Multilayer hierarchical control for robotic coax-helicopter based on the H ∞ robustness and mixing control scheme

Abstract

The control scheme of a robotic coax-helicopter was investigated and a control scheme was proposed to simultaneously satisfy not only nominal performance and robust performance but also the complexity, flexibility, and computation of controller design. The complete dynamics was considered including flapping dynamics and inflow velocities. A robotic coax-helicopter dynamic model was reasonable divided into three subsystems to separately design controller based on the analysis of singularity value of the output and each inputs with fully considering the couplings. Every subsystem controller design was based on the comparisons in the performance and design complexity among the previous $H_{\infty }$ loop-shaping method, the existed linear matrix inequality method, and the proposed method. A sufficient set of solvability conditions are addressed to provide a flexible performance index design for the longitudinal and lateral velocity subsystems controller. Meanwhile, the pitch–roll attitude subsystem controller was designed taking use of the previous full-order $H_{\infty }$ loop-shaping method. The existed linear matrix inequality method was applied the heave–yaw subsystem because of its low computational complexity. Finally, simulations were implemented to show the effectiveness of the selection of each part’s controller law in the proposed control scheme.

Introduction

There has been re-interested in coaxial rotor because of the requirements of high speed and heavy lift.^1–3 It has been validated that coaxial rotor consumes less induced power comparing with the traditional single rotor with same blade properties and solidity condition.^4,5 Most of researches were focused on the optimum performing coaxial rotor. A comprehensive survey has been concluded in most countries considering inflow structure, rotor spacing, and load sharing from the perspectives of computations and experiments.^1,6 In order to seek the highest levels of efficiency, the individual blade geometries were investigated utilizing an optimization method.^7,8 However, vertical distance, blade chord, and rotation were also analyzed to search the optimum coaxial rotor performance by a series of researches. ⁸ Moreover, the dynamic stability characteristics of the coaxial compound helicopter were taken into account and compared with a traditional helicopter in detail. ⁹ However, only few researches considered the complete model foundation and control design of a robotic coaxial rotor helicopter. ³ In addition, a coaxial helicopter is nonlinear uncertainty and severely coupling plant. These characteristics considerably complicate the control scheme design of coaxial rotor helicopter to achieve high-performance design. ¹⁰ Furthermore, not only nominal performance and robust performance but also the complexity, flexibility, and computation of controller design are to be considered seriously, especially in the situation of engineering application. Therefore, following our previous research, ¹⁰ we investigated the control problem of robotic coax-helicopter and proposed a new control scheme to simultaneously satisfy the different requirements and constraints of controller design.

As one type of the rotorcraft, the control scheme of the coaxial rotor helicopter is apparently the same as the conventional helicopter. Many control schemes have been brought to a widely successful applications in actual flight test, ¹¹ for example, longitudinal and lateral controller for the attitude hold of the Bell 205 helicopter adopting the $H_{\infty }$ loop shaping, inner-loop architecture, and outer-loop architecture was constructed using $H_{\infty }$ loop shaping and proportional–integral–derivative (PID). ¹² Three-layer flight control structure is carried out using $H_{\infty }$ control technique for an unmanned helicopter. Heave–yaw and roll–pitch controller structures were designed for a micro coax-helicopter using the $H_{\infty }$ mixing controller. ¹³ However, the order of the $H_{\infty }$ controller in aforementioned scheme is equal to the shaping plant, which may lead to implementation issues. ¹² Therefore, a low-order but equal to the plant controller was proposed using the linear matrix inequality (LMI) considering performance design flexibility. ¹⁴ In summary, a new control structure for a coaxial rotor, which is based on the dynamic characteristic analysis and robust control, will be essential to the control law design for a coaxial rotor helicopter. In addition, it was generally considered as an attractive way of dealing with the inaccuracy and uncertainty of model. A new control scheme was proposed considering the complexity, flexibility, and computation of coaxial helicopter controller design.

In order to address this problem, the dynamic model is needed to be constructed. The integrative approach was implemented to build the dynamic model in combination with the sum of fundamental components, such as the coaxial rotor, rigid body, horizontal empennage, and vertical fin. It has been showed that the model containing the basic 6-degrees-of-freedom (6-DOF) rigid body and force and torque generation mechanism could satisfy the purposes of control design. In addition, a blade element momentum theory (BEMT) was embraced for the inflow model of a two-rotor system to numerically calculate aerodynamic thrust and torque coefficient. This method has less computation complexity but consistent well with the free-vortex method and experiment data. Furthermore, the flapping dynamics model was regarded as separate DOF using the quasi-static assumption because of its rapid response. Finally, the dynamic model was linearized around hover condition. State and control matrices were derived under an assumption of two main rotors of a conventional helicopter taking use of the analytical formulation.

In this article, the prototype of coaxial rotor helicopter is construed in Figure 1. In particular, most parameters were required to be validated by existed experiments or identified by empirical formulation from open literature. Based on singularity response of the each input and output, the control structure was designed as three parts, the pitch–roll subsystem, the heave–yaw subsystem, and the longitudinal and lateral velocities. In addition, we proposed a new control law using the linear matrix equality to satisfy the flexibility design of performance selection. Finally, each subsystem’s control law was based on the comparisons between the performance and design complexity among the previous full-order $H_{\infty }$ loop-shaping method, the existed LMI, and the proposed method. Simulation results show that the good performance was achieved. It provided the evidence of the effectiveness of the control scheme of a robotic coaxial helicopter.

Figure 1.

The prototype of robotic coax-helicopter.

