Abstract
The problem of fault-tolerant control (FTC) for a class of uncertain nonlinear high-order systems with actuator faults is discussed, and an observer-based FTC scheme is proposed. Adaptive fuzzy observers are designed to provide a bank of residuals for fault detection and isolation (FDI). Using a backstepping approach, a novel fault diagnosis algorithm is proposed. Further, an accommodation scheme is proposed to compensate for the effect of the fault. The proposed controller guarantees that all signals of the closed-loop system are semi-globally uniformly ultimately bounded (SGUUB) and converge to a small neighborhood of the origin by appropriately choosing designed parameters. Finally, a numerical example and a practical aircraft longitudinal motion dynamics are used to demonstrate the effectiveness of the proposed FTC approach.
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Keywords
- Semi-globally Uniformly Ultimately Bounded (SGUUB)
- Actuator Faults
- FTC Scheme
- High-order Nonlinear Uncertain Systems
- Fault Tolerant Control Scheme
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.
6.1 Introduction
It is well known that system physical components may become faulty which may cause system performance deterioration or worth, may lead to instability that can further produce catastrophic accidents. The fault effects require to be compensated to enhance the reliability and safety of the system. Accommodating faults to maintain acceptable system performances is particularly important for life-critical systems. In order to improve system reliability and to guarantee system stability in all situations, many effective FTC approaches have been proposed the literature.
Fuzzy logic systems (FLSs), as universal function approximators, have been widely used to model the nonlinearities with arbitrary preciseness. Due to the capability, fuzzy logic systems are also adopted to solve identification and control problems in nonlinear systems [1,2,3,4,5,6]. Various adaptive fuzzy control approaches, based on the feedback linearization, were developed for controlling uncertain nonlinear systems. Robust adaptive backstepping control [1, 5,6,7,8,9,10] and observer-based backstepping control [11,12,13] attracted much attention from many researchers, and many excellent results were obtained during the past decades.
Recently, stable control problems of high-order systems attracted the interest of many researchers [14,15,16,17,18,19]. In [14], the authors presented a continuous feedback solution to the problem of global strong stabilization, for genuine nonlinear systems that may not be stabilized, even locally, by a smooth feedback. The same authors extended their results in [15], where they investigated the reference tracking problem in nonlinear systems with disturbances. However, the control schemes in [14, 15] do not guarantee the closed-loop systems’ stability or better tracking performance under faulty conditions.
In this chapter, we investigate the problem of active FTC for a class of high-order nonlinear uncertain systems with actuator gain faults. Compared with some existing works, the following main contributions are worth to be emphasized:
(1) In literature, results concerning FTC in the literature like [20,21,22,23,24,25,26,27,28,29,30,31] consider the 1-order systems. This chapter extends the results to the more general systems, i.e., so-called high-order systems as [32,33,34,35,36,37], and an observer-based active fault-tolerant backstepping control scheme is proposed.
(2) Differing from the classical backstepping technology, our fault-tolerant control scheme does not need computing the high order derivatives of virtual control signal at each step of backstepping design procedure, which thus reduces the computation complexity.
(3) In general, the denominator of the fault-tolerant control law contains the estimate of the gain fault. If the denominator equals zero, a singularity occurs. In the proposed FTC scheme, the controller singularity is avoided without using a projection algorithm.
(4) In contrast with [20,21,22,23,24,25], the proposed FTC scheme does not require the a priori knowledge of the signs of control gain terms.
The rest of this chapter is organized as follows. In Sect. 6.2, the problem formulation, Nussbaum-type function and mathematical description of FLS, are introduced. Actuator faults are described and the FTC objectives are formulated. In Sect. 6.3, the main technical results of this chapter are given, which include fault detection, isolation, estimation and fault-tolerant control scheme design. The aircraft control application is presented in Sect. 6.4 and simulation results are given and demonstrate the effectiveness of the proposed technique. Finally, Sect. 6.5 draws the conclusion.
6.2 Problem Formulation and Mathematical Description of FLSs
In this section, we will formulate control problem. Then, the FLS description is introduced.
6.2.1 Problem Statement
Considers the following nonlinear systems:
where \(x={{[{{x}_{1}},{{x}_{2}},\ldots ,{{x}_{n}}]}^{T}}\in {{R}^{n}}\) denotes the state vector, \(y={{x}_{1}} \) denotes the system output, \({{u}_{j}}\in R,j=1,2,\ldots ,m\) denote control inputs, \(p\ge 1\) is a known positive odd number, \(f(x)\in R\) denotes an unknown continuous smooth function, \({{g}_{j}}(x)\in R,j=1,\ldots ,m\) are complete unknown control gain functions, i.e., the value and sign of \({{g}_{j}}(x)\) are both unknown.
Remark 6.1
System (6.1) is more general than the considered system in [18] which was described as \({{\dot{x}}_{i}}=x_{_{i+1}}^{p},\) \( i=1,\ldots ,n-1\) and \({{\dot{x}}_{n}}={{u}^{p}}\). In addition, since actuator faults were not considered in [18], only one actuator was used. In this chapter, the FTC problem will be considered. In order to ensure the dependability of the controlled system, redundant actuators are added which leads to an over-actuated system.
In practical application, actuators may become faulty. In this chapter, actuator loss-of-effectiveness failures are considered, which can be modeled as follows.
where unknown function \({{k}_{j}}(x)\) denotes the remaining control rate, \({{t}_{j}}\) is unknown fault occurrence time.
The control objectives, which are valid in normal (no fault) and faulty conditions, are to design the proper control inputs \(u={{[{{u}_{1}},\ldots ,{{u}_{m}}]}^{T}}\) which ensure that the system output can track asymptotically the reference model signal \({{y}_{d}}\) with the tracking error converging to a small neighborhood of the origin and the closed-loop system is uniformly ultimately bounded (SGUUB). Under normal condition (no fault), u is designed to ensure boundedness of the closed-loop signals and asymptotic stability. Meanwhile, the FDI algorithm is working. As soon as actuator faults are detected and isolated, the fault accommodation algorithm is activated and a proper FTC input u is used such that the tracking performance is still maintained stable under faulty situation.
