Control System Fundamentals of Control System

JECRC UDML COLLEGE OF ENGINEERING(DEPARTMENT OF ELECTRONICS &

COMMUNICATION)

Notes

CONTROL SYSTEM(Subject Code: 6EC5)

Prepared By: Shilpi Lavania

Class: B. Tech. III Year, VI Semester

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Control System Fundamentals of Control System

Syllabus

UNIT I: Examples and application of open loop and close loop systems, Brief idea of multivariable control system, Brief idea of Z-transform and digital control systems, Differential equations, Determination of transfer function by block diagram reduction technique & signal flow graph method

Beyond the Syllabus

Laplace transform

Learning ObjectivesKnowledge of Fundamentals of Control System

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UNIT I

Introduction:

In recent years, control systems have gained an increasingly importance in the development and advancement of the modern civilization and technology. Figure 1-1 shows the basic components of a control system. Disregard the complexity of the system, it consists of an input (objective), the control system and its output (result). Practically our day-to-day activities are affected by some type of control systems. There are two main branches of control systems: 1) Open-loop systems and 2) Closed-loop systems.

Fig. 1-1. Basic components of a control system.

1.1 Basic Components of a Control System:-

The basic ingredients of a control system can be described by:

1. Objectives of control.

2. Control-system components.

3. Results or outputs.

1.2 Examples of Control-System Applications

1. Steering Control of an Automobile

2. Idle-Speed Control of an Automobile

3. Sun-Tracking Control of Solar Collectors

1.3 Open-loop systems:

The open-loop system is also called the non-feedback system. This is the simpler of the two systems. A simple example is illustrated by the speed control of an automobile as shown in Figure 1-2. In this open-loop system, there is no way to ensure the actual speed is close to the desired speed automatically. The

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actual speed might be way off the desired speed because of the wind speed and/or road conditions, such as up hill or down hill etc.

Fig. 1-2. Basic open-loop system.

1.4 Closed-loop systems:

The closed-loop system is also called the feedback system. A simple closed-system is shown in Figure 1-3. It has a mechanism to ensure the actual speed is close to the desired speed automatically.

Fig. 1-3. Basic closed-loop system.

1.5 Block Diagrams and Transfer Functions of Multivariable Systems

In this section, we shall illustrate the block diagram and matrix representations (see

Appendix A) of multivariable systems. Two block-diagram representations of a multivariable

system with/? inputs and q outputs are shown in Fig. 1.4 (a) and (b). In Fig.1.4

(a), the individual input and output signals are designated, whereas in the block diagram of

Fig. 1.4 (b), the multiplicity of the inputs and outputs is denoted by vectors. The case of

Fig. 1-4(b) is preferable in practice because of its simplicity.

Fig. 1.4 shows the block diagram of a multivariable feedback control system. The

transfer function relationships of the system are expressed in vector-matrix form

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(a) (b)

Figure 1-4 Block diagram representations of a multivariable system.

where Y(s) is the q x 1 output vector; U(.v), R(s), and B(s) are all p x 1 vectors; and G(s)

and H(s)are q x p and p x q transfer-function matrices, respectively. we get

provided that I + G(s)H(s) is nonsingular. The closed-loop transfer matrix is defined as

1.6 Z-Transform

The definition of the z-Transform is

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where, x(k) is a discrete time sequence (sampled data). When x(k)is defined for k ≥ 0, i.e. causal, one sided z-transform is given by

The variable z is complex, so is X(z).

1.7 Digital Control Systems

Discrete-data control systems differ from the continuous-data systems in that the signals at

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one or more points of the system are in the form of either a pulse train or a digital code.

Usually, discrete-data control systems are subdivided into sampled-data and digital

control systems. Sampled-dala control systems refer to a more general class of discrete-data systems in which the signals are in the form of pulse data. A digital control

system refers to the use of a digital computer or controller in the system so that the signals

are digitally coded, such as in binary code.

In general, a sampled-data system receives data or information only intermittently at

specific instants of time. For example, the error signal in a control system can be supplied

only in the form of pulses, in which case the control system receives no information about

the error signal during the periods between two consecutive pulses. Strictly, a sampled-data

system can also be classified as an ac system, because the signal of the system is pulse

modulated.

Figure 1.5 illustrates how a typical sampled-data system operates. A continuous-data

input signal r{t.) is applied to the system. The error signal e{t) is sampled by a sampling

device, the sampler, and the output of the sampler is a sequence of pulses. The sampling

rate of the sampler may or may not be uniform. There are many advantages to incorporating

sampling into a control system. One important advantage is that expensive equipment used

in the system may be time-shared among several control channels. Another advantage is

that pulse data are usually less susceptible to noise.

Because digital computers provide many advantages in size and flexibility, computer

control has become increasingly popular in recent years. Many airborne systems contain

digital controllers that can pack thousands of discrete elements into a space no larger than

the size of this book. Figure 1.6 shows the basic elements of a digital autopilot for guided missile

control.

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Fig 1.5 Block diagram of a sampled-data control system.

