EG4321/EG7040 Nonlinear Control Dr. Matt Turner
EG4321/EG7040 [An introduction to] Nonlinear Control Dr. Matt Turner
EG4321/EG7040 [An introduction to] Nonlinear [System Analysis] and Control Dr. Matt Turner
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
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
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 12... a 1m A = a 21 a. 22..... A Rn m a n1...... 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
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)
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
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
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
Example: predator prey system Initial condition [ ] [ x 10 = y 0 Exponential instability (Rabbit population explosion) ] Rabbits/Foxes x10 2.5 x 104 2 1.5 1 0.5 Rabbits Foxes 0 0 2 4 6 8 10 Time [months] Rabbits/Foxes x10 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Rabbits Foxes 0 0 2 4 6 8 10 Time [months] Initial condition [ ] x = y [ 0 10 Exponential stability (Fox population extinction) ]
Example predator prey system Initial condition [ ] [ x 10 = y 10 Deadbeat stability (Rabbit/fox populations constant) ] Rabbits/Foxes x10 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 Rabbits Foxes 0 0 2 4 6 8 10 Time [months] Rabbits/Foxes x10 1.25 1.2 1.15 1.1 1.05 1 0.95 0.9 0.85 Rabbits Foxes 0.8 0 2 4 6 8 10 Time [months] Initial condition [ ] [ x 10 = y 12 Oscillation (Fox/rabbit populations ebb and flow) ]
Example predator prey system Initial condition [ ] [ x 20 = y 10 Oscillation (similar to linear oscillator) ] Rabbits/Foxes x10 2 1.8 1.6 1.4 1.2 1 0.8 0.6 Rabbits Foxes 0.4 0 2 4 6 8 10 Time [months] 2 1.8 Foxes x10 1.6 1.4 1.2 1 0.8 Phase portrait Limit cycle (Trajectories converge to an orbit) 0.6 0.4 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Rabbits x10
Example predator prey system Initial condition [ ] [ x 50 = y 50 ] Oscillation (Definitely not linear!) Rabbits/Foxes x10 8 7 6 5 4 3 2 1 Rabbits Foxes 0 0 20 40 60 80 100 Time [months] 8 7 Foxes x10 6 5 4 3 2 Phase portrait Larger amplitude Limit cycle (Trajectories converge to an orbit) 1 0 0 1 2 3 4 5 6 7 8 Rabbits x10
Example predator prey system Initial condition [ ] [ x 100 = y 100 Large initial transient......then decay to oscillation ] Rabbits/Foxes x10 100 90 80 70 60 50 40 30 20 10 Rabbits Foxes 0 0 20 40 60 80 100 Time [months] 18 Foxes x10 16 14 12 10 8 6 4 2 0 0 2 4 6 8 10 Rabbits x10 Phase portrait Same amplitude Limit cycle as before (Trajectories converge to stable limit cycle)
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