Parameter optimization method of iterative learning controller, optimization system thereof and electronic device

Abstract

The application discloses a parameter optimization method of an iterative learning controller, an optimization system thereof and electronic equipment, and belongs to the technical field of repeated operation dynamic control. The parameter optimization method is used for optimizing learning parameters of a learning function in the iterative learning controller. Specifically, convergence conditions of the iterative learning controller in a frequency domain are converted into a semi-positive programming problem, a parameter optimization model is constructed, learning parameters are taken as decision variables, semi-positive symmetric matrix constraints of a limited angle frequency set are introduced, and a weighted function composed of a first-order differential element and a first-order inertia element is added in the convergence conditions in the frequency domain. Under the constraint of the weighted function, the solved learning parameters can obviously improve the learning convergence rate at a low frequency.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of repetitive operation dynamic system control, and more particularly, to a parameter optimization method of an iterative learning controller, an optimization system thereof and an electronic device. BACKGROUND

[0002] Iterative Learning Control (ILC) is a feedback data driven feedforward control method for repetitive operation dynamic system. The core idea is iteration and learning. First, iteration means that ILC is suitable for repetitive operation system, such as industrial robot. Many applications of industrial robot are repetitive operation, which continuously performs repetitive operation according to the given trajectory; second, learning is a big feature of ILC as intelligent control. It is an intelligent operation that simulates the learning behavior of human brain. For industrial robot, according to the deviation between the saved feedback information and the expected trajectory in the previous operation, the control input of the next operation can be adjusted to achieve better motion performance. The approximation optimization brought by learning is the core idea of ILC, that is, through continuous iteration, the motion performance of the robot is getting better and better, and approaches the ideal motion performance. The faster the learning convergence rate, the stronger the learning ability, and the faster the approximation can be achieved.

[0003] At present, in order to ensure the normal operation of iterative learning, the traditional frequency domain design method of iterative learning controller needs to make the iterative learning controller converge at each frequency point, that is, in order to ensure that the convergence condition is met at all frequencies, the learning convergence rate will be affected to a certain extent. Therefore, a new design method is needed to improve the learning convergence rate under the premise of ensuring that the convergence condition is met at all frequencies. SUMMARY

[0004] In view of the above defects or improvement needs of the prior art, the present application provides a parameter optimization method of an iterative learning controller, an optimization system thereof and an electronic device, which aims to improve the learning convergence rate of the iterative learning controller under the premise of ensuring that the convergence condition is met at all frequencies.

[0005] To achieve the above purpose, according to one aspect of the present application, a parameter optimization method of an iterative learning controller is provided, the iterative learning controller is suitable for repetitive operation system, and is used for optimizing the learning parameters of the learning function in the iterative learning controller to speed up the learning convergence rate of the iterative learning controller in the expected low frequency band, the z domain expression L(z) of the learning function is Wherein, s is the order of learning function, l i is the learning parameter of z -i ;

[0006] The parameter optimization method comprises:

[0007] selecting p corresponding normalized angular frequencies in a low frequency band containing a low-pass filter bandwidth to construct a finite angular frequency set Ω ω , p≥10;

[0008] constructing a parameter optimization model, wherein: the decision variable is [γ f 2 , l -s , …, l0, …, l s ]; the target is to minimize the supremum parameter γ f ; the constraint conditions include supremum constraint and semi-positive definite symmetric matrix constraint [Δ] * is the conjugate complex of [Δ], T(z) = Q(z)(1-z m L(z)G(z)), z = e iω , Q(z) is the z-domain expression of the low-pass filter, G(z) is the z-domain impulse transfer function of the controlled system, m is the relative order of the controlled system, H(z) is the z-domain weighting function, and the s-domain weighting function H(s) corresponding to the z-domain weighting function H(z) is the expected amplitude K a , the product of a first-order inertial link transfer function and a first-order differential link transfer function;

[0009] inputting the expected amplitude K a , the frequency f1 of the first-order inertial link, and the frequency f2 of the first-order differential link into the parameter optimization model, wherein the frequency f1 is the upper limit of the expected low frequency band, K a >1, and f2 = K a f1;

[0010] solving the parameter optimization model to output the decision variable.

[0011] In one embodiment, the cut-off frequency of the low-pass filter Q(z) is defined as f b , the sampling frequency of the iterative learning controller is f s , the p normalized angular frequencies in the low frequency band containing the low-pass filter bandwidth are selected to construct the finite angular frequency set Ω ω , which comprises: selecting p corresponding dispersed normalized angular frequencies in the frequency range [f α , f β ], wherein 0<f α ≤0.2Hz, f b <f β <f s / 2.

