CN109901633B - Linear feedback gain scheduling control method based on complex mode - Google Patents

Linear feedback gain scheduling control method based on complex mode Download PDF

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CN109901633B
CN109901633B CN201811645300.XA CN201811645300A CN109901633B CN 109901633 B CN109901633 B CN 109901633B CN 201811645300 A CN201811645300 A CN 201811645300A CN 109901633 B CN109901633 B CN 109901633B
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complex
controller
modal
matrix
controlled object
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CN109901633A (en
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王杰
李东旭
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National University of Defense Technology
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Abstract

The invention provides a linear feedback gain scheduling control method based on a complex mode. The method comprises the following steps: s1, establishing a dynamic model of the controlled object, and acquiring the complex modal characteristics of the controlled object; s2, performing modal reduction on the dynamic model by using a modal truncation method, and designing a linear feedback controller in a complex space based on a linear Riccati equation; and S3, considering the saturation condition of the controller, designing a gain scheduling controller, and utilizing the efficiency of the controller to the maximum extent under the condition of ensuring the stability of a closed loop. The invention solves the problem of controller design when the controlled object adopts complex modal representation.

Description

Linear feedback gain scheduling control method based on complex mode
Technical Field
The invention relates to the field of vibration control research, in particular to a complex-mode-based linear feedback gain scheduling control method.
Background
The vibration control problem is one of the ubiquitous problems of many engineering branches, especially large flexible structures. When excited by internal or external loads, such systems have serious vibration problems, which affect the positioning accuracy, service life, fatigue, safety and other properties of the system. Therefore, designing an efficient controller is one of the important issues facing engineers. In recent decades, for vibration control of flexible structures, researchers have conducted a great deal of research and proposed various control theories, such as direct speed control, strain feedback control, optimal control, Sliding Mode Control (SMC), independent mode space control, robust control, and the like.
However, most of the control theories at present are directed to the design of systems belonging to real space, i.e. the state space and its coefficient matrix belong to the real space system. For systems with structural damping systems or systems considering gyroscopic effects, such as transverse vibration of a rotary shafting system, a high-speed flexible connecting rod system and the like, the state space of the system belongs to a complex space after the modal coordinate representation system is adopted. At present, few researches are carried out on the vibration control of a complex modal system.
Disclosure of Invention
The invention aims to provide a linear feedback gain scheduling control method based on a complex mode, which solves the problem of controller design when a controlled object adopts complex mode representation.
The invention relates to a linear feedback gain scheduling control method based on a complex mode, which comprises the following steps:
s1, establishing a dynamic model of the controlled object, and acquiring the complex modal characteristics of the controlled object;
s2, performing modal reduction on the dynamic model by using a modal truncation method, and designing a linear feedback controller in a complex space based on a linear Riccati equation;
and S3, considering the saturation condition of the controller, designing the gain scheduling controller, and proving the stability of the closed-loop system containing the gain scheduling controller.
Compared with the prior art, the invention has the following advantages:
the invention provides and designs a linear feedback gain scheduling controller aiming at a system expressed in a complex space, expands the controller theory to the complex space, and can be used for a system adopting complex modal representation. The basic idea is to analyze the complex modal characteristics of a controlled object in a complex space, design a linear feedback controller based on the complex modal, consider the saturation condition of the controller, divide the state of a system into a plurality of nested ellipsoid sets according to an attraction domain, and design a gain scheduling controller so as to achieve the purpose of utilizing the efficiency of the controller to the maximum.
Drawings
FIG. 1 flow chart of the method of the present invention
FIG. 2 nested set schematic
Detailed Description
The following description of the embodiments refers to the accompanying drawings.
The invention mainly comprises three steps, as shown in figure 1, and the following detailed description of the specific process is as follows:
s1: establishing a dynamic model of the controlled object, and analyzing and acquiring the complex modal characteristics of the controlled object in a complex space;
substep S11: establishing a dynamic model of a controlled object;
a mathematical model of a controlled object is established based on the Hamiltonian principle, and a control equation is as follows
Figure BDA0001931938900000031
In the formula, δ (N × 1, N system degrees of freedom) is a system degree of freedom; m (NxN), K (NxN) are respectively a mass matrix and a rigidity matrix, and are symmetric matrixes; c (NxN) is the sum of the damping matrix and the gyro matrix. When neglected, the damping matrix, matrix C, is an antisymmetric matrix. P (N × 1) represents a gyro force vector, and F (N × 1) represents an external load vector.
