On Uniform Global Asymptotic Stability of Nonlinear Discrete-Time Systems With Applications

This paper presents new characterizations of uniform global asymptotic stability for nonlinear and time-varying discrete-time systems. Under mild assumptions, it is shown that weak zero-state detectability is equivalent to uniform global asymptotic stability for globally uniformly stable systems. By employing the notion of reduced limiting systems, another characterization of uniform global asymptotic stability is proposed on the basis of the detectability for the reduced limiting systems associated with the original system. As a by-product, we derive a generalized, discrete-time version of the well-known Krasovskii-LaSalle theorem for general time-varying, not necessarily periodic, systems. Furthermore, we apply the obtained stability results to analyze uniform asymptotic stability of cascaded time-varying systems, and show that some technical assumptions in recent papers can be relaxed. Through a practical application, it is shown that our results play a similar role to the classic LaSalle invariance principle in guaranteeing attractivity, noting that reduced limiting systems are used instead of the original system. To validate the conceptual characterizations, we study the problem of sampled-data stabilization for the benchmark example of nonholonomic mobile robots via the exact discrete-time model rather than approximate models. This case study also reveals that in general, sampled-data systems may become non-periodic even though their original continuous-time system is periodic. A novel sampled-data stabilizer design is proposed using the new stability results and is supported via simulation results.     

Characterizations of detectability notions in terms of discontinuous dissipation functions

We consider a new Lyapunov-type characterization of detectability for non-linear systems without controls, in terms of lower-semicontinuous (not necessarily smooth, or even continuous) dissipation functions, and prove its equivalence to the GASMO (global asymptotic stability modulo outputs) and UOSS (uniform output-to-state stability) properties studied in previous work. The result is then extended to provide a construction of a discontinuous dissipation function characterization of the IOSS (input-to-state stability) property for systems with controls. This paper complements a recent result on smooth Lyapunov characterizations of IOSS. The utility of non-smooth Lyapunov characterizations is illustrated by application to a well-known transistor network example.     

Robust ℒ︁2‐gain feedforward control of uncertain systems using dynamic IQCs

We consider the problem of robust ??₂‐gain disturbance feedforward control for uncertain systems described in the standard LFT form. We use integral quadratic constraints (IQCs) for describing the uncertainty blocks in the system. For technical reasons related to the feedforward problem, throughout the paper, we work with the duals of the constraints involved in robustness analysis using IQCs. We obtain a convex solution to the problem using a state‐space characterization of nominal stability that we have developed recently. Specifically, our solution consists of LMI conditions for the existence of a feedforward controller that guarantees a given ??₂‐gain for the closed‐loop system. We demonstrate the effectiveness of using dynamic IQCs in robust feedforward design through a numerical example. Copyright © 2008 John Wiley & Sons, Ltd.     

Input to state stabilizability for parametrized families of systems

This paper studies various stability issues for parametrized families of systems, including problems of stabilization with respect to sets. The study of such families is motivated by robust control applications. A Lyapunov-theoretic necessary and sufficient characterization is obtained for a natural notion of robust uniform set stability; this characterization allows replacing ad hoc conditions found in the literature by more conceptual stability notions. We then use these techniques to establish a result linking state-space stability to ‘input to state’ (bounded-input bounded-state) stability. In addition, the preservation of stabilizability under certain types of cascade interconnections is analysed.     

A new parameterization of stable polynomials

In this note, we develop a new characterization of stable polynomials. Specifically, given n positive, ordered numbers (frequencies), we develop a procedure for constructing a stable degree n monic polynomial with real coefficients. This construction can be viewed as a mapping from the space of n ordered frequencies to the space of stable degree n monic polynomials. The mapping is one-one and onto, thereby giving a complete parameterization of all stable, degree n monic polynomials. We show how the result can be used to generate parameterizations of stabilizing fixed-order proper controllers for unity feedback systems. We apply these results in the development of stability margin lower bounds for systems with parameter uncertainty.     

Time-relevant stability of 2D systems

For many 2D systems, one of the independent variables plays a distinct role in the evolution of the trajectories; since often this special independent variable is time, we call such systems ‘time-relevant’. In this paper, we introduce a stability notion for time-relevant systems described by higher-order difference equations. We give algebraic tests in terms of the location of the zeros of the determinant of a polynomial matrix describing the system. We also give an LMI characterization of time-relevant stability involving only constant matrices.     

