CN116700002A - Piezoelectric actuator Hammerstein modeling and predictive control method based on deep neural network - Google Patents
Piezoelectric actuator Hammerstein modeling and predictive control method based on deep neural network Download PDFInfo
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Abstract
The invention discloses a piezoelectric actuator Hammerstein modeling and predictive control method based on a deep neural network, which comprises the steps of firstly defining a Hammerstein model of the piezoelectric actuator, and fitting the model by using a fully-connected neural network to obtain a discrete time state space model; secondly, a training set of input and output data of the piezoelectric actuator is established, and fitting identification is carried out on the model through training; secondly, designing a performance evaluation function based on the trained piezoelectric nonlinear dynamics model under the condition of considering state constraint and dynamics constraint, and obtaining optimal input by minimizing the performance evaluation function; finally, in order to reduce the complexity of online solving the optimal input, the online solving problem is approximated efficiently by using a deep neural network, so that the optimal input which can be realized offline is obtained, and the optimal input is deployed on a hardware platform to control the piezoelectric actuator in real time.
Description
Technical Field
The invention belongs to the technical field of precision electromechanical control, and relates to a piezoelectric actuator Hammerstein modeling and predictive control method based on a deep neural network.
Background
In recent years, piezoelectric actuators (PEAs) have been widely used in various fields such as microscopic imaging, beam direction adjustment, additive manufacturing, and the like, due to their advantages of high precision, fast response speed, no heat generation, high rigidity, and the like, and are often used for accurately tracking a given reference trajectory. For example, PEA-driven samples enable super-resolution scanning under an atomic force microscope. However, because of the inherent hysteresis nonlinearity, especially in tracking high frequency signals, the hysteresis relationship between the control input and the displacement output can have a great influence on the positioning accuracy of PEA, so how to accurately describe the nonlinearity hysteresis characteristic of the piezoelectric actuator has an important meaning for realizing high-accuracy positioning of PEA, but the work still has a great challenge because the nonlinearity hysteresis characteristic is closely related to the amplitude and frequency of the working point and the input history information. In addition, model predictive control is used as an advanced control strategy, and can calculate the optimal input of a system on the premise of meeting the constraint condition of the system, and is widely used in the fields of process control, unmanned aerial vehicle control, unmanned automobile and the like at present, and the application of the model predictive control to the precise control of a piezoelectric actuator by a technical means is also a research hotspot, however, the model predictive control has higher requirements on the accuracy of a system model and on computational resources on the other hand, so that how to obtain an optimal solution of the optimization problem by utilizing limited computational resources on the premise of obtaining the precise model of the piezoelectric actuator becomes a problem to be solved urgently at present.
Disclosure of Invention
In order to solve the problem that the conventional piezoelectric actuator cannot realize high-precision track tracking in a wide frequency range and a large travel range, the invention provides a Hammerstein modeling and predictive control method for the piezoelectric actuator based on a deep neural network. Firstly, defining a Hammerstein model of a piezoelectric actuator, wherein the model consists of a linear time-invariant model and a static nonlinear hysteresis model, and fitting the Hammerstein model by using a fully-connected neural network to obtain a discrete time state space model; secondly, a training set for input and output data of a piezoelectric actuator of the fully-connected neural network is established, and fitting identification is carried out on the Hammerstein model through training; secondly, designing a performance evaluation function based on the trained piezoelectric nonlinear dynamics model under the condition of considering state constraint and dynamics constraint, and obtaining optimal input by minimizing the performance evaluation function; finally, in order to reduce the complexity of the online solving of the optimal input, the online solving process is approximated efficiently by using a deep neural network, so that the offline solving of the optimal input is realized, and the optimal input is deployed on a hardware platform to control the piezoelectric actuator in real time.
In order to achieve the above purpose, the technical scheme of the invention mainly comprises the following steps:
step 1: establishing a nonlinear dynamics model of the piezoelectric actuator based on a Hammerstein model based on a deep neural network;
step 2: the data acquisition system is used for acquiring data of input voltage and output displacement of the piezoelectric actuator, and then partial data are selected as a training set and training is carried out on the Hammerstein model;
step 3: describing a discrete time state space model of the piezoelectric actuator system based on the trained Hammerstein model, further constructing a performance evaluation function for predictive control and constraint conditions of a dynamic model, and obtaining required optimal input by minimizing the performance evaluation function;
step 4: and (3) approximating the solving process of the optimal input in the step (3) by using a deep neural network, thereby realizing the offline solving of the optimal input, greatly improving the solving speed of the model predictive control problem and ensuring the feasibility of the model predictive control problem in a closed-loop feedback experiment.
The invention has the advantages that:
1. the Hammerstein model established based on the deep neural network is a model which integrates a physical mechanism and a deep learning idea, solves the problem of a black box in deep learning to a certain extent, ensures that the model has a certain interpretability, and can more accurately describe the nonlinear characteristic dynamic characteristics of the piezoelectric actuator.
