Division of Information Technology, Engineering and the Environment
School of Mathematics and Statistics
School of Mathematics and Statistics
Ganti Prasada Rao, International Centre for Water and Energy Systems, PO Box 2623, Abu Dhabi. UAE.
Keywords: Systems, Block diagram, Characteristic equation, Characteristic polynomial, Controller, constitutive relations, Discrete time systems, Effort variable, Feedback, Flow variable
Forced Response, Free response, Frequency response, Interconnective constraints, Laplace transform, Open loop control, Plant, Pole, Sampled data, Signal flow graph, Similarity transformation, SISO, State space, State vector, Time invariant systems, Time response, Time-varying systems, Transfer function, Z-Transform, Zero
3. Mathematical Models of Dynamical Systems
3.1. Differential Equation Models for Lumped Parameter Systems in Continuous Time Domain
3.2. State Space Description of Lumped Parameter Systems
3.3. Linear Time-Invariant Systems
3.4. Discrete-Time Systems or Sampled Data Systems
3.5. Block Diagram Representation and Simplification of Systems:
3.6. Distributed Parameter Systems:
3.7. Deterministic and Stochastic Systems
3.8. Nonlinear Models and Linearization
3.9. Causal and Non-Causal Systems
3.10. Stable and Unstable Systems
3.11. Single-Input-Single-Output (SISO) and Multiple-Input-Multiple-Output (MIMO) Systems
Glossary
System: A system is a set of components, physical or otherwise, which are connected in such a manner as to form and act as an entire unit.
Block diagram: A graphic representation of a system showing the individual elements/subsystems and their interconnections. Based on certain conventions, block diagrams can be manipulated and simplified for ease of analysis.
Canonical form: A canonical form is a compact form of the mathematical model that involves minimal number of parameters.
Characteristic equation: An algebraic equation that portrays the inherent nature of a linear time-invariant dynamical system such as stability. In a rational transfer function this equation is obtained by equating the denominator to zero.
Characteristic polynomial: The denominator of a rational transfer function.
Compensator: The controller in a control system is sometimes referred to as a compensator.
Constitutive relations: The descriptions of the basic physical phenomena and properties of physical elements. They are also known as material relations.
Continuous time systems: Systems described in the continuous time domain.
Control signal: The signal that is applied to a controlled plant in order to make it respond in a certain desired way.
Controller: The device or unit that generates the control signal by considering the error in a control signal in a control system. A computer may act as a controller in a control system.
Discrete time systems: Sampled data systems or systems described in the discrete time domain.
Effort variable: A variable in a system whose product with the so-called flow variable has the sense of power (rate of energy). It is also known as across variable.
Eigenvalue: The eigenvalue of a matrix A is the root of the characteristic equation: sI-A=0
Feedback: Feedback is an arrangement by which the actual output of a system is fed back to the input end for comparison with the desired output.
Flow variable: A variable in a system whose product with the so-called effort variable has the sense of power (rate of energy). It is also known as through variable.
Forced Response: The response of a system due only to the input from outside in the absence of initial conditions.
Free response: The response of a system due only to the initial conditions and no other input from outside.
Frequency response: The steady-state response of a system to sinusoidal signals of unity amplitude and variable frequency. This function in the frequency domain is obtained by setting s = jw in the system transfer function.
Interconnective constraints: Conditions arising out of the connections among the elements within a system that constrain the definition of variables in a system. They are based on Kirchhoffs laws in a generalized setting.
Laplace transform: A mathematical transformation that converts the calculus of time invariant linear differential equations into an algebra thereby lending simplicity to the analysis and design of control systems.
MIMO: Multiple-input-multiple-output
Open loop control: Control without feedback
Plant: The object that is to be controlled.
Pole: The point in the s-plane where the system transfer function attains an infinite value. It is also a root of the characteristic equation of the system.
Sampled data: Signals and information available only at certain sampling instants.
Signal flow graph: A graphical representation of the interconnections of the subsystems in a system in which nodes denote signals and branches represent subsystems.
Similarity transformation: A transformation in state space that changes the state variable coordinate system without altering the system properties. The eigenvalues of a matrix remain unaltered under similarity transformation.
SISO: Single-input-single-output.
