Application to an Inverted Pendulum

Robust Model-Based Fault Tolerant Control of a Mobile Structure - Application to an Inverted Pendulum Alberto Cardoso* and António Dourado** CISUC- Centro de Informática e Sistemas da Universidade de Coimbra Departamento de Engenharia Informática Pólo II da Universidade, Pinhal de Marrocos, 3030 COIMBRA PORTUGAL

Abstract. A model-based approach to design a robust fault tolerant control of a mobile structure is studied. The method considers the application of robust control techniques to integrate the control design and the fault diagnosis. The main purpose of this approach is to design a robust controller insensitive to the effects of faults and a fault detector filter. This approach is applied to a particular unstable mobile structure, an inverted pendulum. Keywords: fault tolerant control; model-based FDI; robust control; uncertain systems; inverted pendulum.

1 Introduction The issues of availability, reliability and operating safety are of the major importance to, not only safetycritical systems, but also mobile systems. For systems that are not safety-critical, on-line robust fault tolerant control techniques can be used to improve plant efficiency and close loop performance in presence of faults and uncertainty. The model-based fault detection and isolation (FDI) considers the analytical redundancy inherent in the dynamic relationships between inputs and outputs of a system. Usually, a mathematical model is used to derive a residual quantity which is supposed to be “small” for an unfaulty plant and “large” whenever a fault occurs. Faults could then be detected if the residual exceeds a given threshold. In order to achieve fault isolation, a number of residuals could be used, each one indicating a different fault. Recent surveys can be found for instance in (Frank, 1997), (Patton, 1997) and (Chen and Patton, 1999). Robustness against disturbances, process uncertainty and modelling errors are an important subject in FDI research. Unsatisfactory diagnostic performance due to neglected interaction between the controller and the FDI module should be avoid. To overcome this problem, an integrated approach using the robust control formulation can be used, where the controller an the model-based fault detection filter are designed simultaneously. This methodology was initially presented in (Nett et al., 1988) and further developed in (Tyler and Morari, 1994), (Isermann, 1994) and (Niemann and Stoustrup, 1997). Some applications of this methodology can be found in (Akesson, 1997), (Eich and Sattler, 1997) and (Stoustrup et al., 1997). The approach presented in this work results from the application of the µ-synthesis method in order to obtain, simultaneously, a robust linear controller and a robust fault diagnosis filter to a mobile structure. Using the D-K iterative method followed by the optimal Hankel norm approximation (Anderson and Liu, 1989) for controller order reduction, a suitable controller is obtained for on-line fault tolerant control purposes. This work claims to give a contribution for the robustness analysis of the control and fault diagnosis system with respect to abrupt faults in actuators and sensors of an unstable mobile plant. The generalised controller, including the robust controller and the fault detection filter, is successfully designed using the robust control framework and the frequency domain representation of the design objectives. The fault detection filter is designed to generate a set of residuals corresponding to the expected faults in the system. The residual signals can be used as inputs of a fault detection supervisor to generate alarm signals and to monitor the closed loop system's performance. * Email: [email protected] , **Email: [email protected]

2 Problem Formulation In the model-based approach a mathematical model should be built in order to monitor the plant. In the case of a non-linear system, this implies a model linearization around an operating point. The deviations between the linear model and the real plant will be considered as uncertainty in the control system formulation. For fault diagnosis purposes, the system is described including actuator and sensor faults which are represented by additive signals as illustrated in Figure 1.

input u(t)

sensor faults

actuator faults

f (t) a

+

actuation u (t)

Plant Dynamics

R

Outputs y (t)

f (t) s

+

measured outputs y(t)

