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We address the regional input-to-state stabilization of discrete-time Takagi-Sugeno fuzzy models under delayed state, saturating actuators, and external energy bounded disturbances. We propose a convex method to design state feedback-based non-parallel distributed compensation, ensuring the fuzzy closed-loop system’s regional stability. Moreover, we introduce a geometrical estimate of the region of attraction in an augmented space, allowing a more extensive set of safe initial sequences. Furthermore, the minimization of the l 2 -gain from the external disturbance to the controlled output ensures closed-loop performance. Also, we provide optimization procedures allowing the maximization of the estimate of the region of attraction or the disturbance energy tolerance. By employing numerical examples, we compare our approach with similar ones found in the literature, showing our proposal’s efficacy, and its outperform concerning the compared methods.

This paper presents a robust Fault-Tolerant Control (FTC) methodology for the design of virtual sensors and virtual actuators for discrete-time Linear Parameter Varying (LPV) systems. Conditions based on Linear Matrix Inequalities (LMIs) are proposed for the synthesis of a reconfiguration block composed of a virtual actuator and a virtual sensor, guaranteeing input-to-state stability (ISS). The main contribution of the proposed FTC approach is to deal with LPV models where input and output matrices can be parameter-dependent. Moreover, differently from results found in the literature, a single reconfiguration block can be designed to be robust to different kinds and magnitudes of faults. Real-time level-control experiments illustrate the efficiency of the proposed procedure; the system used in the experiments consists of a nonlinear two coupled tanks with two inputs and two outputs, represented by an LPV model subject to sensor and actuator faults. Experimental results and comparisons with results in the literature indicate that the proposed approach is able to mitigate the fault effects with better performance indices than the literature approaches.

Although most of the control design methods assume unbounded control signals, real systems do have saturating actuators, which may degenerate closed-loop performance or even lead to unstable behavior. Additionally, the delay is generally almost ubiquitous in processes, that is also imposing performance and stability constraints. Our main contribution is to provide a controller design methodology for the stabilization of delayed systems under saturating actuators. Specifically, we address the design of non-Parallel Distributed Compensation (non-PDC) state feedback fuzzy control laws that locally stabilize a class of nonlinear discrete-time systems with state time-varying delay and saturating actuators. The proposed non-PDC control law depends on the current state xk and the state delayed by d ¯ samples. Based on the Lyapunov–Krasovskii approach, we characterize the safe region of initial conditions through two sets: an ellipsoidal one for the current state vector, and another set for the delayed state vectors. Through two convex optimization procedures, we can maximize the estimate of the region of attraction of the closed-loop control system. Additionally, a relaxation method inspired by the Frank-Wolfe algorithm is introduced, yielding better estimates of the region of attraction. The achievements are compared with other finds in the literature, illustrating the efficiency of this proposal.

A recognized challenge in the Takagi-Sugeno (T-S) fuzzy controller design concerns the use of different membership functions (MF) for controller and system. Most of the conditions available in the literature require that the controller’s MF matche the system’s one. Therefore, the implementation of such controllers may lead to unsafe operational conditions whenever such a match is lost. The main contribution of this paper is to provide new convex formulations for both stability analysis and controller design for T-S fuzzy systems under unmatched MF. We assume the same number of premises, yielding conditions called partially matched premises. To reduce conservatism, we use the Lyapunov approach, and we write the MF of the controller from the MF of the system. Two examples serve to compare our approach with others found in the literature. The achieved results suggest that our method outperforms the others.

This paper is concerned with the design of state-feedback controllers for Linear Parameter Varying (LPV) polynomial continuous-time systems. The vector field presents polynomial dependence on the states. Two synthesis conditions are proposed, the first one considers arbitrary rates of variation in the time-varying parameters, while the second one provides LPV controllers that are constructed based on a smoothed approximation of the time-varying parameter. The sum of squares matrix decomposition is employed to solve the proposed conditions. The L2 gain is also considered to give a robustness measure of the proposed controllers. The results are illustrated with examples from the literature.

