Research Article Adaptive Neural Network Dynamic Inversion . - Hindawi

4Journal of Applied Mathematicswhere 𝜀 is the transformed error and a smooth and strictlyincreasing function 𝑆 has the following properties: 𝛿 𝑆 (𝜀) 𝛿,lim 𝑆 (𝜀) 𝛿,𝜀̇ (23)lim 𝑆 (𝜀) 𝛿,𝜀 The derivative of (28) is𝜀 𝑆 (0) 0.(24) 𝛿𝜌 (𝑡) 𝜌 (𝑡) 𝑆 (𝜀) 𝛿𝜌 (𝑡) .𝑒 ̇ (𝑡) 𝑒 (𝑡) 𝜌̇ (𝑡)𝜆̇ . 𝜌 (𝑡)𝜌2 (𝑡)(26)𝜀̈ 𝜀 (𝑡) 𝑆 1 (𝑒 (𝑡)).𝜌 (𝑡) ( 𝑆 1 / 𝜆) 𝑒 ̇ (𝑡) 𝑒 (𝑡) 𝜌̇ (𝑡) 2 )( 𝜆𝜌 (𝑡)𝜌2 (𝑡) (27)In addition, from the third property in (25),lim𝑡 𝑒(𝑡) 0 can be achieved if lim𝑡 𝜀(𝑡) 0 is satisfied.Then (22) can be described as (b) Stabilization of the transformed system using (28) issufficient to guarantee the prescribed performance.In what follows, an adaptive neural network dynamicinversion method is proposed to stabilize the transformedsystem using (28).Assumption 5. The desired states 𝑥𝑑 (𝑡) are known boundedtime functions, with known bounded derivatives.𝑖 𝑝, 𝑞, 𝑟,(29)where 𝜂𝑖 , 𝑖 𝑝, 𝑞, and 𝑟 are positive constants to be chosen.We define𝜆 (𝑡) 𝑒 (𝑡).𝜌 (𝑡)(30)(34)And the pitch and yaw errors are derived by the similarmethod.Substituting (30)–(33) into (34), we obtain𝐸𝑝̇ 𝑆𝑝 11𝑒 ̈ (𝑡) 𝐸𝑝𝑀,𝜌𝑝 (𝑡) 𝑝 𝜆 𝑝(35)where𝐸𝑝𝑀 ( 𝑆𝑝 1 / 𝜆 𝑝 ) 𝜆 𝑝 [ 𝑆𝑝 1 𝜆 𝑝(𝑒𝑝̇ (𝑡)𝜌𝑝 (𝑡) 𝑒𝑝 (𝑡) 𝜌𝑝̇ (𝑡)𝜌𝑝2 (𝑡)2) 𝑆𝑝 1𝜆̇ 𝑝 𝜆 𝑝2𝑒𝑝̇ (𝑡) 𝜌𝑝̇ (𝑡)𝜌𝑝2 (𝑡)Assumption 6. The states 𝑥(𝑡) of the nonlinear system in (14)are available for measurement.We define the following error function 𝐸𝑖 (𝑡), whichdescribes the dynamics of the new error system using theerror transformation equation (28).𝑒 (𝑡) 𝜌̈ (𝑡) 2𝑒 ̇ (𝑡) 𝜌2̇ (𝑡) ].𝜌2 (𝑡)𝜌3 (𝑡)𝐸𝑝̇ (𝑡) 𝜀𝑝̈ (𝑡) 𝜂𝑝 𝜀𝑝̇ (𝑡) . 𝜂𝑝3.3. Controller Design and Stability Analysis(33)Then we compute the time derivative of 𝐸𝑝 (𝑡) for the rollerror as(a) The system in (14) is invariant under the error transformation equation (22).