阅读说明：本技术 一种非最小相位系统精确跟踪控制方法 (Accurate tracking control method for non-minimum phase system ) 是由 施孟佶 李维豪 祝洋 陈博文 杜文涛 林伯先 秦开宇 于 2021-08-20 设计创作，主要内容包括：本发明公开了一种非最小相位系统精确跟踪控制方法,包括如下步骤：S1、建立非最小相位系统线性模型,基于输出重定义技术求解最小相位输出建立最小相位系统；S2、针对最小相位系统,以原始非最小相位系统的参考轨迹为基础,设计迭代解算方案,求解最小相位相位系统的参考轨迹；S3、利用S2中迭代求解的最小相位系统的参考轨迹,设计基于“反馈+前馈”结构的控制器。本发明在寻找到最小相位输出的基础上,通过迭代算法,利用稳定逆方法精确求解最小相位系统的参考轨迹,为控制系统设计提供先验信息,而后再设计“反馈+前馈”的控制结构,有效解决非最小相位系统现有输出重定义方法存在的近似精度不足问题,改善了此类系统的输出跟踪性能。(The invention discloses a non-minimum phase system accurate tracking control method, which comprises the following steps: s1, establishing a non-minimum phase system linear model, solving minimum phase output based on an output redefinition technology and establishing a minimum phase system; s2, designing an iterative solution scheme for the minimum phase system based on the reference track of the original non-minimum phase system, and solving the reference track of the minimum phase system; and S3, designing a controller based on a feedback + feedforward structure by using the reference track of the minimum phase system iteratively solved in the S2. On the basis of finding out the minimum phase output, the invention accurately solves the reference track of the minimum phase system by using a stable inverse method through an iterative algorithm, provides prior information for the design of a control system, and then designs a control structure of 'feedback + feedforward', effectively solves the problem of insufficient approximate precision of the existing output redefinition method of a non-minimum phase system, and improves the output tracking performance of the system.)

1. A non-minimum phase system accurate tracking control method is characterized by comprising the following steps:

s1, establishing a non-minimum phase system linear model of the system, then solving minimum phase output based on an output redefinition technology, and establishing a minimum phase system;

s2, aiming at the minimum phase system established in S1, an iterative solution scheme is designed on the basis of the reference track of the original non-minimum phase system, and the reference track of the minimum phase system is solved;

and S3, designing a controller based on a feedback + feedforward structure by using the reference track of the minimum phase system iteratively solved in the S2.

2. The non-minimum phase system precise tracking control method according to claim 1, wherein the step S1 includes the following sub-steps:

s11, establishing a single-input single-output non-minimum phase system model as follows:

wherein the system state Represents the first derivative of x; control inputSystem outputAnd is A. B, C are the state matrix, input matrix, and output matrix of the system, respectively; respectively representing one-dimensional, nxn-dimensional and 1 xn-dimensional spaces, wherein n is a system order;

s12, calculating the output relative order of the non-minimum phase system model:

wherein r is the relative output order of the system;

s13, setting the relative order r calculated in S12 to be less than n, and designing a non-singular transformation matrixThe system state in S11 is decomposed into:

wherein the content of the first and second substances,represents an external state, which is closely related to the control input of the system,represents a system internal state that is unaffected by system control inputs;

s14, according to the conversion relation in the step S13, writing the general form of the system in S11 as follows:

wherein the content of the first and second substances,respectively represent an external state z₁,…,z_rThe first derivative of (a); first state z₁Equivalent to the output y of the system in S11, which is also the output of system (4); matrix arrayAndrespectively an internal dynamic system matrix and an input matrix,a first derivative of the state eta representing the internal state of the system;

s15, constructing a transformation matrix based on the system model in the general form obtained in S14 and the transformation relation in S13Comprises the following steps:

Φ＝[C CA … CA^r-1 α₁ α_i … α_n-r]^T

wherein i is not less than 0 and not more than n-r and alpha_iUnder the condition of ensuring that the matrix phi is nonsingular, the equation alpha is solved_iB is 0;

s16, setting the output y-z in the S14 system model₁Constant 0, zero dynamics of the solution system (4) is

(5) Describing the internal dynamics of the system, if the matrix P in (5) has at least one eigenvalue in the left half plane of the frequency domain, i.e. λ (P) > 0, then the system (4) in S14 is called a non-minimum phase system;

s17, based on S14-S16, determining whether the system is a non-minimum phase system, and defining the system output matrix obtained by output redefinition asThereby establishing a minimum phase system of

