Design of Software for Embedded Systems

ENTWURF VON SOFTWARE FR EINGEBETTETE SYSTEME (SWES)

DESIGN OF SOFTWARE FOR EMBEDDED SYSTEMS (SWES)

Dr. Peter TrgerOperating Systems Group, TU Chemnitz

PT 15/16Design of Software for Embedded Systems (SWES)

COURSE ORGANIZATION Weekly lectures (starting today) + tutorials (starting next week)

5 assignment sheets, minimum of 50% of all points needed

OPAL: News, slides (after lecture !), discussion forum, time plans

Written final exam

Module 565050 (adopted from Dr. habil. D. Mller)

Basic concepts and terminology

Control theory for practitioners

Programming for embedded systems in C and Ada

Model-driven development for embedded systems, PLC systems

Safety and security standards

2


THE PROJECT 5 weeks of project work

Work on a piece of embedded (real-time) software

Targeting Raspberry PI with extension board, available in our lab

Project results are submitted as assignment solutions

Teams of 2 persons

Q&A in tutorial sessions and OPAL forum

First project-related task with assignment sheet 3 (mid-November)

Submission system for both non-project and project assignments

You hand-in either code or a PDF document > DEMO

3

BASIC CONCEPTS AND TERMINOLOGY

DESIGN OF SOFTWARE FOR EMBEDDED SYSTEMS (SWES)



EMBEDDED SYSTEM Computer system in a context

Specific dedicated task (vs. all-purpose computer)

Often hardware/software co-design

Optimize design based on application characteristics

Often non-visible for user

Often real-time systems (max. response time deadline) High cost pressure - low memory size, simple CPUs

Energy efficiency

Everywhere: Cell phones, printer,s automobiles, aviation products, household devices, medical devices, children toys,

5


EMBEDDED SYSTEMS

6

[http

://di

dik.b

log.u

ndip.

ac.id

/]

http://didik.blog.undip.ac.id/


SOFTWARE FOR EMBEDDED SYSTEMS

Ebert and Jones, 2009

Worldwide market of 160 billion

~30 embedded processors per person in developed countries

98% of all produced microprocessors for embedded applications

7


CHALLENGES How to minimize power consumption and costs ?

Amount and type of hardware needed

Power-aware software and operating systems

How can you interact with the physical world ?

Hardware sensors and actuators

Real-time software constraints

How to ensure non-functional properties ?

1990-2000: 40% of recalled pacemakers due to firmware errors

New cars with > 20 electronic control units (ECU),1GB of software in premium car

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PROBLEM AREAS

9

Hardware

Software

Non

-func

tiona

l pro

pert

ies


HARDWARE

10

Desired functionality

General purpose Application-specific Single-purpose


HARDWARE

11

General purpose Application-specific Single-purpose

[Vah

id /

Giva

rgis]


HARDWARE General-purpose hardware

Main goal is flexible usage with best-possible performance

Programmable for all use cases

Application-specific hardware

Programmable for restricted performance-optimized uses cases

Examples: Video stream processing, audio encoding

Single-purpose hardware

Integrated circuit to perform exactly one task in the fastest way

Not programmable, algorithms fixed in hardware

12


FLEXIBLE OR EFFICIENT?

13

[Alte

ra]


HARDWARE

14

General purpose Application-specific Single purpose

Programmable in Software Hardware Software -

Performance low medium medium high

Energy Efficiency low medium medium high

Feature Flexibility very high medium low very low

Per unit price savings low low medium very high

Example Microprocessor PLD / FPGA DSP ASIC / ASSP


GENERAL PURPOSE CHIP

Microprocessor: Multi-purpose, programmable device

Central Processing Units (CPUs) + volatile memory + I/O devices

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[Stallings]

Fetch instruction and execute it - typically memory access, computation, and / or I/O

I/O devices and memory controller may interrupt the instruction processing


GENERAL PURPOSE CHIP RISC - Reduced Instruction Set Computer

MIPS, ARM, DEC Alpha, Sparc, IBM 801, Power, etc.

Small number of instructions, few data types in hardware

Instruction size constant, few addressing modes

Relies on optimization in software

CISC - Complex Instruction Set Computer

VAX, Intel X86, IBM 360/370, etc.

Large number of complex instructions, may take multiple cycles

Variable length instructions, smaller code size

Focus on optimization in hardware

RISC designs lend themselves to exploitation of instruction level parallelism

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ARM ARM design started in 1983 by Acorn Computers Ltd. Roger Wilson and Steve Furber

1990 renamed to Advanced RISC Machines Ltd., 1998 renamed to ARM Ltd. 32-Bit and 64-Bit RISC processor architecture

ARM does not manufacture itself, licenses to others: ATMEL, Intel, IBM, Nintendo, ...

ARM targets low-power and low-cost embedded applications

17

[ent

erpr

isete

ch.co

m]

http://enterprisetech.com


GENERAL PURPOSE CHIP Programmable logic device

(PLD)

Type of semiconductor hardware that can be re-wired after production

18

[Ty

Gar

ibay

, Alte

ra]

FPGA Design Cost 3.200.000$

FPGA Starting Unit Price 4x375 $ = 1500$

ASIC Design Cost 85.000.000$

ASIC Starting Unit Price 400$

ASIC market delay 3 months

Early entry device volume 49%

Processor Blade Example [Xilink]


GENERAL PURPOSE CHIP Field Programmable Gate Arrays (FPGA)

Reconfigurable technology

Hardware (!) functionality can be changed

Errors can be corrected/masked

Very short development times, suitable for low number of units

Hardware description languages: VHDL and Verilog

May be used to validate ASIC design

SRAM- and Flash-based FPGAs allow reconfigurable computing

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APPLICATION-SPECIFIC CHIP Digital signal processor (DSP)

Specialized microprocessor for digital processing of analogous signals

A/D converter for input (e.g., camera), D/A converter for output

Often fixed point arithmetics (faster)

Real-time requirements (mostly soft RT), no multitasking

Multimedia applications (image and audio processing)

Cryptography

Programmable in software

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SINGLE PURPOSE CHIP Application-specific

integrated circuit (ASIC)

Hardware device for being used in one product

Application-specific standard part (ASSP)

ASIC for re-use in multiple products (e.g. USB host controller)

21

[xilin

k.com

]

http://xilink.com


MICROCONTROLLER Integrated circuit with processor, memory, and I/O hardware

(C, uC, MCU)

Example: Atmel AT89S8253 8-bit microcontroller

MCS 51 instruction set

12 kByte Flash memory, 2 kByte EEPROM data memory

256 x 8 Bit internal RAM

32 programmable I/O lines

UART serial port

Three 16-bit timers

2.7V - 5.5V operating range

Power saving mode

22

[mikr

oe.co

m]

http://mikroe.com


SYSTEM ON CHIP System on chip (SoC)

Combines functional blocks (IP cores) in one large IC

CPU, ROM, RAM, flash, Ethernet, Bluetooth, audio, USB, time, voltage,

Soft-IP-Core in hardware description language (Verilog, VHDL)

Can be parameterized (cache sizes, bus sizes, ), see opencores.org

Licensed by companies such as ARM and MIPS

Hard-IP-Core already built or integrated in FPGA

Replace multi-chip setup, reduces power and space demands

Typically with debugging interface (JTAG, USB)

23

http://opencores.org


EXAMPLE: APPLE IPHONE 6

24

[tech

insigh

ts.co

m]

http://techinsights.com



25

[tech

insigh

ts.co

m]




