Feedback Stabilization Problem Research Articles

Global output-feedback stabilization for a class of switched uncertain nonlinear systems

This paper is mainly concerned with the problem of global output feedback stabilization for a class of switched nonlinear systems with uncertain control coefficients. First of all, it is shown that the unknown control coefficients are lumped together via a common coordinate transformation, thus the original switched nonlinear system is transformed into a new switched system for which control design is feasible. Second, by constructing a common Lyapunov function relying on the designed state observer, the common output feedback controller independent of switching signals is designed based on the backstepping method so as to guarantee the global asymptotic stability of the resulting closed-loop switched system under arbitrary switchings. Two examples are included for verifying the effectiveness of the method proposed.

Global stabilisation for a class of uncertain nonlinear time-delay systems by dynamic state and output feedback

This paper investigates the design problem of constructing the state and output feedback stabilisation controller for a class of uncertain nonlinear systems subject to time-delay. First, a dynamic linear state feedback control law with an adaptive strategy is developed to globally stabilise the uncertain nonlinear time-delay system under a lower-triangular higher-order growth condition. Then, one more challenging problem of the adaptive output feedback stabilisation is addressed, which can globally stabilise the time-delay system when the unmeasurable states linearly grow with rate functions consisting of higher-order output.

Global output feedback stabilisation of high-order nonlinear systems with multiple time-varying delays

This paper investigates the problem of global output feedback stabilisation for a class of high-order nonlinear systems with multiple time-varying delays. By using backstepping recursive technique and the homogeneous domination approach, a continuous output feedback controller is successfully designed, and the global asymptotic stability of the resulting closed-loop system is proven with the help of an appropriate Lyapunov– Krasovskii functional. Two simulation examples are given to illustrate the effectiveness of the proposed approach.

Stabilization of Markov Jump Linear Systems with Input Quantization

This paper addresses the feedback stabilization problem for Markov jump linear systems with quantized control input. A simple but powerful input controller for eliminating the input quantization in Markov jump linear systems is used. The proposed mode-dependent input controller comprises two parts: the linear part and the nonlinear part, where the linear part determines the fundamental characteristics of the systems, and the nonlinear part eliminates the effect of the input quantization. The stability is analyzed in detail for both the discrete-time and continuous-time Markov jump linear systems with input quantization. The sufficient conditions and the specific mode-dependent input controller to guarantee the stability of the systems are obtained. Numerical examples are given to illustrate the effectiveness of the results.

Global State Control for a Class of Inherently Higher-order Parameterized Nonlinear Time-delay Systems Based on Homogeneous Domination Approach

This paper addresses the problem of global state feedback stabilization for a class of inherently higher-order parameterized nonlinear systems subject to time-delay. By using the homogeneous domination approach, we construct a homogeneous state feedback controller with an adaptive strategy. A constructive design method is developed based an effective coupling of the inductive method and a sign function, leading to adaptive regulators with minimal parameterization. With the aid of a homogeneous Lyapunov-Krasovskii functional, the closedloop systems can be asymptotically stabilized globally.

Exponential Stabilization of the Wave Equation by Dirichlet Integral Feedback

We consider the problem of boundary feedback stabilization of a vibrating string that is fixed at one end and with control action at the other end. In contrast to previous studies that have required $L^2$-regularity for the initial position and $H^{-1}$-regularity for the initial velocity, in this paper we allow for initial positions with $L^1$-regularity and initial velocities in $W^{-1,1}$ on the space interval. It is well known that for a certain feedback parameter, for sufficiently regular initial states the classical energy of the closed-loop system with Neumann velocity feedback is controlled to zero after a finite time that is equal to the minimal time where exact controllability holds. In this paper, we present a Dirichlet boundary feedback that yields a well-defined closed-loop system in the ($L^1$, $W^{-1,1}$) framework and also has this property. Moreover, for all positive feedback parameters our feedback law leads to exponential decay of a suitably defined $L^1$-energy. For more regular initial states with $(L^2, \,H^{-1})$ regularity, the proposed feedback law leads to exponential decay of an energy that corresponds to this framework. If the initial states are even more regular with $H^1$-regularity of the initial position and $L^2$-regularity of the initial velocity, our feedback law also leads to exponential decay of the classical energy.

A dual‐observer approach for global output feedback stabilization of planar nonlinear systems with output‐dependent growth rates

SummaryIn this paper, we investigate the problem of global output feedback stabilization for a class of planar nonlinear systems under a more general growth condition, which encompasses both lower‐order and higher‐order state growths with output‐dependent rates. For more accurate estimation, two new observers with nonlinear gains are constructed to estimate the states on the lower‐order and higher‐order scales, respectively. The estimates produced from the dual‐observer are used delicately in the output feedback control law with both lower‐order and higher‐order modes. The overall stability of the system is guaranteed by rigorously choosing these nonlinear gains in the control law and the dual‐observer.

