Abstract
This chapter investigates the problem of fuzzy adaptive tracking control for a class of uncertain nonlinear strict-feedback systems with actuator fault. The actuator fault is assumed to have not only time-varying gain fault but also time-varying bias fault. Combining command filtered backstepping design with the integral-type Lyapunov function and utilizing Nussbaum-type gain technique, an adaptive fuzzy fault-tolerant control scheme is proposed to guarantee that the resulting closed-loop system is asymptotically bounded with the tracking error converging to a neighborhood of the origin. The control scheme requires only virtual control and its first one derivative instead of them and their higher derivatives in backstepping design procedures. Simulation results demonstrate the effectiveness of the proposed techniques.
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4.1 Introduction
Fuzzy control has found extensive applications for modeling nonlinear systems in the past 10 years. According to the fuzzy approximation theorem of the fuzzy logic systems (FLSs) [1,2,3,4,5,6], researchers proposed many approximation-based adaptive fuzzy control design methods for nonlinear systems (see, e.g., [7,8,9,10,11,12] and the references therein).
It has been proved that adaptive backstepping technique is a powerful tool to solve tracking or regulation control problems of unknown nonlinear systems in or transformable to parameter strict-feedback form [13]. For such systems, many adaptive fuzzy backstepping controllers have been developed (see, e.g., [14,15,16,17,18,19] and the references therein), where FLSs or neural networks are used to approximate unknown nonlinear smooth functions. It is well known that, however, in standard backstepping design procedure, analytic computation of the first derivatives of virtual control signals \({{\alpha }_{i}}\) \((i=1,2,\ldots ,n-1)\), i.e., \({{\dot{\alpha }}_{i}}\), is necessary. Note that, the computation of \({{\dot{\alpha }}_{i}}\) requires the higher derivatives of \({{\dot{\alpha }}_{j}}\), \(j=0,1,\ldots ,i-1\). Obviously, as system dimension, i.e., n, increases, the computation of \({{\dot{\alpha }}_{i}}\) becomes increasingly complicated. This limits the theoretical results’ field of practical applications. Hence, how to reduce the computation of \({{\dot{\alpha }}_{i}}\) is crucial issue in controller design, which is a motivation of this chapter. In addition, the aforementioned approaches required the knowledge of the desired trajectory \({{y}_{d}}(t)\) and the first n derivatives, i.e., \(y_{d}^{(i)}(t),i=1,2,\ldots ,n\) should be available. It is important to note that in some important applications (e.g., land vehicle or aircraft) the desired trajectory may be generated by a planner, an outer-loop, or a user input device that does not provide higher derivatives. Relaxing the assumption motivates us for this work.
On the other hand, actuators, sensors or other system components in practical engineering fail frequently, which can cause system performance deterioration and lead to instability that can further produce catastrophic accidents. Thus, many effective fault tolerant control (FTC) approaches have been proposed to improve system reliability and to guarantee system stability in all situations [20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39].
In this chapter, a bank of command filters (see, e.g., [40, 41] and the references therein) are proposed to respectively generate the first derivations of the desired trajectory and virtual control signals. Then, by using backstepping technique, a robust adaptive fuzzy controller is proposed to guarantee that the tracking error converges to a neighborhood of the origin, where FLSs are utilized to approximate the unknown functions. The contributions form our work are generalized the following aspects:
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(1)
The desired trajectory and only its first derivative are necessary for the control scheme presented in this chapter, which is more reasonable in practical applications. The theoretic results of this chapter are thus valuable in a wide field of practical applications;
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(2)
Compared with the existing literatures concerning the standard backstepping design, the control scheme presented in this chapter does not need to compute the higher derivatives of virtual control signals in backstepping design procedures, which decreases the computation complexity;
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(3)
Different from some results in literature where all system functions are known, the system functions considered in this chapter are unknown. In particular, the signs of control gain functions are also unknown.
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(4)
The actuator fault model that is presented in this chapter integrates not only unknown gain faults, but also unknown bias faults,where both faults are dependent on the system state and will be approximated by FLSs.
The rest of this chapter is organized as follows. Section 4.2 formulates the problem under investigation. Nussbaum type gain and mathematical description of FLSs are also provided. In addition, some basic assumptions and preliminary results are given. In Sect. 4.3, the main technical results of this chapter are given, where command filters and adaptive fuzzy controller are designed, and the closed-loop system’s stability analysis is developed. A numerical example is presented in Sect. 4.4. Simulation results are presented to demonstrate the effectiveness of the proposed technique. Finally, Sect. 4.5 draws the conclusion.
4.2 Problem Statement and Preliminaries
4.2.1 Problem Statement
Considers the following uncertain nonlinear systems:
where \({{\bar{x}}_{i}}={{({{x}_{1}},\ldots ,{{x}_{i}})}^{T}}\in {{R}^{i}},i=1,\ldots ,n\) is the state; y denotes the output; \({{u}}\in R\) is the input; \(f_{i}(\cdot )\in R\) and \(g_{i}(\cdot )\in R\), \(i=1,\ldots ,n\) are the unknown smooth functions; \(d_{i}(\cdot ,t)\), \(i=1,\ldots ,n\), denote the unknown dynamic disturbances.
