CN109917647B - Teaching and learning algorithm improved based on teaching strategy and liquid-filled spacecraft optimization sliding mode control method - Google Patents

Teaching and learning algorithm improved based on teaching strategy and liquid-filled spacecraft optimization sliding mode control method Download PDF

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CN109917647B
CN109917647B CN201910166219.1A CN201910166219A CN109917647B CN 109917647 B CN109917647 B CN 109917647B CN 201910166219 A CN201910166219 A CN 201910166219A CN 109917647 B CN109917647 B CN 109917647B
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sliding mode
liquid
spacecraft
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肖玲斐
何虹兴
申斌
马磊明
叶志锋
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Nanjing University of Aeronautics and Astronautics
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Abstract

The invention provides a teaching and learning algorithm improved based on a teaching strategy and a sliding mode control method for optimizing a liquid-filled spacecraft. The invention refers to a specific teaching process, makes practical adjustment on a teaching strategy in a teaching and learning algorithm through group learning and teacher depth inter-learning, provides a teaching and learning algorithm (SMTLBO) based on teaching strategy improvement, introduces an intermediate variable to decouple the system aiming at a typical underactuated system such as a liquid-filled spacecraft, which has high nonlinearity and high coupling, constructs a time-varying sliding mode surface, inputs a system state variable as the SMTLBO algorithm, obtains a sliding mode surface parameter at the current moment by utilizing the dynamic calculation of the SMTLBO, realizes the dynamic adjustment of the sliding mode surface parameter, and the designed sliding mode controller can ensure the stability of the liquid-filled spacecraft system.

Description

Teaching and learning algorithm improved based on teaching strategy and liquid-filled spacecraft optimization sliding mode control method
Technical Field
The invention relates to a control system optimization technology, in particular to a teaching and learning method based on teaching strategy improvement and a liquid-filled spacecraft optimization sliding mode control method.
Background
At present, the commonly used spacecraft often has the requirements of small volume, light weight and long on-orbit time, so most of the spacecraft adopt liquid fuel with higher heat value as propellant. With the continuous development of the aerospace industry, the spacecraft has more and more tasks, and the specific gravity of the liquid propellant carried by the spacecraft is gradually increased. For a general spacecraft, the liquid fuel inside the spacecraft often generates oscillation in the dynamic process of the spacecraft, so that the spacecraft is impacted, and the stability of spacecraft control is affected. The disturbance and impact of the sloshing of the liquid fuel on the spacecraft can cause the instability of the control system or the damage of the structure once the regulation range of the control system or the upper limit of the structure bearing is exceeded. Therefore, sloshing of the liquid fuel during operation of the spacecraft is an important issue that has to be addressed.
Considering that the overall quality of the spacecraft is often greatly increased by a general passive control means, at present, scholars at home and abroad widely adopt an active control method. The nonlinear feedback control law based on Lyapunov is designed by the Mahmut Reyhanoglu; in consideration of the problem of uncertain parameters caused by fuel consumption in the maneuvering process of the spacecraft, a self-adaptive layered sliding mode controller is designed by the aid of a Schchenjian and the like; hesham shaker and Taogang et al study the equivalent pendulum model of a liquid-sloshing spacecraft and design self-adaptive pole configuration attitude control for the spacecraft in a translation state. However, the active control means cannot directly regulate and control the liquid sloshing, and only indirectly adjust the attitude of the spacecraft to inhibit the sloshing, so the liquid-filled spacecraft system is often regarded as an under-actuated system. The under-actuated system is a nonlinear system with the number of independent control variables of the system smaller than the number of degrees of freedom of the system, and the system has a simple structure and is convenient for integral dynamics analysis and experiment. Due to the reasons of high nonlinearity of the system, requirements such as parameter perturbation and multi-target control, limited control quantity and the like, the underactuated system is sufficiently complex, and the effectiveness of various algorithms is convenient to study and verify. In the literature (X.Rong, U.S. Ozgunner.sliding Mode Control of a class of systematic of exploited Systems [ J ]. Automatica,2008,44: 233-.
A Teaching-Learning Based Optimization (TLBO) is a new swarm intelligence Optimization algorithm proposed by Rao et al in 2010, and the method simulates the Teaching process of a teacher and the Learning process of a student to obtain an optimal solution. The teaching and learning algorithm has the advantages of few parameters, simple structure, concise concept, high solving precision, high convergence speed and strong convergence capability. Compared with some classical intelligent optimization algorithms, such as a particle swarm algorithm, the algorithm is characterized in that the algorithm only has two parameters, namely the group membership and the iterative algebra, and the number of the parameters to be set is small, so that the problems of low calculation efficiency or easy falling into local convergence and the like caused by improper parameter setting can be solved. Teaching and learning algorithms as an emerging intelligent optimization algorithm still have many problems, such as the characteristics of easy premature convergence and rapid loss of algorithm diversity. This is due to the "teaching" process of the algorithm, which is essentially the process by which a student quickly approaches a single teacher. In the process, the diversity of the algorithm is lost rapidly, and the population often shows the characteristics similar to those of individual teachers, so that local convergence is easy to be caused.
