a Sichuan Sanlian New Material Co., Ltd, 610100, Chengdu, China. Email: Xues11@126.com; Jian_jiang1975@163.com
Abstract—In this paper, a class of nonlinear systems with mismatched disturbance and actuator failure is investigated. A disturbance observer is proposed to estimate the disturbance first and the error of the estimation converges to zero exponentially. By introducing an integral sliding mode surface, the disturbance observer-based integral sliding mode fault tolerant control scheme is proposed to attenuate the disturbance and guarantee the stability of the system. In particular, the control law is designed for decoupling the partial disturbance and attenuating the disturbance that cannot be decoupled. Finally, two examples are given to illustrate the effectiveness of the proposed method.
Index Terms—Actuator fault, Fault tolerant control (FTC), Disturbance observer, Adaptive Integral Sliding mode control, Nonlinear system.
1. INTRODUCTION
Faults frequently occur in the engineering system because of the increasing complexity and scalability of industrial applications. Unexpected deviations of performance or system parameters can induce serious damage and even break down the system in the presence of a fault. With the growing demand for higher reliability, safety, and maintainability, it is desired that the fault can be detected at the early stage, determine the location and magnitude of the fault, identify the severity of the fault, and then accommodate the effects on the system and provide an acceptable performance. Abundant of results have been reported on the theme and many achievements have been applied to industrial systems such as aircraft systems [1], electric systems [2], and motor systems [3-4], etc. Many excellent methods were exploited such as robust control [5], sliding mode control [6-7], observer-based control [8], intelligent learning control [9], and adaptive control [10-11].
As aforementioned, the fault is vulnerable to occur in the practical system, which certainly reduces the nominal performance, results in vibration, destroys the stability of the system, and even causes catastrophic accidents. Actuator fault as the most common fault has caused wide attention. In [12], the flight tracking control system with actuator fault was investigated; an adaptive controller was designed. In [13], a class of singular systems with actuator saturation was developed; an adaptive controller was designed to compensate for the fault effects through the linear matrix inequality (LMI) technique. A class of nonlinear large-scale systems with actuator fault was considered and an observer-based fuzzy adaptive control was developed in [14]. The problem of part loss of effectiveness of the actuator was addressed in [15]. In [16], a class of nonlinear systems with partial loss of actuator was considered; a third-order sliding model control was developed to compensate for the actuator faults. These results have considered linear or nonlinear systems, however, the nonlinear function either satisfies the matched condition [16], i.e., the nonlinear function is in the control channel.
It is worth pointing out that disturbance exists widely in many industrial processes, which affects the stability of the system seriously. The basic characteristics of disturbance are its uncertainty, nonlinearity, and complexity. In some cases, the fault is considered as an additional uncertainty, disturbance, or nonlinear function in the system [16], [17]. The common method is to make a compensator, i.e. construct an anti-disturbance mechanism to compensate for the disturbance. As the characteristics of the disturbance, observer design is necessary and popular in the existing literature, i.e., disturbance observer (DO). In [18], a precision positioning table system was studied and a discrete time-tracking controller was designed based on the DO. In [19], the ball mill grinding circuits system was investigated and a DO was designed to estimate the strong disturbance, the controller was designed based on the observer to compensate for the disturbance. In [20], a class of nonlinear systems was investigated, by introducing a disturbance generator; the DO-based control law was established. [21] addressed a class of Markovian jump systems with multiple disturbances; the control law was designed by integrating the DO output information and state feedback control. [22] take an insight into a generic hypersonic vehicle system with modeled and unmodeled disturbances. In these results, the disturbance matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): B_d
and the control matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): B
are assumed to satisfy the match condition, i.e., Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): rank\left(B\right)=rank\left(\left[\begin{array}{cc} B & B_d \end{array}\right]\right)
, this limitation may not be applicable in some control processes.