Substep S12: extracting complex modal characteristics of a controlled object;
the system equation is expressed in a state space form
Figure BDA0001931938900000032
In the formula
Figure BDA0001931938900000033
The conjugated system of formula (2) is
Figure BDA0001931938900000041
Upper labelTThe indication of the rank of the turn is,Hindicating the conjugate transition rank.
The free vibration equation of the above system and its conjugate system can be expressed as
Figure BDA0001931938900000042
Asterisks denote the conjugate of a scalar, vector, or matrix. The solutions for the two systems described above can be expressed as
Figure BDA0001931938900000043
Where phi is the left eigenvector and psi is the right eigenvector, the above formula is substituted into formula (5) to obtain
Figure BDA0001931938900000044
Matrix AδAsymmetric, so the eigenvalue λ and eigenvector can be represented as complex conjugate pairs
Figure BDA0001931938900000045
In the formula
Figure BDA0001931938900000046
The matrix can be partitioned into
Figure BDA0001931938900000047
The complex left eigenvector and the complex right eigenvector satisfy the following orthogonality condition
Figure BDA0001931938900000051
arAnd brIs a scalar quantity, satisfies the following equation
Figure BDA0001931938900000052
Defining a modal matrix phi as a transformation matrix
Figure BDA0001931938900000053
The elements in vector x are all real numbers, so the modal coordinates can be divided into two conjugated sub-vectors.
Substituting equation (13) for equation (2) and considering equation (11), the system equation is
Figure BDA0001931938900000054
In the formula
Figure BDA0001931938900000055
Step S2: performing modal reduction on the dynamic model by using a modal truncation method, and designing a linear feedback controller in a complex space based on a linear Riccati equation;
substep S21: carrying out modal reduction on the dynamic model by using a modal truncation method, and writing the modal reduction into a state space form;
typically, the system is mainly affected by the lower order modes, so only the first 2n order modes Φ are retainedcAnd the external load is ignored. The system is shown as
Figure BDA0001931938900000061
In the formula
Figure BDA0001931938900000062
Substep S22: designing a linear feedback controller in a complex space based on a linear Riccati equation;
for a positive definite matrix Q, there is a unique positive definite solution P (2n × 2n) such that
PHA+AHP-PBBHP+Q＝0 (18)
State feedback controller having the following form
u＝-BHPx (19)
Satisfy u*And stabilizes the system (16).
Step S3: and considering the saturation condition of the controller, designing a gain scheduling controller, and maximally utilizing the efficiency of the controller under the condition of ensuring the stability of a closed loop.
Substep S31: defining a series of attraction domains, and designing a gain scheduling controller;
in the case of saturation of the motion vector u, the non-linear system can be represented as
Figure BDA0001931938900000063
The function sat (u) represents actuator saturation with dimension m × 1, and component sat (u)j) Is defined as follows
Figure BDA0001931938900000064
In the formula uj maxFor the extremum of the jth input, the function sign (·) represents a sign function.
The low gain control law may be expressed as
uL＝FL(ε)x (22)
In the formula
FL(ε):＝-BHPε,ε∈(0,1] (23)
In the formula, PεIs a unique positive solution of the following equation
PHA+AHP-PBBHP+Qε＝0 (24)
For any ε >0, the attraction domain may be defined as a set of ellipsoids
ε(Pε){x∈2n:xHPεx≤c} (25)
Wherein, c >0 is defined as
Figure BDA0001931938900000071
Consideration set
ε＝{ε0,ε1,...,εN}εi∈+ and εi<εi+1(i＝0,1,...,N) (27)
In the formula (I), the compound is shown in the specification,
Figure BDA0001931938900000072
is a positive integer.
As shown in FIG. 2, the corresponding set of ellipsoids is
Figure BDA0001931938900000073
Based on the nested ellipsoid set, define the nested controller as
Figure BDA0001931938900000074
With time-varying feedback gain
Figure BDA0001931938900000075
And is
Figure BDA0001931938900000081
tiIndicating that the system state reaches the ith ellipsoid set epsilon (P)i) Time of boundary, λmin(. cndot.) represents the minimum of the real part of the eigenvalue of the positive definite matrix.
Substep S32: the stability of a closed-loop system with a gain scheduling controller is proved;
a closed loop system including a gain scheduling controller as shown in equation (29) may be represented as
Figure BDA0001931938900000082
Selecting the Lyapunov function
V(Pi,t)＝x(t)HPix(t) (33)
V(PiT) derivative with respect to time of
Figure BDA0001931938900000083
Therefore for anyIntention t e [ t ∈i,ti+1)
Figure BDA0001931938900000084
For any time t e [ t ∈. ]i,ti+1) The following inequality can be obtained
Figure BDA0001931938900000091
When t is equal to tiCoefficient ofiIs zero, the system state is satisfied
Figure BDA0001931938900000098
Namely, it is
Figure BDA0001931938900000092
Introducing intermediate variables
Figure BDA0001931938900000093
The Lyapunov function of the intermediate variable is selected as
Figure BDA0001931938900000094
According to the formula (36), a
Figure BDA0001931938900000095
Thus, the intermediate variable is in the attraction domain, when the system has not reached saturation, i.e.
Figure BDA0001931938900000096
And is
Figure BDA0001931938900000097
The controller input represented by equation (29) therefore satisfies the constraint.
The above description is only a preferred embodiment of the present invention, and is not intended to limit the scope of the present invention, and all modifications and equivalents of the present invention, which are made by the contents of the present specification and the accompanying drawings, or directly/indirectly applied to other related technical fields, are included in the scope of the present invention.

