Implementable Discrete-Time $L_{1}$ Adaptive Control for a Cart Inverted Pendulum System

Elnaggar, Mahmoud M.; Lasheen, Ahmed

doi:10.1109/codit.2019.8820613

Cited by 1 publication

(17 citation statements)

References 21 publications

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“…Note that the proof in general follows prior studies on discrete-time L 1 control [55][56][57], but the critical difference here is that prior studies only consider g 1 (x t ) as a linear state-dependent uncertainty and do not consider the null space uncertainty g 2 (x t ). The key points in the proof are to deal with the nonlinearity in g 1 (x t ) and simultaneously consider the null space uncertainty g 2 (x t ).…”

Section: Theoretical Analyses Of Stability and Robustnessmentioning

confidence: 99%

“…We assume that the control gain uncertainty ∆ω is bounded, the nonlinear state-dependent uncertainty g 1 (x) and g 2 (x) are bounded, and their derivatives with respect to x are also bounded. We note that prior discrete-time L 1 adaptive controllers have only considered linear range space uncertainty [55][56][57], thus are not robust to nonlinearity or the null space uncertainty; they are special cases of our model with g 1 (x t ) being linear and g 2 (x t ) being zero.…”

Section: Nonlinear State-space Brain Network Modelmentioning

confidence: 99%

“…It is critical to ensure the stability of the closedloop system when using the robust adaptive controller. Thus, we next extend the existing discrete-time L 1 adaptive control theory [55][56][57] to include nonlinear range space uncertainty and null space uncertainty and prove the stability of the proposed control algorithm. We also show that there exists a theoretical trade-off between stability and robustness (see section 2.6).…”

Section: Overview Of the Robust Adaptive Controller Designmentioning

confidence: 99%

“…To provide prerequisites for stable operation of the robust adaptive controller in practical applications, it is critical to establish the theoretical stability of the closed-loop system. We thus extend the existing discrete-time L 1 adaptive control theory [55][56][57] to include nonlinear range space uncertainty and null space uncertainty, and derive sufficient conditions for ensuring stability. We summarize our analyses in Theorem 1, which quantitatively establishes a stability condition on the filter D z and the 'amplitude' of the model uncertainty terms g 1 (x t ) and g 2 (x t ).…”

Section: Theoretical Analyses Of Stability and Robustnessmentioning

confidence: 99%

“…L 1 adaptive control theory is a powerful control-theoretic method for developing modern robust adaptive controllers [54]. However, current discrete-time L 1 adaptive control methods have only assumed a specific linear form of model uncertainty [55][56][57], which may not be sufficient to achieve accurate and robust control of the complex nonlinear uncertain dynamics in brain networks (see section 4.2). The third challenge is to establish the stability of the closedloop control algorithm.…”

Section: Introductionmentioning

confidence: 99%

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Designing and validating a robust adaptive neuromodulation algorithm for closed-loop control of brain states

Fang¹,

Yang²

2022

J. Neural Eng.

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Objective. Neuromodulation systems that use closed-loop brain stimulation to control brain states can provide new therapies for brain disorders. To date, closed-loop brain stimulation has largely used linear time-invariant controllers. However, nonlinear time-varying brain network dynamics and external disturbances can appear during real-time stimulation, collectively leading to real-time model uncertainty. Real-time model uncertainty can degrade the performance or even cause instability of time-invariant controllers. Three problems need to be resolved to enable accurate and stable control under model uncertainty. First, an adaptive controller is needed to track the model uncertainty. Second, the adaptive controller additionally needs to be robust to noise and disturbances. Third, theoretical analyses of stability and robustness are needed as prerequisites for stable operation of the controller in practical applications. Approach. We develop a robust adaptive neuromodulation algorithm that solves the above three problems. First, we develop a state-space brain network model that explicitly includes nonlinear terms of real-time model uncertainty and design an adaptive controller to track and cancel the model uncertainty. Second, to improve the robustness of the adaptive controller, we design two linear filters to increase steady-state control accuracy and reduce sensitivity to high-frequency noise and disturbances. Third, we conduct theoretical analyses to prove the stability of the neuromodulation algorithm and establish a trade-off between stability and robustness, which we further use to optimize the algorithm design. Finally, we validate the algorithm using comprehensive Monte Carlo simulations that span a broad range of model nonlinearity, uncertainty, and complexity. Main results. The robust adaptive neuromodulation algorithm accurately tracks various types of target brain state trajectories, enables stable and robust control, and significantly outperforms state-of-the-art neuromodulation algorithms. Significance. Our algorithm has implications for future designs of precise, stable, and robust closed-loop brain stimulation systems to treat brain disorders and facilitate brain functions.

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Section: Theoretical Analyses Of Stability and Robustnessmentioning

confidence: 99%

Section: Nonlinear State-space Brain Network Modelmentioning

confidence: 99%