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A Robust Approach to Fault Tolerant Controller Design Based on

GIMC structure for Non-Minimum Phase systems

Abstract-This paper represents a robust scheme for

designing fault tolerant controller (FTC) based on

Generalized Internal Model Control structure, and extends

it to non-minimum phase LTI systems. In this scheme, first

the nominal controller is designed to have a closed-loop

stability and acceptable performance, and then by

correcting the nominal controller based on the idea of

GIMC, a robustification signal is imposed to compensate the

actuator faults. The proposed scheme, deals with non-

minimum phase systems by utilizing the idea of

transmission zero-assignment. It will be shown that the

problem of zero-assignment problem for state-accessible

systems is equivalent to a pole-placement problem with state

feedback on a reduced-order subspace, which always has asolution. The advantages of the proposed control scheme

have been verified by the simulation on the quadruple-tank

process subjected to the actuator fault.

Keywords: Fault Tolerant Controller (FTC), Non-

Minimum Phase System, Transmission Zero-Assignment,

Generalized Internal Model Control (GIMC).

1. Introduction

In the early stages of control applications, closed-loopperformance was the main objective for the control

engineer. To achieve this goal, the implementations of

these feedback configurations involved sensors, actuators,electronic instrumentation, and signal processors.

However, during a normal operation, these parts could

fail in some degree, and the resulting performance of the

closed-loop system will be largely deteriorated, or even

instability can be observed. So, it is necessary to design

controllers which are capable of not only diagnosing

faults in closed-loop systems, but also maintaining

closed-loop stability and performance in the presence offaults. A closed-loop control system which can tolerate

component malfunctions, while maintaining desirable

performance and stability properties is said to be a Fault-

Tolerant Control system (FTCs) (see, [1]-[2]).

FTC can be approached from two perspectives:passive and active [3]. Passive approaches make use ofrobust control techniques to ensure that a closed-loop

system remains insensitive to certain faults using constant

controller parameters and without use of on-line fault

information [4]-[5]. Active approaches react to the

system component faults actively by reconfiguring

control actions so that the stability and acceptableperformance of the entire system can be maintained. In

such control systems, the controller compensates for the

impacts of the faults either by selecting a pre-computed

control law or by synthesizing a new one on-line [6]-[7].

The survey papers [8] and [9] give the state of the art in

the field of FTC.

One of the approaches in FTC design is control

systems based on Generalized Internal Model Control.

In traditional feedback control systems, it is well

recognized that a robust controller design is usuallyachieved at the expense of performance. This is not hard

to understand since most robust control design techniques

are based on the worst possible scenario which may never

occur in a particular control system. The GIMC structure

was proposed for this problem [10]. This structure is IMC

generalized by introducing outer feedback controller. The

distinguished feature of GIMC is that, this controller

architecture has the potential to overcome the conflict

between performance and robustness in the traditional

feedback framework by showing structurally how the

controller design for performance and robustness may bedone separately. This GIMC structure was used as FTC in

[11]-[12] and [13]. In [11] it is applied to gyroscope

system so that to keep stability and performance of

perturbed plant when one of the sensors breaks down. In

[12], it is applied to dc-motor as speed regulation

problem and experimentally shows good performance ofthis structure in the case of sensor failure. Moreover,

GIMC has been applied to Magnetic Suspension System

in [13] which is unstable system and experiment was

carried out in the case of sensor failure. These research

works have shown that the GIMC is very useful for

system perturbation. However, all those previous resultsare restricted to only minimum phase systems. In other

words, application of GIMC approaches has been limited

to the minimum phase systems. This prohibits applying

the powerful GIMC structure technique into the plants

which are non-minimum phase.

The objective of this paper is to extend the use of

powerful GIMC structure into the case of non-minimum

phase systems. In particular, the paper extends concepts

of GIMC presented in [10]-[11] and [12] into the area ofnon-minimum phase systems with eliminating this

limitation in controller design. In this paper, we applyGIMC structure to quadruple-tank process which has

unstable zeros. Then we utilize the idea of zero-

assignment to place the unstable zeros of the plant in the

stable region [14]. Thus, it will be possible to implement

the GIMC structure on this plant.