26

[tech

insigh

ts.co

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EXAMPLE: APPLE A8 SOC

27

[ww

w.an

andt

ech.c

om]

http://www.anandtech.com


EXAMPLE: RASPBERRY PI Broadcom BCM2835 SoC

ARM11 processor

Videocore 4 GPU

General purpose I/O (GPIO) pins

Audio, USB, HDMI, I2C,UART

Timers

Interrupt controller

28

[rasp

berr

ypi.o

rg]

http://raspberrypi.org


PROBLEM AREAS

29

Hardware

Software

Non

-func

tiona

l pro

pert

ies


EMBEDDED SOFTWARE Embedded software interacts with the physical world

Often written by domain experts, not computer scientists

Timeliness becomes more important (e.g. deadlines)

Concurrency becomes more important (e.g. physical events)

Liveness becomes crucial (e.g. deadlock prevention)

Heterogeneity becomes default

In general: Predictable behavior, from nice-to-have to mission-critical

30

http://ptolemy.eecs.berkeley.edu/publications/papers/02/embsoft/embsoftwre.pdf

http://ptolemy.eecs.berkeley.edu/publications/papers/02/embsoft/embsoftwre.pdf


Controlled Process

TYPICAL STRUCTURE

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SensorsControllerSoftware

Clock

Operator

Envir

onm

ent

Actuators Display


STARTING POINT

32

Understanding of resources

Describe resources available to the application (CPU, memory, OS)

Driven by cost factors and environmental conditions

Understanding of algorithms

Which resources will be used in which way

Relevant resulting performance metrics

Understanding of workload

Must consider control and data dependencies

Driven by environmental conditions

Describe tasks to be handled + timeliness constraints


TIMELINESS Embedded systems are often real-time systems

Hard real-time systems are often embedded systems

A real-time system is one in which the correctness of the computations not only depends on the logical correctness of the computation, but also on the time at which the result is produced (deadline). If the timing constraints of the system are not met, system failure is said to have occurred.

Autopilot in airplane vs. YouTube video player

Position calculation vs. 30 images / s

Do all tasks have to be executed before their deadline ?

How to deal with missed deadlines ?

When is the result produced ?

33


REAL-TIME Hard real-time: Missing a deadline is not acceptable

Aircraft control systems

Nuclear power / chemical plant safety mechanisms

Medical devices

Soft real-time: Missing a deadline is undesirable

Multimedia

Airline reservation

High-speed trading applications

Real-time objectives may change during operation

Example: Grounded airplane vs. flying airplane

34


TASK / VALUE FUNCTIONS

35

Value

deadline

Value

deadline

Value

deadline

Value

deadline

Value

deadline

Value

deadline

Value

deadline

Value

deadline

Value

deadline

Deadline missed

Hard real-time: Task result has no more value

Soft real-time: Task result has reduced value


HARD REAL-TIME

36

ReleaseTime

SchedulingTime

Completiontime

AbsoluteDeadline

Execution Time

Relative Deadline

Zeitliche Parameter (Timing constraints) eines Jobs.

t0

Response Time


SOFT REAL-TIME

37

ReleaseTime

SchedulingTime

Completiontime

t0 AbsoluteDeadline

Tardiness

Relative Deadline

Execution Time

Response Time


REAL-TIME TASKS Periodic tasks

Examples: Sensor data acquisition, action planning, system monitoring

Must be regularly activated (once per period)

Aperiodic tasks

Example: Background services, logging, operator requests

Triggered by well-known event at any time

Sporadic tasks

Example: Collision detection in a roboter, I/O device interrupt

Aperiodic task with minimum inter-arrival time (rate restriction)

38


REAL-TIME SCHEDULING Scheduling: Define order of task execution

Mature theory for real-time schedules on uniprocessors since 1970s

Theory for real-time multiprocessor schedules still under research

On small embedded systems (micro-controller scale)

Only one / a few tasks

Manual scheduling by developer good enough

On larger embedded systems

Real-time operating system

Implements appropriate scheduling concepts

Supports prioritization and synchronization of concurrent tasks

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REAL-TIME SCHEDULING

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Real-Time Scheduling

SoftHard

Dynamic Static

Preemptive Non-Preemptive Preemptive Non-Preemptive



Manual scheduling

Simple round-robin implementation, based on polling

Device C needs periodic attention, A and C are purely event-driven

42

8

Embedded Operating Systems

HPI


HPI

Device A(rain sensor)

Device B(ABS)

Device C(speed display)



Manual scheduling with round robin with interrupts

43

8


HPI


HPI

9


HPI

void Task2(){ while(true){ // wait for signal Y // handle device B }

}


HPI

high priority

low priority

Round Robin Round Robin with Interrupts

RTOS

all code

device A ISR

device B ISR

all other code

device A ISR

device B ISR

Task 1

Task 2

With support from real-time operating system



44

Real-Time Operating System (RTOS) features

Real-time scheduling with priorities

Support for concurrency, preemption and prioritization

Predictable timing behavior of interrupt routines and system calls9


HPI

void Task2(){ while(true){ // wait for signal Y // handle device B }

}


HPI

high priority

low priority

Round Robin Round Robin with Interrupts

RTOS

all code

device A ISR

device B ISR

all other code

device A ISR

device B ISR

Task 1

Task 2


PROBLEM AREAS

45

Hardware

Software

Non

-fun

ctio

nal p

rope

rtie

s


DEPENDABILITY Umbrella term for operational requirements on a system

Laprie: Trustworthiness of a computer system such that reliance can be placed on the service it delivers to the user

Adds a third dimension to system quality

General question: How to deal with unexpected events ?

46


DEPENDABILITY IN EMBEDDED Critical application domains always considered dependability

Aviation industry, power industry, military equipment, But all embedded systems have actuators, people count on them

Dangerous real-world interactions may be less explicit Examples: Heating devices, power / water supply devices

Today more domain experts than software engineering experts New challenging through increasingly interconnected devices

Internet of Things (IoT)

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DEPENDABILITY TREE [LAPRIE]

48


THREATS System failure - ,Ausfall

Event that occurs when the service no longer complies with the specification / deviates from the correct service.

System error - ,Fehler(zustand)

Part of system state that can lead to subsequent failure

Some sources define errors as active faults - not in this course ...

System fault - ,Fehler(ursache)

Adjudged or hypothesized cause of an error Failure occurs when error state alters the provided service Systems are build from connected components, which are again systems Fault is the consequence of a failure of some other system to deliver its service

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Fault Error

Failure


CONSEQUENCES [KNIGHT] Human injury or loss of life Damage to the environment Damage to or loss of equipment Damage to or loss of data Financial loss by theft Financial loss through production of useless or defective products Financial loss through reduced capacity for production or service Loss of business reputation, customer base, or jobs

50


FAULT MODEL Faults can be classified into categories on different abstraction levels

Physics

Circuit level / switching circuit level

Interesting for hardware design research (not this course)

Investigate logical signals on connections

stuck-at-zero, stuck-at-one, bridging faults, stuck-open

Register transfer level

Processor-memory-switch (PMS) level

Hardware system level

... (Software) ...

