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L2-Gain Disturbance Attenuation by P·SPR·D and P·SPR·D+I Control
Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011
L2-Gain Disturbance Attenuation by P·SPR·D and P·SPR·D+I Control Kiyotaka Shimizu ∗ ∗
Keio University, Hiyoshi, Kohokuku Yokohama Kanagawa JAPAN (e-mail: [email protected])
Abstract: This paper is concerned with L2 -gain disturbance attenuation problem for linear multivariable systems. First, high gain output feedback theorem for L2 -gain disturbance attenuation is obtained in case of minimum phase system. Next, we consider L2 -gain disturbance attenuation problem for general (nonminimum phase) systems. As a control law, P·SPR·D and/or P·SPR·D+I control is applied (SPR is an abbreviation for strict positive real). We can solve the problem by changing the general system to minimum phase one with the use of P, SPR, D elements and by applying the above high gain output feedback theorem. 1. INTRODUCTION
The effectiveness of the proposed method is confirmed with the simulation results of position control problem of 3inertia system (Hara et al. (1995)). 2. HIGH GAIN OUTPUT FEEDBACK STABILIZATION
As a method of system stabilization, high gain output feedback control (Isidori (1995); Khalil (2002); Shimizu (2004)) is well known. A system can be asymptotically stabilized by high gain output feedback, provided that the system has relative degree 1 and is minimum phase.
Consider a linear multivariable minimum phase system
We first review high gain output feedback theorem for linear minimum phase system as preliminaries. Next, we study L2 -gain disturbance attenuation problem for MIMO minimum phase system. We derive high gain output feedback theorem for the L2 -gain disturbance attenuation of minimum phase system, applying Lyapunov’s method.
where x(t) ∈ Rn , u(t) ∈ Rm , y(t) ∈ Rm are the state vector, the control input and the output, respectively. And consider an output feedback control
The L2 -gain disturbance attenuation was investigated (Schaft (2000); Marino et al. (1994); Isidori (1996)) and extended (Isidori (1999); Shen (2002)) as a basis of H∞ control, based on the small gain theorem and passivity property. It is also discussed as the generalized K-Y-P lemma in relation to positive boundedness of transfer function, strict positive realness, the Riccati inequality, LMI, etc. and it contributes to a basis of robust control. Meanwhile, we solve it here as high gain output feedback for L2 -gain disturbance attenuation, applying Lyapunov’s direct method (LaSalle and Lefschetz (1961)). Finally, we study the L2 -gain disturbance attenuation problem for general (non-minimum phase) systems. As a control law, P·SPR·D control (Tamura and Shimizu (2008); Shimizu (2009)) is applied. The P·SPR·D and/or P·SPR·D+I control are a structured controller as PID control using a SPR (strict positive real) element. We can solve the L2 -gain disturbance attenuation problem of general systems by making the system be minimum phase with the use of the SPR element and by applying the high gain output feedback theorem obtained for the minimum phase system. It is noted that our approach differs from existing H∞ control theory and it possesses possibility of not only assuring small gain property of a transfer function but also improving control performance to some extent. 978-3-902661-93-7/11/$20.00 © 2011 IFAC
˙ x(t) = Ax(t) + Bu(t)
(1)
y(t) = Cx(t)
(2)
u(t) = −Ky(t)
(3)
m×m
with a gain matrix K ∈ R . Then the following high gain output feedback theorem is obtained. [Theorem 1] (High Gain Output Feedback) Suppose that system (1),(2) has relative degree {1, 1, · · · , 1} (i.e. CB is nonsingular) and is minimum phase. Consider an output feedback control (3). Then there exist constants γi0 > 0, i = 1, 2, · · · , m such that the closed-loop system (1) ∼ (3) is symptotically stable, provided that K is taken as K = (CB)−1 (Q11 + Γ) with Γ = diag(γ1 , γ2 , · · · , γr ), γi ≥ γi0 > 0, where Q11 is a partial matrix in the normal form (see eq.(5)). Namely, ˙ x(t) = Ax(t) − BKy(t) is asymptotically stable. (Proof ) From the assumption of relative degree {1, 1, · · · , 1}, by a nonsingular coordinate transformation C ξ x, T B = O (4) = T η system (1),(2) can be transformed into the normal form (Isidori (1995); Sepulcure (1997)) ;
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ξ˙ = Q11 ξ + Q12 η + CBu
(5)
10.3182/20110828-6-IT-1002.00234
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
η˙ = Q21 ξ + Q22 η
3. L2 -GAIN DISTURBANCE ATTENUATION FOR MINIMUM PHASE SYSTEM
(6)
y=ξ
(7)
m
n−m
m
and y ∈ R . Then, η˙ = where ξ ∈ R , η ∈ R Q22 η is called zero dynamics. If the zero dynamics be asymptotically stable, the system (1),(2) is said minimum phase.
Let us study L2 -gain disturbance attenuation problem in case of existence of disturbances. Consider the following linear multivariable system
By executing the output feedback u(t) = −Ky(t) = −Kξ(t)
(8)
y(t) = Cx(t)
(15)
(9) (10)
Since η˙ = Q22 η is asymptotically stable from the assumption of minimum phase, the Lyapunov inequality
m
u(t) = −Ky(t) with a gain matrix K ∈ R
Now we define a Lyapunov function V (ξ, η) =
ξ η
T
I O O P
ξ η
(12)
V (ξ, η) is a positive definite function, because of P > 0.
m×m
(17)
.
The L2 -gain disturbance attenuation problem is defined to attain a control such that the closed-loop system satisfies the following conditions under the given disturbance attenuation level γ > 0. P1. When w = 0, the closed-loop system is asymptotically stable. p2. When x(0) = 0, the following inequality holds for the arbitrarily given T > 0.
Calculate its time derivative along (9),(10) to get V˙ (ξ, η) T ˙ T I O I O ξ ξ ξ ξ˙ = + O P η η O P η˙ η˙ T Q11 ξ + Q12 η − CBKξ ξ I O = η O P Q21 ξ + Q22 η T ξ I O Q11 ξ + Q12 η − CBKξ + η Q21 ξ + Q22 η O P T T T ξ CBKξ ξ Q11 QT21 P ξ − = η 0 η QT12 QT22 P η T T ξ Q11 Q12 CBKξ ξ ξ + − η 0 η P Q21 P Q22 η T ξ ξ Θ = η η
m
where x(t) ∈ R , u(t) ∈ R , y(t) ∈ R , z(t) ∈ ′ Rm , w(t) ∈ Rl are the state vector, the control input, the measurable output, the controlled output and the disturbance input, respectively. And consider an output feedback control
(11)
holds for some P > 0.
