CN115167116A - Ellipsoid-based nonlinear time-varying interconnection system interval estimation method - Google Patents
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Abstract
The invention provides a non-linear time-varying interconnection system interval estimation method based on an ellipsoid, which comprises the following steps: firstly, establishing a state space dynamic model of a three-intelligent-vehicle system, exchanging information among subsystems by adopting a weighting trial and one-time discarding protocol, then constructing an interval observer and an error dynamic system, and finally solving an observer gain matrix to obtain a state estimation ellipsoid set; on the basis of the state estimation research of the nonlinear time-varying interconnection system, the technical problems of high calculation burden, complex observer design, low estimation precision and strong conservatism of the nonlinear time-varying interconnection system caused by a large number of subsystems and mutual coupling are solved; meanwhile, a 'weighting try once discarding' protocol is adopted among the subsystems for data exchange, so that the communication bandwidth is saved; and finally, taking a three-intelligent-vehicle system formed by three intelligent vehicles as an example, the effectiveness of the method is verified.
Description
Technical Field
The invention belongs to the technical field of intelligent vehicle control, and particularly relates to an ellipsoid-based nonlinear time-varying interconnection system interval estimation method.
Background
With the development of society and the continuous improvement of technological level, the role of interconnected system in the aspect of production and life is increasing day by day, especially in intelligent vehicle control field. The interconnection system is composed of a plurality of subsystems, and the overall control target is achieved through mutual coupling among the subsystems. Meanwhile, in the actual production process, nonlinear links and time-varying terms often exist in the mathematical model of the intelligent vehicle. Therefore, the nonlinear time-varying interconnection system has attracted the attention of a large number of scholars in recent years. Based on the characteristics of the subsystems of the interconnected system, the state of the system is difficult to obtain directly, and a common method is to construct a state observer for the system and estimate the state of the system. Compared with a state observer for estimating an accurate point value, the interval observer has more excellent performance and wider application range. Compared with the interval observer designed based on the positive system theory, the distributed interval observer designed based on the ellipsoid method has low calculation complexity and high accuracy.
The distributed interval observer designed for the nonlinear time-varying interconnection system does not need to collect the states of all subsystems, and the calculation burden is small; and the distributed interval observer can calculate the states of all the subsystems in parallel, and the calculation speed is higher. In addition, the need for information transfer between subsystems in an interconnected system is considered. Therefore, a "weighted attempt one drop" protocol is employed during information transfer. The protocol of 'one-time discarding of weighting attempt' can effectively save communication bandwidth and avoid the occurrence of data collision phenomenon.
Disclosure of Invention
The invention provides an ellipsoid-based nonlinear time-varying interconnection system interval estimation method on the basis of state estimation research of a nonlinear time-varying interconnection system, and solves the technical problems of high calculation load, complex observer design, low estimation precision and strong conservatism of the nonlinear time-varying interconnection system due to the fact that a plurality of subsystems are coupled with one another.
The invention is realized by the following technical scheme:
a non-linear time-varying interconnection system interval estimation method based on an ellipsoid comprises the following steps:
the method specifically comprises the following steps:
step one, establishing a state space, and establishing a state space expression of the three intelligent vehicle systems by using a kinetic equation;
step two, constructing a distributed interval observer of the nonlinear time-varying interconnection system according to the state space expression established in the step one;
step three, constructing an error dynamic system according to the state space expression established in the step one and the distributed observer constructed in the step two;
step four, calculating and solving an observer gain matrix according to the distributed interval observer in the step two and the error dynamic system in the step three;
and step five, outputting a state space estimation result.
Further, in the first step,
the dynamic system is a three-intelligent-vehicle system, and the expression of the system state space is as follows:
wherein A is ii (k),A ij (k),B i (k),C i (k),D i (k) A time-varying matrix of known appropriate dimensions; w is a i (k) Is process noise, v i (k) The noise types are unknown bounded noise for measurable noise; x is the number of i (k) Is the state vector of the ith subsystem; f (x) i (k) ) is a non-linear perturbation.