State space: The higher dimensional space in which the dynamics of a system is studied in terms of the trajectory of the state vector.
State vector: Vector whose elements are the state variables of a dynamical system.
Time invariant systems: Dynamical systems whose properties are time invariant. The parameters of the model of a time-invariant system are constants.
Time response: The time history of the output of a system.
Time-varying systems: Dynamical systems whose properties change in time. The parameters of the model of a time-varying system are independent functions of time.
Transfer function: A mathematical function that characterizes the transfer behaviour of a system. It is the ratio of the Laplace transform of the output in the absence of initial conditions, to the Laplace transform of the input.
Z-Transform: A mathematical transformation that converts the calculus of time invariant discrete time dynamical systems into an algebra thereby lending simplicity to the analysis and design of digital control systems. The relation (a), with T as the sampling period, connects the Laplace and z- transforms.
(a)
Zero: The point in the s-plane where the system transfer function attains a zero value.
Summary
This paper presents a perspective of the elements of control systems. Human engineered control systems form part of automation that is characteristic of our society particularly in the present times. Systems are made as collections of certain individual elements assembled and connected in specific ways to perform functions for which they are intended. Systems are controlled to meet specified needs and control techniques enhance their performance as control systems. We understand systems for their behaviour by modelling, simulation and analysis. Mathematical models of dynamical systems can be obtained either in time domain or in frequency domain. A particular model for a system can be obtained in a chosen form by determining the numerical values of the parameters associated with the model based on input-output data. This process is known as system identification. Feedback control can be designed for a system with a known model with reference to certain performance criteria such as stability, steady-state accuracy, optimality, disturbance rejection, etc. Controller action can be realized in a computer that works with sampled signals. In the presence of uncertainties and unknown disturbances, stochastic estimation and control techniques are to be applied. When the plant characteristics vary during the period of operation adaptive control techniques may be used to render the controller adaptive to the changing conditions. Supported by powerful computational facilities in the control environment features such as learning and decision making can be incorporated to render control as intelligent and control systems can be made fully automatic and autonomous. The history of control dates back to the ancient times but the beginning of an era of theory and practice of automatic control was made in the 18th century following the inception of the governor. Major developments took place in the 20th century.
Systems are sets of components, physical or otherwise, which are connected in such a manner as to form and act as entire units. Control is the effort to make systems act as desired. A process is the action of a system or alternatively, a system in action.
Humans have created control systems as technical innovations to enhance the quality and comfort of their lives. Human engineered control systems are part of automation, which is a feature of our modern life. They are applied in several aspects of our daily life- in heating and air conditioning to control our living environment and in many of our household appliances. They significantly relieve us from the burden of operation of complex systems and processes and enable us to achieve control with desired precision. Control systems enable accurate positioning and control of machine tools in metal cutting operations and automate manufacturing processes. They automatically guide and control space vehicles, aircraft, large sea going vessels, and high-speed ground transportation systems. Modern automation of a plant involves components such as sensors, instruments, computers and application of techniques of data processing and control. The principles and techniques of automatic control may be applied in a wide variety of systems in order to enhance the quality of their performance.
Control systems are not human inventions; they have naturally evolved in the earths living system. The action of automatic control regulates the conditions necessary for life in almost all living things. They possess sensing and controlling systems and counter disturbances. An automatic temperature control system, for example, makes it possible to maintain the temperature of the human body constant at the right value despite varying ambient conditions. The human body is a very sophisticated biochemical processing plant in which the consumed food is processed and glands automatically release the required quantities of chemical substances as and when necessary in the process. The stability of the human body and its ability to move as desired are due to some very effective motion control systems. A bird in flight, a fish swimming in water or an animal on the run- all are under the influence of some very efficient control systems that have evolved in them.
The field of automatic control is very well developed. The established techniques in this field can be applied to the control of a wide range of systems - engineering systems such as machines and complex plants, natural systems such as biological and ecological systems, and non-physical systems such as economic and sociological systems following the understanding of the similarity of the underlying problems.
Understanding a system for its properties is prerequisite to the creation of a control system for it. Before attempting to control a system, it is essential to know how it generally behaves and responds to external stimuli. Such an understanding is possible with the help of a model. The process of developing a model is known as modelling.
