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Approximating Singularly Perturbed Systems: Tikhonov Theorem and Stability Concepts

This lecture discusses the approximation of singularly perturbed systems on compact time intervals, focusing on the Tikhonov Theorem. We will overview the model, its properties, and the boundary layer system's stability. The approximation involves comparing solutions from the reduced system and boundary layer system, ensuring the uniform approximation of perturbed solutions. An illustrative example from DC motor dynamics will clarify the application of the theorem. The significance of exponential stability and necessary conditions for the boundary layer system will also be explored.

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Approximating Singularly Perturbed Systems: Tikhonov Theorem and Stability Concepts

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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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