Design of Anti-disturbance Reliable Control for Fuzzy Networked Control Systems with Multiple Disturbances

Abstract

This research addresses the problems of finite-time boundedness and disturbance rejection for T-S fuzzy networked systems that are vulnerable to actuator faults, linear fractional uncertainties and multiple disturbances through a reliable anti-disturbance control design. In particular, multiple disturbances encompasses two kinds, where the first kind of disturbance is matched, which is produced by fuzzy exogenous systems and the second kind is mismatched disturbance, which is norm-bounded. Moreover, in order to depict the reality more precisely, the actuator faults are taken into account in the system under consideration. In particular, the fuzzy disturbance observer is constructed for closely estimating the matched disturbances. After that, a reliable anti-disturbance controller is designed in accordance with the output of the conceived disturbance observer. Furthermore, the $(\xi _1,\xi _2,\xi _3)-\eta$ dissipative performance is deployed in order to reduce the imprints of mismatched disturbances to the greatest extent feasible. Moreover, by constructing a Lyapunov-Krasovskii functional, a set of adequate criteria in the form of linear matrix inequalities is established, which guarantees that the undertaken system achieves finite-time boundedness. In conclusion, two numerical examples are shown, one of which is a model of a tunnel diode circuit, which reveals the competence and applicability of the suggested control mechanism.

Appendices

Appendix A

Proof of Theorem 1

With a view to achieving the requisite proof of this theorem, we will pick the Lyapunov-Krasovskii functional candidate in the below form:

$$\begin{aligned} {\mathbb {V}}(t)&={\textsf{x}}^T(t) P_1{\textsf{x}}(t)+e^T(t) P_2e(t)+\int _{t-h}^{t}{\textsf{x}}^T(s) Q {\textsf{x}}(s)ds\nonumber \\&+h\int _{-h}^{0}\int _{t+s}^{t}{\textsf{x}}^T(v) R {\textsf{x}}(v)dvds. \end{aligned}$$

(22)

In the subsequent step, by reckoning the derivative of the previously stated relation (21) along the path of the equation (5), (9), we thus arrive at

$$\begin{aligned} \dot{{\mathbb {V}}}(t)=&\ 2{\textsf{x}}^T(t) P_1 \dot{{\textsf{x}}}(t)+2e^T(t)P_2 {\dot{e}}(t)+{\textsf{x}}^T(t) Q {\textsf{x}}(t)\nonumber \\&-{\textsf{x}}^T(t-h) Q {\textsf{x}}(t-h)+h^2{\textsf{x}}^T(t) R {\textsf{x}}(t)\nonumber \\&-h\int _{t-h}^{t}{\textsf{x}}^T(s) R {\textsf{x}}(s)ds. \end{aligned}$$

(23)

Here, we leverage Lemma 1 to recast the single integral term appearing in the equation (22) as follows:

$$\begin{aligned} -h\int _{t-h}^{t}{\textsf{x}}^T(s) R \,{\textsf{x}}(s)ds \le \bar{\phi }^T(t) \begin{bmatrix} -9R{}\, \frac{36}{h}R\, {} \frac{-60}{h^2}R\, \\ *{}\, \frac{-192}{h^2}R \,{}\frac{360}{h^3}R\\ *{}\, *{}\, \frac{-720}{h^4}R\, \end{bmatrix} \bar{\phi }(t) \end{aligned},$$

(24)

where $\bar{\phi }(t)= \begin{bmatrix} \bar{\phi }^T_1(t)&\bar{\phi }^T_2(t)&\bar{\phi }^T_3(t) \end{bmatrix}^T, \bar{\phi }_1(t) = \int _{t-h}^{t}{\textsf{x}}(s) ds, \bar{\phi }_2(t) = \int _{t-h}^{t} ds\int _{t-h}^{s}{\textsf{x}}(w) dw \ \text {and} \bar{\phi }_3(t)= \int _{t-h}^{t} ds\int _{t-h}^{s}dw \int _{t-h}^{w}{\textsf{x}}(u) du.$

On the other hand, the below equality is viable for some invertible matrix ${\mathbb {M}}$:

$$\begin{aligned} 0=&\ 2 \begin{bmatrix} {\textsf{x}}^T(t)+ \dot{{\textsf{x}}}^T(t)+ {\textsf{x}}^T(t-h) \end{bmatrix}{\mathbb {M}}\bigg [\sum _{i=1}^{\gamma }\sum _{j=1}^{\gamma }{\mathfrak {h}}_i({\textsf{x}}){\mathfrak {g}}_j({\textsf{x}})\nonumber \\ [{A_i}{\textsf{x}}(t)+{B_i}{\mathfrak {S}}K_{pj}{\textsf{x}}(t)+B_i{\mathfrak {S}}K_{rj}{\textsf{x}}(t-h)\nonumber \\+B_i{\mathcal {V}}_ie(t)+{C}_{i}\rho _1(t)-\dot{{\textsf{x}}}(t)]\bigg ]. \end{aligned}$$

