CN118192335A - Control law design method conforming to full-drive form system - Google Patents
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Abstract
The invention provides a control law design method conforming to a full-drive form system, and belongs to the field of systems and control. Firstly writing a physical model of the system according to a physical rule, then converting the system into a full-drive form according to a conversion step, and finally designing a full-drive control rule of the system and giving a linear steady closed-loop characteristic equation capable of configuring poles at will. The invention can convert a system with nonlinearity into a full-drive form and give out a corresponding full-drive control law, and simultaneously can eliminate all nonlinearity dynamics of the system, thereby obtaining a required linear steady closed-loop system and having more simplicity in system analysis and design.
Description
Technical Field
The invention belongs to the field of systems and control, and particularly relates to a method conforming to the control law design of a full-drive type system.
Background
All practical systems have a physical background, and when modeling a system by using Lagrangian equations or physical laws such as momentum theorem, a second or higher order model is often obtained, and the original model also reflects the physical background of the system in some aspects. In some cases, the background information may aid in analysis and design. However, in the first order system approach, the system model is simplified into a first order state space model, which results in the background information of the system being removed from the original model.
Because nonlinear systems are widely available, complex nonlinearities can make analysis of Lyapunov functions more difficult, so that the traditional state space method has certain limitations in terms of control of a processing system; although the state space method has certain advantages in dealing with problems such as response analysis and observation, the method does not take control variables of a system as important points; because the full-drive characteristic allows all dynamic characteristics (whether linear or nonlinear) of the open-loop system to be canceled, and meanwhile, a brand new hopeful closed-loop dynamic characteristic is established, when a system conforming to the full-drive form is converted into the full-drive system for processing, the characteristic can be utilized to simplify some complex problems, so that the theory of system analysis and design is more practical.
From a physical perspective, there are many fully driven systems in the world, whose original system models are both second or higher order models built from certain laws of physics. The physical concept of full drive can be generalized mathematically to describe various control systems, linearization systems, feedback linearization systems, and some other more general types of nonlinear systems can all be equivalently converted to full drive system forms. In the system modeling process, a series of subsystems in a second-order system form are obtained by utilizing a certain physical quantity, and on one hand, a system in a first-order state space form can be further obtained by variable expansion or equivalently by defining a state vector; on the other hand, the full-drive system can be finally obtained by a variable elimination method. For such a system, the nonlinearity existing in the system can be eliminated by designing a full-drive controller, so that a constant linear closed-loop system is obtained, and the subsequent system analysis is simplified.
Disclosure of Invention
The invention aims to eliminate all dynamic characteristics of an open-loop system and provide a desired steady linear closed-loop system with any configurable characteristic structure by converting a system conforming to a full-drive form into a full-drive system. The invention provides a conversion step from a physical model of the system to a full-drive form and a design method of a full-drive controller through researching a specific system conforming to the full-drive form, and obtains a hopeful steady linear closed-loop system by utilizing the characteristic that the full-drive characteristic allows to cancel all dynamic characteristics of an open-loop system.