It should be pointed out that the mismatched disturbance, that is, the disturbance enters the system in different channels, is more practical in industrial dynamic systems. In [23], a class of nonlinear systems with actuator fault, sensor fault, and mismatched disturbance was considered. In [24], the disturbance was considered as two parts, the matched disturbance was compensated by the DO while the mismatched disturbance was attenuated by variable structure control. In [25], by constructing a nonlinear disturbance observer (NDO), the sliding mode control scheme was developed to counteract the mismatched disturbance.
Based on the aforementioned analysis, this paper attempts to solve the FTC problem for a class of nonlinear systems with actuator fault and mismatched disturbance. The main works can be summarized as follows. First of all, an improved nonlinear disturbance observer is designed to estimate the mismatched disturbance. Then, the integral sliding mode controller is presented based on the observer, where the reachability of sliding motion is proved. Furthermore, the mismatched disturbance is divided into two parts, in which the matched part is compensated by the disturbance information while the remaining part is attenuated by the adaptive controller. Finally, the adaptive control law is proposed, which can adaptively adjust controller parameters to compensate for the fault and disturbance. The effectiveness of the method is verified through two examples.
The remaining part of this paper is organized as follows. In section Ⅱ, the system description and some assumptions are presented. In section Ⅲ, the construction of the observer and the design of the controller are presented. In section Ⅳ, two simulation examples are used to illustrate the effectiveness of the method, and some conclusions are obtained in section Ⅴ.
2. Problem formulation
Consider the following nonlinear system with actuator fault and mismatched disturbance as
|
(1) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x\left(t\right)\in R^n
is the state vector, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): y\left(t\right)\in R^q
is the output of the system, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u\left(t\right)\in R^p
is the control input. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta f\left(x,t\right)
is a nonlinear function, which can be regarded as the un-modeled uncertainty. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): d\left(t\right)\in R^v
is the unmatched external disturbance. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \xi \left(x,t\right)
is the mismatched nonlinearity. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): A
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): B
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): B_d
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): C
are known constant coefficient matrices with appropriate dimensions.
In this paper, the actuator failure problem is concerned. The actuators can be divided into two parts Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u_H
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u_F
, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u_H
stands for the health actuator and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u_F
represent the actuator in a fault condition. Then system (1) becomes (2)
|
(2) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle B_H}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\textstyle B_F}
denote the healthy and fault matrices.
One can see that proposer designing of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u_H
can guarantee the stability of the system despite the existence of external disturbance.
Assumption 1[17] The actuators in fault condition work abnormally, and the remaining actuators work normally.
With assumption 1, one can use Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\overline{u}}_F
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u_F
represent the actual control and designed control respectively for the actuators in fault condition. Then system (2) becomes (3)
|
(3) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta u_F={\overline{u}}_F-u_F
.
Assumption2 [15] There exists a known function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \overline{f}\left(x\right)
and two unknown constants Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\theta }_0
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\upsilon }_0
, such that the following inequality holds,
|
(4) |
Assumption3 The disturbance and the derivative of the disturbance are bounded, i.e., Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Vert d\left(t\right)\Vert \leq {\epsilon }_1
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Vert \dot{d}\left(t\right)\Vert \leq {\epsilon }_d
, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\epsilon }_1
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\epsilon }_d
are two positive constants.
Assumption4 The nonlinearity function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \xi \left(x,t\right)
is bounded and satisfies the following condition
|
(5) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): R
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Q
are two positive symmetry matrices, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tilde{\xi }\left(x,t\right)=\xi \left(x_1,t\right)-
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \xi \left(x_2,t\right)
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tilde{x}\left(x,t\right)=x_1\left(x,t\right)-x_2\left(x,t\right)
.