Physical systems are modelled by applying the phenomenological laws that govern their behaviour. For example, mechanical systems are described by Newtons laws and electrical systems by Ohms, Faradays and Lenzs laws. These laws form the basis for the constitutive properties of the elements in a system.
Physical systems may be regarded as energy manipulating units and modelling them is based on the distribution and transfer of energy taking place within them. Energy from certain sources enters a system schematically as shown in Figure 1 and is manipulated within the system by the various components and subsystems in accordance with their inherent properties and depending on the manner in which they are connected inside the system. Energy manipulation phenomena are studied in terms of a pair of variables whose product has the sense of power and thereby the meaning of energy. Some elements store energy and some convert it onto another form. When an element converts energy into heat, it is termed as a dissipater. The assignment of the term dissipator to such elements seems to be prejudiced by their association with heat, a form of energy that is degenerate and vulnerable to loss or dissipation, although the generated heat may indeed be intended for use, say for heating.
Fig.1: Physical system as an energy manipulator
The energy manipulations in system elements are studied in terms of effort variables and flow variables whose product corresponds to the rate of energy or power as indicated in Figure 2 in general. For instance, in an electrical system shown in Figure 3, voltage is regarded as an effort variable and current as the flow variable. Because of the manner in which the effort and flow variables occur , for instance, as voltage across an element and current through it, they are also termed as across and through variables respectively.
The elements within a given system may have the property to store or dissipate energy. Energy stores are classified as effort stores and flow stores. For example, in electrical systems, inductors accumulate the effort variable (voltage) and capacitors accumulate the flow variable (electric current). Resistors convert electrical energy into heat and are termed as dissipaters.
It is the presence of stores that renders a system dynamic. Figures 4 and 5 show the representations in fluid and mechanical systems respectively.
Fig.2: Effort and flow variables
Fig.3: A simple electrical system
Fig 4. : A simple fluid system
Fig.5: A simple mechanical system
Mathematical modelling of a system is the process of obtaining a mathematical description that adequately describes the aspects of its behaviour, which are of interest in the context of a study. Modelling is by itself a well-developed field and there are some general approaches that are applicable to a wide variety of systems. The following are some important approaches to physical system modelling:
Network methods
Variational methods
Bond graph methods
The network methods of system modelling are based on generalization of the methods of electrical network theory. First, all the elements in the system are described (modelled) by their constitutive properties in terms of storage, dissipation, and conversion by applying the physical laws governing their behaviour. Next, generalized Kirchhoffs laws are applied to take into account the connections among the elements in the system. These give rise to the so-called continuity and compatibility conditions, which constrain the effort and flow variables in accordance with the system configuration. As a result of these constraints, the effort and flow variables of the individual elements in a system cannot all be assigned independent labels. The variables are bound by the structural configuration of the system or in other words, the manner in which the individual elements are connected in the system. Figure 6 shows how the effort variables in a closed loop are constrained, and Figure 7 shows how the flow variables are constrained. The effort variables in the system of Figure 6 representing a loop are such that their algebraic sum is zero. Likewise, the algebraic sum of the flow variables at a junction is zero. This condition is termed the continuity constraint because this implies continuity, that is, the inflows and the outflows must be equal at a junction.
Fig.6: Compatibility constraint on effort variables
Fig.7: Continuity constraint on flow variables
Graph theoretic methods may be applied as general tools to apply the interconnectivity constraints. These constraints will eliminate the redundancy in the labels chosen to describe the variables. For example, in the loop of Figure 6, only one flow variable is to be defined and it applies to all the components by virtue of the series connection. Furthermore, it is enough if all but one of the effort variables in the loop are labelled. The unlabeled variable is naturally determined by the negative sum of these n-1 variables. Thus application of the interconnectivity constraints brings down the multitude of the system variables to the appropriate number and mutual relationships. The resulting equations are then arranged in the desired form to represent the system model.
The variational methods of Lagrange and Hamilton avoid explicit formulation of both sets of interconnectivity constraints. Only one set needs to be directly known and the other is complementary and implicit in these methods. Complex couplings of different energy handling media are particularly susceptible to the variational approach. In this approach infinitesimal alterations in certain key system effort or flow accumulation variables, without transgressing the related compatibility or continuity constraints, are considered as admissible variations. A scalar function known as the variational indicator has to be zero in a natural configuration. In this approach, variational calculus, Hamiltons principle and Lagranges equation are applied. Lagranges equations, which are in terms of certain energy functions, directly give rise to the differential equations governing the system. This approach is applicable to composite systems containing elements and subsystems belonging to different worlds - electrical, mechanical, etc.
