da Silva Gesser, Rodrigo
Robust Model Predictive Control : A Comparative Study Considering Implementation Issues / Rodrigo da Silva Gesser ; orientador, Julio Elias Normey Rico, coorientador, Daniel Martins Lima, 2018. 127 p.
Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro Tecnológico, Programa de Pós Graduação em Engenharia de Automação e Sistemas, Florianópolis, 2018.
Inclui referências.
1. Engenharia de Automação e Sistemas. 2. Engenharia de Controle. 3. Controle Robusto. I. Normey Rico, Julio Elias. II. Martins Lima, Daniel. III. Universidade Federal de Santa Catarina.
Programa de Pós-Graduação em Engenharia de Automação e Sistemas. IV. Título.
This dissertation is dedicated to my dearest love, whom I admire the most and inspires me to be a better person every day.
ACKNOWLEDGEMENTS
My sincerest thanks to my tutors Prof. Julio Elias Normey-Rico and Prof. Daniel Martins Lima, who accompanied every step of my academic journey and gave me the knowledge and inspiration to ac-complish this work.
A special thanks to my close friends, who raided the university with me and gave me strength to move forward.
My biggest thanks to the support of my family, who taught me that patience and perseverance are keys to a great job.
This master thesis is supported by the Centro de Pesquisas Leo-poldo Américo Miguez de Mello (CENPES) which is the main research center from Petrobras. The work here described was developed in a project by CENPES in conjunction with the Departamento de Automa¸cão e Sistemas (DAS). The name of the project is "Desenvolvimento de Algo-ritmos de Controle Preditivo Não Linear e de Avalia¸cão de Desempenho de Controladores Preditivos para Plataformas de Produ¸cão de Petróleo" and the professors responsible are Prof. Julio E. Normey-Rico and Prof. Leandro B. Becker.
The chances of finding out what’s really go-ing on in the universe are so remote, the only thing to do is hang the sense of it and keep yourself occupied.
ABSTRACT
Robust Model Predictive Control (RMPC) is related to a variety of methods designed to guarantee control performance using optimization algorithms while considering systems with uncertainties. Many RMPC methods were created with different formulations, however there is lack of studies comparing their performances using pre-established metrics. Thus, the objective of this work is to analyze and compare different RMPC methods for different systems. The methods are sorted by Min-max RMPC, Tube-based RMPC, LMI-based RMPC and Cost-contractive RMPC, which were carefully chosen based on their relevance in the cur-rent literature. The metrics used to compare the methods are separated in off-line, related to configuration of the controller prior to its use, and on-line, which are related to the computational cost of the algorithm and difficulties that may arise during its application. Firstly, it is real-ized a thorough comparison of the RMPC methods for SISO systems, observing the behavior of the algorithms for distinct scenarios. Then, the RMPC methods are analyzed for a MIMO case study, the Continuous Stirred Tank Reactor (CSTR), using a realistic non-linear model of the process.
RESUMO EXPANDIDO
Introdu¸cão
Nos últimos anos houve um aumento no número de estudos relacio-nados à aplica¸cão de controladores avan¸cados na indústria. Entre as técnicas estudadas encontra-se o Controle Preditivo baseado em Mo-delo (MPC), cuja aplica¸cão industrial está mais voltada para sistemas petroquímicos. As vantagens de se utilizar um controlador preditivo está no seu conceito intuitivo e na forma natural com que dinâmicas complexas podem ser tratadas, como atrasos de transporte e sistemas com fase não-mínima. Porém, por depender de modelos, o MPC é sus-cetível a incertezas associadas a erros de modelagem e perturba¸cões externas ao sistema, o que pode prejudicar a sua performance quando o controle é projetado apenas considerando o modelo nominal do pro-cesso. Para contornar esse problema foram criados algoritmos de MPC robustos (RMPC), onde modelos das incertezas e perturba¸cões são in-cluídos na formula¸cão do controlador, permitindo a obten¸cão de um controle robusto; o que implica em um sistema de controle que mantém um conjunto de especifica¸cões de malha fechada mesmo para quando se opera com um processo diferente do modelo nominal.
Objetivos
Na literatura existem muitas técnicas de controle preditivo baseado em modelo robustos. Porém, poucos estudos abordam análises comparati-vas entre as diversas técnicas existentes, a maioria das revisões focam nos problemas teóricos de cada técnica e pouco comentam sobre aspec-tos de implementa¸cão. Portanto, tendo em vista a falta de estudos com-parativos na literatura, o principal objetivo dessa disserta¸cão é analisar e comparar métodos RMPC focando nos aspectos de implementa¸cão e estudar o comportamentos dos algoritmos de controle para diferentes cenários de simula¸cão.
Metodologia
Primeiramente foram definidas métricas para auxiliar na compara¸cão dos controladores, de forma que a compara¸cão entre os métodos seja va-lidada de forma justa. As métricas definidas são: métricas off-line, que dizem respeito às configura¸cões do controlador que são necessárias an-tes de iniciá-lo e métricas on-line, que visam aspectos de implementa¸cão durante o tempo de execu¸cão do problema de controle. Os tipos de controlador RMPC foram escolhidos pela sua relevância no âmbito aca-dêmico, analisando bancos de dados com publica¸cões e estudos de cada técnica. Os métodos são: Min-max RMPC, RMPC baseado em tubos,
Resultados e Discussão
A partir dos resultados obtidos com simula¸cões em diversos cenários, observou-se que os métodos sofrem grande influência de diversas va-riáveis. O horizonte de predi¸cão, por exemplo, aumenta considera-velmente o tamanho do problema de otimiza¸cão de diversas técnicas, porém não afeta os RMPC baseados em LMI visto que esta técnica in-depende do horizonte na sua formula¸cão. Além disso, o aumento no número de estados também aumenta o tamanho da otimiza¸cão, po-rém deve-se ficar atento às matrizes que condicionam o problema de otimiza¸cão. Isso significa que, mesmo aumentando o número de esta-dos, podem ocorrer situa¸cões em que a solu¸cão da otimiza¸cão é gerada mais rapidamente, caso as matrizes da otimiza¸cão estejam bem condici-onadas. Ao aplicar as técnicas de RMPC no sistema CSTR, observou-se uma performance satisfatória dos métodos quanto à resposta tempo-ral do sistema. Além disso, o custo computacional, de forma getempo-ral, manteve-se dentro de padrões aceitáveis para aplica¸cões deste porte. Os métodos baseado em LMIs obtiveram pior performance de tempo computacional quando comparado aos outros métodos, enquanto o RMPC com custo contrativo obteve o melhor custo computacional con-siderando apenas os controladores que se beneficiam de otimiza¸cão. Considera¸cões Finais
O estudo comparativo das técnicas RMPC alcan¸cou seu objetivo prin-cipal que era entender e comparar os problemas de implementa¸cão destas técnicas em diferentes situa¸cões. Observou-se a influência de variáveis como o horizonte de predi¸cão e número de estados do sistema em modelos tipo SISO, além de outros fatores que são determinantes para a implementa¸cão dos métodos RMPC. A aplica¸cão das técnicas no caso de estudo refor¸cou os resultados coletados durante a análise dos modelos SISO, estudando o comportamento dos controladores RMPC implementados no CSTR, um reator com dinâmicas não-lineares com-plexas.
Palavras-chave: Controle Robusto, Controle Preditivo baseado em
LISTA DE FIGURAS
2.1 MPC scheme . . . 7
2.2 MPC strategy . . . 9
2.3 The polytopic representation . . . 15
2.4 The multi-plant representation . . . 16
2.5 Inner and outer controller concept . . . 18
2.6 State trajectories for different control policies. . . 30
2.7 Illustration of a tighter constraint . . . 32
3.1 Number of occurrences of computation time for cost-contractive RMPC . . . 54
3.2 Number of occurrences of computation time for different RMPC . . . 55
3.3 Time response for min-max RMPC methods . . . 63
3.4 Time response for tube-based RMPC methods . . . 63
3.5 Time response for LMI-based RMPC methods . . . 64
3.6 Time response for cost-contractive RMPC method . . . . 64
3.7 Robust Control Invariant Set . . . 67
3.8 Time response for min-max RMPC method . . . 73
3.9 Time response for tube-based RMPC method . . . 73
3.10 Time response for LMI-based RMPC method . . . 74
3.11 Time response for cost-contractive RMPC method . . . . 74
3.12 Robust Control Invariant Set . . . 77
3.13 Robust Control Invariant Set . . . 85
3.14 Mean computation time of tube-based method #2 each iteration . . . 90
3.15 Time response of min-max RMPC methods . . . 92
3.16 Time response of tube-based RMPC methods . . . 93
3.17 Time response of LMI-based RMPC methods . . . 93
3.18 Time response of cost-contractive RMPC method . . . . 94
4.1 Representation of the CSTR . . . 97
4.2 Static curves of the CSTR . . . 100
4.3 State time response of min-max RMPC methods - blue indicates min-max #1 and dashed red min-max #2 . . . 110
4.4 Input time response of min-max RMPC methods - blue indicates min-max #1 and dashed red min-max #2 . . . 111 4.5 State time response of tube-based RMPC methods - blue
indicates tube-based #1 and dashed red tube-based #2 . 112 4.6 Input time response of tube-based RMPC methods - blue
4.8 Input time response of LMI-based RMPC methods - blue indicates LMI-based #1 and dashed red LMI-based #2 . 113 4.9 State time response of costcontractive RMPC methods
-blue indicates cost-contractive #1 . . . 114 4.10 Input time response of costcontractive RMPC methods
LISTA DE TABELAS
3.1 Organization of the RMPC methods . . . 42 3.2 Tuning parameters . . . 43 3.3 Tuning parameters defined for the RMPC methods . . . . 49 3.4 Solver . . . 49 3.5 Number of decision variables and constraints for
differ-ent values of N . . . 51 3.6 Increase in the total size of decision variables and
con-straints of the RMPC methods for the change in N . . . . 52 3.7 Computational cost for different values of N . . . 53 3.8 Increase in the mean computation time of the RMPC
methods for the change in N . . . 56 3.9 Number of decision variables and constraints for the
first-order model with and without delay . . . 59 3.10 Increase in the total size of decision variables and
con-straints of the RMPC methods for the first-order model with and without delay . . . 60 3.11 Computational cost for the first-order model with and
without delay . . . 61 3.12 Increase in the mean computation time of the RMPC
methods for the first-order model with and without delay 62 3.13 Tuning parameters defined for the RMPC methods . . . . 68 3.14 Number of decision variables and constraints for
differ-ent values of n . . . 69 3.15 Increase in the total size of decision variables and
con-straints of the RMPC methods for the change in n . . . . 70 3.16 Computational cost for different values of n . . . 71 3.17 Increase in the mean computation time of the RMPC
methods for the change in n . . . 72 3.18 Tuning parameters defined for the RMPC methods . . . . 78 3.19 Number of decision variables and constraints for
differ-ent values of n . . . 79 3.20 Increase in the total size of decision variables and
con-straints of the RMPC methods for the change in n . . . . 80 3.21 Computational cost for different values of n . . . 82 3.22 Increase in the mean computation time of the RMPC
methods for the change in n . . . 83 3.23 Tuning parameters defined for the RMPC methods . . . . 86 3.24 Number of decision variables and constraints for n = 3
with and without zero . . . 87 3.25 Increase in the total size of decision variables and
methods with the addition of the zero . . . 91
4.1 Main variables of the CSTR . . . 98
4.2 Parameters of the CSTR . . . 99
4.3 Operating points of the system . . . 100
4.4 Contraints . . . 102
4.5 Operating points defining the polytopic uncertainty . . . 103
4.6 Tuning parameters defined for the RMPC methods . . . . 107
4.7 Number of decision variables for different values of N . 108 4.8 Computational cost for LMI-based RMPC . . . 109
ACRONYMS
CSTR Continuous Stirred-tank Reactor . . . 113, 115 FIR Finite Impulse Response . . . . 26
GLPK GNU Linear Programming Kit . . . . 73
LMI Linear Matrix Inequalities . . . . 26, 27
LQR Linear Quadratic Regulator . . . . 67
MIMO Multiple-Input Multiple-Output . . . 113 MPC Model Predictive Control . . . . 25–28
MPT Multi-Parametric Toolbox . . . . 68
PID Proportional-Integral-Derivative . . . . 25
RMPC Robust Model Predictive Control . . . . 26–29
SeDuMi Self-Dual Minimization . . . . 73
SISO Single-Input Single-Output . . . 65, 68, 69 ZOH Zero Order Holder . . . . 69
CONTENTS
1 Introduction 1
1.1 Robust Model Predictive Control . . . 2 1.2 Motivation . . . 4 1.3 Objectives and Methodology . . . 5 1.4 Final Comments . . . 6
2 Robust Model Predictive Control - RMPC 7
2.1 Model Predictive Control - MPC . . . 7 2.2 State-space MPC Formulation . . . 10 2.2.1 Linear State-Space Dynamic Models . . . 10 2.2.2 Optimization Problem . . . 11 2.2.3 Uncertainty Description . . . 14 2.3 Min-max Robust Model Predictive Control . . . 16
2.3.1 Min–max feedback model predictive control for constrained linear systems . . . 17 2.3.2 Feedback min–max model predictive control
using a single linear program: robust stability and the explicit solution . . . 21 2.4 LMI-based Robust Model Predictive Control . . . 23 2.4.1 Linear Matrix Inequalities . . . 23 2.4.2 Robust constrained model predictive control
using linear matrix inequalities . . . 24 2.4.3 An improved approach for constrained robust
model predictive control . . . 28 2.5 Tube-based Robust Model Predictive Control . . . 30 2.5.1 Robust model predictive control using tubes . 31 2.5.2 Robust model predictive control of constrained
linear systems with bounded disturbances . . 34 2.6 Cost-contractive Robust Model Predictive Control . . 35
2.6.1 A robust model predictive control algorithm for stable linear plants . . . 36 2.7 Final Comments . . . 38
3 Comparative Study of RMPC Methods 39
3.1 Configuration of the RMPC Controllers . . . 42 3.2 First-order model - Analysis of the prediction horizon 46 3.2.1 Off-line Analysis . . . . 48 3.2.2 On-line Analysis . . . . 52 3.3 First-order model vs First-order model with delay . . 57 3.3.1 Off-line Analysis . . . . 58
3.4.1 Off-line Analysis . . . . 67 3.4.2 On-line Analysis . . . . 70 3.5 Second-order model vs Third-order model . . . 75 3.5.1 Off-line Analysis . . . . 77 3.5.2 On-line Analysis . . . . 80 3.6 Third-order model vs Third-order model with
uncer-tain zero . . . 84 3.6.1 Off-line Analysis . . . . 87 3.6.2 On-line Analysis . . . . 88 3.7 Final Comments . . . 95
4 Case Study 97
4.1 Description of the Process . . . 97 4.2 Uncertainty Description . . . 102 4.3 Analysis of the RMPC Controllers . . . 105 4.3.1 Results . . . 106 4.4 Final Comments . . . 115
5 Conclusion 117
5.1 Future works . . . 121
1 INTRODUCTION
In general, classical controllers, such as Proportional-Integral-Derivative (PID) controllers, are found in large number in process industry [1] because most processes are easily translated to systems with simple and linear dynamics. These controllers use the idea of feedback to reduce the effects of disturbances applied to the system and to guarantee tracking when operating in steady-state, which turns the PID controller sufficient to solve many problems in process industry. However, there are many situations where the interactions between the process variables or variations of the process are high if the classical controllers that handle them are inadequate to fulfill the control requirements, demanding more advanced strategies.
Over the past few decades there were a rising number of stud-ies related to advanced control as an alternative to classical control strategies in scenarios which PID controllers present poor perfor-mance. There are many challenging scenarios to overcome, such as multi-variable systems with high coupling, non-linearities, systems with delay and physical constraints. Therefore, the advanced con-trollers are being developed considering these scenarios within their formulation, increasing the complexity of the control algorithm but offering a better performance [2].
