New Trends in Observer-Based Control: An Introduction to Design Approaches and Engineering Applications

By Magdi S. Mahmoud

New Trends in Observer-Based Control: An Introduction to Design Approaches and Engineering Applications, Volume One presents a clear-and-concise introduction to the latest advances in observer-based control design. It provides a comprehensive tutorial on new trends in the design of observer-based controllers for which the separation principle is well established. In addition, since the theoretical developments remain more advanced than the engineering applications, more experimental results are still needed. A wide range of applications are covered, and the book contains worked examples which make it ideal for both advanced courses and researchers starting in the field.

• Presents a clear-and-concise introduction to the latest advances in observer-based control design
• Offers concise content on the many facets of observer-based control design
• Discusses key applications in the fields of power systems, robotics and mechatronics, and flight and automotive systems

• Language: English
• Publisher: Academic Press
• Release date: Mar 30, 2019
• ISBN: 9780128170397

Book preview

New Trends in Observer-Based Control

An Introduction to Design Approaches and Engineering Applications

First Edition

Olfa Boubaker

Quanmin Zhu

Magdi S. Mahmoud

José Ragot

Hamid Reza Karimi

Jorge Dávila

Table of Contents

Cover image

Title page

Copyright

Contributors

Foreword

Part I: Observer Design

Chapter 1: On Dynamic Observers Design for Descriptor Systems

Abstract

1 Introduction

2 Basic Theory of Descriptor Systems

3 Observers for Descriptor Systems

4 Generalized Dynamic Observer

5 H∞ Generalized Observer Design

6 Concluding Remarks

Chapter 2: Adaptive Observer Design for Nonlinear Interconnected Systems With Applications

Abstract

1 Introduction

2 System Description and Problem Formulation

3 Adaptive Observer Design With Parameters Estimation

4 Stability of the Error Dynamical Systems

5 Case Study Examples

6 Conclusion

Acknowledgments

Chapter 3: On the Observability and Observer Design in Switched Linear Systems

Abstract

1 Introduction

2 Switched Linear Systems and Basic Behavior

3 Observability Analysis for Switched Linear Systems Under Disturbance

4 Observer Design for SLS Under Disturbances

5 Observer Design for SAS Under Bounded Disturbances With Nonautonomous Chaotic Modulation

Chapter 4: On Unknown Input Observer Design for Linear Systems With Delays in States and Inputs

Abstract

1 Introduction

2 Problems Statement and Preliminaries

3 Design of the Delay-Dependent Observer

4 Design of the Delay-Independent Observer

5 Simulation Results

6 Conclusion

Part II: Observer-Based Control Design

Chapter 5: Observer-Based Control Design: Basics, Progress, and Outlook

Abstract

1 Introduction

2 Historical Development

3 Observer-Based Fault-Tolerant Control

4 Robust Control of NCS With Partially Known Transition Matrix

5 Switched Discrete-Time Systems

6 Disturbance Observer-Based Control for Nonlinear Systems

7 NCS With Quantization and Nonstationary Random Delays

8 Outlook of Observer-Based Control

9 Conclusions

Acknowledgments

Chapter 6: Observer-Based Stabilization of Switched Discrete-Time Linear Systems With Parameter Uncertainties: New Scenarios of LMI Conditions

Abstract

1 Introduction

2 Problem Formulation

3 Main Results: New LMI Design Algorithms

4 Numerical Examples and Comparisons

5 Conclusion

Chapter 7: Practical Study of Derivative-Free Observer-Based Nonlinear Adaptive Predictive Control

Abstract

1 Introduction

2 Predictive Control

3 Model Predictive Control-Based State Observer

4 Practical Study of NMPC-Based DDF2

5 Conclusion

Chapter 8: A Robust Decentralized Observer-Based Stabilization Method for Interconnected Nonlinear Systems: Improved LMI Conditions

