State Observation of LTV Systems with Delayed Measurements: A Parameter Estimation-based Approach
Alexey Bobtsov, Nikolay Nikolaev, Romeo Ortega, Denis Efimov
aa r X i v : . [ ee ss . S Y ] A ug State Observation of LTV Systems with DelayedMeasurements: A Parameter Estimation-based Approach
Alexey Bobtsov a , Nikolay Nikolaev a , Romeo Ortega b , Denis Efimov c . a Department of Control Systems and Robotics, ITMO University, Kronverkskiy av. 49, Saint-Petersburg, 197101, Russia b Departamento Acad´emico de Sistemas Digitales, ITAM, Ciudad de M´exico, M´exico c Inria, Univ. Lille, CNRS, UMR 9189 - CRIStAL, F-59000 Lille, France
Abstract
In this paper we address the problem of state observation of linear time-varying systems with delayed measurements, which hasattracted the attention of many researchers—see [7] and references therein. We show that, adopting the parameter estimation-based approach proposed in [3,4], we can provide a very simple solution to the problem with reduced prior knowledge.
Key words:
Linear time-varying systems; state observer; delay system .
Consider a linear time-varying (LTV)system ˙ x ( t ) = A ( t ) x ( t ) + B ( t ) u ( t ) y ( t ) = C ( ϕ ( t )) x ( ϕ ( t )) , (1)for t ≥ x ( t ) ∈ R n , u ( t ) ∈ R m , y ( t ) ∈ R q , where ϕ ( t ) is a known delay function verifying t ≥ ϕ ( t ) ≥ . The generalized parameter estimation-based observer˙ ξ ( t ) = A ( t ) ξ ( t ) + B ( t ) u ( t )˙Φ( t ) = A ( t )Φ( t ) , Φ(0) = I n ˆ x ( t ) = ξ ( t ) − Φ( t )ˆ θ ( t ) , ⋆ This paper was not presented at any IFAC meeting. Cor-responding author N. Nikolaev. Tel. +79213090016.
Email addresses: [email protected] (Alexey Bobtsov), [email protected] (Nikolay Nikolaev), [email protected] (Romeo Ortega), [email protected] (Denis Efimov). with the gradient parameter estimator˙ˆ θ ( t ) = ΓΦ ⊤ ( ϕ ( t )) C ⊤ ( ϕ ( t ))[ C ( ϕ ( t )) ξ ( ϕ ( t )) − y ( t ) − C ( ϕ ( t ))Φ( ϕ ( t ))ˆ θ ( t )] , (2)with Γ >
0, which ensureslim t →∞ | ˆ x ( t ) − x ( t ) | = 0 , ( exp. )provided C ( t )Φ( t ) is persistently exciting (PE) [8], thatis, there exists positive constants T and δ such that Z t + Tt C ( s )Φ( s )Φ ⊤ ( s ) C ⊤ ( s ) ds ≥ δI q , ∀ t ≥ . (3) PROOF.
Define the error signal e ( t ) := ξ ( t ) − x ( t ),which satisfies ˙ e ( t ) = A ( t ) e ( t ) , hence e ( t ) = Φ( t ) θ, with θ := e (0). Consequently, x ( t ) = ξ ( t ) − Φ( t ) θ. (4)The output of the system (1) then satisfies y ( t ) = C ( ϕ ( t )) [ ξ ( ϕ ( t )) − Φ( ϕ ( t )) θ ] . Preprint submitted to Automatica 21 August 2020 rom which we get the linear regression equation C ( ϕ ( t )) ξ ( ϕ ( t )) − y ( t ) = C ( ϕ ( t ))Φ( ϕ ( t )) θ, that, replacing in (2), yields the parameter error equa-tion˙˜ θ ( t ) = − ΓΦ ⊤ ( ϕ ( t )) C ⊤ ( ϕ ( t )) C ( ϕ ( t ))Φ( ϕ ( t ))˜ θ ( t ) , with ˜ θ ( t ) := ˆ θ ( t ) − θ .Invoking standard adaptive control arguments [8, The-orem 2.5.1] we conclude that, the PE assumption (3)ensures lim t →∞ | ˜ θ ( t ) | = 0 , ( exp. )The proof is completed noting thatˆ x ( t ) − x ( t ) = − Φ( t )˜ θ ( t ) . Remark 1
Another, more complex, solution to thisproblem that requires the knowledge of ˙ ϕ ( t ) is reportedin [7] under the classical assumption of existence of anexponentially stable Luenberger observer for the LTVsystem (1) with ϕ ( t ) = t , i.e. [7, Assumption 2]. Thatestimator requires a more sophisticated implementationsince it is based on a PDE representation of the delay,with an observer designed for the coupled LTV-PDEsystem. As is well known [6] the PE assumption madehere is equivalent to uniform complete observability ofthe pair ( C ( t ) , A ( t )) and this, in its turn, is a sufficient condition for the verification of [7, Assumption 2]. Remark 2
The PE assumption made here can be re-laxed by the significantly weaker condition of intervalexcitation [2] using the finite convergence time versionof the dynamic regressor extension and mixing (DREM)estimator proposed in [5], with the additional advantageof ensuring convergence in finite time . Adding fractionalpowers in the estimator, as done in [9,10], it is also possi-ble to achieve convergence in fixed time. The details areomitted for brevity.
