Simultaneous stabilization of a class of nonlinear descriptor systems via Hamiltonian function method

This paper studies simultaneous stabilization of a class of nonlinear descriptor systems via the Hamiltonian function method. Firstly, based on the Hamiltonian realization of the nonlinear descriptor systems and a suitable output feedback, two nonlinear descriptor systems are equivalently transformed into two nonlinear Hamiltonian differential-algebraic systems by a nonsingular transformation, and a sufficient condition for two closed-loop systems to be impulse-free is given. The two systems are then combined to generate an augmented dissipative Hamiltonian differential-algebraic system by using the system-augmentation technique, based on which a simultaneous stabilization controller and a robust simultaneous stabilization controller are designed for the two systems. Secondly, the case of more than two nonlinear descriptor systems is investigated, and two new results are proposed for the simultaneous stabilization and robust simultaneous stabilization, respectively. Finally, an illustrative example is studied by using the results proposed in this paper, and simulations show that the simultaneous stabilization controllers obtained in this paper work very well.

PARALLEL SIMULTANEOUS STABILIZATION OF A SET OF PORT-CONTROLLED HAMILTONIAN SYSTEMS SUBJECT TO ACTUATOR SATURATION

This paper investigates parallel simultaneous stabilization （PSS） of a set of multi-input nonlinear Port-Controlled Hamiltonian （PCH） systems subject to actuator saturation （AS）, and proposes a number of results on the design of PSS controllers for the PCH systems with AS. Firstly, the case of two PCH systems with AS is studied. Exploring the special property of the saturation nonlinearity and the structural properties of dissipative Hamiltonian system, the two systems are combined to generate an augmented PCH system, with which some results on the control design are then obtained. When there are external disturbances in the two systems, a robust PSS controller is designed for the systems. Secondly, the case of more than two PCH systems with AS is investigated, and several new results are proposed for the PSS problem. Finally, two illustrative examples are presented to show that the stabilization controllers obtained in this paper work very well.

Solution to the Generalized Champagne Problem on simultaneous stabilization of linear systems

The well-known Generalized Champagne Problem on simultaneous stabilization of linear systems is solved by using complex analysis and Blonders technique. We give a complete answer to the open problem proposed by Patel et al., which automatically includes the solution to the original Champagne Problem. Based on the recent development in automated inequality-type theorem proving, a new stabilizing controller design method is established. Our numerical examples significantly improve the relevant results in the literature.

Some Open Problems on Simultaneous Stabilization of Linear Systems

Simultaneous stabilization of linear systems is a fundamental issue in the system and control theory, and is of theoretical and practical significance. In this paper, the authors review the recent research progress and the state-of-art results on simultaneous stabilization of single-input single-output linear time-invariant systems. Especially, the authors list the ever best results on the parameters involved in the well known ＂French Champagne Problem＂ and ＂Belgian Chocolate Problem＂ from the point of view of mathematical theoretical analysis and numerical calculation. And the authors observed that Boston claimed the lower bound of 5 can be enlarged to 0.976461 in 2012 is not accurate. The authors hope it will inspire further study on simultaneous stabilization of several linear systems.

Simultaneous Stabilization of Port-Hamiltonian Systems Subject to Actuation Saturation and Input Delay

This paper investigates the simultaneous stabilization of Port-Hamiltonian(PH) systems subject to actuation saturation(AS) and input delay. Firstly, two parallel connecting PH systems subject to the AS and input delay are proposed. Secondly, a simultaneous stabilization control law is designed by a difference between the two feedback control laws containing the input delay.Thirdly, computing a Lyapunov-Krasovskii function assures the simultaneous stabilization of the above systems. Finally, simulation is given to show the correctness of the proposed contents.

Adaptive robust simultaneous stabilization of multiple nn-degree-of-freedom robot systems

In this paper,the adaptive robust simultaneous stabilization problem of uncertain multiple n-degree-of-freedom(n-DOF)robot systems is studied using the Hamiltonian function method,and the corresponding adaptive L2 controller is designed.First,we investigate the adaptive simultaneous stabilization problem of uncertain multiple n-DOF robot systems without external disturbance.Namely,the single uncertain n-DOF robot system is transformed into an equivalent Hamiltonian form using the unified partial derivative operator(UP-DO)and potential energy shaping method,and then a high dimensional Hamiltonian system for multiple uncertain robot systems is obtained by applying augmented dimension technology,and a single output feedback controller is designed to ensure the simultaneous stabilization for the higher dimensional Hamiltonian system.On this basis,we further study the adaptive robust simultaneous stabilization control problem for the uncertain multiple n-DOF robot systems with external disturbances,and design an adaptive robust simultaneous stabilization controller.Finally,the simulation results show that the adaptive robust simultaneous stabilization controller designed in this paper is very effective in stabilizing multi-robot systems at the same time.

Simultaneous quadratic performance stabilization for linear time-delay systems

A newly designed approach of simultaneous stabilization is given for linear discrete time-delay systems. The problem of stabilization for a collection of systems is discussed initially. Adequate condition are obtained in terms of linear matrix inequalities （LMIs） which are independent of time delays such that the resultant collection of discrete time-delay systems are stable with an upper bound of the quadratic performance index. Subsequently, controllers are designed such that the resultant closed-loop discrete time-delay systems are simultaneously stabilized with the upper bound of the quadratic performance index. Finally,a numerical example is given to illustrate the design method.

Simultaneous H_2/H_∞ Stabilization for Chemical Reaction Systems Based on Orthogonal Complement Space

This paper presents a simultaneous H2/H∞ stabilization problem for the chemical reaction systems which can be modeled as a finite collection of subsystems. A single dynamic output feedback controller which simultaneously stabilizes the multiple subsystems and captures the mixed H2/H∞ control performance is designed. To ensure that the stability condition, the H2 characterization and the H∞ characterization can be enforced within a unified matrix inequality framework, a novel technique based on orthogonal complement space is developed. Within such a framework, the controller gain is parameterized by the introduction of a common free positive definite matrix, which is independent of the multiple Lyapunov matrices. An iterative linear matrix inequality （ILMI） algorithm using Matlab Yalmip toolbox is established to deal with the proposed framework. Simulation results of a typical chemical reaction system are exploited to show the validity of the proposed methodology.

Simultaneous pole-placement/stabilization of pairs of linear time-invariant plants using a continuous-time periodic controller

Given a pair of single input single output （SISO）, linear time-invariant （LTI）, and strictly proper plants of relative order r, this paper employs a continuous-time periodic controller to achieve 1） simultaneous pole-placement for r = 1 and 2） guaranteed simultaneous stabilization for r = 2, 3, and 4, which jobs LTI controllers cannot, in general, do. The controller also ensures insignificant output ripples. The analysis is based on averaging principle. The computational steps for controller synthesis are linear algebraic in nature. An example illustrates the design procedure.