Observer-based finite-time fuzzy adaptive control for MIMO non-strict feedback nonlinear systems with errors constraint
Introduction
During the past several years, the neural networks (NNs) and fuzzy logic systems (FLSs) [1], [2], [3], [4], [5], [6], [7] are utilized to solve control problems for uncertain nonlinear systems. Simultaneously, many interesting results have been published, such as [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. Among them, the authors in [8] presented fuzzy adaptive decentralized control scheme for stochastic nonlinear systems, and [10], [11] developed neural or fuzzy adaptive control strategies for nonlinear single-input and single output (SISO) pure-feedback systems, respectively. Furthermore, it should be mentioned that tracking errors are very large in the control objective of the real-world systems, which is not be permitted. Thus, it is significant problem that tracking errors should converge to a prescribed performance bound, which is so-called performance constraint problem. In recent years, performance constraint problem has been paid considerable attention, see [12], [13], [14], [15], [16], [17], [18]. Among them, the authors in [12], [14] investigated the neural or fuzzy adaptive constraint control problems for nonlinear strict-feedback systems by utilizing barrier Lyapunov function (BLF). Furthermore, the authors in [15], [16], [17] proposed the adaptive fuzzy output feedback control schemes for nonlinear systems with errors constraint, and in [18] presented a novel output tracking errors-constraint method for MIMO nonlinear systems by taking performance function and SBLF into account. Note that the whole state variables in [18] are measurable.
However, the controlled systems in Refs. [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18] are pure feedback or strict feedback systems, it can be clearly seen that nonlinear functions in nonstrict feedback systems are functions of state variables which contain the whole states. Supposing that when the above control schemes for strict feedback or pure feedback systems are adopted to solve the control problems of nonstrict feedback systems, “algebraic loop problem” will result. To overcome this problem, some novel control schemes have been presented for non-strict feedback systems, such as [19], [20], [21], [22], [23], [24], [25], [26]. The authors in [19] studied the adaptive neural control problem for nonlinear stochastic non-strict feedback system with unknown dead zone and output constraint, and the authors in [20], [22] investigated the adaptive neural or fuzzy output feedback control problems for SISO nonlinear non-strict feedback systems. Furthermore, the control scheme in [18] has been further extended to the non-strict feedback system in [24], and developed adaptive fuzzy fault-tolerant control strategy for SISO non-strict feedback system with error-constraint. The authors in [25], [26] proposed fuzzy adaptive control methods for MIMO nonlinear systems.
Note that the aforementioned presented results do not considered the reaching time in control process. The presented adaptive fuzzy or NN control schemes only guarantee the system stability when the time variables go to infinite. However, in real-world systems, such as the missile systems and the attitude control systems of the flight vehicle and robot control systems, they expected that the controlled systems are reaching the equilibrium state in a finite-time or fixed-time. Infinite time control scheme cannot be applied to achieve the control objective of high performance, which will cause the long transient response and the perform time may be very long. Due to the controllers embody the exponential power terms, thus, compared with the traditional asymptotic Lyapunov stability theory, finite-time control approaches have the advantages of high precision performance, fast transient performance and better robustness, which can ensure the state variables quickly converge to equilibrium.
In recent decades, finite-time control methods have paid great attention for many scholars and published many interesting results, for example [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40]. Among them, Bhat et al. [27] first proposed finite-time control for nonlinear systems and handled the chattering problems of the adaptive laws caused by terminal sliding mode controller. In addition, Bhat et al. [28] also provided several finite-time stability criterions. Subsequently, based on adaptive backstepping design, the authors in [29], [30], [31], [32], [33], [34], [35] proposed the semi-global practical finite-time stability (SGPFS) for nonlinear systems. Among them, the authors in [29] proposed observer-based adaptive neural finite-time control scheme for SISO quantized nonlinear system, and the authors in [30] presented adaptive fuzzy finite-time control method for pure-feedback system, and [31], [32] are for non-strict feedback systems. Furthermore, the authors in [34] first presented fuzzy adaptive finite-time control scheme for stochastic nonlinear system, and the authors in [35] extends the finite-time control strategies to large-scale systems in non-strict feedback forms. By combining adding a power integrator theory with adaptive backstepping design, the authors in [36], [37], [38], [39] investigated the global finite-time control problems for strict feedback systems. Among them, [38] studied the finite-time output constraint control problem for nonlinear SISO system by adopting BLF. In addition, [40] investigated neural adaptive control problem for high-order nonlinear non-strict feedback systems. Note that the above considered systems do not consider the input nonlinearities.
