Hybrid decentralized maximum entropy control for large-scale dynamical systems

In the analysis of complex, large-scale dynamical systems it is often essential to decompose the overall dynamical system into a collection of interacting subsystems. Because of implementation constraints, cost, and reliability considerations, a decentralized controller architecture is often required for controlling large-scale interconnected dynamical systems. In this paper, a novel class of fixed-order, energy-based hybrid decentralized controllers is proposed as a means for achieving enhanced energy dissipation in large-scale lossless and dissipative dynamical systems. These dynamic decentralized controllers combine a logical switching architecture with continuous dynamics to guarantee that the system plant energy is strictly decreasing across switchings. The general framework leads to hybrid closed-loop systems described by impulsive differential equations. In addition, we construct hybrid dynamic controllers that guarantee that each subsystem–subcontroller pair of the hybrid closed-loop system is consistent with basic thermodynamic principles. Special cases of energy-based hybrid controllers involving state-dependent switching are described, and an illustrative combustion control example is given to demonstrate the efficacy of the proposed approach.     

Consensus seeking in multi-agent systems via hybrid protocols with impulse delays

This paper studies the consensus problem of multi-agent systems with both fixed and switching topologies. A hybrid consensus protocol is proposed to take into consideration of continuous-time communications among agents and delayed instant information exchanges on a sequence of discrete times. Based on the proposed algorithms, the multi-agent systems with the hybrid consensus protocols are described in the form of impulsive systems or impulsive switching systems. By employing results from matrix theory and algebraic graph theory, some sufficient conditions for the consensus of multi-agent systems with fixed and switching topologies are established, respectively. Our results show that, for small impulse delays, the hybrid consensus protocols can solve the consensus problem if the union of continuous-time and impulsive-time interaction digraphs contains a spanning tree frequently enough. Simulations are provided to demonstrate the effectiveness of the proposed consensus protocols.     

On optimal control problems of a class of impulsive switching systems with terminal states constraints

The global optimal control problem is proposed for a special class of hybrid dynamical systems, i.e. impulsive switching systems. Then the necessary condition of the above problem, the minimum principle, is given. Ekeland’s variational principle and the matrix cost functional structure expression are utilized in the process of the proof. Based on the main result, a special linear hybrid impulsive and switching system (HISS) is illustrated and the optimal control algorithm is presented. Moreover, the cases of pure impulsive systems and pure switched systems are included in this paper.     

Hybrid output regulation for linear impulsive systems with aperiodic jumps: A discrete-time feedback controller design approach

This paper is concerned with the problem of hybrid output regulation for a class of linear impulsive systems with aperiodic jumps. Firstly, by leveraging time-dependent Lyapunov function technique and impulsive control theory, sufficient conditions for achieving output regulation are obtained in state feedback case. Then, the results are extended to error feedback case by constructing an impulsive observer. In this framework, two novel hybrid controllers are designed. Such controllers only need the discrete-time system state or error signal for feedback. The complete procedures for controller designs are also presented. Finally, two illustrative examples, including a numerical example and an LC circuit, are given to show the validity and applicability of the proposed control laws.     

Hybrid control for tracking of invariant manifolds

This paper presents a hybrid control method that controls to unstable equilibria of nonlinear systems by taking advantage of systems’ free dynamics. The approach uses a stable manifold tracking objective in a computationally efficient, optimization-based switching control design. Resulting nonlinear controllers are closed-loop and can be computed in real-time. Our method is validated for the cart–pendulum and the pendubot inversion problems. Results show the proposed approach conserves control effort compared to tracking the desired equilibrium directly. Moreover, the method avoids parameter tuning and reduces sensitivity to initial conditions. The resulting feedback map for the cart–pendulum has a switching structure similar to existing energy based swing-up strategies. We use the Lyapunov function from these prior works to numerically verify local stability for our feedback map. However, unlike the energy based swing-up strategies, our approach does not rely on pre-derived, system-specific switching controllers. We use hybrid optimization to automate switching control synthesis on-line for nonlinear systems.     

Stability and robustness of quasi-linear impulsive hybrid systems

Both hybrid dynamical systems and impulsive dynamical systems are studied extensively in the literature. However, impulsive hybrid systems are not yet well studied. Nonetheless, many physical systems exhibit both system switching and impulsive jump phenomena. This paper investigates stability and robust stability of a class of quasi-linear impulsive hybrid systems by using the methods of Lyapunov functions and Riccati inequalities. Sufficient conditions for stability and robust stability of those systems are established. Some examples are given to illustrate the applicability of our results.     