(16)
n
ξ˙ = Q11 ξ + Q12 η − CBKξ
P Q22 + QT22 P < 0
(14)
z(t) = CP x(t)
the closed-loop system (5)∼(7) becomes as follows.
η˙ = Q21 ξ + Q22 η
˙ x(t) = Ax(t) + Bu(t) + Bw(t)
T
2
z(t) dt ≤ γ
2
T
w(t)2 dt
0
0
It is noticed that P2 is equivallent to having L2 -gain below γ when x(0) = 0, that is, z2 ≤ γ 2 w2 . It implies that for all w ∈ L2 [0, T ] and for the supply rate s(z, w) = γ 2 wT w −z T z, the following γ-dissipation inequality holds with a storage function V (x) (Schaft (2000); Shen (2002)). V˙ (x(t)) ≤ γ 2 w(t)T w(t) − z(t)T z(t)
(18)
Now if system (14),(15) has relative degree {1, 1, · · · , 1}, then by the nonsingular coordinate transformation
(13)
where
ξ η
=
C x, T B = O, T B = O T
(19)
system (14)∼(16) can be transformed into the normal form
Q11 − CBK + QT11 − K T B T C T QT21 P + Q12 Θ≡ P Q21 + QT12 P Q22 + QT22 P Accordingly, if we take K = (CB)−1 (Q11 + Γ), then the (1,1) block of Θ becomes −Γ − ΓT , which can be made negative definite, provided that Γ is a positive definite matrix. Hence, if Γ = diag{γ1 , γ2 , · · · , γm }, γi ≥ γi0 > 0 is taken sufficiently large, then Θ becomes negative definite in consideration of (11). Consequently, system (9),(10) becomes asymptotically stable. Therefore, system (1),(2) is asymptotically stable by the output feedback (3). Q.E.D
ξ˙ = Q11 ξ + Q12 η + CBu + CBw
(20)
η˙ = Q21 ξ + Q22 η
(21)
y=ξ
(22)
z = Q31 ξ + Q32 η
(23) ′
where ξ ∈ Rm , η ∈ Rn−m , y ∈ Rm , z ∈ Rm and w ∈ Rl . Note that in general there does not exist a matrix T satisfying T B = 0 and T B = 0. But we consider the case of B and B for which there exists a matrix such that T B = 0 and T B = 0 hold. It will be applied in Chapter 4.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
By executing the output feedback u(t) = −Ky(t) = −Kξ(t)
(24)
the closed-loop system (20)∼(24) becomes as follows. ξ˙ = Q11 ξ + Q12 η − CBKξ + CBw
(25)
η˙ = Q21 ξ + Q22 η
(26)
z = Q31 ξ + Q32 η
(27)
Since η˙ = Q22 η is asymptotically stable from the assumption of minimum phase, the Lyapunov inequality P Q22 + QT22 P < 0 (28) holds for some P > 0. Here let us assume : [Assumption 1] There exists P > 0 such that P Q22 + QT22 P + QT32 Q32 < 0 The following theorem solves the L2 -gain diturbance attenuation problem. [Theorem 2] (High Gain Output Feedback for L2 gain Disturbance Attenuation) Suppose that system (14)∼(16) has relative degree {1,1,· · ·,1 } (i.e. CB is nonsingular) and is minimum phase. Suppose also that there exists a matrix T such that T B = 0 and T B = 0 hold. Further assume Assumption 1 holds. And consider the output feedback control (17). Then, there exist constants γi0 > 0, i = 1, 2, · · · , m such that the closed-loop system (14)∼(17)satisfies condition P2, provided that K is taken T as K = (CB)−1 Q11 + 12 (QT31 Q31 + γ12 CBB C T ) + Γ with Γ = diag{γ1 , γ2 , · · · , γm }, γi ≥ γi0 > 0. Namely, ˙ x(t) = Ax(t) − BKy(t) + Bw(t) possesses L2 -gain less than γ.
ξ QT11 − K T B T C T QT21 P = η QT22 P QT12 T T w ξ I O B CT O + 0 η O P O O T Q11 − CBK Q12 ξ ξ + η P Q21 P Q22 η T ξ w I O CB O + O P η 0 O O T T ξ Q31 Q31 QT31 Q32 ξ + − γ 2 wT w η QT32 Q31 QT32 Q32 η T T ξ Q11 − K T B T C T QT21 P ξ = η η QT12 QT22 P T T Q11 − CBK Q12 ξ ξ +wT B C T ξ + η P Q21 P Q22 η T T ξ Q31 Q31 QT31 Q32 ξ T +ξ CBw + η QT32 Q31 QT32 Q32 η 1 T 1 T T T ξ CB − γw − B C ξ − γw γ γ 1 T T + 2 ξ T CBB C T ξ − wT B C T ξ − ξ T CBw γ T ξ ξ = Θ η η 1 T 1 T T T − ξ CB − γw B C ξ − γw γ γ where we put Θ11 Θ12 Θ≡ , Θ21 Θ22
V (ξ, η) =
ξ η
T
I O O P
ξ η
T
ξ QT31 Q − γ 2wT w ] [Q 31 32 η QT32 T ξ I O Q11 ξ + Q12 η − CBKξ + CBw = η O P Q21 ξ + Q22 η T ξ I O Q11 ξ + Q12 η − CBKξ + CBw + η O P Q21 ξ + Q22 η T T T Q31 Q31 Q31 Q32 ξ ξ + − γ 2 wT w η QT32 Q31 QT32 Q32 η V˙ (ξ, η) +
ξ η
(30)
Θ21 ≡ P Q21 + QT12 + QT32 Q31 , Θ22 ≡ P Q22 + QT22 P + QT32 Q32
(29)
V (ξ, η) is a positive definite function, because of P > 0. To prove that the γ-dissipation inequality V˙ (x(t)) + (z(t)T z(t) − γ 2 w T (t)w(t)) ≤ 0 holds, make the following calculation ;
T
Θ11 ≡ Q11 − CBK + QT11 − K T B T C T 1 T + QT31 Q31 + 2 CBB C T , γ Θ12 ≡ QT21 P + Q12 + QT31 Q32 ,
Furthermore, the closed-loop system (14)∼(17) satisfies condition P1. (Proof ) Define a storage function (which is also a Lyapunov function)
ξ η
Since Θ is a symmetric matrix, it becomes negative definite by Schur complement, if Θ11 is sufficiently small negative definite in consideration of Assumption 1. Here, if we take K as 1 1 T K = (CB)−1 Q11+ (QT31 Q31 + 2 CBB C T )+ Γ (31) 2 γ , then Θ11 becomes −Γ − ΓT , which is negative definite, provided that Γ is a positive definite matrix. Hence, if Γ = diag{γ1 , γ2 , · · · , γm }, γi ≥ γi0 > 0 is sufficiently large, then Θ becomes negative definite under Assumption 1. Accordingly, we have
T T Q31 Q31 QT31 Q32 ξ − γ 2 wT w QT32 Q31 QT32 Q32 η T 1 T T 1 T T <− B C ξ − γw B C ξ − γw ≤ 0 (32) γ γ V˙ (ξ, η) +
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ξ η
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Consequently, the γ-dissipation inequality holds, and so it follows that we have L2 -gain below γ.
