It is common to think and write in time domain quantities, but this is not the best thing to do in creating the mathematical description of the system we are dealing with. Continuous systems- using Laplace or s- transform to create the mathematical description of the system will simplified the process. When dealing with discrete systems, the - transform will be used to simplify the process. 2
The transform is a mathematical tool commonly used for the analysis and synthesis of discrete-time control systems. The transform in discrete-time systems play a similar role as the Laplace transform in continuous-time systems 3
In a linear discrete-time control system, a linear difference equation characteries the dynamics of the system. To determine the system's response to a given input, such a difference equation must be solved. transformation transforms linear time- invariant difference equations into algebraic equations in. 4
To present definitions of the transform, basic theorems associated with the transform, methods for finding the inverse transform. Discussed how to solved the difference equations by using the transform method. 5
Discrete-time signals arise if the system involves a sampling operation of continuous-time signals. The sampled signal is x(0,x(t, x(2t,..., T = sampling period. Such a sequence of values arising from the sampling operation is usually written as x(kt. If the system involves an iterative process carried out by a digital computer, the signal involved is a number sequence x(0, x(, x(2... The sequence of numbers is usually written as x(k, k indicates the order in which the number occurs in the sequence, for example, x(0,x(,x(2... Although x(k is a number sequence, it can be considered as a sampled signal of x(t when the sampling period T is sec. 6
continuous-time signal x, sampled signal x(, and the number sequence x(. 7
The transform of a time function x(t, where t is nonnegative, or of a sequence of values x(kt where k takes ero or positive integers and T is the sampling period, is defined by the following equation: k X( = Z [x(t] = Z [x(kt] = (2. 2 k 0 x( kt For a sequence of numbers x(k, the transform is defined by k X( = Z [x(k] = x( k (2. 3 k 0 The transform defined by Equation (2. 2 or (2. 3 is referred to as the one-sided transform The symbol Z denotes "the transform of." 8
In the one-sided x(t = 0 for t < 0 or x(k = 0 for k < 0. transform, we assume 9
The transform of x(t, where - <t<, or of x(k, where k takes integer values (k = 0, ±, ±2,, is defined by k X( = Z [x(t] = Z [x(kt] = x ( kt (2. 4 k Or X( = Z [x(k] = x ( k (2. 5 k The transform defined by Equation (2. 4 or (2.5 is referred to as the two-sided transform k 0
In the two-sided transform, the time function x(t is assumed to be nonero for t < 0 and the sequence x(k is considered to have nonero values for k < 0
Both the one-sided and two-sided transforms are series in powers of -. (The latter involves both positive and negative powers of -. In our discussion, only the one-sided transform is considered in detail. 2
Expansion of the right-hand side of Equation (2. 2 gives: X( = x(0 + x(t - + x(2t -2 + + x(kt -k + (2. 6 3
4
Let us find the transform of the unit-step function ( t 0 t x( t 0, t 0 In sampling a unit-step function we assume that this function is continuous from the right; that is, (0 =. Then, referring to Equation (2. 2, we have X( = Z [(t] = = + - + -2 + -3 + k k 0 k 0 k It is noted that (k as defined by ( k, k 0, k 0 0,,2,... is commonly called a unit-step sequence. 5
Consider the unit-ramp function x( t t, 0 0, t Notice that x(kt = kt, k = 0,,2,... t 0 Figure 2-8 - sampled unitramp signal. The magnitudes of the sampled values are proporti- onal to the sampling period T 6
transform of the unit-ramp function: X( = Z [t] = x( kt k kt k T k k k 0 k 0 k 0 T ( 2 2 3 3... T ( T 2 ( 2 7
k Obtain the transform of x(k as defined by x( k k a, k 0,,2,... 0, k 0 a = constant 8
k Referring to the definition of the transform given by Equation (2. 3, we obtain X( = Z [a k ] = k 0 x( k k = + a - + a 2-2 + a 3-3 + a a k 0 a k k 9
Find the transform of at e, 0 x( t 0, t t 0 Since x(kt = e -akt, k = 0,, 2, we have X( = Z [e -at ] = x ( kt k 0 k e akt = + e -at - + e -2aT -2 + e -3aT -3 + e at k 0 e at k 20
Consider the sinusoidal function Noting that Sin t, x ( t 0, e j t = cos t + j sin t e -j t = cos t - j sin t we have 0 t t 0 Sin t j t j t 2 j ( e e Cos t j t j t ( e 2 e 2
Since the transform of the exponential function is Z [e -at ] = at Z [e -at ] = X( = Z [Sin t] = Z e at ( t j t j e e X( = Z [Sin t] = Z ( 2 e e j 2 e e j T j T j 2 ( ( 2 e e e e j T j T j T j T j 2 sin T sin 2 T 22 2 cos 2 T cos 2 2 T
EXERCISE Obtain the transform of the cosine function x t Cos t t t 23
Solution X( = Z [cos t] = Z ( t j t j e e X( = Z [cos t] = Z ( e e T j T j 2 e e T j T j ( 2 e e T j T j 2 ( ( 2 2 e e e e T j T j 2 cos 2 cos T T cos 2 cos 2 2 T T 24 cos 2 2 T
EXERCISE 2 (example 2-2 Obtain the transform of X (s s s( s Hint Whenever a function in s is given, one approach for finding the corresponding transform is to convert X(s into x(t and then find the transform of x(t. Another approach is to expand X(s into partial fractions and use a transform table to find the transforms of the expanded terms. 25
Solution The inverse Laplace transform of X(s is x(t = e -t, 0 t Hence, X( = Z [ e -t ] = T e ( ( e T T ( e ( ( e ( T e T 26
Just as in working with the Laplace transformation, a table of transforms of commonly encountered functions is very useful for solving problems in the field of discrete-time systems. Table 2- and table 2.(ctnd is such a table. 27
IMPORTANT PROPERTIES AND THEOREMS OF THE TRANSFORM Will discussed important properties and useful theorems of the transform. Assumption time function x(t is -transformable and x(t = 0 for t < 0. 28
Will discussed properties and theorems:. Multiplication by a Constant 2. Linearity of the Transform 3. Multiplication by a k 4. Shifting Theorem 5. Complex Translation Theorem 6. Initial Value Theorem 7. Final Value Theorem 29