Fig 1.6 Digital autopilot system for a guided missile.

1.8 Difference Equation

Where, y(k) is output sequence, and u(k) is input sequence. In control systems, b0 is often 0, as input u(k) does not immediately affect output y(k). Take the z-transform of this difference equation considering u(k) = 0 and y(k) = 0 for k < 0.

This leads to a transfer function of the difference equation.

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The output Y (z) for a given input U(z) is given by

1.9 Transfer Function

A Transfer Function is the ratio of the output of a system to the input of a system, in the Laplace domain considering its initial conditions to be zero. If we have an input function of X(s), and an output function Y(s), we define the transfer function H(s) to be:

1.9.1Transfer Function, Poles and Zeros

Consider the general linear time-invariant difference equation describing the input output

relationship of a discrete-time linear system

(4.47)

In Eq. (4.47) the signal x(m) is the system input, y(m) is the system output, and ak and

bk are the system coefficients. Taking the z-transform of Eq. (4.47) we obtain

(4.48)

Eq. (4.48) can be rearranged and expressed in terms of the ratio of a numerator

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polynomial Y(z) and a denominator polynomial X(z) as

(4.49)

H(z) is known as the system transfer function. The frequency response of a system

H(ω) may be obtained by substituting in Eq. (4.49).

1.10 Block Diagrams and Signal-Flow Graphs

1-10.1 Block Diagram

Because of their simplicity and versatility, block diagrams are often used by control engineers to describe all types of systems. A block diagram can be used simply to represent the composition and interconnection of a system. Also, it can be used, together with transfer functions, to represent the cause-and-effect relationships throughout the system. Transfer Function is defined as the relationship between an input signal and an output signal to a device.

Three most basic simplifying rules are described in detail as follows.

(a)Series Connection –

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(b)Parallel Connection –

(c)Feedback Control Systems –

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Some basic rules of simplifying block diagrams are tabulated in Table 1-1.

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Example 1: Find the transfer function of the closed-loop system below.

Solution:

Use case 3 to simplify the inner feedback blocks.

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Use case 8 to get the following block diagram.

Use case 1 to combine the two sets of series blocks.

Use 3 to calculate the overall transfer function of the system

Systems with Two Inputs:

Figure 1-4 shows the block diagram of a system with two inputs, 1) the setpoint, R(s) and 2) the load disturbance, D(s). By superposition,

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Fig. 1-4. System with two inputs.

(i) The component C’(s) produced by R(s) exists only (D(s)=0)

(ii)Similarly, the component C”(s) produced by D(s) exists only (R(s)=0)

The total value of C(s)

G1(s)G2(s)H(s) is the open-loop transfer function. The characteristic equation is

1.10.2 SIGNAL-FLOW GRAPHS (SFGs)

A signal-flow graph (SFG) may be regarded as a simplified version of a block diagram. The

SFG was introduced by S. J. Mason [2] for the cause-and-effect representation of linear

systems that are modeled by algebraic equations. Besides the differences in the physical

appearance of the SFG and the block diagram, the signal-flow graph is constrained by more

rigid mathematical rules, whereas the block-diagram notation is more liberal. An SFG may

be defined as a graphical means of portraying the input-output relationships among the

variables of a set of linear algebraic equations.

Consider a linear system that is described by a set of /V algebraic equations:

It should be pointed out that these N equations are written in the form of cause-and-effect

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relations:

or simply

This is the single most important axiom in forming the set of algebraic equations for SFGs.

When the system is represented by a set of integrodifferential equations, we must first

transform these into Laplace-transform equations and then rearrange the latter in the form

of Eq. or

Basic Elements of an SFG

When constructing an SFG, junction points, or nodes, are used to represent variables. The

nodes are connected by line segments called branches, according to the cause-and-effect

equations. The branches have associated branch gains and directions. A signal can transmit

through a branch only in the direction of the arrow. In general, given a set of equations

such as Eq. (3-31) or Eq. (3-47), the construction of the SFG is basically a matter of following through the cause-and-effect relations of each variable in terms of itself and the

others. For instance, consider that a linear system is represented by the simple algebraic

equation

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Figure 1.5 Signal flow graph of

where yi is the input, y2 is the output, and ci\2 is the gain, or transmittance, between the two

variables. The SFG representation of Eq. is shown in Fig.1.5. Notice that the branch

directing from node y\ (input) to node y2 (output) expresses the dependence of y2 on y 1 but not

the reverse. The branch between the input node and the output node should be interpreted as a

unilateral amplifier with gain a, 2, so when a signal of one unit is applied at the input yi, a signal

of strength is delivered at node y2. Although algebraically Eq. can be written as

the SFG of Fig. 1.5 does not imply this relationship. If Eq. is valid as a cause-and effect

equation, a new SFG should be drawn with y2 as the input and j i as the output.

EXAMPLE: As an example on the construction of an SFG, consider the following set of algebraic equations:

The SFG for these equations is constructed, step by step, in Fig.1.6

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Figure1.6 Step-by-step construction of the signal-flow graph in Eq. (3-50).

