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Modern control theory provides us with a spectrum of methods for studying the interconnection of dynamic systems using input-output properties of the interconnected subsystems. Perhaps the most advanced framework for such inputoutput analysis is the use of Integral Quadratic Constraints (IQCs), which considers the interconnection of a nominal linear system with an unmodelled nonlinear or uncertain subsystem with known input-output properties. Although these methods are widely used for Ordinary Differential Equations (ODEs), there have been fewer attempts to extend IQCs to infinitedimensional systems. In this paper, we present an IQC-based framework for Partial Differential Equations (PDEs) and Delay Differential Equations (DDEs). First, we introduce infinitedimensional signal spaces, operators, and feedback interconnections. Next, in the main result, we propose a formulation of hard IQC-based input-output stability conditions, allowing for infinite-dimensional multipliers. We then show how to test hard IQC conditions with infinite-dimensional multipliers on a nominal linear PDE or DDE system via the Partial Integral Equation (PIE) state-space representation using a sufficient version of the Kalman-Yakubovich-Popov lemma (KYP). The results are then illustrated using four example problems with uncertainty and nonlinearity.
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Control of Large-Scale Delayed Networks: DDEs, DDFs and PIEs
Delay-Differential Equations (DDEs) are the most common representation for systems with delay. However, the DDE representation has limitations. In network models with delay, the delayed channels are typically low-dimensional and accounting for this heterogeneity is challenging in the DDE framework. In addition, DDEs cannot be used to model difference equations. In this paper, we examine alternative representations for networked systems with delay and provide formulae for conversion between representations. First, we examine the Differential-Difference (DDF) formulation which allows us to represent the low-dimensional nature of delayed information. Next, we consider the coupled ODE-PDE framework and extend this to the recently developed Partial-Integral Equation (PIE) representation. The PIE framework has the advantage that it allows the H∞-optimal estimation and control problems to be solved efficiently using the recently developed software package PIETOOLS. In each case, we consider a very general class of networks, specifically accounting for four sources of delay - state delay, input delay, output delay, and process delay. Finally, we use a scalable network model of temperature control to show that the use of the DDF/PIE formulation allows for optimal control of a network with 40 users, 80 states, 40 delays, 40 inputs, and 40 disturbances.
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- PAR ID:
- 10483586
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- IFAC-PapersOnLine
- Volume:
- 55
- Issue:
- 30
- ISSN:
- 2405-8963
- Page Range / eLocation ID:
- 97 to 102
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.ifacol.2022.11.035