R

Fig. 1. System description with actuator and sensor faults. The system dynamics can be described by the state space model as: x& ( t ) = Ax ( t ) + Bu ( t ) R y( t ) = y ( t ) + f ( t ) R s

y ( t ) = Cx ( t ) + Du ( t ) R R . u ( t ) = u( t ) + f ( t ) R a

Concerning the control system formulation, instead of using a standard one parameter controller, a two parameter generalised controller (controller and fault detection filter) given by: u( t ) = Ky( t ) =  K 1  y ( t )  K 2   r ( t ) will be considered to integrate the control design (input u(t)) and the fault diagnosis (residual signal r(t)). The design setup uses a generalisation of the standard configuration for robust control (Zhou et al., 1995; Niemann and Stoustrup, 1997) as illustrated in Figure 2. d

∆

e

v

∆f

a

w

∆p

z

d w v

z a

P

∆

e

P u

K

y

u r

K

y

(ii)

(i)

Fig. 2. (i) Generalised setup for robust control (ii) with performance and fault detection specifications represented by fictitious perturbations blocks. In this generalised setup, the model uncertainty is represented by ∆, and the fault detection and the performance specifications by fictitious perturbations blocks ∆f and ∆p, respectively. It is assumed that each block is scaled such that ∆ ≤ 1, ∀ω . The block P is an augmented plant, including the nominal plant model and the uncertainty description, fault detection and performance weighting functions (Lundström et al., 1991). To achieve fault isolation, the fault detection weighting functions must represent the expected behaviour of each fault. In this work, the effects of abrupt faults are investigated. 2

The controller to be designed using the µ-synthesis must achieve nominal stability, nominal performance, robust stability and robust performance. For the purpose of analysis, the controller K is combined with the augmented plant P using a lower linear fractional transformation (LFT) to obtain a M-∆ structure (Balas, et al., 1993; Postlethwaite and Skogestad, 1993). To obtain a robust controller and a robust fault detection filter, the D-K iterative procedure, initially proposed by Doyle and Stein (1981), is applied using the Matlab  µ-Analysis and Synthesis Toolbox (Balas, et al., 1993).

3 Design Example: The Inverted Pendulum The mobile structure - Inverted Pendulum - consists of a cart and an aluminium rod with a cylindrical weight (pendulum) fixed to the cart by an axis. The cart, which can be moved along a guiding bar, is connected by a transmission belt to a drive wheel. The wheel is driven by a current controlled motor which delivers a torque proportional to the acting control voltage (Us) such that the cart is accelerated. A scheme of the plant is illustrated in Figure 3. 7

φ 9 4

3

Us

1

6 2

-p

1- Servo-amplifier 2- Motor 5- Metal guiding bar 6- Cart

8

5

+p

0

3- Drive wheel 7- Pendulum weight

4- Transmission Belt 8- Guide roll 9- Pendulum rod

Fig. 3. Scheme of the Inverted Pendulum. Two output variables are measured: i) the cart position by means of an incremental encoder which is fixed to the driving shaft of the motor; ii) the angle of the pendulum rod by means of an incremental encoder which is fixed to the pivot of the pendulum. The inverted pendulum system can be described by a mathematical model as a system of coupled differential equations. These equations have been derived using the equation of motion for the cart and the angular momentum conservation law for the rotary motion of the rod about the centre of gravity. This model is a non-linear system with some uncertain factors as the dry friction (Coulomb friction) acting on the cart. In order to obtain a suitable linear model a linearization is performed around the main operating point of the plant. The mathematical model is valid as long the following conditions are satisfied: i) a limitation of the control force F ( F ≤ 20 N ); ii) a limitation of the guiding bar ( cart position ≤ 0.5m ); iii) a limitation of the angle φ ( φ ≤ 10 º ). Considering the following state vector:

 x 1   ∆p   cart position perturbation   x   φ   pendulum angle x = 2= =  cart velocity  x 3   r&     x 4   φ&   pendulum angular velocity and the input variable:

u = [Force acting via the transmission belt ] ,

a state and output equations describing the system can be written as: x& = Ax + bu

and 3

y = Cx .

