Robust state estimation and fault diagnosis for uncertain hybrid nonlinear systems

Robust state estimation and fault diagnosis are challenging problems in the research of hybrid systems. In this paper, a novel robust hybrid observer is proposed for a class of uncertain hybrid nonlinear systems with unknown mode transition functions, model uncertainties and unknown disturbances. The observer consists of a mode observer for discrete mode estimation and a continuous observer for continuous state estimation. It is shown that the mode can be identified correctly and the continuous state estimation error is exponentially uniformly bounded. Robustness to unknown transition functions, model uncertainties and disturbances can be guaranteed by disturbance decoupling and selecting proper thresholds. The transition detectability and mode identifiability conditions are rigorously analyzed. Based on the robust hybrid observer, a robust fault diagnosis scheme is presented for faults modeled as discrete modes with unknown transition functions, and the analytical properties are investigated. Simulations of a hybrid three-tank system demonstrate that the proposed approach is effective.     

Robust state estimation and fault diagnosis for uncertain hybrid systems

Robust state estimation and fault diagnosis are challenging problems in the research into hybrid systems. In this paper a novel robust hybrid observer is proposed for a class of hybrid systems with unknown inputs and faults. Model uncertainties, disturbances and faults are represented as structured unknown inputs. The robust hybrid observer consists of a mode observer for mode identification and a continuous observer for continuous state estimation and mode transition detection. It is shown that the mode can be identified correctly and the continuous state estimation error is exponentially uniformly bounded. Robustness to model uncertainties and disturbances can be guaranteed for the hybrid observer by disturbance decoupling. Furthermore, the detectability and mode identifiability conditions are rigorously analyzed. On the basis of the robust hybrid observer, a robust fault detection and isolation scheme is presented also in the paper. Simulations of a hybrid four-tank system show the proposed approach is effective.     

Modeling and Analysis of Modal Switching in Networked Transport Systems

We consider networked transport systems defined on directed graphs: the dynamics on the edges correspond to solutions of transport equations with space dimension one. In addition to the graph setting, a major consideration is the introduction and propagation of discontinuities in the solutions when the system may discontinuously switch modes, naturally or as a hybrid control. This kind of switching has been extensively studied for ordinary differential equations, but not much so far for systems governed by partial differential equations. In particular, we give well-posedness results for switching as a control, both in finite horizon open loop operation and as feedback based on sensor measurements in the system.     

Balanced truncation for linear switched systems

We propose a model order reduction approach for balanced truncation of linear switched systems. Such systems switch among a finite number of linear subsystems or modes. We compute pairs of controllability and observability Gramians corresponding to each active discrete mode by solving systems of coupled Lyapunov equations. Depending on the type, each such Gramian corresponds to the energy associated to all possible switching scenarios that start or, respectively end, in a particular operational mode. In order to guarantee that hard to control and hard to observe states are simultaneously eliminated, we construct a transformed system, whose Gramians are equal and diagonal. Then, by truncation, directly construct reduced order models. One can show that these models preserve some properties of the original model, such as stability and that it is possible to obtain error bounds relating the observed output, the control input and the entries of the diagonal Gramians.     

Unsteady convective boundary layer flow of a viscous fluid at a vertical surface with variable fluid properties

In this paper we present numerical solutions to the unsteady convective boundary layer flow of a viscous fluid at a vertical stretching surface with variable transport properties and thermal radiation. Both assisting and opposing buoyant flow situations are considered. Using a similarity transformation, the governing time-dependent partial differential equations are first transformed into coupled, non-linear ordinary differential equations with variable coefficients. Numerical solutions to these equations subject to appropriate boundary conditions are obtained by a second order finite difference scheme known as the Keller-Box method. The numerical results thus obtained are analyzed for the effects of the pertinent parameters namely, the unsteady parameter, the free convection parameter, the suction/injection parameter, the Prandtl number, the thermal conductivity parameter and the thermal radiation parameter on the flow and heat transfer characteristics. It is worth mentioning that the momentum and thermal boundary layer thicknesses decrease with an increase in the unsteady parameter.     

On the bifurcation of large amplitude solutions for a system of wave and beam equations

We consider the existence of solutions for linearly coupled system of wave and beam equation with a sublinear perturbation. The main concept is the matrix spectrum which is a natural extension of the usual spectrum. Using the standard ‘change of degree’ argument we shall find necessary and sufficient conditions for the existence of asymptotic bifurcation with respect to the matrix spectrum both in one parameter and four parameter cases. Obviously the results can be modified for more general systems of partial differential equations.     

