A linear active disturbance rejection controller design method for stabilizing a first-order non-self-balancing time-delay system

Abstract

This invention discloses a linear active disturbance rejection control (ADRC) design method for stabilizing a first-order non-self-balancing time-delay system, used to solve the controller parameter tuning problem of such systems. The method includes the following steps: First, identifying the first-order non-self-balancing time-delay model of the controlled system; then, designing a corresponding first-order linear ADRC based on the identified model; next, fixing the bandwidth ratio and feeding the identified model parameters into a computational unit derived by the dual-trajectory method to obtain the upper bound of the controller's bandwidth stability domain; finally, selecting appropriate parameters within the stability domain as needed, thus enabling the closed-loop system to achieve better tracking performance. This invention addresses the parameter tuning problem of the first-order linear ADRC for first-order non-self-balancing time-delay systems, resulting in better control performance for the closed-loop system.

Description

Technical Field

[0001] This invention belongs to the field of process control technology, specifically relating to a design method that uses the dual-track method to gradually obtain the closed-loop system stability parameter range of a linear active disturbance rejection controller for a first-order non-self-balancing time-delay system, selects appropriate parameter values ​​within the parameter range as needed, and then implements the application in an industrial control computer. Background Technology

[0002] A non-self-balancing system is a system whose output either increases infinitely or remains consistently low under a step signal excitation in its open-loop state, failing to reach equilibrium. Non-self-balancing systems exhibit open-loop instability. They are common in industrial production, such as boiler drum level control systems and storage tank level control systems in some chemical production processes. Compared to self-balancing systems, research on control schemes for non-self-balancing systems is relatively limited. Furthermore, the presence of integral elements in non-self-balancing systems leads to poles at the origin, making stabilization more challenging than for self-balancing systems. In industrial production, time delays are unavoidable due to limitations imposed by transmission, measurement, and energy conversion equipment, as well as the controlled object itself. Non-self-balancing systems are no exception, typically containing a time delay element. This results in delayed control action, reduced control effectiveness, and potential production losses. The presence of integral and time delay elements in non-self-balancing time-delay systems makes control system design more difficult than for self-balancing systems.

[0003] The integral element in non-self-balancing systems further complicates traditional control methods. In particular, the integral element causes poles at the origin in the controlled non-self-balancing system, making parameter tuning a challenging problem even if a traditional controller can stabilize it. Slight deviations in parameter tuning can cause the closed-loop system to diverge. For time-delay components in industrial production, conventional control methods such as PID control are difficult to handle. A common approach for time-delay components is to approximate them using Taylor expansion and then design the controller based on the approximate model. However, if the time delay is large, the approximate model will have significant errors compared to the actual system, thus affecting controller performance. In conclusion, traditional controllers present greater challenges in controlling non-self-balancing systems with time delays.

[0004] The core idea of ​​Active Disturbance Rejection (ADRF) is to treat internal and external disturbances as a total disturbance, and then use an extended state observer to estimate and compensate for them in real time. This forces the controlled system into a standard integral-series type, which is then controlled by a feedback controller. Researchers have proposed a linear ADRF controller based on this concept and provided a bandwidth parameterization tuning method, further advancing the application of ADRF controllers in industry. The structure of a first-order linear ADRF controller is shown below.Figure 3 As shown, this includes an extended state observer and a linear state error feedback controller. The extended state observer is a two-input, two-output module. The two inputs are the controlled object's input u(t) and output y(t), and the two outputs are the estimated state z1 of the controlled object and the total disturbance z2 of the system. The linear state error feedback controller is a two-input, one-output module. The inputs are the state z1 of the controlled object estimated by the extended state observer and the reference signal r(t), and the output is the virtual control signal u0. For non-self-balancing systems, which inherently contain integral elements, the estimation pressure on the extended state observer in the active disturbance rejection controller (ADRC) is reduced, making it more suitable to use an ADRC for control. However, the parameter tuning and optimization problems in the controller design process are still complex issues that need to be addressed. Summary of the Invention

[0005] The purpose of this invention is to design a first-order linear active disturbance rejection controller (ADRC) for a first-order non-self-balancing time-delay system. After pole placement, the designed ADRC has only two parameters to be tuned: the extended state observer bandwidth and the controller bandwidth. By fixing the bandwidth ratio, these two parameters are transformed into a single controller bandwidth parameter. The bandwidth ratio and the time delay of the controlled object are fed into a calculation unit derived using the dual-trajectory method to obtain the upper limit of the stable controller bandwidth value that ensures the stability of the closed-loop system, thus obtaining the stability region. Within the stability region, an appropriate controller bandwidth is selected as needed to achieve control of the controlled object.

