CN114185358B - Self-adaptive signal tracking method for satellite navigation receiver - Google Patents
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Abstract
The invention discloses a satellite navigation receiver self-adaptive signal tracking method, and relates to the field of signal processing. The invention combines the traditional PID control strategy with the neural network algorithm to generate the novel PID controller with robustness and adaptability. The method utilizes the self-learning capability of the neural network, and can use a gradient descent method to automatically correct on-line and automatically modify robust PID parameters. The satellite navigation receiver tracking result shows that the code tracking loop technology has the advantages of strong adaptability and strong robustness.
Description
Technical Field
The invention relates to the technical field of signal processing, in particular to a satellite navigation receiver self-adaptive signal tracking method.
Background
Baseband signal processing has been an important component of satellite navigation receivers, whose core content is signal acquisition and tracking. The main purpose of tracking is to track the code phase and the doppler frequency of the carrier, which consists of two interoperable feedback loops, a Delay Locked Loop (DLL) for code tracking and a Phase Locked Loop (PLL) for carrier tracking. In code tracking loops, a linear PLL model is often employed.
Typically, the code tracking loop includes a first or second order loop filter and a Voltage Controlled Oscillator (VCO). These systems typically have large overshoots, long settling times, and instability due to the non-linear and time delay characteristics present in the actual code tracking loop. In classical control theory, the smith method may be used to construct a controller if the transfer function of the system is known.
However, the transfer function of an actual system is not easily measured or accomplished. It is well known that Proportional Integral Derivative (PID) control is one of the most important control strategies. PID controllers are widely used in closed loop process control systems, particularly in systems where an accurate mathematical model can be built, due to their simplicity and robustness.
While PID controllers are powerful, they are not suitable for use in long delay and control of nonlinear complex systems where the P, I and D parameters are difficult to select and to accommodate a wide range of time-varying characteristics. Because of unsuitable parameters, the conventional PID controller cannot achieve an ideal control effect. Therefore, the application of conventional PID control is subject to further limitations and challenges.
Disclosure of Invention
In view of this, the invention provides a satellite navigation receiver adaptive signal tracking method, which combines the traditional PID control with the neural network algorithm, has the self-learning capability of the neural network, and can self-adjust and automatically modify robust PID parameters in the network by a gradient descent method.
In order to achieve the above purpose, the technical scheme adopted by the invention is as follows:
a satellite navigation receiver adaptive signal tracking method comprises the following steps:
step 1: the input signal is subjected to closed-loop control through a PID controller;
step 2: sending the signals output by the PID controller into an RBF neural network identifier, identifying a nonlinear system through the RBF neural network, and predicting Jacobian Jacobian matrix information of the controlled object in the system;
step 3: and sending the signals output by the RBF neural network identifier into a neural network controller, wherein the neural network controller adopts a self-adaptive PID controller based on a single neuron, and performs self-adjustment according to the system state so as to achieve the optimal control effect.
Further, the specific mode of the step 1 is as follows:
step 1.1: calculating three independent parameters of an input signal, namely a proportional term P, an integral term I and a differential term D, defining u (t) as the output of the controller, and then, the output equation in the time domain is as follows:
in equation (1), the input e (t) =r (t) -y (t), is the difference between the measurement process output and the reference input, i.e., the error; k (K) p Representing the proportional gain; t (T) i Representing an integration time; t (T) d Representing an integration time; k (K) i ＝K p T is the integral gain; k (K) d ＝K p T d Is a differential gain; t is the current time;
when the backward difference method is adopted, the output of the incremental digital PID algorithm is as follows:
u[k]＝u[k-1]+K P {e[k]-e[k-1]+K i e[k]}+K d {e[k]-2e[k-1]+e[k-2]} (2)
step 1.2: the following transfer function is obtained according to equation (2):
and modeling the PID controller through the transfer function of the formula (3) to realize PID control.
