EE422 HW#23

Due Tuesday, April 20th

1. Recall the continuous-time, state variable model from HW#16: (Please, use Matlab liberally on this first problem!)

a) Is the continuous model stable, marginally stable, or unstable?

b) Discretize the model using with Ts=ln(2) sec. Remember, if A^-1 exists then you can use the trick that

c) Is the discrete model stable, marginally stable, or unstable? Is it controllable (find )?

d) Pick closed-loop eigenvalues in the S-plane of {-2 -4 -5} and map these values into the Z-plane.

e) Using these eigenvalues in the Z-plane, design a feedback regulator, Vin=-Kxk, such that the closed-loop discrete system has these eigenvalues (i.e., the eigenvalues of ) (If you are hazy, follow the procedures given in HW#16 to do this)

f) How does your gain K compare to the analog gain K given in the solution to HW#16?

h) Please check your answer by calculating the eigenvalues . What is the settling time of the closed-loop discrete system.

2. Consider the linear, shift-invariant, single-output single-input system which has the following discrete unit impulse response:

a) Express h(kT) in terms of discrete unit step functions, then use the discrete convolution sum to find the output of the above linear, shift invariant system to the input w(kT)=2[u(kT)-u((k-2)T)]:

b) Express h(kT) in terms of discrete unit impulse functions, then repeat part a)

c) Find the transfer function, H(z), for both your h(kT) form part a) and your h(kT) from part b). Verify that these transfer functions are the same!!

d) Repeat part a) using Z-transforms (i.e., find y(kT) = Z^-1{Y(z)} = Z^-1{H(z)W(z)}

e) Repeat part a) using graphical convolution methods

e) Finally, use your knowledge of linear shift invariant systems to find y[kT] if we use the input w[k] = 2(u[(k-5)T]-u[(k-7)T]) + 8(u[kT]-u[(k-2)T])