Abstract

1 Introduction

2 Notation and Preliminary Useful Lemmas

3 Problem Formulation

4 New LMI Synthesis Condition

5 Illustrative Examples

6 Conclusion

Part III: Observer-Based Fault Detection and Tolerant Control

Chapter 9: Polytopic Models for Observer and Fault-Tolerant Control Designs

Abstract

1 Polytopic Model-Based Modeling

2 Polytopic Model Stability

3 State Feedback Design Based on Polytopic Models

4 Observer State Design Based on polytopic models

5 Active Fault-Tolerant Control

6 Overall Conclusion

Chapter 10: Disturbance Observer-Based Fault-Tolerant Control for a Class of Additive Faults

Abstract

1 Introduction

2 Problem Statement

3 Strong Observability, Strong Detectability, and Some of Their Properties

4 Fault-Tolerant Control

5 Fault-Tolerant Control for a Roll Autopilot Design

6 Conclusions

Chapter 11: Robust State and Fault Estimation for Linear Descriptor and Nonlinear Stochastic Systems With Disturbances

Abstract

1 Introduction

2 The Robust Two-Stage Kalman Filter

3 Robust State and Fault Estimator for Linear Stochastic Systems

4 Robust State and Fault Estimation for Linear Descriptor Stochastic Systems

5 Robust State and Fault Estimation for Nonlinear Descriptor Stochastic Systems With Disturbances

6 Numerical Simulation Results

7 Conclusions

Chapter 12: Fault and Switching Instants Estimation for Switched Linear Systems

Abstract

1 Introduction

2 Problem Statement

3 Discrete State Estimation

4 Continuous State Estimation

5 Fault Estimation

6 Simulation Results

7 Conclusion

Chapter 13: Observer-Based Event-Triggered Attack-Tolerant Control Design for Cyber-Physical Systems

Abstract

1 Introduction

2 Problem Statement

3 Observer-Based Attack-Tolerant Control Design

4 Simulation Results: Application on a Real-Time Laboratory Three-Tank System

5 Conclusion

Index

Copyright

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Contributors

Carlos-Manuel Astorga-Zaragoza     Electronic Engineering Department, Tecnológico Nacional de México/CENIDET, Cuernavaca, Morelos, Mexico

Fazia Bedouhene     Laboratory of Pure and Applied Mathematics, University Mouloud Mammeri, Tizi-Ouzou, Algeria

Cherifa Bennani     Laboratory of Pure and Applied Mathematics, University Mouloud Mammeri, Tizi-Ouzou, Algeria

Talel Bessaoudi     Department of Electrical Engineering, National Higher Engineering School of Tunis, University of Tunis, Tunis, Tunisia

Hamza Bibi     Laboratory of Pure and Applied Mathematics, University Mouloud Mammeri, Tizi-Ouzou, Algeria

Faouzi Bouani     National Engineering School of Tunis, LR11ES20, Laboratory of Analysis, Conception and Control of Systems, University of Tunis El Manar, Tunis, Tunisia

Olfa Boubaker     National Institute of Applied Sciences and Technology, University of Carthage, Tunis, Tunisia

Latifa Boutat-Baddas     Research Center for Automatic Control of Nancy, UMR-CNRS 7039, Université de Lorraine/IUT de Longwy, Cosnes-et-Romains, France

Jérôme Cieslak     IMS-Lab, Automatic Control Group, University of Bordeaux, Talence, France

Jorge Dávila     School of Mechanical and Electrical Engineering, IPN Instituto Politécnico Nacional, Mexico City, Mexico

Mohamed Darouach     Research Center for Automatic Control of Nancy, UMR-CNRS 7039, Université de Lorraine/IUT de Longwy, Cosnes-et-Romains, France

David Gómez-Gutiérrez     Tecnologico de Monterrey, School of Engineering and Science, Jalisco, Mexico

Stefano Di Gennaro     University of L’Aquila, Department of Information Engineering, Computer Science and Mathematics, Center of Excellence DEWS, L’Aquila, Italy

David Henry     IMS-Lab, Automatic Control Group, University of Bordeaux, Talence, France

Fayçal Ben Hmida     Department of Electrical Engineering, National Higher Engineering School of Tunis, University of Tunis, Tunis, Tunisia

