Sliding mode observers - Spring School : sliding mode control

Sliding mode observers - historical background and basic introduction Sarah K. Spurgeon School of Engineering and Digital Arts University of Kent, UK

Spring School, Aussois, June 2015

Sarah K. Spurgeon

Sliding mode observer

Outline of Presentation

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Sliding mode control versus sliding mode observers?

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Sliding mode control versus sliding mode observers? Historical perspective - The Utkin Observer

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Sliding mode control versus sliding mode observers? Historical perspective - The Utkin Observer Tutorial Example

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Sliding mode control versus sliding mode observers? Historical perspective - The Utkin Observer Tutorial Example Further historical milestones

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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

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Introduction - The Control Problem

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

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Sliding mode techniques were perhaps originally best known for their potential as a robust control method, and evolved from pioneering work in the 1960’s in the former Soviet Union. Such a sliding mode control is characterised by a suite of feedback control laws and a decision rule. The decision rule, termed the switching function, has as its input some measure of the current system behaviour and produces as an output the particular feedback controller which should be used at that instant in time.

Sarah K. Spurgeon



Sliding mode techniques were perhaps originally best known for their potential as a robust control method, and evolved from pioneering work in the 1960’s in the former Soviet Union. Such a sliding mode control is characterised by a suite of feedback control laws and a decision rule. The decision rule, termed the switching function, has as its input some measure of the current system behaviour and produces as an output the particular feedback controller which should be used at that instant in time. In sliding mode control, VSCS are designed to drive and then constrain the system state to lie within a neighbourhood of the switching function.

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Introduction - The Control Problem There are a number of advantages to this approach. Firstly, the dynamic behaviour of the system may be tailored by the particular choice of switching function.

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Introduction - The Control Problem There are a number of advantages to this approach. Firstly, the dynamic behaviour of the system may be tailored by the particular choice of switching function. Secondly, the closed-loop response becomes totally insensitive to a particular class of uncertainty in the system.

Sarah K. Spurgeon


Introduction - The Control Problem There are a number of advantages to this approach. Firstly, the dynamic behaviour of the system may be tailored by the particular choice of switching function. Secondly, the closed-loop response becomes totally insensitive to a particular class of uncertainty in the system. . y y

trajectory

sliding surface

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The Control Problem - Disadvantages

A disadvantage of the method in the domain of control applications has been the necessity to implement a fundamentally discontinuous control signal which, in theoretical terms, must switch with infinite frequency to provide total rejection of uncertainty. Control implementation via approximate, smooth strategies is widely reported, but in such cases total invariance is routinely lost.

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Introduction - The Observer Problem In contrast, the application of sliding mode methods to the observer problem is much less mature and has some fundamentally different advantages and disadvantages.

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Introduction - The Observer Problem In contrast, the application of sliding mode methods to the observer problem is much less mature and has some fundamentally different advantages and disadvantages. the ability to generate a sliding motion on the error between the measured plant output and the output of the observer ensures that a sliding mode observer produces a set of state estimates that are precisely commensurate with the actual output of the plant.

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Introduction - The Observer Problem In contrast, the application of sliding mode methods to the observer problem is much less mature and has some fundamentally different advantages and disadvantages. the ability to generate a sliding motion on the error between the measured plant output and the output of the observer ensures that a sliding mode observer produces a set of state estimates that are precisely commensurate with the actual output of the plant. analysis of the average value of the applied observer injection signal, the so-called equivalent injection signal, contains useful information about the mismatch between the model used to define the observer and the actual plant.

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Introduction - The Observer Problem In contrast, the application of sliding mode methods to the observer problem is much less mature and has some fundamentally different advantages and disadvantages. the ability to generate a sliding motion on the error between the measured plant output and the output of the observer ensures that a sliding mode observer produces a set of state estimates that are precisely commensurate with the actual output of the plant. analysis of the average value of the applied observer injection signal, the so-called equivalent injection signal, contains useful information about the mismatch between the model used to define the observer and the actual plant. the discontinuous injection signals which were perceived as problematic for many control applications, have no disadvantages for software based observer frameworks. Sarah K. Spurgeon


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 that B and C are full rank and (A, C ) is observable.

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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 that B and C are full rank and (A, C ) is observable. Canonical form - Utkin Observer Consider the change of coordinates x 7→ Tc x whereby T Nc Tc = C where the columns of Nc ∈ Rn×(n−p) span the null space of C . This transformation is nonsingular Sarah K. Spurgeon


(2)

A Historical Perspective Canonical form for the nominal system x˙ 1 (t) = A11 x1 (t) + A12 y (t) + B1 u(t)

(3)

y˙ (t) = A21 x1 (t) + A22 y (t) + B2 u(t)

(4)

where

Tc x =

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x1 y

ln−p lp


A Historical Perspective Canonical form for the nominal system x˙ 1 (t) = A11 x1 (t) + A12 y (t) + B1 u(t)

(3)

y˙ (t) = A21 x1 (t) + A22 y (t) + B2 u(t)

(4)

where

Tc x =

x1 y

ln−p lp

The Observer xˆ˙ 1 (t) = A11 xˆ1 (t) + A12 yˆ (t) + B1 u(t) + Lν yˆ˙ (t) = A21 xˆ1 (t) + A22 yˆ (t) + B2 u(t) − ν

(5) (6)

where (ˆ x1 , yˆ ) represent the state estimates, L ∈ R(n−p)×p is a gain matrix and νi = M sgn(ˆ yi − yi ) where M ∈ R+ . Sarah K. Spurgeon


A Historical Perspective The error system Define e1 = xˆ1 − x1 and ey = yˆ − y then e˙ 1 (t) = A11 e1 (t) + A12 ey (t) + Lν

(7)

e˙ y (t) = A21 e1 (t) + A22 ey (t) − ν

(8)

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A Historical Perspective The error system Define e1 = xˆ1 − x1 and ey = yˆ − y then e˙ 1 (t) = A11 e1 (t) + A12 ey (t) + Lν

(7)

e˙ y (t) = A21 e1 (t) + A22 ey (t) − ν

(8)

Choice of L Since the pair (A, C ) is observable, the pair (A11 , A21 ) is also observable and L can be chosen to make the spectrum of A11 + LA21 lie in C− . Define a further change of coordinates by In−p L ˜ T = 0 Ip Sarah K. Spurgeon


A Historical Perspective Error dynamics

In−p L 0 Ip With e˜1 = e1 + Ly , he error system becomes ˜ = Define a change of coordinates by T

˜ 11 e˜1 (t) + A ˜ 12 ey (t) e˜˙ 1 (t) = A ˜ 22 ey (t) − ν e˙ y (t) = A21 e˜1 (t) + A

(9) (10)

˜ 11 = A11 + LA21 , A ˜ 12 = A12 + LA22 − A ˜ 11 L and where A ˜ 22 = A22 − A21 L. A In the domain ˜ 22 + A ˜T Ω = {(e1 , ey ) : kA21 e1 k + 12 λmax (A 22 )key k < M − η} (11) where η < M is some small positive scalar, the reachability condition eyT e˙ y < −ηkey k is satisfied. Sarah K. Spurgeon


A Historical Perspective Sliding Motion An ideal sliding motion will take place on the surface So = {(e1 , ey ) : ey = 0}

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(12)

After some finite time ts , for all subsequent time, ey = 0 and e˙ y = 0.

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(12)

After some finite time ts , for all subsequent time, ey = 0 and e˙ y = 0. The corresponding sliding mode dynamics are given by ˜ 11 e˜1 (t) e˜˙ 1 (t) = A which, by choice of L, represents a stable system and so e˜1 → 0 and consequently, xˆ1 → x1 as t → ∞.

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(13)

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=

0 1

C=

The System This represents a simple harmonic oscillator.

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1 1

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=

0 1

C=

1 1

The System This represents a simple harmonic oscillator. For simplicity assume u = 0 and consider the problem of designing a sliding mode observer. Sarah K. Spurgeon


Tutorial Example Canonical Form Define a nonsingular matrix Tc =

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1 0 1 1


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1 0 1 1

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This change of coordinates gives the system triple −1 1 0 −1 Tc A Tc = Tc B = CTc−1 = 0 1 −3 1 1

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1 0 1 1

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This change of coordinates gives the system triple −1 1 0 −1 Tc A Tc = Tc B = CTc−1 = 0 1 −3 1 1 An appropriate choice of observer gain is L = 0.5 which ˜ 11 = −2.5. results in an error system governed by A

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1 0 1 1

(16)

This change of coordinates gives the system triple −1 1 0 −1 Tc A Tc = Tc B = CTc−1 = 0 1 −3 1 1 An appropriate choice of observer gain is L = 0.5 which ˜ 11 = −2.5. results in an error system governed by A The scaling constant M in the discontinuous component has been set to unity. Sarah K. Spurgeon


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 , an ideal sliding motion cannot be maintained; only after approximately 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 takes place. State Estimation Errors

1 0.5 0 -0.5 -1

0

1

2

3

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4

5 Time, sec

6


7

8

9

10

Tutorial Example In the original coordinates perfect tracking occurs after approximately 3 seconds. The dotted lines represent the true states and the solid line the estimates from the observer.

State Evolution

2 1 0 -1 -2

0

1

2

3

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4

5 Time, sec

6


7

8

9

10

Tutorial Example

Discontinuous Component

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

1 0.5 0 -0.5 -1

0

1

2

3

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4

5 Time, sec

6


7

8

9

10

Classical Utkin Observer with Linear Injection The new error system ˜ 11 e˜1 (t) + A ˜ 12 ey (t) − G1 ey (t) e˜˙ 1 (t) = A ˜ 22 ey (t) − G2 ey (t) − ν e˙ y (t) = A21 e˜1 (t) + A

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(17) (18)


(17) (18)

˜ 12 and G2 = A ˜ 22 − As , where As is any By selecting G1 = A 22 22 stable design matrix of appropriate dimension, then ˜ 11 e˜1 (t) e˜˙ 1 (t) = A e˙ y (t) = A21 e˜1 (t) +

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(19) As22 ey (t)


−ν

(20)


(17) (18)

˜ 12 and G2 = A ˜ 22 − As , where As is any By selecting G1 = A 22 22 stable design matrix of appropriate dimension, then ˜ 11 e˜1 (t) e˜˙ 1 (t) = A e˙ y (t) = A21 e˜1 (t) +

(19) As22 ey (t)

−ν

(20)

The error system is asymptotically stable for ν ≡ 0 because the poles of the combined system are given by ˜ 11 ) ∪ λ(As ) and so lie in the open left half complex plane. λ(A 22 Sarah K. Spurgeon


Classical Observer

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

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Classical Observer

Observations In the original Utkin observer, the switching action ν was potentially required to make the error system stable. Thus far the only restriction imposed on the nominal linear system is that the pair (A, C ) is observable.

