Iterative learning control method for aircraft trajectory tracking based on Gaussian process regression

Abstract

The present invention relates to an iterative learning control method for aircraft track tracking based on Gaussian process regression, belonging to the field of civil aircraft control technology. This method solves the problem in the prior art of requiring a system model, which is difficult to accurately establish, as prior information in order to perform iterative learning control of civil aircraft takeoff. The present invention's iterative learning control method for aircraft track tracking based on Gaussian process regression utilizes known input data, output data, and tracking error data from iterative learning control of the controlled system during its climb phase, and constructs an input-output relationship model of the controlled system through Gaussian process regression. This method can achieve the goal of tracking the aircraft track against the target trajectory even when the aircraft dynamics model information is completely unknown.

Description

Technical Field

[0001] The present invention relates to the technical field of civil aviation aircraft control, and in particular to an aircraft track tracking iterative learning control method based on Gaussian process regression. Background Art

[0002] During civil aircraft flight, takeoff requires repeated tracking of the same given trajectory. The target trajectory remains fixed, and the system operates under the same conditions each time. To address this, iterative learning control algorithms mimic human learning behavior, gaining "experience" from the control input and tracking error of the current iteration to guide the design of the control law for the next iteration. Even if the initial tracking results may be unsatisfactory, repeated learning ultimately achieves good tracking of the desired trajectory over the entire time period. Because iterative learning control involves a repetitive learning process, it has important applications in systems with repetitive operation characteristics, such as intelligent transportation networks, motor control, and batch processing systems.

[0003] Existing iterative learning control algorithms used for civil aircraft takeoff cannot completely break away from their reliance on high-precision modeling of the controlled object, especially optimized iterative learning control. However, in practice, traditional identification methods often require a large amount of computation and may result in large identification errors, which in turn affect the execution of control tasks. This limits the application of iterative learning control algorithms, especially those based on optimization. Existing optimization-based iterative learning control algorithms cannot directly solve the optimal control law for the iterative learning system when the system model is unknown. In other words, existing methods require a known system model as prior information. However, due to the highly nonlinear characteristics of actual aircraft dynamic systems, it is often impossible to establish an accurate system model. In this case, the optimization-based iterative learning control algorithm is greatly limited.

[0004] In summary, existing technologies require the use of an aircraft dynamics model, which is difficult to accurately establish, as prior information in order to perform iterative learning control on civil aircraft takeoff. Due to the complex aerodynamic characteristics of aircraft, directly establishing an aircraft dynamics model through system identification methods and then performing iterative learning control based on this model is not only technically challenging but also incurs very high computational costs. Summary of the Invention

[0005] In view of the above problems, the present invention provides an iterative learning control method for aircraft trajectory tracking based on Gaussian process regression, which solves the problem in the existing technology that a system model that is difficult to accurately establish must be used as prior information in order to perform iterative learning control on the takeoff of civil aircraft.

[0006] The present invention provides an iterative learning control method for aircraft track tracking based on Gaussian process regression. The method is characterized in that it is used to control the aircraft track of a civil aircraft during the climb phase to achieve target trajectory tracking when the aircraft dynamic model information is completely unknown. The method comprises the following steps:

[0007] S1. Using a civil aircraft as the controlled system, collect input data, output data, and tracking error data from previous iterations of iterative learning control during the climb phase of the controlled system.

[0008] S2. Based on the input data, output data, and tracking error data collected in S1, the iterative input difference data and the iterative output tracking error difference data are calculated as a training data set;

[0009] S3. Using the training data set, establish an input-output relationship model of the controlled system through Gaussian process regression;

[0010] S4. Based on the input-output relationship model of the controlled system, solve the optimization problem of the optimization index function and obtain the optimized iterative learning control law;

[0011] S5. Input the optimized iterative learning control law into the controlled system to obtain an output;

[0012] S6. Determine whether the output completes tracking of the target trajectory; if so, tracking the target is completed and the iteration ends; if not, return to S1 and continue iterative learning control until tracking the target is completed.