A decoupled adaptive control algorithm for global state feedback stabilization of a class of nonlinear systems

SummaryA decoupled adaptive control algorithm, namely the combined dynamic gain and adaptive homogeneous domination approach, is introduced to solve the global state feedback stabilization problem for a class of uncertain nonlinear systems. Compared with the conventional adaptive backstepping/tuning functions approach, the algorithm differs in the way of constructing the estimator and handling the nonlinear drifts, and allows the adaptive control law that is decoupled via a dynamic gain to be designed only by choosing some appropriate constants. The proposed adaptive controller guarantees that all the states of the closed‐loop system are globally bounded and the system solutions converge to zero asymptotically. Both physical and academic examples are provided to demonstrate the validness of the theory. Copyright © 2014 John Wiley & Sons, Ltd.

State Feedback Stabilization of Stochastic Feedforward Nonlinear Systems with Input Time-delay

In this paper, the problem of state feedback stabilization for stochastic feedforward nonlinear systems with input time-delay is considered for the first time. By introducing a variable transformation, skillfully combining the homogeneous domination method, and constructing an appropriate Lyapunov-Krasovskii functional, a state feedback controller is developed to guarantee the closed-loop system globally asymptotically stable in probability.

Static output feedback stabilization for systems with time-varying delay based on a matrix transformation method

This paper is concerned with the problem of static output feedback stabilization for linear systems with time-varying delay. A novel controller design is proposed based on a matrix transformation method with a new equation condition, which can solve the controller gain more easily and avoid complicated calculations of the non-linear matrix inequality. Then, the corresponding criteria can be obtained by combining the matrix transformation method and the ideas of non-uniformly dividing delay interval, which can guarantee the asymptotic stability of the closed-loop systems and calculate the satisfactory controller gain. Finally, two numerical examples are provided to verify the effectiveness of the proposed design scheme.

Periodically time-varying memory static output feedback control design for discrete-time LTI systems

This paper addresses the problem of static output feedback (SOF) stabilization for discrete-time LTI systems. We approach this problem using the recently developed periodically time-varying memory state-feedback controller (PTVMSFC) design scheme. A bilinear matrix inequality (BMI) condition which uses a pre-designed PTVMSFC is developed to design the periodically time-varying memory SOF controller (PTVMSOFC). The BMI condition can be solved by using BMI solvers. Alternatively, we can apply two-steps and iterative linear matrix inequality algorithms that alternate between the PTVMSFC and PTVMSOFC designs. Finally, an example is given to illustrate the proposed methods.

Stabilization of a non-linear rotating flexible structure under the presence of time-dependent delay term in a dynamic boundary control

This paper is concerned with the feedback stabilization problem for a rotating rigid body-beam system via a torque control exerted on the rigid body and a time-dependent delayed dynamic boundary force control acting at the free end of the beam. Sufficient conditions on the angular velocity of the rigid body and the delay term are derived to assure the existence and uniqueness of solutions to the system. Furthermore, the energy method approach is employed to establish the exponential stability of the closed-loop system under stronger but reasonable assumptions.

Sliding-Mode Predictive Control of Networked Control Systems Under a Multiple-Packet Transmission Policy

This paper addresses the output feedback stabilization problem for a class of networked control systems (NCSs) under a multiple-packet transmission policy. To compensate the time delays and packet losses, a sliding-mode predictive controller is constructed by taking full advantage of network transmission capacity. A Kalman predictor is used to estimate the current and predict the future plant states, while sliding-mode control (SMC) is employed to compute the control sequences. The estimated state errors are transformed into a system disturbance, and the stability of the closed-loop NCSs is guaranteed by constructing the robust stabilizing SMC controller. Simulation and experimental results are given to demonstrate the effectiveness of the proposed approach.

Adaptive Fuzzy Decentralized Output Stabilization for Stochastic Nonlinear Large-Scale Systems With Unknown Control Directions

In this paper, an adaptive decentralized fuzzy output feedback stabilization problem is investigated for a class of uncertain stochastic nonlinear large-scale systems. The addressed stochastic nonlinear systems contain unknown nonlinear functions, unknown control direction, and without the measurements of the states. Fuzzy logic systems are used to identify the unknown nonlinear functions, and a fuzzy state filter observer is designed to estimate the unmeasured states. To solve the problem of the unknown control direction in decentralized control design, Nussbaum-type functions are introduced and new property on Nussbaum-type function is proved. Based on the backstepping recursive design technique and the established Nussbaum function property, a new robust stabilization control approach is developed. It is proved that the proposed control approach can guarantee that all the signals of the resulting closed-loop system are bounded in probability, and the observer errors and system output converge to a small neighborhood of the origin. A simulation example is provided to show the effectiveness of the proposed approach.