International audience; This paper investigates the sampled-data control of continuous-time Takagi-Sugeno (T-S) fuzzy systems. The closed-loop dynamics is rewritten as a T-S system with input time-varying delays. In this context, asynchronous membership functions appears in the closed-loop dynamics. Thus, to reduce the conservatism of design conditions involving mismatch membership functions, a dedicated relaxation scheme is proposed. Then, from a convenient Lyapunov-Krasovskii function and the application of the Finsler's Lemma, new LMI-based conditions are proposed for the design of sampled-data Parallel-Distributed-Compensation (PDC) controllers. An example is provided to illustrate the effectiveness of the proposed design methodology in simulation, as well as to highlight their conservatism improvement regarding to previous related results from the literature.

This paper presents new conditions to design linear parameter-varying (LPV) state-feedback controllers for systems under saturating actuators. The proposed design ensures a minimal rate of contractivity of an associated Lyapunov function. A proportional-integral (PI) like structure is employed to ensure null tracking error for piecewise constant reference signals. Therefore, this proposal fits the design requirements of LPV and quasi-LPV real systems under saturating actuators. Experimental essays conducted on a second-order nonlinear level control illustrate the potential of the proposed approach. Additionally, the tests indicate how the contractivity rate affects the size of the estimate of the region of attraction.

International audience; This paper addresses the problem of emulation-based dynamic output-feedback control for discrete-time linear parameter varying systems under actuator saturation. Sufficient conditions based on linear matrix inequalities (LMI) are derived to design the parameters of two dependent event-triggering rules handling the sensor-to-controller and controller-to-actuator communication and to guarantee the regional asymptotic stability of the closed-loop system. A convex optimization scheme incorporating these conditions is also formulated to reduce the output and control signals' update rate through communication channels. A numerical example illustrates the usefulness of the proposal.

International audience; This paper examines an event-triggered control design approach for discrete-timelinear parameter-varying (LPV) systems under control constraints. A parameter-dependentdynamic output-feedback controller with an event-triggering condition is designed to ensurethe regional asymptotic stability of the closed-loop system while minimizing a quadratic costfunction. Sufficient conditions are derived in terms of linear matrix inequalities (LMIs) thanksto the use of a parameter-dependent quadratic Lyapunov function. Also, convex optimizationschemes are proposed using these conditions to minimize an upper bound of a given cost functionor to maximize the size of the region of closed-loop stability. Examples illustrate the proposal

We address the problem of robust input-to-state stabilization of parameter-varying discrete-time systems with time-varying state delay, saturating actuators, and subject to $\ell _2$ -limited disturbance. It is assumed that the delay belongs to a known interval and its maximum variation between two consecutive instants is taken into account. The proposed convex delay-dependent conditions for the synthesis of robust state feedback controllers ensure local input-to-state stability of the closed-loop system for a set of initial conditions and for energy-bounded disturbance signals. However, the computed controllers do not require the real-time knowledge of the delay. The approach is based on the rewriting of the saturating and delayed system in terms of a switching uncertain augmented delay-free system with a dead-zone nonlinearity and on the application of the generalized sector condition. To illustrate the efficiency of our approach, we compare it by means of numerical examples with others found in the literature.

The nonlinear action of hybrid PI controllers makes possible to achieve almost none overshot with the smallest rise time possible (for the available input energy) and a settling time almost equal to the rise time. However, all of these nice properties require changes on the control loop. Our contribution is the proposition of a plug-in control action that can be added to the classical linear PI to avoid changes in the hardware, and still achieve the same behavior of a recently introduced adaptive hybrid PI controller. Our proposal is illustrated by real-time level control experiments in a nonlinear reservoir. Performance indexes are presented to evaluate the experimental tests.

Applications involving Unmanned Aerial Vehicles (UAVs) have increasingly required faster and more accurate movements to reduce flight time and to improve efficiency in the obstacle avoidance capability. In this context, this work proposes a nonlinear model predictive control (NMPC) strategy formulated on the Special Euclidean group SE(3) for quadrotor trajectory tracking within cluttered environments with unknown obstacles. The approach considers constraints in the states and inputs, with constant disturbance rejection and capable of executing aggressive maneuvers. The UAV attitude is considered as an optimization variable within the control problem thanks to an algebraic ellipsoidal set approach. As a consequence, the collision check takes the UAV attitude into account, allowing aggressive maneuvers. Numerical experiments under realistic conditions allow evaluating the performance of the proposed approach for the UAV. The tested maneuvers are throwing a narrow gap, passing by a nonconvex obstacle gap, avoiding a convex obstacle, and doing slalom movements. In all cases, uncertainties are considered. The achieved results indicate the advantages of executing aggressive maneuvers.

