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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. 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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3. 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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4. 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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5. On infinitesimal contraction analysis for hybrid systems

   Burden, S; Coogan, S (December 2022, IEEE Conference on Decision and Control)

   Infinitesimal contraction analysis, wherein global convergence results are obtained from properties of local dynamics, is a powerful analysis tool. In this paper, we generalize infinitesimal contraction analysis to hybrid systems in which state-dependent guards trigger transitions defined by reset maps between modes that may have different norms and need not be of the same dimension. In contrast to existing literature, we do not restrict mode sequence or dwell time. We work in settings where the hybrid system flow is differentiable almost everywhere and its derivative is the solution to a jump-linear-time-varying differential equation whose jumps are defined by a saltation matrix determined from the guard, reset map, and vector field. Our main result shows that if the vector field is infinitesimally contracting, and if the saltation matrix is non-expansive, then the intrinsic distance between any two trajectories decreases exponentially in time. When bounds on dwell time are available, our approach yields a bound on the intrinsic distance between trajectories regardless of whether the dynamics are expansive or contractive. We illustrate our results using wo examples: a constrained mechanical system and an electrical circuit with an ideal diode. 

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