Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2012 (2012), Article ID 808327, 14 pages
http://dx.doi.org/10.1155/2012/808327
Research Article

Offset-Free Strategy by Double-Layered Linear Model Predictive Control

Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110016, China

Received 29 March 2012; Revised 27 May 2012; Accepted 28 May 2012

Academic Editor: Xianxia Zhang

Copyright © 2012 Tao Zou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In the real applications, the model predictive control (MPC) technology is separated into two layers, that is, a layer of conventional dynamic controller, based on which is an added layer of steady-state target calculation. In the literature, conditions for offset-free linear model predictive control are given for combined estimator (for both the artificial disturbance and system state), steady-state target calculation, and dynamic controller. Usually, the offset-free property of the double-layered MPC is obtained under the assumption that the system is asymptotically stable. This paper considers the dynamic stability property of the double-layered MPC.

1. Introduction

The technique model predictive control (MPC) differs from other control methods mainly in its implementation of the control actions. Usually, MPC solves a finite-horizon optimal control problem at each control interval, so that the control moves for the current time and a period of future time (say, totally 𝑁 control intervals) are obtained. However, only the current control move is applied to the plant. At the next control interval, the same kind of optimization is repeated with the new measurements [1]. The MPC procedures applied in the industrial processes lack theoretical guarantee of stability. Usually, industrial MPC adopts a finite-horizon optimization, without a special weighting on the output prediction at the end of the prediction horizon.

Theoretically, the regulation problem for the nominal MPC can have guarantee of stability by imposing special weight and constraint on the terminal state prediction [2]. The authors in [2] give a comprehensive framework. However, [2] does not solve everything for the stability of MPC. In the past 10 years, the studies on the robust MPC for regulation problem go far beyond [2]. We could say that, for the case of regulation problem when the system state is measurable, the research on MPC is becoming mature (see e.g., [38]). For the case of regulation problem when the system state is unmeasurable, and there is no model parametric uncertainty, the research on MPC is becoming mature (see e.g. [911]). For other cases (output feedback MPC for the systems with parametric uncertainties, tracking MPC, etc.), there are many undergoing researches (see e.g., [1216]).

A synthesis approach of MPC is that with guaranteed stability. However, the industrial MPC adopts a more complex framework than the existing synthesis approaches of MPC. Its hierarchy is shown in, for example [17]. In other words, the synthesis approaches of MPC have not been sufficiently developed to include the industrial MPC. Today, the separation of the MPC algorithm into steady-state target and dynamic control move calculations is a common part of industrial MPC technology [17]. The use of steady-state target calculation is necessary, since the disturbances entering the systems or new input information from the operator may change the location of the optimal steady-state at any control interval (see e.g., [18]). The goal of the steady-state target calculation is to recalculate the targets from the local optimizer every time the MPC controller executes.

In the linear MPC framework, offset-free control is usually achieved by adding step disturbance to the process model. The most widely used industrial MPC implementations assume a constant output disturbance that can lead to sluggish rejections of disturbances that enter the process elsewhere. In [19, 20], some general disturbance models that accommodate unmeasured disturbances entering through the process input, state, or output, have been proposed. In a more general sense, the disturbance model can incorporate any nonlinearity, uncertainty, and physical disturbance (measured or unmeasured). The disturbance can be estimated by the Kalman filter (or the usual observer). The estimated disturbance is assumed to be step-like, that is unchanging in the future, at each control interval (MPC refreshes its solution at each control interval). The estimated disturbance drives the steady-state target calculation, in order to refresh the new target value for the control move optimization.

This paper visits some preliminary results for the stability of double-layered MPC or output tracking MPC. These results could be useful for incorporating the industrial MPC into the synthesis approaches of MPC. The preliminary results for this paper can be found in [21, 22].

Notations 1. For any vector 𝑥 and positive-definite matrix 𝑀, 𝑥2𝑀=𝑥𝑇𝑀𝑥. 𝑥(𝑘+𝑖𝑘) is the value of vector 𝑥 at time 𝑘+𝑖, predicted at time 𝑘. 𝐼 is the identity matrix with appropriate dimension. All vector inequalities are interpreted in an element-wise sense. The symbol induces a symmetric structure in the matrix inequalities. An optimal solution to the MPC optimization problem is marked with superscript . The time-dependence of the MPC decision variables is often omitted for brevity.

2. System Description and Observer Design

Consider the following discrete-time model:𝑑𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐵𝑢(𝑘)+𝐸𝑑(𝑘),(𝑘+1)=𝑑(𝑘)+Δ𝑑(𝑘),𝑦(𝑘)=𝐶𝑥(𝑘)+𝐷𝑑(𝑘),(2.1) where 𝑢𝑚 denotes the control input variables, 𝑥𝑛 the state variables, 𝑦𝑝 the output variables, and 𝑑𝑞 the unmeasured signals including all disturbances and plant-model mismatches.