𝐸𝑖 (𝑡) 𝜀𝑖̇ (𝑡) 𝜂𝑖 𝜀𝑖 (𝑡) , 𝑆 1 𝑒 ̈ (𝑡) 2𝑒 ̇ (𝑡) 𝜌̇ (𝑡)[ 𝜆 𝜌 (𝑡)𝜌2 (𝑡)(28)Lemma 4 (see [9]). Consider system in (14), the transientand steady state tracking error behavior bounds described bythe performance function 𝜌(𝑡) and the error transformationequation (22). The following results hold.(32)And the second derivative of (28) isAccording to (19), we obtain 𝛿𝜌 (𝑡) 𝑒 (𝑡) 𝛿𝜌 (𝑡) .(31)where(25)According to the first property in (23) and 𝜌(𝑡) 0, wehave 𝑆 1 ̇𝜆, 𝜆 𝑒𝑝 (𝑡) 𝜌𝑝̈ (𝑡)𝜌𝑝2 (𝑡) 2𝑒𝑝̇ (𝑡) 𝜌𝑝2̇ (𝑡)𝜌𝑝3 (𝑡)].(36)Then we can derive𝐸̇ 𝐸𝑅 𝑒 ̈ (𝑡) 𝐸𝑀,(37)𝑇̈ [𝑒𝑝̈ (𝑡), 𝑒𝑞̈ (𝑡), 𝑒𝑟̈ (𝑡)]𝑇 , 𝐸𝑀 where 𝐸̇ [𝐸𝑝̇ , 𝐸𝑞̇ , 𝐸𝑟̇ ] , 𝑒(𝑡)𝑇[𝐸𝑝𝑀, 𝐸𝑞𝑀, 𝐸𝑟𝑀] , and 𝐸𝑅 is𝐸𝑝𝑅[𝐸𝑅 [𝐸𝑞𝑅[]],𝐸𝑟𝑅 ](38)

Journal of Applied Mathematics5where 𝐸𝑝𝑅 ( 𝑆𝑝 1 / 𝜆 𝑝 )/𝜌𝑝 (𝑡), 𝐸𝑞𝑅 ( 𝑆𝑞 1 / 𝜆 𝑞 )/𝜌𝑞 (𝑡),𝐸𝑟𝑅 ( 𝑆𝑟 1 / 𝜆 𝑟 )/𝜌𝑟 (𝑡), and 𝑒𝑝̈ (𝑡), 𝑒𝑞̈ (𝑡), 𝑒𝑟̈ (𝑡), are𝑒𝑝̈ (𝑡) 𝜙 ̈ 𝜙𝑑̈ ,𝑒𝑞̈ (𝑡) 𝜃̈ 𝜃𝑑̈ ,(39)The control input of roll channel is 1𝑢𝑝 𝛿𝑎 𝐺𝑝 1 [ 𝐹𝑝 (𝐸𝑝𝑅 ) (𝐸𝑝𝑀 𝑘𝑝 𝐸𝑝 ) 𝜙𝑑̈ 𝑢𝑝𝑎𝑑 ] .(43)The adaptive signal of roll channel is𝑒𝑟̈ (𝑡) 𝜓̈ 𝜓̈𝑑 .𝑢𝑝𝑎𝑑 𝑤𝑝𝑇 𝑔𝑝 .To simplify the controller design progress, we linearize (2)in an equilibrium point which is the steady wings-level flightstate.The neural network weight update law of roll channel is𝑇[𝑝,̇ 𝑞,̇ 𝑟]̇ 𝐴 𝜔 [𝑉0 Δ𝑉, 𝛼0 Δ𝛼, 𝛽0 Δ𝛽, 𝑝0𝑇 Δ𝑝, 𝑞0 Δ𝑞, 𝑟0 Δ𝑟](40)𝑇 𝐵𝜔 [𝛿𝑎0 Δ𝛿𝑎 , 𝛿𝑒0 Δ𝛿𝑒 , 𝛿𝑟0 Δ𝛿𝑟 ] ,where the 𝐴 𝜔 and 𝐵𝜔 are the appropriate dimension constantmatrixes, 𝛽0 𝑝0 𝑞0 𝑟0 𝛿𝑎0 𝛿𝑟0 0. And𝑉0 , 𝛼0 , and 𝛿𝑒0 are the flight velocity, angle of attack andelevator deflection in some equilibrium point, respectively.The symbol Δ represents the small perturbation from theequilibrium value.According to (2), (13), (14), and (40), we can obtain𝑥̈ 𝐹 (𝑥) 𝐺 (𝑥) Δ𝑢 𝜒,(41)𝑇𝛾 (𝑔 (𝐸 ) 𝐸𝑝𝑅 𝜎𝑝 𝑤𝑝 ) ,̃̇ 𝑝 { 𝑝 𝑝 𝑝𝑤0(44) 𝐸𝑝 𝜁𝑝 , 𝐸 𝜁 , 𝑝 𝑝22 𝑅 𝐸𝑝 ℎ𝑝 ( 𝐸𝑝𝑅 ℎ𝑝 ) 𝑘𝑝 𝜎𝑝 (𝑤𝑝max ) 𝜁𝑝 ,2𝑘𝑝where the vector 𝑔𝑝 is a set of basis functions to approximatethe uncertainty and the neural network weight vector 𝑤𝑝 isthe set of coefficients of each basis function in the roll channel.The adaptation gain 𝛾𝑝 determines the learning rate