A data-driven adaptive control method for a piezoelectric micro-motion platform based on the play operator

Abstract

The invention discloses a data-driven adaptive control method for a piezoelectric micro-motion platform based on the play operator, belonging to the technical field of control engineering. The method of the invention considers the obvious hysteretic nonlinearity of the piezoelectric micro-motion platform itself, and introduces an exogenous variable function, namely a weighted superposition expression of multiple play operators, to describe the hysteretic nonlinearity characteristic. A data-driven adaptive control method based on the play operator is designed only relying on the input-output data of the system. It can more accurately describe the hysteretic nonlinearity of the piezoelectric micro-motion platform, and can avoid the influence of the complex hysteretic nonlinearity of the platform itself and the model accuracy on the effectiveness of the controller. The invention considers the typical hysteretic nonlinearity characteristic of the piezoelectric micro-motion platform under the framework of the data-driven control method, reduces the complexity of the pseudo partial derivative matrix estimation, lightens the parameter estimation burden of the improved projection algorithm, further improves the control accuracy of the controller, and enables the piezoelectric micro-motion platform to have a good trajectory tracking effect.

Description

Technical Field

[0001] The present invention belongs to the technical field of control engineering, and more specifically, relates to a data-driven adaptive control method for a piezoelectric micro-motion platform based on the play operator. Background Technique

[0002] With the continuous development and progress of science and technology, the demand for high-precision positioning control in high-tech fields is increasing. Traditional materials and their actuators cannot meet the requirements of high-precision positioning control. Therefore, domestic and foreign scholars tend to choose piezoelectric actuators, which are intelligent material actuators with large mechanical force, strong driving ability, fast response time, and high frequency.

[0003] When performing an open-loop test on a piezoelectric micro-motion platform, there are complex input-output relationships such as hysteresis nonlinearity, rate-dependent characteristics, creep characteristics, and initial periodic displacement residuals between the input voltage signal and the output displacement signal. This complex nonlinear relationship will have a very large impact on the positioning accuracy of the piezoelectric micro-motion platform. Therefore, it brings great challenges to the modeling and control of the piezoelectric micro-motion platform.

[0004] In order to achieve the purpose of precise trajectory tracking control of the piezoelectric micro-motion platform, the traditional method is to first model the piezoelectric micro-motion platform and then establish a model-based controller. However, due to the complex input-output characteristics of the piezoelectric micro-motion platform, its model establishment is very complex, and the modeling accuracy will also have a great impact on the control accuracy, making it difficult to achieve high-precision positioning control. Summary of the Invention

[0005] In view of the dependence of the existing control method of the piezoelectric micro-motion platform on the model and the influence of the hysteresis nonlinear characteristic on the control accuracy, the present invention provides a data-driven adaptive control method for a piezoelectric micro-motion platform based on the play operator. The purpose is to only use the input-output data of the piezoelectric micro-motion platform to avoid the influence of the modeling accuracy on the control accuracy. Therefore, a data-driven adaptive control method that does not require any information on the mechanism model and mathematical model of the piezoelectric micro-motion platform is adopted. Considering the obvious hysteresis nonlinearity of the piezoelectric micro-motion platform, the parameter estimation burden is reduced, thereby improving the control accuracy and enabling the output signal of the piezoelectric micro-motion platform to accurately track the desired signal of the system.

[0006] The technical solution adopted by the present invention is as follows:

[0007] A data-driven adaptive control method for a piezoelectric micro-motion platform based on the play operator, the specific steps are as follows:

[0008] Step 1: Introduce multiple play operator weighted superposition expressions as exogenous variable functions and combine the full-format dynamic linearization method to avoid the influence of the complex hysteresis nonlinearity of the platform itself on the control accuracy, and equivalently transform the piezoelectric micro-motion platform into a dynamic linearization data model with play operators.

[0009] Step 2: Establish a control input criterion function and obtain a data-driven control algorithm using the idea of optimization.

