SINGULARLY PERTURBED DERIVATIVE COUPLING-FILTER: THE SISO CASE

SINGULARLY PERTURBED DERIVATIVE COUPLING-FILTER: THE SISO CASE Hugo M´ endez D. ∗,1 Mois´ es Bonilla E. ∗ Michel Malabre ∗∗

∗ CINVESTAV-IPN. Depto. Control Autom´ atico. AP 14-740. M´exico 07000, MEXICO. [email protected] ∗∗ IRCCyN, CNRS UMR 6597, B.P. 92101, 44321 Nantes, Cedex 03, FRANCE. [email protected].

Abstract: In some control and observations problems, it is convenient, at least from the analysis point of view, to use non proper systems. However, for their implementation, proper approximations have to be designed. We show here how c exponential approximations can be rather easily designed. Copyright 2007 IFAC. Keywords: Linear systems, PD controllers. proper approximations.

NOTATION is the complex Laplace variable.  is the complex √ number −1. k·k2 stands for the Euclidean (L2 ) norm and k·k1 stands for the L1 norm. Given two functions of ε, f and ϕ, with ε > 0 and ϕ(ε) > 0, we write: f (ε) = O(ϕ(ε)) when there exist constants ε∗ > 0 and K > 0 such that |f (ε)| ≤ Kϕ(ε) for ε ∈ (0, ε∗ ) (see Hardy (1975) for details). χik denotes a k × 1 vector whose i-th component is 1 and the others are zero (in the case that i > k all its components are taken equal to zero). Ik denotes a k × k identity matrix, or simply I. BDM {X1 , ..., Xk } denotes a block diagonal matrix whose diagonal blocks are the matrices X1 , . . . , Xk . Tu {vT } (T` {v}) denotes an upper (lower) triangular Toeplitz matrix with first row (column) vT (v).

s

(Jackson, 1960). With the introduction of the socalled generalized systems (also named descriptor, singular and implicit systems) by Rosenbrock (1970), it became possible to study derivative actions from a more structural point of view. Also some control problems have been considered in the framework of the implicit systems using non proper compensators (see for example Willems (1982), Bonilla and Malabre (1999), Bonilla and Malabre (2000) and Bonilla and Malabre (2001)). Then, for their effective implementation, proper approximations must be designed which do not “much” alter the control objective. Let us consider the following problem: Problem 1. Given any non-proper system,
κ+1 T N w(t) ˙ = w(t) − Γv(t) ;

1. INTRODUCTION The use of derivative actions was very familiar to practical control pioneers (see for example Coinsidine (1957)) who proposed useful approximations of such derivative actions, see also 1

PhD student of CINVESTAV–IPN

u(t) = χ

κ+1

Σc :

w(t)

(1)

where N = T` {χ2κ+1 } and Γ = [ γ0 γ1 · · · γκ ]T , with γ0 6= 0. N is a nilpotent matrix with index of nilpotency κ + 1, Γ is a map such that the matrix [N Γ] is epic, and v , u and w are the input, the output and the descriptor variable, respectively. We assume that the first κ derivatives of v exist and v is bounded and that the first κ derivatives of v exist and are bounded Lipschitz continuous:

H1. H2.

|v(t)| ≤ K0 ,

for all

t ≥ 0,

with i K0 > 0. i d v(t1 ) − d v(t2 ) ≤ |v(t1 ) − v(t2 )| ≤ L0 |t1 − t2 |, i i dt dt Li |t1 − t2 |, ∀ t1 , t2 > 0, with Li > 0. i ∈ {0, 1, . . . , κ}.

˙ Find a strictly proper filter, Σf : ζ(t) = A(ε)ζ(t) + B(ε)u(t), y(t) = Cζ(t), and find positive constants β and ε, such that (ε∗ is some positive constant):

Q1)

lim |y(t) − u(t)| ≤ K e−βt , ∀ t > 0.

ε→0 Σf

Q2) is internally stable ∀ ε ∈ (0, ε∗ ). Q3) The overall system, Σf ◦ Σc is externally equivalent 2 to a proper system. Thus, we are looking for a proper filter, Σf , which makes proper the overall system, Σf ◦ Σc , and which output, y(t), exponentially tends to the non proper behaviour of Σc . The interest is to finally synthesize the overall proper system Σf ◦ Σc as a proper approximation of Σc . In Section 2 we propose a two-time-scale filter which aim is to couple the time derivatives of (1), in Section 3 we solve Problem 1 and in Section 4 we conclude. The Lemmas are proved in the Appendix.

