CN113238485A - Control method for air traffic flow input saturation - Google Patents

Control method for air traffic flow input saturation Download PDF

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CN113238485A
CN113238485A CN202110522454.5A CN202110522454A CN113238485A CN 113238485 A CN113238485 A CN 113238485A CN 202110522454 A CN202110522454 A CN 202110522454A CN 113238485 A CN113238485 A CN 113238485A
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traffic flow
air traffic
airspace
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邓宣金
张俊锋
贾祥磊
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Hangzhou Dianzi University
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    • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
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    • G05B13/042—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric involving the use of models or simulators in which a parameter or coefficient is automatically adjusted to optimise the performance
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Abstract

The invention discloses a control method for air traffic flow input saturation, which comprises the following steps: step 1, collecting traffic flow data of an airspace, and establishing a state space model of an air traffic flow control system with input saturation; step 2, constructing event trigger conditions of air route congestion in the airspace; and 3, designing a class of event trigger controllers for the air traffic flow control system based on the positive Markov jump system with input saturation, and controlling the air traffic operation condition in real time to ensure the safe and stable operation of the air traffic. The method can effectively solve the problem that the aircrafts are difficult to allocate in time due to the capacity limit of the aircrafts in the airspace route.

Description

Control method for air traffic flow input saturation
Technical Field
The invention relates to the technical field of automation, in particular to an event trigger control method based on an input saturation positive Markov jump system, which can be applied to flow management of an air traffic system.
Background
The aviation industry is the most rapidly developed industry in China and even the global transportation industry, and the construction of an air traffic management system is the most important core link in the construction of the whole system of the aviation transportation industry. The air traffic management system maintains the air traffic order, ensures the air traffic transport capacity by coordinating the flight time, the sequence and the like, and ensures the smooth operation of the air traffic. The air traffic development speed in China is high, but because the air traffic flow control technology has short development time, the air traffic flow control technology still has a great space for improvement in the aspects of air traffic management system construction, flow management and the like. In order to prevent and correct the situations of excessive concentration of aircrafts, over-specified flow and the like in the airway and the air port area, the air traffic operation order must be scientifically regulated and controlled to ensure the smoothness of the whole flight path. The invention mainly dynamically acquires the air traffic flow value, adopts a more reasonable control method according to the actual condition of the traffic flow, and provides a reliable air traffic flow control method so as to take necessary control measures for the operation of an aircraft and achieve the aim of safe operation of air traffic.
The capacity and the throughput in the airspace traffic are non-negative, so that the flow of the air traffic can be represented by using non-negative variables. Furthermore, it is reasonable to model the air traffic flow control system with a positive system. Fig. 1 shows a schematic diagram of a traffic network in an airspace, wherein, the airport takes A, B, C and four cities as an example, and the change of air traffic flow is random due to phenomena of delayed scheduling when uncertain flow constraint exists, adjustment of routes of aircraft caused by limited capacity of the airspace, and a markov jump system is used to describe a system composed of random variables. The current air traffic flow is almost saturated, the pre-increase of the future air traffic flow is difficult to meet, the congestion of an air traffic network is difficult to avoid, and the problems of ground and air operation delay and the like frequently occur in aircrafts in a capacity limited airspace or an arrival rate limited airport. The event-triggered control strategy is a real-time control method based on events. When the air traffic capacity is saturated, an event trigger control strategy is adopted, the traffic flow in the traffic system is controlled in real time, the airspace capacity limitation can be effectively solved, and the traffic jam is relieved. Therefore, the input saturated positive Markov jump system is adopted to model a type of air traffic flow control system, and the control method of the system based on the event trigger mechanism is designed to control the flow of air traffic in real time so as to ensure the safe and effective operation of the air traffic.
