A LADRC control system based on linear expansion tracking differentiator and its parameter tuning method

Abstract

The present invention discloses a LADRC control system based on a linear expansion tracking differentiator and its parameter tuning method. By designing an n+1-order linear expansion tracking differentiator and adding high-order differential signals, this control system effectively addresses the hysteresis and estimation errors inherent in standard LADRC signal tracking. This also addresses the overshoot problem in the closed-loop transfer function step response of the combination of a linear LADRC and a controlled object. This tuning method, with clear and specific theoretical guidance and clear and reasonable debugging rules, is more adaptable to on-site debugging practices and avoids the issues of ambiguous parameter interpretation and unclear debugging processes associated with traditional empirical tuning formulas.

Description

Technical Field

[0001] The invention belongs to the technical field of industrial process automation and relates to an improved linear active disturbance rejection controller and a parameter tuning method thereof. Background Art

[0002] Active Disturbance Rejection Control (ADRC) technology has been widely used in industrial control due to its superior state estimation and disturbance rejection capabilities. While the linear tracking differentiator (LTD) effectively extracts a smooth derivative signal from noise, avoiding the noise amplification problem of traditional differential control, standard linear ADRC control systems suffer from overshoot in the closed-loop step response. This can cause unnecessary oscillations during rapid system responses, reducing system stability and control quality.

[0003] Based on the inherent connection between the open-loop transfer function and the closed-loop characteristic equation of a feedback control system, W.R. Evans proposed a simple and practical graphical method for obtaining closed-loop characteristic roots—the root locus method. A root locus is the trajectory of the closed-loop characteristic roots on the s-plane as a parameter in the system varies from 0 to positive infinity. The stability of a closed-loop control system can be determined by the poles of the closed-loop transfer function, i.e., the roots of the closed-loop system's characteristic equation. The basic characteristics of the system's transient response are also dominated by the closed-loop poles.

[0004] When studying feedback control systems, the first step is to design the system's closed-loop poles and zeros, given the system's structure and parameters. Secondly, to ensure the desired control performance, it's necessary to consider how variations in the system's structural parameters affect these poles and zeros. A system's closed-loop poles are the roots of its closed-loop characteristic equation. For high-order systems, analytically determining these poles is difficult. In particular, examining the general patterns of how the system's closed-loop poles vary with its structure and parameters requires extensive and complex design.

[0005] In the existing standard LADRC, the absolute value of the real part of the closed-loop transfer function pole is c (ω c ≤ω o ), but there is a zero point equal to -ω c / 2, the zero point is located on the right side of the closed-loop dominant pole of the system close to the imaginary axis. The zero point reduces the system damping, so the step response of the closed-loop system has overshoot.

[0006] LTD technology plays a crucial role in LADRC. By tracking changes in the input signal and generating a smooth derivative signal, LTD effectively suppresses noise interference and provides stable state estimation. LTD essentially reduces input signal fluctuations through filtering, making the signal derivative estimate smoother and more accurate. This effectively improves the system's robustness and anti-interference capabilities, especially in noisy environments. By filtering the system input signal, LTD avoids the high-frequency noise amplification caused by direct differentiation, enabling the system to better track the setpoint and enhance control accuracy. Combining LADRC with LTD can significantly improve the system's anti-interference performance while enhancing the smoothness and stability of signal tracking. However, in industrial practice, LADRC parameter tuning requires extensive field experience and trial and error, and there is the challenge of balancing system output overshoot with fast response and anti-interference performance.

[0007] Therefore, it is urgent to solve the following problems: how to effectively improve the standard LADRC from a theoretical perspective, and how to design an explainable parameter tuning method from a principle perspective. Summary of the Invention

[0008] In order to solve the overshoot problem of closed-loop step response of standard LADRC control system, a LADRC control system based on linear expansion tracking differentiator and its parameter tuning method are proposed.

[0009] The technical solutions of the present invention are as follows:

[0010] A LADRC control system based on a linear expansion tracking differentiator includes an active disturbance rejection controller and a controlled object. The active disturbance rejection controller includes a linear expansion tracking differentiator based on zero-point design, a linear expansion state observer, and a linear error feedback control law.

[0011] For an n-order control system, the order of the linear expansion tracking differentiator based on zero-point design is n+1, and the zero point design of the closed-loop transfer function of the system is -ω c ,ω c is the bandwidth of the linear error feedback control law;

[0012] The deviation between the differential signal outputted by the linear extended tracking differentiator (n+1th path) and the total disturbance estimate outputted by the linear extended state observer (n+1th path) serves as the linear error feedback control law's (n+1th path) input. The error feedback control variable generated by the linear error feedback control law is outputted to the controlled object and the linear extended state observer, respectively.

[0013] The present invention is further designed to:

[0014] The input of the linear expansion tracking differentiator is v0, and the output is v i, i=1,2,…,n+1;

[0015] The output of the linear extended state observer is z i , i=1,2,…,n+1;

[0016] The linear expansion tracks the output of the differentiator v i , i = 1, 2, ..., n + 1 and the output z of the linear extended state observer i , the deviation of i=1,2,…,n+1 is used as the input of the linear error feedback control law e i =v i -z i , i=1,2,…,n+1;

[0017] The output u0 of the linear error feedback control law is used as the controller output u after a gain of 1 / b0. The controller output u is divided into two paths, one path is used as the input of the controlled object, and the other path is used as the first input signal of the linear extended state observer after a gain of b0; the output y of the controlled object is used as the second input signal of the linear extended state observer.