51


FAILURE TYPES Duration of the failure

Permanent failures - no possibility for repairing or replacement

Recoverable failures - back in operation after the system recovered from error state

Transient failures - short duration, no major recovery action

Effect of the failure

Functional failures - system does not operate according to its specification

Performance failures - performance or SLA specifications not met

Scope of the failure

Partial failure - only parts of the system become unavailable

Total failure - all services go down

52


FAILURE SEVERITY Denotes consequences of failure

Benign failures (,unkritische Ausflle)

Failure costs and operational benefits are similar

Sometimes also umbrella term for failures only detected by inspection

A system with only such failures is fail-safe

Catastrophic failures (,kritische Ausflle)

Costs of failure consequences are much larger than service benefit

Significant / serious failures - Intermediate steps expressing reduced service

Grading of failure consequences on overall system depends on application Flying airplane - Catastrophic stopping failure, Train - Benign stopping failure

Criticality - Highest severity of possible failure modes in the system

53


ATTRIBUTES Reliability - Function R(t)

Probability that a system is functioning properly and constantly over time

Assumes that system was fully operational at t=0

Denotes failure-free interval of operation

Availability - Statement if a system is operational at a point in time / fraction of time

Describe system behavior in presence of error treatment mechanisms

Steady-state availability - Probability that a system will be operational at any random point of time,

Fraction of time a system is operational during its expected lifetime: As = Uptime / Lifetime

54


SAFETY Different levels of critical participation for a computer system

Information provisioning to human controller on request

Interpretation of data and presentation to the user

Issues command on behalf of the human controller

Replaces human controller

Trend to realize critical systems with commercial-of-the-shelf components

Driven by budget cuts and performance advantage

Puts sole responsibility on software layer, in contrast to early hardware-only redundancy approaches

55


EXAMPLE: DO-178D Software Considerations in Airborne Systems and Equipment Certification

Mature document, developed for more than 20 years

Definition of severity of failure for airplane, crew, and passengers

Catastrophic - Loss of ability to continue safe flight and landing

Major - Reduced airplane or crew capability to cope with operating conditions

Reduction in safety margins and functional capabilities

Higher workload or physical distress for the crew

Minor - Not significantly reduced airplane safety, slight increase in workload (Example: Change of flight plan)

No effect - Failure results in no loss of operational capabilities and no increase in crew workload

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EXAMPLE: DO-178D

57


SAFETY VS. SECURITY

Different technical foundations, e.g. for recovery from errors Embedded system development may need to consider both aspects

58

Safety Security

Assumes trustworthy operators Assumes fault-free system

Assumes closed system Assumes open, connected system

Existing standards(DO-178C, ISO26262, )

Existing standards(ISO 27002, Common Criteria, )


PROBLEMS AREAS

59

Hardware

Software

Non

-func

tiona

l pro

pert

ies

Microprocessor CISC vs. RISC Microcontroller SoC ASIC vs. PLC ARM

Real-Time Code-driven Model-driven Cross-Compile Control loops

Dependability Safety Security Reliability Availability

CONTROL THEORYDESIGN OF SOFTWARE FOR EMBEDDED SYSTEMS (SWES)



Controlled Process

EMBEDDED SYSTEM

61

SensorsController Software

Clock

Operator

Envir

onm

ent

Actuators Display


CONTROL THEORY Create abstract models for a controlled process

Typical challenge in connected embedded systems

Focus on static and dynamic behavior

Results in the behavioral design of a controller

Abstraction from specific implementation details (e.g. car speed control) creates a dedicated mathematical problem

How to consider measured input to control some regulation output

Mathematical principles as glue between engineering disciplines

Everybody has some kind of control problem

Example: Control unit (Steuergert) in automotive systems

62


CONTROL THEORY Open loop / non-feedback control (Steuerung)

Controller activity based on current state Output of the controlled system is not observed External deviations must be considered at design-time Example: Power supply for electric motor with constant load

63

Example: Branch point (Fig. 6)

A branch point is represented by a point. Here, a line of action splits up intotwo or more lines of action. The signal transmitted remains unchanged.

Example: Signal flow diagram of open loop and closed loop control

The block diagram symbols described above help illustrate the differencebetween open loop and closed loop control processes clearly.

In the open action flow of open loop control (Fig. 7), the operator positionsthe remote adjuster only with regard to the reference variable w. Adjustmentis carried out according to an assignment specification (e.g. a table: set pointw1 = remote adjuster position v1; w2 = v2; etc.) determined earlier.

14

Fundamentals Terminology and Symbols in Control Engineering

SAM

SON

AG

V74

/DK

E

x1

x2x1 = x2 = x3

x3

Fig. 6: Branch point

xw

Fig. 7: Block diagram of manual open loop control

manremoteadjuster system

controlvalve

signal flow diagramof open loop control


CONTROL THEORY Closed loop / feedback control (Regelung)

Feedback from system output used to adjust controller operation

Error between output and reference used to adopt the operation

System can react on external disturbances Instability can happen

64

In the closed action flow of closed loop control (Fig. 8), the controlled varia-ble x is measured and fed back to the controller, in this case man. The con-troller determines whether this variable assumes the desired value of thereference variable w. When x and w differ from each other, the remote ad-juster is being adjusted until both variables are equal.

Device-related representation

Using the symbols and terminology defined above, Fig. 9 shows the typicalaction diagram of a closed loop control system (abbreviations see page 10).

15

Part 1 L101 ENSA

MSO

NA

G0

0/03

xw +_

Fig. 8: Block diagram of manual closed loop control

man

remoteadjuster

controlvalve

system

z

ew +

yr y x

r

Fig. 9: Block diagram of a control loop

controllingelement

measuringequipment

controller

finalcontrolelement

system

actuator

signal flow diagramof closed loop control

elements and signalsof a control loop


CLOSED LOOP CONTROL

65

In the closed action flow of closed loop control (Fig. 8), the controlled varia-ble x is measured and fed back to the controller, in this case man. The con-troller determines whether this variable assumes the desired value of thereference variable w. When x and w differ from each other, the remote ad-juster is being adjusted until both variables are equal.

Device-related representation

Using the symbols and terminology defined above, Fig. 9 shows the typicalaction diagram of a closed loop control system (abbreviations see page 10).

15

Part 1 L101 EN

SAM

SON

AG

00/

03

xw +_

Fig. 8: Block diagram of manual closed loop control

man

remoteadjuster

controlvalve

system

z

ew +

yr y x

r

Fig. 9: Block diagram of a control loop

controllingelement

measuringequipment

controller

finalcontrolelement

system

actuator

signal flow diagramof closed loop control

elements and signalsof a control loop

Regulatory control: Manage with respect to a reference value (w)

Disturbance rejection: Eliminate effect of a disturbance (z)

Optimization: Achieve the best value of the outputs (x) [Sam

son

AG]


Sensor

Controller

System

CLOSED LOOP CONTROL

66

Disturbance (d)

Controlled Variable (y)

Measured Output (yM)

Actuator

Actuating Variable (uS)

Controller Output (u)

Envir

onm

ent

Error / Control deviation:

e=w-yM

Reference / Set Point (w)

Symbols differ in German DIN notation and English notations!


Sensor

Controller

System

DIGITAL CONTROLLER

67

Disturbance (d)

Controlled Variable (y)

Measured Output (yM)

Actuator

Actuating Variable (uS)

Controller Output (u)

Envir

onm

ent

Error / Control deviation:

e=w-yM

Reference / Set Point (w)

A/D

D/A

Symbols differ in German DIN notation and English notations!


CLOSED LOOP CONTROL Set point given by user, or higher-level system, or both

Examples

Biology: Upright walk, body temperature, eye adaption on light

Home devices: Heating based on internal temperature sensor, fridges

Industry: ABS in cars, power grid control, servo systems

Time dependency of set point

Constant set point: Fixed set point control

Example: Boiler temperature control

Varying set point: Follow-up control

Example: Weather-compensated temperature control

68


EXAMPLE

69

18

1.6 Application examples 1.6.1 Heating control In the following examples from the domain of heating, closed-loop and open-loop control will again be compared and the differences indicated.

1.6.1.1 Room temperature control

Fig. 1-16 shows a simple room temperature control system in which, unlike the system shown in Fig. 1-5, controller 1 acts on a mixing valve (3). The room temperature is measured by temperature sensor 2 and passed to the controller as the controlled variable x. The controller compares the reference variable with the controlled variable and, in case of a control difference, the valve is further opened or closed depending on polarity until the controlled variable and the reference variable are equal again. The closed control loop is clearly identifiable, as the corresponding block diagram also shows.