Bond graph methods represent the energetic interactions between systems and their components by single lines termed as energy bonds. Bond graph representation is alternative to the network convention and it is more compact and orderly than the equivalent system graph. It also allows multipart elements to be modelled explicitly and neatly.
Physical system modelling on the basis of the above approaches can be computer aided and software packages are available for this purpose.
(see Mathematical Models, Physical Laws, Electrical Networks, Graph Theory, Variational methods, Bond graphs)
Mathematical models may be in the form of differential, algebraic or logical equations depending on the nature of the system (see General Models of Dynamic Systems). They are useful in providing an understanding of the input-output behaviour and stability studies. They are helpful in the analysis or synthesis of control systems as well as in the simulation studies with the help of analogue, digital or hybrid computers. The mathematical equations are solved in devices, computational or otherwise to display the system behaviour. Through simulation we gain an understanding of the performance of a system under different situations, without the need to run the actual system.
(see Modelling and Simulation, Computational Methods)
Differential equations describing a linear time varying systems may be organized in the form of a set of first order differential equations and written in the form:
(12)
where x is an n-vector (i.e., nx1 matrix) containing the state
variables, u is an r-vector of inputs and y is a
p-vector of outputs. A, B, C, and D are respectively
nxn, nxr, pxn, and pxr matrices. Often D happens to be
a matrix with zeros as its elements so it is not always shown in the
above description. The first equation is called the state equation and
the second is termed as the output equation. If the original
differential equation has constant coefficients, then all these matrices
are also constant. This is known as the state space description.
Techniques of handling linear systems in state space are well
established (see Description and Classification).
One can think of systems assembled from several ideal elements such as resistors, capacitors, inductors, masses, springs, dampers etc. Such systems are called lumped parameter systems. These are described by ordinary differential equations. If a system possesses an infinite number of such infinitesimally small elements that are smoothly distributed, then it becomes a distributed parameter system (DPS) and such systems are described by partial differential equations. A typical example of such a system is an electric transmission line. The voltage on such a line is a function of both distance and time and hence is describable only by a partial differential equation. Distributed parameter systems are often studied by means of lumped approximations. For example, transmission lines are studied with the help of the so called T and P approximations.
(see Partial differential equations)
Uncertainty and disturbances are usual in real systems. In the deterministic case, the signals and the mathematical model of a system are known without uncertainty and the time behavior can be reproduced by repeated experimentation. In the stochastic case this not possible due to uncertainty that exists either in its model parameters or in its signals or in both. The values of the signals or the variables occurring in the system can only be estimated with the help of the methods of probability and statistics. The results are presented as expected values together with the bounds of error (see Probability and Statistics).
Causality usually refers to events in time. A causal or nonanticipatory system is one in which the output xo(t1) at any arbitrary instant t1 depends on its input xi(t) in the past up to and including t = t1. If this property does not exist, then the system is non-causal. All real systems are causal in their temporal behaviour.
A system whose response either oscillates within certain finite bounds or grows without bounds is regarded as unstable. If for every bounded input the output is bounded, the system is said to be I/O stable (see Stability Concepts). If this is not the case, the system is unstable. Stability in linear time invariant systems is easily ascertained by applying well-established criteria. Stability in a nonlinear system is quite complex; it depends on the inputs and the point at which the system is operated. Nonlinear systems can therefore be stabilized by manipulating the input signals acting on them, for example, sometimes by injecting additional high frequency signals. This is not possible in linear systems; its stability cannot be altered by external actions; they have to be stabilized by manipulating their inherent properties, that is, by altering the system parameters. Figuratively therefore we can say that instability in non-linear systems can be cured by medical treatment but in linear systems it requires surgery.
When a system has only one input and one output, it is referred to as a single-input-single-output (SISO) system. When a system has more than one input or more than one output, it is termed as a multi-input multi-output (MIMO) system (see Control of linear Multivariable Systems). The various properties of dynamic systems that are briefly introduced here will be reflected in system models.