The paper is organized as follows. Section 2 describesthe problem formulation. Section 3 presents the structure

of the FTC scheme. First the general methodology is

introduced and the extension to non-minimum phase

system is explored. At last, the structured algorithm for

designing FTC is presented. Simulation results on aquadruple-tank process subjected to the actuator faults aregiven in Section 4. Finally, Section 5 makes some

concluding remarks.

M. Pezeshkian*, H. Ozma**, and M. J. Khosrowjerdi***

*M.S. Graduated from Sahand University of Technology of Tabriz,m_pezeshkian@sut.ac.ir

**M.S. Graduated from Sahand University of Technology ofTabriz, h.ozma@sut.ac.ir

*** Assistant Professor of Sahand University of Technology of

Tabriz, khosrowjerdi@sut.ac.ir

2011 2nd International Conference on Control, Instrumentation and Automation (ICCIA)

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2. Problem Formulation

The objective of this paper is to design FTC to

compensate the faults occurred in the actuators of non-

minimum phase LTI systems. The faults addressed in this

paper are modeled as additive faults. But, in problem

formulation we consider the general system with possible

faults in both actuators and sensors. Consider a system

( )P s affected by possible faults lf R described by

(1)1

2

( )x Ax Bu F f

P sy Cx Du F f

-

where nx R represents the vector of states, mu R the

vector of inputs, andp

y R the vector of outputs. Thus

matrix 1n p

F Ru

stands for the distribution matrix of the

actuator or system faults, and 2p l

F Ru

for sensor faults.

The nominal system ( , , , )A B C D is consideredcontrollable and observable. On the other hand, the

system response y can be analyzed in a transfer matrix

form (frequency domain) as follows:

(2))()()()()( sfsPsusPsy fu

where

(3)

1

1 2

( )

( )

u

f

P C sI A B D

P C sI A F F

A left coprime factorization for each transfer matrix can

be derived as follow [15]

(4)

1

1

u

f f

P M N

P M N

where , , fM N N RHf and can be obtained as follow

fM N N

(5)1 2

2

.A LC L B LD F LF

C I D F

where matrix n pL Ru

is such that A LC be Hurwitz.Now, it is assumed that a nominal controller K stabilizes

the nominal plant uP , and it provides a desired closed-

loop performance. The controller K is considered

observable, and consequently, it can also be expressed bya left coprime factorization as

(6)1K V U

where ,U V RH f and

K K

K K

A BK

C D

(7)> @K K K K K K K

K K

A L C B L D LU V

C D I

The nominal controller can be synthesized following

classical techniques or optimal control: PI , PID ,

2/LQG H , Hf loop shaping design, and so on. The

proposed scheme for designing fault tolerant controller is

depicted in Figure 1. In this structure, the feedback

system is controlled only with nominal controller

1K V U while there are no faults in the systems. Butwhen fault occur in any component of the system, the

compensator signal q is activated to attenuate or

compensate the effect of the fault in the closed-loop

system. In next section, the way to how to produce the

compensator signal q based on GIMC structure is

introduced.

Figure 1: The proposed FTC scheme

3. FTC Structure

In previous section, it has been seen that the proposed

FTC structure is based on the production of the

compensator signal q , see Figure 1. In this section, we

introduce the way to construct the compensator signal q

based on GIMC structure.

3.1 Overall FTC structure

Consider the nominal plant uP and let K be a linear

stabilizing controller for uP . Suppose that K and uP have

the coprime factorizations as Equations (4) and (6). Then

every controller K that internally stabilizes uP can be

written in the following form:

1( ) ( )K V QN U QM

where Q RHf and det( ( ) ( ) ( )) 0V Q Vf f f z [15]. A

new implementation of the controller parameterization

for producing the compensator signal q is shown in

Figure 2.