Remark1 The assumption 1 is general for the actuator failure condition [16], [17], in which the actuator fault can be treated as an additional uncertainty or disturbance. Compared with the results in [26], the assumption 2 in this study has been much more relaxed. Assumption 3 is common in FTC control results [15]. Assumption 4 is more general compared with the traditional Lipschitz condition [16], it should be noted that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): R\mbox{=}I
, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Q=l_f{}^2I
, then assumption 4 degenerates to the normal Lipschitz condition, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): l_f
is the Lipschitz constant.
With assumptions 1 and 2, the object of the paper is to design a control law to compensate for the effects of the disturbance and fault so that the stability and convergence of the system can be guaranteed in normal and fault conditions.
3. Main results
In this part, an observer will be applied to estimate the external disturbance, and an observer-based integral sliding mode fault colorant control scheme will be designed
3.1 Observer design
For the feasibility of the observer, the following assumption is necessary.
Assumption5 [16] The additional fault term Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \varphi \left(t\right)=B_F\Delta u_F\left(t\right)
satisfies the following condition, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Vert B_F\Delta u_F\left(t\right)\Vert \leq {\overline{\omega }}_b
, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\overline{\omega }}_b
is a positive scalar.
For system (3), the following observer is proposed in the form of
|
(6) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): p\left(t\right)
is the internal state of the observer, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \overset{\mbox{ˆ}}{d}\left(t\right)
is the estimation of the disturbance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): d\left(t\right)
. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): q\left(x\right)
is a nonlinear function to be designed. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): T_n
is a parameter matrix with proper dimensions. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): g\left(x\right)
is the observer gain and satisfies the following condition
|
(7) |
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\delta }_n
is the error compensator and is defined by
|
(8) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\delta }_d={\overline{\omega }}_b-\Vert B_F\Delta u_F\left(t\right)\Vert
.
Lemma1 With the assumption 4 and the observer (6), the error of the observer converges to zero exponentially.
Proof Define Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): e_d\left(t\right)=d\left(t\right)-\overset{\mbox{ˆ}}{d}\left(t\right)
. From the observer (6), it is easy to obtain that
|
(9) |
It can be derivate that
|
(10) |
The solution of (9) is given by
|
(11) |
From (10), one can deduce that
|
(12) |
According to the inequality Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Vert e^{At}\Vert \leq ce^{-\rho t}
[3], where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): c
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \rho
are two positive constants. The one can obtain that
|
(13) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): h_0
is a constant bound satisfy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Vert e_d\left(0\right)\Vert \leq h_0
. Define Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \overline{\kappa }=ce^{-\rho t}h_0+\frac{c{\epsilon }_d}{\rho }\left(1-\right.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. e^{-\rho t}\right)
, then (13) can be written as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): e_d\leq \overline{\kappa }
. This completes the proof.
3.2 Adaptive FTC design
In this section, the observer-based integral sliding mode fault tolerant control will be designed. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \overline{\kappa }
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\upsilon }_0
are supposed to be unknown, the adaptive controller is designed as follows.
|
(14) |
where
|
(15) |
|
(16) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): K_1\left(t\right)
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): K_2\left(t\right)
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): K_3\left(t\right)
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): K_{d1}\left(t\right)
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): K_{d2}\left(t\right)
are the parameter functions to be designed
|
(17) |
|
(18) |
|
(19) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\lambda }_1
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\lambda }_2
are two positive scalars, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\upsilon }_1\geq {\upsilon }_0
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\gamma }_1^{{_\ast}}\geq {\gamma }_1
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\gamma }_1
is defined in (23), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\delta }_m
is the bound of the nonlinear function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \xi \left(x,t\right)
. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \overset{\mbox{ˆ}}{\overline{\kappa }}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\overset{\mbox{ˆ}}{\upsilon }}_0
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\overset{\mbox{ˆ}}{\delta }}_m
are the estimate of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \overline{\kappa }
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\upsilon }_0
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\delta }_m
, and the adaptive control laws are designed in the form of
|
(20) |
|
(21) |
|
(22) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\chi }_p
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\chi }_q
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\chi }_r
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\beta }_p
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\beta }_q
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\beta }_r
are positive parameters.
