LADRC control system based on linear expansion tracking differentiator and parameter setting method thereof

Abstract

The invention discloses an LADRC control system based on a linear expansion tracking differentiator and a parameter setting method of the LADRC control system, according to the control system, the n + 1-order linear expansion tracking differentiator is designed, high-order differential signals are increased, and lagging and estimation errors of a standard LADRC during signal tracking are effectively solved. The problem that overshoot exists in closed-loop transfer function step response after linear LADRC and a control object are combined is solved. The setting method has clear and specific theoretical guidance and clear and reasonable debugging rules, is more suitable for field debugging habits, and avoids the problems of fuzzy parameter interpretation and unclear debugging process caused by a traditional experience setting formula.

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 setting method thereof. Background Art

[0002] Active Disturbance Rejection Control (ADRC) has been widely used in the field of industrial control due to its excellent state estimation capability and disturbance suppression performance. Since the linear tracking differentiator (LTD) can effectively extract a smooth derivative signal from the noise, thus avoiding the problem of traditional differential amplification noise, the closed-loop step response of the standard linear ADRC control system has an overshoot problem, which may cause unnecessary oscillations in the system when responding quickly, reducing the stability and control quality of the system.

[0003] Based on the intrinsic relationship between the open-loop transfer function of the feedback control system and its closed-loop characteristic equation, W. R. Evans proposed a simple and practical graphical method for obtaining the closed-loop characteristic roots - the root locus method. The root locus is the trajectory of the closed-loop characteristic roots moving on the s plane when a parameter in the system changes 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, that is, the roots of the closed-loop system characteristic equation. The basic characteristics of the system transient response are also dominated by the closed-loop poles.

[0004] For the research of feedback control system, the first thing is to design the closed-loop poles and closed-loop zeros of the system when the structure and parameters of the system are known; secondly, in order to make the system have the desired control performance, it is necessary to consider the influence of the changes in the system structure parameters on its closed-loop poles and closed-loop zeros. The closed-loop poles of the system are the roots of its closed-loop characteristic equation. For high-order systems, it is difficult to solve its closed-loop poles using analytical methods. In particular, it is necessary to conduct a large number of complex designs to examine the general laws of the closed-loop poles of the system as the structure and parameters change.

[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 vital role in LADRC. By tracking the changes in the input signal and generating a smooth derivative signal, LTD can effectively suppress noise interference and provide stable state estimation. The essence of LTD is to reduce the fluctuation of the input signal through a filtering mechanism, making the derivative estimation of the signal smoother and more accurate, especially in a noisy environment, which can effectively improve the robustness and anti-interference ability of the system. By filtering the system input signal, LTD avoids the problem of high-frequency noise amplification caused by direct differentiation, so that the system can better track the set value and improve control accuracy. Combining LADRC with LTD can significantly improve the anti-interference performance of the system, while enhancing the smoothness and stability of signal tracking. However, in industrial practice, the parameter setting of LADRC requires rich field experience and trial and error, and there is a problem of how to balance the 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 solution of the present invention is as follows:

[0010] A LADRC control system based on a linear expansion tracking differentiator comprises an active disturbance rejection controller and a controlled object, wherein the active disturbance rejection controller comprises 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 output by the n+1th output of the linear extended tracking differentiator and the total disturbance estimate output by the n+1th output of the linear extended state observer is used as the n+1th input of the linear error feedback control law. The error feedback control quantity generated by the linear error feedback control law is output 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 v 0 , the output is vi , 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 v of the differentiator i , i = 1, 2, ..., n + 1 and the output z of the linear extended state observer i , i=1,2,…,n+1 as the input of the linear error feedback control law e i =v i -z i , i=1,2,…,n+1;

[0017] The output u of the linear error feedback control law 0 1 / b 0 The controller output u is divided into two paths, one as the input of the controlled object and the other as the output of the controller. 0 The multiplication gain is used as the first input signal of the linear extended state observer; the output y of the controlled object is used as the second input signal of the linear extended state observer.

