Ganti Prasada Rao, International Centre for Water and Energy Systems,
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
Contents
- Introduction
- System Modeling
- Mathematical Models of Dynamical Systems
3.1 Differential Equation Models for Lumped Parameter Systems in Continuous Time
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
- Systems Control
4.1 Open Loop Control
4.2 Feedback Control
4.3 Closed-Loop Behavior of Control Systems
4.4 Control Strategies
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 Kirchhoff’s laws in a generalized setting.
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 behavior of a system. It is the ratio
of the
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 
, with T as the
sampling period, connects the
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 behavior by modeling, 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.
1. Introduction
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 earth’s 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 modeling.
Physical systems are modeled by applying the
phenomenological laws that govern their behavior. For example, mechanical
systems are described by
2. System Modeling
Physical systems may be regarded as energy manipulating units and modeling 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 dissipator. 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.
Figure1: 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 dissipators.
It is the presence of stores that renders a system ‘dynamic’. Figures 4 and 5 show the representations in fluid and mechanical systems respectively.
Figure 2. Effort and flow variables
Figure 3. A simple electrical system
Figure 4. A simple fluid system
Figure 5. A simple mechanical system
Mathematical modeling of a system is the process of obtaining a mathematical description that adequately describes the aspects of its behavior, which are of interest in the context of a study. Modeling 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 modeling:
Network methods
Variational methods
Bond graph methods
The network methods of system modeling are based on generalization of the methods of electrical network theory. First, all the elements in the system are described (modeled) by their constitutive properties in terms of storage, dissipation, and conversion by applying the physical laws governing their behavior. Next, generalized Kirchhoff’s 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.
Figure 6. Compatibility constraint on effort variables
Figure 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 labeled. 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,
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 multiport elements to be modeled explicitly and neatly.
Physical system modeling 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)
3. Mathematical Models of Dynamical Systems
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 behavior 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 analog, digital or hybrid computers. The mathematical equations are ‘solved’ in devices, computational or otherwise to display the system behavior. Through simulation we gain an understanding of the performance of a system under different situations, without the need to run the actual system.
(see Modeling and Simulation, Computational Methods)
3.1 Differential Equation Models for Lumped Parameter Systems in Continuous Time Domain
Different classes of differential equations describe different types of dynamical systems. Lumped parameter systems are described by ordinary differential equations. Lumped linear continuous-time systems are described by linear differential equations. For instance, the n-th order linear differential equation with the single input x and single output y of the general form:
(1)
satisfies the principle of superposition by virtue of its linear property. If the coefficients in the above are constant as in Eq. (2), it represents a linear time invariant system of the form:
(2)
Working with linear time
invariant systems becomes simplified with the help of 
is defined as
(3)
This transformation converts
a linear differential equation into an algebraic form in the domain s that represents the complex frequency.
Let 
without loss of
generality. Let the constant initial conditions be defined as
(4)
Then the
(5)
and the transform of the output is
(6)
The denominator term 
is called the characteristic
polynomial.
The response Y(s)
consists of two components. The first term is due to the input and therefore it
is referred to as the forced response
or the zero-state response. The set
of initial conditions (4) represents the initial state of the system. The coefficient of X(s) in the first term of the output expression is referred as the transfer function and it has to be
obtained in the absence of initial conditions as the ratio of the
If a unit impulse function or the Dirac delta function
denoted as is considered as the input x(t) =d(t),
X(s) =1, the forced response
component in Eq. (6) happens to be equal to the transfer function itself. Thus,
the transfer function may also be regarded as the (unit) impulse response and
in time domain the unit impulse response is given by the inverse
(7)
and for any function continuous at t,
defined in the ordinary sense
(8)
The inverse
Suppose that the
characteristic polynomial has n1
roots each equal to –p1, n2
roots each equal to –p2,….,nr
roots each equal to -pr such
that 
. Then 
and the function Y(s)/X(s) can be written as
(9)
where bn=0 unless m=n. The coefficients are given by
(10)
These coefficients are also known as the residues of F(s) at –pi , i = 1, 2, …r. Inverse Laplace transformation of (9) gives:
(11)
where d(t) is the unit impulse function, and bn = 0 unless m=n.
