Fault Diagnosis and Accommodation of LTI systems by modified Youla parameterization

In this paper an Active Fault Tolerant Control (FTC) scheme is proposed for Linear Time Invariant (LTI) systems, which achieves fault diagnosis followed by fault accommodation. The fault diagnosis scheme is carried out in two steps; Fault detection followed by Fault isolation. Fault detection filter use the sensor measurements to generate residuals, which have a unique static pattern in response to each fault. Distortion in these static patterns generates the probability of the presence of fault. The fault accommodation scheme is carried out using the Generalized Internal Model Control (GIMC) architecture, also known as modified Youla parameterization. In addition, performance indices are also evaluated to indicate that the resulting fault tolerant scheme can detect, identify and accommodate actuator and sensor faults under additive faults. The DC motor example is considered for the demonstration of the proposed scheme. Keywords –Fault Tolerant Control, Youla parameterization, Generalized Internal Model Control, Residuals. I. INTRODUCTION Nowadays, control systems are present everywhere in life. They are constantly and inexhaustibly working, making life more comfortable and more pleasant. Hence modern technological systems rely on sophisticated control systems to meet increased performance and safety requirements. The existing trade-off between increasing complexity and limited resources is conditioned by operational risk, which is an unavoidable feature of modern industrial systems. Because the components of industrial systems can fail, it is compulsory to prevent and to correct failures in order to minimize both the operational risk and the number of unwanted stops of the system. Thus, it is fundamental to detect faults. Model based failure detection and identification is a common approach to achieve fault diagnosis and constitutes an important domain of research activity [1,2,4]. An issue considered is the fundamental diagnosis problem for systems with parametric faults [15]. The fundamental diagnosis problem is the problem of detection, isolation and/or estimation of faults with zero thresholds. This requires that it is possible to decouple the disturbances exactly. The fundamental detection and isolation problem for systems with additive faults has been investigated in detail in [17]. The estimation problem has been investigated in [18]. The approach applied in [15] can be considered as an algebraic approach to handle fault detection and fault isolation of system including parametric faults. In connection with fault diagnosis, [19] has pointed out that it consists of three tasks, fault detection as the lower task, and fault isolation as the middle task and then fault estimation/identification as the highest task. In the lower task, which is the simpler task, it is detected whether any fault has occurred in the system. In the isolation task, the faults, which have occurred in the system, are identified. In the last task, the quantitative extent of the occurred faults is determined.Based on this, it might not be an optimal method to detect and/or isolate parametric faults based on parameter estimation. Detection and/or isolation of parameter faults might instead be based on detection and/or isolation effects in the system from parametric changes/faults. This requires that it is possible to measure these effects from the measurement signals. If this is possible, fault detection/isolation can then be done in a similar way as detection/isolation of additive faults. The fault detection and identification process consists essentially of two chronologically ordered stages [16]: residual generation and decision-making. The first stage concerns the failure detection and the identification of the failed components. The second stage concerns the reconfiguration. This work looks to extend the ideas initially presented in [6,7,10]. Hence fault detection, isolation and accommodation are discussed in a more general framework under the GIMC control structure for additive faults. The contribution of this paper lies in the following lines. 1773 Fault Diagnosis and Accommodation of LTI systems by modified Youla parameterization ISSN 2277-1956/V1N3-1772-1780 1) A two-step Fault tolerant scheme is proposed for LTI systems under an additive fault scenario. 2) Design strategies are proposed for accommodation based on general optimization criteria. 