u(t) = −KP Cx(t) + KS ζ(t)
When w = 0, condition P1 has been already proved in Theorem 1. Q.E.D [Corollary 1] Let us consider the case of z = y. Suppose that system (14),(15) has relative degree {1,1,· · ·,1 } (i.e. CB is nonsingular) and is minimum phase. Suppose also that there exists a matrix T such that T B = 0 and T B = 0 hold. Further assume there exists P > 0 satisfying P Q22 + QT22 P + I < 0. And consider the output feedback control (17). Then, there exist constants γi0 > 0, i = 1, 2, · · · , m such that the closed-loop system (14),(15),(17) satisfies condition P2, provided that K is T taken as K = (CB)−1 Q11 + 21 (I + γ12 CBB C T ) + Γ with Γ = diag{γ1 , γ2 , · · · , γm }, γi ≥ γi0 > 0. Namely,
by substituting (34),(39) into (38).
−KD C(Ax(t) + Bu(t) + Bw(t)) Furthermore, arranging this equation, we obtain u(t) = −(Ir + KD CB)−1 (KP C + KD CA)x(t) +(Ir + KD CB)−1 KS ζ(t) −(Ir + KD CB)−1 KD CBw(t) = −KX x(t) + KΞ ζ(t) − KW w(t) where
−1
KX ≡ (Ir + KD CB)
−1
KΞ ≡ (Ir + KD CB)
(40)
(KP C + KD CA)
KS
−1
KW ≡ (Ir + KD CB)
KD CB
By substituting (40) into (33), we obtain the closed-loop system
˙ x(t) = Ax(t) − BKy(t) + Bw(t) possesses L2 -gain less than γ.
˙ x(t) = (A − BKX )x(t) + BKΞ ζ(t) − BKW w(t)
Furthermore, the closed-loop system (14),(15),(17) satisfies condition P1.
4.1 P·SPR·D Control Consider the following general MIMO system: ˙ x(t) = Ax(t) + Bu(t) + Bw(t)
(33)
y(t) = Cx(t)
(34)
z(t) = CP x(t)
(35)
where x(t) ∈ Rn , u(t) ∈ Rr , y(t) ∈ Rm , z(t) ∈ ′ Rm , w(t) ∈ Rl are the state vector, the control input, the measurable output, the controlled output and the disturbance input, respectively. We investigate P·SPR·D control for L2 -gain diturbance attenuation, by making the system be minimum phase and applying the high gain output feedback. In case of regulation problem, PID control is given as
u(t) = −KP y(t) + KI
˙ −y(τ )dτ − KD y(t)
Now we define the SPR gain KS as K S ≡ HS L r×m
where KP , KI , KD ∈ R
Consider the followings.
(36)
(44)
′
(45)
Then, by substituting (34),(39) into (45), we have
In this paper, we propose the following P·SPR·D control ˙ ζ(t) = Dζ(t) − y(t), ζ(0) = 0, D < 0
(37)
˙ u(t) = −KP y(t) + KS ζ(t) − KD y(t)
(38)
where (37) represents a SPR element with negative definite D ∈ Rm×m , and KS ∈ Rr×m denotes the SPR gain.
u(t) = −KP Cx(t) + HS ζ ′ (t) −KD C(Ax(t) + Bu(t) + Bw(t)) Arranging this equation as before, we obtain u(t) = −KX x(t) + KΞ′ ζ ′ (t) − KW w(t)
Since it holds from (33),(34) that
we have
′ ζ˙ (t) = LD′ ζ ′ (t) − Ly(t)
˙ u(t) = −KP y(t) + HS ζ (t) − KD y(t)
are P, I, D gains.
˙ y(t) = C(Ax(t) + Bu(t) + Bw(t))
(43)
m×m
and L ∈ R are called the interwhere HS ∈ R mediate parameter matrix and the adjustable parameter matrix, respectively. The parameterization as (43) is made in order to apply high gain output feedback to the system (49)∼(51) below. Namely, L corresponds to the gain of output feedback and HS is a parameter used to asymptotically stabilize the zero dynamics. The high gain output feedback makes controller parameter tuning easy for improving the control performance.
