Abstract
This chapter addresses the problem of fault tolerant control (FTC) for near space vehicle (NSV) attitude dynamics with multiple actuator faults. Firstly, a general actuator fault model is proposed, which integrates state-dependent bias and gain faults. Then, sliding mode observers (SMOs) are designed to provide a bank of residuals for fault detection and isolation (FDI). Based on Lyapunov stability theory, a novel fault diagnostic algorithm is proposed to detect and isolate multiple actuator faults. Further, for the two cases where the state is available or not, two accommodation schemes are proposed to compensate for the effect of the faults. These schemes do not need the condition that the bounds of the time derivative of the faults should be known. Finally, simulation results of NSV are presented to demonstrate the efficiency of the proposed FTC approach.
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Keywords
- Fuzzy Logic System
- Actuator Fault
- Fault Tolerant Control
- Fuzzy Observer
- Parallel Distribute Compensation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
3.1 Introduction
It is well known that the controlled systems in practical applications may become faulty due to various reasons. Hence, FD and FTC have received considerable attention, and obtained significant results in the past decades, see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22] and the references therein. However, most of the existing results on FD and FTC work under the restrictive condition that only one actuator or sensor fault occurs at one time. In real applications, multiple types multiple faults may occur in the controlled system. The faulty cases include: multiple actuator faults, multiple sensors faults and multiple actuator and sensor faults. Up to now, few revelent results are reported in literature [23]. In [23], an actuator fault diagnosis scheme was proposed for a class of affine nonlinear systems with both known and unknown inputs, which was designed by making use of the derived input/output relation and the recently developed high-order sliding-mode robust differentiators. Hence, considering multiple type multiple faults simultaneously occurred in the controlled system is a motivation of this chapter.
Near space hypersonic vehicle, as a class of vehicle flying in near space which offers a promising and new, lower cost technology for future spacecraft. It can advance space transportation and also prompt global strike capabilities. Such complex technological system attracts considerable interests from the control research community and aeronautical engineering in the past couple of decades and significant results were reported [24,25,26,27,28,29,30,31,32,33,34,35]. For such high technological system, it is of course essential to maintain high reliability against possible faults [36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54].
Recently, T-S fuzzy system was used to describe the NSV attitude dynamics which are complex nonlinear, multi-variable and strongly coupled ones [24,25,26,27,28,29,30,31,32,33,34,35]. During the past two decades, the stability analysis for Takagi-Sugeno (T-S) fuzzy systems has attracted increasing attention [25,26,27]. In [55], the authors studied the problem of fault-tolerant tracking control for near-space-vehicle attitude dynamics with bias actuator fault, where the bias fault was assumed to be unknown constant. However, in practical application, the fault may be state-dependent, namely, it is a unknown function of system state. In this chapter, we will propose a more general FTC scheme that handles such state-dependent faults. On the other hand, as a universal approximation, fuzzy logic system (FLS) played an important role in modeling and controlling uncertain systems, see [56,57,58,59,60,61] and the references therein. In this chapter, we use the above FLSs to approximate the unknown state-dependent gain and bias faults.
In this chapter, we investigate the problem of fault tolerant control NSV with multiple state-dependent actuator faults, with the objective to provide an efficient solution for controlling NSV in faulty situations. Compared with existing literatures, the following contributions are worth to be emphasized.
-
(1)
The actuator fault model presented in this chapter integrates state-dependent gain bias faults, which means that a wide class of faults can be handled. The theoretic developments and results of this chapter are thus valuable in a wide field of practical applications.
-
(2)
Differing from some design scheme in literature, the fault-tolerant control scheme does not need the condition that the bounds of the time derivatives of the faults should be known constants, which thus enlarges the practical application scope.
-
(3)
In general, the denominator of the fault-tolerant control input contains the estimation of the gain fault. If the denominator is equal to zero, a controller singularity occurs. In the proposed modified FTC scheme, the controller singularity is avoided without projection algorithm.
The rest of this chapter is organized as follows. In Sect. 3.2, the T-S fuzzy model for NSV attitude dynamics is first briefly recalled. Actuator faults are integrated in such model, and the FTC objective is formulated. In addition, mathematical description of fuzzy logic system is given. In Sect. 3.3, the main technical results of this chapter are given, which include fault detection, isolation, and fuzzy logic system-based fault accommodation in the two cases where system states are available or not. The NSV application is presented in Sect. 3.4. Simulation results of NSV are presented to demonstrate the effectiveness of the proposed technique. Finally, Sect. 3.5 draws the conclusion.
3.2 Problem Statement and Preliminaries
3.2.1 Problem Statement
In this chapter, a NSV attitude dynamics in re-entry phase is given as [62]:
where \(J \in {R^{3 \times 3}}\) is the symmetric positive definite moment of inertia tensor, and \(\omega = {[p,q,r]^T} = {[{\omega _1},{\omega _2},{\omega _3}]^T}\) is the angular rate vector composed of roll p, pitch q and yaw rate r, \(\delta = {[{\delta _e},{\delta _\alpha },{\delta _r}]^T} \in {R^{3 \times 1}}\) is the control surface deflection, \({\delta _e},{\delta _\alpha },{\delta _r}\) are the elevator deflection, the aileron deflection, the rudder deflection, respectively. The skew symmetric matrix \(\varOmega \) is given by:
In the re-entry phase, \(R(\cdot )\) is defined as follows:
where \(\gamma = {[\phi ,\beta ,\alpha ]^T}\) and \(\phi ,\beta ,\alpha \) are the bank, sideslip, and the attack angles, respectively. According to the singular perturbation theory, the above six equations can be expressed in the form of inner loop \(\omega \) and outer loop \(\gamma \); \(\omega \) and \(\gamma \) are also respectively called fast loop and slow loop. The control diagram of NSV attitude dynamics is shown in Fig. 3.1. From the motion law of NSV, it is easy to find that, the response of the angular rate \(\omega \) is faster than the one of the attitude angle \(\gamma \). Based on time scale principle, we define \(\omega \) as fast state and \(\gamma \) as slow state, thus system (3.1) can be divided into the following two subsystems: fast subsystem (3.4a) related to fast state \(\omega \) and slow subsystem (3.4b) related slow state \(\gamma \).
where \(f({x_\omega }) = {J^{ - 1}}\varOmega (\omega )J{x_\omega }\), \(g({x_\omega }) = {J^{ - 1}}\), \(f({x_\gamma }) = R( \cdot )\), \({x_\omega } = \omega ,{x_\gamma } = \gamma \).
