Partial-state feedback multivariable MRAC and reduced-order designs

Abstract

This paper develops a new model reference adaptive control (MRAC) framework using partial-state feedback for solving a multivariable adaptive output tracking problem. The developed MRAC scheme has full capability to deal with plant uncertainties for output tracking and has desired flexibility to combine the advantages of full-state feedback MRAC and output feedback MRAC. With such a new control scheme, the plant-model matching condition is achievable as with an output or state feedback MRAC design. A stable adaptive control scheme is developed based on LDS decomposition of the plant high-frequency gain matrix, which guarantees closed-loop stability and asymptotic output tracking. The proposed partial-state feedback MRAC scheme not only expands the existing family of MRAC, but also provides new features to the adaptive control system, including additional design flexibility and feedback capacity. Based on its additional design flexibility, a minimal-order MRAC scheme is also presented, which reduces the control adaptation complexity and relaxes the feedback information requirement, compared to the existing MRAC schemes. New results are presented for plant-model matching, error model, adaptive law and stability analysis. A simulation study of a linearized aircraft model is conducted to demonstrate the effectiveness and new features of the proposed MRAC control scheme.

Introduction

Adaptive control is an effective control methodology which can deal with system uncertainties. In the past decades, it attracted tremendous attentions (Astolfi et al., 2008, Ding, 2013, Ioannou and Sun, 1996, Krstic et al., 1995, Landau et al., 2011, Lavretsky and Wise, 2013, Morse, 1980, Narendra and Annaswamy, 1989, Ortega et al., 2019). Recently, more results have been developed such as adaptive backstepping control (Chen et al., 2011, Wang and Lin, 2012b, Zhou and Wen, 2008, Zhou et al., 2009), adaptive posicast control (Dydek, Annaswamy, Slotine, & Lavretsky, 2013), adaptive sliding mode control (Chen, 2006), robust adaptive control (Hovakimyan and Cao, 2010, Oliveira et al., 2016, Ortega and Tang, 1989) and other adaptive control designs (Bian et al., 2014, Ortega et al., 2003, Wang and Lin, 2012a, Yucelen and Haddad, 2013).

Model reference adaptive control (MRAC) is a main and mature branch among various adaptive control techniques. One of its essential features is the capability of ensuring asymptotic output or state tracking of a given reference model system and closed-loop signal boundedness in the presence of system uncertainties (Costa, Hsu, Imai, & Kokotovic, 2003), Hoagg and Bernstein (2012). During the recent decades, multivariable MRAC theory for multi-input and multi-output systems, such as aircraft systems, has evolved into a mature branch. Considerable effort has been devoted to the development of multivariable MRAC theory. The existing results include (i) state feedback MRAC for state tracking (Ioannou and Sun, 1996, Narendra and Annaswamy, 1989, Tao et al., 2001, Yuan et al., 2016); (ii) state feedback MRAC for output tracking (Bodson and Groszkiewicz, 1997, Guo et al., 2011); and (iii) output feedback MRAC for output tracking (Goodwin & Sin, 1984 Ch. 6), Weller and Goodwin, 1994, Wen et al., 2016. The controller structure of state feedback MRAC for state tracking is simple, but the plant-model state matching condition is restrictive, which can only be satisfied for system matrices in certain canonical forms. Compared to state feedback MRAC for state tracking, state feedback MRAC for output tracking is suitable for more applications because of its unrestrictive matching condition and simple controller structure (Guo et al., 2011), although the full-state vector requirement may still confine its applications. When the full state vector is hard to be obtained, output feedback MRAC for output tracking attracts more attention, although its controller structure is more complex that may limit its applications.

Open problems. Although multivariable MRAC has been extensively studied, it is still desirable to develop a new control scheme that enjoys a simpler controller structure than the output feedback controller and requires less feedback signals than the state feedback controller as well. In this paper, we will solve the following two problems to achieve this goal.

Problem 1. Partial-state feedback multivariable MRAC. In this paper, we will develop and investigate a new multivariable MRAC scheme by using partial-state signal for output tracking, which can guarantee asymptotic output tracking and closed-loop system stability as well. The solution to this partial-state feedback multivariable MRAC will answer the following three technical questions:

• How to use partial-state signal, instead of the state vector $x\left(t\right)$ or the output signal $y\left(t\right)$, to build up a stable MRAC scheme for achieving multivariable output tracking?

