sliding mode observers - historical background and basic introduction

Sliding mode observers - historical background andbasic introduction

Sarah K. Spurgeon

School of Engineering and Digital ArtsUniversity of Kent, UK

Spring School, Aussois, June 2015

Sarah K. Spurgeon Sliding mode observer

Outline of Presentation

Sliding mode control versus sliding mode observers?

Historical perspective - The Utkin Observer

Tutorial Example

Further historical milestones

Window on the state of the art - Lecture 2


Introduction - The Control Problem

Sliding mode techniques were perhaps originally best knownfor their potential as a robust control method, and evolvedfrom pioneering work in the 1960’s in the former Soviet Union.

Such a sliding mode control is characterised by a suite offeedback control laws and a decision rule. The decision rule,termed the switching function, has as its input some measureof the current system behaviour and produces as an outputthe particular feedback controller which should be used atthat instant in time.

In sliding mode control, VSCS are designed to drive and thenconstrain the system state to lie within a neighbourhood ofthe switching function.



There are a number of advantages to this approach.

Firstly, the dynamic behaviour of the system may be tailoredby the particular choice of switching function.

Secondly, the closed-loop response becomes totally insensitiveto a particular class of uncertainty in the system.

y

y.

trajectory

sliding surface

Figure : A sliding mode



There are a number of advantages to this approach.

Firstly, the dynamic behaviour of the system may be tailoredby the particular choice of switching function.Secondly, the closed-loop response becomes totally insensitiveto a particular class of uncertainty in the system.

y

y.

trajectory

sliding surface



The Control Problem - Disadvantages

A disadvantage of the method in the domain of control applicationshas been the necessity to implement a fundamentally discontinuouscontrol signal which, in theoretical terms, must switch with infinitefrequency to provide total rejection of uncertainty. Controlimplementation via approximate, smooth strategies is widelyreported, but in such cases total invariance is routinely lost.


Introduction - The Observer Problem

In contrast, the application of sliding mode methods to theobserver problem is much less mature and has some fundamentallydifferent advantages and disadvantages.

the ability to generate a sliding motion on the error betweenthe measured plant output and the output of the observerensures that a sliding mode observer produces a set of stateestimates that are precisely commensurate with the actualoutput of the plant.

analysis of the average value of the applied observer injectionsignal, the so-called equivalent injection signal, contains usefulinformation about the mismatch between the model used todefine the observer and the actual plant.

the discontinuous injection signals which were perceived asproblematic for many control applications, have nodisadvantages for software based observer frameworks.


A Historical Perspective

System

Consider initially the linear system described by

x(t) = Ax(t) + Bu(t)y(t) = Cx(t)

(1)

where A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n and p ≥ m. Assume thatB and C are full rank and (A,C ) is observable.

Canonical form - Utkin Observer

Consider the change of coordinates x 7→ Tcx whereby

Tc =

[NTc

C

](2)

where the columns of Nc ∈ Rn×(n−p) span the null space of C .This transformation is nonsingular



Canonical form for the nominal system

x1(t) = A11x1(t) + A12y(t) + B1u(t) (3)

y(t) = A21x1(t) + A22y(t) + B2u(t) (4)

where

Tcx =

[x1y

]ln−plp

The Observer

˙x1(t) = A11x1(t) + A12y(t) + B1u(t) + Lν (5)

˙y(t) = A21x1(t) + A22y(t) + B2u(t)− ν (6)

where (x1, y) represent the state estimates, L ∈ R(n−p)×p is a gainmatrix and νi = M sgn(yi − yi ) where M ∈ R+.



The error system

Define e1 = x1 − x1 and ey = y − y then

e1(t) = A11e1(t) + A12ey (t) + Lν (7)

ey (t) = A21e1(t) + A22ey (t)− ν (8)

Choice of L

Since the pair (A,C ) is observable, the pair (A11,A21) is alsoobservable and L can be chosen to make the spectrum ofA11 + LA21 lie in C−.

Define a further change of coordinates by

T =

[In−p L

0 Ip

]



Error dynamics

Define a change of coordinates by T =

[In−p L

0 Ip

]With e1 = e1 + Ly , he error system becomes

˙e1(t) = A11e1(t) + A12ey (t) (9)

ey (t) = A21e1(t) + A22ey (t)− ν (10)

where A11 = A11 + LA21, A12 = A12 + LA22 − A11L andA22 = A22 − A21L.

In the domain

Ω = (e1, ey ) : ‖A21e1‖+ 12λmax(A22 + AT

22)‖ey‖ < M − η(11)

where η < M is some small positive scalar, the reachabilitycondition eTy ey < −η‖ey‖ is satisfied.



Sliding Motion

An ideal sliding motion will take place on the surface

So = (e1, ey ) : ey = 0 (12)

After some finite time ts , for all subsequent time, ey = 0 andey = 0.

The corresponding sliding mode dynamics are given by

˙e1(t) = A11e1(t) (13)

which, by choice of L, represents a stable system and soe1 → 0 and consequently, x1 → x1 as t →∞.


Tutorial Example

The Model

Consider the second-order linear system

x(t) = Ax(t) + Bu(t) (14)

y(t) = Cx(t) (15)

where

A =

[0 1−2 0

]B =

[01

]C =

[1 1

]The System

This represents a simple harmonic oscillator.

For simplicity assume u = 0 and consider the problem ofdesigning a sliding mode observer.


Tutorial Example

Canonical Form

Define a nonsingular matrix

Tc =

[1 01 1

](16)

This change of coordinates gives the system triple

TcAT−1c =

[−1 1−3 1

]TcB =

[01

]CT−1c =

[0 1

]An appropriate choice of observer gain is L = 0.5 whichresults in an error system governed by A11 = −2.5.

The scaling constant M in the discontinuous component hasbeen set to unity.


Tutorial Example

Canonical Form


Tc =

[1 01 1

](16)


TcAT−1c =

[−1 1−3 1

]TcB =

[01

]CT−1c =

[0 1

]

An appropriate choice of observer gain is L = 0.5 whichresults in an error system governed by A11 = −2.5.



Tutorial Example

Canonical Form


Tc =

[1 01 1

](16)


TcAT−1c =

[−1 1−3 1

]TcB =

[01

]CT−1c =

[0 1

]An appropriate choice of observer gain is L = 0.5 whichresults in an error system governed by A11 = −2.5.



Tutorial Example

The Figure shows the state estimation errors e1(t) and ey (t)resulting from the initial conditions e1 = −1 and ey = 0.Although the error system starts on the sliding surface So , anideal sliding motion cannot be maintained; only afterapproximately 1.2 seconds is sliding established.At this point in the time interval 1.2 to 3.0 seconds, e1(t)exhibits a first-order exponential decay to the origin.After 3.0 seconds almost perfect replication of the states takesplace.

-1

-0.5

0

0.5

1

0 1 2 3 4 5 6 7 8 9 10

Time, sec

Stat

e E

stim

atio

n E

rror

s



Tutorial Example

In the original coordinates perfect tracking occurs afterapproximately 3 seconds.The dotted lines represent the true states and the solid linethe estimates from the observer.

-2

-1

0

1

2

0 1 2 3 4 5 6 7 8 9 10

Time, sec

Stat

e E

volu

tion



Tutorial Example

This Figure shows the value of ν with respect to time andshows switching taking place from 1.2 seconds onwards.

-1

-0.5

0

0.5

1

0 1 2 3 4 5 6 7 8 9 10

Time, sec

Dis

cont

inuo

us C

ompo

nent



Classical Utkin Observer with Linear Injection

The new error system

˙e1(t) = A11e1(t) + A12ey (t)− G1ey (t) (17)

ey (t) = A21e1(t) + A22ey (t)− G2ey (t)− ν (18)

By selecting G1 = A12 and G2 = A22 − As22, where As

22 is anystable design matrix of appropriate dimension, then

˙e1(t) = A11e1(t) (19)

ey (t) = A21e1(t) + As22ey (t)− ν (20)

The error system is asymptotically stable for ν ≡ 0 becausethe poles of the combined system are given byλ(A11)∪ λ(As

22) and so lie in the open left half complex plane.


Classical Observer

Observations

In the original Utkin observer, the switching action ν waspotentially required to make the error system stable.

Thus far the only restriction imposed on the nominal linearsystem is that the pair (A,C ) is observable.


Further Historical Developments

The Slotine Observer (1980s)

The output errors are fed back in both a linear and adiscontinuous manner for nonlinear systems in companionform with the objective of ensuring a ’sliding patch’, whichdefines the region in which it is possible for the dynamicalobserver system to exhibit sliding behaviour, is maximised.

Once only a subset of state information is known, the ability ofany system to attain and maintain sliding will be more limitedthan in the situation where full state information is available.

With the Slotine observer, the linear feedback elements are aLuenberger observer with the role of the magnitude of thediscontinuous element to enhance robustness.

It is shown that a larger discontinuous element can enhancerobustness but this can be at the expense of increasedsensitivity to measurement noise.


Further Historical Developments

The Walcott and Zak Observer, (1980s)

This was instrumental in defining the structural conditions forexistence of sliding mode observers for linear systems and laidimportant foundations for subsequent contributions whichformulated constructive design methodologies and includeduncertainty.

It was key in showing the promise of the methodology forobserver design for nonlinear systems, where methodologiesfor relatively general nonlinear system representations wereconsidered

These ideas will be developed further in the next lecture.


Sliding mode observers: towards a constructivedesign framework

Sarah K. Spurgeon





Walcott and Zak Observer

Introduction to a constructive sliding mode observer designframework

Potential of the discontinuous injection signal - lecture 3


Recap

Thus far the only restriction imposed on the nominal linearsystem is that the pair (A,C ) is observable.

Uncertainty and robustness has not been considered.



System Representation

x(t) = Ax(t) + Bu(t) + f (t, x , u)y(t) = Cx(t)

(1)

A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n and p ≥ m; in addition thematrices B and C are assumed to be of full rank.

The function f : R+ ×Rn ×Rm → Rn is unknown andrepresents the system uncertainty.

