Neural backstepping controller design methods and systems for trajectory tracking control problems

Abstract

The present application relates to a kind of structured neural network controller design method and system for trajectory tracking control problem.The method includes the following steps: according to the information of moving object, the dynamic model of the moving object is constructed, the dynamic characteristics of the moving object are analyzed;Input convex neural network is constructed;Lagrange neural network is constructed;Neural backstepping controller design when system model information is accurately known is executed;The structure of the Lagrange neural network is improved using the input convex neural network, and the neural backstepping controller design when system model is unknown is executed.The system includes the following modules: dynamic characteristics analysis module, convex neural network construction module, Lagrange neural network construction module, configured to construct Lagrange neural network therein, first neural backstepping controller design module, second neural backstepping controller design module.The neural backstepping controller proposed in the present application has stability guarantee and performance guarantee, is stable for any feasible deep neural network parameters, and can improve performance by optimizing parameters, in the case where system exists disturbance or model uncertainty, steady-state tracking error can also be guaranteed to be bounded.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of automatic control, and in particular to a neural backstepping controller design method and system for trajectory tracking control problems. BACKGROUND

[0002] In the process of engineering practice, engineering technicians often need to control complex nonlinear systems such as robots and aircraft. For the control of nonlinear systems, it is generally divided into two categories, namely traditional nonlinear control method and neural network-based control method. Traditional nonlinear control methods, such as sliding mode control and robust adaptive control, are not suitable for situations with state and input constraints. Neural network-based control, such as deep neural network (DNN), has strong approximation ability, but due to its black box nature, it is difficult to provide stability guarantee. For the control of nonlinear systems, there is no structured neural network controller on the market that is safe and has performance guarantee for trajectory tracking control problems. Moreover, when the system model of the nonlinear system is unknown, the performance and stability of the controller will be greatly limited. SUMMARY

[0003] To solve some or all of the problems in the prior art, the present application provides a structured neural network controller design method for trajectory tracking control problems, namely a neural backstepping controller design method, which comprises the following steps:

[0004] According to the information of the moving object, a dynamic model of the moving object is constructed, and the dynamic characteristics of the moving object are analyzed;

[0005] An input convex neural network is constructed;

[0006] A Lagrange neural network is constructed;

[0007] The neural backstepping controller design when the system model information is accurately known is performed;

[0008] The structure of the Lagrange neural network is improved using the input convex neural network, and the neural backstepping controller design when the system model is unknown is performed.

[0009] Further, the construction of the model of the moving object and the analysis of the dynamic characteristics of the moving object comprise:

[0010] According to the Euler-Lagrange equation, a dynamic model of the moving object is constructed,

[0011]

[0012] where L represents the Lagrangian, T represents the kinetic energy, V represents the potential energy, q represents the generalized coordinates, u represents the generalized non-conservative force, τ d represents the disturbance acting on the system.

[0013] The dynamic model (1) can be transformed into the standard mechanical arm dynamics equation,

[0014]

[0015] where, represents the inertia matrix, represents the gravity vector, represents the Coriolis matrix.

[0016] According to the dynamics equation (2), the following two properties can be obtained,

[0017] Property 1: M represents a symmetric and positive definite matrix, and is limited to where a1, a2 represent positive constants, and ‖·‖ represents the Euclidean norm;

[0018] Property 2: is a skew-symmetric matrix,

[0019] Further, the constructing input convex neural network comprises:

[0020] constructing a full-input convex neural network,

[0021]

[0022] where σ i represents the activation function of each layer; is a deep neural network parameter, and y0≡0, The condition for ensuring that f(x; θ) is convex with respect to x is to ensure that is non-negative and the activation function σ i is convex and non-decreasing.

[0023] constructing a partial-input convex neural network,

[0024]

[0025] where σ i and represent the activation function of each layer, y0=0, is a deep neural network parameter, is the Hadamard product; The condition for ensuring that f(x; θ) is convex with respect to x is to ensure that is non-negative and the activation function is convex and non-decreasing.

[0026] Further, the constructing the Lagrange neural network comprises:

[0027] constructing a fully connected neural network with parameter γ

[0028] sampling N+1 states in the actual operation of the system to obtain a training data set,

[0029]

[0030] calculating corresponding constituting a corresponding training set

[0031] training by solving an optimization problem of equation (5),

[0032]

[0033]

[0034] where s.t. means that satisfies, Q = Q T > 0.

[0035] Further, the executing the neural backstepping controller when the system model information is accurately known comprises:

[0036] defining a tracking error as

[0037]

[0038] where q d (t) represents a reference trajectory, and is continuously differentiable; q(t) represents a system state, and φ(t) is a virtual signal designed subsequently.

