243 research outputs found
Digital implementation of hierarchical piecewise-affine controllers
This paper proposes the design of hierarchical piecewise-affine (PWA) controllers to alleviate the processing time or prohibitive memory requirements of large controller structures. The constituent PWA modules of the hierarchical solution have fewer inputs and/or coarser partitions, so that they can reduce considerably the hardware resources required and/or the time response of the controller. A design methodology aided by CAD tools is employed to design the parameters of the controller, implement its architecture in an FPGA, and verify the static and dynamic behavior of the digital implementation by applying hardware-in-the-loop testing.Comunidad Europea FP7-IST-248858Ministerio de Ciencia e Innovación TEC2008-04920 y DPI2008-03847Junta de AndalucÃa P08-TIC-0367
An integral sliding-mode parallel control approach for general nonlinear systems via piecewise affine linear models
The fundamental problem of stabilizing a general nonaffine continuous-time
nonlinear system is investigated via piecewise affine linear models (PALMs) in
this article. A novel integral sliding-mode parallel control (ISMPC) approach
is developed, where an uncertain piecewise affine system (PWA) is constructed
to model a nonaffine continuous-time nonlinear system equivalently on a compact
region containing the origin. A piecewise sliding-mode parallel controller is
designed to globally stabilize the PALM and, consequently, to semiglobally
stabilize the original nonlinear system. The proposed scheme enjoys three
favorable features: (i) some restrictions on the system input channel are
eliminated, thus the developed method is more relaxed compared with the
published approaches; (ii) it is convenient to be used to deal with both
matched and unmatched uncertainties of the system; and (iii) the proposed
piecewise parallel controller generates smooth control signals even around the
boundaries between different subspaces, which makes the developed control
strategy more implementable and reliable. Moreover, we provide discussions
about the universality analysis of the developed control strategy for two kinds
of typical nonlinear systems. Simulation results from two numerical examples
further demonstrate the performance of the developed control approach
Robust nonlinear control of vectored thrust aircraft
An interdisciplinary program in robust control for nonlinear systems with applications to a variety of engineering problems is outlined. Major emphasis will be placed on flight control, with both experimental and analytical studies. This program builds on recent new results in control theory for stability, stabilization, robust stability, robust performance, synthesis, and model reduction in a unified framework using Linear Fractional Transformations (LFT's), Linear Matrix Inequalities (LMI's), and the structured singular value micron. Most of these new advances have been accomplished by the Caltech controls group independently or in collaboration with researchers in other institutions. These recent results offer a new and remarkably unified framework for all aspects of robust control, but what is particularly important for this program is that they also have important implications for system identification and control of nonlinear systems. This combines well with Caltech's expertise in nonlinear control theory, both in geometric methods and methods for systems with constraints and saturations
Approximation of explicit model predictive control using regular piecewise affine functions: an input-to-state stability approach
Abstract Piecewise affine (PWA) feedback control laws defined on general polytopic partitions, as for instance obtained by explicit MPC, will often be prohibitively complex for fast systems. In this work we study the problem of approximating these high-complexity controllers by low-complexity PWA control laws defined on more regular partitions, facilitating faster on-line evaluation. The approach is based on the concept of input-to-state stability (ISS). In particular, the existence of an ISS Lyapunov function (LF) is exploited to obtain a priori conditions that guarantee asymptotic stability and constraint satisfaction of the approximate low-complexity controller. These conditions can be expressed as local semidefinite programs (SDPs) or linear programs (LPs), in case of 2-norm or 1, ∞-norm based ISS, respectively, and apply to PWA plants. In addition, as ISS is a prerequisite for our approximation method, we provide two tractable computational methods for deriving the necessary ISS inequalities from nominal stability. A numerical example is provided that illustrates the main results. The authors are with the Hybrid an
Stability Verification of Neural Network Controllers using Mixed-Integer Programming
We propose a framework for the stability verification of Mixed-Integer Linear
Programming (MILP) representable control policies. This framework compares a
fixed candidate policy, which admits an efficient parameterization and can be
evaluated at a low computational cost, against a fixed baseline policy, which
is known to be stable but expensive to evaluate. We provide sufficient
conditions for the closed-loop stability of the candidate policy in terms of
the worst-case approximation error with respect to the baseline policy, and we
show that these conditions can be checked by solving a Mixed-Integer Quadratic
Program (MIQP). Additionally, we demonstrate that an outer and inner
approximation of the stability region of the candidate policy can be computed
by solving an MILP. The proposed framework is sufficiently general to
accommodate a broad range of candidate policies including ReLU Neural Networks
(NNs), optimal solution maps of parametric quadratic programs, and Model
Predictive Control (MPC) policies. We also present an open-source toolbox in
Python based on the proposed framework, which allows for the easy verification
of custom NN architectures and MPC formulations. We showcase the flexibility
and reliability of our framework in the context of a DC-DC power converter case
study and investigate its computational complexity