Figure 2: FTC structure for producing signal q

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As can be seen from Figure 2

(8)( ) ( ) ( ) ( ) ( )s N s u s M s y sX

In fault detections, this signal is called residual signal,

the difference between the estimated output and the true

output of the system [1]. If system is working in fault free

condition, then 0X and the control system will be

solely controlled by the nominal controller 1K V U . In

the other hand, when fault occur in the system, then

0X z and the compensator signal q will be active to

compensate or attenuate the effect of the occurred fault.

Since the signal X represents the estimated output error

(residual), this signal contains valuable information in

case of a component failure.

Define the following nominal closed-loop transfermatrices:

i. input sensitivity: 1i uS I KP { ii. output sensitivity: 1o uS I P K {

iii. complementary output sensitivity: 1o u uT I P K P K

{

Lemma 1. In the FTC configuration of Figure 2

considering additive faults, the resulting closed-loop

characteristics for the control signal u and output y are

given by

1 1( )( )i i fu S Kr S V UM Q N f

(9)1 1

( )( )o o fy T r S M I NV Q N f

and the closed-loop system is stable, provided that

Q RHf .

Proof. From the Figure 1, for signal u we have:

> @ > @1 1( ) ( )u V q U r y V Q U r yX

> @1 ( ) ( )V Q Nu My U r y

by substitution of Equation (4) in above equality:

1f u fu V QN f Ur UP u UP f

by some arithmetic operation we can deduce following

equation

1 1i i fu S Kr S V UM Q N f

by substitution ofu from above equation in Equation (2),

signaly can be deduced.

By considering Equation (9), variety of scenarios for

designing the compensator signal q can be introduced.

For instance, for minimizing the fault effect at the sensorsignal, the problem 1 can be expressed:

Problem 1: Consider Figure 2, given 0J ! find the

compensator Qsuch that fyT Jf .

For solving Problem 1, according to Equation (9),

following optimization strategy is suggested:

1 1min ( )( )o f

Q RH

S M I NV Q N f

f f

(10)min ( , )L Q

Q RH

F G Qf

f

where QG represents the generalized plant given by (see

Figure 3)

(11)

1

0

o f o u

Q

f

S P S P V G

N

In Equation (10), LF represents lower Linear Fractional

Transformation (lower LFT). Thus, the problem ofdesigning fault tolerant controller leads to the problem of

solving a robust control problem that can be handled

easily using standard softwares such as Matlab [17].

Figure 3: LFT framework for compensatorQ

In a special case, if the nominal plant uP has a stable

inverse, i.e. be a minimum phase system, a complete

output decoupling for faults can be achieved. Thefollowing lemma suggests the best optimal solution to

problem 1, when the nominal plant has a stable inverse.

Lemma 2. If the nominal plant satisfies uP,1

uP RH

f ,

then1

N RH

f , and by considering Equation (9) and

taking1

Q VN RH

f , the resulting output signal is

completely decoupled from the faults, that is

1i fu S Kr N N f

(12)oy T r

Proof. By considering1

Q VN

and substituting it in

Equation (9), the Equation (12) can be easily deduced.

Note that the compensation proposed in Lemma 2 is

particularly useful for an actuator or system faults, since

the output is perfectly decoupled from faults. In addition,

the above Lemma is useful when the nominal plant uP be

the minimum-phase systems.

In the following sub-section, we introduce a method

to overcome to this weakness and expand the above

lemma to non-minimum phase systems.

3.2 Expansion of FTC to non-minimum phasesystems

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In this sub-section, our goal is to introduce the

method to assign the unstable zeros of the non-minimumphase system to the stable region by using the idea of

zero-assignment to overcome the weakness of the Lemma

1 [14]. In this method, we will show that the problem of

zero-assignment problem for state-accessible systems is

equivalent to a pole-placement problem with statefeedback on a reduced-order subspace, which always hasa solution.

