Robust ${H}_{\infty}$ output feedback finite-time control for interval type-2 fuzzy systems with actuator saturation

1.   Introduction

The Takagi-Sugeno (T-S) fuzzy system has a strong ability to approximate the nonlinear systems ^[1,2,3]. Compared with the T-S fuzzy system based on type-1 theory, the superiority of IT2 T-S is that it can solve the parametric uncertainties existing in the systems. Due to this fact, many significant works based on the IT2 T-S scheme were explored in ^{[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]}. Among them, the authors in ^[4] and ^[5] investigated reasonable initialization enhanced Karnik-Mendel algorithms and weighted-based non-iterative algorithms for interval type-2 fuzzy logic systems. The authors in ^[6,7,8,9,10] studied the state feedback control design for the IT2 T-S fuzzy systems. Alternatively, ^[11,12,13] introduced the output feedback fuzzy control schemes to address this issue. The authors in ^{[14,15,16,17,18]} proposed stable fuzzy state feedback controllers for IT2 T-S with the discrete-time form. For IT2 T-S fuzzy interconnected systems, the fuzzy decentralized output feedback controller has been proposed in ^[19]. However, it is obvious that although the control methods mentioned in the above results can make the T-S systems based on IT2 fuzzy theory reach the asymptotically stable performance, they ignored the finite convergence time problem which is eagerly desired in practical situations.

Note that the control target of the finite time stability is that the trajectory of system state can approximate the periodic orbits or equilibrium values within a finite time. Besides, the systems converge rapidly and have better robustness subject to uncertainties. Hence, the concept of finite-time is introduced in ^{[20,21,22,23]} and it has received increasing attention and achieved great progress over past decade ^{[24,25,26,27,28,29,30,31]}. Among them, ^[24,25] respectively studied the ${H_\infty }$ finite-time control design problem subject to the measurable uncertain continuous-time Markovian jump systems and discrete-time Markovian jump systems, respectively. Alternatively, the authors in ^[26] and ^[27] studied the output feedback ${H_\infty }$ finite-time control design problems. For IT2 T-S fuzzy systems, the finite-time control design problems subject to time delays and actuator faults are studied in ^[28] and ^[29]. Besides, the designed approaches in ^[30] and ^[31] have extended the fuzzy decentralized control schemes for IT2 T-S fuzzy interconnected systems while the states of each subsystem is available. However, the aforementioned works are based on the assumption that the system states are measurable, thus they are not suitable for the condition in which the states are immeasurable. Besides, they also ignore the phenomenon of the actuator saturation. In practice, the impact of input saturation will extremely damage the control performance of the systems. Hence, the robust ${H_\infty }$ output feedback finite-time control for IT2 T-S fuzzy systems with input saturation and external disturbances is worthy studying. By far, there are no results on investigating the robust ${H_\infty }$ output feedback finite-time control for IT2 T-S fuzzy systems in presence of external disturbances and input saturation.

This paper develops the robust observer-based ${H_\infty }$ output feedback finite-time controller for the IT2 T-S fuzzy systems in presence of external disturbances and input saturation. In the process of control design, an observer associated with IT2 T-S states is firstly designed, and then a robust fuzzy finite-time controller is developed by using the estimating states. After that, a polyhedron model to represent input saturation is introduced. Based on the Lyapunov function scheme and LMIs theory, the sufficient conditions of finite-time stabilization for system with input saturation and external disturbances are gained. The primary contributions of this paper can be roughly concluded as follows

(i) This paper first studied the fuzzy output feedback finite-time ${H_\infty }$ control design problem for the IT2 T-S fuzzy systems. By developing a state observer, the proposed the fuzzy output feedback finite-time control method can not only guarantee the finite-time stabilization of the closed-loop system, but also solve the state unmeasured problem. Note that although the reference ^{[11,12,13,19]} also developed the fuzzy output feedback control methods for a class of IT2 T-S fuzzy systems, they do not consider the finite-time stability, that is, it investigate output feedback asymptotic control, not the output feedback finite-time control like this paper.

