## Abstract

In this article, a robust attitude controller design for an uncommon quadrotor aerial vehicle is discussed. This aerial vehicle is designed to have two big rotors on the longitudinal axis to increase the lift capacity and flight endurance, and two small tilt rotors on the lateral axis to stabilize the attitude. Similar to other multirotors, linearization of the full nonlinear model and using an appropriate rotor mixing matrix give the approximate diagonal attitude model of this quadrotor around hover. However, this ideal model lacks sensor delays, uncertain parameters, flexible modes of a structure, and inexact decoupling dynamics. Therefore, using this model in the control design limits the achievable attitude control performance. Unlike most studies, a system identification method is applied to estimate a more accurate model and increase the resulting attitude control performance. The aim of this paper is to obtain a suitable nominal model with accompanying uncertainty using robust control criterion in the system identification. In this way, an uncertain model that gives high performance in the subsequent robust control design is obtained. The experimental results show that this combined identification and robust control procedure improves attitude control performance compared to existing classical controller design methods.

## 1 Introduction

Uninhabited aerial vehicles (UAVs) have drawn a lot of interest in both academic research and commercial areas in the last decade. Among all UAVs, quadrotors have become the most common configuration due to their vertical takeoff and landing, hover in place, and maneuverability abilities. The reduction of cost and complexity has increased the use of quadrotors in various military and civil applications [1]. Apart from these favorable properties, quadrotors suffer from high energy consumption. Therefore, the capability of the power system has limited the usage of quadrotors in different applications.

Power source and efficiency of transferring that energy to the air via rotors both limit flight endurance and payload capacity. Typical small commercial quadrotors have less than 20-min flight endurances and payload sizes less than 1 kg [2,3]. This is due to the inadequate stored energy per unit mass of batteries to supply brushless DC motors. For multirotors, the weights of the vehicles are mostly determined by batteries [2]; thus, increasing battery sizes does not contribute much to flight duration since it also increases required power. Therefore, significant progress in flight endurance requires improvements to current battery technology. Since this is a slow process, other methods, such as using an alternative power source and increasing the efficiency of a lifting system, have been studied recently.

For a typical multirotor, the power requirement to maintain thrust over a rotor disc area changes exponentially with total weight and the inverse of the rotor disc area [3]. Consequently, quadrotors suffer from lower efficiency compared to helicopters because of the smaller rotor area in the given unit diameter footprint. On the other hand, the mechanically complex rotor assemblies of helicopters are fragile and require intensive maintenance. Therefore, combining simple and robust quadrotor structure and efficiency of helicopters has also been studied recently.

The triangular quadrotor (Fig. 1(a)) introduced in Ref. [3] uses large rotor in the center to provide lift and canted three small rotors for control and counter torque. Even if a 15% efficiency improvement is reported, this configuration gives degraded hover attitude control performance due to the uncompensated gyroscopic torque of the main rotor compared to a standard quadrotor. In Ref. [4], bi-copter with tilting rotors (Fig. 1(b)) is introduced, and its theoretical efficiency improvement over traditional configurations is reported. For that configuration, roll torque depends on the vertical distance between the center of gravity and the plane of the propellers. Therefore, the roll response of this vehicle is slow since limited torque can be obtained due to this small distance. In addition, roll transients cause linear movement in the *y*-direction [8]. Moreover, in bi-copter, thrust is obtained from rotors placed over tilt mechanisms. This may cause extra vibrations and resonance problems in the structure for larger bi-copter systems.

Using hydrocarbon fuel as an alternative power source significantly outperforms the battery powered propulsion system, and it gives relatively large flight endurance. In Ref. [5], a coaxial inverting thruster using two gasoline engines is placed at the center of a standard quadrotor (Fig. 1(c)). In this case, gyroscopic and counter torques are compensated; however, coaxial rotors lose some of their efficiencies [9]. In Ref. [6], a quadrotor with variable pitch rotors that are powered from a gasoline engine (Fig. 1(d)) is reported. This configuration requires a complex drivetrain and four variable pitch rotors. In Ref. [7], a quadrotor based on four gasoline engines (Fig. 1(e)) is reported. This vehicle can carry large payloads, but the existence of four engines increases the cost dramatically.

