Let's say you are given a 2nd order system. Because we can diagonalize a matrix, let's just use:

A = [a1 0; 0 a2]; (Note the eigenvalues of a diagonal matrix are just the diagonal entries. Here it's a1 and a2).

Under a similarity transform (i.e. a change of variables where we just get a rotated, squashed version of A which is non-diagonal), the eigenvalues of the similar A will stay the same. That is, A represents the dynamics of every possible 2nd order system with just two numbers a1, a2 no matter what coordinate system you choose.

Let's say we have a general B vector B = [b1; b2] that can take on any real values (any single or double input system, not just SISO!). And we have two gains as K = [k1 k2] that we can set to anything we like.

Find the characteristic equation of the system and put it in standard form.

Hint:

syms a1 a2 k1 k2 b1 b2 s A = [a1 0; 0 a2];

B = [b1; b2];

K = [k1 k2]; myEqn = det(s*eye(2) -(A-B*K)) % characteristic equation of A with gains k1, k2 applied

(you can use collect (myEqn, s) to collect terms by s)

Let's say we want a desired closed loop performance of poles at p1 and p2.

(i) Given two gains k1, k2 (that is, 'control knobs' that you can set) and any possible 2nd order system (described by typical, full-rank A and B) can you make the closed loop system achieve any desired closed-loop performance as described by poles p1, p2?

Hints:

Recall that closed loop poles with gains k1 k2 will be set by (A-BK)

This is like the last problem of Midterm1 but in state space.

(ii) What is the equation for k1 and k2 given p1, p2 and A, K?