文档介绍：Chapter 5 System stability
System stability
Basic concepts of system stability
The stability in sense of Lyapunov
The Lyapunov stability theorem
The Lyapunov second method
The Lyapunov stability analysis of linear system
1. Basic concepts of system stability
If the system is subjected to a bounded input or disturbance and the response is bounded in magnitude, the system is said to be stable.
Absolute stable and relative stable
fighter aircraf t is less stable than mercial transport
1) Outer stability-BIBO
Outer stability: For any system which is casual and relax at time , the relationship between input and output is
Where is the system impulse response matrix
If the input is bounded
And the output is also bounded
Where are all proper positive real numbers
Then we call the system is outer stable (or boundary input boundary output stable system)
Review the criterion for system outer stable
(2) If the system transfer function matrix is proper rational fraction matrix
Each pole of all entries of has negative real part
(3) Time-domain criterion for LTI system
Routh criterion
Hurwitz criterion
(4) Frequency-domain criterion for LTI system
Nyquist criterion
A workable system must be stable and only stable systems could be applied in engineering practice.
Stability is the premise and precondition for system design
Inner stability: Considering the following system
With initial state
Set the input , then the state equation solution/state response
If the solution could satisfy
Then we call the system is asymptotically stable (inner stable)
Note: relationship between outer and inner stable
For LTI system, if the system is inner stable, then the system must be outer stable (BIBO stable)
The outer stable LTI system may not be inner stable
System transfer function matrix reflects only the controllable and
observable part of the whole system. The rest part, . the
uncontrollable and unobservable subsystem can not be reflected.
If the system is contr