The control diagram of NSV
The control objectives are,
-
(1)
for the slow subsystem (the outer loop), to design the ideal angular rate \({y_\omega }({=}{\omega _d})\) such that subsystem output \({y_\gamma }\) follows the desired reference signal \({y_d}\) whose 1st derivative are available and bounded;
-
(2)
for the fast subsystem (the inner loop), to design the control u(t) such that the angular rate \({x_\omega }\) follows the ideal angular rate \({y_\omega }({=}{\omega _d})\).
That is to say, the main task is to design proper control input u(t) such that \({\lim _{t \rightarrow \infty }}({x_\omega } - {\omega _d}) = 0 \Rightarrow \) \({\lim _{t \rightarrow \infty }}({\gamma _\gamma } - {\gamma _d}) = 0\).
A fuzzy linear dynamic model has been proposed by Takagi and Sugeno in 1985 to represent a nonlinear system as an aggregation of local linear input/output relations. The fuzzy linear model is described by fuzzy IF-THEN rules and will be employed to deal with the fuzzy control problem for inner loop dynamics described by (3.4a) in this chapter.
Consider the following T-S fuzzy model composed of a set of fuzzy implications, where each implication is expressed by a linear state space model. The ith rule of this T-S fuzzy model is of the following form:
Plant rule i: IF \({z_1}(t)\) is \({M_{i1}}\) and \( \ldots {z_q}(t)\) is \({M_{iq}}\), THEN
where \(i = 1, \ldots ,r\), r is the number of the IF-THEN rules, \({M_{ij}}\), \(j = 1, \ldots ,q\) is the fuzzy set, \(z(t) = {[{z_1}(t), \ldots ,{z_q}(t)]^T}\) are the premise variables which are supposed to be known, \(x(t) = {[{x_1}(t), \ldots ,{x_n}(t)]^T} \in {R^n}\) denotes state vector, \(u(t) \in {R^m}\) denotes control input, \({A_i} \in {R^{n \times n}}\), and \({B_i} \in {R^{n \times m}}\) are local state and control matrices.
The overall fuzzy system is inferred as follows:
where \({h_i}(z(t))\) is defined as
where \({M_{ij}}[z(t)]\) is the grade of membership of \({z_j}(t)\) in \({M_{ij}}\). It is assumed in this chapter that \(\prod \limits _{j = 1}^n {{M_{ij}}[z(t)]} \geqslant 0\) for all t. Therefore, we have \(\sum \limits _{i = 1}^r {{h_i}(z(t))} = 1,{} {} {} {} {} {} {} 0 \leqslant {h_i}(z(t)) \leqslant 1\) for all t.
In this chapter, the state feedback control strategy is chosen as a parallel distributed compensation (PDC), which can be described as follows:
Control rule i: IF \({z_1}(t)\) is \({M_{i1}}\) and \( \ldots {z_q}(t)\) is \({M_{iq}}\), THEN
where \({K_i}\) is the controller gain matrix to be determined later.
The overall fuzzy controller is given as follows:
The control objective under normal conditions is to design a proper state feedback control input u(t) such that \({\lim _{t \rightarrow \infty }}({x_\omega } - {\omega _d}) = 0 \Rightarrow {\lim _{t \rightarrow \infty }}({\gamma _\gamma } - {\gamma _d}) = 0\).
However, in practical application, actuators may become faulty. Two kinds of actuator faults are considered: loss of effectiveness of the actuators and actuator bias faults. The first kind of fault is modeled as follows.
where \(\rho _i^u(x)\) (\(0 \leqslant \rho _i^u(x) < 1\)), which is supposed to be unknown, denotes the remaining control rate, \({t_j}\) is unknown fault occurrence time. The second kind of fault, namely actuator bias fault, can be described as:
where \(d_i^u(x)\) denotes a bounded signal. Therefore, the above two kinds of actuator faults can be uniformly described as
Furthermore, a more general fault model can be given as:
where \(d_{i,j}^u(x),i = 1, \ldots ,m,j = 1, \ldots ,p_i^u\) denotes a bounded signal, \(p_i^u\) is a known positive constant. \(g_{i,j}^u\) denotes an unknown constant. With no restriction, let suppose \(p_1^u = \cdots = p_m^u\) \( = p\), with p a known positive constant. Consider the following notation \(a_{i,j}^u(x) = g_{i,j}^ud_{i,j}^u(x)\), (3.13) can be re-written as follows:
where the nonlinear functions \(\rho _i^u(x),a_{i,j}^u(x)\) and the failure time instant \({t_j}\) are unknown. In this chapter, both bias and gain faults are handled by considering the general fault model (3.14).
Now, the control objective is re-defined as follows. An active fault tolerant control approach is proposed to obtain the above tracking objective in normal and faulty conditions, namely, \({\lim _{t \rightarrow \infty }}({x_\omega } - {\omega _d}) = 0\). Furthermore, \({\lim _{t \rightarrow \infty }}({\gamma _\gamma } - {\gamma _d}) = 0\). Under normal condition (no fault), a state feedback control input u(t) is designed, such that \({\lim _{t \rightarrow \infty }}({x_\omega } - {\omega _d}) = 0\). Meanwhile, the FDI algorithm is working. As soon as actuator faults are detected and isolated, the fault accommodation algorithm is activated and a proper fault-tolerant control input u(t) is used such that the tracking performance (\({\lim _{t \rightarrow \infty }}({x_\omega } - {\omega _d}) = 0\)) is still maintained stable under faulty case.
