A Study on FEM Generated Large Scale Dynamic Systems. Syed Fawad Raza Ali Bokhari

Transcription

1 A Study on FEM Generated Large Scale Dynamic Systems Syed Fawad Raza Ali Bokhari

2 Declaration Undersigned born on 15 th December 1978 in Rawalpindi hereby declare that this submission is my own work and that, to the best of knowledge and belief, it contains no material previously published or written by another person nor material which to a substantial extent has been accepted for the award of any other degree or diploma of the university or other institute of higher learning, except where due acknowledgment has been made in the text. Syed Fawad Raza Ali Bokhari 1

3 Acknowledgment I am so much humbled and thankful to The most merciful and beneficent GOD, Who provided me the energy to study and opportunity to experience the life and to learn. This thesis is the fruit of valuable life time guidance of my father Engr. Syed Ahsan Raza Ali Bokhari and maternal uncle Engr. Syed Sajjad Haider Bokhari; whose personalities has always been a guidance for me in all aspects of life, and prayers & good wishes of my mother M.A. Syeda Nighat Batool Bokhari, that she extended for me all the time. All my achievements are of them. I pay my hearties gratitude to them and pray for their long and happy life. efforts of my younger brother Engr. Syed Farhan Raza Ali Bokhari, that he is giving due love and attention to my parents and giving them support hence providing me the mental piece, here abroad, to excel in my career by doing masters studies. May God give him every success and happiness in life. I am extremely thankful to Prof. Dr. Herbert Werner for facilitating me with the opportunity to exploit my capabilities in the demanding environment of idea generation, work accomplishment and discipline at Institute of Control systems. I am greatly thankful to my supervisor M.Sc. Saulat Chughtai. By the virtue of his esteemed guidance I am able to learn and complete this project successfully. He always made the environment in which I had to learn the independent research attitude, which is really valuable for me for all times to come. I am grateful to my wife Iffat whose moral support has always supported me to achieve accomplishments. In the last I would appreciate all the friends Hossam, Mukhtar, Badar, Hendrik, Aron, Santiago, Meike, Maria, Sven who were really good company in the control department. 2

4 Contents 1 Introduction Motivation Thesis Objectives Thesis Outline Large Scale Dynamic Systems Interconnected Systems Decentralization Restriction for feedback control of Large scale systems Decomposition Models and Structure of Interconnected Systems Unstructured [26] I/O-oriented model [26] Interaction-oriented model (D Siljak Approach) Decomposition into disjoint subsystems Vidyasagar Approach Raffaello D Andrea s Approach Interconnected System & Finite Element Methods (FEM) Finite Element Methods (FEM) Continuous and Discrete Systems FEM in Mechanical Systems Steps of Finite Element Method (FEM) Discretization Element displacement field or shape function The stress strain relationship Element Matrices Coordinate transformation Matrix Assembly & Solution FEM model of Cantilever Beam MATLAB CODE Decomposition of FEM based Large Systems Philosophy of Decomposition Algorithm Indices of G i, i th Diagonal part Indices of G vi, i th interconnection part Applying decomposition function decomp() on FEM generated system Algorithm

5 CONTENTS Decomposing the FEM Model of a Cantilever Beam Active Vibration Control of Beam Structures; Example Design of Decentralized Robust Control Transformation of Augmented large scale dynamic system into form suitable for decentralized controller Problem Formulation for Active Vibration Control of different Beam Structures Control Objective Cantilever Beam (Case # 1) Simply Supported Beam (Case # 2) Special Case (case # 3) Steps involved in Decentralized Robust Controller Synthesis for Active Vibration Control of Beam Structures Results, Discussion & Conclusion Accuracy of decomposition algorithm Systems Response Active Vibration Control of Beam Structure Conclusion Future Directions & Ideas Conversion of FEM generated systems into descriptive form Active Noise Control Decentralized H 2 and H controller for FEM based system A Constraints on Sensitivity and Control Sensitivity 68 A.1 Output Disturbance Sensitivity & Control Sensitivity A.2 Input Disturbance Sensitivity & Control Sensitivity B MatLab Toolbox for decomposition 71 B.1 decomp() function B.2 Example B.2.1 G i and G vi for 1 st subsystem B.2.2 G i and G vi for 2 nd subsystem B.2.3 G i and G vi for 3 rd subsystem

6 List of Figures 1.1 Basic steps in FEM Decentralized Control Scheme Vidyasagar Approach Basic Building Block i th subsystem Force on a cantilever beam GUI program for FEM Block diagram of Algorithm of Decentralized Controller through Homotopy Method Nodes of Cantilever beam Nodes of Simply supported beam Nodes of beam Sum of percentage error between the outputs of interconnected and lumped system Control & Input disturbance sensitivity Three different cases of beam structure Eigen Values of Cantilever & simply supported beam structure Eigen values of special case, Decentralized control scheme Singular value Plots for Cantilever Beam Singular value Plots for simply supported beam Singular value Plots for special case Minimum γ A.1 Input Disturbance A.2 Generalized Plant for Input disturbance rejection

7 List of Tables 4.1 Characteristics of a cantilever beam Proposed gain for generalized plant of different beam structures Inputs & Outputs of beam Characteristics of a cantilever beam Inputs & Outputs of beam Inputs & Outputs of beam Inputs & Outputs of beam

8 Abstract Finite Element Methods (FEM) play a key role in system model development. Models generated from FEM are of very high dimension and are difficult to handle for control system design. In this thesis, an algorithm is presented that converts a large scale dynamic system i.e. system generated from FEM, into the state space model of an interconnected system, thus vastly reducing the complexity of controller synthesis. A previously proposed method, which involves the solution of NMI by using a homotopy method, is used to synthesize the dynamic output feedback decentralized controller. This NMI is solved iteratively, by alternatively fixing some of the variable to convert the NMI into LMI. Structural constraints on matrix variables are introduced to compute the decentralized control. A simple cantilever beam problem has been taken as an example to illustrate the proposed decomposition methods and two other beam structure along with cantilever beam are considered for implementation of decentralized robust controller. A state space representation of these system is obtained through FEM analysis. This model is then converted from lumped state space to interconnected form. Decentralized robust controllers are designed to reject input disturbances by putting appropriate constraint on input disturbance sensitivity and control sensitivity. It was found that marginally stable, coupled, oscillatory system become stable and decoupled. 7

9 Chapter 1 Introduction In many applications the control objective is not only to control any physical variable in time but also in space. One such example can be the control of temperature profile over a large place. One way of looking at these system is to consider it as a large multivariable system. However if one attempts to control these systems using standard control design techniques, severe limitations, in terms of substantial computational burden, sensitivity to failures and modeling errors, will quickly be encountered [2]. Furthermore most optimal control synthesis techniques cannot handle systems of very high dimension and with a large number of inputs and outputs. Model order reduction can be a solution but this may causes the unwanted close loop dynamic behaviour (Control spillover). This necessitates the development of distributed controller. This chapter is organized as follows: Motivation and reference literature are discussed in Section 1.1. Point wise thesis objectives are discussed in Section 1.2. Finally the thesis outline is provided in Section Motivation Engineering systems in which subsystems actively interact with each others, have become of an interest for the scientific and engineering community for last three decades[4]. With the advancement in the design techniques and computational capabilities more complex system models of very high dimensions are being developed. Examples include, automated highway systems [11], airplane formation flight [12], satellite constellations [13], cross-directional control in paper processing applications [14], micro-cantilever array control for massively parallel data storage [15]. One can also consider lumped approximations of partial differential equations (PDEs) examples include the deflection of beams, plates, and membranes, and the temperature distribution of thermally conductive materials [16]. Systems whose dynamics are based on partial differential equation are continuum problems and these are based on assumption that all bodies of interest are continuous. Solution to these type of problems are possible for very simple systems but for practical systems Finite Element Method (FEM) is a very effective tools that can solve these type of problems numerically[7, 8, 9], and converts the systems of infinite dimensions (continuous system) into finite dimensional systems (Finite elements). Areas of their application include structural analysis, heat transfer, fluid mechanics, electromagnetism, bio mechanics, geomechanics, acoustic etc [9] 8

10 CHAPTER 1. INTRODUCTION 9 In FEM a complete system is divided into small elements which are spatially distributed and interacts with each other based on their spatial location with respect to each other. In structural analysis behaviour of system under consideration is described by displacement of elements satisfying material laws (constitutive equation) [7, 8, 9]. All elements are assembled together and requirement of continuity and equilibrium are satisfied between neighboring elements. Unique solution can be found for boundary conditions. Fig 1.1 explains some basic steps to generate FEM based the state space Figure 1.1: Basic steps in FEM An important aspect of many of these systems, i.e. FEM based systems, is that sensing and actuation capabilities exist at every unit. Inherent properties (high level of interconnections and information flow) of these systems of very high dimension and with a large number of inputs and outputs makes them infeasible to control with centralized schemes the typical outcome of most optimal control design techniques as high level of connectivity, imposes a substantial computational burden and in many cases it is difficult to be implemented[2]. Model order reduction is among one of the solutions but it may induces control spillover which is defined as Excitation of unmodeled higher modes, causing unwanted structural vibrations in the closed loop So this reduction may eliminate some aspects of the dynamic behavior which may get exited in the closed loop, thus degrading the achieved performance. In order to avoid control spillover due to reduction in dimensions very fine meshes (more elements) are required which may results in very large number of states for which control system design, using the modern control design techniques, imparts limitations during synthesis and implementation. Large arrays of sensors and actuators introduce large numbers of inputs and outputs which lead to severe limitations as many of the standard multi-variable control techniques are inadequate in handling systems with high dimensions [2]. A control design based on a distributed model has the potential to eliminate the effects of control spillover since it is directly based on the mathematical model that represents the actual physical system [22]. Moreover, a distributed model is suitable for decentralized control schemes due to the existence of distributed and decentralized sensing and actuation. So the necessity to develop the advanced distributed/decentralized control design techniques which not only have the potential to reduce the above mentioned implementation & computational limitations but also to eliminate the control spillover is the motivation behind this thesis.

