Simultaneous Stabilization Problem Research Articles

Simultaneous stabilization for a collection of uncertain time-delay systems using sliding-mode output feedback control

ABSTRACT In this paper, the simultaneous stabilisation problem is addressed for a collection of time-delay nonlinear systems which are perturbed by time-varying norm-bounded uncertainties. The time-delay is considered to change within an interval. A design procedure for constructing a simultaneously stabilising sliding mode output feedback controller is presented. At first, by applying a delay-range-dependent stability analysis, sufficient conditions concerned with the existence of a common sliding surface are derived which are then transformed into the LMI-based conditions. Finally, a robust simultaneously stabilising control law is proposed to drive all the systems to the sliding surface in finite-time. A numerical example is included to demonstrate the effectiveness of the proposed method.

Adaptive Parallel Simultaneous Stabilization of a Class of Nonlinear Descriptor Systems via Dissipative Matrix Method

This paper investigates the adaptive parallel simultaneous stabilization and robust adaptive parallel simultaneous stabilization problems of a class of nonlinear descriptor systems via dissipative matrix method. Firstly, under an output feedback law, two nonlinear descriptor systems are transformed into two nonlinear differential-algebraic systems by nonsingular transformations, and a sufficient condition of impulse-free is given for two resulting closed-loop systems. Then, the two systems are combined to generate an augmented dissipative Hamiltonian differential-algebraic system by using the system-augmentation technique. Based on the dissipative system, an adaptive parallel simultaneous stabilization controller and a robust adaptive parallel simultaneous stabilization controller are designed for the two systems. Furthermore, the case of more than two nonlinear descriptor systems is investigated. Finally, an illustrative example is studied by using the results proposed in this paper, and simulations show that the adaptive parallel simultaneous stabilization controllers obtained in this paper work very well.

Simultaneous Stability of Large-scale Systems via Distributed Control Network with Partial Information Exchange

This paper is concerned with the simultaneous stability of the multi-mode large-scale systems composed of the interaction subsystems. A novel distributed control network consisting of multiple network-based controllers with the partial information exchange is adopted to simultaneously stabilize the large-scale systems in multiple operation modes. In the distributed control network (DCN), a partial state information exchange approach is developed to save the real-time communication and computation resources. To compensate for the effects of dynamic couplings between interaction subsystems, the designed controllers use both the local states and the neighbors’ partial information with packet dropouts for local feedback design. Then, a series of Lyapunov functions are constructed to derive a matrix-inequality-based sufficient condition for the existence of the desired controllers. Based on an orthogonal complement technique, the gains of the controllers in DCN are parameterized. The iterative algorithm for the solution of simultaneous stabilization problem is also developed. Finally, a numerical example is performed to show the relevant feature of the proposed method.

Adaptive robust simultaneous stabilization controller with tuning parameters design for two dissipative Hamiltonian systems

AbstractThis paper investigates the problem of adaptive robust simultaneous stabilization (ARSS) of two dissipative Hamiltonian systems (DHSs), and proposes a number of results on the controller parameterization design. Firstly, an adaptiveH∞control design approach is presented by using the dissipative Hamiltonian structural for the case that there are both external disturbances and parametric uncertainties in two DHSs. Secondly, an algorithm for solving tuning parameters of the controller is proposed using symbolic computation. The proposed controller parameterization method avoids solving Hamilton-Jacobi-Issacs (HJI) equations and the obtained controller is easier as compared to some existing ones. Finally, an illustrative example is presented to show that the ARSS controller obtained in this paper works very well.

Dynamic control of the space tethered system

We discuss the problem of simultaneous dynamical stabilization and suppression of transverse and longitudinal vibrations of the space tethered system deployed along a certain trajectory. The dynamics of the system is described by a system of nonlinear partial differential equations for the longitudinal and transverse waves and we consider a non-classical version of the problem with one moving boundary. We formulate a mathematical model and perform the analytic and numerical analysis of the boundary control problem based on the Lyapunov method. A scheme of the deployment mechanism is suggested. It includes a control torque and transverse displacement of the boundary and ensures stable deployment of the whole system.

A Family of Robust Simultaneous Controllers With Tuning Parameters Design for a Set of Port‐Controlled Hamiltonian Systems

AbstractThis paper investigates the robust simultaneous stabilization (RSS) and robust adaptive simultaneous stabilization (RASS) problem for a set of port‐controlled Hamiltonian (PCH) systems. Some results for designing a family of robust simultaneous controllers with tuning parameters for such systems are proposed. Firstly, using the dissipative Hamiltonian structural properties, these systems are combined to generate an augmented PCH system, and two simultaneous stabilization controllers with parameters are designed for the systems: one is a robust controller and the other is adaptive. Secondly, an algorithm for solving the tuning parameters’ ranges of controller is proposed with symbolic computation. Finally, a numerical example is studied by applying the method obtained in this paper to a set of PCH systems with external disturbance. The effectiveness of the proposed control method is verified by simulations. Compared with conventional simultaneous stabilization control, the controller obtained in this paper not only has strong robustness for a set of systems, but also can optimize the robustness for the systems by adjusting the parameters’ values.