In this study, inspired by these works, an uncommon quadrotor configuration illustrated in Fig. 2 is introduced to address the uncompensated gyroscopic torque problem for a single large rotor, the low efficiency problem for coaxial large rotors, the limited roll torque, possible vibrations, and resonance problems of bi-copter design. Therefore, two large rotors are placed on the longitudinal axis which are counter-rotating to minimize their gyroscopic and rotor drag torques. On the lateral axis, small counter-rotating rotors are used for attitude control. In addition, the distance between the small rotor and body is kept small to avoid a large vehicle horizontal width. In this configuration, large rotors carry a major portion of the vehicle weight. Alternate rotors are not counter-rotating to minimize rotor drag and gyroscopic torques in this configuration unlike a standard case. Therefore, small rotors are equipped with a tilt mechanism to gain the control of yaw motion. In addition, sufficient roll control performance can be obtained with this layout. Herein, main thrust sources are directly attached to the main frame which reduces the possible vibration and resonance problems. Therefore, this configuration is more suitable for larger scale compared to a bi-copter. Moreover, if one aims for longer flight duration, large propellers can easily be powered from gasoline engines. In this way, some drawbacks of the gasoline engine configurations discussed above can be eliminated. In this article, battery powered propulsion system is studied. Theoretical efficiency improvements and earlier flight tests of the proposed vehicle prototype are reported previously in Ref. [10].

For multirotors, controllers are generally designed based on an ideal dynamic model derived from physical laws. Thrust and rotor drag coefficients are obtained from static thrust tests. Moreover, inertia matrices are obtained from pendulum tests [1, Ch. 6]. With measuring mass and distances between rotors and mass center, approximate linear model can easily be constructed. Based on this model, proportional-integral-derivative (PID) type controllers can be designed, and sufficient control performance around hover can be reached [1, Ch. 11]. An alternative way of using the ideal dynamic model directly is to assign uncertainties to parameters. From this perspective, robust adaptive [11] and nonlinear $H\u221e$ attitude controllers [12] are applied to satisfy robustness against uncertainty in mass and inertia values. Similar studies can be added, but they all mainly aim for robustness against parametric uncertainties in ideal models.

If higher control performance is demanded, one needs to obtain a more accurate model using a system identification method. From the frequency response function (FRF), dynamic model can be obtained. In this way, the dynamics of the sensors, flexible modes of the airframe, and couplings of different axes due to an inaccurate rotor mixing matrix can be included in the model. In Ref. [13], model of the aircraft is estimated from FRF obtained by injecting multisine signals. Similarly, system identification of multirotors is studied in Refs. [14,15] previously. However, in these studies, perfect decoupling of axes is assumed, and only diagonal entries of the overall transfer matrix are computed. In this article, in addition to diagonal entries, coupling dynamics are also analyzed, and the FRF of the overall three-input three-output attitude model is estimated.

From a robust control point of view, nominal model and uncertainty structure selections are very important for high performance. It is possible to minimize the robust control criterion during identification of the nominal model and uncertainty if a specific dual-Youla–Kucera structure is used [16]. With this structure, control synthesis can be performed in a non-conservative manner, and similar high performance can be obtained for the entire model set. This method gives promising results in motion control applications [17,18]. Herein, this method is successfully applied for an attitude control problem for the first time.

In the robust control framework, nominal model should be accompanied by minimum uncertainty such that uncertain model and measured outputs from true system are consistent. In this study, the validation-based uncertainty modeling procedure introduced in Ref. [19] is used for that purpose. To the author’s knowledge, this study is the first implementation of this technique to UAV flight control problem. After that, the uncertain model and required robustness are determined. It is known that standard *μ* synthesis optimizes robustness and performance at the same time [20, Ch. 8]. Therefore, this method is not suitable when the required robustness is already known. Hence, uncommon skewed-*μ* synthesis is used for robust controller design.

### 1.1 Contribution and Outline of This Paper.

The main contribution of this paper is to apply the combined identification and robust control procedure for the attitude control problem. To the author’s knowledge, similar methods, which include identification of a multivariable attitude model, validation-based uncertainty quantification, and skewed-*μ* synthesis, are not investigated in the previous studies for UAV flight control problem. In this respect, this article contributes to the studies in this area and shows performance improvements compared to traditional methods.

The outline of this article is as follows. In Sec. 2, a nonlinear dynamic model of the proposed quadrotor configuration is obtained. This model is later used in Sec. 3 to find an approximate linear model around hover. In Sec. 4, robust-control-relevant identification and its main motivation are explained. In Sec. 5, skewed-*μ* synthesis and in Sec. 6, prototype of the vehicle are introduced. Then, in Sec. 7, several experimental results are illustrated, and the attitude control performances of robust and proportional-integral (PI) controllers are compared.

## 2 Dynamical Model

As in the standard case, the proposed quadrotor configuration can be viewed as a connection of five rigid bodies, quadrotor body *B* with four rotor groups *P*_{i} attached to it. However, the vehicle illustrated in Fig. 2 differs in that it consists of big rotors on the longitudinal axis (*i* = 1, 3) and small rotors equipped with tilting mechanisms on the lateral axis (*i* = 2, 4).