11 CHAPTER 1. INTRODUCTION 10 In this thesis an approach has been proposed to decompose the model used for control synthesis into subsystems; of form where each subsystem interacts with other subsystems in terms of information flow, rather than reducing the order of system. This decomposition can facilitate the use of modern controller synthesis techniques for large systems, using the framework of interconnected systems. Then each subsystem has lesser number of states and has its own sensor and actuator. A general interconnected systems composed of N subsystems can be represented in its state space form as, Where 1 = 1, 2,...N. ẋ i (t) = A ii x i (t) + B 1i w i (t) + B 2i u i (t) + z i (t) = C 1i x i (t) + D 12 u i (t) y i (t) = C 2i x i (t) + D 21i w i (t) N j=1,j i A ij x j (t) (1.1) x i R ni, w i R ri, u i R mi, z i R (li+mi) and y i R li are the states, disturbance input, controlled input, controlled output and measured output of the i th subsystem. Here, the matrix A ij represents the interconnection of i th subsystem with j th subsystem. Systems which can be represented like (1.1) are extensively studied and efficient synthesis tools had been proposed to design control systems for such systems. Some earlier work on such systems, is presented in [20]. In [2] authors have shown that well known Kalman-Yaqubovich-Popov lemma can be generalized for class of interconnected systems with identical subsystems having interconnection only with their near neighbours. In [18] a designed method is proposed to design decentralized controller for systems represented by the form (1.1). This method involves the solution of NMI, by using a homotopy method, and is used to synthesize the dynamic output feedback decentralized controller. This NMI is solved iteratively, by alternatively fixing some of the variable to convert the NMI into LMI. Structural constraints on matrix variables are introduced to compute the decentralized control. Structural constraints on matrix variables are introduced to compute the decentralized control. Modification of controller for time-invariant, norm bounded uncertainties is also included by the authors[18]. 1.2 Thesis Objectives The main idea explored in this thesis is to decompose a FEM generated model into a system suitable for control synthesis thus retaining all the dynamics of the systems and to apply suitable decentralized control system design for decomposed interconnected system. The main objectives of the thesis were Understanding of Finite Element Methods (FEM) as a large scale dynamic system To understand the structure in the mathematical equation generated through FEM so that it can be exploited for controller synthesis for FEM based large scale systems. Basic commercial software usually provides end information to user which includes, stresses, displacements, velocity, acceleration etc. In order to get further information basic FEM theory is studied and modelled in MATLAB. Benefits of doing this is to get the flexibility of information excess, which we need.

12 CHAPTER 1. INTRODUCTION 11 Study of interconnected systems and development of tools to convert the large scale dynamic systems. FEM based lumped systems were analyzed. An algorithm is developed to convert an FEM generated system into interconnected form. This algorithm was applied on the beam problem and was found to be accurate up to the numerical precision. Synthesis of decentralized controller for the interconnected systems. By using [18] decentralized controller for the active vibration control of flexible beam, is designed and implemented. A software tool is developed to declare of LMI variable appropriately as per the information of interconnection. Suitable weighting filters are applied on the systems. This system is then used in the algorithm proposed by [18]. 1.3 Thesis Outline 1. Preliminary background & references and motivation of the project are discussed in Chapter 1. Thesis objectives and summary of next coming chapters are discussed in it. 2. Interconnected systems, large scale dynamic systems, their structure, are touched in Chapter FEM are typical of large scale systems, its theory is discussed in Chapter 3 to understand the structure of FEM based mechanical system from control point of view. A cantilever beam is considered as an example and its FEM model is developed. This basic theory of FEM is modelled in MatLab and also discussed in this chapter. 4. Detailed derivation of the algorithm developed for the decomposition of large scale system is discussed and presented in Chapter 4. Implementation of this algorithm on cantilever beam model developed in chapter # 3, is also discussed in this chapter. Decomposition algorithms are applied to convert it into interconnected subsystems. 5. A feasibility solution of proposed NMI [18] is discussed in Chapter 5. Three different cases of beam structure have been taken as examples. State space representation of these systems were developed through the modelled software as per steps of FEM theory. Design of decentralized control scheme, by using homotopy method is implemented for active control of vibrations due to input disturbances. 6. Results & discussions on the accuracy of decomposition algorithm and active vibration control of input disturbances for three beam structures are discussed in Chapter Future outlook is discussed in Chapter There are two appendices (a) Appendix A focuses on some background about the control & sensitivity constraints for both input and output disturbances. (b) Appendix B focuses on MatLab tool for system decomposition, which is developed to implement the decomposition algorithm.

13 Chapter 2 Large Scale Dynamic Systems There are many engineering applications in which system is composed of high dimensional and complex interacting subsystem. These systems are called large scale dynamic systems. Generally it is not possible to make a complete model for an overall large-scale system, however an approximation can be made which may describes only a part of the whole system. Controller synthesis and stability analysis for these systems becomes usually very complex. System are controlled not by a single unit but by several separate feedback loops, each of which deals with only a subset of the measured signal and operate on a subset of the actuators [26], i.e. severe limitation in terms of computational burden will arise which encourages to decompose and divide the analysis & control problems of the overall system into independent subproblems and to deal with the plant uncertainties of the system to be controlled. Furthermore, solutions have to be made, for analysis and design tasks, so that different decision maker (control agent) may only communicate in a restricted way. In this regard it becomes necessary to exploit the structure of decentralized/distributed decision making for the analysis of interconnected systems. Solution can be in not visualizing these system as a whole but having a view from subsystem point of point of view. One can visualize the whole system as interconnected subsystems interacting with each other in such a way, that over all system response is same. This chapter is organized as follows: Basic overview and salient features of interconnected systems are discussed in Section 2.1. An idea of decentralization in large scale dynamic systems is presented in Section 2.2. Restrictions for feedback control of large scale systems are summarized in Section 2.3. Basic motivation and idea of decomposition of overall system into smaller interacting sub-systems is discussed in Section 2.4. Different models and structures of interconnected systems have been briefly touched in Section 2.5. D Siljak, Vidyasagar, and D Andrea s approach for the interconnected system modelling is discussed in Section 2.6, 2.7 and 2.8 respectively. Discussion on the points which leads us to go for D Siljak approach when applying the frame work of interconnected systems on FEM based systems is discussed in Section

14 CHAPTER 2. LARGE SCALE DYNAMIC SYSTEMS Interconnected Systems The theory of large-scale system deals with the problem of large size of the system to be controlled, model uncertainties, and constraints on the information structure. The idea of information structure plays the fundamental role for the system being small or large. The structure describes the way in which a priori and a posteriori information is being communicated between the control agents. Basic idea is to utilizes internal structure of interconnected systems which may be used to synthesize the decentralized controllers and reduce the requirements for accuracy of the plant model. Fundamental approach is to explore the possibility to break down a given control task into manageable sub-tasks which may communicate with each other and may, therefore, be solved by coordinating control agents. [26] In this regard it is important to note that, classical and modern control theory strictly assumes of a unique plant & single unit controller and decision making is based on the whole plant, i.e. all sensor data is available and all input signal is provided for control decision. This unit can be thought of as centralized decision maker. However, this crucial assumption is difficult to satisfy in the case of large scale dynamic system. Too large and complex systems pose a substantial computational burden which causes a problem that these systems cannot be solved efficiently by the principles and methods of multivariable systems and control theory and even the basic assumption of multivariable control are far from being satisfied. The number of subsystems may exceed up to non manageable numbers. As a result it becomes almost impossible to get a complete model (a priori information) and a complete set of measurement data (posteriori information) for a centralized control scheme. One approach will be to break the overall system into different coupled/semi coupled subproblems. Consequently, several independent controllers (depending upon the number of sub-problems), which may interact with each other in a defined and restricted way, will make the decision making instead of a single centralized controller. [26] Three main features of large scale interconnected systems are as follows High Dimensionality These systems 1. Contain large number of inputs and outputs 2. have dynamic and complex interactions 3. Exposed to many external disturbances. As a results the system model has a large dynamical orders comprising of several system parameters. When implemented, the real-time constraints make the dimensionality problem more severe. Therefore it is uneconomical or even impossible to solve, design or control these problems as a whole thus owing to the requirement of simplification or decomposition of overall system into smaller ones.[26] Uncertainties System dynamics, described by some mathematical model, cannot be completely described due to uncertainties. These uncertainties are due to following reason. 1. A large dynamic system cannot be completely identification. It must be simplified to manage analysis and control problems. 2. It impossible to setup a model that is valid for all operating points. So explicit characterization of the model errors and a its consideration, while solving analytical and design tasks, is a requirement. [26]

