Mathematical Problems in Engineering

Volume 2015, Article ID 870189, 11 pages

http://dx.doi.org/10.1155/2015/870189

## Stability Constraints for Robust Model Predictive Control

^{1}Department of Physics and Mathematics, Federal University of São João del-Rei, Campus Alto Paraopeba, Rodovia MG 443, KM 7, 36420-000 Ouro Branco, MG, Brazil^{2}Department of Mathematics, Federal University of Minas Gerais, Campus da Pampulha, Avenida Antônio Carlos 6627, 31270-901 Belo Horizonte, MG, Brazil^{3}Department of Electronic Engineering, Federal University of Minas Gerais, Campus da Pampulha, Avenida Antônio Carlos 6627, 31270-901 Belo Horizonte, MG, Brazil

Received 18 August 2015; Revised 15 October 2015; Accepted 27 October 2015

Academic Editor: Yan-Jun Liu

Copyright © 2015 Amanda G. S. Ottoni et al. 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

This paper proposes an approach for the robust stabilization of systems controlled by MPC strategies. Uncertain SISO linear systems with box-bounded parametric uncertainties are considered. The proposed approach delivers some constraints on the control inputs which impose sufficient conditions for the convergence of the system output. These stability constraints can be included in the set of constraints dealt with by existing MPC design strategies, in this way leading to the “robustification” of the MPC.

#### 1. Introduction

The term Model Predictive Control, MPC, stands for a wide range of control methods which make an explicit usage of the process model to obtain the control signal by minimizing an objective function [1]. It is difficult to determine the exact origin of MPC techniques, but it is known that they were developed in the 70s to solve control problems related to chemical industry and oil refining [2]. Currently, their application has been widespread in other sectors. The main features that have contributed to the growing use of predictive controllers, according to Maciejowski [3], are their ability to deal with time delays. Physical and operational constraints can be incorporated directly in the control design, reducing the number of emergency stops of the system. This makes the MPC controllers efficient and able to operate for long periods without requiring intervention [4].

A critical step in the design of MPC controllers is the determination of plant model to be used in the prediction. Discrepancies between the plant and its model may lead to poor performances or even to system instability. This situation motivated the development of robust MPC techniques that are intended to preserve stability and performance, despite inaccuracies or uncertainties in the model. As pointed out by Mayne [5], “*while major aspects of nominal MPC were well understood by 2000, the presence of uncertainty (…) and the associated topic of robustness against uncertainty, is a major challenge that is still receiving considerable attention.”* According to Mayne, assuming that the decision variable is a control sequence, there are three general approaches that are followed in MPC design for uncertain systems: (i) to take the uncertainty into account by requiring that the control problem constraints are satisfied for all possible realizations of the disturbance sequence; (ii) to employ a local feedback around a nominal trajectory; and (iii) to consider unstructured uncertainty in the system model.

Under the approach (i), the earlier literature employed the nominal value of the objective function of the MPC as a Lyapunov function [6–8]. In [9] a robust invariant terminal set was employed in order to ensure recursive feasibility. Several papers [10–12] addressed the problem of min-max MPC, in which the objective function is defined as the maximum, over the uncertainty set, of a cost function. Such a methodology is theoretically interesting, but it tends to be computationally costly. Reference [13] proposed an approximated min-max approach in order to reduce the computational burden. More recently, the robust stability of MPC has been studied under the viewpoint of the input-to-state stability (ISS) [14, 15].

The approach (ii), also called the* tube-based MPC*, employs simple parameterized local policies in order to approximate the (ideal) optimization over the control policies, instead of performing the optimization over the control sequences. References [16–18] follow this approach.

Finally, the approach (iii) usually relies on the small gain theorem. The first attempt to follow this direction was developed in the works [19–21], employing a frequency domain approach. In [22], the standard model is employed. A related approach is described in [23].

