On the stability of convolution feedback systems with dynamical feedback

This paper considers distributed n-inputn-output convolution feedback systems characterized by $\text{y = G}{}_1\text{1e}$, $\text{z = G}{}_2\text{1y}$ and e = u ? z, where the forward path transfer function $\text{G}\text{?}{}_1$ and the feedback path transfer function $\text{G}\text{?}{}_2$ both contain a real single unstable pole at different locations. Theorem 1 gives necessary and sufficient conditions for both input-error and input-output stability. In addition to usual conditions that guarantee input-error stability a new condition is found which results in the fact that input-error stability will guarantee input-output stability. These conditions require to investigate only the open-loop characteristics. A basic device is the consideration of the residues of different transfer functions at the open-loop unstable poles. An example is given.     

The parallel projection operators of a nonlinear feedback system

This paper defines and studies a pair of nonlinear parallel projection operators associated with a nonlinear feedback system. These operators have been seen to play an important role in the robustness and design of linear systems especially in the theory of the gap metric, the use of weighted gaps in control system design and Glover-McFarlane loop-shaping. We show that the input-output L₂-stability of a feedback system amounts to a ‘coordinatization’ of the input and output spaces, which is also equivalent to the existence of a pair of nonlinear parallel projection operators onto the graph of the plant and the inverse graph of the controller respectively. These projections are shown to have equal norms whenever one of the feedback elements is linear. A bound on this norm is given in the case of passive systems with unity negative feedback.     

Global stability of relay feedback systems

For a large class of relay feedback systems (RFS) there will be limit cycle oscillations. Conditions to check existence and local stability of limit cycles for these systems are well known. Global stability conditions, however, are practically nonexistent. The paper presents conditions in the form of linear matrix inequalities (LMIs) that, when satisfied, guarantee global asymptotic stability of limit cycles induced by relays with hysteresis in feedback with linear time-invariant (LTI) stable systems. The analysis consists in finding quadratic surface Lyapunov functions for Poincare maps associated with RFS. These results are based on the discovery that a typical Poincare map induced by an LTI flow between two hyperplanes can be represented as a linear transformation analytically parametrized by a scalar function of the state. Moreover, level sets of this function are convex subsets of linear manifolds. The search for quadratic Lyapunov functions on switching surfaces is done by solving a set of LMIs. Although this analysis methodology yields only a sufficient criterion of stability, it has proved very successful in globally analyzing a large number of examples with a unique locally stable symmetric unimodal limit cycle. In fact, it is still an open problem whether there exists an example with a globally stable symmetric unimodal limit cycle that could not be successfully analyzed with this new methodology. Examples analyzed include minimum-phase systems, systems of relative degree larger than one, and of high dimension. Such results lead us to believe that globally stable limit cycles of RFS frequently have quadratic surface Lyapunov functions     

输出多采样反馈控制器及系统稳定性

针对由连续被控对象和数字控制器构成的数字控制系统,将现有的线性系统输出多采样线性反馈数字控制器设计方法推广到非线性系统．并相应地研究了非线性输出多采样反馈控制器及摄动非线性系统．给出了这类非线性输出多采样数字控制系统及其摄动系统的稳定性和鲁棒性条件．     

Static output feedback controllers: stability and convexity

The main objective of this paper is to solve the following stabilizing output feedback control problem: given matrices (A; B₂ ; C₂) with appropriate dimensions, find (if one exists) a static output feedback gain L such that the closed-loop matrix A-B₂LC₂ is asymptotically stable. It is known that the existence of L is equivalent to the existence of a positive definite matrix belonging to a convex set such that its inverse belongs to another convex set. Conditions are provided for the convergence of an algorithm which decomposes the determination of the aforementioned matrix in a sequence of convex programs. Hence, this paper provides a new sufficient (but not necessary) condition for the solvability of the above stabilizing output feedback control problem. As a natural extension, we also discuss a simple procedure for the determination of a stabilizing output feedback gain assuring good suboptimal performance with respect to a given quadratic index. Some examples borrowed from the literature are solved to illustrate the theoretical results     

Linear dynamic output feedback: Invariants and stability

A full invariant under linear dynamic output feedback is derived. Some of its applications to the design of internally stable feedback control systems are considered.     

Algebraic stability criteria and symbolic derivation of stability conditions for feedback control systems

This article provides algebraic settings of the stability criteria of Nyquist and Popov and the circle criterion for closed-loop linear control systems with linear or nonlinear feedback whose transfer functions are rational ones with integer coefficients. The proposed settings make use of algebraic methods of parametric curve implicitisation, real root isolation, symbolic integration and quantifier elimination and allow one to derive exact stability conditions for feedback control systems with symbolic computation. An example is presented to illustrate the algebraic approach and its effectiveness. Some numerical stability results obtained previously are confirmed.     

Output feedback of forward-backward periodic systems stability

An n-periodic forward-backward system defined by a collection of nonproper rational matrices is considered. In addition, an n-periodic collection of output feedbacks such that the closed-loop system is n-periodic and forward-backward is constructed. Block-ordered coprime factorizations of a rational matrix are used in the stabilization of that closed-loop system     

The stability of systems with multiplicative feedback

The global phase-portrait concept of Poincaré is used initially to predict the stability of two continuous systems which have multiplicative feedback. Singularities at infinity are revealed which are coalitions of saddle points and nodes. Where stability is not global, Lyapunov functions are generated to define the domain of stability in the general case of each system. Diagrams are presented of the allowable input step for stable response, as a function of the initial input equilibrium state.     