A probabilistic approach to stability and stabilization of networked control systems

SummaryIn this paper, a probabilistic approach is proposed to study the modeling, stability, and stabilization problems for networked control systems (NCSs) with simultaneous random network communication delay and data dropout. Different from existing modeling approaches, the proposed NCSs model naturally captures the stochastic characteristics of the considered NCSs by taking into account the effects of the random network communication delay and data dropout. Based on the developed NCSs model, necessary and sufficient conditions of stability analysis, state, and output feedback stabilization problems are tackled. Moreover, necessary and sufficient stability and stabilization conditions are also presented for the case when the occurrence probabilities of delay and data dropout are partially known. Finally, three numerical examples are provided to demonstrate the effectiveness and the reduced conservatism of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.

H∞ stabilisation of networked control systems with time delays and packet losses

This paper addresses the H∞ state feedback stabilisation problem for networked control systems (NCSs) in the presence of time delays and packet losses. By introducing the concept of an effective sensor packet, the NCS is transformed into a new discrete-time switched model, where the parameters have a clear physical meaning and can be easily determined. In this framework, we derive the stability conditions of the closed-loop system in the H∞ sense, and also provide the corresponding H∞ stabilising controller design method. Finally, we give simulation and experimental results to demonstrate the effectiveness of the proposed approaches.

Nonlinear decentralized control of large-scale systems with strong interconnections

This paper addresses the problem of nonlinear decentralized state feedback stabilization for a class of large-scale systems with strong interconnections. The interconnections and their bounds are general functions of all the states. By developing a new recursive design method, a decentralized state feedback controller is successfully constructed for the large-scale system. The novelty of the proposed method is that a Lyapunov function in an appropriate product integral form is introduced at each step so that the recursive design can be carried out. A numerical example is given to illustrate the effectiveness of the proposed method.

Adaptive neural output feedback control for stochastic nonlinear time-delay systems with unknown control directions

This paper first focuses on the problem of adaptive output feedback stabilization for a more general class of stochastic nonlinear time-delay systems with unknown control directions. By using a linear state transformation, the original system is transformed to a new system for which control design becomes feasible. Then a novel adaptive neural network (NN) output feedback control strategy, which only contains one adaptive parameter, is developed for such systems by combining the input-driven filter design, the backstepping technique, the NN's parameterization, the Nussbaum gain function method and the Lyapunov---Krasovskii approach. The proposed control design guarantees that all signals in the closed-loop systems are 4-moment (or 2-moment) semi-globally uniformly bounded. Finally, two simulation examples are given to demonstrate the effectiveness and the applicability of the proposed control design.

Zero Dynamics and Stabilization for Analytic Linear Systems

We combine different system theoretic concepts to solve the feedback stabilization problem for linear time-varying systems with real analytic coefficients. The algebraic concept of the skew polynomial ring with meromorphic coefficients and the geometric concept of (A,B)-invariant time-varying subspaces are invoked. They are exploited for a description of the zero dynamics and to derive the zero dynamics form. The latter is essential for stabilization by state feedback: the subsystem describing the zero dynamics are decoupled from the remaining system which is controllable and observable. The zero dynamics form requires an assumption close to autonomous zero dynamics; this in some sense resembles the Byrnes-Isidori form for systems with strict relative degree. Some aspects of the latter are also proved. Finally, using the zero dynamics form, we show for square systems with autonomous zero dynamics that there exists a linear state feedback such that the Lyapunov exponent of the closed-loop system equals the Lyapunov exponent of the zero dynamics; some boundedness conditions are required, too. If the zero dynamics are exponentially stable this implies that the system can be exponentially stabilized. These results are to some extent also new for time-invariant systems.

On the relation of delay equations to first-order hyperbolic partial differential equations

This paper establishes the equivalence between systems described by a single first-order hyperbolic partial differential equation and systems described by integral delay equations. System-theoretic results are provided for both classes of systems (among them converse Lyapunov results). The proposed framework can allow the study of discontinuous solutions for nonlinear systems described by a single first-order hyperbolic partial differential equation under the effect of measurable inputs acting on the boundary and/or on the differential equation. An illustrative example shows that the conversion of a system described by a single first-order hyperbolic partial differential equation to an integral delay system can simplify considerably the solution of the corresponding robust feedback stabilization problem.