Nomenclature
-
${x}_{1}$
-
system variable
-
${x}_{2}$
-
system variable
-
${x}_{3}$
-
system extended variable
- h(t)
-
unknown function in system
-
${\sigma (t)}$
-
system uncertainty
-
${c}_{\sigma }{(t)}$
-
unknown derivative of
$\sigma (t)$
-
${L}_{\sigma }$
-
upper bound of
$c_{\sigma}(t)$
-
${y}_{o1}$
-
position measurement
-
${y}_{o2}$
-
velocity measurement
- d(t)
-
bounded disturbance in position
-
${n}_{1}{(t)}$
-
high frequency noise in position
-
${n}_{2}{(t)}$
-
high frequency noise in velocity
-
${L}_{d}$
-
upper bound of d(t)
-
$\widehat{x}_{1}$
-
corrector variable
-
$\widehat{x}_{2}$
-
corrector variable
-
$\widehat{x}_{3}$
-
observer variable
-
$\widehat{x}_{4}$
-
observer variable
-
${\omega }_{nc}$
-
natural frequency of corrector
-
${\omega }_{no}$
-
natural frequency of observer
-
${k}_{1}$
-
corrector parameter
-
${k}_{2}$
-
corrector parameter
-
${k}_{3}$
-
observer parameter
-
${k}_{4}$
-
observer parameter
-
${\varepsilon }_{c}$
-
perturbation parameter in corrector
-
${\varepsilon }_{o}$
-
perturbation parameter in observer
-
${\alpha }_{c}$
-
function power in corrector
-
${\alpha }_{o}$
-
function power in observer
- m
-
mass of UAV
- g
-
gravity acceleration
- l
-
rotor distance to gravity centre
- x
-
position in x-direction
- y
-
position in y-direction
- z
-
position in z-direction
-
${\psi }$
-
yaw angle
-
${\theta }$
-
pitch angle
-
${\phi }$
-
roll angle
-
${J}_{\phi }$
-
moment of inertia about roll
-
${J}_{\theta }$
-
moment of inertia about pitch
-
${J}_{\psi }$
-
moment of inertia about yaw
- b
-
rotor force coefficient
- k
-
rotor torque coefficient
-
${\Delta }_{x}$
-
uncertainty in x-direction
-
${\Delta }_{y}$
-
uncertainty in y-direction
-
${\Delta }_{z}$
-
uncertainty in z-direction
-
${\Delta }_{\psi }$
-
uncertainty in yaw
-
${\Delta }_{\theta }$
-
uncertainty in pitch
-
${\Delta }_{\phi }$
-
uncertainty in roll
-
${k}_{x}$
-
drag coefficient in x-direction
-
${k}_{y}$
-
drag coefficient in y-direction
-
${k}_{z}$
-
drag coefficient in z-direction
-
${k}_{\psi }$
-
drag coefficient in yaw
-
${k}_{\theta }$
-
drag coefficient in pitch
-
${k}_{\phi }$
-
drag coefficient in roll
-
${F}_{i}$
-
thrust force by rotor i
-
${Q}_{i}$
-
reactive torque of rotor i
-
${\omega }_{i}$
-
rotor i rotational velocity
1. Introduction
Usually UAV flight needs information of position, attitude and dynamic model. Global position plays very important roles for large-range navigation and control [Reference Kwak and Sung1, Reference Grip, Fossen, Johansen and Saberi2]. Meanwhile, the uncertainties exist in UAV flight dynamics: aerodynamic disturbance, unmodelled dynamics and parametric uncertainties are inevitable. These uncertainties bring serious challenges for control system design.
GPS (Global positioning system) can provide position information with accuracy of several meters or even tens of meters [Reference Hsu3, Reference Liu, Bianchin and Pasqualetti4]. Also, the adverse circumstances may contaminate signals from GPS [Reference Liu, Bianchin and Pasqualetti4]. Velocity is also necessary for UAV navigation and control. GPS can measure device velocity with two different accuracies: (1) large-error velocity by the difference method with accuracy of a meter per second due to GPS position accuracy and noise effect; (2) precise velocity by Doppler shift measurement with accuracy of a few centimeters even millimeters per second [Reference Freda, Angrisano, Gaglione and Troisi5, Reference Serrano, Kim, Langley, Itani and Ueno6]. Alternatively, accurate velocity of device can be measured by a Doppler radar sensor [Reference Griffiths7]. Except for sensing, velocity can be estimated from position using robust observers [Reference Levant8–Reference Wang, Chen and Yang13]. However, the relatively accurate measurements of position are required for these observer, and position with stochastic disturbance brings much worse velocity estimation.
Without position and velocity sensing, INS (inertial navigation system) can estimate them through integrations from acceleration measurement. However, even small measurement error or very weak non-zero mean noise in acceleration through integrations can cause velocity and position to drift over time. The observer-based INS methods were used to estimate unknown variables in navigation [Reference Rogne, Bryne, Fossen and Johansen14, Reference Hamel, Hua and Samson15]. However, position signals are limited to be local and bounded, but not global.
For attitude information, an IMU (inertial measurement unit) can determine the attitude angles from the measured angular rates through integration, and angle drifts happen. Meanwhile, the outputs of the accelerometers and the magnetometer in IMU can determine the large-error pitch, roll and yaw angles [Reference Ludwig and Jiménez16].
In order to reduce the disturbances/errors in position/attitude angle, KF (Kalman filter) or EKF (extended Kalman filter) is adopted for signals fusion to restrict the defects of individual measurements based on the optimisation of a recursive least mean square error [Reference Deo, Silvestre and Morales17–Reference Stovner, Johansen and Fossen19]. Thus, the accuracy of system outputs is improved. For KF/EKF, the relatively accurate system models are needed. Moreover, existence of uncertainty in noise statistics limits the KF/EKF parameters’ regulation. In addition, for EKF, system model linearisation may cause filter divergence, and the derivation of the Jacobian matrices are nontrivial.
The uncertainties or disturbances in system can be estimated by the extended state observers [Reference Levant8, Reference Khalil11, Reference Wang and Zheng20–Reference Wang, Chen and Yuan22]. However, the accurate position measurements are required as these observer inputs. Even the accurate velocity is used for uncertainty estimation, the disturbance in position cannot be corrected.
In this paper, a class of uncertain systems with large disturbance/error in position measurement are considered. As an example, the relevant problems in UAV navigation and control are also considered. According to the relations between position, velocity and uncertainty in system, as well as the large disturbance/error in position sensing and the relatively accurate measurement in velocity, the position correction and uncertainty estimation are completely decoupled. The independent signal corrector and uncertainty observer are presented according to finite-time stability [Reference Bhat and Bernstein23, Reference Bhat and Bernstein24] and the complete decoupling estimation. In the stable signal corrector, the bounded stochastic disturbance in position sensing can be reduced by the accurate velocity measurement. With the specific nonlinear functions and the perturbation parameter, the corrector can reduce the disturbance/error in sensing further. Moreover, the bounded system gains can avoid peaking phenomenon for the corrector. Importantly, through estimation decoupling, the observer can estimate the system uncertainty using the accurate velocity measurement independently, avoiding the use of large-error position information. Both corrector and observer are the low-order systems, their system parameters are regulated easily. Frequency analysis is used to explain the robustness of the corrector and observer.
The signal corrector and uncertainty observer are applied to an experiment on a quadrotor UAV navigation and control, and the performance is compared to the traditional KF-based navigation [Reference Crassidis25]. In the experiment, the following adverse conditions are considered: large stochastic disturbances/errors in GPS position/IMU attitude angles, uncertainties in position/attitude dynamics, and existence of high-frequency noise. The signal correctors are adopted to correct the disturbances/errors in GPS position/IMU attitude angles, and the observers are used to estimate the uncertainties in the UAV dynamics. Finally, the control laws based on the correction and estimation are formed to stabilize the UAV flight.
Comparing with the existing related studies, the contributions of this paper include:
(1) Completely decoupling estimation: The completely decoupling estimation is implemented for a class of systems with large stochastic disturbance in position sensing and system uncertainty. Thus, using the same accurate velocity measurement but towards the opposite directions, the presented signal corrector can correct the stochastic disturbance in position sensing, and the system uncertainty is estimated by the uncertainty observer. However, the usual observers cannot correct the disturbance/error in position sensing, and the positions are taken as the input for estimation. It brings the large estimation errors, even it makes the system unstable.
(2) Relax requirements for system model and sensing disturbance: For the uncertain systems with large stochastic disturbance in position sensing, the presented method does not need the accurate models and complete noise statistics information. Only the upper bounds of the stochastic disturbance in sensing and system uncertainty are required. The design of the signal corrector is model free, and only two measurement signals in position and velocity are required for the correction. The design of the observer is based on partial system model. For KF/EKF, the relatively accurate system model and noise statistics are necessary, and the parameters’ regulation and estimation performance are affected adversely by uncertainties in system model and noise statistics.
(3) High-precision correction/estimation and strong robustness: The signals fusion in the corrector is implemented according to the finite-time stability with strong robustness against bounded disturbance. With the use of specific continuous nonlinear functions, and due to the strong robustness against bounded stochastic disturbance, the correction error is small enough even for large disturbance. In addition, through frequency analysis, the design corrector and observer have the ability of low-pass filters. Therefore, the corrector and observer provide the accurate and smoothed correction/estimation outputs.
(4) Ease of parameters’ regulation: The implementation of completely decoupling estimation makes both corrector and observer in low order. For the stability, the parameters’ selection is only required to be satisfied with a Routh–Hurwitz Stability Criterion. For improving the robustness, the power of the nonlinear function in the corrector is regulated.
(5) Suitable to the complex and adverse dynamic systems: The corrector and observer can be applied to the complex flight dynamic systems: relatively accurate navigation and control are implemented even large stochastic disturbances/errors in position/attitude sensing and existence of modelling uncertainites/disturbances.
2. Problem description
The technical problems considered in this paper for a class of uncertain systems with large-error sensing include:
(1) large bounded stochastic error/disturbance in position sensing; (2) existence of system uncertainty; (3) overshoot/oscillations existence and difficult parameters selection in high-order estimation systems.
2.1 Correction of large error in position and estimation of uncertainty in position dynamics
Measurement conditions: GPS provides the large error/disturbance position of device, and the relatively accurate velocity can be determined by GPS with Doppler shift measurement or by a Doppler radar sensor. Also, the uncertainties/disturbances exist in system dynamics. Under these conditions, we have:
Question 1: How to correct the bounded stochastic error/disturbance in position and to estimate uncertainty in position dynamics?