Wherein the content of the first and second substances,is a redefined system output which ensures that the system (6) is a minimum phase system; for the convenience of recording, the 'OR' marks on the rest parts represent the variables after the redefined output is implemented;

s18, solving the correlation equation in the following two conditions to make y^ORFor minimum phase output:

condition 1: there is a state feedback variableSo that the matrix (A-BK)_c) N-r characteristic values are positioned on the left half plane of the frequency domain;

condition 2: redefined output matrix C^ORSatisfies the following conditions

S19, defining the minimum phase output matrix selected from the candidate pool asWhich satisfies the following conditions:

wherein n is^ORIs the minimum phase output C obtained by solving the conditions 1,2^ORNumber of (2), n^OR≥1；|| ||₂Is expressed in EuropeThe distance in degrees f.

3. The method as claimed in claim 2, wherein the step S2 adopts an iterative solution algorithm to solve the real reference trajectory of the redefined minimum phase system, and the specific process is as follows:

A. selecting an initial value of an iterationy_dA reference trajectory for an original non-minimum phase system; make the minimum phase outputWhere i ∈ {1,2, …, n^ORAnd let the current iteration period j equal to 0;

B. if j is less than N, based on non-singular transformation matrix phi^IRAnd a stable inverse calculation method, wherein the state reference and the control input reference of the iteration cycle of the calculation system are respectively as follows:

a reference representing the external state and the reference representing the internal state of the j-th iteration after performing the output redefinition, respectively;

then, the calculation is carried out to obtainAnd executing the step C;

if j is more than or equal to N, the iteration is stopped, and output is carried outFinishing the operation; wherein N is the maximum iteration number;

C. if it isWherein epsilon is a preset threshold value; let j equal j +1, calculate the reference trajectory of the minimum phase system for the next iteration cycle:

and returning to step B, k_aIs an iteration gain; otherwise, terminating iteration and outputtingAnd finishing the operation.

4. The non-minimum phase system precise tracking control method according to claim 3, wherein the step S3 includes the following sub-steps:

s31, designing a controller based on the "feedback + feedforward" structure for the system (1) in step S11 according to the reference trajectory of the minimum phase system in step S2 as follows:

u＝u_d+K(x_d-x)

wherein the content of the first and second substances,is a feedback gain matrix, u_dAnd x_dRespectively a control input reference and a state reference; feedforward control term u_dFor realizing accurate tracking, feedback control item K (x)_d-x) for improving system stability and correcting deviations from the state reference trajectory;

and S32, determining a feedback gain matrix K according to the pole arrangement theorem.

Technical Field

The invention relates to a non-minimum phase system accurate tracking control method aiming at a non-minimum phase system existing in engineering.

Background

A salient feature of non-minimum phase systems is that the system contains at least one unstable zero. In particular, for a linear non-minimum phase system, there is at least one zero point for the right half-plane. Non-minimum phase characteristics are widely present in a variety of aircraft, including conventional fixed wing aircraft, VTOL aircraft, hypersonic aircraft, and autonomous helicopters. Since the non-minimum phase system cannot be stabilized by a typical output feedback controller due to the inherent unstable internal dynamics of the non-minimum phase system, it is very challenging to design a suitable controller, and therefore, classical control methods, such as a back-stepping method, a sliding-mode control, and a feedback-based linearization control, cannot be directly applied to the non-minimum phase system. Generally, the non-minimum phase characteristics of the system are closely related to the selection of the system output, so that an approximate minimum phase system can be found by selecting different outputs (i.e. output redefinition), the internal dynamic characteristics of the system are improved, and the design of the control system is facilitated. However, existing output redefinition methods focus on finding the minimum phase output and do not study the tracking control effect of the redefined original non-minimum phase system output.

Disclosure of Invention

The invention aims to overcome the defects of the prior art, and provides a non-minimum phase system accurate tracking control method which can accurately solve the reference track of a minimum phase system by using a stable inverse method through an iterative algorithm on the basis of finding the minimum phase output, provide prior information for the design of a control system, and then design a 'feedback + feedforward' control structure, effectively solve the problem of insufficient approximation accuracy of the existing output redefinition method of the non-minimum phase system, and improve the output tracking performance of the system.

The purpose of the invention is realized by the following technical scheme: a non-minimum phase system accurate tracking control method comprises the following steps:

s1, establishing a non-minimum phase system linear model of the system, then solving minimum phase output based on an output redefinition technology, and establishing a minimum phase system;

s2, aiming at the minimum phase system established in S1, an iterative solution scheme is designed on the basis of the reference track of the original non-minimum phase system, and the reference track of the minimum phase system is solved;

and S3, designing a controller based on a feedback + feedforward structure by using the reference track of the minimum phase system iteratively solved in the S2.

