Nonlinear Systems: Properties and Tests

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Nonlinear Systems Properties and Tests

• M. Sami Fadali
• Professor EBME
• University of Nevada
• Reno

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Outline

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

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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.

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Linear State-space Model

• State equations Set of linear first-order
  differential equations.
• Output equations Set of algebraic equation.

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Linear State-space Model

• Linear equations.
• Can be solved analytically.

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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

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Additivity Homogeneity
Zero-input response.
Zero-state response.
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Nonlinear Systems

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

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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.

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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.

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Examples

• Pendulum Two equilibrium points.
• Bistable Switch
• 3 equilibrium points (0, v0, v1)

?g(v)
v
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Limit Cycles

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

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Example Fitzhugh-Nagumo Model

• Simplified version of H-H model.
• Parameters

Param v0 v1 I k b
Value 0.2 1 1 0.5 0
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Bifurcation

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

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Chaos

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

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Lorenz attractor

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

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Response to Sinusoid

• Linear scale amplitude and phase shift.
• Nonlinear
• Harmonics multiple of input frequency.
• Subharmonics fraction of input frequency.
• Unrelated frequency.
• Examples

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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.

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Example Chi-Square Distribution
fX(x)
x
fY(y)
n4 D.O.F.
y
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System Properties

• Stability
• Controllability
• Observability
• Passivity

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Robustness

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

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Bode Sensitivity
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ExampleBiochemical System

• Metabolite Xi is produced from substrate Xj by an
  enzyme-catalyzed reaction (MM Kinetics)

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Sensitivity Equation

• First-order estimates of the effect of parameter
  variations (near q)

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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.

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Stability
Exponential Stability
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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)
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Stability of Motion

• Stability of equilibrium of the error dynamics

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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.

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Laypunov Stability Theorem

• Asymptotic stability if

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Lyapunov Approach

• Use quadratic Lyapunov function.
• Local stability for v lt 0.2
• Negative definite derivative

f(v)
v
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Method to Obtain a Lyap Function

• Krasovskiis method Use Jacobian (derivative)
  of RHS of state eqn.
• Stable if the derivative is negative near the
  origin.

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Example Metabolic Process

• Use Jacobian (derivative) of RHS of state eqn.

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Example Fitzhugh-Nagumo Model

• System with zero bias has a stable equilibrium
  (stable node) at (0,0).
• Small perturbation return to equilibrium.

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Limitations

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

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Controllability Observability

• Controllability Can go wherever you want no
  matter where you start.
• ?x0, xf , ? control u0,T?U, T lt ?, s.t. x(T
  x0) xf.
• Indistinguishable ? u? U
• x01, x02, y0,T?Y, T lt ?
• 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.

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Graphical Interpretation
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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.

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Passivity

• Supply rate integrate to obtain energy.
• Storage function S
• Dissipative system storage lt supply
• Passive dissipative with bilinear supply rate.

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Example of Passive System

• Spring-mass-damper
• R-L-C circuit.

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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 X3
  behave?

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Stability of Passive Systems

• Zero-state detectable (observable) System with
  zero input has stable zero dynamics (resp. y0 ?
  x0)
• Theorem Zero-state detectable and passive
• a) ? x0 with u0 is stable.
• b) ? x0 with u ?y ? h(x) is asymptotically
  stable.

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Absolute Stability

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

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Example Artificial Neural Networks

• Use passivity to show stability

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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 W such that

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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 W such that

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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.

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Periodic KYP
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Linearization

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

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Linearization
1st order approximation
f(x)
f(x0)
x
x0
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Linearization of Linear Pathway
Equilibrium (1/4, 1/16, 1/64)
Stable Equilibrium (?1/2, ?4, ? 1/8) all in LHP
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Stability

• Stability Condition Eigenvalues in LHP

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Stability of Linear DT Systems

• Eigenvalues inside the unit circle.
• Examples

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Conditions for 2nd-order Case

• Second-order recursion

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Example Dynamic Neural Network

• IIR Filter
• Nonlinear activation function (monotone
  increasing, slope g2 gt0)
• Stable network for any stable matrix A.
• Problem How to minimize error subject to the
  stability constraint?

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Constrained Optimization

• Minimize square error subject to stability
  constraints.
• Consider 2nd-order (explicit constraints)
• Modify stability margins (safety factor)

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Global Linearization

• Find a transformation of the nonlinear system to
  a decoupled linear system (easy transformation is
  special cases).
• Design linear control then transform back.
• Use differential geometry to derive the theory.

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Example Mechanical Systems
q vector of generalized coordinates. D(q)
s?s positive definite inertia matrix s?s
matrix of velocity related terms g(q) s?1
vector of gravitatioinal terms ? vector of
generalized forces
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Global Linearization

• Let the acceleration vector be the input.
• Series of double integrators.
• Choose acceleration for desired behavior.
• Calculate torque from accelerations, positions,
  and velocities.

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Limitations

• Complex mathematical theory (general case) but
  solution is hard to obtain must solve a
  nonlinear partial differential equation
  analytically for transformation.
• Results sensitive to modeling errors.
• Nonlinearity and coupling can be exploited to
  provide desirable behavior.

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Discrete-time Periodic Systems

• All system matrices are governed by
• M(k) M(kK) , k 0, 1, 2, ...
• Model multi-rate sampled systems.
• Time-invariant reformulations lift, cyclic.

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Fuzzy Models

• Include qualitative information.
• Fuzzy sets graded membership.

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Lyapunov Stability of TS Systems

• Linear matrix inequalities (LMI).
• Common Lyapunov function.
• Restrictive system can be stable even if one or
  more local model is unstable!
• Computational load large number of LMIs.

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Hybrid Systems

• Switch between different models
• Includes piecewise-linear.
• Overall behavior can be
• Stable even if each subsystem is unstable.
• Unstable even if each subsystem is stable.
• Piecewise linear Sufficient stability condition
  using common Lyapunov function.

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ExampleGene Regulation

• Effector gene cycles between two alternative
  environments
• H high demand environment (negative control
  mode repressor protein).
• L low demand environment Positive control mode
  activator protein).
• TH (TL)Av. duration in phase H (L).
• Av. Cycle time C TH TL

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Mathematical Model

• Response diverges if an eigenvalue of AHL is
  greater than 1.
• Steady state zero if all eigenvalues are inside
  the unit circle.
• Nonzero for one or more eigenvalue on the unit
  circle.

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Simulation Example
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Fault Detection

• Predict output of system using a mathematical
  model.
• Compare predicted output to measured output
  primary residual.
• Filter the primary residual and use the result to
  detect an error.
• Use state estimator (Kalman filter, observer,
  Bayesian network, fuzzy model, neural network)

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Conclusion

• Mathematical models of physical systems
• Linear.
• Nonlinear.
• Piecewise-linear.
• Properties
• Stability.
• Controllability observability.
• Passivity.
• Applications.

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