On Computing the Worst-case H∞ Performance of Lur'e Systems with Uncertain Time-invariant Delays

This paper presents a worst-case H∞ performance analysis for Lur'e systems with time-invariant delays. The sucient condition to guarantee an upper bound of worst-case performance is developed based on the delay-partitioning Lyapunov-Krasovskii functional containing the integral of sector-bounded nonlinearities. Using Jensen inequality and S-procedure, the delay-dependent criterion is given in terms of linear matrix inequalities. In addition, we extend the criterion to compute the worst-case performance for Lur'e systems subject to norm-bounded uncertainties by using a matrix eliminating lemma. Numerical results show that our criterion provide the least upper bound on the worst-case H∞ performance comparing to the criteria derived based on existing techniques

Stability and stabilization of sampled-data control for lure systems

Este trabalho apresenta um novo método para a análise de estabilidade e estabilização de sistemas do tipo Lure com controle amostrado, sujeitos a amostragem aperiódica e não linearidades que são limitadas em setor e restritas em derivada, em ambos contextos global e regional. Assume-se que os estados da planta estão disponíveis para medição e que as não linearidades são conhecidas, o que leva a uma formulação mais geral do problema. Os estados são adquiridos por um controlador digital que atualiza a entrada de controle em instantes de tempo discretos e aperiódicos, mantendo-a constante entre dois instantes sucessivos de amostragem. A abordagem apresentada neste trabalho é baseada no uso de uma nova classe de looped-functionals e em uma função do tipo Lure generalizada, que leva a condições de estabilidade e estabilização que são escritas na forma de desigualdades matriciais lineares (LMIs) e quasi-LMIs, respectivamente. Com base nestas condições, problemas de otimização são formulados com o objetivo de computar o intervalo máximo entre amostragens ou os limites máximos do setor para os quais a estabilidade assintótica da origem do sistema de dados amostrados em malha fechada é garantida. No caso em que as condições de setor são válidas apenas localmente, a solução desses problemas também fornece uma estimativa da região de atração para as trajetórias em tempo contínuo do sistema em malha fechada. Como as condições de síntese são quasi-LMIs, um algoritmo de otimização por enxame de partículas é proposto para lidar com as não linearidades envolvidas nos problemas de otimização, que surgem do produto de algumas variáveis de decisão. Exemplos numéricos são apresentados ao longo do trabalho para destacar as potencialidades do método.This work presents a new method for stability analysis and stabilization of sampleddata controlled Lure systems, subject to aperiodic sampling and nonlinearities that are sector bounded and slope restricted, in both global and regional contexts. We assume that the states of the plant are available for measurement and that the nonlinearities are known, which leads to a more general formulation of the problem. The states are acquired by a digital controller which updates the control input at aperiodic discrete-time instants, keeping it constant between successive sampling instants. The approach here presented is based on the use of a new class of looped-functionals and a generalized Luretype function, which leads to stability and stabilization conditions that are written in the form of Linear Matrix Inequalities (LMIs) and quasi-LMIs, respectively. On this basis, optimization problems are formulated aiming to compute the maximal intersampling interval or the maximal sector bounds for which the asymptotic stability of the origin of the sampled-data closed-loop system is guaranteed. In the case where the sector conditions hold only locally, the solution of these problems also provide an estimate of the region of attraction for the continuous-time trajectories of the closed-loop system. As the synthesis conditions are quasi-LMIs, a Particle Swarm Optimization (PSO) algorithm is proposed to deal with the involved nonlinearities in the optimization problems, which arise from the product of some decision variables. Numerical examples are presented throughout the work to highlight the potentialities of the method

An improved stability criterion for discrete-time time-delayed Lur’e systemwith sector-bounded nonlinearities

