CN115167116B - Ellipsoid-based nonlinear time-varying interconnection system interval estimation method - Google Patents
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Abstract
The invention provides an ellipsoid-based nonlinear time-varying interconnection system interval estimation method, which comprises the following steps: firstly, a state space dynamic model of a three-intelligent vehicle system is established, information exchange is carried out between subsystems by adopting a weighted try one-time discarding protocol, then an interval observer and an error dynamic system are established, and finally, an observer gain matrix is solved and a state estimation ellipsoid set is obtained; on the basis of state estimation research of a nonlinear time-varying interconnection system, the method solves the technical problems of high calculation load, complex observer design, low estimation precision and strong conservation caused by the fact that the nonlinear time-varying interconnection system has a large number of factor systems and is mutually coupled; meanwhile, the subsystems adopt a weighting try one-time discarding protocol to exchange data, so that the communication bandwidth is saved; 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
Along with the development of society and the continuous improvement of the technological level, the interconnection system has an increasing effect in production and life, and is especially in the field of intelligent vehicle control. Such interconnected systems are composed of a plurality of subsystems, with the overall control objective being achieved by the mutual coupling between the subsystems. Meanwhile, in the actual production process, nonlinear links and time-varying items often exist in a mathematical model of the intelligent vehicle. Therefore, nonlinear time-varying interconnection systems have attracted the eyes of a large number of students in recent years. Based on the characteristics of the interconnected subsystem, it is difficult to directly obtain the state of the system, 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 accurate point values, the interval observer has more excellent performance and wider application range. Compared with a distributed interval observer designed based on the positive system theory, the distributed interval observer designed based on the ellipsoidal 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 moreover, the distributed interval observer can calculate the states of all subsystems in parallel, so that the calculation speed is higher. In addition, the information transmission between subsystems in the interconnected system is considered. Thus, the "weighted attempt one drop" protocol is employed in the information transfer process. The weighted try one-time discard protocol can effectively save communication bandwidth and avoid the occurrence of data collision.
Disclosure of Invention
The invention provides an ellipsoid-based nonlinear time-varying interconnection system interval estimation method based on nonlinear time-varying interconnection system state estimation research, and solves the technical problems of high calculation load, complex observer design, low estimation precision and strong conservation caused by the fact that the nonlinear time-varying interconnection system factor systems are large in number and are mutually coupled.
The invention is realized by the following technical scheme:
An ellipsoid-based nonlinear time-varying interconnection system interval estimation method comprises the following steps:
the method specifically comprises the following steps:
Step one, a state space is established, and a state space expression of a three-intelligent vehicle system is established by using a dynamics 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 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.
Further, in a first step, the first step,
The dynamic system is a three-intelligent vehicle system, and the system state space expression is as follows:
Wherein a ii(k),Aij(k),Bi(k),Ci(k),Di (k) is a time-varying matrix of known appropriate dimensions; w i (k) is process noise, v i (k) is measurable noise, and the noise types are unknown bounded noise; x i (k) is the state vector of the ith subsystem; f (x i (k)) is a nonlinear disturbance.
Further, in the second step, the first step,
The interval observer satisfies the following assumption:
Assume that the first, process noise w i (k) and the measurable noise v i (k) are bounded and belong to an ellipsoid set;
assume that the initial state x i (0) is bounded and belongs to the ellipsoid set;
Suppose that the three intelligent vehicle systems are observable;
The interval observer is:
Wherein, For selecting a data signal matrix for accessing the communication network; k ii(k),Kij (K) is the observer gain matrix that needs to be designed.
Further, in step three:
the error dynamic system is as follows:
Wherein, Is an error; /(I)Taylor's expansion at x i (k) is a higher order infinitesimal term and belongs to the ellipsoid set ε (0, Q i);Hi (k) is a nonlinear function f (x i (k)).
Further, the process noise w i (k), the measurable noise v i (k) and the higher-order infinitely small term of the error dynamic system in the step three are obtainedMerging into an ellipsoidal set/> Is an ellipsoidal shape matrix, and has the following form:
Wherein,
The error dynamic system in step three can be rewritten as follows:
wherein ω i (k) is the combined expression of the process noise, the measurable noise and the higher-order infinitely small term in the error dynamic system.