2. SINGULARLY PERTURBED FILTER Let us consider the following singularly perturbed linear system (SPLS) (Kokotovi´c, 1999): x(t) ˙ = −βx(t) − εκ+1 (χ1 )T z(t) κ εz(t) ˙ = χκ x(t) − (Mκ − Uκ )z(t) + χκ u(t) κ κ y(t) = (χ1 )T z(t)

(2)

κ

R1

where x ∈ , z ∈ Rκ , u ∈ R1 and y ∈ R1 . κ is a given positive integer and β and ε are two positive real numbers. Uκ = Tu {(χ2κ )T } and Mκ is defined as 3 :  BDM {M1 , ..., Mκ/2 } if κ is even Mκ = (3) BDM {M , ..., M , 1} if κ is odd 1

(κ−1)/2

Mi = (sin θi )I2 + T` {(cos θi )χ2 } ; θ1 = π/(2κ) 2 θi+1 = θi + ∆θ , ∆θ = π/κ ; i = 1, . . . , κ − 1

If κ is odd then: zκκ = 1 and z iκ = [ 1 sin θi ]T And i<κ the transfer function of the fast system is:  −1  (7) χκ = 1 ∆B (s) (χ1 )T sIκ + (Mκ − Uκ ) κ κ
Q where ∆B (s) is equal to κ/2 (s + sin θi )2 + cos2 θi , i=1 Q(κ−1)/2 when κ is even, and it is equal to (s + 1) i=1
(s + sin θi )2 + cos2 θi , when κ is odd. An ad hoc model simplification procedure is the following (see section 2.5 of Kokotovi´c (1999)): 4 (1) Doing ε = 0 in (2) and disregarding the initial condition for z we get the slow model: T  zs (t) =

1 sin θ1 · · · 1 sin θ κ−1 1

(xs (t) + u(t))

2

xs (t) = x(0)e−βt ; ys (t) = xs (t) + (χ1 )T zs (t) κ

(2) And defining the fast scaling time, τ get the fast model:

= t/ε,

(8) we


dzf (τ ) = − Mκ − Uκ zf (τ ) dτ
−1 zf (0) = z0 + Mκ − Uκ χκ x(0)

zf (τ ) = z(τ ) − zs (τ ) ;

κ

(9) From the common ad hoc model simplification, we realize that singularly perturbed derivative coupling filter (SPDC-Filter) seems to track the input signal u(t). This input can be for example the output of a time derivative of order κ. The composite system (the time derivative plus the SPDC-Filter) will be proper (under certain conditions). In order to validate the above ad hoc procedure we first obtain a two-time-scale model of (2), we next find their eigenvalues properties, and we finally validate the state approximation (8) and (9). 2.1 Two-time-scale model We need the following two key results:

2

(4)

The slow part of the SPLS (2) is composed by a first order system with an eigenvalue in −β. The fast part of the SPLS (2) is composed by a normalized low pass Butterworth filter (see for example Daniells (1974)). Lemma 2. The inverse matrix of

(Mκ − Uκ )

(Mκ − Uκ )−1 = [zij ]

(5)

zii = (sin θi )I2 − T` {(cos2 θi )χ2 } + Tu {(χ2 )T } z

2

ij i
=



1 sin θi

T 

sin θj 1

2

2

; z

is:

ij i>j

=0

Lemma 3. If  3 ε1 =

0 < ε < min{1, ε1 }, 1

κ 2 (β + 1) + κ 2 + 2

there then exists

p

where:

(β + 1)κ1/2

L(ε) ∈ Rκ+1 ,

−1

(10)

solution of:

χκ + (Mκ − εβIκ − Uκ )L + εκ+2 L(χ1 )T L = 0 κ

κ

(11)

Furthermore, an approximated solution of (11) is:
L(ε) = Iκ + εβ(Mκ − Uκ )−1 L(0) + O(ε2 ) (12)  T 1 and For κ odd: L(0) = − 1 sin θ1 · · · 1 sin θ κ−1 2  T for κ even: L(0) = − 1 sin θ1 · · · 1 sin θ κ2 .

(6)

As introduced by Willems (1983) two models are externally equivalent, if the corresponding sets of all possible trajectories for their external behaviour (in our case the input-output trajectories) are the same. 3 If κ = 1, then: U = 0 and M = 1. κ κ

Lemma 4. Under the same conditions as those of Lemma 3 there exists H(ε) ∈ R1×κ solution of:
H (Mκ − Uκ ) − ε β − εκ+1 (χ1 )T L Iκ κ
(13) κ+2 1 T 1 T κ+1 +ε

4

L(χ ) κ

−ε

(χ ) κ

=0

We assume κ odd. The even case is similar.