Event-triggered control for piecewise affine discrete-time systems
In the present work, we study the problems of stability analysis of piecewise-affine (PWA) discrete-time systems, and trigger-function design for discrete-time event-triggered control systems. We propose a representation for piecewise-affine systems in terms of ramp functions, and we rely on Lyapunov theory for the stability analysis. The proposed implicit piecewise-affine representation prevents the shortcomings of the existing stability analysis approaches of PWA systems. Namely, the need to enumerate regions and allowed transitions of the explicit representations. In this context, we can emphasize two benefits of the proposed approach: first, it makes possible the analysis of uncertainty in the partition and, thus, the transitions. Secondly, it enables the analysis of event-triggered control systems for the class of PWA systems since, for ETC, the transitions cannot be determined as a function of the state variables. The proposed representation, on the other hand, implicitly encodes the partition and the transitions. The stability analysis is performed with Lyapunov theory techniques. We then present conditions for exponential stability. Thanks to the implicit representation, the use of piecewise quadratic Lyapunov functions candidates becomes simple. These conditions can be solved numerically using a linear matrix inequality formulation. The numerical analysis exploits quadratic expressions that describe ramp functions to verify the positiveness of extended quadratic forms. For ETC, a piecewise quadratic trigger function defines the event generator. We find suitable parameters for the trigger function with an optimization procedure. As a result, this function uses the information on the partition to reduce the number of events, achieving better results than the standard quadratic trigger functions found in the literature. We provide numerical examples to illustrate the application of the proposed representation and methods.Ce manuscrit présente des résultats sur l’analyse de stabilité des systèmes affines par morceaux en temps discret et sur le projet de fonctions de déclenchement pour des stratégies de commande par événements. Nous proposons une représentation pour des systèmes affines par morceaux et l’on utilise la théorie de stabilité de Lyapunov pour effectuer l’analyse de stabilité globale de l’origine. La nouvelle représentation implicite que nous proposons rend plus simple l’analyse de stabilité car elle évite l’énumération des régions et des transitions entre régions tel que c’est fait dans le cas des représentations explicites. Dans ce contexte nous pouvons souligner deux avantages principaux, à savoir I) la possibilité de traiter des incertitudes dans la partition qui définit le système et, par conséquent des incertitudes dans les transitions, II) l’analyse des stratégies de commande par événements pour des systèmes affines par morceaux. En effet, dans ces stratégies les transitions ne peuvent pas être définies comme des fonctions des variables d’état. La théorie de stabilité de Lyapunov est utilisée pour établir des conditions pour la stabilité exponentielle de l’origine. Grâce à la représentation implicite des partitions nous utilisons des fonctions de Lyapunov quadratique par morceaux. Ces conditions sont données par des inégalités dont la solution numérique est possible avec une formulation par des inégalités matricielles linéaires. Ces formulations numériques se basent sur des expressions quadratiques décrivant des fonctions rampe. Pour des stratégies par événement, une fonctions quadratique par morceaux est utilisée pour le générateur d’événements. Nous calculons les paramètres de ces fonctions de déclenchement a partir de solutions de problèmes d’optimisation. Cette fonction de déclenchement quadratique par morceaux permet de réduire le nombre de d’événementsen comparaison avec les fonctions quadratiques utilisées dans la littérature. Nous utilisons des exemples numériques pour illustrer les méthodes proposées.No presente trabalho, são estudados os problemas de análise de estabilidade de sistemas afins por partes e o projeto da função de disparo para sistemas de controle baseado em eventos em tempo discreto. É proposta uma representação para sistemas afins por partes em termos de funções rampa, e é utilizada a teoria de Lyapunov para a análise de estabilidade. A representação afim por partes implÃcita proposta evita algumas das deficiências das abordagens de análise de estabilidade de sistemas afins por partes existentes. Em particular, a necessidade de anumerar regiões e transições admissÃveis das representações explÃcitas. Neste contexto, dois benefÃcios da abordagem proposta podem ser enfatizados: primeiro, ela torna possÃvel a análise de incertezas na partição, e, assim, nas transições. Segundo, ela permite a análise de sistemas de controle baseado em eventos para a classe de sistemas afins por partes, já que, para o controle baseado em eventos, as transições não podem ser determinadas como uma função das variáveis de estado. A representação proposta, por outro lado, codifica implicitamente a partição e as transições. A análise de estabilidade é realizada com técnicas da teoria de Lyapunov. Condi- ções para a estabilidade exponencial são então apresentadas. Graças à representação implÃcita, o uso de funções candidatas de Lyapunov se torna simples. Essas condições podem ser resolvidas numéricamente usando uma formulação de desigualdades matriciais lineares. A análise numérica explora expressões quadráticas que descrevem funções de rampa para verificar a postivividade de formas quadráticas extendidas. Para o controle baseado em eventos, uma função de disparo quadrática por partes define o gerador de eventos. Parâmetros adequados para a função de disparo sãoencontrados com um procedimento de otimização. Como resultado, esta função usa informação da partição para reduzir o número de eventos, obtendo resultados melhores do que as funções de disparo quadráticas encontradas na literatura. Exemplos numéricos são fornecidos para ilustrar a aplicação da representação e mé- todos propostos
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