CN101872164B - Method of reducing astrodynamics system state sensitivity - Google Patents

Abstract

The invention provides a method of reducing astrodynamics system state sensitivity, which belongs to the dynamics and control technology field. The invention quantizes the influence of turbulence on system state by introducing the sensitivity index J2 of the state and then obtains the optimal solution of the system by obtaining the minimum value of an optimal performance index J equals to J1 plus cO.J2 considering the system state sensitivity. Compared with the traditional optimal control method based on maximum value principle, the optimal control method considering the system state sensitivity has the following advantages; (1) the previous optimal performance index J1 is almost not influenced and (2) on condition that the previous optimal performance index is ensured, the influence of the turbulence on the system state can be reduced effectively.

Description

A kind of method that reduces astrodynamics system state sensitivity

Technical field:

The present invention relates to a kind of method that reduces astrodynamics system state sensitivity, belong to astrodynamics and control technology field.

Background technology:

Traditional method for optimally controlling based on maximal principle can only the guaranteed performance index optimization; And do not have ability guaranteeing under the optimum precondition of concrete performance index simultaneously disturbance suppression to the influence of system state, and this influence many times all can bring adverse influence to system.For example, reenter in the process of the earth at spacecraft, the state error constantly that leaves the right or normal track all can cause very big influence to reentering process, cause bigger landing error with the uncertainty that reenters aerodynamic parameter in the process.Therefore, a kind of new method for optimally controlling of development, under the optimum prerequisite of guaranteed performance index, to reduce state be a task that ten minutes is urgent to the susceptibility of disturbance.

Summary of the invention

Goal of the invention:

Technical matters to be solved of the present invention is not have influence this shortcoming of ability disturbance suppression to system state in order to overcome traditional method for optimally controlling based on maximal principle; A kind of method that reduces astrodynamics system state sensitivity is provided, and this method reenters various fields such as control and martian atmosphere entering guidance control at the spacecraft earth all have extremely important using value.

Technical scheme:

The present invention adopts following technical scheme for realizing the foregoing invention purpose:

A kind of method that reduces astrodynamics system state sensitivity comprises the steps:

The A step, set up the system dynamics equation:

$\stackrel{\·}{x}=f(x,u,t);---\left(1\right)$

Wherein, x is that dimension is the state variable of n, and u is a control variable, and t is a time variable, and n is a natural number;

The B step is established any t ∈ [t ₀, t _f], t wherein ₀Be initial time, t _fBe the terminal point moment, system state is designated as x (t)=X (t|t ₀, x ₀), x ₀Be initial time t ₀State, X represents from initial time t ₀To any state transitions process of t constantly; The optimal performance index of taking into account system state sensitivity defines as follows:

J＝J ₁+c ₀·J ₂； (2)

Wherein, J ₁Be optimal performance index, J ₂It is the system state sensitivity indexes; c ₀Be penalty factor, be used to regulate the shared weight of sensitivity index;

The c step, the state x of definition any time t is to initial time t ₀State x ₀Susceptibility matrix S (t|t ₀) as follows:

$S\left(t\right|t_0)=\frac{\∂x}{{\∂x}_0}---\left(3\right)$

Be prone to checking susceptibility matrix S (t|t ₀) have a following characteristic:

$\frac{\∂S\left(t\right|t_0)}{\∂t}=(\frac{\∂f\left(x\right)}{\∂x}+\frac{\∂f\left(x\right)}{\∂u}\frac{\∂u}{\∂x})\·S\left(t\right|t_0);---\left(4\right)$

S(t ₀|t ₀)＝I； (5)

S(t ₂|t ₁)＝S(t ₂|t ₀)·S(t ₁|t ₀) ^-1；(6)

Wherein, I is a unit matrix, () ^-1Be inversion operation;

The D step is found the solution the said system state susceptibility of formula (3) matrix S according to formula (4), (5), (6);

The E step is according to system state susceptibility matrix S define system state sensitivity index J ₂As follows:

J ₂＝∑g(S _i，j(t _f，t)) (7)

Wherein g is to be the function of independent variable with the susceptibility matrix, i, and j is natural number;

The F step, convolution (1), formula (4) adopt and join the optimal performance index J that an optimized Algorithm is asked for the defined taking into account system state sensitivity of formula (2).

Beneficial effect:

The present invention compares with traditional method for optimally controlling based on maximal principle, and following advantage is arranged:

(1) original optimal performance index J ₁Influenced hardly.

(2) guaranteeing to effectively reduce the influence of disturbance under the optimized prerequisite of original performance index to system state.

Description of drawings:

Fig. 1 is a process flow diagram of the present invention.