Furthermore, an approximated solution of (13) is: κ+1

H(ε) = ε

κ+2

H(0) + O(ε

)



1 For κ odd: H(0) = sin θ1 1 · · · sin θ κ−1 2  for κ even: H(0) = sin θ1 1 · · · sin θ κ2 1 .

1



(14)

(4) Let us note that the successive total derivatives of (19) and (20), with respect to ε are: 5

and

dψf (sf , ε)/dε = (∂ψf /∂sf )(dsf /dε) + ∂ψf /∂ε = 0 ∂ψf di sf dsf di ∂ψf di ψf (sf , ε) = + i i dε ∂sf dε dεi ∂sf dε   di ∂ψf + i = 0 ; i = 2, ..., n + 1 dε ∂ε Since at ε = 0, the eigenvalues s = λB i (i = 1, . . . , κ)



From Lemmas 3 and 4 we define the variables:


(15)

ξ = 1 − εH(ε)L() x − εH(ε)z ; η = L()x + z

leading to the two-time-scale model of (2):


˙ = − β − εκ+1 (χ1 )T L(ε) ξ − H(ε)χκ u ξ(t) κ


κ

εη(t) ˙ = − (Mκ − Uκ ) + εκ+2 L(ε)(χ1 )T η + χκ u κ




y(t) = (χ1 )T L(ε)ξ + (χ1 )T 1 − εL()H(ε) η κ

κ

(16)

κ

2.2 Eigenvalue properties Let us state the first principal result: n
1 o Theorem 5. If 0 < ε < min 1, ε1 , sin π/(2κ) κ+1 , there then exists ε∗0 > 0, such that (16) (and so (2)) is Hurwitz stable, and for all ε ∈ (0, ε∗0 ) its eigenvalues λi , are approximated as:
¯ (2i−1) (ε) λ0 (ε) = − β + εκ+1 + O(εκ+2 ), λ2i (ε) = λ
κ+2 λ(2i−1) (ε) = (1/ε) − cos θi +  sin θi + O(ε

)

2.3 Approximation of the SPDC-Filter We need the following two results: Lemma 6. If

PROOF. This proof is based on Theorem 3.1 and Corollary 3.1 of (Kokotovi´c, 1999):

Lemma 7. Let us define:



ψs (s, ε) = (s + β) − εκ+1 χ1

κ

ψf (sf , ε) = det



T

(19) 

L(ε)




sf Iκ + Mκ − Uκ − εκ+2 L(ε)(χ1 )T κ

(20) where ψs and ψf are the characteristic polynomials of the slow and fast subsystems, respectively, with the high-frequency scale sf = εs. (2) With respect to the slow subsystem characteristic polynomial, we have from (19) and (12), for the odd case: λ0 (ε) = − β +

eλ0 (ε)t = e−(β+ε

−β −

−

εκ+2 β

2

P(2κ−1)/2 i=1




sin θi + 1

+O(εκ+3 )

=

−β − εκ+1 + O(εκ+2 ); and for the even case: λ0 (ε)
Pκ/2 = = −β − εκ+1 − εκ+2 β 2 i=1 sin θi + O(εκ+3 ) = −β − εκ+1 + O(εκ+2 ).

(3) With respect to the fast subsystem characteristic polynomial, let us first note that the κ eigenvalues, λB i , of −(Mκ − Uκ ) are the roots of the Butterworth polynomial ∆B (s) and thus they are all distinct.

)t

+ O(εκ ) ,



⇒ ∃ ε∗1 > 0

Z

κτ

s.t.

for ε ∈ (0, ε∗1 )

Λκ (ε) := − (Mκ − Uκ ) + εκ+2 L(ε)(χ1 )T κ hκ (τ, ε) := (χ1 )T eΛκ (ε)τ χκ

(21)


(22)

κ

hκ (τ − σ, ε)dσ

ακ (τ, ε) := 0


1/(κ+1) √ If 0 < ε < min 1, ε1 , (1/ 2) sin π/(2κ) , there then exists k0 = O(1), k1 = O(1), and ε∗2 > 0 s.t.
Λκ (ε) = − (Mκ − Uκ ) + O(εκ+2 )
√ − sin θ1 − 2εκ+1 τ (23) |hκ (τ, ε)| ≤ k1 e


√ − sin θ1 − 2εκ+1 τ

|ακ (τ, ε) − 1| ≤ k0 e

for

+ εκ+1

ε ∈ (0, ε∗2 ).

