This paper characterizes the asymptotic convergence properties of the primal-dual dynamics to the solutions of a constrained concave optimization problem using classical notions from stability analysis. We motivate our study by providing an example which rules out the possibility of employing the invariance principle for hybrid automata to analyze the asymptotic convergence. We understand the solutions of the primal-dual dynamics in the Caratheodory sense and establish their existence, uniqueness, and continuity with respect to the initial conditions. We employ the invariance principle for Caratheodory solutions of a discontinuous dynamical system to show that the primal-dual optimizers are globally asymptotically stable under the primal-dual dynamics and that each solution of the dynamics converges to an optimizer.
The paper is concerned with two-person games with saddle point. We investigate the limits of value functions for long-time-average payoff, discounted average payoff, and the payoff that follows a probability density. Most of our assumptions restrict the dynamics of games. In particular, we assume the closedness of strategies under concatenation. It is also necessary for the value function to satisfy Bellman's optimality principle, even if in a weakened, asymptotic sense. We provide two results. The first one is a uniform Tauber result for games: if the value functions for long-time-average payoff converge uniformly, then there exists the uniform limit for probability densities from a sufficiently broad set; moreover, these limits coincide. The second one is the uniform Abel result: if a uniform limit for self-similar densities exists, then the uniform limit for long-time average payoff also exists, and they coincide.
This paper investigates the control problem for a class of highly nonlinear-coupled partial differential equations that describe radiative-conductive heat transfer systems. Thanks to the special structure of the obtained state system, using the Galerkin method for the semi-discretization of PDE and to the differential mean value theorem (DMVT), a new LMI condition is provided for the observer-based controller design. The observer and controller gain are computed simultaneously by solving linear matrix inequality (LMI), i.e a convex problem. Also we provide a reduced order observer based controller that assures global asymptotic stability.
This paper presents an inversion-based fault reconstruction approach for a wide class of nonlinear systems subject to an actuator or plant fault. If the nonlinear system has finite relative order with respect to the fault signal, the inverse system as an observer-based filter, reproduces the fault at its output. A simulation for a continuous-stirred tank reactor (CSTR) model include flow rate fault is used to illustrate the effectiveness of the proposed method.
Optimal control theory has gained increasing importance in biomedical applications, e.g., in the automatic administration of anesthetics during general anesthesia. In this context, one of the features that needs to be monitored is the depth of anesthesia. This is usually achieved by the joint administration of hypnotics and analgesics. The depth of anesthesia is quantified by the bispectral index that varies between 97.7% and 0%. This index should usually be kept at a reference level between 40% and 60% during surgeries with general anesthesia. In this contribution, we consider an open-loop control strategy to achieve this goal. In order to determine a suitable controller, we formulate a nonlinear optimal control problem and we solve it using direct methods. These methods have become increasingly useful when computing the numerical solution of an optimal control problem. Moreover, they are known to provide a very robust and general approach.
Model predictive control (MPC) anticipates future events to take appropriate control actions. Nonlinear MPC (NMPC) describes systems with nonlinear models and/or constraints. A Continuation/GMRES Method for NMPC, suggested by T. Ohtsuka in 2004, uses the GMRES iterative algorithm to solve a forward difference approximation $Ax=b$ of the Continuation NMPC (CNMPC) equations on every time step. The coefficient matrix $A$ of the linear system is often ill-conditioned, resulting in poor GMRES convergence, slowing down the on-line computation of the control by CNMPC, and reducing control quality. We adopt CNMPC for challenging minimum-time problems, and improve performance by introducing efficient preconditioning, utilizing parallel computing, and substituting MINRES for GMRES.
