Robust H _∞ output feedback control with pole placement constraints for uncertain discrete-time fuzzy systems

Abstract

This paper investigates the problem of multi-objective control for a class of uncertain discrete-time fuzzy systems. The state-space Takagi–Sugeno T–S fuzzy model with linear fractional parameter uncertainties is adopted. Based on a linear matrix inequality approach and via so-called dynamic parallel distributed compensation, a fuzzy full-order dynamic output feedback controller is developed such that the L ₂ gain performance from the exogenous input signals to the controlled output is less than or equal to some prescribed value and, for all admissible uncertainties, the closed-loop poles of each local system are within a pre-specified sub-region of complex plane. Two numerical examples are provided to illustrate the effectiveness of the proposed design method.

Appendices

Appendix 1: Proof of Theorem 3.1

Considering (19) for the closed-loop system (15), the following inequality holds

$$ \left[ \begin{array}{llll} -P & * &*&*\\ 0&-\gamma I&*&*\\ \tilde{A}_{zz}(k) & \tilde{B}_z(k)&-P^{-1} &*\\ \tilde{C}_z & \tilde{D}_{z}&0&-\gamma I\\ \end{array} \right]<0 $$

(48)

and it could be be written as

$$ \left[\begin{array}{llll} -P & * &*&*\\ 0&-\gamma I&*&*\\ \tilde{A}_{zz} & \tilde{B}_z&-P^{-1} &*\\ \tilde{C}_z & \tilde{D}_{z}&0&-\gamma I\\ \end{array} \right] +\hbox{sym} \left\{ \left[\begin{array}{l} 0\\ 0\\ \tilde H^{T}\\ 0 \end{array} \right] \Updelta (k) \left[\begin{array}{llll} \tilde{E}_{z}&\tilde{E}_{1z}&0&0 \end{array}\right] \right\} <0 $$

(49)

Then, from Lemma 2.3, one has

$$ \left[\begin{array}{llllll} -P & * &*&*&*&*\\ 0&-\gamma I&*&*&*&*\\ \tilde{A}_{zz} & \tilde{B}_{z}&-P^{-1}&*&*&*\\ \tilde{C}_{z} & \tilde{D}_{z}&0&-\gamma I&*&*\\ \tilde{E}_{z}&\tilde{E}_{1z}&0&0&-\varepsilon I&*\\ 0&0&\varepsilon \tilde{H}^{T}&0&\varepsilon J&-\varepsilon I \end{array} \right] <0 $$

(50)

Let G a nonsingular matrix of appropriate dimension. Checking a congruence transformation by diag{I, I, G, I, I, I} to (50), and taking into account the fact P − G − G ^T ≤  − G ^T P ⁻¹ G, which is derived from (P − G)^T P ⁻¹(G − P) ≥ 0, yields

$$ \left[\begin{array}{llllll} -P & * &*&*&*&*\\ 0&-\gamma I&*&*&*&*\\ G^{T}\tilde{A}_{zz} & G^{T}\tilde{B}_{z}&P-G-G^{T}&*&*&*\\ \tilde{C}_{z} & \tilde{D}_{z}&0&-\gamma I&*&*\\ \tilde{E}_{z}&\tilde{E}_{1z}&0&0&-\varepsilon I&*\\ 0&0&\varepsilon \tilde{H}^{T}G &0&\varepsilon J&-\varepsilon I \end{array} \right] <0 $$

(51)

Under the conditions of Theorem 3.1, a feasible solution of the inequalities (31) satisfies the condition $Q-\Upsigma-\Upsigma^T<0,$ then we have

$$ \Upsigma+\Upsigma^T= \left[\begin{array}{ll} Y+Y^T&Z+I\\ Z^T+I&X+X^T \end{array} \right] >Q>0 $$

(52)

It follows that X + X ^T > 0 and Y + Y ^T > 0. So, X and Y are nonsingular. Checking a congruence transformation to the previous inequality by [Y ^−T, −I]^T, we get easily

$$ (X-Y^{-T}Z) + (X-Y^{-T}Z)^T>0 $$

(53)

which implies that Z − Y ^T X is also nonsingular. Hence, there exist nonsingular matrices M ₁ and N ₁ such that (35) holds.

