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Simultaneous state and unknown inputs estimation with PI and PMIobservers for Takagi Sugeno model with unmeasurable premise
variables
Dalil Ichalal, Benoît Marx, José Ragot and Didier Maquin
Centre de Recherche en Automatique de Nancy (CRAN)Nancy-University, CNRS
17th Mediterranean Conference on Control and Automation MED’09June 24-26, 2009, Makedonia Palace, Thessaloniki, Greece
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 1 / 29
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Motivations
Objective
Estimate simultanuously the state and unknown inputs of nonlinear systems described by aTakagi-Sugeno model using a Proportional Integral Observer (PIO) and Proportional MultipleIntegral Observer (PMIO)
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 2 / 29
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Motivations
Objective
Estimate simultanuously the state and unknown inputs of nonlinear systems described by aTakagi-Sugeno model using a Proportional Integral Observer (PIO) and Proportional MultipleIntegral Observer (PMIO)
DifficultiesUnmeasurable premise variables
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 2 / 29
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Outline
1 Takagi-Sugeno approach for modelingTakagi-Sugeno principleTakagi-Sugeno model
2 Proportional Integral Observer design
3 State and unknown input estimation
4 Numerical example
5 Conclusions
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 3 / 29
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Outline
1 Takagi-Sugeno approach for modelingTakagi-Sugeno principleTakagi-Sugeno model
2 Proportional Integral Observer design
3 State and unknown input estimation
4 Numerical example
5 Conclusions
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 3 / 29
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Outline
1 Takagi-Sugeno approach for modelingTakagi-Sugeno principleTakagi-Sugeno model
2 Proportional Integral Observer design
3 State and unknown input estimation
4 Numerical example
5 Conclusions
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 3 / 29
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Outline
1 Takagi-Sugeno approach for modelingTakagi-Sugeno principleTakagi-Sugeno model
2 Proportional Integral Observer design
3 State and unknown input estimation
4 Numerical example
5 Conclusions
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 3 / 29
![Page 8: Simultaneous state and unknown inputs estimation with PI](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fd95b20a3977bdc0226f6/html5/thumbnails/8.jpg)
Outline
1 Takagi-Sugeno approach for modelingTakagi-Sugeno principleTakagi-Sugeno model
2 Proportional Integral Observer design
3 State and unknown input estimation
4 Numerical example
5 Conclusions
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 3 / 29
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Takagi-Sugeno approach for modeling
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 4 / 29
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Takagi-Sugeno principle
◮ Operating range decomposition in several local zones.◮ A local model represent the behavior of the system in each zone.◮ The overall behavior of the system is obtained by the aggregation of the sub-models with
adequate weighting functions.
ξ1(t)
ξ2(t)
ξ1(t)
ξ2(t)
Operating
space
zone 1
zone 2
zone 3
zone 4
Multiple model representationNonlinear system
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 5 / 29
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Takagi-Sugeno approach for modeling
Interests of Takagi-Sugeno approach◮ Simple structure for modeling complex nonlinear systems.◮ The specific study of the nonlinearities is not required.◮ Possible extension of the theoretical LTI tools for nonlinear systems.
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 6 / 29
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Takagi-Sugeno model
The considered system is given by the following equations
x(t) =r∑
i=1µi (ξ (t))(Ai x(t)+Bi u(t))
y(t) =r∑
i=1µi (ξ (t))(Ci x(t)+Di u(t))
• Interpolation mechanism :r∑
i=1µi (ξ (t)) = 1 and 0 ≤ µi (ξ (t)) ≤ 1, ∀t , ∀i ∈ {1, ..., r}
• The premise variable ξ (t) can be measurable (input u(t),output y(t),...) or unmeasurable(state x(t),...).
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 7 / 29
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Takagi-Sugeno model
The considered system is given by the following equations
x(t) =r∑
i=1µi (ξ (t))(Ai x(t)+Bi u(t))
y(t) =r∑
i=1µi (ξ (t))(Ci x(t)+Di u(t))
• Interpolation mechanism :r∑
i=1µi (ξ (t)) = 1 and 0 ≤ µi (ξ (t)) ≤ 1, ∀t , ∀i ∈ {1, ..., r}
• The premise variable ξ (t) can be measurable (input u(t),output y(t),...) or unmeasurable(state x(t),...).
