Tracking control of MIMO nonlinear systems under full state constraints: A Single-parameter adaptation approach free from feasibility conditions

doi:10.1016/j.automatica.2019.05.032

Automatica

Volume 107, September 2019, Pages 52-60

https://doi.org/10.1016/j.automatica.2019.05.032 Get rights and content

Abstract

This paper investigates the tracking control problem of a class of multi-input multi-output (MIMO) nonlinear systems under asymmetric full-state constraints. By integrating a nonlinear transformed function (NTF) with the upper bound estimation technique, we develop a robust adaptive dynamic surface control (DSC) method that exhibits several attractive features: 1) both symmetric and asymmetric state constraints are directly addressed without the need for converting the state constraints into new constraints on tracking errors (as used in the Barrier Lyapunov Function (BLF) in most existing works). Such control scheme (with its structure unchanged) is also applicable to the systems free from state constraints; 2) the undesirable feasibility conditions on virtual control laws commonly involved in most existing works are completely eliminated; and 3) the utilization of NTF-based DSC and upper bound estimation technique not only avoids the explosion of complexity problem in traditional backstepping design, but also allows a structurally simple and computationally inexpensive control scheme to be developed, in which only one single parameter updating is needed, significantly reducing the design complexity and online computation. The effectiveness of the proposed scheme is verified via simulation.

Introduction

It is nontrivial to control a dynamic system involving unknown parameters, unmodeled nonlinearities, non-vanishing disturbances, as well as unknown multiple input couplings. The underlying problem becomes even more challenging and interesting if the system is subject to partial or full state constraints. Although various control methods have been developed during the past decades in addressing some or all of those issues (Chen et al., 2011, Ge and Tee, 2007, Jin, 2016, Jin, 2017, Liu and Tong, 2016, Liu and Tong, 2017, Tee et al., 2009, Zhao and Song, 2018a, Zhao and Song, 2018b), to just name a few), there still remain certain important problems that have not been adequately addressed. For instance, computational burden is always an issue in most control methods, where the number of adaptive parameters to be updated normally gets larger as the order or input–output dimension of the MIMO system becomes higher, dramatically limiting the practicality of the algorithms (Chen et al., 2011, Ge and Tee, 2007, Jin, 2016, Jin, 2017, Meng et al., 2015). In recent works by Chen et al. (2017) and Wang and Lin (2012), new adaptive control algorithms are proposed, in which only one adaptive parameter is involved, substantially reducing the overall computation burden. However, the plant considered is in linear time-invariant (LTI) form. Thus far, an effective solution for reducing computational burden for nonlinear MIMO systems has not been established.

On the other hand, output or state constraints are always imposed in many practical applications. Barrier Lyapunov Function (BLF) method proposed by Ngo, Mahony, and Jiang (2005) is a typical approach to deal with output or full-state constraints (Liu and Tong, 2017, Liu, Tong et al., 2017, Tee and Ge, 2011, Tee and Ge, 2012, Tee et al., 2009, to just name a few). However, most studies are focused on single-input single-output (SISO) systems. In Meng et al. (2015), an adaptive neural control of MIMO nonlinear systems with asymmetric constraints was proposed, the author in Jin (2017) proposed a tan-type BLF-based fault-tolerant control algorithm for MIMO nonlinear systems with symmetric constraints. However, only the output constraints (rather than full-state constraints) were considered in these works. As demonstrated by Tee and Ge (2011), Tang, Ge, Tee, and He (2016) and Zhao and Song (2018a), full state constraints are more difficult to handle than output constraints because the virtual control laws need to satisfy feasibility conditions. To cope with such conditions in SISO nonlinear systems, the offline optimization has to be carried out by using MATLAB routine fmincon.m to verify and to obtain the optimal design parameters (Liu and Tong, 2016, Liu and Tong, 2017, Tang et al., 2016, Tee and Ge, 2011), which increases the complexity of the design procedure. It is noted that, compared with SISO nonlinear systems, online verification of the feasibility conditions on virtual control laws for MIMO systems is much more complicated and demanding. It is thus highly desirable (although challenging) to eliminate the feasibility conditions on constrained MIMO nonlinear systems.

