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# Real-time Systems_Jane.W.S.Liu

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非常经典的实时系统调度教程，用于操作系统，和嵌入式。基础知识，由浅入深。阐明了基本的概念，解决了只读论文带来的概念模糊。非常值得阅读

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Integre Technical Publishing Co., Inc. Liu January 13, 2000 8:45 a.m. chap1 page 1

CHAPTER 1

Typical Real-Time Applications

From its title, you can see that this book is about real-time (computing, communication, and

information) systems. Rather than pausing here to deﬁne the term precisely, which we will

do in Chapter 2, let us just say for now that a real-time system is required to complete its

work and deliver its services on a timely basis. Examples of real-time systems include digital

control, command and control, signal processing, and telecommunication systems. Every day

these systems provide us with important services. When we drive, they control the engine

and brakes of our car and regulate trafﬁc lights. When we ﬂy, they schedule and monitor the

takeoff and landing of our plane, make it ﬂy, maintain its ﬂight path, and keep it out of harm’s

way. When we are sick, they may monitor and regulate our blood pressure and heart beats.

When we are well, they can entertain us with electronic games and joy rides. Unlike PCs and

workstations that run nonreal-time applications such as our editor and network browser, the

computers and networks that run real-time applications are often hidden from our view. When

real-time systems work correctly and well, they make us forget their existence.

For the most part, this book is devoted to real-time operating systems and communica-

tion protocols, in particular, how they should work so that applications running on them can

reliably deliver valuable services on time. From the examples above, you can see that mal-

functions of some real-time systems can have serious consequences. We not only want such

systems to work correctly and responsively but also want to be able to show that they indeed

do. For this reason, a major emphasis of the book is on techniques for validating real-time

systems. By validation, we mean a rigorous demonstration that the system has the intended

timing behavior.

As an introduction, this chapter describes several representative classes of real-time ap-

plications: digital control, optimal control, command and control, signal processing, tracking,

real-time databases, and multimedia. Their principles are out of the scope of this book. We

provide only a brief overview in order to explain the characteristics of the workloads gener-

ated by the applications and the relation between their timing and functional requirements. In

later chapters, we will work with abstract workload models that supposely capture the rele-

vant characteristics of these applications. This overview aims at making us better judges of

the accuracy of the models.

In this chapter, we start by describing simple digital controllers in Section 1.1. They are

the simplest and the most deterministic real-time applications. They also have the most strin-

gent timing requirements. Section 1.2 describes optimal control and command and control

applications. These high-level controllers either directly or indirectly guide and coordinate

1

Integre Technical Publishing Co., Inc. Liu January 13, 2000 8:45 a.m. chap1 page 2

2 Chapter 1 Typical Real-Time Applications

digital controllers and interact with human operators. High-level controllers may have signif-

icantly higher and widely ﬂuctuating resource demands as well as larger and more relaxed

response time requirements. Section 1.3 describes signal processing applications in general

and radar signal processing and tracking in particular. Section 1.4 describes database and

multimedia applications. Section 1.5 summarizes the chapter.

1.1 DIGITAL CONTROL

Many real-time systems are embedded in sensors and actuators and function as digital con-

trollers. Figure 1–1 shows such a system. The term plant in the block diagram refers to a

controlled system, for example, an engine, a brake, an aircraft, a patient. The state of the plant

is monitored by sensors and can be changed by actuators. The real-time (computing) system

estimates from the sensor readings the current state of the plant and computes a control output

based on the difference between the current state and the desired state (called reference input

in the ﬁgure). We call this computation the control-law computation of the controller. The

output thus generated activates the actuators, which bring the plant closer to the desired state.

1.1.1 Sampled Data Systems

Long before digital computers became cost-effective and widely used, analog (i.e., continuous-

time and continuous-state) controllers were in use, and their principles were well established.

Consequently, a common approach to designing a digital controller is to start with an analog

controller that has the desired behavior. The analog version is then transformed into a digi-

tal (i.e., discrete-time and discrete-state) version. The resultant controller is a sampled data

system. It typically samples (i.e., reads) and digitizes the analog sensor readings periodically

and carries out its control-law computation every period. The sequence of digital outputs thus

produced is then converted back to an analog form needed to activate the actuators.

