# Design and Assessment of a Multiple Sensor Fault Tolerant Robust Control System

Authors: S.S. Yang; J. Chen
Journal: Journal of Control Science and Engineering
 (2008-01-20)

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## Abstract

This paper presents an enhanced robust control design structure to realise fault tolerance towards sensor faults suitable for multi-input-multi-output (MIMO) systems implementation. The proposed design permits fault detection and controller elements to be designed with considerations to stability and robustness towards uncertainties besides multiple faults environment on a common mathematical platform. This framework can also cater to systems requiring fast responses. A design example is illustrated with a fast, multivariable and unstable system, that is, the double inverted pendulum system. Results indicate the potential of this design framework to handle fast systems with multiple sensor faults.


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## Body

## 1. Introduction

Growing demands
for plant or system availability, reliability, and survivability have prompted active research in fault tolerant
control systems (FTCSs) [1, 2]. FTCSs are
designed to accommodate component faults automatically by ensuring overall
system stability and acceptable performance. A typical FTCS design
incorporating separate control and fault detection elements can achieve fault
tolerance objectives, but without due considerations given to significant
interactions between the elements such as those described in [3, 4].
In addition, addressing issues concerning uncertainties is crucial as practical
problems associated with variations in actual plant operating range are undesirable.Fault detectors
are typically based upon the use of process models [5–7]. Data from the monitored plant is input
to these algorithms and the outputs are compared with the corresponding plant
outputs. If there are discrepancies, then it is an indication that at least one
fault has occurred. The model-based approach to designing sensor FTCS employs
mathematical manipulation of available signals, that is, analytical redundancy, via suitably designed
controllers to accommodate for faults rather than using extra hardware (sensors/actuators).


### 1.1. Integrating Control and Fault Detection in FTCS

An integrated
approach [8–11] where fault detection and controller
elements are designed with consideration to the overall system stability or
interaction is favourable as the reliability of operation can be determined in
a mathematically sound setting offering fast control responses in addition to
the availability of the established solution for incorporating robustness
towards uncertainties.In this paper, a robust
controller-based MIMO FTCS which integrates the fault detection and controller
elements in a single design is presented. A fault indicating residual is
utilised as a function of control. The residual signals act as weighting
factors, which put corresponding emphasis on nominal controller and fault
accommodating controller. The FTCS structure proposed allows the plant to be
controlled by a nominal controller that ensures the achievement of best performance
objectives, when sensor faults and uncertainties are not present, while
preserving the stability at a lower degree of system performance in the
presence of major sensor faults [11, 12]. The proposed structure can handle systems with fast
responses, multiple sensor faults, and modelling uncertainties.Note that purely
robust control-based FTCS such as described in [13, 14] ensures robustness
towards minor faults only; faults are modelled as very small perturbations on
the system. As demonstrated by [13, 14], it is
not possible for a purely robust control structure to maintain high performance,
when faults are not present as they are designed using worst case criterion.


## 1.1. Integrating Control and Fault Detection in FTCS

An integrated
approach [8–11] where fault detection and controller
elements are designed with consideration to the overall system stability or
interaction is favourable as the reliability of operation can be determined in
a mathematically sound setting offering fast control responses in addition to
the availability of the established solution for incorporating robustness
towards uncertainties.In this paper, a robust
controller-based MIMO FTCS which integrates the fault detection and controller
elements in a single design is presented. A fault indicating residual is
utilised as a function of control. The residual signals act as weighting
factors, which put corresponding emphasis on nominal controller and fault
accommodating controller. The FTCS structure proposed allows the plant to be
controlled by a nominal controller that ensures the achievement of best performance
objectives, when sensor faults and uncertainties are not present, while
preserving the stability at a lower degree of system performance in the
presence of major sensor faults [11, 12]. The proposed structure can handle systems with fast
responses, multiple sensor faults, and modelling uncertainties.Note that purely
robust control-based FTCS such as described in [13, 14] ensures robustness
towards minor faults only; faults are modelled as very small perturbations on
the system. As demonstrated by [13, 14], it is
not possible for a purely robust control structure to maintain high performance,
when faults are not present as they are designed using worst case criterion.


