Robust Stability Under Mixed Time Varying, Time Invariant and Parametric Uncertainty

Robustness analysis is considered for systems with structured uncertainty involving a combination of linear time-invariant and linear time-varying perturbations, and parametric uncertainty. A necessary and sufficient condition for robust stability in terms of the structured singular value μ is obtained, based on a finite augmentation of the original problem. The augmentation corresponds to considering the system at a fixed number of frequencies. Sufficient conditions based on scaled small-gain are also considered and characterized

Robust stability of fractional-order linear time-invariant systems: Parametric versus Unstructured Uncertainty Models

The main aim of this paper is to present and compare three approaches to uncertainty modeling and robust stability analysis for fractional-order (FO) linear time-invariant (LTI) single-input single-output (SISO) uncertain systems. The investigated objects are described either via FO models with parametric uncertainty, by means of FO unstructured multiplicative uncertainty models, or through FO unstructured additive uncertainty models, while the unstructured models are constructed on the basis of appropriate selection of a nominal plant and a weight function. Robust stability investigation for systems with parametric uncertainty uses the combination of plotting the value sets and application of the zero exclusion condition. For the case of systems with unstructured uncertainty, the graphical interpretation of the utilized robust stability test is based mainly on the envelopes of the Nyquist diagrams. The theoretical foundations are followed by two extensive, illustrative examples where the plant models are created; the robust stability of feedback control loops is analyzed, and obtained results are discussed.European Regional Development Fund under the project CEBIA-Tech Instrumentation [CZ.1.05/2.1.00/19.0376]; Ministry of Education, Youth and Sports of the Czech Republic within the National Sustainability Programme [LO1303 (MSMT-7778/2014)

Polynomial Optimization with Applications to Stability Analysis and Control - Alternatives to Sum of Squares

In this paper, we explore the merits of various algorithms for polynomial optimization problems, focusing on alternatives to sum of squares programming. While we refer to advantages and disadvantages of Quantifier Elimination, Reformulation Linear Techniques, Blossoming and Groebner basis methods, our main focus is on algorithms defined by Polya's theorem, Bernstein's theorem and Handelman's theorem. We first formulate polynomial optimization problems as verifying the feasibility of semi-algebraic sets. Then, we discuss how Polya's algorithm, Bernstein's algorithm and Handelman's algorithm reduce the intractable problem of feasibility of semi-algebraic sets to linear and/or semi-definite programming. We apply these algorithms to different problems in robust stability analysis and stability of nonlinear dynamical systems. As one contribution of this paper, we apply Polya's algorithm to the problem of H_infinity control of systems with parametric uncertainty. Numerical examples are provided to compare the accuracy of these algorithms with other polynomial optimization algorithms in the literature.Comment: AIMS Journal of Discrete and Continuous Dynamical Systems - Series

Robustness Analysis and Optimally Robust Control Design via Sum-of-Squares

A control analysis and design framework is proposed for systems subject to parametric uncertainty. The underlying strategies are based on sum-of-squares (SOS) polynomial analysis and nonlinear optimization to design an optimally robust controller. The approach determines a maximum uncertainty range for which the closed-loop system satisfies a set of stability and performance requirements. These requirements, de ned as inequality constraints on several metrics, are restricted to polynomial functions of the uncertainty. To quantify robustness, SOS analysis is used to prove that the closed-loop system complies with the requirements for a given uncertainty range. The maximum uncertainty range, calculated by assessing a sequence of increasingly larger ranges, serves as a robustness metric for the closed-loop system. To optimize the control design, nonlinear optimization is used to enlarge the maximum uncertainty range by tuning the controller gains. Hence, the resulting controller is optimally robust to parametric uncertainty. This approach balances the robustness margins corresponding to each requirement in order to maximize the aggregate system robustness. The proposed framework is applied to a simple linear short-period aircraft model with uncertain aerodynamic coefficients

Consensus Synthesis of Robust Cooperative Control for Homogeneous Leader-Follower Multi-Agent Systems Subject to Parametric Uncertainty

