1. Nonlinear Systems: Properties and Tests M. Sami Fadali Professor EBME University of Nevada Reno

2. Outline • Linear versus nonlinear. • Nonlinear behavior. • Controllability and observability. • Stability. • Passivity.

3. State Variables • Minimal set of variables that completely describe the system. • State: set of numbers (initial conditions) that allows us to solve for the response for a given input. • State variable: variables obtained by letting the state evolve with time. • Example: position, velocity.

4. Linear State-space Model • State equations: Set of linear first-order differential equations. • Output equations: Set of algebraic equation.

5. Linear State-space Model • Linear equations. • Can be solved analytically.

6. Linear Systems • Additivity: add responses for added effects. • Homogeneity: scale responses for scaled effects. • Zero-input Response: Due to initial conditions. • Zero-state Response: Due to the input. • Total response = zero-input response + zero-state response

7. Additivity & Homogeneity Zero-input response. Zero-state response.

8. Nonlinear Systems • No additivity or homogeneity. • Dependent responses due to initial conditions and input. • More complex behavior

9. Examples of Nonlinear Behavior • Multiple equilibrium points. • Limit cycles: fixed period without external input. • Bifurcation: drastic changes of behavior with small changes in parameter values. • Chaos: aperiodic deterministic behavior which is very sensitive to its initial conditions. • Response to sinusoid: harmonics, subharmonics or unrelated frequencies.

10. Multiple Equilibrium Points • Equilibrium: stay there if you start there. • Stability of equilibrium not system. • No change: time derivative is zero. • Solve for equilibrium points.

11. Examples • Pendulum: Two equilibrium points. • Bistable Switch • 3 equilibrium points (0, v0, v1) g(v) v

12. Limit Cycles • Unlike linear system oscillations • Amplitude does not depend on initial state. • Stable or unstable limit cycle.

13. Example: Fitzhugh-Nagumo Model • Simplified version of H-H model. • Parameters

14. Bifurcation • Bifurcation point: Behavior changes drastically as parameter changes slightly. • Example: As parameter changes: periodic oscillations • period doubling chaos. • Example: Pitchfork • Undamped Duffing Equation

15. Chaos • Behavior is extremely sensitive to initial conditions. • Behavior is deterministic but looks random. • Example: cardiac arythmia (irregular beating patterns)

16. Lorenz attractor • Two unstable equilibrium points. • Model turbulent convection in fluids (weather patterns).

17. Response to Sinusoid • Linear: scale amplitude and phase shift. • Nonlinear: • Harmonics: multiple of input frequency. • Subharmonics: fraction of input frequency. • Unrelated frequency. • Examples

18. Response to Noise • Linear Systems • Gaussian input gives Gaussian output. • Completely characterized by mean and covariance matrix (variance). • Total response = zero-input response + zero-state response • Nonlinear systems • Gaussian input gives non-Gaussian output. • Need higher order statistics.

19. Example: Chi-Square Distribution fX(x) x fY(y) n=4 D.O.F. y

20. System Properties • Stability • Controllability • Observability • Passivity

21. Robustness • Property holds over a specified subset of parameter space. • Sensitivity: local measure of robustness. • Robustness w.r.t. noise and disturbances.

22. Bode Sensitivity

23. Example:Biochemical System • Metabolite Xi is produced from substrate Xj by an enzyme-catalyzed reaction (MM Kinetics)

24. Sensitivity Equation • First-order estimates of the effect of parameter variations (near q*)

25. Stability • Local or global • Lyapunov stability: continuity w.r.t. the initial conditions. • Asymptotic stability: Lyapunov stability plus asymptotic convergence to the equilibrium. • Exponential stability: ||x|| trajectory bounded above by an exponential decay.

26. Stability Exponential Stability

27. Example: Model of Linear Pathway • Specify kinetic orders, independent variables • Determine equilibrium: (1/4, 1/16, 1/64) • Solve differential equations (separation of variables): asymptotically stable. Equilibrium: (0, 0, 0)

28. Stability of Motion • Stability of equilibrium of the error dynamics

29. Lyapunov Stability Theory • Generalized energy function (positive definite). • Energy min at a stable equilibrium, energy max at an unstable equilibrium. • Trajectories converge to equilibrium if energy is decreasing in its vicinity (negative definite). • Design: choose control to make energy decreasing along trajectories.

30. Laypunov Stability Theorem • Asymptotic stability if

31. Lyapunov Approach • Use quadratic Lyapunov function. • Local stability for v < 0.2 • Negative definite derivative f(v) v

32. Method to Obtain a Lyap Function • Krasovskii’s method: Use Jacobian (derivative) of RHS of state eqn. • Stable if the derivative is negative near the origin.

33. Example: Metabolic Process • Use Jacobian (derivative) of RHS of state eqn.

34. Example: Fitzhugh-Nagumo Model • System with zero bias has a stable equilibrium (stable node) at (0,0). • Small perturbation: return to equilibrium.

35. Limitations • Sufficient conditions for stability and instability: if condition fails, no conclusion. • Necessary and sufficient for the linear case only.

36. Controllability & Observability • Controllability: Can go wherever you want no matter where you start. • x0, xf , control u:[0,T]U, T < , s.t. x(T;x0) = xf. • Indistinguishable:  u U • x01, x02,y:[0,T]Y, T <  • y(T, x01) = y(T, x02) • Observability: Can determine the initial state from the measurements (no two are indistinguishable). • x01, x02, y(T, x01) = y(T, x02)  x01= x02.

37. Graphical Interpretation

38. Example • Identical tanks with identical connections to a water source. • Not observable: Measuring the difference gives zero regardless the two levels. • Not controllable.: Filling the two tanks from one source gives the same level.

39. Passivity • Supply rate: integrate to obtain energy. • Storage function:S • Dissipative system: storage < supply • Passive: dissipative with bilinear supply rate.

40. Example of Passive System • Spring-mass-damper • R-L-C circuit.

41. Zero Dynamics • Internal dynamics of the system when the output is kept identically zero by the input. • Example: Metabolite Concentrations • Select X4 such that X1 = 0 how do X2 & X3behave?

42. Stability of Passive Systems • Zero-state detectable (observable) System with zero input has stable zero dynamics (resp. y=0  x=0) • Theorem: Zero-state detectableand passive a)  x=0 with u=0 is stable. b)  x=0 with u= y= h(x) is asymptotically stable.

43. Absolute Stability • Stable for any sector-bound nonlinearity. • G linear passive

44. Example: Artificial Neural Networks • Use passivity to show stability

45. Passivity of Linear Systems (CT) • A minimal state-space realization (A, B, C, D) is passive if and only if there exist real matrices P, L, and Wsuch that

46. Passivity of Linear Systems (DT) • A minimal state-space realization (A, B, C, D) is passive if and only if there exist real matrices P, L, and Wsuch that

47. Passivity of Periodic System • (F, G, H, E) =DT minimal cyclic reformulation of a periodic system. • System is passive if and only if it satisfies the following conditions with • a positive definite symmetric matrix P • real matrices W and L.

48. Periodic KYP

49. Linearization • Local behavior in the vicinity of an equilibrium. • Stability. • Controllability. • Observability. • Passivity: KYP lemma.

50. Linearization 1st order approximation f(x) f(x0) x x0