## 2. Problem Statement

Assuming that the
MIMO plants and controllers are described mathematically in state-space form as
follows:(1)x˙(t)=Ax(t)+Bu(t),y(t)=Cx(t)+Du(t),
where x∈Rn is state vector, u∈Rl is the input vector, while y∈Rm is the measured output vector.A,B,C, and D are known matrices with appropriate dimensions related to the
system dynamics. In addition, σ¯(M) denotes the largest singular value of M. H∞ denotes the Banach space of bounded analytic
functions with the ∞ norm defined as ∥F∥∞=supωσ¯(F(jω)) for any F∈H∞.Definition 1.
All MIMO transfer matrix representations have appropriate
dimensions and are proper real-rational matrices, stabilisable, and detectable.
A state space rational proper transfer function is denoted by(2)G(s)=[ABCD]=C(sI−A)−1B+D. Furthermore, let P
be a block matrix,(3)P=[P11P12P21P22]. Therefore, the
linear fractional transformation of P over F is defined as
(4)Fl(P,F)=P11+P12F(I−P22F)−1P21, where F is
assumed to have appropriate dimensions and (I−P22F)−1 is well defined.


### 2.1. Sensor Faults Defined

Sensor fault
symptoms can be observed as measurements that are unavailable, incorrect, or unusually
noisy. These faults may occur individually or concurrently or simultaneously,
resulting in total system failure or degradation in performance. Significant
information about the influence of faults on a process cannot be known without
the inclusion of its model in the design. Additive faults provide a suitable
framework for sensor faults and are modelled as additional input signals to a system [5],
(5)x˙(t)=Ax(t)+Bu(t),y′(t)=Cx(t)+Du(t)+fs(t),
where fs(t)∈Rm denote sensor faults. Hence
(6)y(s)=Gp(s)u(s). The variable y(s) denotes all available sensor outputs. When
output sensor faults occur in the plant as shown in (5), the measured outputs
become
(7)y′(s)=y(s)+fs(s). Due to the existence
of fault represented by fs(s),
a conventional controller cannot usually satisfy required performance and the
closed-loop control system may even become unstable. A sensor
fault-compensating controller can be introduced to augment a nominal controller
designed for best performance. However, since the structure of the system as
seen in Figure 1 is virtually an internal model controller [15], conditions for physical realizability need to be observed. To ensure
that the fault-compensating controller, Q is well defined and proper, the transfer matrix representation from fs(s) to controller output u(s) must exist and is also proper. Therefore,
(8)fs(s)=Ws(s)fs′(s).
By appropriate
use of input weight, Ws(s),
the input fs′(s) can be normalised and transformed into the
physical input, fs(s).
Consideration of such sensor fault models has been shown to be suitable for use
in formulating the FTCS objectives for the rejection of sensor faults as an
optimisation problem. Uncertainties affecting the sensors can also be classified
as a subset of fs(s).
Figure 1 shows the block diagram illustrating the interconnections assumed for the
formulation H∞ problem associated with the proposed FTCS
design.Figure 1
Block diagram representation ofH∞ problem formulation for the proposed FTCS
design.


### 2.2. Fault Indicating Residuals

The presence of sensor
faults and uncertainty vectors defined in Section2.1 can be reflected by a
fault indicating residual, since a filtered estimation can be obtained via
coprime factorisation of the plant model, Gp(s) [11, 12]. Let
(9)Gp(s)=M−1(s)N(s). Hence, from (8)
and (9), the fault indicating residual denoted by fr(s) can be defined as
(10)fr(s)=N(s)u(s)−M(s)y′(s)=N(s)u(s)−M(s)[y(s)+fs(s)]=−M(s)Ws(s)f′(s).


### 2.3. Integrating the Controller Element

Now, sincefr(s) reflects the presence of faults and uncertainty,
it can be utilised as an input to the fault compensating controller. The
perturbations caused can then be minimised by control actions due to the nominal
controller and fault compensating controller. The control signal vector can be
expressed as follows:
(11)u(s)=uk(s)+uq(s),
where (12)uk(s)=K(s)e(s)
and uk(s) denotes nominal controller (K(s)) output, and uq(s) denotes sensor fault compensator (Q(s)) output. Error from feedback is
denoted by e(s) whereby r(s) denotes input demand. Thus, from (10), fr(s) is utilised in the following manner:
(13)uq(s)=Q(s)fr(s)=−Q(s)M(s)Ws(s)fs′(s). From (6), (7),
and (8), e(s) can be expressed as
(14)e(s)=r(s)−y′(s)=r(s)−y(s)−fs(s)=r(s)−Gp(s)u(s)−Ws(s)fs′(s). By substituting (12),
(13), and (14) into (11), the following is derived:
(15)u(s)=(I+K(s)Gp(s))−1×{K(s)r(s)−(K(s)+[Q(s)M(s)])Ws(s)f′s(s)}. Thus,
(16)y′(s)=Gp(s)(I+K(s)Gp(s))−1×{K(s)r(s)−(K(s)+[Q(s)M(s)])Ws(s)f′s(s)}. The plant output
expression in (16) shows that in the absence of sensor faults and
uncertainties, the output closed-loop system is only reliant on the nominal
controller K(s), allowing for high
performance during healthy operation. Note that the fault detection scheme
generating the above-mentioned fault indicating residual does not need to be
made robust, since the fault indicating residual is mainly used as an
activating signal for Q(s). It is
thus not essential to identify nor to estimate the source of the faults, hence
even if the presence of fr(s) is due to uncertainties and not faults in the
sensors, Q(s) will still provide the
necessary control signals to compensate for such perturbations thereby
introducing robustness to the system.