Robust synchronization of uncertain complex systems with time-varying delays

Synchronization analysis and design problems for uncertain time-delayed high-order complex systems with dynamic output feedback synchronization protocols are investigated. By stating projection on the synchronization subspace and the complement synchronization subspace, synchronization problems are transformed into simultaneous stabilization problems of multiple subsystems related to eigenvalues of the Laplacian matrix of the interaction topology of a complex system. In terms of linear matrix inequalities (LMIs), sufficient conditions for robust synchronization are presented, which include only five LMI constraints. By the changing variable method, sufficient conditions for robust synchronization in terms of LMIs and matrix equalities are given, which can be checked by the cone complementarily linearization approach. The effectiveness of theoretical results is shown by numerical examples.

Some computational aspects of simultaneous stabilization problems

Some computational aspects of simultaneous stabilization problems

Synchronization of fractional-order linear complex networks

In this paper, we concentrate on the synchronization problem of fractional-order complex networks with general linear dynamics under connected topology. By introducing a pseudo-state transformation, the problem is converted into an equivalent simultaneous stabilization problem of independent subsystems, which is characterized by nonzero eigenvalues of the Laplacian matrix. Then, sufficient conditions in terms of linear matrix inequalities (LMIs) for synchronization are established, which can be easily solved by efficient convex optimization algorithms. Finally, three examples are provided to illustrate the effectiveness of the proposed method.

A Simultaneous Stabilization Problem of Linear Systems and their Discretized Models

We present a method to design a feedback controller which stabilizes a continuous-time linear system and its discretized models simultaneously. These discretized models include the Euler approximate discrete-time model and the exact discrete-time model. We present some conditions on solvability of the simultaneous stabilization problem, and a procedure for designing such a controller. We also give three examples to illustrate our research results.

Simultaneous stabilization for a set of Boolean control networks

This paper investigates simultaneous stabilization of a collection of Boolean control networks (BCNs) by using the semi-tensor product method, and presents a number of new results. First, an algebraic expression of the BCNs is obtained by the semi-tensor product, based on which some necessary and sufficient conditions are presented to solve the simultaneous stabilization problem by a free control sequence, a state-feedback control, and an output-feedback control, respectively. Second, using the column stacking form of matrices, a new procedure is established to design both state-feedback and output-feedback controllers for the simultaneous stabilization problem. The study of two illustrative examples shows that the new results obtained in this paper are very effective in solving simultaneous stabilization of a collection of BCNs.

Cyclic condition of simultaneous stabilizability for three linear systems in a behavioral framework

In this paper, we consider the simultaneous stabilization problem for a triple of linear systems in a behavioral framework. Here, we provide a new sufficient condition for a triple of linear systems to be simultaneous stabilizable. The new condition we provide here has a cyclic form.

Output regulation for non‐square linear multi‐rate systems

SUMMARYWe address the problem of regulating a subset of outputs of a linear time‐invariant plant with multi‐rate measurements so as to achieve asymptotic tracking of an exogenous signal generated by the free motion of a linear time‐invariant system, denoted by exosystem. A solution to this problem is required to yield closed‐loop stability and should be such that output regulation is achieved even in the presence of small plant uncertainties and exogenous disturbances also generated by the exosystem. Contrarily to previous works, we propose a solution to the general case where the plant may have more measured outputs than inputs. We show that this solution allows us to solve simultaneous stabilization and output regulation problems that are not possible to solve through the previous works. Besides incorporating an internal model of the exosystem, the key feature of our proposed controller is that it includes a system that blocks signals generated by the exosystem arriving to the controller from the non‐regulated outputs. Copyright © 2012 John Wiley & Sons, Ltd.

Consensus of multi-agent systems: a simultaneous stabilisation approach

This study considers the state consensus problem of multi-agent systems by virtue of novel simultaneous stabilisation techniques, where the distributed controller is proposed using relative outputs of the agents. A new consensus analysis condition is derived by studying the corresponding simultaneous stabilisation problem. Based on the analysis result, a necessary and sufficient condition is then established to ensure the consensus achievement of multi-agent systems, where the controller gain matrix is parameterised by a positive definite matrix, which is independent of the Lyapunov matrices. Iterative approaches are also developed to solve the design conditions, and to optimise the initial values with the solvability of the original problem improved. Further results are obtained for the dynamic output-feedback case. Finally, a numerical example is given to show the effectiveness of the obtained results.

