1.1 A General View on Control System Design

Control loops exist extensively in biological systems, engineering systems, financial systems, social systems, and so on [1]. Whereas feedback is an essential characteristic in control systems, factors like developments of technologies, performance requirement strengthening, and so on make analysis and synthesis of a control system very different in distinct stages. In a classic control system, both its input and output are scalars. A system with this property is usually called single-input single-output system, which is often abbreviated as SISO. For these systems, transfer functions and differential equations are widely utilized in describing their input–output properties. Methods like Bode diagram, root locus, and so on, play central roles in the analysis and synthesis of a linear and time-invariant SISO system. The most extensively adopted controller in classic control systems is the so called PID controller, with its abbreviations P, I, and D standing respectively for proportion, integration, and differentiation. This controller is a special kind of the so-called lead-lag controller, its realization has been well supported by standard industrial products, and various methods have been developed for the adjustments of this controller to satisfy design specifications of a control system. Methods like description functions, phase plane, and so on have also been developed to analyze systems with some special types of nonlinearities.

Around the 1960s, with the requirements from astronautics and aeronautics, plants began to emerge in which there are multiple variables that are to be controlled simultaneously. These plants are usually called multi-input multi-output systems, which are often abbreviated as MIMO, and state space models are invented to describe their dynamics. For these systems, the Luenberger observer and the Kalman filter have been developed to estimate their states, among many other methods. Moreover, a method called the linear quadratic Gaussian control, usually abbreviated as LQG, has been developed to construct a controller. In addition to this, many other methods, such as decoupling control, model predictive control, and so on, have been developed to control an industrial MIMO process.

When modeling errors are explicitly and directly taken into account in system analysis and synthesis, robust control theory has been developed to handle this problem. One of the well-known robust controller design methods is the $H_{\infty }$ control, in which the $H_{\infty }$-norm of a transfer function matrix is minimized guaranteeing internal stability of the feedback control system. This norm is originally introduced by Zames [2] and found to be appropriate in specifying both the size of modeling errors and the induced gain from a disturbance input vector to an error output vector in control systems. It is extensively believed that one of the motivations for introducing this induced norm into system analysis and synthesis is to bring modeling errors in a plant dynamics description, which is described in the frequency domain, back to the central stage in system analysis and synthesis. Note that in classic control theories, robustness of a feedback control system is reflected by its gain margin and phase margin, which is measured by the frequency response of its open-loop transfer function, whereas in the control theories developed around 1960s, whose representative results include LQG control, pole placements, and so on, an explicit and accurate model is required, which is usually given in a state space form. Two of the most important results associated with this model error description seem to be the so-called small gain theorem and two Riccati-equation-based formulas for the $H_{\infty }$ control problem. Although the $H_{\infty }$-norm is widely regarded to be suitable in describing unmodeled dynamics, it usually introduces conservativeness in dealing with parametric modeling errors. To overcome this drawback, a block diagonal structure is suggested for modeling error descriptions, and structured singular values (SSV) are defined to measure the robustness of a feedback system with both parametric modeling errors and unmodeled dynamics [3,4]. SSV computations, however, are proven to be generally NP-hard [5]. This computational difficulty greatly hampers applications of the SSV to system analysis and synthesis.

In all these theories, communications have not been explicitly taken into account, and it is implicitly assumed that all data transmissions in a feedback control system, which include those among plant subsystems, those between a plant and its controller, and so on, are performed with an infinite precision in value, an infinite communication bandwidth, and an infinite speed. This implicit assumption makes state estimation algorithms, control algorithms, and so on completely independent of communications and greatly simplifies system analysis and synthesis. The associated results work well for traditional engineering systems. Due to technology developments in sensors, communications, and so on, as well as more complicated and demanding tasks expected for a system, the number of subsystems increases significantly, and they are expected to cooperate to achieve a much higher performance level. In addition, digital communication networks are expected to be used in transmitting information among different components of a control system in order to reduce hardware costs of the system and to improve maintenance capabilities. Systems with these characteristics are becoming ubiquitous, with applications ranging from electricity power systems to rescue robot teams, remote surgery, and so on.

However, with the increment of the number of subsystems and the introduction of public communication channels into a control system, some essentially new and challenging issues arise in system analysis and synthesis. For example, the structure of a plant becomes an important factor, as well as communication qualities. More precisely, in a networked system, data transmission and information processing cannot be performed instantaneously in general. Data may be delayed, be out of order, and even be dropped out, noting that data transmissions through communication networks are usually corrupted due to noises in the communication medium, congestion of a communication network, and protocol malfunctions and that communication channels may change from time to time, and so on. To make things worse, there may even exist some attackers who inject malicious disturbances into the system with an objective of destroying its functions. All these bring new challenging issues in system analysis and synthesis.