### 2.1 Preliminary Definitions.

Let $FE:{OE;xE,yE,zE}$ be an earth inertial frame and $FB:{OB;xB,yB,zB}$ be a quadrotor body frame attached to its center of gravity. Moreover, let fixed rotor frames $FPi:{OPi;xPi,yPi,zPi},i=1,2,3,4$, be aligned with each other and body frame. The orientation of second and forth rotors are changed by rotation around $yPi$ axis by an angle of *α*_{i}. This rotation creates a new frame for small rotors as shown in Fig. 2, and they are represented by $FP\xafi:{OP\xafi;xP\xafi,yP\xafi,zP\xafi},i=2,4$.

To balance gyroscopic and counter torques in hover, rotating directions of similar type motors must be opposite. Therefore, rotors 1 and 2 revolve in clockwise direction, and rotors 3 and 4 revolve in counter clockwise direction.

Let the vector $\xi =[xyz]T$ denote translational coordinates of *O*_{B} relative to $FE$, $\eta =[\varphi \theta \psi ]T$ denote the orientation of the quadrotor determined by roll angle *ϕ*, pitch angle *θ*, and yaw angle *ψ* ($Z\u2212Y\u2212X$ Euler angles), $RBE$ denote the overall rotation matrix from body frame to inertial frame, and $RP\xafiB$ denote the rotation matrix from rotating rotor frame $FP\xafi$ to body fixed frame $FB$ [21].

### 2.2 Equations of Motion.

*G*

^{E}= [0 0 −

*g*]

^{T}is the gravity vector, $TB=\u2211i=14FiB$ is the total thrust vector,

*m*is the mass of the quadrotor,

*τ*

^{B}is the total torque applied to the center of mass, Ω = [

*p q r*]

^{T}is the body frame angular rate, and $I=diag(Ixx,Iyy,Izz)$ is the moments of inertia about the body-fixed frame $FB$ of symmetric quadrotor structure. In Eq. (2),

*τ*

^{B}term consists of actuators torque $\tau AB$, gyroscopic torque $\tau GB$, and rotor drag torque $\tau DB$, i.e., ($\tau B=\tau AB+\tau DB\u2212\tau GB$). Herein, $VE=\xi \u02d9$ is the translational velocity of the vehicle in the earth frame $FE$ [21].

## 3 Motion Control for the Proposed Quadrotor

The full nonlinear model derived in Eqs. (1) and (2) is useful for simulation purpose despite some neglected aerodynamic effects. However, especially rotational motion equation is fairly complex for control purpose. Therefore, suitable reduced model is needed. First, it is assumed that bandwidth of rotor speed control is sufficiently high such that transients on rotor speed *w*_{i} are neglected. Therefore, *w*_{i}’s are considered as control inputs. For a quadrotor in slow flight conditions, the second-order inertial and gyroscopic terms are smaller than the forces and torques generated by propeller actuation. Therefore, these second-order terms are considered as disturbances that need to be canceled by the controller. Therefore, simplified rotational motion equation $\Omega \u02d9=I\u22121\tau B$ is used instead of Eq. (2) for controller design purpose where *τ*^{B} is now equal to $\tau B=\tau AB+\tau DB$.

### 3.1 Linearization of the Model in Hover.

Approximate linear model around hover is obtained using Eq. (1) and $\Omega \u02d9=I\u22121\tau B$, relation between rates of Euler angles and body rates, manipulated variables $u=[wT\alpha T]T$, and states $x=[(VE)T\eta T\Omega T]T$. Herein, $w=[w12w22w32w42]T$ includes square of the rotor speeds and $\alpha =[\alpha 2\alpha 4]T$ includes tilt angles.

*x*

_{eq}and

*u*

_{eq}is obtained by dropping all but first-order terms in the Taylor series expansion. Linearization is obtained around hover position where $u=[wT\alpha T]T=[weqT00]T$ and states $x=[(VE)T\eta T\Omega T]T=[00000\psi 000]T$. Overall linearization procedure is not given here due to space limit, but one can investigate [22] for details. Similar to a common configuration, stabilization of the proposed quadrotor around hover requires control of translational motion in $z$-axis and rotational motions. Then, the following reduced equation can be obtained:

Herein, $kfb$, $kfs$ are the thrust factors, *l*_{s} is the small arm length, *l*_{sh} is the distance between mass center and small rotor along $z$-axis, $kdb$, $kds$ are the rotor drag factors, and *w*_{eqi} is the square of the hover speed of the *i*th rotor. In addition, $\xi \xaf=z$ and *δ* denotes the perturbation of the parameter from the equilibrium value. Then, nonlinear quadrotor model is approximated as $x\xaf=x\xafeq+\delta x\xaf$ and *u* = *u*_{eq} + *δu* where $\delta x\xaf=[\delta z\u02d9\delta \Omega T]T$ and $\delta u=[\delta wT\delta \alpha T]T$.

