Elsevier

Journal of Process Control

Volume 88, April 2020, Pages 78-85
Journal of Process Control

Performance improvement by optimal reset dynamic output feedback control based on model predictive strategy

https://doi.org/10.1016/j.jprocont.2020.02.008Get rights and content

Highlights

  • A systematic approach for designing a reset dynamic output feedback controller is presented.

  • The D-stability approach is used to design a DOFC for the base system.

  • The after reset value at each reset time is designed by solving an optimization problem based on model prediction strategy.

  • The best times for resetting action are derived by genetic algorithm based on minimizing a multivariable cost function which contains certain transient performances.

  • The merits of the proposed controller are demonstrated by applying it on a continuous stirred tank reactor and a distillation column.

Abstract

This paper proposes a systematic approach for designing a reset dynamic output feedback controller (DOFC). The reset controller is designed in three steps. First, the D-stability approach is used to design the gain matrices of the DOFC for the base system such that the poles of the closed-loop system are placed in a predefined region. This region is chosen based on the control objectives. Moreover, the sufficient conditions for existence of base controller are formulated in terms of linear matrix inequalities. Second, the reset times are derived by applying genetic algorithm to minimize a novel cost function of certain transient performances. Third, an optimization problem based on model prediction strategy is solved for finding the after reset value at each reset time. The proposed method is applied to two different systems: continuous stirred tank reactor and distillation column. The effectiveness of the proposed reset controller is shown by comparing the results with other approaches.

Introduction

Reset control is a special kind of hybrid control which can be used to improve the transient performance. A reset controller is a dynamical system whose states are reset to zero or other specified values called after reset value, whenever some certain conditions hold [1]. Clegg in [2] initially introduced the concept of reset control which was named Clegg Integrator (CI). The state vector of this controller resets to zero when its input and output have opposite signs. This controller was generalized in [3], where the First Order Reset Elements (FORE) was developed. One important question in reset systems is: under what conditions does the reset system confer a well-defined solution. The well-posedness of reset systems is dependent on the selection of the reset condition and the after reset value. If these functions are chosen improperly, deadlock and beating phenomena may occur which destroys the existence of solutions [4]. These phenomena have been avoided by assuming that the after reset states are not elements of the reset condition. These two phenomena are automatically avoided in reset systems by predefining reset times. Besides the two mentioned phenomena, the solution of a reset control system may contain an infinite number of reset actions within a compact time interval which is called Zeno solution. A common solution in practice to avoid the Zeno problem is to use time regularization.

In recent years, reset control has been employed in the control system structures to achieve better transient response. For instance, the exponential stability was proved for a class of time-delay reset systems in [5]. In [6], a delay dependent stability analysis of reset control systems was proposed for certain and uncertain linear time invariant system. In [7] and [8], a Generalized First Order Reset Element (GFORE) reset controller was used to overcome the Overshoot Performance Limitation (OPL). In [9], a novel reset control synthesis method was proposed to overcome waterbed effect. In [10], a new type of Unknown Input Observer (UIO) called Reset-UIO was designed for state estimation of linear systems. In [11], the synchronization problem for multi-agent systems by using the reset control was addressed. The proposed reset controller guarantees the asymptotic convergence and improves transient response in multi-agent systems.

There are three basic problems in a reset control system design scheme: stability analysis, base system design and reset law design [12]. In other words, designing a reset mechanism is dependent on the suitable design of the base controller for the system. This means that an appropriate control law for the base system should exist prior to the design of reset law [13]. The stability of reset systems was proposed in the literature [14], [15], [16], however the problem of designing a strong base controller in reset control systems was not widely considered. There are a number of articles [17], [18], [19], [20], [21], [22], [23], [24], [25] that considered the problem of designing reset law containing after reset value and reset condition. The reset controllers could be designed in a full offline procedure. In this method, several state variables of the controller are reset to zero [17], [18] or reset to other values which have to be designed [12], [19], [20]. The after reset value can be found by minimizing a cost function. For instance, in [12] a quadratic performance index of states and predefined reset times is minimized to achieve after reset value.

Online design of the after reset value for a stable base system was proposed in [13], [21], [22], [23], [24], [25]. For the first time, a model predictive strategy was proposed in [13] that finds the after reset values. In this paper, the after reset values for a single–input single-output plant with a nonlinear term satisfying the Lipschitz constraint, was designed by minimizing a quadratic cost function. In [21], a model prediction based framework was proposed to determine a suitable after reset value for the uncertain linear control system. Discrete time after reset value for continuous time linear system based on MPC strategy was designed in [22]. In [23], a systematic approach for reset control of a time-delay system with a Lipchitz nonlinear term was presented. A systematic approach for designing a model predictive based reset dynamic state feedback controller for a class of nonlinear systems represented by LPV models was presented in [24]. In [25], the after reset value was designed for two typical types of head-positioning systems with nonlinearities in hard disk drives.

