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

|θ( t) |

s, σs =

σ

− 1

if M

(8.4.18)



0

0 ≤ |θ( t) | ≤ 2 M 0



M 0

σ 0

if |θ( t) | > 2 M 0

shown in Figure 8.4 where the design constants M 0 , σ 0 are as defined in

the discontinuous switching σ-modification. In this case the adaptive law is

given by

˙ θ = γ 1 u − γσsθ, 1 = y − θu

and in terms of the parameter error

˙˜ θ = γ 1 u − γσsθ, 1 = −˜ θu + η

(8.4.19)

The time derivative of V (˜

θ) = ˜ θ 2 along the solution of (8.4.18), (8.4.19) is

2 γ

given by

2

˙

d 2

V = − 2

˜

1

˜

0

1 + 1 η − σsθθ ≤ −

− σ θθ +

(8.4.20)

2

s

2

Now

σ ˜

sθθ

= σs( θ − θ∗) θ ≥ σs|θ| 2 − σs|θ||θ∗|

≥ σs|θ|( |θ| − M 0 + M 0 − |θ∗|)

8.4. MODIFICATIONS FOR ROBUSTNESS: SIMPLE EXAMPLES 561

i.e.,

σ ˜

sθθ ≥ σs|θ|( |θ| − M 0) + σs|θ|( M 0 − |θ∗|) ≥ 0

(8.4.21)

where the last inequality follows from the fact that σs ≥ 0 , σs( |θ| − M 0) ≥ 0

and M

˜

0 > |θ∗|. Hence, −σsθθ ≤ 0. Therefore, in contrast to the fixed σ-

modification, the switching σ can only make ˙

V more negative. We can also

verify that

−σ ˜

˜

sθθ ≤ −σ 0 θθ + 2 σ 0 M 2

0

(8.4.22)

which we can use in (8.4.20) to obtain

2

˙

d 2

V ≤ − 1 − σ ˜

θθ + 2 σ

0

(8.4.23)

2

0

0 M 2

0 + 2

The boundedness of V and, therefore, of θ follows by using the same pro-

cedure as in the fixed σ case to manipulate the term σ ˜

0 θθ in (8.4.23) and

express ˙

V as

˙

d 2

|θ∗| 2

V ≤ −αV + 2 σ

0

0 M 2

0 +

+ σ

2

0

2

where 0 ≤ α ≤ σ 0 γ which implies that ˜

θ converges exponentially to the

residual set

γ

Ds = ˜

θ |˜

θ| 2 ≤

( d 2

α 0 + σ 0 |θ∗| 2 + 4 σ 0 M 20)

The size of Ds is larger than that of Dσ in the fixed σ-modification case

because of the additional term 4 σ 0 M 20. The boundedness of ˜ θ implies that

θ, ˙ θ, 1 ∈ L∞ and, therefore, property (i) of the unmodified adaptive law in

the ideal case is preserved by the switching σ-modification when η = 0 but

η ∈ L∞.

As in the fixed σ-modification case, the switching σ cannot guarantee

the L 2 property of 1 , ˙ θ in general when η = 0. The bound for 1 in m.s.s.

follows by integrating (8.4.20) to obtain

t+ T

t+ T

2

σ ˜

2

sθθdτ +

1 dτ ≤ d 2

0 T + c 1

t

t

where c

˜

1 = 2 sup t≥ 0[ V ( t) − V ( t + T )] , ∀t ≥ 0 , T ≥ 0. Because σsθθ ≥ 0 it follows that

σ ˜

˜

sθθ, 1 ∈ S( d 2

0). From (8.4.21) we have σsθθ ≥ σs|θ|( M 0 −

|θ∗|) and, therefore,

( σ ˜

θθ)2

σ 2

s

˜

s |θ| 2 ≤

≤ cσ θθ

( M

s

0 − |θ∗|)2

562

CHAPTER 8. ROBUST ADAPTIVE LAWS

for some constant c that depends on the bound for σ 0 |θ|, which implies that

σs|θ| ∈ S( d 20). Because

| ˙ θ| 2 ≤ 2 γ 2 21 u 2 + 2 γ 2 σ 2 s|θ| 2

it follows that | ˙ θ| ∈ S( d 20). Hence, the adaptive law with the switching- σ

modification given by (8.4.19) guarantees that

(ii)

1 , ˙

θ ∈ S( d 20)

In contrast to the fixed σ-modification, the switching σ preserves the ideal

properties of the adaptive law, i.e., when η disappears ( η = d 0 = 0), equation

(8.4.20) implies (because −σ ˜

˜

sθθ ≤ 0) that 1 ∈ L 2 and

σsθθ ∈ L 2, which,

in turn, imply that ˙ θ ∈ L 2. In this case if ˙ u ∈ L∞, we can also establish

as in the the ideal case that 1( t) , ˙ θ( t) → 0 as t → ∞, i.e., the switching

σ does not destroy any of the ideal properties of the unmodified adaptive

law. The only drawback of the switching σ when compared with the fixed

σ is that it requires the knowledge of an upper bound M 0 for |θ∗|. If M 0 in

(8.4.18) happens not to be an upper bound for |θ∗|, then the adaptive law

(8.4.18) has the same properties and drawbacks as the fixed σ-modification

(see Problem 8.7).

