An active fault tolerant control method based on support vector machines

Abstract

In this study, an active Fault Tolerant Control (FTC) method based on Support Vector Machines (SVM) is presented. The proposed FTC method is not limited to certain faults in the reconfiguration manner and but it also includes a reconfiguration mechanism with direct on-line controller calculation. Here, PID type controller is utilized within the method as a reconfiguration sub-system. The reconfiguration mechanism and the diagnosis unit work independently within the method. Therefore, there is no need for the isolation of faults before tolerating them. In diagnosis and reconfiguration stages of the method, support vector regression machines are used. This FTC technique uses the real-time data generated by the system and it produces the appropriate gains of the controller in an on-line manner. The PID controller coefficients or the gains to be used in the training stage for faulty and non-faulty cases are all obtained by using the Genetic Algorithm optimization approach in an off-line manner. Moreover, it has also been shown that the proposed method can handle multiple and simultaneous occurrences of various types of faults. The performance of the proposed method is tested on a simulation model of two tank level control system for various fault scenarios.

Keywords

Fault tolerant control fault diagnosis support vector machines PID controllers two tank liquid level system

1 Introduction

Since there is an increasing need for reliability, safety, maintainability and survivability in technological systems, FTC systems have become extremely important in last three decades. FTC systems have been developed to overcome some of the weaknesses in the conventional feedback control design, such as instability and unsatisfactory performance in faulty cases. In complex systems such as aircrafts, nuclear power plants, chemical plants etc., the results of a minor fault in the system can be destructive [1]. Therefore, it is necessary to design control systems that are able to tolerate potential faults in such systems. A control system with this kind of fault tolerance capability is defined as a FTC system [2].

Generally speaking, FTC systems are classified into two types: Passive Fault Tolerant Control (PFTC) and Active Fault Tolerant Control (AFTC) [3 –5]. In AFTC systems, reconfiguration mechanism can be classified as on-line controller selection and calculation techniques [6]. In on-line controller selection approach, the controllers associated with certain/ predetermined faulty conditions are computed in an off-line manner in the design stage and they are selected in an on-line manner based on real time information from fault detection and diagnosis algorithm [7]. In the on-line controller calculation approach, the controller parameters are calculated in an on-line manner right after the occurrence of fault [6, 8]. The reconfiguration is generally activated after a fault has been detected and isolated. A fault detection and diagnosis system is a unit that obtains the occurrence of faults and determines their features in terms of type, location, size and/ortime.

In this study, the proposed method includes an on-line controller calculation type reconfiguration mechanism in which the reconfiguration mechanism and the diagnosis unit work independently. The proposed method developed here works with real time system outputs and is not limited to certain or predetermined faults in the reconfiguration manner. Support vector regression (SVR) machines are used in detection, diagnosis and reconfiguration stages of the method. The proposed method utilizes PID type controllers in reconfiguration sub-system and can handle multiple and simultaneous occurrences of various types of faults.

In this paper, SVR mechanism is described in Section 2 and the AFTC method is presented in Section 3. Simulations related to this method are carried out on a two tank liquid level system and the related results are given and discussed in Section 4. Final discussions and conclusions are presented in Section 5.

2 Support vector machines

Support vector machine (SVM) has become one of the most popular intelligent learning machines and a very good alternative to neural networks. This method has been developed by Vapnik [9]. SVM’s are supervised learning models with associated learning algorithms that analyze data and recognize patterns, used for classification and regression analysis [10 –12]. Since SVM is a popular and well-known method used for solving many problems in various areas such as control, communication and signal processing [13].

2.1 Support Vector Regression (SVR)

The SVM can be used for the regression problems successfully [13 –18]. Let us consider the problem of approximating the set of data: $D = {(x_{1}, y_{1}), (x_{2}, y_{2}) \dots \dots (x_{l}, y_{l})}, x \in R^{n}, y \in R$ (1) where x is the input and y is the output of the regression problem. Input-output relation can be modeled with a linear regression model in feature space as below: $f (x, w) = w^{T} x + b$ (2)

SVM find the function to be estimated as f (x). In the SVR, the approach error is used. In practice, several loss functions exist. ɛ-tolerance loss function is one of the most used one (3): $Y_{ɛ} = {\begin{matrix} 0 if | y - f (x, w) | < ɛ \\ | y - f (x, w) | - ɛ other \end{matrix}$ (3)

The thought of SVR is that a tube or a band, with radius ɛ, is defined around estimating function f(x,w). If the value f is inside the tube, that means there is no loss. In other words, it does not matter about errors as long as they are less than ɛ, but will not be acceptable any deviation larger than this. Vapnik’s loss function for ɛ= 0 is equal to absolute loss function.

