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Lecture 22

Lecture 22. Singularly perturbed systems. Recommended reading. Khalil Chapter 9 (2 nd edition). Outline:. Overview of the model Approximation using the reduced system Properties of the boundary layer system Tikhonov Theorem (approximation on compact time intervals) Example Summary.

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Lecture 22

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  1. Lecture 22 Singularly perturbed systems

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

  3. Outline: • Overview of the model • Approximation using the reduced system • Properties of the boundary layer system • Tikhonov Theorem (approximation on compact time intervals) • Example • Summary

  4. Overview of the model • We consider the singularly perturbed system on the time interval [t0,t1]. • We denote solutions of  as x(t,) and z(t,). • All functions are sufficiently smooth. • The system is in standard form!

  5. Approximation using r

  6. Overview of the approximating model • We use the reduced model: • Denote solution of r as xr(t). Suppose that:

  7. Comments • We first want to see how well the solutions of the reduced system are good in approximation (other choices possible). • Boundary layer system has to satisfy certain conditions for this approach to be valid. • Solution xr(t) of r can approximate x(t,) uniformly on [t0,t1] since x(t0,)-xr(t0)=O(). • We expect to obtain:

  8. Comments • Solution zr(t) CAN NOT approximate z(t,) uniformly on [t0,t1] since z(t0,)-zr(t0) can be arbitrarily large! • We expect that for arbitrary t1>tb>t0: • This holds under reasonable conditions!

  9. Required properties of bl

  10. Boundary layer system • Introducing y:=z-h(t,x), we use the boundary layer system in time scale  = (t-t0)/: • We assume that for t  [0,t1] we have xr(t)  Br. Then, (t,x)  [0,t1]  Br are regarded in the above equation as frozen parameters. This is justified since they are slowly varying compared to y when  is small.

  11. Stability of bl • Similar to slowly time-varying systems, we require that there exist positive constants k,  and 0 such that for all (t,x)  [0,t1]  Br • bl is a parameterized family of systems that is exponentially stable uniformly in (t,x). • We denote the solution of bl as ybl(.).

  12. Comments • Exponential stability uniform in (t,x) can be checked either via the linearization or via Lyapunov analysis. • If we use the former, then we require that there exists c>0 such that  (t,x)  [0,t1]  Br

  13. Comments • If we use Lyapunov analysis, then we require that there exist positive c1,c2, c3,  such that for all (t,x,y)  [0,t1]  Br B.

  14. Tikhonov Theorem

  15. Conditions: Let z=h(t,x) be an isolated root of g(t,x,z,0)=0. Suppose that for all (t,x,z-h(t,x),)  [0,t1]  Br, B [0,0] we have: • f, g, h, (e), (e) are sufficiently smooth; • r has a unique solution defined on [t0,t1] and |xr(t)|  r1<r for all t  [t0,t1]; • The origin of bl is exponentially stable, uniformly in (t,x). In particular, we assume that 0/k.

  16. Conclusions: • There exist *, >0 such that for all  (0,*) and |(0)-h(t0,(0))|<, has a unique solution on [t0,t1] and the following holds: • tb (t0,t1), **>0 such that for  (0,**)

  17. Comments • Tikhonov Theorem is a result on closeness of solutions between  and its approximating systems (r, bl) on compact time intervals. • If we do not use solutions of bl, then we can not have uniform approximation of z(t,) by zr(t)=h(t,xr(t)). • If we use solutions of bl, then we can obtain a uniform approximation of z(t,). • xr(t) uniformly approximates x(t,).

  18. Example (DC motor) • Consider the singular perturbation model of a DC motor: • We approximate the solutions using the Tikhonov Theorem on the interval t  [0,1].

  19. Step 1: solving –x-z+t=0 yields h(t,x)=-x+t • Step 2: g(t,x,y+h(t,x),0)=-x-(y-x+t)+t=-y and is UGES for all (t,x). • Step 3: f(t,x,h(t,x),0) = -x+t and has a unique solution

  20. Step 4: The boundary layer problem: has a unique solution: • From the Tikhonov Theorem we conclude:

  21. Summary: • Solutions of a singularly perturbed system  can be approximated well by solutions of r and bl on compact time intervals if bl is exponentially stable uniformly in (t,x). • We use results for slowly time-varying systems to analyse bl. • Results for closeness of solutions on infinite intervals and stability of  via stability of r and bl can also be stated.

  22. Next lecture: • Stability of singularly perturbed systems Homework: read Chapter 9 in Khalil

  23. Thank you for your attention!

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