0
r×m
(41)
Accordingly, by combining (41) and (37), the closed-loop system by the P·SPR·D control becomes ˙ x(t) A − BKX BKΞ x(t) = ˙ ζ(t) −C D ζ(t) −BKW + B + w(t) (42) O
4. L2 -GAIN DISTURBANCE ATTENUATION BY P·SPR·D AND/OR P·SPR·D+I CONTROL
t
+Bw(t)
where (39)
KΞ′
−1
≡ (Ir + KD CB)
(46)
HS
By substituting (46) into (33), we obtain the closed-loop system
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
˙ x(t) = (A − BKX )x(t) + BKΞ′ ζ ′ (t) − BKW w(t) +Bw(t) Accordingly, by combining (47) and (44), we have ˙ x(t) x(t) A − BKX BKΞ′ ′ = −LC LD′ ζ ′ (t) ζ˙ (t) −BKW + B w(t) + O
′
(47)
(48)
(t) ∈ Rm and z (t) ∈ Rm . (t) ∈ RN , v(t) ∈ Rm , y where x
B If system (54),(55) has relative degree {1, 1, · · · , 1} (i.e. C is nonsingular), then the system can be transformed into the normal form. That is, by the nonsingular coordinate transformation C ξ =O = O, TB , TB = x (57) η T
(54)∼(56) can be transformed into the normal form
Next let us consider the following hypothetical system based on the closed-loop system (48) ; ˙ x(t) x(t) A − BKX BKΞ′ ′ = O O ζ ′ (t) ζ˙ (t) O −BKW + B + v(t) + w(t) Im O x(t) + Bv(t) (49) := A + Bw(t)
x(t) x (t) = C −D′ (t) y := C (50) ζ ′ (t) x(t) P x (t) (t) = [ CP O ] ′ (51) z := C ζ (t) ′
(t) ∈ Rn+m , v(t) ∈ Rm , y ∈ Rm , z ∈ Rm where x denote the state vector, the input, the measurable output and the controlled output of the hypothetical system C, B, B, C P }, respectively. And let the input v(t) be {A, given by the output feedback
x(t) ′ v(t) = −L y (t) = −L C −D (52) ζ ′ (t)
Bv +C Bw ξ˙ = Q11 ξ + Q12 η + C
(58)
η˙ = Q21 ξ + Q22 η
(59)
=ξ y
(60)
= Q31 ξ + Q32 η z m
where ξ ∈ R , η ∈ R
N −m
∈R , y
(61) m
∈R and z
m′
.
Here we assume : [Assumption 2] There exists P > 0 such that P Q22 + QT22 P + QT32 Q32 < 0
Applying Theorem 2 to system (54)∼(56) we have : C, B, B, C P } [Propositon 1] Suppose that system {A, given by (54)∼(56) has relative degree {1, 1, · · · , 1} (i.e. B is nonsingular) and is minimum phase. Suppose also C =0 = 0 and TB that there exists a matrix T such that TB hold. Further assume Assumption 2 holds. And consider an output feedback control
where L ∈ Rm×m is the output feedback gain. Then, the closed-loop system of the hypothetical system becomes ˙ x(t) x(t) A − BKX BKΞ′ ′ = −LC LD′ ζ ′ (t) ζ˙ (t) −BKW + B w(t) (53) + O which is the same as (48).
. Then, there exist constants with a gain matrix L ∈ R γi0 > 0, i = 1, 2, · · · , m such that the closed-loop system (54)∼(56),(62) satisfies condition P2, provided that L is B TC B T + B) −1 Q11 + 1 QT Q31 + 12 C taken as L = (C 31 2 γ Γ with Γ = diag(γ1 , γ2 , · · · , γm ), γi ≥ γi0 > 0. Namely,
Now if we put D′ = DL−1 and ζ ′ = Lζ, then the lower column of (53) becomes equal to (37). Therefore, if we set KS = HS L, D = D′ L, (53) coincides with (42).
x(t) − BL y (t) + Bw(t) ˙ (t) = A x
Consequently, (53) becomes equal to the closed-loop error system with the P·SPR·D control (33),(34),(37),(38), provided that the output feedback gain L is taken equal to the adjustable parameter of (43).
v(t) = −L y (t)
(62)
m×m
possesses L2 -gain below γ.
Furthermore, the closed-loop system (54)∼(56),(62) satisfies condition P1. (Proof ) Obvious from Theorem 2.
Q.E.D
4.2 Design of P·SPR·D Controller for L2 Gain Disturbance Attenuation
Now let us consider to apply Proposition 1 to the hypothetical system (49)∼(51) to design the P·SPR·D controller. So check first the relative degree with respect to v. Since
Let us apply Theorem 2 to design the P·SPR·D controller for L2 gain disturbance attenuation. Consider the following general MIMO system ;
O ˙ (t) ∂y B = C −D′ =C = −D′ Im ∂v(t)
x(t) + Bv(t) ˙ (t) = A x + Bw(t) x (t) = C (t) y P x (t) = C (t) z
(54) (55) (56)
, this matrix has to be nonsingular in order to apply Proposition 1. So let D′ be a nonsingular matrix. Accordingly, we can transform system (49)∼(51) into the normal form to obtain its zero dynamics. Consider the following transfomation from (57) ;
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
ξ η
=
x C C −D′ x = Tn O ζ′ T
(63)
= O, = O, TB T = [ Tn O ] , TB
where
and so η ∈ Rn . Note We know that ξ ∈ Rm from ξ = y that the inverse matrix of (63) becomes
C −D′ Tn O
−1
=
O Tn−1 ′ ′ −1 −1 −D D CTn−1
Transform the hypothetical system (49)∼(51) by (63) to get A − BKX BKΞ′ ξ˙ = C −D′ × Tn O O O η˙ O Tn−1 ξ ′ ′ −1 −1 −1 η −D D CTn O C −D′ + v Im Tn O C −D′ −BKW + B + w (64) Tn O O Then, we can obtain the normal form of the hypothetical system (49)∼(51) ;
4.3 Determination of Controller Parameters An important task in our method is to satisfy Assumption 3, that is, to determine KP , KD and HS such that the zero dynamics (69), i.e. η˙ = Tn A − B(Ir + KD CB)−1 (KP C + KD CA ′ −HS D −1 C) Tn−1 η (70) is asymptotically stable.
Since eigenvalues of zero dynamics (70) are the same with those of the matrix ′
A − B(Ir + KD CB)−1 (KP C + KD CA − HS D −1 C)(71) we consider a method of finding KP , KD , HS such that this matrix be asymptotically stable. We first change the matrix of (71) as follows. ′
A − B(Ir + KD CB)−1(KP C + KD CA − HS D −1 C)
C ′ =A − B(Ir + KD CB)−1 KP − HS D −1 KD (72) CA
ξ˙ = −CBKΞ′ D′−1 ξ ′ +C (A − BKX )Tn−1 + BKΞ′ D −1 CTn−1 η − D′ v +C(−BKW + B)w
Then, the closed-loop system (33)∼(35),(37),(38) satisfies the conditions P1 and P2. Also, from the property of high gain output feedback, we can improve the convergence speed by adjusting the high gain L to some extent.
