Source-linked AI summary
An unscented Kalman filter method for real time input-parameter-state estimation
Marios Impraimakis, Andrew W. Smyth
TL;DR
The paper examines whether systems can be identified from output information while estimating system states, parameters, and inputs jointly. It develops a novel unscented Kalman filter approach and uses perturbation analysis to show potential unique identification under known zero or non-zero input conditions.
Problem
The paper asks whether the system is identifiable in joint input-parameter-state estimation.
Method
The method uses a novel unscented Kalman filter to estimate system response, inputs, and model parameters simultaneously, including erroneous input and parameter components in its analysis.
Results
A system with at least one zero or non-zero known input can potentially be uniquely identified using perturbation analysis.
Takeaways & Limitations
The methodology jointly estimates dynamic states, parameters, and inputs from output information.
Takeaways & Limitations
High-frequency noise is the hardest type of noise to tackle in the output information.
Abstract
from arXiv · showhide
The input-parameter-state estimation capabilities of a novel unscented Kalman filter is examined herein on both linear and nonlinear systems. The unknown input is estimated in two stages within each time step. Firstly, the predicted dynamic states and the system parameters provide an estimation of the input. Secondly, the corrected with measurements states and parameters provide a final estimation. Importantly, it is demonstrated using the perturbation analysis that, a system with at least a zero or a non-zero known input can potentially be uniquely identified. This output-only methodology allows for a better understanding of the system compared to classical output-only parameter identification strategies, given that all the dynamic states, the parameters, and the input are estimated jointly and in real-time.
1. Introduction
The introduction motivates joint input-parameter-state estimation from output information and positions a novel UKF as a real-time solution for linear and nonlinear systems.
- Motivation: Output-only monitoring is motivated when inputs cannot be measured reliably, including traffic and wind loads on large structural systems.
- Prior limitations: Existing unknown-input methods commonly assume known system parameters, conflicting with joint estimation from response-only sensors.
- Prior limitations: Prior approaches reported limitations including unsatisfactory displacement estimates, problematic Jacobian derivatives, and divergence during continued vibration.
- Contribution: The proposed unscented Kalman filter estimates unknown inputs in two stages using predicted, then measurement-corrected, states and system parameters.
- Contribution: The method is presented for linear and nonlinear systems with at least one zero or non-zero known input, without requiring Jacobian derivatives or least-square introductions.
2. Formulation of the standard unscented Kalman lter
The standard UKF formulation augments dynamic states with system parameters and propagates them through nonlinear state-transition and observation functions under additive Gaussian noise.
- State-space model: The augmented state vector contains dynamic states and system parameters, while the observation vector records measured outputs.
- State-space model: The state-transition and observation functions incorporate the input vector, process noise, and measurement noise with covariance matrices Q(t) and R(t).
- UKF formulation: The standard UKF is discretized at sampling interval ∆t and its steps are specified in Table 1A.
- UKF formulation: The parameter λ depends on α, κ, and state dimension L, while α ∈ [10^-4, 1] controls sigma-point spread around the state estimate.
3. An unknown input-parameter-state unscented Kalman lter
The IPS-UKF jointly estimates unknown inputs, parameters, and states by refining input estimates within each time step using predicted and measurement-updated quantities.
- Input estimation: The unknown input is first estimated from predicted states and parameters, then corrected using states and parameters updated with measurements.
- Model update: The dynamic model may be linear or nonlinear, and its estimated parameters are updated at every step.
- Input estimation: Predicted states use the prior input, after which known input rows are replaced by known zero or non-zero inputs for measurement updating.
- Input estimation: The initial input estimate is erroneous because predicted states and system parameters have not yet incorporated the current measurements.
- Iteration and errors: The procedure repeats at the next time step, while modeled process error prevents acceleration measurement error from endangering full estimation success.
4. Identi ability discussion for the joint input-parameter-state es-
The identifiability discussion shows why jointly estimating unknown inputs and parameters is difficult, and how known inputs can restore potential identifiability in MDOF systems. Perturbation analysis explains ambiguities in SDOF and unknown-input cases, while a known input at one DOF can distinguish the real parameters and remaining input.
- SDOF and unknown-input ambiguity: The SDOF linear system is not identifiable in the time domain even with exact mass and complete dynamic-state knowledge.The unknown input creates ambiguity that standard identifiability tests do not resolve.
- Perturbation analysis: Perturbing parameters and input can produce alternative combinations that satisfy the equation of motion exactly.The analysis replaces c, k, and u(t) with c + ∆c, k + ∆k, and u(t) + ∆u(t), revealing equivalent explanations.
