Users Often Want . . . At Least the Users . . . Description of the . . . Simple Case Study . . . Perfect Reproducibility Is What Control . . . Not Always Algorithmically Control Strategy (cont-d) Comment Possible: A Pedagogical Can We Achieve . . . Proof Observation Home Page Jake Lasley, Salamah Salamah, and Vladik Kreinovich Title Page ◭◭ ◮◮ Department of Computer Science University of Texas at El Paso, El Paso, Texas 79968, USA, ◭ ◮ jlasley@miners.utep.edu, isalamah@utep.edu, vladik@utep.edu Page 1 of 16 Go Back Full Screen Close Quit
Users Often Want . . . At Least the Users . . . 1. Users Often Want Perfect Reproducibility Description of the . . . • Users of software and, more generally, users of computer- Simple Case Study . . . based systems often want perfect reproducibility : that What Control . . . Control Strategy (cont-d) – if we place the system in the exact same situation, Comment – it should react the exact same way. Can We Achieve . . . • Of course, if a real-life system includes sensors and Proof measurements, we cannot have exact reproducibility. Home Page • If we measure the same value several times, we may Title Page get different results. ◭◭ ◮◮ • As a result, e.g., when we have a computer-controlled ◭ ◮ thermoregulation system, then: Page 2 of 16 – even for the exact same temperature, Go Back – the sensors readings will be slightly different and Full Screen thus, the system’s reaction may be different. Close Quit
Users Often Want . . . At Least the Users . . . 2. At Least the Users Want Perfect Reproducibil- Description of the . . . ity in the Ideal Situation Simple Case Study . . . • The above measurement uncertainty is well known. What Control . . . Control Strategy (cont-d) • So what the users want is that: Comment – the system’s behavior be perfectly reproducible in Can We Achieve . . . the idealized situation, Proof – when we can measure each quantity with any given Home Page accuracy. Title Page • We provide simple arguments that even in this ideal- ◭◭ ◮◮ ized case, perfect reproducibility is not always possible. ◭ ◮ Page 3 of 16 Go Back Full Screen Close Quit
Users Often Want . . . At Least the Users . . . 3. Description of the Simple Case Study Description of the . . . • Let us consider a very simple control situation, when Simple Case Study . . . we want to keep some quantity q at a given level q 0 . What Control . . . Control Strategy (cont-d) • To perform this task, we measure q . Comment • We consider an idealized case when: Can We Achieve . . . – for every integer n , Proof – we can measure q with accuracy of n binary digits Home Page (i.e., with an accuracy 2 − n ). Title Page • In such a measurement, we get a measurement result ◭◭ ◮◮ q n which is 2 − n -close to q : | q − q n | ≤ 2 − n . ◭ ◮ • Let us also consider a very simplified version of a con- Page 4 of 16 troller, with only two options: Go Back – we can switch on a device that increases q , Full Screen – or we can switch on a device that decreases q , – or we can do nothing. Close Quit
Users Often Want . . . At Least the Users . . . 4. Simple Case Study (cont-d) Description of the . . . • Example: regulating room temperature: Simple Case Study . . . What Control . . . – if the temperature is above a certain threshold, Control Strategy (cont-d) switch on the air conditioner, Comment – if the temperature is below a certain threshold, Can We Achieve . . . switch on the heater, and Proof – if the temperature is comfortable, do nothing. Home Page • Keeping a satellite at a given height above earth: Title Page – if the height decreases, we switch on an engine that ◭◭ ◮◮ pushes the orbit up; ◭ ◮ – if the height increases, we switch on another engine Page 5 of 16 that pushes the orbit down; and Go Back – if the height is close to desired one, do nothing. Full Screen Close Quit
Users Often Want . . . At Least the Users . . . 5. What Control Strategy We Can Apply Description of the . . . • We would like to design a computer-based control sys- Simple Case Study . . . tem for this setting. What Control . . . Control Strategy (cont-d) • This system can start by measuring the value of the de- Comment sired quantity q with some initial accuracy of n 0 binary Can We Achieve . . . digits. Proof • Based on the result q n 0 of this measurement, we can Home Page make four possible decisions: Title Page – we can switch on the device that increases q ; we ◭◭ ◮◮ will denote the corresponding decision by +; – we can switch on the device that decreases q ; we ◭ ◮ will denote the corresponding decision by − ; Page 6 of 16 – we can decide to do nothing at this point; we will Go Back denote the corresponding decision by 0; or Full Screen – we can select to perform a more accurate measure- ment. Close Quit
