1 Introduction

In the past decades, many approximation-based adaptive backstepping control approaches have been developed to deal with uncertain nonlinear strict-feedback systems via fuzzy-logic-systems (FLSs) or neural-networks (NNs) approximators; see, for example, [1–16] and references herein. Adaptive fuzzy or Neural network backstepping control approaches in [1–10] are for single-input and single-output (SISO) nonlinear systems, and in [11, 12] are for multiple-input and multiple-output (MIMO) nonlinear systems, while those in [13–16] are for SISO/MIMO nonlinear systems with immeasurable states. Adaptive fuzzy or neural network backstepping control approaches can provide a systematic methodology of solving tracking or regulation control problems for a larger of unknown nonlinear systems, where FLSs or NNs are used to approximate unknown nonlinear functions, and the backstepping design technique is applied to construct adaptive controllers and the adaptation adjusted laws of the parameters. Two of the main features of these adaptive approaches are (i) they can be used to deal with those nonlinear systems without satisfying the matching conditions, and (ii) they do not require the unknown nonlinear functions being linearly parameterized. Therefore, the approximator-based adaptive fuzzy or neural network backstepping control becomes one of the most popular design approaches to a large class of uncertain nonlinear systems.

Despite that many developments have been achieved for the adaptive backstepping control of uncertain nonlinear strict-feedback systems using FLSs or NNs, the mentioned above adaptive control approaches are only applied to the deterministic nonlinear strict-feedback systems without stochastic disturbances. It is well known that stochastic disturbances often exist in many practical systems. Their existence is a source of instability of the control systems, thus, the investigations on stochastic systems modeling and control have received considerable attention in recent years [17]. Authors in [18] first proposed an adaptive backstepping control design approach for strict-feedback stochastic systems by a risk-sensitive cost criterion. Authors in [19] solved the output feedback stabilization problem of strict-feedback stochastic nonlinear systems by using the quartic Lyapunov function, while authors in [20] and [21] developed backstepping control design approaches for nonlinear stochastic systems with Markovian switching. Meanwhile, by using the linear reduced-order state observer, several different output-feedback controllers are developed in [22–24] for strict-feedback nonlinear stochastic systems with unmeasured states. However, these schemes are only suitable for those nonlinear stochastic systems with nonlinear dynamics models known exactly or with the unknown parameters appearing linearly with respect to known nonlinear functions.

To handle the above the problems, authors in [25] and [26] first developed adaptive output feedback control approaches for a class of uncertain nonlinear stochastic systems by using neural networks and the stability proofs of the control systems are given on the stochastic stability theory [27]. Afterward, authors in [28] extended the results of [25] and [26] to a class of uncertain large-scale nonlinear stochastic systems and developed adaptive NN decentralized output feedback control schemes. The adaptive NN backstepping control approaches in [25, 26], and [28] can control a class of nonlinear stochastic systems with immeasurable states, however, the nonlinear uncertainties in the nonlinear stochastic systems are only the functions of the system output, not related with the other states variables. Moreover, the mentioned above approaches are only limited to those SISO or large-scale nonlinear stochastic systems. To our best knowledge, to date, there are few results on MIMO stochastic nonlinear systems with immeasurable states.

Motivated by the above observations, in this paper, an observer-based adaptive fuzzy backstepping output feedback DSC approach is proposed for a class of MIMO stochastic nonlinear strict-feedback systems. In the design, the FLSs are first used to approximate the unknown functions, and a nonlinear fuzzy state observer is designed to estimate the unmeasured states. Combining the adaptive backstepping design along with the DSC technique, an observer-based adaptive fuzzy backstepping control approach is developed. It is proved that this control approach can guarantee that all the signals of the closed-loop system are semiglobally uniformly ultimately bounded (SUUB) in probability, and the observer errors and the output of the system converge to a small neighborhood of the origin by appropriate choice of the design parameters. Compared with the existing results, the main advantages of the proposed control schemes are as follows: (i) by designing a fuzzy nonlinear state observer, the proposed adaptive control method does not require that all the states of the system are measured directly. Meanwhile, the designed state observer can achieve the better estimation results for the unmeasured states than the linear reduced-order state observer in [25, 26, 28]. (ii) DSC technique is incorporated in adaptive fuzzy backstepping control design, thus the proposed adaptive control method can overcome the problem of “explosion of complexity” inherent in the methods of [25, 26, 28].

