Unknown input matrix estimation via adaptive random search algorithm Rafal J´ ozefowicz
Marcin Witczak
Marcin Ka´ zmierczak
University of Zielona G´ ora, Institute of Control and Computation Engineering, Podg´ orna 50, 65-246 Zielona G´ ora, Poland. e-mail: {R.Jozefowicz,M.Kazmierczak}@weit.uz.zgora.pl,
[email protected] Abstract: This paper deals with the estimation of an unknown input distribution matrix for the extended unknown input observer. In particular, the estimation problem is formulated as a global optimisation one and to solve it the adaptive random search algorithm is employed. The final part of the paper presents and illustrative example concerning an induction motor which confirms the performance of the proposed strategy. Keywords: Nonlinear systems, observers, model-based control, fault detection. 1. INTRODUCTION
Contemporary technical systems have become more complex, more complicated and automated. The safety and reliability of these real systems is of great importance for overall costs of production and products quality. The faultless functioning of modern industrial systems has also a comprehensive impact for the protection of the natural environment, human health and life. The sufficiently early detection of all occurring faults is significant in avoiding system performance degradation or even its complete damage, what can involve danger of human life. Correct technical diagnosis can help to make the right decisions on fault occurrence, emergency actions and repairs. Generally, simple technical systems can be inspected by the human expert but complex industrial systems require automated processes of diagnostics techniques in order to determine the occurrence, location and the reason for the fault fast and precisely. This real world’s development pressure has transformed the fault diagnosis, initially perceived as the art of designing the satisfactorily safe systems (e.g. nuclear reactors, aircrafts), into the modern science that it is today. A precise model of a given system can be a tool providing knowledge or prediction about the system behavior. Nowadays, a lot of advanced fault diagnosis systems are based on models of supervised systems. This is the explanation why the application of the model-based fault diagnosis has received considerable attention during the last few decades. In fault diagnosis schemes a model of a real system is able to provide estimates of measured and/or unmeasured signals. These estimated signals can be compared with original (measured) signals to calculate the so-called residual signal. The residual is defined as a difference between measured signal and its estimate obtained by means of a model. The fault diagnosis systems also known as Fault Detection and Isolation (FDI) systems employ this signals in fault detection stage Witczak [2006].
To manage such challenging problems like technical fault diagnosis many Fault-Tolerant Control Systems (FTCS) have been developed. The FTCS is the system that possess the ability to accommodate for the supervised system faults automatically. It should have the ability to maintain overall system stability and acceptable performance in the event of faults. An example of a contemporary control
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Fig. 1. Modern Fault-Tolerant Control System. system is presented in Fig. 1 Zhang and Jiang [2003]. The supervised system which consists of actuators, system components and sensors is the main part of the scheme. This controlled system may by affected by faults. A fault is defined as unexpected change of at least one characteristic property or parameter of the system from the standard (nominal) condition, e.g. a sensor malfunction. All the unpredicted changes that lead to degrade the entire system performance can also be defined as faults. On the contrary the term failure can be defined as a complete breakdown of the system. It is possible the system to recover after the fault occurrence without activation of safety systems. Generally, as shown in Fig. 1 faults can be split into three categories Witczak [2007] i.e: actuators faults, system components faults, sensors faults.