2.2 Correction of large error in attitude angle and estimation of uncertainty in attitude dynamics
Measurement conditions: The gyroscopes in IMU provide the direct measurement of relatively accurate angular rate. The large-error attitude angles can be determined by the outputs of the accelerometer and magnetometer in IMU. Then:
Question 2: How to correct the bounded error/disturbance in attitude angle and to estimate uncertainty in attitude dynamics?
2.3 Difficult parameters selection and oscillations existence for high-order estimation system
For multivariate estimation/correction, a high-order observer can be used. However, many system parameters need to be adjusted cooperatively, and oscillations are prone to occur. The oscillations can amplify the noise in the estimation outputs. Therefore, we hope the decoupling low-order estimate systems can be designed to overcome these issues instead of a single high-order observer.
3. General form of decoupling corrector and observer for uncertain systems with large error sensing
3.1 Uncertain system with large error/disturbance sensing
The following uncertain system has a minimum number of states and inputs, but it retains the features that is considered for many applications:
\begin{align}\dot{x}_{1} & =x_{2} \nonumber \\\dot{x}_{2} & =h(t)+\sigma (t) \nonumber \\y_{o1} & =x_{1}+d(t) \nonumber \\y_{o2} & =x_{2}+n(t),\end{align}
where,
$x_{1}$
and
$x_{2}$
are the states; h(t)
$\in \Re $
is the known function including the controller and the other unknown terms;
$\sigma(t)\in \Re $
is the system uncertainty;
$y_{o1}$
and
$y_{o2}$
are the sensing outputs; d(t) is the unknown bounded stochastic error/disturbance in measurement, it may be in the low, intermediate or high frequency bands, and
$\sup_{t\in \lbrack 0,\infty )}\left\vert d(t)\right\vert \leq L_{d}<+\infty $
; n(t) is the high-frequency noise. The design missions include: error correction in
$y_{o1}$
; estimation of
$\sigma (t)$
.
3.2 System extension
Assumption 3.1. Suppose uncertainty
$\sigma (t)$
in system (1) satisfies
\begin{equation}\dot{\sigma}(t)=c_{\sigma }(t),\end{equation}
where,
$c_{\sigma }(t)$
is unknown and bounded, and
$\sup_{t\in \lbrack0,\infty )}\left\vert c_{\sigma }(t)\right\vert \leq L_{\sigma }<+\infty $
. Actually, this assumption holds for many applications, e.g., crosswind dynamics.
In system (1), in order to estimate the uncertainty
$\sigma (t)$
, we define it as a new variable, i.e.,
$x_{3}=\sigma (t)$
. Therefore,
$\dot{x}_{3}=\dot{\sigma}(t)=c_{\sigma }(t)$
holds. Then, second-order system (1) is extended equivalently into a third-order system, i.e.,
\begin{align}\dot{x}_{1} & =x_{2} \nonumber \\\dot{x}_{2} & =x_{3}+h(t) \nonumber \\\dot{x}_{3} & =c_{\sigma }(t) \nonumber \\y_{o1} & =x_{1}+d(t) \nonumber \\y_{o2} & =x_{2}+n(t).\end{align}
3.3 System decoupling according to the accurate measurement
The estimations of
$x_{1}$
and
$x_{3}$
are in the opposite directions from the relatively accurate measurement
$y_{o2}$
. Then, system (3) can be decoupled into the following two systems:
(1) the unobservable (from
$y_{o2}$
) system
\begin{align}\dot{x}_{1} & =x_{2} \nonumber \\y_{o1} & =x_{1}+d(t) \nonumber \\y_{o2} & =x_{2}+n(t).\end{align}
(2) and the observable system
\begin{align}\dot{x}_{2} & =x_{3}+h(t) \nonumber \\\dot{x}_{3} & =c_{\sigma }(t) \nonumber \\y_{o2} & =x_{2}+n(t).\end{align}
3.4 General form of completely decoupling correction and estimation
We give the following assumptions before the correction and estimation systems are constructed.
Assumption 3.2. Suppose the origin is the finite-time-stable equilibrium of system
\begin{align}\dot{z}_{1} & =z_{2} \nonumber \\\dot{z}_{2} & =f_{c}(z_{1},k\cdot z_{2}),\end{align}
where,
$f_{c}()$
is continuous and
$f_{c}(0,0)=0$
, and
$k>1$
.
Assumption 3.3. For (6), there exist
$\rho \in (0,1]$
and a nonnegative constant a a such that
\begin{equation}\left\vert f_{c}(\tilde{z}_{1},k\cdot z_{2})-f_{c}(\bar{z}_{1},k\cdot z_{2})\right\vert \leq a\left\vert \tilde{z}_{1}-\bar{z}_{1}\right\vert^{\rho },\end{equation}
where,
$\tilde{z}_{1},\bar{z}_{1}\in \Re $
.
Remark 3.1. There are many types of functions satisfying this assumption. For example, one such function is
$\left\vert \tilde{x}^{\rho }-\overline{x}^{\rho }\right\vert \leq 2^{1-\rho }\left\vert \tilde{x}-\overline{x}\right\vert ^{\rho },\rho \in \left( 0,1\right] $
.
Assumption 3.4. Suppose the origin is the finite-time-stable equilibrium of system
\begin{align}\dot{z}_{3} & =z_{4}+f_{o1}(z_{3}) \nonumber \\\dot{z}_{4} & =f_{o2}(z_{3}),\end{align}
where,
$f_{o1}()$
and
$f_{o2}()$
are continuous, and
$f_{o1}(0)=0$
and
$f_{o2}(0)=0$
.
Usually high-frequency noise does not determine (but affects) system stability. Thus, the noise n(t) is ignored when we consider the system stability. When considering the system robustness in frequency analysis, we will analyze the effect of noise n(t).
Theorem 3.1 (General form of decoupling signal corrector and uncertainty observer):
The following uncertain system, for which Assumptions 3.1
$\sim $
3.4 hold, is considered:
\begin{align}\dot{x}_{1} & =x_{2} \nonumber \\\dot{x}_{2} & =h(t)+\sigma (t) \nonumber \\y_{o1} & =x_{1}+d(t) \nonumber \\y_{o2} & =x_{2},\end{align}
where,
$x_{1}$
and
$x_{2}$
are the states; h(t)
$\in \Re $
is the known function including the controller;
$\sigma (t)\in \Re $
is the system uncertainty,
$\dot{\sigma}(t)=c_{\sigma }(t)$
, and
$c_{\sigma }(t)$
is bounded with
$\sup_{t\in \lbrack 0,\infty )}\left\vert c_{\sigma}(t)\right\vert \leq L_{\sigma }<+\infty $
;
$y_{o1}$
and
$y_{o2}$
are the measurement outputs; d(t) is the bounded stochastic disturbance/error in measurement
$y_{o1}$
, and
$\sup_{t\in \lbrack 0,\infty )}\left\vert d(t)\right\vert \leq L_{d}<+\infty $
. In order to correct large error in measurement
$y_{o1}$
and to estimate uncertainty
$\sigma (t)$
(i.e.,
$x_{3}$
), the completely decoupling second-order corrector and observer are designed, respectively, as follows:
(1) Signal corrector
\begin{align}\dot{\widehat{x}}_{1} & =\widehat{x}_{2} \nonumber \\\varepsilon _{c}^{3}\dot{\widehat{x}}_{2} & =f_{c}(\varepsilon _{c}(\widehat{x}_{1}-y_{o1}),\widehat{x}_{2}-y_{o2}),\end{align}
where,
$\varepsilon _{c}\in \left( 0,1\right) $
; and
(2) Uncertainty observer
\begin{align}\varepsilon _{o}\dot{\widehat{x}}_{3} & =\varepsilon _{o}\widehat{x}_{4}+f_{o1}\left(\widehat{x}_{3}-y_{o2}\right)+\varepsilon _{o}h(t) \nonumber \\\varepsilon _{o}^{2}\dot{\widehat{x}}_{4} & =f_{o2}\left(\widehat{x}_{3}-y_{o2}\right),\end{align}
where,
$\varepsilon _{o}\in \left( 0,1\right) $
. Then, there exist
$\gamma_{c}>\frac{3}{\rho }$
,
$\gamma _{o}>1$
and
$t_{s}>0$
, such that, for
$t\geq t_{s}$
,
\begin{align}\widehat{x}_{1}-x_{1} & =O\left(\varepsilon _{c}^{\rho \gamma _{c}-1}\right)\text{; }\widehat{x}_{2}-x_{2}=O\left(\varepsilon _{c}^{\rho \gamma _{c}-2}\right)\text{;}\nonumber \\[5pt]\widehat{x}_{3}-x_{2} & =O\left(\varepsilon _{c}^{2\gamma _{o}}\right)\text{; }\widehat{x}_{4}-\sigma (t)=O\left(\varepsilon _{c}^{2\gamma _{o}-1}\right),\end{align}
where,
$O\left(\varepsilon _{c}^{\rho \gamma _{c}-1}\right)$
means that the error between
$\widehat{x}_{1}$
and
$x_{1}$
is of order
$O\left(\varepsilon_{c}^{\rho \gamma _{c}-1}\right)$
[Reference Kahlil26]. The proof of Theorem 3.1 is presented in the Appendix.
Remark 3.2. In the signal corrector (10), the input signals include the measurements
$y_{o1}$
and
$y_{o2}$
, and the states
$\widehat{x}_{1}$
and
$\widehat{x}_{2}$
estimate the system states
$x_{1}$
and
$x_{2}$
, respectively. Importantly, the bounded stochastic disturbance/error d(t) in measurement
$y_{o1}$
is rejected. In the observer (11), the input signal is the measurement
$y_{o2}$
, and
$\widehat{x}_{3}$
and
$\widehat{x}_{4}$
estimate
$x_{2}$
and uncertainty
$\sigma (t)$
, respectively. Two independent low-order estimation systems are designed to correct the large error in measurement and to estimate the uncertainty, and the completely decoupling estimations are implemented.
4. Implementation of completely decoupling corrector and observer for uncertain systems
In the following, we implement: For a class of uncertain systems with bounded stochastic error/disturbance sensing, the completely decoupling low-order corrector and observer are designed to implement signal correction and uncertainty estimation, respectively.
