Exponential Convergence Bounds using Integral Quadratic Constraints

The theory of integral quadratic constraints (IQCs) allows verification of stability and gain-bound properties of systems containing nonlinear or uncertain elements. Gain bounds often imply exponential stability, but it can be challenging to compute useful numerical bounds on the exponential decay rate. In this work, we present a modification of the classical IQC results of Megretski and Rantzer that leads to a tractable computational procedure for finding exponential rate certificates

From classical absolute stability tests towards a comprehensive robustness analysis

In this thesis, we are concerned with the stability and performance analysis of feedback interconnections comprising a linear (time-invariant) system and an uncertain component subject to external disturbances. Building on the framework of integral quadratic constraints (IQCs), we aim at verifying stability of the interconnection using only coarse information about the input-output behavior of the uncertainty

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

Stability Analysis of Piecewise Affine Systems with Multi-model Model Predictive Control

Constrained model predictive control (MPC) is a widely used control strategy, which employs moving horizon-based on-line optimisation to compute the optimum path of the manipulated variables. Nonlinear MPC can utilize detailed models but it is computationally expensive; on the other hand linear MPC may not be adequate. Piecewise affine (PWA) models can describe the underlying nonlinear dynamics more accurately, therefore they can provide a viable trade-off through their use in multi-model linear MPC configurations, which avoid integer programming. However, such schemes may introduce uncertainty affecting the closed loop stability. In this work, we propose an input to output stability analysis for closed loop systems, consisting of PWA models, where an observer and multi-model linear MPC are applied together, under unstructured uncertainty. Integral quadratic constraints (IQCs) are employed to assess the robustness of MPC under uncertainty. We create a model pool, by performing linearisation on selected transient points. All the possible uncertainties and nonlinearities (including the controller) can be introduced in the framework, assuming that they admit the appropriate IQCs, whilst the dissipation inequality can provide necessary conditions incorporating IQCs. We demonstrate the existence of static multipliers, which can reduce the conservatism of the stability analysis significantly. The proposed methodology is demonstrated through two engineering case studies.Comment: 28 pages 9 figure

Exponential input-to-state stability for Lur’e systems via Integral Quadratic Constraints and Zames–Falb multipliers

Absolute stability criteria that are sufficient for global exponential stability are shown, under a Lipschitz assumption, to be sufficient for the a priori stronger exponential input-to-state stability property. Important corollaries of this result are as follows: (i) absolute stability results obtained using Zames–Falb multipliers for systems containing slope-restricted nonlinearities provide exponential input-to-state-stability under a mild detectability assumption; and (ii) more generally, many absolute stability results obtained via Integral Quadratic Constraint methods provide, with the additional Lipschitz assumption, this stronger property

The Variable Gradient Method of Generating Liapunov Functions with Application to Automatic Control Systems

The contribution of this thesis is the introduction and development of the variable gradient method of generating Liapunov functions. A Liapunov function, V, is considered to be generated if the form of V is not known before the generating procedure is applied. Two previous attempts at the generation of Liapunov functions to prove global asymptotic stability for nonlinear autonomous systems have been made. These attempts are summarized and evaluated in some detail, as they form the basis for the variable gradient approach proposed in this thesis. It is assumed that the system whose stability is being investigated is represented by n first order, ordinary, nonlinear differential equations in state variable form The particular state variables used throughout the thesis are the phase variables. This was done for convenience. The problem of finding a scalar V(x) to satisfy a particular Liapunov theorem is recast into the problem of finding a vector function, \nabla V, having suitable properties. As the name implies, \nabla V is assumed to be a vector of n elements, \nabla Vi, each of which has n arbitrary coefficients. These coefficients, designated as α ij may be constants or functions of the state variables, In its most general form, the variable gradient is assumed to be V may be determined as a line integral of \nabla V if the following (n-l)n/2 partial differential equations are satisfied. Here \nabla V^ are the elements of the vector \nabla V. The equations (3) are referred to as generalized curl equations. dv/dt may also be determined from \nabla V. An outline of the procedure by which a suitable V and dY/dt may be determined for a particular problem, starting from the variable gradient of (2) is as follows, 1. Assume a gradient of the form (2), 2. From the variable gradient, determine dV/dt by equation (4). 3. In conjunction with and subject to the requirements of the generalized curl equations (3), constrain dV/dt to be at least negative semi- definite, 4. From the now known \nabla V, determine V, 5. Invoke the necessary theorem to establish stability, Numerous examples are worked to illustrate the procedure outlined above, V functions are generated that involve higher order terms in x, integrals, and terms involving three state variables as factors. The problem of determining Hurwitz like criteria for nonlinear systems is considered in some detail. The last chapter attempts to extend .the variable gradient approach to nonautonosnous systems. The results of this chapter, though somewhat marginal, are of interest from the point of view of further researc

Convolutional Neural Networks as 2-D systems

This paper introduces a novel representation of convolutional Neural Networks (CNNs) in terms of 2-D dynamical systems. To this end, the usual description of convolutional layers with convolution kernels, i.e., the impulse responses of linear filters, is realized in state space as a linear time-invariant 2-D system. The overall convolutional Neural Network composed of convolutional layers and nonlinear activation functions is then viewed as a 2-D version of a Lur'e system, i.e., a linear dynamical system interconnected with static nonlinear components. One benefit of this 2-D Lur'e system perspective on CNNs is that we can use robust control theory much more efficiently for Lipschitz constant estimation than previously possible