Chandrasekhara Bharath Panathula

Continuous second-order sliding mode control: Convergence time estimation

The contribution of this paper is threefold. First, a vector super-twisting algorithm is designed to provide a direct extension of the conventional scalar supertwisting control, without any additional terms. An upper estimate of its convergence time is calculated. Second, a fixed time convergent continuous vector super-twisting-like algorithm is presented and its fixed convergence time is estimated. Third, an estimate of the finite convergence time of the scalar supertwisting algorithm is obtained as a particular case of the vector super-twisting one, which occurs to be less conservative than the one derived specially for the scalar case. The proposed theory is applied to F-16 jet-fighter flight control.

Continuous Finite-Time Higher Order Output Regulators for Systems With Unmatched Unbounded Disturbances

This paper presents two continuous finite-time convergent control algorithms driving an output (the highest relative degree state) of an n-dimensional integrator to zero for a finite time using a scalar input in the presence of both matched and unmatched unbounded disturbances. No knowledge of all system states is required: only the output should be available for the control design. No knowledge or reconstruction of individual disturbances is assumed as well. This paper concludes with a case study of controlling an industrial benchmark DC motor, whose third-order mathematical model is perturbed by matched and unmatched disturbances.

Adaptive uniform finite-/fixed-time convergent second-order sliding-mode control

ABSTRACT This paper presents an adaptive gain algorithm for second-order sliding-mode control (2-SMC), specifically a super-twisting (STW)-like controller, with uniform finite/fixed convergence time, that is robust to perturbations with unknown bounds. It is shown that a second-order sliding mode is established as exact finite-time convergence to the origin if the adaptive gain does not have the ability to get reduced and converge to a small vicinity of the origin if the adaptation algorithm does not overestimate the control gain. The estimate of fixed convergence time of the studied adaptive STW-like controller is derived based on the Lyapunov analysis. The efficacy of the proposed adaptive algorithm is illustrated in a tutorial example, where the adaptive STW-like controller with uniform finite/fixed convergence time is compared to the adaptive STW controller with non-uniform finite convergence time.

Chattering Analysis of HOSM Controlled Systems: Frequency Domain Approach

In this article, a methodology of chattering analysis of Sliding Mode/Higher Order Sliding Mode (SM/HOSM) control systems in the frequency domain is presented. A numerical method for computing the Describing Functions (DFs) of HOSM control algorithms is given. The algorithm for predicted chattering parameters in dynamically perturbed system via the Describing Function-Harmonic Balance (DF-HB) technique is proposed. The stability conditions for limit cycles in dynamically perturbed HOSM control systems are presented. The concepts of Practical Stability Phase Margin ( PSPM) and Practical Stability Gain Margin (PSGM) as the robustness metrics to unmodeled dynamics in HOSM control systems are introduced. The methodologies for computing PSPM and PSGM via DF-HB technique are provided. The accuracy of the proposed chattering analysis techniques is confirmed via computer simulations.

Frequency domain analysis of HOSM systems

The frequency domain analysis of dynamically perturbed Higher Order Sliding Mode (HOSM) systems is tackled using Describing Function (DF) and Harmonic Balance (HB) techniques. The goal of this analysis is to study possible limit cycles in such systems. DFs of Nested 3rd and 4th order algorithms are obtained for the first time. Then, HB equation is used to analyze the real sliding motion in the HOSM system, where the sliding set converges to a limit cycle. Chattering (limit cycle) in the HOSM systems is studied, and the chattering parameters (amplitude and frequency) are computed. A definition of Tolerance Limits, which characterizes the acceptable performance in the real HOSM system, is applied to verify if the chattering parameters fit the amplitude and frequency limits. Next, Performance Phase Margin and Performance Gain Margin definitions, which give the metrics for robustness of real HOSM to unmodeled dynamics, are applied to assess the robustness of the limit cycle emerged in the real HOSM system. Examples and simulations that validate the obtained results are presented.