4.1 Design of decoupling low-order corrector and observer for uncertain system with large-error sensing
Theorem 4.1: The following uncertain system is considered:
\begin{align}\dot{x}_{1} & =x_{2} \nonumber \\\dot{x}_{2} & =h(t)+\sigma (t) \nonumber \\y_{o1} & =x_{1}+d(t) \nonumber \\y_{o2} & =x_{2},\end{align}
where,
$x_{1}$
and
$x_{2}$
are the states; h(t)
$\in \Re $
is the known function including the controller;
$\sigma (t)\in \Re $
is the system uncertainty,
$\dot{\sigma}(t)=c_{\sigma }(t)$
, and
$c_{\sigma }(t)$
is bounded with
$\sup_{t\in \lbrack 0,\infty )}\left\vert c_{\sigma}(t)\right\vert \leq L_{\sigma }<+\infty $
;
$y_{o1}$
and
$y_{o2}$
are the measurement outputs; d(t) is the bounded stochastic disturbance/error in measurement
$y_{o1}$
, and
$\sup_{t\in \lbrack 0,\infty )}\left\vert d(t)\right\vert \leq L_{d}<+\infty $
. In order to correct large error in measurement
$y_{o1}$
and to estimate uncertainty
$\sigma (t)$
, the completely decoupling second-order corrector and observer are designed, respectively, as follows:
(1) Signal corrector
\begin{align}\dot{\widehat{x}}_{1} & =\widehat{x}_{2} \nonumber \\\varepsilon _{c}^{3}\dot{\widehat{x}}_{2} & =-k_{1}\left\vert \varepsilon_{c}\left(\widehat{x}_{1}-y_{o1}\right)\right\vert ^{\frac{\alpha _{c}}{2-\alpha _{c}}}\text{sign}\left( \widehat{x}_{1}-y_{o1}\right) -k_{2}\left\vert \widehat{x}_{2}-y_{o2}\right\vert ^{\alpha _{c}}\text{sign}\left( \widehat{x}_{2}-y_{o2}\right),\end{align}
where,
$k_{1}>0$
,
$k_{2}>0$
,
$\alpha _{c}\in (0,1)$
, and time-scale parameter
$\varepsilon _{c}\in \left( 0,1\right) $
; and
(2) Uncertainty observer
\begin{align}\varepsilon _{o}\dot{\widehat{x}}_{3} & =\varepsilon _{o}\widehat{x}_{4}-k_{4}\left\vert \widehat{x}_{3}-y_{o2}\right\vert ^{\frac{\alpha _{o}+1}{2}}\text{sign}\left( \widehat{x}_{3}-y_{o2}\right) +\varepsilon_{o}h\left( t\right) \nonumber \\\varepsilon _{o}^{2}\dot{\widehat{x}}_{4} & =-k_{3}\left\vert \widehat{x}_{3}-y_{o2}\right\vert ^{\alpha _{o}}\text{sign}\left( \widehat{x}_{3}-y_{o2}\right),\end{align}
where,
$k_{3}>0$
,
$k_{4}>0$
,
$\alpha _{o}\in (0,1)$
, and time-scale parameter
$\varepsilon _{o}\in \left( 0,1\right) $
. Then, there exist
$\gamma _{c}>\frac{6-3\alpha _{c}}{\alpha _{c}}$
,
$\gamma _{o}>1$
and
$t_{s}>0 $
, such that, for
$t\geq t_{s}$
,
\begin{align}\widehat{x}_{1}-x_{1} & =O\left(\varepsilon _{c}^{\frac{\alpha _{c}}{2-\alpha _{c}}\gamma _{c}-1}\right)\text{; }\widehat{x}_{2}-x_{2}=O\left(\varepsilon _{c}^{\frac{\alpha _{c}}{2-\alpha _{c}}\gamma _{c}-2}\right)\text{;} \nonumber \\[5pt]\widehat{x}_{3}-x_{2} & =O\left(\varepsilon _{c}^{2\gamma _{o}}\right)\text{; }\widehat{x}_{4}-\sigma (t)=O\left(\varepsilon _{c}^{2\gamma _{o}-1}\right),\end{align}
where,
$O\left(\varepsilon _{c}^{\frac{\alpha _{c}}{2-\alpha _{c}}\gamma _{c}-1}\right)$
means that the error between
$\widehat{x}_{1}$
and
$x_{1}$
is of order
$O\left(\varepsilon _{c}^{\frac{\alpha _{c}}{2-\alpha _{c}}\gamma _{c}-1}\right)$
. The proof of Theorem 4.1 is presented in the Appendix.
4.2 Analysis of stability and robustness
Here, the describing function method is used to analyse the nonlinear behaviours of the corrector and observer. Although it is an approximation method, it inherits the desirable properties from the frequency response method for nonlinear systems. We will find that the corrector and observer lead to perform accurate estimation and strong rejection of noise under the condition of the bounded estimation gains.
In signal corrector (14) and uncertainty observer (15), for the nonlinear function
$\left\vert \ast \right\vert ^{\alpha _{i}}sign\left( \ast \right) $
, by selecting
$\ast =A_{i}\sin (\omega t)$
, its describing function can be expressed by
$N_{i}(A_{i})=\frac{\Omega (\alpha _{i})}{A_{i}^{1-\alpha _{i}}}$
, where,
$\Omega (\alpha _{i})=\frac{2}{\pi }\int_{0}^{\pi }\left\vert \sin(\omega \tau )\right\vert ^{\alpha _{i}+1}d\omega \tau $
, and
$\Omega(\alpha _{i})\in \left[ 1,\frac{4}{\pi }\right) $
when
$\alpha _{i}\in (0,1)$
.
For the corrector (14), define:
$\alpha _{c2}=\alpha _{c}\in (0,1)$
and
$\alpha _{c1}=\dfrac{\alpha _{c2}}{2-\alpha _{c2}}$
;
$\widehat{x}_{1}-y_{o1}=A_{c1}\sin (\omega t)$
and
$\widehat{x}_{2}-y_{o2}=A_{c2}\sin(\omega t)$
. Here,
$A_{c1}$
is the error magnitude in position sensing, and
$A_{c2}$
is the error magnitude in velocity sensing. Therefore, the approximations of signal corrector (14) and uncertainty observer (15) through the describing function method are given, respectively, by
\begin{align}\dot{\widehat{x}}_{1} & =\widehat{x}_{2} \nonumber \\\varepsilon ^{3}\dot{\widehat{x}}_{2} & =-\frac{k_{1}\Omega (\alpha _{c1})}{A_{c1}^{1-\alpha _{c1}}}\varepsilon \left( \widehat{x}_{1}-y_{o1}\right) -\frac{k_{2}\Omega (\alpha _{c2})}{A_{c2}^{1-\alpha _{c2}}}\left( \widehat{x}_{2}-y_{o2}\right),\end{align}
and
\begin{align}\dot{\widehat{x}}_{3} & =\widehat{x}_{4}-\frac{k_{4}\Omega (\dfrac{1+\alpha_{o}}{2})}{\varepsilon _{o}A_{o}^{\frac{1-\alpha _{o}}{2}}}\left( \widehat{x}_{3}-y_{o2}\right) +h\left( t\right) \nonumber \\[5pt]\dot{\widehat{x}}_{4} & =-\frac{k_{3}\Omega (\alpha _{o})}{\varepsilon_{o}^{2}A_{o}^{1-\alpha _{o}}}\left( \widehat{x}_{3}-y_{o2}\right).\end{align}
Define the Laplace transforms
$\widehat{X}_{1}(s)=L\left[ \widehat{x}_{1}\right] $
,
$\widehat{X}_{2}(s)=L\left[ \widehat{x}_{2}\right] $
, and
$Y_{02}(s)=L\left[ y_{02}\right] $
, for (17), the following transfer functions are determined:
\begin{equation}\frac{\widehat{X}_{j}(s)}{Y_{02}(s)}=\frac{k_{2}\dfrac{\Omega \left( \alpha_{2}\right) }{A_{c2}^{1-\alpha _{2}}}s^{j-1}+\varepsilon k_{1}\dfrac{\Omega\left( \alpha _{1}\right) }{A_{c1}^{1-\alpha _{1}}}s^{j-2}}{\varepsilon^{3}s^{2}+k_{2}\dfrac{\Omega \left( \alpha _{2}\right) }{A_{c2}^{1-\alpha_{2}}}s+\varepsilon k_{1}\dfrac{\Omega \left( \alpha _{1}\right) }{A_{c1}^{1-\alpha _{1}}}},\end{equation}
where,
$j=1,2$
.
$\frac{\widehat{X}_{2}(s)}{Y_{02}(s)}$
means the velocity filtering output
$\widehat{x}_{2}$
; and
$\frac{\widehat{X}_{1}(s)}{Y_{02}(s)}$
means the effect of the corrector output
$\widehat{x}_{1}$
from the velocity measurement
$y_{02}$
.
We get the natural frequency of the corrector by
\begin{equation}\omega _{nc}=\frac{\sqrt{k_{1}}\sqrt{\Omega (\alpha _{1})}}{\varepsilon_{c}A_{c1}^{\dfrac{1-\alpha _{c1}}{2}}},\end{equation}
and similarly the natural frequency of the observer is
\begin{equation}\omega _{no}=\frac{\sqrt{k_{3}}\sqrt{\Omega (\alpha _{o})}}{\varepsilon_{o}A_{o}^{\dfrac{1-\alpha _{o}}{2}}}.\end{equation}
The effects of the parameters on the corrector robustness are analysed as follows.
Frequency characteristic with different
$\varepsilon _{c}$
and
$\alpha _{c2}$
. For the corrector, We select:
$k_{1}=1$
,
$k_{2}=30$
;
$\alpha _{c2}=\alpha =0.8,0.5,0.3$
;
$\varepsilon _{c}=\varepsilon=0.8,0.4,0.25$
, respectively. With different selections of
$\alpha _{c2}$
, we get the other parameter values shown in Table 1.
Table 1. Values of
$\Omega (\alpha _{c2})$
and
$\Omega(\alpha _{c1})$
with different
$\alpha _{c2}$
The Bode plots of (19) with different
$\varepsilon _{c}$
and
$\alpha _{c2}$
are presented in Fig. 1: Fig. 1(a) and (b) describe the frequency characteristics of (
$y_{02}\rightarrow \widehat{x}_{2}$
) and (
$y_{02}\rightarrow \widehat{x}_{1}$
), respectively.
