Finite-time fuzzy output-feedback control for $p$-norm stochastic nonlinear systems with output constraints

1.   Introduction

Over the past decades, a variety of control design strategies have been proposed for different nonlinear systems ^{[1,2,3,4,5,6,7]}. Especially, many approximated-based control schemes have been developed for uncertain nonlinear systems by using neural networks (NNs) or the fuzzy logic systems (FLSs) ^{[8,9,10,11,12,13,14,15,16,17,18,19]}. Among these studies, the research of stochastic systems is much more attracted (see, e.g. ^{[16,17,18,19]} and the references therein), due to their wide application. It is worth noting that the aforementioned NNs-based or FLSs-based control strategies haven't taken output constraint into account. In fact, many practical systems are usually required to satisfy an output constraint in the operation for considering the performance specifications or safety ^[20,21]. It is well known that, the BLF-based approaches are useful tools to settle controller design problems of output-constrained nonlinear systems, see references ^{[22,23,24,25,26,27]} for instances. In the latest research progress of constrained control, many kinds of adaptive neural or fuzzy control design methods have been presented by combining the different BLFs with NNs or FLSs approximators for various stochastic nonlinear systems subject to output constraint and unknown nonlinearities in ^{[28,29,30,31,32]}. Nevertheless, the above-mentioned works have mainly considered strict-feedback stochastic systems whose fractional powers are all equal to one, rather than $p$-norm stochastic nonlinear systems in which the fractional powers are the positive odd rational numbers and at least one of the fractional powers is greater than one.

As an important class of nonlinear systems, the p-norm stochastic nonlinear systems which are more general and complex, have received increased attention in recent years. For such a kind of systems, many control design problems have been well solved by utilizing the adding a power integrator technique under some strong or weaker growth conditions ^{[33,34,35,36,37]}. Subsequently, references ^[38,39,40] have taken these growth conditions away and considered the adaptive NNs control for switched and large-scale stochastic high-order nonlinear systems. Meanwhile, the control problems have been investigated for $p$-norm nonlinear systems with output/states constraints under some nonlinear growth conditions in a few literatures, such as ^{[41,42,43,44]}. In these works, different state-feedback controllers have been mainly constructed and finite-time stability has been obtained. Worth noting that an output-feedback controller has been designed for high-order planar deterministic nonlinear systems in ^[44]. On these basics, references ^{[45,46,47,48]} have further taken both unknown nonlinearities and constraints into accounts, and designed some fuzzy controllers for $p$-norm stochastic nonlinear systems with output/states constraints. However, it should be pointed out that the above-mentioned results of $p$-norm stochastic nonlinear systems have mainly focused on addressing the asymptotical convergence rather than the finite-time convergence in the case of unknown nonlinearities. As a matter of fact, the finite-time convergence has only been investigated for $p$-norm stochastic nonlinear systems with known nonlinearities satisfying some nonlinear growth conditions. On the other hand, it can be observed that the existing constrained controllers of $p$-norm stochastic nonlinear systems are mainly based on the assumption that full-state measurements are available. In other words, when only the system output can be accurately measured, the problem of finite-time output-feedback control for $p$-norm stochastic nonlinear systems with unknown nonlinearities and output constraints, to our best knowledge, has never been considered in the literature.

Motivated by above discussions, this paper will investigate how to design the finite-time output-feedback controller for a kind of $p$-norm stochastic nonlinear systems subject to output constraint. The main contributions can be summarized as follows: 1) It is first time to consider $p$-norm stochastic nonlinear systems with output constraints and unmeasurable states. Note that the existing studies have mainly addressed the constrained control problems under the the assumption that all the states are measurable (e.g., ^{[33,34,35,36,37,41,43]}), while this paper investigates the constrained control design in the case of that the states are all unmeasurable except the system output. What's more, the system nonlinearities are completely unknown. Thus, this work will extend and develop the existing control design theory for $p$-norm stochastic nonlinear systems; 2) A finite-time controller is designed. The developed scheme not only ensures the system output is constrained in a given compact set, but also enables the closed-loop system is semi-global finite-time stable in probability (SGFSP).

