Controllability and accessibility analysis of nonlinear biosystems

Background: We address the problem of determining the controllability and accessibility of nonlinear biosystems. We consider models described by affine-in-inputs ordinary differential equations, which are adequate for a wide array of biological processes. Roughly speaking, the controllability of a dynamical system determines the possibility of steering it from an initial state to any point in its neighbourhood; accessibility is a weaker form of controllability. Methods: While the methodology for analysing the controllability of linear systems is well established, its generalization to the nonlinear case has proven elusive. Thus, a number of related but different properties – including different versions of accessibility, reachability or weak local controllability – have been defined to approach its study, and several partial results exist in lieu of a general test. Here, leveraging the applicable results from differential geometric control theory, we source sufficient conditions to assess nonlinear controllability, as well as a necessary and sufficient condition for accessibility. Results: We develop an algorithmic procedure to evaluate these conditions efficiently, and we provide its open source implementation. Using this software tool, we analyse the accessibility and controllability of a number of models of biomedical interest. While some of them are fully controllable, we find others that are not, as is the case of some models of EGF and NF����B signalling networks. Conclusions: The contributions in this paper facilitate the accessibility and controllability analysis of nonlinear models, not only in biomedicine but also in other areas in which they have been rarely performed to date.Agencia Estatal de Investigación | Ref. PID2020-113992RA-I00Agencia Estatal de Investigación | Ref. RYC-2019-027537-IXunta de Galicia | Ref. ED431F 2021/00

Exact control of genetic networks in a qualitative framework: the bistable switch example

International audienceA qualitative method to control piecewise affine differential systems is proposed and explored for application to genetic regulatory networks. This study considers systems whose outputs and inputs are of a qualitative form, well suited to experimental devices: the measurements indicate whether the variables are "strongly" or "weakly" expressed, that is, only the region of the state space where trajectories evolve at each instant can be known. The control laws are piecewise constant functions in each region and in time, and are only allowed to take three qualitative values corresponding to no control (u=1u=1), high synthesis rates (View the MathML sourceu=umax) or low synthesis rates (View the MathML sourceu=umin). The problems of controlling the bistable switch to each of its steady states is considered. Exact solutions are given to asymptotically control the system to either of its two stable steady states. Two approximate solutions are suggested to the problem of controlling the system to the unstable steady state: either control to a neighborhood of the state, or in the form of a periodic cycle that passes through the state

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 implícita 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 admissíveis das representações explícitas. Neste contexto, dois benefícios da abordagem proposta podem ser enfatizados: primeiro, ela torna possível 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 implícita, 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

Modeling and Analyzing Biomolecular Networks

The authors argue for the need to model and analyze biological networks at molecular and cellular levels. They propose a computational toolbox for biologists. Central to their approach is the paradigm of hybrid models in which discrete events are combined with continuous differential equations to capture switching behavior

Stability of Uncertainty Piecewise Affine Time-Delay Systems with Application to All Wheel Drive Clutch Control.

Piecewise affine (PWA) systems provide good flexibility and traceability for modeling a variety of nonlinear systems. The stability of PWA systems is an important but challenging problem since the stability of the sub-systems does not directly imply the stability of the global system. Meanwhile, time delays and uncertainty exist in many practical systems in engineering and introduce various complex behaviors such as oscillation, instability and poor performance. To ensure the stability of the practical control systems developed via the PWA system framework, the stability of uncertain PWA time-delay systems is investigated. In addition, a quantitative description of asymptotic behavior for time-delay systems is also studied. First, the stability problem for uncertain piecewise affine time-delay systems is investigated. It is assumed that there exists a constant time delay in the system and the uncertainly is norm-bounded. Sufficient conditions for the stability of nominal systems and the stability of systems subject to uncertainty are derived using the Lyapunov-Krasovskii functional with a triple integration term. This approach handles switching based on the delayed states (in addition to the states) for a PWA time-delay system, considers structured as well as unstructured uncertainty, and reduces the conservativeness of previous approaches. Second, an application of the PWA system framework to the modeling and control of an automotive all wheel drive clutch system is presented. The open-loop system is modeled as a PWA system, followed by the design of a piecewise linear feedback controller. The stability of the closed-loop system is examined using the proposed stability method. Finally, a new Lambert W function based approach for estimation of the decay function for time-delay systems is presented. Using this solution form, a decay function estimate, which is less conservative than existing methods, is obtained.Ph.D.Mechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/86297/1/duansm_1.pd

