EE611 Old Exam II (July 17, 1995)

Problem 1 - 20 pts.

a) Consider the following controller architecture for the system shown. Assume that x is nx1, y is mx1, and we have the same number of inputs as outputs. Derive a design equation just involving Nx such that xref = xk and thus yk=yref in steady-state (hint: start with yk=Cxk).

b) The role of the feedforward gain, Nu, is to permit ek to go to zero without requiring wk to be zero ( if wk is zero and we have a stable system, then xk will go to zero, not xref!). Derive a second design equation involving both Nx and Nu such that e(t) can go to zero and xk can go to xref = constant in steady-state (Hint: xk+1 = xk if x is a constant).

c) What are the dimensions of Nx? What are the dimensions of Nu?

d) Find K where wk=-Kxk such that the closed-loop eigenvalue is 1/2 for the scalar system,

e) Now use your answers to parts a) and b) to find values for Nx, Nu, and yref if we want the system output in part d) to go to 7

Problem 2 - 20 points

Suppose the rank of the controllability matrix of a sixth-order system is 4. I have found a similarity transformation, x=Tccfz, to put the system into controllable canonical form. The results of this transformation is:

a) Explain how we found the similarity transformation,Tccf.

b) Find the controllable and uncontrollable eigenvalues of the original system.

c) Is the system stabilizable?

e) What is the settling time of your closed-loop system?

Problem 3 - 15 pts.

a) Given the TIME-VARYING discrete linear system, , find expressions for x₁, x₂, and x₃.

b) From your answer to part a), find an expression for the discrete state transition matrix, if the solution to the time varying discrete state equation is

c) Find if

d) Solve the system in part c) for x₆ if w_k is a unit impulse.

Problem 4 - 15 pts

a) For the time-invariant system, , give a definition of controllability in terms of altering the eigenvalues of A-BK.

b) Define the controllability matrix, M, for the above system and give an equivalent test for controllability in terms of M

c) Now define stabilizability for the system in part a)

d) Now consider the time-varying system, . Give a definition of controllability for this system.

e) Define the controllability grammian for the above system

f) Finally, show that your answer to part e) is sufficient by deriving a control which statisfies your definition in part d)

Problem 5 - 15 points

Given the following unit impulse response for a 2nd order discrete SISO system: y₁= {0,3,0,6,-6,...}

a) Find the Markov parmeters, m₁, m₂, m₃, m₄,

b) Find the state representation for the system,

c) Is your model a minimal realization (i.e., completely observable and completely controllable)?

d) Find the transfer function, H(z)=Y(z)/W(z) for the discrete system

e) Suppose the given data actually came from continuous time system, . Assuming that we used the approximation, and the data samples are 2 seconds apart, find the continuous time system.

Problem 6 - 15 pts.

Given the system,

a) Derive a reduced-order observer procedure for the above system (Hint: Think about how your could rewrite the output equation to resemble our standard output)

b) Use your procedure to design a reduced-order Deadbeat observer for

c) Draw a block diagram of your reduced-order observer.