Instituto Tecnológico y de Estudios Superiores de Monterrey

Instituto Tecnológico y de Estudios Superiores de Monterrey

Campus Monterrey

School of Engineering and Sciences

An MPC/LQR Controller for LPV Systems with Scheduling Parameter Prediction and Terminal Ellipsoid Set

A thesis presented by

Daniel Orlando Rodríguez Guevara

Submitted to the

School of Engineering and Sciences

in partial fulfillment of the requirements for the degree of Master of Science

In

Engineering Science

Monterrey Nuevo León, December 3^rd, 2020

With much love and affection to my mother, my father and my little brother who always support every decision I make and give me a helping hand every time I need it. To my girlfriend Sharon, who always listen to me and help me to be a better person everyday, for being my motivation and strength to finish this important accomplishment in my life.

Acknowledgement

I would like to express my deepest gratitude to Tecnologico de Monterrey, for giving me the opportunity of studying a Master degree. For the tuition support given to me along these 2 years and for all the tools and the facilities which made my Master studies possible.

Also, to my thesis committee, specially to my thesis advisor Dr. Antonio Ramón Xicoténcatl Favela Contreras who always helped me to develop my research work and support me with his knowledge and advices.

Last, but not least, I would like to thank CONACyT for giving me an economic support throughout these 2 years in order to focus on my research work.

by

Daniel Orlando Rodríguez Guevara

Abstract

A LQR-Model Predictive Control (MPC) algorithm for Linear Parameter Varying (LPV) systems is developed herein. The proposed algorithm applies a MPC approach whenever the system is outside a terminal ellipsoid defined as a stability zone where a LPV-LQR feedback gain is implemented to reach the equilibrium point. In order to cope with the uncertainties of the future values of the scheduling parameters, a recursive least squares algorithm is performed before the prediction of future states is made in order to be suitable and feasible for a more simple online optimization problem.

Quadratic stability is ensured by the means of working with the LPV system as an uncertain parametric system with the uncertainty being the error of the recursive least squares prediction and considering the discrete algebraic Riccati Equations in a Linear Matrix Inequality form. This method also includes terminal set attraction to steer the predictions of the states into the terminal ellipsoid. This thesis also includes two successful simulations of this technique in order to show the benefits and applications of the algorithm.

INDEX

Executive Summary ···5

1- Introduction···8

1.1 Introduction to Recursive Least Squares algorithms ···8

1.2 Introduction to Linear Parameter Varying Systems ···9

1.3 Introduction to Optimal Control and Linear Quadratic Regulator ···14

1.4 Introduction to Model Predictive Control ···18

1.5 Justification ···18

1.6 Hypothesis ···18

2- Background ···19

2.1 LPV modeling and optimal control design ···19

2.2 Model Predictive Control approaches ···21

2.3 LPV-MPC approaches ···23

2.4 Terminal sets in MPC ···27

2.5 Quadratic stability through LMI in LPV control and MPC ···29

2.6 Limitations ···31

3- The LQR-MPC for LPV systems with parameter prediction and terminal ellipsoidal sets ···32

3.1 LPV formulation ···32

3.2 Parameter prediction using recursive least squares ···33

3.3 MPC formulation for LPV systems and a definite prediction horizon ···39

3.4 Quadratic Stability in LPV-MPC ···45

3.5 Attraction sets and Terminal Stability Set for LQR control ···49

4 - Simulation tests and results for the proposed MPC/LQR-LPV approach ···55

4.1 Semi-Active Suspension of a Quarter-Car ···55

4.2 Two Coupled-Nonlinear Tanks system ···66

5- Conclusions and Future Work ···76

5.1 Conclusions ···76

5.2 Future Work ···77

References ···79

LIST OF FIGURES

Figure 1.- Flowchart of the recursive least squares algorithm for scheduling parameter

prediction. ···40

Figure 2.- Flowchart of the MPC-LPV using RLS scheduling parameter prediction ···45

Figure 3.- Flowchart of the LQR/MPC-LPV algorithm with scheduling parameter prediction and quadratic conditions with attraction sets and terminal set ···54

Figure 4.- Schematic of a Quarter Car Semi Active Suspension System ···56

Figure 5.- Road disturbance ···57

Figure 6.- Output response (Acceleration of the chassis) for all three different approaches ····59

Figure 7.- Output Response (Acceleration of the suspension unit) for all three different approaches ···59

Figure 8.- Input to the system (Force exerted by the actuator) ···60

Figure 9.- Scheduling parameter evolution (Speed differential between chassis and suspension unit) ···61

Figure 10.- Scheduling parameter comparison ···62

Figure 11.- State 1 (Position of the chassis) ···63

Figure 12.- State 2 (Velocity of the chassis) ···63

Figure 13.- State 3 (Position of the suspension unit) ···64

Figure 14.- State 4 (Velocity of the suspension unit) ···64

Figure 15.- Two Coupled-Nonlinear Tanks System designed by (Apkarian, 1999) ···67

Figure 16.- Output Response (Height of liquid in Tank 2) ···69

Figure 17.- Voltage Input supplied ···70

Figure 18.- Scheduling parameter evolution ···71

Figure 19.- Scheduling parameter evolution ···71

Figure 20.- State one evolution (height of liquid in tank 1) ···72

Figure 21.- State two (height of liquid in tank 2) ···73

Figure 22.- Scheduling parameter comparison ···74

Figure 23.- Scheduling parameter comparison ···74

ρ₁ ρ₂

ρ₁ ρ₂

LIST OF TABLES

Table 1.- Variable Definition for Semi-Active Suspension of a Quarter-Car ···57

Table 2.- Performance Comparison of the algorithms for example 1 ···65

Table 3.- Variable definition for the two nonlinear-coupled tanks system ···68

Table 4.- Performance Comparison of the algorithms for example 2 ···73

Executive Summary

This research work presents a novel strategy of a Linear Quadratic Regulator- Model Predictive Controller (LQR-MPC) for Linear Parameter Varying (LPV) systems.

This approach ensures Quadratic Stability at every iteration step, while maintaining good performance due to the inclusion of terminal attraction sets in order to steer the states into a desired final setpoint.

Nowadays, industrial processes need an accurate control system in order to attain to the quality and safety specifications of their products. However, achieving this is not a simple task, due to the nonlinear dynamics that are often found in this kind of processes. Usually, these processes are modeled as a linear system and controlled by a linear controller, which results in a poor performance or non-optimal control results.

