SoSe 2018 M. Werner 2 / 29 osg.informatik.tu-chemnitz.de

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1 Real-Time Systems Summer term 2018 Real-Time Systems 2 nd Chapter Requirements Prof. Matthias Werner For a system to be designed, a specification has to be provided Specifications (and the resulting system) have to meet a number of requirements Usually, these requirements describe the functional properties of the system In addition to normal (functional) requirements, there exist for real- application timing requirements Professur Betriebssysteme SoSe 2018 M. Werner 2 / 29 osg.informatik.tu-chemnitz.de Timing Requirements (cont.) Kindes of Deadlines Different kinds of deadlines value deadline There exists several kinds of timing requirements firm example: cell phone Most frequently: specify, till when a certain event must happen deadline Typically, the event is the end of a calculation, or the output of a result Deadlines may be absolute or relative In case of deadline violation, task will be canceled value deadline More general: value function (TVF) Gives the benefit, if an event happens at certain (absolute or relative) instant of Even the concept of TVF is more general, in real life mostly deadlines are used hard Task has to meet deadline soft value deadline example: airbag example: video streaming Correspondingly, the real- systems are (a bit inexactly) called hart or soft SoSe 2018 M. Werner 3 / 29 osg.informatik.tu-chemnitz.de SoSe 2018 M. Werner 4 / 29 osg.informatik.tu-chemnitz.de

2 Other Timing Requirements Other Timing Requirements (cont.) Beside of deadlines, there are other timing requirements Somes, also premature events are useless: value useful interval Some a task may compensate the failing/delay of another task Requirements for the task set T = {T 1,..., T n } f i (t): value function of task T i f i (t c,i )! v T T t c,i : completion of task T i v T : minimal required value of task set T Instead of sum, any other arbitrary function can be used Mostly harmless buffering Jitter SoSe 2018 M. Werner 5 / 29 osg.informatik.tu-chemnitz.de SoSe 2018 M. Werner 6 / 29 osg.informatik.tu-chemnitz.de We consider the semantics of real- systems Frequently, there are periodic incidences Then there may special timing requirements, e.g.: Jitter: deviation from period Actually, all real- systems do control Also computer games or muldia applications; however, the system to be controlled (control plant) can not be always easily identified Structure of a real- system: event jitter sensors computer system actuators period environment SoSe 2018 M. Werner 7 / 29 osg.informatik.tu-chemnitz.de SoSe 2018 M. Werner 8 / 29 osg.informatik.tu-chemnitz.de

3 Feedback Control Loop Refined structure: Example: Simple Control Program user reference sensors computer system actuators measuring environment disturbance controlled process controlling 1 while ( true ) { 2 in [ port1 ], y; // read process variable 3 in [ port2 ], w; // read set point 4 x := func(w y); // calculate manipulated variable 5 out [ port3 ], x; // write manipulated variable 6 sleep ( ); // between control commands 7 } Similar to classical feedback control loop reference w( t) + disturbance control deviation actuating variable d( t) e( t) u( t) - controller system controlled variable y( t) If the calculation (line 4) fluctuates, period of control commands also does Increases jitter SoSe 2018 M. Werner 9 / 29 osg.informatik.tu-chemnitz.de Alternative Control Program 1 set r to interrupt periodically with period ; 2 3 at each r interrupt do { 4 in [ port1 ], y; // read process variable 5 in [ port2 ], w; // read set point 6 x := func(w y); // calculate manipulated variable 7 out [ port3 ], x; // write manipulated variable 8 } Jitter-free call Selection of parameter is important design decision Term: sampling period Typical dimensions: milliseconds til second Processing should be in sequential order next interrupt sets deadline for the previous one SoSe 2018 M. Werner 11 / 29 osg.informatik.tu-chemnitz.de SoSe 2018 M. Werner 10 / 29 osg.informatik.tu-chemnitz.de Transfer Elements Problem How does one determine the actual requirements for a real- system? To derive timing requirements, we take a look at the classical control theory (maybe recap) Ideas: Dynamic physical systems are regarded as information transfer systems Abstract from physical impact and medium Consider transfer elements Transfer elements constitute a relation between physical entities SoSe 2018 M. Werner 12 / 29 osg.informatik.tu-chemnitz.de

