Elements of Control Systems


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




  1. Introduction
  2. System Modeling
  3. 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

  1. Systems Control

4.1  Open Loop Control

4.2  Feedback Control

4.3  Closed-Loop Behavior of Control Systems

4.4  Control Strategies




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.

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 behavior 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  , with T as the sampling period, connects the Laplace and z- transforms.

Zero: The point in the s-plane where the system transfer function attains a zero value.




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 Newton’s laws and electrical systems by Ohm’s, Faraday’s and Lenz’s laws. These laws form the basis for the constitutive properties of the elements in a system.

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, Hamilton’s principle and Lagrange’s equation are applied. Lagrange’s 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 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:




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:




Working with linear time invariant systems becomes simplified with the help of Laplace transforms. The Lapalce transform of a function  is defined as


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




Then the  Laplace transform of Eq. (2) is given by





and the transform of the output is




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 Laplace transforms of the output to the input. The expression for output in Eq. (6) also contains another term that depends on the initial conditions only and not on the input. This component of the system output is known as the free response or the zero-input response.


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 Laplace transform of the system transfer function. The impulse function is not an ordinary function of time. That is, the value of this function is not definitively defined at a given time. A unit impulse function d(t) is indirectly defined by the following properties:




and for any function  continuous at t, defined in the ordinary sense




The inverse Laplace transform of Eq. (6) for y(t) is obtained be the method of partial fractions as follows.


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




where bn=0 unless m=n. The coefficients are given by




These coefficients are also known as the residues of F(s) at –pi , i = 1, 2, …r. Inverse Laplace transformation of (9) gives:




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



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

Laplace transformation applied to the state variable model of a general linear time invariant system with lumped parameters in the general state variable form







The Laplace transform of the vector of outputs is given by




If we let D=0 which is the common case,




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.



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.




They can be studied by applying the methods of z-transform, which for a discrete time signal f(nT), n=0,1,2, …, where T is the sampling interval, is defined as




z-transformation of Eq. (17) gives




Discrete time systems are described in state space as




Stability conditions for discrete time systems are discussed with reference to the z-plane. A linear time invariant discrete-time system described as above is stable if the eigenvalues of the A matrix ( poles of the transfer function or roots of the characteristic polynomial) lie within the unit circle centered at the origin of the z-plane.


Strips parallel and symmetric to the real axis in the s-plane will have to be recognized due to sampling and the width of these strips is proportional to the sampling frequency 1/T.  The first of these is the primary strip that is of significance. The z-transform is related to the Laplace transform through the relation . According to this relation, the entire left half of the s-plane is transformed into the inside of the unit circle and the right half into the region outside the unit circle in the z-plane. The jw axis winds itself into the unit circle itself with its various segments in the horizontal strips coinciding with the same circle periodically.


If continuous time systems are to be discretized, the minimum sampling frequency that is necessary to preserve the information in the sampled signal and to avoid aliasing effects is twice the highest frequency occurring in the signal spectrum. This criterion is called Shannon’s sampling criterion. However, one rule of thumb in practice is to select T such that lm T£ 0.5, where lm is the magnitude of the largest eigenvalue of the system. In actual practice it is desirable to make the sampling interval much smaller than the value specified by this rule. The result of such a choice is to force all poles to lie in a small lens shaped region in the z-plane as shown in Figure 10.

Figure10. The region of normal operation in the z-plane

3.5         Block Diagram Representation and Simplification of Systems:

Systems are denoted as transfer elements or blocks. Transfer elements possess a unique direction of action indicated by arrows; their action is not reversible. Every controllable and observable transfer element has at least one input and at least one output. The output of a transfer element depends only on its own input but not on the loading effect of the following connections. Transfer processes are described by block diagrams with appropriate connections among the blocks. Within each block, the mathematical description of the transfer element is written; when it is too large and complex to be accommodated, a symbol denoting the transfer relation is shown. In the case of static nonlinear elements, the description in the block symbol is either in the form of the functional description of the nonlinear characteristic or the graph of the nonlinear function.


Linear time invariant system models transformed into the s- and z-domains attain simple algebraic properties as has been already observed above enabling system models to be manipulated algebraically. For example in the s-domain, if a system G(s) is excited by an input signal U(s), the response is given by G(s).U(s). If two systems G1(s) and G2(s) are in cascade, the transfer function of the overall system is given by their product G1(s).G2(s) as shown in Figure 11.

Figure 11. Simplification of systems in cascade

A system with feedback can be simplified as shown in Figure 12.