### 2.4. Sensor Fault Compensator Realisation

The sensor fault
compensatorQ(s) is integrated into the
framework by utilising fr(s) as a function of control. The design Q(s) is achieved with the H∞ technique. A performance weights Wftc(s) can be defined to establish postfault
performance requirements, which emphasise on stability rather than high
performance. The corresponding solution for achieving Q(s) is by minimising the following optimisation criterion:
(17)γ=minQ(s)‖Fl[Pf(s),Q(s)]‖∞. Therefore, the standard H∞ problem is specified in (17) for which the
corresponding transfer functions from fs′(s) to z(s) must satisfy. If the controller Q(s) in (17) is found, then the closed-loop
system is said to have robust performance towards uncertainty and sensor
faults; it is well known that a system satisfies robust performance if and only
if it is robustly stable with respect to norm-bounded matrix perturbation [16].
The equivalent linear fractional transformation (LFT) block diagram for the H∞ problem stated above is shown in Figure 2.Figure 2
The LFT representation of the proposed FTCS.Thus,(18)[z(s)fr(s)]=[P11(s)P12(s)P21(s)P22(s)]︸Ps(s)[fs′(s)uq(s)]. From (10), P21 and P12 can be derived as
(19)P21(s)=−M(s)Ws(s),P12(s)=0.
Now, note that
(20)uk(s)=K(s)(r(s)−y′(s))=K(s)(r(s)−Gp(s)u(s)−Ws(s)fs′(s))=K(s)r(s)−K(s)Gp(s)[uk(s)+uq(s)]−K(s)Ws(s)fs′(s),
and thus,
(21)uk(s)=(I+K(s)Gp(s))−1K(s)r(s)−(I+K(s)Gp(s))−1K(s)Ws(s)fs′(s)−(I+K(s)Gp(s))−1K(s)Gp(s)uk(s).Also(22)z(s)=Wftc(s)y′(s)=Wftc(s)[Gp(s)u(s)+Ws(s)fs′(s)]=Wftc(s)Gp(s)uk(s)+Wftc(s)Gp(s)uq(s)+Wftc(s)Ws(s)fs′(s). Substituting (21)
into (22),
(23)z(s)=Wftc(s)Gp(s)(I+K(s)Gp(s))−1K(s)r(s)−Wftc(s)Gp(s)(I+K(s)Gp(s))−1K(s)Gp(s)uq(s)+Wftc(s)Gp(s)uq(s)−Wftc(s)Gp(s)(I+K(s)Gp(s))−1×K(s)Ws(s)fs′(s)+Wftc(s)Ws(s)fs′(s). Ignoring the
reference input r(s), we have
(24)P11(s)=−Wftc(s)Gp(s)(I+K(s)Gp(s))−1×K(s)Ws(s)+Wftc(s)Ws(s)=Wftc(s){I−Gp(s)(I+K(s)Gp(s))−1K(s)}Ws(s)=Wftc(s)(1+Gp(s)K(s))−1Ws(s),P12(s)=−Wftc(s)Gp(s)(I+K(s)Gp(s))−1×K(s)Gp(s)+Wftc(s)Gp(s)=Wftc(s){I−Gp(s)(I+K(s)Gp(s))−1K(s)}Gp(s)=Wftc(s)(1+Gp(s)K(s))−1Gp(s).
Note that the following matrix operation (Zhou, Doyle & Glover,
1996, page 23) has been used in the derivation of (24):
(25)A11−1+A11−1A12(A22−A21A11−1A12)−1A21A11−1=(A11−A12A22−1A21)−1. With the conditions
laid out, the closed-loop system shown above is guaranteed to be tolerant to
sensor faults and modelling uncertainty, stable for any nonlinear, time varying,
and stable K(s) and Q(s) due to the minimisation of the
transfer matrix between fault-generating signal fs′(s) to the performance evaluation signal z(s).