Simultaneous observability and stabilization of some uncoupled wave equations

We consider uncoupled wave equations with different speed of propagation in a bounded domain. Using a combination of the Bardos–Lebeau–Rauch observability result for a single wave equation and a new unique continuation result for uncoupled wave equations, we prove an observability estimate for that system. Applying Lionsʼ Hilbert uniqueness method (HUM), one may derive simultaneous exact controllability results for the uncoupled system; the controls being locally distributed, with their supports satisfying the geometric control condition of Bardos, Lebeau and Rauch. Afterwards, we discuss the related simultaneous stabilization problem; this latter problem is solved by a combination of the new observability inequality, and a result of Haraux establishing an equivalence between observability and stabilization for second order evolution equations with bounded damping operators. Our observability and stabilization results generalize to higher space dimensions some earlier results of Haraux established in the one-dimensional setting.

LMI 기법을 이용한 방향성 통신 토폴로지를 갖는 고차 선형시스템들의 상태일치

This paper deals with the consensus problem for multi-agent linear dynamic systems with directed communication topology. All the agents are identical high-order linear systems and their state information is exchanged through a communication network with directed graph. It is shown that a consensus is achieved if there exists a feasible solution to a set of linear matrix inequalities obtained for a simultaneous stabilization problem for multiple systems. Examples are presented to show the effectiveness of the proposed method.

Network Topology and Communication Data Rate for Consensusability of Discrete-Time Multi-Agent Systems

This paper investigates the joint effect of agent dynamic, network topology and communication data rate on consensusability of linear discrete-time multi-agent systems. Neglecting the finite communication data rate constraint and under undirected graphs, a necessary and sufficient condition for consensusability under a common control protocol is given, which explicitly reveals how the intrinsic entropy rate of the agent dynamic and the communication graph jointly affect consensusability. The result is established by solving a discrete-time simultaneous stabilization problem. A lower bound of the optimal convergence rate to consensus, which is shown to be tight for some special cases, is provided as well. Moreover, a necessary and sufficient condition for formationability of multi-agent systems is obtained. As a special case, the discrete-time second-order consensus is discussed where an optimal control gain is designed to achieve the fastest convergence. The effects of undirected graphs on consensability/formationability and optimal convergence rate are exactly quantified by the ratio of the second smallest to the largest eigenvalues of the graph Laplacian matrix. An extension to directed graphs is also made. The consensus problem under a finite communication data rate is finally investigated.

Simultaneous stabilization for uncertain descriptor systems with input saturation

SUMMARYIn this paper, the simultaneous stabilization problem for r uncertain descriptor systems is investigated. First, r uncertain closed‐loop descriptor systems with an input saturation controller are introduced and a sufficient condition for these systems to be simultaneously stabilized is derived by a common generalized Lyapunov function. Meanwhile, an input saturation controller for these uncertain descriptor systems is designed via using linear matrix inequalities. Second, a simultaneous stability region for the r uncertain stabilized descriptor systems is investigated and described via system parameters. Furthermore, for the r different uncertain descriptor systems, the r different generalized Lyapunov functions are used to discuss the simultaneous stabilization problem with the advantage of relatively small conservativeness compared with the common generalized Lyapunov function. Of course, the advantages of the proposed method still include the numerical reliability of algorithm. Finally, a numerical example is given to illustrate the effectiveness of the proposed design method.Copyright © 2011 John Wiley & Sons, Ltd.

Simultaneous stabilization: Universal controller synthesis

We consider the simultaneous stabilization problem for linear time-invariant stationary objects of arbitrary order. For such objects, we formulate numerically verifiable necessary and sufficient simultaneous stabilization conditions and show a constructive algorithm for the construction of a stabilizing controller with minimal restrictions on the structure and parameters of the stabilized objects. We give a general scheme of studying simultaneous stabilizability of a family of objects and analyze in detail a numerical algorithm for finding a stabilizing controller with applied interval analysis. To solve the problem, we use an approach based on the study of affine transformations of controller parameter spaces into the spaces of coefficients of denominators of closed-loop objects.

Consensus analysis and design for high-order linear swarm systems with time-varying delays

Consensus analysis and design problems for high-order linear time-invariant swarm systems with time-varying delays are dealt with. First, a consensus subspace and a complement consensus subspace are introduced. By the state projection onto the two subspaces, consensus problems are converted into simultaneous stabilization problems of multiple time-delayed subsystems with low dimensions, and a method to analyze and design the consensus function is given. Then, sufficient conditions for consensus and consensualization are presented, which include only four linear matrix inequality constraints. Finally, theoretical results are applied to deal with cooperative control problems of multi-agent supporting systems.