Figure 1. Bode plot of corrector with different parameter selections. (a)
$y_{02}\rightarrow \widehat{x}_{2}$
(
$\varepsilon =0.8,0.4,0.25$
;
$\alpha =0.8,0.5,0.3$
). (b)
$y_{02}\rightarrow\widehat{x}_{1}$
(
$\varepsilon =0.8,0.4,0.25$
;
$\alpha =0.8,0.5,0.3$
).
Conclusions on the correction and estimation
From the proof of Theorem 1, the systems are finite time stable, and their approximations are asymptotically stable according to (17) and (18). According to the analysis in time and frequency domains, the system stability and robustness have the following properties:
(1) Large error rejection in sensing:
In time domain, from (16), in spite of the large error/disturbance in position sensing, the estimate errors are always small enough after a finite time. In addition, we find that even for unbounded position navigation, no drift exists in position due to the small bound of estimate errors. In frequency domain, from (20) and Fig. 1(a), when the error magnitude
$A_{c1}$
in position sensing is relatively large, the corrector natural frequency
$\omega _{nc}$
is smaller, and much noise is rejected.
(2) Strong correction from accurate
$y_{02}$
:
From the system (13), we know that,
$\dot{x}_{1}=x_{2}$
. According to Theorem 1,
$\widehat{x}_{2}-y_{02}=$
$\widehat{x}_{2}-x_{2}$
is small enough. Thus
$\widehat{x}_{2}$
approaches
$x_{2}$
. According to
$\dot{\widehat{x}}_{1}=\widehat{x}_{2}$
and system stability, the corrector output
$\widehat{x}_{1}$
approaches the actual position
$x_{1}$
. Therefore, the large error/disturbance in position is corrected sufficiently. Furthermore, from the Bode plot of
$\frac{\widehat{X}_{1}(s)}{Y_{02}(s)}$
in Fig. 1(b), we find that, when
$y_{02}$
is in low frequency band,
$y_{02}$
contributes the correction very well, and the corrector output
$\widehat{x}_{1}$
approaches the integral of
$y_{02}$
; while the noise in
$y_{02}$
is rejected in high frequency band.
(3) No peaking (bounded gains of corrector): If the large gains are selected, they make the bandwidth very large, and it is sensitive to high-frequency noise. Moreover, peaking phenomenon happens. It means that the maximal value of system output during the transient increases infinitely when the gains tend to infinity. For the presented corrector and observer, the system gains do not need to be large, and no peaking phenomenon happens. In fact, in the estimate errors,
$\gamma _{c}>1$
and
$\gamma _{o}>1$
are sufficiently large. Therefore, for any
$\varepsilon _{c}\in (0,1)$
and
$\varepsilon _{o}\in (0,1)$
, the estimate errors are sufficiently small. Thus,
$\varepsilon _{c}$
and
$\varepsilon _{o}$
do not need small enough in the estimation systems. Meanwhile, from (17) and (18), near the neighborhood of equilibrium,
$1/A_{c1}^{1-\alpha _{c1}}$
and
$1/A_{c2}^{1-\alpha _{c2}}$
in the corrector and
$1/A_{o}^{\frac{1-\alpha _{o}}{2}}$
in the observer are large enough, and these large terms make the feedback effect still strong. Therefore, the large parameter gains are unnecessary, and peaking is avoided.
(4) No chattering: Both corrector and observer are continuous, and their system outputs are smoothed. Therefore, the corrector and observer can provide smoothed estimations to reduce high-frequency chattering.
(5) Robustness against noise: If the errors magnitudes
$A_{c1}$
and
$A_{o}$
are relatively large, according to (20) and (21), the natural frequencies
$\omega _{nc}$
and
$\omega _{no}$
for the corrector and observer are relatively small. Thus, more disturbances/errors are rejected, and
$A_{c1}$
and
$A_{o}$
become small. Furthermore, the corrector and observer are continuous, and the estimate outputs are smoothed. Therefore, the high-frequency noise in the estimations is rejected.
(6) Performance by parameter selections:
$\varepsilon _{c}$
affects the frequency bandwidth: Decreasing
$\varepsilon _{c}$
, the frequency bandwidth becomes larger; increasing
$\varepsilon _{c}$
, the frequency bandwidth becomes smaller, and much noise is rejected.
$\alpha _{c2}\in(0,1) $
affects the estimate precision: smaller
$\alpha _{c2}\in (0,1)$
can obtain more precise estimations; on the other hand, relatively larger
$\alpha _{c2}\in (0,1)$
can reduce much noise.
4.3 Parameters selection rules of corrector and observer
Because the corrector and observer are completely decoupling, their parameters can be regulated independently. According to stability of nonlinear continuous systems [Reference Bhat and Bernstein23], we have:
(1) Parameters selection for system stability (Routh–Hurwitz Stability Criterion):
Signal corrector (14): For any
$\varepsilon _{c}\in (0,1)$
and
$\alpha_{c}\in (0,1)$
,
$s^{2}+\frac{k_{2}}{\varepsilon _{c}^{2\alpha _{c}}}s+k_{1}$
is Hurwitz if
$k_{1}>0$
and
$k_{2}>0$
. Furthermore, in order to avoid oscillations, we select:
$k_{1}>0$
,
$k_{2}>0$
,
$k_{2}^{2}\geq 4\varepsilon_{c}^{4\alpha _{c}}k_{1}$
,
$\varepsilon _{c}\in (0,1)$
and
$\alpha _{c}\in(0,1)$
.
Uncertainty observer (15):
$s^{2}+k_{4}s+k_{3}$
is Hurwitz if
$k_{3}>0$
and
$k_{4}>0$
. Furthermore, in order to avoid oscillations, we select:
$k_{3}>0$
,
$k_{4}>0$
, and
$k_{4}^{2}\geq 4k_{3}$
,
$\varepsilon _{o}\in (0,1)$
and
$\alpha _{o}\in (0,1)$
.
Sensing error rejection: When the sensing error d(t) in
$y_{o1}$
increases, i.e.,
$L_{d}$
becomes larger, in order to reduce the error effect
$k_{1}L_{d}^{\frac{\alpha _{c}}{2-\alpha _{c}}}$
of
$\delta _{c}=2^{1-\frac{\alpha _{c}}{2-\alpha _{c}}}k_{1}L_{d}^{\frac{\alpha _{c}}{2-\alpha _{c}}}+L_{p}$
in (64), parameter
$k_{1}>0$
should decrease. Meanwhile,
$\alpha_{c}\in (0,1)$
can decrease to make
$L_{d}^{\frac{\alpha _{c}}{2-\alpha _{c}}}$
smaller.
(2) Parameters selection for filtering:
$\varepsilon _{c}$
(or
$\varepsilon _{o}$
) affects the frequency band of the corrector (or observer). If much noise exists,
$\varepsilon _{c}\in (0,1)$
(or
$\varepsilon _{o}\in (0,1)$
) should increase, and/or
$\alpha _{c}\in(0,1)$
(or
$\alpha _{o}\in (0,1)$
) increases, to make the low-pass frequency bandwidth narrow. Thus, noise can be rejected sufficiently. Also, from (20),
$k_{1}$
decreases, the frequency band also decreases, and much noise will be rejected.
$\alpha _{c}\in (0,1)$
(or
$\alpha _{o}\in (0,1)$
) guarantees the finite-time stability of corrector (or observer), and it can avoid the selection of sufficiently small
$\varepsilon _{c}$
(or
$\varepsilon _{o}$
).
5. UAV navigation and control based on decoupling estimations
A UAV navigation and control with large-error sensing in position/attitude angle are considered. The UAV forces and torques are explained in Fig. 2, and the system parameters are introduced in Table 2.
Figure 2. Forces and torques of quadrotor UAV.
Table 2. UAV parameters
5.1 Quadrotor UAV dynamics
The inertial and fuselage frames are denoted by
$\Xi _{g}=\left(E_{x},E_{y},E_{z}\right) $
and
$\Xi _{b}=\left(E_{x}^{b},E_{y}^{b},E_{z}^{b}\right) $
, respectively;
$\psi $
,
$\theta $
and
$\phi $
are the yaw, pitch and roll angles, respectively.
$F_{i}=b\omega_{i}^{2}$
is the thrust force by rotor i, and its reactive torque is
$Q_{i}=k\omega _{i}^{2}$
. The sum of the four rotor thrusts is
$F=\sum\limits_{{i=1}}^{{4}}F_{i}$
. The motion equations of the UAV flight dynamics are expressed by
\begin{equation}\ddot{x}_{i}=h_{i}(t)+\sigma _{i}(t),\end{equation}
where,
$i=1,\cdots ,6$
;
$x_{1}=x$
,
$x_{2}=y$
,
$x_{3}=z$
,
$x_{4}=\psi $
,
$x_{5}=\theta $
,
$x_{6}=\phi $
;
$h_{1}(t)=\frac{u_{x}}{m}$
,
$h_{2}(t)=\frac{u_{y}}{m}$
,
$h_{3}(t)=\frac{u_{z}}{m}-g$
,
$h_{4}(t)=\frac{u_{\psi }}{J_{\psi}}$
,
$h_{5}(t)=\frac{u_{\theta }}{J_{\theta }}$
,
$h_{6}(t)=\frac{u_{\phi }}{J_{\phi }}$
;
$\sigma _{1}(t)=m^{-1}(-k_{x}\dot{x}+\Delta _{x})$
;
$\sigma_{2}(t)=m^{-1}(-k_{y}\dot{y}+\Delta _{y})$
;
$\sigma _{3}(t)=m^{-1}(-k_{z}\dot{z}+\Delta _{z})$
;
$\sigma _{4}(t)=J_{\psi }^{-1}(-k_{\psi }\dot{\psi}+\Delta _{\psi })$
;
$\sigma _{5}(t)=J_{\theta }^{-1}(-lk_{\theta }\dot{\theta}+\Delta _{\theta })$
;
$\sigma _{6}(t)=J_{\phi }^{-1}(-lk_{\phi }\dot{\phi}+\Delta _{\phi })$
;
$k_{x}$
,
$k_{y}$
,
$k_{z}$
,
$k_{\psi }$
,
$k_{\theta }$
and
$k_{\phi }$
are the unknown drag coefficients;
$(\Delta _{x},\Delta_{y},\Delta _{z})$
and
$(\Delta _{\psi },\Delta _{\theta },\Delta _{\phi })$
are the uncertainties in position and attitude dynamics, respectively;
$J=diag\{J_{\psi },J_{\theta },J_{\phi }\}$
is the matrix of three-axial moment of inertias;
$c_{\theta }$
and
$s_{\theta }$
are expressed for
$\cos\theta $
and
$\sin \theta $
, respectively; and
\begin{align}u_{x} & =(c_{\psi }s_{\theta }c_{\phi }+s_{\psi }s_{\phi })F,\text{ }u_{y}=(s_{\psi }s_{\theta }c_{\phi }-c_{\psi }s_{\phi })F,\text{ }u_{z}=c_{\theta }c_{\phi }F, \nonumber \\u_{\psi } & =\frac{k}{b}\sum\limits_{{i=1}}^{{4}}(-1)^{i+1}F_{i},\text{ }u_{\theta }=(F_{3}-F_{1})l,\text{ }u_{\phi }=(F_{2}-F_{4})l.\end{align}
5.2 Sensing
GPS provides the global position, and a microwave radar sensor measures velocity. An IMU gives the attitude angle and angular rate. The sensing outputs are:
\begin{equation}y_{i,1}=x_{i}+d_{i}(t),\text{ }y_{i.2}=\dot{x}_{i}+n_{i}(t),\end{equation}
where,
$d_{i}(t)$
is the bounded stochastic error/disturbance in sensing, and
$\sup_{t\in \lbrack 0,\infty )}\left\vert d_{i}(t)\right\vert \leq L_{i}<\infty $
;
$n_{i}(t)$
is the high-frequency noise;
$i=1,\cdots ,6$
.