Besides the known/measured inputs each part of the supervised system can also be impaired by so-called unknown inputs (additional inputs of unknown/unmeasurable values) which represent the process and measurement noise, as wall as external disturbances acting on a system, e.g. abrupt changes of outward temperatures, unknown changes of input voltages Korbicz et al. [2004]. When a model-based control and an analytical redundancy-based fault diagnosis are employed Chen and Patton [1999], Korbicz et al. [2004], then these unknown inputs can be extended by a so-called model uncertainty, i.e. the mismatch between the system and its model behavior. Consequently, to perform an exact model-based fault diagnosis the most proper model should be employed. However, by reason of noise and unavoidable disturbances a perfectly accurate and complete mathematical model of physical system is never available. Moreover, in reality also the parameters of the system may vary with time in uncertain manner, and the characteristics of mentioned disturbances and noise are unknown, so that they cannot be modeled accurately. Therefore there is the need to designs robust fault detection schemes, i.e. insensitive or even invariant to modeling uncertainty without losing fault sensitivity. The robustness is typically accomplished in fault detection stage. The lack of robustness to model unreality may cause very undesirable situations during fault detection task, i.e. undetected faults or false alarms. In this case such fault detection system is useless in real applications. The effect of modeling uncertainties is therefore the most crucial point in the model-based FDI concept, and the solution of this problem is the key for its practical applicability Chen and Patton [1999]. The importance of robustness in modelbased FDI has been widely recognized by both academia and industry. The development of robust model-based FDI methods has been a key research topic during the last ten years. A number of methods have been proposed to tackle this important problem (see Chen and Patton [1999], Patton et al. [2000] and the references therein). However, the research is still under the way to develop the practically applicable methods. As can be observed in the literature Chen and Patton [1999], Korbicz et al. [2004], Patton et al. [2000], the most common approach to robust fault diagnosis is to use robust observers such us very well-known Unknown Input Observer (UIO), which can tolerate a degree of model uncertainty and hence increase the reliability of fault diagnosis Witczak [2007]. xk+1 = g (xk ) + h(uk ) + E k dk + L1,k f k , (1) y k+1 = C k+1 xk+1 + L2,k+1 f k+1 , (2) It corresponds to the system description (1)–(2) in which model uncertainty is modeled by an unknown input additive term, i.e. where dk ∈ Rq is an unknown input, and E k ∈ Rn×q denotes its distribution matrix (xk ∈ Rn stands for the state vector, y k ∈ Rm is the output, uk ∈ Rr is the input, g (·), h (·) are non-linear functions, L1,k , L2,k are fault distribution matrices and f k ∈ Rs is a fault vector). The most complicated task related to design the UIO is concerned with the problem of estimating the unknown input distribution matrix (E). There are approaches that can be used to tackle this problem for linear systems (Chen and Patton [1999]), but for a general class of non-linear systems the problem still remains
open. Taking into account the above discussion, the main purpose of this paper is to propose a direct approach that can be used for the unknown input distribution matrix E estimation for the so-called Extended Unknown Input Observer (EUIO) Korbicz et al. [2004], Witczak [2007]. The remaining part of the paper is organised as follows: Section 2 presents an elementary background regarding the Extended Unknown Input Observer. In Section 3 the approach for the unknown input distribution matrix estimation is described in details. In Section 4 there is a brief description of Adaptive Random Search (ARS) algorithm. Section 5 presents an experimental study regarding estimation of an unknown input distribution matrix using the ARS algorithm for an induction motor. Finally, the last part concludes the paper. 2. EXTENDED UNKNOWN INPUT OBSERVER Let us consider a non-linear discrete-time system described by (1)–(2). The main problem is to design an observer that is insensitive to the influence of the unknown input. The necessary condition for the existence of a solution to the unknown input de-coupling problem is as follows: rank(C k+1 E k ) = rank(E k ) = q, (3) (see [Chen and Patton, 1999, p. 72, Lemma 3.1] for a comprehensive explanation). If condition (3) is satisfied, then it is possible to calculate H k+1 = (C k+1 E k )+ , where (·)+ stands for the pseudo-inverse of its argument. Thus, by multiplying (2) by H k+1 and then inserting (1) it is straightforward to show that " # dk = H k+1 y k+1 − C k+1 [g (xk ) + h(uk )] .
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Substituting (4) into (1) gives ¯ (uk ) + E¯k y ¯ (xk ) + h xk+1 = g (5) k+1 , where ¯ (·) = G¯k h(·) ¯ (·) = G¯k g (·) , g h ¯ Gk = I − E k H k+1 C k+1 , E¯k = E k H k+1 . (6) Thus, the unknown input observer for (1)–(2) is given as: ˆ k+1 = x ˆ k+1/k + K k+1 (y k+1 − C k+1 x ˆ k+1/k ), x where ¯ (uk ) + E¯k y ˆ k+1/k = g ¯ (ˆ x xk ) + h (7) k+1 . As a consequence, the algorithm used for the state estimation of (1)–(2) can be given as follows (Witczak [2007]): ¯ (uk ) + E¯k y ˆ k+1/k = g ¯ (ˆ x xk ) + h (8) k+1 , T P k+1/k = A¯k P k A¯k + Qk , (9) −1 K k+1 = P k+1/k C Tk+1 C k+1 P k+1/k C Tk+1 + Rk+1 (10) (11) (12)
ˆ k+1 = x ˆ k+1/k + K k+1 (y k+1 − C k+1 x ˆ k+1/k ), x P k+1 = [I − K k+1 C k+1 ] P k+1/k , where ∂g (xk ) ∂¯ g (xk ) ¯ ¯ = Gk = G¯k Ak . Ak = ∂xk xk =ˆxk ∂xk xk =ˆxk The algorithm (8)–(12) can be perceived as an Extended Kalman Filter (EKF) for non-linear systems with an unknown input. Another important property is that the proposed algorithm is used for deterministic systems (1)– (2), which is clearly justified in Witczak [2007].
