Method for constructing complex coefficient repetitive control system

Abstract

The application provides a construction method of a complex coefficient repetitive control system, comprising the following steps: constructing a complex coefficient filter, extracting and processing key frequency components of a periodic signal; constructing a complex coefficient repetitive controller, realizing real-time response and accurate tracking of the periodic signal; constructing a structure of the complex coefficient repetitive control system, connecting the complex coefficient repetitive controller and a state feedback controller in series and forming a closed loop system with a controlled object, introducing a feedback loop to ensure dynamic performance; setting parameters of complex coefficient repetitive control coefficients according to system stability conditions; and using the complex coefficient repetitive control system to realize real-time tracking of a to-be-tracked periodic signal. The application integrates complex coefficient filtering, complex coefficient repetitive control strategy, system structure optimization and stability analysis, provides an effective solution for high-precision tracking of a periodic signal, and ensures stable operation of the system through strict parameter setting.

Description

Technical Field

[0001] This invention relates to the field of repetitive control, and in particular to a method for constructing a complex coefficient repetitive control system. Background Technology

[0002] In control practice, there are many systems that need to perform cyclic control tasks, such as industrial robots, CNC machine tools, and disk and optical disc drives. These systems require high-precision tracking of given cyclic signals.

[0003] The theory of repetitive control provides an effective way to achieve high-precision tracking of periodic signals in systems. The theoretical basis of repetitive control is the internal model principle. This principle states that if a signal generator is included in a closed-loop control system, the system can achieve high-precision tracking or effective suppression of that signal. Based on this, Inoue et al. proved that a time-delay positive feedback loop can be a generator of any known periodic signal; this loop is also called a repetitive controller. Due to its advantages such as high control accuracy and simple controller structure, the repetitive controller has attracted much attention from scholars since its inception, resulting in significant theoretical research achievements and successful applications in engineering practice.

[0004] However, the introduction of the repetitive controller results in the control system containing a time-delay positive feedback loop, which also transforms the control system into a neutral time-delay system, making it difficult to guarantee the stability of the system. To solve this problem, some scholars have solved the problem by connecting a low-pass filter in series after the time-delay link, thus transforming the neutral time-delay system into a time-delay system. However, since the filter causes the signal phase in the repetitive controller to lag, the tracking performance of the system is sacrificed to some extent. Although the system performance can be improved by increasing the filter bandwidth or using a higher-order filter, it may also amplify the influence of noise and even destroy the stability of the system [9]. This shows that there is a trade-off between stability and control performance in repetitive control systems. Summary of the Invention

[0005] The main objective of this invention is to provide a method for constructing a complex coefficient repetitive control system, thereby solving the problem of the trade-off between stability and control performance in existing repetitive control systems.

[0006] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is: a method for constructing a complex coefficient repetitive control system, comprising the following steps:

[0007] S1. Construct a complex coefficient filter to extract and process the key frequency components of the periodic signal;

[0008] S2. Based on the complex coefficient filter, a complex coefficient repetitive controller is constructed. Through complex signal construction, first-order hysteresis element, filter processing and time delay positive feedback, a frequency domain expression is formed to realize real-time response and accurate tracking of periodic signals.

[0009] S3. Based on the complex coefficient repetitive controller, construct the structure of the complex coefficient repetitive control system, connect the complex coefficient repetitive controller and the state feedback controller in series and form a closed-loop system with the controlled object, and introduce a feedback loop to ensure dynamic performance.

[0010] S4. Based on the system stability conditions, set the parameters of the complex coefficient repeatability control coefficient;

[0011] S5. A complex coefficient repetitive control system is used to achieve real-time tracking of the signal to be tracked in the cycle period.

[0012] The above steps complete the construction of the complex coefficient repetitive control system.

[0013] In the preferred scheme, the expression for the complex coefficient filter is:

[0014]

[0015] Where H(s) represents the complex coefficient filter, s represents the Laplace operator, and ω c ω represents the filter time constant. m represents the imaginary part of the complex coefficient, and j represents the imaginary unit.

[0016] In the preferred embodiment, the specific process of constructing a complex coefficient repetitive controller includes the following sub-steps:

[0017] S21. Construct a complex signal λ(t), where the real part of the complex signal is denoted as v(t), and the imaginary part of the complex signal leads or lags v(t) by 90 degrees.

