问题描述

参考文献：[1] Meditch J. On the problem of optimal thrust programming for a lunar soft landing[J]. IEEE Transactions on Automatic Control, 1964, 9(4): 477-484.

考虑一类月球探测器着陆最优控制问题，$x$为月球探测器与月心的矢径，其动力学方程为

$$\{\begin{array}{rl}\ddot{x}& =-k\frac{\text{d}}{\text{d}t}(\ln m)-g,\\ \dot{x}& =-k\ln \frac{m(t)}{m(0)}-gt+x(0).\end{array}$$

令$x_1=h$，$x_2=v$，$x_3=m$，可以得到

$${\dot{x}}_1=v,{\dot{x}}_2=-\frac{k}{x_3}u-g,{\dot{x}}_3=m,u=\dot{m}.$$

性能指标如下：

$$J={\int }_{t_0}^{t_f}\dot{m}(t)\text{d}t=m(0)-m(t_f)$$

化简之后，月球探测器落地动力学方程为

$$\{\begin{array}{rl}{\dot{x}}_1& =v,\\ {\dot{x}}_2& =u-g.\end{array}$$

化简之后的性能指标为

$$J={\int }_{t_0}^{t_f}u\text{d}t$$

GPOPS代码

main function

%% 01.初始参数设置
%-------------------------------------------------------------------------%
%----------------------- 设置问题的求解边界 ------------------------------%
%-------------------------------------------------------------------------%
% 设置时间边界
t0min = 0;
t0max = 0;
tfmin = 0;
tfmax = 10;% 设置状态初值
h0 = 10;
v0 = -2;
% 设置状态终值
hf = 0;
vf = 0;% 设置状态边界
hmin = 0;
hmax =  20;
vmin = -10;
vmax =  10;% 设置控制量边界
umin = 0;
umax = 3;% 月球重力常数
setup.auxdata.g = 1.6;
%% 02.边界条件设置
%-------------------------------------------------------------------------%
%------------------------ 将求解边界设置于问题中 -------------------------%
%-------------------------------------------------------------------------%
setup.bounds.phase.initialstate.lower = [h0, v0];
setup.bounds.phase.initialstate.upper = [h0, v0];
setup.bounds.phase.state.lower = [hmin, vmin];
setup.bounds.phase.state.upper = [hmax, vmax];
setup.bounds.phase.finalstate.lower = [hf, vf];
setup.bounds.phase.finalstate.upper = [hf, vf];
setup.bounds.phase.control.lower = umin;
setup.bounds.phase.control.upper = umax;
setup.bounds.phase.integral.lower = -100;
setup.bounds.phase.integral.upper = 100;
setup.bounds.phase.initialtime.lower = t0min;
setup.bounds.phase.initialtime.upper = t0max;
setup.bounds.phase.finaltime.lower = tfmin;
setup.bounds.phase.finaltime.upper = tfmax;%% 03.初值猜测
%-------------------------------------------------------------------------%
%------------------------------- 初值猜想 --------------------------------%
%-------------------------------------------------------------------------%
h_guess = [h0; hf];
v_guess = [v0; vf];
u_guess = [umin; umin];
t_guess = [t0min; 5];
setup.guess.phase.state   = [h_guess, v_guess];
setup.guess.phase.control = u_guess;
setup.guess.phase.time    = t_guess;
setup.guess.phase.integral = 10;%% 04.设置GPOPS求解器参数
%-------------------------------------------------------------------------%
%---------------------------- 设置求解器参数 -----------------------------%        
%-------------------------------------------------------------------------%
setup.name = 'Moon-Lander-OCP';
setup.functions.continuous  = @mlocpContinuous;
setup.functions.endpoint   	= @mlocpEndpoint;
setup.nlp.solver            = 'ipopt';
setup.derivatives.supplier  = 'sparseCD';
setup.derivatives.derivativelevel = 'second';
setup.mesh.method           = 'hp1';
setup.mesh.tolerance        = 1e-4;
setup.mesh.maxiteration     = 45;
setup.mesh.colpointsmax     = 4;
setup.mesh.colpointsmin     = 10;
setup.mesh.phase.fraction   = 0.1*ones(1,10);
setup.mesh.phase.colpoints  = 4*ones(1,10);
setup.method = 'RPMintegration';
% setup.method = 'RPMdifferentiation';%% 05.求解
%-------------------------------------------------------------------------%
%----------------------- 使用 GPOPS2 求解最优控制问题 --------------------%
%-------------------------------------------------------------------------%
tic;
output = gpops2(setup);
toc;%% 06.画图
solution = output.result.solution;% 状态量
figure('Color',[1,1,1]);
plot(solution.phase.time, solution.phase.state,'-x','LineWidth',1.5);
xlabel('Time',...'FontWeight','bold');
ylabel('State',...'FontWeight','bold');
legend('Altitude','Velocity')
set(gca,'FontName','Times New Roman',...'FontSize',15,...'LineWidth',1.3);
print -dpng moonlander_state.png% 控制量
figure('Color',[1,1,1]);
plot(solution.phase.time, solution.phase.control,'-o','LineWidth',1.5);
xlabel('Time',...'FontWeight','bold');
ylabel('Control',...'FontWeight','bold');
set(gca,'FontName','Times New Roman',...'FontSize',15,...'LineWidth',1.3);
print -dpng moonlander_control.png

continuous function

function phaseout = mlocpContinuous(input)g = input.auxdata.g;t = input.phase.time;h = input.phase.state(:,1);v = input.phase.state(:,2);u = input.phase.control(:,1);dh = v;dv = -g + u;phaseout.dynamics = [dh, dv];phaseout.integrand = u;
end

endpoint function

function output = mlocpEndpoint(input)J = input.phase.integral;output.objective = J;
end

仿真结果

状态量：

控制量

最后

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