3. ESTIMATION OF THE UNKNOWN INPUT DISTRIBUTION MATRIX As can be observed in the literature (see Chen and Patton [1999]), the matrix E is usually obtained in such a way as to minimize some measure of the state estimation error. From the fault diagnosis viewpoint, this matrix should be designed so as to minimize some measure of the residual signal, which is defined as a difference between the system output and its estimate obtained with the EUIO, i.e. ˆ k (E), zk = yk − y (13) ˆ k (E) = C k x ˆ k where (E) means that y ˆ k depends with y on E. Indeed, the performance of the EUIO-based residual generator very strongly depends on the correct form of the matrix E. Unfortunately, there is no estimation technique for linear systems that can be efficiently extended to the considered class of non-linear systems (1)–(2). Taking into account this unappealing property, the following straightforward approach for estimation the unknown input distribution matrix is proposed, i.e. an optimization strategy of the form ˆ = arg min J(E), E (14) E∈Rn×q
where J(E) stands for some cost function formulated with the use of (13). The general algorithm for computing the value of J(E) can be described in the following form: Step 1: Generate the value of E matrix. Step 2: Check the optimization constraints: 2.1 The rank condition (3). 2.2 The rank observability condition of the system described by (2) and (7). Step 3: If both constraints are satisfied then go to Step 4, otherwise go to Step 1. Step 4: Run the EUIO for a given matrix E. Step 5: Calculate the residual z k , k = 1, . . . , nt . Step 6: Obtain the value of the cost function J(E). The implementation of step 6 requires a precise definition of the cost function. In this paper it is proposed to use the objective function which is based on the idea of minimizing the mean-square error of the residual, the special term (kE − L1 k−1 ) was introduced to ensure the proper faults sensitivity of EUIO-based residual generator. m 1 X 1 ˆ i (E), k2 , J(E) = α +β ky i − y kE − L1 k nt m i=1 (15) where α is the so-called weighting factor chosen experiˆ i (E) = mentally, β = (1 − α), y i = [yi,1 , . . . , yi,nt ]T and y [ˆ yi,1 , . . . , yˆi,nt ]T , m is the number of outputs, nt is the number of input-output measurements. The objective function of the form (15) is used in the proposed optimization process where the norm kE − L1 k is maximized in order to the direction of the unknown input (E) was not consistent with the direction of faults (L1 ). If the above condition is satisfied it is assured that EUIO is insensitive to the influence of the unknown input (unknown input is uncoupled from the residual signal) and still remains sensitive to faults. The effectiveness of the proposed objective function (15) was check during the experiments where the estimated by ARS algorithm value of unknown input distribution matrix assures that EUIO
is able to uncouple the unknown input from the residual and is still sensitive to faults. The above proposed strategy is simply to implement but the main problem is that the well known and widely used gradient-based optimization algorithms cannot be used. The first reason is that it is evident that the cost function is non-differentiable as it involves a direct run of the EUIO to compute its value. Another reason is that, the optimization problem can be a global one and hence the gradientbased algorithms are not adequate tools for such task. Taking into account the above discussion in this paper it is proposed to use the ARS global optimization algorithm, which has successfully been used in many engineering problems, e.g. the design of the robust EUIO (Witczak [2007]). The subsequent section will present the ARS strategy in details. 4. ADAPTIVE RANDOM SEARCH ALGORITHM The search process of the ARS algorithm Walter and Pronzato [1996] can be split into two phases. The first phase (variance-selection phase) consists in selecting an element from the sequence {σ (i) },
i = 1, . . . , imax
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where σ stands for an initial standard deviation selected by the designer, and σ (i) = 10(−i+1) σ (1) . (17) In this way, the range of σ ensures both proper exploration properties over the search space and a sufficient accuracy of optimum localization. Larger values of σ decrease the possibility of getting stuck in a local minimum. The second phase (variance-exploration phase) is dedicated to exploring the search space with the use of σ obtained from the first phase and consists in repetitive random perturbation of the best point obtained in the first phase. A scheme of the ARS algorithm used in further simulations is as follows: 0. Input data: • σ (1) – the initial standard deviation; • jmax – the number of iterations in each phase; • imax – the number of standard deviations (σ i ) changes; • kmax – the global number of algorithm runs; • E (0) – the initial value of unknown input distribution matrix; 1. Initialize (1.1) E best → E 0 . (1.2) k → 1. (1.3) i → 1. 2. Variance-selection phase (2.1) j → 1, E (j) → E (0) and σ (i) → 10(−i+1) σ (1) . (j) (2.2) Perturb E (j) to get a new trial point E + . (j) (j) (2.3) If J(E + ) ≤ J(E (j) ) then E (j+1) → E + else (j+1) (j) E →E . (j) (j) (2.4) If J(E + ) ≤ J(E best ) then E best → E + , ibest → i.