### 3.2 Quadrotor Control.

A hierarchical control strategy commonly used for quadrotors is also applicable to the proposed configuration. At the lowest level, according to commands sent from the flight controller, rotor speeds and angles of the tilt mechanisms are controlled by related electronics. Generally, these loops have very high bandwidth, and their transients can be neglected. In the second level, attitude control loops exist to track desired angles. At the top level, position controllers exist that determine required angles according to desired position commands [1, Ch. 11]. This cascaded control structure, which includes these levels, is illustrated in Fig. 3.

*T*

_{y}=

*I*

_{4}, ideal decoupled model in Eq. (4) can be obtained which facilitates the controller design where

*δu*=

*T*

_{u}

*δu*′ and $\delta y\u2032=Ty\delta y=\delta y=$$[\delta z\u02d9\delta \Omega T]T$.

However, due to inexact decoupling, gyro sensor delay, and other negligible dynamics, the model differs from the ideal. In this article, a more accurate plant model will be obtained using a system identification procedure.

To control the attitude of quadrotors, the cascaded control structure in Fig. 4 is used. Therefore, the accuracy of the attitude control is mainly determined from inner rate control loop performance. Herein, designing high performance rate loop which minimizes the effects of neglected and nonlinear dynamics is the main motivation. Since model-based robust control technique will be used, the equivalent model *P*_{o} in Fig. 4 for the inner loop is needed. This model will be obtained from flight tests when the quadrotor is stabilized by manually tuned PI rate and proportional (P) angle controllers.

## 4 Combined Identification and Robust Control

### 4.1 Problem Formulation.

*P*denotes the linear time invariant system to be controlled, and

*C*denotes the feedback controller. Minimization of a control criterion

*J*(

*P*,

*C*) for the actual plant

*P*

_{o}, i.e., min

_{C}

*J*(

*P*

_{o},

*C*), is the main goal of corresponding optimal control problem. Following criterion is used throughout the article:

*W*= diag (

*W*

_{y},

*W*

_{u}) and

*V*= diag (

*V*

_{2},

*V*

_{1}) are bistable weighting filters and transfer matrix

*T*(

*P*,

*C*) is the mapping defined as Eq. (6) which corresponds to the closed loop in Fig. 5.

*T*(*P*, *C*) includes sensitivity, process sensitivity, control sensitivity, and complementary sensitivity functions. Compared to standard mixed sensitivity problems, this four-block problem eliminates unwanted pole/zero cancelations during controller synthesis. In addition, this problem also appears in $H\u221e$ loop shaping method in Ref. [23] where the shaped plant is robustly stabilized with respect to coprime factor uncertainty [24]. Since similar design method and uncertainty structure are targeted, this four-block problem is used in this article.

### 4.2 Robust Control.

*P*

_{o}is unknown. An optimal controller based on a single approximate model $P^$

*J*(

*P*

_{o},

*C*

_{NP}), and it may result in unbounded value due to instability. Therefore, an uncertain model set $P$ which is structured using upper linear fractional transformation (LFT)

_{u}around the nominal model $P^$, and it includes the actual plant, i.e., $Po\u2208P$. $H^$ contains the nominal model $P^$ and the uncertainty structure. In addition, $H\u221e$-norm-bounded perturbation equation (9) is considered [16].

*P*

_{o}is guaranteed to hold. In other words,

*C*

_{RP}stabilizes and gives certain level of performance for the true system

*P*

_{o}after implementation [17].

### 4.3 System Identificiation for Robust Control.

*C*

_{RP}is implemented, a certain performance bound is obtained. However, exact performance level, the value of $JWC(P,CRP(P))$, depends on the shape and size of $P$. Construction of the model set $P$ that includes

*P*

_{o}and leads to small performance bound is the aim of robust-control-relevant identification. However, minimization of $JWC(P,CRP(P))$ is not explicit due to complex dependency between $CRP(P)$ with $P$. Therefore, the following identification criterion selection is reasonable:

*C*

_{exp}denotes the controller which stabilizes the quadrotor during identification experiment. Herein, selecting

*C*

_{exp}close to $CRP(P)$ increases the accuracy of the approximation equation (11). This closeness can be obtained by iterations over $P\u27e8k\u27e9$ and $CRP\u27e8k\u27e9(P)$, and using the robust controller of the last iteration as $Cexp\u27e8k+1\u27e9=CRP\u27e8k\u27e9$ [16]. In this article, one iteration is performed, and Eq. (11) is only minimized for the initial PI controller $Cexp\u27e81\u27e9$.

