阅读说明：本技术 一种自耦pd协同控制理论新方法 (Novel self-coupling PD cooperative control theory method ) 是由 曾喆昭 于 2019-10-31 设计创作，主要内容包括：针对PD及其各类改进型PD增益鲁棒性差、抗扰动鲁棒性也差的问题,发明了一种不依赖于被控对象模型的自耦PD(ACPD)协同控制理论新方法。该方法将系统所有未知内外复杂因素定义为总和扰动,从而将未知非线性复杂系统映射为等价的未知线性系统,进而构建了总和扰动反相激励下的受控误差系统；根据未知被控对象的动态特性测试获得过渡过程时间的取值范围,据此设计最小速度因子模型和自适应速度因子模型。理论分析了ACPD协同控制系统的全局鲁棒稳定性。本发明为现有运行中的各类PD控制器的技术评估与技术升级提供了科学的理论依据和技术保障,在电力、机械、化工、轻工、交通、航空、航天等领域具有广泛的应用价值。(Aiming at the problems of poor gain robustness and poor disturbance resistance robustness of PD and various improved PD types thereof, the novel self-coupling PD (ACPD) cooperative control theory method independent of a controlled object model is invented. According to the method, all unknown internal and external complex factors of a system are defined as sum disturbance, so that an unknown nonlinear complex system is mapped into an equivalent unknown linear system, and a controlled error system under sum disturbance inverse excitation is constructed; and obtaining the value range of the transition process time according to the dynamic characteristic test of the unknown controlled object, and designing a minimum speed factor model and a self-adaptive speed factor model according to the value range. The overall robust stability of the ACPD cooperative control system is theoretically analyzed. The invention provides scientific theoretical basis and technical guarantee for technical evaluation and technical upgrading of various PD controllers in the prior operation, and has wide application value in the fields of electric power, machinery, chemical industry, light industry, traffic, aviation, aerospace and the like.)

1. The new self-coupling PD cooperative control theory method is characterized by comprising the following specific steps of:

step A: measuring unit step response characteristics of an unknown nonlinear complex system, and obtaining the transition process time T of the controlled object according to the dynamic change characteristics_rThe value range of (a) is in seconds;

and B: the transition process time T obtained according to the step A_rEstablishing a minimum speed factor model as follows:

z_cm＝20α/T_r

wherein 1 is<α≤10，T_rIs the transition time of the controlled object from the dynamic state to the steady state.

2. The ACPD cooperative control theory new method as claimed in claim 1, characterized by comprising the following specific steps:

and C: according to a given desired output y_dAnd differential information thereof

e₁＝y_d-y，

wherein the content of the first and second substances,

step D: according to the stepsB and step C separately obtain z_cmAnd e₂Thereafter, to avoid overshoot, an adaptive speed factor z is established_cComprises the following steps:

z_c＝z_cmexp(-β|e₂|)

wherein z is_cmIs the minimum velocity factor, β ═ 1+0.1 α;

step E: obtaining e according to step C and step D respectively₁、e₂And z_cAnd then establishing the ACPD cooperative control law or cooperative control force as follows:

step F: after the ACPD cooperative control force u is obtained according to step E, the cooperative control force u is required to be limited in consideration of the limited input condition of the actual physical system, which is specifically as follows:

|u|≤u_m

wherein u is_mIs the maximum magnitude of the ACPD cooperative control force u.

Technical Field

The invention relates to unknown nonlinear time-varying system control, in particular to a novel Auto-Coupling PD (ACPD) cooperative control method.

Background

For more than 80 years, particularly for nearly half a century, modern scholars, whether engaged in classical control theory or modern control theory, pay attention to the method research of controller gain adjustment, neglect the physical properties of controller gain and control force, and disregard the objective existence of the actual controlled object. Although various control methods can control individual case objects through gain setting, the method lacks wide scientific guiding significance and is only in the technical level of repeated application of the existing intelligent calculation method and control technology. The following is mainly to comprehensively and deeply analyze the limitations of the widely used PID control theory in the field of actual control, and aims to invent an advanced new control theory method, namely an ACPD cooperative control theory new method. In order to facilitate understanding of the theoretical defects of PID, first, the physical properties of the control input u of the controlled object are known from the controlled object.

1) First-order controlled object

Setting a controlled object of a certain order:

where x is the measurable state, u and y are the actual outputs of the control inputs of the system, respectively, f (x, ξ) is a known or unknown linear or non-linear smooth function, and ξ is the model parameter.

