Developments in Control Theory Towards Glocal Control
2: City University of Hong Kong, China
3: University of California, Los Angeles, USA
4: Kyoto University, Japan
Glocal control, a term coined by Professor Shinji Hara at The University of Tokyo, represents a new framework for studying behaviour of complex dynamical systems from a feedback control perspective. A large number of dynamical components can be interconnected and interact with each other to form an integrated system with certain functionalities. Such complex systems are found in nature and have been created by man, including gene regulatory networks, neuronal circuits for memory, decision making, and motor control, bird flocking, global climate dynamics, central processing units for computers, electrical power grids, the World Wide Web, and financial markets. A common feature of these systems is that a global property or function emerges as a result of local, distributed, dynamical interactions of components. The objective of 'glocal' (global + local) control is to understand the mechanisms underlying this feature, analyze existing complex systems, and to design and create innovative systems with new functionalities. This book is dedicated to Professor Shinji Hara on the occasion of his 60th birthday, collecting the latest results by leading experts in control theories to lay a solid foundation towards the establishment of glocal control theory in the coming decades.
Inspec keywords: nonlinear dynamical systems; networked control systems; robust control; optimal control
Other keywords: optimal control; robust control; control theory; networked dynamical system; mathematical system; global control
Subjects: Control theory
- Book DOI: 10.1049/PBCE076E
- Chapter DOI: 10.1049/PBCE076E
- ISBN: 9781849195331
- e-ISBN: 9781849195348
- Page count: 224
- Format: PDF
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Front Matter
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Part I: Robust and Optimal Control
1 Measurement-based control design for unknown systems
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This chapter outlines an approach to control design for an unknown system, that is, a system whose model or parameters are unknown. We show that a few strategic measurements processed appropriately can determine the control parameters for the otherwise unknown system. A linear circuit and a block diagram example are presented to illustrate the approach.
2 Quantized linear quadratic Gaussian control for scalar systems
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This chapter considers the quantized linear quadratic Gaussian (QLQG) control problem, which is generalized from the classical LQG problem but with the constraint that the feedback signal is quantized by a fixed-rate quantizer. It turns out that the well-known separation principle for LQG control fails to generalize to QLQG, and this is caused by the fact that minimizing the quantization error at each time instant separately does not lead to a minimum cost globally. Here we consider the QLQG problem for a scalar system and present an adaptive quantization scheme. Using this scheme, we show that the quantization distortion order is R2-2R for a large bit rate R. This means that the separation principle holds approximately when the bit rate is sufficient. More importantly, this adaptive quantization scheme guarantees mean-square stability for the closed-loop system.
3 Robust H∞ filter design for nonuniformly sampled systems
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We consider robust H∞ filtering for a sampled-data system whose measurements are sampled at uncertain and nonuniform sampling instants. A discrete-time, fixed-structure filter is considered. The resulting error system is time-varying, which makes the filter design difficult. A procedure is presented to design the filter so that the error system remains robustly stable with an H∞ performance level γ for all possible variations of sampling periods. The effectiveness of the proposed method is demonstrated through a numerical example and a comparison with existing work.
4 Analysis of frequency response across switching
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This chapter deals with systems with switching which occurs only once. For those systems, a method to analyze the gain from the sinusoidal input injected before switching to the output produced after switching is proposed. The obtained gain information will help us to design the frequency weighting functions used in synthesis of compensators to suppress undesirable responses due to disturbances and switching.
5 Optimal tracking and power allocation over AWN feedback channels
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In this chapter, we study the fundamental limitations of multi-input, multi-output (MIMO) networked feedback systems in tracking reference input signals. We adopt an additive white noise (AWN) model for the communication channel, and consider as our performance measure the mean-square error for a system's output to track a random signal with finite power. We show that in this setting the AWN channel power constraint imposes fundamental limits to the system's stabilizability and tracking performance, which depend on the unstable poles and nonminimum phase zeros of the system. In particular, for MIMO systems, these limits are seen to be dependent on the directions of the unstable poles and nonminimum phase zeros. Moreover, we also show that to achieve the optimal tracking performance, the total channel power must be allocated to individual channels proportional to their degrees of difficulty to control, a scheme that departs fundamentally from the Shannon's classical 'water-filling' strategy.
6 Stability analysis for a class of Hamiltonian systems with digital control
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Asymptotic stability of port-controlled Hamiltonian systems with digital static feedback control is considered. Sufficient conditions are derived based on characterization of the gap between continuous-time and digital control. The effectiveness of the proposed method is demonstrated by numerical examples.
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Part II: Mathematical System and Control Theory
7 Maximizing mutual information between random variables and applications to order reduction of stochastic processes
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In this chapter, we define a metric distance between probability distributions of unequal dimensions. Using this metric, we then address the problem of optimally approximating a high-order distribution by another one of a lower, prespecified order. It is shown that both the problem of computing the distance and of finding the optimal reduced-order approximation can be formulated as extensible bin- packing problems, and are thus NP-hard. Polynomial-time suboptimal algorithms are provided for both problems.
