Mixed sensitivity design
In the previous chapter we mentioned that there are several ways to capture uncertainty in the model and analyze robustness with respect to the uncertainty. We have chosen the worst-case approach based on the small gain theorem for analysis of robust stability, which in the case of linear systems has an intuitive frequency-domain interpretation.
This chosen framework has two benefits. First, having being formulated in frequency domain, it offers us to take advantage of the insight developed in introductory courses on automatic control, that typically invest quite some effort into developing frequency doman concepts such as magnitude and phase Bode plots, Nyquist plot, and sensitivity and complementary sensitivity functions. Generations of control engineers have contributed to the collective know-how carried by these classical concepts and techniques.
Second, by formulating the requirements on robust stability, nominal stability and robust performance as constraints on \mathcal H_\infty norms of some closed-loop systems, an immediate extension from analysis to automated synthesis (control design) is enabled by availability of numerical methods for \mathcal H_\infty optimization. This enhances the classical frequency-domain control design techniques in that while the classical methods require that we know what we want and we also know how to achieve it, the \mathcal H_\infty optimization based methods require that we only know what we want (and express our requirements in frequency domain). We don’t have to bother with how to achieve it because there are numerical solvers that will do the job for us.
Let’s introduce the first instance of such methodology. We have learnt that the robust performance condition in presence of multiplicative uncertainty is formulated as a bound on the \mathcal H_\infty norm of the mixed sensitivity function \begin{bmatrix}W_pS\\WT\end{bmatrix}, namely \left\| \begin{bmatrix} W_pS\\WT \end{bmatrix} \right\|_{\infty} < \frac{1}{\sqrt{2}}.
Evaluating this condition can be done in a straightforward way, either at a grid of frequencies (inefficient) or by invoking a method for computing the norm.
But the major god news of this chapter is that we can also turn this into an optimization problem \operatorname*{minimize}_{K \text{ stabilizing}} \left\| \begin{bmatrix} W_pS\\WT \end{bmatrix} \right\|_{\infty}.
In words, we are looking for a controller K that guaranees stability of the closed-loop system and it also minimizes the \mathcal H_\infty norm of the mixed sensitivity function.
Such optimization solvers are indeed available.
In anticipation of what is to come, we note here that the above minimization of the \mathcal H_\infty norm of the mixed sensitivity function is a special case of the more general \mathcal H_\infty optimization problem (minimization of the norm of a general closed-loop transfer function). Therefore, even if your software tools does not have a specific function for mixed sensitivity optimization, chances are that a solver for the general \mathcal H_\infty optimization function is available. And we will soon see how to reformulate the mixed sensitivity minimization as the general \mathcal H_\infty optimization problem.
Having derived the bound on the norm of the mixed sensitivity function (equal to 1/\sqrt{2} in the SISO case), it may now be tempting to conclude that the only goal of the optimization is to find a controller that satisfies this bound. However, it turns out that the optimization has another useful property – it is called self-equalizing property. We are not going to prove it, we will be happy just to interpret it: it means that with the optimal controller the frequency response of the considered (weighted and possibly mixed) sensitivity function is flat (constant over all frequencies).
In order to understand the impact of this property, let us consider the problem of minimizing just \|WT\|_\infty. We choose this problem even though practically it is not really useful to require just (robust) stability. For \gamma = \min_{K}\|WT\|_\infty, the flatness of the frequency response |W(j\omega)T(j\omega)| means that the magnitude frequency response |T(j\omega)| is proportional to 1/|W(j\omega)|, that is,
|T(j\omega)| = \frac{\gamma}{|W(j\omega)|},\qquad \gamma \in \mathbb R, \gamma > 0.
This gives another motivation for our \mathcal{H}_\infty optimization endeavor – through minimization we shape the closed-loop magnitude frequency responses. This automatic/automated loopshaping is the second benefit promised at the beginning of this section. But we emphasize that for practical pursposes it is only useful to minimize the norm of the mixed sensitivity function, in which case more than just simultaneous shaping of W_\mathrm{p}S and WT must be achieved.
With this new interpretation, we can feel free to include other terms in the optimization criterion. In particular, the criterion can be extended to include the control effort as in (after reindexing the weighting filters) \boxed {\operatorname*{minimize}_{K \text{ stabilizing}} \left\| \begin{bmatrix} W_1S\\W_2KS\\W_3T \end{bmatrix} \right\|_{\infty}.}
The middle term penalizes control (similarly as R term in LQ optimality criterion \int(x^TQx+u^TRu)dt). Typically it is sufficient to set it equal to a nonnegative constant.
An important property of this method is that it extends to the multiple-input-multiple-output (MIMO) case. Nothing needs to be changes in the formal problem statement as the \mathcal H_\infty norm is defined for MIMO systems as well.
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