The control law will be used in the integral sliding mode control, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u_m
is used to make the system asymptotically stable, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u_n
is used to compensate for the effects of the actuator fault, disturbance estimation error, and nonlinear factors. In this paper, the sliding mode switching function is designed as follows
|
(23) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): D\in R^{p\times n}
is a designed matrix that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): DB
is invertible. In the next section, the reaching ability will be verified.
Theorem 1 With the controller in the (14), the state strategies of the system will drive onto the sliding surface Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): s\left(t\right)
in finite time.
Proof Denote Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tilde{\overline{\kappa }}=\overline{\kappa }-\overset{\mbox{ˆ}}{\overline{\kappa }}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\tilde{\upsilon }}_0={\upsilon }_0-{\overset{\mbox{ˆ}}{\upsilon }}_0
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\tilde{\delta }}_m={\delta }_m-{\overset{\mbox{ˆ}}{\delta }}_m
. Consider the following Lyapunov function candidate
|
(24) |
From the expression (20), the time derivative of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): s\left(t\right)
is
|
(25) |
From assumption 2, one has
|
(26) |
From (23), one has
|
(27) |
The time derivative of (35) can be determined as
|
(28) |
By introducing (17)-(27), (28) can be rewritten as
|
(29) |
Thus, the reaching ability is satisfied, this completes the proof.
Remark 2 The controller proposed in (14) is discontinuous, to reduce chattering in the practical implementation, the discontinuous function can be replaced, for example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): u_1\left(t\right)
can be replaced by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): -\frac{{\left(DB\right)}^{-1}DB_dB_d{}^TD^Ts\left(t\right)}{\Vert s^T\left(t\right)DB_d\Vert +\alpha }\left(\overset{\mbox{ˆ}}{\overline{\kappa }}+\right.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. {\lambda }_1\right)
, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \alpha
is a small positive constant.
3.3 Stability analysis
In this section, the stability of the closed-loop system will be analyzed. By solving the equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \dot{s}\left(t\right)=0
in (25), the equivalent control law can be obtained as
|
(30) |
Substituting (26) and (30) into the system (3) yields
|
(31) |
By defining Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): K_d{}_1={\left(DB\right)}^{-1}DB_d
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\overline{B}}_d=B_d-B{\left(DB\right)}^{-1}DB_d
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \overline{M}=I-B{\left(DB\right)}^{-1}D
, (31) can be rewritten as
|
(32) |
As before mentioned, the disturbance-matched condition is not satisfied, i.e. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): rank\left(B\right)<rank\left(B,B_d\right)
. From the definition of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\overline{B}}_d
, one can easily check that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): rank\left(B\right)<rank\left(B,{\overline{B}}_d\right)
. In this paper, the disturbance is divided into two parts, i.e., Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\overline{B}}_d=\left[\begin{array}{cc} {\overline{B}}_{d_1} & {\overline{B}}_{d_2} \end{array}\right]
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): d={\left[\begin{array}{cc} d_1 & d_2 \end{array}\right]}^T
, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\overline{B}}_dd\left(t\right)
can be written as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\overline{B}}_dd\left(t\right)={\overline{B}}_{d1}d_1\left(t\right)+
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\overline{B}}_{d2}d_2\left(t\right)
, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\overline{B}}_{d_1}=\left[\begin{array}{cccc} {\overline{B}}_{d_{\kappa 1}} & {\overline{B}}_{d_{\kappa 2}} & ... & {\overline{B}}_{d_{\kappa m1}} \end{array}\right]\in R^{n\times m_1}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\overline{B}}_{d_2}=\left[\begin{array}{cccc} {\overline{B}}_{d_{\theta 1}} & {\overline{B}}_{d_{\theta 2}} & ... & {\overline{B}}_{d_{\theta m2}} \end{array}\right]\in R^{n\times m_2}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): d_1={\left[\begin{array}{cccc} d_{\kappa 1} & d_{\kappa 2} & ... & d_{\kappa m_1} \end{array}\right]}^T\in R^{m_1\times 1}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): d_2={\left[\begin{array}{cccc} d_{\theta 1} & d_{\theta 2} & ... & d_{\theta m_1} \end{array}\right]}^T\in R^{m_2\times 1}
, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): m_1+m_2=\nu
.
Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): rank\left(B\right)=rank\left(B,{\overline{B}}_{d\mbox{1}}\right)
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): rank\left(B\right)<rank\left(B,{\overline{B}}_{d2}\right)
and the parameter Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): K_{d2}
in (15) can be chosen as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): K_{d2}=\left[\begin{array}{cccc} K_{s1} & K_{s1} & ... & K_{s\upsilon } \end{array}\right]\in R^{p\times \upsilon }
, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): BK_{si}={\overline{B}}_{d\kappa i}
for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): i\leq m_1
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): K_{si}=0
for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): i>m_1
. Then the equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): BK_{d2}={\overline{B}}_{d\mbox{1}}
is solvable. Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \overline{M}=B\tilde{M}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tilde{M}=\left(B^{\dagger }-{\left(DB\right)}^{-1}D\right)
, where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): B^{\dagger }=B^T{\left(BB^T\right)}^{-1}
. Then (32) can be rewritten as
|
(33) |
Remark 3 One can see that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\overline{B}}_{d2}=0
, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): rank\left(B\right)=rank\left(B,{\overline{B}}_d\right)
, so the matched condition is a special case, i.e., this paper considers a more general case. In addition, one can also obtain that one of the solutions of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): K_{d2}
is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): K_{si}=B^{\dagger }{\overline{B}}_{d\kappa i}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): B^{\dagger }
is the general inverse of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): B
.
The object of the next part is to design Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): K_1\left(t\right)
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): K_2\left(t\right)
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): K_3\left(t\right)
such that the stability of the system can be guaranteed. The control laws are designed as follows
|
(34) |
|
(35) |
|
(36) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\overset{\mbox{ˆ}}{w}}_1
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \overset{\mbox{ˆ}}{\varpi }
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\overset{\mbox{ˆ}}{\gamma }}_1
are estimations of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): w_1
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \varpi
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\gamma }_1
respectively, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): w_1
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \varpi
are designed in (59) and (60). Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): P
is a positive symmetry matrix. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \sigma \left(t\right)
is a continuous function, and satisfies
|
(37) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\sigma }^{{_\ast}}
is a positive scalar.
The adaptive updating control laws are given by
|
(38) |
|
(39) |
|
(40) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\beta }_1
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\beta }_2
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\beta }_3
are three positive constants.
Denote Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tilde{\varpi }=\varpi -\overset{\mbox{ˆ}}{\varpi }
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\tilde{w}}_1=w_1-{\overset{\mbox{ˆ}}{w}}_1
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\tilde{\gamma }}_1={\gamma }_1-{\overset{\mbox{ˆ}}{\gamma }}_1
, one can obtain the following dynamics
|
(41) |
|
(42) |
|
(43) |
Theorem 2 : With the controller (34)-(36) and the adaptive control laws (38)-(40), the closed-loop system is stable if there exist two positive symmetry matrices Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): P
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Q
, such that the following condition holds
|
(44) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): P_1=PB\tilde{M}
, Q is defined in (5).
Proof Design the Lyapunov function candidate as
|
(45) |
Then the time derivative of (51) can be obtained as
|
(46) |
From (41)-(46), it can be derivative that
|
(47) |
|
(48) |
|
(49) |
|
(50) |
|
(51) |
|
(52) |
By assumptions 2, 4, 5, and lemma 1, one can derivative that the following inequalities hold
|
(53) |
|
(54) |
|
(55) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \varpi
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): w_1
are positive scalars.