[0039] defining a neural backstepping controller and a virtual signal as

[0040]

[0041]

[0042] where represents an inertia matrix, represents a gravity vector, represents a Coriolis matrix.

[0043] defining Φ as:

[0044] Φ(z1(t); θ1, S) = ψ(z1(t); θ1) + z1(t) T Sz1(t)

[0045] where S represents a hyper-parameter, is a positive definite matrix; ψ represents a fully input convex neural network with input z1 and parameter θ1.

[0046] Set the bias term b in equation (3) to 0 to guarantee ψ(0) = 0. i

[0047] Use the smooth ReLU function as the activation function to guarantee ψ(z1) ≥ 0.

[0048] D(z2; θ2, m) represents a deep neural network with parameter θ2, hyper-parameter m, and input z2.

[0049] Optimize the parameters of the neural network to improve the control performance, and the optimization equation is,

[0050]

[0051] s.t. model: (2)

[0052] Error: (6)

[0053] Controller: (7)

[0054] Initial state:

[0055] where s.t. means that, l t represents the stage cost at time t, and α is a normal number; the corresponding optimal parameter can be realized by solving the optimization equation (8).

[0056] Further, the use of the input convex neural network to improve the structure of the Lagrange neural network to perform neural backstepping controller design when the system model is unknown comprises:

[0057] Use a partially input convex neural network to learn Use a fully connected neural network L V (q(t); γ2) to learn V(q(t)), and the improved Lagrange neural network is,

[0058]

[0059] where γ = {γ1, γ2} is the neural network parameter, and the activation function is the softplus function.

[0060] Take the Hessian matrix and Jacobian matrix of the output of the improved Lagrange neural network to obtain the estimates of M, C, and G,

[0061] ​

[0062] The improved Lagrange neural network is taken as an uncertain system, and the equation of the system is

[0063]

[0064] wherein, is an uncertain term; according to the Taylor first-order expansion, it is set that the uncertain term satisfies:

[0065]

[0066] wherein, a, b, c are normal numbers, and ‖·‖ takes the Euclidean norm.

[0067] The neural backstepping controller when the system model is unknown is

[0068]

[0069]

[0070] The control performance is improved by optimizing the parameters of the neural network, and the optimization equation is

[0071]

[0072] s.t. model: (9)

[0073] Error: (6)

[0074] Controller: ((11)

[0075] Initial state:

[0076] wherein, s.t. means that, l t represents the stage cost at t, and α is a normal number.

[0077] Further, for the model of formula (2), it is assumed that there is no disturbance, formula (7) is used as the controller; Φ(z1) is strongly convex to the input z1, and there is only one minimum point at z1=0 and Φ(0)=0; D(z2) is positive definite; the system is globally asymptotically stable at z1=0, z2=0.

[0078] Further, for the model of formula (2), it is assumed that the boundary of the disturbance is ‖τ d ‖ 2 ≤d, formula (7) is used as the controller, and if The tracking error of the system converges to

[0079] For the model of equation (2), assuming the bound of disturbance is ‖τ d ‖ 2 ≤d, using equation (7) as the controller, if and the tracking error of the system converges to

[0080] Further, for the model of equation (9), assuming the bound of disturbance is ‖τ d || 2 ≤d, the uncertain term satisfies equation (10), using equation (11) as the controller, if and the tracking error of the system converges to wherein,

[0081] The application also provides a neural backstepping controller design system for a trajectory tracking control problem, which comprises the following modules:

[0082] A dynamics characteristic analysis module is configured to construct a dynamic model of a moving object according to information of the moving object, and analyze the dynamics characteristic of the moving object;

[0083] A convex neural network construction module is configured to construct an input convex neural network therein;

[0084] A Lagrange neural network construction module is configured to construct a Lagrange neural network therein;

[0085] A first neural backstepping controller design module is configured to perform neural backstepping controller design when the system model information is accurately known therein;

[0086] A second neural backstepping controller design module is configured to improve the structure of the Lagrange neural network using the input convex neural network, and perform neural backstepping controller design when the system model is unknown.

[0087] Compared with the prior art, the application has the following advantages:

[0088] 1. The application proposes a neural backstepping controller with stability guarantee and performance guarantee for a trajectory tracking control problem.

[0089] 2. The neural backstepping controller proposed by the application is stable for any feasible deep neural network parameters, and can improve the performance by optimizing the parameters.