[0012] In one embodiment, f α=0.1Hz, f β =5f b , 10≤p≤20.

[0013] In one embodiment, the desired amplitude K a The range of values ​​can be The value range of frequency f1 can be [0.1Hz, 5Hz].

[0014] In one embodiment, the desired amplitude K a The method for determining it is as follows: under the premise that the parameter optimization model has a solution, select the largest expected amplitude K. a .

[0015] In one embodiment, the parameter optimization model is solved using the LMI solution method.

[0016] According to another aspect of the present invention, a parameter optimization system for an iterative learning controller is provided. The iterative learning controller is suitable for repeatedly running systems. The system optimizes the learning parameters of the learning function in the iterative learning controller to accelerate the learning convergence rate of the iterative learning controller in the expected low-frequency range. The z-domain expression L(z) of the learning function is: Where s is the order of the learning function, l i For z -i Learning parameters;

[0017] The parameter optimization system includes:

[0018] The weighting function parameter input unit is used to obtain the desired amplitude K. a The frequencies f1 of the first-order inertial element and f2 of the first-order differential element, where frequency f1 is the upper limit of the expected low-frequency band, K a >1, f2 = K a f1;

[0019] An angular frequency selection unit is used to obtain p corresponding normalized angular frequencies selected within the low-frequency band containing the low-pass filter bandwidth to construct a finite angular frequency set Ω. ω p≥10;

[0020] The parameter optimization model building unit is used to construct a parameter optimization model, where the decision variable is [γ]. f 2 ,l -s ,……,l0,……,l s The objective is to make the supremum parameter γ f Minimum; constraints include supremum constraints. semidefinite symmetric matrix constraints [Δ] *is the conjugate complex of [Δ], T(z) = Q(z)(1-z m L(z)G(z)), z = e iω , Q(z) is the z-domain expression of the low-pass filter, G(z) is the z-domain impulse transfer function of the controlled system, m is the relative order of the controlled system, H(z) is the z-domain weighting function, the s-domain weighting function H(s) corresponding to the z-domain weighting function H(z) is the desired amplitude K a , the product of the first-order inertia link transfer function and the first-order differential link transfer function;

[0021] a solving unit configured to solve the parameter optimization model and output a decision variable.

[0022] In one embodiment, the cut-off frequency of the low-pass filter Q(z) is defined as f b , the sampling frequency of the iterative learning controller is f s , the finite angular frequency set Ω ω is selected from the frequency range [f α , f β ], wherein 0 < f α ≤ 0.2 Hz, f b < f β < f s / 2.

[0023] In one embodiment, the solving unit solves the parameter optimization model by using an LMI solving method.

[0024] According to another aspect of the present application, there is provided an electronic device comprising a memory and a processor, wherein the memory stores a computer program, and the processor implements the steps of the above method when executing the computer program.

[0025] Overall, compared with the prior art, the above technical solutions conceived by the present application can achieve the following beneficial effects:

[0026] This invention improves the frequency design method of the learning function in iterative learning controllers. By constructing a parameter optimization model, the convergence condition of the iterative learning controller in the frequency domain is transformed into a semidefinite programming (SDP) problem, and a corresponding frequency domain linear matrix inequality (FLMI) is established for solution. When constructing the constraints, this invention adds a weighted function composed of a first-order differential element and a first-order inertial element to the frequency domain convergence condition. Under the influence of the weighted function constraint, the solved learning parameters significantly improve the learning convergence rate at low frequencies. Simultaneously, during the solution calculation, the size and range of the finite frequency set are determined based on the low-pass filter bandwidth, eliminating the need for excessive debugging. This method ensures that all normalized frequencies satisfy the convergence condition, optimizing the learning parameters of higher-order learning strategies while maintaining convergence across all frequencies. In summary, this invention not only guarantees that all normalized frequencies satisfy the convergence condition but also significantly improves the expected learning convergence rate in the low-frequency range. Attached Figure Description

[0027] Figure 1 A flowchart illustrating the steps of a parameter optimization method for an iterative learning controller according to one embodiment;

[0028] Figure 2 Here is a graph of the amplitude-frequency response of the weighting function in one embodiment;

[0029] Figure 3 Here is a simulation diagram of robot motion before iteration in one embodiment, where (a) is a comparison diagram of the expected position and feedback position of the link, and (b) is a diagram of the change in link trajectory tracking error;

[0030] Figure 4 This is a schematic diagram of the trajectory tracking error of each link without low-frequency optimization.

