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This article approaches deterministic filtering via an application of the min-plus linearity of the corresponding dynamic programming operator. This filter design method yields a set-valued state estimator for discrete-time nonlinear systems (nonlinear dynamics and output functions). The energy bounds in the process and the measurement disturbances are modeled using a sum quadratic constraint. The filtering problem is recast into an optimal control problem in the form of a Hamilton-Jacobi-Bellman (HJB) equation, the solution to which is obtained by employing the min-plus linearity property of the dynamic programming operator. This approach enables the solution to the HJB equation and the design of the filter without recourse to linearization of the system dynamics/ output equation.
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The design of deterministic filters can be cast as a problem of minimizing an associated cost function for an optimal control problem. Employing the min-plus linearity property of the dynamic programming operator (associated with the control problem) results in a computationally feasible approach (while avoiding linearization of the system dynamics/output). This article describes the salient features of this approach and a specific form of pruning/projection, based on clustering, which serves to facilitate the numerical efficiency of these methods.
A robust (deterministic) filtering approach to the problem of optimal sensor selection is considered herein. For a given system with several sensors, at each time step the output of one of the sensors must be chosen in order to obtain the best state estimate. We reformulate this problem in an optimal control framework which can then be solved using dynamic programming. In order to tackle the numerical computation of the solution in an efficient manner, we exploit the preservation of the min-plus structure of the optimal cost function when acted upon by the dynamic programming operator. This technique yields a grid free numerical approach to the problem. Simulations on an example problem serve to highlight the efficacy of this generalizable approach to robust multi-sensor state estimation.
We present a new model of neural networks called Min-Max-Plus Neural Networks (MMP-NNs) based on operations in tropical arithmetic. In general, an MMP-NN is composed of three types of alternately stacked layers, namely linear layers, min-plus layers and max-plus layers. Specifically, the latter two types of layers constitute the nonlinear part of the network which is trainable and more sophisticated compared to the nonlinear part of conventional neural networks. In addition, we show that with higher capability of nonlinearity expression, MMP-NNs are universal approximators of continuous functions, even when the number of multiplication operations is tremendously reduced (possibly to none in certain extreme cases). Furthermore, we formulate the backpropagation algorithm in the training process of MMP-NNs and introduce an algorithm of normalization to improve the rate of convergence in training.
In this paper, we study the problem of designing a simultaneous mode, input, and state set-valued observer for a class of hidden mode switched nonlinear systems with bounded-norm noise and unknown input signals, where the hidden mode and unknown inputs can represent fault or attack models and exogenous fault/disturbance or adversarial signals, respectively. The proposed multiple-model design has three constituents: (i) a bank of mode-matched set-valued observers, (ii) a mode observer, and (iii) a global fusion observer. The mode-matched observers recursively find the sets of compatible states and unknown inputs conditioned on the mode being the true mode, while the mode observer eliminates incompatible modes by leveraging a residual-based criterion. Then, the global fusion observer outputs the estimated sets of states and unknown inputs by taking the union of the mode-matched set-valued estimates over all compatible modes. Moreover, sufficient conditions to guarantee the elimination of all false modes (i.e., mode-detectability) are provided and the effectiveness of our approach is demonstrated and compared with existing approaches using an illustrative example.
We study Min-Max affine approximants of a continuous convex or concave function $f:Deltasubset mathbb R^kxrightarrow{} mathbb R$ where $Delta$ is a convex compact subset of $mathbb R^k$. In the case when $Delta$ is a simplex we prove that there is a vertical translate of the supporting hyperplane in $mathbb R^{k+1}$ of the graph of $f$ at the vertices which is the unique best affine approximant to $f$ on $Delta$. For $k=1$, this result provides an extension of the Chebyshev equioscillation theorem for linear approximants. Our result has interesting connections to the computer graphics problem of rapid rendering of projective transformations.
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