Disclosure of Invention
In order to solve the limitation and the deficiency of the capacity of the existing airspace system and realize the aim of continuously increasing air traffic flow demand, the invention adopts the following technical scheme:
a control method for air traffic flow input saturation is characterized by comprising the following steps:
step 1, establishing a state space model of an air traffic flow control system with input saturation;
step 2, constructing an event trigger control condition of air route congestion in an airspace;
step 3, designing a controller of the air traffic flow control system;
step 4, verifying the positive performance of the constructed air traffic flow control system under the controller;
and 5, verifying the random stability of the constructed air traffic flow control system under the controller.
Further, step 1 is specifically as follows:
1.1 firstly, collecting traffic flow data of an airspace, and establishing a state space model of the number of aircrafts in an air traffic flow control system by using the collected data, wherein the form is as follows:
x(k+1)＝Ar(k)x(k)+Br(k)sat(u(k)),
wherein the content of the first and second substances,
Figure BDA0003064510230000021
representing the number of aircraft in the airspace at the kth sampling instant, n representing the number of routes in the airspace,
Figure BDA0003064510230000022
for control signals of airspace traffic flow, m represents the number of airports, and the function sat (·):
Figure BDA0003064510230000023
is a saturation function and is defined as sat (u) ═ sat (u)1),sat(u2),…,sat(um)]TR (k) is a markov jump process that takes values in a finite set S ═ 1,2, …, N,
Figure BDA0003064510230000024
in the interior of said container body,
Figure BDA0003064510230000025
and
Figure BDA0003064510230000026
is a known system matrix, for r (k) i, i ∈ S, there are
Figure BDA0003064510230000027
1.2 Markov jump signal r (k) is designed, and the transfer rate thereof meets the following conditions:
P(r(k+1)＝j|r(k)＝i)＝πij,
wherein, for i, j belongs to S, and is not less than 0 and not more than piijIs less than or equal to 1, and
Figure BDA0003064510230000028
further, the event triggering conditions for establishing the airline congestion in the airspace in the step 2 are as follows:
‖e(k)‖1＞β‖x(k)‖1,
where β is a constant and satisfies 0 < β < 1, e (k) is a sampling error and satisfies
Figure BDA0003064510230000029
Figure BDA00030645102300000210
Represents the sampling state | · |1Represents the 1 norm of the vector, i.e., the sum of the absolute values of all the elements in the vector.
Further, the step 3 of designing the event trigger controller of the air traffic flow control system comprises the following steps:
step 3.1, a symmetric polyhedron
Figure BDA00030645102300000211
Is defined as:
Figure BDA00030645102300000212
wherein the content of the first and second substances,
Figure BDA00030645102300000213
hsis the s-th row of the matrix H.
Meanwhile, considering the capacity limit of the airspace, a cone domain is introduced, which is specifically as follows:
Figure BDA0003064510230000031
where T is the transposed symbol,
Figure BDA0003064510230000032
is an n-dimensional real column vector and satisfies
Figure BDA0003064510230000033
I.e. each element in the vector is a positive number.
Step 3.2, the air traffic flow control system adopts an input saturation method to analyze the problems, namely the requirements of the air traffic flow control system are met
Figure BDA0003064510230000034
Wherein v satisfies the condition | | v | | non-woven phosphor∞≤1,
Figure BDA0003064510230000035
Figure BDA0003064510230000036
Is the set of all m x m diagonal matrices with diagonal elements of 1 or 0,
Figure BDA0003064510230000037
and is
Figure BDA0003064510230000038
Step 3.3, designing a state feedback law of event trigger control as follows:
Figure BDA0003064510230000039
wherein the content of the first and second substances,
Figure BDA00030645102300000310
is a controller gain matrix, and
Figure BDA00030645102300000311
the concrete form is as follows:
Figure BDA00030645102300000312
1ma column vector in which all elements of m-dimension are 1,
Figure BDA00030645102300000313
an m-dimensional column vector representing that the iota-th element is 1 and the rest is 0,
Figure BDA00030645102300000314
is an n-dimensional column vector.