Let y be a generalized "displacement" quantity, such as: temperature in a temperature-dependent system, flow in a flow system, angle in a rotating system, position in a moving system, etc. Obviously, the control input u in the system (1) is a physical quantity with a generalized "speed" dimension, so for any first order system, the control input u has a generalized "speed" dimension, and therefore the control force u formed by a PI controller or other type of controller should also have a generalized "speed" dimension.

2) Second-order controlled object

Setting a controlled object of a certain second order:

wherein, y₁,y₂Is two measurable states, u and y are the control input and actual output of the system, respectively, f (y)₁,y₂ξ) is a linear or nonlinear smooth function, known or unknown, and ξ is a model lumped parameter.

Let y be y₁Is a generalized "displacement" quantity, then, y₂It is a physical quantity of a broad sense of "speed",

it is a generalized "acceleration" physical quantity. It is clear that the control input u in the system (2) should be of a generalized "acceleration" dimensionThe physical quantities and, therefore, the control input u for any second order system have a generalized "acceleration" dimension, and thus it is required that the PID or PD or other type of controller form a control force u that should also have a generalized "acceleration" dimension.

By analogy, for any three-order system, the control input u has a generalized "jerk" dimension, and it is required that the control force u formed by each type of controller should also have a generalized "jerk" dimension.

3) The physical property of the gain of the controller is generally ignored by the existing control theory

After the above control force input u to each order system grasps the information of its respective physical attributes, it is required that the control force u formed by each type of controller should also have the corresponding matching physical attributes. However, because the existing control theory methods neglect the physical properties of the gains and control forces u of various controllers, the classical control theory represented by the "control theory" and the modern control theory represented by the "model theory" both disregard the objective existence of the actual controlled object, so that the existing various controllers are only suitable for the application technology level of the specific controlled object and lack extensive scientific guiding significance. Detailed analysis is made below only around the widely used PID control strategy, finding the root cause of PID limitations and its elimination.

Let r and y be the expected output and the actual output of the controlled object, respectively, and the tracking error and its integral and differential are: e.g. of the type₁＝r-y，e₀＝∫e₁dt，

According to e₀、e₁And e₂Then, there is a PID control law model:

u＝k_p(e₁+e₀/T_i+T_de₂) (3)

or

u＝k_pe₁+k_ie₀+k_de₂ (4)

Wherein k is_p>0、k_i＝k_p/T_i、k_d＝k_pT_dProportional, integral and derivative gains, respectively; t is_iAnd T_dRespectively, an integration time constant and a differentiation time constant.

The information available from the PID control law model (3) or (4) is very limited, only k is known_p>0，k_i＝k_p/T_i、k_d＝k_pT_dOr k_d＝k_iT_iT_dEtc., and thus the PID also has several historical legacy problems:

proportional gain lacks physical property

PID is published so far, only k is given_p>0, for which no well-defined physical property is defined, and therefore the proportional gain k is often given_pTreated as dimensionless independent variables;

② PID control force has generalized displacement dimension only

T is introduced into a PID control law prototype (3)_iAnd T_dThese two time constants, the purpose of which is to make the following expression:

u₀＝e₁+e₀/T_i+T_de₂

all have the same dimension (generalized displacement) so as to satisfy the basic arithmetic operation rule, and then the sum result u is summed₀Amplification of k_pTo form the control force u. If k is_pDimensionless, the control force u formed by the PID prototype (3) or (4) only has a generalized displacement dimension, and conflicts with the control input u required by any first-order system to have a generalized velocity dimension or the control input u required by any second-order system to have a generalized acceleration dimension;

two time constants for real time

In PID prototype (3), T_iAnd T_dHow to determine? T_iAnd T_dWhether there is an internal relationship? T between_iOr T_dThe modern scholars at home and abroad, who are? related to the controlled object, pay little attention to the question problems, and in any case, the PID control law model in the form of (4) is generally usedType, only concerning k_p、k_iAnd k_dThe online optimization algorithm with the three gains is rarely concerned about T_iAnd T_dThus often leading to two time constants known as real life;

PID three gains are mutually independent

Although k is_p、k_iAnd k_dThe mutual relationship is established between the following components: k is a radical of_i＝k_p/T_i、k_d＝k_pT_dAnd k_d＝k_iT_iT_d. However, if k_pIs an independent variable without attributes, T_iAnd T_dAre two independent time variables, then k_p、k_i、k_dThe above-mentioned relationship between the three is very uncertain, and belongs to a very loose relationship. In fact, in pair k_p、k_iAnd k_dIn the online optimization process of (a), it is usually treated as three gain variables independent of each other.