[0010] Step 3: Use an improved projection algorithm for online estimation when estimating the pseudo-partial derivative matrix.

[0011] Step 4: Integrate Step 2 and Step 3, introduce a reset algorithm to strengthen the time-varying tracking ability of the parameters, and obtain a data-driven adaptive control method for the piezoelectric micro-motion platform based on play operators.

[0012] The specific process of Step 1 is as follows:

[0013] Describe the piezoelectric micro-motion platform in the form of a general single-input single-output discrete-time nonlinear system as follows:

[0014] y(k + 1) = f(y(k), (k - 1), …, y(k - n y ), u(k - 1), …, u(k - n u )) (1)

[0015] where y(k) ∈ R and u(k) ∈ R are the output signal and input signal of the system at time k, respectively, n y , n u ∈ Z + are the orders of the system output signal and input signal, and f(·) is the nonlinear function used to describe the piezoelectric micro-motion platform.

[0016] For the piezoelectric micro-motion platform, the partial derivatives of the system (1) with respect to all variables are continuous and satisfy the following generalized Lipschitz condition:

[0017]

[0018] where M ∈ Z + , x m > 0 are the variables of the system, the Lipschitz constant ζ m > 0, and m ∈ {1, …, M}.

[0019] When ||△J(k)|| ≠ 0, there must exist pseudo-partial derivatives ξ i (k), i = 1, …, L y + L u+n, such that the system (1) can be equivalently represented as the following dynamic linearization data model with a play operator:

[0020]

[0021] where Ξ p (k) ∈ R n , L(k) = [l1(k), l2(k), …, l n (k)] T , L(k) ∈ R n , ΔL(k) = L(k) - L(k - 1), at any time k, ξ i (k), i = 1, …, L y +L u +n are all bounded, |ξ i (k)| ≤ β, where β is a constant, l i (k), i = 1, 2, …, n are the values of the play operator.

[0022] The following play operator expression is selected:

[0023] l i (k) = max{u(k) - r i , min{u(k), l i (k - 1)}}, i = 1, 2, … n (4)

[0024] where the number of play operators is n, and the threshold is r i , and the threshold expression is The expected displacement signal of the system at time k is y d (k).

[0025] For the convenience of subsequent control law design, equation (4) can be rewritten as the following expression:

[0026]

[0027] where the threshold expression is defined as u m is a constant close to the maximum value of y d (k).

[0028] Equation (5) can be rewritten as:

[0029] l i (k) = d i (k)Δu(k) + a i (k - 1) (6)

[0030] Among them,

[0031]

[0032] Define \(A(k)=[a_1(k),a_2(k),\cdots,a n (k)] T , \(A(k)\in R n , \(D(k)=[d_1(k),d_2(k),\cdots,d n (k)] T , \(D(k)\in R n , then there is:

[0033] L(k)=D(k)\Delta u(k)+A(k - 1)\ (7)

[0034] The specific process of step two is as follows:

[0035] To avoid the adverse effects of the rapid change of the control quantity \(u(k)\) on the system, the control input criterion function of the system is defined as follows:

[0036] J(\Delta u(k))=(y d (k + 1)-y(k + 1)) 2 +k(u(k)-u(k - 1)) 2 \ (8)

[0037] Among them, \(k\gt0\) is the weight factor.

[0038] Combined with the dynamic linearization data model with the play operator and taking the derivative of the control quantity \(u(k)\), let The control law can be obtained as follows:

[0039]

[0040] Among them, in order to make the control algorithm universal, a step size factor \(\theta\) is added to the control law (9) i \in(0,1],i = 0,1,\cdots,L y +L u +n.

[0041] The specific process of step three is as follows:

[0042] Define Then equation (3) can be written as the following expression:

[0043] y(k + 1)=y(k)+\Xi T (k)\Delta J(k)\ (10)

[0044] Among them, \Delta J(k)=J(k)-J(k - 1),\Xi(k), and Ξ p (k) are both pseudo-partial derivative matrices.

