Adaptive Tracking Control for Stochastic Nonlinear Systems with Full-State Constraints and Unknown Covariance Noise
Introduction
As is well-known that nonlinear system control plays a crucial role in practical engineering field such as the single-link robot, the tunnel diode circuit, the chemical reactor systems, and the complex networked systems [1], [2], [3], etc.. Based on nonlinear stability theory [4], Lyapunov function control method [5], the backstepping technique [6], and other control methods, numerous control results have been emerged for nonlinear systems in [7], [8], [9], [10], [11], [12], [13], [14] and the references therein. On the other hand, stochastic disturbances and observation noise including the parameter perturbations, system’s control input variations and stochastic errors, widely occur in physical systems and complex networked systems [15], [16], [17], [18], [19], [20], [21]. This makes practical nonlinear systems sometimes can be modeled by stochastic nonlinear differential equations and called as stochastic nonlinear systems. By generalizing the backstepping technique to stochastic nonlinear systems [22] and using the basic stochastic stability theory [23], [24], the controller design for different classes of stochastic nonlinear systems are fully considered; e.g., large-scale stochastic nonlinear systems [25], stochastic nonlinear systems with high-order [26], [27], stochastic nonlinear systems with time delays [28], [29], [30], stochastic switched nonlinear systems [31], [32] and inverse dynamic stochastic nonlinear systems [33], etc.
Note that constraints with the form physical stoppage, performance and safety specifications frequently appear in practical systems, which sometimes deteriorates system performance and stability. It is proven that the barrier Lyapunov function (BLF) proposed by [34] is an effective tool since it can yield a value that approaches infinity while its arguments approach some limits. Since then, BLF together with backstepping technique, neural networks or fuzzy approximation methods has witnessed remarkable control results for deterministic nonlinear systems; see, e.g., [35], [36], [37], [38] and the related references. In contrast to nonlinear systems in deterministic forms, the full-state constrained stochastic nonlinear systems are more difficult to be controlled and the design procedure is more complicated. Until now, for stochastic nonlinear systems with full-state constraints, the related design results are very few and remain to be further addressed. This enlightens us to further take stochastic nonlinear system into accountwhere and are the system measurable states, output and control input, respectively, which are constrained within compact sets as |xi| < kbi with kbi being known constants; ; constant vectors are unknown and called as the parametric uncertainties; denotes the known smooth functions with ; the unknown smooth function satisfies ; nonnegative definite matrices is an unknown bounded function and the infinitesimal covariance ΣΣT is a function of Σdω; ω which is defined on a complete probability space, is an r-dimensional standard Wiener process with incremental covariance ΣΣTdt, this means .
When and |xi| < kbi, some related results have been obtained for system (1). With the help of BLF, backstepping, fuzzy and neural network approximation methods, [39], [40], [41] investigated the feedback control problems for system (1) in various structures. But [39], [40], [41] did not take the parametric uncertainties into account. Note that parametric uncertainties exhibit more complex dynamics commonly exist, such as the machines with friction and the biochemical processes. It is significant to consider the parametric uncertainties when stabilizing the stochastic nonlinear systems. In [42], by constructing BLF in symmetric and asymmetric forms and using the adaptive design, a control scheme was presented for stochastic nonlinear systems with unknown parameters. For stochastic nonlinear systems with parametric uncertainties, based on the tan-type BLF and finite-time backstepping design, an adaptive finite-time controller was constructed in [43].
When ΣΣT ≠ I, the stabilization problem for system (1) has been studied in [44], [45], [46] by using stochastic controller design methods and theory without considering the full-state constraints |xi| < kbi. To the authors’ knowledge, the control for full-state constrained stochastic nonlinear systems with ΣΣT ≠ I is still unsolved, not mention to considering the parametric uncertainties.
In this paper, we focus on the mentioned control problem that is unsolved. The main work and distinctive features are listed as follows.
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This note makes the first try to address the adaptive control problem of system (1) in more general form containing full-state constraints, uncertain parameters and unknown covariance noise.
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By using the BLF and adaptive control technique to overcome the negative effect brought by state constraints, parametric uncertainties and unknown covariance noise, an adaptive state-feedback controller is designed. It can be shown that the closed-loop system is uniformly ultimately bounded (UUB), the states are not violated and the system output can be used to track a reference signal.
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Compared with the related references [42]-[43], the distinctive features of this paper are three folds: the considered system is more general with the covariance noise being considered; the exactly known knowledge of gi is removed; a practical stochastic simple pendulum system instead of a completely numerical example is skillfully established and used to illustrate the effectiveness of the proposed schemes.
The rest of this paper is organized as follows. Section 2 gives some mathematical preliminaries. In Section 3, an adaptive controller is given. Section 4 shows the stability analysis of the controller. A simulation example is provided in Section 5. This is followed by Section 6 to conclude this paper.
Notations: Throughout the whole paper, denotes all the non-negative real numbers set; stands for n-dimensional Euclidean space; represents the family of all the functions with continuous ith partial derivations; XT denotes the transpose of a given vector or matrix X and ‖ · ‖ denotes its Euclidean norm with Tr{X} being its trace when X is square; | · | is the absolute value of a number or a function; a function is called as a class function if it is continuous, strictly increasing and ; is the set of all functions which are of class and γ(x) → ∞ if x → ∞.
Section snippets
Mathematical Preliminaries
For the normal stochastic nonlinear systemwhere the state vector ; ω is defined as those in (1); f(x): and g: satisfying and are locally Lipschitz.
Definition 1 [24]. For any function along system (2), denote the differential operator of V as , then, along system (2) iswhere the term is called the Herssian term. Lemma 1 [25]. Consider system (2), and assume there exist a
Design of Adaptive State-Feedback Controller
This section gives the controller design procedure by the backstepping technique, which is based on the state coordinate transformation with the formwhere yr is the reference signal satisfying Assumption 1 and α2, α3, ⋅⋅⋅, αn are virtual control laws need to be designed. In addition, one further defines the adaptive parameter Θ with the following form to handle the unknown covariance noise
Then, using (1)-(4) and the Itô’s differentiation
Stability Analysis
The theorem contains the main results is concluded as follows. Theorem 1 Considering system (1) under Assumptions 1-2, if the virtual control laws are designed as (12), (20), (30), the adaptive laws are given by (13), (21), (31) and (34)-(35), the actual controller is constructed as (33) and the initial conditions are chosen in where with and then, it holds (i). all the signals in
A Simulation Example
Take the simple pendulum system shown in [6] into consideration. The equation of motion in the tangential direction is given bywhere l denotes the length of the rod; m denotes the mass of the bob; θ is the angle subtended by the rod and the vertical axis through the pivot point; k is a coefficient of friction. Applying a torque T and viewed it as a control input. Then, define and one has
Following [46], the Gaussian white noise
Conclusions
This paper has studied the state-feedback control for full-state constrained stochastic nonlinear systems in more general forms. The BLF is used to handle the full-state constraints and the adaptive control technique is applied to deal with the covariance noise and parametric parameters. Through the backstepping procedure, an adaptive controller is designed and guarantees the closed-loop system to be UUB, the tracking error is within a compact set of the origin and the full-states are not
Conflict of Interest
We declare that none of the authors have a conflict of interests.
Acknowledges
This work was supported in part by the Project funded by China Postdoctoral Science Foundation under grant 2019M661692; in part by NSFC of China under Grant 61673215, Grant 61473151, Grant 61573172, Grant 61977037 and Grant 61703059.
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