یک الگوریتم برنامه ریزی پویا برای برآورد ورودی در سیستم های نوع زمان خطی

A time domain input estimation algorithm for linear systems with general time-varying parameters is developed. The algorithm is an extension of an existing approach for time-invariant state space models and several new features, such as higher order input approximations and an extended time-variant output relation including direct input influence, are introduced. Numerical examples are given to illustrate the new features and show that the algorithm is valid in a general time-variant setting. In particular, excellent results are obtained for an ill-posed moving force identification problem with noise-contaminated data, treated with Tikhonov regularization.

Knowledge of time-varying excitation is of importance in the design of a wide range of engineering applications, from spacecraft and processing plants to electronic circuits. Regardless of the actual application or the underlying physics, the expected input will play a key role in the determination of adequate system properties or parameters. In a trivial case, the desired information may be obtained by direct measurement of the input. When this is not possible, the analyst may resort to indirect measurement techniques, which means that the unknown input is established as the solution to an inverse problem, based on measurements of system response. This process is known as input estimation. In mechanical engineering, the concept of input estimation is mainly associated with determination of unknown dynamic forces acting on some kind of mechanical system. The nature of such forces will in many cases imply practical difficulties that prevent them from being measured directly. The loading positions may be inaccessible, the spatial distribution may be complicated, or the application of force transducers may intrude on the load path or alter system properties in an undesired manner. Various methods for solving the inverse problem associated with indirect force measurement have been proposed, see e.g. [1], [2] and [3] for an overview. With few exceptions (e.g. [4] and [5]), these methods are concerned with linear systems. Several of the methods operate in the frequency domain or utilize modal system descriptions for computational efficiency, see e.g. [6] and [7], which is convenient for time-invariant systems. However, it is not very efficient to treat problems with time-varying system parameters with methods based on a modal approach, since the modal descriptions will change over time. Thus, it is often more convenient to treat such problems with time domain techniques. Recent work concerned with time-variant problems includes [8], where the conjugate gradient method is used to identify the excitation force on a single-degree-of-freedom system with time-varying stiffness and damping parameters, and [9] where dynamic programming is used to identify moving forces from strain and velocity measurements on a bridge model. In this paper a non-iterative recurrence algorithm for input estimation on fully time-variant linear systems is presented. The algorithm is based on dynamic programming and is an extension of a time domain approach for time-invariant systems presented in [10]. The extended algorithm applies to problems where the explicit time-dependence of system and output coefficient matrices is known. For force estimation problems in mechanics, this implies that the force locations may vary but must be well-defined at all times. The method sets out from a system of ordinary differential equations in time, suggesting that any spatial variations in the physical problem have been parameterized, e.g. through finite element discretization. Thus, unknown excitations with spatial distribution should be discretized accordingly and consequently estimated in terms of the corresponding parameters. A general restriction, implied by the adopted temporal parametrization of the input, is that the number of sought inputs cannot exceed the number of sensors used.

A non-iterative input estimation algorithm for linear, time-variant systems on first-order state space form has been derived. The estimation process may comprise both inputs variables and initial conditions, such that optimal values for unknown initial state variables may be calculated. For an ideal system description and noise-free measurement data, the algorithm will identify true inputs and initial conditions exactly. The algorithm presented here is a generalization of a dynamic programming approach for time-invariant systems, and the new formulation extends the field of application to problems with time-varying system descriptions, such as moving load problems and problems with varying sampling frequency. It is suggested that discretization of time-variant systems be carried out by means of numerical quadrature. Thus, direct manipulation of the integrands may be avoided and integration may be kept consistent with the approximation of the input variation on each time interval. The new formulation allows for an extended output relation, such that incorporation of measurements with direct influence from sought inputs will not require expansion of the state vector. For large systems, this means that the considerable increase in computational cost following from such expansions may be avoided. In addition, the algorithm is no longer restricted to the use of system descriptions based on zeroth-order-hold input approximations. It is shown in the validation examples that a considerable improvement of estimation results may be obtained if first-order-hold approximations are used. Some basic validation examples are used to demonstrate the most important features of the extended algorithm. The results show that no errors are introduced by the algorithm itself. Application of the algorithm to a moving force problem involving noise-contaminated strain measurements shows that excellent estimations results can be obtained if higher order Tikhonov regularization is used. However, it is shown that the L-curve maximum curvature criterion may provide a regularization parameter that oversmooths the solution. Thus, a more robust criterion for choosing the parameter is needed, in particular for higher noise levels. Future work will focus on the solution of larger problems, to investigate how the algorithm behaves for increasing system size. Different ways to stabilize the descending sweep and reduce the effects of noise in measurement data will be studied, e.g. time-varying regularization. An alternative or complementary way of handling noise would be to shift sensors from one time instance to another, which would require time-varying output relations and residual weighting. Such an approach might reduce noise effects, since sudden changes in data from individual sensors will be less significant.