Time-varying state-feedback stabilisation of stochastic feedforward nonlinear systems with unknown growth rate

ABSTRACT We consider the time-varying state-feedback stabilisation problem for a class of stochastic feedforward nonlinear systems with unknown growth rate in this paper. A new LaSalle-type theorem for stochastic time-varying systems is firstly established by using the generalized weakly positive definite function. As an application, to deal with serious uncertainties in the unknown growth rate, a time-varying approach, rather than an adaptive one, is adopted to design the scheme of a state-feedback controller for stochastic feedforward systems. Based on the established LaSalle-type theorem, it is shown that all signals of the resulting closed-loop system converge to zero almost surely. Illustrative examples are given to verify the theoretical findings.


Introduction
Since stochastic integral was first introduced by Itô in 1949, there has been a quick development of the theory of stochastic differential equations. The powerful Lyapunov method has been used to deal with stochastic stability by many authors, see, e.g., Khasminskii (2011), Kushner (1972), Deng, Krstić, and Williams (2001), Wu, Cui, Shi, and Karimi (2013), Chen, Zheng, and Shen (2009), Wu, Zheng, and Gao (2013), Kang, Zhai, Liu, Zhao, and Zhao (2014), Chen and Zheng (2016), Kang, Zhai, Liu, and Zhao (2016) and the references therein. Note that these results are only applied to time-invariant systems. Recently, a new type of stability theorem for stochastic systems with arbitrary variability in the time was established in Zhao and Deng (2015) by the extension of positive definite functions. However, these results cannot be used to locate the limit sets of stochastic systems and the definition of weakly positive definite functions seems to be restrictive. This problem was firstly dealt with in Mao (1999), where the LaSalle-type theorem for stochastic delay systems was proposed. Then, better results were obtained in Yu, Xie, and Duan (2010), Li and Mao (2012) by removing the linear growth condition in Mao (1999).
Feedforward systems, also called upper-triangular systems, have been widely applied to model many physical devices, such as the cart-pendulum system (Mazenc & Bowong, 2003) and the induction heater circuit system (Jo, Choi, & Lim, 2014). Based on the fundamental stability theory, the controller design and analysis for CONTACT Wei Xing Zheng w.zheng@westernsydney.edu.au feedforward systems have been a hot topic recently, see, e.g., Teel (1992), Mazenc and Praly (1994), Tsinias and Tzamtzi (2001), , Du, Qian, and Li (2013), Qian and Du (2012) and the references therein. More precisely, the problem of asymptotic tracking for feedforward systems was addressed in Mazenc and Praly (2000) by constructing time-varying state feedback. In Shang, Liu, & Zhang (2015), Ye and Unbehauen (2004), the adaptive stabilizing problem for feedforward systems with unknown linear growth rates was investigated by introducing a dynamic gain and a switching logic, respectively. By introducing the dynamic gain approach to time-delay systems, constructive control techniques were proposed for controlling feedforward nonlinear time-delay systems in . Although noticeable progresses have been made for deterministic feedforward systems (see, e.g., Frye, Trevino, & Qian, 2007;Tsinias & Tzamtzi, 2001;, up to now there has been very few literature considering stabilisation of stochastic feedforward systems. In Liu and Xie (2013), a state-feedback controller was designed to globally stabilise a class of stochastic feedforward nonlinear systems. Subsequently, for stochastic high-order feedforward systems, Zhao and Xie (2014) examined the problem of state controller design and then (Jiao et al., 2014) studied the problem of decentralised stabilisation for a class of large-scale cases. However, these works are limited in the sense that the nonlinearities depending on the states grow at a known constant rate. For stochastic feedforward systems with unknown growth rates, how to design a stabilizing controller as in Shang et al. (2015),  still remains as an open and challenging problem. The purpose of this paper is to develop a global timevarying stabiliser for a class of stochastic feedforward systems possessing an unknown growth rate. The main contributions of this paper are highlighted as follows: (1) In almost all the literature, the global stability analysis of controller designs of stochastic nonlinear systems heavily relies on the stochastic theories in Deng et al. (2001), Khasminskii (2011). However, it is not applicable to more general stochastic systems, particularly to stochastic time-varying cases. On the other hand, although Zhao and Deng (2015) provided a good solution to this problem, the condition of weakly positive definition in Zhao and Deng (2015) is still restrictive (see the motivating example in the next section). In this paper, we will generalise the work of Zhao and Deng (2015) under more common assumptions. Thus, a new definition of the weakly positive definite function is given, based on which we provide a new LaSalle-type theorem for stochastic time-varying systems.
(2) With the stochastic LaSalle-type theorem, a timevarying state-feedback controller (instead of an adaptive one) for stochastic feedforward nonlinear systems with an unknown growth rate is designed to guarantee that all signals of the closed-loop system are bounded almost surely and the system states converge to zero almost surely.
The remainder of this paper is organised as follows. Section 2 introduces some preliminaries on stochastic theory and gives a LaSalle-type theorem. In Section 3 a time-varying state-feedback design scheme is developed for a class of stochastic feedforward systems with an unknown growth rate. Section 4 illustrates our proposed control design through one simulation example and one practical example, after which the paper is concluded in Section 5.
Notation. The following standard notation is used throughout this paper. For a vector x, |x| stands for its usual Euclidean norm and x T denotes its transpose. A represents the 2-norm of a matrix A. EX(t) denotes the expectation of the stochastic process X(t). K stands for the set of all functions α(s) : R + → R + , which are continuous, strictly increasing and vanishing at zero; K ∞ denotes the set of all functions which are of class K and unbounded; KL denotes the set of all functions β(s, t ) : R + × R + → R + which are of class K for each fixed t, and decrease to zero as t → Ý for each fixed s. L 1 (R + ; R + ) denotes the family of all Borel measurable functions l : R + → R + such that ∞ 0 l(s)ds < ∞. (R + ; R + ) represents the family of function σ (t) satisfying that ∞ k=1 t k +δ t k σ (s) ds = ∞ for any δ > 0 and any increasing sequence {t k } k ࣙ 1 (t k → Ý as k → Ý).