of neuralnetwork. The 𝜎𝑝 is a modification term to limit the growth ofthe neural network weights. The constant 𝑘𝑝 is positive. And thepositive constant ℎ𝑝 is the neural network approximate errorwhich is bounded. The neural network weight error is̃𝑝 𝑤𝑝 𝑤𝑝 ,𝑤𝑇where Δ𝑢 [Δ𝛿𝑎 , 𝛿𝑒0 Δ𝛿𝑒 , Δ𝛿𝑟 ] , and(45)(46)where the 𝑤𝑝 is the true value of the neural network weight inthe roll channel.𝑇𝐹 (𝑥) [𝐹𝑝 , 𝐹𝑞 , 𝐹𝑟 ] 𝑔 (𝜙, 𝜃, 𝜓, 𝜙,̇ 𝜃,̇ 𝜓)̇𝑇 𝐿 (𝜙, 𝜃) 𝐴 𝜔 [𝑉0 Δ𝑉, 𝛼0 Δ𝛼, Δ𝛽, Δ𝑝, Δ𝑞, Δ𝑟] ,𝑇𝐺 (𝑥) [𝐺𝑝 , 𝐺𝑞 , 𝐺𝑟 ] 𝐿 (𝜙, 𝜃) 𝐵𝜔𝑇𝜒 [𝜒𝑝 , 𝜒𝑞 , 𝜒𝑟 ] ,(42)where 𝜒 is the model error which will be analyzed inSection 3.4.The formula 𝑥̈ 𝐹(𝑥) 𝐺(𝑥)Δ𝑢 in (41) is namedas the design model in some equilibrium point, which isdifferent from the real nonlinear model in (14). And thedifference between the design model and the nonlinear modelis represented by the symbol 𝜒, which will be compensated bythe adaptive neural network.Because there are three channels in the attitude controland the form of each channel is the same, consider thefollowing Theorem 7 for the roll channel. And the pitch andyaw channels are similar.Theorem 7. Considering Assumption 5, Assumption 6, andthe nonlinear system in (14), all the signals are bounded, andthe tracking error 𝑒(𝑡) satisfies the performance described bythe performance function 𝜌(𝑡), if the control input of systemsatisfies the following formula.Proof. A suitable Lyapunov function of roll channel will be𝑇𝑇11̃𝑝 ,{(̃𝑤𝑝 ) 𝑤(𝐸 ) 𝐸𝑝 {{2 𝑝2𝛾𝑝𝑉𝑝 { 1𝑇𝑇1{{ (𝐸𝑝0 ) 𝐸𝑝0 ̃𝑝 ,(̃𝑤𝑝 ) 𝑤2𝛾𝑝{2 𝐸𝑝 𝜁𝑝 , 𝐸𝑝 𝜁𝑝 , (47)where ‖𝐸𝑝0 ‖ 𝜁𝑝 and 𝜁𝑝 is to be defined later.Firstly, if ‖𝐸𝑝 ‖ 𝜁𝑝 is satisfied, then the time derivative of(47) is given by𝑇𝑇1̃̇ 𝑝 .𝑉𝑝̇ (𝐸𝑝 ) 𝐸𝑝̇ (̃𝑤𝑝 ) 𝑤𝛾𝑝(48)Substituting (37) into (48), we derive𝑇𝑇1̃̇ 𝑝 .𝑉𝑝̇ (𝐸𝑝 ) [𝐸𝑝𝑀 𝐸𝑝𝑅 (𝜙 ̈ 𝜙𝑑̈ )] (̃𝑤 ) 𝑤𝛾𝑝 𝑝(49)Considering (41)-(42) and (49), we have𝑇𝑉𝑝̇ (𝐸𝑝 ) [𝐸𝑝𝑀 𝐸𝑝𝑅 (𝐹𝑝 𝐺𝑝 𝑢𝑝 𝜒𝑝 𝜙𝑑̈ )] 𝑇1̃̇ 𝑝 .(̃𝑤𝑝 ) 𝑤𝛾𝑝(50)