On the other hand, for most of large-scale networked systems, rather than utilizing a single processing unit, it is preferable to distribute the control tasks and/or estimation tasks among several processing units. In this task division, sparseness of the system is an important factor, which needs to be taken into account, as well as the topology of subsystem connections. In addition, these processing units may not be triggered by a common clock pulse, which makes the sampling, holding, and computation activities of these processing units not synchronized. In other words, different processing units may have different sampling periods. When communication channels are shared by several networked systems, the sampling period of each processing unit may have some irregularities, as time sharing of a communication channel often hampers a precise scheduling for data transmissions in a particular plant.

Generally speaking, analysis and synthesis of a networked system is a relatively new and challenging field in system theories, which requires knowledge on feedback control systems, communication theory, information theory, and so on, which previously belong to different engineering disciplines. The purposes of this book are to investigate some important issues and summarize some important recent works in this area.

1.2 Communication and Control

In a control system, both information and energy are transmitted. Information transmission is necessary as a deviation of the plant output from a desirable trajectory and must be recorded and processed by a signal processing unit, which is usually called a controller or a regulator, and the processed deviations or plant outputs must be sent back to the plant as its input to make the plant work properly. In other words, information transmission is essential in the improvement of plant performances. Different from signal processing, energy transmission is also necessary in a control system, noting that power is required in accomplishing any task in a mechanical system, a biology system, an electrical system, and so on. Particularly, energy transmissions usually cannot be performed exactly in a control system due to technology difficulties in accurately generating the required amount of energies for accomplishing the tasks, which will introduce model errors into the system. A general situation is that the bigger the transmitted energy, the larger the model error. As feedback is usually adopted in a control system, the influence of modeling errors may be amplified if the associated controller has not been well designed. This makes robustness a much more essential issue in control system designs comparing to those of other fields like signal processing, mechanical system design, and so on.

Traditionally, information transmissions in a control system are assumed to be instantaneous and to have an infinite precision. As mentioned in the previous section, with the introduction of public communication channels into a control system, these conditions are no longer satisfied. In addition to this, communication costs must also be taken into account in the design of a networked system. Naturally, it will be appreciated if less data is sent with a lower precision in a networked system, provided that the required tasks can be accomplished satisfactorily by the associated plant. In other words, in the analysis and synthesis of a networked system, one of the major concerns is on the description of the minimum amount of information transfer required for satisfactory system performances, in which stability is usually the most important factor. Information transfer may happen among various parts of the system, for example, between different subsystems, between a local controller and a subsystem, and so on. Another important issue is about the influences on system performances from the data transmission rate and the data expression precision. It is also interesting to investigate effective coordinations among different subsystems with minimum requirements on information exchanges.

When a system consists of a great amount of subsystems and these subsystems are spatially far from each other, and when a system has a great amount of states, it is usually not appreciative, and even prohibitive/impossible, in actual applications either to estimate all the plant states by a single centralized estimator or to control all the subsystems with a single centralized controller. In addition to computational costs, the reasons also include considerations from maintenances, robustness against failures of a subsystem, and so on. On the other hand, if each subsystem is independently estimated or controlled, then performances of estimations or control may be significantly deteriorated. In the worst case, even stability of the closed-loop system may not be reached although the whole system is controllable. This means that some coordinations, which is sometimes also called cooperations, are necessary among the estimators/controllers designed for each individual subsystem. To realize these coordinations/cooperations, communications among these estimators/controllers are also necessary. In other words, the estimator/controller itself also consists of several subestimators/subcontrollers, and these subestimators/subcontrollers may be connected through public communication channels, which may cause data missing, data disordering, and so on.

In communication networks, source signals are usually sampled and encoded into a sequence of channel input symbols, which is then transmitted through some communication media, for example, antenna, satellites, optical fibers, and so on and received by an equipment that gives a sequence of channel output symbols. A perfectly designed communication network intends to completely recover the original source signals from the channel output symbols. Since external disturbances are unavoidable during transmissions and quantification is widely adopted in communications, a completely perfect source signal recovery is usually impossible, noting that under these situations, two different resource signals may lead to a completely equal channel output sequence. To describe the capability of a communication channel in reliably transmitting signals, a concept called channel capacity is suggested by Shannon during World War II, which provides a mathematical model that can be accurately computed. Intuitively, the capacity of a communication channel is the tightest upper bound about the rate at which a signal can be reliably transmitted over it. Alternatively, a channel capacity can also be explained using the noisy-channel coding theorem as the highest information rate (in units of information per unit time) that can be achieved by a communication channel with an arbitrarily small error probability. Mathematically, the capacity of a communication channel is given by the maximal value of the mutual information between the input signal and output signal of the channel, in which the maximization is taken over all possible probability density functions of the input signal.