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Further Historical Developments The Slotine Observer (1980s) The output errors are fed back in both a linear and a discontinuous manner for nonlinear systems in companion form with the objective of ensuring a ’sliding patch’, which defines the region in which it is possible for the dynamical observer system to exhibit sliding behaviour, is maximised.

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Further Historical Developments The Slotine Observer (1980s) The output errors are fed back in both a linear and a discontinuous manner for nonlinear systems in companion form with the objective of ensuring a ’sliding patch’, which defines the region in which it is possible for the dynamical observer system to exhibit sliding behaviour, is maximised. Once only a subset of state information is known, the ability of any system to attain and maintain sliding will be more limited than in the situation where full state information is available.

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Further Historical Developments The Slotine Observer (1980s) The output errors are fed back in both a linear and a discontinuous manner for nonlinear systems in companion form with the objective of ensuring a ’sliding patch’, which defines the region in which it is possible for the dynamical observer system to exhibit sliding behaviour, is maximised. Once only a subset of state information is known, the ability of any system to attain and maintain sliding will be more limited than in the situation where full state information is available. With the Slotine observer, the linear feedback elements are a Luenberger observer with the role of the magnitude of the discontinuous element to enhance robustness.

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Further Historical Developments The Slotine Observer (1980s) The output errors are fed back in both a linear and a discontinuous manner for nonlinear systems in companion form with the objective of ensuring a ’sliding patch’, which defines the region in which it is possible for the dynamical observer system to exhibit sliding behaviour, is maximised. Once only a subset of state information is known, the ability of any system to attain and maintain sliding will be more limited than in the situation where full state information is available. With the Slotine observer, the linear feedback elements are a Luenberger observer with the role of the magnitude of the discontinuous element to enhance robustness. It is shown that a larger discontinuous element can enhance robustness but this can be at the expense of increased sensitivity to measurement noise. Sarah K. Spurgeon


Further Historical Developments

The Walcott and Zak Observer, (1980s) This was instrumental in defining the structural conditions for existence of sliding mode observers for linear systems and laid important foundations for subsequent contributions which formulated constructive design methodologies and included uncertainty.

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The Walcott and Zak Observer, (1980s) This was instrumental in defining the structural conditions for existence of sliding mode observers for linear systems and laid important foundations for subsequent contributions which formulated constructive design methodologies and included uncertainty. It was key in showing the promise of the methodology for observer design for nonlinear systems, where methodologies for relatively general nonlinear system representations were considered

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The Walcott and Zak Observer, (1980s) This was instrumental in defining the structural conditions for existence of sliding mode observers for linear systems and laid important foundations for subsequent contributions which formulated constructive design methodologies and included uncertainty. It was key in showing the promise of the methodology for observer design for nonlinear systems, where methodologies for relatively general nonlinear system representations were considered These ideas will be developed further in the next lecture.

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Sliding mode observers: towards a constructive design framework Sarah K. Spurgeon School of Engineering and Digital Arts University of Kent, UK


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Walcott and Zak Observer

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Walcott and Zak Observer Introduction to a constructive sliding mode observer design framework

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Walcott and Zak Observer Introduction to a constructive sliding mode observer design framework Potential of the discontinuous injection signal - lecture 3

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Recap

Thus far the only restriction imposed on the nominal linear system is that the pair (A, C ) is observable. Uncertainty and robustness has not been considered.

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Walcott and Zak Observer 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 the matrices B and C are assumed to be of full rank.

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(1)

A ∈ Rn×n , B ∈ Rn×m , C ∈ Rp×n and p ≥ m; in addition the matrices B and C are assumed to be of full rank. The function f : R+ × Rn × Rm → Rn is unknown and represents the system uncertainty.

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(1)

A ∈ Rn×n , B ∈ Rn×m , C ∈ Rp×n and p ≥ m; in addition the matrices B and C are assumed to be of full rank. The function f : R+ × Rn × Rm → Rn is unknown and represents 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 kξ(t, x, u)k ≤ r1 kuk + α(t, y )

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


(3)

Walcott and Zak Observer Problem Statement Estimate the states of the uncertain system so that the error system e(t) = xˆ(t) − x(t) is quadratically stable despite the presence of the uncertainty.

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(4)

Walcott and Zak Observer Problem Statement Estimate the states of the uncertain system so that the error system e(t) = xˆ(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 stable eigenvalues and there exists a Lyapunov pair (P, Q) for A0 such that the structural constraint C T F T = PB is satisfied for some F ∈ Rm×p . Sarah K. Spurgeon


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Walcott and Zak Observer The Observer xˆ˙ (t) = Aˆ x (t) + Bu(t) − G (C xˆ(t) − y (t)) + P −1 C T F T ν

( ν=

FCe −ρ(t, y , u) kFCek 0

if FCe 6= 0 otherwise

(6)

(7)

ρ(·) is any function satisfying ρ(t, y , u) ≥ r1 kuk + α(t, y ) + η for some positive scalar η.

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(8)

Walcott and Zak Observer Error System e(t) ˙ = (A − GC )e(t) − Bξ(t, x, u) + Bν

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Walcott and Zak Observer Error System e(t) ˙ = (A − GC )e(t) − Bξ(t, x, u) + Bν Quadratic Stability of the Error System Consider V (e) = e T Pe as a candidate Lyapunov function.

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Quadratic Stability of the Error System Consider V (e) = e T Pe as a candidate Lyapunov function. Evaluate the derivative along the system trajectories V˙

= e T (PA0 + A0 P)e − 2e T PBξ + 2e T PBν T

T

≤ −e Qe − 2e PBξ − 2ρ(t, y , u)kFCek

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(10)


(9)

Quadratic Stability of the Error System Consider V (e) = e T Pe as a candidate Lyapunov function. Evaluate the derivative along the system trajectories V˙

= e T (PA0 + A0 P)e − 2e T PBξ + 2e T PBν T

(10)

T

≤ −e Qe − 2e PBξ − 2ρ(t, y , u)kFCek Using the structural constraint C T F T = PB: V˙

≤ −e T Qe − 2e T C T F T ξ − 2ρ(t, y , u)kFCek (11) ≤ −e T Qe − 2kFCek(ρ(t, y , u) − kξk) ≤ −e T Qe − 2ηkFCek Sarah K. Spurgeon


Walcott and Zak Observer Conclusion There exists a domain in which a sliding motion is induced on the surface in the state error space given by Swz = { e ∈ Rn

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: FCe = 0 }


(12)


: FCe = 0 }

(12)

A constructive result? Relies on establishing whether there exists a gain matrix G such that, for the resulting closed-loop matrix A0 , there exists a Lyapunov matrix P for A0 satisfying C T F T = PB for some F ∈ Rm×p .

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: FCe = 0 }

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A constructive result? Relies on establishing whether there exists a gain matrix G such that, for the resulting closed-loop matrix A0 , there exists a Lyapunov matrix P for A0 satisfying C T F T = PB for some F ∈ Rm×p . Walcott and Zak suggested the application of symbolic computation tools.

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: FCe = 0 }

(12)

A constructive result? Relies on establishing whether there exists a gain matrix G such that, for the resulting closed-loop matrix A0 , there exists a Lyapunov matrix P for A0 satisfying C T F T = PB for some F ∈ Rm×p . Walcott and Zak suggested the application of symbolic computation tools. Can the essence of their result be used to determine a constructive framework? Sarah K. Spurgeon


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 with p ≥ q. Assume that the matrices B, C and D are full rank and the function ξ : R+ × Rn × Rm → Rq is unknown but bounded.

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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 with p ≥ q. Assume that the matrices B, C and D are full rank and the function ξ : R+ × Rn × Rm → Rq is unknown but bounded. Let (A, D, C ) represent the nominal part of (13). Define an observer for the uncertain system (13) z(t) ˙ = Az(t) + Bu(t) − Gl Ce(t) + Gn ν

(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 to be determined. Sarah K. Spurgeon


Sliding mode observers for linear uncertain systems Existence Conditions

A sliding mode observer of the form (14 ) which rejects the uncertainty class in (13 ) exists if and only if the nominal linear system defined by the matrices (A, D, C ) satisfies rank (CD) = q any invariant zeros of (A, D, C ) must lie in C− .

Sarah K. Spurgeon



A sliding mode observer of the form (14 ) which rejects the uncertainty class in (13 ) exists if and only if the nominal linear system 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.

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A sliding mode observer of the form (14 ) which rejects the uncertainty class in (13 ) exists if and only if the nominal linear system 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 uncertainty channel and the observer design will be directly determined by the uncertainty distribution matrix.

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Sliding mode observers for linear uncertain systems Canonical Form for Design A change of coordinates exists so that the triple with respect to ¯ D, ¯ C ¯ ) has the following structure: the new coordinates (A, The system matrix can be written as   A12 A11 ¯ =  A211  (16) A A22 A212 where A11 ∈ R(n−p)×(n−p) and A211 ∈ R(p−q)×(n−p) . When partitioned these matrices have the structure o A11 Ao12 A11 = and A211 = 0 Ao21 (17) 0 Ao22 where Ao11 ∈ Rr ×r and Ao21 ∈ R(p−q)×(n−p−r ) for some r ≥ 0 and the pair (Ao22 , Ao21 ) is completely observable. Furthermore, the eigenvalues of Ao11 are the invariant zeros of (A, D, C ). Sarah K. Spurgeon


Sliding mode observers for linear uncertain systems Canonical Form for Design

The matrix distributing any forcing functions has the form 0 ¯ D = (18) D2 where D2 ∈ Rq×q is nonsingular.

