1. 서 론

For the last decade, the control problems of unmanned aerial vehicle have received a lot of attention and currently remain as active research areas in control engineering field [1-3,5,7-10, 12,16-18]. For example, in ⁽¹⁾, a model predictive control algorithm deploying fewer prediction points and less computational requirement is presented in controlling unmanned quadrotor helicopter; in ^(7,8), the robust attitude control problem of quadrotor is addressed under parameter perturbations and external disturbance as uncertainties; and in ⁽¹⁸⁾, an adaptive controller for quadrotor in the presence of time-varying aerodynamic effect and bounded external disturbance is proposed.

However, the previous results did not consider some measurement sensitivity or noise in feedback sensor. As addressed in ^(4,6,13,14), in practice, there can often be uncertain measurement sensitivity and sensor noise in feedback channel. Furthermore, there can be various uncertainties such as unknown parameters, uncertain dynamics, and external disturbance in the actual system model. So, under these conditions, the stability or boundedness of the closed-loop system of quadrotor should be further considered carefully. With respect to measurement sensitivity, there are some recent results ^(4,15). In ⁽⁴⁾, they newly addressed a system stabilization problem where there is an uncertain measurement sensitivity issue in feedback channel. However, they considered a single measurement sensitivity issue in the output feedback. Although in ⁽¹⁵⁾, the robust state estimation problem for two-dimension systems with measurement noise was investigated, their aim was to minimize the upper bound of the error covariance by using the proposed filter. Also, they just considered unknown measurement noise in an additive form.

In this paper, we consider a robust attitude control problem for quadrotor under unknown measurement sensitivity and external disturbance. Since we are addressing a state feedback case, there is measurement sensitivity issue for each state in feedback channel. Here, we formally state of our control problem.

Problem 1. We aim to show that in the presence of measurement sensitivity and external disturbance, (i) some non-trivial measurement sensitivity in each state can be allowed; (ii) the boundedness of controlled system states can be obtained; (iii) the ultimate bounds of controlled system states can be reduced.

Our approach to solving our control problem is organized as: (i) a new determination process of a compact set with allowed measurement sensitivity is presented by utilizing Lyapunov equation; (ii) a newly designed robust attitude controller with gain-scaling factor is proposed for boundedness of all system states; (iii) the ultimate bounds of controlled states are shown to be made arbitrarily small by utilizing the gain-scaling factor; (iv) both numerical simulation and experiment results are presented to illustrate the effectiveness of the proposed control scheme in the attitude control of quadrotor.

2. Quadrotor System Description

As shown in 그림 1(a), the body axes coordinate system of a quadrotor is denoted $(x,\: y,\: z)$.

The attitude of quadrotor is described using Euler angle and

$(\phi ,\:\theta ,\:\psi)$ represents roll, pitch, and yaw angles, which are defined as the rotation angle about the $x$, $y$ and $z$ axis, respectively. The model of the yaw axis is shown in 그림 1(b) where the directions of the propeller motion are indicated. The motors and propellers are configured so that the front and back motors spin clockwise and the left and right motors spin counter-clockwise when viewed from the top.

The mathematical attitude model of quadrotor with external disturbance is shown as

where the distance between the mass center and motor of quadrotor is represented by $l$, and uncertain drag coefficients are described by $\kappa_{1}$, $\kappa_{2}$, $\kappa_{3}$ associated with the aerodynamic drag force. The moment of inertia about the $x$-axis, $y$-axis, $z$-axis, are represented by $I_{x}$, $I_{y}$, $I_{z}$, respectively. The wind disturbances are $w_{\phi}(t)$, $w_{\theta}(t)$, and $w_{\psi}(t)$. For convenience, the input $U_{\phi}$, $U_{\theta}$ and $U_{\psi}$ corresponding to the roll, pitch, and yaw moments, respectively are defined as

where thrust and drag factors are symbolized by $K_{t}$ and $K_{y}$, respectively.

The symbols are summarized in 그림 1.