Assumption 2.1. The augmented pair ,𝐶𝐷𝐴𝐸0𝐼(2.2) is detectable, and the following condition holds: rank𝐼𝐴𝐸𝐶𝐷=𝑛+𝑞.(2.3)

The augmented observer is =+𝐵0𝐹̂𝑥(𝑘+1)𝑑(𝑘+1)𝐴𝐸0𝐼̂𝑥(𝑘)𝑑(𝑘)𝑢(𝑘)+1𝑠𝐹2𝑠𝐶̂𝑥(𝑘)+𝐷𝑑(𝑘)𝑦(𝑘),(2.4) where 𝐹𝑠=[(𝐹1𝑠)𝑇,(𝐹2𝑠)𝑇]𝑇 is the prespecified observer gain. Define the estimation error ̃𝑥(𝑘)=𝑥(𝑘)̂𝑥(𝑘) and 𝑑(𝑘)=𝑑(𝑘)𝑑(𝑘); then one has the following observer error dynamic equation: =+𝐹̃𝑥(𝑘+1)𝑑(𝑘+1)𝐴𝐸0𝐼1𝑠𝐹2𝑠+0𝐼𝐶𝐷̃𝑥(𝑘)𝑑(𝑘)Δ𝑑(𝑘).(2.5)

Assumption 2.2. Δ𝑑(𝑘) is an asymptotically vanishing item, and the observer error dynamics is asymptotically stable, that is, lim𝑘{Δ𝑑(𝑘),̃𝑥(𝑘),𝑑(𝑘)}={0,0,0}.

3. Double-Layered MPC with Off-Set Property

For the system (2.1), its steady-state state and input target vectors, 𝑥𝑡(𝑘) and 𝑢𝑡(𝑘), can be determined from the solution of the following quadratic programming (QP) problems (steady-state target calculation, steady-state controller): min𝑥𝑡,𝑢𝑡𝑢𝑡𝑢𝑟2𝑅𝑡,𝑥(3.1)s.t.𝐼𝐴𝐵𝐶0𝑡𝑢𝑡=𝐸𝑑(𝑘)𝑦𝑟𝑢𝐷𝑑(𝑘)min𝑢𝑡𝑢max(3.2)min𝑥𝑡,𝑢𝑡𝑦𝑡𝑦𝑟2𝑄𝑡,𝑥(3.3)s.t.𝐼𝐴𝐵𝐶0𝑡𝑢𝑡=𝐸𝑑𝑦(𝑘)𝑡𝑢𝐷𝑑(𝑘)min𝑢𝑡𝑢max,(3.4) where 𝑦𝑟 is the desired steady-state output (e.g., from the local optimizer), 𝑢𝑟 is the desired steady-state input, and (𝑢min,𝑢max) are the input bounds. Problems (3.1) and (3.2) is solved; when (3.1) and (3.2) is feasible, 𝑦𝑡=𝑦𝑟 and (3.3) and (3.4) is not solved; when (3.1) and (3.2) is infeasible, (3.3) and (3.4) is solved.

When this target generation problem is feasible, one has 𝑥𝑡(𝑘)=𝐴𝑥𝑡(𝑘)+𝐵𝑢𝑡𝑑𝑦(𝑘)+𝐸(𝑘),𝑡(𝑘)=𝐶𝑥𝑡(𝑘)+𝐷𝑑(𝑘).(3.5) Subtracting (3.5) from (2.1) and utilizing (2.5) yield 𝜒(𝑘+1,𝑘)=𝐴𝜒(𝑘,𝑘)+𝐵𝜔(𝑘)𝐹1𝑠𝐶̃𝑥(𝑘)+𝐷𝑑(𝑘),(3.6) where the shifted variables 𝜒(,𝑘)=̂𝑥()𝑥𝑡(𝑘), 𝜔=𝑢𝑢𝑡. The following nominal model of the transformed system (3.6) is used for prediction 𝜒(𝑘+𝑖+1𝑘)=𝐴𝜒(𝑘+𝑖𝑘)+𝐵𝜔(𝑘+𝑖𝑘).(3.7) Its infinite horizon predictive control performance cost is defined as 𝐽0(𝑘)=𝑖=0𝑊,𝜒(𝑘+𝑖𝑘),𝜔(𝑘+𝑖𝑘)(3.8) where 𝑊(𝜒(𝑘+𝑖𝑘),𝜔(𝑘+𝑖𝑘))=𝜒(𝑘+𝑖𝑘)2𝑄+𝜔(𝑘+𝑖𝑘)2𝑅.