15 CHAPTER 2. LARGE SCALE DYNAMIC SYSTEMS 14 Complex Solution for large scale systems must be reasonable or advantageous rather than being optimal with respect to single objective, i.e. satisfying different and partly contradictory requirement, are more important than optimality. Different directions of conceptual difficulties, that is peculiar of these system to be controlled, and the scope of the required aims make them complex. Analytical and design problems for large scale systems are not simply larger in size but more complex, i.e., they cannot be solved by using faster computers with larger memories, but they raise new questions which cannot answered by means of the methods developed for small systems. The complexity of large scale systems makes the direct application of multivariable control methods unreasonable or even impossible and necessities new analysis and design philosophies. [26] 2.2 Decentralization In decentralized control, the control agents are completely independent or atleast almost independent. It is known that communication among the decision-making units is necessary in order to achieve the overall goal. In decentralized structures, such a coordination is impossible or restricted in accordance with the information exchange that is permitted, so decentralized structures reduce the quality of the solution.[26] See fig 2.1. Figure 2.1: Decentralized Control Scheme Only a limited amount of information is required by control agent, rather than the whole process model and objective function. No synchronization of the activities of the autonomous units has to be implemented. This is advantageous with respect not only to the time and effort required to solved the problems, but also to the flexibility of the control units when perturbations with in the plant occur. Since the independently received decision may not be compatible, decentralization is more involved than decomposition. The subproblems have not only temporarily exempted from there interdependancies and a prescription of the order in which they have to work, but also to be made completely independent of each other, which may not give desired results.[26] 2.3 Restriction for feedback control of Large scale systems In addition to the assumptions from classical and multivariable control theory, large scale systems theory assumes the follows [26]:

16 CHAPTER 2. LARGE SCALE DYNAMIC SYSTEMS Control structure is required to avoid the unreasonable costs or delays in data transmission. 2. Control structure has to be chosen for reliability reasons. The closed loop system has to remain stable even if some control stations or some subsystems go out operation. 3. The interactions of the controllers through the common plant must be taken into account in the design phase. 2.4 Decomposition On the basis of the fact that computational requirement, for the analysis and controller synthesis, grows faster than the size of the systems, it is better to breakdown the whole problem into smaller sub-problems. These sub-problems are solved separately, and then these solutions are combined to get a global results for the original problem [26]. As sub-problems are not independent so some modification of these solution is necessary to consider the interrelationships between the sub-problems. In this regard, the concepts and techniques for reformulating a control problem as a set of interdependent sub-problems and solving these sub-problems are often referred to as distributed control. Decomposition of the control problems is based on the information of the internal structure of the system to be controlled. In this connection the system is not considered as a single unit but as a compound of different interacting sub-systems. Decomposition methods exploit the system structure, which can be obtained from the building block of the systems. Decomposition replace the overall problem by set of subproblems that can be solved by units with limited capability and under time restrictions. Instead of using a unique control agent for the whole problem, decompositions produces a structure of several control agents. Decomposition makes the information, about the control aims and the system, distributed over local and supremal units instead of making them available at central unit. Large scale systems are often controlled by a network of control agents whose tasks results from mixed multilevel-multilayer decomposition of the overall system. [26] 2.5 Models and Structure of Interconnected Systems Unstructured [26] Its state space representation of an overall dynamical system with input vector u R m and output vector y y l has the form ẋ(t) = Ax(t) + Bu(t) x(0) = x 0 (2.1) y(t) = Cx(t) + Du(t) Where x R n are the states of the overall system. For LTI system, the matrices A, B, C and D will have constant elements. This model has been used extensively and successfully in multi variable system theory, but it loses importance for large scale systems because it provides no information for the subsystems of the overall system.

17 CHAPTER 2. LARGE SCALE DYNAMIC SYSTEMS I/O-oriented model [26] For decentralized control, the sensors and actuators are grouped into u i R mi and y i R li (i = 1,..., N), where i th control station gets y i as its input and calculates u i as its output, i.e. the overall system input and output is decomposed into subvectors u = [u T i ut 2... u T N ]T and y = [y1 T y2 T... yn T ]T. Instead of (2.1) the model N ẋ(t) = Ax(t) + B si u i (t) x(0) = x 0 (2.2) y i (t) = C si x(t) + i=1 N D ij u j (t) (i = 1,..., N) j=1 is used which makes the structural constraints of the decentralized control perceptible. The matrices of (2.2) can be obtained from (2.1) by decomposing B, C, and D into submatrices, the dimensions of which are compatible with the dimensions of u i and y i. The model (2.2) exhibits the structure of the inputs and outputs but does not show how the over all system dynamics depends on the subsystems Interaction-oriented model Many large scale systems emerge as a result of interactions between different sub-systems. These couplings can have the nature of energy, material or information flows [26]. They are represented by signals s i and z i through which the i th subsystem interacts with other subsystems. These additional input and output signals of the subsystems are internal signals of the overall systems. Since every subsystem represents a dynamical system of its own, it can be described, [26], by a stats space model x i (t) = A i x i (t) + B i u i (t) + E i s i (t) x i (0) = x 0 (2.3) y i (t) = C i x i (t) + D i u i (t) + F i s i (t) z i (t) = C zi x i (t) + D zi u i (t) + F zi s i (t) where x i R ni state vector of the i th sub-system. Equation (2.3) Will be referred to as the i th subsystem. If the interactions between the subsystems are neglected (s i (t) = 0), equation (2.3) yields the model of the isolated subsystem. ẋ i (t) = A i x i (t) + B i u i (t) (2.4) Further extension of this form is as follows: ẋ i (t) = A ii x i (t) + y i (t) = C i x i (t) + D i u i (t) j=1, j i y i (t) = C i x i (t) A ij x j (t) + B i u i (t) (2.5) 2.6 (D Siljak Approach) Decomposition into disjoint subsystems The disjoint decomposition into input-decentralized or output-decentralized subsystems has been proposed by Siljak and Vukcevic (1976). Sezer and Siljak (1984) gave a decomposition where the interactions of the resulting subsystems are lower than the prescribed threshold. In this regard, the overall system description (2.1) was formed from the subsystem models (2.3) and the interaction relation. This bottom-up way can be used if the subsystem models are given in isolation

18 CHAPTER 2. LARGE SCALE DYNAMIC SYSTEMS 17 from each other and overall system model has to be found, [26]. In the following, the top-down way from the overall system ẋ(t) = Ax(t) + Bu(t) x(0) = x 0 (2.6) y(t) = Cx(t) to the subsystem models is considered. The partition to the state vector x into subvector x i yields ẋ i (t) = A ii x i (t) + N j=1, j= i y i (t) = A ij x j (t) + N B ij u i (t) x i (0) = x i0 (2.7) j=1 N C ij x i (t) j=1 where the matrices B and C have been partitioned into blocks B ij and C ij (i = 1,..., N; j = 1,..., N) according to the partition of x, u and y.obviously, the subsystem state x i depends on all inputs u i and subsystem output y i on all states x i. Since the aim is to get weakly coupled subsystem the partition of x should be done in such a way that the dependencies of y i on u j (i j) are zeros or weak. Systems of the form N ẋ i (t) = A ii x i (t) + A ij x j (t) + B i u i (t) x i (0) = x i0 (2.8) j=1, j= i y i (t) = are said to be input decentralized and systems ẋ i (t) = A ii x i (t) + N j=1, j= i N C ij x i (t) j=1 A ij x j (t) + N B ij u i (t) x i (0) = x i0 (2.9) j=1 y i (t) = C i x i (t) are called output decentralized. The decomposition of the overall systems into composite systems with a specific interconnected structure is an important means of deriving special analytical and design methods, which are tailored to particular classes of systems and are more efficient than general purpose methods.[26] 2.7 Vidyasagar Approach In this section an outline of the class of large scale interconnected systems as proposed by M. Vidyasagar, is presented. He proposed the following Where i = 1,..., m e i = u i m H ij y j (2.10) j=1 y i = G i e i (2.11)