## 2.1. Sensor Faults Defined

Sensor fault
symptoms can be observed as measurements that are unavailable, incorrect, or unusually
noisy. These faults may occur individually or concurrently or simultaneously,
resulting in total system failure or degradation in performance. Significant
information about the influence of faults on a process cannot be known without
the inclusion of its model in the design. Additive faults provide a suitable
framework for sensor faults and are modelled as additional input signals to a system [5],
(5)x˙(t)=Ax(t)+Bu(t),y′(t)=Cx(t)+Du(t)+fs(t),
where fs(t)∈Rm denote sensor faults. Hence
(6)y(s)=Gp(s)u(s). The variable y(s) denotes all available sensor outputs. When
output sensor faults occur in the plant as shown in (5), the measured outputs
become
(7)y′(s)=y(s)+fs(s). Due to the existence
of fault represented by fs(s),
a conventional controller cannot usually satisfy required performance and the
closed-loop control system may even become unstable. A sensor
fault-compensating controller can be introduced to augment a nominal controller
designed for best performance. However, since the structure of the system as
seen in Figure 1 is virtually an internal model controller [15], conditions for physical realizability need to be observed. To ensure
that the fault-compensating controller, Q is well defined and proper, the transfer matrix representation from fs(s) to controller output u(s) must exist and is also proper. Therefore,
(8)fs(s)=Ws(s)fs′(s).
By appropriate
use of input weight, Ws(s),
the input fs′(s) can be normalised and transformed into the
physical input, fs(s).
Consideration of such sensor fault models has been shown to be suitable for use
in formulating the FTCS objectives for the rejection of sensor faults as an
optimisation problem. Uncertainties affecting the sensors can also be classified
as a subset of fs(s).
Figure 1 shows the block diagram illustrating the interconnections assumed for the
formulation H∞ problem associated with the proposed FTCS
design.Figure 1
Block diagram representation ofH∞ problem formulation for the proposed FTCS
design.


## 2.2. Fault Indicating Residuals

The presence of sensor
faults and uncertainty vectors defined in Section2.1 can be reflected by a
fault indicating residual, since a filtered estimation can be obtained via
coprime factorisation of the plant model, Gp(s) [11, 12]. Let
(9)Gp(s)=M−1(s)N(s). Hence, from (8)
and (9), the fault indicating residual denoted by fr(s) can be defined as
(10)fr(s)=N(s)u(s)−M(s)y′(s)=N(s)u(s)−M(s)[y(s)+fs(s)]=−M(s)Ws(s)f′(s).


## 2.3. Integrating the Controller Element

Now, sincefr(s) reflects the presence of faults and uncertainty,
it can be utilised as an input to the fault compensating controller. The
perturbations caused can then be minimised by control actions due to the nominal
controller and fault compensating controller. The control signal vector can be
expressed as follows:
(11)u(s)=uk(s)+uq(s),
where (12)uk(s)=K(s)e(s)
and uk(s) denotes nominal controller (K(s)) output, and uq(s) denotes sensor fault compensator (Q(s)) output. Error from feedback is
denoted by e(s) whereby r(s) denotes input demand. Thus, from (10), fr(s) is utilised in the following manner:
(13)uq(s)=Q(s)fr(s)=−Q(s)M(s)Ws(s)fs′(s). From (6), (7),
and (8), e(s) can be expressed as
(14)e(s)=r(s)−y′(s)=r(s)−y(s)−fs(s)=r(s)−Gp(s)u(s)−Ws(s)fs′(s). By substituting (12),
(13), and (14) into (11), the following is derived:
(15)u(s)=(I+K(s)Gp(s))−1×{K(s)r(s)−(K(s)+[Q(s)M(s)])Ws(s)f′s(s)}. Thus,
(16)y′(s)=Gp(s)(I+K(s)Gp(s))−1×{K(s)r(s)−(K(s)+[Q(s)M(s)])Ws(s)f′s(s)}. The plant output
expression in (16) shows that in the absence of sensor faults and
uncertainties, the output closed-loop system is only reliant on the nominal
controller K(s), allowing for high
performance during healthy operation. Note that the fault detection scheme
generating the above-mentioned fault indicating residual does not need to be
made robust, since the fault indicating residual is mainly used as an
activating signal for Q(s). It is
thus not essential to identify nor to estimate the source of the faults, hence
even if the presence of fr(s) is due to uncertainties and not faults in the
sensors, Q(s) will still provide the
necessary control signals to compensate for such perturbations thereby
introducing robustness to the system.