On the other hand, some nonlinear models have been implemented along nonlinear controllers with good results, but are often limited to slow processes due to the computational effort needed to compute a nonlinear optimal control law.

Model predictive control (MPC) is a control optimization strategy based on the prediction of the behavior of the states of a system based on a model of the process, then, an optimization process is performed in order to minimize a cost function while complying with constraints in order to reach stability in the system. MPC is not a new technique, in fact, it started in 1980 with the introduction on Dynamic Matrix Control (DMC) (Cutler and Ramaker, 1980) and rapidly gained popularity due to its tunable performance between control speed and actuator energy consumed; as well as the inclusion of constraints. Nowadays, several commercial MPC solutions for industry exist, with excellent results in process control that can be accurately represented by linear models like: AspenTech DMCplus, RMPCT by Honeywell and SMOC by Shell.

(SK Lahiri, 2017).

Thus, when the system is not suitable for being represented by a linear model, several solutions have been proposed. Nonlinear Model Predictive Control (NMPC) is a nonlinear version of the traditional MPC, where the system is represented by nonlinear differential equations and the prediction of the behavior of the system is also treated as being nonlinear. Several NMPC have been proposed in the literature (Olivieri et. al.,

2017; Diangelakis et. al. 2018; Metzler et. al, 2018). All of these algorithms proved to be efficient in some applications, however, the optimization time is large and therefore, not suitable for fast real-time implementations. The solution to this limitation is to compute an explicit optimization solver, which solves every possible optimization problem previously offline and only perform a look-up procedure online. However, as the number of states increases, the size of the look-up tables increases resulting in the requirement of large data storage. Other solutions to NMPC have been suboptimal methods in order to increase solver speed (Neunert, 2016; Stella, 2017). Nevertheless, the problem must remain unconstrained and may result into non-desirable performance.

In order to diminish the computational load of NMPC several methods have been proposed to make systems more suitable for typical MPC approaches. One of them is represent the non-linear system as a Linear Parameter Varying (LPV) system. In which the state-space model matrices are linear but are dependent on a scheduling variable . However, the main limitation of this strategy is that the prediction of the system states becomes uncertain because the future values of are not available at each time step.

To solve this problem, several authors have chosen to maintain the value of the scheduling parameter frozen over the prediction of the future states (Morato et. al, 2018;

Alcalá et. al, 2019; Calderón et. al. 2019). This may results into inaccurate predictions for the system dynamics, mostly when the scheduling parameter may vary very fast.

Another strategy is to consider the rate of change of the scheduling parameter to be in a range . Therefore, the future states consider the admissible rate of change at each iteration step in order to predict the behavior of the system at future steps (Bumroongsri, 2014; Huang et. al, 2014; Ding et. al, 2016; Longge et al, 2017; Hu et. al, 2020). However, as the future states can be inside a region where an infinite set of states can be found, the worst case analysis is the one that is considered in this approaches, and therefore, the optimization problem becomes a min-max optimization problem in order to minimize the cost while maximizing the variation of the scheduling parameters. This leads to a conservative performance and a high computational load in order to ensure robustness to the system.

In order to overcome some limitations of the previously mentioned approaches, this thesis present a novel LQR-LPV controller for LPV system, with scheduling

ρ ρ

{Δρ

, Δρ

}

parameter prediction using an online recursive least squares algorithm. LMI conditions are also included into the optimization problem in order to ensure quadratic stability in the system, as well as the addition of terminal attraction sets in order to improve the speed to steer the states to a desired setpoint to overcome the conservative performances that are present in other applications.

The main contributions of this thesis are: 1) The LPV system formulation into a MPC approach considering the availability of the future scheduling parameters in a compact matrix form to treat the future states and inputs in vector sets resulting suitable for optimization. 2) The inclusion of a modified recursive least squared (RLS) algorithm improve the behavior of the predictions and the control performance. 3) The inclusion of terminal attraction sets and quadratic stability LMI’s to a LPV-MPC algorithm with scheduling parameter prediction. This is proposed in order to cope with the prediction error and still guarantee quadratic stability without the loss of control performance and relax the computational complexity that robust LPV-MPC approaches have. Finally, a terminal region around the desired setpoint is defined in which the control strategy switch from MPC to a LPV-LQR in order to reach the setpoint faster than it has been reported in previous works in the literature.

Chapter 1- Introduction

1.1 Introduction to Recursive Least Squares algorithms

Predicting the future behavior of systems and signals have been always of great interest in many professional areas. When previous data is available, usually models are build based on that data using different strategies. One of those strategies is the Least Squares (LS) method, which was introduced in the early XIX century by Carl Friedrich Gauss and demonstrated its advantages in the Gauss-Markov Theorem. Its main characteristic is that it is a linear unbiased estimator and has a low sampling variance when compared with other estimators of the same kind.

The LS approach have several applications in the academia, one of the most important is system identification, in which, by a set of input and output data, a model can be defined through the minimization of sum of the squared errors that the defined model has over the original data. Through this thesis work only linear LS is considered in which the parameters to be solved are linearly independent.

The LS strategy in system identification has been widely used and different models can be defined by using this approach like state-space (SS) models, Box- Jenkins models, Autoregressive-moving-average model (ARMAX) and Autoregressive Exogenous Model (ARX) (The LS application to this kind of models and other can be found in the book Filtering and System Identification: A Least Squares Approach by Verhaegen and Verdult, 2007). The ARX is useful for representing a linear dynamic system in discrete time and define its behavior as a function of both the previous inputs and outputs of the system in the following form.

Previous equation can be extended to with nm being the maximum value between and as the following:

y_k = a₁y_k−1+ a₂y_k−2+ … + a_nay_k−na+ b₁u_k−1+ b₂u_k−2+ … + b_nbu_k−nb+ e_k (1)

k = nm, …, N

nb na

With being a matrix containing the previous values of inputs and outputs and being the vector containing the values of all and . Therefore, the goal of the LS algorithm is to find the solution for all and that minimizes the cost function

which represents the sum of the squared errors.

However, the LS approach have some drawbacks. If there is only a few data for the formulation of the ARX model the system may be inaccurate for some operational points where no data was available during the modeling. Another downside is that systems may change over time due to actuator deterioration, changes on the external operating conditions, electronic measurement devices malfunction, etc. Also, previous data for system is not available and can be only measured online while the process is running. Therefore, the necessity of the adjustments of the parameters and becomes necessary as the data obtained online changes its behavior.

