CN107966905B - Consistency control method and device for multi-trolley single-stage inverted pendulum system - Google Patents
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Abstract
The invention discloses a consistency control method and a device of a multi-trolley single-stage inverted pendulum system, wherein the method comprises the steps of S1, setting the first trolley single-stage inverted pendulum as a pilot, and setting N trolley single-stage inverted pendulums as followers of the pilot; s2, determining the feedback control force u of the ith follower in the N followers according to the following algorithmi：
1,2, N, wherein xiState vector, x, representing the ith followerjRepresenting the state vector, x, of the jth dolly single-stage inverted pendulumlState vector representing the pilot, aijDenotes xjAnd xiA weight coefficient ofilDenotes xlAnd xiK represents a controller gain matrix; a if the ith follower can acquire the state vector of the jth dolly single-stage inverted pendulum from the jth dolly single-stage inverted pendulumij1, otherwise aij＝0。
Description
[ technical field ] A method for producing a semiconductor device
The invention relates to a consistency control method and device for a multi-trolley single-stage inverted pendulum system.
[ background of the invention ]
In the cooperative control problem of a multi-agent system, a plurality of agents interact by means of information communication or the like to achieve a desired overall objective. Compared with a single intelligent agent system, the multi-intelligent agent system has stronger flexibility, higher reliability and the like. The cooperative control of the multi-agent system has important application prospects in the fields of satellite formation control, sensor network information fusion, synchronous control of complex dynamic networks, congestion control, intelligent traffic control and the like. The consistency problem is a fundamental problem in the cooperative control of multi-agent systems, the objective being to drive some state quantities of all agents towards the same value. The goal of coherency control is to make it possible for each agent to achieve some state of consistency for all agents based on local information by designing appropriate coherency algorithms.
In order to achieve consistency, the agents need to exchange necessary status information (e.g., position, speed, etc. of vehicles in formation control), and thus reliable transmission and reception of information is a prerequisite to ensure that the status of all agents converge to the same value. When information of a plurality of agents is transmitted through a wireless network, the performance and reliability of the system are affected by network environment factors due to network time lag or data packet loss and other problems which may occur. In addition, multiple agents are required to share a wireless communication channel to increase spectrum usage, thereby causing allocation and contention issues for wireless networks. One solution to this problem is to design a media access control protocol to ensure that the necessary state information for multiple agents can be successfully transmitted and received within a sample period.
The inverted pendulum is an underactuated unstable system, has high requirements on the real-time performance of information, and is a common experimental device for designing and testing control strategies in the control field. The study on the consistency control problem of a plurality of inverted pendulums sharing a wireless network can effectively verify the effectiveness of a consistency algorithm and the reliability of a medium access control protocol on solving the information conflict problem.
[ summary of the invention ]
In order to overcome the defects of the prior art, the invention provides a consistency control method and a consistency control device of a multi-trolley single-stage inverted pendulum system, so that each inverted pendulum keeps a stable state and achieves consistent states.
A consistency control method of a multi-trolley single-stage inverted pendulum system,
s1, setting the first trolley single-stage inverted pendulum as a pilot, and setting N trolley single-stage inverted pendulums as followers of the pilot;
s2, determining the feedback control force u of the ith follower in the N followers according to the following algorithmi：
Wherein x isiState vector, x, representing the ith followerjRepresenting the state vector, x, of the jth dolly single-stage inverted pendulumlState vector representing the pilot, aijDenotes xjAnd xiA weight coefficient ofilDenotes xlAnd xiK represents a controller gain matrix; a if the ith follower can acquire the state vector of the jth dolly single-stage inverted pendulum from the jth dolly single-stage inverted pendulumij1, otherwise aij＝0；
S3 controlling force u by feedbackiThe ith follower is controlled.
Preferably, the first and second liquid crystal materials are,
the communication between the pilot and the follower and between the pilot and the pilot adopts a medium access control protocol of a protocol sequence constructed by a method based on a generalized prime sequence.