Defining a quadratic function 𝑉(𝜒(𝑘+𝑖𝑘))=𝜒(𝑘+𝑖𝑘)2𝑃, if one can show that 𝑉𝜒(𝑘+𝑖+1𝑘)𝑉𝜒(𝑘+𝑖𝑘)𝑊𝜒(𝑘+𝑖𝑘),𝜔(𝑘+𝑖𝑘),(3.9) then it can be concluded that 𝑉(𝜒(𝑘+𝑖𝑘))0 as 𝑖. Furthermore, summing (3.9) from 𝑖=𝑁 to yields the upper bound of 𝐽𝑁 as 𝑖=𝑁𝑊.𝜒(𝑘+𝑖𝑘),𝜔(𝑘+𝑖𝑘)𝑉𝜒(𝑘+𝑁𝑘)(3.10) By substituting (3.10) into (3.8), one can get 𝐽0(𝑘)𝑁1𝑖=0𝑊𝜒(𝑘+𝑖𝑘),𝜔(𝑘+𝑖𝑘)+𝑉𝜒(𝑘+𝑁𝑘)=𝐽.𝜒(𝑘),𝜋(𝑘)(3.11) Here 𝐽(𝜒(𝑘),𝜋(𝑘)) gives an upper bound of 𝐽0(𝑘); so we can formulate the MPC as an equivalent minimization problem on 𝐽(𝜒,𝜋) with respect to the optimal control sequence 𝜋(𝑘)=[𝜔(𝑘𝑘)𝑇,𝜔(𝑘+1𝑘)𝑇,,𝜔(𝑘+𝑁1𝑘)𝑇]𝑇.(3.12) When ̂𝑥(𝑘+𝑁𝑘) lies in the terminal region, 𝜔(𝑘+𝑖𝑘)=𝐾𝜒(𝑘+𝑖𝑘), 𝑖𝑁. From the definition of 𝐽(𝜒(𝑘),𝜋(𝑘)), at time instant 𝑘+1, one has 𝐽=𝜒(𝑘+1),𝜋(𝑘+1)𝑁𝑖=1𝑊𝜒(𝑘+𝑖𝑘+1),𝜔(𝑘+𝑖𝑘+1)+𝑉𝜒(𝑘+𝑁+1𝑘+1)(3.13) with the shifted control sequence 𝜔𝜋(𝑘+1)=(𝑘+1𝑘)+𝑢𝑡(𝑘)𝑢𝑡(𝑘+1)𝑇𝜔,,(𝑘+𝑁1𝑘)+𝑢𝑡(𝑘)𝑢𝑡(𝑘+1)𝑇,𝐾𝜒(𝑘+𝑁𝑘)+𝑢𝑡(𝑘)𝑢𝑡(𝑘+1)𝑇𝑇.(3.14) We can explicitly derive the multi-step-ahead state and output prediction: 𝜒(𝑘+𝑁𝑘)=𝐴𝑁𝐴𝜒(𝑘)+𝐵𝑌𝜋(𝑘),(3.15)𝜒𝑇(𝑘)=𝐴𝑇𝜒(𝑘)+𝐵𝜋(𝑘),(3.16) where 𝐴𝐵=𝐴𝑁1,𝑌𝐵,,𝐴𝐵,𝐵𝜒𝑇(𝑘)=𝜒(𝑘𝑘)𝜒(𝑘+1𝑘)𝜒(𝑘+𝑁1𝑘),(3.17)𝐴=𝐼𝐴𝐴𝑁1,𝑇𝐵=𝐴000𝐵00𝑁2𝐵𝐵0.(3.18)

Lemma 3.1. For a quadratic function 𝑊(𝑥,𝑢)=𝑥𝑇𝑄𝑥+𝑢𝑇𝑅𝑢, 𝑄,𝑅>0, there exist finite Lipschitz constants 𝑥,𝑢>0 such that 𝑊𝑥1,𝑢1𝑥𝑊2,𝑢2𝑥𝑥1𝑥2+𝑢𝑢1𝑢2(3.19) for all 𝑥1,𝑥2𝒳, 𝑢1,𝑢2𝒰, where 𝒳, 𝒰 are bounded regions. Similarly, for a quadratic function 𝑉(𝑥)=𝑥T𝑃𝑥, 𝑃>0, there exists a finite Lipschitz constant 𝑉>0 such that 𝑉𝑥1𝑥𝑉2𝑉𝑥1𝑥2(3.20) for all 𝑥1,𝑥2𝒳.

Clearly, 𝑥, 𝑢, 𝑉 depend on 𝒳, 𝒰. However, it is unnecessary to specify 𝒳, 𝒰 in the following. Moreover, 𝑉 depends on 𝑃, which is time varying; this paper assumes taking 𝑉 for all possible 𝑃.

Lemma 3.2. Consider the prediction model (3.7). Then, with the shifted control sequence 𝜋(𝑘+1), 𝜒(𝑘+𝑖𝑘+1)𝜒(𝑘+𝑖𝑘)𝐴𝑖1𝐹1𝑠𝐶𝐹̃𝑥(𝑘)+1𝑠𝐷+𝑥𝑑(𝑘)𝑡(𝑘)𝑥𝑡+(𝑘+1)𝑖2𝑗=0𝐴𝑗𝑢𝐵𝑡(𝑘)𝑢𝑡.(𝑘+1)(3.21)

