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- Publisher
- Hindawi Publishing Corporation
- Copyright
- Copyright © 2018 Rui Ma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- ISSN
- 1687-9600
- eISSN
- 1687-9619
- DOI
- 10.1155/2018/9897684
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- See Article on Publisher Site
Abstract
Hindawi Journal of Robotics Volume 2018, Article ID 9897684, 12 pages https://doi.org/10.1155/2018/9897684 Research Article Dynamic Decoupling Control Optimization for a Small-Scale Unmanned Helicopter 1 2 1 Rui Ma , Li Ding , and Hongtao Wu College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China College of Mechanical Engineering, Jiangsu University of Technology, Changzhou 213001, China Correspondence should be addressed to Li Ding; nuaadli@163.com Received 22 January 2018; Revised 27 March 2018; Accepted 22 April 2018; Published 27 June 2018 Academic Editor: Huosheng Hu Copyright © 2018 Rui Ma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 oers ff an appealing alternative to existing control approaches for helicopters by combining decoupling and disturbance rejection without a detailed plant dynamics. eTh tuning of the control system is formulated as a function optimization problem to capture various design considerations. In comparison with several dieff rent 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. 1. Introduction methods are implemented in [5–7]. In these researches, the controller is optimized based on the identified model In recent years, rotary-wing Unmanned Aerial Vehicles established rfi stly for a small helicopter. Nevertheless, many (UAVs) including quadrotors, helicopters, and ducted fans researchers have recognized that the complexity nature are attractive to industries and academia [1, 2]. With the of helicopter and better control capability requires more unique features such as hovering, good maneuverability, and advanced technologies. Cases include direct adaptive neu- low costs, they have been applied to diverse domains by ral command controller [8], adaptive control methods [9], installing different sensors, cameras, or other payloads on nonlinear control methods [10], vision-based guidance con- the platform [3, 4]. However, due to the complexity of flight trol techniques [11], and intelligent control methods like dynamics, it is still challenging to design an appropriate flight fuzzy logic approach [12] and neural network [13]. And control system that satisfies the requirement for autonomous the most pervasive choice in practical applications is the flight. robust control approach: for example, the Kalman filter-based Small-scale unmanned helicopter is a representative of linear quadratic integral (LQI) approach [14], linear quadratic the rotary-wing UAVs. It is considered as an inherently regulator (LQR) [15], and the 𝐻 control approach [16, 17]. unstable, highly nonlinear, and underactuated system with ∞ These methods provide a reasonable countermeasure for significant dynamic coupling. With the small size and agile both disturbances and multivariable effects of the helicopter. maneuverability, it is more susceptible to gust disturbance However, the effectiveness of the prevailing model-based than those full-sized counterpart. Furthermore, the dynamic control approaches is exceedingly dependent on the exact parameters change with the load and flight conditions. These model and aerodynamic coefficients of the plant. Facing the factors cause serious challenges in dealing with concerns complicated control problem of the helicopter, the effective about the robustness, disturbance rejection, decoupling, and other control problems. To address the above problems, the solution is to compensate the disturbance immediately and classical single-input/single-output (SISO) feedback control reduce the dependence of plant model. 