The corrector (14) and observer (15) are used to estimate (x, y, z,
$\psi $
,
$\theta $
,
$\phi $
) and the system uncertainties, respectively.
5.3 Control law design
The control laws are designed to stabilise the UAV flight. For the desired trajectory (
$x_{d},y_{d},z_{d}$
) and attitude angle (
$\psi _{d},\theta_{d},\phi _{d}$
), the error systems of position and attitude dynamics can be expressed, respectively, by
\begin{equation}\ddot{e}_{p}=m^{-1}\left(u_{p}+\Xi _{p}+\delta _{p}\right),\end{equation}
and
\begin{equation}\ddot{e}_{a}=J^{-1}(u_{a}+\Xi _{a}+\delta _{a}),\end{equation}
where,
$e_{p1}=x-x_{d}$
,
$e_{p2}=\dot{x}-\dot{x}_{d}$
,
$e_{p3}=y-y_{d}$
,
$e_{p4}=\dot{y}-\dot{y}_{d}$
,
$e_{p5}=z-z_{d}$
,
$e_{p6}=\dot{z}-\dot{z}_{d}$
;
$e_{a1}=\psi -\psi _{d}$
,
$e_{a2}=\dot{\psi}-\dot{\psi}_{d}$
,
$e_{a3}=\theta-\theta _{d}$
,
$e_{a4}=\dot{\theta}-\dot{\theta}_{d}$
,
$e_{a5}=\phi -\phi_{d}$
,
$e_{a6}=\dot{\phi}-\dot{\phi}_{d}$
;
\begin{equation}{e}_{p}{=}\left[\begin{array}{c}{e}_{p1} \\[3pt]{e}_{p3} \\[3pt]{e}_{p5}\end{array}\right] ,\text{ }u_{p}=\left[\begin{array}{c}u_{x} \\[3pt]u_{y} \\[3pt]u_{z}\end{array}\right] ,\text{ }{\delta _{p}}{=}\left[\begin{array}{c}{\Delta }_{x}-k_{x}\dot{x} \\[3pt]{\Delta }_{y}-k_{y}\dot{y} \\[3pt]{\Delta }_{z}-k_{z}\dot{z}\end{array}\right] ,{\Xi _{p}}{=}\left[\begin{array}{c}-m\ddot{x}_{d} \\[3pt]-m\ddot{y}_{d} \\[3pt]-m\ddot{z}_{d}-mg\end{array}\right],\end{equation}
and
\begin{equation}e_{a}=\left[\begin{array}{c}e_{a1} \\[3pt]e_{a3} \\[3pt]e_{a5}\end{array}\right] ,\text{ }u_{a}=\left[\begin{array}{c}u_{\psi } \\[3pt]u_{\theta } \\[3pt]u_{\phi }\end{array}\right] ,\text{ }\Xi _{a}=\left[\begin{array}{c}-J_{\psi }\ddot{\psi}_{d} \\[3pt]-J_{\theta }\ddot{\theta}_{d} \\[3pt]-J_{\phi }\ddot{\phi}_{d}\end{array}\right] ,\delta _{a}=\left[\begin{array}{c}{\Delta }_{\psi }-k_{\psi }\dot{\psi} \\[3pt]{\Delta }_{\theta }-lk_{\theta }\dot{\theta} \\[3pt]{\Delta }_{\phi }-lk_{\phi }\dot{\phi}\end{array}\right].\end{equation}
5.3.1 Position dynamics control:
In the position dynamics, for the desired trajectory (
$x_{d},y_{d},z_{d}$
), the control law
\begin{equation}u_{p}=-\Xi _{p}-\widehat{\delta }_{p}-m\left(k_{p1}\widehat{e}_{p}+k_{p2}\widehat{\dot{e}}_{p}\right),\end{equation}
is designed to make position error vectors
$e_{p}\rightarrow \vec{0}$
and
$\dot{e}_{p}\rightarrow \vec{0}$
as
$t\rightarrow \infty $
, where
$\widehat{e}_{p1}=\widehat{x}-x_{d}$
,
$\widehat{e}_{p2}=\widehat{\dot{x}}-\dot{x}_{d}$
,
$\widehat{e}_{p3}=\widehat{y}-y_{d}$
,
$\widehat{e}_{p4}=\widehat{\dot{y}}-\dot{y}_{d}$
,
$\widehat{e}_{p5}=\widehat{z}-z_{d}$
,
$\widehat{e}_{p6}=\widehat{\dot{z}}-\dot{z}_{d}$
and
$\widehat{\delta }_{p}$
are estimated by the correctors;
$k_{p1},k_{p2}>0$
; and
\begin{equation}\widehat{e}_{p}=\left[\begin{array}{ccc}\widehat{e}_{p1} & \widehat{e}_{p3} & \widehat{e}_{p5}\end{array}\right] ^{T},\widehat{\dot{e}}_{p}=\left[\begin{array}{ccc}\widehat{e}_{p2} & \widehat{e}_{p4} & \widehat{e}_{p6}\end{array}\right] ^{T}.\end{equation}
5.3.2 Attitude dynamics control:
In the attitude dynamics, for the desired attitude angle (
$\psi _{d},\theta _{d},\phi _{d}$
), the control law
\begin{equation}u_{a}=-\Xi _{a}-\widehat{\delta }_{a}-J\left(k_{a1}\widehat{e}_{a}+k_{a2}\widehat{\dot{e}}_{a}\right),\end{equation}
is designed to make attitude error vectors
$e_{a}\rightarrow \vec{0}$
and
$\dot{e}_{a}\rightarrow \vec{0}$
as
$t\rightarrow \infty $
, where,
$\widehat{e}_{a1}=\widehat{\psi }-\psi _{d}$
,
$\widehat{e}_{a2}=\widehat{\dot{\psi}}-\dot{\psi}_{d}$
,
$\widehat{e}_{a3}=\widehat{\theta }-\theta _{d}$
,
$\widehat{e}_{a4}=\widehat{\dot{\theta}}-\dot{\theta}_{d}$
,
$\widehat{e}_{a5}=\widehat{\phi }-\phi _{d}$
,
$\widehat{e}_{a6}=\widehat{\dot{\phi}}-\dot{\phi}_{d}$
and
$\widehat{\delta }_{a}$
are estimated by the observers;
$k_{a1},k_{a2}>0$
; and
\begin{equation}\widehat{e}_{a}=\left[\begin{array}{ccc}\widehat{e}_{a1} & \widehat{e}_{a3} & \widehat{e}_{a5}\end{array}\right] ^{T},\widehat{\dot{e}}_{a}=\left[\begin{array}{ccc}\widehat{e}_{a2} & \widehat{e}_{a4} & \widehat{e}_{a6}\end{array}\right] ^{T}.\end{equation}
6. Experiment on uav navigation and control
In this section, a UAV flight experiment is presented to demonstrate the proposed method. The UAV flight platform is explained in Fig. 3. The UAV navigation and control based on the decoupling corrector and observer are implemented in the platform setup. The control system hardware is described in Fig. 4, whose elements include: A Gumstix and Arduino Mega 2560 (16MHz) are selected as the driven boards; Gumstix is to collect data from measurements; Arduino Mega is to run algorithm of estimation and control, and it sends out control commands; A XsensMTI AHRS (10kHz) provides the 3-axial accelerations, the angular rates and the earth’s magnetic field.
Figure 3. Platform of UAV flight control system.
Figure 4. Control system hardware.
Figure 5. UAV navigation based on corrector and observer. (a) Navigation trajectories. (b) Position estimation.
Real position acquisition for comparison: In order to get the real position for comparison with the estimation by the corrector, the output of the Vicon system with sub-millimeter accuracy is taken as the real position.
Large-error position from GPS: A low-cost GPS receiver proivdes intermittent position signals with accuracy of 10
$\sim $
20m. When a intermittence happens, the most recent valid readings from GPS are taken as the measured position signals.
Accurate velocity sensing: A 24GHz microwave Doppler radar sensor is adopted to measure the velocity.
Desired flight trajectory: The UAV desired trajectory includes: (1) take off and climb; (2) then fly in a circle with the radius of 5m, the velocity of 1m/s and the altitude of 3m. The 3D desired trajectory is shown in Fig. 5(a).
The corrected positions from the signal correctors and the uncertainty estimations from the observers are used in the controllers. Controllers (29) and (31) drive the UAV to track the desired trajectory. The corrector parameters:
$k_{1,i}=1$
,
$k_{2,i}=30$
,
$1/\varepsilon _{c,i}=1.2$
,
$\alpha_{c,i}=0.1$
,
$i=1,2,3$
. The observer parameters:
$k_{3,i}=20$
,
$k_{4,i}=4$
,
$1/\varepsilon _{c,i}=1.1$
,
$\alpha _{o,i}=0.6$
,
$i=1,2,3$
. The control law parameters:
$k_{p1}=2.5$
,
$k_{p2}=4$
,
$k_{a1}=2.5$
,
$k_{a2}=4$
. The position-correction performance of corrector is compared with the EKF-based GPS/radar sensor integration.