The first aspect of the invention provides a control law design method conforming to a full-drive form system, which comprises the following steps:
(1) The conversion from the system physical model to the full-drive system is completed, and the full-drive system form of the system is obtained, which comprises the following steps:
step 101, obtaining a physical model corresponding to the system based on a known physical law;
Wherein x and u are the state vector and the input vector of the system, respectively; θ is the virtual control amount of the system, and
Step 102, converting the physical model of the system in step 101 into a full-drive system model;
converting the first equation in the formula (1) to obtain the following formula:
Where x is the state vector of the system; θ is the virtual control amount of the system;
step 103, the formula (2) obtained in the step 102 is subjected to derivative and then is arranged to obtain the following formula:
Where x is the state vector of the system; θ is the virtual control amount of the system;
step 104, substituting the formula (3) obtained in the step 103 into the second formula of the formula (1) in the step 101, and finishing to obtain a full-drive form of the system:
wherein x and u are the state vector and the control input vector of the system, respectively;
Step 105, order And/>Substituting it into step 104; then equation (4) is rewritten as follows:
x(4)＝F(·)+L(·)u (5)
Wherein L (·) is a continuous matrix function, and detL (·) is not equal to 0; f (·) is a continuous vector function;
(2) The design of the system full-drive control law is completed, a required steady linear closed loop characteristic equation is obtained, and the steady coefficient of the closed loop system equation is assigned, and the method comprises the following steps:
step 201, for any given linear steady matrix, giving a control law of the system in a full-drive mode:
where x (i), i=0, 1,2,3 is the state vector of the system; v is a set value; l (·) is a continuous matrix function; f (·) is a continuous vector function;
step 202, obtaining a linear steady closed loop characteristic equation expected by the system according to the full-drive control law written in step 201, wherein the linear steady closed loop characteristic equation is as follows:
Wherein x (i), i=0, 1,2,3,4 is a state vector; v is a set value; a i, i=0, 1,2,3 is the constant coefficient of the desired closed loop system equation;
Step 203, assigning the equation coefficients a i, i=0, 1,2,3 by introducing a fourth-order expansion of the differential link to the linear steady closed-loop characteristic equation obtained in step 202;
wherein T is a differential time constant; s is a differential operator;
step 204, the linear steady closed loop characteristic equation in step 202 corresponds to the differential fourth-order expansion obtained in step 203, and a i, i=0, 1,2,3 values are obtained;
and 205, assigning a value to T to complete the design of a control law of the full-drive system and obtain a linear steady closed loop characteristic equation hoped by the system.
The second aspect of the present invention provides a control law design system conforming to a full-drive system, comprising:
the full-drive system form conversion module is used for completing conversion from a system physical model to a full-drive system to obtain a full-drive system form of the system;
the full-drive system form conversion module realizes the functions thereof by the following steps:
step 101, obtaining a physical model corresponding to the system based on a known physical law;
Wherein x and u are the state vector and the input vector of the system, respectively; θ is the virtual control amount of the system, and
Step 102, converting the physical model of the system in step 101 into a full-drive system model;
converting the first equation in the formula (1) to obtain the following formula:
Where x is the state vector of the system; θ is the virtual control amount of the system;
step 103, the formula (2) obtained in the step 102 is subjected to derivative and then is arranged to obtain the following formula:
Where x is the state vector of the system; θ is the virtual control amount of the system;
step 104, substituting the formula (3) obtained in the step 103 into the second formula of the formula (1) in the step 101, and finishing to obtain a full-drive form of the system:
wherein x and u are the state vector and the control input vector of the system, respectively;
Step 105, order And/>Substituting it into step 104; then equation (4) is rewritten as follows:
x(4)＝F(·)+L(·)u (5)
Wherein L (·) is a continuous matrix function, and detL (·) is not equal to 0; f (·) is a continuous vector function;
the full-drive control law design module is used for completing the design of the full-drive control law of the system, obtaining a required steady linear closed-loop characteristic equation and assigning a steady coefficient of the closed-loop system equation;
The full-drive control law design module realizes the functions by the following steps:
step 201, for any given linear steady matrix, giving a control law of the system in a full-drive mode:
where x (i), i=0, 1,2,3 is the state vector of the system; v is a set value; l (·) is a continuous matrix function; f (·) is a continuous vector function;
step 202, obtaining a linear steady closed loop characteristic equation expected by the system according to the full-drive control law written in step 201, wherein the linear steady closed loop characteristic equation is as follows:
Wherein x (i), i=0, 1,2,3,4 is a state vector; v is a set value; a i, i=0, 1,2,3 is the constant coefficient of the desired closed loop system equation;
Step 203, assigning the equation coefficients a i, i=0, 1,2,3 by introducing a fourth-order expansion of the differential link to the linear steady closed-loop characteristic equation obtained in step 202;
wherein T is a differential time constant; s is a differential operator;
step 204, the linear steady closed loop characteristic equation in step 202 corresponds to the differential fourth-order expansion obtained in step 203, and a i, i=0, 1,2,3 values are obtained;
and 205, assigning a value to T to complete the design of a control law of the full-drive system and obtain a linear steady closed loop characteristic equation hoped by the system.
