using linear observer on a nonlinear model

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liam 2019-5-2

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/459914-using-linear-observer-on-a-nonlinear-model

I'm trying to use a linear luenberger observer

to approximate the motion of an inverted pendulum on a cart grpahically with ODE45:

With states θ and

and controller baed on the estimated states:

. The controller was designed from linearisation about

. The dynamics of my true state and estimated state should converge when the released from an initial displacement near π but this isn't happening. The code is:

A=[0 1; 1 0];

B=[0; 1];

C=[1 0];

T0=pi+0.1; %The initial angle of displacement

s1=[-1,-2]; %poles for the true dynamics

s2=[-2,-4]; %poles for the observer

K=place(A,B,s1)

L=place(A',C',s2)'

f= @(t,x) [x(2); -sin(x(1))+[K(1,1)*(x(3)-pi)+K(1,2)*x(4)]*cos(x(1))

;L(1,1)*(x(1)-pi)-L(1,1)*(x(3)-pi)+x(4);

L(2,1)*(x(1)-pi)+(1-L(2,1)-K(1,1))*(x(3)-pi)-K(1,2)*x(4)];

[t,x]= ode45(f,[0 20],[T0 0 0 0]);

plot(t,x(:,1),t,x(:,3),'--')

Where x(1) is θ, x(2) is

, x(3) is

and x(4) is

. The ODE45 was obatined from:

$\begin{bmatrix} \dot{x} \\ \dot{\hat{x}} \end{bmatrix} = \begin{bmatrix} A & -BK \\ LC & A-LC-BK \end{bmatrix} \begin{bmatrix} x \\ \hat{x} \end{bmatrix}$