Integrated Fault Tolerant Scheme for a DC Speed ... - Semantic Scholar

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Integrated Fault Tolerant Scheme for a DC Speed Drive D.U. Campos-Delgado, Member, IEEE, S. Mart´ınez-Mart´ınez, and K. Zhou, Fellow, IEEE,

Abstract— In this paper, an active fault tolerant control (FTC) scheme is presented with disturbance compensation. Faultdetection and compensation are merged together to propose a robust algorithm against model uncertainties. The GIMC control architecture is used as a feedback configuration for the active fault-tolerant scheme. The synthesis procedure for the parameters of the fault-tolerant scheme is carried out by using tools of robust control theory. A detection filter is designed for fault isolation taking into account uncertainties and disturbances in the mathematical model. Finally, the fault compensation strategy incorporates an estimate of the disturbances into the system to improve the performance of the closed-loop systems after the fault is detected. In order to illustrate these ideas, the speed regulation of a DC motor is selected as a case study, and experimental results are reported. Index Terms— Fault tolerant control, robust control, H∞ design, DC motor.

I. I NTRODUCTION In many industrial applications, costly equipment is managed and human operators are involved. In these conditions, it is desirable to provide some safety degree into the automated process. Thus, the human operator must receive an indication of the possible faults into the process in order to take proper action. For certain types of faults, it is possible that the nominal control system could be designed to tolerate or maintain some of the performance for those faults (passive approach) [1]–[3]. However, this strategy tends to be conservative in the practice since the controllers have to be designed taking into account the worst-case scenario. One way of synthesizing these fault-tolerant controllers is by appealing to H∞ robust design techniques [4]–[6]. Another approach for FTC relies on the detection of a faultcase in the control process, in order to introduce a proper compensation to the feedback system (active approach) [7]. In this scheme, it is first necessary to detect a fault scenario, and next, to design an algorithm to identify the type fault occurred (fault isolation). Based on the fault isolation block, an external compensation signal for the nominal control signal is introduced, or the parameters of the controller are updated [8], [9]. Three main types of faults are recognized: actuator, sensor and plant faults [4], [10], [11]. The first two are modelled as Manuscript received October 20, 2003; revised April 19, 2004. This research is supported in part by a grant from PROMEP (PTC-28), CONACYT, NASA and LEQSF. D.U. Campos-Delgado is with the Universidad Autónoma de of San Luis Potos´ı, Facultad de Ciencias, Av. Salvador Nava s/n, Zona Universitaria, C.P. 78290, S.L.P., México. S. Mart´ınez-Mart´ınez is with the Universidad Autónoma de of San Luis Potos´ı, CIEP, Facultad de Ingenier´ıa, México. K. Zhou is with Louisiana State University, Electrical and Computer Engineering Department, Baton Rouge, LA 70803.

external signals that are added to the nominal ones (additive faults). Meanwhile, the plant faults are related to mechanical wear down of the plant elements, or intrinsical changes in the dynamics of the system. These faults are usually modelled as parameter variations in the mathematical model of the plant. The problem of additive faults will be addressed in this paper. Hence for an active FTC configuration, the first challenge is fault detection and isolation (FDI). The paradigm in FDI is to detect and isolate a fault condition despite possible disturbances, noise, and model uncertainty in the system [4], [7], [10], [11]. Thus, filters are designed such that the effect of faults is maximized at the outputs while the effect of disturbances and model uncertainty is minimized. Several approaches have been suggested: robust detection and isolation based on eigenstructure assignment [12], estimation based on H2 and H∞ optimization [13], [14], detection and isolation by frequency domain optimization [15], unknown input observers [4], parity space approaches [16], robust observed-based detection [17], etc. Most of the existing research is focused on linear systems, but extensions to FDI of nonlinear systems have also been proposed in [18] and [19]. Furthermore, the applications of fuzzy logic [20] and wavelet transforms [21] to fault detection have been recently introduced. Next, in the active approach for FTC, once the fault has been detected and isolated, the control strategy is reconfigured by a supervisory system. Recently, inspired by the Youla parameterization used in robust control theory, a reconfigurable control structure for fault compensation has been suggested in [22]. This scheme applies the GIMC (Generalized Internal Model Control) structure introduced in [23] to design a control compensation signal after a fault is detected. On the other hand, reconfigurable FTC structures have also been studied where the model-matching strategy is used to design linear controllers [24], and adaptive schemes for nonlinear systems [25]. The problem of adaptive compensation for actuator failures was recently addressed in [26]. Thus, due to the practical applications in industry processes and theoretical challenges, the research in FTC has increased in recent years, pursuing to provide certain safety degree into the automated processes. The paper is structured as follows. Section 2 describes the problem formulation. The details about the fault detection and compensation strategies are shown in Section 3. Section 4 gives a description of the case study: speed regulation of a dcmotor, and experimental results are reported. Finally, Section 5 gives some concluding remarks.


2

The open-loop system response y of (1) can be analyzed in the transfer matrix form:

II. P ROBLEM F ORMULATION A. System Description

y(s) = Puy (s)u(s) + Pf y (s)f (s) + Pdy (s)d(s) f

d

ref

Controller -

where

P

Puy (s) = Pf y (s) = Pdy (s) =

y

u

q

Fig. 1.