8 On compact sets in the graph topology
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The graph topology plays a central role in characterizing the robustness of feedback systems. In particular, it provides necessary and sufficient conditions for the continuity properties of the transfer matrices of stabilized closed-loop systems. It is possible to derive stronger conclusions by confining our attention to a compact set of controllers. Specifically, if a family of plants is stabilized by each controller belonging to a compact set of controllers, then the closed-loop transfer matrix is uniformly continuous, and uniform a priori estimate of the performance can be given. However, at present a precise characterization of compactness in the graph topology is not available. That is the topic of the present paper. In general it appears difficult to give a necessary and sufficient condition for a set to be compact. Hence we give a necessary condition and a sufficient condition, and discuss the gap between the two. The necessary condition is standard, while the proof of the sufficient condition is based on two major theorems in analysis: Montel's theorem on normal families of analytic functions, and the corona theorem for coprimeness in H∞. Finally, it is shown how the notion of a compact set of controllers can be applied to the problem of approximate design and performance estimation for sampled-data control systems.
9 Matrix pencils in time and frequency domain system identification
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We give an overview of system identification methods that employ matrix pencils constructed from the data. The main tools are Loewner matrix pencils in the frequency domain and Hankel matrix pencils in the time domain.
10 Identification of nonparametric piecewise affine models via data compression
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In this chapter, a new identification method for nonparametric piecewise affine (PWA) models is introduced. The method is based on the nonparametric data-based representation of PWA maps and the data compression with the ℓ1 optimization technique, which enable the method to deal with large data sets. This method can be applied to a wide range of modeling problems, and an example with a DC motor system is shown to verify the usability of the method. Also, a method for choosing appropriate compression ratio is discussed.
11 Performance benefits in two-axle railway vehicle suspensions employing inerters
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The aim of this chapter is to investigate the possibility of improving the ride quality of a two-axle railway vehicle with single-stage suspension by using passive suspensions employing inerters. The goal is to improve the ride quality in vertical motion in response to track irregularities while keeping the suspension deflection within acceptable limits. Performance benefits for several simple passive suspension structures are demonstrated here in comparison with a conventional passive suspension.
12 Stabilization of quantum spin systems via continuous feedback control
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In this chapter, we introduce our recent results on feedback control of quantum spin systems. The quantum systems are composed of atoms held in a cavity under continuous measurement by photons and a magnetic field that is applied to atoms for controlling the quantum states. The quantum states can be estimated by quan- tum filtering through the continuous measurement, and the intension of magnetic field is controlled depending on the estimations. We show that our proposing control input attains the global stability on the assigned eigenstates.
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Part III: Networked Dynamical Systems and Glocal Control
13 Combining distance-based formation shape control with formation translation
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Steepest descent control laws can be used for formation shape control based on specified interagent distances, assuming point agents with single integration of the control action to yield velocity. Separately, it is known how to achieve equal velocity for a collection of agents in a formation using consensus ideas, given appropriate properties for the graph describing information flows. This work shows how the two concepts of formation shape control and flocking behavior can be combined when one changes from an agent with single integration to one with double integration, and our new contribution is to do this when, as is common, there is a leader in the formation.
14 Energy management in wireless sensor networks
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This chapter is concerned with system-level energy management in wireless sensor networks. The network is designed to conduct certain tasks that require information from individual sensors to be sent to a base station. Sensors cease functioning when they deplete their energy reserves or may fail abruptly due to random malfunctions. Sensor selection refers to the number of times sensors are interrogated, while sensor scheduling refers to the sequence in which these interrogations are conducted. A sensor management layer that isolates the system objectives from selection/ scheduling is proposed. In the metric of expected network lifetime, it is shown that sensor selection reduces to integer linear programming (ILP). Sensor scheduling is necessary only when random sensor failures are considered or when the task definition is not stationary. Some general principles emerge. If all sensors are equally reliable, the optimal policy is to use the most energetic sensors first. If all sensors are equally energetic, the optimal policy is to use the least reliable sensor first.
15 Distributed randomized PageRank algorithms over unreliable channels
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The PageRank algorithm, employed at Google assigns a measure of importance to each web page for ranking purposes. Recently, we have proposed a distributed randomized approach for this algorithm, where web pages compute their own PageRank by communicating over selected links. Here, the focus is on the effects of unreliability in communication, where random data losses are modeled as an outcome of Markov chains. We provide a generalization of the distributed scheme along with analysis on its convergence.
16 Stabilization of multi-input networked control systems over additive white Gaussian noise channels
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In this chapter, we study stabilization of multi-input networked control systems over additive white Gaussian noise (AWGN) channels. Different from the single-input case, which is available in the literature and boils down to a typical H2 optimal control problem, the multi-input case involves a judicious allocation of the total capacity among the input channels in addition to the design of the feedback con- troller. With this channel-controller codesign, we successfully show that a net- worked multi-input system over AWGN channels can be stabilized by state feedback under channel resource allocation, if and only if the total channel capacity is greater than the topological entropy of the plant. A numerical example is given to demonstrate our result.
17 Clustering of large-scale dynamical networks for glocal control
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A novel state-aggregation method for a dynamical bidirectional network with an input, which we call network clustering, is introduced to generate a multilayered network model. In addition, the concept of a hierarchical distributed observer based on this model is briefly discussed as a possible framework for realizing glocal control.
18 Glocal control for natural oscillations
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When controlling a mechanical system to achieve a periodic trajectory, its resonances, or natural oscillations, may be exploited for increased efficiency. This chapter considers a class of mechanical systems with collocated actuators and sensors and develops a 'glocal' control theory for natural entrainment. More specifically, we propose a control design method for achieving a global property a prescribed mode of natural oscillations - via local actions of distributed control units without direct communications to each other.
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Back Matter
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