[0018] S22. Let v(t) pass through a first-order lag element; its transfer function is:

[0019]

[0020] Where ε and κ are adjustment parameters;

[0021] Its time-domain expression is:

[0022]

[0023] Where z(t) is the signal obtained by passing v(t) through a first-order lag element, ε is chosen such that z(t) lags v(t) by 90 degrees when ω > ω0. At this time, z(t) is taken as the imaginary part of the complex signal, and the expression of the complex signal is obtained as follows:

[0024] λ(t)=v(t)-jz(t) (4)

[0025] S23. Let the signal after λ(t) is processed by the complex coefficient filter H(s) be:

[0026]

[0027] Combining (1), (4), and (5), we get:

[0028]

[0029] in, They represent The Laplace transform yields:

[0030]

[0031] S24, will Through a time-delay positive feedback loop, a complex coefficient repetitive controller is obtained, and simultaneously...

[0032]

[0033] Where C represents a constant matrix, x(t) represents the system state, and T represents the period of the periodic signal to be tracked.

[0034] In the preferred embodiment, the frequency domain expression of the complex coefficient repetitive controller is:

[0035]

[0036] Where T represents the period of the periodic signal to be tracked, e -Ts This indicates a time delay.

[0037] The specific process of constructing the structure of the complex coefficient repetitive control system in the preferred scheme is as follows:

[0038] By connecting the complex coefficient repetitive controller, the state feedback controller, and the controlled object in sequence, and adding a feedback loop, the structure of the complex coefficient repetitive control system is obtained.

[0039] In the preferred scheme, the expression of the controlled object is as follows:

[0040]

[0041] Where x(t) represents the system state, u(t) represents the control input, y(t) represents the system output, and A, B, and C represent constant matrices;

[0042] u(t) is specifically represented as:

[0043]

[0044] Among them, K RC It is the gain of the repetitive controller, K P It is the state feedback gain.

[0045] In the preferred scheme, the specific process of setting the parameters of the complex coefficient repetition control coefficient includes the following sub-steps:

[0046] S41. Let the periodic signal to be tracked, r(t), be 0. The expression for the complex coefficient repetitive control system is:

[0047]

[0048] S42. Construct the Lyapunov function based on the expression of the complex coefficient repetitive control system;

[0049] S43. Differentiate the Lyapunov function and find parameters that satisfy the condition that the derivative of the Lyapunov function is less than zero as system parameters.

[0050] In the preferred scheme, the controlled object is a speed control system, and the parameters of the controlled object are:

[0051]

[0052] This invention provides a method for constructing a complex coefficient repetitive control system. A complex coefficient filter is constructed to extract and process key frequency components of a periodic signal. Then, a complex coefficient repetitive controller is designed based on the filter. Through complex signal construction, first-order lag elements, filter processing, and time-delay positive feedback, a frequency domain expression is formed, achieving real-time response and accurate tracking of the periodic signal. Based on the complex coefficient repetitive controller, a system structure is constructed by connecting the complex coefficient repetitive controller in series with a state feedback controller to form a closed-loop system with the controlled object. A feedback loop is introduced to ensure the system's dynamic performance. Furthermore, based on Lyapunov stability theory, the complex coefficient repetitive control coefficient parameters are set. By constructing a Lyapunov function and taking its derivative, parameters that make the function's derivative less than zero are selected to ensure the system's global stability.

[0053] The beneficial effects of this invention are as follows:

[0054] This invention constructs a complex coefficient repetitive controller, and thus a system based on it, solving the trade-off between stability and control performance inherent in traditional repetitive control systems. It increases the open-loop gain of the system at the frequency of the periodic signal, thereby improving the system's tracking performance for periodic signals. Attached Figure Description

[0055] The present invention will be further described below with reference to the accompanying drawings and embodiments:

[0056] Figure 1This is an execution flowchart of a method for constructing a complex coefficient repetitive control system according to the present invention;

[0057] Figure 2 This is a block diagram illustrating the implementation of the complex coefficient filter of this invention;

[0058] Figure 3 This is a system framework diagram of a traditional repetitive control system (RC).