4) Localization analysis of PID control strategies

As can be seen from the historical legacy problem with PID above, if T_iAnd T_dAre two independent time variables, and the proportional gain k_pIs an independent variable without physical properties, which can cause the PID control law model to have the following principle error or uncoordinated control mechanism:

principle error of dimension mismatch between PID control force and controlled object control input

Under the above-mentioned assumption of reality, the proportional control force u of PID_p＝k_pe₁Integral control force u_i＝k_ie₀And a differential control force u_d＝k_de₂Are all control forces in a generalized "displacement" dimension, so PID control forces: k ═ u_p(e₁+e₀/T_i+T_de₂) Or u ═ k_pe₁+k_ie₀+k_de₂And are also control forces in a generalized "displacement" dimension. However, control of first order systemsThe braking force input u requires a generalized "speed" dimension; the control force input u of the second order system then requires a generalized "acceleration" dimension. Therefore, if a PID (including PI and PD) controller is used to control a first-order or second-order system, a principle error of dimension mismatch between the PID control force u and the controlled object control force input u is caused, or in other words, if a PID control force with a low-order dimension is used to control a controlled object with a high-order dimension control input, it is difficult for the PID control capability to exert a good control effect;

② uncoordinated control mechanism reduces dynamic quality and steady state performance

If k is_pIs an independent variable without physical property, T_iAnd T_dAre two independent time variables, then k_p、k_i、k_dThe three are actually independent of each other, so that the proportional control force u of the PID is caused_p＝k_pe₁Integral control force u_i＝k_ie₀And a differential control force u_d＝k_de₂Mutually independent and independent uncoordinated control mechanisms are presented in the control process. This is true of the fact that this uncoordinated control mechanism makes it difficult to achieve good dynamic quality and steady state performance for the PID control system.

In summary, if k_pIs an independent variable without physical property, T_iAnd T_dThe PID controller is two independent time variables, so that not only can a principle error of mismatching of control force classes occur and the control capability of the PID be reduced, but also three physical links of the PID can show an uncoordinated control mechanism in the control process, and therefore, the PID control system is difficult to obtain good dynamic quality and steady-state performance.

In addition, an uncoordinated control mechanism can only ensure that a local transient steady state exists in the PID control system, and once an expected track sudden change, a working condition state sudden change, a model parameter time change or external disturbance exists and the like, the PID gain must be re-set to enter another local transient steady state, which is a root cause of poor PID gain robustness, poor time change robustness and poor disturbance resistance robustness.

Because the integral link is an inert link, the main function of the integral link is to eliminate static deviation and improve the steady-state control precision, but the integral saturation causes overshoot and oscillation, so that the PID control can be degenerated to PD control under the condition of neglecting the integral link. In fact, Sliding Mode Control (SMC) and Active Disturbance Rejection Control (ADRC) usually ignore the effect of the integral element and also correspond to the PD control category, so the invention also considers the new method of self-coupled PD cooperative control theory with only proportional and differential elements.

Disclosure of Invention

The invention aims to solve the technical problem of overcoming the defects in the prior art and provide a novel ACPD cooperative control theory method which is simple in model structure, easy to set and good in dynamic quality and steady-state performance.

The technical scheme adopted for solving the technical problems is that the novel ACPD cooperative control theory method is characterized by comprising the following steps:

1. the ACPD cooperative control theory new method is characterized by comprising the following specific steps of:

step A: measuring unit step response characteristics of an unknown nonlinear complex system, and obtaining the transition process time T of the controlled object according to the dynamic change characteristics_rThe value range of (a) is in seconds;

and B: the transition process time T obtained according to the step A_rEstablishing a minimum speed factor model as follows:

z_cm＝20α/T_r

wherein 1 is<α≤10，T_rIs the transition time of the controlled object from the dynamic state to the steady state.