International audience; This paper deals with discrete time-varying systems with state-delayed and saturating actuators. A robust state feedback control gain is designed through convex methods ensuring the local stability of the time-varying system with interval time-varying state delay. The estimate of the set of admissible initial conditions is characterized by three convex sets allowing less conservative estimates. Through two numerical examples, we compare our approach with others in the literature and illustrate the better behavior of the proposed methods.

International audience; The command governor (CG) is a useful strategy to avoid constraints violation in case of control reconfiguration or operational conditions changes, like switched systems mode changes. In general, the structure of the CG specializes in producing an alternative reference signal to drive a closed-loop system to the desired reference while avoiding constraints violations in the control signal, state-space, or combinations of them. In this paper, we propose a simple strategy that modifies the conventional structures of the CG and supervisor, yielding a switched rule based on the estimated controllers' region of attraction. We present some simulations to illustrate the proposal's potential and compare it with a competitor scheme exploiting a dwell-time approach. The results suggest that our approach adds new possibilities of CG and supervisor design, reducing the transition time between system modes and improving the closedloop performance indexes.; Le régulateur de commande (CG) est une stratégie utile pour éviter la violation des contraintes en cas de reconfiguration des commandes ou de changements de conditions opérationnelles, comme les changements de mode des systèmes commutés. En général, la structure du CG se spécialise dans la production d'un signal de référence alternatif pour conduire un système en boucle fermée vers la référence souhaitée tout en évitant les violations de contraintes dans le signal de commande, l'espace d'états ou leurs combinaisons. Dans cet article, nous proposons une stratégie simple qui modifie les structures conventionnelles du CG et du superviseur, produisant une règle commutée basée sur la région d'attraction estimée des contrôleurs. Nous présentons quelques simulations pour illustrer le potentiel de la proposition et le comparer avec un schéma concurrent exploitant une approche temps de séjour. Les résultats suggèrent que notre approche ajoute de nouvelles possibilités de conception de CG et de superviseur, réduisant le temps de transition entre les modes du système et améliorant les indices de performance en boucle fermée.

We propose convex conditions for stabilization of nonlinear discrete-time systems with time-varying delay in states through a fuzzy Takagi-Sugeno (T-S) modeling. These conditions are developed from a fuzzy Lyapunov-Krasovskii function and they are formulated in terms of linear matrix inequalities (LMIs). The results can be applied to a class of nonlinear systems that can be exactly represented by T-S fuzzy models inside a specific region called the region of validity. As a consequence, we need to provide an estimate of the set of safe initial conditions called the region of attraction such that the closed-loop trajectories starting in this set are assured to remain in the region of validity and to converge asymptotically to the origin. The estimate of the region of attraction is done with the aid of two sets: one dealing with the current state, and the other concerning the delayed states. Then, we can obtain the feedback fuzzy control law depending on the current state, x k , and the maximum delayed state vector, xk−d. It is shown that such a control law can locally stabilize the nonlinear discrete-time system at the origin. We also develop convex optimization procedures for the computation of the fuzzy control gains that maximize the estimates of the region of attraction. We present two examples to demonstrate the efficiency of the developed approach and to compare it with other approaches in the literature.

In this paper we consider sampled-data systems controlled by hybrid PI controllers. In these systems, the sample time may induce some undesirable overshoot due to the postponed reset action on the control integral state, which deteriorates the performance of the closed loop by leading to output overshoot as well as to larger settling time values. We propose a methodology to compute a bounded compensating signal to be added to the control input that minimizes the overshoot and the settling time. The action of the compensating signal synchronizes the reset instants of the hybrid PI controller with the sample-time for a class of systems. The proposed methodology is tested in a laboratory setup system composed of two connected water reservoirs with capacity of 200 liters each, actioned by a variable velocity pump. The control objective is to regulate the level of the first tank and to reject disturbances. We compare the performances achieved by two different hybrid PI schemes with and without the proposed additive compensating signal. The experimental tests illustrate the performance, robustness and disturbance rejection achieved by our approach when applied to uncertain systems.