This paper presents a design of robust consensus for homogeneous leader-follower multiagent systems (MAS). Each agent of MAS is described by a linear time-invariant dynamic model subject to parametric uncertainty. The agents are interconnected through a static interconnection matrix over an undirected graph to cooperate and share information with their neighbours. The consensus design of MAS can be transformed to stability analysis by using decomposition technique. We apply Lyapunov theorem to derive the sufficient condition to ensure the consensus of all independent subsystems. In addition, we design a robust distributed state feedback gain based on linear quadratic regulator (LQR) setting. Controller gain is computed via solving a linear matrix inequality. As a result, we provide a robust design procedure of a cooperative LQR control to achieve consensus objective and maximize the admissible bound of the uncertainty. Finally, we give numerical examples to illustrate the effectiveness of the proposed consensus design. The results show that the response for MAS in presence of uncertainty using robust consensus design follows the response of the leader and is better than that of the existingnominal consensus design

H 2 And H ∞ Filtering Design Subject To Implementation Uncertainty

This paper presents new filtering design procedures for discrete-time linear systems. It provides a solution to the problem of linear filtering design, assuming that the filter is subject to parametric uncertainty. The problem is relevant, since the proposed filter design incorporates real world implementation constraints that are always present in practice. The transfer function and the state space realization of the filter are simultaneously computed. The design procedure can also handle plant parametric uncertainty. In this case, the plant parameters are assumed not to be exactly known but belonging to a given convex and closed polyhedron. Robust performance is measured by the H 2 and H ∞ norms of the transfer function from the noisy input to the filtering error. The results are based on the determination of an upper bound on the performance objectives. All optimization problems are linear with constraint sets given in the form of LMI (linear matrix inequalities). Global optimal solutions to these problems can be readily computed. Numerical examples illustrate the theory. © 2005 Society for Industrial and Applied Mathematics.442515530Gevers, M., Li, G., (1993) Parametrizations in Control, Estimation and Filtering Problems, , Springer-Verlag, LondonWilliamson, D., Finite wordlength design of digital Kalman filters for state estimation (1985) IEEE Trans. Automat. Control, 30, pp. 930-939Williamson, D., Kadiman, K., Optimal finite wordlength linear quadratic regulators (1989) IEEE Trans. Automat. Control, 34, pp. 1218-1228Liu, K., Skelton, R.E., Grigoriadis, K., Optimal controllers for finite wordlength implementation (1992) IEEE Trans. Automat. Control, 37, pp. 1294-1304Hwang, S.Y., Minimum uncorrelated unit noise in state-space digital filtering (1977) IEEE Trans. Acoustics Speech Signal Process, 25, pp. 273-281Amit, G., Shaked, U., Minimization of roundoff errors in digital realizations of Kalman filters (1989) IEEE Trans. Acoustics Speech Signal Process, 37, pp. 1980-1982De Oliveira, M.C., Skelton, R.E., Synthesis of controllers with finite precision considerations (2001) Digital Controller Implementation and Fragility: A Modern Perspective, pp. 229-251. , R. S. H. Istepanian and J. F. Whidborne eds., Springer-Verlag, New YorkKeel, L.H., Bhattacharyya, S.P., Robust, fragile or optimal (1997) IEEE Trans. Automat. Control, 42, pp. 1098-1105Keel, L.H., Bhattacharyya, S.P., Authors' reply to: "Comments on 'Robust, fragile or optimal' " by P. M. Mäkilä (1998) IEEE Trans. Automat. Control, 43, p. 1268Dorato, P., Non-fragile controller design: An overview (1998) Proceedings of the 1998 American Control Conference, 5, pp. 2829-2831. , Philadelphia, IEEE, Piscataway, NJFamularo, D., Dorato, P., Abdallah, C.T., Haddad, W.H., Jadbabaie, A., Robust non-fragile LQ controllers: The static state feedback case (2000) Internat. J. Control, 73, pp. 159-165Yang, G.H., Wang, J.L., Robust nonfragile Kalman filtering for uncertain linear systems with estimator gain uncertainty (2001) IEEE Trans. Automat. Control, 46, pp. 343-348Haddad, W.M., Corrado, J.R., Robust resilient dynamic controllers for systems with parametric uncertainty and controller gain variations (2000) Internat. J. Control, 73, pp. 1405-1423Keel, L.H., Bhattacharyya, S.P., Stability margins and digital implementation of controllers (1998) Proceedings of the 1998 American Control Conference, 5, pp. 2852-2856. , (Philadelphia), IEEE, Piscataway, NJGeromel, J.C., Optimal linear filtering under parameter uncertainty (1999) IEEE Trans. Signal Process, 47, pp. 168-175Nesterov, Y., Nemirovskii, A., (1994) Interior-Point Polynomial Algorithms in Convex Programming, , SIAM, PhiladelphiaGeromel, J.C., Bernussou, J., Garcia, G., De Oliveira, M.C., H 2 and H ∞ robust filtering for discrete-time linear systems (2000) SIAM J. Control Optim., 38, pp. 1353-1368Geromel, J.C., De Oliveira, M.C., Bernussou, J., Robust filtering of discrete-time linear systems with parameter dependent Lyapunov functions (2002) SIAM J. Control Optim., 41, pp. 700-711De Oliveira, M.C., Bernussou, J., Geromel, J.C., A new discrete-time robust stability condition (1999) Systems Control Lett., 37, pp. 261-265Sayed, A.H., A framework for state-space estimation with uncertain models (2001) IEEE Trans. Automat. Control, 46, pp. 998-1013Balakrishnan, V., Huang, Y., Packard, A., Doyle, J.C., Linear matrix inequalities in analysis with multipliers (1994) Proceedings of the 1994 American Control Conference, 2, pp. 1228-1232. , Baltimore, MD, IEEE, Piscataway, NJGeromel, J.C., Peres, P.L.D., Bernussou, J., On a convex parameter space method for linear control design of uncertain systems (1991) SIAM J. Control Optim., 29, pp. 381-40