2. The ACPD cooperative control theory new method as claimed in claim 1, characterized by comprising the following specific steps:

and C: according to a given desired output y_dAnd differential information thereof

And

combining actual output y of unknown complex nonlinear object to establish tracking error e₁And its differential e₂Respectively as follows:

e₁＝y_d-y，

wherein the content of the first and second substances,

is the rate of change of the state output of the controlled object;

step D: obtaining z according to step B and step C, respectively_cmAnd e₂Then, in order to avoid overshoot due to differential peak, an adaptive velocity factor z is established_cComprises the following steps:

z_c＝z_cmexp(-β|e₂|)

wherein z is_cmIs the minimum velocity factor, β ═ 1+0.1 α;

step E: obtaining e according to step C and step D respectively₁、e₂And z_cAnd then establishing the ACPD cooperative control law or cooperative control force as follows:

step F: after the ACPD cooperative control force u is obtained according to step E, the cooperative control force u is required to be limited in consideration of the limited input condition of the actual physical system, which is specifically as follows:

|u|≤u_m

wherein u is_mIs the maximum magnitude of the ACPD cooperative control force u.

The method defines all unknown uncertain complex factors such as unknown controlled system dynamics, internal uncertainty, external disturbance and the like as a total disturbance, establishes a controlled error system under the reverse excitation of the total disturbance according to the error between the given expected output and the actual output, further establishes an ACPD cooperative controller model, determines the value range of the transition process time of a controlled system by measuring the dynamic response characteristic of an unknown complex nonlinear system, further establishes a minimum speed factor model and an adaptive speed factor model, and analyzes the overall robust stability of the ACPD closed-loop control system from a complex frequency domain.

The important significance of considering the concept of sum perturbation proposed by mr. hangjing, han, scholars in china in ADRC is: the system classification concepts of linearity and nonlinearity, certainty and uncertainty, time variation and time invariance and the like are completely diluted, any complex nonlinear system can be mapped into an equivalent integral series unknown linear system, the complex problem is simplified, and the limitation of researching corresponding complex control strategies around various complex systems is effectively avoided. However, ADRC requires the use of a high gain ESO to observe the sum disturbance and feed forward the observations to the control input to cancel the sum disturbance as much as possible for auto-immunity purposes, thus increasing the complexity of ACRD.

The ACPD cooperative controller of the invention not only saves an ESO function module due to good internal disturbance rejection robustness, so that the ACPD cooperative controller has a simple structure, but also the only speed factor is completely set by the transition process time of a controlled system, thereby facilitating practical application, effectively promoting zero-distance track connection of an ACPD cooperative control theory and an actual control project, effectively solving the problem of setting of PD, and providing scientific theoretical basis for evaluation and upgrade of the existing PD control technology.

Drawings

Fig. 1 is a block diagram of an ACPD coordinated control system.

Fig. 2 shows the dynamic characteristics of an unknown controlled object, (a) unit step response, and (b) dynamic change speed.

FIG. 3 is an external perturbation.

Fig. 4 shows the sinusoidal tracking control result of the unknown nonlinear system, (a) track tracking, (b) control input, (c) tracking control error, and (d) error local amplification effect.

Fig. 5 shows the step tracking control result of the unknown nonlinear system, (a) track tracking, (b) control input, (c) tracking control error, and (d) error local amplification effect.

Detailed Description

1. Mapping idea from unknown nonlinear time-varying system to linear uncertain system

1) Problem background

Let a certain second order unknown nonlinear time varying system be:

wherein, y₁,y₂Is two states of the system, u and y being the control input and the actual output of the system, respectively, f (y)₁,y₂) Is the system unknown smooth function, b (t) is the control channel time varying gain, d is the externally bounded perturbation.

Definition 1, b (t) and b₀B and b₀Not equal to 0 is an estimated value (not required to be accurate) in the variation range of the time-varying gain b (t) of the control channel, if all unknown uncertain complexity factors of the unknown nonlinear time-varying system (5) are disturbed by using a lumped state, namely the sum y₃To represent, a sum perturbation y can be defined₃(also referred to as the expanded state) are:

y₃＝f(y₁,y₂)+d+△bu (6)

from equation (6), the sum of the disturbances y₃Not only contains the unknown internal dynamics f (y)₁,y₂) And an external disturbance d, but also uncertain control force information deltabu.

From the sum perturbation (6), the unknown nonlinear time-varying system (5) can be mapped to an equivalent unknown linear system:

wherein, b₀Not equal to 0 is some estimate (not required to be accurate) of the variation range of the time-varying gain b (t) of the control channel.

Since the unknown linear system (7) is equivalently mapped by the unknown nonlinear time-varying system (5), the control force u constructed by the system (7) can be directly applied to the control input of the unknown nonlinear system (5).

