Dynamical Systems Analysis III: Phase Portraits By Peter Woolf University of Michigan Michigan Chemical Process Dynamics and Controls Open Textbook version 1.0 Creative commons

Questions answered & questions remaining.. 1)Create model of physical process and controllers 2)Find fixed points 3)Linearize your model around these fixed points 4)Evaluate the stability around these fixed points Questions: What about all of the other points? What happens when we are not at a fixed point? If there are multiple stable fixed points, how large are their ‘basins of attraction’? Is there a way to visualize this? Is there a way to automatically do all of this?

Nonlinear model From last class… Linear approximation at A=0, B=0 Linear approximation at A=0, B=1 Linear approximation at A=3, B=0 Linear approximation at A=4, B=-1 unstable saddle stable unstable saddle

Linear approximation at A=0, B=0 Linear approximation at A=0, B=1 Linear approximation at A=3, B=0 Linear approximation at A=4, B=-1 unstable saddle stable unstable saddle A B ?

What happens at A=3, B=1 A B ? (Not steady state) Check derivatives of nonlinear model

A B Trajectories A time B Phase Portrait

Fixed points Vector field Trajectory Stable and unstable orbits I: converge to fixed point II: diverge III: diverge IV: diverge

Other possibilities Another nonlinear system (Default example in PPLANE) stable unstable Basin of attraction I Basin of attraction II.1 Basin of attraction II.2

Other possibilities Another nonlinear system (FitzHugh-Nagumo model) Limit cycle unstable Region I Region II Note: Locally unstable systems can be globally stable!

Other possibilities Another nonlinear system (Lorenz equations) Chaotic system: 3+ dimensions Never converges to a point or cycle Image from java app at Unstable fixed point

Other possibilities Image from java app at Unstable fixed point (Same system shown in 3D with white balls following the trajectories)

Concepts from phase portraits extend to higher dimensions Fixed points, trajectories, limit cycles, chaos, basins of attraction Many real chemical engineering systems are high dimensional and very nonlinear. Example: CSTR with cooling jacket, multiple reactions, and one PID controller

What does this have to do with controls? Control systems modify the dynamics of your process to: –Move fixed points to desirable places –Make unstable points stable –Modify boundaries between basins –Enlarge basins of attraction

–Move fixed points to desirable places –Make unstable points stable –Modify boundaries between basins –Enlarge basins of attraction

How can a control system change the dynamics? Adding new relationships between variables Adding new variables (I in PID control) Adding or countering nonlinearity Providing external information

Take Home Messages Phase portraits allow you to visualize the behavior of a dynamic system Control actions can be interpreted in the context of a phase portrait Local stability analysis works locally but can’t always be extrapolated for a nonlinear system.