In order to design an appropriate controller, the following lemmas are introduced.
Lemma 6.1
([38]) \(\forall q>1\), being an odd integer, \(a,b\in R\), the following inequality holds:
Lemma 6.2
([38]) \(\forall m>0\in R,\forall n>0\in R\) and \(r(x,y)>0\in R\), the following inequality holds:
Lemma 6.3
([11]) For \(\alpha \in {{R}^{{{n}_{a}}}}\), \(\beta \in {{R}^{{{n}_{b}}}}\), \(M\in {{R}^{{{n}_{a}}\times {{n}_{b}}}}\), and arbitrary matrices \(X\in {{R}^{{{n}_{a}}\times {{n}_{a}}}}\), \(Y\in {{R}^{{{n}_{a}}\times {{n}_{b}}}}\), \(Z\in {{R}^{{{n}_{b}}\times {{n}_{b}}}}\), if \(\left[ \begin{matrix} X &{} Y \\ {{Y}^{T}} &{} Z \\ \end{matrix} \right] >0\), then
6.2.2 Nussbaum Type Gain
Any continuous function \(N(s):R\rightarrow R\) is a function of Nussbaum type if it has the following properties:
For example, the continuous functions \({{\varsigma }^{2}}\cos (\varsigma )\), \({{\varsigma }^{2}}\sin (\varsigma )\), and \({{e}^{{{\varsigma }^{2}}}}\cos ((\pi /2)\varsigma )\) verify the above properties and are thus Nussbaum-type functions [39]. The even Nussbaum function \({{e}^{{{\varsigma }^{2}}}}\cos ((\pi /2)\varsigma )\) is used throughout this chapter.
Lemma 6.4
([40, 41]) Let \(V(\cdot )\) and \(\varsigma (\cdot )\) be smooth functions defined on \([0,{{t}_{f}})\) with \(V(t)\ge 0,\forall t\in [0,{{t}_{f}})\), and \(N(\cdot )\) be an even smooth Nussbaum-type function. If the following inequality holds:
where \(\underline{g}\ne 0\) is a constant, and \({{c}_{0}}\) represents a suitable constant, then \(V(t),\varsigma (t)\) and \(\int _{0}^{t}{\underline{g}N(\varsigma )\dot{\varsigma }}d\tau \) must be bounded on \([0,{{t}_{f}})\).
Lemma 6.5
([41]) Let \(V(\cdot )\) and \(\varsigma (\cdot )\) be smooth functions defined on \([0,{{t}_{f}})\) with \(V(t)\ge 0,\forall t\in [0,{{t}_{f}})\), and \(N(\cdot )\) be an even smooth Nussbaum-type function. For \(\forall t\in [0,{{t}_{f}})\), if the following inequality holds,
where constant \({{c}_{1}}>0\), \(\underline{g}(\cdot )\) is a time-varying parameter which takes values in the unknown closed intervals \(I:=[{{l}^{-1}},{{l}^{+1}}]\) with \(0\notin I\), and \({{c}_{0}}\) represents some suitable constant, then V(t), \(\varsigma (t)\) and \(\int _{0}^{t}{\underline{g}(\tau )N(\varsigma )\dot{\varsigma }}d\tau \) must be bounded on \([0,{{t}_{f}})\).
6.2.3 Mathematical Description of FLSs
A fuzzy logic system 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 \(\underline{x}={{\left[ {{x}_{1}},\ldots ,{{x}_{n}} \right] }^{T}}\subset {{R}^{n}}\) 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 [42], 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 \(\xi (x)=[{{\xi }_{^{1}}}(x),\ldots ,{{\xi }_{^{M}}}(x)]^{T}\), then the above FLS can be rewritten as:
The stability results obtained in FLS control literature are semi-global in the sense that, as long as the input variable of the FLS remains within some pre-fixed compact set, where the compact set can be made as large as desired, there exist controllers with sufficiently large number of FLS rules such that all the signals in the closed-loop remain bounded.
Lemma 6.6
([5, 6]) Let f(x) be a continuous function defined on a compact set \(\varOmega \). Then for any constant \(\varepsilon >0\), there exists a FLS such as
In this chapter, using FLS, the unknown functions f(x), \({{g}_{j}}(x)\) and \({{g}_{kj}}(x),j=1,2,\ldots ,m\), are approximated as
Let define the optimal parameter vector \(\theta _{f}^{*},\theta _{gj}^{*}\) and \(\theta _{gkj}^{*}\) as
where \({{\varOmega }_{f}},{{\varOmega }_{gj}},{{\varOmega }_{gkj}}\), U and \(\hat{U}\) are compact regions for \({{\hat{\theta }}_{f}},{{\hat{\theta }}_{gj}},{{\hat{\theta }}_{gkj}}\), x and \(\hat{x}\), respectively; \({{\hat{\theta }}_{f}},{{\hat{\theta }}_{gj}},{{\hat{\theta }}_{gkj}}\) and \(\hat{x}\) are the estimates of \(\theta _{f}^{*},\theta _{gj}^{*},\theta _{gkj}^{*}\) and x, respectively. Similar to [11,12,13], The FLS minimum approximation errors and actual approximation errors are defined as
Now, the following assumptions are made.
Assumption 6.1
There exist unknown positive real constants \(\varepsilon _{^{f}}^{*},\delta _{^{f}}^{*},\varepsilon _{^{gj}}^{*},\delta _{^{gj}}^{*},\varepsilon _{^{gkj}}^{*},\delta _{^{gkj}}^{*}\) and known positive real constants \({{\bar{M}}_{\varepsilon f}},{{\bar{M}}_{\delta f}},\) \({{\bar{M}}_{\varepsilon gj}}\), \({{\bar{M}}_{\varepsilon gkj}}\), such that \(|{{\varepsilon }_{f}}|\le \varepsilon _{^{f}}^{*},\varepsilon _{^{f}}^{*}\le {{\bar{M}}_{\varepsilon f}},|{{\delta }_{f}}|\le \delta _{^{f}}^{*},\delta _{^{f}}^{*}\le {{\bar{M}}_{\delta f}},\) \(|{{\varepsilon }_{gj}}|\le \varepsilon _{^{gj}}^{*}\), \(\varepsilon _{^{gj}}^{*}\le {{\bar{M}}_{\varepsilon gj}}\), \(|{{\varepsilon }_{gkj}}|\le \varepsilon _{gkj}^{*},\varepsilon _{^{gkj}}^{*}\le {{\bar{M}}_{gkj}}\).