In practical applications, actuators may fail. The fault model considered in this chapter can be described as follows:
where \(g_f(\bar{x}_n)\) and \(b_f(\bar{x}_n)\) are smooth functions, which denote unknown gain fault and bias fault, respectively; \(t_F\) is an unknown fault occurrence time.
Control objective is to design an adaptive fuzzy controller by backstepping with command filter for system (4.1) such that output y can track accurately the desired trajectory \({{y}_{d}}\) as possible regardless of actuator fault and unknown dynamic disturbances.
To design appropriate controller, the following lemma and some assumptions are given.
Lemma 4.1
([42]) For \(\forall x\in R\), \(|x|-\tanh (x/\delta )x\le 0.2785\delta \), where \(\delta >0\in R\).
Assumption 4.1
There exist known constants \(g_{i0}>0\in R\) and \(g_{i1}>0\in R\) such that \({g_{i1}} \geqslant \left| {{g_i}({{\bar{x}}_i})} \right| \geqslant {g_{i0}} > 0,\forall {\bar{x}_i} \in {R^i},i = 1,2, \ldots ,n\).
Assumption 4.2
There exist unknown constant \(p^*_i\) and known smooth positive function \(\phi _i(\bar{x}_i)\) such that \(|d_i(\cdot ,t)|\le p^*_i \phi _i(\bar{x}_i)\).
Assumption 4.3
The desired trajectory \({{y}_{d}}(t)\) and its first derivative are bounded and available.
Assumption 4.4
\(g_f(\bar{x}_n)\) is bounded, i.e., there exist known constants \(g_{f0}>0\in R\) and \(g_1>0\in R\) such that \(g_{f1}\ge |g(\bar{x}_n)|\ge g_{f0}\).
Remark 4.1
In literature, the existing results concerning the trajectory tracking problems of the strict-feedback systems require the classical assumption that the desired trajectory \({{y}_{d}}(t)\) and the first n derivatives, i.e., \(y_{d}^{(i)}(t),i=0,1,\ldots ,n\) should be available. Just stated in Introduction, in some important applications (e.g., land vehicle or aircraft) the desired trajectory may be generated by a planner, an outer-loop, or a user input device that does not provide higher derivatives. Thus, in such case, these results do not work. Assumption 4.3 in this chapter is more reasonable in practical applications.
4.2.2 Nussbaum Type Gain
Any continuous function \(N(s):R\rightarrow R\) is a function of Nussbaum type if it has the following properties:
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(1)
\(\underset{s\rightarrow +\infty }{\mathop {\lim }}\,\sup \frac{1}{s}\int _{0}^{s}{N(\varsigma )d\varsigma }=+\infty \);
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(2)
\(\underset{s\rightarrow -\infty }{\mathop {\lim }}\,\inf \frac{1}{s}\int _{0}^{s}{N(\varsigma )d\varsigma }=-\infty \)
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 [43].The even Nussbaum function \({{e}^{{{\varsigma }^{2}}}}\cos ((\pi /2)\varsigma )\) is used throughout this chapter.
Lemma 4.2
([44]) 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 4.3
([45]) 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 \([0,{{t}_{f}})\).
4.2.3 Mathematical Description of Fuzzy Logic Systems
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, 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:
Lemma 4.4
([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
By Lemma 4.4, we know, FLS can approximate any smooth function on a compact space. Due to this approximation capability, we can assume that the nonlinear function 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:
From Lemma 4.4, the following assumption is made.
Assumption 4.5
There exist an unknown real bounded constant \(\varepsilon ^*>0\) such that \(|\varepsilon |\le \varepsilon ^*\) on compact sets \(\varOmega \) and U.
In this chapter, we use the above FLS to approximate the unknown function \(h_i(z_i),(i=1,\ldots ,n)\) will defined later, namely, there exists \({\theta } _{i}^{*}\) and \({{\varepsilon }_{i}}\) such that
From Assumption 4.5, there exists an unknown positive real constant \(\varepsilon _i\) such that \(|\varepsilon _i|\le \varepsilon ^*_i\).
For notational simplicity, we use \(\bullet \) to denote \(\bullet (\cdot )\). For example, \(f_{i}\) is the abbreviation of \(f_{i}({{\bar{x}}_{i}})\).
4.3 Design of Adaptive Fuzzy Controller and Stability Analysis
Define
where \({{\alpha }_{\text {0}}}\text {=}{{y}_{d}}\), \({{\alpha }_{i-1}}\) (\(i=\text {2},\ldots ,n\)) is a virtual control which will be designed at each step, \({{\alpha }_{n}}=u\) is actual control input. The recursive design procedure contains n steps. From Step 1 to Step \(n-1\), \(\alpha _i~(i=1,\ldots ,n-1)\) is designed at each step. Finally an overall control law \(u(\alpha _n)\) is constructed at Step n.