19 CHAPTER 2. LARGE SCALE DYNAMIC SYSTEMS 18 u i, e i, y i all belong to the extended space L ni pe for a fixed p [1, ] and some positive integer n i, the operator G i maps L ni pe into itself, and the operator H ij maps L nj pe into L ni pe. We can refer to u i, e i, y i as the i th input, error, and output, respectively. Where convenient, we use the symbol u to denote the m-tuple (u 1,..., u m ), e to denote (e 1,..., e m ) and y to denote (y 1,..., y m ). Note that u, e, y all belong to product space L n pe, where n = m n i. Similarly the symbols G and H to denote operator from L n pe into itself defined by i=1 G = G (2.12) 0 G m H 11 H 1m H =..... (2.13) H m1 H mm e = u Hy (2.14) y = Ge (2.15) The system description (2.10 & 2.11) is quite general and has the potential to represent different types of systems. (2.10 & 2.11) can be thought of as several isolated or decoupled subsystems, corresponding to the operators G 1,..., G m, such that the inputs to G i is a linear combination of an external input u i and several interaction signals H ij y j. This is depicted in fig 2.2. Figure 2.2: Vidyasagar Approach For this reason, we refer to m as the number of subsystems, G i,..., G m as the subsystem operators, and H 11,... H mm as the interconnection operators. Thus for a large scale interconnected system one can either represent the system in the decomposed form (2.10 & 2.11) and analyze it at subsystem level, or one can represent it in the overall form (2.14 & 2.15) and analyze it as single loop system. All standard principles and methods of standard modern control can be applied if latter is chosen. Vidyasagar in [3], proposed methods to analyze larges scale interconnected system at the subsystem level by taking the full advantage of the fact that the system is composed of subsystems which are in coordination with each other and sharing information.

20 CHAPTER 2. LARGE SCALE DYNAMIC SYSTEMS Raffaello D Andrea s Approach In some cases interacting systems are distributed spatially and their interactions depends on the spatial location of one subsystem with respect to another. In [2], a detailed analysis and synthesis framework is presented to deal with systems which consists of identical interconnected subsystems located at the nodes of a fixed lattice. D Andrea discussed a framework of analysis, synthesis, and implementation of distributed controllers, designed for spatially interconnected systems. He developed a state space framework for posing problems of spatially interconnected systems, and focus on systems whose model is spatially discrete.for any optimal control technique to be successful, the structure of the system must be exploited in order to obtain tractable algorithms. In this regard it is important to note that D Andreas s approach aims at the controller synthesis for highly structured interconnected topology. It is important to note that, in this approach a frame work for the periodic and infinite interconnections is discussed. Consider the diagram in Fig Figure 2.3: Basic Building Block It assumes of a finite dimensional, linear time-invariant system governed by the following state-space equations: Tx(t, s) A T T A T S B T x Sv(t, s) = A ST A SS B S v x(0, s) = x 0 (s) R z(t, s) C T C S D w(t, s) Where Tx(t, s) = (x) t S i v(t, s) = x(t, s 1,... s i, s i+1,... s L ) s = [s 1,... s L ] represents the spatial co- ordinates. v = vector of all states that are going to i th subsystem from other subsystems j, j = 1,..., i 1, i+1,... N as an internal inputs which are equal to x j and in the case of fully interconnected system it will be equal to = [x T 1,... x T i 1, xt i+1,..., xt N ]T. A T S = N j=1,j i A ij x j (t)

21 CHAPTER 2. LARGE SCALE DYNAMIC SYSTEMS 20 and v(t, s) = [ v+ (t, s) v (t, s) ] Sv(t, s) = [ SI 0 0 SI ] [ v+ (t, s) v (t, s) Here it is assumed that v + (t, s) and w + (t, s) are of the same size, and that v (t, s) and w (t, s) are of the same size. Here I is an identity matrix of appropriate dimensions. 2.9 Interconnected System & Finite Element Methods (FEM) Large scale systems may be treated well by considering them as interconnected systems. In interconnected systems subsystems are in interaction with the other subsystems through some coupling represented by an operator or matrix. Initially these interacting subsystems can be considered independently then the interconnection constraints can be applied on the solution of these independent subsystems to get a solution of an over all large scale systems. The idea proposed by Vidyasagar (2.10, 2.11) has been discussed in detail in [3]. However this approach requires to introduce new operators, i.e. H, G etc which are different from the operators or mapping matrices that are being used commonly, and effectively in modern control theory, i.e. system, input, output matrices etc. State space based representation in system theory is being used very successfully for the optimal and robust centralized controller synthesis of multivariable systems, which motivates us to go for the frame works that involves state space representations, i.e. D Siljak s or D Andrea s approach. So by using these two approaches, all big system may be visualized by N subsystem interacting with other in such a way that overall dynamic behaviour is same as that of a whole system. The subsystem i is not only getting external input (controller input u and input disturbances w) from environment, but also receiving internal inputs (inputs from the interacting subsystems j i, j = 1,..., i 1, i + 1,... N, N in terms of affine combination of states). So for i th subsystem, the internal inputs will be A ij x j (t). ] j=1, j i In D Siljak approach, this internal input is the affine combinations of local states of all other sub-systems j, j i. In other words we can say that, the local internal output will be the local state x i itself and going to other subsystems in affine combination with other local states. Figure 2.4: i th subsystem However D Andrea s approach considers that, the internal output of i th subsystem is an affine combination of the local states x i of the i th subsystem i.e. A ST x i. Where A ST defines the matrix

22 CHAPTER 2. LARGE SCALE DYNAMIC SYSTEMS 21 containing the elements which are in linear combination of the local i th i th output. So for D Siljak approach, state x i, to produce the internal A ST = Idendity S i v(t, s) = x i A SS = B S = O Where Ois the null matrix of appropriate dimension. In this connection it is important to note that D Andrea s approach to design a distributed control design for spatially interconnected system is on a crucial assumption that only neighboring and identical subsystems are interacting with each other, which may not be the case in most of the numerical solution of mechanical systems i.e. FEM based systems. In FEM based systems, there exists lots of information flow among most of the subsystems. There exist a high level of complex interconnections. Inputs and state of one subsystem may not only effect the neighboring system but also other subsystems i.e. system and input matrices may be fully populated. This leads us to go for an approach that not only incorporates the generic assumption to consider high interconnectivity but for which decentralized/distributed controller synthesis technique [18] is available, thus letting us to go for D Siljak Approach. Besides, going for this controller techniques that incorporates D Siljak approach and can cope with the issues of FEM based large scale systems (high interconnectivity), it is also important to pose a question If system matrices are fully populated, i.e almost every subsystems is interacting with other subsystem, then will it still be feasible to use a generalized frame work of interconnected system to design distributed/decentralized controller when there already exists well established theory on the centralized controller synthesis of multivariable systems? The answer is yes. The structure of large scale dynamic system written in interconnected form may be utilized to implement decentralized/distributed control scheme which will make the controller implementation practical, reasonable and economical. As it is known now that a centralized control scheme for high dimensional, complex, uncertain system can result in severe problems, pron to failures, thus making the implementation impossible or infeasible so, Even if we have a complex & highly interconnected system, i.e. making it a multivariable system, we can still exploit the structure of FEM based system and proposed controller methods [18] (most suited for highly interconnected system i.e. D Siljak approach) and make it possible to implement a feasible, reasonable and economical distributed/decentralized controller. In this regard [18] proposed a designed method, based on the bounded real lemma, to develop a dynamic output feedback decentralized controller for systems represented through D Siljak approach. The method requires the solution of a non-linear matrix inequality [1] by using the homotopy method and results in a decentralized controller that can stabilize the overall large scale system and minimizes the gain of the transfer function from disturbance w to fictitious outputs z. So by using the D Siljak approach, for which w i is the disturbance and z i is fictitious output, the i th subsystem can be written as ẋ i (t) = A ii x i (t) + B 1i w i (t) + B 2i u i (t) + z i (t) = C 1i x i (t) + D 12 u i (t) y i (t) = C 2i x i (t) + D 21i w i (t) N j=1,j i A ij x j (t) (2.16)

23 CHAPTER 2. LARGE SCALE DYNAMIC SYSTEMS 22 Where 1 = 1, 2,...N. x i R ni, w i R ri, u i R mi, z i R (li+mi) and y i R li are the states, disturbance input, controlled input, fictitious output and measured output of the i th subsystem. Models in the form of (2.16) have been extensively studied, and efficient synthesis tools have been proposed to design controllers for such systems. As one goal of this thesis is to explore the possibility to utilize the frame work of interconnected system for FEM based system, so it important the understand the structure of the FEM based state space representation (Next Chapter).

