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Design of Full-Order Neural Observer with Nonlinear Filter Techniques for State Estimation of a Three-Tank Process Control System


A novel model-based approach to design a full-order state observer for estimating the states of a three-tank process has been attempted in this research study. State estimation has been a methodology that integrates the prediction from exact models pertaining to the system and achieves consistent estimation of the non-measurable variables. This study has attempted to develop a full-order observer for estimation of non-measurable variables of the considered three-tank process control system. Neural observer is designed with the nonlinear state update equation that is structured as the neural network employing radial basis function (RBF) model. Also, nonlinear full-order state observer is designed based on a new recursive likelihood synthesizer (RLS) of the extended Kalman filter (EKF) and classic unscented Kalman filter (UKF) and finally the states are estimated. The likelihood synthesizer determines the covariance and Kalman gains so as to match the real-time process measurements. Three-tank process system (TTPS) is represented by its mathematical model and the developed state estimation techniques are applied for estimating the non-measurable variables. Likelihood synthesizer tends to evaluate the covariance of the initial states and simulation tests confirm the attainment of better results using the new nonlinear filtering techniques. RBF neural observer has resulted in an ARMSE of 4.1629 × 10–3, 0.3963 × 10–3 and 0.1085 × 10–3 for the measured heights h1, h2 and h3, respectively. The new RLS-EKF observer with its recursive determination of the maximum likelihood has attained ARMSE of 2.1982 × 10–6, 0.1512 × 10–6 and 0.0261 × 10–7 for the measured heights h1, h2 and h3, respectively. This novel RLS-EKF has proved to be highly robust and has higher precision than the RBF neural observer and UKF technique as applied for the TTPS model.

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Radial basis function


Recursive likelihood synthesizer


Extended Kalman filter


Unscented Kalman filter


Three-tank process system

UI :

Unknown inputs


Intelligent adaptive unscented Kalman filter

EM :

Expected maximum


Average root mean square error

List of symbols

h :

Height of liquid/fluid level in the tank

fw :

Flow rate within and from the three tanks

R :

Resistance of the flow

g :

Gravitational force

a :

Area of connecting pipes

A :

Area of cylindrical tanks

D :

Inner diameter of the tanks

h 1 :

Steady-state fluid level of tank 1

h 2 :

Steady-state fluid level of tank 2

h 3 :

Steady-state fluid level of tank 3

C 1 :

Discharge of inlet orifice

C 2 :

Discharge of interconnecting orifice—coupler

C 3 :

Discharge of outlet orifice


System’s unobserved state


Observed state


Input signal

v(q) and z(q):

Gaussian white noise

L and M:

Covariance matrices

G and H:

Functional vectors


Past and present observations

ψ RBF :

Nonlinear function

α y :

Number of past output corresponding to the measured state

α e :

Number of past output corresponding to the observed state


Error term used in RBF full-order neural observer

χ 0 :

Initial covariance

x q |Q :

Smoothing estimate state value

J q :

Jacobian matrix of the measured vector

C q :

Conditional covariance of xq

C q|Q :

Covariance estimate

w :



Linked weights

σ :

Spread of the mean of the sigma point

Q :

Secondary scaling coefficient

ζ :

Prior knowledge of the probability distribution

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Suguna, A., Ranganayaki, V. & Deepa, S.N. Design of Full-Order Neural Observer with Nonlinear Filter Techniques for State Estimation of a Three-Tank Process Control System. Iran J Sci Technol Trans Electr Eng (2022).

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  • Neural observer
  • State estimator
  • Likelihood synthesizer
  • Three-tank process system
  • Extended Kalman filter
  • Unscented Kalman filter
  • Observer design