Mathematical preliminaries
In this section, the LaSalle-type theorem for stochastic time-varying systems is presented.
Consider the following stochastic nonlinear system where x(t ) ∈ R n denotes the state vector, x 0 is a deterministic initial value, B(t) is an m-dimensional standard Wiener process defined on the complete probability space ( , F, {F t } t≥0 , P) with being a sample space, F being a σ -field, {F t } t≥0 being a filtration, and P being a probability measure. Functions f (·) ∈ R n and g(·) ∈ R m×n satisfy the following assumption.
Assumption 2.1 (Zhao & Deng, 2015): The generalised local Lipschitz condition. For all K > 0, there exists a continuous function L(K, t) > 0 such that, t ࣙ 0 and |x|ࢴ|y| ࣘ K, Let us give an example to show the necessity of introducing LaSalle-type theorem for stochastic time-varying systems.
A motivating example: Consider a simple onedimensional stochastic system (2) ( From the fact that system (2) and function −3cos 2 (t)x 4 (t) are time varying, it follows that only by means of (3) and Lemma 1 in , one cannot obtain the global asymptotic stability in probability of system (2). On the other hand, although Zhao and Deng (2015) proposed a new type of stability theorem for stochastic time-varying systems, many functions do not belong to the family of weakly positive definite functions defined in Zhao and Deng (2015), for example, cos 2 (t), sin 2 (t) and |cos (t)|, etc. From the viewpoint of control theory, it is interesting and necessary to extend the conclusion of Theorem 1 in Zhao and Deng (2015) for more general class of stochastic time-varying systems. The main aim of this section is to give a very positive answer.
Definition 2.2 (Deng et al., 2001): The equilibrium x = 0 of system (1) is said to be r globally stable in probability if ϵ, there exists a r globally asymptotically stable in probability if it is globally stable in probability and