[0090] 3. The neural backstepping controller proposed by the application can guarantee that the steady-state tracking error is bounded even in the presence of disturbance or model uncertainty.

[0091] 4. When the system model is unknown or difficult to obtain, the neural backstepping controller proposed by the present application can also be further improved in combination with the improved Lagrange neural network, and the stability and performance of trajectory tracking can be guaranteed. BRIEF DESCRIPTION OF DRAWINGS

[0092] To further clarify the above and other advantages and features of the present embodiments, a more particular description of embodiments of the application will be rendered by reference to specific embodiments thereof which are illustrated in the appended drawings. It is appreciated that these drawings depict only typical embodiments of the application and are therefore not to be considered limiting of its scope. The same or corresponding elements in the drawings are denoted by the same or similar reference signs.

[0093] Figure 1 a flowchart of a neural backstepping controller design method for a trajectory tracking control problem in an embodiment of the present application is shown;

[0094] Figure 2 a structural diagram of a neural backstepping controller when the system model information is accurately known in an embodiment of the present application is shown;

[0095] Figure 3 a diagram of the angle evolution process of a two-link robot arm when the neural backstepping controller is untrained in an embodiment of the present application is shown;

[0096] Figure 4 a diagram of the angle evolution process of a two-link robot arm after the neural backstepping controller is trained in an embodiment of the present application is shown;

[0097] Figure 5 a diagram of the maximum steady-state tracking error under different α after the neural backstepping controller is trained in an embodiment of the present application is shown;

[0098] Figure 6 a diagram of the angle evolution process of a two-link robot arm based on the neural backstepping controller based on LNNs in an embodiment of the present application is shown;

[0099] Figure 7 a diagram of the angle evolution process of a three-link robot arm under the MuJoCo simulation platform based on the neural backstepping controller based on LNNs in an embodiment of the present application is shown; and

[0100] Figure 8 a diagram of a neural backstepping controller design system for a trajectory tracking control problem in an embodiment of the present application is shown. DETAILED DESCRIPTION

[0101] In the following description, reference is made to the accompanying drawings which form a part hereof, and which are shown by way of illustration of the embodiments of the present application. However, it is understood that embodiments of the present application can be practiced without many of the specific details that are set forth in the following description. In other instances, well-known structures and operational aspects of the devices or methodologies have not been shown or described in order to avoid obscuring the application.

[0102] In this specification, reference to "one embodiment" or "the embodiment" means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment of the application. The appearances of the phrase "in one embodiment" in various places in the specification are not necessarily all referring to the same embodiment.

[0103] It should be noted that the embodiments of the present application are described in a specific order in this specification, however, this is just for the purpose of describing the specific embodiment, and does not limit the order of the steps. On the contrary, in different embodiments of the present application, the order of the steps can be adjusted according to the actual needs of adjustment.

[0104] In the present application, each module of the system according to the present application can be implemented using software, hardware, firmware or a combination thereof. When the module is implemented using software, the function of the module can be implemented by a computer program flow, for example, the module can be implemented by a code segment (such as a code segment in C, C++, etc.) stored in a storage device (such as a hard disk, a memory, etc.), wherein when the code segment is executed by a processor, the corresponding function of the module can be implemented. When the module is implemented using hardware, the function of the module can be implemented by setting a corresponding hardware structure, for example, the function of the module can be implemented by hardware programming on a programmable device such as a field programmable logic gate array (FPGA), or the function of the module can be implemented by designing an application specific integrated circuit (ASIC) including a plurality of transistors, resistors and capacitors, etc. electronic devices. When the module is implemented using firmware, the function of the module can be written in the form of program code in the read-only memory such as EPROM or EEPROM of the device, and when the program code is executed by the processor, the corresponding function of the module can be implemented. In addition, some functions of the module can need to be implemented by a separate hardware or cooperated with the hardware, for example, the detection function is implemented by a corresponding sensor (such as a proximity sensor, an acceleration sensor, a gyroscope, etc.), the signal transmitting function is implemented by a corresponding communication device (such as a Bluetooth device, an infrared communication device, a baseband communication device, a Wi-Fi communication device, etc.), the output function is implemented by a corresponding output device (such as a display, a speaker, etc.), and so on.

