ADRC ACCWorkshop Sira-Ramirez

Active Disturbance Rejection and the Control of Differentially Flat Systems

Active Disturbance Rejection and the Control of

Differentially Flat Systems

Hebertt Sira-Ramırez †

† Center for Research and Advanced Studies of the National Polytechnic InstituteMexico, D.F., Mexico

May 2013


Contents of Presentation

1 Introduction to ADRC

2 Definition of SISO Flat Systems and Examples

3 Definition of MIMO Flat Systems and Examples

4 Conclusions


Introduction to ADRC

Introduction

Active Disturbance Rejection Control (ADRC) is based on:

Retain input gain (matrix) and (multi-index) order of the systemoutput as the only required design elements.

Simultaneously estimate, from the available outputs,a.- unmeasured states or output phase variables andb.- lumped effects of endogenous (state-dependent) uncertaintiesexogenous disturbances, unmodeled dynamics or neglected higherorder terms (h.o.t.) affecting the input-output dynamics.Key idea: Treat these lumped effects as if they were unknowntime-varying functions (i.e., signals).

Devise a controller that cancels all uncertain effects, while using theestimated states to complete a suitable static, or dynamic, feedbackcontroller of the (hopefully linear) remaining, well known, part ofthe system.



The ADRC approach is naturally fitted for input-output systemmodels with no zero dynamics and no gain singularities.Nonlinear SISO systems of the form:

y (n) = ψ(t, y , ..., y (n−1))u + φ(t, y , y , ..., y (n−1))

are envisioned as the simpler system:

y (n) = µ(t)u + ξ(t)

with ξ(t) being an unstructured time signal, assumed to beuniformly absolutely bounded, causing no finite escape time of yfor a smooth bounded u. and µ(t) is an on-line estimate of thenonlinear gain. Fact: ξ(t) is observable. Simplest cases:ψ(t, y , ..., y (n−1)) =constant, or ψ(t, y , ..., y (n−1)) = ψ(t, y).



ADRC Trajectory tracking controller for y → y∗(t).

u =−ξ(t) + [y∗(t)](n) −

∑n−1j=0 κj(yj − [y∗(t)](j))

ψ(t, y , y1..., yn−1)

y0 = y1 + λm+n−1(y − y0)y1 = y2 + λm+n−2(y − y0)...yn−1 = ψ(t, y , y1..., yn−1)u + z1 + λm(y − y0)

z1 = z2 + λm−1(y − y0)...

zm = λ0(y − y0), ξ(t) = z1



A two tank system

Consider the following mathematical model of a two tank system

x1 = − c

A

√x1 +

1

Au

x2 = − c

A

√x2 +

c

A

√x1

y = x2

It is desired to regulate the output y(t) of the system, so that ittracks a desired given smooth trajectory y∗(t). System parametersc , and A, are assumed to be known.



x1

x2

u

y =

A

A

c

c

A two tank system



The two tank system admits the following input-output model:

y =1

2A

[

c2/A2

y + cA

√y

]

u − c

2A

y√y− c2

2A2

Consider the simplified system with a nonlinear input gain

y =1

2A

[

c2/A2

y + cA

√y

]

u + ξ(t)

where

ξ(t) = − c

2A

y(t)√

y(t)− c2

2A2



We propose the following observer based output feedbackcontroller:

u = 2A

[

A2

c2

]

(

y1 +c

A

√y)

v

v = −ξ(t) + y∗(t)− 2ζcωnc(y1 − y∗(t))− ω2nc(y − y∗(t))

where y1 is the estimated output velocity which is synthesized bythe following GPI observer:

y0 = y1 + γ5(y − y0)

y1 =1

2A

[

c2/A2

y1 +ca

√y

]

u + z1 + γ4(y − y0)

z1 = z2 + γ3(y − y0)

z2 = z3 + γ2(y − y0)

z3 = z4 + γ1(y − y0)

z4 = γ0(y − y0), ξ(t) = z1



0 10 20 30 40 50 602

4

6

0 10 20 30 40 50 600

5

10

0 10 20 30 40 50 600

1

2

y(t), y∗(t)

x1(t)x1(t)

u(t)

Response of ADRC controlled two tank system



0 10 20 30 40 50 60−0.07

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01

0

z1s(t)

Estimation of state-dependent perturbation for ADRC controlled



0 10 20 30 40 500

5

10

15

time

Liquid heights

0 10 20 30 40 50−0.4

−0.2

0

0.2

0.4

time

Disturbance and lumped estimate of uncertainties

0 10 20 30 40 500

5

10

15

time

observed input gain (red) and actual input gain (black)

0 10 20 30 40 500

1

2

3

4

time

control input

ξ(t)

u(t)ψ(y, y)

ψ(y, y1)

p(t)x1(t)

y(t) = x2(t)

Performance of ADRC controlled perturbed two tank system



0 5 10 15 20 25 30 35 40 45 502.5

3

3.5

4

4.5

5

5.5

time

y∗(t)

y(t) = x2(t)

Output trajectory tracking for ADRC controlled two tank system


Definition of SISO Flat Systems and Examples

Flat SISO systems

Consider a nonlinear system

x = f (x , u), x ∈ Rn, u ∈ R

The system is flat if there exists an independent output functiony ∈ R , such that:

y = θ(x) ∈ R

and all system variables are differentially parameterizable in termsof y :

x = ψ(y , y , ..., y (n−1)), u = φ(y , y , ..., y (n))

In general, y is non-unique.