Chattering analysis of uniform fixed-time convergent Sliding Mode Control

Abstract In this paper, chattering in systems controlled by a super-twisting based fixed-time convergent controller is analyzed. The analysis is performed by introducing dynamical perturbations into the system driven by the fixed-time convergent controller that yields self-sustained oscillations with a finite frequency and a finite amplitude. Describing Function-Harmonic Balance technique is used to estimate the amplitude and the frequency of those self-sustained oscillations. Then, the stability of the self-sustained oscillations is analyzed. Furthermore, considering a DC motor as a case study, chattering of super-twisting based fixed-time convergent controller is compared with chattering of the traditional super-twisting controller having the same gains.

Practical Stability Phase and Gain Margins Concept

A new concept of chattering characterization for the systems driven by finite-time convergent controllers (FTCC) in terms of practical stability margins is presented. Unmodeled dynamics of order two or more incite chattering in FTCC driven systems. In order to analyze the FTCC robustness to unmodeled dynamics the novel paradigm of Tolerance Limits (TL) is introduced to characterize the acceptable emerging chattering. Following this paradigm a new notions of Practical Stability Phase Margin (PSPM) and Practical Stability Gain Margin (PSGM) as a measure of robustness to cascade unmodeled dynamics is introduced. Specifically, PSPM and PSGM are defined as the values that have to be added to the phase and gain of dynamically perturbed system driven by FTCC so that the characteristics of the emerging chattering reach TL. For practical calculation of PSPM and PSGM, the Harmonic Balance (HB) method is employed and a numerical algorithm to compute Describing Functions (DFs) for families of FTCC (specifically, for nested, and quasi-continuous Higher Order Sliding Mode (HOSM) controllers) was proposed. A database of adequate DFs was developed. A numerical algorithm for solving HB equation using the Newton–Raphson method was suggested to obtain predicted chattering parameters. Finally, computational algorithms that identify PSPM and PSGM for the systems driven by FTCC were proposed. The algorithm of a cascade linear compensator design that corrected the FTCC, making the values of PSPM and PSGM to fit the prescribed quantities, was presented. In order to design the flight-certified FTCC for attitude for the F-16 jet fighter, the proposed technique was employed in a case study. The prescribed robustness to cascade unmodeled dynamics was achieved.

Closing gaps for aircraft attitude Higher Order Sliding Mode Control Certification

Prior to be implemented in aircraft flight control system, Higher Order Sliding Mode (HOSM) controllers must be certified for robustness to unmodeled dynamics. There exist certain standards imposed on Phase Margin (PM) and Gain Margin (GM) for linear controller to be certified for controlling aircrafts. In this work, the conventional control system certification approach based on PM and GM is extended to HOSM controllers. The proposed Practical Stability Phase Margin (PSPM) and Practical Stability Gain Margin (PSGM) are used to quantify the robustness of an attitude F-16 aircraft control against unmodeled dynamics. The algorithms for the identification of PSPM and PSGM are developed using Describing Function-Harmonic Balance method. HOSM control is cascaded with linear compensator to satisfy required PSPM and PSGM. The presented HOSM control-based technology that includes enforcement of PSPM and PSGM is closing the gap for certification of aircraft HOSM controllers.

Continuous finite-time output-feedback control for systems with unmatched unbounded disturbances

This paper presents two continuous finite-time convergent control algorithms driving an output (the highest relative degree state) of an n-dimensional integrator to zero for a finite time using a scalar input in the presence of both matched and unmatched unbounded disturbances. No knowledge of all system states is required: only the output should be available for the control design. No knowledge or reconstruction of individual disturbances is assumed as well. The paper concludes with a case study of controlling a DC motor, whose 3D mathematical model is perturbed by matched and unmatched disturbances.

Adaptive fixed-time convergent super-twisting-like control

This paper presents an adaptive gain algorithm for second order sliding mode control (2-SMC), specifically a super-twisting (STW)-like controller, with fixed convergence time, against perturbations with unknown bounds. The Lyapunov-based 2-SMC adaptive algorithm guarantees reduction of the adaptive gain and provides minimal gain overestimation of the perturbation bounds. The estimate of fixed convergence time of the studied adaptive STW-like controller is derived based on the Lyapunov analysis. The efficacy of the proposed adaptive algorithm is illustrated in a tutorial example, where the adaptive STW-like controller with fixed convergence time is compared to the adaptive STW controller with finite convergence time.