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1. Representing and Computing the B-Derivative of the Piecewise-Differentiable Flow of a Class of Nonsmooth Vector Fields

   https://doi.org/10.1115/1.4054481  

   Council, George; Revzen, Shai; Burden, Samuel A. (September 2022, Journal of Computational and Nonlinear Dynamics)

   Abstract This paper concerns first-order approximation of the piecewise-differentiable flow generated by a class of nonsmooth vector fields. Specifically, we represent and compute the Bouligand (or B-)derivative of the piecewise-differentiable flow generated by a vector field with event-selected discontinuities. Our results are remarkably efficient: although there are factorially many “pieces” of the derivative, we provide an algorithm that evaluates its action on a tangent vector using polynomial time and space, and verify the algorithm's correctness by deriving a representation for the B-derivative that requires “only” exponential time and space to construct. We apply our methods in two classes of illustrative examples: piecewise-constant vector fields and mechanical systems subject to unilateral constraints. 

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2. Sensitivity and Optimal Control Theory for Linear Complementarity Systems

   https://doi.org/10.1007/978-3-031-84869-8_44  

   Stechlinski, Peter; Abdelfattah, Hesham; Eisa, Sameh A (January 2025, Springer Nature Switzerland)

   Kilgour, D_M; Kunze, H; Makarov, R_N; Melnik, R; Wang, X (Ed.)

   This article focuses on sensitivity and control theory for linear complementarity systems (LCSs), a type of dynamical system that experiences hybrid continuous/discrete behavior and is therefore nonsmooth. In particular, a sensitivity theory is given that characterizes generalized derivative information of solutions of LCSs with respect to parametric perturbations. With this theory in hand, a computationally-relevant open-loop optimal control theory is provided using a direct method (i.e., the control is parametrically discretized and generalized gradients of the objective function are described). The approach here is based on lexicographic directional differentiation theory, a relatively new tool in nonsmooth analysis, being applied to nonlinear complementarity systems (NCSs). The optimal control theory is illustrated with an example. As a byproduct of the sensitivity theory, well-posedness results for a new class of hybrid dynamical system, called the lexicographic linear complementarity system (LexLCS), are also established. 

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3. Identification of Error-in-Variables Switched Systems using a Riemannian Embedding

   https://doi.org/10.1109/TAC.2023.3275973  

   Sznaier, M.; Zhang, X.; Camps, O. (May 2023, IEEE Transactions on Automatic Control)

   This paper considers the problem of error in variables identification for switched affine models. Since it is well known that this problem is generically NP hard, several relaxations have been proposed in the literature. However, while these approaches work well for low dimensional systems with few subsystems, they scale poorly with both the number of subsystems and their memory. To address this difficulty, we propose a computationally efficient alternative, based on embedding the data in the manifold of positive semidefinite matrices, and using a manifold metric there to perform the identification. Our main result shows that, under dwell-time assumptions, the proposed algorithm is convergent, in the sense that it is guaranteed to identify the system for suitably low noise. In scenarios with larger noise levels, we provide experimental results showing that the proposed method outperforms existing ones. The paper concludes by illustrating these results with academic examples and a non-trivial application: action video segmentation. 

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4. Efficient Identification of Error-in-Variables Switched Systems via a Sum-of-Squares Polynomial Based Subspace Clustering Method

   https://doi.org/10.1109/CDC40024.2019.9029570  

   Ozbay, B.; Camps, O.; Sznaier, M. (December 2019, 2019 IEEE 58th Conference on Decision and Control (CDC))

   This paper addresses the problem of identification of error in variables switched linear models from experimental input/output data. This problem is known to be generically NP hard and thus computationally expensive to solve. To address this difficulty, several relaxations have been proposed in the past few years. While solvable in polynomial time these (convex) relaxations tend to scale poorly with the number of points and number/order of the subsystems, effectively limiting their applicability to scenarios with relatively small number of data points. To address this difficulty, in this paper we propose an efficient method that only requires performing (number of subsystems) singular value decompositions of matrices whose size is independent of the number of points. The underlying idea is to obtain a sum-of-squares polynomial approximation of the support of each subsystem one-at-a-time, and use these polynomials to segment the data into sets, each generated by a single subsystem. As shown in the paper, exploiting ideas from Christoffel's functions allows for finding these polynomial approximations simply by performing SVDs. The parameters of each subsystem can then be identified from the segmented data using existing error-in-variables (EIV) techniques. 

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5. Modified Legendre-Gauss Collocation Method for Optimal Control Problems with Nonsmooth Solutions

   Abadia-Doyle, G; Rao, A V (December 2024, IEEE)

   A modified form of Legendre-Gauss orthogonal direct collocation is developed for solving optimal control problems whose solutions are nonsmooth due to control discon- tinuities. This new method adds switch time variables, control variables, and collocation conditions at both endpoints of a mesh interval, whereas these new variables and collocation con- ditions are not included in standard Legendre-Gauss orthogonal collocation. The modified Legendre-Gauss collocation method alters the search space of the resulting nonlinear programming problem and optimizes the switch point of the control solution. The transformed adjoint system of the modified Legendre- Gauss collocation method is then derived and shown to satisfy the necessary conditions for optimality. Finally, an example is provided where the optimal control is bang-bang and contains multiple switches. This method is shown to be capable of solving complex optimal control problems with nonsmooth solutions. 

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