Feedback Stabilization Of Linear Systems Research Articles

Memoryless Dual-Observer-Based Output Feedback Stabilization of Linear Systems With Input and Output Delays.

This article focuses on the dual-observer-based stabilization of linear systems with delays in both the inputs and outputs. By a model reduction approach the system can be converted to an equivalent linear system without delays, for which a dual-observer-based stabilizing controller can be designed. However, such a controller is infinite dimensional or memory-based. To solve such a problem, a modified memoryless dual-observer-based stabilizing controller is designed, and the closed-loop stability is proven under some additional conditions. Compared with the reduced-order observer-based controller, the dimension of the dual-observer-based controller is smaller if the system has more inputs than outputs. At the same time, the design approach is more challenging in proving closed-loop stability, as for example a more intricate Lyapunov-Krasovskii functional has to be constructed associated with the proposed approach. The proposed approach is applicable to both continuous-time and discrete-time systems. Numerical simulations validate the effectiveness of the proposed method.

PDE-based observation and predictor-based control for linear systems with distributed infinite input and output delays

This paper considers output feedback stabilization of linear systems with both distributed infinite input and output delays. We first propose a PDE-based observer to deal with distributed infinite output delays. The main idea is to transform infinite-delayed output into a transport PDE with a semi-infinite domain. It is proved that the resulted error systems are exponentially stable and thus the observer is effective. Furthermore, relying on the PDE-based observer, a predictor-based output feedback stabilization controller is provided. This work removes a restrictive condition on eigenvalues of open-loop system in the previous work, which is a distinctive advantage of this work. A simulation example is given to illustrate the effectiveness of our proposed observer and controller.

Functional and dual observer based prescribed-time control of linear systems by periodic delayed feedback

In this paper the prescribed-time output feedback stabilization of linear systems is addressed using both a functional observer, which includes the reduced-order observer as a special case, and a dual observer. The approach is built upon smooth periodic delayed feedback which is a special case of linear periodic time-varying feedback. In contrast to existing periodic control approaches, the proposed feedback achieves prescribed-time convergence of the state to zero rather than asymptotic convergence. At the same time its synthesis requires solving algebraic equations rather than periodic differential equations. Compared with full-order observer based output feedback, the dimensions of the dynamic equations in the reduced-order observer and dual observer based output feedback are respectively n−p and n−m, where n,m and p are respectively the number of state variables, inputs and outputs, which indicates that the established controllers are less expensive than the full-order observer based controllers in implementations. A numerical example is given to verify the effectiveness of the proposed methods.

Robust dynamic feedback stabilization for linear systems with disturbances

In this paper, we are concerned with the dynamic feedback design for linear systems with input disturbance. We propose a new approach to design dynamic feedback such that the disturbance in the control channel can be compensated effectively. An interesting characteristic about the disturbance robustness of dynamic feedback is found. Thanks to this interesting characteristic, we just need to consider the disturbance-free system to make the dynamic feedback design. This fact is well validated by several dynamic feedback laws proposed in this paper. Our results are available for the regular linear system with harmonic disturbance. Moreover, the stabilization of an unstable heat equation with general periodic disturbance is also investigated. In order to validate the theoretical results visually, some numerical simulations are given to show the effectiveness of the proposed dynamic feedback.

Output feedback stabilization of linear systems with infinite distributed input and output delays

This paper considers the output feedback stabilization problem of linear systems with infinite distributed delays in their inputs and outputs. A low gain observer is proposed to estimate the state of the concerned system with infinite distributed delays in its inputs and outputs. The observer based output feedback controller is then developed. It is shown that the resulting closed loop system is globally asymptotically stable. It should be pointed out that the output feedback control problem of linear systems with infinite distributed input and output delays is first time investigated to the best of our knowledge. Finally, some numerical examples are presented to demonstrate the effectiveness of the proposed controller.

On the fixed-time stabilization of input delay systems using act-and-wait control

Abstract We address the state feedback stabilization of linear systems with input delay using the so-called act-and-wait control strategy. The latter approach induces an analysis of the closed-loop stability based on a finite-dimensional monodromy operator, and rendering this matrix nilpotent results in fixed-time stability. We show that any controllable planar system can be stabilized in a fixed time, and the constructive proof reveals that always two isolated solutions for the corresponding controller gain co-exist. Next, for the fixed-time stabilization of general linear systems, we present both a novel numerical approach for the computation of the controller gain, and an analytic approach relying on incorporating a predictor and an appropriately defined feedback transformation in the control scheme. Several illustrations complete the presentation.

