Stability Analysis Of Feedback Systems Research Articles

Stability Analysis of Feedback Systems with ReLU Nonlinearities via Semialgebraic Set Representation

This paper is concerned with the stability/instability analysis problem of feedback systems with rectified linear unit (ReLU) nonlinearities. Such feedback systems arise when we model dynamical (recurrent) neural networks (NNs) and NN-driven control systems where all the activation functions of NNs are ReLUs. In this study, we focus on the semialgebraic set representation characterizing the input-output properties of ReLUs. This allows us to employ a novel copositive multiplier in the framework of the integral quadratic constraint and, thus, to derive a new linear matrix inequality (LMI) condition for the stability analysis of the feedback systems. However, the infeasibility of this LMI does not allow us to obtain any conclusion on the system's stability due to its conservativeness. This motivates us to consider its dual LMI. By investigating the structure of the dual solution, we derive a rank condition on the dual variable certificating that the system at hand is unstable. In addition, we construct a hierarchy of dual LMIs allowing for improved instability detection. We illustrate the effectiveness of the proposed approach by several numerical examples.

Strengthened Circle and Popov Criteria for the Stability Analysis of Feedback Systems With ReLU Neural Networks

This paper considers the stability analysis of a Lurie system with a static repeated ReLU (rectified linear unit) nonlinearity. Properties of the ReLU function are leveraged to derive new tailored quadratic constraints (QCs) which are satisfied by the repeated ReLU. These QCs are used to strengthen the Circle and Popov Criteria for this specialised Lurie system. It is shown that the criteria can be cast as a set of linear matrix inequalities (LMIs) with less restrictive conditions on the matrix variables. Many systems involving a neural network (NN) with ReLU activations are important instances of this specialised Lurie system; for example, a continuous time recurrent neural network (RNN) or the interconnection of a linear system with a feedforward NN. Numerical examples show the strengthened criteria strike an appealing balance between reduced conservatism and complexity, compared to existing criteria.

Errata on “Strengthened Circle and Popov Criteria for the Stability Analysis of Feedback Systems With ReLU Neural Networks”

Errata on “Strengthened Circle and Popov Criteria for the Stability Analysis of Feedback Systems With ReLU Neural Networks”

Formula omitted] induced norm analysis of discrete-time LTI systems for nonnegative input signals and its application to stability analysis of recurrent neural networks

In this paper, we focus on the “positive” l2 induced norm of discrete-time linear time-invariant systems where the input signals are restricted to be nonnegative. To cope with the nonnegativity of the input signals, we employ copositive programming as the mathematical tool for the analysis. Then, by applying an inner approximation to the copositive cone, we derive numerically tractable semidefinite programming problems for the upper and lower bound computation of the “positive” l2 induced norm. This norm is typically useful for the stability analysis of feedback systems constructed from an LTI system and nonlinearities where the nonlinear elements provide only nonnegative signals. As a concrete example, we illustrate the usefulness of the “positive” l2 induced norm for the stability analysis of recurrent neural networks with activation functions being rectified linear units.

Full‐block multipliers for repeated, slope‐restricted scalar nonlinearities

SummaryThis paper provides a comprehensive treatment of full‐block multipliers within the integral quadratic constraints framework for stability analysis of feedback systems containing repeated, slope‐restricted scalar nonlinearities. We develop a novel stability result that offers more flexibility in its application because it allows for the inclusion of general Popov and Yakubovich criteria in combination with the well‐established Circle and Zames‐Falb stability tests within integral quadratic constraint theory. A particular focus lies on the formulation of stability criteria in terms of full‐block multipliers, some of which are new, and thus typically involve less conservatism than current methods. Furthermore, a new asymptotically exact parametrization of full‐block Zames‐Falb multipliers is given that allows to exploit the complete potential of this stability test. Copyright © 2017 John Wiley & Sons, Ltd.

Stability analysis of feedback systems with dead-zone nonlinearities by circle and Popov criteria

A method of global stability analysis is proposed for a feedback system with dead-zone nonlinearities. Using a global property that the output of a saturation function is bounded, the bound on the input to the saturation function is estimated using the L∞ norm of a linear subsystem. The feedback system can be treated as a feedback system with a narrower sector bound using this method, and a sharper global stability condition is obtained by applying the circle or Popov criterion to the system.