Remark 3
Notice that if the state transition matrixconverges to zero, e.g. , for a constant, Hurwitz matrix A , the estimation error converges to zero independentlyof the excitation conditions. In this case, the observerbehaves like an open-loop emulator. Remark 4
Following [7] assume that the function ϕ ( t )admits a (piece-wise) continuous time derivative, thenthe PE condition can be rewritten as follows: there exist T > δ > Z ϕ ( t + T ) ϕ ( t ) ˙ ϕ − ( s )Φ ⊤ ( s ) C ⊤ ( s ) C ( s )Φ( s ) ds ≥ δI n , ∀ t ≥ , which is equivalent to the previous formulation if ˙ ϕ ( t ) > t ≥
0, and also provides an additional degreeof freedom if ˙ ϕ ( t ) = 0 is allowed for some instants orintervals of time. Consider the LTV system (1) with m = q = 1, n = 2 and A = " − sin ( t ) 0 , B = " , C = " For the estimation of θ we use the DREM approach [1]with Γ = γI . We consider three cases: C1 ϕ ( t ) = t (Fig. 1 and Fig. 2); C2 ϕ ( t ) = ϕ ( t − τ ), τ = 1 (Fig. 3 and Fig. 4); C3 ϕ ( t ) = ϕ ( t − τ ), τ = 1+0 . t ) (Fig. 5 and Fig. 6); Fig. 1. Error transients x ( t ) − ˆ x ( t ) for diffrerent γ and case C1 Fig. 2. Error transients x ( t ) − ˆ x ( t ) for diffrerent γ and case C1 Acknowledgements
This work was supported by the Ministry of Scienceand Higher Education of Russian Federation, passportof goszadanie no. 2019-0898, and by Government of Rus-sian Federation (Grant 08-08).
References [1] Stanislav Aranovskiy, Alexey Bobtsov, Romeo Ortega, andAnton Pyrkin. Performance enhancement of parameter Fig. 3. Error transients x ( t ) − ˆ x ( t ) for diffrerent γ and case C2 Fig. 4. Error transients x ( t ) − ˆ x ( t ) for diffrerent γ and case C2 Fig. 5. Error transients x ( t ) − ˆ x ( t ) for diffrerent γ and case C3 Fig. 6. Error transients x ( t ) − ˆ x ( t ) for diffrerent γ and case C3 estimators via dynamic regressor extension and mixing. IEEETransactions on Automatic Control , 62(7):3546–3550, 2016.(See also arXiv:1509.02763 for an extended version.).[2] G. Kreisselmeier, and G. Rietze-Augst. Richness andexcitation on an interval-with application to continuous-timeadaptive control.
IEEE Transactions on Automatic Control , 35(2):165–171, 1990.[3] Romeo Ortega, Alexey Bobtsov, Nikolay Nikolaev,Johannes Schiffer, and Denis Dochain. Generalizedparameter estimation-based observers: Application to powersystems and chemical-biological reactors. arXiv preprintarXiv:2003.10952 , 2020.[4] Romeo Ortega, Alexey Bobtsov, Anton Pyrkin, and StanislavAranovskiy. A parameter estimation approach to stateobservation of nonlinear systems.
Systems & Control Letters ,85:84–94, 2015.[5] Romeo Ortega, Dmitry N Gerasimov, Nikita E Barabanov,and Vladimir O Nikiforov. Adaptive control of linearmultivariable systems using dynamic regressor extensionand mixing estimators: Removing the high-frequency gainassumptions.
Automatica , 110:108589, 2019.[6] Wilson J Rugh.
Linear System Theory . Prentice-Hall, Inc.,1996.[7] Ricardo Sanz, Pedro Garcia, and Miroslav Krstic.Observation and stabilization of LTV systems with time-varying measurement delay.
Automatica , 103:573–579, 2019.[8] S. Sastry and M. Bodson. Adaptive control: Stability,Convergence and Robustness.
Prentice Hall, USA , 1989.[9] Jian Wang, Denis Efimov, Stanislav Aranovskiy, andAlexey A Bobtsov. Fixed-time estimation of parametersfor non-persistent excitation.
European Journal of Control ,55:24–32, 2019.[10] Jian Wang, Denis Efimov, and Alexey A Bobtsov. On robustparameter estimation in finite-time without persistence ofexcitation.
IEEE Transactions on Automatic Control ,65(4):1731–1738, 2019.,65(4):1731–1738, 2019.