Consider the nominal system described by Equation

(1). In this paper, we deal with the actuator or system

faults, so 2 0F . And, without loss of generality, assume

also that 0D and pair ( , )A B is controllable. Assume

further that matrix CB is non-singular, i.e. ( )rank CB m .

This system is written here for simplicity

(13)1x Ax Bu F f

y Cx

It can be shown that there exist an orthogonal matrixn n

T Ru such that

2[0 ]T

TB B

where 2m m

B Ru and is non-singular. The matrix T can

be computed via QR decomposition on the matrix B .

By using the coordinate transformation x Txl , the

distribution matrix of the actuator faults 1F from

Equation (13) has the following structure

(14)11

112

c

FF TF

F

In addition, the nominal system described by Equation

(13), when 0f , has the following representation

(15)11 12 11

21 22 2 22

0

ccBA

A A xxu

A A x Bx

11 22

xy C C

x

(16)

where 1n m

Rx , 2m

Rx and other matrices have

appropriate dimension. In addition, the matrix 2m m

RC u

is non-singular. Transmission zeros of this system is

given by [14]

111 12 2 1 0n mI A A C CO

.

Define new output for system (15) and (16) as

(17)1y y Mx

where( )n n m

M Ru is a design matrix. Substituting for

1x and y from Equations (15) and (16) in Equation (17)

yields:

11 22

S

xy S

xS

(18)

where

(19)

1 1 11

2 2 12

S C MA

S C MA

Now, transmission zeros of the transformed system (15)

with a new output (18) is given by:

(20)111 12 2 1 0n mI A A S SO

Equation (20) is equivalent to the pole-placement for

the pair 11 12( , )A A with the state feedback matrix

12 1zK S S . It can be easily shown that this pole-

placement problem, and in fact, this transmission zero-

assignment problem have always a solution. For this,

consider the following lemma.

Lemma 3. The matrix pair 11 12( , )A A is controllable if

and only if the pair ( , )A B is controllable.

Proof. Because of the special form of the plant (15) and

using the fact that 2det( ) 0B z , it follows that

11 12

21 22 2

[ ]0

rank sI A B rank sI A A

A sI A B

> @11 12rank sI A A m for all s

This implies

11 12[ ] [ ]rank sI A B n rank sI A n m $

and from the Popov-Belevitch-Hautus (PBH) rank test it

follows that ( , )A B is controllable if and only if the pair

11 12( , )A A is controllable.

By assumption, the pair ( , )A B is controllable. Thus

according to Lemma 3 and appropriate selection of the

gain matrix1

2 1zK S S , there is always solution to

transmission zero-assignment problem. Therefore, we can

place right half-plane zeros of the plant, if there was any,

in the left half-plane and convert non-minimum phase

system to minimum phase system.By substituting for 1S and 2S from Equation (19) in

the gain matrix zK , we have

12 12 1 11( ) ( ) zC MA C MA K

or

11 12 2 1( )z zM A A K C K C

by appropriate selection of zK , the matrix 11 12( )zA A K is

non-singular and we have

(21)1

2 1 11 12( )( )z zM C K C A A K

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By finding M from Equation (21) and substituting it in

Equation (17), the compensated plant with y as a new

output has transmission zeros placed in an appropriate

place.

Remark. The remarkable notice in this method is that it

is clear that 1lim ( ) 0x t

as to f

. Thus, according toEquation (17)

lim ( ) ( )t

y t y t o f

which shows that this method has no impact on closed-

loop system tracking problem.

FTC design algorithm:

Data: given plant with Equation (1)

Assumption: ( , )A B is controllable and ( , )A C is detectable

2 0D F

( )rank CB m

x Step1) For nominal plant in Equation (1), designnominal controller to achieve desired performance

x Step2) find left coprime factorization of nominalplant and nominal controller as Equations (4) and