To deal with analysis and synthesis of a networked system, it appears necessary to introduce an appropriate model of communication channels into the descriptions of its dynamics. With the concept of channel capacity, various models are expected to be developed to meet this requirement, which characterize the communication constraints in a networked system depending on the underlying channel characteristics and information pattern.

More precisely, when data missing is concerned, a Bernoulli process model and a Markov chain model are extensively adopted. When the data missing is described by a Bernoulli process, the communication channel is usually called an erasure channel in communications, whereas in the case of a Markov chain, it is called a Gilbert–Elliott channel. In an erasure channel, data loss is assumed to be independent of each other, and its influences on state estimation and system control are relatively easy to be analyzed. However, the Gilbert–Elliott channel appears to be a more realistic model in the description of data losses due to imperfect communications, since it takes influences of the previous states of a communication network on its current states, which is closer to actual situations. A cost is that this model may make system analysis more complicated. With these data missing models, the capacity of a communication channel can be simply characterized. For example, when the Bernoulli process is used, if information is contained in the input signal to indicate that it is a signal, then the channel capacity can be proved to be equal to the probability that a data packet is not lost.

When external disturbances are taken into account, they are usually treated as an additive noise, which is simple and yet representative in various communication channels. Under such a situation, the channel capacity constraint arises as a bound that is usually put on the power of channel input signals with the purpose to reduce interferences among different communication users and to meet hardware requirements. When the external disturbance has a Gaussian distribution, the channel capacity can be simply computed using the signal-to-noise ratio of the channel.

However, when a channel capacity is derived using the mutual information, this capacity is achievable under several assumptions that in general cannot be easily satisfied in practical applications. These assumptions include that the capacity-achieving code can be arbitrarily long, there does not exist any restriction on the coding complexity, and so on. In addition, causality has not been explicitly taken into account in this derivation. Note that a control system usually requires feedbacks, and a long code often leads to significant time delays that are usually not very appreciative in control system designs. This means that in an actual engineering problem, the channel capacity usually cannot be achieved. On the other hand, when a wireless communication channel is used, due to the effects of multipaths and shadowing in a wireless channel, a signal may experience fluctuations in its transmissions. This phenomenon is still difficult to be modeled satisfactorily in a general setting, and only some simple statistical models have been proposed, such as the Rayleigh model, the Rician model, and so on, which depend heavily on the particular signal propagation environments and transmission scenarios. Moreover, for a large-scale networked system, a multiple-input multiple-output communication network appears to be necessary. However, there is still no mature theory that successfully deals with information transmissions in such a communication network facing channel inferences and external noises.

In summary, when public communication channels are adopted in a networked control system, various efforts are still required for establishing an appropriate model for a communication channel that satisfies requirements raised by controller analysis and synthesis.

1.3 Book Contents

Being aware of the importance of communications in a networked system, as well as that centralized estimation and/or centralized control is not very appropriate for a large-scale system and systems that are constituted from subsystems that are geometrically far away from each other, this book investigates five important issues in the analysis and synthesis of a large-scale networked system, which are listed as follows.

These topics are dealt with respectively in the following Chapter 3 to Chapter 12.

1.4 Bibliographic Notes

It appears that systematic investigations on a large-scale dynamic system emerged around the beginning of the 1960s. Various research literature and research monographs have been published after that time, in which a state space model is extensively adopted with an emphasis on subsystem interconnections through their states. Other models include hierarchical model, graph-based model, and so on. Relations among system stability, regulation performances, and system structure have been extensively studied from various different aspects. Major results can be found in [6] and the references therein.

Revealing causal relations between different time series record sets is also a long and attractive topic. A historic review on major achievements and recent advances in this field can be found in [7], [8], and the references included in these monographs. The former is more focused on relations between the recorded data of two time series, whereas the latter has put many efforts on data generated from a dynamic system that consists of a large amount of subsystems.

Recent interests in networked systems were mainly triggered by the introduction of a public communication channel into the system used to transfer signals between a plant and a controller or from a plant to a state estimator and to coordinate a group of subsystems with neighbor information to construct a desired formation. Imperfect signal transfers make it necessary to introduce a model of communication channels in system analysis and synthesis, whereas variations of neighbors ask to take system structure switchings into account in the design of a control law. The monograph [9] appears to be the first research that summarizes major research results in a unified way on estimation and control with communication channels, whereas [10] places special emphasis on the influences of model errors in the analysis and synthesis of a networked system. In [11], a game theory-based approach is adopted in the investigation of the interactions between information and control in a networked system, which consists of several subsystems, has heterogeneous communication media, uses decentralized and distributed measurements, and acquires information with possible delays. Unified studies are also given in [12] on the effects of imperfect communications on the stability and performances of a networked system.