Inverted pendulum for simulation

We use an inverted pendulum for simulation, the physical parameters are shown in the following figure:

Based on the knowledge of theoretical mechanics, one obtains the inverted pendulum system dynamic:

Rewrite it as a non-linear state space model :

, where $b$ is the coefficient of friction for cart movement and is set as $0$ in the simulation.

Reformulate the system as and linearize system at . Then one obatins a continuous time LTI system: $\dot{x}(t)=A_c x(t) +B_c u(t)$. Sample the continuous time system with period $T_s$, then the discrete system is $x(k+1)=Ax(k)+Bu(k)$ where

The system output equation is

where $v(k)$ is the output noise and $a(k)$ is the injected attack. There are 3 sensors monitoring the cart position and 1 sensor monitoring the pendulum angle.

The system is simulated with sampling time $T_s=0.1 s$.

The system noise process has covariance and the measurement noise is , i.e., the noise is scaled according to sampling time $T_s$. The initial state is set as $x_0=[0,1,0,1]'$, and the estiamtor is assumed to know the initial state.

Remark on state transformation

In order to better analyze the observability structure, the matrix $A$ is transformed into a Jordan canonical form:

And the simulation is performed on the following system where $\bar{x}=U^{-1}x$:

The estimation of $x(k)$ can be obtained by performing transformation $U$ on the estimation of $\hat{x}(k)$.