In order to estimate the virtual control \(\alpha _{i-1}\) (\(i=2,\ldots , n\)), define the following command filter
where \(\eta _\omega >0\) is a design parameter. Let us define the estimation error signal \(v_i\) as
Remark 4.2
The command filter (4.4) is constructed to avoid the computation of the higher derivatives of \(\alpha _{i-1}\), \(i=2,\ldots ,n\). It should be pointed out that the error \(v_i\) will be compensated at Step n in this chapter.
Step 1:
Now, consider \({{z}_{1}}\)-subsystem: \({{z}_{1}}={{x}_{1}}-{{\alpha }_{0}}\). Form (4.1) and (4.3), one has
Define the following function
From the integral-type mean value theorem, it can be known that, there exists a constant \({\lambda _1} \in (0,1)\) such that \({V_{{z_1}}} = {z_1}^2/2g({\lambda _1}{z_1} + {y_d})\). Hence, from Assumption 4.1, we have
which means that, \(V_{{z_1}}\) is a positive definite function of variable \({z_1}\).
Since \(\frac{{\partial \left| {{g^{ - 1}}(\sigma + {y_d})} \right| }}{{\partial {y_d}}} =\frac{{\partial \left| {{g^{ - 1}}(\bar{x},\sigma + {y_d})} \right| }}{{\partial \sigma }}\), we can obtain
Let \({\bar{z}_1} = {({x_1},{\omega _1},{\dot{\omega }_1})^\mathrm{T}}\) and
Note that, \(h_i(\bar{z}_1)\) will be approximated by FLSs on a compact set \(\varOmega _{z_1}\) as: \(h_1(z_1)=\theta _1^{*T}{\xi _1}({\bar{z}_1})+\varepsilon _1(\bar{z}_1)\). From Assumption 4.5, we know, there exists an unknown constant \(\varepsilon _1^*\) such that \(|\varepsilon _1(\bar{z}_1) |\le \varepsilon _1^*\).
Then, we have
Virtual control \(\alpha _1\) is defined as follows:
where \(k_1>1\) is a design parameter; \(h_1(z_1,\hat{\theta }_1)=\hat{\theta }^T_1\xi _1(\bar{z}_1)\) and \(\hat{\theta }_1\) are estimates of \(\theta _1^{*T}{\xi _1}({\bar{z}_1})\) and \({\theta }^*_1\), respectively; \({\hat{b}_1}\) is an estimate of \(b_{_1}^*=\max \{ \varepsilon _{_1}^*,\frac{{p_1^*}}{{{g_{10}}}}\}\), \({\bar{\varphi }_1}({\bar{x}_1}) = 1 + {\varphi _1}({\bar{x}_1})\).
Hence, from Lemma 4.1 and Assumptions 4.1 and 4.2, (4.7) can be further developed as follows:
where \({\tilde{\theta }_1} = \theta _1^* - {\theta _1},{\tilde{b}_1} = b_1^* - {b_1}\).
Consider the following function
Adaptive laws are defined as follows:
where \(\varGamma _1\) is a positive matrix with appropriate dimensions, \({\sigma _1} > 0\), \({\sigma _{b1}} > 0\), \({\eta _1} > 0\) and \({\lambda _1} > 0\) are design parameters.
Differentiating \(V_1\) with respect to time t and considering (4.9)–(4.12), we have
where Lemma 4.1 is used, namely, \(0 \leqslant \left| x \right| - x\tanh (\frac{x}{\varepsilon }) \leqslant 0.2785\varepsilon ,{\text { }}\forall \varepsilon > 0, \forall x\in R\).
Since
then (4.17) can be derived as
where
Further, we have
Let \( {\rho _1} = {c_{\varepsilon 1}}/{c_1}\), and integrating both the sides of the above inequality (4.20), it yields
Obviously, if there are not \({e^{ - {c_1}t}}\int _0^t {\frac{1}{4}{e^{{c_1}t}}z_{_2}^2d\tau }\) and \({e^{ - {c_1}t}}\int _0^t {{e^{{c_1}t}}\varDelta _1d\tau }\) in (4.21), then, from Lemmas 4.2 and 4.3, it can be obtained that \({V_1}(t),{\varsigma _1},{\hat{\theta }_1},{\hat{b}_1}\) are bounded in \([0,{{t}_{f}})\). On the other hand, if it can be proved that \(z_2(t)\) is bounded in \([0,t_f)\), from the following inequality
we can obtain that \({e^{ - {c_1}t}}\int _0^t {\frac{1}{4}{e^{{c_1}t}}z_{_2}^2d\tau }\) is bounded. From Lemmas 2 and 3, we further obtain that \({V_1}(t),{\varsigma _1},{\hat{\theta }_1},{\hat{b}_1}\) also are bounded in \([0,t_f)\).