*C*

_{exp}to LFT representation of $P$ as in Fig. 6. Then, Eq. (12) gives the worst-case performance where $M^$ is partitioned suitably.

*C*

_{exp}, and bounded performance can be obtained. Coprime factorization-based dual-Youla–Kucera uncertainty structure (Eq. (13)) can be used for that purpose [16].

Right coprime factorization (RCF) of $P^$ is denoted by the pair ${N^,D^}$ if $D^$ is invertible, $N^,D^\u2208RH\u221e$, $P^=N^D^\u22121$, and $\u2203Xr,Yr\u2208RH\u221e$ satisfying Bezout identity $XrD^+YrN^=I$. Similarly, it is assumed that the stabilizing controller *C*_{exp} has a RCF {*N*_{c}, *D*_{c}}. Then, using $PDY$ in Fig. 6 gives the LFT representation (Eq. (14)).

_{u}and becomes bounded.

### 4.4 Nominal Model Identification.

*P*

_{o}can be related using the following triangle equality [25]:

*C*is designed for $P^$, rather than for

*P*

_{o}. Here, the performance degradation term is used to measure the control-relevance of the model. Hence, nominal model identification minimizes this term when it is evaluated for

*C*

_{exp}as Eq. (17).

*C*

_{exp}

*V*

_{2}

*V*

_{1}] with a co-inner numerator, i.e., $N~eN~e*=I$, where $N~e=[N~e,2N~e,1]$. Then, let ${N^,D^}$ be described by Eq. (18).

*P*

_{o}[16]. By using appropriate parametrization

*ρ*for $[N^T(\rho )D^T(\rho )]T$, Lawson’s algorithm for $L\u221e$ approximation and Sanathanan–Koerner (SK) iterations, Eq. (19) is recast as a linear least squares (LS) problem at a discrete frequency grid. During parametrization, specific orthonormal polynomials are used that gives optimal numerical conditioning of the corresponding LS problem. Accurate parametric coprime factors are obtained after subsequent SK and Gauss-Newton iterations [21]. Nonparametric estimate of ${No,Do}$ can be obtained as

### 4.5 Robust-Control-Relevant Model Set.

## 5 Robust Controller Synthesis

*W*

_{γ}denotes the identified uncertainty bound which is embedded in Eq. (25) to normalize the uncertainty, i.e., $\sigma \xaf(\Delta u)\u22641$. Standard

*μ*synthesis optimizes robustness and performance at the same time. Therefore, it is not suitable for problems where the uncertainty and required robustness are already determined as in this case. Therefore, following skewed-

*μ*definition is used:

_{t}is a fictitious uncertainty block introduced for robust performance measure [20, Ch. 8].

Solution technique of skewed-*μ* synthesis is *D*–*K* iterations as in the standard *μ* case. However, the criterion of the *D*–*K* iterations is slightly modified as $minK,D\Vert DUM^D\u22121\Vert \u221e$ by introducing a new matrix $U=diag(In,(1/\mu s)Inp)$ with appropriate dimensions that is updated at each iteration [20, Ch. 8].

## 6 Prototype of an Uncommon Quadrotor Configuration

Most of the electrical and mechanical parts of the first prototype are constructed using low-cost off-the-shelf devices available in the UAV laboratory of the METU EEE department. While two T-Motor KV400 motors driving 16 in. (40.6 cm) propellers are used for the large rotors, two T-Motor KV830 motors driving 9 in. (22.9 cm) propellers correspond to the small rotors. Two 3D printed tilt mechanisms together with two Savöx SA-1256 servo motors are used to tilt the small rotors as depicted in Fig. 8. A Hobbywing Quattro four-channel electronic speed controller (ESC) and a Gens ace 4S (14.8 V) Lipo battery are used to power the brushless motors. An Ublox Neo 7 GPS-Compass module, a 3DR 433 Mhz radio telemetry kit, and a Futaba 2.4 Ghz R2008SB receiver are also attached to the top frame. Moreover, a clone mRo Pixhawk flight controller board compatible with the Pixhawk/PX4 open source flight control software is mounted with a vibration damper in the middle of the top frame. This prototype given in Fig. 8 has a nominal mass of 2.7 kg. The desired motor speeds and tilt angles are regulated with pulse width modulation (PWM) signals sent to the ESC and servo motors. In this study, to reach a convenient solution, available PX4 flight control software is customized for this uncommon quadrotor configuration.