(65)
By defining the following matrices
′
′
η˙ = −Tn BKΞ′ D −1 ξ ′ + Tn (A − BKX + BKΞ′ D −1 C)Tn−1 η =ξ y
= [CP z
ξ O Tn−1 O] = CP Tn−1 η −D′−1 D′−1 CTn−1 η
Fη1 = (Ir + KD CB)−1 (KP − HS D −1 )
η˙ = Tn
′ A − B(KX − KΞ′ D −1 C) Tn−1 η
Fη2 = (Ir + KD CB)
(74)
(67)
Fη = [ Fη1
(75)
(68)
(69)
In order to satisfy the minimum phase requirement, the zero dynamics (69) has to be asymptotically stable. Thus we assume : [Assumption 3] There exist parameter matrices KP ,HS , KD such that the zero dynamics (69) is asymptotically stable. Corresponding to Assumption 2, we assume also : [Assumption 4] There exists P > 0 such that
where
Q22 ≡ Tn (A − B(KX − KΞ′ D
−1
KD C Fη2 ] , Cη = CA
(72) can be expressed as A − BFη Cη (76) This can be regarded as a matrix of the closed-loop system due to the output feedback −Fη Cη η for system {A, B, Cη }. Accordingly, in order to get matrices KP , KD , HS asymptotically stabilizing the zero dynamics, we can apply any existing method of output feedback stabilization (e.g. Tamura and Shimizu (2006)) to the system {A, B, Cη }. After determining the output feedback gain Fη stabilizing (76) , we can obtain KP , KD , HS stabilizing (71) from the relations (73) and (74). Besides, we can make it by trial and error, checking eigenvalues of matrix (71) directly. From Proposition 1 the high gain L can be attained :
P Q22 + QT22 P + QT32 Q32 < 0 ′
(73)
(66)
Accordingly, from (66) the zero dynamics is expressed as
−1
C)Tn−1 ,
Q32 ≡ CP Tn−1 Consequently, we can apply Proposition 1 under Assumptions 3 and 4 in order to obtain L of the output feedback (52) which solves the L2 -gain disturbance attenuation problem for (49)∼(51). Accordingly, by setting SPR parameter matrices as KS = HS L, D = D′ L, we obtain the P·SPR·D controller (37),(38) for the system (33)∼(35).
B TC B) −1 Q11 + 1 QT Q31 + 1 C T + Γ B L = (C 31 2 γ2 1 1 C(−BKW + B) = −D′−1 − CBKΞ′ D′−1 + 2 γ2 ×(−BKW + B)T C T + Γ = −D′−1 − CB(Ir + KD CB)−1 HS D′−1 1 1 + C(−B(Ir + KD CB)−1 KD CB + B) 2 γ2
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
×(−B(Ir + KD CB)−1 KD CB + B)T C T + Γ
(77)
1.4 1.2
controlled output z = θ
3
A design procedure is arranged as below. [Algorithm 1] ′ [Step 1] Give a nonsingular matrix D ∈ Rm×m . [Step 2] Detemine gain matrices KP , HS , KD ∈ Rr×m which make the zero dynamics matrix (71) be asymptotically stable and satisfy Assumption 4. [Step 3] Choose L as (77), based on Proposition 1, and determine KS = HS L and D = D′ L. [Step 4] The P·SPR·D controller is given by (37),(38).
1 0.8 0.6 0.4 0.2 0 0
0.5
1 time t
1.5
2
Fig.1 Step Desired Output Response for z ∗ = 1 0.16
As a practical design policy, it may be effective to decide Γ i.e. L subjectively, observing the output responses concretely, since D, KP , KS , KD can be computed immediately given L in the proposed method.
0.14
controlled output z = θ3
0.12
4.4 P·SPR·D+I Coontrol
0.1 0.08 0.06 0.04 0.02
The P·SPR·D control certainly satisfies condition P1 (i.e. the closed-loop system converges to the origin x = 0 when w = 0). But an off-set arises when w = 0. Hence we necessitate to add I action to P·SPR·D control to get rid of it. Moreovere, in order to cope with servo problem, the I action is also needed. Our method can handle the set-point servo problem easily by adding the I action.
0 −0.02 0
1.5
2
1.4
controlled output z = θ
3
1.2 1 0.8 0.6 0.4 0.2 0 0
(78)
0.5
1 time t
1.5
2
∗
Fig.3 Step Desired Output Response for z = 1 0.09 0.08 0.07
controlled output z = θ3
˙ u(t) = −KP y(t) + KS ζ(t) − KD y(t) t +KI (z ∗1 − z 1 (τ ))dτ
1 time t
Fig.2 Step Disturbance Response Against τd1 = 1
For that purpose let us assume that among controlled output variables z, variables z 1 to track the desired value z ∗1 is a partial vector of z (i.e. z 1 ⊂ z) and can be measured to use for integral control. In such a case consider the following P·SPR·D+I control adding the integral action. ˙ ζ(t) = Dζ(t) − y(t), ζ(0) = 0, D < 0
0.5
(79)
0
Since stability and L2 -gain disturbance attenuation are guaranteed sufficiently by the high gain feedback, we devise here only a countermove for a steady state error.
0.06 0.05 0.04 0.03 0.02 0.01 0 0
0.5
1 time t
1.5
2
Fig.4 Step Disturbance Response Against τd1 = 1
Incidentally, in a special case where the controlled output z equals to the measurable output y, i.e., y ≡ z, the prososed methods can be simplified to a great extent.