Specific embodiments:

Below in conjunction with accompanying drawing the present invention is made further detailed description:

As shown in Figure 1, step of the present invention is following:

(1) set up the system dynamics equation:

$\stackrel{\·}{x}=f(x,u,t);---\left(1\right)$

Wherein, x is that dimension is the state variable of n, and u is a control variable, and t is a time variable, and n is a natural number;

(2) establish any t ∈ [t ₀, t _f], t wherein ₀Be initial time, t _fBe the terminal point moment, system state is designated as x (t)=X (t|t ₀, x ₀), x ₀Be initial time t ₀State, X represents from initial time t ₀To any state transitions process of t constantly; The state x of definition any time t is to initial time t ₀State x ₀Susceptibility matrix S (t|t ₀) as follows:

$S\left(t\right|t_0)=\frac{\∂x}{{\∂x}_0}---\left(2\right)$

Be prone to checking susceptibility matrix S (t|t ₀) have a following characteristic:

$\frac{\∂S\left(t\right|t_0)}{\∂t}=(\frac{\∂f\left(x\right)}{\∂x}+\frac{\∂f\left(x\right)}{\∂u}\frac{\∂u}{\∂x})\·S\left(t\right|t_0);---\left(3\right)$

S(t ₀|t ₀)＝I； (4)

S(t ₂|t ₁)＝S(t ₂|t ₀)·S(t ₁|t ₀) ^-1；(5)

Wherein, I is a unit matrix, () ^-1Be inversion operation;

(3) find the solution the said system state susceptibility of formula (2) matrix S according to formula (3), (4), (5);

(4) according to system state susceptibility matrix S define system state sensitivity index J ₂As follows:

J ₂＝∑g ₂(S _i，j(t _f，t)) (6)

G wherein ₂Be to be the function of independent variable with the susceptibility matrix, i, j is natural number;

(5) according to concrete engineering background definition optimal performance index J ₁, the shortest such as the time, burnup is the most excellent.

J ₁＝g ₁(x，u，t) (7)

G wherein ₁Be with state, control and time are the function of independent variable;

(6) definition of the optimal performance index of taking into account system state sensitivity is as follows:

J＝J ₁+c ₀·J ₂； (8)

Wherein, J ₁Be optimal performance index, J ₂It is the system state sensitivity indexes; c ₀Be penalty factor, be used to regulate the shared weight of sensitivity index;

(7) convolution (1), formula (3) adopt and join the optimal performance index J that an optimized Algorithm is asked for the defined taking into account system state sensitivity of formula (8).That is, transfer formula (1)-(8) described optimal control problem to nonlinear programming problem, find the solution a kind of method that reduces astrodynamics system state sensitivity through SQP by means of point collocation.

Claims (1)

1. method that reduces astrodynamics system state sensitivity is characterized in that: comprises the steps,

The A step, set up the system dynamics equation:

$\stackrel{\·}{x}=f(x,u,t);---\left(1\right)$

Wherein, x is that dimension is the state variable of n, and u is a control variable, and t is a time variable, and n is a natural number;

The B step is established any t ∈ [t ₀, t _f], t wherein ₀Be initial time, t _fBe the terminal point moment, system state is designated as x (t)=X (t|t ₀, x ₀), x ₀Be initial time t ₀State, X represents from initial time t ₀To any state transitions process of t constantly; The optimal performance index of taking into account system state sensitivity defines as follows:

J＝J ₁+c ₀·J ₂；(2)

Wherein, J ₁Be optimal performance index, J ₂It is the system state sensitivity indexes; c ₀Be penalty factor, be used to regulate the shared weight of sensitivity index;

The C step, the state x of definition any time t is to initial time t ₀State x ₀Susceptibility matrix S (t|t ₀) as follows:

$S\left(t\right|t_0)=\frac{\∂x}{{\∂x}_0}---\left(3\right)$

Be prone to checking susceptibility matrix S (t|t ₀) have a following characteristic:

$\frac{\∂S\left(t\right|t_0)}{\∂t}=(\frac{\∂f\left(x\right)}{\∂x}+\frac{\∂f\left(x\right)}{\∂u}\frac{\∂u}{\∂x})\·S\left(t\right|t_0);---\left(4\right)$

S(t ₀|t ₀)＝I； (5)

S(t ₂|t ₁)＝S(t ₁|t ₀)·S(t ₁|t ₀) ^-1；(6)

Wherein, I is a unit matrix, () ^-1Be inversion operation;

The D step is found the solution the said system state susceptibility of formula (3) matrix S according to formula (4), (5), (6);

The E step is according to system state susceptibility matrix S define system state sensitivity index J ₂As follows:

J ₂＝∑g(S _i，j(t _f，t)) (7)

Wherein g is to be the function of independent variable with the susceptibility matrix, i, and j is natural number;

The F step, convolution (1), formula (4) adopt and join the optimal performance index J that an optimized Algorithm is asked for the defined taking into account system state sensitivity of formula (2).

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