24 Chapter 3 Finite Element Methods (FEM) This chapter focuses on the brief introduction of the basic theory behind finite element methods. FEM s main feature is to replace a continuous and complex system by an equivalent discrete one. In FEM a complete system is divided into small finite elements which are spatially distributed and interacts with each other based on their spatial location with respect to each other. In structural analysis, behaviour of system under consideration is described by displacement of elements satisfying material laws (constitutive equation) [7, 8, 9]. All elements are assembled together and requirement of continuity and equilibrium are satisfied between neighboring elements. Unique solution can be found for boundary conditions. Model generated from FEM may contain thousands of states, thus making it a typical large scale dynamic system. This chapter is organized as follows: Introduction of continuous and discrete system as a preliminary background is discussed in Section 3.1. Basic steps of FEM and important numerical steps involved in converting continuous and complex mechanical system into discrete lumped form is discussed in Section 3.2. A cantilever beam example has been considered and its FEM model is developed in Section 3.3. Basis of theory discussed in section 3.2 is modelled in MatLab, whose brief overview is discussed in Section Continuous and Discrete Systems Continuous systems are the systems modeled as a continuum, which is represented by partial differential equations. It ignores the fact that body is discrete and cannot completely fills the space it occupies. The continuum concept assumes that matter is composed of homogeneous structure (distributed uniformly) rather that heterogeneous microstructure allowing the approximation of physical quantities, such as energy and momentum, at the infinitesimal limit [23]. Modeling of such systems ends up in differential equations, solution (exact solution) of which describes the behaviour of the system under consideration. Vast majority of of continuous systems lead to eigenvalue problems that do not lend themselves to closed-form solution, owing to nonuniform mass or stiffness distribution. Hence it is practical to seek approximate solution. The approximate methods consists of scheme for the discretization of continuous systems, i.e. procedures for replacing a continuous system by an equivalent discrete one. It is important 23

25 CHAPTER 3. FINITE ELEMENT METHODS (FEM) 24 to note that discrete and continuous systems represents two different model of same physical system and hence same dynamic behaviour[24]. Similarly continuum mechanics deals with the dynamic analysis of mechanical systems modeled as a continuum, e.g., solids and fluids. A continuum can be continually sub-divided into infinitesimal small elements with properties being those of the bulk material. Solution to these type of problems are possible for very simple systems but for practical systems Finite Element Method (FEM) is a very effective tool that can solve these type of problems numerically[7, 8, 9], and converts the systems of infinite dimensions (continuous system) into finite dimensional system (Finite elements). The basic concept is the mesh discretization of a continuous domain into a set of discrete sub-domains. The solution approach is based either on OR Eliminating the differential equation completely Rendering the PDE into an approximating system of ordinary differential equations, which are then solved using standard techniques such as finite differences, Runge-Kutta, etc. Resulting equation should be numerically stable, meaning that errors in the input data and intermediate calculations do not accumulate. The Finite Element Method is a good choice for solving partial differential equations through FEM is appropriate when Analyzing or modeling complex domains (complex structure i.e. Aerospace, cars, bridges, fluid flow etc) Domain changes Desired precision varies over the entire domain. i.e Some regions in the solution requires accurate results than other. For instance, in complex geometry the places where geometry profile changes there is higher level of stress concentration so accurate results near these region is required than the region of less stress concentration. [23] 3.2 FEM in Mechanical Systems FEM is a powerful technique originally developed for numerical solution of complex problems in structural mechanics, and it remains the method of choice for complex systems. It must be regarded as the most successful techniques of structural analysis. Turner, clough, Martin and Topp originally conceived the ides in mid 1950s as a procedure of static analysis of complex structure. With the development of powerful digital computers FEM became phenomenon successful. [25]Areas of application include structural analysis, heat transfer, fluid mechanics, electromagnetism, bio mechanics, geomechanics, acoustic etc [9] In FEM, discretization of continuous system is performed locally over small regions of simple but arbitrary shape, i.e. finite elements. In structure analysis it is represented as assemblage of discrete truss and beam elements. The structural system is modeled by a set of appropriate finite elements spatially interconnected at points called nodes. Elements may have physical properties such as thickness, coefficient of thermal expansion, density, Young s modulus, shear modulus and Poisson s ratio. This process converts the partial differential equation into matrix equation relating the input at specified points in the elements to the output at these same points in such a way that compatibility and constitutive relationships are satisfied. To solve equations over large regions, the matrix equations for the smaller sub regions are summed node by node to yield global matrix equations.

26 CHAPTER 3. FINITE ELEMENT METHODS (FEM) 25 The elements are interconnected only at the exterior nodes, and altogether they should cover the entire domain as accurately as possible. Nodes will have nodal (vector) displacements or degrees of freedom which may include translations, rotations, and for special applications, higher order derivatives of displacements. When the nodes displace, they will drag the elements along in a certain manner dictated by the element formulation. In other words, displacements of any point in the element will be interpolated from the nodal displacements, and this is the main reason for the approximate nature of the solution Steps of Finite Element Method (FEM) The forces at nodes, which are usually called nodal forces, are primary input to the structural system while the displacements at the nodes, which are usually called nodal displacements, describing the configuration of system during and after deformation, are the primary responses of the structural system. Following basic steps are involved in FEM [10] 1. The discretization of the continuum into elements. [10] 2. The representation of the displacement Field of an element in terms of its nodal displacements.[10] 3. The expression of the internal stresses and strains in terms of the nodal displacements. (In case of structural strength analysis)[10] 4. The derivation of the element matrices (static and dynamic)[10] 5. Coordinate Transformation [8] 6. The assembly of element matrices and solution of resulting equation.[8] Discretization The continuum is separated or distcretized by fictitious points, lines, or surfaces into a number of finite elements, the properties of which can be determined relatively easily. The elements are assumed to be interconnected only at discrete number of nodes situated on the boundaries, although internal nodes are also permitted.the nodes are allowed to displace independently and nodal displacements are the basic unknowns of the problem.[10] Element displacement field or shape function A set of shape function, N(x, y, z) is chosen top relate the state of displacement with in each element. i.e. u(x, y, z) = [N(x, y, z)]{u e (t)} Where u e are the generalized displacements which may include translational, rotational displacement etc. This equation is only approximation and accuracy of final results depends greatly on the choice of shape function.[10] The stress strain relationship With in each element, the state of the strain ε by the mean of strain-displacement relationship in the theory of elasticity, i.e. ε = (u e) (r) = [L][N]{u e} = [B]{u e }

27 CHAPTER 3. FINITE ELEMENT METHODS (FEM) 26 and the stress strain relationship σ = [D]ε = [D][B]{u e } Where [L] is a problem dependent matrix of differential operators, [B] is called the strain matrix and [D]the property or elasticity matrix.[10] Element Matrices Static Case If the element is subjected to external concentrated forces {f c } acting at nodes only, then the work done is W = {u e } T {f c } and the strain energy is U = 1 ˆ {ε} T {σ}dv 2 In the state of equilibrium the total potential energy, U W, should be stationary, i.e. ˆ 1 2 {u e} T [B] T [D][B]dv{u e } {u e } T {f c } = stationary v v As the components of {u e } are the only parameters which can be adjusted to satisfy the stationary conditions, the equation is differentiated with respect to each parameter in turn and equated to zero, resulting in ˆ [B] T [D][B]dv{u e } = {f c } where [k] = v [B]T [D][B]dv{u e } v [k]{u e } = {f c } which is known as stiffness matrix. Furthermore, if there are other forces {ψ(x, y, z)} distributed over the element, in addition to the concentrated forces {f c }, then the work done becomes ˆ W = {u e } T {f c } + {u(x, y, z)} T {ψ(x, y, z)}dv v ˆ = {u e } T {f c } + {u e } T [N] T {ψ}dv v. = {u e }({f c } + {f d }) Where {f d } is the equivalent nodal force vector for distributed forces, i.e., ˆ {f d } = [N] T {ψ}dv (3.1) v And the final equation becomes [k]{u e } = {f c } + {f d } (3.2)

28 CHAPTER 3. FINITE ELEMENT METHODS (FEM) 27 Dynamic Case For dynamic system, even if only the nodal forces are present, the distributed inertia forces must be taken into account. Such inertia forces are given by {ψ(x, y, z)} = [ρ]{ü(x, y, z, t)} where [ρ] is mass density matrix. (3.1) for equivalent dynamic nodal forces, can then be written as ˆ {f d } = [N] T [ρ]{ü e (x, y, z)}dv (3.3) ˆ = v v [N] T [ρ][n] dv {ü e } = [m]{ü e } } {{ } [m] Where [m] is defined as consistent mass matrix. So (3.2) becomes [m]{ü e } + [k]{u e } = {f c } When damping is present in the system then (??) can be re-written as ˆ ˆ {f d } = [N] T [ρ][n]{ü e (x, y, z)}dv [N] T [d][n]{ u e (x, y, z)}dv (3.4) v } {{ } [m] v } {{ } [c] Where [d] is damping coefficient matrix and [c] is the damping matrix. So the final equation become [m]{ü e } + [c]{ u e } + [k]{u e } = {f e } (3.5) Where {f e } is equivalent nodal force vector of externally applied forces. It is important to note that formulation of [c] is empirical because, matrix [b] is difficult to define in practice. The damping effects are incorporated by applying the modal damping ration, which is beyond the scope of this thesis report.[10] Coordinate transformation The direction cosine matrix for general three-dimensional case is define as l x m x n x λ = l y m y n y (3.6) l z m z n z in which, e.g. l x is the cosine of the angle measured from global X to local x axis traversing in the counter clockwise direction with similar definition for the other angles in (3.6). Then the following relationships can easily be derived that define the relationship of variable between local and global coordinates. Displacement u e = Λq e or q e = Λ T u e Element stiffness [ k] = Λ T [k]λ Element inertia [ m] = Λ T [m]λ Element damping [ c] = Λ T [c]λ