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Sliding mode observers for linear uncertain systems Canonical Form for Design

The matrix distributing any forcing functions has the form 0 ¯ D = (18) D2 where D2 ∈ Rq×q is nonsingular. The output distribution matrix has the form ¯ = 0 T C where T ∈ Rp×p and is orthogonal.

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Sliding mode observers for linear uncertain systems Canonical Form for Design In order to ensure compatibility in the partition of the state-space matrices, let A211 0 ¯ A21 = and D = (20) ¯2 A212 D ¯ 2 is defined as where D ¯2 = D

0 D2

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lp−q lq


(21)

Sliding mode observers for linear uncertain systems Canonical Form for Design In order to ensure compatibility in the partition of the state-space matrices, let A211 0 ¯ A21 = and D = (20) ¯2 A212 D ¯ 2 is defined as where D ¯2 = D

0 D2

lp−q lq

(21)

The necessary and sufficient conditions for existence of a sliding mode observer together with the canonical form provide a pathway to a constructive method for observer design.

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Sliding mode observers - a pathway to design

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

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As the invariant zeros are stable by assumption, Ao11 is stable. As a consequence there exists a matrix L ∈ R(n−p)×(p−q) such that A11 + LA211 is stable.

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As the invariant zeros are stable by assumption, Ao11 is stable. As a consequence there exists a matrix L ∈ R(n−p)×(p−q) such that A11 + LA211 is stable. Define a nonsingular transformation as ¯ In−p L TL = 0 T

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where ¯= L

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L 0(n−p)×q


Sliding mode observers - a pathway to design After changing coordinates with respect to TL , the new output distribution matrix becomes ˜ = CT −1 = 0 Ip C L

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Sliding mode observers - a pathway to design After changing coordinates with respect to TL , the new output distribution matrix becomes ˜ = CT −1 = 0 Ip C L ¯ and D ¯2 From the definition of L ¯D ¯2 = L

L 0

0 D2

=0

and so the uncertainty distribution matrix is given by ¯D ¯2 L 0 ˜ D = TL D = ¯2 = T D ¯2 TD

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Sliding mode observers - a pathway to design After changing coordinates with respect to TL , the new output distribution matrix becomes ˜ = CT −1 = 0 Ip C L ¯ and D ¯2 From the definition of L ¯D ¯2 = L

L 0

0 D2

=0

and so the uncertainty distribution matrix is given by ¯D ¯2 L 0 ˜ D = TL D = ¯2 = T D ¯2 TD ˜ = TL A T −1 , it can be shown by direct evaluation Finally, if A L that ˜ 11 = A11 + LA211 A which is stable by choice of L. Sarah K. Spurgeon



˜ D, ˜ C ˜ ) is now in the form The system triple (A, x˙ 1 (t) = A11 x1 (t) + A12 y (t) + B1 u(t) y˙ (t) = A21 x1 (t) + A22 y (t) + B2 u(t) + D2 ξ where x1 ∈ R(n−p) , y ∈ Rp and the matrix A11 is stable.

Sarah K. Spurgeon


(23)


˜ D, ˜ C ˜ ) is now in the form The system triple (A, x˙ 1 (t) = A11 x1 (t) + A12 y (t) + B1 u(t) y˙ (t) = A21 x1 (t) + A22 y (t) + B2 u(t) + D2 ξ

(23)

where x1 ∈ R(n−p) , y ∈ Rp and the matrix A11 is stable. Define the corresponding observer by xˆ˙ 1 (t) = A11 xˆ1 (t) + A12 yˆ (t) + B1 u(t) − A12 ey (t) yˆ˙ (t) = A21 xˆ1 (t) + A22 yˆ (t) + B2 u(t) − (A22 − As22 )ey (t) + ν where As22 is a stable design matrix and ey = yˆ − y .

Sarah K. Spurgeon


Sliding mode observers - a pathway to design Let P2 ∈ Rp×p be symmetric positive definite Lyapunov matrix for As22 then the discontinuous vector ν is defined by ( P e −ρ(t, y , u)kD2 k kP22 eyy k if ey 6= 0 ν= (24) 0 otherwise where the scalar function ρ : R+ × Rp × Rm → R+ satisfies ρ(t, y , u) ≥ r1 kuk + α(t, y ) + γo and γo is a positive scalar.

Sarah K. Spurgeon


(25)

Sliding mode observers - a pathway to design Let P2 ∈ Rp×p be symmetric positive definite Lyapunov matrix for As22 then the discontinuous vector ν is defined by ( P e −ρ(t, y , u)kD2 k kP22 eyy k if ey 6= 0 ν= (24) 0 otherwise where the scalar function ρ : R+ × Rp × Rm → R+ satisfies ρ(t, y , u) ≥ r1 kuk + α(t, y ) + γo

(25)

and γo is a positive scalar. If the state estimation error e1 = xˆ1 − x1 , then it is straightforward to show e˙ 1 (t) = A11 e1 (t) e˙ y (t) = A21 e1 (t) + Sarah K. Spurgeon

(26) As22 ey (t)

+ ν − D2 ξ


(27)

Sliding mode observers - a pathway to design Define Q1 ∈ R(n−p)×(n−p) and Q2 ∈ Rp×p as symmetric positive definite design matrices and define P2 ∈ Rp×p as the unique symmetric positive definite solution to the Lyapunov equation P2 As22 + (As22 )T P2 = −Q2

Sarah K. Spurgeon


(28)


(28)

Let P1 ∈ R(n−p)×(n−p) be the unique symmetric positive definite solution to the Lyapunov equation −1 T P1 A11 + AT 11 P1 = −(A21 P2 Q2 P2 A21 + Q1 )

Sarah K. Spurgeon


(29)


(28)

Let P1 ∈ R(n−p)×(n−p) be the unique symmetric positive definite solution to the Lyapunov equation −1 T P1 A11 + AT 11 P1 = −(A21 P2 Q2 P2 A21 + Q1 )

(29)

Taking the quadratic form V (e1 , ey ) = e1T P1 e1 + eyT P2 ey

(30)

as a candidate Lyapunov function it can be shown that the error system is quadratically stable. Sarah K. Spurgeon



Further, consideration of the quadratic form Vs (ey ) = eyT P2 ey

(31)

shows that an ideal sliding motion takes place on (15) and the output error ey enters Ω = {(e1 , ey ) : kA21 e1 k < kD2 kγo − η} where η is a small positive scalar in finite time and remains there.

Sarah K. Spurgeon


The Sliding Mode Observer If xˆ represents the state estimate for x and e = xˆ − x then the robust observer can conveniently be written as xˆ˙ (t) = Aˆ x (t) + Bu(t) − Gl Ce(t) + Gn ν where

A12 Gl = A22 − As22 0 −1 Gn = kD2 kTo Ip To−1

(32)

(33) (34)

and ( ν=

−ρ(t, y , u) kPP22 Ce Cek 0

if Ce 6= 0 otherwise

(35)

A key development in this formulation of the sliding mode observer design framework is that there is no requirement for (A, C ) to be observable. Sarah K. Spurgeon


Tutorial Example Pendulum System x˙ 1 = x2 x˙ 2 = − sin(x1 )

The state and control matrices are 0 1 0 A = and B = 0 0 1 The matched ‘uncertain’ bounded function ξ(t, x1 , x2 ) = − sin(x1 ). The output distribution matrix C = 1 1 .

Sarah K. Spurgeon


(36)

Tutorial Example Canonical Form Representation Change coordinates with respect to Tc = the output is a state variable.

Sarah K. Spurgeon


1 0 1 1

so that

Tutorial Example Canonical Form Representation

1 0 1 1

= CTc−1 =

Change coordinates with respect to Tc =

so that

the output is a state variable. The system triple becomes ˜= A

Tc A Tc−1

=

−1 1 −1 1

and ˜ = Tc B = B

0 1

and

Sarah K. Spurgeon

˜ C


0 1

Tutorial Example Canonical Form Representation

1 0 1 1

= CTc−1 =

Change coordinates with respect to Tc =

so that

the output is a state variable. The system triple becomes ˜= A

Tc A Tc−1

=

−1 1 −1 1

and ˜ = Tc B = B

0 1

and

˜ C

0 1

˜ 11 = −1, A robust observer exists for this system because A which is stable. This is the transmission zero of the system. Sarah K. Spurgeon


Tutorial Example Observer design Let the design matrix As22 = −1 so that λ(A0 ) = {−1, −1}.

Sarah K. Spurgeon


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.

Sarah K. Spurgeon


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 1 0 Gl = and Gn = 1 1

Sarah K. Spurgeon


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 1 0 Gl = and Gn = 1 1 The observer becomes d xˆ1 −1 0 xˆ1 1 0 = + y+ ν −1 −1 xˆ2 1 1 dt xˆ2

Sarah K. Spurgeon


Tutorial Example Demonstrates the nonlinear observer tracking the output from the pendulum when the initial conditions of the true states and observer states are deliberately set to different values

2

Outputs

1 0 -1 -2

0

2

4

6

Sarah K. Spurgeon

8

10 12 Time, sec


14

16

18

20

Tutorial Example A comparison of the true and estimated states. After approximately 4 seconds, visually perfect replication of the states is taking place. 2

First State

1 0 -1 -2

0

2

4

6

8

10 12 Time, sec

14

16

18

20

2

4

6

8

10 12 Time, sec

14

16

18

20

Second State

2 1 0 -1 -2

0

Sarah K. Spurgeon


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 the outputs of the system and the outputs of the observer; this is due to the presence of the nonlinear sine term.

4

Output

2 0 -2 -4

0

2

4

6

Sarah K. Spurgeon

8

10 12 Time, sec


14

16

18

20

Tutorial Example The role of the applied discontinuous injection nu On average this replicates the mismatch between the plant and the model assumed for observer design How can we2 use this for practical applications - lecture 3 nu

1 0 −1

filtered nu, sin term in plant

−2 0

2

4

6

8

10 12 Time,sec

14

16

18

20

2

4

6

8

10 12 Time,sec

14

16

18

20

2 1 0 −1 −2 0

Sarah K. Spurgeon


Sliding mode observers: a robust condition monitoring and fault detection tool Sarah K. Spurgeon School of Engineering and Digital Arts University of Kent, UK


Sarah K. Spurgeon


Sliding Mode Observers


Sarah K. Spurgeon



Outline of Presentation Principle of the equivalent injection - Fault detection and isolation

Sarah K. Spurgeon



Outline of Presentation Principle of the equivalent injection - Fault detection and isolation Fault detection and isolation - a case study

Sarah K. Spurgeon



Outline of Presentation Principle of the equivalent injection - Fault detection and isolation Fault detection and isolation - a case study A sampled framework for practical application? - lecture 4

Sarah K. Spurgeon



Recap We have developed constructive frameworks for design We have noted in the tutorial example that when an observer designed based on the dynamics of the double integrator is applied to a nonlinear pendulum system, the discontinuous signal when in the sliding mode, on average, reconstructs the mismatch between the model used for design and the plant used for implementation

Sarah K. Spurgeon


Sliding mode observers for fault detection and fault reconstruction Historical Perspective One of the first papers designed an observer so that the observer error moves from the switching surface and sliding ceases in the presence of a fault.