2 Journal of Robotics Table 1: Physical descriptions of the state and input variables of the helicopter dynamical model. Variable Physical significance Unit 𝑃 = [,𝑌𝑋 ,𝑍] Position in Earth-fixed coordinate frame m 𝑉=[,𝑢 V,𝑤] Velocity vector along body frame X-, Y-, and Z-axes m/s Ω= [,𝑝 𝑞,𝑟] Roll, pitch, and yaw angular rates rad/s Θ=[,𝜑 ,𝜃 𝜓] Euler angles rad 𝑎 ,𝑏 Longitudinal and lateral main rotor flapping angle rad 𝑠 𝑠 𝑟 Intermediate state in yaw rate feedback controller dynamics N/A 𝑢 lateral cyclic rotor control input N/A 𝑢 longitudinal cyclic rotor control input N/A 𝑢 collective pitch control input N/A 𝑢 tail rotor pedal control input N/A 𝑝𝑒𝑑 predetermined input-output pairing of helicopter’s dynamic The main contribution of this paper is introducing a dynamic decoupling control (DDC) strategy [18] and its model, (ii) the effect of one input to all other outputs that optimization method to a small-scale helicopter. This strategy is not paired with, namely, the cross channel interference is viewed as a ‘disturbance’ to be actively estimated and is rooted in active disturbance rejection control (ADRC) that was recently proposed by Han [19]. The key idea of the canceled out in DDC framework, and (iii) the parameter method is to treat the total disturbance (incorporating the tuning problem is transformed into a functional optimization interactions among control loops and the unknown external problem defined by a combination of different control perfor- disturbances) as a state variable, which can be estimated by an mance indexes, and ABC algorithm is introduced to calculate extended state observer (ESO) through the input-output data the optimal solution. Using this approach, we can optimize of the plant in real time. Consequently, unlike most existing controllers with different control requirement. model-dependent control methods, very little information of The paper is organized as follows. In Section 2, the helicopter dynamical model under consideration is briefly the model is required for ADRC [20]. ADRC oer ff s a practical solution to decoupling control problems in the presence of introduced. Section 3 describes how to use DDC strategy large uncertainties and has been successfully applied in many to decouple the helicopter. Section 4 formulates the param- engineering applications, e.g., aircraft flight control [21] and eter optimization problem. Simulation tests on the model the chemical processes [22]. Moreover, in order to simplify are shown in Section 5. Finally, the main conclusions are the implementation of ADRC, Gao proposed the linear active summarized in Section 6. disturbance rejection control (LADRC), which oer ff s much better performance and needs few parameters to tune, and 2. System Model for Unmanned detailed comparison studies can be found in [23]. Small-Scale Helicopter Since the performance of LADRC depends on the con- vergence speed of state observer, the bandwidth of which is Based on the first principle approach, the dynamics of the most important tuning parameter. Obviously, the trade- helicopter is regarded as a six-degrees-of-freedom rigid-body offs between the robustness and performance have always dynamics augmented with a simplified main rotor flapping been difficult, especially for the helicopter. This problem can dynamics and a factory-installed yaw rate gyro dynamics. As be solved by using a multiobjective optimization algorithm illustrated in Figure 1 and summarized in Table 1, this model in the simulated environment. Artificial bee colony algo- contains fifteen states and four inputs. Detailed information rithm (ABC) was rs fi t proposed by Karaboga in 2005 [24] of the physical parameters and modeling structure can be and successfully applied to control optimization, including found in [28]. A brief overview of the flight dynamical model optimal tuning of PID controller in [25], optimized LQR is presented next. controller in [26], and robust fuzzy PSS design [27]. As we The translational motion and rotational motion of the have known, usual optimization algorithms conduct only helicopter [15] are described as one search operation in one iteration, but ABC algorithm 𝑃= 𝑅 (Θ) 𝑉, can conduct both local search and global search in each iteration; as a result, the probability of finding the optimal Θ=𝑆 (Θ) Ω, parameters is