2. The method has the advantages that the model predictive control solving problem is approximated by using the deep neural network, so that the solving speed is greatly improved, and the model predictive control is enabled to be feasible for a real-time feedback system, thereby being successfully applied to the precise control of the piezoelectric actuator.
Drawings
FIG. 1 is a flow chart of a method for modeling and predictive control of a piezoelectric actuator based on a deep neural network.
FIG. 2 is a block diagram of a deep neural network based piezoelectric actuator Hammerstein modeling architecture of the method of the present invention.
Detailed Description
The invention will now be further described with reference to specific examples, figures:
the invention discloses a piezoelectric actuator Hammerstein modeling and predictive control method based on a deep neural network, which specifically comprises the following steps according to a flow diagram of FIG. 1:
a piezoelectric displacement platform P16.XY80S with two degrees of freedom developed by the Ming-Tian science and technology company of the Harbin core, a matched driving controller and PXIe-4300 and PXIe-6738 data acquisition board cards produced by the NI company are selected to serve as a platform for experimental data acquisition and verification test.
Step 1: establishing a nonlinear dynamics model of the piezoelectric actuator based on a Hammerstein model based on a deep neural network;
first, the input voltage and output displacement of the piezoelectric actuator at the k moment are known to be u (k) and y (k), respectively, and a Hammerstein model of the piezoelectric actuator is defined, and the Hammerstein model consists of a linear time-invariant model and a static nonlinear hysteresis model:
h(k)＝f(u(k),y(k-1),u(k-1),...)
where h (k) =f (u (k), y (k-1), u (k-1), and.+ -.) is a static nonlinear hysteresis model, a (z) and B (z) satisfy an n-th order polynomial,
A(z)＝1+a 1 z -1 +...+a n z -n ，
B(z)＝b 1 z -1 +...+b n z -n
wherein ,z-k Representing a delay of k sampling periods, a i and bi The model coefficients for the linear time invariant model of the piezoelectric actuator, n is the order of the linear time invariant model, where i=1, 2.
Next, the Hammerstein model is fitted and identified by using a fully connected neural network, and nonlinear fitting is performed on the static nonlinear hysteresis model part by using the neural network as shown in fig. 2. First, an input sequence (x (k), u (k)) of the fully-connected neural network is defined, where x (k) represents a set consisting of input voltages and output displacements at different times of the piezoelectric actuator:
x(k)＝[y(k),...,y(k-n+1),u(k-1),...,u(k-n+1)] T ,
and defining an intermediate static nonlinear hysteresis output sequence of the fully-connected neural network:
wherein , and />Respectively representing the intermediate static nonlinear hysteresis output at the kth moment and the intermediate static nonlinear hysteresis output at the kth-n+1 moment.
Input sequence (x (k), u (k)) and intermediate static nonlinear hysteresis output sequenceThe mapping of (c) can be expressed as:
wherein ,φh (-) is an affine function combination of the neural network, θ h For affine function parameters that need to be optimized.
Based on static nonlinear hysteresis prediction output, the prediction output under the linear time-invariant model can be further representedThe method comprises the following steps:
wherein ,ai and bi Model coefficients for a linear time-invariant model of a piezoelectric actuator, where i=1, 2.
Step 2: the input voltage and the output displacement of the piezoelectric actuator are subjected to data acquisition through a data acquisition system, partial data are selected as a training set, and the Hammerstein model is trained; the method comprises the following steps:
firstly, carrying out parameter identification on a linear time-invariant model, utilizing a frequency response analyzer PSM1700 to generate a fixed 0.2V small-amplitude sweep voltage signal for a piezoelectric displacement platform P16.XY80S, carrying out frequency sweep of a frequency range of 1Hz to 6kHz on a piezoelectric actuator, then carrying out parameter identification on the piezoelectric actuator by utilizing a system identification kit of MATLAB, thereby obtaining a proper model order n, and obtaining n=4 through calculation;
next, in order to perform fitting identification on the Hammerstein model by using the fully connected neural network, a section of sinusoidal signal with amplitude ranging from 0.1V to 8V randomly and frequency ranging from 1Hz to 500Hz randomly needs to be designed as input voltage of the piezoelectric actuator, and the input voltage and the output displacement of the sinusoidal signal are subjected to data acquisition by using a PXIe-4300 and PXIe-6738 real-time data acquisition system, so that a data set for the fully connected neural network is constructed:
where x (k), u (k) is the input of the dataset,for the output of the dataset, +.>Is the training set length.