3.2.2 Mathematical Description of Fuzzy Logic System
FLS consists of four parts: the knowledge base, the fuzzifier, the fuzzy inference engine working on fuzzy rules, and the defuzzifier. The knowledge base for FLS comprises a collection of fuzzy if-then rules of the following form:
where \(\underline{x} = {\left[ {{x_1},{x_2}, \ldots ,{x_n}} \right] ^T} \in U \subset {R^n}\) and y are the FLS input and output, respectively. Fuzzy sets \(A_i^l\) and \({B^l}\) are associated with the fuzzy functions \({\mu _{A_i^l}}({x_i}) = \exp ( - {(\frac{{{x_i} - a_i^l}}{{b_i^l}})^2})\) and \({\mu _{{B^l}}}({y^l}) = 1\), respectively. M is the rules number. Through singleton function, center average defuzzification and product inference, the FLS can be expressed as
where \({\bar{y}^l} = {\max _{y \in R}}{\mu _{{B^l}}}\). Define the fuzzy basis functions as
and define \({\theta ^T} = [{\bar{y}^1},{\bar{y}^2}, \ldots ,{\bar{y}^M}] = [{\theta _1},{\theta _2}, \ldots ,{\theta _M}]\) and \(\xi = {[{\xi ^1},{\xi ^2}, \ldots ,{\xi ^M}]^T}\), then FLS (3.15) can be rewritten as
Lemma 3.1
(Boulkroune et al. [60]) Let f(x) be a continuous function defined on a compact set \(\varOmega \). Then for any constant \(\varepsilon > 0\), there exists an FLS (3.17) such as
By Lemma 3.1, FLSs are universal approximations, i.e., they can approximate any smooth function on a compact space. Due to this approximation capability, we can assume that the nonlinear term f(x) can be approximated as
Define the optimal parameter vector \({\theta ^*}\) as
where \(\varOmega \) and U are compact regions for \(\theta \) and x, respectively. Also the FLS minimum approximation error is defined as
In this chapter, we use the above fuzzy logic system to approximate the unknown functions \(\rho _i^u(x),a_{i,j}^u(x)\), namely, there exist \(\theta _{\rho ,i}^*,\theta _{\alpha ,i,j}^*\), \({\varepsilon _{\rho ,i}},{\varepsilon _{\alpha ,i,j}}\) such that \(\rho _i^u(x) = \theta _{\rho ,i}^*{\xi _{\rho ,i}}(x) + {\varepsilon _{\rho ,i}}a_{i,j}^u(x) = \theta _{\alpha ,i,j}^*{\xi _{\alpha ,i,j}}(x) + {\varepsilon _{\alpha ,i,j}}\). Now, the following assumptions are made.
Assumption 3.1
There exist unknown constants \(\varepsilon _{^{\rho ,i}}^*> 0,\varepsilon _{^{\alpha ,i,j}}^* > 0\) and two known constants \({\bar{M}_{\rho ,{s_k}}},{\bar{M}_{\alpha ,{s_k},j}}\) such that \(|{\varepsilon _{\rho ,i}}| \leqslant \varepsilon _{^{\rho ,i}}^*,|{\varepsilon _{\alpha ,{s_k},j}}| \leqslant \varepsilon _{^{\alpha ,i,j}}^*,\varepsilon _{^{\rho ,i}}^* \leqslant {\bar{M}_{\rho ,{s_k}}},\varepsilon _{^{\alpha ,i,j}}^* \leqslant {\bar{M}_{\alpha ,{s_k},j}}\).
Assumption 3.2
There exist known constants \({M_{\rho ,{s_k}}},{M_{\alpha ,{s_k},j}}\) such that \(||\theta _{_{\rho ,{s_k}}}^*|| \leqslant {M_{\rho ,{s_k}}},||\theta _{_{\alpha ,{s_k},j}}^*|| \leqslant {M_{\alpha ,{s_k},j}}\).
3.3 Fault Diagnosis and FLS-Based Fault Accommodation
In this section, the main technical results of this chapter are given. We will first formulate the fault diagnosis and accommodation problems of the above T-S fuzzy system. We will then design a bank of SMOs to generate residuals, investigate the FDI algorithm based on the SMOs, and propose a FTC scheme to tolerate the faults by compensating for faults.
3.3.1 Preliminary
Consider the T-S fuzzy faulty system described in (3.6). We assume that only actuator faults occur and no sensor fault is involved. The following assumptions are considered.
Assumption 3.3
Matrix \({B_i}\) is of full column rank and the pair \(({A_i},{C_i})\) is observable.
We first design the fault diagnosis observers to detect and isolate the faults, and then, propose a FTC method to compensate the faults.
3.3.2 Fault Detection
In order to detect the actuator faults, we design a fuzzy state-space observer for the system (3.6), which is described as:
Observer rule i: IF \({z_1}(t)\) is \({M_{i1}}\) and \( \ldots {z_q}(t)\) is \({M_{iq}}\), THEN
where \({L_i},i = 1,\ldots ,r\) is the observer gain for the ith observer rule.
The overall fuzzy system is inferred as follows:
Denote
then the error dynamics is described by
Lemma 3.2
The estimation error \({e_x}\) converges asymptotically to zero if there exist common matrices \(P = {P^T} > 0\) and \(Q > 0\) with appropriate dimensions such that the following linear matrix inequality is satisfied:
Proof
Consider the following Lyapunov function
Differentiating \({V_1}\) with respect to timet, one has
Because \({V_D}(t) \in {L_\infty }\) is a monotonous and non-increasing bounded function, \({V_D}( + \infty )\) exists. Hence, we have
which means taht \({e_x}(t) \in {L_2}\). Since \({e_x}(t),{\dot{e}_x}(t) \in {L_\infty }\), using the Lyapunov stability theory, we obtain \(\mathop {\lim }\limits _{t \rightarrow \infty } {e_x}(t) = 0\). Furthermore, we have \(\mathop {\lim }\limits _{t \rightarrow \infty } {e_y}(t) = 0\). The proof is completed.
From Lemma 3.2, we have
where \(\kappa = {\lambda _{\min }}(Q)/{\lambda _{\max }}(P) \in R\). Hence,
Furthermore, we have
Therefore the norm of the error vector satisfies
Furthermore, the detection residual can be defined as
From (3.30), it can be seen that the following inequality holds in the healthy case:
Then, the fault detection can be performed using the following mechanism:
where threshold \({T_d}\) is defined as follows.
3.3.3 Fault Isolation
Since the system has m actuators, which maybe become faulty, we have \(C_m^1 + C_m^2 + \cdots + C_m^m\) possible faulty cases, where \(C_m^i\) denotes the number of faulty cases where there are i faulty actuators within m actuators. Let us define the following symbol, \(j_i^k \left( {i = 1,2, \ldots ,m; k = 1,2, \ldots ,i} \right) \) which denotes the situation where the ith actuator fails when there are k possible faulty actuators among the m actuators. Fault patterns can be described in details as follows.