Figure 12. Simplification of systems in a feedback loop

Signal flow diagrams are close in spirit to the block diagrams. In a signal flow graph the signals are denoted by nodes and the transfer relations by branches between nodes. A block diagram and its signal flow graph are shown in Figure 13. In manipulating and simplifying a signal flow graph, Mason’s rule offers a general procedure:


The overall transfer function G of a system represented as a signal flow graph is given by



k   = the number of forward (from the input end of the graph towards  the output end) paths

Tk = the transfer function of the forward path given by the product of the transfer functions of the

        cascaded elements.

D = 1 – sum of loop transfer functions + sum of non-touching loop transfer functions taken two at a

       time – sum of loop transfer functions taken three at a time +  ……..

Dk = D - sum of the loop transfer functions touching the k-th forward path.

Figure 13. The block diagram and signal flow diagram for a system

Referring to the signal flow graph shown in Figure 13(b) we identify the following:


Transfer function of the forward path = G1G2G3G4G5

Loop transfer functions:G2H1, G4H2, G7H4, G2G3G4G5G6G7G8

Non-touching loops taken two at a time: G2H1G4H2, G2H1G7H4, G4H2G7H4

Non-touching loops taken three at a time: G2H1G4H2G7H4


D = 1 – [G2H1 + G4H2+ G7H4  + G2G3G4G5G6G7G8] + [G2H1G4H2 + G2H1 G7H4+ G4H2 G7H4] –[G2H1G4H2 G7H4]


D1 = 1 – G7H4


G = TkD1/ D

3.6         Distributed Parameter Systems:

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)

3.7         Deterministic and Stochastic Systems

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).

3.8         Nonlinear Models and Linearization

Most natural systems are nonlinear. An important criterion that distinguished nonlinear systems from linear systems is the principle of superposition. If this principle holds good as it happens in linear systems, the sum of all the individual outputs due to several individual inputs, each considered to be acting alone on the system is equal to the output due to all the inputs acting simultaneously on a system. Nonlinear systems do not obey the principle of superposition. The response of a linear system due to a sinusoidal input signal remains sinusoidal with an amplitude modification and phase shift, whereas a non-linear system produces distortion that gives rise to harmonic components of the input signal frequency in its output.   In stochastic situations, the output of a linear system due to Gaussian random inputs preserves the Gaussian property and in nonlinear systems this is not the case. Linear systems are studied by a fairly general framework of techniques. Although there exist some methods to study nonlinear systems, there is no general methodology, which is universally applicable to nonlinear systems. For this reason nonlinear systems are in general quite complex.


Nonlinear differential equations are not easy to handle. System operation about an operating point is often very relevant in practice and an understanding of such a situation is provided by studying the model of the system obtained by linearizing it about the chosen operating point. The resulting linear model is studied with help of the well-established techniques for linear systems by considering deviations about the point and the related signals as sufficiently small.


Consider the problem of linearizing a function f(x) about a point xo. Expanding the function in Taylor series about the point


f(x)- f(xo) = (df/dx)½         (x - xo)

        x =xo


which is a linear relationship in the form:


df(x) = mo dx


The procedure for linearization for a general nonlinear equation in state space is as follows:






x  (t) = [x1(t) ……..xn(t)]T

u (t) = [u1(t) ……..ur(t)]T


and f(x,u) is a vector nonlinear function.

Linearization of this leads to the linear vector differential equation




Where A and B are Jacobians containing the various partial derivative terms as follows:


A =               B =                                  (23)


If the system parameters vary with time, it is called a time-varying (often referred to as time-variable or non stationary) system. For instance, a rocket represents a time-varying system as its mass changes with time during the course of its flight due to the expense of fuel.

3.9         Causal and Non-Causal Systems

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 behavior.

3.10     Stable and Unstable Systems

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.

3.11     Single-Input-Single-Output (SISO) and Multiple-Input-Multiple-Output (MIMO) Systems

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 behavior of the process and the other is called black-box modeling 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 modeling 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 behavior of a system can be quickly obtained without many details by means of experiment.

4.         Systems Control

4.1         Open Loop Control

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

4.2         Feedback Control

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

q       Open loop operation

q       The effects of known disturbances alone can be countered. Other disturbances cannot be taken into account.

q       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

q       Closed loop operation using negative feedback

q       The effects of disturbances are countered by virtue of negative feedback.

q       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. 

4.3         Closed-Loop Behavior of Control Systems

The performance of feedback control systems is assessed in terms of the following aspects of the closed-loop behavior:


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 behavior: The tracking behavior 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 behavior.


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.

4.4         Control Strategies

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.


Figure16. 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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