Dynamical Systems Analysis II: Evaluating Stability, Eigenvalues By Peter Woolf University of Michigan Michigan Chemical Process Dynamics and Controls Open Textbook version 1.0 Creative commons

Problem: Given a large and complex system of ODEs describing the dynamics and control of your process, you want to know: (1)Where will it go? (2)What will it do? Is there anything fundamental you can say about it? E.g. With my control architecture, this process will always ________. Solution: Stability Analysis Steady state from last lecture. Topic for today!

Exponential increaseIncrease w/ oscillation Stable oscillation Periodic solutionNon-periodic solution (chaotic) Only possible for nonlinear systems Decay w/ oscillationExponential decay What will your system do?

How can we know where the system will go? Possible approaches: 1.Simulate system and observe Disadvantages: Can’t provide guaranteed behavior, just samples of possible trajectories. Requires simulations starting from many points Assumes we have all variables defined, thus hard to use to design controllers. Advantages: Works for any system you can simulate Intuitive--you see the results

How can we know where the system will go? Possible approaches: 1.Simulate system and observe 2.Stability Analysis (this class) Disadvantages: Only works for linear models Linear approximations of nonlinear models break down away from the point of linearization Advantages: Provides strong guarantees for linear systems General

Nonlinear model Linear approximation at A=0, B=0 Jacobian Or in a different format From last class… Intuitively, what will the linear system do if A is perturbed slightly from 0? Increase in A above 0 yields a positive derivative Increase in A Increase in slope of A Exponential increase

But what if our model is more complex? E.g. (note: example below is made up) Or in a different format What will happen if A or B are increased slightly from the steady state value of A=1, B=3? Result: increase A, A and B increase! Result: increase B, A and B decrease! Increase A by  :Increase B by  :

Observations: 1.It is easy to predict where a linear system will go if the variables are decoupled 2.Coupling between variables makes it harder to predict what will happen 3.Coupling is determined by the Jacobian A only influences A, B only influences B. -> Variables are decoupled Changes in A influence changes in A and B. Changes in B influence changes in A and B. --> Variables are coupled

Is it possible to change a coupled system to a decoupled one? ?? Can we find a value that satisfies this relationship? Written differently.. This is an eigenvalue

expand Solve for B

Solve for Observations: 1)Yes! There is always a way decouple a coupled linear system 2)Direct approach involves lots of algebra There is an easier way..

A bit of linear algebra background Goal: solve this system for Determinant: a property of any square matrix that describes the degree of coupling between the equations. Determinant equals zero when the system is not linearly independent, meaning one of the equations can be cast as a linear combination of the others.

A bit of linear algebra background Goal: solve this system for Determinant: a property of any square matrix that describes the degree of coupling between the equations. Determinant equals zero when the system is not linearly independent, meaning one of the equations can be cast as a linear combination of the others. Revised Goal: find that satisfies

Similar Analysis can be done in Mathematica: Det[{a,b},{c,d}] :Find the determinant of a matrix Solve [{eqn1, eqn2,..},{var1, var2,..} ] : Solve algebraically Eigenvalues[{a,b},{c,d}] : Automatically find the eigenvalues

What do eigenvalues tell us about stability? Eigenvalues tell us the exponential part of the solution of the differential equation system Three possible values for an eigenvalue 1)Positive value: system will increase exponentially 2)Negative value: system will decay exponentially 3)Imaginary value: system will oscillate (note combinations of the above are possible)

What do eigenvalues tell us about stability? Effect: If any eigenvalue has a positive real part, the system will tend to move away from the fixed point

Marble Analogy Small perturbations left or right will cause the marble to decay back to the steady state position Negative real eigenvalue Small perturbations left or right will cause the marble to decay away from the steady state position (x ss ) Positive real eigenvalue Small perturbations in y are stable, while perturbations in x are unstable (saddle point), thus overall point is unstable! Positive and negative real eigenvalues x ss x Case I: stable x ss x Case II: unstable x ss,,y ss x y Case III: Saddle point

Revisit our example: What will happen here? 1)Calculate eigenvalues Eigenvalues: 1 =2, 2 = -1 2) Classify stability: At least one eigenvalue is positive, so the point is unstable and a saddle point. Exponential increase

A more complex example: What will happen here? 1)Calculate eigenvalues Force Mathematica to find a numerical value using N[ ] Using the Eigenvalue[ ] function in Mathematica Given these eigenvalues what will it do?

2) Classify stability: The real component of at least one eigenvalue is positive, so the system is unstable. There are imaginary eigenvalue components, so the response will oscillate. Increase w/ oscillation A more complex example: What will happen here?

Exponential increaseIncrease w/ oscillation Stable oscillation Decay w/ oscillationExponential decay What will your system do? (according to eigenvalues) All s are real and negative All s are real and at least one positive All s have negative real parts, some imaginary parts At least one has positive real parts, some imaginary parts All s have zero real parts and nonzero imaginary parts

Take Home Messages Stability of linear dynamical systems can be determined from eigenvalues Complicated sounding terms like eigenvalues and determinant can be derived from algebra alone--fear not! Stability of nonlinear dynamical systems can be locally evaluated using eigenvalues