Methodology

Model construction

The model is essential for the control scheme. The basic 6-DOF rigid-body model and generation mechanism of aerodynamic force and torque are considered. In addition, the flapping dynamics model was regarded as separate DOF using the quasi-static assumption because of its rapid response. This body-fixed axis has three orthogonal axes. The applied aerodynamic forces and moments of various components were transformed to act along the body-fixed axis. The Newton–Euler equations of motion are commonly applied taking the form of nonlinear differential equations as follows

$$\left[\begin{array}{c}\stackrel{·}{u}\hfill \\ \multicolumn{1}{c}{\stackrel{·}{v}}\\ \multicolumn{1}{c}{\stackrel{·}{w}}\end{array}\right]=\left[\begin{array}{c}-g\sin \theta \hfill \\ \multicolumn{1}{c}{g\sin \varphi \cos \theta }\\ \multicolumn{1}{c}{g\cos \varphi \sin \theta }\end{array}\right]+\frac{1}{m}\left[\begin{array}{c}X\hfill \\ \multicolumn{1}{c}{Y}\\ \multicolumn{1}{c}{Z}\end{array}\right]-{\omega }^{*}\left[\begin{array}{c}u\hfill \\ \multicolumn{1}{c}{v}\\ \multicolumn{1}{c}{w}\end{array}\right]$$

(1)

$$\left[\begin{array}{c}\stackrel{·}{p}\hfill \\ \multicolumn{1}{c}{\stackrel{·}{q}}\\ \multicolumn{1}{c}{\stackrel{·}{r}}\end{array}\right]=-I^{-1}{\omega }^{*}I\left[\begin{array}{c}p\hfill \\ \multicolumn{1}{c}{q}\\ \multicolumn{1}{c}{r}\end{array}\right]+I^{-1}\left[\begin{array}{c}R\hfill \\ \multicolumn{1}{c}{M}\\ \multicolumn{1}{c}{N}\end{array}\right]$$

(2)

Here, ${\omega }^{*}=\left[\begin{array}{ccc}0\hfill & -r\hfill & q\hfill \\ \multicolumn{1}{c}{r}& 0\hfill & -p\hfill \\ \multicolumn{1}{c}{-q}& p\hfill & 0\hfill \end{array}\right]$ and $I=\left[\begin{array}{ccc}{I}_{\mathrm{xx}}\hfill & 0\hfill & -I_{\mathrm{xz}}\hfill \\ \multicolumn{1}{c}{0}& I_{\mathrm{yy}}\hfill & 0\hfill \\ \multicolumn{1}{c}{-I_{\mathrm{xz}}}& 0\hfill & I_{\mathrm{zz}}\hfill \end{array}\right]$ . $(X,Y,Z)$ and $(R,M,N)$ are the resultant forces and moments given as follows, respectively

$$\left[\begin{array}{c}X=X_{\mathrm{up}}+X_{\mathrm{low}}+X_H+X_V+X_F\hfill \\ \multicolumn{1}{c}{Y=Y_{\mathrm{up}}+Y_{\mathrm{low}}+Y_V+Y_F}\\ \multicolumn{1}{c}{Z=Z_{\mathrm{up}}+Z_{\mathrm{low}}+Z_H+Z_V+Z_F}\end{array}\right]$$

(3)

$$\left[\begin{array}{c}R=R_{\mathrm{up}}+Y_{\mathrm{up}}{h}_{\mathrm{up}}+Z_{\mathrm{up}}{y}_{\mathrm{up}}+R_{\mathrm{low}}+Y_{\mathrm{low}}{h}_{\mathrm{low}}\hfill \\ \multicolumn{1}{c}{+Z_{\mathrm{low}}{y}_{\mathrm{low}}+Y_V{h}_V+Y_F{h}_F+R_F}\\ \multicolumn{1}{c}{M=M_{\mathrm{up}}-X_{\mathrm{up}}{h}_{\mathrm{up}}+Z_{\mathrm{up}}{l}_{\mathrm{up}}+M_{\mathrm{low}}}\\ \multicolumn{1}{c}{-X_{\mathrm{low}}{h}_{\mathrm{low}}+Z_{\mathrm{low}}{l}_{\mathrm{low}}-X_H{h}_H+Z_H{l}_H}\\ \multicolumn{1}{c}{-X_V{h}_V+M_F+Z_F{l}_F-X_F{h}_F}\\ \multicolumn{1}{c}{N=N_{\mathrm{up}}-Y_{\mathrm{up}}{l}_{\mathrm{up}}-X_{\mathrm{up}}{y}_{\mathrm{up}}-N_{\mathrm{low}}-Y_{\mathrm{low}}{l}_{\mathrm{low}}}\\ \multicolumn{1}{c}{-X_{\mathrm{low}}{y}_{\mathrm{low}}-Y_V{l}_V+N_F-Y_F{l}_F}\end{array}\right]$$

(4)

where subscripts up, low, H, V, and F indicate upper one, lower one, empennage, fin, and fuselage, respectively. $I_{\mathrm{xx}}$ , $I_{\mathrm{yy}}$ , and $I_{\mathrm{zz}}$ are fuselage moments inertia referred to the body-fixed axes. Each component’s distance respecting to the center of the axes is stood by l, y, and h in three axes.

The relation between the inertia and the body-fixed frame was described by the kinematic equations as follows

$$\left[\begin{array}{c}\stackrel{·}{\theta }\hfill \\ \multicolumn{1}{c}{\stackrel{·}{\varphi }}\\ \multicolumn{1}{c}{\stackrel{·}{\psi }}\end{array}\right]=\left[\begin{array}{ccc}0\hfill & \cos \varphi \hfill & -\sin \varphi \hfill \\ \multicolumn{1}{c}{1}& \sin \varphi \tan \theta \hfill & \cos \varphi \tan \theta \hfill \\ \multicolumn{1}{c}{0}& \sin \varphi \sec \theta \hfill & \cos \varphi \sec \theta \hfill \end{array}\right]\left[\begin{array}{c}p\hfill \\ \multicolumn{1}{c}{q}\\ \multicolumn{1}{c}{r}\end{array}\right]$$

(5)

The upper and lower rotors’ inflow holds the key ingredient to pursue the force and torque of the rotor in the simulation model. There are complex, inherent, and reciprocal aerodynamic interferences between rotors. However, in predicting the force and moment coefficients, a parsimonious method called the BEMT has been validated to nearly consistent with the free-vortex prediction method, which offers a good approximation of the experimental data. ⁵ Therefore, this method was adopted to model the inflow with a few adaptations. In addition, it is assumed that the slipstream inflow of upper one interferes the inflow of the lower one, but the lower one does not affect the wake of the upper one. The advancing blade of the upper was on the right side and the lower rotor rotated clock-wise. Therefore, an efficient BEMT is addressed to calculate the coefficients of the rotors as follows ⁸

$$\{\begin{array}{l}{\lambda }_{up}=\sqrt{{\left(\frac{{\sigma }_{up}{C}_l}{16F}\right)}^2+\frac{{\sigma }_{up}{C}_l}{8F}{\theta }_{up}r}-\left(\frac{{\sigma }_{up}{C}_l}{16F}\right)\hfill \\ {\lambda }_{low}=\sqrt{{\left(\frac{{\sigma }_{low}{C}_l}{16F}-\frac{{\lambda }_{up}}{2a^2}\right)}^2+\frac{{\sigma }_{low}{C}_l}{8F}{\theta }_{low}r}-\left(\frac{{\sigma }_{up}{C}_l}{16F}-\frac{{\lambda }_{up}}{2a^2}\right)\left(r\le a\right)\hfill \\ {\lambda }_{low}=\sqrt{{\left(\frac{{\sigma }_{low}{C}_l}{16F}\right)}^2+\frac{{\sigma }_{low}{C}_l}{8F}{\theta }_{low}r}-\left(\frac{{\sigma }_{low}{C}_l}{16F}\right)\left(r>a\right)\hfill \end{array}$$

