Stability Analysis Of Sampled-data Systems Research Articles

Stability Analysis of Sampled-Data Systems Based on Sawtooth-Characteristic-Based Hierarchical Integral Inequality.

The aim of this article is to investigate the stability of sampled-data systems (SDSs) by introducing a sawtooth-characteristic-based hierarchical integral inequality (SCBHII) and to obtain the maximum allowable sampling period that maintains the stability of the system. First, by associating the sawtooth characteristics of the input delay in SDSs with free matrices, an SCBHII is proposed; its accuracy improves as the hierarchy increases. Subsequently, a high-order two-sided looped-functional, which considers both the sampling multi-integral states and the sawtooth pattern, is introduced to cater to the aforementioned inequality. In addition, the system variables are augmented by sawtooth pattern-related terms, which eliminates the need for additional secondary processing when determining the negative-definiteness of derivatives with high-order terms. By combining the high-order two-sided looped-functional with the proposed SCBHII, a stability criterion for SDSs with reduced conservatism is achieved, presented in the form of linear matrix inequalities. The proposed inequality technique and the stability criterion are shown to be effective and superior through three numerical examples and a real-world simplified power market model.

An enhanced looped-functional framework for stability analysis of sampled-data systems

This paper enhances the looped-functional framework for sampled-data systems by using continuous Lyapunov functionals instead of xT(t)Px(t). The continuous Lyapunov functionals exploit the maximum allowable sampling interval compared with the traditional Lyapunov function xT(t)Px(t) in the conventional looped-functional framework. The enhanced looped-functional framework ensures the negativeness of the continuous Lyapunov functionals concerning the sampling instants and guarantees the stability of the sampled-data systems. Based on these continuous Lyapunov functionals, this paper constructs novel Lyapunov functionals with the looped-functional which not only exploits all available vectors related to sampling intervals but also introduces an additional interval which is zero at t=tk+1. Also, this paper proposes the stability criteria based on the enhanced looped-functional framework for the sampled-data systems with the asynchronous and synchronous samplings, respectively. The numerical examples show the effectiveness of the proposed framework.

Robust Stability Analysis of Sampled-Data Systems With Uncertainties Characterized by the ${\mathcal {L}}_\infty$-Induced Norm: Gridding Treatment With Convergence Rate Analysis

This paper tackles the problem of stability analysis for uncertain sampled-data systems consisting of the feedback interconnection between a nominal linear time-invariant (LTI) sampled-data system and a structured linear time-varying (LTV) model uncertainty. When the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathcal {L}}_\infty$</tex-math></inline-formula> norm and its induced norm are taken into account to characterize such uncertain systems, it has been shown that the necessary and sufficient condition for robust stability of such uncertain sampled-data systems is described through the supremum of the spectral radii of a certain nonnegative matrix defined on the sampling interval <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$[0,h)$</tex-math></inline-formula> . However, a direct computation of the supremum for practical uncertain sampled-data systems is quite difficult since it intrinsically involves an infinite number of computations of spectral radii over the continuous-time domain <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$[0,h)$</tex-math></inline-formula> . To solve this problem, we introduce a gridding approach, by which an upper bound, as well as a lower bound on the supremum of the spectral radii, are obtained. The upper and lower bounds also lead to a sufficient condition and a necessary condition for the stability of uncertain sampled-data systems, respectively. Furthermore, the gap between the original supremum of the spectral radii and the upper bound (as well as the lower bound) is shown to converge to 0 at the rate no smaller than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(1/M)^{1/n}$</tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M$</tex-math></inline-formula> is the gridding parameter and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> is the number of perturbation blocks in the structured model uncertainty. To put it another way, both the upper bound used in the sufficient condition and the lower bound taken in the necessary condition tend to the exact supremum of the spectral radii by taking <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M$</tex-math></inline-formula> larger. Finally, a numerical example is given to demonstrate the effectiveness of the overall arguments.

Stability analysis of sampled-data systems via affine canonical Bessel-Legendre inequalities

The looped-functional approach has been extensively used for stability analysis of sampled-data systems. For this approach, this paper introduces two new functionals to minimise the conservatism of stability conditions. The first one is called a non-negative integral functional (NIF) that is added to the derivative of a common Lyapunov functional. The second one is a general looped-functional (GLF), whose values at sampling instants are traditionally not well-defined, and are defined here in terms of their limits. Leveraging further affine canonical Bessel-Legendre inequalities expressed by simplified polynomials, a polynomial-based NIF, a polynomial-based GLF, and a polynomial-based zero equality are tailored for obtaining hierarchical types of stability conditions in the form of linear matrix inequalities. Finally, numerical examples show that: (i) the proposed functionals lead to less conservatism when compared with several state-of-the-art methods; and, (ii) the resulting stability conditions exhibit a hierarchical characteristic in the sense that the higher level of hierarchy, the less conservatism of the criteria.