## 2.4. Sensor Fault Compensator Realisation

The sensor fault
compensatorQ(s) is integrated into the
framework by utilising fr(s) as a function of control. The design Q(s) is achieved with the H∞ technique. A performance weights Wftc(s) can be defined to establish postfault
performance requirements, which emphasise on stability rather than high
performance. The corresponding solution for achieving Q(s) is by minimising the following optimisation criterion:
(17)γ=minQ(s)‖Fl[Pf(s),Q(s)]‖∞. Therefore, the standard H∞ problem is specified in (17) for which the
corresponding transfer functions from fs′(s) to z(s) must satisfy. If the controller Q(s) in (17) is found, then the closed-loop
system is said to have robust performance towards uncertainty and sensor
faults; it is well known that a system satisfies robust performance if and only
if it is robustly stable with respect to norm-bounded matrix perturbation [16].
The equivalent linear fractional transformation (LFT) block diagram for the H∞ problem stated above is shown in Figure 2.Figure 2
The LFT representation of the proposed FTCS.Thus,(18)[z(s)fr(s)]=[P11(s)P12(s)P21(s)P22(s)]︸Ps(s)[fs′(s)uq(s)]. From (10), P21 and P12 can be derived as
(19)P21(s)=−M(s)Ws(s),P12(s)=0.
Now, note that
(20)uk(s)=K(s)(r(s)−y′(s))=K(s)(r(s)−Gp(s)u(s)−Ws(s)fs′(s))=K(s)r(s)−K(s)Gp(s)[uk(s)+uq(s)]−K(s)Ws(s)fs′(s),
and thus,
(21)uk(s)=(I+K(s)Gp(s))−1K(s)r(s)−(I+K(s)Gp(s))−1K(s)Ws(s)fs′(s)−(I+K(s)Gp(s))−1K(s)Gp(s)uk(s).Also(22)z(s)=Wftc(s)y′(s)=Wftc(s)[Gp(s)u(s)+Ws(s)fs′(s)]=Wftc(s)Gp(s)uk(s)+Wftc(s)Gp(s)uq(s)+Wftc(s)Ws(s)fs′(s). Substituting (21)
into (22),
(23)z(s)=Wftc(s)Gp(s)(I+K(s)Gp(s))−1K(s)r(s)−Wftc(s)Gp(s)(I+K(s)Gp(s))−1K(s)Gp(s)uq(s)+Wftc(s)Gp(s)uq(s)−Wftc(s)Gp(s)(I+K(s)Gp(s))−1×K(s)Ws(s)fs′(s)+Wftc(s)Ws(s)fs′(s). Ignoring the
reference input r(s), we have
(24)P11(s)=−Wftc(s)Gp(s)(I+K(s)Gp(s))−1×K(s)Ws(s)+Wftc(s)Ws(s)=Wftc(s){I−Gp(s)(I+K(s)Gp(s))−1K(s)}Ws(s)=Wftc(s)(1+Gp(s)K(s))−1Ws(s),P12(s)=−Wftc(s)Gp(s)(I+K(s)Gp(s))−1×K(s)Gp(s)+Wftc(s)Gp(s)=Wftc(s){I−Gp(s)(I+K(s)Gp(s))−1K(s)}Gp(s)=Wftc(s)(1+Gp(s)K(s))−1Gp(s).
Note that the following matrix operation (Zhou, Doyle & Glover,
1996, page 23) has been used in the derivation of (24):
(25)A11−1+A11−1A12(A22−A21A11−1A12)−1A21A11−1=(A11−A12A22−1A21)−1. With the conditions
laid out, the closed-loop system shown above is guaranteed to be tolerant to
sensor faults and modelling uncertainty, stable for any nonlinear, time varying,
and stable K(s) and Q(s) due to the minimisation of the
transfer matrix between fault-generating signal fs′(s) to the performance evaluation signal z(s).


## 3. A Numerical Simulation Example

An experimental
study of the FTCS implementation on a double inverted pendulum system for
tolerance towards sensor faults is shown next to illustrate the feasibility of
the proposed design method. The implementation is tested for fault tolerance
towards sensors in nominal and under plant uncertainty conditions.