Chien-Shu Hsieh     Department of Electrical and Electronic Engineering, Ta Hwa University of Science and Technology, Qionglin, Hsinchu, Taiwan, ROC

Dalil Ichalal     IBISC, Univ Evry, Université Paris Saclay, 91020 Courcouronnes, France

Bin Jiang     College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China

Hamid Reza Karimi     Department of Mechanical Engineering, Politecnico di Milano, Milan, Italy

Houria Kheloufi     Laboratory of Pure and Applied Mathematics, University Mouloud Mammeri, Tizi-Ouzou, Algeria

Khaled Laboudi     Université de Reims Champagne Ardenne, CRESTIC EA 3804, Reims, France

Mihai Lungu     Faculty of Electrical Engineering, University of Craiova, Craiova, Romania

Magdi S. Mahmoud     Systems Engineering Department, KFUPM, Dhahran, Saudi Arabia

Noureddine Manamanni     Université de Reims Champagne Ardenne, CRESTIC EA 3804, Reims, France

Zehui Mao     College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, China

Didier Maquin     CRAN, Université de Lorraine-CNRS, 54000 Nancy, France.

Benoît Marx     CRAN, Université de Lorraine-CNRS, 54000 Nancy, France.

Mamadou Mboup     Université de Reims Champagne Ardenne, CRESTIC EA 3804, Reims, France

Nadhir Messai     Université de Reims Champagne Ardenne, CRESTIC EA 3804, Reims, France

Mokhtar Mohamed     Instrumentation, Control and Embedded Systems Research Group, School of Engineering and Digital Arts, University of Kent, Canterbury, United Kingdom

Rodolfo Orjuela     IRIMAS, Université de Haute-Alsace, 68093, Mulhouse, France

Gloria-Lilia Osorio-Gordillo     Electronic Engineering Department, Tecnológico Nacional de México/CENIDET, Cuernavaca, Morelos, Mexico

José Ragot     CRAN, Université de Lorraine-CNRS, 54000 Nancy, France.

Antonio Ramírez-Teviño     CINVESTAV, Jalisco, Mexico

Souad Bezzaoucha Rebaï     Automatic Control Research Group, Interdisciplinary Centre for Security, Reliability and Trust (SnT), University of Luxembourg, Luxembourg City, Luxembourg

Hichem Salhi     National Engineering School of Tunis, LR11ES20, Laboratory of Analysis, Conception and Control of Systems, University of Tunis El Manar, Tunis, Tunisia

Abdulaziz Sherif     Electrical Engineering Department, University of Tripoli, Tripoli, Libya

Hieu Trinh     School of Engineering, Faculty of Science Engineering and Built Environment, Deakin University, Geelong, VIC, Australia

Carlos Renato Vázquez     Tecnologico de Monterrey, School of Engineering and Science, Jalisco, Mexico

Holger Voos     Automatic Control Research Group, Interdisciplinary Centre for Security, Reliability and Trust (SnT), University of Luxembourg, Luxembourg City, Luxembourg

Seifeddine Ben Warrad     National Institute of Applied Sciences and Technology, University of Carthage, Tunis, Tunisia

Xing-Gang Yan     Instrumentation, Control and Embedded Systems Research Group, School of Engineering and Digital Arts, University of Kent, Canterbury, United Kingdom

Ali Zemouche

CRAN UMR CNRS 7039, University of Lorraine, Cosnes et Romain, France

EPI Inria DISCO, Laboratoire des Signaux et Systèmes, CNRS-Centrale Supelec, Gif-sur-Yvette, France

Quan Min Zhu     Department of Engineering Design and Mathematics, University of the West of England, Bristol, United Kingdom

Foreword

José Ragot, Université de Lorraine, CNRS, CRAN, F54000 Nancy, France

For more than half a century, considerable efforts have been made in the field of system control and monitoring. These actions are, obviously, crucial in the presence of technological risks directly impacting human health and the environment. Techniques to be developed aim, in general, to better understand, at each moment, the state of a system. The estimation phase is, of course, insufficient, and should be completed by state analysis to evaluate its normal or abnormal character. In the latter case, the analysis is further refined to accurately localize where the anomaly is to be found, to specify which part of the system, sensor, or actuator, is faulty. In order to judge the importance of the anomaly, its magnitude should be estimated. The ultimate phase of diagnosis seeks to specify the cause of the anomaly. In some cases, although this remains marginal because of great difficulty at the moment, the future evolution estimation of the anomaly is made. All these steps can contribute to considering how to react to anomalies in order to reduce their effects by means of appropriate control laws.

