Nonlinear Dynamics and Control I

One Dimensional Flows

One Dim Flows

• interpreting a diff equation as a vector field.
• compared to exact solution, vector field is clear and simple
• fixed points: points where there is no flow
  • stable: attractors/sink as flow converges to them
  • unstable: repellers/sources as flow diverges from them
• this representation allows us to find equilibrium points and visualize trajectory from any IC.
• it CANNOT tell us quanitative things:
  • abstracting away time
  • time at which speed is the greatest
• phase potrait: shows all the qualitatively different trajectories of the (1D flow) system
  • appearance controlled by fixed points, defined by $f(x^*) = 0$
  • stagnation points of the flow
• stablity: small distrufaces dont lead to divergence, “dampening” of disturbances
• unstability: disturbances grow with time
• ex of 1D flows: population growth (using logistic equation) $\dot{N} = r N(1-\frac{N}{K})$

Linear Stability Analysis

• quantitaive measure of stability:
  • rate of decay
• get this by linearizing around fixed point
  • if the $f(x^*) > 0$: unstable, disturbances exponentially diverges
  • if the $f(x^*) < 0$: stable, disturbances exponentially decays
  • $\mid f(x^*) \mid$ is the magnitude of the decay/growth.

Existence and uniqueness

• Solutions to IVP are unique
  • Given that $f(x)$ and $f’(x)$ are continuous.
  • When uniqueness fails, geometric approach does not work as the phase point doesn’t know how to move.
• Some can be of infinite time and some can be finite time
• Even finite time solution can “blow up” to infinity in fintie time

Oscillations are impossible

• Trajectories can only converge to fixed points or diverge to infinity in the first order system. No oscillations can occur.
• Geometrically, phase point never reverses direction. Hence, no periodic solutions.
  • Corresponds to flow on a line, if you flow on a line, you will never come back to the starting place.
  • Mechanical analog: overdamped system

Potentials

• Visualize dynamics of 1d system using the idea of potential energy
  • $f(x) = -\frac{dV}{dx}$
  • Going “downhill”
  • $V(t)$ by the above definition, decreases along trajectories, and so the particle always moves towards lower potential.

Numerically solving dynamics

• Euler’s method: $x_{n+1} = x_{n} + f(x_n)\triangle t
  • not recommended in practice
  • estimates derivates only at the left end of the time interval
  • use average derivative across the interval (take a trial step, take derivative at that step and use that for estimation)
• Industry practice: Runge-Kutta Method (4th order)
• Round-off error: doing excessively many calculations
  • occur every calculation and accumulate (can be catastropic if $\triangle t$ is too small)

Bifurcations

• Bifurcations (“splitting into two branches”) are qualitative changes in dynamics (such as creation/destruction of fixed points, stability, etc.)
  • Parameter values at which these changes occur are called bifurcation points.
• Ex: buckling of a beam
  • If the load is small, beam can support a weight on top, if it is too heavy the beam “buckles”
  • Instability arises

Saddle Node Bifurcation

• Basic mechanism to create or destroy fixed point
• As a parameter is varied, two fixed point move toward each other, collide and mutually annihilate.
  • Ex: $\dot{x} = r + x^2$
    • $r \leq 0$: two fixed points (one stable one unstable)
    • $r = 0$: one fixed point (saddle point), one side of the fixed point is converging and the other side is diverging.
    • $r \geq 0$: no fixed points
    • Bifurcation happens at $r=0$
• Bifurcation Diagram: indep axis the control parameter (which can bifurcate), dep axis the state which has fixed point determined by the control parameter.
• Saddle node bifur usually show the $r+x^2$ behavior
  • Called “normal forms” for the saddle node bifur
• Tangential condition needed with the x axis for the bifurcation to occur.