Stability of complex-valued impulsive and switching system and application to the Lü system

In this paper, the stability of complex-valued impulsive and switching system is addressed. By using switched Lyapunov functions on a complex field, some new stability criteria of complex-valued impulsive and switching systems are established, which not only generalize some known results in the literature, but also greatly reduce the complexity of analysis and computation. As an application, a new hybrid impulsive and switching feedback controller for the complex-valued chaotic Lü system is designed.     

Control vector Lyapunov functions for large-scale impulsive dynamical systems

Vector Lyapunov theory has been developed to weaken the hypothesis of standard Lyapunov theory in order to enlarge the class of Lyapunov functions that can be used for analyzing system stability. In this paper, we provide generalizations to the recent extensions of vector Lyapunov theory for continuous-time systems to address stability and control design of impulsive dynamical systems via vector Lyapunov functions. Specifically, we provide a generalized comparison principle involving hybrid comparison dynamics that are dependent on the comparison system states as well as the nonlinear impulsive dynamical system states. Furthermore, we develop stability results for impulsive dynamical systems that involve vector Lyapunov functions and hybrid comparison inequalities. Based on these results, we show that partial stability for state-dependent impulsive dynamical systems can be addressed via vector Lyapunov functions. Furthermore, we extend the recently developed notion of control vector Lyapunov functions to impulsive dynamical systems. Using control vector Lyapunov functions, we construct a universal hybrid decentralized feedback stabilizer for a decentralized affine in the control nonlinear impulsive dynamical system that possesses guaranteed gain and sector margins in each decentralized input channel. These results are then used to develop hybrid decentralized controllers for large-scale impulsive dynamical systems with robustness guarantees against full modeling and input uncertainty.     

A LMI approach to stability analysis and synthesis of impulsive switched systems with time delays

This paper studies the asymptotic stability problem for a class of impulsive switched systems with time invariant delays based on linear matrix inequality (LMI) approach. Some sufficient conditions, which are independent of time delays and impulsive switching intervals, for ensuring asymptotical stability of these systems are derived by using a Lyapunov–Krasovskii technique. Moreover, some appropriate feedback controllers, which can stabilize the closed-loop systems, are constructed. Illustrative examples are presented to show the effectiveness of the results obtained.     

Finite-time stabilization of nonlinear impulsive dynamical systems

Finite-time stability involves dynamical systems whose trajectories converge to a Lyapunov stable equilibrium state in finite time. For continuous-time dynamical systems finite-time convergence implies nonuniqueness of system solutions in reverse time, and hence, such systems possess non-Lipschitzian dynamics. For impulsive dynamical systems, however, it may be possible to reset the system states to an equilibrium state achieving finite-time convergence without requiring non-Lipschitzian system dynamics. In this paper, we develop sufficient conditions for finite-time stability of impulsive dynamical systems using both scalar and vector Lyapunov functions. Furthermore, we design hybrid finite-time stabilizing controllers for impulsive dynamical systems that are robust against full modelling uncertainty. Finally, we present a numerical example for finite-time stabilization of large-scale impulsive dynamical systems.     

Stability analysis of a reduced model of the lac operon under impulsive and switching control

The study of the stability is an important motif in biological systems. Stability plays a significant role in some of the basic processes of life. In this paper, we provide a theoretical method for analyzing the exponential stability of a reduced model of the lac operon under the impulsive and switching control. Under the hybrid control, the reduced model becomes a nonlinear hybrid impulsive and switching system, and we get the sufficient conditions of its exponential stability. Finally, we give examples to illustrate the effectiveness of our method.     

Stabilization of complex network with hybrid impulsive and switching control

This paper studies the asymptotic stability properties of a class of complex dynamical networks under a hybrid impulsive and switching control. By utilizing the concept of impulsive control and the stability results for impulsive systems, some new criteria for global and local stability are established for this model. Some numerical examples and simulations are included to illustrate the effectiveness of the theoretical results.     

Hybrid Impulsive Control of Stochastic Systems with Multiplicative Noise Under Markovian Switching

A problem of state output feedback stabilization of discrete-time stochastic systems with multiplicative noise under Markovian switching is considered. Under some appropriate assumptions, the stability of this system under pure impulsive control is given. Further under hybrid impulsive control, the output feedback stabilization problem is investigated. The hybrid control action is formulated as a combination of the regular control along with an impulsive control action. The jump Markovian switching is modeled by a discrete-time Markov chain. The control input is simultaneously applied to both the stochastic and the deterministic terms. Sufficient conditions based on stochastic semi-definite programming and linear matrix inequalities (LMIs) for both stochastic stability and stabilization are obtained. Such a nonconvex problem is solved using the existing optimization algorithms and the nonconvex CVX package. The robustness of the stability and stabilization concepts against all admissible uncertainties are also investigated. The parameter uncertainties we consider here are norm bounded. Two examples are given to demonstrate the obtained results.     