In order to obtain a simple mathematical model of an actual relationship in a tractable, but sufficiently accurate form, the structure as well as the parameters should be identified. System identification can be accomplished by two approaches. One is based on the physical principles underlying the phenomenological behaviour of the process and the other is called black-box modelling in which a discrete time model is chosen and its parameters are estimated by fitting the input-output data. In the former approach, applying the physical laws governing the process, the basic relations are written as equations representing balance of certain physical entities in the process. Physical system modelling gives rise to generic models which are native to the continuous-time domain and the numerical values of the parameters in such models can be directly estimated from input-output data using the techniques of identification that are specially developed for continuous-time models in the recent decades. In certain situations, the essential features of the behaviour of a system can be quickly obtained without many details by means of experiment.
To understand the development of control concepts let us consider a SISO system for the sake of simplicity. The basic action in the control of a system is the application of input (control signal). Given a general understanding of the system response (controlled variable) to inputs, a specific input may be applied to give rise to the desired response. This is called open loop control because of the nature of the diagram representing such an action that is shown in Figure 14. The controlled system is also referred to as the plant. Open loop control has obvious limitations. For instance, if there is a disturbance on the output side of the process, control action does not take it into consideration. In order to remove this limitation, feedback has to be provided.
Figure 14: Open loop control system
Figure 15 shows a typical feedback control system. In this system, the actual output is fed back and compared with the desired response. The resulting error is the basis for the application of a control signal to the plant. The controller generates the control signal on the basis of the error. If a mechanical signal has to be applied to the plant, it is generated by an actuator (not explicitly shown in the figure) from the output of the controller.
In this arrangement, the control signal takes the actual controlled variable into account including disturbances if any. The plant is driven (by the control signal) until the error is reduced. This is the principle of feedback control in which feedback is negative.
Figure 15: Feedback control system
A comparison would show the following differences between open loop and closed loop control schemes.
Open loop Control
Open loop operation
The effects of known disturbances alone can be countered. Other disturbances cannot be taken into account.
As long as the controlled plant is itself stable, the control system cannot become unstable, that is the controlled variable cannot oscillate or grow beyond bounds
In open loop control the controller is blind to what actually takes place at the output end and goes on driving the plant in a fixed and predetermined manner.
Close loop control
Closed loop operation using negative feedback
The effects of disturbances are countered by virtue of negative feedback.
Closed loop operation can be unstable even if the plant is stable.
In closed loop or feedback control the controller notices what actually takes place at the output end and drives the plant in such a way as to obtain the desired output.
There can be two different cases of feedback control. One is to reduce the effect of disturbances. Certain variables of a process such as the controlled variable should be maintained at given fixed values despite disturbances. Such a control is called set point control, or regulation, or control for disturbances rejection. The other is tracking, that is, the controlled variable (output) is made to follow, as closely as possible, the desired command (reference) signal. In both the cases, the controlled variables (or outputs) should be measured continuously and compared with the respective reference signals. The resulting error signal has to be made to vanish as much as possible by control action. The control action involves the use of the error signal itself in generating suitable input signal to drive the plant. It may be manual or automatic. The steering of a vehicle along a street manually by the driver is an example of manual control.
The performance of feedback control systems is assessed in terms of the following aspects of the closed-loop behaviour:
Disturbance rejection: The closed loop system design may be specifically addressed to the rejection of disturbances if the situation specifically warrants. For this purpose it is necessary to characterize the disturbances for their nature and the point of occurrence in the system. Then, from the point at which disturbance enters the system, the transfer function of the system may be evaluated towards the output end. The design of a suitable compensator that yields the desired disturbance rejection properties may be obtained and inserted in the system.
Tracking behaviour: The tracking behaviour is important if the output of a system has to follow the input faithfully in time. This requires that the system has good transient response behaviour.
Steady-state accuracy: The accuracy with which a feedback control system responds to inputs is governed by the steady-state error constants, which are evaluated with reference to inputs in the form of polynomials in time. The simplest is the zero degree polynomial, or the unit step function and the other two are the unit ramp and the unit parabola.
Unit step applied at t=0: u(t), whose Laplace Transform is 1/s
Unit ramp applied at t=0: r(t), whose Laplace Transform is 1/s2 .