If a system does not contain dead time elements (delay elements) the transfer function F(s) is rational, that is, a ratio of two polynomials. The roots of the numerator polynomial are referred to as the zeros and the roots of the characteristic polynomial, or that of the denominator are termed the poles of the transfer function. These terms are suggestive of the nature of the function F(s) with reference to the complex frequency variable s. If F(s) is viewed as a potential function on the s-plane, the value of the function is zero at the zeros. At the points representing the zeros of the characteristic polynomial the value soars to infinity making the profile of the potential function F(s) at these points in the s-plane appear as poles. For this reason, the roots of the characteristic polynomial are called the poles. In the s-plane, a pole is shown as x and a zero as o.
The
response as 
, is called the steady state response. The system
response as a function of time before it reaches the steady state is called the
transient response. The steady state value can be determined by applying
the final value theorem: 
, if the limit exists.
Notice that nature of f(t) depends on the values of pi, the poles of the transfer function. When a pole is real the response component due to it is purely exponential. If it is negative, the response decays asymptotically in time and when it is positive, the response grows. Complex poles appear as conjugate pairs and the response due to such a pair is sinusoidal in nature. If the pair has a negative real part, the oscillations decay in time and when they have a positive real part, the oscillations grow in amplitude without limit. Referring to the complex s-plane, these conditions are interpreted as conditions for stability for linear time invariant dynamical systems in the following manner. If all the poles of the transfer function lie inside the left half of the s-plane, the system is asymptotically stable. If any pole lies on the imaginary axis, the system is critically stable and if any pole lies in the right half of the s-plane, then the system is unstable. These criteria are illustrated in Figure 8. Routh-Hurwitz stability criteria are used to detect the location of the roots, without actually solving the characteristic equation for its roots. Nyquist criterion ascertains the stability of a closed loop system by examining the transfer function of the open loop system. A more detailed discussion on the stability theory of dynamical systems is given elsewhere (see Stability Concepts).
Figure 8. Stability criteria for linear time invariant dynamic systems
The transfer function F(jw) evaluated along the jw-axis of the s-plane is of significance as it represents the steady state response of the system to a sinusoidal input of frequency w. It is a complex number with magnitude representing the amplification/attenuation and a phase angle that is the phase shift between the input and output signals.
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:
3.2 State Space Description of Lumped Parameter Systems
(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).
3.3 Linear Time-invariant Systems
(13)
gives
The
(14)
If we let D=0 which is the common case,
(15)
The denominator term on the right hand side of the above is the characteristic polynomial. The eigenvalues of the system matrix A are the poles. The coefficient of the term U(s) is the transfer function matrix whose (i,j)-th element happens to be transfer between the i-th input and the j-th output in the multi-input-multi-output system described by Eq. (15).
Solution of the state equation in time domain is direct by analogy with the first order scalar equation dx/dt=ax+bu.
(16)
The state variable representation is not unique; it depends on the choice of the set of state variables, which correspond, to the coordinate system in the n-dimensional space, which is referred to as the state space. Similarity transformation brings about a change in the state variable description without actually influencing the properties of the system. Certain state variable representations are termed as canonical because they involve minimal number of system parameters. State variable representation permits examination of additional properties such as controllability and observability of a system (see System Characteristics). Figure 9 illustrates these properties.
Figure 9. System controllability and observability with respect to segregated subsystems
Clearly, the controllable and observable part of the system is reflected in the input-output behavior and the transfer function of the overall system is given by the controllable and observable part only.
The state variable representation is a more complete description than the transfer function representation. It presents system behavior both internal and external while the transfer function gives the external (input-output) behavior only. The state space description is very appropriate for finite dimensional systems, that is, systems described by ordinary differential equations of a finite degree. If a system has time delays, the resulting delay differential equations cannot be represented easily in state space form. However, if a state space representation is desired, the delay terms have to be represented, in some sense of approximation, as finite dimensional elements. Thus the presence of delay terms in a differential equation gives rise to an arbitrary enlargement of the dimension of the state space.
3.4 Discrete-Time Systems or Sampled Data Systems
If a system variable (signal) y, at any arbitrary instant of time can be varied within known limits continuously, it is called “continuous”. If a signal can take only known discrete amplitude values, then it is called a “quantized signal”. If a signal is known only at certain discrete instants of time, then it is known as a discrete-time (or discrete) signal. If the signal values are given at uniformly sampled instants of time separated by an interval T, T is referred to as the sampling period. The signal itself is referred to as ‘sampled’. Systems, in which such signals occur, are called discrete-time systems, or discrete systems or sampled-data systems. In general, if digital computers are employed in control systems, for instance to act as controllers, only quantized discrete-time data is processed. Linear time invariant discrete time systems are described by difference equations.
(17)
They can be studied by ap