Disclosure of Invention
The purpose of the invention is as follows: in order to promote the research on the theory and the application of the intelligent optimization algorithm in the control system, the invention aims at the attitude control problem of the liquid-filled spacecraft,
the technical scheme is as follows:
a teaching and learning algorithm improved based on a teaching strategy comprises the following steps:
1) initializing the class: randomly generating each student in a class in a search space
Figure BDA0001986376420000021
j ═ 1,2, …, NP; the generation method is carried out according to the following formula:
Figure BDA0001986376420000022
wherein the content of the first and second substances,
Figure BDA0001986376420000023
and
Figure BDA0001986376420000024
respectively, as the upper and lower bounds of each dimension, where i ═ 1,2, …, d
2) Dividing the class into num groups, evaluating the adaptive value of the kth group, and selecting the optimal groupThe body is taken as the teacher individual of the kth group
Figure BDA0001986376420000025
The teaching process of the individual i in the kth group member is represented by the following formula (2) and formula (3):
Figure BDA0001986376420000026
Figure BDA0001986376420000027
in the formula:
Figure BDA0001986376420000028
and
Figure BDA0001986376420000029
respectively representing the values before and after learning of the ith student of the k groups; k is 1,2, …, mum;
wherein the content of the first and second substances,
Figure BDA00019863764200000210
is the mean value of the kth subgroup, TF i k1+ rand (0,0.5) is the teaching factor of the teacher, ri kRand (0.5,1) is the learning step length of the student;
3) for students
Figure BDA00019863764200000211
Randomly selecting a member from the group
Figure BDA00019863764200000212
Analyze oneself with
Figure BDA00019863764200000213
The difference between the two is used for learning adjustment;
Figure BDA0001986376420000031
wherein the content of the first and second substances,
Figure BDA0001986376420000032
in order to learn the values of the trainee after adjustment,
Figure BDA0001986376420000033
adjusting the values of the trainees prior to learning;
4) for individual teacher
Figure BDA0001986376420000034
Randomly selecting another teacher individual
Figure BDA0001986376420000035
Carrying out learning adjustment;
4.1) let k equal to 1 for individual teachers
Figure BDA0001986376420000036
Randomly selecting another teacher individual
Figure BDA0001986376420000037
4.2) let y equal to 1, use
Figure BDA0001986376420000038
Is substituted by the y-component
Figure BDA0001986376420000039
The y-th dimension component of (a) to generate a test solution if the adaptation value of the test solution is better than that of the test solution
Figure BDA00019863764200000310
It is replaced;
4.3) making y equal to y +1, repeating the step 4.2) until y equal to d, and finishing the inter-learning;
4.4) let k be k +1, repeat steps 4.2) and 4.3) until k is num, then the learning exchange of all teachers is finished;
4.5) comparing the adaptive values of all teachers, and taking the teacher with the optimal adaptive value as a global optimal solutionXbest；
5) Self-learning by the whole staff: for any individual XiThe operation steps of the self-learning process are as follows:
Figure BDA00019863764200000311
wherein randn is a normal distribution, and wherein,
Figure BDA00019863764200000312
XU,XLrespectively representing the maximum value and the minimum value of the actual individual of the population; t is the maximum allowable iteration number, and T is the current iteration number;
6) judging whether a termination condition is met, if so, terminating the algorithm and outputting an optimal solution; otherwise, jumping to the step 2), and continuing the iterative computation.
A liquid-filled spacecraft optimization sliding-mode control method based on teaching strategy improved teaching and learning algorithm comprises the following steps:
1) the liquid-filled spacecraft is equivalent to a simple pendulum model, the x axis and the z axis are inertial coordinate systems, the liquid-filled spacecraft has constant thrust T and axial speed v along the bodyx(ii) a The dry weight of the liquid-filled spacecraft is m, and the mass of the fuel part in the storage tank is mf(ii) a The moment of inertia of the liquid-filled spacecraft relative to the center of the tank is I, and the moment of inertia of the fuel relative to the center of the tank is If(ii) a The distance from the mass center of the spacecraft to the suspension point of the simple pendulum is b, and the pendulum length of the simple pendulum is a; the attitude angle of the spacecraft is theta, and the liquid oscillation angle, namely the equivalent simple pendulum swing angle is theta
Figure BDA00019863764200000313
The controlled inputs of the system are transverse control force F and pitching moment M of the mass center of the spacecrafty，vzA lateral velocity generated for influence by a control input;
Figure BDA00019863764200000314
is the energy dissipation factor;
2) obtaining a liquid-filled spacecraft dynamics equation of liquid sloshing:
Figure BDA0001986376420000041
Figure BDA0001986376420000042
Figure BDA0001986376420000043
Figure BDA0001986376420000044
equation (6) is simplified to:
Figure BDA0001986376420000045
order to
Figure BDA0001986376420000046
Comprises the following steps:
Figure BDA0001986376420000047
wherein the content of the first and second substances,
Figure BDA0001986376420000048
the control inputs F and M of the system are transformed via equation (11)yConversion to u1And u2Accordingly, formulae (7) to (9) are converted into the following forms:
Figure BDA0001986376420000049
Figure BDA00019863764200000410
Figure BDA00019863764200000411
wherein the content of the first and second substances,
Figure BDA00019863764200000412
3) designing a sliding mode controller with the controller designed to target system state variables
Figure BDA00019863764200000413
Reach zero in a finite time; divide the system into
Figure BDA00019863764200000414
And an under-actuated subsystem (15) comprising
Figure BDA00019863764200000415
Full drive subsystem (16):
Figure BDA0001986376420000051
Figure BDA0001986376420000052
for the subsystem (16), a control law is designed according to a general sliding mode control method:
Figure BDA0001986376420000053
Figure BDA0001986376420000054
in the formula, c2Is a subsystemSliding mode surface parameter of system (16) < rho >2And2for subsystem sliding mode surface s2Corresponding exponential and isovelocity approach terms, and satisfies c2,ρ2,2＞0；
For the sub-system (15), a new state variable eta is introduced instead of
Figure BDA0001986376420000055
Figure BDA0001986376420000056
The two sides of the formula (19) are subjected to derivation to obtain:
Figure BDA0001986376420000057
thereby converting the subsystem (15) into the form:
Figure BDA0001986376420000058
for the subsystem (21), define
Figure BDA0001986376420000059
The sliding mode function is designed as follows
s1＝μ1-Mμ2 (23)
Wherein M is [ M ]1,m2]Is a sliding mode surface matrix
Order to
Figure BDA00019863764200000510
Then
Figure BDA00019863764200000511
The control law of the subsystem (21) is designed as follows:
Figure BDA00019863764200000512
in the formula, ρ1，1Is s is1Corresponding exponential approach term and constant velocity approach term, and satisfies rho1,1＞0；
Define the lyapunov function as:
Figure BDA0001986376420000061
then
Figure BDA0001986376420000062
The sliding mode arrival condition is met, and the time t exists1For t ≧ t1，s1＝μ1-Mμ 20, i.e. at the slip-form face s1On 0, there is μ1＝Mμ2；
Then:
Figure BDA0001986376420000063
wherein, DeltaμIs a very small real number vector, substituted
Figure BDA0001986376420000064
It is possible to obtain,
Figure BDA0001986376420000065
in the formula (I), the compound is shown in the specification,
Figure BDA0001986376420000066
the solution to M is:
Figure BDA0001986376420000067
4) and solving the value of M based on a teaching and learning algorithm improved by the teaching strategy.
