CHAPTER 2:

Discrete Data Control Systems

Introduction to Digital ControlIn most modern engineering systems,

There is a need to control the evolution with time of one or more of the system variables.

Controllers are required to ensure satisfactory transient and steady-state behavior for these engineering systems.

To guarantee satisfactory performance in the presence of disturbances and model uncertainty, most controllers in use today employ some form of negative feedback.

A sensor is needed to measure the controlled variable and compare its behaviorto a reference signal.

Control action is based on an error signal defined as the difference between the reference and the actual values.

The controller that manipulates the error signal- To determine the desired control action has classically been an analog system, which includes

oelectrical, fluid,

opneumatic, or

o mechanical components.

These systems all have analog inputs and outputs

(i.e., their input and output signals are defined over a continuous time interval andhave values that are defined over a continuous range of amplitudes).

In the past few decades, analog controllers have often been replaced by digital controllers whose inputs and outputs are defined at discrete time instances.

The digital controllers are in the form of digital circuits, digital computers, or microprocessors.

Why Digital Control? OR Discrete Control Systems

Accuracy. Digital signals are represented in terms of zeros and ones with typically12 bits or more to represent a single number. This involves a very small error as compared to analog signals where noise and power supply drift are always present.

Digital control offers distinct advantages over analog control. Here are some of its many advantages:

Implementation errors. Digital processing of control signals involves additionand multiplication by stored numerical values. The errors that resultfrom digital representation and arithmetic are negligible. By contrast, the processing of analog signals is performed using components such as resistors and capacitors with actual values that vary significantly from the nominal design values.Flexibility. An analog controller is difficult to modify or redesign once implemented in hardware. A digital controller is implemented in firmware or software, and its modification is possible without a complete replacement of the original controller.

Speed. The speed of computer hardware has increased exponentially since the 1980s. This increase in processing speed has made it possible to sample andprocess control signals at very high speeds. Because the interval between samples, the sampling period, can be made very small, digital controllers achieve performance that is essentially the same as that based on continuous monitoring of the controlled variable.Cost. Although the prices of most goods and services have steadily increased, thecost of digital circuitry continues to decrease. Advances in very large scaleintegration (VLSI) technology have made it possible to manufacture better,faster, and more reliable integrated circuits and to offer them to the consumerat a lower price.

The Structure o f a Digital Control System

To control a physical system or process using a digital controller, the controller must receive measurements from the system, process them, and then send control signals to the actuator that effects the control action.In almost all applications, both the plant and the actuator are analog systems.

Structure o f a Digital Control System

DIGITAL COMPUTER CONTROL SYSTEM APPLICATIONS

Example1:A block diagram of a computer control system,

includingthe signal converters. The signal is indicatedas digital or analog

The development of INTEL microprocessors measured in millions oftransistors

Automatic computer-controlled systems are used for purposes as diverse as measuring the objective refraction of the human eye and controlling the engine spark timing

Example1: Digital Control Systems

Closed-Loop Drug Delivery System

Example2: Computer Control of an Aircraft Turbojet Engine

Example3: Robotic manipulator control system. (a) 3-D.O.F. robotic manipulator. (b) Block diagram of a manipulator control system.

Sampling is necessary for the processing of analog data using digital elements.Successful digital data processing requires that the samples reflect the nature of the analog signal and that analog signals be recoverable, at least in theory, from a sequence of samples.

The Sampling Theorem

Figure shows two distinct waveforms with identical samples.

Obviously, faster sampling of the two waveforms would produce distinguishable sequences. Thus, it is obvious that sufficiently fast sampling is a prerequisite for successful digital data processing.

where wm is the bandwidth of the signalTherefore, the spectrum of the sampled waveform is a periodic function of frequency ws. Assuming that f(t) is a real valued function, then it is well known that the magnitude |F( jw)| is an even function of frequency, whereas the phase ∠F( jw) is an odd function.

