as below

EEL 5613 Modern Control (Fall 2017)

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 an observer:

 xyuxx 121 1

1

1

300

520

041



  





  







  





  











 

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

and checking that the system is fully controllable, design a state feedback control to

place the closed loop poles at {-2+2j, -2-2j, -20}. Follow the step-by-step SISO control

design based on mapping to a phase canonical representation.

Use MATLAB CST to do the following verifications: Using the “ctrb” command

construct the controllability matrix. Using the “acker” and “place” commands check

the vector of gains that you have received, then find the eigenvalues of the A+bk matrix

(or A-bk, depending how you defined k).

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

{-1.5, -2.5, -3.5}.

Simulate the system and its observer in CST or Simulink and compare the estimated

vector )( ^

tx to the actual state vector x(t), for a unity step input 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?

1.4. Coming back to the design task of (1.1), find the vector of gains k using Ackermann’s

formula.

1.5. Do the pole placement of (1.1) using Lyapunov equation.

1.6. Design the full order observer of (1.2) using Lyapunov equation.

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

   





   







   





   





 



01

10

10

01

4000

0300

0220

0001

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, -4}.

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: Tuesday 11/14/2015.