CN115454122A - Adjacent convex optimization method for high-speed aircraft pursuit escape differential game - Google Patents
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Abstract
The invention provides a method for optimizing an adjacent convex of a high-speed aircraft pursuit differential game, which comprises the following specific steps: 1. establishing a high-speed aircraft pursuit differential game minimum and maximum problem model; 2. establishing a high-speed aircraft pursuit escaping differential game proximity optimization sub-problem; 3. establishing a high-speed aircraft pursuit differential game adjacent convex optimization sub-problem; 4. and (4) giving an initial guess, and iteratively solving the adjacent convex optimization sub-problem of the high-speed aircraft pursuing differential game. Through the steps, the high-speed aircraft pursuit differential game can be realized, the method can be applied on line, and better stability and universality are achieved.
Description
Technical Field
The invention provides a proximity convex optimization method for a high-speed aircraft pursuit escape differential game, which is a differential game method for solving a pursuit escape strategy of a high-speed aircraft in space and belongs to aerospace; guidance, navigation and control techniques; the field of differential gaming.
Background
In recent years, the demand for aerospace resources is increasing in countries around the world, so that high-speed aircrafts need to perform various tasks; the high-speed aircraft sometimes needs to capture a certain object or avoid space debris impact in the task execution process, and the tasks can be regarded as a typical escape following differential game problem;
in practical application, aiming at the pursuit and evasion differential game problem, the existing method mainly adopts an approximate analytical formula to solve a two-point boundary value problem or a dynamic planning problem, or adopts an off-line numerical method to solve the pursuit and evasion game problem; this solution is often difficult to adapt to multi-constraint situations, or does not have the ability to be used online; therefore, the research of a method capable of solving the pursuit differential game problem on line becomes a key and difficult problem of the research in the aerospace field;
in conclusion, in order to solve the problem of the conventional high-speed aircraft pursuit differential game, the invention provides a method for solving and designing the adjacent convex optimization aiming at the high-speed aircraft pursuit differential game under the constraint condition, and the method is widely suitable for pursuit differential game tasks and has a certain originality.
Disclosure of Invention
Objects of the invention
The invention provides a proximity convex optimization method for a high-speed aircraft pursuit differential game, which aims at solving the difficulty in solving the pursuit differential game problem under the constraint condition aiming at a high-speed aircraft pursuit differential game task with fixed flight time in a vacuum environment, so that the pursuit differential game problem is converted into a convex optimal control problem to be solved by using the proximity convex optimization method, and the problems of poor generality, difficulty in online and the like in the prior art are solved.
(II) technical scheme
The invention discloses a high-speed aircraft pursuit differential game adjacent convex optimization method, which comprises the following specific steps:
step one, establishing a high-speed aircraft pursuit escaping differential game minimum and maximum problem model;
according to task requirements, dynamics and constraints of a high-speed aircraft pursuit differential game pursuit chaser and an escaper are given, a payment function of a problem is specified, and the minimum and maximum problems of the high-speed aircraft pursuit differential game are established;
step two, establishing a high-speed aircraft pursuit escaping differential game proximity optimization sub-problem;
according to the minimum and maximum problem model of the high-speed aircraft pursuit differential game established in the step one, introducing a proximity operator into the minimum and maximum problem model, and establishing a proximity optimization sub-problem of the high-speed aircraft pursuit differential game about a chaser and an escaper;
step three, establishing a high-speed aircraft pursuit differential game adjacent convex optimization sub-problem;