Proof . It is easy to show that 𝜒(𝑘+1,𝑘+1)=̂𝑥(𝑘+1)𝑥𝑡(𝑘+1)=𝜒(𝑘+1𝑘)𝐹1𝑠𝐶̃𝑥(𝑘)+𝐷𝑑(𝑘)+𝑥𝑡(𝑘)𝑥𝑡(𝑘+1).(3.22) Then, =𝐹𝜒(𝑘+1𝑘+1)𝜒(𝑘+1𝑘)𝜒(𝑘+1)𝜒(𝑘+1𝑘)1𝑠𝐶𝐹̃𝑥(𝑘)+1𝑠𝐷+𝑥𝑑(𝑘)𝑡(𝑘)𝑥𝑡,=𝐴𝑢(𝑘+1)𝜒(𝑘+2𝑘+1)𝜒(𝑘+2𝑘)𝜒(𝑘+1𝑘+1)𝜒(𝑘+1𝑘)+𝐵(𝜔(𝑘+1𝑘+1)𝜔(𝑘+1𝑘))𝐴𝜒(𝑘+1𝑘+1)𝜒(𝑘+1𝑘)+𝐵𝑡(𝑘)𝑢𝑡.(𝑘+1)(3.23) By induction, one can easily show the claimed result, and thus the proof is completed.

Theorem 3.3. For the system (2.1) subject to the input constraints 𝑢min𝑢𝑢max,(3.24) under Assumptions 2.1-2.2, the closed-loop output feedback model predictive control system, with objective function 𝐽(𝜒(𝑘),𝜋(𝑘)), augmented observer (2.4), and target generation procedure (3.1)–(3.4), achieves the offset-free reference tracking performance if the following three conditions are satisfied. (a) There exist feasible solutions (𝑥𝑡(𝑘),𝑢𝑡(𝑘)) to the target generation problem (3.1)–(3.4), at each time 𝑘.(b) There exist feasible solutions, including a control sequence 𝜋(𝑘), a positive-definite matrix 𝑋, and any matrix 𝑌, at each time 𝑘, to the dynamic optimization problem (dynamic control move calculation problem)min𝛾1,𝛾2,𝜋,𝑋,𝑌𝛾1+𝛾2,(3.25)subject to the linear matrix inequalities 𝛾1𝑇𝐴𝑇𝜒(𝑘)+𝐵𝜋𝑄1𝑅𝜋01𝐴0,(3.26)1𝑁𝐴𝜒(𝑘)+𝐵𝜋𝑋𝐴𝑌0,(3.27)𝑋𝑋+𝐵𝑋𝑋0𝛾2𝑄1𝑌00𝛾2𝑅10,(3.28)𝑢2𝑗𝑌T𝑈T𝑗𝑋𝐼0,𝑗=1,,𝑚,(3.29)𝑚×𝑁𝐼𝑚×𝑁Π𝜋𝑚𝑢max𝑢𝑡(𝑘)Π𝑚𝑢min𝑢𝑡(𝑘),(3.30)where 𝑈𝑗 is the 𝑗th row of the 𝑚-ordered identity matrix, 𝑄=𝐼𝑁𝑄, 𝑅=𝐼𝑁𝑅, Π𝑚=[𝐼𝑚,,𝐼𝑚]𝑇, and 𝑢𝑗=min{(𝑢max𝑢𝑡(𝑘))𝑗,(𝑢𝑡(𝑘)𝑢min)𝑗}. (c) By applying 𝑢(𝑘)=𝑢𝑡(𝑘)+𝜔(𝑘𝑘), where 𝜔(𝑘𝑘) is obtained by solving (3.25)—(3.30), the closed-loop system is asymptotically stable.

Proof. The matrix inequality (3.28) implies that (𝐴+𝐵𝐾)𝑇𝑃(𝐴+𝐵𝐾)𝑃+𝑄+𝐾𝑇𝑅𝐾0.(3.31) By referring to [23], it is easy to prove that (3.9) holds for all 𝑖𝑁. Then, 𝑉(𝜒(𝑘+𝑁𝑘))𝜒(𝑘+𝑁𝑘)2𝑃. Let 𝜒(𝑘+𝑁𝑘)2𝑃𝛾2(𝑘), which is guaranteed by (3.27), where 𝑃=𝛾2𝑋1. Meanwhile, it is easy to show that, by applying (3.26), the optimal 𝛾1(𝑘) is exactly the optimal value of 𝐽0𝑁1(𝑘)=𝑁1𝑖=0𝑊𝜒(𝑘+𝑖𝑘),𝜔.(𝑘+𝑖𝑘)(3.32)
Now we check if each element of the predictive control inputs satisfies the constraints 𝑢𝑗,min𝑢𝑗(𝑘+𝑖𝑘)𝑢𝑗,max, 𝑖0, 𝑗=1,,𝑚. For any 𝑖 within the finite horizon 𝑁, the input constraints are satisfied since Π𝑚(𝑢min𝑢𝑡(𝑘))𝜋Π𝑚(𝑢max𝑢𝑡(𝑘)), as shown in (3.30). Otherwise, beyond the finite horizon 𝑖𝑁, 𝜒(𝑘+𝑖𝑘) belongs to the constraint set ={𝑧𝑛𝑧𝑇𝑋1𝑧1}, which is guaranteed by (3.27). In this case, by referring to [23], it is easy to show that, (3.27)–(3.29) guarantee that the feedback control law 𝜔(𝑘+𝑖𝑘)=𝐾𝜒(𝑘+𝑖𝑘), 𝑖𝑁, 𝑌𝑋𝐾=1 satisfies the input constraints.
Since point (c) is assumed, the offset-free property can be referred to as in [19, 20, 22].