Multiplication by a Constant. If X( is the transform of x(t, then Z [ax(t] = az [x(t] = ax( where a is a constant. To prove this, note that by definition Z [ax(t] = k 0 ax ( kt k a k 0 x ( kt k ax ( 30
Linearity of the Transform. The transform possesses an important property: linearity. This means that, if f(k and g(k are -transformable and and are scalars, then x(k formed by a linear combination x(k = f(k + g(k has the transform X( = F( + G( where F( and G( are the transforms of f(k and g(k, respectively. Prove this property!!! 3
Multiplication by a k If X( is the transform of x(k, then the transform of a k x(k can be given by X(a - : Z [a k x(k] = X(a - This can be proved as follows: Z [a k x(k] = k a k x( k 0 k k 0 x( k( a k = X(a - 32
Shifting Theorem Also call real translation theorem. If x(t = 0 for t < 0 and x(t has the transform X(, then Z [x(t-nt] = -n X( (2.8 and n k Z [x(t+nt] = X ( x ( kt (2.9 n = ero or a positive integer Prove eqn (2.8 and eqn (2.9 n k 0 33
Example 2-3 Find the transforms of unit-step functions that are delayed by sampling period and 4 sampling periods, respectively, as shown in figure (a and (b below 34
Solution Using the shifting theorem given by Equation (2. 8, we have Z [(t-t] = - Z [(t] = Also, Z [(t-4t] = -4 Z [(t] = 4 4 (Note that - represents a delay of sampling period T, regardless of the value of T. 35
Example 2-4 Obtain the transform of f ( a a k 0,, k,2,3,... Solution: Referring to Equation (2. 8, we have Z [x(k-] = - X( The transform of a k is Z [a k ] = a and so Z [f(a] = Z [a k- ] = where k =,2,3,... k 0 a a 36
Example 2-5 Consider the function y(k, which is a sum of functions x(h, where h = 0,, 2,..., k, such that y( k 0 x( h, where y(k = 0 for k < 0. Obtain the transform of y(k. h k k 0,,2,... 37
Solution First note that y(k = x(0 + x( + y(k- = x(0 + x( +. + x(k- + x(k. + x(k- Hence, y(k y(k- = x(k, k = 0,, 2, Therefore, Z [y(k y(k-] = Z[x(k] or Y( - Y( = X( Which yield, Y ( X ( Where, X( = Z [x(k] 38
Complex Translation Theorem If x(t has the transform X(, then the transform of e -at x(t can be given by X(e at. To prove Z [e -at x(t] x( kt k 0 e akt k k 0 x( kt ( e at k X ( e at Thus, we see that replacing in X( by e at gives the transform of e -at x(t. 39
Example 2-6 By using the complex translation theorem, obtain the transforms of: e -at sin t and e -at cos t, respectively, 40
Solution Noting that Z [sin t ] = sin T 2 cos T Using the complex translation theorem e sin cos T T e at at Z [ e sin t ] at 2aT 2 2e 2 4
Solution Z [cos t ] cos T 2 cos T 2 Using the complex translation theorem Z [e -at cos t ] at e cos T at 2 at 2e cos T e 2 42
Exercise By using the complex translation theorem, Obtain the transform of te -at. Hint: T Z [t] = X ( 2 ( 43
Initial Value Theorem. If x(t has the transform X( and if lim X ( exists, then the initial value x(0 of x(t or x(k is given by x ( 0 lim X ( 44
Initial Value Theorem (ctnd. The initial value theorem is convenient for checking transform calculations for possible errors. Since x(0 is usually known, a check of the initial value by lim X ( can easily spot errors in X(, if any exist. 45
Example 2-8 Determine the initial value x(0 if the transform of x(t is given by T ( e X ( T ( ( e By using the initial value theorem, we find T ( e x( 0 lim 0 T ( ( e Referring to Example 2-2, notice that this X( was the transform of t x ( t e and thus x(0 = 0, which agrees with the result obtained earlier. 46
Final Value Theorem. The final value of x(k, that is, the value of x(k as k approaches infinity, can be given by lim x( k k lim[( X ( ] (2. 27 Proved? 47
Example 2-9 Determine the final value x( of X (, a at e by using the final value theorem. By applying the final value theorem to the given X(, we obtain x( lim ( X ( lim ( 0 at e lim e at 48
In this section we have presented important properties and theorems of the transform that will prove to be useful in solving many transform problems. For the purpose of convenient reference, these important properties and theorems are summaried in Table 2. 2. 49
The transformation serves the same role for discrete-time control systems that the Laplace transformation serves for continuous-time control systems. The notation for the inverse transform is Z -. The inverse transform of X( yields the corresponding time sequence x(k. 50
An obvious method for finding the inverse transform is to refer to a transform table. However, unless we refer to an extensive transform table, we may not be able to find the inverse transform of a complicated function of. 5
Other methods: Direct division method Computational method Partial-fraction-expansion method Inversion integral method 52
In obtaining the inverse transform, we assume, as usual, that the time sequence x(kt or x(k is ero for k 0. Before we present the four methods, however, a few comments on poles and eros of the pulse transfer function are in order. 53
Poles and Zeros in the Plane. In engineering applications of the transform method, X( may have the form or m m b0 b... bm X ( ( m n n n a... a b0 ( ( 2...( m X ( ( m n ( p ( p...( p ( 2 pn n (2.28 where the p i s (i =,2,...,n are the poles of X( and the j s (j =,2,...,m the eros of X(. 54
Note that in control engineering and signal processing X( is frequently expressed as a ratio of polynomials in -, as follows: X b b ( n m ( n m 0 ( a a 2 2...... b a n m n n (2.29 where - is interpreted as the unit delay operator. 55
In finding the poles and eros of (, it is convenient to express ( as a ratio of polynomials in. For example, X 2 2 Clearly, ( has poles at = - and = -2 and eros at = 0 and = -0.5. 56
If X( is written as a ratio of polynomials in -, however, the preceding X( can be written as 0.5 0.5 X ( 3 2 2 ( ( 2 Although poles at = -l and = -2 and a ero at = -0.5 are clearly seen from the expression, a ero at = 0 is not explicitly shown, and so the beginner may fail to see the existence of a ero at = 0. Therefore, in dealing with the poles and eros of X(, it is preferable to express X( as a ratio of polynomials in, rather than polynomials in - 57
Direct Division Method. Obtain the inverse transform by expanding ( into an infinite power series in -. Useful when it is difficult to obtain the closed-form expression for the inverse transform or it is desired to find only the first several terms of x(k. 58