Problem Formulation for Control.

The problem addressed in this paper is formulated as follows. Consider an LTI system P (s) affected by disturbances d ∈ Rr and possible faults f ∈ Rl (additive), see Fig. 1, described by x˙ y

(2)

= Ax + Bu + F1 f + E1 d = Cx + Du + F2 f + E2 d

(3)

Hence Puy (s) represents the nominal plant (mapping from u to y). Furthermore assume that there exists knowledge about the model uncertainties in the description of the nominal plant [6]. Two possible scenarios can be seen: structured or unstructured uncertainty according with robust control theory. • If the uncertainty can be derived from certain parameters of the model, where a range of variation can be deduced, then a structured uncertainty is adopted. As a consequence, the real plant can be represented by an upper linear fractional transformation (LFT): Pûy (s) = Fu (P, ∆) = P22 + P21 ∆(I − P11 ∆)−1 P12 (4) where · ¸ P11 P12 P= (5) P21 P22

(1)

where x ∈ Rn represents the vector of state, u ∈ Rm the vector of input, and y ∈ Rp the vector of output. Thus, the matrix F1 ∈ Rn×l stands for the distribution matrix of the actuator faults, and F2 ∈ Rp×l for the sensor faults. Assume that a nominal controller K(s) stabilizes the nominal plant and it provides a desired closed-loop performance. Consequently, the control objective of the active FTC scheme is presented as: design an integrated detection/compensation scheme such that it detects the occurrence of a fault in the closed-loop system, and provides an appropriate compensation signal q to the control system in order to maintain some closed-loop performance, see Fig. 1. Two assumptions are made in the problem formulation: • The fault is non-repetitive. • The disturbances are known or partially known. Remark 1: The first assumption establishes that the faults studied have a permanent effect in the system. As a result, once the fault is detected, the compensation signal q is switchedon in the feedback configuration and remains active. Thus the process will have to be stopped to replace the faulty device and reset the FTC scheme. On the other hand, the information of the disturbances into the system is important to be able to provide the correct compensation signal q in the FTC scheme. In case that the disturbance information could not be retrieved, the performance after the fault will be deteriorated. If there was a disturbance change at the same time of the fault, the algorithm will not be able to distinguish it immediately and the control signal will not have the correct information of the actual disturbance in the compensation. In this way, the resulting compensation after the fault will be affected. Moreover, if two or more independent faults act on the system simultaneously, the fault detection will be triggered but according with the time constant of each fault, the transient behavior will be affected after the compensation.

C(sI − A)−1 B + D C(sI − A)−1 F1 + F2 C(sI − A)−1 E1 + E2

•

∆ = diag[δ1 δ2 . . . δk ] with δi ∈ (−1, 1) i = 1, . . . , k, and the index k represents the number of uncertain parameters. In this case, the generalized plant P is derived by pulling out the variation parameters δi from the nominal plant. Note that for the nominal plant Puy (s) = P22 (s). In the case that the uncertainty could be considered unstructured, the real plant Pûy is given by Pûy (s) = Puy (s) + W2 (s)∆(s)W1 (s) = Fu (P, ∆) (6) where W1 , W2 ∈ RH∞ are weighting functions for the uncertainty, ∆ ∈ RH∞ with k∆k∞ < 1, and in this specific case, the generalized plant P is given by · ¸ 0 W2 P= (7) W1 Puy

B. Residual Design In FDI theory, based on the input-output description of the system, a signal that contains information of faults has to be designed: residual [4], [10], [11]. Using the representation (1), the nominal plant can be expressed by a left coprime ˜ −1 (s)N ˜ (s) where N ˜, M ˜ ∈ factorization, i.e. Puy (s) = M RH∞ . Then a residual signal r can be constructed by: ˜ (s)u(s) + M ˜ (s)y(s)] r(s) = H(s)[−N

(8)

where H ∈ RH∞ is known as detection filter. By substituting (2) and using the model uncertainty description, it is obtained that ˜ (s)∆uy (s)u(s) + N ˜d (s)d(s) + N ˜f (s)f (s)] r(s) = H(s)[M (9)


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where ∆uy = W2 (s)∆(s)W1 (s) or ∆uy = P21 ∆(I − P11 ∆)−1 P12 for unstructured and structured uncertainty respectively. Moreover, using the problem formulation in (1), then ˜ −1 (s)N ˜f (s), Pf y (s) = M

˜ −1 (s)N ˜d (s) (10) Pdy (s) = M

˜d , N ˜f ∈ RH∞ . As a consequence, the residual signal r with N is affected by the control signal through the model uncertainty, the disturbances and the faults. Therefore, the aim of the filter H(s) must be to isolate the effect of faults f in the residual r, i.e. ˜ (s)∆uy (s) ≈ 0 and H(s)N ˜d (s) ≈ 0, • H(s)M ˜ • H(s)Nf (s) 6= 0 and as large as possible in some sense. ˜f (s) ≈ For perfect fault isolation, it is then needed H(s)N I. In order to detect a fault-scenario, the following residual evaluation criteria are commonly followed sZ t

krk

=

r∗ (τ )r(τ )dτ

krk2,t,T =

(11)

t−T

krk

=

krk∞ = sup krk2 t

sup

krk

(17)

∀d,u

Consequently, the optimal filter H(s) must maximize the size of Υ [17], i.e. max dim Υ (18) H(s)∈RH∞

in order to obtain the best tradeoff between fault sensitivity and robustness. III. FAULT-T OLERANT S CHEME In this section, the active FTC strategy is described in detail. A. Generalized Internal Model Control ref e -d - U ˜ 6 −

- d - V˜ −1 6q Q 6

Fig. 2.

fe

u

- P

y -

? ˜ N ? d¾ −

˜ ¾ M

Generalized Internal Model (GIMC) Control Structure.