Exponential mixing properties of stochastic PDEs through asymptotic coupling

 We consider parabolic stochastic partial differential equations driven by white noise in time. We prove exponential convergence of the transition probabilities towards a unique invariant measure under suitable conditions. These conditions amount essentially to the fact that the equation transmits the noise to all its determining modes. Several examples are investigated, including some where the noise does not act on every determining mode directly. Received: 20 September 2001 / Revised version: 21 April 2002 / Published online: 10 September 2002     

The renormalizing series of some integral equations

We consider integral equations for which the perturbation expansion gives a power series in a parameter h whose coefficients are divergent integrals. We eliminate the divergent integrals by introducing a renormalizing Z(t, h) series in the minimal subtraction scheme. We investigate the convergence of the formal Z series in relation to the kernels of the integral equations. We find a relation of the renormalizing series to the Lagrange inversion series and also some other relations.     

The Nonparallel Evolution of Nonlinear Short Waves in Buoyant Boundary Layers

Buoyant boundary-layer flows, typified by the flow over a heated flat plate, have the curious property that they can exhibit regions of "overshoot" in which the streamwise velocity exceeds its free-stream value. A consequence of this is the streamwise velocity develops a local maximum and is inflectional in nature. It is therefore inviscidly unstable, and the fastest growing wave mode is known to be one whose wavelength is short compared to the boundary-layer thickness. In this work we consider the nonparallel evolution of these short waves and show that they can be described in terms of the solution of a system of ordinary differential equations. Numerical and asymptotic studies enable us to explain the ultimate fate of the wave and show, depending on a key parameter which is a function of the underlying boundary layer, that two possibilities can arise. Nonparallelism may be sufficiently stabilizing so as to extinguish the linearly unstable waves or, in other cases, the mode may intensify but concentrate itself in a very thin zone surrounding the maximum in the streamwise velocity. These findings enable us to give some indication of the part these modes play in the transition to turbulence in buoyant boundary layers.     

On Bayes Sequential Estimation of the Location Parameter with Nonsmooth Density

We consider the problem of sequential estimation of the location parameter for a density with irregular behavior at some points (discontinuity, infinite values of the derivative, and so on). Thus, for our problem we have no finite Fisher information. In this situation, sequential estimation is usually more preferable compared to estimation based on samples of fixed size. In this paper, we establish the asymptotic efficiency of the Bayes sequential estimation plans and find their limit distribution. Bibliography: 14 titles.     

An infeasible nonmonotone SSLE algorithm for nonlinear programming

In this paper, we propose a new nonmonotone algorithm using the sequential systems of linear equations, which is an infeasible QP-free method. We use neither a penalty function nor a filter. Therefore, it is unnecessary to choose a problematic penalty parameter. The new algorithm only needs to solve three systems of linear equations with the same nonsingular coefficient matrix. Under some suitable conditions, the global convergence is established. Some numerical results are also presented.     

Invariants of Characteristics of Some Systems of Partial Differential Equations

We introduce the notion of an invariant of characteristics for a system of first-order partial differential equations. We prove that the existence of invariants is connected with passiveness of some systems. We describe a few methods for construction of new invariants from those already known. We give a scheme for application of the invariants to reduction and integration of systems of partial differential equations. As an application we consider the equation of gas dynamics.     

Method of limiting equations for the stability analysis of equations with infinite delay in the Carathéodory conditions: II

We consider systems of nonautonomous nonlinear differential equations with the infinite delay. We study the stability properties and the limiting equations whose right-hand sides are defined as the limit points of some sequence in the introduced function space. By using the method of limiting equations, we obtain new sufficient conditions for the asymptotic stability of the zero solution of the considered class of equations.     