Furthermore, from [43], the same results can be obtained when \({{t}_{f}}=+\infty \).
Notice that, the boundedness of \({{z}_{2}}\) will be considered in the next step, and the error \({e^{ - {c_1}t}}\int _0^t {{e^{{c_1}t}}\varDelta _1d\tau }\) will be compensated in Step n.
Remark 4.3
In [41], the error between \(\omega -1\) and \(\alpha _0\) is not considered in the stability analysis of the overall closed-loop system. Since there exists a difference between them, the effect of the error should be considered in the closed-loop system stability analysis. If not, the stability analysis is not complete.
Remark 4.4
It is valuable to point out, the signs of the control gain functions considered in this chapter are unknown as well as the control coefficients, which means that the system model is more general and the results obtained in this chapter thus have a great significance both on theory and on practical implication.
Step i (\(i=2, 3, \ldots , n-1\)):
In this step, consider the subsystem: \(z_i=x_i-\alpha _{i-1}\). From (4.1) and (4.3), we have
Define the following Lyapunov function
Similar to the analysis in the first step, it can be easily seen that \(V_{{z_i}}\) is a positive definite function of \(z_i\). Since
and from the derivation rule of compound function, we have
From the definition of the error between the command filter’s state and virtual control, we know, \(\alpha _{i-1}=\omega _i-v_i\). Replacing \(\alpha _{i-1}\) in (4.26) by \(\omega _i-v_i\), from (4.1) and (4.26), we have
where \({\bar{z}_i} = {(\bar{x}_{_i}^T,{\omega _{i}},{\dot{\omega }_i})^\mathrm{T}} \in {\varOmega _{{{\bar{z}}_i}}} \subset {R^{i + 2}}\),
Note that, \(h_i(\bar{z}_i)\) will be approximated by FLSs on a compact set \(\varOmega _{z_i}\) as: \(h_i(z_i)=\theta _i^{*T}{\xi _i}({\bar{z}_i})+\varepsilon _i(\bar{z}_i)\). From Assumption 4.5, we know, there exists an unknown constant \(\varepsilon _i^*\) such that \(|\varepsilon _i(\bar{z}_i) |\le \varepsilon _i^*\).
The following virtual control is designed as follows:
where \(k_i>1\frac{1}{4}\) is a design parameter; \({h_i}({\bar{z}_i},{\hat{\theta }_i}) = \hat{\theta }_i^T{\xi _i}({\bar{z}_i})\) is an estimate of \(\theta _i^{*T}{\xi _i}({\bar{z}_i})\); \(\hat{b}_i\) is an estimate of \(b_i^*\), \(b_{_i}^* = \max \{ \varepsilon _{_i}^*,\frac{{p_i^*}}{{{g_{10}}}}\}\), \({\bar{\varphi }_i}({\bar{x}_i}) = 1 + {\varphi _i}({\bar{x}_i})\).
Remark 4.5
It seems strange that \(k_i\) is set to be \(k_i>1\frac{1}{4}\). The purpose of “\(\frac{1}{4}\)” is to compensate for the term \(\frac{1}{4}z^2_i\) which derived in the previous step.
Similar to (4.13), substituting (4.30) and (4.31) into (4.27) and re-arranging it, we have
where \({\tilde{\theta }_i} = \theta _{_i}^* - {\hat{\theta } _i}\) and \({\tilde{b} _i} = b _{_i}^* - {\hat{b} _i}\).
Consider the following Lyapunov function
The following adaptive laws are designed as follows:
where \({\varGamma _i}\) is a positive definite matrix, and \({\eta _i}> 0,{\sigma _i}> 0,{\sigma _{bi}} > 0\) and \({\lambda _i} > 0\) are design parameters.
Similar Step 1, differentiating \(V_i\) with respect to time t and considering (4.34) and (4.35), from Lemma 4.1, one has
Since \({\sigma _i}\tilde{\theta }_i^\mathrm{T}{\hat{\theta }_i} \leqslant - \frac{{{\sigma _i}{{\left\| {{{\tilde{\theta }}_i}} \right\| }^2}}}{2} + \frac{{{\sigma _i}{{\left\| {\theta _i^ * } \right\| }^2}}}{2}\) and \({\sigma _{bi}}{\tilde{b}_i}{\hat{b}_i} \leqslant - \frac{{{\sigma _{bi}}{{\tilde{b}}_i}^2}}{2} + \frac{{{\sigma _{bi}}b{{_i^ * }^2}}}{2}\), then let \({c_{\varepsilon i}} =( 0.2785{\eta _i}b_i^* + \frac{{{\sigma _i}{{\left\| {\theta _i^ * } \right\| }^2}}}{2} + \frac{{{\sigma _{bi}}b{{_i^ * }^2}}}{2}\), \({c_i} = \min \{ 2({k_i} - 1\frac{1}{4}){g_{i0}},\frac{{{\sigma _i}}}{{{\lambda _{\min }}(\varGamma _i^{ - 1})}},\frac{{{\sigma _{bi}}}}{{{\lambda _i}}}\}\) and considering (4.17), then (4.36) can be developed as follows:
Further, we have
As doing in the first step, integrating both the sides of (4.38), we have
where \({\rho _i} = \frac{\sum \nolimits ^i_{j=1}{c_{\varepsilon j}}}{{c_i}}\).