[0018] A further design is that the expression of the linear extended state observer is as follows:

[0019]

[0020] Where y is the output of the controlled object; u is the controller output; b0 is the control gain of the ADRC; z i , i=1,2,…,n+1, is the output of the linear extended state observer; β i , i=1,2,…,n+1, is the coefficient polynomial of the linear extended state observer, satisfying s n+1 +β1s n +…+β n s+β n+1 =(s+ω o ) n+1 ,in, i=1,2,…,n+1,ω o is the bandwidth of the linear extended state observer;

[0021] The expression of the linear expansion tracking differentiator is as follows,

[0022]

[0023] Where v0 is the output signal; v i , i=1,2,…,n+1, is the output of the linear expansion tracking differentiator; r is the parameter of the linear expansion tracking differentiator;

[0024] The expression of the linear error feedback control law is as follows:

[0025]

[0026] Where, v i is the output of the tracking differentiator; i is the output of the extended state observer; k i is the parameter of the error feedback control law, satisfying s n +k1s n-1 +…+k n-1 s+k n =(s+ω c ) n ,in, i=1,2,…,n,ω c is the bandwidth of the linear error feedback law.

[0027] Further design is that when the system order is first order,

[0028] The expression of the second-order extended state observer is as follows:

[0029]

[0030] The expression of the second-order linear expansion tracking differentiator is as follows,

[0031]

[0032] Based on the above second-order extended state observer and second-order linear extended tracking differentiator, the corresponding linear error feedback control law is expressed as follows:

[0033] u0=(v2-z2)+ω c (v1-z1)

[0034] u=u0 / b0=((v2-z2)+ω c (v1-z1)) / b0;

[0035] Where, the bandwidth of the linear error feedback law ω c .

[0036] Further design is that when the system order is second order,

[0037] The expression of the third-order extended state observer is as follows:

[0038]

[0039] The expression of the third-order linear expansion tracking differentiator is as follows,

[0040]

[0041] Based on the above third-order linear extended tracking differentiator and third-order extended state observer, the corresponding linear error feedback control law is expressed as follows:

[0042]

[0043] Where, the bandwidth of the linear error feedback law ω c .

[0044] Further design is that the bandwidth ω of the linear error feedback law in the system c , the bandwidth of the extended state observer ω o and controller bandwidth ω a The setting relationship is:

[0045]

[0046] Where,

[0047] The setting relationship of the control quantity gain b0 is:

[0048] when

[0049] Where K is the steady-state gain of the controlled object, T is the inertia time constant, τ is the delay time, and γ is the fine-tuning factor of the control quantity gain b0.

[0050] The parameter tuning method of the LADRC control system based on the linear expansion tracking differentiator includes the following specific steps:

[0051] Step 1: Using the soaring curve method to obtain the second-order simplified object parameters of the controlled object through field data, including the steady-state gain K, inertia time constant T and delay time τ of the controlled object;

[0052] Step 2: Set the initial value of λ and obtain the bandwidth ω of the linear error feedback law according to the above parameters and parameter tuning formula. c The initial value of the extended state observer, the bandwidth ω o The initial value of the controller bandwidth ω a The initial value of and the initial value of the control gain b0 of the active disturbance rejection control system are obtained, and b0 satisfies K·b0>0, and debugging is performed based on the simulation platform to obtain the closed-loop response curve under the initial parameter value state;

[0053] The parameter tuning formula includes:

[0054] Controller bandwidth in the system ω a With the linear extended state observer ω o and linear error feedback control law bandwidth ω c The setting relationship of the formula is:

[0055]

[0056] Where,

[0057] The tuning relationship between the control quantity gain b0 is:

[0058] when

[0059] Where, γ is the fine-tuning factor of the control gain b0;

[0060] Step 3: To obtain a faster adjustment time, the control quantity gain b0 is parameterized: when K>0, the control quantity gain b0 is gradually reduced from the initial value so that the closed-loop control meets the performance index; when K<0, the control quantity gain b0 in the control system is gradually increased from the initial value so that the closed-loop control meets the performance index;

[0061] Step 4: To obtain the best adjustment time while meeting the performance index requirements, adjust the bandwidth ω of the linear error feedback control law. c Parameter tuning is performed on the control quantity gain b0: First, the bandwidth ω of the linear error feedback control law is reduced from the initial value. c Then adjust b0 according to step 3); repeat step 4) to adjust the parameter ω c and b0 tuning steps until the peak-to-valley ratio in the performance index drops to a value of 2:1, and the optimal linear error feedback control law bandwidth ω is obtained. c And the control quantity gain b0.

[0062] A further design is that, in step 2, the value range of the setting parameter λ is 0<λ≤1, preferably λ=0.1.

[0063] The further design is that in step 2, when the initial value of the control amount gain b0 is set, When, preferably when When γ=3, it is preferred.

[0064] The further design is that the performance indicators include a response curve overshoot of the closed-loop system not exceeding 5%, and a response curve peak-to-valley ratio not less than 2:1;

[0065] In steps 3 and 4, the adjustment range of the control quantity gain b0 is 0.0001~0.01, and the bandwidth of the linear error feedback law ω c The adjustment range is 0.001 to 0.01.