[0031] Figure 5 A schematic diagram of the L2 norm of the trajectory tracking error of each link when low-frequency optimization is not performed;

[0032] Figure 6 This is a schematic diagram illustrating the tracking error of each link trajectory during low-frequency optimization.

[0033] Figure 7 A schematic diagram of the 2-norm of the tracking error of each link trajectory during low-frequency optimization. Detailed Implementation

[0034] In order to make the objects, technical solutions and advantages of the present application clearer, the present application will be further described in detail below with reference to the drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application and should not be used to limit the present application. In addition, the technical features involved in each embodiment of the present application described below can be combined with each other as long as they do not conflict with each other.

[0035] In order to facilitate the understanding of the present application, the conventional learning strategy of the iterative learning controller suitable for repetitive motion control will be briefly introduced.

[0036] It is considered that the repetitive motion execution system is a single-input single-output linear time-invariant system in discrete time, that is, the input and output satisfy:

[0037] y j (k)=G(q)u j (k)+d j (k) (1)

[0038] Wherein, k is the time index, representing the discrete time within a limited time. j is the iteration index, representing the jth running of the system, and a new iteration is performed and the time index is reinitialized every time the system is re-run. y j (k) represents the output signal of the system at time k during the jth running, u j (k) represents the input signal of the system at time k during the jth running, d j (k) represents an initial correlation signal related to the initial state at time k during the jth running. q represents the time shift operator, which satisfies qy j (k)≡y j (k+1). G(q) is a rational function about the time shift operator q, which describes the model of the system.

[0039] For example, for an industrial robot, taking the mechanical and servo system of the robot as the repetitive motion execution system, the input can be the calculated command position of the connecting rod (i.e. the position that the connecting rod needs to reach), and the output is the actual feedback position of the connecting rod (i.e. the actual position that the connecting rod reaches) through the repetitive motion execution system. It can be understood that the input and output can be flexibly set according to the specific control in different scenarios, for example, the input can also be force or torque, and the output can also be speed or acceleration.

[0040] The general learning strategy of ILC can be described as:

[0041] u j+1 (k)=Q(q)(u j (k)+L(q)e j (k)) (2)

[0042] where Q(q) and L(q) are called filter function and learning function respectively; e j (k) represents the error between the actual output y d (k) of the system and the desired output y j (k) of the system, e j (k) = y d (k) - y j (k). This learning strategy means that the error of the last iteration is learned by using the learning function and then filtered.

[0043] For example, 1000 command positions of a connecting rod are initially planned and sent to the robot motion, during the robot motion, the feedback positions of the connecting rod are also 1000, then the command positions and the feedback positions are subtracted to obtain the corresponding error input to learn in the iterative learning controller, the result of the iterative learning is the new command position of the connecting rod, and then the new command position is sent to the robot (i.e. the repeated motion execution system) to run, which is equivalent to the second running of the robot, and the cycle is repeated.

[0044] Formula (1) and formula (2) are a description from the time domain. Z-transform is performed on each signal, and the system can be described by an impulse transfer function from the Z frequency domain:

[0045] Y j (z) = G(z)U j (z) + D j (z) (3)

[0046] U j+1 (z) = Q(z)(U j (z) + z m L(z)E j (z)) (4)

[0047] where Y j (z), U j (z), D j (z), E j (z) represent the Z-transform of the discrete signals of the system output signal y j (k), the system input signal u j (k), the initial correlation signal d j (k) and the error e j (k), G(z) represents the impulse transfer function of the system, which is the Z-transform of G(q), Q(z), L(z) are respectively a low-pass filter and a learning function, which are respectively the Z-transforms of Q(q) and L(q), and the essence is also two impulse transfer functions. m represents the relative order of the system.

[0048] where a learning function of an s order can be expressed as:

[0049] L(z) = (l -s z s +…l0z 0 +…+l s z -s ) (5)

[0050] wherein, l i is a learning parameter of z -i , also a variable to be decided subsequently. It is to be noted that the learning function can be a non-causal learning function, or a learning function satisfying causal relationship. In the non-causal learning function, l -s ,……,l -1 are not all equal to 0, while in the learning function satisfying causal relationship, l -1 =…=l -s =0. The specific type of learning function can be selected in advance according to the need. If it is a non-causal learning function, each learning parameter in [l -s ,……,l0,……,l s ] needs to be planned, if it is a learning function satisfying causal relationship, l -1 =…=l -s =0, and the actual planning is each learning parameter in [l0,……,l s ].

[0051] For the above ILC system, the convergence condition in the frequency domain can be described as:

[0052] ||Q(z)(1-z m L(z)G(z))|| ∞ <1 (6)

[0053] wherein, ‖*‖ ∞ represents the infinite norm.