(25)

At the next step, it is straightforward to arrive at the following expression by blending the preceding relations (22, 23, (24):

$$\begin{aligned}\dot{{\mathbb {V}}}(t)- \varpi {\mathbb {V}}(t)-{\mathfrak {D}}^T(t) \varpi {\mathfrak {D}}(t)\nonumber \\ & \\\quad\le \sum _{i=1}^{\gamma }\sum _{j=1}^{\gamma }{\mathfrak {h}}_i({\textsf{x}}){\mathfrak {g}}_j({\textsf{x}})\Lambda^T(t)[\Xi _{ij}]\Lambda (t), \end{aligned}$$

(26)

where$\Lambda^T(t)= \big [ {\textsf{x}}^T(t) \ \ \dot{{\textsf{x}}}^T(t) \ \ e^T(t) \ \ {\textsf{x}}^T(t-h) \ \ \bar{\phi }^T_1(t) \ \ \bar{\phi }^T_2(t) \ \ \bar{\phi }^T_3(t) \ \ {\mathfrak {D}}^T(t) \big ]$ and the matrix $[\Xi _{ij}]$ is specified in the statement of the theorem.

Besides, we present the following criteria, all of which are grounded in the characteristics of the membership function:

$$\begin{aligned} \sum _{i=1}^{\gamma }\sum _{j=1}^{\gamma }{\mathfrak {h}}_i({\textsf{x}})({\mathfrak {h}}_j({\textsf{x}})-{\mathfrak {g}}_j({\textsf{x}}))S_{i}=0 \end{aligned},$$

(27)

where $S_i = diag \{S_{i1},S_{i2},...,S_{i9}\}$ is any positive definite slack matrix. Then,

$$\begin{aligned} \sum _{i=1}^{\gamma }&\sum _{j=1}^{\gamma }{\mathfrak {h}}_i({\textsf{x}}){\mathfrak {g}}_j({\textsf{x}})\Xi _{ij}\\&=\sum _{i=1}^{\gamma }\sum _{j=1}^{\gamma }{\mathfrak {h}}_i({\textsf{x}}){\mathfrak {g}}_j({\textsf{x}})\Xi _{ij}+\sum _{i=1}^{\gamma }\sum _{j=1}^{\gamma }{\mathfrak {h}}_i({\textsf{x}})({\mathfrak {h}}_j({\textsf{x}})-{\mathfrak {g}}_j({\textsf{x}}))S_i \nonumber \\&=\sum _{i=1}^{\gamma }\sum _{j=1}^{\gamma }{\mathfrak {h}}_i({\textsf{x}})({\mathfrak {g}}_j({\textsf{x}})-\beta _j{\mathfrak {h}}_j({\textsf{x}})+\beta _j{\mathfrak {h}}_j({\textsf{x}}))\Xi _{ij}\nonumber \\&\,\,\,\,+\sum _{i=1}^{\gamma }\sum _{j=1}^{\gamma }{\mathfrak {h}}_i({\textsf{x}})({\mathfrak {h}}_j({\textsf{x}})-{\mathfrak {g}}_j({\textsf{x}})-\beta _j{\mathfrak {h}}_j({\textsf{x}})+\beta _j{\mathfrak {h}}_j({\textsf{x}}))S_i\nonumber \\&=\sum _{i=1}^{\gamma }\sum _{j=1}^{\gamma }{\mathfrak {h}}_i({\textsf{x}})({\mathfrak {g}}_j({\textsf{x}})-\beta _j{\mathfrak {h}}_j({\textsf{x}})+\beta _j{\mathfrak {h}}_j({\textsf{x}}))\Xi _{ij}\nonumber \\&\,\,\,\,+\sum _{i=1}^{\gamma }\sum _{j=1}^{\gamma }{\mathfrak {h}}_i({\textsf{x}})({\mathfrak {h}}_j({\textsf{x}})-\beta _j{\mathfrak {h}}_j({\textsf{x}}))S_i\nonumber \\&\,\,\,\,-\sum _{i=1}^{\gamma }\sum _{j=1}^{\gamma }{\mathfrak {h}}_i({\textsf{x}})({\mathfrak {g}}_j({\textsf{x}})-\beta _j{\mathfrak {h}}_j({\textsf{x}}))S_i\nonumber \\&=\sum _{i=1}^{\gamma }\sum _{j=1}^{\gamma }{\mathfrak {h}}_i({\textsf{x}}){\mathfrak {h}}_j({\textsf{x}})(\beta _j\Xi _{ij}-\beta _jS_i+S_i)\nonumber \\&\,\,\,\,+\sum _{i=1}^{\gamma }\sum _{j=1}^{\gamma }{\mathfrak {h}}_i({\textsf{x}})({\mathfrak {g}}_j({\textsf{x}})-\beta _j{\mathfrak {h}}_j({\textsf{x}}))(\Xi _{ij}-S_i)\nonumber \\&=\sum _{i=1}^{\gamma }\sum _{j=1}^{\gamma }{\mathfrak {h}}_i({\textsf{x}})({\mathfrak {g}}_j({\textsf{x}})-\beta _j{\mathfrak {h}}_j({\textsf{x}}))(\Xi _{ij}-S_i)\nonumber \\&\,\,\,\,+\sum _{i=1}^{\gamma }{{\mathfrak {h}}_i^2({\textsf{x}})}(\beta _i\Xi _{ii}-\beta _iS_i+S_i)\nonumber \\&\,\,\,\,+\sum _{i=1}^{\gamma }\sum _{j=i+1}^{\gamma }{\mathfrak {h}}_i({\textsf{x}}){\mathfrak {h}}_j({\textsf{x}})(\beta _j\Xi _{ij}-\beta _jS_i+S_i\nonumber \\&\,\,\,\,+\beta _i\Xi _{ji}-\beta _iS_j+S_j), \end{aligned}$$