Figure 5(a) shows the comparison of flight trajectories in 3D space, including the measured from GPS, the real from the Vicon, the desired trajectories, and the estimations by the corrector and the EKF. Meanwhile, the trajectory comparisons in the three directions are shown in Fig. 5(b): The measurement errors in position from GPS are about 20m. The estimate errors by the corrector are less than 0.04m, while the estimate errors by the EKF are about 5m. Thus, the large errors/disturbances in position measurements are rejected by the corrector, and the effect of noises is reduced sufficiently. In addition, during a 1,000s-duration flight test, no drift happened.
Uncertainties estimation: The unexpected uncertainties exist in the UAV flight, and we cannot read these uncertainties. Therefore, the real uncertainties cannot be determined to compare with the estimate results. Here, we use a simulation to illustrate the uncertainty estimations by the observers. The unknown drag coefficients in the UAV model are supposed to be:
$k_{x}=k_{y}=k_{z}=0.01\mathrm{N}\cdot \mathrm{s/m}$
,
$k_{\psi }=k_{\theta }=k_{\phi}=0.012\mathrm{N}\cdot \mathrm{s/rad}$
. The unmodelled uncertainties are assumed as:
$\Delta_{x}=0.3\sin (t)+0.2\cos (0.5t)$
,
$\Delta _{y}=0.2\sin (0.5t)+0.5\cos (t)$
,
$\Delta _{z}=0.4\sin (0.6t)+0.2\cos (t)$
. Then, we can determine the real uncertainty vector
${\delta _{p}}$
according to (27). All the parameters in the system model, correctors, observers and controllers are selected the same as those in the above experiment. Figure 6 shows that the observers can get the accurate estimation of uncertainties although much noise exists.
Figure 6. Uncertainty estimations.
7. Conclusions
For a class of uncertain systems with large-error sensing, according to the completely decoupling, the low-order signal corrector and observer have been developed to reject the large error in sensing and to estimate the system uncertainty. The proposed corrector and observer have been demonstrated by a UAV navigation-control experiment: It succeeded in rejecting the large errors/disturbances in position sensing, and the system uncertainties were estimated accurately. It confirms the merits of the corrector and observer: They can provide accurate and smoothed estimation of position and uncertainty even simultaneous existence of large sensing error and system uncertainty.
A. Appendix
Proof of Theorem 3.1
Proof of the general signal corrector (10) in Theorem 3.1: Define the corrector error as
$e_{1}=\widehat{x}_{1}-x_{1}$
and
$e_{2}=\widehat{x}_{2}-x_{2}$
. Then, the error system of signal corrector (10) and decoupled system (4) can be described by:
\begin{align}\dot{e}_{1} & =e_{2}; \nonumber \\\varepsilon _{c}^{3}\dot{e}_{2} & =f_{c}(\varepsilon_{c}(e_{1}-d(t)),e_{2})-\varepsilon _{c}^{3}\ddot{x}_{2}(t),\end{align}
and Equation (33) can be rewritten as
\begin{align}\frac{d\varepsilon _{c}e_{1}}{dt/\varepsilon _{c}} & =\varepsilon_{c}^{2}e_{2}; \nonumber \\[5pt]\frac{d\varepsilon _{c}^{2}e_{2}}{dt/\varepsilon _{c}} & =f_{c}\left(\varepsilon_{c}e_{1}-\varepsilon _{c}d(t),\frac{1}{\varepsilon _{c}^{2}}\varepsilon_{c}^{2}e_{2}\right)-\varepsilon _{c}^{3}\ddot{x}_{2}(t).\end{align}
By choosing the following coordinate transform
\begin{equation}\tau _{c}=t/\varepsilon _{c},z_{1}(\tau )=\varepsilon _{c}e_{1},z_{2}(\tau_{c})=\varepsilon _{c}^{2}e_{2},z_{c}=\left[\begin{array}{cc}z_{1}(\tau _{c}) & z_{2}(\tau _{c})\end{array}\right] ^{T};\overline{d}\left( \tau \right) =\varepsilon _{c}d\left(t\right) ;\,\overline{p}\left( \tau \right) =\varepsilon _{c}^{3}\ddot{x}_{2}(t),\end{equation}
we get
$z_{c}=\Xi (\varepsilon _{c})e_{c}$
, where,
$\Xi (\varepsilon_{c})=diag\{\varepsilon _{c},\varepsilon _{c}^{2}\}$
and
$e_{c}=[\begin{array}{cc}e_{1} & e_{2}\end{array}]^{{T}}$
. It is rational the system acceleration is bounded, and we can assume that
$\left\vert \ddot{x}_{2}(t)\right\vert \leq L_{p}<+\infty $
. Then, (34) becomes
\begin{align}\frac{dz_{1}}{d\tau _{c}} & =z_{2}; \nonumber \\\frac{dz_{2}}{d\tau _{c}} & =f_{c}\left(z_{1}-\overline{d}\left( \tau _{c}\right),\frac{1}{\varepsilon _{c}^{2}}z_{2}\right)-\overline{p}\left( \tau _{c}\right).\end{align}
Define
$k=\frac{1}{\varepsilon _{c}^{2}}$
and
\begin{equation}g(\tau _{c},z_{c}(\tau _{c}))=f_{c}\left(z_{1}-\overline{d}\left( \tau_{c}\right) ,\frac{1}{\varepsilon _{c}^{2}}z_{2}\right)-f_{c}\left(z_{1},\frac{1}{\varepsilon _{c}^{2}}z_{2}\right)-\overline{p}\left( \tau _{c}\right),\end{equation}
then, (36) can be rewritten as
\begin{align}\frac{dz_{1}}{d\tau _{c}} & =z_{2}; \nonumber \\\frac{dz_{2}}{d\tau _{c}} & =f_{c}(z_{1},k\cdot z_{2})+g(\tau_{c},z_{c}(\tau _{c})).\end{align}
From Assumption 3.3, the contraction mapping rule
$\left\vert f_{c}\left(z_{1}-\overline{d}\left( \tau _{c}\right) ,\frac{1}{\varepsilon _{c}^{2}}z_{2}\right)-f_{c}\left(z_{1},\frac{1}{\varepsilon _{c}^{2}}z_{2}\right)\right\vert \leq a\left\vert \overline{d}\left( \tau _{c}\right) \right\vert ^{\rho }$
holds, where,
$\rho \in \left( 0,1\right] $
. Then, we get
\begin{equation}\delta \overset{\text{define}}{=}\underset{(\tau _{c},z_{c})\in R^{3}}{\sup }\left\vert g(\tau _{c},z_{c}(\tau _{c}))\right\vert \leq a\left\vert\varepsilon _{c}L_{d}\right\vert ^{\rho }+\varepsilon _{c}^{3}L_{p}\leq \varepsilon ^{\rho }\delta _{c},\end{equation}
where
$\delta _{c}=aL_{d}^{\rho }+L_{p}$
. From Assumption 3.2, the unperturbed system
\begin{align}\frac{dz_{1}}{d\tau _{c}} & =z_{2}; \nonumber \\\frac{dz_{2}}{d\tau _{c}} & =f(z_{1},k\cdot z_{2}),\end{align}
is finite-time stable. Furthermore, from Proposition 8.1 in [Reference Bhat and Bernstein23], Theorem 5.2 in [Reference Bhat and Bernstein24] and (39), for (38), there exist the bounded constants
$\mu_{c}>0 $
and
$\Gamma \left( z_{c}\left( 0\right) \right) >0$
, such that, for
$\tau _{c}\geq \Gamma \left( z_{c}\left( 0\right) \right) $
,
\begin{equation}\left\Vert z_{c}\left( \tau \right) \right\Vert \leq \mu _{c}\delta ^{\gamma_{c}}\leq \mu _{c}(\varepsilon _{c}^{\rho }\delta _{c})^{\gamma _{c}}.\end{equation}
Therefore, from (35) and (41), we get
\begin{equation}\Vert\begin{array}{cc}\varepsilon _{c}e_{1} & \varepsilon _{c}^{2}e_{2}\end{array}\Vert \leq \mu _{c}\left( \varepsilon _{c}^{\rho }\delta _{c}\right)^{\gamma _{c}},\end{equation}
for
$t\geq \varepsilon _{c}\Gamma \left( {\Xi (\varepsilon }_{c}{)e}_{c}\left( {0}\right) \right) $
. Thus, for
$\forall t\in \lbrack \varepsilon_{c}\Gamma \left( {\Xi (\varepsilon }_{c}{)e}_{c}\left( {0}\right) \right),\infty )$
, the following relations hold:
\begin{equation}\left\vert e_{1}\right\vert \leq L_{c}\varepsilon ^{\rho \gamma_{c}-1},\left\vert e_{2}\right\vert \leq L_{c}\varepsilon ^{\rho \gamma_{c}-2},\end{equation}
where,
$L_{c}=\mu _{c}\delta _{c}^{\gamma }$
. Then, (43) can be written as
\begin{equation}e_{1}=O\left(\varepsilon _{c}^{\rho \gamma _{c}-1}\right),e_{2}=O(\varepsilon_{c}^{\rho \gamma _{c}-2}).\end{equation}
From Theorems 4.3 and 5.2 in [Reference Bhat and Bernstein24],
$\gamma _{c}$
can be chosen to be arbitrarily large, and
\begin{equation}\gamma _{c}>3/\rho,\end{equation}
is not restrictive. Accordingly, we can get
$\rho \gamma _{c}-i>1$
for
$i=1,2$
. It implies that, for
$\varepsilon _{c}\in \left( 0,1\right) $
, the estimate error in (44) is of higher order than the small perturbation. Consequently, the corrector can make the estimate errors sufficiently small.
