LQR、LQR-MPC、GP-MPC控制倒立摆

Robert--cao

2024-07-11 帮助1人

LQR控制倒立摆：

倒立摆状态方程：

学新通

目标任务：

学新通

模型参数：

学新通

%% LQR for the cart-pole system
load('cp_params.mat');
syms phi phid x xd I u
fr(1) = 0.2; % friction magnitude
Ts = 0.01;
Duration = 2.5;
Q = [100 0 1 0]; % LQ weights
R = 0.0001;
q1 = [phi; phid; x; xd];
q1 = cp_dynmodel(q1,u,par,fr);
Asym = jacobian(q1,[phi;phid;x;xd]);
A = double(subs(Asym,[phi;phid;x;xd;u],[0;0;0;0;0]));
Bsym = jacobian(q1,u);
B = double(subs(Bsym,[phi;phid;x;xd;u],[0;0;0;0;0]));
A = eye(4) Ts*A;
B = Ts*B;
clear phi phid x xd I u k1 k2 k3 k4 q1
%% LQR
C = diag([1,1,1,1]);
D = zeros(4,1);
K = dlqr(A,B,diag(Q),R);
%% Simulation
qLQR = zeros(5,Duration/Ts 1);
q = [0.2;0;0;0];
qLQR(:,1) = [q; 0];
for i = 1 : Duration/Ts
u = -K*q;
[~,q] = ode45(@(t,q) cp_dynmodel(q,u,par,fr),[0 Ts/2 Ts],q);
q = q(3,:)';
qLQR(:,i 1) = [q; u];
end
%% Plots
plotTraj(qLQR,Ts)

首先非线性模型通过雅可比矩阵线性化：

Asym = jacobian(q1,[phi;phid;x;xd]);
A = double(subs(Asym,[phi;phid;x;xd;u],[0;0;0;0;0]));
Bsym = jacobian(q1,u);
B = double(subs(Bsym,[phi;phid;x;xd;u],[0;0;0;0;0]));

离散化模型：

A = eye(4) Ts*A;
B = Ts*B;

通过LQR计算出反馈矩阵K值：

K = dlqr(A,B,diag(Q),R);

仿真，通过

%% Simulation
qLQR = zeros(5,Duration/Ts 1);
q = [0.2;0;0;0];
qLQR(:,1) = [q; 0];
for i = 1 : Duration/Ts
u = -K*q;
[~,q] = ode45(@(t,q) cp_dynmodel(q,u,par,fr),[0 Ts/2 Ts],q);
q = q(3,:)';
qLQR(:,i 1) = [q; u];
end
%% Plots
plotTraj(qLQR,Ts)

结果：

学新通

LQR-MPC控制倒立摆：

这里就是使用mpc产生输入补偿LQR输入：

input = fmincon(@(u) cost_func_lin(u,horizon,q,Q,R,A2,B),input,[],...
[],[],[],(K*q-40)*ones(horizon,1),(K*q 40)*ones(horizon,1),[],options);
u = -K*q input(1);

%% MPC for the cart-pole system
load('cp_params.mat');
syms phi phid x xd I u
fr(1) = 0.2; % friction magnitude
Ts = 0.01;
horizon = 25;
Duration = 10;
Q = [100 0 1 0]; % MPC weights
R = 0.0001;
q1 = [phi; phid; x; xd];
q1 = cp_dynmodel(q1,u,par,fr);
Asym = jacobian(q1,[phi;phid;x;xd]);
A = double(subs(Asym,[phi;phid;x;xd;u],[0;0;0;0;0]));
Bsym = jacobian(q1,u);
B = double(subs(Bsym,[phi;phid;x;xd;u],[0;0;0;0;0]));
A = eye(4) Ts*A;
B = Ts*B; %离散化
clear phi phid x xd I u k1 k2 k3 k4 q1
%% LQ feedback gain
C = diag([1,1,1,1]);
D = zeros(4,1);
K = dlqr(A,B,diag(Q),R);
%% Simulation
A2 = A-B*K;
u = 0;
qlinMPC = zeros(5,Duration/Ts 1);
q = [0.2; 0; 0; 0]; % initial condition
qlinMPC(:,1) = [q; u];
input = zeros(horizon,1);
options = optimoptions('fmincon','Algorithm','sqp');
for i = 1: Duration / Ts
input = fmincon(@(u) cost_func_lin(u,horizon,q,Q,R,A2,B),input,[],...
[],[],[],(K*q-40)*ones(horizon,1),(K*q 40)*ones(horizon,1),[],options);
u = -K*q input(1);
[~,q] = ode45(@(t,q) cp_dynmodel(q,u,par,fr),[0 Ts/2 Ts],q);
q = q(3,:)';
qlinMPC(:,i 1) = [q; u];
end
%% Plots
plotTraj(qlinMPC,Ts)