It is also worth mentioning MPC schemes for nonlinear systems. For instance, [24] presents a methodology which is based on an adaptive neural network. In [25], a neural network is also employed in order to achieve the robustness of the MPC against uncertainties in the control input matrix. The paper [26] performs an MPC that deals with nonlinear systems with dead-zone input. Other related references are [27, 28].

This paper proposes a methodology for the robust stabilization of uncertain SISO systems with parametric uncertainty which can be situated in the approach (i) above. The issue of stabilization is stated here from its first principles, as a feasibility problem related to the convergence of a sequence. This stability condition can be treated as a constraint in the MPC synthesis, taking advantage of the easy management of constraints within MPC. This defines a problem of optimal control synthesis with a constraint of robust stability which, itself, is calculated as a solution of a nonlinear optimization problem. It should be noticed that the algorithm for solving such a problem involves the solution of a subproblem of nonlinear optimization within an algorithm step which requires some computation time that may prevent the application of the proposed scheme to systems with small sampling times. In relation to this issue, the following points should be mentioned: (i) Several important plants have sampling times that are within the range of several minutes to several hours. In those cases, it is better to apply a more precise and less conservative algorithm that runs in some seconds than an approximate algorithm that runs in some milliseconds. (ii) Any proposal of robust MPC that adopts approximated formulae and conservative bounds in order to achieve fast computation should be benchmarked against a less conservative version of the same strategy. The proposed approach can be used as a benchmark for the performance of other MPC design procedures.

The proposed methodology is compared with the design technique of class (i) presented in [13] and with the design technique of class (iii) presented in [19–21]. Computational experiments show that the proposed methodology leads to the stabilization of plants belonging to larger uncertainty sets, considering parametric uncertainties represented by interval sets. The results suggest that the proposed methodology can be less conservative in the case of uncertainty sets of that type.

The remainder of this paper is structured as follows. Section 2 presents the problem statement. Section 3 describes the proposed formulation. Section 4 shows simulation results. Section 5 closes the paper with some conclusions.

#### 2. Problem Statement

Consider a system described by a CARIMA (*Controlled Auto-Regressive Integrating Moving-Average*) model, as follows [1]:where the term corresponds to an integral action that enables deleting the static error andis the transfer function of the model for the parameter vector , with .

Suppose that the time delay of the system is given by ; thus (1) is equivalent to

A large diversity of real processes can be modeled using CARIMA models, for instance, unmanned seaplanes [29], stirred tank reactors [30], vehicle yaw [31], gas engines [32], and distillation columns [33, 34]. Therefore, the development of control design techniques that are suitable for those models is relevant.

It is assumed that the process to be controlled is described by the model (1), subject to uncertainties on the coefficients of the transfer function, with each parameter and belonging to its respective uncertainty interval and . The box set given bydescribes the possible instances of the system parameter vector. It is assumed here that and .

System (1) with uncertain parameters described by (4) will be controlled by a* Model Predictive Control* (MPC) scheme, for which the following notation is employed: is the reference which should track; is the predicted value of the output increment on time , calculated on time , for , where is the prediction horizon; is the future control signal increment at time , used in the output increment predictions, for , where is the control horizon .

Different cost functions can be adopted for the definition of an MPC strategy which works along with the stability constraints that are proposed in this paper. For simplicity, the cost function of the MPC is assumed to be the traditional quadratic function that considers the error between the prediction and a known reference and the control input increment :where and are the weighting sequences of the error and the control effort, respectively. The following min-max problem is considered in the MPC, for the plant model with parameter uncertainty:An MPC employing the min-max objective function (6) was proposed in [13]. That control, in its unconstrained version, is named here as the RMPC (MPC with robust min-max approach). The MPC to be considered in this paper will be based on the RMPC, because the min-max index in (6) is suitable for dealing with set-bounded uncertainties. However it should be noticed that other indices could be defined, for instance, based on the expectation of the values of the uncertain parameters. Those alternative indices are expected to lead to better performances, since they are not assigned to the role of ensuring the system robustness in the context of the proposed methodology, which will enforce stability using constraints. The definition of those alternative indices is left as a theme for future research.