- Perturbation analysis: With unknown input, systems having erroneous input can share exactly the same mass and dynamic states as the true system.An equivalent input can compensate for erroneous quantities in the equation of motion.
- Relation to prior approaches: Frequency-domain decomposition can handle some no-input cases, but it does not estimate inputs or dynamic states and is limited to linear systems.The cited approach is also described as offline and difficult to automate.
- MDOF identifiability: A known zero or non-zero input at one DOF can potentially make an MDOF system identifiable.For the 2-DOF example, an equivalent input may match the second row, but it fails at the DOF whose input is known to be zero.
- MDOF identifiability: The analysis states that the known input can enable correct identification of the real parameters and the remaining input, including for nonlinear systems.The same conclusion is reported for nonlinear systems.
5. Applications
The applications evaluate the methodology on linear and nonlinear MDOF systems using synthetic measurements with pulse, ambient, and noise-type excitations. The reported estimates converge satisfactorily for the linear 3-DOF and nonlinear 2-DOF examples.
- Linear MDOF system: The linear 3-DOF application applies a 100N pulse for 0.01s at 5s, with the pulse time unknown beforehand.The system starts from zero displacement and velocity, and measurements are generated over 30s using fourth-order Runge–Kutta integration.
- Linear MDOF system: The linear application estimates responses, stiffness and damping parameters, and the input and its error at DOF 3.The process uses full-state measurements and updates estimated system matrices at every step.
- Linear MDOF system: The linear 3-DOF estimates show satisfactory convergence under the pulse excitation.The reported results compare true and estimated responses, parameters, and input.
- Linear MDOF system: A second linear application uses an ambient noise-type input with mean 0 and variance 4.The results again report response, stiffness, damping, and input estimation at DOF 3.
- Nonlinear MDOF system: The nonlinear application considers a Duffing nonlinear 2-DOF system with pulse and noise-type excitations applied at DOF 2.The nonlinear springs are extensively excited in the reported simulation.
- Nonlinear MDOF system: The nonlinear 2-DOF results show satisfactory convergence for the true and estimated response, parameters, input, and input error at DOF 2.The application uses full-state measurements, with the measurement-assumption discussion deferred to another section.
6. Output information sensitivity analysis
The sensitivity analysis evaluates how reduced output measurements affect joint input, parameter, and state estimation in a nonlinear 2-DOF system. Non-displacement and non-velocity measurements remain adequate, whereas acceleration-only measurements are unreliable.
- Measurement requirements: Full state measurements are recommended, but sensitivity analysis shows they are not mandatory.At least two dynamic states at each DOF may be adequate for identification.
- Noise effects: High-frequency noise is hardest to handle when differentiation of an unmeasured dynamic-state derivative is required.Low-frequency noise can instead cause integration error or loss of drift information when integration is required.
- Nonlinearity and drift: For highly nonlinear systems without displacement measurements, drift-information loss can endanger the identification procedure.The investigation of non-acceleration measurement cases is therefore constrained by the required derivative or integration operations.
- Reduced measurements: Non-displacement and non-velocity measurement cases adequately estimate the response, parameters, and input.These two cases are characterized as satisfying in the nonlinear 2-DOF example.
- Acceleration-only measurements: Acceleration-only measurements produce misleading and unreliable estimates for most parameters and dynamic states.The input and response show a trend toward continuously higher error, with different convergence values or divergence across IPS-UKF runs.
- System size: More degrees of freedom can provide more known zero inputs, potentially enabling faster and more accurate identification.This observation is made in the context of the sensitivity investigation for the nonlinear 2-DOF system.
7. Conclusions
The paper examines a novel IPS-UKF for real-time joint estimation of unknown inputs, model parameters, and dynamic states. It uses two input-estimation stages and shows potential identification under limited known-input conditions, while covariance calibration remains outside scope.
- Conclusions: The unknown input is estimated twice within each time step, first from predicted states and parameters and then from measurement-corrected quantities.The second stage produces the final input estimate.
- Conclusions: A system with at least one zero or non-zero known input can potentially be uniquely identified.This conclusion is supported by perturbation analysis.
- Conclusions: Process and measurement covariance matrices play an important role in correct convergence, but their proper calibration is outside the study's scope.Covariance calibration is identified as an unresolved scope boundary.
- Conclusions: Full estimation recommends displacement, velocity, and acceleration measurements, although data fusion may relax this requirement.The recommendation is not mandatory according to the reported sensitivity analysis.
- Conclusions: The IPS-UKF estimates system response, inputs, and model parameters simultaneously in real time.The method is examined for input-parameter-state estimation using an unscented Kalman filter.