(5)

x Step3) assume the overall structure of the FTC asdepicted in Figure 2

x Step4) solve Problem 1 and find compensator Q If the nominal plant was stable and also was minimum

phase, then go to step 10. But if the plant was stable and

non-minimum phase, continue

x Step5) using coordinate transformation T ,construct cF as Equation (14). Furthermore,

transform the nominal plant uP to new coordinate

system as Equations (15) and (16)

x Step6) find zK such that the poles of the matrix11 12( )zA A K is placed in appropriate place

x Step7) find M according to Equation (21)x Step8) construct the compensated plant (15) and

(18)

x Step9) replace 1( , , , )A B F C in Equation (1) with( , , ),c c cA B F S and construct new uP and fP in

Equation (3). Then go to step 2

x Step10) Find the compensator Q as Lemma 2.The above algorithm is constructive and can be

implemented using standard softwares such as Matlab

[16].

4. Numerical example: the quadruple-tank process

To demonstrate the effectiveness of the proposed FTC

scheme, we consider the quadruple-tank process [17]. A

linearized dynamic model of the quadruple-tank system is

given in the state space formulation as [17]:

( )

0.50 00

0 0.500

0.0159 0 0.0419 0 0.035700 0.0111 0 0.0333 0 0.01050 0 0.0419 0 0 0.10770 0 0 0.0333 0.07280

x x u f

y x

where 4x R is the level of water in each tank,

1 2( )T

u u u is the voltage applied to pumps 1 and 2,

1 2( )T

f f f is the actuator faults associated with pump 1

and 2. The linearized system is stable and transmission

zeros of the system is located at 1 0.1336z and

2 0.2088z . So the system has one unstable zero that

made it non-minimum phase system. According to

method described in section 3.2, first by coordinate

transformation x Txl , we transfer the system into new

coordinate as Equations (14) and (15). Then, we find gain

matrix zK so that to locate the transmission zero of the

system in 1 0.1336z , and 2 0.2088z . For this, the

calculated M according to Equation (20) is

6.8943 3.21674.0387 1.2166

M

. Now, for the compensated plant

with new output as in Equation (17), implement the

algorithm. For designing fault tolerant controller, we take

the nominal controller K as a Hf -controller, which

calculated in [18]. In order to analyze the effectiveness of

the proposed FTC, the actuator fault is applied as

depicted in Figure 4, and also assumed that only the

actuator 1 is faulty. The output of the system with

nominal Hf

-controller is shown in Figure 5. As shown

in this figure, the output can perfectly track the reference

input. The output of the system with nominal controller

when fault occur in the actuator is shown in Figure 6. As

can be seen, the output cannot track the reference inputproperly with the occurrence of the actuator fault. In

Figure 7, the output y by using the proposed FTC

structure has been shown. The simulation was carried out

by assuming fault detections delay of about 2 second. As

shown in this figure, although the open-loop system is

subjected to actuator fault, one can see that the closed-loop system is internally stable and tracking performance

is accurate. The same results can be achieved with

constant faults (bias) and other kind of time-varyingactuator faults that show the effectiveness and good

performance of the proposed FTC scheme.

Figure 4: Occurred fault in actuator 1

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Figure 5: Outputs of the system with nominal controller without

actuator fault

Figure 6: Outputs of the system with nominal controller when actuator

fault occur at t=300s

Figure7: Output signals with FTC, when actuator fault occur in t=300s

5. Conclusion

In this paper a robust scheme for designing fault

tolerant controller (FTC) based on Generalized Internal

Model Control structure, and its extension to non-

minimum phase LTI systems is presented. In this scheme,

first the nominal controller is designed to achieve closed-loop stability and acceptable performance, and then by

correcting the nominal controller based on the idea ofGIMC, a robustification signal is imposed to compensate

the actuator faults. In particular, we extend the concepts

of GIMC into the area of non-minimum phase systems.

The proposed scheme, deals with non-minimum phasesystems by utilizing the idea of transmission zero-

assignment. It has been shown that the problem of zero-

assignment problem for state-accessible systems is

equivalent to a pole-placement problem with state

feedback on a reduced-order subspace, which always has

a solution. At last, a constructive algorithm for FTC

design has been proposed and simulation results on the

quadruple-tank process, which has unstable zeros,

subjected to the actuator fault have shown satisfactorylevel of control performance.