[0006] This invention is achieved through the following technical means: First, the first-order non-self-balancing time-delay system to be controlled is identified, and the gain b and lag time τ of the first-order non-self-balancing time-delay system are obtained, thereby obtaining the model of the first-order non-self-balancing system; based on the obtained model, a first-order linear active disturbance rejection controller is designed, and the parameters in the designed first-order linear active disturbance rejection controller are simplified to ω according to the bandwidth parameterization theory. c and ω o Choose a fixed bandwidth ratio k such that ω o =kω c Thus, the parameters of the first-order linear active disturbance rejection controller are simplified to a single parameter ω. c Then, through a computational unit derived using the dual-trajectory method, the ω that stabilizes the first-order non-self-balancing time-delay system is obtained. c upper limit That is, the stability region of the entire closed-loop system is Finally, select the parameter ω within the stability region as needed. c The designed first-order linear active disturbance rejection controller with optimized values ​​is implemented in an industrial computer. The specific technical solution is as follows:

[0007] A method for designing a linear active disturbance rejection controller for stabilizing a first-order non-self-equilibrium time-delay system includes the following steps:

[0008] Step 1: Identify the first-order non-self-balancing time-delay model of the controlled system;

[0009] Step 2: Input the identified parameters of the controlled system into a calculation unit to obtain the upper limit of the parameters of the first-order linear active disturbance rejection controller that makes the entire closed-loop system stable, that is, to obtain the stability region.

[0010] Step 3: Select the controller parameters as needed within the stability region and apply them in a first-order linear active disturbance rejection controller.

[0011] Furthermore, step 1 involves: performing system identification on the controlled object to obtain the following first-order non-self-balancing time delay model:

[0012]

[0013] Where s is the Laplace operator, G p Let b be the gain of the first-order non-self-balancing time-delay system model, and τ be the time delay.

[0014] Furthermore, step 2 includes:

[0015] Step (2.1): For the first-order non-self-balancing time-delay model system identified in Step 1, design a first-order linear active disturbance rejection controller (ADRC). The designed ADRC includes an extended state observer (hereinafter referred to as the observer) and a linear state error feedback controller (hereinafter referred to as the controller). The controller and the identified system model constitute the forward path, and the observer, as the feedback path, forms a closed-loop system with the controller and the system model. After bandwidth parameterization, the observer takes the following form:

[0016]

[0017] Here, y represents the system output, u represents the system input, and z1 and z2 represent the state estimates of the controlled first-order non-self-balancing system and the total disturbance estimate of the closed-loop system, respectively. and These are the first derivatives of z1 and z2, respectively, ω o This represents the observer bandwidth.

[0018] The controller is in the following form:

[0019] u0=ω c (r-z1) (3)

[0020] Here, r is the system reference input signal, ω c Where u is the controller bandwidth and u0 is the virtual control variable.

[0021] Step (2.2): Select a fixed positive bandwidth ratio k, i.e., ω o =kω c Then the observer is written in the following form:

[0022]

[0023] Step (2.3): Calculate the value of the auxiliary function g(k), which has the following form:

[0024]

[0025] Here, i is the imaginary unit. Since the bandwidth ratio k is positive, the imaginary part of g(k) is 0, that is, g(k) is a real number, and the superscript of k represents the power.

[0026] Step (2.4): Calculate the ω that stabilizes the system using the following equation. c upper limit

[0027]

[0028] Where π is the mathematical constant pi, and arctan is the arctangent operator.

[0029] Furthermore, step 3 specifically involves determining the parameter stability region obtained in step 2. Select the appropriate parameter ω as needed. c The designed first-order linear active disturbance rejection controller is programmed in an industrial control computer and used to obtain the control quantity.

[0030] For a given first-order non-self-balancing time-delay system, the first-order linear active disturbance rejection controller designed by this method can stabilize the system, and by selecting appropriate parameters within the parameter stability domain, it can achieve good control performance and meet control requirements. At the same time, compared with other methods, this method can directly obtain the parameter stability domain, which greatly reduces the workload of parameter tuning. Attached Figure Description

[0031] Figure 1 This is a flowchart of the process of the present invention;

[0032] Figure 2 This is a diagram of the boiler steam-water system in the embodiment;

[0033] Figure 3 The block diagram of a first-order linear active disturbance rejection controller for a first-order non-self-balancing time-delay system is shown.

[0034] Figure 4 The step response of a first-order non-self-balancing system under the control of a first-order linear active disturbance rejection controller with critical stability parameters;

[0035] Figure 5 This represents the step response of a first-order non-self-balancing time-delay system under the control of a first-order linear active disturbance rejection controller whose parameters are within the stability domain. Detailed Implementation

[0036] The present invention will be further described below with reference to the accompanying drawings and embodiments:

[0037] Figure 1 This is a flowchart illustrating the design method of a linear active disturbance rejection controller for a stabilized first-order non-self-balancing time-delay system according to the present invention. First, the model is identified to obtain the first-order non-self-balancing time-delay model of the system. Then, a first-order linear active disturbance rejection controller is designed for the identified model. After determining a suitable bandwidth ratio, the upper limit of the controller bandwidth that makes the system stable is obtained through a calculation unit to obtain the stability domain of the parameters. Within the stability domain of the parameters, a suitable controller bandwidth is selected as needed and control is executed.