(c) 1 -modification [172]. Another attempt to eliminate the main drawback

of the fixed σ-modification led to the following modification:

w( t) = | 1 |ν 0

where ν 0 > 0 is a design constant. The adaptive law becomes

˙ θ = γ 1 u − γ| 1 |ν 0 θ, 1 = y − θu

and in terms of the parameter error

˙˜ θ = γ 1 u − γ| 1 |ν 0 θ, 1 = −˜ θu + η

(8.4.24)

The logic behind this choice of w is that because in the ideal case 1 is

guaranteed to converge to zero (when ˙ u ∈ L∞), then the leakage term w( t) θ

will go to zero with 1 when η = 0; therefore, the ideal properties of the

adaptive law (8.4.24) when η = 0 will not be affected by the leakage.

8.4. MODIFICATIONS FOR ROBUSTNESS: SIMPLE EXAMPLES 563

The time derivative of V (˜

θ) = ˜ θ 2 along the solution of (8.4.24) is given

2 γ

by

˜

˙

θ 2

|θ∗| 2

V = − 2

˜

1 + 1 η − | 1 |ν 0 θθ ≤ −| 1 |

| 1 | + ν 0

− ν

− d

2

0

2

0

(8.4.25)

where the inequality is obtained by using ν ˜

˜

θ 2

|θ∗| 2

0 θθ ≤ −ν 0

+ ν

. It is

2

0 2

clear that for |

˜

θ 2

|θ∗| 2

|θ∗| 2

1 | + ν 0

≥ ν

+ d

( ν

+ d

2

0 2

0, i.e., for V ≥ V 0 =

1

γν

0

0)

0

2

we have ˙

V ≤ 0, which implies that V and, therefore, ˜

θ, θ ∈ L∞. Because

1 = − ˜

θu + η and u ∈ L∞ we also have that 1 ∈ L∞, which, in turn, implies

that ˙ θ ∈ L∞. Hence, property (i) is also guaranteed by the 1-modification

despite the presence of η = 0.

Let us now examine the L 2 properties of 1 , ˙ θ guaranteed by the unmod-

ified adaptive law ( w( t) ≡ 0) when η = 0. We rewrite (8.4.25) as

2

2

˜

˙

d 2

|

θ 2

|θ∗| 2

d 2

V ≤ − 1 + 0 − |

˜

θ 2 − |

1 − 1 |ν 0

+ |

+ 0

2

2

1 |ν 0

1 |ν 0 θ∗ ˜

θ ≤ − 2

2

1 |ν 0

2

2

by using the inequality −a 2 ± ab ≤ − a 2 + b 2 . If we repeat the use of the

2

2

same inequality we obtain

2

˙

d 2

|θ∗| 4

V ≤ − 1 + 0 + ν 2

(8.4.26)

4

2

0

4

Integrating on both sides of (8.4.26), we establish that 1 ∈ S( d 20 + ν 20).

Because u, θ ∈ L∞, it follows that | ˙ θ| ≤ c| 1 | for some constant c ≥ 0 and

therefore ˙ θ ∈ S( d 20 + ν 20). Hence, the adaptive law with the 1-modification

guarantees that

(ii)

1 , ˙

θ ∈ S( d 20 + ν 20)

which implies that 1 , ˙ θ are of order of d 0 , ν 0 in m.s.s.

It is clear from the above analysis that in the absence of the disturbance

i.e., η = 0, ˙

V cannot be shown to be negative definite or semidefinite unless

ν

˜

0 = 0. The term | 1 |ν 0 θθ in (8.4.25) may make ˙

V positive even when η = 0

and therefore the ideal properties of the unmodified adaptive law cannot

be guaranteed by the adaptive law with the 1-modification when η = 0

unless ν 0 = 0. This indicates that the initial rationale for developing the

1-modification is not valid. It is shown in [172], however, that if u is PE,

564

CHAPTER 8. ROBUST ADAPTIVE LAWS

then 1( t) and therefore w( t) = ν 0 | 1( t) | do converge to zero as t → ∞ when η( t) ≡ 0 , ∀t ≥ 0. Therefore the ideal properties of the adaptive law can be

recovered with the 1-modification provided u is PE.