Further, the matrix and the vector from the first step to the third step are amplified, and the expression after the amplification is:
e(k)＝colN{ei(k)},H(k)＝diagN{Hi(k)},A(k)＝[Aij(k)]N×N,
B(k)＝diagN{Bi(k)},C(k)＝diagN{Ci(k)},D(k)＝diagN{Di(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 an ellipsoid set In (2), the conditions are as follows:
Wherein, For k+1 time interval observer ellipsoidal shape matrix,/>Combining the amplified ellipsoidal shape matrix for the process noise, the measurable noise and the high-order infinitely small term in the error dynamic system,
Further, the error dynamic system in step three satisfies the input to state stabilization, i.e. the following matrix inequality should be satisfied:
Wherein, 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 dynamic system in step three satisfies the L ∞ performance, i.e., the following matrix inequality should be satisfied:
Wherein I is an identity matrix, I is more than 0 and less than 1, and ρ ω is more than 1;
and solving an 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:
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 a state space expression of the three-intelligent vehicle system is established by using a dynamic 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 established 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 interconnecting a plurality of subsystems, and the subsystems are subjected to information exchange by adopting a weighted try one-time discard protocol.
Further, the protocol of "weighted try once discard" is:
Wherein ζ i (k) is the selected data access communication network flag; An ith component of the ith subsystem state vector at time k; /(I) Is the last information sent before time k; q li is a given weight matrix; /(I)To select a data signal matrix for access to a communication network.
The invention has the beneficial effects that
The invention designs an ellipsoidal-based distributed interval observer aiming at a nonlinear time-varying interconnection system, and compared with a positive system theory, the ellipsoidal-based distributed interval observer reduces the computational complexity;
the subsystems of the invention adopt a weighting try one-time discarding protocol to exchange information, thereby saving communication bandwidth and avoiding 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 load 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 terms of the nonlinear terms by an ellipsoid set instead of neglecting, thereby improving the estimation precision.
Drawings
FIG. 1 is a flow chart of the technical scheme of the invention;
FIG. 2 is a schematic diagram of a three-intelligent vehicle system of the present invention;
FIG. 3 is an estimated state center of subsystem 1 of the present invention;
FIG. 4 is a graph of the upper and lower estimated limits of x {11} (k) of the present invention;
FIG. 5 is a graph of the upper and lower estimated limits of x {12} (k) of the present invention;
FIG. 6 is an error dynamics system of subsystem 1 of the present invention;
FIG. 7 is an estimated state center of subsystem 2 of the present invention;
FIG. 8 is an upper and lower estimated limit of x {21} (k) of the present invention;
FIG. 9 is a graph of the upper and lower estimated limits of x {22} (k) of the present invention;
FIG. 10 is an error dynamics system of subsystem 2 of the present invention;
FIG. 11 is an estimated state center of subsystem 3 of the present invention;
FIG. 12 is a graph of the upper and lower estimated limits of x {31} (k) of the present invention;
FIG. 13 is a graph of the upper and lower estimated limits of x {32} (k) of the present invention;
fig. 14 is an error dynamic system of subsystem 3 of the present invention.
Detailed Description
The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
With reference to fig. 1 to 14, the method designs the interval observer based on the method of describing the estimated state set by using ellipsoids, reduces the computational complexity compared with the positive system theory, and further improves the accuracy by representing the higher-order infinite small terms of the nonlinear terms by using ellipsoids instead of neglecting.
An ellipsoid-based nonlinear time-varying interconnection system interval estimation system:
the system comprises a building 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 a state space expression of the three-intelligent vehicle system is established by using a dynamic 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 established 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 interconnecting a plurality of subsystems, and the subsystems are subjected to information exchange by adopting a weighted try one-time discard protocol.
The protocol of the weighted try once discard is as follows:
Wherein ζ i (k) is the selected data access communication network flag; An ith component of the ith subsystem state vector at time k; /(I) Is the last information sent before time k; q li is a given weight matrix; /(I)To select a data signal matrix for access to a communication network.
An ellipsoid-based nonlinear time-varying interconnection system interval estimation method comprises the following steps:
the method specifically comprises the following steps:
Step one, a state space is established, and a state space expression of a three-intelligent vehicle system is established by using a dynamics 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 constructed 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.
In a first step of the process, the process 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),Aij(k),Bi(k),Ci(k),Di (k) is a time-varying matrix of known appropriate dimensions; w i (k) is process noise, v i (k) is measurable noise (noise is unknown bounded noise); x i (k) is the state vector of the ith subsystem; f (x i (k)) is a nonlinear disturbance.
In step two, the interval observer satisfies the following assumption:
Assume that the first, process noise w i (k) and the measurable noise v i (k) are bounded and belong to an ellipsoid set;
assume that the initial state x i (0) is bounded and belongs to the ellipsoid set;
Suppose that the three intelligent vehicle systems are observable;
The interval observer is:
Wherein, For selecting a data signal matrix for accessing the communication network; k ii(k),Kij (K) is the observer gain matrix that needs to be designed.
In step three: the error dynamic system is as follows:
Wherein, Is an error; /(I)Taylor's expansion at x i (k) is a higher order infinitesimal term and belongs to the ellipsoid set ε (0, Q i);Hi (k) is a nonlinear function f (x i (k)).
