# Dynamic Decoupling Control Optimization for a Small-Scale Unmanned Helicopter

Authors: Rui Ma; Li Ding; Hongtao Wu
Journal: Journal of Robotics
 (2018)

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## Abstract

This article presents design and optimization results from an implementation of a novel disturbance decoupling control strategy for a small-scale unmanned helicopter. Such a strategy is based on the active disturbance rejection control (ADRC) method. It offers an appealing alternative to existing control approaches for helicopters by combining decoupling and disturbance rejection without a detailed plant dynamics. The tuning of the control system is formulated as a function optimization problem to capture various design considerations. In comparison with several different iterative search algorithms, an artificial bee colony (ABC) algorithm is selected to obtain the optimal control parameters. For a fair comparison of control performance, a well-designed LQG controller is also optimized by the proposed method. Comparison results from an attitude tracking simulation against wind disturbance show the significant advantages of the proposed optimization control for this control application.


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## Body

## 1. Introduction

In recent years, rotary-wing Unmanned Aerial Vehicles (UAVs) including quadrotors, helicopters, and ducted fans are attractive to industries and academia [1, 2]. With the unique features such as hovering, good maneuverability, and low costs, they have been applied to diverse domains by installing different sensors, cameras, or other payloads on the platform [3, 4]. However, due to the complexity of flight dynamics, it is still challenging to design an appropriate flight control system that satisfies the requirement for autonomous flight.Small-scale unmanned helicopter is a representative of the rotary-wing UAVs. It is considered as an inherently unstable, highly nonlinear, and underactuated system with significant dynamic coupling. With the small size and agile maneuverability, it is more susceptible to gust disturbance than those full-sized counterpart. Furthermore, the dynamic parameters change with the load and flight conditions. These factors cause serious challenges in dealing with concerns about the robustness, disturbance rejection, decoupling, and other control problems. To address the above problems, the classical single-input/single-output (SISO) feedback control methods are implemented in [5–7]. In these researches, the controller is optimized based on the identified model established firstly for a small helicopter. Nevertheless, many researchers have recognized that the complexity nature of helicopter and better control capability requires more advanced technologies. Cases include direct adaptive neural command controller [8], adaptive control methods [9], nonlinear control methods [10], vision-based guidance control techniques [11], and intelligent control methods like fuzzy logic approach [12] and neural network [13]. And the most pervasive choice in practical applications is the robust control approach: for example, the Kalman filter-based linear quadratic integral (LQI) approach [14], linear quadratic regulator (LQR) [15], and the H∞ control approach [16, 17]. These methods provide a reasonable countermeasure for both disturbances and multivariable effects of the helicopter. However, the effectiveness of the prevailing model-based control approaches is exceedingly dependent on the exact model and aerodynamic coefficients of the plant. Facing the complicated control problem of the helicopter, the effective solution is to compensate the disturbance immediately and reduce the dependence of plant model.The main contribution of this paper is introducing a dynamic decoupling control (DDC) strategy [18] and its optimization method to a small-scale helicopter. This strategy is rooted in active disturbance rejection control (ADRC) that was recently proposed by Han [19]. The key idea of the method is to treat the total disturbance (incorporating the interactions among control loops and the unknown external disturbances) as a state variable, which can be estimated by an extended state observer (ESO) through the input-output data of the plant in real time. Consequently, unlike most existing model-dependent control methods, very little information of the model is required for ADRC [20]. ADRC offers a practical solution to decoupling control problems in the presence of large uncertainties and has been successfully applied in many engineering applications, e.g., aircraft flight control [21] and the chemical processes [22]. Moreover, in order to simplify the implementation of ADRC, Gao proposed the linear active disturbance rejection control (LADRC), which offers much better performance and needs few parameters to tune, and detailed comparison studies can be found in [23].Since the performance of LADRC depends on the convergence speed of state observer, the bandwidth of which is the most important tuning parameter. Obviously, the trade-offs between the robustness and performance have always been difficult, especially for the helicopter. This problem can be solved by using a multiobjective optimization algorithm in the simulated environment. Artificial bee colony algorithm (ABC) was first proposed by Karaboga in 2005 [24] and successfully applied to control optimization, including optimal tuning of PID controller in [25], optimized LQR controller in [26], and robust fuzzy PSS design [27]. As we have known, usual optimization algorithms conduct only one search operation in one iteration, but ABC algorithm can conduct both local search and global search in each iteration; as a result, the probability of finding the optimal parameters is significantly increased, which efficiently avoids local optimum to a large extent.This paper considers a design and optimization of the DDC controller used in our TREX-600 helicopter. As a controlled plant, the dynamical model is obtained through the system identification method in previous work [26]. The main idea can be characterized as follows: (i) in the design of decoupling control, all the information needed is only the predetermined input-output pairing of helicopter’s dynamic model, (ii) the effect of one input to all other outputs that is not paired with, namely, the cross channel interference is viewed as a ‘disturbance’ to be actively estimated and canceled out in DDC framework, and (iii) the parameter tuning problem is transformed into a functional optimization problem defined by a combination of different control performance indexes, and ABC algorithm is introduced to calculate the optimal solution. Using this approach, we can optimize controllers with different control requirement.The paper is organized as follows. In Section2, the helicopter dynamical model under consideration is briefly introduced. Section 3 describes how to use DDC strategy to decouple the helicopter. Section 4 formulates the parameter optimization problem. Simulation tests on the model are shown in Section 5. Finally, the main conclusions are summarized in Section 6.


