An LPV Pole-placement approach to friction compensation as an FTC Problem(Patton).pdf

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An LPV Pole-placement approach to friction compensation as an FTC

problem

Lejun Chen, Ron Patton IEEE Fellow and Supat Klinkhieo

Abstract The concept of combining robust fault estimationwithin a controller system to achieve active Fault TolerantControl (FTC) has been the subject of considerable interestin the recent literature. The current study is motivated by theneed to develop model-based FTC schemes for systems thathave no unique equilbria and are therefore difficult to linearise.Linear Parameter Varying (LPV) strategies are well suitedto model-based control and fault estimation for such systems.This contribution involves pole-placement within suitable LMIregions, guaranteeing both stability and performance of a multi-fault LPV estimator employed within an FTC structure. Theproposed design strategy is illustrated using a non-linear two-link manipulator system with friction forces acting simulta-neously at each joint. The friction forces, considered as aspecial case of actuator faults, are estimated and their effectcompensated within a polytope controller system, yielding arobust form of active FTC that is easy to apply to real robotsystems.

I. INTRODUCTION

Friction phenomena are widely encountered in mechan-

ical/mechatronic systems and the problem of control of

systems with friction presents interesting challenges. In

recent years there has been a substantial literature on this

subject focussed on modelling of friction dynamics [1]. From

a control point of view, friction compensation strategies

that require detailed models of friction characteristics havelimitations arising from the representation of non-smooth

non-linearity and the fact that friction modelling remains an

imprecise subject, thereby resulting in difficult robustness

issues.

Friction forces degrade the performance of a mechanical

system and the concept of compensating their effect in a

closed-loop system can be viewed as an FTC problem in

which the friction forces actually act on the system as com-

ponent faults [2]. In [2],the sliding mode observer theory is

combined with a sliding mode controller to achieve a robust

method of friction force estimation and FTC compensation

for an inverted pendulum system. In a similar way to [2]

a recent investigation [3] based on a two-link manipulator

study, has shown that the fault estimation can be achieved

using an LPV estimation approach, based on a combination

of LPV estimation and LPV control within an active or

direct FTC framework. In [3] simple perturbation faults

are considered, whilst the current paper revisits the friction

compensation problem for the same manipulator system.

The LPV modelling methodology has been widely adopted

in control system design especially related to vehicle and

L Chen and R J Patton ([email protected]) are with the Departmentof Engineering, University of Hull, Cottingham, Hull, HU6 7RX, UK andS Klinkhieo is with the Synchrotron Light Research Institute, Thailand

aerospace control [4], [5]. The essential idea is that a time-

varying system can be represented via an LPV model system,

for example using a polytopic structure with the control

for each vertex system computed via efficient interior-point

algorithms and LMIs. LPV modelling can also be used to

design residual-based fault detection and diagnosis (FDD)

estimators that provide robustness to modelling uncertainty

[6], [7]. FTC studies based on the LPV concept have also

focussed on the use of FDD residuals and system reconfig-

uration [5].

The current paper extends the theoretical approach in [8]

by (a) providing a proof for the LPV estimator stability,

via pole-placement design in an LMI region, (b) including

the use of an LPV stabilizing controller and (c) introducing

simultaneously acting friction forces based on the two-

link manipulator model. Section II outlines the theoretical

foundations of LPV estimator design and introduces the

LMI-based pole-placement design utilised to guarantee the

performance and the stability of the fault estimator is deter-

mined efficiently via the solution of a set of LMIs. Section

III describes the two-link manipulator case study example.

The Stribeck friction model is outlined prior to constructing

the corresponding active LPV FTC scheme. The LMI-based

LPV pole-placement strategy is developed from the time-varying joint angle measurements and the estimated joint

friction forces are used in an active FTC compensation

mechanism in each control signal, based on the concept of

fault effect factors, first defined in [9]. Section IV gives a

brief conclusion to the work.

II. THEORETICAL FOUNDATION OF ROBUST LPV

ESTIMATOR

The LPV system with faults is described as follows:

xp(t) = Ap()xp(t) + Bp()u(t) + Ep()d(t) + Fp()f(t)yp(t) = Cp()xp(t) + Dp()u(t) + Gp()d(t) + Hp()f(t)

(1)

where, xp(t) Rn, u(t) Rr, yp(t) Rm and d(t) R

q are the states, control inputs, outputs, and disturbances.

f(t) Rg is the fault vector. Rs is a varying parametervector, and Ap(), Bp(), Cp(), Dp(), Ep(), Fp(),Gp() and Hp() are matrices with appropriate dimensions.