In the step 4), the sliding mode controller of the liquid-filled spacecraft is at a balance point
Figure BDA0001986376420000068
Designed on a nearby basis to obtain
Figure BDA0001986376420000069
The value of M is taken as the solution target,
Figure BDA00019863764200000610
as input, dependent on the current time
Figure BDA00019863764200000611
Calculating to obtain M (t) such that
Figure BDA00019863764200000612
Can be approximated to Aμ→0μ2(t); definition of
Figure BDA00019863764200000613
The fitness function is taken as
Figure BDA0001986376420000071
Setting the iteration number G to be 20 and the population size NP to be 20, averagely dividing the population into num to be 3 groups, and obtaining the optimal solution M (t) at the current time.
Has the advantages that: according to the method, a specific teaching process is referred, a teaching and learning algorithm improved based on the teaching strategy is provided through grouping learning and teacher deep learning, an intermediate variable is introduced for the liquid-filled spacecraft to decouple the system, a time-varying sliding mode surface is constructed, a system state variable is used as an SMTLBO algorithm input, sliding mode surface parameters at the current moment are obtained through SMTLBO dynamic calculation, dynamic adjustment of the sliding mode surface parameters is achieved, and the designed sliding mode controller can guarantee stability of the liquid-filled spacecraft system.
Drawings
Fig. 1 is a box diagram of the results of SMTLBO calculations on the rosenblock function.
Fig. 2 is a box diagram of the calculation result of SMTLBO on the Griewank function.
Fig. 3 is a box diagram of the results of SMTLBO calculations on the rastigin function.
Fig. 4 is a distribution diagram of the optimal solution calculated by SMTLBO when the number of iterations G is 25 in a certain iteration.
Fig. 5 is a liquid-filled spacecraft model.
FIGS. 6 to 12 are views of
Figure BDA0001986376420000079
And (5) a time simulation result graph.
Fig. 13 is a graph showing the variation of M calculated by SMTLBO in a dynamic process.
FIGS. 14 to 22 are views showing
Figure BDA00019863764200000710
And
Figure BDA00019863764200000711
and (5) a time simulation result graph.
Detailed Description
The invention is further elucidated with reference to the drawings and the embodiments.
Basic TLBO algorithm:
for one optimization problem:
Figure BDA0001986376420000072
search space
Figure BDA0001986376420000073
Any search point in space X ═ X1,x2,…xd) Where d represents the dimension of the dimensional space (number of decision variables),
Figure BDA0001986376420000074
and
Figure BDA0001986376420000075
expressed as the upper and lower bounds of each dimension, respectively, and f (x) is the objective function. Is provided with
Figure BDA0001986376420000076
In order to search for a point in the space,
Figure BDA0001986376420000077
is a point XjNP is the number of spatial search points (i.e., population size). Corresponding to the basic TLBO algorithm is: 1) class: in the TLBO algorithm, the set of all search particles in the search space is called class (class); 2) the student: any individual in a class
Figure BDA0001986376420000078
Called a student; 3) a teacher: student X with best performance in classbestCalled teacher, in the invention using XteacherAnd (4) showing.
Thus, a class may be represented in the form:
Figure BDA0001986376420000081
wherein, Xj(j-1, 2, …, NP) represents a class student, Xteacher＝argmaxf(Xj) (j ═ 1,2, …, NP). NP is the number of students, d is the number of subjects learned by the students.
The teaching and learning optimization algorithm (TLBO) is as follows:
1) initializing the class: randomly generating each student in a class in a search space
Figure BDA0001986376420000082
Figure BDA0001986376420000083
The generation method is carried out according to the following formula:
Figure BDA0001986376420000084
2) the "teach" stage: in the teaching stage of TLBO algorithm, selecting the trainee X with the most excellent performancebestAs teacher Xteacher. According to the subject scores of the students, the students can learn according to the difference value between the teacher and the average Mean of the students, and the score of each student is improved to a certain extent, so that the class average is improved. It should be noted that the amount of knowledge that the trainee can acquire depends on the difference between the teacher and the average value of the class, the teaching factor of the teacher and the learning ability of the student, and therefore, the promotion space of the teaching stage is limited.
Assuming that the subject performance of the student follows normal distribution, the average performance of the class is Mean at the beginningA30, the average performance is low and the distribution is wide. After the teacher's multiple teaching' process, the class average score is gradually improved to MeanBAt 80, the performance improved and the distribution concentrated. The specific teaching method is shown in the formulas (2) and (3).
Figure BDA0001986376420000085
difference＝ri×(Xteacher-TFi×Mean)(3)
Wherein the content of the first and second substances,
Figure BDA0001986376420000086
and
Figure BDA0001986376420000087
respectively represent the values before and after learning of the ith student,
Figure BDA0001986376420000088
represents the average of all trainees. In addition, there are two important parameters in the formula: teaching factor TF of teacheri＝round[1+rand(0,1)]And learning step length r of studentiRand (0, 1). The former characterizes the teaching ability of teachers, and the latter characterizes the learning ability of students.