## 7 Quadrotor Attitude Control: System Identification and Robust Control

*T*

_{u}given in (29) is used

*δτ*

_{(.)}denote perturbations of the thrust and torque commands. In this way, the yaw axis is controlled only by tilting the small rotors. This implementation also gives small couplings between the yaw and other axes that need to be compensated in the final control design. Euler angles and translational dynamics are controlled by PID and P controllers, which give sufficient performance. Since

*P*

_{o}is unknown in the beginning, PI rate and P angle controllers are tuned manually in flight by investigating the command tracking performances. Gains of the controllers are adjusted, and the ones that give step responses with small overshoots and rise times are selected. Then, from flight tests data, estimate of

*P*

_{o}is obtained. Next, coprime factors ${N^,D^}$ and uncertainty bound

*W*

_{γ}are found. After that, a robust controller is designed, and its performance is compared with the PI controller having sufficient stability margins.

### 7.1 Frequency Response Function Identification.

*r*

_{2},

*u*, and

*y*are measured, and

*r*

_{2}excitation signal is applied and

*r*

_{1}is kept at zero. Therefore, the following equation is obtained where the discrete Fourier transforms (DFTs) of measured signals $R2\u27e8j\u27e9\u2208C3\xd71$, $U\u27e8j\u27e9\u2208C3\xd71$, and $Y\u27e8j\u27e9\u2208C3\xd71$ on the DFT grid $\Upsilon $ are obtained where 〈

*j*〉 represents the

*j*th test. Performing three tests to span the required space leads to

*T*(

*P*

_{o},

*C*

_{exp}) is obtained as

*ω*

_{k},

*a*

_{k}, $\u03d1k$ and index

*k*denote corresponding frequency, amplitude, phase, and

*k*th frequency, respectively. When $\Omega \xafref$ is constant and full rank

*Q*is selected, $\omega k\u2208\Upsilon id$ is satisfied. Due to cascaded loops, $\Omega \xafref$ is not constant here, but

*Q*=

*I*gives approximately diagonal $[R2\u27e81\u27e9R2\u27e82\u27e9R2\u27e83\u27e9]$, and $\omega k\u2208\Upsilon id$ is not violated. For identification experiments,

*Q*=

*I*corresponds to sequential excitation in the roll, pitch, and yaw axes of the quadrotor. Video of the excitation tests is available online.

^{2}Finally, the estimate of

*P*

_{o}is obtained by $P~o=T~11T~21\u22121$ on the $\Upsilon id$ and depicted in Fig. 9.

*T*

_{u}, small couplings between different axes can be observed in the off-diagonal elements. Especially in roll and pitch axes, these couplings are very small in low frequencies and around the crossover region compared to the diagonal elements. Therefore, their effects on the control design may be insignificant, and diagonal controller can give high performance in these axes. Around the crossover region, roll and yaw axes coupling and diagonal element can be in comparable magnitudes, reducing performance for a single-input single-output (SISO) controller. If higher performance is desired, a more advanced controller that decouples these dynamics is required. The robust controller discussed here can be used for that purpose. Then, nonparametric estimate of {

*N*

_{o},

*D*

_{o}} on $\Upsilon id$ is obtained from Eq. (20). In this article, the data sets are obtained based on the following frequency grid $\Upsilon id$ during identification tests:

The input signal is periodic with a period of 4 s. Moreover, the phases are selected according to Schröer rule [26, Ch. 4]. The identification tests take nearly 120 s which correspond to 30-period input excitation tests, which gives sufficiently accurate FRFs and covariance estimate of the disturbances for the model validation procedure [19].

#### 7.1.1 Weight Determination.

In classical control theory, robustness is often specified by considering limits on gain margin (GM) and phase margin (PM). According to common rules of thumb, 2 < GM < 5 and 30 deg < PM < 60 deg are often aimed in practice [20, Ch. 2]. Previously, the PI rate controller was tuned manually in flight, and these margins were not known since the estimate of *P*_{o} had not been available. Now, estimate $P~o$ is found using identification, and using the diagonal entries of it and the PI controller, gain and phase margins are investigated. In roll axis, 2 Hz bandwidth, 37 deg PM, and 3.9 GM are obtained. In the pitch axis, 2.65 Hz bandwidth, 34 deg PM, and 2.6 GM are achieved. For yaw axis, approximately 0.28 Hz bandwidth and 2 GM are reached. Phase loss due to the finite bandwidth of motor speed control and gyro sensor delay does not allow any gain increase in the PI controller for roll and pitch axes to maintain sufficient margins. For the yaw axis, resonance around 20 Hz limits the achievable gain and bandwidth for a standard PI controller. Adding extra lead and lag filters is necessary to improve the performance, but it is not in the scope of this study. Hence, the PI rate controller that is available during the identification test is also used in the performance evaluation part. This diagonal controller which gives stability margins above is denoted by *C*_{exp} in this section.

