Parameter-dependent Nonlinear Systems Research Articles

An Exponential Stabilization Analysis for Switched Interval Type-2 Fuzzy Sampled-Data Control Systems

This work is focused on designing the admissible edge-dependent average dwell time (AE-DADT) based fuzzy sampled-data control problem for a class of switched nonlinear systems. The upper and lower grade of membership functions is used to characterize uncertain parameter-dependent nonlinear systems in the interval type-2 fuzzy (IT2F) model. Unlike from the existing literature, a new fuzzy dependent refined looped-Lyapunov functional (LF) is introduced which involves the new exponent terms and state information such as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\frac{e^{2\gamma _{v}(\varrho -\varrho _{\mathscr {k}})}-1}{2\gamma _{v}}, \ \frac{e^{2\gamma _{v}(\varrho _{\mathscr {k}+1}-\varrho)}-1}{2\gamma _{v}}, \ \mathscr {z}(\varrho _{\mathscr {k}})$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathscr {z}(\varrho _{\mathscr {k}+1})$</tex-math></inline-formula> with tuning parameters and few slack variables. The AE-DADT switching property has also been taken into account for nonlinear systems, and it contributes to getting more information about previous modes. Based on the proposed new refined looped-LFs, new sufficient conditions are derived in terms of linear matrix inequalities (LMI), which ensure the global uniform exponential stability (GUES) of the nominated switched IT2F systems. The feasibility of the obtained results is verified by the numerical example, and the tunnel diode circuit system model and Lorenz chaotic system are established to illustrate the feasibility and efficacy of the theoretical findings.

Analysis of shape optimization problems for unsteady fluid-structure interaction

Shape optimization via the method of mappings is investigated for unsteady fluid-structure interaction (FSI) problems that couple the Navier–Stokes equations and the Lamé system. Building on recent existence and regularity theory we prove Fréchet differentiability results for the state with respect to domain variations. These results form an analytical foundation for optimization und inverse problems governed by FSI systems. Our analysis develops a general framework for deriving local-in-time continuity and differentiability results for parameter dependent nonlinear systems of partial differential equations. The main part of the paper is devoted to conducting this analysis for the FSI problem, transformed to a shape reference domain. The underlying shape transformation—actually we work with the corresponding shape displacement instead—represents the shape and the main result proves the Fréchet differentiability of the solution of the FSI system with respect to the shape transformation.

A C0 interior penalty discontinuous Galerkin method for fourth‐order total variation flow. I: Derivation of the method and numerical results

We consider the numerical solution of a fourth‐order total variation flow problem representing surface relaxation below the roughening temperature. Based on a regularization and scaling of the nonlinear fourth‐order parabolic equation, we perform an implicit discretization in time and a C0 Interior Penalty Discontinuous Galerkin (C0IPDG) discretization in space. The C0IPDG approximation can be derived from a mixed formulation involving numerical flux functions where an appropriate choice of the flux functions allows to eliminate the discrete dual variable. The fully discrete problem can be interpreted as a parameter dependent nonlinear system with the discrete time as a parameter. It is solved by a predictor corrector continuation strategy featuring an adaptive choice of the time step sizes. A documentation of numerical results is provided illustrating the performance of the C0IPDG method and the predictor corrector continuation strategy. The existence and uniqueness of a solution of the C0IPDG method will be shown in the second part of this paper.

Output Feedback Control of a Dual‐Excenter Vibration Actuator via qLPV Model and TP Model Transformation

AbstractVibration actuators are widely used in handheld devices to provide vibrotactile feedback or silent notification to the users. In most cases, miniature DC motors with eccentric rotors or the so‐called coin‐type shaftless vibration motors are utilized. The common disadvantage of the single rotor designs is that the frequency and the intensity of the generated vibration cannot be adjusted separately. The construction composed of two independently driven coaxial eccentric rotors – which makes a strongly coupled nonlinear system – allows for the separate control of the frequency and amplitude by the adjustment of the angular speed and the total eccentricity. This paper presents a complete control design approach based on quasi‐linear parameter varying (qLPV) modelling and linear matrix inequality (LMI)‐based synthesis utilizing the tensor product (TP) model transformation to determine the convex polytopic representation of the parameter dependent nonlinear system. The design approach is demonstrated via a concrete numerical example using the parameters of a real dual‐excenter prototype device. The control performance is validated through numerical simulations. This case study goes through a complete nonlinear control problem from the modelling phase to the implementation‐ready controller, while drawing generalizable conclusions for qLPV modelling.

Uniform modeling of parameter dependent nonlinear systems

This paper addresses the problem of approximating parameter dependent nonlinear systems in a unified framework. This modeling has been presented for the first time in the form of parameter dependent piecewise affine systems. In this model, the matrices and vectors defining piecewise affine systems are affine functions of parameters. Modeling of the system is done based on distinct spaces of state and parameter, and the operating regions are partitioned into the sections that we call ‘multiplied simplices’. It is proven that this method of partitioning leads to less complexity of the approximated model compared with the few existing methods for modeling of parameter dependent nonlinear systems. It is also proven that the approximation is continuous for continuous functions and can be arbitrarily close to the original one. Next, the approximation error is calculated for a special class of parameter dependent nonlinear systems. For this class of systems, by solving an optimization problem, the operating regions can be partitioned into the minimum number of hyper-rectangles such that the modeling error does not exceed a specified value. This modeling method can be the first step towards analyzing the parameter dependent nonlinear systems with a uniform method.