(28)

where ${\mathfrak {g}}_j({\textsf{x}})-\beta _{j}{\mathfrak {h}}_j({\textsf{x}}) \ge 0,$ for all j.

Now, if the criteria (10, 11, 12) are fulfilled, we can easily derive the following relation by combining the inequality (25, 26, 27, 28):

$$\begin{aligned} \dot{{\mathbb {V}}}(t) - \varpi {\mathbb {V}}(t)-{\mathfrak {D}}^T(t) \varpi {\mathfrak {D}}(t) <0. \end{aligned}$$

Considering the foregoing inequality and integrating it from 0 to $T_f$, we yield

$$\begin{aligned} {\mathbb {V}}(t)e^{-\varpi T_f}<&\ {\mathbb {V}}(0)+\varpi \int _{0}^{T_f}e^{-\varpi s}{\mathfrak {D}}^T(s) {\mathfrak {D}}(s)ds, \nonumber \\ {\mathbb {V}}(t)\le&\ e^{\varpi T_f}[{\mathbb {V}}(0)+\varrho (1-e^{-\varpi T_f})] . \end{aligned}$$

(29)

Following this, by supposing the criteria $\breve{P_1}=L_f^{-\frac{1}{2}}P_1L_f^{-\frac{1}{2}}$, $\breve{P_2}=L_f^{-\frac{1}{2}}P_2L_f^{-\frac{1}{2}}$, $\breve{Q}=L_f^{-\frac{1}{2}}QL_f^{-\frac{1}{2}}$ and $\breve{R}=L_f^{-\frac{1}{2}}RL_f^{-\frac{1}{2}}$, it is simple to compute the expression that is displayed below for some positive definite matrix $L_f$:

$$\begin{aligned} {\mathbb {V}}(t) \ge&\ {\textsf{x}}^T(t)P_1{\textsf{x}}(t) =\ {\textsf{x}}^T(t) L_f^{\frac{1}{2}}\breve{P_1}L_f^{\frac{1}{2}}{\textsf{x}}(t)\nonumber \\ \ge&\iota _{min}(\breve{P_1}){\textsf{x}}^T(t) L_f{\textsf{x}}(t)\nonumber \\ =&\iota _{1}{\textsf{x}}^T(t) L_f{\textsf{x}}(t) . \end{aligned}$$

(30)