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

[0019]

[0020] In the formula, y is the output of the controlled object; u is the controller output; b 0 is the control quantity gain of the ADRC; 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 +β 1 s 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] In the formula, v 0is 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] In the formula, 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 +k 1 s 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] u 0 =(v 2 -z 2 )+ω c (v 1 -z 1 )

[0034] u=u 0 / b 0 =((v 2 -z 2 )+ω c (v 1 -z 1 )) / b 0 ;

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

[0036] The 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 is c .

[0044] A 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] In the formula,

[0047] Control quantity gain b 0 The setting relationship is:

[0048] when

[0049] In the formula, 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 control gain b 0 The fine-tuning factor.

[0050] The parameter setting 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 setting formula. cThe initial value of the extended state observer, the bandwidth ω o The initial value of the controller bandwidth ω a The initial value of the control quantity gain b of the ADRC system 0 The initial value of b 0 Satisfy K·b 0 >0, and debug based on the simulation platform to obtain the closed-loop response curve under the initial value state of the parameters;

[0053] The parameter setting formula includes:

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

[0055]

[0056] In the formula,

[0057] and control quantity gain b 0 The setting relationship of:

[0058] when

[0059] In the formula, γ is the control gain b 0 The fine-tuning factor of

[0060] Step 3: To obtain faster adjustment time, adjust the control gain b 0 Perform parameter tuning: When K>0, gradually reduce the control gain b from the initial value 0 , so that the closed-loop control meets the performance index; when K < 0, gradually increase the control gain b in the control system from the initial value 0 , so that the closed-loop control meets the performance indicators;

[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 and control quantity gain b 0 Perform parameter tuning: First, reduce the bandwidth ω of the linear error feedback control law from the initial value c , and then adjust b according to step 3) 0 ; Repeat step 4) for parameter ω c and b 0 The tuning steps are as follows, 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 control quantity gain b 0 .

[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] A further design is that in step 2, the control amount gain b 0 When the initial value of When when When γ=3,

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

[0065] In steps 3 and 4, the control gain b 0 The adjustment range is 0.0001~0.01, and the bandwidth of the linear error feedback law is ω c The adjustment range is 0.001~0.01.

[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 setting 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 dominant pole of the closed-loop system, 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.

[0068] 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 characteristic equation. The basic characteristics of the system transient response are also dominated by the closed-loop poles, and the closed-loop zero point affects the form of the system transient response. The present invention improves the response speed and reduces the action delay by designing the closed-loop zero point.

[0069] The parameter setting method of the present invention adopts the LADRC control system based on the linear expansion tracking differentiator, which is suitable for parameter setting when a high-order object is reduced to a second-order object. a With the linear extended state observer ω o and the linear error feedback control law bandwidth ω c The setting relationship of the control quantity and the design control gain b 0 The setting relationship avoids blind trial and error in setting and saves time in parameter adjustment.

[0070] The existing trial-and-error method for parameter tuning relies on rich field experience and repeated adjustments. The process is cumbersome and time-consuming, and it is difficult to find the best control parameters in a short time. Although the existing improved LADRC can improve the response speed, stability and robustness of the system, in actual industrial applications, parameter tuning still faces the contradiction between overshoot, fast response and anti-disturbance performance. Specifically, the trial-and-error method often finds it difficult to strike a balance in the controller's gain parameters. If the gain is not properly selected, the system may respond too quickly, causing overshoot, or respond too slowly, affecting the system's dynamic performance, and even making the system unstable in extreme cases. Therefore, the trial-and-error method is not only inefficient, but also easily leads to unstable and unsatisfactory system performance.

[0071] The parameter setting method of the present invention systematically determines the zero-pole configuration of the system through theoretical analysis, avoiding the subjectivity and uncertainty of empirical trial and error. It can accurately balance overshoot, response speed and anti-interference performance, optimize the control effect while ensuring system stability. The method reduces the time and cost of test debugging and improves the setting efficiency.

