This paper presents a joint input-state estimation algorithm that can be used for the identification of forces applied to a structure and for the extrapolation of the measured data to unmeasured response quantities of interest. The estimation of the input and the system states is performed in a minimum-variance unbiased way, based on a limited number of response measurements and a system model. No prior information on the system input is required, permitting online application of the algorithm. The paper also presents a novel approach for quantification of the estimation uncertainty originating from measurement errors and unknown stochastic excitation, that is acting on the structure besides the forces that are to be identified. The joint input-state estimation algorithm and the uncertainty quantification approach are verified using numerical simulations.

a b s t r a c tIdentified modal characteristics are often used as a basis for the calibration and validation of dynamic structural models, for structural control, for structural health monitoring, etc. It is therefore important to know their accuracy. In this article, a method for estimating the (co) variance of modal characteristics that are identified with the stochastic subspace identification method is validated for two civil engineering structures. The first structure is a damaged prestressed concrete bridge for which acceleration and dynamic strain data were measured in 36 different setups. The second structure is a mid-rise building for which acceleration data were measured in 10 different setups. There is a good quantitative agreement between the predicted levels of uncertainty and the observed variability of the eigenfrequencies and damping ratios between the different setups. The method can therefore be used with confidence for quantifying the uncertainty of the identified modal characteristics, also when some or all of them are estimated from a single batch of vibration data. Furthermore, the method is seen to yield valuable insight in the variability of the estimation accuracy from mode to mode and from setup to setup: the more informative a setup is regarding an estimated modal characteristic, the smaller is the estimated variance.

In structural dynamics, the forces acting on a structure are often not well known. System inversion techniques may be used to estimate these forces from the measured response of the structure. This paper first derives conditions for the invertibility of linear system models that apply to any instantaneous input estimation or joint input-state estimation algorithm. The conditions ensure the identifiability of the dynamic forces and system states, their stability and uniqueness. The present paper considers the specific case of modally reduced order models, which are generally obtained from a physical, finite element model, or from experimental data. It is shown how in this case the conditions can be directly expressed in terms of the modal properties of the structure. A distinction is made between input estimation and joint input-state estimation. Each of the conditions is illustrated by a conceptual example. The practical implementation is discussed for a case study where a sensor network for a footbridge is designed.

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This paper presents a geometric algorithm to investigate the theoretical observability of nonlinear systems with partially measured inputs and outputs. The algorithm is based on Lie algebra and applies to systems whose state and measurement equations are analytical and affine in all inputs. It investigates whether the system satisfies a necessary observability condition that is named the Observability Rank Condition for systems with Direct Feedthrough (ORC-DF). The presented algorithm allows to assess the observability of the dynamic system states, the identifiability of constant-in-time parameters, and the ability to track unmeasured inputs, which is referred to as system invertibility. It is also shown how the developed methodology can be extended to investigate the observability of non-smooth systems that can be broken into different smooth branches, often encountered in mechanical applications related to sliding and damage. Possible applications are illustrated with several examples from structural engineering.