[0066] Compared with the prior art, the present invention has the following beneficial effects:

[0067] The present invention proposes a LADRC control system based on a linear expansion tracking differentiator and a parameter tuning method thereof. The control system adopts a linear expansion tracking differentiator based on zero point design. By designing an n+1 order linear expansion tracking differentiator LTD and adding a high-order differential signal, the lag and estimation error of the standard LADRC in signal tracking are effectively solved. The design sets the zero point of the closed-loop transfer function of the system to -ω c , and near the closed-loop dominant pole of the system, the influence of the zero point on the system is eliminated, so the step response of the system will not oscillate, solving the system overshoot problem.

[0068] The stability of a closed-loop control system is determined by the poles of the closed-loop transfer function, i.e., the roots of the closed-loop system's characteristic equation. The basic characteristics of the system's transient response are also dominated by the closed-loop poles, while the closed-loop zeros influence the shape of the system's transient response. This invention improves response speed and reduces operational delay by designing the closed-loop zeros.

[0069] The parameter tuning method of the present invention adopts the LADRC control system based on the linear expansion tracking differentiator, which is suitable for parameter tuning when a high-order object is reduced to a second-order object. a With the linear extended state observer ω o and linear error feedback control law bandwidth ω c The setting relationship of , and the setting relationship of the design control quantity gain b0, avoids blind trial and error in the setting and saves the parameter adjustment time.

[0070] Existing trial-and-error parameter tuning relies on extensive field experience and repeated adjustments. This process is cumbersome and time-consuming, making it difficult to find the optimal control parameters quickly. Although existing improved LADRCs can improve system response speed, stability, and robustness, parameter tuning in actual industrial applications still faces a conflict between overshoot, fast response, and interference rejection. Specifically, trial-and-error methods often struggle to strike a balance between controller gain parameters. If the gain is improperly selected, the system may respond too quickly, causing overshoot, or too slowly, affecting the system's dynamic performance and even, in extreme cases, causing system instability. Therefore, trial-and-error methods are not only inefficient but also prone to unstable and suboptimal system performance.

[0071] The parameter tuning method of this invention systematically determines the system's zero-pole configuration through theoretical analysis, avoiding the subjectivity and uncertainty of empirical trial and error. It accurately balances overshoot, response speed, and interference rejection, optimizing control effectiveness while ensuring system stability. This method reduces the time and cost of experimental debugging and improves tuning efficiency.

[0072] Compared with the existing second-order linear active disturbance rejection control system, the parameters that need to be tuned are: tracking differentiator parameter r, linear extended state observer ω o and its coefficient, linear error feedback control law bandwidth ω c and its coefficients; the parameters that need to be adjusted in the present invention are simplified to only the controller bandwidth ω a and control quantity gain b0, and has clear and specific theoretical guidance and clear and reasonable debugging rules, which are more suitable for on-site debugging habits and avoid the problems of vague parameter interpretation and unclear debugging process brought by traditional empirical tuning formulas.

[0073] As can be seen from the above, this invention solves the existing system's inability to balance output overshoot, fast response, and anti-interference performance. From an engineering application perspective, it can be widely applied to industrial process control, especially in energy and power, automation equipment, petrochemicals, metal smelting, and intelligent driving, showing significant advantages in process control. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] Figure 1 2. It is a system block diagram of a LADRC control system based on a linear expansion tracking differentiator in an embodiment;

[0075] Figure 2 This is a system block diagram of a first-order system in the embodiment;

[0076] Figure 3 This is a system block diagram of a second-order system in the embodiment;

[0077] Figure 4 This is a closed-loop response curve effect diagram of a controlled object with no delay time using the control system and parameter tuning method of the present invention;

[0078] Figure 5 A comparison chart of closed-loop response curves of the control system and parameter tuning method of the present invention and those of standard LADRC and PID;

[0079] Figure 6 This is a comparison chart of the control system and parameter tuning method of the present invention with the standard LADRC and PID control output curves;

[0080] Figure 7 This is a comparison chart of the anti-interference performance of the control system and parameter tuning method of the present invention and the standard LADRC and PID closed-loop response curves;

[0081] Figure 8 This is a comparison chart of the anti-interference performance of the control system and parameter tuning method of the present invention and the standard LADRC and PID control output curves;

[0082] Figure 9This is a diagram showing the effect of adjusting the closed-loop response curve of the main steam temperature system model using the parameter adjustment method of the present invention;

[0083] Figure 10 Comparison chart of closed-loop response curves of the main steam temperature system model using the control system and parameter tuning method of the present invention and standard LADRC and PID;

[0084] Figure 11 A comparison chart of the output curves of the main steam temperature system model using the control system and parameter tuning method of the present invention and standard LADRC and PID control. DETAILED DESCRIPTION

[0085] The present invention will be further described below with reference to specific embodiments and corresponding drawings.

[0086] The linear tracking differentiator (LTD) used in the standard linear active disturbance rejection control (LADRC) has certain hysteresis and estimation errors when tracking signals. Especially during fast response, this can easily lead to excessive overshoot of the system output, thereby generating unnecessary oscillations and reducing the control quality of the system.