Sarah K. Spurgeon


Sliding mode observers for fault detection and fault reconstruction Historical Perspective One of the first papers designed an observer so that the observer error moves from the switching surface and sliding ceases in the presence of a fault. This approach is difficult to implement in practice - the choice of gain to maintain sliding motion from the theory is often conservative and therefore it is difficult to ensure a fault induces a break in sliding.

Sarah K. Spurgeon


Sliding mode observers for fault detection and fault reconstruction Historical Perspective One of the first papers designed an observer so that the observer error moves from the switching surface and sliding ceases in the presence of a fault. This approach is difficult to implement in practice - the choice of gain to maintain sliding motion from the theory is often conservative and therefore it is difficult to ensure a fault induces a break in sliding. Observers, when exhibiting sliding motion, enable faults and/or values of immeasurable system parameters to be reconstructed using the principle of the equivalent injection signal.

Sarah K. Spurgeon


Sliding mode observers for fault detection and fault reconstruction 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 < n and 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 and only the signals u(t) and y (t) are available.

Sarah K. Spurgeon


The Observer The design objective To synthesise an observer to generate a state estimate xˆ(t) such that a sliding mode is established in which the output error ey (t) = yˆ (t) − y (t) is forced to zero in finite time.

Sarah K. Spurgeon


(3)

The Observer The design objective To synthesise an observer to generate a state estimate xˆ(t) such that a sliding mode is established in which the output error ey (t) = yˆ (t) − y (t)

(3)

is forced to zero in finite time. The observer structure xˆ˙ (t) = Aˆ x (t) + Bu(t) − Gl ey (t) + Gn ν 0 where Gl is a linear gain and Gn = To−1 with Ip ( ν=

P e

−ρ(t, y , u)kD2 k kP22 eyy k 0 Sarah K. Spurgeon

if ey 6= 0 otherwise


(4)

(5)

The Error Dynamics

Error dynamics in the canonical form used for observer design e˙ 1 (t) = A11 e1 (t) e˙ y (t) = A21 e1 (t) +

Sarah K. Spurgeon

(6) As22 ey (t)

+ ν − D2 fi (t)


(7)

Fault Reconstruction Input Signals Consider initially the case when fo = 0.

Sarah K. Spurgeon


Fault Reconstruction Input Signals Consider initially the case when fo = 0. Assume that an appropriate observer has been designed and a sliding motion has been established so that ey = 0 and e˙ y = 0.

Sarah K. Spurgeon


Fault Reconstruction Input Signals Consider initially the case when fo = 0. Assume that an appropriate observer has been designed and a sliding motion has been established so that ey = 0 and e˙ y = 0. In appropriate coordinates 0 = A21 e1 (t) − D2 fi (t) + νeq

(8)

where νeq is the equivalent injection applied during the sliding mode.

Sarah K. Spurgeon


Fault Reconstruction Input Signals Consider initially the case when fo = 0. Assume that an appropriate observer has been designed and a sliding motion has been established so that ey = 0 and e˙ y = 0. In appropriate coordinates 0 = A21 e1 (t) − D2 fi (t) + νeq

(8)

where νeq is the equivalent injection applied during the sliding mode. From the stability of the e1 subsystem, it follows that e1 (t) → 0 and therefore νeq → D2 fi (t)

Sarah K. Spurgeon


(9)

Fault Reconstruction Input Signals A commonly used approach to reconstruct the equivalent injection is to use a low pass filter. Alternatively replace the discontinuous injection by the continuous approximation P e

y νδ = −ρkD2 k kP2 e2y k+δ

where δ is a small positive scalar.

Sarah K. Spurgeon


(10)



(10)

where δ is a small positive scalar. The equivalent control can be approximated by (10) to any required accuracy by a small enough choice of δ. Since rank(D2 ) = q it follows that P e (t)

y fi (t) ≈ −ρkD2 k(D2T D2 )−1 D2T kP2 e2y (t)k+δ

Sarah K. Spurgeon


(11)



(10)

where δ is a small positive scalar. The equivalent control can be approximated by (10) to any required accuracy by a small enough choice of δ. Since rank(D2 ) = q it follows that P e (t)

y fi (t) ≈ −ρkD2 k(D2T D2 )−1 D2T kP2 e2y (t)k+δ

(11)

The equivalent injection signal can be computed on-line and depends only on the output estimation error ey ; thus the fault fi can be approximated to any degree of accuracy. Sarah K. Spurgeon


Detection of faults at the output Methodology Assume fi = 0 and consider the effect of fo (t).

Sarah K. Spurgeon


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 that ey (t) = Ce(t) − fo (t) and the state estimation error is given by e˙ 1 (t)

= A11 e1 (t) + A12 fo (t)

e˙ y (t)

As22 ey (t)

= A21 e1 (t) +

Sarah K. Spurgeon

(12) ˙ − fo (t) + A22 fo (t) + ν (13)



= A11 e1 (t) + A12 fo (t)

e˙ y (t)

As22 ey (t)

= A21 e1 (t) +

(12) ˙ − fo (t) + A22 fo (t) + ν (13)

fo (t) and f˙o (t) are output disturbances; ρ must be chosen sufficiently large to maintain sliding

Sarah K. Spurgeon



= A11 e1 (t) + A12 fo (t)

e˙ y (t)

As22 ey (t)

= A21 e1 (t) +

(12) ˙ − fo (t) + A22 fo (t) + ν (13)

fo (t) and f˙o (t) are output disturbances; ρ must be chosen sufficiently large to maintain sliding During sliding 0 = A21 e1 − f˙o (t) + A22 fo (t) + νeq and for slowly varying faults νeq ≈ −(A22 − A21 A−1 11 A12 )fo Sarah K. Spurgeon


(14)

Output fault detection Methodology The equivalent control νeq can be calculated and consequently, if (A22 − A21 A−1 11 A12 ) is nonsingular, the fault signal can be obtained from νeq ≈ −(A22 − A21 A−1 11 A12 )fo .

Sarah K. Spurgeon


Output fault detection Methodology The equivalent control νeq can be calculated and consequently, if (A22 − A21 A−1 11 A12 ) is nonsingular, the fault signal can be obtained from νeq ≈ −(A22 − A21 A−1 11 A12 )fo . From the Schur expansion det(A) = det(A11 ) det(A22 − A21 A−1 11 A12 )

(15)

and thus (A22 − A21 A−1 11 A12 ) is nonsingular if and only if det A 6= 0.

Sarah K. Spurgeon


Output fault detection Methodology The equivalent control νeq can be calculated and consequently, if (A22 − A21 A−1 11 A12 ) is nonsingular, the fault signal can be obtained from νeq ≈ −(A22 − A21 A−1 11 A12 )fo . From the Schur expansion det(A) = det(A11 ) det(A22 − A21 A−1 11 A12 )

(15)

and thus (A22 − A21 A−1 11 A12 ) is nonsingular if and only if det A 6= 0. Even if (A22 − A21 A−1 11 A12 ) is singular, useful information about sensor faults fo can still be potentially obtained, depending on the structure of the rank deficiency.

Sarah K. Spurgeon


Example: Inverted Pendulum with a Cart System Assume the pendulum rotates in the vertical plane and the cart is to be manipulated so that the pendulum remains in an upright position. The cart is linked by a transmission belt to a drive wheel which is driven by a DC motor.

θ x

Sarah K. Spurgeon


Example: Inverted Pendulum with a Cart Equations of Motion (M + m)¨ x + Fx x˙ + ml(θ¨ cos θ − θ˙2 sin θ) = u J θ¨ + Fθ θ˙ − mlg sin θ + ml x¨ cos θ = 0 where the values of the physical parameters used are given by Table : Model parameters for the inverted pendulum

M m J l

(kg) (kg) (kg m2 ) (m)

3.2 0.535 0.062 0.365

Sarah K. Spurgeon

Fx Fθ g

(kg/sec) (kg m2 ) (m/sec2 )


6.2 0.009 9.807

(16) (17)

Linear model Linearisation about the origin 

A =

C

=

0  0   0 0  

0 0 −1.9333 36.9771 1 0 0

0 1 0

1 0 −1.9872 6.2589  0 0 0 0  1 0

  0 0   1 0  B =   0.3205 0.0091  −0.1738 −1.0095

  

(18)

Assumptions x, θ, x˙ and θ˙ are the system states A sliding mode control law is used to control the system

Sarah K. Spurgeon




Linear model Linearisation about the origin 

A =

C

=

0  0   0 0  

0 0 −1.9333 36.9771 1 0 0

0 1 0

1 0 −1.9872 6.2589  0 0 0 0  1 0

  0 0   1 0  B =   0.3205 0.0091  −0.1738 −1.0095

   

(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 fault distribution matrix D = B. Sarah K. Spurgeon


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,:);

Sarah K. Spurgeon


Numerical methods for design Current triple  Ac

=

Cc

=

    

−0.1738 0 1 0.0091 0 0 0

1 0 0

0 0 0 0 0 1 0

36.9771 0 0 −1.9333  0 0  1

   −0.0095 6.2589    0 1   Bc =     0 0 0.3205 −1.9872 (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


Numerical Methods for Design Current Triple  Aa

=

Ca

=

    

−0.1451 0 1 −0.0091 0 0 0

0 0 0 0

−1 0 0

30.8877 0 0 1.9333 0 1 0

  0 −0.4568   0 1  Ba =   0 3.1498  −0.3205 −2.0159

 0 0  −1

   

(20)

The system has no transmission zeros and thus the transformation to separate the unobservable modes is not required Determine a gain matrix so that the top left sub-system has the desired 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)’; Sarah K. Spurgeon


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

=

C

=

   

−10.0000 0 1.0000 −0.0091

0  0 0

−1 0 0

0 0 0 0 0 1 0

−67.6603 0 9.8548 1.8437  0 0  −1

Sarah K. Spurgeon

 −31.4960  1.0000  3.1496  −2.0158

  D =  

 0  0   0 −0.3205 (21)


Recap: Sliding mode observers - a pathway to design System Triple Given a system triple in the form x˙ 1 (t) = A11 x1 (t) + A12 y (t) + B1 u(t) y˙ (t) = A21 x1 (t) + A22 y (t) + B2 u(t) + D2 ξ where x1 ∈ R(n−p) , y ∈ Rp and the matrix A11 is stable.