Fig. 1-16 Room temperature control

1 Temperature controller 2 Temperature sensor 3 Mixing valve 4 Circulation pump 5 Radiator 6 Boiler 7 Controlled system 8 Controlling equipment

w x

T1

2T

5

y

3 4

6

w

x

8

7y

z B31-12

. 1 Temperature controller

. 2 Temperature sensor

. 3 Mixing valve

. 4 Circulation pump

. 5 Radiator

. 6 Boiler

Closed loop clearly identifiable

Valve is opened and closed tobring controlled variable andreference variable together

[Siemens]


EXAMPLE 1 - Outside temperature sensor

2 - Heating curve setting

3 - Supply temperature sensor

4 -Supply temperature controller 5 -Control valve (actuator + control element) w - Supply temperature setpoint

(open-loop control variable 1)

x1 - Supply temperature actual value(closed-loop controlled variable)

x2 - Room temperature actual value (open-loop control variable 2)

z1 - Interference variable 1(external heat gain)

z2 -Interference variable 2 (fluctuating boiler water temperature) y - Manipulated variable

70

20

Fig. 1-17 Weather-compensated supply temperature control

1 Outside temperature sensor 2 Heating curve setting 3 Supply temperature sensor 4 Supply temperature controller 5 Control valve (actuator + control element) a Open-loop control equipment 1 b Closed-loop control equipment (supply) c Closed-loop controlled system (supply) b+c Open-loop control equipment 2 (room) d Open-loop controlled system 2 (room) A Open control loop 1 B Closed-loop control C Open control loop 2 w Supply temperature setpoint (open-loop control variable 1) x1 Supply temperature actual value (closed-loop controlled variable) x2 Room temperature actual value (open-loop control variable 2) y Manipulated variable z1 Interference variable 1 (external heat gain) z2 Interference variable 2 (fluctuating boiler water temperature)

z1

x2

1

y 4

2

3

5

z2

x1

a b

d c

w

x2

z1 z2

B31-13

w

x

y

1x2

A B C

[Siemens]


VARIATIONS Disturbance feed-forward control

Disturbance is directly used by controller, allows faster reaction

Disturbance must be locatable and measurable

Cascade control

One controller feeding another one

Robust control, adaptive control

Modification of controller through self-feedback

Example: Flying missile with decreasing fuel mass

71


STABILITY Control loop is an oscillatory system

Controlled and manipulated variable influence each other

System therefore may escalate into instability

Bounded input / bounded output (BIBO) stability

Output signal should not grow indefinitely when the input is limited

Mandatory condition for every serious control system

72

81

For a given controlled system, therefore, the loop gain VO is decisive for the control quality. Since the transfer coefficient KS of the system can be considered as given, the necessary loop gain VO is set simply by adjusting the transfer coefficient KR of the P-controller.

4.2 Stability investigations As already known, the degree of difficulty S of the controlled system and the loop gain VO are decisive for the oscillatory characteristics of a control loop.

In stability investigations, a distinction is made between the response to interference and the response to setpoint changes:

The response to interference of a control loop is obtained by investigating the reaction of the controlled variable x to a sudden interference or interference variable change.

The response to setpoint changes of the control loop describes the reaction of the controlled variable x to a setpoint step w.

4.2.1 Response to interference If, without external intervention, the controlled variable in the control loop fails to assume a constant value even after an infinite period of time since the occurrence of the last interference (sustained oscillation), the term self-excitation is used, and the control loop is said to be unstable. This state must be avoided at all costs by correct adjustment of the controller.

In unstable control loops two different types of oscillation can be distinguished periodically undamped oscillation and periodically excited oscillation:

In the case of periodically undamped oscillation, the amplitude and frequency of the sustained oscillation occurring after occurrence of the interference z are constant (Fig. 4-3). The control element must not travel to either the upper or lower limit position. The time required for one full oscillation is referred to as the period of oscillation TP. The loop gain that gives rise to undamped oscillation in the control loop is referred to as the critical loop gain Vokrit. The proportional band Xp and the transfer coefficient KR KP of the P-controller that produce this undamped oscillation are referred to as Xp krit. and KR KP krit. respectively.

TPx

t

z

Fig. 4-3 Undamped oscillation

Tp Period of oscillation z Interference

82

Periodically excited oscillation is characterised by a continuously increasing amplitude of the sustained oscillation that occurs (Fig. 4-4). It occurs if the loop gain VO of the control loop is set higher than its critical value.

x

z

t Fig. 4-4 Excited oscillation

z Interference

In practice, the amplitude of oscillation continues to increase as described until it is limited by the end stops, e.g. valve fully open or valve closed.

In a stable control loop, oscillations of the controlled variable also occur after interference, but their amplitude decreases over time until they disappear completely. This is referred to as damped oscillation. A steady state, i.e. rest, is achieved (Fig. 4-5).

z

x

t Fig. 4-5 Damped oscillation

z Interference

A steady state, and therefore rest, occurs if the loop gain VO of the control loop is set lower than its critical value:

In order to achieve sufficient damping of a control process, a simple, generally applied rule of thumb recommends that the loop gain VO should be set to half of the critical loop gain.

In higher order controlled systems and in systems with dead time the loop gain VO should not be too high, otherwise the control process will exhibit poor damping or can even become unstable.

On the other hand, too low a loop gain makes the control loop sluggish and imprecise.

In many cases, fast correction is desirable. Therefore, a certain amount of overshoot is accepted. So there is always a trade-off between speed and overshoot.

Undamped oscillation Damped oscillation


CONTROL QUALITY Places of disturbance may vary

Influence on measured output, system input or system output Control deviation should be minimized over time

Especially relevant for real-time systems

73

11/10/13Dirk Mller: Design of SW for embedded Sys., Winter term 2013/14

14/61

Relationship to RT (2/2)

t

r

Latency or settling time (e.g. until deviation below 3%)

y

e(t3): here overshoot

(as maximum)

t3

t 1

t 2

r (t) y (t)dt

t1 t2 Error e and integral give inspiration for first (simple)

control path and controller types=> P controller and I controller


CONTROLLER How to create a reasonable controller ?

Desired value of controlled variable given by set point

Compute new value of a control / actuating variable to correct the error between measured variable and set point

Often, controlled variable is physical, while measured variable is the representation of it

Create a control path from basic elements Proportional (P) element

Integral (I) element

Derivative (D) element

74

[Wiik

iped

ia]


ELEMENTS Proportional (P) element

Perform immediate proportional reaction to error at output

Adjusted by gain factor kP Integral (I) element

Accumulate past errors and react on them

Adjusted by gain factor kI Derivative (D) element

Based on current rate of change of the error, predict the future

Adjusted by gain factor kD

75


PROPORTIONAL ELEMENT Output is proportional

to the current error

Gain factor too low:Reaction not good enough to fix the error

Gain factor too high:Reaction to error is too extreme (stability)

Only instantaneous reaction, therefore steady-state error

76


P CONTROLLER Example: Mechanical water level

control systems

Adjustable inflow with valve 1

Variable outflow with valve 2

Float sensor 3

Gate valve as control element

Controlled variable is water level x

Output variable y is position of gate valve

Keep water level constant, regardless of water drawing

Desired water level (set point) defined by lever attachment height h

77

49

3.3 The P, PI, PID response characteristic 3.3.1 The proportional-action controller (P-controller) 3.3.1.1 Operating principle

The operating principle of a P-controller can be explained most simply using the example of a mechanical water level control system as per Fig. 3.5. The illustration shows an open water tank with adjustable inflow (gate valve 1) and variable outflow (gate valve 2). The control system consists of float 3 (sensor), lever a + b (comparator and amplifier) and gate valve 1 (control element). The controller's input variable, controlled variable x, is the water level measured by the float. The output variable, manipulated variable y, is the position of gate valve 1 in the inlet pipe. The variable water outlet (depending on the position of gate valve 2) represents the load Q on the system. The control task is to keep the water level in the tank as constant as possible in spite of varying load (drawing of water). The desired water level (setpoint w) is set by the attachment height h of the lever.

