Composite anti-disturbance model reference adaptive control for switched systems
Introduction
As an important subclass of hybrid systems, switched systems have received much attention, in part due to the challenges they present to the control theorists and in part due to the powerful ability to approximate many practical processes with complex nonlinearity. Recently, the study of switched systems has been one of the most active research fields [13], [19], [24], [32], [38], [41]. Meanwhile, some analysis and design methods have been exploited [2], [13], [40]. However, if switched systems suffer uncertainties, which is an ubiquitous problem for a practical system, the conventional stability analysis and design methods are not applied directly, mainly because the co-existence of the switching signals and the uncertainties makes the analysis design of the switched systems with uncertainties more complex. At present, a few of results have been made to study this topic [14], [16], [25].
To handle parametric uncertainties, the adaptive control plays a significant role owing to its superior adapting ability to achieve a desired level of performance by combining parameter estimation with control. Among the various types of adaptive control strategies [1], [18], model reference adaptive control (MRAC) is one of the most active subjects [7], [10]. The objective of the MRAC is to design a controller with an adaptive law forcing the state or the output of the uncertain system to track the state or the output of a reference model with desirable performance. In recent years, MRAC has been extended to switched systems [3], [21], [28], [30], [31]. For conventional state tracking MRAC problem, the controller is designed relying on the system state, and the adaptive law is constructed relying on the state tracking error. While if the system sate is unmeasurable, obviously, the controller and the adaptive law can not be designed by the conventional design method. To deal with the state unmeasurable problem, for a certain system, usually, a state observer is constructed. Then, based on the estimation state, the controller is designed. And yet, if the system with parametric uncertainties, the problem is difficult. First of all, in MRAC scheme, constructing an observer to estimate the state of the system is not easy due to the existence of the parametric uncertainties. Then, it is very hard to design a controller and an adaptive law dependent on the estimation state to achieve the control objective. Moreover, for the state tracking MRAC problem of switched systems, the problem becomes rather complicated and difficulty. Beside the above difficulties, another difficulty has to be faced— the design of the switching signals, which also can not rely on the system state or the state tracking error. The co-design of the controllers, adaptive laws and switching laws leads to the difficulty increasing greatly. Thus, if the system state is infeasible, we must explore the other design methods to achieve the state tracking for switched systems. In our previous paper [31], under unmeasurable system state, the H∞ state tracking MRAC problem was investigated, in which the output of the system and the state of the reference model were used in the design of the controllers and adaptive laws. Besides, a new state was constructed to design the switching law.
On the other hand, external disturbances extensively exist in most practical controlled processes, which may result in poor system performance, oscillation, or even instability. Thus, how to attenuate and reject the disturbances becomes a crucial problem. Several effective disturbance rejection methods have been proposed, such as output regulation theory [8], [15], H∞ control [22], [39], disturbance observer based control (DOBC) strategy [17] and so on. But, when the systems encounter multiple types of disturbances, the control effects of the methods mentioned above are all unsatisfactory. Motivated by this, a composite anti-disturbance control strategy consisting of DOBC and other control such as robust control [26], adaptive control [5], H∞ control [35] and so on, has been proposed to achieve the anti-disturbance performance. During the past two decades, the composite anti-disturbance control strategy has been successfully developed for linear and non-linear systems [27], [35]. Inevitably, multiple types of disturbances also exist in switched systems, and the composite anti-disturbance control strategy has further been extended to switched systems. By the average dwell time method, Sun and Hou [23] solved the composite anti-disturbance control problem for a discrete-time time-varying delay system with actuator failures. Yang and Zhao [36] investigated the disturbance attenuation properties of switched liner systems via mixed state-dependent and time-driven switching. It is worth mentioning that the results above all assume that the state of the systems is measurable. However, in practice, the state of the system is hard to be measured. For the unavailable state, only a few results have been reported. In Han et al. [9], the composite anti-disturbance control problem was solved by designing a state and disturbance observers-based polynomial fuzzy controller. It is note that, in Han et al. [9], the bounded disturbances and the non-switched systems were considered. Then, for switched systems, if the state is not measurable, for the composite anti-disturbance control strategy, the disturbance observer can not be constructed depending on the state of the systems. Meanwhile, the design of the controllers and the switching laws can not depend on the system state also. In this situation, how to design the disturbance observer, the controllers and the switching laws to achieve the control objective? Obviously, this increases the difficulty of the control design. Moreover, apart from multiple types of disturbances, if switched systems suffer from parametric uncertainties simultaneously, under the unmeasurable state, how to design the composite anti-disturbance control to attenuate and reject the disturbances? Apparently, this is a challenge in the switched systems field. So far, to our knowledge, no relevant results have been reported.