[0087] The improved linear active disturbance rejection control (LADRC) system of this invention improves the traditional linear tracking differentiator (LTD) by adopting a linear expansion tracking differentiator. This effectively solves the overshoot and oscillation problems that occur during the traditional LTD's rapid response. The LADRC, through its optimized structural design, enhances its ability to accurately track the system's input signal, thereby improving the system's response speed, stability, and robustness.

[0088] The following is a comparative analysis of the present invention and the standard linear active disturbance rejection control (LADRC):

[0089] The standard n-order LADRC control system includes a controlled object and an active disturbance rejection controller, which includes a linear tracking differentiator, a linear extended state observer, and a linear error feedback control law.

[0090] The LADRC control system based on the linear expansion tracking differentiator designed in the present invention is different from the standard LADRC control system in that the problem of overshoot in the step response of the closed-loop transfer function after the LADRC and the control object are solved by transforming the n-order linear tracking differentiator into an n+1-order linear expansion tracking differentiator.

[0091] After the standard n-order LADRC expansion, the controller output u=G v (s)v0-G y The closed-loop transfer function after (s)y is combined with the controlled object is:

[0092] Where v0 is the input of the linear tracking differentiator; y is the output of the controlled object; is the transfer function of the controlled object;

[0093] The poles of the closed-loop transfer function are the closed-loop characteristic equation 1+G y (s)G(s)=0. The closed loop stability is only related to G y (s) related to G v (s) is irrelevant. To further analyze the theoretical characteristics of the closed-loop transfer function, the bandwidth ω of the linear error feedback law is used. c and the bandwidth ω of the extended state observer o Define D1-D4, let

[0094] D1=s n+1 +β1s n +…+β n s+β n+1 =(s+ω o ) n+1

[0095] D2=s n +k n s n-1 +…+k2s+k1=(s+ω c ) n

[0096] D1D2=D3s n+1 +D4

[0097] but

[0098] In the formula, b0 is the control gain, G TD is the transfer function of the linear tracking differentiator TD;

[0099] Assuming that a linear tracking differentiator TD is used in the standard LADRC, the transfer function of the linear tracking differentiator TD can be expressed as The parameter of the linear expansion tracking differentiator r = ω o ,but

[0100]

[0101] In particular, when n = 2, that is, the system order is second order, we have

[0102] D1=(s+ω o ) 3 , D2=(s+ω c ) 2

[0103]

[0104] Assuming a second-order object

[0105] Among them, parameter ζ is the damping ratio, K is the steady-state gain, T is the inertia time constant, and τ is the delay time;

[0106] Then there is,

[0107]

[0108] It can be proved that the absolute value of the real part of the closed-loop transfer function pole is at ω c (ω c ≤ω o ), but there is a zero point equal to -ω c / 2, the zero point is located on the right side of the closed-loop dominant pole of the system (close to the imaginary axis). The zero point will reduce the system damping, so it can be seen that the step response of the closed-loop system has overshoot.

[0109] In fact, if LADRC can fully compensate for internal and external disturbances, the object will be transformed into an integral series type, and the ideal closed-loop transfer function must be When its step response is n≥2, the closed-loop transfer function introduces zero points. These zero points destroy the pure attenuation characteristics of the system, introduce oscillation components into the system response, and inevitably cause overshoot.

[0110] Compared with the existing

[0111] The LADRC control system based on the linear expansion tracking differentiator of the present invention transforms the n-order LTD into the n+1-order LTD. The transfer function of the n+1-order LTD is: Then the closed-loop transfer function generated by the combination of the improved LADRC of the present invention and the controlled object is obtained, that is, the closed-loop transfer function of the system Since the zero point of D2 is -ω c And near the dominant pole of the system closed loop, the zero point of D2 is the zero point of the system closed loop transfer function. The influence of the zero point on the system is eliminated, so the step response of the system will not oscillate, thereby solving the overshoot problem.

[0112] Compared with the existing

[0113] The LADRC control system based on the linear expansion tracking differentiator of the present invention designs the controller bandwidth ω a With the linear extended state observer ω o and linear error feedback control law bandwidth ω c The tuning relationship of and the tuning formula of the control quantity gain b0 avoid blind trial and error and save parameter adjustment time. The details are as follows:

[0114] Still for the above second-order objects: Then there is

[0115]

[0116] D1=(s+ω o ) 3 , D2=(s+ω c ) 2

[0117]

[0118] therefore,

[0119]

[0120]

[0121] make get

[0122] Therefore, through theoretical analysis, the controller bandwidth ω is obtained a With the linear extended state observer ω o and linear error feedback control law bandwidth ω c The setting relationship avoids blind trial and error during parameter setting and saves parameter adjustment time.

[0123] Furthermore, G y1 ,have to

[0124]

[0125] Furthermore, the constant term is denoted as

[0126]

[0127] Therefore, when choosing When

[0128] Therefore, through theoretical analysis, the tuning relationship of the control quantity gain b0 was obtained, which avoided blind trial and error and saved parameter adjustment time.

[0129] The parameter tuning method of this invention systematically determines the zero-pole configuration through theoretical analysis, avoiding the subjectivity and uncertainty of empirical trial and error. It accurately balances overshoot, response speed, and interference rejection, optimizing control effectiveness while ensuring system stability. This method reduces the time and cost of experimental debugging and improves tuning efficiency.