A Simple Example. As an example, we consider an analog single-input/single-output

PID (Proportional, Integral, and Derivative) controller. This simple kind of controller is com-

monly used in practice. The analog sensor reading y(t) gives the measured state of the plant

at time t.Lete(t) = r (t) − y(t) denote the difference between the desired state r(t) and the

measured state y(t) at time t. The output u(t) of the controller consists of three terms: a term

A/D

A/D

Sensor

Plant

Actuator

controller

control-law

computation

D/A

reference

input

r(t)

y(t) u(t)

r

k

y

k

u

k

FIGURE 1–1 A digital controller.

Integre Technical Publishing Co., Inc. Liu January 13, 2000 8:45 a.m. chap1 page 3

Section 1.1 Digital Control 3

that is proportional to e(t), a term that is proportional to the integral of e(t) and a term that is

proportional to the derivative of e(t ).

In the sampled data version, the inputs to the control-law computation are the sampled

values y

k

and r

k

,fork = 0, 1, 2,..., which analog-to-digital converters produce by sam-

pling and digitizing y(t) and r (t ) periodically every T units of time. e

k

= r

k

− y

k

is the kth

sample value of e(t). There are many ways to discretize the derivative and integral of e(t).For

example, we can approximate the derivative of e(t) for (k − 1)T ≤ t ≤ kT by (e

k

− e

k−1

)/T

and use the trapezoidal rule of numerical integration to transform a continuous integral into a

discrete form. The result is the following incremental expression of the kth output u

k

:

u

k

= u

k−2

+ αe

k

+ βe

k−1

+ γ e

k−2

(1.1)

α, β,andγ are proportional constants; they are chosen at design time.

1

During the kth sam-

pling period, the real-time system computes the output of the controller according to this

expression. You can see that this computation takes no more than 10–20 machine instruc-

tions. Different discretization methods may lead to different expressions of u

k

, but they all are

simple to compute.

From Eq. (1.1), we can see that during any sampling period (say the kth), the control

output u

k

depends on the current and past measured values y

i

for i ≤ k. The future measured

values y

i

’sfori > k in turn depend on u

k

. Such a system is called a (feedback) control loop

or simply a loop. We can implement it as an inﬁnite timed loop:

set timer to interrupt periodically with period T ;

at each timer interrupt, do

do analog-to-digital conversion to get y;

compute control output u;

output u and do digital-to-analog conversion;

end do;

Here, we assume that the system provides a timer. Once set by the program, the timer gener-

ates an interrupt every T units of time until its setting is cancelled.

Selection of Sampling Period. The length T of time between any two consecutive

instants at which y(t) and r(t) are sampled is called the sampling period. T is a key design

choice. The behavior of the resultant digital controller critically depends on this parameter.

Ideally we want the sampled data version to behave like the analog version. This can be done

by making the sampling period small. However, a small sampling period means more frequent

control-law computation and higher processor-time demand. We want a sampling period T

that achieves a good compromise.

In making this selection, we need to consider two factors. The ﬁrst is the perceived

responsiveness of the overall system (i.e., the plant and the controller). Oftentimes, the system

is operated by a person (e.g., a driver or a pilot). The operator may issue a command at any

time, say at t. The consequent change in the reference input is read and reacted to by the digital

1

The choice of the proportional constants for the three terms in the analog PID controller and the methods for

discretization are topics discussed in almost every elementary book on digital control (e.g., [Leig]).

Integre Technical Publishing Co., Inc. Liu January 13, 2000 8:45 a.m. chap1 page 4

4 Chapter 1 Typical Real-Time Applications

controller at the next sampling instant. This instant can be as late as t + T . Thus, sampling

introduces a delay in the system response. The operator will feel the system sluggish when the

delay exceeds a tenth of a second. Therefore, the sampling period of any manual input should

be under this limit.