significantly increased, which efficiently avoids local optimum to a large extent. (1) This paper considers a design and optimization of the 𝑉= + −Ω×𝑉 𝑚 𝑚 DDC controller used in our TREX-600 helicopter. As a −1 controlled plant, the dynamical model is obtained through Ω=𝐼 [𝑀 −Ω × (𝐼 Ω)] the system identification method in previous work [26]. The main idea can be characterized as follows: (i) in the design where𝐹 =[−𝑔𝑚 sin 𝜃, 𝑔𝑚 sin 𝜙 cos 𝜃, 𝑔𝑚 cos 𝜃 cos 𝜙] is the of decoupling control, all the information needed is only the gravity force vector projected onto the body frame (BF); 𝑚 𝑐𝑜 𝑙𝑜 𝑙𝑎 𝑓𝑏 Journal of Robotics 3 The main rotor afl pping dynamics, which are common to all small-scale helicopters, is described by the following two coupled rfi st-order differential equations [29]: −𝑞 − +𝐴 𝑏+ 𝐴 𝑢 𝑏 𝑙𝑜𝑛 𝑙𝑜𝑛 𝑎 ̇ [ 𝜏 ] Y ,,,q 𝑠 B 𝑓 [ ] [ ]= (6) [ ] ̇ 𝑠 −𝑝 + 𝐵 𝑎 − +𝐵 𝑢 𝑎 𝑠 𝑙𝑎𝑡 𝑙𝑎𝑡 E 𝜏 [ 𝑓 ] where 𝜏 is the main rotor time constant; 𝐴 and 𝐵 are 𝑓 𝑏 𝑎 X ,u,,p cross coupling derivatives that influence the longitudinal Z ,w,,r Z B E and lateral afl pping motions; and 𝐴 and 𝐵 are eeff ctive 𝑙𝑜𝑛 𝑙𝑎𝑡 Figure 1: Illustration for helicopter states and coordinates. linkage gains. Since the high sensitivity of the bare yaw channel dynam- ics, a feedback yaw rate controller is widely used in small- scale helicopters. The UAV system reserved this feature for the is the helicopter mass;𝐼 = diag{𝐼 ,𝐼 ,𝐼 } is the inertial convenience of manual control. Accordingly, the augmented moment matrix about the reference axes;𝐹 and𝑀 denote 𝑏 𝑏 yaw dynamics are modeled as a first-order bare airframe thecombined aerodynamic forceand moment vectors acting dynamics with a yaw rate feedback represented by a simple on the helicopter center of gravity (CG), respectively. The first-order low-pass filter [30]. The corresponding differential transformation matrices 𝑅 and 𝑆 are, respectively, given as equations are given as 𝑐 𝑐 𝑠 𝑠 𝑐 −𝑐 𝑠 𝑐 𝑠 𝑐 +𝑠 𝑠 𝜃 𝜓 𝜙 𝜃 𝜓 𝜙 𝜓 𝜙 𝜃 𝜓 𝜙 𝜓 [ ] [ ] 𝑅= 𝑐 𝑠 𝑠 𝑠 𝑠 +𝑐 𝑐 𝑐 𝑠 𝑠 +𝑠 𝑐 , (2) 𝜃 𝜓 𝜙 𝜃 𝜓 𝜙 𝜓 𝜙 𝜃 𝜓 𝜙 𝜓 𝑟 ̇ 𝑁 𝑟+𝑁 (𝑢 −𝑟 ) [ ] 𝑟 𝑝𝑒𝑑 𝑝𝑒𝑑 [ ]=[ ] (7) −𝑠 𝑠 𝑐 𝑐 𝑐 𝜃 𝜙 𝜃 𝜙 𝜃 𝑟 ̇ −𝐾 𝑟 +𝐾 𝑟 [ ] 𝑏 𝑟 1𝑡 𝑠 𝑡 𝑐 𝜃 𝜙 𝜃 𝜙 where 𝑁 , 𝑁 , 𝐾 ,and 𝐾 are the parameters to be 𝑟 𝑝𝑒𝑑 𝑟 𝑏 [ ] [0𝑐 −𝑠 ] identified. 𝜙 𝜙 𝑆= [ ] , (3) [ ] 𝑠 𝑐 𝜙 𝜙 𝑐 𝑐 [ ] 3. Dynamic Decoupling Control (DDC) 𝜃 𝜃 of the Helicopter in which the compact notations 𝑠 , 𝑐 ,and 𝑡 denote (∗) (∗) (∗) sin(∗),cos(∗),and tan(∗), respectively. 3.1. LESO Based Dynamic Decoupling Control Method. Linear In the helicopter system,𝐹 and 𝑀 are generated by Active Disturbance Rejection Controller (LADRC) is a novel 𝑏 𝑏 the aerodynamic forces of the fuselage and the control control method which is parameterized from ADRC [20] forces which originate from the main rotor thrust and tail to simplify the tuning process. In this work, LADRC-based rotor thrust. Generally, we calculate𝐹 and𝑀 without the DDC approach [18] is implemented to tackle the decoupling 𝑏 𝑏 consideration of the aerodynamic forces of fuselage due to the problem for helicopter attitude dynamics. Define 𝑓 as the relatively small influence on the model. Therefore, the force combined effect of the internal coupling dynamics and and moment components in the BF are, respectively, given by external disturbances in each channel: (1) (𝑛 −1) 𝑓 =ℎ (𝑥, 𝑥 ,⋅⋅⋅ 𝑥 ,𝑤 )+ ∑ 𝑏 𝑢 −𝑏 𝑢 𝐹 −𝑇 sin 𝑎 (8) 𝑋 𝑚𝑟 𝑠 𝑖 𝑖 𝑖 𝑗 𝑖𝑖 𝑖 𝑗=1 [ ] [ ] 𝐹 = 𝐹 = 𝑇 sin 𝑏 −𝑇 , (4) [ ] [ ] 𝑏 𝑌 𝑚𝑟 𝑠 𝑡𝑟 𝐹 −𝑇 cos 𝑎 cos 𝑏 Then, the helicopter model can be seen as a set of coupled [ ] [ ] 𝑍 𝑚𝑟 𝑠 𝑠 input-output equations with a predetermined relationship: 𝐿 (𝐾 +𝑇 𝐻 ) sin 𝑏 −𝑇 𝐻 𝛽 𝑚𝑟 𝑚𝑟 𝑠 𝑡𝑟 𝑡𝑟 [ ] (𝑛 ) [ ] 𝑦 =𝑓 +𝑏 𝑢 [ ] 𝑀 = [ 𝑀 ] = (5) 1 1 11 1 (𝐾 +𝑇 𝐻 ) sin 𝑎 [ ] 𝛽 𝑚𝑟 𝑚𝑟 𝑠 [ ] . 𝑁 +𝑇 𝐷 [ 𝑚𝑟 𝑡𝑟 𝑡𝑟 ] where 𝑇 , 𝑁 ,and 𝑇 are the main rotor thrust and 𝑚𝑟 𝑚𝑟 𝑡𝑟 (𝑛 ) (9) 𝑦 =𝑓 +𝑏 𝑢 𝑖 𝑖𝑖 𝑖 moment and the tail rotor thrust, respectively; 𝐷 is the 𝑖 𝑡𝑟 distance from the CG to the tail rotor hub, along the 𝑥 direction; 𝐻 and 𝐻 are the distance from the CG to 𝑚𝑟 𝑡𝑟 the main rotor and tail rotor hub, along the 𝑧 direction, (𝑛 ) respectively, and 𝐾 is the main rotor spring constant. 