Previous chapter Next chapter Full AccessProceedings 2015 Proceedings of the Conference on Control and its Applications (CT)Diffusive Realization of a Lyapunov Equation Solution, and Parallel Algorithms ImplementationHuu-Quan Do, Michel Lenczner, Raphaël Couturier, and Youssef YakoubiHuu-Quan Do, Michel Lenczner, Raphaël Couturier, and Youssef Yakoubipp.68 - 75Chapter DOI:https://doi.org/10.1137/1.9781611974072.10PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract In Yakoubi [11] and Lenczner et al. [5] the authors developed a theoretical framework of diffusive realization for state-realizations of some linear operators. Those are solutions to certain linear operator differential equations posed in one-dimensional bounded domains. They illustrate the theory on a Lyapunov equation arising from the optimal control theory of the heat equation. In principle their method might be very efficient for real-time computations, however it suffers from strong limitations. Here, we present significant improvements and report numerical results. A method of contour optimization is provided. It is based on a theoretical error estimate of the solution. Finally, we discuss expected gains if the method is implemented on different parallel computer topologies. The envisioned applications are for real-time distributed control on distributed computing architectures. Previous chapter Next chapter RelatedDetails Published:2015eISBN:978-1-61197-407-2 https://doi.org/10.1137/1.9781611974072Book Series Name:ProceedingsBook Code:PRCT15Book Pages:1-490
We consider controller design for robust output tracking and disturbance rejection for continuous-time periodic linear systems with periodic reference and disturbance signals. As our main results we present four different controllers: A feedforward control law and a discrete-time dynamic error feedback controller for output tracking and disturbance rejection, a robust discretetime feedback controller, and finally a discrete-time feedback controller that achieves approximate robust output tracking and disturbance rejection. The presented constructions are also new for time-invariant finite and infinite-dimensional systems. The results are illustrated with two examples: A periodically timedependent system of harmonic oscillators and a nonautonomous two-dimensional heat equation with boundary disturbance.
The control design of an artificial pancreas, a hot research topic in diabetology, is tackled via the newly introduced model-free control and its corresponding "intelligent" proportional controller, which were already quite successful in many concrete and diverse situations. It results in an insulin injection for type 1 diabetes which displays via constant references a good nocturnal/fasting response, but unfortunately a poor postprandial behavior due to long hyperglycemia. When a variable reference is introduced, that switches between a constant one, when glycemia is more or less normal or moderate, and an exponential decay reference path, when a high glycemia rate indicates a meal intake, the results in silico, which employ real clinical data, become excellent. We obtain a bolus-shaped insulin injection rate during postprandial phases. The hyperglycemic peaks are therefore lowered a lot.
Performance of optimization algorithms based on metaheuristics and/or based on derivative-free methods is highly dependent on its parameters. hen, in order to reach a quality solution as fast as possible, the algorithm has to be tuned adequately. A detailed statistical analysis of the system response quality found by Ant Colony Optimization (ACO) based algorithm with respect to discretization of the search space and the number of ants is presented for tuning 4 nonlinear controller structures. he resulting sensitivity curves permit to determine appropriate ACO parameter values to initiate the Nelder-Mead (NM) algorithm. A statistical study of NM convergence is also presented. Using the results of ACO and NM convergence studies has permited to reduce the average ACO-NM computation time by up to 7 times for an equivalent system response quality as compare to the previous
We consider a networking infrastructure, upon which various "large" users (e.g., Telecom Operators, data centers, etc.) have multiple paths to deliver an aggregated entry flow to a certain destination. The flow of each user can be split among the different paths that traverse energy-aware routers. The routers adopt a specific strategy to minimize the power–delay product for each link, which gives rise to quadratic link (in the aggregated link flows) cost functions. We seek person–by–person satisfactory (p.b.p.s.) strategies stemming from a team optimal control problem of the users. The team optimization problem is defined among Decision Makers (DMs – one for each user) that try to minimize a common aggregate cost function of their routes, each one acting solely on the basis of the knowledge of the amount of flow to be routed. We derive piecewise linear p.b.p.s. solutions, which are characterized by a set of parameters. The latter can be found by solving a set of nonlinear fixed point equations.
In this paper, the problem of finite-time boundary stabilization of two strings connected by point mass is investigated. Based on the so-called Riemann invariant transformation, the vibrating strings are transformed in two hybridhyperbolic systems, and leads to the posedness of our system. In order to act in the system, it is desirable to choose boundary feedbacks, in this case, Hölderien stabilizing feedback laws to vanish in finite-time the right and the left of the solutions are considered.MSC codesFinite-time stabilizationfeedback lawspoint massRiemann transformationstrings
A game is considered where the communication network of the first player is explicitly modeled. The second player may induce delays in this network, while the first player may counteract such actions. Costs are modeled through expectations over idempotent probability measures. Idempotent algebras are used to obtain an algorithm for solution of the game.
