Robust Adaptive Control by Petros A. Ioannou, Jing Sun - HTML preview

Setting β = 0, (4.3.78), (4.3.79) become

˙ θ = P φ

˙

P φφ P

P

= −

,

P (0) = P

m 2

0

(4.3.80)

In terms of the P − 1 we have

d

φφ

P − 1 =

dt

m 2

which implies that d( P − 1) ≥ 0, and, therefore, P − 1 may grow without bound.

dt

In the matrix case, this means that P may become arbitrarily small and slow

down adaptation in some directions. This is the so-called covariance wind-up

problem that constitutes one of the main drawbacks of the pure least-squares

algorithm.

Despite its deficiency, the pure least-squares algorithm has the unique

property of guaranteeing parameter convergence to constant values as de-

scribed by the following theorem:

Theorem 4.3.4 The pure least-squares algorithm (4.3.80) guarantees that

(i)

, ns, θ, ˙ θ, P ∈ L∞.

(ii)

, ns, ˙ θ ∈ L 2 .

(iii) lim t→∞ θ( t) = ¯

θ, where ¯

θ is a constant vector.

(iv) If ns, φ ∈ L∞ and φ is PE, then θ( t) converges to θ∗ as t → ∞.

Proof From (4.3.80) we have that ˙

P ≤ 0, i.e., P ( t) ≤ P 0. Because P ( t) is nonin-

creasing and bounded from below (i.e., P ( t) = P ( t) ≥ 0 , ∀t ≥ 0) it has a limit, i.e.,

lim P ( t) = ¯

P

t→∞

where ¯

P = ¯

P

≥ 0 is a constant matrix. Let us now consider

d

φφ ˜

θ

( P − 1 ˜

θ) = −P − 1 ˙

P P − 1 ˜

θ + P − 1 ˙˜

θ =

+ φ = 0

dt

m 2

where the last two equalities are obtained by using ˙ θ = ˙˜

θ, d P − 1 = −P − 1 ˙

P P − 1

dt

˜

and

= − θ φ = − φ ˜ θ. Hence, P − 1( t)˜

θ( t) = P − 1 ˜

θ(0), and, therefore, ˜

θ( t) =

m 2

m 2

0

P ( t) P − 1 ˜

˜

˜

0

θ(0) and lim t→∞ θ( t) = ¯

P P − 1

0

θ(0), which implies that lim t→∞ θ( t) = θ∗ +

¯

P P − 1 ˜

0

θ(0) = ¯

θ.

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

Because P ( t) ≤ P

˜

0 and ˜

θ( t) = P ( t) P − 1

0

θ(0) we have θ, ˜

θ ∈ L∞, which, together

˜

with φ ∈ L

θ φ and , n

m

∞, implies that

m = − m

s ∈ L∞. Let us now consider the

function

˜

θ P − 1( t)˜

θ

V (˜

θ, t) =

2

The time derivative ˙

V of V along the solution of (4.3.80) is given by

˜

2

2

˙

θ φφ ˜

θ

m 2

m 2

V = ˜

θ φ +

= − 2 m 2 +

= −

≤ 0

2 m 2

2

2

which implies that V ∈ L∞, m ∈ L 2; therefore, , ns ∈ L 2. From (4.3.80) we have

|φ|

| ˙ θ| ≤ P

| m|

m

Because P, φ , m ∈ L

m

∞ and

m ∈ L 2, we have ˙ θ ∈ L∞

L 2, which completes the

proof for (i), (ii), and (iii). The proof of (iv) is given in Section 4.8.

✷

Remark 4.3.16

(i) We should note that the convergence rate of θ( t) to θ∗ in Theorem 4.3.4

is not guaranteed to be exponential even when φ is PE. As shown in the

proof of Theorem 4.3.4 (iv) in Section 4.8, P ( t), ˜

θ( t) satisfy

¯

m

P − 1

P ( t) ≤

I, |˜

θ( t) | ≤

0

¯

m |˜ θ(0) |, ∀t > T

( t − T

0

0) α 0

( t − T 0) α 0

where ¯

m = sup t m 2( t), i.e., |˜ θ( t) | is guaranteed to converge to zero with

a speed of 1 .

t

(ii) The convergence of θ( t) to ¯

θ as t → ∞ does not imply that ˙ θ( t) → 0 as

t → ∞ (see examples in Chapter 3).

