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# State and parameter estimation in linear systems with delays

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### https://hal.inria.fr/hal-02558797

Preprint submitted on 29 Apr 2020

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## State and parameter estimation in linear systems with delays

### Qinghua Zhang (

Email: Qinghua.Zhang@inria.fr)

### August 2, 2017

Consider systems in the form of

˙x(t) = A(t)x(t) + B(t)x(t − τ ) + Φ(t)θ (1a) y(t) = C(t)x(t) + D(t)x(t − τ ) (1b) where θ is an unknown constant vector. The matrices A(t), B(t), C(t), D(t), Φ(t) are known, bounded and piecewise continuous. The delay τ is known.

Remark: Φ(t)θ can be generalized to Φ1(t)θ1 + Φ2(t − τ )θ2: let Φ(t) = [Φ1(t) Φ2(t − τ )] and

θT = [θT1 θ2T] so that Φ(t)θ = Φ1(t)θ1+ Φ2(t − τ )θ2. More generally, it is possible to consider

Φ(t)θ = Φ1(t − τ1)θ1+ · · · + Φm(t − τm)θm,

provided the delays are all known.

Assumption 1. In the particular case Φ(t)θ ≡ 0, a state observer is available in the form of

˙ˆx0(t) = A(t)ˆx0(t) + B(t)ˆx0(t − τ ) + L(t)[y(t) − C(t)ˆx0(t) − D(t)ˆx0(t − τ )], (2)

or in other words, the error dynamics ˙ η(t) = [A(t) − L(t)C(t)]η(t) + [B(t) − L(t)D(t)]η(t − τ ), (3) is such that lim t→+∞η(t) = 0 exponentially. (4)  The adaptive observer for system (1):

˙ Υ(t) = [A(t) − L(t)C(t)]Υ(t) + [B(t) − L(t)D(t)]Υ(t − τ ) + Φ(t) (5a) ˙ˆx(t) = A(t)ˆx(t) + B(t)ˆx(t − τ) + Φ(t)ˆθ(t) + L(t)[y(t) − C(t)ˆx(t) + D(t)ˆx(t − τ)] + Υ(t)˙ˆθ(t) +[B(t) − L(t)D(t)]Υ(t − τ )[ˆθ(t) − ˆθ(t − τ )] (5b) ˙ˆθ(t) = Γ[C(t)Υ(t) + D(t)Υ(t − τ)]T ny(t) − C(t)ˆx(t) + D(t)ˆx(t − τ )−D(t)Υ(t − τ )[ˆθ(t) − ˆθ(t − τ )]o (5c) where Γ is either a constant positive definite matrix or a recursively computed time varying matrix.

The red terms are unusual. They will be helpful for error dynamics analysis. Define the estimation errors:

˜

x(t) , x(t) − ˆx(t) (6)

˜

θ(t) , θ − ˆθ(t). (7)

(3)

Then (recall that θ is a constant vector) ˙˜x(t) = A(t)˜x(t) + B(t)˜x(t − τ) + Φ(t)˜θ − L(t)C(t)˜x(t) − L(t)D(t)˜x(t − τ ) + Υ(t)θ(t) + [B(t) − L(t)D(t)]Υ(t − τ )[˜˙˜ θ(t) − ˜θ(t − τ )]. (8) Now define η(t) , ˜x(t) − Υ(t)˜θ(t), (9) then ˙ η(t) = [A(t) − L(t)C(t)]η(t) + [B(t) − L(t)D(t)]η(t − τ ) + [A(t) − L(t)C(t)]Υ(t)˜θ(t) + [B(t) − L(t)D(t)]Υ(t − τ )˜θ(t − τ ) + Φ(t)˜θ(t) + Υ(t)θ(t)˙˜ + [B(t) − L(t)D(t)]Υ(t − τ )[˜θ(t) − ˜θ(t − τ )] − ˙Υ(t)˜θ(t) − Υ(t)θ(t).˙˜ (10)

In this last equation, the first occurence of [B(t) − L(t)D(t)]Υ(t − τ )˜θ(t − τ ) cancels out with a later term, so does Υ(t)θ(t). Hence˙˜

˙

η(t) = [A(t) − L(t)C(t)]η(t) + [B(t) − L(t)D(t)]η(t − τ )

+ {[A(t) − L(t)C(t)]Υ(t) + [B(t) − L(t)D(t)]Υ(t − τ ) + Φ(t) − ˙Υ(t)}˜θ(t), (11) which is then simplified, by taking into account (5a), to

˙

η(t) = [A(t) − L(t)C(t)]η(t) + [B(t) − L(t)D(t)]η(t − τ ). (12) According to Assumption 1, η(t) → 0.

Now consider the error dynamics ˜θ(t). It follows from (5c) that ˙˜ θ(t) = −Γ[C(t)Υ(t) + D(t)Υ(t − τ )]T nC(t)˜x(t) + D(t)˜x(t − τ ) − D(t)Υ(t − τ )[ˆθ(t) − ˆθ(t − τ )]o (13) = −Γ[C(t)Υ(t) + D(t)Υ(t − τ )]T[C(t)η(t) + D(t)η(t − τ )] − Γ[C(t)Υ(t) + D(t)Υ(t − τ )]T nC(t)Υ(t)˜θ(t) + D(t)Υ(t − τ )˜θ(t − τ ) + D(t)Υ(t − τ )[˜θ(t) − ˜θ(t − τ )]o = −Γ[C(t)Υ(t) + D(t)Υ(t − τ )]T[C(t)η(t) + D(t)η(t − τ )] − Γ[C(t)Υ(t) + D(t)Υ(t − τ )]T[C(t)Υ(t) + D(t)Υ(t − τ )]˜θ(t). (14)

The homogeneous part of this error dynamics (corresponding to η(t) ≡ 0) is stable under the following assumption.

Assumption 2. There exist T > 0 and α > 0 such that, for all t ≥ t0,

Z t+T

t

[C(s)Υ(s) + D(s)Υ(s − τ )][C(s)Υ(s) + D(s)Υ(s − τ )]Tds ≥ αI. (15)  The remaining analysis is then similar to already published cases.

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