Unit parabola applied at t=0: r(t), whose Laplace Transform is 1/s3 .
A feedback system whose overall loop transfer function has m poles at the origin of the s-plane is known as a type-m system. That is, system type number denotes the number of pure integrating elements within the feedback loop and as this number increases the steady state behaviour improves provided the system stability does not deteriorate with the increased number of poles at the origin of the s-plane.
The steady state error in the case of a type-0 system is finite for a step input and becomes infinity for ramp and parabolic inputs. In the case of a type-1 system the steady-state error for step inputs is always zero but remains finite for ramp inputs and becomes infinite for parabolic inputs. A type-2 system has no steady-state error for step and ramp inputs but has a finite error for parabolic inputs. The steady-state errors are evaluated in terms of the error constants (see Closed Loop Behavior).
Comparison of the desired and actual output of a system by feedback should ideally provide the error or disparity information over all time, that is, past, present and future with reference to any point in time. The aim of most feedback control strategies is to generate a control input to the plant that would reduce the disparity as far as possible.
Information or knowledge over all time including the future is complete and is referred to in the orient as 'trikaalagnaanam' (In Sanskrit it means tri=three, kaala=time, gnanam=knowledge). The physical world permits us to know the first two of these 'three times', while the future is left for us only to ponder. Due to uncertainty in information and irreversibility of certain physical processes, the symmetry of time is lost; and the so-called Arrow of Time becomes a reality. Nevertheless, efforts aimed at capturing information as far as possible over all time continue by manifesting themselves in the field of estimation, in which smoothing, filtering and prediction are concerned with the past, present and the future respectively.
The controller is the element that implements the desired control strategy; it takes the error between the desired and the actual system response and generates the control input to the plant based on feedback. The control strategy is the result of consideration of one among a wide variety of techniques of control.
A control strategy is the basis on which the control signal is generated from the error signal in a feedback system. In other words, the controller embodies the control strategy as its characteristic.
One of the simplest forms of controller is a relay, which is a simple, rugged, and robust power amplifier. For any positive/negative value of the error the control signal has its full positive/negative value. This is known as bang-bang control and it leads to a non-linear control system. This simple strategy is in the spirit of the advice: use the standard stick to deal even with a small snake.
To realize a simple linear feedback control strategy, the error itself may be made to act directly as the control signal to drive the plant. For more rapid action, the error may be amplified to become the control signal. That is, the control signal u(t) = Kp e(t), where Kp is the amplification factor or gain and e(t) the error signal. Since this strategy relates u(t) with e(t), it is capable of handling only the present with reference to the instant of time t. This is the proportional control action, which is part of a more general proportional-integral-derivative (PID) scheme. Presentation of an amplified version of the error to the plant as control signal makes the plant overact or quick-acting. Such a strategy drives the plant harder, and beyond a certain limit may drive it crazy, that is, into instability. It is possible to determine this limit of stability using the Nyquist criterion in frequency domain. The extent to which one can provide amplification in the feedback loop without causing instability, is known as the gain margin.
The integral strategy
takes the error history from the beginning to the instant of time t into consideration in generating the control signal.
The derivative strategy
probes slightly into the future with respect to t in generating the control signal.
Thus, the well-known PID control strategy may be viewed as an attempt to take into account, the error information over the three-times additively together in some way. Therefore, in general, the controller may be regarded as a dynamic system. The controller is also referred to as the compensator.
In the more general field of systems engineering, the so-called inactive, reactive, interactive and proactive approaches for development are control strategies in a similar spirit. The first approach is tantamount to open loop control taking nothing into account. The others are feedback-based approaches that take into account the past, present and the future respectively in the strategy.
Optimal control: The system performance may be optimized with respect to the controller parameters in a chosen structure employing the techniques of optimal control. A quadratic functional of the state and/or input is defined as performance index. This is optimized with respect the control input. Often the result is converted into a control law in terms of the controller parameters.
The problem of designing a control system for a process, with a given precise model, is straightforward. The Linear Quadratic Regulator (LQR) problem is a typical example of this class of problems for optimal control (see Design of State Controllers). However, control problems in the real world are not so ideal. The problem of control in the presence of noise in measurements is the stochastic control problem, which is characterized by the application of estimation methods (see Control of Stochastic Systems). The Linear Quadratic Gaussian (LQG) problem is for optimal control in the presence of noise.