Similar to step 1, if \(z_{i+1}\) is proved to be bounded and \(\sum \nolimits ^i_{j=1}{\varDelta _j}=0\), then, from Lemmas 4.2 and 4.3, one has, \({e^{ - {c_i}t}}\int _0^t {\frac{1}{4}{e^{{c_i}t}}z_{_{i + 1}}^2d\tau }\) is bounded, and \({V_i}(t),{\varsigma _i},{\hat{\theta }_i},{\hat{b}_i}\) further are bounded in \([0,+\infty )\).
Note that, the boundedness of \({{z}_{i+1}}\) will be considered in the next step while \(\sum \nolimits ^i_{j=1}{\varDelta _j}=0\) will be compensated in the last step.
Remark 4.6
From the aforementioned analysis, it is easily seen that virtual control laws \({{\alpha }_{i}}\) are continuous functions of variables \({{\bar{x}}_{i}}\), \({{\bar{z}}_{i}}\), \(\omega _1\), \(\dot{\omega }_1\) and \(\hat{\theta }_i\). Since these variables are available, the first derivative of \({{\alpha }_{i}}\), i.e., \({{\dot{\alpha }}_{i}}\), can be obtained by analytical computation. However, just stated in Introduction section, as system dimension, i.e., n, increases, the computation of the higher derivatives of \({{{\alpha }}_{i}}\) becomes increasingly complicated. In this chapter, by using command filter (4.4), only its first derivative is utilized, which reduce such computation complexity.
Step n:
Now, consider \({{z}_{n}}\)-subsystem: \({{z}_{n}}={{x}_{n}}-{{\alpha }_{n-1}}\). Form (4.1)–(4.3), one has
where \(\bar{f}_n({\bar{x}_n})={f_n}({\bar{x}_n}) + {g_n}({\bar{x}_n})b_f({\bar{x}_n})\) and \({\bar{g}_n}({\bar{x}_n})={g_n}({\bar{x}_n}){g_f}({\bar{x}_n})\).
Define the following Lyapunov function
From the analysis in the previous step, \(V_{z_n}\) is a positive definite function of \(z_n\).
Similar to the previous steps, differentiating \({{V}_{{{z}_{n}}}}\) with respect to time t, one has
where
Adding and subtracting \(\sum \nolimits ^{n-1}_{j=1}{\varDelta _j}\) in the right side of (4.42), we have
Remark 4.7
The purpose of “adding and subtracting \(\sum \nolimits ^{n-1}_{j=1}{\varDelta _j}\)” is to remove the error terms \(\sum \nolimits ^{n-1}_{j=1}{\varDelta _j}\) (4.37), which is introduced by command filter (4.4) in the previous \(n-1\) steps.
It is easily seen that \(\varDelta _j(j=1,\ldots ,n)\) is a function of variables \(\bar{x}_j\), \(\bar{z}_j\), \(\bar{\alpha }_j\), \(\dot{\bar{ \alpha }}_j\), \(\bar{\omega }_j\) and \(\dot{\bar{\omega }}_j\), where \(\bar{x}_j=(x_1,\ldots ,x_j)^T\), \(\bar{z}_j=(z_1,\ldots ,z_j)^T\), \(\bar{\alpha }_j=(\alpha _0,\ldots ,\alpha _{j-1})^T\), \(\dot{\bar{\alpha }}_j=(\dot{\alpha }_0,\ldots ,\dot{\alpha }_{j-1})^T\), \(\bar{\omega }_j=(\omega _1,\ldots ,\omega _j)^T\), \(\dot{\bar{\omega }}_j=(\dot{\omega }_1,\ldots ,\dot{\omega }_j)^T\). Let
where \(\bar{Z}_n=(\bar{x}_n^T, \bar{z}_n^T, \bar{\alpha }_n^T, \dot{\bar{ \alpha }}_n^T, \bar{\omega }_n^T, \dot{\bar{\omega }}_n^T )^T\).
From the previous analysis, it is seen that \(h'(\bar{Z}_n)\) and \(\varDelta _j\) are smooth, which means that \(h(\bar{Z}_n)\) also is smooth. Hence, FLSs can be utilized to approximate it in the form: \(h(\bar{Z}_n)=\theta ^{*T}_n\xi _n(\bar{Z}_n)+\varepsilon _n(\bar{Z}_n)\). From Assumption 5, we know, there exists an unknown constant \(\varepsilon _n^*\) such that \(|\varepsilon _n(\bar{Z}_n)|\le \varepsilon _n^*\).
