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.

Optimization Problem

An optimization problem has the general form:

$$\underset{\text{variables}}{\mathrm{minimize}}\text{ cost/objective}$$

subject to some $\text{constraints}$.

In our case, we want to:

$$\underset{w,x,\hat{x}}{\mathrm{minimize}}\text{ cost/objective}$$

and our constraints are the IOP + specs equations above.

Example

1. Choose poles

First, we choose poles $\{p_1,\dots ,p_m\}$. Following our method for choosing poles for IOP and SPA , we choose poles along a spiral initially, but we don’t want the spiral to go all the way out to the boundaries of the unit disk.

Consider a pole of the form $p=r{e}^{j\theta }$. Then, $p^k=r^k{e}^{jk\theta }$, where the $r^k$ term controls the rate of decay and the $e^{jk\theta }$ controls the rate of oscillation. If we pick poles where $|r|$ is close to $1$, the rate of decay will be very slow. Furthermore, we may create numerical issues. Thus, we select some $r_{max}<1$ (typically between 0.8 and 0.95) for our spiral.

We also need to choose the number of poles; we typically start with a larger number of poles ($m=20-30$) and then try to reduce it later if desired. With more poles, it’s more likely we will find a feasible design.

2. Calculate Variables

Then, we calculate $\alpha ,\beta ,\gamma ,\hat{\gamma },A,b,\text{steady\_state},\text{step\_ru},\text{step\_ry}$.

3. Define objective and constraints

This is done in YALMIP - we can pretty much directly write constraints

4. Solve the optimization problem

Then, we solve the optimization problem with YALMIP (language)/MOSEK(solver).

5. Recover controller

Then we recover

$$D[z]=\frac{W[z]}{X[z]}$$

where

$$\begin{array}{rl}W[z]& =\sum _{i=1}^m\frac{w_i}{z-p_i}\\ X[z]& =\sum _{i=1}^m\frac{x_i}{z-p_i}+\sum _{k=1}^{\hat{n}}\frac{{\hat{x}}_k}{z-q_k}\end{array}$$

Design Implications

• Safety
  • Stability margins for improved robustness
  • Simplicity of controllers (reducing the number of poles)
  • More robust to physical or engineered limits (can tighten limits $u[k]\le C$ or $y[k]\le C$) further in specs
• Economics
  • Less expensive sensors; increase sampling time $T$, which limits the performance (response time) of our control system
  • Reducing actuator wear and tear
    • (i) By minimizing the size of the control signal
    • (ii) By minimizing the variability of the control signal

$$\begin{array}{rl}\text{objective}& =\underset{0<k\le K}{\max }(u[k])=\text{max}(\text{step\_ru}\ast w)\\ \text{objective}& =\text{norm}(u[2:K]-u[1:(K-1)],\text{‘inf’})\end{array}$$

• Note that here we are using the Infinity Norm

• Sustainability

  • Minimizing the energy use of our controller

$$\begin{array}{rl}\text{objective}& =\text{norm}(y[1:K],2{)}^2\\ \text{objective}& =\text{norm}(u[1:K],2{)}^2\end{array}$$