[0105] For the trajectory tracking control problem of nonlinear systems, the existing technology does not have a safe and performance guaranteed structured neural network controller. Moreover, when the system model of the nonlinear system is unknown, the performance and stability of the controller will be greatly limited. In view of the problems existing in the prior art, a structured deep neural network controller is designed on the basis of backstepping control, which is called neural backstepping controller (NBS controller). The NBS controller provided by the present application is designed for Lagrange mechanics, and the system model information is integrated. The Lagrange neural network is used to learn the model of the complex system from the data, and the design of the NBS controller when the system model information is accurately known is performed. In the case that the model information of the complex system is difficult to obtain, the input convex neural network is used to improve the structure of the existing Lagrange neural network, and then the improved Lagrange neural network is used to learn the model of the complex system and perform the design of the NBS controller. When the system model is unknown or difficult to obtain, the NBS controller proposed in the present application can be further improved in combination with the improved Lagrange neural network, which can guarantee the stability and performance of the trajectory tracking, so as to expand the application of the NBS controller in complex systems.

[0106] The technical solutions in the embodiments of the present application will be clearly and completely described below in combination with the drawings in the embodiments of the present application.

[0107] Figure 1 The flowchart of the neural backstepping controller design method for the trajectory tracking control problem of one embodiment of the present application is shown. As shown in Figure 1 The neural backstepping controller design method for the trajectory tracking control problem includes the following steps:

[0108] First, according to the information of the moving object, the dynamic model of the moving object is constructed, and the dynamic characteristics of the moving object are analyzed. According to the Euler-Lagrange equation, the dynamic model of the moving object is constructed,

[0109]

[0110] Wherein, L represents the Lagrange quantity, T represents the kinetic energy, V represents the potential energy, q represents the generalized coordinate, u represents the generalized non-conservative force, τ d represents the disturbance acting on the system.

[0111] The dynamic model (1) can be converted into a standard mechanical arm dynamics equation,

[0112]

[0113] Wherein, represents the inertia matrix, represents the gravity vector, M represents a Coriolis matrix.

[0114] According to the dynamic equation (2), the following two properties can be obtained,

[0115] Property 1: M represents a symmetric and positive definite matrix, and is limited to where a1, a2 represent positive constants, and ‖·‖ represents the Euclidean norm.

[0116] Property 2: is a skew-symmetric matrix,

[0117] Next, the input convex neural network is constructed. The input convex neural network (ICNN) is a neural network architecture with constraints on the parameters of the deep neural network, mainly divided into two types: a fully input convex neural network (FICNN) that guarantees that the output of the network is a convex function for all inputs, and a partially input convex neural network (PICNN) that is a convex function for part of the input.

[0118] The basic structure of FICNN is,

[0119]

[0120] where σ i represents the activation function of each layer; is a deep neural network parameter, and y0≡0, The condition for guaranteeing that f(x; θ) is convex for x is to guarantee is non-negative and the activation function σ i is convex and non-decreasing.

[0121] The basic structure of PICNN is,

[0122]

[0123] where σ i and represent the activation function of each layer, y0=0, is a deep neural network parameter, is the Hadamard product; The condition for guaranteeing that f(x; θ) is convex for x is to guarantee is non-negative and the activation function is convex and non-decreasing.

[0124] Note that ICNNs do not guarantee strong convexity for inputs x.

[0125] Next, we construct the Lagrangian neural network. Lagrangian neural networks (LNNs) are used to learn system- effective conservation laws and Lagrangian quantities from data. LNNs are generally fully connected neural networks with parameters

[0126] We sample the N+1 states during the actual running of the system to obtain the dataset for training LNNs,

[0127]

[0128] We compute the corresponding We construct the corresponding training set

[0129] We train by solving the optimization problem of equation (5),

[0130]

[0131]

[0132] where s.t. means that satisfies, Q = Q T > 0.

[0133] Since we need to compute the Hessian matrix, the second derivative of the activation function should not be 0, and we choose softplus as the activation function here.

[0134] Next, we perform the neural backstepping controller design when the system model information is known exactly. Assume that the given reference trajectory q d (t) is continuously differentiable, and the tracking goal of the controller is to let the system state q(t) track q d (t). Define the tracking error as,

[0135]

[0136] where q d (t) represents the reference trajectory and is continuously differentiable; q(t) represents the system state, and φ(t) is a virtual signal designed later.

[0137] Define the NBS controller and virtual signal as,

[0138]

[0139]

[0140] where, represents the inertia matrix, represents the gravity vector, represents the Coriolis matrix.

[0141] Φ(z1(t);θ1,S) is defined in the NBS controller as:

[0142] Φ(z1(t);θ1,S)=ψ(z1(t);θ1)+z1(t) T Sz1(t)

[0143] where S represents a hyper-parameter, which is a positive definite matrix; ψ represents a FICNN with input z1 and parameter θ1. The activation function of ReLU is used to guarantee

[0144] The bias term b in equation (3) is set to 0 to guarantee ψ(0)=0. i

[0145] The smooth ReLU function is used as the activation function to guarantee ψ(z1)≥0. By such definition, Φ is strongly convex with respect to the input z1 and Φ(0)=0 holds if and only if z1=0.

