February 8, 2006
Mathematical models of real-world systems are often too difficult to build based on first principles alone .
Figure by MIT OCW.
Figure by MIT OCW.
System Ident cation;
“Let the data speak about the system”.
Image removed for copyright reasons.
HVAC
Courtesy of Prof. Asada. Used with permission.
1. Passive elements: mass, damper, spring
2. Sources
3. Transducers
4. Junction structure
Physically meaningful parameters
G ( s ) =
Y
U
(
( s s
)
)
= b
0 s s n m
+
+ b a
1
1 s s m − 1 n − 1 +
+ L
L +
+ a b n m a i b i
= a i
( M , B , K )
= b i
( M , B , K )
1

Input u( t)
Output y( t)
G ( s ) =
Y
U
(
( s s
)
)
= b
0 s s n m
+
+ b a
1
1 s s m − 1 n − 1
+ L + b
+ L + a n m
Physical modeling
Pros
1.
Physical insight and knowledge
2.
Modeling a conceived system before hardware is built
Cons
1.
Often leads to high system order with too many parameters
2.
Input-output model has a complex parameter structure
3.
Not convenient for parameter tuning
4.
Complex system; too difficult to analyze
Pros
1.
Close to the actual input-output behavior
2.
Convenient structure for parameter tuning
3.
Useful for complex systems; too difficult to build physical model
Cons
1.
No direct connection to physical parameters
2.
No solid ground to support a model structure
3.
Not available until an actual system has been built
2

b
2 b
1 b
3 u(t ) y(t )
FIR
Finite Impulse Response Model t y ( t ) = b
1 u ( t − 1) + b
2 u ( t − 2) + ⋅ ⋅ ⋅ ⋅ ⋅⋅ + m u ( − m )
Define θ : = [ b
1
, b
2
, ⋅ ⋅ ⋅ , b m
] T ∈ R m unknown ϕ ( t ) : = [ u ( t − 1 ), u ( t − 2 ), ⋅ ⋅ ⋅ , u ( t − m )] T ∈ R m known
Vector θ collectively represents model parameters to be identified based on observed data y(t) and ϕ ( t ) for a time interval of 1 ≤ t ≤ N .
Observed data: y ( ), ⋅ ⋅ ⋅ ⋅ ⋅⋅ , y ( N )
Estimate θ y t ) = ϕ ( t ) T θ
This predicted output may be different from the actual y(t) .
Find θ that minimize V
N
( θ )
V
N
( θ ) =
1
N
N ∑
( t t = 1 y ( ) − ˆ( t )) 2
θ ˆ = avg min
θ
V
N
( θ ) dV
N
( θ ) d θ
= 0
V
N
( θ ) =
1
N
N
∑
( t t = 1 y ( ) − ϕ T ( t ) θ ) 2
2
N
N
∑
( t t = 1 y ( ) − ϕ T ( t ) θ )( − ϕ ) = 0
N
∑ t t = 1 y ( ) ϕ ( t ) =
N
∑
( ϕ T ( t ) θ ) ϕ ( t ) t = 1
}
3



 t
N
∑
= 1
( ϕ ( t ) ϕ T ( t )



θ = t
N
∑
= 1 y ( t ) ϕ ( t )
∴
R
N
θ ˆ
N
= R − 1
N
N
∑ t t = 1 y ( ) ϕ ( t )
Question1
What will happen if we repeat the experiment and obtain θ ˆ
N again?
Consider the expectation of θ ˆ
N when the experiment is repeated many times?
Average of θ ˆ
N
Would that be the same as the true parameter θ
0
?
Let’s assume that the actual output data are generated from y ( t ) = ϕ T ( t ) θ
0
+ e ( t )
θ
0 is considered to be the true value.
Assume that the noise sequence {e(t)} has a zero mean value, i.e. E[e(t)]=0, and has no correlation with input sequence {u(t)}.
θ ˆ
N
= R − 1
N
N
∑ t t = 1 y ( ) ϕ ( t ) = R − 1
N
N
∑ t = 1
[ ( ϕ T ( t ) θ
0
+ e ( t )
) ϕ ( t )
]
= R − 1
N


 t
N
∑
= 1 ϕ ( t ) ϕ T ( t ) 


θ
0
+ R − 1
N
N
∑ ϕ ( t t = 1 t ) ( )
∴
R
N
θ ˆ
N
− θ
0
= R
N
− 1
N
∑ ϕ ( t t = 1 t ) ( )
Taking expectation
E [ θ ˆ
N
− θ
0
] = E 


R − 1
N t
N
∑
= 1 ϕ ( t ) ( t ) 


= R − 1
N
N
∑ ϕ ( t ) ⋅ E [ t t = 1 e ( )] = 0
Question2
Since the true parameter θ
0 is unknown, how do we know how close
θ ˆ
N will be to θ
0
? How many data points, N , do we need to reduce the error
θ ˆ
N
− θ
0
to a certain level?
4

Consider the variance (the covariance matrix) of the parameter estimation error.
P
N
= E θ
N
− θ
0
)( ˆ
N
− θ
0
) ]
= E




R − 1
N t
N
∑
= 1 ϕ ( t ) ( t ) ⋅ 

R − 1
N
N s
∑
= 1 ϕ ( s ) ( s )
 
 
= E



R − 1
N t
N
=
N
∑∑
1 s = 1 ϕ ( t e t e s ) ϕ T ( s ) R − 1
N



= R − 1

N

 t
N N
∑∑
= 1 s = 1 ϕ ( t ) E
[ e t e s )
] ϕ T ( s )



R − 1
N
Assume that {e(t)} is stochastically independent
E
[ e ( ) ( s )
]
=
 E [ e
E [ t e 2 ( t s
)]
)]
=
=
λ
0 t t
=
≠ s s
Then P
N
= R − 1
N


 t
N
∑
= 1 ϕ ( t ) λϕ T ( t )



R − 1
N
= λ R − 1
N
As N increases, R
N
tends to blow out, but R
N
/N converges under mild assumptions. lim
N → ∞
1
N
N
∑ ϕ ( t ) ϕ T ( t ) = lim t = 1
N → ∞
1
N
R
N
= R
For large N , R
N
≅ N R , R − 1
N
≅
1
N
R
θ ˆ
N
θ
0
N
P
N
=
λ
N
R − 1 for large N .
5

I. The covariance P
N
decays at the rate 1/N .
Parameters approach he limiting value at the rate of
1
N
II. The covariance is inversely proportional to
P
N
∝
λ magnitude R
R
N
=



 r
M
11 r m 1
K
O
K r r
1 m
M mm



 r ij
=
N
∑ t t = 1 u ( − i ) ( t − j )
III. The convergence of θ ˆ
N to θ
0 may be accelerated if we design inputs such that R is large.
IV.
The covariance does not depend on the average of the input signal. Only the second moment
A) How to best estimate the parameters
What type of input is maximally informative?
• Informative data sets
• Persistent excitation
• Experiment design
• Pseudo Random Binary signals, Chirp sine waves, etc.
How to best tune the model / best estimate parameters
How to best use each data point
• Covariance analysis
• Recursive Least Squares
• Kalman filters
• Unbiased estimate
• Maximum Likelihood
6

B). How to best determine a model structure
How do we represent system behavior? How do we parameterize the model? i. Linear systems
• FIR, ARX, ARMA, BJ,…..
• Data compression: Laguerre series expansion ii. Nonlinear systems
• Neural nets
• Radial basis functions iii. Time-Frequency representation
• Wavelets
Model order: Trade-off between accuracy/performance and reliability/robustness
• Akaike’s Information Criterion
• MDL
7