A third aspect of the invention provides an apparatus comprising:
one or more processors;
A memory for storing one or more programs,
The one or more programs, when executed by the one or more processors, cause the one or more processors to perform a class of control law design methods consistent with a fully-driven form of the system as described.
A fourth aspect of the present invention provides a computer-readable storage medium storing a computer program which, when executed by a processor, implements a control law design method conforming to a full-drive form system as described.
The invention has outstanding substantive features and significant advances over the prior art, in particular:
the invention can convert a system conforming to a full-drive form into a full-drive system, and eliminates all dynamic characteristics of an open-loop system by utilizing the full-drive characteristic, so that a hopeful steady linear closed-loop system can be obtained even under the condition of a nonlinear system, and the nonlinear control problem which cannot be solved by a plurality of Liidernofv methods is solved.
Drawings
Fig. 1 is a block diagram of a type of system consistent with a full drive format.
FIG. 2 is a flow chart of a system conversion full-drive system conforming to the full-drive form.
FIG. 3 is a simulation diagram of a type of full-drive controller conforming to a full-drive form system.
Detailed Description
Various exemplary embodiments of the invention will be described in detail below in connection with specific embodiments. The description of the exemplary embodiments is merely illustrative, and is not intended to limit the invention, its application, or uses. The present invention may be embodied in many different forms and is not limited to the embodiments described herein. These embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the invention to those skilled in the art. It should be noted that: the relative arrangement of the components and steps set forth in these embodiments should be construed as exemplary only and not as limiting unless specifically stated otherwise.
All terms (including technical or scientific terms) used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs, unless specifically defined otherwise. It will be further understood that terms, such as those defined in commonly used dictionaries, should be interpreted as having a meaning that is consistent with their meaning in the context of the relevant art and will not be interpreted in an idealized or overly formal sense unless expressly so defined herein.
Techniques, methods, and apparatus known to one of ordinary skill in the relevant art may not be discussed in detail, but are intended to be part of the specification where appropriate.
The following is a further detailed description of the embodiments and the accompanying drawings.
Taking a physical system conforming to a full-drive form as an example, which is shown in fig. 1, and describing in detail a conversion flow of the system of fig. 2 into a full-drive system, the specific implementation manner is as follows:
Step 101, obtaining a physical model corresponding to the system based on a known physical law; the formula (1) is as follows:
Wherein x and u are the state vector and input vector of the system, respectively; θ is the virtual control amount of the system and:
Step 102, converting the physical model of the system in step 101 into a full-drive system model, and firstly converting a first equation in the formula (1); the equation (2) is obtained as follows:
Where x is the state vector of the system; θ is the virtual control amount of the system;
Step 103, the formula (2) obtained in the step 102 is subjected to derivation and then is arranged; the equation (3) is obtained as follows:
Where x is the state vector of the system; θ is the virtual control amount of the system;
step 104, substituting the formula (3) obtained in the step 103 into the second formula of the formula (1) in the step 101 and finishing, so as to obtain a system full-drive form; equation (4) is as follows:
wherein x and u are the state vector and the control input vector of the system, respectively;
Step 105, order And/>Substituting it into step 104, equation (4) may be rewritten as follows:
x(4)＝F(·)+L(·)u (5)
Wherein L (·) is a continuous matrix function and detL (·) is not equal to 0; f (·) is a continuous vector function;
step 101) -step 105), the conversion from the system physical model to the full-drive system is completed; according to the form of the full-drive system obtained in the steps, the design of the full-drive control law of the system is completed, and a required steady linear closed-loop characteristic equation is obtained; the method specifically comprises the following steps:
Step 201, for any given linear steady matrix, a control law of the system in a full-drive mode can be given; as in formula (6):
Where x (i), i=0, 1,2,3 is the state vector of the system; v is a set value; l (·) is a continuous matrix function; f (·) is a continuous vector function;
step 202, a linear steady closed loop characteristic equation expected by a system can be obtained by the full-drive control law given in step 201; as in formula (7):
Where x (i), i=0, 1,2,3,4 is a state vector; v is a set value; a i, i=0, 1,2,3 is the constant coefficient of the desired closed loop system equation;
Step 203, assigning the equation coefficients a i, i=0, 1,2,3 by introducing a fourth-order expansion of the differential link to the linear steady closed-loop characteristic equation obtained in step 202; as in formula (8):
wherein T is a differential time constant; s is a differential operator;
Step 204, the linear steady closed loop characteristic equation in step 202 corresponds to the differential fourth-order expansion obtained in step 203, so as to obtain a i, i=0, 1,2,3 values; as in formula (9):
and 205, assigning a value to T to complete the design of a control law of the full-drive system and obtain a linear steady closed loop characteristic equation hoped by the system.