Remark 8.4.1

(i) Comparing the three choices for the leakage term w( t), it is clear that

the fixed σ- and 1-modification require no a priori information about

the plant, whereas the switching- σ requires the design constant M 0 to

be larger than the unknown |θ∗|. In contrast to the fixed σ- and 1-

modifications, however, the switching σ achieves robustness without

having to destroy the ideal properties of the adaptive scheme. Such

ideal properties are also possible for the 1-modification under a PE

condition[172].

(ii) The leakage −wθ with w( t) ≥ 0 introduces a term in the adaptive law

that has the tendency to drive θ towards θ = 0 when the other term

(i.e., γ 1 u in the case of (8.4.9)) is small. If θ∗ = 0, the leakage term

may drive θ towards zero and possibly further away from the desired

θ∗. If an a priori estimate ˆ

θ∗ of θ∗ is available the leakage term −wθ

may be replaced with the shifted leakage −w( θ − ˆ

θ∗), which shifts the

tendency of θ from zero to ˆ

θ∗, a point that may be closer to θ∗. The

analysis of the adaptive laws with the shifted leakage is very similar to

that of −wθ and is left as an exercise for the reader.

(iii) One of the main drawbacks of the leakage modifications is that the es-

timation error 1 and ˙ θ are only guaranteed to be of the order of the

disturbance and, with the exception of the switching σ-modification, of

the order of the size of the leakage design parameter, in m.s.s. This

means that at steady state, we cannot guarantee that 1 is of the order

of the modeling error. The m.s.s. bound of 1 may allow 1 to exhibit

“bursting,” i.e., 1 may assume values higher than the order of the mod-

eling error for some finite intervals of time. One way to avoid bursting

is to use PE signals or a dead-zone modification as it will be explained

later on in this chapter

8.4. MODIFICATIONS FOR ROBUSTNESS: SIMPLE EXAMPLES 565

(iv) The leakage modification may be also derived by modifying the cost

2

function J( θ) = 1 = ( y−θu)2 , used in the ideal case, to

2

2

( y − θu)2

θ 2

J( θ) =

+ w

(8.4.27)

2

2

Using the gradient method, we now obtain

˙ θ = −γ∇J = γ 1 u − γwθ

which is the same as (8.4.9). The modified cost now penalizes θ in

addition to 1 which explains why for certain choices of w( t) the drifting

of θ to infinity due to the presence of modeling errors is counteracted.

8.4.2

Parameter Projection

An effective method for eliminating parameter drift and keeping the param-

eter estimates within some a priori defined bounds is to use the gradient pro-

jection method to constrain the parameter estimates to lie inside a bounded

convex set in the parameter space. Let us illustrate the use of projection for

the adaptive law

˙ θ = γ 1 u, 1 = y − θu

We like to constrain θ to lie inside the convex bounded set

g( θ) = θ θ 2 ≤ M 20

where M 0 ≥ |θ∗|. Applying the gradient projection method, we obtain





 γ

˙

1 u

if |θ| < M 0

˜

θ =

˙ θ =

or if |θ| = M

(8.4.28)



0 and θ 1 u ≤ 0

 0

if |θ| = M 0 and θ 1 u > 0

1

= y − θu = −˜

θu + η

which for |θ(0) | ≤ M 0 guarantees that |θ( t) | ≤ M 0 , ∀t ≥ 0. Let us now

analyze the above adaptive law by considering the Lyapunov function

˜

θ 2

V = 2 γ

566

CHAPTER 8. ROBUST ADAPTIVE LAWS

whose time derivative ˙

V along (8.4.28) is given by





 − 21 + 1 η

if |θ| < M 0

˙

V =

or if |θ| = M

(8.4.29)



0 and θ 1 u ≤ 0

 0

if |θ| = M 0 and θ 1 u > 0

Let us consider the case when ˙

V = 0, |θ| = M 0 and θ 1 u > 0. Using the

expression 1 = −˜

θu + η, we write ˙

V = 0 = − 21 + 1 η − ˜ θ 1 u. The last term

in the expression of ˙

V can be written as

˜

θ 1 u = ( θ − θ∗) 1 u = M 0sgn( θ) 1 u − θ∗ 1 u

Therefore, for θ 1 u > 0 and |θ| = M 0 we have

˜

θ 1 u = M 0 | 1 u| − θ∗ 1 u ≥ M 0 | 1 u| − |θ∗|| 1 u| ≥ 0

where the last inequality is obtained by using the assumption that