2 Problem formulation and some preliminaries

2.1 Problem formulation

Consider the following MIMO uncertain strict-feedback stochastic nonlinear system

(1)

where \(\underline{x}_{j,i_{j}} = (x_{j,1}, \ldots,x_{j,i_{j}})^{T} \in R^{i_{j}}\), i j =1,2,…,m j is the state vector for the first i j differential equations of the jth subsystem, u j and y j are the input and output of the first j subsystems. \(f_{j,i_{j}}( \cdot) \) is an unknown smooth nonlinear function. \(X = (x_{1}^{T}, \ldots,x_{n}^{T})^{T} \) with \(x_{j} = (x_{j,1},\ldots,x_{j,m_{j}})^{T}\). w is an independent r-dimensional standard Wiener process. In this paper, it is assumed that the only output variable y j =x j,1 is available for measurement.

Assumption 1

\(\phi_{j,i_{j}}(\underline{x}_{j,i_{j}}) =g_{j,i_{j}}(y_{j})\), where \(g_{j,i_{j}}(y_{j})\) is a smooth function satisfying locally Lipschitz condition.

Write (1) in the state space form

(2)

where

Choose vector K j such that matrix A j is a strict Hurwitz, therefore, given \(Q_{j} = Q_{j}^{T} > 0\), there exists a positive definite matrix \(P_{j} = P_{j}^{T} \) such that

$$ A_{j}^{T}P_{j} +P_{j}A_{j} = - Q_{j}$$
(3)

Control objective: Using fuzzy logic systems to determine an output feedback controller and parameters adaptive laws such that all the signals involved in the closed-loop system are SUUB in probability and the observer errors and the output of the system are as small as the desired.

2.2 Stochastic system and stability

To establish stochastic stability as preliminary, we consider the following stochastic nonlinear system:

$$ d\chi(t) = f \bigl(\chi(t) \bigr)dt + g \bigl(\chi(t) \bigr)d\omega(t)$$
(4)

where χ∈R n is the state, ω is an r-dimensional independent standard Wiener process, and f(⋅):R n→R n and g(⋅):R n→R n×r are locally Lipschitz and satisfy f(0)=0, g(0)=0.

Define a differential operator ℓ for twice continuously differentiable function V(χ) as follows:

$$ \ell V(\chi) = \frac{\partial V}{\partial\chi} f(\chi) + \frac{1}{2}\mathit{Tr}\biggl\{ g^{T}(\chi)\frac{\partial^{2}V}{\partial\chi^{2}}g(\chi) \biggr\}$$
(5)

Recall two stability notions for nonlinear stochastic system (4).

Definition 1

[27]

Consider system (4) with f(0)=0 and g(0)=0. The solution χ(t)=0 is said to be asymptotically stable in the large if for any ε>0,

$$\mathop{\lim}_{\chi(0) \to0}P \Bigl\{ \mathop{\sup}_{t \ge 0} \bigl \Vert \chi(t) \bigr \Vert \ge\varepsilon \Bigr\} = 0$$

And for any initial condition χ(0),

$$P \Bigl\{ \mathop{\lim}_{t \to\infty} \chi(t) = 0 \Bigr\} = 1$$

Definition 2

[27]

The solution process {χ(t),t≥0} of stochastic differential system (4) is said to be bounded in probability, if

$$\mathop{\lim}_{c \to\infty} \mathop{\sup}_{0 \le t \le \infty} P \bigl\{ \bigl \Vert \chi(t) \bigr \Vert \ge c \bigr\} = 0$$

Lemma 1

Consider the stochastic nonlinear system (4). If there exists a positive definite, radially unbounded, twice continuously differentiable Lyapunov V:R n→R, and constants ρ>0 and μ≥0, such that

$$ \ell V(\chi) \le- \rho V(\chi) + \mu $$
(6)

then the following conclusions are true:

  1. (1)

    the system has a unique solution almost surely;

  2. (2)

    the system is bounded in probability;

  3. (3)

    in addition, if f(0)=0 and g(0)=0 and μ=0. Then the system is asymptotically stable in the large.