6Journal of Applied MathematicsLet the control input 𝑢𝑝 satisfy (43), then (50) can bedescribed as𝑇𝑇1̃̇ 𝑝 . (51)𝑉𝑝̇ (𝐸𝑝 ) [ 𝑘𝑝 𝐸𝑝 𝐸𝑝𝑅 (𝜒𝑝 𝑢𝑝𝑎𝑑 )] (̃𝑤 ) 𝑤𝛾𝑝 𝑝Substituting (44)–(46) into (51), we obtain𝑇𝑇𝑉𝑝̇ 𝑘𝑝 (𝐸𝑝 ) 𝐸𝑝 (𝐸𝑝 ) 𝐸𝑝𝑅𝑇(52)𝑇 (𝜒𝑝 (𝑤𝑝 ) 𝑔𝑝 ) 𝜎𝑝 (̃𝑤𝑝 ) 𝑤𝑝 .By using the norms of the terms on the right side of (52),we obtain the following inequality: 2 𝑉𝑝̇ 𝑘𝑝 𝐸𝑝 𝐸𝑝 𝐸𝑝𝑅 ̃̇ 𝑝 0, andHere the weight update law is 𝑤̇ 𝑝 𝑤𝑉𝑝̇ 0. Therefore, the system is stable, and all the signalsare bounded. Considering Lemma 4, the tracking error 𝑒(𝑡)satisfied the performance described by the performancefunction 𝜌(𝑡).This completes the proof.3.4. Analysis of the Model Error. According to (2)–(5), themoment model is nonlinear, complicated, and must becontinuously varying with the flight condition. For simplicity,the linear model of (40) in an equilibrium point is used toreplace the nonlinear model of (2).We define the model error Λ [Λ 𝑝 , Λ 𝑞 , Λ 𝑟 ]𝑇 , whichis the error between the linear model Equation (40) and thenonlinear model equation (6). And the Λ is(53) 𝑇 ̃𝑝 𝑤𝑝 . 𝜒𝑝 (𝑤𝑝 ) 𝑔𝑝 𝜎𝑝 𝑤 𝑇Λ [𝑓𝑝 , 𝑓𝑞 , 𝑓𝑟 ] 𝐴 𝜔 [𝑉0 Δ𝑉, 𝛼0 Δ𝛼, Δ𝛽, Δ𝑝, Δ𝑞, Δ𝑟]In addition, the approximate error of neural network isbounded, so the following equation is satisfied: 𝜒𝑝 (𝑤 )𝑇 𝑔𝑝 ℎ𝑝 .𝑝 (54)The maximum weight of ideal neural network in the rollchannel is 𝑤𝑝max , so we have 𝑤𝑝 𝑤𝑝max . (55)𝑇 2 2𝑉𝑝̇ 𝑘𝑝 𝐸𝑝 𝐸𝑝 𝐸𝑝𝑅 ℎ𝑝 𝜎𝑝 (𝑤𝑝max 𝑤𝑝 𝑤𝑝 ) .(56)(61)Then (6) can be rewritten as𝑇[𝑝,̇ 𝑞,̇ 𝑟]̇ 𝐴 𝜔 [𝑉0 Δ𝑉, 𝛼0 Δ𝛼, Δ𝛽, Δ𝑝, Δ𝑞, Δ𝑟]𝑇𝑥̈ 𝐹 (𝑥) 𝐺 (𝑥) Δ𝑢 𝐿 (𝜙, 𝜃) Λ.2) 𝜎𝑝 (𝑤𝑝max22(57)).If the system is stable, then 𝑉𝑝̇ 0. And (57) can betransformed to the following formula:max2𝑤𝑝 2 𝑘𝑝 𝐸𝑝 𝐸𝑝 𝐸𝑝𝑅 ℎ𝑝 𝜎𝑝 () 0.2(58)Then we can derive22 𝑅 𝐸 ℎ ( 𝐸𝑝𝑅 ℎ𝑝 ) 𝑘𝑝 𝜎𝑝 (𝑤𝑝max ) 𝑝 𝑝(59) 𝐸𝑝 𝜁𝑝 . 2𝑘𝑝Next, if ‖𝐸𝑝 ‖ 𝜁𝑝 is satisfied, then the time derivative of(47) is derived by𝑇1̃̇ 𝑝 .