[0130] Example 1:

[0131] like Figure 1As shown, the present invention provides a LADRC control system based on a linear expansion tracking differentiator, including an active disturbance rejection controller and a controlled object. The active disturbance rejection controller includes a linear expansion tracking differentiator based on zero-point design, a linear expansion state observer, and a linear error feedback control law.

[0132] For an n-order control system, the order of the linear expansion tracking differentiator based on zero-point design is n+1, and the zero-point design of the closed-loop transfer function of the system is -ω c ,ω c is the bandwidth of the linear error feedback control law;

[0133] The deviation between the differential signal outputted by the linear extended tracking differentiator (n+1) and the total disturbance estimate outputted by the linear extended state observer (n+1) serves as the linear error feedback control law's (n+1) input. The error feedback control variable generated by the linear error feedback control law is outputted to the controlled object and the linear extended state observer, respectively.

[0134] The present invention effectively solves the lag and estimation error of the standard LADRC in signal tracking by designing an n+1 order linear expansion tracking differentiator LTD and adding a high-order differential signal. The design sets the zero point of the closed-loop transfer function of the system to -ω c , and near the closed-loop dominant pole of the system, the influence of the zero point on the system is eliminated, so the step response of the system will not oscillate, solving the system overshoot problem.

[0135] Example 2:

[0136] like Figure 1 As shown, the present invention provides a LADRC control system based on a linear expansion tracking differentiator.

[0137] The input of the linear expansion tracking differentiator is v0 and the output is v i , i=1,2,…,n+1;

[0138] The output of the linear extended state observer is z i , i=1,2,…,n+1;

[0139] The linear expansion tracks the output v of the differentiator i , i = 1, 2, ..., n + 1 and the output z of the linear extended state observer i , the deviation of i=1,2,…,n+1 is used as the input of the linear error feedback control law e i =v i -z i , i=1,2,…,n+1;

[0140] The output u0 of the linear error feedback control law is multiplied by a gain of 1 / b0 as the controller output u. The controller output u is divided into two paths, one as the input of the controlled object, and the other as the first input signal of the linear extended state observer after a gain of b0; the output y of the controlled object serves as the second input signal of the linear extended state observer.

[0141] By designing an n+1 order linear expansion tracking differentiator LTD and adding high-order differential signals, the zero point of the closed-loop transfer function of the system is set to -ω c , the influence of the zero point on the system is eliminated, so the step response of the system will not oscillate, solving the problem of system overshoot.

[0142] The expression of the linear extended state observer is as follows,

[0143]

[0144] Where y is the output of the controlled object; u is the controller output; b0 is the control gain of the ADRC; z i , i=1,2,…,n+1, is the output of the linear extended state observer; β i , i=1,2,…,n+1, is the coefficient polynomial of the linear extended state observer, satisfying s n+1 +β1s n +…+β n s+β n+1 =(s+ω o ) n+1 ,in, i=1,2,…,n+1,ω o is the bandwidth of the linear extended state observer;

[0145] The expression of the linear expansion tracking differentiator is as follows,

[0146]

[0147] Where v0 is the output signal; v i , i=1,2,…,n+1, is the output of the linear expansion tracking differentiator; r is the parameter of the linear expansion tracking differentiator;

[0148] The expression of the linear error feedback control law is as follows,

[0149]

[0150] Where, v i is the output of the tracking differentiator; i is the output of the extended state observer; k i is the parameter of the error feedback control law, satisfying s n +k1sn-1 +…+k n-1 s+k n =(s+ω c ) n ,in, i=1,2,…,n,ω c is the bandwidth of the linear error feedback law.

[0151] Example 3:

[0152] like Figure 2 As shown, based on the first embodiment, this example proposes a first-order LADRC control system based on a linear expansion tracking differentiator.

[0153] The expression of the second-order extended state observer is as follows:

[0154]

[0155] The expression of the second-order linear expansion tracking differentiator is as follows,

[0156]

[0157] Based on the above second-order extended state observer and second-order linear extended tracking differentiator, the corresponding linear error feedback control law is expressed as follows:

[0158] u0=(v2-z2)+ω c (v1-z1)

[0159] u=u0 / b0=((v2-z2)+ω c (v1-z1)) / b0;

[0160] Where, the bandwidth of the linear error feedback law ω c .

[0161] Example 4:

[0162] like Figure 3 As shown, based on the first embodiment, this example proposes a second-order LADRC control system based on a linear expansion tracking differentiator.

[0163] The expression of the third-order extended state observer is as follows:

[0164]

[0165] The expression of the third-order linear expansion tracking differentiator is as follows,

[0166]

[0167] Based on the above third-order linear extended tracking differentiator and third-order extended state observer, the corresponding linear error feedback control law is expressed as follows:

[0168]

[0169] Where, the bandwidth of the linear error feedback law ω c .

[0170] Embodiment 5:

[0171] The LADRC control system based on the linear expansion tracking differentiator of the present invention further designs a parameter setting relationship. The bandwidth ω of the linear error feedback law in the system c , the bandwidth of the extended state observer ω o and controller bandwidth ω a The setting relationship is:

[0172]

[0173] Where,

[0174] The setting relationship of the control quantity gain b0 is:

[0175] when

[0176] Where K is the steady-state gain of the controlled object, T is the inertia time constant, τ is the delay time, and γ is the fine-tuning factor of the control quantity gain b0.