Defining the main operating point as x = [x 1 = p 0

x2 = 0

x3 = 0

x4 = 0

]T ,

the state matrices are

given by:

0 0 A= 0 0 

0 1 0  0 0 1  − 0.757 − 2.47 0.00068   20.346 4.7569 − 0.0185

b

 0   0  = 0.247  − 0.475  

1 0 0 0  C = . 0 1 0 0

Because the system is unstable, the controller must guarantee the stability of the closed loop system, even if noise, disturbances or faults, limited to a given range, are present.

4 The Framework for Robust Control To design a robust controller for the inverted pendulum, the framework of the control system with uncertainties, which will allow to apply the µ-synthesis and analysis, is shown in Figure 4.

∆ e

wi -

d

Inverted Pendulum n1

n2

w

ref fa fs1

v

fs2

ref-y1

wp1

y2

z

wp2

wn1

wu

wn2 wr1 wref

-

wfa

wr3

wfs1 wfs2

a

wr2 ref u y1 y2

y1

Inverted Pendulum Observer

x2o y2 x4o

u r1

K

r2 r3

Fig. 4. Setup for robust control and fault detection for the inverted pendulum plant. 4

In this figure the structure of the augmented plant P is built using the weighting transfer functions used to describe the objectives and to weight the input signals. In the framework, the external input w includes the measurement noise in each sensor and the cart position reference. The three considered faults (fa, fs1, fs2) are represented by the input variable v. The performance and the fault detection objectives are represented by the external outputs z and a, respectively. The inputs of the generalised controller are the cart position reference, the two plant's measured outputs (the cart position and the pendulum angle) and the other two state variables (cart and pendulum angular velocities), which are generated by an observer system. Concerning the outputs, the control action is generated by the controller and the residual signals are obtained by the fault detection filter. To approximate the control problem into a µ-problem suitable for D-K iteration, the uncertainty is represented by a norm-bounded perturbation and a weighting transfer function, and the specifications for the closed loop system are expressed in the frequency domain as weighting transfer functions. The design objectives for the closed loop system can be formulated as: i) the output signal y1 should track the cart position reference, ref, and the output signal y2 should tend towards zero; ii) the two outputs should be insensitive to noise and faults; iii) the control action at high frequencies should be restricted, avoiding rapid variations; iv) the residual signals, r1, r2 and r3, should be large only when a fault has occurred, fa, fs1 and fs2, respectively; v) these objectives should hold in the presence of a bounded uncertainty, ∆. To achieve these objectives, the weighting transfer functions of the augmented plant P must be chosen appropriately. The selection of these transfer functions is a fundamental task to address the trade-offs between control and fault detection. 4.1 Uncertainty description The inverted pendulum plant is represented by the nominal plant model, Gnom(s), with an uncertainty description defining the set of all possible plant variations. In this case, the plant uncertainty is described by structured multiplicative input uncertainty. The set of all possible plants G(s) is then described by the following equation: G ( s ) := G nom ( s )(1 + ∆( s )w i ( s )) : ∆( s ) stable , ∆( s ) ∞ ≤ 1 .

{

}

Essentially, the plant uncertainty is due to modelling errors and nonlinear effects. The most significant nonlinearities are the dry friction (Coulomb friction) and the static friction acting on the cart. Therefore, the plant uncertainty is described by the following weighting transfer function: wi (s ) =

0.05( s + 0.2 ) s

2

.

+ 0.07s + 0.49

4.2 Performance specifications To achieve the performance objectives (i, ii, and iii design objectives) the transfer functions from the inputs w and v to the output z are shaped using the weighting transfer functions wp1, wp2 and wu. The transfer functions wn1, wn2, wref, wfa, wfs1 and wfs2 are chosen to express the knowledge about the measurement noise in each sensor, reference and faults, respectively. Therefore, the weighting transfer functions are chosen in order to achieve: w

T w ≤ 1, for all ∆ satisfying ∆ ≤ 1. ∞ out out , in in ∞

The input signals are assumed to be bounded according to: i) the measurement noise is white noise with low power; ii) the reference signal satisfies ref ≤ 0.3m ; iii) the fault on the control action satisfies f a ≤ 2 N ; 5

iv) the faults on the measurements are given by f s 1 ≤ 0.2 m and f s 2 ≤ 2 º . Given these bounds, the weighted transfer functions for the inputs are given by: w n1 ( s ) =

0.005( s + 1) 0.5s + 1

, w n 2 (s ) =

0.0025( 2 s + 1) s +1

, w ref ( s ) = 0.3 , w fa ( s ) = 2 , w fs 1 ( s ) = 0.2 and w fs 2 ( s ) = 0.035 .

