Revision:

1. Revision: If the average system is exponentially stable, what can we conclude about stability of the actual system?

2. Lecture 21 Singular perturbations

3. Recommended reading • Khalil Chapter 9 (2nd edition)

4. Outline: • Motivation • Model set-up • Reduced (slow) model • Boundary layer (fast) model • Example: DC motor • Summary

5. Motivation: • Often a response of our system looks like: Fast transient Slow convergence to steady-state

6. Motivation • Electro-mechanical systems often contain a fast electrical part and a slow mechanical part. Often we ignore the fast part and instead of modelling this part as a dynamical system, we model it as an algebraic equation. Singular perturbations provide a rigorous procedure for this kind of approximations. • Numerically integrating systems that have very fast and very slow modes is notoriously hard – “stiff problems”.

7. Model set-up • Consider the parameterized system: where >0 is a small parameter. When we set =0, the second system degenerates into:

8. Comments • The above model is in the so called “singular perturbation form”. • Setting the parameter  equal to zero changes the order of the system (singular perturbation). • In previous lectures we dealt only with perturbations that do not change the order of the system (regular perturbations). • The discontinuity in  can be avoided if an appropriate time-scale separation is used.

9. Reduced (slow) model

10. Standard form: • We say that the singularly perturbed model is in standard form if has k  1 isolated real roots for each (t,x) in the domain of interest:

11. Reduced model • When =0, we obtain the reduced model (assuming only one root): • The above model is sometimes called the “quasi steady-state” model.

12. Preliminary analysis: • Note that we have the following trend:  0  |dz/dt| =|g(t,x,z,)/| • Hence, z changes very rapidly. • z is the “fast variable”.

13. Boundary layer (fast) model

14. Change of time scales • Introduce  = (t-t0)/. Then, the model is • When we set =0, we have

15. Preliminary analysis • Note that we have the following trend  0  dx/d 0 • Hence, x changes very slowly in  scale. • x is the “slow” variable. It is “frozen” when =0. • We will analyse the z system as a system with slowly time varying inputs.

16. Graphical interpretation z=h(t,x) z Fast system: Slow system x

17. Comments • Using the reduced (slow) model and the fast model in different time scales, we can show that they approximate the behaviour of the original system in an appropriate sense. • Appropriate stability of reduced and fast models guarantees stability of the original system. • Again we will simplify the original problem in order to analyse it.

18. Example: DC motor • Ad hoc approach to modelling: • Mechanical part is slow • Electrical transients die out quickly (ignored) • We provide a step by step modelling that leads to the reduced model under certain assumptions – singular perturbations approach is crucial in this rigorous analysis.

19. Full model is given by • i is the armature current; • u is the armature voltage; • R and L are the armature resistance and inductance; • J is the moment of inertia; • is the angular velocity.

20. Suppose that L  0. Then, L can play the role of  and when =0 we have: whose unique solution is: • The reduced model becomes:

21. Summary: • Singular perturbations can be used when we have that a part of the state is fast and the other part is slow (slow and fast modes). • Modelling of singularly perturbed systems is more involved than that of regularly perturbed systems and we need to introduce time scale separation in order to analyse the system. • A reduced model and the fast model will be used to approximate solutions of the original system.

22. Next lecture: • Singularly perturbed systems. Homework: read Chapter 9 in Khalil

23. Thank you for your attention!