Further, the specific mode of the step 2 is as follows:
step 2.1: let x= [ X ] 1 ，x 2 ，…，x n ] T Is the input vector of the RBF neural network, the neurons in the second layer of the RBF neural network are activated by radial basis functions, and the radial vector of the RBF neural network is expressed as H= [ H ] 1 ，h 2 ，…，h m ] T Wherein h is j J=1, 2, …, m is a multiple gaussian function, specifically:
in the formula (4) of the present invention, sign I representation of euclidean norms; c (C) j ＝[c j1 ，c j2 ，…，c jm ] T Is a neural network j th A center vector of the node; b j Is the basic width parameter of node j in the hidden layer, b j > 0, the resulting base width vector b= [ B ] 1 ，b 2 ，…，b m ] T In a hidden layer of the neural network;
step 2.2: the weight vector of the RBF neural network is expressed as w= [ W ] 1 ，W 2 ，…W m ] T The output of the RBF neural network is made up of a linear weighted sum of radial basis functions in the hidden layer as shown in equation (5):
in the formula (5), the subscript I indicates that the control system adopts an RBF neural network as an identifier, w j Representing output neurons and hidden layer neurons J th Weight of the two;
step 2.3: the performance function of the identification process is defined as follows:
step 2.4: in order to make RBF neural network output y I (k) Error e between the actual output y (k) of the controlled object I (k) Minimum, adopting a gradient descent method to adjust the weight between the output layer and the hidden layer, the node center of the hidden layer and the node radial width parameter; specifically, iterative operation is performed by formulas (7) (8) (9):
w j (k)＝w j (k-1)+η[y(k)-y I (k)]h j +α[w j (k-1)-w j (k-2)] (7)
b j (k)＝b j (k-1)+ηΔb j +α[b j (k-1)-b j (k-2)] (8)
c ji (k)＝c ji (k-1)+ηΔc ji +α[c ji (k-1)-c ji (k-2)] (9)
wherein:
in the above formula, η is the slope rate and α is the momentum factor;
step 2.5: acquiring Jacobian Jacobian matrix information, namely the sensitivity of controlled object output to control input, through an RBF neural network online identification process; the expression of Jacobian matrix information is as follows:
further, the specific mode of the step 3 is as follows:
step 3.1: the output of the adaptive PID controller corresponds to the NCO output in the GNSS receiver code tracking loop as shown in equation (13):
Δu(k)＝v 1 (k)xc 1 (k)+v 2 (k)xc 2 (k)+v 3 (k)xc 3 (k) (13)
in formula (13), xc i (i=1, 2, 3) is input, v i (i=1, 2, 3) is the corresponding weight of an individual neuron;
step 3.2: combining equation (2), the following equation is obtained:
step 3.3: in a PID controller based on a neural network, by adjusting the weight v i (i=1, 2, 3) squaring the systematic error J C (k) Decreasing to zero; j (J) C (k) Is defined as follows:
in equation (16), e (k) =r (k) -y (k) is the error between the actual output of the control system and its currently required reference input;
step 3.4: in order to obtain optimal performance of the adaptive PID controller of the single neuron, the value of the weight is adjusted and updated online by adopting a gradient descent method according to the Hebb criterion, and the algorithm is as follows:
namely, it is equivalent to:
in the middle ofJacobian Jacobian matrix information, which is a controlled object in the system, can be obtained from RBF neural network identification equipment in formula (12).
Compared with the prior art, the invention has the following advantages:
1. the invention combines the traditional PID control strategy with the neural network algorithm to generate the novel PID controller with robustness and adaptability.
2. The method of the invention obviously improves the performance of the code tracking loop when the GNSS receiver works in nonlinear and time-varying signal environments.
3. The method has the self-learning capability of the neural network, so that the robust PID parameters in the network can be self-set and automatically modified by a gradient descent method.
Drawings
FIG. 1 is a block diagram of a generic GNSS receiver code tracking loop.
FIG. 2 is a block diagram of an online identification process of an adaptive PID control system based on RBF neural network.
Fig. 3 is a structure of an RBF neural network.
Fig. 4 is a structure of a single neuron PID controller.
FIG. 5 is a scatter plot of an adaptive code tracking loop based on RBF neural network identification in a GNSS receiver.
FIG. 6 is a diagram showing the rapid accumulation of phases of an adaptive code tracking loop in a GNSS receiver based on RBF neural network identification.
FIG. 7 is a correlation envelope of early, immediate, and late adaptive code tracking loops based on RBF neural network identification in a GNSS receiver.
FIG. 8 is a graph of Jacobian information identified online by a GNSS receiver adaptive code tracking loop RBF neural network.
FIG. 9 is a block diagram illustrating the self-tuning of adaptive code tracking loop parameters based on RBF neural network identification in a GNSS receiver.
Detailed Description
The following describes in detail the embodiments of the present invention, which are implemented on the premise of the technical solution of the present invention, and give detailed embodiments and specific operation procedures, but the scope of protection of the present invention is not limited to the following embodiments.