[0045] To implement the control law (9), consider obtaining the value of the time-varying parameter Ξ(k) using the input-output data of the system, and estimating the value used to describe the hysteresis characteristics of the system. The algorithm criterion function for designing the estimator is:

[0046]

[0047] where τ > 0 is the weight factor.

[0048] Taking the extreme value of equation (11) with respect to Ξ(k), the estimate of Ξ(k) can be obtained as:

[0049]

[0050] where is the estimated value of Ξ(k). To make the control algorithm universal, a step size factor

[0051] The specific process of step four is as follows:

[0052] To make the estimation algorithm of the pseudo-partial derivative matrix Ξ(k) have a stronger tracking ability for time-varying parameters, a reset algorithm mechanism is introduced as follows:

[0053] If or ||△J(k)|| ≤ σ, or when, then make where σ is a very small positive number.

[0054] Combined with the estimation algorithm of the pseudo-partial derivative matrix Ξ(k), a data-driven adaptive control method for a piezoelectric micro-motion platform based on the play operator is as follows:

[0055]

[0056] where is the estimated value of, is the estimated value of Ξ p (k).

[0057] The beneficial effects of the present invention are:

[0058] The present invention provides a data-driven adaptive control method for a piezoelectric micro-motion platform based on the play operator, that is, the high-precision positioning control of the piezoelectric micro-motion platform is realized by using the input voltage and output displacement data of the piezoelectric micro-motion platform over a period of history, avoiding the influence of unmodeled dynamics and model accuracy on the control effect. At the same time, the typical hysteresis nonlinear characteristics of the piezoelectric micro-motion platform are considered within the framework of the data-driven control method, making the dynamic linearization data model have a certain physical meaning, reducing the complexity of the pseudo-partial derivative matrix and the estimation burden, and improving the control accuracy and control effect of the controller. Description of the Drawings

[0059] Figure 1 It is an experimental device diagram of the piezoelectric micro-motion platform control system of the present invention;

[0060] Figure 2 It is the relationship between the input voltage signal with different frequencies and the output actual displacement signal of the piezoelectric micro-motion platform under the open-loop test of the present invention;

[0061] Figure 3 It is the working principle diagram of the piezoelectric micro-motion platform of the present invention;

[0062] Figure 4 It is the block diagram of the data-driven adaptive control system of the piezoelectric micro-motion platform based on the play operator of the present invention;

[0063] Figure 5 It is the output displacement result curve of the system under the step signal of the present invention;

[0064] Figure 6 It is the output displacement result curve of the system under the 1Hz sine wave input signal of the present invention;

[0065] Figure 7 It is the output displacement result curve of the system under the 20Hz sine wave input signal of the present invention;

[0066] Figure 8 It is the output displacement result curve of the system under the 50Hz sine wave input signal of the present invention;

[0067] Figure 9 It is the output displacement result curve of the system under the 100Hz sine wave input signal of the present invention;

[0068] Figure 10 It is the output displacement result curve of the system under the mixed wave signal of the present invention;

[0069] Figure 11 It is the tracking error curve of the system under the step signal of the present invention;

[0070] Figure 12This is the system tracking error curve of the present invention under a 1 Hz sine wave input signal;

[0071] Figure 13 This is the system tracking error curve of the present invention under a 20 Hz sine wave input signal;

[0072] Figure 14 This is the system tracking error curve of the present invention under a 50 Hz sine wave input signal;

[0073] Figure 15 This is the system tracking error curve of the present invention under a 100 Hz sine wave input signal;

[0074] Figure 16 This is the system tracking error curve of the present invention under a mixed wave signal; Detailed implementation manners

