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1. Cadence Tracking for Switched FES-Cycling With Unknown Time-Varying Input Delay

   https://doi.org/10.1115/DSCC2019-9078  

   Allen, B.; Cousin, C.; Rouse, C.; Dixon, W. (October 2019, ASME 2019 Dynamic Systems and Control Conference)

   Functional electrical stimulation (FES) induced exercise, such as motorized FES-cycling, is commonly used in rehabilitation for lower limb movement disorders. A challenge in closed-loop FES control is the presence of an input delay between the application (and removal) of the electrical stimulus and the production of muscle force. Moreover, switching between motor control and FES control of various muscle groups can be destabilizing. This paper examines the development of a control method and state-dependent trigger condition to account for the time-varying input delayed response. Uniformly ultimately bounded tracking for a switched uncertain nonlinear dynamic system with input delays is achieved. 

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2. Saturated Control of a Switched FES-Cycle with an Unknown Time-Varying Input Delay

   Allen, B. C.; Stubbs, K. J.; Dixon, W. E. (December 2020, IFAC Conference on Cyber-Physical Human-Systems)

   A common rehabilitation for those with lower limb movement disorders is motorized functional electrical stimulation (FES) induced cycling. Motorized FES-cycling is a switched system with uncertain dynamics, unknown disturbances, and there exists an unknown time-varying input delay between the application/removal of stimulation and the onset/removal of muscle force. This is further complicated by the fact that each participant has varying levels of sensitivity to the FES input, and the stimulation must be bounded to ensure comfort and safety. In this paper, saturated FES and motor controllers are developed for an FES-cycle that ensure safety and comfort of the participant, while likewise being robust to uncertain parameters in the dynamics, unknown disturbances, and an unknown time-varying input delay. A Lyapunov-based stability analysis is performed to ensure uniformly ultimately bounded cadence tracking. 

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3. Robustness analysis of uncertain time‐varying systems using integral quadratic constraints with time‐varying multipliers

   https://doi.org/10.1002/rnc.5299  

   Fry, J. Micah; Abou Jaoude, Dany; Farhood, Mazen (December 2020, International Journal of Robust and Nonlinear Control)

   Summary This article presents a dissipativity approach for robustness analysis using the framework of integral quadratic constraints (IQCs). The derived results apply for linear time‐varying nominal systems with uncertain initial conditions. IQC multipliers are used to describe the sets of allowable uncertainty operators, and signal IQC multipliers are used to describe the sets of allowable disturbance signals. The novel concepts of dichotomic nodes and their corresponding factorizations are introduced, which allow for the aforementioned multipliers to be general time‐varying operators. The results are illustrated via the robustness analysis of a flight controller for an unmanned aircraft system tasked to perform a Split‐S maneuver. 

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4. Integral Quadratic Constraints with Infinite-Dimensional Channels

   https://doi.org/10.23919/ACC55779.2023.10156335  

   Talitckii, Aleksandr; Seiler, Peter; Peet, Matthew M. (May 2023, Proceedings of the American Control Conference)

   Modern control theory provides us with a spectrum of methods for studying the interconnection of dynamic systems using input-output properties of the interconnected subsystems. Perhaps the most advanced framework for such inputoutput analysis is the use of Integral Quadratic Constraints (IQCs), which considers the interconnection of a nominal linear system with an unmodelled nonlinear or uncertain subsystem with known input-output properties. Although these methods are widely used for Ordinary Differential Equations (ODEs), there have been fewer attempts to extend IQCs to infinitedimensional systems. In this paper, we present an IQC-based framework for Partial Differential Equations (PDEs) and Delay Differential Equations (DDEs). First, we introduce infinitedimensional signal spaces, operators, and feedback interconnections. Next, in the main result, we propose a formulation of hard IQC-based input-output stability conditions, allowing for infinite-dimensional multipliers. We then show how to test hard IQC conditions with infinite-dimensional multipliers on a nominal linear PDE or DDE system via the Partial Integral Equation (PIE) state-space representation using a sufficient version of the Kalman-Yakubovich-Popov lemma (KYP). The results are then illustrated using four example problems with uncertainty and nonlinearity. 

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5. Robust Analysis of Linear Systems with Uncertain Delays using PIEs

   https://doi.org/10.1016/j.ifacol.2021.11.133  

   Wu, S.; Peet, M.; Sun, F.; Hua, C. (January 2021, IFACPapersOnLine)

   Appeared in the proceedings of the 2021 IFAC Workshop on Time-Delay Systems This paper establishes a PIE (Partial Integral Equation)-based technique for the robust stability and H∞ performance analysis of linear systems with interval delays. The delays considered are time-invariant but uncertain, residing within a bounded interval excluding zero. We first propose a structured class of PIE systems with parametric uncertainty, then propose a Linear PI Inequality (LPI) for robust stability and H∞ performance of PIEs with polytopic uncertainty. Next, we consider the problem of robust stability and H∞ performance of multidelay systems with interval uncertainty in the delay parameters and show this problem is equivalent to robust stability and performance of a given PIE with parametric uncertainty. The robust stability and H∞ performance of the uncertain time-delay system are then solved using the LPI solver in the MATLAB PIETOOLS toolbox. Numerical examples are given to prove the effectiveness and accuracy of the method. This paper adds to the expanding field of PIE approach and can be extended to linear partial differential equations. 

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