3) The "learning" stage: the "learning" stage refers to mutual learning among students, and learning is performed by comparing and analyzing differences among students. For each student Xi(i ═ 1,2, …, NP), randomly choosing a learning object X in the classj(j＝1,2,…,NP,j≠i)，XiThe learning adjustment is carried out by analyzing the difference between the self and the Xj, and the method for improving the learning is similar to a difference mutation operator in a difference algorithm. The difference is that the learning step r in the teaching and learning algorithm uses a different learning factor for each different learner. Student XiAnd XjBy comparing the respective objective function values (namely, the learning results), the inferior one gets close to the superior one, and in this way, the inter-learning and progress among students are realized. The specific adjustment process can be represented by the following formula (4):
Figure BDA0001986376420000091
wherein r isiIs the learning step size of the ith student, and ri＝U(0,1)。
4) The "update" operation: the student performs the updating operation when going through the stages of 'teaching' and 'learning'. The purpose of the update operation is to replace the inferior individuals with the better ones after learning to achieve an improvement in the average performance of all trainees. The update operation is as follows:
Figure BDA0001986376420000092
Figure BDA0001986376420000093
End.
in a standard teaching and learning algorithm, a teaching process is a process that population members approach to a single good individual, and the process easily causes loss of diversity of the population and falls into local convergence. Considering that a single teacher individually performs teaching operation on a whole member, the method is not only low in efficiency, but also easy to fall into local convergence, so that the invention adopts a grouping teaching method and provides SMTLBO. When the main parameters of the algorithm are initialized, the population members are averagely divided into num groups, and on the basis, the subsequent teaching process is developed.
1) Grouped teaching process
Evaluating the adaptive value of the kth subgroup, and selecting the individual with the optimal adaptive value as the teacher individual of the kth subgroup
Figure BDA0001986376420000094
Thus, the teaching process for individual i among the kth group members can be represented by equations (5) and (6):
Figure BDA0001986376420000095
Figure BDA0001986376420000096
in the formula:
Figure BDA0001986376420000097
and
Figure BDA0001986376420000098
respectively representing the values before and after learning of the ith student of the k groups. Unlike TLBO, the membership of each team is highly different due to grouping, and if the average value of the students in TLBO is used for processing, the teaching results are influenced. Meanwhile, the number of students which need to be responsible for by each group of teachers is reduced along with the generation of grouping, and the teaching process can be more targeted. Therefore, it is not allowed to:
Figure BDA0001986376420000101
and teaching factor TF for the other two parametersi kAnd learning step length ri kCorresponding changes are also made. Considering that students have stronger receptivity and comprehension ability and stronger learning willingness under the targeted teaching, the following steps are not required: TFi k＝1+rand(0,0.5)，ri k＝rand(0.5,1)。
2) Group learning process
The grouping learning process is mainly divided into two parts: first, the members in the group learn each other; the second is the joint progress represented by the groups. The inter-study of the members in the group can be carried out by referring to basic teaching and learning algorithms for the members
Figure BDA0001986376420000102
Randomly selecting a member from the group
Figure BDA0001986376420000103
Figure BDA0001986376420000104
Analyze oneself with
Figure BDA0001986376420000105
The difference between the two values is used for learning adjustment, and a specific adjustment method can be performed by referring to equation (4).
The inter-group representative is the inter-study of teachers in each group. Since the quality of the individual teacher is one of the most important factors influencing the quality of the group fitness value to which the individual teacher belongs, the individual teacher must learn each other comprehensively and deeply. For individual teacher
Figure BDA0001986376420000106
Randomly selecting another teacher individual
Figure BDA0001986376420000107
And (5) carrying out learning adjustment. Unlike the individual study of students, the study among teachers is not limited to twoThe differences between the users also include the mutual exchange of knowledge and experience. Therefore, the invention introduces the idea of co-evolution, exchanges internal information between the teacher individuals and components in the same dimension, and optimizes the current teacher individuals in a dimension-by-dimension mode. The specific operation steps are as follows:
2.1) let k equal to 1 for individual teachers
Figure BDA0001986376420000108
Randomly selecting another teacher individual
Figure BDA0001986376420000109
2.2) let y equal to 1, use
Figure BDA00019863764200001010
Is substituted by the y-component
Figure BDA00019863764200001011
The y-th dimension component of (a) to generate a test solution if the adaptation value of the test solution is better than that of the test solution
Figure BDA00019863764200001012
It is replaced.
2.3) let y be y +1, repeat step 2.2) until y is d, then the iteration ends.
2.4) let k be k +1, repeat steps 2.2) and 2.3) until k is num, and the learning communication ends for all teachers.
2.5) comparing the adaptive values of all teachers, and taking the teacher with the optimal adaptive value as a global optimal solution Xbest。
3) Full-member self-learning process
Compared with basic teaching and learning algorithms, the self-learning process is added so that the algorithms do not lose the diversity of the population prematurely in the teaching and mutual learning processes. Considering that the adaptive value of the population is continuously improved and the distribution is more concentrated along with the progress of the algorithm, the adaptive learning step length is adopted to adjust the local search capability so as to eliminate the defect that the fixed step length local search causes low algorithm convergence precision in the later stage of the algorithm.For any individual XiThe operation steps of the self-learning process are as follows:
Figure BDA0001986376420000111
wherein randn is a normal distribution, and wherein,
Figure BDA0001986376420000112
XU,XLrespectively representing the maximum value and the minimum value of the actual individual of the population. T is the maximum number of iterations allowed, and T is the current number of iterations.
In order to prevent the self-learning process from causing the deterioration of the population adaptive value, the teacher individual of the current iteration is compared with the teacher individual of the previous generation, and the teacher individual with the better adaptive value is reserved.
