Matrix-Scheduling of QSR-Dissipative Systems

Abstract

This paper considers gain-scheduling of QSR-dissipative subsystems using scheduling matrices. The corresponding QSR-dissipative properties of the overall matrix-gain-scheduled system, which depends on the QSR properties of the subsystems scheduled, are explicitly derived. The use of scheduling matrices is a generalization of the scalar scheduling signals used in the literature, and allows for greater design freedom when scheduling systems, such as in the case of gain-scheduled control. Furthermore, this work extends the existing gain-scheduling results to a broader class of QSR-dissipative systems. The matrix-scheduling of important special cases, such as passive, input strictly passive, output strictly passive, finite L2 gain, very strictly passive, and conic systems are presented. The proposed gain-scheduling architecture is used in the context of controlling a planar three-link robot subject to model uncertainty. A novel control synthesis technique is used to design QSR-dissipative subcontrollers that are gain-scheduled using scheduling matrices. Numerical simulation results highlight the greater design freedom of scheduling matrices, leading to improved performance.

Figures (7)

• Figure 1: Gain-scheduling of $N$ subsystems using two different gain-scheduling architectures. The scalar scheduling signals $s_{{\it i}}(t)$ and the scheduling matrices $\boldsymbol{\Phi}_{u, {\it i}}(t)$ and $\boldsymbol{\Phi}_{y, {\it i}}(t)$ are designed to interpolate between the subsystems to achieve acceptable performance.
• Figure 2: Gain-scheduled system $\bar{\bm{\mathcal{G}}}$, composed of $N$ parallel QSR-dissipative subsystems. The input and output of each subsystem are scheduled as per \ref{['eq:GS_QSR_io']} via the matrix multiplication between the scheduling matrices $\boldsymbol{\Phi}_{u, {\it i}}(t)$ and $\boldsymbol{\Phi}_{y, {\it i}}(t)$, and their corresponding signals $\mbf{u}(t)$ and $\mbf{y}_{{\it i}}(t)$, respectively.
• Figure 3: Gain-scheduling of the $i^{th}$ subsystem, $\bm{\mathcal{G}}_i$. The input and output of $\bm{\mathcal{G}}_i$ are scheduled as per \ref{['eq:GS_QSR_io']} via the matrix multiplication between the scheduling matrices $\boldsymbol{\Phi}_{u, {\it i}}(t)$ and $\boldsymbol{\Phi}_{y, {\it i}}(t)$, and their corresponding signals $\mbf{u}(t)$ and $\mbf{y}_{{\it i}}(t)$, respectively.
• Figure 4: Planar rigid three-link robotic manipulator with a fixed base, damped joints, no forces acting on the end-effector, and no potential energy due to gravity
• Figure 5: Gain-scheduled feedback control of the plant to be controlled. The input and output of each subcontroller are scheduled as per \ref{['eq:GS_QSR_io']} via the matrix multiplication between the scheduling matrices $\boldsymbol{\Phi}_{u, {\it i}}(t)$ and $\boldsymbol{\Phi}_{y, {\it i}}(t)$, and their corresponding signals $\mbf{u}(t)$ and $\mbf{y}_{{\it i}}(t)$, for ${\it i} \in \mleft\{ 1, 2, 3 \mright\}$.

Theorems & Definitions (25)

• 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: QSR-Dissipativity QSRHill_Moylan_QSR
• Definition 6: Active Scheduling Matrices
• Definition 7: Pseudo-Commuting Scheduling Matrices
• Lemma 1
• proof
• Theorem 1