[0177] Example 6:

[0178] Based on the LADRC control system based on the linear expansion tracking differentiator of the present invention, the specific parameter setting method includes the following specific steps:

[0179] Step 1: Using the soaring curve method to obtain the second-order simplified object parameters of the controlled object through field data, including the steady-state gain K, inertia time constant T and delay time τ of the controlled object;

[0180] Step 2: Set the initial value of λ and obtain the bandwidth ω of the linear error feedback law according to the above parameters and parameter tuning formula. c The initial value of the extended state observer, the bandwidth ω o The initial value of the controller bandwidth ω a The initial value of and the initial value of the control gain b0 of the active disturbance rejection control system are obtained, and b0 satisfies K·b0>0, and debugging is performed based on the simulation platform to obtain the closed-loop response curve under the initial parameter value state;

[0181] The parameter tuning formula includes:

[0182] Controller bandwidth in the system ω a With the linear extended state observer ω o and linear error feedback control law bandwidth ω c The setting relationship of the formula is:

[0183]

[0184] Where,

[0185] The tuning relationship between the control quantity gain b0 is:

[0186] when

[0187] Where, γ is the fine-tuning factor of the control gain b0;

[0188] Step 3: To obtain a faster adjustment time, the control quantity gain b0 is parameterized: when K>0, the control quantity gain b0 is gradually reduced from the initial value, that is, γ is reduced, so that the closed-loop control meets the performance index; when K<0, the control quantity gain b0 in the control system is gradually increased from the initial value, that is, γ is reduced, so that the closed-loop control meets the performance index;

[0189] Step 4: To obtain the best adjustment time while meeting the performance index requirements, adjust the bandwidth ω of the linear error feedback control law. c Parameter tuning is performed on the control quantity gain b0: First, the bandwidth ω of the linear error feedback control law is reduced from the initial value. c , that is, reducing the controller bandwidth ω a Then adjust b0 according to step 3); repeat step 4) to adjust the parameter ω c and b0 tuning steps until the peak-to-valley ratio in the performance index drops to a value of 2:1, and the optimal linear error feedback control law bandwidth ω is obtained. c And the control quantity gain b0.

[0190] In step 2, the value range of the parameter λ is set to 0<λ≤1, preferably λ=0.1.

[0191] In step 2, when the initial value of the control quantity gain b0 is set, When, preferably when When γ=3, it is preferred.

[0192] Performance indicators include the closed-loop system response curve overshoot not exceeding 5%, and the response curve peak-to-valley ratio not less than 2:1; Test Example 1:

[0193] For the second-order time-delay-free model The second-order LADRC control system based on the linear expansion tracking differentiator proposed in the fourth embodiment of the present invention (hereinafter referred to as the improved LADRC) and the parameter tuning method given in the sixth embodiment are used for simulation verification.

[0194] The simulation is solved using the fixed-step Euler method with a step size of 0.01.

[0195] Select standard LADRC and The open-loop response curve of is used for comparison.

[0196] Determine the system parameters T=1, K=1, since τ=0, take Get And let λ = 0.1, and then we get

[0197] The step response curve obtained by running is as follows Figure 4 As shown, the simulation results show that the overshoot of the standard LADRC is about 10% under the premise of maintaining b0=b, which is much higher than that of the improved LADRC. The results show that the control effect is significantly improved after the improvement, which proves the necessity of the present invention, that is, the overshoot problem of the standard LADRC is solved without time delay.

[0198] Test Example 2:

[0199] This example is for the second-order inertial delay model The second-order LADRC control system based on the linear expansion tracking differentiator (hereinafter referred to as the improved LADRC) proposed in Example 4 of the present invention and the parameter tuning method given in Example 6 were used for simulation verification. Standard LADRC and PID were selected for comparison.

[0200] The simulation is solved using the fixed-step Euler method with a step size of 0.01.

[0201] Determine the system parameters T = 1, K = 1, since τ = 1, take γ = 3, Get And let λ = 0.1, and then we get

[0202]

[0203] Among them, λ is the setting parameter; K is the steady-state gain, T is the inertia time constant, τ is the delay time; ω a is the controller bandwidth, ω o is a linear extended state observer, ω c is the bandwidth of the linear error feedback control law.

[0204] The above parameters are used as initial values ​​and parameter tuning is performed according to the parameter tuning method of the present invention.

[0205] During the parameter adjustment process, gradually reduce the value of b0 so that the closed-loop control meets the performance indicators; then reduce ω c , gradually reduce b0 from the current value, repeat the above operation until the adjustment time reaches the minimum, the adjustment range of b0 is 0.0001~0.01, ω c The adjustment range is 0.001 to 0.01. At the same time, the PID parameter tuning method is used to adjust the PID parameters to the optimal performance.

[0206] In PID, K c is the proportional constant, K p is the proportional gain, K d is the differential gain, T d is the differential time, T i is the integral time constant. The parameters adjusted to the optimal parameters of the present invention are ω o1 、ω c1 and b o1 .

[0207] The simulation results show that the step response curve and control output curve of the closed-loop system are as follows: Figure 5 、 Figure 6 As shown in the figure, the maximum overshoots of the improved LADRC, standard LADRC, and PID are 4.88%, 4.90%, and 4.96%, respectively; and the adjustment times are 2.95s, 3.54s, and 3.84s, respectively. The specific parameters after tuning are as follows:

[0208]

[0209]

[0210] Simulation results show that the LADRC control system based on the linear expansion tracking differentiator of the present invention has the smallest maximum overshoot and the shortest adjustment time.