Fig. 3-5 Model of a P-controller

1 Gate valve in inlet 2 Gate valve in outlet 2 Float (sensor) 4,5 Float attachment points V1 Inlet volume V2 Outlet volume a+b Lever h Attachment height of the lever w Reference variable x Controlled variable (water level) Xp Proportional band Yh Positioning range

V2

V1

h

4 5

a b1

Yh

Xp

3

x230 cm

200

170

w

2

B33-4

[Siemens]


INTEGRAL ELEMENT

Output is proportional to past error behavior

Delayed reaction to accumulated errors

Can fix steady-state error resulting from proportional reaction

May produce output that exceeds the target(overshoot)

78


I CONTROLLER Integral-action controller

Output variable formed from sum of consecutive input variables over time

Example: Tank with inlet and outlet

Flow rate at inlet as input variable

Level at output as output variable

With constant flow rate, level remains constant

With increased incoming flow, output flow is the same

With input greater than discharge rate, level will raise

Proportional reaction no longer suitable

Needs integral reaction

79

[Siemens]

55

3.3.2 The integral-action controller (I-controller) 3.3.2.1 General Example 1

Integrate, in this context has the sense of combining or adding. In control loop elements with integral action, the output variable is formed from the sum of consecutive input

variables over a period of time.

This can be demonstrated with a simple example

Fig. 3-10 Container as level controller, Example of a control loop with integral action response

The controlled system in the example is used to control level, represented by a tank

with an inlet and an outlet. The input variable is the flow rate at the inlet, and the output

variable is the level (not the flow rate) at the output. If we assume that the flow rate at

the inlet is equal to theat at the outlet, then the level will remain constant. To record the

step rsponse, we need mentally to increase the flow by a sudden large amount. For

example, the inlet valve could be opended manually, so that now, the flow rate at the

inlet is 300l/h instead of 200l/h. The discharge rate will remain unchanged. The output

variable, i.e. the level, will increase continuously until the tank overflows.

We can now no longer maintain that there is a given output variable for every input

variable, since in our example, for any input variable which is greater than the

discharge rate, the level will rise until it overflows. This type of response is defined as

integral action. The contents of the tank are given by the sum of the liquid flowing into

the tank minus the quantity discharged. Only the length of the time taken to fill the tank

will be affected by the input variable. In other words, it is not the output variable itself,

but the rate of change of the outut variable, which depends on the input variable. This is

expressed by the formula:

va = KI . xe

KI is the so called integral control action constand and in this example u.a. is

dependent upon the container.

For an integral action controller the speed of the output variable is dependent upon the

input variable. For a step change in the input variable then the output variable changes

consecutively.

N.B.

S0169

xe2

t

xa

xe

xe1

xa1

xa2

xa

xe

berlauf

t

200 l/h

200 l/h

200 l/h

300 l/h

55

3.3.2 The integral-action controller (I-controller) 3.3.2.1 General Example 1

Integrate, in this context has the sense of combining or adding. In control loop elements with integral action, the output variable is formed from the sum of consecutive input

variables over a period of time.

This can be demonstrated with a simple example

Fig. 3-10 Container as level controller, Example of a control loop with integral action response

The controlled system in the example is used to control level, represented by a tank

with an inlet and an outlet. The input variable is the flow rate at the inlet, and the output

variable is the level (not the flow rate) at the output. If we assume that the flow rate at

the inlet is equal to theat at the outlet, then the level will remain constant. To record the

step rsponse, we need mentally to increase the flow by a sudden large amount. For

example, the inlet valve could be opended manually, so that now, the flow rate at the

inlet is 300l/h instead of 200l/h. The discharge rate will remain unchanged. The output

variable, i.e. the level, will increase continuously until the tank overflows.

We can now no longer maintain that there is a given output variable for every input

variable, since in our example, for any input variable which is greater than the

discharge rate, the level will rise until it overflows. This type of response is defined as

integral action. The contents of the tank are given by the sum of the liquid flowing into

the tank minus the quantity discharged. Only the length of the time taken to fill the tank

will be affected by the input variable. In other words, it is not the output variable itself,

but the rate of change of the outut variable, which depends on the input variable. This is

expressed by the formula:

va = KI . xe

KI is the so called integral control action constand and in this example u.a. is

dependent upon the container.

For an integral action controller the speed of the output variable is dependent upon the

input variable. For a step change in the input variable then the output variable changes

consecutively.

N.B.

S0169

xe2

t

xa

xe

xe1

xa1

xa2

xa

xe

berlauf

t

200 l/h

200 l/h

200 l/h

300 l/h


CONTROLLERS P controllers are load-dependent

Controlled variable cannot be kept constant at all loads Output variable is proportional to the input variable Quick reaction Residual derivation: Steady-state deviation

I controllers are load-independent Control action builds up slowly, therefore time-dependent Manipulated variable changes as long as deviation is given

80


Combine best of both worlds: PI controller

Remove steady-state error, get medium speed of reaction

Fast P controller + load-independent I controller

Can be implemented in different ways Mechanical / hydraulic /

electrical / electronic solution

PI CONTROLLER

81

59

3.3.3 The proportional-plus-integral controller (PI-controller)

3.3.3.1 Operating principle and step response

A PI-controller can be imagined as a P-controller and an I-controller connected in

parallel (Fig. 3-15). This combines the advantages of the P-controller (fast) with those

of the I-controller (load-independent).

Fig. 3-15 Combination of P-controller and I-controller to a PI-controller

Therefore, the step response of a PI-controller (Fig. 3-16) is the sum of the step

responses of a P-controller and an I-controller, i.e. it consists of a proportional

component (P-component) with a superimposed integral-action component (I-

component).

As in the P-controller, the P-component initially gives rise to an adjustment yP of the manipulated variable y that is proportional to the control difference e:

yP = eP

K

However, due to the I-component, the valve does not remain in that position but

continues to travel at a positioning speed vy that is dependent on the integral-action

coefficient Kl of the I-controller and proportional to the control difference e until it is

limited by the end stops, e.g. valve stroke = 0...20 mm.

vy = eI

K

Therefore, after a certain time (t0t1), the manipulated variable change yPI is made up of the sum of the two variables:

( ) ( )tI

xI

KeP

KI

yP

yPI

y =+=

Fig. 3-16 Step response of the PI-controller

yp P-component yI I-component yPI P-component and I-component

e

t

t

y

PI

I

P yP I

y I

yP

0

t0

t1 B33-14

e y

P

I

PI

+

+

59

3.3.3 The proportional-plus-integral controller (PI-controller)


A PI-controller can be imagined as a P-controller and an I-controller connected in

parallel (Fig. 3-15). This combines the advantages of the P-controller (fast) with those

of the I-controller (load-independent).

Fig. 3-15 Combination of P-controller and I-controller to a PI-controller

Therefore, the step response of a PI-controller (Fig. 3-16) is the sum of the step

responses of a P-controller and an I-controller, i.e. it consists of a proportional

component (P-component) with a superimposed integral-action component (I-

component).