(13)

f =0,∀d,u

The threshold Jth , in general, can be also a function of time [4]. However, this approach is not pursued in this paper. Once the residual signal is constructed, a fault can then be detected according to the criterion: krk ≥ Jth

Υ = {f | inf krk ≥ Jth }

(12)

where T is the window length or horizon of evaluation. In some special cases, the filter H(s) can cancel completely the effect of uncertainty and disturbances in the residual. Therefore, in the practice to prevent a false alarm in the evaluation, a threshold value Jth is selected such that Jth =

conservativeness to this type of bound. As a result, define the set of strongly detectable faults

(14)

where the threshold for detection can be calculated by using the size of the uncertainty, the disturbance information and the maximum value of the control signal. Thus, by applying the triangle inequality to (9) °· ¸° ° £ ¤° ° u ° ° ° ° ° ˜ ˜ krk ≤ H(s) M (s)∆uy (s) Nd (s) ° d °+ ˜f (s)k kf k. kH(s)N (15) a threshold Jth can be chosen, where the relation between the L1 norm of systems and the l∞ norm of signals [27] can be exploited, i.e. °· ¸° ° £ ¤° ° u ° ° ° ° ° ˜ ˜ Jth = H(s) M (s)∆uy (s) Nd (s) 1 ° (16) d °∞ Alternatively, the relation between the H∞ norm of systems and the l2 norm of signals could be used instead [4], [6]. However, the threshold using the L1 norm tend to be more appropriate, since the signals in the feedback loop are in general bounded in time. Moreover, the l2 norm has to be approximated using a finite window length, which adds

The FTC architecture proposed in this work is derived from robust control theory [6], where a new implementation of the Youla parameterization called Generalized Internal Model Control (GIMC) is used [22], [23], see Fig. 2. In this configuration, the nominal controller K(s) is represented by ˜ (s) such its left coprime factorization, i.e. K(s) = V˜ −1 (s)U ˜ , V˜ ∈ RH∞ . It is observed that fe represents the filtered that U error between the estimated output and the true output of the system. Thus if fe = 0 (⇒ q = 0) that will represent that there is no model uncertainties, external disturbances or faults into the system, then the feedback system will be solely controlled by the nominal controller K(s). Consequently, from the GIMC configuration in Fig. 2, the control signal u has a component due to the control tracking error and will have another from the filtered error fe through the compensator Q in the faultscenario : h i ˜ (s)e(s) + q(s) u(s) = V˜ −1 (s) U (19) h i ˜ (s) {ref (s) − y(s)} + Q(s)fe (s) = V˜ −1 (s) U Note that if disturbances d are affecting the nominal system then the filtered error fe will be drastically altered. However, it is assumed that the disturbance d is known or partially known. Therefore, this information can be feedforward into the estimation process to cancel its effect from the filtered error fe . As a consequence, a new implementation of the GIMC architecture is suggested in Fig. 3. The residual signal proposed in the previous section, see (8), can be constructed by taking the signal fe and process it through the detection filter


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H(s), i.e. r(s) = −H(s)fe (s). Now, once the fault is detected, the compensation signal q is fed back into the controller structure, where q is also constructed from the filtered error fe but through the robustification controller Q(s). Therefore, the signal q has to compensate the control signal by the missing/erroneous information due to the fault. Consequently, an integrated FTC scheme can be achieved, as shown in Fig. 3. Note that the GIMC structure allows to design the nominal controller and the fault detection/compensation independently. Thus, the nominal controller K(s) can have any given structure that satisfies the performance specification, such as PID or observer-based, and the detection will operate in parallel with the system until a fault is detected, at this time, the compensation enters in the feedback loop. The FTC scheme d

~ ~ [ I-V U]

ref

u^

f

u

P

where Tef ν = Fl (Fu (GH , ∆) , H) with 





P11 z  ef  =  0 ˜ P21 fe M

0 0 ˜d N

-H

w f d u r

     

(21)

the true plant is given by Pûy = Fu (P, ∆) according with the uncertainty representations of (4) and (6), ν = [f d u]T is the vector of inputs, and GH the generalized plant in the LFT format. The design problem in (20) can be tackled with tools from robust control theory (µ-synthesis) [6]. The synthesis procedure using µ-synthesis is carried out in an iterative algorithm that performs a two-parameter minimization in sequential fashion to reach the optimal solution: D − K iteration [6].

y

w

Q fe

0   I   0 

 w , GH  ν  r

u

~ ~ ~ [[Nd N] -M ]

d

f

Pdy

Pfy

z

∆

residual evaluation

r

P12 0 ˜ P22 − N ˜ M



q

-

0 −I ˜f N

 

y

P ~ N

~ M

^ P uy -

Fig. 3.