A satellite navigation receiver adaptive signal tracking method comprises the following steps:
step 1: the input signal enters a traditional PID controller, which forms closed loop control and ensures the stability of the whole system;
step 2: the signals from PID are divided into several modules and enter RBF Neural Network Identifier (NNI), which uses RBF neural network to realize nonlinear system identification function for predicting Jacobian information of controlled object in system, i.e. the system is provided with a function of identifying the controlled object
Step 3: the signals from RBFNNI enter a Neural Network Controller (NNC), the NNC adopts a self-adaptive PID controller based on a single neuron, and performs self-adjustment capacity according to the system state so as to achieve the optimal control effect;
step 4: the result of the GNSS receiver adaptive code tracking loop is comprehensively optimized.
In the step 1, a traditional PID controller is entered, and the specific process is as follows:
step 1.1: in a first step, three independent constant parameters of the input signal are calculated, namely proportional, integral and derivative terms, denoted P, I and D, respectively.
These values can be explained in time: p depends on the current error, I depends on the accumulation of past errors, and D is a prediction of future errors based on the current rate of change.
Defining u (t) as the controller output, the output equation in the time domain is:
in equation (1), the input is e (t) =r (t) -y (t), which is the difference (error) between the measurement process output and the reference input; k (K) p Representing the proportional gain; t (T) i Representing an integration time; t (T) d Representing an integration time; k (K) i ＝K p T is the integral gain; k (K) d ＝K p T d Is a differential gain; t is time or instantaneous time (current).
Step 1.2: when the backward differencing method is employed, the output of the digital PID algorithm is incremented:
u[k]＝u[k-1]+K P {e[k]-e[k-1]+K i e[k]}+K d {e[k]-2e[k-1]+e[k-2]} (2)
step 1.3: second, the PID controller is modeled by the transfer function of equation (3):
the process of selecting controller parameters to meet a given performance specification is referred to as controller tuning. The tuning control loop adjusts its control parameters (proportional band/gain, integral gain/reset, differential gain/rate) to the optimum value of the desired control response.
Stability (bounded oscillations) is a fundamental requirement, but in addition to this, different systems have different behavior, different applications have different requirements, and conflicts between requirements may occur.
In order to obtain better PID control performance, the proportional, integral and derivative parameters of the PID controller need to be well adjusted to form a cooperative constraint relation of control quantity. Because the neural network has arbitrary approximation capability to the nonlinear function, the control system based on the neural network has strong self-adaptive capability, and the optimal combined PID control effect can be achieved through a self-learning process.
In step 2, an RBF Neural Network Identifier (NNI) is entered, and the specific process of predicting Jacobian information of the controlled object in the system is as follows:
step 2.1: let x= [ X ] 1 ，x 2 ，…，x n ] T Is the input vector of the neural network, the neurons in the second layer (hidden layer) are activated by radial basis functions, and the radial vector of the neural network is denoted as h= [ H ] 1 ，h 2 ，…，h m ] T Wherein h is j (j=1, 2, …, m) should be a multiple gaussian function, which can be written in particular as:
the sign of equation (4) |·|| representing euclidean norms; c (C) j ＝[c j1 ，c j2 ，…，c jm ] T Is a neural network j th A center vector of the node; b j Is the basic width parameter of the node j in the hidden layer, which has a value greater than zero (b j > 0), a basic width vector b= [ B ] is formed 1 ，b 2 ，…，b m ] T In the hidden layer of the neural network.
Step 2.2: the weight vector of the neural network is expressed as w= [ W ] 1 ，W 2 ，…W m ] T The output of the RBF neural network is formed by the linear weighted sum of the number of radial basis functions in the hidden layer, as shown in a formula (5):
wherein the output subscript I indicates that the control system adopts RBF neural network as an identifier, w j Representing output neurons and hidden layer neurons J th Weight of the same.
Step 2.3: in this section, the performance function of the identification process may be defined as follows:
step 2.4: is thatMake RBF neural network output y I (k) Error e between the actual output y (k) of the controlled object I (k) And (3) minimum, adjusting the weight between the output layer and the hidden layer by adopting a gradient descent method, wherein the node center and the node radial width parameters of the hidden layer. The following iterative algorithm was applied:
w j (k)＝w j (k-1)+η[y(k)-y I (k)]h j +α[w j (k-1)-w j (k-2)] (7)
b j (k)＝b j (k-1)+ηΔb j +α[b j (k-1)-b j (k-2)] (8)
c ji (k)＝c ji (k-1)+ηΔc ji +α[c ji (k-1)-c ji (k-2)] (9)
where η is the slope rate and α is the momentum gene.
