Tsypkin Criterion Research Articles

Круговой критерий и критерий Цыпкина для систем с несколькими нелинейностями без использования $S$-процедуры

Круговой критерий и критерий Цыпкина для систем с несколькими нелинейностями без использования $S$-процедуры

The circle criterion and Tsypkin's criterion for systems with several nonlinearities without the use of the $S$-procedure

The circle criterion and Tsypkin's criterion for systems with several nonlinearities without the use of the $S$-procedure

Tsypkin and Jury–Lee Criteria for Synchronization and Stability of Discrete-Time Multiagent Systems

Dynamics of complex networked systems consisting of a large number of interconnected components (agents) has attracted considerable attention of researchers in control theory and dynamical systems. One of the most interesting questions in this area concerns the emergence of a coherent joint behavior of nodes caused by couplings between them, such as, e.g., their synchronization. We study a network where each node is a linear dynamical system of arbitrary order in discrete time, and couplings between the nodes are nonlinear and unknown but satisfy the sector inequality with known bounds. We obtain frequency-domain criteria of synchronization and stability of the network, which are generalizations of the Tsypkin criterion and the Jury–Lee criterion for discrete-time Lurie systems.

A direct proof of the discrete-time multivariate circle and Tsypkin criteria

This technical note presents a new proof of the circle criterion for multivariate, discrete-time systems with time-varying feedback nonlinearities. A new proof for the multivariate Tsypkin criterion for time-invariant monotonic feedback nonlinearities is derived as well. Both integrator- and non-integrator systems are considered. The proofs are direct in the sense that they do not resort to any existing result in systems theory, such as Lyapunov theory, passivity theory or the small-gain theorem. Instead, the proofs refer to the asymptotic properties of block-Toeplitz matrices. One major advantage of the new proof is that it elegantly handles integrator systems without resorting to loop transformation/pole shifting techniques. Additionally, less conservative stability bounds are derived by making stronger assumptions on the sector bound conditions on the feedback nonlinearities. In particular, it is exemplified how this technique relaxes stability conditions of (i) a model predictive control (MPC) rule and (ii) an integrator system.

A novel extended Tsypkin criterion for discrete-time descriptor systems with ferromagnetic hysteresis nonlinear physical phenomenon

This paper concerns the absolute stability problem of discrete-time descriptor systems with feedback connected ferromagnetic hysteresis nonlinearities. The ferromagnetic hysteresis model satisfies the passivity conditions of hysteresis operator, that is the input-output relation of the transformed operator is passive. The bound condition of the solution of the ferromagnetic hysteresis model is given. Through the framework of loop transformation, an augmented discrete-time descriptor system model is established for the stability analysis. A new extended Tsypkin criterion for the absolute stability of discrete-time descriptor systems with hysteresis is presented based on the linear matrix inequalities technique. A numerical example is given to illustrate the effectiveness of the extended criterion.

Absolute stability of uncertain discrete Lur’e systems and maximum admissible perturbed bounds

The robust absolute stability problem for norm uncertain and structured uncertain discrete Lur’e systems is considered in this paper by using Lyapunov function method. A sufficient condition of absolute stability for discrete Lur’e systems is established in terms of linear matrix inequalities (LMIs) or the equivalent frequency-domain condition. We compare the result with the Popov-like criterion (Tsypkin criterion) and extended strictly positive real (ESPR) lemma. Furthermore, sufficient conditions on absolute stability for discrete Lur’e systems with norm and structured uncertainties are also presented based on linear matrix inequalities. Estimates of the maximum bounds of all admissible perturbations are given by generalized eigenvalue problems . Finally, several numerical examples are worked out to illustrate the efficiency of the main results.