(6)

Furthermore, it is nature to undertake the blade element theory. In the meanwhile, two-dimensional (2D) and steady airfoil were assumed to calculate the thrust and torque coefficients of the rotors

$$\{\begin{array}{c}d{\left(C_T\right)}_{\mathrm{up}}=\frac{{\sigma }_{\mathrm{up}}{C}_l}{2}({\theta }_{\mathrm{up}}{r}^2-{\lambda }_{\mathrm{up}}r)\mathrm{dr}\hfill \\ \multicolumn{1}{c}{d{\left(C_T\right)}_{\mathrm{low}}=\frac{{\sigma }_{\mathrm{low}}{C}_l}{2}({\theta }_{\mathrm{low}}{r}^2-{\lambda }_{\mathrm{low}}r)\mathrm{dr}}\end{array}$$

(7)

Blade is assumed to discretize into a series of small radial constant elements. ⁸ Thrust and torque coefficients of the upper rotor can be numerically integrated as follows

$$\{\begin{array}{c}{T}_i=\rho A{\left(\Omega R\right)}^2{C}_{T_i}\hfill \\ \multicolumn{1}{c}{Q_i=\rho A{\left(\Omega R\right)}^{2}R{C}_{Q_i}}\end{array}$$

(8)

Here, $T_i$ and $Q_i$ stand for thrust and torque, respectively; air density is representation of $\rho$ ; disk area is written in terms of A; rotational speed of rotors is indicated by $\Omega$ ; and R stands for radius of the rotor. $C_T$ and $C_Q$ are thrust coefficient and torque coefficient, respectively. i stands for up rotor or low rotor.

During the process of control design, flapping dynamics of two rotors is also a fundamental factor. Thus, flapping equation for each rotor is constructed as follows

$${\stackrel{··}{\beta }}_i=\frac{1}{J_{{\beta }_i}}\underset{0}{\overset{R-e}{\int }}{\left(M_h\right)}_i\mathrm{dr}$$

(9)

where $\beta$ is flapping angle. $J_{\beta }$ stands for the blade moment of inertia. R and e are the rotor radius and the blade hinge offset from the center of the rotor, respectively. $M_h$ is the incremental moment coefficient along the blade. The solution of flapping equation was assumed to be periodic. ¹⁵ Therefore, first term was formulated under the quasi-static assumption as follows

$${\beta }_i(t)=a_{i0}(t)-a_{i1}(t)\cos \Psi +b_{i1}(t)\sin \Psi$$

(10)

The three terms stand for the coning, the pitching, and the rolling motion, respectively. ⁵ The first harmonic coefficients relationship between the flapping motion and blade pitching motion is given as follows ¹¹

$${\stackrel{·}{a}}_i=\left[\begin{array}{cc}0\hfill & I\hfill \\ \multicolumn{1}{c}{-K_i}& -D_i\hfill \end{array}\right]a_i+f_i$$

(11)

where $a_i=[a_{i0},a_{i1},b_{i1},{\stackrel{·}{a}}_{i0},{\stackrel{·}{a}}_{i1},{\stackrel{·}{b}}_{i1}]$ , $K_i$ , $D_i$ , and $f_i$ are the coefficients of the first-order differential dynamic equation of two rotors.

Except for the coaxial system, the dynamic model is also combined with the other fundamental components model, such as the rigid body, horizontal empennage, and vertical fin. Hence, the method of modeling conventional helicopter’s fuselage was expanded to model fuselage of coaxial rotor helicopter. The thrust and torque of the fuselage were supposed that are only affected by downwash of lower rotor. Therefore, an empirical formulation was taken to account the effects of the lower one wake in taking use the isolated fuselage shape. ¹⁵ Empirical parameter was applied to calculate force and moment in the dimensions of the rigid body of the constructed coaxial rotor helicopter.

As we all know, the torques and forces of the fin and empennage play key roles in yawing moment and longitudinal equilibrium equation, respectively. However, this effect was evident in the flight, but not well predicted by simulation. ¹⁶ Hence, the model of two parts was neglected during modeling.

Linearize

The nonlinear differential equation of the robotic coaxial rotor helicopter is expressed by

$$\stackrel{·}{x}=f(\mathbf{x},\mathbf{u})$$

(12)

The state vector is

$$\mathbf{x}=[u,v.w,p,q,r,\theta ,\varphi ,\psi ,a_{i0},a_{i1},b_{i1},{\stackrel{·}{a}}_{i0},{\stackrel{·}{a}}_{i1},{\stackrel{·}{b}}_{i1}]$$

(13)

And the input vector is

$$\mathbf{u}=\left[\begin{array}{cccccc}{\left(B_1\right)}_{\mathrm{up}}\hfill & {\left({\theta }_{\mathrm{col}}\right)}_{\mathrm{up}}\hfill & {\left(B_1\right)}_{\mathrm{low}}\hfill & {\left(A_1\right)}_{\mathrm{up}}\hfill & {\left(A_1\right)}_{\mathrm{low}}\hfill & {\left({\theta }_{\mathrm{col}}\right)}_{\mathrm{low}}\hfill \end{array}\right]$$

(14)

However, it is necessary to trim the nonlinear differential equation at operating condition in order to extract the linear mode. Trim analysis is always complex because coaxial rotor is the underactuated vehicle. In order to establish a square equilibrium equation, the same cyclic pitch inputs were implemented to trim the nonlinear equation of the coaxial helicopter, ¹⁷ while the terms of differential collective pitch are not used. In the end, the nonlinear differential equation was solved by the analytical method under small-angle assumption.