We address the problem of input-to-state stabilization of linear parameter-varying (LPV) discrete-time systems with delayed state and subject to saturated actuators and to l2-limited disturbance. It is assumed that the time-varying delay belongs to a known interval and its maximum variation between two consecutive instants has limited rate. We propose convex delay-dependent conditions for the synthesis of LPV state feedback controllers that do not require the real-time knowledge of the delay and ensure the local input-to-state stability of the closed-loop system for a set of initial conditions and subject to energy bounded disturbance input signals. The approach is based on rewriting the delayed system as a switched uncertain augmented delay-free system with a dead-zone non-linearity and on the application of the generalized sector condition to deal with the saturating actuators. To illustrate the efficiency of our approach, we present a numerical example.

In this paper, we consider the implementation and the robust stability analysis of a new hybrid PI controller. Two laboratory processes are used, one to control the level in water reservoir system with capacity of 200 liters, and the other to control the temperature in a continuous oven of one meter long. The robust stability analysis conditions are proposed in terms of convex optimization procedures and can be used to certify the robust closed-loop stability. The dynamics of the processes are approximated by a first order system plus delay and, because of this, a Smith predictor is jointly used with all controllers. Concerning the hybrid PI implementation, we compare the performances achieved by a classical linear PI, by a hybrid controller found in the literature and by a new hybrid controller with adaptation of the reset value. We illustrate the performance, robustness and disturbance rejection of all three controllers in a prototype by control real time tests. The achieved performance of each controller is quantified by means of typical industrial indexes.

This paper deals with the non-quadratic stabilization of Takagi–Sugeno (T-S) models with D -stability constraints. Based on a recently proposed Non-Quadratic Lyapunov Function (NQLF), which involves the mean values of the membership functions (MFs) over a given time interval, three theorems are proposed for the design of non-Parallel Distributed Compensation (non-PDC) controllers satisfying closed-loop D -stability specifications. Despite previous non-quadratic approaches and thanks to the nature of the considered NQLF, it is highlighted that the proposed LMI-based procedures not only apply for the global non-quadratic D -stabilization of T-S models, but also for a larger class of T-S models with piecewise membership functions (i.e. a class of switching nonlinear systems), since no requirement is needed regarding to the bounds of the MFs derivatives. The effectiveness of the proposed LMI-based conditions and their relative degrees of conservatism, compared with previous quadratic D -stabilization results, are illustrated through an academic example involving piecewise membership functions.

Exact feedback linearization is a method for nonlinear control which amounts to cancel the nonlinearities of a nonlinear system such that the resulting closed-loop dynamics is linear. The effectiveness of exact feedback linearization relies on a precise description of the system nonlinearities. This paper suggests a novel robust control approach for adaptive control of nonlinear systems called robust granular feedback linearization. The approach employs an instance of evolving the participatory learning algorithm to continuously estimate unknown nonlinearities and cancel their effects in the control loop. Under mild conditions, the robust granular feedback linearization is ensured to be Lyapunov stable by using convex methods. Simulation experiments with a surge tank is used to evaluate and to compare the performance of the robust granular feedback linearization against exact feedback linearization and an adaptive controller based on bacterial foraging. The results indicate that the robust granular feedback linearization outperforms both, the exact and the adaptive foraging controllers. The effectiveness of robust granular feedback linearization is further testified in an actual surge tank control system application.

The robust local stabilization of uncertain discrete-time systems with time-varying state delayed and subject to saturating actuators is investigated in this work. A convex optimization method is proposed to compute robust state feedback control law such that the uncertain closed-loop is locally asymptotically stable if the initial condition belongs to an estimate of the region of attraction for the origin. The proposed procedure allows computing estimates of the region of attraction through the intersection of ellipsoidal sets in an augmented space, reducing the conservatism of the estimates found in the literature. Also, the conditions can handle the amount of delay variation between two consecutive samples, which is new in the literature for the discrete-time case. Although the given synthesis conditions are delay dependent, the proposed control law is delay independent, yielding to easier real time implementations. A convex procedure is proposed to maximize the size of the set of safe initial conditions. Numerical examples are provided to illustrate the effectiveness of our approach and also to compare it with other conditions in the literature. Fil: Silva, J. V. V.. Centro Federal de Educação Tecnológica de Minas Gerais; Brasil Fil: Silva, L. F. P.. Centro Federal de Educação Tecnológica de Minas Gerais; Brasil Fil: Rubio Scola, Ignacio Eduardo Jesus. Cefetmg/ufjs; Fil: Leite, V. J. S.. Cefetmg/ufjs