Assumption 6.2
There exist known positive real constants \({{M}_{\theta f}}\), \({{M}_{\theta gj}}\) and \({{M}_{gkj}}\) such that \(||\theta _{_{f}}^{*}||\le {{M}_{\theta f}}\), \(||\theta _{_{gj}}^{*}||\le {{M}_{\theta gj}}\) and \(||\theta _{_{gkj}}^{*}||\le {{M}_{\theta gkj}}\).
In order to facilitate the descriptions, in the following, f(x), g(x), \({{g}_{kj}}(x)\), \(\hat{f}(\hat{x})\), \(\hat{g}(\hat{x})\), \({{\hat{g}}_{kj}}(\hat{x})\), \({{\xi }_{f}}(\hat{x})\), \({{\xi }_{gj}}(\hat{x})\) and \({{\xi }_{gkj}}(\hat{x})\) are abbreviated to f, g, \({{g}_{kj}}\), \(\hat{f}\), \(\hat{g}\), \({{\hat{g}}_{kj}}\), \({{\xi }_{f}}\), \({{\xi }_{gj}}\) and \({{\xi }_{gkj}}\), respectively.
6.3 Main Results
In this section, the main technical results of this chapter are given. We will first consider the stability control problem of system (6.1) under normal conditions, design a bank of observers to generate residuals, investigate the FDI algorithm based on the observers, and propose a FTC scheme to tolerate the fault using estimated fault information.
6.3.1 Fault Detection
In order to detect the fault, the following observer is constructed.
where \({{l}_{i}},i=1,\ldots ,n\) are constant parameters that will be designed later.
Let \(\hat{x}={{[{{\hat{x}}_{1}},{{\hat{x}}_{2}},\ldots ,{{\hat{x}}_{n}}]}^{T}}\) and define observer errors \({{e}_{i}}={{x}_{i}}-{{\hat{x}}_{i}},i=1,\ldots ,n\), then observer error dynamics can be described as follows:
Using the notation \(e=x-\hat{x}\), the above error dynamics can be re-written as:
where \({{e}_{p}}={{[e_{1}^{p},\ldots ,e_{n}^{p}]}^{T}}\), \({{d}_{i}}=\sum \nolimits _{l=1}^{p}{C_{p}^{l}e_{i+1}^{l}\hat{x}_{i+1}^{p-l}}\), \(i=1,\ldots n-1\), \({{d}_{f}}=f-\hat{f}={{\delta }_{f}}\), \({{d}_{g}}=\sum \nolimits _{j=1}^{m}{({{g}_{j}}}-{{\hat{g}}_{j}}-{{\hat{\varepsilon }}_{gj}})u_{^{i}}^{p}\), and
In the following we will use the backstepping technique to design the fault-tolerant controller.
Define
where \({{\alpha }_{0}}=0,{{z}_{n+1}}=0\), and \({{\alpha }_{i-1}},i=1,\ldots ,n-1\) are virtual controls which will be designed at each step, \({{\alpha }_{n}}=u\) is the actual control input. The recursive design procedure contains n steps. From Step 1 to Step \(n-1\), virtual control \({{\alpha }_{i-1}}\) is designed at each step. Finally an overall control law u is constructed at step n.
Step 1:
From \({{z}_{1}}={{\hat{x}}_{1}}-{{y}_{d}}\), one has
Define
where \(P={{P}^{T}}>0\) denotes a matrix with appropriate dimensions. Differentiating \({{V}_{11}}\)with respect to time t leads to
Notice that, \(p+1\ge 2\) is an even number. Differentiating \({{V}_{e}}\) with respect to time t, from Lemma 6.3, it leads to
where X, Y, Z denote matrices with appropriate dimensions, and \(\left[ \begin{matrix} X &{} Y \\ {{Y}^{T}} &{} Z \\ \end{matrix} \right] >0\).
From Lemma 6.2, one has
where \({{w}_{e1}}=\left[ \sum \nolimits _{k=1}^{p}{C_{p}^{k}\frac{k}{p}\sigma } \right] ,{{w}_{e2}}=\left[ \sum \nolimits _{k=1}^{p}{C_{p}^{k}\frac{p-k}{p}{{\sigma }^{-\left( \frac{k}{p-k} \right) }}} \right] \).
Define
where \(\lambda >1\) is a design parameter. Since \(0<\sigma \le 1\), one has \(\text { }{{w}_{e1}}|{{e}_{2}}{{|}^{p}}\le \frac{1}{\lambda }|{{e}_{2}}{{|}^{p}}\). Therefore,
Further one has
Similarly, one has
Hence,
From Young’s inequality, one has
Further, one has
where \({{\bar{\varDelta }}_{\text {0}}}=-{{({{e}_{1}}^{p})}^{2}}+\sum \nolimits _{i=2}^{n}{2{{({{w}_{e2}})}^{2}}{{(|{{{\hat{x}}}_{i}}{{|}^{p}})}^{2}}}+{{({{\bar{M}}_{\delta f}})}^{2}}\), I denotes identity matrix with appropriate dimensions.