2.   Problem and Preliminaries

In this paper, we consider the following class of $p$-norm stochastic nonlinear systems

where $\omega$ is a $r$-dimension standard Wiener process; $x = (x_1, \cdots, x_n)^{T}\in R^{n}$ is system state vector; $u\in R\; {\rm and} \; y\in R$ are respectively control input and output; the fractional power $p_\in R^{\geq1}_{odd}: = \{m/k| m\geq k, \; m\; {\rm and}\; k \; {\rm are \; positive\; odd \; integers}\}$; for $i = 1, \cdots, n$, $\bar{x}_i = (x_1, \cdots, x_i)^{T}\in R^{i}$; $\phi_i: R^{i}\rightarrow R$ and $g_i: R^{i}\rightarrow R^{r}$ are unknown continuous functions satisfying $\phi_i(0) = 0, \; g_i(0) = 0$. The system output $y = x_{1}$ is measurable and constrained in $\Pi_1 = \{y(t)\in R, |y(t)| < \varepsilon\}$ with a constant $\varepsilon > 0$, while the other states $x_2, \cdots, x_n$ are all unmeasurable.

The objective of this paper is to design a finite-time fuzzy output-feedback controller for system (2.1) such that: 1) the output don't violate the given constrained boundary; 2) all the signals of the closed-loop system converge to a small compact of the original point in finite-time in probability in presence of unknown nonlinearities and unmeasured states $x_i(i = 2, \cdots, n)$.

Firstly, some concepts and lemmas are presented for preliminaries. Consider the following stochastic system

where $\phi(x)$ and $g(x)$ are continuous functions with satisfying $\phi(0) = g(0) = 0$.

Definition 1. ^[1] For any given $V(x)\in C^{2}(R^n)$, associated with system (2.2), the second-order differential operator $\ell$ is defined as follows:

Definition 2. ^[17] The equilibrium $x = 0$ of stochastic nonlinear system (2.2) is semi-global finite-time stable in probability (SGFSP) if for all $x(t_0, \omega) = x(0)$, there exist a constant $c > 0$ and a settling time $T^*(c, x_0, \omega) < \infty$ to make $E[\|x(t, \omega)\|] < c$, for all $t\geq t_0 + T^*$.

Lemma 1. ^[17] Consider the stochastic system (2.2) and assume that $f(0)$, $h(0)$ are bounded uniformly in $t$. If there exist a $C^2$ Lyapunov function $V:R^{n}\rightarrow R^+$, functions $\varrho_1, \varrho_2\in\mathcal{K}_\infty$, and constants $\bar{\mu}_0 > 0$, $0 < \bar{\varsigma}_0 < 1$ and $\bar{\nu}_0 > 0$ such that

Then, the stochastic nonlinear system (2.2) is SGFSP.

Remark 1. As stated in ^[17], the Eq (2.4) implies that there exists the stochastic setting time function $T^*(x, \omega) = \frac{1}{l_0\bar{\mu}_0(1-\bar{\varsigma}_0)}\left[E[V^{1-\bar{\varsigma}_0}(x(0))]-\left(\frac{\bar{\nu}_0}{(1-l_0)\bar{\mu}_0}\right)^{\frac{\bar{\varsigma}_0}{1-\bar{\varsigma}_0}}\right]$, such that $E[V^{\bar{\varsigma}_0}(x)]\leq\frac{\bar{\nu}_0}{(1-l_0)\bar{\mu}_0}$ for all $t\geq t_0+ T^*(x, \omega)$, where $0 < l_0\leq1$ is an arbitrary constant.

Lemma 2. ^[7] Let $a, b\in R^{+} \; {\rm with}\; a\geq1.$ For any $\zeta, \eta\in R$, the following inequalities hold:

(i) $|\zeta^{a}-\eta^{a}|\leq a(2^{a-2}+2)|\zeta-\eta|(|\zeta-\eta|^{a-1}+\eta^{a-1}),$

(ii) $\mid \zeta^{\frac{b}{a}}-\eta^{\frac{b}{a}}\mid\leq2^{1-\frac{1}{a}}\mid\lceil \zeta\rceil^{b}-\lceil \eta\rceil^{b}\mid^{\frac{1}{a}},$

(iii) $(|\zeta|+|\eta|)^{\frac{1}{a}}\leq|\zeta|^{\frac{1}{a}}+|\eta|^{\frac{1}{a}}\leq2^{1-\frac{1}{a}}(|\zeta|+|\eta|)^{\frac{1}{a}}.$

Lemma 3. ^[42] For any constants $k_1, k_2, \vartheta, \varsigma\in R^+$ and any variables $\zeta_1, \zeta_2\in R$, we have the following inequality

Lemma 4. ^[8] Let $p\in (0, \infty)$, for any $\zeta_i\in R, i = 1, \cdots, n,$ one has

where $b = \max\{n^{p-1}, 1\}$.

