c such that the system will be robustly stable. (c will be a function of a.)

3.      Now suppose that E is still an uncertainty whose infinity norm is bounded by c, but the only nonzero element of E is the upper left element. Use the max singular value bound to find an upper bound for the SSV of Nydud. Use the spectral radius bound to find a lower bound for the SSV of Nydud. These two bounds will be equal, which gives an exact value of the SSV. Use the SSV value to find the largest value of c such that the system will be robustly stable. (c will not be a function of a.)

4.      Use Matlab to plot the max singular value of Nydud for part 2 with a=1. This will give some numerical confirmation of your analytical result.

5.      Use Matlab to plot the SSV of Nydud for part 3 with a=1. This will give some numerical confirmation of your analytical result.

Chapter 8 (part 3) - due November 28

1.      Take the system that you defined for your term project, and find a stabilizing controller. It can be any linear controller – you can use a decoupled PI controller, or pole placement, or Matlab’s “mixsyn” or “hinfsyn” function, or anything else, as long as the closed loop system is stable.

2.      Prove that your closed loop system is internally stable.

3.      Define an unstructured uncertainty that affects some part of your system. Find the smallest infinity norm of the uncertainty that will cause your system to become unstable.

4.      Define a parametric uncertainty in your system. Pull the parametric uncertainty out so that it can be written as a structured uncertainty. Find the smallest infinity norm of the uncertainty that will cause your system to become unstable.

5.      Define a frequency-dependent performance requirement for your system. Make sure that the nominal system meets the performance requirement. (If necessary, relax the requirement.)

6.      For this problem you can assume that your system is affected by either the unstructured uncertainty that you defined in problem 3, or the structured uncertainty that you defined in problem 4. In this problem you will find out if the uncertain system meets the performance requirement of problem 5. Use Theorem 8.7 to find the smallest infinity norm of the augmented uncertainty that will cause your system to violate the performance requirement.

Term Project

The format for the term project is based on the suggestion of Appendix B in the text. Each student is responsible to to choose and conduct a research project based on some nonlinear MIMO system that they are interested in. The project will be graded on the basis of written and oral status reports, a written report handed in at the end of the semester, and an oral presentation given during the last week of classes. The written project reports are due at the end of the last regularly scheduled class of the semester. Late project reports will not be accepted. The written reports can be based on the template at http://academic.csuohio.edu/simond/courses/Report Template.doc although this is not required. For each of the tasks below, an in-class presentation of between 5 and 10 minutes is required, and a written report is required.

1.      Problem Definition - due September 12

2.      System Model - due October 3

3.      System Analysis - due October 17

4.      Controller Designs - due October 31

5.      Simulations - due November 14

6.      Final Report - due December 7 - The final report (both oral and written) should be comprehensive.