The aim is to minimize the experimental and observational risk expressing total error in the formulation of SVM algorithm and ∥w ∥ ² simultaneously. This problem can be written as a convex optimization problem:

$\begin{matrix} Minimize : R (w, ξ, ξ^{*}) = \frac{1}{2} {∥ w ∥}^{2} + C \sum_{i = 1}^{1} (ξ + ξ^{*}) \\ subject to : y_{i} - w^{T} x_{i} - b \leq ɛ + ξ i = 1, \dots, l \\ w^{T} x_{i} + b - y_{i} \leq ɛ + ξ^{*} i = 1, \dots, l \\ ξ \geq 0 i = 1, \dots, l \\ ξ^{*} \geq 0 i = 1, \dots, l \end{matrix}$ (4) where C is the tradeoff parameter between the flatness of f and the amount up to which deviations larger than ɛ are tolerated. These parameters are selected by the user by trial-and-error. ξ and ξ ^* are slack variables. This constrained optimization problem is solved by establishing the primary Lagrangian

$\begin{matrix} L : & = & \frac{1}{2} {∥ w ∥}^{2} + C (\sum_{i = 1}^{n} ξ_{i} + \sum_{i = 1}^{n} ξ_{i}^{*}) \\ - & \sum_{i = 1}^{n} α_{i} (y_{i} - w^{T} x_{i} - b + ɛ + ξ_{i}) \\ - & \sum_{i = 1}^{n} α_{i}^{*} (y_{i} - w^{T} x_{i} - b + ɛ + ξ_{i}^{*}) \\ - & \sum_{i = 1}^{n} (β_{i}^{*} ξ_{i}^{*} + β_{i} ξ_{i}) \end{matrix}$ (5)

The partial derivatives of L with respect to the primal variables (w, b, ξ and ξ ^*) have to disappear for optimality. Then the problem becomes:

$\begin{matrix} \max_{α, α^{*}} W (α, α^{*}) = - ɛ \sum_{i = 1}^{n} (α_{i}^{*} + α_{i}) + \sum_{i = 1}^{n} (α_{i}^{*} - α_{i}) y_{i} \\ - \frac{1}{2} \sum_{i = 1, j = 1}^{n} (α_{i}^{*} - α_{i}) (α_{j}^{*} - α_{j}) x_{i}^{T} x_{j} \end{matrix}$ (6) $\begin{matrix} subject to \sum_{i = 1}^{n} α_{i}^{*} = \sum_{i = 1}^{n} α_{i} \\ 0 \leq α_{i}^{*} \leq C i = 1, \dots, n \\ 0 \leq α_{i} \leq C i = 1, \dots, n \end{matrix}$ (7)

This secondary Lagrangian, W, is indicated by Lagrangian multipliers α ve α ^* and maximized applying Karush-Kuhn-Tucker (KKT) optimality condition. Solving Equation 7 with constraints determines the Lagrange multipliers, and the regression function is given by, $f (x, w) = w^{T} x + b = \sum_{support vectors} (α_{i}^{*} - α_{i}) < x_{i} x > + b$ (8) where $w_{0} = \sum_{i = 1}^{n} (α_{i}^{*} - α_{i}) x_{i}$ (9) $b_{0} = \frac{1}{n} \sum_{i = 1}^{n} (y_{i} - x_{i}^{T} w_{0})$ (10)

w vector is a linear combination of the input data, b is the bias term, α and α ^* are Lagrange multipliers. Among the training data, x vectors are called support vectors, whose (α _i - $α_{i}^{*}$ ) coefficient is different from zero. Using the Kernel function the optimum regression function becomes, $f (x, w) = \sum_{i = 1}^{SVs} (α_{i}^{*} - α_{i}) K (x_{i} x) + b$ (11)

Kernel functions need to satisfy the Mercer condition [9] and two of most commonly used Kernel functions are;

Polynomial kernel function: + 1) ^p Radial basis function:

3 Proposed method: Fault tolerant control for uncertain faulty cases

In general, reconfiguration mechanism of a FTC system utilizes the information from fault detection and diagnosis unit at the decision stage. In this FTC method, the reconfiguration mechanism and diagnosis unit work independently. Both of them use only real time system outputs. A powerful and fast learning algorithm is needed for this purpose. In this method, SVR machines have been used in the fault detection and diagnosis processes and also in the reconfigurable controller unit. PID type controllers have been used in the reconfiguration sub-system. The PID controller coefficients or gains of faulty and non-faulty cases to be used in the training stage are obtained by the Genetic Algorithm (GA) optimization approach in an off-line manner. In the reconfiguration mechanism, for each PID controller coefficient or gain, one SVR machine is set up. Faulty and non-faulty system outputs are collected for different input signals. In the training stage, faulty system outputs are taken as the input and corresponding PID controller parameters are taken as the output of the SVR machines. Thus, the training phase of SVM is completed in an off-line manner. In the normal operation phase, the outputs of the system are sent to a decision unit periodically. Three of SVR machines simultaneously evaluate the data sent by the system and produce coefficients or gains of the PID controller. The controller, of which its gains are reconfigured, takes the duty so as to maintain proper system performance in an on-line manner.

Here, the fault diagnosis unit runs in parallel with the intelligent fault tolerant controller. The proposed FTC method does not need any fault detection and diagnosis operation to reconfigure the faulty response. Figure 1 shows an intelligent control mechanism in this system. The performances of these knowledge-based fault diagnosis and AFTC methods are illustrated on a simulation example involving a two-tank liquid level control system under faulty conditions.

3.1 Determination of the PID controller gains

The PID controller gains of faulty and non-faulty cases to be used in the training stage are obtained by using the GA optimization approach in an offline manner. In the reconfiguration mechanism, for each PID controller gain one SVR machine is set up as illustrated in Fig. 1. Details of the training and operating phases of the method are given below:

(a) Training phase (offline);

The most proper PID gains for non-faulty and pre-defined faulty cases are obtained by GA optimization method and the performance index to be minimized is chosen to be Integral Time Square Error (ITSE),

For each PID gain one SVR machine is set up; i.e, SVM_kp, SVM_ki, SVM_kd,

Faulty and non-faulty system outputs are collected for a variety of input signals. The diversity of input signals raises the generalization capacity of the method,

The inputs of SVMs are the collected responses of faulty cases and the outputs are the corresponding PID controller coefficients or gains,

(b) Operating phase (online);

The system has been run for the reference input signal and the outputs of the system are sent to the decision unit periodically,

Three of SVR machines simultaneously evaluate the data sent by the system and produce coefficients or gains of the PID controller. The re-configurable PID controller is composed by these three gains.

3.2 Fault detection and diagnosis via SVR

In order to determine the type (the magnitude or degree of the type) of the fault, a similar process is exploited using one SVR machine. In the training phase, inputs of this SVR machine are faulty and non-faulty system outputs and the outputs are the features of the faults. This fault diagnosis unit runs parallel with intelligent fault tolerant controller as given in Fig. 2.

4 Simulations on two tank liquid level control system

The performances of these fault diagnosis and AFTC methods are illustrated on a simulation example involving a two-tank liquid level control system under faulty conditions described in Section 4.1.

Any significant switching time or stability problem has not occurred in the running of the simulation examples. Proposed method can be seen as one kind of gain scheduling. However, further precautions should be considered in real plant applications [19, 20].

4.1 The model and the parameters of the system

The proposed FTC algorithms will be employed on a well-known benchmark problem; two-tank liquid level system [21, 22] that is given in Fig. 3. The material balance equations for the two-tank liquid level system are; $\begin{matrix} {\dot{h}}_{1} (t) = \frac{1}{A_{s}} (- K_{p 1} sign (h_{1} (t) - h_{2} (t)) \sqrt{2 g | h_{1} (t) - h_{2} (t) |} + q_{1} (t)) \\ {\dot{h}}_{2} (t) = \frac{1}{A_{s}} (K_{p 1} sign (h_{1} (t) - h_{2} (t)) \sqrt{2 g | h_{1} (t) - h_{2} (t) |} - K_{p 2} \sqrt{2 {gh}_{2} (t)}) \end{matrix}$ (12)

where, K _p1 = a ₁ S _p1, K _p2 = a ₂ S _p2 and a ₁ = a ₂ = 1 for simplicity. The parameters of the benchmark system are given in Table 1.