Lemma 5. ^[36] If $\zeta, \eta\in R$ and $p > 1$ is an odd number, then

Lemma 6. ^[12] With any $a_0 > 0, c_0 > 0$, and $\zeta(t) > 0$, for $\dot{\eta}(t) = a_0\zeta(t)-c_0 \eta(t)$, if $\eta(0)\geq0$ can be satisfied, then one has $\eta(t)\geq0$ for $\forall t\geq0$.

In this paper, the nonlinear functions $\phi_i(\cdot)$ and $g_i(\cdot)$ are all unknown. The unknown functions will be approximated by the FLSs based on the following presented lemma.

Lemma 7. ^[49] Let $F(X)$ be a continuous function defined on a compact set $\Pi_0$. Then, for a given desired level of accuracy $\delta > 0$, there exists a fuzzy logic system $\Upsilon^T \Psi(X)$ such that

$\Upsilon = (\upsilon_1, \cdots, \upsilon_N)^T$ is the ideal constant weight vector, and $\Psi(X) = \frac{\left(\psi_1(X), \cdots, \psi_N(X)\right)^T}{\sum_{j = 1}^{N}\psi_j(X)}$ is the basis function vector, with $N > 1$ being the number of the fuzzy rules and $\psi_j(X)$ being chosen as Gaussian functions, i.e., for $j = 1, \cdots, N$

where $\lambda_j = (\lambda_{j1}, \cdots, \lambda_{jn})^T$ and $\vartheta_j$ respectively denote the center vector and the width of the Gaussian function.

Remark 2. In view of Lemma 7, any function $F(X)$ which is defined and continuous on a compact set $\Pi_0$, can be approximated by

where $\epsilon(X)$ is the FLS approximation error satisfying $|\epsilon(X)| < \delta$.

3.   Main results

3.1.   State-feedback controller design

In this section, a fuzzy state-feedback controller will be explicitly designed for system (2.1) by combining a tan-type BLF and the FLSs into the adding a power integrator technique.

First of all, we introduce a coordinate transformation as follow

where ${q_1 = 0}$, $q_j = \frac{q_{j-1}+1}{p}(j = 2, \cdots, n+1)$, and $H > 1$ is a constant to be determined later. Based on (3.1), system (2.1) turns into

where $\bar{\chi}_i = (\chi_1, \cdots, \chi_i)$, $f_1 = \phi_1$, $h_1 = g_1$, $f_j = \frac{\phi_j}{H^{q_j}}$, and $h_j = \frac{g_j}{H^{q_j}}, {(j = 2, \cdots, n)}$.

In what follows, a fuzzy state-feedback controller will be designed through $n$ steps based on the equivalent system (3.2).

Define

where $\beta_i$'s are the virtual signals being constructed later.

Step 1. From (3.1), we can get

Choose the first Lyapunov function

where $b_1 > 0$ is an adjustment parameter, $\tilde{\alpha}_1 = \alpha_1-\hat{\alpha}_1$ is the estimate error and $\hat{\alpha}_1$ is the estimator of the parameter $\alpha_1$.

Remark 3. Clearly, $V_B(\xi_1) = \frac{\varepsilon^{4}}{2\pi}\tan\left(\frac{\pi\xi_1^4}{2\varepsilon^{4}}\right)$ is a tan-type BLF adopted to deal with the system output constraint. Compared to the log-type BLF, $V_B(\xi_1)$ possesses the following characteristic:

which implies that the proposed method is also applicable to the system without output constraints.

Then, one can easily get from the definition of $V_B(\xi_1)$ that

where $S_1(\xi_1) = \sec^2\left(\frac{\pi\xi_1^4}{2\varepsilon^{4}}\right)$.

In view of (2.3), (3.5) and (3.6), it is not hard to gain

From Lemma 3, one obtains

Then, we have

where $Z_1 = \chi_1$,

and $\varrho_1 > 0$ is an adjustment parameter.

Thus, by Lemma 7, one can approximate $F_1(Z_1)$ by

where $|\epsilon_1(Z_1)|\leq\delta_1$ and $\delta_1 > 0$ is a given constant.