Sarah K. Spurgeon


(22)

Recap: Sliding mode observers - a pathway to design System Triple Given a system triple in the form x˙ 1 (t) = A11 x1 (t) + A12 y (t) + B1 u(t) y˙ (t) = A21 x1 (t) + A22 y (t) + B2 u(t) + D2 ξ

(22)

where x1 ∈ R(n−p) , y ∈ Rp and the matrix A11 is stable. Corresponding Observer xˆ˙ 1 (t) = A11 xˆ1 (t) + A12 yˆ (t) + B1 u(t) − A12 ey (t) yˆ˙ (t) = A21 xˆ1 (t) + A22 yˆ (t) + B2 u(t) − (A22 − As22 )ey (t) + ν where As22 is a stable design matrix and ey = yˆ − y .

Sarah K. Spurgeon


Example: Inverted Pendulum with a Cart Canonical Form Representation  A

=

C

=

   

−10.0000 0 1.0000 −0.0091

0  0 0

−1 0 0

0 0 0 0 0 1 0

−67.6603 0 9.8548 1.8437  0 0  −1

 −31.4960 1.0000   3.1496  −2.0158

  D =  

 0  0   0 −0.3205 (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 the closed-loop poles of the controlled plant. The symmetric positive definite matrix P2 satisfies P2 A22 + AT 22 P2 = −I The scalar function ρ = 75 Sarah K. Spurgeon


Nonlinear simulation testing Fault Reconstruction It can be verified that the eigenvalues of A are {0, 5.8702, − 6.3965, − 1.6347} and the steady-state gain from fo to νeq is singular. In fact 

 0 0 −1  0 −3.0888 0  (A22 − A21 A−1 11 A12 ) = 0 1.9052 1.9872 which is clearly rank deficient.

Sarah K. Spurgeon


(24)

Nonlinear simulation testing Fault Reconstruction It can be verified that the eigenvalues of A are {0, 5.8702, − 6.3965, − 1.6347} and the steady-state gain from fo to νeq is singular. In fact 

 0 0 −1  0 −3.0888 0  (A22 − A21 A−1 11 A12 ) = 0 1.9052 1.9872

(24)

which is clearly rank deficient. If νeq,i and fo,i denote the ith components of νeq and fo in νeq ≈ −(A22 − A21 A−1 11 A12 )fo and using the distribution matrix above νeq,1 ≈ fo,3

(25)

νeq,2 ≈ 3.0888fo,2

(26)

Sarah K. Spurgeon


Nonlinear simulation testing Fault Reconstruction Any fault in the first output channel has no direct long-term effect on νeq Because of the structure of D2 , it can be verified that (D2T D2 )−1 D2T = [ 0 0 3.1200 ] and so from (9) νeq,3 ≈ 0.3205fi

Sarah K. Spurgeon


(27)

Nonlinear simulation testing Fault Reconstruction Any fault in the first output channel has no direct long-term effect on νeq Because of the structure of D2 , it can be verified that (D2T D2 )−1 D2T = [ 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 be used as detector signals

Sarah K. Spurgeon


Nonlinear simulation testing Response of the detection signals to a fault in the input channel As predicted by the theory, the third detector signal reproduces the fault signal whilst not affecting the other two signals 1

Input Fault Signal

0.1

0.8

Detector Signal 1

0.05

0.6 0 0.4 -0.05

0.2 0

0

5

10

15

-0.1 0

0.1

Detector Signal 2

5

10

15

Time

Time 1

Detector Signal 3

0.05 0.5

0 -0.05 -0.1 0

0 5

10 15 Time K. Spurgeon Sarah

0 5 10 Time Sliding mode observer

15

Nonlinear simulation testing Response to a fault in the first output channel The detector signals do not reproduce the fault signal - the second signal approximates the gradient 0.1

Output Fault 1

0.1

0.08

Detctor Signal 1

0.05

0.06 0 0.04 -0.05

0.02 0 0

5

10

15

-0.1 0

0.1

Detector Signal 2

5

10

15

Time

Time 1

Detector Signal 3

0.05 0.5

0 -0.05 -0.1 0

0 5



15

Nonlinear simulation testing Response to a fault in the second output channel The appropriate detector signal reproduces the ramp fault signals in channel 2 - other channels are also affected 0.1

Output Fault 2

0.1

0.08

Detector Signal 1

0.05

0.06 0 0.04 -0.05

0.02 0 0

5

10

15

-0.1 0

0.1

Detector Signal 2

5

10

15

Time

Time 1

Detector Signal 3

0.05 0.5

0 -0.05 -0.1 0

0 5



15

Nonlinear simulation testing Response to a fault in the third output channel The appropriate detector signal reproduces the ramp fault signals in channel 3 - other channels are also affected 0.1

Output Fault 3

0.1

0.08

Detector Signal 1

0.05

0.06 0 0.04 -0.05

0.02 0 0

5

10

15

-0.1 0

0.1

Detector Signal 2

5

10

15

Time

Time 1

Detector Signal 3

0.05 0.5

0 -0.05 -0.1 0

0 5



15

Sampling effects?

2

reconstructed fault signal input fault signal

1

0

−1

−2 0

5

10 time (s)

15

20

Figure : Fault reconstruction using the classical observer designed with no a priori knowledge of output sampling characteristics

Sarah K. Spurgeon


Concluding remarks

Fault detection and isolation (FDI) using sliding mode observers The principle of the equivalent injection is a strong result to under pin the development of FDI schemes

Sarah K. Spurgeon


Concluding remarks

Fault detection and isolation (FDI) using sliding mode observers The principle of the equivalent injection is a strong result to under pin the development of FDI schemes Constructive design approach and can run in real time

Sarah K. Spurgeon


Concluding remarks

Fault detection and isolation (FDI) using sliding mode observers The principle of the equivalent injection is a strong result to under 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 process measurements may be sampled at a rate which is not quasi-continuous and the sampling can impact on the fidelity of the reconstruction

Sarah K. Spurgeon


Sliding mode observers: implementation as a condition monitoring tool Sarah K. Spurgeon School of Engineering and Digital Arts University of Kent, UK


Sarah K. Spurgeon




Sarah K. Spurgeon



Outline of Presentation Parameter Estimation - an industrial case study

Sarah K. Spurgeon



Outline of Presentation Parameter Estimation - an industrial case study A sampled framework for the design of sliding mode observers

Sarah K. Spurgeon



Recap We have developed constructive frameworks for observer design We have seen that sliding mode observers are a natural tool for the development of FDI schemes We have also seen via an example that sampling of the signals used to drive the observer can affect the fidelity of the reconstruction

Sarah K. Spurgeon


Fault detection via parameter monitoring - An Example A sliding mode observer can be used to estimate or monitor the variation in system parameters.

Sarah K. Spurgeon


Fault detection via parameter monitoring - An Example A sliding mode observer can be used to estimate or monitor the variation in system parameters. The development and implementation of a simple condition monitoring system for a high speed rotating machine will be considered.

Sarah K. Spurgeon


Fault detection via parameter monitoring - An Example A sliding mode observer can be used to estimate or monitor the variation in system parameters. The development and implementation of a simple condition monitoring system for a high speed rotating machine will be considered. As such machines become more complex and valuable, there is a greater need to protect them, and the systems they support, from the consequences of breakdown. This is relevant in the semiconductor industry, for example, where machine failure can cause the loss of a valuable batch of wafers.

Sarah K. Spurgeon


Fault detection via parameter monitoring - An Example A sliding mode observer can be used to estimate or monitor the variation in system parameters. The development and implementation of a simple condition monitoring system for a high speed rotating machine will be considered. As such machines become more complex and valuable, there is a greater need to protect them, and the systems they support, from the consequences of breakdown. This is relevant in the semiconductor industry, for example, where machine failure can cause the loss of a valuable batch of wafers. Dry vacuum pumps have been successfully used in the semiconductor industry but the harsh nature of many semiconductor processes creates a challenge for condition monitoring. Any diagnostic scheme should be able to operate under all conditions and be able to detect faults, creating an alarm to warn the user that maintenance must be conveniently scheduled before catastrophic loss occurs. Sarah K. Spurgeon


Consider a model for monitoring cooling water flow in a dry vacuum pump in the absence of a flow transducer. Heat transfer through the system is illustrated below:

Convection to atmosphere Electrical Power

Dry Vacuum Pump Coolant outlet

Coolant inlet

Sarah K. Spurgeon


A simple heat transfer model The heat transfer model for the coolant can be expressed through the following pair of equations. mB CB T˙ B = KI (t) − m ˙ c Cc (To − Ti ) − hB AB (TB − Ta )

(1)

where mB is the mass of the pump body, CB is the specific heat coefficient of the pump body material, m ˙ c 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 a scalar, Ta is the ambient temperature, TB is the temperature of the pump body, Ti and To are the coolant inlet and outlet temperatures respectively, and mc Cc T˙ o = −m ˙ c Cc (TB − To ) + hc Ac (TB − To )

(2)

where mc is the mass of the coolant, Cc is the specific heat coefficient, hc is the convective heat transfer coefficient and Ac is the coolant heat transfer area. Sarah K. Spurgeon


Parameterisation of the Model In equations (1-2), the signals I , TB , Ta , Ti and To and parameters 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 m ˙c can be approximated to parameterise the nominal operating conditions of the pump. Condition monitoring is effected by investigating changes in the coolant mass flow rate, m ˙ c , and the heat transfer ˆ˙ c + ∆m coefficient, hc . Defining m ˙c =m ˙ c and hc = hˆc + ∆hc ˆ˙ c and hˆc are the nominal parameter values where m representing normal system operation and ∆m ˙ c and ∆hc represent deviations in the parameter values as could be caused by malfunction of the machine.