[0075] The present invention designs a data-driven adaptive control method for a piezoelectric micro-motion platform based on the play operator. First, the piezoelectric micro-motion platform is described in the form of a general single-input single-output discrete-time nonlinear system. Multiple play operator weighted superposition expressions are introduced as exogenous variable functions to describe the hysteresis nonlinearity of the piezoelectric micro-motion platform. Combining with the full-format dynamic linearization data model, the piezoelectric micro-motion platform is equivalently transformed into a dynamic linearization data model with play operators, making the data model have partial physical meanings. Then, the input criterion function is reasonably set, and the optimal idea is adopted to obtain a data-driven adaptive control algorithm based on the platform dynamic linearization data model. Considering that the pseudo partial derivative matrix in the dynamic linearization data model is unknown, an improved projection algorithm is used to realize the online estimation of the pseudo partial derivative parameters, and the control law of the data-driven adaptive control method based on the play operator can be obtained. At the same time, the hysteresis characteristics of the platform can also be estimated. Since the exogenous variable function, that is, multiple play operator weighted superposition expressions, is introduced, the complexity of the behavior of the pseudo partial derivative matrix is greatly reduced, and the control effect of the control method is effectively improved. Finally, in order to make the estimation algorithm of the pseudo partial derivative matrix have stronger tracking ability for time-varying parameters, a reset algorithm mechanism is introduced into the data-driven adaptive control method based on the play operator.

[0076] The following further elaborates the present invention in detail with reference to the accompanying drawings and specific embodiments.

[0077] The experimental device diagram of the piezoelectric micro-motion platform control system of the present invention is as Figure 1 shown, which includes:

[0078] A computer (model: Dell Vostro 3900) that has completed the writing, debugging, and experimentation of the control algorithm, with the controlled object being a piezoelectric micro-motion platform (model: MPT-2MRL102A), a multi-functional data acquisition card (model: PCI-1716) that collects experimental signals and performs A / D and D / A conversions on the collected signals, an integrated controller (model: PPC-2CR0120) that integrates a voltage amplification module and a displacement sensing module, and a vibration isolation table (model: SPFO-I-B-750-750) that can reduce external environmental interference.

[0079] An open-loop test was conducted on the piezoelectric micro-motion platform. The experimental results showed that the relationship between the input voltage signal and the output displacement signal under sinusoidal input signals with amplitudes of 36μm and frequencies of 1Hz, 10Hz, 20Hz, 50Hz, and 100Hz is as Figure 2 shown. It can be seen that the platform has extremely strong and complex nonlinearities, including hysteresis nonlinearity, rate-dependent characteristics, initial periodic displacement residuals, etc. The most obvious one is hysteresis nonlinearity. Therefore, only the input voltage signal and the output displacement signal within a certain period of the system's history are considered, and combined with the hysteresis nonlinearity of the piezoelectric micro-motion platform, a data-driven control algorithm is designed.

[0080] To achieve high-precision positioning control of the piezoelectric micro-motion platform, a control system as Figure 1 shown is used to perform closed-loop control on the piezoelectric micro-motion platform. The control process is as follows:

[0081] The desired displacement signal and the control algorithm are written in the MATLAB / Simulink software of the computer. At this time, the signal output is a digital signal, which is converted to an analog voltage signal through the D / A conversion of the multi-functional data acquisition card. Then, the signal is amplified 15 times by the voltage amplification module of the integrated controller to drive the deformation of the piezoelectric actuator in the piezoelectric micro-motion platform. The actual displacement signal of the platform is collected by the displacement sensor of the piezoelectric micro-motion platform, and the collected displacement signal is reduced by 6 times through the displacement sensing module of the integrated amplifier and transmitted to the multi-functional data acquisition card, and then converted to a digital signal through A / D conversion and sent back to the computer. The working principle diagram of the piezoelectric micro-motion platform is as Figure 3 shown.

[0082] The specific control method designed and used for the piezoelectric micro-motion platform is a data-driven adaptive control method based on the play operator. Its control system block diagram is as Figure 4 shown, including a data-driven adaptive control algorithm, a play operator, and an improved projection algorithm.