Lemma 2

(Young’s inequality)

For any vectors x,y∈R n, there is inequality, \(x^{T}y \le\frac{a^{p}}{p}\Vert x \Vert^{p} +\frac{1}{qa^{q}}\Vert y \Vert^{q}\), where a>0,p>1,q>1, and (p−1)(q−1)=1.

2.3 Fuzzy logic systems

A FLS consists of four parts: the knowledge base, the fuzzifier, the fuzzy inference engine, and the defuzzifier. The knowledge base is composed of a collection of fuzzy. If-then rules of the following form:

(7)

where x=(x 1,x 2,…,x n )T and y are FLS input and output, respectively, \(\mu_{F_{i}^{l}}(x_{i})\) and \(\mu_{G^{l}}(y) \) are the membership function of fuzzy sets \(F_{i}^{l}\) and G l,N is the number of inference rules.

Through singleton fuzzifier, center average defuzzification and product inference [29], the FLS can be expressed as

$$ y(x) = \frac{\sum_{l = 1}^{N} \bar{y}_{l}\prod_{i = 1}^{n} \mu_{F_{i}^{l}}(x_{i})}{\sum_{l = 1}^{N} [\prod_{i = 1}^{n} \mu_{F_{i}^{l}}(x_{i})]}$$
(8)

where \(\bar{y}_{l} = \max_{y \in R}\mu_{G^{l}}(y)\).

Define the fuzzy basis functions as

$$ \varphi_{l} = \frac{\prod_{i = 1}^{n}\mu_{F_{i}^{l}}(x_{i})}{\sum_{l= 1}^{N} [\prod_{i = 1}^{n} \mu_{F_{i}^{l}}(x_{i}) ]}$$
(9)

Denoting \(\theta^{T} = [ \bar{y}_{1},\bar{y}_{2}, \ldots,\bar{y}_{N} ] =[ \theta_{1},\theta_{2}, \ldots,\theta_{N} ]\) and φ(x)=[φ 1(x),φ 2(x),…,φ N (x)]T, then fuzzy logic system (8) can be rewritten as

$$ y(x) = \theta^{T}\varphi(x)$$
(10)

Lemma 3

[29]

Let f(x) be a continuous function defined on a compact set Ω. Then for any constant ε>0, there exists a fuzzy logic system (10) such as

$$ \mathop{\sup}_{x \in \varOmega } \bigl \vert f(x) -\theta^{T}\varphi(x) \bigr \vert \le\varepsilon $$
(11)

By Lemma 3, we can assume that the nonlinear functions in (1) can be approximated by the following fuzzy logic systems as

(12)

where 1≤j≤n,i j =1,2,…,m j . \(\hat{X}_{j,i_{j}} \) is the estimation of state vector \(X_{j,i_{j}}\).

The optimal parameter vector \(\theta_{j,i_{j}}^{ *}\) is defined as

(13)

where \(\varOmega _{j,i_{j}},U_{j,i_{j}}\), and \(\hat{U}_{j,i_{j}} \) are compact regions for \(\theta_{j,i_{j}},X_{j,i_{j}}\), and \(\hat{X}_{j,i_{j}}\), respectively. The fuzzy minimum approximation errors \(\varepsilon_{j,i_{j}} \) and approximation errors \(\delta_{j,i_{j}} \) are defined as

(14)

Assumption 2

[9, 15, 30]

There are unknown positive constants \(\varepsilon_{j,i_{j}}^{ *}\) and \(\delta_{j,i_{j}}^{ *}\) such that \(\vert \varepsilon_{j,i_{j}} \vert \le\varepsilon_{j,i_{j}}^{ *}\) and \(\vert \delta_{j,i_{j}} \vert \le\delta_{j,i_{j}}^{ *}\).

Denote \(\omega_{j,i_{j}} = \varepsilon_{j,i_{j}} - \delta_{j,i_{j}}\), by Assumption 2, one has \(\vert \omega_{j,i_{j}} \vert \le\varepsilon_{j,i_{j}}^{*} + \delta_{j,i_{j}}^{ *} = \omega_{j,i_{j}}^{ *}\), where \(\omega_{j,i_{j}}^{ *}\) is also an unknown constant. \(\varepsilon_{j,i_{j}}^{ *}\) and \(\omega_{j,i_{j}}^{ *}\) can be estimated by the parameters adaptation laws to be designed in the next section.