Sarah K. Spurgeon


Sliding mode observer Define a pair of sliding mode observers by ˆ˙ m B CB T B ˆ˙ mc Cc T o

ˆB − Ta − β1 eb + νb ˆ˙ c Cc (To − Ti ) − hB AB T = KI (t) − m ô + hˆc Ac (TB − To ) − β2 eo + νo ˆ˙ c Cc TB − T = −m

where ei nui

î = Ti − T ei = Ki kei k

(3)

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


Sliding mode analysis Forming the error dynamics and noting that the sliding mode condition yields ei = e˙ i = 0, it follows that ˆ˙ c Cc (To − Ti ) − νb 0 = ∆m ô − ∆hˆc Ac (TB − To ) − νo ˆ˙ c Cc TB − T 0 = ∆m The applied observer injection signals can be readily used to monitor deviations in the nominal parameters. Note that this is a unique property of sliding mode observers that can only be realised by application of a discontinuous injection signal. Other observer formulations would not attain and maintain an identically zero output error but would seek to minimise the error. This latter paradigm is not useful for parameter reconstruction.

Sarah K. Spurgeon


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

Sarah K. Spurgeon


Observer performance in the presence of a changed mass flow rate

Sarah K. Spurgeon


Fault reconstruction under output sampling - a time delay approach

A sliding mode observer in the presence of sampled output information and its application to fault reconstruction is studied.

Sarah K. Spurgeon



A sliding mode observer in the presence of sampled output information and its application to fault reconstruction is studied. The observer is designed by using the delayed continuous-time representation of the sampled-data system, for which a set of Linear Matrix Inequalities (LMIs) provide conditions for the ultimate boundedness, where the bound is proportional to the sampling time and the magnitude of the switching gain.

Sarah K. Spurgeon



A sliding mode observer in the presence of sampled output information and its application to fault reconstruction is studied. The observer is designed by using the delayed continuous-time representation of the sampled-data system, for which a set of Linear Matrix Inequalities (LMIs) provide conditions for the ultimate boundedness, where the bound is proportional to the sampling time and the magnitude of the switching gain. It is shown that, for a sufficiently small value of µ, a perturbation parameter, a transducer or sensor fault can be reconstructed reliably from the output error dynamics.

Sarah K. Spurgeon


Problem Formulation Consider the linear, time-invariant system with sampled outputs 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 vector respectively and y ∈ Rp is a discrete-time output measurement generated by zero-order hold functions with a sequence of hold times 0 = t0 < t1 · · · < tk < · · · , where limk→∞ tk = ∞.

Sarah K. Spurgeon



(4)

where x ∈ Rn , u ∈ Rm are the state and the input vector respectively and y ∈ Rp is a discrete-time output measurement generated by zero-order hold functions with a sequence of hold times 0 = t0 < t1 · · · < tk < · · · , where limk→∞ tk = ∞. fi ∈ Rq represents an unknown actuator fault which is assumed to be bounded by kfi (t)k ≤ ∆.

Sarah K. Spurgeon



(4)

where x ∈ Rn , u ∈ Rm are the state and the input vector respectively and y ∈ Rp is a discrete-time output measurement generated by zero-order hold functions with a sequence of hold times 0 = t0 < t1 · · · < tk < · · · , where limk→∞ tk = ∞. fi ∈ Rq represents an unknown actuator fault which is assumed to be bounded by kfi (t)k ≤ ∆. It is assumed q ≤ p < n and A, B, C , D are constant matrices of appropriate dimensions.

Sarah K. Spurgeon


Problem Formulation Following the approach in Mikheev, Sobolev and Fridman and Fridman, Seuret and Richard, system (4) with sampled output can be presented as a continuous-time system with a known output measurement 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 any two sequential sampling times is not greater than some pre-chosen h > 0, then τ (t) ∈ (0, h] with τ˙ (t) = 1 for t 6= tk is known. It is assumed that 1

rank (CD) = q.

2

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

Sarah K. Spurgeon


Canonical form for design Under these assumptions the system (5) can be transformed into: x˙ 1 (t) = A11 x1 (t) + A12 x2 (t) + B1 u(t) x˙ 2 (t) = A21 x1 (t) + A22 x2 (t) + B2 u(t) + D1 fi (t) (6) y (t) = Tx2 (t − τ (t)) 0 n−p p ¯ 1 ∈ Rq×q , A11 where x1 ∈ R , x2 ∈ R , D1 = ¯ , D D1 has stable eigenvalues and T is an orthogonal matrix.

Sarah K. Spurgeon


Canonical form for design Under these assumptions the system (5) can be transformed into: x˙ 1 (t) = A11 x1 (t) + A12 x2 (t) + B1 u(t) x˙ 2 (t) = A21 x1 (t) + A22 x2 (t) + B2 u(t) + D1 fi (t) (6) y (t) = Tx2 (t − τ (t)) 0 n−p p ¯ 1 ∈ Rq×q , A11 where x1 ∈ R , x2 ∈ R , D1 = ¯ , D D1 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 , xˆ2 ∈ Rp : se (t) = T x2 (t−τ (t))−ˆ x2 (t−τ (t)) = 0} (7) where xˆ2 (t − τ (t)) is the corresponding component of the estimated states from an observer.

Sarah K. Spurgeon


Canonical form for design Under these assumptions the system (5) can be transformed into: x˙ 1 (t) = A11 x1 (t) + A12 x2 (t) + B1 u(t) x˙ 2 (t) = A21 x1 (t) + A22 x2 (t) + B2 u(t) + D1 fi (t) (6) y (t) = Tx2 (t − τ (t)) 0 n−p p ¯ 1 ∈ Rq×q , A11 where x1 ∈ R , x2 ∈ R , D1 = ¯ , D D1 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 , xˆ2 ∈ Rp : se (t) = T x2 (t−τ (t))−ˆ x2 (t−τ (t)) = 0} (7) where xˆ2 (t − τ (t)) is the corresponding component of the estimated states from an observer. An ideal sliding mode can be achieved with h = 0 under assumptions 1, 2. Sarah K. Spurgeon


The Observer

Consider the observer xˆ˙ (t) = Aˆ x (t) + Bu(t) − Gl e¯2 (t − τ (t)) + Gn v (t − τ (t)) yˆ (t) = C xˆd (tk ), tk ≤ t < tk+1 (8) where Gl ∈ Rn×p , Gn ∈ Rn×p and e¯2 (t) = T x2 (t) − xˆ2 (t) . The discontinuous injection term v is given by v (t) = −(kTD1 k + δ)∆[sign e¯21 (t), . . . , sign e¯2p (t)]T where δ > 0 is a positive number.

Sarah K. Spurgeon


(9)

The Observer ¯ 0 with Assume there exists L ∈ R(n−p)×p where L = L ¯ ∈ R(n−p)×(p−q) such that a coordinate change T0 yields L xˆ˙ 1 (t) = A11 xˆ1 (t) + A12 xˆ2 (t) + B1 u(t) −( µ1 L + A11 L)(x2 (t − τ (t)) − xˆ2 (t − τ (t))) + LT T v (t − τ (t)) xˆ˙ 2 (t) = A21 xˆ1 (t) + A22 xˆ2 (t) + B2 u(t) −(A21 L − µ1 Ip )(x2 (t − τ (t)) − xˆ2 (t − τ (t)) − T T v (t − τ (t)) yˆ (t) = T xˆ2 (t − τ (t)) (10) where Gl = T0−1

1 L µ

+ A11 L A21 L − µ1 Ip

,

Gn = T0−1

with µ > 0.

Sarah K. Spurgeon


LT T −T T

(11)

The error dynamics Defining the state estimation error as e1 (t) = x1 (t) − xˆ1 (t) and e2 (t)= x2 (t) − xˆ2 (t), and performing a change of coordinates such e¯1 (t) e1 (t) In−q L that = TL with TL = . Since e¯2 (t) e2 (t) 0 T LD1 = 0, one obtains e¯˙ 1 (t) = (A11 + LA21 )¯ e1 (t) − (A11 L + LA21 L − A12 −LA22 )T T e¯2 (t) + (A11 + LA21 )¯ e2 (t − τ (t))

(12)

e¯˙ 2 (t) = TA21 e¯1 (t) − (TA21 LT T − TA22 T T )¯ e2 (t) +TA21 LT T e¯2 (t − τ (t)) − µ1 e¯2 (t − τ (t)) +v (t − τ (t)) + TD1 fi (t)

(13)

with initial condition e¯(t0 ) = e¯0 ,

e¯(t) = 0,

Sarah K. Spurgeon

t < t0


(14)

Boundedness of e¯1 (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 columns zero, such that the LMI

PA11 + AT11 P + YA21 + AT21 Y T + αP ∗

−P −bI

0

(19)

where P1 ∈ Rn−p , and choose the Lyapunov-Krasovskii functional: V (t) = e¯(t)T Pµ e¯(t) + (h − µξ(t))

Z

t ¯ e α(s−t) e¯˙ 2 (s)U e¯˙ 2 (s)ds

t−µξ(t)

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


(20)

Lemma ¯ α ¯ let there exist a n × n matrix Pµ > 0 in Given positive scalars µ, ξ, ¯ and b, (19), p × p matrices U > 0, P4 , P5 and (n − p) × (n − p) matrices P6 , P7 such that the following LMIs   ¯ T21 P5 + µP3T µA 0 θ11 θ12 θ13 T  ∗ θ22 θ23 θ24 P4    T (21) Θµ0 =  0 0  ∗ ∗ −P 7   0

e2 ≥ −k1 + α1 e1 if e1 < 0

Now revisit the error dynamics imposing the sliding condition. Sarah K. Spurgeon


Slotine Nonlinear Observer

Sliding Mode Dynamics when e1 = 0 It follows from the e˙ 1 dynamic equation that e2 − k1 sgn(e1 ) = 0 and therefore e˙ 2 = e3 − ..

..