The main purpose of this paper is to propose a systematic method to design a reset dynamic output feedback controller (DOFC) for linear systems. To do this, the base DOFC without reset action is designed first, so that the poles of the closed-loop system are placed in a predefined region. This region is chosen such that the exponential stability and small reaching time of the closed-loop system is guaranteed. Then, the after reset value is designed in reset times by minimizing a suitable cost function for achieving better transient performance. In the mentioned papers [13], [21], [22], [23], [24], [25], the after reset value was designed with the assumption that all of the state variables of the system were available. In this paper, for the first time the after reset value is specified only by the information of the system output. This paper proposes a new pattern for reset times which has some advantages in comparison with alternative pattern [12]. While in the approach that was used in [23], it is required to find the time when transient response terminates and then the reset condition changes to no reset to avoid enormous number of reset actions, this step is not needed by using our proposed pattern and a few reset actions after termination of transient response might be done instead. Moreover, while the temporal regularization was used in [13], [21], [22], [23], [24], [25] to avoid Zeno solution, there is no need to apply temporal regularization in our approach and the Zeno solution is avoided by properly choosing two parameters of the proposed pattern. The issue of how to choose the reset time instants has not been considered in [12]. In this paper, the reset times are computed by minimizing a novel cost function of certain transient performances. A genetic algorithm is adopted to solve the optimization problem. To the best of the authors’ knowledge, this method has not been used to design the reset DOFC.

The rest of this paper is organized as follows. In Section 2, the system description and reset controller are given. In Section 3, dynamic output feedback controller without reset action is designed based on the D–stability concept. In Section 4, the reset condition is derived. Section 5 details finding the after reset value by minimizing a quadratic cost function based on the model predictive strategy. The simulation results are presented in Section 6. Finally, Section 7 draws the conclusion.

Section snippets

Problem formulation

Consider a multi–input multi–output linear plant described by the following state space representation:{x˙p=Apxp+Bpuy=Cpxpwhere xpRnp is the state vector, uRp is the input vector and yRq is the output vector of the plant (1). Ap,  Bp, and Cp are known matrices with appropriate dimensions. Without loss of generality, it is assumed that Cp is full row rank.

The following reset dynamic output feedback controller is used to control the given plant (1):{x˙r=Arxr+Bre,ttkxr+=ρr(xr,e),t=tku=Crxr+Dre

Dynamic output feedback controller design

In this section, the base controller without resetting action is designed based on D–stability concept. The LMI region is chosen such that the obtained controller guarantees the exponential stability and short reaching time of the closed-loop system. In the following, a theorem is proposed for D-stability of the closed-loop base system.

Theorem 1

The base system (7) without resetting action is D-stable with the region matrices defined in (9), if there exist matrices X=XT, Y=YT, A^r,B^r,C^r,D^r with

Reset condition design

As it was mentioned, in this paper the reset time instants tk are pre-specified. In this section, an algorithm is proposed to specify the reset times based on minimization of a cost function. The aim of adding reset mechanism on a stable base system is improving transient response. Therefore, a cost function which contains certain parameters of transient response can be chosen. These parameters can be settling time, maximum overshoot, peak. The proposed cost function is shown in (17).OF(tk)=αTs+

After reset value design

Consider the plant (1), and assume that a base controller (2) is designed based on Theorem 1 such that the closed-loop system (7) without resetting action is d-stable. Now, the after reset values of the controller states at times t=tk are designed such that the transient performance of the closed-loop system (7) with reset improves. For this reason, the cost function (19) is introduced and the after reset values are derived such that this cost function will be minimized:J(tk)=tk+ηT(tk++τ|tk)Qη

Simulation results

In this section, the proposed reset control is applied to two examples to show the effectiveness of our proposed method for designing a reset controller. First, in Example 1 the proposed controller with and without resetting actions is applied to a continuous stirred tank reactor. The results of applying these controllers are compared with the reset controllers in [12] and [13]. Then, in Example 2 the problem of tracking a constant reference is considered. The proposed method is applied to a

Conclusion

In this paper, a systematic approach is introduced to design a tracking Reset DOFC. First, a DOFC is designed based on d-stability concept such that exponential stability and small reaching time of the closed-loop system are obtained. The sufficient conditions are derived in terms of LMIs. Then the best times for resetting action are derived by genetic algorithm based on minimizing a multivariable cost function which contains certain transient performances. The after reset value at each reset

CRediT authorship contribution statement

Sepide Yazdi: Conceptualization, Methodology, Software, Validation, Writing - original draft. Alireza Khayatian: Conceptualization, Supervision, Writing - review & editing.

Declaration of Competing Interest

We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

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