4. Improved Procedure for Double-Layered MPC

At each time 𝑘+10, we consider the following constraints: 𝑥𝐼𝐴𝐵𝐶0𝑡𝑢𝑡=𝐸𝑑(𝑘+1)𝑦𝑟,𝑢𝐷𝑑(𝑘+1)min𝑢𝑡𝑢max,𝑥(4.1)𝐼𝐴𝐵𝐶0𝑡𝑢𝑡=𝐸𝑦𝑑(𝑘+1)𝑡,𝑢𝐷𝑑(𝑘+1)min𝑢𝑡𝑢max,1(4.2)𝜐(𝐴𝑘)𝑁̂𝑥(𝑘+1)𝑥𝑡+𝐴𝐵𝜋(𝑘+1)𝑋(𝑘)0,(4.3)𝜐(𝑘)𝑈𝑗𝑌(𝑘)𝑋(𝑘)1𝑌(𝑘)𝑇𝑈𝑇𝑗1/2(𝑢max𝑢𝑡)𝑗,𝜐(𝑘)𝑈𝑗𝑌(𝑘)𝑋(𝑘)1𝑌(𝑘)𝑇𝑈𝑇𝑗1/2(𝑢𝑡𝑢min)𝑗,𝑗=1,,𝑚,(4.4) where 𝜐(𝑘)=𝛾2(𝑘)/(𝛾2(𝑘)𝑊(𝜒(𝑘+𝑁𝑘),𝜔(𝑘+𝑁𝑘))+(1𝜚)𝑊(𝜒(𝑘𝑘),𝜔(𝑘𝑘))), with 𝜚(0,1] being a given design parameter. Equation (4.1) is utilized for (3.1); (4.2) is utilized for (3.3).

Theorem 4.1. For the system (2.1) subject to the input constraints under Assumptions 2.1-2.2, the closed-loop output feedback model predictive control system, with objective function 𝐽(𝜒(𝑘),𝜋(𝑘)), augmented observer (2.4), target generation procedure (at 𝑘=0, (3.1)–(3.4); at any 𝑘+1, (3.1), (3.3), (4.1)–(4.4)), and dynamic optimization problem (3.25)–(3.30), is input-to-state (ISS) stable if the following two conditions are satisfied. (a)There exist feasible solutions (𝑥𝑡(𝑘),𝑢𝑡(𝑘)) to the target generation problem, at each control interval. (b)There exist feasible solutions, including a control sequence 𝜋(𝑘), a positive-definite matrix 𝑋, and any matrix 𝑌, at time 𝑘=0, to the dynamic optimization problem (3.25)–(3.30).