The main motivation is to reach approximately 3 Hz bandwidth in the roll and pitch axes and 0.5 Hz bandwidth in the yaw axis to increase the disturbance rejection performance. It is assumed that coupling effects between different axes are small in the low-frequency range. Therefore, both *P* and *C* are approximately diagonal which leads to three different bandwidths definition corresponding to each axis. In this study, three different bandwidths *f*_{roll}, *f*_{pitch}, and *f*_{yaw} refer to crossover frequencies in the corresponding axes satisfying |*P*^{rr}*C*^{rr}(2*πf*_{roll})| = 1, |*P*^{pp}*C*^{pp}(2*πf*_{pitch})| = 1, and |*P*^{yy}*C*^{yy}(2*πf*_{yaw})| = 1 where *P*^{rr}, *P*^{pp}, *P*^{yy} and *C*^{rr}, *C*^{pp}, *C*^{yy} terms indicate the diagonal elements of *P* and *C* for the roll, pitch, and yaw axes, respectively. In addition, to use similar formulation, the notation *f*_{bw} is used which may represent *f*_{roll}, *f*_{pitch}, or *f*_{yaw} according to axis selection.

In this article, weighting functions are selected to shape the loop similar to the method in Ref. [23]. These weighting filters are designed such that *W*_{2}*P W*_{1} has desired open loop shape of *PC*. Since $P^$ is not found yet, the estimate $P~o$ can be used directly. In this study, *W*_{1} is selected to have an integrator for good disturbance rejection, and its cutoff is at *f*_{bw}/3 as shown in Fig. 10 where *f*_{bw} = 3 Hz for the roll and pitch axes and *f*_{bw} = 0.5 for the yaw axis. *W*_{2} is selected to satisfy 0 slope around *f*_{bw} such that desired open loop shape has −1 slope in this region for sufficient robustness. Moreover, high frequency roll-off beyond 4*f*_{bw} is also enforced with *W*_{2}. Finally, *W*_{2} and shaped open loop *W*_{2}*P W*_{1} are depicted in Fig. 10.

The two-block problem considered in Ref. [23] is equivalent to four-block problem considered in this article in terms of control criterion if the weighting filters are selected as $W=diag(Wy,Wu)=diag(W2,W1\u22121)$ and $V=diag(V2,V1)=diag(W2\u22121,W1)$ in Eq. (5). In this article, this method is followed.

#### 7.1.2 Coprime Factorization.

*z*. The detailed procedure for estimating coprime factors ${N^,D^}$ is not in the scope of this paper, and it is discussed in the separate study [21]. The results therein show that the estimated coprime factors are appropriate for subsequent robust control synthesis.

### 7.2 Construction of the Model Set.

*γ*is calculated on the discrete frequency grid. Since a continuous uncertainty bound is essential for robust control design, a dynamic overbound

*W*

_{γ}is obtained and illustrated in Fig. 11. This dynamic bound also reduces possible conservatism compared to static overbound

*γ*during controller synthesis. To see this, two model sets are constructed using static overbound

*γ*and dynamic overbound

*W*

_{γ}as

Herein, although the performance bound in (24) is not affected with tight dynamic overbound *W*_{γ}, possible conservatism during controller synthesis can be reduced since $Pdyn\u2208Psta$ is satisfied. These model sets are visualized using the method introduced in Ref. [28], and elementwise singular values are depicted in Fig. 12. It shows that $Pdyn$ is narrower in the frequency domain compared to $Psta$, reducing possible conservatism. In addition, both uncertain model sets get narrower around the crossover region which is very essential for high robust performance. Conversely, model sets are uncertain in both low and high frequency regions since they are relatively large there.

### 7.3 Controller Design and Implementation.

The uncertain model set $Pdyn$ is used to synthesize a robust controller. Gains of the resulting robust controller *C*_{RP} and PI controller *C*_{exp} are illustrated in Fig. 13 where the input of the controller is error $\epsilon =r2\u2212y$, and the output of the controller is control signal *u* when *r*_{1} is zero. In Fig. 13, controllers are mostly single variate in roll and pitch axes since off-diagonal elements are relatively small compared to diagonal entries in large frequency interval. However, in yaw axis, *C*_{RP} shows multivariable characteristics. This may improve the performance in this axis. In addition, better disturbance rejection capability can be expected from *C*_{RP} due to larger controller gains in the low frequencies compared to *C*_{exp}. Next, nominal and worst-case performances of these controllers for $Pdyn$ are given in Table 1. It shows that *C*_{RP} gives improved performance compared to *C*_{exp}. Herein, smaller performance criterion indicates that achieved bandwidth is closer to the desired one. As reported in Ref. [23], *J*(*P*, *C*) < 4 is an indicator of successful loop shape. Therefore, *C*_{RP} has closer properties to desired loop shape. Table 1 also indicates that all candidate models in the uncertain model set give similar high performance for *C*_{RP} since *J* and *J*_{WC} are almost equal in this case. Therefore, $Pdyn$ is indeed robust control relevant. In other words, this tight model set shown in Fig. 12 enables non-conservative controller synthesis as expected.

