Robust state and fault estimation observer for discrete-time D-LPV systems with unmeasurable gain scheduling functions. Application to a binary distillation column.

ROBUST STATE AND FAULT ESTIMATION OBSERVER
FOR DISCRETE-TIME D-LPV SYSTEMS
WITH UNMEASURABLE GAIN SCHEDULING FUNCTIONS.
APPLICATION TO A BINARY DISTILLATION COLUMN
F. R. LÓPEZ ESTRADA (TecNM - Instituto Tecnológico de Tuxla Gutiérrez, Chiapas, México)
J.C. PONSART, D. THEILLIOL (CRAN – University of Lorraine, Nancy, France)
C. M. ASTORGA-ZARAGOZA, M. FLORES-MONTIEL (CENIDET, Cuernavaca, México)
frlopez@ittg.edu.mx
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OBSERVER DESIGN
Introduction
Fault detection and estimation problem Observer design
Sensor fault estimation
Actuator fault estimation
Application to a binary distillation column - Simulation results
Conclusions

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1
INTRODUCTION

2
INTRODUCTION: SYSTEMS REPRESENTATIONS
Nonlinear systems (ODE, DAE)
• Commonly approximate
by LTI systems

2
Nonlinear approximation
by LTI systems

2
LPV systems, Q-LPV systems (TS)
by LTI systems
Continuous time

2
by LTI systems
Continuous time
*E is a singular matrix
representing ODE (dynamic)
and algebraic equations
(non-dynamic)

2
by LTI systems
Continuous time
(non-dynamic)
* Local models

2
by LTI systems
Continuous time
(non-dynamic)
* Local models
* Gain scheduling functions
depending on the state

Discrete-time D-LPV systems under noise and disturbances
where:
3
DISCRETE-TIME D-LPV SYSTEMS

Discrete-time D-LPV systems under noise and disturbances
where:
3
DISCRETE-TIME D-LPV SYSTEMS
The scheduling functions are defined through the following convex set
Note that the scheduling functions are depending on the state,
which is considered unmeasurable

In addition the system can be affected by sensor faults
4
FAULT DETECTION AND ESTIMATION PROBLEM
Or actuator faults

In addition the system can be affected by sensor faults
4
FAULT DETECTION AND ESTIMATION PROBLEM
Or actuator faults
The challenge is to
detect and estimate
these faults, despite
the problem of
unknown gain
scheduling functions
depending on the
system's states
Then, a suitable needs the estimation of the
scheduling functions

5
OBSERVER DESIGN
In order to estimate the unmeasurable scheduling functions, the first challenge
is to design a state observer by considering the system without faults

5
OBSERVER DESIGN
Then, for the discrete system

5
OBSERVER DESIGN
The following observer is proposed
are unknown gain matrices to be computed

5
OBSERVER DESIGN
are unknown gain matrices to be computed
The following observer is proposed

To deal with the UGF, the D-LPV system is transformed into a perturbed system
with estimated scheduling function as follows:
6
OBSERVER DESIGN
with:

To deal with the UGF, the D-LPV system is transformed into a perturbed system
with estimated scheduling function as follows:
6
OBSERVER DESIGN
with:
Then, the estimation error is computed as

The error becomes
7
OBSERVER DESIGN

The error becomes
7
OBSERVER DESIGN
After some algebraic manipulations,
the following error system is obtained
with
dk  dk
T
k
T



T

The error becomes
7
OBSERVER DESIGN
After some algebraic manipulations,
the following error system is obtained
with
Then, in order to guarantee asymptotic stability of (15) and robustness,
the following criterion performance is considered
dk  dk
T
k
T



T

Theorem 1: Robust state observer
8
OBSERVER DESIGN
then, the estimation error is quadratically stable with
Proof: Details in the paper.
If there exist matrices T1 and T2, a common matrix P = PT
> 0, matrices i and
a scalar , such that i [1, 2, …, h] 

9
SENSOR FAULT ESTIMATION
The same observer design can be considering to estimate jointly the states
and sensor faults

9
The same observer design can be considering to estimate jointly the states
and sensor faults
For instance, let us consider a D-LPV system under sensor fault as

In order to estimate sensor faults, the system (18) can be rewritten as an
augmented system with , such that the following augmented system is obtained
10
where:
xk  xk
T
fsk
T



T

10
where:
The same observer is considered to estimate the augmented system
In order to estimate sensor faults, the system (18) can be rewritten as an
augmented system with , such that the following augmented system is obtainedxk  xk
T
fsk
T