Since $\Psi^T_1(\cdot)\Psi_1(\cdot)\leq1$, it is easily obtained from Lemma 3 that

where $\alpha_1 = {\|\Upsilon_1\|}^{\frac{p+3}{3}}$ and $\sigma_{11} > 0$ is an adjustment parameter.

Substituting (3.9) into (3.7) gets

Then, one could design

and

where $M_1\geq\frac{3}{p+3}(S_1(\xi_1))^{\frac{p}{3}}[\sigma_{11}\hat{\alpha}_1+1]+\frac{\rho_1}{S_1(\xi_1)}+\varrho_1 > 0$; $\varrho_1, d_1 > 0$ are adjustment parameters; and the value of $\rho_1 > 0$ will be given in the next step.

Substituting (3.11) and (3.12) into (3.10), gets

In addition, it is easily obtained that

Therefore, we can get

where $Q_1 = \frac{3}{4}+\frac{pH}{p+3}{\sigma_{11}^{-\frac{3}{p}}}+\frac{pH}{p+3}\delta_1^{\frac{p+3}{p}}+\frac{d_1\alpha_1^2}{2b_1}$.

Remark 4. Notice that

which means that the continuity of $F_1(Z_1)$ is ensured.

Remark 5. According to Lemma 6, one gets $\hat{\alpha}_1\geq0$, for $\forall t\geq0$. In each design step, this characteristic will be always applied.

Step 2. From (3.3) and It$\hat{o}$'s formula, we have

where $\ell\beta_1 = \frac{\partial\beta_1}{\partial \chi_1}(H\chi_2^p+f_1) + \frac{\partial\beta_1}{\partial\hat{\alpha}_1}\dot{\hat{\alpha}}_1+\frac{1}{2}\frac{\partial^2\beta_1}{\partial \chi_1^2}h_1^Th_1$. Combining the definition of $\beta_1$ with the properties of $f_1(\chi_1)$ and $h_1(\chi_1)$, implies that $\ell\beta_1$ is valid and continuous.

Choose the second Lyapunov function as

with

where $b_2 > 0$ is an adjustment parameter, $\tilde{\alpha}_2 = \alpha_2-\hat{\alpha}_2$ is the estimate error and $\hat{\alpha}_2$ is the estimator of the parameter $\alpha_2$.

Applying (2.3), (3.14) and (3.16), it can be gotten that

Besides, applying Lemma 3 renders

On the other hand, we gets

where $D = (2^{p-2}+2)p$; $\rho_1 = \frac{1}{p+3}\left[3(DS_1(\xi_1))^{\frac{p+3}{3}}\sigma^{-\frac{p}{3}}_{12}+(p+2)S_1(\xi_1)\right]$, $\tau_2 = \frac{S_1(\xi_1)\varphi^{p(p+3)}_1}{p+3}$, and $\sigma_{12} > 0$ is an adjustment parameter.

Thus, from (3.13), (3.17) and (3.18), we have

where $Z_2 = (\bar{x}_2, \hat{\alpha}_1)^T$,

Obviously, the continuity of $F_2(Z_2)$ can be directly proved by the fact that the functions $f_2(\bar{\chi}_2)$, $h_2(\bar{\chi}_2)$ and $\ell\beta_1$ are all continuous. Thus, $F_2(Z_2)$ can be approximated as

where $|\epsilon_2(Z_2)|\leq\delta_2$ and $\delta_2 > 0$ is a given constant.

In view of the fact $\Psi^T_2(\cdot)\Psi_2(\cdot)\leq1$ and Lemma 3, we obtain

where $\alpha_2 = {\|\Upsilon_2\|}^{\frac{p+3}{3}}$ and $\sigma_{21} > 0$ is an adjustment parameter.

Substituting (3.21) into (3.19) yields

Then, one can design

and

where $M_2\geq\frac{3}{p+3}[\sigma_{21}\hat{\alpha}_2+1]+\frac{p\sigma_{12}}{p+3}+\rho_2+\varrho_2 > 0$; $\varrho_2 > 0, \; d_2 > 0$ are adjustment parameters. The value of $\rho_2 > 0$ will be given in the next step.