Semiclassical determination of exponentially small intermode transitions for 1 + 1 spacetime scattering systems

We consider the semiclassical limit of systems of autonomous PDEs in 1 + 1 spacetime dimensions in a scattering regime. We assume the matrix‐valued coefficients are analytic in the space variable, and we further suppose that the corresponding dispersion relation admits real‐valued modes only with one‐dimensional polarization subspaces. Hence a BKW‐type analysis of the solutions is possible. We typically consider time‐dependent solutions to the PDE that are carried asymptotically in the past and as x → ?∞ along one mode only and determine the piece of the solution that is carried for x → +∞ along some other mode in the future. Because of the assumed nondegeneracy of the modes, such transitions between modes are exponentially small in the semiclassical parameter; this is an expression of the Landau‐Zener mechanism. We completely elucidate the spacetime properties of the leading term of this exponentially small wave, when the semiclassical parameter is small, for large values of x and t, when some avoided crossing of finite width takes place between the involved modes. © 2006 Wiley Periodicals, Inc.     

A Minimax Estimation by Incomplete Data in the Right-Hand Sides of Elliptic Second-Order Partial Differential Equations with Discontinuous Coefficients

We consider systems described by boundary-value problems for elliptic second-order partial differential equations with discontinuous coefficients. These systems arise in the study of steady-state processes of a fluid filtration in multicomponent schistous mediums with nonuniform conditions for a nonideal contact. By observing the state of the systems, we find minimax estimates for functionals of the right-hand sides of these equations. Here we suppose that the right-hand sides of the equations, the boundary conditions, the junction conditions, and the error in the measurements are not known precisely, but we know only the sets to which they belong. We show that finding the minimax estimates is reduced to solving some integro-differential equations.     

Existence of invariant measures of stochastic systems with delay in the highest order partial derivatives

In this note, we shall consider the existence of invariant measures for a class of infinite dimensional stochastic functional differential equations with delay whose driving semigroup is eventually norm continuous. The results obtained are applied to stochastic heat equations with distributed delays which appear in such terms having the highest order partial derivatives. In these systems, the associated driving semigroups are generally non eventually compact.     

Approximability of nonlinear affine control systems

This paper focuses on a strong approximability property for nonlinear affine control systems. We consider control processes governed by ordinary differential equations (ODEs) and study an initial system and the associated generalized system. Our theoretical approach makes it possible to prove a strong approximability result for the above dynamical systems. The latter can be effectively applied to some classes of variable structure and hybrid control systems. In particular, this paper deals with applications of the strong approximability property obtained to the conventional sliding mode processes and to hybrid control systems with autonomous location transitions. We also take into consideration some optimal control problems for the above class of hybrid systems.     

Parameter estimation in moving boundary problems

We describe an approximation scheme which can be used to estimate unknown parameters in moving boundary problems. The model equations we consider are fairly general nonlinear diffusion/reaction equations of one spatial variable. Here we give conditions on the parameter sets and model equations under which we can prove that the estimates obtained using the approximations will converge to best-fit parameters for the original model equations. We conclude with a numerical example.     

Identification and estimation of parameters defining a class of hybrid systems

In this paper, we consider the problem of parameter estimation in an air brake system. In an air brake system, the pressure of air in the brake chamber and the displacement of the pushrod and their derivatives form a set of states that characterize the system. The position of a valve or mass flow rate of air is an input and the pressure is the measured variable or the output. The pressure acting on the pushrod of the brake chamber causes motion, and the mode in which the system operates depends on the displacement of the pushrod. The mode-dependent nature of the system is a result of different sets of spring compliances associated with the piston in different ranges of its displacement. The mode to mode transition in the air brake system is governed by a parameter which is the clearance between the brake pads and the drum. The clearance between the brake pads and the drum can vary due to a variety of factors — for example, brake pad wear or brake fade. In these applications, characterizing the transition from one mode to another requires a lot of constitutive assumptions, and it can be difficult to calibrate the parameters associated with the constitutive assumptions. We therefore treat the air brake system as a system in which the parameter governing the transition from one mode to another (clearance between the brake pads and the drum) is not known exactly. Clearly, this parameter dictates the time delay and lag between the command and delivery of the brake torque at the wheels and affects the stopping distance of the vehicles considerably. The problem of identification considered in this paper is as follows. Suppose that the pressure of the fluid were to be measured and that the motion of the piston is not measured. Is it possible to estimate the final displacement of the piston without knowing the parameters that govern the system to transition from one mode to another?     

高阶椭圆型偏微分方程奇异摄动问题一致收敛差分格式

本文,我们讨论了一类高阶椭圆型偏微分方程奇异摄动问题。给出了连续问题解的先验估计。另外,我们还提供了一种数值求解该类问题的指数型差分格式。最后,证明了差分问题的解在能量范数意义下关于小参数一致收敛到连续问题的解。