### 3.1. The Double Inverted Pendulum System

The double
inverted pendulum system is an example of a chaotic system. The system is a
fast, multivariable, nonlinear, and unstable process. The pendulum system is a
standard classical control test rig for the verification of different control
methods, and is among the most difficult systems to control in the field of
control engineering. Similar to the single inverted pendulum problem, the
control task for the double inverted pendulum is to stabilise the two pendulums.
The position of the carriage on the track is controlled quickly and accurately,
so that the pendulums are always erected in their inverted position during such
movements.The double inverted
pendulum system is made up of two aluminium arms connected to each other with
the lower arm attached to a cart placed on a guiding rail, as illustrated in
Figure3. Data used in this case study has been obtained from [9]. The aluminium arms are constrained to rotate within a
single plane and the axis
of rotation is perpendicular to the direction of the force acting on the
cart motion f. The cart can move
along a linear low-friction track and is moved by a belt driven by a servo
motor system. Sensors providing measurements of cart position xc,
the pendulums angles θ1 and θ2,
controller output, u, and motor
current i are assumed available for
the purpose of control. The control law has to regulate the lower-arm angle and
upper-arm angle, θ1 and θ2,
respectively, from an initial condition, and the control of the position of the
cart xc from an initial position.Figure 3
Schematic diagram of the pendulum system.


### 3.2. Nominal High-Performance Controller

AnH∞ loop shaping controller,
as high-performance nominal controller K for
the MIMO system, is designed using the MATLAB command ncfsyn.m. The specification
function Wp is augmented
to K in the manner shown in Figure 4.
Sensors for detecting ex (cart
positional error), θ1 and θ2, are fault prone sensors. Motor
voltage and current are denoted by u and i, respectively. The controller
output variable is the corresponding motor voltage demand u. The controller performance was tested on the SIMULINK model of the double inverted
pendulum. Initial conditions are with θ1=0.05rad and θ2=−0.04rad.
The cart movement command signal rc is initiated at 0.5 m and at −0.5 m after 50 seconds, is shown in Figure  5, while system responses are shown
in Figure  6. It is observed that the output responses are within limits of
specifications, and the cart position set points have been achieved in a stable
and smooth manner.Figure 4
TheH∞ loop-shaping controller K with specification function.Figure 5
Command
signal requiring the cart to move from 0.5 m to−0.5 m.Figure 6
System
responses withK implementation (position
of cart xc is shown instead of cart position error ex).


### 3.3. FTCS Design and Implementation

The nominal model
of the double inverted pendulum model is described by its left coprime factors
to ensure well posedness. The double inverted pendulum model without modelling
uncertainty is considered for the representation of the nominal plant in the
fault indicating residual generator setup. Fault indicating residuals are
denoted byfθ1, fθ2 and fex for faults in the
corresponding sensors.The
interconnection of the system is setup and the design of the controller sensor
fault compensating controller,Q is
automated with the command hinfsyn.m provided in MATLAB’sμ-analysis and synthesis
toolbox [17], which iteratively solves the
optimisation criterion set out in (17). When γ value of below 1 is
obtained, the solution of a satisfactory Q is used. This condition is only met with relaxations to the effects of additive
faults, as it is obvious that total failure cannot be handled. Note that the
performance weights Wftc(s) (shown in the appendix) to establish postfault
performance requirements reuse the elements in the original specification
function Wp, which are
related to the fault prone sensors, that is, sensors providing measurements of cart
position xc,
the pendulums angles θ1 and θ2. The
block diagram showing the augmentation of Q to nominal controller K is
illustrated in Figure 7.Figure 7
Block
diagram of sensor fault compensatorQ augmented to nominal controller K in
the FTCS structure.