[0038] Example

[0039] like Figure 2 The boiler steam-water system shown controls the boiler drum water level by controlling the feedwater flow. With the feedwater flow as input and the boiler drum water level as output, the controlled object is identified as a first-order non-self-balancing time-delay object, and its transfer function is as follows:

[0040]

[0041] For the identified first-order non-self-balancing time-delay system, a first-order linear active disturbance rejection controller is designed as follows: Figure 3 As shown, the first-order non-self-balancing time-delay system is the boiler steam-water system. u refers to the feedwater flow rate, y refers to the boiler drum water level, and r is the reference signal, specifically the setpoint of the boiler drum water level.

[0042] The extended state observer is expressed as follows:

[0043]

[0044] The linear state error feedback controller is expressed as follows:

[0045] u0=ω c (r-z1)

[0046] Choose a fixed positive bandwidth ratio of 3, i.e., ω o =6ω c Here, the engineering experience value is 3-10, and theoretically there is no limit; then the extended state observer is written in the following form:

[0047]

[0048] Calculate the value of the auxiliary function g(k), which has the following form:

[0049]

[0050] Substituting k=3, we get:

[0051] g(3)=5.47898517470586

[0052] The ω that makes the system stable can be calculated using the following equation. c upper limit

[0053]

[0054] Substituting k=3 and τ=18, we get:

[0055]

[0056] This value is the upper limit for system stability, i.e., the system at ω c The system output y at position = 0.0260813733805969 appears as follows: Figure 4 The constant amplitude oscillation is shown; therefore, the parameter ω c The stability region is (0, 0.0260813733805969).

[0057] To achieve a faster settling time, ω is selected within the stability region. c =0.01956. Substituting this parameter into the designed first-order linear active disturbance rejection controller, the system output y is as follows: Figure 5 As shown.

[0058] In this embodiment, k is specific to this embodiment only; the other choices also apply.

[0059] This invention is a design method for a linear active disturbance rejection controller (ADRC) for stabilizing a first-order non-self-balancing time-delay system. The method designs a corresponding first-order linear ADRC based on the identified first-order non-self-balancing time-delay system model, fixes the bandwidth ratio, and obtains the upper bound of the stability domain of the parameter controller bandwidth through a computing unit, thereby obtaining the stability domain of the parameters. Within the stability domain, appropriate parameters are selected as needed to obtain faster settling time and better tracking performance.

Claims

1. A method for designing a linear active disturbance rejection controller for a stabilized first-order non-self-balancing time-delay system, characterized in that, Includes the following steps: Step 1: Identify the first-order non-self-balancing time-delay model of the controlled system; Step 2: Input the identified parameters of the controlled system into a calculation unit to obtain the upper limit of the parameters of the first-order linear active disturbance rejection controller that makes the entire closed-loop system stable, that is, to obtain the stability region. Step 3: Select the controller parameters as needed within the stability region and apply them in a first-order linear active disturbance rejection controller; System identification is performed on the controlled object, and the following first-order non-self-balancing time-delay model is obtained: Where s is the Laplace operator, G p For a first-order non-self-balancing time-delay system model, b is the gain of the first-order non-self-balancing time-delay system, and τ is the time delay. Step (2.1): For the first-order non-self-balancing time-delay model system identified in Step 1, design a first-order linear active disturbance rejection controller (ADRC). The designed ADRC includes an extended state observer and a linear state error feedback controller. The controller and the identified system model form a forward path, and the observer, as a feedback path, forms a closed-loop system with the controller and the system model. After bandwidth parameterization, the observer takes the following form: y represents the system output, u represents the system input, and z1 and z2 represent the state estimates of the controlled first-order non-self-balancing system and the total disturbance estimate of the closed-loop system, respectively. and These are the first derivatives of z1 and z2, respectively, ω o For observer bandwidth; The controller is in the following form: u0=ω c (r-z1) (3) r is the system reference input signal, ω c Where u0 is the controller bandwidth and u0 is the virtual control variable; Step (2.2): Select a fixed positive bandwidth ratio k, i.e., ω o =kω c Then the observer is written in the following form: Step (2.3): Calculate the value of the auxiliary function g(k), which has the following form: i is the imaginary unit, and the bandwidth ratio k is a positive value, so the imaginary part of g(k) is 0, that is, g(k) is a real number, and the superscript of k represents the power. Step (2.4): Calculate the ω that stabilizes the system using the following equation. c upper limit Where π is the mathematical constant pi, and arctan is the arctangent operator.

2. The design method for a linear active disturbance rejection controller for a stabilized first-order non-self-balancing time-delay system as described in claim 1, characterized in that, Step 3 is: the parameter stability region obtained in step 2. Select the appropriate parameter ω as needed. c The designed first-order linear active disturbance rejection controller is programmed in an industrial control computer and used to obtain the control quantity.

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