Definition 2.3:
A time-varying function W (x, t ) : R n × R + → R + is said to be generalised weakly positive definite, if for any δ > 0 and any increasing sequence Remark 2.1: Obviously, the generalised weakly positive definite function covers that defined in Zhao and Deng (2015). For the special case W(x, t) = σ (t)φ(x), where φ( · ) is positive definite, the generalised weakly positive definiteness of W(t, x) can be described by σ (t ) ∈ (R + ; R + ).
We now state the following LaSalle-type theorem.
Then (i) There exists a unique and bounded almost surely strong solution to system (1) in [0, Ý); , and l(t) = 0, the equilibrium x = 0 is globally asymptotically stable in probability.
Proof: For any r > 0, define the stopping time as Thus, there exists a constant M > 0 such that EV(x(tࢳζ r ), tࢳζ r ) ࣘ M for r, t ࣙ 0, which leads to the first conclusion i) by Lemma 2 in . Next, we give a proof of conclusion ii). When W(x, t) = ρ(t)w(x), from (6), by letting r → Ý, t → Ý and applying Fatou's lemma, one gets Based on the proof of Theorem 2.1 in Li and Mao (2012), it follows from (7) that Since ω is a continuous function, we have from which we obtain that y ࢠ Ker(ω) and Ker(ω) contains all of the limit points of x(t, ϖ 0 ). By the continuity of x(t; ϖ 0 ), we arrive at which, together with P( 0 ) = 1, implies conclusion ii).
It follows from (4)-(5) and l(t) = 0 that the equilibrium x = 0 is globally stable in probability by Theorem 2.1 in Deng et al. (2001).
Next we show that To this end, we first prove that If this is not true, then we can find a pair of positive constants κ, ε such that According to the continuity of V(x(t), t) and the generalised weak positive definiteness of W(x(t), t), we know that there exists an instant t * > 0 such that, t > t * , Then, by (4), we obtain Therefore, with (6), (8) and the weakly positive definiteness of W(x(t), t), by choosing t k = t * + k − 1 and δ = 1, we get This leads to a contradiction, thus it follows that Hence, by Definition 2.2, the equilibrium x = 0 is globally asymptotically stable in probability.

Remark 2.2:
Given that the LaSalle theorems are powerful in the study of stability for locating limit sets of timeinvariant systems, here we have presented a new LaSalletype theorem for time-varying systems, which not only can be applied much more easily but also can cover a much wider class of stochastic time-varying systems. To see this point, let us return our attention to the motivating example (2). It is easy to see that cos 2 (t ) ∈ (R + ; R + ). By Lemma 2.1, we can conclude that the equilibrium x = 0 of system (2) is globally asymptotically stable in probability. Figure 1 shows the simulation result of the system state with x(0) = 2, which also implies the effectiveness of the proposed result in Lemma 2.1.

Problem formulation
In this paper, consider the following stochastic feedforward nonlinear system where x = [x 1 , · · · , x n ] T ∈ R n denotes the measurable state vector and u ∈ R is the control input. B(t) is an m-dimensional standard Wiener process defined on the complete probability space ( , F, {F t } t≥0 , P) with being a sample space, F being a σ -field, {F t } t≥0 being a filtration, and P being a probability measure. f i (·) ∈ R n , g i (·) ∈ R m×n are generalised locally Lipschitz functions and satisfy f i (0, t) = 0, g i (0, t) = 0. For system (9), the following assumption is needed:

Assumption 3.1: There is an unknown constant
Remark 3.1: It follows from Assumption 3.1 that system (9) is indeed a stochastic feedforward nonlinear system possessing linear state-dependent growth with unknown growth rates. The system considered in the paper is more common than some existing ones, for instance, the diffusion terms in Du, Qian, He, and Cheng This paper aims to find a time-varying state-feedback control u = μ(x, t) such that the resulting closed-loop system has a unique and almost surely bounded strong solution x(t) in [0, Ý), and P(lim t → Ý |x(t)| = 0) = 1.

Time-varying state-feedback controller
In what follows, we present a time-varying state-feedback controller design that globally stabilises system (9), which consists of the following two parts.

.. Transformation of coordinates
First, let us introduce the following time-scaling coordinate change to transform system (9) into a system with a time gain in the nonlinearities: where b > 0 is a constant. Hence, by (10), system (9) can be expressed as and η n + 1 (t) = 0.

Recursive
Step k (k = 3, , n): Suppose that steps 1, … , k − 1 have been completed and there is a C 2 , positive definite and proper Lyapunov function where Next we show that a similar conclusion still holds true for Step k. Let ϕ k = η k − η * k and then define the k-th Lyapunov function V k = V k−1 + 1 2 ϕ 2 k . It follows from (11) and (17) that With the aid of Young's inequality, we have where l k11 > 0, l kj2 > 0 are constants.
Combining (18) and (19) with (20) results in By choosing the following virtual controller and defining Hence, from the last step of the above design procedure, we obtain the actual controller which leads to where V n (ϕ 1 , . . . , ϕ n ) = n k=1 1 2 ϕ 2 k , andᾱ i = α n · · · α i and c n, j are positive constants.