To overcome this limitations, a recursive approach can be used and it is known as recursive least squares (RLS). This method was also proposed by Gauss but was used until the mid XX century (Plackett, 1950). This approach used a filter-like algorithm in order to adjust the and parameters based on the prediction error find in the previous approximation and the new data acquired at each time step. A more detailed explanation of LS and RLS algorithms can be find in Section 3.2 in order to estimate the future scheduling parameters.

1.2 Introduction to Linear Parameter Varying Systems

Most of real-life industrial processes present nonlinear behavior in their normal operating conditions, some of them, can be modeled as linear systems in order to design a linear controller to be used on them. However, there are some more complex systems that cannot be modeled by linear models and need to be modeled by using

Y_N = ξ_NΩ_N + e_N (2)

ξ

Ω

a

b

a

b

J(N ) = ∑

k=nm

e

W

e

a

b

a

b

nonlinear models. Thus, having a nonlinear system often requires the design of a nonlinear controller, which makes the control design a more difficult task.

Often, nonlinear controllers tend to be complex to design and several modeling solutions have been developed in order to represent the nonlinear systems as linear systems without losing the accuracy of the nonlinear model. One of the solutions is to define a nonlinear model as a Linear Parameter-Varying (LPV) model.

LPV models are defined as “linear dynamical systems whose mathematical description depends on parameters that change values over time.” (Briat, 2015) A general LPV model represented in a discrete state-space form as the following:

In which all matrices , , and are linear, but depend on a scheduling variable which can vary at every time instant . Therefore, having a varying parameter embedded in a state-space representation, nonlinear behaviors can be represented as a set of linear systems in which the scheduling variable act as a switch between those linear systems. However, the accuracy of the LPV systems depend on the number of different values that the scheduling parameter can take and therefore, the number of linear systems contained on the set.

In order to approximate nonlinear systems more appropriately, the nonlinearities present on the process can be embedded by the scheduling parameter in order to represent the nonlinearities that can be present on the states . Therefore, the scheduling parameter can be defined as a function of the observable states at every time as the following:

When representing nonlinear systems in a LPV form where the scheduling variable is a function of the current states, the LPV system is defined as a quasi-LPV (qLPV) model. qLPV models represent nonlinear systems in a more accurate way, while

x(k + 1) = A(ρ(k))x(k) + B(ρ(k))u(k) (3)

y(k) = Cx(k) + Du(k)

A B C D

ρ(k) k

ρ(k) x(k) ρ(k)

k

ρ(k) = f (x(k)) (4)

preserving the linear form of the state-space representation; and also, the stability and robustness analysis for LPV systems can be extended to qLPV systems.

There exists four different main representations of LPV systems often found in the literature: 1) Generic LPV systems, 2) Polytopic LPV systems, 3) LPV systems in Linear Fractional Transformation Form, and 4) LPV systems in Input/Output Form. All of the previous 4 kinds of LPV representations are presented and described in the book Linear Parameter-Varying and Time-Delay Systems (Briat, 2015). The different representations of LPV systems are suitable for certain systems and more convenient in certain kind of process that is described for each kind in the next paragraphs.

Generic LPV systems are the most general kind of LPV representations, this kind of systems require no preprocessing to the scheduling parameter and the dependence of system matrices to the scheduling parameter can be represented in almost every form, for example, polynomial, exponential, rational, trigonometrical, etc. The only considerations made in order to build a generic LPV representation is that the scheduling parameter needs to be bounded, and often, the boundaries are defined by the physical constraints of the system states.

The main advantages of generic LPV systems is that they are suitable for almost every kind of system, due to the flexibility of the dependency of the scheduling parameter, however, this may became more complex at the moment of design an LPV controller. Also, since the generic form of this kind of LPV’s system make possible to generate multiple LPV representations for the same system, and therefore, choosing the best one is a task that still under investigation.

The second kind of LPV representation is the polytopic LPV system. Contrary to the generic representation, polytopic LPV’s require some a-priori modifications to the system in order to represent the system’s dynamics in order to represent the LPV variations of the systems as a Linear Time-Invariant (LTI) system with a set of variations that can be represented with a convex unit polytope, which vertex depend on the values of the scheduling variable, which is restricted to values

ρ ∈ [−1,1]

The polytopic LPV’s have the main disadvantage that not every system can be embedded into a convex polytope, and therefore, they cannot be represented in the polytopic LPV form. Also, encoding a system into a polytopic LPV representation may be difficult and require some experience on LPV systems. Besides, polytopic representation may increase the number of scheduling parameters needed for representing a system when compared to the generic LPV models.

Thus, polytopic LPV representations offer many advantages when dealing with LPV controllers design. Due to its convex formulation structure, stability and robustness conditions can be proved easily using convex optimization techniques. Also, for optimal control approaches, the optimization process for finding the optimal control action is done in a similar approach as in LTI systems. Therefore, even when the formulation may be more difficult that when using generic LPV systems, the controller design and the stability and robustness conditions are easier to run.

Another advantage of polytopic LPV systems is that they may be represented as uncertain LPV systems, in which the system dynamics are defined to be inside a convex polytope (not necessary a unit convex polytope). Therefore, in order to evaluate the system stability and design a controller, the system needs to be evaluated around its vertex. This kind of analysis of LPV systems have been widely used in recent control approaches that are described in section 2.3.

The third kind of LPV representation is the Linear-Fractional Transformation (LFT) form. In this kind of approach, LPV systems are represented as uncertain systems constructed by the interconnection of two systems, the known part of the system, which is usually constant and time-invariant, and the uncertain part of the system, where the LPV part is often represented. This kind of systems have been used widely for robustness analysis and to build robust controllers due to the uncertain part of the system being always present and with the availing of representing the dynamics of the system when noise or disturbances are present.

The necessity of LFT-LPV systems grew in order to develop robust optimal controllers, such as gain-scheduling

H

design of LPV robust optimal controllers. However, the uncertain part of this interconnected systems is difficult to model, and the scheduling parameter dependency is not immediate and requires a deep knowledge of the system that is being modeled.

Also, the formulation of the uncertain part is often complex and not suitable for complex optimization problems with constraints in which the running time is a critical aspect.

