Dynamical Analysis and Robust Control Synthesis for Water Treatment Processes

Nowadays, water demand and water scarcity are very urgent issues due to population growth, drought and poor water quality all over the world. Therefore, water treatment plants are playing a vital role for good living condition of human. Water area needs more concentration study to increase water productivity and decrease water cost. This dissertation presents the analysis and control of water treatment plants using robust control techniques. The applied control algorithms include H∞, gain scheduled and observer-based loop-shaping control technique. They are modern control algorithms and very powerful in robust controlling of systems with uncertainties and disturbances. The water treatment plants include a desalination system and a wastewater process. Since fresh water scarcity is getting more serious, the desalination plants are to produce drinking water and wastewater treatment plants give the reusable water. The desalination system is a RO one used to produce drinking water from seawater and brackish water. The nonlinear behaviors of this system is carefully analyzed before the linearization. Due to the uncertainty caused by concentration polarization, the system is linearized using linear state-space parametric uncertainty framework. The system also suffer from many disturbances which water hammer is one of the most influential one. The mixed robust H∞ and μ-synthesis control algorithm is applied to control the RO system coping with large uncertainties, disturbances and noises. The wastewater treatment process is an activated sludge process. This biological process use microorganisms to convert organic and certain inorganic matter from wastewater into cell mass. The process is very complex with many coupled biological and chemical reactions. Due to the large variation in the influent flow, the system is modelized and linearized as a linear parametric varying system using affine parameter-dependent representation. Since the influent flow is quickly variable and easily to be measured, the robust gain scheduled robust controller is applied to deal with the large uncertainty caused by the scheduled parameter. This control algorithm often gives better performances than those of general robust H∞ one. In the wastewater treatment plant, there exist an anaerobic digestion, which is controlled by the observer-based loop-shaping algorithm. The simulations show that all the controllers can effectively deal with large uncertainties, disturbances and noises in water treatment plants. They help improve the system performances and safeties, save energy and reduce product water costs. The studies contribute some potential control approaches for water treatment plants, which is currently a very active research area in the world.Contents ······················································································· iv List of Tables ··············································································· viii List of Figures ··············································································· ix Chapter 1. Introduction ···································································· 1 1.1 Reverse osmosis process ···································································· 2 1.2 Activated sludge process ···································································· 6 1.3 Robust H∞ and gain scheduling control ··················································· 10 Chapter 2. Robust H∞ controller ······················································· 13 2.1 Introduction ·················································································· 13 2.2 Uncertainty modelling ······································································ 13 2.2.1 Unstructured uncertainties ···························································· 14 2.2.2 Parametric uncertainties ······························································· 15 2.2.3 Structured uncertainties ································································ 16 2.2.4 Linear fractional transformation ······················································ 16 2.3 Stability criterion ············································································ 17 2.3.1 Small gain theorem ····································································· 17 2.3.2 Structured singular value (muy) synthesis brief definition ·························· 19 2.4 Robustness analysis and controller design ··············································· 20 2.4.1 Forming generalised plant and N-delta structure ····································· 20 2.4.2 Robustness analysis ···································································· 24 2.5 Reduced controller ·········································································· 26 2.5.1 Truncation ··············································································· 27 2.5.2 Residualization ········································································· 29 2.5.3 Balanced realization···································································· 29 2.5.4 Optimal Hankel norm approximation ················································ 31 Chapter 3. Robust gain scheduling controller ······································· 37 3.1 Introduction ·················································································· 37 3.2 Linear parameter varying (LPV) system ·················································· 39 3.3 Matrix Polytope ·············································································· 40 3.4 Polytope and affine parameter-dependent representation ······························· 41 3.4.1 Polytope representation ································································ 41 3.4.2 Affine parameter-dependent representation ········································· 42 3.5 Quadratic stability of LPV systems and quadratic (robust) H∞ performance ········· 43 3.6 Robust gain scheduling ····································································· 44 3.6.1 LPV system linearization ······························································ 44 3.6.2 Polytope-based gain scheduling ······················································ 45 3.6.3 LFT-based gain scheduling ··························································· 48 Chapter 4. Mixed robust H∞ and μ-synthesis controller applied for a reverse osmosis desalination system ····························································· 52 4.1 RO principles ················································································ 52 4.1.1 Osmosis and reverse osmosis ························································· 52 4.1.2 Dead-end filtration and cross-flow filtration ········································ 53 4.2 Membranes ··················································································· 54 4.2.1 Structure and material ································································· 54 4.2.2 Hollow fine fiber membrane module ················································ 55 4.2.3 Spiral wound membrane module ····················································· 57 4.3 Nonlinear RO modelling and analysis ···················································· 58 4.3.1 RO system introduction ······························································· 58 4.3.2 Modelling ··············································································· 60 4.3.3 Nonlinear analysis ······································································ 62 4.3.4 Concentration polarization ···························································· 64 4.4 Water hammer phenomenon ······························································· 66 4.4.1 Water hammer, column separation and vaporous cavitation ······················ 66 4.4.2 Water hammer analysis and simulation ·············································· 69 4.4.3 Prevention of water hammer effect··················································· 78 4.5 RO linearization ············································································· 79 4.5.1 Nominal linearization ·································································· 79 4.5.2 Uncertainty modeling ·································································· 81 4.5.3 Parametric uncertainty linearization ················································· 83 4.6 Robust H∞ controller design for RO system ·············································· 85 4.6.1 Control of uncertain RO system ······················································ 85 4.6.2 Robustness analysis and H∞ controller design ······································ 86 4.7 Simulation result and discussion··························································· 90 4.8 Conclusion ··················································································· 95 Chapter 5. Robust gain scheduling control of activated sludge process ······· 96 5.1 Introduction about activated sludge process ············································· 96 5.1.1 State variables ·········································································· 98 5.1.2 ASM1 processes ······································································ 100 5.1.3 The control problem of activated sludge process ································· 102 5.2 System modelling ········································································· 104 5.3 Model linearization ········································································ 107 5.4 Robust gain-schedule controller design for activated sludge process ··············· 109 5.5 Simulation result and discussion························································· 115 5.6 Conclusion ················································································· 120 Chapter 6. Observer-based loop-shaping control of anaerobic digestion ···· 121 6.1 Introduction ················································································ 121 6.1.1 Control problem in anaerobic digestion ··········································· 122 6.2 System modelling ········································································· 123 6.3 Controller design ·········································································· 124 6.3.1 H∞ loop-shaping controller ························································· 125 6.3.2 Coprime factor uncertainty ·························································· 126 6.3.3 Control synthesis ····································································· 127 6.4 Simulation result ··········································································· 131 6.5 Conclusion ················································································· 133 Chapter 7. Conclusion ··································································· 134 References ·················································································· 136 Appendices ················································································· 144Docto