In addition, it is evident that

Substituting (3.25) into (3.22), one gets

where $Q_2 = \frac{3}{4}+\frac{pH}{p+3}{\sigma_{12}^{-\frac{3}{p}}}+\frac{pH}{p+3}\delta_2^{\frac{p+3}{p}}+\frac{d_2\alpha_2^2}{2b_2}.$

Inductive Step $(3\leq k\leq n)$. In view of above two steps, we can deduce the following similar property whose proof can be found in the Appendix.

Proposition 1. For the $k$th Lyapunov function $V_{k}:\Pi_k\rightarrow R^+$ as

with

there exists a virtual controller $\beta_k$ and the adaptive law of $\hat{\alpha}_k$ of the following forms

such that

where $M_k\geq\frac{3}{p+3}[\sigma_{k1}\hat{\alpha}_k+1]+\frac{p\sigma_{k-1\; 2}}{p+3}+\rho_k+\varrho_k > 0$; $b_k, \varrho_k, d_k > 0$ are adjustment parameters; and the value of $\rho_k > 0$ will be given in the next step.

Step n According to above steps, there exist a series of virtual controllers and adaptive parameter laws ($\beta_k, \hat{\alpha}_k)(k = 1, \cdots, n)$ make that Eq (3.31) holds when $k = n$ with $\chi_{n+1} = \nu$. Therefore, the fuzzy adaptive control law can be designed as

Apparently, one can further get

By the definitions of $\xi_k(k = 1, \cdots, n)$, one gets the fuzzy state-feedback controller

where $\bar{\varphi}_k = \prod_{j = k}^n\varphi_j$ for $k = 1, \ldots, n.$

3.2.   Output-feedback controller design

In this subsection, a fuzzy output-feedback controller will be designed by combining the above-constructed state-feedback controller with the state-observer constructed later. Since $\chi_2, \cdots, \chi_n$ are unmeasurable, a reduced-order observer is required. Firstly, define a series of new variables as follows:

Then, it directly gets

where $\gamma_i\geq1(i = 2, \cdots, n)$ are gain parameters to be determined. Hence, the $(n-1)$-dimensional observer can be constructed as ^[36]

According to (3.36), the estimate $\hat{\chi}_i$ of $\chi_i$ can be gotten by

Thus, one constructs the implementable controller of system (2.4) by using Eq (3.35) and the certainty equivalence principle as below:

Therefore, the output-feedback controller of origin system (2.1) is

3.3.   Selection of the observer gains

In this section, we will analyze the appropriate values of the gains $\gamma_i (i = 2, \cdots, n)$ and some constant parameters in output-feedback controller.

To determine the observer gains $\gamma_2, \cdots, \gamma_n$, we first define the error dynamics

Further, the following coordinate transformation is introduced

It can easily infer from (3.40) and (3.41) that

Now, a proposition is provided for helping to determine gain constants, whose proof will be given in Appendix.

Proposition 2. For the Lyapunov function

by utilizing Lemmas 2-5, Eqs (2.3) and (3.42), it is not difficult to obtain

where $\gamma > 0$ is an adjustment constant; $\bar{c}_i = \bar{c}_i(\gamma_{i+1}, \cdots, \gamma_n) > 0, (i = 2, \cdots, n-1)$ are constants independent of $H$; and $\bar{c}_n$ is a constant independent of $H$ and $\gamma_{i}(i = 2, \cdots, n)$; and $\widetilde{F}_1(\chi) = H\sum_{i = 2}^n[\frac{1}{4}(f_i-\gamma_if_{i-1})^4+\frac{3\gamma^{\frac{2}{3}}}{4H^{\frac{2}{3}}}\|h_i-\gamma_ih_{i-1}\|^4+\frac{2p-1}{p+3}\chi_{i}^{p+3}]$ is an unknown continuous function.

Besides, we can obtain from Lemmas 2-4 that

where $\widetilde{F}_2(\xi_n) = \tau_n\xi_n^{p+3}$; $\tau_n = \frac{3}{p+3}\sum_{i = 2}^n\left[D(n-1)^p\bar{\varphi}^p_i\right]^{\frac{p+3}{3}} +\frac{p+2}{p+3}\sum_{i = 2}^n3\left[D(n-1)^p\varphi_n^{p-1}\bar{\varphi}_i\right]^{\frac{p+3}{p+2}}$; $\tilde{\iota}_n = n^{p+2}\frac{p+1}{p+3}$ and $\tilde{\iota}_i = \tilde{\iota}_i(\gamma_{i+1}, \cdots, \gamma_n)(i = 2, \cdots, n-1)$ are positive constants.

