as below

EEL 5613 Modern Control (Fall 2015)

Homework 4

Problem 1: The following linear time invariant SISO system is to be controlled using

state feedback, comparing the performance with and without the use of observer:

[ ]xyuxx 011 1 0 2

101 110 221

=   





  



 +

  





  





− −

−−− =

•

1.1. Assuming that the state vector x is fully accessible and can be measured in real time,

design a state feedback control to place the closed loop poles at {-1, -2, -2}. Follow

the step-by-step SISO control design, converting the system to controllability

canonical form (that is, phase canonical form), find the vector of gains in that

representation and then converting it back to the original system. Use MATLAB

CST to find the location of the closed-loop poles and closed-loop zeros.

1.2. Verify that the given system is observable. Then design an observer that has poles at

{-2, -2, -3}. Simulate the system and its observer in Simulink and compare the

estimated vector )( ^

tx to x(t), for a unity step u(t) and initial conditions x(0) =

[1,2,1]’. For the observer take arbitrary initial conditions (such as zero).

1.3. In this part of the homework we assess how the separation principle works. We use

the designs of (1.1) and (1.2) to do outputs comparison via Simulink simulation.

Compare the state feedback system of (1.1) (with the input and initial conditions of

(1.2)) to a system that combines (1.1) and (1.2). That is, the state feedback is

implemented using the estimated state vector created by the observer. Do the two y(t)

signals match right from the start (t=0), or is there some transient period?

Problem 2: Consider the following linear time invariant MIMO system which has two

control signals u = [u1, u2]’ . In this problem the output vector y is irrelevant.

uxBuAxx   





  



 +

  





  



 −=+=

•

10 10 01

300 220

001

Please verify first the following quick observations, and explain each briefly: a) The open

loop system is unstable, b) The system can be stabilized (why?), c) It is not possible to

carry out a SISO state feedback design, using directly either one of the scalar control

inputs u1(t) or u2(t) (why?).

In this problem the stabilization is done by state feedback, assigning arbitrary stable

closed-loop poles such as {-1, -2, -3}.

2.1. Follow the step-by-step design procedure given in the original proof to Heymann’s

Lemma. Verify, using MATLAB CST on the original system (with the state feedback

control law that you found), that the closed loop poles are at the designated locations.

2.2. Follow the step-by-step design procedure given in Hautus’ proof to Heymann’s

Lemma. Again, verify that the design works correctly.

Submission Deadline: Monday 11/23/2015 (by either a printed/written version and/or e-

mail submission).