[0146] D(z2;θ2,m) represents a specially designed neural network with parameter θ2, hyper-parameter m and input z2. Two fully connected neural networks are used to output the diagonal elements and off-diagonal elements of a lower triangular matrix T respectively. The activation function of each fully connected neural network is tanh. The non-negativity of the diagonal elements is guaranteed by the ReLU activation function, and an extra constant m is added to guarantee the positive definiteness of the diagonal elements. Meanwhile, m also affects the minimum value of the output matrix. T is multiplied by itself to produce the required output positive definite matrix D(z2;θ2,m)≥0.

[0147] The structure diagram of the neural backstepping controller of the system model information accurate known embodiment of the present application is shown in Figure 2 where All of them can be obtained by automatic differentiation (AutoDiff). The neural backstepping controller designed by the present application is stable for any feasible neural network parameters, so the performance can be guaranteed by optimizing the neural network parameters, and the optimization equation is

[0148]

[0149] s.t. model: (2)

[0150] error: (6)

[0151] controller: (7)​

[0152] Initial state:

[0153] where s.t. means such that, l t represents the stage cost at time t, and a is a positive constant; the corresponding optimal parameter This can be achieved by solving the optimization equation (8). The solution of this optimization equation can be achieved by first discretizing the model and cost function, and then numerically solving the corresponding discrete-time part.

[0154] Finally, the structure of the improved Lagrangian neural network is used to improve the Lagrangian neural network, and the neural backstepping controller design is performed when the system model is unknown.

[0155] In the design process of the above NBS controller, M, C, G are assumed to be known and are designed into the structure of the controller. However, for complex systems, these model information is sometimes difficult to obtain, so it is necessary to first learn the model from data using Lagrangian neural networks (LNNs).

[0156] The general LNNs use fully connected neural networks, which cannot guarantee is a positive definite matrix. Since For is a convex function, a PICNN is used to learn A fully connected neural network L V (q(t);γ2) is used to learn V(q(t)), and the improved Lagrangian neural network is

[0157]

[0158] where γ={γ1,γ2} is the neural network parameter, and the activation function is the softplus function. Through the above structure improvement, it can be guaranteed that for any neural network parameter, is a positive definite matrix and satisfies properties 1 and 2.

[0159] The Hessian matrix and Jacobian matrix of the output of the improved Lagrangian neural network are obtained, and the estimates of M, C, G are

[0160]

[0161] The improved Lagrangian neural network is regarded as an uncertain system, and the equation of the system is

[0162]

[0163] where is uncertain term; according to Taylor's first expansion, set uncertain term to satisfy:

[0164]

[0165] Wherein, a, b, c are normal numbers, ||·|| is Euclidean norm;

[0166] The neural backstepping controller when system model is unknown is,

[0167]

[0168]

[0169] The control performance is improved by optimizing the parameters of neural network, and the optimization equation is,

[0170]

[0171] s.t.model:(9)

[0172] Error:(6)

[0173] Controller:(11)

[0174] Initial state:

[0175] Wherein, s.t. means that, l t Represent the stage cost at t, and alpha is a normal number.

[0176] The above controller, the inventor gives the following four theorems from theory.

[0177] Theorem one: for the model of formula (2), assuming no disturbance, using formula (7) as the controller; Φ(z1) is strongly convex to input z1, and there is only one minimum point at z1=0 and Φ(0)=0; D(z2) is positive definite; The system is globally asymptotically stable at z1=0, z2=0.

[0178] The proof process of theorem one is as follows:

[0179] According to formula (6) and (7), formula (2) can be transformed into:

[0180]

[0181] For the following Lyapunov function:

[0182]

[0183] It can be concluded that only if When ‖z1‖, ‖z2‖→∞,

[0184]

[0185] V(z1,z2)→∞. Then take the derivative of V:

[0186] because is a skew-symmetric matrix, D(z2) is positive definite, and according to formula (6) we can deduce Then it can be deduced that:

[0187]

[0188] Therefore, it can be deduced that the system will converge to Since Φ(z1) is strongly convex to input z1, if and only if z1=0, Established, we can launch:

[0189]

[0190] From Theorem 1, it can be concluded that the stability of the controller is independent of the parameters of the deep neural network.