Preferably, the first and second liquid crystal materials are,
k is calculated according to the following algorithm:
K＝c(R+HTPH)-1HTPG；
where the matrix P is the solution of the following algebraic ricati equation:
P-GTPG+(1-ζ2)GTPH(R+HTPH)-1HTPG+Q＝0；
wherein R controls a weight value of energy consumption; h and G are model parameter matrixes of the inverted pendulum system of the trolley; c is a weight factor of the coupling strength between the inverted pendulums of the trolleys, Q is a positive definite matrix or a semi-positive definite real symmetric matrix, R is a positive definite real symmetric matrix, and c satisfies the following inequality:
wherein,
j is 1,2
L is a graph Laplace matrix describing the relationship among all following trolley inverted pendulum systems;
j 1,2, 4 represents the unstable characteristic root of the system parameter matrix G;
wherein the parameters
Preferably, the first and second liquid crystal materials are,
if the state vector of the follower at the k +1 th moment and the state vector x [ k ] of the follower at the k-th moment satisfy the following relation, the consistency between the N followers and the pilot is judged to be achieved:
The invention also provides a consistency control device of the multi-trolley single-stage inverted pendulum system,
the first processing unit is used for setting the first trolley single-stage inverted pendulum as a pilot and setting the N trolley single-stage inverted pendulums as followers of the pilot;
a second processing unit for determining the feedback control force u of the ith follower of the N followers according to the following algorithmi：
Wherein x isiState vector, x, representing the ith followerjRepresenting the state vector, x, of the jth dolly single-stage inverted pendulumlState vector representing the pilot, aijDenotes xjAnd xiA weight coefficient ofilDenotes xlAnd xiK represents a controller gain matrix; a if the ith follower can acquire the state vector of the jth dolly single-stage inverted pendulum from the jth dolly single-stage inverted pendulumij1, otherwise aij＝0；
A third processing unit for controlling the force u by feedbackiThe ith follower is controlled.
Preferably, the first and second liquid crystal materials are,
the communication between the pilot and the follower and between the pilot and the pilot adopts a medium access control protocol of a protocol sequence constructed by a method based on a generalized prime sequence.
Preferably, the first and second liquid crystal materials are,
k is calculated according to the following algorithm:
K＝c(R+HTPH)-1HTPG；
where the matrix P is the solution of the following algebraic ricati equation:
P-GTPG+(1-ζ2)GTPH(R+HTPH)-1HTPG+Q＝0
wherein R controls a weight value of energy consumption; h and G are model parameter matrixes of the inverted pendulum system of the trolley; c is a weight factor of the coupling strength between the inverted pendulums of the trolleys, Q is a positive definite matrix or a semi-positive definite real symmetric matrix, R is a positive definite real symmetric matrix, and c satisfies the following inequality:
wherein,
j is 1,2
L is a graph Laplace matrix describing the relationship among all following trolley inverted pendulum systems;
j 1,2, 4 represents the unstable characteristic root of the system parameter matrix G;
wherein the parameters
Preferably, the first and second liquid crystal materials are,
if the state vector of the follower at the k +1 th moment and the state vector x [ k ] of the follower at the k-th moment satisfy the following relation, the consistency between the N followers and the pilot is judged to be achieved:
the invention has the beneficial effects that:
the states of the inverted pendulums of the plurality of trolleys tend to be consistent, and simultaneously the stability of the inverted pendulums needs to be controlled, so that the inverted pendulums keep stable states and reach consistent states.
The method adopts a medium access control protocol based on a protocol sequence and applies a Generalized prime sequence (Generalized prime sequences) method to construct the protocol sequence so as to ensure that the state information of the inverted pendulum of each trolley can be successfully transmitted and received in a sampling period, ensure the certainty and reliability of a system communication topological structure and further ensure the effectiveness of the proposed consistency algorithm.
[ detailed description ] embodiments
The preferred embodiments of the invention are described in further detail below.
Consistency control of multiple cart inverted pendulum system
The first-order trolley inverted pendulum system is a closed loop system consisting of an embedded controller, a motor driving circuit, a servo motor, a swing rod and a photoelectric coded disc. The photoelectric coded disc feeds back the angle and angular speed signals of the connecting rod to the embedded controller, and the angle and angular speed signals of the oscillating bar are also fed back to the embedded controller by the photoelectric coded disc. The embedded controller receives the real-time data and calculates corresponding control quantity according to the control decision so as to drive the servo motor to rotate, thereby driving the connecting rod to move and keeping the balance of the swing rod in the vertical direction. After air resistance and various friction forces are neglected, the linear primary inverted pendulum system can be abstracted into a system consisting of a trolley and a homogeneous rod.