In this paper, aiming at eliminating the demanding feasibility conditions and mitigating the computational burden, we propose a single-parameter-estimation based robust adaptive control for MIMO nonlinear systems under asymmetric state constraints. The proposed method is partially motivated by but is essentially different from the works by Chen et al. (2017) and Zhao and Song (2018a). In fact,

$(1)$
to our best knowledge, it is the first time to use the one parameter estimation based control to handle asymmetric full-state constraints directly without the need for piecewise BLF for MIMO nonlinear systems. Such control is also able to address the case of no constraints with unchanged controller and little reprogramming of implementation;
$(2)$
based on the nonlinear transformed function, we convert the original constrained system into a new “non-constrained” one, and then we introduce a suitably defined coordinate transformation into DSC so that the feasibility conditions on virtual controllers are completely eliminated, rendering the control undemanding in design and user-friendly in implementation; and
$(3)$
by using the upper bound estimation technique, the proposed approach involves only one adaptive parameter, significantly reducing the computational burden.

The remainder of this paper is organized as follows. The problem formulation and preliminaries are given in Section 2.1. A nonlinear transformed function based system transformation is developed in 2.2, where we convert the original constrained MIMO nonlinear system into an equivalent non-constrained one, whose stability is sufficient to solve the asymmetric state constraint problem. In Section 3, the controller design methodology is proposed. The detailed stability analysis is stated in Section 4 which shows that the developed control law can guarantee the boundedness of all signals in the closed-loop system and the cases with and without constraints can be handled simultaneously. Simulation examples are presented for illustration in Section 5. Conclusions are drawn in Section 6.

Section snippets

Dynamics model

Consider the following MIMO dynamic equation (Meng et al., 2015): $\{\begin{matrix} {\dot{x}}_{j} = F_{j} ({\bar{x}}_{j}, p_{j}) + G_{j} ({\bar{x}}_{j}, η_{j}) x_{j + 1}, j = 1, \dots, n - 1, \\ {\dot{x}}_{n} = F_{n} ({\bar{x}}_{n}, p_{n}) + G_{n} ({\bar{x}}_{n}, η_{n}) u, \\ y = x_{1}, \end{matrix}$ where $x_{i} = {[x_{i 1}, \dots, x_{i m}]}^{T} \in R^{m}$ , $i = 1, \dots, n$ , are the system states and ${\bar{x}}_{i} = {[x_{1}^{T}, \dots, x_{i}^{T}]}^{T} \in R^{m i}$ , which are required to satisfy the following asymmetric constraints: $- {\underset{̲}{F}}_{i k} < x_{i k} < {\bar{F}}_{i k}, i = 1, \dots, n; k = 1, \dots, m,$ provided $- {\underset{̲}{F}}_{i k} < x_{i k} (0) < {\bar{F}}_{i k}$ , with ${\underset{̲}{F}}_{i k}$ and ${\bar{F}}_{i k}$ being positive constants. $u = {[u_{1}, \dots, u_{m}]}^{T} \in R^{m}$ and $y \in R^{m}$ are the control input vector and output vector of the system, respectively; $F_{i} (\cdot) \in R^{m}$

One parameter estimation based DSC design

In this paper, the DSC technique is employed to design the robust adaptive controller (Swaroop et al., 2000, Wang and Lin, 2010, Xia and Zhang, 2018). As one of the key steps for control design, instead of using the normally used coordinate transformations $z_{i} = x_{i} - α_{i f} (i = 1, \dots, n)$ in DSC design, where $x_{i}$ are the system states and $α_{i f}$ are the outputs of first-order filter (which will be given later), we construct the following coordinate transformations by employing $ζ_{i}$ : $z_{i} = ζ_{i} - α_{i f}, i = 1, \dots, n,$ where $α_{1 f} = [α_{1 f}$

Stability analysis

We begin with deriving the analytic expressions of the closed-loop system in the new coordinates: the surface errors $z_{i}$ as defined in (13) and the following boundary layer errors $y_{i}$ : $y_{i} = α_{i f} - β_{i}^{- 1} α_{i - 1}, i = 2, \dots, n .$ Here, we define that ${\bar{y}}_{i} = {[y_{2}^{T}, \dots, y_{i}^{T}]}^{T}, i = 2, \dots, n$ , with $y_{i}$ are the boundary layer errors as defined in (32). Considering (13), (32), we have $ζ_{i} = z_{i} + y_{i} + β_{i}^{- 1} α_{i - 1}, i = 2, \dots, n .$ Differentiating (13), with the help of the definitions of $ζ_{i}$ as defined in (33), one has ${\dot{z}}_{i} = μ_{i} (F_{i} + G_{i} β_{i + 1} (z_{i + 1} + y_{i + 1}) + G_{i} α_{i}) - {\dot{α}}_{i f},$ $\dot{z}$