[0072] Compared with the existing second-order linear active disturbance rejection control system, the parameters that need to be adjusted are: tracking differentiator parameter r, linear extended state observer ω o and its coefficients, linear error feedback control law bandwidth ω c and its coefficient; the parameters that need to be adjusted in the present invention are simplified to only the controller bandwidth ω a and control quantity gain b 0 , and has clear and specific theoretical guidance and clear and reasonable debugging rules, which is more suitable for on-site debugging habits and avoids the problems of vague parameter interpretation and unclear debugging process brought about by traditional empirical setting formulas.

[0073] As can be seen from the above, the invention solves the problem that the existing system cannot take into account the output overshoot, fast response and anti-interference performance. From the perspective of engineering application, it can be widely used in industrial process control, especially in the process control of energy and power, automation equipment, petrochemical, metal smelting and intelligent driving, showing significant advantages. BRIEF DESCRIPTION OF THE DRAWINGS

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

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

[0076] Figure 3 It 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 without delay time using the control system and parameter setting method of the present invention;

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

[0079] Figure 6 A comparison chart of the control system and parameter setting method of the present invention with the standard LADRC and PID control output curves;

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

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

[0082] Fig. 9 This 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] Fig.10 Comparison chart of closed-loop response curves of the main steam temperature system model using the control system and parameter setting method of the present invention and the standard LADRC and PID;

[0084] Fig.11 The main steam temperature system model adopts the control system and parameter setting method of the present invention and the standard LADRC and PID control output curve comparison diagram. DETAILED DESCRIPTION

[0085] The present invention will be further described below in conjunction with 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 error when tracking signals. Especially in fast response, it is easy to cause excessive overshoot of system output, thus generating unnecessary oscillation and reducing the control quality of the system.

[0087] The improved linear active disturbance rejection control (LADRC) of the present invention improves the traditional linear tracking differentiator (LTD) and adopts a linear expansion tracking differentiator, which effectively solves the overshoot and oscillation problems generated by the traditional LTD in the fast response process. The linear expansion tracking differentiator enhances the ability to accurately track the input signal of the system by optimizing its own structural design, thereby improving the response speed, stability and robustness of the system.

[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 by 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 combined is 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)v 0 -G y (s)y, the closed-loop transfer function after combining with the controlled object is:

[0092] In the formula, v 0 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 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] D 1 =s n+1 +β 1 s n +…+β n s+β n+1 =(s+ω o ) n+1

[0095] D 2 =s n +k n s n-1 +…+k 2 s+k 1 =(s+ω c ) n

[0096] D 1 D 2 =D 3 s n+1 +D 4

[0097] but

[0098] Where b 0 is the control quantity 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] D 1 =(s+ω o ) 3 , D 2 =(s+ω c ) 2

[0103]

[0104] Assuming the 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, which destroy the pure attenuation characteristics of the system and introduce oscillation components into the system response, which will 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. Then 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 is Due to D 2 The zero point is -ω c And near the dominant pole of the closed loop of the system, D 2 The zero point is the zero point of the closed-loop transfer function of the system. The influence of the zero point on the system is eliminated, so the step response of the system will not oscillate, thus 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 the linear error feedback control law bandwidth ω c The setting relationship of the control quantity gain b 0 The setting formula avoids blind trial and error and saves parameter adjustment time. The details are as follows:

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

[0115]

[0116] D 1 =(s+ω o ) 3 , D 2 =(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 the linear error feedback control law bandwidth ω c The setting relationship avoids blind trial and error in the parameter setting process and saves parameter adjustment time.

[0123] Furthermore, by introducing 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 control gain b is obtained 0 The setting relationship avoids blind trial and error and saves parameter adjustment time.