The process noise w i (k) of the error dynamic system in the step three, the measurable noise v i (k) and a higher-order infinitely small termMerging into an ellipsoidal set/> Is an ellipsoidal shape matrix, and has the following form:
Wherein,
The error dynamic system in step three can be rewritten as follows:
wherein ω i (k) is the combined expression of the process noise, the measurable noise and the higher-order infinitely small term in the error dynamic system.
And (3) amplifying the matrix and the vector in the first to third steps, wherein the amplified expression is:
e(k)＝colN{ei(k)},H(k)＝diagN{Hi(k)},A(k)＝[Aij(k)]N×N,
B(k)＝diagN{Bi(k)},C(k)＝diagN{Ci(k)},D(k)＝diagN{Di(k)},
the augmented error dynamic system expression is:
All state vectors of the three-intelligent vehicle system in the third step should be contained in the ellipsoid set The conditions were as follows:
Wherein, For k+1 time interval observer ellipsoidal shape matrix,/>Combining the amplified ellipsoidal shape matrix for the process noise, the measurable noise and the high-order infinitely small term in the error dynamic system,The following was demonstrated:
According to hypothesis 2 in the second step, there are Further, e (k) ∈ε (0, X (k)) is derived. From the definition of e (k) in step three, it can be deduced thatThus can be obtained
Wherein, From this we can deduce that all states are contained in one ellipsoid set, proving pich.
The error dynamic system in step three satisfies the input stabilization, i.e. the following matrix inequality should be satisfied:
Wherein, 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 the L ∞ performance, i.e., the following matrix inequality should be satisfied:
Wherein I is an identity matrix, I is more than 0 and less than 1, and ρ ω is more than 1;
and solving an observer gain matrix K (K) by solving the matrix inequality, and further outputting a state space estimation result.
The following was demonstrated:
Constructing Lyapunov function V (k) =e T (k) Pe (k), and further solving for DeltaV (k) =η T (k) Φη (k), wherein
Η (k) = [ e T(k),ωT(k)]T, defining w=pk (k), multiplying diag { I, P -1 } and its transpose about the inequality can get a new linear matrix inequalityBy using the Shull's complement theory, we can obtain phi+diag { iota P, -sigma I } < 0, then the left and right co-products eta T (k) and its transpose can deduceThe establishment can obtain that the input of the error dynamic system is stable;
The linear matrix inequality can be converted into the following equation by applying the Shul's complement theory to the linear matrix inequality To the left and right of which are multiplied eta T (k) and the transpose thereof,
And can then be deduced
Therefore, the error dynamic system in the third step can be obtained to meet the L ∞ performance, and the completion of the error dynamic system is verified.
Step four: and (3) solving the linear matrix inequality obtained in the step (III) to obtain an observer gain matrix K (K).
In the environment 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:
The system matrix a ij is as follows:
matrix B i is as follows:
matrix C i,Di is as follows:
the process noise and the measurement noise w i(k),Vi (k) follow a normal distribution over the interval [ -0.002,0.002 ].
The nonlinear term is as follows:
The initial state of the subsystem is as follows:
The initial values are 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 a section calculation process, then solving an ellipsoid shape matrix of the estimated state, substituting the ellipsoid shape matrix into an ellipsoid equation, and finally obtaining a state estimation set. Fig. 3 to fig. 14 are simulation results, where fig. 3 to fig. 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 boundary of the corresponding ellipsoid estimation set, the blue "type is the center of the actual state, fig. 4 and fig. 5 are the upper and lower estimation 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 upper estimation limit, the green curve is the lower estimation 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 in the results of fig. 4,5, 8, 9, 12, and 13, the estimated state upper and lower limits are very close to the actual state, indicating that the section observer designed by the present invention has sufficient accuracy. As shown in the results of fig. 3, 7 and 11, all states of the three subsystems are contained in the corresponding ellipsoid set, 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 one of the methods described above when the processor executes the computer program.
A computer readable storage medium storing computer instructions which, when executed by a processor, implement the steps of the method of any of the preceding claims.
The invention overcomes the technical problems of high calculation load, complex observer design, low estimation precision and strong conservation caused by the fact that the nonlinear time-varying interconnection system factor systems are large in number and mutually coupled on the basis of the state estimation research of the nonlinear time-varying interconnection system. Meanwhile, the subsystems adopt a weighting try one-time discard protocol to exchange data, so that the communication bandwidth is saved. Finally, taking a three-intelligent vehicle system formed by three intelligent vehicles as an example, the effectiveness of the method is verified.