[0059] Figure 4 This is a system framework diagram of a complex coefficient repetitive control system (CRC) constructed according to the present invention;

[0060] Figure 5 This invention compares the simulation results of a traditional repetitive control system (RC) and a complex coefficient repetitive control system (CRC). Detailed Implementation

[0061] Example 1

[0062] like Figures 1-5 As shown, a method for constructing a complex coefficient repetitive control system includes the following steps:

[0063] S1. Construct a complex coefficient filter to extract and process the key frequency components of the periodic signal;

[0064] S2. Based on the complex coefficient filter, a complex coefficient repetitive controller is constructed. Through complex signal construction, first-order hysteresis element, filter processing and time delay positive feedback, a frequency domain expression is formed to achieve real-time response and accurate tracking of periodic signals.

[0065] S3. Based on the complex coefficient repetitive controller, construct the structure of the complex coefficient repetitive control system, connect the complex coefficient repetitive controller and the state feedback controller in series and form a closed-loop system with the controlled object, and introduce a feedback loop to ensure dynamic performance.

[0066] S4. Based on the system stability conditions, set the parameters of the complex coefficient repeatability control coefficient;

[0067] S5. A complex coefficient repetitive control system is used to achieve real-time tracking of the signal to be tracked in the cycle period.

[0068] The above steps complete the construction of the complex coefficient repetitive control system.

[0069] The above steps complete the construction of the complex coefficient repetitive control system.

[0070] This construction method details a complex coefficient repetitive control system designed to accurately track periodic signals and ensure system stability. The specific steps include: constructing a complex coefficient filter to extract and process key frequency components of the periodic signal, effectively suppressing noise and aperiodic interference; constructing a complex coefficient repetitive controller, which uses complex signal construction, first-order hysteresis, filter processing, and time-delay positive feedback to form a frequency domain expression, achieving real-time response and accurate tracking of the periodic signal; constructing the system structure, connecting the complex coefficient repetitive controller and a state feedback controller in series with the controlled object to form a closed-loop system, introducing a feedback loop to ensure dynamic performance; and finally, setting the complex coefficient repetitive control coefficient parameters based on Lyapunov stability theory, constructing a Lyapunov function and selecting parameters that make the function's derivative less than zero to ensure global system stability. This method has significant meaning and value for high-precision tracking of periodic signals, enhancing system stability, and its wide application in power electronics, robot control, and communication systems.

[0071] In the preferred scheme, the expression for the complex coefficient filter is:

[0072]

[0073] Where H(s) represents the complex coefficient filter, s represents the Laplace operator, and ω c ω represents the filter time constant. m represents the imaginary part of the complex coefficient, and j represents the imaginary unit.

[0074] In the preferred embodiment, the specific process of constructing a complex coefficient repetitive controller includes the following sub-steps:

[0075] S21. Construct a complex signal λ(t), where the real part of the complex signal is denoted as v(t), and the imaginary part of the complex signal leads or lags v(t) by 90 degrees.

[0076] To implement a complex coefficient filter, a complex signal λ(t) must first be constructed. The real part of the complex signal is denoted as v(t), and the imaginary part of the complex signal should lead or lag v(t) by 90 degrees. Although differentiators or integrators can achieve this, it may reduce the stability margin of the control system.

[0077] S22. Let v(t) pass through a first-order lag element; its transfer function is:

[0078]

[0079] Where ε and κ are adjustment parameters;

[0080] Its time-domain expression is:

[0081]

[0082] Where z(t) is the signal obtained by passing v(t) through a first-order lag element, ε and κ are adjustment parameters. ε is mainly used to adjust the phase of z(t), and κ is mainly used to adjust the amplitude of z(t). A sufficiently small ε is chosen such that when ω > ω0, z(t) lags v(t) by 90 degrees. In this case, z(t) is treated as the imaginary part of the complex signal, thus the expression for the complex signal is obtained as follows:

[0083] λ(t)=v(t)-jz(t) (4)

[0084] S23. Let the signal after λ(t) is processed by the complex coefficient filter H(s) be:

[0085]

[0086] Combining (1), (4), and (5), we get:

[0087]

[0088] in, They represent The Laplace transform yields:

[0089]

[0090] According to formula (7), a complex coefficient filter can be obtained by... Figure 2 The structure shown is implemented, where v(t) to The transfer function is named

[0091] S24, will Through a time-delay positive feedback loop, a complex coefficient repetitive controller is obtained, and simultaneously...

[0092]

[0093] Where C represents a constant matrix, x(t) represents the system state, and T represents the period of the periodic signal to be tracked.

[0094] In the preferred embodiment, the frequency domain expression of the complex coefficient repetitive controller is:

[0095]

[0096] Where T represents the period of the periodic signal to be tracked, e -Ts This indicates a time delay.