The direct division method stems from the fact that if X( is expanded into a power series in -, that is, if or X ( k 0 x( kt k x(0 x( T x(2t 2... x( kt k... X ( k 0 x( k k x(0 x( x(2 2... x( k k... then x(kt or x(k is the coefficient of the -k term. Hence, the values of x(kt or x(k for k = 0,,2,... can be determined by inspection. 59
If X( is given in the form of a rational function, the expansion into an infinite power series in increasing powers of - can be accomplished by simply dividing the numerator by the denominator, where both the numerator and denominator of X( are written in increasing powers of -. If the resulting series is convergent, the coefficients of the -k term in the series are the values x(kt of the time sequence or the values of x(k of the number sequence. 60
Although the present method gives the values of x(0, x(t, x(2t,... or the values of x(0, x(, x(2,... in a sequential manner, it is usually difficult to obtain an expression for the general term from a set of values of x(kt or x(k. 6
Example 2-0 Find x(k for k = 0,,2,3,4 when X( is given by X ( 0 ( ( 5 0.2 First, rewrite X( as a ratio of polynomials in -, as follows: X ( 0 5.2 2 0.2 2 62
Dividing the numerator by the denominator, we have have... 8.68 8.4 7 0 5 0 0.2.2 4 3 2 2 2 2 7 2 2 0 5 0 0.2.2 3 2 3 2 2 2 3.4 20.4 7 2 7 4 3 4 3 2 3 2 3.68 22.08 8.4 3.4 8.4 5 4 5 4 3 4 3 3.736 22.46 8.68 3.68 8.68 6 5 4 5 4 63
Thus, X ( 0 7 2 8.4 3 8.68 By comparing this infinite series expansion of X( with X( = k 0 x( k x(0 = 0 x( = 0 x(2 = 7 x(3 = 8.4 x(4 = 8.68 k, we obtain 4... 64
As seen from this example, the direct division method may be carried out by hand calculations if only the first several terms of the sequence are desired. In general, the method does not yield a closed-form expression for x(k 65
Example 2- Find x(k when X( is given by X ( Solution 66
Example 2-2 Obtain the inverse transform of X( = + 2 - + 3-2 + 4-3 The transform X( is already in the form of a power series in -. Since X( has a finite number of terms, it corresponds to a signal of finite length. By inspection, we find x(0 = x( = 2 x(2 = 3 x(3 = 4 All other x(k values are ero. 67
Two computational approaches to obtain the inverse transform. MATLAB approach Difference equation approach Consider a system G( defined by 2 0.4673 0.3393 G( 2.5327 0.6607 (2.30 68
In finding the inverse transform, we utilie the Kronecker delta function 0 (kt where, 0(kT =, for k = 0 = 0 for k 0 Assume that x(k, the input to the system G(, is the Kronecker delta input, or x(k =, for k = 0 = 0, for k 0 The transform of the Kronecker delta input is X( = 69
Using the Kronecker delta input, Equation (2. 30 can be rewritten as 2 Y ( 0.4673 0.3393 G ( 2 X (.5327 0.6607 2 0.4673 0.3393.5327 0.6607 (2.3 70
MATLAB Approach. MATLAB can be used for finding the inverse transform. Referring to Equation (2. 3, the input X( is the transform of the Kronecker delta input. In MATLAB the Kronecker delta input is given by x = [ eros(,n] where N corresponds to the end of the discretetime duration of the process considered. 7
Since the transform of the Kronecker delta input X( is equal to unity, the response of the system to this input is 2 0.4673 0.3393 Y ( G( 2 2.5327 0.6607 0.4673 0.3393.5327 0.6607 Hence the inverse transform of G( is given by y(0,y(l,y(2,... Let us obtain y(k up to k = 40. 72
To obtain the inverse transform of G( with MATLAB, we proceed as follows: Enter the numerator and denominator as follows: num = [0 0.4673-0.3393] den = [ -.5327 0.6607] Enter the Kronecker delta input. x = [ eros(,40] Then enter the command y = filter(num,den,x to obtain the response y(k from k = 0 to k = 40. 73
MATLAB Program 2- % Finding inverse transform % ***** Finding the inverse transform of C( is the same as % finding the response of the system Y(/X( = G( to the % Kronecker delta input ***** % ***** Enter the numerator and denominator of C( ***** num = [0 0.4673-0.3393]; den = [ -.5327 0.6607]; % ***** Enter the Kronecker delta input x and filter command % y = filter(num,den,x ***** x=[ eros(,40]; y = filter(num,den,x 74
Computational Method MATLAB (ctnd If this program is executed, the screen will show the output y(k from k = 0 to 40 as follows: y = Columns through 7 0 0.4673 0.3769 0.2690 0.632 0.0725 0.0032 Columns 8 through 4-0.0429-0.0679-0.0758-0.072-0.059-0.0436-0.0277 Columns 5 through 2-0.037-0.0027 0.0050 0.0094 0.0 0.008 0.0092 Columns 22 through 28 0.0070 0.0046 0.0025 0.0007-0.0005-0.003-0.006 Columns 29 through 35-0.006-0.004-0.00-0.0008-0.0004-0.0002 0.0000 Columns 36 through 4 0.0002 0.0002 0.0002 0.0002 0.0002 0.000 75
Computational Method-MATLAB (ctnd (Note that MATLAB computations begin from column and end at column 4, rather than from column 0 to column 40. These values give the inverse transform of G(. That is, y(0 = 0 y( = 0.4673 y(2 = 0.3769 y(3 = 0.2690 y(40 = 0.000 76
To plot the values of the inverse transform of G(, follow the procedure given in the following. Plotting Response to the Kronecker Delta Input. Consider the system given by Equation (2. 3. A possible MATLAB program to obtain the response of this system to the Kronecker delta input is shown in MATLAB Program 2-2. The corresponding plot is shown in Figure 2-2. 77
MATLAB Program 2-2 % Response to Kronecker delta input ------------------- num = [0 0.4673-0.3393]; den = [ -.5327 0.6607]; x = [ eros(,40]; k = 0:40; y = filter(num,den,x; plot(k,y, o v=[0 40 - ]; axis(v; grid title ( Response to Kronecker Delta Input xlabel( k ylabel( y(k 78
Figure 2-2 Response of the system defined by Equation (2. 3 to the Kronecker delta input. 79
Difference Equation Approach Noting that Equation (2. 3 can be written as ( 2 -.5327 + 0.6607Y( = (0.4673-0.3393X( Convert this equation into the difference equation as follows: y(k + 2 -.5327y(k + + 0.6607y(k = 0.4673x(k +- 0.3393x(k ---- (2. 32 where x(0 = and x(k = 0 for k 0, and y(k = 0 for k < 0. [x(k is the Kronecker delta input.] 80
