H1 controller design for the vibration reduction in the central zone 

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Smart materials for active vibration control

Smart materials are materials that are able to generate strain based on the change of external physical environment such as temperature, electric field or magnetic field, etc. This is the result of the coupling effects described by constitutive equations. According to [9], there exist several popular smart materials such as Shape Memory Alloys (SMA), Magnetostrictive materials, Magneto-rheological (MR) and Piezoelectric materials. Here, we will give a brief introduction of these smart materials as well as their properties which gives the choice of the type of material used in this research.
Shape Memory Alloys (SMA) are able to ‘remember’ its original shape and return to its predeformed shape when heated. They are only sensitive at low frequencies with low precision. As a result, they are little used in active vibration control. However, we could still find some applications, for example in [37].
Magnetostrictive materials generates strain under magnetic field. They can be used in compress situation as load carrying mechanisms [38] because it has maxi-mum response when it is subjected to compress load. TERFENOL-D is the most popular magnetostrictive material. In some applications like sonar, TERFENOL-D can be an alternative choice of PZT.
Magneto-rheological (MR) is a particular viscous fluid that contains particles of magnetic material in micron-size. They are mainly used in semi-active vibration control of suspension systems [39].
Piezoelectric materials can be used as actuators as well as sensors because of its bidirectional piezoelectric effect. Ceramics and Polymers are two classes of piezoelectric materials mainly used in vibration control. The piezopolymers are not usually used as actuators because of the high voltage requirement and the limited control authority. Thus, they are mostly used as sensors. The best known piezopolymer is the Polyvinylidene Fluoride (PVDF). The application of PVDF in active vibration control can be seen for example in [40, 41]. The piezoceramics can be used as both actuators and sensors. They are applicable for a wide frequency band, till ultrasonic frquencies in some applications. They also have high precision up to the nanometer range1 [9]. The most popular piezoceramic is the Lead Zirconate Titanate (PZT). PZT patches can be glued or co-fired on the target structures which makes it very applicable for light structures. Piezoelectric materials are the most commonly used smart materials in active vibration control [10].
Considering the properties of these smart materials, we conclude that the piezo-electric materials, especially the PZT patches, are the most suitable for active vibration of light weighted flexible structure such as an aluminum beam consid-ered in this research. It has advantages over other materials in many aspects such as mechanical simplicity, small volume and lightweight, wide working frequency band, high precision, and the ability of high level integration in the structure. As mentioned above, piezoelectric materials can be used to control the vibration because of its bidirectional piezoelectric effect which is first discovered by the Curie brothers in 1880 [42, 43]. In particular, they find that squeezing particular materials leads to the change of electric charge. This phenomenon allows piezo-electric materials to be used as strain sensors. On the other hand, the converse 1Up to 1nm = 10−9m effect is also possible which means that it is able to generate a mechanical strain under the application of an electric voltage. This allows the use of piezoelectric materials as actuators. Active vibration control using piezoelectric materials has attracted considerable interest of the researches during the past few decades. To design efficient piezoelectric smart structures for active vibration control, both structural dynamics and control laws have to be investigated as introduced in the next section.

Active vibration control of flexible structures

As far as the researches about active vibration (centralized) control of flexible structures, various methods have been developed in the past few decades. Gen-erally, the design process for an active vibration control problem involves many steps. A typical scenario is as follows [44]:
1. Analyze the property of the to-be-controlled structure.
2. Deduce a reliable mathematical model of the to-be-controlled structure by using techniques such as Finite Element Analysis or data-based modeling method (e.g. the identification techniques).
3. Reduce the model (if necessary) such that it has less degree of freedoms and lower dimension2. 4. Analyze properties, dynamic characteristics of the resulting model. Properly define the disturbance.
5. Quantify requirements for sensors and actuators and decide their types as well as their locations on the structure.
6. Analyze the effect of the chosen actuators and sensors on the global dynamics of the structure.
7. According to the control objectives, quantify performance criterion and sta-bility trade-offs.

Distributed control of flexible structures

For the active vibration control problem, the general objective will always be controlling as many modes as possible with the best performance. Such objec-tive could be achieved by giving smart structures more actuation and sensing capabilities which requires the use of more actuators and sensors. However, in the presence of a large number of actuators and sensors, designing a centralized controller will require high computational cost and high level of physical connec-tivity. An alternative way is to design distributed controllers which is referred to as the distributed control. Distributed controllers work in a different way from centralized controllers in that one controller only reacts with a local part of the structure, for example connects with a few number of actuators and sen-sors that are close to each other which forms a local control unit. Several such controllers working together to achieve a global control effect. For this purpose, the smart structure will be spatially discretized and thus can be considered as a spatially interconnected system (or spatially distributed system) [85]. Each subsystem will have actuation and sensing capabilities which allow it to be con-nected with a distributed controller. Moreover, neighbor controllers will change information between each other which helps them to achieve global control ob-jectives. Fig. 2.1 [85] shows the architecture of such spatially distributed control system where G denotes the interconnected subsystems of the smart structure and K the distributed controllers. There also exists a similar control architecture which is the so-called decentralized control [86] where different controllers do not communicate with each other. However, it has been proven on an actuated beam in [85, 87] that the distributed control can achieve better performance than the decentralized control.