[0054] If the ILC system satisfies formula (6), the ILC system is asymptotically stable and monotonically convergent.

[0055] Through analysis, the frequency domain convergence condition expressed by formula (6) can be converted into a semidefinite programming (SDP) problem, and a corresponding frequency domain linear matrix inequality (FLMI) is established to solve, and the learning parameters [l -s ,……,l0,……,l s ] of the learning function are optimized under the premise of satisfying the convergence condition in the full frequency. The specific conversion process is as follows:

[0056] Introducing the upper bound parameter γf Equation (6) can be described as follows:

[0057] ||Q(z)(1-z m L(z)G(z))|| ∞ ≤γ f (γ f <1) (7)

[0058] Let:

[0059] T(z)=Q(z)(1-z m L(z)G(z)) (8)

[0060] The relationship between the symbol z and the normalized angular frequency ω can be represented by Euler's formula:

[0061] z=e iω =cos(ω)+isin(ω) (9)

[0062] According to equations (8) and (9), equation (7) can be equivalent to:

[0063]

[0064] where the matrix +0, indicates that the matrix is a semi-positive definite symmetric matrix. T * (e iω ) is the conjugate complex of T(e iω ), and ω represents the normalized angular frequency.

[0065] The normalized angular frequency ω is calculated as:

[0066]

[0067] where f represents a certain frequency of interest in the frequency domain convergence condition, with the unit of Hz; f s represents the sampling frequency of the repetitive motion execution system during the execution of the action, with the unit of Hz.

[0068] When the system model G(z) and the selected low-pass filter Q(z) are known, the learning parameters [l -s ,……,l0,……,l s ] of the learning function L(z) are optimized. For a certain specific frequency f0, equation (11) is transformed into the corresponding normalized angular frequency ω0, and then ω0 is calculated according to equation (9) It can be found that is actually the learning parameter l _s ,…,l sAn affine function of. Moreover, the form of equation (10) is a semi-definite symmetric matrix, so the convergence condition in frequency domain can be actually converted into a standard SDP problem. Since the frequency convergence condition of formula (7) is converted into formula (10), it is required that all normalized angular frequencies in the interval [0, 2π] satisfy the semi-definite matrix in formula (10).

[0069] Based on this, a learning parameter optimization model can be constructed, all frequencies satisfying the semi-definite matrix as a constraint condition to form the FLMI constraint, and the goal of the model is to minimize f , that is, min f Therefore, the smaller f is, the faster the convergence rate is. The purpose of the present application is to improve the learning convergence rate under the premise of satisfying the convergence condition of all frequencies. The learning parameter with the fastest convergence rate under the condition of satisfying the convergence condition can be obtained by solving the model through the LMI solving method provided by the Matlab simulation software. However, since the model involves numerous semi-definite matrix constraints, it is difficult to solve in reality.

[0070] Therefore, in actual application, a limited frequency set ω is selected to solve under the limited frequency set. That is, the learning parameter optimization model is described as:

[0071]

[0072] It should be noted that min c T x is actually min , which is equivalent to min f . The smaller f is, the faster the convergence rate is.

[0073] However, directly solving the learning parameter with the model in formula (12), the f decided -s is still not small enough, that is, the learning parameter obtained is not the best parameter, so there is still optimization space.

[0074] Based on the above introduction and analysis, a new parameter optimization method of iterative learning controller is proposed, a first-order derivative element and a first-order inertia element are introduced, and a new SDP problem is reconstructed for solving, and the more optimized learning parameters [l s ,……,l0,……,l, to accelerate the learning convergence rate of the expected low frequency band under the premise of meeting the full frequency convergence condition. It should be noted that the learning convergence rate of the low frequency band is accelerated, and the learning convergence rate of the remaining high frequency band is basically not affected, because for a repetitive motion system, the convergence performance of the low frequency band is often concerned, and therefore the learning of the low frequency band needs to be strengthened, and the high frequency band may have more noise, and the learning requirement is not high. The range of the low frequency band that needs to be improved can be set according to the actual situation, and the low frequency band that needs to be strengthened for learning convergence rate is different for different repetitive motion systems.

[0075] As shown in Figure 1 , in an embodiment, the parameter optimization method of the iterative learning controller comprises the following steps:

[0076] Step S100: selecting p corresponding normalized angular frequencies in the low frequency band containing the low pass filter bandwidth to construct a finite angular frequency set Ω ω .

[0077] Wherein, p≥10.