Let

$$ G=\left[\begin{array}{ll} X& (I-XY)N_{1}^{-1}\\ M_{1}&- M_{1}Y N_{1}^{-1} \end{array}\right],\quad \Uppi_{2}= \left[\begin{array}{ll} Y&I\\ N_{1}&0 \end{array}\right],\quad \Uppi=\hbox{diag}\{\Uppi_{2},I,\Uppi_{2},I,I,I\}. $$

(54)

Clearly, G is nonsingular. Pre- and post-multiplying (51) by $\Uppi^{T}$ and $\Uppi,$ respectively, and letting $Q=\Uppi_{2}^{\rm T}P\Uppi_{2}$ and $\Upsigma=\Uppi_{2}^{\rm T}G \Uppi_{2},$ yields

$$ \Upupsilon_{zz}= \left[\begin{array}{llllll} -Q&*&*&*&*&*\\ 0&-\gamma I&*&*&*&*\\ \Upphi_{Azz}&\Upphi_{Bz}&Q-\Upsigma-\Upsigma^T&*&*&*\\ \Upphi_{Cz}&\Upphi_{Dz}&0&-\gamma I&*&*\\ \Upphi_{Ez}& \Upphi_{E1z}&0&0&-\varepsilon I&*\\ 0&0&\varepsilon \Upphi_{H}^T&0&\varepsilon J&-\varepsilon I \end{array} \right]<0 $$

(55)

where

$$ \begin{aligned} \Upphi_{Azz}&= \left[\begin{array}{ll} AzY+ B_{2z}{\mathcal{C}}_{z}& A_{z}+ B_{2z}{\mathcal{D}} C_{2z}\\ {\mathcal{A}}_{zz}&X^{T} A_{z}+{\mathcal{B}}_{z} C_{2z} \end{array}\right] \quad \Upphi_{Bz}= \left[\begin{array}{ll} B_{1z}+B_{2z}{\mathcal{D}} D_{21z}\\ X^{T} B_{1z}+{\mathcal{B}}_{z} D_{21z} \end{array}\right]\\ \Upphi_{Cz}&= \left[\begin{array}{ll} C_{1z}Y+D_{12z}{\mathcal{C}}_{z}& C_{1z}+ D_{12z}{\mathcal{D}} C_{2z} \end{array}\right]\quad \Upphi_{Dz}= D_{12z}{\mathcal{D}} D_{21z}\\ \Upphi_{Ez}&= \left[\begin{array}{ll} E_{z}Y+E_{2z}{\mathcal{C}}z& E_{z}+ E_{2z}{\mathcal{D}} C_{2z} \end{array}\right]\quad \Upphi_{E1z}= E_{1z}+ E_{2z}{\mathcal{D}} D_{21z}\\ \Upphi_{H}^{T}&= \left[\begin{array}{ll} H^{T}&H^{T}X \end{array}\right] \end{aligned} $$

(56)

with

$$ \begin{aligned} A_{zz}&=X^{T} A_{z}Y+X^{T}\left(B_{2z}\hat{{\mathbf{D}}} C_{2z}Y+ B_{2z}\hat{{\mathbf{C}}}_{z} N_{1}\right) + M_{1}^T \hat{{\mathbf{B}}}_{z} C_{2z}Y+ M_{1}^{T} \hat{{\mathbf{A}}}_{zz} N_{1}\\ {\mathcal{B}}_{z}&=X^{T} B_{2z}\hat{{\mathbf{D}}}+ M_{1}^{T}\hat{{\mathbf{B}}}_{z}\\ {\mathcal{C}}_{z}&=\hat{{\mathbf{D}}} C_{2z}Y+\hat{{\mathbf{C}}}_{z} N_{1}\\ {\mathcal{D}}&=\hat{{\mathbf{D}}} \end{aligned} $$

(57)

Now, using the properties (9) and expanding (55), we have

$$ {\sum_{i=1}^{r}\sum_{j=1}^{r}h_ih_j} \Upupsilon {ij}<0 $$

(58)

Based on the parameterized linear matrix inequality (PLMI) (Tuan et al. 2001), (58) yields feasible conditions that are less conservative and computationally efficient. The relaxation result is written as (31).□