## 2. System Model for Unmanned Small-Scale Helicopter

Based on the first principle approach, the dynamics of helicopter is regarded as a six-degrees-of-freedom rigid-body dynamics augmented with a simplified main rotor flapping dynamics and a factory-installed yaw rate gyro dynamics. As illustrated in Figure1 and summarized in Table 1, this model contains fifteen states and four inputs. Detailed information of the physical parameters and modeling structure can be found in [28]. A brief overview of the flight dynamical model is presented next.Table 1
Physical descriptions of the state and input variables of the helicopter dynamical model.
Variable Physical significance Unit P = [ X , Y , Z ] T Position in Earth-fixed coordinate frame m V = [ u , v , w ] T Velocity vector along body frame X-, Y-, and Z-axes m/s Ω = [ p , q , r ] T Roll, pitch, and yaw angular rates rad/s Θ = [ φ , θ , ψ ] T Euler angles rad a s , b s Longitudinal and lateral main rotor flapping angle rad r f b Intermediate state in yaw rate feedback controller dynamics N/A u l a t lateral cyclic rotor control input N/A u l o n longitudinal cyclic rotor control input N/A u c o l collective pitch control input N/A u p e d tail rotor pedal control input N/AFigure 1
Illustration for helicopter states and coordinates.The translational motion and rotational motion of the helicopter [15] are described as (1)P˙=RΘV,Θ˙=SΘΩ,V˙=Fbm+Fgm-Ω×VΩ˙=I-1Mb-Ω×IΩwhere Fg=-mgsin⁡θ,mgsin⁡ϕcos⁡θ,mgcos⁡θcos⁡ϕT is the gravity force vector projected onto the body frame (BF); m is the helicopter mass; I=diagIXX,IYY,IZZ is the inertial moment matrix about the reference axes; Fb and Mb denote the combined aerodynamic force and moment vectors acting on the helicopter center of gravity (CG), respectively. The transformation matrices R and S are, respectively, given as(2)R=cθcψsϕsθcψ-cϕsψcϕsθcψ+sϕsψcθsψsϕsθsψ+cϕcψcϕsθsψ+sϕcψ-sθsϕcθcϕcθ,(3)S=1tθsϕtθcϕ0cϕ-sϕ0sϕcθcϕcθ,in which the compact notations s(∗), c(∗), and t(∗) denote sin⁡(∗), cos⁡(∗), and tan⁡(∗), respectively.In the helicopter system,Fb and Mb are generated by the aerodynamic forces of the fuselage and the control forces which originate from the main rotor thrust and tail rotor thrust. Generally, we calculate Fb and Mb without the consideration of the aerodynamic forces of fuselage due to the relatively small influence on the model. Therefore, the force and moment components in the BF are, respectively, given by(4)Fb=FXFYFZ=-Tmrsin⁡asTmrsin⁡bs-Ttr-Tmrcos⁡ascos⁡bs,(5)Mb=LMN=Kβ+TmrHmrsin⁡bs-TtrHtrKβ+TmrHmrsin⁡asNmr+TtrDtrwhere Tmr, Nmr, and Ttr are the main rotor thrust and moment and the tail rotor thrust, respectively; Dtr is the distance from the CG to the tail rotor hub, along the x direction; Hmr and Htr are the distance from the CG to the main rotor and tail rotor hub, along the z direction, respectively, and Kβ is the main rotor spring constant.The main rotor flapping dynamics, which are common to all small-scale helicopters, is described by the following two coupled first-order differential equations [29]:(6)a˙sb˙s=-q-asτf+Abb+Alonulon-p+Baas-bsτf+Blatulatwhere τf is the main rotor time constant; Ab and Ba are cross coupling derivatives that influence the longitudinal and lateral flapping motions; and Alon and Blat are effective linkage gains.Since the high sensitivity of the bare yaw channel dynamics, a feedback yaw rate controller is widely used in small-scale helicopters. The UAV system reserved this feature for the convenience of manual control. Accordingly, the augmented yaw dynamics are modeled as a first-order bare airframe dynamics with a yaw rate feedback represented by a simple first-order low-pass filter [30]. The corresponding differential equations are given as(7)r˙r˙fb=Nrr+Npeduped-rfb-Krfbrfb+Krrwhere Nr, Nped, Kr, and Krfb are the parameters to be identified.