This book does not attempt to address all these problems, but it can be a good introduction to some of the techniques designed to estimate the system states in different situations. In particular, the design of an observer to reconstruct the system states from partial measurements, how to use a state observer to detect and locate anomalies, and how to adjust a control law to counter the effect of anomalies on the behavior of a system will be discussed. In addition, a number of difficulties resulting from realistic physical constraints are considered: the presence of uncertainties and delays in system models; the influence of unmeasured exogenous inputs on system dynamics; the nonlinear behavior of systems; switching systems, or systems with several operating modes and interconnected systems. This diversity of objectives, systems, and constraints is covered in complementary chapters, and addresses a large part of the problem of observer state estimation and its applications.

This book is a timely and comprehensive reference guide for graduate students, researchers, engineers, and practitioners in the areas of control theory. The content has been written for investigators acting in the fields of electrical, mechanical, aerospace, or mechatronics engineering. With contributions by eminent scientists in the field of control theory and systems engineering from 22 countries, this book covers the latest advances in observer-based control, from new design approaches to control engineering applications. Readers will find the fundamentals and applications related to this topical issue. The book contains examples that make it ideal for advanced courses, as well as for researchers starting to work in the field, or engineers wishing to enter the field quickly and efficiently.

The authors of the various chapters have tried to clearly present the theoretical concepts underlying the proposed solutions, and to illustrate them with pedagogical examples of modest dimensions; but allowing us to clearly see the implementation of these solutions and to assess their relevance through numerical data. In some cases, short Matlab programs complete the formulation. The book is structured in 13 chapters, and the organization is given as follows.

Chapter 1 is dedicated to the class of descriptor systems. After some reminders on proportional and proportional-integral observers, the authors propose a new dynamical observer called general observer structure, and its extension to the case of systems with disturbing input. The stability analysis of the observer is proved via a Lyapunov method and solved via a set of linear matrix inequalities (LMIs). Several academic examples illustrate the performance of the proposed structure.

Chapter 2 proposes an observer design technique for nonlinear interconnected systems with uncertain variable parameters. The observer has adaptive parameters that are adjusted from a stability study of the reconstruction error. The two examples that are given, coupled reverse pendulums and a quarter vehicle system, illustrate the implementation.

The case of a linear switching system is discussed in Chapter 3, incorporating two difficulties: the presence of unknown inputs, and a lack of knowledge of the switching law. The observer is then designed to estimate the continuous and discrete states of the system. The proposed technique is applied to a modulation/demodulation procedure in a secure communication system with chaotic behavior.

Another interesting situation is the subject of Chapter 4: state estimation for linear systems with unknown inputs and delays affecting their states and inputs. A first method proposes the design of a delay-dependent unknown input observer (UIO), whereas a second one suggests the design of a delay-independent UIO. The numerical example of the quadruple-tank benchmark is used to illustrate the efficiency of the two proposed methods for the case study.

Chapter 5 presents the basics, progress, and outlook for the observer-based control design problem in dynamical systems. After reviewing the roots and needs of the problem, the authors have provided complete analytical results pertaining to dynamic modeling, control design, and computer simulation of several distinct approaches. The authors have also investigated issues regarding robust stability and robust performance of control design for different system configurations.

In Chapter 6, the authors have developed new sufficient LMI conditions for the problem of stabilization of discrete-time uncertain switched linear systems under arbitrary switching rules. Different scenarios of the use of Finsler’s lemma are proposed to reduce the conservatism of existing results in the literature. Numerical examples and simulation results are presented to demonstrate the effectiveness of the proposed methods.

