杭州电子科技大学现代控制理论基础课件第四章 控制系统的稳定性（2）.ppt

Modern Control Theory Lecture 12 Modern Control Theory Letures Notes Coryrighr@Xiaodong zhao 4.3 Lyapunov Stability Analysis of Linear System 4.3.1 Lyapunov first method LTI system Theorem 4-6: LTI system is asymptotically stable iff each eigenvalue of A has negative real part: Proof: 2 Linear time varying system Definition : Theorem 4-7: for Linear time varying system 4.3.2 Application of Lyapnov second method LTI system Theorem 4-8: LTI system is asymptotically stable at iff Example 4-8 2. Linear time varying system Theorem 4-9: 3. LDTI system Theorem 4-10: Example 4-10 4. Linear time varying discrete system Theorem 4-11: 5. Adjustment of LTI system using Lyapnov second method Example 4-11 Stabilize of double integrator 6. System dynamic performance Definition: Assume the origin is an equilibrium point and the system is asymptotically stable. The rapidity index is defined as The slowest speed : Transient process time estimate: 7. Optimal control problem Example 4-13 4.4 Nonlinear system analysis 4.4.1 Lyapunov first method (linearlization) Proof: case (3): can not test stability 4.4.2 Lyapunov second method 1. Krasovski method Example 4-14 2. Linear approximation method Example 4-16 * * * * *