Then, substituting (3.44) into (3.34) yields

Then, define the entire Lyapunov function $V = V_n+U_n$. Apparently, it directly infers from Proposition 2 and Eq (3.45) that

where $\widetilde{F}(\cdot) = \widetilde{F}_1(\cdot)+\widetilde{F}_2(\cdot)$.

Using FLS to deal with the unknown function $\widetilde{F}(\cdot)$, one deduces from Lemma 7 that $\widetilde{F}(\cdot) = \Upsilon_0^T\Psi_0+\epsilon_0(\cdot)\leq\|\Upsilon_0\| \|\Psi_0 \|+\delta_0 \leq \frac{3}{p+3}\alpha_0+\frac{p}{p+3}+ \delta_0$ where $\alpha_0 = \|\Upsilon_0\|^{\frac{p+3}{3}}$.

Therefore, the observer gains $\gamma_2, \cdots, \gamma_n$ and constant $H$ can be chosen in the following recursive manner

Then, Eq (3.46) turns into

where $\theta_i(i = 2, \cdots, n)$ are positive constants, and $Q = \sum_{i = 1}^{n}Q_j+\frac{(p-1)H}{p+3}+\frac{3}{p+3}\alpha_0+\frac{p}{p+3}+ \delta_0$.

On the other hand, it can be verified from Lemma 3 that

which renders

where $i = 2, \cdots, n$, $\bar{\theta}_i = (p+3) H\theta_i$ and $\bar{\varrho}_i = (p+3) H\varrho_i$. Further, we get

where $\bar{Q} = Q+\frac{p-1}{4}H\sum_{i = 2}^n\theta_i+\frac{p-1}{4}H\sum_{i = 2}^n\varrho_i$.

3.4.   Stability analysis

To state the main result, the following theorem is presented.

Theorem 1. For the $p$-norm stochastic nonlinear system (2.1) and a given constant, there exists a finite-time fuzzy output-feedback controller (3.39) together with the parameter adaptive laws (3.12), (3.24), (3.30), and (3.33) such that

i) the system output isn't violated in the sense of probability, i.e., $P\{|y(t)| < \varepsilon\} = 1$.

ii) all the signals in the closed-loop stochastic nonlinear system (2.1) are SGFSP.

Proof. ⅰ) Let $\mu_0 = \min\{{\varrho_1}, d_1, \cdots, d_n, \bar{\varrho}_2, \cdots\bar{\varrho}_n, \frac{\bar{\theta}_2}{\gamma}, \cdots\frac{\bar{\theta}_n}{\gamma}\}$ and $\pi_0 = \bar{Q}$. Then, Eq (3.51) can be expressed as

We can easily get from Eq (3.52) that

For $x(0) = (x_1(t_0), \cdots, x_n(t_0))^{T}$ satisfying $x_1(t_0)\in \Pi_1$, it easily obtains that the mean of $V(t)$ is bounded, which implies that $V$ is bounded in probability. It can be directly deduced from the definition of $V$ that

Consequently, it is clear that $P\{|y(t)| < \varepsilon\} = P\{|\xi_1(t)| < \varepsilon\} = 1$, which demonstrates that the output constraint of system (2.1) is not violated in the sense of probability.

ⅱ) For $\forall\; 0 < \bar{\varsigma}_0 < 1$, it is easy to get from Lemma 3 that

Further, one has

Then, substituting (3.55) into (3.52) drives

where $\bar{\mu}_0 = \frac{\mu_0}{\bar{\varsigma}_0}$ and $\bar{\pi}_0 = \frac{(1-\bar{\varsigma}_0)\mu_0}{\bar{\varsigma}_0}+\pi_0$.