As to teetering two-blade configuration for two rotors whose blade are identical in NACA0012 blade sections, radius, planform, twisted rates, and cut-out are shown as follows.

Airfoil parameters

Airfoil NACA0012: the number of blades is 2 (upper rotor and lower rotor), the rotor spacing of 0.1R, chord length of 0.15 m, radius of 1.8 m, deflection amount of 0.1 m, twist rate of −6°, no taper, steer speed of 800 r/min, and teetering rotors.

The distance between the original of the rotor hub and the center of body: 0.3 m, paddle degree: double rotor is 0.106, single rotor is 0.053, the number of Locke is 4, the Mach number is 0.44, the quality factor is 0.7316 and 0.4961. Coaxial quality factor is 0.7770.

Body parameters

The thrust and state variables of upper and lower rotor are outlined in hovering flight. In this case, the transverse and longitudinal periodic variance is the first-order coefficient of the waving coefficient. M = 200 kg, $I_{\mathrm{xx}}=293.96\mathrm{kg}{\mathrm{m}}^2$ , $I_{\mathrm{yy}}=915.34\mathrm{kg}{\mathrm{m}}^2$ , $I_{\mathrm{zz}}=881.4\mathrm{kg}{\mathrm{m}}^2$ , and $I_{\mathrm{xz}}=34.37\mathrm{kg}{\mathrm{m}}^2$ . The distance between the origin of the lower rotor hub and the origin of body gravity is 0.5 m. Body size: height of 0.5 m, width of 0.35 m, and length of 3 m. Body origin of gravity from the bottom of the fuselage is 0.25. Landing gear from the bottom is 0.25. The fuselage on the surface from the lower wing is 0.07. The ground vertical height is 1 m.

Trim values in hovering state

Trim values of input are $a_{1s}=-0.0317\mathrm{rad}=-1.82\circ$ , $\Theta =0.0327$ , $b_{1s}=0.0331$ , and $\Phi =-0.0342$ . After trimming, it is implemented to extract linear model at operating condition in this procedure. It was supposed that a change of the dynamic model can be depicted by superposition of individual effects of which “two-main-rotor” conduces in one helicopter. Moreover, it is impressive hypothesis that linear effects are added up. Furthermore, small perturbation theory is embraced. ¹⁵ Therefore, the linear model is constructed as follows

$$\stackrel{·}{\mathbf{x}}=\mathbf{Ax}+\mathbf{Bx}$$

(15)

where the state and control matrices were written in two terms. The elements of the state and control matrices summed the effects of the two rotors, respectively. Thus, the stability and control derivatives are calculated as follows

$$A=\left[\begin{array}{cccccccc}-0.002089\hfill & 0\hfill & 0.46\hfill & -9.7948\hfill & -0.00025\hfill & -0.01875\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -0.519\hfill & 0\hfill & -0.3203\hfill & 0\hfill & 0\hfill & 0.335\hfill & 0\hfill \\ \multicolumn{1}{c}{0.0095}& 0.00567\hfill & -0.04582\hfill & 0\hfill & 0.000128\hfill & 0.001875\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0.9994\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & -0.0342\hfill \\ \multicolumn{1}{c}{0.00025}& 0\hfill & -0.01875\hfill & 0.01038\hfill & -0.002089\hfill & -0.46\hfill & 9.799\hfill & 0\hfill \\ \multicolumn{1}{c}{0.0004}& -0.01975\hfill & -0.03153\hfill & 0\hfill & -0.0298\hfill & 0.7739\hfill & 0\hfill & 0.0791\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -0.00106\hfill & 0\hfill & 0\hfill & 1\hfill & 0\hfill & 0.03268\hfill \\ \multicolumn{1}{c}{0.00002}& -0.01793\hfill & 0\hfill & 0\hfill & -0.00113\hfill & 0.02942\hfill & 0\hfill & 0.69365\hfill \end{array}\right]$$

(16)

$$B=\left[\begin{array}{cccccc}5.8115\hfill & 1.8408\hfill & 2.409\hfill & 1.00415\hfill & -1.14425\hfill & -0.47595\hfill \\ \multicolumn{1}{c}{0}& -58.0734\hfill & 0\hfill & -31.6764\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-2.4474}& 0.4301\hfill & -1.9704\hfill & 0.2802\hfill & 0.48187\hfill & 0.38806\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{1.14425}& -1.92225\hfill & 0.47595\hfill & -1.0485\hfill & 5.8115\hfill & 2.409\hfill \\ \multicolumn{1}{c}{1.5085}& -2.3701\hfill & 1.2148\hfill & -1.0261\hfill & 7.6611\hfill & 6.1679\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0.05734}& 2.2179\hfill & 0.0462\hfill & 2.2703\hfill & 0.29121\hfill & 0.23445\hfill \end{array}\right]$$

(17)

Control scheme design

The typical control scheme of unmanned helicopter controller

Unmanned helicopter is a kind of system with multiple inputs/outputs and underactuation. However, the helicopter is supposed to accomplish the control state variables, such as attitude, altitude, heading, and velocity. Therefore, arranging the corresponding relation between input and output is the key role in flight control structure design. According to the matching number I/Os, the structures can be divided into the following forms.

The multiloop control structure with cascade single I/O

The characteristic of the kind of flight controller is to divide the system with multiple I/O into several submodules with single I/O and then to design the flight controllers of each submodules. The typical application is Project AVATAR of University of Southern California Institute of Technology and Berkeley’s Maxima. ¹⁸ This flight structure is used in widespread open-source flight control, such as Ardu Pilot Mega (APM) and Pixhawk.^19,20 Its control structure is shown in Figure 2.

Figure 2.

The schematic diagram of the multiloop control system with cascade single I/O.

This kind of structure is most widely adopted at present, and it has been proved by actual tests. It highlights the easy-adjusted controller parameters and has clear physical meanings. However, its control effect, specially the robustness of the controller, is limited because its coupling characteristics between models are ignored, such as the strong coupling between the pitching and rolling channels.