4 Transfer Elements (cont.) Mathematical Description physical value A transfer element Input and output may be arbitrary entities physical value B Mechanical systems (speed, acceleration, etc.) Electrical systems (voltage, current, charge, etc.) Thermodynamic systems (pressure, temperature, heat, etc.)... Arbitrary combinations The relation beween input and output of transfer elements 1 can be described by differential equations Let Then: u(t) input signal y(t) output signal d n y a n dt + a d n 1 y n n 1 dt + + a dy n 1 1 dt + a 0y(t) = d q u b q dt + b d q 1 u q q 1 dt + + b du q 1 1 dt + b 0u(t) 1 More precise: linear, -invariant transfer elements (LTI elements) SoSe 2018 M. Werner 13 / 29 osg.informatik.tu-chemnitz.de Examples Examples for electrical systems transfer element symbol relation resistor inductor capacitor u(t) = i(t) R u(t) = L i(t) = C d dt i(t) d dt u(t) SoSe 2018 M. Werner 14 / 29 osg.informatik.tu-chemnitz.de Step Response and Impulse Response LTI transfer elements are unambiguously defined by their or impulse response Step response Output (response) of a transfer element 0 t < 0 to the input: u(t) = 1 t 0 (Heaviside step function) 1,25 1 0,75 Impulse Response Output (response) of a transfer element to the input: 1 t [0, t] u(t) = lim t t 0 0 sonst (Dirac delta function) 1,25 1 0,5 0,75 0,25 0,5-0,5-0,25 0 0,25 0,5 0,75 1 1,25 1,5 1,75 0,25 SoSe 2018 M. Werner 15 / 29 osg.informatik.tu-chemnitz.de -0,25-0,5-0,25 0 0,25 0,5 0,75 1 1,25 1,5 1,75 Heaviside function -0,25 Dirac function SoSe 2018 M. Werner 16 / 29 osg.informatik.tu-chemnitz.de

5 General Response Typical Transfer Elements name transfer function schematic diagram If the impuls response is known, one can determine the answer to an arbitrary signal P-element y(t) = K u(t) K General Response A transfer element with the impulse response g(t) reacts to a signal x(t) with the following response: y(t) = x(t) g(t) D-element y(t) = d dt u(t) Ks I-element d K dt y(t) = u(t) s Remarks: is the convolution operator: a(t) b(t) def = a(t τ) b(τ)dτ To avoid integration, one unsually applies the Laplace transformation T t -element y(t) = u(t T 0 ) T t PT 1 -element T 9y(t) + y(t) = u(t) PT 1 SoSe 2018 M. Werner 17 / 29 osg.informatik.tu-chemnitz.de Superposition SoSe 2018 M. Werner 18 / 29 osg.informatik.tu-chemnitz.de Combination Due to the superposition, transfer elements can easily be combined For LTI elements, the following is true: concatenation u(t) = a u 1 (t) + b u 2 (t) y(t) = a y 1 (t) + b y 2 (t) split With other words: one can deal with each signal component separately This is called superposition principle combine Arrows describe signal flow Transfer elements are reactionless Filled circles are signal splittings and empty circles are signal combinations SoSe 2018 M. Werner 19 / 29 osg.informatik.tu-chemnitz.de SoSe 2018 M. Werner 20 / 29 osg.informatik.tu-chemnitz.de