(ii) Note that ^{[28,29,30,31]} investigated the finite-time control of IT2 T-S fuzzy system, however, the limitation is that the system state must be completely measurable which are not suitable for systems with immeasurable states like this study. Besides, the impact of input saturation is also investigated in this paper.

The rest of this paper is comprised such that: Section 2 gives the descriptions of the IT2 T-S fuzzy systems, state observer design, actuator saturation functions and necessary definitions and lemmas. The robust finite-time $\mathit{H}_{\infty}$ controller design, the finite-time stabilization and the finite-time boundedness conditions of the considered system are derived in Section 3. In order to verify the feasibility of the developed scheme, two practical examples are given in Section 4. Section 5 derives the conclusion.

2.   Preliminaries

In this section, the descriptions of the IT2 T-S fuzzy system and input saturation functions will be shown.

2.1.   IT2 T-S fuzzy system

Consider the IT2 T-S fuzzy system with continue-time form as follows.

Plant rule p: If $\theta_{1} \left(x \left(t \right) \right)$ is $F_{p1}$ and $\cdots$ and $\theta_{i} \left(x \left(t \right) \right)$ is $F_{pi}$, then

where ${\theta _s}(x(t)) = {[{\theta _1}(x(t)), {\theta _2}(x(t)), \ldots, {\theta _i}(x(t))]^T}$ represents the premise vector and ${F}_{ps}$ represents the IT2 fuzzy set. ${x} \left({t} \right) \in \Re ^{n}$, ${sat} \left({u} \left({t} \right) \right) \in \Re ^{m}$ and ${w} \left({t} \right) \in \Re ^{h}$ are system state vector, saturated control input vector and bounding external disturbances satisfying ${L}_{2} \left({0}, \infty \right]$, respectively. ${z} \left({t} \right) \in \Re ^{g}$ is the controlled output and ${y} \left({t} \right) \in \Re ^{l}$ is the measured output. ${A}_{p}$, ${B}_{p}$, ${D}_{p}$, ${E}_{p}$, ${F}_{p}$ and ${C}_{p}$ are known constant matrices with appropriate dimensions.

Based on the IT2 fuzzy logic systems theory, the firing strength of the $p{\rm{th}}$ fuzzy rule can be defined within the following interval sets

Through utilizing the method of singleton fuzzifier, product fuzzy inference, and weighted average defuzzifier, the IT2 T-S fuzzy system (2.1) is inferred as follows

in which

the above $\vartheta_{P}^{L}\left(\theta \left({x} \left({t} \right) \right)\right)$, $\vartheta_{P}^{U}\left(\theta \left({x} \left({t} \right) \right)\right)$ are the lower and upper grades of memberships, respectively. ${\underline \mu_{{F_{ps}}}}(x(t))$ and ${\bar \mu _{{F_{ps}}}}(x(t))$ are the lower and upper membership functions of IT2 fuzzy sets ${F_{ps}}$. $\upsilon _{P}^{L}\left(\theta \left({x} \left({t} \right) \right)\right)$, $\upsilon _{P}^{U}\left(\theta \left({x} \left({t} \right) \right)\right)$ are known weighting coefficient nonlinear functions.

2.2.   Saturation functions

The aforementioned $u(t)$ is the control signal given by the ideal output feedback fuzzy controller and $sat(u(t)) = {[sat({u_1}), sat({u_2}), \ldots, sat({u_m})]^T}$ is the practical control signal, which is used to address saturation problem.

In order to model better saturation effect, the polytopic model introduced in ^[32,33] will be utilized. Referring from ^[32,33], we can obtain such that

in which ${sat} \left({u}_{a}\right) = {sign}\left({u}_{a}\right)min\left \{ \overline{u}_{a}, \left |{u}_{a} \right |\right \}, \overline{u}\in \Re ^{m}$ is the saturation level and ${u} \left(t \right) \in \Re ^{m}$ represents the control input, $\overline{u}_{a}$ and ${u}_{a}$ are the ${a}$th element of the saturation level and the control input, respectively.