The invention selects a Rosenbrock function, a Griewank function and a Rastrigin function as test functions. The population size of the algorithm is set to be NP which is 12, the maximum iteration number G which is 100 and the subgroup number num which is 3, each function is independently calculated for 20 times, and the result is counted and compared with the previous result. The statistical results are shown in table 1:
table 1: statistical table of calculation results of three test functions
Figure BDA0001986376420000113
As can be seen from the statistical table 1 and fig. 1 to fig. 3, SMTLBO is significantly better than other algorithms in calculation of the rosenblock function, and the algorithm completely converges on the global optimal solution, which proves that SMTLBO has strong convergence and high calculation efficiency. The statistical results on the Griewank function and the rasstrigin function of the multi-peak value also demonstrate the effectiveness of SMTLBO. Although the magnitude of the adaptive value is within the acceptable range, it cannot be ignored that the standard deviation of the statistical result of the SMTLBO algorithm is relatively large, which indicates that SMTLBO is not stable in the 20 times of calculation, and the results of several times of calculation are relatively poor. Considering the strong convergence of the algorithm, the algorithm is not repeated for the number of iterationsIn the case of feet, there is a problem that local convergence tends to be easily caused. In fig. 4, a distribution diagram of the optimal solution calculated by SMTLBO when the number of iterations G is 25 in a certain iteration is plotted. From the figure we can see that the solution of the local optimal solution is distributed in the neighborhood of the global optimal solution X ═ 0, and the components with more dimensions are trapped in X i1 on the local optimum solution. This shows that SMTLBO risks getting into local convergence when dealing with the multi-peak function problem, and the local search capability of the algorithm is relatively poor. As can be seen from fig. 1 to fig. 3, most of the 20 calculations converge to the global optimal solution, and the optimal adaptation values of only a few calculations are relatively large. This also illustrates that SMTLBO is in most cases reliable, but still has the disadvantage of being prone to local convergence when dealing with the multiple peak problem.
Fig. 5 is a liquid-filled spacecraft model. Considering that the liquid surface is mainly acted by surface tension under the microgravity condition, the shaking of the liquid surface can be approximately equivalent to a simple pendulum model, and the liquid shaking angle is equivalent to a simple pendulum swinging angle if the suspension point of the simple pendulum is the center of the spherical cavity, as shown in fig. 5.
The fuel part of the liquid-filled spacecraft equivalent to a simple pendulum model is assumed to be rigid. The x-axis and z-axis are inertial coordinate systems. The liquid-filled spacecraft has constant thrust T and axial speed v along the machine bodyx(ii) a The dry weight of the liquid-filled spacecraft is m, and the mass of the fuel part in the storage tank is mf(ii) a The moment of inertia of the liquid-filled spacecraft relative to the center of the tank is I, and the moment of inertia of the fuel relative to the center of the tank is If(ii) a The distance from the mass center of the spacecraft to the suspension point of the simple pendulum is b, and the pendulum length of the simple pendulum is a; the attitude angle of the spacecraft is theta, and the liquid oscillation angle, namely the equivalent simple pendulum swing angle is theta
Figure BDA0001986376420000128
The controlled inputs of the system are transverse control force F and pitching moment M of the mass center of the spacecrafty，vzA lateral velocity generated for influence by a control input;
Figure BDA0001986376420000129
for energy dissipationAnd (4) the coefficient.
The spacecraft dynamics equation of the liquid sloshing can be obtained:
Figure BDA0001986376420000121
Figure BDA0001986376420000122
Figure BDA0001986376420000123
Figure BDA0001986376420000124
considering that the thrust T is large, the liquid sloshing has a negligible effect on the acceleration in the x-axis direction, but the effect is small, so v can be set to be smallxLooking at the external variables, equation (8) can be simplified to:
Figure BDA0001986376420000125
order to
Figure BDA0001986376420000126
Comprises the following steps:
Figure BDA0001986376420000127
wherein the content of the first and second substances,
Figure BDA0001986376420000131
the control inputs F and M of the system are transformed via equation (13)yConversion to u1And u2Thus, the formulae (9) to (11) can be convertedIn the form:
Figure BDA0001986376420000132
Figure BDA0001986376420000133
Figure BDA0001986376420000134
wherein the content of the first and second substances,
Figure BDA0001986376420000135
thus, a controller can be designed on a simplified system, the controller being designed with the goal of making the system state variables
Figure BDA0001986376420000136
The zero point is reached within a limited time, so that the requirements of inhibiting liquid from shaking and keeping the attitude of the spacecraft stable are met.
As can be seen from the equations (14) - (16), the independent control variables of the system are smaller than the number of degrees of freedom of the system, so that the liquid-filled spacecraft is an under-actuated system, and the general design method cannot effectively control all the variables of the system.
Consider the system as comprising
Figure BDA0001986376420000137
And an under-actuated subsystem (17) comprising
Figure BDA0001986376420000138
The full drive subsystem (18):
Figure BDA0001986376420000139
Figure BDA00019863764200001310
for the subsystem (18), a control law can be designed according to a general sliding mode control method:
Figure BDA00019863764200001311
Figure BDA00019863764200001312
in the formula, c2Is a sliding mode surface parameter, rho, of the subsystem (18)2And2for subsystem sliding mode surface s2Corresponding exponential and isovelocity approach terms, and satisfies c2,ρ2,2Is greater than 0. Easy to verify, the control law meets the sliding mode arrival condition of the subsystem (18), and
Figure BDA00019863764200001313
Figure BDA00019863764200001314
the zero point can be reached in a limited time.
For the subsystem (17), due to the designed control law u2Is only one of
Figure BDA00019863764200001315
Related item, and satisfy when
Figure BDA0001986376420000141
When u is turned on2→ 0. Then in the subsystem (17) it is possible to convert
Figure BDA0001986376420000142
And u2The subsystem (17) can be regarded as a single-input and multi-output under-actuated system processing by being regarded as an external variable.