The multiloop control structure with submodules of multiple I/O

The controlled objects are divided into several submodules with multiple I/Os. Then, the multiple-variable flight controller aimed at submodules is designed, and the multiple-variable $H_{\infty }$ algorithm is widely adopted as its control algorithm. ²¹ The typical example of this structure is Project “Shelion” of Singapore Institute of Technology.^13,22 Its control system has accomplished the flight test successfully. It is capable of autonomous flight and autonomous taking off and landing. Its scheme structure is depicted in Figure 3.

Figure 3.

The schematic diagram of the multiloop control system with submodules of multiple inputs.

The state variables $[X,Y,Z,U,V,W,p,q,r,\psi ,\phi ,\theta ]$ and $[{\mathrm{Z}}_f,Y_f,X_f,{\psi }_f]$ are reference inputs, and $[{\delta }_{\mathrm{col}},{\delta }_{\mathrm{ped}},{\delta }_{\mathrm{lat}},{\delta }_{\mathrm{lon}}]$ is the control matrix. $K_{\mathrm{in}}$ and $K_o$ are controller transfer matrices, which divide the controlled object G into two submodules to construct the controller, respectively. One of the submodules controls the attitude - yaw velocity. The other controls are position, yaw angle, and rate.

This structure highlights the comprehensive consideration of the models, such as the coupling interference between states. Its control accuracy is more effective. But the order of controller is a bit higher, and it is not flexible to adjust the design variables of the controller. Besides, the control accuracy is directly related to the model precision, even a limited-precision model will lead to an unstable circumstance.

The hybrid multiloop control structure adopted both single and multiple I/Os

This method is a combination of the advantages of the above two kinds of structures, which divides the system into submodules of multiple I/O and single I/O. Then, the flight controllers are designed to aim at each submodules. The typical example is the Project R-50 of Carnegie Mellon University, which adopts multiple-variable $H_{\infty }$ controller in the inner loop and PID in the outer loop.^23,24 Its control structure has successfully accomplished the flight tests of autonomous flight and autonomous taking off and landing. Its schematic diagram is depicted in Figure 4.

Figure 4.

The schematic diagram of $H_{\infty }-\mathrm{based}$ inner-loop and outer-loop hybrid control structure.

In this structure, the inner loop consists of pitching rolling angle submodule, yaw velocity, and vertical velocity submodule, while the outer loop consists of velocity loop, position control loop, and heading control submodule. The inner loop adopts cascade structure in order to select the appropriate feedback value and reference value. This structure remains strong interference coupling, such as pitching and rolling angles. It uses single-variable controller to design the performance figures flexibly. It highlights the full consideration of the dynamic characteristics of the model so that it is convenient to adjust the design parameters of the controller.

Overall, it is necessary to pay attention to the following key points in designing flight control architecture:

1. The order of submodules don't need to be exorbitanted, and the submodules need to remain the main coupling interference of the model to adjust the performance easily;

2. The multiple-variable control is suitable to be adopted in pitching and rolling channels because of the strong axisymmetric coupling between them;

3. The state variable of attitude and velocity can be divided into submodules to design their own controllers, respectively, because the period of lateral motion and the longitudinal motion is longer than that of attitude angle response. In addition, the coupling interference between velocity and attitude angle can be ignored.

4. The multiple-variable control is suitable to be adopted to control the height and heading because the degree of coupling between the height channels and the pitching rolling channels is quite small. The coupling interference between the height channel and the attitude angle channel is also small.

Singularity response analysis of robotic coaxial rotor helicopter

Multilayer hierarchical control is based on the time scale operating on different time scales and different granularities. ²⁵ Obviously, the control structure is designed based on the model of the manipulation and coupling characteristics. Then, control effect, design complexity of the designer, and easy to implement factor have been considered. Design philosophy based on singularity response.

Singularity response analysis at low frequencies

At low frequencies, the amplitude of the velocity response is significantly higher than the angle. At high frequencies, the gap between the two becomes relatively small. Output singular value responses of the height and angle channel are also at low frequency.

Singularity response analysis under cycle pitch input of the lower rotor

During the vertical cycle pitch input of the lower one, there is a significant coupling between the transverse and longitudinal velocities at the low frequency. The values of pitch and roll are smaller. The singularity response of the height and heading are relatively small at the high and low frequencies, while the vertical cycle pitch input of the upper rotor.

Singularity response between velocity and angle channel

The upper and lower rotor linkage cycle pitch can be adopted. However, the singular value response of pitch and roll angle under upper rotor cycle pitch is higher than the lower rotor cycle pitch. The singular value response value is larger than the use of linkage cycle pitch, and the crossing frequency is higher. Therefore, the pitch and roll angle are controlled by cycle pitch of the upper rotor, while velocity is controlled by cycle pitch of the lower rotor.

Singularity response between the height and the heading channel

There is interference between the height and heading and is affected by the velocity at the same time. In the singular value response curve of the upper rotor, the height channel is relatively obvious at the low frequency. While the heading rate at high frequency is more obvious than velocity channel in the singular value response curve of the lower rotor. The singular value of height channel is relatively obvious for both inputs.

Architectural design

The unmanned coax-helicopter is a multiple I/Os system and there are strong coupling interferences between each of input channels. If the multivariable feedback control architecture is adopted directly, rich debugging experience^12–13 of loop shaping is necessary to design the best performance figures. Even a minor adjustment can seriously impact the performance of time domain and frequency domain, and it will also seriously impact the stability of robustness. The architecture of full-order multivariable controller is adopted, which brings better effects to the overshoot and stability margin, but its rise time and decoupling performance still need improvement. Therefore, the architecture analysis and design of the coax-helicopter are quite indispensable.

Based on the analyses and summaries of the typical unmanned helicopter flight control structures, the flight performance figures, the stabilities of robustness, and the engineering implementation requirements are seriously considered. According to the singular value response and the coupling characteristic, a “cascaded double-layer three-submodule” flight control architecture of unmanned coax-helicopter is concluded and shown in Figure 5.

Figure 5.

The control architecture schematic diagram of the unmanned coax-helicopter.

In this unmanned coax-helicopter, the inner layer includes the attitude submodule, and the outer layer consists of the velocity submodule and vertical velocity–yaw velocity submodule. Here are the control strategies:

1. The pitching angle $\theta$ and the rolling angle $\phi$ (including the angular rate p and q) are adopted as the feedback quantity of the pitching and rolling submodule;

2. The vertical velocity w and yaw velocity r are adopted as the feedback quantity of the vertical velocity–yaw velocity submodule. The lateral velocity u and longitudinal velocity v are adopted as the feedback quantity of the velocity submodule.