[0083] The piezoelectric micro-motion platform is described in the form of a general single-input single-output discrete-time nonlinear system as follows:

[0084] y(k + 1) = f(y(k), (k - 1), …, y(k - n y ), u(k - 1), …, u(k - n u )) (1)

[0085] where y(k) ∈ R and u(k) ∈ R are the output signal and input signal of the system at time k, respectively, and n y , n u ∈ Z + is the order of the system output signal and input signal, and f(·) is a nonlinear function used to describe the piezoelectric micro - motion platform.

[0086] For the piezoelectric micro - motion platform, the system (1) can be equivalently transformed into the following dynamic linearization data model with a play operator:

[0087]

[0088] where Ξ p (k) ∈ R n , L(k) = [l1(k), l2(k), …, l n (k)] T , L(k) ∈ R n , △L(k) = L(k) - L(k - 1), and at any time k, ξ i (k), i = 1, …, L y + L u + n are bounded, |ξ i (k)| ≤ β, where β is a constant, and h i (k), i = 1, 2, …, n are the values of the play operator; L y , L u is the pseudo - order of the system.

[0089] Select the play operator expression as follows:

[0090] l i (k) = max{u(k) - r i , min{u(k), l i (k - 1)}}, i = 1, 2, … n (3)

[0091] where the number of play operators is n, the threshold is r i , and the threshold expression is the expected displacement signal of the system at time k is y d (k).

[0092] For the convenience of subsequent control law design, Equation (3) can be rewritten as the following expression:

[0093]

[0094] Among them, the threshold expression is defined as u m is a constant close to the maximum value of y d (k).

[0095] Equation (4) can be rewritten as:

[0096] l i (k) = d i (k)△u(k) + a i (k - 1) (5)

[0097] Among them,

[0098]

[0099] Define A(k) = [a1(k), a2(k), …, a n (k)] T , A(k) ∈ R n , D(k) = [d1(k), d2(k), …, d n (k)] T , D(k) ∈ R n , then there is:

[0100] L(k) = D(k)△u(k) + A(k - 1) (6)

[0101] Define Then equation (2) can be written as the following expression:

[0102] y(k + 1) = y(k) + Ξ T (k)△J(k) (7)

[0103] Among them, △J(k) = J(k) - J(k - 1), Ξ(k), and Ξ p (k) are both pseudo - partial derivative matrices.

[0104] To avoid the adverse effects of the rapid change of the control variable u(k) on the system, the control input criterion function of the system is defined as follows:

[0105] J(△u(k)) = (y d (k + 1) - y(k + 1)) 2 + k(u(k) - u(k - 1)) 2 (8)

[0106] Among them, k > 0 is the weight factor.

[0107] Derive the control variable u(k) by combining with the dynamic linearized data model with the play operator, and let The data-driven adaptive control law can be obtained as follows:

[0108]

[0109] Among them, in order to make the control algorithm universal, a step size factor θ is added to the control law (9) i ∈(0, 1], i = 0, 1, …, L y +L u +n.

[0110] To implement the control law (9), consider obtaining the value of the time-varying parameter Ξ(k) by using the input-output data of the system (1), and estimate the value used to describe the hysteresis characteristics of the system. The algorithm criterion function for designing the estimator is:

[0111]

[0112] Among them, τ > 0 is the weight factor.

[0113] Taking the extreme value of equation (10) with respect to Ξ(k), the estimate of Ξ(k) can be obtained as:

[0114]

[0115] Among them, is the estimated value of Ξ(k). In order to make the control algorithm universal, a step size factor is added to the estimation algorithm

[0116] Combined with the estimation algorithm of the pseudo partial derivative matrix Ξ(k), a data-driven adaptive control method for a piezoelectric micro-motion platform based on the play operator is as follows:

[0117]

[0118] Among them, is the estimated value of is the estimated value of Ξ p (k).