[0129] The parameter setting method of the present invention systematically determines the zero-pole configuration through theoretical analysis, avoiding the subjectivity and uncertainty of empirical trial and error. It can accurately balance overshoot, response speed and anti-interference performance, optimize the control effect while ensuring system stability. The method reduces the time and cost of test debugging and improves the setting efficiency.

[0130] Embodiment 1:

[0131] like Figure 1 As 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, wherein 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 output by the n+1th output of the linear extended tracking differentiator and the total disturbance estimate output by the n+1th output of the linear extended state observer is used as the n+1th input of the linear error feedback control law. The error feedback control quantity generated by the linear error feedback control law is output 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 dominant pole of the closed-loop system, 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.

[0135] Embodiment 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 v 0 , 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 , i=1,2,…,n+1 as the input of the linear error feedback control law e i =v i -z i , i=1,2,…,n+1;

[0140] The output u of the linear error feedback control law 0 1 / b 0 The controller output u is divided into two paths, one as the input of the controlled object and the other as the output of the controller. 0 The multiplied gain is used as the first input signal of the linear extended state observer; the output y of the controlled object is used 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 a high-order differential signal, 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 have oscillation, solving the problem of system overshoot.

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

[0143]

[0144] In the formula, y is the output of the controlled object; u is the controller output; b 0 is the control quantity gain of the ADRC; 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 +β 1 s 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] In the formula, v 0 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] In the formula, 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 +k 1 s 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.

[0151] Embodiment three:

[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] u 0 =(v 2 -z 2 )+ω c (v 1 -z 1 )

[0159] u=u 0 / b 0 =((v 2 -z 2 )+ω c (v 1 -z 1 )) / b 0 ;

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

[0161] Embodiment 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 is c .

[0170] Embodiment five:

[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] In the formula,

[0174] Control quantity gain b 0 The setting relationship is:

[0175] when

[0176] In the formula, 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 control gain b 0 The fine-tuning factor.

[0177] Embodiment six:

[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 setting 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 the control quantity gain b of the ADRC system 0 The initial value of b 0 Satisfy K·b 0 >0, and debug based on the simulation platform to obtain the closed-loop response curve under the initial value state of the parameters;

[0181] The parameter setting formula includes:

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

[0183]

[0184] In the formula,

[0185] and control quantity gain b 0 The setting relationship of:

[0186] when

[0187] In the formula, γ is the control gain b 0 The fine-tuning factor of

[0188] Step 3: To obtain faster adjustment time, adjust the control gain b 0 Perform parameter tuning: When K>0, gradually reduce the control gain b from the initial value 0 , that is, reduce γ so that the closed-loop control meets the performance index; when K < 0, gradually increase the control gain b in the control system from the initial value 0 , that is, reduce γ 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 and control quantity gain b 0 Perform parameter tuning: First, reduce the bandwidth ω of the linear error feedback control law from the initial value c , that is, reduce the controller bandwidth ω a , and then adjust b according to step 3) 0 ; Repeat step 4) for parameter ω c and b 0 The tuning steps are as follows, 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 control quantity gain b 0 .

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

[0191] In step 2, the control gain b 0 When the initial value of When when When γ=3,

[0192] 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; Test Example 1:

[0193] For the second-order model without time delay 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 improved LADRC) and the parameter setting method given in the sixth embodiment are used for simulation verification.

[0194] The simulation uses the fixed-step Euler method to solve the problem, with a step size of 0.01.

[0195] Select standard LADRC and The open-loop response curve 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 get

[0197] The step response curve obtained by running is as follows Figure 4 As shown in the simulation results, the standard LADRC maintains b 0 =b, the overshoot is about 10%, which is much higher than 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 improved LADRC) proposed in the fourth embodiment of the present invention and the parameter setting method given in the sixth embodiment are used for simulation verification. Standard LADRC and PID are selected for comparison.

[0200] The simulation uses the fixed-step Euler method to solve the problem, 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 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] Taking the above parameters as initial values, the parameters are set according to the parameter setting method of the present invention.