International audience; This paper addresses the boundary control problem of fluid transport in a Poiseuille flow taking the actuator dynamics into account. More precisely , sufficient stability conditions are derived to guarantee the exponential stability of a linear hyperbolic differential equation system subject to non-linear quadratic dynamic boundary conditions by means of Lyapunov based techniques. Then, convex optimization problems in terms of linear matrix inequality constraints are derived to either estimate the closed-loop stability region or synthesize a robust control law ensuring the local closed-loop stability while estimating an admissible set of initial states. The proposed results are then applied to application-oriented examples to illustrate local stability and stabilization tools.

A convex condition in terms of linear matrix inequalities (LMIs) is developed for the synthesis of stabilizing fuzzy state feedback controllers for nonlinear discrete-time systems with time-varying delays. A Takagi-Sugeno (T-S) fuzzy model is used to represent exactly the nonlinear system in a restricted domain of the state space, called region of validity. The proposed stabilization condition is based on a Lyapunov-Krasovskii (L-K) function and it takes into account the region of validity to determine a set of initial conditions for which the actual closed-loop system trajectories are asymptotically stable and do not evolve outside the region of validity. This set of allowable initial conditions is determined from the level set associated to a fuzzy L-K function as a Cartesian product of two subsets: one characterizing the set of states at the initial instant and another for the delayed state sequence necessary to characterize the initial conditions. Finally, we propose a convex programming problem to design a fuzzy controller that maximizes the set of initial conditions taking into account the shape of the region of validity of the T-S fuzzy model. Numerical simulations are given to illustrate this proposal.

We address the design of parameter-dependent dynamic output feedback controllers for discrete-time systems with time-varying parameters and subject to actuators saturations. The particularity of the proposed controller is that all its matrices are dependent of the varying parameters, differently from a previous proposed compensator that was partially dependent of the parameters. The proposed LMI stability and stabilization results are based on the use of a parameter dependent Lyapunov function and on the λ-contractivity of the associated level set to guarantee the local stability and some time performance. Two control strategies are addressed: avoiding saturation and allowing saturation, in both cases taking into account the realistic consideration of local validity of the plant model. The proposed numerical example shows the control design for a nonlinear system represented by a Takagi-Sugeno Fuzzy model.

Neste trabalho são apresentadas condições convexas na forma de desigualdades matriciais lineares (LMIs) que resultam em condições sucientes para teste de estabilidade robusta de sistemas discretos no tempo com atrasos variantes no tempo. Na seqüência, são obtidas condições LMI para a síntese de ganhos robustos para realimentação de estados. Dois tipos de condições são propostas: dependentes do atraso e independentes do atraso. Ambas são convexas e empregam matrizes extras para reduzir o conservadorismo dos resultados. Exemplos da literatura são utilizados para comparar o desempenho das condições aqui propostas com os resultados obtidos em outros trabalhos. É apresentado um estudo de um modelo com incertezas politópicas obtido de um forno industrial elétrico. Nesse estudo, o sistema incerto é estabilizado por meio das condições LMI aqui propostas. In this paper some convex conditions are presented in the form of linear matrix inequalities (LMIs), yielding sufcient conditions for testing the robust stability of uncertain discrete-time systems with time-varying delay. In the sequel, LMI conditions for design robust state feedback gains are obtained. Two types of delay conditions are proposed: delay dependent and delay independent. Both of them are convex and employ extra matrices to achieve less conservative results. Examples from the literature are used to compare the performance of the conditions proposed here with the results obtained in other papers. It is presented a study of a model with polyopic uncertainty obtained from an industrial electric heater. In this study, the uncertain system is stabilized by means of the LMI conditions proposed here.