Special Case: matched uncertainty

f (t, x , u) = Bξ(t, x , u) (2)

where ξ(t, x , u) is unknown, but bounded

‖ξ(t, x , u)‖ ≤ r1‖u‖+ α(t, y) (3)

with r1 a known scalar and α(t, y) a known function.



System Representation

x(t) = Ax(t) + Bu(t) + f (t, x , u)y(t) = Cx(t)

(1)

A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n and p ≥ m; in addition thematrices B and C are assumed to be of full rank.

The function f : R+ ×Rn ×Rm → Rn is unknown andrepresents the system uncertainty.

Special Case: matched uncertainty

f (t, x , u) = Bξ(t, x , u) (2)

where ξ(t, x , u) is unknown, but bounded

‖ξ(t, x , u)‖ ≤ r1‖u‖+ α(t, y) (3)

with r1 a known scalar and α(t, y) a known function.Sarah K. Spurgeon Sliding mode observer


Problem Statement

Estimate the states of the uncertain system so that the errorsystem

e(t) = x(t)− x(t) (4)

is quadratically stable despite the presence of the uncertainty.

Assumption - Constrained Lyapunov Problem

There exists a G ∈ Rn×p such that A0 = A− GC has stableeigenvalues and there exists a Lyapunov pair (P,Q) for A0 suchthat the structural constraint

CTFT = PB (5)

is satisfied for some F ∈ Rm×p.



The Observer

˙x(t) = Ax(t) + Bu(t)− G (Cx(t)− y(t)) + P−1CTFTν (6)

ν =

−ρ(t, y , u) FCe

‖FCe‖ if FCe 6= 0

0 otherwise(7)

ρ(·) is any function satisfying

ρ(t, y , u) ≥ r1‖u‖+ α(t, y) + η (8)

for some positive scalar η.



Error System

e(t) = (A− GC )e(t)− Bξ(t, x , u) + Bν (9)

Quadratic Stability of the Error System

Consider V (e) = eTPe as a candidate Lyapunov function.

Evaluate the derivative along the system trajectories

V = eT(PA0 + A0P)e − 2eTPBξ + 2eTPBν (10)

≤ −eTQe − 2eTPBξ − 2ρ(t, y , u)‖FCe‖Using the structural constraint CTFT = PB:

V ≤ −eTQe − 2eTCTFTξ − 2ρ(t, y , u)‖FCe‖ (11)

≤ −eTQe − 2‖FCe‖(ρ(t, y , u)− ‖ξ‖)≤ −eTQe − 2η‖FCe‖



Error System

e(t) = (A− GC )e(t)− Bξ(t, x , u) + Bν (9)





≤ −eTQe − 2eTPBξ − 2ρ(t, y , u)‖FCe‖

Using the structural constraint CTFT = PB:





Error System

e(t) = (A− GC )e(t)− Bξ(t, x , u) + Bν (9)





≤ −eTQe − 2eTPBξ − 2ρ(t, y , u)‖FCe‖Using the structural constraint CTFT = PB:





Conclusion

There exists a domain in which a sliding motion is induced on thesurface in the state error space given by

Swz = e ∈ Rn : FCe = 0 (12)

A constructive result?

Relies on establishing whether there exists a gain matrix Gsuch that, for the resulting closed-loop matrix A0, there existsa Lyapunov matrix P for A0 satisfying CTFT = PB for someF ∈ Rm×p.

Walcott and Zak suggested the application of symboliccomputation tools.

Can the essence of their result be used to determine aconstructive framework?


Sliding mode observers for linear uncertain systems

Consider the uncertain dynamical system

x(t) = Ax(t) + Bu(t) + Dξ(t, y , u)y(t) = Cx(t)

(13)

where A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n and D ∈ Rn×q withp ≥ q. Assume that the matrices B,C and D are full rank and thefunction ξ : R+ ×Rn ×Rm → Rq is unknown but bounded.

Let (A,D,C ) represent the nominal part of (13). Define anobserver for the uncertain system (13)

z(t) = Az(t) + Bu(t)− GlCe(t) + Gnν (14)

where e = z − x , ν is discontinuous about the hyperplane

So = e ∈ Rn : Ce = 0 (15)

and Gl ,Gn ∈ Rn×p are gain matrices whose precise structure is tobe determined.


Sliding mode observers for linear uncertain systems

Consider the uncertain dynamical system

x(t) = Ax(t) + Bu(t) + Dξ(t, y , u)y(t) = Cx(t)

(13)

where A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n and D ∈ Rn×q withp ≥ q. Assume that the matrices B,C and D are full rank and thefunction ξ : R+ ×Rn ×Rm → Rq is unknown but bounded.Let (A,D,C ) represent the nominal part of (13). Define anobserver for the uncertain system (13)

z(t) = Az(t) + Bu(t)− GlCe(t) + Gnν (14)

where e = z − x , ν is discontinuous about the hyperplane

So = e ∈ Rn : Ce = 0 (15)

and Gl ,Gn ∈ Rn×p are gain matrices whose precise structure is tobe determined.


Sliding mode observers for linear uncertain systems -Existence Conditions

A sliding mode observer of the form (14 ) which rejects theuncertainty class in (13 ) exists if and only if the nominal linearsystem defined by the matrices (A,D,C ) satisfies

rank (CD) = q

any invariant zeros of (A,D,C ) must lie in C−.

For a square system the above two conditions require the triple(A,D,C ) to be relative degree one and minimum phase.These conditions depend upon a specific selection of uncertaintychannel and the observer design will be directly determined by theuncertainty distribution matrix.




rank (CD) = q


For a square system the above two conditions require the triple(A,D,C ) to be relative degree one and minimum phase.

These conditions depend upon a specific selection of uncertaintychannel and the observer design will be directly determined by theuncertainty distribution matrix.




rank (CD) = q


For a square system the above two conditions require the triple(A,D,C ) to be relative degree one and minimum phase.These conditions depend upon a specific selection of uncertaintychannel and the observer design will be directly determined by theuncertainty distribution matrix.


Sliding mode observers for linear uncertain systems -Canonical Form for Design

A change of coordinates exists so that the triple with respect tothe new coordinates (A, D, C ) has the following structure:

The system matrix can be written as

A =

A11 A12

A211

A212A22

(16)

where A11 ∈ R(n−p)×(n−p) and A211 ∈ R(p−q)×(n−p). Whenpartitioned these matrices have the structure

A11 =

[Ao11 Ao

12

0 Ao22

]and A211 =

[0 Ao

21

](17)

where Ao11 ∈ Rr×r and Ao

21 ∈ R(p−q)×(n−p−r) for some r ≥ 0and the pair (Ao

22,Ao21) is completely observable. Furthermore,

the eigenvalues of Ao11 are the invariant zeros of (A,D,C ).



The matrix distributing any forcing functions has the form

D =

[0D2

](18)

where D2 ∈ Rq×q is nonsingular.

The output distribution matrix has the form

C =[

0 T]

(19)

where T ∈ Rp×p and is orthogonal.



In order to ensure compatibility in the partition of the state-spacematrices, let

A21 =

[A211

A212

]and D =

[0D2

](20)

where D2 is defined as

D2 =

[0D2

]lp−qlq

(21)

The necessary and sufficient conditions for existence of a slidingmode observer together with the canonical form provide a pathwayto a constructive method for observer design.


Sliding mode observers - a pathway to design

As the invariant zeros are stable by assumption, Ao11 is stable.

As a consequence there exists a matrix L ∈ R(n−p)×(p−q) suchthat A11 + LA211 is stable.

Define a nonsingular transformation as

TL =

[In−p L

0 T

](22)

whereL =

[L 0(n−p)×q

]



After changing coordinates with respect to TL, the newoutput distribution matrix becomes

C = CT−1L =[

0 Ip]

From the definition of L and D2

LD2 =[L 0

] [ 0D2

]= 0

and so the uncertainty distribution matrix is given by

D = TLD =

[LD2

TD2

]=

[0

TD2

]Finally, if A = TLAT−1L , it can be shown by direct evaluationthat

A11 = A11 + LA211

which is stable by choice of L.




C = CT−1L =[

0 Ip]


LD2 =[L 0

] [ 0D2

]= 0


D = TLD =

[LD2

TD2

]=

[0

TD2

]

Finally, if A = TLAT−1L , it can be shown by direct evaluationthat

A11 = A11 + LA211





C = CT−1L =[

0 Ip]


LD2 =[L 0

] [ 0D2

]= 0


D = TLD =

[LD2

TD2

]=

[0

TD2

]Finally, if A = TLAT−1L , it can be shown by direct evaluationthat

A11 = A11 + LA211




The system triple (A, D, C ) is now in the form

x1(t) = A11x1(t) +A12y(t) + B1u(t)y(t) = A21x1(t) +A22y(t) + B2u(t) +D2ξ

(23)

where x1 ∈ R(n−p), y ∈ Rp and the matrix A11 is stable.

Define the corresponding observer by

˙x1(t) = A11x1(t) +A12y(t) + B1u(t)−A12ey (t)˙y(t) = A21x1(t) +A22y(t) + B2u(t)− (A22 −As

22)ey (t) + ν

where As22 is a stable design matrix and ey = y − y .



The system triple (A, D, C ) is now in the form


(23)

where x1 ∈ R(n−p), y ∈ Rp and the matrix A11 is stable.Define the corresponding observer by


22)ey (t) + ν




Let P2 ∈ Rp×p be symmetric positive definite Lyapunov matrix forAs

22 then the discontinuous vector ν is defined by

ν =

−ρ(t, y , u)‖D2‖ P2ey

‖P2ey‖ if ey 6= 0

0 otherwise(24)

where the scalar function ρ : R+ ×Rp ×Rm → R+ satisfies

ρ(t, y , u) ≥ r1‖u‖+ α(t, y) + γo (25)

and γo is a positive scalar.

