Specs for Control Design

How do we design controllers to meet performance specifications using IOP/SPA?

1. Closed-loop stability

Closed loop stability $⟺$ $W,X,V$ stable $⟹$ already guaranteed by choosing $\{p_i{\}}_{i=1}^m\subset \mathbb{D}$ and satisfying Equations ($\ast$) and ($\ast \ast$) from Simple Pole Approximation.

2. Steady-state error

($e_{ss}$) is given by \begin{align} e_{ss} = T_{re}[1] & =X[1] \quad \quad [\text{IOP theorem part b}] \\[2ex] & = 1+ \sum_{i=1}^{m} \frac{x_{i}}{1-p_{i}}+\sum_{k=1}^{\hat{n}} \frac{\hat{x}_{k}}{1-q_{k}} \end{align} - Case 1. $e_{ss}=0$

$$e_{ss}=1+\sum _{i=1}^m\frac{x_i}{1-p_i}+\frac{{\sum }_{k=1}^{\hat{n}}{\hat{x}}_k}{1-q_k}=0$$

- Case . $| e_{ss} |\leq C$

$$\begin{array}{rl}{e}_{ss}& \le C\\ e_{ss}& \ge -C\end{array}$$

Then the step response of a simple pole:

$$Y[z]=\frac{1}{z-p}\underset{\text{unit step}}{\underbrace{\frac{z}{z-1}}}=\frac{1}{1-p}\left(\frac{1}{z-1}-p\frac{1}{z}\frac{z}{z-p}\right)$$

• Partial fraction decomposition

3. Limit on control effort

$$\begin{array}{rl}& u[k]\le C\forall k>0⟹\underset{=W\text{(IOP theorem part b)}}{\underbrace{\text{step}(T_{ru})}}[k]\le C\forall k\ge 0\\ & ⟹\text{step}(W)[k]\le C\forall k\ge 0\\ & ⟺\text{step}\left(\sum _{i=1}^m\frac{w_i}{z-p_i}\right)[k]\le C\forall k\ge 0\\ & ⟺\sum _{i=1}^m{w}_i\text{step}\left(\frac{1}{z-p_i}\right)[k]\le C\\ & ⟺\sum _{i=1}^m\frac{1-p_i^k}{1-p_i}{w}_i\le C\forall k\ge 0\end{array}$$

Then, we have

$$\begin{array}{rl}e=r-y& ⟹y=r-e\\ & ⟹Y=R-E=R-T_{re}R=(1-T_{re})R\end{array}$$

• $T_{ry}=1-T_{re}=1-X$

4. Overshoot

We want some ($\%\text{OS}\le C$).

Then, we have

$$\begin{array}{rl}\text{step}(T_{ry})[j]& \le (1+C)y_{ss}\\ & =(1+C)(1-e_{ss})\\ & =(1+C)(1-X[1])\end{array}$$

which means

$$\underset{j\ge 0}{\max }\left(-\sum _{i=1}^m\frac{1-p_i^j}{1-p_i}{x}_i-\sum _{k=1}^{\hat{n}}\frac{1-q_k^j}{1-q_k}{\hat{x}}_k\right)\le (1+C)\left(-\sum _{i=1}^m\frac{x_i}{1-p_i}-\sum _{k=1}^{\hat{n}}\frac{{\hat{x}}_k}{1-q_k}\right)$$

5. Settling Time (within 2%)

Let $T$ be the sample time and let $\hat{j}=\text{min}\{j:jT\ge C\}$.

Then:

$$\begin{array}{r}\text{step}(T_{ry})[j]\le 1.02y_{ss}\\ \text{step}(T_{ry})[j]\ge 1.02y_{ss}\end{array}$$

• where $T_{ry}=1-X$

Thus, we have

$$\underset{j\ge 0}{\max }\left(-\sum _{i=1}^m\frac{1-p_i^j}{1-p_i}{x}_i-\sum _{k=1}^{\hat{n}}\frac{1-q_k^j}{1-q_k}{\hat{x}}_k\right)\le 1.02\left(-\sum _{i=1}^m\frac{x_i}{1-p_i}-\sum _{k=1}^{\hat{n}}\frac{{\hat{x}}_k}{1-q_k}\right)$$