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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Computing Optimal Upper Bounds on the H2-norm of ODE-PDE Systems using Linear Partial Inequalities
Recently, a broad class of linear delayed and ODE-PDEs systems was shown to have
an equivalent representation using Partial Integral Equations (PIEs). In this paper, we use this PIE
representation, combined with algorithms for convex optimization of Partial Integral (PI) operators to
bound the H2-norm for input-output systems of this class. Specifically, the methods proposed here apply
to delayed and ODE-PDE systems (including delayed PDE systems) in one or two spatial variables
where the disturbance does not enter through the boundary.
For such systems, we define a notion of H2-norm using an initial state-to-output framework and
show that this notion reduces to more traditional concepts under the assumption of existence of a
strongly continuous semigroup. Next, we consider input-output systems for which there exists a PIE
representation and for such systems show that computing a minimal upper bound on the H2-norm of
delayed and PDE systems can be equivalently formulated as a convex optimization problem subject to
linear PI operator inequalities (LPIs). We convert, then, these optimization problems to Semi-Definite
Programming (SDP) problems using the PIETOOLS toolbox. Finally, we apply the results to several
numerical examples – focusing on time-delay systems (TDS) for which comparable H2 approximation
results are available in the literature. The numerical results demonstrate the accuracy of the computed
upper bound on the H2-norm.
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- Award ID(s):
- 1935453
- NSF-PAR ID:
- 10483579
- Publisher / Repository:
- IFAC papers online
- Date Published:
- Journal Name:
- IFAC-PapersOnLine
- Volume:
- 56
- Issue:
- 2
- ISSN:
- 2405-8963
- Page Range / eLocation ID:
- 6994 to 6999
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract This article investigates the ‐optimal estimation problem of a class of linear system with delays in states, disturbance input, and outputs. The estimator uses an extended Luenberger estimator format which estimates both the present and history states. The estimator is designed using an equivalent Partial Integral Equation (PIE) representation of the coupled nominal system. The advantage of the resulting PIE representation is compact and delay free—obviating the need for commonly used bounding technique such as integral inequalities which typically introduces conservatism into the resulting optimization problem. The ‐optimal estimator synthesis problem is then reformulated as a Linear Partial Inequality (LPI)—a form of convex optimization using operator variables and inequlities. Such LPI‐based optimization problems can be solved using semidefinite programming via the PIETOOLS toolbox in Matlab. Compared with previous work, the proposed method simplifies the analysis and computation process and resulting in observers which are non‐conservtism to 4 decimal places when compared with Pad‐based ODE observer design methodologies. Numerical examples and simulation results are given to illustrate the effectiveness and scalability of the proposed approach.
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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.more » « less
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In this paper, we present the Partial Integral Equation (PIE) representation of linear Partial Differential Equations (PDEs) in one spatial dimension, where the PDE has spatial integral terms appearing in the dynamics and the boundary conditions. The PIE representation is obtained by performing a change of variable where every PDE state is replaced by its highest, well-defined derivative using the Fundamental Theorem of Calculus to obtain a new equation (a PIE). We show that this conversion from PDE representation to PIE representation can be written in terms of explicit maps from the PDE parameters to PIE parameters. Lastly, we present numerical examples to demonstrate the application of the PIE representation by performing stability analysis of PDEs via convex optimization methods.more » « less
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The Partial Integral Equation (PIE) framework provides a unified algebraic representation for use in analysis, control, and estimation of infinite-dimensional systems. However, the presence of input delays results in a PIE representation with dependence on the derivative of the control input, u˙. This dependence complicates the problem of optimal state-feedback control for systems with input delay – resulting in a bilinear optimization problem. In this paper, we present two strategies for convexification of the H∞-optimal state-feedback control problem for systems with input delay. In the first strategy, we use a generalization of Young's inequality to formulate a convex optimization problem, albeit with some conservatism. In the second strategy, we filter the actuator signal – introducing additional dynamics, but resulting in a convex optimization problem without conservatism. We compare these two optimal control strategies on four example problems, solving the optimization problem using the latest release of the PIETOOLS software package for analysis, control and simulation of PIEs.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.ifacol.2023.10.538