Step 3.4, which can be obtained from step 2, step 3.2 and step 3.3:
Figure BDA00030645102300000315
wherein, x (k) -e γ (v)(i),1)，
Figure BDA00030645102300000316
HiIs a controller auxiliary gain matrix, an
Figure BDA00030645102300000317
The concrete form is as follows:
Figure BDA00030645102300000318
step 3.5, designing the condition that the air traffic flow control system with input saturation stably runs under an event trigger mechanism, wherein the condition is as follows:
design constants μ > 0, e > 1, κ > 1, if there is an n-dimensional vector
Figure BDA00030645102300000319
Figure BDA00030645102300000320
Such that the following inequality holds:
Figure BDA0003064510230000041
Figure BDA0003064510230000042
Figure BDA0003064510230000043
Figure BDA0003064510230000044
Figure BDA0003064510230000045
Figure BDA0003064510230000046
Figure BDA0003064510230000047
Figure BDA0003064510230000048
Figure BDA0003064510230000049
wherein i ∈ S, S ═ 1,2, …,2m,Φ＝I-β1n×n,Ψ＝I+β1n×nThen in the law of state feedback control
Figure BDA00030645102300000410
And controller auxiliary gain matrix HiThe closed loop system is positive and randomly stable.
Further, the positive verification process in step 4 is as follows:
according to step 1 and step 3.4:
Figure BDA00030645102300000411
according to step 1, step 2 and step 3.5, for any
Figure BDA00030645102300000412
The following is derived according to a recursive method:
Figure BDA00030645102300000413
and is
Figure BDA00030645102300000414
Thus, the closed loop system is positive.
Further, the verification process of random stability in step 5 is as follows:
step 5.1, designing a random linear complementary plum Jacobov function:
V(x(k),r(k)＝i)＝xT(k)v(i),
the mathematical expectation of its difference is:
Figure BDA0003064510230000051
combining the step 4 to obtain:
Figure BDA0003064510230000052
step 5.2, according to step 3.4 and step 3.5, has:
Figure BDA0003064510230000053
Figure BDA0003064510230000054
Figure BDA0003064510230000055
consider DisNot equal to I and DisWith ≠ 0, the expectation of the lyapunov function difference satisfies the inequality in step 5.1.
Step 5.3, according to the conditions proposed in step 5.2, obtaining:
Figure BDA0003064510230000056
step 5.4, from step 3.5 and step 5.3, the following inequality can be obtained:
E{ΔV(x(k),r(k)＝i)}≤-μ||x(k)||1.
consider DisIs I and DisWhen 0, the above formula is valid through step 3.5, step 5.2 and step 5.3.
And 5.5, summing two sides of the inequality in the step 5.4 from 0 to ∞simultaneouslyto obtain:
Figure BDA0003064510230000057
thus, it is possible to provide
Figure BDA0003064510230000058
Thus, the closed loop system is randomly stable.
Further, the method also comprises the following steps:
step 5.6, proving the invariance of the considered system, which is concretely as follows:
order to
Figure BDA0003064510230000061
And is
Figure BDA0003064510230000062
Get xT(k0)v(r(0))Less than or equal to 1. According to step 3.5, there are
Figure BDA0003064510230000063
Further: Δ V (x (k)0) Is less than 0, i.e.
Figure BDA0003064510230000064
Thus, x (k)0)∈Υ(v(i),1)。
By mathematical induction, can be obtained
Figure BDA0003064510230000065
The following inequalities are further derived:
-1≤-xT(k0)v(r(0))≤-xT(k)v(r(k))≤Hisx(k),
Hisx(k)≤xT(k)v(r(k))xT(k0)v(r(0))≤1.
therefore, the temperature of the molten metal is controlled,
Figure BDA0003064510230000066
i.e., x (k) -e γ (v)(i),1)。
The invention has the advantages and beneficial effects that:
aiming at the problems of airspace aircraft capacity limitation and aircraft operation conflict in the current air traffic management system, a state space model of the air traffic flow control system established by using the modern control theory technology is provided, the positivity and the stability of the state space model are analyzed, an event trigger controller is designed, the system design cost is reduced, and the safe and stable operation of air traffic is ensured.
