*2.4. Stabilization*

We are now interested in designing a controller that is not only able to regulate the system variables to a desired set-point, but that can also guarantee stability during disturbances and events that are typical in an electrical system.

In terms of linear difference systems, the equations of the plant and the controller can be represented as in (2), that is, by *P*(*σ*)*w* = 0 and *C*(*σ*)*w* = 0, respectively. Moreover, the interconnected (closed-loop) system can be represented by:

$$\underbrace{\begin{bmatrix} P(\sigma) \\ \mathbb{C}(\sigma) \end{bmatrix}}\_{R(\sigma)} w = 0 \,, \tag{5}$$

where plant *P*(*σ*)*w* = 0 and controller *C*(*σ*)*w* = 0 laws must be simultaneously satisfied by *w*. This means that, by selecting a suitable controller, we are able to restrict the trajectories of the system to those that are asymptotically stable and discard those that are undesirable, for example, unstable, highly oscillatory, too slow, and so forth.

The design of controller *C*(*σ*) can impose the stability on (5). For this, it must be guaranteed that, having a partition *w* = col(*u*, *y*), the stability conditions recalled in

Section 2.3 for a Lyapunov function candidate *Q*Ψ, hold for all *w* satisfying (5). Notice that, if the coefficient matrix satisfies Ψ > 0, then *Q*<sup>Ψ</sup> ≥ 0 prevails. Then it is still necessary to guarantee that ∇*Q*<sup>Ψ</sup> < 0 ∀*w* satisfying (5). For this, the description of the closed-loop system can be introduced in the inequality, by considering a polynomial matrix *V*(*σ*), which is non zero, and has the same dimensions as *R*(*σ*) in (5). In this form, the symmetry necessary to satisfy the inequality is preserved, that is,

$$\underbrace{\sigma Q \Psi(w) - Q \Psi(w)}\_{\nabla Q \Psi(w)} + \underbrace{w^\top V(\sigma)^\top \begin{bmatrix} P(\sigma) \\ \mathcal{C}(\sigma) \end{bmatrix} w + w^\top \begin{bmatrix} P(\sigma) \\ \mathcal{C}(\sigma) \end{bmatrix}^\top V(\sigma) w}\_{\text{Symmetrie component}} < 0 \,\,. \tag{6}$$

Notice that the condition imposed by inequality (6) is interpreted as follows. If a QdF *Q*<sup>Ψ</sup> ≥ 0 (i.e., Ψ > 0) exists and is such that (6) is satisfied, then asymptotic stability is guaranteed for the interconnected system (5). This follows from the fact that every trajectory *w* satisfying the interconnected system laws, will cancel out the additional symmetric component (because *P*(*σ*)*w* = *C*(*σ*)*w* = 0), which meets the condition ∇*Q*<sup>Ψ</sup> < 0, concerning the trajectories *w* produced by the closed-loop system (5).

Next, we introduce a numerical solution to this apparently algebraically complex condition. For this, we use a candidate controller whose gains are unknown that will be eventually computed using measurement data, rather than a model of the system. The plant mathematical model will be ultimately substituted by a condition on coefficient matrices built entirely from data.
