EG4321/EG7040. Nonlinear Control. Dr. Matt Turner

Transcription

1 EG4321/EG7040 Nonlinear Control Dr. Matt Turner

2 EG4321/EG7040 [An introduction to] Nonlinear Control Dr. Matt Turner

3 EG4321/EG7040 [An introduction to] Nonlinear [System Analysis] and Control Dr. Matt Turner

4 Limitations with linear systems All real systems are - to some extent - nonlinear Linear systems theory cannot capture the variety/complexity of the behaviour of some systems Local and global behaviour are rarely identical (Although local regions can be large) Implicitly, linear systems theory implies - for each static input - precisely one equilibrium point - Often not the case

5 Representation of Nonlinear Systems Nonlinear System: Cannot simply use Laplace/frequency domain methods x 0 u(t) G y(t) Interpretation: G: mapping from input+initial condition to output State-space G ẋ = f(x,u,t) [ ] u(t) G y = h(x,u,t) x 0 y(t) x(0) = x 0 f(.,.,.) and g(.,.,.) are vector-valued functions

6 Some quick short-hand Vector sizes x = x 1 x 2. x n x Rn u = u 1 u 2. u m u Rm... Matrix sizes a 11 a a 1m A = a 21 a A Rn m a n a nm Vector-valued functions f(.,.) : R n R m R p f maps an n-vector and and m-vector to a p-vector

7 Some special cases Generic Nonlinear state-space system ẋ = f(x,u,t) G y = h(x,u,t) x(0) = x 0 Assumption: the solution (x, y) exists and is unique (not necessarily the case) Linear State-space systems ẋ = Ax +Bu y = Cx +Du x R n, u R m, y R p Stability: entirely dependent on eigenvalues of A R(λ i (A)) < 0 i Autonomous Nonlinear Systems x R n, y R p ẋ = f(x) y = h(x) No input/time-dependency Stability: basically study of differential equation ẋ = f(x)

8 Notes on nonlinear system stability Linear systems: Internal stability Bounded-Input-Bounded-Output (BIBO) stability Nonlinear systems: Internal stability Bounded-Input-Bounded-Output (BIBO) stability Example : ẋ = x +2xu 1. No input: u(t) = 0 t 0 x,u R ẋ = x lim x(t) = 0 asymptotically stable t 2. Let u(t) = 1 t 0 ẋ = x lim x(t) = not BIBO stable t

9 Multiple Equilibria Linear systems: local global Linear systems: If u = 0 (no input): only one equilibrium Nonlinear systems: multiple equilibria: Initial conditions determine long-term characteristics of system

10 Example: Predator Prey System Predator: Foxes, y(t) Prey: Rabbits, x(t) b Birth-rate of rabbits d Death-rate of foxes p Greediness of foxes r Nutritional value of rabbits Innocuous looking system: ẋ = (b py)x ẏ = (rx d)y y 0 ẋ = bx (Linear System) x 0 ẏ = dy (Linear System) Dynamics surprisingly complex

11 Example: predator prey system Initial condition [ ] [ x 10 = y 0 Exponential instability (Rabbit population explosion) ] Rabbits/Foxes x x Rabbits Foxes Time [months] Rabbits/Foxes x Rabbits Foxes Time [months] Initial condition [ ] x = y [ 0 10 Exponential stability (Fox population extinction) ]

12 Example predator prey system Initial condition [ ] [ x 10 = y 10 Deadbeat stability (Rabbit/fox populations constant) ] Rabbits/Foxes x Rabbits Foxes Time [months] Rabbits/Foxes x Rabbits Foxes Time [months] Initial condition [ ] [ x 10 = y 12 Oscillation (Fox/rabbit populations ebb and flow) ]

13 Example predator prey system Initial condition [ ] [ x 20 = y 10 Oscillation (similar to linear oscillator) ] Rabbits/Foxes x Rabbits Foxes Time [months] Foxes x Phase portrait Limit cycle (Trajectories converge to an orbit) Rabbits x10

14 Example predator prey system Initial condition [ ] [ x 50 = y 50 ] Oscillation (Definitely not linear!) Rabbits/Foxes x Rabbits Foxes Time [months] 8 7 Foxes x Phase portrait Larger amplitude Limit cycle (Trajectories converge to an orbit) Rabbits x10

15 Example predator prey system Initial condition [ ] [ x 100 = y 100 Large initial transient......then decay to oscillation ] Rabbits/Foxes x Rabbits Foxes Time [months] 18 Foxes x Rabbits x10 Phase portrait Same amplitude Limit cycle as before (Trajectories converge to stable limit cycle)

16 Noteworthy points Linear system stability purely determined by the system parameters: These determine poles (eigenvalues) Eigenvalues determine stability Nonlinear system stability is a function of System parameters Initial conditions (where system starts from) External inputs Nonlinear systems can exhibit a much wider variety of behaviour than linear systems Characterising this behaviour is not trivial Phase Portraits provide useful graphical information about the stability of the system......but usefulness is limited to planar (two-state) systems