Control Theory: A Mathematical Perspective on Cyber-Physical Systems

Control theory is an interdisciplinary field that is located at the crossroads of pure and applied mathematics with systems engineering and the sciences. Recently the control field is facing new challenges motivated by application domains that involve networks of systems. Examples are interacting robots, networks of autonomous cars or the smart grid. In order to address the new challenges posed by these application disciplines, the special focus of this workshop has been on the currently very active field of Cyber-Physical Systems, which forms the underlying basis for many network control applications. A series of lectures in this workshop was devoted to give an overview on current theoretical developments in Cyber-Physical Systems, emphasizing in particular the mathematical aspects of the field. Special focus was on the dynamics and control of networks of systems, distributed optimization and formation control, fundamentals of nonlinear interconnected systems, as well as open problems in control

Modelling and control of a two-link flexible manipulator using finite element modal analysis

This thesis focuses on Finite Element (FE) modeling and robust control of a two-link flexible manipulator based on a high resolution FE model and the system vibration modes. A new FE model is derived using Euler-Bernoulli beam elements, and the model is validated using commercial software Abaqus CAE. The frequency and time domain analysis reveal that the response of the FE model substantially varies with changing the number of elements, unless a high number of elements (100 elements in this work) is used. The gap between the complexity of the high order FE model capable of predicting dynamics of the multibody system, and suitability of the model for controller design is bridged by designing control schemes based on the reduced order models obtained using modal truncation/H8 techniques. Two reduced order multi-input multi-output modal control algorithms composed of a robust feedback controller along with a feed-forward compensator are designed. The first controller, Inversion-based Two Mode Controller (ITMC), is designed using a mixed-sensitivity H8 synthesis and a modal inversion-based compensator. The second controller, Shaping Two-Mode Controller (STMC), is designed with H8 loopshaping using the modal characteristics of the system. Stability robustness against unmodelled dynamics due to the model reduction is shown using the small gain theorem. Performance of the feedback controllers are compared with Linear Quadratic Gaussian designs and are shown to have better tracking characteristics. Effectiveness of the control schemes is shown by simulation of rest-to-rest maneuver of the manipulator to a set of desired points in the joint space. The ITMC is shown to have more precise tracking performance, while STMC has higher control over vibration of the tip, at the expense of more tracking errors

Optimisation of condition number for eigenstructure problems

This thesis deals with a new solution for the problem of eigenstructure assignment in control systems design. A wide range of challenging issues is examined involving the problem of eigenstructure assignment and associated system properties through different forms of complexity which are strongly related to control system design. In this thesis, specific attention has been given to the issue of skewness of the closedloop eigenframe of the state matrix. In fact, the aim is to develop a new methodology for determining the best angle between closed-loop eigenvectors by optimising the minimal condition number of the closed-loop eigenvector matrix. This problem is strongly linked to sensitivity of eigenvalues to parameter uncertainty, perturbations to model parameter uncertainty. The importance of this methodology can be expressed in terms of results related to the Sensitivity of eigenvalues, Relative measures of controllability and observability and also deviations from strong stability to overshooting behaviour. Among this variety of eigenstructure assignment methods, special consideration has been paid to Geometric Theory, which introduces an alternative solution to the assignability of spectrum of controllability subspaces (cs) based on an eigenvector approach and then develops a new pole assignment algorithm based on openloop/ closed-loop spectra as a practical application of this approach. In order to tackle the problem of measuring the skewness of angles between closed-loop eigenvalues, some measures for Eigenframe skewness have been defined in general and so the necessary and efficient conditions have been derived for the angle between some subspaces in a direct sum decomposition to be maximized. This has been done via three metrics; Condition Number, Determinant of Gram Matrix and Singular Values. The thesis presents the parametrisation of closed-loop eigenframes result by the method generated in [4]. Within this thesis, a non-smooth algorithm has been developed in order to select the most orthogonal closed-loop eigenframe and so the influence of selected closed-loop spectra. Also, the parametrisation of controllability subspaces in a standard direct sum decomposition using matrix fraction description (MFD) has been derived. Within this thesis the construction and the existence of controllability subspaces connected to (A,B)-invariant subspaces, has also been studied. In addition, an algebraic description of the total system behaviour which leads to an algebraic characterisation of the total input, state and output behaviour in an implicit formulation is given based on properties of MFD descriptions, a topic which remains open for future studies