𝑦 =𝑓 +𝑏 𝑢 𝑚 𝑚 𝑚𝑚 𝑚 𝑖𝑗 𝑟𝑓 𝑓𝑏 𝑟𝑓 𝑓𝑏 𝑓𝑏 𝑍𝑍 𝑌𝑌 𝑋𝑋 4 Journal of Robotics where 𝑤 and 𝑥 are external disturbances and state vector, u u y 0,i i i 1/b respectively; 𝑢 and 𝑦 are the dominant input and output of PD Plant 0,i 𝑖 𝑖 𝑡ℎ the 𝑖 loop (𝑖 = 1,2,⋅⋅ ⋅ ,𝑚) , respectively; 𝑏 is the input gain; superscript (𝑛 ) denotes the 𝑛 𝑡ℎ order derivative. Assuming 𝑖 𝑖 LESO the order 𝑛 are given, the numbers of inputs and outputs are [x , x ,···, x ] 1 2 n ,i 𝑖 LADRC the same. Most existing decoupling control approaches assume Figure 2: eTh block diagram of LADRC. the knowledge of the elaborate plant model or disturbance model, which is a considerable challenge in practice. LADRC makes a breakthrough that realistically estimates 𝑓 in real Based on the state-space model, a linear extended state time from input-output data instead of identifying an accu- observer (LESO) is designed to estimate𝑥 : rate mathematical model. The idea is introduced next. (𝑗−1) Define an enlarged state vector 𝑥 =(𝑦 , 𝑦 ̇ ,⋅⋅ ⋅ ,𝑦 , 𝑖 𝑖 𝑖 𝑖 𝑥̂ =𝐴 𝑥̂ +𝐵 𝑢 +𝐿 (𝑦 − 𝑦 ̂ ) 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑓 ), 𝑗 = 1, 2, ⋅ ⋅ ⋅ , 𝑛 ,in which 𝑓 is added as an extended state. 𝑖 𝑖 𝑖 (12) Assume 𝑓 is differentiable and V = 𝑓 is bounded. Then the 𝑖 𝑖 𝑖 𝑦̂ =𝑥̂ 𝑖 𝑖 𝑡ℎ augmented state-space form of 𝑖 loop in (9) is represented as where𝐿 =[𝑙 ,𝑙 ,⋅⋅⋅ ,𝑙 ] is the observer gain needed 𝑖 1,𝑖 2,𝑖 𝑛 +1,𝑖 to be chosen. With a properly selected observer gain, the system states 𝑥 =𝐴 𝑥 +𝐵 𝑢 +𝐺 V and 𝑓 will be accurately estimated by the LESO in real time. 𝑖 𝑖 𝑖 (10) 𝑡ℎ The following control law for 𝑖 loop can be designed to 𝑦 =𝐶 𝑥 reduce the closed-loop system approximately to a unit gain (𝑛 ) cascaded integrator plant 𝑦 =𝑓 +𝑏 𝑢 ≈𝑢 : 𝑖 𝑖 𝑖𝑖 𝑖 0,𝑖 where 𝑢 − 𝑓 0,𝑖 𝑖 (13) 𝑢 = 𝑖,𝑖 01 0⋅⋅⋅ 0 [ ] It is a relatively simple control problem, which is solved [ 00 1⋅⋅⋅ 0] [ ] by using a PD controller with a feedforward term: [ ] . . . . [ ] . . . . 𝐴 = , . . . d . [ ] (𝑛 −1) (𝑛 ) 𝑖 𝑖 [ ] 𝑢 =𝑘 (𝑟 − 𝑥 ̂ )+ ⋅ ⋅⋅+ 𝑘 (𝑟 − 𝑥 ̂ )+ 𝑟 (14) 𝑖,0 1,𝑖 𝑖 1,𝑖 𝑛 ,𝑖 𝑖 𝑛 ,𝑖 𝑖 𝑖 𝑖 [ ] 00 0⋅⋅⋅ 1 [ ] where (𝑘 ,𝑘 ,⋅⋅ ⋅ ,𝑘 )are the controller gains to be selected 1,𝑖 2,𝑖 𝑛,𝑖 00 0⋅⋅⋅ 0 [ ] and 𝑟 is the trajectory reference. The structure of LADRC is shown in Figure 2. [ ] [ 0 ] [ ] 3.2. Parameterization of LADRC. For simplicity and practi- [ ] [ ] cality, both of the LESO and PD controller are parameterized 𝐵 = , [ . ] [ ] in a special case as suggested in [18], where all the observer [ ] poles and controller poles are placed at −𝜔 and −𝜔 , [ ] 0,𝑖 0,𝑖 c,𝑖 respectively. The characteristic polynomials of (12) and (14) [ ] are constituted, respectively, as (11) [ ] 𝑛 +1 𝑛 +1 𝑛 𝑖 0 𝑖 𝑖 [ ] (15) 𝜆 (𝑠 ) =𝑠 +𝑙 𝑠 + ⋅⋅⋅ + 𝑙 =(𝑠 + 𝜔 ) 𝑜,𝑖 1,𝑖 𝑛 +1,𝑖 0,𝑖 [ ] 𝑖 [ ] . 𝑛 𝑛 𝑛 𝐺 = , 𝑖 𝑖 𝑖 [ . ] 𝜆 (𝑠 ) =𝑠 +𝑘 𝑠 + ⋅⋅⋅ + 𝑘 =(𝑠 + 𝜔 ) (16) 𝑐,𝑖 𝑛 ,𝑖 1,𝑖 c,𝑖 [ ] [ ] [ ] with [ ] (𝑛 +1)!𝜔 0,𝑖 (17) 𝑙 = , 𝑗 = 1, 2,⋅⋅ ⋅ ,𝑛 +1 𝑗,𝑖 𝑖 𝑗! (𝑛 +1− )𝑗 ! [ ] [ ] 𝑛 −𝑗+1 [ ] 𝑛 !𝜔 [ ] 𝑐,𝑖 . (18) 𝑘 = , 𝑗 = 1,2, ⋅⋅⋅ , 𝑛 [ ] 𝑗,𝑖 𝑖 𝐶 = . [ . ] (𝑗 − 1)! (𝑛 −𝑗 + 1)! [ ] [ ] [ ] It makes 𝜔 , 𝜔 the bandwidth and the only tuning 0,𝑖 c,𝑖 parameters for LESO and the PD controller, respectively. In [ ] 𝑖𝑗 Journal of Robotics 5 where 𝑏 , 𝑏 ,and 𝑏 are the input gains of lateral cyclic, 01 02 03 Attitude Decoupling Controller u col longitudinal cyclic, and tail rotor collective pitch, respectively. In the LADRC design, these input gains are treating as lat LADRC(3) another tuning parameter besides 𝜔 and 𝜔 to improve 0,𝑖 c,𝑖 the performance of the reduced order closed-loop system. Plant Note that the orders of each loop are 𝑛 =𝑛 =3 and r u lon 1 2 LADRC(3) 𝑛 =2, and the LADRC-based DDC controller can be realized by designing the LESO and PD controller for each ped loop, accordingly. LADRC(3) 4. Optimization Problem Formulation 4.1. eTh Objective Function. The proposed LADRC tuning Figure 3: eTh block diagram of the attitude decoupling controller. method using ABC is schematically shown in Figure 5, where the plant is the identified model of the TREX-600 helicopter. As stated above, the primary concern in the implementation general, higher bandwidth corresponds to better transient of LADRC is