Control system design subject to SNR constraints

This paper deals with networked control systems comprising LTI plants controlled over scalar additive noise channels subject to signal-to-noise ratio (SNR) constraints. We present a general framework, based upon convex optimization concepts, that can accommodate several situations of interest. Our results make explicit the fact that exploiting feedback around the channel plays a key role in reducing the minimal SNR that is compatible with stability. The results also provide a characterization of the best achievable performance subject to an SNR constraint. We apply the results to specific networked control architectures, and provide a numerical example.     

A sufficient condition for asymptotic stability for a class of time-varying parameterized systems

We derive sufficient conditions for local asymptotic stability for a class of parameterized time-varying systems, whose dynamics are in general unbounded in time. Our main result generalizes a well known limiting-dynamics based approach due to Peuteman and Aeyels (1999) in [9] for fast time-varying systems whose dynamics are bounded in time. Its proof is based on a new result providing a Lyapunov characterization for exponential stability.     

Stability analysis of linear systems subject to regenerative switchings

This paper investigates the stability of switched linear systems whose switching signal is modeled as a stochastic process called a regenerative process. We show that the mean stability of such a switched system is characterized by the spectral radius of a matrix. The matrix is obtained by taking the expectation of the transition matrix of the system on one cycle of the underlying regenerative process. The characterization generalizes Floquet’s theorem for the stability analysis of linear time-periodic systems. We illustrate the result with the stability analysis of a linear system with a failure-prone controller under periodic maintenance.     

Preservation of stability under system perturbations in nonlinear control: A simple characterization

A simple characterization of the perturbations a nonlinear closed-loop control system can tolerate before losing its stability is derived. The characterization is given in purely algebraic terms, with the topological aspects being automatically incorporated through the theory of fraction representations of nonlinear systems. The implications of system perturbations on internal stability are also discussed. The presentation is for the case of discrete-time nonlinear systems.     

The problem of absolute stability: a dynamic programming approach

We consider the problem of absolute stability of a feedback system composed of a linear plant and a single sector-bounded nonlinearity. Pyatnitskiy and Rapoport used a variational approach and the Maximum Principle to derive an implicit characterization of the “most destabilizing” nonlinearity. In this paper, we address the same problem using a dynamic programming approach. We show that the corresponding value function is composed of simple building blocks which are the generalized first integrals of appropriate linear systems. We demonstrate how the results can be used to design stabilizing switched controllers.     

Robust Stability and Stabilization of Fractional-Order Interval Systems: An LMI Approach

This technical note presents necessary and sufficient conditions for the stability and stabilization of fractional-order interval systems. The results are obtained in terms of linear matrix inequalities. Two illustrative examples are given to show that our results are effective and less conservative for checking the robust stability and designing the stabilizing controller for fractional-order interval systems.      

Stability and stabilization of periodic piecewise positive systems: A time segmentation approach

This paper is concerned with the stability analysis and stabilization of periodic piecewise positive systems. By constructing a time-scheduled copositive Lyapunov function with a time segmentation approach, an equivalent stability condition, determined via linear programming, for periodic piecewise positive systems is established. Based on the asymptotic stability condition, the spectral radius characterization of the state transition matrix is proposed. The relation between the spectral radius of the state transition matrix and the convergent rate of the system is also revealed. An iterative algorithm is developed to stabilize the system by decreasing the spectral radius of the state transition matrix. Finally, numerical examples are given to illustrate the results.     

Nonlinear observer design for a general class of nonlinear systems with real parametric uncertainty

This paper is a geometric study of the local observer design for a general class of nonlinear systems with real parametric uncertainty. Explicitly, we study the observer design problem for a general class of nonlinear systems with real parametric uncertainty and with an input generator (exosystem). In this paper, we show that for the classical case, when the state equilibrium does not change with the parametric uncertainty, and when the plant output is purely a function of the state, there is no local asymptotic observer for the plant. Next, we show that in sharp contrast to this case, for the general case of problems where we allow the state equilibrium to change with the parametric uncertainty, there typically exist local exponential observers even when the plant output is purely a function of the state. We also present a characterization and construction procedure for local exponential observers for the general class of nonlinear systems with real parametric uncertainty under some stability assumptions. We also show that for the general class of nonlinear systems considered, under some stability assumptions, the existence of local exponential observers in the presence of inputs implies, and is implied by, the existence of local exponential observers in the absence of inputs. Finally, we generalize our results to a general class of nonlinear systems with input generator, and with exogenous disturbance.     

