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This paper provides a rigorous derivation for what is known in the literature as the Lie bracket approximation of control-affine systems in a more general and sequential framework for higher-orders. In fact, by using chronological calculus, we show that said Lie bracket approximations can be derived, and considered, as higher-order averaging terms. Hence, the theory provided in this paper unifies both averaging and approximation theories of control-affine systems. In particular, the Lie bracket approximation of order (n) turns out to be a higher-order averaging of order (n + 1). The derivation and formulation provided in this paper can be directly reduced to the first and second order. Lie bracket approximations available in the literature. However, we do not need to make many of the assumptions that were needed/provided in the literature and show that they are in fact natural corollaries from our work. Moreover, we use our results to show that important and useful information about control-affine extremum seeking systems can be obtained and used for significant performance improvement, including a faster convergence rate influenced by higher-order derivatives. We provide multiple numerical simulations to demonstrate both the conceptual elements of this work as well as the significance of our results on extremum seeking with comparison against the literature. 

In this paper, we provide a novel framework that enables a sensitivity-based observability test and state estimation algorithm for wind turbine power systems (WTPSs). The provided framework is the first of its kind in the literature, as it is able to deal with state-of-the-art WTPS models that are non-reduced, highly nonlinear differential–algebraic equation systems. Moreover, the framework includes nonsmoothness in both the dynamics and output functions to unify the operational conditions over different wind speed regions. We demonstrate the effectiveness of the proposed framework (thanks to the underlying tools from generalized derivatives theory) on different wind speed profiles, including real-world wind data. We also illustrate how the proposed framework, by the utilization of robust observability analysis during nonsmooth transitions, enables accurate state estimation for cases when the conventional Extended Kalman Filter approach fails. 

The problem of hovering in flapping insects/hummingbirds, and potential bio-mimicry by micro aerial vehicles (MAVs), have been studied for decades by scientists and engineers. Said communities often study hovering in flapping systems as either an open-loop or closed-loop system to analyze stability and/or propose control designs. Recently, a fundamentally novel result has been achieved in the literature of the hovering problem. That is, hovering in flapping insects/hummingbirds can be characterized/mimicked as a stable, model-free, real-time extremum seeking control (ESC) feedback system. In this paper we aim at two contributions: (i) provide a novel open-loop, optimal control characterization of hovering; and (ii) compare the performance of closed-loop, real-time ESC in hovering vs. the provided open-loop, non-real-time optimal control in hovering. 

In this letter, we extend the sensitivity-based rank condition (SERC) test for local observability to another class of systems, namely smooth and nonsmooth differential-algebraic equation (DAE) systems of index-1. The newly introduced test for DAEs, which we call the lexicographic SERC (L-SERC) observability test, utilizes the theory of lexicographic differentiation to compute sensitivity information. Moreover, the newly introduced L-SERC observability test can judges which states are observable and which are not. Additionally, we introduce a novel sensitivity-based extended Kalman filter (S-EKF) algorithm for state estimation, applicable to both smooth and nonsmooth DAE systems. Finally, we apply the newly developed S-EKF to estimate the states of a wind turbine power system model. 

New sensitivity-based methods are developed for determining identifiability and observability of nonsmooth input-output systems. More specifically, lexicographic derivatives are used to construct nonsmooth sensitivity rank condition (SERC) tests, which we call lexicographic SERC (L-SERC) tests. The introduced L-SERC tests are practically implementable, accurate, and analogous to (and indeed recover) their smooth counterparts. To accomplish this, a novel first-order Taylor-like approximation theory is developed to directly treat nonsmooth (i.e., continuous but nondifferentiable) functions. An L-SERC algorithm is proposed that determines partial structural identifiability or observability, which are useful characterizations in the nonsmooth setting. Lastly, the theory is illustrated through an application in climate modeling. 

Optimal control theory extending from the calculus of variations has not been used to study the wind turbine power system (WTPS) control problem, which aims at achieving two targets: (i) maximizing power generation in lower wind speed conditions; and (ii) maintaining the output power at the rated level in high wind speed conditions. A lack of an optimal control framework for the WTPS (i.e., no access to actual optimal control trajectories) reduces optimal control design potential and prevents competing control methods of WTPSs to have a reference control solution for comparison. In fact, the WTPS control literature often relies on reduced and linearized models of WTPSs, and avoids the nonsmoothness present in the system during transitions between different conditions of operation. In this paper, we introduce a novel optimal control framework for the WTPS control problem. We use in our formulation a recent accurate, nonlinear differential–algebraic equation (DAE) model of WTPSs, which we then generalize over all wind speed ranges using nonsmooth functions. We also use developments in nonsmooth optimal control theory to take into account nonsmoothness present in the system. We implement this new WTPS optimal control approach to solve the problem numerically, including (i) different wind speed profiles for testing the system response; (ii) real-world wind data; and (iii) a comparison with smoothing and naive approaches. Results show the effectiveness of the proposed approach. 

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. 

We extend the sensitivity rank condition (SERC), which tests for identifiability of smooth input-output systems, to a broader class of systems. Particularly, we build on our recently developed lexicographic SERC (L-SERC) theory and methods to achieve an identifiability test for differential-algebraic equation (DAE) systems for the first time, including nonsmooth systems. Additionally, we develop a method to determine the identifiable and non-identifiable parameter sets. We show how this new theory can be used to establish a (non-local) parameter reduction procedure and we show how parameter estimation problems can be solved. We apply the new methods to problems in wind turbine power systems and glucose-insulin kinetics.