### 3.4. Tests and Results

The following
responses have been recorded from testing the FTCS by simulating the occurrence
of faults in the relevant sensors. Sensor effectiveness indicating faults are
simulated as deterioration of performance; 0%: no fault, 100%: total failure.
Results are shown for conditions with and without modelling uncertainty.
Responses of the inverted double pendulum system performances with the proposed
FTCS,H∞, and μ controllers are recorded for
comparison purposes.Nominal Response, without Modelling Uncertainties and Sensor Faults
Nominal performances
of all controllers for healthy system are recorded in Figure8. Apparently the
proposed FTCS produces faster cart positioning response compared to all other
control system responses, initiating slightly higher overshoots in θ1 and θ2.Nominal
double inverted pendulum system responses of all controllers under healthy
conditions. (a)— FTCS, (b) …H∞ controller, and
(c) - - -μ controller.
(a)(b)(c)Multiple Sensor Faults without Plant Uncertainty
Multiple sensor faults are assumed to occur
at 2, 4, and 6 seconds after the simulation has been initiated (ex at 90% deterioration, θ1 at 20% deterioration, and θ2 at 10% deterioration, resp.).
The output responses are shown in Figure 9. Observe that the proposed FTCS and the μ controller handled the faults and managed to achieve satisfactory control
responses. However, stability could not be maintained by the H∞ controller.System
responses of all controllers under multiple sensor fault condition, without
modelling uncertainty. (a)— FTCS,
(b) …H∞ controller,
and (c) - - -μ controller.
(a)(b)(c)Multiple Sensor Faults with Plant Uncertainty
Tests for control systems to handle system
uncertainty and multiple sensor faults were also performed. Conditions were
made similar to the tests performed for the nominal system with multiple sensor
faults. The supremacy of the proposed FTCS to accommodate for faults even under
the influence of system uncertainties is seen in Figure10.System
responses of all controllers under multiple sensor fault condition with
modelling uncertainty. (a)— FTCS,
(b) …H∞ controller,
and (c) - - -μ controller.
(a)(b)(c)TheH∞ controller could not
handle this mode of fault and oscillates beyond control as shown. Meanwhile,
both the proposed FTCS and the μ controller
handled the fault satisfactorily.Further Discussion
Overall, the proposed FTCS has managed to
handle all pre- and postfault conditions satisfactorily, while maintaining the
highest level of stability in all test scenarios. Although it seems that theμ controller could handle faults and
modelling uncertainty as well as the proposed FTCS, it could not handle certain
cases of single faults such as the cases shown in Figure 11 for the effect of θ2 sensor fault at 10% deterioration. Responses of μ control system is too oscillatory and
unstable.System
responses of all controllers underθ2 sensor fault at 10% deterioration, without uncertainties. (a) — FTCS,
(b) …H∞ controller,
and (c) - - -μ controller.
(a)(b)(c)


## 3.1. The Double Inverted Pendulum System

The double
inverted pendulum system is an example of a chaotic system. The system is a
fast, multivariable, nonlinear, and unstable process. The pendulum system is a
standard classical control test rig for the verification of different control
methods, and is among the most difficult systems to control in the field of
control engineering. Similar to the single inverted pendulum problem, the
control task for the double inverted pendulum is to stabilise the two pendulums.
The position of the carriage on the track is controlled quickly and accurately,
so that the pendulums are always erected in their inverted position during such
movements.The double inverted
pendulum system is made up of two aluminium arms connected to each other with
the lower arm attached to a cart placed on a guiding rail, as illustrated in
Figure3. Data used in this case study has been obtained from [9]. The aluminium arms are constrained to rotate within a
single plane and the axis
of rotation is perpendicular to the direction of the force acting on the
cart motion f. The cart can move
along a linear low-friction track and is moved by a belt driven by a servo
motor system. Sensors providing measurements of cart position xc,
the pendulums angles θ1 and θ2,
controller output, u, and motor
current i are assumed available for
the purpose of control. The control law has to regulate the lower-arm angle and
upper-arm angle, θ1 and θ2,
respectively, from an initial condition, and the control of the position of the
cart xc from an initial position.Figure 3
Schematic diagram of the pendulum system.


## 3.2. Nominal High-Performance Controller

AnH∞ loop shaping controller,
as high-performance nominal controller K for
the MIMO system, is designed using the MATLAB command ncfsyn.m. The specification
function Wp is augmented
to K in the manner shown in Figure 4.
Sensors for detecting ex (cart
positional error), θ1 and θ2, are fault prone sensors. Motor
voltage and current are denoted by u and i, respectively. The controller
output variable is the corresponding motor voltage demand u. The controller performance was tested on the SIMULINK model of the double inverted
pendulum. Initial conditions are with θ1=0.05rad and θ2=−0.04rad.
The cart movement command signal rc is initiated at 0.5 m and at −0.5 m after 50 seconds, is shown in Figure  5, while system responses are shown
in Figure  6. It is observed that the output responses are within limits of
specifications, and the cart position set points have been achieved in a stable
and smooth manner.Figure 4
TheH∞ loop-shaping controller K with specification function.Figure 5
Command
signal requiring the cart to move from 0.5 m to−0.5 m.Figure 6
System
responses withK implementation (position
of cart xc is shown instead of cart position error ex).


