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DIGITAL CONTROL

SYSTEMS Modeling of Digital Control Systems

ADC Model

• Assumptions:

� quantization errors are negligible

� conversion time is negligible

� Sampling is perfectly uniform

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DAC Model

• Assumptions:

�DAC outputs are exactly equal in magnitude to their inputs.

�The DAC yields an analog output instantaneously.

�DAC outputs are constant over each sampling period.

• The input-output relationship of the DAC

The Transfer Function of the ZOH

• The transfer function of the ZOH is

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• Magnitude of the frequency response of the zero-order hold with T = 1 s

Effect of the Sampler on the Transfer

Function of a Cascade • In a discrete-time system including several analog

subsystems in cascade and several samplers, the

location of the sampler plays an important role in

determining the overall transfer function.

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• Inverse Laplace transforming gives the time response

• The equivalent impulse response for the cascade is given

by the convolution of the cascaded impulse responses.

• Cascading results in a new form for the impulse

response. So if the output of the system is sampled to

obtain

it is not possible to separate the three time functions

that are convolved to produce it.

• For n blocks not separated by samplers:

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• For a linear time-invariant (LTI) system with impulse-

sampled input, the output is given by

• Changing the order of summation and integration gives

• Sampling the output yields the convolution summation

• if the cascade is separated by samplers, then each block

has a sampled output and input as well as a z-domain

transfer function.

• For n blocks separated by samplers:

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• Find the equivalent sampled impulse response sequence and

the equivalent z-transfer function for the cascade of the two analog

systems with sampled input

1. If the systems are directly connected.

2. If the systems are separated by a sampler.

• 1

• 2

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DAC, Analog Subsystem, and ADC

Combination Transfer Function

• Because both the input and the output of the cascade are sampled, it is possible to obtain its z-domain transfer function in terms of the transfer functions of the individual subsystems.

• The transfer function of the DAC and analog subsystem

cascad:

• The corresponding impulse response

• The sampled impulse response

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• z-transfer function of the DAC (zero-order hold), analog

subsystem, and ADC (ideal sampler) cascade:

• Find GZAS(z) for the cruise control system for the vehicle shown in

Figure, where u is the input force, v is the velocity of the car, and b is

the viscous friction coefficient.

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• The corresponding partial fraction expansion:

• The z-transform:

• Find GZAS(z) for the vehicle position control system, where u is the

input force, y is the position of the car, and b is the viscous friction

coefficient.

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The Closed-loop Transfer Function

• The closed-loop transfer function for the system:

The closed-loop characteristic equation:

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• Find the Laplace transform of the analog and sampled output

for the block diagram

• which involves three multiplications in the s-domain. In the time

domain, x(t) is obtained after three convolutions.

• Thus, the impulse-sampled variable has the Laplace transform

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• Thus, the sampled output is

• With the transformation

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Analog Disturbances in a Digital

System • Disturbances are variables that are not included in the

system model but affect its response.

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Steady-state Error and Error Constants

• standard inputs: sampled step, sampled ramp, sampled

parabolic

• The tracking error:

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• Applying the final value theorem yields the steady-state

error:

• The limit exists if all (z − 1) terms in the denominator cancel. This

depends on the reference input as well as on the loop gain.

• rewrite the loop gain in the form

• where N(z) and D(z) are numerator and denominator polynomials,

respectively, with no unity roots.

• Definition type Number. The type number of the

system is the number of unity poles in the system z-

transfer function.

• Note that s-domain poles at zero play the same role as z-

domain poles at 1.

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• steady-state error for sampled step input

• Or

• the steady-state error for a sampled unit step input:

position error constant

• steady-state error for sampled ramp input

• the steady-state error for a sampled unit ramp input:

velocity error constant

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• Find the steady-state position error for the digital position control

system with unity feedback and with the transfer functions

1. For a sampled unit step input.

2. For a sampled unit ramp input.

The loop gain of the system is given by

it has zero steady-state error for a sampled step input and a finite

steady-state error for a sampled ramp input given by

• Find the steady-state error for the analog system

1. For proportional analog control with a unit step input.

2. For proportional digital control with a sampled unit step input.

the position error constant for analog control is K/a, and the steady-

state error is

• the DAC-plant-ADC z- transfer function

• the position error constant for digital control is

• and the associated steady-state error is the same as that of the

analog system with proportional control.

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MATLAB commands

• For a system with a zero-order hold and sampler (DAC and ADC),

we use

• For a first-order hold, we use