Overall Active FTC Strategy.

fe

presents then two free parameters to be designed: 1) H(s): the fault detection filter that must diminish the effect of the disturbances or uncertainty into the residual signal, and maximize the effect of the faults [4], [7]. 2) Q(s): the robustification controller that must provide robustness into the closed-loop system in order to maintain acceptable performance against faults [23]. The design strategies for these two parameters are presented next. Due to the structure of the GIMC architecture, both parameters H(s) and Q(s) must be stable and proper transfer matrices. B. Fault Diagnosis To design a robust detection filter H(s), the information of the model uncertainties and disturbances have to be incorporated. Since the information of the uncertainty is assumed to be estimated in the worst case, it has to be considered in the design stage of the detection filter H(s) to improve the fault sensitivity [4]. Assume that it is desired to isolate the vector of faults, i.e. r ≈ f , and define the estimation error for the faults ef = r − f . Then, the performance objective of the filter can be stated as to minimize the effect of uncertainty and disturbances in the fault estimation (see Fig. 4), i.e. min

H(s)∈RH∞

kTef ν k∞

s.t. k∆k∞ < 1

(20)

H e

f

r

Fig. 4.

Fault Detection Synthesis Diagram.

C. Fault Compensation In the design of the fault compensation signal q, the transfer matrix Q(s) is chosen to maintain stability against any fault. Thus, the problem of robust stabilization is addressed. However, some performance specification could also be incorporated in the design of Q(s). If there exist Ki (s) controllers that have a desired performance against certain type of faults ˜ , then the fi , different from the nominal controller K = V˜ −1 U architecture GIMC allows to design specific compensators for each one Qi (s), according with the relation: h i−1 ˜ (s)Ki (s) + M ˜ (s) Qi (s) = V˜ (s) [Ki (s) − K(s)] N (22) ˜ −1 (s)N ˜ (s). This approach for the nominal plant Puy (s) = M will be pursued in future work. Only one limitation on the transfer matrix Q(s) has to be considered, according with the Youla parameterization, it


5

has to be stable, i.e. Q ∈ RH∞ . The synthesis process is again carried out through the philosophy of robust control. In general, the sensor and actuator faults can be modelled in a multiplicative form y˜(s) u ˜(s)

= [I + ∆s ]y(s) = [I + ∆a ]u(s)

where ∆s , ∆a ∈ RH∞ represent the sensor and actuator perturbations due to the faults [22]. Consequently, if these terms are appended to the nominal plant Puy (s), then the faulted input-output mapping P˜uy (s) can be represented as P˜uy (s) = Puy (s)[I + ∆a ] (actuator fault) P˜uy (s) = [I + ∆s ]Puy (s) (sensor fault)

Puy (s) that satisfies the closed-loop performance requirements, then ˜ (s)M ˜ −1 (s) Q(s) = −U (27) is the optimal solution that decouples the fault f signal from the control signal u. Moreover, if Q(s) is chosen in this way, and the compensation scheme of Fig. 5 is implemented after the fault is detected, then the control signal u will have contributions from the disturbance d and reference ref : u(s) = [I +K(s)Puy (s)]−1 [−K(s)Pdy (s)d(s)+K(s) ref (s)] (28)

(23)

Thus the sensor or actuator faults can be modelled as output or input model uncertainties respectively. As mentioned before, we shall consider a basic robustness requirement in this paper, i.e. the closed-loop stability. Hence our objective is to design Q(s) to maximize the failure tolerance in the closed-loop system, i.e. min kTzw k∞ (24)

Pdy ref -

~ V -1

~ U

min

Q(s)∈RH∞

kFl (GQ , Q) k∞

(25)

and internal stability is guaranteed if k∆k∞ < 1/γ. If an output uncertainty (sensor fault) is considered (i.e. P˜uy = [I + ∆s ]Puy ), the generalized plant GQ will be given by ¸ · ˜ ˜ ˜ −1 (s) −S(s)P uy (s)K(s) S(s)Puy (s)V GQ = ˜ (s) −M 0 (26) ˜ where S(s) = [I + Puy (s)K(s)]−1 . Note that in this case γ ≥ 1 since this will represent that the maximum tolerable uncertainty is always k∆s k∞ < 1. Otherwise, the uncertainty could take the value ∆s = −I (sensors outage) and the closed-loop will become unstable. In an alternative way, the optimal Q(s) can be derived by following the problem formulation presented in (1) and (3), and considering the setup of Fig. 5. This applies only to the ˜ −1 ∈ RH∞ . case Puy ∈ RH∞ =⇒ M Theorem 1: Suppose Puy ∈ RH∞ and K(s) = ˜ (s) is a stabilizing controller of the nominal plant V˜ −1 (s)U

Puy

y

Q

fe

where Tzw is the closed-loop transfer function from signals w to z. Two design scenarios can be presented according with the stability of the nominal plant Puy (s) [22]: 1) Puy ∈ RH∞ ⇒ the optimal compensator is given ˜ (s)M ˜ −1 (s), for any plant and type of by Q(s) = −U ˜ (s) and M ˜ (s) uncertainty description [22]. Recall that U are related to the coprime factorizations of the nominal controller K(s) and plant Puy (s). 2) Puy ∈ / RH∞ ⇒ a weighted H∞ approximation has to be solved. For this purpose, the synthesis problem can be put into an LFT framework. Hence, Q(s) is chosen according to

u

Pfy

q

Q(s)∈RH∞

γ=

f

d

Fig. 5.