The included angle theta (radian) between the swing rod and the vertical upward direction is assumed to be very small compared with 1, namely theta is less than 1, so that the approximate processing is performed: cos theta is approximately equal to 1, sin theta is approximately equal to theta,
according to the assumption, the dynamic model of the first-order inverted pendulum system of the trolley is linearized near the balance point. Let xi＝[θ,θ,x,x]TThe state space equation of the ith trolley inverted pendulum system in the continuous time domain is as follows:
that is, the kinetic model of the system is:
wherein M is the mass of the trolley, M is the mass of the oscillating bar, L is the length from the rotating axis of the oscillating bar to the mass center of the bar, I is the inertia of the oscillating bar, u is the mass center of the oscillating bariFor the force applied to the trolley i, p is the trolley position, theta is the included angle between the oscillating bar and the vertical upward direction (considering that the initial position of the oscillating bar is vertical upward), b is the friction coefficient of the trolley, and bp is the friction force between the trolley and the ground.
The system is stabilized by a digital controller, so that we sample a continuous time system (2) with a sampling time TsAnd then the discrete time state equation of the ith trolley inverted pendulum system is as follows:
wherein,
is a parameter matrix of a discretized inverted pendulum system of the small vehicle uiIs a control quantity, xi[k]＝[θ,Δθ,p,Δp]TThe state quantities theta, delta theta, p and delta p of the system respectively represent the angle of the swing rod, the angular speed of the swing rod, the displacement of the trolley and the speed of the trolley, and y is outputi＝[θp]TIncluding pendulum rod angle and dolly displacement. Consider the system to be fully controllable.
According to equation of state (3) for a single inverted pendulum of a dolly, the equation of state for N inverted pendulum systems of dollies can be written as:
wherein
uS[k]＝[u1,u2,L,uN]T，
G, assuming that the N dollies are identicali＝Gj，Hi＝Hj，Ci＝Cj，
And N is added. This is achieved byWhen it is, let Gi＝G，Hi＝H，Ci＝C。
Considering a system comprising N +1 same inverted pendulums of trolleys, the scheme aims to ensure that the states of the N +1 inverted pendulum systems of the trolleys are consistent after the inverted pendulum systems of the trolleys enter a stable state, namely xi[ks]＝xj[ks]，i≠j，ksThe time after the system reaches steady state. Because the inverted pendulums of the trolleys are under-actuated unstable systems, the states of the inverted pendulums of the trolleys tend to be consistent, and simultaneously the stability of the inverted pendulums needs to be controlled, so that the inverted pendulums keep stable states and reach consistent states. Therefore, a piloting-following scheme is adopted, one inverted pendulum of the trolley in the system is used as a pilot, the other N inverted pendulums of the trolley are used as followers, one or more followers receive state information sent by the pilot through a wireless communication channel, and the followers receive and transmit the respective state information through the wireless communication channel. And designing a consistent control law to enable the state of each follower to be consistent with that of a pilot, so that the consistent control of the N +1 trolley inverted pendulum system is realized.
The inverted pendulum of the trolley as a pilot adopts a linear control law. The controller is designed so that when the initial position of the system is not at the equilibrium position, the trolley moves left and right, the swing rod swings left and right along with the trolley, and then the trolley still returns to the vertical position. The angular speed of the pendulum and the translation speed of the trolley can be measured through the encoder, and the angle of the pendulum and the displacement of the trolley can be obtained by integrating the angular speed of the pendulum rod and the translation speed of the trolley, so that the trolley can realize full-state feedback, and the feedback control law K is determined. Consider the objective function of a linear quadratic regulator:
wherein Q is a positive definite matrix or a semi-positive definite real symmetric matrix and represents the weight of the system state offset; and R is a positive definite real symmetric matrix and represents the weight of control consumption.
Considering the discrete time system (3), solving the control law which minimizes the linear quadratic regulator objective function J to obtain the optimal controllerCorresponding feedback state vector Ki. If the inverted pendulum of the vehicle as the pilot is marked as l, the corresponding linear feedback control amount ul[k]Comprises the following steps:
ul[k]＝Kl·xl[k](6)
wherein, KlFeedback state vector, x, for optimal controllerl[k]The state vector of the inverted pendulum of the vehicle as a pilot is shown.
