Lec2.pptx.pdf

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DIGITAL CONTROLSYSTEMSDiscrete-Time Systems

Analog Systems with Piecewiseconstant inputs The plant is analog, the control is piecewise constant


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difference

equation


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Difference Equations

The nonlinear difference equation

is said to be of order n.

Linear difference equation:


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The Z-transform

The z-transform is an important tool in the analysisand design of discrete-time systems. It simplifies thesolution of discrete-time problems by converting LTIdifference equations to algebraic equations andconvolution to multiplication. Thus, it plays a role similarto that served by Laplace transformsin continuous-timeproblems.

Because we are primarily interested in application todigital control systems, this brief introduction to the z-

transform is restricted to causal signals (i.e., signals withzero values for negative time) and the one-sided z-transform.

Definition 1

Given the causal sequence {u0, u1, u2, , uk, }, its z-transform is defined as

The variable in the preceding equation can be

regarded as a time delay operator.


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Definition 2

Given the impulse train representation of a discrete-timesignal

the Laplace transform:

Then substituting in the Laplace transform yieldsthe z-transform.

Unit Impulse

Definition 1:

Definition 2:


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Sampled Step

Exponential


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Properties of the z-Transform

Linearity

Time Delay

Time Advance

Multiplication by Exponential

Sampled step


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Complex Differentiation

The final Value Theorem

If a sequence approaches a constant limit as k tends toinfinity, then the limit is given by


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Inversion of the z-Transform

Complex Integral

Long Division

Using long division, expand F(z) as a series to obtain

Write the inverse transform as the sequence

Obtain the inverse z-transform of the function


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Partial Fraction Expansion

Find the partial fraction expansion of F(z)/z or F(z). Obtain the inverse transform f(k) using the z-transform tables.

Three types of z-domain functions F(z):

functions with simple (nonrepeated) real poles

functions with complex conjugate and real poles

functions with repeated poles

Case 1: Simple Real RootsThe most convenient method to obtain the partial fraction expansion

of a function with simple real roots is the method of residues. Theresidue of a complex function F(z) at a simple pole ziis given by

This is the partial fraction coefficient of the ith term of the expansion

Because most terms in the z-transform tables include a z in the

numerator, it is often convenient to expand F(z)/z and then to multiplyboth sides by z to obtain an expansion whose terms have a z in thenumerator.


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Obtain the inverse z-transform of the function

Example

Table Lookup

Note that f(0)=0 so the time sequence can be rewritten as


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Solve the same problem without dividing by z

Example

Find the inverse z-transform of the functionExample


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Case 2: Complex Conjugate And Simple Real Roots

With complex conjugate poles, we obtain the partial fraction expansion

We then inverse z-transform to obtain

where and are the angle of the pole p and the angle of the partial

fraction coefficient A, respectively. We use the exponential expressionfor the cosine function to obtain

Case 2: Complex Conjugate And Simple Real RootsAn alternative approach:

Assuming that F(z) has real coefficients, then its complex roots occur in

complex conjugate pairs and can be combined to yield a function withreal coefficients and a quadratic denominator. To inverse-transform

such a function, use the following z-transforms:

The denominators of the two transforms are identical and havecomplex conjugate roots. The numerators can be scaled andcombined to give the desired inverse transform.


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Find the inverse z-transform of the function

To evaluate the remaining coefficients, we multiply the equation bythe denominator and equate coefficients to obtain

Example

Table Lookup

The first two terms of the partial fraction expansion can be easily foundin the z-transform tables. The third term resembles the transforms of a

sinusoid multiplied by an exponential if rewritten as

Referring to the z-transform tables, we obtain the inversetransform

The sinusoidal terms can be combined using the trigonometricidentities


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The sinusoidal terms can be combined using the trigonometricidentities

This gives

Solve the same problem with residues methodExample


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Case 3: Repeated Roots

For a function F(z) with a repeated root of multiplicity r, r partial

fraction coefficients are associated with the repeated root. The partialfraction expansion is of the form

The coefficients for repeated roots are governed by

The coefficients of the simple or complex conjugate roots can beobtained as before.

Obtain the inverse z-transform of the functionExample


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Table Lookup

We can therefore rewrite the inverse transform as

z-Transform Solution of DifferenceEquations The equations are first transformed to the z-domain.

Then the variable of interest is solved for and z-transformed.

To transform the difference equation, we typically use the time delay

or the time advance property.

Inverse z-transformation is performed

Solve the linear difference equation

with the initial conditions x(0)=1, x(1)=5/2.

z-Transform


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Solve for X(z)

Partial Fraction Expansion

Thus

Inverse z-Transformation


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The Time Response of a Discrete-time

System

The time response of a discrete-time linear system is thesolution of the difference equation governing the system.For the linear time-invariant (LTI) case, the responsedue to the initial conditions and the response due tothe input can be obtained separately and then addedtoobtain the overall response of the system. The responsedue to the input, or the forced response, is theconvolution summationof its input and its response to a

unit impulse.

Convolution Summation

impulse response sequence

The impulse response sequence can be used to represent theresponse of a linear discrete-time system to an arbitrary input

sequence

To derive this relationship, we first represent the input sequence interms of discrete impulses as follows

For a linear system, the principle of superposition applies andthe system output due to the input is the following sum of impulse

response sequences:


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Hence, the output at time k is given by

where (*) denotes the convolution operation.

For a causal system, the response due to an impulse at time i

is an impulse response starting at time i and the delayed responseh(k-i) satisfies

In other words, a causal system is one whose impulse response is a

causal time sequence. Thus, the summation reduces to

convolution summation

Theorem: response of an LTI system. The response of an LTIdiscrete-time system to an arbitrary input sequence is given by the

convolution summation of the input sequence and the impulseresponse sequence of the system.

For example


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The Convolution Theorem

The following theorem shows how the convolution summation can beavoided by z-transformation.

Theorem: the Convolution theorem. The z-transform of theconvolution of two time sequences is equal to the product of their z-

transforms.

The function H(z) is known as the z-transfer function or simply thetransform function.

Given the discrete-time system

find the impulse response of the system h(k)

1. From the difference equation

2. Using z-transformation

Example


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