Quantized feedback stabilization of linear systems

This paper addresses feedback stabilization problems for linear time-invariant control systems with saturating quantized measurements. We propose a new control design methodology, which relies on the possibility of changing the sensitivity of the quantizer while the system evolves. The equation that describes the evolution of the sensitivity with time (discrete rather than continuous in most cases) is interconnected with the given system (either continuous or discrete), resulting in a hybrid system. When applied to systems that are stabilizable by linear time-invariant feedback, this approach yields global asymptotic stability     

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.     

State feedback stabilization of linear impulsive systems

We address the fundamental problem of state feedback stabilization for a class of linear impulsive systems featuring arbitrarily-spaced impulse times and possibly singular state transition matrices. Specifically, we show that a strong reachability property enables a state feedback law to be constructed that yields a uniformly exponentially stable closed-loop system. The approach adopts a receding horizon strategy involving a weighted reachability gramian in a manner reminiscent of well-known results for time-varying linear systems for both continuous and discrete-time cases.     

On feedback stabilization of time-varying discrete linear systems

Results are given for stabilizing time-varying discrete linear systems by means of a feedback control stemming from a receding-horizon concept and a minimum quadratic cost with a fixed terminal constraint. The results parallel those recently obtained for continuous-time systems [8] and extend a well-known method of Kleinman for stabilizing discrete fixed linear systems [7].     

Simultaneous stabilization of linear single-input systems by linear state feedback control

A necessary and sufficient condition for the existence of a linear state feedback controller which will simultaneously stabilize a finite collection of single-input linear systems is presented. This condition can be checked numerically through the solution of a non-linear programming problem presented here and can be used to determine whether or not a particular simultaneous stabilization problem has a solution. If the stabilization problem has a solution, the method provides a means of constructing a suitable controller. The proposed method is very versatile. It is a simple matter to limit the size of elements of the feedback gain vector and thus ensure that the control action is not excessive. In addition, for simultaneous stabilization problems which have a range of possible solutions, the proposed method makes it possible to select one which yields good performance.     

Semiglobal stabilization of multi-input linear systems with saturated linear state feedback

For multi-input linear systems with eigenvalues in the closed left-half comlex plane, we address the problem of stabilization by a linear state feedback subject to saturation. Using an eigenvalue-generalized eigenvector assignment technique, we prove that such systems can be semiglobally stabilized with a saturated linear state feedback. Based on this result, we propose an algorithm to calculate an -parametrized family of state feedback gain matrices that semiglobally stabilize the system. Two examples are used to illustrate the results.     

Global stabilization of linear discrete-time systems with bounded feedback

This paper deals with the problem of global stabilization of linear discrete time systems by means of bounded feedback laws. The main result proved is an analog of one proved for the continuous time case by the authors, and shows that such stabilization is possible if and only if the system is stabilizable with arbitrary controls and the transition matrix has spectral radius less than or equal to one. The proof provides in principle an algorithm for the construction of such feedback laws, which can be implemented either as cascades or as parallel connections (“single hidden layer neural networks”) of simple saturation functions.     

Finite data-rate feedback stabilization of switched and hybrid linear systems

We study the problem of asymptotically stabilizing a switched linear control system using sampled and quantized measurements of its state. The switching is assumed to be slow enough in the sense of combined dwell time and average dwell time, each individual mode is assumed to be stabilizable, and the available data rate is assumed to be large enough but finite. Our encoding and control strategy is rooted in the one proposed in our earlier work on non-switched systems, and in particular the data-rate bound used here is the data-rate bound from that earlier work maximized over the individual modes. The main technical step that enables the extension to switched systems concerns propagating over-approximations of reachable sets through sampling intervals, during which the switching signal is not known; a novel algorithm is developed for this purpose. Our primary focus is on systems with time-dependent switching (switched systems) but the setting of state-dependent switching (hybrid systems) is also discussed.     

区间线性脉冲系统的鲁棒耗散性及其反馈镇定

主要讨论了区间线性脉冲系统相对于二次供给率的鲁棒耗散性问题, 得到了该系统是鲁棒耗散的充分条件, 而且还给出了这样的鲁棒耗散系统能被一个状态反馈控制律所镇定的条件, 最后, 给出了一个例子.     

