Simultaneous stabilization using stable system inversion

This paper studies the generation of stable transfer functions for which the real or imaginary part takes prescribed values at discrete uniformly spaced points on the unit circle. Formulas bounding the error between a particular interpolating function and any function consistent with the data are presented, which have the desirable property that the error goes to zero exponentially fast with the number of interpolating points. The paper also examines generation of stable minimum phase transfer functions for which the magnitude takes prescribed values at uniformly spaced points on the unit circle, and presents error bounds for this problem. Connection with the discrete Hilbert transform is made. The effect of uncertainty in the original data is also examined.

Using stable factorization theory and state-space realization, this paper studies the simultaneous stabilization problem for a class of singular system. Stable right inverse and stable right null space are introduced into a class of linear systems via stable factorization theory at first. Then, a new design approach of all simultaneous stabilizing general controllers for a class of singular system has been proposed in term of stable inverses and stable null spaces and parameterized. A numerical example is given to illustrate the design procedure.

A new method to design asymptotically stabilizing and adaptive control laws for nonlinear systems is presented. The method relies upon the notions of system immersion and manifold invariance and, in principle, does not require the knowledge of a (control) Lyapunov function. The construction of the stabilizing control laws resembles the procedure used in nonlinear regulator theory to derive the (invariant) output zeroing manifold and its friend. The method is well suited in situations where we know a stabilizing controller of a nominal reduced order model, which we would like to robustify with respect to higher order dynamics. This is achieved by designing a control law that asymptotically immerses the full system dynamics into the reduced order one. We also show that in adaptive control problems the method yields stabilizing schemes that counter the effect of the uncertain parameters adopting a robustness perspective. Our construction does not invoke certainty equivalence, nor requires a linear parameterization, furthermore, viewed from a Lyapunov perspective, it provides a procedure to add cross terms between the parameter estimates and the plant states. Finally, it is shown that the proposed approach is directly applicable to systems in feedback and feedforward form, yielding new stabilizing control laws. We illustrate the method with several academic and practical examples, including a mechanical system with flexibility modes, an electromechanical system with parasitic actuator dynamics and an adaptive nonlinearly parameterized visual servoing application.

This paper overviews the research investigations pertaining to stability and stabilization of control systems with time‐delays. The prime focus is the fundamental results and recent progress in theory and applications. The overview sheds light on the contemporary development on the linear matrix inequality (LMI) techniques in deriving both delay‐independent and delay‐dependent stability results for time‐delay systems. Particular emphases will be placed on issues concerned with the conservatism and the computational complexity of the results. Key technical bounding lemmas and slack variable introduction approaches will be presented. The results will be compared and connections of certain delay‐dependent stability results are also discussed.

In this paper, the stabilization control and tracking control for a bilinear system of u-integration type are investigated. By applying the Zhang-gradient (ZG) method, a ZG stabilization controller is designed and proposed to realize the stabilization control of the bilinear system. In order to show the superiority of the proposed ZG stabilization controller for the stabilization control of the bilinear system, the comparison with the controller designed by means of the back-stepping (BS) technique is conducted via a simulation. Besides, we design and propose two ZG tracking controllers for the tracking control of the bilinear system with one output and two outputs. In addition, simulation results verify the efficacy of the proposed ZG controllers in fulfilling the stabilization and tracking control of the bilinear system.

The finite-time stabilization and robust control of stochastic nonlinear systems was discussed in this paper. First, by transforming the considered system to its equivalent Hamiltonian system, we showed that the stochastic dissipative Hamiltonian system was finite-time stable in probability if the Hamiltonian function and internal damping matrix possessed some structures. Then, we utilized the energy shaping and damping injection techniques to reformulate the internal structures and Hamiltonian function of the stochastic Hamiltonian system and constructed a finite-time stabilization controller. Finally, a robust finite-time controller was put forward for uncertain stochastic nonlinear systems by using the Hamiltonian function to construct a solution of Hamiltonian-Jacobi Inequality. Numerical simulations results demonstrated the effectiveness of the proposed stabilization and robust stabilization methods.

Conditions for a Class of Rational Approximants of Fractional Differentiators/Integrators to Enjoy the Interlacing Property

On Stochastic Finite-Time Stabilization with Continuous State-Feedback Controllers

It is shown that a p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</sup> m transfer function G(s) with p⩾m can be decomposed as G = (I-ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> ...(I-ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r+2</sub> Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r+2</sub> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> (I-ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r=1</sub> Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r=1</sub> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> G <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</sub> where Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> are stable all-pass transfer functions, 1=ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ..=ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> ≫ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r=1</sub> ..≫ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> ≫0 are the Hankel-singular-values of GW* <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> where G*G=WW* with W stable and minimum phase. Results on the McMillan degree of G <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Λ</sup> :=(I-ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> )..(I-ν <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> Δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sub> )G then show that G <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Λ</sup> gives a good low order approximation to G in the sense of relative error.

Quantized stabilization of wireless networked control systems with packet losses

Local feedback stabilization and bifurcation control of nonlinear systems are studied for the case in which the linearized system possesses a simple zero eigenvalue. Sufficient conditions are obtained for local stabilizability of the equilibrium point at criticality, and for local stabilizability of bifurcated equilibria. These conditions involve asumptions on the controllability of the critical mode for the linearized system. Explicit stabilizing feedback controls are constructed. The Projection Method of analysis of stationary bifurcations is employed. This work complements an earlier study by the same authors (Syst. Cont. Lett., 7 (1), 11-17, 1986) of stabilization and bifurcation control in the (Hopf bifurcation) case of two pure imaginary eigenvalues of the linearized system at criticality.

This paper addresses the stabilization and optimization problem of networked control systems (NCSs) with long time delays. According to the actual network conditions, the network time delay is divided into the fixed time delay and the random time delay. Then, the stabilization controller is constructed, where the stabilizing control parameters are obtained with cone complementary linearization (CCL) approach and the optimizing control parameters are solved with estimation of distribution algorithm (EDA) algorithm. Simulation results demonstrate the effectiveness of the proposed methods.

This paper considers the problem of simultaneously stabilizing a finite collection of linear time-varying systems via a stable linear time-varying controller. The class of controllers proposed in this paper are linear, time varying and infinite dimensional. However, the computer implementation of these controllers would not be computationally more demanding than the computer implementation of a finite-dimensional linear time-varying controller.

This paper investigates the problem of parameter-dependent state feedback control of continuous-time systems in the context of finite time stability. The controller is designed in order to guarantee that the closed-loop system is finite time stable. The system is considered time varying with the parameters modeled within a unit simplex. The design conditions obtained by means of Lyapunov functions are expressed as linear matrix inequalities. The finite time stability is assessed by using homogeneous polynomially parameter-dependent state feedback gains with arbitrary degree g. LMI relaxations are proposed based on Pólya's theorem. A controller is obtained by the solution of a factibility problem and the effect of the relaxation procedures analyzed by an optimization problem. Numerical examples are provided.

The problem of stabilization of networked control systems (NCSs) with time-varying transmission delays is investigated. The time-varying delays lie in any two adjoining sampling instants of the plant. A technique is proposed for designing stabilizing controllers by converting NCSs with time-varying transmission delays into parameter uncertain systems with constant input delay, then the theory of robust stabilization of delay systems can be applied to solving this problem. Sufficient condition on stabilization of such NCSs is developed in terms of linear matrix inequality.