(3.1) j → 1, E (j) → E best , i → ibest and σ (i) → 10(−i+1) σ (1) . (j) (3.2) Perturb E (j) to get a new trial point E + . (j) (j) (j) (j+1) (3.3) If J(E + ) ≤ J(E ) then E → E + else E (j+1) → E (j) . (j) (j) (3.4) If J(E + ) ≤ J(E best ) then E best → E + (3.5) If (j ≤ jmax ) then j → j + 1 and go to Step 3.2. (3.6) If (k → kmax ) then STOP. (3.7) k → k + 1, E (0) → E best and resume from (2.1). The subsequent part of the paper presents the results obtained during numerical experiments. 5. EXPERIMENTAL RESULTS The purpose of this section is to show the reliability and effectiveness of the proposed estimation strategy. The numerical example considered here is a fifth-order two-phase non-linear model of an induction motor, which has already been the subject of a large number of various control design applications (Chenafa et al. [2005]). A detailed description of the above non-linear system can be found in Boutayeb and Aubry [1999] and the references therein. A complete discrete-time model of this system is as follows: K x1,k+1 =x1,k + h(−γx1,k + x3,k + Kpx5,k x4,k Tr 1 u1,k ) + E1,1 dk , (18) + σLs K x2,k+1 =x2,k + h(−γx(2, k) − Kpx5,k x3,k + x4,k T 1 + u2,k ) + E2,1 dk , (19) σL M 1 x3,k+1 =x3,k + h( x(1, k) − x3,k − px5,k x4,k ) Tr Tr + E3,1 dk , (20) 1 M x4,k+1 =x4,k + h( x2,k + px5,k X3,k − x4,k ) Tr Tr + E4,1 dk , (21) pM TL x5,k+1 =x5,k + h( (x3,k x2,k − x4,k x1,k ) − ) JLr J + E5,1 dk , (22) y1,k+1 =x1,k+1 , y2,k+1 = x2,k+1 , (23) where xk = [x1,k , . . . , xn,k ]T = [isak , isbk , ψrak , ψrbk , ωk ]T represents the currents, the rotor fluxes, and the angular speed, respectively, while uk = [usak , usbk ]T is the stator voltage control vector, p is the number of pole pairs, and TL is the load torque. The rotor time constant Tr and the remaining parameters are defined as: Lr M2 Tr = , σ =1− , Rr LS Lr M RS Rr M 2 K= , , γ= + σLS Lr σLS σLS L2r where RS , Rr and LS , Lr are stator and rotor per-phase resistances and inductances, respectively, and J is the
x0 = [0, 0, 0, 0, 0]T (25) The unknown input for system described by (18)–(22) is defined as: E = [1.2, 0.2, 2.4, 1, 1.6]T (26) dk = 0.3 sin(0.5πk) cos(0.03πk) (27) 5.1 Unknown input decoupling The system (1)–(2) is described by (18)–(22). Since the system description is given it is possible to design Extended Unknown Input Observer and Extended Kalman Filter (EKF). First let us start the simulation with the unknown input-free case, i.e. E = 0 and EKF as a system observer. Classical EKF is not a robust observer, it does not utilize any information about unknown input or its distribution matrix. For this case the residuals are almost zero what confirms that standard EKF can be used to observe systems which are not influenced by unknown input what is rarely meet in practice. Next let us introduce unknown input (26) into system configuration and observe the system with EKF. Figures 2 and 3 show the residual signals 0.5
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rotor moment inertia. The numerical values of the above parameters are as follows: RS = 0.18Ω, Rr = 0.15Ω, M = 0.068H, LS = 0.0699H, Lr = 0.0699H, J = 0.0586kgm2 , TL = 10N m, p = 1, h = 0.0001s. The input signals and initial conditions are: u1,k = 350 cos(0.03k), u2,k = 300 sin(0.03k), (24)
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observe systems which are influenced by unknown input (the most of real systems) and eliminates such observers from practical applications.