## 3. Dynamic Decoupling Control (DDC) of the Helicopter

### 3.1. LESO Based Dynamic Decoupling Control Method

Linear Active Disturbance Rejection Controller (LADRC) is a novel control method which is parameterized from ADRC [20] to simplify the tuning process. In this work, LADRC-based DDC approach [18] is implemented to tackle the decoupling problem for helicopter attitude dynamics. Define fi as the combined effect of the internal coupling dynamics and external disturbances in each channel:(8)fi=hix,x1,⋯xni-1,wi+∑j=1mbijuj-biiuiThen, the helicopter model can be seen as a set of coupled input-output equations with a predetermined relationship:(9)y1n1=f1+b11u1⋮yin2=fi+biiui⋮ymnm=fm+bmmumwhere wi and x are external disturbances and state vector, respectively; ui and yi are the dominant input and output of the ith loop (i=1,2,⋯,m), respectively; bij is the input gain; superscript (ni) denotes the nith order derivative. Assuming the order ni are given, the numbers of inputs and outputs are the same.Most existing decoupling control approaches assume the knowledge of the elaborate plant model or disturbance model, which is a considerable challenge in practice. LADRC makes a breakthrough that realistically estimatesfi in real time from input-output data instead of identifying an accurate mathematical model. The idea is introduced next.Define an enlarged state vectorx-i=(yi,y˙i,⋯,yi(j-1),fi),j=1,2,⋯,ni, in which fi is added as an extended state. Assume fi is differentiable and vi=f˙i is bounded. Then the augmented state-space form of ith loop in (9) is represented as(10)x-˙i=Ax-+Bui+Gviy=Cx-iwhere(11)A=010⋯0001⋯0⋮⋮⋮⋱⋮000⋯1000⋯0,B=00⋮b0,i0,G=00⋮01,C=10⋮00T.Based on the state-space model, a linear extended state observer (LESO) is designed to estimatex-i:(12)x^˙i=Ax^i+Bui+Liyi-y^iy^i=x^iwhere Li=[l1,i,l2,i,⋯,lni+1,i]T is the observer gain needed to be chosen.With a properly selected observer gain, the system states andfi will be accurately estimated by the LESO in real time. The following control law for ith loop can be designed to reduce the closed-loop system approximately to a unit gain cascaded integrator plant yi(ni)=fi+biiui≈u0,i:(13)ui=u0,i-f^ibi,iIt is a relatively simple control problem, which is solved by using a PD controller with a feedforward term:(14)ui,0=k1,iri-x^1,i+⋯+kni,irini-1-x^ni,i+riniwhere (k1,i,k2,i,⋯,kn,i) are the controller gains to be selected and ri is the trajectory reference. The structure of LADRC is shown in Figure 2.Figure 2
The block diagram of LADRC.


### 3.2. Parameterization of LADRC

For simplicity and practicality, both of the LESO and PD controller are parameterized in a special case as suggested in [18], where all the observer poles and controller poles are placed at -ω0,i and -ωc,i, respectively. The characteristic polynomials of (12) and (14) are constituted, respectively, as(15)λo,is=sni+1+l1,isni+⋯+lni+1,i=s+ω0,ini+1(16)λc,is=sni+kni,isni+⋯+k1,i=s+ωc,iniwith(17)lj,i=ni+1!ω0,ijj!ni+1-j!,j=1,2,⋯,ni+1(18)kj,i=ni!ωc,ini-j+1j-1!ni-j+1!,j=1,2,⋯,niIt makes ω0,i, ωc,i the bandwidth and the only tuning parameters for LESO and the PD controller, respectively. In general, higher bandwidth corresponds to better transient response, disturbance estimation, and rejection. However, too large a value of ω0,i would cause oscillation in states. The measurement noise and excessive increase of ωc,i make the control signal oversize in magnitude and change rate. On the other hand, an appropriate selection of ω0,i and ωc,i should be subjected to physical limits and dynamic characteristics of the plant.As seen above, the LADRC approach is a practical method for decoupling control of MIMO system. The satisfied performance will be obtained by tuning only two parametersω0,i and ωc,i. Furthermore, it works without the detail model of the original plant, except the orders of each input-output pairs and input gains bii. The proofs of stability are given in [31, 32].


