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Delay-Adaptive Linear Control

Yang Zhu and Miroslav Krstic
Princeton University Press
Publication Date: 
Number of Pages: 
Princeton Series in Applied Mathematics
[Reviewed by
Bill Satzer
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Actuator and sensor delays are among the most challenging issues for control theorists in engineering practice, and they have the most adverse effects on stability and transient performance of systems under feedback control. The authors of this monograph focus their attention on control systems governed by ordinary differential equations (ODEs). They consider finite-dimensional linear time-invariant systems with discrete or distributed input delays. The focus is on uncertainties that arise because of delays, unmeasured actuator state, unknown parameters of the plant, and unmeasured state of the finite-dimensional part of the plant. The word “plant” to control engineers means the system that needs to be controlled. It includes a combination of processes, sensors, and actuators.
Control of fluid level in an industrial manufacturing process provides a simple example. Suppose fresh liquid enters a reaction chamber through an inlet, and waste fluid is drained through an outlet. A valve regulates the outlet. The plant state \( X(t) \) represents the liquid present in the container, and the control input \( u(t) \) governs the valve’s opening and closing. Typically the reaction chamber is located on the floor of the factory but the control commands come from a remote control room. A long distance communication network with a controller-to-actuator delay connects the two. Delays can mean spills because of overfilling the reaction chamber.
Applications like this arise very broadly in the chemical, biomedical, electrical, mechanical, and traffic control worlds. Most often there are multiple sensors and actuators. Stabilization of systems with actuator delays is especially challenging, particularly when information about the system is incomplete. One of the most effective approaches to deal with uncertain ODE systems is adaptive control, which has been extensively treated in the control theory literature. Adaptive control operates in real-time to estimate model parameters and provide feedback control in the presence of uncertainties and changing parameter values. An important advance in treating actuator delay has been to treat the actuator delay via a first-order hyperbolic transport partial differential equation (PDE) to convert linear time-invariant ODE systems with time delays into ODE-PDE cascades. That makes it possible to apply adaptive control of PDEs to linear time-invariant ODEs. The authors develop this idea in stages starting with the simplest case. They focus throughout on adaptive control of systems with large input delays.
This book is intended for mathematicians and specialists in control theory who have interests in the broad area of feedback of linear systems and adaptive control of systems with uncertainty. The three parts of the book address progressively more challenging delay conditions. The first considers adaptive control of single-input systems with a discrete input delay. Uncertain multi-input systems with discrete input delays are treated in the second part. The third part looks at uncertain systems with distributed input delays. 
The approach offered here may be of special interest to both researchers and practitioners because it investigates many different uncertainty conditions using quite a variety of techniques. The authors provide an extensive supporting bibliography, and it is used many times to point to necessary background material. This is clearly a book intended for those with a solid background in control theory. The authors implicitly assume that the reader is familiar with concepts like predictor feedback, integrator backstepping, and ODE-PDE cascade systems with boundary control, and they provide very little in the way of background material in the book itself.


Bill Satzer (, now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications ranging from speech recognition and network modeling to optical films and ceramic fiber-reinforced aluminum composites. Along the way he learned more about ceramics and alloys of aluminum than he ever would have imagined in graduate school. He did his PhD work in dynamical systems.
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