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 7 / 29
![Page 14: Simultaneous state and unknown inputs estimation with PI](https://reader033.vdocuments.us/reader033/viewer/2022061607/629fd95b20a3977bdc0226f6/html5/thumbnails/14.jpg)
Takagi-Sugeno model
The considered system is given by the following equations
x(t) =r∑
i=1µi (ξ (t))(Ai x(t)+Bi u(t))
y(t) =r∑
i=1µi (ξ (t))(Ci x(t)+Di u(t))
• Interpolation mechanism :r∑
i=1µi (ξ (t)) = 1 and 0 ≤ µi (ξ (t)) ≤ 1, ∀t , ∀i ∈ {1, ..., r}
• The premise variable ξ (t) can be measurable (input u(t),output y(t),...) or unmeasurable(state x(t),...).
Interests of T-S model with unmeasurable premise variables◮ Exact representation of the model x(t) = f (x(t),u(t)).
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 7 / 29
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Takagi-Sugeno model
The considered system is given by the following equations
x(t) =r∑
i=1µi (ξ (t))(Ai x(t)+Bi u(t))
y(t) =r∑
i=1µi (ξ (t))(Ci x(t)+Di u(t))
• Interpolation mechanism :r∑
i=1µi (ξ (t)) = 1 and 0 ≤ µi (ξ (t)) ≤ 1, ∀t , ∀i ∈ {1, ..., r}
• The premise variable ξ (t) can be measurable (input u(t),output y(t),...) or unmeasurable(state x(t),...).
Interests of T-S model with unmeasurable premise variables◮ Exact representation of the model x(t) = f (x(t),u(t)).◮ Only one T-S model for FDI with observer banks.
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 7 / 29
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Takagi-Sugeno model
The considered system is given by the following equations
x(t) =r∑
i=1µi (ξ (t))(Ai x(t)+Bi u(t))
y(t) =r∑
i=1µi (ξ (t))(Ci x(t)+Di u(t))
• Interpolation mechanism :r∑
i=1µi (ξ (t)) = 1 and 0 ≤ µi (ξ (t)) ≤ 1, ∀t , ∀i ∈ {1, ..., r}
• The premise variable ξ (t) can be measurable (input u(t),output y(t),...) or unmeasurable(state x(t),...).
Interests of T-S model with unmeasurable premise variables◮ Exact representation of the model x(t) = f (x(t),u(t)).◮ Only one T-S model for FDI with observer banks.◮ Exemple of application: security improvement in cryptography systems.
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 7 / 29
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Proportional Integral Observer design
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 8 / 29
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Proportional Integral Observer design
Let us consider the T-S system represented by
x(t) =r∑
i=1µi (ξ (t))(Ai x(t)+Bi u(t)+Ei d(t)+Wi ω(t))
y(t) = Cx(t)+Gd(t)+W ω(t)
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 9 / 29
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Proportional Integral Observer design
Let us consider the T-S system represented by
x(t) =r∑
i=1µi (ξ (t))(Ai x(t)+Bi u(t)+Ei d(t)+Wi ω(t))
y(t) = Cx(t)+Gd(t)+W ω(t)
Assumption 1
Assume that the following assumptions hold:◮ A1. The system is stable◮ A2. The signals u(t), d(t) and ω(t) are bounded.
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 9 / 29
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Proportional Integral Observer design
Let us consider the T-S system represented by
x(t) =r∑
i=1µi (ξ (t))(Ai x(t)+Bi u(t)+Ei d(t)+Wi ω(t))
y(t) = Cx(t)+Gd(t)+W ω(t)
Assumption 1
Assume that the following assumptions hold:◮ A1. The system is stable◮ A2. The signals u(t), d(t) and ω(t) are bounded.
Assumption 2
The unknown inputs d(t) are assumed to be constant:
◮ A3. d = 0
(This assumption will be relaxed in the PMIO design).
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 9 / 29
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Proportional Integral Observer design
Let us denote the estimated state by x .