The PageRank algorithm is used by Google as a way of hierarchically indexing web pages in order to provide relevant and reputable search results. Fundamentally, this algorithm relies on the hypertextual nature of the World Wide Web; indeed, the PageRank vector can be computed based simply on the hyperlink structure of every page in the web. In this paper, we consider a model for PageRank whose dynamics are described by a stochastic system and we establish strong consistency of the least squares estimator of an unknown parameter in this system. Furthermore, motivated by recent work on distributed randomized methods for computing PageRank, we show that the least squares estimator remains strongly consistent within a distributed framework.
A new max-plus fundamental solution semigroup is presented for a class of lossless wave equations. This new semigroup is developed by employing the action principle to encapsulate the propagation of all possible solutions of a given wave equation in the evolution of the value function of an associated optimal control problem. The max-plus fundamental solution semigroup for this optimal control problem is then constructed via dynamic programming, and used to formulate the fundamental solution semigroup for the original wave equation. An application of this semigroup to solving twopoint boundary value problems is discussed via an example.
In this paper, we consider repeated routing games with piecewise-constant congestion taxing in which a central planner sets and announces the congestion taxes for fixed windows of time in advance. Specifically, congestion taxes are calculated using marginal congestion pricing based on the flow of the vehicles on each road prior to the beginning of the taxing window (and, hence, there is a time-varying delay in setting the congestion taxes). We motivate the piecewise-constant taxing policy by that users or drivers may dislike fast-changing prices and that they also prefer prior knowledge of the prices. We prove for this model that the multiplicative update rule and the discretized replicator dynamics converge to a socially optimal flow when using vanishing step sizes. Considering that the algorithm cannot adapt itself to a changing environment when using vanishing step sizes, we propose adopting constant step sizes in this case. Then, however, we can only prove the convergence of the dynamics to a neighborhood of the socially optimal flow (with the size of the neighbourhood being of the order of the selected step size). The results are illustrated on a nonlinear version of Pigou's famous routing game.
Previous chapter Next chapter Full AccessProceedings 2015 Proceedings of the Conference on Control and its Applications (CT)Aircraft Preliminary Design Using Nonlinear Inverse DynamicsMarco Torres-Reyna, Daniel Martinez-Vazquez, and Eduardo Liceaga-CastroMarco Torres-Reyna, Daniel Martinez-Vazquez, and Eduardo Liceaga-Castropp.297 - 302Chapter DOI:https://doi.org/10.1137/1.9781611974072.41PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract What shape should an aircraft have to give certain desirable properties? Nonlinear Inverse Dynamics (NID) may be one of the necessary tools needed to find an answer to this question. In flight dynamics NID is usually applied to define flight trajectories calculations and flight control systems design. The underlying concept behind inverse dynamics applications is the definition of a desired manoeuvre, usually defined by a dynamical model or a pre-established trajectory. By forcing a aircraft whose dynamics are described by a set of nonlinear differential equations to behave like a prescribed model -non necessarily linear- flight control systems are designed. This technique is referred to as nonlinear model matching. As follows NID is used to assist the preliminary design of aircraft. From a set of flight characteristics defined by customer specifications the parameters which define the shape and size, such as: wing span, weight, wing aerofoil selection, engine characteristics and wing polar are estimated. Previous chapter Next chapter RelatedDetails Published:2015eISBN:978-1-61197-407-2 https://doi.org/10.1137/1.9781611974072Book Series Name:ProceedingsBook Code:PRCT15Book Pages:1-490
Large electrical power networks viewed as continuum systems have been studied under constant voltage magnitude assumptions. The continuum system phase behavior was proved to follow the dynamics of a second order nonlinear wave equation. The latter represents electromechanical wave propagation in large electric power networks. In this paper, we generalize this work to time and space variant voltage magnitudes which is the case in real world applications. The resulting partial differential equations (PDEs) are also wave equations but include more nonlinearity terms. Optimal control theory is used to derive optimality conditions for two optimal control problems. The first problem is when the mechanical power is the control input where the constraint is a constant voltage PDE, while the second problem is when the variant voltage magnitude is the control input under a generalized variant voltage PDE as the optimization constraint. Numerical results are presented to illustrate the performance of the resulting closed loop control systems for large power networks. Due to page size limits we present the optimal control results for the variant voltage swing PDE in a different paper.