Finally, there are the LPV systems in Input/Output (LPV-I/O) form. In this kind of LPV representation, the dynamic of the system are represented as a function of the input, the output and the scheduling variable. Therefore, this kind of representations doesn’t use states as a description of the dynamics of the system which may be a limitation for certain standard control approaches. However, since the dynamics of the system are dependent on both input and output and the model considers a bounded variation of both of them, controllers based on LPV-I/O representations are well received in industrial applications where maintaining the input and the outputs in some operational range is a necessity.

LPV-I/O representations are a new emerging system identification strategy that is growing rapidly. Therefore, stability and robustness analysis still in investigation and several approaches mix the controller design and the stability analysis by using two different LPV representations (LPV-I/O and a different one). However, the LPV-I/O representation have proved to be efficient when representing nonlinear systems as LPV transfer functions, which can be useful in the design of several LPV controllers in the future.

As shown in this section, LPV models are useful to represent nonlinear dynamics of complex systems as linear systems with dependence of a scheduling parameter . Therefore, a linear parameter varying controller can be designed rather than a more complex nonlinear controller. In this section, four different LPV representations were mentioned. Along this research work, LPV models in generic form and in polytopic form are used in order to develop a MPC-LQR controller for LPV systems.

ρ

1.3 Introduction to Optimal Control and Linear Quadratic Regulator

In the past, classic controllers often rely on the correction of an error signal found in the process. Whenever an error appears in the system, a control action serves as an input to the plant in order to diminish that error. However, most of these controllers rely on a trial and error methodology based on the response of the system present against several input signals. Therefore, the tuning of this kind of controllers were usually done by empirical laws or iterative methods of analysis in order to find an acceptable performance that was defined only by the control engineer.

Although, since the necessity of more reliable and robust control algorithms arose on the industrial area, the optimal control theory started to arise. As stated by Donald E. Kirk on his book Optimal Control Theory, the objective of optimal control is to:

“determine the control signals that will cause a process to satisfy the physical constraints and at the same time minimize (or maximize) some performance criterion” (Kirk, 2004). With the availability of optimal control methods, industrial process had the advantage to include physical constraints into the control algorithm and also, the possibility to define a desired performance that was often based on the product quality requirements, the energy or resource saving and/or the actuator effort.

As mentioned before, the main advantages of optimal control is the mathematical definition of a desired control performance defined in a cost function . The objective of optimal control is to find a control action that minimizes the cost function, while dealing with defined constraints which are specified by the system dynamics and physical limitations. However, in contrast to classic controllers, optimal controllers need time to compute the optimal control action due to the solving of the optimization problem, and as more complex the system is, the more complex the optimization problem becomes.

However, with the new optimization techniques that are now available, this is no longer a problem for LTI systems.

One of the most popular optimal control methods is Linear-Quadratic (LQ) optimal control. It receives this name because the system dynamics are represented by a set of linear differential equations, for example as a linear state-space representation,

J

and the cost function is defined as a quadratic function. The Linear-Quadratic Regulator (LQR) is one of this kind of controllers. The LQR minimizes the value of the states of a system while minimizing the control actions held into the system over a finite or infinite time horizon. This is done by calculating an optimal state feedback gain that is used to calculate the control action .

The goal of the infinite LQR controller is to minimize the following cost function:

Where , and are weighing matrices in order to define the desired performance of the LQR controller. Therefore, the LQR is suitable for multiple linear systems using the same structure, but there needs to be adjusted to the desired performance for each one by tuning the weighing matrices. Other advantage of LQR controllers is that are designed to be stable due to the solution of the optimization problem is found by the Riccati equations. However, this is only true when the system is an LTI system, otherwise, stability conditions cannot be ensure and need to be checked at every simulation step. Another disadvantage of LQR controllers is that they do not consider constraints in the optimization process, which may became a limitation in some processes. LQR has been adapted in several approaches to be used in nonlinear systems that can be represented as linear systems (often by LPV representations), however, the stability and robustness analysis needs to be performed additionally in order to ensure both of them.

Although LQR has been used for a long time in industry, new research works using LQR controllers is becoming available. However, several new optimal control strategies have been develop more recently, since the computational resources available are considerably better than in the previous years, new complex optimal control algorithms like Model Predictive Control (MPC) have become more popular in recent years. On section 1.4, the MPC control strategy is described in order to give an introduction to this control approach.

K

u(k) = K

x(k)

J = ∑

k=0

[x(k)

Q

x(k) + u(k)

R

u(k) + 2x(k)

N

u(k)] (5)

Q

R

N

1.4 Introduction to Model Predictive Control

Model Predictive Control (MPC) is an optimal control strategy that consists on the prediction of the behavior of the future states in a prediction horizon based on a linear model of the system. The trajectories predicted by the model of the systems are optimized based on the set of future inputs to the plant subject to constraints in the outputs, the inputs and the rate of change of the inputs. After the optimization problem, the first optimal control action is introduced to the system, while the other ones are discarded and the prediction of the next future states is performed again.

MPC strategy started in 1980 with the introduction of Dynamic Matrix Control (DMC) (Cutler and Ramaker, 1980). Then it rapidly gain popularity in industrial chemical processes where systems can be well modeled with linear dynamics. One of the reasons that MPC has generated interest among both researchers and industries is the ability to deal with constraints not only for the current control action and its effects, but also, considering the future behavior of the system to remain within those boundaries.

However, the main limitation found in the MPC strategy is that if the system is complex, the optimization problem may take a long amount of time to obtain the optimal control action, and therefore, the controller is not able to determine the proper control action at every sampling time.

As stated before, the optimization problem is the key part of the MPC strategy, therefore, maintaining a simple cost function is an important part in order to solve the optimization problem in an adequate time. The general form of the cost function for MPC is similar to the LQR one:

Whenever the system is a linear system, the cost function is a Quadratic- Programming (QP) optimization problem subject to constraints that is only dependent

on the set of inputs . However, when the system

N

J

J = ∑

i=1

[x(k + i)

Qx(k + i) + u(k + i − 1)

Ru(k + i − 1)] (6)

[u(k), u(k + 1), …, u(k + N

− 1)]

cannot be fully represented by a linear model and instead, a nonlinear model is used, the cost function becomes nonlinear, which increases the optimization time exponentially by using typical MPC approaches.