Step 2.5: jacobian information, namely the sensitivity of controlled object output to control input, can be obtained through the RBF neural network online identification process. The expression of Jacobian information is as follows:
step 2.6: the Jacobian information obtained will be sent to a single neuron based PID controller described below to achieve the best control effect.
The specific process of re-entering the Neural Network Controller (NNC) from the RBFNNI in step 3 is as follows:
step 3.1: the output of the single neuron (i.e., the output of the adaptive PID controller) corresponds to the NCO output in the GNSS receiver code tracking loop as shown in equation (13):
Δu(k)＝v 1 (k)xc 1 (k)+v 2 (k)xc 2 (k)+v 3 (k)xc 3 (k) (13)
in the above formula, xc i (i=1, 2, 3) is an input; v i (i=1, 2, 3) is the corresponding weight of an individual neuron.
Compared to equation (2), the following equation can be obtained:
in a PID controller based on a neural network, one of the basic tasks is to adjust the weight v i (i=1, 2, 3) squaring the systematic error J C (k) Decreasing to zero. J (J) C (k) Is defined as follows:
in the above equation, e (k) =r (k) -y (k) is the error between the actual output of the control system and its currently required reference input.
In order to obtain the optimal performance of the adaptive PID controller of the single neuron, the value of the weight is adjusted and updated online by adopting a gradient descent method according to the Hebb criterion, and the algorithm is as follows:
corresponding to:
in the middle ofJacobian information, which is the controlled object in the system, can be obtained from the RBF neural network identification device in formula (12).
In the step 4, the comprehensive optimization process of the GNSS signal tracking result is as follows: simulation is carried out on Galileo EI OS BOC (1, 1) signals, and simulation results show that the method has good performance in a code tracking loop of a GNSS receiver, and particularly has strong robustness on nonlinear and time-varying characteristics based on a PID control strategy of a neural network.
FIG. 1 is a general block diagram of a GNSS code tracking loop including a programmable pre-detection integrator, a code loop discriminator, and a loop filter.
These three functions play a key role in determining the two most important performance characteristics of the receiver code tracking loop design: code loop thermal noise error and maximum LOS dynamic stress threshold.
The idea of the DLL is to correlate the input signal with the three codes shown in fig. 1. The method comprises the following specific steps:
step 1: the input signal is multiplied with a perfectly aligned local replica of the carrier, thereby letting the PRN code be converted to baseband.
Step 2: the signal obtained in step 1 is multiplied by three code copies (early, immediate and late), respectively. These three replica codes are generated in principle at + -0.5 chip intervals.
Step 3: after the second multiplication, the six outputs enter an I-D (integrator) circuit. The output of these integrals is a numerical value indicating the degree of correlation of the particular code replica with the code in the input signal.
The receiver has six correlator outputs in total, namely:
i in the formula ES ，Q ES ，I LS ，Q LS ，I PS And Q PS Is the six correlator outputs shown in fig. 1.
This has the advantage that it is independent of the phase on the local carrier. If the local carrier is in phase with the input signal, then all the energy will be in the in-phase arm.
FIG. 2 is a block diagram of an online identification process of an adaptive PID control system based on RBF neural network. It consists of three units:
conventional PID controllers: the closed loop control is formed, and the stability of the whole control system is ensured;
RBF Neural Network Identifier (NNI): it uses RBF neural network to realize nonlinear system identification function for predicting Jacobian information of controlled object in system, i.e.Neural Network Controller (NNC): NNC uses an adaptive PID controller based on a single neuron to perform self-tuning according to the system state to achieve the best control effect.
Fig. 3 is a schematic diagram of a typical RBF neural network, consisting of three layers: an input layer, a hidden layer and an output layer.
In RBF neural networks, the mapping from input to output is nonlinear, but from hidden layer to output layer is linear. Therefore, the learning speed of the neural network can be greatly improved.
It has been demonstrated that RBF neural networks have the ability to approximate any continuous function with arbitrary precision.
In an RBF neural network, each input neuron corresponds to an element of an input vector and is fully connected to a neuron in a hidden layer. Similarly, each neuron of the hidden layer is also connected to an output layer neuron.
