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Register a new userQuantifying uncertainty in state and parameter estimation
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Added by
Jan Schumann-Bischoff
Publication date
2015
fields
Physics
and research's language is
English
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Observability of state variables and parameters of a dynamical system from an observed time series is analyzed and quantified by means of the Jacobian matrix of the delay coordinates map. For each state variable and each parameter to be estimated a measure of uncertainty is introduced depending on the current state and parameter values, which allows us to identify regions in state and parameter space where the specific unknown quantity can (not) be estimated from a given time series. The method is demonstrated using the Ikeda map and the Hindmarsh-Rose model.
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Most data based state and parameter estimation methods require suitable initial values or guesses to achieve convergence to the desired solution, which typically is a global minimum of some cost function. Unfortunately, however, other stable solutions (e.g., local minima) may exist and provide suboptimal or even wrong estimates. Here we demonstrate for a 9-dimensional Lorenz-96 model how to characterize the basin size of the global minimum when applying some particular optimization based estimation algorithm. We compare three different strategies for generating suitable initial guesses and we investigate the dependence of the solution on the given trajectory segment (underlying the measured time series). To address the question of how many state variables have to be measured for optimal performance, different types of multivariate time series are considered consisting of 1, 2, or 3 variables. Based on these time series the local observability of state variables and parameters of the Lorenz-96 model is investigated and confirmed using delay coordinates. This result is in good agreement with the observation that correct state and parameter estimation results are obtained if the optimization algorithm is initialized with initial guesses close to the true solution. In contrast, initialization with other exact solutions of the model equations (different from the true solution used to generate the time series) typically fails, i.e. the optimization procedure ends up in local minima different from the true solution. Initialization using random values in a box around the attractor exhibits success rates depending on the number of observables and the available time series (trajectory segment).
We present a method to determine the relative parameter mismatch in a collection of nearly identical chaotic oscillators by measuring large deviations from the synchronized state. We demonstrate our method with an ensemble of slightly different circle maps. We discuss how to apply our method when there is noise, and show an example where the noise intensity is comparable to the mismatch.
Accurate tracking of the internal electrochemical states of lithium-ion battery during cycling enables advanced battery management systems to operate the battery safely and maintain high performance while minimizing battery degradation. To this end, techniques based on voltage measurement have shown promise for estimating the lithium surface concentration of active material particles, which is an important state for avoiding aging mechanisms such as lithium plating. However, methods relying on voltage often lead to large estimation errors when the model parameters change during aging. In this paper, we utilize the in-situ measurement of the battery expansion to augment the voltage and develop an observer to estimate the lithium surface concentration distribution in each electrode particle. We demonstrate that the addition of the expansion signal enables us to correct the negative electrode concentration states in addition to the positive electrode. As a result, compared to a voltage only observer, the proposed observer can successfully recover the surface concentration when the electrodes stoichiometric window changes, which is a common occurrence under aging by loss of lithium inventory. With a 5% shift in the electrodes stoichiometric window, the results indicate a reduction in state estimation error for the negative electrode surface concentration. Under this simulated aged condition, the voltage based observer had 9.3% error as compared to the proposed voltage and expansion observer which had 0.1% error in negative electrode surface concentration.
Parameter estimation for spatiotemporal dynamics for coupled map lattices and continuous time domain systems is shown using a combination of multiple shooting, Karhunen-Loeve decomposition and Galerkins projection methodologies. The resulting advantages in estimating parameters have been studied and discussed for chaotic and turbulent dynamics using small amounts of data from subsystems, availability of only scalar and noisy time series data, effects of space-time parameter variations, and in the presence of multiple time-scales.
The accuracy of any physical scheme used to estimate the parameter describing the strength of a single qubit Pauli channel can be quantified using standard techniques from quantum estimation theory. It is known that the optimal estimation scheme, with m channel invocations, uses initial states for the systems which are pure and unentangled and provides an uncertainty of O[1/m^(1/2)]. This protocol is analogous to a classical repetition and averaging scheme. We consider estimation schemes where the initial states available are not pure and compare a protocol involving quantum correlated states to independent state protocols analogous to classical repetition schemes. We show, that unlike the pure state case, the quantum correlated state protocol can yield greater estimation accuracy than any independent state protocol. We show that these gains persist even when the system states are separable and, in some cases, when quantum discord is absent after channel invocation. We describe the relevance of these protocols to nuclear magnetic resonance measurements.
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