Control objectives and considerations

The general objective is to use the smart materials as actuators and sensors to particularly reduce the vibration at a specific zone of a flexible structure by only using the measurements in other zones, which can be interpreted as to tackle unmeasured performance variables (see e.g. [102] for an example in another context). As to our experimental setup (Fig. 3.1), we aim at designing a feedback controller (centralized or distributed) that allows to significantly reject the vibration energy from the central zone (the part with length L2 shown in Fig. 3.2) when the beam is subject to an external force disturbance at one end of the beam.
The disturbance is applied along z-axis and thus only the vibration in z-axis is of interest. As to the frequency band of the disturbance, if we forget the complexity of the feedback controller, the ideal goal is of course to reject a dis-turbance with a power spectrum density (PSD) which covers the largest possible frequency band, in other words, to control as many vibration modes as possible. However, the general control ability and the maximum possible performance will necessarily depend on the sensitivity of the actuator, which means that we can only effectively control the modes within the working range (sensitive range) of the actuator. In our case, the used piezoelectric material is Pz26 (see, Table 3.1) which has a very large working range. The force disturbance applied by the elec-trodynamical shaker is chosen to have high PSD in (600, 3000)1 rad/s which is within the working range of the PZT. This bandwidth covers 11 vibration modes2 and the modes outside this bandwidth will thus not be controlled. Consequently, the disturbance is chosen to have large PSD in (600, 3000) rad/s. To the best of our knowledge, this dissertation is the first one proposing a technique that aims to significantly reduce a vibration covering a frequency band with as many as 11 modes in a specific zone free of actuating and sensing transducers.
As to the controller design, the challenging objective of controlling such many modes requires the use of modern multi-variable control design methods. The first reason for this is that modern control design methods allow us to tackle unmeasured performance variables such as the vibration in the central zone where there is no sensor. Second, as it will be discussed in Section 3.3, in order to successfully control the 11 modes in (600, 3000) rad/s, at least two sensors and two actuators are necessary for a centralized controller. We thus have to design a multi-variable (MIMO) controller. As for distributed control, all the 20 PZT pairs should be used which requires even more complex control design methods.
Some other considerations must also be taken into account. First, in order to guarantee the designed control effect on the actual setup, an accurate model is of crucial importance, especially for active vibration control. There are ways to build the theoretical model knowing the values of the physical parameters and geometry of the materials (i.e. Table 3.1). However, there are always errors and it will be further illustrated in Section 4.1.6 that the theoretical model is far from being accurate enough. Therefore, model correction is necessary, for example, using the identification technique which corrects the model using the measurement data from the actual setup and yields an updated model which will have much better accuracy. Second, to ensure a good feasibility of the implementation, the order of the controller should be relatively low, which implies that a low-order model containing only the modes in (600, 3000) rad/s should be used to compute the controller because, as we will see in the sequel, the identified model will have to contain modes outside the frequency band of interest (called the full-order model in the sequel). This introduces a robust stability requirement that the low-order model based controller must also guarantee the 1About (95, 477) Hz stability when applied on the full-order model. This is also to avoid the so-called spill-over problem3. Third, the controller should have reasonably high magnitude in (600, 3000) rad/s to ensure a high vibration reduction rate while relatively low magnitude outside (600, 3000) rad/s to limit energy consumption. For the same reason, we should also take care that sensor/measurement noise (usually located at high frequencies) has limited effect on the control output.

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Centralized controller implementation

The designed controller will be a centralized controller (a model-based feedback controller) which collects together all the measurements (sensing voltage) from the PZT pairs used as sensors and then computes control signals (actuation volt-age) for all the PZT pairs used as actuators. Thus, the more PZT pairs we use, the more complex the controller will be, which increases computational burden and energy consumption of the control board. Thus, we need a minimum number of PZT pairs to obtain a maximum accessible performance. Considering the large bandwidth of the disturbance (i.e. (600, 3000) rad/s), we show in Appendix A that we need at least two sensors and two actuators (a SISO controller is there-fore not sufficient). In Appendix A, we also show that an appropriate choice for these two actuators and these two sensors is to select the 10th and 16th PZT pairs as actuators, the 5th and 11th PZT pairs as sensors (see Fig. 3.4 for the location of these PZT pairs).
The type of all the control devices are listed in Table 3.2 and their functions are detailed as follows. The designed controller will be implemented in a pro-grammable digital control board which should be able to simulate the input-output behavior of transfer functions. Like all the other digital signal processors, this control board is equipped with ADC/DAC card in each input/output chan-nel for sampling/constructing the actual input/output signal. However, it is not equipped with internal anti-aliasing filters. Therefore, extra anti-aliasing filters are necessarily used to sample the outputs of the PZT sensors and the filtered voltage signals are collected by the control board through its own ADC cards.