Further, in the second step, the first step,
the interval observer satisfies the following assumptions:
hypothesis one, process noise w i (k) And measurable noise v i (k) Bounded and belonging to a set of ellipsoids;
assume two, initial state x i (0) Bounded and belonging to a set of ellipsoids;
assuming that the system of the three intelligent vehicles can be observed;
the interval observer is then:
wherein the content of the first and second substances,
selecting a data signal matrix for accessing a communication network; k is ii (k),K ij (k) An observer gain matrix is designed for the needs.
Further, in step three:
the error dynamic system is as follows:
wherein the content of the first and second substances,
is an error;
is a high-order infinite item and belongs to the ellipsoid set epsilon (0 i )；H i (k) As a non-linear function f (x) i (k) In x) i (k) And (4) performing Taylor expansion.
Further, the process noise w of the error dynamic system in the third step is measured i (k) Measurable noise v i (k) And high order infinity terms
Merge into one ellipsoid set
Is an ellipsoidal matrix, and has the following form:
wherein, the first and the second end of the pipe are connected with each other,
the error dynamics in step three can be rewritten as follows:
wherein, ω is i (k) And the expression is obtained by combining process noise, measurable noise and high-order infinite small terms in the error dynamic system.
Further, the matrix and the vector of the first step to the third step are augmented, and the augmented expression is as follows:
e(k)＝col N {e i (k)},H(k)＝diag N {H i (k)},A(k)＝[A ij (k)] N×N ,
B(k)＝diag N {B i (k)},C(k)＝diag N {C i (k)},D(k)＝diag N {D i (k)},
the augmented error dynamic system expression is:
further, all state vectors of the three-intelligent-vehicle system in step three should be contained in the ellipsoid set
The conditions are as follows:
wherein, the first and the second end of the pipe are connected with each other,
an ellipsoid-shaped matrix of an observer at the k +1 time interval,
an ellipsoid-shaped matrix which is obtained by combining and amplifying process noise, measurable noise and high-order infinitesimal terms in the error dynamic system,
further, the error dynamics system in step three satisfies the input-to-state stability, that is, the following matrix inequality should be satisfied:
wherein the content of the first and second substances,
p and W are given positive definite matrices, W = PK (K), I is an identity matrix, and K (K) is an observer gain matrix to be solved.
Further, the error dynamics in step three satisfies L ∞ Performance, i.e. the following matrix inequality should be satisfied:
wherein I is an identity matrix, iota is more than 0 and less than 1, and rho ω ＞1；
And solving the observer gain matrix K (K) by solving the matrix inequality, and further outputting a state space estimation result.
An ellipsoid-based nonlinear time-varying interconnection system interval estimation system comprises:
the system comprises a state subsystem, an interval observation subsystem, an error analysis subsystem and a calculation output subsystem;
the state subsystem is used for establishing a state space and establishing a state space expression of the three intelligent vehicle systems by using a kinetic equation;
the interval observation subsystem is used for constructing a distributed interval observer of the nonlinear time-varying interconnection system according to the established state space expression;
the error analysis subsystem is used for constructing an error dynamic system according to the established state space expression and the constructed distributed interval observer;
the calculation output subsystem is used for calculating and obtaining an observer gain matrix according to the interval observation module and the error analysis module; and outputting a state space estimation result;
the interconnection system is formed by interconnection of a plurality of subsystems, and information exchange is carried out between the subsystems by adopting a weighted attempt once-discard protocol.
Further, the protocol of "weighted attempt drop once" is:
therein, ζ i (k) Accessing a communication network designation for the selected data;
is the ith component of the ith subsystem state vector at time k;
the information which is sent last before the time k; q li Is a given weight matrix;
to select a matrix of data signals for access to a communication network.
The invention has the beneficial effects that
The distributed interval observer based on the ellipsoid is designed aiming at the nonlinear time-varying interconnection system, and compared with a positive system theory, the distributed interval observer based on the ellipsoid reduces the calculation complexity;
the subsystems of the invention exchange information by adopting a weighting try once discarding protocol, thereby saving the communication bandwidth and avoiding the occurrence of data collision;
the distributed interval observer does not need to collect information of all subsystems like a centralized interval observer, so that the calculation burden is reduced;
the invention adopts the distributed interval observer to calculate the states of all subsystems in parallel, thereby improving the calculation speed;
the distributed interval observer designed in the invention expresses the high-order infinite small term of the nonlinear term by an ellipsoid set instead of neglecting, thereby improving the estimation precision.