For a band-limited function, the amplitude and phase in the frequency range 0 to ws/2 can be recovered by an ideallow-pass filter as shown in Fig

Selection of the Sampling Frequency

In practice, the sampling rate chosen is often larger than the lower bound specified in the sampling theorem. A rule of thumb is to choose ws as ωs = kωm, 5 ≤ k ≤ 10

The choice of the constant k depends on the application. In many applications, the upper bound on the sampling frequency is well below the capabilities of state of- the-art hardware. A closed-loop control system cannot have a sampling period below the minimum time required for the output measurement; that is, the sampling frequency is upper-bounded by the sensor delay.

For a linear system, the output of the system has a spectrum given by theproduct of the frequency response and input spectrum. Because the input is not known a priori, we must base our choice of sampling frequency on the frequency response.

The frequency response of a first-order system is

where A is a constant amplitude, and f is a phase angle. Thus, the choice of sampling frequency of ω s is sufficiently fast for oscillations of frequency wd and time to first peak p/wd.

Example 1Given a first-order system of bandwidth 10 rad/s, select a suitable sampling frequency and find the corresponding sampling period.

SolutionA suitable choice of sampling frequency is ws = 60, wb = 600 rad/s. The correspondingsampling period is approximately T = 2p/ws ≅ 0.01 s.

Example 2A closed-loop control system must be designed for a steady-state error not to exceed 5 percent, a damping ratio of about 0.7, and an undamped natural frequency of 10 rad/s. Select a suitable sampling period for the system if the system has a sensor delay of1. 0.02 s2. 0.03 sSolutionLet the sampling frequency be

SAMPLED-DATA SYSTEMS

Computers used in control systems are interconnected to the actuator and the process by means of signal converters.

The output of the computer is processed by a digital-to-analog converter. We will assume that all the numbers that enter or leavethe computer do so at the same fixed period T, called the sampling period.

Thus, for example, the reference input shown in Figure is a sequence of sample values r(kT).The variables r(kT), m(kT), and u(kT) are discrete signals in contrast to m(t) and y(t), which are continuous functions of time.

Sampled data (or a discrete signal) are data obtained for the system variables only at discrete intervals and are denoted as x{kT).

A system where part of the system acts on sampled data is called a sampled-data system. A sampler is basically a switch that closes every T seconds for one instant of time

Consider an ideal sampler, as shown in Figure The input is r(t), and theoutput is r*(t), where nT is the current sample time, and the current value of r*(t) is r(nT).We then have r*(t) = r(nT)8(t — nT), where 8 is the impulse function

Let us assume that we sample a signal r(r), as shown in Figure, and obtain r*(t). Then, we portray the series for r*(t) as a string of impulses starting at t = 0,spaced at T seconds, and of amplitude r(kT).For example, consider the input signal r(t) shown in Figure . The sampled signal is shown in Figure) with an impulse represented by a vertical arrow of magnitude r(kT).

A digital-to-analog converter serves as a device that converts the sampled signal r*(t) to a continuous signal p(t). The digital-to-analog converter can usually be represented by a zero-order hold circuit, as shown in Figure. The zero-order hold takes the value r(kT) and holds it constant for kT < t < (k + 1)7, as shown in Figure for k = 0. Thus, we use r(kT) during the sampling period.A sampler and zero-order hold can accurately follow the input signal if T is small compared to the transient changes in the signal

The response of a sampler and zero-order hold for a ramp input is shown in Figure.Finally, the response of a sampler and zero-order hold for an exponentially decaying signal is shown in Figure for two values of the sampling period.

Clearly, the output p(t) will approach the input r{t) as T approaches zero, meaning that we sample frequently.

The impulse response of a zero-order hold is shown in Figure 13.8. The transfer function of the zero-order hold is

The z-TransformThe z-transform is an important tool in the analysis and design of discrete-time systems.

It simplifies the solution of discrete-time problems by converting LTIdifference equations to algebraic equations and convolution to multiplication.Thus, it plays a role similar to that served by Laplace transforms in continuous-time problems.

The following are two alternative definitions of the z-transform

Properties of the z-Transform

The z-transform can be derived from the Laplace transform as shown in Definition 2.2. Hence, it shares several useful properties with the Laplace transform, which can be stated without proof.

Inversion of the z-Transform