Transcritical Bifurcation

• certain situations where a fixed point must exist for all values of a parameter and can never be destroyed.
  • However, fixed point may change it’s stability as the parameter is varied.
  • This is done via transcritical bifurcations.
• Normal form for transcrit bifur:
  • $\dot{x} = rx - x^2$
  • Doesn’t matter what the value of $r$ is, there will always be a fixed point at $x = 0$. It will just change it’s stability based on the value of $r$
  • “Exhange of stabilities” after bifurcation.
• If we can approximately change a dynamics equation into the normal forms of the bifurcations, we can find if a bifurcation is possible to the system. – Add Image of Fig 3.2.1– – Add Image of 3.2.2 –

Pitchfork Bifurcation

• Common in physical problems that have a symmetry.
  • Fixed points tend to appear and disappear in symmetrical pairs.
• Two types of pitchfork bifurcation

Supercritical PF Bifurcation

• Normal Form: $\dot{x} = rx - x^3$
  • $r \leq 0 $: origin is stable
  • $r = 0$: origin is weakly stable, solution does not decay exponentially fast. Decay much slower function of time. Called critical slowing down
  • $r \geq 0$: origin is unstable, two new stable fixed point appear on wither side at $x = \sqrt{r}$
• Appears a pitchfork in the bifurcation diagram. – Image of normal form – Image of bifur diagram

Imperfect Bifurcations and Catastrophes

• If have multiple tuning parameters like: \(\dot{x} = h + rx - x^3\)
• We have a “codimension-2” bifurcation: we have to tune two params, to achieve bifurcation.
• Special diagrams to visualize bifurcation: stability diagrams.
  • Shows different type of behavior as we move around in the parameter space (r,h) plane.
  • Also bifurcation diagrams make sense too, have to fix one param constant tho.
  • 3D visualization: plotting $x^*$ above the (r,h) plane, we ge tthe cusp catastrophe surface.
    • THe surface folds over on itself in certain places.
    • Projection of these folds onto the (r,h) planes yields the bifurcation curves (stability diagram).
    • Catastrophe here means that as params change,the state of system can be carried over the edge of the uppersurface and drop discontinuously to the lower surface. This jump can be catastriohic for equilibrium of the bridge or building for example.

• Appears in $\beta tanh$ functions, used in neural networks!!!

• Dependence on the control parameter is usually simpler then the dependence on x
• The cubic term is “stabilizing”
  • Pulls $x(t)$ towards $x=0$

Subcritical PF Bifurcation

• Normal form: $\dot{x} = rx + x^3$
• THe cubic term is “destabilizing”
  • pushes x(t) away from origin
• The nonzero fixed points are unstable and exist below the bifurcation ($r \leq 0$)
  • Hence, the name subcritical

– Bifur Diagram

• Hysterisis: Lack of reversibility as a parameter is varied.
  • If multiple stable fixed points exist, the jump to different stable state is possible as $r$ is varied. The jump from one to another stable point due to bifurcations, can lead to leaving one stable point for another permanently – Add image 3.4.8

Flows on the circle

• Consider: $\dot{theta} = f(\theta)$
• Vector field on a circle.
  • $\theta$ is a point on the circle, and its time derivative is the velocity vector at that point.
  • 1D like a line, but a particle can return back to start point.
  • Periodic solution possible.
• Vec field on circle provide the most basic model of systems that can oscillate.
• Need to consider non-uniqueness when defining the vector field on a circle. For example: $\dot \theta = sin(\theta)$ is a valid smooth vector field while $\dot \theta = \theta$ is not as this gives the velocity multiple values (non-unique) for the same point (0, 2pi).

Uniform Oscillator

• Point on circle called angle (or phase)
• Simplest oscillator: $\dot \theta = \omega$
• $\omega$ is constant
• Solution: $\theta(t) = \omega t + \theta_0$
• Solution is periodic.
• Period: $T = 2\pi/ \omega$
• Can’t comment on the amplitude, we will have to move to 2D for that.
• Beat phenomenon: noninteracting oscillators with diff freqs will peridoically go in and out of phase with each other.

Non-uniform oscillator

• $\dot \theta = \omega - \alpha sin \theta$
• Ex: overdamped pendulum driven by a constant torque, firefly flashing rhythm
• Param $a$ causes nonuniformity, increases with higher a
• Bottleneck: slow passage in the phase plot ($a < \omega$ slightly)

Non - Linear Control

• Analysis of a given non-linear system to deduce it’s behavior and characteristics
• In design, given a plant and some specifications, construct a controller.

Why nonlinear control?