### 3.3. Attitude Controller Design for the Helicopter

As shown in Figure3, the decoupling controller is designed to have the form of three LADRC controllers. Moreover, we selected their orders according to the relative degrees of the dynamical model. It is assumed that the controlled output Θ=[φ,θ,ψ]T can be measured directly and that the trim value r=[φr,θr,ψr]T is within the physical limitation of helicopter flight.Figure 3
The block diagram of the attitude decoupling controller.To use the DDC approach, the order of input-output pairs in the model must be explicit. Figure4 displays the interconnection of the helicopter subsystems, which offers a more physically meaningful design. Note that the helicopter attitude dynamics can be separated in two interconnected subsystems [6], i.e., the lateral and longitudinal subsystem and yaw dynamics. The cyclic commands ulon and ulat control the pitch and roll moment, and the pedal command uped manipulates the heading of the helicopter. In this case, we set the lateral and longitudinal subsystem as two third-order systems and the yaw dynamics as a second-order system.Figure 4
Interconnection of the helicopter dynamics model. The terms associated with the gravity force are disregarded.According to aforementioned discussion and analysis, we definef1, f2, and f3 as the total disturbance in each channel and rewrite (9) as (19)ϕn1=f1+b01ulatθn2=f2+b02ulonψn3=f3+b03upedwhere b01, b02, and b03 are the input gains of lateral cyclic, longitudinal cyclic, and tail rotor collective pitch, respectively. In the LADRC design, these input gains are treating as another tuning parameter besides ω0,i and ωc,i to improve the performance of the reduced order closed-loop system. Note that the orders of each loop are n1=n2=3 and n3=2, and the LADRC-based DDC controller can be realized by designing the LESO and PD controller for each loop, accordingly.


## 3.1. LESO Based Dynamic Decoupling Control Method

Linear Active Disturbance Rejection Controller (LADRC) is a novel control method which is parameterized from ADRC [20] to simplify the tuning process. In this work, LADRC-based DDC approach [18] is implemented to tackle the decoupling problem for helicopter attitude dynamics. Define fi as the combined effect of the internal coupling dynamics and external disturbances in each channel:(8)fi=hix,x1,⋯xni-1,wi+∑j=1mbijuj-biiuiThen, the helicopter model can be seen as a set of coupled input-output equations with a predetermined relationship:(9)y1n1=f1+b11u1⋮yin2=fi+biiui⋮ymnm=fm+bmmumwhere wi and x are external disturbances and state vector, respectively; ui and yi are the dominant input and output of the ith loop (i=1,2,⋯,m), respectively; bij is the input gain; superscript (ni) denotes the nith order derivative. Assuming the order ni are given, the numbers of inputs and outputs are the same.Most existing decoupling control approaches assume the knowledge of the elaborate plant model or disturbance model, which is a considerable challenge in practice. LADRC makes a breakthrough that realistically estimatesfi in real time from input-output data instead of identifying an accurate mathematical model. The idea is introduced next.Define an enlarged state vectorx-i=(yi,y˙i,⋯,yi(j-1),fi),j=1,2,⋯,ni, in which fi is added as an extended state. Assume fi is differentiable and vi=f˙i is bounded. Then the augmented state-space form of ith loop in (9) is represented as(10)x-˙i=Ax-+Bui+Gviy=Cx-iwhere(11)A=010⋯0001⋯0⋮⋮⋮⋱⋮000⋯1000⋯0,B=00⋮b0,i0,G=00⋮01,C=10⋮00T.Based on the state-space model, a linear extended state observer (LESO) is designed to estimatex-i:(12)x^˙i=Ax^i+Bui+Liyi-y^iy^i=x^iwhere Li=[l1,i,l2,i,⋯,lni+1,i]T is the observer gain needed to be chosen.With a properly selected observer gain, the system states andfi will be accurately estimated by the LESO in real time. The following control law for ith loop can be designed to reduce the closed-loop system approximately to a unit gain cascaded integrator plant yi(ni)=fi+biiui≈u0,i:(13)ui=u0,i-f^ibi,iIt is a relatively simple control problem, which is solved by using a PD controller with a feedforward term:(14)ui,0=k1,iri-x^1,i+⋯+kni,irini-1-x^ni,i+riniwhere (k1,i,k2,i,⋯,kn,i) are the controller gains to be selected and ri is the trajectory reference. The structure of LADRC is shown in Figure 2.Figure 2
The block diagram of LADRC.