Proof. By applying the shifted control sequence 𝜋(𝑘+1), at time 𝑘+1, one has 𝛾1(𝑘+1)𝛾1(𝑘)=𝐽0𝑁1(𝑘+1)𝐽0𝑁1+(𝑘)=𝑊𝜒(𝑘+𝑁𝑘+1),𝜔(𝑘+𝑁𝑘+1)𝑁1𝑖=1𝑊𝜒(𝑘+𝑖𝑘+1),𝜔(𝑘+𝑖𝑘+1)𝑊𝜒(𝑘+𝑖𝑘),𝜔(𝑘+𝑖𝑘)𝑊𝜒(𝑘𝑘),𝜔.(𝑘𝑘)(4.5) By applying Lemmas 3.1-3.2, it is shown that 𝛾1(𝑘+1)𝛾1(𝑘)𝑊𝜒(𝑘+𝑁𝑘+1),𝜔(𝑘+𝑁𝑘+1)+𝑥𝑁1𝑖=1𝐴𝑖1𝐹1𝑠𝐶𝐹̃𝑥(𝑘)+1𝑠𝐷+𝑥𝑑(𝑘)𝑡(𝑘)𝑥𝑡+(𝑘+1)𝑖2𝑗=0𝐴𝑗𝑢𝐵𝑡(𝑘)𝑢𝑡(𝑘+1)+(𝑁1)𝑢𝑢𝑡(𝑘)𝑢𝑡(𝑘+1)𝑊𝜒(𝑘𝑘),𝜔.(𝑘𝑘)(4.6) By further applying 𝜒(𝑘+𝑁𝑘+1)=𝜒(𝑘+𝑁𝑘)+𝐴𝑁1𝐹1𝑠𝐶̃𝑥(𝑘)+𝐷𝑑(𝑘)+𝑥𝑡(𝑘)𝑥𝑡+(𝑘+1)𝑁2𝑖=0𝐴𝑖𝐵𝑢𝑡(𝑘)𝑢𝑡,(𝑘1)(4.7) it is shown that 𝛾1(𝑘+1)𝛾1(𝑘)𝑊𝜒(𝑘+𝑁𝑘),𝜔+(𝑘+𝑁𝑘)𝑥𝐹1𝑠𝐶𝐹̃𝑥(𝑘)+1𝑠𝐷+𝑥𝑑(𝑘)𝑡(𝑘)𝑥𝑡+(𝑘+1)𝑢𝑢𝑡(𝑘)𝑢𝑡(𝑘+1)𝑊𝜒(𝑘𝑘),𝜔(,𝑘𝑘)(4.8) where 𝑥,𝑢>0 are appropriate scalars.
On the other hand, at time 𝑘+1, since the target generation problem is feasible, it is feasible to choose 𝛾2(𝑘+1)=𝛾2(𝑘)𝑊(𝜒(𝑘+𝑁𝑘),𝜔(𝑘+𝑁𝑘)+(1𝜚)𝑊(𝜒(𝑘𝑘),𝜔(𝑘𝑘)).
Then, 𝛾1(𝑘+1)+𝛾2𝛾(𝑘+1)1(𝑘)+𝛾2(𝑘)𝜚𝑊𝜒(𝑘𝑘),𝜔(+𝑘𝑘)𝑥𝐹1𝑠𝐶(𝐹̃𝑥𝑘)+1𝑠𝐷+𝑥𝑑(𝑘)𝑡(𝑘)𝑥𝑡(+𝑘+1)𝑢𝑢𝑡(𝑘)𝑢𝑡(𝑘+1)𝜚𝜆min+(𝑄)𝜒(𝑘𝑘)𝑥𝐹1𝑠𝐹𝐶̃𝑥(𝑘)+1𝑠𝐷+𝑥𝑑(𝑘)𝑡(𝑘)𝑥𝑡+(𝑘+1)𝑢𝑢𝑡(𝑘)𝑢𝑡.(𝑘+1)(4.9) Hence, 𝛾1(𝑘)+𝛾2(𝑘) can serve as an ISS (for the definition of this term, see [22]) Lyapunov function, and the closed-loop system is input-to-state stable.

If we use the terminal equality constraint, rather than the terminal inequality constraint, then (3.27) should be revised as 𝐴𝑁𝐴𝜒(𝑘)+𝐵𝜋=0(4.10) and (3.28), (3.29) should be removed; moreover, (4.3) should be revised as 𝐴𝑁̂𝑥(𝑘+1)𝑥𝑡+𝐴𝐵𝜋(𝑘+1)=0(4.11) with the shifted control sequence 𝜔𝜋(𝑘+1)=(𝑘+1𝑘)+𝑢𝑡(𝑘)𝑢𝑡(𝑘+1)𝑇,,(𝜔(𝑘+𝑁1𝑘)+𝑢𝑡(𝑘)𝑢𝑡(𝑘+1))𝑇,𝑢𝑡(𝑘)𝑢𝑡(𝑘+1)𝑇𝑇,(4.12) and (4.4) should be removed.

Theorem 4.2. For the system (2.1) subject to the input constraints under Assumptions 2.12.2, the closed-loop output feedback model predictive control system, with objective function 𝐽(𝜒(𝑘),𝜋(𝑘)), augmented observer (2.4), target generation procedure (at 𝑘=0, (3.1)–(3.4); at any 𝑘+1, (3.1), (3.3), (4.1), (4.2), (4.11)), and dynamic optimization problem (3.25), (3.26), (4.10), (3.30), is input-to-state stable if the following two conditions are satisfied. (a)There exist feasible solutions (𝑥𝑡(𝑘),𝑢𝑡(𝑘)) to the target generation problem, at each time 𝑘. (b)There exist feasible solutions 𝜋(𝑘), at time 𝑘=0, to the dynamic optimization problem (3.25), (3.26), (4.10), (3.30).

Proof. By applying the shifted control sequence 𝜋(𝑘+1), at time 𝑘+1, one has 𝛾1(𝑘+1)𝛾1+(𝑘)=𝑊𝜒(𝑘+𝑁𝑘+1),𝜔(𝑘+𝑁𝑘+1)𝑁1𝑖=1𝑊𝜒(𝑘+𝑖𝑘+1),𝜔(𝑘+𝑖𝑘+1)𝑊𝜒(𝑘+𝑖𝑘),𝜔(𝑘+𝑖𝑘)𝑊𝜒(𝑘𝑘),𝜔=𝑊(𝑘𝑘)𝜒(𝑘+𝑁𝑘+1),𝜔(𝑘+𝑁𝑘+1)𝑊+𝜒(𝑘+𝑁𝑘),𝜔(𝑘+𝑁𝑘)𝑁1𝑖=1𝑊𝜒(𝑘+𝑖𝑘+1),𝜔(𝑘+𝑖𝑘+1)𝑊𝜒(𝑘+𝑖𝑘),𝜔(𝑘+𝑖𝑘)𝑊𝜒(𝑘𝑘),𝜔.(𝑘𝑘)(4.13) By analogy to Theorem 4.1, it is shown that 𝛾1(𝑘) can serve as an ISS Lyapunov function, and the closed-loop system is input-to-state stable.