The assumptions that apply to system (1) are [8] that the:

(A1) System (1) is stable.

(A2) The vector (t) varies in a polytope with vertices

2010 Conference on Control and Fault Tolerant SystemsNice, France, October 6-8, 2010

WeA4.4

978-1-4244-8154-5/10/$26.00 2010 IEEE 100


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wudf(t) e

P()

F()

u

yp

f

Fig. 1. Polytopic LPV Estimation System Structure

1, 2, . . . , j (j = 2s), i.e,:

(t) : = Co{1, 2, . . . , j }

= {j

i=1

aii : ai 0,j

i=1

= 1}(2)

(A3) As the state-space matrices depend affinely on (t),system (1) is assumed to be polytopic, i.e.

Ap() Bp() Ep() Fp()Cp() Dp() Gp() Hp()

Co

Ap() Bp() Ep() Fp()

Cp() Dp() Gp() Hp()

i = 1, . . . , j

(3)

(A4) Cp(), Dp(), Gp(), and Hp() are parameter inde-pendent, i.e.

Cp() = Cp, Dp() = Dp, Gp() = Gp, Hp() = Hp (4)

The design of a polytopic estimator can be written as

xf(t) = Af()xf(t) + Bf() u(t)

yp(t)

f(t) = Cf()xf(t) + Df()

u(t)

yp(t)

(5)

Therefore, the estimated error vector ef(t) = f(t) f(t) R

g, is minimised. Here u(t) and yp(t) are defined in (1), andxf(t) Rn is the state vector of the estimator, f(t) is theestimation of the fault f(t). Af(), Bf(), Cf() and Df()are matrices with appropriate dimensions, to be designed.

The LPV estimator (5) can be rewritten as:

F() :=

Af() Bf()

Cf() Df()

Co

Af(i) Bf(i)

Cf(i) Df(i)

i = 1, . . . , j

(6)The system structure is shown in Figure 1.

Define xpf(t) and wudf(t) to be [xp xf]T

and

[u(t) d(t) f(t)]T respectively. Rewrite (1) as

xp = Axp + B1wudf + B2f (7)

e = C1xp + D11wudf + D12f (8)

yp = C2xp + D12wudf + D22f (9)

Above system in Figure 1 can then be expressed as

xpf = A()xpf(t) + B()wudf(t)

ef = C()xpf(t) + D()wudf(t)(10)

where,

A() = A0 + BF()C B() = B0 + BF()D21

C() = D12F()C D() = D11 + D12F()D21

A0 =

A 0

0 0

B0 =

B1

0

B=

0 B2

I 0

C =

0 I

C2 0

D12 =

0 D12

D21 =

0

D21

The definition of the LMI region and the existence theorem

for assignable poles inside this region are given as follows:

Definition (LMI Region): A subset D of the complex planeis called an LMI region if there exist a symmetric matrix

= [kl] and a matrix = [kl] such thatD = {z C : fD < 0} (11)

with

fD := + z + zT = [kl + klz + lkz]l


3/6

where,i = 1, . . . , j. The main result of this section is statedin Theorem 0.2, which provides the solution of the Problem

1.Theorem 0.2: Consider the LPV system in (1) with as-

sumption (A1)-(A3). Let NR = [NT1 NT2 ]

T and NS =[VT1

VT2

]T denote bases of the null spaces of [BT2

DT12

]and [C2 D21] respectively. There exists a polytopic LPVestimator that can determine the solution of Problem 1 if

matrices 0 < R = RT Rnn, 0 < S = ST Rnn canbe found that:

NR 0

0 I

TAp(i)R + RA

Tp (i) 0 B1(i)

0 I D11

BT1 (i) DT11 I

NR 0

0 I

< 0

(16)NS 0

0 I

TSAp(i) + A

Tp (i)S SB1(i) 0

BT1 (i)S I DT11

0 D11 I

NS 0

0 I

< 0

(17)

UT1

(klS+ klAp(i)S+ lkSAT

p (i))U1 < 0 (18)

VT1 (klR + klRAp(i) + lkAT

p (i)R)V1 < 0 (19) R II S

0 i = 1, . . . , j (20)

Proof: Based upon Theorem 0.1, the poles of the LPV

system (10) lie in LMI region D if and only if there existsX, such that,

MD(A(), X) = [klX+ klA()X+ lkXAT

()] < 0(21)

By Lemma 1, Theorem 0.1 and considering the notations

in (5-14), there exists a polytopic LPV fault estimator (5)

which solves the Problem 1 if:

(i) + UT

xF(i)V + V

TFT(i)Ux < 0 (22)

(i) + PTF(i)Q + Q

TFT(i)P < 0 (23)

where,

(i) =

XA0(i) + AT0 (i)X XB(i) 0

BT(i)X I DT110 D11 I

(24)

(i) = klX+ klA0X+ lkXAT0

(25)

Ux = [BTX 0 DT12] (26)

V = [C D21 0] (27)

P = lkBT, Q = CX (28)

Based on the projection lemma, the LMIs of (22) and (23)

hold for some F(i) if and only if:

WTUx

(i)WUx < 0 (29)

WTV(i)WV < 0 (30)

WTP(i)WP < 0 (31)

WTQ(i)WQ < 0 (32)

where, WUx

, WV, WP and WQ denote any bases of thenull spaces of Ux, V, P and Q, respectively.

Observing that:

Ux =

BT 0 DT12 X 0 00 I 0

0 0 I

(33)U = B

T 0 DT12 (34)Q = C (35)The basis for the null space of Ux and Q is given by:

WUx

=

X1 0 0

0 I 0

0 0 I

WU (36)WQ = X

1WQ (37)

where, WU and WQ denote any basis of the null space of Uand Q. Hence, the inequalities (29) and (32) can be rewritten

as:WTU (i)WU < 0 (38)

WTQ(i)WQ < 0 (39)

with

(i) =

A0(i)X1 + X1AT

0(i) B0(i) 0

BT0

(i) I DT110 D11 I

(40)

(i) = (X1)Tkl + (X

1)TklA0 + lkAT0 X

1 (41)

X and X1

can be partitioned as:

X =

S N

NT

X1 =

R M

MT

(42)

where S,R,M,N Rnn and S,R,M > 0, and standsfor the matrix entries which are not of interest.

Then, X and X1 are substituted into (24), (25), (40) and(41), yielding (i), (i), (i) and (i).

(i) =

SAp(i) + ATp (i)S A

Tp (i)N SB1(i) 0

NTAp(i) 0 NTB1(i) 0

BT1 (i) BT1 (i)N I D

T11

0 0 D11 I

(43)

(i) =

Ap(i)R + RATp (i) Ap(i)M B1(i) 0

MTATp (i) 0 0 0

BT1 (i)S 0 I DT11

0 0 D11 I

(44)

(i) = kl

S N

NT

+ kl

Ap(i)S Ap(i)N

0 0

+ lk

SATp (i) 0

NTATp (i) 0

(45)

102


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(i) = kl

R M

MT

+ kl

RAp(i) 0

MTAp(i) 0

+ lk

ATp (i)R A

Tp (i)M

0 0

(46)

LetNR =

U1

U2

andNS =

V1

V2

denote bases of the

null spaces of [BT

2 DT

12] and [C2 D21] respectively, where,U1 and V1 span the null spaces of BT2

and C2. Then, thebases of the null spaces of U, V, P and Q are given by:

WU =

U1 0

0 0

0 I

U2 0

, WV =

V1 0

0 0

0 I

V2 0

(47)

WP =

U1 0

0 0

, WQ =

V1 0

0 0

(48)

Substituting (47), (48), (43), (44), (45) and (46) into (30),(31), (38) and (39), to yield

NR 0

0 I

TAp(i)R + RA

Tp (i) 0 B1(i)

0 I D11

BT1 (i) DT11 I

NR 0

0 I

< 0

(49)NS 0

0 I

TSAp(i) + A

Tp (i)S SB1(i) 0

BT1 (i)S I DT11

0 D11 I

NS 0

0 I

< 0

(50)

UT1

(klS+ klAp(i)S+ lkSAT

p (i))U1 < 0 (51)

VT1

(klR + klRAp(i) + lkAT

p (i)R)V1 < 0 (52)

Based on the matrix completion result, the condition X >

0 is equivalent to: R I

I S

0 (53)

which completes the proof of Theorem 0.2.

Once the matrices R and S are obtained, the LPV estimatorcan be constructed as following Algorithm 1:

Step1. Computing the full rank matrices M, N using SVDsuch that:

M NT = I RS (54)

Step2. Computing X as the unique solution of the linearmatrix equation:

X I R

0 MT

= S I

NT 0

T(55)

Step3.Compute F(i) by solving (22) and (23).Step4.Solve the polytopic LPV estimator:

F() =j

i=1

aipF(i) (56)

where aip is any solution of the convex decompositionproblem:

=

j

i=1aipi (57)

III. TWO-LINK MANIPULATOR CASE STUDY

A. Robust LPV Fault Estimator for Two-Link Manipulator

A two-link manipulator demonstrator system with actuator

faults is illustrated. The actuator faults are considered as the

friction forces acting at the joints of system, changing in

terms of the angular velocity of the joints. Four types of

dynamic torques that arise from the motion of the manipula-tor: Inertial, Centripetal, Coriolis and Friction torques. The

model is shown in Fig.2. The position of the two arms are

defined by the joint angles = [1 2]T. The system inputsare u = [u1 u2]

T, which are the torques actinging onto the

joints.