In a state-space representation, the quadrotor model 식(1) can be written as

or in a matrix form as

where $x=[\phi ,\:\dot\phi ,\:\theta ,\:\dot\theta ,\:\psi ,\:\dot\psi]^{T}$$=[x_{1},\:x_{2},\:x_{3},\:x_{4},\:x_{5},\:x_{6}]^{T}\in R^{6}$ is the state, $u=\left[\dfrac{l U_{\phi}}{I_{x}},\:\dfrac{l U_{\theta}}{I_{y}},\:\dfrac{U_{\psi}}{I_{z}}\right]^{T}$$=[u_{\phi},\:u_{\theta},\:u_{\psi}]^{T}\in R^{3}$ is the control input, uncertain perturbed term is represented by $\eta(t,\:x)=$$\left[0,\:-\dfrac{\kappa_{1}}{I_{x}}x_{2},\: 0,\: -\dfrac{\kappa_{2}}{I_{y}}x_{4},\: 0,\: -\dfrac{\kappa_{3}}{I_{z}}x_{6}\right]^{T}$$=[0,\:\eta_{2}(t,\:x,\:u),\:0,\:\eta_{4}(t,\:x,\:u),\:$$0,\:\eta_{6}(t,\:x,\:u)]^{T}: R\times R^{6}\times R\to R^{6}$, and external disturbance is represented by $w(t)=[0,\:w_{\phi}(t),\:0,\:w_{\theta}(t),\:0,\:w_{\psi}(t)]^{T}$$=[0,\:w_{2}(t),\:$$0,\:w_{4}(t),\:0,\:w_{6}(t)]^{T}\in R^{6}$. The system matrices $A\in R^{6\times 6}$, $B\in R^{6\times 3}$ are

Regarding the measurement sensitivity associated with state, the following condition is imposed.

Assumption 1. There exist uncertain measurement sensitivities $\rho_{i}(t)$ in feedback state $x_{i}$, $i=1$, $\cdots$, $6$ such that $\rho_{i}(t)x_{i}$ are available instead of $x_{i}$ where $\rho_{i}(t)>0$ are bounded and uncertain continuous functions of time. Moreover, $\rho_{i}(t)$ are not necessarily differentiable.

Note that under Assumption 1, the existing results ^(4,13,14) have carried out system analysis by using norm-bound condition, which often yields somewhat conservative results regarding the allowed measurement sensitivity. In our case, we take a matrix inequality approach to obtain more relaxed measurement sensitivity.

In accordance with Assumption 1, if there is no measurement sensitivity, each measurement sensitivity is $\rho_{i}(t)=1$, $i=1$, $\cdots$, $6$. So, each $\rho_{i}(t)$ can be represented as $\rho_{i}(t)=$$1+\rho_{i,\:\delta}(t)$, $\rho_{i,\:\delta}(t)> -1$. All measurement sensitivity $\rho_{i}(t)$ are contained in a compact set $\Omega_{\rho}$.

그림 2 shows the nominal values of quadrotor systems parameters.

Some notations are provided to be used throughout the paper for convenience.

Notations: For any matrix $M^{T}=M$, $\lambda_{\min}(M)$ denotes the minimum eigenvalue of $M$. $||E ||$ denotes the Euclidean norm. Other norms will be denoted by their subscripts. $I_{n}$ denotes an $n\times n$ identity matrix. Define $K=K(1)$, $A_{K}=A_{K(1)}$, $A_{K}^{\rho(t)}$$≡ A_{K(1)}^{\rho(t)}$, $\left. A_{K}=A_{K}^{\rho(t)}\right |_{\rho(t)=I_{6}}$, $\rho_{\delta}(t)={diag}[\rho_{1,\:\delta}(t),\:\cdots ,\:\rho_{6,\:\delta}(t)]$, $\rho(t)=I_{6}+\rho_{\delta}(t)$, and $A_{K}^{\rho(t)}=A_{K}+BK\rho_{\delta}(t)$. The determinant of any matrix $M$ is denoted by $\det(M)$.

