Observer Based Control System Design

Question

Consider the discrete-time system described by

$$ x(k+1) = \begin{pmatrix}0 & 1 \\ -2 & -3\end{pmatrix}x(k) + \begin{pmatrix}0\\1\end{pmatrix}u(k) $$

a) Find a feedback state vector that assigns the closed-loop poles to \$0.5\$ and \$1/4\$

b) Design an observer so that the error between the state and its estimate goes to zero faster than 0.5.

c) Write down the equations of an observer-based control system.

Attempt:

For the first part of the question, I have set up the feadback matrix (A - BK) and computed the state feed back vector K = \$[-11/6 -23/6]\$.

For the second part I know the question is looking for the observer L, which is obtained by setting up the matrix (A - LC), but I dont know how to make sure it goes faster than 0.5.How would I do this?

For the last part I think I just need to multiply the determinants of the matrices (A - BK) and (A - LC) (so I need both K and L ) because K and L are supposed to be able to be designed independently, is this right?

Any Help Appreciated.

Note: Exam Revision, not Homework.

Answer 1

Your feedback matrix K seems to be off; it yields poles at {1/3, 1/2}

For single-input observable systems, you can apply pole placement to place the observer poles wherever you like (as with the state feedback poles).

You might want to review just what a state observer is so you can better understand what you're building. The equations of an observer-based control system will include the both the system state equations and the observer state equations.