On the stability of multiloop feedback systems

Some extensions on recent work in passive stability theory by Desoer and the author are presented. The stability of feedback systems is considered where the forward-loop and return-loop subsystems each have been subdivided into several systems operating in parallel. Results are obtained in the areas of Lyapunov stability, L2input L2output stability, and bounded-input bounded-state stability. Before applying these results to a specific multiloop feedback system, a pseudoenergy analysis must be done on each sub sub-system. Two specific types of systems are so analyzed. The first is a general linear first-order time-varying system. The second is a linear time-varying infinite-dimensional system; the analysis of this system takes the form of a Kalman-Yacubovich-type lemma. By using these two new systems, along with several others that have been analyzed previously, stability theorems for many specific multiloop feedback systems can be proven. One such example is given.     

Output feedback MPC with guaranteed robust stability

This paper focuses on the issues of robust stability of model predictive control (MPC). The control problem is formulated as linear matrix inequalities (LMI) optimization problem. A suboptimal solution for the output feedback control problem is proposed. The size of the resulting MP controller is reduced by using a suitable state-space representation of the process. Guaranteed stability conditions for the output feedback MPC are enforced via a Lyapunov type constraint. An iterative algorithm is developed resulting in a pair of coupled LMI optimization problems which provide a robustly stable output feedback gain. Model uncertainties are considered via a polytopic set of process models. The methodology is illustrated with the simulation of the control problem of two chemical processes. The results show that the proposed strategy eliminates the need to detune the MP controller improving the performance for most of the cases considered.     

A stability criterion for nonlinear feedback systems

A frequency domain criterion is formulated to prove global asymptotic stability of feedback control system with more than one nonlinear element. The input-output characteristics of the nonlinearities are confined to sectors of the input-output planes. Sufficient conditions for global asymptotic stability are derived in terms of a number of inequalities equal to the number of nonlinear elements.     

Robust stability of Metzler operators under parameter perturbations

In this paper we study how the spectral bound of Metzler operators changes under parameter perturbations. Characterizations of the stability radii of Metzler operator with respect to this type of disturbances are established. The results generalize those obtained in (Vietnam J. Math. 2006; 34 :357–368; Vietnam J. Math. 1998; 26 :147–163). Copyright © 2009 John Wiley & Sons, Ltd.     

Robust stability of discrete-time systems using delta operators

The robust stability of discrete-time systems formulated in terms of the delta (δ) operator is discussed. That is, given the nominal characteristic equation P(δ) of a discrete-time system, it is of interest to know how much the coefficients can be perturbed while preserving stability. A procedure to obtain the maximum intervals for a perturbed polynomial P(δ) to still be stable is presented     

Output–input stability implies feedback stabilization

We study the recently introduced notion of output–input stability, which is a robust variant of the minimum-phase property for general smooth nonlinear control systems. This paper develops the theory of output–input stability in the multi-input, multi-output setting. We show that output–input stability is a combination of two system properties, one related to detectability and the other to left-invertibility. For systems affine in controls, we derive a necessary and sufficient condition for output–input stability, which relies on a global version of the nonlinear structure algorithm. This condition leads naturally to a globally asymptotically stabilizing state feedback strategy for affine output–input stable systems.     

Input-output stability of time-varying nonlinear multiloop feedback systems

Sufficient conditions for the input-output stability (BIBO stability) of time-varying nonlinear multiloop feedback systems are established. The present objective is to analyze large-scale systems in terms of their lower order subsystems (subloops) and in terms of their interconnecting structure. Both time-domain and frequency-domain results are presented. In order to demonstrate the usefulness of the present approach, two specific examples are considered.     

Frequency domain stability and boundedness criteria of pulse-modulated feedback systems

Stability and boundedness conditions of pulse modulated feedback systems are derived in this paper. These conditions, based on the frequency domain approach, extend the range of applicability of this approach to systems without the strict sector non-linearity. Geometric interpretations of the results in the Popov plane are also presented.     

Structural stability of unsupervised learning in feedback neuralnetworks

Structural stability is proved for a large class of unsupervised nonlinear feedback neural networks, adaptive bidirectional associative memory (ABAM) models. The approach extends the ABAM models to the random-process domain as systems of stochastic differential equations and appends scaled Brownian diffusions. It is also proved that this much larger family of models, random ABAM (RABAM) models, is globally stable. Intuitively, RABAM equilibria equal ABAM equilibria that vibrate randomly. The ABAM family includes many unsupervised feedback and feedforward neural models. All RABAM models permit Brownian annealing. The RABAM noise suppression theorem characterizes RABAM system vibration. The mean-squared activation and synaptic velocities decrease exponentially to their lower hounds, the respective temperature-scaled noise variances. The many neuronal and synaptic parameters missing from such neural network models are included, but as net random unmodeled effects. They do not affect the structure of real-time global computations     

Mean square stability criteria for stochastic feedback systems

In this paper several stability criteria are obtained for a class of stochastic feedback systems containing a multiplicative white noise element ; continuous time as well as discrete time systems are dealt with. Mean square stability properties are derived by means of Lyapunov theory ; the Lyapunov functions are generated by means of the Lyapunov equation, the path integral technique, and the Kalman-Yacoboviteh lemma. Results similar to the Routh-Hurtwitz conditions and the circle criteria for deterministic feedback systems are discussed. Different interpretations of the white noise element (Itò, Stratonovitch, etc.) are considered.