If the state estimation error e1 = x1 − x1, then it is straightforwardto show

e1(t) = A11e1(t) (26)

ey (t) = A21e1(t) +As22ey (t) + ν −D2ξ (27)



Let P2 ∈ Rp×p be symmetric positive definite Lyapunov matrix forAs

22 then the discontinuous vector ν is defined by

ν =

−ρ(t, y , u)‖D2‖ P2ey


0 otherwise(24)

where the scalar function ρ : R+ ×Rp ×Rm → R+ satisfies

ρ(t, y , u) ≥ r1‖u‖+ α(t, y) + γo (25)

and γo is a positive scalar.If the state estimation error e1 = x1 − x1, then it is straightforwardto show

e1(t) = A11e1(t) (26)

ey (t) = A21e1(t) +As22ey (t) + ν −D2ξ (27)



Define Q1 ∈ R(n−p)×(n−p) and Q2 ∈ Rp×p as symmetric positivedefinite design matrices and define P2 ∈ Rp×p as the uniquesymmetric positive definite solution to the Lyapunov equation

P2As22 + (As

22)TP2 = −Q2 (28)

Let P1 ∈ R(n−p)×(n−p) be the unique symmetric positive definitesolution to the Lyapunov equation

P1A11 +AT11P1 = −(AT

21P2Q−12 P2A21 + Q1) (29)

Taking the quadratic form

V (e1, ey ) = eT1 P1e1 + eTy P2ey (30)

as a candidate Lyapunov function it can be shown that the errorsystem is quadratically stable.



Further, consideration of the quadratic form

Vs(ey ) = eTy P2ey (31)

shows that an ideal sliding motion takes place on (15) and theoutput error ey enters Ω = (e1, ey ) : ‖A21e1‖ < ‖D2‖γo − ηwhere η is a small positive scalar in finite time and remains there.


The Sliding Mode Observer

If x represents the state estimate for x and e = x − x then therobust observer can conveniently be written as

˙x(t) = Ax(t) + Bu(t)− GlCe(t) + Gnν (32)

where

Gl = T−1o

[A12

A22 −As22

](33)

Gn = ‖D2‖T−1o

[0Ip

](34)

and

ν =

−ρ(t, y , u) P2Ce

‖P2Ce‖ if Ce 6= 0

0 otherwise(35)

A key development in this formulation of the sliding mode observerdesign framework is that there is no requirement for (A,C ) to beobservable.


Tutorial Example

Pendulum System

x1 = x2

x2 = − sin(x1) (36)

The state and control matrices are

A =

[0 10 0

]and B =

[01

]The matched ‘uncertain’ bounded functionξ(t, x1, x2) = − sin(x1).

The output distribution matrix C =[

1 1].


Tutorial Example

Canonical Form Representation

Change coordinates with respect to Tc =

[1 01 1

]so that

the output is a state variable.

The system triple becomes

A = TcAT−1c =

[−1 1−1 1

]and

B = TcB =

[01

]and C = CT−1c =

[0 1

]A robust observer exists for this system because A11 = −1,which is stable. This is the transmission zero of the system.


Tutorial Example



[1 01 1

]so that



A = TcAT−1c =

[−1 1−1 1

]and

B = TcB =

[01

]and C = CT−1c =

[0 1

]

A robust observer exists for this system because A11 = −1,which is stable. This is the transmission zero of the system.


Tutorial Example



[1 01 1

]so that



A = TcAT−1c =

[−1 1−1 1

]and

B = TcB =

[01

]and C = CT−1c =

[0 1

]A robust observer exists for this system because A11 = −1,which is stable. This is the transmission zero of the system.


Tutorial Example

Observer design

Let the design matrix As22 = −1 so that λ(A0) = −1,−1.

Defining Q2 = 2 and solving the Lyapunov equation for As22

and Q2 gives P2 = 1.

In the original coordinates the gain matrices become

Gl =

[11

]and Gn =

[01

]

The observer becomes

d

dt

[x1x2

]=

[−1 0−1 −1

] [x1x2

]+

[11

]y +

[01

]ν


Tutorial Example

Demonstrates the nonlinear observer tracking the output fromthe pendulum when the initial conditions of the true statesand observer states are deliberately set to different values

-2

-1

0

1

2

0 2 4 6 8 10 12 14 16 18 20

Time, sec

Out

puts


Tutorial Example

A comparison of the true and estimated states.After approximately 4 seconds, visually perfect replication ofthe states is taking place.

-2

-1

0

1

2

0 2 4 6 8 10 12 14 16 18 20

Time, sec

Firs

t Sta

te

-2

-1

0

1

2

0 2 4 6 8 10 12 14 16 18 20

Seco

nd S

tate

Time, secSarah K. Spurgeon Sliding mode observer

Tutorial Example

If the nonlinear component is removed by setting ρ to zero,the resulting Luenberger Observer behaves as shown.There appears to be a distinct phase discrepancy between theoutputs of the system and the outputs of the observer; this isdue to the presence of the nonlinear sine term.

-4

-2

0

2

4

0 2 4 6 8 10 12 14 16 18 20

Time, sec

Out

put


Tutorial Example

The role of the applied discontinuous injection nuOn average this replicates the mismatch between the plantand the model assumed for observer designHow can we use this for practical applications - lecture 3

0 2 4 6 8 10 12 14 16 18 20−2

−1

0

1

2

Time,sec

nu

0 2 4 6 8 10 12 14 16 18 20−2

−1

0

1

2

Time,sec

filte

red

nu, s

in te

rm in

pla

nt


Sliding mode observers: a robust conditionmonitoring and fault detection tool

Sarah K. Spurgeon




Sliding Mode Observers


Principle of the equivalent injection - Fault detection andisolation

Fault detection and isolation - a case study

A sampled framework for practical application? - lecture 4



Recap

We have developed constructive frameworks for design

We have noted in the tutorial example that when an observerdesigned based on the dynamics of the double integrator isapplied to a nonlinear pendulum system, the discontinuoussignal when in the sliding mode, on average, reconstructs themismatch between the model used for design and the plantused for implementation


Sliding mode observers for fault detection and faultreconstruction

Historical Perspective

One of the first papers designed an observer so that theobserver error moves from the switching surface and slidingceases in the presence of a fault.

This approach is difficult to implement in practice - the choiceof gain to maintain sliding motion from the theory is oftenconservative and therefore it is difficult to ensure a faultinduces a break in sliding.

Observers, when exhibiting sliding motion, enable faultsand/or values of immeasurable system parameters to bereconstructed using the principle of the equivalent injectionsignal.


Sliding mode observers for fault detection and faultreconstruction

Nominal linear system subject to input/actuator and sensor faults

x(t) = Ax(t) + Bu(t) + Dfi (t) (1)

y(t) = Cx(t) + fo(t) (2)

A ∈ Rn×n, B ∈ Rn×m, C ∈ Rp×n, D ∈ Rn×q with q ≤ p < nand the matrices B,C and D are of full rank.

The function fi (t) represents an actuator fault.

fo(t) represents and sensor faults

It is assumed that the states of the system are unknown andonly the signals u(t) and y(t) are available.


The Observer

The design objective

To synthesise an observer to generate a state estimate x(t) suchthat a sliding mode is established in which the output error

ey (t) = y(t)− y(t) (3)

is forced to zero in finite time.

The observer structure

˙x(t) = Ax(t) + Bu(t)− Gley (t) + Gnν (4)

where Gl is a linear gain and Gn = T−1o

[0Ip

]with

ν =

−ρ(t, y , u)‖D2‖ P2ey


0 otherwise(5)


The Error Dynamics

Error dynamics in the canonical form used for observer design

e1(t) = A11e1(t) (6)

ey (t) = A21e1(t) +As22ey (t) + ν −D2fi (t) (7)


Fault Reconstruction

Input Signals

Consider initially the case when fo = 0.

Assume that an appropriate observer has been designed and asliding motion has been established so that ey = 0 and ey = 0.

In appropriate coordinates

0 = A21e1(t)−D2fi (t) + νeq (8)

where νeq is the equivalent injection applied during the slidingmode.

From the stability of the e1 subsystem, it follows thate1(t)→ 0 and therefore

νeq → D2fi (t) (9)



Input Signals

A commonly used approach to reconstruct the equivalentinjection is to use a low pass filter. Alternatively replace thediscontinuous injection by the continuous approximation

νδ = −ρ‖D2‖ P2ey‖P2ey‖+δ (10)

where δ is a small positive scalar.

The equivalent control can be approximated by (10) to anyrequired accuracy by a small enough choice of δ. Sincerank(D2) = q it follows that

fi (t) ≈ −ρ‖D2‖(DT2 D2)−1DT

2P2ey (t)

‖P2ey (t)‖+δ (11)

The equivalent injection signal can be computed on-line anddepends only on the output estimation error ey ; thus the faultfi can be approximated to any degree of accuracy.


Detection of faults at the output

Methodology

Assume fi = 0 and consider the effect of fo(t).

Since y(t) = Cx(t) + fo(t) it follows thatey (t) = Ce(t)− fo(t) and the state estimation error is givenby

e1(t) = A11e1(t) +A12fo(t) (12)

ey (t) = A21e1(t) +As22ey (t)− fo(t) +A22fo(t) + ν (13)

fo(t) and fo(t) are output disturbances; ρ must be chosensufficiently large to maintain sliding

During sliding 0 = A21e1 − fo(t) +A22fo(t) + νeq and forslowly varying faults

νeq ≈ −(A22 −A21A−111 A12)fo (14)


Output fault detection

Methodology

The equivalent control νeq can be calculated andconsequently, if (A22 −A21A−111 A12) is nonsingular, the faultsignal can be obtained from νeq ≈ −(A22 −A21A−111 A12)fo .

From the Schur expansion

det(A) = det(A11) det(A22 −A21A−111 A12) (15)

and thus (A22 −A21A−111 A12) is nonsingular if and only ifdet A 6= 0.

Even if (A22 −A21A−111 A12) is singular, useful informationabout sensor faults fo can still be potentially obtained,depending on the structure of the rank deficiency.


Example: Inverted Pendulum with a Cart

System

Assume the pendulum rotates in the vertical plane and thecart is to be manipulated so that the pendulum remains in anupright position.

The cart is linked by a transmission belt to a drive wheelwhich is driven by a DC motor.