Further, the step S1 includes the following sub-steps:

s11, establishing a single-input single-output non-minimum phase system model as follows:

wherein the system state Represents the first derivative of x; control inputSystem outputAnd is A. B, C are the state matrix, input matrix, and output matrix of the system, respectively; respectively representing one-dimensional, nxn-dimensional and 1 xn-dimensional spaces, wherein n is a system order;

s12, calculating the output relative order of the non-minimum phase system model:

wherein r is the relative output order of the system;

s13, setting the relative order r calculated in S12 to be less than n, and designing a non-singular transformation matrixThe system state in S11 is decomposed into:

wherein the content of the first and second substances,represents an external state, which is closely related to the control input of the system,

represents a system internal state that is unaffected by system control inputs;

s14, according to the conversion relation in the step S13, writing the general form of the system in S11 as follows:

wherein the content of the first and second substances,respectively represent an external state z₁，...，z_rThe first derivative of (a); first state z₁Equivalent to the output y of the system in S11, which is also the output of system (4); matrix arrayAndrespectively an internal dynamic system matrix and an input matrix,a first derivative of the state eta representing the internal state of the system;

s15, constructing a transformation matrix based on the system model in the general form obtained in S14 and the transformation relation in S13Comprises the following steps:

Φ＝[C CA … CA^r-1 α₁ … α_i … α_n-r]^Twherein i is not less than 0 and not more than n-r and alpha_iIs obtained by solving equation alpha, while ensuring that the matrix phi is non-singular (full rank)_iB is 0;

s16, setting the output y-z in the S14 system model₁Constant 0, zero dynamics of the solution system (4) is

(5) Describing the internal dynamics of the system, if the matrix P in (5) has at least one eigenvalue in the left half plane of the frequency domain, i.e. λ (P) > 0, then the system (4) in S14 is called a non-minimum phase system;

s17, based on S14-S16, determining whether the system is a non-minimum phase system, and defining the system output matrix obtained by output redefinition asThereby establishing a minimum phase system of

Wherein the content of the first and second substances,is a redefined system output which ensures that the system (6) is a minimum phase system; for the convenience of recording, the 'OR' marks on the rest parts represent the variables after the redefined output is implemented;

s18, solving the correlation equation in the following two conditions to make y^ORFor minimum phase output:

condition 1: there is a state feedback variableSo that the matrix (A-BK)_c) N-r characteristic values are positioned on the left half plane of the frequency domain;

condition 2: redefined output matrix C^ORSatisfies the following conditions

S19, defining the minimum phase output matrix selected from the candidate pool asWhich satisfies the following conditions:

wherein n is^ORIs the minimum phase output C obtained by solving the conditions 1,2^ORNumber of (2), n^OR≥1；|| ||₂Representing the euclidean distance.

Further, the step S2 adopts an iterative solution algorithm to solve the real reference trajectory of the minimum phase system after the output redefinition, and the specific process is as follows:

A. selecting an initial value of an iterationy_dA reference trajectory for an original non-minimum phase system; make the minimum phase outputWhere i ∈ {1,2, …, n^ORAnd let the current iteration period j equal to 0;

B. if j is less than N, based on non-singular transformation matrix phi^IRAnd a stable inverse calculation method, wherein the state reference and the control input reference of the iteration cycle of the calculation system are respectively as follows:

a reference representing the external state and the reference representing the internal state of the j-th iteration after performing the output redefinition, respectively;

then, the calculation is carried out to obtainAnd executing the step C;

if j is more than or equal to N, the iteration is stopped, and output is carried outFinishing the operation; wherein N is the maximum iteration number;

C. if it isWherein epsilon is a preset threshold value; let j equal j +1, calculate the reference trajectory of the minimum phase system for the next iteration cycle:

and returning to step B, k_aIs an iteration gain; otherwise, terminating iteration and outputtingAnd finishing the operation.

Further, the step S3 includes the following sub-steps:

s31, designing a controller based on the "feedback + feedforward" structure for the system (1) in step S11 according to the reference trajectory of the minimum phase system in step S2 as follows:

u＝u_d+K(x_d-x)

wherein the content of the first and second substances,is a feedback gain matrix, u_dAnd x_dRespectively a control input reference and a state reference; feedforward control term u_dFor realizing accurate tracking, feedback control item K (x)_d-x) for improving system stability and correcting deviations from the state reference trajectory; and S32, determining a feedback gain matrix K according to the pole arrangement theorem.