Let $V$ be the set of $m \times m$ diagonal matrices whose diagonal elements are either 1 or 0. For example, when $m = 2$, then

There are ${2^m}$ elements in $V$. Suppose that each that element of $V$ is labeled as ${H_l}, \; (l = 1, 2, \ldots, {2^m})$, and denote $H_l^ - = I - {H_l}$. Clearly, $H_l^ -$ is also an element of $V$ when ${H_l} \in V$.

For development, the following lemma associated with input saturation is required.

Lemma 1 ^[32,33]. Assume $\left| {{v_a}} \right| \le {\bar u_a}$, in which ${v_a}$ and ${u_a}$ denote the $a{\rm{th}}$ element of $v \in {\Re ^m}$ and $u \in {\Re ^m}$, respectively. If $x(t)$ is inside $\mathop \cap \limits_{j = 1}^r \left\{ {x \in {\Re ^n}\left| {\; \left| {h_p^jx} \right| \le {{\bar u}_p}} \right.} \right\}$, then

in which $0 \le {\delta _l} \le 1\, (l = 1, \ldots, {2^m})$ are scalars and $\sum\nolimits_{l = 1}^{{2^m}} {{\delta _l}} = 1$.

2.3.   Observer-based controller design

In this subsection, a state observer will be designed to estimate immeasurable states. Subsequently, the fuzzy controller will be proposed.

The identical membership functions, which rely on estimated states rather than system states, are used to design the fuzzy observer.

Observer Rule $q$:If ${\theta _1}(\hat x(t))$ is $F _{q1}$ and $\cdots$ and ${\theta _q}(\hat x(t))$ is $F _{qi}$, then

where $\hat{x}\left({t} \right) \in \Re ^{n}$ denotes the estimated state vector, $\hat{z}\left({t} \right) \in \Re ^{l}$ is the estimation output and $\hat{y}\left({t} \right) \in \Re ^{l}$ is the observer output, $L_{p}$ is observer gain matrix to be solved.

After that, the final state observer is described such that

Due to that the membership functions of fuzzy observer rely on the estimated state variable rather than $x(t)$, thereby, $\hat x(t)$ is selected as the membership function representations.

For the purpose of improving the flexibly of fuzzy controller design, by utilizing non-PDC design algorithm, the controller possesses its exclusive membership functions which are different from fuzzy system (2.3) and fuzzy observer (2.7). The details are shown such that:

Controller Rule $j$:If ${g_1}(\hat x(t))$ is $M _{j1}$ and $\cdots$ and ${g_j}(\hat x(t))$ is ${M_{j\tau }}$, then

where ${K_j}(j = 1, 2, \ldots, r)$ are the controller gains to be solved later. Similarly, the $j{\rm{th}}$ firing strength can be expressed as

in which

Then, the fuzzy controller based on IT2 fuzzy theory can be obtained

where

the above ${\underline \mu_{{M_{j\hbar }}}}({g_\hbar }(\hat x(t)))$, ${\bar \mu _{{M_{j\hbar }}}}({g_\hbar }(\hat x(t)))$, $\eta _j^L(\hat x(t))$, $\eta _j^U(\hat x(t))$, $b_j^L(\hat x(t))$ and $b_j^U(\hat x(t))$ respectively denote the lower and upper membership functions, the lower and upper grades of membership, the tradeoff coefficient nonlinear functions.

Then, from Lemma 1, the observer-based controller in presence of input saturation is given as

where ${Z_j}$ is $m \times n$ matrix and $h_p^j$ represents the $p{\rm{th}}$ row of ${Z_j}$.