3 Nonlinear fuzzy adaptive observer design

Note that the states \((x_{j,2}, \ldots,x_{j,m_{j}})^{T} \) in system (1) are not available for measurement, thus a state observer should be designed to estimate the unmeasured states. A fuzzy adaptive observer is designed for (1) as

(15)

Rewrite (15) in state space form

(16)

Let \(e_{j} = \underline{x}_{m_{j}} - \underline{\hat{x}}_{m_{j}} \) be state estimation error vector. From (1) and (16), one has a composite error dynamic equation

(17)

where \(\delta_{j} = (\delta_{j,1},\delta_{j,2},\ldots,\delta_{j,m_{j}})^{T}\).

Consider the following Lyapunov candidate V j,0 as

$$ V_{j,0} = \frac{1}{2} \bigl(e_{j}^{T}P_{j}e_{j}\bigr)^{2}$$
(18)

Using (6) and (17), one has

(19)

where λ j =λ min(P j )⋅λ min(Q j ),λ min(P j ), and λ min(Q j ) are the smallest eigenvalues of the matrices P j and Q j , respectively.

Choosing an appropriate constant η j,0>0 such that

$$p_{j,0} = \lambda_{j} - \frac{3}{2}\eta_{j,0}^{\frac{4}{3}}\Vert P_{j} \Vert ^{\frac{8}{3}}- 3m_{j}\sqrt{m_{j}} \eta_{j,0}^{2}\Vert P_{j} \Vert ^{4} > 0$$

By using the well-known mean value theorem in [26], \(g_{j,i_{j}}(y_{j}) \) can be expressed as \(g_{j,i_{j}}(y_{j}) =y_{j}\psi_{j,i_{j}}(y_{j})\), thus

$$ G_{j}(y_{j}) = y_{j} \bigl[\psi_{j,1}(y_{j}) \cdots\psi_{j,m_{j}}(y_{j})\bigr] = y_{j}\psi_{j}(y_{j})$$
(20)

By Lemma 2, one can obtain the following inequalities:

(21)

where \(\delta_{j}^{ *} = (\delta_{j,1}^{ *},\delta_{j,2}^{ *},\ldots,\delta_{j,m_{j}}^{ *} )^{T}\).

Substituting (20) and (21) into (19) results in

(22)

where \(\varXi _{j} = \frac{1}{2\eta_{j,0}^{4}}\Vert \delta_{j}^{ *} \Vert^{4}\).

Remark

Note that if the stochastic disturbance dw=0 (the third term is zero) and fuzzy logic systems \(\hat{f}_{j,i_{j}}(\hat{X}_{j,i_{j}}\vert \theta_{j,i_{j}} ) \) can well approximate \(f_{j,i_{j}}(X_{j,i_{j}}) \) in system (1), then Ξ j will be small, and by (22), it is concluded that the design state observer (15) is asymptotically stable. It should be pointed that the linear reduced-order state observer used in [22–26], even if dw=0, it cannot be concluded that the state observer is asymptotically stable.

4 Control design and stability analysis

In this section, a fuzzy controller and parameter adaptive laws are to be developed by using the backstepping design and DSC technique so that all the signals in the closed-loop systems are SUUB, the observer errors and the system outputs are as small as the desired.

The m j -steps adaptive fuzzy output-feedback backstepping design is based on the following changes of coordinates:

(23)
(24)
(25)

where \(\chi_{j,i_{j}} \) is called the error surface, \(z_{j,i_{j}}\) (j=1,…,n;i j =2,…,m j ) is called the output error of the first-order filter.