[0211] Test Example 3:

[0212] The run time of Test Example 2 was set to 50 seconds. A step disturbance with an amplitude of 0.2 was added at the 30th second to test the disturbance rejection performance of the second-order LADRC control system based on a linear expansion tracking differentiator (hereinafter referred to as the improved LADRC) proposed in Example 4 of the present invention. Standard LADRC and PID were selected for comparison.

[0213] The simulation is solved using the fixed-step Euler method with a step size of 0.01.

[0214] The expression used for the configuration of the PID controller is

[0215] The anti-disturbance simulation is performed using the parameters obtained in Test Example 2. The running time is set to 50 seconds and a step disturbance with an amplitude of 0.2 is added at the 30th second. The step response curve and control output curve of the closed-loop system are obtained as follows: Figure 7 、 Figure 8 As shown in the figure, the improved LADRC has a faster adjustment speed and better anti-disturbance capability than the standard LADRC and PID. The detailed parameters of the anti-disturbance effect of each controller after adding a step disturbance are shown below:

[0216] In PID,

[0217]

[0218] Simulation results show that the LADRC control system based on the linear expansion tracking differentiator of the present invention has the smallest maximum overshoot and the shortest adjustment time.

[0219] Test Example 4:

[0220] This example is for a main steam temperature system model, where is the leading zone model (℃ / %),

[0221] is the inert zone model (℃ / ℃). This model can be simplified to a second-order model A second-order LADRC control system based on a linear expansion tracking differentiator (hereinafter referred to as improved LADRC) proposed in the fourth embodiment of the present invention was used for simulation tests, and standard LADRC and PID were selected for comparison.

[0222] The simulation is solved using the fixed-step Euler method with a step size of 0.01.

[0223] The expression used for the configuration of the PID controller is

[0224] In PID, K c is the proportional constant, K p is the proportional gain, K d is the differential gain, T d is the differential time, T i is the integration time constant.

[0225] During the parameter adjustment process, first determine the system parameters T = 50, K = -2.51, and obtain And let λ = 0.1, and then we get

[0226] Since τ=60, take γ=3, Debugging is performed with the above parameters as initial values. At this point, the closed-loop control meets the performance indicators, but the adjustment time is not optimal.

[0227] Gradually reduce the absolute value of b0 to obtain a new set of parameters ω c1 =0.0409,ω o1 =0.409,b 01 =-0.0174, at this time the step response curve of the closed-loop system has met the performance index. Finally, reduce ω c The absolute value of b0 is gradually reduced, and the adjustment range of b0 is 0.0001~0.01, ω c The adjustment range is 0.001~0.01. The optimal parameter ω is obtained c2 =0.0209,ω o2 =0.209,b 02 =-0.0066, the closed-loop system step response curves obtained by running the above three sets of parameters are compared. Figure 9 shown.

[0228] The step response curve comparison diagram and control output curve comparison diagram of the main steam temperature system model closed-loop system are obtained by using the optimal parameters of improved ADRC, standard LADRC and PID, respectively, as shown in Figure 2. Figure 10 、 Figure 11 As shown in the figure, the improved LADRC significantly improves control performance. The maximum overshoot of the improved LADRC, standard LADRC, and PID is 4.99%, 4.90%, and 4.84%, respectively. The adjustment time is 184.90s, 210.86s, and 217.31s, respectively. A comparison of specific parameters and control performance is shown below.

[0229]

[0230]

[0231] The test results show that the LADRC control system based on the linear expansion tracking differentiator of the present invention has the smallest maximum overshoot, the shortest adjustment time, and a significantly improved control effect.

Claims

1. A LADRC control system based on a linear expansion tracking differentiator, comprising an active disturbance rejection controller and a controlled object, characterized in that: The active disturbance rejection controller includes a linear expansion tracking differentiator based on zero-point design, a linear expansion state observer and a linear error feedback control law. For an n-order control system, the order of the linear expansion tracking differentiator based on zero-point design is n+1, and the zero point design of the closed-loop transfer function of the system is -ω c ,ω c is the bandwidth of the linear error feedback control law; The deviation between the differential signal outputted by the linear extended tracking differentiator (n+1th path) and the total disturbance estimate outputted by the linear extended state observer (n+1th path) serves as the linear error feedback control law's (n+1th path) input. The error feedback control variable generated by the linear error feedback control law is outputted to the controlled object and the linear extended state observer, respectively.

2. The LADRC control system based on the linear expansion tracking differentiator according to claim 1, characterized in that: The input of the linear expansion tracking differentiator is v0, and the output is v i , i=1,2,…,n+1; The output of the linear extended state observer is z i , i=1,2,…,n+1; The linear expansion tracks the output of the differentiator v i , i = 1, 2, ..., n + 1 and the output z of the linear extended state observer i , the deviation of i=1,2,…,n+1 is used as the input of the linear error feedback control law e i =v i -z i , i=1,2,…,n+1; The output u0 of the linear error feedback control law is used as the controller output u after a gain of 1 / b0. The controller output u is divided into two paths, one path is used as the input of the controlled object, and the other path is used as the first input signal of the linear extended state observer after a gain of b0; the output y of the controlled object is used as the second input signal of the linear extended state observer.