Reducing Engine Idle Speed Deviations Using the Internal Model Principle

Presented in this paper is a multivariable linear feedback controller design methodology for idle speed control of spark-ignition engines. The engine is modeled as a multi-input, single-output system. The proposed feedback control system employs both throttle and ignition timing to control engine speed and engine roughness. Throttle is used to attenuate low frequency components of the speed error and reject mean speed errors. Spark advance is used to reduce cylinder-to-cylinder differences in torque production by limiting high frequency speed deviations. The algorithm is executed in the crank-angle domain, and the internal model principle serves as the basis for cylinder torque balancing. The nonlinear relationship between ignition timing and torque production is explicitly incorporated into the design process using a sector bound. A loop shaping approach is proposed to design the feedback controller, and absolute stability of the nonlinear closed-loop system is guaranteed through the Tsypkin Criterion. Experimental results from implementation on a Ford 4.6L V-8 engine are provided.

A brief look at the Tsypkin criterion: from analysis to design

AbstractTsypkin was among the first to recognize the importance of the Popov criterion and to extend it to discrete‐time systems in the form now known as the Tsypkin criterion. This paper briefly reviews the evolution of the Tsypkin criterion from its original use in absolute stability analysis to its current application as a design tool. Copyright © 2001 John Wiley & Sons, Ltd.

A Revisited Tsypkin Criterion for Discrete-Time Nonlinear Lur’e Systems with Monotonic Sector-Restrictions

This paper revisits a well-known Tsypkin criterion for stability analysis of discrete-time nonlinear Lur’e systems. When nonlinearities are monotonic and sector restricted by [0, Δ ̄ ] , where Δ ̄ is positive definite, it is shown by Kapila and Haddad that the system is absolutely stable if a function G 0(z)= Δ ̄ -1+{I+(1−z -1) K +}G(z) is strictly positive real, where K + is nonnegative diagonal and G( z) represents a transfer function of the linear part of the Lur’e system with invertible or identically zero G(0). This paper extends this criterion when Δ ̄ is positive diagonal, by choosing a new Lyapunov function to obtain an LMI criterion. From a frequency-domain interpretation of this LMI criterion, another sufficient criterion is generated which establishes that the system is absolutely stable if a function G 0(z)= Δ ̄ -1+{I+(1−z -1) K ++(1−z) K -}G(z) is strictly positive real, where K + and K - are nonnegative diagonal and orthogonal to each other.

A multivariable extension of the Tsypkin criterion using a Lyapunov-function approach

For analyzing the stability of discrete-time systems containing a feedback nonlinearity, the Tsypkin criterion is the closest analog to the Popov criterion which is used for analyzing such systems in continuous time. Traditionally, the proof of this criterion is based upon input-output properties and function analytic methods. In this paper we extend the Tsypkin criterion to multivariable systems containing an arbitrary number of monotonic sector-bounded memoryless time-invariant nonlinearities, along with providing a Lyapunov function proof for this classical result.

Criteria for Determining if a High-Order Digital Filter Using Saturation Arithmetic Is Free of Overflow Oscillations

Recently we found that, among recursive digital filters using saturation arithmetic to contend with overflow, a fundamental difference exists between second and higher order filters: the latter may sustain large-amplitude overflow oscillations. In this paper we have derived a new criterion expressly designed for determining when a given high-order recursive system using saturation arithmetic is free of overflow oscillations. The new criterion, which is easy to use, follows from this result: we associate with the given system two trigonometric polynomials in θ of degree equal to the order of the given system; if any linear combination of the polynomials with nonnegative weights is positive for all θ in [0, π] then the system is free of all nontrivial periodic oscillations. We prove that the new criterion subsumes certain well-known criteria, such as Tsypkin's criterion, from the literature on nonlinear systems. To illustrate, three classes of special systems are investigated, and in each case the new criterion gives substantial improvements. Finally, the new test is applied in the synthesis of high order sections for a realistic eighth-order system.

Root-locus method for the Tsypkin and Jury and Lee criteria†

A method is developed for application of the Tsypkin criterion and the Jury and Lee stability criteria for certain classes of non-linear sampled data systems The method, which uses root-locus techniques, is based on the root-locus interpretation of the Popov criterion for continuous data non-linear systems. This method was developed by Ramapriyan et al. (1966).