θ

x



Equations of Motion

(M + m)x + Fx x + ml(θ cos θ − θ2 sin θ) = u (16)

J θ + Fθθ −mlg sin θ + mlx cos θ = 0 (17)

where the values of the physical parameters used are given by

Table : Model parameters for the inverted pendulum

M (kg) 3.2 Fx (kg/sec) 6.2m (kg) 0.535 Fθ (kg m2) 0.009J (kg m2) 0.062 g (m/sec2) 9.807l (m) 0.365


Linear model

Linearisation about the origin

A =

0 0 1 00 0 0 10 −1.9333 −1.9872 0.00910 36.9771 6.2589 −0.1738

B =

00

0.3205−1.0095

C =

1 0 0 00 1 0 00 0 1 0

(18)

Assumptions

x , θ, x and θ are the system states

A sliding mode control law is used to control the system

Actuator faults will occur in the input channel, hence the faultdistribution matrix D = B.


Numerical methods for design

Preliminaries

Consider (A,D,C ) and evaluate the size of the matrices

Change coordinates so the output distribution matrix is [0 I ]

Partition the input distribution matrix conformably

Matlab Commands

[nn,qq]=size(D); [pp,nn]=size(C)

nc = null(C); Tc=[nc’; C]; Ac=Tc*A*inv(Tc);Dc=Tc*D; Cc=C*inv(Tc);

Dc1=Dc(1:nn-pp,:); Dc2=Dc(nn-pp+1:nn,:);


Numerical methods for design

Current triple

Ac =

−0.1738 0 36.9771 6.2589

0 0 0 11 0 0 0

0.0091 0 −1.9333 −1.9872

Bc =

−0.0095

00

0.3205

Cc =

0 1 0 00 0 1 00 0 0 1

(19)

It is necessary to impose the required structure on Cc and Bc

Matlab Commands

Dc1=Dc(1:nn-pp,:); Dc2=Dc(nn-pp+1:nn,:);[T,temp]=qr(Dc2); T=(flipud(T’))’;

Tb=[eye(nn-pp) -Dc1*inv(Dc2’*Dc2)*Dc2’;zeros(pp,nn-pp) T’];

Aa=Tb*Ac*inv(Tb); Da=Tb*Dc; Ca=Cc*inv(Tb);Sarah K. Spurgeon Sliding mode observer

Numerical Methods for Design

Current Triple

Aa =

−0.1451 0 30.8877 −0.4568

0 0 0 11 0 0 3.1498

−0.0091 0 1.9333 −2.0159

Ba =

0

00

−0.3205

Ca =

0 − 1 0 00 0 1 00 0 0 −1

(20)

The system has no transmission zeros and thus the transformationto separate the unobservable modes is not requiredDetermine a gain matrix so that the top left sub-system has thedesired poles

Matlab Commands

tzero(Aa,Da,Ca,zeros(pp,qq))A22o=Aa(1:nn-pp,1:nn-pp);A21o=Aa(nn-pp+1:nn-qq,1:nn-pp);L=place(A22o’,A21o’,-10)’;


Numerical Methods for Design

Matlab Commands

Lbar=[-L zeros(nn-pp,qq)];TL=[eye(nn-pp) Lbar ; zeros(pp,nn-pp) eye(pp)];Acal=TL*Aa*inv(TL); Dcal=TL*Da; Ccal=Ca*inv(TL);

Final Form

A =

−10.0000 0 −67.6603 −31.4960

0 0 0 1.00001.0000 0 9.8548 3.1496−0.0091 0 1.8437 −2.0158

D =

000

−0.3205

C =

0 − 1 0 00 0 1 00 0 0 −1

(21)


Recap: Sliding mode observers - a pathway to design

System Triple

Given a system triple in the form


(22)

where x1 ∈ R(n−p), y ∈ Rp and the matrix A11 is stable.

Corresponding Observer


22)ey (t) + ν





A =

−10.0000 0 −67.6603 −31.4960

0 0 0 1.00001.0000 0 9.8548 3.1496−0.0091 0 1.8437 −2.0158

D =

000

−0.3205

C =

0 − 1 0 00 0 1 00 0 0 −1

(23)

Observer Design

By design A11 = −10

As22 = diag(−11,−12,−13) - the linear component of the

observer poles are approximately three times faster than theclosed-loop poles of the controlled plant.

The symmetric positive definite matrix P2 satisfiesP2A22 +AT

22P2 = −I

The scalar function ρ = 75Sarah K. Spurgeon Sliding mode observer

Nonlinear simulation testing


It can be verified that the eigenvalues of A are0, 5.8702, − 6.3965, − 1.6347 and the steady-state gainfrom fo to νeq is singular.

In fact

(A22 −A21A−111 A12) =

0 0 −10 −3.0888 00 1.9052 1.9872

(24)

which is clearly rank deficient.

If νeq,i and fo,i denote the ith components of νeq and fo inνeq ≈ −(A22 −A21A−111 A12)fo and using the distributionmatrix above

νeq,1 ≈ fo,3 (25)

νeq,2 ≈ 3.0888fo,2 (26)




It can be verified that the eigenvalues of A are0, 5.8702, − 6.3965, − 1.6347 and the steady-state gainfrom fo to νeq is singular.

In fact

(A22 −A21A−111 A12) =

0 0 −10 −3.0888 00 1.9052 1.9872

(24)

which is clearly rank deficient.

If νeq,i and fo,i denote the ith components of νeq and fo inνeq ≈ −(A22 −A21A−111 A12)fo and using the distributionmatrix above

νeq,1 ≈ fo,3 (25)

νeq,2 ≈ 3.0888fo,2 (26)Sarah K. Spurgeon Sliding mode observer



Any fault in the first output channel has no direct long-termeffect on νeq

Because of the structure of D2, it can be verified that

(DT2 D2)−1DT

2 = [ 0 0 3.1200 ]

and so from (9)νeq,3 ≈ 0.3205fi (27)

Three components of the equivalent control, properly scaled,provide estimates of fo,3, fo,2 and fi respectively and may beused as detector signals



Response of the detection signals to a fault in the input channel

As predicted by the theory, the third detector signal reproducesthe fault signal whilst not affecting the other two signals

0

0.2

0.4

0.6

0.8

1

0 5 10 15

Input Fault Signal

Time

-0.1

-0.05

0

0.05

0.1

0 5 10 15

Detector Signal 1

Time

-0.1

-0.05

0

0.05

0.1

0 5 10 15

Detector Signal 2

Time

0

0.5

1

0 5 10 15

Detector Signal 3

TimeSarah K. Spurgeon Sliding mode observer


Response to a fault in the first output channel

The detector signals do not reproduce the fault signal - thesecond signal approximates the gradient

0

0.02

0.04

0.06

0.08

0.1

0 5 10 15

Output Fault 1

Time

-0.1

-0.05

0

0.05

0.1

0 5 10 15

Detctor Signal 1

Time

-0.1

-0.05

0

0.05

0.1

0 5 10 15

Detector Signal 2

Time

0

0.5

1

0 5 10 15

Detector Signal 3



Response to a fault in the second output channel

The appropriate detector signal reproduces the ramp faultsignals in channel 2 - other channels are also affected

0

0.02

0.04

0.06

0.08

0.1

0 5 10 15

Output Fault 2

Time

-0.1

-0.05

0

0.05

0.1

0 5 10 15

Detector Signal 1

Time

-0.1

-0.05

0

0.05

0.1

0 5 10 15

Detector Signal 2

Time

0

0.5

1

0 5 10 15

Detector Signal 3



Response to a fault in the third output channel

The appropriate detector signal reproduces the ramp faultsignals in channel 3 - other channels are also affected

0

0.02

0.04

0.06

0.08

0.1

0 5 10 15

Output Fault 3

Time

-0.1

-0.05

0

0.05

0.1

0 5 10 15

Detector Signal 1

Time

-0.1

-0.05

0

0.05

0.1

0 5 10 15

Detector Signal 2

Time

0

0.5

1

0 5 10 15

Detector Signal 3


Sampling effects?

0 5 10 15 20−2

−1

0

1

2

time (s)

reconstructed fault signalinput fault signal

Figure : Fault reconstruction using the classical observer designed withno a priori knowledge of output sampling characteristics


Concluding remarks

Fault detection and isolation (FDI) using sliding mode observers

The principle of the equivalent injection is a strong result tounder pin the development of FDI schemes

Constructive design approach and can run in real time

There is a conflict between theory and practice in that processmeasurements may be sampled at a rate which is notquasi-continuous and the sampling can impact on the fidelityof the reconstruction


Sliding mode observers: implementation as acondition monitoring tool

Sarah K. Spurgeon






Parameter Estimation - an industrial case study

A sampled framework for the design of sliding mode observers



Recap

We have developed constructive frameworks for observerdesign

We have seen that sliding mode observers are a natural toolfor the development of FDI schemes

We have also seen via an example that sampling of the signalsused to drive the observer can affect the fidelity of thereconstruction


Fault detection via parameter monitoring - An Example

A sliding mode observer can be used to estimate or monitorthe variation in system parameters.

The development and implementation of a simple conditionmonitoring system for a high speed rotating machine will beconsidered.As such machines become more complex and valuable, there isa greater need to protect them, and the systems they support,from the consequences of breakdown. This is relevant in thesemiconductor industry, for example, where machine failurecan cause the loss of a valuable batch of wafers.Dry vacuum pumps have been successfully used in thesemiconductor industry but the harsh nature of manysemiconductor processes creates a challenge for conditionmonitoring. Any diagnostic scheme should be able to operateunder all conditions and be able to detect faults, creating analarm to warn the user that maintenance must beconveniently scheduled before catastrophic loss occurs.



A sliding mode observer can be used to estimate or monitorthe variation in system parameters.The development and implementation of a simple conditionmonitoring system for a high speed rotating machine will beconsidered.

As such machines become more complex and valuable, there isa greater need to protect them, and the systems they support,from the consequences of breakdown. This is relevant in thesemiconductor industry, for example, where machine failurecan cause the loss of a valuable batch of wafers.Dry vacuum pumps have been successfully used in thesemiconductor industry but the harsh nature of manysemiconductor processes creates a challenge for conditionmonitoring. Any diagnostic scheme should be able to operateunder all conditions and be able to detect faults, creating analarm to warn the user that maintenance must beconveniently scheduled before catastrophic loss occurs.



