Krasovskii Passivity and μ-Synthesis Controller Design for Quasi-Linear Affine Systems

Abstract

This paper presents an end-to-end method to design passivity-based controllers (PBC) for a class of input-affine nonlinear systems, named quasi-linear affine. The approach is developed using Krasovskii’s method to design a Lyapunov function for studying the asymptotic stability, and a sufficient condition to construct a storage function is given, along with a supply-rate function. The linear fractional transformation interconnection between the nonlinear system and the Krasovskii PBC (K-PBC) results in a system which manages to follow the provided input trajectory. However, given that the input and output of the closed-loop system do not have the same physical significance, a path planning is mandatory. For the path planning component, we propose a robust controller designed using the μ-synthesis mixed-sensitivity loop-shaping for the linearized system around a desired equilibrium point. As a case study, we present the proposed methodology for DC-DC converters in a unified manner, giving sufficient conditions for such systems to be Krasovskii passive in terms of Linear Matrix Inequalities (LMIs), along with the possibility to compute both the K-PBC and robust controller alike.

Sampling rate selection for multi‐loop cascade control systems in an optimal manner

“…Robust synthesis methods allows the design of both structured or unstructured regulators alike [35,36]. For nonlinear systems, the gain-scheduling technique is widely-used [37], while a nonlinear component proposed by the authors in [38] can be induced, resulting in a cascade controller structure. As such, assuming an LTI framework, the resulting continuous-time and amplitude controllers have the state-space representation:…”

“…If the abandonment counter for a specific employed bee exceeds a predefined threshold, such an employed bee becomes a scout bee and a new initialization as in (38) is performed. Moreover, to avoid searching in an area of domain  where the cost function  indicates that the inner or the outer closed-loop system is unstable, the ith employed bee directly becomes scout bee if  (p…”

“…Robust synthesis methods allows the design of both structured or unstructured regulators alike [35, 36]. For nonlinear systems, the gain‐scheduling technique is widely‐used [37], while a nonlinear component proposed by the authors in [38] can be induced, resulting in a cascade controller structure. As such, assuming an LTI framework, the resulting continuous‐time and amplitude controllers have the state‐space representation: $$$$\begin{equation} {\left(K(s)\right)}: {\begin{cases} \dot{ \mathbf {x}}_c = A_K \mathbf {x}_c + B_K \mathbf {u}_c\\ \mathbf {y}_c = C_K \mathbf {x}_c + D_K \mathbf {u}_c \end{cases}}.$$…”

μ-Synthesis FO-PID for Twin Rotor Aerodynamic System

“…As a future iteration, we propose to find a decentralized controller having a nonlinear component and a linear and time-invariant (LTI) robust component. The nonlinear component needs to be designed such that the lower linear fractional transform interconnection between the plant and such a component is asymptotically stable, as in [28], where the passivity-based control framework has been extended for quasi-linear input-affine systems. Additionally, the LTI robust controller can be designed using the proposed method, the decentralized controller managing to ensure robust stability and performance for the nonlinear system.…”