[0078] The low pass filter Q(z) is known, and it is assumed that the cutoff frequency of the low pass filter Q(z) is f b , and the corresponding normalized angular frequency is ω b , then the frequency range 0~f b is called the bandwidth of the low pass filter. Assuming that Q(z) is an ideal low pass filter, then the amplitude is 1 in the frequency range 0~f b , and the amplitude is 0 in the frequency range f b ~f s / 2. This means that for frequencies greater than the cutoff frequency f b , the learning performance will be significantly inhibited or even closed. Therefore, the normalized frequency set corresponding to the cutoff frequency f b greater than the cutoff frequency f b , the low pass filter Q(z)=0, then the convergence condition must be met. Therefore, only a plurality of frequencies in the low frequency band containing the bandwidth of the low pass filter need to be selected, the corresponding normalized angular frequencies are calculated, and the angular frequency set Ω ω

[0079] In an embodiment, considering that there is no ideal low pass filter in practice, the selection range can be appropriately expanded, and the maximum frequency is greater than the cutoff frequency of the low pass filter. Specifically, the frequency range of the selected frequency is defined as [f α , f β ], and 0<f α ≤0.2Hz, fb <f β <f s / 2. Furthermore, f α =0.1Hz, f β =1.5f b At this point, the maximum value of the finite frequency set can be chosen to be 1.5ω. b This is the normalized angular frequency. The minimum value can be selected as the normalized angular frequency corresponding to 0.1Hz. In addition, the number of frequencies included in the finite frequency set is selected to be 10-20. Too few frequencies may result in some frequencies not being constrained, thus causing the convergence condition not to be satisfied across the entire frequency range; too many frequencies will make the LMI optimization calculation more complex. Therefore, the range and number of the finite frequency set based on the low-pass filter bandwidth proposed in this embodiment are selected as follows:

[0080]

[0081] A simple method based on the low-pass filter bandwidth is used to determine the size and range of a finite frequency set without much debugging. This method can guarantee that all normalized frequencies meet the convergence condition.

[0082] Step S200: Construct a parameter optimization model, where the decision variable is [γ]. f 2 ,l -s ,……,l0,……,l s The objective is to make the supremum parameter γ f Minimum; constraints include supremum constraints. semidefinite symmetric matrix constraints H(z) is the z-domain weighting function, and the corresponding s-domain weighting function H(s) is the expected magnitude K. a The product of the transfer function of the first-order inertial element and the transfer function of the first-order differential element.

[0083] Where, T(z)=Q(z)(1-z) m L(z)G(z)), the relationship has been introduced in formula (8) above.

[0084] Here, H(z) is a newly introduced weighting function that differs from formula (12).

[0085] Based on the initial convergence condition (7), the weighted function can be expressed as:

[0086] ||H(z)Q(z)(1-zL(z)G(z))|| ∞ ≤γ f (γ f <1) (14)

[0087] where H(z) is the designed weighting function. Similarly, equation (14) is equivalent to:

[0088]

[0089] Considering the constraint of the weighting function, equation (15) can be reformulated as a new SDP problem for solving:

[0090]

[0091] The weighting function H(z) is explained as follows.

[0092] The role of the weighting function H(z) is to reduce the value of (1-z m L(z)G(z) without changing the learning performance in the high frequency band. The smaller the value, the faster the learning convergence rate at these frequencies. Therefore, the designed weighting function is expected to have an amplitude of 0 dB in the high frequency band and an adjustable desired amplitude, such as 10 dB, in the low frequency band of interest. This desired amplitude needs to be appropriately selected, as a too large desired amplitude can lead to no optimal solution for the LMI.

[0093] By analyzing the properties of several typical elements, it is found that the combination of a first-order derivative element and an inertia element at different frequencies can achieve the desired effect, i.e., introducing a double-frequency derivative inertia element, which is designed from the continuous transfer function:

[0094]

[0095] where H(s) is the continuous transfer function corresponding to the weighting function H(z); the coefficient K a is the desired amplitude of the adjustable low frequency band; f1 represents the frequency of the first-order inertia element, is the transfer function of the first-order inertia element, f2 represents the frequency of the first-order derivative element, is the transfer function of the first-order derivative element, and the relationship between the frequencies is set as:

[0096] f2 = K a f1 (18)

[0097] where K dB is the desired amplitude K a corresponding to the decibel value, and the conversion relationship is

[0098] To obtain the weighting function H(z), a bilinear transformation is performed on H(s), and the bilinear transformation formula is:

[0099]

[0100]

[0101] wherein f s is the sampling frequency.

[0102] In combination with equations (17), (19) and (20), the expression of the weighting function H(z) can be obtained as follows:

[0103]

[0104] wherein σ1=tan(πf1 / f s ); σ2=tan(πf2 / f s ).

