Removing the feasibility conditions on adaptive fuzzy decentralized tracking control of large-scale nonlinear systems with full-state constraints

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Abstract

This work is dedicated to solving the adaptive fuzzy decentralized tracking control issue of large-scale nonlinear systems with full-state constraints. Different with barrier Lyapunov function, the main difference is that a novel nonlinear state-dependent function (NSDF) is introduced to prevent the state constraints being overstepped. Based on NSDF, the necessary feasibility conditions for virtual controllers are completely removed. Then, the prior knowledge of the unknown virtual control coefficients is no longer required since the original system is transformed via the new affine variable. Under the control strategy, three objectives on system performance are achieved: (a) all signals of the closed-loop system are bounded; (b) the subsystem output closely tracks the reference trajectory and original error is ultimately uniformly bounded; (c) the full-state constraints are not violated for all the time. At the end, two simulation examples are shown to verify the effectiveness of the control method.

Introduction

The issue of the output tracking control is one of the basic problems of control theory [1], [2], [3], [4]. Most actual industrial processes can be boiled down to tracking control problems, such as missiles, satellites, ships, vehicles, and so on. Some significant progress has been made in the research work of linear [6], [7] or nonlinear systems [5], [8]. Meanwhile, more and more scholars realize that many actual systems cannot be simply modeled as a single-modal system, but multi-modal or even large-scale systems. This type of system is widely valued by researchers due to its strong nonlinearity and strong coupling [9], [10], [11], [12], [13]. Particularly, for the large-scale interconnected systems, there exist two main control methods: centralized control and decentralized control [14], [15], [16]. Compared with centralized control, decentralized control, as a control method that only requires local signals, is used extensively to address interconnected nonlinear systems because of its simple structure and less calculation. In [17], the adaptive decentralized control issue was addressed for interconnected nonlinear systems with the prescribed performance via the bound estimation method and two continuous functions. With the help of graph theory, an adaptive decentralized control method was proposed in Li and Yang [18] for nonlinear systems subject to strong interconnections, where the strong interconnection term is set to be bounded. Additionally, the decentralized control method has been extended to stochastic nonlinear systems, and an output-feedback control method for this interconnected systems with partially unknown interactions was developed in Fan et al. [19]. It is worth pointing out that the above masterpiece does not involve the issue of state constraints. However, in actual industrial applications, due to the requirements for system performance and safety considerations, it is often necessary to keep the actuator in a certain operating range to avoid safety accidents.

Barrier Lyapunov function (BLF) and integral BLF (IBLF) have been favored by researchers as two effective methods to solve the problem of state constraints [20], [21], [22], [23], [24], [25], [26], [27], [28]. However, it should be noted that no matter which BLF is selected, the feasibility condition naturally exists in the virtual controllers, i.e., the virtual controllers satisfy pre-given constrained region during the process of backstepping design. As some works [29], [30] have explained, it is difficult to find the design parameters which meet the conditions. Obviously, the existence of design parameters that meet the feasibility conditions is essential for most published control schemes based on BLF or IBLF methods. And the selection of design parameters will directly affect the results of the feasibility analysis after offline calculation. In [31], it is also implied that if the constrained region is smaller, the optimal design parameters may be non-existent, which means that these control schemes are invalid. To eliminate this feasibility condition, some excellent results have emerged. Based on a new nonlinear mapping, in Guo and Wu [32], the original system was transformed into a new system without constraint conditions, while ensuring that the two systems have the same stability and convergence. In [33], BLF is no longer needed for the converted system, and the constraints are established. Then, the method in Zhang et al. [33] was extended to a stochastic system in Li et al. [34], removing the feasibility conditions. Different from these works, to directly address the asymmetric state constraints, nonlinear state-dependent function (NSDF) completely based on the system state is introduced in this work.

Although the research work on nonlinear systems with state constraints has made remarkable achievements [35], [36], [37], [38], it should be pointed out that the common point of most of the above works is that the virtual control coefficients in the system are considered to be known constants or nonlinear functions with known boundaries. In [39], the control coefficients of the system are set as a piecewise known nonlinear function. Unfortunately, these control schemes may fail to handle the unknown virtual control coefficients. To solve this problem, Li proposed a BLFs-based adaptive asymptotic tracking control scheme in Li [40] by assuming that the signs and boundaries of the unknown virtual control coefficients are known. Hence, reducing conservatism and removing assumptions about virtual control coefficients have become the fun of this work.

