Lec_13.pdf
- 2. Introduction 2 The purpose of this chapter is to present the design, analysis, and simulation of algorithms that can be used for online parameter identification. This involves three steps: Step 1 (Parametric model ). Express the form of the parametric model SPM, DPM, B-SPM, or B-DPM. Step 2 (Parameter Identification Algorithm). The estimation error is used to drive the adaptive law that generates online. The adaptive law is a differential equation of the form ( ) ( ) t H t ( ) H t where is a time-varying gain vector that depends on measured signals. Step 3 (Stability and Parameter Convergence). Establish conditions that guarantee * ( ) t
- 3. Example: One-Parameter Case 3 Consider the first-order plant model Step 1: Parametric Model
- 4. Example: One-Parameter Case 4 Step 2: Parameter Identification Algorithm parameter error Adaptive Law The simplest adaptive law for In scalar form may be introduced as provided . In practice the effect of noise especially when is close to zero, may lead to erroneous parameter estimates.
- 5. Example: One-Parameter Case 5 Step 2: Parameter Identification Algorithm Another approach is to update in a direction that minimizes a certain cost of the estimation error. As an example, consider the cost criterion: where is a scaling constant or step size which we refer to as the adaptive gain and where is the gradient of J with respect to . We ill have 0 , (0) adaptive law
- 6. The adaptive law should guarantee that: parameter estimate and speed of adaptation are bounded and estimation error gets smaller and smaller with time. Example: One-Parameter Case 6 Step 3: Stability and Parameter Convergence ( ) t Note that these conditions still do not imply that unless some conditions on the vector referred to as the regressor vector. * ( ) t ( ) t
- 7. Example: One-Parameter Case 7 Step 3: Stability and Parameter Convergence Analysis 1. Solving 2. Lyapunov Solving ( ) 0 t * ( ) t
- 8. and are bounded Example: One-Parameter Case 8 Step 3: Stability and Parameter Convergence is always bounded for any ( ) t ( ) t is bounded ( ) ( ) t t is bounded ( ) t ( ) t
- 9. Analysis by Lyapunov Example: One-Parameter Case 9 Step 3: Stability and Parameter Convergence or
- 10. Example: One-Parameter Case 10 is uniformly stable (u.s.) is uniformly bounded (u.b.) asymptotic stability So, we need to obtain additional properties for asymptotic stability
- 12. Example: One-Parameter Case 12 adaptive law summary (i) (ii)
- 13. Example: One-Parameter Case 13 The PE property of is guaranteed by choosing the input u appropriately. Appropriate choices of u: and any bounded input u that is not vanishing with time.
- 15. Example: Two-Parameter Case 15 Consider the first-order plant model Step 1: Parametric Model Step 2: Parameter Identification Algorithm Estimation Model: Estimation Error: A straightforward choice: where is the normalizing signal such that
- 16. Example: Two-Parameter Case 16 Adaptive Law: Use the gradient method to minimize the cost,
- 17. Example: Two-Parameter Case 17 Step 3: Stability and Parameter Convergence Stability of the equilibrium will very much depend on the properties of the time-varying matrix , which in turn depends on the properties of .
- 18. Example: Two-Parameter Case 18 For simplicity let us assume that the plant is stable, i.e., . If we choose at steady state 2 2 0 1 0 ( ) ( ) A c c is only marginally stable 0 e is bounded but does not necessarily converge to 0. constant input does not guarantee exponential stability. e
- 19. Persistence of Excitation and Sufficiently Rich Inputs 19 Definition Since is always positive semi-definite, the PE condition requires that its integral over any interval of time of length is a positive definite matrix. Definition
- 20. Persistence of Excitation and Sufficiently Rich Inputs 20 Let us consider the signal vector generated as where and is a vector whose elements are strictly proper transfer functions with stable poles. Theorem
- 21. Persistence of Excitation and Sufficiently Rich Inputs 21 Example: in the last example we had: In this case n = 2 and is nonsingular For , is PE.
- 22. Persistence of Excitation and Sufficiently Rich Inputs 22 Example: Possible u
- 23. Example: Vector Case 23 Consider the first-order plant model Parametric Model
- 24. Example: Vector Case 24 Filtering with where, a monic Hurwitz polynomial