A equivalência entre resultados para análise de estabilidade baseados no método de Lyapunov de sistemas com atraso precisamente conhecidos e resultados de estabilidade robusta de sistemas de comparação sem atraso, empregando condições do teorema do ganho pequeno escalonado, foi apresentada recentemente na literatura. A extensão desses resultados para tratar sistemas com atraso e incerteza em domínios politópicos é a principal contribuição deste artigo. A partir da definição de uma realização genérica para um sistema, são apresentadas desigualdades matriciais lineares com variáveis extras, equivalentes a condições de ganho pequeno escalonadas. Devido à inserção dessas variáveis extras, condições independentes e dependentes do atraso para análise de estabilidade de sistemas com atraso em domínios politópicos podem ser elaboradas empregando matrizes de Lyapunov dependentes de parâmetro, resultando em condições menos conservadoras que outras já apresentadas na literatura, como ilustrado por exemplos numéricos.

Condições suficientes na forma de desigualdades matriciais lineares são dadas para a análise de estabilidade robusta de sistemas lineares incertos do tipo neutral, nos quais assume-se que o vetor de estados possui atrasos variantes no tempo. Todas as matrizes do sistema são supostas invariantes no tempo, incertas porém pertencentes a um politopo cujos vértices são conhecidos. A estabilidade robusta do sistema incerto neutral é assegurada por meio de um funcional de Lyapunov-Krasovskii dependente de parâmetro. Além disso, mostra-se como condições para a estabilidade robusta de sistemas incertos contínuos no tempo com e sem atraso nos estados podem ser recuperadas a partir das condições propostas no trabalho. Exemplos numéricos ilustram os resultados obtidos.

International audience; In this paper a flow inside a pipe is modeled by a hyperbolic system (written in terms of a partial differential equation). One boundary condition is given by the coupling with a finite-dimensional dynamic model of heating column and a static modeling of ventilator. Then, the classical finite-dimensional technique is applied for the linearization of first order hyperbolic systems with dynamics associated to the boundary conditions. The discretization of the infinite-dimensional system is used and an augmented discrete linear system with dimension depending on the step size of discretization in space is obtained. The results are illustrated on simulations considering a Poiseuille flow experimental setup.

This paper addresses the problem of state feedback control of continuous-time switched linear systems with arbitrary switching rules. A quadratic Lyapunov function with a common matrix is used to derive a stabilizing switching control strategy that guarantees: (i) the assignment of all the eigenvalues of each linear subsystem inside a chosen circle in the left-hand half of the complex plane; (ii) a minimum disturbance attenuation level for the closed-loop switched system. The proposed design conditions are given in terms of linear matrix inequalities that encompass previous results based on quadratic stability conditions with fixed control gains. Although the quadratic stability based on a fixed Lyapunov matrix has been widely used in robust control design, the use of this condition to provide a convex design method for switching feedback gains has not been fully investigated. Numerical examples show that the switching control strategy can cope with more stringent design specifications than the fixed gain strategy, being useful to improve the performance of this class of systems.

Uma condição suficiente para a existência de uma realimentação de estados robusta dependente de parâmetros é apresentada neste trabalho. A condição é verificada através do teste de factibilidade de um conjunto de desigualdades matriciais lineares. A realimentação obtida garante ainda a alocação robusta dos pólos de malha fechada de sistemas lineares incertos em uma região circular do plano complexo, através da existência de uma função de Lyapunov dependente de parâmetros. Exemplos ilustram os resultados.