Let $T^* = \frac{1}{l_0\bar{\mu}_0(1-\bar{\varsigma}_0)}\left[E\left(V^{1-\bar{\varsigma}_0}(\chi(0), \tilde{e}(0), \hat{\alpha}(0))\right)-\left(\frac{\bar{\pi}_0}{\bar{\mu}_0(1-\bar{\varsigma}_0)}\right)^{\frac{1-\bar{\varsigma}_0}{\bar{\varsigma}_0}}\right]$ where $\chi(0) = (\chi_1(t_0), \cdots, \chi_n(t_0))^T$, $\tilde{e}(0) = (\tilde{e}_2(t_0), \cdots, \tilde{e}_n(t_0))^T$, $\hat{\alpha}(0) = (\hat{\alpha}_1(t_0), \cdots, \hat{\alpha}_n(t_0))$, $0 < l_0 < 1$ is a constant. Then it follows from Lemma 1 that for $\forall t\geq t_0+T^*$, $E\left(V^{1-\varsigma}(\chi, \tilde{e}, \hat{\alpha})\right)\leq\frac{\bar{\pi}_0}{\bar{\mu}_0(1-\bar{\varsigma}_0)}$, which means that all the signals in the closed-loop systems are semi-global finite-time stable in probability.

Remark 6. In this paper, we construct an output-feedback controller rather than the designed state-feedback controllers in existing results about output constraints. On the other hand, it should be pointed out the considered constraint is symmetric rather than asymmetric, which leads that the proposed scheme can not be directly employed or further extended to the case of asymmetric constraints. However, a control scheme based on a new BLF can be developed for asymmetric output constraints in a similar way to this paper. In addition, another limitation is that all of the fractional powers are equal to $p$. If $p_i$'s are taken different values, the proposed strategy seems not applicable. In the future, we will address the two issues.

4.   Simulation example

The validation of the proposed strategy will be testified by the following system.

where the output $y = x_1$ is measurable and constrained by $\Pi_1 = \{y(t)\in R, |y(t)| < 1\}$, and the state $x_2$ is unmeasurable.

According to the controller design procedure, we can respectively design the finite-time output-feedback controller, the adaptive laws and the observer as follows:

where $M_1 = \frac{15}{28}(S_1{(x_1)})^{\frac{13}{15}}\left[\sigma_{11}\hat{\alpha}_1+1\right]+\frac{\rho_1}{S_1{(x_1)}}+\varrho_1$ and $M_2 = \frac{15}{28}[\sigma_{21}\hat{\alpha}_2+1]+\frac{13\sigma_{12}}{28}+\rho_2+\varrho_2$.

Now, we choose the related parameters as $\gamma = 1.5, \sigma_{11} = 22, \; \sigma_{21} = 20, \; \sigma_{12} = 1$, $\theta_2 = 3$, $b_1 = b_2 = 40$, $d_1 = d_2 = 2$, $\varrho_1 = \varrho_2 = 1$ and infer $H = 1.6, \; \gamma_2 = 8.1$. Next, the initial states and adaptive parameters are selected as $[x_1(0), x_2(0), \hat{x}_2(0), \hat{\alpha}_1, \hat{\alpha}_2]^T = [0.4, 8, 8, 10, 10]^T$ the simulation results are displayed in Figures 1–4.

Figure 1 provides the trajectory of $x_1(t)$, which indicates that the system output constraint is not violated under controller (4.2). Meanwhile, the trajectories of $x_2(t)$ and $\hat{x}_2(t)$ are given in Figure 2, which shows that $x_2(t)$ is well estimated by $\hat{x}_2(t)$. Moreover, the trajectory of the controller $u$ is displayed in Figure 3. Finally, Figure 4 expresses the curves of the adaptive parameter vector under the developed strategy. Also, one could evidently observe from these figures that all the signals of system (4.1) are semi-global finite-time stable in probability under controller (4.2).

5.   Conclusion

In this paper, the output-feedback controller design problem is investigated for a class of $p$-norm stochastic nonlinear systems with output constraints. Through using a tan-type BLF, an adaptive fuzzy state-feedback controller is proposed by the adding a power integrator technique. Then, a finite-time fuzzy output-feedback controller is constructed by combining the proposed state-feedback controller and a reduced-order observer. Both rigorous proof and the simulation example verify that the designed controller can ensure the achievement of the system output constraint and semi-global finite-time stability of all the signals in probability. In the future, we will consider the situations of asymmetric constraints, different fractional powers, or multi-input multi-output stochastic nonlinear systems.

Acknowledgments

This work was supported by supported by the National Science Foundation of China under Grant 61973142, the Jiangsu Natural Science Foundation for Distinguished Young Scholars under Grant BK20180045, the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the Project of Anhui Province Outstanding Young Talent Support Program under Grant gxyq2018089.

Conflict of interest

The authors declare that there are no conflicts of interest.