3) Performance indices are suggested in order to evaluate the fault diagnosis scheme. II.PROBLEM FORMULATION The problem addressed in this paper is fault detection, isolation and accommodation for LTI systems under additivefaults and perturbations. In this way, consider a system ( )affected by disturbances ∈ R and possible faults ∈ R ,as shown in Figure 1, described by ( ) = + + + = + + + (1) where ∈ R represents the vector of states, ∈ R the vector of inputs, and ∈ R the vector of outputs. Thus matrix ∈ R × stands for the distribution matrix of the actuator or system faults, and ∈ R × for sensor faults. Let ∈ R and ∈ R with ! = 1, ... , %the columns of the fault signature matrices and , respectively, that is, = & ... ', = & ... ' (2) Thus matrices ( , ) will represent the signature of the !th component in the fault vector . The nominal system ( , , , ) is considered controllable and observable. On the other hand, the system response can be analyzed in a transfer matrix form (frequency domain) as follows: ( ) = *+ ( ) + ,+ ( ) + -+ ( )(3) where *+ = (./ − )1 + ,+ = (./ − )1 + -+ = (./ − )1 + (4) A left coprime factorization for each transfer matrix can be derived by obtaining matrix 2 ∈ R × such that R45 ( + 2 )6 < 0[11,12], as it is shown in equation (5) 9:;<;:;,:;-= = > ?@AB B C D@AE A FG@AFH IG@AIH E J FH IH K(5) Consequently, the LTI systems in equation (4) can be written as *+ = <;1 :;, ,+ = <;1 :;, , -+ = <;1 :;-, (6) where:;<;:;,:;∈ RHN. Figure 1. Problem formulation Now, it is assumed that a nominal controller P stabilizes the nominal plant *+, and it provides a desired closedloop performance in terms of robustness, transient, and steady state responses. The controller P is considered observable and consequently, it can also be expressed by a left coprime factorization, that is, P = QR 1 S; where S;, QR ∈ RHN, P = > ?T DTC BT ETKwhich implies 9S;QR= = > ?T@ATBT BT C DT@ATETAT ET JK (7) The nominal controller can be synthesized following classical techniques or optimal control: lead/lag compensator, PI, PID, LQG/H2, H∞ loop shaping design, and so on. IJECSE, Volume1, Number 3 Minupriya A et al. ISSN 2277-1956/V1N3-1772-1780 III.FAULT DIAGNOSIS SCHEME The proposed scheme relies on a fault diagnosis and isolation (FDI) algorithm. The scheme adopted in this work is motivated by a new implementation called generalized internal model control (GIMC) [6,7,10]. In this configuration, the nominal controller P is represented by its left coprime factorization, that is, P = QR 1 S;. In addition, the GIMC configuration allows performing the FDI process, where the process is carried out by selecting the design parameter U ∈ RHN, as depicted in Figure 2. Consequently, selecting the detection/isolation filter U( ) in such a way that it diminishes the effect of the disturbances or uncertainty into the residual signal, and maximize the effect of the faults, generates the residual V. Figure 2.Fault accommodation scheme using GIMC structure From Figure 2, it can be observed that W ∈ R contains information of perturbations and faults as follows: W( ) = −:;, ( ) − :;( ) (8) Hence a residual Vis naturally constructed by using the information of the coprime factorization of the nominal plant through W; V( ) = −U W( ) = U&:;, ( ) + :;( )' (9) In order to improve the accuracy of the FDI stage, it is proposed to carry out this task in two consecutive steps: a) Fault detection b) Fault isolation. As a result, the FDI algorithm is designed in two parts as follows. 1) A detection filter UEis first synthesized to determine a general fault scenario. 2) Next, an isolation filter UJis computed to identify the faults affecting the system. First, the detection filter UEis constructed to obtain a scalar residual, that is, UE is a % × Xtransfer matrix such that it attenuates the contribution from the perturbations while maximizing the faults effect. Hence the following design criteria are suggested: Y* Z[∈ RH\ ]Z[ ;̂_]T ‖Z[ ;̂a‖b (10) c, drepresent the performance indexes. Alternatively, it can be solved using well-known algorithms through a characterization by a linear fractional transformation (LFT) [11]; min Z[∈ RH\]&0 h' − UE9:;, :;-=]i = min Z[∈ RH\] (jZ[ , UE)]i , d = 2, ∞. (11) whereh ∈ RHNis a 1 × %transfer matrix that describes the faults frequency bandwidth, (. , . ) represents a lower LFT [11], and jZ[ the generalized plant (see Figure 3) given by 1775 Fault Diagnosis and Accommodation of LTI systems by modified Youla parameterization ISSN 2277-1956/V1N3-1772-1780 jZ[ = m 0 h −1 :;, :;0 n (12) One