Assumption 1. if and only if a globally valid control strategy is used, the sum perturbation defined by equation (6) is bounded: | y₃|<∞。

And (3) proving that: from equation (6), y is disturbed due to the sum₃Uncertain control force information deltabu is included, so that as long as a globally effective control strategy is used to form a globally effective control force u, the total disturbance is guaranteed to be bounded: | y₃|<And if not, indicating that the control strategy used is not effective.

2) Physical property analysis of control input u

As known from the unknown linear system (7), y is assumed to be y₁Is a generalized displacement, y₂It is the speed in the broad sense that,

it is a generalized acceleration. Obviously, the sum perturbation y₃And b₀u all have a generalized acceleration dimension. If b is₀Is the reciprocal of the generalized mass, with a dimension of 1/kg, then u should have a dimension of the generalized force, and is also referred to as the control force. In summary, for any second order system, b₀u should have a generalized acceleration dimension.

3) Limitation analysis of PD control law model and solution idea thereof

For a PD controller, the control law is: k ═ u_p(e₁+T_de₂)/b₀Thus, b₀u＝k_p(e₁+T_de₂). Obviously, if the proportional gain k is_pIs a dimensionless variable, derived from the PD control law₀u is only generalized displacement dimension and is connected with the control input b of any second-order system₀u has a generalized acceleration dimension and physical property conflicts occur. Or, using PD control force b having only a generalized displacement dimension₀u has a control requirement of input b of generalized acceleration dimension₀u, any second order object, is theoretically unrealistic,at least a good control effect cannot be obtained. If k is_pWith the generalized acceleration dimension, the limitation problem of PD can be solved easily.

4) Sources of sum perturbation concepts

Summation disturbance is a creative concept proposed by researchers in hangjing, korea, in 20 years ago, and achieves the purpose of auto-disturbance rejection by using an Extended State Observer (ESO) to perform observation estimation on the summation disturbance, feeding forward an estimation value to a control input end so as to counteract the influence of the summation disturbance as much as possible, and inventing an auto-disturbance rejection controller (ADRC). However, ADRC has a complex structure, involves too many parameters, and is computationally expensive. In addition, although robust and stable, the closed loop control system formed by ADRC has difficulty in theoretically analyzing the robust and stable. Therefore, the invention designs the ACPD cooperative controller with global internal disturbance rejection, which omits a high-gain ESO functional module and simplifies the structure of the ACPD cooperative controller.

5) Theoretical significance of sum perturbation concept

From definition 1 of the sum perturbation: the sum perturbation definition has a general meaning since any unknown nonlinear complex system can be mapped into the form of an equivalent unknown linear system (7). Moreover, the sum disturbance definition also completely weakens the concepts of system classification such as linearity and nonlinearity, determination and uncertainty, time variation and time invariance and the like, so that various problems of how to design an effective control strategy for a controlled system of different types are always entangled by two control theory systems of a control theory and a model theory for a long time can be effectively solved.

How to apply globally effective control forces to an unknown nonlinear system (5) or equivalent unknown linear system (7) is the core control strategy of the present invention, i.e. the ACPD cooperative control strategy.

ACPD cooperative control strategy

1) ACPD cooperative controller design

Aiming at the control problem of the linear uncertain system (7), the expected track is set as y_dAnd defining a tracking control error e₁And its differential e₂Respectively as follows:

e₁＝y_d-y₁ (8)

differentiating equation (9) and according to the linear uncertainty system (7) there are:

according to equations (9) and (10), a controlled error system under the sum-perturbation inverse excitation can be established:

it is apparent that the system (11) is a disturbance y in unknown sum₃A second order controlled error system under inverse excitation.

In order to stabilize the controlled error system (11), the ACPD control law (control force) u is defined as:

wherein z is_c>0 is the speed factor of the ACPD cooperative controller, b₀Not equal to 0 is an estimated value within the variation range of the time-varying gain b (t) of the control channel, the same applies hereinafter.

From the ACPD control force (12), the velocity factor z_cThe two different physical links of the proportional link and the differential link of the error are closely coupled together, so that the two different physical links show a cooperative control mechanism with different functions and consistent targets in the control process, and the PD control force is corrected

Two different links in the control process are independent from each other and respectively form an array of uncoordinated control behaviors. Thus, the advent of the ACPD synergistic controller (12) will be a significant revolution in the control theory system.

2) ACPD setting rules

Compared with a PD controller, the ACPD cooperative controller has the setting rule that:

the velocity factor z is known from the ACPD cooperating with the tuning rule (13)_cAnd at the same time is the proportional gain k_pAnd a differential gain k_dThereby ensuring the proportional control force shown in equation (12)And a differential control force u_d＝2z_ce₂/b₀All having the same generalized control force dimension.