A sliding mode observer can be used to estimate or monitorthe variation in system parameters.The development and implementation of a simple conditionmonitoring system for a high speed rotating machine will beconsidered.As such machines become more complex and valuable, there isa greater need to protect them, and the systems they support,from the consequences of breakdown. This is relevant in thesemiconductor industry, for example, where machine failurecan cause the loss of a valuable batch of wafers.

Dry vacuum pumps have been successfully used in thesemiconductor industry but the harsh nature of manysemiconductor processes creates a challenge for conditionmonitoring. Any diagnostic scheme should be able to operateunder all conditions and be able to detect faults, creating analarm to warn the user that maintenance must beconveniently scheduled before catastrophic loss occurs.



A sliding mode observer can be used to estimate or monitorthe variation in system parameters.The development and implementation of a simple conditionmonitoring system for a high speed rotating machine will beconsidered.As such machines become more complex and valuable, there isa greater need to protect them, and the systems they support,from the consequences of breakdown. This is relevant in thesemiconductor industry, for example, where machine failurecan cause the loss of a valuable batch of wafers.Dry vacuum pumps have been successfully used in thesemiconductor industry but the harsh nature of manysemiconductor processes creates a challenge for conditionmonitoring. Any diagnostic scheme should be able to operateunder all conditions and be able to detect faults, creating analarm to warn the user that maintenance must beconveniently scheduled before catastrophic loss occurs.


Consider a model for monitoring cooling water flow in a dryvacuum pump in the absence of a flow transducer. Heat transferthrough the system is illustrated below:

Electrical Power Dry Vacuum Pump

Convection to

atmosphere

Coolant inlet Coolant

outlet

Figure : A schematic of heat transfer in the dry vacuum pump


A simple heat transfer model

The heat transfer model for the coolant can be expressed throughthe following pair of equations.

mBCB TB = KI (t)− mcCc (To − Ti )− hBAB (TB − Ta) (1)

where mB is the mass of the pump body, CB is the specific heatcoefficient of the pump body material, mc is the mass flow rate,hB is the convective heat transfer coefficient from the pump body,AB is the pump surface area, I is the pump inverter current, K is ascalar, Ta is the ambient temperature, TB is the temperature ofthe pump body, Ti and To are the coolant inlet and outlettemperatures respectively, and

mcCc To = −mcCc (TB − To) + hcAc (TB − To) (2)

where mc is the mass of the coolant, Cc is the specific heatcoefficient, hc is the convective heat transfer coefficient and Ac isthe coolant heat transfer area.


Parameterisation of the Model

In equations (1-2), the signals I , TB , Ta, Ti and To andparameters mC , mB , AB and Ac are routinely measurable.

Parameters Cc and CB are known and assumed constant,whilst nominal values of the parameters hB , hc , K and mc

can be approximated to parameterise the nominal operatingconditions of the pump.

Condition monitoring is effected by investigating changes inthe coolant mass flow rate, mc , and the heat transfercoefficient, hc . Defining mc = ˆmc + ∆mc and hc = hc + ∆hc

where ˆmc and hc are the nominal parameter valuesrepresenting normal system operation and ∆mc and ∆hc

represent deviations in the parameter values as could becaused by malfunction of the machine.


Sliding mode observer

Define a pair of sliding mode observers by

mBCBˆTB = KI (t)− ˆmcCc (To − Ti )− hBAB

(TB − Ta

)− β1eb + νb

mcCcˆTo = − ˆmcCc

(TB − To

)+ hcAc (TB − To)− β2eo + νo

where

ei = Ti − Ti

nui = Kiei‖ei‖

(3)

for i = o,B where β1, β2,Ki > 0.The parameters β1, β2 provide asymptotic error decay and thediscontinuous terms nui ensure a sliding mode is attained onei = 0 so that the output error signals between the observer andthe measurement are kept identically at zero.


Sliding mode analysis

Forming the error dynamics and noting that the sliding modecondition yields ei = ei = 0, it follows that

0 = ∆ ˆmcCc (To − Ti )− νb0 = ∆ ˆmcCc

(TB − To

)−∆hcAc (TB − To)− νo

The applied observer injection signals can be readily used tomonitor deviations in the nominal parameters.Note that this is a unique property of sliding mode observers thatcan only be realised by application of a discontinuous injectionsignal. Other observer formulations would not attain and maintainan identically zero output error but would seek to minimise theerror. This latter paradigm is not useful for parameterreconstruction.


Observer performance under the start-up phase of normaloperation of the dry vacuum pump

Figure : Observer performance under nominal operation of the dryvacuum pump


Observer performance in the presence of a changed massflow rate

Figure : Observer performance in the presence of a changed mass flowrate

A valve is used to reduce the mass of coolant flowing into thesystem at approximately 3200 seconds, and the correspondingchange in mass flow rate as predicted by the sliding mode observersystem


Fault reconstruction under output sampling - a time delayapproach

A sliding mode observer in the presence of sampled outputinformation and its application to fault reconstruction isstudied.

The observer is designed by using the delayed continuous-timerepresentation of the sampled-data system, for which a set ofLinear Matrix Inequalities (LMIs) provide conditions for theultimate boundedness, where the bound is proportional to thesampling time and the magnitude of the switching gain.

It is shown that, for a sufficiently small value of µ, aperturbation parameter, a transducer or sensor fault can bereconstructed reliably from the output error dynamics.


Problem Formulation

Consider the linear, time-invariant system with sampledoutputs

x(t) = Ax(t) + Bu(t) + Dfi (t)y(t) = Cxd(tk), tk ≤ t < tk+1

(4)

where x ∈ Rn, u ∈ Rm are the state and the input vectorrespectively and y ∈ Rp is a discrete-time outputmeasurement generated by zero-order hold functions with asequence of hold times 0 = t0 < t1 · · · < tk < · · · , wherelimk→∞ tk =∞.

fi ∈ Rq represents an unknown actuator fault which isassumed to be bounded by ‖fi (t)‖ ≤ ∆.

It is assumed q ≤ p < n and A, B, C , D are constantmatrices of appropriate dimensions.


Problem Formulation

Following the approach in Mikheev, Sobolev and Fridman andFridman, Seuret and Richard, system (4) with sampled output canbe presented as a continuous-time system with a known outputmeasurement delay

x(t) = Ax(t) + Bu(t) + Dfi (t)y(t) = Cx(t − τ(t)), t ∈ [tk , tk+1), τ(t) = t − tk

(5)

Assume that tk+1 − tk ≤ h, ∀ k ≥ 0, i.e. the time between anytwo sequential sampling times is not greater than some pre-chosenh > 0, then τ(t) ∈ (0, h] with τ(t) = 1 for t 6= tk is known. It isassumed that

1 rank (CD) = q.

2 any invariant zeros of (A,D,C ) lie in the left half plane.


Canonical form for design

Under these assumptions the system (5) can be transformedinto:

x1(t) = A11x1(t) + A12x2(t) + B1u(t)x2(t) = A21x1(t) + A22x2(t) + B2u(t) + D1fi (t)y(t) = Tx2(t − τ(t))

(6)

where x1 ∈ Rn−p, x2 ∈ Rp, D1 =

[0

D1

], D1 ∈ Rq×q, A11

has stable eigenvalues and T is an orthogonal matrix.

An observer will be designed which, for sufficiently large t,induces motion in the h∆-neighbourhood of the surface

E = x2, x2 ∈ Rp : se(t) = T(x2(t−τ(t))−x2(t−τ(t))

)= 0

(7)where x2(t − τ(t)) is the corresponding component of theestimated states from an observer.An ideal sliding mode can be achieved with h = 0 underassumptions 1, 2.





(6)


[0

D1

], D1 ∈ Rq×q, A11

has stable eigenvalues and T is an orthogonal matrix.An observer will be designed which, for sufficiently large t,induces motion in the h∆-neighbourhood of the surface


)= 0

(7)where x2(t − τ(t)) is the corresponding component of theestimated states from an observer.

An ideal sliding mode can be achieved with h = 0 underassumptions 1, 2.





(6)


[0

D1

], D1 ∈ Rq×q, A11

has stable eigenvalues and T is an orthogonal matrix.An observer will be designed which, for sufficiently large t,induces motion in the h∆-neighbourhood of the surface


)= 0

(7)where x2(t − τ(t)) is the corresponding component of theestimated states from an observer.An ideal sliding mode can be achieved with h = 0 underassumptions 1, 2.


The Observer

Consider the observer

˙x(t) = Ax(t) + Bu(t)− Gl e2(t − τ(t)) + Gnv(t − τ(t))y(t) = C xd(tk), tk ≤ t < tk+1

(8)where Gl ∈ Rn×p, Gn ∈ Rn×p and e2(t) = T

(x2(t)− x2(t)

).

The discontinuous injection term v is given by

v(t) = −(‖TD1‖+ δ)∆[sign e21(t), . . . , sign e2p(t)]T (9)

where δ > 0 is a positive number.


The Observer

Assume there exists L ∈ R(n−p)×p where L =[

L 0]

with

L ∈ R(n−p)×(p−q) such that a coordinate change T0 yields

˙x1(t) = A11x1(t) + A12x2(t) + B1u(t)−( 1

µL + A11L)(x2(t − τ(t))− x2(t − τ(t))) + LTT v(t − τ(t))˙x2(t) = A21x1(t) + A22x2(t) + B2u(t)−(A21L− 1

µ Ip)(x2(t − τ(t))− x2(t − τ(t))− TT v(t − τ(t))

y(t) = T x2(t − τ(t))(10)

where

Gl = T−10

[ 1µL + A11L

A21L− 1µIp

], Gn = T−1

0

[LTT

−TT

](11)

with µ > 0.