(iii) We can establish that , ˙ θ → 0 as t → ∞ if we assume that ˙ φ/m, ˙

m/m ∈

L∞ as in the case of the gradient algorithm based on the instantaneous

cost.

Pure Least-Squares with Covariance Resetting

The so called wind-up problem of the pure least-squares algorithm is avoided

by using various modifications that prevent P ( t) from becoming singular.

4.3. ADAPTIVE LAWS WITH NORMALIZATION

197

One such modification is the so-called covariance resetting described by

˙ θ = P φ

˙

P φφ P

P

= −

,

P ( t+

m 2

r ) = P 0 = ρ 0 I

(4.3.81)

where tr is the time for which λmin( P ( t)) ≤ ρ 1 and ρ 0 > ρ 1 > 0 are some design scalars. Because of (4.3.81), P ( t) ≥ ρ 1 I ∀t ≥ 0; therefore, P is

guaranteed to be positive definite for all t ≥ 0.

Strictly speaking, (4.3.81) is no longer the least-squares algorithm that

we developed by setting ∇J( θ) = 0 and β = 0. It does, however, behave as

a pure least-squares algorithm between resetting points. The properties of

(4.3.81) are similar to those of the gradient algorithm based on the instan-

taneous cost. In fact, (4.3.81) may be viewed as a gradient algorithm with

time-varying adaptive gain P .

Theorem 4.3.5 The pure least-squares with covariance resetting algorithm

(4.3.81) has the following properties:

(i)

, ns, θ, ˙ θ ∈ L∞.

(ii)

, ns, ˙ θ ∈ L 2 .

(iii) If ns, φ ∈ L∞ and φ is PE then θ( t) converges exponentially to θ∗.

Proof The covariance matrix P ( t) has elements that are discontinuous functions

of time whose values between discontinuities are defined by the differential equation

(4.3.81). At the discontinuity or resetting point tr, P ( t+

r ) = P 0 = ρ 0 I ; therefore,

P − 1( t+

r ) = ρ− 1

0 I . Between discontinuities d P − 1( t) ≥ 0 , i.e., P − 1( t

dt

2) −P − 1( t 1) ≥ 0

∀t 2 ≥ t 1 ≥ 0 such that tr ∈ [ t 1 , t 2], which implies that P − 1( t) ≥ ρ− 1

0 I , ∀t ≥ 0.

Because of the resetting, P ( t) ≥ ρ 1 I, ∀t ≥ 0. Therefore, (4.3.81) guarantees that

ρ 0 I ≥ P ( t) ≥ ρ 1 I, ρ− 1

1 I ≥ P − 1( t) ≥ ρ− 1

0 I ,

∀t ≥ 0

Let us now consider the function

˜

θ P − 1 ˜

θ

V (˜

θ) =

(4.3.82)

2

where P is given by (4.3.81). Because P − 1 is a bounded positive definite symmetric

matrix, it follows that V is decrescent and radially unbounded in the space of ˜

θ.

Along the solution of (4.3.81) we have

˙

1

d( P − 1)

1

d( P − 1)

V =

˜

θ

˜

θ + ˜

θ P − 1 ˙˜

θ = − 2 m 2 + ˜

θ

˜

θ

2

dt

2

dt

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

Between resetting points we have from (4.3.81) that d( P − 1) = φφ ; therefore,

dt

m 2

2

˙

1 (˜

θ φ)2

m 2

V = − 2 m 2 +

= −

≤ 0

(4.3.83)

2

m 2

2

∀t ∈ [ t 1 , t 2] where [ t 1 , t 2] is any interval in [0 , ∞) for which tr ∈ [ t 1 , t 2].

At the points of discontinuity of P , we have

1

V ( t+

˜

r ) − V ( tr) =

θ ( P − 1( t+

2

r ) − P − 1( tr)) ˜

θ

Because P − 1( t+

r ) = 1 I , P − 1( t

I, it follows that V ( t+

ρ

r) ≥ 1

r ) − V ( tr) ≤ 0, which

0

ρ 0

implies that V ≥ 0 is a nonincreasing function of time for all t ≥ 0. Hence, V ∈ L∞

and lim t→∞ V ( t) = V∞ < ∞. Because the points of discontinuities tr form a set

of measure zero, it follows from (4.3.83) that m,

∈ L 2. From V ∈ L∞ and

ρ− 1

1 I ≥ P − 1( t) ≥ ρ− 1

0 I we have ˜

θ ∈ L∞, which implies that , m ∈ L∞. Using

m ∈ L∞

L 2 and ρ 0 I ≥ P ≥ ρ 1 I we have ˙ θ ∈ L∞

L 2 and the proof of (i) and

(ii) is, therefore, complete.

