Energy-based control for hybrid port-controlled Hamiltonian systems

In this paper we develop an energy-based hybrid control framework for hybrid port-controlled Hamiltonian systems. In particular, we obtain constructive sufficient conditions for hybrid feedback stabilization that provide a shaped energy function for the closed-loop system, while preserving a hybrid Hamiltonian structure at the closed-loop level. Furthermore, an inverse optimal hybrid feedback control framework is developed that characterizes a class of globally stabilizing energy-based controllers that guarantee hybrid sector and gain margins to multiplicative input uncertainty of hybrid Hamiltonian systems.     

Hybrid dynamical systems: robust control synthesis problems

The paper presents a new approach to robust control synthesis problems for hybrid dynamical systems. The hybrid system under consideration is a composite of a continuous plant and a discrete event controller. State and output feedback problems are considered. The main results are given in terms of the existence of suitable solutions to a dynamic programming equation and a Riccati differential equation of the H^∞ filtering type. These results show a connection between the theories of hybrid dynamical systems and robust and nonlinear control.     

Dynamic buffer management using optimal control of hybrid systems

This paper studies a dynamic buffer management problem with one buffer inserted between two interacting components. The component to be controlled is assumed to have multiple power modes corresponding to different data processing rates. The overall system is modeled as a hybrid system and the buffer management problem is formulated as an optimal control problem. Different from many previous studies, the objective function of the proposed problem depends on the switching cost and the size of the continuous state space, making its solution much more challenging. By exploiting some particular features of the problem, the best mode sequence and the optimal switching instants are characterized analytically using a variational approach. Simulation results based on real data shows that the proposed method can significantly reduce the energy consumption compared with another heuristic scheme in several typical situations.     

Output-to-state stability for hybrid systems

Output-to-state stability (OSS) is a dual notion of input-to-state stability for dynamical systems. This paper presents Lyapunov and asymptotic characterizations of OSS for hybrid dynamical systems, emphasizing that a globally detectable (i.e. nonuniformly OSS) hybrid system admits a smooth OSS-Lyapunov function.     

Fractals in neurodynamics: From hybrid dynamical systems point of view

Recently, hybrid dynamical systems have attracted considerable attention in the automatic control domain. In this article, a theory for recurrent neural networks is presented from a hybrid dynamical systems point of view. The hybrid dynamical system is defined by a continuous dynamical system discretely switched by external temporal inputs. The theory suggests that the dynamics of continuous-time recurrent neural networks, which are stochastically excited by external temporal inputs, are generally characterized by a set of continuous trajectories with a fractal-like structure in hyper-cylindrical phase space. This work was presented, in part, at the 7th International Symposium on Artificial Life and Robotics, Oita, Japan, January 16–18, 2002     

Event-driven optimization-based control of hybrid systems with integral continuous-time dynamics

In this paper we introduce a class of continuous-time hybrid dynamical systems called integral continuous-time hybrid automata (icHA) for which we propose an event-driven optimization-based control strategy. Events include both external actions applied to the system and changes of continuous dynamics (mode switches). The icHA formalism subsumes a number of hybrid dynamical systems with practical interest, e.g., linear hybrid automata. Different cost functions, including minimum-time and minimum-effort criteria, and constraints are examined in the event-driven optimal control formulation. This is translated into a finite-dimensional mixed-integer optimization problem, in which the event instants and the corresponding values of the control input are the optimization variables. As a consequence, the proposed approach has the advantage of automatically adjusting the attention of the controller to the frequency of event occurrence in the hybrid process. A receding horizon control scheme exploiting the event-based optimal control formulation is proposed as a feedback control strategy and proved to ensure either finite-time or asymptotic convergence of the closed-loop.     

Stabilization of max-plus-linear systems using model predictive control: The unconstrained case

Max-plus-linear (MPL) systems are a class of event-driven nonlinear dynamic systems that can be described by models that are “linear” in the max-plus algebra. In this paper we derive a solution to a finite-horizon model predictive control (MPC) problem for MPL systems where the cost is designed to provide a trade-off between minimizing the due date error and a just-in-time production. In general, MPC can deal with complex input and states constraints. However, in this paper we assume that these are not present and it is only required that the input should be a nondecreasing sequence, i.e. we consider the “unconstrained” case. Despite the fact that the controlled system is nonlinear, by employing recent results in max-plus theory we are able to provide sufficient conditions such that the MPC controller is determined analytically and moreover the stability in terms of Lyapunov and in terms of boundedness of the closed-loop system is guaranteed a priori.     