𝑉𝑝̇ (̃𝑤𝑝 ) 𝑤𝛾𝑝(62)(64)Therefore, the model error mainly depends on the different equilibrium points, attitude angles, actuator deflections,and so on.3.5. Neural Network Structure. The first step in determiningthe appropriate network structure is identifying the networkinputs. Based on the analysis of model error sources describedin Section 3.3, there are three main categories of inputs: theattitude angles, attitude angle rates, and actuator deflections.A Sigma-Pi neural network [18] is used to compensate themodel error 𝜒, and the basis function of the pitch channel 𝑔𝑞is𝑔𝑞 kron (kron (𝐶1𝑞 , 𝐶2𝑞 ) , 𝐶3𝑞 ) ,(65)where kron( , ) represents the Kronecker products and isdefined as follows:2 𝑇(60)(63)Comparing (63) to (41), we obtain𝜒 𝐿 (𝜙, 𝜃) Λ. 2 𝑉𝑝̇ 𝑘𝑝 𝐸𝑝 𝐸𝑝 𝐸𝑝𝑅 ℎ𝑝2𝑇 𝐵𝜔 [Δ𝛿𝑎 , 𝛿𝑒0 Δ𝛿𝑒 , Δ𝛿𝑟 ] Λ.Considering (56), we obtain 𝜎𝑝 ( 𝑤𝑝 𝑇 𝐺𝑢 [𝛿𝑎 , 𝛿𝑒 , 𝛿𝑟 ] 𝐵𝜔 [Δ𝛿𝑎 , 𝛿𝑒0 Δ𝛿𝑒 , Δ𝛿𝑟 ] .Substituting (62) into (10), we haveSubstituting (46) and (54)-(55) into (53), we get𝑤𝑝max𝑇𝐶1𝑞 [1, 𝜃, 𝜃 ] ,𝑇𝐶2𝑞 [1, 𝑞] ,2 𝑇𝐶3𝑞 [1, 𝛿𝑒 , 𝛿𝑒 ] ,(66)

Journal of Applied Mathematics7𝜙̈ d1Π̂qw𝜃𝜃𝜙g2𝜙̇ d𝜔𝜙2 1/s 1/s𝜙dΠ1Σ.q2𝜉𝜙 𝜔𝜙uqad𝜔𝜙21Figure 3: Command filter.𝛿eΠ2𝛿eTable 1: Performance parameters.Figure 2: Neural network structure.𝜙where 𝜃, 𝑞, 𝛿𝑐 and 𝛿𝑒 are normalized variables between 1 and1. The normalization function is𝑦 𝑓 (𝑥) 2 1,1 𝑒 0.1𝑥(67)where 𝑥 is the input parameter and 𝑦 is the output parameter.And a general description of the neural network is shownin Figure 2.And the basis function of roll channel 𝑔𝑝 and the basisfunction of yaw channel 𝑔𝑟 can be derived similarly as follows:𝑔𝑘 kron (kron (kron (kron (𝐶1 , 𝐶2 ) , 𝐶3 ) , 𝐶4 ) , 𝐶𝑘 ) , (68)𝜓𝜌0𝜙𝜌 𝑙𝜙𝛿𝜙 12 0.3 0.70.6𝜌0𝜓𝜌 𝑙𝜓𝛿𝜓 8 0.2 0.70.5𝜌0𝜃𝜃𝜌 𝑙𝜃𝛿𝜃 10 0.2 0.70.6𝛿𝜙1𝛿𝜓1𝛿𝜃1Table 2: Controller 0500.3where 𝑘 𝑝, 𝑟. Then 𝐶𝑖 , 𝑖 1, 2, 3, 4, 𝑘 is2 𝑇𝐶1 [1, 𝜙, 𝜙 ] ,2 𝑇𝐶4 [1, 𝜓, 𝜓 ] ,𝑇𝐶2 [1, 𝑝] ,𝐶3 [1, 𝑟]𝑇 ,2 𝑇𝐶𝑝 [1, 𝛿𝑎 , 𝛿𝑎 ] ,2 𝑇𝐶𝑟 [1, 𝛿𝑟 , 𝛿𝑟 ] .