Summary of the Basic Properties of SFG

1. SFG applies only to linear systems.

2. The equations for which an SFG is drawn must be algebraic equations in the form

of cause-and-effect.

3. Nodes are used to represent variables. Normally, the nodes are arranged from left

to right, from the input to the output, following a succession of cause-and-effect

relations through the system.

4. Signals travel along branches only in the direction described by the arrows of the

branches.

5. The branch directing from node represents die dependence of upon

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but not the reverse.

6. A signal traveling along a branch between and is multiplied by the gain of

the branch so a signal is delivered at

Mason Rule:

where

Gi(s) = path gain of the ith forward path,

Δ = 1-Σall individual loop gains + Σ gain products of all possible two loops which do not touch − Σ gain products of all possible three loops that do not touch + ·····,

Δi = the ith forward path determinant = the value of Δ for that part of the block diagram that does not touch the ith forward path.

A forward path is a path from the input to the output such that no node is included more than once. Any closed path that returns to its starting node is a loop, and a path that leads from a given variable back to the same variable is defined as a loop path. A path is a continuous sequence of nodes, with direction specified by the arrows, with no node repeating.

Example 2-1: A block diagram of control canonical form is shown below. Find the transfer function of the system.

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Solution:

The determinants are

Applying Mason’s rule, the transfer function is

Example 2-2: Find the transfer function of the following system.

Solution:

The determinants are

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Applying Mason’s rule, the transfer function is

Beyond the syllabus:

Laplace transform 

The Laplace transform is a widely used integral transform with many applications

in physics and engineering. Denoted  , it is a linear operator of a function f(t) with a real argument t (t ≥ 0) that transforms it to a function F(s) with a complex argument s. This transformation is essentially bijective for the majority of practical uses; the respective pairs of f(t) and F(s) are matched in tables. The Laplace transform has the useful property that many relationships and operations over the originals f(t) correspond to simpler relationships and operations over the images F(s). It is named after Pierre-Simon Laplace, who introduced the transform in his work on probability theory.The Laplace transform is related to the Fourier transform, but whereas the Fourier transform expresses a function or signal as a series of modes of vibration (frequencies), the Laplace transform resolves a function into its moments. Like the Fourier transform, the Laplace transform is used for solving differential and integral equations. In physics and engineering it is used for analysis of linear time-invariant systems such as electrical circuits, harmonic oscillators, optical devices, and mechanical systems. In such analyses, the Laplace transform is often interpreted as a transformation from the time-domain, in which inputs and outputs are functions of time, to the frequency-domain, where the same inputs and outputs are functions of complex angular frequency, in radians per unit time. Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications.

The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace, who used a similar transform (now called z transform) in his work on probability theory. The current widespread use of the transform came about soon after World War II although it had been used in the 19th century by Abel, Lerch, Heaviside and Bromwich. The older history of similar transforms is as follows. From 1744, Leonhard Euler investigated integrals of the form

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as solutions of differential equations but did not pursue the matter very far. [2] Joseph Louis Lagrange was an admirer of Euler and, in his work on integrating probability density functions, investigated expressions of the form

which some modern historians have interpreted within modern Laplace transform theory. These types of integrals seem first to have attracted Laplace's attention in 1782 where he was following in the spirit of Euler in using the integrals themselves as solutions of equations. [5] However, in 1785, Laplace took the critical step forward when, rather than just looking for a solution in the form of an integral, he started to apply the transforms in the sense that was later to become popular. He used an integral of the form:

akin to a Mellin transform, to transform the whole of a difference equation, in order to look for solutions of the transformed equation. He then went on to apply the Laplace transform in the same way and started to derive some of its properties, beginning to appreciate its potential power.[6]

Laplace also recognised that Joseph Fourier's method of Fourier series for solving the diffusion equation could only apply to a limited region of space as the solutions were periodic. In 1809, Laplace applied his transform to find solutions that diffused indefinitely in space.

The Laplace transform of a function f(t), defined for all real numbers t ≥ 0, is the function F(s), defined by:

The parameter s is a complex number:

 with real numbers σ and ω.

The meaning of the integral depends on types of functions of interest. A necessary condition for existence of the integral is that f must be locally integrable on [0,∞). For locally integrable functions that decay at infinity or are of exponential type, the integral can be understood as a (proper) Lebesgue integral. However, for many applications it is necessary to regard it as a conditionally convergent improper integral  at ∞. Still more generally, the integral can be understood in a weak sense, and this is dealt with below.

One can define the Laplace transform of a finite Borel measure μ by the Lebesgue integral

An important special case is where μ is a probability measure or, even more specifically, the Dirac delta function. In operational calculus, the Laplace transform of a measure is often treated

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as though the measure came from a distribution function f. In that case, to avoid potential confusion, one often writes

where the lower limit of 0− is shorthand notation for

This limit emphasizes that any point mass located at 0 is entirely captured by the Laplace transform. Although with the Lebesgue integral, it is not necessary to take such a limit, it does appear more naturally in connection with the Laplace–Stieltjes transform.

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