Users Often Want . . . At Least the Users . . . 6. Control Strategy (cont-d) Description of the . . . • In the last case: Simple Case Study . . . What Control . . . – the system will generate an integer n > n 0 , Control Strategy (cont-d) – perform the measurement with accuracy 2 − n , Comment – based on the new measurement result q n , again se- Can We Achieve . . . lect one of these four options. Proof Home Page • After one or several iterations, we produce: Title Page – a plus decision + (increase q ), ◭◭ ◮◮ – a minus decision − (decrease q ), or – a 0 decision (do nothing). ◭ ◮ Page 7 of 16 • When the value q is sufficiently large ( q ≥ q for some q > q 0 ), we should make a minus decision. Go Back • When the value q is sufficiently small ( q < q for some Full Screen q < q 0 ), we should make a plus decision. Close Quit
Users Often Want . . . At Least the Users . . . 7. Comment Description of the . . . • In real life, we often have the option of performing a Simple Case Study . . . more accurate measurement; for example: What Control . . . Control Strategy (cont-d) – if a person has fallen down and hurt himself, and Comment an X-ray picture is inconclusive, Can We Achieve . . . – a doctor may order an MRI image to get a more Proof accurate picture of the damage. Home Page • The main difference between such real-life situations Title Page and our idealized situation is that: ◭◭ ◮◮ – in real life, there is always a limit of how accurately ◭ ◮ we can measure, while Page 8 of 16 – in our idealized setting, we assume that we can per- form the measurement with an arbitrary accuracy. Go Back Full Screen Close Quit
Users Often Want . . . At Least the Users . . . 8. Can We Achieve Perfect Reproducibility in Such Description of the . . . a Situation? Simple Case Study . . . • Is it possible, in such an idealized situation, to achieve What Control . . . perfect reproducibility? Control Strategy (cont-d) Comment • In other words, is it possible to design a control strat- Can We Achieve . . . egy in such a way that: Proof – for the same actual value of the parameter q , Home Page – the system would make the exact same decision Title Page when this value is encountered the next time? ◭◭ ◮◮ • We prove that such a perfectly reproducible control ◭ ◮ algorithm is impossible. Page 9 of 16 Go Back Full Screen Close Quit
Users Often Want . . . At Least the Users . . . 9. Proof Description of the . . . • We will prove this impossibility by contradiction. Simple Case Study . . . What Control . . . • Let us assume that such a perfectly reproducible con- Control Strategy (cont-d) trol strategy is possible. Comment • Then, for each actual value q of the corresponding Can We Achieve . . . quantity, this control algorithm returns +, − , or 0. Proof • For values q ≥ q , all the decisions are minus decisions. Home Page Title Page • Thus, the set S − of all the values q for which the algo- rithm produces a minus recommendation is non-empty. ◭◭ ◮◮ • For q ≤ q , all the decisions are + recommendations. ◭ ◮ • Thus, for these values q , we never make a minus deci- Page 10 of 16 sion. Go Back • So, S − only contains values which are larger than q . Full Screen • Therefore, the set S − is bounded from below. Close Quit
Users Often Want . . . At Least the Users . . . 10. Proof (cont-d) Description of the . . . • The set S − is non-empty and bounded from below. Simple Case Study . . . What Control . . . • Thus, this set has the greatest lower bound (infimum) Control Strategy (cont-d) def = inf( S − ). s Comment • One can see that for each n , in the 2 − n -vicinity of the Can We Achieve . . . value s , there exist: Proof – a point s − n for which the algorithm does not produce Home Page minus, and Title Page – a point s + n for which the algorithm does produce ◭◭ ◮◮ minus. ◭ ◮ • As s − n , we can simply take s − n = s − 2 − n . Page 11 of 16 • Since s − n < s , and s is the infimum of S − , the system Go Back cannot return minus for the value s − n . Full Screen • The existence of the value s + n ∈ S − for which s + n ≤ s + 2 − n is also easy to show. Close Quit
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