Even with the long optimization problems that nonlinear MPC’s (NMPC’s) have inherently, several approaches have been developed, but most of them rely on offline methods, where the optimization problem for every admissible point is computed offline and stored in a lookup table. Then the online algorithm just search for the solution at every iteration step and apply it as input to the system. This kind of approaches offer a solution to the long online optimization time present in MPC’s when dealing with nonlinear systems. However, the time to compute the whole set of admissible solutions still very long and may take days or even more depending on the complexity of the systems. Also, the storage space needed to store the set of solutions may be of concern when using embedded controllers. In the future, as the computational power increases, NMPC may offer a suitable solution for real time systems.

In order to overcome the actual limitations of NMPC, several alternatives oriented into avoid the nonlinear representations of the system have been proposed. One of them is to represent the nonlinear system as an LPV system as the one described in section 1.2. Therefore, the cost function becomes a QP problem, but is not only dependent on the optimal control actions, but also to the future values of the scheduling parameter which are considered to be unknown. The unavailability of the future scheduling parameter is a problem to solve the optimization problem and therefore there exist several approaches in order to solve this limitation.

As shown along this section, the MPC strategy is useful when dealing with a constrained system, where an optimal solution needs to be found based on a specific performance index determined by the cost function. However, when dealing with systems that cannot be represented by linear models, the optimization time becomes a limitation for this strategy. Several solutions have been proposed, with all of them having its advantages and limitations. A more detailed description of this kind of MPC approaches and several recent works on this area is described in section 2.2 and 2.3.

1.5 Justification

Nowadays, the inclusion of an optimal reliable control algorithm is a necessity in almost every industrial process in order to ensure product quality while ensuring efficient resource management and meeting safety specifications of the process. Model predictive control has proven to be an efficient solution to deal with this kind of necessities, however, when the system is cannot be properly represented by a linear model, traditional MPC approaches may not deal properly with systems that have a short sampling time where an optimal solution is needed every fraction of a second.

Several solutions to MPC for nonlinear systems have been proposed, but execution times, performance quality and stability and robustness theory still a limitation for those approaches. Therefore, in order to overcome some of the limitations of model predictive control when dealing with nonlinear systems, this thesis presents an MPC/

LQR switched controller for LPV systems in order to deal with the nonlinear dynamics present in a variety of process. A prediction of the scheduling parameter behavior along the prediction horizon is done using a recursive least squares algorithm in order to simplify the optimization problem.

Stability conditions are also considered in the presented approach, taking advantage of the polytopic LPV structure in order to deal with the uncertainty of the prediction of the scheduling parameters and using the discrete algebraic Riccati equations. Also the inclusion of attraction sets and terminal sets in order to steer the states to the desired setpoint is considered in order to improve performance and to overcome the conservativeness given by the stability conditions.

1.6 Hypothesis

A MPC/LQR Controller for LPV systems with scheduling parameter prediction along the prediction horizon and, attraction and terminal ellipsoid sets can be developed in order to ensure quadratic stability while maintaining good control performance and short execution times in order to be suitable for real-life complex processes found in several industries.

Chapter 2 - Background

2.1 LPV modeling and optimal control design

Linear Parameter Varying modeling approaches have been increasing in recent research works due to the possibility to deal with nonlinear systems, or systems with discontinuities in their behavior as linear systems. However, there exist some limitations when using LPV systems and designing controllers based on this kind on models that still being researched in order to improve the performance and stability of this type of controllers. Along this section, several LPV approaches on both modeling of systems and control design are reviewed.

As mentioned in section 1.2, there are multiple ways of representing a system as an LPV systems, and choosing the best one is not a simple or unique process.

Bruzelius et. Al. presented LPV systems as a linear differential inclusion (LDI), which means that the LPV system covers a set of different trajectories including the one of the nonlinear system. Therefore, their goal is to minimize the influence of the scheduling parameter on those trajectories to obtain a closer performance of the LPV model and the nonlinear model using convex Linear Matrix Inequalities (LMI) and an optimization approach. The results showed better approximation to the nonlinear systems and proved to be suitable for convex optimization. (Bruzelius, Pettersson, Breitholtz, 2004).

Convex optimization has been an issue when using LPV models for optimal control approaches, and therefore, the design of LPV models suitable for convex optimization has been a popular research line. Sala presented a framework to transform the LDI’s to convex differential inclusions (CDI) for LPV systems in order to represent nonlinear systems with uncertainties. The results showed more accurate representation of nonlinear systems due to less conservative design on the modeling of uncertainties/

nonlinearities. However, the computational complexity increases in its quasi-convex generalization approach and high-order models this becomes an applicability issue (Sala, 2017). Several approaches have used a similar CDI representation with similar

results: (Ghasemi & Afzalian, 2018; Robles, Sala & Bernal, 2019; Sala, Ariño & Robles, 2019; Tan, Olaru, Roman & Xu, 2019)

LPV models have been widely used in optimal control methods in order to alleviate the computational effort needed to apply this kind of control methods on nonlinear systems. Robust controllers for LPV systems have been a major line of research. Márquez et. Al. presented a stable, non-quadratic Lyapunov functional controller for qLPV systems with LMI optimization for optimal performance. However, the control law and optimization problem is complex and require a large quantity of LMI and decision variables (Márquez, Guerra, Bernal & Kruszewski, 2016). A similar approach is presented by Gonzalez et. Al. in which a nonlinear system is modeled as a LPV representation with singular descriptor in order to consider both algebraic constraints and their differential behavior. Then a gain-scheduled (GS) suboptimal controller is designed in order to enhance performance and reduce computational complexity (Gonzalez, Estrada-Manzo & Guerra, 2017). Other optimal LPV controllers have been presented in the following research works (White, Zhu & Choi, 2016; Ilka &

Vesely, 2017; Koelewijn, Cisneros, Werner & Toth, 2018; He, Zhu, Al-Jiboory, Swei &

Su, 2018; Alcala, Puig, Quevedo & Escobet, 2018).

Convex observers for LPV systems are also an important part of LPV control. In many systems state variables may not be available at every time instant, and therefore, should be estimated using an observer. Houimli et. Al. presented an Adaptive LPV observer using polytopic representation for fault detection and state estimator. The proposed observer provides adaptive response for state estimation based on quadratic/

polytopic stability conditions. The adaptive observer proved to be efficient and to track the real states in a short amount of time. (Houimli, Bedioui & Besbes, 2018). Similar approaches have been developed for both fault detection and state estimation, some of them are the following: (Chandra, Alwi & Edwards, 2017; Chen, Edwards & Alwi, 2019;

Hamdi, Rodrigues, Mechmeche & Benhadj, 2019; Bernardi & Adam, 2020)

As reviewed throughout this section, the LPV systems offer a solution to alleviate the computational complexity of control algorithms for nonlinear system. However, using

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ℋ

LPV representations results into other limitations like building an exact representation and stability and robustness analysis. Also, the several possible LPV representations for nonlinear systems need to be defined carefully, specially when an optimal control approach is being designed because convexity of the solution may be a necessity for the correct behavior of the control strategy.