## 3.2. Parameterization of LADRC

For simplicity and practicality, both of the LESO and PD controller are parameterized in a special case as suggested in [18], where all the observer poles and controller poles are placed at -ω0,i and -ωc,i, respectively. The characteristic polynomials of (12) and (14) are constituted, respectively, as(15)λo,is=sni+1+l1,isni+⋯+lni+1,i=s+ω0,ini+1(16)λc,is=sni+kni,isni+⋯+k1,i=s+ωc,iniwith(17)lj,i=ni+1!ω0,ijj!ni+1-j!,j=1,2,⋯,ni+1(18)kj,i=ni!ωc,ini-j+1j-1!ni-j+1!,j=1,2,⋯,niIt makes ω0,i, ωc,i the bandwidth and the only tuning parameters for LESO and the PD controller, respectively. In general, higher bandwidth corresponds to better transient response, disturbance estimation, and rejection. However, too large a value of ω0,i would cause oscillation in states. The measurement noise and excessive increase of ωc,i make the control signal oversize in magnitude and change rate. On the other hand, an appropriate selection of ω0,i and ωc,i should be subjected to physical limits and dynamic characteristics of the plant.As seen above, the LADRC approach is a practical method for decoupling control of MIMO system. The satisfied performance will be obtained by tuning only two parametersω0,i and ωc,i. Furthermore, it works without the detail model of the original plant, except the orders of each input-output pairs and input gains bii. The proofs of stability are given in [31, 32].


## 3.3. Attitude Controller Design for the Helicopter

As shown in Figure3, the decoupling controller is designed to have the form of three LADRC controllers. Moreover, we selected their orders according to the relative degrees of the dynamical model. It is assumed that the controlled output Θ=[φ,θ,ψ]T can be measured directly and that the trim value r=[φr,θr,ψr]T is within the physical limitation of helicopter flight.Figure 3
The block diagram of the attitude decoupling controller.To use the DDC approach, the order of input-output pairs in the model must be explicit. Figure4 displays the interconnection of the helicopter subsystems, which offers a more physically meaningful design. Note that the helicopter attitude dynamics can be separated in two interconnected subsystems [6], i.e., the lateral and longitudinal subsystem and yaw dynamics. The cyclic commands ulon and ulat control the pitch and roll moment, and the pedal command uped manipulates the heading of the helicopter. In this case, we set the lateral and longitudinal subsystem as two third-order systems and the yaw dynamics as a second-order system.Figure 4
Interconnection of the helicopter dynamics model. The terms associated with the gravity force are disregarded.According to aforementioned discussion and analysis, we definef1, f2, and f3 as the total disturbance in each channel and rewrite (9) as (19)ϕn1=f1+b01ulatθn2=f2+b02ulonψn3=f3+b03upedwhere b01, b02, and b03 are the input gains of lateral cyclic, longitudinal cyclic, and tail rotor collective pitch, respectively. In the LADRC design, these input gains are treating as another tuning parameter besides ω0,i and ωc,i to improve the performance of the reduced order closed-loop system. Note that the orders of each loop are n1=n2=3 and n3=2, and the LADRC-based DDC controller can be realized by designing the LESO and PD controller for each loop, accordingly.


## 4. Optimization Problem Formulation

### 4.1. The Objective Function

The proposed LADRC tuning method using ABC is schematically shown in Figure5, where the plant is the identified model of the TREX-600 helicopter. As stated above, the primary concern in the implementation of LADRC is maximizing the bandwidth ω0,i and ωc,i, and identifying a suitable value of bii while satisfying the system constraints and design objective. It can be accomplished by forming a functional optimization problem. Also, the design specifications are comprehensively represented by a new objective function. In the optimization procedure, by changing the closed-loop step responses according to its automatically selected controller parameters and calculating the objective function value at every generation, the iterative algorithm searches the optimal parameters for the controller subjected to the design specifications.Figure 5
LADRC tuning scheme with ABC.In the tuning of the controller, the objective function can be formed by different performance index that considers the step responses of the entire system. Typical performance index in the time domain includes integral square error (ISE), integral of absolute error (IAE), integral time absolute error (ITAE) [33], rise time (Tr), settling time (Ts), overshoot (OS), and steady-state error (Ess). The selection of these factors and form of the function can be determined depending on the design requirements. In this work, the desired control performance should have a small or no overshoot in the step response with a minimal settling time, and the control signal should be smooth within the physical limit. Hence, we defined the objective function F in this work as a linear combination of the ISE, integral of the square of the control signal, the overshoot OS, and the one percent settling time Ts [34]:(20)F=∑i=13α∫yi-r2+β∫ui2+σOSi+εTs,iwhere the variables of α, β, σ, and ε are the adjustment parameters. The values of these parameters are generally selected by using trial-and-error method. During the minimization of the objective function, all of the performance indexes are minimized and all of the disadvantaged controller parameters caused to system unstable or a poor performance will be eliminated by the algorithm.Using the proposed objective function (20), the parameter tuning for the controller becomes a function optimization problem. This method combines a variety of performance indices which can be selected and weighted as required. Then the desired control performance and its parameter setting can be found by minimizing the value of (20). It makes this method different to the traditional optimal control method. Since (20) is nonlinear and discontinuous, simple search methods are usually lost in local optimum, as shown in [35]. Advanced search methods like GA, PSO, and ABC provide us with efficient solutions for solving this problem.