To eliminate
Figure BDA0001986376420000143
Control amount u in1Introducing a new state variable eta instead of
Figure BDA0001986376420000144
Figure BDA0001986376420000145
And (3) obtaining the following results by performing derivation on two sides of the step (21):
Figure BDA0001986376420000146
the subsystem (17) can thus be converted into the following form:
Figure BDA0001986376420000147
for the subsystem (23), define
Figure BDA0001986376420000148
The sliding mode function is designed as follows
s1＝μ1-Mμ2 (25)
Wherein M is [ M ]1,m2]Is a sliding mode surface matrix
Order to
Figure BDA0001986376420000149
Then
Figure BDA00019863764200001410
The control law of the subsystem (23) can be designed as follows:
Figure BDA00019863764200001411
in the formula, ρ1，1Is s is1Corresponding exponential approach term and constant velocity approach term, and satisfies rho1,1＞0。
Define the lyapunov function as:
Figure BDA00019863764200001412
then
Figure BDA00019863764200001413
The sliding mode arrival condition is met, and the time t exists1For t ≧ t1，s1＝μ1-Mμ 20, i.e. at the slip-form face s1On 0, there is μ1＝Mμ2。
If appropriate, M is selected so that when μ2→ 0,. mu.l1→ 0, then the control objective is achieved. Taking into account the point of equilibrium
Figure BDA0001986376420000151
In the vicinity of the location of the mobile station,
Figure BDA0001986376420000152
then:
Figure BDA0001986376420000153
wherein, DeltaμIs a very small real number vector, substituted
Figure BDA0001986376420000154
It is possible to obtain,
Figure BDA0001986376420000155
in the formula (I), the compound is shown in the specification,
Figure BDA0001986376420000156
in view of
Figure BDA0001986376420000157
And u2Are all considered as external variables and have a time t1For t ≧ t1Is provided with
Figure BDA0001986376420000158
Then if the vector M can be designed such that the matrix a isμHurwitz, then μ2Can reach zero point and keep stable in a limited time, thereby leading to mu1→ 0, subsystem (17) is exponentially stable.
In view of
Figure BDA0001986376420000159
If A is takenμHas a characteristic value of-2, then is composed of (s +2)2＝s2+4s +4, the correspondence can be obtained as follows:
Figure BDA00019863764200001510
thus, a solution for M is obtained:
Figure BDA00019863764200001511
3. solving the value of M based on a teaching and learning algorithm improved by a teaching strategy;
the sliding mode controller of the liquid-filled spacecraft is at a balance point
Figure BDA00019863764200001512
Designed on a near basis, with small deviationsA very good quality, but an effective control performance is not necessarily guaranteed within a large deviation range. This is because the matrix A cannot necessarily be used for large deviations of M calculated according to the small deviation modelμIs Hurwitz. Therefore, in order to make the under-actuated system stable under more severe conditions, the control law should be designed in consideration of the design of M to make μ when the deviation is large2Is exponentially stable.
Can obtain the product
Figure BDA0001986376420000161
Known as a Hurwitz matrix, then the design for M is not as Aμ→0As a reference. In order to obtain a proper M in the whole control process, the invention adopts a teaching strategy to improve a teaching and learning algorithm for calculation and dynamically adjusts the M1,m2]To ensure mu1，μ2Is index stable globally.
The value of M is taken as the solution target,
Figure BDA0001986376420000162
as input, a strategy improvement teaching and learning algorithm is utilized for solving. The purpose being dependent on the current time
Figure BDA0001986376420000163
Calculating a suitable M (t) such that
Figure BDA0001986376420000164
Can be approximated to Aμ→0μ2(t) of (d). Definition of
Figure BDA0001986376420000165
The fitness function may be taken as
Figure BDA0001986376420000166
Setting the iteration number G as 20 and the population size NP as 20, dividing the population into 3 groups on average, and applying the strategyAnd slightly improving the teaching and learning algorithm to obtain the optimal solution M (t) at the current moment.
Considering that the variation curve of M calculated by SMTLBO is not necessarily two smooth curves, but has certain error and high-frequency oscillation around two photochemical curves. Therefore, after M at the current time is obtained through SMTLBO, an averaging filtering unit should be added to reduce or even eliminate the oscillation phenomenon.
The specific embodiment is as follows:
simulation parameters in the spacecraft dynamics equations (8) to (11) of the liquid sloshing are selected as follows:
m＝600kg，mf＝100kg，I＝720kg·m-2
If＝90kg·m-2，a＝0.32m，b＝0.25m
T＝500N，
Figure BDA0001986376420000168
assume initial values of:
θ0＝2°，
Figure BDA0001986376420000167
vx0＝25m/s
vz0＝1.1m/s，
Figure BDA0001986376420000171
the controller parameters are set as:
c2＝5，ρ2＝5，2＝0.5，ρ1＝2，1＝0.2
when in use
Figure BDA0001986376420000172
The simulation results are shown in fig. 6 to 12. Wherein, fig. 6 is vxGraph of variation, v in FIG. 7zGraph of variation, FIG. 8 is θ and
Figure BDA0001986376420000173
graph of change of (1), figure9 is
Figure BDA0001986376420000174
And
Figure BDA0001986376420000175
FIG. 10 is a graph showing a control amount u1Fig. 11 shows the sliding mode quantity s1In FIG. 12 is s1Phase plan view of (a). From these figures we can see that the designed control law can perform well the control task. v. ofxThe acceleration is regarded as an external variable and acts for acceleration movement under the thrust along the x axis, and the magnitude of the acceleration is basically not influenced, so that the simulation result is consistent with the theoretical analysis. Since will contain
Figure BDA0001986376420000176
The subsystem (18) is regarded as a sliding mode controller of a full-drive system design, and the system has simple structure, so that the sum of theta and theta is equal to the sum of theta
Figure BDA0001986376420000177
The control effect of (2) is good. For containing
Figure BDA0001986376420000178
The system reaches a longer equilibrium state because the controller is designed to have (mu) in addition to the sliding mode arrival process1，μ2) The process of (1) stabilization. In the control process, the state variables
Figure BDA0001986376420000179
The deviation is first off the equilibrium point, but still within an acceptable range, and does not compromise the stability of the control. The state variable change curves of fig. 6-9 demonstrate the effectiveness of this control method. And from s1Graph 11 and phase plan 12 show that the sliding mode number s is at the sliding mode arrival stage because M calculated by SMTLBO has a certain oscillation1There is also an oscillation phenomenon, but s1There is still a major trend and oscillations are attenuated as the slip-form surfaces approach. And from phase plane 12It can be seen that the slip form surface s of the present invention is designed1Is a non-linear sliding mode surface, and (mu)1,μ2) Finally, the sliding mode can fall on the nonlinear sliding mode surface, and the sliding mode reaching condition of the subsystem (17) is met.
