CN114690635B - State estimation and control method and system of mass spring damping system - Google Patents
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Abstract
The invention discloses a state estimation and control method and a system of a mass spring damping system, which comprises the steps of establishing a switching system complying with a half Markov process aiming at the switching of two different working states of the spring damping system, wherein compared with the traditional random switching process, the process is more realistic, so that the application value is higher; aiming at the switching characteristics, a multi-ellipse surrounding technology and an accessible set analysis theory are applied, and an estimation algorithm of the state of the open-loop system is designed, and the estimation precision of the technology on the accessible set is higher; by the Lyapunov stability analysis principle, a state feedback controller depending on a system mode is designed, a control algorithm for the state of a closed-loop system is provided, the designed controller can better control a multi-mode system, and the control conservatism is reduced. The invention can effectively solve the problems of state estimation and control of the mass spring damping system under the condition that the system model obeys the half Markov switching process.
Description
Technical Field
The invention relates to the technical field of system state estimation and control, in particular to a state estimation and control method and system of a mass spring damping system.
Background
The mass spring damping system was first proposed in the field of mechanical engineering. Due to the characteristics of simple model structure and accuracy close to reality, the system is frequently used in the field of mechanical engineering, such as a bridge dynamic model, a human body exoskeleton back frame model and the like. Meanwhile, the system is ubiquitous in our lives. For example, a car bumper is a device that can reduce kinetic energy, and is a necessary device for ensuring driver safety. Furthermore, dampers are introduced into the anti-seismic. The reinforcing measures of the building change the natural vibration characteristic of the structure, increase the structural damping, absorb the earthquake energy and reduce the influence of the earthquake action on the building.
In recent years, due to the rapid development of computer technology, the demand of computers for realistically restoring scenes of real objects is increasing day by day, and the spring mass damping system provides a simple and practical method for modeling real objects. Mass spring damping systems have been widely used in computer graphics in the past 15 years and are still very popular. These systems are easier to implement and faster than finite element methods, allowing animation of dynamic behavior. They have been applied to animation of inanimate objects such as cloth or soft materials, etc., and animation of organic active objects such as muscles in character animation, etc.
Therefore, it is very meaningful to develop a mass spring damping system. Obtaining a mathematical model which is more in line with reality is a precondition for researching a mass spring damping system. In the modeling method, because the mass spring damping system has a plurality of system modes, a proper system theory is needed to fit the reality. It can be seen from the existing literature that most of them are modeled by a simple switching theory, which is not practical. To improve this, some learners have resorted to markov process modeling. In fact, the markov process has a lingering time following an exponential distribution, which is characterized by a memoryless character, which is not true to reality. In addition, in terms of research problems, most researchers focus on the research on the stability and the calmness of the mass spring damping system under disturbance. In fact, for the system under disturbance, much attention needs to be paid to ensure that the system state is maintained in an ideal (or safe) area (i.e. reachable set of problems), rather than stability problems, such as a circuit system with disturbance, as long as the current and voltage are maintained in a safe range, rather than a fixed value, is sufficient. Therefore, it is more practical to study the problem of maintaining the displacement range of the whole system in a safe range by using the physical mass spring damping system. Meanwhile, the reachable set is researched, so that the system performance is guaranteed, the control resources are saved, and the control efficiency is improved.
Disclosure of Invention
The invention aims to provide a state estimation and control method and system of a mass spring damping system, which solve the problem that the modeling of the mass spring damping system is not fit in the prior art, and on the basis, the state of the system is estimated more quickly, and meanwhile, the state of the system is effectively controlled to be kept in a safe area.
In order to solve the technical problem, the invention provides a state estimation and control method of a mass spring damping system, which comprises the following steps:
s100, establishing a switching system obeying a semi-Markov process according to the working state of a mass spring damping system, and determining the mode transition probability of the switching system, wherein the transition probability is determined by the lingering time of a switching signal of the switching system, and the lingering time obeys non-exponential distribution;
s200, determining a bounded value of disturbance of the switching system and upper and lower bounds of transfer rate, and calculating a matrix in an ellipse set according to a dynamic equation of a half Markov jump system, thereby determining the ellipse set surrounding the state of the switching system and realizing the state estimation of the switching system in an open loop state;
s300, an expected ellipse set of the switching system in the closed-loop state is given, a state feedback controller depending on the mode of the switching system is established, a gain matrix of the state feedback controller is calculated according to a kinetic equation of the switching system, and the state of the switching system in the closed-loop state is controlled through the gain matrix.