[0191] Theorem 2: For the model of formula (2), assuming that the perturbation limit is ‖τ d ‖ 2 ≤d, use equation (7) as the controller, if The tracking error of the system converges to

[0192] The proof of Theorem 2 is as follows:

[0193] Considering the Lyapunov function V above, the derivative of V becomes,

[0194]

[0195] Since D(z2)≥I / 2, we can get

[0196]

[0197] Then we can find that z1 will converge to This proves that the controller proposed in the present invention can achieve a bounded steady-state tracking error.

[0198] Theorem 3: For the model of formula (2), assuming that the perturbation limit is ‖τ d ‖ 2 ≤d, use equation (7) as the controller, if and The tracking error of the system converges to

[0199] The proof of Theorem 3 is as follows:

[0200] The Taylor expansion at z1=0 is:

[0201]

[0202] where δ represents the higher order terms. One can approximate by Since one can deduce i.e.

[0203] According to Theorem 3, the inventors added a regularization term

[0204] Theorem 4: For the model of equation (9), assuming the bound of disturbance is ‖τ d ‖ 2 ≤ d, the uncertain term satisfies equation (10), and equation (11) is used as the controller, if and the tracking error of the system converges to where The proof steps of Theorem 4 are similar to those of Theorems 2 and 3, and are not repeated here.

[0205] The following simulation experiment is used to illustrate the effect of the neural backstepping controller proposed by the present application.

[0206] A two-link robot arm is selected for the simulation experiment, and pytorch and Adam optimizer are used in the experiment. The training number of rounds and the learning rate of the optimizer parameters are generally determined according to the experience method combined with the actual effect, and in the following simulation process, the training number of rounds selected is 200 rounds, and the learning rate is a decay learning rate starting from 1e-3.

[0207] It is assumed that the mass of the two-link model is concentrated at the two ends of the rod, and the mass of each connecting point is 1 kg, and the length is 1 m. In this simulation experiment, the simulation step is selected to be 0.01. Equation (7) is selected as the controller, where ψ in Φ(z1; θ1, S) has 3 hidden layers, each hidden layer has 32 neurons, and the hyperparameter S = I. Each fully connected neural network in D(z2; θ2, m) has two layers, each layer has 32 neurons, and the hyperparameter m = 0.001.

[0208] The angles of the robot arm connecting rod are respectively tracked sin(0.1t) and cos(0.1t), i.e. β1 and β2 in the following figure. First consider the case without disturbance. ​​

[0209] Figure 3 FIG. 1 shows a schematic diagram of the angle evolution process of a two-link manipulator when the neural backstepping controller according to an embodiment of the present invention is not trained. Figure 3 As shown in Figure 2, the neural backstepping controller can effectively track the reference trajectory, and the steady-state tracking error ‖z1‖≤1.29e -6 , which also proves that the neural backstepping controller proposed in this invention is stable for all feasible parameters.

[0210] Figure 4 The following figure shows the evolution of the angle of a two-link manipulator after training a neural backstepping controller according to an embodiment of the present invention. After training the neural backstepping controller and solving equation (8) 200 times, it was found that optimizing the deep neural network parameters can effectively improve the controller's performance.

[0211] Then verify that the tracking error is also limited in the presence of limited disturbances. Consider τ d =[1.0,1.0], select different α and measure the maximum steady-state tracking error after training. Figure 5 The maximum steady-state tracking error at different α after training the neural backstepping controller according to one embodiment of the present invention is shown. According to Theorem 3, the steady-state tracking error ‖z1‖ 2 ≤d / (2α 2 ).like Figure 5 As shown in Figure 3, the neural backstepping controller can achieve a small steady-state tracking error after training, and the error is limited. Increasing the value of α can improve tracking accuracy, but it also makes training more difficult, so it is necessary to choose a suitable value of α.

[0212] Assuming that the model of the two-link robot arm is unknown, first use To learn the two-link robotic arm model, we get in, It is a PICNN with three hidden layers and 32 neurons in each hidden layer; L V (q(t);γ2) is a fully connected neural network with three hidden layers, each with 32 neurons; the activation function is softplus. Then, automatic differentiation is used to combine the learned To obtain the estimation of the model information required by the neural backstepping controller. First randomly take 2048 points for training 300 epochs, and then train the controller for 200 epochs. Figure 6 FIG1 shows a schematic diagram of the angle evolution process of a two-link manipulator based on a neural backstepping controller of LNNs according to an embodiment of the present invention. Figure 6 As shown, the steady-state tracking error ‖z1‖≤3.1e -4This demonstrates the effectiveness of the LNNs-based approach and breaks the limitation of the controller in the case of unknown complex system model.

