LMIs in Control/pages/Quadratic Stability

The System:
A TS fuzzy model allows the representation of a non-linear model as a set of local LTI (Linear Time Invariant) models $$[1, p. 10]$$, each one called subsystem. A subsystem is the local representation of the system in the space of premise variables $$z(t)$$ = $$[z1(t) z2(t). . . zp(t)]$$ which are known and could depend on the state variables and input variables.

The Optimization Problem:
Let consider an autonomous system $$x$$=$$Ax$$ with $$A$$ being a constant matrix. If we define the Lyapunov function $$V(x)$$=$$x^T P x$$, then the system is stable if there exist $$P > 0$$ such that condition is satisfied.

$$A^T P + P A < 0 $$

If we have a family of matrices $$A(\delta(t))$$ (where $$\delta(t)$$ is a parameter that is bounded by a polytope ∆) instead of a single matrix A, then the system equation becomes $$x$$=$$A(\delta(t))x$$ and condition should be satisfied for all possible values of $$\delta(t)$$. If exists $$P>0$$ such that following condition is satisfied then the system is quadratically stable.

$$A(\delta(t))^T P + P A(\delta(t)) < 0$$ ∀$$\delta(t)$$ ∈ ∆.

Since there are an infinite number of matrices A(δ(t)) there is also an infinite number of constraints like that for quadratic stability mentioned previously that should be fulfilled. From a practical point of view this makes the problem impossible to be solved. Let consider now that the system $$x = A(\delta(t))x$$ can be written in a polytopic form as a Takagi-Sugeno (TS) polytopic system with premise variables $$z(t)$$ and a set of r subsystems $$Ai$$ for $$i = {1,. . ., r}$$.

$$x(t)=\sum_{i=1}^{r}(h_i(z(t)) A_i x(t)$$.

It can be proven that a polytopic autonomous system is quadratically stable if previous condition  is satisfied in the vertices (subsystems) of the polytope. Therefore there is no need to check stability in an infinite number of matrices, but only in subsystems matrices $$A_i$$.

$$A^T_i P + P Ai < 0$$ ∀i = 1,. . ., r.

Stability conditions can be applied to the closed-loop system  and the following set of conditions are obtained.

$$G^T_{ii} P + P G_{ii} < 0$$ ∀i = 1,. . ., r.

$$((G_{ij} + G_{ji})/2)^T P + P((G_{ij} + G_{ji})/2)<= 0$$ ∀i, j ∈ {1,. . ., r}, i < j.

where $$G_{ij} = A_i + B_i K_j $$and $$ hi(z(t))hj (z(t)) \neq 0$$.

In the special case where matrices Bi are constant (i.e. $$B_i = B$$), the first set of inequalities are enough to prove stability. Therefore, assuming constant B for all the subsystems, if there exist P > 0 such that conditions are fulfilled, then the polytopic TS model (2.2) with state feedback control is quadratically stable inside the polytope.

$$(A_i + B K_i)^T P + P (A_i + B K_i) < 0$$ ∀i = 1,. . ., r.

The assumption of constant B can be achieve using a prefiltering of the input. This change is not restrictive and the main consequence is the addition of some new state variables (the ones from the filter) to the TS model.

The LMI:
The design of the controller that stabilizes the closed-loop system boils down to solve the Linear Matrix Inequality (LMI) problem of finding a positive definite matrix P and a set of matrices $$K_i$$ such that conditions are fulfilled. However, since the constraints should be linear combinations of the unknown variable, the following change of variables is applied: $$W_i = K_i Q$$ where $$ Q = P^-1$$. The solution of the LMI problem is the set of matrices $$W_i$$ such that conditions are fulfilled.

$$Q > 0$$

$$ A_i Q + Q A^T_i + B W_i + W^T_i B^T < 0$$. ∀i = 1,. . ., r.

The i-th controller is computed from the solution as $$K_i$$ = $$W_i$$ $$Q^{-1}$$

Conclusion:
The LMI is feasible.