MODERN CONTROL ENGINEERING LAB 3

MODERN CONTROL ENGINEERING LAB 3 January 28th 2008

FULL STATE FEEDBACK Consider a SISO system in its ss representation: i

x = Ax + Bu y = Cx The system response – governed by We want to use eigenvalues!

to change the position of these

We assume where is some reference signal and the gain matrix If

this controller is caller regulator.

We obtain: i

x = Ax + Bu = Ax + B(r − Kx ) = ( A − BK ) x + Br = ACL x + Br y = Cx

such that has the desired eigenvalues! Objective: Pick Condition: States are accessible and the system is controllable. This are the first steps that need to be checked before designing a full state controller. Problem 1. Consider the following system: ⎛1 1 ⎞ ⎛1⎞ x =⎜ x + ⎟ ⎜ ⎟u ⎝1 2 ⎠ ⎝0⎠ y = (1 0 ) x i

a. Design a full state controller such that the closed loop system has the following poles: Solve first analytically and then with the use of MATLAB.

b. Redo the operations for the case in which the changed to:

matrix has

⎛1 1⎞ A=⎜ ⎟ 0 2 ⎝ ⎠ c. Check the tracking performances in the case of (unit step). Construct a SIMULINK diagram for our control system. Do we have zero steady state error for the unit-step reference signal?

Conclusion: The output does not track our unit-step reference signal. How do we solve this problem??

One obvious solution is to scale the reference input u = Nr − Kx , where N is an extra-gain. The open-loop system is:

so that

i

x = Ax + Bu y = Cx + Du

For a unit-step input:

r = rSS u (t )

i

At steady state: x = 0 ⇒ x = xSS , u = uSS . For good tracking we want:

y = ySS = rSS Solve for:

−1

⎛ xSS ⎞ ⎛ A B ⎞ ⎛ 0 ⎞ ⎜ ⎟=⎜ ⎟ ⎜r ⎟ u C D ⎠ ⎝ SS ⎠ ⎝ SS ⎠ ⎝

Let us define: xSS = N x rSS , uss = N u rSS. Also:

u = Nr − Kx ⇒ uSS = NrSS − KxSS ⇒ NrSS = uSS + KxSS Again,

u = uSS − K ( x − xSS ) = N u rSS − K ( x − N x rSS ) = ( N u + KN x )rSS − Kx Such that:

N = N u + KN x Introduce this gain in the SIMULINK Model and check the tracking again!!