[0097] Substituting the expression (1) of the complex coefficient filter H(s) into the frequency domain expression (9) of the complex coefficient repetitive controller, we can obtain the frequency characteristics of the complex coefficient repetitive controller satisfying:

[0098]

[0099] For ω=kω0, we have:

[0100]

[0101] Therefore, at the fundamental frequency (k=1), we can obtain:

[0102] |C CRC (jω0)|=∞

[0103] This means that a complex coefficient repetitive controller can enable the system to fully track the fundamental frequency harmonic.

[0104] The specific process of constructing the structure of the complex coefficient repetitive control system in the preferred scheme is as follows:

[0105] By sequentially connecting the complex coefficient repetitive controller, the state feedback controller, and the controlled object, and adding a feedback loop, the structure of the complex coefficient repetitive control system is obtained, as shown below. Figure 4 As shown.

[0106] In the preferred scheme, the expression of the controlled object is as follows:

[0107]

[0108] Where x(t) represents the system state, u(t) represents the control input, y(t) represents the system output, and A, B, and C represent constant matrices;

[0109] u(t) is specifically represented as:

[0110]

[0111] Among them, K RC It is the gain of the repetitive controller, K P It is the state feedback gain.

[0112] In the preferred scheme, the specific process of setting the parameters of the complex coefficient repetition control coefficient includes the following sub-steps:

[0113] S41. Let the periodic signal to be tracked, r(t), = 0, according to... Figure 2 and Figure 4 The expression for a complex coefficient repetitive control system is:

[0114]

[0115] S42. Construct the Lyapunov function based on the expression of the complex coefficient repetitive control system;

[0116] S43. Differentiate the Lyapunov function and find parameters that satisfy the condition that the derivative of the Lyapunov function is less than zero as system parameters.

[0117] definition The expression for the complex coefficient repetitive control system then becomes:

[0118]

[0119] in,

[0120]

[0121] The stability condition of a complex coefficient repetitive control system can be given by the following theorem.

[0122] Theorem 1: For a given positive scalar a1, a2, a3, a4, κ, ε, ω0 and ω c If there exist positive definite matrices P1, P2, P3, P4, Q1, Q2, Q3, Q4 and suitable-dimensional matrices W1, W2 such that the following linear matrix inequality holds:

[0123]

[0124] in,

[0125]

[0126] The complex coefficient repetitive control system is asymptotically stable, and the feedback control gain K P Repetitive control gain K RC Designed by the following formula:

[0127] K P =W1(a1P1) -1 K RC =W2(a2P2) -1 (20)

[0128] Proof: Consider the following Lyapunov function:

[0129]

[0130] in,

[0131] P=diag{a1P1,a2P2,a3P3,a4P4} (22)

[0132] Furthermore, a1(>0), a2(>0), a3(>0), and a4(>0) are the adjustment parameters of the linear matrix inequality.

[0133] Differentiating the Lyapunov function along the complex coefficient repetitive control system:

[0134]

[0135] in,

[0136]

[0137] For the expression of the derivative of the Lyapunov function (22), if Φ < 0, the complex coefficient repetitive control system is asymptotically stable. According to Schur's complement lemma, Φ < 0 is equivalent to:

[0138]

[0139] Will Substitute into the above equation and let

[0140] W1=a1K P P1, W2 = a2K RC P2 (26)

[0141] Then we can obtain (19), (20), and (21) in Theorem 1.

[0142] Q.E.D.

[0143] In the preferred embodiment, to demonstrate the effectiveness of the method proposed in this invention, a simulation verification was performed using the following speed control system as the controlled object, with the controlled object parameters as follows:

[0144]

[0145] Given a periodic reference input:

[0146] r(t) = sinπt + 0.5cos2πt

[0147] Therefore, we can obtain T = 2s, ω0 = πrad / s. Without loss of generality, we set the cutoff frequency of the filter to 5 times the fundamental frequency, that is:

[0148] ω c =5ω0=5πrad / s

[0149] And make

[0150] ε=1, a=1000, b=0.1, c=0.0001, d=0.1, κ=π

[0151] Solving the linear matrix inequality (17), we obtain...