Stability analysis of sampled-data systems with sector-bounded input nonlinearity

This work proposes a method for the stability analysis of aperiodic sampled-data control systems with sector and slope bounded input nonlinearities. The stability conditions are derived by using a hybrid system representation and a timer-dependent Lur'e type Lyapunov function. Considering a polynomial timer-dependence, the stability conditions are cast in sum-of-squares optimization problems aiming at computing the largest range of sampling intervals or the largest sector bounds on the nonlinearity for which the origin of the closed-loop system is globally asymptotically stable.

A sampling-period-partitioning approach for stability analysis of sampled-data systems with constant delay

This paper is concerned with the stability of sampled-data systems with constant delay. Firstly, by dividing the interval of sampling periods in two subintervals, two separate looped functionals are employed in each of these subintervals. Then, a new Lyapunov functional that combines classical Lyapunov functionals and looped-functionals is constructed. Furthermore, several zero equalities which consider the intrinsic relationships of state vectors in the system are introduced into the derivative of the constructed functional, and some stability criteria with less conservatism are obtained in forms of linear matrix inequalities (LMIs). Finally, two numerical examples are carried out as to verify the effectiveness and advantages of our method.

Stability analysis of sampled-data systems via novel Lyapunov functional method

This study focuses on the stability analysis of sampled-data systems. The idea is to propose a novel Lyapunov functional method using a new framework. In the functional method, the new discontinuous term satisfying V1(tk-)⩾0 and V1(tk)=0 or V2(tk-)=0 and V2(tk)≤0 is different from and further extends existing methods; the common term V0(t)=xT(t)Px(t) has a more relaxed condition than the positive definiteness generally required in the literature. Based on this method, a series of stability criteria in terms of linear matrix inequalities (LMIs) are obtained and demonstrated to be theoretically less conservative than some recent results. Finally, two numerical examples are provided to show the effectiveness and advantages of our results compared with those of previous studies.

Improved Discontinuous Lyapunov Functionals for Stability Analysis of Sampled-Data Systems

This paper is devoted to proposing improved discontinuous Lyapunov functionals for the stability analysis of sampled-data systems. Specifically, by relaxing the existing looped-functional, this paper makes the first attempt to integrate $\frac {1}{t-t_{k}}\int _{t_{k}}^{t} x(s)ds$ and $\frac {2}{(t\!-\!t_{k})^{2}} \int _{t_{k}}^{t}\int _{t_{k}}^{s} x(\tau)d\tau ds$ into the quadratic terms of the discontinuous Lyapunov functionals, which offer the possibility to obtain less conservative stability criteria formulated in terms of linear matrix inequalities (LMIs) than other previous studies. Finally, the effectiveness of the proposed Lyapunov functionals and corresponding approach are demonstrated by several numerical examples.

Improved results on H∞ stability analysis of sampled-data systems via looped-functionals and zero equalities

This paper revisits the H∞ stability analysis for sampled-data system by employing Lyapunov functional and loop functionals. Through existing methods such as the looped-functional approach and zero equality in sampled-data systems, an improved result is derived in terms of linear matrix inequalities. Based on the proposed result, by utilizing the relation of state vectors belonging to augmented vectors, less conservative criteria are introduced. The main objective of this study is to increase a maximum sampling value in both periodic and aperiodic sampling cases according to minimum sampling value. Using the proposed results, H∞ stability for sampled-data systems with disturbance is studied. Lastly, some numerical examples are utilized to confirm the superiority and effectiveness of the results by comparing them with the existing ones.

Improved stability criteria for sampled-data systems using modified free weighting matrix

This paper investigates the stability analysis of sampled-data systems in the looped-functional framework. A modified free-weighting matrix inequality with quadratic-type is proposed to reduce conservatism of the integral term. Based on new looped-functional, improved conditions are derived in terms of linear matrix inequalities (LMIs) by utilizing the proposed integral inequality. Numerical examples show the superiority of the proposed condition through comparisons with the most recent results.

Stability Analysis of Sampled-Data Systems via Free-Matrix-Based Time-Dependent Discontinuous Lyapunov Approach

In this paper, a new time-dependent discontinuous Lyapunov functional, namely, free-matrix-based time-dependent discontinuous (FMBTDD) Lyapunov functional is introduced for stability analysis of sampled-data systems. First, a modified free-matrix-based integral inequality (MFMBII) is derived based on the existing free-matrix-based integral inequality [1] and it is applied to develop a stability criterion for sampled-data systems. And then, inspired by MFMBII, FMBTDD term is established that leads to efficient stability conditions. Four numerical examples are given to demonstrate the effectiveness of the proposed methods.

A new looped-functional for stability analysis of sampled-data systems

In this paper, a new two-sided looped-functional is introduced for stability analysis of sampled-data systems. The functional fully utilizes the information on both the intervals x(t) to x(tk) and x(t) to x(tk+1). Based on the two-sided functional, an improved stability condition is derived in the form of linear matrix inequality (LMI). Numerical examples show that the result computed by the presented condition approximates nearly the theoretical bound (bound obtained by eigenvalue analysis) and outperforms substantially others in the existing literature.