As a further improvement of the present invention, the model of the mass spring damping system is:
wherein m represents mass, F f And F s Representing friction and return force, respectively, u representing control input, friction
C is greater than 0, restoring force F s Involving linear portions and hardening spring forces, i.e. F f ＝kx+ka 2 x 3 K, a are constants, and x represents the displacement from the reference point.
As a further improvement of the invention, said establishing a handover system subject to a semi-markov process comprises:
is provided with
Meanwhile, a ≠ 0 and a =0 follow the half-markov process, the state space equation of the switching system:
wherein x is t Represents the system state, u t Representing a control input, ω t Which is indicative of a process disturbance,
and
each represents a coefficient matrix of a mass-spring damping system.
As a further development of the invention, the process disturbance ω t Peak bounded is satisfied:
as a further development of the invention, the switching signal γ of the switching system t Following the half Markov process, its dwell time τ satisfies the state transition probability matrix as follows:
wherein σ ij (τ) represents the probability of transition from modality i to modality j, and
delta denotes the increment of time t and
as a further improvement of the present invention, the step S200 specifically includes the following steps:
s201, determining a limit value of disturbance of a switching system;
s202, determining a non-exponential distribution type obeyed by the stay time of the switching system;
s203, calculating the upper bound of the lingering time according to the 99% confidence level
Lower bound of residence timeτ；
S204, determining an upper bound of the transfer rate of the corresponding mode according to the boundary value result
Lower bound of transfer rateσ ij And solve for
And
s205, solving a specific matrix P according to the following algorithm i ：P i Once determined, the surrounding system state x t Set of ellipses
Namely, determining; if the system cannot be solved, the state estimation of the system cannot be carried out.
As a further improvement of the present invention, for step S205, if there is a symmetric matrix P i ＞0，
And a constant α > 0, such that the following linear matrix inequality holds:
the reachable set of the switching system is elliptically set in the open loop state
Mean square bounded enclosure in which symbols
As a further improvement of the present invention, the resulting ellipse set is minimized by the optimization problem in step S205: exists in ρ i ≤P i And maximizes the positive parameter ρ i I.e. by
As a further improvement of the present invention, the step S300 specifically includes the following steps:
s301, determining a desired safety area according to the actual requirement of the switching system, wherein the safety area is set in an ellipse
Is expressed in terms of form;
s302, designing a state feedback controller u depending on a mode t ＝K i x t In which K is i Is the controller gain;
s303, solving K according to the following controller gain solving algorithm i ：
For a symmetric matrix f i > 0 and a constant alpha > 0, if any, of the symmetric matrix P i ＞0，T ij ＞0，
And matrix F i So that the following linear matrix inequality holds:
then the feedback controller u is in a modality dependent state t ＝K i x t Given a given set of ellipses to which the reachable set of the closed-loop system is given
Mean square bounded enclosure in which symbols
A state estimation and control system of a mass spring damping system adopts the method to estimate and control the state of the mass spring damping system.
The invention has the beneficial effects that: the invention adopts the memorability half Markov process to model the mass spring damping system, and is more practical compared with the traditional random switching process and Markov process, thereby having higher application value; in addition, the method adopts a multi-ellipse surrounding technology to estimate the reachable set, and compared with the traditional single-ellipse surrounding technology, the multi-ellipse surrounding technology has higher estimation precision on the reachable set; the invention also adopts a state feedback controller depending on the mode, so that the multi-mode system can be better controlled, and the control conservatism is reduced.
Drawings
FIG. 1 is a schematic flow diagram of the present invention;
fig. 2 is a schematic structural view of the mass spring damping system of the present invention.
Detailed Description
The present invention is further described below in conjunction with the following figures and specific examples so that those skilled in the art may better understand the present invention and practice it, but the examples are not intended to limit the present invention.