As can be seen from fig. 13, although the averaging filter is used to eliminate the high frequency oscillation, the obtained M still has an oscillation phenomenon, and has a certain error compared with the M calculated by equation (31). The oscillation phenomenon is mainly generated by the input at each moment
Figure BDA00019863764200001710
In contrast, the algorithm itself has a small accuracy error, and therefore, the high frequency oscillation is represented by the value of M. And for m with larger amplitude2This is due to the adaptation value pair m2Is not sensitive to the change in size of m, thereby affecting m2The calculation accuracy of (2). The error is generated because the mean filtering method cannot achieve the ideal filtering effect for M calculated by SMTLBO. The high frequency oscillation in M has a certain limitation, not the conventional noise, so the generation of error is inevitable. But even if there is a certain oscillation and error, A calculated from MμStill Hurwitz, mu1And mu2The self-stabilization process can still be completed, which also indicates that the method adopted by the invention is feasible.
To verify at
Figure BDA00019863764200001711
Under the condition of large deviation from the balance point, the control method adopted by the invention can still complete the control target, and the invention respectively makes
Figure BDA00019863764200001712
And
Figure BDA00019863764200001713
the simulation results of the simulation experiments of (1) are shown in fig. 14 to 22.
FIGS. 14 to 18 are graphs showing differences
Figure BDA0001986376420000181
In this case, the change profile of each system variable is shown. Wherein, FIG. 14 is different
Figure BDA0001986376420000182
In case vzA graph of variation of (d); FIG. 15 is a drawing showing the difference
Figure BDA0001986376420000183
Under the circumstances
Figure BDA0001986376420000184
The variation curve of (d); FIG. 16 is a drawing showing the difference
Figure BDA0001986376420000185
Under the circumstances
Figure BDA0001986376420000186
The variation curve of (d); FIG. 17 is a variation
Figure BDA0001986376420000187
In case of s1The variation curve of (d); FIG. 18 is a drawing showing the difference
Figure BDA0001986376420000188
In case u1The variation curve of (d); FIG. 19 is a drawing showing
Figure BDA0001986376420000189
The variation curve of M; FIG. 20 is a drawing showing
Figure BDA00019863764200001810
When s is1A phase plane view of; FIG. 21 is a drawing showing
Figure BDA00019863764200001811
The variation curve of M; FIG. 22 is a drawing showing
Figure BDA00019863764200001812
When s is1Phase plan view of (a). It can be seen from the figure that the difference is
Figure BDA00019863764200001813
Under the condition, the designed control law can complete the control target. Except that s1Because the relation of the sliding mode surface parameter M can generate oscillation, the other variables are not subjected to oscillation in the dynamic response process. The state variables deviate from the equilibrium point first, but the deviation is not much within an acceptable range, and the stability of the system is not affected. FIGS. 19 to 22 are graphs showing the difference
Figure BDA00019863764200001814
In this case, the variation curve of M and the phase plane diagram. It can be seen that the change of M is regular and finally converges
Figure BDA00019863764200001815
Nearby. While the phase plan still reflects: although buffeting is generated on the slip form surface in the dynamic response process, the state is not influenced to reach the slip form surface and the slip form motion is kept, and meanwhile, the buffeting of the slip form surface is smaller as the dynamic slip form surface is closer to the dynamic slip form surface. The numerical simulation result shows that the designed controller can effectively control the system under the condition of large deviation, the effectiveness of the method for dynamically designing the control law by utilizing the SMTLBO algorithm is proved, and the method can be suitable for the design of the control law of the complex underactuated system.
Although the preferred embodiments of the present invention have been described in detail, the present invention is not limited to the details of the foregoing embodiments, and various equivalent changes (such as number, shape, position, etc.) may be made to the technical solution of the present invention within the technical spirit of the present invention, and the equivalents are protected by the present invention.

Claims (2)

1. A liquid-filled spacecraft optimization sliding mode control method based on teaching strategy improved teaching and learning algorithm is characterized in that: the method comprises the following steps:
1) the liquid-filled spacecraft is equivalent to a simple pendulum model, the x axis and the z axis are inertial coordinate systems, and the liquid-filled spacecraft has constant thrustForce T, axial speed v along the machine bodyx(ii) a The dry weight of the liquid-filled spacecraft is m, and the mass of the fuel part in the storage tank is mf(ii) a The moment of inertia of the liquid-filled spacecraft relative to the center of the tank is I, and the moment of inertia of the fuel relative to the center of the tank is If(ii) a The distance from the mass center of the spacecraft to the suspension point of the simple pendulum is b, and the pendulum length of the simple pendulum is a; the attitude angle of the spacecraft is theta, and the liquid oscillation angle, namely the equivalent simple pendulum swing angle is theta
Figure FDA0002700281010000019
The controlled inputs of the system are transverse control force F and pitching moment M of the mass center of the spacecrafty，vzA lateral velocity generated for influence by a control input;
Figure FDA00027002810100000110
is the energy dissipation factor;
2) obtaining a liquid-filled spacecraft dynamics equation of liquid sloshing:
Figure FDA0002700281010000011
Figure FDA0002700281010000012
Figure FDA0002700281010000013
Figure FDA0002700281010000014
equation (6) is simplified to:
Figure FDA0002700281010000015
order to
Figure FDA0002700281010000016
Comprises the following steps:
Figure FDA0002700281010000017
wherein the content of the first and second substances,
Figure FDA0002700281010000018
the control inputs F and M of the system are transformed via equation (11)yConversion to u1And u2Accordingly, formulae (7) to (9) are converted into the following forms:
Figure FDA0002700281010000021
Figure FDA0002700281010000022
Figure FDA0002700281010000023
wherein the content of the first and second substances,
Figure FDA0002700281010000024