Define the estimation error $\tilde{x}\left (t \right) = x\left (t \right)-\hat{x}\left (t \right)$, controlled output error $\tilde{z}\left (t \right) = z\left (t \right)-\hat{z}\left (t \right)$ and considering equations (2.3), (2.7) and (2.11), one can obtain

where

$\xi (t) = {\left[{\begin{array}{*{20}{c}} {x(t)} & {\tilde x(t)} \\ \end{array}} \right]^T}$, ${A_{pqj}} = \left[{\begin{array}{*{20}{c}} {{A_p} + {\delta _l}{B_p}{H_l}{K_j} + {\delta _l}{B_p}H_l^ - {Z_j}} & { - {\delta _l}{B_p}{H_l}{K_j} - {\delta _l}{B_p}H_l^ - {Z_j}} \\ {{A_p} - {A_q} - {L_q}{C_p} - {L_q}{C_q}} & {{A_q} - {L_q}{C_q}} \\ \end{array}} \right]$, ${D_{pqj}} = \left[{\begin{array}{*{20}{c}} {{D_p}} \\ {{D_p}} \\ \end{array}} \right]$, ${C_{pqj}} = \left[{\begin{array}{*{20}{c}} {{E_p} - {E_q}} & {{E_q}} \\ \end{array}} \right]$.

Control objective: This study will design a robust ${H_\infty }$ finite-time output feedback controller for the IT2 T-S fuzzy systems (2.3). The developed controller can make the T-S systems based on IT2 fuzzy theory reach the finite-time stable, in which the input saturation and external disturbances are considered.

Remark 1: Note that the authors in ^[19] studied the output feedback control design problem for IT2 T-S fuzzy system, the developed controller can make the controlled system achieve asymptotic stability. However, the authors in ^[19] do not consider the finite-time stability problem. Unlike the above work, in this paper, the robust output feedback finite-time control design problem for T-S systems with input saturation based on IT2 theory is studied. Therefore, the control design and stability analysis are more difficult and challenging.

2.4.   Main definitions and lemmas

For development, the following definitions and lemmas are needed to achieve the control objective.

Definition 1 ^[20,21]. If for given constants $a_{2} > a_{1}\geq 0$ and a positive matrix $R > 0$ satisfying

Then the system (2.3) can be robust finite-time stable within $\left (a_{1}, a_{2}, R, T \right)$.

Definition 2 ^[34]. System (2.3) is of robust $H_{\infty }$ finite-time boundedness performance if there exist positive constants $\bar{\rho} > 0$ and $h > 0$, and the closed–loop system based on IT2 T-S fuzzy theory satisfies (2.13) and following $H_{\infty }$ performance index

where ${\left\| {w(t)} \right\|^2} < h$, and $T$ is a given positive scalar.

Lemma 2 ^[29]. For full rank matrix $rank\left (C \right) = l, F\in \Re ^{l\times n}$ with $l < n$, there exist a singular value decomposition (SVD) for $C = C_{p}$ can be described as $C = O\begin{bmatrix} S & 0 \end{bmatrix}U^{T}$, in which $V\in \Re ^{l\times l}$, $O\in \Re ^{l\times n}$, $S\in \Re ^{n\times n}$, $U\cdot U^{T} = I$ and $O \cdot {O^T} = I$. Let matrices $X > 0$, ${X_{11}} \in {\Re ^{l \times l}}$, ${X_{22}} \in {\Re ^{(n - l) \times (n - l)}}$. Then, there exists $\bar X$ satisfies $CX = \bar XC$ if and only if the following Eigen-value decomposition (EVD) condition holds

3.   Stability analysis

The robust finite-time stability analysis and the robust finite-time $H_{\infty }$ boundedness of the fuzzy system (2.3) will be respectively proved in this section. Based on the Lyapunov function method and LMIs theory, several sufficient conditions will be derived.

