Identification of dynamic networks in the presence of algebraic
loops
Citation for published version (APA):
Weerts, H. H. M., Van den Hof, P. M. J., & Dankers, A. G. (2016). Identification of dynamic networks in the presence of algebraic loops.
Document status and date: Published: 01/01/2016
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Direct identification with an old predictor and a new parameterization
Identification of dynamic networks in the
presence of algebraic loops
Harm H.M. Weerts (h.h.m.weertstue.nl), Paul M.J. Van den Hof, Arne G. Dankers
/ Department of Electrical Engineering - Control Systems
What is an algebraic loop?
Transforming from continuous to discrete time can lead to algebraic loops in dynamic networks as in Figure 1. In such a case dynamic relations that connect the internal variables do not have a delay
lim z→∞G12(z)G21(z) 6= 0. w 2(t) w1(t) G∞ 12 G∞ 21 t w(t) e1(t) e 2(t)
Figure 1: Discrete time approximation can lead to algebraic loops. Typical direct and joint-IO methods are biased, and IV isn’t minimum variance, so we must appropriately deal with thee components in w.
Goal: Obtain a consistent and minimum variance estimate of the network.
The system has 2 faces
There is a dynamic network, the closed-loop in Figure 3, given by w = Gw + Rr + He or expanded " w1 w2 # = " 0 G12 G21 0 # " w1 w2 # + " 1 0 0 1 # " r1 r2 # + " H1 0 0 H2 # " e1 e2 # . (1) Denote limz→∞G(z) = G∞. Different perspectives can make you
per-ceive an object in different ways, e.g. in Figure 2.
Figure 2: A different perspective can make you see different things. A closed-loop can be perceived as an equivalent open-loop system as shown in Figure 3 with equivalence relations P := (I − G)−1R,
Q := (I − G)−1H(I − G∞
), ˆe := (I − G∞)−1e.
Figure 3: The closed-loop system (left) is equivalent to some open-loop system (right).
Just 1 predictor
For both perspectives define the multivariable predictor ˆ
w(t|t − 1) := E{w(t)|w(t − 1)−, r(t)−} which is evaluated in Figure 4.
I
− Q
−1w
+ Q
−1P r
I
− (I − G
∞)
−1H
−1(I − G)
w
+ (I − G
∞)
−1H
−1Rr
E{ w(t) | w(t − 1)
−, r
(t)
−}
=
=
G
∞drops out!
G
∞estimated by r
Figure 4: Predictor expressions for open-loop and dynamic network.
Parameterization of the predictor with model set M :=
{G(q, θ), H(q, θ), R(q, θ), θ ∈ Θ} leads to the prediction error ˆε(θ) = I − G∞(θ)−1H−1(θ) (I − G(θ))w − R(θ)r.
Under standard conditions a least-squares criterion applied to ˆε leads to a consistent and minimum variance estimate.
Result: A combination of the direct and joint-IO methods.
Simulations
A simulation compares the introduced method to a (consistent) instru-mental variable method. The true and parameterized G12 and G21 are FIR filters of order 3, process noise is white, and the power of the ex-ternal excitation r is 100 times smaller than the power of the noise e, i.e. σ2 r = 0.01σe2. Network predictor IV 0 0.2 0.4 0.6
Boxplot of Best Fit Ratio combined over w1 and w2
Figure 5: The best fit ratio on a validation data set is plotted for the true system (black horizontal line), the proposed method (left) and the
