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This paper examines an electrochemistry-based lithium-ion battery model developed by Doyle, Fuller, and Newman. The paper makes this model more tractable and conducive to control design by making two main contributions to the literature. First, we adaptively solve the model's algebraic equations using quasi-linearization. This improves the model's execution speed compared to solving the algebraic equations via optimization. Second, we reduce the model's order by deriving a family of analytic Padé approximations to the model's spherical diffusion equations. The paper carefully compares these Padé approximations to other published methods for reducing spherical diffusion equations. Finally, the paper concludes with battery simulations showing the significant impact of the proposed model reduction approach on the battery model's overall accuracy and simulation speed. This paper examines the problem of developing reduced, electrochemistry-based models of the dynamics of charging and discharging of lithium-ion batteries. The overarching goal of the paper is to develop lithium-ion battery models that satisfy two important but potentially conflicting objectives. First, the models must have the ability to accurately predict the performance of lithium-ion batteries in applications involving potentially complex and rapid charge/discharge cycles, e.g., hybrid vehicle applications. Second, the models must run with sufficient speed to enable battery system design, optimization, and control.Several models have been used to monitor battery state of charge ͑SOC͒ and state of health. 1-4 While these models are very desirable for control and estimation, they do not capture all of the high rate dynamics associated with hybrid vehicle drive cycles. For this one can use an electrochemical battery model. One such model is provided by Doyle, Fuller, and Newman, with the addition of a potential degradation mechanism provided by Ramadass et al. [5][6][7][8] There are two major numerical difficulties with this electrochemical model. The first is the large number of state variables: a finite difference discretization of the model with M points along the width of the cell and N points in the pseudospherical direction has approximately ͑2/3͒ * M * N state variables. The second challenge is the model's nonzero index, represented by approximately ͑2/3͒ * M * N algebraic equations, most of them involving a hyperbolic sine nonlinearity. This results in a large set of differential algebraic equations ͑DAEs͒. Ideally, one would like a model that: ͑i͒ runs quickly with a low number of state variables to enable optimal design and control studies, while ͑ii͒ still retaining the ability to accurately model complex, high rate charge/discharge cycles.Applying model reduction techniques to the above electrochemical battery model can bring it closer to the ideal speed and fidelity goals. Several reductions of this model are already presented in the literature. Some of these reductions pay special attention to the spherical diffusion submodel because it ap...

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A set of memory-based properties is employed in this paper for modeling multiple-path hysteresis response of piezoelectric actuators. These properties, namely, targeting turning points, curve alignment, and wiping-out effect, are applied in a linear mapping strategy to develop a mathematical framework for modeling the hysteresis phenomenon. More specifically, the locations of turning points are detected and recorded for the prediction of future hysteresis trajectory. An internal trajectory is assumed to follow a multiple-segmented path via a continuous connection of several curves passing through every two consequent turning points. These curves adopt their shapes via a linear mapping strategy from the reference hysteresis curves with polynomial configurations. Experimental implementation of the proposed method demonstrates slight improvement over the widely used Prandtl–Ishlinskii hysteresis operator. However, to maintain the level of precision during the operation, a sufficient number of memory units must be included to record the turning points. Otherwise, in the event of memory saturation, two memory-allocation modes, namely, “open” and “closed” strategies, can be implemented. It is shown that the closed memory-allocation strategy demonstrates better performance by keeping the most important target points. The proposed modeling framework is adopted in an inverse model-based control scheme for feedforward compensation of hysteresis nonlinearity. The controller is experimentally implemented on a three-dimensional nanopositioning stage for surface topography tracking, a problem typically encountered in scanning probe microscopy applications.

Nanomechanical cantilever (NMC) active probes have recently received increased attention in a variety of nanoscale sensing and measurement applications. Current modeling practices call for a uniform cantilever beam without considering the intentional jump discontinuities associated with the piezoelectric layer attachment and the NMC cross-sectional step. This paper presents a comprehensive modeling framework for modal characterization and dynamic response analysis of NMC active probes with geometrical discontinuities. The entire length of the NMC is divided into three segments of uniform beams followed by applying appropriate continuity conditions. The characteristics matrix equation is then used to solve for system natural frequencies and mode shapes. Using an equivalent electromechanical moment of a piezoelectric layer, forced motion analysis of the system is carried out. An experimental setup consisting of a commercial NMC active probe from Veeco and a state-of-the-art microsystem analyzer, the MSA-400 from Polytec, is developed to verify the theoretical developments proposed here. Using a parameter estimation technique based on minimizing the modeling error, optimal values of system parameters are identified. Mode shapes and the modal frequency response of the system for the first three modes determined from the proposed model are compared with those obtained from the experiment and commonly used theory for uniform beams. Results indicate that the uniform beam model fails to accurately predict the actual system response, especially in multiple-mode operation, while the proposed discontinuous beam model demonstrates good agreement with the experimental data. Such detailed and accurate modeling framework can lead to significant enhancement in the sensitivity of piezoelectric-based NMC sensors for use in variety of sensing and imaging applications.