Proof of the general uncertainty observer (11) in Theorem 3.1:
The decoupled system (5) from (1) can be rewritten by
\begin{align}\varepsilon _{o}\dot{x}_{2} & =\varepsilon _{o}x_{3}+\varepsilon _{o}h(t)\nonumber \\\varepsilon _{o}^{2}\dot{x}_{3} & =\varepsilon _{o}^{2}c_{\sigma }(t).\end{align}
Define the observer error as
$e_{3}=\widehat{x}_{3}-x_{2}$
and
$e_{4}=\widehat{x}_{4}-x_{3}$
. Then, the error system of the observer (11) and the equivalent decoupled system (46) can be described by:
\begin{align}\varepsilon _{o}\dot{e}_{3} & =\varepsilon _{o}e_{4}+f_{o1}(e_{3}) \nonumber \\\varepsilon _{o}^{2}\dot{e}_{4} & =f_{o2}(e_{3})-\varepsilon_{o}^{2}c_{\sigma }(t),\end{align}
and Equation (47) can be rewritten as
\begin{align}\frac{de_{3}}{dt/\varepsilon _{o}} & =\varepsilon _{o}e_{4}+f_{o1}(e_{3})\nonumber \\[4pt]\frac{d\varepsilon _{o}e_{4}}{dt/\varepsilon _{o}} & =f_{o2}(e_{3})-\varepsilon _{o}^{2}c_{\sigma }(t).\end{align}
By choosing the following coordinate transform
\begin{equation}\tau _{o}=t/\varepsilon _{o},z_{3}(\tau _{o})=e_{3},z_{4}(\tau_{o})=\varepsilon _{o}e_{4},z_{o}=\left[\begin{array}{cc}z_{3}(\tau _{o}) & z_{4}(\tau _{o})\end{array}\right] ^{T};\bar{c}\left( \tau _{o}\right) =\varepsilon _{o}^{2}c_{\sigma}(t),\end{equation}
we get
$z_{o}=\Xi (\varepsilon _{o})e_{o}$
, where,
$\Xi (\varepsilon_{o})=diag\{1,\varepsilon _{o}\}$
and
$e_{o}=[\begin{array}{cc}e_{3} & e_{4}\end{array}]^{{T}}$
. Then, (48) becomes
\begin{align}\frac{dz_{3}}{d\tau _{o}} & =z_{4}+f_{o1}(z_{3}) \nonumber \\[4pt]\frac{dz_{4}}{d\tau _{o}} & =f_{o2}(z_{3})-\bar{c}\left( \tau _{o}\right).\end{align}
From (49), we can get
\begin{equation}\delta _{o}\overset{\text{define}}{=}\underset{\tau _{o}\in R^{+}}{\sup }\left\vert \bar{c}\left( \tau _{o}\right) \right\vert \leq \varepsilon_{o}^{2}L_{\sigma }.\end{equation}
From Assumption 3.4, the unperturbed system
\begin{align}\frac{dz_{3}}{d\tau _{o}} & =z_{4}+f_{o1}(z_{3}) \nonumber \\[4pt]\frac{dz_{4}}{d\tau _{o}} & =f_{o2}(z_{3}),\end{align}
is finite-time stable. Furthermore, from Proposition 8.1 in [Reference Bhat and Bernstein23], Theorem 5.2 in [Reference Bhat and Bernstein24] and (51), for (50), there exist the bounded constants
$\mu_{o}>0 $
and
$\Gamma \left( z_{o}\left( 0\right) \right) >0$
, such that, for
$\tau _{o}\geq \Gamma \left( z_{o}\left( 0\right) \right) $
,
\begin{equation}\left\Vert z_{o}\left( \tau _{o}\right) \right\Vert \leq \mu _{o}\delta_{o}^{\gamma _{o}}\leq \mu _{o}\left(\varepsilon _{o}^{2}L_{\sigma }\right)^{\gamma_{o}}.\end{equation}
Therefore, from (49) and (53), we get
\begin{equation}\Vert\begin{array}{cc}e_{3} & \varepsilon _{o}e_{4}\end{array}\Vert \leq \mu _{o}\left( \varepsilon _{o}^{2}L_{\sigma }\right) ^{\gamma_{o}},\end{equation}
for
$t\geq \varepsilon _{o}\Gamma \left( {\Xi (\varepsilon }_{o}{)e}_{o}\left( {0}\right) \right) $
. Thus, for
$\forall t\in \lbrack \varepsilon_{o}\Gamma \left( {\Xi (\varepsilon }_{o}{)e}_{o}\left( {0}\right) \right),\infty )$
, the following relations hold:
\begin{equation}\left\vert e_{3}\right\vert \leq L_{o}\varepsilon _{o}^{2\gamma_{o}},\left\vert e_{4}\right\vert \leq L_{o}\varepsilon _{o}^{2\gamma _{o}-1},\end{equation}
where,
$L_{o}=\mu _{o}L_{\sigma }^{\gamma _{o}}$
. Then, (55) can be written as
\begin{equation}e_{3}=O(\varepsilon _{o}^{2\gamma _{o}}),e_{4}=O\left(\varepsilon _{o}^{2\gamma_{o}-1}\right).\end{equation}
From Theorems 4.3 and 5.2 in [Reference Bhat and Bernstein24],
$\gamma _{o}$
can be chosen to be arbitrarily large, and
\begin{equation}\gamma _{o}>1,\end{equation}
is not restrictive. Accordingly, we can get
$2\gamma _{o}-i>1$
for
$i=0,1$
. It implies that, for
$\varepsilon _{o}\in \left( 0,1\right) $
, the estimate error in (56) is of higher order than the small perturbation. Consequently, the uncertainty observer can make the estimate errors sufficiently small.
Proof of Theorem 4.1
Proof of the signal corrector (14) in Theorem 4.1:
Define the corrector error as
$e_{1}=\widehat{x}_{1}-x_{1}$
and
$e_{2}=\widehat{x}_{2}-x_{2}$
. Then, the error system of signal corrector (14) and decoupled system (4) can be described by:
\begin{align}\dot{e}_{1} & =e_{2}; \nonumber \\\varepsilon _{c}^{3}\dot{e}_{2} & =-k_{1}\left\vert \varepsilon_{c}(e_{1}-d(t))\right\vert ^{\frac{\alpha _{c}}{2-\alpha _{c}}}\text{sign}\left( e_{1}-d(t)\right) -k_{2}\left\vert e_{2}\right\vert ^{\alpha _{c}}\text{sign}\left( e_{2}\right) -\varepsilon _{c}^{3}\ddot{x}_{2}(t),\end{align}
and Equation (58) can be rewritten as
\begin{align}\frac{d\varepsilon _{c}e_{1}}{dt/\varepsilon _{c}} & =\varepsilon_{c}^{2}e_{2}; \nonumber \\[5pt]\frac{d\varepsilon _{c}^{2}e_{2}}{dt/\varepsilon _{c}} & =-k_{1}\left\vert\varepsilon _{c}e_{1}-\varepsilon _{c}d(t)\right\vert ^{\frac{\alpha _{c}}{2-\alpha _{c}}}\text{sign}\left( e_{1}-d(t)\right) -\frac{k_{2}}{\varepsilon_{c}^{2\alpha _{c}}}\left\vert \varepsilon _{c}^{2}e_{2}\right\vert ^{\alpha_{c}}\text{sign}\left( e_{2}\right) -\varepsilon _{c}^{3}\ddot{x}_{2}(t).\end{align}
By choosing the following coordinate transform
\begin{equation}\tau _{c}=t/\varepsilon _{c},z_{1}(\tau _{c})=\varepsilon_{c}e_{1},z_{2}(\tau _{c})=\varepsilon _{c}^{2}e_{2},z_{c}=\left[\begin{array}{cc}z_{1}(\tau _{c}) & z_{2}(\tau _{c})\end{array}\right] ^{T};\overline{d}\left( \tau _{c}\right) =\varepsilon _{c}d\left(t\right) ;\overline{p}\left( \tau _{c}\right) =\varepsilon _{c}^{3}\ddot{x}_{2}(t),\end{equation}
we get
$z_{c}=\Xi (\varepsilon _{c})e_{c}$
, where,
$\Xi (\varepsilon_{c})=diag\{\varepsilon _{c},\varepsilon _{c}^{2}\}$
and
$e_{c}=[\begin{array}{cc}e_{1} & e_{2}\end{array}]^{{T}}$
. Then, (59) becomes
\begin{align}\frac{dz_{1}}{d\tau _{c}} & =z_{2}; \nonumber \\[5pt]\frac{dz_{2}}{d\tau _{c}} & =-k_{1}\left\vert z_{1}-\overline{d}\left( \tau_{c}\right) \right\vert ^{\frac{\alpha _{c}}{2-\alpha _{c}}}\text{sign}\left( z_{1}-\overline{d}\left( \tau _{c}\right) \right) -\frac{k_{2}}{\varepsilon _{c}^{2\alpha _{c}}}\left\vert z_{2}\right\vert ^{\alpha _{c}}\text{sign}\left( z_{2}\right) -\overline{p}\left( \tau _{c}\right).\end{align}
Define
\begin{equation}g(\tau _{c},z(\tau _{c}))=-k_{1}\left\{ \left\vert z_{1}-\overline{d}(\tau_{c})\right\vert ^{\frac{\alpha _{c}}{2-\alpha _{c}}}\text{sign}\left( z_{1}-\overline{d}(\tau _{c})\right) -\left\vert z_{1}\right\vert ^{\frac{\alpha_{c}}{2-\alpha _{c}}}\text{sign}\left( z_{1}\right) \right\} -\overline{p}\left( \tau _{c}\right),\end{equation}
then, (61) can be rewritten as
\begin{align}\frac{dz_{1}}{d\tau _{c}} & =z_{2}; \nonumber \\[5pt]\frac{dz_{2}}{d\tau _{c}} & =-k_{1}\left\vert z_{1}\right\vert ^{\frac{\alpha _{c}}{2-\alpha _{c}}}\text{sign}\left( z_{1}\right) -\frac{k_{2}}{\varepsilon _{c}^{2\alpha _{c}}}\left\vert z_{2}\right\vert ^{\alpha _{c}}\text{sign}\left( z_{2}\right) +g(\tau _{c},z(\tau _{c})).\end{align}
Since the contraction mapping rule
$\left\vert x^{\rho }-\overline{x}^{\rho}\right\vert \leq 2^{1-\rho }\left\vert x-\overline{x}\right\vert ^{\rho},\rho \in \left( 0,1\right] $
, we obtain
\begin{equation}\delta \overset{\text{define}}{=}\underset{(\tau _{c},z_{c})\in R^{3}}{\sup }\left\vert g(\tau _{c},z_{c}(\tau _{c}))\right\vert \leq 2^{1-\frac{\alpha_{c}}{2-\alpha _{c}}}k_{1}L_{d}^{\frac{\alpha _{c}}{2-\alpha _{c}}}\varepsilon _{c}^{\frac{\alpha _{c}}{2-\alpha _{c}}}+\varepsilon_{c}^{3}L_{p}\leq \varepsilon _{c}^{\frac{\alpha _{c}}{2-\alpha _{c}}}\delta_{c},\end{equation}
