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## Poles and Zeros: Theoretical framework

Before developing any methodology to solve the Disturbance Rejection Problem, modelling, analysis and observer/control synthesis of the dynamical system is required (Integrated DesignApproach). These are necessary steps in the design phase of the observer as well as for th synthesis of control laws. The selected representation (Transfer Function, Space-State representation, BG, : : : ) must allow us to understand the studied physical phenomena with an accurate mathematical model. First of all, in order to study the structure of systems, a model must be chosen. System dynamics are here represented through a differential equation of order n. This differential equation can be written in matrix form and this representation is called State-Variables Model or State-Space Representation. A State-Space representation is also a particular type of Rosenbrock representation [Bourlès, 2010, Fliess, 1990]. Consider the system described by the state-space equation (1.2). ˙ x(t) = Ax(t)+Bu(t) y(t) =Cx(t)+Du(t)

### Control System: Disturbance rejection problem

Some classical control approaches are recalled in the following sections in order to compare them with the Derivative State Feedback – Bond Graph one. This last approach is then the methodology developed in this thesis. One important feature is that a structural analysis of the model must be achieved before control synthesis. We will show that very similar concepts developed for Unknown Input Observer (UIO) property analysis are as well expressed in chapter 2 in term of finite and infinite structures of the model.

#### PID (PI) Control

The birth and large-scale deployment of the PID control technology can be traced back to the period of the 1920s-1940s in response to the demands of industrial automation before World War II. Its dominance is evident even today across various sectors of the entire industry. In process control applications, more than 95% of the controllers are of the PID type, [Åmström and Hägglund, 1995, Guo and Zhao, 2016, Guo and Zhao, 2016]. The PI and PID controllers have been studied since many years and they are the most common control strategies. The Proportional-Integral (PI) Controllers have different expressions and they have the ability to eliminate steady state offsets through the integral action. Their technology has greatly changed, from analogue to digital electronics, with digital versions (discrete time) and the possibility to use micro-controllers [Chen and Seborg, 2002]. There are many methods for tuning PID controllers, most of these methods are based on the classical Ziegler-Nichols methods, [Cominos and Munro, 2002].

The classical PID control is defined in (1.11), where e(t) is the error variable, difference between the reference input signal v(t) and the output variable y(t). It is rewritten in Laplace domain as a transfer function described in (1.12). The PID controller is described by three parameters (Kp-Proportional Gain, Ti-Integral Gain and Td- Derivative Gain). With proportional control, there is normally a control error in the steady state behaviour. The main action of the integral function is to make sure that the process output agrees with the set-point in the steady state behaviour. With an integral action, a small positive error will always lead to an increasing control signal, and a negative error will give a decreasing control signal no matter how small the error is. The purpose of the derivative action is to improve the closed-loop stability. Several properties of the PID control are well studied and analysis are shown in many books in the literature as [Åmström and Hägglund, 1995].

**Static State Feedback (SSF) control**

The state-space formalism is very useful in providing both a simple and complete system representation. This type of representation is indeed simpler than “Rosenbrock representation”: see

[Bourlès, 2010]. On the other hand, within this formalism, a complete description of the system is possible (if the latter’s “structure at infinity” is left aside [Bourlès and Marinescu, 2011]). We recall the study of control by an “elementary” state feedback that is well studied and applied in different fields. Some of these works and historical research of the feedback control are in [Mayr, 1970, Chen, 1998, WilliamsII and Lawrence, 2007]. We will then extend the state feedback knowledge to the (derivative) state feedback control (next section) in order to solve the Disturbance Rejection (DR) control problem. The connection between the ability to arbitrarily place the closed-loop eigenvalues by proper choice of the state feedback gain matrix K and controllability property of the open-loop state equation, i:e:, the pair (A;B) of the system (C;A;B) is well established for systems without disturbance. In that case we can speak of non-controllable modes (poles) or as well as input decoupling zeros. This problem has received also much attention when a disturbance exists. In that case, when the disturbance rejection problem has a solution with static state feedback, the ability to arbitrarily place the closed-loop eigenvalues is related to non-controllable modes but also to some of the invariant zeros. Finite structures of models (C;A;B) and (C;A;B;F) must be compared. A structural approach from the bond graph representation has been proposed in case of control with state feedback. The algebraic approach is also well-known. When the disturbance rejection problem with state feedback has no solution, an alternative control with derivative state feedback can be proposed. We show in the following that the pole placement with the disturbance rejection problem is similar to the state feedback problem in its formulation. A similar approach is developed in terms of structural analysis and then in terms of control synthesis.