3)z_cAnd T_dInter-related relationship between

According to the relationship between two gains of PD: k is a radical of_d＝k_pT_dAnd considering the ACPD cooperating setting rule (13), the speed factor z can be obtained_cAnd T_dThe relationship between them is:

z_c＝2/T_d (14)

wherein, T_dIs the differential time constant of the PD.

Equation (14) shows the speed factor z of the ACPD cooperative controller_cDifferential time constant T with PD_dThe internal relationship between them. T is_dThe smaller the velocity factor z_cThe larger the size, otherwise the opposite is true. However, T_dHow to determine? T_dWhether? two problems related to a controlled object are key scientific problems which are ignored by modern scholars at home and abroad, the two key scientific problems to be solved by the invention are as follows:

4)z_cexternal connection with controlled object

Although formula (14) indicates z_cCan be composed of T_dTo date, however, scholars at home and abroad have little attention to T_dHow to adjust. In view of being controlledThe smaller the time scale tau of the image, the faster the dynamic change speed of the controlled object, otherwise, the reverse is true. Therefore, the inventors consider that: provided that the ACPD co-operates with the speed factor z of the controller_c＝2/T_dThe dynamic change speed of the controlled object is more than 2/tau, the controlled object can be effectively controlled, namely z_c＝2/T_d>2/tau. To this end, a minimum speed factor model may be defined as:

z_cm＝2α/τ (15)

wherein alpha is more than 1 and less than or equal to 10, the same applies below; τ is the time scale of the controlled object.

By the inequality z_c＝2/T_d>2/τ is known, T_d<τ, indicating the differential time constant T of the ACPD-only cooperative controller_dThe time scale tau of the controlled object is smaller, the controlled object can be effectively controlled, and therefore, the ACPD cooperative control strategy effectively solves the problem of T_dAnd the external connection problem between the controlled object is solved.

Since the time scale τ is an abstract concept, it is difficult to obtain τ for nonlinear systems, and therefore it is difficult to use τ to set the minimum velocity factor z_cm. However, considering that the dynamic characteristics of any known or unknown controlled object can be measured, the transient process time of the controlled object from the dynamic state to the steady state is assumed to be T_rAnd is provided with T_rWith 10 τ, according to equation (15), a minimum speed factor may be defined as:

z_cm＝20α/T_r (16)

from equation (16), the minimum speed factor z of the ACPD cooperative controller_cmCan be composed of T_rTo adjust. Such as: if the controlled system is required to enter a stable control state within 1 second, T can be set_r1 second, and z _cm20 α; if the steady control state is entered within 0.1 second, T can be set_r0.1 second, and z_cm200 α; if the steady state control is entered within 10 seconds, T can be set_r10 seconds, z _cm2 α; and so on.

Obviously, by testing the dynamic characteristic (unit step response characteristic) of the unknown controlled system, the transition process time can be determinedInter T_rSo that the ACPD minimum velocity factor z can be set according to equation (16)_cmThe value range of (A) is convenient for practical operation. Due to 1<Alpha.ltoreq.10, thus, z_cmMinimum value z of_cm＝20/T_rAnd a maximum value z_cm＝200/T_rThe difference is 10 times, has great setting elasticity and is usually taken as the middle value, namely z_cm＝100/T_r。

5) Adaptive speed factor model

Considering the large tracking error during the dynamic response, a velocity factor z is required_cShould not be too large, otherwise, the proportional control force will be applied

Too large to cause overshoot. Considering the sensitive nature of the error rate of change, the invention therefore defines an adaptive speed factor model:

z_c＝z_cmexp(-β|e₂|) (17)

wherein z is_cm＝20α/T_r，1<α≤10，β＝1+0.1α。

The expression (17) is the velocity factor z_c＝2/T_dTime T of transition process with controlled object_rThe external connection between them. Obviously, as long as T is determined_rCan adjust z_c. And T can be easily obtained because the dynamic characteristics of any known or unknown controlled object can be obtained through testing_rTo obtain the minimum velocity factor z_cm＝20α/T_rAnd an adaptive velocity factor z_c＝z_cmexp(-β|e₂And | to facilitate zero-distance track connection between ACPD cooperative control theory and actual control engineering, and to provide scientific theoretical basis for evaluation and upgrade of current PD control technology.