4.2 The Performance illustration of the proposed method

4.2.1 Example 1

As it has been pointed out earlier, the FTC method explained in Section 3 is suitable for both expected or predefined and unexpected faults. The types of faults are not needed to be known for the operation of the control reconfiguration mechanism in this structure. In this example, a leakage in tank1 with unknown height h _x is considered. The effect of the leakage to the system can be calculated using the following equation: $q_{x} (t) = a_{1} π (r_{x})^{2} \sqrt{2 {gh}_{x} (t)}$ (13)

The method has been tested for leakages in 10 cm, 30 cm, 50 cm. Table 2 shows the PID coefficients or gains obtained by the FTC as well as the height estimated by the SVR in an online manner. The method can estimate the heights of leakages quite accurately as illustrated in Table 2. The PID gains obtained by the GA optimization approach in an offline manner are shown in Table 3 in order to compare the results of online adaptive FTC method for same leakages.

The system response with no FTC mechanism is shown in Fig. 4a and it is obvious that the system can recover to provide a plausible performance but that performance is really very sluggish. The results of the system responses for the non-faulty case, the proposed online FTC method case and an optimal system response case (as if it were possible to compute) are illustrated in Fig. 4b. It is obvious that the FTC method provides very fine results that are quite close to the virtually designed optimal system response that might only be computed in an offline manner even though the proposed operates in an online manner. This is highly important since it has been demonstrated that the performance of the proposed online method is comparable to an optimally calculated offline design method. The similar results for the leakage of h _x= 30 cm and h _x= 50 cm are given inFigs. 5 and 6.

4.2.2 Example 2

Another important characteristic of the proposed FTC method is that it has large generalization ability. This method is also very successful in faulty cases which are not used in training. In this example, a leakage in tank1 with unknown height h_x and with unknown radius r _x is considered. Although only one of these two parameters is changed in the training phase, the method works just as well enough even when both parameters changed at the same time. In training phase, leakages at the heights of h _x= 0 cm, 40 cm, 80 cm or leakages at the bottom with radius of r _x= 5 mm, 10 mm, 18 mm are utilized.

For the operating phase, some faulty cases given in Table 4 are considered. Table 4 shows the PID coefficients or gains obtained by the FTC. The PID gains obtained by the GA in an offline manner are shown in Table 5 in order to compare the results of an online adaptive FTC method for the same leakages.

The results of the system responses for the non-faulty case, the faulty case without any FTC, the proposed online FTC method case and an optimal system response case (as if it were possible to compute) are illustrated in Figs. 7–10.

5 Conclusion

In this study, an active SVM-based FTC method is presented. It is well known fact that the SVM is a popular machine learning technique with a very high generalization capacity and even though SVMs are trained by a small data set they can scan a very wide region. In the proposed method, the gains of the controller are tuned by a SVM-based intelligent system in an online manner. This method is appropriate for uncertain/ non-predetermined faults. In this FTC method, the reconfiguration mechanism and the diagnosis unit work independently and it does not require the isolation of faults before tolerating them. Furthermore, it is possible to obtain the exact location or degree of the fault with an additional independent FDD unit. This FTC technique uses the real-time data generated by the system and produces the appropriate gains of the PID controller. Then, the controller with the reconfigured coefficients or gains gets into action and maintains system performance in an online manner.

The performance of the proposed FTC technique is illustrated by simulations done over the two-tank liquid level control system. The PID controller parameters and therefore the performance results of this method are very close to the parameters and the performance results obtained via GA optimization in an offline manner. Moreover, there is no need to take a precaution for bump-less transfer problem that may appear in the other online controller selection type methods.

This active online FTC method is not limited to certain or predetermined types of faults in the reconfiguration phase, which is very important feature of this approach. Another major advantage of this method is that it can easily handle the simultaneous and multiple fault occurrence cases with a quitesatisfactory performance as illustrated in the final simulation example.

Footnotes

Acknowledgments

This research is supported by a project given to the scientific Research project (SPR-BAP 31658) of Institute of Science and Technology of Istanbul Technical University. All of support is appreciated.

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