[0013] Furthermore, S1, the controlled system is a discrete-time nonlinear system, which can be expressed as:

[0014]

[0015] Where k = 0, 1, ... is the number of iterative learning; u k (t) is the input of the controlled system at the kth iteration time t; y k (t) is the output of the controlled system at the kth iteration time t; x k (t) is the system state of the controlled system at the kth iteration time t; f(·) and C both contain system model information;

[0016] The control goal of the controlled system is to make the output of the controlled system in a finite time series track the target trajectory y after iterative learning control. d (t), expressed as:

[0017]

[0018] Among them, T r is the total duration of the target trajectory being tracked;

[0019] Then the tracking error e of the controlled system at the kth iteration time t is k (t) is expressed as:

[0020] e k (t) = y d (t)-y k (t);

[0021] The input data U of the previous k times of iterative learning control of the controlled system during the climbing phase k , output data Y k and tracking error data E k Respectively expressed as:

[0022]

[0023]

[0024]

[0025] Among them, m and n represent the input data U k And output data Y k The dimension of .

[0026] Further, S2, iteratively input difference data to form an input training set of the training data set, and iteratively output tracking error difference data to form an output training set of the training data set; wherein,

[0027] The iterative input difference is the difference between each iterative input and the previous iterative input; for the k+1th and kth iterative learning control, the iterative input difference is expressed as:

[0028] ΔU k =U k+1 -U k ;

[0029] The iterative output tracking error difference is the difference between the output tracking error of each iteration and the output tracking error of the previous iteration. For the k+1th and kth iterative learning control, the iterative output tracking error difference is expressed as:

[0030] ΔE k =E k+1 -E k ;

[0031] The iterative input difference of the first k iterative learning control is used as the element to form the iterative input difference matrix [ΔU0 ΔU1... ΔU k-1 ]=T U , using the iterative input difference matrix T U As input training set X;

[0032] The iterative output tracking error difference of the first k iterative learning control is used as the element to form the iterative output tracking error difference matrix T E =[ΔE0 ΔE1 ... ΔE k-1 ]=T E T , and the iterative output tracking error difference matrix T E As the output training set Y, Y=[y1y2...y n ], where y i Denotes ΔE k The training data corresponding to the i-th element.

[0033] Furthermore, S3 specifically includes:

[0034] Using the input training set X and the output training set Y, the ΔU to be found k As the test input, Gaussian process regression is performed to obtain the value of ΔE k The predicted mathematical expectation μ and the predicted variance σ 2 ; Since Gaussian process regression can only fit a single-output system, For each output dimension ΔE k (i), i = 1, 2, ..., m, respectively, establish the following Gaussian process models:

[0035] ΔE k (i) = f i (ΔU k )+εε~N(0,σ ε 2 );

[0036] Among them, f i (·) is an unknown nonlinear mapping; ε is a Gaussian distribution with a mean of 0 and a variance of σ ε 2 noise;

[0037] Set the test input x * =ΔU k , according to the calculation process of Gaussian process regression, set the mean of the prior distribution to 0 and select the kernel function Where l is an adjustable length scale, w and v are arbitrary vectors in the training data; after Gaussian process regression calculation, the input-output relationship model of the controlled system is obtained:

[0038]

[0039] Furthermore, the output of the controlled system is n-dimensional, where n is a positive integer not less than 2. In the Gaussian process regression of S3, a Gaussian process regression is performed on each dimension of the test output to obtain μ1…μ n and For the i-th dimension, the pair ΔE of the i-th dimension is obtained as follows: k The predicted mathematical expectation μ i and the prediction variance

[0040]

[0041]

[0042] in, K i The element corresponding to the hth row and jth column of the matrix is

[0043] Furthermore, S4, the optimization index function is expressed as:

[0044]

[0045] in, stands for Calculate Mathematical Expectation;

[0046] The mean and variance obtained in S3 are used to calculate the optimization index function, and the gradient descent method is used to solve the optimization problem. Then the optimized iterative learning control law of the controlled system is obtained:

[0047] U k+1 =U k +ΔU k .