Drawings
FIG. 1 is a schematic view of an airspace traffic network;
FIG. 2 is a schematic diagram of an event-triggered control framework for a positive Markov jump system based on input saturation for system modeling.
Detailed Description
The present invention is further illustrated by the following examples, which are not intended to limit the invention to these embodiments. It will be appreciated by those skilled in the art that the present invention encompasses all alternatives, modifications and equivalents as may be included within the scope of the claims.
As shown in fig. 2, the embodiment provides an event trigger control method based on an air traffic flow control system with a input saturated positive markov jump system, which includes the following specific steps:
step 1, firstly, collecting traffic flow data in an air domain, and establishing a state space model of the number of aircrafts in an air traffic flow control system by using the collected data, wherein the form is as follows:
x(k+1)＝Ar(k)x(k)+Br(k)sat(u(k)),
wherein the content of the first and second substances,
Figure BDA0003064510230000067
representing the number of aircraft in the space domain at the kth sampling instant, nIndicating the number of routes in the airspace,
Figure BDA0003064510230000068
for control signals of airspace traffic flow, m represents the number of airports, and the function sat (·):
Figure BDA0003064510230000069
is a saturation function and is defined as sat (u) ═ sat (u)1),sat(u2),…,sat(um)]TR (k) is a markov jump process that takes values in a finite set S ═ 1,2, …, N,
Figure BDA00030645102300000610
in the interior of said container body,
Figure BDA00030645102300000611
and
Figure BDA00030645102300000612
is a known system matrix, for r (k) i, i ∈ S, there are
Figure BDA00030645102300000613
Designing a Markov jump signal r (k) with a transfer rate satisfying the following condition:
P(r(k+1)＝j|r(k)＝i)＝πij,
wherein, for i, j belongs to S, and is not less than 0 and not more than piijIs less than or equal to 1, and
Figure BDA0003064510230000071
step 2, establishing event triggering conditions of air space middle route congestion:
‖e(k)‖1＞β‖x(k)‖1,
where β is a constant and satisfies 0 < β < 1, e (k) is a sampling error and satisfies
Figure BDA0003064510230000072
Figure BDA0003064510230000073
Represents the sampling state | · |1Represents the 1 norm of the vector, i.e., the sum of the absolute values of all the elements in the vector.
Step 3, designing an event trigger controller of the air traffic flow control system, wherein the construction form is as follows:
step 3.1, a symmetric polyhedron
Figure BDA0003064510230000074
Is defined as:
Figure BDA0003064510230000075
wherein the content of the first and second substances,
Figure BDA0003064510230000076
hsis the s-th row of the matrix H.
Meanwhile, considering the capacity limit of the airspace, a cone domain is introduced, which is specifically as follows:
Figure BDA0003064510230000077
where T is the transposed symbol,
Figure BDA0003064510230000078
is an n-dimensional real column vector and satisfies
Figure BDA0003064510230000079
I.e. each element in the vector is a positive number.
Step 3.2, the air traffic flow control system adopts an input saturation method to analyze the problems, namely the requirements of the air traffic flow control system are met
Figure BDA00030645102300000710
Wherein v satisfiesCondition
Figure BDA00030645102300000711
Figure BDA00030645102300000712
Is the set of all m x m diagonal matrices with diagonal elements of 1 or 0,
Figure BDA00030645102300000713
and is
Figure BDA00030645102300000714
Step 3.3, designing a state feedback law of event trigger control as follows:
Figure BDA00030645102300000715
wherein the content of the first and second substances,
Figure BDA00030645102300000716
is a controller gain matrix, and
Figure BDA00030645102300000717
the concrete form is as follows:
Figure BDA00030645102300000718
1ma column vector in which all elements of m-dimension are 1,
Figure BDA00030645102300000719
an m-dimensional column vector representing that the iota-th element is 1 and the rest is 0,
Figure BDA0003064510230000081
is an n-dimensional column vector.
