Robust Near-Hovering Flight Controller for Model-Scale Helicopters Via Parametric Approach

（Center for Control Theory and Guidance Technology，Harbin Institute of Technology，Harbin 150001，China）

Robust Near-Hovering Flight Controller for Model-Scale Helicopters Via Parametric Approach

Zhigang Zhou and Yongan Zhang∗

（Center for Control Theory and Guidance Technology，Harbin Institute of Technology，Harbin 150001，China）

This paper aims to provide a parametric design for robust flight controller of the model-scale helicopter.The main contributions lie in two aspects.Firstly，under near-hovering condition，a procedure is presented for simplification of the highly nonlinear and under-actuated model of the model-scale helicopter.This nonlinear system is linearized around the trim values of the chosen flight mode，followed by decomposing this high-order linear model into three lower-order subsystems according to the coupling properties among channels. After decomposition，the three subsystems are obtained which include the coupling subsystem between the roll（pitch）motion and the lateral（longitudinal）motion，the subsystem of the yaw motion and the subsystem of the vertical motion.Secondly，by using eigenstructure assignment，the problem of flight controller design can be converted into solving two optimization problems and the linear robust controllers of these subsystems are designed through solving these optimization problems.Besides，this paper contrasts and analyzed the performances of the LQR controller and the parametric controller.The results demonstrate the effectiveness and the robustness against the parametric perturbations of the parametric controller.

model-scale helicopter；near-hovering；robust flight controller；robust eigenstructure assignment

1 Introduction

In recent years，the research and development of unmanned-aerial-vehicle helicopters have gained much attention in academic and military communities worldwide，as they have potential applications in military and civilian areas.Advances in the embedded system technology and micro-electronic mechanical system technique have made it possible to build a light yet powerful avionic system，which integrates all necessary data processing units，sensors and wireless communication devices，to be installed on a hobby helicopter［1］.As a result，it is economical and realistic to build a model-scale helicopter as an application platform.It is urgent to enable the model-scale helicopter to hover and flight autonomously before putting the platform into applications.

Generally，the model-scale helicopter is an underactuated system with four inputs and nine outputs，and the states are highly coupled.To describe this system and make it flight autonomously，many modelling approaches have been presented in Refs.［2-4］and various control approaches have been exploited by many researchers to design the flight controller in Refs.［4-7］.In Ref.［8］，it is proved that the dynamic properties of the model helicopter in the near-hovering flight can be predicted by an accurate，high-bandwidth，linear state space model. There have existed a variety of linear controllers to make the helicopter flight autonomously.According to the structure of the controller，most of these linear controllers can be divided into the following two categories：

1）The multi-loop control architecture，e.g.the inner／outer control loop structure has been designed asH∞controller／PI controller［6］，H∞controller／state feedback controller［7］and PID controller／PID controller［9］，etc.

2）The multi-input multi-output（MIMO）control architecture，e.g.LQR controller［10］，H∞linear feedback controller［11］，nonlinear MIMO controller［12-14］etc.

In the multi-loop architecture，the flight controller system is divided into the attitude control loop and the position control loop.The position controller is designed to track the desired position via adjusting the roll and pitch angles.And the attitude control loop tracks the desired roll and pitch angles which as the desired input of the position control loop.So，the coupling properties between the attitude and the position of the helicopter cannot be taken into account on the design of the controller.To address these coupling effects，the MIMO approach is considered by many researchers.Most of MIMO approaches are optimization-based，which havebeen exploited by researchers.Thereby the MIMO approach has its own challenges，for example，the stability of the numerical calculation of optimization algorithm.

Different to the currently exiting approaches，this paper aims to develop a MIMO flight controller for a model-scale helicopter via the robust parametric eigenstructure assignment.In order to design the control flight controller，a complete nonlinear model is developed for a model-scale helicopter.Since the stability of numerical calculation is not very well，some necessary simplifications to the nonlinear model will be made to facilitate the linear robust controller design.And then a design procedure of the complete robust hover controller is given，followed by numerical simulations which verify the effectiveness and robustness of this controller.

2 Mathematical Modelling of Model-scale Helicopter

As shown in Fig.1，a body frameOB-XBYBZBis attached to the fuselage at the center of gravity（CG）of the helicopter.The linear and angular velocities of helicopter in term of the body frame are denoted byvBandwB，respectively.The dynamic equations of motion of the helicopter are summarized as follows：

wheremis the total mass of the helicopter；the inertia matrix of the fuselagein body frame，is

and，fBandτBare respectively external forces and torques expressed in the body frame.The external forcefBincludes the main and tail rotor thrustsTmrandTtr，and the gravitational force mg.The external torquesτBinclude the three main directional torquesMφ，Mθ，Mψ，as well as a torque due to the offset of the rotor hingeTmrℓr，and the motor torqueτm．Under the condition of near-hovering flight，the aerodynamic drag can be ignored.Hence，fBandτBcan be expressed as follows：

Hereℓris the longitudinal offset between the axis of the main rotor and the CG of the helicopter and the rotation matrixRIBtransforms the body frame to the inertial frame via the Euler＇s roll，pitch，yaw rotations（φ，θ，ψ），that is to say，

where cxand sxrepresent the cos（x）and sin（x）respectively.