3. The LADRC control system based on the linear expansion tracking differentiator according to claim 2, characterized in that: The expression of the linear extended state observer is as follows: Where y is the output of the controlled object; u is the controller output; b0 is the control gain of the ADRC; z i , i=1,2,…,n+1, is the output of the linear extended state observer; β i , i=1,2,…,n+1, is the coefficient polynomial of the linear extended state observer, satisfying s n+1 +β1s n +…+β n s+β n+1 =(s+ω o ) n+1 ,in, ω o is the bandwidth of the linear extended state observer; The expression of the linear expansion tracking differentiator is as follows, Where v0 is the output signal; v i , i=1,2,…,n+1, is the output of the linear expansion tracking differentiator; r is the parameter of the linear expansion tracking differentiator; The expression of the linear error feedback control law is as follows: Where, v i is the output of the tracking differentiator; i is the output of the extended state observer; k i is the parameter of the error feedback control law, satisfying s n +k1s n-1 +…+k n-1 s+k n =(s+ω c ) n ,in, ω c is the bandwidth of the linear error feedback law.

4. The LADRC control system based on the linear expansion tracking differentiator according to claim 3, characterized in that: When the system order is first order, The expression of the second-order extended state observer is as follows: The expression of the second-order linear expansion tracking differentiator is as follows, Based on the above second-order extended state observer and second-order linear extended tracking differentiator, the corresponding linear error feedback control law is expressed as follows: u0=(v2-z2)+ω c (v1-z1) u=u0 / b0=((v2-z2)+ω c (v1-z1)) / b0; Where, the bandwidth of the linear error feedback law ω c .

5. The LADRC control system based on the linear expansion tracking differentiator according to claim 3, characterized in that: When the system order is second order, The expression of the third-order extended state observer is as follows: The expression of the third-order linear expansion tracking differentiator is as follows, Based on the above third-order linear extended tracking differentiator and third-order extended state observer, the corresponding linear error feedback control law is expressed as follows: Where, the bandwidth of the linear error feedback law ω c .

6. The LADRC control system based on a linear expansion tracking differentiator according to any one of claims 1 to 5, characterized in that: The bandwidth ω of the linear error feedback law in the system c , the bandwidth of the extended state observer ω o and controller bandwidth ω a The setting relationship is: Where, The setting relationship of the control quantity gain b0 is: when Where K is the steady-state gain of the controlled object, T is the inertia time constant, τ is the delay time, and γ is the fine-tuning factor of the control quantity gain b0.

7. The parameter tuning method for the LADRC control system based on the linear expansion tracking differentiator according to any one of claims 1 to 5, characterized in that: The specific steps include: Step 1: Using the soaring curve method to obtain the second-order simplified object parameters of the controlled object through field data, including the steady-state gain K, inertia time constant T and delay time τ of the controlled object; Step 2: Set the initial value of λ and obtain the bandwidth ω of the linear error feedback law according to the above parameters and parameter tuning formula. c The initial value of the extended state observer, the bandwidth ω o The initial value of the controller bandwidth ω a The initial value of and the initial value of the control gain b0 of the active disturbance rejection control system are obtained, and b0 satisfies K·b0>0, and debugging is performed based on the simulation platform to obtain the closed-loop response curve under the initial parameter value state; The parameter setting formula includes: Controller bandwidth in the system ω a With the linear extended state observer ω o and linear error feedback control law bandwidth ω c The setting relationship of the formula is: Where, The tuning relationship between the control quantity gain b0 is: Where, γ is the fine-tuning factor of the control gain b0; Step 3: To obtain a faster adjustment time, the control quantity gain b0 is parameterized: when K>0, the control quantity gain b0 is gradually reduced from the initial value so that the closed-loop control meets the performance index; when K<0, the control quantity gain b0 in the control system is gradually increased from the initial value so that the closed-loop control meets the performance index; Step 4: To obtain the best adjustment time while meeting the performance index requirements, adjust the bandwidth ω of the linear error feedback control law. c Parameter tuning is performed on the control quantity gain b0: First, the bandwidth ω of the linear error feedback control law is reduced from the initial value. c , then adjust b0 according to step 3; repeat step 4 to adjust the parameter ω c and b0 tuning steps until the peak-to-valley ratio in the performance index drops to a value of 2:1, and the optimal linear error feedback control law bandwidth ω is obtained. c And the control quantity gain b0.

8. The parameter tuning method of the LADRC control system based on the linear expansion tracking differentiator according to claim 7 is characterized in that: In step 2, the value range of the setting parameter λ is 0<λ≤1.

9. The parameter tuning method of the LADRC control system based on the linear expansion tracking differentiator according to claim 7 is characterized in that: In step 2, when the initial value of the control quantity gain b0 is set, hour, when When γ=3, 10. The parameter tuning method of the LADRC control system based on the linear expansion tracking differentiator according to claim 8, characterized in that: The performance indicators include that the overshoot of the closed-loop system response curve does not exceed 5%, and the peak-to-valley ratio of the response curve is not less than 2:1; In steps 3 and 4, the adjustment range of the control quantity gain b0 is 0.0001~0.01, and the bandwidth of the linear error feedback law ω c The adjustment range is 0.001 to 0.01.

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