~ ~ ~ [[Nd N] -M ]

General Fault Compensation Setup.

Proof: From Fig. 5, it can be seen that h n ˜ (s) u(s) + N ˜d (s)d(s)− u(s) = V˜ −1 (s) Q(s) N o i ˜ y(s) + U ˜ (s) {ref (s) − y(s)} M (29) and by substituting (3) and (10) in the previous equation, it is obtained after simplification h n o ˜ (s) + U ˜ (s) Pf y (s)f (s)− u(s) = V˜ −1 (s) − Q(s)M i ˜ (s) {Puy (s)u(s) − Pdy (s)d(s) + ref (s)} (30) U Therefore, to decouple the fault signal f from the control signal u is needed ˜ (s) + U ˜ (s) = 0 Q(s)M

˜ (s)M ˜ −1 (s) =⇒ Q = −U

(31)

˜ −1

Note that since M ∈ RH∞ , then Q ∈ RH∞ . Finally, by substitution of Q(s) into the control signal u in (30), it is obtained (28). ¤ Remark 2: From the proof of the previous theorem, it is important to mention that if the disturbance information d is not feedforward into the fault-tolerant scheme, then the control signal u will not adjust its value according the disturbance. Hence, the tracking performance of the resulting closed-loop system will be largely affected, but the closed-loop stability is always guaranteed since the nominal controller internally stabilizes the nominal plant [23], and the disturbances are additive in the loop.


6

Remark 3: It is common that some types of faults are more easy to detect than others, for example abrupt and incipient [11]. In the case of the abrupt faults, the detection time is almost instantaneous, since a large peak is observed in the residual. On the other hand, for an incipient fault, it is more difficult to detect its presence in the residual. Thus, it is possible to have a delay in the detection of the fault triggering or could not be detected at all. Furthermore, in the presence of control constraints as saturation, it is possible that the system could not be stabilized if there is a significant delay in the detection process, since the system could enter in the nonlinear part of the saturation. This delay is argued to be a factor of the speed of response of the nominal controller, as it was seen in simulation. Hence, if the controller has fast dynamics, the tolerable delay is reduced. IV. C ASE S TUDY: S PEED R EGULATION OF A DC M OTOR To illustrate the mentioned FTC technique, the application to a dc-motor speed drive is presented next [28]. The test-bed setup is shown in Fig. 6, and it consists of a 1 HP dc-motor connected to a 3/4 HP permanent magnet motor. The latter one acts as a generator in order to provide a load to the shunt dc-motor. In this setup, the load torque varies according with the angular frequency. The active FTC algorithm was implemented in a dSpace DS1103 system under the environment of MATLAB/Simulink r. The algorithms were run under a sampling frequency of 10kHz. In the implementation, there are measurements of electrical currents made by Hall-effect sensors, and angular velocity by a tacogenerator of 50V /1000 RPM with ±5% tolerance.

TABLE I DC M OTOR PARAMETERS .

Ra = 3.72 Ω La = 7.83 mH B = 2.59 × 10−4 N m/rad/s J = 2.72 × 10−3 kg m Kb = 0.31 V /rad/s Rf = 87.3 Ω Lf = 9.22 H ω0 = 1500 rpm Vr = 90 V Imax = 10 A

Armature Resistance Armature Inductance Friction Coefficient Inertia Electromagnetic Constant Field Resistance Field Inductance Nominal Velocity Rated Armature Voltage Maximum Armature Current

measurements are available for feedback purposes: armature current ia and angular velocity ω. Thus, the dc-motor is modelled as a system with one-input (va ) and two-outputs (ia and ω): dia dt dω dt

= =

Ra Kb 1 ia − ω+ va La La La Kb B 1 ia − ω − Tl J J J

−

where the parameters of the dc-motor were obtained through a systematic experimental process [28], and they are shown in Table I. In the motor description, the load torque Tl is represented as a disturbance into the system, however this variable cannot be measured in real-time. Nevertheless, during the experiments, it is assumed to behave as a constant or present a very slow variation. In this way, its steady state value can be calculated from the angular velocity and armature current measurements by the relation: T l = Kb ia − Bω

T

MOTOR

GENERATOR

RESISTOR

Ga (s) = Test-bed Setup.