Condições suficientes para a análise de estabilidade de sistemas lineares com incertezas politópicas são apresentadas neste trabalho. A estabilidade robusta é garantida a partir da existência de uma função de Lyapunov dependente de parâmetros obtida do teste de factibilidade de um conjunto de desigualdades matriciais lineares (em inglês, LMIs - Linear Matrix Inequalities) formuladas nos vértices do politopo de incertezas. Três condições são apresentadas para sistemas contínuos no tempo, e três para sistemas a tempo discreto. Os resultados são também comparados com a análise baseada na estabilidade quadrática (mesma função de Lyapunov para todo o conjunto de incertezas), tanto para o caso contínuo quanto para o caso discreto. A primeira condição explora o uso de variáveis (matrizes) adicionais em LMIs, e a segunda utiliza um número maior de LMIs. Essas duas condições foram recentemente publicadas e são menos conservadoras que a estabilidade quadrática. A terceira condição, proposta neste trabalho, mescla as duas idéias e apresenta resultados bem mais abrangentes, contendo as condições anteriores como casos particulares. Vários exemplos são apresentados, ilustrando o desempenho numérico das formulações LMI em termos de eficiência e de complexidade computacional. Sufficient conditions for the analysis of stability of linear systems with polytopic uncertainties are presented in this paper. The robust stability is guaranteed by the existence of a parameter dependent Lypaunov function obtained from the feasibility test of a set of linear matrix inequalities (LMIs) formulated at the vertices of the uncertainty polytope. Three conditions are presented, and the results are also compared with the analysis based on quadratic stability (same Lyapunov function for the entire set of uncertainties), for continuous as well as discrete-time systems. The first condition exploits the use of some extra variables (matrices) in the LMIs, and the second one uses a larger number of LMIs. These two conditions have recently appeared in the literature and are less conservative than quadratic stability. The third condition, proposed in this paper, combines the two ideas, yielding better results, and contains the previous conditions as particular cases. Several examples are presented to illustrate the numerical performance of the LMI conditions in terms of efficiency and computational complexity.

Na determinação dos gradientes hidráulicos das águas subterrâneas em aquíferos estuarinos, as ondas de pressão decorrentes das flutuações das marés e corpos de água devem ser consideradas. Para este estudo, monitorou-se por 45 dias as variações de nível de água do Rio Cubatão e carga hidráulica do aquífero raso (em 3 níveis de profundidade) numa área industrial adjacente ao rio, em Cubatão, SP, a fim de se determinar as cargas hidráulicas médias e gradientes médios pelo método conforme proposto por Serfes (1991). Este monitoramento foi contínuo e realizado em dois cenários: com o sistema de remediação (“Air Sparging”) ativado e desativado. Os resultados mostraram a interferência do sistema de remediação e da variação da maré nas cargas hidráulicas dos poços monitorados nos poços próximos ao rio Cubatão, e somente a variação da maré nos poços mais distantes. No entanto, na porção superior do aquífero monitorado mostrou a condição de influência do rio Cubatão nos dois cenários. Condições de efluência e influência na porção intermediária e efluência predominante desse rio na porção mais profunda foram observadas. Nas porções intermediária e profunda, os gradientes hidráulicos calculados apresentam valores baixos (-0,0027 a 0,0030), de modo que a análise sobre o sentido de fluxo deve ser criteriosa.

Neste trabalho são investigados sistemas lineares incertos e discretos no tempo com atraso variante no tempo afetando o vetor de estados. É considerado que as incertezas são representadas em um domínio politópico e que elas podem estar presentes em todas as matrizes do modelo do sistema. Condições expressas como Desigualdades Matriciais Lineares (LMIs, do inglês Linear Matrix Inequalities) são propostas para o cômputo do custo garantido H∞ e para a síntese de ganhos robustos de realimentação de estados que minimizam a norma H∞ entre a entrada de pertubação e a saída do sistema. Essas condições são estabelecidas com a utilização de funções de Lyapunov-Krasovskii (L-K) dependentes de parâmetro. Variáveis matriciais de folga - via Lema de Finsler - são empregadas para desacoplar as matrizes do sistema das matrizes da função de L-K. A "desigualdade de Jensen" é usada para manipular os termos cruzados que aparecem no desenvolvimento das condições, fornecendo uma majoração menos conservadora que outras encontradas na literatura. As condições propostas são dependentes do atraso. Exemplos numéricos são apresentados para ilustrar a eficácia da proposta e para estabelecer comparações com outras encontradas na literatura. This paper deals with uncertain discrete-time systems with time varying delay affecting the state vector. It is considered that the uncertainties are represented in a polytopic domain and they may be present in all matrices of the model of the system. Conditions expressed as Linear Matrix Inequalities (LMIs) are proposed for the H∞ guaranteed cost computation and for the design of robust state feedback control gains that minimize the H∞ norm from the perturbation input to the system output. These conditions are established by using parameter dependent Lyapunov-Krasovskii (L-K) functions. Slack matrix variables - via Finsler's Lemma - are employed to decouple the matrices of the system from the L-K function ones. The "Jensen's inequality" is used to handle crossed terms in the development of the conditions, yielding a less conservative over bound w.r.t. other approaches in the literature. The provided conditions are delay-dependent. Numerical examples are presented to illustrate the eficacy of the proposed conditions and they are used to establish comparisons with other techniques available in the literature.