Feedback stabilization of linear systems with multiple controllers via bumpless transfer inputs and combined switching

SummaryFor a linear system with multiple controllers, we address the problem of stabilization by means of bumpless transfer inputs and combined switching. The main idea is to design combined switching signals consisting of both state‐feedback logic and time‐driven mechanism. The discontinuities of the control signal during controller switching is compensated over the dwell time intervals, and the compensator is designed so that switching signal keeps unchanged. The state trajectories of the compensated system are proved to be bounded and convergent. A numerical example is presented to verify the effectiveness of the proposed scheme.

Numerical optimization for feedback stabilization of linear systems with distributed delays

This paper investigates the feedback stabilization of linear systems with distributed delays. By means of the state transition matrix (fundamental matrix) of the delay systems, an optimization problem is formulated for solving the feedback stabilization problem. We prove a necessary and sufficient condition for feedback gain matrices to stabilize the delay systems. Then a numerical method based on the condition is presented to determine the feedback gain matrices. Furthermore, numerical examples show that the presented algorithm is efficient for stabilization of the delay systems.

Parallel-triggered observer-based output feedback stabilization of linear systems

Based on the parallel-triggered scheme, the stabilization problem is investigated for linear systems. By utilizing both the absolute and relative errors, a continuous parallel-triggered scheme is proposed. The proposed scheme reduces the number of data transmissions as well as maintains control performance. Under the proposed scheme, Zeno behavior can be excluded. Then, by using Lyapunov theory, sufficient criteria are established for the existence of state feedback controller and the observer-based controller. Next, a co-design approach is provided to obtain both weighting matrix of the triggered scheme and the gain of controller. Finally, the superiority of the proposed scheme is demonstrated by a numerical example.

LMI approaches to input and output quantized feedback stabilization of linear systems

This paper investigates the problem of quantized feedback stabilization for linear systems. In the controlled systems, the measurement output and control input signals are transmitted via the digital communication link, and the quantization errors are treated as sector bound uncertainties. Two different approaches to designing output feedback are developed and the corresponding design conditions in terms of solutions to linear matrix inequalities (LMIs) are presented. The resulting design is such that the homologous closed-loop system is asymptotically stable with respect to the quantization effects. Finally, we illustrate the efficiency of our main results by a numerical example.

State Feedback Stabilization of Linear Systems with Unknown Input Time Delay

This paper investigates the control problem of linear systems affected by an unknown constant input delay by means of a finite-dimensional state feedback. The proposed solution extends an approach used in the case of known delays by means of a suitably developed delay identifier. A more general result about the convergence to zero of the controlled system when the delay estimation error only converges to some neighborhood of zero is provided. Numerical examples show the effectiveness of the proposed approach.

Observer-based feedback stabilization of linear systems with event-triggered sampling and dynamic quantization

We consider the problem of output feedback stabilization in linear systems when the measured outputs and control inputs are subject to event-triggered sampling and dynamic quantization. A new sampling algorithm is proposed for outputs which does not lead to accumulation of sampling times and results in asymptotic stabilization of the system. The approach for output sampling is based on defining an event function that compares the difference between the current output and the most recently transmitted output sample not only with the current value of the output, but also takes into account a certain number of previously transmitted output samples. This allows us to reconstruct the state using an observer with sample-and-hold measurements. The estimated states are used to generate a control input, which is subjected to a different event-triggered sampling routine; hence the sampling times of inputs and outputs are asynchronous. Using Lyapunov-based approach, we prove the asymptotic stabilization of the closed-loop system and show that there exists a minimum inter-sampling time for control inputs and for outputs. To show that these sampling routines are robust with respect to transmission errors, only the quantized (in space) values of outputs and inputs are transmitted to the controller and the plant, respectively. A dynamic quantizer is adopted for this purpose, and an algorithm is proposed to update the range and the centre of the quantizer that results in an asymptotically stable closed-loop system.

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.