Referring to fig. 1, the present invention provides a state estimation and control method of a mass spring damping system, comprising the steps of:
s100, establishing a switching system obeying a semi-Markov process according to the working state of a mass spring damping system, and determining the mode transition probability of the switching system, wherein the transition probability is determined by the lingering time of a switching signal of the switching system, and the lingering time obeys non-exponential distribution;
s200, determining a bounded value of disturbance of the switching system and upper and lower bounds of transfer rate, and calculating a matrix in an ellipse set according to a dynamic equation of a half Markov jump system, thereby determining the ellipse set surrounding the state of the switching system and realizing the state estimation of the switching system in an open loop state;
namely, estimating the reachable set of the open-loop system of the model; the method adopts a multi-ellipse technology to estimate the reachable set of the system, the multi-ellipse is expressed in a mathematical form of an ellipse set union set, and the minimum multi-ellipse is obtained through an optimization method;
s300, an expected ellipse set of the switching system in a closed loop state is given, a state feedback controller depending on the mode of the switching system is established, a gain matrix of the state feedback controller is calculated according to a kinetic equation of the switching system, and the state of the switching system in the closed loop state is controlled through the gain matrix;
i.e. the reachable set of the closed-loop system of the model as above. For a given desired region represented in the form of an ellipse set, a modality-dependent state feedback controller is designed that can effectively control the state of the system to lie within the desired ellipse set.
The invention adopts a half Markov process to model the mass spring damping system so as to solve the defect that the modeling of the system is not fit with the reality in the existing modeling form. On the basis, the reachable set theory is used for performing reachable set analysis and control research on the mass spring damping system so as to ensure that the system state is kept in an ideal (or safe) area. Meanwhile, compared with the traditional sliding mode control and state estimation, the system state can be effectively saved and the control efficiency can be improved by researching the system state through the reachable set analysis theory.
Specifically, the invention relates to three parts of half Markov modeling of a mass spring damping system, reachable set estimation of an open loop system and reachable set control of a closed loop system:
1. half Markov modeling of a mass spring damping system:
as shown in fig. 2, a mass spring damping system was modeled. Applying Newton's law, there are:
wherein m represents mass, F f And F s Representing frictional and restoring forces, respectively, and u representing an external input. Frictional force
Restoring force F s Involving linear portions and hardening spring forces, i.e. F f ＝kx+ka 2 x 3 (k, a is a constant) where x represents the displacement from a reference point. Further, it is provided with
Meanwhile, considering disturbance factors existing in a real environment and two conditions that a ≠ 0 and a =0 follow a half-Markov process, the method can be converted into a unified switching system state space equation:
wherein the content of the first and second substances,
and
all represent a coefficient matrix of a mass spring damping system, t represents time, a switching signal gamma t Following the half Markov process, the linger time τ satisfies the transition probability as follows:
for convenience, γ is t = i and γ t+δ = j. Wherein σ ij (τ) represents the rate of transfer from modality i to modality j, and
delta represents a small increment with respect to time t and
different from common state average models (such as random switching process and Markov process), the switching system based on the half Markov process considers the linger time tau of each system mode, and can describe the dynamic characteristics of the mass spring damping system more accurately.
Disturbance ω in a system (2) t Satisfying peak bounding, i.e.
Wherein the content of the first and second substances,
being constant, this assumption is also realistic.
Further, before studying the reachable set, the reachable set of system (2) is defined as follows:
in fact, it is often impossible to obtain the reachable set accurately
The expression form of (2). Thus, the reachable set is typically approximated using as small an area as possible
The boundary of (2). The present invention defines the reachable set using an ellipsoid bounding technique. For the random case, the set of ellipses that encloses the reachable set is defined as follows:
where symmetric matrix M > 0 is the matrix to be determined. Once the matrix M is determined, a specific set of ellipses is determined.
Before the estimation and control content of the reachable set, lemma 1 and lemma 2 need to be mentioned, which will be used later.
Introduction 1: if multiple Lyapunov functions exist
And a constant alpha > 0, such that
If true, then for t * Greater than 0 always has
2, leading: given arbitrary constants epsilon and square matrix
The inequality as follows:
ε(S+S T )≤ε 2 P+SP -1 S T
for arbitrary symmetric positive definite matrix
Both are true.