[0119] In order to make the estimation algorithm of the pseudo partial derivative matrix Ξ(k) have stronger tracking ability for time-varying parameters, a reset algorithm mechanism is selected as follows:

[0120] If or ||△J(k)|| ≤ σ, or when, then make Among them, σ is a very small positive number.

[0121] Next, verify the feasibility and effectiveness of the data-driven adaptive control method of the piezoelectric micro-motion platform based on the play operator through Figure 1 the experimental device shown.

[0122] Write the control algorithm in the computer with MATLAB / Simulink software, and give different desired displacement signals each time, namely step signals, sine wave signals of 1 Hz, 20 Hz, 50 Hz, 100 Hz, and mixed wave signals.

[0123] Among them, the step signal is 12 μm, the amplitude of the sine signal is 36 μm, and the mixed wave signal is y d (k)=13.2sin(10π - 1.7)+6sin(2π + 4)+19.2.

[0124] The parameter settings of the data-driven adaptive control method based on the play operator are: θ1 = 0.1, θ2 = 0.5, θ3 = 0.7, θ4 = 0.3, θ5 = 0.2, θ6 = 0.1, θ7 = 0.1, θ8 = 0.1, n = 4, u m =18, L u =2, L y =2, k = 0.5, τ = 0.1, and the initial value of the pseudo partial derivative matrix is set to L(1)=[0.4 0.35 0.3 0.3] T ,

[0125] As Figure 5 shown, the output displacement result curve of the system of the present invention under the step signal, as Figure 6 , Figure 7 , Figure 8 and Figure 9 are respectively the output displacement result curves of the system of the present invention under the sine wave input signals of 1 Hz, 20 Hz, 50 Hz, and 100 Hz, Figure 10 and

[0126] As Figure 11 shown, the tracking error curve of the system of the present invention under the step signal, as Figure 12 , Figure 13 , Figure 14 and Figure 15 are respectively the tracking error curves of the system of the present invention under the sine wave input signals of 1 Hz, 20 Hz, 50 Hz, and 100 Hz, Figure 16 and

[0127] The overshoot, adjustment time, root mean square error, and maximum absolute error rate after stabilization are selected as the evaluation indicators for the data-driven adaptive control method of the piezoelectric micro-motion platform based on the play operator.

[0128] From Figure 5 , Figure 11 and the corresponding experimental data, it can be obtained that under the step signal, the method proposed in the present invention acts on the piezoelectric micro-motion platform, the overshoot of the tracking curve is 1.4%, the adjustment time is 0.0022 s (2% error band), and the tracking error is 0.0222 μm.

[0129] From Figure 6 , Figure 7 , Figure 8 , Figure 9 , Figure 10 , Figure 12 , Figure 13 , Figure 14 , Figure 15 , Figure 16 and the corresponding experimental data, it can be obtained that under the sine signals and mixed wave signals with different frequencies (1 Hz, 20 Hz, 50 Hz, 100 Hz), the method proposed in the present invention acts on the piezoelectric micro-motion platform, and its root mean square error and maximum absolute error rate after stabilization are shown in the following table:

[0130]

[0131] According to the above experimental results, it can be concluded that the data-driven adaptive control method of a piezoelectric micro-motion platform based on the play operator in the present invention can achieve high-precision positioning control, and has great advantages in terms of rapidity and stability. The present invention can achieve good control effects and trajectory tracking effects. The trajectory tracking results of the piezoelectric micro-motion platform under different input signals fully prove the effectiveness of the data-driven adaptive control method based on the play operator.

[0132] The above-described embodiments are not used to limit the protection scope of the present invention, but only to illustrate the technical solutions of the present invention in detail; the method of the present invention can still be applied to other controlled objects, and at the same time, synonymous modifications, replacements, and improvements are made to some or all of the technical features in the present invention, which cannot make the essence of the corresponding technical solutions deviate from the protection scope of the present invention.

