## 5 Simulation Results and Its Discussion  
In order to study the proposed degree of controllability for specific state variables, we used an example of the stabilization models of a pendulum on a carriage. We studied the value of the degree of controllability and the control performance of the system under different system model parameters. The inverted pendulum model is shown in Figure 1.  
Figure 1 introduces the following symbols: \(m\)--pendulum mass; \(M\)--carriage mass; \(f\)--force applied by the motor to the carriage through the belt; \(2l\)--pendulum length; \(l\)--distance from the pivot point to the center of mass of the pendulum; \(I\)--moment of inertia of the pendulum; \(\theta\left(t\right)\)--the deviation angle of the pendulum from the vertical, in the position of unstable equilibrium is zero, increases clockwise; \(x\left(t\right)\)--carriage position. Near the equilibrium point, the nonlinear model can be linearized as  
\[ \begin{array}{l}\left(M+m\right)\ddot{x}+ml\ddot{\theta}\approx f;\\ lm\ddot{x}+\left(I+ml^{2}\right)\ddot{\theta}\approx lmg\theta\end{array}  \tag{19}\]  
The discrete state-space model of the system has the form  
\[ \begin{array}{l}\mathbf{x}_{k+1}=\mathbf{F}\mathbf{x}_{k}+\mathbf{B}f_{k};\\ \mathbf{y}_{k+1}=\mathbf{C}\mathbf{x}_{k}\end{array}  \tag{20}\]  
Figure 1: Inverted pendulum model. where \(F\) is state transition matrix, \(B\) is control matrix, \(C\) is output matrix, \(f_{k}\) is input,  
\[\mathbf{x}_{k}=\left[ \begin{array}{c}x_{k}\\ \dot{x}_{k}\\ \theta_{k}\\ \dot{\theta}_{k}\end{array} \right],\mathbf{F}= \begin{bmatrix}1&\Delta T&0&0\\ 0&1&\frac{m^{2}l^{2}g\Delta T}{D}&0\\ 0&0&1&\Delta T\\ 0&0&\frac{ml(M+m)\Delta T}{D}&1\end{bmatrix} ,\mathbf{B}= \begin{bmatrix}0\\ \frac{l+ml^{2}}{D}\Delta T\\ 0\\ \frac{ml}{D}\Delta T\end{bmatrix} ,\mathbf{C}= \begin{bmatrix}1&0&0&0\\ 0&0&1&0\end{bmatrix} .\]  
Here \(\Delta T\) is sample time, \(\Delta T=1\)s [37, 38]; \(D= \begin{vmatrix}M+m&ml\\ ml&I+ml^{2}\end{vmatrix} =\left(I+ml^{2}\right)(M+m)-m^{2}l^{2}\); \(I=\frac{4}{3}ml^{2}\). The degree of controllability for each component:  
\[  
Then, we can check the control performance under different conditions (different parameters of the system model \(m\), \(l\)):  
(1) Carriage position \(x\) response process (Figure 4)  
(2) Pendulum angle \(\theta\) response process (Figure 5)  
The simulation results confirm the relationship between the degree of controllability \(DoC^{i}\) and the change in the parameter \(m\). When \(m\) increases, the degree of controllability \(DoC^{i}\) becomes smaller, the initial states of the system \(x_{0}\) converges more slowly and the amplitude becomes larger. Figure 4: Carriage position \(x\) at different \(m\). Figure 3: Change in the degree of controllability for \(\theta\), \(\dot{\theta}\) when increasing the mass of the pendulum \(m\). Figure 2: Change in the degree of controllability for \(x\), \(\dot{x}\) when increasing the mass of the pendulum \(m\). If other parameters are fixed, and only \(l\) changes, then  
Figures 6 and 7 show that with an increase in the length of the pendulums \(l\), the value \(\frac{R}{R^{x}l}\) decreases, degree of controllability \(DoC^{i}\) decreases, i.e., it is harder to control the components of the system state vector. Figure 5: Pendulum angle \(\theta\) at different \(m\). Figure 6: Change in the degree of controllability for \(x\), \(\dot{x}\) when increasing the length of the pendulum \(l\). Figure 7: Change in the degree of controllability for \(\theta\), \(\dot{\theta}\) when increasing the length of the pendulum \(l\). The control performance is shown in Figures 8 and 9. (1) Carriage position x response process  
Obviously, with a relatively small length of the rod \(l\), the initial state \(x_{0}\) of the system would quicker converge to 0 and the amplitude of the response process would be smaller, i.e., the system becomes easier to control. The settling time and the rise time of the system become shorter. The simulation results could be explained by real physical principles.