## 3.3. FTCS Design and Implementation

The nominal model
of the double inverted pendulum model is described by its left coprime factors
to ensure well posedness. The double inverted pendulum model without modelling
uncertainty is considered for the representation of the nominal plant in the
fault indicating residual generator setup. Fault indicating residuals are
denoted byfθ1, fθ2 and fex for faults in the
corresponding sensors.The
interconnection of the system is setup and the design of the controller sensor
fault compensating controller,Q is
automated with the command hinfsyn.m provided in MATLAB’sμ-analysis and synthesis
toolbox [17], which iteratively solves the
optimisation criterion set out in (17). When γ value of below 1 is
obtained, the solution of a satisfactory Q is used. This condition is only met with relaxations to the effects of additive
faults, as it is obvious that total failure cannot be handled. Note that the
performance weights Wftc(s) (shown in the appendix) to establish postfault
performance requirements reuse the elements in the original specification
function Wp, which are
related to the fault prone sensors, that is, sensors providing measurements of cart
position xc,
the pendulums angles θ1 and θ2. The
block diagram showing the augmentation of Q to nominal controller K is
illustrated in Figure 7.Figure 7
Block
diagram of sensor fault compensatorQ augmented to nominal controller K in
the FTCS structure.


## 3.4. Tests and Results

The following
responses have been recorded from testing the FTCS by simulating the occurrence
of faults in the relevant sensors. Sensor effectiveness indicating faults are
simulated as deterioration of performance; 0%: no fault, 100%: total failure.
Results are shown for conditions with and without modelling uncertainty.
Responses of the inverted double pendulum system performances with the proposed
FTCS,H∞, and μ controllers are recorded for
comparison purposes.Nominal Response, without Modelling Uncertainties and Sensor Faults
Nominal performances
of all controllers for healthy system are recorded in Figure8. Apparently the
proposed FTCS produces faster cart positioning response compared to all other
control system responses, initiating slightly higher overshoots in θ1 and θ2.Nominal
double inverted pendulum system responses of all controllers under healthy
conditions. (a)— FTCS, (b) …H∞ controller, and
(c) - - -μ controller.
(a)(b)(c)Multiple Sensor Faults without Plant Uncertainty
Multiple sensor faults are assumed to occur
at 2, 4, and 6 seconds after the simulation has been initiated (ex at 90% deterioration, θ1 at 20% deterioration, and θ2 at 10% deterioration, resp.).
The output responses are shown in Figure 9. Observe that the proposed FTCS and the μ controller handled the faults and managed to achieve satisfactory control
responses. However, stability could not be maintained by the H∞ controller.System
responses of all controllers under multiple sensor fault condition, without
modelling uncertainty. (a)— FTCS,
(b) …H∞ controller,
and (c) - - -μ controller.
(a)(b)(c)Multiple Sensor Faults with Plant Uncertainty
Tests for control systems to handle system
uncertainty and multiple sensor faults were also performed. Conditions were
made similar to the tests performed for the nominal system with multiple sensor
faults. The supremacy of the proposed FTCS to accommodate for faults even under
the influence of system uncertainties is seen in Figure10.System
responses of all controllers under multiple sensor fault condition with
modelling uncertainty. (a)— FTCS,
(b) …H∞ controller,
and (c) - - -μ controller.
(a)(b)(c)TheH∞ controller could not
handle this mode of fault and oscillates beyond control as shown. Meanwhile,
both the proposed FTCS and the μ controller
handled the fault satisfactorily.Further Discussion
Overall, the proposed FTCS has managed to
handle all pre- and postfault conditions satisfactorily, while maintaining the
highest level of stability in all test scenarios. Although it seems that theμ controller could handle faults and
modelling uncertainty as well as the proposed FTCS, it could not handle certain
cases of single faults such as the cases shown in Figure 11 for the effect of θ2 sensor fault at 10% deterioration. Responses of μ control system is too oscillatory and
unstable.System
responses of all controllers underθ2 sensor fault at 10% deterioration, without uncertainties. (a) — FTCS,
(b) …H∞ controller,
and (c) - - -μ controller.
(a)(b)(c)


## 4. Conclusion

The proposed FTCS
has been observed to have managed all faults simulated in the nominal
performance tests, while the two other control systems could not consistently
maintain stability in a majority of fault scenario. Robust performance
assessments showing the performance of the control systems when faced with
system uncertainty in addition to sensor faults were also simulated. Again, it
is observed that fault tolerance capability of the proposed FTCS has been
maintained. The proposed improvement to the model-based FTCS structure provides
a potential framework for the realisation of an integrated MIMO FTCS. This
design framework is suitable as it inherently incorporates fault residuals as
feedback and allows the application of established robust MIMO control design
concept. The test results show the capability of the proposed FTCS to maintain
availability and an acceptable level of performance for multiple deteriorated
sensor conditions.



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*Source: 152030-2008-03-12.xml*