The second factor is the dynamic behavior of the plant. We want to keep the oscillation

in its response small and the system under control. To illustrate, we consider the disk drive

controller described in [AsWi]. The plant in this example is the arm of a disk. The controller

is designed to move the arm to the selected track each time when the reference input changes.

At each change, the reference input r(t) is a step function from the initial position to the

ﬁnal position. In Figure 1–2, these positions are represented by 0 and 1, respectively, and

the time origin is the instant when the step in r (t) occurs. The dashed lines in Figure 1–2(a)

y(t)

510

1

0

0

51015

u

max

u(t)

0

-u

max

510

1

0

0

51015

0

y(t)

u

max

u(t)

-u

max

510

1

0

0

51015

0

y(t)

u

max

u(t)

-u

max

(a)

(b) (c)

FIGURE 1–2 Effect of sampling period.

Integre Technical Publishing Co., Inc. Liu January 13, 2000 8:45 a.m. chap1 page 5

Section 1.1 Digital Control 5

give the output u(t) of the analog controller and the observed position y(t) of the arm as a

function of time. The solid lines in the lower and upper graphs give, respectively, the analog

control signal constructed from the digital outputs of the controller and the resultant observed

position y(t) of the arm. At the sampling rate shown here, the analog and digital versions are

essentially the same. The solid lines in Figure 1–2(b) give the behavior of the digital version

when the sampling period is increased by 2.5 times. The oscillatory motion of the arm is more

pronounced but remains small enough to be acceptable. However, when the sampling period

is increased by ﬁve times, as shown in Figure 1–2(c), the arm requires larger and larger control

to stay in the desired position; when this occurs, the system is said to have become unstable.

In general, the faster a plant can and must respond to changes in the reference input,

the faster the input to its actuator varies, and the shorter the sampling period should be. We

can measure the responsiveness of the overall system by its rise time R. This term refers to

the amount of time that the plant takes to reach some small neighborhood around the ﬁnal

state in response to a step change in the reference input. In the example in Figure 1–2, a small

neighborhood of the ﬁnal state means the values of y(t) that are within 5 percent of the ﬁnal

value. Hence, the rise time of that system is approximately equal to 2.5.

A good rule of thumb is the ratio R/T of rise time to sampling period is from 10 to

20 [AsWi, FrPW].

2

In other words, there are 10 to 20 sampling periods within the rise time.

A sampling period of R/10 should give an acceptably smooth response. However, a shorter

sampling period (and hence a faster sampling rate) is likely to reduce the oscillation in the

system response even further. For example, the sampling period used to obtain Figure 1–2(b)

is around R/10, while the sampling period used to obtain Figure 1–2(a) is around R/20.

The above rule is also commonly stated in terms of the bandwidth, ω, of the system.

The bandwidth of the overall system is approximately equal to 1/2R Hz. So the sampling

rate (i.e., the inverse of sampling period) recommended above is 20 to 40 times the system

bandwidth ω. The theoretical lower limit of sampling rate is dictated by Nyquist sampling

theorem [Shan]. The theorem says that any time-continuous signal of bandwidth ω can be

reproduced faithfully from its sampled values if and only if the sampling rate is 2ω or higher.

We see that the recommended sampling rate for simple controllers is signiﬁcantly higher than

this lower bound. The high sampling rate makes it possible to keep the control input small and

the control-law computation and digital-to-analog conversion of the controller simple.

Multirate Systems. A plant typically has more than one degree of freedom. Its state

is deﬁned by multiple state variables (e.g., the rotation speed, temperature, etc. of an engine

or the tension and position of a video tape). Therefore, it is monitored by multiple sensors and

controlled by multiple actuators. We can think of a multivariate (i.e., multi-input/multi-output)

controller for such a plant as a system of single-output controllers.

Because different state variables may have different dynamics, the sampling periods

required to achieve smooth responses from the perspective of different state variables may

be different. [For example, because the rotation speed of a engine changes faster than its

2

Sampling periods smaller than this range may have an adverse effect. The reason is that quantization error

becomes dominant when the difference in analogy sample readings taken in consecutive sampling periods becomes

comparable or even smaller than the quantization granularity.

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