To attain to the performance objectives the following requirements can be assumed: i) for the error signal given by (ref - y1): steady-state error lower than 8%; amplification at highfrequencies lower than 4dB; closed-loop bandwidth higher than 0.2rad/s; ii) for the output signal y2: steady-state error lower than 0.036rad; attenuation at high-frequencies lower than -20dB; closed-loop bandwidth higher than 0.3rad/s; iii) for the control action u: control action at low frequencies should be lower than 10N; action for frequencies higher than 0.05rad/s should be lower than 2N. Given these performance requirements, the corresponding weighted transfer functions are given by: w p1 ( s ) =

12.5( 5s + 1) 100s + 1

,

w p2 (s ) =

27.8( 3.3s + 1) 10s + 1

and

w u (s ) =

10s + 1 20s + 10

.

4.3 Fault detection specifications In order to achieve the diagnosis performance objectives (iv design objective), the transfer functions from the inputs w and v to the residual output r are shaped using wr1, wr2 and wr3. Assuming the boundaries defined above for the input signals and expressing the desire of good diagnostic performance at low frequencies, the weighted transfer functions for the residual signals are given by: w r1(s ) =

10( s + 1) 10 s + 1

,

wr 2 (s ) =

4( 0.2 s + 1) 2s + 1

and

w r 3 (s ) =

10( 0.5s + 1) s +1

.

4.4 Controller Design The desired generalised controller for the inverted pendulum, satisfying the design objectives, is obtained using the µ-synthesis method. Applying this method, and considering the M-∆ structure (M = F(P, K)) and the structured singular value µ, the generalised controller must satisfy the following condition: Fu ( M , ∆ ) ∞ = µ ( M ) ≤ 1 . Applying the D-K iterative procedure, and after two iterations, a generalised controller K of order 15 is found, giving µ=0.9804. Using the optimal Hankel norm technique, the controller order is then reduced to a seven order state-space representation. To implement this controller, a discrete-time representation of the controller is obtained by zero order sampling with a sampling time of 30ms.

5 Results The generalised controller was tested in a simulation environment, considering periodic step changes in the cart position reference. The responses of the closed loop system to the cart position reference, considering no faults, are represented in Figure 5. As can be seen, the reference tracking and the pendulum stability are good, and the residuals are almost without effect. In order to analyse the responses of the system in presence of faults, three different faults have been applied on the inverted pendulum plant, from t=40s to t=70s. The responses to the fault on the control action (fa=-2N) are represented in Figure 6 and the responses to the faults on the cart position measurement (fs1=0.2m) and on the pendulum angle measurement (fs2=2º) are represented in Figure 7 and Figure 8, respectively. These responses shown that the closed loop system has a satisfactory behaviour even if faults are present. 6

Concerning the residual signals, generated by the fault detection filter, they present a clear response to the representative fault and might be used to detect and to isolate each occurred fault. Reference and Cart Position

0.4 0.2 m

0.2

0

º

-0.2 -0.4

0 -0.2

0

20

40

60

-0.4

80

Force (control action)

2

0.2

N 0

0

-1

-0.2 0

20

40 60 time [s]

0

20

40

-0.4

80

60

80

Residuals

0.4

1

-2

Pendulum Angle

0.4

r1

r3

r2 0

20

40 60 time [s]

80

Fig. 5. Closed loop responses to step changes in the cart position reference (at t=0s, t=30s and t=60s), without faults.