Theorem 1: For given positive scalars $T > 0$, $a_{1} > 0$, $h > 0$, $\beta > 0$, $\delta _{l} > 0$, $\bar \rho > 0$ and a positive symmetric $R > 0$, system (2.3) can achieve robust finite-time stability within $({a_1}, {a_2}, R, T, \bar \rho, h)$, if there exist matrix $P = {P^T} > 0$ satisfying the following matrix inequalities

in which

Proof: Construct the following Lyapunov function

For convenience, one define $P = {\rm{diag\{ }}{P_1}{\rm{, }}{P_1}{\rm{\} }}$ and then calculate the time derivative of (3.6), one has

Define $\chi (t) = {\left[{\begin{array}{*{20}{c}} {\xi (t)} & {w(t)} \\ \end{array}} \right]^T}$, and then substituting the definition of ${A_{pqj}}$, ${C_{pqj}}$, ${D_{pqj}}$ into (3.7), (3.7) can be rewritten as

According to ^[28], the slack matrix is introduced to avoid conservative results as shown such that

Then, from (3.8) and (3.9), one has

Then, based on Theorem 1, one gets

Multiplying (3.11) by ${e^{ - \beta t}}$ and integrating both sides of (24) from 0 to $t$, then since ${\tilde z^T}(t)\tilde z(t) > 0$ and ${\Pi _{pqj}} < 0$, one can obtain

Then, multiplying (3.12) by ${e^{\beta t}}$, (3.12) can be represented as

Recalling to $V(\xi (t)) = {\xi ^T}(t)P\xi (t)$, $\tilde P = {R^{ - \frac{1}{2}}}P{R^{ - \frac{1}{2}}}$ and ${\left\| {w(t)} \right\|^2} < h$, (3.13) leads to

in which ${\lambda _1} = {\lambda _{\max }}(\tilde P)$.

Define ${\lambda _2} = {\lambda _{\min }}(\tilde P)$, since $V(\xi (t)) = {\xi ^T}(t)P\xi (t) \ge {\lambda _2}{\xi ^T}(t)R\xi (t)$, then form (3.14), one has

Suppose ${\xi ^T}(t)R\xi (t) < \frac{{{e^{\beta T}}({\lambda _1}{a_1} + \frac{{{{\bar \rho }^2}{e^{ - \beta T}}h}}{\beta }(1 - {e^{ - \beta T}}))}}{{{\lambda _2}}}, \; \; \forall t \in [0, T]$, then from Definition 1, finite-time stability of fuzzy system (2.3) is ensured within $({a_1}, {a_2}, R, T, \bar \rho, h)$.

In what follows, we will give the proof of the robust ${H_\infty }$ output feedback finite-time boundedness performance of the fuzzy system (2.3).

From the inequality (3.7), on can easily find that

Similarly, multiplying (3.16) by ${e^{ - \beta t}}$, and then integrating it from 0 to $T$, one has

Under the condition of $V(\xi (0)) = 0$ and $V(\xi (T)) > 0$, (3.17) can be rewritten such that

Since ${e^{ - \beta T}}\int_0^T {{e^{ - \beta t}}{w^T}(t)w(t)dt} < \int_0^T {{e^{ - \beta t}}{{\tilde z}^T}(t)\tilde z(t)dt}$, ${\bar \rho ^2}{e^{ - \beta T}}\int_0^T {{e^{ - \beta t}}{w^T}(t)w(t)dt} < {\bar \rho ^2}{e^{ - \beta T}}\int_0^T {{w^T}(t)w(t)dt}$, then from (3.18), one can get

Further, the robust ${H_\infty }$ finite-time boundedness performance index has been obtained. Hence, from Definition 2, the fuzzy system (2.3) is finite-time and can achieve a robust ${H_\infty }$ performance.

The solution of positive controller gains ${K_j}$, observer gains ${L_q}$ and positive definite symmetric matrix $P$ are given in following.