The state of coupling between pitching and rolling angles is specially outstanding so that they are chosen to constitute a submodule, and the coupling between vertical and yaw velocities also behaves in a similar way. Thus, they are designed into a submodule. The motion period of the lateral and longitudinal rates is much longer than that of the other state variables. Therefore, they are regarded as the same submodule when designing the controllers. The above control architecture retains not only the main characteristics of interference but also the flexibility of controller design with full consideration of the coupling interferences among states of the unmanned coax-helicopter.

According to the coupling characteristics and time characteristics of the analysis results, the specific structure of control system will be split out. Relative gain arrangement (RGA) shows the pairing which is the allocation of control strategy. It is because the period of transverse and longitudinal velocities is greater than the pitch–roll angle. The characteristic curve of the output singular response can be used to illustrate that the attitude of the velocity and the inner can be separated.

The singular value response characteristics curves of a certain input were analyzed for the coupling characteristics. The control system can be put together when the coupling is relatively large and split when the coupling is relatively small. It is illustrated that the heading and height can be put together, as well as the angle and velocity. The key reason is that singular value response are different in the same frequency band.

Therefore, the above can be split into three structures: the upper rotor cycle pitch corresponds to attitude angle (including the angular rate of the two), the lower rotor cycle pitch corresponds to the transverse and longitudinal velocities, and the total distance of the upper and lower rotor corresponds to height and heading rate. Attitude angles are stood for q, θ, p, φ:

• State matrix

  $$A_1=\left[\begin{array}{cccc}-0.04582\hfill & 0\hfill & 0.001875\hfill & 0\hfill \\ \multicolumn{1}{c}{0.9994}& 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-0.03153}& 0\hfill & 0.7739\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -0.00106\hfill & 1\hfill & 0\hfill \end{array}\right]$$

  (18)

• Control matrix

  $$B_1=\left[\begin{array}{cc}-2.4474\hfill & 0.48187\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \\ \multicolumn{1}{c}{1.5085}& 7.6611\hfill \\ \multicolumn{1}{c}{0}& 0\hfill \end{array}\right]$$

  (19)

• Output matrix

  $$C_1=\left[\begin{array}{cccc}0\hfill & 1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]$$

  (20)
  $$D_1=\mathrm{zeros}(2,2)$$

  (21)

When the step response is analyzed, the reference instruction and output matrix are expanded to fourth-order matrix. The reference of the angular rate channel is always zero and does not need to be set to 1. The corresponding angular rate is zero from output matrix, and with the reference instruction, the output of the angular rate channel follows the zero. The longitudinal and lateral velocities are given as follows:

• State matrix is

  $$A_2=\left[\begin{array}{cc}-0.002089\hfill & -0.00025\hfill \\ \multicolumn{1}{c}{0.00025}& -0.002089\hfill \end{array}\right]$$

  (22)

• Control matrix is

  $$B_2=\left[\begin{array}{cc}2.409\hfill & -0.47595\hfill \\ \multicolumn{1}{c}{0.47595}& 2.409\hfill \end{array}\right]$$

  (23)

• Output matrix is

  $$C_2=\left[\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right]$$

  (24)

The state and control matrices of height and heading are given as follows:

• State matrix is

  $$A_3=\left[\begin{array}{cc}-0.519\hfill & 0\hfill \\ \multicolumn{1}{c}{-0.01793}& 0.69365\hfill \end{array}\right]$$

  (25)

• Control matrix is

  $$B_3=\left[\begin{array}{cc}-58.0743\hfill & -31.6764\hfill \\ \multicolumn{1}{c}{2.2179}& 2.2703\hfill \end{array}\right]$$

  (26)

• Output matrix is

  $$C_3=\left[\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right]$$

  (27)

Comparison Simulation

Previous works

There are a variety of control programs for different application purposes. Scheme for inner $H_{\infty }$ loop controller and outer-loop PID control is constructed by Walker. ²⁶ There are unstable points, and weighted function selection is not easy when $H_{\infty }$ mixed-sensitive is used. ²⁷ The $H_{\infty }$ is adopted in inner loop and outer loop, but lacks flexibility in the Cornell NanoScale Science & Technology Facility project. ²² Forming $H_{\infty }$ loop shaping is good, but it is needed to adjust weights for several times, which is complex, and it is not easy to implement. ²³ $H_{\infty }$ loop shaping with hybrid control structure is used to reduce the number of iteration calculations based on LMI. But the effect is relatively poor and lacked design flexibility. ¹²

Simulation comparisons

Under the requirements of performance index, robust stability, and easy engineering implementation, the controller of three submodules is designed. The classical $H_{\infty }$ loop shaping, $H_{\infty }$ loop based on LMI, and proposed algorithm are adopted for the proposed scheme. Then, the simulation experiment is carried out and compared with the unit step response using the full-order control structure.

The contrast analysis between the classical loop shaping and the proposed algorithm of attitude submodule is completed. When the same pre-weighting and post-weighting matrices are adopted, the attitude submodule controller of $H_{\infty }$ loop forming based on LMI is

$$K_{\mathrm{LMI}}=\left[\begin{array}{cccc}0.8784\hfill & 1.1363\hfill & -0.1820\hfill & -0.2399\hfill \\ \multicolumn{1}{c}{1.1363}& 0.5387\hfill & -0.2553\hfill & -0.1358\hfill \\ \multicolumn{1}{c}{-0.1820}& -0.2553\hfill & -0.8524\hfill & -1.2309\hfill \\ \multicolumn{1}{c}{-0.2399}& -0.1358\hfill & -1.2309\hfill & -0.8182\hfill \end{array}\right]$$

(28)

The controller design and performance simulation of velocity submodule and height-heading submodule using the classic loop forming, based on the LMI loop forming and the proposed algorithm, are compared and analyzed as follows.