Case 1: only an actuator is faulty
In this case, there are \(C_m^1\) fault patterns.
Case 2: only two actuators are faulty
where the number of fault patterns reached a total of \(C_m^2\).
Case i: only i actuators are faulty
where the total fault pattern is \(C_m^i,i = 1,2, \ldots ,m\).
Case m: all m actuators are faulty
Here, there is only one fault pattern (\(C_m^m = 1\)).
Now, let \({\aleph _m} = \{ \aleph _{_1}^1, \ldots ,\aleph _{_1}^{C_m^1}, \ldots ,\aleph _{_i}^1, \ldots ,\aleph _{_i}^{C_m^i}, \ldots ,\aleph _{_m}^1, \ldots ,\aleph _{_m}^{C_m^m}\} \). Obviously, there are \(C_m^1 + C_m^2 + \cdots + C_m^m\) possible fault patterns that are numbered as the 1st, 2nd, Nth fault pattern, where \(N = C_m^1 + C_m^2 + \cdots + C_m^m\).
In this chapter, it is assumed that there d actuators became faulty whose pattern s is \(\aleph _{_d}^q\), namely, \(s = \aleph _{_d}^q\). We also assume that the d actuators are the \({s_1}{\text {th}},{s_2}{\text {th}}, \ldots ,{s_d}{\text {th}}\) actuators, where \(1 \leqslant {s_1}< {s_2}< \cdots < {s_d} \leqslant m\). Then the faulty model can be described as:
where \({B_i} = [{b_{i,1}},{b_{i,2}}, \ldots ,{b_{i,m}}]\), \({b_{i,l}} \in {R^{n \times 1}},1 \leqslant l \leqslant m\), \(\rho _{_{{s_k}}}^u(x)\), \(a_{_{s,{s_k}}}^u(x),j = 1,2, \ldots ,p\) denote the time profiles of the \({s_k}\)th actuator fault, which are described by (3.14), \(u_{_{{s_k}}}^s(t)\) is the desired controller when the \({s_k}\)th actuator is healthy.
Inspired by the SMOs in [63], we are ready to present one of the results of this chapter. It is assumed that fuzzy observer and fuzzy control systems have the same premise variables z(t), then the following fuzzy observers are proposed to isolate the actuator fault.
Isolation Observer Rule i: IF \({z_1}(t)\) is \({M_{i1}}\) and \( \ldots {z_q}(t)\) is \({M_{iq}}\), THEN
where \({\hat{x}_{is}}(t),{\hat{y}_{is}}(t)\) are the sth fuzzy observer’s state and output, respectively. \({L_i}\) is the observer’s gain matrix for ith observer. The global fuzzy observer is represented as:
where \({F_{i{s_k}}} \in {R^{1 \times n}}\) is the \({s_k}\)th row of \({F_i} \in {R^{m \times n}}\), which will be defined later, \({L_i} \in {R^{n \times n}}\) is chosen such that \({A_i} - {L_i}{C_i}\) is Hurwitz, \({e_{xs}}(t) = {x_s}(t) - {\hat{x}_s}(t)\) and \({e_{ys}}(t) = y(t) - {\hat{y}_s}(t)\) are respectively the state error and output error between the plant and the sth SMO observer. Let l denotes the practical fault pattern where the faulty actuators are the \({l_1}{\text {th}},{l_2}{\text {th}}, \ldots ,{l_{{d^*}}}{\text {th}}\) actuators, \(1 \leqslant {l_1}< {l_2}< \cdots < {l_{{d^*}}} \leqslant m\).
For \(s = l\), namely, \(d = {d^*},{l_1} = {s_1},{l_2} = {s_2}, \ldots ,{l_{{d^*}}} = {s_d}\), the error dynamics is obtained from (3.35) and (3.36).
For \(s \ne {\text {l}} \), namely, \(d \ne {d^*} or d = {d^*}\) and at least there exists \({l_i}\) such that \({\text { }}{l_i} \ne {s_i},i = 1,2, \ldots ,d\), we have
The stability of the state error dynamics is guaranteed by the following theorem.
Theorem 3.1
Under Assumptions 3.1–3.3, if there exist a common symmetric positive definite matrix P and matrices \({L_i},{F_i},Q > 0\), \(i = 1, 2, \ldots , r\) with appropriate dimensions, such that the following conditions hold
Then, when the lth pattern is the actual fault pattern i.e., \(s = l\), we have \(\mathop {\lim }\limits _{t \rightarrow \infty } {e_{xs}} = 0\), and for \(s \ne l\), we have \(\mathop {\lim }\limits _{t \rightarrow \infty } {e_{xs}} \ne 0\).
Proof
(1) For \(s = l\), according to (3.38), we have
Define the following Lyapunov function
Differentiating \({V_2}\) with respect to time t, and using (3.40), one has
From \({\mu _{{s_k}}} = - \sum \limits _{i = 1}^r {{h_i}(z(t))} {F_{i{s_k}}}{e_{y{s_k}}}(t)/||\sum \limits _{i = 1}^r {{h_i}(z(t))} {F_{i{s_k}}}{e_{y{s_k}}}(t)||\) and (3.41), one has
Hence,
Because \({V_I}(t) \in {L_\infty }\) is a monotonous and non-increasing bounded function, \({V_I}( + \infty )\) exists. Hence, we have \({V_I}(0) - \) \({V_I}( + \infty ) \geqslant - \int _{{\text { }}0}^{ + \infty } {e_{xs}^T} (t)Q{e_{xs}}(t)\), i.e., \({e_{xs}}(t) \in {L_2}\). Since \({e_{xs}}(t)\) and \({\dot{ e}_{xs}}(t) \in {L_\infty }\), using the Lyapunov stability theory, we have \(\mathop {\lim }\limits _{t \rightarrow \infty } {e_{xs}}(t) = 0\). Thus, we have \(\mathop {\lim }\limits _{t \rightarrow \infty } {e_{ys}}(t) = 0\).