Step j.1 (j=1,2,…,n) Using (1), (15), (24), and (25), one has

(26)

Consider the following Lyapunov function candidate:

$$ V_{j,1} = V_{j,0} + \frac{1}{4}\chi_{j,1}^{4} + \frac{1}{2\gamma_{j,1}}\tilde{\theta}_{j,1}^{T}\tilde{\theta}_{j,1} +\frac{1}{2\bar{\gamma}_{j,1}}\tilde{\varepsilon}_{j,1}^{2}$$
(27)

where γ j,1>0 and \(\bar{\gamma}_{j,1} > 0\) are design parameters. \(\tilde{\theta}_{j,1} = \theta_{j,1}^{*} - \theta_{j,1} \) and \(\tilde{\varepsilon}_{j,1} =\varepsilon_{j,1}^{ *} - \hat{\varepsilon}_{j,1} \) are the parameter errors. θ j,1 and \(\hat{\varepsilon}_{j,1}\) are the estimates of \(\theta_{j,1}^{*}\) and \(\varepsilon_{j,1}^{*}\), respectively.

From (22) and (26), one has

(28)

By Lemma 2, the following inequalities can be obtained:

(29)
(30)

where η j,1>0 is a design parameter. Substituting (29)–(30) into (28), one has

(31)

Design the intermediate control function α j,1 and the adaptation functions θ j,1 and \(\hat{\varepsilon}_{j,1}\) as

(32)
(33)
(34)

where σ j,1>0 and \(\bar{\sigma}_{j,1} > 0 \) are design parameters, and \(\theta_{j,1}(0) = \hat{\varepsilon}_{j,1}(0) = 0\).

Substituting (32)–(34) into (31) and utilizing the inequalities

(31) becomes

(35)

where \(p_{j,1} = p_{j,0} - \frac{1}{4\eta_{j,1}^{4}}\).

Introduce a new state variable z j,2 and let α j,1 pass through a first-order filter with the constant τ j,2 to obtain z j,2

$$ \tau_{j,2}\dot{z}_{j,2} + z_{j,2} =\alpha_{j,1},\quad z_{j,2}(0) = \alpha_{j,1}(0)$$
(36)

Step j.i j (j=1,2,…,n; i j =2,…m j −1)

From (24) and (25), the time derivative of χ i is

(37)

To avoid repeatedly differentiating \(\alpha_{j,i_{j}} \) in the traditional backstepping design, which leads to the so-called “explosion of complexity,” we can incorporate the DSC technique proposed by [30–32] into the following backstepping design.

Introduce a new state variable z i,j+1 and let \(\alpha_{j,i_{j}} \) pass through a first-order filter with the constant \(\tau_{j,i_{j} + 1}\) to obtain \(z_{j,i_{j} + 1} \)

$$ \tau_{j,i_{j} + 1}\dot{z}_{j,i_{j} + 1} +z_{j,i_{j} + 1} = \alpha_{j,i_{j}},z_{j,i_{j} + 1}(0)= \alpha_{j,i_{j}}(0)$$
(38)

(37) can be rewritten as

(39)

By the definition of \(\xi_{j,i_{j} + 1} = z_{j,i_{j} + 1} -\alpha_{j,i_{j}}\), it yields \(\dot{z}_{j,i_{j} + 1} = - \frac{\xi_{j,i_{j} + 1}}{\tau_{j,i_{j} + 1}}\) and

(40)

where \(\underline{\chi}_{j,i_{j}} = [\chi_{j,1}\cdots\chi_{j,i_{j}}]^{T},\underline{\xi}_{j,i_{j} + 1} = [\xi_{j,2}\cdots \xi_{j,i_{j} + 1}]^{T} \) and

(41)
$$ C_{j,i_{j} + 1}( \cdot) = - \frac{\partial\alpha_{j,i_{j}}}{\partial y_{j}}g_{j,1}(y_{j})$$
(42)

Consider the following Lyapunov function candidate:

(43)

where \(\gamma_{j,i_{j}} > 0\) and \(\bar{\gamma}_{j,i_{j}} > 0\) are design parameters. \(\tilde{\theta}_{j,i_{j}} = \theta_{j,i_{j}}^{*} -\theta_{j,i_{j}}\) and \(\tilde{\omega}_{j,i_{j}} = \omega_{j,i_{j}}^{ *}- \hat{\omega}_{j,i_{j}} \) are the parameters errors. \(\theta_{j,i_{j}} \) and \(\hat{\omega}_{j,i_{j}}\) are the estimates of \(\theta_{j,i_{j}}^{*} \) and \(\omega_{j,i_{j}}^{ *}\), respectively.