Among the advanced strategies there is the Model Predictive Control (MPC), which is a wide range of methods from the optimal control theory that uses models to represent the system and cal-culate the control signals. It is commonly associated with the pro-cess industry, however it is also used in different contexts, such as robotics and clinical anesthesia [3]. Basically, the MPC formulation tries to predict the system’s future output behavior using dynamic models over a moving horizon (or receding horizon), then optimize the predictions to calculate the best control actions [4] minimizing the error between the predicted output and the desired reference [5].
In general, mathematical equations are used to model a sys-tem, usually in the form of ordinary differential equations and then used by the MPC to calculate the predictions. However, in practice these models are usually associated with uncertainties, such as es-timation errors, changes in operating points or unmeasured distur-bances and then a control signal must be calculated so that the desired performance is maintained and that the stability of the con-trolled variables are guaranteed. Thus, the system needs a good characterization of the model uncertainties, as detailed by Bempo-rad (1999) [5].
A controller that is able to find optimal control actions regard-less of uncertainties and guarantees stability is called robust [3]. Within the MPC methods there are different formulations worried with robustness when dealing with uncertainties and these methods are often called Robust Model Predictive Control (RMPC) [4]. Each RMPC formulation depends highly on the uncertainty descriptions and how the optimization problem is stated, however the main fea-ture of all methods is a guaranteed robust stabilization of the uncer-tain system considering input and output constraints.
1.1 ROBUST MODEL PREDICTIVE CONTROL
The first RMPC method proposed is difficult to trace back, however one of the pioneers with great contribution using concepts of MPC extended to uncertain systems is from 1987 when Campo and Morari developed a RMPC using min-max optimization [6]. The results were interesting and increased the optimism over MPC meth-ods that considered uncertainties as a field of research, thus more publications appeared further developing the min-max RMPC, such as Allwright (1992) [7], that proposed a min-max RMPC using a set of Finite Impulse Response (FIR) models and Zheng (1993) [8] which improved the algorithm guaranteeing asymptotic stability.
The approach for min-max RMPC started to change when Lee and Yu (1997) [9] first proposed a method using state-space mod-els for worst-case formulations of MPC. Then, Scokaert (1998) [10] devised an algorithm using a dual-mode control law with feedback properties and guaranteeing stability using robust invariant sets. Kerrigan further improved this algorithm by changing the optimiza-tion problem to eliminate the dual-mode consideraoptimiza-tions [11]. Re-cently, a few new researches started to appear proposing small vari-ations of the min-max RMPC, such as Ding (2010) which suggested an algorithm for parameter-dependent models with polytopic un-certainty descriptions [12], Yan and Wang (2014) that presented a neural network approach for min-max RMPC [13] and Calafiore (2013) [14] that developed a RMPC via scenario optimization.
An alternative algorithm was proposed by Kothare (1996) [15] that changed the min-max formulation using a different ap-proach called Linear Matrix Inequalities (LMI). Considering a state-space model, the idea is to rewrite an infinite horizon optimization problem and translate it to the LMI framework, which is reduced to a convex optimization problem that is solved in polynomial time.
1.1. Robust Model Predictive Control 3 However, Kothare’s algorithm assumes quadratic stabilizability of the uncertain system, which is not required by the algorithm from Cuzzola (2002) [16], changing the procedure incorporating new LMI’s.
Many LMI-based RMPC started arising as a consequence, pre-senting a variety of methods improving this technique. For example, Kouvaritakis (2000) [17] proposes an efficient algorithm that per-forms off-line the demanding computations, leaving only a simple optimization problem to be solved on-line; Wan (2002) [18] and Feng (2004) [19] also developed an off-line algorithm for synthe-sizing an RMPC using the concept of asymptotically stable invari-ant ellipsoids, reducing on-line computation significinvari-antly. Recently, more studies applied LMI-based RMPC to different subjects, such as Al-Gherwi (2011) [20] which proposed a robust distributed model predictive control and Ahn (2015) [21] which created a robust con-troller for neural networks with time-varying delay, both papers in-spired by Kothare’s controller.
A new solution to the uncertainty problem was proposed by Langson (2004) [22] and described by Mayne [4] called tube-based RMPC, using a different approach to describe the uncertainties whe-re a group of trajectories corwhe-responding to each possible whe-realizations of the system’s uncertainty is known as a tube. For a state-space model, the main objective is to find an optimal control action that keeps the state inside the tube for all realizations, knowing that, after choosing the proper tube, the control action calculated will satisfy all the system’s constraints. Seron and Mayne (2005) [23] improved the method by treating the initial state as decision vari-able for the optimization problem, adding a degree of freedom to the controller.
The tube-based RMPC continued to be researched and im-proved for different purposes, such as output tracking problems, with great contributions by Alvarado (2007) [24] using state es-timation with a robustly stabilizing tube-based MPC, and Limon (2009) [25] that developed a controller that steers the uncertain system to the target if there is an admissible solution, otherwise it steers the system to the nearest operating point. Another exam-ple is the proposal of Falugi (2014) [26], which extended the con-cept to nonlinear systems with unmodeled dynamics. Some publi-cations integrated tube-based MPC with other techniques, such as Aswani (2013) [27] that mixed it with economic nonlinear MPC and compared with other methods, and Lucia (2014) [28] using it in learning-based MPC for safety issues.
Furthermore, Badgwell (1997) [29] proposed a new algo-rithm for control of uncertain systems, called cost-contractive RMPC, which consisted of a MPC with restrictions on future behavior of the controller cost function and it achieves robust stability when adding limits that prevent the cost function of the controller from increasing at each time step. Odloak (2004) [30] extended the cost-contractive RMPC for a state-space model with incremental inputs, resulting in a controller with a offset free control law, and Marcio (2016) [31] proposed a new formulation to Odloak’s method for
stable and unstable systems.
The RMPC methods in this dissertation are separated by dif-ferent categories which defines how each method computes the op-timization problem when calculating the control signal. Since there are many different implementations for each category, specific algo-rithms were selected by evaluating their relevance on publications databases, leading to:
1. Min-max model predictive control (based on [10], [11]) 2. LMI-based model predictive control (based on [4], [22], [23]) 3. Tube-based model predictive control (based on [15], [16]) 4. Cost-contractive model predictive control (based on [29])
1.2 MOTIVATION
Clearly, there are many RMPC methods with distinct formu-lations, however, the literature about RMPC, when not proposing a new method, is limited to surveys ([32], [33], [34], [5]) with few quantitative analysis comparing each method. Most surveys are con-cerned with describing each formulation and showing qualitative differences. Therefore, RMPC is a field of important research for control applied to uncertain systems, with many substantial contri-butions in the literature but lacking proper analysis and guidelines. Thus a deep comparison will further emphasize the value of those formulations and increase the comprehension of what method is more appropriate to use at each situation.
Before comparing them, it is important to define what are the main issues related to RMPC strategies, as they are fundamental to understand the flaws of the methods, but also the potential for different situations. These issues could be separated as:
1.3. Objectives and Methodology 5 1. theoretical issues, which is concerned with stability,
robust-ness and time performance.
2. implementation issues that are more connected with the
al-gorithm problems, such as computational cost and scalability. Much of the existing literature and publications already ad-dress the theoretical issues when describing the method, however implementation issues are not as discussed as the first one. More-over, there are hardly any contrast using pre-established metrics for a variety of scenarios and for systems with different dynamics.
1.3 OBJECTIVES AND METHODOLOGY
Considering the previous discussion, this dissertation intends to compare and analyze the RMPC strategies for different scenarios to provide a meaningful understanding about the effects of each strategy, focusing on implementation issues arising from them. The-refore, relevant metrics must be defined to validate the comparison and obtain a trustworthy outcome, which resulted in two classes of metrics:
1. Off-line metrics, which are related to the configurations that must be done before starting the controller.
a) Number of tuning parameters of the controller; b) Number of decision variables;
c) Number of constraints;
d) Scalability of the algorithm related to the size of the opti-mization problem.
2. On-line metrics, which incorporates aspects of implementation during the controller’s execution time.
a) Computational cost (measured in computation time); b) Scalability of the algorithm related to computational cost;
c) Analysis of time response.