A. Model Description The dc-motor is considered in a separated excitation configuration. Thus, the field (stator) voltage vf is fixed to a constant value, and the armature (rotor) voltage va is varied in order to control the angular velocity ω of the motor. Consequently, the electromagnetic field provided by stator is held constant. Two

(33)

The armature voltage va is controlled by dc-dc chopper working under a PWM scheme (switching frequency 50kHz), where the control parameter is the duty cycle. The chopper was selected as control actuator due to its fast response and linear dynamics. The construction of the actuator was carried out in our lab, and it is designed such that it is controlled by a voltage signal u in the interval [0, 5V ]. This saturation in the control signal u did not limited the performance of the system, due to the fast dynamics of the actuator related to the time constant of the dc-motor. The control actuator (dc-dc chopper) was modelled as a first order system:

computer

Fig. 6.

(32)

va (s) s+a = Ka u(s) s+b

(34)

where the parameters of the model (Ka , a, b) were obtained experimentally by applying the theory of algebraic identification [29], [30]. Six different experiments were carried out and the average values of the parameters Ka , a and b are given in Table II. In the overall mathematical model, there exist some uncertainty in the description of some of the parameters including motor and actuator. Thus, for Ra , La , J, Ka , a and b, they can be


7

TABLE II C ONTROL ACTUATOR PARAMETERS .

Ka = 20.71 a = −6.66 b = 5.43

Actuator Gain zero location pole location

associated with a parametric uncertainty description: â R â L Jˆ â K

= = = = a ˆ = ˆb =

Ra (1 + 0.7δR ) La (1 + 0.2δL ) J(1 + 0.1δJ ) Ka (1 + 0.015δK ) a(1 + 0.09δa ) b(1 + 0.23δb )

(35)

where “ ˆ· ” denotes the real value of the variables, and −1 ≤ δi ≤ 1. As a consequence, a structured uncertainty description can be arranged for the open-loop system. This uncertainty description will be used to design the fault detection filter H(s). B. Design of Fault Tolerant Scheme Note that with the parameters given to the nominal plant (32) and actuator (34), the open-loop system is stable. Next, the nominal controller K(s) is designed to guarantee good step tracking capabilities. For this purpose, an LQG controller [6] is synthesized where integral action is appended to the controller to improve closed-loop tracking and disturbance rejection. Using a structured uncertainty description for the plant, the optimization problem formulated in (20) was carried out to design H(s). Thus, according with the diagram of Fig. 4, the LFT is constructed to formulate the robust performance filter problem using µ-synthesis. The D − K iteration was run and the resulting filter of order 21th was reduced through balance truncation to 9th . Finally, since the nominal plant is stable, then the compensation controller is selected Q(s) = ˜ (s)M ˜ −1 (s). −U C. Experimental Implementation Next, the integrated FTC structure shown in Fig. 3 was implemented first in simulation by using MATLAB/Simulink r, and experimentally in the dSpace system. The disturbance estimation T l in (33) was used to feedforward this information into the active FTC scheme. Two abrupt faults were considered for the sensors: 1) Case 1: the angular velocity sensor ω is completely disconnected from the system at a given time, 2) Case 2: there is an outage of the armature current ia sensor. The failure cases were simulated by software in the system and not directly in the experiment. In both scenarios, the fault-tolerant system was able to compensate properly the control signal for these faults. The experimental results for the tacogenerator fault are shown in Fig. 7. Note that in the

case of angular velocity fault, the uncompensated closed-loop system becomes unstable. Since, after the fault, the controller detects a sudden drop in the angular velocity measurement and due to its integral action, the nominal control signal starts to grow looking to increase the armature voltage in order to compensate the velocity. Therefore, without the compensation, the control signal will keep increasing due to the erroneous information of the velocity sensor. This behavior will be present when the nominal controllers contain integral action, as in PI and PID compensators, which are a common choice in industrial processes. As a result, the control compensation is necessary in these cases to avoid damage to the motor or accidents in the process. The angular velocity reference for both fault cases was set to 1500 RPM. The plots illustrate that the compensation is capable to reconfigure the control strategy to maintain some of the closed-loop performance. However, it is noticeable that the reference is not maintained, this is due to the estimation of the load torque Tl (disturbance) which is an approximation of the real value. As a consequence, the error in the estimation of the load torque affects the tracking capabilities of the overall system. Therefore, if this signal could be acquired by a direct measurement then the compensation will be more accurate. This behavior was seen during simulation. Now, in Fig. 8, the experimental testing for a change of reference after a fault scenario (Case 1) is illustrated. This plot shows that the system still is able to follow a reference signal after the fault, but with a certain error. In order to investigate further the performance of the FTC scheme, an exponential type of fault was tested experimentally. Thus, the angular velocity fault was modelled as an exponential drop in the sensor gain, with the following characteristics: ½ ω ˆ=

ω£ ¡ ¢¤ t < ton ω 1 − 0.5 1 − e−15(t−ton ) t ≥ ton

(36)

where ton is the activation time of the fault, and ω ˆ denotes the angular velocity measurement by the sensor, and ω its real value. The system response is shown in Fig. 9. Hence the system without compensation is unstable and the compensated system still is able to recover stability, but the reference tracking performance is deteriorated. Remark 4: An slower incipient type of fault, compared to (36), was also tested in simulation. Thus, the sensor measurement was slowly decaying after the triggering time. For this type of fault, the main problem is fault detection, since according to the decaying rate of the measurement, the residual will take more or less time to overcome the detection threshold. However, the FTC algorithm can also compensate this fault, but there is some deterioration of the performance due to the delay time in the detection process. The implementation for this fault was only carried out in simulation, since there was a risk of damaging the dc-motor in the actual test-bed setup. Consequently, other decision algorithms have to be applied to improve the size of the set of strongly detectable faults Υ, for example based on wavelet analysis and fuzzy logic.