Reference and Cart Position

0.4 0.2 m

0.2

0

º

-0.2 -0.4

0 -0.2

0

20

40

60

-0.4

80

Force (control action)

3

0

N 1

-1

0 20

40 60 time [s]

20

40

60

80

40 60 time [s]

80

Residuals r3

r1

r2

-2

fa=-2N 0

0

1

2

-1

Pendulum Angle

0.4

-3

80

0

20

Fig. 6. Closed loop responses to step changes in the cart position reference 7

and a step fault (fa=-2N) in the control action, from t=40s to t=70s.

Reference and Cart Position

Pendulum Angle

0.4

0.5

0.2 m

0

0

º

-0.2

fs1=0.2m

-0.5 0

20

40

60

-0.4

80

Force (control action)

2

0.2

N 0

0.1

-1

0 0

20

40 60 time [s]

20

-0.1

80

40

60

80

Residuals

0.3

1

-2

0

r2

0

20

r1

r3

40 60 time [s]

80

Fig. 7. Closed loop responses to step changes in the cart position reference and a step fault (fs1=0.2m) in the cart position measurement, from t=40s to t=70s.

Reference and Cart Position

0.4 0.2 m

2

0

º

-0.2 -0.4

1 0

0

20

40

60

-1

80

Force (control action)

2

2

N 0

1

-1

0 0

20

40 60 time [s]

fs2=2º 0

20

-1

80

40

60

80

40 60 time [s]

80

Residuals

3

1

-2

Pendulum Angle

3

r3

r1 r2 0

20

Fig. 8. Closed loop responses to step changes in the cart position reference 8

and a step fault (fs2=2º) in the pendulum angle measurement, from t=40s to t=70s.

6 Conclusions Control and fault detection systems for dynamic systems are often designed independently. This methodology may lead to unnecessarily poor diagnosis performance due to interaction between the controller and the fault detection filter, especially in the case of uncertain plants. To address this problem, the design of the controller and the fault detection filter should be integrated into the same framework. An approach relying on robust control methods has been studied in this paper, to achieve a fault tolerant control system for the inverted pendulum plant. The simultaneous design of the control system and a model based fault detection system has been converted into a robust control problem. The inverted pendulum is described by a state space representation and a norm-bounded transfer function is used to represent knowledge of process uncertainty. The performance and fault detection specifications are expressed using weighting transfer functions. A generalised controller, integrating the controller and the fault detection filter, has been obtained using the µ-synthesis and the D-K iterative method. The results shown a good performance of the controller and the residual signals, generated by the fault detection filter, shown its ability to diagnose and isolate abrupt faults on the control action and on the measurement signals. A similar performance is expected for the application of this approach to a laboratory inverted pendulum. The obtained results show that the approach described in this work can be an useful and suitable strategy to robust fault tolerant control of mobile structures.

Acknowledgments This work was partially financed by FCT/PRAXIS XXI Program.

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Patton, R. J., (1997). Fault-Tolerant Control: The 1997 Situation. Proceedings of IFAC Symposium on Fault Detection, Supervision and Safety for Technical Processes (SAFEPROCESS’97), Pergamon 1998, Univ. of Hull, UK, 1033-1055. Postlethwaite, I. and S. Skogestad (1993). Robust Multivariable Control using H∞ Methods: Analysis, Design and Industrial Applications, "Essays on Control: Perspectives in the Theory and its Applications" (Editors: H. L. Trentelman e J. C. Willems), Birkhäuser. Stoustrup, J., Grimble, M. J. and Niemann, H. H. (1997). Design of Integrated Systems for Control and Detection of Actuator/Sensor Faults. Sensor Review, 17, 157-168. Tyler, M. L. and Morari, M. (1994). Optimal and robust design of integrated control and diagnostic modules. Proceedings of the 1994 American Control Conference, 2060-2064. Zhou, K., J.C. Doyle and K. Glover (1996), Robust and Optimal Control, Prentice Hall.

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