Selecting a partial data setTraining the Hammerstein model based on the fully connected neural network as a training set, wherein +.>For the predicted output of the neural network at the time k+1, the trained optimization objective function is as follows:
and needs to satisfy:
in the above-mentioned problems, in the above-mentioned,for training set length, phi H (-) is an affine function combination of the neural network, θ H For affine function parameters that need to be optimized.
Step 3: describing a discrete time state space model of the piezoelectric actuator system based on a trained Hammerstein model, further constructing a performance evaluation function and dynamic constraint conditions for predictive control, and obtaining required optimal input by minimizing the performance evaluation function; the method comprises the following steps:
first, the Hammerstein model is described as a discrete-time state space model:
x(k+1)＝φ(x(k),u(k)),
y(k)＝Cx(k)
wherein phi (-) is a nonlinear state space model,
constructing a performance evaluation function of predictive control:
wherein ,indicating the prediction of future N at time k p An input voltage set of time steps, where y (i|k) represents the output displacement of i time steps after the k moment, r (i|k) represents the reference signal of i time steps after the k moment, Δu (k) =u (k) -u (k-1) represents the increment of the input voltage, and β is a weight constant.
The objective function with dynamic constraints is expressed as:
u * (·|k)＝argminJ(u(·|k))
the method meets the following conditions:
x(0|k)＝x(k)
u(-1|0)＝u(k-1)
y(i|k)∈[y min ,y max ]
u(i|k)∈[u min ,u max ]
wherein ,ymin ,y max Respectively, the minimum value and the maximum value of the output displacement, u min ,u max The minimum value and the maximum value of the input voltage are respectively, and the optimal control input u can be obtained by minimizing the performance cost function * (·|k)。
The nonlinear model predictive control problem is solved on line through a nlmpmove function of MATLAB, so that the optimal control input u can be obtained * (·|k)。
Step 4: approximating the optimal input solving process in the step 3 by using a deep neural network, thereby realizing the offline solving of the optimal input; the method comprises the following steps:
first, a dataset for optimal input approximation is constructed:
wherein s (k) = [ x ] T (k),r(k+1),...,r(k+N)] T R (k+N) is a reference signal to be tracked for N time steps in the future at time k, u * (0|k) represents the optimal input at time k, being the output of the dataset.
Next, an optimization objective function of the deep neural network is constructed:
the method meets the following conditions:
wherein ,φP (-) is an affine function combination of the neural network, θ P In order to require optimized affine function parameters,is the training set length.
The method has the advantages that an approximation strategy for offline solving of optimal input is obtained, the online calculation speed of a model predictive control algorithm is greatly increased, the algorithm is conveniently used for closed loop feedback experiments of piezoelectric actuators, the algorithm is deployed on an FPGA development board of ZynqUltraScale+MPSoC ZCU106, and relevant hardware in-loop experiment verification is carried out.
It will be readily appreciated by those skilled in the art that the foregoing description is merely a preferred embodiment of the invention and is not intended to limit the invention, but any modifications, equivalents, improvements or alternatives falling within the spirit and principles of the invention are intended to be included within the scope of the invention.
Claims (5)
1. A piezoelectric actuator Hammerstein modeling and predictive control method based on a deep neural network is characterized by comprising the following steps:
step 1: establishing a nonlinear dynamics model in the form of a piezoelectric actuator Hammerstein based on a deep neural network;
step 2: the data acquisition system is used for acquiring data of input voltage and output displacement of the piezoelectric actuator, and then partial data are selected as a training set and training is carried out on the Hammerstein model;
step 3: describing a discrete time state space model of the piezoelectric actuator based on a trained Hammerstein model, further constructing a performance cost function and dynamic constraint conditions for predictive control, and obtaining required optimal input by minimizing the performance cost function;
step 4: and (3) approximating the optimal input solving process in the step (3) by using a deep neural network, so as to realize the off-line solving of the optimal input in the step (3).
2. The deep neural network-based piezoelectric actuator Hammerstein modeling and predictive control method according to claim 1, wherein the method comprises the following steps:
step 1: establishing a nonlinear dynamics model in the form of a piezoelectric actuator Hammerstein based on a deep neural network; the method comprises the following steps:
(1) The input voltage and output displacement of the piezoelectric actuator at the moment k are known as u (k) and y (k), respectively, and a Hammerstein model of the piezoelectric actuator is defined, wherein the Hammerstein model consists of a linear time-invariant model and a static nonlinear hysteresis model:
h(k)＝f(u(k),y(k-1),u(k-1),...)