Aiming at N small vehicle inverted pendulums as followers, the corresponding control quantity is as follows:
wherein if the ith trolley inverted pendulum can obtain information from the jth (j ═ l,1,2, K, N) inverted pendulum, aij1 is ═ 1; otherwise, aij＝0。
The controller gain matrix K in equation (7) is:
K＝c(R+HTPH)-1HTPG (8)
wherein the matrix R is from equation (5), H and G are system parameter matrices from equation (4), and the weighting factor c satisfies the following inequality:
wherein,
j is 1,2
L is a graph Laplace matrix describing the relationship among all following trolley inverted pendulum systems;
j 1,2, 4 represents the root of the instability feature of the system parameter matrix G. The matrix P in equation (8) is the solution of the following algebraic ricati equation:
P-GTPG+(1-ζ)GTPH(R+HTPH)-1HTPG+Q＝0 (10)
wherein,
matrices H and G are from equation (4) and R and Q are from equation (5).
Based on a consistency control algorithm (7), the closed loop kinetic equation of N inverted pendulums of the trolleys as followers is as follows:
when k is large enough, the inverted pendulum of the trolley can be judged whether to reach the consistent state or not by observing the value of x [ k ].
(II) medium access control protocol based on protocol sequence
In order to effectively utilize wireless spectrum resources, the inverted pendulums of the trolleys share the same communication channel, namely, information among the inverted pendulums of the trolleys is transmitted through the wireless communication channel of the same frequency band. The medium access control protocol can provide an effective solution for the problem of information transmission collision caused by the shared channel. Traditional distributed medium access control protocols, such as the Aloha protocol and the carrier sense multiple access/collision detection method, have the advantage of high throughput, but cannot ensure that information is successfully transmitted and received within a certain set time threshold, the delay of the information is time-varying, and even data packet loss occurs, which makes the communication topology structure of the whole system a time-varying topology structure, thereby affecting the control performance of a consistency algorithm, or the system cannot achieve consistency due to data packet loss.
Considering the problems of the conventional distributed medium access control protocol, the protocol sequence (protocol sequence) has the characteristic of ensuring successful transmission of information in a limited time, so the medium access control protocol based on the protocol sequence is adopted to ensure that the state information of each inverted pendulum of the trolley can be successfully transmitted and received in one sampling period.
First, the basic concept of the protocol sequence and its application in the medium access control protocol are described by taking an example of three users sharing a wireless communication channel. Allocating PS to three users sharing a wireless communication channel1，PS2And PS3Three protocol sequences:
PS1:1 0 0 1 0 0 1 0 0,
PS2:1 0 0 0 1 0 0 0 1,
PS3:1 0 0 0 0 1 0 1 0. (12)
the protocol sequence is a periodic sequence, and each period of the protocol sequence is divided into LSPAnd a time slot. In the current time slot, if the corresponding numerical value of the protocol sequence allocated to the kth user is 1, the user transmits signals; if the corresponding value is "0", the user does not transmit a signal in the time slot. Each user transmits information according to the allocated protocol sequence, and by constructing and designing an effective protocol sequence, all users in the system can be ensured to successfully transmit information at least once in one period.
We construct protocol Sequences using the method of Generalized Prime Sequences. Let rem (z, p) denote the remainder of z/p, where z is a positive integer and p is a prime number. The prime number p is called the hamming weight and represents the number of "1" s in the protocol sequence. Given Hamming weight p and positive integer q, wherein q is more than or equal to p, the period is LSPThe protocol sequence p · q can be configured as follows:
Ig＝{rem(gl,p)+lq:l＝0,1,2,...,p-1} (13)
wherein, g is 0,1, p-1 represents the g-th sequence of p sequences, IgRepresents the set of positions of "1" in the g-th sequence.
The protocol sequence in equation (12) is constructed by the method in equation (13), where hamming weight p is 3, q is 3, and L isSP＝9，I0＝{0,3,6}，I1＝{0,4,8}，I2＝{0,5,7}。
When q ≧ 2p-1, no matter how large the relative shift of the values between the sequences exists, it can be proved that the p sequences constructed by equation (13) can ensure that the probability of successful transmission of the information corresponding to each sequence in one period is 1. Therefore, the time period of the protocol sequence is set to be equal to the sampling period of the inverted pendulum system of the trolley, so that the protocol sequence constructed by the equation (13) can ensure that the state information of each inverted pendulum of the trolley can be successfully transmitted and received in one sampling period, and the certainty and the reliability of the communication topological structure of the system are ensured, thereby ensuring the effectiveness of the consistency algorithm provided by the invention.
The foregoing is a more detailed description of the invention in connection with specific preferred embodiments and it is not intended that the invention be limited to these specific details. To those skilled in the art to which the invention relates, numerous changes, substitutions and alterations can be made without departing from the spirit of the invention, and these changes are deemed to be within the scope of the invention as defined by the appended claims.