Claims (1)

1. A linear feedback gain scheduling control method based on complex mode is characterized by comprising the following steps:
s1, establishing a dynamic model of the controlled object, and acquiring the complex modal characteristics of the controlled object;
s2, performing modal reduction on the dynamic model by using a modal truncation method, and designing a linear feedback controller in a complex space based on a linear Riccati equation;
s3, considering the saturation condition of the controller, designing a gain scheduling controller, and proving the stability of a closed-loop system containing the gain scheduling controller;
the step S1 includes the following steps:
s11, establishing a dynamic model of the controlled object as follows:
Figure FDA0002760315640000011
wherein δ (N × 1) is a system degree of freedom; m (NxN), K (NxN) are respectively a mass matrix and a rigidity matrix, and are symmetric matrixes; c (NxN) is the sum of the damping matrix and the gyro matrix; p (N multiplied by 1) represents a gyro force vector, F (N multiplied by 1) represents an external load vector, and N is a system degree of freedom;
s12, extracting the complex modal characteristics of the controlled object,
the system equation is expressed in a state space form
Figure FDA0002760315640000012
In the formula
Figure FDA0002760315640000013
Figure FDA0002760315640000014
The state transformation is carried out through a complex eigenvector phi, and the system equation is
Figure FDA0002760315640000021
Wherein pi is a complex matrix, λ is a eigenvalue vector, and
Figure FDA0002760315640000022
；
the step S2 includes the following steps:
s21, performing modal reduction on the dynamic model by using a modal truncation method, and writing the dynamic model into a state space form
Figure FDA0002760315640000023
In the formula
Figure FDA0002760315640000024
u is a control quantity;
s22, the linear feedback controller in complex space is:
u＝-BHPx
PHA+AHP-PBBHP+Qε＝0
wherein u is a control quantity, QεIs a positive definite matrix, and P is a positive definite solution;
the step S3 includes the following steps:
s31, designing the gain scheduling controller as:
Figure FDA0002760315640000025
in the formula (I), the compound is shown in the specification,
Figure FDA0002760315640000026
to nest sets, αiIs a gain factor;
s32, proving stability of closed loop system: lyapunov function V (P)iT) the derivative with respect to time satisfies
Figure FDA0002760315640000031
In the formula etaiIs a positive real number, tiIndicating that the system state reaches the ith ellipsoid set
Figure FDA0002760315640000032
The time of the boundary.
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