• Linear methods rely on the assumption of small range operation to be valid.
  • When operation range is large, linear controller can be unstable/poor as nonlinearities are not compensated
  • Ex: robot motion control problems
• Linear model assume system model is linearizable
  • many nonlinearities who discontinuous nature does not allow linear approximation.
  • Columb friction, saturation, deadzones, backlash, etc.
• Linear models assume parameters of the models are well known
  • Many problems have uncertainties in model param
  • Nonlinearities intentionally included so uncertainties can be tolerated

Nonlinear System Behavior

• If operating range of control system is small, and if nonlinearities are smooth, then the control system can be approximated by a linearized system.
• Nonlinearities can be: continuous or discontinuous
  • Discontinuous nonlinearities cannot be locally approximated by linear functions

Common nonlinear behavior

• multiple equlibrium points
• existence of limit cycles: oscillations with a fixed period without external excitation
  • Van der Pol Oscillator
  • amplitude of oscillation independent of IC (unlike linear systems)
  • not easily affected by parameter changes
• bifurcations
  • number of equilibrium points can change with parameters of the system
  • values of the parameter where it leads to qualitative change of system properties is called critical or bifurcation values.
  • informally, bifurcation values change the number of equlibria (pitch fork bifurcation); emergence of limit cycles (Hopf bifurcation)
• chaos:
  • small differences in IC can cause vastly different outcomes in some nonlinear systems
  • “unpredictability of the system output”
  • deterministic
  • ex: turbulence in fluid mechanics, atmospheric dynamics.
  • occurs mostly in strongly nonlinear system
  • cannot occur in linear systems
  • active research: when a nonlinear system will go into a chaotic mode, and in case it does how to recover from it.

Misc.

• Laplace and Fourier does not apply to nonlinear systems

Stabilization vs Tracking

• Given a physical system to be controlled and the specs, construct a feedback control law to make the closed loop system display the desired behavior.
• Two types of nonlinear control problems
  • Regulation (stabilization): design a controller where the closed-loop system to be stabilized around an equlibrium point.
    • Temperature control
    • Altitude control
    • Position control
  • Tracking (servo): design controller so the system output tracks a given time-varying trajectory
    • find a control law for the input $u$ such that start for any Ic in a region $\omega$, the tracking errors $y(t) - y_d(t)$ go to zero.
• Relations between Stabilization and tracking
  • Tracking more difficult as the controller has to stabilize and drive thes ystem output toward desired ouput.
  • Stabilization special case of tracking, constant trajectory.

Specifying the desiered behavior

• Usually, look for qualitative specs of the desired behavior in ROIs
• Computer simulation very important
• Stability must be guranteed (local or global sense); region of stability also important
• Robustness: sensitivity to effects not considered in the design like noise, disturbances, unmodelled dynamics, etc.
• Cost

Control Design process

• Typical control design
  1. Specify desired behavior, select actuators and sensors
  2. Model physical plant by set of Diff equations
  3. Design a control law for the system
  4. Analyze and simulate the system
  5. Implement on hardware

Feedback and Feedforward

• Feedforward is much more important in nonlinear control
  • Used to cancel out the effects of known distrubances and provide anticipative actions in tracking tasks
  • Model of the plant is always required for feedforward compensation (need to be accurate)
• Controllers $u = feedforward+feedback$
  • Feedforward cancels out known distrubances
  • Feedback stabilizes

Available Methods

• Trial and Error: Use analyis tools to guide search; phase plane method, Lyapunov analysis.
  • Needs experience and intuition
  • Fails for complex system
• Feedback Linearization: techniques for transforming original system models into equivalent models of a simpler form
  • transform a nonlinear system into (fully or partially) linear system
  • Then use linear design techniques
  • Applies to input-state linearizable or minimum phase systems, requires full state measurement
  • Does not gurantee robustness.
  • Can also be used as model-simplifying devices for robust/adaptive controllers.
• Robust Control: controller is designed based on both the nominal modela and some characterization of model uncertaineties
  • ex: robot is carrying load between 2kg to 10kg.
  • Ex: sliding control
  • Require state measurements
• Adaptive Control: applys to system with known dynamic structure but unknown constant/slowly varying.
  • Inherently nonlinear.
  • Mostly, for SISO but can be applied to MIMO.
• Gain-scheduling: selecting number of operating points which cover range of system operation points
  • for each point, designer makes an LTI approx of the dynamics and designs a linear controller.
  • Between operating points, parameters of the controller (compensators) are interpolators or “scheduled”, resulting in a global compensator
  • Simple and practicaly, but limited theoretical gurantees of stability.
  • Computational expensive