3) designing a sliding mode controller with the controller designed to target system state variables
Figure FDA00027002810100000211
Reach zero in a finite time; divide the system into
Figure FDA00027002810100000212
And an under-actuated subsystem (15) comprising
Figure FDA00027002810100000213
Full drive subsystem (16):
Figure FDA0002700281010000025
Figure FDA0002700281010000026
for the subsystem (16), a control law is designed according to a general sliding mode control method:
Figure FDA0002700281010000027
Figure FDA0002700281010000028
in the formula, c2Is a sliding mode surface parameter, rho, of the subsystem (16)2And2for subsystem sliding mode surface s2Corresponding exponential and isovelocity approach terms, and satisfies c2，ρ2，2＞0；
For the sub-system (15), a new state variable eta is introduced instead of
Figure FDA00027002810100000214
Figure FDA0002700281010000029
The two sides of the formula (19) are subjected to derivation to obtain:
Figure FDA00027002810100000210
thereby converting the subsystem (15) into the form:
Figure FDA0002700281010000031
for the subsystem (21), define
Figure FDA0002700281010000038
The sliding mode function is designed as follows
s1＝μ1-Mμ2 (23)
Wherein M is [ M ]1，m2]Is a sliding mode surface matrix
Order to
Figure FDA0002700281010000039
Then
Figure FDA0002700281010000032
The control law of the subsystem (21) is designed as follows:
Figure FDA0002700281010000033
in the formula, ρ1，1Is s is1Corresponding exponential approach term and constant velocity approach term, and satisfies rho1，1＞0；
Define the lyapunov function as:
Figure FDA0002700281010000034
then
Figure FDA0002700281010000035
The sliding mode arrival condition is met, and the time t exists1For t ≧ t1，s1＝μ1-Mμ20, i.e. at the slip-form face s1On 0, there is μ1＝Mμ2；
Then:
Figure FDA0002700281010000036
wherein, DeltaμIs a very small real number vector, substituted
Figure FDA00027002810100000310
It is possible to obtain,
Figure FDA0002700281010000037
in the formula (I), the compound is shown in the specification,
Figure FDA0002700281010000041
the solution to M is:
Figure FDA0002700281010000042
4) the teaching and learning algorithm improved based on the teaching strategy for solving the value of M comprises the following steps:
41) initializing the class: randomly generating each student in a class in a search space
Figure FDA0002700281010000043
The generation method is carried out according to the following formula:
Figure FDA0002700281010000044
wherein the content of the first and second substances,
Figure FDA00027002810100000419
and
Figure FDA00027002810100000420
respectively, as the upper and lower bounds of each dimension, where i ═ 1,2, …, d
42) Dividing the class member into num groups, evaluating the adaptive value of the kth group, selecting the individual with the optimal adaptive value as the teacher individual of the kth group
Figure FDA00027002810100000421
The teaching process of the individual i in the kth group member is represented by the following formula (2) and formula (3):
Figure FDA0002700281010000045
Figure FDA0002700281010000046
in the formula:
Figure FDA0002700281010000047
and
Figure FDA0002700281010000048
respectively representing the values before and after learning of the ith student of the k groups; k is 1,2, …, num;
wherein the content of the first and second substances,
Figure FDA0002700281010000049
is the mean value of the kth subgroup, TFi k＝1+rand(0，0.5) Is a teaching factor of the teacher, ri kRand (0.5,1) is the learning step length of the student;
43) for students
Figure FDA00027002810100000410
Randomly selecting a member from the group
Figure FDA00027002810100000411
Figure FDA00027002810100000412
Analyze oneself with
Figure FDA00027002810100000413
The difference between the two is used for learning adjustment;
Figure FDA00027002810100000414
wherein the content of the first and second substances,
Figure FDA00027002810100000415
in order to learn the values of the trainee after adjustment,
Figure FDA00027002810100000416
adjusting the values of the trainees prior to learning;
44) for individual teacher
Figure FDA00027002810100000417
Randomly selecting another teacher individual
Figure FDA00027002810100000418
Carrying out learning adjustment;
44.1) let k equal 1 for individual teacher
Figure FDA0002700281010000051
Randomly selecting another oneIndividual teacher
Figure FDA0002700281010000052
44.2) let y equal 1, use
Figure FDA0002700281010000053
Is substituted by the y-component
Figure FDA0002700281010000054
The y-th dimension component of (a) to generate a test solution if the adaptation value of the test solution is better than that of the test solution
Figure FDA0002700281010000055
It is replaced;
44.3) making y equal to y +1, repeating the step 44.2) until y equal to d, and finishing the inter-learning;
44.4) let k be k +1, repeat steps 44.2) and 44.3) until k is num, then the learning exchange of all teachers is finished;
44.5) comparing the adaptive values of all teachers, and taking the teacher with the optimal adaptive value as the global optimal solution Xbest；
5) Self-learning by the whole staff: for any individual XiThe operation steps of the self-learning process are as follows:
Figure FDA0002700281010000056
wherein randn is a normal distribution, and wherein,
Figure FDA0002700281010000057
XU，XLrespectively representing the maximum value and the minimum value of the actual individual of the population; e is the maximum allowable iteration number, and t is the current iteration number;
6) judging whether a termination condition is met, if so, terminating the algorithm and outputting an optimal solution; otherwise, jumping to the step 2), and continuing the iterative computation.
2. The liquid-filled spacecraft optimization sliding-mode control method according to claim 1, characterized in that: in the step 4), the sliding mode controller of the liquid-filled spacecraft is at a balance point
Figure FDA00027002810100000515
Designed on a nearby basis to obtain
Figure FDA0002700281010000058
The value of M is taken as a solving target, vz，
Figure FDA0002700281010000059
As input, v according to the current timez(t)，
Figure FDA00027002810100000510
Figure FDA00027002810100000511
Calculating to obtain M (t) such that
Figure FDA00027002810100000512
Can be approximated to Aμ→0μ2(t); definition of
Figure FDA00027002810100000513
The fitness function is taken as
Figure FDA00027002810100000514
Setting the iteration number G to be 20 and the population size NP to be 20, averagely dividing the population into num to be 3 groups, and obtaining the optimal solution M (t) at the current time.
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