Digital Control Systems

z-Plane Analysis of Discrete Time Control Systems

Digital Control System

A control system which uses a digital computer as a controller or compensator is known as digital control system. The advantages of using a digital computers for compensation include: accuracy, reliability, economy and most importantly, flexibility.

Sampler

Converts an anolog signal to a digital signal (train of pulses)

A practical sampler acts like a switch closing every T seconds for a short duration of p seconds. Therefore, sampled signal can represented as follows:

The output of an ideal sampler is given by

(u(t) is unit step function)

Sampler

• In practice p is much smaller and can be neglected. This leads to the Ideal Sampler.

• The Ideal Sampling process can be considered as the multiplication of a pulse train with a continuous signal

Signal r(t) after samling Signal r(t) after ideal samling

r* (t) = P(t) r(t)

Sampler

r(t) = 0, for t < 0

Hold Device

The function of hold device is to convert sampled signal into continuous signal. The values of continuous time signal in between the sampling instants are calculated by extrapolation.

The continuous signal h(t) during the time interval t may be approximated by a polynomial in :

Since the continuous time output signal, at sampling instants, must equal input signal, :

Therefore:

0

The hold device is called n-th order hold if it uses an n-th order polynomial extrapolator.n=1 first order holdn=0 zero order hold

Hold Device

Zero Order Hold (ZOH)

Input-Output signals for a ZOH

The output of ZOH:

Hold Device

Zero Order Hold (ZOH)

The output of ZOH:

We know from Laplace transforms that

ZOH Transfer function

Hold Device

Zero Order Hold (ZOH)

Suppose the transfer function G(s) follows the ZOH. Then the product of ZOH and G(s) becomes:

Obtain the z transform of X(s):

Let

Hold Device

Zero Order Hold (ZOH)

Hold Device

Zero Order Hold (ZOH)

Example:The response of a sampler and a ZOH to a ramp input for two different values of a sampling period.

A sampler and ZOH can accurately follow the input signal if the sampling time T is small compares to the transient changes in the signal

Example: Ideal sampler followed by a ZOH

Pulse Transfer Function

Transfer function of a continuous time system relates the Laplace transform of the continuous-time output to that of the continuous time input.

Pulse transfer function relates the z-transform of the output at the sampling instants to that of the sampled input.

Starred Laplace Transform

Assumption: Zero initial conditions.

• While taking the starred Laplace tranform of a product of transforms, where some are ordinary Laplace transforms and others are starred Laplace transform, the functions already in starred Laplace transforms can be factored out of the starred Laplace transform.

• The output of a sampled-data system is continuous with respect to time. However the pulse transform (starred LT) of the output, Y (s) and z-transform of the output Y (z), gives the values of output y(t) only at sampling instants.

Pulse Transfer Function

Starred Laplace Transform

Since the z transform is starred Laplace transform with esT replaced by z,

Pulse Transfer Function

Starred Laplace Transform

The presence or absence of the input sampler is crucial in determining the pulse transfer function of a system

Example:

Pulse Transfer Function

Pulse transfer function of cascaded elements

Taking starred Laplace transforms on both sides of above equations

Pulse Transfer Function

Pulse transfer function of cascaded elements

where

Pulse Transfer Function

Pulse transfer function of closed loop systems

Pulse Transfer Function

Example:

Pulse Transfer Function

Pulse transfer function of a digital controller

Pulse tranfer function of a digital controller can be easily obtained from its input-ouput characteristic which is specified by means of difference equation:

where m(k) and e(k) are the output and input signals respectively. Taking z-transforms and simplifying we get the pulse transfer function of the digital controller:

Pulse Transfer Function

Closed-loop Pulse transfer function of a digital control system

A general block diagram of a digital control system

Mathematical block diagram of a digital control system

Pulse Transfer Function

Closed-loop Pulse transfer function of a digital control system

Realization of Digital Controllers and Digital Filters

The general form of the pulse transfer function between the output Y(z) and X(z) is given by

• Direct programming

• Standart programming

In these programmings, coefficients appear as multipliers in the block diagram realization. Those block diagram schemes where the coefficients appear directly as multipliers are called direct structures.

Realization of Digital Controllers and Digital Filters

Direct programming

Realize the numerator and denominator of the pulse transfer function using separate sets of delay elements.

Total number of delay elementsin direct programming=n+m

Digital filter:

Block diagram realization:

Realization of Digital Controllers and Digital Filters

Standart programming

The number of delay elements required in direct programming can be reduced.

The number of delay elements used in realizing the pulse transfer function can be reduced from n+m to n (where n≥m) by rearranging the block diagram.

Realization of Digital Controllers and Digital Filters

Standart programming