The invention has the beneficial effects that: aiming at the defects of the output redefinition method of the existing non-minimum phase system, the invention accurately solves the reference track of the minimum phase system by using a stable inverse method through an iterative algorithm on the basis of finding the minimum phase output, provides prior information for the design of a control system, and then designs a control structure of 'feedback + feedforward', effectively solves the problem of insufficient approximation precision of the existing output redefinition method of the non-minimum phase system, and improves the output tracking performance of the system.

Drawings

FIG. 1 is a flow chart of a non-minimum phase system precision tracking control method of the present invention;

fig. 2 is a flow chart of an iterative redefinition method of a non-minimum phase system of the present invention.

Detailed Description

The technical scheme of the invention is further explained by combining the attached drawings.

As shown in fig. 1, a non-minimum phase system precise tracking control method of the present invention includes the following steps:

s1, establishing a non-minimum phase system linear model of the system, then solving minimum phase output based on an output redefinition technology, and establishing a minimum phase system;

the method comprises the following substeps:

s11, establishing a single-input single-output non-minimum phase system model as follows:

wherein the system state Represents the first derivative of x; control inputSystem outputAnd is A. B, C are the state matrix, input matrix, and output matrix of the system, respectively; respectively representing one-dimensional, nxn-dimensional and 1 xn-dimensional spaces, wherein n is a system order;

s12, calculating the output relative order of the non-minimum phase system model:

wherein r is the output relative order of the non-minimum phase system model;

s13, setting the relative order r calculated in S12 to be less than n, and designing a non-singular transformation matrixThe system state in S11 is decomposed into:

wherein the content of the first and second substances,represents an external state, which is closely related to the control input of the system,represents a system internal state that is unaffected by system control inputs;

s14, according to the conversion relation in the step S13, writing the general form of the system in S11 as follows:

wherein the content of the first and second substances,respectively represent an external state z₁，…，z_rThe first derivative of (a); first state z₁Equivalent to the output y of the system in S11, which is also the output of system (4); matrix arrayAndrespectively an internal dynamic system matrix and an input matrix,a first derivative of the state eta representing the internal state of the system;

s15, constructing a transformation matrix based on the system model in the general form obtained in S14 and the transformation relation in S13Comprises the following steps:

Φ＝[C CA … CA^r-1 α₁ … α_i … α_n-r]^T

wherein i is more than or equal to 0 and less than or equal to n-r, alpha_iIs obtained by solving equation alpha, while ensuring that the matrix phi is non-singular (full rank)_iB is 0;

s16, setting the output y-z in the S14 system model₁Constant 0, zero dynamics of the solution system (4) is

(5) Describing the internal dynamics of the system, if the matrix P in (5) has at least one eigenvalue in the left half plane of the frequency domain, i.e. λ (P) > 0, then the system (4) in S14 is called a non-minimum phase system;

s17, based on S14-S16, determining whether the system is a non-minimum phase system, and defining the system output matrix obtained by output redefinition asThereby establishing a minimum phase system of

Wherein the content of the first and second substances,is a redefined system output which ensures that the system (6) is a minimum phase system; for the convenience of recording, the 'OR' marks on the rest parts represent the variables after the redefined output is implemented;

s18, solving the correlation equation in the following two conditions to make y^ORFor minimum phase output:

condition 1: there is a state feedback variableSo that the matrix (A-BK)_c) N-r characteristic values are positioned on the left half plane of the frequency domain;

condition 2: redefined output matrix C^ORSatisfies the following conditions

S19, it should be noted that the system minimum phase output matrix C obtained by solving the conditions 1 and 2^ORMore than one output matrix is needed to be selected, so that the subsequent operation is facilitated. Defining a minimum phase output matrix selected from the candidate pool asWhich satisfies the following conditions:

wherein n is^ORIs the most obtained by solving the conditions 1,2Small phase output C^ORNumber of (2), n^OR≥1；|| ||₂Representing the euclidean distance. Therefore, based on the steps of S11-S19, the approximate minimum phase system (6) and minimum phase output of the original non-minimum phase system (1) can be solved

S2, aiming at the minimum phase system established in S1, an iterative solution scheme is designed on the basis of the reference track of the original non-minimum phase system, and the reference track of the minimum phase system is solved; the method comprises the following substeps:

s21, for the original non-minimum phase system (1), the reference track is y_dDefining a tracking error asThe tracking control target is as follows: the control input u is designed such that the tracking error satisfies the following inequality:

wherein the content of the first and second substances,represents the tracking errorWhen ε → 0, satisfyε > 0, ε is an adjustable parameter; t is t_b> 0 is the corresponding settling time of the system;

s22, on the basis of the reference track defined in S21, defining the reference system of the system (1) as follows:

wherein x is_dRepresents a reference state of the system and,denotes x_dThe first derivative of (a); u. of_dA reference control input representative of a system; combining (10) and (1), defining a state tracking error asHaving a first derivative of