In the next section, we will propose a robust controller and present determination process of a compact set $\Omega_{\rho}$ where allowed measurement sensitivity is contained in $\Omega_{\rho}$.

3. Robust Attitude Controller and Determination of a Compact Set with Allowed Measurement Sensitivity

The gain-scaling feedback controller under uncertain measure- ment sensitivity takes the following form

where $\rho(t)={diag}[\rho_{1}(t),\:\cdots ,\:\rho_{6}(t)]$,

From 식(4) and 식(6), the closed-loop system is obtained as

where $A_{K(\epsilon)}^{\rho(t)}=A+BK(\epsilon)\rho(t)$.

The configuration of the robust attitude control schematic is shown in 그림 2. The closed-loop system can be divided into roll, pitch, and yaw subsystems. So, each robust attitude control gain of subsystems can be separately designed.

Now, the following process is introduced in order to determine the allowed range of measurement sensitivity.

Determination process of a compact set $\Omega_{\rho}$:

(i) Select $K$ such that $A_{K}$ is Hurwitz. Let $M={diag}[m_{1},\:$$\cdots ,\: m_{6}]$$>0$.

(ii) Compute a positive definite symmetric matrix $P=[p_{i,\:j}]$, $\in R^{6\times 6}$, $i=1$, $\cdots$, $6$, $j=1$, $\cdots$, $6$ from the following Lyapunov equation

where $P={diag}[P_{1},\: P_{2},\: P_{3}]$, $P_{1}=[p_{i,\:j}]\in R^{2\times 2}$, $i=1$, $2$, $j=1$, $2$, $P_{2}=[p_{i,\:j}]\in R^{2\times 2}$, $i=3$, $4$, $j=3$, $4$, and $P_{3}=[p_{i,\:j}]$ $\in R^{2\times 2}$, $i=5$, $6$, $j=5$, $6$.

(iii) Compute a symmetric matrix $Q(t)=[q_{i,\:j}(t)]\in R^{6\times 6}$ $i=1$, $\cdots$, $6$, $j=1$, $\cdots$, $6$ by following

(iv) Determine the compact set $\Omega_{\rho}$ with allowed measurement sensitivity as long as $Q(t)>0$, $\forall t\ge 0$. Then, allowed measurement sensitivity $\rho(t)$ is contained in $\Omega_{\rho}$.

Note that in order for $Q(t)$ to be positive definite for all times, each of sub-matrices $Q_{i}(t)$, $i=1$, $\cdots$, $6$ of $Q(t)$ should be satisfied with $\det(Q_{i}(t))>0$ where sub-matrices $Q_{i}(t)$ are defined as ⁽¹⁹⁾:

So, a compact set $\Omega_{\rho}$ with the allowed $\rho_{i}(t)$ can be actually obtained from 식(11). However, if there are two or more variables $\rho_{i}(t)$ in the sub-matrices $Q_{i}(t)$ in 식(11), the calculation is considerably complicated and a compact set $\Omega_{\rho}$ may be varied by the ranges of $\rho_{i}(t)$. For example, from~ 식(10), the obtained $Q(t)$ is

From 식(11), the following conditions should be satisfied to obtain a compact set $\Omega_{\rho}$.

Then, the range of allowed $\rho_{2}(t)$ in 식(14) can be changed according to the upper-bound of the range $-1<\rho_{1}(t)<\infty$ in 식(13), because $\rho_{1}(t)$ and $\rho_{2}(t)$ are interactive with each other in the conditions 식(13)- 식(14). In other words, a compact set $\Omega_{\rho}$ can be varied by the range of allowed $\rho_{1}(t)$. In this regard, we introduce a relatively simple and easier way to estimate $\Omega_{\rho}$ regardless of each range of allowed $\rho_{i}(t)$ in the following.

A simplified algorithm in obtaining an estimated $\widetilde\Omega_{\rho}$ from $Q(t)$: Steps (iii)-(iv) of determination process are replaced by the following steps 식(1) to 식(4).

