Abstract
An adaptive controller is constructed for a class of stochastic manipulator nonlinear systems in this paper. The states are constrained in the compact set. A tan-type Barrier Lyapunov Function (BLF) is employed to deal with state constraints. The proposed control scheme guarantees the output error convergence to a small neighbourhood of zero. All the signals in the closed-loop system are bounded. The simulation results illustrate the validity of the proposed method.
1. Introduction
Adaptive control of strict-feedback nonlinear systems has received a lot of attention since the appearance of recursive backstepping design [1] and feedforward design [2, 3]; a great deal of work has been done for this class of systems in the past decades; for example, see [4–9]. Constraints are widespread in many real systems such as robotic manipulators and physical engineering systems. Therefore robotic manipulators have received increasing attention over the last few years. For this reason, many methods have been used to handle the issue of constraints, for example, [10–16]. In detail, in [10], the author studies the neural network adaptive control design for robotic systems with the velocity constraints and input saturation. An adaptive finite-time controller is considered in [11] for a class of strict-feedback nonlinear systems with parametric uncertainties and full state constraints. In [12], the problem of position control of manipulators operating in the task space under state constraints has been addressed. Literatures [15, 16] discussed the trajectory and tracking control problems of the mobile manipulator with constrained end-effector and adaptive controllers are proposed.
The practical systems are inevitable to contain the stochastic disturbance and it can cause instability of system. So the stability of stochastic nonlinear systems has attracted great attention [17–19]. In [20], the author has proposed the state equations of the stochastic dynamics of an open-chain manipulator in a fluid environment. This paper has given an algorithm for the discretization of the state equations and explained how the interaction between a fluid and a manipulator can be taken into account in the control of manipulators. The work [21] studies the problem of output feedback stabilization for a class of stochastic feedforward nonlinear systems with state and input delays and the unknown output function. In [22], the stochastic response of a mobile robotic manipulator has been investigated. This paper has studied the sensitivities of the joints responses to base velocity, the surface roughness coefficients, manipulator configuration, and damping in detail.
In this paper, an adaptive tracking controller will be designed for stochastic manipulator nonlinear systems with full state constraints. A backstepping technique with a tan-type Barrier Lyapunov Function (BLF) will be constructed to address the state constraints problem and all the states in stochastic nonlinear systems are not violated. The error signals have converged to an arbitrarily small neighbourhood of zero and all the signals in the closed-loop system are bounded.
2. System Description
2.1. Problem Statement and Preliminary Results
Consider the one-link manipulator which contains motor dynamics and stochastic disturbances. The model is described as [9]
where are the link position, velocity, and acceleration, respectively; denotes the torque produced by the electrical system; is the known torque stochastic disturbance; is the electromechanical torque control input; is a known mechanical inertia; is unknown coefficient of viscous friction at the joint; is a known constant related to the mass of the load and the coefficient of gravity; is the known armature inductance; is the known armature resistance; is the known back electromotive force coefficient; is the link mass; and is the length of the link which is known. Then, following change of coordinates , we can get
All the states are constrained in the compact set as , where , are positive constants.
Given a reference trajectory , the control objective is to design an adaptive control algorithm such that tracks the desired trajectory as much as possible; all the signals in closed-loop systems are bounded; the state constraint requirements are not violated. To facilitate control system design, the following assumption and lemma are proposed.
Assumption 1. The reference trajectory and its derivatives up to the n-th ones are continuous and bounded. That is, for any constant , there exist positive constants , such that , .
Remark 2. This assumption is reasonable. Assumption 1 is the worst case one. The requirement on derivatives is widely fixed in backstepping control [23]. This is because the standard backstepping technique requires the reference signal to be continuous and derivable to design the desired controller. in assumption is always true in practice for the requirement of output tracking control. A similar assumption is also considered in [24, 25].
Next, consider the following stochastic nonlinear system:
where is the system state vector; and satisfy the locally Lipschitz functions and the linear growth condition and ; is an r-dimensional standard Wiener process.
Definition 3 (see [2]). For any Lyapunov function , we define the differential operator L as follows:
where is the matrix trace.
Lemma 4 (see [26]). There exist a function , two constants , and class functions satisfying and ; then, there is a unique strong solution which satisfies
3. Control Design and Stability Analysis
Step 1. Define the tracking error , where are the virtual controllers and with are positive constants, and with . We can get Consider a candidate BLF as follows: Based on Definition 3, one hasDesign the virtual controller aswhere is a design parameter. Substituting (9) into (8), we can obtain
Step 2. Define the tracking error ; the following can be obtained: Choose a candidate BLF as follows: Here , denotes the estimation of . Based on Definition 3, we can obtainDesign the virtual controller aswhere is a design parameter. An tuning function is chosen as . Substituting (14) into (13) yields
Step 3. Defining the tracking error , we have Construct a candidate BLF as follows: Based on Definition 3, computing , we can figure outDesign the controller as follows:where is a design parameter. Choosing and substituting (18) and (19) into (15) result inThe adaptive law is given as with a design parameter . Then, we can obtainChoose ; then (21) can be rewritten aswhere ; that is,
Theorem 5. Consider system (1) under Assumption 1; the controller is given in (19), and the adaption law . Then, the following are guaranteed:(1)The full state constraints are not violated.(2)All the signals in the closed-loop system are bounded.(3)The error signals will converge to .
Proof. Define a candidate BLF as follows: From (23), we can get that Based on Lemma 4, we know that Then, the following inequality holds: Hence, Hence, the size of can be made small enough by choosing appropriate parameters. On the other hand, from the above inequality, we can obtain that is limited by Then, we know that is bounded. So is bounded and is bounded. Then we know that , , , and are bounded. From and , we know that From and , we can obtain that . In a similar way, we can obtain . Thus, the full state constraints are not violated.
4. Simulation
In this section, simulation is introduced to demonstrate the effectiveness of the proposed scheme.
where , , , , , and . All the states are constrained in the compact set as . The reference signal is chosen as .
In the simulation, the design parameters are chosen as , , , , , , and . The results of the simulation are shown in Figures 1–4. The output tracking and is illustrated in Figure 1. It can be seen that the position state can primely track the desired trajectory . It is shown in Figure 2 that all the states are strictly constrained in . The parameter updating law and input are all bounded as shown in Figures 3 and 4. The simulation results demonstrate the effectiveness of the proposed adaptive control scheme.
5. Conclusions
This study carries out the adaptive tracking control for a class of stochastic manipulator nonlinear systems. An adaptive controller is proposed to ensure that the mean square of the tracking error can be made arbitrarily small. Simulation results are presented to illustrate the effectiveness of the proposed control strategy.
Data Availability
In this paper, only numerical simulation is given; all the data are produced by Matlab program. No other data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was supported by the National Natural Science Foundation of China under Grants 61603170, 61573177, and 61773191; the Natural Science Foundation of Shandong Province for Outstanding Young Talents in Provincial Universities under Grant ZR2016JL025; the Natural Science Foundation of Shandong Province of China ZR2018MF028; and Special Fund Plan for Local Science and Technology Development led by Central Authority.