(see Optimization)
Adaptive control: Quite often, control problems have to be tackled with no process models readily served to the designer. Control design has to be carried out on the basis of knowledge of the process, which is either developed off-line or on-line on the basis of available measurements that are usually subject to uncertainty. This is accomplished by self-tuning. Control design may follow a separate modeling exercise that provides estimates of an approximate plant model together with the limits of uncertainty associated with it. The control is then designed to be robust against such uncertainty. (see Adaptive Control)
If the onus of 'understanding or modeling' the process rests on the designer, and if it has to be taken up while the process is in operation, control techniques will have to be rendered comprehensive by encompassing some estimation method that is capable of providing on-line, on the basis of the available measurements, a process model that is adequate for the purpose of control. Predictive ability is considered to be a desirable feature for a process model for the purpose of control and a class of control techniques based on modeling and prediction are of considerable importance. Modeling of real world processes based on the so called black-box approach, i.e., without the use of physical laws, is of considerable importance in the fields of control and signal processing. Black-box approaches are motivated by circumstances in which the methods employing physical principles are either very complex or surrounded by high uncertainty. These are invariably associated with estimation - a process that is specifically referred to as smoothing, filtering and prediction respectively according to its focus on the past, present and future. These techniques support decision and control in a significant way.
In certain situations in practice, a chosen set of controller parameters may not remain valid over the entire range of operating conditions of a plant. The plant dynamics, which is the basis for controller design, may change thereby necessitating redesign and adaptation of the controller. This is the main principle of adaptive control, which is illustrated in Figure 16.
Fig.16: An adaptive control system
Robust control: In reality, despite efforts by identification and parameter estimation, system models are neither precisely known nor are guaranteed to remain the same under the different conditions of operation. While adaptive techniques automatically tune the control action to meet mainly the latter contingency, the issue of uncertainty is tackled by robust control techniques. Here, the controller is designed for a nominally specified plant model by taking uncertainties and unmodelled plant dynamics such that the resulting control guarantees satisfactory control under the limitations of knowledge of the plant model (see Robust Control).
Intelligent control: The term intelligent control cannot be defined precisely as it encompasses many unusual features and capabilities that characterize the control as intelligent. An important feature of intelligent control is the presence of a body of knowledge on various aspects of control coded and made available in a computer system to aid decisions and actions together with a learning capability. Fuzzy logic control and neural network methods are used in such systems (see Fuzzy control Systems, Neural Control Systems, Expert Control systems).
The fields of systems, control and information processing are closely related to the science of cybernetics which attempts to understand the behavior of systems in nature. This understanding leads to the knowledge towards improving the performance of natural or man-made processes. In recent years, techniques of systems, control and information processing, are handled with less reference to machines and other man-made physical processes, in the general field of Systems Science.
More detailed presentation of the Elements of Control Systems may be found under this topic (see Introduction to Basic Elements, General Models of Dynamic Systems, System Description in Time-Domain, Description in Frequency Domain, Closed-loop Behavior).
The author is grateful to Prof. H. Unbehauen for the opportunity to contribute to the EOLSS and for the helpful suggestions in the preparation of the manuscript.
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Ljung L. (1987). System Identification - Theory for the User, Englewood Cliffs: Prentice Hall.[This presents the theoretical aspects of identification of discrete time models]
Middleton R.H. and Goodwin G.C. (1990). Digital Control and Estimation - A Unified Approach, Englewood Cliffs: Prentice Hall. [This book presents a new perspective of digital control and related methods introducing alternative methods of discretization of continuous time systems]
Papoulis A. (1991). Probability, Random Variables, and Stochastic Processes, New York: McGraw-Hill. [This book presents the background that is needed for handling stochastic processes]
Unbehauen H. and Rao G.P. (1987). Identification of Continuous Systems, Amsterdam: North Holland. . [This presents all essential aspects of identification of continuous time models. The CONTSID toolbox of Matlab contains several methods that are presented in this book]
Wellstead P.E. (1979). Introduction to Physical Systems Modelling, New York: Academic Press. [ This book presents all essential aspects and approaches to physical systems modeling in a general and comprehensive way]