Model Predictive Control Strategies for Advanced Battery Management Systems

Consumer electronics, wearable and personal health devices, power networks, microgrids, and hybrid electric vehicles (HEVs) are some of the many applications where Lithium-ion (Li-ion) batteries are employed. From a manufacturer point of view, the optimal design and management of such electrochemical accumulators are important aspects for ensuring safe and profitable operations. The adoption of mathematical models can support the achievement of the best performance, while saving time and money. In the literature, all the models used to describe the behavior of a Li-ion battery belong to one of the two following families: (i) Equivalent Circuit Models (ECMs), and (ii) Electrochemical Models (EMs). While the former family represents the battery dynamics by means of electrical circuits, the latter resorts to first principles laws of modeling. As a first contribution, this Thesis provides a thorough investigation of the pseudo-two-dimensional (P2D) Li-ion battery EM. In particular, the objectives are to provide: (i) a detailed description of the model formulation, (ii) the Li-ION SIMulation BAttery (LIONSIMBA) toolbox as a finite volume Matlab implementation of the P2D model, for design, simulation, and control of Li-ion cells or battery packs, (iii) a validation of the proposed tool with respect to the COMSOL MultiPhysics commercial software and the Newman's DUALFOIL code, and (iv) some demonstrative simulations involving thermal dynamics, a hybrid charge-discharge cycle emulating the throttle of an HEV, and a battery pack of series connected cells. The second contribution is related to the development of several charging strategies for Advanced Battery Management Systems (ABMSs), where predictive approaches are employed to attain optimal control. Model Predictive Control (MPC) refers to a particular family of control algorithms that, according to a mathematical model, predicts the future behavior of a plant, while considering inputs and outputs constraints. According to this paradigm, in this Thesis different ABMSs strategies have been developed, and their effectiveness shown through simulations. Due to the complexity of the P2D model, its inclusion within an MPC context could prevent the online application of the control algorithm. For this reason, different approximations of the P2D dynamics are proposed and their MPC formulations carefully explained. In particular, finite step response, autoregressive exogenous, piecewise affine, and linear time varying approximations are presented. For all the aforementioned reformulations, the closed-loop performance are evaluated considering the P2D implementation of LIONSIMBA as the real plant. The closed-loop simulations highlight the suitability of the MPC paradigm to be employed for the development of the future ABMSs. In fact, its ability to predict the future behavior of the cell while considering operating constraints can help in preventing possible safety issues and improving the charging performance. Finally, the reliability and efficiency of the proposed Matlab toolbox in simulating the P2D dynamics, support the idea that LIONSIMBA can significantly contribute in the advance of the battery field.Consumer electronics, wearable and personal health devices, power networks, microgrids, and hybrid electric vehicles (HEVs) are some of the many applications where Lithium-ion (Li-ion) batteries are employed. From a manufacturer point of view, the optimal design and management of such electrochemical accumulators are important aspects for ensuring safe and profitable operations. The adoption of mathematical models can support the achievement of the best performance, while saving time and money. In the literature, all the models used to describe the behavior of a Li-ion battery belong to one of the two following families: (i) Equivalent Circuit Models (ECMs), and (ii) Electrochemical Models (EMs). While the former family represents the battery dynamics by means of electrical circuits, the latter resorts to first principles laws of modeling. As a first contribution, this Thesis provides a thorough investigation of the pseudo-two-dimensional (P2D) Li-ion battery EM. In particular, the objectives are to provide: (i) a detailed description of the model formulation, (ii) the Li-ION SIMulation BAttery (LIONSIMBA) toolbox as a finite volume Matlab implementation of the P2D model, for design, simulation, and control of Li-ion cells or battery packs, (iii) a validation of the proposed tool with respect to the COMSOL MultiPhysics commercial software and the Newman's DUALFOIL code, and (iv) some demonstrative simulations involving thermal dynamics, a hybrid charge-discharge cycle emulating the throttle of an HEV, and a battery pack of series connected cells. The second contribution is related to the development of several charging strategies for Advanced Battery Management Systems (ABMSs), where predictive approaches are employed to attain optimal control. Model Predictive Control (MPC) refers to a particular family of control algorithms that, according to a mathematical model, predicts the future behavior of a plant, while considering inputs and outputs constraints. According to this paradigm, in this Thesis different ABMSs strategies have been developed, and their effectiveness shown through simulations. Due to the complexity of the P2D model, its inclusion within an MPC context could prevent the online application of the control algorithm. For this reason, different approximations of the P2D dynamics are proposed and their MPC formulations carefully explained. In particular, finite step response, autoregressive exogenous, piecewise affine, and linear time varying approximations are presented. For all the aforementioned reformulations, the closed-loop performance are evaluated considering the P2D implementation of LIONSIMBA as the real plant. The closed-loop simulations highlight the suitability of the MPC paradigm to be employed for the development of the future ABMSs. In fact, its ability to predict the future behavior of the cell while considering operating constraints can help in preventing possible safety issues and improving the charging performance. Finally, the reliability and efficiency of the proposed Matlab toolbox in simulating the P2D dynamics, support the idea that LIONSIMBA can significantly contribute in the advance of the battery field