سیستمهای کنترل خطی پاییز 1389 بسم ا... الرحمن الرحيم دکتر حسين بلندي- دکتر سید مجید اسما عیل زاده

Recap. Transient Response: First order system transient response –Step response specs and relationship to pole location Second order system transient response 2

Error

The error signal: e(t) = 1-y(t)=e -at us(t) Normalized time t/ 

General First-order system Step response starts at y(0+)=k, final value kz/p 1/p =  is still time constant; in every , y(t) moves 63.2% closer to final value

Unit ramp response:

Note: In step response, the steady-state tracking error = zero.

Unit impulse response:

Performance of a second-order system 10

Prototype 2 nd order system:

Characteristic equation 12

Unit step response: 1) Under damped, 0 < ζ < 1

cos  =  =-Re/|root|  = cos -1 (Re/|root|)  = tan -1 (-Re/Im) d  =Im =-Re

To find y(t) max:

Response to unit step input 21

Natural frequency  n - the frequency of natural oscillation that would occur for two complex poles if the damping were equal to zero Damping ratio  - a measure of damping for second-order characteristic equation 22

Characteristic equation 23

Finding  n and  for a second-order system 24

Second-order responses for  underdamped 25

Unit impulse response R(s)=1 T(s)=Y(s) 26

27

Standard performance measures Peak time Settling time Percent overshoot Peak response 28

fig_04_14 29

Settling time The settling time is defined as the time required for a system to settle within a certain percentage of the input amplitude. 30

Settling time 31

Rise time The time it takes for a signal to go from 10% of its value to 90% of its final value 32

Rise time 33

Peak time Peak time is the time required by a signal to reach its maximum value. 34

Peak time 35

Percent overshoot Percent Overshoot is defined as: P.O. = [(M pt – fv) / fv] * 100% M pt = The peak value of the time response fv = Final value of the response 36

Percent overshoot 37

Finding transient response 38

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Gain design for transient response 40

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 Controllable Canonical Form

 Observable Canonical Form  Diagonal Form

 Jordan Form

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سؤال: چگونه ماتريس را محاسبه كنيم؟ (2). (3). (1).

Example 1)

Example2) If

روش سوم: تابع يك ماتريس مربعي ) Function of a square matrix) يك تابع باشد كه در Spectrum (دامنه يا طيف) و اگر نيز يك چند جمله اي باشد كه داراي مقادير مساوي مانند در دامنه باشد سپس تابع مي باشد، بشكل زير تعريف مي گردد: اگر تعريف شده باشد كه يك تابع مقدار-ماتريسي ( Matrix Valued )

يعني كه اگر يك ماتريس باشد، اگر كه مقدار از روي شود، ما مي توانيم يك چندجمله اي پيدا كنيم كه داراي درجه داده مي باشد بطوريكه: بطوريكه برابر با در دامنه كه تمامي توابع عبارتند از: مي باشد. از اين تعريف مي دانيم

طريقه عمل : 1. مقادير ويژه را محاسبه نمائيد مثال : اگر را محاسبه نمائيد.

1) 2)

مثال:.1.2

مثال: (1

From

مثال:

حل معادلات حالت

حل : (1 (2 (3 (4

مثال: حل:

حل:

مثال : حل :