29 CHAPTER 3. FINITE ELEMENT METHODS (FEM) 28 In which u e and q e are the element nodal displacement in local and global coordinate systems. Similarly [ m], [ c] and [ k] are in global coordinate systems, in which λ 1 λ 2 0 Λ =. 0.. λne2 is the direction cosine matrix for each element, the order of which is equal to the total number of element degree of freedom (NE2 = 2 number of nodes in the element).[8] Matrix Assembly & Solution Once the element matrices have been obtained in the global coordinate system, they may be assembled appropriately to yield the matrices for the entire structure, written symbolically as [K] = [ k] [M] = [ m] [C] = [ c] The global equation for the whole assembly can be obtained by combining the matrix contribution of all individual elements, such that coefficients belonging to common nodes are added together. This is achieved easily by adopting a systematic assembly process in which each element of typical [ k] matrix, with its row location identified as nodal degree of freedom where as column location indicative of similar identification of coupled node, is inserted in an exactly similar location of the assembled matrix.[8] Let us define then K = k (1) 11 k (1) 12 k (1) 21 k (1) 22 + k(2) 11 [ k] (i) =... [ k (i) 11 k (i) 12 k (i) 21 k (i) 22 ] k (i 1) 22 + k (i) 11 k (i) 12 k (i) 21 k (i) 22 + k(i+1) (3.7) Similarly global mass matrix M can also be found. M = m (1) 11 m (1) 12 m (1) 21 m (1) 22 + m(2) m (i 1) 22 + m (i) 11 m (i) 12 m (i) 21 m (i) 22 + m(i+1) (3.8)

30 CHAPTER 3. FINITE ELEMENT METHODS (FEM) 29 Similarly the damping matrix can be obtained. With M, C and K available to us (3.5) can be re-written as Mẍ + Cẋ + Kx = F(t) (3.9) Which is the general equation of motion mechanical system for MIMO system for lumped masses at discrete points of continuous systems and its state space form is [ ] [ ] [ ] [ ] ẋ O I x O = ẍ M 1 K M 1 + C ẋ M 1 F(t) (3.10) }{{}}{{} Ã B Where M = Mass matrix K = Stiffness matrix C = Damping matrix y = Cx + Du x = state of the system as nodal displacement in assembled form and in global coordinate system 3.3 FEM model of Cantilever Beam Cantilever can be considered as a beam with clamped boundary condition at one end and free boundary condition at other as shown in Fig 3.1. In order to illustrate the generation of FEM model of a cantilever beam of single element with nodes of a clamped free uniform beam is taken. We would assume a case with no damping and we will see later that this will bring the eigen values at imaginary axis, thus making the system oscillatory, which gives us an opportunity to define an oscillatory problem for our objective of disturbance rejection. Let a force f act downwards at the node 2 as shown in Fig 3.1. This force result in d(x, t) and angular rotation as θ = w(x,t) x. The equation of motion can be written as, ρ 2 w(x, t) t 2 + EI 4 w(x, t) x 4 = f(x, t) (3.11) Figure 3.1: Force on a cantilever beam Where ρ,e, I are density, Young modulus of elasticity and second moment of inertia.

31 CHAPTER 3. FINITE ELEMENT METHODS (FEM) 30 Behaviour of structural system (displacement) is approximated in terms of interpolation function (linear, quadratic, or higher order) such that w 1 (t) w(x, t) = [ N 1 (x) N 2 (x) N 3 (x) N 4 (x) ] θ 1 (t) w 2 (t) θ 2 (t) By approximating interpolation function N i (x) for i th function as N i (x) = c 0 + c 1 x + c 2 x 2 + c 3 x 3 i = 1, 2, 3, 4 Values of these coefficients can be found out by applying geometric boundary condition w and θat node 1 and 2. Let us define and substituting in (3.11) we get ρn T (x) d2 q(t) dt 2 Solving this equation would become N T (x) = [ N 1 (x) N 2 (x) N 3 (x) N 4 (x) ] + EI d4 N T (x)q(t) dx 4 = f(x, t) ˆl ρn(x)n T d 2 q(t) ˆl (x)dx dt 2 + EI d2 N(x) d 2 N T (x) ˆl dx 2 dx 2 q(t) = N(x)f(x, t)dx + F }{{}}{{}}{{} M This equation is equivalent to K M q + Kq = F d + F Where F d are the applied nodal forces. F is the boundary conditions in terms of shear force f and moment m as follows x=0 F = f 1 m 1 f 2 m 2 = = EI 3 w(x,t) x 3 EI 2 w(x,t) x=0 x 2 EI 3 w(x,t) x=l x 3 EI 2 w(x,t) x=l x 2 f fl f 0 For the above interpolation function following local Mass and stiffness matrix have been calculated F d

32 CHAPTER 3. FINITE ELEMENT METHODS (FEM) 31 [m] = ρ 13l 35 11l l 70 13l [k] = EI 11l l l l l 70 13l l 35 11l l l l l l l l l 2 2 l 12 2 l 6 l 12 2 l 6 l 3 l 2 l 2 l l 2 l 6 4 l 2 l These local expression, representing local element in FEM, of mass and stiffness matrices are transformed and then assembled in a global mass and stiffness matrix to generate lumped matrices [19]. (See 3.7, 3.8, 3.9, 3.10) 3.4 MATLAB CODE Steps discussed in section 3.2 are coded and a graphical user interface (GUI) is made for easy interaction between the coded file and user input. This program can be solve for truss and beam elements by providing the following information. 1. Nodal Information (a) Degree of freedom in local frame (b) Degree of freedom in global frame (c) No of nodes 2. Material Properties (a) Modulus of elasticity (b) Poisson ratio (c) Area 3. Force Details (a) Force value (b) Vector of three angle in 3D space (c) node number on which force is being applied 4. Connection Sequence (Rows of matrix equal to number of connections) (a) start point (b) end point point (c) relative angle and axis of rotation with respect to global frame 5. Coordinates (a) Element wise entry of 3D coordinate of nodes of elements

33 CHAPTER 3. FINITE ELEMENT METHODS (FEM) Boundary condition in global frame (a) Node number (b) axis number in which node is constrained The output of the program provides the user with the mass and stiffness matrices, which are eventually used to compute the state space representation for different simple mechanical structures. GUI interface is shown in fig 3.2. The code is made available at the institute of control department of Hamburg university of technology. Figure 3.2: GUI program for FEM

34 Chapter 4 Decomposition of FEM based Large Systems In previous chapter steps were discuss to obtain the state space of FEM based systems. This state space represents the systems of infinite dimensions (continuous system) through finite dimensional system (Finite elements). Modern control system design techniques have been applied to the state space mentioned above by reducing the order of system. However reducing the dimension may cause the the controller to excite unmodeled higher order modes, causing unwanted structural vibration (control spillover). Very fine meshes (more elements) are required to avoid these problems resulting in very large number of states for which control system design, using the modern control design techniques, imparts limitations during synthesis and implementation, which encourages the decomposition of system. In this chapter an algorithm is presented that can decompose large scale dynamic systems i.e. FEM generated model into the interconnected form of (2.16), thus making it suitable for control system design and vastly reducing the complexity of the controller synthesis. This chapter is distributed as follows: Basic philosophy and steps in the development of decomposition algorithm is discussed in Section 4.1. Algorithm for the software tool which converts the system generated through FEM into the form suitable for controller synthesis is presented in Section 4.2. A cantilever beam example has been considered and decomposition algorithm is applied in Section Philosophy of Decomposition Algorithm In this section the basic function decomp() is introduced. The function decomposes any matrix G R (p q) into the block structure G 11 G 12 G 1n G 21 G 22 G 2n..... (4.1). G n1 G n2 G nn In the following G i will represent i th diagonal block G(i, i) where, G i R (f hi f ji) such that 33