Model predictive control (MPC) based on state observers for nonlinear multivariable systems is the subject of Chapter 7. To overcome classical limits, the authors developed an adaptive MPC-based observer for nonlinear multivariable systems. The implementation of the proposed approach to a three-tank benchmark system is performed, with a comparison between linear and nonlinear predictive controllers.

The authors of Chapter 8 propose a new decentralized observer-based controller design method for nonlinear discrete-time interconnected systems with nonlinear interconnections. An enhanced linear matrix inequality design condition is provided to guarantee asymptotic stability for systems with both known and unknown interconnection bounds. Two numerical examples illustrate the effectiveness of the design approach.

Chapter 9 presents results of the polytopic model (PM) approach to cope with the modeling, stability analysis, state feedback control, state, and unknown input estimation, and finally, fault-tolerant control of nonlinear systems. The backbone of all presented results is the capacity of the PM structure to represent nonlinearities in a selected operating range of the system. It is proposed to design a fault-tolerant controller fed with the simultaneous state and unknown input estimates. The benefits of the active fault-tolerant controller based on the PM approach are illustrated in an example consisting of the stabilization of the lateral dynamics of a vehicle.

Chapter 10 studies the application of high-order sliding mode observers for the estimation of faults and their later compensation in linear systems. The study is restricted to the systems with strongly observable faults. The main idea is to exploit the finite-time convergence of the high-order sliding mode-based observers to estimate the states, and also the dynamic effects, of the faults, showing that these are powerful tools not only to estimate states, but also unknown signals. The methodology is illustrated with the design of a fault-tolerant control of the roll autopilot for a missile mode.

The authors of Chapter 11 consider the problem of simultaneous state and fault estimation of linear descriptor and nonlinear descriptor discrete-time stochastic systems with arbitrary unknown disturbances. The study is based on input filtering and the use of a robust two-stage Kalman filter.

Chapter 12 investigates the problem of the simultaneous estimation of discrete state, continuous state, and faults of a class of switched linear systems with measurement noise. A new algebraic approach is developed in order to estimate, in real time, and with a negligible delay, the switching times, and to reconstruct the discrete state. The proposed strategy is illustrated by a system with three operating modes whose dynamics are affected by two faults.

Chapter 13 introduces an appropriate model associating paradigms from control theory and computer science to deal with the system subject to both physical attacks and sensor/actuator attacks via the connected network. Inspired by a combination of the classical fault-tolerant control approach and the event-triggered control, an observer-based, attack-tolerant control solution is proposed. The control design is applied to a laboratory benchmark including a three-tank system subject to physical attacks.

Finally, on behalf of all the editors, I would like to express my gratefulness to all the authors of the book for their valuable contributions, and all reviewers for their helpful and professional efforts to provide valuable comments and feedback.

Part I

Observer Design

Chapter 1

On Dynamic Observers Design for Descriptor Systems

Gloria-Lilia Osorio-Gordillo*; Mohamed Darouach†; Latifa Boutat-Baddas†; Carlos-Manuel Astorga-Zaragoza*    * Electronic Engineering Department, Tecnológico Nacional de México/CENIDET, Cuernavaca, Morelos, Mexico

† Research Center for Automatic Control of Nancy, UMR-CNRS 7039, Université de Lorraine/IUT de Longwy, Cosnes-et-Romains, France

Abstract

dynamical observer is also presented to deal with disturbed descriptor systems. The stability analysis of each observer is proved by the Lyapunov method, and it is guaranteed by solving a set of linear matrices inequalities. Particular cases of the GDO as the PIO and PO were developed in order to compare their performances in simulation. A practical example is also presented to show the performance of our approach.

Keywords

Descriptor systems; Linear systems; Generalized dynamic observer; H-infinity; LMI

1 Introduction

Descriptor systems were introduced by Luenberger in 1977 to represent the systems for which the results obtained in the domain of control and observation of standard systems cannot be applied. This class of systems can represent the physical phenomena that the model by ordinary differential equations cannot describe. Descriptor systems have found their application in modeling the motion of aircraft, and in chemical processes, the mineral industry, electrical circuits, economic systems, and robotics [1]. Some control techniques, such as those that are based on state-feedback control, do not have all the states available for their measurement. Either by technical or economic reasons it is difficult, or even impossible, to measure all the system state variables. Therefore, it is necessary to estimate the state of the system. Observers are one of the principal tools to provide the estimation of system state variables. The first work on the problem of reconstruction of state variables was devoted to standard linear systems [2]. Since then, many theoretical results have been presented, and they are widely used in control and fault diagnosis.