maximizing the bandwidth 𝜔 and 𝜔 ,and 0,𝑖 c,𝑖 response, disturbance estimation, and rejection. However, identifying a suitable value of 𝑏 while satisfying the system 𝑖𝑖 too large a value of 𝜔 would cause oscillation in states. The 0,𝑖 constraints and design objective. It can be accomplished measurement noise and excessive increase of 𝜔 make the c,𝑖 by forming a functional optimization problem. Also, the controlsignaloversize in magnitude and changerate. On the design specifications are comprehensively represented by a other hand, an appropriate selection of 𝜔 and 𝜔 should 0,𝑖 c,𝑖 new objective function. In the optimization procedure, by be subjected to physical limits and dynamic characteristics of changing the closed-loop step responses according to its the plant. automatically selected controller parameters and calculating As seen above, the LADRC approach is a practical method the objective function value at every generation, the iterative for decoupling control of MIMO system. The satisefi d perfor- algorithm searches the optimal parameters for the controller mance will be obtained by tuning only two parameters 𝜔 0,𝑖 subjected to the design specifications. and 𝜔 . Furthermore, it works without the detail model of c,𝑖 In the tuning of the controller, the objective function the original plant, except the orders of each input-output pairs can be formed by different performance index that considers and input gains 𝑏 . The proofs of stability are given in [31, 32]. 𝑖𝑖 the step responses of the entire system. Typical performance index in the time domain includes integral square error (ISE), 3.3. Attitude Controller Design for the Helicopter. As shown integral of absolute error (IAE), integral time absolute error in Figure 3, the decoupling controller is designed to have (ITAE) [33], rise time (𝑇 ), settling time (𝑇 ), overshoot (𝑂𝑆 ), 𝑟 𝑠 the form of three LADRC controllers. Moreover, we selected and steady-state error (𝐸 ). The selection of these factors their orders according to the relative degrees of the dynamical and form of the function can be determined depending on model. It is assumed that the controlled output Θ = [,𝜃𝜑 ,𝜓] the design requirements. In this work, the desired control performanceshould havea smallor noovershoot in the can be measured directly and that the trim value 𝑟 = step response with a minimal settling time, and the control [𝜑 ,𝜃 ,𝜓 ] is within the physical limitation of helicopter 𝑟 𝑟 𝑟 signal should be smooth within the physical limit. Hence, flight. we den fi ed the objective function 𝐹 in this work as a linear To use the DDC approach, the order of input-output combination of the ISE, integral of the square of the control pairs in the model must be explicit. Figure 4 displays the signal, the overshoot 𝑂𝑆 , and the one percent settling time 𝑇 interconnection of the helicopter subsystems, which oer ff s a [34]: more physically meaningful design. Note that the helicopter attitude dynamics can be separated in two interconnected 2 2 subsystems [6], i.e., the lateral and longitudinal subsystem 𝐹= ∑ 𝛼∫ (𝑦 −𝑟) +𝛽 ∫ 𝑢 +𝜎(𝑂𝑆 )+𝜀 (𝑇 ) (20) 𝑖 𝑖 𝑖 𝑠, and yaw dynamics. The cyclic commands 𝑢 and 𝑢 control 𝑙𝑜𝑛 𝑙𝑎𝑡 𝑖=1 the pitch and roll moment, and the pedal command 𝑢 𝑝𝑒𝑑 where the variables of 𝛼 , 𝛽 , 𝜎 ,and 𝜀 are the adjustment param- manipulates the heading of the helicopter. In this case, we eters. The values of these parameters are generally selected set the lateral and longitudinal subsystem as two third-order by using trial-and-error method. During the minimization systems and the yaw dynamics as a second-order system. of the objective function, all of the performance indexes are According to aforementioned discussion and analysis, we minimized and all of the disadvantaged controller parameters define 𝑓 , 𝑓 ,and 𝑓 as the total disturbance in each channel 1 2 3 caused to system unstable or a poor performance will be and rewrite (9) as eliminated by the algorithm. (𝑛 ) Using the proposed objective function (20), the parame- 𝜙 =𝑓 +𝑏 𝑢 1 01 𝑙𝑎𝑡 ter tuning for the controller becomes a function optimization (𝑛 ) (19) problem. This