The initial data y(0 and y( can be determined as follows: By substituting k = -2 into Equation (2.32, we find y(0 -.5327y(-l + 0.6607y(-2 = 0.4673x(- - 0.3393x(-2 from which we get y(0 = 0 Next, by substituting k = - into Equation (2. 32, we obtain y( -.5327y(0 + 0.6607y(- = 0.4673x(0-0.3393x(- from which we get: y( = 0.4673 8
To find the inverse transform of Y( - solve the following difference equation for y(k: y(k + 2 -.5327y(k + + 0.6607y(k = 0.4673x(k+- 0.3393x(k ---- (2. 33 with the initial data y(0 = 0, y( = 0.4673, x(0 =, and x(k = 0 for k 0. Equation (2. 33 can be solved easily by hand, or by use of BASIC, FORTRAN, or others. 82
The partial-fraction expansion method presented here, which is parallel to the partial-fraction- expansion method used in Laplace transformation, is widely used in routine problems involving transforms. The method requires that all terms in the partial fraction expansion be easily recogniable in the table of transform pairs. 83
To find the inverse transform, if X( has one or more eros at the origin ( = 0, then X(/ or X( is expanded into a sum of simple first or second-order terms by partial fraction expansion, and a transform table is used to find the corresponding time function of each expanded term. It is noted that the only reason that we expand X(/ into partial fractions is that each expanded term has a form that may easily be found from commonly available transform tables 84
Example 2-3 Before we discuss the partial-fraction-expansion method, we shall review the shifting theorem. Consider the following X(: X ( a By writing X( as Y(, we obtain X ( Y ( a 85
Referring to Table 2-, the inverse transform of Y( can be obtained as follows: Z - [ Y ( ] y( k k a Hence, the inverse transform of X( = - Y( is given by Z - [ X ( ] x( k y( k Since y(k is assumed to be ero for all k < 0, we have k y( k a, k,2,3,... x( k 0, k 0 86
Consider X( as given by m m b 0 b... b m b m X (, m n n n a... a a To expand X( into partial fractions, we first factor the denominator polynomial of X( and find the poles of X(: m m b b... bm X ( ( p ( p...( p 0 m 2 n n n b 87
Expand X(/ into partial fractions so that each term is easily recogniable in a table of transforms. If the shifting theorem is utilied in taking inverse transforms, however, X( instead of X(/, may be expanded into partial fractions. The inverse transform of X( is obtained as the sum of the inverse transforms of the partial fractions. 88
A commonly used procedure for the case where all the poles are of simple order and there is at least one ero at the origin (that is, b m = 0 is to divide both sides of X( by and then expand X(/ into partial fractions. Once X(/ is expanded, it will be of the form X ( a p a 2 p 2... a n p n 89
The coefficient a i can be determined by multiplying both sides of this last equation by - p i and setting = p i. This will result in ero for all the terms on the right-hand side except the a i term, in which the multiplicative factor p i has been cancelled by the denominator. Hence, we have a ( pi i X ( Note that such determination of a i is valid only for simple poles. p i 90
If X(/ involves a multiple pole, for example, a double pole at = p and no other poles, then X(/ will have the form X ( c c2 2 ( p ( p The coefficients c and c 2 are determined from 2 X ( c ( p It is noted that if X(/ involves a triple pole at = p, then the partial fractions must include a term ( + p /( p 3. p 9
Example 2-4 Given the transform at ( e X ( ( e ( at where a is a constant and T is the sampling period, determine the inverse transform x(kt by use of the partial-fraction-expansion method. 92
Partial-Fraction-Expansion Method (ctnd ( at B A e X ( ( ( at at at e B A e e X ( ( ( ( ( at at e e X A ( ( ( ( ( at at at e e X e B e ( ( ( ( ( at at e at e e e e e e B at at ( ( at at at at e e e e at e X ( 93 e
The partial fraction expansion of X( is found to be X ( at e From Table 2- we find - Z Z - e at e akt Hence, the inverse transform of X( is akt x ( kt e, k 0,,2,... 94
Difference equations can be solved easily by use of a digital computer, provided the numerical values of all coefficients and parameters are given. However, closed-form expressions for x(k cannot be obtained from the computer solution, except for very special cases. The usefulness of the transform method is that it enables us to obtain the closed-form expression for x(k. 95
Consider the linear time-invariant discrete-time system characteried by the following linear difference equation: x ( k a x ( k... a x ( k n b u( k b u( k... 0 n n b u( k n (2.34 where u(k and x(k are the system's input and output, respectively, at the kth iteration. 96
Let us define Z [x(k] = X( Then x( k, x( k 2, x( k 3,... and x ( k, x ( k 2, x ( k 3,... can be expressed in terms of X( and the initial conditions. Their transforms were summaried in Table 2.3. 97
Table 2.3 transform of x(k+m and x(k-m 98
Example 2-8 Solve the following difference equation by use of the transform method: x(k + 2 + 3x(k + + 2x(k = 0, x(0 = 0, x( = Solution First note that the transforms of x(k + 2, x(k +, and x(k are given, respectively, by Z [x(k + 2] = 2 X( - 2 x(0 - x( Z [x(k + ] = X( - x(0 Z [x(k] = X( 99
Taking the transforms of both sides of the given difference equation, we obtain 2 X( - 2 x(0 - x( + 3X( - 3x(0 + 2X( = 0 Substituting the initial data and simplifying gives X ( 2 3 2 ( ( 2 2 2 00
Noting that k Z - (, Z - 2 ( k k we have x( k ( ( 2, k 0,,2,... 2 k 0
Example 2-9 Obtain the solution of the following difference equation in terms of x(0 and x(: x(k + 2 + (a + bx(k + + abx(k = 0 where a and b are constants and k = 0,,2,... The transform of this difference equation can be given by [ 2 X( - 2 x(0 - x(l] + (a + b[x( - x(0] + abx( = 0 02
Solving this last equation for X( gives [ 2 ( a b ] x (0 x ( X ( 2 ( a b ab Notice that constants a and b are the negatives of the two roots of the characteristic equation. We shall now consider separately two cases: (a a b and (b a = b. 03
(a For the case where a b, expanding X(/ in to partial fractions, we obtain X ( bx (0 x ( ax (0 x (, a b a a a b b from which we get bx(0 x( ax(0 x( X ( b a a a b b The inverse transform of X( gives bx (0 x ( k ax (0 x ( k x ( k ( a ( b, b a a b where k = 0,,2,.. b a b 04