..

k2 e2 k1

e˙ n = ∆f −

Sarah K. Spurgeon

kn e2 k1


Slotine Nonlinear Observer

Sliding Mode Dynamics when e1 = 0 It follows from the e˙ 1 dynamic equation that e2 − k1 sgn(e1 ) = 0 and therefore e˙ 2 = e3 − ..

..

..

k2 e2 k1

e˙ n = ∆f −

kn e2 k1

What is the role of the observer gains?

Sarah K. Spurgeon


Slotine Nonlinear observer The sliding patch The αi only affect the dynamic performance prior to the reaching of the so-called sliding patch - patch dynamics are  k  − k21 1 0 ... 0  k3   − k1 0 1 ... 0    (10) λIn−1 −   = 0 .    ... 1  kn − k1 0 0 ... 0

Sarah K. Spurgeon


Slotine Nonlinear observer The sliding patch The αi only affect the dynamic performance prior to the reaching of the so-called sliding patch - patch dynamics are  k  − k21 1 0 ... 0  k3   − k1 0 1 ... 0    (10) λIn−1 −   = 0 .    ... 1  kn − k1 0 0 ... 0 Assuming kn is selected as a constant ratio with k1 and that the poles defining the dynamics on the patch are critically damped i.e. are real and equal to some constant value γ, then (i) i = 0, ..., n − 2 (11) e2 ≤ (2γ)i k1 from which the precision of the state estimates can be determined. Sarah K. Spurgeon


Nonlinear observers- Further historical developments Further Developments of Slotine The effect of measurement noise on sliding mode observers was formulated. The system does not attain a sliding mode in the presence of noise, but remains within a region of the sliding patch determined by the bound on the noise. Moreover, it was demonstrated that the average dynamics can be modified by selection of the ki which in turn can tailor the contribution of the noise to the state estimates.

Sarah K. Spurgeon


Nonlinear observers- Further historical developments Further Developments of Slotine The effect of measurement noise on sliding mode observers was formulated. The system does not attain a sliding mode in the presence of noise, but remains within a region of the sliding patch determined by the bound on the noise. Moreover, it was demonstrated that the average dynamics can be modified by selection of the ki which in turn can tailor the contribution of the noise to the state estimates. Next steps The equivalent injection design concept - Drakunov Important as it is possible to develop an observer without using input derivatives Further developed by several authors Sarah K. Spurgeon


Nonlinear step-by-step observer design

Nonlinear system in triangular input form ξ˙1 = ξ2 + g¯1 (ξ1 , u) ξ˙2 = ξ3 + g¯2 (ξ1 , ξ2 , u) .. = .. ˙ξn−1 = ξn + g¯n−1 (ξ1 , ξ2 , ..., ξn−1 , u) ξ˙n = f¯n (ξ1 , ..., ξn ) + g¯n (ξ1 , ..., ξn , u) where y = ξ1 , g¯i (., 0) = 0 for i = 1, ..., n and the system is assumed bounded input, bounded state in finite time.

Sarah K. Spurgeon


Nonlinear step-by-step observer design The Observer ˙ ξˆ1 = ξˆ2 + g¯1 (ξ1 , u) + λ1 sign(ξ1 − ξˆ1 ) ˙ ξˆ2 = ξˆ3 + g¯2 (ξ1 , ξ˜2 , u) + λ2 sign(ξ˜2 − ξˆ2 ) .. = .. ˙ ˆ ξn−1 = ξˆn + g¯n−1 (ξ1 , ξ˜2 , ..., ξñ−1 , u) + λn−1 sign(ξñ−1 − ξˆn−1 ) ξ˙n = f¯n (ξ1 , ξ˜2 , ..., ξñ ) + g¯n (ξ1 , ..., ξñ , u) + λn sign(ξñ − ξˆn ) where for i = 2, ...., n − 1 ξ˜i = ξî + λi−1 sign(ξi−1 − ξî−1 )

Sarah K. Spurgeon


(12)

Nonlinear step-by-step observer design The Observer ˙ ξˆ1 = ξˆ2 + g¯1 (ξ1 , u) + λ1 sign(ξ1 − ξˆ1 ) ˙ ξˆ2 = ξˆ3 + g¯2 (ξ1 , ξ˜2 , u) + λ2 sign(ξ˜2 − ξˆ2 ) .. = .. ˙ ˆ ξn−1 = ξˆn + g¯n−1 (ξ1 , ξ˜2 , ..., ξñ−1 , u) + λn−1 sign(ξñ−1 − ξˆn−1 ) ξ˙n = f¯n (ξ1 , ξ˜2 , ..., ξñ ) + g¯n (ξ1 , ..., ξñ , u) + λn sign(ξñ − ξˆn ) where for i = 2, ...., n − 1 ξ˜i = ξî + λi−1 sign(ξi−1 − ξî−1 )

(12)

Observation error information is not used before the corresponding sliding manifold is reached The manifolds are reached sequentially and ξ˜i − ξî converges to zero if the ξ˜j − ξˆj with j < i have already converged to zero. Sarah K. Spurgeon


Nonlinear step-by-step observer design The error dynamics e˙ 1 = e2 − λ1 sign(ξ1 − ξˆ1 ) e˙ 2 = e3 + g¯2 (ξ1 , ξ2 , u) − g¯2 (ξ1 , ξ˜2 , u) − λ2 sign(ξ˜2 − ξˆ2 ) .. = .. e˙ n−1 = ξˆn − g¯n−1 (ξ1 , ξ˜2 , ..., ξñ−1 , u) − λn−1 sign(ξñ−1 − ξˆn−1 ) e˙ n = f¯n (ξ1 , ..., ξn ) − f¯n (ξ1 , ξ˜2 , ..., ξñ ) + g¯n (ξ1 , ξ2 , ..., ξn−1 , u) − g¯n (ξ1 , ..., ξñ , u) − λn sign(ξñ − ξˆn )

Sarah K. Spurgeon


Nonlinear step-by-step observer design The error dynamics e˙ 1 = e2 − λ1 sign(ξ1 − ξˆ1 ) e˙ 2 = e3 + g¯2 (ξ1 , ξ2 , u) − g¯2 (ξ1 , ξ˜2 , u) − λ2 sign(ξ˜2 − ξˆ2 ) .. = .. e˙ n−1 = ξˆn − g¯n−1 (ξ1 , ξ˜2 , ..., ξñ−1 , u) − λn−1 sign(ξñ−1 − ξˆn−1 ) e˙ n = f¯n (ξ1 , ..., ξn ) − f¯n (ξ1 , ξ˜2 , ..., ξñ ) + g¯n (ξ1 , ξ2 , ..., ξn−1 , u) − g¯n (ξ1 , ..., ξñ , u) − λn sign(ξñ − ξˆn )

For sufficiently large λ1 , a sliding mode is attained on e1 = 0 in finite time so that e2 = λ1 sign(ξ1 − ξˆ1 ) which with (12) yields ξ˜2 = ξ2 . Sarah K. Spurgeon


Nonlinear step-by-step observer design Revised error dynamics e˙ 1 = 0 e˙ 2 = e3 − λ2 sign(ξ˜2 − ξˆ2 ) .. = .. e˙ n−1 = ξˆn − g¯n−1 (ξ1 , ξ2 , ..., ξñ−1 , u) − λn−1 sign(ξñ−1 − ξˆn−1 ) e˙ n = f¯n (ξ1 , ..., ξn ) − f¯n (ξ1 , ξ2 , ..., ξñ ) + g¯n (ξ1 , ξ2 , ..., ξn−1 , u) − g¯n (ξ1 , ..., ξñ , u) − λn sign(ξñ − ξˆn )

Sarah K. Spurgeon


Nonlinear step-by-step observer design Revised error dynamics e˙ 1 = 0 e˙ 2 = e3 − λ2 sign(ξ˜2 − ξˆ2 ) .. = .. e˙ n−1 = ξˆn − g¯n−1 (ξ1 , ξ2 , ..., ξñ−1 , u) − λn−1 sign(ξñ−1 − ξˆn−1 ) e˙ n = f¯n (ξ1 , ..., ξn ) − f¯n (ξ1 , ξ2 , ..., ξñ ) + g¯n (ξ1 , ξ2 , ..., ξn−1 , u) − g¯n (ξ1 , ..., ξñ , u) − λn sign(ξñ − ξˆn ) 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 it follows that e3 = λ2 sign(ξ2 − ξˆ2 ) which yields ξ˜3 = ξ3 . Sarah K. Spurgeon


Nonlinear step-by-step observer design Error dynamics at the nth stage Proceeding sequentially e˙ 1 = 0 e˙ 2 = 0 .. = .. e˙ n−1 = 0 e˙ n = −λn sign(ξñ − ξˆn ) and it follows trivially that a sliding mode is finally attained on en = 0 in finite time.

Sarah K. Spurgeon


Nonlinear step-by-step observer design Error dynamics at the nth stage Proceeding sequentially e˙ 1 = 0 e˙ 2 = 0 .. = .. e˙ n−1 = 0 e˙ n = −λn sign(ξñ − ξˆn ) and it follows trivially that a sliding mode is finally attained on en = 0 in finite time. Many application specific studies of this step-by-step observer framework appear in the literature The results were generalised to the MIMO case by Floquet and co-workers. Sarah K. Spurgeon


Step-by step observer design and higher order sliding modes Step-by-step observer The step-by-step procedure uses successive filtered values of the so-called equivalent output injections obtained from recursive first order sliding mode observers The approximation of the equivalent injections by low pass filters at each step will typically introduce some delays that lead to inaccurate estimates or to instability for high order systems To overcome this problem, the discontinuous first order sliding mode output injection can be replaced by a continuous second order sliding mode injection

Sarah K. Spurgeon


Higher order sliding modes The concept of Higher Order Sliding Modes (HOSM) generalise the sliding mode concept so that the discontinuity acts on higher order derivatives of the sliding variable and the applied injection is smooth In general, if the control appears on the r th derivative of s, the rth order ideal sliding mode is defined by: s = s˙ = ¨s = .... = s (r −1) = 0

Sarah K. Spurgeon


(13)

Higher order sliding modes The concept of Higher Order Sliding Modes (HOSM) generalise the sliding mode concept so that the discontinuity acts on higher order derivatives of the sliding variable and the applied injection is smooth In general, if the control appears on the r th derivative of s, the rth order ideal sliding mode is defined by: s = s˙ = ¨s = .... = s (r −1) = 0 A second order sliding mode injection - The super twisting algorithm 1

ν(s) = φ(s) + λs |s| 2 sign(s) ˙ φ(s) = αs sign(s) λs , αs

> 0

Sarah K. Spurgeon


(13)

Decentralised observation Many results consider observer design for interconnected systems but very few adopt a decentralised approach.