Fig.1 Illustrations of the coordinates frames

The linear velocity of the helicopter with respect to the inertial frame iszI）is the coordinates of the CG of the helicopter in the inertial frame.This velocity can also be expressed as follows：

The angular velocity of the helicopter with respect to the inertial frame，expressed in the body frame，is denoted bywBand can be computed as

Choosing the state vectorxas

and the input vector to the helicopteruas

and then，the state equation of the helicopter can be expressed as follows：

where，txrepresents tan（x）；ℓtris the distance between the main and tail rotor axes；KgwBzis a simple linear model of the electronic gyro.

3 Model Simplifications

Some necessary simplifications to the state Eq.（7）of the helicopter must be made to facilitate the design for the linear robust controller via a parametric eigen-structure assignment approach.In this section，a general procedure of simplifications is given to this nonlinear model，followed by a numerical example. Due to space limitation，only the hover controller is taken as an example to present the procedure of the design in remaining section of this paper.In the case of forward flight controller，a similar presentation may pertain.

3.1 Procedure of Simplifications

3.1.1 Linearization around the operation point

For model-scale helicopters，there are in general two distinct trim conditions：hover and forward flight. The hover condition is characterized by zero linear and angular velocities，i.e.，

and the trim condition for forward flight is characterized as follows：

whereci，i＝1，2 are constants.

These trim conditions can be used to solve the state equation（1）for the trim value at the chosen flight mode.And a linearization of the nonlinear model（7）is obtained by deriving the first-order Taylor series expansion in the trim conditions.The small-signal linear model is given as

whereA∈R9×9，B∈R9×4．It is difficult to design the controller directly for this high-order linear model.For this reason，we must reduce the order by analyzing the coupling properties of this nonlinear system（7）.

3.1.2 Decomposing the linear model of helicopter

Noting thatTtr／Tmris small around the operation point，it is reasonable to decompose the linear model（8）into three subsystems by neglecting coupling properties between the yaw motion and the lateral motion.TheAandBcan be represented by partitioned matrixes as follows，when the couplings between the yaw motion and the lateral motion，that isb23，are neglected.

According to Ref.［8］，theAij，i，j＝1，2，3，are sparse matrices and this fact can also be demonstrated by the following numerical example.As a result，we can decompose the high order linear system into three subsystems which include the coupling subsystem between the roll（pitch）motion and the lateral（longitudinal）motion，or simply Subsystem I for short：

the subsystem of the yaw motion（Subsystem II）：

and the subsystem of the vertical motion（Subsystem III）：

3.2 Numerical Example

As mentioned above，we will take the hover controller as an example to present the procedure of simplifications.The geometric and aerodynamic parameters of a model-scale helicopter are given in Ref.［2］.The trim values at hover are summarized as follows：

The system matrixAand the control matrixBof this linear system（8）are separately shown as follows：

After decomposition，the parameter matrices of the Subsystem I-III are given by

theA13，A23andA32are zero matrixes with appropriate dimension.

The intrinsic coupling properties between the translational subsystem and the rotational subsystem are demonstrated by numerical simulation.The reasonability of this decomposition can be also revealed by these simulation results.The responses of the translational velocities and the attitude angles of the nonlinear system（7）are shown in Fig.2，when the projection ofvBalong theXBdirection，i.e.vBx，has a deviation of 0.1 m／s.Besides，the responses of the nonlinear system（7）induced by the deviations ofvByandvBzare shown in Figs.3 and 4，respectively.The responses of the translational velocities and the attitude angles of the nonlinear system（7）are shown in Fig.5，when the roll angleφhas a deviation of 5°.And，the responses of the nonlinear system（7）induced by the deviations of pitch angleθand yaw angleψare shown in Figs.6 and 7，respectively.

Under the condition of hover fight，these following results can be easily obtained from Figs.2-7：

1）The longitudinal（lateral）motion has stronger coupling properties with the pitch（roll）motion than with other channels.So，the longitudinal（lateral）motion can be controlled through adjusting the pitch（roll）angle.

2）The coupling properties between the vertical motion with other channels can be neglected，so the altitude of the helicopter can be controlled by the lift of the main rotor.

3）The translational subsystem is a slowly varying system and the rotational subsystem is a fast varying system，and the latter can be regarded as the input of the former.

4）There exist weak coupling properties between the lateral motion and the yaw motion.

Hence，this decomposition is reasonable and the controller design for the model-scale helicopter is converted into the controller design for these three subsystems.

For the subsystem（12），the controller can be simply designed as

wheres9is the closed-loop pole of the vertical subsystem.To finish the hover controller design，the robust parameterized controllers for subsystems（10）and（11）will be given in Section 4.