35 CHAPTER 4. DECOMPOSITION OF FEM BASED LARGE SYSTEMS 34 p = q = n f hi (4.2) i=1 n f ji (4.3) i=1 The function requires two further inputs from the user. These are fh T = [ ] T f h1 f h2 f hn (4.4) Let us define G vi as f T j = [ f j1 f j2 f jn ] T (4.5) G vi = [ G (i,1) G (i,i 1), G (i,i+1) G (i,n) ] In order to extract G i and G vi from G we need upper and lower row and column indices in terms of the elements of vectors f h and f j. Next we will find these indices for G i and G vir, where r = 1,..., i 1, i + 1,... n Indices of G i, i th Diagonal part Let us define e h and e j as lower index of row and column of G corresponding to G i respectively, then these can be found from vectors f h and f j as (4.6) then, i 1 e h = f hk + 1 k=1 i 1 e j = f jk + 1 k=1 G i = G(e h+ l i, e j+ g i ) Where l i = 0, 1,..., (f hi 1) g i = 0, 1,..., (f ji 1) (4.7)

36 CHAPTER 4. DECOMPOSITION OF FEM BASED LARGE SYSTEMS Indices of G vi, i th interconnection part Let, then. n ( G vir = G(l vir, g vir ) l vir = e h + l i As ((G vi R (f hi jr f ji)) r=1f ) Gvri ) G m 0, then for the case when r < i, and for the case when r > i, g vir i 1 = e j f jk + {0, 1,..., f jr 1} (4.8) k=r r 1 g vir = e j + f jk + {0, 1,..., f jr 1} (4.9) k=i Above can be coded into basic function decomp(g, f h, f j, i, m) where, the binary input m can be used to get G i (i th diagonal block G(i, i)) if m = 0 or G vi (4.6) if m Applying decomposition function decomp() on FEM generated system Re-writing (3.10), [ x x ] [ ] O I = M 1 K M 1 C }{{} Ã y = Cx + Du [ ẋ x where, x, u, and y are same vectors as defined in previous chapter. State vector in (3.10) is like which must be transformed into ] [ ] O + M 1 F(t) }{{} B x T = [ x T 1 x T 2 x T n ẋ T 1 ẋ T 2 ẋ T n ] T x T = [ x T 1 ẋ T 1 x T 2 ẋ T 2 x T n ẋ T n ] T This can be done by defining unitary transformation T r such that x = T r x then A = T r ÃTr 1, B = T r B, C = and D = D, then for state vector x, f hi = f xi + fẋi. New state space representation will be CT 1 r ẋ = Ax + Bu y = Cx + Du (4.10) As A, B, C and D defined in (4.10) are similar to G, so the function decomp(), defined in previous section can be applied effectively to convert (4.10) into (1.1) for i th interconnection. Next we will show

37 CHAPTER 4. DECOMPOSITION OF FEM BASED LARGE SYSTEMS 36 how to decompose the transformed system into the form (1.1) with N subsystems. subsystem, ẋ i, x i, R fxi, z i R fzi and w i R fωi. Then, define the following vectors, Let for the i th F x = (f x1,..., f xn ) F z = (f zi,..., f zn ) F w = (f w1,..., f wn ) The function decomp() can then be applied on each of A, B, C, and D to decompose them to get the state space representation (1.1) Algorithm For an FEM generated state space in the form (3.10) Step 1: D. Find the transformation matrix T r and apply similarity transformation to get A, B, C and Step 2: Define the vector F x, F z and F w. Step 3: Get the other matrices by A i = decomp(a, F x, F x, i, 0); B vi = decomp(b vi, F x, F x, i, 1) B i = decomp(a, F x, F w, i, 0) C i = decomp(a, F z, F x, i, 0) D i = decomp(a, F z, F xw, i, 0) 4.3 Decomposing the FEM Model of a Cantilever Beam In this section an example is presented, in which the proposed decomposition algorithm is applied on the FEM based state space of cantilever beam. The algorithm converts the FEM based large scale dynamic system into the interconnected form as per (2.16). So that the techniques of decentralized interconnected controller synthesis can be made possible. The algorithm discussed in this section has been accepted by the IFAC world Congress FEM model for a cantilever beam is developed by using the MatLab program developed by the author of thesis as per the per guideline mentioned in section 3.2. The global mass matrix and stiffness matrix are used in (3.10) to generate the state space representation. A cantilever beam of 04 nodes has been considered. i.e. each element has two nodes. Each node has four states (w,θ, ẇ, θ) and has two inputs and four outputs. The inputs are the force and torque, The outputs are the linear & angular displacements and corresponding velocities. Table 4.1 shows the characteristics of beam that were considered for modelling. So, in our case for f xi = 4, f zi = 4 and f wi = 2. Applying the algorithm

38 CHAPTER 4. DECOMPOSITION OF FEM BASED LARGE SYSTEMS 37 Field Length Length of element Values 0.3 m 0.1 m Modulus of Elasticity N m 2 Table 4.1: Characteristics of a cantilever beam Step 1: D. Choose Find the transformation matrix T r and apply similarity transformation to get A, B, C and T r = I o o o o o o o o o o o I o o o o I o o o o o o o o o o o I o o o o I o o o o o o o o o o o I o o o o I o o o o o o o o o o o I (4.11) Where I R 2 2 o R 2 2 Step 2: Define the vector F x, F z and F w. F x = F z = [4, 4, 4, 4] T F w = [2, 2, 2, 2] T Step 3: Get the other matrices by A i = decomp(a, F x, F x, i, 0); B vi = decomp(b vi, F x, F x, i, 1) B i = decomp(a, F x, F w, i, 0) C i = decomp(a, F z, F x, i, 0) D i = decomp(a, F z, F xw, i, 0) By applying the decomposition, on this FEM based large scale dynamic system an interconnected form (4 subsystem) is created. Both systems are simulated. It is observed that percentage error between the outputs of both systems model remain zeros with in the numerical precision, which shows that the decomposition retains all the dynamics of the system. Detailed discussion about these results is in chapter # 6.

39 Chapter 5 Active Vibration Control of Beam Structures; Example In chapter # 2 large scale dynamic systems whose subsystems are in interaction with other subsystems in terms of information flow have been considered. Chapter # 3 focuses on the systems generated through FEM. These systems are typical example of large scale dynamic systems with high level of interconnection. In chapter # 1 and 4 it was discussed that if one attempts to control these systems using standard control design techniques, severe limitations will quickly be encountered as most optimal control techniques cannot handle systems of very high dimension and with a large number of inputs and outputs. Order reduction can result in unwanted close loop vibrations (control spill over), which encourages the decomposition of system. In chapter # 4 an algorithm was presented that can decompose large scale dynamic systems i.e. FEM generated model into the form of (2.16), thus making it suitable for control system design and may vastly reducing the complexity of the controller synthesis. In this chapter active vibration control (disturbance rejection) of three different beam structures has been taken as examples. Three different problem of vibrating and oscillatory beam will be defined. State space representation of generalized beam structure, obtained through FEM will be developed as per steps discussed in chapter # 3. A propose method based on bounded real lemma; involving solution of NMI using homotopy method, to synthesize dynamic output feedback decentralized controller for interconnected systems is discussed in this chapter. This decentralized controller will be implemented on three different oscillatory beam structures to show the active vibration control (input disturbance rejection). This chapter is organized as follows: Synthesis of decentralized Robust H controller for large scale dynamic systems is discussed in Section 5.1. Transformation of Augmented large scale dynamic system into form suitable for decentralized controller i s discussed in Section 5.2. Problem formulation for active vibration control of three different beam structures are presented in Section 5.3. Application of decentralized controller for active vibration control; (disturbance rejection), is shown in Section

40 CHAPTER 5. ACTIVE VIBRATION CONTROL OF BEAM STRUCTURES; EXAMPLE Design of Decentralized Robust Control It is known now that a wide variety of problems arising in system and control theory may be reduced to a handful of standard convex and quasi convex optimization problems that involve matrix inequalities. It has been established that linear matrix inequality (LMI) based approaches are very powerful for centralized controller design for which these tools of convex optimization, i.e. Interior-point methods, have been found to be extremely efficient in practice[1]. Moreover, for synthesis of robust centralized controller, it is required that close loop system from disturbance w to fictitious output z should be stabilizable with desired attenuation. The system is stabilizable with the disturbance attenuation level γ if the closed loop system satisfies T zw < γ [1], where γ is specified a positive number. Lemma 1 (Bounded Real Lemma), [6, 1]: The following statements are equivalent: 1. Closed loop system matrix A cl is a stable matrix and T zw < γ. 2. There exist a positive definite matrix P which satisfies the LMI: AT cl P + P A cl P B cl Ccl T Bcl T γi 0 < 0 (5.1) C cl 0 γi However it is not true in the decentralized case, where controller design problem cannot be reduced to a feasibility problem for LMI because of the structural constraint on the controller, i.e., block-diagonal form of coefficient matrices. Decentralized controller for nominal system is computed by imposing structural constraints on the controller gradually. In [18] a designed method based on the bounded real lemma is developed for dynamic output feedback controller, which is reduced to a feasibility problem for nonlinear matrix inequality (NMI). A homotopy method was used to solve the NMI iteratively by alternatively fixing some of the variable to convert the NMI into LMI. Structural constraints on matrix variables are introduced to compute the decentralized control. In this thesis same is considered as controller synthesis techniques for decentralized control of beam structure.. In this regard, decentralized controller synthesis techniques for large scale dynamic system; which also incorporates the D Siljak approach, requires that large scale system should be re-arranged or transformed to the form as described in (5.2), [18]. The detailed method for converting the state space representation of large scale system into (5.2) is discussed in section 5.2. ẋ z y = A A 1N..... } A N1... {{ A NN } A C O..... } O... {{ C 1N } C1 C O..... } O... {{ C 2N } C2 B B 11N..... } B 1N1... {{ B 1NN } B1 O z D O..... } O... {{ D 21N } D21 B B 21N..... B 2N1... B 2NN }{{} B2 D O..... O... D 12N }{{} D12 O y x w u (5.2)