As in the P-controller, the P-component initially gives rise to an adjustment yP of the manipulated variable y that is proportional to the control difference e:

yP = eP

K

However, due to the I-component, the valve does not remain in that position but

continues to travel at a positioning speed vy that is dependent on the integral-action

coefficient Kl of the I-controller and proportional to the control difference e until it is

limited by the end stops, e.g. valve stroke = 0...20 mm.

vy = eI

K

Therefore, after a certain time (t0t1), the manipulated variable change yPI is made up of the sum of the two variables:

( ) ( )tI

xI

KeP

KI

yP

yPI

y =+=

Fig. 3-16 Step response of the PI-controller

yp P-component yI I-component yPI P-component and I-component

e

t

t

y

PI

I

P yP I

y I

yP

0

t0

t1 B33-14

e y

P

I

PI

+

+

[Siem

ens]


DERIVATIVE ELEMENT Compute derivative

(slope of change) of the error

Again adjusted by gain factor

Initially, the error changes dramatically, so D-element compensates directly

Afterwards, influence reduces and P+I parts are taking over

82

PT14/15Design of Software for Embedded Systems (SWES)

PID CONTROLLER

Initial charge by D

Reduction almost to P level

Rises again from I influence

83

[Siemens]

68

3.3.6 The proportional-plus-integral-plus-derivative controller (PID-controller)


A PID-controller can be imagined as a proportional-action controller (P-controller), an integral-action controller (I-controller) and a derivative-action element (D-element) connected in parallel (Fig. 3-25a). The controller's output signal (manipulated variable change) is generated not only from the magnitude of the control difference e but, due to the D-element, also its rate of change. As already known, with a D-component, the magnitude of the manipulated variable change is proportional to the rate of change of the controlled variable or control difference.

Fig. 3-25 PID-controller a) Model b) Step response P P-component I I-component D D-component

The step response of the PID-controller (Fig. 3-25b) consists of the additive superimposition of the P-component, I-component and D-component. The illustration shows that an initial large change of the manipulated variable is produced by the D-element. This means that interference will not give rise to too great a control difference. The manipulated variable y is then reduced almost to the level of the P-controller, from where it rises again in a linear fashion under the influence of the I-component. Due to the I-component, the PID controller compensates for setpoint changes and disturbances with no residual deviation and is faster than a PI or P-controller. It is especially suitable for difficult controlled systems.

e y

P

I

D

PID

DI

Tn

y

t

a)

b)

B33-22

1XP

= KP

68

3.3.6 The proportional-plus-integral-plus-derivative controller (PID-controller)


A PID-controller can be imagined as a proportional-action controller (P-controller), an integral-action controller (I-controller) and a derivative-action element (D-element) connected in parallel (Fig. 3-25a). The controller's output signal (manipulated variable change) is generated not only from the magnitude of the control difference e but, due to the D-element, also its rate of change. As already known, with a D-component, the magnitude of the manipulated variable change is proportional to the rate of change of the controlled variable or control difference.

Fig. 3-25 PID-controller a) Model b) Step response P P-component I I-component D D-component

The step response of the PID-controller (Fig. 3-25b) consists of the additive superimposition of the P-component, I-component and D-component. The illustration shows that an initial large change of the manipulated variable is produced by the D-element. This means that interference will not give rise to too great a control difference. The manipulated variable y is then reduced almost to the level of the P-controller, from where it rises again in a linear fashion under the influence of the I-component. Due to the I-component, the PID controller compensates for setpoint changes and disturbances with no residual deviation and is faster than a PI or P-controller. It is especially suitable for difficult controlled systems.

e y

P

I

D

PID

DI

Tn

y

t

a)

b)

B33-22

1XP

= KP

PT14/15Design of Software for Embedded Systems (SWES)

PID CONTROLLER Date back to 1890 governor design and ship steering

Based on observation of human ship controller

Compensates based on current error, past error and change rate

Optimum behavior

Disturbance rejection - Stay at a given setpoint

Command tracking - Implement setpoint changes

Rise time - How fast going close to the final value

Settling time - How fast settling into some range around the final value

84


DIGITAL PID CONTROLLER Digital microcontrollers implement control function

Easier realization of non-linear behavior and adaptive control

Reconfiguration of software possible

Time-discrete, quantized behavior

A/D and D/A conversion requires sampling

Danger of instability through phase shift

Restricted data word length

Accumulation of rounding errors

Quantization errors

Overflow in calculation with wrap-around

85

7

Institut fr Robotik und Prozessinformatik Technische Universitt Braunschweig- 37/64 -

Einfhrung in die diskrete Signalverarbeitung

A/D-Wandlung

D/A-Wandlung


Abhngigkeit der Abtastperiode T


berblick - Einfhrung in die Regelungstechnik II

1. Wiederholung

2. Reglerentwurf fr kontinuierliche Systeme

3. Beispiel zum Reglerentwurf

4. Einfhrung in die diskrete Signalverarbeitung

5. Z-Transformation und digitale Regelung

6. Beispiel zur digitalen Regelung

7. Zusammenfassung

8. Literatur

Reglerentwurf und diskrete Systeme


Z-Transformation

DA

diskretkontinuierlich

Z-Transformation

Inverse Z-Transformation

T Abtastperiode


Korrespondenz-tabelle zur Z-

Transformation


Z-Transformation


ADJUSTING THE CONTROLLER Controller adjustment by determining gain factors / coefficients

Heuristic method by Ziegler / Nichols for P / PI / PID controllers

Only suitable for stable systems, focus on disturbance compensation

I-gain and D-gain set to zero, increase P-gain until oscillation

Table lookup for gain parameters, based on oscillation period

Empirical method

Response too slow:-> Increase influence of P component, reduce influence of I component afterwards

Response oscillates slowly towards goal signal -> Increase influence of P component, reduce influence of D component afterwards

Overshooting -> Reduce influence of P component, increase influence of I component afterwards

86

Design of Software for Embedded Systems

WS15/16

Design of Software for Embedded Systems

Chair of Operating Systems

Jafar Akhundov

Mathematical Theory of Systems

and Control

Chapter 3

http://osg.informatik.tu-chemnitz.de

Purpose of this Extended Lecture

Control system engineering is a complex task What (I hope) you will take away from this lecture: Necessary basics to:

Be able to understand simple systems Read further literature (if necessary) Communicate with engineers Build better systems

Motivation to use math and modelling more frequently Appreciate another view on systems

WS15/16

Design of Software for Embedded SystemsChapter 3: Mathematical Theory of Systems and Control

88



Contents1.Control System Design Process 2.Math Background

1.Complex Variables 2.Differential Equations 3.Laplace Transform

3.Modeling in Frequency Domain 1.Transfer Functions 2.Use case: translational mechanical systems 3.Block Diagram Composition

4. Time Response Analysis 1. Poles and Zeros of Transfer Function 2.1st-order Systems 3.2nd-order Systems

5. System Stability Analysis 6. PID-control Tuning

WS15/16


89



1.1 Control System Definition (refreshment)

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90

Definition: A control system consists of subsystems and processes (or plants) assembled for the purpose of obtaining a desired output with desired performance, given a specified input.

Two major goals of performance: Transient response Steady-State Error

Source: [2]


Input Command

2

1


Example: Elevator

WS15/16


91

t

h

Elevator Response

Steady-state error

Transient Response

Source: [2]



1.2 Open-Loop vs. Closed-Loop (refreshment)

Open-Loop Stable No feedback Not precise Cannot compensate disturbances Simple Cheap

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92

Closed-Loop Can become unstable Has a feedback loop Precise Compensates disturbances Complex Expensive



1.3 Objectives of Analysis and Design Three major goals: Producing the desired transient response Reducing steady-state error Achieving Stability

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93

Definition: Control system is stable if its natural response a) eventually approaches zero or b) oscillates. If the natural response of the system grows without bound and becomes much greater than forced response, the system is considered unstable.