Assume that 𝐴 is nonsingular. Then, applying (4.11) yields 𝑥𝑡=̂𝑥(𝑘+1)+𝐴𝑁𝐴𝐵𝜋(𝑘+1).(4.14) Further applying (4.3) yields 𝑦𝑡=𝐶̂𝑥(𝑘+1)+𝐶𝐴𝑁𝐴𝐵𝜋(𝑘+1)+𝐷𝑑(𝑘+1) and 𝐵𝑢𝑡=(𝐼𝐴)̂𝑥(𝑘+1)+𝐴𝑁𝐴𝐵𝜋(𝑘+1)𝐸𝑑(𝑘+1).(4.15) Hence, by applying (4.10)-(4.11), an analytical solution of the steady-state controller may be obtained.

5. Numerical Example

Let us consider the heavy fractionator, which is a Shell standard problem, with the following model: 𝐺𝑈(𝑠)=4.05𝑒27𝑠50𝑠+11.77𝑒28𝑠60𝑠+15.88𝑒27𝑠50𝑠+15.39𝑒18𝑠50𝑠+15.72𝑒14𝑠60𝑠+16.90𝑒15𝑠40𝑠+14.38𝑒20𝑠33𝑠+14.42𝑒22𝑠44𝑠+17.2019𝑠+1,𝐺𝐹(𝑠)=1.2𝑒27𝑠45𝑠+11.44𝑒27𝑠60𝑠+11.52𝑒18𝑠25𝑠+11.83𝑒15𝑠20𝑠+11.1427𝑠+11.2632𝑠+1,(5.1) where 𝐺𝑈(𝑠) is the transfer function matrix between inputs and outputs, and 𝐺𝐹(𝑠) between disturbances and outputs. The three inputs of the process are the product draw rates from the top and side of the column (𝑢1, 𝑢2), and the reflux heat duty for the bottom of the column (𝑢3). The three outputs of the process represent the draw composition (𝑦1) from the top of the column, the draw composition (𝑦2), and the reflux temperature at the bottom of the column (𝑦3). The two disturbances are the reflux heat duties for the intermediate section and top of the column (𝑑1, 𝑑2).

The inputs are constrained between 0.5 and 0.5, while the outputs between 0.5 and 0.5. The weighting matrices are identity matrices. 𝑁=3. The sampling interval is 3 seconds. With the algorithm as in Theorem 3.3 applied, the simulation results are shown in Figure 1. The steady-state calculation begins running at instant 𝑘=20, when the optimizer finds the optimum target 𝑦𝑡=[0.5,0.5,0.4269]𝑇. The objective value is 0.3538, indicating that 0.3538 unit benefits are obtained. During time 𝑘=200300, the disturbances 𝑑1=1.3 and 𝑑2=1 are added. The simulation verifies our theoretical results.

808327.fig.001
Figure 1: The closed-loop output trajectories, the corresponding control input signals, and the disturbances.

6. Conclusions

We have given some preliminary results for the stability of double-layered MPC. The results cannot be seen as the strict synthesis approaches; rather, they are endeavors towards this kind of approaches. Instead of asymptotic stability, we obtain the input-to-state stability, as in [22]. The results are inspired by [22]; but they are much different, as shown in Remarks 1–11 of [21].

We believe that several works need to be continued. Indeed, assuming feasibility of the target generation problem at each control interval is very restrictive, and overlooking the uncertainties in the prediction model brings difficulties for proving both the asymptotic stability and offset-free property. It may be necessary to develop a whole procedure, where the target generation problem is guaranteed (rather than assumed) to be feasible at each control interval and an augmented system is used for the stability analysis.

Acknowledgments

This work is supported by the Innovation Key Program (Grant KGCX2-EW-104) of the Chinese Academy of Sciences, by the Nature Science Foundation of China (NSFC Grant no. 61074059), by the Foundation from the State Key Laboratory of Industrial Control Technology (Grant no. ICT1116), by the Public Welfare Project from the Science Technology Department of Zhejiang Province (Grant no. 2011c31040) and by the Nature Science Foundation of Zhejiang Province (Grant no. Y12F030052).