2.2 Model Predictive Control approaches

As previously described, Model Predictive Control (MPC) is an optimization control strategy based on the prediction of the behavior of the future dynamics of a system and find the set of future control actions in order to optimize a desired performance subject to constraints. Since its inception in 1980 by Cutler and Ramaker in their research work on Dynamic Matrix Control (DMC), MPC has became a popular control strategy for systems where the constraints need to be considered over a prediction horizon . Several works have been proposed and developed since then and they are mentioned and described throughout this section.

Several MPC approaches have been proposed, one of them was presented by Alamir in his book "A pragmatic story of model predictive control: self-contained algorithms and case-studies” (Alamir, 2013) and in previous research papers. In this approach, the prediction of the future states and outputs is performed using a matrix form to store all the future values and perform and easier optimization process to obtain the optimal set of inputs. The precision of this approach depends on the accuracy of the model with respect to the real system. However, the complexity of building the matrix increase with the size of the prediction horizon . Also, if the system is not well modeled, the prediction of the states and outputs deviates from the real behavior of the system along the prediction horizon.

There exist other MPC’s representations, one of them is the Discrete Model Predictive Control (DMPC) presented by Wang in the book “Model Predictive Control System Design and Implementation Using MATLAB®” (Wang, 2009). In this strategy, the behavior of the system dynamics is estimated using Laguerre functions, in order to

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have the values of the future states and outputs as a function of the future set of inputs and then, solve the optimization problem using a Hildreth’s Quadratic Programming (QP) solution subject to constraints which simplifies the optimization procedure due to the absence of matrix inversion in the solution. However, the use of Laguerre functions rely on an identification process in order to predict the behavior of the system.

Therefore, there exist the necessity of adding terms to the Laguerre function in order to model the system properly. This process may be inefficient in system with complex dynamics or dynamics that may change rapidly. Other MPC’s approaches have been presented as the Generalized predictive controller (Clarke, Mohtadi & Tuffs, 1987) and Adaptive Predictive Control (Martín & Rodellar, 1996).

All the previous mentioned MPC approaches have proven to be efficient on linear systems for real-time implementation. However, when the system needs to be modeled as a nonlinear system, the algorithm becomes a nonlinear system and the optimization problem is no longer a QP-optimization problem. Therefore, the control strategy becomes a nonlinear model predictive control (NMPC). Several NMPC can be found in the literature, mostly in recent approaches. Metzler et. Al. presented an explicit NMPC control strategy in which both the system and the optimization problem are formulated as nonlinear. This arises real-time implementation issues, because the optimization problem becomes nonlinear. In order to solve this, they propose an explicit controller were all the possible solutions for the admissible operation zones are computed offline and stored in a lookup table. Then, the online algorithm search for that solution and apply it to the system (Metzler, Tavernini, Sorniotti & Gruber, 2018). Several explicit NMPC have been proved to be efficient and stable (Chakrabarty, Dinh, Buzzard, Zak &

Rundell, 2014; Liu, Lu & Chen, 2015; Zhang, Chakrabarty, Ayoub, Buzzard &

Sundaram, 2016), with the limitation of being restricted to an operational zone and the necessity of having a large space of memory available for the storage of the set of solutions.

Several real-time online NMPC have been also proposed in the literature. Favela et. Al. presented a discrete-time NMPC in which a continuous nonlinear system is discretized using Backward Euler implicit scheme, therefore, the output prediction is

done by Taylor Series expansion considering a fixed space where the trajectories of the future behavior of the outputs is computed by the use of Lie Derivatives. However, the optimization problem presented only considers the system performance, while constraints and stability conditions are left out in order to simplify the optimization procedure (Favela, Beltrán, Sotelo, Dieck, Sotelo, Rodriguez & Sotelo, 2018). Other kinds of NMPC have been also developed, algorithmic NMPC approaches (Fan & Han, 2012; Xu, Chen, Gong & Mei, 2015; Ichihara, Dórea & Xavier de Souza, 2017) and real- time implementations using advanced step optimization process (Yu & Biegler, 2018;

Nurkanovic, Zanelli, Albrecht & Diehl, 2019).

As shown along this section, MPC strategies have proven to be optimal when dealing with linear processes subject to constraints. However, when the system needs to be modeled as a nonlinear model the optimization problem becomes a major problem due to the complexity of it. Several nonlinear approaches have been proposed in order to deal with nonlinear systems, however, real-time implementation still an issue.

Algorithmic approaches or different prediction of the future output/states have been proposed in NMPC approaches in order to allow real-time implementation.

Nevertheless, the stability and robustness conditions are often set aside in this kind of algorithms in order to reduce the optimization problem complexity.

An alternative to NMPC approaches is to represent the nonlinear system as a set of linear systems or a linear system depending on a varying parameter (a LPV system).

LPV-MPC approaches are described and reviewed in the following section in order to introduce the state-of-art research works to control this kind of systems by means of a MPC strategy.

2.3 LPV-MPC approaches

As mentioned on the previous section, the nonlinearities present on many industrial processes represent a limitation for the implementation of MPC algorithms.

Therefore, in order to reduce the complexity of nonlinear systems in MPC paradigms, LPV-MPC approaches have became popular in recent years. Instead of having a nonlinear model to predict the behavior of the system, an LPV or qLPV model can be

defined in order to preserve the linear properties and to cope with the traditional MPC strategy. However, other limitations arise, being the unavailability of the future scheduling parameter one of the most relevant. To solve this problem, several strategies have been developed. Some of them are:

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✴ Frozen approaches

✴ One step approaches

✴ Parameter prediction approaches

All the previous mentioned LPV-MPC approaches are described throughout this section and several research works are also mentioned.

The worst case LPV-MPC approaches are one of the most researched brand on this kind of control strategies. In this approach, the future scheduling parameter is considered to be unknown, but its rate of change is bounded at every time step.