Hence, one has
Obviously, if matrices X, Y, Z, \(Q>0\) and \(P={{P}^{T}}>0\) are chosen appropriately such that \(\left[ \begin{matrix} X &{} Y \\ {{Y}^{T}} &{} Z \\ \end{matrix} \right] >0\) and
where I denotes identity matrix with appropriate dimensions, then,
Let \({{\varDelta }_{\text {0}}}={{z}_{1}}{{l}_{1}}(y-\hat{y})-{{z}_{1}}{{\dot{y}}_{d}}+{{\bar{\varDelta }}_{\text {0}}}\), one has
Thus, virtual control \({{\alpha }_{\text {1}}}\) can be modified as
Remark 6.2
In general, virtual control \({{\alpha }_{\text {1}}}\) can be chosen as follows
Just as pointed out in [41], for the above virtual control (6.23), controller singularity may occur since \(\frac{{{\varDelta }_{0}}}{{{z}_{1}}}\) is not well defined at \({{z}_{1}}=0\). Therefore, care must be taken to guarantee the boundedness of the control. It is noted that the controller singularity takes place at the point \({{z}_{1}}=0\), where the control objective is supposed to be achieved. From a practical point of view, once the system reaches its origin, no control action should be taken for less power consumption. As \({{z}_{1}}=0\) is hard to detect owing to the existence of measurement noise, it is more practical to relax our control objective of convergence to a “ball” rather than to the origin.
Similar to [41], let define \({{\varOmega }_{{{c}_{{{z}_{i}}}}}}\subset \varOmega \) and \(\varOmega _{{{c}_{{{z}_{i}}}}}^{0}\) s.t.
where \({{c}_{{{z}_{i}}}}>0\) is a constant that can be chosen arbitrarily small and “-" is used to denote the complement of set B in set A as \(A-B:=\{x|x\in A\text { and }x\notin B\}\). Thus, virtual control \({{\alpha }_{\text {1}}}\) can be modified as (6.23).
Step 2.
Since \({{z}_{2}}={{\hat{x}}_{2}}-{{\alpha }_{1}}\), one has
Define
Differentiating \({{V}_{2}}\) with respect to time t, leads to
Let
Similarly, choose a virtual control as follows
Substituting \({{\alpha }_{\text {2}}}\) into (6.27), it yields
Step k:
Since \({{z}_{k}}={{\hat{x}}_{k}}-{{\alpha }_{k-1}}\), one has
Define
Differentiating \({V_k}\) with respect to time t, leads to
where
Just as \({\alpha _{k - 1}}\), virtual control \({\alpha _k}\) is chosen as follows
Substituting \({\alpha _k}\) into (6.28), yields
Step n:
Since \({z_n} = {\hat{x}_n} - {\alpha _{n - 1}}\), one has
Define
where \(\gamma _f^* = \varepsilon _f^* + \delta _f^*,{\tilde{\gamma }_f} = \gamma _f^* - {\hat{\gamma }_f}\), \({\tilde{\theta }_f} = \theta _f^* - {\hat{\theta }_f},{\tilde{\gamma }_f} = \gamma _f^* - {\hat{\gamma }_f},{\tilde{\theta }_{gj}} = \theta _{gj}^* - {\hat{\theta }_{gj}},{\tilde{\varepsilon }_{gj}} = \varepsilon _{gj}^* - {\hat{\varepsilon }_{gj}}\), \({\hat{\theta }_f},{\hat{\gamma }_f}\), \({\hat{\theta }_{gj}},{\hat{\varepsilon }_{gj}}\) are the estimates of \(\theta _f^*,\gamma _f^*\), \(\theta _{gj}^*,\varepsilon _{gj}^*\), and \({\eta _1}> 0,{\eta _2}> 0,{\eta _3} > 0\) are adaptive rates.
Differentiating \({V_n}\) with respect to time t, leads to
where
Since
from the above inequality, one has
Choose control law \({\alpha _{n,i}},i = 1,2, \ldots ,m\) and adaptation functions \({\dot{\hat{\theta }} _f},{\dot{ \hat{ \gamma }} _f}\), \({\dot{\hat{\theta }} _{gj}},{\dot{ \hat{\varepsilon }} _{gj}}\) as follows:
where \(\dot{\varsigma }= - \frac{1}{2}z_n^2 - {\varDelta _{n - 1}}\),
and \({\eta _f}> 0,{\eta _\gamma }> 0,{\eta _{gj}} > 0\) are design parameters, \({u_j}\) is a bounded control input which is applied simultaneously to the ith actuator in the system (6.1) and the observer (6.13).
Applying Young’s inequality, one has
Substituting the above inequalities into (6.36), it yields
where
The above control design procedures and analysis are summarized in the following theorem.
Theorem 6.1
Consider nonlinear system (6.1) under Assumptions 6.1 and 6.2, control law (6.37) and adaptive laws (6.38–6.41). If matrices X, Y, Z, \(Q > 0\) and \(P = {P^T} > 0\) are such that \(\left[ {\begin{array}{*{20}{c}} X&{}Y \\ {{Y^T}}&{}Z \end{array}} \right] > 0\) and
we can guarantee the following properties under bounded initial conditions
(1) all signals in the closed-loop system are semi-globally uniformly ultimately bounded;
(2) the vectors \({z_i}\) remain in the compact set \(\varOmega _{{z_i}}^0\), \(i = 1,2, \ldots ,n\) specified as
whose size is \(\bar{\mu }= \frac{\mu }{g} + {c_g} + {V_n}(0) > 0\), which can be adjusted by appropriately choosing the design parameters \({\eta _1},{\eta _2},{\eta _3},{\eta _f},{\eta _\gamma },{\eta _{g,1}}, \ldots ,{\eta _{g,m}}\).