[0105] According to equations (18) and (21), only one of the two frequencies f1 and the expected amplitude K a of the low frequency band need to be determined, and then a specific H(z) can be designed.

[0106] Step S300: inputting the expected amplitude K a , the frequency f1 of the first-order inertia link and the frequency f2 of the first-order differential link into the parameter optimization model, wherein the frequency f1 is the upper limit of the expected low frequency band, K a >1, and f2=K a f1.

[0107] wherein the expected amplitude K a , the frequency f1 of the first-order inertia link and the frequency f2 of the first-order differential link are all known parameters selected in advance.

[0108] The expected amplitude K a satisfies K a >1, and in the present application, the expected amplitude is positively correlated with the learning rate, the greater the expected amplitude, the faster the learning rate, but it will also lead to more difficult model solving, and an excessively large expected amplitude K a may lead to no solution of the model. Therefore, the expected amplitude K a can be determined by trial and error, if the selected expected amplitude K a leads to no solution of the model, it indicates that the expected amplitude K a is set too large, and the expected amplitude K a needs to be reduced, in other words, the expected amplitude K a needs to be greater than 1 and needs to make the model have a solution. Further, on the basis of the model having a solution, the maximum expected amplitude K a can be selected.

[0109] Similarly, the greater the frequency f1 is set, the more difficult the model is to solve, and too large will cause the model to have no solution. Therefore, after determining the expected low frequency band [0, f1], the frequency of the first-order inertia link is directly determined as f1, and then the frequency f2 is determined based on the determined expected amplitude K a and the frequency f1, f2=K a f1. If the model has no solution regardless of the value of the expected amplitude K a , it indicates that the range of the expected low frequency band [0, f1] is too large, that is, the frequency f1 is too large, and the expected low frequency band needs to be reduced, that is, the frequency f1 needs to be reduced. It can be understood that the frequency f1 should be less than or equal to the cut-off frequency f b of the low-pass filter.

[0110] In summary, the values of the frequency f1 and the expected amplitude K a need to make the model have a solution.

[0111] According to the experience analysis, the value range of the expected amplitude K a may be , which is equivalent to the range of K dB being [1, 20]; and the value range of the frequency f1 may be [0.1Hz, 5Hz].

[0112] As shown in Figure 2 , f1=5Hz, K dB =12dB, the amplitude-frequency curve of the corresponding H(z) is obtained.

[0113] Step S400: solving the parameter optimization model to output the decision variable.

[0114] After outputting the expected amplitude K a , the frequency f1 of the first-order inertia link and the frequency f2 of the first-order differential link, the parameter optimization model can be solved to obtain the learning parameters [l -s , ……, l0, ……, l s ].

[0115] Correspondingly, the application also relates to a parameter optimization system of an iterative learning controller, comprising:

[0116] a weighting function parameter input unit configured to acquire an expected amplitude K a , a frequency f1 of a first-order inertia link and a frequency f2 of a first-order differential link, wherein the frequency f1 is an upper limit of an expected low frequency band, K a >1, and f2=K a f1;

[0117] an angular frequency selection unit configured to acquire p corresponding normalized angular frequencies selected in a low frequency band containing a bandwidth of a low-pass filter to construct a finite angular frequency set Ω ω , p>10.

[0118] a parameter optimization model construction unit configured to construct a parameter optimization model, wherein: decision variables are [gamma f 2 ,l -s ,……,l0,……,l s ]; an objective is to minimize a supremum parameter gamma f ; constraint conditions include supremum constraint and semi-positive definite symmetric matrix constraint [Delta] * is a conjugate complex of [Delta], T(z) = Q(z)(1-z m L(z)G(z)), z = e iω , Q(z) is a z-domain expression of a low-pass filter, G(z) is a z-domain impulse transfer function of an iterative learning controller, m is a relative order of the iterative learning controller, H(z) is a z-domain weighting function, an s-domain weighting function H(s) corresponding to the z-domain weighting function H(z) is a desired amplitude K a , a product of a first-order inertia link transfer function and a first-order differential link transfer function;

[0119] a solution unit configured to solve the parameter optimization model and output the decision variables.

[0120] The parameter optimization system of the above iterative learning controller can be used for the parameter optimization method of the iterative learning controller introduced in the foregoing, and the related units thereof can be used to perform the related steps corresponding to the functions thereof, and details can be referred to the foregoing, which will not be described herein.