Firstly, define $X = P_1^{ - 1}$, ${Y_j} = {K_j}X$, ${S_j} = {Z_j}X$ and ${T_q} = {L_q}\bar X$, by left and right multiplying ${\rm{diag}}\{ X, X, I\}$ on both sides of (3.2) and (3.3), thus one can get the following LMIs

where

with

Subsequently, by solving LMIs (3.19) and (3.20), one can obtain $X$, ${Y_j}$, ${S_j}$ and ${T_q}$, the it follows ${P_1}$, ${K_j} = {Y_j}{X^{ - 1}}$ and ${S_j} = {H_j}{X^{ - 1}}$. According to Lemma 2, the SVD of the output matrix $C \in {\Re ^{l \times n}}$, for $C = {C_p}$ with a full rank $C = O[S, 0]{U^T}$, and there exists a matrix $\bar X \in {\Re ^{n \times n}}$ satisfying $CX = \bar XC$, if there exists EVD $X = U\left[{\begin{array}{*{20}{c}} {{X_{11}}} & 0 \\ 0 & {{X_{22}}} \\ \end{array}} \right]{U^T}$. Then, one can find that $O[S, 0]{U^T}U\left[{\begin{array}{*{20}{c}} {{X_{11}}} & 0 \\ 0 & {{X_{22}}} \\ \end{array}} \right]{U^T} = \bar XO[S, 0]{U^T}$. By calculation, one gets $\bar X = OS{X_{11}}{S^{ - 1}}{O^{ - 1}}$, thus, ${\bar X^{ - 1}} = OSX_{11}^{ - 1}{S^{ - 1}}{O^{ - 1}}$. For bilinear terms ${L_q}CX = {L_q}\bar XC$, let ${T_q} = {L_q}\bar X$, then one can obtain ${L_q} = {T_q}{\bar X^{ - 1}} = {T_q}OSX_{11}^{ - 1}{S^{ - 1}}{O^{ - 1}}$.

From ^[28,29], we know that the constraint $\mathop \cap \limits_{j = 1}^r \left\{ {x \in {\Re ^n}\left| {\; \left| {h_p^jx} \right| \le {{\bar u}_p}} \right.} \right\}$ is equivalent to

By Schur complement, (3.21) is equivalent to the following LMI

The design procedures are organized:

Step1: Choose the fuzzy rules, and then give ${\bar u_p}(p = 1, \ldots, r)$, $T > 0$, $a_1 > 0$, $h > 0$, $\bar \rho > 0$, $\beta > 0$ and select $0 \le {\delta _l} \le 1\, (l = 1, \ldots, {2^m})$ which satisfies $\sum\nolimits_{l = 1}^{{2^m}} {{\delta _l}} = 1$, and diagonal matrices ${H_l}$, $H_l^ -$, which satisfies ${H_l} + H_l^ - = {I_n}$.

Step 2: Solve the LMI (3.19), (3.20) to get $X$, ${K_j}$, ${Z_j}$ and ${L_q}$.

Step 3: Substitute the solved gains into (3.1), (3.4) and (3.5) to verify whether they are established.

Step 4: In case of the condition (3.1), (3.4) and (3.5) are not satisfied, increase $\bar \rho$, ${a_2}$ and ${\bar u_p}(p = 1, \ldots, r)$ and increase $\beta$ until the conditions are established. Else, decrease $\bar \rho$, ${a_2}$ and ${\bar u_p}(p = 1, \ldots, r)$ and increase $\beta$ and go to Steps 1-4 until $X$ cannot be obtained.

Step 5: Represent the fuzzy state observer (2.7) and fuzzy controller (2.11).