### 4.2. ABC Algorithm

In order to introduce the search mechanism of ABC algorithm, we should define three essential components: employed bees, unemployed bees, and food source [36]. And the unemployed bees are divided into the following bees and scout bees. The population of the colony bees is Ns, the number of employed bees is Ne, and the number of unemployed bees is Nu, which satisfies the relation Ns=2Ne=2Nu. We also define D as the dimension of solution vector, i.e., the number of the unknown parameters. ABC algorithm treats each solution vector as a food source and combines the global search of unemployed with the local search of employed bees. The detailed procedure of executing the proposed algorithm is described as follows.Step 1.
Randomly initialize a set of possible solutions(x1,⋯,xNs), and the particular solution xi can be governed by(21)xij=xminj+rand0,1xmaxj-xminjwhere j∈{1,⋯,D} denotes the jth dimension of the solution vector. xminj and xmaxj mean the lower and upper bounds, respectively.Step 2.
Apply a specific function to calculate the fitness of the solutionxi according to the following equations and select the top Ne best solutions as the number of the employed bees:(22)fiti=11+Fiwhere fiti is the fitness function and Fi is objective function depicted in (20).Step 3.
Each employed bee searches new solution in the neighborhood of the current position vector in thenth iteration as follows:(23)vij=xij+λijxij-xkjwhere k∈{1,⋯,D}, k≠i, both k and j are randomly generated, and λij is a random parameter in the range from -1 to 1. In order to ensure that the algorithm evolves to the global optimal, we apply the greedy selection equation (22) to choose the better solution between vij and xij into the next generation:(24)xij=vij,fitvij>fitxijxij,fitvij≤fitxijStep 4.
Each following bee selects an employed bee to trace according to the parameter of probability value. The formula of the probability method is described as(25)pi=fiti∑i=1NefitiStep 5.
The following bee searches in the neighborhood of the selected employed bee’s position to find new solutions. Update the current solution according to their fitness.Step 6.
If the search timetrial is larger than the predetermined thresholdlimit and the optimal value cannot be improved, the location vector can then be reinitialized randomly by scout bees according to the following equation:(26)xin+1=xmin+rand0,1xmax-xmin,trial>limitxin,trial≤limitStep 7.
Output the best solution parameters achieved at the present time, and go back to Step3 until termination criteria Tmax are met.


## 4.1. The Objective Function

The proposed LADRC tuning method using ABC is schematically shown in Figure5, where the plant is the identified model of the TREX-600 helicopter. As stated above, the primary concern in the implementation of LADRC is maximizing the bandwidth ω0,i and ωc,i, and identifying a suitable value of bii while satisfying the system constraints and design objective. It can be accomplished by forming a functional optimization problem. Also, the design specifications are comprehensively represented by a new objective function. In the optimization procedure, by changing the closed-loop step responses according to its automatically selected controller parameters and calculating the objective function value at every generation, the iterative algorithm searches the optimal parameters for the controller subjected to the design specifications.Figure 5
LADRC tuning scheme with ABC.In the tuning of the controller, the objective function can be formed by different performance index that considers the step responses of the entire system. Typical performance index in the time domain includes integral square error (ISE), integral of absolute error (IAE), integral time absolute error (ITAE) [33], rise time (Tr), settling time (Ts), overshoot (OS), and steady-state error (Ess). The selection of these factors and form of the function can be determined depending on the design requirements. In this work, the desired control performance should have a small or no overshoot in the step response with a minimal settling time, and the control signal should be smooth within the physical limit. Hence, we defined the objective function F in this work as a linear combination of the ISE, integral of the square of the control signal, the overshoot OS, and the one percent settling time Ts [34]:(20)F=∑i=13α∫yi-r2+β∫ui2+σOSi+εTs,iwhere the variables of α, β, σ, and ε are the adjustment parameters. The values of these parameters are generally selected by using trial-and-error method. During the minimization of the objective function, all of the performance indexes are minimized and all of the disadvantaged controller parameters caused to system unstable or a poor performance will be eliminated by the algorithm.Using the proposed objective function (20), the parameter tuning for the controller becomes a function optimization problem. This method combines a variety of performance indices which can be selected and weighted as required. Then the desired control performance and its parameter setting can be found by minimizing the value of (20). It makes this method different to the traditional optimal control method. Since (20) is nonlinear and discontinuous, simple search methods are usually lost in local optimum, as shown in [35]. Advanced search methods like GA, PSO, and ABC provide us with efficient solutions for solving this problem.