Proof
Since \({\dot{V}_n} \leqslant - g{V_n} + \mu + hN(\varsigma ) + 1)\dot{\varsigma }\), one has
Applying Lemma 6.5, we can conclude that, \({V_n}(t)\), \(\int \limits _0^t {(hN(\varsigma ) + 1){e^{ - g\tau }}\dot{\varsigma }d\tau } \) and \(\varsigma (t)\) are SGUUB on \([0,{t_f})\). According to Proposition 2 in [39], if the solution of the closed-loop system is bounded, then \({t_f} = + \infty \). Let \({c_g}\) be the upper bound of \(\int \limits _0^t {(hN(\varsigma ) + 1){e^{ - g\tau }}\dot{\varsigma }d\tau } \), we have the following inequalities:
Thus, (6.44) becomes
Hence, if matrices X, Y, Z, Q and positive definite symmetric matrices P are chosen appropriately such that \(\left[ {\begin{array}{*{20}{c}} X&{}Y \\ {{Y^T}}&{}Z \end{array}} \right] > 0\) and (6.38) holds, then, the proposed control input (6.37) can ensure that \({V_n}(t)\) is bounded, namely, the closed-loop system is semi-globally uniformly ultimately bounded. Noting the definitions of \({V_n}(t)\) and \({z_i},i = 1,2, \ldots ,n\), we have \(\frac{1}{2}z_i^2 \leqslant {V_n}(t) \leqslant \bar{\mu }\) and \(\frac{1}{{2{\eta _1}}}\tilde{\theta }_f^T{\tilde{\theta }_f} \leqslant \bar{\mu }\). Furthermore, we have \(|{z_i}| \leqslant \sqrt{2\bar{\mu }} ,||{\tilde{\theta }_f}|| \leqslant \sqrt{2{\eta _1}\bar{\mu }} \). Similarly, we have \(|{\tilde{\gamma }_f}| \leqslant \sqrt{2{\eta _2}\bar{\mu }} ,||{\tilde{\theta }_{g,i}}|| \leqslant \sqrt{2{\eta _3}\bar{\mu }} ,|{\tilde{\varepsilon }_{g,i}}| \leqslant \sqrt{2{\eta _3}\bar{\mu }} ,||e|| \leqslant \sqrt{\frac{{\bar{\mu }}}{{{\lambda _{\min }}(P)}}} \). From the above analysis, we can conclude that there do exist compact sets \(\varOmega _{{z_i}}^0\) such that \({z_i} \in \varOmega _{{z_i}}^0,\forall t \geqslant 0\). The proof is completed.
From Theorem 6.1, one has
Furthermore, the detection residual can be defined as
From (6.46), 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:
6.3.2 Fault Isolation and Estimation
Since the system has m actuators and it is assumed that only one actuator becomes faulty at one time, we have m possible faulty cases in total. When the sth (\(1 \leqslant s \leqslant m\)) actuator is faulty, the faulty model can be described as:
The faulty system (6.1) can be described as follows:
After a fault has been detected, the isolation scheme is activated. Now, the following m nonlinear fault isolation observers are designed as follows:
where \({{l}_{si}},i=1,2,\ldots ,n,s=1,2\cdots .m\) are constants, which will be designed later, \(\hat{\theta }_{g\rho ,r}^{T}{{\xi }_{g\rho ,r}}({{\hat{x}}_{s}},v)\) is the estimate of \({{g}_{r}}(x,v)\rho _{r}^{p}({{x}_{r}})\), \(r=1,\ldots ,m\).
Let \({{\hat{x}}_{s}}={{[{{\hat{x}}_{s,1}},{{\hat{x}}_{s,2}},\ldots ,{{\hat{x}}_{s,n}}]}^{T}}\), the error terms \({{e}_{s}}={{x}_{s}}-{{\hat{x}}_{s}}\) and \({{e}_{ys}}={{y}_{s}}-{{\hat{y}}_{s}}\) are respectively the state error and output error between the faulty plant and the sth observer. The above error dynamics can be re-written as:
where \({{e}_{sp}}={{[e_{s,1}^{p},\ldots ,e_{s,n}^{p}]}^{T}}\), \({{d}_{f}}=f-\hat{\theta }_{f}^{T}{{\xi }_{f}}\), \({{\rho }_{s}}={{g}_{s}}k_{s}^{p}u_{s}^{p}-[\hat{\theta }_{gkr}^{T}{{\xi }_{gkr}}+{{\hat{\varepsilon }}_{gkr}}]u_{r}^{p}\), \({{d}_{g}}= \) \( \sum \nolimits _{{\begin{matrix} i=1 \\ i\ne s,i\ne r \end{matrix}}}^{m}{({{g}_{j}}-{{{\hat{g}}}_{j}}-{{{\hat{\varepsilon }}}_{gj}})u_{j}^{p}}\) and
Similar to the previous subsection, differentiating \({{V}_{se}}=e_{s}^{T}{{P}_{s}}{{e}_{s}}\) with respect to time t and using (6.20) and (6.54), it leads to
From Young’s inequality, one has
Furthermore, one has
where \({{\varDelta }_{\text {0}}}=-{{(e_{s,1}^{p})}^{2}}+\sum \nolimits _{i=2}^{n}{2{{({{w}_{e2}})}^{2}}{{(|{{{\hat{x}}}_{s,i}}{{|}^{p}})}^{2}}}+{{({{\bar{M}}_{\delta f}})}^{2}}\).
In the following, stability analysis will be given at two cases, i.e., \(s=r\) or \(s\ne r\).
Case 1: \(s=r\)
Since
where \({{P}_{sn}}\) is the nth column of \({{P}_{s}}\).
Similar to the above subsection, define
and choose a virtual control \({{\alpha }_{s,i}},i=1,2,\ldots ,n-1\) and practical control \({{\alpha }_{s,nj}},j=1,\ldots ,m\) as follows
where \(\dot{\varsigma }=-\frac{1}{2}z_{s,n}^{2}-{{\varDelta }_{n-1}}\), \({{\varOmega }_{{{c}_{s,{{z}_{i}}}}}},i=1,\ldots ,n\) are defined as \({{\varOmega }_{{{c}_{{{z}_{k}}}}}}\) in the previous subsection. The adaptive laws are designed as follows:
where \({{u}_{j}}\) is a bounded control input which is applied simultaneously to the jth actuator in the system (6.1) and the observer (6.53), and \({{\eta }_{1}}>0\), \({{\eta }_{2}}>0\), \({{\eta }_{3}}>0,{{\eta }_{4}}>0,{{\eta }_{f}}>0,{{\eta }_{\gamma }}>0,{{\eta }_{gks}}>0,{{\eta }_{gj}}>0,{{\eta }_{gks}}>0\) are design parameters.