T

11
SENSOR ESTIMATION ERROR
The error equation is computed as
with
dk  dk
T
fsk
T
k
T



T

11
SENSOR ESTIMATION ERROR
with
Note that the error equation has similar form that previous error equation,
then Theorem 1 can be considered to obtain a solution,
which guarantee convergence of the observer
and therefore, robust state and sensor fault estimation.
dk  dk
T
fsk
T
k
T



T

12
ACTUATOR FAULT ESTIMATION
Under actuator faults and disturbance, the D-LPV system is represented by

12
We assume that the faults are slow variation

12
We assume that the faults are slow variation
A perturbed augmented system with

13
ACTUATOR ESTIMATION ERROR
with

13
ACTUATOR ESTIMATION ERROR
with
Note that the error equation has similar form that previous error equation,
then Theorem 1 can be considered to obtain a solution,
which guarantee convergence of the observer
and therefore, robust state and actuator fault estimation.

A 5 trays distillation column located at the Process Control
Laboratory of CENIDET in Cuernavaca, is considered.
The distillation column was operated under liquid-vapor-(LV)
configuration and without feed flow (F = 0).
The states variables are x = [x1 x2 x3 x4 x5], which represent the
liquid compositions of the light component from the top to
the bottom, respectively.
14

For a mixture ethanol-water, the system inputs is u = [Vr Lr]T
.
These inputs are functions of state x5, the reflux rv and the heating power on the boiler Qb
such as:
are the vaporization enthalpy
of ethanol and water
15

D-LPV system with 5 states, 4 local models, and 3 measured outputs
16
D-LPV MODEL

The gain scheduling functions are
Note that the scheduling functions depend on Vr and Lr that are
inputs depending on the state x5, which is unmeasurable.
Therefore a suitable observer design is required for the estimation of x5.
17
D-LPV MODEL

The observer gains for jointly state and sensor fault estimation are obtained
by solving Theorem 1 with the YALMIP toolbox.
The following gain matrices are computed
18
OBSERVER GAIN COMPUTATION

19
SIMULATION RESULTS
The disturbance signal included
in the system is a zero mean random
signal bounded by 0.02
Simultaneous bias fault are
considered on sensors 2 & 3
Simulation conditions

20
SIMULATION RESULTS
State estimation errors and system inputs
- the observer converges fast and asymptotically
- small deviation from zero of x4 related to changes in the system inputs which
consequently generate a change of the operation regions

21
SIMULATION RESULTS
Estimated gain scheduling functions
- the evolutions of the estimated gain scheduling functions illustrate how the system and
consequently the observer are constantly changed the operating region.

22
SIMULATION RESULTS
Estimated simultaneous sensor faults
- the observer is able to detect and estimate simultaneous occurring faults, despite the
small fault magnitudes.
- all simulation results show that the state and fault estimation are well performed despite
the disturbance and the error provided by the unmeasurable gain scheduling functions.

23
CONCLUSIONS
 A discrete-time state fault estimation (sensor or actuator) observer for D-LPV
systems was proposed
 Unmeasurable gain scheduling functions (UGF) are considered. This consideration
increases the level of abstraction, but also the applicability
 To solve the problem of UGF, the system was transformed into a perturbed D-LPV
system with estimated gain scheduling functions
 Robustness and asymptotic stability, of the estimation error are guaranteed by
considering a quadratic criteria and a Lyapunov equation
 Feasible LMIs are obtained to compute the observer gains
 The method was extended to estimate jointly the state and the faults
 The methodology was successfully applied to a realistic model of a binary distillation
column.

ROBUST STATE AND FAULT ESTIMATION OBSERVER
FOR DISCRETE-TIME D-LPV SYSTEMS
WITH UNMEASURABLE GAIN SCHEDULING FUNCTIONS.
F. R. LÓPEZ ESTRADA (TecNM - Instituto Tecnológico de Tuxla Gutiérrez, Chiapas, México)
J.C. PONSART, D. THEILLIOL (CRAN – University of Lorraine, Nancy, France)
C. M. ASTORGA-ZARAGOZA, M. FLORES-MONTIEL (CENIDET, Cuernavaca, México)
Jean-Christophe.Ponsart@univ-lorraine.fr

Robust state and fault estimation observer for discrete-time D-LPV systems with unmeasurable gain scheduling functions. Application to a binary distillation column.

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