(69)4. Simulation ResultsIn this section, we consider the attitude angles controlproblem for a fixed-wing aircraft, and the initial flight stateis the wings-level flight. Then the attitude angles commandsin three channels will be tracked, respectively.In the following simulation, the initial flight height andvelocity are 6000 m and 190 m/s, and the initial attitude anglesand angular rates including 𝜙, 𝜃, 𝜓, 𝑝, 𝑞, and 𝑟 are zeros. Inaddition, all the initial actuator deflections are zeros.The error transformation function [19] in the simulationis described as𝑆 (𝜀) 𝛿𝑒(𝜀 𝑦) 𝛿𝑒 (𝜀 𝑦),𝑒(𝜀 𝑦) 𝑒 (𝜀 𝑦)(70)where 𝑦 ln(𝛿/𝛿)/2. It can be shown that 𝑆(𝜀) satisfies theproperties in (23)–(25).The attitude angles commands of three channels aretransformed into the desired attitude angles commandsthrough the command filters. And the structure of commandfilter for the roll channel is shown in Figure 3. In addition, thedesired attitude angles commands for yaw and pitch channelscan be obtained by the similar command filters.The command filter parameters are set as 𝜉𝑖 1, 𝜔𝑖 2.5,and 𝑖 𝜙, 𝜓, 𝜃.Design the control inputs with prescribed performancefor three channels through the procedures in Section 3.2. Theperformance and controller parameters are shown in Tables 1and 2.Remark 8. For the performance function 𝜌(𝑡), 𝜌0 is derivedby subtracting the attitude command from the initial attitudeangle. 𝜌 is the allowable attitude tracking error at the steadystate. And the decrement of tracking error 𝑒(𝑡) will decreasewhen the parameter 𝑙 decreases.Remark 9. For the controller parameters, the adaptation gain𝛾 will improve the attitude tracking performance, especially,when there are much larger model errors. The 𝜎𝑝 is amodification term to limit the growth of the neural networkweights; therefore, it is small. The transient performanceof attitude tracking error can be improved by increasingthe parameter 𝑘; however, the increase will increase themagnitude of the control input. Then a compromise must bereached.