Furthermore, the goal is to compare the issues in the imple-mentation itself, mainly concerned with the coding effort and diffi-culties with the statement of the control problem, such as the chal-lenges of finding robust invariant sets or using the LMI framework.
1.4 FINAL COMMENTS
To present the aforementioned analysis, Chapter 2 starts defin-ing the state-space formulation of MPC, focusdefin-ing on uncertain sys-tems, and describe the four categories of RMPC chosen for this study. Then, in Chapter 3 each method for different SISO systems is com-pared, using predetermined metrics to analyze the performance of each controller. Chapter 4 presents a case study for the RMPC meth-ods, using a realistic non-linear model of a CSTR. Finally, Chapter 5 concludes this work with an overall analysis of the results.
2 ROBUST MODEL PREDICTIVE CONTROL - RMPC
This chapter begins with an overview of the fundamental def-initions of MPC, using the state-space formulation, explaining the control algorithm and some important properties. Then, the follow-ing sections introduce uncertainties and how to describe them for different situations, ending with a detailed explanation of the RMPC categories studied in this work.
2.1 MODEL PREDICTIVE CONTROL - MPC
The MPC is a wide range of optimal control strategies that are characterized by the application of dynamic models, which are used to predict the future behavior of the system, and also by the minimization of a cost function to calculate the optimal control ac-tion to be applied at the process during the control loop. The main differences from each formulation are the type of model used to represent the system, the disturbances considered during the calcu-lation of the future predictions and the cost function used during the minimization [35].
Figure 2.1: MPC scheme.
Although there are many formulations, the basic scheme is the same for all MPC algorithms as demonstrated in the diagram shown in Figure 2.1. This scheme is the foundation of every MPC and it might be summarized as:
\bullet For a given system, the first step is to read the current out-put through a sensor or estimate it in the case of unavailable measurements;
\bullet Then, there is the explicit use of dynamic models to predict the system output at future time instants. For instance, the following model is a linear discrete state-space representation of a process (as shown in Figure 2.1):
xk+1= Axk+ Buk
yk= Cxk+ Duk
(2.1) where xk, ukand ykare the system’s states, input and output
at instant k, respectively, and xk+1 is the successor state at
instant k + 1. Using this representation, the states might be predicted during the optimization problem;
\bullet A optimal control action is calculated by minimizing a cost function and considering physical constraints of the system, which might be represented in many forms, such as:
\BbbP (\^uk+j) = \mathrm{m}\mathrm{i}\mathrm{n} \^ u N - 1 \sum j=0 \{ L(xk+j, uk+j)\} subject to Axx \geq bx Auu \geq bu (2.2)
where L(x, u) is a cost function, \BbbP is the optimal control prob-lem, \^uk+j is the optimal predicted control action, Axx \geq bx
and Auu \geq bu are state and input constraints, respectively
[4];
\bullet In (2.2) the use of the parameter N , commonly known as re-ceding (or moving) horizon, can be observed. It represents the prediction time used by the optimization problem. It is called moving horizon because each time instant the "window" of prediction is shifted and only the first control action \^u(0) cal-culated during the optimization \BbbP (\^uk)is applied to the plant.
\bullet Figure (2.2) shows the predicted time behavior for a certain
horizon. The MPC concept is that, given a reference trajec-tory, the MPC controller uses the data from past control tions and measured output to propose future control ac-tions, such that the predicted output converges to the
2.1. Model Predictive Control - MPC 9
Figure 2.2: MPC strategy.
There are many advantages of using MPC when comparing to classical control techniques. Firstly, the main idea is very intuitive needing limited information about control theory, indicating that the algorithm is appealing to users without strong control knowl-edge. The algorithm allows to naturally define the problem for mul-tivariable systems and complex dynamics, such as dead-time, non-minimum phase and unstable poles, are intrinsically dealt with by the controller without any changes.
Also, the tuning of MPC controllers is in the time-domain, which is intuitive to understand, and constraints are easily incor-porated in the optimization problem. A controller that incorporates constraints is advantageous because usually the most profitable erating point is near the constraint region and the MPC finds op-timal control actions to operate in this situation. A controller that operates in an economical optimum value, while considering physi-cal constraints, is highly desirable and, for that reason, the visibility of MPC controllers in industry is increasing.
However, all these advantages have a cost to the controller. Solving a optimization problem is not an easy task and require high computational power depending on the situation, resulting in a con-troller with much more computational cost compared to classical controllers that do not need any optimization. Also, the tuning of the MPC is not as intuitive as the concept of the MPC formulation and is usual for the operators to start the controllers with arbitrary
tuning parameters and finishing with a "fine-adjustment" to improve the performance.
The next section details the state-space formulation of MPC, explaining the most important variables used.
2.2 STATE-SPACE MPC FORMULATION
In a vast majority, the nominal system’s dynamics are nonlin-ear and the most common descriptions for them are represented by continuous nonlinear differential equations such as:
dx
dt = f (x, u, t) y = h(x, u, t)
(2.3)
where x \in \BbbR n is an n-dimensional state vector, u \in \BbbR m is an
m-dimensional input vector, y \in \BbbR p is a p-dimensional output
vec-tor and t \in \BbbR is the time. Using nonlinear equations to represent the system increases the complexity of the controller, therefore it is usual to approximate the problem using linear models in an op-erating point, which reduces the calculation needed for advanced controllers such as MPC.
2.2.1 Linear State-Space Dynamic Models
Although the system in 2.3 is written in continuous-time, mod-els are, in general, sampled at discrete times. The reason to describe the model in discrete-form is that the controllers are digitally imple-mented most of the time and, if the sampling time is properly cho-sen, the behavior of the sampled model is a good approximation of the system.
\bullet Time-varying discrete model. Using this model the system is
represented more generically by:
xk+1= Akxk+ Bkuk
yk= Ckxk+ Dkuk
(2.4)
where Ak \in \BbbR n\times n is the state transition matrix, Bk \in \BbbR n\times m
is the input distribution matrix, Ck \in \BbbR p\times n is the output
ma-trix and Dk \in \BbbR p\times m is matrix relating directly the input and
2.2. State-space MPC Formulation 11 \bullet Time-invariant discrete model. Considering that the system
is invariant at each sample time, the model is redefined as: xk+1= Axk+ Buk
yk= Cxk+ Duk
(2.5)
Typically, the time-invariant model is used for simplicity of representation. As expected, approximating nonlinear systems with linear models yields errors, referred to as model uncertainty, which is mentioned later in this chapter.
2.2.2 Optimization Problem
As mentioned previously, the optimal control action applied to the system is calculated by solving an optimization problem \BbbP , which basically consists of minimizing a cost function, as observed in (2.6): \BbbP (x, u) = \mathrm{m}\mathrm{i}\mathrm{n} \^ uk+j \infty \sum j=0 L(xk+j, uk+j) s.t. x \in \bfX u \in \bfU (2.6)
where \bfX is the state constraint set and \bfU the input constraint set, \^
uk+j is optimal control sequence and L is called the stage cost.
For practical reasons, the cost is defined over a horizon N , to ensure an optimal control problem that is solved fast enough and efficiently. Since the problem is reduced to N steps, a terminal cost Vf might be included to approximate the semi-infinite solution [4]
resulting in (2.7). \BbbP (x, u) = \mathrm{m}\mathrm{i}\mathrm{n} \^ uk+j N - 1 \sum j=0 L(xk+i, uk+i) + Vf(xN) s.t. x \in \bfX u \in \bfU xN \in \bfX \bfN (2.7)
where xN is the state at instant N and \bfX \bfN is the terminal
con-straint set. The optimization problem from (2.7) is quite complex, with many aspects and variables that influences the outcome of the optimization, such as:
\bullet Prediction horizon
The horizon N is a tuning parameter of the MPC controller that determines the window which the predicted behavior of the system is calculated. Choosing a large horizon compared to the settling time of the system indicates a preference over the steady-state response, otherwise decreasing the value of N changes the priority to transient response.