The actual control is defined as follows:
where \(k_n>\frac{1}{4}\) is a design parameter; \(h_n(\bar{Z}_n,\hat{\theta }_n)=\hat{\theta }_n^T\xi _n(\bar{Z}_n)\) is an estimate of \(\theta _n^{*T}\xi _n(\bar{Z}_n)\); \(\hat{b}_n\) is an estimate of \(b_{_n}^* = \max \{ \varepsilon _{_n}^*,\frac{{p_n^*}}{{{g_{10}}}}\}\); \({\bar{\varphi }_n}({\bar{x}_n}) = 1 + {\varphi _n}({\bar{x}_n})\).
Substituting (4.46) and (4.47) into (4.45), it yields
where \(\tilde{\theta }_n=\theta ^*_n-\hat{\theta }_n\) and \(\tilde{b}_n=b^*_n-\hat{b}_n\).
Define the following Lyapunov function
The following adaptive laws are defined as:
where \(\varGamma _n\) is a positive definite matrix, \({\eta _n}> 0,{\sigma _n}> 0,{\sigma _{bn}} > 0\) and \({\lambda _n} > 0\) are design parameters.
Differentiating \({{V}_{{{n}}}}\) with respect to time t and considering (4.50), (4.51) and Lemma 4.1, similar to the previous steps, one has
From Young’s inequality, we have
Let \({c_{\varepsilon n}} = 0.2785{\eta _n}b_n^* + \frac{{{\sigma _n}{{\left\| {\theta _n^ * } \right\| }^2}}}{2} + \frac{{{\sigma _{bn}}b{{_n^ * }^2}}}{2}\), then (4.52) can be derived as
Let
from the analysis in the previous steps, then (4.54) can be further developed as follows:
Further, we have
where \(\bar{g}_i(\cdot )=g_i(\cdot )\), \(i=1,\ldots , n-1\).
Let \({\rho _n} = \frac{\sum \nolimits ^n_{j=1}{c_{\varepsilon j}}}{c_n}\). Similar to the previous steps, integrating both the sides of the above inequality, we have
From Lemmas 4.2 and 4.3, it is easily seen that \({V_n}(t),{\varsigma _n},{\hat{\theta }_n},{\hat{b}_n}\) are bounded in \([0,t_f)\). From [43], the same results can be obtained in \([0,+\infty )\). Thus, it can be obtained that \(z_n\) is bounded in \([0,+\infty )\), which means that \(z_{n-1}\) in \((n-1)\)th step is bounded. Doing the same reasoning, we finally obtained that all \({z_i}(t), i = 1,2, \ldots n\) are bounded.
From the definitions of \(V_{z_{i}}\) and \(V_i\), \(i=1,\ldots ,n\), we known
From the previous analysis, we have
Hence, from (4.57–4.59), we have
where \({\mu } = 2{\bar{g}_{\max }}(\rho {}_n + {V_n}(0) + {N_n})\), \({\tilde{g}_{\max }} = \mathop {\max }\limits _{1 \leqslant i \leqslant n} {\bar{g}_{i1}} > 0\), \(\bar{g}_{i1}={g}_{i1}\), \(i=1,\ldots ,n-1\), \(\bar{g}_{n1}={g}_{n1}g_{f1}\),
The above design procedures and analysis are summarized in the following theorem.
Theorem 4.1
Consider system (4.1) and fault (4.2). If Assumptions 4.1–4.5 hold, command filters (4.4), actual control defined by (4.46) and (4.47), and the adaptation laws (4.15), (4.16), (4.34), (4.35), (4.50) and (4.51) are employed, then the closed-loop system is asymptotically bounded with the tracking error converging to a neighborhood of the origin.
Proof
From the aforementioned analysis, it is easy to obtain the conclusion. The detailed proof is omitted here.
4.4 Illustrative Example
In this example, a class of nonlinear systems are described as follows:
From (4.61), it is easily seen that, \(g_{10}=0.5\), \(g_{11}=1.5\), \(g_{20}=2\), \(g_{21}=4\), \(p^*_{1}=0.2\), \(\varphi _{1}=x_1\), \(p^*_{2}=0.1\) and \(\varphi _{2}=1\), which means that Assumptions 4.1 and 4.2 hold. In this work, the desired trajectory \({{y}_{d}}=0.1\sin (t)\). Obviously, Assumption 4.3 holds. The actuator fault considered in this simulation research is described as follows:
Obviously, \(g_{f0}=0.5\) and \(g_{f1}=1.5\), which means that Assumption 4.4 holds.
The control objective is to construct an adaptive state feedback controller for nonlinear system (4.61) such that the system output y tracks the desired reference signal \({{y}_{d}}\) with all the signals in the resulting closed-loop system being asymptotically bounded.