Simulation experiment
The simulation is performed after assigning the constant coefficients of the closed loop system equation in this embodiment, that is, when t=5, 10, 15, and 20 are set, respectively. As shown in fig. 3, the simulation result shows that the simulation process is that the system starts to be 0, the set value of the solid line is stepped from 0 to 0.5 at 10s, and the thick dotted line, the thin dotted line, the dotted line and the dash-dot line are respectively the system outputs when t=5, 10, 15 and 20 are selected; it can be seen that the full-drive control law designed based on the above conversion steps can ensure tracking performance.
As a further embodiment of the present invention, there is provided a control law design system conforming to a full-drive form system, including:
the full-drive system form conversion module is used for completing conversion from a system physical model to a full-drive system to obtain a full-drive system form of the system;
and the full-drive control law design module is used for completing the design of the full-drive control law of the system, obtaining a required steady linear closed-loop characteristic equation and assigning a steady coefficient of the closed-loop system equation.
The specific implementation method of the control law design system in this embodiment refers to a control law design method conforming to a full-drive system, and is not described herein.
As still another embodiment of the present invention, there is provided an apparatus including:
one or more processors;
A memory for storing one or more programs,
The one or more programs, when executed by the one or more processors, cause the one or more processors to perform one of the methods of control law design consistent with a fully-driven form of the system described above.
As a further embodiment of the present invention, there is provided a computer-readable storage medium storing a computer program which, when executed by a processor, implements a control law design method conforming to a full-drive-form system of one of the above inventions.
Claims (4)
1. A control law design method conforming to a full-drive form system is characterized by comprising the following steps:
(1) The conversion from the system physical model to the full-drive system is completed, and the full-drive system form of the system is obtained, which comprises the following steps:
step 101, obtaining a physical model corresponding to the system based on a known physical law;
Wherein x and u are the state vector and the input vector of the system, respectively; θ is the virtual control amount of the system, and
Step 102, converting the physical model of the system in step 101 into a full-drive system model;
converting the first equation in the formula (1) to obtain the following formula:
Where x is the state vector of the system; θ is the virtual control amount of the system;
step 103, the formula (2) obtained in the step 102 is subjected to derivative and then is arranged to obtain the following formula:
Where x is the state vector of the system; θ is the virtual control amount of the system;
step 104, substituting the formula (3) obtained in the step 103 into the second formula of the formula (1) in the step 101, and finishing to obtain a full-drive form of the system:
wherein x and u are the state vector and the control input vector of the system, respectively;
Step 105, order And/>Substituting it into step 104; then equation (4) is rewritten as follows:
x(4)＝F(·)+L(·)u (5)
Wherein L (·) is a continuous matrix function, and detL (·) is not equal to 0; f (·) is a continuous vector function;
(2) The design of the system full-drive control law is completed, a required steady linear closed loop characteristic equation is obtained, and the steady coefficient of the closed loop system equation is assigned, and the method comprises the following steps:
step 201, for any given linear steady matrix, giving a control law of the system in a full-drive mode:
where x (i), i=0, 1,2,3 is the state vector of the system; v is a set value; l (·) is a continuous matrix function; f (·) is a continuous vector function;
step 202, obtaining a linear steady closed loop characteristic equation expected by the system according to the full-drive control law written in step 201, wherein the linear steady closed loop characteristic equation is as follows:
Wherein x (i), i=0, 1,2,3,4 is a state vector; v is a set value; a i, i=0, 1,2,3 is the constant coefficient of the desired closed loop system equation;
Step 203, assigning the equation coefficients a i, i=0, 1,2,3 by introducing a fourth-order expansion of the differential link to the linear steady closed-loop characteristic equation obtained in step 202;
wherein T is a differential time constant; s is a differential operator;
step 204, the linear steady closed loop characteristic equation in step 202 corresponds to the differential fourth-order expansion obtained in step 203, and a i, i=0, 1,2,3 values are obtained;
and 205, assigning a value to T to complete the design of a control law of the full-drive system and obtain a linear steady closed loop characteristic equation hoped by the system.
