robust state and fault estimation observer for discrete-time d-lpv systems with unmeasurable gain...

ROBUST STATE AND FAULT ESTIMATION OBSERVERFOR 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

INTRODUCTION: SYSTEMS REPRESENTATIONS

Nonlinear systems (ODE, DAE)

• Commonly approximate by LTI systems

2

Nonlinear approximation

LPV systems, Q-LPV systems (TS)

INTRODUCTION: SYSTEMS REPRESENTATIONS

Nonlinear systems (ODE, DAE)

• Commonly approximate by LTI systems

Continuous time

2

Nonlinear approximation

LPV systems, Q-LPV systems (TS)

INTRODUCTION: SYSTEMS REPRESENTATIONS

Nonlinear systems (ODE, DAE)

• Commonly approximate by LTI systems

Continuous time

*E is a singular matrix representing ODE (dynamic) and algebraic equations (non-dynamic)

2

Nonlinear approximation

LPV systems, Q-LPV systems (TS)

INTRODUCTION: SYSTEMS REPRESENTATIONS

Nonlinear systems (ODE, DAE)

• Commonly approximate by LTI systems

Continuous time

*E is a singular matrix representing ODE (dynamic) and algebraic equations (non-dynamic)

* Local models

2

Nonlinear approximation

LPV systems, Q-LPV systems (TS)

INTRODUCTION: SYSTEMS REPRESENTATIONS

Nonlinear systems (ODE, DAE)

• Commonly approximate by LTI systems

Continuous time

*E is a singular matrix representing ODE (dynamic) and algebraic equations (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 challengeis to design a state observer by considering the system without faults

5

OBSERVER DESIGN

Then, for the discrete system

In order to estimate the unmeasurable scheduling functions, the first challengeis to design a state observer by considering the system without faults

5

OBSERVER DESIGN

Then, for the discrete system

The following observer is proposed

are unknown gain matrices to be computed

In order to estimate the unmeasurable scheduling functions, the first challengeis to design a state observer by considering the system without faults

5

OBSERVER DESIGN

Then, for the discrete system

are unknown gain matrices to be computed

In order to estimate the unmeasurable scheduling functions, the first challengeis to design a state observer by considering the system without faults

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

dk dkT 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

dk dkT 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 statesand sensor faults

9

SENSOR FAULT ESTIMATION

The same observer design can be considering to estimate jointly the statesand sensor faults

For instance, let us consider a D-LPV system under sensor fault as

9

SENSOR FAULT ESTIMATION

The same observer design can be considering to estimate jointly the statesand 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

SENSOR FAULT ESTIMATION

where:

xk xkT fsk

T

T

10

SENSOR FAULT ESTIMATION

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 obtained xk xkT fsk

T

T

11

SENSOR ESTIMATION ERROR

The error equation is computed as

with

dk dkT fsk

T kT

T

11

SENSOR ESTIMATION ERROR

The error equation is computed as

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.

dk dkT fsk

T kT

T

12

ACTUATOR FAULT ESTIMATION

Under actuator faults and disturbance, the D-LPV system is represented by

12

ACTUATOR FAULT ESTIMATION

Under actuator faults and disturbance, the D-LPV system is represented by

We assume that the faults are slow variation

12

ACTUATOR FAULT ESTIMATION

Under actuator faults and disturbance, the D-LPV system is represented by

We assume that the faults are slow variation

A perturbed augmented system with

13

ACTUATOR ESTIMATION ERROR

The error equation is computed as

with

13

ACTUATOR ESTIMATION ERROR

The error equation is computed as

with

13

ACTUATOR ESTIMATION ERROR

The error equation is computed as

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 tothe bottom, respectively.

14

APPLICATION TO A BINARY DISTILLATION COLUMN

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

APPLICATION TO A BINARY DISTILLATION COLUMN

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 obtainedby 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

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

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

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

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

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

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 OBSERVERFOR 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)

Jean-Christophe.Ponsart@univ-lorraine.fr