(b For the case where a = b, the transform X( becomes 2 ( 2a x(0 x( X ( 2 2 2 a a x(0 [ ax(0 x(] 2 a ( a Entry 8 x(0 [ ax(0 a ( a x(] 2 The inverse transform of X( gives x(k = x(0(-a k + [ax(0 + x(l]k(-a k-, a = b where k = 0,,2,... Entry 9 05
RECONSTRUCTING ORIGINAL SIGNALS FROM SAMPLED SIGNALS If the sampling frequency is sufficiently high compared with the highest-frequency component involved in the continuous-time signal, the amplitude characteristics of the continuous-time signal may be preserved in the envelope of the sampled signal. To reconstruct the original signal from a sampled signal, there is a certain minimum frequency that the sampling operation must satisfy. Such a minimum frequency is specified in the sampling theorem. We shall assume that a continuous-time signal x(t has a frequency spectrum as shown in Figure 2-3. This signal x(t does not contain any frequency components above radians per second. 06
RECONSTRUCTING ORIGINAL SIGNALS FROM SAMPLED SIGNALS (ctnd Figure 2-3 A frequency spectrum. 07
RECONSTRUCTING ORIGINAL SIGNALS FROM SAMPLED SIGNALS (ctnd Sampling Theorem. If s, defined as 2 /T, where T is the sampling period, is greater than 2 ; or s>2 is the highest-frequency component present in the continuous-time signal x(t, then the signal x(t can be reconstructed completely from the sampled signal x*(t. The theorem implies that if s>2 then, from the knowledge of the sampled signal, it is theoretically possible to reconstruct exactly the original continuous-time signal. 08
Sampling Theorem (ctnd. To show the validity of this sampling theorem, we need to find the frequency spectrum of the sampled signal x*(t. The Laplace transform of x*(t is given by X *( s T k X ( s j sk (2.35 X ( X ( s j s k T k s ( / T ln (2.36 09
Sampling Theorem (ctnd. To obtain the frequency spectrum, we substitute j for s in Equation (2. 35. Thus, X *( j T k X ( j j s k T X ( j( s T X ( j T X ( j( s (2.37 0
Sampling Theorem (ctnd. Equation (2. 37 gives the frequency spectrum of the sampled signal x*(t. The frequency spectrum of the impulse-sampled signal is reproduced an infinite number of times and is attenuated by the factor /T. Thus, the process of impulse modulation of the continuous-time signal produces a series of sidebands. Since X*(s is periodic with period 2 / s or X*(s = X*(s ± j sk, s k = 0,,2,.. if a function X(s has a pole at s = s, then X*(s has poles at s = s ± j sk (k= 0,,2,...
Sampling Theorem (ctnd. Figure 2-4(a and (b show plots of the frequency spectra X*(j versus for two values of the sampling frequency s. Figure 2-4(a corresponds to s > 2 while Figure 2-4(b corresponds to s < 2. Each plot of X*(j versus consists of X(j /T repeated every s = 2 /T rad/sec. In the frequency spectrum of X*(j the component X(j /T is called the primary component, and the other components, X(j( ± s k /T, are called complementary components. 2
Figure 2-4 Plots of the frequency spectra X*(j versus for two values of sampling frequency s: (a s > 2 ; (b s< 2 3
Sampling Theorem (ctnd. If s>2, no two components of X*(j will overlap, and the sampled frequency spectrum will be repeated every s rad/sec. If s < 2, the original shape of X(j no longer appears in the plot of X*(j versus because of the superposition of the spectra. Therefore, we see that the continuous-time signal x(t can be reconstructed from the impulse-sampled signal x*(t by filtering if and only if s > 2 4
Sampling Theorem (ctnd. It is noted that although the requirement on the minimum sampling frequency is specified by the sampling theorem as s>2 where is the highest-frequency component present in the signal, practical considerations on the stability of the closed-loop system and other design considerations may make it necessary to sample at a frequency much higher than this theoretical minimum. (Frequently, s is chosen to be 0 to 20. 5
Ideal Low-Pass Filter. The amplitude frequency spectrum of the ideal lowpass filter G I (j is shown in Figure 2-5. The magnitude of the ideal filter is unity over the frequency range and is ero outside this frequency range. 6
Ideal Low-Pass Filter (ctnd. Figure 2-5 Amplitude frequency spectrum of the ideal low-pass filter. 7
Ideal Low-Pass Filter (ctnd. The sampling process introduces an infinite number of complementary components (sideband components in addition to the primary component. The ideal filter will attenuate all complementary components to ero and will pass only the primary component, provided the sampling frequency s is greater than twice the highest-frequency component of the continuous-time signal. Ideal filter reconstructs the continuous-time signal represented by the samples. 8
Ideal Low-Pass Filter (ctnd. Figure 2-6 shows the frequency spectra of the signals before and after ideal filtering. Figure 2-6 Frequency spectra of the signals before and after ideal filtering. 9
The frequency spectrum at the output of the ideal filter is /T times the frequency spectrum of the original continuous-time signal x(t. Since the ideal filter has constant-magnitude characteristics for the frequency region, 2 s 2 s, there is no distortion at any frequency within this frequency range. there is no phase shift in the frequency spectrum of the ideal filter. (The phase shift of the ideal filter is ero. 20
Ideal Low-Pass Filter (ctnd. It is noted that if the sampling frequency is less than twice the highest-frequency component of the original continuous-time signal, then because of the frequency spectrum overlap of the primary component and complementary components, even the ideal filter cannot reconstruct the original continuoustime signal. (In practice, the frequency spectrum of the continuous-time signal in a control system may extend beyond 2 s even though the amplitudes at the higher frequencies are small. 2
Ideal Low-Pass Filter (ctnd. Let find the impulse response function of the ideal filter. Since the frequency spectrum of the ideal filter is given by 2 s 2 s GI ( j 0 elsewhere the inverse Fourier transform of the frequency spectrum gives g I ( t sin( T s t t / 2 / 2 s (2.38 22