2. Estimation of the reachable set:
the reachable set of the open-loop system is estimated, that is, a sufficient condition that an elliptic set surrounds the reachable set is provided for determining the bounded region of the reachable set.
Considering an open loop system as above, if a matrix is present
And a constant α > 0, such that the following linear matrix inequality holds:
the reachable set of the open-loop system is set by the ellipse
The mean square is bounded. Wherein the symbol
And (3) proving that: constructing a Lyapunov function
Wherein the cumulative distribution function G here i (τ) is a function of τ in the i-mode, q ij Representing the probability density of the system from modality i to modality j.
X in (9) t+δ First order Taylor expansion to [ A i δ+I]x t +C i ω t δ + o (δ), and, at the same time,
further, define
Is provided with
Based on the theory 1, if
Wherein the content of the first and second substances,
then
It holds, that means that the reachable set of the open-loop system is elliptically set
The mean square bounded surrounds.
At this point, the resulting matrix inequality is non-linear, and then processing is done as a linear matrix inequality.
In the step (11), the first step is carried out,
further, according to lemma 2, there is a correlation to all | Δ σ ij |≤o ij Existence of a symmetric matrix T ij Greater than 0 has
This means that (11) can be obtained if the following inequality holds:
wherein the content of the first and second substances,
by Schur supplement, (14) can be rewritten as (7). From this, the certification is complete.
At the same time, the ellipse set needs to be optimized so that it is as small as possible. Conditions need to be added on the basis of the above: rho i ≤P i And maximizes the positive parameter ρ i That is to say that
3. And (3) control of the reachable set:
the reachable set of the closed-loop system as above is controlled, i.e. a mode-dependent state feedback controller is designed such that the reachable set of the system is located within a given elliptical area.
Considering a closed-loop system as above, for a given symmetric matrix f i > 0 and a constant alpha > 0, if any, of the symmetric matrix P i ＞0，T ij ＞0，
And matrix F i The following linear matrix inequality holds:
then the feedback controller u is in a modality dependent state t ＝K i x t Given a set of ellipses, the reachable set of the closed-loop system
The mean square is bounded. Wherein the symbols
And (3) proving that: a in (11) i Is replaced by A i +B i K i Can obtain
Multiplying both sides of (18) by diag { P -1 I }, order
Is provided with
It is noted that
Thus, these two terms are discussed separately below.
For the previous item
By introducing a relaxation variable symmetric matrix Q ij (j ≠ i) makes (16) stand, have
For the latter item
According to lemma 2, for all | Δ σ | ij |≤o ij There is a symmetric positive definite matrix T ij So that
Taking the above two aspects into consideration, (19) holds if the following matrix inequality holds:
wherein
By Schur supplement, (22) can be rewritten as (15). From this, the certification is complete.
The invention relates to an accessible set estimation and control method of a mass spring damping system, which adopts a memorable half Markov process to model the mass spring damping system aiming at different working states of the mass spring damping system and constructs a uniform state space equation. Compared with the traditional random switching process and Markov process, the method is more practical and has higher application value. Based on the above, in order to determine the bounded region of the reachable set, the method adopts the multi-ellipse surrounding technology to estimate the reachable set, and compared with the traditional single-ellipse surrounding technology, the multi-ellipse surrounding technology has higher estimation precision on the reachable set. In addition, the present invention designs a modality-dependent state feedback controller such that the reachable set of the closed-loop system is within a given elliptical region. The controller designed can better control the multi-modal system, and conservatism of control is reduced.
The invention also provides a state estimation and control system of the mass spring damping system, and the method is adopted to estimate and control the reachable set of the mass spring damping system.
The principle part is the same as the method, and the description is omitted here, namely, a memorability half Markov process is adopted to model the mass spring damping system, and compared with the traditional random switching process and Markov process, the method is more practical and has higher application value. In addition, the method adopts the multi-ellipse surrounding technology to estimate the reachable set, and compared with the traditional single-ellipse surrounding technology, the multi-ellipse surrounding technology has higher estimation precision on the reachable set. The invention also adopts a state feedback controller depending on the mode, so that the multi-mode system can be better controlled, and the control conservatism is reduced.