On yet another note, it can be drawn from the relationship (21) that

$$\begin{aligned} {\mathbb {V}}(0)=&\ {\textsf{x}}^T(0) P_1 {\textsf{x}}(0)+e^T(0) P_2 e(0)+\int _{-h}^{0}{\textsf{x}}^T(s) Q {\textsf{x}}(s)ds\nonumber \\&\,\,\,\,+h\int _{-h}^{0}\int _{s}^{0}{\textsf{x}}^T(v) R {\textsf{x}}(v)dvds \nonumber \\ \le&[\ \iota _{max}(\breve{P_1})+\iota _{max}(\breve{P_2})+h \iota _{max}(\breve{Q})\nonumber \\&\,\,\,\,+\frac{h^3}{2}\iota _{max}(\breve{R})] \ {\textsf{x}}^T(\phi ) L_f{\textsf{x}}(\phi )\nonumber \\ \le&(\iota _2 + \iota _3+h \iota _4 + \frac{h^3}{2}\iota _5 )c_1\nonumber \\ =&{\hat{\tau }}_1 c_1. \end{aligned}$$

(31)

In a further, the next inequality could well be readily generated with the use of the expressions (29, 30, 31):

$$\begin{aligned} {\textsf{x}}^T(t) L_f{\textsf{x}}(t) \le&\frac{{\mathbb {V}}(t)}{\iota _1} \nonumber \\ <&\frac{e^{\varpi T_f} [{\hat{\tau }}c_1 + \varrho (1-e^{-\varpi T_f})}{\iota _1}. \end{aligned}$$

(32)

Therefore, if the criterion (13) is met, then it is conspicuous that ${\textsf{x}}^T(t) L_f{\textsf{x}}(t)< c_2, \forall \ t \in [0,T_f].$ Accordingly, taking into account the Definition 1, we are able to state that the closed-loop TSFNS (9) is FT bounded with respect to $(c_1, c_2, T_f, \varrho ,L_f)$. And with this, the proof is concluded. $\square$

Appendix B

Proof of Theorem 2

To provide the proof of the theorem for the events of unknown actuator fault ${{\mathfrak {S}}}={{\mathfrak {S}}}_0 + {{\mathfrak {S}}}_1\Sigma$ and $\Delta {A_i}(t)\ne 0$, the same Lyapunov-Krasovskii functional in (21) is adopted. Using these factors and the concept of dissipative performance, we reach the following expression:

$$\begin{aligned}&\dot{{\mathbb \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!{V}}}(t) - \varpi {\mathbb {V}}(t)-J(z,{\mathfrak {D}},T_f)+\eta {\mathfrak {D}}^T(t){\mathfrak {D}}(t) \,\,\,\,\,\,\,\,\,\,\,\,\,\nonumber \,\\\le \Lambda^T(t){\tilde{\Xi }}_{ij}\Lambda (t), \end{aligned}$$

(33)

where${\tilde{\Xi }}^{1,1}_{ij}=sym\{{\mathbb {M}} ({A}_i+\Delta {A_i}(t))+{\mathbb {M}}B_i({{\mathfrak {S}}}_0 + {{\mathfrak {S}}}_1\Sigma )K_{pj}\}+Q+h^2 R-D_i^T\xi _1D_i-\varpi P_1,\ {\tilde{\Xi }}^{1,2}_{ij}=P_1-{\mathbb {M}}+({A}_i+\Delta {A_i}(t))^T{\mathbb {M}}+K^T_{pj}({{\mathfrak {S}}}_0 + {{\mathfrak {S}}}_1\Sigma )^TB^T_i{\mathbb {M}},\ {\tilde{\Xi }}^{1,3}_{ij}={\mathbb {M}} B_i {\mathcal {V}}_i-\Delta {A_i}^T(t){\L }^T_iP_2,\ {\tilde{\Xi }}^{1,4}_{ij}={\mathbb {M}} B_i ({{\mathfrak {S}}}_0 + {{\mathfrak {S}}}_1\Sigma ) K_{rj}+({A}_i+\Delta {A_i}(t))^T{\mathbb {M}}+K^T_{pj}({{\mathfrak {S}}}_0 + {{\mathfrak {S}}}_1\Sigma )^TB^T_i{\mathbb {M}},\ {\tilde{\Xi }}^{1,8}_{ij}={\mathbb {M}} C_i-D^T_i \xi _2,\ {\tilde{\Xi }}^{2,2}_{ij}=sym\{-M\},\ {\tilde{\Xi }}^{2,3}_{ij}={\mathbb {M}} B_i{\mathcal {V}}_i,\ {\tilde{\Xi }}^{2,4}_{ij}={\mathbb {M}}B_i({{\mathfrak {S}}}_0 + {{\mathfrak {S}}}_1\Sigma )K_{rj}-{\mathbb {M}}^T,\ {\tilde{\Xi }}^{2,8}_{ij}={\mathbb {M}}C_i,\ {\tilde{\Xi }}^{3,3}_{ij}= sym\{P_2{\mathfrak {E}}_i+P_2{\L } _iB_i{\mathcal {V}}_i\},\ {\tilde{\Xi }}^{3,4}_{ij}={\mathcal {V}}^T_iB^T_i{\mathbb {M}},\ {\tilde{\Xi }}^{3,8}_{ij} =P_2{\L } _iC_i,\ {\tilde{\Xi }}^{3,9}_{ij}=P_2\aleph _i,\ {\tilde{\Xi }}^{4,4}_{ij}=sym\{{\mathbb {M}}B_i{\mathfrak {S}}K_{rj}\}-Q,\ {\tilde{\Xi }}^{4,8}_{ij}={\mathbb {M}} C_i,\ {\tilde{\Xi }}^{5,5}_{ij}=-9R,\ {\tilde{\Xi }}^{5,6}_{ij}=\frac{36}{h}R,\ {\tilde{\Xi }}^{5,7}_{ij}=\frac{-60}{h^2}R,\ {\tilde{\Xi }}^{6,6}_{ij}=\frac{-192}{h^2}R,\ {\tilde{\Xi }}^{6,7}_{ij}=\frac{360}{h^3}R,\ {\tilde{\Xi }}^{7,7}_{ij}=\frac{-720}{h^4}R, \ {\tilde{\Xi }}^{8,8}_{ij}=-(\xi _3-\eta I)$ and ${\tilde{\Xi }}^{9,9}_{ij}=-(\xi _3-\eta I).$