The two tank system

The two tank system,

x1 = − c

A

√x1 +

1

Au, x2 = − c

A

√x2 +

c

A

√x1

is flat, with flat output y = x2.Indeed

x2 = y

x1 =A2

c2

(

y +c

A

√y)2

u = A(

y +c

A

√y)

[

1 +2A2

c2

(

y +c

2A

y√y

)]



Slosh control in Liquid containers



x1 = x2

x2 = −2ζωx2 − ω2x1 +aω2

2gu

x3 = x4

x4 = u

x1: liquid height, above the resting levelx2: corresponding rate of the liquid heightx3: position of the containerx4: its associated velocityu: applied force input

ω, ζ: liquid natural oscillation frequency and damping factor.



Flat output

F =

[

2

ω4a

(

−1 + 4ζ2)

g

]

x1 +

(

4ζ

aω5g

)

x2

+

(

1

ω2

)

x3 −(

2

ω3ζ

)

x4

A differential parametrization of all system variables, in terms ofthe flat outputs, is given by:

x1 =aω2

2gF

x2 =aω2

2gF (3)

x3 = ω2F + 2ζωF + F

x4 = ω2F + 2ζωF + F (3)

u = ω2F + 2ζωF (3) + F (4)



For an ADRC treatment of the slosh control problem one wouldtake:

F (4) = u + ξ(t)

The observer design is substantially alleviated thanks to thestructure revealed by flatness


Definition of MIMO Flat Systems and Examples

Flat MIMO systems

Consider a nonlinear system

x = f (x , u), x ∈ Rn, u ∈ R

m

The system is differentially flat if there exists an independentm-dimensional output vector y and a set of finite multi-indices: p,q and r , such that:

y = θ(x , u, u, ..., u(p)) ∈ Rm

and all system variables are differentially parameterizable in termsof y :

x = ψ(y , y , ..., y (q)), u = φ(y , y , ..., y (r))

In general, y is non-unique.



System with non-holonomic constraint

Consider the system

x1 = u1, x2 = u2, x3 = u1u2

Define the flat outputs as1

F = x1, L = x3 − u1x2

One readily obtains the following differential parametrization:

x1 = F , x2 = − L

F, x3 = L−

(

L

F

)

F

u1 = F , u2 = − LF − LF (3)

(F )2

1This output represents the integrable part of the nonintegrable constraintx3 − x1x2 = 0, portrayed by the last equation.



The inputs to flat outputs relation is non-invertible, clearlyindicating there is an obstruction to static feedback linearization.Dynamic feedback, through input extension, may be achieved bysetting the following (extended) input coordinate transformation:

v1 = u(3)1 = F (4), v2 = u2

One now gets the invertible extended inputs-to-flat-outputs relation

[

F (4)

L

]

=

[

1 0

0 −F

] [

v1v2

]

+

[

0LF (3)

F

]



The batch reactor model

Consider the following dynamic model of a batch reactor:

x1 =

[

µx2Ks + x2

]

x1 − u1x1x3

x2 = −[

µx2Ks + x2

]

1

Ysx

+u1(u2 − x2)

x3x3 = u1

The system is flat, with flat outputs given by:

F = x1, R = x3

representing, respectively, the product yield and the reactor volume.



Indeed, all variables are differentially parameterizable in terms of Fand R,

u1 = R , x1 = F , x3 = R, x2 = Ks

[

FR + FR

µFR − (F R + FR)

]

u2 = Ks

F R + 2F R + FR[

µFR − (FR + FR)]

−

(

FR + FR) [

µ(

FR + FR)

−(

FR + 2F R + FR)]

[

µFR − (FR + FR)]2

R

R

+1

Ysx

(

F R + RF

FR

)

R

R+ Ks

[

F R + FR

µFR − (FR + FR)

]



The extended model is simply given by,

x1 =

[

µx2Ks + x2

]

x1 − x4x1x3

x2 = −[

µx2Ks + x2

]

1

Ysx

+x4(v2 − x2)

x3x3 = x4

x4 = x5

x5 = v1

which now exhibits a well defined vector relative degree for R andF given by:

(3, 2)

Notice that the additional states x4, x5 are readily available asintegrals of the auxiliary control input v1.



We now have the following differential parametrization,

v1 = R (3), x3 = R , x4 = R , x5 = R , x1 = F ,

x2 = Ks

[

F R + FR

µFR − (FR + FR)

]

v2 = Ks

FR + 2F R + FR[

µFR − (F R + FR)]

−

(

F R + FR) [

µ(

FR + FR)

−(

F R + 2F R + FR)]

[

µFR − (F R + FR)]2

R

R

+1

Ysx

(

F R + RF

FR

)

R

R+ Ks

[

FR + FR

µFR − (F R + FR)

]



After tedious, but elementary, manipulations, one obtains thefollowing extended input-output dynamics:

R (3) = v1

F =

[


R

KsµFR3

v2 + φ(R , R, R ,F , F )

φ(R, R, R,F , F ) =

[


R

KsµFR3×

{

−

2F R + F R

µFR − (FR + FR)+

(FR + FR)(

µ(FR + FR)− 2F R − FR

)

[


RKs

R

−

R

Ysx

(

FR + FR

FRR

)

− Ks

(

FR + FR

µFR − (FR + FR)

)

}


Conclusions

Conclusions

The controller design task for nonlinear uncertain, perturbed,flat systems may be substantially simplified thanks to the useof a combination of high gain extended Luenberger observersand linearizing output feedback controllers. The key idea is toretain input gain and order (gain matrix and Kroneckerindices) as design elements, relegating the lumped effect of alladditive uncertainties and disturbances to jointly act asunknown time signals disturbances that may be estimated.

The scheme is easily extendable to systems with known inputdelays, systems with fractional derivatives, discrete systems,etc.

ADRC ACCWorkshop Sira-Ramirez

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