LMI Relaxations for $\mathcal{H }_{\infty }$ and $\mathcal{H }_{2}$ Static Output Feedback of Takagi–Sugeno Continuous-Time Fuzzy Systems

Abstract

This paper presents new results concerning the problem of static output feedback $\mathcal{H }_{\infty }$ and $\mathcal{H }_{2}$ control design for continuous-time Takagi–Sugeno (T–S) fuzzy systems. A fuzzy line integral Lyapunov function with arbitrary polynomial dependence on the premise variables is used to certify closed-loop stability with a bound to the $\mathcal{H }_{\infty }$ and $\mathcal{H }_{2}$ norms, allowing the membership functions to vary arbitrarily (i.e., no bounds on the time-derivative of the membership functions are assumed). The static output feedback fuzzy controller is obtained through a two-step procedure: first, a fuzzy state feedback control gain is determined by means of linear matrix inequalities (LMIs). Then, the state feedback gain matrices are used in the LMI conditions of the second step that, if satisfied, provide the fuzzy static output feedback control law. The proposed approach also allows the output feedback gains to have independent and arbitrary polynomial dependence on some specific premise variables, selected by the designer, with great advantages for practical applications. The efficiency of the proposed strategy is demonstrated by means of numerical examples and time domain simulations.

Appendix

This Appendix illustrates how to construct numerical tractable conditions from infinite dimension LMIs represented by homogeneous polynomials in multi-simplexes. For that, as stated in Sect. 4, the homogeneous polynomials are given in terms of their coefficients.

As a simple example, consider matrix $M_g(\mu )$ given in (33). The inequality $M_g(\mu ) > 0$, for all $\mu \in \mathcal U$, holds if all terms of (33) are positive definite, that is,

$$\begin{aligned}&M_{((1,0),(2,0))} > 0, ~~ M_{((1,0),(1,1))} > 0, ~~ M_{((1,0),(0,2))} > 0,\\&M_{((0,1),(2,0))} > 0, ~~ M_{((0,1),(1,1))} > 0, ~~ M_{((0,1),(0,2))} > 0. \end{aligned}$$

Using the notations introduced in Sect. 4, one can write

$$\begin{aligned} M_{k_1 k_2} > 0, ~~ \forall k_1 \in \mathcal K _{2}(1), ~~ \forall k_2 \in \mathcal K _{2}(2) \end{aligned}$$

or simply

$$\begin{aligned} M_{k} > 0, ~~ \forall k \in \mathbf K _r(g)=\mathcal K _{2}(1) \times \mathcal K _{2}(2), ~~ g=(1,2). \end{aligned}$$

When other matrices and variables are involved, all terms of the parameter-dependent LMI matrix must be in the same degree to allow the construction of finite dimension LMIs. The advantage of handling multi-simplexes is that each simplex is homogenized to the same degree independently.

Before presenting details of how to construct numerical tractable conditions, from the multinomial theory (generalization of the binomial theorem to polynomials) define

$$\begin{aligned} U_g(\mu ) \triangleq \prod \limits _{i=1}^{n} \underbrace{\left( \sum \limits _{j=1}^{r_i} \mu _{ij} \right) ^{g_i}}_{\sum \limits _{k_i \in \mathcal K _{r_i}(g_i)} \frac{g_i!}{k_i!} \mu _i^k} \triangleq \sum \limits _{k \in \mathbf K _r(g)} \dfrac{g!}{k!} \mu ^k. \end{aligned}$$

(38)

As an example, considering LMI (22), the maximum degree of the polynomial matrix is given by $w=\max \{g,q+\sigma \},\,\sigma = \text{ ones }(1,n)$. Therefore, all terms of (22) are homogenized to the same degree $w$, that is

$$\begin{aligned}&\begin{bmatrix} \overline{A}(\mu )^{\prime }S_q(\mu )^{\prime } +S_q(\mu )\overline{A}(\mu )&\star \\ P_g(\mu ) -S_q(\mu )^{\prime } +G_q(\mu )\overline{A}(\mu )&-G_q(\mu ) -G_q(\mu )^{\prime } \end{bmatrix}\nonumber \\&= \begin{bmatrix} \mathcal M _{11}(\mu )&\mathcal M _{12}(\mu )\\ \star&-\mathcal M _{22}(\mu ) \end{bmatrix}\nonumber \\&= \sum \limits _{k \in \mathbf K _r(w)} \mu ^k \begin{bmatrix} \mathcal M _{11_k}&\mathcal M _{12_k}\\ \star&-\mathcal M _{22_k} \end{bmatrix} < 0 \end{aligned}$$

(39)

where

$$\begin{aligned} \mathcal M _{11}(\mu )&= U_{w-q-\sigma }(\mu ) (\overline{A}(\mu )^{\prime }S_q(\mu )^{\prime } +S_q(\mu )\overline{A}(\mu ))\\ \mathcal M _{12}(\mu )&= U_{w-g}(\mu ) P_g(\mu ) - U_{w-q}(\mu ) S_q(\mu )\\&\quad +\, U_{w-q-\sigma }(\mu ) G_q(\mu )\overline{A}(\mu )\\ \mathcal M _{22}(\mu )&= U_{w-q}(\mu ) (G_q(\mu ) +G_q(\mu )^{\prime }) \end{aligned}$$

and

$$\begin{aligned} \mathcal M _{11_k}&= \displaystyle \sum _{\stackrel{\tilde{k} \in \mathcal K _r(w-s-\sigma )}{\tilde{k} \preceq k}} \displaystyle \sum _{\stackrel{\hat{k} \in \mathcal K _r(\sigma )}{\tilde{k}+\hat{k} \preceq k}} \dfrac{(w-s-\sigma )!}{\tilde{k}!}~~\overline{A}_{\hat{k}}^{\prime }S_{k-\tilde{k}-\hat{k}}^{\prime }\\&\quad +\, S_{k-\tilde{k}-\hat{k}}\overline{A}_{\hat{k}} \end{aligned}$$

$$\begin{aligned} \mathcal M _{12_k}&= \displaystyle \sum _{\stackrel{\tilde{k} \in \mathcal K _r(w-g)}{\tilde{k} \preceq k}} \dfrac{(w-g)!}{\tilde{k}!} P_{k-\tilde{k}} -\displaystyle \sum _{\stackrel{\tilde{k} \in \mathcal K _r(w-q)}{\tilde{k} \preceq k}} \dfrac{(w-q)!}{\tilde{k}!} S_{k-\tilde{k}}\\&\quad + \displaystyle \sum _{\stackrel{\tilde{k} \in \mathcal K _r(w-s-\sigma )}{\tilde{k} \preceq k}} \displaystyle \sum _{\stackrel{\hat{k} \in \mathcal K _r(\sigma )}{\tilde{k}+\hat{k} \preceq k}} \dfrac{(w-s-\sigma )!}{\tilde{k}!} G_{k-\tilde{k}-\hat{k}} \overline{A}_{\hat{k}} \end{aligned}$$

$$\begin{aligned} \mathcal M _{22_k}&= \displaystyle \sum _{\stackrel{\tilde{k} \in \mathcal K _r(w-q)}{\tilde{k} \preceq k}} \dfrac{(w-q)!}{\tilde{k}!} G_{k-\tilde{k}} + G_{k-\tilde{k}}^{\prime }. \end{aligned}$$

As can be noted, the last term of (39) is written as a homogeneous polynomial with LMI coefficients. If all the coefficients are imposed to be negative definite (numerically tractable conditions), the feasibility of (22) is assured.