Inspired by the above discussion, we discuss in depth a value issue of adaptive decentralized control, that is, removing the feasibility conditions of virtual controllers and the assumption of unknown virtual control coefficients of large-scale nonlinear systems, and realizing the systems output tracking the reference signals. The main contributions are as follows:

  • (1)

    The original systems are converted to new systems by utilizing new affine variables, such that the convergence and stability of the two systems are consistent. This means that the new systems will no longer be affected by the unknown virtual control coefficients, and the conservative assumption about the unknown virtual control coefficients is removed.

  • (2)

    Compared with the previous BLFs or IBLFs-based works [26], [27], [28], [37], [38], [39] that are subject to state constraints, in this work, in order to directly deal with asymmetric state constraints, NSDF completely based on the system state is introduced to remove the feasibility conditions on the virtual controllers.

  • (3)

    Only one adaptive parameter is updated live for each subsystem to lessen the controller computing burden.

Section snippets

System description

Consider the following large-scale non-lower-triangular interconnected nonlinear systems:{x˙i,j=fi,j(x¯i)+gi,j(x¯i,j)xi,j+1+hi,j(y¯)x˙i,ni=fi,ni(x¯i)+gi,ni(x¯i,ni)ui+hi,ni(y¯)yi=xi,1subject to full-state constraints described asxi,jDi,j:={xi,jR:Li,j<xi,j(t)<Ri,j,1iN,1jni},where Li,jRi,j are known positive constants and the initial condition satisfies Li,j<xi,j(0)<Ri,j. From the above system model Eq. (1), x¯i,j=[xi,1,xi,2,,xi,j]T, x¯i=[x1,x2, ,xi]T, and xi=[xi,1,xi,2,,xi,ni]T is the

Nonlinear state-dependent function (NSDF)

In the traditional tracking control research of nonlinear systems based on the BLFs control method [20], [21], [22], [23], [24], [25], [26], [27], [28], the state constraint is transformed into a new boundary of tracking error, which imposes some restrictions on the initial value of the system and virtual controllers. More generally, NSDF ξi,j purely based on the system state is constructed to directly address the asymmetric state constraints, and ξi,j is defined asξi,j=xi,j(t)(Li,j+xi,j(t))(Ri,

Main results

Firstly, the new variable is designed asξd,i=yd,i(t)(Li,1+yd,i(t))(Ri,1yd,i(t)).

Considering Assumption 2, it can be judged that ξd,i are bounded in the compact set Dd and ξ˙d,i=ψd,iy˙d,i with ψd,i=Li,1Ri,1+yd,i2(t)(Li,1+yd,i(t))2(Ri,1yd,i(t))2 being bounded and continuous in the set Dd.

For some BLFs- or IBLFs-based nonlinear system control methods [26], [27], [28], [37], [38], [39], there exists the so-called feasibility conditions Li,j<αi,j1(t)<Ri,j,(1iN,2jni) on the virtual

Stability analysis

Theorem 1

For the nonlower-triangular large-scale interconnected nonlinear systems Eq. (1) subject to asymmetric full-state constraints Eq. (2), by designing the virtual controllers, the actual controllers and the parameter adaptation laws Eq. (11), then the developed fuzzy decentralized tracking control scheme guarantees that all signals of the closed-loop system are bounded and the full-state constraints Eq. (2) are removed from the need for feasibility conditions on virtual controllers.

Proof

In view of Eqs.

Simulation studies

Example 1

Consider the numerical model below to further confirm the effectiveness of the developed control scheme:{x˙i,1=gi,1xi,2+fi,1+hi,1,i=1,2,x˙i,2=gi,2ui+fi,2+hi,2,yi=xi,1.

When i=1, the system Eq. (39) means subsystem 1 and the system functions are selected as g1,1=1, f1,1=sin(x1,1x1,2), h1,1=x1,1x2,1, g1,2=1+0.1x1,12, f1,2=0.1x1,1x1,2, h1,2=x1,1+x2,1; when i=2, it means subsystem 2 and the systems functions are selected as g2,1=1+0.5sin(x21), f2,1=x2,1sin(x2,2), h2,1=x1,12x2,1, g2,2=1+x2,12, f2,2=x2

Conclusion

In this work, an adaptive fuzzy decentralized tracking control frame has been constructed for large-scale nonlinear systems with full-state constraints and unknown virtual control coefficients. The primary obstacle is how to use NSDF to make the proposed control scheme independent on the feasibility conditions for the virtual controllers, so that the design of the controller is freer. By using the new affine variables to transform the system and the FLS to handle unknown nonlinearities, the

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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    This work was supported in part by the National Natural Science Foundation of China under Grant 62073094, in part by the Natural Science Foundation of Heilongjiang Province of China under Grant YQ2019F004, in part by the China Postdoctoral Science Foundation under Grants 2018M63034, and 2018T110275.

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