Intro Single-elements Interconnected systems Hammerstein system NARMAX system Summary Identication of Interconnected Systems by Instrumental Variables Method Grzegorz Mzyk Institute of Computer Engineering, Control and Robotics Wroc aw University of Technology Poland 3th-May-2012
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Intro Single-elements Interconnected systems Hammerstein system NARMAX system Summary
Identification of Interconnected Systemsby Instrumental Variables Method
Grzegorz Mzyk
Institute of Computer Engineering, Control and RoboticsWrocław University of Technology
Poland
3th-May-2012
Intro Single-elements Interconnected systems Hammerstein system NARMAX system Summary
Structure of the presentation
1 Identification of single-element systems—MISO linear static element—SISO linear dynamic elementLeast squares (LS) method and instrumental variables(IV) method
2 Interconnected linear static systems—LS-based estimate and limit properties— IV-based estimate and limit properties—generation of instrumental variables
3 Nonlinear dynamic block-oriented systems—Hammerstein system—NARMAX system
Intro Single-elements Interconnected systems Hammerstein system NARMAX system Summary
Structure of the presentation
1 Identification of single-element systems—MISO linear static element—SISO linear dynamic elementLeast squares (LS) method and instrumental variables(IV) method
2 Interconnected linear static systems—LS-based estimate and limit properties— IV-based estimate and limit properties—generation of instrumental variables
3 Nonlinear dynamic block-oriented systems—Hammerstein system—NARMAX system
Intro Single-elements Interconnected systems Hammerstein system NARMAX system Summary
Structure of the presentation
1 Identification of single-element systems—MISO linear static element—SISO linear dynamic elementLeast squares (LS) method and instrumental variables(IV) method
2 Interconnected linear static systems—LS-based estimate and limit properties— IV-based estimate and limit properties—generation of instrumental variables
3 Nonlinear dynamic block-oriented systems—Hammerstein system—NARMAX system
Intro Single-elements Interconnected systems Hammerstein system NARMAX system Summary
MISO linear static block
(1)x(2)x
( )sx
y*a
z
Figure: MISO linear static block
a∗ =
a∗1a∗2...a∗s
Assumptions:
Ez = 0, varz < ∞x (i ), z — independent !!!
XN =
xT1xT2...xTN
=x (1)1 x (2)1 .. x (s)1x (1)2 x (2)2 .. x (s)2...
.........
x (1)N x (2)N .. x (s)N
Intro Single-elements Interconnected systems Hammerstein system NARMAX system Summary
MISO linear static block (continued)
YN =
y1y2...yN
, ZN =z1z2...zN
Measurement equation
YN = XNa∗ + ZN
Model
YN (a) = XNa
Least squares criterion∥∥YN −YN (a)∥∥22 → mina
Normal equation
XTNXNa = XTNYN
Uniqueness of the solution
rankXN = s
LS estimate
aN=(XTNXN
)−1XTNYN = X
+NYN
aNp.1→ a∗, as N → ∞
Intro Single-elements Interconnected systems Hammerstein system NARMAX system Summary
FIR linear dynamics
kyku kvkε
)( 1−qB
Figure: Linear dynamic object MA(s)
vk = b∗0uk + ...+ b
∗s uk−s
yk = vk + εk
yk = b∗0uk + ...+ b
∗s uk−s + zk
zk = εk
Intro Single-elements Interconnected systems Hammerstein system NARMAX system Summary
FIR linear dynamics (2)
ku
ky*b
kz1ku −
k su −
Figure: MA object
b∗ =
b∗0b∗1...b∗s
Assumptions:
Ez = 0, varz < ∞{uk} , {zk} — independent !!!
ΦN =
φT1
φT2
...φTN
=u1 u0 .. u1−su2 u1 .. u2−s...
.........
uN uN−1 .. uN−s
YN = ΦNb∗ + ZN bN=
(ΦTNΦN
)−1ΦTNYN
Intro Single-elements Interconnected systems Hammerstein system NARMAX system Summary
IIR linear dynamics
kyku ( )( )
1
1
B q
A q
−
−
kvkε
Figure: Linear dynamic object ARMA(s,p)
vk = b∗0uk + ...+ b
∗s uk−s + a
∗1vk−1 + ....+ a
∗pvk−p
yk = vk + εk
yk = b∗0uk + ...+ b
∗s uk−s + a
∗1yk−1 + ....+ a
∗pyk−p + zk
zk = εk − a∗1εk−1 − ...− a∗pεk−p
Intro Single-elements Interconnected systems Hammerstein system NARMAX system Summary
IIR linear dynamics (2)
ku
ky*θ
kz1ku −
k su −
1ky −
k py −
Figure: ARMA object
θ∗ =
b∗0b∗1...b∗sa∗1a∗2...a∗p
Ez = 0, varz < ∞
uk−i , zk — independentyk−i , zk — correlated !!!
Intro Single-elements Interconnected systems Hammerstein system NARMAX system Summary