[0048] Compared to existing technologies, the present invention's iterative learning control method for aircraft trajectory tracking based on Gaussian process regression utilizes known input, output, and tracking error data from iterative learning control of the controlled system during its climb phase. Through Gaussian process regression, it constructs a model of the controlled system's input-output relationship. This method can achieve the goal of tracking the aircraft's trajectory against the target trajectory even when the aircraft's dynamic model information is completely unknown. This method eliminates the existing reliance on aircraft dynamic model information, effectively reducing computational and modeling costs while completing the target trajectory tracking task. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] The drawings are only for purposes of illustrating particular embodiments and are not to be considered limiting of the invention.

[0050] Figure 1 This is a flowchart of the aircraft track tracking iterative learning control method based on Gaussian process regression disclosed in the present invention;

[0051] Figure 2This is a flowchart of an aircraft track tracking iterative learning control method based on Gaussian process regression disclosed in an embodiment of the present invention;

[0052] Figure 3 This is a rendering of the four-dimensional trajectory tracking control of an aircraft disclosed in an embodiment of the present invention;

[0053] Figure 4 Schematic diagram of the convergence speed of the iterative learning control method for aircraft track tracking based on Gaussian process regression according to an embodiment of the present invention;

[0054] Figure 5 Schematic diagram of the convergence speed of the D-type iterative learning control algorithm of the comparative example of the present invention. DETAILED DESCRIPTION

[0055] In order to more clearly understand the above-mentioned objects, features and advantages of the present invention, the present invention is further described in detail below with reference to the accompanying drawings and specific embodiments. It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments can be combined with each other. In addition, the present invention can also be implemented in other ways different from those described herein. Therefore, the scope of protection of the present invention is not limited by the specific embodiments disclosed below.

[0056] The aircraft trajectory tracking iterative learning control method based on Gaussian process regression is used to achieve the goal of controlling the aircraft trajectory to track the target trajectory during the climb phase of a civil aircraft when the aircraft dynamic model information is completely unknown. Figure 1 As shown, the following steps are included:

[0057] S1. Using a civil aircraft as the controlled system, collect the input data, output data, and tracking error data from all previous iterative learning control phases of the controlled system during the climb phase.

[0058] Specifically, the controlled system is a discrete-time nonlinear system, which can be expressed as:

[0059]

[0060] Where k = 0, 1, ... is the number of iterative learning; u k (t) is the input of the controlled system at the kth iteration time t; y k (t) is the output of the controlled system at the kth iteration time t; x k (t) is the system state of the controlled system at the kth iteration time t; f(·) and C both contain system model information.

[0061] The control goal of the controlled system is to make the output of the controlled system in a finite time series track the target trajectory y after iterative learning control. d (t), expressed as:

[0062]

[0063] Among them, T r is the total duration of the tracked target trajectory.

[0064] Then the tracking error e of the controlled system at the kth iteration time t is k (t) is expressed as:

[0065] e k (t) = y d (t)-y k (t), (3).

[0066] The input data U of the previous k times of iterative learning control of the controlled system during the climbing phase k , output data Y k and tracking error data E k Respectively expressed as:

[0067]

[0068]

[0069]

[0070] Among them, m and n represent the input data U k And output data Y k The dimension of .

[0071] S2. Based on the input data, output data and tracking error data collected in S1, iterative input difference data and iterative output tracking error difference data are calculated as a training data set.

[0072] Specifically, the iterative input difference data constitutes the input training set of the training data set, and the iterative output tracking error difference data constitutes the output training set of the training data set.

[0073] The iterative input difference is the difference between each iterative input and the previous iterative input; for the k+1th and kth iterative learning control, the iterative input difference is expressed as:

[0074] ΔU k =U k+1 -U k , (7).

[0075] The iterative output tracking error difference is the difference between the output tracking error of each iteration and the output tracking error of the previous iteration. For the k+1th and kth iterative learning control, the iterative output tracking error difference is expressed as:

[0076] ΔEk =E k+1 -E k , (8).

[0077] The iterative input difference of the first k iterative learning control is used as the element to form the iterative input difference matrix [ΔU0 ΔU1... ΔU k-1 ]=T U , using the iterative input difference matrix T U The training set X is used as input.

[0078] The iterative output tracking error difference of the first k iterative learning control is used as the element to form the iterative output tracking error difference matrix T E =[ΔE0 ΔE1 ... ΔE k-1 ]=T E T , and the iterative output tracking error difference matrix T E As the output training set Y, Y=[y1y2...y n ], where y i Denotes ΔE k The training data corresponding to the i-th element.