Velocity subsystem controller design

According to the state and control matrices derived, the velocity submodule state and control matrices are

$$A_2=\left[\begin{array}{cc}-0.002089\hfill & -0.00025\hfill \\ \multicolumn{1}{c}{0.00025}& -0.002089\hfill \end{array}\right]$$

(29)

$$B_2=\left[\begin{array}{cc}2.409\hfill & -0.47595\hfill \\ \multicolumn{1}{c}{0.47595}& 2.409\hfill \end{array}\right]$$

(30)

The output matrix is $C_2=\left[\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right]$ , the submodule state matrix $x_2=[u,v]$ , and the input matrix is the transverse cycle pitch and longitudinal cycle pitch of the lower rotor. The calculation shows that the rank sum of the matrix $(A_2,B_2)$ and $(A_2,C_2)$ is 4. It indicates that the velocity subsystem is controllable and observable. Thus, the classic $H_{\infty }$ loop forming, LMI-based $H_{\infty }$ loop shaping, and the proposed algorithm can be used for velocity submodule controller design.

The pre and post-weighted matrices are selected referring to summary of the selection experience. The integral item is used to improve the performance of the controller p, for example, the descent rate of the crossing frequency and the gain at low frequency. The proportional item is used to adjust the lead and lag of the phase.^10,21 By adjusting the pre-weighted matrix coefficient, the final pre-weighted matrix is

$$W_{21}=\mathrm{diag}\left[\begin{array}{cc}\frac{0.018s+0.0015}{s}\hfill & \frac{0.01s+0.0025}{s}\hfill \end{array}\right]$$

(31)

The post-weighted matrix $W_{22}$ is the unit matrix

$$W_{22}=\mathrm{diag}\left[\begin{array}{cc}1\hfill & 1\hfill \end{array}\right]$$

(32)

The controller for the velocity submodule using the classic $H_{\infty }$ loop forming, ²⁸ LMI-based $H_{\infty }$ loop forming, ¹² and the proposed algorithm ¹⁰ is as follows, respectively

$$\begin{array}{c}{K}_{normal}:A=\left[\begin{array}{cccc}-9.636\hfill & -0.8316\hfill & 0.09087\hfill & -0.0943\hfill \\ -0.09044\hfill & -18.58\hfill & 0.008414\hfill & 0.1269\hfill \\ -0.2038\hfill & -0.1004\hfill & -0.0813\hfill & -0.00152\hfill \\ 0.1302\hfill & -1.411\hfill & -0.000788\hfill & -0.238\hfill \end{array}\right]\hfill \\ B=\left[\begin{array}{ll}-7.02\hfill & -0.0209\hfill \\ -0.00687\hfill & -4.536\hfill \\ -0.1471\hfill & -0.0206\hfill \\ 0.09704\hfill & -0.3427\hfill \end{array}\right]\hfill \\ C=\left[\begin{array}{llll}1.109\hfill & 0.9093\hfill & 2.602\hfill & 0.0487\hfill \\ -0.2981\hfill & 3.377\hfill & 0.0126\hfill & 3.808\hfill \end{array}\right]\hfill \\ D=zeros\left(2,2\right)\hfill \end{array}$$

(33)

$$K_{\mathrm{LMI}}=\left[\begin{array}{cc}-1.0319\hfill & -0.0039\hfill \\ \multicolumn{1}{c}{-0.0039}& -0.9828\hfill \end{array}\right]$$

(34)

$$K_{\mathrm{proposed}}=\left[\begin{array}{cc}-1.0742\hfill & 0.0566\hfill \\ \multicolumn{1}{c}{0.0566}& -1.0308\hfill \end{array}\right]$$

(35)

Performance indicators are $\gamma =1.6303$ , $\gamma =1.6432$ and $\gamma =1.8575$ , respectively. The unknown crossbreeding uncertainties which the closed-loop system can tolerate are 61.33%, 60.86%, and 53.84%, respectively. These parameters indicate that the uncertainties tolerance indicators of the three are close to each other in terms of robust stability. The closed-loop system can maintain a certain degree of robust stability.

Height and heading submodule controller design

According to the state matrix and control matrix derived, the height and heading state matrix and control matrix are

$$A_3=\left[\begin{array}{cc}-0.519\hfill & 0\hfill \\ \multicolumn{1}{c}{-0.01793}& 0.69365\hfill \end{array}\right]$$

(36)

$$B_3=\left[\begin{array}{cc}-58.0734\hfill & -31.6764\hfill \\ \multicolumn{1}{c}{2.2179}& 2.2703\hfill \end{array}\right]$$

(37)

$$C_3=\left[\begin{array}{cc}1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill \end{array}\right]$$

(38)

The calculation shows that rank sum of the matrix $(A_3,B_3)$ and $(A_3,C_3)$ is 4. It indicates that the subsystem is controllable and observable. Thus, the classic $H_{\infty }$ loop forming, LMI-based $H_{\infty }$ loop forming and the proposed algorithm can be used for controller design.

The pre-weighted and post-weighted matrices are selected based on the selection experience. Thus the integral item is adopted to improve the performance of the controller. The proportional item is used to adjust the lead and lag of the phase. By adjusting the pre-weighted matrix coefficient, the final pre-weighted matrix is selected as follows

$$W_{31}=\mathrm{diag}\left[\begin{array}{cc}\frac{0.015s+0.0027}{s}\hfill & \frac{0.017s+0.0032}{s}\hfill \end{array}\right]$$

(39)

The post-weighted matrix is

$$W_{32}=\mathrm{diag}\left[\begin{array}{cc}1\hfill & 1\hfill \end{array}\right]$$

(40)

The controller of the height and heading submodule using the classic $H_{\infty }$ loop forming, simplified LMI control algorithm and the proposed algorithm is derived as follows, respectively

$$\begin{array}{c}\multicolumn{1}{c}{K_{\mathrm{normal}}:A=\left[\begin{array}{cccc}-132\hfill & 317.2\hfill & -5.054\hfill & 7.366\hfill \\ \multicolumn{1}{c}{6.04}& -43.72\hfill & 0.2631\hfill & -0.4088\hfill \\ \multicolumn{1}{c}{0.1865}& 0.871\hfill & -0.1748\hfill & -0.00623\hfill \\ \multicolumn{1}{c}{0.08227}& -2.592\hfill & 0.0067\hfill & -0.2004\hfill \end{array}\right]}\\ \multicolumn{1}{c}{B=\left[\begin{array}{cc}-111.5\hfill & 36.09\hfill \\ \multicolumn{1}{c}{5.323}& -5.983\hfill \\ \multicolumn{1}{c}{0.1515}& 0.1339\hfill \\ \multicolumn{1}{c}{0.08272}& -0.3578\hfill \end{array}\right]}\\ \multicolumn{1}{c}{C=\left[\begin{array}{cccc}-0.8383\hfill & -1.084\hfill & 2.797\hfill & 0.0996\hfill \\ \multicolumn{1}{c}{-0.1451}& 7.122\hfill & -0.1069\hfill & 3.206\hfill \end{array}\right]}\\ \multicolumn{1}{c}{D=\mathrm{zeros}(2,2)}\end{array}$$