Topics in Automotive Rollover Prevention: Robust and Adaptive Switching Strategies for Estimation and Control

The main focus in this thesis is the analysis of alternative approaches for estimation and control of automotive vehicles based on sound theoretical principles. Of particular importance is the problem rollover prevention, which is an important problem plaguing vehicles with a high center of gravity (CG). Vehicle rollover is, statistically, the most dangerous accident type, and it is difficult to prevent it due to the time varying nature of the problem. Therefore, a major objective of the thesis is to develop the necessary theoretical and practical tools for the estimation and control of rollover based on robust and adaptive techniques that are stable with respect to parameter variations. Given this background, we first consider an implementation of the multiple model switching and tuning (MMST) algorithm for estimating the unknown parameters of automotive vehicles relevant to the roll and the lateral dynamics including the position of CG. This results in high performance estimation of the CG as well as other time varying parameters, which can be used in tuning of the active safety controllers in real time. We then look into automotive rollover prevention control based on a robust stable control design methodology. As part of this we introduce a dynamic version of the load transfer ratio (LTR) as a rollover detection criterion and then design robust controllers that take into account uncertainty in the CG position. As the next step we refine the controllers by integrating them with the multiple model switched CG position estimation algorithm. This results in adaptive controllers with higher performance than the robust counterparts. In the second half of the thesis we analyze extensions of certain theoretical results with important implications for switched systems. First we obtain a non-Lyapunov stability result for a certain class of linear discrete time switched systems. Based on this result, we suggest switched controller synthesis procedures for two roll dynamics enhancement control applications. One control design approach is related to modifying the dynamical response characteristics of the automotive vehicle while guaranteeing the switching stability under parametric variations. The other control synthesis method aims to obtain transient free reference tracking of vehicle roll dynamics subject to parametric switching. In a later discussion, we consider a particular decentralized control design procedure based on vector Lyapunov functions for simultaneous, and structurally robust model reference tracking of both the lateral and the roll dynamics of automotive vehicles. We show that this controller design approach guarantees the closed loop stability subject to certain types of structural uncertainty. Finally, assuming a purely theoretical pitch, and motivated by the problems considered during the course of the thesis, we give new stability results on common Lyapunov solution (CLS) existence for two classes of switching linear systems; one is concerned with switching pair of systems in companion form and with interval uncertainty, and the other is concerned with switching pair of companion matrices with general inertia. For both problems we give easily verifiable spectral conditions that are sufficient for the CLS existence. For proving the second result we also obtain a certain generalization of the classical Kalman-Yacubovic-Popov lemma for matrices with general inertia

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

Stability and Performance Analysis of Systems Under Constraints

All real world control systems must deal with actuator and state constraints. Standard conic sector bounded nonlinearity stability theory provides methods for analyzing the stability and performance of systems under constraints, but it is well-known that these conditions can be very conservative. A method is developed to reduce conservatism in the analysis of constraints by representing them as nonlinear real parametric uncertainty