Therefore, the optimization problem becomes a min-max problem, where the objective is to minimize the cost function while maximizing the variation of the scheduling parameter. Several works have been developed using this strategy, Huang et. Al.

presented a quasi min-max MPC algorithm to control an I/O LPV system with robust stability constraints (Huang, He & Chen, 2014). The algorithm proved to be efficient on systems with small number of states. Similar approaches considering worst case optimization and stability conditions have been presented in the following references:

(Besselmann, Lofberg & Morari, 2012; Bumroongsri, 2014; Ding & Pan, 2016; Longge &

Yan, 2017; Ping, Wang & Zhang, 2018; Abbas, Hanema, Tóth, Mohammadpour &

Meskin, 2018; Hu & Ding, 2020).

LPV-MPC worst case approaches have proven to be robust and effective in several industrial processes. However, the main limitations found in this kind of approaches is that the worst-case consideration on the variation of the scheduling

parameter may led to a conservative performance. Also, the robustness and stability conditions added to the optimization problem make the real-time implementation a difficult task on systems having complex dynamics or in algorithms considering a long prediction horizon.

Another kind of approach is the tube-based LPV-MPC approaches. In this kind of approaches, the stability and robustness conditions are considered along the prediction horizon in order to determine all the possible admissible trajectories that the LPV system may take considering the bounded variations of the scheduling parameter and bounded disturbance that may affect the system. Then, a tube-like set along the prediction horizon is defined containing all the possible trajectories previously described. Then, the optimization problem searches for the optimal trajectory contained in the tube in order to determine the optimal control action for the MPC control law. This kind of approach is relatively new and some of the recent research works are the following: (Brunner, Lazar & Allgöwer, 2013; Abbas, Männel, né Hoffmann & Rostalski, 2019; Ismail & Liu, 2020; Hanema, Lazar & Tóth, 2020; Pipino, Morato, Bernardi, Adam

& Normey-Rico, 2020; Alcalá, Puig, Quevedo & Sename, 2020).

Tube-based LPV-MPC approaches have proven to be efficient and robust when dealing with bounded disturbance and the uncertainty of the future values of the scheduling parameter. Also, the performance of the system tends to be better than min- max approaches as not always the worst case is the chosen. However, the main limitations of tube-based MPC’s is the construction of the tube itself. This process is complex and if the tube is not well designed, the overall performance of the algorithm deteriorates, as well as, the optimization time tends to increase, which is an important issue for real-time implementations. The tube-based MPC still in a developing stage of research and with new concepts available in the future this strategy may be one of the most promising in LPV-MPC control.

There exist some LPV-MPC approaches that consider that the future scheduling parameter is available. This is, in most of the times, not true in real-life industrial processes, but is suitable in several processes where the variation of the scheduling parameter is small and/or slow. One way to assume that the scheduling parameter is

always available is to have a frozen scheduling parameter. In this kind of approach, the scheduling parameter is measured at every time instant and is assumed to maintain that value along the prediction horizon. Therefore, the formulation of the MPC paradigm is the same as in the traditional MPC and the optimization problem is a simple QP problem. However, this kind of approaches are limited to short prediction horizons due to the deviation of the scheduling parameter may increase significantly when the prediction horizon is large. Also, stability and robustness is not guaranteed along the prediction of the dynamics due to the uncertainty of the deviation of the future scheduling parameters are not quantifiable. This kind of approach can be find in the following research works: (Morato, Sename & Dugard, 2018; Cisneros, Sridharan &

Werner, 2018; Alcala, Puig & Quevedo, 2019; Morato, Sename & Normey-Rico, 2020).

In order to avoid the unavailability of the future values of the scheduling parameter, one-step MPC-LPV approaches have been developed. This kind of approach implements a prediction horizon , therefore, the current scheduling parameter is known at every time step. Then, the optimization problem is similar to a constrained LQR optimization problem. Therefore, stability and robustness conditions are easy to implement. The main disadvantage of this kind of approach is that there is not trajectory planning minimization, thus, the control action may vary rapidly in system with fast dynamics. Some of the research works that use this strategy are the following:

(Nguyen, Olaru & Gutman, 2013; Franze, Lucia & Tedesco, 2014).

Another solution given to the unavailability of the future scheduling parameters for the computation of the MPC-LPV optimization problem is the estimation of the future scheduling parameters using different identification methods. Then, previous to every step optimization problem, a prediction algorithm is used to estimate the future scheduling parameter values. Thus, the optimization problem becomes a QP problem only depending on the set of control actions to be implemented to the system. One of this approaches was presented by Sename et. Al. In this research work, the future values of the scheduling parameter are estimated using a recursive least squares algorithm by using an ARX model depending on the past values of the scheduling parameter itself and the previous inputs and output (Sename, Morato & Normey-Rico,

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2019). Other research works with prediction of the behavior of the scheduling parameter are the following: (Cavanini, Cimini & Ippoliti, 2017; Pour, Puig & Cembrano, 2018;

Alcalá, Puig, Quevedo & Rosolia, 2020). This kind of approaches depends on the efficiency of the estimation of the scheduling parameter. Therefore, if the behavior of the scheduling parameter is erratic or random, this kind of approaches may not behave properly. Also, in order to ensure stability and robustness, the prediction error should be measured and considered when dealing with stability tests.

2.4 Terminal sets in MPC

As mentioned in the previous sections, LPV-MPC have a limitation of having conservative control performances due to the uncertainty of the future values of the scheduling parameter. Therefore, worst case approaches or parameter prediction approaches considers that variation and apply a conservative control action in order to preserve stability for the LPV system in future steps. To reduce the conservativeness of LPV-MPC approaches, terminal sets have been widely used in several applications.

The idea of terminal sets in MPC is to design a robust positively invariant set around the origin of the system states where a feedback control action can be applied to the system while ensuring both stability and robustness. Also, the set is designed so that the feedback gain steers the states to the origin following a trajectory that remains inside the positively invariant set. This kind of strategy is used in the following research works: (Suzuki & Sugie, 2006; Franzè, Famularo, Garone & Casavola, 2009; Hamouda, Bennouna, Ayadi & Langlois, 2013; Manrique, Fiacchini, Chambrion & Millerioux, 2016).

This kind of strategies have proven to be efficient even in complex nonlinear systems.

However, even when the state feedback control action decreases the average computational time of the algorithm, when the states are outside the robust positively invariant set, the MPC optimization problem can still require considerable amount of time. Also, the definition of a robust positively invariant set is not a simple process and requires broad knowledge of the system or process that is being controlled.