Ideal Low-Pass Filter (ctnd. Equation (2. 38 gives the unit-impulse response of the ideal filter. Figure 2-7 shows a plot of g I (t versus t Notice that the response extends from t = - to t =. This implies that there is a response for t < 0 to a unit impulse applied at t = 0. (That is, the time response begins before an input is applied. This cannot be true in the physical world. Hence, such an ideal filter is physically unrealiable. 23
Figure 2-7 Impulse response g I (t of ideal filter. 24
Ideal Low-Pass Filter (ctnd. In practice ideal low-pass filter is not physically realiable. Because of the ideal filter is unrealiable and signals in practical control systems generally have higher-frequency components and are not ideally band limited, it is not possible, in practice, to exactly reconstruct a continuous-time signal from the sampled signal, no matter what sampling frequency is chosen. In other words, it is not possible to reconstruct exactly a continuous-time signal in a practical control system once it is sampled 25
Frequency-Response Characteristics of the Zero- Order Hold. The transfer function of a ero-order hold is Ts ( e Gh0 s (2.39 s To compare the ero-order hold with the ideal filter, obtain the frequency-response characteristics of the transfer function of the ero-order hold. 26
Frequency-Response Characteristics of the Zero- Order Hold (ctnd. frequency-response x-tics of the ZOH.(Figure 2-8(a Undesirable gain peaks at frequencies of 3 s /2, 5 s /2 and so on. The magnitude is more than 3 db down (0.637 = -3.92 db at 2 frequency. s Because the magnitude decreases gradually as the frequency increases, the complementary components gradually attenuate to ero. Since the magnitude characteristics of the ero-order hold are not constant, if a system is connected to the sampler and eroorder hold, distortion of the frequency spectra occurs in the system. 27
Figure 2.8 (a Frequency-response curves for the eroorder hold; (b equivalent Bode diagram when T = sec. 28
Frequency-Response Characteristics of the Zero-Order Hold (ctnd. The phase-shift characteristics of the ZOH Note that sin ( T/2 alternates positive and negative values as increases from 0 to s, s to 2 s, 2 s to 3 s, and so on. Thus, the phase curve [Figure 2-8(a, bottom] is discontinuous at = k s = 2 k/t where k =,2,3,... Such a discontinuity or a switch from a positive value to a negative value, or vice versa, may be considered to be a phase shift of ±80. In Figure 2-8(a, phase shift is assumed to be -80 29
Frequency-Response Characteristics of the Zero- Order Hold (ctnd. Summary: The frequency spectrum of the output of the ZOH includes complementary components, since the magnitude characteristics show that the magnitude of G h0 (j is not ero for, except at points 2 s where = ± s, = ± 2 s, = ± 3 s,. In the phase curve there are phase discontinuities of ±80 at frequency points that are multiples of s. Except for these phase discontinuities, the phase characteristic is linear in. 30
Figure 2-9 shows the comparison of the ideal filter and the ero-order hold Figure 2-9 Comparison of the ideal filter and the ero-order hold 3
Folding. The phenomenon of the overlap in the frequency spectra is known as folding. Figure 2-20 shows the regions where folding error occurs. 2 The frequency s is called the folding frequency or Nyquist frequency N. That is, N 2 s T 32
Figure 2-20 Diagram showing the regions where folding error occurs 33
Folding(ctnd. In practice, signals in control systems have high- frequency components, and some folding effect will almost always exist. For example, in an electromechanical system some signal may be contaminated by noises. The frequency spectrum of the signal, therefore, may include low- frequency components as well as high-frequency noise components (that is, noises at 60 or 400 H. Since sampling at frequencies higher than 400 H is not practical, the high frequency will be folded in and will appear as a low frequency. Remember that all signals with frequencies higher than 2 s appear as signals of frequencies between 0 and. 2 s 34
Aliasing. In the frequency spectra of an impulse-sampled signal x* (t, where s < 2 as shown in Figure 2-2, consider an arbitrary frequency point 2 that falls in the region of the overlap of the frequency spectra. The frequency spectrum at = 2 comprises two components, X*(j 2 and X*(j( s- 2. The latter component comes from the frequency spectrum centered at = s. 35
Aliasing (ctnd. Figure 2-2 Frequency spectra of an impulsesampled signal x*(t. 36
Aliasing (ctnd. Thus, the frequency spectrum of the sampled signal at = 2 includes components not only at frequency 2 but also at frequency s - 2 (in general, at s ± 2, where n is an integer. When the composite spectrum is filtered by a lowpass filter, such as a ero-order hold, some higher harmonics will still be present in the output. The frequency component at = n s ± 2 (where n is an integer will appear in the output as if it were a frequency component at = 2 It is not possible to distinguish the frequency spectrum at = 2 from that at = n s ± 2. 37
Aliasing (ctnd. As shown in Figure 2-2, the phenomenon that the frequency component s - 2 (in general, n s ± 2, where n is an integer shows up at frequency 2 when the signal x(t is sampled is called aliasing. This frequency s - 2 (in general, n s ± 2 is called an alias of 2. It is important to remember that the sampled signals are the same if the two frequencies differ by an integral multiple of the sampling frequency s. s If a signal is sampled at a slow frequency such that the sampling theorem is not satisfied, then high frequencies are "folded in" and appear as low frequencies. 38
Aliasing (ctnd. To avoid aliasing, choose the sampling frequency high enough s > 2, where is the highest-frequency component present in the signal or use a prefilter ahead of the sampler to reshape the frequency spectrum of the signal (so that the frequency spectrum for 2 s is negligible before the signal is sampled. 39