S23, based on the general form system model established in S13, on the basis of the reference track of the original non-minimum phase system of the known system, obtaining the reference of the external state as follows:

(y_d)′…(y_d)^(r-1)respectively represent y_dFirst to r-1 order derivatives of;

thus, further based on the relationship of the internal dynamics to the external dynamics:

calculating a reference eta of the internal state by solving a differential equation of (12)_dComprises the following steps:

wherein eta is_d(0) Represents eta_dAn initial value of (d);

s24, further calculating the reference state of the system based on the external state and the internal state obtained by the calculation of S23 as follows:

and reference to control input:

the steps given in the above equations (13) - (15) are essentially to find the system stability inverse. However, the equation (12) for solving the internal dynamics is unstable due to the characteristic inherent to the non-minimum phase, so that the equations given by (13) to (15) cannot be solved directly in practice. Therefore, only by output redefinition and finding the minimum phase system, the state reference track and the control input reference of the system can be calculated through the formulas (13) to (15), and the basis is laid for designing the controller.

S25, on the basis of solving the minimum phase output, applying the stable inverse solving method of (13) to (15) to the minimum phase system (6) after output redefinition, wherein the state reference and the control input reference of the computing system (6) are respectively as follows:

s26, calculating a quantization approximation error:

in the formula (18), the reaction mixture,is a reference y from the original non-minimum phase system_dCalculated minimum phase output due to searchNot equal to the original non-minimum phase output matrix C, so calculated based on an output redefinition methodCan only be approximated by x_dThis approximation is the root cause for the output tracking error;

s26, according to the analysis process in S25, in order to solve the approximate error brought by (18), an iterative solution algorithm is adopted to solve the real reference track of the minimum phase system after output redefinition, and the specific process is given by the following algorithm 1:

algorithm 1: outputting a reference trajectoryIterative redefinition of

Data: reference trajectory y of the original non-minimum phase system_dCurrent iteration cycle j, threshold epsilon, maximum iteration number N, and iteration gain k_a；

And (3) outputting: outputting the reference trajectory of the redefined minimum phase system

As shown in fig. 2, the specific process is as follows:

A. selecting an initial value of an iterationy_dA reference trajectory for an original non-minimum phase system; make the minimum phase outputWhere i ∈ {1,2, …, n^ORAnd let the current iteration period j equal to 0;

B. if j is less than N, based on non-singular transformation matrix phi^IRAnd a stable inverse calculation method for outputting the minimum phase C^IRInto the formula (1)3) - (15) calculating the state reference and the control input reference for this iteration cycle of the system respectively:

a reference representing the external state and the reference representing the internal state of the j-th iteration after performing the output redefinition, respectively;

then, the calculation is carried out to obtainAnd executing the step C;

if j is more than or equal to N, the iteration is stopped, and output is carried outFinishing the operation; wherein N is the maximum iteration number;

C. if it isWherein epsilon is a preset threshold value; let j equal j +1, calculate the reference trajectory of the minimum phase system for the next iteration cycle:

and returning to the step B; otherwise, terminating iteration and outputtingAnd finishing the operation.

By the iterative algorithm, the reference trajectory of the output redefined minimum phase system can be calculatedThe output tracking error is effectively reduced, and the control precision is improved.

S3, designing a controller based on a feedback and feedforward structure by using the reference track of the minimum phase system iteratively solved in S2; the method comprises the following substeps:

s31, designing a controller based on the "feedback + feedforward" structure for the system (1) in step S11 according to the reference trajectory of the minimum phase system in step S2 as follows:

u＝u_d+K(x_d-x) (22)

wherein the content of the first and second substances,is a feedback gain matrix, u_dAnd x_dRespectively a control input reference and a state reference; feedforward control term u_dFor realizing accurate tracking, feedback control item K (x)_d-x) for improving system stability and correcting deviations from the state reference trajectory; and S32, determining a feedback gain matrix K according to the pole arrangement theorem.

It will be appreciated by those of ordinary skill in the art that the embodiments described herein are intended to assist the reader in understanding the principles of the invention and are to be construed as being without limitation to such specifically recited embodiments and examples. Those skilled in the art can make various other specific changes and combinations based on the teachings of the present invention without departing from the spirit of the invention, and these changes and combinations are within the scope of the invention.