Nearly optimal control laws for nonlinear systems with saturating actuators using a neural network HJB approach

The Hamilton-Jacobi-Bellman (HJB) equation corresponding to constrained control is formulated using a suitable nonquadratic functional. It is shown that the constrained optimal control law has the largest region of asymptotic stability (RAS). The value function of this HJB equation is solved for by solving for a sequence of cost functions satisfying a sequence of Lyapunov equations (LE). A neural network is used to approximate the cost function associated with each LE using the method of least-squares on a well-defined region of attraction of an initial stabilizing controller. As the order of the neural network is increased, the least-squares solution of the HJB equation converges uniformly to the exact solution of the inherently nonlinear HJB equation associated with the saturating control inputs. The result is a nearly optimal constrained state feedback controller that has been tuned a priori off-line.     

Dynamic programming for constrained optimal control of discrete-time linear hybrid systems

In this paper we study the solution to optimal control problems for constrained discrete-time linear hybrid systems based on quadratic or linear performance criteria. The aim of the paper is twofold. First, we give basic theoretical results on the structure of the optimal state-feedback solution and of the value function. Second, we describe how the state-feedback optimal control law can be constructed by combining multiparametric programming and dynamic programming.     

Output feedback control of switched nonlinear systems using multiple Lyapunov functions

This work presents a hybrid nonlinear control methodology for a broad class of switched nonlinear systems with input constraints. The key feature of the proposed methodology is the integrated synthesis, via multiple Lyapunov functions, of “lower-level” bounded nonlinear feedback controllers together with “upper-level” switching laws that orchestrate the transitions between the constituent modes and their respective controllers. Both the state and output feedback control problems are addressed. Under the assumption of availability of full state measurements, a family of bounded nonlinear state feedback controllers are initially designed to enforce asymptotic stability for the individual closed-loop modes and provide an explicit characterization of the corresponding stability region for each mode. A set of switching laws are then designed to track the evolution of the state and orchestrate switching between the stability regions of the constituent modes in a way that guarantees asymptotic stability of the overall switched closed-loop system. When complete state measurements are unavailable, a family of output feedback controllers are synthesized, using a combination of bounded state feedback controllers, high-gain observers and appropriate saturation filters to enforce asymptotic stability for the individual closed-loop modes and provide an explicit characterization of the corresponding output feedback stability regions in terms of the input constraints and the observer gain. A different set of switching rules, based on the evolution of the state estimates generated by the observers, is designed to orchestrate stabilizing transitions between the output feedback stability regions of the constituent modes. The differences between the state and output feedback switching strategies, and their implications for the switching logic, are discussed and a chemical process example is used to demonstrate the proposed approach.     

Kalman filter for controlled hybrid systems

The linear partially observed discrete-continuous (hybrid) stochastic controllable system described by differential equations with measures is considered. The optimal filtering equations in the form of generalized Kalman filter are obtained in the case of non-anticipating control. This result could be a theoretical basis for the optimal control in stochastic hybrid systems with incomplete information.     

Robust-optimal control of electromagnetic levitation system with matched and unmatched uncertainties:experimental validation

The electromagnetic levitation system(EMLS)serves as the most important part of any magnetic levitation system.However,its characteristics are defined by its highly nonlinear dynamics and instability.Furthermore,the uncertainties in the dynamics of an electromagnetic levitation system make the controller design more difficult.Therefore,it is necessary to design a robust control law that will ensure the system's stability in the presence of these uncertainties.In this framework,the dynamics of an electromagnetic levitation system are addressed in terms of matched and unmatched uncertainties.The robust control problem is translated into the optimal control problem,where the uncertainties of the electromagnetic levitation system are directly reflected in the cost function.The optimal control method is used to solve the robust control problem.The solution to the optimal control problem for the electromagnetic levitation system is indeed a solution to the robust control problem of the electromagnetic levitation system under matched and unmatched uncertainties.The simulation and experimental results demonstrate the performance of the designed control scheme.The performance indices such as integral absolute error(IAE),integral square error(ISE),integral time absolute error(ITAE),and integral time square error(ITSE)are compared for both uncertainties to showcase the robustness of the designed control scheme.     

Regularizability of complex switched server queueing networks modelled as hybrid dynamical systems

The paper considers the problem of the qualitative analysis of complex switched server queueing networks. Such networks can be used to model various flexible manufacturing, communications, and computer systems. We introduce the concept of regularizability for such systems and obtain a necessary and sufficient condition for a switched server queueing network to be regularizable.     

