Controller Design

Stabilizing controller design: Given the plant $P(s)$, design the controller $C(s)$ so that the closed-loop system is internally stable.

The idea of pole placement is to design $C(s)$ so that the poles closed-loop system are the roots of the characteristic polynomial, such that the roots of the closed-loop characteristic polynomial are placed in the pre-specified desirable location.

Proper Controllers

Let a plant be given by

$$P(s)=\frac{b(s)}{a(s)}=\frac{b_0{s}^n+b_1{s}^{n-1}+\cdots +b_n}{a_0{s}^n+a_1{s}^{n-1}+\cdots +a_n},a_0\ne 0$$

where $a(s)$ and $b(s)$ are coprime.

Consider proper controllers:

$$C(s)=\frac{q(s)}{p(s)}=\frac{q_0{s}^m+q_1{s}^{m-1}+\cdots +q_m}{p_0{s}^m+p_1{s}^{m-1}+\cdots +p_m},p_0\ne 0$$

where $p(s)$ and $q(s)$ are coprime.

The closed-loop characteristic polynomial is

$$c(s):=a(s)p(s)+b(s)q(s)=c_0{s}^{n+m}+c_1{s}^{n+m-1}+\cdots +c_{n+m}$$

Given $a(s),b(s),c(s)$ and unknown $p(s),q(s)$, it is also called the polynomial Diophantine equation.

Stabilizing controller design:

• Specify an $(n+m)$th order desired stable $c(s)$.
• Choose $p(s)$ and $q(s)$ so that the Diophantine equation is satisfied.
• Set $C(s)=\frac{q(s)}{p(s)}$.

Define two $(n+m)\times m$ Toeplitz matrices:

Strictly Proper Controller