(41)

$$K_{\mathrm{LMI}}=\left[\begin{array}{cc}0.7989\hfill & 0.1547\hfill \\ \multicolumn{1}{c}{0.1547}& -1.2717\hfill \end{array}\right]$$

(42)

$$K_{\mathrm{proposed}}=\left[\begin{array}{cc}0.7268\hfill & 0.1078\hfill \\ \multicolumn{1}{c}{0.1078}& -1.2633\hfill \end{array}\right]$$

(43)

Performance indicators are $\gamma =1.745$ , $\gamma =1.7743$ , and $\gamma =2.2464$ , respectively. The unknown crossbreeding uncertainties which the closed-loop system can tolerate are 57.31%, 56.36%, and 44.52%, respectively. These indicate that the uncertainties tolerance indicators of the three are close to each other in terms of robust stability.

Simulation results analysis

According to dynamic characteristics and performance indicators, the different control algorithms can be selected for each subsystem’s flexibility. Three controllers are compared in terms of stable margin rise time, stability time, and overshoot step performance indicators.

The performance of the submodule using the low-order controller has been described. Compared with Figures 6 and 7, the controller with the attitude submodule has no overshoot.

Figure 6.

Speed subsystem step response characteristics with full order: (a) lateral velocity and (b) longitudinal velocity.

Figure 7.

Speed subsystem step response characteristics: (a) lateral velocity and (b) longitudinal velocity.

The rise time is more consistent with the model motion period, and the decoupling performance is also significantly better than full-order design. However, the order is significantly lower than the full-order one. In addition, the controller performance design and adjustment are more flexible and more convenient.

It can be seen from Figure 7 that the three algorithms are consistent with the overshoot and decoupling performance of the unit step response in the closed-loop velocity submodule. The transverse velocity rise time is about 1 s. The longitudinal velocity rise time is about 1.5 s. These two time parameters conform to the movement cycle of the velocity state. The proposed algorithm is faster to reach the stable time than the other two algorithms, but the rise time is shorter. The decoupling performance of classical $H_{\infty }$ loop forming is more ideal than the other two algorithms. The performance indicator of LMI-based $H_{\infty }$ loop-forming algorithm is between the other two algorithms. However, it lacks flexibly adjustment of performance indicators.

The low-order parametric $H_{\infty }$ loop-forming algorithm can change the system step response performance by adjusting the numerical value of the parameters. It also provides the designer with the extra DOF for adjusting the performance index. It will facilitate the flexible design of the performance index. Compared with the whole-order system controller in Figure 6, the controller with velocity submodule design has no overshoot. And the rise time is more consistent with the model motion period. However, the decoupling performance is weaker than the whole-order controller. The controller order is significantly lower than full-scale controller. Therefore, design and adjustment of performance indicators are more flexible, and selection of control algorithm is also more convenient.

It can be seen from Figure 8 that the three algorithms are almost identical to the unit step response overshoot, rise time, settling time, and decoupling performance of the closed-loop height-heading submodule. The vertical velocity rise time is about 0.5 s. The rise time of the yaw rate response is about 1 s, which conforms to the motion period of the coaxial double rotor. There is still some coupling interference between the vertical rate and the yaw rate.

Figure 8.

Height-heading subsystem step response: (a) vertical rate and (b) yaw rate.

Overshoot of yaw rate step response less than 20%. This is mainly because there is a strong coupling between the two rotor interferences. Adjusting the weight function coefficient is suggested a good way to adjust the performance of closed systems.

Compared with Figure 8, the controller is smaller and the rise time is more consistent with the model’s motion period in Figure 9. The decoupling performance is weaker than the whole-order controller, but the controller order is significantly lower than the full-order controller. The other controllers’ performance design and adjustment are more flexible, and selection of the control algorithm is more convenient.

Figure 9.

Height-heading subsystem step response with full order: (a) vertical rate and (b) yaw rate.

Considering the performance index, robust stability, and easy engineering implementation, it is determined that the inner loop adopts the classical loop forming and the outer layer adopts the proposed algorithm. The order of the three subsystem controllers of the cascade three-module flight structure is 8th order, 4th order, and 4th order, respectively. The order of the whole flight controller is 16th order, and the order of the controller adopted full order and classic $H_{\infty }$ loop forming is 24th order. It indicated that the use of cascade double three-module flight structure reduces the order of the controller significantly and improves the ease of implementation of the engineering application.

Conclusion

The main work is that the control scheme of a robotic coax-helicopter was investigated to simultaneously satisfy not only nominal and robust performance but also the complexity, flexibility, and computation of controller design. The key points of the flight control architecture design are concluded by the analyses to the typical flight control structures of unmanned helicopters. According to the singular value response and the coupling characteristic of the constructed model, a “cascaded double-layer three-submodule” flight control architecture is concluded. The scheme consisted of the submodules of attitude, height-heading, and velocity. The design of the submodules adopts $H_{\infty }$ loop shaping, LMI-based $H_{\infty }$ method, and the proposed algorithm. The performance indexes such as stability margin, rise time, decoupling characteristics, and overshoot are also analyzed by the comparison simulation. The flight controller, which adopts $H_{\infty }$ loop shaping in the inner layer and proposed algorithm in the outer layer, is concluded with comprehensive consideration of the performance figures, stability of robustness, and engineering implementation requirements. The order of the “cascaded double-layer three-submodule” flight controller is much smaller than that of the multivariable full-order flight control architecture. The input and output of high-order controls are simplified into a number of low-order controls. The number of iterations is avoided reducing the difficulty of the selection function in the whole order. High model accuracy and reduced order control can be used to adjust the parameters of the controller. The high-level design process of the impact of uncertainty is weakened. The proposed scheme reduces the difficulty of engineering implementation and improves the flexibility of flight controller design. This article has made a great progress for practical control design of robotic coax-helicopter.