Stability analysis
Now we are ready to state the main result.
Theorem 3.1: Under Assumption 3.1, there is a global unique solution for the closed-loop system consisting of the system (9) and the time-varying state-feedback controller u = ν (t+b) n . Moreover, P(lim t → Ý |x(t)| = 0) = 1. Proof: The proof can be divided into the following two steps.
Step 1: We first prove that the closed-loop system consisting of (11) and (22) has a global unique solution, which converges to zero almost surely.
For illustration convenience, the closed-loop system (11) and (22) is written in the following compact form as . . .
Obviously, the controller ν defined in (22) is C 1 . On the other hand, the nonlinear functionsf i andg i are generalised locally Lipschitz. Therefore, the generalised locally Lipschitz condition of the closed-loop system (24) can be guaranteed.
From (28), we obtain which implies the existence of global solution for system (24) by Theorem 3.5 in Khasminskii (2011). We then prove lim t → Ý |η(t)| = 0 a.s. By (28), there is a sufficiently large T * > 0 such that On the other hand, from which it yields that c 01 t + b ∈ (R + ; R + ).
Step 2: From (10), we see that the coordinate transformation is equivalent. Hence, the closed-loop system (9) with u = ν (t+b) n has the same properties as systems (11) and (22). Therefore, Theorem 3.1 holds true. Remark 3.2: Compared with the work by Liu and Xie (2013) and Zhao and Xie (2014), we have only got the almost sure convergence of the closed-loop system, rather than the asymptotic stability. In fact, since c 01 , c 02 and c 03 in (28) are dependent on the unknown parameter θ, the right-hand side of (28) may be positive during a small initial time interval. Hence, the asymptotic stability cannot be deduced for the closed-loop system from (28). Finally, from (32), it is concluded that system (31) can be globally stabilised by the time-varying state-feedback control For simulation purposes, the parameters are chosen as a 1 = −12, a 2 = −6, b = 6, c 11 = 0.5, c 22 = 0.7, c 33 = 0.8, l 211 = l 212 = l 311 = 0.1, l 312 = l 322 = 0.3, the sampling period 0.001 s and the initial conditions x 10 = 0.2, x 20 = −3.5, x 30 = 0.5. Simulation results plotted in Figures 2  and 3 show the usefulness of the obtained results. Example 4.2: Consider an induction heater circuit system (Lander, 1987) (see Figure 4 ). It is known that the input voltage V may change due to the external noises. That is, as proposed in Ugrinovskii and Petersen (1999), one may suppose that the input voltage V is given by V = V 1 + aV 1Ḃ (t ), whereḂ(t ), the formal derivative of a standard Wiener process B(t), is the white noise and a is an unknown constant. Thus, by defining x 1 = V c , x 2 = i L , the state equation of the circuit system can be expressed as Then, if we choose the parameters L = 1H, C = 1F, d = 2 A/V, R 1 = R 2 = 1 , and the unit of the current is A, the unit of the voltage in V, by the invertible transformatioñ x 1 = x 1 ,x 2 = x 1 − x 2 , u = V 1 , system (41) is rewritten as dx 1 =x 2 dt, dx 2 = u + au dB(t ). (42) Obviously, system (42) where c 21 = c 11 − l 211 − l 212 > 0. From (43), system (41) can be globally stabilised by the time-varying state-feedback control whereᾱ 1 = α 1 α 1 ,ᾱ 2 = α 2 . In the simulation, the parameters are chosen as a = 0.5, b = 10, c 11 = 3, c 22 = 2, l 211 = l 212 = 1, the sampling period 0.001 s and the initial conditions x 10 = 2.2 V, x 20 = 1.8 A. Figures 5 and 6 plot the responses of the closedloop system, which implies the validity of the proposed control strategy.

Concluding remarks
In this paper, the stabilisation problem has been studied for a class of stochastic feedforward nonlinear systems whose growth rate is not necessarily known. To effectively compensate the serious uncertainty, we have proposed a time-varying state-feedback controller. With the help of the new type of LaSalle-type theorem, it has been proved that the closed-loop system has a global unique solution, which converges to zero almost surely. Two examples have been provided to show the effectiveness of the proposed control scheme. Finally, our future research will seek to investigate the stochastic feedforward nonlinear system with state-dependent growth rates, which will be an interesting and challenging topic.

Disclosure statement
No potential conflict of interest was reported by the authors.

Funding
This work was supported in part by the Australian Research Council [grant number DP120104986].