Multirate sampling and input-to-state stable receding horizon control for nonlinear systems

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Abstract

In this paper, a robust receding horizon control for multirate sampled-data nonlinear systems with bounded disturbances is presented. The proposed receding horizon control is based on the solution of Bolza-type optimal control problems for the approximate discrete-time model of the nominal system. “Low measurement rate” is assumed. It is shown that the multistep receding horizon controller that stabilizes the nominal approximate discrete-time model also practically input-to-state stabilizes the exact discrete-time system with disturbances.

Introduction

Stabilization of controlled systems is one of the central topics in control theory that has led to a wealth of different stabilization methods. Among these methods, receding horizon control (RHC) strategies, also known as model predictive control (MPC), have become quite popular (see for example [5], [16] for an overview). The receding horizon method obtains the feedback control by solving a finite horizon optimal control problem at each time instant using the current state of the system as the initial state for the optimization and applying “the first part” of the optimal control. The use of digital computers in the implementation of the controllers necessitated the investigation of sampled-data systems. A sampled-data control system consists of a continuous-time plant controlled by a discrete-time (digital) controller. An overview and analysis of existing approaches for the stabilization of sampled-data systems can be found in [19], [21], [22] (see also the references of these papers).

Two main approaches of sampled-data control design can be distinguished: the first one consists in the implementation of a continuous-time stabilizing control law at a sufficiently high sampling rate, while the second way is to discretize the continuous-time model and design a stabilizing controller on the basis of the approximate discrete-time model.

Stabilization of disturbance-free sampled-data nonlinear systems via their approximate discrete-time models was presented in the recent papers [21], [22]. In these papers, sufficient conditions are presented which guarantee that the same family of controllers as stabilizes the approximate discrete-time model also practically stabilizes the exact discrete-time model of the plant. These results yield a general framework for the control design. A significant amount of work is also devoted to the study of control design methods which results in controllers satisfying the above mentioned sufficient conditions (see e.g. [6], [10] and [11]). Input-to-state stabilization and integral versions of input-to-state stabilization of sampled-data nonlinear systems with disturbances are studied by [20], [18], respectively. All of these investigations deal with the case when the sampling rates of the control function and the state measurement coincide, i.e. a single-rate approach is presented. In some applications, the sampling rates for the control function and the state measurement are different (see e.g. [1], [2], [3] and [23]). In [3], we presented a multirate version of the receding horizon algorithm for the stabilization of sampled-data nonlinear systems under a “low measurement rate” in the presence of measurement and computational delays. In this paper, we do not consider the delays; instead we consider the presence of disturbances.

When the disturbances are present, several receding horizon control approaches can be proposed. Some of them are based on the nominal prediction [17], [24], [13], [4]; others are based on the H technique [9] or on min–max MPC [14], [15]. The drawback of H or the min–max technique is that it becomes much more difficult to solve the optimization problem needed to generate the feedback law. In the first approach, the Lipschitz (or at least) continuity of the value function is needed. An important fact is that, in the presence of an explicit final state constraint, the continuity of the value function is not generic. One way to overcome this problem is to make an additional assumption of Lipschitz (continuity) of the value function but it is difficult to verify this condition. Therefore, in the present paper, the proposed receding horizon control method is based on the nominal prediction with an implicit final state constraint. A great majority of works deal either with continuous-time systems with sampling [4] or without taking into account any sampling [7], [8] or with discrete-time systems considering the model given directly in discrete time [9]. In [4], a sampled-data control is applied to the continuous-time system without taking into account any approximation in the plant model.

The aim of the present paper is to derive a multirate version of the receding horizon algorithm based on the nominal approximate discrete-time model, and establish that the receding horizon controller which stabilizes the nominal approximate discrete-time model also practically input-to-state stabilizes the exact discrete-time system with disturbances. The basic idea of handling the disturbances is very similar to that of [4], but, in contrast to this work, the design of the controller is based on the approximate model of the nominal system in the present paper. The importance of taking this fact into account is supported by a series of counter-examples (see e.g. [22], [21] and [10]), which show that even for disturbance-free systems one can design a controller to stabilize the approximate model, but the exact discrete-time model is destabilized by the same controller. In what follows, the notation BΔ={zRl:zΔ} will be used both in Rn and Rm; and K, K and KL denote the usual class-K, class-K and class-KL functions (see e.g. [22]).

Section snippets

Preliminaries and problem statement

Consider a continuous-time nonlinear control system with disturbances given by ż(t)=f(z(t),u(t),w(t)),z(0)=z0 where z(t)Rn, u(t)URm, w(t)WRp are the state, control input and exogenous disturbance, respectively, f:Rn×U×WRn, with f(0,0,0)=0, U is closed and 0U, W is compact and 0W. Let μ>0 be such that WBμ. Let ΓRn be a given compact set containing the origin and consisting of all initial states to be taken into account. The system is to be controlled digitally using piecewise constant

Nominal stability of -step RHC

Consider the nominal system ẋ(t)=f(x(t),u(t),0),x(0)=z(0). Let ϕE(.,x0,u¯) be the solution of (4) with given u(t)u¯, and x(0)=x0.

The nominal exact discrete-time system is given by xi+1=FTE(xi,ui), where FTE(x,u)ϕE(T,x,u). We note that ϕE(.,x,u)ΦE(.,x,u,0).

Remark 2

If Assumption A1 is valid, then FTE is continuous in x and u and it is locally Lipschitz.

We note that, since f is typically nonlinear, FTE in (5) is not known in most cases, and therefore the controller design can be carried out by means

Input-to-state stability

The concept of input-to-state stability has been widely used in stability analysis and control synthesis of nonlinear systems. Recently, considerable attention has been paid to the practical input-to-state stabilization of the general parametrized family of discrete-time nonlinear systems, which commonly arises in sampled-data control design (see [12], [20]).

Definition 2

System (2) is practically input-to-state stabilizable (P-ISS) in Γ about the origin with the parametrized family Uh if there exist a β(.,.

Acknowledgment

The author is grateful to Prof. E. Gyurkovics (School of Mathematics, Budapest University of Technology and Economics) for her valuable ideas and comments, and to the anonymous reviewer for constructive suggestions, which helped to improve the presentation of the paper.

References (24)

  • A.M. Elaiw, É. Gyurkovics, Stabilization of HIV/AIDS model using an ℓ-step receding horizon control based treatment...
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