The error dynamics

Defining the state estimation error as e1(t) = x1(t)− x1(t) ande2(t) = x2(t)− x2(t), and performing a change of coordinates such

that

[e1(t)e2(t)

]= TL

[e1(t)e2(t)

]with TL =

[In−q L

0 T

]. Since

LD1 = 0, one obtains

˙e1(t) = (A11 + LA21)e1(t)− (A11L + LA21L− A12

−LA22)TT e2(t) + (A11 + LA21)e2(t − τ(t))(12)

˙e2(t) = TA21e1(t)− (TA21LTT − TA22TT )e2(t)+TA21LTT e2(t − τ(t))− 1

µ e2(t − τ(t))

+v(t − τ(t)) + TD1fi (t)

(13)

with initial condition

e(t0) = e0, e(t) = 0, t < t0 (14)


Boundedness of e1(t)

The dynamics of the switching manifold is governed by equation(12), where (A11,A21) is detectable from assumptions 1, 2.

Lemma

Given scalars α > 0, b > 0, if there exists an (n − p)× (n − p)matrix P > 0 and a matrix Y ∈ R(n−p)×p with last q columnszero, such that the LMI[

PA11 + AT11P + YA21 + AT

21YT + αP −P

∗ −bI

]< 0 (15)

holds, then the solution of (12) with L = P−1Y and with theinitial condition (14) is bounded by

eT1 (t)Pe1(t) < e−α(t−t0)eT1 (t0)Pe1(t0) + bα

(‖(A11L + LA21L

−A12 − LA22)TT‖2 + ‖A11 + LA21‖2)‖e2[t0,t]

(t)‖2∞

(16)


Input-to-state stability of the error dynamics: a singularperturbation approach

The closed-loop system (12), (13) can be expressed as

˙e1(t) = A11e1(t) + A12e2(t) + A11LTT e2(t − µξ(t)) (17)

µ ˙e2(t) = µA21e1(t) + µA22e2(t) + (µAd22 − Ip)·e2(t − µξ(t)) + µfi (t)

(18)

for appropriately defined A11, A12, A21, A22, Ad22 withµξ(t) = τ(t), µξ = h, 0 ≤ ξ(t) ≤ ξ andfi (t) = v(t − µξ(t)) + TD1fi (t), i.e.‖fi (t)‖ ≤

((‖TD1‖+ δ)

√p + ‖TD1‖

)∆.

Define positive definite Pµ ∈ Rn×n

Pµ =

[P1 µPT

3

∗ µP2

]> 0 (19)

where P1 ∈ Rn−p, and choose the Lyapunov-Krasovskii functional:

V (t) = e(t)TPµe(t) + (h − µξ(t))

∫ t

t−µξ(t)

eα(s−t) ˙e2(s)U ˙e2(s)ds (20)

for (17), (18), where U ∈ Rp is a positive matrix.Sarah K. Spurgeon Sliding mode observer

Lemma

Given positive scalars µ, ξ, α and b, let there exist a n × n matrix Pµ > 0 in(19), p × p matrices U > 0, P4, P5 and (n − p)× (n − p) matrices P6, P7

such that the following LMIs

Θµ0 =

θ11 θ12 θ13

∗ θ22 θ23

∗ ∗ −PT7

∗ ∗ ∗∗ ∗ ∗

µAT21P5 + µPT

3 0θ24 PT

4

0 0−µPT

5 + µξU PT5

∗ −bI

< 0 (21)

Θµ1 =

θ11 θ12 θ13 µAT21P5 + µPT

3 −µξPT6 A11LT

T 0∗ θ22 θ23 θ24 −µξPT

4 (µAd22 − Ip) PT4

∗ ∗ −PT7 0 −µξPT

7 A11LTT 0

∗ ∗ ∗ −µPT5 −µξPT

5 (µAd22 − Ip) PT5

∗ ∗ ∗ ∗ −µξe−αµξU 0∗ ∗ ∗ ∗ ∗ −bI

< 0

(22)are feasible, then solutions of (12)-(13) with initial condition (14) satisfy

eT (t)Pµe(t) < e−α(t−t0)eT (t0)Pµe(t0) + µ2 bα‖fi [t0,t]‖2

∞ (23)

for all µξ(t) ∈ [0, h] with µξ(t) = 1, thus (12)-(13) is input-to-state stable.Sarah K. Spurgeon Sliding mode observer

LMIs for switching gain design

Conditions will now be derived that guarantee the followingbounds:

lim supt→∞ ‖[

A21 A22

]e(t)‖ ≤ k1δ∆,

lim supt→∞ ‖[

0 Ad22

]e(t − τ(t))‖ ≤ k2δ∆

(24)

with some k1, k2 ≥ 0 such that k1 + k2 = 1.(24) holds if the following inequalities are satisfied for t →∞:

µ2eT (t)[A21 A22]T [A21 A22]e(t) <αeT (t)Pµe(t)k2

1 δ2

b(

(‖TD1‖+δ)√p+‖TD1‖

)2

µ2eT (t − µξ(t))[0 Ad22]T [0 Ad22]e(t − µξ(t))

<αeT (t−µξ(t))Pµe(t−µξ(t))k2

2 δ2

b(

(‖TD1‖+δ)√p+‖TD1‖

)2


LMIs for switching gain design

Hence, the inequalities −k21 M1P1 −µk2

1 M1PT3 µAT

21

∗ −µk21 M1P2 µAT

22

∗ ∗ −Ip

< 0 −k22 M1P1 −µk2

2 M1PT3 0

∗ −µk22 M1P2 µAT

d22

∗ ∗ −Ip

< 0

(25)

where M1 = αδ2

b(

(‖TD1‖+δ)√p+‖TD1‖

)2 , guarantee that the solutions

of (12), (13) satisfy the bound (24).


Ultimate boundedness of the error dynamics

Theorem

Given positive constants µ, ξ, α, b and k1, k2, let there exist an × n-matrix Pµ > 0, positive p × p-matrices U > 0 and p × pmatrices P4, P5, (n − p)× (n − p) matrices P6, P7 such thatLMIs (19), (21), (22) and (25) are feasible. Let e(t) be a solutionto (12), (13), then every component of e2(t) satisfies the bound

lim supt→∞

|e2i (t)| ≤ 2M0µξ (26)

where M0 = 2(δ + ‖TD1‖)∆, i = 1, . . . , p denotes the i-thcomponent of e2 for all µξ(t) ∈ [0, h] with µξ(t) = 1.


Reconstruction of the input and output fault signals

For sufficiently small µ

0 ≈ − 1

µe2(t − τ(t)) + v(t − τ(t)) + TD1fi (t) (27)

Since rank(D1) = q it follows from (27) that

fi (t) ≈((TD1)T (TD1)

)−1(TD1)T

( 1

µe2(t − τ(t))− v(t − τ(t))

(28)To reconstruct the fault signal fi , replace the discontinuouscomponent v(t) by the continuous approximation

vr = −(‖TD1‖+ δ)∆[e21

|e21 |+ r, . . . ,

e2p

|e2p |+ r]T (29)

where r ≥ 0 is chosen to be small enough.


Reconstruction of the input and output fault signals

Now consider the case when fi = 0 and consider the effect of afault f0(t) at the output. In this situation, x2 is replaced byx2 → x2 + f0 and ey = e2 + f0. For sufficiently small µ, it can beobtained that

e1(t) ≈ −A−111 ( L

µ + A11L)f0

ey (t) = A21e1(t) + A22ey (t)− A22f0

+(A21L− Ipµ )ey (t − τ(t)) + TT v(t − τ(t)) + f0

(30)

The fault can be approximated by

f0 ≈W−1((A22L− Ip

µ )ey (t − τ(t)) + TT v(t − τ(t)))

(31)

if W = A21A−111 ( L

µ + A11L) + A22 is invertible.


Simulation Example

An inverted pendulum system is considered which is linearizedabout the equilibrium at the origin

A =

0 0 1 00 0 0 10 −1.9333 −1.9872 0.00910 36.9771 6.2589 −0.1738

,

B =

00

0.3205−1.0095

, C =

1 0 0 00 1 0 00 0 1 0

(32)

A compensator approach is designed to stabilize the pendulum.It is assumed that D = B and an input fault is bounded by‖fi‖ ≤ ∆ = 2. The sampled data outputs are implemented in thesimulation using the zero-order-hold function.


Example continued

In the LMI (15), L = [0 1.526 0] is obtained. LMIs (21) and (22)are feasible with α = 8, µ = 0.019, ξ = 0.524, i.e. the samplingperiod is given by µξ = 0.01s. LMI (25) is feasible with δ = 77and k1 = 0.8, k2 = 0.2. Hence the observer (8) with gains in (11)and (9) has been chosen which ensures the error variable isbounded in the range |e2i (t)| ≤ 6.2 according to the estimate (26).


Example continued

This figure is plotted using the sign function where every errorvariable is stabilized into a bound |e2i | ≤ 1.1, which is within theestimate.Note that the high degree of switching is acceptable for anobserver error signal; this is not present in the reconstruction ofthe fault signals.

0 2 4 6 8 10 12 14 16 18 20−2

0

2

e2

1

0 2 4 6 8 10 12 14 16 18 20−10

0

10

e2

2

0 2 4 6 8 10 12 14 16 18 20−2

0

2

time (s)

e2

3

Figure : Error response e2 with sampling period h = 0.01s at outputs.


Example continued

Suppose the input fault is fi (t) = 2sin(t), while the output faultfo = 0. The fault is reconstructed in Figure 30 according to (28).

0 5 10 15 20−2

−1

0

1

2

time (s)


Figure : Input fault reconstruction


Example continued

From the fault distribution structure

W =

0 0 10 −4.77 −3.150 −1.97 −2.01

only the third output fault can be reconstructed. This can beapproximated as the equivalent injection signal at the firstchannel of (A22L− Ip

µ )ey (t − τ(t)) + TT v(t − τ(t)).

Suppose the third output fault f03 = 5sin(t), the fault signal isreconstructed accurately under the sampled output withsampling period h = 0.01s.