Load

m g2

m g1

lc2

lc1

l1

fric2

u2

fric1

u1

g

1

2

.

.

Fig. 2. Two-link Manipulator Structure

The dynamic of the manipulator [11], [12] is shown as

() + O(, ) + g() = u fric() (58)

where, () R22 is the manipulator inertia tensor matrix,O(, ) R2 is the function containing the Centripetal andCoriolis torques. g() R2 and fric() the gravitationaland friction torques,respectively. The equations of motion

and the physical system parameters are described in [3].

The friction forces acting on the joints is described by the

discontinuous Stribeck friction model [13].

ffric = g(xp)Sign(xp) (59)

where, Sign(xp) =

1 if xp < 0

[1, 1] if xp = 0

1 if xp > 0

and g(xp) = Fc +(FsFc)exp(|xp|/vs) is the Stribeckfriction function with Fc and Fs are the Coulomb and staticfriction levels, respectively and vs, > 0 are the Stribeckvelocity and shaping parameters, respectively. In the friction

simulation, the following parameter values are used: Fc =5N, vs = 0.15ms

1, Fs = 2.5N and s = 1.22. Here, thequadratic terms O(, ) are not considered since they arenot bounded. (58) can be simplified as shown in (60).

() + g() = u fric() (60)

where,

() =

m1lc

21 + m2lc

21 + I1 m2l1lc2 cos(1 2)

m2l1lc2 cos(1 2) m2lc22 + I2

103


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g() =

(m1lc1 + m2l1)g sin(1)

m2glc2 sin(2)

The nonlinear term in () is clearly a bounded function

1() = cos(1 2) [1 1] (61)

Hence, () can be represented by a polytope whosevertices are defined by:

() Co{1 2} (62)

To construct a state-space formulation, the vector field

gg() with R2 can be arranged in the form of Gg ()and function 2() can now be defined which is bounded.

sin(1) = (sin(1)

1)1 = 2()1 (63)

where, 0.2 2 1.From the bound of function 2() in terms of the angle

, Gg () is considered as a polytope as follows:

Gg() Co{Gg1

, Gg2

, Gg3

, Gg4

} (64)

To define the state space representation of the two-link

system, let: x(t) = [1(t) 2(t) 1(t) 2(t)]

.

The LMI constraints with nonlinear system dynamics are

then given by the following descriptor system:I 0

0 ()

x(t) =

0 I

Gg() 0

x(t)+Wbu(t) (65)

The state space equation can be expressed as follows:

x(t) = A()x(t) + B()u(t) (66)

where, A() = 1 0 I

Gg() 0

and B() = 1Wb.

With the faults,

x(t) = A()x(t) + B()u(t) + Fafa(t)

= Aijx(t) + Biu(t) + Fafa(t)(67)

where, Fa is fault distribution matrix and fa representsactuator faults which represent the friction forces acting on

each joint. The actuator fault estimate fa(t) in system (67)can be implemented by using Algorithm1.

The open-loop two-link manipulator system is unstable.

Therefore, a stable closed-loop system has to be configured

to satisfy (A1). In this case, a common constant gain matrix

K is build to stabilise the fault-free open-loop system oneach vertex. Let Lc = KSc, then the quadratic stabilityconditions for 8-vertex system can be written as:

AijSc + BiLc + ScATij + L

Tc B

Ti 0 (68)

The gain matrix can be calculated based upon solving

above LMIs.

The desired LMI region D is forced to lie in the half-planex < 0.7. Also, choosing to be 2.7550, after 15 iterations,the estimator Fi() is calculated, corresponding to each ver-tex. The LPV estimator is then built through the combination

of Fi(), i = 1, 2, . . . , 8. Fig.3 shows the results of thefriction estimation, with a Gaussian random disturbance d(t)of zero-mean and variance 0.02. Simulation results show thatrobust LPV fault estimator provides a good online estimation

performance for both friction forces simultaneously. The blue

curve represents the fault signal and the red curve depicts

the estimated fault signal. However, the estimation errors

are existed since the two faults affect the system dynamicssimultaneously. The estimation performance for each fault is

affected by the existence of another fault.