## 4.2. ABC Algorithm

In order to introduce the search mechanism of ABC algorithm, we should define three essential components: employed bees, unemployed bees, and food source [36]. And the unemployed bees are divided into the following bees and scout bees. The population of the colony bees is Ns, the number of employed bees is Ne, and the number of unemployed bees is Nu, which satisfies the relation Ns=2Ne=2Nu. We also define D as the dimension of solution vector, i.e., the number of the unknown parameters. ABC algorithm treats each solution vector as a food source and combines the global search of unemployed with the local search of employed bees. The detailed procedure of executing the proposed algorithm is described as follows.Step 1.
Randomly initialize a set of possible solutions(x1,⋯,xNs), and the particular solution xi can be governed by(21)xij=xminj+rand0,1xmaxj-xminjwhere j∈{1,⋯,D} denotes the jth dimension of the solution vector. xminj and xmaxj mean the lower and upper bounds, respectively.Step 2.
Apply a specific function to calculate the fitness of the solutionxi according to the following equations and select the top Ne best solutions as the number of the employed bees:(22)fiti=11+Fiwhere fiti is the fitness function and Fi is objective function depicted in (20).Step 3.
Each employed bee searches new solution in the neighborhood of the current position vector in thenth iteration as follows:(23)vij=xij+λijxij-xkjwhere k∈{1,⋯,D}, k≠i, both k and j are randomly generated, and λij is a random parameter in the range from -1 to 1. In order to ensure that the algorithm evolves to the global optimal, we apply the greedy selection equation (22) to choose the better solution between vij and xij into the next generation:(24)xij=vij,fitvij>fitxijxij,fitvij≤fitxijStep 4.
Each following bee selects an employed bee to trace according to the parameter of probability value. The formula of the probability method is described as(25)pi=fiti∑i=1NefitiStep 5.
The following bee searches in the neighborhood of the selected employed bee’s position to find new solutions. Update the current solution according to their fitness.Step 6.
If the search timetrial is larger than the predetermined thresholdlimit and the optimal value cannot be improved, the location vector can then be reinitialized randomly by scout bees according to the following equation:(26)xin+1=xmin+rand0,1xmax-xmin,trial>limitxin,trial≤limitStep 7.
Output the best solution parameters achieved at the present time, and go back to Step3 until termination criteria Tmax are met.