6) ACPD cooperative control force limiting

Because the excessive response speed and differential peak value easily cause the overshoot phenomenon, and the input limitation condition of the actual physical system is considered, the ACPD cooperative control force u is required to be carried outAnd (6) limiting. Let the maximum amplitude of the ACPD cooperative control force be u_mThe clipping conditions are as follows:

|u|≤u_m (18)

ACPD control system block diagram, as in FIG. 1.

3. Closed loop control system stability analysis

Theorem 1. from assumption 1, as long as the sum perturbation is bounded: | y₃|<Infinity, then if and only if z_c>At 0, the closed-loop control system composed of the ACPD controller shown in the formula (12) is globally asymptotically stable and has good robustness against disturbance.

And (3) proving that:

1) stability analysis

The ACPD control law (12) is substituted for the controlled error system shown in the formula (11), and an ACPD closed-loop control system is as follows:

it is apparent that the closed loop control system (19) is actually a disturbance y in the sum₃Second order error system under inverse excitation. Considering the initial state:

taking a single-sided laplace transform of an ACPD closed loop control system (19) results in:

finishing to obtain:

it is apparent that the first term of the closed loop control system (21) is a zero input response and the second term is a zero state response. The transfer function defining the closed loop control system is:

according to the complex frequency domain analysis theory, if and only if z_c>At 0, the system transfer function (22) has a dual pole of-z on the real axis of the left half complex plane_cAnd thus the error transmission system (22) and thus the closed loop control system (21) is stable. And because of z_cIndependent of the model parameters of the controlled object, the closed-loop control system (21) is thus globally asymptotically stable.

2) Robust analysis of disturbance rejection

By substituting system (22) into system (21), the closed loop control system can be represented as:

since the unit impulse response of the system (22) is:

and is

The time domain solution available from the closed loop control system (23) is:

wherein "+" denotes a convolution integral operation.

When z is_c>At 0 time, due to

Thus, as long as the sum perturbation is bounded: | y₃|<Infinity, then must be:

i.e. the tracking error e of the controlled system₁(t) and the differential thereof

The stable balance point origin (0,0) can be approached gradually from any non-zero initial state, and theoretically, accurate control can be realized. And because e₁(t) → 0 and e₂(t) → 0 and y only₃|<Infinity, and with the sum perturbation y₃The ACPD closed-loop control system has good total disturbance robustness resistance, including model robustness, time-varying robustness, external disturbance robustness and the like, and the ACPD closed-loop control system is proved to be complete.

Performance testing and analysis of ACPD cooperative control system

In order to verify the effectiveness of the novel ACPD cooperative control theory method, the following simulation experiment is carried out aiming at the control problem of an unknown nonlinear time-varying object.

Let a certain unknown nonlinear time-varying system be:

wherein the content of the first and second substances,

is a system unknown model, b (t) ═ 1+ sin²(t) is the time-varying gain of the control channel, and 1 ≤ b (t) ≦ 2, u and y are the control input and actual output of the system, respectively, and d is the external disturbance.

1) Unknown nonlinear time-varying system dynamic characteristic test

Setting the sampling frequency f_s1000Hz, initial state: y is₁(0)＝0.5、y₂(0) The dynamic characteristics when u is 1 are as shown in fig. 2. As can be seen from FIG. 2, the state output of the object changes smoothly until 4.383 seconds, howeverAt the moment of 4.383 seconds, the state output of the unknown object changes abruptly, and the unknown system (29) is an unstable system.

As can be seen from the dynamics test information of FIG. 2, a transient time T is required for effective control of the unknown system (29)_rLess than or equal to 4 seconds. Due to T_rThe smaller the speed factor, the larger the response speed, otherwise, the reverse is true, therefore, in order to increase the response speed of the control system, T is taken_rAt 1 second, the minimum velocity factor that can be obtained according to equation (16) is: z is a radical of_cm＝20α/T_rSince 20 α is used, the adaptive speed factor model is expressed by equation (17)

z_c＝20αexp(-β|e₂|) (30)

Wherein 1< alpha is less than or equal to 10, and beta is 1+0.1 alpha.

2) ACPD cooperative controller related parameters

When the unknown nonlinear time-varying system (29) is controlled, if α is 5 and β is 1+0.1 α is 1.5, the adaptive speed factor is as follows according to equation (30):

z_c＝100exp(-1.5|e₂|) (31)

according to equation (12), the ACPD co-controller is:

since 1. ltoreq. b (t) 2, it is preferable that b is₀1. In all the following simulation experiments, the initial state of the controlled object is as follows: y is₁(0)＝0.5、y₂(0) 0; the relevant parameters of ACPD are the same, and the control force clipping conditions are the same, namely: the u is less than or equal to 5.