Fig. 4 is a structure of a single neuron, which is a multiple-input single-output information processing unit forming an essential part of a neural network, having self-learning and self-adaptation capabilities. Simple structure and easy realization.
Therefore, the single neuron is combined with the traditional PID controller to form the single neuron self-adaptive PID controller with simple design, strong adaptability and strong robustness.
The controller utilizes Jacobian information obtained by the RBF neural network identification unit to carry out on-line adjustment on control parameters, and realizes the self-adaptive setting of the PID controller under different real-time conditions.
Fig. 5 is a scatter diagram of an adaptive code tracking loop based on RBF neural network identification in a GNSS receiver, which is a simulation of Galileo EI OS BOC (1, 1) signals in an embodiment of the present invention.
The input layer consists of 4 neurons, so the input vector of RBF neural network can be selected as X= [ u (k), u (k-1), y (k), y (k-1)] T Where u is the NCO output in the code tracking loop, the actual output y of the code tracking system can be selected to be according to equation (19) At the hidden layer, 8 neurons were used. Considering equation (5), the output layer consists of only one neuron, the output of which corresponds to y in equation (5) I 。
Its carrier to noise ratio (C/N) 0 ) At a sampling frequency f of 46dB-Hz s 16.3676MHz, intermediate frequency f IF The front-end quantization level is 4 bits for 4.1304 MHz.
In the simulation, the pre-detection time of the integrating circuit block of the code tracking loop is 8ms.
In the Jacobian information online identification device (namely NNI) based on the RBF neural network, the learning rate eta of the gradient descent algorithm is 0.44, and the momentum factor alpha is 0.12.
In a single neuron based adaptive PID controller (i.e., NNC), the learning rate is set to η 1 ＝η 2 ＝η 3 =0.2, momentum factor is set to α 1 ＝α 2 ＝α 3 ＝0.1。
In fig. 5, a scatter plot of in-phase (I-instant) and quadrature (Q-instant) components is shown, with two bubbles occurring due to navigation bit transitions.
Fig. 6 shows the accumulation of I-moments indicating navigation bits in an input signal.
In fig. 7, the early, immediate and late (E-P-L) correlation envelope of an adaptive code tracking loop based on RBF neural network on-line system identification is depicted.
FIG. 8 shows Jacobian information of RBF neural network identifiers in a GNSS receiver adaptive code tracking loop to be sent to NNCs for weighting adjustment to achieve optimal control.
FIG. 9 is a schematic diagram of adaptive code tracking loop parameter self-tuning based on RBF neural network identification in a GNSS receiver, in which PID tuning parameter K can be adaptively tuned by using Jacobian information obtained from RBF neural network identification equipment in a code tracking system in a single neuron-based adaptive PID controller p ，K i And K d 。
When the moment of time reaches about 0.48s, these tuning parameters almost become constant.
Simulation results show that the method has good performance in a code tracking loop of the GNSS receiver, and particularly has strong robustness on nonlinear and time-varying characteristics based on a PID control strategy of a neural network. The proposed adaptive code tracking technology enables the GNSS receiver to have robustness, self-adaption and self-learning, and can adapt to work under different signal conditions.
In a word, the invention combines the traditional PID control strategy with the neural network algorithm to generate a novel PID controller with robustness and adaptability. The method utilizes the self-learning capability of the neural network, and can use a gradient descent method to automatically correct on-line and automatically modify robust PID parameters. The satellite navigation receiver tracking result shows that the code tracking loop technology has the advantages of strong adaptability and strong robustness.
The foregoing description of the preferred embodiments of the invention is not intended to be limiting, but rather is intended to cover all modifications, equivalents, and alternatives falling within the spirit and principles of the invention.
