Robustness analysis and synthesis for nonlinear uncertain systems

The stability and performance robustness analysis for a class of uncertain nonlinear systems with bounded structured uncertainties are characterized in terms of various types of nonlinear matrix inequalities (NLMIs). As in the linear case, scalings or multipliers are used to find Lyapunov functions that give sufficient conditions, and the resulting NLMIs yield convex feasibility problem. For these problems, robustness analysis is essentially no harder than stability analysis of the system with no uncertainty. Sufficient conditions for the solvability of related robust synthesis problems are developed in terms of NLMIs as well. >

Stability and state-feedback stabilization of linear systems with uncertain coefficients and uncertain time-varying delays are considered. The system under consideration may be unstable without delay, but it becomes asymptotically stable for positive values of the delay. A new descriptor discretized Lyapunov-Krasovskii functional (LKF) method is introduced, which combines the application of the complete LKF and the discretization method of K. Gu with the descriptor model transformation. For the first time, the new method allows to apply the discretized LKF method to synthesis problems. Moreover, the analysis of systems with polytopic time-invariant uncertainties is less restrictive by the new discretized method. Sufficient conditions for robust stability and stabilization of uncertain neutral type systems are derived in terms of linear matrix inequalities (LMIs) via input-output approach to stability. Numerical examples illustrate the efficiency of the new method.

Analysis of Nonlinear Systems with Stochastic Input and Stochastic Parameters—II

This paper presents a design scheme of a minimal order observer-based guaranteed cost controller for uncertain neutral systems, in which some state variables can not be measured. The uncertainties are assumed to be norm-bounded. The initial state is assumed unknown but their mean and covariance are assumed known. A sufficient condition for robust stability analysis and robust stabilization are derived via linear matrix inequalities (LMIs). To show the advantage of the proposed method, a numerical example is given.

Theory of nonlinear systems

The identification of block-oriented systems represented by interconnections of linear dynamic systems and non-memory nonlinear elements have been of great importance for control and for analysis of biological nonlinear systems. In this study, two new algorithms to obtain structural models of nonlinear systems using numerically-given (experimentally obtained) Volterra kernels are proposed. One is to determine the parallel factorable base model; a sum of subsets, each of which has a parallel structure of linear dynamic systems whose outputs are multiplied. Another is called the Sm model; linear dynamic systems and nonlinear power elements in cascade. The results of numerical experiments showed that both methods could be practically useful for identifications and analysis of nonlinear systems.

This paper focuses on the stability analysis of linear fractional-order systems with fractional-order 0<α<2, in the presence of time-varying uncertainty. To obtain a robust stability condition, we first derive a new upper bound for the norm of Mittag-Leffler function associated with the nominal fractional-order system matrix. Then, by finding an upper bound for the norm of the uncertain fractional-order system solution, a sufficient non-Lyapunov robust stability condition is proposed. Unlike the previous methods for robust stability analysis of uncertain fractional-order systems, the proposed stability condition is applicable to systems with time-varying uncertainty. Moreover, the proposed condition depends on the fractional-order of the system and the upper bound of the uncertainty matrix norm. Finally, the offered stability criteria are examined on numerical uncertain linear fractional-order systems with 0<α<1 and 1<α<2 to verify the applicability of the proposed condition. Furthermore, the stability of an uncertain fractional-order Sallen–Key filter is checked via the offered condition.

In this note, a new result that extends the scaled small gain theorem is presented. As is well known, the scaled small gain theorem gives necessary and sufficient conditions for robust stability of a nominal linear time-invariant system in feedback with structured operators of norm less than or equal to unity. We propose alternative linear matrix inequality conditions that give necessary and sufficient conditions for robust stability against the class of structured unitary operators (invertible operators of exactly unit norm). It is shown that this result, besides being a less conservative version of the scaled small gain theorem, has connections to several recent results on the control of spatially interconnected systems and serves to unify and quantify the conservatism of those results.

Analysis of fast-sampled non-linear systems: generalised frequency response functions for δ-operator models

Enhanced robust stability analysis of large hydraulic control systems via a bifurcation-based procedure

Observability and observable degree, which are derived from modern control theory and relative to control and estimation performances, are both important concepts for state estimation. For linear time-invariant systems, judgement of observability and evaluation of observable degree are irrelevant to filtering process and parameters. Thereby, it has limited application ability. Different from linear systems, observability and observable degree becomes more complex and are seldom studied deeply and integrally for nonlinear systems. Due to the complexity of nonlinear filtering, the observable degree depends strongly on filtering process, filter class and associated parameters. Thereby, we study the observable degree analysis of nonlinear estimation systems by using unscented information filter (UIF). A way to evaluate observable degree using singular value decomposition (SVD) of observability matrix is presented based on the UIF and the influencing factors are deeply analyzed. Moreover, differences of observable degree analysis are distinctly discussed between linear and nonlinear systems. The results reveal that evaluation of the observable degree for nonlinear systems can be affected by filtering estimate at last time, initial estimate and filter's parameters such as covariances of process and measurement noises. Some simulations are demonstrated to validate the results.

Stability plays an important role in both control theory and system identification. Furthermore, the stability issue is of crucial importance in relation to training algorithms adjusting the parameters of neural networks. If the predictor is unstable for certain choices of neural model parameters, serious numerical problems can occur during training. Stability criteria should be universal, applicable to as broad a class of systems as possible and at the same time computationally efficient. The majority of well-known approaches are based on Lyapunov’s method [163, 77, 164, 165, 166, 167]. Fang and Kincaid applied the matrix measure technique to study global exponential stability of asymmetrical Hopfield type networks [168]. Jin et. al [169] derived sufficient conditions for absolute stability of a general class of discrete-time recurrent networks by using Ostrowski’s theorem. Recently, global asymptotic as well exponential stability conditions for discrete-time recurrent networks with globally Lipschitz continuous and monotone nondecreasing activation functions were introduced by Hu and Wang [170]. The existence and uniqueness of an equilibrium were given as a matrix determinant problem. Unfortunately, most of the existing results do not consider the stabilization of the network during training. They allow checking the stability of the neural model only after training it. Literature about the stabilization of neural network during training is rather scarce. Jin and Gupta proposed two training methods for a discrete-time dynamic network: multiplier and constrained learning rate algorithms. Both algorithms utilized stability conditions derived by using Lyapunov’s first method and Gersgorin’s theorem. In turn, Suykens et al. [172] derived stability conditions for recurrent multi-layer network using linearisation, robustness analysis of linear systems under non-linear perturbations and matrix inequalities. The elaborated conditions have been used to constrain the dynamic backpropagation algorithm. These solutions, however, are devoted to globally recurrent networks.

Robust analysis and control for a class of uncertain nonlinear discrete-time systems

This chapter considers different techniques for robustness analysis of uncertain systems modeled by linear fractional representations, based on the use of parameter-dependent Lyapunov functions and multipliers. Several sufficient conditions for robust stability, relying on the choice of simplified structures for the Lyapunov function and the multiplier matrices, are proposed. These conditions provide a useful tool for trading off conservatism and computational burden, which is a key issue when addressing robustness analysis of high dimensional uncertain systems arising from clearance problems.

Theory of nonlinear control