4.   Simulation results

Example 1 (Mass-spring-damping system) A mass-spring-damping system example as shown in Figure 1 is employed to verify the feasibility of the developed scheme in this section. The corresponding dynamical equation is shown as follows

where $M$ is the mass, $F_{c1}$ and $F_{c2} = r\dot{x}$ denote the friction force, $F_{c2}$ is the restoring force of the spring, and $u(t)$ denotes the control input. $F_{c2} = r\dot{x}$ with $r > 0$ and ${F_{c1}} = d(1 + {\varepsilon ^2}{x^2})x$. Then, (4.1) can be described as

where $x$ is the displacement from a reference point. Define $x(t) = {\left[{\begin{array}{*{20}{c}} {{x_1}(t)} & {{x_2}(t)} \end{array}} \right]^T} = {\left[{\begin{array}{*{20}{c}} x & {\dot x} \end{array}} \right]^T}$ and $\tilde f(t) = (- d - d{\varepsilon ^2}x_1^2(t))/M$. The operating domain are ${x_1}(t) \in [-2, 2]$ and $d \in [5,8]N/M$. Let $M = 1\; {\rm{kg}}$, $r = 2\; {\rm{N}} \cdot {\rm{m/s}}$, and $\varepsilon = 0.3\; {{\rm{m}}^{ - 1}}$. Then, we can calculate that ${\tilde f_{\max }} = - 5$ in which $d = 5$ and ${x_1}(t) = 0$. ${\tilde f_{\min }} = - 10.88$, $d = 8$ and $x_1^2(t) = 4$.

System (4.2) is approximated by the following two fuzzy models

Rule 1: If ${\theta _1}(x(t))$ is $F_{11}$, then:

Rule 2: If ${\theta _1}(x(t))$ is $F_{21}$, then:

where $x(t) = {\left[{\begin{array}{*{20}{c}} {{x_1}(t)} & {{x_2}(t)} \end{array}} \right]^T} = {\left[{\begin{array}{*{20}{c}} x & {\dot x} \end{array}} \right]^T}$, in which ${x_1}(t)$ represents the premise vector.$F_{11}$ and $F_{21}$ represent the IT2 fuzzy sets of the known function ${x_1}(t)$.

Finally, the system (4.3) and (4.4) based on IT2 fuzzy theory is represented such that

The fuzzy state observer is designed such that

Then the IT2 T-S fuzzy ${H_\infty }$ output feedback finite-time controller can be obtained as

where ${H_1} = 1$, $H_1^ - = 0$, ${H_2} = 0$, $H_2^ - = 1$, ${\delta _1} = 0.6$ and ${\delta _2} = 0.4$.

The disturbance input is selected as

The normalized membership functions of fuzzy system, state observer and controller are selected by Table 1, Table 2 and Table 3, respectively.

For given ${a_1} = 5.8426$, ${a_2} = 12.8613$, ${h} = 3$, ${\beta} = 0.01$, $\bar \rho = 0.3$, ${T} = 10$, and then by solving LMIs in Theorem 1 with the saturation levels $\bar u = 0.18$, we are able to get the constants and positive definite matrices as follows:

The results are carried out by choosing the initial conditions $x(0) = {\left[{\begin{array}{*{20}{c}} {0.7} & { - 0.6} \end{array}} \right]^T}$, $\hat x(0) = {\left[{\begin{array}{*{20}{c}} {0.6} & { - 0.5} \end{array}} \right]^T}$. Then, Figures 2-6 show the responses of closed-loop system. In addition, by Figures 2-3, from which we can see that all the estimations converge to zero asymptotically. Besides, the precise condition ${\xi ^T}(0)R\xi (0) \le {a_1} = 5.8426$ is verified, and operating domain of the membership functions is established, i.e, ${x_1}(t) \in [-2, 2]$. Figure 4 is the response of the saturated control input $u$ which satisfies the conditions that $\bar u \le 0.18$. From Figure 5, one can find that ${\xi ^T}(t)R\xi (t)$ stays within the bound of ${a_2} = 12.8613$, which implies that the definition of robust ${H_\infty }$ finite-time stability is satisfied. Figure 6 shows the distance attenuation for the fuzzy system, from which one can know that the robust ${H_\infty }$ finite-time boundedness is satisfied.