学新通

关于GPR参考资料：

Gaussian process 的重要组成部分——关于那个被广泛应用的Kernel的林林总总 - 知乎

Documentation for GPML Matlab Code

%% GPMPC for cart-pole dynamic system

addpath '..\TDK_2020\gp' % write path of the containing folder
addpath '..\TDK_2020\util'
load('cp_params.mat');
load('cp_trainingpoints.mat');
fr(1) = 0.2;
Ts = 0.01;
horizon = 25;
Duration = 2.5;
Q = [100 0 1 0];
R = 0.0001;

%% GP training
gpmodel.inputs = xtrain';
gpmodel.targets = ytrain';
[gpmodel2, nlml] = train(gpmodel, 1, 100);
%% Precomputation of GP matrices and vectors
N = length(xtrain(1,:));
L = zeros(4,4,4);
Kxx = zeros(N,N,4);
Kinv = Kxx;
alpha = zeros(N,1,4);
sf = 0.1;
sn = 0.01;
for k = 1:4
L(:,:,k) = diag(1./exp(2*gpmodel2.hyp(1:4,k)));

for i=1:N
for j=1:N
Kxx(i,j,k)=sf^2*kernel(xtrain(:,i),xtrain(:,j),L(:,:,k));
end
end
alpha(:,:,k) = (Kxx(:,:,k) sn^2*eye(N))\ytrain(k,:)';
Kinv(:,:,k) = inv(Kxx(:,:,k) sn^2*eye(N));
end

%% Discretisation, linearisation
syms phi phid x xd I u

q1 = [phi; phid; x; xd];
q1 = cp_dynmodel(q1,u,par,fr);

Asym = jacobian(q1,[phi;phid;x;xd]);
A = double(subs(Asym,[phi;phid;x;xd;u],[0;0;0;0;0]));
Bsym = jacobian(q1,u);
B = double(subs(Bsym,[phi;phid;x;xd;u],[0;0;0;0;0]));
A = eye(4) Ts*A;
B = Ts*B;
clear phi phid x xd I u k1 k2 k3 k4 q1

%% LQR
C = diag([1,1,1,1]);
D = zeros(4,1);
K = dlqr(A,B,diag(Q),R);

%% GPMPC Simulation
A2 = A-B*K;
q = [0.2; 0; 0; 0]; % initial condition
qGPMPC = zeros(5,Duration/Ts 1);
qGPMPC(1:4,1) = q;
input = zeros(horizon,1);
options = optimoptions('fmincon','Algorithm','sqp');
for i = 1: Duration / Ts
input = fmincon(@(u) cost_func_GP(u,horizon,q,Q,R,A2,B,xtrain,L,alpha,Kinv),...
input,[],[],[],[],(K*q-40)*ones(horizon,1),(K*q 40)*ones(horizon,1),[],options);
u = -K*q input(1);
[~,q] = ode45(@(t,q) cp_dynmodel(q,u,par,fr),[0 Ts/2 Ts],q);
q = q(3,:)';
qGPMPC(:,i 1) = [q; u];
end

%% Plots
plotTraj(qGPMPC,Ts)

%% Plot the prediction at a certain time instant and the simulation
% results at that time interval

niter = horizon;
start = 12; % t = 0.12 s
mean = zeros(4,niter 1);
mean(:,1) = qGPMPC(1:4,start);
u = qGPMPC(5,start);

Ktest = zeros(1,length(xtrain(1,:)));
M = zeros(4,niter);
S = M;
N = length(xtrain(1,:));

for iter = 1:niter
for k = 1:4
for j=1:N
Ktest(1,j,k)=sf^2*kernel(mean(:,iter),xtrain(:,j),L(:,:,k));
end
M(k,iter) = Ktest(:,:,k)*alpha(:,:,k);
S(k,iter) = sf^2 - Ktest(:,:,k)*Kinv(:,:,k)*Ktest(:,:,k)';
end
mean(:,iter 1) = A*mean(:,iter) B*u M(:,iter);
end
tsimu = start*Ts:Ts:(start horizon-1)*Ts;
figure()
subplot(2,1,1)
hold on
plot(tsimu,qGPMPC(1,start:start niter-1),'r','LineWidth',1)
errorbar(tsimu,mean(1,1:25),2*sqrt(S(1,:)),'k','LineWidth',1)
xlabel('$t$ (s)')
ylabel('$\varphi$ (rad)')
legend('Simulation result','Prediction at $t=0.12$ s')
hold off

subplot(2,1,2)
hold on
plot(tsimu,qGPMPC(3,start:start niter-1),'r','LineWidth',1)
errorbar(tsimu,mean(3,1:25),2*sqrt(S(3,:)),'k','LineWidth',1)
xlabel('$t$ (s)')
ylabel('$x$ (m)')
hold off
具体代码：

MPC/LQR. LQR-MPC.GP-MPC.rar at main · caokaifa/MPC · GitHub

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LQR、LQR-MPC、GP-MPC控制倒立摆

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