The above-mentioned embodiments are merely preferred embodiments for fully illustrating the present invention, and the scope of the present invention is not limited thereto. The equivalent substitution or change made by the technical personnel in the technical field on the basis of the invention is all within the protection scope of the invention. The protection scope of the invention is subject to the claims.
Claims (5)
1. A state estimation and control method of a mass spring damping system is characterized in that: the method comprises the following steps:
s100, establishing a switching system which obeys a half Markov process according to the working state of a mass spring damping system, and determining the mode transition probability of the switching system, wherein the transition probability is determined by the lingering time of a switching signal of the switching system, and the lingering time obeys non-exponential distribution;
the model of the mass spring damping system is as follows:
wherein m represents mass, F f And F s Representing friction and return force, respectively, u representing control input, friction
C is greater than 0, restoring force F s Involving linear portions and hardening spring forces, i.e. F s ＝kx+ka 2 x 3 K and a are constants, and x represents the displacement relative to a reference point;
the establishing of a handover system subject to a semi-Markov process comprises:
is provided with
Meanwhile, a ≠ 0 and a =0 follow the half-markov process, the state space equation of the switching system:
wherein x is t Represents the system state, u t Representing a control input, ω t Which is indicative of a process disturbance,
and
all represent coefficient matrixes of the mass spring damping system;
the process disturbance ω t The peak-bounded is satisfied:
switching signal gamma of the switching system t Following the half Markov process, its dwell time τ satisfies the state transition probability matrix as follows:
wherein σ ij (τ) represents the probability of transition from modality i to modality j, and
delta denotes the increment of time t and
s200, determining a bounded value of disturbance of the switching system and upper and lower bounds of transfer rate, and calculating a matrix in an ellipse set according to a dynamic equation of a half Markov jump system, thereby determining the ellipse set surrounding the state of the switching system and realizing the state estimation of the switching system in an open loop state;
the step S200 specifically includes the following steps:
s201, determining a limit value of disturbance of a switching system;
s202, determining a non-exponential distribution type obeyed by the stay time of the switching system;
s203, calculating the upper bound of the lingering time according to the 99% confidence level
Lower bound of residence timeτ；
S204, determining an upper bound of the transfer rate of the corresponding mode according to the boundary value result
Lower bound of transfer rateσ ij And solve for
And
s205, solving a specific matrix P according to the following algorithm i ：P i Once determined, the surrounding system state x t Set of ellipses
Namely, determining; if the system cannot be solved, the state estimation of the system cannot be carried out;
s300, an expected ellipse set of the switching system in the closed-loop state is given, a state feedback controller depending on the mode of the switching system is established, a gain matrix of the state feedback controller is calculated according to a kinetic equation of the switching system, and the state of the switching system in the closed-loop state is controlled through the gain matrix.
2. A state estimation and control method of a mass spring damping system as claimed in claim 1, characterized in that: for step S205, if there is a symmetric matrix P i ＞0,T ij ＞0，
And a constant α > 0, such that the following linear matrix inequality holds:
the reachable set of the switching system is set by ellipses in the open loop state
Mean square bounded enclosure in which symbols
3. A state estimation and control method of a mass spring damping system as claimed in claim 2, characterized in that: in step S205, the resulting ellipse set is minimized by an optimization problem: exists in ρ i ≤P i And maximizes the positive parameter ρ i I.e. by
4. A state estimation and control method of a mass spring damping system as claimed in claim 1, characterized in that: the step S300 specifically includes the following steps:
s301, determining a desired safety zone according to the actual requirement of the switching system, wherein the safety zone is set by ellipses
Is expressed in terms of form;
s302, designing a state feedback controller u depending on a mode t ＝K i x t In which K is i Is the controller gain;
s303, solving K according to the following controller gain solving algorithm i ：
For the symmetric matrix Γ i > 0 and a constant alpha > 0, if any, of the symmetric matrix P i ＞0，T ij > 0, matrix Q ij And matrix F i ，
Such that the following linear matrix inequality holds:
then the feedback controller u is in a modality dependent state t ＝K i x t Given a given set of ellipses to which the reachable set of the closed-loop system is given
Mean square bounded enclosure in which symbols
5. A state estimation and control system for a mass spring damping system, characterized by: the method according to any of claims 1-4 is used for estimating and controlling the state of a mass spring damping system.
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