This chapter is about techniques for robust stability analysis and robust stabilization of discrete-time systems with delay in the state vector. The relevance of this study is mainly due to the unavoidable presence of delays in dynamic systems. Even small time-delays can reduce the performance of systems and, in some cases, lead them to instability. Examples of such systems are robotics, networks, metal cutting, transmission lines, chemical and thermal processes among others as can be found in the books from Gu et al. (2003), Richard (2003), Niculescu (2001) and Kolmanovskii & Myshkis (1999). Studies and techniques for dealing with such systems are not new. Since the beginning of control theory, researchers has been concerned with this issue, either in the input-output approach or in state-space approach. For the input-output approach, techniques such as Pade approximation and the Smith predictor are widely used, mainly for process control. The use of state space approach allows to treat both cases. For both approaches delays can be constant or time-varying. Besides, both the delay and the systems can be precisely known or affected by uncertainties. In this chapter the class of uncertain discrete-time systems with state delay is studied. For these systems, the techniques for analysis and design could be delay dependent or delay independent, can lead with precisely known or uncertainty systems (in a polytopic or in a norm-bonded representation, for instance), and can consider constant or time-varying delays. For discrete-time systems with constant and known delay in the state it is always possible to study an augmented delay-free system Kapila & Haddad (1998), Leite & Miranda (2008a). However, this solution does not seem to be suitable to several cases such as time-varying delay or uncertain systems. For these systems,most of the applied techniques for robust stability analysis an robust control design are based on Lyapunov-Krasovskii (L-K) approach, which can be used to obtain convex formulation problems in terms of linear matrix inequalities (LMIs). In the literature it is possible to find approaches based on LMIs for stability analysis, most of them based on the quadratic stability (QS), i.e., with the matrices of the Lyapunov-Krasovskii function being constant and independent of the uncertain parameters. In the context of QS, non-convex formulations of delay-independent type have been proposed, for example, in Shi et al. (2003) where the delay is considered time-invariant. In Fridman & Uncertain Discrete-Time Systems with Delayed State: Robust Stabilization with Performance Specification via LMI Formulations 17

Sufficient linear matrix inequality (LMI) conditions to verify the robust stability and to design robust state feedback gains for the class of linear discrete-time systems with time-varying delay and polytopic uncertainties are presented. The conditions are obtained through parameter-dependent Lyapunov-Krasovskii functionals and use some extra variables, which yield less conservative LMI conditions. Both problems, robust stability analysis and robust synthesis, are formulated as convex problems where all system matrices can be affected by uncertainty. Some numerical examples are presented to illustrate the advantages of the proposed LMI conditions.

This paper investigates the robust stability of continuous-time, time-invariant linear uncertain systems in polytopic domains. Robust stability is checked by constructing a quadratic parameter-dependent Lyapunov function. The matrix associated with this quadratic Lyapunov function is a polynomial function of the uncertain parameters, expressed as a particular polynomial matrix involving powers of the dynamic matrix of the system and one symmetric matrix to be determined. The degree of this polynomial matrix function is arbitrary. Finsler’s Lemma is used to lift the obtained stability conditions into a larger space in which sufficient stability tests can be developed in the form of Linear Matrix Inequalities (LMIs), which must be verified at the vertices of the uncertainty polytopic domain. Examples illustrate the method, which is compared with similar results in the literature by means of random numerical experiments.

Linear matrix inequality conditions are given for the robust stability of neutral systems with time-varying delays in the state vector. All the system matrices are supposed to be time-invariant, belonging to a known polytope. The robust stability is assured by means of a parameter-dependent Lyapunov-Krasovskii functional, independently of the size of the time-varying delays. A numerical example illustrates the results.