Substituting (47)-(55), (46) can be rewritten as
|
(56) |
Note that the following equalities
|
(57) |
|
(58) |
|
(59) |
Substituting (57)-(59) into (56) yields
|
(60) |
For any positive scalar Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\lambda }_m
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\lambda }_n
, the following equality holds Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0\leq \frac{{\lambda }_m{\lambda }_n}{{\lambda }_m+{\lambda }_n}\leq {\lambda }_n
. Then one can obtain that
|
(61) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\zeta }_{\kappa }=2+\frac{1}{4}\left({\varpi }^2+\right.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \left. w^2{}_1+{\gamma }^2{}_1\right)
.
According to (45), by integrating (61) yields
|
(62) |
which means the system described in (33) is bounded. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): {\lambda }_{min}\left(Q{}_{}^1\right)
denotes the minimum eigenvalue of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): Q_1
, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): -Q_1=A^TP+PA+P_1{}^TP_1+Q
. (68) also implies
|
(63) |
According to Barbalat Lemma, one has Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \underset{t\rightarrow \infty}{lim}{\Vert x\left(t\right)\Vert }^2=
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): 0
. This completes the proof.
Remark 4 Compared with the results in [15], where the nonlinear function is matched, i.e., Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta f\left(x,t\right)
is in the control channel. In this paper, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta f\left(x,t\right)
exists in the different channel from the control input, i.e., which means that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta f\left(x,t\right)
is more general.
4. Numerical examples
In this section, two examples are simulated to illustrate the effectiveness of the proposed method.
Example1 In this example, the linearized longitudinal dynamic of the VTOL aircraft which is borrowed from [23] is considered. It is assumed that the system is subjected to unmodeled dynamics, actuator fault, and external disturbance. Then, the system can be described as system (1), where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x\left(t\right)={\left[\begin{array}{cccc} x_1\left(t\right) & x_2\left(t\right) & x_3\left(t\right) & x_4\left(t\right) \end{array}\right]}^T
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x_1\left(t\right)
is the horizontal velocity ( Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): knot
), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x_2\left(t\right)
represents the vertical velocity ( Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): knot
), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x_3\left(t\right)
is the pitch rate ( Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): degree/s
), and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x_4\left(t\right)
expresses the pitch angle ( Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): degree
). The parameters of the system are given as follows
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): A=\left[\begin{array}{cccc} -9.9477 & -0.7476 & 0.2632 & 5.0337\\ 52.1659 & 2.7452 & 5.5532 & -24.4221\\ 26.0922 & 2.6361 & -4.1975 & -19.2774\\ 0 & 0 & 1 & 0 \end{array}\right]
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): B=\left[\begin{array}{cc} 0.4422 & 0.1761\\ 3.5446 & -7.5922\\ -5.5200 & 4.4900\\ 0 & 0 \end{array}\right]
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): C=\left[\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 1 & 1 & 1 \end{array}\right]
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): B_d=\left[\begin{array}{cc} 0.4422 & 0.116\\ 3.5446 & -0.102\\ -5.5200 & 0\\ 0 & 0 \end{array}\right]
,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): B_F=\left[\begin{array}{c} \\ \\ \\ \end{array}0.1767.5924.4900\right]
.
The nonlinear unmodeled uncertainty and mismatched nonlinearity are assumed to be: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta f\left(x,t\right)=sin\left(x_4\right)
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \xi \left(x,t\right)=sin\left(x_1\right)
. The actuator fault and external are supposed to be as follows
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): f\left(t\right)=\lbrace \begin{array}{c} 0\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\left(0\leq t<8\right)\\ 0.5sin\left(2t+1\right)\mbox{ }\mbox{ }\left(8\leq t<10\right)\\ 0\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\left(10\leq t<15\right)\mbox{ }\mbox{ } \end{array}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \lbrace \begin{array}{c} d_1\left(t\right)=2sin\left(2t+1\right)\mbox{ }\mbox{ }\mbox{ }\mbox{ }\left(0\leq t<15\right)\\ d_2\left(t\right)=0.5sin\left(2t-3\right)\mbox{ }\left(0\leq t<15\right) \end{array}
.