10 15 20 25 30 35 4010

0

10

Time [s]

FrictionForce1[N]

10 15 20 25 30 35 4010

0

10

Time [s]

FrictionForce2[N]

Fig. 3. Fault Estimation provided by the Polytopic LPV Estimator

B. Active LPV FTC Scheme

The dynamic system of in (67) includes an additive term

of the actuator faults. However, the faults can have a multi-

plicative effect in the system representation. A multiplicative

actuator fault representation can be defined as:

x(t) = Aijx(t) + Bi[Ir a

(t)]u(t) (69)

where, a is the so-called fault-effect factor, and a =diag[a

1, a

2, . . . , ar ], and 0

ai < 1 represents a fault in

the ith actuator and ai = 0 means that ith actuator operates

normally (fault-free), whilst ai > 0 means that some degreeof fault effect occurs in the actuator [9]. The distribution

matrix Fa is equal to the matrix B in an actuator fault case.The estimation offault-effect factora(t) is determined fromthe fault estimation f(t) provided by the LPV fault estimator.In Section III-A, a constant controller is developed to achieve

P()

KF T C(a, )

LPV Estimator

uF T C(t)

a

Fig. 4. Active LPV Fault-tolerant Control Scheme

the stability of the 8-vertex system to satisfy assumption

104


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(A1) that the framework of the LPV estimator design can

be implemented. The LPV controller can be expressed by:

Klpv = K+ K() (70)

where, K is the developed constant controller. The structureof the active LPV FTC system is shown in Fig.4, wherein,

KF T C(a, ) is the adaptive LPV controller for the FTC

mechanism, depending on the on-line estimation a andmeasurement .

Theorem 0.3: From a design consideration consider the

system in (69) with i = 1, . . . , r actuator faults (a = 0)acting independently within the control system with LPV

gain matrix Klpv . Define = [I a(t)]. The new controlaction(assuming non-zero fault effects) is given as:

uF T C(t) = ( + (I )Z)Klpv x(t) = KF T C(

a, )x(t)(71)

where is required to be full row rank. represents thePseudo-Inverse. Z is the free matrix. Degree of freedom ofdesigning the LPV controller can be fully utilised through

choosing various Z [14].Proof: Based upon the system state equation with

faults (69) and the new control input uF T C(t), the faultcompensation system is given by

x(t) = Ax(t) + B uF T C(t)

= Ax(t) + B( + (I )Z)Klpv x(t)

= Ax(t) + Bunom(t)

(72)

It can be seen that the term (I a) acting on the system of(69) can be removed through replacing u(t) with uF T C(t),which completes the proof.

Fig.5 shows the system outputs after the fault being

compensated. The nominal performance of system outputs

generated by the LPV controller can be effectively recovered

after the fault compensation being activated at 40s. Thisdemonstrates very well the fault-tolerance of the active LPV

FTC system.

0 20 40 60 803.5

3

2.5

2

1.5

1

0.5

0

0.5

Time[s]

RotationAngle1[rad]

0 20 40 60 800.5

0

0.5

1

1.5

2

2.5

3

3.5

Time[s]

RotationAngle2[rad]

1

2

LPV Controllerbeing activated

at 40s

LPV Controllerbeing activated

at 40s

Fig. 5. Nonlinear System Output Responses with Active FTC

IV. CONCLUSIONS

This study concerns the development of an LPV fault

estimation scheme within an active or direct FTC for a tradi-

tional non-linear robot manipulator problem with challenging

control requirements. Friction forces at the manipulator joints

are considered as actuator faults acting in direct competition

with the control system. As a part of the structure of an LPV

fault compensation scheme, the principle of LPV robust fault

estimation with pole placement has been introduced through

via a set of LMIs using efficient interior point algorithms. It

is shown that not only can the robustness of the estimation

error be improved, corresponding to the system controlinputs, disturbances and the faults, but also the structure of

the robust LPV estimator can be modified online through

the measurement of the varying parameters. The friction

forces acting in the manipulator joints are generated using

the Stribeck friction model. The simulation results show that

the simultaneously acting friction forces can be estimated

effectively using the polytopic estimator and the actuator

faults can be compensated through the developed active LPV

FTC scheme.

V. ACKNOWLEDGMENTS

The authors wish to thank the European Commission

for research funding in the contract FP7-233815, AdvancedFault Diagnosis for Safer Flight Guidance and Control

(ADDSAFE).

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An LPV Pole-placement approach to friction compensation as an FTC Problem(Patton).pdf