Stability analysis of sampled-data systems via open-/closed-loop characteristic polynomials contraposition

ABSTRACTIn this paper, a class of sampled-data systems with double feedbacks are scrutinized for internal/external stability through the open-/closed-loop characteristic polynomials contraposition. A sampled-data system with double feedbacks consists of a continuous-time plant and two feedback control loops: a discrete-time one for digital control objectives with sampler and zero-order hold synchronized, and a continuous-time one for analog control performances. Contraposition stability criteria are worked out by exploiting what we call the contraposition return difference relationship in between the open- and closed-loop characteristic polynomials defined via the lifted model. The criteria for asymptotical stability are necessary and sufficient, independent of open-loop poles distribution of the lifted model and locus orientation specification. Moreover, the criteria for Lp-stability are sufficient, while retaining the technical merits of the criteria for asymptotical stability. All criteria present stability conditions in the standard discrete-time sense that are implementable either graphically with loci plotting, or numerically without loci plotting. Examples are included to illustrate the results.

Stability of aperiodic sampled-data systems — An augmented looped Lyapunov functional with wirtinger inequality approach

This paper is concerned with stability analysis of sampled-data systems with non-uniform sampling patterns. The stability problem is solved based on an augmented looped Lyapunov functional with Wirtinger inequality approach. The effectiveness of this approach depends on proper selecting looped Lyapunov functional parts and less conservative inequality techniques. The method developed here is an extension of previous results by using novel augmented functional and Wirtinger inequality. Numerical examples are given to illustrate the result.

Stability analysis of sampled-data systems considering time delays and its application to electric power markets

This paper investigates the stability of linear control systems with aperiodic sampled data and communication delays. A systematic analysis method is presented and then it is applied to an electric power market. Firstly, the sampled-data system is transformed into a system with a special time-varying delay via the input delay method. Secondly, a less conservative stability criterion is derived based on Lyapunov theory. Several augmented terms and an extra integral term are introduced during the constructing of candidate Lyapunov–Krasovskii functional (LKF); and an improved free-weighting matrix approach is used to handle with the LKF itself and its derivative for obtaining the relaxed conditions ensuring the positive and decreasing requirements of the LKF. The benefit of those treatments on the conservativeness-reducing is analyzed and verified based on a simple numerical example. Finally, the application of the proposed method to a simplified electric power market is investigated, including modeling the system with market clearing time and communication delay, and determining the stability region. The application also shows the practical significance of the reducing of the conservativeness.

Stability Analysis of Sampled-Data Systems Using Sum of Squares

This technical brief proposes a new approach to stability analysis of linear systems with sampled-data inputs or channels. The method, based on a variation of the discrete-time Lyapunov approach, provides stability conditions using functional variables subject to convex constraints. These stability conditions can be solved using the sum of squares methodology with little or no conservatism in both the case of synchronous and asynchronous sampling. Numerical examples are included to show convergence.

Stability Analysis for Sampled-Data Systems with Input Delay

This article proposes a novel approach to assess the stability of linear systems with delayed and sampled-data inputs. It proposes an extension of existing results on the stability of sampled-data systems to the case where a delay is introduced in the control loop. The method is based on a continuous-time modelling of the systems together with the discrete-time Lyapunov theorem, which provides easy tractable sufficient conditions for asymptotic stability. Those conditions cope the problem of stability under asynchronous samplings and time-varying delays. The period and delay-dependent conditions are expressed using computable linear matrix inequalities. Several examples show the efficiency of the stability criteria.

Stability Analysis of Sampled-Data Systems with Static Sector Nonlinearities via Causal Linear Periodically Time-Varying Scaling

This paper is concerned with causal linear periodically time-varying (LPTV) scaling for stability analysis. The effectiveness of such a scaling technique was first shown for the robust stability an...

Discontinuous Lyapunov Functional for Linear Systems with Sawtooth Delays*

Abstract Exponential stability of linear systems with time-varying piecewise-continuous delays is studied. It is assumed that the delay function has a form of a sawtooth with a constant delay derivative ≠ 0. In the recent paper (Fridman, 2009) piecewise-continuous (in time) Lyapunov-Krasovskii Functionals (LKFs) have been suggested for the stability analysis of sampled-data systems (with ≠ = 1) in the framework of input delay approach. Differently from the existing time-independent LKFs for systems with time-varying delays, the discontinuous ones can guarantee the stability under the sampling which may be greater than the analytical upper bound on the constant delay that preserves the stability. The objective of the present paper is to extend the piecewise-continuous LKF method to systems with a general sawtooth delay. The discontinuous terms of LKFs improve the results for all values of, though the most essential improvement corresponds to = 1.

Positive-realness analysis of sampled-data systems and its applications

In this paper, we first study strong positive-realness of sampled-data systems and introduce a measure called positive-realness gap index. We show that this index can be computed efficiently with a bisection method, and provide state space formulas for its computation. The importance of this index lies in that it is useful for robust stability analysis of sampled-data systems. An iterative procedure for computing an exact robust stability margin is given and illustrated through a numerical example.