41 CHAPTER 5. ACTIVE VIBRATION CONTROL OF BEAM STRUCTURES; EXAMPLE 40 Where x R (l+m+n), w R ω, u R m, y R l, and z R (l+m) are the states (when augmentation with first order weighting filters), disturbance input, controlled input, controlled output and measured output of the system. Where n denotes the actual number of state that appear in the development of state space representation of generalized plant. When generalized plant is augmented with single order weighting filters the new state increases by l + m. However if simple gains are used instead of weighting filters then final number of states remain unchanged. w = [ w T 1... w T N u = [ u T 1... u T N ] and ]. The matrices A, B1, B2, C1, O z, D12, C2, D21 and O y are of appropriate dimensions. For FEM generated systems B 1 and B 2 are fully populated, which also poses future work to reduce the system and input matrix population So the whole large scale system will be ẋ = Ax + B 1 w + B 2 u z = C 1 x + D 12 u y = C 2 x + D 21 w (5.3) From above i th subsystem can be written as Where 1 = 1, 2,...N. ẋ i (t) = A ii x i (t) + B 1i w i (t) + B 2i u i (t) + z i (t) = C 1i x i (t) + D 12 u i (t) y i (t) = C 2i x i (t) + D 21i w i (t) N j=1,j i A ij x j (t) (5.4) x i R (li+mi+ni), w i R ri, u i R mi, z i R (li+mi) and y i R li are the states, disturbance input, controlled input, controlled output and measured output of the i th subsystem. The matrices A ii, B 1i, B 2i, C 1i, C 2i, D 12i and D 21i are of appropriate dimensions. Strictly proper i th output feedback controller for system in (5.3) can be defined by ẋ ci = A ci x ci + B ci y i u i = C ci x ci, i = 1, 2,..., N (5.5) In this regard the closed loop system (5.6) from input disturbances w to fictitious output z is of the following form. ẋ cl = A cl x cl + B cl w z = C cl x cl (5.6) Where x cl = [x T x T c ], x c = [x T cl... xt cn ],

42 CHAPTER 5. ACTIVE VIBRATION CONTROL OF BEAM STRUCTURES; EXAMPLE 41 [ A cl = ] [ A B 2 C c, B B c C 2 A cl = c ] B 1, B c D 21 C cl = [C 1 D 12 C c ] In this connection synthesis of decentralized control requires the enforcement of structural constraints on system defined by (5.1)i.e. A c, B c, and C c matrices of decentralized controller will be decoupled and block diagonal. So decentralized control for the system in (5.3) requires that A c = diag{a c1,... A cn }, B c = diag{b c1,... B cn }, C c = diag{c c1,..., C cn }. This can be done as follows: Let us define positive definite block diagonal symmetric matrices X, Y, such that UV T = I XY. Then by defining [ ] X I Π 1 = U T 0 We can pre and post multiply (5.1) by ΠT I I, Π I I to obtain Π T 1 (A T cl P + P A cl)π 1 Π T 1 P B cl Π T 1 Ccl T Bcl T P Π 1 γi 0 < 0 (5.7) C cl Π 1 0 γi [ ] Y V Now by defining P = V T U 1 XY XU T U 1 XU T, and block diagonal matrices Q = V A c U T, L = V B c, and F = C c U T (5.7) will convert into following form [18]. J 11 J21 T B 1 XC 1 J 21 J 22 Y B 1 + LD 21 C1 T B1 T B1 T Y + D21L T T γi 0 C 1 X + D 12 F C 1 0 γi [ ] X I > 0 I Y < 0 (5.8) Positive definite block diagonal matrices X and Y and diagonal matrices F, L, Q are obtained by solving the above matrix inequality. From this block diagonal coefficient matrices are given by Where J 11 = AX + XA T + B 2 F + F T B T, J 21 = A T + Y AX + LC 2 X + Y B 2 F + Q, A c = V 1 QU T, B c = V 1 L, C c = F U T (5.9)

43 CHAPTER 5. ACTIVE VIBRATION CONTROL OF BEAM STRUCTURES; EXAMPLE 42 J 22 = Y A + A T Y + LC 2 + C T 2 L T. The size of sub matrices in the block diagonal are compatible with the dimensions of the state, input, and output vectors of subsystems. It is clear from that J 21 is a NMI in variable Y, X, L, F. Solving NMI (5.8) involves fixing a group of variables to make it an LMI. First variables L, and Y are fixed then NMI (5.8) becomes LMI in X, F and Q. Then variables X, F are fixed and then NMI (5.8) becomes another LMI. Solving these two LMIs alternately can lead to a solution.[18]. However choosing or fixing these variables is equivalent to finding a solution of NMI. In this regard a homotopy method can be applied in which path from centralized controller to decentralized controller is divided into number of parts M. structural constraints are gradually applied, by defining a real number λ i.e. λ = k M where is k is gradually increased from 0 to M, which defines a matrix function H(X, Y, F, L, Q, λ) = T (X, Y, F, L, (1 λ)q F + λq) < 0 Which is same as (5.8) except that J 21 is replaced by A T + Y AX + LC 2 X + Y B 2 F + (1 λ)q F + λq. Then { T (X, Y, F, L, Q F ), λ = 0 H(X, Y, F, L, Q, λ) = T (X, Y, F, L, Q), λ = 1 So solution is in problem H(X, Y, F, L, Q, λ) < 0, λ [0, 1] (5.10) Initially we compute the solution X o, Y o, L o, F o by making NMI (5.8) into a LMI by setting J 21 = J T 21 = 0 or by setting Q F = (A T + Y o AX o + L o CX o + Y o B 2 F o ) (5.11) Now considering a homotopy path to transform this initial solution at λ = 0 (centralized control), to a solution at λ = 1 (decentralized control), [18] proposed following algorithm to solve the above mentioned family of problems Step 1: 1. At k = 0, compute block diagonal solution X o, Y o, L o, F o of BMI (5.8), with Q F as per (5.11). 2. Initialize the M Z up to certain positive value and also set certain upper bound M max. Step 2: 1. Set k = k + 1 and compute λ k by k M. 2. Compute block diagonal solutions X, F, Q, of H(X, Y k 1, F, L k+1, Q, λ k ) < 0 with Y = k If it is not feasible go to step If it is feasible, set X k = X, and F k = F and compute block diagonal solutions Y, L, Q of H(X k, Y, F k, L, Q, λ k ) < If it is feasible, then set Y k = Y, L k = L, and Q k = Q and go to step 5. Step 3: 1. Compute block diagonal solutions Y, L, Q of H(X k 1, Y, F k 1, L, Q, λ k ) < If it is not feasible go to step 4.

44 CHAPTER 5. ACTIVE VIBRATION CONTROL OF BEAM STRUCTURES; EXAMPLE If it is feasible, set Y k = Y, L k = L and compute block diagonal solutions of X, F, Q of H(X, Y k, F, L k, Q, λ k ) < 0 and set X k = X, F k = F and Q K = Q and go to step 5. Step 4: 1. set M = 2M under the constraint M M max, set X 2(k 1) = X k 1, Y 2(k 1) = Y k 1, F 2(k 1) = F k 1, L 2(k 1) = L k 1, k = 2(k 1) and go to step If we cannot increase M any more, we conclude that this algorithm does not converge for upper bound M max. Step 5: 1. If k < M, go to step 2. If k = M, the obtained matrices X M, Y M, F M, L M are the solution of BMI (5.8). Step 6: 1. Compute block-diagonal matrices U and V such that UV T = I X M Y M. 2. Define the coefficient matrices of decentralized H controllers as A c = V 1 QU T, B c = V 1 L, C c = F U T. A block diagram of the algorithm is shown in Fig 5.1 for better comprehension. Note: It is important to note that (5.3) requires that the state space of large scale dynamic system, i.e. FEM based systems, should be transformed to the form of (5.2), particularly when system is augmented with weighting filters. The details of this transformation is discussed in the next section.

45 CHAPTER 5. ACTIVE VIBRATION CONTROL OF BEAM STRUCTURES; EXAMPLE 44 Figure 5.1: Block diagram of Algorithm of Decentralized Controller through Homotopy Method