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Experimental Response with Exponential Angular Velocity Fault .

V. C ONCLUSIONS An integrated active FTC scheme was introduced. The strategy relies on information of the process and potential uncertainties. The GIMC control architecture is used as a feedback configuration for the fault-tolerant scheme. The synthesis procedures were carried out by using robust control theory. The disturbances entering the system affect the performance of the compensated control system, consequently this information is estimated and feed back into the FTC architecture to cancel their effect. The speed regulation of a dc motor was selected as a case study, and the experimental results show success in the detection and compensation for the fault scenarios tested. However, the fault detection/decision scheme limits the overall performance of the FTC strategy, and more advanced techniques should be explored in future work. In summary, from the results obtained in the paper, it is observed that the

area of FTC presents challenging theoretical problems: • Robust disturbance decoupling, • Detection of incipient faults, • Delay in fault detection and compensation, • Fault Compensation with performance specifications that have to be addressed for advanced practical applications. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for helping to improve the quality of the paper. R EFERENCES [1] A. J. Alos, “Stabilization of a class of plants with possible loss of outputs or actuator failures,” IEEE Trans. Automat. Contr., vol. 28, pp. 231–233, 1983.


[2] D. Kim and Y. Kim, “Robust variable structure controller design for fault tolerant flight control,” Journal of Guidance, Control and Dynamics, vol. 23, no. 3, pp. 430–437, 2000. [3] J. Stoustrup and V. D. Blondel, “Fault tolerant control: A simultaneous stabilization results,” IEEE Trans. Automat. Contr., vol. 49, no. 2, pp. 305–310, 2004. [4] J. Chen and R. J. Patton, Robust Model-Based Fault Diagnosis for Dynamic Systems. Kluwer Academic Publishers, 1999. [5] S. H. Huang, J. Lam, G. Yang, and S. Zhang, “Fault tolerant decentralized H∞ control for symmetric composite systems,” IEEE Trans. Automat. Contr., vol. 44, no. 11, pp. 2108–2114, 1999. [6] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control. Upper Saddle River, New Jersey: Prentice Hall Inc., 1996. [7] M. Blanke, M. Kinnaert, J. Lunze, and M. Staroswiecki, Diagnosis and Fault-Tolerant Control. Springer-Verlag, 2003. [8] H. Noura, D. Sauter, F. Hamelin, and D. Theilliol, “Fault-tolerant control in dynamic systems: Application to a winding machine,” IEEE Control Syst. Mag., vol. 20, no. 1, pp. 33–49, 2000. [9] P. Zhang, S. X. Ding, G. Z. Wang, T. Jeinsch, and D. H. Zhou, “Application of robust observer-based fdi systems to fault tolerant control,” in 15th Trinennial World Congress of IFAC, Barcelona, Spain, June 2002. [10] L. H. Chiang, E. L. Russell, and R. D. Braatz, Fault Detection and Diagnosis in Industrial Systems. Springer-Verlag, 2001. [11] R. Isermann, “Supervision, fault-detection and fault-diagnosis methods -an introduction,” Control Engineering Practice, vol. 5, no. 5, pp. 639– 652, 1997. [12] L. C. Shen, S. K. Chang, and P. Hsu, “Robust fault detection and isolation with unstructured uncertainty using eigenstructure assignment,” Journal of Guidance, Control and Dynamics, vol. 21, no. 1, pp. 50–57, 1998. [13] E. G. C. Jr. and T. Song, “Robust H∞ estimation and fault detection of uncertain dynamic systems,” Journal of Guidance, Control and Dynamics, vol. 23, no. 5, pp. 857–864, 2000. [14] Z. Qiu and J. Gertler, “Robust FDI systems and H∞ optimization,” in Proc. IEEE Conf. Decision Control, San Antonio, Texas, Dec. 1993, pp. 1710–1715. [15] D. Sauter and F. Hamelin, “Frequency-domain optimization for robust fault detection and isolation in dynamic systems,” IEEE Trans. Automat. Contr., vol. 44, no. 4, pp. 878–882, 1999. [16] X. Ding, L. Guo, and T. Jeinsch, “A characterization of parity space and its application to robust fault detection,” IEEE Trans. Automat. Contr., vol. 44, no. 2, pp. 337–343, 1999. [17] S. X. Ding, P. M. Frank, and E. L. Ding, “Observer-based fault detection schemes for linear uncertain systems,” in 15th Trinennial World Congress of IFAC, Barcelona, Spain, June 2002. [18] C. D. Persis and A. Isidori, “A geometric approach to nonlinear fault detection and isolation,” IEEE Trans. Automat. Contr., vol. 46, no. 6, pp. 853–865, 2001. [19] H. Hammouri, M. Kinnaert, and E. H. E. Yaagoubi, “Observer-based approach to fault detection and isolation for nonlinear systems,” IEEE Trans. Automat. Contr., vol. 44, no. 10, pp. 1879–1884, 1999. [20] S. Altug, M. Y. Chow, and H. J. Trussell, “Fuzzy inference systems implemented on neural architectures for motor fault detection and diagnosis,” IEEE Trans. Ind. Electron., vol. 46, no. 6, pp. 1063–1079, 1999. [21] H. Ye, S. X. Ding, and G. Wang, “Integrated design for fault detection systems in time-frequency domain,” IEEE Trans. Automat. Contr., vol. 47, no. 2, pp. 384–390, 2002. [22] D. U. Campos-Delgado and K. Zhou, “Reconfigurable fault tolerant control using GIMC structure,” IEEE Trans. Automat. Contr., vol. 48, no. 5, pp. 832–838, 2003. [23] K. Zhou and Z. Ren, “A new controller architecture for high performance, robust and fault tolerant control,” IEEE Trans. Automat. Contr., vol. 46, no. 10, pp. 1613–1618, 2001. [24] Z. Yang and J. Stoustrup, “Robust reconfigurable control for parametric and additive faults with fdi uncertainties,” in Proc. IEEE Conf. Decision Control, Sidney, Australia, Dec. 2000. [25] X. Zhang, T. Parisini, and M. M. Polycarpou, “Adaptive fault-tolerant control of nonlinear uncertain systems: An information-based diagnostic approach,” IEEE Trans. Automat. Contr., vol. 49, no. 8, pp. 1259–1274, 2004. [26] G. Tao, S. Chen, and S. M. Joshi, “An adaptive actuator failure compensation controller using output feedback,” IEEE Trans. Automat. Contr., vol. 47, no. 3, pp. 506–511, 2002.

