Projection Operator in Adaptive Systems

These notes started in [2] as a personal communication from Eugene to colleagues in the field of adaptive control and summarized results from [5,3,1,4]. Properties of the projection operator are explored in detail in the following section.

Remark. Essentially, a convex set has the following property. For any two points x, y ∈ E where E is convex, all the points on the connecting line from x to y are also in E.

Choose θ b as a boundary point so that f (θ b ) = δ. Then the following holds:

where ∇f

For any 0 < λ ≤ 1:

and taking the limit as λ → 0 yields (1).

3 Projection Definition 5. The Projection Operator for two vectors θ, y ∈ R k is now introduced as

where

. Note that the following are notationally equivalent Proj(θ, y) = Proj(θ, y, f ) when the exact structure of the convex function f is of no importance.

Remark. A geometrical interpretation of (2) follows. Define a convex set Ω 0 as

and let Ω 1 represent another convex set such that

From ( 3) and ( 4)

From the definition of the projection operator in (7) θ is not modified when θ ∈ Ω 0 . Let

represent an annulus region. Within Ω A the projection algorithm subtracts a scaled component of y that is normal to boundary θ|f (θ) = λ}. When λ = 0, the scaled normal component is 0, and when λ = 1, the component of y that is normal to the boundary Ω 1 is entirely subtracted from y, so that Proj(θ, y, f ) is tangent to the boundary θ|f (θ) = 1 . This discussion is visualized in Figure 1. Remark. Note that (∇f (θ)) T Proj(θ, y) = 0∀θ when f (θ) = 1 and that the general structure of the algorithm is as follows

for some time varying α when the modification is triggered. Multiplying the left hand side of the equation by (∇f (θ)) T and solving for α one finds that

and thus the algorithm takes the form

where the modification is active. Notice that the f (θ) has been added to the definition, making (7) continuous.

Lemma 6. One important property of the projection operator follows. Given θ * ∈ Ω 0 ,

Proof. Note that

and using Lemma 4

Definition 7 (Projection Operator). The general form of the projection operator is the n × m matrix extension to the vector definition above.

The application of the projection algorithm in adaptive control is explored below.

Lemma 9. If an initial value problem, i.e. adaptive control algorithm with adaptive law and initial conditions, is defined by:

1. θ = P roj(θ, y, f )

Proof. Taking the time derivative of the convex function

Substitution of (9) into (2) leads to

Remark. Given θ 0 ∈ Ω 0 , θ may increase up to the boundary where f (θ) = 1. However, θ never leaves the convex set Ω 1 .

Example 10 (Projection Algorithm in Adaptive Control Law). Let Θ(t) : R + → R m×n represent a time varying feedback gain in a dynamical system. This feedback gain is implemented as:

where u ∈ R n represents the control input and x ∈ R m the state vector. The time varying feedback gain is adjusted using the following adaptive law Θ = Proj(Θ, -xe T P B, F ) where e ∈ R m is an error signal in the state vector space, P ∈ R m×m is a square matrix derived from a Lyapunov relationship and B ∈ R m×n is the input Jacobian for the LTI system to be controlled and

The projection algorithm operates with the family of convex functions

Then, the components of the convex vector function F are chosen as

Each i-th component of F is associated with two constant scalar quantities ϑ i and ε i . From (10), f i (θ i ) = 0 when θ i = ϑ i , and f i (θ i ) = 1 when θ i = ϑ i + ε i . If the initial condition for Θ is such that Θ(t = 0) ∈ Θ 0 = [θ 0,1 . . . θ 0,m ] where {θ 0,i |f i (θ i ) ≤ 0 i = 1 to m}, then each θ i satisfies all three conditions for Lemma 9. Thus θ i (t) ≤ ϑ i + ǫ i ∀t ≥ 0.

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