Simulation verification

To demonstrate the effectiveness of the proposed control scheme, a numerical example has been carried out on a second-order MIMO system described by (1) and the detailed expressions for $F_{i}$ and $G_{i}, i = 1, 2$ , are given as: $F_{1} (x_{1}, p_{1}) = [\begin{matrix} p_{11} x_{11}^{2} \\ x_{11} x_{12} e^{- p_{12} t} \end{matrix}], F_{2} ({\bar{x}}_{2}) = [\begin{matrix} x_{11} x_{21} cos (p_{21} t) \\ p_{22} x_{11} x_{22}^{3} \end{matrix}],$ $G_{1} (x_{1}) = [\begin{matrix} 3 + sin (x_{11}) & tanh (x_{12}) \\ tanh (x_{12}) & 2 + cos (x_{12}) \end{matrix}],$ $G_{2} ({\bar{x}}_{2}) = [\begin{matrix} 2 + x_{21}^{2} & arctan (x_{11} x_{21}) \\ arctan (x_{11} x_{21}) & 3 + x_{22}^{2} \end{matrix}],$ where $p_{1} = {[p_{11}, p_{12}]}^{T} = {[1, 0.5]}^{T}$ and $p_{2} = {[p_{21}, p_{22}]}^{T} = {[0.1, 0.5]}^{T}$ . The objective is to ensure that all signals in the closed-loop

Conclusion

In this paper, a robust adaptive DSC approach is proposed for a class of uncertain nonlinear MIMO systems with asymmetric full-state constraints. The key features of the proposed algorithm are that only one parameter updating is needed and the prescribed limits on virtual control laws are completely removed such that the control scheme is less undemanding in design, less expensive in computation and more user-friendly in implementation. Extension of the method to MIMO nonlinear systems with

Kai Zhao received the M.S. degree from the School of Automation, Chongqing University, Chongqing, China, in 2015. He is currently pursuing the Ph.D. degree with the School of Automation, Chongqing University. His current research interests include intelligent control, robust adaptive control, prescribed performance control, and state-constrained control.

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Yongduan Song received the Ph.D. degree in Electrical and Computer Engineering from Tennessee Technological University, Cookeville, TN, USA, in 1992. He held a tenured Full Professor with North Carolina A&T State University, Greensboro, NC, USA, from 1993 to 2008 and a Langley Distinguished Professor with the National Institute of Aerospace, Hampton, VA, USA, from 2005 to 2008. He is currently the Dean of the School of Automation, Chongqing University, Chongqing, China. He was one of the six Langley Distinguished Professors with the National Institute of Aerospace (NIA), Hampton, VA, USA, and the Founding Director of Cooperative Systems with NIA. His current research interests include intelligent systems, guidance navigation and control, bio-inspired adaptive and cooperative systems, rail traffic control and safety, and smart grid.

Prof. Song was a recipient of several competitive research awards from the National Science Foundation, the National Aeronautics and Space Administration, the U.S. Air Force Office, the U.S. Army Research Office, and the U.S. Naval Research Office. He is an Associate Editor of a number of journals, including the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, the IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, and the IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS.

Zhirong Zhang received the B.S. degree from the Southwest University, Chongqing, China, in 2016. She is currently pursuing the Ph.D. degree with the School of Automation, Chongqing University, Chongqing, China. Her current research interests include adaptive control under constraints with application to multiple unmanned systems.

^☆: This work was supported in part by the National National Natural Science Foundation of China under Grant 61860206008, 61773081 and in supported by the Fundamental Research Funds for the Central Universities under Project No. 2018CDKYGL0011, 2018CDPTCG0001/43. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Shuzhi Sam Ge under the direction of Editor Miroslav Krstic.

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Brief paperTracking control of MIMO nonlinear systems under full state constraints: A Single-parameter adaptation approach free from feasibility conditions☆

Abstract

Introduction

Section snippets

Dynamics model

One parameter estimation based DSC design

Stability analysis

Simulation verification

Conclusion

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Automatica

Systems & Control Letters

Automatica

Automatica

Automatica

Automatica

Automatica

IEEE Transactions on Automatic Control

Systems & Control Letters

Robust adaptive control of feedback linearizable MIMO nonlinear systems with prescribed performance

IEEE Transactions on Automatic Control

Incorporating a class of constraints into the dynamics of optimal control problems

Optimal Control Applications & Methods

Brief paper
Tracking control of MIMO nonlinear systems under full state constraints: A Single-parameter adaptation approach free from feasibility conditions☆