where h (k) =f (u (k), y (k-1), u (k-1), and.+ -.) is a static nonlinear hysteresis model, a (z) and B (z) satisfy an n-th order polynomial,
wherein ,z-k Representing a delay of k sampling periods, a i and bi Model coefficients for a linear time-invariant model of a piezoelectric actuator, n being the order of the linear time-invariant model, where i=1, 2,n；
(2) Performing fitting identification on the Hammerstein model in the step (1) by using a fully connected neural network, and performing nonlinear fitting on a static nonlinear hysteresis model part by using the neural network; first, an input sequence (x (k), u (k)) of the fully-connected neural network is defined, where x (k) represents a set consisting of input voltages and output displacements at different times of the piezoelectric actuator:
x(k)＝[y(k),...,y(k-n+1),u(k-1),...,u(k-n+1)] T ,
and defining an intermediate static nonlinear hysteresis output sequence of the fully-connected neural network:
wherein , and />Respectively represents the intermediate static nonlinear hysteresis output at the kth moment and the intermediate static nonlinear hysteresis output at the kth-n+1 moment,
input sequence (x (k), u (k)) and intermediate static nonlinear hysteresis output sequenceThe mapping of (c) can be expressed as:
wherein ,φh (-) is an affine function combination of the neural network, θ h Affine function parameters to be optimized;
based on static nonlinear hysteresis prediction output, the prediction output under the linear time-invariant model can be further representedThe method comprises the following steps:
wherein ,ai and bi Model coefficients for a linear time-invariant model of a piezoelectric actuator, where i=1, 2.
3. The deep neural network-based piezoelectric actuator Hammerstein modeling and predictive control method according to claim 2, wherein the method comprises the following steps:
step 2: the input voltage and the output displacement of the piezoelectric actuator are subjected to data acquisition through a data acquisition system, partial data are selected as a training set, and the Hammerstein model is trained; the method comprises the following steps:
(1) Carrying out parameter identification on the linear time-invariant model, generating a sweep frequency voltage signal with a fixed small amplitude by utilizing a frequency response analyzer to sweep the frequency of the piezoelectric actuator, and then carrying out parameter identification to obtain a proper model order n;
(2) In order to utilize the fully-connected neural network to perform fitting identification on the Hammerstein model, a section of sine signal with amplitude and frequency randomly changed is designed to serve as input voltage of the piezoelectric actuator, and data acquisition is performed on the input voltage and output displacement of the sine signal through a real-time data acquisition system, so that a data set for the fully-connected neural network is constructed:
where x (k), u (k) is the input of the dataset,for the output of the dataset, +.>Is the length of the training set;
(3) Selecting a partial data setTraining the Hammerstein model based on the fully connected neural network as a training set, wherein +.>For the predicted output of the neural network at time k+1, the objective function of the training network is:
and needs to satisfy:
wherein ,for training set length, phi H (-) is an affine function combination of the neural network, θ H For affine function parameters that need to be optimized.
4. The deep neural network-based piezoelectric actuator Hammerstein modeling and predictive control method according to claim 3, wherein:
step 3: describing a discrete time state space model of the piezoelectric actuator based on a trained Hammerstein model, further constructing a performance evaluation function and dynamic constraint conditions for predictive control, and obtaining required optimal input by minimizing the performance evaluation function; the method comprises the following steps:
(1) Describing the Hammerstein model as a discrete-time state space model:
x(k+1)＝φ(x(k),u(k)),
y(k)＝Cx(k)
wherein phi (-) is a nonlinear state space model,
(2) Constructing a performance evaluation function of predictive control:
wherein ,indicating the prediction of future N at time k p A set of input voltages for each time step, wherein y (i|k) represents the output displacement for i time steps after the k moment, r (i|k) represents the reference signal for i time steps after the k moment, Δu (k) =u (k) -u (k-1) represents the increment of the input voltage, and β is a weight constant;
the objective function with dynamic constraints is expressed as:
u * (·|k)＝argminJ(u(·|k))
the method meets the following conditions:
x(0|k)＝x(k)
u(-1|0)＝u(k-1)
y(i|k)∈[y min ,y max ]
u(i|k)∈[u min ,u max ]
wherein ,ymin ,y max Respectively, the minimum value and the maximum value of the output displacement, u min ,u max Respectively minimum and maximum values of the input voltage, and obtaining the optimal control input u by minimizing the performance cost function * (·|k)。
5. The deep neural network-based piezoelectric actuator Hammerstein modeling and predictive control method according to claim 4, wherein the method comprises the following steps:
step 4: approximating the optimal input solving process in the step 3 by using a deep neural network, thereby realizing the offline solving of the optimal input; the method comprises the following steps:
(1) Constructing a dataset for optimal input approximation:
wherein s (k) = [ x ] T (k),r(k+1),...,r(k+N)] T For the input of the data set, r (k+N) is the reference signal to be tracked for the next N time steps at time k, u * (0|k) representing the optimal input at time k as the output of the dataset;
(2) Constructing an optimized objective function of the deep neural network:
the method meets the following conditions:
wherein ,φP (-) is an affine function combination of the neural network, θ P In order to require optimized affine function parameters,is the training set length.
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