Note that the mismatched disturbance and the condition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): rank\left(B\right)\not =rank\left(B,{\overline{B}}_d\right)
hold, hence that the traditional method will be failed in this example. Choosing the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): D=\left[\begin{array}{cccc} 1 & 0 & 1 & 0\\ -1 & 1 & 0 & 1 \end{array}\right]
, one can check that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): DB
is invertible. By solving (50), one can obtain that
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): P=\left[\begin{array}{cccc} 14843 & 161.7 & 226.9 & -745.0\\ 161.7 & 25.50 & 12.80 & -101.8\\ 226.9 & 12.80 & 74.80 & -47.40\\ 745.0 & -101.8 & -47.40 & 896.5 \end{array}\right]
.
Choose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \sigma \left(t\right)=2e^{-t}
, the results of the simulation are as follows
Fig. 1 Response of state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x\left(t\right)
Fig. 2. Estimation of disturbance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): d\left(t\right)
Fig. 3 Response of state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x\left(t\right)
without disturbance compensatorFig.1 shows the trajectory of the system, Fig.2 illustrates the estimation of the disturbance signals, the solid line is the original signal and the dashed line is the estimated value. From Fig.1, one can know that the states of the system have a fast response with the proposed method. In addition, the controller can ensure the stability of the system in the presence of the actuator fault and mismatched disturbance. Fig.2 characterizes that the observer has a good performance of the disturbance.
In order to illustrate the importance of the disturbance observer, the responses of the system are shown in Fig.3 without the disturbance observer. From the figure, one can see that the closed-loop system becomes unstable when removing the disturbance observer and compensator.
Example2 In this section, the two-cart system which borrowed form [27] is provided to illustrate the effectiveness of the proposed method.
As shown in Fig.4, the first cart is connected to a rigid wall via a damper, and is connected to a second cart by a spring. The external force is applied to a second cart via an actuator. Both carts have a nominal mass of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): a=1kg
, the damper has a constant of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): b_0=1N/m
, and the spring constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): c_0=1N/m
. The time constant of the actuator Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \tau =0.2
. The states are the force, velocities, and positions of the two carts. The actuator fault and mismatched disturbance are considered. The system parameters are given as follows
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): A=\left[\begin{array}{ccccc} -\frac{1}{\tau } & 0 & 0 & 0 & 0\\ 0 & -\frac{b_0}{a} & 0 & -\frac{c_0}{a} & \frac{c_0}{a}\\ \frac{1}{a} & 0 & 0 & \frac{c}{a} & -\frac{c_0}{a}\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 \end{array}\right]
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): B=\left[\begin{array}{c} \frac{1}{\tau }\\ 0\\ 0\\ 0\\ 0 \end{array}\right]
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): B_F=\left[\begin{array}{c} \frac{1}{\tau }\\ 0\\ 0\\ 0\\ 0 \end{array}\right]
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): C=\left[\begin{array}{ccccc} 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 1 \end{array}\right]
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): B_d=\left[\begin{array}{cc} 1 & 1\\ 0 & 0.2\\ 0 & 0.2\\ 0 & 0.1\\ 0 & 0.2 \end{array}\right]
.