[0121] Correspondingly, the present application also protects an electronic device including a memory and a processor, the memory stores a computer program, and the processor implements the steps of the above parameter optimization method when executing the computer program. Specifically, the electronic device can be a desktop computer, a notebook computer, a palm computer, a cloud server and other computing devices.

[0122] Generally, the present application adds a weighting function composed of a first-order differential link and a first-order inertia link to the frequency domain convergence condition, and reconstructs it as a new SDP problem for solving. Under the constraint of the weighting function, the learning parameters solved will significantly improve the learning convergence rate at low frequencies. At the same time, in the solving process, a simple method based on the bandwidth of the low-pass filter is proposed to determine the size and range of the finite frequency set, which does not require excessive debugging. This method can ensure that all normalized frequencies meet the convergence condition.

[0123] In the following, the effect of the present application is illustrated by a specific embodiment of an industrial robot trajectory tracking control.

[0124] The numerical simulation experimental platform is the HSR-Co610, a six-joint collaborative robot with dual encoders. The simulation tracks the robot's motion trajectory. Due to the presence of dual encoders—a motor-side encoder and a link-side encoder—the primary focus is on the operational performance of each joint link. In the experiment, the link trajectory tracking error is calculated by acquiring the feedback position from the link-side encoder. Figure 3 The figure shows the trajectory of the robot's motion before iteration, where (a) represents a comparison of the expected position and the actual feedback position of each joint link, and (b) represents the trajectory tracking error of each joint link. J1 to J6 represent joints 1 to 6, respectively.

[0125] This experiment aims to improve the link trajectory tracking accuracy after each iteration through iterative learning control. The learning convergence rate is measured by the 2-norm of the link trajectory tracking error.

[0126] In the simulation, the low-pass filter Q(z) was selected as a zero-phase fourth-order Butteworth low-pass filter with a cutoff frequency of 20Hz. The learning function L(z) was designed as a second-order learning function that satisfies causality.

[0127] L(z) = l0 + l1z -1 +l2z -2 (1)

[0128] Since a low-pass filter with a cutoff frequency of 20Hz was chosen, the finite frequency set is selected according to the proposed simple method based on the low-pass filter bandwidth as follows:

[0129] Ω ω = [0.1,1,5,8,10,12,15,18,20,25,30,35]×2π / 8000 (2)

[0130] Where 8000 is the sampling frequency.

[0131] The model expressed by formula (12) represents the model without using the proposed low-frequency optimization algorithm, while the model expressed by formula (16) represents the model using the proposed low-frequency optimization algorithm. Frequency domain design was performed in both cases. The linkage trajectory tracking error L2 norm was compared after 10 iterations in both cases to verify the effectiveness of the low-frequency optimization algorithm in improving the learning convergence rate.

[0132] Method 1: Iterate 10 times without using the low-frequency optimization algorithm; the link trajectory tracking error is as follows: Figure 4 As shown, its 2-norm is as follows Figure 5 As shown, the specific values ​​are shown in Table 1.

[0133] Table 1: 2-norm of link trajectory tracking error (without low frequency optimization)

[0134]

[0135] Method two: set the expected amplitude K of the improved low frequency band dB = 12 dB, and the frequency is set as f1= 0.5 Hz. After using the low frequency optimization algorithm for 10 times of iteration, the link trajectory tracking error is as shown in Figure 6 , the 2-norm is as shown in Figure 7 , and the specific values are shown in Table 2.

[0136] Table 2: 2-norm of link trajectory tracking error (after low frequency optimization)

[0137]

[0138]

[0139] Comparison Figure 5 and Figure 7 , it can be found from Table 1 and Table 2 that, for the frequency domain design method, the learning convergence rate is obviously improved by improving the method through the proposed low frequency optimization algorithm based on the double frequency differential inertia link.

[0140] Those skilled in the art can easily understand that the above is only a preferred embodiment of the present application, and is not intended to limit the present application, and any modification, equivalent replacement and improvement made within the spirit and principle of the present application shall be included in the protection scope of the present application.