2. A control law design system conforming to a full drive form system, comprising:
the full-drive system form conversion module is used for completing conversion from a system physical model to a full-drive system to obtain a full-drive system form of the system;
the full-drive system form conversion module realizes the functions thereof by the following steps:
step 101, obtaining a physical model corresponding to the system based on a known physical law;
Wherein x and u are the state vector and the input vector of the system, respectively; θ is the virtual control amount of the system, and
Step 102, converting the physical model of the system in step 101 into a full-drive system model;
converting the first equation in the formula (1) to obtain the following formula:
Where x is the state vector of the system; θ is the virtual control amount of the system;
step 103, the formula (2) obtained in the step 102 is subjected to derivative and then is arranged to obtain the following formula:
Where x is the state vector of the system; θ is the virtual control amount of the system;
step 104, substituting the formula (3) obtained in the step 103 into the second formula of the formula (1) in the step 101, and finishing to obtain a full-drive form of the system:
wherein x and u are the state vector and the control input vector of the system, respectively;
Step 105, order And/>Substituting it into step 104; then equation (4) is rewritten as follows:
x(4)＝F(·)+L(·)u (5)
Wherein L (·) is a continuous matrix function, and detL (·) is not equal to 0; f (·) is a continuous vector function;
the full-drive control law design module is used for completing the design of the full-drive control law of the system, obtaining a required steady linear closed-loop characteristic equation and assigning a steady coefficient of the closed-loop system equation;
The full-drive control law design module realizes the functions by the following steps:
step 201, for any given linear steady matrix, giving a control law of the system in a full-drive mode:
where x (i), i=0, 1,2,3 is the state vector of the system; v is a set value; l (·) is a continuous matrix function; f (·) is a continuous vector function;
step 202, obtaining a linear steady closed loop characteristic equation expected by the system according to the full-drive control law written in step 201, wherein the linear steady closed loop characteristic equation is as follows:
Wherein x (i), i=0, 1,2,3,4 is a state vector; v is a set value; a i, i=0, 1,2,3 is the constant coefficient of the desired closed loop system equation;
Step 203, assigning the equation coefficients a i, i=0, 1,2,3 by introducing a fourth-order expansion of the differential link to the linear steady closed-loop characteristic equation obtained in step 202;
wherein T is a differential time constant; s is a differential operator;
step 204, the linear steady closed loop characteristic equation in step 202 corresponds to the differential fourth-order expansion obtained in step 203, and a i, i=0, 1,2,3 values are obtained;
and 205, assigning a value to T to complete the design of a control law of the full-drive system and obtain a linear steady closed loop characteristic equation hoped by the system.
3. An apparatus, comprising:
one or more processors;
A memory for storing one or more programs,
The one or more programs, when executed by the one or more processors, cause the one or more processors to perform a class of control law design method conforming to a fully-driven form system as recited in claim 1.
4. A computer-readable storage medium storing a computer program, which when executed by a processor implements a control law design method according to claim 1, in accordance with a full-drive form system.
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