Output Constrained Adaptive Controller Design for Nonlinear Saturation Systems

Yongliang Yang, Member, IEEE, Zhijie Liu, Member, IEEE, Qing Li, and Donald C. Wunsch, Fellow, IEEE

Abstract—This paper considers the adaptive neuro-fuzzy control scheme to solve the output tracking problem for a class of strict-feedback nonlinear systems. Both asymmetric output constraints and input saturation are considered. An asymmetric barrier Lyapunov function with time-varying prescribed performance is presented to tackle the output-tracking error constraints. A high-gain observer is employed to relax the requirement of the Lipschitz continuity about the nonlinear dynamics. To avoid the “explosion of complexity”, the dynamic surface control (DSC) technique is employed to filter the virtual control signal of each subsystem. To deal with the actuator saturation, an additional auxiliary dynamical system is designed.It is theoretically investigated that the parameter estimation and output tracking error are semi-global uniformly ultimately bounded. Two simulation examples are conducted to verify the presented adaptive fuzzy controller design.

I. INTRODUCTION

ADAPTIVE control theory is widely applied to deal with the engineering applications, including robotic manipulators [1], [2], power systems [3]–[6], chemical process [7], [8], mechanical systems [9], circuit systems [10],[11], etc. Fuzzy logic systems (FLS) and neural networks(NN) can efficiently compensate for the typical nonlinearity and uncertainty in modeling and control, such as saturation[12]–[14], deadzones [15], and hysteresis backlash [16], to name a few. It is usually desirable to drive the plant to behave with predetermined behavior. Traditional neuro-adaptive control methods only consider the convergence of the parameter adaptation and the boundedness of the closed-loop signals. It is critical in real-world applications to guarantee the variables under consideration within some certain region. This paper investigates adaptive fuzzy control of uncertain systems with predefined transient tracking performance and input saturation simultaneously.

The backstepping technique is widely used in adaptive control of strict feedback nonlinear systems, where the virtual control input of each subsystem is designed in each step [17].The backstepping method suffers from an “ explosion of complexity” due to the repeatedly differentiation of the virtual control input [18]. The complexity of external control input design grows drastically as the system order increases. To avoid this issue, the dynamic surface control (DSC) method with first-order and second-order filters are developed to obviate the virtual control differentiation [19]–[21]. Adaptive control designs with DSC with output consideration of constraints have been investigated in [22], [23]. However,these results are based on full-state measurement, which might be a strong requirement and impossible for some applications.Nonlinear observer design provides an efficient estimation of the full-state to obviate the requirement of full-state measurement, which is widely used in adaptive constrained output feedback design [24], [25]. Nevertheless, the input constraints are not considered in these literature. In addition,in existing observer-based neuro-adaptive controller design,[26]–[28], the nonlinear systems are restricted to be Lipschitz continuous captured by a Lipschitz constant. In this paper, we modified this requirement by presenting a weak Lipschitz condition, where the Lipschitz constant is relaxed to be a nonnegative function. On this basis, we combine the high-gain observer with the DSC method to deal with the constraints on both input and output.

In stability analysis, the Lyapunov function is usually not unique and the common selection for the Lyapunov candidate is in quadratic form [29]. However, only asymptotic behavior of the closed-loop system can be guaranteed using the quadratic Lyapunov candidate. To consider the transient behavior of dynamical systems, the concept of a “barrier function” in constrained optimization [30] has been adopted in Lyapunov candidate selection [31], which is referred to as the barrier Lyapunov function (BLF) [32]. BLF based adaptive control design has been applied to deal with multiple problems, such as constrained differential games [33], fullstate constraints of a robotic manipulator [34] and purefeedback systems [35], and the constrained tracking problem of nonlinear systems with unknown control direction [36]. A BLF based method was recently extended to deal with different types of constraints, including with asymmetric and symmetric constraints [37]. However, the constraints considered in the BLF method are usually restricted to be time-invariant. To consider a time-varying constraint, the prescribed performance adaptive control (PPAC) method is developed to establish both the transient response rate and the ultimate residual level set using a set of user-defined performance functions [15], [38]. However, the Lyapunov candidate construction in the PPAC method is still in quadratic form. In this paper, we combine a BLF with PPAC by using an asymmetric BLF with time-varying constraints to guarantee the steady and the transient performance.

Fig.1. Overall design scheme. ① Input saturation design: the input saturation is considered using additional dynamical systems (34) and the Nussbaumfunction related dynamical system (35). ② Adaptive high-gain observer design: the output estimation error feedback gain is time varying by developing the adaptive high-gain (18). ③ Output constraint guarantee: the transient output constraint is captured by the presented BLF with prescribed tracking performance which results in the special form of virtual control input α1 in (27).

The main contributions of this paper are threefold. First, a BLF with an asymmetric time-varying constraint is presented to ensure the prescribed transient performance on the output tracking error. Second, to estimate the unmeasured states, the high-gain observer with adaptive feedback gain is designed with a relaxed continuity assumption. Finally, the input saturation is solved by introducing an additional auxiliary system. The Nussbaum function-based method is used for stability analysis.

The remainder of this paper is described as follows. The preliminaries and background knowledge are given in Section II. Section III provides the problem formulation. In Section IV, the high-gain observer with adaptive high-gain is presented to estimate the unmeasured states. The asymmetric BLF with prescribed transient performance is designed in Section V. On this basis, the adaptive backstepping design is developed. Section VI provides the simulation examples to verify the effectiveness of the presented adaptive fuzzy backstepping controller design. The concluding remarks are given in Section VII.

II. BACKGROUND AND PRELIMINARIES

A. Nussbaum Function

Definition 1 [39]: The function N(ω) is called a Nussbaum gain function if it has the following properties:

B. Fuzzy Logic Systems For Function Approximation

A fuzzy logic system consists of the knowledge base, the fuzzifier, the fuzzy inference engine and the defuzzifier. The knowledge base for FLS is composed of a series of fuzzy If-Then inference rules of the following form

Fig.8. The system output y(t)=x1(t), reference signal xd(t) and tracking error z1(t) with prescribed output tracking performance.

Fig.9. The state estimation with high-gain observer.

VII. CONCLUSIONS

In this paper, the tracking problem with prescribed performance for uncertain nonlinear strict-feedback systems have been solved by a high-gain observer-based adaptive output feedback design. The unmeasured states are estimated by the observer with an adaptive high-gain. The input saturation has been solved by an additional dynamical system with Nussbaum function-based design. The output constraints are obtained by the BLF with asymmetric and time-varying tracking performance. The calculation burden of the virtual control is reduced by employing the DSC filter design. It has been discussed in theory that the overall design scheme can guarantee the boundedness of all the closed-loop signals.

Fig.10. The adaptive parameter update.

Fig.11. The control input design u(t) with saturation.

Future works aim to investigate some other type of BLF and extend to the case with event-triggered mechanism [51].

APPENDIX A

PROOF OF LEMMA 6

Proof: The proof consists of three parts.

1) We will prove by contradiction. Supposes that r(t)<1; it follows that r˙>0. Furthermore, one can obtain r(t)≥r(0)=1,which contradicts the fact r(t)<1. Therefore, one can conclude that r(t)≥1 for t ∈[0,+∞).