In order to go further, pre and post multiply the above matrix by $diag\{X,X,I,X,X,X,X,I,I\}$ and apply Lemma  2, Schur complement lemma, the requirements ${{\mathbb {M}}}^{-1}=X,{\hat{P}}_1=XP_1X,{\hat{Q}}=XQX,{\hat{R}}=XRX$, $P_2{\L } _i = Z_{i}$, $X{K}_{pj}={\mathcal {Y}}_{pj}$, $X{K}_{rj}={\mathcal {Y}}_{rj}$ on the resulting matrix. In doing so, we can have the matrix ${\hat{\Xi }}_{ij}$, which is specified in the theorem statement.

Moreover, by following similar steps as in (26), (28) for ${\hat{S}}_i = diag \{S_{i1},S_{i2},...,S_{i30}\}$ and with the aid of the constraints (14, 15, 16), we get $\dot{{\mathbb {V}}}(t) - \varpi {\mathbb {V}}(t)-J(z,{\mathfrak {D}},T_f)+\eta {\mathfrak {D}}^T(t){\mathfrak {D}}(t) < 0.$ As of right now, we are able to say that the closed-loop TSFNS (9) is asymptotically stable in conformity with the Lyapunov stability theory.

In the next stage, with the intent to prove FT boundedness, let us define $X=L^{-\frac{1}{2}}_f\breve{X}L^{\frac{1}{2}}_f,P_2=L^{\frac{1}{2}}_f{\tilde{P}}_2L^{\frac{1}{2}}_f, {\hat{Q}}=L^{\frac{1}{2}}_f{\tilde{Q}}L^{\frac{1}{2}}_f,{\hat{R}}=L^{\frac{1}{2}}_f{\tilde{R}}L^{\frac{1}{2}}_f$. On the other hand, by the relation $\iota _{max}(\breve{X})=\frac{1}{\iota _{min}(\breve{{\mathbb {M}}})}$, it follows that $I<L^{-\frac{1}{2}}_f{\mathbb {M}}L^{-\frac{1}{2}}_f<\sigma _1I$ which implies $\iota _1=\iota _{min}(\breve{{\mathbb {M}}})>1$ and $\iota _2=\iota _{max}(\breve{{\mathbb {M}}})>\sigma _1.$ Further, from (16) and with the settings $\iota _{max}({\tilde{P}}_2)\le \sigma _2,\iota _{max}({\tilde{Q}})\le \sigma _3, \iota _{max}({\tilde{R}})\le \sigma _4,$ (13) can be rewritten as

$$\begin{aligned} \bigg [\frac{1}{\sigma _1}+\sigma _2 + h\sigma _3 + \frac{h^3}{2}\sigma _4\bigg ]c_1+\varrho (1- e^{-\varpi T_f}) < c_2 e^{-\varpi T_f}. \end{aligned}$$

Subsequently, by employing Schur complement in the preceding relation, we can easily deduce the relation (18). In this way and with Definitions 1 and 2, it is assured that the closed-loop TSFNS (9) is FT $(\xi _1, \xi _2, \xi _3)-\eta$ dissipative, which concludes the theorem’s proof.