References

[1] J. Chen and R. J. Patton,Robust Model-Based Fault Diagnosis forDynamic Systems. Norwell, MA: Kluwer, 1999.[2] M. Blanke, M. Kinnaert, J. Lunze, and M. Staroswiecki,Diagnosis and Fault Tolerant Control, Berlin, Germany: Springer-

Verlog, 2003.[3] R. J. Patton, "Fault Tolerant Control: The 1997 situation," In theProceeding of the 3

rd

IFAC symposium on fault detection, supervisionand safety for technical processes, pp. 1033-1055, 1997.[4] Sh. Suryanarayanam, M. Tomizuka, and T. Suzuki, " Design ofSimultaneously Stabilizing Controllers and Its Application to Fault-

Tolerant Lane-Keeping Controller Design for Automated Vehicles,"IEEE Transaction on Control Systems Technology, Vol. 12, NO. 3, pp.

329-339, 2004.

[5] H. Niemen, and J. Stoustrup, "Passive Fault Tolerant Control ofDouble Inverted Pendulum a case study," Control Engineering

Practice, Vol. 13, pp. 1047-1059, 2005.

[6] B. Jiang, M. Staroswiecki, and V. Cocquempot, "FaultAccommodation for Nonlinear Dynamic Systems," IEEE Transaction

on Automatic Control, Vol. 51, pp. 1578-1583, 2006.

[7] D. Ye and G. H. Yang, "Adaptive Fault-Tolerant TrackingControl Against Actuator Faults with Application to Flight Control,"

IEEE Transaction on Automatic Control, Vol. 14, pp. 1068-1096, 2006.

[8]

M. Blanke, M. Staroswieki, and N. E. Wu, "Concepts andMethods in Fault Tolerant Control," In the Proceeding of American

Control Conference, Arlington, VA, pp. 2602-2620, 2001.

[9] Y. Zhang and J. Jiang, "Bibliographical review on reconfigurablefault-tolerant control systems," Annual reviews in control, Vol 32, pp.

229-252, 2008.

[10] K. Zhou and Z. Ren, "A new controller architecture for highperformance, robust, and fault-tolerant control," IEEE Transaction on

Automatic Control, Vol. 46, NO. 10, pp. 1613-1618, 2001.

[11] D. U. Campos-Delgado and K. Zhou, "Reconfigurable faulttolerant control using GIMC structure,"IEEE Transaction on AutomaticControl, Vol. 48, NO. 5, pp. 832-838, 2003.[12] D. U. Campos-Delgado, S. M. Martinez, and K. Zhou, "IntegratedFault Tolerant Scheme with Disturbance Feedforward," Proceeding ofthe 2004 American Control Conference. Boston, Massachusetts, pp.1799-1804, 2004.[13] Y. Nakaso, and T. Namerikawa, "GIMC-based Fault Detectionand Its Application to Magnetic Suspension System,"Proceeding of the17th World Congress, the International Federation of AutomaticControl. Seoul, Korea, pp. 7363-7368, 2008.[14] A. K. Sedigh, Analysis of Multivariable Control System, K. N.Toosi University of Technology, 1996.[15] K. Zhou, and J. C. Doyle, Essential of Robust Control, UpperSaddle River, NJ: Prentic-Hall, 1998.[16] G. Balas, R. Chiang, A. Packard, and M. Safonov,Robust ControlToolbox 3: User Guides. The Math Works, Inc, 2007.[17] K. H. Johansson, The Quadruple-Tank Process: A MultivariableLaboratory Process with an Adjustable Zero, IEEE Transaction onAutomatic Control, Vol, 8, NO. 3, pp. 456-465, 2000.[18] M. Pezeshkian, and M. J. Khosrowjerdi, Active Fault TolerantController Implemented on The Quadruple-Tank Process, 17thInternational Conference on Electrical and Engineering, Tehran, Iran,May 2009.

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