(2) For \(s \ne l\), it follows from (3.35) and (3.39) that:
Because matrix \({B_i}\) is of full column rank (Assumption 3.1), we know that \({b_{i{s_k}}}\) and \({b_{i{l_{k1}}}}\) are linearly independent. Therefore,
Thus, we have \(\mathop {\lim }\limits _{t \rightarrow \infty } {e_{xs}}(t) \ne 0\) and \(\mathop {\lim }\limits _{t \rightarrow \infty } {e_{ys}}(t) \ne 0\).
From (1) and (2), we obtain the conclusion. This ends the proof.
Now, we denote the residuals between the real system and SMOs as follows:
According to Theorem 3.1, when the actual fault pattern is \(s = l\), the residual \({J_s}(t)\) will tend to zero; while for any \(s \ne l\), \({J_s}(t)\) does not equal zero. Furthermore, from Lemma 3.2, we have, if \(l = s\),
and if \(l \ne s\), then
Hence, the isolation law for actuator fault can be designed as
where threshold \({T_I}\) is defined as follows.
Notice that, the denominator of \({\mu _{{s_k}}} = - \sum \limits _{i = 1}^r {{h_i}(z(t))} {F_{i{s_k}}}{e_{y{s_k}}}(t)/||\sum \limits _{i = 1}^r {{h_i}(z(t))} {F_{i{s_k}}}{e_{y{s_k}}}(t)||\) in (3.37), contains \({e_{ys}}(t)\). Just as pointed out in [63], the chattering phenomenon occurs when \({e_{ys}}(t) \rightarrow 0\) in practice. Inspired by [63], in order to reduce this chattering in practical applications, we modify SMOs (3.37) by introducing a positive constant\(\delta \) as follows:
where \(\delta > 0 \in R\) is a constant. Obviously, the denominator of \({\mu '_{{s_k}}}\) will converge asymptotically to \(\delta \) when \({e_{ys}} \rightarrow 0\), which reduces this chattering phenomenon.
3.3.4 Fuzzy Logic Systems-Based Fault Accommodation with Available System State
After fault isolation, the next task is fault accommodation. Before this task, we investigate firstly the following normal systems (fault-free), and drive the ideal control \({u^{_{^s}}}{(t)}\) when all actuators are healthy.
The parallel distributed compensation (PDC) technique offers a procedure to design a fuzzy control law from a given T-S fuzzy model. In the PDC design, each control rule is designed from the corresponding rule of T-S fuzzy model. The designed fuzzy controller has the same fuzzy sets as the considered fuzzy system.
Control Rule i: IF \({z_1}(t)\) is \({M_{i1}}\) and \( \ldots {z_q}(t)\) is \({M_{iq}}\), THEN
and the overall fuzzy controller is given as follows:
where the controller gain matrix \({K_i}\) is determined by solving the following condition:
where \(P = {P^T} > 0\), \(Q> 0,{S_1}> 0,{S_2} > 0\) are matrices with appropriate dimensions.
Define tracking error \(\bar{e} = y - {\omega _d}\). The tracking error dynamics is obtained from the above equations,
Because all the states are supposed to be available, we have \({C_i} = {I_{m \times m}}\). The tracking error dynamics can be simplified as follows:
Define the following Lyapunov function
where \(P = {P^T} > 0\).
Differentiating \({V_0}\) with respect to time t, leads to
Since
(3.54) can be re-written as follows:
where \({\varDelta _1} = P({A_i} + {K_i}{B_i}) + {({A_i} + {K_i}{B_i})^T}P + {({A_i} + {K_i}{B_i})^T}P{S_1}P({A_i} + {K_i}{B_i}) + P{S_2}P\).
Obviously, if
then
where \({\mu _0} = \sum \limits _{i = 1}^r {{h_i}(z(t)){\text {[}}{{({\omega _d} - {{\dot{ \omega } }_d})}^T}(S_1^{ - 1} + S_2^{ - 1} + 2P)({\omega _d} - {{\dot{ \omega }}_d})]} \), \({\lambda _0} = \frac{{{\lambda _{\min }}(Q)}}{{{\lambda _{\max }}(P)}}\), \(Q = {Q^T} > 0\).
Then, one has \(\frac{d}{{dt}}({V_0}(t){e^{{\lambda _0}t}}) \leqslant {e^{{\lambda _0}t}}{\mu _0}\). Furthermore,
Therefore, the error system (3.53) is asymptotically stable. Moreover, \(\bar{e}(t)\) is semi-globally uniformly ultimately bounded, converging asymptotically to a small neighborhood of zero, namely, \(|\bar{e}| \leqslant \sqrt{\alpha /{\lambda _{\min }}(P)} \), where \(\alpha = {\mu _0}/{\lambda _0} + {V_0}(0)\).
After obtaining the desired control \({u^s}(t)\), we will design fault-tolerant control u(t) such that the same control objective can be achieved in spite of actuator faults.
On the basis of the desired control \({u^s}(t)\), the fault tolerant controller is constructed as
where \({\hat{\theta } _{\rho ,{s_k}}},{\hat{\theta } _{\alpha ,{s_k},j}},{\hat{\rho } _{\rho ,{s_k}}}(x,{\hat{\theta } _{\rho ,{s_k}}}),{\hat{a}_{i,j}}(x,{\hat{\theta } _{\alpha ,{s_k},j}})\) are the estimations of \(\theta _{\rho ,{s_k}}^*\), \(\theta _{\alpha ,{s_k},j}^*\), \({\rho _{{s_k}}}(x,\theta _{\rho ,{s_k}}^*)\), \({a_{{s_k},j}}(x,\theta _{\alpha ,{s_k},j}^*)\), which are used to compensate for the gain and bias faults \({\rho _{{s_k}}}(x)\), \({\alpha _{{s_k},j}}(x)\), and \({\rho _{{s_k}}}(x) = {\rho _{{s_k}}}(x,\theta _{\rho ,{s_k}}^*) + {\varepsilon _{{s_k}}}\), \({a_{{s_k},j}}(x) = {a_{{s_k},j}}(x,\theta _{\alpha ,{s_k},j}^*) + {\varepsilon _{{s_k},j}}\), \({\varepsilon _{{s_k}}},{\varepsilon _{{s_k},j}}\) are approximation errors, \(\theta _{\rho ,{s_k}}^*,\theta _{\alpha ,{s_k},j}^*\) are optimal vectors.