Observers can be classified by order: full-order, reduced-order, and partial-order observers, and by structure: proportional, proportional-integral, and generalized observer. In the estimation by a proportional observer (PO), there always exists a static estimation error in the presence of disturbance. In order to deal with this disadvantage of the PO, proportional-integral observers (PIOs) were introduced with an integral gain of the output error (difference between the estimated output and the measured output) in their structure, this form of observer achieves steady-state accuracy in the state estimation. Then, it is apparent that a modified structure with additional degrees of freedom in the observer can provide the best estimation and robustness in the presence of parameter variation and disturbances. A new structure of observer was developed by Goodwin and Middleton [3] and Marquez [4], known as the generalized dynamic observer (GDO). This structure presents an alternative state estimation that can be considered to be more general than the PO and the PIO.

2 Basic Theory of Descriptor Systems

2.1 Definition of Descriptor Systems

A linear descriptor system with constant coefficients can be represented by the following set of equations:

   (1.1)

is the measured output. E is a singular matrix with constant parameters, it is assumed that rank(E) ≤ n.

Descriptor systems, also known as singular systems or differential-algebraic systems, are a class of systems that can be considered a generalization of dynamical systems. The descriptor system representation is a powerful modeling tool because it can describe processes governed by both differential equations and algebraic equations. So it represents the physical phenomena that the model, by ordinary differential equations, cannot describe. These systems were introduced by Luenberger [5] from a control theory point of view, and since, great efforts have been made to investigate descriptor systems theory and its applications.

2.2 Some Properties of Descriptor Systems

Consider the following descriptor system

   (1.2)

are real, and x(t), u(t), and y(t) are vectors of appropriate dimensions.

2.2.1 Regularity of Descriptor Systems

The regularity property in descriptor systems guarantees the existence and uniqueness of solutions. We can give the following definition for regularity.

Definition 1

(Yip and Sincovec [6])

System (such that

or equivalently, the polynomial det(sE − A) is not identically zero. In this case, we also say that the pair (E, A), or the matrix pencil sE − A, is regular.

Remark 1

In [7, 8] the authors show that the regularity of the matrix pair (E, A) is a property not needed for observer design, instead of the preceding regularity property for square systems, this property is replaced by

for the rectangular descriptor systems. Where the normal−rank of the matrix pencil sE − A is defined as the rank of (sE − Aand U(s) is the Laplace transformation of u(t) (see [7, 9] and references therein).

2.2.2 Stability of Descriptor Systems

Stability of a dynamical system describes the response behavior of the system at infinity time with respect to initial condition disturbances, and is well regarded as one of the most important properties of dynamical systems.

Definition 2

(Duan [10])

The regular descriptor linear system (1.2) is asymptotically stable if, and only if,

where eig(E, A) is defined as the roots of det(sE − A) = 0, which must lie in the stable region, that is, the open left-half plane for the continuous-time systems.

2.2.3 Impulse-Free Behavior

Definition 3

(Duan [10])

If the state response of a descriptor linear system, starting from an arbitrary initial value, does not contain impulse terms, then the system is called impulse-free.

The following statements are equivalent:

•The pair (E, A) is impulse-free.

•deg(det(sE − A)) = rank(E).

.

2.2.4 Admissibility

Definition 4

The pair (E, A) is said to be admissible if it is regular, impulse-free, and stable.

Theorem 1

System (1.2) or the pair (E, A) is admissible if and only if there exists a nonsingular matrix Θ such that ETΘ = ΘTE ≥ 0 and ATΘ + ΘTA < 0.

2.2.5 C-Observability

It is assumed that system (1.2) has the following slow and fast subsystems

   (1.3)

and

   (1.4)

are a nilpotent matrix.