Stability analysis of genetic regulatory network with time delays and parameter uncertainties

The effect of parameter uncertainties on stability of delayed genetic regulatory networks is investigated. With the presence of parameter uncertainties, the delayed coupled genetic regulatory network is a parameter-dependent non-linear system, in which not only the dynamics are indefinite but also its equilibrium point is unknown. The authors first derive a sufficient condition to guarantee robust stability of uncertain genetic regulatory networks by introducing convergence region. Then, the estimation of convergence region bound is obtained by using Lyapunov stability theory and linear matrix inequality. It is worth noting that these results are derived in the case that the equilibrium point of parameter-dependent non-linear system is unknown. Finally, two examples are given to illustrate the effectiveness of the proposed results.

A Bisection-Like Algorithm for Branch Switching at a Simple Branch Point

At a Simple Branch Point x 0 of a parameter-dependent nonlinear system, two distinct solutions branches S 1 and S 2 intersect. A popular method of branch switching from S 1 to S 2 at x 0 is to use an orthogonal vector to the tangent vector to S 1 at x 0 as an initial guess for the tangent vector to S 2 at x 0. We describe an improved version of this method using a bisection-like procedure, which is especially useful for large problems.

Stability analysis for point delay fractional description models via linear matrix inequalities

This paper is devoted to linear parameter systems under linear fractional representations (LFR) of parameter-dependent nonlinear systems with real–rational nonlinearities and point-delayed dynamics. The robust global asymptotic stability of the system either independent of or dependent on the delay sizes is investigated. The associate matrix inequalities are related to the time-derivatives of appropriate Lyapunov functions at all the vertices of the polytope which contains the parameterized uncertainties.

Stability Analysis and Design for Polynomial Nonlinear Systems Using SOS with Application to Aircraft Flight Control

In this paper, aircraft with control surface impairment faults is modelled as a set of affine parameter-dependent nonlinear systems with polynomial vector fields. Based on the Lyapunov stability theorem, sufficient conditions to test the stability of the closed-loop system are presented. The synthesis problem of stabilizing feedback controllers to enlarge the region of attraction is converted to an optimization problem based on sum-of-squares polynomial. Design and simulation results for the longitudinal model of an F-8 aircraft are presented to illustrate the effectiveness of proposed approach.

Path-following primal-dual interior-point methods for shape optimization of stationary flow problems

We consider shape optimization of Stokes flow in channels where the objective is to design the lateral walls of the channel in such a way that a desired velocity profile is achieved. This amounts to the solution of a PDE constrained optimization problem with the state equation given by the Stokes system and the design variables being the control points of a Bezier curve representation of the lateral walls subject to bilateral constraints. Using a finite element discretization of the problem by Taylor-Hood elements, the shape optimization problem is solved numerically by a path-following primal-dual interior-point method applied to the parameter dependent nonlinear system representing the optimality conditions. The method is an all-at-once approach featuring an adaptive choice of the continuation parameter, inexact Newton solves by means of right-transforming iterations, and a monotonicity test for convergence monitoring. The performance of the adaptive continuation process is illustrated by several numerical examples.

Robust stability analysis and dynamic gain-scheduled controller design for point time-delay systems with parametrical uncertainties

This paper discusses linear fractional representations (LFR) of parameter-dependent nonlinear systems with real-rational nonlinearities and point-delayed dynamics. Sufficient conditions for robust global asymptotic stability independent of the delays and the existence of a robust stabilizing gain-scheduled dynamic controller are investigated via linear matrix inequalities. Such inequalities are obtained from the values of the time-derivatives of appropriate Lyapunov functions at all the vertices of the polytope which contains the parametrized uncertainties. The synthesized stabilizing controller consists of an interpolation being performed with the stabilizing controllers at the set of vertices of a certain polytope where the nonlinear-rational parametrization belongs to. Some extensions are also given concerning robust global asymptotic stability dependent of the delays. Numerical examples corroborate the usefulness of the proposed formalism and its applicability to practical related problems.

Global analysis of limit cycles in networks of oscillators

This paper is concerned with the global stability of limit cycle oscillations for a particular class of systems and networks. In previous work, we defined a class of parameter-dependent nonlinear systems exhibiting an almost globally asymptotically stable limit cycle. The results were proven for values of the parameter in the vicinity of a bifurcation value. In the present paper we restrict ourselves to a piecewise linear version of this class of systems and adapt numerical tools recently proposed in the literature to prove global stability of the limit cycle for a fixed value of the parameter above the bifurcation value. Furthermore, we show how the global stability results for one isolated oscillator are useful to prove the existence of a globally synchrone oscillation in particular networks of identical oscillators.

Improved stability analysis and gain-scheduled controller synthesis for parameter-dependent systems

We present new algorithms for robust stability analysis and gain-scheduled controller synthesis for linear systems affected by time-varying parametric uncertainties. These new techniques can also be applied to parameter-dependent nonlinear systems with real-rational nonlinearities. Sufficient conditions for robust stability, as well as conditions for the existence of a robustly stabilizing gain-scheduled controller, are given in terms of a finite number of linear matrix inequalities (LMIs); explicit formulas for constructing robustly stabilizing gain-scheduled controllers are given in terms of the feasible set of these LMIs. The improvement offered by our approach over existing methods for stability analysis and gain-scheduled controller synthesis for parameter-dependent linear systems are analyzed in theory. Numerical examples demonstrate that our approach can offer significant improvement in practice.

An estimate for the small parameter in the asymptotic analysis of non-linear systems by the method of averaging

The Method of Averaging is an asymptotic method that can be used to obtain approximate solutions for many parameter dependent non-linear systems. The resulting approximate solutions are as accurate as desired provided the system parameter is sufficiently small. In this paper, a general technique is developed to obtain a relationship between the magnitude of the small parameter and the permissible error in the approximate solution. The technique is then demonstrated by its application to two examples.