Control Design with IOP and SPA

We have a vectorized form IOP with SPA equation:

$$A\left[\begin{array}{c}w\\ x\\ \hat{x}\end{array}\right]=b$$

We also have vectorized forms of the specs for control design:

• Steady-state error $e_{ss}=0$ (or could be something like $e_{ss}\le C$):

$$1+\text{Steady state}\left[\begin{array}{c}x\\ \hat{x}\end{array}\right]=0$$

• Control effort $u[k]\le C_1$:

$$\text{max}(\text{step\_ru}\ast w)\le C_1$$

• Overshoot $\%OS\le C_2$:

$$\text{max}\left(\text{step\_ry}\left[\begin{array}{c}x\\ \hat{x}\end{array}\right]\right)\le (1+C_2)\left(-\text{steady\_state}\left[\begin{array}{c}x\\ \hat{x}\end{array}\right]\right)$$

• Settling time (within 2%) $T_s\le C_3$ with $\hat{j}=\text{min}\{j:jT\ge C_3\}$:

$$\begin{array}{r}\text{max}\left(\text{step\_ry}(\hat{j}:\text{end, :})\left[\begin{array}{c}x\\ \hat{x}\end{array}\right]\right)\le 1.02\left(-\text{steady state}\left[\begin{array}{c}x\\ \hat{x}\end{array}\right]\right)\\ \text{min}\left(\text{step\_ry}(\hat{j}:\text{end, :})\left[\begin{array}{c}x\\ \hat{x}\end{array}\right]\right)\ge 0.98\left(-\text{steady state}\left[\begin{array}{c}x\\ \hat{x}\end{array}\right]\right)\end{array}$$

We can turn this into a numerical optimization problem.