Example 2 (Tunnel diode circuit system) A nonlinear tunnel diode circuit system example as shown in Figure 7 is employed to verify the feasibility of the developed scheme in this subsection.

where ${i_L}(t)$ is the inductor current, and ${i_D}(t)$ depicts the nonlinear volt-ampere characteristic of tunnel diode. ${V_C}$ is the voltage of the capacitor $C$. ${V_D}$ is the voltage of the tunnel diode $D$, $R$ and $L$ are electrical resistance.

The corresponding dynamical equation is shown as follows

in which $\tau \in [0.01, 0.03]$ is an uncertain parameter. Let ${x_1}(t) = {V_D}(t)$, ${x_2}(t) = {i_D}(t)$ and $g({V_D}(t)) = 0.002 + \tau V_D^2(t)$ as nonlinear function composed of the terminal voltage across the tunnel diode. Here, state variable is supposed to be ${x_1}(t) \in [-3, 3]$. Then, one can find that ${g_{\max }} = 0.0272$ with $\tau = 0.03$ and ${x_1}(t) = 3$. ${g_{\min }} = 0.002$ with $\tau = 0.01$ and ${x_1}(t) = 0$.

Then, (4.9) can be described as

Then system (4.10) is approximated by the following two fuzzy models

Rule 1: If ${\theta _1}(x(t))$ is $F_{11}$, then:

Rule 2: If ${\theta _1}(x(t))$ is $F_{21}$, then:

where $x(t) = {\left[{\begin{array}{*{20}{c}} {{x_1}(t)} & {{x_2}(t)} \\ \end{array}} \right]^T}$, in which ${x_1}(t)$ represents the premise vector.$F_{11}$ and $F_{21}$ represent the IT2 fuzzy sets of the known function ${x_1}(t)$.

The normalized membership functions of fuzzy system, state observer and controller are selected by Table 4, Table 5 and Table 6, respectively.

For given ${\delta _1} = 0.7$, ${\delta _2} = 0.3$, ${a_1} = 2.9403$, ${a_2} = 10.4953$, ${h} = 0.3$, ${\beta} = 0.2$, $\bar \rho = 0.3$, ${T} = 6$, and then by solving LMIs in Theorem 1 with the saturation levels $\bar u = 0.82$, we are able to get the constants and positive definite matrices as follows:

The results are carried out by choosing the initial conditions the same as Example 1. Then, Figures 8-12 show the responses of closed-loop system. In this example, ${\xi ^T}(0)R\xi (0) \le {a_1} = 2.9403$, saturated control input $\bar u \le 0.82$ and ${\xi ^T}(t)R\xi (t) \le {a_2} = 10.4953$.

To further verify the feasibility of the proposed fuzzy output feedback finite-time controller, we apply the fuzzy controller in ^[19], which is designed based on asymptotic stability theory to control system (4.9). Especially pointed out that, the controlled IT2 T-S interconnected system in ^[19] has been revised as IT2 T-S fuzzy system, thus can be compared further. In the simulation, we utilize the same IT2 T-S fuzzy system and the same initial conditions of $x(t)$ and $\hat x(t)$. The controller and observer gain matrices are as follows:

The simulation results are also depicted by Figs. 8-9. From Figs. 8-9, we clearly know that the variables and their estimations of the controlled system in this study have faster convergent rates than those by ^[19].

5.   Conclusions

The robust ${H_\infty }$ observer-based finite-time control problem for IT2 T-S fuzzy system in presence actuator saturation has been studied. In order to obtain the finite-time stability and robust ${H_\infty }$ finite-time boundedness for the controlled plant, a fuzzy state observer and a fuzzy robust ${H_\infty }$ finite-time controller with saturation limitation have been proposed. Then, on the basis of the Lyapunov function method and LMIs theory, the finite-time stability sufficient conditions are derived to ensure of IT2 T-S fuzzy systems in presence of input saturation limitation. At last, two practical examples of nonlinear mass-spring-damping and tunnel diode circuit systems is given to verify the feasibility of the developed fuzzy control scheme.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (No. 61903167).

Conflict of interest

The authors declare that there is no conflicts of interest.