Define
Similar to the previous subsection, differentiating \({{V}_{s}}\) with respect to time t, one has
It is obvious that if
where X, Y, Z denote matrices with appropriate dimensions, respectively, and \(\left[ \begin{matrix} X &{} Y \\ {{Y}^{T}} &{} Z \\ \end{matrix} \right] >0\), matrix \({{Q}_{s}}>0\), then from (6.69), one has
Similar to (6.42) in the above subsection, considering (6.62–6.67), from (6.71), one has
where
Since \({{\dot{V}}_{s}}\le -{{g}_{s}}{{V}_{s}}+{{\bar{\mu }}_{s}}+(\bar{h}(\hat{x})N(\varsigma )\dot{\varsigma }+\dot{\varsigma })\), one has
Applying Lemma 6.5, we can conclude that, \({{V}_{n}}(t)\), \(\int \limits _{0}^{t}{(\bar{h}(\hat{x})N(\varsigma )+1)\dot{\varsigma }{{e}^{-g\tau }}\dot{\varsigma }d\tau }\) and \(\varsigma (t)\) are SGUUB on \([0,{{t}_{f}})\). According to Proposition 2 in [39], if the solution of the closed-loop system is bounded, then \({{t}_{f}}=+\infty \). Let \({{c}_{g}}\) be the upper bound of \(\int \limits _{0}^{t}{\bar{h}(\hat{x})(N(\varsigma )+1)\dot{\varsigma }{{e}^{-{{g}_{s}}\tau }}\dot{\varsigma }d\tau }\), we have the following inequalities:
Thus, (6.73) becomes
Hence, if matrices \({{X}_{s}}\), \({{Y}_{s}}\), \({{Z}_{s}},\) \({{Q}_{s}}\) and the positive definite symmetric matrix \({{P}_{s}}\) are chosen appropriately such that \(\left[ \begin{matrix} {{X}_{s}} &{} {{Y}_{s}} \\ Y_{s}^{T} &{} {{Z}_{s}} \\ \end{matrix} \right] >0\) and (6.74) holds, then, the proposed control input (6.61) and adaptive laws (6.62–6.67) can ensure that \({{V}_{s}}(t)\) is bounded, namely, the closed-loop system is semi-globally uniformly ultimately bounded. That is to say, all signals of the closed-loop system remain the following compact set \({{\varOmega }_{1}}\),
Case 2: \(s\ne r\)
Since \(s\ne r\), from the faulty (6.52) and the observer (6.53), one has
From the adaptive laws (6.64–6.67), one has
It is noted that \(2e_{s}^{T}{{P}_{s}}{{B}_{s}}[({{g}_{ks}}-{{\hat{g}}_{s}}-{{\hat{\varepsilon }}_{gs}})u_{s}^{p}+({{g}_{r}}-{{\hat{g}}_{kr}}-{{\hat{\varepsilon }}_{gkr}})u_{r}^{p}]\) varies infinitely since \({{\dot{\hat{\theta }}}_{gs}}\ne {{\dot{\hat{\theta }}}_{gks}},\) \({{\dot{\hat{\theta }}}_{gr}}\ne {{\dot{\hat{\theta }}}_{gkr}}\), \({{\dot{\hat{\varepsilon }}}_{gs}}\ne {{\dot{\hat{\varepsilon }}}_{gks}}\) and \({{\dot{\hat{\varepsilon }}}_{gr}}\ne {{\dot{\hat{\varepsilon }}}_{gkr}}\), which further cause that \({{V}_{s}}(t)\) varies infinitely. As a result, basically, all signals of the closed-loop systems such as \({{e}_{si}}\) do not remain \({{\varOmega }_{1}}\) using the above control law and adaptive laws.
The above design procedure and analysis are summarized in the following theorem.
Theorem 6.2
Consider the faulty system (6.52) under Assumptions 6.1 and 6.2, with virtual controls (6.58–6.60), control law (61) and adaptive laws (6.62–6.67). If matrices \({{X}_{s}}\), \({{Y}_{s}}\), \({{Z}_{s}}\), \({{Q}_{s}}>0\) and \({{P}_{s}}=P_{s}^{T}>0\) are such that \(\left[ \begin{matrix} {{X}_{s}} &{} {{Y}_{s}} \\ Y_{s}^{T} &{} {{Z}_{s}} \\ \end{matrix} \right] >0\) and
then, we can guarantee the following properties under bounded initial conditions, when the rth actuator is faulty,
(1) for \(s=r\), the closed-loop system is semi-globally uniformly ultimately stable, and all signals involved in the closed-loop systems remain a small neighborhood of the origin, i.e., \({{\varOmega }_{1}}\) specified as
(2) \(s\ne r\), all signals of the closed-loop systems do not remain the compact set \({{\varOmega }_{1}}\).
Remark 6.3
It is valuable to point out that, if the design parameters such as \({{\eta }_{i}},i=1,\ldots ,4\), \({{\eta }_{f}},{{\eta }_{\gamma }},{{\eta }_{gks}},{{\eta }_{gj}}\), \(j=1,\ldots ,m\) are appropriately chosen, \({{\mu }_{s}}\) is small enough, and all signals of the closed-loop system converge to a smaller neighborhood of the origin, which means that better control performance is obtained.
Now, we denote the residuals between the real system and isolation estimators as follows:
According to Theorem 6.2, when the rth actuator is faulty, i.e., \(s=r\), the residual \({{e}_{s}}(t)\) must tend to \({{\varOmega }_{1}}\); while for any \(s\ne r\), basically, \({{e}_{s}}(t)\) does not belong to \({{\varOmega }_{1}}\). Hence, the isolation law for actuator fault can be designed as
where threshold \({{T}_{I}}\) is defined as follows.