[0205] During the parameter adjustment process, gradually reduce b 0 The value of makes the closed-loop control meet the performance index; then reduce ω c , gradually decrease from the current value b 0 Repeat the above operation until the adjustment time reaches the minimum, b 0 The adjustment range 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 proportionality 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 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. The maximum overshoot of the improved LADRC, standard LADRC and PID are 4.88%, 4.90% and 4.96% respectively; the adjustment time is 2.95s, 3.54s and 3.84s respectively. The specific parameters after adjustment are as follows:

[0208]

[0209]

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

[0211] Test Example 3:

[0212] The running time of the second test example is set to 50 seconds, and a step disturbance with an amplitude of 0.2 is added at the 30th second to test the anti-disturbance performance of the second-order LADRC control system based on the linear expansion tracking differentiator (hereinafter referred to as improved LADRC) proposed in the fourth embodiment of the present invention. The standard LADRC and PID are selected for comparison.

[0213] The simulation uses the fixed-step Euler method to solve the problem, with a step size of 0.01.

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

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

[0216] In PID,

[0217]

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

[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 is used for simulation test, and a standard LADRC and PID are selected for comparison.

[0222] The simulation uses the fixed-step Euler method to solve the problem, 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 proportionality 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 get

[0226] Since τ=60, we take γ=3, The above parameters are used as initial values ​​for debugging. At this time, the closed-loop control has met the performance indicators, but the adjustment time is not optimal;

[0227] Gradually reduce b 0 The absolute value of 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 indicators, and finally, reduce ω c and gradually reduce the value of b 0 The absolute value of b 0 The adjustment range 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 step response curves of the closed-loop system obtained by running the above three sets of parameters are compared. Fig. 9 shown.

[0228] The optimal parameters of improved ADRC, standard LADRC and PID are used to obtain the step response curve comparison diagram and control output curve comparison diagram of the closed-loop system of the main steam temperature system model, as shown in Figure 2. Fig.10 , Fig.11 As shown in the figure, the control effect of the improved LADRC is significantly improved. The maximum overshoot of the improved LADRC, standard LADRC and PID are 4.99%, 4.90% and 4.84% respectively; the adjustment time is 184.90s, 210.86s and 217.31s respectively. The specific parameters and control performance comparison are 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 extended tracking differentiator based on zero-point design, a linear extended 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 output by the n+1th output of the linear extended tracking differentiator and the total disturbance estimate output by the n+1th output of the linear extended state observer is used as the n+1th input of the linear error feedback control law. The error feedback control quantity generated by the linear error feedback control law is output 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 v of the differentiator i , i = 1, 2, ..., n + 1 and the output z of the linear extended state observer i , i=1,2,…,n+1 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: In the formula, 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; 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: In the formula, 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 is 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 is 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 is 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 of the linear error feedback law in the system is ω c , the bandwidth of the extended state observer ω o and controller bandwidth ω a The setting relationship is: In the formula, 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 setting method of 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 setting 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 ADRC 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 value state of the parameters; The parameter setting formula includes: The controller bandwidth in the system ω a With the linear extended state observer ω o and the linear error feedback control law bandwidth ω c The setting relationship of the formula is: In the formula, And the setting relationship of control quantity gain b0: when In the formula, γ is the fine-tuning factor of the control gain b0; Step 3, in order 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 And control quantity gain b0 are adjusted: First, the bandwidth ω of the linear error feedback control law is reduced from the initial value c , and then adjust b0 according to step 3); repeat step 4) for parameter ω c and b0 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 setting 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, preferably λ=0.

1.

9. The parameter setting 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 gain b0 is set, When when When γ=3, 10. The parameter setting method of the LADRC control system based on the linear expansion tracking differentiator according to claim 8 is characterized in that: The performance indicators include that the overshoot of the response curve of the closed-loop system does not exceed 5%, and the peak-to-valley ratio of the response curve is not less than 2:1; In step 3 and step 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 0.001~0.01.

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