according to the high-speed aircraft pursuit differential game adjacent optimization sub-problem established in the second step, a linearization method is adopted to carry out convexity on process constraint, terminal constraint and performance indexes of the problem to form a high-speed aircraft pursuit differential game adjacent convex optimal control sub-problem, and the high-speed aircraft pursuit differential game adjacent convex optimization sub-problem is established through equal time interval discretization;
step four, giving an initial guess, and iteratively solving the high-speed aircraft pursuit differential game adjacent convex optimization sub-problem;
wherein, the dynamics and constraints of the high-speed aircraft pursuit differential game pursuit player and the escaper in the step one are as follows:
in the formula x p And u p The state quantity and the control quantity x of the chaser respectively e And u e The state quantity and the controlled quantity, p, of the escaper p And p e Process constraints of chaser and escaper, respectively,. Psi p And psi e Terminal constraints of chasers and escapes, respectively, x p,0 And x e,0 Initial states of the chaser and the escaper are respectively;
wherein, the "payment function of the question" in the step one is:
wherein, the "high-speed aircraft pursuit differential game minimum and maximum problems" in the step one is:
wherein, the "proximity operator" in step two is: i x (t) -x (k) (t)|| 2 (4)
Wherein x is the state quantity of the chaser or the escaper, x (k) Initial guesses of state quantities for chasers or escapes in the k-th iteration process;
wherein, the sub-problem of the high-speed aircraft pursuit differential game about the adjacent optimization of the pursuit player and the escaper in the step two is as follows:
in the formula (I), the compound is shown in the specification,
and
respectively estimating states of a chaser and an escaper in the kth iteration process, wherein gamma is a penalty factor which is usually 1-1000 according to experience;
wherein, the "linearization method" described in step three is the classic method in guidance, navigation and control technology, and the technology is the known technology in the field;
wherein, the high-speed aircraft pursuit differential game adjacent convex optimal control subproblem in the step three is as follows:
in the formula (I), the compound is shown in the specification,
and
estimate the amount of control of the chaser and the escaper in the kth iteration process, eta, respectively 1 And η 2 Is a relaxation variable;
the "equal time interval discretization method" described in step three is a classical method in guidance, navigation and control technologies, and the technology is well known in the art;
wherein, in the third step, "the high-speed aircraft pursuit differential game adjacent convex optimization subproblem" is:
in the formula, A p,i 、
z p,i And A e,i 、
z e,i State transition matrices for the chaser and the escaper, respectively;
wherein the "initial guess" in the fourth step is the estimation of the state quantity and the control quantity of the chasers and the evacuees
And with
Wherein, the "iterative solution" described in step four means that: solving the adjacent convex optimization subproblem of the high-speed aircraft pursuit differential game according to the initial guess, taking the solved result as the initial guess of the next solving, and when the deviation between the solved result and the initial guess is generated
When the error is smaller than the allowable error, stopping solving; according to the experience, the tolerance error is 0.001-0.0001;
through the steps, the high-speed aircraft pursuit differential game can be realized, the method can be applied on line, and better stability and universality are achieved.
(III) the advantages and effects of the invention
(1) According to the invention, the pursuit differential game problem is converted into the convex optimal control problem for solving by using the adjacent convex optimization method, so that the pursuit differential game strategy can be obtained and can be used on line;
(2) The method of the invention is scientific, has good manufacturability and has wide popularization and application value.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
FIG. 2 is a schematic diagram of a flight trajectory of a high-speed aircraft in an embodiment of the invention.
FIG. 3 is a schematic representation of the results of high speed aircraft control in an embodiment of the invention.
Detailed Description
The invention will be further explained in detail with reference to the drawings and the embodiments.
