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Dead-zone effect on the performance of state estimators for hydraulic actuators

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Abstract

State estimation in hydraulic actuators is a fundamental technique for fault detection and it is also a valid tool useful to reduce the installation of sensors. The performance of the linear/linearization based techniques for the state estimation is strongly limited due to hard nonlinearities that characterize hydraulic actuator. One of the most common hard nonlinearities in hydraulic actuator is the dead-zone. This paper focuses on an alternative nonlinear estimation method that is able to fully take into account dead-zone hard nonlinearity and measurement noise. The estimator is based on the state-dependent-Riccati-equation (SDRE). A fifth order nonlinear model is derived and employed for the synthesis of the estimator. Several simulations have been conducted in order to analyse the effect of the dead-zone characteristic on the novel estimator performance, showing comparisons with the largely used extended Kalman filter (EKF). Numerical results demonstrate the effectiveness of SDRE based technique in applications characterized by extended dead-zone for which the EKF method provides poor results.

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Abbreviations

P T :

Tank pressure

y :

Movable mass displacement

m :

Load mass

F f :

Friction force

A p :

Annulus area of the actuator piston

P L :

Load pressure

P A , P B :

Pressures in the chambers of the cylinder

F c :

Coulomb friction force

F c0 :

Static value of the Coulomb friction force

g :

Gravitational acceleration

Z :

Net tangential force

V 0 :

Volume of each chamber for the centred position of the piston

Q L :

Load flow

Q A , Q B :

Load flow in the two chambers of the cylinder

v e :

Displacement signal of the spool valve

v en , v ep , k q0, k qn , k qp :

Parameters of the dead-zone function

v e0 :

Spool position bias

k e :

Input gain

u :

Valve command

x :

State vector

\(\tilde{v}_{e}\) :

Auxiliary variable

\(\Delta v_{e}\) :

Auxiliary constant

\({\mathbf{u}}\) :

Input vector

f, h :

Nonlinear functions

\({\mathbf{Q}}_{k}\) :

Covariance of the process noise for the EKF

z :

Measurement vector

\({\mathbf{g}}_{k}\) :

Gaussian white measurement noise for the EKF

\({\mathbf{R}}_{k}\) :

Covariance of the measurement noise for the EKF

\({\mathbf{K}}_{k}\) :

Filter gain of the EKF

t :

Time

\(\left({\hat{\bullet}} \right)\) :

Estimate

E :

Expected value

\({\mathbf{x}}_{0}\) :

Initial condition on the state vector

\(P_{0}^{{}}\) :

Initial condition on the error covariance

P k :

Error covariance for the EKF

A k−1, L k−1, H k , M k :

Partial derivative matrices for the EKF

\({\mathbf{F}}({\mathbf{x}}(t),{\mathbf{u}}(t))\), \({\mathbf{H}}\left( {{\mathbf{x}}(t),{\mathbf{u}}(t)} \right)\) :

Input and state dependent matrices of the SDC form

Q :

Covariance of the process noise for the SDREF

g :

Gaussian white measurement noise for the SDREF

R :

Covariance of the measurement noise for the SDREF

\({\mathbf{K}}_{{\mathbf{f}}}\) :

Filter gain of the SDREF

P :

Solution of the algebraic Riccati equation

σ:

Viscous friction coefficient

μ :

Coulombian friction coefficient of the linear guide

μ 0 :

Static value of the Coulombian friction coefficient

β :

Effective Bulk modulus

Ψ :

Dead-zone function

ω nv :

Natural frequency of the proportional valve

ξ v :

Damping ratio of the proportional valve

\({\varvec{\uppsi}}_{k}\) :

Process noise for the EKF

\({\varvec{\uppsi }}\) :

Process noise for the SDREF

ε :

Auxiliary function

−:

A priori estimate

+:

A posteriori estimate

k − 1:

Related to time tk−1

k :

Related to time tk

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Authors and Affiliations

  1. Department of Industrial Engineering, University of Naples Federico II, Naples, Italy

    Carmelina Abagnale, Salvatore Strano & Mario Terzo

  2. Department of Mechanical and Industrial Engineering, Università degli Studi di Brescia, Brescia, Italy

    Francesco Aggogeri & Alberto Borboni

Authors
  1. Carmelina Abagnale
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  2. Francesco Aggogeri
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  3. Alberto Borboni
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  4. Salvatore Strano
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  5. Mario Terzo
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Correspondence to Mario Terzo.

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Abagnale, C., Aggogeri, F., Borboni, A. et al. Dead-zone effect on the performance of state estimators for hydraulic actuators. Meccanica 52, 2189–2199 (2017). https://doi.org/10.1007/s11012-016-0563-3

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