˙ ζ(t) = Dζ(t) + (y ∗ − y(t)), ζ(0) = 0, D < 0
(80)
∗
˙ u(t) = KP (y − y(t)) + KS ζ(t) − KD y(t) t + KI (y ∗ − y(τ ))dτ
Meanwhile, when w = 0, at an equilibrium state xe holding the controlled output z at z ∗ must satisfy the following relation;
(81)
0
∗
0 = Axe + Bu, z = CP xe Since this relation consists of (n + m′ ) equations and (n + r) variables, when r ≥ m′ , (r − m′ ) state variables xeN can be set arbitrary value x∗eN , but the remained dependently. state variables xeB and u are determined ∗ x eN Putting such an equilibrium as x∗ = and xeB (x∗eN , z ∗ ) u∗ = u(x∗eN , z ∗ ), we have
5. SIMULATION RESULTS Let us consider a position control problem of 3-inertia system (Hara et al. (1995)). Letting θi , i = 1, 2, 3, be the rotating angle of each rotating mass and τ the manipulated torque and τdi , i = 1, 2, 3, the disturbance torque, an equation of motion of the system is represented as
0 = Ax∗ + Bu∗ , z ∗ = CP x∗ Then, the measurable output y should be y ∗ = Cx∗ . Thus we can consider the following P·SPR·D+I control : 7509
J1 θ¨1 = d1 θ˙1 − ka (θ1 − θ2 ) − da (θ˙1 − θ˙2 ) + τ + τ d1 J2 θ¨2 = ka (θ1 − θ2 ) + da (θ˙1 − θ˙2 ) − d2 θ˙2 − kb (θ2 − θ3 ) −db (θ˙2 − θ˙3 ) + τ d2
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
J2 θ¨3 = kb (θ2 − θ3 ) + db (θ˙2 − θ˙3 ) − d3 θ˙3 + τ d3 where Ji , i = 1, 2, 3, denotes the inertia moment, di , i = 1, 2, 3, a, b, the coefficient of viscosity friction and ki , i = a, b, the spring constant. Let x = (θ1 , θ2 , θ3 , θ˙1 , θ˙2 , θ˙3 )T , u = τ and w = (τd1 , τd2 , τd3 )T , then a state space representation of the system is given as x˙ = Ax + Bu + Bw
that convergence speed for the set-point servo problem got improved very much compared to (78),(79), though step disturbance response became a little slower. It can be said that satisfactory control performances were obtained for both cases. Note also that even if gains KP , KS , KD , KI change around the above values to some degree, the closed-loop system maintains to be asymptotically stable.
y = Cx
6. CONCLUDING REMARKS
z = CP x where A= 0
0 0 −ka J1 ka J2 0
0 0 0 ka 1 J −ka − kb J2 kb J3
0 0 0 B= , 1/J1 0 0
0 0 0
1 0 0 −d1 − da 0 J1 kb da J2 J2 −kb 0 J3 0 0 0 0 0 0 1 B = J1 0 0 1 J2 0 0
C = [1 0 0 0 0 0]
0 0 1 0 0 1 da 0 J1 −da − d2 − db db J2 J2 −db − d3 db J3 J3 0 0 0 0 0 1 J3
CP = [ 0 0 1 0 0 0 ] where plant parameters are given as J1 = J2 = 0.001, J3 = 0.002, d1 = 0.05, d2 = 0.001, d3 = 0.007, da = db = 0.001, ka = 920, kb = 80. Using Algorithm 1, we can solve L2 -gain disturbance attenuation problem. We set γ = 1. We first determine KP , KD , HS asymptotically stabilizing the zero dynamics (71) and then set controller parameters of P·SPR·D+I control (78),(79) as follows. D′ = −1, HS = 1, KP = 4, KD = 1, KI = 10 Γ = 5, L = 5.5, KS = 5.5 The simulation results of set-point servo problem for z ∗ = 1 and x(0) = 0, w = 0 is shown in Fig.1. The simulatioin results against a step disturbance w = (τd1 , τd2 , τd3 )T = ( 1 0 0 )T is shown in Fig.2, when z ∗ = 0 and x(0) = 0. Next, we consider P·SPR·D+I control (80),(81), setting controller parameters as follows. D′ = −1, HS = 1, KP = 10, KD = 0.5, KI = 10 Γ = 5, L = 5.5, KS = 5.5 The simulation results of set-point servo problem for z ∗ = 1 and x(0) = 0, w = 0 is shown in Fig.3. In this case we set y ∗ = 1 corresponding to z ∗ = 1. The simulatioin results against a step disturbance w = [ 1 0 0 ]T is shown in Fig.4, when we set z ∗ = 0, y ∗ = 0 and x(0) = 0. We see
We investigated the P·SPR·D and/or P·SPR·D+I control for the L2 -gain disturbance attenuation problem, based on the idea that one makes a system have relative degree {1, 1, · · · , 1} and be minimum phase by use of the SPR element such that high gain output feedback can be applied. The use of SPR element contributes powerfully to stabilizing the closed-loop system. The P·SPR·D or P·SPR·D+I control is very practical because of its simple structure. Compared to H∞ control, our method can handle the set-point servo prblem easily by using I action. Besides, we do not have to solve the Riccati inequality or LMI for designing the controller. Although we studied only to stabilize the system and to attain L2 -gain below γ against disturbances, how to determine KP , KS , KD , D giving better control performance is considered a future topic. REFERENCES T.Hara,Y.Chida,M.Saeki,K.Nonami, Bench Mark Problem for Robust Control (I), SICE Jjournal, Vol.34, No.5, pp.403-409, 1995 (in Japanese) A.Isidori, Nonlinear Control Systems, Third Edition, Springer-Verlag, 1995 A.Isidori, Global Almost Disturbance Decoupling with Stability for Nonminimum Phase Systems, Syst. Contr. Lett., Vol.27, pp.191-194, 1996 A.Isidori, Nonlinear Control Systems II, Chap. 13, Disturbance Attenuation, Springer-Verlag, 1999 H.K. Khalil, Nonlinear Systems, 3rd ed., Prentice Hall, 2002 J.LaSalle and S.Lefschetz, Stability by Liapunov’s Direct Method, Academic Press, 1961 R.Marino, W.Rosenblock, A. van der Schaft, P.Tomei, Nonlinear H∞ Almost Disturbance Decoupling, Syst. Contr. Lett., Vol.23, pp.159-168, 1994 A. van der Schaft, L2 -Gain and Passivity Techniques in Nonlinear Control, 2nd ed., Springer-Verlag, 2000 T.Shen, Robust Control of Nonlinear System Based on Dissipativity, SICE, 2002 (in Japanese) R.Sepulcure, M.Jankovic and P.V.Kokotovic, Constructive Nonlinear Control, Springer-Verlag, 1997 K.Shimizu, Generalization and Proof of High Gain output Feedback Theorem, Trans. SICE, Vol.40, No.12, pp.1247-1249, 2004 (in Japanese) K.Shimizu, P·SPR·D and P·SPR·D·I Control for Linear Multi-Variable Systems-Stabilization Based on High gain Output Feedback-, Proceedings of 2009 American Control Conference, pp.4666-4672, St. Louis, 2009 K.Tamura and K.Shimizu, Eigenvalue-Assignment Method by PID Control for MIMO Systems, Systems, Control and Informatioin, Vol.19, No. 5, pp.193-202, 2006 (in Japanese) K.Tamura and K.Shimizu, A Control Method for MIMO Systems(P+quasi·I+D Control)–Stabilization Based on High Gain Output Feedback–, Trans.SICE, Vol.44,No.5, pp.434-443, 2008 (in Japanese)