advantage of the LFT characterization is that it can be augmented to include model uncertainty in the problem formulation. Figure 3. LFT formuation for detection filter UE Meanwhile, the isolation filter UJ(% × Xtransfer matrix) is designed to isolate the fault vector and decouple the perturbations , that is, (i) UJ:;,( ) ≈ 0, (ii) UJ:;-( ) ≈ p, (13) wherep ∈ RHNis a diagonal transfer matrix. Transfer matrixpis a design parameter, and it should be chosen according to the frequency response of :;, in order to achieve the isolation and decoupling objectives.Once more, the design criterion can be proposed by combining both objectives measured by a system norm d = 2, ∞ as follows: min Zq∈ RH\]&0 p' − UJ&:;, :;-' ]i = min Zq∈ RH\] (jZq , UJ)]i , (14) wherejZq stands for the generalized plant associated to the LFT formulation given by jZq = m 0 p −1 :;, :;0 n (15) Hence once a fault is detected, in the isolation stage, the filter UJhas to provide a good estimate of the fault affecting the system. Therefore, it is fundamental that UJcould render diagonally the product UJ:;-, or at least diagonally dominant. In fact, this issue has to be verified after UJ is designed for correct fault identification. III.FAULT ACCOMMODATION SCHEME In this configuration, a free parameter r ∈ RHN is selected to achieve the fault compensation, with the assurance that closed-loop stability is achieved after the fault accommodation. In this fashion, the accommodation scheme adopted in this work is motivated by a new implementation of the Youla parameterization called Generalized Internal Model Control (GIMC) as in Fig 2. In this configuration, the nominal controller P is represented by its left coprime factorization, that is, P = QR 1 S;. In addition, the GIMC configuration allows performing accommodation by selecting design parameter, r ∈ RHN.The accommodation signal s generated by the compensator r, using the filtered signal W, is generated with the choice of r( ). r( )is the robustification controller, that must provide robustness into the closed-loop system in order to maintain acceptable performance against faults.The Theorem 1 characterizes the dynamic behavior of the compensated control input and output of the closed-loop system. Theorem 1 In the GIMC configuration of Fig 2 considering additive faults, the resulting closed-loop characteristics for the control signal and output are given by ( ) = . P Vt ( ) − . QR 1 (S;<;1 + r)(:;, ( ) + :;( )), ( ) = pu Vt ( ) − .u<;1 (/ + :;QR 1 r)(:;, ( ) + :;( )(16) The resulting closed-loop system is stable, provided that r ∈ RHN, since the nominal controller P internally stabilizes the nominal plant *+ . For this reason, by analyzing (16), if it is desired to minimize the faults effect at the control signal, while reducing the perturbations contribution at the output, the compensator r should be designed by following the optimization strategy. The cost function is designed as, IJECSE, Volume1, Number 3 Minupriya A et al. ISSN 2277-1956/V1N3-1772-1780 min v∈RZ\ wmx,.y *+ 0 0 x-. P -+n + z−x,.y *+QR 1 −x-. QR 1 { r 9:;, :;-=wi = minv∈RZ\] (jv| , r)]i (17) here the controller ‘Q’ can be obtained by optimizing the cost function (17 ) as minv∈RZ\] (jv| , r)]i (18) where d represents the H or HNnorms, ∝,, ∝-∈ [0, 1] are two weighting factors to balance the tradeoff between perturbations and faults reduction.where,jv| , represents the generalized plant is given by, jv| = ~x,.y *+ 0 −x,.y *+QR 1 0 x-. P -+ −x-. QR 1 :;, :;0 (19) Meanwhile, if it is desired to attenuate both faults and perturbations at the output , then the next optimization scheme is suggested as, min v∈RZ\].u<;1 (/ + :;QR 1 r)&x,:;,x-:;-']i = min v∈RZ\ € jv‚ , rƒ€i (20) wherejv‚ is given by jv‚ = zx,.y ,+ x-.y *+ −.y *+QR 1 x,:;, x-:;0 {(21) the overall active FTC algorithm consists of three stages according to the information of the FDI block as follows: 1) In the fault-free case, just the nominal control loop is active; 2) After a fault scenario is detected into the system, a general compensator r designed by (17) is activated; 3) Finally, after the fault is detected, an specific compensator r is selected. In a general fault condition, it is then decided to decouple (if possible) or attenuate the effect of faults at the control signal , until the fault is well characterized during the detection stage. As a result, after the fault is detected, the specific compensation is injected into the closed-loop configuration to improve the post fault performance. V.PERFORMANCE