For this work, the following parameters are given as follows: \(k_1=k_2=3\), \(\varGamma _1=\varGamma _2=diag{1,1,1,1,1,1,1,1,1,1}\), \(\lambda _1=\lambda _2=1\), \({{\eta }_{1 }}={{\eta }_{2}}=0.01\), \({{\sigma }_{b1}}={{\sigma }_{b1}}=0.1\), \({{\theta }_{i}}\in {{R}^{10}}\), \(i=1,2\) are taken randomly in interval (0,1]. Initial state x(0) is set as \({{(0.2,0.1)}^{T}}\). The sample time is 0.08s.
Simulation results are shown in Figs. 4.1, 4.2 and 4.3. From Fig. 4.1, we can find that system (4.61) has good tracking performance. Figure 4.2 shows that the tracking error converges to a neighborhood of the origin. Meanwhile, the boundedness of control input signal is shown in Fig. 4.3.
4.5 Conclusions
In this chapter, an adaptive fuzzy tracking fault-tolerant control problem of a class of uncertain strict-feedback nonlinear systems with actuator fault has been investigated. FLSs are used to approximate the unknown nonlinear functions. By applying adaptive command filtered backstepping recursive design, integral-type Lyapunov function method and Nussbaum-type gain technique, an adaptive fuzzy control scheme is proposed to guarantee that the closed-loop system is asymptotically bounded with the tracking error converging to a neighborhood of the origin.
References
Wang, L.X., Mendel, J.M.: Fuzzy basis functions, universal approximation and orthogonal least-squares learning. IEEE Trans. Neural Netw. 3(5), 807–814 (1992)
Ying, H.: Sufficient conditions on general fuzzy systems as function approximators. Automatica 30(3), 521–525 (1994)
Wang, L.X.: Stable adaptive fuzzy control of nonlinear system. IEEE Trans. Fuzzy Syst. 1(2), 146–155 (1993)
Driankov, D., Hellendoom, H., Reinfrank, M.: An Introduction to Fuzzy Control. Springer, New York (1993)
Boulkroune, A., Tadjine, M., Saad, M.M., Farza, M.: How to design a fuzzy adaptive controller based on observers for uncertain affine nonlinear systems. Fuzzy Sets Syst. 159(8), 926–948 (2008)
Wang, L.X.: Adaptive Fuzzy Systems and Control: Design and Stability Analysis. Prentice-Hall, Englewood Cliffs (1994)
Zhang, Y.P., Peng, P.Y., Jiang, Z.P.: Stable neural controller design for unknown nonlinear systems using backstepping. IEEE Trans. Neural Netw. 11(6), 1347–1360 (2000)
Wang, M., Chen, B., Shi, P.: Adaptive neural control for a class of perturbed strict-feedback nonlinear time-delay systems. IEEE Trans. Syst. Man Cybern. Part B Cybern. 38(3), 721–730 (2008)
Liu, J., Wang, W., Tong, S.C.: Robust adaptive tracking control for nonlinear systems based on bounds of fuzzy approximation parameters. IEEE Trans. Syst. Man Cybern. Part A Syst. Hum. 40(1), 170–184 (2010)
Lee, H.: Robust adaptive fuzzy control by backstepping for a class of MIMO nonlinear systems. IEEE Trans. Fuzzy Syst. 19(2), 265–275 (2011)
Ge, S.S., Tee, K.P.: Approximation-based control of nonlinear MIMO time-delay systems. Automatica 43(1), 31–43 (2007)
Zhang, T.P., Ge, S.S.: Adaptive neural network tracking control of MIMO nonlinear systems with unknown dead zones and control directions. IEEE Trans. Neural Netw. 20(3), 483–497 (2009)
Krstic, M., Kanellakopoulos, I., Kokotovic, P.: Nonlinear and Adaptive Control Design. Wiley, Hoboken (1995)
Lin, T.C., Lee, T.Y.: Chaos synchronization of uncertain fractional-order chaotic systems with time delay based on adaptive fuzzy sliding mode control. IEEE Trans. Fuzzy Syst. 19(4), 623–635 (2011)
Li, Z.J., Cao, X.Q., Ding, N.: Adaptive fuzzy control for synchronization of nonlinear teleoperators with stochastic time-varying communication delays. IEEE Trans. Fuzzy Syst. 19(4), 745–757 (2011)
Pan, Y.P., Er, M.J., Huang, D.P., Wang, Q.R.: Adaptive fuzzy control with guaranteed convergence of optimal approximation error. IEEE Trans. Fuzzy Syst. 19(5), 807–818 (2011)
Cara, A.B., Pomares, H., Rojas, I.: A new methodology for the online adaptation of fuzzy self-structuring controllers. IEEE Trans. Fuzzy Syst. 19(3), 449–464 (2011)
Lemos, A., Caminhas, W., Gomide, F.: Multivariable gaussian evolving fuzzy modeling system. IEEE Trans. Fuzzy Syst. 19(1), 91–104 (2011)
Hsueh, Y.C., Su, S.F., Tao, C.W., Hsiao, C.C.: Robust L2-gain compensative control for direct-adaptive fuzzy-control-system design. IEEE Trans. Fuzzy Syst. 18(4), 661–673 (2010)