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The objective of presenting the next example is to show how important is the knowledge of unknown input distribution matrix in order to use the EUIO in fault detection stage. Let us start with the case when unknown input distribution matrix used by EUIO (8) has exactly the same value as unknown input distribution matrix (26) used in system configuration (18)–(22). It can be stated as E EUIO = E SYS . The residual signals for this case are shown in Fig. 4 and Fig. 5. It can be seen that both residuals are almost zero. Now let us continue simulations with
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Fig. 7. Residual z2,k for E EUIO 6= E SYS case, EUIO as an observer. from real applications since it is sensitive to unknown input.
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and f 2,k = 0. Thus, the system is now described by (1)–(2) with (18)–(22), (26), (27), f k = [f 1,k , f 2,k ]T , and " 1 #T 0 0 0 0 L1,k = σLs , (29) 0 0 0 0 0 0 0 L2,k = . (30) 0 0
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The objective of presenting the next example is to show the effectiveness of the proposed strategy (in the presence of both the unknown input (27) and faults) of estimation of the unknown input distribution matrix. For that purpose the following scenario of abrupt fault of u1,k actuator was considered: −0.2u1,k , 3000 < k < 4000, f 1,k = , (28) 0, otherwise.
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Fig. 5. Residual z2,k for E EUIO = E SYS case, EUIO as an observer. the case when unknown input distribution matrix used by EUIO (8) was randomly generated (E EUIO 6= E SYS ). Figures 6 and 6 show the residual signals for this case. 0.1
The parameters related to the ARS algorithm for the minimization of (15) were (after many experiments) selected as follows: j = 250, i = 4, k = 4, σ = 0.1. Estimated by ARS algorithm value of unknown input distribution matrix was E = [−3.20, −0.54, −6.30, −2.63, −2.67]T (31) Figures 8 and 9 show the residual signals for fault free case. It can be observed that residual signals are almost zero, it means that estimated by ARS algorithm value of E matrix causes that EUIO is able almost completely decuple unknown input from the residual signal. Now
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Fig. 6. Residual z1,k for E EUIO 6= E SYS case, EUIO as an observer. It can be observed that both residuals significantly differ from zero. It means that the lack of knowledge about unknown input distribution matrix disqualify such EUIO
Fig. 8. Residual z1,k for E EUIO estimated by ARS, EUIO as an observer. let us introduce fault term (29) and (28) into system configuration. From Figs. 10 and 11 it can be observed
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Fig. 9. Residual z2,k for E EUIO estimated by ARS, EUIO as an observer.
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REFERENCES
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Fig. 11. Residual z2,k for an actuator u1,k fault that the residual signal is sensitive to the faults under consideration. This, together with an effective unknown input decoupling confirms the rightness of the proposed unknown input matrix estimation strategy for EUIO. It must be emphasized that the term (kE − L1 k)−1 in 15 is very important. Without using this term in objective function (15) it is possible to obtain E matrix for which EUIO will decouple both unknown input and also faults from the residual signal. Such a case is shown in figures 12 and 13. 6. CONCLUSIONS The main objective of this paper was to propose a direct strategy of unknown input distribution matrix estimation for extended unknown input observer. In particular, the paper proposes the estimation strategy with the use of global optimization ARS algorithm and especially defined objective function. The paper also experimentally confirms the usefulness of the proposed method for unknown input decoupling in the presence of faults.
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