[0133] The symbol comparison table involved in the present invention is as follows:

[0134]

[0135]

Claims

1. A data-driven adaptive control method for a piezoelectric micro-motion platform based on the play operator, characterized in that The specific steps of this method are as follows: Step 1: Introduce multiple weighted superposition expressions of the play operator as exogenous variable functions to describe the hysteresis nonlinearity of the piezoelectric micro-motion platform. Only considering the input and output data of the platform at historical moments, establish an equivalent dynamic linearization data model of the piezoelectric micro-motion platform; where \(y(k)\in\mathbb{R}\) and \(u(k)\in\mathbb{R}\) are the output signal and input signal of the system at time \(k\), respectively. Ξ p \((k)\in\mathbb{R}\) n , \(L(k)=[l_1(k),l_2(k),\cdots,l\) n \((k)]\) T , \(L(k)\in\mathbb{R}\) n , \(\Delta L(k)=L(k)-L(k - 1)\). At any time \(k\), \(\xi\) i \((k)\), \(i = 1,\cdots,L\) y +L u +n are bounded, \(|\xi\) i \((k)|\leq\beta\), where \(\beta\) is a constant, \(l\) i \((k)\), \(i = 1,2,\cdots,n\) are the values of the play operator; \(L\) y , \(L\) u is the pseudo - order of the system. The expression of the play operator is as follows: Among them, the threshold value u m is a constant close to the maximum value of y d (k), where y d (k) is the expected displacement signal of the system at time k; Rewrite Equation (2) as: l i y(k) = d i y(k)Δu(k) + a i y(k - 1) (3) It should be noted that in the original text, there may be some unclear or incorrect notations. For example, it's not clear what the "(3)" in "y(k - 1) (3) " specifically means. The above translation is based on the best understanding of the provided content. Among them, Define \(A(k)=[a_1(k),a_2(k),\ldots,a n (k)] T , \(A(k)\in R n , \(D(k)=[d_1(k),d_2(k),\ldots,d n (k)] T , D(k) ∈ R n , then there is: L(k) = D(k)△u(k) + A(k - 1) (4); Step 2: Establish a control input criterion function and obtain a data-driven control algorithm using the idea of optimization; To avoid the adverse effects of the rapid change of the control quantity u(k) on the system, design the control input criterion function of the system, J(△u(k))=(y d (k + 1)-y(k + 1)) 2 +κ(u(k)-u(k - 1)) 2 (5) where κ > 0 is the weight factor; Derive the control variable u(k) of the criterion function by combining with the dynamic linearized data model with the play operator, and let The corresponding data-driven control law can be initially obtained; Among them, in order to make the control algorithm universal, a step size factor θ is added to the control law (6). i ∈(0,1], i = 0, 1, …, L y +L u +n; Step 3: Use an improved projection algorithm for online estimation when estimating the pseudo-partial derivative matrix; Definition Then, Equation (1) can be written as the following expression: y(k + 1) = y(k) + Ξ T (k)△J(k) (7) wherein, △J(k) = J(k) - J(k - 1), Ξ(k), and Ξ p (k) are all pseudo partial derivative matrices; To implement the control law (6), consider obtaining the value of the time-varying parameter Ξ(k) by using the input-output data of the system, and estimating the value. The algorithm criterion function of the estimator is designed as follows: where τ > 0 is the weight factor; Taking the extreme value of Equation (8) with respect to Ξ(k), the estimation of Ξ(k) can be obtained as: Among them, is the estimated value of Ξ(k). To make the control algorithm universal, a step-size factor is added to the estimation algorithm Step 4: Combining Step 2 and Step 3, introduce a reset algorithm to enhance the time-varying tracking ability of the parameters, If either ||△J(k)|| ≤ σ, or then make where σ is a very small positive number; Combined with the estimation algorithm of the pseudo-partial derivative matrix Ξ(k), the final control law is as follows: Among them, is the estimated value of, is the estimated value of Ξ p (k).

Images