8Journal of Applied Mathematics15Yaw angle (deg)Roll angle (deg)151050 50102030Time (s)40501050 5600102030405060Time (s)(b)Pitch angle (deg)(a)151050 50102030Time (s)405060Attitude angle commandResponse of method [20]Response of proposed method(c)Figure 4: Responses of the attitude angles.10Yaw error (deg)Roll error (deg)20100 10 200102030Time (s)40506050 5 100102030Time (s)Pitch error (deg)(a)405060(b)1050 5 100102030Time (s)Error of method [20]Error of proposed method405060The upper bound of errorThe lower bound of error(c)Figure 5: Tracking errors of the attitude angles.The design model I is derived at the trimmed flightcondition of 6000 m and 190 m/s, and the model error issmall.The aircraft tracks the attitude angles commands from theinitial flight state. And the attitude angles tracking responsesand tracking errors are shown in Figures 4 and 5.The two methods have achieved the attitude angles command tracking. Figure 4 shows the better attitude responsesare achieved by the proposed method compared to themethod in [20]. And the coupling among different channelsis smaller when the proposed method is used. For example,the roll angle response has a less change when the aircrafttracks the yaw command. In Figure 5, the attitude anglestracking errors satisfy the prescribed performance boundwith the proposed method in the dynamic and steady state.The main reason is that the method in [20] does not considerthe performance bound defined by the performance function𝜌(𝑡) in the design process.

Journal of Applied Mathematics910Yaw angle (deg)Roll angle (deg)151050 50102030Time (s)405050 560010Roll commandRoll response2030Time (s)405060Yaw commandYaw response(a)(b)Pitch angle (deg)1050 50102030Time (s)405060Pitch commandPitch response(c)10105Yaw error (deg)200 10 200102030Time (s)40500 5 1060010Roll tracking error𝜌p (t) 0.6𝜌p (t)2030Time (s)Yaw tracking error𝜌r (t) 0.5𝜌r (t)(a)(b)10Pitch error (deg)Roll error (deg)Figure 6: Responses of the Attitude angles with model error.50 5 100102030405060Time (s)Pitching tracking error𝜌q (t) 0.6𝜌q (t)(c)Figure 7: Tracking errors of the attitude angles with model error.405060

10Journal of Applied Mathematics105Rudder (deg)Aileron (deg)100 5 10010203040500 10 20600102030Time (s)405060405060Time (s)(a)(b)Elevator (deg)100 10 200102030405060Time (s)Design model IDesign model II(c)Figure 8: Deflections of the control actuators in two design models.Yaw channel uad50 20 400102030Time (s)40500 5 106001020(a)30Time (s)(b)20Pitch channel uadRoll channel uad200 20 40 600102030Time (s)4050Design model IDesign model II(c)Figure 9: The outputs of neural network in three channels.60

Journal of Applied MathematicsIn the real flight control system, there must be the modelerror. In order to verify that the similar tracking performanceis also achieved when there is the large model error, we haveconducted the following simulation study.The flight condition is the same, and the initial flightheight and velocity are 6000 m and 190 m/s. However, thedesign model II used to design the attitude angles controllersis derived at the trimmed flight condition of 4000 m and150 m/s. Apparently, the model error is large.And the attitude angles tracking responses and trackingerrors are shown in Figures 6 and 7.Figures 6 and 7 show the attitude angles tracking errorsstill satisfy the prescribed performance bound, although themodel error is large in this situation. In addition, Figures6 and 7 show the track performance is similar when thedifferent design models are used.The control actuators deflections for three channels arecompared in Figure 8 when the two design models are used.Figure 8 shows the deflections of the control actuatorsusing the design model II increase to derive the desiredattitude angles tracking performance. In addition, the outputsof neural network in three channels are shown in Figure 9.Figure 9 shows the outputs of neural network