时变滞后Lurie型系统的改进稳定性准则

In this technical note, we present a new stability analysis procedure for ascertaining the delay-dependent stability of a class of Lurie systems with time-varying delay and sector-bounded nonlinearity using Lyapunov-Krasovskii (LK) functional approach. The proposed analysis, owing to the candidate LK functional and tighter bounding of its time-derivative, yields less conservative absolute and robust stability criteria for nominal and uncertain systems respectively. The effectiveness of the proposed criteria over some of the recently reported results is demonstrated using a numerical example.     

A Personal Resource for Technology Interaction: Development and Validation of the Affinity for Technology Interaction (ATI) Scale

Successful coping with technology is relevant for mastering daily life. Based on related conceptions, we propose affinity for technology interaction (ATI), defined as the tendency to actively engage in intensive technology interaction, as a key personal resource for coping with technology. We present the 9-item ATI scale, an economical unidimensional scale that assesses ATI as an interaction style rooted in the construct need for cognition (NFC). Results of multiple studies (n > 1500) showed that the scale achieves good to excellent reliability, exhibits expected moderate to high correlations with geekism, technology enthusiasm, NFC, self-reported success in technical problem-solving and technical system learning success, and also with usage of technical systems. Further, correlations of ATI with the Big Five personality dimensions were weak at most. Based on the results, the ATI scale appears to be a promising tool for research applications such as the characterization of user diversity in system usability tests and the construction of general models of user-technology interaction.     

On uniform asymptotic stability of time-varying parameterized discrete-time cascades

Recently, a framework for controller design of sampled-data nonlinear systems via their approximate discrete-time models has been proposed in the literature. In this paper, we develop novel tools that can be used within this framework and that are useful for tracking problems. In particular, results for stability analysis of parameterized time-varying discrete-time cascaded systems are given. This class of models arises naturally when one uses an approximate discrete-time model to design a stabilizing or tracking controller for a sampled-data plant. While some of our results parallel their continuous-time counterparts, the stability properties that are considered, the conditions that are imposed, and the the proof techniques that are used, are tailored for approximate discrete-time systems and are technically different from those in the continuous-time context. A result on constructing strict Lyapunov functions from nonstrict ones that is of independent interest, is also presented. We illustrate the utility of our results in the case study of the tracking control of a mobile robot. This application is fairly illustrative of the technical differences and obstacles encountered in the analysis of discrete-time parameterized systems.     

Output feedback controllers for systems with structured uncertainty

This paper deals with the design of compensators that provide a guaranteed level of performance, in the sense of L₂ gain, for systems with time-varying structured uncertainty. For stability, the notion of quadratic stability, using a single Lyapunov function, is used. The uncertainty is assumed to be polytopic (e.g., the uncertainty vector enters the state-space representation of the systems affinely). Earlier results in the characterization of H_∞ controllers are used to show that quadratic stability with performance is equivalent to a bi-affine problem. A set of conditions, under which the problem becomes convex, is discussed     

Nonlinear observer design for a general class of discrete-time nonlinear systems with real parametric uncertainty

This paper is a geometric study of the local observer design for a general class of discrete-time nonlinear systems with real parametric uncertainty. Explicitly, we study the observer design problem for a general class of discrete-time nonlinear systems with real parametric uncertainty and with an input generator (exosystem). In this paper, we show that for the classical case, when the state equilibrium does not change with the parametric uncertainty, and when the plant output is purely a function of the state, there is no local asymptotic observer for the plant. Next, we show that in sharp contrast to this case, for the general case of problems where we allow the state equilibrium to change with the parametric uncertainty, there typically exist local exponential observers even when the plant output is purely a function of the state. We also present a characterization and construction procedure for local exponential observers for the general class of discrete-time nonlinear systems with real parametric uncertainty under some stability assumptions. We also show that for the general class of discrete-time nonlinear systems considered, under some stability assumptions, the existence of local exponential observers in the presence of inputs implies, and is implied by the existence of local exponential observers in the absence of inputs. Finally, we generalize our results to a general class of discrete-time nonlinear systems with input generator, and with exogenous disturbance.     

Disturbance decoupling in a class of linear systems

The disturbance decoupling problem by state feedback (DDP) is solved for a class of linear systems where the left-invertible systems are included. Also obtained is a complete characterization of the solution as well as the parameterization of all corresponding state feedback matrices. Such characterization enables one to conclude about the solvability of DDP with stability and/or pole placement. A numerical method is proposed for computation of the controllers in an analytical form, suitable to the treatment of complementary design specifications. The solution of the disturbance decoupling problem by measurement feedback (DDPM) is also discussed and, based on the results obtained for DDP, a procedure for design of a reduced-order compensator is proposed