Claims (1)
1. The adaptive signal tracking method for the satellite navigation receiver is characterized by comprising the following steps of:
step 1: the input signal is subjected to closed-loop control through a PID controller; the specific mode of the step 1 is as follows:
step 1.1: calculating three independent parameters of an input signal, namely a proportional term P, an integral term I and a differential term D, defining u (t) as the output of the PID controller, and then obtaining an output equation in a time domain as follows:
in the formula (1), the input is e (t) =r (t) -y (t), which is the difference between the actual output y (t) and the reference input r (t) in the measurement process, namely, the error; k (K) p Representing the proportional gain; t (T) i Representing an integration time; t (T) d Representing differential time; k (K) i ＝H p /T i Is the integral gain; k (K) d ＝K p T d Is a differential gain; t is the current time;
when the backward difference method is adopted, the output of the incremental digital PID algorithm is as follows:
u[k]＝u[k-1]+K p {e[k]-e[k-1]+K i e[k]}+K d {e[k]-2e[k-1]+e[k-2]} (2)
step 1.2: the following transfer function is obtained according to equation (2):
modeling the PID controller through the transfer function of the formula (3) to realize PID control;
step 2: sending the signals output by the PID controller into an RBF neural network identifier, identifying a nonlinear system through the RBF neural network, and predicting Jacobian Jacobian matrix information of the controlled object in the system; the specific mode of the step 2 is as follows:
step 2.1: let x= [ X ] 1 ，x 2 ，…，x n ] T Is the input vector of RBF neural network identifier, RBF godNeurons in the second layer of the network are activated by radial basis functions, and the radial vector of the RBF neural network is denoted as h= [ H ] 1 ，h 2 ，…，h m ] T Wherein h is j J=1, 2, …, m is a multiple gaussian function, specifically:
in the formula (4) of the present invention, sign I representation of euclidean norms; c (C) j ＝[c j1 ，c j2 ，…，c jm ] T Is the center vector of the j-th node in the neural network; b j Is the radial width parameter of the node j in the hidden layer, b j > 0, the resulting base width vector b= [ B ] 1 ，b 2 ，…，b m ] T In a hidden layer of the neural network;
step 2.2: the weight vector of the RBF neural network is expressed as w= [ W ] 1 ，W 2 ，…W m ] T The output of the RBF neural network identifier is made up of a linear weighted sum of radial basis functions in the hidden layer as shown in equation (5):
in the formula (5), the subscript I indicates that the control system adopts an RBF neural network as an identifier, w j Representing a weight between the output neuron and the jth neuron of the hidden layer;
step 2.3: the performance function of the identification process is defined as follows:
step 2.4: to make the output y of RBF neural network identifier I (k) Error e between the actual output y (k) of the controlled object I (k) Minimum, gradient descent methodTo adjust the weight w j (k) Node center c of hidden layer ji (k) And radial width parameter b j (k) The method comprises the steps of carrying out a first treatment on the surface of the Specifically, iterative operation is performed by formulas (7) (8) (9):
w j (k)＝w j (k-1)+η[y(k)-y I (k)]h j +α[w j (k-1)-w j (k-2)](7)
b j (k)＝b j (k-1)+ηΔb j +α[b j (k-1)-b j (k-2)] (8)
c ji (k)＝c ji (k-1)+ηΔc ji +α[c ji (k-1)-c ji (k-2)] (9)
wherein:
in the above formula, η is the slope rate and α is the momentum factor;
step 2.5: acquiring Jacobian Jacobian matrix information, namely the sensitivity of controlled object output to control input, through an RBF neural network online identification process; the expression of Jacobian matrix information is as follows:
step 3: the signals output by the RBF neural network identifier are sent to a neural network controller, and the neural network controller adopts a self-adaptive PID controller based on a single neuron and carries out self-adjustment according to the system state so as to achieve the optimal control effect; the specific mode of the step 3 is as follows:
step 3.1: the output of the adaptive PID controller based on a single neuron corresponds to the NCO output in the GNSS receiver code tracking loop as shown in equation (13):
Δu(k)＝v 1 (k)xc 1 (k)+v 2 (k)xc 2 (k)+v 3 (k)xc 3 (k) (13)
in formula (13), xc i Is input, v i Is the corresponding weight of a single neuron, i=1, 2,3;
step 3.2: combining equation (2), the following equation is obtained:
step 3.3: by adjusting the weights v in an adaptive PID controller based on a single neuron i Square J of systematic error C (k) Decreasing to zero, i=1, 2,3; j (J) C (k) Is defined as follows:
in equation (16), e (k) =r (k) -y (k) is the error between the actual output y (k) of the controlled object and its currently required reference input r (k);
step 3.4: in order for an adaptive PID controller based on a single neuron to achieve optimal performance, the value of the weight is adjusted and updated online according to the Hebb criterion using the gradient descent method, the algorithm is as follows:
namely, it is equivalent to:
in the middle ofJacobian Jacobian matrix information, which is the object controlled in the system, can be obtained from equation (12).
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