Drawings
FIG. 1 is a flow chart of the technical solution of the present invention;
FIG. 2 is a schematic diagram of a three smart car system of the present invention;
FIG. 3 is the estimated state center of the subsystem 1 of the present invention;
FIG. 4 shows x of the present invention {11} (k) Estimating an upper limit and a lower limit;
FIG. 5 shows x in the present invention {12} (k) Estimating an upper limit and a lower limit;
FIG. 6 is an error dynamics system of the subsystem 1 of the present invention;
FIG. 7 is an estimated state center of the subsystem 2 of the present invention;
FIG. 8 shows x of the present invention {21} (k) Estimating an upper limit and a lower limit;
FIG. 9 shows x in the present invention {22} (k) Estimating an upper limit and a lower limit;
FIG. 10 is an error dynamics system of subsystem 2 of the present invention;
FIG. 11 is an estimated state center of the subsystem 3 of the present invention;
FIG. 12 shows x of the present invention {31} (k) Estimating an upper limit and a lower limit;
FIG. 13 shows x of the present invention {32} (k) Estimating an upper limit and a lower limit;
fig. 14 shows an error dynamics system of the subsystem 3 according to the invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be described below clearly and completely with reference to the accompanying drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be obtained by a person skilled in the art without making any creative effort based on the embodiments in the present invention, belong to the protection scope of the present invention.
With reference to fig. 1 to 14, the interval observer is designed based on the method for describing and estimating the state set by using the ellipsoid, compared with the positive system theory, the computation complexity is reduced, the high-order infinite terms of the nonlinear term are represented by using the ellipsoid rather than being ignored, and the precision is further improved.
An ellipsoid-based nonlinear time-varying interconnected system interval estimation system comprises:
the system comprises a state establishing subsystem, an interval observation subsystem, an error analysis subsystem and a calculation output subsystem;
the state subsystem is used for establishing a state space and establishing a state space expression of the three intelligent vehicle systems by using a kinetic equation;
the interval observation subsystem is used for constructing a distributed interval observer of the nonlinear time-varying interconnection system according to the established state space expression;
the error analysis subsystem is used for constructing an error dynamic system according to the established state space expression and the constructed distributed interval observer;
the calculation output subsystem is used for calculating and obtaining an observer gain matrix according to the interval observation module and the error analysis module; and outputting a state space estimation result;
the interconnection system is formed by interconnection of a plurality of subsystems, and information exchange is carried out between the subsystems by adopting a weighted attempt once-discard protocol.
The protocol for the "weighted attempt one drop" is:
therein, ζ i (k) Accessing a communication network designation for the selected data;
is the ith component of the ith subsystem state vector at time k;
the information which is sent last before the time k; q li Is a given weight matrix;
to select a matrix of data signals for access to a communication network.
A non-linear time-varying interconnection system interval estimation method based on an ellipsoid comprises the following steps:
the method specifically comprises the following steps:
step one, establishing a state space, and establishing a state space expression of the three intelligent vehicle systems by using a kinetic equation;
step two, constructing a distributed interval observer of the nonlinear time-varying interconnection system according to the state space expression established in the step one;
step three, constructing an error dynamic system according to the state space expression established in the step one and the distributed interval observer established in the step two;
step four, calculating and solving an observer gain matrix according to the distributed interval observer in the step two and the error dynamic system in the step three;
and step five, outputting a state space estimation result.
In the first step, the first step is carried out,
the dynamic system is a three-intelligent-vehicle system, and the system state space expression is as follows:
wherein, A ii (k),A ij (k),B i (k),C i (k),D i (k) A time-varying matrix of known appropriate dimensions; w is a i (k) Is process noise, v i (k) Measurable noise (noise is unknown bounded noise); x is the number of i (k) Is the state vector of the ith subsystem; f (x) i (k) ) is a non-linear perturbation.