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 10 / 29
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Proportional Integral Observer design
Let us denote the estimated state by x .The system can be re-written as follows:
x =r
∑i=1
µi (x)(Ai x +Bi u +Ei d +Wi ω +ν)
where:
ν =r
∑i=1
(µi (x)−µi (x))(Ai x +Bi u +Ei d +Wi ω)
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 10 / 29
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Proportional Integral Observer design
Let us denote the estimated state by x .The system can be re-written as follows:
x =r
∑i=1
µi (x)(Ai x +Bi u +Ei d +Wi ω +ν)
where:
ν =r
∑i=1
(µi (x)−µi (x))(Ai x +Bi u +Ei d +Wi ω)
The assumption A3. allows to write the system in the following augmented form:
xa =r∑
i=1µi (x)
(
Ai xa + Bi u +Γi ω)
y = Cxa + Dω
where:
Ai =
[
Ai Ei0 0
]
, Bi =
[
Bi0
]
, Γi =
[
I Wi0 0
]
, ω =
[
νω
]
C =[
C G]
, D =[
0 W]
, xa =
[
xd
]
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 10 / 29
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Proportional Integral Observer design
The proposed PI observer is given by the following equations:
˙x =r∑
i=1µi (x)
(
Ai x +Bi u +Ei d +KPi (y − y))
y = Cx +Gd˙d =
r∑
i=1µi (x)KIi (y − y)
where x and d are the estimates of x and d .
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 11 / 29
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Proportional Integral Observer design
The proposed PI observer is given by the following equations:
˙x =r∑
i=1µi (x)
(
Ai x +Bi u +Ei d +KPi (y − y))
y = Cx +Gd˙d =
r∑
i=1µi (x)KIi (y − y)
where x and d are the estimates of x and d .
A similar reasoning makes it possible to transform the proposed PI observer in the followingaugmented form:
˙xa =r∑
i=1µi (x)
(
Ai xa + Bi u + Ki (y − y))
y = Cxa
where:
Ki =
[
KPiKIi
]
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 11 / 29
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Proportional Integral Observer design
Augmented state estimation errorea(t) = xa(t)− xa(t)
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 12 / 29
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Proportional Integral Observer design
Augmented state estimation errorea(t) = xa(t)− xa(t)
ea =r
∑i=1
µi (x)(
(Ai − Ki C)ea +(Γi − Ki D)ω)
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 12 / 29
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Proportional Integral Observer design
Augmented state estimation errorea(t) = xa(t)− xa(t)
ea =r
∑i=1
µi (x)(
(Ai − Ki C)ea +(Γi − Ki D)ω)
Objectives
The goal is to determine the gain matrices Ki such that:◮ The system which generates the state and unknown inputs errors is stable.◮
limt→∞
e(t) = 0, pour ‖ω‖2 = 0
‖e(t)‖2 < γ ‖ω‖2 pour ‖ω‖2 6= 0
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 12 / 29
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Proportional Integral Observer design
Lemma 1 [Tanaka and Wang 2001]
Consider the continuous-time TS-system defined by:
x(t) =r∑
i=1µi (x(t))(Ai x(t)+Bi u(t))
y(t) = Cx(t)
The system is stable and verifies the L2-gain condition:
||y(t)| |2 < γ ||u(t)| |2
if there exists a symmetric positive definite matrix P such that the following LMIs are satisfied fori = 1, ..., r :
[
ATi P +PAi +CT C PBi
BTi P −γ2I
]
< 0
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 13 / 29
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Proportional Integral Observer design
Theorem 1: PIOThe PI observer for the system is determined by minimizing γ under the following LMI constraintsin the variables P = PT > 0, Mi and γ for i = 1, ..., r :
[
ATi P +PAi −Mi C − CT Mi
T + I PΓi −Mi DΓT
i P − DT MiT −γI
]
< 0
The gains of the observer are derived from:
Ki = P−1Mi
and the attenuation level is calculated by:
γ =√
γ
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 14 / 29
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Proportional Multiple Integral Observer design
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 15 / 29
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Proportional Multiple Integral Observer design
Let us consider the T-S system represented by
x(t) =r∑
i=1µi (ξ (t))(Ai x(t)+Bi u(t)+Ei d(t)+Wi ω(t))
y(t) = Cx(t)+Gd(t)+W ω(t)
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 16 / 29
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Proportional Multiple Integral Observer design
Let us consider the T-S system represented by
x(t) =r∑
i=1µi (ξ (t))(Ai x(t)+Bi u(t)+Ei d(t)+Wi ω(t))
y(t) = Cx(t)+Gd(t)+W ω(t)
Assumption 3
The unknown input is assumed to be a bounded time varying signal with null qth derivative:◮ A4. d (q)(t) = 0
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 16 / 29
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Proportional Multiple Integral Observer design
Consider the generalization of the proportional multiple-integral observer to T-S systems of thePMI observer:
˙x =r∑
i=1µi (x)(Ai x +Bi u +Ei d0 +KPi (y − y))
y = Cx +Gd˙d0 =
r∑
i=1µi (x)K 0
Ii (y − y)+ d1
˙d1 =r∑
i=1µi (x)K 1
Ii (y − y)+ d2
...˙dq−2 =
r∑
i=1µi (x)K q−2
Ii (y − y)+ dq−1
˙dq−1 =r∑
i=1µi (x)K q−1
Ii (y − y)
where di , i = 1,2, ...,(q−1) are the estimation of the (q−1) first derivatives of the unknowninput d(t).