References

  1. S. J. Qin and T. A. Badgwell, “A survey of industrial model predictive control technology,” Control Engineering Practice, vol. 11, no. 7, pp. 733–764, 2003. View at Publisher · View at Google Scholar · View at Scopus
  2. D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert, “Constrained model predictive control: stability and optimality,” Automatica, vol. 36, no. 6, pp. 789–814, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. A. Bemporad, F. Borrelli, and M. Morari, “Min-max control of constrained uncertain discrete-time linear systems,” Institute of Electrical and Electronics Engineers, vol. 48, no. 9, pp. 1600–1606, 2003. View at Publisher · View at Google Scholar
  4. L. Chisci, J. A. Rossiter, and G. Zappa, “Systems with persistent disturbances: predictive control with restricted constraints,” Automatica, vol. 37, no. 7, pp. 1019–1028, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. B. Ding, “Properties of parameter-dependent open-loop MPC for uncertain systems with polytopic description,” Asian Journal of Control, vol. 12, no. 1, pp. 58–70, 2010. View at Publisher · View at Google Scholar
  6. B. Kouvaritakis, J. A. Rossiter, and J. Schuurmans, “Efficient robust predictive control,” Institute of Electrical and Electronics Engineers, vol. 45, no. 8, pp. 1545–1549, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. D. Li and Y. Xi, “The feedback robust MPC for LPV systems with bounded rates of parameter changes,” Institute of Electrical and Electronics Engineers, vol. 55, no. 2, pp. 503–507, 2010. View at Publisher · View at Google Scholar
  8. H. Huang, D. Li, Z. Lin, and Y. Xi, “An improved robust model predictive control design in the presence of actuator saturation,” Automatica, vol. 47, no. 4, pp. 861–864, 2011. View at Publisher · View at Google Scholar · View at Scopus
  9. Y. I. Lee and B. Kouvaritakis, “Receding horizon output feedback control for linear systems with input saturation,” IEE Proceedings of Control Theory and Application, vol. 148, pp. 109–115, 2001. View at Google Scholar
  10. C. Løvaas, M. M. Seron, and G. C. Goodwin, “Robust output-feedback model predictive control for systems with unstructured uncertainty,” Automatica, vol. 44, no. 8, pp. 1933–1943, 2008. View at Publisher · View at Google Scholar
  11. D. Q. Mayne, S. V. Raković, R. Findeisen, and F. Allgöwer, “Robust output feedback model predictive control of constrained linear systems: time varying case,” Automatica, vol. 45, pp. 2082–2087, 2009. View at Google Scholar
  12. B. Ding, “Constrained robust model predictive control via parameter-dependent dynamic output feedback,” Automatica, vol. 46, no. 9, pp. 1517–1523, 2010. View at Publisher · View at Google Scholar · View at Scopus
  13. B. Ding, Y. Xi, M. T. Cychowski, and T. O'Mahony, “A synthesis approach for output feedback robust constrained model predictive control,” Automatica, vol. 44, no. 1, pp. 258–264, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. B. Ding, B. Huang, and F. Xu, “Dynamic output feedback robust model predictive control,” International Journal of Systems Science, vol. 42, no. 10, pp. 1669–1682, 2011. View at Publisher · View at Google Scholar
  15. B. Ding, “Dynamic output feedback predictive control for nonlinear systems represented by a Takagi-Sugeno model,” IEEE Transactions on Fuzzy Systems, vol. 19, no. 5, pp. 831–843, 2011. View at Publisher · View at Google Scholar · View at Scopus
  16. B. C. Ding, “Output feedback robust MPC based-on direct input-output model,” in Proceedings of the Chinese Control and Decision Conference, Taiyuan, China, 2012.
  17. D. E. Kassmann, T. A. Badgwell, and R. B. Hawkins, “Robust steady-state target calculation for model predictive control,” AIChE Journal, vol. 46, no. 5, pp. 1007–1024, 2000. View at Google Scholar · View at Scopus
  18. T. Zou, H. Q. Li, X. X. Zhang, Y. Gu, and H. Y. Su, “Feasibility and soft constraint of steady state target calculation layer in LP-MPC and QP-MPC cascade control systems,” in Proceedings of the 4th International Symposium on Advanced Control of Industrial Processes (ADCONIP '11), pp. 524–529, Hangzhou, China, 2011. View at Scopus
  19. K. R. Muske and T. A. Badgwell, “Disturbance modeling for offset-free linear model predictive control,” Journal of Process Control, vol. 12, no. 5, pp. 617–632, 2002. View at Publisher · View at Google Scholar · View at Scopus
  20. G. Pannocchia and J. B. Rawlings, “Disturbance models for offset-free model-predictive control,” AIChE Journal, vol. 49, no. 2, pp. 426–437, 2003. View at Publisher · View at Google Scholar · View at Scopus
  21. B. C. Ding, T. Zou, and H. G. Pan, “A discussion on stability of offset-free linear model predictivecontrol,” in Proceedings of the Chinese Control and Decision Conference, Taiyuan, China, 2012.
  22. T. Zhang, G. Feng, and X. J. Zeng, “Output tracking of constrained nonlinear processes with offset-free input-to-state stable fuzzy predictive control,” Automatica, vol. 45, no. 4, pp. 900–909, 2009. View at Publisher · View at Google Scholar · View at Scopus
  23. M. V. Kothare, V. Balakrishnan, and M. Morari, “Robust constrained model predictive control using linear matrix inequalities,” Automatica, vol. 32, no. 10, pp. 1361–1379, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH