Passivity-Based Gain-Scheduled Control with Scheduling Matrices

TL;DR

This work extends passivity-based gain scheduling to matrix scheduling signals, enabling richer design freedom for MIMO systems while preserving stability via the passivity framework. It proves that a matrix-gain-scheduled controller composed of VSP subcontrollers remains VSP when scheduling matrices are strongly active and bounded, generalizing prior scalar-signal results. SPR subcontrollers are synthesized using LQR with the Kalman–Yakubov-Popov (KYP) lemma, with multiple linearization points for a rigid two-link planar manipulator; a constant feedthrough ensures VSP behavior. Numerical simulations show that matrix scheduling outperforms scalar scheduling by achieving substantially lower RMS tracking errors, while torque demands remain comparable, highlighting practical gains in performance and flexibility for uncertain nonlinear plants.

Abstract

This paper considers gain-scheduling of very strictly passive (VSP) subcontrollers using scheduling matrices. The use of scheduling matrices, over scalar scheduling signals, realizes greater design freedom, which in turn can improve closed-loop performance. The form and properties of the scheduling matrices such that the overall gain-scheduled controller is VSP are explicitly discussed. The proposed gain-scheduled VSP controller is used to control a rigid two-link robot subject to model uncertainty where robust input-output stability is assured via the passivity theorem. Numerical simulation results highlight the greater design freedom, resulting in improved performance, when scheduling matrices are used over scalar scheduled signals.

Figures (4)

• Figure 1: Gain-scheduled controller $\bar{\bm{\mathcal{G}}}$, composed of $N$ parallel VSP subcontrollers. The node $\otimes$ performs matrix multiplication between the scheduling matrices $\boldsymbol{\Phi}_i(\bm{\zeta }(t), \mbf{x}(t),t)$ and the signals $\mbf{u}_{\text{c}}(t)$ and $\mbf{y}_i(t)$ resulting in \ref{['eq:GS i/o']}. The positive constants $\alpha_i$ are used to scale the gain.
• Figure 2: Rigid two-link robotic manipulator with joint angles $\theta_1$ and $\theta_2$ and joint torques $\tau_1$ and $\tau_2$.
• Figure 3: Scalar scheduling signals $s_1(t)$, $s_2(t)$, and $s_3(t)$ defined in \ref{['eq:scalar_scheduling_signals']}.
• Figure 4: Gain-scheduled feedback control of the plant to be controlled $\bm{\mathcal{G}}_0$, prewrapped with proportional control, and the gain-scheduled controller $\bar{\bm{\mathcal{G}}}$.

Theorems & Definitions (12)

• Definition 1: Induced Matrix Norm Zhou_Robust_Control
• Definition 2: Truncated Signal Marquez
• Definition 3: Truncated Inner Product Marquez
• Definition 4: $\mathcal{L}_p$ Signal Spaces Marquez
• Definition 5: Very Strictly Passive (VSP) Marquez
• Definition 6: Active Scheduling Matrices
• Lemma 1
• proof
• Theorem 1
• proof