In step two, the interval observer satisfies the following assumptions:
hypothesis one, process noise w i (k) And measurable noise v i (k) Bounded and belonging to a set of ellipsoids;
suppose twoInitial state x i (0) Bounded and belonging to a set of ellipsoids;
thirdly, the system of the three intelligent vehicles can be observed;
the interval observer is then:
wherein, the first and the second end of the pipe are connected with each other,
selecting a data signal matrix for accessing a communication network; k ii (k),K ij (k) An observer gain matrix is designed for the requirements.
In step three: the error dynamic system is as follows:
wherein the content of the first and second substances,
is an error;
is a high-order infinite item and belongs to the ellipsoid set epsilon (0 i )；H i (k) As a non-linear function f (x) i (k) In x) i (k) And (4) performing Taylor expansion.
Dynamic systematic process noise w of the error in the third step i (k) Measurable noise v i (k) And high order infinity terms
Merge into one ellipsoid set
Is a matrix in the shape of an ellipsoid,the form is as follows:
wherein the content of the first and second substances,
the error dynamics in step three can be rewritten as follows:
wherein, ω is i (k) And combining process noise, measurable noise and high-order infinitesimal terms in the error dynamic system.
And (3) amplifying the matrixes and vectors in the first step to the third step, wherein the expression after amplification is as follows:
e(k)＝col N {e i (k)},H(k)＝diag N {H i (k)},A(k)＝[A ij (k)] N×N ,
B(k)＝diag N {B i (k)},C(k)＝diag N {C i (k)},D(k)＝diag N {D i (k)},
the augmented error dynamic system expression is:
all state vectors of the three-smart car system in step three should be contained in the ellipsoid set
The conditions were as follows:
wherein, the first and the second end of the pipe are connected with each other,
an ellipsoid-shaped matrix of an observer at the k +1 time interval,
an ellipsoid-shaped matrix which is formed by combining and amplifying process noise, measurable noise and high-order infinite small terms in the error dynamic system,
the following was demonstrated:
according to the assumption 2 in the second step, there are
Further, e (k) is estimated to be epsilon (0, X (k)). According to the definition of e (k) in step three, it can be deduced
Thus can obtain
Wherein, the first and the second end of the pipe are connected with each other,
therefore, all states can be deduced to be contained in an ellipsoid set, and the verification is finished.
The error dynamics in step three satisfies the input stability, i.e. the following matrix inequality should be satisfied:
wherein the content of the first and second substances,
p and W are given positive definite matrices, W = PK (K), I is an identity matrix, and K (K) is an observer gain matrix to be solved.
The error dynamic system in step three satisfies L ∞ Performance, i.e. the following matrix inequality should be satisfied:
wherein I is an identity matrix, iota is more than 0 and less than 1, and rho ω ＞1；
And solving the observer gain matrix K (K) by solving the matrix inequality, and further outputting a state space estimation result.
The following was demonstrated:
construction of Lyapunov function V (k) = e T (k) Pe (k), and further solved to Δ V (k) = η T (k) Φ η (k) wherein
η(k)＝[e T (k),ω T (k)] T Define W = PK (k), multiplying diag { I, I, P equally around the inequality -1 The new linear matrix inequality can be obtained by the transposition of the first and second linear matrix inequalities
By applying the Sull's complement theory we can get phi + diag { iota P, -sigma I } < 0, and then multiply eta by the left and right T (k) And its transposition can be deduced
If the error is established, the input stability of the error dynamic system can be obtained;
applying schur complement to linear matrix inequalityThe lemma can be obtained by converting the linear matrix inequality
On its left and right sides by η T (k) And the transpose thereof,
further, can deduce
Step four: and solving the linear matrix inequality obtained in the third step to obtain an observer gain matrix K (K).
In the environment of MatlabR2017a, the method designed by the invention is verified by taking a three-intelligent-vehicle system consisting of three intelligent vehicles as an example, and the related parameters of the intelligent vehicles are as follows:
system matrix A ij The following were used:
matrix B i The following were used:
matrix C i ,D i The following:
process and measurement noise w i (k),V i (k) In the interval [ -0.002,0.002]Obeying a normal distribution.