6.3.3 Fault Accommodation
After that the fault information is obtained, we will consider the fault-tolerant control problem of system (6.1), and design a fault-tolerant control law to recover the control system’s dynamics performance when an actuator fault occurs. Firstly, we consider the fuzzy control problem for the following nominal system without actuator faults:
Consider matrices X, Y, Z, \(Q>0\) and \(P={{P}^{T}}>0\) such that \(\left[ \begin{matrix} X &{} Y \\ {{Y}^{T}} &{} Z \\ \end{matrix} \right] >0\) and
virtual control laws (6.58–6.60), control input (6.61) and adaptive laws (6.62–6.67).
From Theorem 6.1, under Assumptions 6.1 and 6.2, the closed-loop system is semi-globally uniformly ultimately stable, and all signals involved in the closed-loop systems converge to a small neighborhood of the origin.
On the basis of the estimated actuator fault, the fault tolerant controller is constructed as
where \({{\varepsilon }_{u}}>0\) is a design parameter, \(u_{s}^{N}\) is the sth desired control input under healthy condition, \({{\hat{\rho }}_{s}}\) is the estimate of \({{g}_{s}}{{k}_{s}}\), which is used to compensate for the gain fault \({{k}_{s}}\).
Theorem 6.3
Consider the high-order system (6.1) under Assumptions 6.1 and 6.2, fault model (6.2), virtual and practical control laws (6.58–6.61) and adaptive laws (6.62–6.67). If there exist matrices X, Y, Z, \(Q>0\) and \(P={{P}^{T}}>0\) with appropriate dimensions, such that \(\left[ \begin{matrix} X &{} Y \\ {{Y}^{T}} &{} Z \\ \end{matrix} \right] >0\) and
then, the faulty system (6.1) is asymptotically stable under the feedback FTC (6.79) and all signals involved in the closed-loop system are semi-globally uniformly ultimately bounded, converging asymptotically to a small neighborhood of zero, i.e. \(||{{\tilde{\theta }}_{f}}||\le \sqrt{2{{\eta }_{sf}}{{\mu }_{s}}}\), \(||{{\tilde{\theta }}_{gj}}||\le \sqrt{2{{\eta }_{gj}}{{\mu }_{s}}}\), \(||{{\tilde{\theta }}_{g\rho ,s}}||\le \sqrt{2{{\eta }_{gks}}{{\mu }_{s}}}\), \(||e||\le \sqrt{\frac{2{{\mu }_{s}}}{{{\lambda }_{\min }}({{P}_{s}})}}\), where
Proof
Similar to the proof of Theorem 6.1, it is easy to obtain the conclusions of Theorem 3. The detailed proof is thus omitted here.
6.4 Simulation Results
In this section, a practical aircraft longitudinal motion dynamics, which can be described as a 1-order nonlinear system, namely \(p=1\), and a high-order numerical example where \(p=3\), are taken to show the effectiveness of the proposed fault tolerant control scheme.
6.4.1 An Application to Aircraft Longitudinal Motion Dynamics
In this subsection, we apply the proposed FTC scheme to diagnose and accommodate failures in an aircraft longitudinal motion dynamics. The aircraft longitudinal motion dynamics of the twin otter [43] can be described as 1-order nonlinear system as follows:
where Vis the velocity, \(\alpha \)is the angle of attack, \(\theta \) is the angle of pitch andq is the pitch rate, m is the mass, \({{I}_{y}}\) is the moment of inertia, and \({{F}_{x}}=\bar{q}S{{C}_{x}}(\alpha ,q,{{\delta }_{e1}},{{\delta }_{e2}})+{{T}_{1}}\cos {{\gamma }_{1}}+{{T}_{2}}\cos {{\gamma }_{2}}-mg\sin (\theta )\), \({{F}_{z}}=\bar{q}S{{C}_{z}}(\alpha ,q,\) \({{\delta }_{e1}},{{\delta }_{e2}})+{{T}_{1}}\cdot \) \(\sin {{\gamma }_{1}}+{{T}_{2}}\sin {{\gamma }_{2}}-mg\cos (\theta )\), \(M=\bar{q}cS{{C}_{m}}(\alpha ,q,{{\delta }_{e1}},{{\delta }_{e2}})\), where \(\bar{q}=\frac{1}{2}\rho {{V}^{2}}\) is the dynamic pressure, \(\rho \) is the air density, S is the wing area, c is the mean chord, \({{T}_{1}}\) and \({{T}_{2}}\) are independent thrusts with corresponding thrust misalignments \({{\gamma }_{1}}\) and \({{\gamma }_{2}}\). The functions \({{C}_{x}},{{C}_{z}},{{C}_{m}}\) are of the polynomial form: \({{C}_{x}}={{C}_{x1}}\alpha +{{C}_{x2}}{{\alpha }^{2}}+\) \({{C}_{x3}}+{{C}_{x4}}\left( {{d}_{1}}{{\delta }_{e1}}+{{d}_{2}}{{\delta }_{e2}} \right) \),\({{C}_{z}}={{C}_{z1}}\alpha +{{C}_{x2}}{{\alpha }^{2}}+{{C}_{z3}}+{{C}_{z4}}\left( {{d}_{1}}{{\delta }_{e1}}+{{d}_{2}}{{\delta }_{e2}} \right) +{{C}_{x5}}q\), \({{C}_{m}}={{C}_{m1}}\alpha +{{C}_{m2}}{{\alpha }^{2}}+{{C}_{m3}}+\) \({{C}_{m4}}\left( {{d}_{1}}{{\delta }_{e1}}+{{d}_{2}}{{\delta }_{e2}} \right) +{{C}_{m5}}q\), where \({{\delta }_{e1}}\) and \({{\delta }_{e2}}\) are the elevator angles of an augmented two-pieces elevators used as two actuators \({{u}_{1}}\) and \({{u}_{2}}\) for failure compensation study. Choosing \(V,\alpha ,\theta \) and q as the states \({{x}_{1}},{{x}_{2}},{{x}_{3}}\) and \({{x}_{4}}\), and \({{\delta }_{e1}},{{\delta }_{e2}},{{T}_{1}},{{T}_{2}}\) as the inputs \({{u}_{1}},{{u}_{2}},{{u}_{3}},{{u}_{4}}\), (6.81) will be put into the state form:
where