Sarah K. Spurgeon


Decentralised observer design for nonlinear interconnected systems Nonlinear interconnected system with n ni -th order subsystems x˙ i

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

(14)

yi

= hi (xi ),

(15)

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

x ⊂ Rni , ui ⊂ Rmi and yi ⊂ Rmi are the states, inputs and outputs of the i-th subsystem kξi (t, xi )k ≤ 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: kui k ≤ γui Sarah K. Spurgeon


Decentralised observer design for nonlinear interconnected systems Design Objective To design n ni -th order dynamical systems xˆ˙ i = Φi (t, xî , yi , ui ),

i = 1, 2, . . . , n

(16)

where xî ∈ Rni , such that the solutions xî (t) of system (16) are convergent to xi (t) exponentially for i = 1, 2, . . . , n, that is, there exist constants αi > 0 and βi > 0 such that kxi (t) − xî (t)k ≤ αi exp{−βi t},

i = 1, 2, . . . , n

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

Sarah K. Spurgeon


Decentralised observer design for nonlinear interconnected systems Design Objective To design n ni -th order dynamical systems xˆ˙ i = Φi (t, xî , yi , ui ),

i = 1, 2, . . . , n

(16)

where xî ∈ Rni , such that the solutions xî (t) of system (16) are convergent to xi (t) exponentially for i = 1, 2, . . . , n, that is, there exist constants αi > 0 and βi > 0 such that kxi (t) − xî (t)k ≤ 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 the interconnected system (14)–(15). Sarah K. Spurgeon


Decentralised observer design for nonlinear interconnected systems

i-th isolated subsystem of the interconnected system x˙ i

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

yi

= hi (xi ),

Sarah K. Spurgeon

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



i-th isolated subsystem of the interconnected system x˙ i

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

yi

= hi (xi ),

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

i-th nominal isolated subsystem of the interconnected system x˙ i

= fi (xi ) + gi (xi )ui

yi

= hi (xi ),

Sarah K. Spurgeon

(17) i = 1, 2, . . . , n,


(18)


Definition 1 The i-th nominal isolated subsystem has a uniform relative degree vector (ρi1 , ρi2 , · · · , ρimi ) and the distribution Gi (xi ) = span{gi1 (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 be linearisable - only this relative degree requirement

Sarah K. Spurgeon


Decentralised observer design for nonlinear interconnected systems Exploiting structure - Canonical Form for Design 1 P i Let ρi := m j=1 ρij for i = 1, 2, . . . n. From Assumption 1, the ρ −1

differentials dhij (xi ), dLfi hij (xi ), · · · , dLfiij hij (xi ) are linearly independent for j = 1, . . . , mi and i = 1, . . . , n. Let   hij (xi )  Lfi hij (xi )    zij =  (19) ..  := zij (xi ), j = 1, 2, . . . ρi   . ρ −1 Lfiij hij (xi ) for i = 1, . . . , n. Since the Gi (xi ) are involutive, there exist ni − ρi functions wi1 , · · · , wi(ni −ρi ) such that the Jacobian matrix of the following mapping is nonsingular: Ti : xi 7→ col(zi1 , · · · , ziρi , wi1 , · · · , wi(ni −ρi ) ) Sarah K. Spurgeon


(20)


Exploiting structure - Canonical Form for Design 2 The transformations col(zi , wi ) = Ti (xi ) are diffeomorphisms. Let    T (x) :=  

T1 (x1 ) T2 (x2 ) .. .

    

(21)

Tn (xn )

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

Sarah K. Spurgeon


Decentralised observer design for nonlinear interconnected systems Assumption 2 - required for exponential stability of the error dynamics The interconnection terms ψi (x) satisfy the following Lψi (x) hij (xi )

=

0

(22)

Lψi (x) Lfi (xi ) hij (xi )

=

0

(23)

=

0

(24)

······ ρ −2 Lψi (x) Lfi ij(xi ) hij (xi )

there exist constants

γψi such that for any x, ρij −1

Lψi (x) Lfi (xi ) hij (xi ) ≤ γψi , h

i

∂Ti (xi ) ∂xi ψi (x)

xi =Ti−1 (zi ,wi )

P (ni − m j=1 ρij )

Φi (·) ∈ R uniformly for zi

=

? Φi (zi , wi )

where

are Lipschitz with respect to wi

Sarah K. Spurgeon


Decentralised observer design for nonlinear interconnected systems Canonical Form Under Assumptions 1-2, the interconnected system becomes z˙ i1

=

z˙ i2

=

Ai1 zi1 + Bi1 ui1 + ηij (t, zi ) + Lψi (x) Lρfi i1(x−1 h (x ) i1 i i) ρi2 −1 Ai2 zi1 + Bi2 ui2 + ηij (t, zi ) + Lψi (x) Lfi (xi ) hi2 (xi )

(25) (26)

.. . z˙ imi

=

ρim −1 Aimi zimi + Bimi uimi + ηij (t, zimi ) + Lψi (x) Lfi (xii ) himi (xi ) (27)

w˙ i

=

qi (zi , wi ) + Φi (zi , wi )

(28)

yij

=

Cij zij

(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 Sarah K. Spurgeon


Decentralised observer design for nonlinear interconnected systems HOSM-based decentralised observer design The i-th subsystem of (25)-(27) can be written as z˙ ij yij (t)

ρ −1 = Aij zi + Bij uij + ηij (t, zi ) + Lψi (x) Lfi ij(xi ) hij (xi ) ,

(30)

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

(31)

Sarah K. Spurgeon


Decentralised observer design for nonlinear interconnected systems HOSM-based decentralised observer design The i-th subsystem of (25)-(27) can be written as z˙ ij yij (t)

ρ −1 = Aij zi + Bij uij + ηij (t, zi ) + Lψi (x) Lfi ij(xi ) hij (xi ) ,

(30)

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

(31)

Consider the higher order sliding surfaces defined by ρ

sij = Lgi sij = L2gi sij = · · · = Lgiij sij = 0 where sij (t) := zîj1 (t) − yij (t) and zîj1 is determined by a HOSM differentiator scheme

Sarah K. Spurgeon


Decentralised observer design for nonlinear interconnected systems HOSM Differentiator zˆ˙ ij1 νij1 zˆ˙ ij(ρij −1) νij(ρij −1)

= νij1 = −λij1 |sij | .. .

(32) ρij −1 ρij

sign(sij ) + zij1

(33)

= νij(ρij −1)

(34)

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

1 2

sign(zij(ρij −1) − νij(ρij −2) ) + zijρij zˆ˙ ijρij

= −λijρij sign(ˆ zijρij − νij(ρij −1) )

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


(35) (36)

Decentralised observer design for nonlinear interconnected systems Remarks It follows that by choosing appropriate parameters λijk , zijk (k) will converge to the k − th derivative of yij (t), yij in finite time Tij . Choose T0 > Tij and let zîj := col zîj1 , zîj2 , · · · , zîjρij for j = 1, 2, . . . , mi and i = 1, 2, . . . , n. Considering the structure of (Aij , Bij ) in (30)–(31), when t ≥ T0 , zî1 = zi1 ,

zî2 = zi2 ,

··· ,

zîmi = zimi

for i = 1, 2, . . . n. Consider the interconnected system. The analysis above shows that zîj produced by the differentiator (32)–(36), is an estimate of zij . The objective now is to estimate the variables wi for which the following assumptions are imposed. Sarah K. Spurgeon


Decentralised observer design for nonlinear interconnected systems 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 positive functions ki (zi ) such that for any variables ϑi ∈ Rni −ρi , zi and ∂qi (zi ,wi ) (zi ,wi ) wi ϑT ϑi ≤ −ki (zi )kϑi k2 where ∂qi∂w denote the i Pi ∂wi i Jacobian matrices of qi (·) with respect to the variables wi . Remarks Condition ii), Assumption 2 and condition i), Assumption 3 are fundamental in the local case and hold in any bounded compact set. If (zi ,wi ) the matrix ∂qi∂w at col(zi , wi ) = 0 is Hurwitz, then condition ii) in i Assumption 3 holds in a neighbourhood of the origin. Sarah K. Spurgeon



The Observer w ˆ˙ i = qi (ˆ zi , w ˆ i ) + Φi (ˆ zi , w ˆ i ),

i = 1, 2, · · · , n

(37)

where zî := (ˆ zi11 , zî12 , · · · , zî1ρi1 , · · · , zîmi 1 , · · · , zîmi ρi1 ) and zîjk are given by (32)–(36). Clearly the n systems defined in (37) are decoupled from each other. Let ei (t) = wi (t) − w ˆ i (t). It follows from Assumption 2 that the error dynamics are described by e˙ i

= qi (zi , wi ) − qi (ˆ zi , w ˆ i ) + Φi (zi , wi ) − Φj (ˆ zi , w î )

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

Sarah K. Spurgeon


(38)


Theorem Under Assumptions 1–3, the error dynamical system (38) is exponentially stable if for i = 1, 2, . . . , n n o inf ki (zi ) − kPi k LΦi (zi ) := βi > 0 zi

(39)

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

Sarah K. Spurgeon



Remarks From Levant HOSM algorithm, the variables zî converge to zi in finite time. Theorem 1 shows that the variables w î converges to w exponentially. The HOSM differentiator and the designed dynamics together form the exponential observer. From the structure of the designed dynamical system and the structure of Levant differentiator, it is straightforward to see that the designed dynamics are decoupled and thus the designed observers are decentralised.

Sarah K. Spurgeon



Concluding Remarks We have developed an exponential observer which exploits structure in the design including mismatched uncertainty HOSM and sliding mode differentiators have been introduced Demonstrated that the sliding mode observer paradigm translates to nonlinear and complex system scenarios

Sarah K. Spurgeon


Sliding mode observers - Spring School : sliding mode control

Sliding mode observers - Spring School : sliding mode control

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