Chen, J., Patton, R.J.: Robust Model-Based Fault Diagnosis For Dynamic Systems. Kluwer Academic, Boston (1999)
Mahmoud, M.M., Jiang, J., Zhang, Y.: Active Fault Tolerant Control Systems. Springer, NewYork (2003)
Yang, H., Jiang, B., Cocquempot, V.: Fault Tolerant Control Design For Hybrid Systems. Springer, Berlin (2010)
Wang, D., Shi, P., Wang, W.: Robust Filtering and Fault Detection of Switched Delay Systems. Springer, Berlin (2013)
Du, D., Jiang, B., Shi, P.: Fault Tolerant Control for Switched Linear Systems. Springer, Cham, Heidelberg (2015)
Shen, Q., Jiang, B., Cocquempot, V.: Fault diagnosis and estimation for near-space hypersonic vehicle with sensor faults. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng. 226(3), 302–313 (2012)
Shen, Q., Jiang, B., Cocquempot, V.: Adaptive fault tolerant synchronization with unknown propagation delays and actuator faults. Int. J. Control Autom. Syst. 10(5), 883–889 (2012)
Shen, Q., Jiang, B., Cocquempot, V.: Fuzzy logic system-based adaptive fault tolerant control for near space vehicle attitude dynamics with actuator faults. IEEE Trans. Fuzzy Syst. 21(2), 289–300 (2013)
Shen, Q., Jiang, B., Cocquempot, V.: Adaptive fault-tolerant backstepping control against actuator gain faults and its applications to an aircraft longitudinal motion dynamics. Int. J. Robust Nonlinear Control 20(10), 448–459 (2013)
Astrom, K.J.: Intelligent control. In: Proceedings of 1st European Control Conference, Grenoble, pp. 2328–2329 (1991)
Gertler, J.J.: Survey of model-based failure detection and isolation in complex plants. IEEE Control Syst. Mag. 8(6), 3–11 (1988)
Frank, P.M., Seliger, R.: Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy—a survey and some new results. Automatica 26(3), 459–474 (1990)
Frank, P.M.: Analytical and qualitative model-based fault diagnosis—a survey and some new results. Eur. J. Control 2(1), 6–28 (1996)
Garcia, E.A., Frank, P.M.: Deterministic nonlinear observer-based approaches to fault diagnosis: a survey. IFAC Control Eng. Pract. 5(6), 663–670 (1997)
Patton, R.J.: Fault-tolerant control: the 1997 situation (survey). In: Proceedings of the IFAC Symposium on Fault Detection, Supervision and Safety for Technical Processes: SAFEPROCESS, pp. 1029–1052 (1997)
Isermann, R., Schwarz, R., Stolzl, S.: Fault-tolerant drive-by-wire systems-concepts and realization. In: Proceedings of the IFAC Symposium Fault Detection, Supervision and Safety for Technical Processes: SAFEPROCESS, pp. 1–5 (2000)
Frank, P.M.: Online fault detection in uncertain nonlinear systems using diagnostic observers: a survey. Int. J. Syst. Sci. 25(12), 2129–2154 (1994)
Patton, R.J.: Robustness issues in fault-tolerant control. In: Proceedings of the International Conference on Fault Diagnosis, Toulouse, France, pp. 1081–1117 (1993)
Song, Q., Hu, W.J., Yin, L., Soh, Y.C.: Robust adaptive dead zone technology for fault-tolerant control of robot manipulators. J. Intell. Robot. Syst. 33(1), 113–137 (2002)
Shen, Q.K., Jiang, B., Shi, P.: Novel neural networks-based fault tolerant control scheme with fault alarm. IEEE Trans. Cybern. 44(11), 2190–2201 (2014)
Zuo, Z.: Trajectory tracking control design with command-filtered compensation for a quadrotor. IET Control Theory Appl. 4(11), 2343–2355 (2012)
Farrell, J.A., Polycarpou, M., Sharma, M., Dong, W.: Command filtered backstepping. IEEE Trans. Autom. Control 54(6), 1391–1395 (2009)
Polycarpou, M.M., Ioannou, P.A.: A robust adaptive nonlinear control design. Automatica 31(3), 423–427 (1995)
Ryan, E.P.: A universal adaptive stabilizer for a class of nonlinear systems. Syst. Control Lett. 16(91), 209–218 (1991)
Ye, X., Jiang, J.: Adaptive nonlinear design without a priori knowledge of control directions. IEEE Trans. Autom. Control 43(11), 1617–1621 (1998)
Ge, S.S., Hong, F., Lee, T.H.: Adaptive neural control of nonlinear time-delay systems with unknown virtual control coefficients. IEEE Trans. Syst. Man Cybern. Part B Cybern. 34(1), 499–516 (2004)
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Shen, Q., Jiang, B., Shi, P. (2017). Command Filtered Adaptive Fuzzy Backstepping FTC Against Actuator Fault. 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_4
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