1. Calculate $M^{\rho_{\delta}(t)}$ as

where $P$ from 식(9).

2. Let

3. Obtain compact sets $\widetilde\Omega_{\rho_{i}}$, $i=1$, $2$ that satisfy

when $\widetilde\Omega_{\rho_{i}}$ are non-empty set. If $\rho_{1,\:\delta}(t)=\rho_{3,\:\delta}(t)=\rho_{5,\:\delta}(t)=0$, then $\widetilde\Omega_{\rho_{1}}=\varnothing$ (empty set). Similarly, $\widetilde\Omega_{\rho_{2}}=\varnothing$ when $\rho_{2,\:\delta}(t)$$=$$\rho_{4,\:\delta}(t)=\rho_{6,\:\delta}(t)=0$.

4. Obtain the estimated set $\widetilde\Omega_{\rho}$ as

Lemma 1. After the steps (i)-(ii) of the process of obtaining a compact set $\Omega_{\rho}$, if each of $M+ M_{1}^{\rho_{\delta}(t)}>0$ and $M+ M_{2}^{\rho_{\delta}(t)}$ $>0$ in 식(18) holds, then $Q(t)$ becomes positive definite.

Proof. Given that the steps (i)-(ii) are completed, using the relation $A^{\rho(t)}_{K}=A_{K}+A^{\rho_{\delta}(t)}_{K}$ and 식(9)- 식(10), 식(15)- 식(16), we can obtain

Note that if any each square symmetric matrix $A_{1}$, $A_{2}$ is positive definite, then $A_{1}+A_{2}$ is positive definite. Thus, since each of $M+ M_{1}^{\rho_{\delta}(t)}$ and $M+ M_{2}^{\rho_{\delta}(t)}$ is positive definite, $Q(t)$ becomes positive definite from 식(20). ꟃ

So, in accordance with Lemma 1, $M_{1}^{\rho_{\delta}(t)}$ and $M_{2}^{\rho_{\delta}(t)}$ are obtained by using 식(16).

Then, using 식(21)- 식(22), the following inequalities must be satisfied for $Q(t)>0$, $\forall t\ge 0$.

Thus, with the conditions 식(23), the estimated set $\widetilde\Omega_{\rho}$ of an allowed measurement sensitivity can be obtained. In Section 5, we will show the actual calculations in obtaining a compact set of the allowed measurement sensitivity.

4. System Analysis

We present the robust analysis of the system matrix against uncertain measurement sensitivity $\rho(t)$ and external disturbance $w(t)$ in the following.

Theorem 1. Suppose that the determination process of $\Omega_{\rho}$ is followed. There exists $\epsilon >0$ such that the following holds:

where $\sigma =4\sqrt{6}\alpha^{-1}||P ||$, $\alpha ={i nf}_{t\ge 0}\left\{\lambda_{\min}(Q(t))\right\}$ is a positive constant, $\gamma\dfrac{=}{l\in}e\kappa /\underline l\in e I$, $\dfrac{}{l\in}e\kappa =\max\left\{|\kappa_{1}|,\:|\kappa_{2}|,\:|\kappa_{3}|\right\}$, and $\underline l\in e I =\min\left\{|I_{x}|,\:\right.$ $\left. |I_{y}|,\:|I_{z}|\right\}$ is a positive constant associated with perturbed terms in 식(3), $P$ is a positive definite matrix in 식(9), and $Q(t)$ is a positive definite matrix function in 식(10). Also, there always exists $\epsilon^{*}=1/(\sigma\gamma)>0$ such that for $\epsilon\in(0,\:\epsilon^{*})$ and the closed-loop system 식(8) remains bounded with the controller 식(6).

Moreover, the ultimate bounds of all system states are as follows and can be made arbitrarily small by adjusting $\epsilon$.

where $p_{2i-1,\:2i}$ and $p_{2i,\:2i}$ are elements of $P$.