The invention discloses a method for optimizing adjacent convex of a high-speed aircraft pursuit differential game, the flow chart of which is shown in figure 1, and the method comprises the following steps:
step one, establishing a high-speed aircraft pursuit differential game minimum and maximum problem model;
in this embodiment, the dynamics and constraints of the high-speed aircraft pursuit differential game pursuit and evacuee are given:
wherein, the dynamics and constraints of the high-speed aircraft pursuit differential game pursuit chaser and escaper in the step one are as follows:
in the formula x p And u p The state quantity and the control quantity x of the chaser respectively e And u e The state quantity and the control quantity, p, of the escaper p And p e Process constraints of chaser and escaper, respectively,. Psi p Phi and phi e Terminal constraints, x, for chasers and escapes, respectively p,0 And x e,0 In this embodiment, the state quantities of the high-speed aircraft pursuing the differential game pursuit and the escaper are vectors composed of speed and position, the controlled quantity is acceleration, and the dynamics and the constraints thereof are as follows:
in the formula u p,max And u e,max The upper limit of the controlled variable amplitude, A, of the chaser and the escaper respectively c And B c Respectively as follows:
the payment function for the specified question is:
establishing a minimum and maximum problem of the high-speed aircraft pursuit differential game:
step two, establishing a high-speed aircraft pursuit escaping differential game adjacent optimization sub-problem;
according to the high-speed aircraft pursuit differential game minimum and maximum problem model established in the first step, a proximity operator is introduced into the model:
||x(t)-x (k) (t)|| 2 (16)
wherein x is the state quantity of the chaser or the escaper, x (k) Is the initial guess of the state quantity during the kth iteration for the chaser or escaper. Establishing a high-speed aircraft pursuit differential game and optimizing a sub-problem about the proximity of a chaser and an escaper:
in the formula (I), the compound is shown in the specification,
and
respectively estimating states of a chaser and an escaper in the kth iteration process, wherein gamma is a penalty factor, and in the embodiment, the penalty factor is 10;
step three, establishing a high-speed aircraft pursuit differential game adjacent convex optimization sub-problem;
according to the high-speed aircraft pursuit differential game adjacent optimization sub-problem established in the step two, a linearization method is adopted to carry out convexity on process constraint, terminal constraint and performance indexes of the problem, and a high-speed aircraft pursuit differential game adjacent convex optimal control sub-problem is formed:
in the formula (I), the compound is shown in the specification,
and
estimates of the control quantities, eta, of the chasers and escapes during the kth iteration, respectively 1 And η 2 Is the relaxation variable. And (3) establishing a high-speed aircraft pursuit differential game adjacent convex optimization sub-problem through equal time interval discretization treatment:
in the formula, A and B + 、B - And the following steps:
wherein Δ t = t f N is a discrete time interval, t f For the flight time, N is the number of discrete points, and in this embodiment, N =1 is taken;
step four, giving estimation of the state quantity and the control quantity of the chasers and the escapes
And with
Sub-problem of pursuing evasion differential game adjacent convex optimization of high-speed aircraftSolving, using the solution result as the initial guess of the next solution, and calculating the deviation between the solution result and the initial guess
When the error is smaller than the allowable error, stopping solving; in this embodiment, the allowable error is 0.0001;
simulation case:
the part is demonstrated by taking a numerical simulation case as a method, and is not an actual flight task;
the dimensionless initial states of the chaser and the escaper are x respectively p,0 ＝[-2,0] T And x e,0 ＝[0,1] T Dimensionless flight time of t f And =2, the upper limit of the amplitude of the control quantity is:
u p,max ＝8,u e,max ＝5
the initial guess given for the first iteration to solve is:
according to the implementation process of the method, a schematic diagram of flight tracks of the chasers and the escapes is obtained and is shown in fig. 2, a schematic diagram of control results of the chasers and the escapes is shown in fig. 3, and the flight strategy of the chasers and the escapes in the high-speed aircraft chasing differential game can be given by using the method.
