This thesis consists of eight chapters. A brief overview of the content in each chapter is given below:
Chapter 2 reviews the framework of spatially-interconnected systems. Definitions on the multidimensional signal and system norms are given as a preliminary. The multidi-mensional state space representations, that are employed throughout the work, proposed in [8] for LTSI systems, in [21] for LTSV systems, as well as their correspondent con-trollers, which inherit the distributed nature of the plant, are presented. The definitions of the well-posedness, exponential stability, and quadratic performance in the context
of distributed systems are discussed. The analysis conditions for an LTSI system to be well-posed, exponentially stable, and with the imposed performance criteria satisfied are provided for both continuous and discrete systems.
Chapter 3 takes physical aspects of the experimental structure into consideration, re-viewing the functionality of piezoelectric patches as actuator and sensor, respectively.
Linear constitutive equations are applied to analyse the linear dynamics of both the piezo actuators and sensors. The application of a piezoelectric FE modelling approach yields a theoretical FE model characterized in terms of mass and stiffness matrices, based on known and assumed knowledge on the physical properties of the actuated beam. To reduce the deviation between the theoretical FE model and test structure, experimen-tal modal analysis is performed to update the mass and stiffness matrices at first, then the proportional-assumed damping matrix. Meanwhile, a direct feed-through effect is observed from actuators to collocated sensors.
Instead of exploring the inherent physics of a flexible structure using the FE mod-elling, Chapter 4 identifies a structure through identifying its FRF matrix from the input/output measurements. It is demonstrated step by step, that even for a structure comprised of identical subsystems, its FRF matrix exhibits spatially-varying character-istics. A local LPV identification technique for temporal systems is extended to spatio-temporal systems to capture the spatially-varying properties of FRFs. Actuating and sensing at selected locations results in a set of measured FRFs, each being estimated as an LTI model using a least-squares-based identification technique. The application of the extended local LPV approach parametrizes the set of estimated LTI models as a spa-tial LPV model by defining the spaspa-tial coordinates of actuating and sensing locations as spatial scheduling parameters. The proposed approach allows to perform identification ex-periments at a small number of selected actuating and sensing locations, and parametrize a spatial LPV model. Then unknown FRFs at other locations can be easily approximated through interpolation. The proposed approach is tested experimentally.
Both the obtained FE model in Chapter 3and the identified FRF matrix in spatial LPV representation in Chapter 4treat the plant as a MIMO lumped system. Chapter 5deals with the identification problem in the context of spatially-distributed systems. A two-dimensional input/output model induced by the temporal and spatial discretization of governing PDEs is considered as the mathematical model for identification. It describes the dynamics of a spatially-discrete subsystem interacting with nearby subsystems. Black-box identification techniques for the identification of LTSI and LTSV models are briefly reviewed, and experimentally implemented. To improve the model accuracy, especially at resonant peaks, a new identification procedure which makes use of the FE model obtained in Chapter 3is proposed. Both the identified LTSI and spatial LPV models preserve the two-dimensional input/output structure, and suggest a better representation of the plant dynamics than black-box identification.
Based on the input/output models identified in Chapter 5, Chapter 6 solves the con-troller design problem for both the LTSI and LTSV systems. In order to employ the well-developed state-space based analysis and synthesis conditions, the experimentally identified input/output models are first converted into their multidimensional state space
realizations. The construction of a multidimensional generalized plant for shaping the mixed sensitivity of the closed-loop system is discussed. The synthesis conditions of a distributed LTSI controller are briefly reviewed. Both a distributed and a decentralized controller are designed and implemented, with their performance compared experimen-tally. The synthesis conditions of temporal/spatial LPV controllers for LTSV systems are derived with the application of the full block S-procedure (FBSP), using both the CLFs and PDLFs. The experimental results demonstrate a superior performance of the LPV controller designed using PDLFs.
Chapter 7 addresses a two-step distributed AW compensator design in the presence of actuator saturation in physical systems. A lumped AW scheme is first revisited. The definition of a mathematical tool – integral quadratic constraints (IQCs) [42] – and its application to the robust analysis of an LFT model with a nonlinear uncertainty is shortly recapped. Inspired by the lumped setup, a distributed AW scheme, which preserves the distributed nature of the plant and the controller, is proposed. The stability of the closed-loop subsystem in LFT form, with the nonlinear deadzone operator as uncertainty, is analysed using IQCs. The synthesis conditions are derived after applying the elimina-tion lemma. The performance of the distributed AW compensator is illustrated using a simulation example, in comparison with a decentralized AW scheme.
In Chapter 8, conclusions to this thesis are drawn; and an outlook for future research is given.
Spatially-Interconnected Systems
2.1 Introduction
In this chapter, relevant preliminary materials regarding spatially-interconnected systems are briefly reviewed. In Section 2.2, signal and system norms and shift operators in the context of spatially-interconnected systems are extended from their lumped counterparts.
Instead of considering the distributed-parameter system as a large-scale lumped MIMO system, the distributed framework proposed in [8], where a spatially-distributed system can be seen as an array of interconnected subsystems, is presented in Section 2.3. The system dynamics are defined at the subsystem level using a multidimensional state space representation. Depending on the physical properties of subsystems, such a system can be either LTSI or LTSV, where subsystems in an LTSI system share identical dynamics, whereas the varying dynamics of an LTSV system can be captured using temporal/spatial-LPV models. It is desired that the controller inherits the distributed feature of the plant.
The controller structures for both parameter-invariant and parameter-varying systems are given in Section 2.4. In Section 2.5, the well-posedness, exponential stability and quadratic performance are defined for spatially-interconnected systems, respectively. The analysis conditions that establish well-posedness, stability and performance specifications are stated in terms of LMIs.