Proof. Define $E_{\epsilon}={diag}[1,\:\epsilon ,\:1,\:\epsilon ,\:1,\:\epsilon]$. Then, the following relation holds for all $\epsilon > 0$:

By substituting 식(26) into 식(10), we can derive a Lyapunov equation such as

where $P_{\epsilon}=E_{\epsilon}P E_{\epsilon}>0$.

Now, we set a Lyapunov function as $V(x)=x^{T}P_{\epsilon}x$. Along the trajectory of 식(8), we have the time-derivative of the Lyapunov function using 식(27) as

Note that there is the following relation

where $\alpha ={i nf}_{t\ge 0}\left\{\lambda_{\min}[Q(t)]\right\}$ is a positive real constant due to $Q^{T}(t)=Q(t)>0$, $\forall t\ge 0$.

With 식(29) and noting that $| E_{\epsilon}x |^{T}| E_{\epsilon}x | = || E_{\epsilon}x ||^{T}|| E_{\epsilon}x ||$, we have

Here, we investigate the norm bound of $||E_{\epsilon}\eta(t,\:x)||$ as follows. For $i=1$, $\cdots$, $6$, there exists a constant $\gamma\ge 0$ such that for $\epsilon >0$,

From 식(30)- 식(31), we have the following inequality

Then, there always exists $\epsilon^{*}$ such that for $0<\epsilon <\epsilon^{*}$,

where $\sigma =4\sqrt{6}\alpha^{-1}||P ||$.

For $0<\epsilon <\epsilon^{*}$, we can select $\epsilon$ such that $\Delta(\epsilon)>0$ in 식(33). With 식(32)- 식(33), we obtain

For $\dot V(e,\:x)$ to be negative definite, we need to have

Given that $\Delta(\epsilon)>0$ holds from 식(33), the ultimate bound of each system state $x_{i}$ is summarized as

Thus, the ultimate bounds of states $x_{1}$, $\cdots$, $x_{6}$ can be made arbitrarily small by decreasing $\epsilon$.

5. Numerical Simulation

In this section, we will show the validity of our control method via numerical simulation. For simulation, the initial conditions are set as $x_{0}=[0.3,\: 0,\: -0.15,\: 0,\: 0.2,\: 0]^{T}$ and aerodynamic drag force coefficients are set as $\kappa_{1}=\kappa_{2}=\kappa_{3}$$=0.012$. Let measurement sensitivities be

External disturbances are set as

The values of $K$ is selected as $K=[-6,\:-5,\:-6,\:-5,\:-2,\:-3]$ such that $A_{K}$ is Hurwitz. Let $M=I_{6}$. Then, using the condition 식(23), the ranges of allowed measurement sensitivity are obtained as

Then, we obtain the estimated set $\widetilde\Omega_{\rho}=\widetilde\Omega_{\rho_{1}}$$\cup$$\widetilde\Omega_{\rho_{2}}$=$\left\{\rho_{1}(t):\right.$ $\rho_{1}(t)\in[0.64,\:2.37]$, $\rho_{2}(t):\rho_{2}(t)\in[0.62,\:4.73]$, $\rho_{3}(t):\rho_{3}(t)$$\in[0.64,\:2.37]$, $\rho_{4}(t):\rho_{4}(t)\in[0.62,\:4.73]$, $\rho_{5}(t):\rho_{5}(t)$$\in$$[0.59,\:$$3.40]$, $\left.\rho_{6}(t):\rho_{6}(t)\in[0.73,\:2.60]\right\}$. Using 식(31), we obtain that $\gamma =0.4$ and $\sigma =4\sqrt{6}\alpha^{-1}||P ||=23.2081$. So, this yields the selection range of $\epsilon$ as $0<\epsilon <0.1077$ such that $\Delta(\epsilon)>0$ in 식(33).