[0213] Next, we use MuJoCo simulation platform to do the simulation verification of three-link in the case of unknown model. In this simulation experiment, we assume that all the model information is unknown, use MuJoCo physical simulation software to sample data, and then use the improved LNNs to learn the Lagrange neural network of three-link. Among them, is a PICNN, which has three hidden layers, each with 32 neurons; L V (q(t);γ2) is a fully connected neural network with three hidden layers, each with 32 neurons; the activation function is softplus. Make the MuJoCo simulator run without any control input, obtain the state information of the model to form the data set for training L. The simulation step is set to 0.001, the initial state is the origin, and the control quantity u=0. 10000 points are sampled, the batch size is 10, and γ * is obtained after solving equation (5) for 200 rounds.

[0214] After training to obtain , use the LNNs-based NBS controller (11). Among them, ψ in Φ(z1;θ1,S) has 3 hidden layers, each with 32 neurons, and the hyperparameter S=I; each fully connected neural network in D(z2;θ2,m) has two layers, each with 32 neurons, and the hyperparameter m=1.0. Discretize the optimization problem equation (12), the time step is 0.01s, the simulation step T=1, Use a decaying learning rate starting from 1e-3, and train for 200 rounds. Figure 7 The figure shows the angle evolution process of the three-link robot arm under the LNNs-based neural backstepping controller of one embodiment of the application in the MuJoCo simulation platform. As Figure 7 shown, the NBS controller can effectively achieve high-precision tracking and ensure that the steady-state tracking error ‖z1‖ 2 ≤1.5e -3 . This experiment further demonstrates the effectiveness of the LNNs-based NBS controller, especially in the case of unknown or difficult-to-obtain system model.

[0215] For the trajectory tracking control problem, the application proposes a neural backstepping controller with stability guarantee and performance guarantee. The neural backstepping controller proposed by the application is stable for any feasible deep neural network parameters, and can improve the performance by optimizing the parameters. In the case of system disturbance or model uncertainty, the steady-state tracking error can also be guaranteed to be bounded. When the system model is unknown or difficult to obtain, the neural backstepping controller proposed by the application can also be further improved by combining the improved Lagrange neural network, and the stability and performance of trajectory tracking can be guaranteed.

[0216] The application also provides a neural backstepping controller design system for the trajectory tracking control problem, Figure 8 The neural backstepping controller design system for the trajectory tracking control problem of one embodiment of the application is shown in the schematic diagram. As shown in the figure, Figure 8 The system includes the following modules:

[0217] The dynamic characteristic analysis module is configured to construct a dynamic model of the moving object according to the information of the moving object, and analyze the dynamic characteristics of the moving object;

[0218] The convex neural network construction module is configured to construct an input convex neural network therein;

[0219] The Lagrange neural network construction module is configured to construct a Lagrange neural network therein;

[0220] The first neural backstepping controller design module is configured to perform neural backstepping controller design when the system model information is accurately known therein;

[0221] The second neural backstepping controller design module is configured to use the input convex neural network to improve the structure of the Lagrange neural network, and perform neural backstepping controller design when the system model is unknown.

[0222] Although the above describes the embodiments of the application, it should be understood that they are only presented as examples and not as limitations. It is obvious to those skilled in the relevant art that various combinations, modifications and changes can be made without departing from the spirit and scope of the application. Therefore, the width and scope of the application disclosed herein should not be limited by the above disclosed exemplary embodiments, but should only be defined according to the appended claims and their equivalent replacements.