**Table of contents :**

Résumé

Acknowledgements

List of Abbreviations

Table of Contents

List of Tables

List of Figures

**General Introduction **

Thesis Layout and Summary of Thesis

Contributions of the Thesis

Journals

International conferences

**1 Disturbance Rejection Problem **

1.1 Introduction

1.2 Poles and Zeros: Theoretical framework

1.3 Control System: Disturbance rejection problem

1.3.1 PID (PI) Control

1.3.2 Static State Feedback (SSF) control

1.3.3 Static State Feedback (SSF) Control with disturbance rejection

1.4 Disturbance Rejection with Derivative State Feedback (DSF)

1.4.1 DR-DSF without pole placement

1.4.2 DR-DSF with pole placement

1.5 Experimental System

1.5.1 System description

1.5.2 State-Space Equation

1.5.3 Model Validation

1.6 Disturbance Rejection for the T-B System

1.6.1 Structural properties of the BG model: Disturbance Rejection problem .

1.6.2 Proportional-Integral-Derivative (PID) (Proportional-Integral (PI)) Control

1.6.3 Disturbance Rejection with Derivative State Feedback (DSF)

1.7 Conclusion

**2 Unknown Input Observer (UIO): Background and new developments **

2.1 Introduction

2.2 Comparison between UIO approaches

2.2.1 Some classical approaches

2.2.2 UIO: Bond Graph (BG) Approach

2.2.3 UIO Single Input – Single Output (SISO) case: Simulations

UIO: Bond Graph Approach

PI Observer

UIO: Inverted Matrices

UIO: Algebraic Approach

2.2.4 Remarks

2.3 UIO-BG: Multiple Input – Multiple Output (MIMO) case

2.3.1 Non-Square Model

2.3.2 Square Model: UIO without Null Invariant Zeros

2.3.3 Square Model: UIO with Null Invariant Zeros

2.4 Conclusion

**3 DR with estimation and I/O Decoupling with DSF **

3.1 Introduction

3.2 Disturbance Rejection – Three approaches

3.2.1 Disturbance Observed-Based Control (DOBC)

3.2.2 Active Disturbance Rejection Control (ADRC)

3.2.3 DSF-UIO-BG

3.3 Disturbance Rejection: Simulations

3.3.1 Disturbance Observer-Based Control

3.3.2 Active Disturbance Rejection Control

3.3.3 Disturbance Rejection by Derivative State Feedback using UIO

3.3.4 Analysis of the results

3.4 Input-Output decoupling with DSF

3.4.1 Regular Static State Feedback (RSSF): some properties

3.4.2 DSF for Input-Output decoupling

3.4.3 Properties of the controlled model with pole placement

3.4.4 Comparison between RSSF and DSF

3.4.5 Case study: simple mechanical system

3.4.6 Concluding remarks on Input-Output decoupling with DSF

3.5 Conclusion

**4 DR – DSF – UIO – BG: Study case **

4.1 Introduction

4.2 Real Torsion-Bar Description

4.3 Disturbance Rejection: Applications

4.3.1 PID control

4.3.2 Disturbance Observed-Based Control (DOBC)

4.3.3 Active Disturbance Rejection Control (ADRC)

4.3.4 DSF-UIO-BG

4.4 Concluding remarks

**5 Future Works: Renewal energy **

5.1 Introduction

5.2 Renewable sources: Brief summary

5.3 Hydroelectric plant

5.3.1 Hydroelectric model: Word Bond Graph

5.3.2 Mathematical description

5.3.3 Simulations

5.4 Conclusion

General Conclusion and Perspectives

**Bibliography **