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L2-Gain Disturbance Attenuation by P·SPR·D and P·SPR·D+I Control Kiyotaka Shimizu ∗ ∗
Keio University, Hiyoshi, Kohokuku Yokohama Kanagawa JAPAN (e-mail: [email protected])
Abstract: This paper is concerned with L2 -gain disturbance attenuation problem for linear multivariable systems. First, high gain output feedback theorem for L2 -gain disturbance attenuation is obtained in case of minimum phase system. Next, we consider L2 -gain disturbance attenuation problem for general (nonminimum phase) systems. As a control law, P·SPR·D and/or P·SPR·D+I control is applied (SPR is an abbreviation for strict positive real). We can solve the problem by changing the general system to minimum phase one with the use of P, SPR, D elements and by applying the above high gain output feedback theorem. 1. INTRODUCTION
The effectiveness of the proposed method is confirmed with the simulation results of position control problem of 3inertia system (Hara et al. (1995)). 2. HIGH GAIN OUTPUT FEEDBACK STABILIZATION
As a method of system stabilization, high gain output feedback control (Isidori (1995); Khalil (2002); Shimizu (2004)) is well known. A system can be asymptotically stabilized by high gain output feedback, provided that the system has relative degree 1 and is minimum phase.
Consider a linear multivariable minimum phase system
We first review high gain output feedback theorem for linear minimum phase system as preliminaries. Next, we study L2 -gain disturbance attenuation problem for MIMO minimum phase system. We derive high gain output feedback theorem for the L2 -gain disturbance attenuation of minimum phase system, applying Lyapunov’s method.
where x(t) ∈ Rn , u(t) ∈ Rm , y(t) ∈ Rm are the state vector, the control input and the output, respectively. And consider an output feedback control
The L2 -gain disturbance attenuation was investigated (Schaft (2000); Marino et al. (1994); Isidori (1996)) and extended (Isidori (1999); Shen (2002)) as a basis of H∞ control, based on the small gain theorem and passivity property. It is also discussed as the generalized K-Y-P lemma in relation to positive boundedness of transfer function, strict positive realness, the Riccati inequality, LMI, etc. and it contributes to a basis of robust control. Meanwhile, we solve it here as high gain output feedback for L2 -gain disturbance attenuation, applying Lyapunov’s direct method (LaSalle and Lefschetz (1961)). Finally, we study the L2 -gain disturbance attenuation problem for general (non-minimum phase) systems. As a control law, P·SPR·D control (Tamura and Shimizu (2008); Shimizu (2009)) is applied. The P·SPR·D and/or P·SPR·D+I control are a structured controller as PID control using a SPR (strict positive real) element. We can solve the L2 -gain disturbance attenuation problem of general systems by making the system be minimum phase with the use of the SPR element and by applying the high gain output feedback theorem obtained for the minimum phase system. It is noted that our approach differs from existing H∞ control theory and it possesses possibility of not only assuring small gain property of a transfer function but also improving control performance to some extent. 978-3-902661-93-7/11/$20.00 © 2011 IFAC
˙ x(t) = Ax(t) + Bu(t)
(1)
y(t) = Cx(t)
(2)
u(t) = −Ky(t)
(3)
m×m
with a gain matrix K ∈ R . Then the following high gain output feedback theorem is obtained. [Theorem 1] (High Gain Output Feedback) Suppose that system (1),(2) has relative degree {1, 1, · · · , 1} (i.e. CB is nonsingular) and is minimum phase. Consider an output feedback control (3). Then there exist constants γi0 > 0, i = 1, 2, · · · , m such that the closed-loop system (1) ∼ (3) is symptotically stable, provided that K is taken as K = (CB)−1 (Q11 + Γ) with Γ = diag(γ1 , γ2 , · · · , γr ), γi ≥ γi0 > 0, where Q11 is a partial matrix in the normal form (see eq.(5)). Namely, ˙ x(t) = Ax(t) − BKy(t) is asymptotically stable. (Proof ) From the assumption of relative degree {1, 1, · · · , 1}, by a nonsingular coordinate transformation C ξ x, T B = O (4) = T η system (1),(2) can be transformed into the normal form (Isidori (1995); Sepulcure (1997)) ;
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ξ˙ = Q11 ξ + Q12 η + CBu
(5)
10.3182/20110828-6-IT-1002.00234
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
η˙ = Q21 ξ + Q22 η
3. L2 -GAIN DISTURBANCE ATTENUATION FOR MINIMUM PHASE SYSTEM
(6)
y=ξ
(7)
m
n−m
m
and y ∈ R . Then, η˙ = where ξ ∈ R , η ∈ R Q22 η is called zero dynamics. If the zero dynamics be asymptotically stable, the system (1),(2) is said minimum phase.
Let us study L2 -gain disturbance attenuation problem in case of existence of disturbances. Consider the following linear multivariable system
By executing the output feedback u(t) = −Ky(t) = −Kξ(t)
(8)
y(t) = Cx(t)
(15)
(9) (10)
Since η˙ = Q22 η is asymptotically stable from the assumption of minimum phase, the Lyapunov inequality
m
u(t) = −Ky(t) with a gain matrix K ∈ R
Now we define a Lyapunov function V (ξ, η) =
ξ η
T
I O O P
ξ η
(12)
V (ξ, η) is a positive definite function, because of P > 0.
m×m
(17)
.