Claims

1. A parameter optimization method for an iterative learning controller, wherein the iterative learning controller is suitable for repeatedly running systems, characterized in that, The parameter optimization method is used to optimize the learning parameters of the learning function in the iterative learning controller to accelerate the learning convergence rate of the iterative learning controller in the expected low-frequency range. The z-domain expression L(z) of the learning function is: Where s is the order of the learning function, l i For z -i Learning parameters; The parameter optimization method includes: Within the low-frequency band encompassing the bandwidth of the low-pass filter, select p corresponding normalized angular frequencies to construct a finite angular frequency set Ω. ω p≥10; Construct a parameter optimization model, where the decision variable is [γ]. f 2 ,l -s ,……,l0,……,l s The objective is to make the supremum parameter γ f Minimum; constraints include supremum constraints. semidefinite symmetric matrix constraints [Δ] * For the conjugate complex number of [Δ], T(z) = Q(z)(1-z) m L(z)G(z)), z=e iω Q(z) is the z-domain expression of the low-pass filter, G(z) is the z-domain pulse transfer function of the controlled system, m is the relative order of the controlled system, H(z) is the z-domain weighting function, and the s-domain weighting function H(s) corresponding to the z-domain weighting function H(z) is the desired amplitude K. a The product of the transfer function of the first-order inertial element and the transfer function of the first-order differential element; The expected amplitude K a The frequencies f1 of the first-order inertial element and f2 of the first-order differential element are input into the parameter optimization model, where frequency f1 is the upper limit of the expected low-frequency band, and K... a >1, f2 = K a f1; Solve the parameter optimization model and output the decision variables.

2. The parameter optimization method for the iterative learning controller as described in claim 1, characterized in that, The cutoff frequency of the low-pass filter Q(z) is defined as f. b The sampling frequency of the iterative learning controller is f. s The step involves selecting p normalized angular frequencies within a low-frequency band that includes the bandwidth of a low-pass filter to construct a finite angular frequency set Ω. ω Including, in the frequency range [f α ,f β Within the range, select p corresponding dispersed normalized angular frequencies, where 0 <f α ≤0.2Hz, f b <f β <f s / 2.

3. The parameter optimization method for the iterative learning controller as described in claim 2, characterized in that, f α =0.1Hz,f β =1.5f b ,10≤p≤20。 4. The parameter optimization method for the iterative learning controller as described in claim 1, characterized in that, Expected amplitude K a The range of values ​​can be The value range of frequency f1 can be [0.1Hz, 5Hz].

5. The parameter optimization method for the iterative learning controller as described in claim 4, characterized in that, Expected amplitude K a The method for determining it is as follows: under the premise that the parameter optimization model has a solution, select the largest expected amplitude K. a .

6. The parameter optimization method for the iterative learning controller as described in claim 1, characterized in that, The parameter optimization model is solved using the LMI solution method.

7. A parameter optimization system for an iterative learning controller, wherein the iterative learning controller is suitable for repeatedly running systems, characterized in that, The learning parameters of the learning function in the iterative learning controller are used to optimize the learning convergence rate of the iterative learning controller in the expected low-frequency range. The z-domain expression L(z) of the learning function is: Where s is the order of the learning function, l i For z -i Learning parameters; The parameter optimization system includes: The weighting function parameter input unit is used to obtain the desired amplitude K. a The frequencies f1 of the first-order inertial element and f2 of the first-order differential element, where frequency f1 is the upper limit of the expected low-frequency band, K a >1, f2 = K a f1; An angular frequency selection unit is used to obtain p corresponding normalized angular frequencies selected within a low-frequency band containing the bandwidth of a low-pass filter to construct a finite angular frequency set Ω. ω p≥10; The parameter optimization model building unit is used to construct a parameter optimization model, where the decision variable is [γ]. f 2 ,l -s ,……,l0,……,l s The objective is to make the supremum parameter γ f Minimum; constraints include supremum constraints. semidefinite symmetric matrix constraints [Δ] * For the conjugate complex number of [Δ], T(z) = Q(z)(1-z) m L(z)G(z)), z=e iω Q(z) is the z-domain expression of the low-pass filter, G(z) is the z-domain pulse transfer function of the controlled system, m is the relative order of the controlled system, H(z) is the z-domain weighting function, and the s-domain weighting function H(s) corresponding to the z-domain weighting function H(z) is the desired amplitude K. a The product of the transfer function of the first-order inertial element and the transfer function of the first-order differential element; The solver unit is used to solve the parameter optimization model and output decision variables.

8. The parameter optimization system for the iterative learning controller as described in claim 7, characterized in that, The cutoff frequency of the low-pass filter Q(z) is defined as f. b The sampling frequency of the iterative learning controller is f. s finite angular frequency set Ω ω For the frequency range [f α ,f β Within the range, select p corresponding dispersed normalized angular frequencies, where 0 <f α ≤0.2Hz, f b <f β <f s / 2.

9. The parameter optimization system for the iterative learning controller as described in claim 7, characterized in that, The solving unit uses the LMI solution method to solve the parameter optimization model.

10. An electronic device comprising a memory and a processor, wherein the memory stores a computer program, characterized in that, When the processor executes the computer program, it implements the steps of the method according to any one of claims 1 to 6.

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