From (39), (40), and (43), one has

(44)

Choose intermediate control function \(\alpha_{j,i_{j}} \) and adaptation functions \(\theta_{j,i_{j}} \) and \(\hat{\omega}_{j,i_{j}} \) as:

(45)
(46)
(47)

with \(\theta_{j,i_{j}}(0) = 0,\hat{\omega}_{j,i_{j}}(0) = 0\).

Substituting (45)–(47) into (44) and utilizing the inequalities

one can obtain

(48)

Step j.m j (j=1,2,…,n) In the final step, the actual control input u j appears. From (15) and (24), one has

(49)

Choose the following Lyapunov function candidate:

(50)

where \(\tilde{\theta}_{j,m_{j}} = \theta_{j,m_{j}}^{*} -\theta_{j,m_{j}}\) and \(\tilde{\omega}_{j,m_{j}} = \omega_{j,m_{j}}^{ *}- \hat{\omega}_{j,m_{j}} \) are the parameter errors, \(\theta_{j,m_{j}} \) and \(\hat{\omega}_{j,m_{j}} \) are the estimates of \(\theta_{j,m_{j}}^{*},\omega_{j,m_{j}}^{ *}\), respectively.

Design controller u j and adaptation functions \(\theta_{j,m_{j}} \) and \(\hat{\omega}_{j,m_{j}} \) as

(51)
(52)
(53)

with \(\theta_{j,m_{j}}(0) = \hat{\omega}_{j,m_{j}}(0) = 0\).

Similar to the derivations in step j.i j , one has

(54)

Applying Young’s inequality, one has

(55)
(56)
(57)
(58)
(59)
(60)
(61)

where

Assumption 3

[33]

For a given \(p_{j,i_{j}} > 0\), for all initial conditions satisfying \(V_{j,i_{j}}(t) \le p_{j,i_{j}}\), where

Since for any \(p_{j,i_{j}} > 0\), the sets \(\prod_{j,k} = \{ V_{j,i_{j}}\le2p_{j,i_{j}}\}\) (j=1,…,n,k=2,…,i j ) is a compact set in \(R^{\sum_{i_{j} = 1}^{k} N_{i_{j}} + j + 3k}\) where \(N_{i_{j}} \) is the dimension of \(\tilde{\theta}_{i,i_{j}}\). Since \(B_{j,i_{j} + 1}(\cdot) \) and \(\mathit{Tr}\{ C_{j,i_{j} + 1}( \cdot)^{T}C_{j,i_{j} + 1}( \cdot)\} \) are continuous functions, there exists the positive constants \(M_{j,i_{j} + 1}( \cdot),N_{j,i_{j} + 1}( \cdot)\) such that |B j,k+1(⋅)|≤M j,k+1(⋅),|Tr{C j,k+1(⋅)T C j,k+1(⋅)}|≤N j,k+1(⋅) on ∏ j,k .

Substituting (55)–(61) into (54) results in

(62)

Choose the design parameters \(\eta_{j,0},\eta_{j,1},c_{j,i_{j}},\upsilon_{j,i_{j}}\) and \(\rho_{j,i_{j}}\) (j=1,…,n;i j =1,…,m j ) such that

(63)
(64)
(65)
(66)
(67)

Substituting (63)–(67) into (62), one has

(68)

Denote λ max(P j ) is the largest eigenvalue of P j , and let

then (68) becomes

$$ \ell V_{j,m_{j}} \le- \rho_{j}V_{j,m_{j}} +\mu_{j}$$
(69)

Finally, choose the whole Lyapunov function candidate as

$$ V = \sum_{j = 1}^{n}V_{j,m_{j}}$$
(70)

Combining (69) and (70), one has

(71)

where ρ=min{ρ 1,ρ 2⋯ρ n },μ=μ 1+μ 2+⋯+μ n .

By Lemma 1 and inequality (74), and using the same arguments as [23, 26, 28], one can obtain that all the signals of the closed-loop system are bounded by μ/ρ, that is, e j and \(\chi_{j,i_{j}} \) are SUUB in probability. \(\tilde{\theta}_{j,i_{j}},\tilde{\varepsilon}_{j,1}\), and \(\tilde{\omega}_{j,k}\) are also SUUB in probability (j=1,2,…,n,i j =1,2,…,m j ,k=2,…,m j ). Moreover, choosing appropriate design parameters, the states observer errors and the outputs of the control system can be made as small as the desired [25, 26].