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 17 / 29
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Proportional Multiple Integral Observer design
The state and unknown inputs estimation errors are:
e = x − x , e0 = d −˙d0, . . . , eq−1 = dq−1 −
˙dq−1
Their dynamics are given in the following form:
e =r∑
i=1µi (x)((Ai −KPi C)e +(Γi −KPi W )ω +(Ei −KPi G)e0)
e0 =r∑
i=1µi (x)(−K 0
Ii Ce +e1 −K 0Ii W ω −K 0
Ii Ge0)
e1 =r∑
i=1µi (x)(−K 1
Ii Ce +e2 −K 1Ii W ω −K 1
Ii Ge0)
...
eq−2 =r∑
i=1µi (x)(−K 0
Ii Ce +eq−1 −K q−2Ii W ω −K q−2
Ii Ge0)
eq−1 =r∑
i=1µi (x)(−K q−1
Ii Ce−K 0Ii W ω −K q−1
Ii Ge0)
where:Γi =
[
In Wi]
, W =[
0 W]
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 18 / 29
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Proportional Multiple Integral Observer design
In the augmented form, we have:
˙e =r
∑i=1
µi (x)((Ai − Ki C)e +(Γi − Ki W )ω)
[
ee0
]
= Ce
where:
e =
ee0e1
...eq−2eq−1
, Ai =
Ai Ei 0 · · · 0 00 0 Is · · · 0 0
0 0 0. . . 0 0
......
......
......
0 0 0 0 0 Is0 0 0 0 0 0
,
Ki =[
K TPi K 0
IiT
K 1Ii
T. . . K q−2
Ii
TK q−1
Ii
T ]T
C =[
C G 0 · · · 0 0]
Γi =[
ΓTi 0 ... 0
]T
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 19 / 29
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Proportional Multiple Integral Observer design
In the following, we are only interested with particular components e and e0 of e:[
ee0
]
= Ce
where:
C =
[
In 00 Is
0 · · · 0]
0 represents null matrix with appropriate dimensions.
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 20 / 29
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Proportional Multiple Integral Observer design
Theorem 2
The PMI observer for the system that minimizes the transfer from ω(t) to [e(t)T e0(t)T ] isobtained by finding the matrices P = PT > 0, Mi and γ that minimize γ under the following LMIconstraints for i = 1, ..., r :
[
ATi P +PAi −Mi C − CT MT
i + CT C PΓi −Mi WΓT
i P −W T MTi −γI
]
< 0
The gains of the observer are derived from:
Ki = P−1Mi
and the attenuation level is calculated by:
γ =√
γ
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 21 / 29
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Numerical example
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 22 / 29
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Numerical example
Consider a continuous-time T-S system defined by:
A1 =
−2 1 11 −3 02 1 −8
, A2 =
−3 2 −25 −3 01 2 −4
,
B1 =
15
0.5
, B2 =
31−7
, E1 =
0 70 50 2
,
E2 =
0 60 30 1
, W1 = W2 =
111
, W =
[
0.50.5
]
,
and
C =
[
1 1 11 0 1
]
, G =
[
5 01 0
]
d(t) = [d1(t)T d2(t)T ]T .