EVALUATION One important question, after the design stage is completed is the resulting performance of the fault detection and isolation algorithms. To address this problem, different quantification indices will be proposed using the system performance indexes in [11,12] of the resulting transfer functions. The selection of the applied performance index in [11,12] will depend on the a priori fault information’s or the desired interpretation of the quantification index. A. Fault evaluation The capability of the detection filter UEof reducing the perturbations frequency content compared to increasing the faults sensitivity is evaluated by equation (22) /IE ≜ ]Z[ ;̂_]T ‖Z[ ;̂a‖b (22) Hence a large value of /IE will indicate good evaluation characteristics. B. Fault isolation This index is constructed by analyzing the property of UJof diagonalizing : …-while attenuating the disturbances frequency content given by equation (23) /IJ ≜ †9Zq ;̂_=a‡ˆ‰†T †9Zq ;̂_=Š‹Ša‡ˆ‰† @‖Zq ;̂a‖b (23) where&. ', Œdenotes the diagonal part of the transfer matrix, and &. ' y , Œthe off diagonal structure. In fact, /IJ is usually denoted as signal-to-noise and interference ratio (SNIR) in the signal processing community. Thus for large value of /IJ, fault isolation is achieved. C. Fault accommodation The fault accommodation isquantified in terms of the property of reducing the effect offaults and perturbations 1777 Fault Diagnosis and Accommodation of LTI systems by modified Youla parameterization ISSN 2277-1956/V1N3-1772-1780 simultaneously into the closed-loopsystem. The accommodation performance criteria is defined as /I? ≜ x-]Ž? :;] + x,].u<;1 (/ − :;QR 1 r ):;,]i(24) where the weighting Ž? is selected according to the fault effect as suggested in equation (25) Ž? ≜ .u<;1 (/ − :;QR 1 r ) actuator fault, . QR 1 (S;<;1 + r ) sensor fault. (25) where (x, , x-) are the positive weighting factors to judge the importance of perturbations or faults attenuation. Now, a small value of /I? will indicate good fault accommodation. It is to be noted that this value is related to a worst-case performance degradation level expected in the FTC scheme. V.ILLUSTRATIVE EXAMPLE In order to illustrate the ideas presented in the paper, the design of an active FTC scheme for a separately excited DC motor is considered. The dynamics of a second-order actuator are appended to the motor description. To have a more realistic simulation, the actuator gain is limited by a saturation function. Thus a system with single input and single output (angular velocity ω) is studied [10,14]. The load torque is modeled as an unknown constant or slowly time-varying external disturbance into the system. The control objective is defined as the regulation of the angular velocity to a prescribed reference. Note that since there is only one measurements and one unknown perturbation, then only the effect of two different faults could be analyzed simultaneously [2]. The studied faults are actuator (gain of the dc drive) and sensor (angular velocity measurement). The parameters of the dc motor are shown in Table I. The mathematical model of the studied system is presented next ~ › œ = ž žž Ÿ− ˆ Aˆ − ¡¢ Aˆ 0 Aˆ ¡¢ £ − D£ 0 0 0 0 0 −1 0 0 ¤Œ −2¥Œ¤Œ¦§ §§̈ ~ › œ + ~ 00 PŒ 0  + žž Ÿ 0 − £ 00 ¦§ §̈ + ~ 0 0 0 0 PŒ 0 0 0 m n (26) ¤ = &0 1 0 0' ~ › œ + © 0 0 0 1 0 0a m n (27) The nominal controller is designed following a PID structure with respect to the velocity reference error for which the gain values are obtained as, c = 5.95, p = 0.8, p, = 0.2. The detection and isolation filters (UE, UJ) were designed following the optimization indices (11) and (14) with j = ∞, and selecting h( ) = &1 1' × Y ®® ̄ @°± (28) p( ) = Y ®2 ̄ @ >1 0 0 1K (29) The transfer matrix p( ) was chosen as a diagonal low-pass filter since the frequency content of :;allows perfect decoupling in the low frequency.The performance indices in (22) and (23) were computed takingthe ∞ norm (c = d = ∞), and they are listed in Table II. Hence the results in Table 2 reflect that the active FTC willprovide good performance in the diagnosis and accommodation stages. Furthermore, no degradation is expected in a steady state. Table I DC motor parameters Parameter Description Value 3 ́ Armature resistance μ. ¶·· ̧ 1 ́ Armature inductance μ. o·» 1⁄4 1⁄2 Friction coefficient 3⁄4. ¿3⁄43⁄4 × Àμ1ÁNm/rad/s  Inertia o. »· × Àμ1Ákg m ÃÄ Electromagnetic constant μ. »3⁄4¶ V/rad/s à ́ Actuator gain 3⁄4μ Å ́ Actuator natural frequency oÆ × Á¶μrad/s Ç ́ Actuator damping factor μ. »