6 Control Objective of a feedback control loop An output signal (y(t)) should be controlled with respect to a target signal (w(t)). Time-Discrete Control Loop The use of a computer enforces a -discrete control disturbance d( t) d( t) reference w( t) controller u( t) u ( t) actuator R adjustment characteristic disturbance characteristic y S( t) y( t) w( t) + e( t) u( t) holding controller element - system y( t) system measuring element Classical controllers consist of analogous transfer elements In this course: controller = computer + software y m( t) - random noise r( t) SoSe 2018 M. Werner 21 / 29 osg.informatik.tu-chemnitz.de Additional sampling elements and holding element The controlled value is measured at discrete s only (sampling instance) Time between two samples is generally constant sampling SoSe 2018 M. Werner 22 / 29 osg.informatik.tu-chemnitz.de Nyquist Shannon Sampling Theorem Dead Time Computerization implies another impact: dead s Consider event chain: Known from Computer Networks or Communication Theory : Nyquist Shannon sampling theorem A bandlimited signal (highes frequency: f max ) must be sampled with a sampling frequency f sample that is at least twice f max to allow a perfect reconstruction of the original signal. event effect delay read sensor SuC interrupt hardware A/D conversion software ISR t sense Correspondingly, a discrete controller can not react to signal parts f max > f sample 2 control task ready Signal determines (minimum) sampling rate and thus, maximum deadline control task t CPU t dead control event D/A conversion t act SoSe 2018 M. Werner 23 / 29 osg.informatik.tu-chemnitz.de SoSe 2018 M. Werner 24 / 29 osg.informatik.tu-chemnitz.de

7 Precision of the Real-Time Image 0#&%'&*+#(1#234&56#&% The accuracy interval of real- image mapping correspondents to a dead "#$%#&'%#()*+%,#&%-.-%#/# Impact of Dead Times Additional dead s can render a controller inoperable System becomes unstable >? 8$32- y z dead = 3** y x % 9:- % &;6 % &;!((((( % &<% <% 4-# 7#3'(8&/# P-controller 5-B426%('#%,%#(C&-%9$&#(D$#*#2% Figure: accuracy +&-%9$. interval (from 7CE(=#$(:#9:3*+%#%#2(78()2%&%. [Kop97]),4/(0#&%B426%(%(&-%(#&2#(5#9$=2#%#(F#25#(G92(0#&%B426%#2(H% & I(% &;! I(JJJI(% &;6 K SoSe 2018 M. Werner 25 / 29 osg.informatik.tu-chemnitz.de M25#(= 3** <,D% & E;,D% &;6 E(5&:%(,#&%'J(1#234&56#&%(32 /35#(&-%(,#&%'J(5#234(,4/(0#&%B426%(% & I(O#22(#PJ(% Q!(7C & R("3'4#D78 N/35#(3%(% & E( '4#D78 )2%&%. 3%(% Q E( G3''(=#$(,#&%'J(1#234&56#&%(&-%(=4$*+(=&#(S.23/&6(=#$(78()2%&%. Impact of Dead Times (cont.) :#-%&//% Further Requirements w Source: Samal u. Becker: Grundriss der praktischen Regelungstechnik SoSe 2018 M. Werner 26 / 29 osg.informatik.tu-chemnitz.de %&'( )* +(, -*./( )0"#1$ (,'2 3"##453"#1$ %,6 "# $%&'( 72 &66 8+(*&'2((69:*;+'<0=(*&*,,(*>?&*>.'@ Stability can determined by different methods, e.g.: Criterion by NYNQUIST Criterion by HURWITZ Even for stable systems, dead s may limit the control quality or restrict the (controllable) state space Stability and quality requirements can influence timing requirements Beside timing requirement, real- systems have usually other non-functional requirements, too, e.g.!! reliability availability safety security... Out of scope for this class However: such kind of requirements may interact with timing requirements SoSe 2018 M. Werner 27 / 29 osg.informatik.tu-chemnitz.de SoSe 2018 M. Werner 28 / 29 osg.informatik.tu-chemnitz.de

8 References References [Liu00] [WB05] [Kop97] Jane W. S. Liu. Real-Time Systems. Prentice Hall, 2000, Chapter 1 and 2 Heinz Wörn and Uwe Brinkschulte. Echtzeitsysteme. Grundlagen, Funktionsweisen, Anwendungen. Springer, 2005, Chapter 1 Hermann Kopetz. Real-Time Systems Design Principles for Distributed Embedded Applications. Kluver Academic, 1997, Section 1.3 SoSe 2018 M. Werner 29 / 29 osg.informatik.tu-chemnitz.de