The relations between the coefficient matrices of the two systems are given by

   (1.5)

where the matrices Q and P are the left and right transformation matrices, respectively.

Definition 5

(Duan [10])

The regular system (1.2) is called completely observable (C-observable), if the initial condition x(0) of the system can be uniquely determined from the output data y(t. Alternatively, the system (1.2) is C-observable if the zero output y(t) ≡ 0 with u(t) ≡ 0 implies that the system has only the trivial solution x(t) ≡ 0. Consider the regular system (1.2), with its slow subsystem (1.3) and fast subsystem (1.4).

1.The slow subsystem (1.3) is C-observable if and only if

2.The fast subsystem (1.4) is C-observable if and only if

3.System (1.2) is C-observable is and only if conditions of statements 1 and 2 hold or,

2.2.6 R-Observability

The system (1.2) is called observable within the reachable set (R-observable) if any state in the reachable set can be uniquely determined by y(t) and u(t) for t ≥ 0.

Definition 6

(Duan [10])

The regular descriptor system (1.2) is R-observable if and only if

2.2.7 Impulse Observability

Definition 7

(Boukas [11])

Impulse observability guarantees the ability to uniquely determine the impulse behavior in x(t) from information of the impulse behavior in the output y(t). System (1.2) is called impulse observable if

2.2.8 Detectability

Definition 8

(Dai [1])

The system (1.2) is detectable if and only if all its output’s transmission zeros are stable, that is

2.2.9 Stabilizability

Definition 9

(Duan [10])

System (1.2) is stabilizable if there exists a state feedback controller u(t) such that the resulted closed-loop system is stable.

The regular system (1.2) is stabilizable if and only if

2.3 Practical Examples

In the following sections some descriptor models of practical systems are shown.

2.3.1 Hydraulic System

Model of the change of the height of water in three tanks, with an input flow in the first tank and with the third tank leaking (Fig. 1.1).

Fig. 1.1 Hydraulic system.

The pressures at the bottom of the tanks 1, 2, and 3 are represented as p1, p2, and p3, respectively. The pipe from tank 1 branches off to tanks 2 and 3. The pressure at the pipe branch is given as pB.

By using the Hagen-Poiseuille equation, the flow rates between the tanks and the pipe branch can be written as:

   (1.6a)

   (1.6b)

   (1.6c)

where η is the dynamic viscosity, Li, ∀i ∈ [1, 2, B] are the lengths of pipes, and dp is the diameter of the pipe.

All the fluid leaving tank 1 should enter into tanks 2 and 3. This is presented as a constraint.

   (1.7)

The pressure in each tank is given by:

   (1.8a)

   (1.8b)

   (1.8c)

where hi, ∀i ∈ [1, 2, 3] is the depth in each tank, ρ is the density of the liquid, and g is the gravity acceleration.

The rate at which the fluid leaves or enters the tank is directly proportional to the rate of change of the height of the fluid in the tank. This relation comes through the analysis of mass conservation of incompressible fluids:

   (1.9a)

   (1.9b)

   (1.9c)

where a1, a2, and a3 are the cross-sectional areas of tanks 1, 2, and 3, respectively. Fin represents the leaking in tank 3.

Using Eqs. (1.6)–(1.9) the state-space representation of the three interconnected tanks can be constructed as

   (1.10)

.

2.3.2 Mechanical System

Model of the mass-spring-damper system, which includes a rigid bar that can prevent the motion of the second mass. The exciting force is applied to mass 1 (Fig. 1.2).

Fig. 1.2 Mechanical system.

The positions of masses m1 and m2 are represented by x1 and x2, respectively. The spring coefficients are given as k1 and k2. The damper coefficients are given as b1 and b2.

Using Newton’s second law, the forces acting on each mass can be written as:

   (1.11)

   (1.12)

and the constraint equation

   (1.13)

where α represents the state of the switch 1 = closed and 0 = open, and μ(t) is the force absorbed.

Using Eqs. (1.11)–(1.13), the state-space representation of the mechanical system can be constructed as

   (1.14)

2.3.3 Electric System

Model of an electric system controlled by the voltage v (Fig. 1.3).