method combines a variety of performance 𝜃 =𝑓 +𝑏 𝑢 2 02 𝑙𝑜𝑛 indices which can be selected and weighted as required. Then (𝑛 ) 𝜓 =𝑓 +𝑏 𝑢 the desired control performance and its parameter setting 3 03 𝑝𝑒𝑑 𝑠𝑠 6 Journal of Robotics Flapping Dynamics Rigid Body Dynamics lon p K +T H ＭＣＨ b −T H p p mr mr s tr tr ȧ −q−a /+ A b+ A u s s f b lon lon a, b −1 = q̇ q q = K +T H ＭＣＨ a − × mr mr s u ̇ −p+B a −b / +B u lat b a ss f lat lat ṙ r r N +T D mr tr tr q, p Yaw Dynamics Attitude Dynamics 1t s t c p ped ṙ N r+ N (u −r ) r ped ped fb r = 0c −s · q ṙ −K r +K r fb rfb fb r ̇ 0s /c c /c r Figure 4: Interconnection of the helicopter dynamics model. The terms associated with the gravity force are disregarded. where 𝑗 ∈ {1, ⋅⋅⋅ ,𝐷} denotes the 𝑗 th dimension of the System Specifications solution vector. 𝑥 and 𝑥 mean the lower and upper min max bounds, respectively. ABC Step 2. Apply a specific function to calculate the tness fi of the Optimization solution 𝑥 according to the following equations and select the optimized parameters top 𝑁 best solutions as the number of the employed bees: , ,b , i = 1,2,3 0,i ＝,i 0,i r u LADRC Attitude 𝑡 = (22) Plant (1 + 𝐹 ) Controller sensor noise where 𝑡 is the tness fi function and 𝐹 is objective function 𝑖 𝑖 depicted in (20). Figure 5: LADRC tuning scheme with ABC. Step 3. Each employed bee searches new solution in the neighborhood of the current position vector in the 𝑛 th iteration as follows: can be found by minimizing the value of (20). It makes this method different to the traditional optimal control method. 𝑗 𝑗 𝑗 𝑗 𝑗 V =𝑥 +𝜆 (𝑥 −𝑥 ) (23) 𝑖 𝑖 𝑖 𝑖 𝑘 Since (20) is nonlinear and discontinuous, simple search methods are usually lost in local optimum, as shown in [35]. where 𝑘 ∈ {1,⋅⋅⋅ ,𝐷} , 𝑘 =𝑖 ̸ ,both 𝑘 and 𝑗 are randomly Advanced search methods like GA, PSO, and ABC provide us generated, and 𝜆 is a random parameter in the range from with efficient solutions for solving this problem. -1 to 1. In order to ensure that the algorithm evolves to the global optimal, we apply the greedy selection equation (22) 4.2. ABC Algorithm. In order to introduce the search mech- 𝑗 𝑗 to choose the better solution between V and 𝑥 into the next 𝑖 𝑖 anism of ABC algorithm, we should define three essential generation: components: employed bees, unemployed bees, and food source [36]. And the unemployed bees are divided into the 𝑗 𝑗 𝑗 V,𝑓𝑖𝑡( V )> 𝑖𝑡𝑓(𝑥 ) 𝑖 𝑖 𝑖 following bees and scout bees. The population of the colony 𝑥 = (24) 𝑗 𝑗 𝑗 bees is 𝑁 , the number of employed bees is 𝑁 ,and the 𝑥 ,𝑓𝑖𝑡( V )≤ 𝑖𝑡𝑓(𝑥 ) 𝑠 𝑒 𝑖 𝑖 𝑖 number of unemployed bees is 𝑁 , which satisfies the relation Step 4. Each following bee selects an employed bee to trace 𝑁 =2𝑁 =2𝑁 . We also define 𝐷 as the dimension of 𝑠 𝑒 𝑢 solution vector, i.e., the number of the unknown parameters. according to the parameter of probability value. The formula of the probability method is described as 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 𝑝 = (25) the proposed algorithm is described as follows. ∑ 𝑡 𝑖=1 Step 5. The following bee searches in the neighborhood of Step 1. Randomly initialize a set of possible solutions the selected employed bee’s position to find new solutions. (𝑥 , ⋅⋅⋅ ,𝑥 ), and the particular solution 𝑥 can be governed 1 𝑁 𝑖 Update the current solution according to their tfi ness. by Step 6. If the search time trial is larger than the pre- 𝑗 𝑗 𝑗 (21) 𝑥 =𝑥 + rand (0, 1) (𝑥 −𝑥 ) min max min determined threshold limit and the optimal value cannot 𝑓𝑖 𝑁𝑒 𝑓𝑖 𝑓𝑖 𝑓𝑖 Journal of Robotics 7 Table 2: Tuning performance of trial-and-error method, GA, PSO, and ABC algorithm. Time domain performance Applied process 𝑇 /s 𝑂𝑆/ % 𝑇 /𝑠 𝐸 /rad 𝐹 𝑟 𝑠 𝑠𝑠 0.70 2.33 1.17 0 trial-and-error 1.2351 0.55 1.30 0.77 0 0.37 1.86 0.65 0 0.596 2.13 0.97 0 GA 1.013 0.514 1.74 0.80 0 0.303 1.27 0.48 0 0.62 2.01 0.94 0 PSO 0.9713 0.53 1.82 0.81 0 0.30 1.11 0.40 0 0.60 2.08 0.96 0 ABC 0.9315 0.57 1.60 0.86 0 0.35 0.72 0.31 0 be improved, the location vector can then be reinitialized randomly by scout bees according to the following equation: 𝑥 (𝑛+1 ) 1.4 (26) 𝑥 + rand (0, 1) (𝑥 −𝑥 ), 𝑡𝑟𝑖𝑙𝑎 > 𝑙𝑖𝑚𝑖𝑡 min max min { 1.3 𝑥 (𝑛 ) , 𝑡𝑟𝑖𝑎𝑙 ≤ 𝑙𝑖𝑚𝑖𝑡 1.2 1.1 Step 7. Output the best solution parameters achieved at the present time, and go back to Step 3 until termination criteria 𝑇 are met. max 0.9 35 40 45 50 5. Simulation Tests For best performance of optimization, we compare the results 0 1020304050 of ABC with the existing search iterative algorithm methods, Iteration including the trial-and-error method, GA, and PSO. We set the population size as 20 and iteration numbers as 50 for ABC GA each algorithm. The adjustment parameters ( 𝛼 , 𝛽 , 𝜎 , 𝜀 )are PSO 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 Figure 6: Evolutionary curves of the three algorithms. 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 other techniques in terms of rising time, settling time, and parameters, i.e., social, individual and inertia weight, are set quadratic performance index. to 2, 2, and 0.8, respectively. Finally, for ABC the threshold To assess the improvements of the proposed controller, is set to limit =5. Theresults arepresented in Table 2 and the closed-loop performance of helicopter attitude control Figure 8. with ABC-based LADRC and LQG is compared and analyzed Figure 6 shows the evolution curves of ABC, GA, and by attitude tracking test under wind disturbance. The LQG PSO. The figure demonstrates that the objective function controller is designed based on the linear model of Trex- reduces as the generation iterates with time, gradually 600 helicopter; it is also implemented experimentally in [13]. converging to an optimal result. Compared with GA and As shown in Figure 7, the LQG controller consists of a PSO, ABC achieves a better result with smaller objective state estimator based on Kalman lfi ter and a MIMO state- function aer ft 28 iterations. Table 2 indicates that all of the feedback controller, which ensures that the output Θ tracks controllers have no steady-state error and that the trial- the reference command𝑟 and rejects process disturbances and-error method gets the largest time domain index. The and measured output noise. The Kalman lfi ter produces esti- ABC-optimized LADRC responds to the input and stabilizes mates𝑥̂ of the plant. The observer gain 𝐿 and optimal state the system faster than other three methods. In summary, feedback gain𝐾 are achieved by solving two independent the results suggest that our proposed method outperforms Riccati equations [14] Objective Function F 8 Journal of Robotics Table 3: Parameters of LADRC and LQG. Controllers Results LADRC LQG 𝑤 = 126.50 0,1 𝑤 = 11.81 𝑐,1 𝑏 = 658.66 𝑄 = diag(0, ⋅ ⋅ ⋅ , 0, 79.11, 242.98, 72.78) 𝑤 = 116.67 0,2 𝑅 = diag(1,1,1) Optimal parameters 𝑤 = 12.40 c,2 𝑄 = diag([0.01, 0.01, 0.01]); 𝑏 = 638.15 𝑅 = diag([0.01, 0.01, 0.01]); 𝑤 = 160.45 0,3 𝑤 = 19.32 c,3 𝑏 = −163.33 𝐹 1.1355 1.7086 LQG Servo Controller col 1/s ＣＨ ＣＨ - Plant ＣＨ ＣＨ - 1/s ＣＨ Kalman Figure 7: LQG controller. Figure 8: Wind disturbance vector. −1 𝐾 =𝑅 𝐵 𝑃 (27) modeled by independently exciting of the correlated Gauss- T T −1 𝐴 𝑃 +𝑃 𝐴 −𝑃 𝐵 𝑃 +𝑄 =0 Markov processes is chosen for the wind components: T −1 𝐿 =𝑃 𝐶 𝑅 𝑓 f in f − 00 (28) 𝑊 [ ] 𝑊 𝑑 𝑠 𝑋 𝑋 T T −1 T [ ] [ ] 𝑃 𝐴 +𝐴 𝑃 −𝑃 𝐶 𝑅 𝐶 𝑃 + 𝐺 = 0 [ ] [ ] [ ] f in f f in f f in in f [ ̇ ] 0− 0 𝑊 = 𝑊 +𝜌 𝐵 𝑑 (29) [ ] [ ] [ ] 𝑌 𝑌 𝑊 𝑌 [ ] [ 𝜏 ] [ ] ̇ 1 𝑊 𝑑 𝑊 [ ] [ ] 𝑍 𝑍 [ ] 00 − [ ] where 𝑄 and 𝑅 are symmetric weight matrices; Q and R are covariance matrices of process disturbances 𝑤 and where 𝜏 is the correlation time of the wind; 𝜌 is the scalar f 𝑠 measured output noise V, respectively. It is obvious that the weighting factor;𝐵 is the turbulence input identity matrix; choice of weighting matrices (𝑄 ,𝑅 ) dominates the closed- 𝑑 , 𝑑 ,and 𝑑 are independent with zero mean. 