Emulation Design

There are two ways to control sampled-data systems:

1. Direct control of $D[z]$ in DT, such as IOP with SPA
2. Emulation design: Design $C(s)$ in CT, and then approximate $C(s)$ with $D[z]$ in DT

Assume we have a continuous time system of the form:

We use a running assumption that $C(s)$ has only simple poles:

$$\begin{array}{rl}C(s)=\sum _{i=1}^n\frac{c_i}{s-p_i}& =\left[\begin{array}{ccc}1& \dots & 1\end{array}\right]\underset{=(sI-A{)}^{-1}}{\underbrace{\left[\begin{array}{ccc}\frac{1}{s-p_1}& & \\ & \ddots & \\ & & \frac{1}{s-p_n}\end{array}\right]}}\underset{B}{\underbrace{\left[\begin{array}{c}{c}_1\\ ⋮\\ c_n\end{array}\right]}}\\ & ={\left(sI-\underset{A}{\underbrace{\left[\begin{array}{ccc}{p}_1& & \\ & \ddots & \\ & & p_n\end{array}\right]}}\right)}^{-1}B\end{array}$$

Note that if we assume $d=0$, we also have

$$\begin{array}{r}U(s)=C(s)E(s)\end{array}$$

which gives

$$\begin{gathered} \begin{array}{r}\dot{x}=\underset{A}{\underbrace{\left[\begin{array}{ccc}{p}_1& & \\ & \ddots & \\ & & p_n\end{array}\right]}}x+\underset{B}{\underbrace{\left[\begin{array}{c}{c}_1\\ ⋮\\ c_n\end{array}\right]}}e\end{array} \\ u=\left[\begin{array}{ccc}1& \dots & 1\end{array}\right]x \end{gathered}$$

Recall that we can write

$$\begin{array}{rl}x((k+1)T)& =x(kT)+\Delta [k+1]\\ & =x(kT)+{\int }_{kT}^{(k+1)T}(Ax(s)+Be(s))ds\end{array}$$

We can use the left-side rule for numerical integration:

$$\begin{array}{rl}\Delta [k+1]& \approx T(Ax(kT)+Be(kT))=T(Ax[k]+Be[k])\\ & ⟹x[k+1]=x[k]+T(Ax[k]+Be[k])\end{array}$$

Taking the $z$-transform gives

$$\begin{array}{rl}& zX[z]=X[z]+T[AX[z]+BE[z]]\\ & ⟹((z-1)I-TA)X[z]=TBE[z]\\ & ⟹X[z]=((z-1)I-TA{)}^{-1}TBE[z]\end{array}$$

… … …

Finally, we can do

$$U[z]=\sum _{i=1}^n\underset{=:D[z]}{\underbrace{\frac{c_i}{\frac{1}{T}(z-1)-p_i}E}}[z]⟹D[z]=C(s){|}_{s=\frac{1}{T}(z-1)}$$

With the right side rule