Methodology overview for centralized controller design

In order to design a centralized controller that achieve the control objectives described in Section 3.2 with a performance well guaranteed when applied on the actual setup, we propose the following methodology (based on H∞ control and all other modern control design techniques).
Fig. 3.6 gives a general overview of this methodology and each step will be detailed in the following chapters. First, the theoretical model of the beam-piezo system is derived using the commercial software COMSOL which performs Finite Element Analysis and modal decomposition to a virtual 3D beam model. Only a finite number of vibration modes (chosen by the user) can be tackled. This results in a state-space model that is only valid up to a certain frequency band (corresponds to the user-chosen maximum vibration mode) which is here chosen slightly larger than the maximal frequency of the disturbance i.e. 3000 rad/s. In this state-space model, the output vector is not only made up of the voltages at the PZT pairs selected as sensors, but also of the (vibration) velocities at a number of locations in the central zone. An expression for the vibration energy in the central zone can indeed be derived from these velocities. Then, the model parameters, for which COMSOL gives a rough initial estimate, are tuned using grey-box identification in order to obtain a model with better accuracy4, yielding the so-called full-order model of the system. The effect of all the control devices is also considered in the identification process. This (full-order) model covers a frequency band that is larger than the frequency band of interest (i.e.

Table of contents :

I Dissertation 
1 Introduction 
1.1 Motivation of this research
1.2 Approaches of this research
1.3 Publications
1.4 Organization of this dissertation
2 Background 
2.1 Smart materials for active vibration control
2.2 Active vibration control of flexible structures
2.3 Distributed control of flexible structures
2.4 Summary
3 System description 
3.1 Experimental setup
3.2 Control objectives and considerations
3.3 Centralized controller implementation
3.4 Methodology overview for centralized controller design
3.5 Overview of modeling for distributed control
3.6 Summary
4 Modeling for centralized control 
4.1 State-space modeling of beam-piezo system
4.1.1 Governing equation deduced with COMSOL
4.1.1.1 Finite Element Modeling
4.1.1.2 Modal decomposition and truncation — Modal Displacement Method
4.1.1.3 Piezoelectric capacitance correction—Static correction
4.1.2 Damping effect
4.1.3 Measurement circuit
4.1.4 Determination of the central energy
4.1.5 State-space representation
4.1.6 Application to the experimental setup
4.2 Model improvement using Grey-box identification
4.2.1 Grey-box identification theory
4.2.2 Grey-box optimization of the beam-piezo system model
4.2.3 Application
4.3 Multi-variable model reduction
4.3.1 Overview of model reduction methods for mechanical structures
4.3.2 The proposed method
4.3.3 Modal form truncation
4.3.4 Relative error minimization
4.3.4.1 Relative error selection
4.3.4.2 Formulation and solution of the minimization problem
4.3.5 Application to the beam-piezo system model
4.4 Summary
5 Centralized controller design 
5.1 H1 control approach
5.1.1 Performance and criterion
5.1.2 Robust stability
5.2 H1 controller design for the vibration reduction in the central zone
5.2.1 Vibration reduction problem statement
5.2.2 Augmented system and control criterion
5.2.3 Controller reduction and discretization
5.2.4 Application to the beam-piezo system
5.2.4.1 Augmented system
5.2.4.2 Weightings and results
5.2.4.3 Reduced-order controller
5.2.4.4 Performance — Central energy reduction rate
5.2.4.5 Numerical simulation
5.3 Experimental validation
5.4 Discussion in single-variable case — Limitation of SISO controllers
5.5 Summary
6 Modeling for distributed control 
6.1 Distributed modeling overview
6.2 Governing equation simplification for structural cells
6.2.1 Guyan condensation
6.2.2 Euler-Bernoulli kinematic assumption
6.3 Application to the structural cells
6.3.1 Cells with piezos
6.3.1.1 Simplified governing equation
6.3.1.2 COMSOL solutions
6.3.2 Homogeneous cells
6.3.3 Assembly
6.4 State-space modeling of interconnected LTI subsystems
6.4.1 Discussions
6.4.2 Architecture of the interconnected system
6.4.3 Construction of LTI subsystems
6.5 Appendix
6.6 Summary
7 Conclusions and future research 
II Résumé en français 
1 Introduction 
1.1 Motivation de cette recherche
1.2 Contexte
1.3 Organisation de la thèse
2 Description du système 
2.1 Installation expérimentale
2.2 Objectifs et considérations
2.3 Implémentation du contrôleur
3 Contrôle centralisé 
3.1 Méthodologie
3.2 Modélisation
3.3 Correction du modèle
3.4 Réduction du modèle multi-variable
3.4.1 Troncature de la forme modale
3.4.2 Minimisation de l’erreur relative
3.4.3 Application
3.5 Conception du contrôleur par contrôle H1
3.5.1 Cahier des charges
3.5.2 Critère H1
3.5.3 Réduction et discrétisation du contrôleur
3.5.4 Application et résultats
4 Modélisation pour le contrôle distribué 
4.1 Modélisation des cellules
4.1.1 Condensation de Guyan
4.1.2 Hypothèse cinématique d’Euler-Bernoulli
4.1.3 Application
4.2 Modélisation des sous-systèmes
5 Conclusions et recherches futures 
A PZT selection 
B Construction of the weighting for robust stability 
References 

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