In this paper, an adaptive fuzzy finite-time output feedback control problem is investigated for MIMO nonlinear non-strict feedback systems with error constraint and unknown dead zone. FLSs are adopted to approximate the unknown nonlinear functions and a fuzzy state observer is established to estimate the unmeasurable states. By using the certainty equivalence dead zone inverse, the problem of dead-zone problem is solved. Combining SBLF with prescribed performance theory, an observer-based adaptive fuzzy finite-time control scheme is presented. Compared with existing works, the major contributions are as follows: (i) this paper studied fuzzy adaptive finite-time output feedback control problem for MIMO nonlinear non-strict feedback systems with unknown dead zone and errors-constraint. Note that similar problem is solved in [18] and [24], but the controlled system in [18] are strict feedback form and all state variable are measurable. In addition, nonlinear system in [24] is SISO ones and not consider the input nonlinearities; (ii) The presented control strategy cannot only guarantees that all signals of the closed-loop system are SGPFS, but also ensure that tracking errors converge to a small neighborhood of the zero and not violate the prescribed performance bound in a finite time.
Section snippets
System description
Consider the MIMO non-strict feedback nonlinear systems aswhere and yi are the state vectors and output of the i-th subsystems, respectively. are the unknown nonlinear functions and satisfy . Di(ui) is the output of the unknown dead zone. In addition,
Fuzzy state observer design
Note that the only available variables are the outputs therefore, a state observer needs to be constructed to estimate the unmeasurable state variables . The following FLS is adopted to approximate the nonlinear function fi,j(xi) aswhere is the estimation of state .
Defining the optimal parameter vectors as
Fuzzy adaptive finite-time control design and stability analysis
In this section, an observer-based adaptive fuzzy finite-time control strategy is proposed with errors constraint and unknown dead zone. Defining coordinates transformation as follows:where zi,1 is the tracking error, Si,1 is the transformation error and are the intermediate control functions.
Step i, 1: According to (1) and (28), define we have
Simulation example
In this section, the following numerical example is shown to elaborate the effectiveness of the presented control scheme.
Consider the MIMO nonlinear systems aswhere and
Conclusion
In this paper, the problem of adaptive fuzzy finite-time output feedback control has been solved for MIMO non-strict feedback nonlinear systems with errors-constraint and unknown dead zone. By using the certainty equivalence dead zone inverse, the problem of dead-zone has been solved. Combining backstepping design with prescribed performance theory, adaptive fuzzy finite-time control strategy has been proposed. Based on Lyapunov theory of finite-time stability, the stability analysis has been
Acknowledgement
This work was supported in part by the National Natural Science Foundation of China under Grant 61773188.
Kewen Li received the B.S. degree in Information and Computing Science from Liaoning University of Technology, Jinzhou, China, in 2016. He is currently pursuing the M.E. degree in Applied Mathematics from Liaoning University of Technology, Jinzhou, China. His current research interests include finite time control, fuzzy control, and adaptive control.
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Kewen Li received the B.S. degree in Information and Computing Science from Liaoning University of Technology, Jinzhou, China, in 2016. He is currently pursuing the M.E. degree in Applied Mathematics from Liaoning University of Technology, Jinzhou, China. His current research interests include finite time control, fuzzy control, and adaptive control.
Shaocheng Tong (SM’15) received the B.S. degree in Mathematics from Jinzhou Normal College, Jinzhou, China, the M.S. degree in Fuzzy Mathematics from Dalian Marine University, Dalian, China, and the Ph.D. degree in Fuzzy Control from the Northeastern University, Shenyang, China, in 1982, 1988, and 1997, respectively. He is currently a Professor with the College of Science, Liaoning University of Technology, Jinzhou, China. His current research interests include fuzzy and neural networks control, and nonlinear adaptive control.