Terminal sets have been also used in MPC-LPV for reducing the conservative control performance. This is done by defining a terminal cost in which the final predicted

states of a system are desired to be inside a terminal set (also know as terminal ellipsoid set or terminal contractive set) after several steps ahead which are often greater or equal than the prediction horizon . Therefore, this terminal sets define a desired trajectory for the states towards the origin, in order to push the conservative performance towards a less conservative one. This terminal sets are built considering the variation of the scheduling parameter and the stability conditions defined for the system in order to avoid unstable control actions along the prediction horizon. Some of the research works that use this strategy are the following: (Angeli, Casavola, Franzé &

Mosca, 2008; Ferramosca, Limón, Alvarado, Alamo & Camacho, 2009; Bumroongsri, Kheawhom, 2012; Mate, Kodamana, Bhartiya & Nataraj, 2019). Also, the previous mentioned approach (Sename, Morato & Normey-Rico, 2019) incorporates this strategy along a contractive horizon .

The definition of size and form of the terminal sets is a crucial aspect when using them in LPV-MPC approaches. If the sets are too big, the control algorithm will still present a conservative performance. Conversely, if the sets are defined to be too small, the optimization problem may present difficulties to find an optimal control sequence that can reach the terminal sets at every iteration step. In order to avoid designing terminal conditions that can result into unfeasible optimization problems in the MPC algorithm, Löfberg presented a method to detect a-priori the lost of recursive feasibility in both undisturbed and disturbed MPC controllers by using a bilevel optimization problem based on the Farka’s lemma (Löfberg, 2012).

As mentioned throughout this section, the inclusion of terminal sets into the MPC-LPV algorithm can improve the overall control performance. However, the construction of the terminal sets needs to be properly done in order to enhance the controller performance without losing feasibility when the optimization problem is performed. Also, the constructions of the sets are often done online in order to reduce the computational time needed to build, and the solve, the optimization problem. The terminal sets can be stored in a lookup table for all admissible states or embedded into a piecewise function depending on both the states and the scheduling parameter. In section 3.5, a strategy of attraction sets based on terminal sets is presented in order to

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steer the states to a desired performance and a terminal stability set is defined in order to switch the LPV-MPC control law to a LPV-LQR control approach.

2.5 Quadratic stability through LMI in LPV control and MPC

Ensuring stability in control systems is a crucial part for industrial controllers.

Even though optimal control algorithms based their performance on stability laws such as Lyapunov techniques, some controllers need additional stability conditions to be considered. In the MPC-LPV scheme, stability conditions need to be considered not only based on the control action that is computed to be optimal, but also, considerations about the uncertain future values of the scheduling parameter need to be included in the algorithm. Through this section, quadratic stability approaches used in both LPV control and LPV-MPC approaches are described.

Quadratic stability of a LTV or LPV system implies exponential stability is ensured for an infinite time horizon for every admissible time realizations of the system for a specific control action and/or set of control actions. This will ensure that a varying system can be feed by a constant control action for an infinite amount of time and the system will reach a stable value within the admissible region of the varying parameters.

If the admissible region for the varying parameters are not available, quadratic stability cannot be ensured for LPV/LTI systems. Also, if the admissible range for the states of the system is too large, the stable control actions available may be conservative in terms of control performance.

Ensuring quadratic stability in LPV systems can be seen as an extension of the quadratic stability conditions for LTI systems based on the Lyapunov stability and the discrete algebraic Riccati equations (DARE). Several books have presented approaches to adapt the quadratic stability conditions for LPV systems into a limited number of linear matrix inequalities (LMI). Francesco Amato in his book “Robust Control of Linear Systems Subject to Uncertain Time-Varying Parameters” describe a several approaches for dealing with quadratic stability analysis performed on systems with time- varying parameters. One of the approaches is to represent the LTV systems as polytopic LPV systems. Afterwards, stability analysis is performed only of the vertex of

ρ

the convex polytope defined as the limit admissible values of the scheduling parameters embedded into the system (Amato, 2006).

Another research works related to stability analysis for LPV systems was developed by Franco Blanchini in the book “Lyapunov methods in robustness—an introduction”. This book present several stability and robustness conditions for LTI, LTV, LPV and nonlinear systems. Quadratic stability is also reviewed with application to LPV systems represented as polytopic systems (Blanchini, 2009). However, the quadratic stability polytopic representations is conservative because there exist systems which are not quadratic stabilizable but the system is stabilizable and vice-versa.

The use of quadratic stability on LPV control has been widely used in several controllers. Ilka et. Al. presented a robust gain-scheduled controller for an uncertain LPV system using the affine quadratic stability approach previously described. In order to obtain a good control performance, a LQ-based LPV controller is designed, however, the optimization problem is suboptimal, guaranteeing only to be in a range of certain performance but not finding the optimal solution explicitly. (Ilka & Vesely, 2014). Similar approaches to robust LPV control can be found in the following research works: (Cherifi, Guelton & Arcese, 2015; Liu, Theilliol, Gu, He, Yan & Han, 2017; Hypiusová &

Rosinová, 2019).

Quadratic stability approaches to ensure robustness has been also used in LPV- MPC approaches. However, the inclusion of stability LMI conditions to the MPC optimization problem increases the required time to compute an optimal solution that copes with all the imposed constraints that the robust LPV-MPC may have. Some of the approaches that guarantee quadratic stability in the MPC paradigm are: (Cisneros &

Werner, 2017; Ping, 2017; Ping, Yang, Ding, Raïssi & Li, 2020).

As mentioned along this section, quadratic stability conditions are widely used in LPV control strategies due to the possibility of considering uncertain parameters in the control algorithm for dealing with varying parameters inside a convex polytope.

However, quadratic stability often brings conservative performances which is not the case when using other stability approaches. The main advantage of quadratic stability is

that robustness test can be performed along the quadratic stability conditions by adding the effects of bounded disturbances to the stability LMI’s.

2.6 Limitations

Several MPC-LPV approaches previously review along this section 2 have proven to be efficient for specific LPV systems. However, there are some limitations for this kind of control strategy. The main limitations found in the published research works are the following:

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In order to counter some of the previous limitations, section 3 presents a novel LQR/MPC for LPV systems with scheduling parameter prediction along the prediction horizon and the inclusion of terminal ellipsoidal sets and terminal attraction sets while ensuring quadratic stability conditions for every computed input by the MPC law.