The above design procedures and stable analysis are summarized in the following theorem.

Theorem 1

For stochastic nonlinear system (1), under Assumptions 1–3, the state observer (15) and the controller (51), with the intermediate control (32), (45) and parameter laws (33)–(34), (46)–(47), and (52)–(53) guarantee that all the signals in the closed-loop system is semiglobally uniformly ultimately bounded in probability. Moreover, the states observer errors and the outputs of the control system can be made as small as the desired by choosing appropriate design parameters.

5 Simulation example

In this section, the proposed adaptive fuzzy control approach is applied to the following example to verify its effectiveness.

Example

Consider a two-continuous stirred tank reactor process with stochastic disturbances, which is described by the following differential equation [16, 34]:

$$ \begin{cases} \dot{x}_{1,1} = b_{11}x_{1,2} + y_{1}^{3}dw\\\dot{x}_{1,2} = b_{12}u_{1} + \frac{1}{2}y_{1}^{2}dw \\y_{1} =x_{1,1} \\\dot{x}_{2,1} = b_{21}x_{2,2} + \phi_{21}(x_{1,1},x_{2,1})+ \varPhi x_{2,1} + y_{2}^{2}dw \\ \dot{x}_{2,2} = b_{22}u_{2} +\phi_{22}(x_{2,1},x_{2,2}) + y_{2}^{2}\cos(y_{2}^{2})dw \\y_{2} = x_{2,1} \\\dot{x}_{3,1} = b_{31}x_{3,2} +\phi_{31}(x_{1,1},x_{2,1},x_{2,2},x_{3,1})\\\ \ \qquad{} + \varPsi \omega+ 2y_{3}^{2}dw\\\dot{x}_{3,2} = b_{32}u_{3} + \phi_{32}(x_{3,1},x_{3,2}) +y_{3}^{5}dw \\y_{3} = x_{3,1}\end{cases} $$
(72)

as the described [34], cooling water is added to the cooling jackets around both reactors at flow rates F j1 and F j2, temperatures T j,1 and T j,2, respectively. Denote \(x_{1,1} =C_{A2} - C_{A2}^{d}\), x 1,2=F 2, \(x_{2,1} = T_{2} - T_{2}^{d}\), \(x_{2,2} = T_{j2} -T_{j2}^{d}\), \(x_{3,1} = T_{1} - T_{j1}^{d}\), with V j1=V j2=V j ,V 1=V 2=V,F 0=F 2=F. w is an independent r-dimensional standard Wiener process, and the parameters in (72) are

where α,E, and λ denote the reaction rate constant, activation energy, and heat generation rate; ρ and ρ j are the densities of liquid in the reactors and in the jackets; c p and c j stand for heat capacities. The values of the process parameters are provided in Table 1.

Table 1 The values of the process parameters
Full size table

The objective is to control C A2,T 1, and T 2 by manipulating C A0,T j10, and T j20. The deviation \(T_{0} - T_{0}^{d}\) of the inlet temperature T 0 from the steady-state value \(T_{0}^{d} \) is assumed to be an unmeasurable disturbance.

Define the following coordinate changes: \(\bar{x}_{1,1} =x_{1,1},\bar{x}_{1,2} = b_{11}x_{1,2},\bar{x}_{2,1} =x_{2,1},\bar{x}_{2,2} = b_{21}x_{2,2}\), \(\bar{x}_{3,1} = x_{3,1} \) and \(\bar{x}_{3,2} = b_{31}x_{3,2}\), then the system (72) is of the same form as in system (1)