\bullet Cost function
The stage cost L is used to obtain the control law and might be defined as:
L(x, u) = (x - xr)\prime Q(x - xr) + (u - ur)\prime R(u - ur) (2.8)
which depends on the input vector u, an input vector reference ur, the state vector x and state vector reference xr. The
tun-ing parameters are the weights Q and R. When Q have higher values than R, the controller is focused on steering the state to the reference value xrat the expense of a higher control
ac-tion. However, for higher R values compared to Q, the control action is reduced and the rate at which the state approaches xris slowed down [4].
\bullet Terminal cost
The terminal cost Vf is commonly used to guarantee stability,
by approximating the infinite horizon MPC to a finite horizon. The stability is guaranteed by choosing a terminal weight P , such that:
Vf(xN) = x\prime NP xN (2.9)
and P is the solution of the Ricatti equation that proves the controller’s stability [36]. The Ricatti equation is derived from the Lyapunov stability theory, which affirms that, given the function Vf from (2.9), the stability is guaranteed if P > 0
and [37]:
Vf(xN) > 0,
V (xN) - V (xN - 1) < 0
(2.10) \bullet Constraints
In general, most systems are physically restricted and con-straints are used to represent these limitations. For example,
2.2. State-space MPC Formulation 13 the control signal u applied to the system is bounded by a given setU which establishes the field of actuation permitted.
These constraints are related to physical limitations of actu-ators, safety or environmental issues and even the range of sensors [3].
x \in \bfX
u \in \bfU (2.11)
\bullet Terminal set constraint
A terminal constraint is used to steer the system for a desirable region at the end of the horizon during the prediction. It adds another condition to stabilize the system if the terminal set \bfX \bfN is chosen correctly.
xN \in \bfX \bfN (2.12)
Usually, the terminal set constraint is composed by a large number of inequality equations, which increases the optimiza-tion problem’s complexity and the computaoptimiza-tional effort to solve it. A solution is to reduce the number the terminal constraint, as proposed in [38], which suggests to define the terminal constraint only to the unstable states of the system.
Thus, the basic algorithm of the MPC can be summarized as:
Basic Algorithm
(a) Initialize xk = x(0), where x(0) is the current state of the
system;
(b) Calculate the predictions \^xk+j of the system;
(c) Solve the optimization problem \BbbP (\^xk, u)as in (2.7) and obtain
the optimal control sequence \^uk+j;
(d) At time k, apply the first input uk of the optimal control
se-quence to the plant;
(e) Measure xkand return to Step (b).
Previously, it was noted that real processes are modeled us-ing mathematical equations. In fact, mathematical models are ap-proximations of the reality that essentially attempts to contemplate
as many aspects of reality as possible. Simplifications are assumed when using models for control objectives to keep the MPC structure simple for real-time applications, and therefore, much of the pro-cess information is lost due to approximations and the dynamic of the system is not completely described.
Therefore, the design and performance of the control depends completely on the model. It is usual to apply a nominal model struc-ture in most control techniques where no external disturbances are considered, such as in (2.4) and (2.5). But the condition where control model matches the process and there are no external dis-turbances affecting is not true in the majority of the situations, re-quiring a robust controller that is capable of preserving stability and performance in spite of model inaccuracies or uncertainties in real processes [3].
The dependency to model descriptions is a great issue and the controller’s performance is highly sensitive to errors between the real process and the model used. These uncertainties are criti-cal in control applications. Thus, as mentioned before, a controller capable of performing well regardless of uncertainties is desired to counter the need of a model. There are different approaches for de-scribing uncertainties depending mainly on the type of technique used to design the controllers.
2.2.3 Uncertainty Description
The uncertainty description is added to represent mismatches between the nominal system and model. These uncertainties ap-pear from different sources, for instance, the system might be dis-turbed by additive signals that arise from unknown sources, the states may not be completely measured or the model might be inac-curate. Therefore, they might be separated in classes of uncertainty, as described in [5].
\bullet Polytopic Uncertainty
This formulation enables the use of the time-variant model when it is guaranteed to stay within the polytope.
xk+1= Aixk+ Biuk
yk= Cxk
[Ai Bi] \in \Omega = Co\{ [A1 B1], . . . , [Ap Bp]\}
2.2. State-space MPC Formulation 15 where \Omega is defined as the convex hull, Co, of all vertex within the polytope [Ai Bi]. For example, consider an uncertain
sys-tem composed by a polytope with three vertices, such that: [Ai Bi] \in \Omega = Co\{ (A1, B1), (A2, B2), (A3, B3)\} (2.14)
thus, the uncertain system is described by every possible con-vex combination of (A, B) that is within the polytope com-posed by \Omega . Figure (2.3) illustrates this example, where the blue region defines the convex hull of \Omega with vertices (A1, B1),
(A2, B2)and (A3, B3).
Figure 2.3: The polytopic representation.
\bullet Multi-Plant
The multi-plant description is used when the plant is not kno-wn exactly, however there is a set \Omega of p possible plants wherein the plant lies.
xk+1= Aixk+ Biuk, i = 1, . . . , p
yk= Cxk
[Ai Bi] \in \Omega = \{ [A1, B1], . . . , [Ap, Bp]\}
(2.15)
As observed, this description is similiar to the polytopic, whe-reas the multi-plant defines a set composed by the vertex of the polytope instead of the convex hull of [Ai Bi]as the set.
In Figure (2.4) is presented an example of multi-plant descrip-tion, noticing that the set system is only described by the ver-tex of \Omega .
Figure 2.4: The multi-plant representation.
\bullet Additive Disturbance
The description for a bounded additive disturbance describes an unknown uncertainty w whose bounds are known, and which is defined by a set \bfW where all possible disturbances lie.
xk+1= Axk+ Buk+ wk
yk= Cxk+ Duk
(2.16)
where wk\in \bfW . The set \bfW is generally bounded wmin\geq wk\geq
wmax or it is defined by a compact and convex polyhedron.
This description is interesting because it may easily be used to represent nonlinear models with additive disturbances and it characterizes any type of uncertainties, thus it is globally used to represent model uncertainty [39].
The category of MPC that is preoccupied with controlling sys-tems with uncertainties is called robust MPC. The next sections in this chapter provide a detailed definition of each RMPC algorithm that is used in this dissertation.
2.3 MIN-MAX ROBUST MODEL PREDICTIVE CONTROL
The first approach of min-max RMPC detailed in this section was proposed by Scokaert and Mayne [10], followed by a second approach devised by Kerrigan and Maciejowski [11].
2.3. Min-max Robust Model Predictive Control 17
2.3.1 Min–max feedback model predictive control for constrai-ned linear systems
A linear time-invariant discrete-time model is described by the equation:
xk+1= Axk+ Buk+ wk (2.17)
where x \in \BbbR n, u \in \BbbR mare state and input vectors at time k, A and
Bare state transition and input distribution matrices and (A, B) is assumed controllable. Finally, w \in \bfW \subset \BbbR n is an unknown
distur-bance bounded by the compact and convex set \bfW .
The controller’s objective is to steer the states to the origin. Standard min-max MPC strategies optimize an open-loop control se-quence and, therefore, the controller cannot steer the system to the origin due to the disturbance w and the absence of feedback dur-ing the optimization. Scokaert’s proposal introduces the notion of feedback at each sample time during the optimization, resulting in a control law computationally more demanding but that improves performance.
Control Law
Considering that the system is never steered to the origin due to the presence of external disturbances, the objective of the con-troller is to drive the state to a robust control invariant set, denomi-nated \bfX \bfzero , while satisfying state and input constraints in the form:
x \in \bfX u \in \bfU (2.18)
where \bfX \bfzero \subseteq \bfX \subseteq \BbbR n and \bfU \subseteq \BbbR m are compact, convex sets that
contains an open neighborhood of the origin. A set \bfX \bfzero is defined as
a robust control invariant for xk+1= f (x, u, w), w \in \bfW if, for every
x \in \bfX \bfzero , there exists a u \in \bfU such that f (x, u, \bfW ) \subseteq \bfX \bfzero . Hence,
if the state is within \bfX \bfzero , any control action derived from the set
\bfU will always keep the state inside the robust control invariant set regardless of any disturbance in \bfW affecting the system.