Reachability for partially observable discrete time stochastic hybrid systems

When designing optimal controllers for any system, it is often the case that the true state of the system is unknown to the controller. Imperfect state information must be taken into account in the controller’s design in order to preserve its optimality. The same is true when performing reachability calculations. To estimate the probability that the state of a stochastic system reaches, or stays within, some set of interest in a given time horizon, it is necessary to find a controller that drives the system to that set with maximum probability, given the controller’s knowledge of the true state of the system. To date, little work has been done on stochastic reachability calculations with partially observable states. The work that has been done relies on converting the reachability optimization problem to one with an additive cost function, for which theoretical results are well known. Our approach is to preserve the multiplicative cost structure when deriving a sufficient statistic that reduces the problem to one of perfect state information. Our transformation includes a change of measure that simplifies the distribution of the sufficient statistic conditioned on its previous value. We develop a dynamic programming recursion for the solution of the equivalent perfect information problem, proving that the recursion is valid, an optimal solution exists, and results in the same solution as to the original problem. We also show that our results are equivalent to those for the reformulated additive cost problem, and so such a reformulation is not required.     

Gradient geodesic and Newton geodesic HMP algorithms for the optimization of hybrid systems

This paper provides algorithms for the optimization of autonomous hybrid systems based on the geometrical properties of switching manifolds. By employing the notion of geodesic curves on switching manifolds, the Hybrid Maximum Principle (HMP) algorithm introduced in Shaikh and Caines (2007) is extended to the so-called gradient geodesic and Newton geodesic algorithms. The convergence analysis for the algorithms is based upon the Lasalle Invariance Principle and simulation results illustrate their efficacy.     

On absolute stability of Lur''''e control systems with multiple non-linearities using linear matrix inequalities

Necessary and suffcient conditions for the existence of a Lyapunov function in the Lur'e form to guarantee the absolute stability of Lur'e control systems with multiple non-linearities are discussed in this paper. It simplifies the existence problem to one of solving a set of linear matrix inequalities (LMIs), If those LMIs are feasible, free parameters in the Lyapunov function,such as the positive definite matrix and the coefficients of the integral terms, are given by the solution of the LMIs. Otherwise, this Lyapunov function does not exist. Some sufficient conditions are also obtained for the robust absolute stability of uncertain systems.A numerical example is provided to demonstrate the effectiveness of the proposed method.     

Predictive control of hybrid systems: Input-to-state stability results for sub-optimal solutions

This article presents a novel model predictive control (MPC) scheme that achieves input-to-state stabilization of constrained discontinuous nonlinear and hybrid systems. Input-to-state stability (ISS) is guaranteed when an optimal solution of the MPC optimization problem is attained. Special attention is paid to the effect that sub-optimal solutions have on ISS of the closed-loop system. This issue is of interest as firstly, the infimum of MPC optimization problems does not have to be attained and secondly, numerical solvers usually provide only sub-optimal solutions. An explicit relation is established between the deviation of the predictive control law from the optimum and the resulting deterioration of the ISS property of the closed-loop system. By imposing stronger conditions on the sub-optimal solutions, ISS can even be attained in this case.     

Optimal distributed real-time scheduling of flexible manufacturing networks modeled as hybrid dynamical systems

The paper considers a class of flexible manufacturing networks. We employ hybrid dynamical systems to model such networks. The main and new achievement of the paper is that we propose a distributed implementable in real time scheduling rule such that the corresponding closed-loop system is stable and optimal. In stable systems the processes converge to periodic ones. The paper gives computing relations for the determination of the parameters of the periodic processes. These are very much suitable for planning purposes. On this basis—and this, we consider, is also a new, significant result—optimal arrival (demand) rates determination method is proposed. Quality characteristics are outlined. Field of application of hybrid dynamical approach for FMS scheduling is analyzed. The results open perspectives for MRP level task planning. Example and simulation results are presented.     

Geometric synthesis of a hybrid limit cycle for the stabilizing control of a class of nonlinear switched dynamical systems

This paper proposes a new constructive method for synthesizing a hybrid limit cycle for the stabilizing control of a class of switched dynamical systems in R², switching between two discrete modes and without state discontinuity. For each mode, the system is continuous, linear or nonlinear. This method is based on a geometric approach. The first part of this paper demonstrates a necessary and sufficient condition of the existence and stability of a hybrid limit cycle consisting of a sequence of two operating modes in R² which respects the technological constraints (minimum duration between two successive switchings, boundedness of the real valued state variables). It outlines the established method for reaching this hybrid limit cycle from an initial state, and then stabilizing it, taking into account the constraints on the continuous variables. This is then illustrated on a Buck electrical energy converter and a nonlinear switched system in R². The second part of the paper proposes and demonstrates an extension to Rⁿ for a class of systems, which is then illustrated on a nonlinear switched system in R³.