Fig. 1.3 Electric system.

The currents i1 and i2 are measured through the resistors R1 and R2. The electric charge in the capacitor C is denoted as q, and L represents the inductance.

The following relations are obtained by using current and voltage Kirchhoff’s laws

   (1.15)

   (1.16)

and the constraint equation

   (1.17)

Using Eqs. (1.15)–(1.17), the state-space representation of the electrical circuit can be expressed as

   (1.18)

3 Observers for Descriptor Systems

A descriptor system is a set of equations that are the result of modeling a system. These equations represent a general class of phenomena that is evolving in time, where some of the variables are related in a dynamical way, whereas others are purely static [5].

The so-called state observer is reconstructing the state variables of the system asymptotically. Based on this, an observer should satisfy the following two necessary conditions [1]:

1.The inputs of the observer should be the control input and the measured output of system.

, where e(t, the difference between the state and its estimate.

The general scheme of an observer is shown in Fig. 1.4.

Fig. 1.4 Observer’s general scheme.

Some applications of observers for descriptor systems can be found in [12], where a nonlinear observer for descriptor systems is designed to estimate the state variables and unknown inputs in a wastewater treatment plant. Discrete-time descriptor systems are used in [13] to design observers for state estimation in an experimental hydraulic tank system. The observers design for descriptor systems have been widely studied in [8, 14–16].

As for observers, design for standard and descriptor systems with unknown inputs has been treated in [17–21]. The design of unknown input observers is a crucial problem because, in many practical cases, all input signals cannot be known. Moreover, this class of observers is widely used in the area of fault diagnosis, even if all the inputs are known (see [19, 22], and references therein). All these results use the PO. In the estimation by a PO there always exists a static error estimation in the presence of disturbances. In order to deal with the disadvantage of PO, PIOs were introduced with an integral gain in their structure, which achieves steady-state accuracy in their estimations. The first results on the PIO were presented by Wojciechowski [23] for single-input, single-output (SISO) systems. Its extension to multivariable systems was presented in [24–26], where the authors show the performances of the PIO compared with PO in the presence of disturbances and uncertainties. Some recent results on the PIO for systems with unknown inputs are presented in [17, 27].

As shown, numerous works have focused on the design and implementation of algorithms for estimation and control for descriptor systems. Different techniques for state estimation have been developed, and these depend largely on the structure of the model, the information available in the process, and the relations that can be established between them.

3.1 Descriptor State Observer

Descriptor state observers are typically derived from the system representation. An additional term is included in order to ensure that the estimated state converges to the real state. Specifically, the additional term consists of the difference between the system output and the estimated output, and is then multiplied by a matrix of correction. This is added to the state of the observer to produce the so-called descriptor state observer. Some results of descriptor state observers in descriptor systems are found in [28–30].

In [1] a singular observer for singular systems is presented. The following system is considered:

   (1.19)

are constant matrices. It is assumed that system (1.19) is regular and nr = rank(E) < n.

Considering that system (1.19) is detectable, the following full-order observer is proposed:

   (1.20a)

   (1.20b)

.

be the estimation error between real and estimated states. Its derivative is described by:

   (1.21)

converges asymptotically toward x(t).

If rank(E) < n, observer (1.20) is called descriptor state observer. Otherwise, when rank(E) = n, E = In is assumed without loss of generality, so the observer (1.20) is called a Luenberger observer.

3.2 Proportional Observers

Commonly, the estimation of the different variables of interest is performed through a Luenberger observer, also known as a PO. Some results on PO for descriptor systems are shown in [8, 20, 28, 31].

In [8], a functional PO design for linear descriptor systems is presented. The approach is based on a new definition of partial impulse observability. The following system is considered:

   (1.22)

are known constant matrices.

Definition 10

The system (1.22) with u(t) = 0, or the triplet (C, E, A) is said to be partially impulse observable with respect to L if y(t) is impulse free for t ≥ 0, only if Lx(t) is impulse free for t ≥ 0.

The reduced-order observer for the singular system (1.22) is given by:

   (1.23a)

   (1.23b)