Consider the following faulty system
Submitting the fault-tolerant control law (3.55) to the faulty system (3.56), it yields
where \({\tilde{\theta } _{\alpha ,{s_k},j}} = {\hat{\theta } _{\alpha ,{s_k},j}} - \theta _{_{\alpha ,{s_k},j}}^*\), \({\kappa _k} = (\frac{{u_{{s_k}}^s - \sum \limits _{j = 1}^p {[\hat{\alpha } _{_{\alpha ,{s_k},j}}^u(x,{{\hat{\theta } }_{\alpha ,{s_k},j}}) - {{\hat{\varepsilon } }_{\alpha ,{s_k},j}}]} }}{{1 - {{\hat{\rho } }_{{s_k}}}(x,{{\hat{\theta } }_{\rho ,{s_k}}}) - {{\hat{\varepsilon } }_{\rho ,{s_k}}}}})\), \({\tilde{ \theta } _{\rho ,{s_k}}} = {\hat{\theta } _{\rho ,{s_k}}} - \theta _{{\rho ,{s_k}}}^*\). Further, the error dynamics is obtained:
Now, an adaptive fault accommodation algorithm is proposed to control the faulty system. The stability of the error dynamics is guaranteed by the following theorem.
Theorem 3.2
Under Assumptions 3.1–3.3, if there exist a common symmetric positive definite matrix P, real matrices \({K_i}\) and \(Q > 0\), \(i = \) \(1,2, \ldots ,r\) with appropriate dimensions, such that the following conditions hold
Consider the control law (3.55) and the adaptive laws given as follows:
where \({\eta _i} > 0,i = 1, \ldots ,4\) denote the adaptive rates, then the error system (3.59) is asymptotically stable. Moreover, \(\bar{e}(t)\), \({\tilde{ \theta } _{\rho ,{s_k}}}\) and \({\tilde{ \theta } _{\alpha ,{s_k},j}}\) are semi-globally uniformly ultimately bounded, converging asymptotically to a small neighborhood of zero, namely, \(||\bar{e}|| \leqslant \sqrt{\alpha /{\lambda _{\min }}(P)} \), \(||{\tilde{\theta }_{\rho ,{s_k}}}|| \leqslant \sqrt{2{\eta _1}\alpha } \), and \(||{\tilde{\theta }_{\alpha ,{s_k},j}}|| \leqslant \) \(\sqrt{2{\eta _2}\alpha } \), where \(\alpha = \) \(\frac{{{\mu _0}}}{{{\lambda _0}}} + V(0)\), \(\lambda = \min \{ \frac{{{\lambda _{\min }}(Q)}}{{{\lambda _{\max }}(P)}},\frac{1}{{2{\eta _1}}},\frac{1}{{2{\eta _2}}}\} \), \(\mu = \sum \limits _{i = 1}^r {{h_i}(z(t))(\frac{2}{{{\eta _2}}}\bar{\theta }_{_{\rho ,{s_k}}}^2 + \sum \limits _{j = 1}^p {\frac{2}{{{\eta _2}}}\theta _{_{\alpha ,{s_k},j}}^2)} } + {\mu _0}\), and \({\mu _0} = \sum \limits _{i = 1}^r {{h_i}(z(t)){\text {[}}{{({\omega _d} - {{\dot{\omega } }_d})}^T}(S_1^{ - 1}} + S_2^{ - 1} + 2P)({\omega _d} - {\dot{ \omega } _d})]\).
Proof
Define the following smooth function
where
Differentiating V with respect to time t, it leads to
where
Since \({u_{{s_k}}} = \frac{{(u_{{s_k}}^s - \sum \nolimits _{j = 1}^p {{{\hat{a}}_{{s_k},j}}(x,{{\hat{\theta } }_{\alpha ,{s_k},j}})} - {{\hat{\varepsilon }}_{\alpha ,{s_k},j}})}}{{1 - {{\hat{\rho } }_{{s_k}}}(x,{{\hat{\theta } }_{\rho ,{s_k}}}) - {{\hat{\varepsilon } }_{\rho ,{s_k}}}}}\), then
where \(\varDelta = \frac{{u_{{s_k}}^s - \sum \nolimits _{j = 1}^p {{{\hat{a}}_{{s_k},j}}(x,{{\hat{\theta } }_{\alpha ,{s_k},j}})} - {{\hat{\varepsilon } }_{\rho ,{s_k},j}}}}{{1 - {{\hat{\rho }}_{{s_k}}}(x,{{\hat{\theta } }_{\rho ,{s_k}}}) - {{\hat{\varepsilon } }_{\rho ,{s_k}}}}}\). Furthermore, one has
Substituting the adaptive laws (3.60–3.63) into the above equation, it yields
Since \(||{\hat{\theta } _{\rho ,{s_k}}}|| \leqslant {M_{\rho ,{s_k}}},||{\hat{\theta } _{\alpha ,{s_k},j}}|| \leqslant {M_{\alpha ,{s_k},j}}\), which can be guaranteed by using the adaptive laws (3.60) and (3.61), when Assumptions 3.1 and 2.2 (i.e., \(||\theta _{_{\rho ,{s_k}}}^*|| \leqslant {M_{\rho ,{s_k}}}\), \(||\theta _{_{\alpha ,{s_k},j}}^*|| \leqslant {M_{\alpha ,{s_k},j}}\)) are satisfied, one has
where \(\mu = \sum \limits _{i = 1}^r {{h_i}(z(t))[\frac{4}{{{\eta _1}}}M_{_{\rho ,{s_k}}}^2 + \sum \limits _{j = 1}^p {\frac{4}{{{\eta _2}}}M_{_{\alpha ,{s_k},j}}^2} + \frac{4}{{{\eta _3}}}\bar{M}_{\rho ,{s_k}}^2 + \sum \limits _{j = 1}^p {\frac{4}{{{\eta _4}}}\bar{M}_{_{\alpha ,{s_k},j}}^2} )]} \, + \, {\mu _0}\), \(\lambda = \min \{ \frac{{{\lambda _{\min }}(Q)}}{{{\lambda _{\max }}(P)}},\frac{1}{{2{\eta _1}}},\frac{1}{{2{\eta _2}}}\}\).