Fuzzy two-term controllers with multi-fuzzy sets: mathematical models and analysis

This paper first presents mathematical models for fuzzy two-term (PI/PD) controllers which employ N1 (³3) number of symmetric fuzzy sets for the input variable displacement, N2 (³3) number of symmetric fuzzy sets for the input variable velocity and N1 + N2 – 1 number of symmetric fuzzy sets for the output variable. These models are derived via triangular membership functions for fuzzification of the inputs and output variables, linear control rules, minimum/algebraic product triangular norm, different triangular conorms, different inference methods and centre of sums (COS) defuzzification method. Properties of such models are investigated. Using the well-known small-gain theorem bounded-input bounded-output (BIBO) stability analysis of feedback systems involving fuzzy PD controller as a subsystem is presented. Next, mathematical models with N1 and N2 number of asymmetric input fuzzy sets and N1 + N2 – 1 number of asymmetric output fuzzy sets (both triangular and trapezoidal) are also presented. Finally, s...

Positivity Preservation Properties of the Rantzer Multipliers

The Rantzer multipliers are known to preserve the positivity of certain aberrations of memoryless monotone positive nonlinearities. We show that if the nonlinearity input is constrained to be positive valued for all time instants, these multipliers are positivity preserving for a larger class of nonlinearities. As a result, it follows that the Rantzer multipliers are useful in reducing the conservatism inherent in the multiplier theoretic stability analysis of feedback systems featuring a larger class of nonlinearities than the one these multipliers were originally intended for, so long as the nonlinearity input is positive-valued for all time instants.

A note on the Laplace transform method for initial value problems

This note provides a tutorial treatment of the basic (unilateral) Laplace transform method for solving initial value problems for linear constant coefficient differential equations with terms that contain derivatives of the input. The input is allowed to contain jump discontinuities (steps) and so it would first appear that one needs the advanced machinery of generalized functions (distributions) to apply the Laplace transform method here. This is, however, not the case. It suffices to use the basic Laplace transform for ordinary functions, where the integral is interpreted as the most basic integral notion in calculus, i.e. as a Riemann integral. There is thus no need to specify any convention on the lower limit of integration as they will all give the same transform. Furthermore, it suffices to use the derivative rule in the basic form L[f′](s)=sL[f](s)−f(0) for smooth enough differentiable functions. Finally, we point out important limitations of the bilateral Laplace transform method in input–output stability analysis of feedback systems.

多項式計画による飽和系の局所安定解析

This paper considers the local stability analysis of feedback systems with saturation nonlinearities. First, considering the detailed structure of the saturation nonlinearity, we derive a local stability condition with the aid of S-Procedure of polynomial with higher degrees of freedom. Then, we obtain a Lyapunov function using Sum of Squares decomposition. Furthermore, it is shown that this method can be easily extended to the case where piecewise quadratic Lyapunov functions are adopted. Finally we demonstrate its effectiveness through numerical examples.

Degenerate bifurcations and catastrophe sets via frequency analysis

We present some bifurcation conditions using the well-known stability analysis of feedback systems. A general ordinary differential equation system is formulated in two parts: one that considers the linear part and the other that includes the memoryless nonlinear part, in a similar way as the describing function. The bifurcation conditions are obtained using the results of the generalized Nyquist stability criterion (GNSC) with some explicit formulae derived from some properties of the complex variable We analyse simultaneously both static and dynamic (Hopf) bifurcations and their degeneracies in a rich example, a continuous stirred-tank reactor (CSTR), in which two consecutive, irreversible, first-order reactions A→B→C occur

Generalized factorizability conditions for stability multipliers

Generalized conditions are given for the factorization of a linear convolution operator in a Banach algebra into a cascade of two operators, one causal and the other anticausal, which has application in the input-output stability analysis of feedback systems. These conditions are an improvement over the earlier ones presented by the authors.

Passivity of a class of non-monotone non-linearities†

Abstract Conditions for the passivity of certain classes of non-monotone time-varying non-linearities are presented which are useful in stability analysis of feedback systems.