where
$\delta _{c}=2^{1-\frac{\alpha _{c}}{2-\alpha _{c}}}k_{1}L_{d}^{\frac{\alpha _{c}}{2-\alpha _{c}}}+L_{p}$
. From [Reference Bhat and Bernstein23], we know that the unperturbed system
\begin{align}\frac{dz_{1}}{d\tau _{c}} & =z_{2}; \nonumber \\[5pt]\frac{dz_{2}}{d\tau _{c}} & =-k_{1}\left\vert z_{1}\right\vert ^{\frac{\alpha _{c}}{2-\alpha _{c}}}\text{sign}\left( z_{1}\right) -\frac{k_{2}}{\varepsilon _{c}^{2\alpha _{c}}}\left\vert z_{2}\right\vert ^{\alpha _{c}}\text{sign}\left( z_{2}\right),\end{align}
is finite-time stable. Furthermore, from Proposition 8.1 in [Reference Bhat and Bernstein23], Theorem 5.2 in [Reference Bhat and Bernstein24] and (64), for (63), there exist the bounded constants
$\mu_{c}>0 $
and
$\Gamma \left( z_{c}\left( 0\right) \right) >0$
, such that, for
$\tau _{c}\geq \Gamma \left( z_{c}\left( 0\right) \right) $
,
\begin{equation}\left\Vert z_{c}\left( \tau _{c}\right) \right\Vert \leq \mu _{c}\delta_{c}^{\gamma _{c}}\leq \mu _{c}\left(\varepsilon _{c}^{\frac{\alpha _{c}}{2-\alpha _{c}}}\delta _{c}\right)^{\gamma _{c}}.\end{equation}
Therefore, from (60) and (66), we get
\begin{equation}\Vert\begin{array}{cc}\varepsilon _{c}e_{1} & \varepsilon _{c}^{2}e_{2}\end{array}\Vert \leq \mu _{c}\left( \varepsilon _{c}^{\frac{\alpha _{c}}{2-\alpha _{c}}}\delta _{c}\right) ^{\gamma _{c}},\end{equation}
for
$t\geq \varepsilon _{c}\Gamma \left( {\Xi (\varepsilon }_{c}{)e}_{c}\left( {0}\right) \right) $
. Thus, for
$\forall t\in \lbrack \varepsilon_{c}\Gamma \left( {\Xi (\varepsilon }_{c}{)e}_{c}\left( {0}\right) \right),\infty )$
, the following relations hold:
\begin{equation}\left\vert e_{1}\right\vert \leq L_{c}\varepsilon _{c}^{\frac{\alpha _{c}}{2-\alpha _{c}}\gamma _{c}-1},\left\vert e_{2}\right\vert \leq L_{c}\varepsilon _{c}^{\frac{\alpha _{c}}{2-\alpha _{c}}\gamma _{c}-2},\end{equation}
where,
$L_{c}=\mu _{c}\delta _{c}^{\gamma _{c}}$
. Then, (68) can be written as
\begin{equation}e_{1}=O\left(\varepsilon ^{\frac{\alpha _{c}}{2-\alpha _{c}}\gamma_{c}-1}\right),e_{2}=O\left(\varepsilon ^{\frac{\alpha _{c}}{2-\alpha _{c}}\gamma_{c}-2}\right).\end{equation}
From Theorems 4.3 and 5.2 in [24],
$\gamma _{c}$
can be chosen to be arbitrarily large, and
\begin{equation}\gamma _{c}>\frac{6-3\alpha _{c}}{\alpha _{c}},\end{equation}
is not restrictive. Accordingly, we can get
$\frac{\alpha _{c}}{2-\alpha _{c}}\gamma _{c}-i>1$
for
$i=1,2$
. It implies that, for
$\varepsilon _{c}\in\left( 0,1\right) $
, the estimate error in (69) is of higher order than the small perturbation. For
$\varepsilon _{c}\in (0,1)$
, according to the Routh-Hurwitz Stability Criterion,
$s^{2}+\frac{k_{2}}{\varepsilon_{c}^{2\alpha _{c}}}s+k_{1}$
is Hurwitz if
$k_{1}>0$
and
$k_{2}>0$
.
Proof of the uncertainty observer (15) in Theorem 4.1:
The decoupled system (5) from (1) can be rewritten by
\begin{align}\varepsilon _{o}\dot{x}_{2} & =\varepsilon _{o}x_{3}+\varepsilon _{o}h(t)\nonumber \\\varepsilon _{o}^{2}\dot{x}_{3} & =\varepsilon _{o}^{2}c_{\sigma }(t).\end{align}
Define the observer error as
$e_{3}=\widehat{x}_{3}-x_{2}$
and
$e_{4}=\widehat{x}_{4}-x_{3}$
. Then, the error system of observer (15) and decoupled system (71) can be described by:
\begin{align}\varepsilon _{o}\dot{e}_{3} & =\varepsilon _{o}e_{4}-k_{4}\left\vert e_{3}\right\vert ^{\frac{\alpha _{o}+1}{2}}\text{sign}\left( e_{3}\right)\nonumber \\[3pt]\varepsilon _{o}^{2}\dot{e}_{4} & =-k_{3}\left\vert e_{3}\right\vert^{\alpha _{o}}\text{sign}\left( e_{3}\right) -\varepsilon _{o}^{2}c_{\sigma}(t),\end{align}
and Equation (72) can be rewritten as
\begin{align}\frac{de_{3}}{dt/\varepsilon _{o}} & =\varepsilon _{o}e_{4}-k_{4}\left\vert e_{3}\right\vert ^{\frac{\alpha _{o}+1}{2}}\text{sign}\left( e_{3}\right)\nonumber \\[3pt]\frac{d\varepsilon _{o}e_{4}}{dt/\varepsilon _{o}} & =-k_{3}\left\vert e_{3}\right\vert ^{\alpha _{o}}\text{sign}\left( e_{3}\right) -\varepsilon_{o}^{2}c_{\sigma }(t).\end{align}
By choosing the following coordinate transform
\begin{equation}\tau _{o}=t/\varepsilon _{o},z_{3}(\tau _{o})=e_{3},z_{4}(\tau_{o})=\varepsilon _{o}e_{4},z_{o}=\left[\begin{array}{cc}z_{3}(\tau _{o}) & z_{4}(\tau _{o})\end{array}\right] ^{T};\bar{c}\left( \tau _{o}\right) =\varepsilon _{o}^{2}c_{\sigma}(t),\end{equation}
we get
$z_{o}=\Xi (\varepsilon _{o})e_{o}$
, where,
$\Xi (\varepsilon_{o})=diag\{1,\varepsilon _{o}\}$
and
$e_{o}=[\begin{array}{cc}e_{3} & e_{4}\end{array}]^{{T}}$
. Then, (73) becomes
\begin{align}\frac{dz_{3}}{d\tau _{o}} & =z_{4}-k_{4}\left\vert z_{3}\right\vert ^{\frac{\alpha _{o}+1}{2}}\text{sign}\left( z_{3}\right) \nonumber \\[3pt]\frac{dz_{4}}{d\tau _{o}} & =-k_{3}\left\vert z_{3}\right\vert ^{\alpha _{o}}\text{sign}\left( z_{3}\right) -\bar{c}\left( \tau _{o}\right).\end{align}
From (74), we can get
\begin{equation}\delta _{o}\overset{\text{define}}{=}\underset{\tau _{o}\in R^{+}}{\sup }\left\vert \bar{c}\left( \tau _{o}\right) \right\vert \leq \varepsilon_{o}^{2}L_{\sigma }.\end{equation}
From Theorem 1 in [Reference Wang and Lin27], we know that the unperturbed system
\begin{align}\frac{dz_{3}}{d\tau _{o}} & =z_{4}-k_{4}\left\vert z_{3}\right\vert ^{\frac{\alpha _{o}+1}{2}}\text{sign}\left( z_{3}\right), \nonumber \\[3pt]\frac{dz_{4}}{d\tau _{o}} & =-k_{3}\left\vert z_{3}\right\vert ^{\alpha _{o}}\text{sign}\left( z_{3}\right),\end{align}
is finite-time stable. Furthermore, from Proposition 8.1 in [Reference Bhat and Bernstein23], Theorem 5.2 in [Reference Bhat and Bernstein24] and (76), for (75), there exist the bounded constants
$\mu_{o}>0 $
and
$\Gamma \left( z_{o}\left( 0\right) \right) >0$
, such that, for
$\tau _{o}\geq \Gamma \left( z_{o}\left( 0\right) \right) $
,
\begin{equation}\left\Vert z_{o}\left( \tau _{o}\right) \right\Vert \leq \mu _{o}\delta_{o}^{\gamma _{o}}\leq \mu _{o}\left(\varepsilon _{o}^{2}L_{\sigma }\right)^{\gamma_{o}}.\end{equation}
Therefore, from (74) and (78), we get
\begin{equation}\Vert\begin{array}{cc}e_{3} & \varepsilon _{o}e_{4}\end{array}\Vert \leq \mu _{o}\left( \varepsilon _{o}^{2}L_{\sigma }\right) ^{\gamma_{o}},\end{equation}
for
$t\geq \varepsilon _{o}\Gamma \left( {\Xi (\varepsilon }_{o}{)e}_{o}\left( {0}\right) \right) $
. Thus, for
$\forall t\in \lbrack \varepsilon_{o}\Gamma \left( {\Xi (\varepsilon }_{o}{)e}_{o}\left( {0}\right) \right),\infty )$
, the following relations hold:
\begin{equation}\left\vert e_{3}\right\vert \leq L_{o}\varepsilon _{o}^{2\gamma_{o}},\left\vert e_{4}\right\vert \leq L_{o}\varepsilon _{o}^{2\gamma _{o}-1},\end{equation}
where,
$L_{o}=\mu _{o}L_{\sigma }^{\gamma _{o}}$
. Then, (80) can be written as
\begin{equation}e_{3}=O(\varepsilon _{o}^{2\gamma _{o}}),e_{4}=O\left(\varepsilon _{o}^{2\gamma_{o}-1}\right).\end{equation}
From Theorems 4.3 and 5.2 in [Reference Bhat and Bernstein24],
$\gamma _{o}$
can be chosen to be arbitrarily large, and
\begin{equation}\gamma _{o}>1,\end{equation}
is not restrictive. Accordingly, we can get
$2\gamma _{o}-i>1$
for
$i=0,1$
. It implies that, for
$\varepsilon _{o}\in \left( 0,1\right) $
, the estimate error in (81) is of higher order than the small perturbation. According to the Routh-Hurwitz stability criterion,
$s^{2}+k_{4}s+k_{3}$
is Hurwitz if
$k_{3}>0$
and
$k_{4}>0$
.
This concludes the proof.
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