Let us assume that the input signal is a bounded Lipschitz continuous function. Namely, there exist two positive constants K 0 and L0 such that:

εκ+1 (χ1 )T L(0) + εκ+2 β(Mκ − Uκ )−1 L(0) +O(εκ+3 ) = κ εκ+1



0 < ε < min 1, β/2 κ+1

(1) In the same way as Section 2.3 of (Kokotovi´c, 1999) the characteristic equation of the SPLS (16) (and so the SPLS (2), ψ(s, ε), is factorized as: (18)




 are all distinct, it is guaranteed that ∂ψ λB ∂sf i ,0 n+1 6= 0. This implies the existence of dn+1 λB i (ε)/dε for all ε ∈ (0, ε∗0 ), for some ε∗0 > 0, which implies B the existence of the Taylor series: λB i (ε) = λi (0) n B B n n +εdλi (0)/dε + · · · +(ε /n!)d λi (0)/dε + Rn+1 , with Rn+1 = O(εn+1 ). (5) Given the existence of the Taylor series, around ε = 0, of the roots of ψs (s, ε) and ψf (sf , ε) and the approximation (12) of L(ε), we get (17). (6) Let us finally note that the inequalities 0 < ε < 1 and εκ+1 < εκ < (1/ε) sin θ1 guarantee that the nearest eigenvalues to the imaginary axis, λ1 (ε) and λ2 (ε), remain in the open left half complex = sin θ1 all the plane. Furthermore, since sin ∆θ 2 eigenvalues remain also distinct. 2

(17) where for κ even: i = 1, . . . , κ/2, and for κ odd:
i = 1, . . . , (κ − 1)/2, with λκ (ε) = (1/ε) −1 + O(εκ+2 ) .

1 ψ(s, ε) = κ ψs (s, ε)ψf (sf , ε) = 0 ε



(24)

|u(t1 )| ≤ K 0 , |u(t1 ) − u(t2 )| ≤ L0 |t1 − t2 | ∀ t1 , t2 ≥ 0.

Let us state the second principal result:

Theorem 8. Let  0 < ε < min

such that p1 + 1   κ+1

p, q ∈ Z+

1, ε1 ,

β , 2

1 √ 2

sin

π (2κ)

,

1 q

= 1.

If: 

1 , 1q K0 L0

(25) 5

We work with ψf . ψs is done in the same way.


there then exists ε∗ ∈ 0, min{ε∗0 , ε∗1 , ε∗2 } , s.t. ∀ ε ∈ (0, ε∗ ), the output of (2) is approximated by:

N w(t) ˙ = w(t) − Γv(t) ; x(t) ˙ = −βx(t) − εκ+1 (χ1 )T z(t) κ κ κ+1 T κ ) w(t) + χ x(t) − (Mκ − Uκ )z(t) εz(t) ˙ = χ (χ κ

y(t) = u(t) + e

√ κ+1
x(0) e−(sin θ1 − 2ε )t/ε + O(ε1/p ) (26) ∀ t ≥ 0. Moreover, the “boundary layer” correction (9) is significant in the interval [0, t∗ ], where: 6

+ L(0)k0 + O(1)

t∗

= O((ε/(sin θ1 −

√

2εκ+1 )) ln(K

0

/ε1/p ))

(27)

after which κ+1

y(t) = u(t) + e−(β+ε

)t x(0)

+ O(ε1/p ) ∀ t ≥ t∗

(28)

PROOF. From (16), (12), (14), (21), and Theorem 5, we get (recall that εK 0 < 1): ξ(t) = eλ0 (ε)t ξ(0)




− εκ+1 H(0) + O(εκ+2 ) χκ

t

Z

eλ0 (ε)(t−σ) u(σ)dσ

κ

κ+1

= e−(β+ε η(t) = e y(t) =

Λκ (ε)t/ε

0 )t

ξ(0) + O(εκ ) t/ε

Z

Λκ (ε)(t/ε−σ) κ χ u(εσ)dσ κ

e

η(0) + 0




=e

e

Z

e

κ

κ

(29) Let us now define the following constants: Pκ−1 Pκ K 0 = |γκ | K0 +

i=0

|γi | Lκ−1−i ; L0 =

i=0

|γi | Lκ−i

(30) Then, under the same conditions as those of Theorem 8, the overall system (29) is Hurwitz stable for all ε ∈ [0, ε∗ ) and: lim y(t) − γ0 dκ /dtκ + γ1 dκ−1 /dtκ−1 + · · · ε→0 (31)
+ γκ−1 d/dt + γκ v(t) ≤ |x(0)| e−βt For solving Problem 1, it only remains to show that the overall system (29) is externally equivalent to a proper system, namely, the input–output trejectories can be described by a proper model: (1) Let us first define two invertible matrices: " # " # Q=




1 + O(ε) ξ(t) + (χ1 )T + O(εκ+2 ) η(t) κ −(β+εκ+1 )t ξ(0) + (χ1 )T Λκ (ε)t/ε η(0) κ t/ε + (χ1 )T Λκ (ε)(t/ε−σ) χκ u(εσ)dσ + O(ε) κ κ 0