The nonlinear unmodeled uncertainty and mismatched nonlinearity are assumed to be: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \Delta f\left(x,t\right)=sin\left(x_4\right)
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \xi \left(x,t\right)=sin\left(x_1\right)
. The actuator fault and external are supposed to be as follows
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): f\left(t\right)=\lbrace \begin{array}{c} 0\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\left(0\leq t<\mbox{25}\right)\\ \mbox{0}\mbox{.5}sin\left(0.2t\right)\mbox{ }\left(\mbox{25}\leq t<4\mbox{0}\right)\\ \mbox{1}\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\left(4\mbox{0}\leq t<\mbox{80}\right) \end{array}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): d_1\left(t\right)\mbox{=}\lbrace \begin{array}{c} 0\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\left(0\leq t<15\right)\\ 0.5sin\left(t\right)\mbox{ }\mbox{ }\mbox{ }\left(15\leq t<50\right)\mbox{ }\mbox{ }\mbox{ }\\ \mbox{1}\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\left(\mbox{5}0\leq t<80\right) \end{array}
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): d_2\left(t\right)=\lbrace \begin{array}{c} 0.5sin\left(\mbox{3}t\right)\mbox{ }\mbox{ }\mbox{ }\left(0\leq t<20\right)\mbox{ }\\ 0.5sin\left(\mbox{3}t\right)\mbox{ }\mbox{ }\mbox{ }\left(20\leq t<50\right)\\ 1\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\mbox{ }\left(50\leq t<80\right)\mbox{ } \end{array}
.
Choosing the matrix Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): D=\left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \end{array}\right]
, one can check that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): DB
is invertible. By solving (50), one can obtain that
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): P=\left[\begin{array}{ccccc} \mbox{0}\mbox{.0799} & -0.000 & -0.000 & -0.000 & -0.000\\ -0.000 & 1.1046 & 0.000 & -0.000 & -0.000\\ -0.000 & 0.000 & 1.1046 & 0.000 & -0.000\\ -0.000 & -0.000 & 0.000 & 1.1046 & 0.000\\ -0.000 & -0.000 & -0.000 & 0.000 & 1.1046 \end{array}\right]
.
Choose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): \sigma \left(t\right)=2e^{-t}
, the results of the simulation are as follows
Fig. 5 Response of state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x_1\left(t\right)
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x_2\left(t\right)
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x_3\left(t\right)
Fig. 6 Response of state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x_4\left(t\right)
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x_5\left(t\right)
Fig. 7 Estimation of disturbance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): d\left(t\right)
Fig. 8 Response of state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x_1\left(t\right)
, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x_2\left(t\right)
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x_3\left(t\right)
without controller
Fig. 9 Response of state Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x_\mbox{4}\left(t\right)
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://mathoid.scipedia.com/localhost/v1/":): x_\mbox{5}\left(t\right)
without controllerFig.5 and Fig. 6 express the trajectories of the system, from Figs.5-6, one can see that the stability of the positions and velocities of the first and second carts can be guaranteed. Fig.7 shows the estimation of the external disturbance, one can check that the proposed method performs better than the intermediate method proposed in [27], precisely, the method proposed responds faster than the method in [27], and the proposed method has less chattering.
In order to illustrate the effectiveness of the proposed method, the responses of the system are shown in Fig.8 and Fig. 9 without the controller. From the figure, one can infer from this that the closed-loop system becomes unstable when removing the disturbance observer and compensator.
5. Conclusion
In this paper, the problem of a general Lipschitz nonlinear system with actuator fault and unmatched disturbance is investigated. Specifically, a disturbance observer is designed to estimate the mismatched disturbance first. Then, an observer-based integral sliding mode fault tolerant control scheme is proposed. In order to guarantee the stability of the system, three adaptive control laws are constructed because of the unknown nonlinear function parameters and the unmodeled uncertainty. Finally, two examples are given to illustrate the effectiveness of the proposed method. In our future work, we would like to focus on the fault-tolerant control methods for multiple faults and disturbances and their applications.
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Document information
Published on 19/03/24
Accepted on 28/02/24
Submitted on 15/02/24
Volume 40, Issue 1, 2024
DOI: 10.23967/j.rimni.2024.03.002
Licence: CC BY-NC-SA license
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