The nonlinear term is as follows:
the initial state of the subsystem is as follows:
the initial values were estimated as follows:
the observer gain matrix solved at K =3, 5, 7 is as follows:
substituting the solved observer gain matrix K (K) into an interval calculation process, then solving an ellipsoid shape matrix of the estimation state and substituting the ellipsoid shape matrix into an ellipsoid equation, and finally obtaining a state estimation set. Fig. 3 to 14 are simulation results, where fig. 3 to 6 correspond to the subsystem 1, the red diamond shown in fig. 3 is the center of the state estimation ellipsoid of the subsystem 1, the dotted line around the red diamond represents the corresponding ellipsoid estimation set boundary, the blue "×" type is the center of the actual state, fig. 4 and 5 are the estimation upper and lower limits of the first and second components of the state vector of the subsystem 1, respectively, where the blue curve is the actual value, the red curve is the estimation upper limit, the green curve is the estimation lower limit, and fig. 6 is the error dynamic system curve of the subsystem 1. The remaining fig. 7-10 and 11-14 correspond to subsystem 2 and subsystem 3, respectively.
As shown by the results of fig. 4, 5, 8, 9, 12, and 13, the upper and lower limits of the estimated state are very close to the actual state, indicating that the interval observer designed by the present invention has sufficient accuracy. The results according to fig. 3, 7 and 11 show that all the states of the three subsystems are contained in the corresponding sets of ellipsoids, indicating that the interval observer designed by the present invention is reliable.
An electronic device comprising a memory storing a computer program and a processor implementing the steps of any of the above methods when the computer program is executed by the processor.
A computer readable storage medium storing computer instructions which, when executed by a processor, implement the steps of any of the above methods.
On the basis of the state estimation research of the nonlinear time-varying interconnection system, the invention overcomes the technical problems of high calculation burden, complex observer design, low estimation precision and strong conservatism of the nonlinear time-varying interconnection system caused by a large number of subsystems and mutual coupling. Meanwhile, a 'one-time discarding in weighting attempt' protocol is adopted among the subsystems for data exchange, so that the communication bandwidth is saved. And finally, taking a three-intelligent-vehicle system formed by three intelligent vehicles as an example, the effectiveness of the method is verified.
The method for estimating the interval of the non-linear time-varying interconnection system based on the ellipsoid, which is provided by the invention, is introduced in detail, and the principle and the implementation mode of the invention are explained, and the description of the above embodiment is only used for helping to understand the method and the core idea of the invention; meanwhile, for a person skilled in the art, according to the idea of the present invention, there may be variations in the specific embodiments and the application scope, and in summary, the content of the present specification should not be construed as a limitation to the present invention.
Claims (10)
1. A non-linear time-varying interconnection system interval estimation method based on an ellipsoid is characterized in that:
the method specifically comprises the following steps:
step one, establishing a state space, and establishing a state space expression of the three intelligent vehicle systems by using a kinetic equation;
step two, constructing a distributed interval observer of the nonlinear time-varying interconnection system according to the state space expression established in the step one;
step three, constructing an error dynamic system according to the state space expression established in the step one and the interval observer established in the step two;
step four, calculating and obtaining an observer gain matrix according to the distributed interval observer in the step two and the error dynamic system in the step three;
and step five, outputting a state space estimation result.
2. The method of claim 1, wherein: in the first step of the method,
the dynamic system is a three-intelligent-vehicle system, and the expression of the system state space is as follows:
wherein A is ii (k),A ij (k),B i (k),C i (k),D i (k) A time-varying matrix of known appropriate dimensions; w is a i (k) Is process noise, v i (k) The noise types are unknown bounded noise for measurable noise; x is a radical of a fluorine atom i (k) Is the state vector of the ith subsystem; f (x) i (k) ) is a non-linear perturbation.