Robust stabilisation of multivariable systems: A super-optimisation approach

The work aims to derive extended robust stability results for the case of unstructured uncertainty models of multivariable systems. More specifically, throughout the thesis, additive and coprime unstructured perturbation models are considered. Refined robust stabilisation problems of MIMO systems are defined and maximally robust controllers are synthesised in a state-space form. Unstructured perturbations which destabilise the feedback system for every optimal (maximally robust) controller are identified on the boundary of the optimal ball, i.e. the set of all admissible perturbations with norm equal to the maximum robust stability radius. Boundary perturbations are termed "uniformly destabilising" if they destabilise the closed-loop system for every optimal controller and it is shown that they all share a common characteristic, i.e. a projection of magnitude equal to the maximal robust stability radius, along a fixed direction defined by a pair of maximising vectors (scaled Schmidt pair) of a Hankel operator related to the problem. By imposing a directionality constraint it is shown that it is possible to increase the robust stability radius in every other direction by a subset of all optimal controllers. In order to solve this problem, super-optimisation techniques are developed. Independently a natural extension of Hankel norm approximations, the so-called super­ optimisation problem is posed and solved explicitly for the case of one-block problems in a state-space setting. It is thus shown that a subset of all maximally robust controllers, namely the class of super-optimal controllers, stabilises all perturbed plants within an extended stability radius 11,*(b), subject to a directionality constraint. In addition, the work is related to robust stabilisation subject to structured perturbations. The notions of structured robust stabilisation problem, and structured set approximation are defined in connection with the maximised set of permissible perturbations. It is further shown that µ*(J) can serve as an upper bound the structured robust stabilisation problem. The effect of µ*(J) as an upper bound depends on the compatibility between the two structures, the true structure and the artificial structure of the extended permissible set. [Look inside the thesis' abstract for an exact version of formulas and equations

A fluid power application of alternative robust control strategies

This thesis presents alternative methods for designing a speed controller for a hydrostatic power transmission system. Recognising that such a system, comprising a valve controlled motor supplied by the laboratory ring main and driving a hydraulic pump as a load, contains significant non-linearities, the thesis shows that robust 'modern control' approaches may be applied to produce viable controllers without recourse to the use of a detailed model of the system. In its introduction, it considers why similar approaches to the design of fluid power systems have not been applied hitherto. It then sets out the design and test, in simulation and on a physical rig, of two alternative linear controllers using H∞ based methods and a 'self organising fuzzy logic' controller (SOFLC). In the linear approaches, differences between the characteristics of the system and the simple models of it are accommodated in the controller design route as 'perturbations' or 'uncertainties'. The H∞ based optimisation methods allow these to be recognised in the design. “Mixed sensitivity” and “Loop shaping” methods are each applied to design controllers which are tested successfully on the laboratory rig. The SOFLC in operation does not rely on a model, but instead allows fuzzy control rules to evolve. In the practical tests, the system is subjected to a range of disturbances in the form of supply pressure fluctuations and load torque changes. Also presented are test results for proportional and proportional plus integral (PI) controllers, to provide a reference. It is demonstrated qualitatively that performance using the linear controllers is superior to that using proportional and PI controllers. An increased range of stable operation is achieved by the controller designed using “loop shaping” – performance is enhanced by the use of two controllers selected automatically according to the operating speed, using a “bumpless” transfer routine. The SOFLC proved difficult to tune. However, stable operation was achieved.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Systems and control problems in early systems design

This thesis is concerned with the evaluation of properties of early design models, the control structure selection and the representation and properties of composite system models. The work is motivated by the need to introduce a Systems and Control Theory based framework for early design stages of the overall system design of engineering processes, and in particular chemical processes, such as process synthesis and control structure selection. The overall spirit of the work is that engineering design is an evolutionary process, the different stages of which shape the structure of the resulting system models and precondition the potential for design at the next stage. The work identifies a number of key problems in the overall design, which are of a generic, systemic character, and then deals with working out solutions for such problems. The results contribute in the development of a framework for systems integration using as criteria and tools, those provided by Systems and Control Theory. The work aims to provide a control theoretic dimension to the rules and practices currently used in the specific application areas. The thesis contributes in the development of a Systems and Control conceptual and tools framework for integrated design of engineering processes by providing results in the following areas: - Specification of a number of generic problems in the field of integrated design and identification of relevant control theoretic concepts and tools. - Study of Model Orientation for linear implicit state-space models and definition of classes of oriented realisations. - Development of solutions to two problems of Structural Identification for uncertain early process models related to infinite zero structure and McMillan degree. - Development of a generic representation of composite systems that allows the study of transition from the aggregate to composite system properties, as a generalised Control Design and characterisation of some key system properties. -Specification of a framework for integrated Control Structure Selection and development of software for many approaches of the "interaction analysis" indicators. The results contribute in the area of development of the systems and control ideas for the problems of systems integration and early design. The work emphasises the strong links between Modelling, Selection of Control Structures and Control Structural methodologies