using thedesign model II are larger than the one using the designmodel I. The main reason is that the model error is largerwhen the design model II is used, and the larger outputs ofneural network are used to compensate the large model error.5. ConclusionIn this paper, an adaptive neural network dynamic inversion with prescribed performance method is proposed foraircraft attitude control. By incorporating the adaptive neuralnetwork dynamic inversion with the prescribed performanceconcept, the proposed method guarantees the system tracking error satisfies the prescribed performance bound in thetransient and steady behavior. The nonlinear simulation ofthe aircraft also verifies the effectiveness of the proposedapproach.Further investigation is needed for the situations in thepresence of the external wind disturbance and unmodeleddynamics. And, these design parameters in this methodshould be decreased and optimized to achieve a real application.AcknowledgmentsThis work was supported by the Program for New CenturyExcellent Talents in University (Grant no. NCET-10-0032)and by the National Natural Science Foundation of China(Grant no. 61175084).References[1] S. A. Snell, D. F. Enns, and W. L. Garrard Jr., “Nonlinear inversion flight control for a supermaneuverable aircraft,” Journal ofGuidance, Control, and Dynamics, vol. 15, no. 4, pp. 976–984,1992.11[2] D. Enns, D. Bugajski, R. Hendrick, and G. Stein, “Dynamicinversion: an evolving methodology for flight control design,”International Journal of Control, vol. 59, no. 1, pp. 71–91, 1994.[3] J. Leitner, A. Calise, and J. V. R. Prasad, “Analysis of adaptiveneural networks for helicopter flight control,” Journal of Guidance, Control, and Dynamics, vol. 20, no. 5, pp. 972–979, 1997.[4] R. Rysdyk and A. J. Calise, “Robust nonlinear adaptive flightcontrol for consistent handling qualities,” IEEE Transactions onControl Systems Technology, vol. 13, no. 6, pp. 896–910, 2005.[5] C. Cao and N. Hovakimyan, “Design and analysis of a novel𝐿 1 adaptive control architecture with guaranteed transientperformance,” IEEE Transactions on Automatic Control, vol. 53,no. 2, pp. 586–591, 2008.[6] W. Gai, H. Wang, and D. Li, “Trajectory tracking for automatedaerial refueling based on adaptive dynamic inversion,” Journalof Beijing University of Aeronautics and Astronautics, vol. 38, no.5, pp. 585–590, 2012 (Chinese).[7] H. Xu and P. A. Ioannou, “Robust adaptive control for a class ofMIMO nonlinear systems with guaranteed error bounds,” IEEETransactions on Automatic Control, vol. 48, no. 5, pp. 728–742,2003.[8] V. Stepanyan and K. Krishnakumar, “Adaptive control withreference model modification,” Journal of Guidance, Control,and Dynamics, vol. 35, no. 4, pp. 1370–1374, 2012.[9] C. P. Bechlioulis and G. A. Rovithakis, “Adaptive control withguaranteed transient and steady state tracking error bounds forstrict feedback systems,” Automatica, vol. 45, no. 2, pp. 532–538,2009.[10] C. P. Bechlioulis and G. A. Rovithakis, “Prescribed performanceadaptive control for multi-input multi-output affine in thecontrol nonlinear systems,” IEEE Transactions on AutomaticControl, vol. 55, no. 5, pp. 1220–1226, 2010.[11] A. Kostarigka and G. Rovithakis, “Adaptive dynamic outputfeedback neural network control of uncertain MIMO nonlinearsystems with prescribed performance,” IEEE Transactions onNeural Networks and Learning Systems, vol. 23, no. 1, pp. 138–149, 2012.[12] C. P. Bechlioulis, Z. Doulgeri, and G. A. Rovithakis, “Guaranteeing prescribed performance and contact maintenance via anapproximation free robot force

by dynamic model inversion is well known and has been applied to the control of high angle of attack ghter aircra [ , ]. e primary drawback of dynamic inversion for aircra ight control is the need for high- delity nonlinear model which must be inverted in real time. However, it is di cult to obtain the exact aircra dynamic model in practice. e .