Fig.2 Responses of the system，whenvBxhas a deviation of 0.1 m／s

Fig.3 Responses of the system，when vByhas a deviation of 0.1 m／s

Fig.4 Responses of the system，when vBzhas a deviation of 0.1 m／s

Fig.5 Responses of the system，when φ has a deviation of 5°

Fig.6 Responses of the system，whenθhas a deviation of 5°

Fig.7 Responses of the system，whenψhas a deviation of 5°

4 Robust Hover Controller Design

As mentioned in previous section，we will design the robust parameterized controllers for the subsystems（10）and（11）to enable the helicopter to hover in this section.These requisite concepts of the parametric design for a general linear system can be obtained in Ref.［15］.Here，we propose the procedure of robust hover controller design via the parametric approach.

The nominal system of the subsystem（10）is

Then the closed-loop system can be obtained as

Supposing that all desired closed-loop eigenvalues of the matrixAc1are distinct and the eigenvalues set and the eigenvector matrix associated with the eigenvalues set is denoted respectively bys1andV1（s1），the following Sylvester matrix equation can be established

According to the result of Ref.［15］on the solutions to the Sylvester matrix equation（15），the general parametric expressions for the closed-loop eigenvector matrix and the corresponding matrixcan be obtained as

In order to ensure that the matrixK1is real and the independence between the closed-loop eigenvectors，the parameter vector6，satisfy the following constraints

When the above constraints are met，the parametric form of the gain matrixK1∈R2×6is given by

The gain matrixK1designed for the nominal system（A11，B11），will impact the desired close-loop poles，in that the uncertain factors in modeling make the nominal system be not consistent with the actual system.According to Ref.［16］，we chooseκ2（V1）＝as the robust performance index such that the close-loop poles are insensitive to perturbations as possible.The index function is chosen as follows：

whereαi，i＝1，2，are the weighting coefficients.This index function can be parameterized as a function offands1．Then，the problem to controller design is converted into a constrained optimization problem as follows：

The desired gain matrixK1can be obtained by solving the above optimization problem（21）via the MATLAB sloptim toolbox.The robust parameterized controllerK2for the subsystem（11）can also be designed in a similar way.

5 Numerical Simulations

When the weighting coefficients，αi，i＝1，2，are 0.3 and 0.7，and the weighting matrices，Qi，i＝1，2，are chosen as

After calculation，the close-loop poles of the subsystem（10）are

and the gain matrixK1is

Analogously，the close-loop poles and the gain matrixK2of the subsystem（11）can be obtained as follows：

where the initial deviations of the lateral velocity and yaw angle are chosen as 0.05 m／s and 1°，because of the existence of gyro.Under the condition of nonparameter perturbations，the trajectories of the linear velocities and the attitude angles of this nonlinear system（7）are separately shown in Figs.8-13.When the perturbations of inertial moments are 30%of nominal values，Figs.14-19 show the trajectories of the linear velocities and the attitude angles of this nonlinear system（7），respectively.The solid line and dash line represent respectively the parameterized controller and LQR controller.Obviously，whether or not there are parametric perturbations，the parametric controller can rapidly make the linear velocities converge to zeroes，compared to the LQR method.This result can also be demonstrated by the fact that the attitude angles have the bigger overshoot in that the linear velocities are controlled by varies of the roll and pitch angles.

Fig.8 Trajectories of thevBxwithout parametric perturbations

Fig.9 Trajectories of the vBywithout parametric perturbations

Fig.10 Trajectories of the vBzwithout parametric perturbations

Fig.11 Trajectories of the φ without parametric perturbations

Fig.12 Trajectories of the θ without parametric perturbations

Fig.13 Trajectories of theψwithout parametric perturbations

Fig.14 Trajectories of the vBxwith parametric perturbations

Fig.15 Trajectories of the vBywith parametric perturbations

Fig.16 Trajectories of the vBzwith parametric perturbations

Fig.17 Trajectories of the φ with parametric perturbations

Fig.18 Trajectories of the θ with parametric perturbations

Fig.19 Trajectories of the ψ with parametric perturbations

6 Conclusions

A robust near-hovering flight controller has been designed for a model-scale helicopter via eigenstructure assignment in this paper.According to the knowledge of the eigenstructure assignment，the problem of flight controller design has been converted into solving optimization problems.In order to ensure the numerical stability of optimization algorithm，a simplification procedure for the complex nonlinear model has been proposed.After simplifications，the controller design for a high-order model is also reduced into the controllers design for three lower-order subsystems. Numerical simulations have demonstrated the performance of parameterized controller and robustness against the parameter perturbations.

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TP273

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：1005-9113（2015）05-0069-09

10.11916／j.issn.1005-9113.2015.05.011

2014-06-14.

∗Corresponding author.E-mail：zya＠hit.edu.cn.