[0152] K P = [-0.2050, 0, 10.10]K RC =37.89

[0153] To be fair, this invention will compare the traditional repetitive control system ( Figure 3 The parameters are set to be exactly the same as those of the complex coefficient repetitive control system, and the simulation results are as follows: Figure 5As shown, the complex coefficient repetitive control system has a smaller tracking error, with a peak-to-peak tracking error of 0.0276 rpm, which is only 54.01% of the output of the traditional repetitive control system.

[0154] This invention provides a complex coefficient repetitive controller that uses a complex coefficient filter to improve the tracking error of a repetitive control system, and implements the complex coefficient filter in the real domain based on frequency domain analysis. Simulation results show that the complex coefficient repetitive control system improves the tracking performance by 47.78% compared to the traditional repetitive control system, demonstrating significant advantages.

[0155] The above embodiments are merely preferred technical solutions of the present invention and should not be considered as limitations on the present invention. The scope of protection of the present invention should be limited to the technical solutions described in the claims, including equivalent substitutions of the technical features described in the claims. That is, equivalent substitutions and improvements within this scope are also within the scope of protection of the present invention.

Claims

1. A method for constructing a complex coefficient repetitive control system, characterized in that, Includes the following steps: S1. Construct a complex coefficient filter to extract and process the key frequency components of the periodic signal; S2. Based on the complex coefficient filter, a complex coefficient repetitive controller is constructed. Through complex signal construction, first-order hysteresis element, filter processing and time delay positive feedback, a frequency domain expression is formed to realize real-time response and accurate tracking of periodic signals. S3. Based on the complex coefficient repetitive controller, construct the structure of the complex coefficient repetitive control system, connect the complex coefficient repetitive controller and the state feedback controller in series and form a closed-loop system with the controlled object, and introduce a feedback loop to ensure dynamic performance. S4. Based on the system stability conditions, set the parameters of the complex coefficient repeatability control coefficient; S5. A complex coefficient repetitive control system is used to achieve real-time tracking of the signal to be tracked in the cycle period. The above steps complete the construction of the complex coefficient repetitive control system; The expression for the complex coefficient filter is: （1）； in, Let represent a complex coefficient filter, and s represent the Laplace operator. Represents the filter time constant. represents the imaginary part of the complex coefficients, and j represents the imaginary unit; The specific process of constructing the complex coefficient repeating controller includes the following sub-steps: S21. Construct a complex signal The real part of the complex signal is denoted as The imaginary part of the complex signal leads or lags. 90 degrees; S22, Order Through a first-order lag element, its transfer function is: （2）； in, and It refers to adjusting parameters; Its time-domain expression is: （3）； in, for The signal obtained through the first-order hysteresis circuit is selected. When hour Lag 90 degrees, at this time, As the imaginary part of the complex signal, the expression for the complex signal is obtained as follows: （4）； S23, Order After complex coefficient filter The processed signal is: （5）； Combining (1), (4), and (5), we get: （6）； in, , They represent , The Laplace transform yields: （7）； S24, will Through a time-delay positive feedback loop, a complex coefficient repetitive controller is obtained, and simultaneously... （8）； Where C represents a constant matrix, The system state is represented by T, which represents the period of the signal to be tracked. The frequency domain expression of the complex coefficient repetitive controller is: （9）； Where T represents the period of the periodic signal to be tracked. This indicates a time delay.

2. The method for constructing a complex coefficient repetitive control system according to claim 1, characterized in that: The specific process of constructing the structure of the complex coefficient repetitive control system is as follows: By connecting the complex coefficient repetitive controller, the state feedback controller, and the controlled object in sequence, and adding a feedback loop, the structure of the complex coefficient repetitive control system is obtained.

3. The method for constructing a complex coefficient repetitive control system according to claim 2, characterized in that: The expression of the controlled object is as follows: （10）； in, Indicates the system status. Indicates control input, Indicates system output, A, B, and C represent... Indicator constant matrix: Specifically, it is expressed as follows: （11）； in, It is the gain of the repetitive controller. It is the state feedback gain.

4. The method for constructing a complex coefficient repetitive control system according to claim 3, characterized in that: The specific process of setting the parameters of the complex coefficient repetition control coefficient includes the following sub-steps: S41, Set the period signal to be tracked The expression for a complex coefficient repetitive control system is: （12）； S42. Construct the Lyapunov function based on the expression of the complex coefficient repetitive control system; S43. Differentiate the Lyapunov function and find parameters that satisfy the condition that the derivative of the Lyapunov function is less than zero as system parameters.

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