Appendix

Proof of Proposition 1. Firstly, suppose there exist $\beta_{k-1}(3\leq k\leq n)$ such that

At the same time, from It$\hat{o}$'s formula and (3.1), one gets

where $\ell\beta_{k-1} = \sum_{j = 1}^{k-1}\frac{\partial\beta_{k-1}}{\partial \chi_j}(H\chi_{j+1}^p+f_j)+\Sigma_{j = 1}^{k-1}\frac{\partial\beta_{k-1}}{\partial\hat{\alpha}_j}\dot{\hat{\alpha}}_j +\frac{1}{2}\sum_{j, l = 1}^{k-1}\frac{\partial^2\beta_{k-1}}{\partial \chi_j\partial \chi_l}h_j^Th_l$. Clearly, $\ell\beta_{k-1}$ is valid and continuous, which can be illustrated in a similar way by following the lines to obtain $\ell\beta_{1}$.

We choose the Lyapunov function as

with

where $b_k > 0$ is an adjustment parameter, $\tilde{\alpha}_k = \alpha_k-\hat{\alpha}_k$ is the parameter error, and $\hat{\alpha}_k$ is the estimation of the unknown parameter $\alpha_k$.

Applying (2.3), (A.2) and (A.4), one can obtain

It is not difficult to get from Lemma 3 that

On the other hand, from Lemmas 2 and 3, one can verify

where $\rho_{k-1} = \frac{1}{p+3}\left[3D^{\frac{p+3}{3}}\sigma^{-\frac{p}{3}}_{12}+p+2\right]$, $\tau_k = \frac{\varphi^{(p-1)(p+3)}_1}{p+3}$, and $\sigma_{k-1\; 2} > 0$ is an adjustment parameter.

From (A.1)-(A.7), it can be deduced that

where $Z_k = (\bar{\chi}_k^T, \bar{\hat{\alpha}}_{k-1}^T)^T$, $\bar{\hat{\alpha}}_{k-1} = (\hat{\alpha}_1, \cdots, \hat{\alpha}_{k-1})^T$ and

Similar to the first two steps, $F_k(Z_k)$ can also be approximated as

where $|\epsilon_k(Z_k)|\leq\delta_k$ and $\delta_k > 0$ is a given constant.

One directly obtains from Eq (A.9) and Lemma 3 that

where $\alpha_k = {\|\Upsilon_k\|}^{\frac{p+3}{3}}$ and $\sigma_{k1} > 0$ is an adjustment parameter.

Substituting (A.10) into (A.8) renders

Then, we can design

where $M_k\geq\frac{3}{p+3}\left[\sigma_{k1}\hat{\alpha}_k+1\right]+\frac{p\sigma_{k-1\; 2}}{p+3}+\rho_k+\varrho_k$; $\varrho_k, d_k > 0$ are adjustment parameters; and the value of $\rho_k > 0$ will be given in $(k+1)$th step.

Moreover, one has

From (A.1)-(A.13), it can be deduced that

where $Q_k = \frac{3}{4}+\frac{pH}{p+3}{\sigma_{k1}^{-\frac{3}{p}}}+\frac{pH}{p+3}\delta_k^{\frac{p+3}{p}}+\frac{d_k\alpha_k^2}{2b_k}.$ The proof of Proposition 1 is completed.

Proof of Proposition 2. By the definition of $U_n$, one has

By Lemma 5, one can infer

Since $\chi_i-\hat{\chi}_i-\tilde{e}_i = \sum_{j = 2}^{i-1}\gamma_i\cdots\gamma_{j+1}\tilde{e}_j$, we can get that through applying Lemmas 2-5

where $\tilde{a}_{ij} = \tilde{a}_{ij}(\gamma_i, \cdots, \gamma_{j+1})$ is a constant independent of $H$.

Noting that $e_{i} = \sum_{j = 2}^{i-1}\gamma_i\cdots\gamma_{j+1}\tilde{e}_j+\tilde{e}_i$, one has

where $\tilde{c}_{ij} = \tilde{c}_{ij}(\gamma_i, \cdots, \gamma_{j+1})$ is a constant independent of $H$.

In addition, we have

Substituting (A.16)-(A.19) into (A.15) yields

where $\bar{c}_i = \sum_{j = 2}^i(\tilde{a}_{ji}+\tilde{c}_{ji})$. The proof of Proposition 2 is completed.