$$ \begin{cases} \dot{\bar{x}}_{1,1} = \bar{x}_{1,2} +y_{1}^{3}dw \\ \dot{\bar{x}}_{1,2} = \bar{u}_{1} +\frac{1}{2}y_{1}^{2}dw \\ y_{1} = \bar{x}_{1,1} \\ \dot{\bar{x}}_{2,1}= \bar{x}_{2,2} + \bar{\phi}_{21}(\bar{x}_{1,1},\bar{x}_{2,1}) +\varPhi \bar{x}_{2,1} + y_{2}^{2}dw \\ \dot{\bar{x}}_{2,2} = \bar{u}_{2} +\bar{\phi}_{22}(\bar{x}_{2,1},\bar{x}_{2,2}) +y_{2}^{2}\cos(y_{2}^{2})dw \\ y_{2} = \bar{x}_{2,1} \\\dot{\bar{x}}_{3,1} = \bar{x}_{3,2} + \bar{\phi}_{31}(\bar{x}_{2,1},\bar{x}_{3,1}) + \varPsi \omega+ 2y_{3}^{2}dw \\\dot{\bar{x}}_{3,2} = \bar{u}_{3} + \bar{\phi}_{32}(\bar{x}_{3,1},\bar{x}_{3,2}) + y_{3}^{5}dw \\ y_{3} =\bar{x}_{3,1} \\\end{cases} $$
(73)

where \(\bar{u}_{1} = b_{11}b_{12}u_{1},\bar{u}_{2} =b_{21}b_{22}u_{2},\bar{\phi}_{21}(\bar{x}_{2,1},\bar{x}_{2,1}) =\phi_{21}(x_{1,1},x_{2,1})\), \(\bar{\phi}_{22}(\bar{x}_{2,1},\bar{x}_{2,2})= b_{21}\phi_{22}(x_{2,1}, x_{2,2})\), \(\bar{u}_{3} = b_{31}b_{32}u_{3}\) and \(\bar{\phi}_{32}(\bar{x}_{3,1},\bar{x}_{3,2}) =b_{31}\phi_{32}(x_{3,1}, b_{31}x_{3,2})\).

In the simulation study, eleven fuzzy set are defined over interval [−10,10] for all \(\bar{x}_{1,1},\bar{x}_{1,2},\bar{x}_{2,1}, \bar{x}_{2,2},\bar{x}_{3,1}\), and \(\bar{x}_{3,2}\), and by choosing partitioning points as −10,−8,−6,−4,0,2,4,6,8, and 10, their fuzzy membership functions are given as follows:

Let

Then

The fuzzy controllers and parameters of adaptive law are constructed as

The design parameters are chosen as

The initial conditions are chosen as

The simulation results are shown in Figs. 1–9.

Fig. 1
figure 1

The trajectories of x 1,1 “solid line” and \(\hat{x}_{1,1} \) “dash-dotted”

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Fig. 2
figure 2

The trajectories of x 1,2 “solid line” and \(\hat{x}_{1,2} \) “dash-dotted”

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Fig. 3
figure 3

The trajectories of x 2,1 “solid line” and \(\hat{x}_{2,1} \) “dash-dotted”

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Fig. 4
figure 4

The trajectories of x 2,2 “solid line” and \(\hat{x}_{2,2} \) “dash-dotted”

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Fig. 5
figure 5

The trajectories of x 3,1 “solid line” and \(\hat{x}_{3,1} \) “dash-dotted”

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Fig. 6
figure 6

The trajectories of x 3,2 “solid line” and \(\hat{x}_{3,2} \) “dash-dotted”

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Fig. 7
figure 7

The trajectory of u 1

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Fig. 8
figure 8

The trajectory of u 2

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Fig. 9
figure 9

The trajectory of u 3

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From the above simulation results, it is clear that even though the exact information on the nonlinear functions in the system is not available and the state variables are immeasurable, the proposed adaptive fuzzy output feedback control approaches guarantee the stability of the closed-loop adaptive control system and achieve good control performance.

6 Conclusions

In this paper, an observer-based adaptive fuzzy output feedback control approach has been proposed for a class of uncertain MIMO stochastic nonlinear system with immeasurable states. Fuzzy logic systems are used to approximate the unknown nonlinear functions and a fuzzy state observer is designed to estimate those immeasurable states. By combining the adaptive backstepping design with the DSC technique, a novel adaptive fuzzy output feedback backstepping control approach is developed. It is proved that all the signals of the closed-loop control system are semiglobally uniformly ultimately bounded (SUUB) in probability; the observer errors and the system outputs can be made as small as the desired by appropriate choice of the design parameters. Simulation results are provided to show the effectiveness of the proposed approach.