Non-linear control Flashcards
What is asymptotic stability
x(t) tends to zero and t tends to infinity
What is local asymptotic stability
x(t) tends to zero and t tends to infinity, provided the initial state x(0) was within some range (known as the zone of attraction)
What is global asymptotic stability
x(t) tends to zero and t tends to infinity, from any initial state x(0)
What is lyapunov stability
System state will remain in some region near to the origin without necessarily reaching the origin (see limit cycles)
What is exponential stability
Asymptotically stable and some weird shit
What is the zone of attraction
The region in which a system can start and remain stable, if the system only has local stability
What are limit cycles and how can they be useful
Occur when a system oscillates continuously over some finite range (example of this is the Van der Pol eqn.)
Can be used to make a system more robust (any initial condition can be made to follow a set of periodic characteristics)
What are the limitations of the ‘Linearisation method’
Becomes less accurate as the system moves away from the equilibrium state
If a system is marginally stable (one point on origin, rest on LHP) it cannot be used
When is V(x) positive definite
V(0) = 0 and V(x) > 0 for all of (x =/= 0)
When is V(x) positive semi-definite
V(x) >= 0 for all x
When is V(x) negative definite
V(0) = 0 and V(x) < 0 for all of (x =/= 0)
When is V(x) negative semi-definite
V(x) <= 0 for all x
What are the 3 conditions needed for global asymptotic stability
- Positive definite
- Negative derivative
- Radially unbounded ( V(x) tends to infinity as x goes to infinity )
What are the 4 steps of gain scheduling
- Determine a family of linear approximations to non-linear plant
- Design linear controllers (eg PID or P+DFB)
- Implement controllers
- Assess performance of system (theory, simulation or physical testing)
What are scheduling variables
Observable parameters which are used to select the
most appropriate gain values (eg a vehicle’s speed)