In order to verify the disturbance rejection capability of the ACPD cooperative control system, the same external disturbance was used in the following simulation experiments, namely, a square wave disturbance with the amplitude of + -1 existed in the period of (9 s-11 s), as shown in FIG. 3.

Simulation experiment 1: sine tracking control experiment

In order to verify the sinusoidal tracking control performance of the ACPD cooperative control theory new method, a sinusoidal tracking control experiment is carried out on an unknown controlled system (29).

Given a desired output trajectory of y_dSin (t), the control method of the present invention was used, and the test results are shown in fig. 4. FIG. 4 shows that the ACPD cooperative control system not only has a fast response speed (entering a steady state in about 0.5 seconds) but also has a high control precision (the maximum absolute error is less than 2.8 multiplied by 10)^-4) And the method has good time-varying robustness and disturbance-resistant robustness, thereby being an effective control method. In addition, it was found in the experiment that: at b₀The same control effect can be obtained by taking any value within the range of 1-2, which shows that the estimated value of the control channel gain is not required to be accurate; any value within the range of 1 ≦ α ≦ 10 can be effectively controlled, and the larger α is, the higher steady-state accuracy is, and the stronger disturbance rejection capability is, however, a transient oscillation phenomenon occurs in the control input, so that it is generally appropriate to take α to 5 (intermediate value).

Simulation experiment 2: step tracking control experiment

In order to verify the step tracking control capability of the novel ACPD cooperative control theory method, a step tracking control experiment is carried out on an unknown controlled system (29).

The expected output track is set as a unit step signal, and the transition process time of the unknown system (29) is set as T _r1 second, therefore, the desired output transition is set as: y is_d(t)＝1-exp(-10t)，

The simulation results are shown in fig. 5 by using the control method of the present invention. FIG. 5 shows that the ACPD cooperative control system of the present invention not only has a fast response speed (about 1.0 second, the system can enter a stable state) but also has a high control precision (the maximum absolute error is less than 5.5 multiplied by 10)^-5) And the method also has good time-varying robustness and disturbance-resistant robustness, and further shows that the ACPD cooperative control theory new method is a globally stable strong disturbance-resistant control method. In addition, it was found in the experiment that: b₀The same control effect can be obtained by taking any value within the range of 1-2, and further shows thatThe estimated value of the control channel gain does not require accuracy; any value within the range of 1 ≦ α ≦ 10 can be effectively controlled, and the larger α is, the higher steady-state accuracy is, and the stronger disturbance rejection capability is, however, a transient oscillation phenomenon occurs in the control input, so that it is generally appropriate to take α to 5 (intermediate value).

5. Conclusion

Although the PID, SMC and ADRC based on the control theory strategy are three main flow controllers widely used in the field of control engineering at present, the limitations of poor gain robustness and poor disturbance rejection robustness exist in the PID and various improved PIDs thereof; although the robust stability performance of the SMC is good, an irreconcilable contradiction exists between high-frequency buffeting and disturbance rejection capability; although the ADRC has good stability and disturbance robustness, the ADRC has the limitations of excessive gain parameters, large calculation amount, complex controller structure and the like.

Compared with the existing three main flow controllers, the novel ACPD cooperative control theory method of the invention integrates the advantages of the three main flow controllers and eliminates the limitations of the three main flow controllers, namely: the method has the advantages of simple PID structure, good SMC robustness and stability, and good ADRC robustness against sum disturbance; the problem of difficulty in PID setting is effectively avoided, the problem that SMC is not adjustable between high-frequency buffeting and disturbance resisting capacity is effectively solved, and the problems of excessive ADRC gain parameters and large calculated amount are effectively avoided.

Because the dynamic characteristics of the controlled object of any known or unknown model can be predicted and tested, the value range of the transition process time can be easily obtained through the dynamic characteristics, and the minimum speed factor and the self-adaptive speed factor model can be determined according to the value range, so that the development of the ACPD cooperative control theory taking the speed factor as the core coupling factor is a subversive change of the existing control theory system, corrects the indifferent objective physical properties of the classical control theory and the modern control theory, and limits the unscientific control thought of the paper-based soldiers.

The invention not only has wide application prospect in the field of unknown nonlinear complex system control, but also can provide scientific theoretical basis and technical guarantee for the evaluation and upgrade of the existing PD control technology.