κ+1

y(t) = (χ1 )T z(t)

−(β+εκ+1 )t

Iκ+1 0 0 0 1 0 Q0 0 Iκ

;

Iκ+1 0 0 0 1 0 R0 0 Iκ

R=

(32)

where Q0 is the solution of the following algebraic equation (N is the nilpotent matrix of system (1)):

From Lemma 7, we get: Q0

y(t) − e−(β+εκ+1 )t ξ(0) − (χ1 )T eΛκ (ε)t/ε η(0) − u(t) κ  Z t/ε ≤ O(ε) + u(t) (χ1 )T eΛκ (ε)(t/ε−σ) χκ dσ − 1 κ κ 0 Z t/ε
1 T Λκ (ε)(t/ε−σ) κ + (χ ) e χ u(εσ) − u(t) dσ κ κ 0

= O(ε) + |u(t)| |ακ (t/ε, ε) − 1|

Z

t/ε

+

h


|u(εσ) − u(t)| dσ
√

(t/ε − σ), ε

0

κ+1

− sin θ1 − 2ε

= O(ε) + k0 K 0 e Z t 1 + k1 L0 e− ε 0

t/ε

√ − sin θ1 − 2εκ+1 t/ε

+ εL0 k1




sin θ1 −

≤ O(ε1/p ) + k0 K 0 e

1 N ε



+ (Mκ − Uκ )−1 Q0 = −(Mκ − Uκ )−1 χκ (χκ+1 )T κ

κ+1

(33) (34)

R0 = −(1/ε)Q0 N

(2)

Let us next note that: Spectrum N ∩ then (33) has a unique solution (see Ch. VIII §3 of Gantmacher (1959)). (3) Let us now apply matrices Q and R to the matrices of the overall system (29): " # " # " # " # Spectrum (Mκ − Uκ )−1 = ∅,

Q

N 0 0 0 1 0 0 0 εIκ

N 0 0 0 1 0 0 0 εIκ

R=

Γ 0 0

; Q

Γ 0 Q0 Γ

=




≤ O(ε) + k0 K 0 e



+ εκ+1 K 0

√ sin θ1 − 2εκ+1 (t−¯ σ )/ε



√

(t − σ ¯ )d¯ σ

+ εκ+1 K 0

κ+1 2

2ε

Iκ+1 0 0 0 −β −εκ+1 (χ1 )T  R = Q κ χκ (χκ+1 )T χκ −(Mκ − Uκ ) κ




√ − sin θ1 − 2εκ+1 t/ε




=

From (15),
(12) and (14), we have: ξ(0)
κ+2 1 − εH(ε)L(ε) x(0) − εH(ε)z(0) =

1 + O(ε

= ) x(0)

+ O(εκ+2 )z(0) and η(0) = L(ε)x(0) + z(0) = (L(0) + O(ε)) x(0) + z(0). And from Theorem 5, we have (where k3 = O(1)): (χ1 )T eΛκ (ε)t/ε η(0) ≤ √ κ+1 2ε )t/ε





κ

k3 e−(sin θ1 −

. which implies (26). (28) is directly derived. 2 3. PROPER APPROXIMATION

"κ+1

κ

Iκ+1 0 0 −εκ+1 (χ1 )T R0 −β −εκ+1 (χ1 )T κ κ 0 χκ −(Mκ − Uκ )

#

κ



0 0 (χ1 )T κ



R=



(χ1 )T R0 0 (χ1 )T κ



κ

(4) Let us glimpse which shape has the matricial product (χ1κ )T R0 . For this, let us denote by ri and q the column vectors of R0 and Q0 , respectably. i Thus from (34) we get:   r 1 · · · rκ rκ+1 =   2  χ · · · χκ+1 0 = −(1/ε) q 1 · · · q κ q κ+1 κ+1 κ+1   = −(1/ε) q 2 · · · q κ+1 0

(35) We are now in position for solving Problem 1. For this, let us take together systems (1) and (2): 6

If K 0 ≤ ε1/p just do t∗ = 0.

And from (33) and (34) we get: Q0 + χκ (χκ+1 )T = (Mκ − Uκ )R0 κ κ+1     q ··· q q + 0 · · · 0 χκ = 1 κ κ+1 κ   0 = −(1/ε)(Mκ − Uκ ) q · · · q 2

κ+1

(36)

From (35), (36), and (A.1), we realize that: Pκ i q = j

where

q(i,j)

q χ i=j−1 (i,j) κ

is the entry (χ1 )T R0 κ

=

j = 3, · · · , κ + 1

;

(i, j)

of matrix

Q0 .