0 5 10 15 20−5

0

5

time (s)

reconstructed fault signaloutput fault signal

Figure : Output fault reconstruction in the third component of theoutput vector


Example continued

From the fault distribution structure

W =

0 0 10 −4.77 −3.150 −1.97 −2.01

only the third output fault can be reconstructed. This can beapproximated as the equivalent injection signal at the firstchannel of (A22L− Ip

µ )ey (t − τ(t)) + TT v(t − τ(t)).Suppose the third output fault f03 = 5sin(t), the fault signal isreconstructed accurately under the sampled output withsampling period h = 0.01s.

0 5 10 15 20−5

0

5

time (s)

reconstructed fault signaloutput fault signal

Figure : Output fault reconstruction in the third component of theoutput vector


Example continued

The fault reconstruction is good under the sampled output withsampling period h = 0.01s used for design. The observer preservesthe construction accuracy even when it is operating under a largersampling time h = 0.03s as seen below.

0 5 10 15 20−2

−1

0

1

2

time (s)


Figure : Fault reconstruction using the proposed observer approach; theoutput sampling period is greater than that assumed in the design


Example continued

0 5 10 15 20−2

−1

0

1

2

time (s)


Figure : Fault reconstruction using the classical observer designed withno a priori knowledge of output sampling characteristics


Conclusions

The use of sliding mode observers in estimating faults andunknown parameters has been demonstrated.

A constructive fault reconstruction technique has beendemonstrated whereby the fault can be reconstructed reliablywhen the measured output is sampled.


Sliding mode observers: higher order slidingmodes, complex systems and nonlinearity

Sarah K. Spurgeon






Sliding mode observer design for nonlinear systems - historicalperspective

Higher order sliding modes

Complex systems - decentralisation vs centralisation and theimpact on the observer design paradigm



Recap

We have developed constructive frameworks for observerdesign, particularly when systems can be approximated by alinear state-space model

We have yet to see translation of the core ideas into acomplex, system of system framework

We have not said anything about the implications of higherorder sliding modes on the observer paradigm


Nonlinear sliding mode observers - historical perspective

Nonlinear system in companion form

x (n) = f (x , t) (1)

where f (x , t) is a nonlinear, uncertain function and x1 is themeasurement.

Slotine’s Observer

˙x1 = −α1e1 + x2 − k1sgn(e1) (2)˙x2 = −α2e1 + x3 − k2sgn(e1) (3)

.. .. .. (4)˙xn = −αne1 + f − knsgn(e1) (5)

αi chosen as for a Luenberger observer to ensure asymptoticerror decay when ki = 0

e1 = x1 − x1, f is an estimate of f (x , t)


Slotine Nonlinear Observer

Error Dynamics

e1 = −α1e1 + e2 − k1sgn(e1) (6)

e2 = −α2e1 + e3 − k2sgn(e1) (7)

.. .. .. (8)

en = −αne1 + ∆f − knsgn(e1) (9)

∆f = f − f is assumed bounded and kn ≥| ∆f |

The sliding condition ddt (e1)2 < 0 is satisfied in the region

e2 ≤ k1 + α1e1 if e1 > 0

e2 ≥ −k1 + α1e1 if e1 < 0

Now revisit the error dynamics imposing the sliding condition.



Error Dynamics

e1 = −α1e1 + e2 − k1sgn(e1) (6)

e2 = −α2e1 + e3 − k2sgn(e1) (7)

.. .. .. (8)

en = −αne1 + ∆f − knsgn(e1) (9)

∆f = f − f is assumed bounded and kn ≥| ∆f |The sliding condition d

dt (e1)2 < 0 is satisfied in the region

e2 ≤ k1 + α1e1 if e1 > 0

e2 ≥ −k1 + α1e1 if e1 < 0

Now revisit the error dynamics imposing the sliding condition.



Sliding Mode Dynamics when e1 = 0

It follows from the e1 dynamic equation that

e2 − k1sgn(e1) = 0

and therefore

e2 = e3 −k2

k1e2

.. .. ..

en = ∆f − knk1

e2

What is the role of the observer gains?


Slotine Nonlinear observer

The sliding patch

The αi only affect the dynamic performance prior to the reachingof the so-called sliding patch - patch dynamics are∣∣∣∣∣∣∣∣∣∣∣

λIn−1 −

−k2

k11 0 ... 0

−k3k1

0 1 ... 0

.... 1

−knk1

0 0 ... 0

∣∣∣∣∣∣∣∣∣∣∣= 0 (10)

Assuming kn is selected as a constant ratio with k1 and that thepoles defining the dynamics on the patch are critically damped i.e.are real and equal to some constant value γ, then∣∣∣e(i)

2

∣∣∣ ≤ (2γ)ik1 i = 0, ..., n − 2 (11)

from which the precision of the state estimates can be determined.


Slotine Nonlinear observer

The sliding patch

The αi only affect the dynamic performance prior to the reachingof the so-called sliding patch - patch dynamics are∣∣∣∣∣∣∣∣∣∣∣

λIn−1 −

−k2

k11 0 ... 0

−k3k1

0 1 ... 0

.... 1

−knk1

0 0 ... 0

∣∣∣∣∣∣∣∣∣∣∣= 0 (10)

Assuming kn is selected as a constant ratio with k1 and that thepoles defining the dynamics on the patch are critically damped i.e.are real and equal to some constant value γ, then∣∣∣e(i)

2

∣∣∣ ≤ (2γ)ik1 i = 0, ..., n − 2 (11)

from which the precision of the state estimates can be determined.Sarah K. Spurgeon Sliding mode observer

Nonlinear observers- Further historical developments

Further Developments of Slotine

The effect of measurement noise on sliding mode observerswas formulated.

The system does not attain a sliding mode in the presence ofnoise, but remains within a region of the sliding patchdetermined by the bound on the noise.

Moreover, it was demonstrated that the average dynamics canbe modified by selection of the ki which in turn can tailor thecontribution of the noise to the state estimates.

Next steps

The equivalent injection design concept - Drakunov

Important as it is possible to develop an observer withoutusing input derivatives

Further developed by several authors


Nonlinear step-by-step observer design

Nonlinear system in triangular input form

ξ1 = ξ2 + g1(ξ1, u)

ξ2 = ξ3 + g2(ξ1, ξ2, u)

.. = ..

ξn−1 = ξn + gn−1(ξ1, ξ2, ..., ξn−1, u)

ξn = fn(ξ1, ..., ξn) + gn(ξ1, ..., ξn, u)

where y = ξ1, gi (., 0) = 0 for i = 1, ..., n and the system isassumed bounded input, bounded state in finite time.



The Observer

˙ξ1 = ξ2 + g1(ξ1, u) + λ1sign(ξ1 − ξ1)

˙ξ2 = ξ3 + g2(ξ1, ξ2, u) + λ2sign(ξ2 − ξ2)

.. = ..˙ξn−1 = ξn + gn−1(ξ1, ξ2, ..., ξn−1, u) + λn−1sign(ξn−1 − ξn−1)

ξn = fn(ξ1, ξ2, ..., ξn) + gn(ξ1, ..., ξn, u) + λnsign(ξn − ξn)

where for i = 2, ...., n − 1

ξi = ξi + λi−1sign(ξi−1 − ξi−1) (12)

Observation error information is not used before thecorresponding sliding manifold is reached

The manifolds are reached sequentially and ξi − ξi convergesto zero if the ξj − ξj with j < i have already converged to zero.



The error dynamics

e1 = e2 − λ1sign(ξ1 − ξ1)

e2 = e3 + g2(ξ1, ξ2, u)− g2(ξ1, ξ2, u)− λ2sign(ξ2 − ξ2)

.. = ..

en−1 = ξn − gn−1(ξ1, ξ2, ..., ξn−1, u)− λn−1sign(ξn−1 − ξn−1)

en = fn(ξ1, ..., ξn)− fn(ξ1, ξ2, ..., ξn) + gn(ξ1, ξ2, ..., ξn−1, u)−gn(ξ1, ..., ξn, u)− λnsign(ξn − ξn)

For sufficiently large λ1, a sliding mode is attained on e1 = 0 infinite time so that

e2 = λ1sign(ξ1 − ξ1)

which with (12) yields ξ2 = ξ2.



Revised error dynamics

e1 = 0

e2 = e3 − λ2sign(ξ2 − ξ2)

.. = ..

en−1 = ξn − gn−1(ξ1, ξ2, ..., ξn−1, u)− λn−1sign(ξn−1 − ξn−1)

en = fn(ξ1, ..., ξn)− fn(ξ1, ξ2, ..., ξn) + gn(ξ1, ξ2, ..., ξn−1, u)−gn(ξ1, ..., ξn, u)− λnsign(ξn − ξn)

Proceeding as before it can be shown that for sufficiently large λ2,a sliding mode is then attained on e2 = 0 in finite time and itfollows that

e3 = λ2sign(ξ2 − ξ2)

which yields ξ3 = ξ3.



Error dynamics at the nth stage

Proceeding sequentially

e1 = 0

e2 = 0

.. = ..

en−1 = 0

en = −λnsign(ξn − ξn)

and it follows trivially that a sliding mode is finally attained onen = 0 in finite time.

Many application specific studies of this step-by-step observerframework appear in the literature

The results were generalised to the MIMO case by Floquetand co-workers.


Step-by step observer design and higher order slidingmodes

Step-by-step observer

The step-by-step procedure uses successive filtered values ofthe so-called equivalent output injections obtained fromrecursive first order sliding mode observers

The approximation of the equivalent injections by low passfilters at each step will typically introduce some delays thatlead to inaccurate estimates or to instability for high ordersystems

To overcome this problem, the discontinuous first order slidingmode output injection can be replaced by a continuous secondorder sliding mode injection


Higher order sliding modes

The concept of Higher Order Sliding Modes (HOSM)generalise the sliding mode concept so that the discontinuityacts on higher order derivatives of the sliding variable and theapplied injection is smooth

In general, if the control appears on the rth derivative of s,the rth order ideal sliding mode is defined by:

s = s = s = .... = s(r−1) = 0 (13)

A second order sliding mode injection - The super twistingalgorithm

ν(s) = φ(s) + λs |s|12 sign(s)

φ(s) = αssign(s)

λs , αs > 0


Decentralised observation

Many results consider observer design for interconnected systemsbut very few adopt a decentralised approach.


