EE611

Homework #28

Due Friday, November 26

1. Given the single-input 3rd-order discrete-time system: with the following impulse response:

[0 1.0000 4.0833 7.0486 7.1464 5.2689 3.3217 2.3050 2.0817 2.0818 1.9332 1.6111 ...]

a) Find the Markov Parameters, m₁, m₂ , ... , m_2n-1

b) Find the Hankel matrices, H₁ and H₂, from these parameters

c) Find a discrete system, , which produced this impulse response

d) Check your answer by simulating the discrete impulse response on MATLAB. What is y_k at k=11?

e) Suppose that the impulse response data actually came from a continuous state variable model that we sampled at T=1/10 seconds. Use your answer to part c) to find the original continuous time state variable model, What is the is the settling-time of the system? (Hint: assume that the discrete model came from)

2. Consider the time-varying, discrete system, (*).

Let us define a discrete controllability grammian,

a) From your definition of controllabiilty for continuous time-varying systems, give a definition for the controllability of system (*) over the period [0, i].

b) Find an equivalent necessary and sufficient condition for the controllability of system (*) in terms of the discrete controllability grammian.

c) Prove sufficiency by finding an expression for a control which will drive any initial state, x₀, to any final state, x_i.

d) Determine the controllability of the following discrete time-varying system over the period [0,3]

e) Find an expression for w_k which will drive x₀ = [1 0]^T to x₃ = [0 0]^T