سیستمهای کنترل خطی پاییز 1389 بسم ا... الرحمن الرحيم دکتر حسين بلندي - دکتر سید مجید اسما عیل زاده
Recap. Transient Response: Second order system transient response State space equation 2
سؤال: چگونه ماتريس را محاسبه كنيم؟ (2). (3). (1). 3
4 1) When ζ = 1, ωd = 0
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The tracking error: 6
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8 2) Over damped: ζ > 1
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3) 2 nd order General: Recall 1 st order system step response: 2 nd order General case: 11
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Pole location determines transient 13
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All closed-loop poles must be strictly in the left half planes Transient dies away Dominant poles: those which contribute the most to the transient Typically have dominant pole pair – (complex conjugate) – Closest to jω-axis (i.e. the least negative) – Slowest to die away 17
4) Putting all things together: Settling time: 18
Typical design specifications Steady-state: e ss to step ≤ # % t s ≤ · · · Speed (responsiveness) t r ≤ · · · t d ≤ · · · Relative stability M p ≤ · · · % 19
These specs translate into requirements on ζ, ωn or on closed-loop pole location : Find ranges for ζ and ωn so that all 3 are satisfied. 20
Find conditions on σ and ωd. 21
In the complex plane : 22
Constant σ : vertical lines σ > # is half plane 23
or ts=5 or ts < 2.5 ts < … corresponds to a half plane to the left of a vertical line Constant σ : vertical lines σ > # is half plane Any poles on the same vertical line have the same and the same settling time t s 24
Constant ωd : horizontal line 25
Constant ωn : circles ωn < · · · inside of a circle ωn > · · · outside of a circle 26
Constant ωn : circles ωn < · · · inside of a circle ωn > · · · outside of a circle Any poles on the same circle have the same n and similar rise time/delay time Inside circle corresponds to tr> … Outside a circle corresponds to tr < … 27
Constant ζ : φ = cos -1 ζ constant Constant ζ = ray from the origin ζ > · · · is the cone ζ < · · · is the other part 28
In the complex plane : p=- + j d -- jdjd Any pole on the ray have the same Mp < … or > … corresponds to a conic region about the neg real axis 29
If more than one requirement, get the common (overlapped) area e.g. ζ > 0.5, σ > 2, ωn > 3 gives Sometimes meeting two will also meet the third, but not always. 30
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Try to remember these: 32
+ - Example: When given unit step input, the output looks like: Q: estimate k and τ. 33
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Effects of additional zeros Suppose we originally have: i.e. step response Now introduce a zero at s = -z The new step response: 36
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Effects: Increased speed, Larger overshoot, Might increase ts 38
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Effects of additional pole Suppose, instead of a zero, we introduce a pole at s = -p, i.e. 41
L.P.F. has smoothing effect, or averaging effect Effects: Slower, Reduced overshoot, May increase or decrease ts 42
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افزودن يك قطب به تابع تبديل حلقه - باز فرض مي كنيم قطب واقع در به تابع تبديل مرتبة دوم نمونه اضافه شود. تابع تبديل حلقه - بسته سيستم شكل زير پاسخ هاي پله اي واحد سيستم حلقه - بسته پاسخ هاي پله اي واحد سيستم حلقه - بسته
افزودن قطبي به تابع تبديل حلقه – باز به طور كلي بر افزايش فراجهش ماكسيمم پاسخ پله اي حلقه - بسته اثر دارد. با افزايش مقدار Tp ، قطب واقع به مبدأ صفحة s نزديك مي شود، و فراجهش ماكسيمم افزايش مي يابد. اين پاسخها همچنين نشان مي دهند كه قطب افزوده زمان خيز پاسخ پله اي را افزايش مي دهد 45
افزودن يك صفر به تابع تبديل حلقه- باز فرض مي‌كنيم صفري واقع در -1/Tz به يك تابع تبديل حلقه- باز افزوده مي‌شود. عبارت 1+Tz در صورت مي‌آيد، لذا فراجهش ماكسيمم را افزايش مي‌دهد، اما Tz در مخرج به صورت ضريب جملة S مي‌آيد و باعث بهبود ميرايي، يا كاهش فراجهش ماكسيمم مي‌شود. 47
PERFORMANCE INDICES AND OPTIMAL SYSTEMS
To select one or more parameters to give the best performance If a measure or index of performance can be expressed mathematically, the problem can be solved for the best choice of the adjustable parameters. 50
the selection criteria. The resulting system is termed optimal with respect to the selection criteria. 51 system Parameter Adjuster Performance index
Optimization: Maximization Minimization 52
Optimization: on-line optimization; The parameters may vary as the system operates Off-line optimization; Optimization occur before the system begins operation 53
Optimum system A control system is optimum when the elected performance index is minimized. The optimum value of the parameters depends directly upon the definition of optimum, that is, the performance index. 54
Simplified description of a control system 55
Performance Indices Elevator 56
Elevator input and output When the fourth floor button is pressed on the first floor, the elevator rises to the fourth floor with a speed and floor level accuracy designed for passenger comfort. 57
Push of the fourth-floor button is an input that represent a desired output, shown as a step function. 58
Transient response Passenger comfort and passenger patience are dependent upon the transient response. If this response is too fast, passenger comfort is sacrificed; if too slow, passenger patience is sacrificed. 59
Steady-state error Passenger safety and convenience would be sacrificed if the elevator is not properly level. 60
Performance Indices A performance index is a quantitative measure of the performance of a system and is chosen so that emphasis is given to the important system specifications. 61
Response of the system 62
ISE - Integral of Square of Error 63
The Integral Squared Error 64
Mason’s Rule: Example 1: Error transmittance: Error to step input:
Error signal: Error square: Integral square error :
Example 2:
IAE - Integral of the Absolute Magnitude of the Error 70
ITAE - Integral of Time Multiplied by Absolute Error 71
ITSE - Integral of Time Multiplied by Squared Error 72
General form of the performance integral 73
In general: 74
Performance criteria 75
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