Grammatical Evolution of Robust Controller Structures Using Wilson Scoring and Criticality Ranking

Abstract

In process control it is essential that disturbances and parameter uncertainties do not affect the process in a negative way. Simultaneously optimizing an objective function for different scenarios can be solved in theory by evaluating candidate solutions on all scenarios. This is not feasible in real-world applications, where the scenario space often forms a continuum. A traditional approach is to approximate this evaluation using Monte Carlo sampling. To overcome the difficulty of choosing an appropriate sampling count and to reduce evaluations of low-quality solutions, a novel approach using Wilson scoring and criticality ranking within a grammatical evolution framework is presented. A nonlinear spring mass system is considered as benchmark example from robust control. The method is tested against Monte Carlo sampling and the results are compared to a backstepping controller. It is shown that the method is capable of outperforming state of the art methods.

Appendix: Stability Analysis

We calculate the Jacobian of the system from Eq. (9), substituting u with the GE controller from Eq. (14). The Jacobian for this system is given by $\varvec{A}(\varvec{x}) = \begin{bmatrix} \frac{\partial \varvec{f}}{\partial \varvec{x}} \end{bmatrix}$. The characteristic polynomial can be calculated by

$$\begin{aligned} p(\lambda ) = \det (\lambda \varvec{I} - \varvec{A}) = \lambda ^4 + \frac{3}{m_1}\lambda ^3 + \frac{k_1(m_1 + m_2) + 3m_2}{m_1m_2}\lambda ^2 + \frac{3k_1}{m_1m_2} \lambda + \frac{k_1}{m_1m_2}\,. \end{aligned}$$

(15)

Looking at the intervals from Eq. (10) it is trivial to see that all coefficients of this polynomial are strictly positive. We can thus continue the analysis by looking at the Hurwitz matrix of the polynomial:

$$\begin{aligned} \varvec{H} = \begin{bmatrix} 1&H_{1,2}&\frac{k_1}{m_1m_2}&0 \\ 0&\frac{3}{m_1}&\frac{4k_1}{m_1m_2}&0 \\ 0&1&H_{3, 3}&\frac{k_1}{m_1m_2} \\ 0&1&\frac{3}{m_1}&\frac{3k_1}{m_1m_2} \end{bmatrix},\, H_{1, 2} = H_{3, 3} = \frac{k_1m_1 + k_1m_2 + 3m_2}{m_1m_2} \,. \end{aligned}$$

(16)

From the 4 principal minors of $\varvec{H}$ we get the Hurwitz conditions for stability

$$\begin{aligned} \det (\varvec{H}_1)&= 1 \overset{!}{>} 0\,,\quad \det (\varvec{H}_2) = \frac{3}{m_1} \overset{!}{>} 0 \\ \det (\varvec{H}_3)&= \frac{3(k_1 + 9)}{m_1^2} \overset{!}{>} 0, \,\quad \det (\varvec{H}_4) = \frac{9k_1^2 + 18k_1}{m_1^3m_2} \overset{!}{>} 0. \end{aligned}$$

Again since these conditions only depend on $k_1, m_1$ and $m_2$ which are strictly positive, the stability conditions hold for any parameter combination $\varvec{q} \in Q$. The controller $u_{\text {GE}}(\varvec{x})$ is thus a (local) robust stabilizer of the system.