MAPPING BETWEEN THE s PLANE AND THE PLANE The absolute stability and relative stability of the linear time-invariant continuous-time closed-loop control system are determined by the locations of the closed-loop poles in the s plane. For example, complex closed-loop poles in the left half of the s plane near the j axis will exhibit oscillatory behavior, and closed-loop poles on the negative real axis will exhibit exponential decay. 40
MAPPING BETWEEN THE s PLANE AND THE PLANE (ctnd Complex variables and s relationship = e Ts pole and ero locations in the plane are related to the pole and ero locations in the s plane. The stability of the linear time-invariant discrete-time closed-loop system can be determined in terms of the locations of the poles of the closed-loop pulse transfer function. 4
MAPPING BETWEEN THE s PLANE AND THE PLANE (ctnd Dynamic behavior of the discrete-time control system depends on the sampling period T. Locations of poles and eros in the plane depend on the sampling period T. Therefore, a change in the sampling period T modifies the pole and ero locations in the plane and causes the response behavior to change. 42
MAPPING BETWEEN THE s PLANE AND THE PLANE (ctnd Mapping of the Left Half of the s Plane into the Plane. In the design of a continuous-time control system, the locations of the poles and eros in the s plane are very important in predicting the dynamic behavior of the system. In designing discrete-time control systems, the locations of the poles and eros in the plane are very important. Will investigate how the locations of the poles and eros in the s plane compare with the locations of the poles and eros in the plane. 43
MAPPING BETWEEN THE s PLANE AND THE PLANE (ctnd Mapping of the Left Half of the s Plane into the Plane (ctnd complex variable s has real part and imaginary part, we have s = + j and e T ( j T jt T j( T 2 k e e e e 44
MAPPING BETWEEN THE s PLANE AND THE PLANE (ctnd Mapping of the Left Half of the s Plane into the Plane (ctnd Since is negative in the left half of the s plane, the left half of the s plane corresponds to = et < The j axis in the s plane corresponds to =. That is, the imaginary axis in the s plane (the line = 0 corresponds to the unit circle in the plane, The left half of the s plane corresponds to the interior of the unit circle 45
MAPPING BETWEEN THE s PLANE AND THE PLANE (ctnd Conclusion. The j axis in the s plane corresponds to = 2. The left half of the s plane corresponds to the interior of the unit circle 3. The right half of the s plane corresponds to the exterior of the unit circle 46
MAPPING BETWEEN THE s PLANE AND THE PLANE (ctnd j Unit circle s plane Figure : Mapping s plane plane plane 47
48
OPEN-LOOP DISCRETE-TIME SYSTEMS The Relationship Between E( and E*(s -The -transform of the number sequence e(k was defined as: Z [e(k] = E( = e(0 + e( - + e(2-2 + (4- -The starred transform for the time function e(t was defined as: E*(s = e(0 + e(te -Ts + e(2te -2Ts +... (4-2 -If e(k of (4- is equal to e(kt of (4-2, and if e st = in (4-2, then (4-2 becomes the -transform. Hence in this case E ( E *( s e st...(4.3 49
The Pulse Transfer Function To develop an expression for the -transform of the output of open-loop sampled-data systems. Consider the open-loop system of Figure 4-la, Gp(s - plant transfer fn. G(s - product of the plant transfer fn and the ero-order hold transfer fn, that is Ts e G( s Gp ( s s Hence this system can be represented like in Figure 4-lb. ZOH Figure 4- Open-loop sampled-data system 50
The Pulse Transfer Function (cont In Figure 4-, C(s = G(sE*(s.(4.5 Assume that c(t is continuous at all sampling instants. From defination, C * ( s C( s jn s [ G( s E * ( s]*...(4.6 T n Thus, from (4-5 and (4-6, C * ( s G( s jn s E * ( s jn s T n...(4.7 From property of E*(s, E*(s+jn s = E*(s 5
The Pulse Transfer Function (cont Thus (4-7 becomes C *( s E *( s G( s jn s E *( s G *( s T n...(4.8 and then from (4-3, C( = E(G(.(4-9 G( is called the pulse transfer function and is the transfer function between the sampled input and the output at the sampling instants. 52
The derivation above is completely general. Thus given any function that can be expressed as A (s = B(sF*(s... (4-0 where F*(s must be expressible as F*(s = f 0 + f e -Ts + f 2 e -2Ts +. then, from the preceding development, A* (s = B*(sF*(s. (4- Hence, from (4-3, A( = B(F(.(4-2 where B(s is a function of s and F*(s is a function of e Ts ; that is, in F*(s, s appears only in the form e Ts. Then, in (4-2, B( = Z [B(s], F( = F*(s Ts e = (4-3 The following examples illustrate this procedure. 53
Example 4.2 Find the -transform of e A ( s s( s Solution Ts From (4-0, we consider : s( s ( e Ts and B( s s(s( s F *( s e Ts F ( F *( s e Ts 54
The Pulse Transfer Function (cont Then, from the -transform tables, The Pulse Transfer Function (cont Then, from the -transform tables, ( ( T e Z B ( ( ( ( ( T e e s s Z B and T T T T T T e e e e F B A ( ( ( ( ( ( e e ( ( 55
Example 4.3 Given the system shown in Figure 4-2, with input e(t a unit step function, determine the output function C(. Figure 4.2 56
Solution C( s Ts e E *( s s(s( s G( s E *( s In Example 4.2 it was shown that G( Z e s( s Ts In addition, from the table E ( Z ( t Thus C( ENTRANCE E 8 G( E( ( ( e T e e T T ( e T e T 57
and the inverse -transform of this function yields c(nt = e -nt Entrance 8 This response is plotted in Figure 4-2b. 58
The Pulse Transfer Function (cont Investigate open-loop systems of other configurations in Fig 4.3. Figure 4.3 Open-loop sampled-data systems 59
The Pulse Transfer Function (cont Figure 4-3a - two plants, both G (s and G 2 (s contain the transfer functions of the data holds. Then, C( s G2( s A*( s C( G2( A( A ( s G ( s E *( s A( G ( E( from (4.5 and (4.6 (4.5 (4.6 C ( G ( G 2 ( E ( (4.7 and the total transfer function is the product of the pulse transfer functions. 60
The Pulse Transfer Function (cont Figure 4-3b - G 2 (s not contain a data-hold transfer function. Then C( s G ( s G2( s E *( s C( GG 2( E( where GG 2( Z G ( s G2( s (4.8 bar above a product term indicates that the product must be performed in the s-domain before the - transform is taken. GG 2( G ( G2( ( 4.9 6
The Pulse Transfer Function (cont For the system of Figure 4-3c, Thus C( s G2( s A* ( s G2( s GE *( s C( G2( GE ( (4.20 62
63