Applications of numerical optimal control to nonlinear hybrid systems

This paper develops a technique for numerically solving hybrid optimal control problems. The theoretical foundation of the approach is a recently developed methodology by S.C. Bengea and R.A. DeCarlo [Optimal control of switching systems, Automatica. A Journal of IFAC 41 (1) (2005) 11–27] for solving switched optimal control problems through embedding. The methodology is extended to incorporate hybrid behavior stemming from autonomous (uncontrolled) switches that results in plant equations with piecewise smooth vector fields. We demonstrate that when the system has no memory, the embedding technique can be used to reduce the hybrid optimal control problem for such systems to the traditional one. In particular, we show that the solution methodology does not require mixed integer programming (MIP) methods, but rather can utilize traditional nonlinear programming techniques such as sequential quadratic programming (SQP). By dramatically reducing the computational complexity over existing approaches, the proposed techniques make optimal control highly appealing for hybrid systems. This appeal is concretely demonstrated in an exhaustive application to a unicycle model that contains both autonomous and controlled switches; optimal and model predictive control solutions are given for two types of models using both a minimum energy and minimum time performance index. Controller performance is evaluated in the presence of a step frictional disturbance and parameter uncertainties which demonstrates the robustness of the controllers.     

Unified impulsive fuzzy-model-based controllers for chaotic systems with parameter uncertainties via LMI

In this paper, a novel technique based on impulsive fuzzy T–S model is proposed for controlling chaotic systems with parameter uncertainties. According to this new model, a unified methodology for establishing robust stability, asymptotic stability and exponential stability of impulsive controllers is developed. Various robust stability conditions are presented in the form of linear matrix inequalities (LMI). A simple iterative algorithm is also provided for calculating design parameters based on LMI techniques. Finally, a typical design procedure is developed by using well-known chaotic systems for illustration, accompanied by several numerical simulations to demonstrate the validity of the proposed methodology.     

Impulsive consensus of one-sided Lipschitz nonlinear multi-agent systems with Semi-Markov switching topologies

Many real systems involve not only parameter changes but also sudden variations in environmental conditions, which often causes unpredictable topologies switching. This paper investigates the impulsive consensus problem of the one-sided Lipschitz nonlinear multi-agent systems (MASs) with Semi-Markov switching topologies. Different from the existing modeling methods of the Markov chain, the Semi-Markov chain is adopted to describe this kind of randomly occurring changes reasonably. To cope with the communication and control cost constraints in the multi-agent systems, the distributed impulsive control method is applied to address the leader–follower consensus problem. Beyond that, to obtain a wider nonlinear application range, the one-sided condition is delicately developed to the controller design, and the results are different from the ones obtained in the traditional method with the Lipschitz condition (note that the existing results are usually only applicable to the case with small Lipschitz constant). Based on the characteristics of cumulative distribution functions, the theory of Lyapunov-like function and impulsive differential equation, the asymptotically mean square consensus of multi-agent systems is maintained with the proposed impulsive control protocol. Finally, an explanatory simulation is presented to validate the correctness of the proposed approach conclusively.     

Robust exponential stabilization for large-scale uncertain impulsive systems with coupling time-delays

In this paper, we aim to study robust exponential stabilization for a large-scale uncertain impulsive system with coupling time-delays. Furthermore, we also provide an estimation of the rate of convergence of exponential stabilization. By utilizing the Lyapunov method and Razumikhin technique, we shall design the feedback hybrid controllers in terms of linear matrix inequalities under which the robust exponential stability is achieved for a closed-loop large-scale uncertain impulsive system with coupling time-delays. Moreover, we shall also use the results obtained to design impulsive controllers for a large-scale uncertain continuous system under which the closed-loop continuous system achieves robust and exponential stability. To illustrate our results, one example is solved.     

Stability and stabilization of impulsive switched system with inappropriate impulsive switching signals under asynchronous switching

The main objective of this paper is to study the stability and stabilization problems for a class of impulsive switched systems with inappropriate impulsive switching signals under asynchronous switching. Here, “inappropriate” means that the impulse jump moment may be inconsistent with the asynchronous switching moment or the system switching moment. And “asynchronous” implies that the switching of controller modes lags behind that of system modes. The hybrid case of stable or unstable subsystems combining with stable and unstable impulses is explored. A novel Lyapunov-like function is constructed, which is discontinuous at some special instants, including the switching instants, the instants when the system modes and filter modes are matched, and the impulse jump instants. Based on the novel multiple Lyapunov-like function, the sufficient conditions for the closed loop system to be globally uniformly exponentially stable (GUES) are obtained with admissible edge-dependent switching signals. Furthermore, by excogitating the state-feedback switching controller, the gain matrix of the controller can be obtained by solving the linear matrix inequalities. Finally, two numerical examples and simulation results are given to prove the effectiveness of our main results.