Then, one has, \(\frac{d}{{dt}}(V(t){e^{\lambda t}}) \leqslant {e^{{\lambda _{}}t}}\mu \). Furthermore,
Let \(\alpha = \frac{\mu }{\lambda } + V(0)\), one has \(|\bar{e}| \leqslant \sqrt{\frac{\alpha }{{{\lambda _{\min }}(P)}}} \), \(|{\tilde{\theta }_{\rho , {s_k}}}| \leqslant \sqrt{2{\eta _1}\alpha } \), and \(|{\tilde{\theta }_{\alpha ,{s_k},j}}| \leqslant \sqrt{2{\eta _2}\alpha } \). This ends the proof.
3.3.5 Modified Fault Accommodation with Available System State
In the above subsection, the fault tolerant controller was constructed as
Unfortunately, there may exist controller singularity when \(1 - {\hat{\rho } _{{s_k}}}(x,{\hat{\theta } _{\rho ,{s_k}}}) - {\hat{\varepsilon } _{\rho ,{s_k}}} = 0\).
In order to avoid such singularity, the fault tolerant controller is modified as follows
where \(\varepsilon > 0 \in R\) is a design constant. Correspondingly, the adaptive laws in Theorem 3.2 are re-designed as follows
where \(\kappa ' = \frac{{(1 - {{\hat{\rho } }_{{s_k}}}(x,{{\hat{\theta } }_{\rho ,{s_k}}}) - {{\hat{\varepsilon } }_{\rho ,{s_k}}})[u_{{s_k}}^s - \sum \limits _{j = 1}^p {\hat{\theta } _{_{\alpha ,{s_k},j}}^T\xi _{_{\alpha ,{s_k},j}}^u(x)} ]}}{{{{(1 - {{\hat{\rho } }_{{s_k}}}(x,{{\hat{\theta } }_{\rho ,{s_k}}}) - {{\hat{\varepsilon } }_{\rho ,{s_k}}})}^2} + \varepsilon }}\), \({\eta _l} > 0,l = 1, \ldots ,4\) denote the adaptive rates.
Now, a modified adaptive fault accommodation algorithm is proposed to control the faulty system. The stability of the error dynamics is guaranteed by the following theorem.
Theorem 3.3
Under Assumptions 3.1–3.3, if there exist a common symmetric positive definite matrix P, real matrices \({K_i}\) and \(Q > 0\), \(i = 1,2, \ldots ,r\) with appropriate dimensions, such that the following conditions hold
when the control law (3.64) and adaptive laws (3.65–3.68) are applied, the error system (3.58) is asymptotically stable. Moreover \(\bar{e}(t)\), \({\tilde{\theta }_{\rho ,{s_k}}}\) and \({\tilde{\theta }_{\alpha ,{s_k},j}}\) are semi-globally uniformly ultimately bounded, converging asymptotically to a small neighborhood of zero, namely, \(||\bar{e}|| \leqslant \sqrt{\alpha /{\lambda _{\min }}(P)} \), \(||{\tilde{\theta }_{\rho ,{s_k}}}|| \leqslant \sqrt{2{\eta _1}\alpha } \), and \(||{\tilde{\theta }_{\alpha ,{s_k},j}}|| \leqslant \sqrt{2{\eta _2}\alpha } \) where \(\lambda = \min \{ \frac{{{\lambda _{\min }}(Q)}}{{{\lambda _{\max }}(P)}},\frac{1}{{2{\eta _1}}},\frac{1}{{2{\eta _2}}}\},\) \(\mu = \sum \limits _{i = 1}^r {{h_i}(z(t))(\frac{2}{{{\eta _2}}}\bar{\theta }_{_{\rho ,{s_k}}}^2 + \sum \limits _{j = 1}^p {\frac{2}{{{\eta _2}}}\bar{\theta }_{_{\alpha ,{s_k},j}}^2} )} + {\mu _0}\), and \({\mu _0} = \sum \limits _{i = 1}^r {{h_i}(z(t))} {\text {[}}{({\omega _d} - {\dot{ \omega } _d})^T}(S_1^{ - 1} + S_2^{ - 1} +2P)({\omega _d} - {\dot{ \omega }_d}) + \omega ]\), \(\alpha = \frac{\mu }{\lambda } + \) V(0).
Proof
Similar to the proof of Theorem 3.2, it is easy to obtain the conclusion. The detailed proof is omitted.
3.3.6 FLSs-Based Fault Accommodation with Unavailable System State
Notice that, the FTC (3.55) and the modified FTC (3.64) are designed under the condition that system states are measurable. In fact, in some situations, system state may be unavailable, and the above FTC (3.55) and (3.64) do not work. In this case, observers (3.21) and (3.22) may be used to obtain the estimation \(\hat{x}\) of system state x, and design the following observer-based FTC.
Correspondingly, the adaptive laws in Theorem 3.3 are re-designed as follows:
where \(\omega = \frac{{(1 - {{\hat{\rho }}_{{s_k}}}(x,{{\hat{\theta } }_{\rho ,{s_k}}}) - {{\hat{\varepsilon }}_{\rho ,{s_k}}})[u_{{s_k}}^s - \sum \limits _{j = 1}^p {\hat{\theta } _{_{\alpha ,{s_k},j}}^T\xi _{_{\alpha ,{s_k},j}}^u(\hat{x})} ]}}{{{{(1 - {{\hat{\rho } }_{{s_k}}}(\hat{x},{{\hat{ \theta } }_{\rho ,{s_k}}}) - {{\hat{\varepsilon } }_{\rho ,{s_k}}})}^2} + \varepsilon }}\), \({\eta _l} > 0,l = 1, \ldots ,4\) denote the adaptive rates.
Now, an observer-based adaptive fault accommodation algorithm is proposed to control the faulty system. The stability of the error dynamics is guaranteed by the following theorem.