[0079] S3. Using the training data set, establish an input-output relationship model for the controlled system through Gaussian process regression. In subsequent control and analysis processes, this input-output relationship model of the controlled system replaces the original model, eliminating reliance on the original model information.

[0080] Specifically, using the input training set X and the output training set Y, the ΔU to be found k As the test input, Gaussian process regression is performed to obtain the value of ΔE k The predicted mathematical expectation μ and the predicted variance σ 2 .

[0081] Since Gaussian process regression can only fit a single-output system, For each output dimension ΔE k (i), i = 1, 2, ..., m, respectively, establish the following Gaussian process models:

[0082] ΔE k (i) = f i f(ΔU k )+ε,ε~N(0,σ ε 2 ), (9).

[0083] Among them, f i (·) is an unknown nonlinear mapping; ε is a Gaussian distribution with a mean of 0 and a variance of σ ε 2 noise.

[0084] Set the test input x * =ΔU k According to the calculation process of Gaussian process regression, the mean of the prior distribution is set to 0. Since the system is nonlinear, a suitable kernel function is selected. Where l is an adjustable length scale, w and v are arbitrary vectors in the training data. After Gaussian process regression calculation, the input-output relationship model of the controlled system is obtained:

[0085]

[0086] It should be noted that the output of the controlled system is n-dimensional, where n is a positive integer not less than 2. In the Gaussian process regression of S3, a Gaussian process regression is performed on each dimension of the test output to obtain μ1…μ n and For the i-th dimension, the pair ΔE of the i-th dimension is obtained as follows: k The predicted mathematical expectation of

[0087] μ i

[0088] and the prediction variance

[0089]

[0090]

[0091]

[0092] in, K i The element corresponding to the hth row and jth column of the matrix is

[0093] It should be noted that, except for the known input training set X and output training set Y, the regression results are only related to the test input.

[0094] S4. Based on the input-output relationship model of the controlled system, the optimization problem of the optimization index function is solved to obtain the optimized iterative learning control law.

[0095] Specifically, the optimization index function is expressed as:

[0096]

[0097] in, Stands for calculating mathematical expectation.

[0098] The mean and variance obtained in S3 are used to calculate the optimization index function, and the gradient descent method is used to solve the optimization problem. Then the optimized iterative learning control law of the controlled system is obtained:

[0099] U k+1 =U k +ΔU k , (14).

[0100] S5. Input the optimized iterative learning control law into the controlled system to obtain the output.

[0101] S6. Determine whether the output completes tracking of the target trajectory; if so, tracking the target is completed and the iteration ends; if not, return to S1 and continue iterative learning control until tracking the target is completed.

[0102] In order to illustrate the effectiveness of the method proposed in the present invention, the above technical solution of the present invention is described in detail below through a specific embodiment, and the specific embodiment is as follows:

[0103] Example 1

[0104] Commercial aircraft, especially those operating on fixed routes, will exhibit repetitive characteristics during takeoff, climb, cruise, and landing. In this embodiment of the present invention, a civil aircraft is selected as the controlled system, and its climb phase is used as a representative of the TBO (Trajectory Based Operation) problem. The trajectory selected for tracking is continuous climb operation. As the name suggests, this is an operation technology that allows the aircraft to continue climbing according to the designed trajectory after takeoff. This technology can significantly reduce fuel consumption and emissions and has been widely used.

[0105] The process of the iterative learning control method for aircraft track tracking based on Gaussian process regression in the embodiment of the present invention is as follows: Figure 2 As shown; first set the parameters.

[0106] This embodiment of the present invention uses a common three-degree-of-freedom dynamic model to describe the motion of a variable mass point on a non-rotating flat-Earth model with an international standard atmosphere, specifically considering symmetric flat-wing flight on a vertical plane. It is assumed that the aircraft has a constant heading angle, no sideslip or pitch, and all forces act on the aircraft's plane of symmetry. The discretized aircraft dynamic and kinematic equations are as follows:

[0107]

[0108]

[0109] x e (t+1)=xe (t)+V(t)cosγ(t);

[0110] h e (t+1)=h e (t)+V(t)sinγ(t);

[0111] m(t+1)=m(t)-T(t)η(V(t)).

