SUMMARYReal-time testing with dynamic substructuring is a novel experimental technique capable of assessing the behaviour of structures subjected to dynamic loadings including earthquakes. The technique involves recreating the dynamics of the entire structure by combining an experimental test piece consisting of part of the structure with a numerical model simulating the remainder of the structure. These substructures interact in real time to emulate the behaviour of the entire structure. Time integration is the most versatile method for analysing the general case of linear and non-linear semi-discretized equations of motion. In this paper we propose for substructure testing, L-stable real-time (LSRT) compatible integrators with two and three stages derived from the Rosenbrock methods. These algorithms are unconditionally stable for uncoupled problems and entail a moderate computational cost for real-time performance. They can also effectively deal with stiff problems, i.e. complex emulated structures for which solutions can change on a time scale that is very short compared with the interval of time integration, but where the solution of interest changes on a much longer time scale. Stability conditions of the coupled substructures are analysed by means of the zero-stability approach, and the accuracy of the novel algorithms in the coupled case is assessed in both the unforced and forced conditions. LSRT algorithms are shown to be more competitive than popular Runge-Kutta methods in terms of stability, accuracy and ease of implementation. Numerical simulations and real-time substructure tests are used to demonstrate the favourable properties of the proposed algorithms.

SUMMARYIn this paper, Rosenbrock-based algorithms originally developed for real-time testing of linear systems with dynamic substructuring are extended for use on nonlinear systems. With this objective in mind and for minimal overhead, both two-and three-stages linearly implicit real-time compatible algorithms were endowed with the Jacobian matrices requiring only one evaluation at the beginning of each time step. Moreover, these algorithms were improved with subcycling strategies. In detail, the paper briefly introduces Rosenbrock-based L-Stable Real-Time (LSRT) algorithms together with linearly implicit and explicit structural integrators, which are now commonly used to perform real-time tests. Then, the LSRT algorithms are analysed in terms of linearized stability with reference to an emulated spring pendulum, which was chosen as a nonlinear test problem, because it is able to exhibit a large and relatively slow nonlinear circular motion coupled to an axial motion that can be set to be stiff. The accuracy analysis on this system was performed for all the algorithms described. Following this, a coupled spring-pendulum example typical of real-time testing is analysed with respect to both stability and accuracy issues. Finally, the results of representative numerical simulations and real-time substructure tests, considering nonlinearities both in the numerical and the physical substructure, are explored. These tests were used to demonstrate how the LSRT algorithms can be used for substructuring tests with strongly nonlinear components.

We study the convergence properties of a direct model reference adaptive control system by applying techniques from numerical analysis. In particular, a first-order discrete system coupled to a minimal control synthesis algorithm discretized by the one-step one-stage zero-order-hold sampling is studied. This results in a strongly non-linear dynamic system owing to the adaptive mechanism where stability at steady state, i.e. at the operating point, equates to successful control. This paper focuses on the convergence analysis of the overall dynamical system for understanding accuracy, stability and performance at steadystate. The local stability of the steady state solution is considered by linearizing the system in the neighbourhood of an operating point when the input is a step function. This analysis allows us to specify two gain space domains which define the region of local stability. Moreover, both the accuracy and the frequency-domain analyses give insight into the range of adaptive control weightings that results in optimal performance of the minimal control synthesis algorithm and also highlights a possible approach to a priori selection of the time step and adaptive weighting values.The effectiveness of the proposed analysis is further demonstrated by simulations and experiments on a first-order plant.Adaptive control methods are generally divided into (i) direct and (ii) indirect methods. In the first case the adjustment rules provide directly how to update the controller parameters; in the indirect methods, the parameters of the unknown plant are estimated on-line, and the controller parameters are calculated on the base of these estimates. In the context of control theory some recent books and papers relevant to direct methods can be found [2,3]. In the context of structural control several applications have been made by several researches as well [4][5][6].When a direct reference model controller is used, it can be also applied to systems where the details of the plant cannot be fully known a priori or are varying with time. Using this type of algorithms without the knowledge of plant parameters, such that we assume zero initial conditions for the controller gains, has become known as the minimal control synthesis (MCS) approach [7]. Basing adaptive control schemes on a reference model enables the system to be controlled to behave like the model itself. This type of approach [8,9] has been applied to a wide range of systems including non-linear and chaotic systems [3,10]. As the approach is based primarily on linear control theory being the reference model usually a linear one, the effect of non-linearities and/or disturbances in non-linear systems is compensated for by the adaptive nature of the controller. Very recently in the field of dynamic and seismic testing [11], the MCS method was used in real-time substructuring, in order to minimize the error between displacements imposed by the actuators on the substructure and displacements at the interfaces of the numerical model [12,13]; and it was adop...

SUMMARYAdaptive control techniques can be applied to dynamical systems whose parameters are unknown. We propose a technique based on control and numerical analysis approaches to the study of the stability and accuracy of adaptive control algorithms affected by time delay. In particular, we consider the adaptive minimal control synthesis (MCS) algorithm applied to linear time-invariant plants, due to which, the whole controlled system generated from state and control equations discretized by the zero-order-hold (ZOH) sampling is nonlinear. Hence, we propose two linearization procedures for it: the first is via what we term as physical insight and the second is via Taylor series expansion. The physical insight scheme results in useful methods for a priori selection of the controller parameters and of the discrete-time step. As there is an inherent sampling delay in the process, a fixed one-step delay in the discrete-time MCS controller is introduced. This results in a reduction of both the absolute stability regions and the controller performance. Owing to the shortcomings of ZOH sampling in coping with high-frequency disturbances, a linearly implicit L-stable integrator is also used within a two degree-of-freedom controlled system. The effectiveness of the methodology is confirmed both by simulations and by experimental tests.

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