Appendix 2: Proof of Theorem 3.2

Using Lemma 2.2, we learn that the closed-loop poles of each sub-system are all within the given D _R region ${\mathfrak{D}, }$ if there exist matrices P > 0 and G such that

$$ \left[\begin{array}{ll} R_{11} \otimes P&*\\ R_{12}^{\rm T}\otimes P+R_{22} \otimes {G^{\rm T}\tilde{A}_{zz}(k)}& R_{22} \otimes (P-G-G^{\rm T})\\ \end{array}\right] <0 $$

(59)

Since $\left[\begin{array}{ll} I&I \otimes \tilde{A}_{zz}^{T}(k) \end{array}\right]$ has full rank, (59) implies that

$$ \left[\begin{array}{ll} I \\ {I}\otimes{\tilde{A}_{zz}(k)} \end{array}\right]^{\rm T} \left[\begin{array}{ll} R_{11} \otimes P&*\\ R_{12}^{\rm T} \otimes P+R_{22} \otimes {G^{\rm T}\tilde{A}_{zz}(k)}&R_{22} \otimes {(P-G-G^{\rm T})}\\ \end{array}\right] \left[\begin{array}{l} I \\ {I}\otimes{\tilde{A}_{zz}(k)} \end{array}\right] <0 $$

(60)

which is

$$ R_{11} \otimes P+ R_{22} \otimes \left(\tilde{A}_{zz}^{T}(k)P\tilde{A}_{zz}(k)\right) +Sym\left(R_{12} \otimes \left(P\tilde{A}_{zz}(k)\right)\right)<0 $$

(61)

Inequality (59) can be written also as

$$ \left[\begin{array}{ll} R_{11} \otimes P&*\\ R_{12}^{\rm T} \otimes P+ R_{22} \otimes G^{\rm T}\tilde A_{zz}&R_{22} \otimes (P-G-G^{\rm T})\\ \end{array}\right]+ sym\left\{ \left[\begin{array}{l} 0 \\ R_{22} \otimes \tilde H^T G \end{array}\right] \tilde{\Updelta} (k) [I \otimes \tilde{E}_{z}\, 0] \right\}<0 $$

(62)

where

$$ \begin{aligned} \tilde{\Updelta}(k)&={I\otimes{\Updelta(k)}}=\tilde{F}(k)\big(I-\tilde {J} \tilde {F}(k)\big)^{-1},\qquad \tilde {J}&=\big ({I\otimes{J}}\big)=\hbox{diag}(J,\ldots,J),\\ \tilde{F}k&=\big ({I\otimes{F(k)}}\big)=\hbox{diag}(F(k),\ldots,F(k)) \end{aligned} $$

(63)

From (5) and (6), it is easy to verify that $\tilde{J}^{\rm T}\tilde{J}- I<0$ and $\tilde{F}(k)^{\rm T}\tilde{F}(k)\leq I.$

According to Lemma 2.3, it holds that

$$ \left[\begin{array}{lllll} R{11}\otimes P&*&*&*\\ R_{12}^{\rm T}\otimes P+R_{22}G^{\rm T} \tilde A_{zz} & R_{22}{(P-G-G^{\rm T})}&*&*\\ I \otimes \Phi_{Ez}&{0}&-\varepsilon_{0}I&*\\ {0}&\varepsilon_{0} R_{22}\otimes \Upphi_{H}^{T}&\varepsilon_{0} I \otimes J-\varepsilon_{0}I \end{array} \right]<0 $$

(64)

Let $\Upxi=\hbox{diag}(I\otimes {\Pi_2},I\otimes{\Pi_2},I,I).$ Pre- and post-multiplying (64) by $\Upxi^{T}$ and $\Upxi$ respectively, yields

$$ \Upgamma_{zz}=\left[ \begin{array}{llll} R_{11}\otimes Q&*&*&*\\ R_{12}^T Q+R_{22}\otimes \Phi_{Azz} & R_{22}{(Q-\Upsigma-\Upsigma^T)}&*&*\\ {I \otimes \Phi_{Ez}}&{0}&-\varepsilon_{0}I&*\\ {0}&\varepsilon_{0} R_{22}\otimes \Upphi_{H}^{\rm T}&\varepsilon_{0}{I}\otimes J-\varepsilon_{0}I \end{array} \right]<0 $$

(65)

Based on PLMI (Tuan et al. 2001), the relaxation result is written as (36).□