基于LMI 的线性时滞系统输出动态反馈镇定

考虑线性时滞系统的输出动态反馈镇定问题.利用自由参数矩阵对闭环系统进行适当变换,并结合相应的Lyapunov-Krasovskii泛函得到了时滞相关的控制器存在性判据.利用控制器参数化方法,将控制器参数与泛函参数的求解归结为线性矩阵不等式解的形式,从而克服了时滞无关性及求解非凸优化问题所导致的保守性.仿真算例验证了结论的有效性.     

Optimal simultaneous stabilization of linear single-input systems via linear state feedback control

Simultaneous stabilization of a collection of linear time-invariant systems is considered. The aim is to develop a practical method for the design of a single feedback controller such that all the systems involved are stabilized with good transient responses. Hurwitz's necessary and sufficient conditions are used as the required set of constraints for simultaneous stability. For transient behaviour, we introduce the usual quadratic objective function as performance index for each system. Their sum is the objective function for the collection of systems and the problem is to minimize it by choosing a feedback gain vector subject to boundedness and the Hurwitz stability constraints. A computational technique is proposed for solving this problem. The results obtained give good transient behaviour. For illustration, two numerical examples are solved.     

On pole assignment of linear systems by gain output feedback

In this paper, the Geometric Approach is used to derive in a straightforward way a sufficient condition for pole assignability by gain output feedback. This result leads to a pole assignment procedure which reduces to solving a system of min (n - m, n - p) polynomial equations where n is the number of states, m the number of inputs, p the number of outputs. In the case where m + p > n, this system clearly appears to be linear. The degrees of freedom related to the pole assignment problem are expressed in terms of (right or left) eigenvectors.     

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.     

Nonlinear versus linear control in the direct output feedback stabilization of linear systems

This paper investigates the use of nonlinear direct output feedback in the control of linear time-invariant systems. Inparticular, it is concerned with those nonlinear controllers which result in a closed loop system which is quadratically stable. That is, the closed loop nonlinear system is asymptotically stable and furthermore, a quadratic Lyapunov function can be used to establish this stability. The main result of this paper can be stated roughly as follows: Any linear system which can be made quadratically stable using nonlinear direct output feedback control can also be stabilized using linear direct output feedback control.     

带传输滞后的线性离散系统的状态反馈镇定

针对存在传输滞后的线性离散系统的状态反馈镇定问题,给出了系统可镇定的一个内部限制条件.为克服这一限制条件,提出了两种方法:一种是充分利用滞后状态的信息,另一种是设计带有递推动态的状态反馈控制器.研究结果表明,若系统在没有传输滞后时能通过状态反馈被镇定,则存在传输滞后时一定也能通过设计新的控制器使系统被镇定.     

Output feedback stabilization of nonlinear MIMO systems having uncertain high-frequency gain matrix

The purpose of this paper is to provide a method for (semi-global) asymptotic stabilization of a nonlinear minimum-phase MIMO system, under a mild hypothesis of the so-called “high-frequency gain” matrix. This result is based on a non-trivial extension, to the MIMO setting, of the approach based on the use of extended observers. As a byproduct, a dynamic output feedback control is obtained, that asymptotically stabilizes the equilibrium of the closed-loop system, in spite of uncertainties in the high-frequency gain matrix.     

Diagonal decoupling of linear multivariable systems with constant gain measurement feedback

This note derives a necessary and sufficient condition to diagonally decouple a linear multivariable system through constant gain measurement feedback. This condition is represented in terms of the controllability subspaces and the kernel of a measurement matrix. The number of assignable poles of the resulting closed-loop system is given.     

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.     

A theorem on global stabilization of nonlinear systems by linear feedback

In this paper we investigate the global stabilizability problem for a wide class of single-input nonlinear systems whose the linearization at the equilirrium is controllable. We show that under general assumptions there exists a linear feedback law which globally exponentially stabilizes the system at its equilibrium. The proof of our main theorem is based on some ideas from a previous paper. We use the theorem to recover a recent result of Gauthier et al. concerning the observer design problem.