**5. Conclusion**

46 Numerical Simulation – From Theory to Industry



20


20


0

0

e2

e2


e2

0

e2 dot [rad/s2]

e2 dot [rad/s2]

e2 dot [rad/s2]

**Figure 6.** Error Dynamics of the Pendulum (a) 2*e t* ( ) in (22), (b) 2*e t* ( ) in (34), and (c) 2*e t* ( ) in (35)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

(b)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

(c)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time [sec]

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

(b)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

(c)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time [sec]

(a)

(a)

**Figure 7.** Velocity Errors from (a) 2*e t*( ) from (22), (b) 2*e t*( ) in (34), and (c) 2*e t*( ) in (35)

A tracking control of a model-based linear time-varying system is developed in application to the nonlinear inverted pendulum model. A novelty of this paper is that not only found a gramian matrix which is difficult to find or compute but also utilized to the linear timevarying tracking controller which satisfies the necessary and sufficient of the global stability of the system. Another is that the linear time-varying system is further complicated by parametric uncertainty where the combined parameters are unknown. The suggested adaptive control approach and update laws are applied for estimating the parameters while preserving the system to be stable and converging the tracking error close to zero. Numerical simulation results are demonstrated the validity of the proposed system.
