Basic identification methods. ⢠Frequency domain approach. ⢠Time domain approach. â Example. Ë ( ). Ë. ( ). Ë(
ECE 2680: Adaptive Control (3 Credits, Fall 2016)
Lecture 5: System Identification (I)
September 29, 2016 Zhi-Hong Mao Associate Professor of ECE and Bioengineering University of Pittsburgh, Pittsburgh, PA 1
Outline • Homework 2 • Basic identification methods
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Homework problems • Homework 2 (due next Thursday Oct. 6) – Problems 0.3 and 2.2(a)-(b)
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Basic identification methods • Frequency domain approach – Frequency response: steady-state response of systems to sinusoidal inputs The figure compares the output response of a system (red solid line) with a sinusoidal input (black dashed line) Both the magnitude and the phase shift of a system will change with the frequency of the input into the system
Phase Shift 1.5 1
A
0.5 Output 0
B
-0.5 -1 -1.5 0
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20
30
Time
Amplitude Ratio = B / A
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Basic identification methods • Frequency domain approach –
Frequency response: steady-state response of systems to sinusoidal inputs
– Frequency response function ^ its frequency • Given a system with transfer function P(s), ^ response function is P(jω) • The steady-state gain of a system for a sinusoidal input sin(ω0t) is the magnitude of the transfer function evaluation at s = jω0, and the phase shift of the output sinusoid relative to ^ 0) the input sinusoid is the angle of P(jω
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Basic identification methods • Frequency domain approach –
Frequency response: steady-state response of systems to sinusoidal inputs
–
Frequency response function
– Example
yˆ p ( s ) ˆ kp P( s ) rˆ( s ) s ap
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Basic identification methods • Frequency domain approach –
Frequency response: steady-state response of systems to sinusoidal inputs
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Frequency response function
–
Example
Question: How about identification of a system of higher order?
yˆ p ( s ) ˆ s n 1 1 P( s ) n n n 1 ˆ( r s) s n s 1
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Basic identification methods •
Frequency domain approach
• Time domain approach – Example
yˆ p ( s ) ˆ kp P( s ) rˆ( s ) s ap Question: What is the differential equation for the above frequency-domain description?
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Basic identification methods •
Frequency domain approach
• Time domain approach – Example
yˆ p ( s ) ˆ kp P( s ) rˆ( s ) s ap y p ( t ) a p y p ( t ) k p r (t ) Question: If we have measurement of r(t) and dyp(t)/dt at t1 and t2, how can we estimate ap and kp?
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Basic identification methods •
Frequency domain approach
• Time domain approach – Example
yˆ p ( s ) ˆ kp P( s ) rˆ( s ) s ap y p ( t ) a p y p ( t ) k p r (t )
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Basic identification methods •
Frequency domain approach
• Time domain approach – Example
yˆ p ( s ) ˆ kp P( s ) rˆ( s ) s ap
y p ( t ) a p y p ( t ) k p r (t )
Question: Why we avoid the measurement of dyp(t)/dt and how to do that?
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Basic identification methods •
Frequency domain approach
• Time domain approach – Example
yˆ p ( s ) ˆ kp P( s ) rˆ( s ) s ap Define
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Basic identification methods •
Frequency domain approach
• Time domain approach – Example
yˆ p ( s ) ˆ kp P( s ) rˆ( s ) s ap Define
y p (t ) *T w(t ) wT (t ) *
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Basic identification methods •
Frequency domain approach
• Time domain approach – Example
y p (t ) *T w(t ) wT (t ) *
Nominal identifier parameter
e1 (t ) T w(t ) y p (t ) T *T w(t ) Identification error
Adaptive identifier parameter
Linear error equation
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Basic identification methods •
Frequency domain approach
• Time domain approach –
Example
– Gradient algorithm
e1 (t ) T w(t ) y p (t ) T *T w(t ) Objective: minimize e12 (t ) Gradient:
2 e1 2e1 e1 2e1w
Parameter update law:
It is a vector!
d ge1w dt 15
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Basic identification methods •
Frequency domain approach
• Time domain approach –
Example
– Gradient algorithm • Advantages of using the gradient algorithm over using the following calculation 1
1 w1 (t1 ) w2 (t1 ) y p (t1 ) w (t ) w (t ) y (t ) 2 1 2 2 2 p 2
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Basic identification methods •
Frequency domain approach
• Time domain approach – –
Example Gradient algorithm
– Least-squares algorithm Objective: minimize the integral-squared-error (ISE) t
ISE = T ( )w( ) y p ( ) d 2
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Least-squares estimate
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Basic identification methods •
Frequency domain approach
• Time domain approach – –
Example Gradient algorithm
– Least-squares algorithm Recursive formulation
(t ) = P(t )w(t ) T (t )w(t ) y p (t ) P(t ) = P(t )w(t )wT (t ) P(t )
(0) 0
P(0) PT (0) P0
Remark: It can be shown that (t) converges t asymptotically to * if w( )wT ( )d is unbounded as 0 t∞. 18
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Basic identification methods •
Frequency domain approach
• Time domain approach – – –
Example Gradient algorithm Least-squares algorithm
– Model reference identification yˆ p ( s ) ˆ kp P( s ) rˆ( s ) s ap
a0
uˆ( s )
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Reference model
yˆ m ( s ) ˆ k M ( s) m uˆ( s ) s am yˆ m ( s ) Mˆ ( s )
rˆ( s )
_+
b0 Pˆ ( s )
yˆ p ( s )
eˆ1 ( s )
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References • • • • •
K. J. Astrom and B. Wittenmark, Adaptive Control, 2nd Edition, Addison-Wesley, 1995. C.-T. Chen, Linear System Theory and Design, 4th Edition, Oxford University Press, 2013. R. M. Murray, Z. Li, and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation, CRC, 1994. S. Sastry and M. Bodson, Adaptive Control: Stability, Convergence, and Robustness, Prentice-Hall, 1989. G. Strang, Introduction to Applied Mathematics, Wellesley-Cambridge Press, 1986.
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