The controller is separated in two different configurations, of-ten called the dual-mode controller. The idea is to design an "inner" and "outer" controllers, such that:
\bullet Inner controller: Operates when the state is inside the robust
in this set in spite of the external disturbance. If the inner con-troller is defined by a linear equation, such as u = Kx, then it must be guaranteed that the constraint is never violated; \bullet Outer controller: Operates when the state is outside the
in-variant set and steers the system to \bfX \bfzero .
The use of the inner and outer controllers are illustrated in Figure (2.5), where the blue region defines the robust invariant set \bfX \bfzero and the white region defines the state constraint \bfX . Thus,
while the states are outside the blue region, the outer controller is activated and steers the system to \bfX \bfzero . When it reaches the blue
re-gion, the controller is switched to the inner controller and, if the set is robust and invariant, any control signal calculated by the inner controller will keep the states inside \bfX \bfzero regardless of any external
disturbance.
Figure 2.5: Inner and outer controller concept.
Since the controller’s objective is to steer the state to the ori-gin or as close to it as possible due to disturbances, a fine solution for the inner controller is to be able to guarantee that xk+1 = wk.
2.3. Min-max Robust Model Predictive Control 19 linear gain, such that:
xk+1= Axk+ Buk+ wk
= Axk - BKxk+ wk
= (A - BK)xk+ wk
(2.19)
therefore, K must be chosen to ensure that with (A - BK) the state xk+1is close to wk.
For the outer controller, a fixed horizon feedback min-max MPC was proposed as the solution to steer the state to the robust control invariant set. At time k, let \{ wk+j\} represent all possible
realizations of the disturbance. Also, let \{ uk+j\} represent a control
action sequence related to a realization and \{ xk+j\} denote the
so-lution to:
xk+1= Axk+ Buk+ wk (2.20)
Then, the purpose of the min-max optimization is to calculate a control signal to minimize the cost function for the worst possible disturbance. Hence, the proposed optimization is:
\mathrm{m}\mathrm{i}\mathrm{n} \{ \^uk+j\} \mathrm{m}\mathrm{a}\mathrm{x} wk N - 1 \sum j=0 L(xk+j, uk+j) s.t. xk+j \in \bfX , j > 0
uk+j \in \bfU , j \geq 0
xN \in \bfX \bfzero
(2.21)
where N is the prediction horizon and the stage cost L(xk, uk) =
| | Qxk| | p+ | | Ruk| | p : \BbbR n\times \BbbR m \rightarrow \BbbR is a positive semidefinite
func-tion and | | . | | p denotes the p-norm of a function; the weights are
Q \in Rn\times n and R \in Rm\times m both used as tuning parameters for the
controller; the terminal constraint xN \in \bfX \bfzero is required to ensure
stability. The outer controller solution is acquired by solving this optimization problem over a moving horizon.
The Control Algorithm
The algorithm to control the system is as follow:
(a) Initialize xk = x(0), where x(0) is the initial state of the
sys-tem;
(c) If xk \in \bfX \bfzero then set uk = - Kxk. Otherwise, solve the
opti-mization problem (2.21) and obtain \^uk+j;
(d) Apply the first input ukof the optimal control sequence \^uk+j;
(e) Measure xkand return to Step (b).
Calculation of the Robust Control Invariant Set
A set \bfX \bfzero is called robust control invariant set if, for a
predeter-mined control law uk= - Kxk, the system (2.17) satisfies the state
constraints and input constraints, and, for a disturbance wk\in \bfW :
(A - BK)\bfX \bfzero + \bfW \subset \bfX \bfzero (2.22)
Thus, for any state xkwithin the robust control invariant set,
the control law uk = - Kxk will maintain all subsequent states of
the uncertain system inside \bfX \bfzero .
Consider that a nominal system is given by:
zk+1= Azk+ Buk (2.23)
where zk\in \BbbR nis the state of the nominal system. The error between
the nominal state and the actual state is defined as:
ek= xk - zk (2.24)
The first step to derive \bfX \bfzero is done by iterating recursively
(2.24). Let the successor state xk+1be given by (2.19) and, similarly,
the successor nominal state be given by zk+1 = (A - BK)zk, the
future prediction errors are: ek+1= xk+1 - zk+1 = (A - BK)xk+ wk - (A - BK)zk = (A - BK)(xk - zk) + wk = (A - BK)ek+ wk ek+2= xk+2 - zk+2 = (A - BK)2ek+ (A - BK)wk+1+ wk .. . ek+i= (A - BK)iek+ i - 1 \sum j=0 (A - BK)jwk+j (2.25)
2.3. Min-max Robust Model Predictive Control 21 Therefore, the disturbance is accumulated throughout the fu-ture prediction errors. Using this result, the prediction error is lim-ited by:
\bfE = \bfW \oplus Ak\bfW \oplus \cdot \cdot \cdot \oplus Ai - 1k \bfW = i - 1
\sum
j=0
Ajk\bfW (2.26)
where \oplus is the Minkowski sum operator1, and A
k = A - BK. The
importance of the set E follows from the property:
zk\in \bfX \ominus \bfE (2.27)
where \ominus is the Minkowski difference operator2.
This property implies that if the nominal state belongs to a set that, for the given limits of the prediction error and the accumulated disturbances, never exceeds the state constraints for any control law uk = - Kxk, the actual state is within \bfX . Thus, the robust control
invariant set \bfX \bfzero is calculated by using the result from (2.26).
The next section shows the solution proposed by Kerrigan and Maciejowski [11] which extends the algorithm of Scokaert.
2.3.2 Feedback min–max model predictive control using a sin-gle linear program: robust stability and the explicit solu-tion
The stage cost defined by Scokaert separated the controller’s action into two different control laws: the inner and outer controller. When analyzing the stage cost of the RMPC proposed by Scokaert, it is clear that, while the outer controller optimizes the prediction us-ing a stage cost | | Qxk| | p+ | | Ruk| | p, the inner controller uses a linear
control law uk = - Kxk, hence the stage cost could be interpreted
as L(xk, uk) = 0, because no optimization is performed. Thus, this
concept is summarized as:
L(xk, uk, rk) =
\Biggl\{
| | Qxk| | p+ | | Ruk| | p if xk\not \in \bfX \bfzero
0 if xk\in \bfX \bfzero
(2.28) where the outer controller is activated when the state xkis outside
the robust control invariant set \bfX \bfzero and the inner controller when
the state is inside the robust control invariant set.
1A \oplus B := \{ a + b | a \in A, b \in B\} 2A \ominus B := \{ x \in \BbbR n| \{ x\} \oplus B \subseteq A\}
Using a discontinuous stage cost such as in (2.28) is problem-atic to implementation considering standard solvers. Thus, Kerrigan proposed a solution, which changes this stage cost to a continuous stage cost formulation that allows using smooth, convex program-ming solvers, while keeping the same robust stability of the system guaranteed by Scokaert’s solution. The new stage cost is:
L(xk, uk) = | | Q(xk - rk)| | p+ | | R(uk+ Kxk)| | p (2.29)
where rk \in \BbbR n is a reference trajectory defined as a decision
vari-able. The idea behind (2.29) is that deviations of the state trajectory xkfrom the robust invariant set \bfX \bfzero as well as deviations from some
control law uk = - Kxk are penalized, instead of penalizing the
deviation from the origin (as usually considered in state-space MPC formulations). The robust control invariant set is assumed as a new target set of the system, which leads the controller to optimize the reference trajectory rkin order to steer the system to \bfX \bfzero .
This new optimization problem calculates the control action, while keeping the reference trajectory inside the target set. Consid-ering the stage cost (2.29), the optimization is defined as:
\mathrm{m}\mathrm{i}\mathrm{n} \{ \^uk+j,rk\} \mathrm{m}\mathrm{a}\mathrm{x} wk N - 1 \sum j=0 L(xk+j, uk+j, rk+j) s.t. xk+j\in \bfX , j > 0
uk+j\in \bfU , j \geq 0
xN \in \bfX \bfzero ,
rk+j\in \bfX \bfzero , j \geq 0
(2.30)
This new solution proposed by Kerrigan is a direct extension to the min-max RMPC of Scokaert, which uses a different approach but essentially uses the same concept. Both methods are guarantee-ing robustness by calculatguarantee-ing a optimal control law considerguarantee-ing the worst possible disturbances that might affect the system. This ap-proach could be quite conservative in cases that the worst-case sce-nario is uncommon for normal operating conditions, but for critical conditions these techniques guarantee that the system will never violate any hard constraints.