Stability Analysis of a Closed-Loop System. Will discuss the stability of linear time-invariant single-input-single-output discrete-time control systems. Consider the following closed-loop pulse-transfer function system: C( R( G( GH ( (2.43 64
Stability Analysis of a Closed-Loop System (ctnd The stability of the system defined by Equation (2. 42, as well as of other types of discrete-time control systems, may be determined from the locations of the closed- loop poles in the plane, or the roots of the characteristic equation as follows: P( = + GH( = 0 65
Stability Analysis of a Closed-Loop System (ctnd. For the system to be stable, the closed-loop poles or the roots of the characteristic equation must lie within the unit circle in the plane. Any closed- loop pole outside the unit circle makes the system unstable. 2. If a simple pole lies at =, then the system becomes critically stable. Also, the system becomes critically stable if a single pair of conjugate complex poles lies on the unit circle in the plane. Any multiple closed-loop pole on the unit circle makes the system unstable. 3. Closed-loop eros do not affect the absolute stability and therefore maybe located anywhere in the plane. 66
Stability Analysis of a Closed-Loop System (ctnd Thus, a linear time-invariant single-input-single- output discrete-time closed-loop control system becomes unstable if any of the closed-loop poles lies outside the unit circle and/or any multiple closed-loop pole lies on the unit circle in the plane. 67
Stability Analysis of a Closed-Loop System (ctnd Example 4-2 Consider the closed-loop control system shown in Figure 2-34. Determine the stability of the system when K =. The open-loop transfer function G(s of the system is G( s s e s s( s 68
Figure 2-34 Closed-loop control system of Example 4-2. 69
Stability Analysis of a Closed-Loop System (ctnd Solution The transform of G(s is G( = Z e s 0.3679 0.2642 s s( s ( 0.3679( (2.44 Since the closed-loop pulse transfer function for the system is C( G( R ( G ( the characteristic equation is + G( = 0 70
Stability Analysis of a Closed-Loop System (ctnd which becomes ( 0.3679( - + 0.3679 + 0.2642 = 0 Or 2 + 0.632 = 0 The roots of the characteristic equation are found to be = 0.5 + j0.68, 2 = 0.5 j0.68 Since = 2 < the system is stable. 7
Methods for Testing Absolute Stability. Three stability tests can be applied directly to the characteristic equation P( = 0 without solving for the roots. Schur-Cohn stability test Jury stability test. Routh stability criterion Will only discussed Jury stability test 72
The Jury Stability Test. In applying the Jury stability test to characteristic equation P( = 0, we construct a table whose elements are based on the coefficients of P(. Assume that the characteristic equation P( is a polynomial in as follows: n n P( a a... a n 0 n a (2.45 where a 0 > 0. Then the Jury table becomes as given in Table 2. 4. 73
Table 2. 4 General Form Of The Jury Stability Table 74
First row consist of the coefficients in P( arranged First row consist of the coefficients in P( arranged in the ascending order of powers of. Second row consist of the coefficients of P( arranged in the descending order of powers of. The elements for rows 3 through 2n - 3 are given by the following determinants: 0,,2,...,, 0 n k a a a a b k k n n k 2 0,,2,...,, 0 2 n k b b b b c k k n n k 2 3 0 p p b b k k 75 0,,2, 0 2 3 k p p p p q k k k
Stability Criterion by the Jury Test. A system with the characteristic equation P( = 0 given by Equation (2. 44, rewritten as n n P( a0 a... an an where a 0 > 0, is stable if the following conditions are all satisfied:. a n < a 0 2. P( 0 ( 0 for n even P ( 3. 0 for n odd 76
b n b 4. n 0 c n 2 c 0 q 2 q 0 77
Example 4-3 Construct the Jury stability table for the following characteristic equation: P 4 3 2 ( a0 a a2 a3 a4 where a 0 > 0. Write the stability conditions. 78
Solution Referring to the general case of the Jury stability table given by Table 2. 4, a Jury stability table for the fourth-order system may be constructed as shown in Table 2. 5. This table is slightly modified from the standard form and is convenient for the computations of the b's and c's. The determinant given in middle of each row gives the value of b or c written on the right-hand side of the same row. 79
Solution The stability conditions are as follows:. a 4 < a 0 2. P( a0 a a2 a3 a4 0 3. P( a0 a a2 a3 a4 4. b 3 b 0 0, n 4 even c 2 c 0 It is noted that the value of c (or, in the case of the nth-order system, the value of q is not used in the stability test, and therefore the computation of c (or q may be omitted. 80
Table 2. 5 Jury Stability Table For The Fourth-Order System 8
Example 4-4 Examine the stability of the following characteristic equation: 4 3 2 P(.2 0.07 0.3 0.08 0 Solution Notice that for this characteristic equation a 0 a a a 2 3 a 4.2 0.07 0.3 0.08 82
Solution (ctnd Examine st condition for stability, a 4 < a 0, -0.08 < satisfied. 2nd condition for stability P ( 0 : ( P(l = -.2 + 0.07 + 0.3-0.08 = 0.09 > 0 satisfied. 3rd condition for stability (P(- > 0, n= even P(- = +.2 + 0.07-0.3-0.08 =.89 > 0, satisfied. 83
Solution (ctnd 4 th condition, construct the Jury stability table. Referring to Example 4-3, we compute the values of b 3, b 2, b, and b 0 and c 2 and c 0. The result is shown in Table 2.6. From table 2.6, we get b 3 = 0.994 > 0.204 = b 0 c 2 = 0.946 > 0.35 = c 0 Thus both parts of the fourth condition given in Example 4-3 are satisfied. Since all conditions for stability are satisfied, the given characteristic equation is stable, or all roots lie inside the unit circle in the plane. 84
Table 2. 6 JURY STABILITY TABLE FOR THE SYSTEM OF EXAMPLE 4-4 85
Solution (ctnd As a matter of fact, the given characteristic equation P( can be factored as follows: P( = ( - 0.8( + 0.5( - 0.5( - 0.4 As a matter of course, the result obtained above agrees with the fact that all roots are within the unit circle in the plane. 86
87
This document was created with Win2PDF available at http://www.win2pdf.com. The unregistered version of Win2PDF is for evaluation or non-commercial use only. This page will not be added after purchasing Win2PDF.