The absolute stability problem of discrete-time time-delayed Lur\u27e systems with sector bounded nonlinearities is investigated in this paper. Firstly, a modified Lyapunov-Krasovskii functional (LKF) is designed with augmenting additional double summation terms, which complements more coupling information between the delay intervals and other system state variables than some previous LKFs. Secondly, some improved delay-dependent absolute stability criteria based on linear matrix inequality form (LMI) are proposed via the modified LKF and the relaxed free-matrix-based summation inequality technique application. The stability criteria are less conservative than some results previously proposed. The reduction of the conservatism mainly relies on the full use of the relaxed summation inequality technique based on the modified LKF. Finally, two common numerical examples are presented to show the effectiveness of the proposed approach

Stability Conditions for a Class of Nonlinear Systems with Delay

This chapter presents an extension and offers a more comprehensive overview of our previous paper entitled “Stability conditions for a class of nonlinear time delay systems” published in “Nonlinear Dynamics and Systems Theory” journal. We first introduce a more complete approach of the nonlinear system stability for the single delay case. Then, we show the application of the obtained results to delayed Lur’e Postnikov systems. A state space representation of the class of system under consideration is used and a new transformation is carried out to represent the system, with delay, by an arrow form matrix. Taking advantage of this representation and applying the Kotelyanski lemma in combination with properties of M-matrices, some new sufficient stability conditions are determined. Finally, illustrative example is provided to show the easiness of using the given stability conditions

Quasilinear Control of Systems with Time-Delays and Nonlinear Actuators and Sensors

This thesis investigates Quasilinear Control (QLC) of time-delay systems with nonlinear actuators and sensors and analyzes the accuracy of stochastic linearization for these systems. QLC leverages the method of stochastic linearization to replace each nonlinearity with an equivalent gain, which is obtained by solving a transcendental equation. The idea of QLC is to stochastically linearize the system in order to analyze and design controllers using classical linear control theory. In this thesis, the existence of the equivalent gain for a closed-loop time-delay system is discussed. To compute the equivalent gain, two methods are explored. The first method uses an explicit but complex algorithm based on delay Lyapunov equation to study the time-delay, while the second method uses Pade approximant. It is shown that, under a suitable criterion, Pade approximant can be effectively applied for QLC of time-delay systems. Furthermore, the method of Saturated-Root Locus (S-RL) is extended to nonlinear time-delay systems. It turns out that, in a time-delay system, S-RL always terminates prematurely as opposed to a delay-free system, which may or may not terminate prematurely. Statistical experiments are performed to investigate the accuracy of stochastic linearization compared to a system without time-delay. The impact of increasing the time-delay in the approach of stochastic linearization is also investigated. Results show that stochastic linearization effectively linearizes a nonlinear time-delay system, even though delays generally degrade accuracy. Overall, the accuracy remains relatively high over the selected parameters. Finally, this approach is applied to pitch control in a wind turbine system as a practical example of a nonlinear time-delay system, and its performance is analyzed to demonstrate the efficacy of the approach

Data-driven stabilization and safe control of nonlinear systems

The recent successes of machine learning solutions have inspired the research of new control algorithms derived directly from the available data without any intermediate step. Being able to design a stabilizing controller directly from data has the main advantage that, since it does not rely on a model of the system to control, the controller design is not influenced by any modeling error.Most of the time real systems are simplified with linear models to reduce the overall complexity in the controller design discarding all the complex nonlinear behaviors. A linear approximation could be an excessive simplification for complex system where the presence of nonlinear dynamics are important to understand those processes and nonlinearities can not be ignored. However, the analysis and control of a nonlinear model is often challenging. This thesis investigates data-based control methods for continuous and discrete-time nonlinear systems that do not require to model the system. In particular, we have developed a solution to obtain a stabilizing state feedback controller for the case of nonlinear systems. Stabilizing a closed-loop system is critical, but sometimes it is not enough. Safety is another important criteria considered in the design of a controller. We were able to formulate a new data-driven procedure to find a stabilizing controller that can also guarantee that the state of the system never violates the safety constraints.For all the solutions presented, we discussed how to handle real noisy measurements