[0112] The above kinematic equations are expressed in the ground reference system, while the dynamic equations are expressed in the fuselage coordinate system. The state vector is x(t) = (V(t), γ(t), x e (t), h e (t), m(t)) T , where V(t) is the true airspeed, γ(t) is the track angle, and x e (t) and h e (t) represents the horizontal and vertical displacements, respectively, and m(t) is the mass of the aircraft. F(t) = T(t) - m(t)gsinγ(t) is the vertical external force, where T(t) is the time-varying engine thrust. The drag coefficient is assumed to be C D (t) = C D0 +KC L (t) 2 , where C L (t) is the lift coefficient. Then, as a component of aerodynamic force, drag lift That is the dynamic pressure, ρ(h e (t))V(t) 2 is the air density in relation to altitude. w is the wing area. η(V(t)) is the fuel efficiency.

[0113] For the above controlled system, y(t) = (V(t), γ(t)) T As output, the expected output y d (t)=(V d (t), γ d (t)) T is the velocity and angle corresponding to the target trajectory. The input is the lift coefficient u(t) = C L (t), which is a known function of the angle of attack and the Mach number. The parameters of the controlled system and its constraints can be obtained from the aircraft database BADA developed by EUROCONTROL.

[0114] The control task of the controlled system is to find a series of optimal control inputs u after several iterations (the number of iterations is recorded as k) k (t), so that the output y of civil aircraftk (t) realize the y of target trajectory d (t) tracking.

[0115] This simulation example is a nonlinear system, and it is assumed that the prior mean of the Gaussian process is 0 to simplify the model. The kernel function selected in the process of modeling and controlling the nonlinear system is The optimization index function is selected as J k+1 =E(E k+1 T E k+1 )+ΔU k T ΔU k During the iteration process, the tracking error of each iteration is recorded as Error = E k T E k .

[0116] Then analyze the results.

[0117] Depend on Figure 3 It is easy to see that under the action of the iterative learning control method for aircraft trajectory tracking based on Gaussian process regression, the actual output of the controlled system can achieve tracking of the target trajectory.

[0118] To demonstrate the superiority of the algorithm, it can be compared with the D-type iterative learning control algorithm. Select the same system and simulation conditions, and the control law is u k+1 (t) = u k (t)+Γ(e k (t)-e k (t-1)), the gain Γ is selected as [-0.05 0.05]. In the simulation, the tracking error Error = E k T E k The curve of the number of iterations k changes as follows Figure 4 and Figure 5 As shown. By comparison, Figure 4 The Gaussian process regression-based iterative learning control method for aircraft trajectory tracking converges rapidly. Figure 5 The D-type iterative learning control algorithm shown converges significantly slower. Therefore, the Gaussian process regression-based iterative learning control method for aircraft trajectory tracking is superior to the D-type iterative learning control algorithm.

[0119] Compared with the existing technology, the aircraft trajectory tracking iterative learning control method based on Gaussian process regression of the present invention utilizes the known input data, output data and tracking error data of the iterative learning control of the controlled system in the climbing phase, and constructs the input-output relationship model of the controlled system through Gaussian process regression. It can achieve the goal of tracking the target trajectory of the aircraft trajectory when the aircraft dynamic model information is completely unknown.

[0120] The above description is only a preferred specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions that can be easily thought of by any technician familiar with this technical field within the technical scope disclosed by the present invention should be covered by the scope of protection of the present invention.