The proof of (iii) is very similar to the proof of Theorem 4.3.2 (iii) and is

omitted.

✷

Modified Least-Squares with Forgetting Factor

When β > 0, the problem of P ( t) becoming arbitrarily small in some direc-

tions no longer exists. In this case, however, P ( t) may grow without bound

since ˙

P may satisfy ˙

P > 0 because βP > 0 and the fact that P φφ P is only

m 2

positive semidefinite.

One way to avoid this complication is to modify the least-squares algo-

rithm as follows:

˙ θ = P φ

˙

P

=

βP − P φφ P

if P ( t) ≤ R

m 2

0

(4.3.84)

0

otherwise

where P (0) = P 0 = P 0 > 0, P 0 ≤ R 0 and R 0 is a constant that serves as an upper bound for P . This modification guarantees that P ∈ L∞

and is referred to as the modified least-squares with forgetting factor. The

above algorithm guarantees the same properties as the pure least-squares

with covariance resetting given by Theorem 4.3.5. They can be established

4.3. ADAPTIVE LAWS WITH NORMALIZATION

199

by choosing the same Lyapunov-like function as in (4.3.82) and using the

identity d P − 1 = −P − 1 ˙

P P − 1 to establish

dt

dP − 1 = −βP− 1 + φφ

if

P ≤ R

m 2

0

dt

0

otherwise

where P − 1(0) = P − 1

0

, which leads to

˜

˙

− 2 m 2 − β θ P − 1 ˜

θ if

P ≤ R

V =

2

2

0

− 2 m 2

otherwise

2

Because ˙

V ≤ − 2 m 2 ≤ 0 and P ( t) is bounded and positive definite ∀t ≥ 0,

2

the rest of the analysis is exactly the same as in the proof of Theorem 4.3.5.

Least-Squares with Forgetting Factor and PE

The covariance modifications described above are not necessary when ns,

φ ∈ L∞ and φ is PE. The PE property of φ guarantees that over an interval of

time, the integral of −P φφ P is a negative definite matrix that counteracts

m 2

the effect of the positive definite term βP with β > 0 in the covariance

equation and guarantees that P ∈ L∞. This property is made precise by the

following corollary:

Corollary 4.3.2 If ns, φ ∈ L∞ and φ is PE then the recursive least-squares

algorithm with forgetting factor β > 0 given by (4.3.78) and (4.3.79) guar-

antees that P, P − 1 ∈ L∞ and that θ( t) converges exponentially to θ∗.

The proof is presented in Section 4.8.

The use of the recursive least-squares algorithm with forgetting factor

with φ ∈ L∞ and φ PE is appropriate in parameter estimation of stable

plants where parameter convergence is the main objective. We will address

such cases in Chapter 5.

Let us illustrate the design of a least-squares algorithm for the same

system considered in Example 4.3.1.

Example 4.3.4 The system

z = W ( s) A sin( ωt + ϕ)

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CHAPTER 4. ON-LINE PARAMETER ESTIMATION

where A, ϕ are to be estimated on-line is rewritten in the form

z = θ∗ φ 0

where θ∗ = [ A 1 , A 2] , φ 0 = W ( s) φ, φ = [sin ωt, cos ωt] . The least-squares algorithm for estimating θ∗ is given by

˙ θ = P φ 0

˙

φ

P

= βP − P 0 φ 0 P,

P (0) = ρ

m 2

0 I

where = z−θ φ 0 , m 2 = 1 + φ

m 2

0 φ 0 and β ≥ 0, ρ 0 > 0 are design constants. Because

φ 0 is PE, no modifications are required. Let us simulate the above scheme when

A = 10 , ϕ = 16 ◦ = 0 . 279 rad , ω = 2 rad/sec , W ( s) = 2 . Figure 4.10 gives the s+2

time response of ˆ

A and ˆ

ϕ, the estimate of A and ϕ, respectively, for different values

of β. The simulation results indicate that the rate of convergence depends on the

choice of the forgetting factor β. Larger β leads to faster convergence of ˆ

A, ˆ

ϕ to

A = 10 , ϕ = 0 . 279, respectively.

4.3.7