and \(\theta ,{{p}_{1}},{{a}_{1}},{{a}_{2}},,{{b}_{1}},{{b}_{2}},{{c}_{1}},{{c}_{2}},{{d}_{1}},{{d}_{2}},{{\gamma }_{1}},{{\gamma }_{2}}\) are unknown constant parameters while \({{p}_{0}}\) is the gravity constant which is known. There exists a diffeomorphism \({{[\xi ,x]}^{T}}=T(\chi )=[{{T}_{1}}(\chi ),{{T}_{2}}(\chi ),{{x}_{3}},{{x}_{4}}{{]}^{T}}\) such that (6.82) can be transform into the parameter-strict-feedback form, where the positive odd number \(p=1\)
and the zero dynamics \(\dot{\xi }=\phi (\xi ,\chi )+\varPhi (\xi ,\chi )\vartheta \), where \(\vartheta \in {{R}^{4}}\) is an unknown constant vector. The relative degree o equals 2. The aircraft parameters in the simulation study are chosen based on the data sheet in [44]: \(m=4600\text { }kg\), \({{I}_{y}}=31027\text { }kg\text { }{{m}^{2}}\), \(S=39.2\text { }{{m}^{2}}\), \(c=1.98\text { }m\), \({{T}_{x}}=4864\text { }N\), \(\text { }{{T}_{z}}=212\text { }N\), \(\rho =0.7377\text { }kg/{{m}^{3}}\) at the altitude of 5000 m, and for the \({{\text {0}}^{\circ }}\) flap setting. In addition, \({{C}_{x1}}=0.39,{{C}_{x2}}=2.9099,{{C}_{x3}}=-0.0758,{{C}_{x4}}=0.0961\), \({{C}_{z1}}=-7.0186\), \({{C}_{z2}}=4.1109\), \({{C}_{z3}}=-0.3112\), \({{C}_{z4}}=\) \(-0.2340\), \({{C}_{z5}}=-0.1023,\) \({{C}_{m1}}\) \(=-0.8789\), \({{C}_{m2}}=-3.852\), \({{C}_{m3}}=\) \(-0.0108,{{C}_{m4}}=-1.8987\), \({{C}_{m5}}=-0.6266\) are unknown constants. Reference signal \({{y}_{d}}\)is set as \({{y}_{d}}={{e}^{-0.05t}}\cdot \) \(\sin (0.2t)\). The initial states and estimates are set as \(\chi (0)={{[75,0,0,15,0]}^{T}}={{e}^{-0.05t}}\sin (0.2t),\hat{\vartheta }(0)=\) \([0,0,-0.004,0]\). It is assumed that the zero dynamics \(\dot{\xi }=\phi (\xi ,\chi )+\varPhi (\xi ,\chi )\vartheta \) is input-to-state stable with respect tox taken as the input. In addition, \({{b}_{i}},i=1,\ldots ,m\) are assumed to be complete unknown, i.e., these values and signs are both unknown.
The fault case considered in this example is modeled as
where \({{\rho }_{1}}(x)=0.4\cos ({{x}_{3}})\).
Firstly, the matrices inequality (6.43) are transformed to LMI, then by using Matlab toolbox to solve the matrices inequalities, one can obtain symmetric matrix \(X,Y,Z,P,Q,{{X}_{s}},{{Y}_{s}},{{Z}_{s}},{{P}_{s}},{{Q}_{s}}\) and the nominal controller gains \({{K}_{i}}\). Therefore, one can design the desired control (6.37). Using this desired control, we can design fault-tolerant controller (6.79). In this example, we assume that the system state is not fully measured and thus the observer (6.53) is used to estimate the system state. Consequently, the observer-based fault-tolerant control (6.79) is used to control the faulty system. The simulation results are presented in Figs. 6.1, 6.2, 6.3, 6.4, 6.5 and 6.6. From Figs. 6.1 and 6.2, it is seen that, under normal operating condition, the system states globally asymptotically converge to a small neighborhood of the origin. Figures 6.3 and 6.4 show that, when an actuator fault occurs, when keeping the normal controller, the system states deviate significantly from the neighborhood. However, as shown in Figs. 6.5 and 6.6, using the proposed FTC (6.79), better tracking performance is obtained, again.
Time response of the velocity without fault
Time response of the attack of angle, the pitch angle and the pitch rate without fault
Time response of the velocity without FTC
Time response of the attack of angle, the pitch angle and the pitch rate without FTC
Time response of the velocity with FTC
Time response of the attack of angle, the pitch angle and the pitch rate with FTC
State response under normal condition
State response under faulty condition without FTC
6.4.2 A High-Order Numerical Example
Consider the following high-order nonlinear system
The fault case considered in this example is modeled as
where \({{\rho }_{1}}(x)=0.8\cos (2+{{x}_{1}}+{{x}_{2}})\), the fault occurs at time \(t=10s\). As expected, we can find that system output y follows well \({{y}_{d}}=0\) as shown in Fig. 6.7. Meanwhile, Figs. 6.8 and 6.9 illustrate that, under the faulty condition, the system output y does not converge to the desired reference signal without FTC, however, using FTC, the system has better tracking performance.
State response under faulty condition with FTC
6.5 Conclusions
In this chapter, the fault-tolerant control problem for a class of uncertain nonlinear systems in presence of actuator faults is discussed. We first design a bank of observers to detect, isolate and estimate the fault. Then a sufficient condition for the existence of an FDI observer is derived. Simulation show that the designed fault detection, isolation and estimation algorithms and fault-tolerant control scheme have better dynamic performances in the presence of actuator faults.
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Shen, Q., Jiang, B., Shi, P. (2017). Adaptive Fault Tolerant Backstepping Control for High-Order Nonlinear Systems. 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_6
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