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[27] N. Elia and M. A. Dahleh, Computational Methods for Controller Design, lecture notes in control and information sciences 238 ed. Springer-Verlag, 1998. [28] R. Krishnan, Electric Motor Drives: Modeling, Analysis, and Control. Prentice-Hall, 2001. [29] M. Fliess and H. Sira-Ram´ırez, “An algebraic framework for linear identification,” ESAIM, Control, Optimization and Calculus of Variations, vol. 9, pp. 151–168, 2003. [30] H. Sira-Ram´ırez, E. Fossas, and M. Fliess, “An algebraic, on-line, parameter identification approach to uncertain dc-to-ac power conversion,” in Proc. IEEE Conf. Decision Control, Las Vegas, Nevada, Dec. 2002.

D.U. Campos-Delgado was born in San Luis Potos´ı, México, on October 14, 1973. He received the B.S. in electronics engineering from the Autonomous University of San Luis Potos´ı, México, in 1996, the PLACE M.S.E.E. and the Ph.D. degrees from Louisiana State PHOTO University in 1999 and 2001, respectively. In May HERE 2001, The College of Engineering of LSU granted him the ”Exemplary Dissertation Award”. Since 2001, he is an associate professor at the College of Sciences of the Autonomous University of San Luis Potos´ı, México. His research interests include robust control, fault-tolerant control, optimization, modelling and control of dynamic systems.

PLACE PHOTO HERE

S. Mart´ınez-Mart´ınez was born in San Luis Potos´ı, México, on January 6, 1979. He received the B.E. in mechanical and electrical engineering, and the M.Sc. in electrical engineering from the Autonomous University of San Luis Potos´ı, México, in 2001 and 2004 respectively. He is currently pursuing the Ph.D. degree with the Centre on Automatic Control of Nancy in the University Henri Poincaré of Nancy, France. His research interests include fault-tolerant control, control reconfiguration and structural systems analysis

K. Zhou was born in Wuhu, China, on May 7, 1962. He received the B.S. degree in Automatic Control from Beijing University of Aeronautics and Astronautics, Beijing, in 1982, the M.S.E.E. and PLACE the Ph.D. degrees from University of Minnesota, PHOTO Minneapolis, in 1986 and 1988, respectively. HERE He was a Research Fellow and Lecturer at California Institute of Technology from 1988 to 1990. Since 1990, he has been with the Department of Electrical and Computer Engineering, Louisiana State University. He was promoted to Associate Professor in 1996 and Full Professor in 2000. He is currently the Chair and the holder of the Voorhies Distinguished Professorship in Electrical Engineering. He is the leading author of two popular books in the field: Robust and Optimal Control (Prentice Hall, 1995), which has been used worldwide as graduate textbook and research references and has been translated into Japanese (1997) and Chinese (2002), and Essentials of Robust Control (Prentice Hall, 1997). He is currently an Associate Editor of Automatica, Systems and Control Letters, Journal of System Sciences and Complexity, and Journal of Control Theory and Applications. He was also an Associate Editor of IEEE Transactions on Automatic Control and SIAM Journal on Control and Optimization. He was named the Oskar R. Menton Endowed Professor in Electrical Engineering in 1998. He was given a certificate of special congressional recognition in 2003, recognition for outstanding accomplishments and contributions to Louisiana State University by the House of Representatives of the Louisiana Legislature in 2004, received outstanding young investigator award from Natural Science Foundation of China in 2004. He was elected to IEEE Fellow in 2003 for contributions to the robust control system theory and applications.