Then:

−(q(1,2) /ε)(χ1 )T κ

(37)

(5) Finally, doing in (29) the change of variable: (38)

z(t) = z¯(t) + R0 w(t)

and adding its third row with the pre-multiplication of its first row by Q0 , we get the externally equivalence with the following proper system (recall (33) and (34)): 7        κ+1 1 T x˙ εz¯˙

−β −ε (χ ) κ χκ −(Mκ − Uκ )

=

x z¯

κ

y =



0 (χ1 )T κ



x z¯

T

+

h

+

−

εκ q(1,2) γ0 −Q0 Γ

v

1 q(1,2) γ0 v ε

i

(39)

4. CONCLUSION In this paper we have solved the problem of approximating in an exponential way non proper SISO controllers by the use of a a Butterworth low pass filter. The solution is a modification of the proposition of (Pacheco et al, 2003), which consists in the substitution of the heavy filter 1/(εs + 1)κ by a Butterworth low pass filter, which amounts to a nice application of the solid results of (Kokotovi´c, 1999). Our solution for designing the adequate filters nicely separates the quality of the approximation, given by fast subsystem, parameterized by the inverse of the positive ε, from the convergence ratio, given by the slow subsystem parameterized by the positive β . The parameter ε is chosen in such a way that: (i) it guarantees the separation of the two time scales, the slow one and the fast one, by diagonalizing the fast and slow subsystems (16), (ii) it guarantees the stability of the SPLS (2) (see Theorem 5), and (iii) it takes into account the functional characteristics of the bounded Lipschitz continuous signal u(t) (reflected by its L∞ norm, K 0 , and its Lipschitz constant, L0 ) for solving the Problem 1 (see Theorem 8).

Bonilla, M. and M. Malabre (2000). Proportional and Derivative State Feedback Decoupling of Linear Systems. IEEE Transactions on Automatic Control, AC-45(4), 730–733. Bonilla, M. and M. Malabre (2001). Structural Conditions for Disturbance Decoupling with Stability using Proportional and Derivative Control Laws. IEEE Transactions on Automatic Control, AC-46(1), 160–165. Coinsidine, D.M. (1957). Process Instruments and Controls Handbook. New York: McGraw-Hill. Daniells, R. W. (1974). Approximation Methods for electronic Filter Design. New York: McGraw-Hill. Gantmacher, F.R. (1959). The theory of matrices, Vol. I. New York: Chelsea Pub. Co. Hardy, G.H. (1975). A course of pure mathematics. Cambridge: Cambridge University Press, 10th edition. Jackson, A.S. (1960). Analog Computation. New York: McGraw-Hill. Kokotovi´c, P. (1999). Singular Perturbation Methods in Control: Analysis and Design. London: SIAM. Pacheco, J., M. Bonilla, and M. Malabre (2003). Proper Exponential Approximation of Non Proper Compensators: The MIMO Case. In: 42nd IEEE Conference on Decision and Control. 110–115. Rosenbrock, H.H. (1970). State–Space and Multivariable Theory. London: Nelson. Willems, J.C. (1982). Feedforward control, PID control laws, and almost invariant subspaces. Systems & Control Letters, 1(4), 277–282. Willems, J.C. (1983). Input–output and state space representation of finite-dimensional linear time-invariant systems. Linear Algebra and its Applications, 50, 581–608.

All the proofs are done for κ odd, the even cases are treated in the same way.

Appendix A. PROOF OF LEMMA 2 The Matrix

(Mκ − Uκ )



REFERENCES Bonilla, M. and M. Malabre (1999). Necessary and Sufficient Conditions for Disturbance Decoupling with Stability using PID Control Laws. IEEE Transactions on Automatic Control, AC-44(6), 1311–1315.

X1  . (Mκ − Uκ ) =  0 0

Let us note that (χ1κ )T N = 0, then from (29) we get = (χ1 )T Γv = γ0 v . κ

(χ1 )T w κ

0 . . .

0 . . .

··· ··· ··· ···

0 0 0 . . . 0 Xκ−1 Yκ−1 0 0 Xκ

   (A.1)

Xi = (sin θi )I2 + T` {(cos2 θi )χ2 } − Tu {(χ2 )T }, i 2 2 = 1, . . . , κ − 1, Xκ = 1, Yi = −T` {χ2 }, i = 1, . . . , κ − 2, 2 and Yκ−1 = −χ2 . Then: (Mκ − Uκ )−1 = [Zij ], where:

where

2

Zii = Xi−1 , Z ij = (−1)i+j i
7

Y1 . . .

is:

Qj−1 k=i

Xk−1 Yk Xj−1 , Z ij = 0, i>j

which implies (6). For the transfer function:
−1 κ (−1)1+κ 1 T Qκ−1 χ = (s+1) (χ ) (χ1 )T sIκ + (Mκ − Uκ ) i=1 κ κ 2
(sI2 + Xi )−1 Yi = 1/∆B (s).