Step 3.4, which can be obtained from step 2, step 3.2 and step 3.3:
Figure BDA0003064510230000082
wherein, x (k) -e γ (v)(i),1)，
Figure BDA0003064510230000083
HiIs a controller auxiliary gain matrix, an
Figure BDA0003064510230000084
The concrete form is as follows:
Figure BDA0003064510230000085
step 3.5, designing the condition that the air traffic flow control system with input saturation stably runs under an event trigger mechanism, wherein the condition is as follows:
design constants μ > 0, e > 1, κ > 1, if there is an n-dimensional vector
Figure BDA0003064510230000086
Figure BDA0003064510230000087
Such that the following inequality holds:
Figure BDA0003064510230000088
Figure BDA0003064510230000089
Figure BDA00030645102300000810
Figure BDA00030645102300000811
Figure BDA00030645102300000812
Figure BDA00030645102300000813
Figure BDA00030645102300000814
Figure BDA00030645102300000815
Figure BDA00030645102300000816
wherein i ∈ S, S ═ 1,2, …,2m,Φ＝I-β1n×n,Ψ＝I+β1n×nThen in the law of state feedback control
Figure BDA00030645102300000817
And controller auxiliary gain matrix HiThe closed loop control system is positive and randomly stable.
And 4, verifying the positive performance of the constructed air traffic flow control system under the controller, wherein the verification process is as follows:
according to step 1 and step 3.4:
Figure BDA0003064510230000091
according to step 1, step 2 and step 3.5, for any
Figure BDA0003064510230000092
The following is derived according to a recursive method:
Figure BDA0003064510230000093
and is
Figure BDA0003064510230000094
Thus, the closed loop system is positive.
And 5, verifying the random stability of the constructed air traffic flow control system under the controller, wherein the verification process is as follows:
step 5.1, designing a random linear complementary plum Jacobov function:
V(x(k),r(k)＝i)＝xT(k)v(i),
the mathematical expectation of its difference is:
Figure BDA0003064510230000095
combining the step 4 to obtain:
Figure BDA0003064510230000096
step 5.2, according to step 3.4 and step 3.5, has:
Figure BDA0003064510230000097
Figure BDA0003064510230000098
Figure BDA0003064510230000099
consider DisNot equal to I and DisNot equal to 0, the expectation of the Lyapunov function difference satisfies stepInequality in step 5.1.
Step 5.3, according to the conditions proposed in step 5.2, obtaining:
Figure BDA0003064510230000101
step 5.4, from step 3.5 and step 5.3, the following inequality can be obtained:
E{ΔV(x(k),r(k)＝i)}≤-μ||x(k)||1.
consider DisIs I and DisWhen 0, the above formula is valid through step 3.5, step 5.2 and step 5.3.
And 5.5, summing two sides of the inequality in the step 5.4 from 0 to ∞simultaneouslyto obtain:
Figure BDA0003064510230000102
thus, it is possible to provide
Figure BDA0003064510230000103
Thus, the closed loop system is randomly stable.
In addition, the invariance of the system needs to be considered, which is specifically as follows:
order to
Figure BDA0003064510230000104
And is
Figure BDA0003064510230000105
Get xT(k0)v(r(0))Less than or equal to 1. According to step 3.5, there are
Figure BDA0003064510230000106
Further: Δ V (x (k)0) Is less than 0, i.e.
Figure BDA0003064510230000107
Thus, x (k)0)∈Υ(v(i),1). By mathematical induction, can be obtained
Figure BDA0003064510230000108
The following inequalities are further derived:
-1≤-xT(k0)v(r(0))≤-xT(k)v(r(k))≤Hisx(k),
Hisx(k)≤xT(k)v(r(k))xT(k0)v(r(0))≤1.
therefore, the temperature of the molten metal is controlled,
Figure BDA0003064510230000109
i.e., x (k) -e γ (v)(i),1)。

Claims (7)

1. A control method for air traffic flow input saturation is characterized by comprising the following steps:
step 1, establishing a state space model of an air traffic flow control system with input saturation;
step 2, constructing an event trigger control condition of air route congestion in an airspace;
step 3, designing a controller of the air traffic flow control system;
step 4, verifying the positive performance of the constructed air traffic flow control system under the controller;
and 5, verifying the random stability of the constructed air traffic flow control system under the controller.