Motivated by the aforementioned discussion, in this paper, we study the problem of composite anti-disturbance MRAC (CADMRAC) for the switched systems via the multiple Lyapunov functions technique, where the switched systems are subjected to parametric uncertainties and multiple types of disturbances at the same time. The measurability of the system state and the disturbance generated by the exosystem and the solvability of the CADMRAC of each subsystem are not required. The contributions of this note are summarized as follows:
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Unavailable state. A switched adaptive state-disturbance observer is designed to estimate the system state and the disturbance simultaneously. In the most of existing literatures [6], [24], [35], [42], for the measurable state, the disturbance observer is designed depending on the state. Thus, from practical point of view, our result is more applicable for physical systems. Moreover, the systems we studied suffer uncertainties, which are not considered in the most of existing literatures.
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New design technique. In term of design technique, almost all existing design techniques depend on exact information state. As mentioned earlier, the state often unavailable in practice. Then, the existing design techniques cannot be applied directly to handle the CADMRAC problem for switched uncertain systems. What we do is to design a composite switched adaptive controller and a switching law depending on the state estimation and the output of the system.
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No solvability of the CADMRAC problem for any subsystem is required, which greatly relaxes the conditions in theory. To the best of our knowledge, for switched uncertain systems with unstable subsystems, the CADMRAC problem has not been explored yet in the existing literatures.
The remainder of this paper is organized as follows. A motivating example is described in Section 2. Section 3 gives the problem formulation and preliminaries. The main results are shown in Section 4. An example is presented to illustrate the effectiveness of the proposed strategy in Section 5.
Section snippets
Motivating example
As described in Fig. 1. the electrohydraulic system is made up of a hydraulic pump, a hydraulic arm, two accumulators, a MOOG760 torque motor/flapper-operated four-way double-acting servovalve (East Aurora, NY, USA) and a single rod actuator. In this paper, we select two operating conditions with respect to two different supply pressures, 11.0 MPa and 1.4 MPa. Different supply pressures will make plant parameters variation [20]. According to Yuan et al. [37] and Wu et al. [28], the
Problem formulation and preliminaries
Consider the following switched system with parametric uncertaintieswhere x(t) is the system state, d1(t) is the unknown external disturbance which can represent the constant and harmonic noises and d1(t) is generated from an exosystemwhere w(t) is the state of the exosystem, is the additional disturbance from the perturbations and uncertainties in the exosystem. d2(t) is
Main results
In this section, a composite switched adaptive controller and a state-dependent switching law are designed to solve the CADMRAC problem of the system (4). Firstly, a switched adaptive state-disturbance observer is constructed. Then, based on the switched adaptive state-disturbance observer, a composite switched adaptive controller and a state-dependent switching law are designed. Finally, we give a sufficient condition ensuring that the problem of CADMRAC of the system (4) is solvable.
Example
In this section, we apply the CADMRAC scheme to an electrohydraulic system described in Motivating example.
Conclusions
In this paper, we have studied the CADMRAC problem for the switched systems with parametric uncertainties and multiple types of disturbances. Multiple types of disturbances include the norm-bounded disturbance and the disturbance generated by the exosystem. Under the unmeasurable state and the unmeasurable disturbance generated by the exosystem, by design of the controllers and the switching law, achieve the state tracking and the anti-disturbance performance. Firstly, a switched adaptive
Acknowledgments
This work was supported by the National Natural Science Foundation of China under grants 61803225, 61773098, the 111 Project(B16009), and the General Project of Scientific Research of the Education Department of Liaoning Province under grant L2016020.
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