The problem to be solved in this paper is defined as follows [35].

*Definition 1 (asymptotic convergence problem). *Consider system (3) with uncertain parameters as described by (4), with any given initial condition , and a given constant reference signal , with . Find a sequence of control input increments , for , such thatfor all .

For the sake of simplicity, the reference input is assumed to be zero in the development that follows. More general reference signals can be tackled according to the guidelines presented, for instance, in [21].

#### 3. Convergence Constraints

The results to be established in the sequel rely on the following lemma.

Lemma 2. *Suppose the output sequence of a discrete-time system satisfies the following condition:where is a positive integer. Then, when .*

*Proof. *This comes directly from .

This lemma states a standard sufficient condition for the convergence of a sequence. The basic idea here is to impose condition (8) to the plant output, propagating it as a constraint for the values of the input increments such that the predicted output satisfies (8). The plant output should stay as close as possible to a reference signal . The following convergence condition is imposed:where and is a positive integer, which leads to . Considering the current time instant, a sequence of future input increments should be determined such that (9) is satisfied. The computation of such a sequence will be performed using a state-space description of the system. Let the following matrices be defined: matrix with rows and columns, with all entries equal to . : identity matrix with rows and columns. Define also the vectors:In addition, the matrices and are defined in different ways, depending on the value of the time delay . For , For ,

The system represented by the transfer function (2) is equivalent to the space-state system given byNow define the matrixand the decision variable vectorThe following lemma can be stated as a result of the recursive application of (13).

Lemma 3. *The -step-ahead prediction model of system (2) is given by*

*Proof. *Expression (16) comes as a direct composition of (13) applied on consecutive steps, with the replacement of expressions (14) and (15) in the suitable places.

The matrices and depend on the uncertain parameters and . On the instant , the vector is composed of known scalars (past values of and , up to instant ). The variable depends on , and on the decision variable vector , which should be specified such that (9) holds. The following lemma, stated under the assumption that there is no system uncertainty, constitutes a well-known result related to deadbeat controllers which is presented here as a bound for the achievable system performance.

Lemma 4. *Assume that system (13) is controllable. In this case, for any initial condition , there exists a decision variable vector with such that .*

*Proof. *It should be noticed that becomes equivalent to the controllability matrix of the system when . In the case of a controllable system, the controllability matrix is a full-rank square matrix of size . Therefore, when it will be possible to state such thatIn some special cases, it might be possible that a number of steps smaller than become enough, which completes the proof.

Considering the uncertain parameter case, a control strategy that would be analogous to the deadbeat control may be represented by Lemma 4 suggests that the choice would be reasonable, even for the case of systems with uncertainty, because with a smaller value for the control action might be unable to produce an arbitrary contraction of the output error. A more relaxed design formulation may be stated, requiring only the feasibility of condition (9):Expression (19) reduces to (18) when reaches its minimum value that still results in a feasible problem. In order to solve (19), it is worthy to note thatUsing (16), (20) can be rewritten aswhere is the matrix:

ConsiderWith the use of optimization tools, it is possible to obtain

Suppose, without loss of generality, that the reference signal is zero. So, the stability condition (9) can be rewritten asEquivalently,which leads to

The coefficients of the vector are limited byConsider the matrix , whose rows correspond to all possible combinations of minimum and maximum values of the components of vector :Note that

On this point, it is possible to state Theorem 5, which constitutes the main result of this paper.

Theorem 5. *Let denote the vector of the next control input increments to be applied to system (2) after instant . Ifholds, then the conditionis satisfied, for all .*

*Proof. *The proof of this theorem is stated as the sequence of expressions, from expression (25) to expression (31).

The RMPC, with objective function stated in (6), jointly with the stability constraints (32), will be called the RMPC-SC (RMPC with Stability Constraints). The RMPC-SC algorithm is presented in Algorithm 1.