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 23 / 29
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Numerical example
The weighting functions depend on the first component x1 of the state vector x and are definedas follows:
{
µ1(x) = 1−tanh(x1)2
µ2(x) = 1−µ1(x)(1)
The weighting functions obtained without perturbations and unknown inputs are shown in figure1. This figure shows that the system is clearly nonlinear since µ1 and µ2 are not constantfunctions.
0 5 10 15 20 25 30 35 40 45 500
0.2
0.4
0.6
0.8
1µ
1(t)
µ2(t)
Figure: Weighting functions µ1 and µ2
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 24 / 29
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Numerical example
0 5 10 15 20 25 30 35 40 45 50−3
−2
−1
0
1
2
3original d
1estimated d
1
0 5 10 15 20 25 30 35 40 45 50−3
−2
−1
0
1
2
3original d
2estimated d
2
Figure: Unknown input estimation with a PI observer
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 25 / 29
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Numerical example
0 5 10 15 20 25 30 35 40 45 50−3
−2
−1
0
1
2
3original d
1estimated d
1
0 5 10 15 20 25 30 35 40 45 50−3
−2
−1
0
1
2
3original d
2estimated d
2
Figure: Unknown input estimation with a PMI observer
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Numerical example
0 5 10 15 20 25 30 35 40 45 50−1.5
−1
−0.5
0
0.5e
1e
2e
3
Figure: State estimation error with a PI observer
0 5 10 15 20 25 30 35 40 45 50−1.5
−1
−0.5
0
0.5e
1e
2e
3
Figure: State estimation error with a PMI observer
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 27 / 29
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Conclusions and perspectives
Conclusions◮ PI and PMI observers design for nonlinear systems represented by a Takagi-Sugeno
structure.
Perspectives
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 28 / 29
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Conclusions and perspectives
Conclusions◮ PI and PMI observers design for nonlinear systems represented by a Takagi-Sugeno
structure.◮ Study of the case where the premise variables are unmeasurable
Perspectives
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 28 / 29
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Conclusions and perspectives
Conclusions◮ PI and PMI observers design for nonlinear systems represented by a Takagi-Sugeno
structure.◮ Study of the case where the premise variables are unmeasurable◮ Comparison between the results obtained by PI and PMI observers
Perspectives
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 28 / 29
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Conclusions and perspectives
Conclusions◮ PI and PMI observers design for nonlinear systems represented by a Takagi-Sugeno
structure.◮ Study of the case where the premise variables are unmeasurable◮ Comparison between the results obtained by PI and PMI observers◮ Pole assignement in a specific region (see the paper).
Perspectives
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 28 / 29
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Conclusions and perspectives
Conclusions◮ PI and PMI observers design for nonlinear systems represented by a Takagi-Sugeno
structure.◮ Study of the case where the premise variables are unmeasurable◮ Comparison between the results obtained by PI and PMI observers◮ Pole assignement in a specific region (see the paper).
Perspectives
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 28 / 29
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Conclusions and perspectives
Conclusions◮ PI and PMI observers design for nonlinear systems represented by a Takagi-Sugeno
structure.◮ Study of the case where the premise variables are unmeasurable◮ Comparison between the results obtained by PI and PMI observers◮ Pole assignement in a specific region (see the paper).
Perspectives◮ Study and reduction of the conservatism when searching a common Lyapunov matrix P,
which satisfies r LMIs.
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 28 / 29
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Conclusions and perspectives
Conclusions◮ PI and PMI observers design for nonlinear systems represented by a Takagi-Sugeno
structure.◮ Study of the case where the premise variables are unmeasurable◮ Comparison between the results obtained by PI and PMI observers◮ Pole assignement in a specific region (see the paper).
Perspectives◮ Study and reduction of the conservatism when searching a common Lyapunov matrix P,
which satisfies r LMIs.◮ Extension to fault tolerant control of nonlinear systems.
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 28 / 29
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Thank you for your attention!
Ichalal, Marx, Ragot, Maquin (CRAN) State estimation MED 2009 29 / 29