## 5. Simulation Tests

For best performance of optimization, we compare the results of ABC with the existing search iterative algorithm methods, including the trial-and-error method, GA, and PSO. We set the population size as 20 and iteration numbers as 50 for each algorithm. The adjustment parameters (α, β, σ, ε) are selected as 1, 1, 0.1, and 0.2, respectively. The step command with the value of 0.2618 rad is applied to each of the input channels. 0.1% measurement white noise is added to the plant. For GA, the crossover probability and mutation probability are chosen as 0.8 and 0.2, respectively. For PSO, the optimal parameters, i.e., social, individual and inertia weight, are set to 2, 2, and 0.8, respectively. Finally, for ABC the threshold is set tolimit =5. The results are presented in Table 2 and Figure 8.Table 2
Tuning performance of trial-and-error method, GA, PSO, and ABC algorithm.
Applied process Time domain performance T r / s O S / % T s / s E s s / r a d F trial-and-error 0.70 2.33 1.17 0 1.2351 0.55 1.30 0.77 0 0.37 1.86 0.65 0  GA 0.596 2.13 0.97 0 1.013 0.514 1.74 0.80 0 0.303 1.27 0.48 0  PSO 0.62 2.01 0.94 0 0.9713 0.53 1.82 0.81 0 0.30 1.11 0.40 0  ABC 0.60 2.08 0.96 0 0.9315 0.57 1.60 0.86 0 0.35 0.72 0.31 0Figure6 shows the evolution curves of ABC, GA, and PSO. The figure demonstrates that the objective function reduces as the generation iterates with time, gradually converging to an optimal result. Compared with GA and PSO, ABC achieves a better result with smaller objective function after 28 iterations. Table 2 indicates that all of the controllers have no steady-state error and that the trial-and-error method gets the largest time domain index. The ABC-optimized LADRC responds to the input and stabilizes the system faster than other three methods. In summary, the results suggest that our proposed method outperforms other techniques in terms of rising time, settling time, and quadratic performance index.Figure 6
Evolutionary curves of the three algorithms.To assess the improvements of the proposed controller, the closed-loop performance of helicopter attitude control with ABC-based LADRC and LQG is compared and analyzed by attitude tracking test under wind disturbance. The LQG controller is designed based on the linear model of Trex-600 helicopter; it is also implemented experimentally in [13]. As shown in Figure 7, the LQG controller consists of a state estimator based on Kalman filter and a MIMO state-feedback controller, which ensures that the output Θ tracks the reference command r and rejects process disturbances and measured output noise. The Kalman filter produces estimates x^ of the plant. The observer gain Lf and optimal state feedback gain Kf are achieved by solving two independent Riccati equations [14] (27)Kf=R-1B-TPA-TP+PA--PB-R-1B-TP+Q=0(28)Lf=PfCinTRf-1PfAinT+AinPf-PfCinTRf-1CinPf+GQfGT=0where Q and R are symmetric weight matrices; Qf and Rf are covariance matrices of process disturbances w and measured output noise v, respectively. It is obvious that the choice of weighting matrices (Q,R) dominates the closed-loop performance.Figure 7
LQG controller.Figure 8
Wind disturbance vector.Using the proposed method to optimize LQG, Table3 summarizes the optimal parameters and objective values of these two controllers. It is observed that objective value of LQG is 1.5 times larger than that of LADRC. Hence, LADRC achieves better performance than LQG.Table 3
Parameters of LADRC and LQG.
Results Controllers LADRC LQG Optimal parameters w 0,1 = 126.50 w c , 1 = 11.81 b 01 = 658.66 w 0,2 = 116.67 w c , 2 = 12.40 b 02 = 638.15 w 0,3 = 160.45 w c , 3 = 19.32 b 03 = - 163.33 Q = d i a g ( 0 , ⋯ , 0,79.11,242.98,72.78 ) R = d i a g ( 1,1 , 1 ) Q f = d i a g ( [ 0.01,0.01,0.01 ] ) ; R f = d i a g ( [ 0.01,0.01,0.01 ] ) ;  F 1.1355 1.7086To simulate the measurement noise of the helicopter, the white noise is included in the output of the plant. The wind turbulence disturbances (WX, WY, WZ), as shown in Figure 8, are also injected to the velocity vector V along body frame XB-, YB-, and ZB-axes. Here, a shaping filter [37] modeled by independently exciting of the correlated Gauss-Markov processes is chosen for the wind components:(29)W˙XW˙YW˙Z=-1τs000-1τs000-1τsWXWYWZ+ρBWdXdYdZwhere τs is the correlation time of the wind; ρ is the scalar weighting factor; BW is the turbulence input identity matrix; dX, dY, and dz are independent with zero mean.Figures9 and 10 show the roll and pitch responses and tracking error of the two control systems. We can observe that LADRC controller has more advanced performance in attitude tracking and disturbance resisting as compared to its counterpart. Figure 11 shows that LADRC controller responds faster and has smaller interfere between channels. For the LQG controller, however, the oscillation of ulon increases apparently when there is a change in ulat. Figures 12 and 13 show the angular velocities versus their estimates of both controllers. LESO has more effective estimation performance than Kalman filter, which means faster compensating for the coupling effects and disturbances. All the simulation results indicate that the proposed ABC-optimized LADRC is the perfect control optimization strategy in terms of both the control performance and efficiency of design and parameter tuning. It obtains the lowest objective value and fastest convergence speed. But above all, LQG relies on the precise linear model of the plant, while LADRC only needs the input gains bii that can be even considered as the tuning parameter.Figure 9
Attitude responses of both controllers with wind disturbance.Figure 10
Tracking error of both controllers.Figure 11
Control signals of both controllers.Figure 12
Performance of the LESO.Figure 13
Performance of the Kalman filter.


## 6. Conclusion

In this paper, the ABC algorithm is first applied to tune the controller parameters of LADRC-based DDC controller for a small-scale unmanned helicopter. With the proposed method, the decoupling control of small helicopters is reformulated as a disturbance rejection one, with only the orders of each input-output pairs of the system. The controller optimization is formulated as a function optimization problem and an objective function is proposed for multiple conflicting performance specifications. Four different optimization algorithms are investigated and evaluated in the search of global optimum. The proposed controller is also compared with the traditional LQG technique on the performance of state estimation and disturbance rejection. The simulation results verify the robustness and effectiveness of the ABC-optimized DDC strategic. As future works, the presented strategic will be utilized to design a path following controller for our helicopter and test its reliability in real flight experiments.



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*Source: 9897684-2018-06-27.xml*