𝑋 𝑌 𝑧 loop performance. Figures 9 and 10 show the roll and pitch responses and Using the proposed method to optimize LQG, Table 3 tracking error of the two control systems. We can observe summarizes the optimal parameters and objective values of that LADRC controller has more advanced performance in these two controllers. It is observed that objective value of attitude tracking and disturbance resisting as compared to its LQG is1.5 timeslarger than that ofLADRC. Hence, LADRC counterpart. Figure 11 shows that LADRC controller responds achieves better performance than LQG. faster and has smaller interfere between channels. For the To simulate the measurement noise of the helicopter, LQG controller, however, the oscillation of 𝑢 increases 𝑙𝑜𝑛 the white noise is included in the output of the plant. The apparently when there is a change in 𝑢 .Figures 12 and 13 𝑙𝑎𝑡 wind turbulence disturbances (W ,W ,W ), as shown in show the angular velocities versus their estimates of both 𝑋 𝑌 𝑍 Figure 8, are also injected to the velocity vector 𝑉 along controllers. LESO has more effective estimation performance body frame X -, Y -, and Z -axes. Here, a shaping filter [37] than Kalman filter, which means faster compensating for the B B B 𝐺𝑄 𝐵𝑅 Journal of Robotics 9 0.3 0.25 0.2 0.15 0.1 0.05 −0.05 0123456789 10 time (s) LADRC LQG Reference 0.3 0.25 0.2 0.15 0.1 0.05 −0.05 0123456789 10 time (s) LADRC LQG Reference Figure 9: Attitude responses of both controllers with wind disturbance. 0.2 0.1 −0.1 −0.2 −0.3 01 1234567890 time (s) LQG LADRC 0.3 0.2 0.1 −0.1 −0.2 −0.3 0 123456789 10 time (s) LQG LADRC Figure 10: Tracking error of both controllers. coupling effects and disturbances. All the simulation results method, the decoupling control of small helicopters is refor- indicate that the proposed ABC-optimized LADRC is the mulated as a disturbance rejection one, with only the orders perfect control optimization strategy in terms of both the of each input-output pairs of the system. The controller opti- control performance and efficiency of design and parameter mization is formulated as a function optimization problem tuning. It obtains the lowest objective value and fastest and an objective function is proposed for multiple conflicting convergence speed. But above all, LQG relies on the precise performance specifications. Four different optimization algo- linear model of the plant, while LADRC only needs the input rithms are investigated and evaluated in the search of global gains 𝑏 that canbe evenconsidered as the tuning parameter. optimum. The proposed controller is also compared with 𝑖𝑖 the traditional LQG technique on the performance of state estimation and disturbance rejection. The simulation results 6. Conclusion verify the robustness and effectiveness of the ABC-optimized In this paper, the ABC algorithm is rfi st applied to tune DDC strategic. As future works, the presented strategic will be utilized to design a path following controller for our the controller parameters of LADRC-based DDC controller for a small-scale unmanned helicopter. With the proposed helicopter and test its reliability in real flight experiments. tracking error of (rad) (rad) (rad) tracking error of (rad) 10 Journal of Robotics 0.2 0.1 −0.1 −0.2 0 246 8 10 time (s) LADRC LQG 0.2 0.1 −0.1 −0.2 0 246 8 10 time (s) LADRC LQG Figure 11: Control signals of both controllers. 0.5 −0.5 −1 0 246 8 10 time (s) LESO Measurement 0.5 −0.5 −1 0 246 8 10 time (s) LESO Measurement Figure 12: Performance of the LESO. Acronyms LADRC: Linear active disturbance rejection control LESO: Linear extended state observer ABC: Artificial bee colony LQG: Linear quadratic Gaussian ADRC: Active disturbance rejection control LQI: Linear quadratic integral BF: Body frame LQR: Linear quadratic regulator CG: Center of gravity MIMO: Multi-input multi-output DDC: Dynamic decoupling control PD: Proportional derivative ESO: Extended state observer PID: Proportional integral derivative GA: Genetic algorithm PSO: Particle swarm optimization IAE: Integral of absolute error SISO: Single-input/single-output ISE: Integral square error UAVs: Unmanned aerial vehicles. ITAE: Integral time absolute error q (rad/s) p (rad/s) u (-1,1) u (-1,1) lat lon Journal of Robotics 11 0.5 −0.5 −1 0 2468 10 time [s] Kalman Filter Measurement 0.5 −0.5 −1 0 2468 10 time [s] Kalman Measurement Figure 13: Performance of the Kalman filter. Data Availability [6] B. 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