Other goals:

Finances Robust design

Source: [2]



Control System Design Process: Overview

WS15/16


94

Determine a physical

system and specifications

from the requirements.

Draw a functional

block diagram.

Transform the physical system

into a schematic.

If multiple blocks, reduce the block

diagram to a single block or

closed-loop system.

Analyze, design, and test to see

that requirements and

specifications are met.

Use the schematic to obtain a

block diagram, signal-flow diagram,

or state-space representation.

Source: [2]



Control System Design Process: Stages

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95

1.Transform requirements into a physical system 1.Determine physical dimensions, e.g. position, mass 2.Using the requirements, define specifications for the design, e.g. transient response, steady-state accuracy

2.Functional block diagram 1.Translate qualitative description into functional subsystems 2.Define interconnections between subsystems

3.Create schematic 1.Make approximations of the system 2.Make assumptions about subsystems 3.Simplify, but not oversimplify 4.Instantiate the modules from functional block diagram

Controller

Engine Air Intake

User Input

Velocity of a car

Source: [2]



Control System Design Process: Stages (2)

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96

4.Develop a mathematical model (block diagram) 1.From schematic and physical laws (Kirchhoff, Newton's) 2.Many systems can be described by a linear ODE which relates input with the output

3.Assumptions and approximations may simplify this ODE 4.Often other representations are used instead:

1.For linear, time-invariant (LTI) systems, Laplace transform derives a transfer function 2.Alternatively, one can-use state-space representation (possible for non-LTI systems)

5.Reduce Block Diagram 1.Single block 2.Only system input and output present

andnc(t)

dtn+ an1

dn1c(t)

dtn1+ a0c(t) = bm

dmr(t)

dtm+ bm1

dm1r(t)

dtm1+ b0r(t)

Input Output

r(t) c(t)System

Source: [2]



Control System Design Process: Stages (3)

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97

6.Analyze and Design 1.Mostly testing and verification 2.Redesign if necessary 3.Test signals commonly used:

Input Function Use

Impulse Transient response

Step Transient response Steady-state errorRamp Steady-state error

Parabola Steady-state error

Sinusoid Transient response Steady-state error

(t)

u(t)

u(t)t1

2u(t)t2

sin!t

Source: [2]



2.1. Complex Numbers (refreshment) Complex numbers were introduced to solve equations like:

A complex number written in rectangular form:

j is called the imaginary unit. Alternatively,

R is the magnitude and is the phase of

WS15/16


98

z = x+ jy, j =p1

x

2 = 1

x = R cos

y = R sin

Real

Imaginary

z

= x jy

z = x+ jy

R

RzSource: [1]



Complex Numbers Properties (refreshment) Complex numbers written in the Euler form:

Addition/Subtraction is easier to do in the rectangular form:

Multiplication/Division are more convenient in the Euler form:

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99

z = R cos + jR sin

= R(cos + j sin )

= Rej

z1 z2 = (x1 x2) j(y1 y2)

z1z2 = (R1R2)ej(1+2)

z1z2

= (R1R2

)ej(12)

Source: [1]



2.2 Differential Equations (refreshment)

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100

Definition: Differential equation is any equation which contains derivatives, either ordinary derivatives or partial.

Definition: Solution to a differential equation on an interval is any function which satisfies the equation on this interval.

< t < y(t)

Definition: Order of differential equation is the larges derivative present in it.

Source: [3]




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101

Example (Newton's Second Law):F = ma

F = md

2x

dt

2

Definition: Linear differential equation is any differential equation that can be written in the following form:

an(t)y(n)(t) + an1(t)y

(n1)(t) + ...+ a1(t)y0(t) + a0(t)y(t) = g(t)

Note: there are no products of the function and its derivatives and neither the the function or its derivatives occur to any power other than the first power.

Source: [3]




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102

Definition: Differential equation is time-invariant if it does not depend explicitly on time.

Many techniques for solving ODEs: Linear 1st order ODEs Separable ODEs Exact ODEs Bernoulli ODEs

...

N(y)dy

dx

= M(x)

dy

dt+ p(t)y = g(t)

M(x, y) +N(x, y)dy

dx

= 0

y

0 + p(x)y = q(x)yn

Source: [3]



2.3 Laplace Transform

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103

Motivation: Differential equation relates input to output, yes But: All system parameters, input and output are mixed up in the equation Hence: it is not a satisfying representation from a system perspective

Note: the notation for the lower limit means that even if the function is discontinuous at t=0, it is possible to start integration prior to discontinuity.

Definition: The Laplace transform is defined as

where is a complex variable. Condition for existence:

s = + j!

L|f(t)| = F (s) =Z 1

0f(t)estdt

Z 1

0

f(t)ektdt


2.3 Laplace Transform Simple Example

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104

Problem: Find the Laplace transform of

Solution:

f(t) = Aeatu(t)

F (s) =

Z 1

0f(t)estdt

=

Z 1

0Aeatestdt

= A

Z 1

0e(s+a)tdt

= As+ a

e(s+a)t1

t=0

=A

s+ a

This integral converges only if

(s+ a) < 0s+ a > 0

Source: [2]



2.3 Laplace Transform table

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105

Function Transform

(t)

u(t)

u(t)t

eatu(t)

sin(!t)u(t)

cos(!t)u(t)

1

1

s1

s2

1

s+ a!

s2 + !2s

s2 + !2

tnn!

sn+1

...



2.3 Laplace Transform Another Example

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107


Solution:

f(t) = 6e5t + e3t + 5t3 9

F (s) = 61

s (5) +1

s 3 + 53!

s3+1 91

s

=6

s+ 5+

1

s 3 +30

s4 9

s

Source: [3]



2.3 Laplace Transform Complex Example

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108


Solution:

f(t) = t2sin(2t)

F (s) =4s

(s2 + 4)2

F 0(s) = 12s2 16

(s2 + 4)3

H(s) =12s2 16(s2 + 4)3

L|tf(t)| = F 0(s) L|tsin(at)| =2as

(s2 + a2)2Using the fact that and

Source: [3]



2.3 Inverse Laplace Transform

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109

Definition: The inverse Laplace transform is defined as

where is a complex variable, is real and u(t) is a

unit-step function. s = + j!

L1|F (s)| = 12j

Z +j1

j1F (s)estds = f(t)u(t)

ExampleProblem: Find inverse Laplace transform of F (s) = 1

(s+ 3)2

Solution: We use frequency shift theorem and the transform of L|u(t)t| =1

s2

L1| 1(s+ a)2

| = eatu(t)tSo, and f(t) = e3tu(t)t

Source: [2]



2.3 Partial-Fraction Expansion

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110

Problem: how to find an ILT of a complex function? Solution: we can convert the function into a sum of simpler terms, for which we know

the ILT. This method is called partial fraction expansion Assume that , where the order of is lower than that of If this is not the case, divide polynomials until this condition is fulfilled Three cases are possible: Roots of the denominator are real and distinct Roots of the denominator are real and repeated Roots of the denominator are complex or imaginary (not considered here)

F (s) =N(s)

D(s)N(s) D(s)

D(s)

Source: [2]



2.3 Partial-Fraction Expansion (Case 1)

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111

Roots of denominator are real and distinct It is possible to write the PFE as a sum of terms Constants, called residues, form numerators For example,

To find , we multiply this equation by (s+1) which isolates

Then we let s approach -1 which eliminates last term and we get Similarly solving this for the second residue, we get The resulting function can be transformed now:

F (S) =2

(s+ 1)(s+ 2)=

K1s+ 1

+K2s+ 2

K1 K1

2

s+ 2= K1 +

(s+ 1)K2s+ 2

K1 = 2

K2 = 2f(t) = (2et 2e2t)u(t)

Source: [2]



2.3 Partial-Fraction Expansion (Case 2)

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Case 2: Roots of denominator are real and repeated

To find the residues for the roots of multiplicity greater than unity:

For all other residues, use the method of PFE for the case 1

F (s) =N(s)

D(s)=

N(s)

(s+ p1)r(s+ p2)...(s+ pn)

K1(s+ p1)r

+K2

(s+ p1)r1+ ...+

Krs+ p1

+Kr+1s+ p2

+ ...+Kn

s+ pn

Ki =1

(i 1)!di1F (s)

dsi1

s!p1

, i = 1, .., r

Source: [2]



2.3 Using Laplace Transform To Solve Linear ODE

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Problem: Solve this differential equation if all initial conditions are zero.d2y

dt2+ 12

dy

dt+ 32y = 32u(t)

s2Y (s) + 12sY (s) + 32Y (s) =32

s

Y (s) =32

s(s2 + 12s+ 32)=

32

s(s+ 4)(s+ 8)

Y (s) =32

s(s+ 4)(s+ 8)=

K1s

+K2s+ 4

+K3s+ 8

K1 =32

(s+ 4)(s+ 8)

s!0

= 1

K2 =32

s(s+ 8)

s!4

= 2

K3 =32

s(s+ 4)

s!8

= 1

Y (s) = (1 2e4t + e8t)u(t)

Solution: Using differentiation properties and Laplace transform of unit-step response, we get:

Y (s) =1

s 2

s+ 4+

1

s+ 8

Source: [2]



2.3 Using Laplace Transform To Solve Linear ODE

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In general: 1. Having a linear ODE, take a Laplace transform of the ODE using Laplace transform rules

and initial conditions 2. Algebraically solve the resulting equation 3. Take the inverse Laplace transform using PFE

Source: [2]



3.1 Transfer Functions

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It is now possible for us to give a definition of a function that algebraically relates output of the system to its input

Take a general n-th order linear time-invariant ODE:

Here, a, b are constants and c(t) and r(t) are output and input, respectively Assuming that all initial conditions are =0 and taking Laplace transform of both sides:

The ratio

is called the transfer function. It is evaluated with zero initial conditions.The output can be found by:

andnc(t)

dtn+ an1

dn1c(t)

dtn1+ ...+ a0c(t) = bm

dmr(t)

dtm+ bm1

dm1r(t)

dtm1+ ...+ b0r(t)

(ansn + an1s

n1 + ...+ a0)C(s) = (bmsm + bm1s

m1 + ...+ b0)R(s)

C(s)

R(s)= G(s) =

(bmsm + bm1sm1 + ...+ b0)

ansn + an1sn1 + ...+ a0

C(s) = G(s)R(s)Source: [2]



3.1 Transfer Functions - Example

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Problem: Find the transfer function represented by ODE:

Solution: Taking Laplace transform of both sides and assuming zero initial conditions,

If a system response is needed for a unit-step input:

Using PFE and taking inverse Laplace transform:

dc(t)

dt+ 2c(t) = r(t)

sC(s) + 2C(s) = R(s)

G(s) =1

s+ 2

C(s) = R(s)G(s) =1

s(s+ 2)

C(s) =1/2

s 1/2

s+ 2c(t) =

1

2 1

2e2t

Source: [2]



3.2 Mechanical Translational Transfer Functions

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Component Force-Displacement Transfer Function

Viscous damperf(t), x(t)

b

f(t) = bdx(t)

dt

bs

M

Mass

f(t), x(t)f(t) = M

d

2x(t)

dt

2Ms2

f(t), x(t)

K

Springf(t) = Kx(t) K

Source: [2]


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118

M

f(t), x(t)

Problem: Derive a transfer function for this system

Solution: First, write down the differential equation describing the system

M

d

2x(t)

dt

2+ b

dx(t)

dt

+Kx(t) = f(t)

Taking the Laplace Transform:

Ms2X(s) + bsX(s) +KX(s) = F (s)

After some algebraic manipulations:

(Ms2 + bs+K)X(s) = F (s)

1

Ms2 + bs+K

X(s)F (s)

3.2 Mechanical Translational Transfer Functions Example

X(s)

F (s)=

1

Ms2 + bs+K

Source: [2]



Summary of the last lecture

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Previous lecture was about step 4 of control design: creating a mathematical model How do we do that?

Describe a system as a differential equation If the differential equation is ordinary, time-invariant and linear, we use Laplace

transform

As a result, we get representation of the system in complex domain (transfer function) Why do we do that?

Easy solution of differential equations Easy reduction of complex systems (step 5): discussed in this lecture Easy reasoning about time response, stability (step 6): discussed in this lecture

119



3.3 Block diagrams

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Complex systems are represented by interconnections of many subsystems Our goal: represent response of the whole system as a single transfer function Block diagrams are used for frequency-domain analysis and design Signal flow graphs are used for state-space methods Subsystem is represented as a block with: input, output and transfer function

Input Output

R(s) C(s)G(s)

System

++

R1(s)

R2(s)R3(s)

C(s) = R1(s) +R2(s)R3(s)

Summing junction Pickoff point

R(s)R(s)

R(s)

R(s)

For building more complex systems, two more elements are needed:

Source: [2]

120



3.3 Cascade Form of Transfer Functions

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121

R(s) C(s)G3(s)G2(s)G1(s)

Source: [2]

R(s)

G1(s) G2(s) G3(s)

G1(s)R(s) G2(s)G1(s)R(s) G3(s)G2(s)G1(s)R(s)

C(s) =



3.3 Parallel Form of Transfer Functions

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R(s)

G1(s)R(S)G1(s)

G2(s)

G3(s)

G2(s)R(s)

G3(s)R(s)

C(s) = (G1(s)G2(s)G3(s))R(s)

R(s) C(s)G1(s)G2(s)G3(s)

Source: [2]



3.3 Feedback Form of Transfer Functions

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123

R(s) +

C(s)E(s)G(s)

H(s)

error

Plant and Controller

Input Output

Feedback

R(s) C(s)

G(s)H(s) is called loop gain or open loop transfer function

G(S)

1G(s)H(s)

E(s) = R(s) C(s)H(s)

Using

C(s) = E(s)G(s)) E(s) = C(s)G(s)

and

G(S)

1G(s)H(s)we get:

and solving for C(s)

R(s)= Ge(s)

Source: [2]



3.3 Restructuring Topologies (Summing Junction)

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R(s) +

G(s)

X(s)

R(s) +

G(s)

X(s)

G(s)C(s) C(s)

Source: [2]

R(s) +

G(s)

X(s)

R(s) +

X(s)

G(s)C(s) C(s)

1

G(s)



3.3 Restructuring Topologies (Summing Junction)

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125

R(s)

G(s)

1

G(s)

R(s)

R(s)

R(s)G(s)

R(s)

R(s)

R(s)

R(s)G(s)

G(s)1

G(s)

R(s)G(s)

R(s)G(s)

R(s)G(s)

R(s)G(s)

R(s)

R(s)G(s)

G(s)

G(s)

G(s)

R(s)G(s)

R(s)G(s)

Source: [2]



3.3 Example

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Problem: reduce the block diagram and find the transfer function of the system:

Source: [2]

R(s) + C(s)G1(s) G2(s)

H2(s)

H1(s)

H3(s)

G3(s)

+

+ +



3.3 Example

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Design of Software for Embedded SystemsChapter 3: Mathematical Theory of Systems and Contro