Claims

1. A neural backstepping controller design method for trajectory tracking control problems, characterized by: The steps include: Constructing a dynamic model of the moving object based on the information of the moving object and analyzing the dynamic characteristics of the moving object; Construct an input convex neural network; Construct Lagrangian neural network; Design of neural backstepping controllers when the execution system model information is precisely known; and Using the input convex neural network to improve the structure of the Lagrangian neural network, and designing a neural backstepping controller when the system model is unknown; The steps of constructing a model of the moving object and analyzing the dynamic characteristics of the moving object include: The dynamic model of the moving object is constructed according to the Euler-Lagrange equation. （1） in, represents the Lagrangian, Represents kinetic energy, Represents potential energy, represents the generalized coordinates, represents a generalized non-conservative force, represents the disturbance acting on the system; The dynamic model (1) can be transformed into the standard manipulator dynamics equation, （2） in, represents the inertia matrix, represents the gravity vector, represents the Coriolis matrix; According to the kinetic equation (2), the following two properties can be obtained: Property 1: represents a symmetric and positive definite matrix, and is restricted to ,in, represents a positive constant, stands for Euclidean norm; Property 2: is a skew-symmetric matrix, ; The method of using the input convex neural network to improve the structure of the Lagrangian neural network and designing a neural backstepping controller when the execution system model is unknown includes: Using a partially input convex neural network Come learn , using a fully connected neural network Come learn , the improved Lagrangian neural network is, in, is the neural network parameter, and the activation function is the softplus function; The Hessian matrix and Jacobian matrix of the output of the improved Lagrangian neural network are obtained. The estimate is, ， ， ； The improved Lagrangian neural network is regarded as an uncertain system, and the equation of the system is: （9） in, is an uncertain term; according to Taylor's first-order expansion, the uncertain term is set to satisfy: （10） in, a, b, c They are all normal numbers. Take the Euclidean norm; The neural backstepping controller when the system model is unknown is: （11） By optimizing the parameters of the neural network to improve the control performance, the optimization equation is: （12） Among them, st means satisfying, Representatives in The stage price of time, is a positive constant.

2. The neural backstepping controller design method for trajectory tracking control problem according to claim 1 is characterized in that: The constructing of the input convex neural network includes: Construct a full-input convex neural network, （3） in, Represents the activation function of each layer; are the deep neural network parameters, and ; Guarantee The condition for being convex is to ensure is non-negative and the activation function is convex and non-decreasing; Construct a partial input convex neural network, （4） in, and represents the activation function of each layer, , , , are the deep neural network parameters, It is Hadamard; Guarantee The condition for being convex is to ensure is non-negative and the activation function is convex and non-decreasing.

3. The neural backstepping controller design method for trajectory tracking control problem according to claim 1 is characterized in that: The Lagrangian neural network construction includes: The construction parameters are Fully connected neural network , During the actual operation of the system The training data set is obtained by sampling the states. Calculate the corresponding , forming the corresponding training set , Training is performed by solving the optimization problem of formula (5). （5） Among them, st means satisfying, .

4. The neural backstepping controller design method for trajectory tracking control problems according to claims 1 and 2 is characterized in that: The design of the neural backstepping controller when the execution system model information is precisely known includes: Define the tracking error as, （6） in, represents the reference trajectory and is continuously differentiable; Represents the system status, It is a virtual signal designed later; Define the neural backstepping controller and virtual signal as, （7） in, represents the inertia matrix, represents the gravity vector, represents the Coriolis matrix; definition for: in, represents the hyperparameter, which is a positive definite matrix; Represents an input , the parameters are Fully input convex neural network; The bias term in equation (3) Set to 0 to ensure ; Use the smooth ReLU function as the activation function to ensure ; Represents a deep neural network parameter , hyperparameters ,enter Specially customized neural networks; By optimizing the parameters of the neural network to improve the control performance, the optimization equation is: （8） Among them, st means satisfying, Representatives in The stage price of time, is a positive constant; the corresponding optimal parameter This can be achieved by solving the optimization equation (8).

5. The neural backstepping controller design method for trajectory tracking control problems according to claims 1-4 is characterized in that: For the model of equation (2), assuming there is no disturbance, use equation (7) as the controller; Input Strongly convex, there is only one minimum point at and ; is positive definite; the system is globally asymptotically stable at .

6. The neural backstepping controller design method for trajectory tracking control problems according to claims 1-4 is characterized in that: For the model of formula (2), it is assumed that the perturbation limit is , using equation (7) as the controller, if , the tracking error of the system converges to ; For the model of formula (2), it is assumed that the perturbation limit is , using equation (7) as the controller, if ,and , the tracking error of the system converges to .

7. The neural backstepping controller design method for trajectory tracking control problems according to claims 1-4 is characterized in that: For the model of formula (9), it is assumed that the perturbation limit is , the uncertainty term satisfies equation (10), and equation (11) is used as the controller. If , and , the tracking error of the system converges to ;in, .

8. A neural backstepping controller design system for trajectory tracking control problems, characterized by: Includes the following modules: a dynamic characteristics analysis module configured to construct a dynamic model of the moving object based on the information of the moving object and analyze the dynamic characteristics of the moving object; a convex neural network building module configured to build an input convex neural network therein; a Lagrangian neural network building module configured to build a Lagrangian neural network therein; a first neural backstepping controller design module configured to perform neural backstepping controller design when system model information is precisely known; and The second neural backstepping controller design module is configured to use the input convex neural network to improve the structure of the Lagrangian neural network and perform neural backstepping controller design when the system model is unknown.

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