Decentralised observer design for nonlinear interconnectedsystems

Nonlinear interconnected system with n ni -th order subsystems

xi = fi (xi ) + gi (xi ) (ui + ξi (t, xi )) + ψi (x) (14)

yi = hi (xi ), i = 1, 2, . . . , n, (15)

x ⊂ Rni , ui ⊂ Rmi and yi ⊂ Rmi are the states, inputs andoutputs of the i-th subsystem

‖ξi (t, xi )‖ ≤ γξi , i = 1, 2, . . . , n

for some positive constants γξi .

The terms ψi (x) are interconnections of the i-th subsystem.

The control signals are bounded: ‖ui‖ ≤ γui



Design Objective

To design n ni -th order dynamical systems

˙xi = Φi (t, xi , yi , ui ), i = 1, 2, . . . , n (16)

where xi ∈ Rni , such that the solutions xi (t) of system (16) areconvergent to xi (t) exponentially for i = 1, 2, . . . , n, that is, thereexist constants αi > 0 and βi > 0 such that

‖xi (t)− xi (t)‖ ≤ αi exp−βi t, i = 1, 2, . . . , n

where xi (t) are the solutions of the interconnected systems(14)–(15).

The systems in (16) comprise an exponential observer for theinterconnected system (14)–(15).



Design Objective

To design n ni -th order dynamical systems

˙xi = Φi (t, xi , yi , ui ), i = 1, 2, . . . , n (16)

where xi ∈ Rni , such that the solutions xi (t) of system (16) areconvergent to xi (t) exponentially for i = 1, 2, . . . , n, that is, thereexist constants αi > 0 and βi > 0 such that

‖xi (t)− xi (t)‖ ≤ αi exp−βi t, i = 1, 2, . . . , n

where xi (t) are the solutions of the interconnected systems(14)–(15).The systems in (16) comprise an exponential observer for theinterconnected system (14)–(15).



i-th isolated subsystem of the interconnected system

xi = fi (xi ) + gi (xi )(ui + ξi (t, xi ))

yi = hi (xi ), i = 1, 2, . . . , n,

i-th nominal isolated subsystem of the interconnected system

xi = fi (xi ) + gi (xi )ui (17)

yi = hi (xi ), i = 1, 2, . . . , n, (18)



Definition 1

The i-th nominal isolated subsystem has a uniform relative degreevector

(ρi1, ρi2, · · · , ρimi)

and the distribution Gi (xi ) = spangi1(xi ), .., gimi(xi ) is involutive

in the domain Xi for i = 1, 2, . . . , n.

Note that there is no requirement for the isolated subsystems to belinearisable - only this relative degree requirement



Exploiting structure - Canonical Form for Design 1

Let ρi :=∑mi

j=1 ρij for i = 1, 2, . . . n. From Assumption 1, the

differentials dhij(xi ), dLfihij(xi ), · · · , dLρij−1fi

hij(xi ) are linearlyindependent for j = 1, . . . ,mi and i = 1, . . . , n. Let

zij =

hij(xi )

Lfihij(xi )...

Lρij−1fi

hij(xi )

:= zij(xi ), j = 1, 2, . . . ρi (19)

for i = 1, . . . , n. Since the Gi (xi ) are involutive, there exist ni − ρifunctions wi1, · · · , wi(ni−ρi ) such that the Jacobian matrix of thefollowing mapping is nonsingular:

Ti : xi 7→ col(zi1, · · · , ziρi ,wi1, · · · ,wi(ni−ρi )) (20)Sarah K. Spurgeon Sliding mode observer


Exploiting structure - Canonical Form for Design 2

The transformations col(zi ,wi ) = Ti (xi ) are diffeomorphisms. Let

T (x) :=

T1(x1)T2(x2)

...Tn(xn)

(21)

It is clear that T (x) defines a new coordinate system col(z1, w1,z2,w2, · · · , zn,wn).



Assumption 2 - required for exponential stability of the errordynamics

The interconnection terms ψi (x) satisfy the following

Lψi (x)hij(xi ) = 0 (22)

Lψi (x)Lfi (xi )hij(xi ) = 0 (23)

· · · · · ·Lψi (x)L

ρij−2fi (xi )

hij(xi ) = 0 (24)

there exist constants γψisuch that for any x ,∥∥∥Lψi (x)L

ρij−1

fi (xi )hij(xi )

∥∥∥ ≤ γψi,[

∂Ti (xi )∂xi

ψi (x)]xi=T−1

i (zi ,wi )=

[?

Φi (zi ,wi )

]where

Φi (·) ∈ R(ni−∑m

j=1 ρij ) are Lipschitz with respect to wi

uniformly for zi Sarah K. Spurgeon Sliding mode observer


Canonical Form

Under Assumptions 1-2, the interconnected system becomes

zi1 = Ai1zi1 + Bi1

(ui1 + ηij(t, zi ) + Lψi (x)L

ρi1−1fi (xi )

hi1(xi ))

(25)

zi2 = Ai2zi1 + Bi2

(ui2 + ηij(t, zi ) + Lψi (x)L

ρi2−1fi (xi )

hi2(xi ))

(26)

...

zimi = Aimi zimi + Bimi

(uimi + ηij(t, zimi ) + Lψi (x)L

ρimi−1

fi (xi )himi (xi )

)(27)

wi = qi (zi ,wi ) + Φi (zi ,wi ) (28)

yij = Cijzij (29)

where zi := col (zi1, zi2, · · · , zimi) with zij ∈ Rρij and

wi := col(wi1,wi2, · · · ,wi(ni−ρi )

)∈ Rni−ρi , the triples

(Aij ,Bij ,Cij) have the Brunovsky standard form



HOSM-based decentralised observer design

The i-th subsystem of (25)-(27) can be written as

zij = Aijzi + Bij

(uij + ηij(t, zi ) + Lψi (x)L

ρij−1fi (xi )

hij(xi )), (30)

yij(t) = zij1(t) := hij(xi (t)) (31)

Consider the higher order sliding surfaces defined by

sij = Lgi sij = L2gisij = · · · = L

ρijgi sij = 0

where sij(t) := zij1(t)− yij(t) and zij1 is determined by a HOSMdifferentiator scheme



HOSM Differentiator

˙zij1 = νij1 (32)

νij1 = −λij1 |sij |ρij−1

ρij sign(sij) + zij1 (33)

...˙zij(ρij−1) = νij(ρij−1) (34)

νij(ρij−1) = −λij(ρij−1) |zij(ρij−1) − νij(ρij−2)|12

sign(zij(ρij−1) − νij(ρij−2)) + zijρij (35)

˙zijρij = −λijρij sign(zijρij − νij(ρij−1)) (36)

where λijk are positive parameters for i = 1, 2, . . . , n,j = 1, 2, . . . ,mi and k = 1, 2, . . . , ρij .



Remarks

It follows that by choosing appropriate parameters λijk , zijk

will converge to the k − th derivative of yij(t), y(k)ij in finite

time Tij .

Choose T0 > Tij and let zij := col(zij1, zij2, · · · , zijρij

)for

j = 1, 2, . . . ,mi and i = 1, 2, . . . , n. Considering the structureof (Aij ,Bij) in (30)–(31), when t ≥ T0,

zi1 = zi1, zi2 = zi2, · · · , zimi= zimi

for i = 1, 2, . . . n.

Consider the interconnected system. The analysis aboveshows that zij produced by the differentiator (32)–(36), is anestimate of zij . The objective now is to estimate the variableswi for which the following assumptions are imposed.



Assumption 3

The nonlinear functions qi (zi ,wi ) satisfy the following

i) qi (zi ,wi ) are Lipschitz with respect to the variables wi

uniformly for zi ;

ii) there exist Pi > 0 (Pi ∈ R(ni−ρi )×(ni−ρi )) and positivefunctions ki (zi ) such that for any variables ϑi ∈ Rni−ρi , zi and

wi ϑTi Pi

∂qi (zi ,wi )∂wi

ϑi ≤ −ki (zi )‖ϑi‖2 where ∂qi (zi ,wi )∂wi

denote theJacobian matrices of qi (·) with respect to the variables wi .

Remarks

Condition ii), Assumption 2 and condition i), Assumption 3 arefundamental in the local case and hold in any bounded compact set. If

the matrix ∂qi (zi ,wi )∂wi

at col(zi ,wi ) = 0 is Hurwitz, then condition ii) inAssumption 3 holds in a neighbourhood of the origin.



The Observer

˙wi = qi (zi , wi ) + Φi (zi , wi ), i = 1, 2, · · · , n (37)

where zi := (zi11, zi12, · · · , zi1ρi1 , · · · , zimi1, · · · , zimiρi1) and zijkare given by (32)–(36). Clearly the n systems defined in (37) aredecoupled from each other. Let ei (t) = wi (t)− wi (t). It followsfrom Assumption 2 that the error dynamics are described by

ei = qi (zi ,wi )− qi (zi , wi ) + Φi (zi ,wi )− Φj(zi , wi ) (38)

for i = 1, 2, . . . , n.



Theorem

Under Assumptions 1–3, the error dynamical system (38) isexponentially stable if for i = 1, 2, . . . , n

infzi

ki (zi )− ‖Pi‖LΦi

(zi )

:= βi > 0 (39)

where Pi and ki (zi ) satisfy Assumption 3 and Φi (zi ) are given inAssumption 2.



Remarks

From Levant HOSM algorithm, the variables zi converge to ziin finite time. Theorem 1 shows that the variables wi

converges to w exponentially. The HOSM differentiator andthe designed dynamics together form the exponential observer.

From the structure of the designed dynamical system and thestructure of Levant differentiator, it is straightforward to seethat the designed dynamics are decoupled and thus thedesigned observers are decentralised.



Concluding Remarks

We have developed an exponential observer which exploitsstructure in the design including mismatched uncertainty

HOSM and sliding mode differentiators have been introduced

Demonstrated that the sliding mode observer paradigmtranslates to nonlinear and complex system scenarios


sliding mode observers - historical background and basic introduction

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