Controller | Minimized criterion | $J(P^,C)$ | f_{roll} | f_{pitch} | f_{yaw} | $JWC(Pdyn,C)$ |
---|---|---|---|---|---|---|

C_{exp} | None (PI) | 18.87 | 2.0 | 2.65 | 0.28 | 18.88 |

C_{RP} | $JWC(Pdyn,C)$ | 3.04 | 2.42 | 2.31 | 0.35 | 3.06 |

Controller | Minimized criterion | $J(P^,C)$ | f_{roll} | f_{pitch} | f_{yaw} | $JWC(Pdyn,C)$ |
---|---|---|---|---|---|---|

C_{exp} | None (PI) | 18.87 | 2.0 | 2.65 | 0.28 | 18.88 |

C_{RP} | $JWC(Pdyn,C)$ | 3.04 | 2.42 | 2.31 | 0.35 | 3.06 |

Main duties of the designed attitude rate controllers are to follow angular rate references from the angle controller and to attenuate exogenous disturbances such as wind. Since attitude error is an important performance indicator, effects of these controllers to attitude error is investigated instead of angular rate. During hovering, small attitude reference signals are generated from the outer local velocity controller. Therefore, reference tracking problem is considered where the performance variable $\epsilon =r2\u2212y$ denotes the attitude error. When the vehicle is stabilized around hover, attitude errors are obtained for the two controllers. In Fig. 14, attitude errors in roll, pitch, and yaw axes are depicted. It shows that attitude errors are attenuated with *C*_{RP} due to higher controller gains in the low-frequency region for roll and pitch axes and higher achieved bandwidth for yaw axis. More importantly, *C*_{RP} handles the roll and yaw axis coupling more accurately, thanks to its multivariable structure. Therefore, as depicted in Fig. 14, *C*_{RP} gives smaller attitude errors in all channels compared to *C*_{exp}. Standard deviations of the attitude errors in hover are given in Table 2. It shows that *C*_{RP} improves the attitude tracking performances around 40% in the roll, 55% in the pitch, and 62% in the yaw axes relative to the PI controller (*C*_{exp}) baseline. Moreover, good disturbance rejection capability is also essential. Therefore, to analyze the disturbance rejection performance, a signal is injected at the plant input for each axis. This torque disturbance corresponds to 1/3 of the maximum available torque, i.e., *τ*_{d} = *τ*_{max}/3, for each axis. In Fig. 15, attitude errors and torque disturbance injection points are illustrated, and standard deviations of the attitude errors are given in Table 3. These results show that disturbance rejection performances are improved around 57% in the roll, 56% in the pitch, and 35% in the yaw axes over the baseline. In view of these findings, attitude control performance is significantly improved with the robust rate controller *C*_{RP} compared to the PI rate controller *C*_{exp} around hover.

## 8 Conclusions

In this article, an uncommon efficient quadrotor configuration is proposed. An uncertain model set of this vehicle which facilitates subsequent high performance robust control is constructed, and a robust controller is designed with skewed-*μ* synthesis. First, the proposed vehicle is stabilized using PID type controllers with sufficient attitude control performance. Second, performance improvement is studied with the robust controller. Results of several flight experiments illustrate that the proposed control method significantly outperforms the classical method. Both command following and disturbance rejection performances are improved with the proposed design. In conclusion, the proposed quadrotor configuration can be very useful over the conventional configuration, and it is worthy of further research. In addition, the robust control technique can be helpful to improve the performance achieved with common control methods. Some parts of this study can be improved with further research. Experimental efficiency of this vehicle can be calculated. Alternatively, large rotors that are powered from gasoline engines can be used to improve the flight endurance. Rotor speed and tilt angle control dynamics can be added to the ideal model. Larger deviations from hover position can also be considered as future works.

## Footnotes

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The authors attest that all data for this study are included in the paper.

## References

*h*

_{∞}Control Structure for a Quadrotor Helicopter

*h*

_{∞}Synthesis

*h*

_{∞}/

*μ*Feedback Control Design for High-Precision Wafer Stage Motion