The above detailed description of the nonlinear time-varying interconnection system interval estimation method based on ellipsoids, which is provided by the invention, explains the principle and the implementation of the invention, and the above description of the embodiment is only used for helping to understand the method and the core idea of the invention; meanwhile, as those skilled in the art will have variations in the specific embodiments and application scope in accordance with the ideas of the present invention, the present description should not be construed as limiting the present invention in view of the above.
Claims (9)
1. An ellipsoid-based nonlinear time-varying interconnection system interval estimation method is characterized in that:
the method specifically comprises the following steps:
Step one, a state space is established, and a state space expression of a three-intelligent vehicle system is established by using a dynamics 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;
The interval observer satisfies the following assumption:
Assume that the first, process noise w i (k) and the measurable noise v i (k) are bounded and belong to an ellipsoid set;
assume that the initial state x i (0) is bounded and belongs to the ellipsoid set;
Suppose that the three intelligent vehicle systems are observable;
The interval observer is:
Wherein Φ ζi(k)＝diag[ι(ζi(k)-1)I ι(ζi(k)-2)I … ι(ζi (k) -n) I is a data signal matrix for selecting access to the communication network; k ii(k),Kij (K) is an observer gain matrix to be designed;
step three, constructing an error dynamic system according to the state space expression established in the step one and the interval observer constructed 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 according to claim 1, wherein: in a first step of the process, the process 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),Aij(k),Bi(k),Ci(k),Di (k) is a time-varying matrix of known appropriate dimensions; w i (k) is process noise, v i (k) is measurable noise, and the noise types are unknown bounded noise; x i (k) is the state vector of the ith subsystem; f (x i (k)) is a nonlinear disturbance.
3. The method according to claim 2, characterized in that: in step three:
the error dynamic system is as follows:
Wherein, Is an error; /(I)Taylor's expansion at x i (k) is a higher order infinitesimal term and belongs to the ellipsoid set ε (0, Q i);Hi (k) is a nonlinear function f (x i (k)).
4. A method according to claim 3, characterized in that:
The process noise w i (k) of the error dynamic system in the step three, the measurable noise v i (k) and a higher-order infinitely small term Merging into an ellipsoidal set/> Is an ellipsoidal shape matrix, and has the following form:
Wherein,
The error dynamic system in step three can be rewritten as follows:
wherein ω i (k) is the combined expression of the process noise, the measurable noise and the higher-order infinitely small term in the error dynamic system.
5. The method according to claim 4, wherein:
And (3) amplifying the matrix and the vector in the first to third steps, wherein the amplified expression is:
e(k)＝colN{ei(k)},H(k)＝diagN{Hi(k)},A(k)＝[Aij(k)]N×N,
B(k)＝diagN{Bi(k)},C(k)＝diagN{Ci(k)},D(k)＝diagN{Di(k)},
the augmented error dynamic system expression is:
6. the method according to claim 5, wherein:
All state vectors of the three-intelligent vehicle system in the third step should be contained in the ellipsoid set The conditions are as follows:
Wherein, For k+1 time interval observer ellipsoidal shape matrix,/>Combining the amplified ellipsoidal shape matrix for the process noise, the measurable noise and the high-order infinitely small term in the error dynamic system,
The error dynamic system should satisfy the input to state stabilization, i.e. the following matrix inequality should be satisfied:
Wherein, 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.
7. The method according to claim 6, wherein:
The error dynamic system in step three satisfies the L ∞ performance, i.e., the following matrix inequality should be satisfied:
Wherein I is an identity matrix, I is more than 0 and less than 1, and ρ ω is more than 1;
and solving an observer gain matrix K (K) by solving the matrix inequality, and further outputting a state space estimation result.
8. An estimation system for performing the ellipsoid-based nonlinear time-varying interconnection system interval estimation method according to any one of claims 1 to 7, characterized by:
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 a state space expression of the three-intelligent vehicle system is established by using a dynamic 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 interval observer satisfies the following assumption:
Assume that the first, process noise w i (k) and the measurable noise v i (k) are bounded and belong to an ellipsoid set;
assume that the initial state x i (0) is bounded and belongs to the ellipsoid set;
Suppose that the three intelligent vehicle systems are observable;
The interval observer is:
Wherein, For selecting a data signal matrix for accessing the communication network; k ii(k),Kij (K) is an observer gain matrix to be designed;
the error analysis subsystem is used for constructing an error dynamic system according to the established state space expression and the established 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 interconnecting a plurality of subsystems, and the subsystems are subjected to information exchange by adopting a weighted try one-time discard protocol.
9. The system according to claim 8, wherein:
The protocol of the weighted try once discard is as follows:
Wherein ζ i (k) is the selected data access communication network flag; An ith component of the ith subsystem state vector at time k; /(I) Is the last information sent before time k; q li is a given weight matrix; /(I)To select a data signal matrix for access to a communication network.
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