• How to build a unified multivariable MRAC scheme to build up a bridge between the two existing MRAC schemes? and

• What are the control design flexibility and performance improvement by using partial-state feedback multivariable MRAC?

Problem 2. Minimal-order multivariable MRAC. In this paper, based on the developed partial-state feedback multivariable MRAC scheme, we will present an observer-based minimal-order multivariable MRAC scheme. The solution to this new scheme will answer the following two technical questions:

• What is the least number of feedback signal for $M$-output tracking? and

• How the system adaptation complexity can be reduced by the minimal-order multivariable MRAC scheme?

The study in this paper gives complete answers to the above five questions for the multivariable MRAC framework and improves the understanding of multivariable MRAC.

Research background. Besides state feedback control and output feedback control, as we mentioned above, research focusing on partial-state feedback control has been reported in the literature. In Krstic et al. (1995), a partial-state feedback design is developed for nonlinear systems in a canonical form to achieve asymptotic output tracking by using a vector with a subset of state variables. In Lavretsky and Wise (2013), by using a full-order Luenberger-based state observer, an adaptive model reference controller using system measurements of dimension greater than the number of inputs is developed for bounded output tracking of multi-input-multi-output systems with $(A,B,C,C_z)$ known whose dynamics may have high relative degree and are not necessarily minimum-phase. In Ling and Tao (1997), an adaptive backstepping output feedback control scheme was designed and analyzed for linear multivariable plants to ensure desired asymptotic output reference tracking plus virtual signal tracking properties. In Freeman and Kokotovic (2008, Ch.7), the backstepping technique is utilized to construct a controller to achieve global convergence, whose design procedure may become complex when the plant order is high. In Dixon, Zergeroglu, Dawson, and Hannan (2000), link position tracking is guaranteed by a partial-state feedback controller since the requirement of the full state signal is removed by a set of filters, which is only achievable for the robotic systems under some conditions. In addition, some partial-state feedback control designs for special applications (plants) without adaptation have been developed. In Uchiyama (2009), partial-state feedback control is studied for a rotary crane system. In Do, Jiang, and Pan (2005), partial-state feedback control is studied for an underactuated ship.

As we have seen, a rigorous and systematic partial-state feedback multivariable MRAC for general linear multi-input multi-output time-invariant system has not been developed yet. The new partial-state feedback multivariable MRAC scheme to be developed in this paper has less restrictive matching conditions, less state information requirement, more design flexibilities and more feedback capacities. It provides an additional and complete theoretical framework to guarantee asymptotic output of a given reference model system and closed-loop signal boundedness, in the presence of constant parameter uncertainties. It not only makes an addition to the existing family of multivariable MRAC designs and bridges the existing multivariable MRAC schemes, but also reveals some new feedback capacity of multivariable adaptive control systems. Based on the new capacity, a minimal-order MRAC system is presented. Such a new multivariable MRAC scheme requires the least number of feedback signal for multivariable output feedback control, which reduces the computation burden for control scheme implementation.

The new technical contributions of this work include:

• developing an adaptive multivariable MRAC scheme by using partial-state feedback signal which can guarantee asymptotic output tracking and closed-loop signal boundedness in the presence of plant parameter uncertainties;

• conducting a complete analysis of plant-model output matching for the nominal control design, and a complete analysis of stability and tracking performance for the adaptive control design;

• presenting a complete system computation complexity analysis of the partial-state feedback reduced-order multivariable MRAC scheme; and

• providing a minimal-order MRAC scheme which enjoys minimum feedback signal requirement and reduces the system adaptation complexity, compared to the other observer-based MRAC designs.

The rest of the paper is organized as follows. The partial-state feedback multivariable MRAC problem and the minimal-order multivariable MRAC problem are formulated in Section 2. In Section 3, the LDS decomposition-based adaptive partial-state feedback multivariable MRAC design is developed, together with the system stability and tracking performance analysis. In Section 4, some unique features of partial-state feedback multivariable MRAC are discussed, including the inherent unification it brings to the MRAC schemes and the system complexity reduction it brings to MRAC implementation. In Section 5, the observer-based minimal-order MRAC scheme is provided. In Section 6, simulation results on an aircraft system model are presented to confirm the desired control system performance with the partial-state feedback reduced-order multivariable MRAC scheme.