The Control Algorithm The new algorithm is:
2.4. LMI-based Robust Model Predictive Control 23 (a) Initialize xk = x(0), where x(0) is the initial state of the
sys-tem;
(b) Calculate the predictions \^xk+j of the system;
(c) Solve the optimization problem (2.30) and obtain \^uk+j;
(d) Apply the first input ukof the optimal control sequence \^uk+j;
(e) Measure xkand return to Step (b).
2.4 LMI-BASED ROBUST MODEL PREDICTIVE CONTROL
Although inspired by a min-max RMPC, the approach pre-sented in this section uses a distinct solution to robust MPC, solely based on linear matrix inequalities. Therefore, it is important to pro-vide a brief definition of LMI as a tool for control optimization and what advantages does LMI provides for MPC applications.
2.4.1 Linear Matrix Inequalities
A linear matrix inequality is expressed by: F (x) = F0+
m
\sum
i=1
xiFi > 0 (2.31)
where Fi \in \BbbR n\times n are symmetrical matrices, xi \in \BbbR m are variables
and the inequality F (x) > 0 is used to inform that F (x) is a positive definite matrix [40].
Many of the existing MPC problems can be translated to the LMI framework, even nonlinear expressions such as quadratic in-equalities commonly used in control. This translation is performed by using mathematical results such as Schur complement, S proce-dure [41] and D-G scaling [42]. For example, given Q(x) = Q(x)T,
R(x) = R(x)T and S(x) matrices affinely dependent on x. Then,
using Schur complement the nonlinear inequality: R(x) > 0, Q(x) - S(x)R(x) - 1S(x)T > 0
and Q(x) > 0, R(x) - S(x)TQ(x) - 1S(x) > 0 (2.32) is expressed as the LMI
\biggl[ Q(x) S(x)
S(x)T R(x)
\biggr]
The function F (x) defined in (2.31) is affine in x and there-fore is a convex function. About the convexity of LMI, Boyd [40] observed that
"The LMIs that arise in system and control theory can be formulated as convex optimization problems that are amenable to computer solution."
This observation leads to the conclusion that much of the con-trol theory problems could also be solved using the LMI formulation. Also, many improvements are being made towards the solution of convex optimization problems efficiently with the development of interior-point algorithms (see [43] and [44] for more references). These improvements enable much of the existing MPC theory to be rewritten as LMI using efficient software tools.
The first method was proposed by Kothare [15] and is dis-cussed in the next section.
2.4.2 Robust constrained model predictive control using lin-ear matrix inequalities
Using LMIs are also appealing for robust MPC because they allow to explicitly incorporate model uncertainties and therefore the plant model is more reliable. The model is given by:
xk+1= Aixk+ Biuk
[Ai Bi] \in \Omega = Co\{ [A1 B1], . . . , [Ap Bp]\}
= p \sum j=1 \lambda k[Aj Bj] (2.34)
where x \in \BbbR n, u \in \BbbR m are state and input vectors at time k, p
is the number of vertex of the polytope defined by \Omega , Co is the convex hull and \lambda \in \BbbR p is an unknown but bounded time-varying
parameter with\sum p
j=1\lambda k= 1.
Control Law
The proposed algorithm modifies an unconstrained worst-case infinite horizon MPC to a reduced linear objective minimization problem, where this linear objective is written as an LMI. After changing the optimization, constraints are added using LMIs and
2.4. LMI-based Robust Model Predictive Control 25 the RMPC solution is ready. Thus, consider the optimization prob-lem:
\mathrm{m}\mathrm{i}\mathrm{n}
u [A\mathrm{m}\mathrm{a}\mathrm{x}iBi]\in \Omega L\infty L\infty = \infty \sum j=0 [xTk+jQ1xk+j+ uTk+jRuk+j] (2.35)
Consider the Lyapunov function V (x) = xT
kP xk, P > 0 with
V (0) = 0. The system is stabilized if the Lyapunov function V (x) > 0and \.V (x) < 0. Suppose V (x) satisfies the inequality:
V (xk+j+1) - V (xk+j)
\leq - [(xk+j)TQ1(xk+j)
+ (uk+j)TR(uk+j)]
(2.36)
Then, the robust stability requires that, for k \rightarrow \infty , x\infty = 0
and therefore V (x\infty ) = 0. Summing (2.36) from j = 0 to j = \infty
results in:
- V (xk) \leq - L\infty (2.37)
This results in the following property: \mathrm{m}\mathrm{a}\mathrm{x}
[AiBi]\in \Omega
L\infty \leq V (xk) (2.38)
which gives an upper bound to the robust objective function. This upper bound is important because it enables to guarantee robust performance with the use of an LMI optimization problem by re-defining the cost function L\infty to a minimization that satisfies the
following performance requirement:
V (xk) = xTkP xk< \gamma (2.39)
where \gamma is a suitable nonnegative coefficient to be minimized that is characterizing the upper bound. Thus, a state-feedback control law uk+i = - Kxk+iis calculated and the state-feedback matrix K
is given by:
K = - Y Q - 1 (2.40)
where K is determined such that it minimizes the upper bound V (xk), which is done defining Q > 0 and Y as the solution of the
\mathrm{m}\mathrm{i}\mathrm{n}
\gamma ,Q,Y\gamma
s.t. \Biggl[ 1 xTk xk Q \Biggr] \geq 0 \left[
Q \ast \ast \ast
AiQ + BiY Q 0 0 Q1/21 Q 0 \gamma I 0 R1/2Y 0 0 \gamma I \right] \geq 0, i = 1, 2, . . . , p. (2.41)
where * transposes the symmetric element in the matrix, Q1 and
R are the weight matrices of the objective function and p is the number of vertex from a polytope \Omega representing all possible com-binations of [AiBi]
The robust stability is guaranteed because V (xk) = xTkP xk,
where P = Y - 1 is the optimal solution, is a strictly decreasing
Lya-punov function for the closed loop [18].
Derivation of the optimization (2.41):
Consider that V (xk) = xTkP xk< \gamma is equivalent to:
\mathrm{m}\mathrm{i}\mathrm{n}
\gamma ,Q,Y\gamma
s.t. xTkP xk<\gamma
(2.42)
Defining Q = \gamma P - 1 and using the LMI property (2.32), this is equivalent to:
\mathrm{m}\mathrm{i}\mathrm{n}
\gamma ,Q,Y\gamma
s.t. \Biggl[ 1 xT k xk Q \Biggr] \geq 0 (2.43)
Now assuming the control law uk = Kxkand the inequality
(2.36) becomes:
xTk+j+1((Ak+j+ Bk+jK)TP (Ak+j+ Bk+jK) - P
+ KTRK + Q1)xk+j+1 \leq 0
2.4. LMI-based Robust Model Predictive Control 27 This inequality is satisfied if, for all j \geq 0:
(Ak+j+Bk+jK)TP (Ak+j+Bk+jK) - P +KTRK +Q1\leq 0 (2.45)
Substituting P = \gamma Q - 1, Q > 0, Y = KQ in (2.45) and
multi-plying both sides of the equation by Q, the result is the LMI: \left[
Q \ast \ast \ast
AiQ + BiY Q 0 0
Q1/21 Q 0 \gamma I 0 R1/2Y 0 0 \gamma I
\right]
(2.46)
Input and State Constraints
Considering the optimization problem in (2.41), the input and state constraints are given by:
\Biggl[ u2max Y YT Q \Biggr] \geq 0 (2.47) \Biggl[ x2maxI \ast C(AiQ + BiY ) Q \Biggr] \geq 0 (2.48)
where | | uk+j| | 22 \leq u2max is the bound for uk+j and | | xk+j| | 22\leq x2max
is the bound for xk+j. The derivation of these LMI follows the same
logic from before. The Control Algorithm
The algorithm to control the system is as follows:
(a) Initialize xk = x(0), where x(0) is the initial state of the
sys-tem;
(b) Calculate the predictions \^xk+j of the system;
(c) Solve the optimization problem (2.41) with constraints (2.47) and (2.48) to obtain Q and Y ;
(d) Calculate K = - Y Q - 1 and set the control action u
k= Kxk;
(e) Apply the control uk;
The algorithm proposed by Kothare requires the quadratic sta-bilizability, xTP x, of the system to find robustly the optimal control.