Appendix B. PROOF OF LEMMA 3 This Lemma follows from Lemma 2.1 of (Kokotovi´c, 1999) by doing: A11 = −β, A12 (ε) = −εκ+1 (χ1κ )T , A21 = χκ , and A22 = −(Mκ − Uκ ). κ For proving (12), let us set: A0 (ε) = −βIκ + εκ+1 (χ1 )T L(0) in Lemma 2.2 of (Kokotovi´ c, 1999), κ we then get from the proof of lemma 2.2 of (Kokotovi´c, 1999) (recall  Lemma 2): f (D) = −ε(Mκ − Uκ )−1

− β + εκ+1 (χ1 )T (L(0) +

κ

 D(ε)) (L(0) + D(ε)), where: c = ε(Mκ − Uκ )−1 ≤ 2

√ √ √ ε κ (Mκ − Uκ )−1 = ε κ k[Zij ]k1 < ε κκ = εκ3/2 , 1



a = −β + εκ+1 (χ1 )T L(0) = −β − εκ+1 , = β + κ 2 2

εκ+1 , b = εκ+1 (χ1 )T = εκ+1 , and ` = kL(0)k2 . κ 2

Since

√

√

, we get:

κ sin θ1 < ` < κ √ εκ+1 ) sin θ1 κ/εκ+1 <

√ a`/b < (β + εκ+1 ) κ/εκ+1 , √ κ+1 and + 2 ab` < (β + ε ) + εκ+1 κ + p also: a + b` √ √ 2 (β + εκ+1 )εκ+1 κ, i.e. : (β + εκ+1 ) + εκ+1 κ + p √
−1 √
−1 2 (β + εκ+1 )εκ+1 κ < a + b` + 2 ab` . (β +

Since

√

√ c < ε κ3 ,

√ εκ+1 κ

p

3

we then need: εκ 2 √
−1

<

(β + εκ+1 ) +

Applying (12) in (22.a) we get (23.a). From Lemma 2 and Theorem 5,
we get: −1 κ χ hκ (τ, ε) = L−1 (χ1 )T sf Iκ − Λκ (ε) κ κ 
 Pκ−1 j = L−1 1 + ρ (ε)s ∆ B,f (sf , ε) and f j=0 j
Q(κ−1)/2  ∆B,f (sf , ε) = sf + 1 + aκ (ε) (sf + sin θi i=1  +bi (ε))2 + (cos θi + ci (ε))2 , where: aκ (ε), bi (ε), ci (ε), ρj (ε) ∈ R, aκ (ε) = O(εκ+2 ), bi (ε) = O(εκ+2 ), ci (ε) = O(εκ+2 ), and ρj (ε) = O(εκ+2 ); with i = 1, · · · , (κ − 1)/2 and j = 0, · · · , κ − 1. That is to say, there exist Kh > 0 and ε∗h > 0 such that |aκ (ε)|
such that Kh ε∗2 < 1, viz , |aκ (ε)| < , |bi (ε)| < , and |ci (ε)| < εκ+1 , for ε ∈ (0, ε∗2 ), with i = 1, · · · , (κ − 1)/2. εκ+1

0 < ε < ε1 and 0 < εκ+1 p < ε <√1,
imply: √ −1 εκ < (β + 1) + κ + 2 (β + 1) κ p √ √
−1 < (β + εκ+1 ) + εκ+1 κ + 2 (β + εκ+1 )εκ+1 κ ,

But 3 2

which proves (12). Appendix C. PROOF OF LEMMA 4 Doing:

Appendix E. PROOF OF LEMMA 7

Let

(β + εκ+1 )εκ+1 κ

+ 2

apply the Taylor Series inequality, i.e   to the above κ+1 ) ∆(t∗ ) < e−β/(2ε 1 − (1 − 2εκ+1 /β + R2 ) where R2 = (1 + 2aκ+1 /β)−3 (2εκ+1 /β)2 , with a ∈ (0, ε), viz κ+1 ) [2εκ+1 /β + O(1)(2εκ+1 /β)2 ] = ∆(t∗ ) < e−β/(2ε κ O(ε ), which implies (21).