First, we select $\epsilon =8$ which is outside the selection range of $\epsilon$ and observe that the system states tend to diverge as shown in 그림 3(a). Then, as consistent with our analysis, we reduce the value of $\epsilon$ to $0.1$ and observe that system states are bounded and moreover the ultimate bounds are decreased as shown in 그림 4(a). Meanwhile, we note that the input magnitudes tend to increase as the gain-scaling factor $\epsilon$ gets reduced as shown in 그림 3(b) and 그림 4(b). Thus, in practice, the gain-scaling factor $\epsilon$ should be selected within the controller

capacity and any excessive large control input should be avoided. In the next section, we will show the validity of our control scheme via actual experiment.

6. Experiment results

In 그림 5(a), the hardware configuration of quadrotor is depicted and 그림 5(b) shows the experimental setup used in this study. The experiment is carried out on the Quanser's Qball2 ⁽¹¹⁾.

In quadrotor, the brushless motor (BLDC) uses E-Flite Park 480 (1020 Kv) motors fitted with paired counter-rotating APC 10$\times$4.7 propellers. The motors are mounted to quadrotor frame along the $x$ and $y$ axes and connected to the four speed controllers, which are also mounted on the frame. The electronic speed controllers (ESCs) receive commands from the controller in the form of PWM outputs from 1ms to 2 ms. There are avionics sensors such as 3-axis gyroscope and 3-axis accelerometer. To measure on-board sensors and drive the motors, quadrotor utilizes avionics data acquisition device and a wireless Gumstix DuoVero embedded computer. The quadrotor uses 2700 mAh lithium-polymer(Li-Po) batteries for two 3-cell.

Measurement sensitivity in the actual quadrotor system:

In practice, there always exists measurement sensitivity error on sensors in feedback due to the limitations of physical structures, manufacturing reasons, and etc ^(6,14). In our actual test, when quadrotor is in a stationary position on the ground, some non-ignorable amount of measurement sensitivity is observed as shown in 그림 6. Thus, under measurement sensitivity and external disturbance, our proposed control scheme is designed and applied to the attitude control of quadrotor. In accordance with our proposed control method, $K$ is first selected as $K=[-110,\: -21,\: -110,\: -21,\: -30,\: -11]$. The

following two cases are considered in the experiment.

Cases 1: The roll, pitch, yaw desired angles $\phi_{d}$, $\theta_{d}$, $\psi_{d}$ are set as 0 [rad]. As it can be seen from 그림 7, when $\epsilon =1$, the controlled system angles remains bounded but rather large deviations from the desired angles are noticed due to measurement sensitivity and external disturbance. In comparison with 그림 7, the ultimate bound of each angle is clearly reduced by decreasing $\epsilon$ from 1 to 0.7 as shown in 그림 8. In this experiment, it is observed that our control method offers more accurate performance as summarized in 그림 3.

That is, the ultimate bounds of each of roll, pitch, yaw are reduced as much as 61\%, 67\%, 76\%, respectively. Thus, our proposed control scheme is clearly valid.

Case 2: From Case 1, we notice that our controller with $\epsilon =0.7$ yields the much reduced ultimate bounds. So, we further continue to test the performance of controller in tracking the time-varying reference signals. In this regard, the roll, pitch, yaw desired angles $\phi_{d}$, $\theta_{d}$, $\psi_{d}$ are set as $0$ [rad], the sine wave with $0.12$ [rad] amplitude and $2$ [Hz] frequency, the pulse wave with $0.2$ [rad] amplitude and $0.1$ [Hz] frequency, respectively.

We observe that our designed controller with $\epsilon =0.7$ provides good tracking performance as shown in 그림 9.

7. Conclusions

We have proposed a robust attitude control method for quadrotor system where there are measurement sensitivity and external disturbance. The novelty of our control scheme is that (i) a new determination process of a compact to deal with non-trivial measurement sensitivity in each state is provided; (ii) a newly designed robust attitude controller is proposed for boundedness of systems states under external disturbance. We have shown that the ultimate bounds of controlled states can be reduced by adjusting the gain-scaling factor. The validity of our control scheme is verified via numerical simulation and experiment. The current work can be further developed for the output feedback control problems in future.