Claims (10)
1. A high-speed aircraft pursuit differential game adjacent convex optimization method is characterized in that: the method comprises the following specific steps:
step one, establishing a high-speed aircraft pursuit escaping differential game minimum and maximum problem model;
according to task requirements, dynamics and constraints of a high-speed aircraft pursuit differential game pursuit player and an escaper are given, a payment function of a problem is specified, and the minimum and maximum problems of the high-speed aircraft pursuit differential game are established;
step two, establishing a high-speed aircraft pursuit escaping differential game adjacent optimization sub-problem;
according to the minimum and maximum problem model of the high-speed aircraft pursuit differential game established in the step one, introducing a proximity operator into the minimum and maximum problem model, and establishing a proximity optimization sub-problem of the high-speed aircraft pursuit differential game about a chaser and an escaper;
step three, establishing a high-speed aircraft pursuit differential game adjacent convex optimization sub-problem;
according to the high-speed aircraft pursuit differential game adjacent optimization sub-problem established in the step two, process constraint, terminal constraint and performance indexes of the problem are emphasized by adopting a linearization method to form a high-speed aircraft pursuit differential game adjacent convex optimal control sub-problem, and the high-speed aircraft pursuit differential game adjacent convex optimization sub-problem is established through equal time interval discretization treatment;
and step four, giving an initial guess, and iteratively solving the high-speed aircraft pursuit differential game adjacent convex optimization sub-problem.
2. The method for optimizing a neighbor convex of a high-speed aircraft pursuit differential game according to claim 1, wherein the method comprises the following steps: the dynamics and constraints of the high-speed aircraft pursuit differential game pursuit chaser and escaper in the step one are as follows:
in the formula x p And u p The state quantity and the control quantity x of the chaser respectively e And u e The state quantity and the control quantity, p, of the escaper p And p e Process constraints of chasers and escapes, psi p And psi e Terminal constraints, x, for chasers and escapes, respectively p,0 And x e,0 The initial states of the chaser and the escaper are respectively.
3. The method for optimizing a neighbor convex of a high-speed aircraft pursuit differential game according to claim 1, wherein the method comprises the following steps: the "payment function for the question" described in step one is:
4. the method for optimizing a neighbor convex of a high-speed aircraft pursuit differential game according to claim 1, wherein the method comprises the following steps: the "minimum and maximum problems of high-speed aircraft pursuit of differential game" in the step one is as follows:
5. the method for optimizing a neighbor convex of a high-speed aircraft pursuit differential game according to claim 1, wherein the method comprises the following steps: the "neighbor operator" in step two is:
||x(t)-x (k) (t)|| 2 (4)
wherein x is the state quantity of the chaser or the escaper, x (k) Is the initial guess of the state quantity during the kth iteration for the chaser or the escaper.
6. The method for optimizing adjacent convex of high-speed aircraft pursuit differential gaming according to claim 1, wherein: the sub-problem of the high-speed aircraft pursuit differential game about the adjacent optimization of the chaser and the escaper in the step two is as follows:
in the formula (I), the compound is shown in the specification,
and with
Respectively estimating states of a chaser and an escaper in the kth iteration process, taking gamma as a penalty factor and taking 1-1000.
7. The method for optimizing adjacent convex of high-speed aircraft pursuit differential gaming according to claim 1, wherein: the high-speed aircraft pursuit differential game adjacent convex optimal control subproblem stated in step three is as follows:
in the formula (I), the compound is shown in the specification,
and with
Estimates of the control quantities, eta, of the chasers and escapes during the kth iteration, respectively 1 And η 2 Is the relaxation variable.
8. The method for optimizing adjacent convex of high-speed aircraft pursuit differential gaming according to claim 1, wherein: the high-speed aircraft pursuit differential game adjacent convex optimization sub-problem in the third step is as follows:
in the formula, A p,i 、
z p,i And A e,i 、
z e,i State transition matrices for the chaser and the escaper, respectively.
9. The method for optimizing adjacent convex of high-speed aircraft pursuit differential game according to claim 1 or 5, characterized in that: the "initial guess" in step four is an estimate of the state quantity and control quantity of the chasers and escapes
And
10. the method for optimizing a neighboring convex of a high-speed aircraft pursuit differential game according to claim 1 or 8, wherein the method comprises the following steps: the "iterative solution" described in step four means that: solving the adjacent convex optimization sub-problem of the high-speed aircraft pursuit differential game according to the initial guess, taking the solved result as the initial guess of the next solving, and when the deviation between the solved result and the initial guess is generated
When the error is smaller than the allowable error, stopping solving; according to experience, the tolerance is 0.001-0.0001.
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