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Due to its simplicity and inexpensive computation, the gradient descent algorithm is one of the most used tools in adaptive control and system identification. Although it has been studied for decades, little has been achieved in terms of tuning methods in the presence of disturbances. One of the main difficulties in its analysis is the time-varying nature of the algorithm. In this work, we contribute in such direction by providing LMI tools for tuning the gradient descent algorithm gain such that a guaranteed upper bound on the L2 -gain with respect to parameter variations and measurement noise is achieved. Two academic examples are provided to illustrate the efficient application of the method.
In this work, an observer for a linear time-varying system with delayed measurements is developed. The delay is assumed to be unknown, bounded, and it can be time-varying with no restriction on its rate of change. The observer uses auxiliary signals related to the constructibility Gramian of the system and it contains nonlinearities that provide a uniform fixed-time convergence to a bounded region in the estimation error coordinates. This means that the convergence time can be bounded by a positive constant which is independent from the initial conditions and the initial time. This property is new for the addressed class of systems. The ultimate bound of the estimation error depends on the maximum difference between the nominal output and the delayed one, and not directly on the delay size or its time derivative. These properties are illustrated in a numerical simulation.
Strong Lyapunov functions for two classical problems in adaptive control and parameter identification are presented. These Lyapunov functions incorporate in their structure the classical persistency of excitation conditions, allowing to show global uniform asymptotic stability of the associated adaptive systems under sufficient and necessary conditions.
The worldwide transition to climate-friendly energy systems entails the substitution of conventional energy generation based on synchronous generators by renewable energy sources based on power electronics. As a consequence, the overall inertia of the grid decreases, resulting in high volatility of the frequency and posing new challenges for the estimation of the latter quantity. To address this problematic, in the present paper a phase-locked-loop (PLL) based on the super-twisting algorithm is considered for the estimation of the phase angle and time-varying frequency of a symmetric three-phase signal. A rigorous proof of the algorithm’s exact convergence in the presence of a fast-varying frequency together with tuning rules for its gains are derived by means of Lyapunov theory. Additionally, an estimate of the region of attraction is provided. The effectiveness of the proposed tuning method is illustrated in numerical simulations, while comparing its performance against a standard synchronous reference frame PLL.
Robust State and Input Estimation with Enhanced Convergence Rate for Monitoring Anaerobic Digestion
(2022)
In this study, an observer for estimating the state and unknown inputs is proposed for monitoring anaerobic digestion processes. This estimator is based on a dynamic model considering acidogenesis and methanogenesis, and consists of three sub-observers: (a) a gramian-based fixed-time convergent observer for the inlet chemical oxygen demand (COD) and the acidogenic bacteria population, (b) an asymptotic observer for the methanogenic bacteria population, and (c) a super-twisting observer for systems with time-varying parameters to estimate the inlet volatile fatty acid (VFA) concentration. These sub-observers can be designed independently, which greatly simplifies the tuning process. Proofs of convergence are developed and simulation tests show the performance of the estimation scheme as compared to classical extended Kalman filtering.