3. The method of claim 2, wherein: in the second step of the method, the first step of the method,
the interval observer satisfies the following assumptions:
hypothesis one, process noise w i (k) And measurable noise v i (k) Bounded and belonging to a set of ellipsoids;
assume two, initial state x i (0) Bounded and belonging to a set of ellipsoids;
assuming that the system of the three intelligent vehicles can be observed;
the interval observer is then:
wherein, the first and the second end of the pipe are connected with each other,
selecting a data signal matrix for accessing a communication network; k ii (k),K ij (k) An observer gain matrix is designed for the needs.
4. The method of claim 3, further comprising: in the third step:
the error dynamic system is as follows:
wherein, the first and the second end of the pipe are connected with each other,
is an error;
is a high-order infinite item and belongs to the ellipsoid set epsilon (0 i )；H i (k) As a non-linear function f (x) i (k) In x) i (k) And (4) performing Taylor expansion.
5. The method of claim 4, further comprising:
dynamic systematic process noise w of the error in the third step i (k) Measurable noise v i (k) And high order infinity terms
Merge into one ellipsoid set
Is an ellipsoidal matrix, and has the following form:
wherein the content of the first and second substances,
the error dynamics in step three can be rewritten as follows:
wherein, ω is i (k) And the expression is obtained by combining process noise, measurable noise and high-order infinite small terms in the error dynamic system.
6. The method of claim 5, further comprising:
and (5) amplifying the matrix and the vector from the first step to the third step, wherein the expression after the amplification is as follows:
e(k)＝col N {e i (k)},H(k)＝diag N {H i (k)},A(k)＝[A ij (k)] N×N ,
B(k)＝diag N {B i (k)},C(k)＝diag N {C i (k)},D(k)＝diag N {D i (k)},
ω(k)＝col N {ω i (k)},
K(k)＝[K ij (k)] N×N ,
y(k)＝col N {y i (k)},
the augmented error dynamic system expression is:
7. the method of claim 6, wherein:
all state vectors of the three-smart car system in step three should be contained in the ellipsoid set
The middle conditions are as follows:
wherein the content of the first and second substances,
an ellipsoid-shaped matrix of an observer at the k +1 time interval,
an ellipsoid-shaped matrix which is obtained by combining and amplifying process noise, measurable noise and high-order infinitesimal terms in the error dynamic system,
the error dynamics should satisfy the input to state stability, i.e. the following matrix inequality should be satisfied:
wherein the content of the first and second substances,
p and W are given positive definite matrices, W = PK (K), I is an identity matrix, and K (K) is an observer gain matrix to be solved.
8. The method of claim 7, further comprising:
the error dynamic system in step three satisfies L ∞ The performance, i.e. the following matrix inequality should be satisfied:
wherein I is an identity matrix, iota is more than 0 and less than 1, and rho ω ＞1；
And solving the observer gain matrix K (K) by solving the matrix inequality, and further outputting a state space estimation result.
9. A non-linear time-varying interconnection system interval estimation system based on an ellipsoid method is characterized in that:
the system comprises a state subsystem, an interval observation subsystem, an error analysis subsystem and a calculation output subsystem;
the state subsystem is used for establishing a state space and establishing a state space expression of the three intelligent vehicle systems by using a kinetic equation;
the interval observation subsystem is used for constructing a distributed interval observer of the nonlinear time-varying interconnection system according to the established state space expression;
the error analysis subsystem is used for constructing an error dynamic system according to the established state space expression and the constructed distributed interval observer;
the calculation output subsystem is used for calculating and solving an observer gain matrix according to the interval observation module and the error analysis module; and outputting a state space estimation result;
the interconnection system is formed by interconnection of a plurality of subsystems, and information exchange is carried out between the subsystems by adopting a weighted attempt once-discard protocol.
10. The system of claim 9, wherein:
the protocol for the "weighted attempt one drop" is:
therein, ζ i (k) Accessing a communication network designation for the selected data;
is the ith component of the ith subsystem state vector at time k;
the information which is sent last before the time k; q li Is a given weight matrix;
to select a matrix of data signals for access to a communication network.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
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| CN202210584321.5A CN115167116B (en) | 2022-05-27 | Ellipsoid-based nonlinear time-varying interconnection system interval estimation method |
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