Claims (6)
1. A consistency control method of a multi-trolley single-stage inverted pendulum system is characterized by comprising the following steps:
s1, setting the first trolley single-stage inverted pendulum as a pilot, and setting N trolley single-stage inverted pendulums as followers of the pilot;
s2, determining the feedback control force u of the ith follower in the N followers according to the following algorithmi：
Wherein x isiState vector, x, representing the ith followerjRepresenting the state vector, x, of the jth dolly single-stage inverted pendulumlState vector representing the pilot, aijDenotes xjAnd xiA weight coefficient ofilDenotes xlAnd xiK represents a controller gain matrix; a if the ith follower can acquire the state vector of the jth dolly single-stage inverted pendulum from the jth dolly single-stage inverted pendulumij1, otherwise aij＝0；
S3 controlling force u by feedbackiControlling the ith follower;
wherein K is calculated according to the following algorithm:
K＝c(R+HTPH)-1HTPG；
where the matrix P is the solution of the following algebraic ricati equation:
P-GTPG+(1-ζ2)GTPH(R+HTPH)-1HTPG+Q＝0；
wherein R controls a weight value of energy consumption; h and G are model parameter matrixes of the inverted pendulum system of the trolley; c is a weight factor of the coupling strength between the inverted pendulums of the trolleys, Q is a positive definite matrix or a semi-positive definite real symmetric matrix, R is a positive definite real symmetric matrix, and c satisfies the following inequality:
wherein,
is a matrix
L is a graph Laplace matrix describing the relationship among all following trolley inverted pendulum systems;
an unstable characteristic root representing a system parameter matrix G;
wherein the parameters
2. The method for controlling the consistency of a multi-trolley single-stage inverted pendulum system as claimed in claim 1, wherein:
the communication between the pilot and the follower and between the pilot and the pilot adopts a medium access control protocol of a protocol sequence constructed by a method based on a generalized prime sequence.
3. The consistency control method of the multi-trolley single-stage inverted pendulum system according to claim 1, wherein if the state vector of a follower at the k +1 th moment and the state vector x [ k ] of the follower at the k th moment satisfy the following relationship, it is determined that N followers and a pilot have reached consistency:
wherein, INDenotes an NxN identity matrix.
4. The utility model provides a uniformity control device of many dollies single-stage inverted pendulum system which characterized by:
the first processing unit is used for setting the first trolley single-stage inverted pendulum as a pilot and setting the N trolley single-stage inverted pendulums as followers of the pilot;
a second processing unit for determining the feedback control force u of the ith follower of the N followers according to the following algorithmi：
Wherein x isiState vector, x, representing the ith followerjRepresenting the state vector, x, of the jth dolly single-stage inverted pendulumlState vector representing the pilot, aijDenotes xjAnd xiA weight coefficient ofilDenotes xlAnd xiK represents a controller gain matrix; a if the ith follower can acquire the state vector of the jth dolly single-stage inverted pendulum from the jth dolly single-stage inverted pendulumij1, otherwise aij＝0；
A third processing unit for controlling the force u by feedbackiTo the ith followerPerforming row control;
wherein K is calculated according to the following algorithm:
K＝c(R+HTPH)-1HTPG；
where the matrix P is the solution of the following algebraic ricati equation:
P-GTPG+(1-ζ2)GTPH(R+HTPH)-1HTPG+Q＝0
wherein R controls a weight value of energy consumption; h and G are model parameter matrixes of the inverted pendulum system of the trolley; c is a weight factor of the coupling strength between the inverted pendulums of the trolleys, Q is a positive definite matrix or a semi-positive definite real symmetric matrix, R is a positive definite real symmetric matrix, and c satisfies the following inequality:
wherein,
is a matrix
L is a graph Laplace matrix describing the relationship among all following trolley inverted pendulum systems;
an unstable characteristic root representing a system parameter matrix G;
wherein the parameters
5. The consistency control device of a multiple-trolley single-stage inverted pendulum system as defined in claim 4, wherein:
the communication between the pilot and the follower and between the pilot and the pilot adopts a medium access control protocol of a protocol sequence constructed by a method based on a generalized prime sequence.
6. The consistency control device of the multi-trolley single-stage inverted pendulum system according to claim 4, wherein if the state vector of a follower at the k +1 th time and the state vector x [ k ] of the follower at the k th time satisfy the following relationship, it is determined that the N followers and the pilot have reached consistency:
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