A11 = −β , A12 (ε) = −εκ+1 (χ1 )T ,

and A22 κ in equation (4.3) of (Kokotovi´c, 1999) we get (13). From (13)   and (12) follows: H (Mκ − Uκ ) − εβIκ + O(εκ+2 ) = εκ+1 (χ1 )T . κ = −(Mκ − Uκ )

Let us note that (recall that 0 < sin θi < 1 and that 0 < ε < ε1 < 1/β ): det (Mκ − Uκ ) − εβIκ =  Q(κ−1)/2  (1 − εβ) (1 − εβ)2 + 2εβ(1 − sin θi ) 6= 0. We i=1 then get from the inversion Lemma 8 : H(ε) = εκ+1 (χ1 )T (Mκ − Uκ )−1 + O(εκ+2 ), which together κ with Lemma 2 imply (14). Appendix D. PROOF OF LEMMA 6

εκ+1

Now, if εκ+1 < √12 sin ∆θ/2 and εκ+1 < √12 sin θ1 , there is no intersection between the neighbour√ hoods of radii 2εκ+1 and centered at the eigenvalues of (Mκ − Uκ ), namely at −1 and at − cos θi ±  sin θi , i = 1, · · · , (κ − 1)/2; and they all lie on the left half complex plan. And so, all the poles of the Laplace transform of hκ (τ, ε) are distincts and Hurwitz, thus:
√ − sin θ1 − 2εκ+1 τ

hκ (τ, ε) = k1 e fκ (τ, ε) =

+

1 k1

Aκ (ε)

P κ−1 2

e−

Xi (ε)

i=1

fκ (τ, ε)

e−(1−sin θ1 +

sin θi −sin θ1 +

∆(t ) =

1+

−β/(2εκ+1 )

< e Now, since

1 − (1 + 2ε

β



2εκ+1 /β)−1

1 − (1 + ε < β/2 and 2εκ+1 /β < εκ ε < 1,

8 (A − εβI)−1 = A−1 +

Pκ+1 i=1

/β)

 .

we can

(εβ)i A−(2i−1) + O(εκ+2 )

√




2εκ+1 +bi (ε) τ

cos (cos θi + ci (ε))τ + ϕi (ε)




,

where: k1 , Aκ (ε), Xi (ε), ϕi (ε) ∈ R, k1 > 0, Aκ (ε) = O(1), and Xi (ε) = ; with i = 1, . . . , (κ − 1)/2. Doing PO(1) (κ−1)/2 k1 = |Aκ | + |Xi | we get (23.b). i=1 With respectn to ακ (τ, ε) =

L−1

we have:  Pκ−1 j

ακ (τ, ε)

1+

j=0

ρj (ε)sf √

κ+1

Let: f (t, ε) = e−(β+ε )+g(ε)t where g(ε) = O(εκ+2 ), i.e. |g(ε)| < Kf εκ+2 for ε ∈ (0, ε∗f ) with Kf > 0 and ε∗f > 0. Let ε∗1 ∈ (0, ε∗f ) such that Kf ε∗1 < 1, viz κ+1 )t < |g(ε)| < εκ+1 for ε ∈ (0, ε∗1 ). Then: e−(β+2ε −βt f (t, ε) < e . κ+1 Let: ∆(t) = e−βt − e−(β+2ε )t , which
maximum occurs at: t∗ = (2εκ+1 )−1 ln 1 + 2εκ+1 /β , i.e
β  − κ+1   2ε 2εκ+1 κ+1 −1 ∗

and

√ κ+1 2ε +aκ )τ

κ+1

sf ∆B,f (sf , ε)


o ,




− sin θ1 − 2ε τ ¯ ακ (τ, ε) = A0 (ε) + k0 e fκ (τ, ε), √ κ+1 1 +aκ )τ f¯κ (τ, ε) = Aκ (ε)e−(1−sin θ1 + 2ε k0

+

P(κ−1)/2 i=1

X i (ε)

e−

sin θi −sin θ1 +

√

cos (cos θi + ci (ε))τ + ϕ ¯i (ε) A0 (ε) =

1+ρ0 (ε) ∆B,f (0,ε)




2εκ+1 +bi (ε) τ




, and

= 1 + d(ε)

where:

k0 , Aκ (ε), X i (ε), ϕ¯i (ε), d(ε), ∈ R, k0 > 0, Aκ (ε) = O(1), X i (ε) = O(1); with i = 1, . . . , (κ − 1)/2. And d(ε) = O(εκ+2 ), that is to say, there exist Kα > 0 and ε∗α > 0 such that |d(ε)| < Kα εκ+2 for ε ∈ (0, ε∗α ). Let ε∗2 ∈ (0, min{ε∗h , ε∗α }) such that ∗ Kα ε∗2 < 1, viz , |d(ε)| < εκ+1 for ε ∈ (0, ε2 ). Doing

now

k0 = Aκ +

P(κ−1)/2 i=1

Xi

we get (23.c).