Cuzzola mentions in [16] that this approach is conservative because the quadratic stabilizability of the system is required, and presents an improved technique that reduces conservativeness of the algo-rithm.
2.4.3 An improved approach for constrained robust model pre-dictive control
The algorithm suggested by Cuzzola is different from the LMI-based RMPC from Kothare in the sense that, instead of using a sin-gle constant quadratic Lyapunov function V (xk) to synthesize the
problem, he proposes a parameter-dependent Lyapunov function, Vk(xk), for each vertex of the uncertainty polytope, thus reducing
the conservativeness. This new approach stabilizes the system for all admissible uncertain parameters \lambda if guaranteed that Vk(k) is
a strictly decreasing function for all possible \lambda that describes the uncertain system.
Considering that Pk = \sum pj=1\lambda kPj, where p corresponds to
the number of vertices in the polytope, the Lyapunov function Vkis
a quadratic parameter-dependent function given by:
Vk(xk) = xTkPkxk (2.49)
The goal is of this LMI-based controller is similar to the dis-cussed in the previous section: defining an optimal feedback control law uk+j = Kxk+j that stabilizes the system by minimizing an
up-per bound that is guaranteed to satisfy the inequality:
Vk(xk) = xTkPkxk< \gamma (2.50)
where V (xk)is a parameter-dependent Lyapunov function.
Control Law
This approach finds a state feedback control law synthesized from a LMI optimization problem considering the same infinite hori-zon MPC problem (2.35) and minimizing the upper bound \gamma .
The difference from Kothare’s algorithm is that the LMIs from (2.41) are extended for each vertex of the polytope, considering
that:
2.4. LMI-based Robust Model Predictive Control 29 and:
Pk= Q - 1k (2.52)
Thus, the system described by (2.34) is robustly stabilizable if there exist p symmetric matrices Qj with j = 1, . . . , p and a pair
of matrices (Y, G) that minimizes an upper bound \gamma by satisfying the following optimization:
\mathrm{m}\mathrm{i}\mathrm{n} \gamma ,G,Y,Qj \gamma s.t. \Biggl[ 1 xT k xk Qj \Biggr] \geq 0 \left[ G + GT - Q
j \ast \ast \ast
AjG + BjY Qj 0 0 Q1/2G 0 \gamma I 0 R1/2Y 0 0 \gamma I \right] \geq 0, \Biggl[ u2 max Y YT G + GT - Qj \Biggr] \geq 0 \Biggl[ x2 maxI \ast C(AiQ + BiY ) G + GT + Q - j \Biggr] \geq 0 j = 1, 2, . . . , p. (2.53)
where the matrices (Y, G) are the solution from (2.53), G+GT > Q j
and the feedback control law is defined by uk = - Kxk, with K =
- Y G - 1. Recalling that P
j = Q - 1j , the problem (2.53) guarantees
that the system is robustly stabilizable if Vk(xk) = xTkPkxkis strictly
decreasing. The derivation of (2.53) is similar to (2.41), thus it is omitted.
The Control Algorithm
The algorithm to control the system is as follow:
(a) Initialize xk = x(0), where x(0) is the initial state of the
sys-tem;
(c) Solve the optimization problem (2.53) to obtain Qj, G and Y ;
(d) Calculate K = - Y G - 1 and set the control action u
k= Kxk;
(e) Apply the control uk;
(f) Measure xkand return to Step (b).
2.5 TUBE-BASED ROBUST MODEL PREDICTIVE CONTROL
A different approach is proposed by Langson in [22], which introduces the notion of tubes to create a new strategy for RMPC. Basically, the tubes are sequences of sets that define all the possi-ble states and input trajectories, that are induced by the system’s dynamics and bounded uncertainties, given a control law.
Figure 2.6: State trajectories for different control policies.. For instance, consider the two scenarios shown in Figure (2.6), where it is illustrated the state trajectories that arise from three dif-ferent disturbances applied to the system:
\bullet The left scenario shows the state trajectories xiwhen an
open-loop control law is used, for each disturbance applied to the system. The set composed by all possible values of xiat instant
k is called \bfX \bfk . Thus, as the tube defines all sequences that
compose the possible state trajectories, the tube for the open-loop control law is specified by \{ \bfX \bfone , \bfX \bftwo , \bfX \bfthree \} .
\bullet However, if the control law is changed to a feedback policy, as shown in the right scenario from Figure (2.6), the tube \{ \bfX \bfone , \bfX \bftwo , \bfX \bfthree \} changes, because the state trajectory for three
2.5. Tube-based Robust Model Predictive Control 31 possible disturbances alters the sequences that define each \bfX \bfk .
The main idea is to design an RMPC controller capable of controlling the uncertain system by defining a RMPC control law, ensuring that all realizations of the state trajectory lie inside a pre-defined tube.
The tube-based RMPC presented in the next section is de-tailed by Mayne in [4], which is based on the article introduced by Langson [22].
2.5.1 Robust model predictive control using tubes
Mayne used the notion of tubes to guarantee that state and control constraints are satisfied for every realization of the distur-bance sequence. However, the solution of the exact tube is quite complex, so it is replaced by approximating the exact tube to a outer-bounding tube.
Basically, the center of the outer-bounding tube is optimized using conventional MPC with tighter constraints on the nominal sys-tem, reducing the extent of the tube. Also, a local feedback con-troller is applied to drive all trajectories of the uncertain system to the central trajectory [4].
Control Law
Consider the following system:
xk+1= Axk+ Buk+ wk (2.54)
where x \in \BbbR n, u \in \BbbR mare state and input vectors at time k. Finally,
w \in \bfW \subset \BbbR n is an unknown disturbance bounded by the compact and convex set \bfW . Hard constraints are assumed for state, input and disturbances, assuming a set \bfU as convex compact polytope and \bfX a convex closed polytope.
xk\in \bfX , uk\in \bfU , wk\in \bfW (2.55)
Then, let the nominal system be given by:
zk+1= Azk+ Bvk (2.56)
where zk\in \BbbR nand vk\in \BbbR mare the nominal state and input vectors.
of the constraints \bfX and \bfU to guarantee that, for all disturbances, the actual state remains inside the center trajectory. These tighter constraints are defined by:
zk\in \bfZ =X \ominus \bfX \bfzero
vk\in \bfV =U \ominus K\bfX \bfzero
(2.57) in which \bfX \bfzero \subseteq \bfX is a robust control invariant set and K a state
feedback gain. The satisfaction of the state constraints at the end of the horizon N is guaranteed if a terminal constraint \bfZ \bff is also
satisfied by the nominal system:
zN \in \bfZ \bff \subseteq \bfZ (2.58)
In Figure (2.7) is shown an example of a tighter constraint. The inner region in green defines a tighter constraint \bfZ , that is re-duced to a set that, for every admissible disturbance wk, the actual
state of the system will remain inside the set \bfX , illustrated by the outer region with red line, if the nominal state zk\in \bfZ .
Figure 2.7: Illustration of a tighter constraint.
Thus, the tube-based RMPC objective is to find an optimal nominal trajectory, which is done by optimizing a finite horizon op-timal control problem in the form:
\mathrm{m}\mathrm{i}\mathrm{n} \{ \^vk+j\} N - 1 \sum j=0 L(zk+j, vk+j) + Vf(zN) s.t. zk+j \in \bfZ , j > 0 vk+j\in \bfV , j \geq 0 zN \in \bfZ \bff (2.59)