Theorem 3.4
Under Assumptions 3.1–3.3, if there exist a common symmetric positive definite matrix P, real matrices \({K_i}\) and \(Q > 0\), \(i = 1,2, \ldots ,r\) with appropriate dimensions, such that the following conditions hold
when the control law (3.70) and adaptive laws (3.71–3.74) are applied, then the error system (3.58) is asymptotically stable. Moreover \(\bar{e}(t)\), \({\tilde{\theta }_{\rho ,{s_k}}}\) and \({\tilde{\theta }_{\alpha ,{s_k},j}}\) are semi-globally uniformly ultimately bounded, converging asymptotically to a small neighborhood of zero, namely, \(||\bar{e}|| \leqslant \sqrt{\alpha /{\lambda _{\min }}(P)} \), \(||{\tilde{\theta }_{\rho ,{s_k}}}|| \leqslant \sqrt{2{\eta _1}\alpha } \), and \(||{\tilde{\theta }_{\alpha ,{s_k},j}}|| \leqslant \sqrt{2{\eta _2}\alpha } \), where \(\alpha = \mu /\lambda + V(0)\), \(\lambda = \min \{ \frac{{{\lambda _{\min }}(Q)}}{{{\lambda _{\max }}(P)}},\frac{1}{{2{\eta _1}}},\frac{1}{{2{\eta _2}}}\}\), \(\mu = \sum \limits _{i = 1}^r {{h_i}(z(t))(\frac{2}{{{\eta _2}}}\bar{\theta }_{_{\rho ,{s_k}}}^2 + \sum \limits _{j = 1}^p {\frac{2}{{{\eta _2}}}\bar{\theta }_{_{\alpha ,{s_k},j}}^2} )} + {\mu _0}\), and \({\mu _0} = \sum \limits _{i = 1}^r {{h_i}(z(t))} {\text {[}}({\omega _d} - {\dot{ \omega } _d}{)^T} \cdot (S_1^{ - 1} + S_2^{ - 1}+ 2P)({\omega _d} - {\dot{ \omega } _d}) + \omega ]\).
Proof
Similar to the proof of Theorem 3.2, it is easy to obtain the conclusion. The detailed proof is omitted.
3.4 Simulation Results
To verify the effectiveness of the proposed method, we consider the re-entry phase of a NSV with the altitude \(H = 40\,\mathrm{km}\) and speed \(V = 2500\,\mathrm{m/s}\) as the initial states. The symmetric, positive definite moment of inertia tensor is given as follows:
Consider that the nonlinearity of NSV re-entry attitude dynamics mainly comes from attack angle \(\alpha \) and attitude angular velocity \(\omega \). In NSV re-entry phase \(\alpha \in [0,\pi /4]\), we assume that \(\alpha \) has two related fuzzy sets \(\{ \alpha = 0\} \) and \(\{ \alpha = \pi /4\} \), the corresponding membership functions are given by:
We choose six operating points:
Under the membership functions and the six operating points, six plant rules and six control rules can be defined. All \(A_i\) and \(B_i\) can be obtained by substituting the six operating points to \(f({x_\omega }),~g({x_\omega })\). The detailed matrix parameters are given in [62].
The initial conditions are taken as follows: \(\omega (0) = {[0,0,0]^T},\gamma (0) = {[0,0,0]^T}\) and the tracking command is chosen as \({\omega _d} = {[0,0,0]^T},{\gamma _d} = {[1,0,2]^T}\) during the re-entry phase. The parameters are taken as in [62] and will not be described in detail here. We consider the case where only two actuators fail at one time:
The observer errors time responses: \({e_1},~{e_2},~{e_3}\)
State responses of NSV attitude dynamics under normal conditions
State responses of NSV attitude dynamics without FTC
State responses of NSV attitude dynamics with observer-based FTC (3.70)
where \({\rho _1}(x) = 0.4\cos ({x_1}),p = 1,{g_{1,1}} = 0.4,{f_{1,1}}(x) = \cos ({x_3})\), \({\rho _2}(x) = 0.4\sin ({x_2}),{g_{2,1}} = 0.4\), \({f_{2,1}}(x) = \cos ({x_3})\). By using Matlab toolbox to solve the matrices inequalities (3.25), one can obtain the fault diagnostic observer gains \({L_i}\). By solving (3.52), one can obtain the positive definite symmetric matrix P and the nominal controller gains \({K_i}\). Therefore, one can design the ideal control (3.51). Using the ideal control input (3.51), we can design fault-tolerant controller (3.55), the modified fault-tolerant (3.64) and the observer-based fault-tolerant control (3.70). In this example, we assume that the system state is not fully measured and thus the observer (3.22) is used to estimate the system state. Consequently, the observer-based fault-tolerant control input (3.70) is used to control the faulty system. The simulation results are presented in Figs. 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8 and 3.9. From Fig. 3.2, it is seen that, under normal operating condition, observation errors globally asymptotically converge to zero. If no actuator fails, the system states globally asymptotically converge to zero, as shown in Fig. 3.3. Figure 3.4 shows that, when an actuator fault occurs, when keeping the normal controller, the system states deviate significantly from zero. However, as shown in Fig. 3.5, using the proposed FTC (3.70), the system states globally asymptotically converge to zero. From Figs. 3.6, 3.7, 3.8 and 3.9, we can clearly draw the conclusion that both gain faults and bias faults can be approximated accurately and promptly by FLSs.
The estimation error of bias fault \({g_{1,1}}{f_{1,1}}(x)\)
The estimation error of gain fault \({\rho _1}(x)\)
The estimation error of bias fault \({g_{2,1}}{f_{2,1}}(x)\)
The estimation error of gain fault \({\rho _2}(x)\)
3.5 Conclusions
In this chapter, the problem of fault tolerant control for NSV with multiple state-dependent faults was studied. We first designed a bank of SMOs to detect and estimate the fault. Compared with some results in literature, the proposed fault accommodation scheme is designed to online approximate not only bias faults but also gain faults. Moreover, it can accommodate multiple actuator faults simultaneously. In addition, the adaptive fault accommodation algorithm removes the classical assumption that the time derivative of the output errors should be known. Simulation results of NSV show that the designed fault detection, isolation and estimation algorithms as well as the fault-tolerant control scheme have good dynamic performances in the presence of multiple actuator faults.
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Shen, Q., Jiang, B., Shi, P. (2017). Fuzzy Logic System-Based Adaptive FC for NSV Attitude Dynamics with Multiple Faults. In: Fault Diagnosis and Fault-Tolerant Control Based on Adaptive Control Approach. Studies in Systems, Decision and Control, vol 91. Springer, Cham. https://doi.org/10.1007/978-3-319-52530-3_3
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