Claims

1. An iterative learning control method for aircraft trajectory tracking based on Gaussian process regression, characterized in that: It is used to control the flight path of a civil aircraft during its climb phase and track the target trajectory when the aircraft dynamics model information is completely unknown. The process includes the following steps: S1. Using a civil aircraft as the controlled system, collect input data, output data, and tracking error data from previous iterations of iterative learning control during the climb phase of the controlled system. S2. Based on the input data, output data, and tracking error data collected in S1, the iterative input difference data and the iterative output tracking error difference data are calculated as a training data set; S3. Using the training data set, establish an input-output relationship model of the controlled system through Gaussian process regression; S4. Based on the input-output relationship model of the controlled system, solve the optimization problem of the optimization index function and obtain the optimized iterative learning control law; S5. Input the optimized iterative learning control law into the controlled system to obtain an output; S6. Determine whether the output completes tracking of the target trajectory; if so, the tracking target is completed and the iteration ends; if not, return to S1 and continue iterative learning control until the tracking target is completed; S2, iteratively input difference data to form the input training set of the training data set, iteratively output tracking error difference data to form the output training set of the training data set; wherein, The iterative input difference is the difference between each iterative input and the previous iterative input; for the k+1th and kth iterative learning control, the iterative input difference is expressed as: ΔU k =U k+1 -U k ; The iterative output tracking error difference is the difference between the output tracking error of each iteration and the output tracking error of the previous iteration. For the k+1th and kth iterative learning control, the iterative output tracking error difference is expressed as: ΔE k =E k+1 -E k ; The iterative input difference of the first k iterative learning control is used as the element to form the iterative input difference matrix [ΔU0 ΔU1 …ΔU k-1 ]=T U , using the iterative input difference matrix T U As input training set X; The iterative output tracking error difference of the first k iterative learning control is used as the element to form the iterative output tracking error difference matrix T E =[ΔE0 ΔE1 … ΔE k-1 ]=T E T , and the iterative output tracking error difference matrix T E As the output training set Y, Y=[y1 y2 … y n ], where y i Denotes ΔE k The training data corresponding to the i-th element; S3 specifically includes: Using the input training set X and the output training set Y, the ΔU to be found k As the test input, Gaussian process regression is performed to obtain the value of ΔE k The predicted mathematical expectation μ and the predicted variance σ 2 ; Since Gaussian process regression can only fit a single-output system, For each output dimension ΔE k (i), i = 1, 2, ..., m, respectively, establish the following Gaussian process models: DE k (i)=f i (ΔU k )+ε,ε~N(0,σ ε 2 ); Among them, f i (·) is an unknown nonlinear mapping; ε is a Gaussian distribution with a mean of 0 and a variance of σ ε 2 noise; Set the test input x * =ΔU k , according to the calculation process of Gaussian process regression, set the mean of the prior distribution to 0 and select the kernel function Where l is an adjustable length scale, w and v are arbitrary vectors in the training data; after Gaussian process regression calculation, the input-output relationship model of the controlled system is obtained: The output of the controlled system is n-dimensional, where n is a positive integer not less than 2. In the Gaussian process regression of S3, a Gaussian process regression is performed on each dimension of the test output to obtain μ1…μ n and For the i-th dimension, the pair ΔE of the i-th dimension is obtained as follows: k The predicted mathematical expectation μ i and the prediction variance m i =k i* T (K i +s ε 2 I) -1 y i in, K i The element corresponding to the hth row and jth column of the matrix is 2. The aircraft trajectory tracking iterative learning control method based on Gaussian process regression according to claim 1, characterized in that: S1, the controlled system is a discrete-time nonlinear system, expressed as: Where k = 0, 1, ... is the number of iterative learning; u k (t) is the input of the controlled system at the kth iteration time t; y k (t) is the output of the controlled system at the kth iteration time t; x k (t) is the system state of the controlled system at the kth iteration time t; f(·) and C both contain system model information; The control goal of the controlled system is to make the output of the controlled system in a finite time series track the target trajectory y after iterative learning control. d (t), expressed as: Among them, T r is the total duration of the target trajectory being tracked; Then the tracking error e of the controlled system at the kth iteration time t is k (t) is expressed as: e k (t)=y d (t)-y k (t); The input data U of the previous k times of iterative learning control of the controlled system during the climbing phase k , output data Y k and tracking error data E k Respectively expressed as: Among them, m and n represent the input data U k And output data Y k The dimension of .

3. The aircraft trajectory tracking iterative learning control method based on Gaussian process regression according to claim 1, characterized in that: S4, the optimization index function is expressed as: in, stands for Calculate Mathematical Expectation; The mean and variance obtained in S3 are used to calculate the optimization index function, and the gradient descent method is used to solve the optimization problem. Then the optimized iterative learning control law of the controlled system is obtained: U k+1 =U k +ΔU k 。

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