2. The method as claimed in claim 1, wherein the step 1 is as follows:
1.1 firstly, collecting traffic flow data of an airspace, and establishing a state space model of the number of aircrafts in an air traffic flow control system by using the collected data, wherein the form is as follows:
x(k+1)＝Ar(k)x(k)+Br(k)sat(u(k))，
wherein the content of the first and second substances,
Figure FDA0003064510220000011
representing the number of aircraft in the airspace at the kth sampling instant, n representing the number of routes in the airspace,
Figure FDA0003064510220000012
for control signals of airspace traffic flow, m represents the number of airports, function
Figure FDA0003064510220000013
Is a saturation function and is defined as sat (u) ═ sat (u)1)，sat(u2)，…，sat(um)]TR (k) is a Markov jump process that takes values in a finite set
Figure FDA0003064510220000014
In the interior of said container body,
Figure FDA0003064510220000015
and
Figure FDA0003064510220000016
is a known system matrix with a for r (k) i, i e Si≥0，Bi≥0；
1.2 Markov jump signal r (k) is designed, and the transfer rate thereof meets the following conditions:
P(r(k+1)＝j|r(k)＝i)＝πij，
wherein, for i, j belongs to S, and is not less than 0 and not more than piijIs less than or equal to 1, and
Figure FDA0003064510220000017
3. the method of claim 2 in which the event triggering condition for establishing flight path congestion in airspace in step 2 is:
||e(k)||1＞β||x(k)||1，
where β is a constant and satisfies 0 < β < 1, e (k) is a sampling error and satisfies
Figure FDA0003064510220000018
Figure FDA0003064510220000019
Represents the sampling state, | · | non-woven vision1Represents the 1 norm of the vector, i.e., the sum of the absolute values of all the elements in the vector.
4. A method of controlling saturation of an air traffic flow input according to claim 3, wherein: step 3, designing an event trigger controller of the air traffic flow control system comprises the following steps:
step 3.1, a symmetric polyhedron
Figure FDA00030645102200000110
Is defined as:
Figure FDA0003064510220000021
wherein the content of the first and second substances,
Figure FDA0003064510220000022
hsis row s of matrix H;
meanwhile, considering the capacity limit of the airspace, a cone domain is introduced, which is specifically as follows:
Figure FDA0003064510220000023
where T is the transposed symbol,
Figure FDA0003064510220000024
is an n-dimensional real column vector and satisfies v(i)> 0, that is, each element in the vector is a positive number;
step 3.2, the air traffic flow control system adopts an input saturation method to analyze the problems, namely the requirements of the air traffic flow control system are met
Figure FDA0003064510220000025
Wherein v satisfies the condition | | v | | non-woven phosphor∞≤1，
Figure FDA0003064510220000026
Figure FDA0003064510220000027
Is the set of all m x m diagonal matrices with diagonal elements of 1 or 0,
Figure FDA0003064510220000028
Figure FDA0003064510220000029
and is
Figure FDA00030645102200000210
Step 3.3, designing a state feedback law of event trigger control as follows:
Figure FDA00030645102200000211
wherein the content of the first and second substances,
Figure FDA00030645102200000212
is a controller gain matrix, and
Figure FDA00030645102200000213
the concrete form is as follows:
Figure FDA00030645102200000214
1ma column vector in which all elements of m-dimension are 1,
Figure FDA00030645102200000215
an m-dimensional column vector representing that the iota-th element is 1 and the rest is 0,
Figure FDA00030645102200000216
is an n-dimensional column vector;
step 3.4, which can be obtained from step 2, step 3.2 and step 3.3:
Figure FDA00030645102200000217
wherein x (k) e γ (v)(i)，1)，
Figure FDA00030645102200000218
HiIs a controller auxiliary gain matrix, an
Figure FDA00030645102200000219
The concrete form is as follows:
Figure FDA00030645102200000220
step 3.5, designing the condition that the air traffic flow control system with input saturation stably runs under an event trigger mechanism, wherein the condition is as follows:
design constant
Figure FDA00030645102200000222
If there is an n-dimensional vector v(i)＞0，
Figure FDA00030645102200000221
Figure FDA0003064510220000031
Such that the following inequality holds:
Figure FDA0003064510220000032
Figure FDA0003064510220000033
Figure FDA0003064510220000034
Figure FDA0003064510220000035
Figure FDA0003064510220000036
Figure FDA0003064510220000037
Figure FDA0003064510220000038
Figure FDA0003064510220000039
Figure FDA00030645102200000310
wherein i ∈ S, S ═ 1,2, …,2m，Φ＝I-β1n×n，Ψ＝I+β1n×nThen in the law of state feedback control
Figure FDA00030645102200000311
And controller auxiliary gain matrix HiThe closed loop system is positive and randomly stable.
5. A method of controlling saturation of an air traffic flow input according to claim 4, wherein: the positive validation process in step 4 is as follows:
according to step 1 and step 3.4:
Figure FDA00030645102200000312
according to step 1, step 2 and step 3.5, for any x (0) > 0, the following is derived by a recursive method:
x(k+1)≥0，
Figure FDA00030645102200000313
and is
-β1n×nx(k)≤e(k)≤β1n×nx(k).
Thus, the closed loop system is positive.
6. A method of controlling saturation of an air traffic flow input according to claim 5, wherein: the verification process of random stability in step 5 is as follows:
step 5.1, designing a random linear complementary plum Jacobov function:
V(x(k)，r(k)＝i)＝xT(k)v(i)，
the mathematical expectation of its difference is:
Figure FDA0003064510220000041
combining the step 4 to obtain:
Figure FDA0003064510220000042
step 5.2, according to step 3.4 and step 3.5, has:
Figure FDA0003064510220000043
Figure FDA0003064510220000044
Figure FDA0003064510220000045
consider DisNot equal to I and DisWhen not equal to 0, the expectation of the lyapunov function difference satisfies the inequality in step 5.1;
step 5.3, according to the conditions proposed in step 5.2, obtaining:
Figure FDA0003064510220000046
step 5.4, from step 3.5 and step 5.3, the following inequality can be obtained:
E{ΔV(x(k)，r(k)＝i)}≤-μ||x(k)||1.
consider DisIs I and DisWhen 0, the above formula is valid by step 3.5, step 5.2 and step 5.3;
and 5.5, summing two sides of the inequality in the step 5.4 from 0 to ∞simultaneouslyto obtain:
Figure FDA0003064510220000047
thus, it is possible to provide
Figure FDA0003064510220000048
Thus, the closed loop system is randomly stable.
7. A method of controlling saturation of an air traffic flow input according to claim 6, further including the steps of:
step 5.6, proving the invariance of the considered system, which is concretely as follows:
order to
Figure FDA0003064510220000051
And x (k)0) > 0, get xT(k0)v(r(0))Less than or equal to 1; according to step 3.5, there is x (k)0+1) is not less than 0, further: Δ V (x (k)0) Is less than 0, i.e.
Figure FDA0003064510220000052
Thus, x (k)0)∈Υ(v(i)，1)；
Obtaining x (k) less than or equal to x (k) by a mathematical induction method0) Further, the following inequality is derived:
-1≤-xT(k0)v(r(0))≤-xT(k)v(r(k))≤Hisx(k)，
Hisx(k)≤xT(k)v(r(k))xT(k0)v(r(0))≤1.
therefore, the temperature of the molten metal is controlled,
Figure FDA0003064510220000053
i.e., x (k) e γ (v)(i)，1)。
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