Pseudo-predictor feedback stabilization of linear systems with time-varying input delays

This paper is concerned with stabilization of (time-varying) linear systems with a single time-varying input delay by using the predictor based delay compensation approach. Differently from the traditional predictor feedback which uses the open-loop system dynamics to predict the future state and will result in an infinite dimensional controller, we propose in this paper a pseudo-predictor feedback (PPF) approach which uses the (artificial) closed-loop system dynamics to predict the future state and the resulting controller is finite dimensional and is thus easy to implement. Necessary and sufficient conditions guaranteeing the stability of the closed-loop system under the PPF are obtained in terms of the stability of a class of integral delay operators (systems). Moreover, it is shown that the PPF can compensate arbitrarily large yet bounded input delays provided the open-loop (time-varying linear) system is only polynomially unstable and the feedback gain is well designed. Comparison of the proposed PPF approach with the existing results is well explored. Numerical examples demonstrate the effectiveness of the proposed approaches.

Stabilization and robustness of constrained linear systems

In this paper, we consider the feedback stabilization of linear systems in a Hilbert state space. The paper proposes a class of nonlinear controls that guarantee exponential sta- bility for linear systems. Applications to stabilization with saturating controls are provided. Also the robustness of constrained stabilizing controls is analyzed.

Robust quantized feedback stabilization of linear systems based on sliding mode control

SUMMARYThis paper investigates the robust quantized feedback stabilization problem for a class of single‐input linear uncertain systems on the basis of sliding mode control schemes. It is assumed that the system signals including the system state and control input are quantized before being transmitted through digital communication channels. A static adjustment policy of the quantization parameter for dynamic quantizers is developed. With the adjustment of the quantization parameter, a sliding mode control strategy is designed to ensure the reachability of the sliding manifold, and then, the system is induced into the stable sliding motion thereafter. Finally, an example is provided to illustrate the effectiveness of the proposed approach. Copyright © 2012 John Wiley & Sons, Ltd.

REDUCED-ORDER MODELS FOR FEEDBACK STABILIZATION OF LINEAR SYSTEMS WITH A SINGULAR PERTURBATION MODEL

The problem of output feedback stabilization of linear systems based on a reduced-order model is addressed in this paper. New reduced-order models are proposed for the output feedback design of linear systems with a singular perturbation model. An output feedback controller with a zero steady-state gain matrix is proposed for stabilizing this kind of system. It is shown that with the proposed controller the reduced-order model based feedback design can guarantee the actual closed-loop stability for the sufficiently small perturbation parameter. This approach can overcome the difficulties in the existing design method using the so-called zeroth-order approximation model, whose validity is highly dependent on the value of the perturbation parameter.

Robust quantized feedback stabilization of linear systems

This paper investigates the feedback stabilization problem for SISO linear uncertain control systems with saturating quantized measurements. In the fixed quantization sensitivity framework, we propose a time varying control law able to effectively account for the presence of saturation, which is often the main source of instability, designed using sliding mode techniques. Such controller is proved able to stabilize the plant both in the presence and in the absence of quantization.

Output Feedback Stabilization of Linear Systems with Unknown Additive Output Sinusoidal Disturbances

The output feedback exponential stabilization problem is addressed for known linear stabilizable and detectable systems when the measured output is affected by sinusoidal disturbances generated by an unknown exosystem and only an upper bound on the exosystem order is supposed to be known.Necessary and sufficient conditions are given: in particular a solution to the problem exists if and only if the set of exosystem eigenvalues is disjoint from the set of system eigenvalues. Two adaptively stabilizing control algorithms are proposed: the first one drives the state of the given system exponentially to zero when the actual disturbances are exactly modelled by the exosystem. When the exosystem overmodels the actual disturbances an on-line detector of the number of excited frequencies is included in the second more complex algorithm: the exponentially converging estimates of the system state variables are then used to drive the state of the given system exponentially to zero. An illustrative example with a disturbance containing a variable number of frequencies is worked out in details and simulated.

Output Feedback Stabilization of Linear Systems With Actuator Saturation

The note presents a method for designing an output feedback law that stabilizes a linear system subject to actuator saturation with a large domain of attraction. This method applies to general linear systems including strictly unstable ones. A nonlinear output feedback controller is first expressed in the form of a quasi-LPV system. Conditions under which the closed-loop system is locally asymptotically stable are then established in terms of the coefficient matrices of the controller. The design of the controller (coefficient matrices) that maximizes an estimate of the domain of attraction is then formulated and solved as an optimization problem with LMI constraints