The L2 -gain disturbance attenuation problem is defined to attain a control such that the closed-loop system satisfies the following conditions under the given disturbance attenuation level γ > 0. P1. When w = 0, the closed-loop system is asymptotically stable. p2. When x(0) = 0, the following inequality holds for the arbitrarily given T > 0.
Calculate its time derivative along (9),(10) to get V˙ (ξ, η) T ˙ T I O I O ξ ξ ξ ξ˙ = + O P η η O P η˙ η˙ T Q11 ξ + Q12 η − CBKξ ξ I O = η O P Q21 ξ + Q22 η T ξ I O Q11 ξ + Q12 η − CBKξ + η Q21 ξ + Q22 η O P T T T ξ CBKξ ξ Q11 QT21 P ξ − = η 0 η QT12 QT22 P η T T ξ Q11 Q12 CBKξ ξ ξ + − η 0 η P Q21 P Q22 η T ξ ξ Θ = η η
m
where x(t) ∈ R , u(t) ∈ R , y(t) ∈ R , z(t) ∈ ′ Rm , w(t) ∈ Rl are the state vector, the control input, the measurable output, the controlled output and the disturbance input, respectively. And consider an output feedback control
(11)
holds for some P > 0.
(16)
n
ξ˙ = Q11 ξ + Q12 η − CBKξ
P Q22 + QT22 P < 0
(14)
z(t) = CP x(t)
the closed-loop system (5)∼(7) becomes as follows.
η˙ = Q21 ξ + Q22 η
˙ x(t) = Ax(t) + Bu(t) + Bw(t)
T
2
z(t) dt ≤ γ
2
T
w(t)2 dt
0
0
It is noticed that P2 is equivallent to having L2 -gain below γ when x(0) = 0, that is, z2 ≤ γ 2 w2 . It implies that for all w ∈ L2 [0, T ] and for the supply rate s(z, w) = γ 2 wT w −z T z, the following γ-dissipation inequality holds with a storage function V (x) (Schaft (2000); Shen (2002)). V˙ (x(t)) ≤ γ 2 w(t)T w(t) − z(t)T z(t)
(18)
Now if system (14),(15) has relative degree {1, 1, · · · , 1}, then by the nonsingular coordinate transformation
(13)
where
ξ η
=
C x, T B = O, T B = O T
(19)
system (14)∼(16) can be transformed into the normal form
Q11 − CBK + QT11 − K T B T C T QT21 P + Q12 Θ≡ P Q21 + QT12 P Q22 + QT22 P Accordingly, if we take K = (CB)−1 (Q11 + Γ), then the (1,1) block of Θ becomes −Γ − ΓT , which can be made negative definite, provided that Γ is a positive definite matrix. Hence, if Γ = diag{γ1 , γ2 , · · · , γm }, γi ≥ γi0 > 0 is taken sufficiently large, then Θ becomes negative definite in consideration of (11). Consequently, system (9),(10) becomes asymptotically stable. Therefore, system (1),(2) is asymptotically stable by the output feedback (3). Q.E.D
ξ˙ = Q11 ξ + Q12 η + CBu + CBw
(20)
η˙ = Q21 ξ + Q22 η
(21)
y=ξ
(22)
z = Q31 ξ + Q32 η
(23) ′
where ξ ∈ Rm , η ∈ Rn−m , y ∈ Rm , z ∈ Rm and w ∈ Rl . Note that in general there does not exist a matrix T satisfying T B = 0 and T B = 0. But we consider the case of B and B for which there exists a matrix such that T B = 0 and T B = 0 hold. It will be applied in Chapter 4.
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
By executing the output feedback u(t) = −Ky(t) = −Kξ(t)
(24)
the closed-loop system (20)∼(24) becomes as follows. ξ˙ = Q11 ξ + Q12 η − CBKξ + CBw
(25)
η˙ = Q21 ξ + Q22 η
(26)
z = Q31 ξ + Q32 η
(27)
Since η˙ = Q22 η is asymptotically stable from the assumption of minimum phase, the Lyapunov inequality P Q22 + QT22 P < 0 (28) holds for some P > 0. Here let us assume : [Assumption 1] There exists P > 0 such that P Q22 + QT22 P + QT32 Q32 < 0 The following theorem solves the L2 -gain diturbance attenuation problem. [Theorem 2] (High Gain Output Feedback for L2 gain Disturbance Attenuation) Suppose that system (14)∼(16) has relative degree {1,1,· · ·,1 } (i.e. CB is nonsingular) and is minimum phase. Suppose also that there exists a matrix T such that T B = 0 and T B = 0 hold. Further assume Assumption 1 holds. And consider the output feedback control (17). Then, there exist constants γi0 > 0, i = 1, 2, · · · , m such that the closed-loop system (14)∼(17)satisfies condition P2, provided that K is taken T as K = (CB)−1 Q11 + 12 (QT31 Q31 + γ12 CBB C T ) + Γ with Γ = diag{γ1 , γ2 , · · · , γm }, γi ≥ γi0 > 0. Namely, ˙ x(t) = Ax(t) − BKy(t) + Bw(t) possesses L2 -gain less than γ.
ξ QT11 − K T B T C T QT21 P = η QT22 P QT12 T T w ξ I O B CT O + 0 η O P O O T Q11 − CBK Q12 ξ ξ + η P Q21 P Q22 η T ξ w I O CB O + O P η 0 O O T T ξ Q31 Q31 QT31 Q32 ξ + − γ 2 wT w η QT32 Q31 QT32 Q32 η T T ξ Q11 − K T B T C T QT21 P ξ = η η QT12 QT22 P T T Q11 − CBK Q12 ξ ξ +wT B C T ξ + η P Q21 P Q22 η T T ξ Q31 Q31 QT31 Q32 ξ T +ξ CBw + η QT32 Q31 QT32 Q32 η 1 T 1 T T T ξ CB − γw − B C ξ − γw γ γ 1 T T + 2 ξ T CBB C T ξ − wT B C T ξ − ξ T CBw γ T ξ ξ = Θ η η 1 T 1 T T T − ξ CB − γw B C ξ − γw γ γ where we put Θ11 Θ12 Θ≡ , Θ21 Θ22
V (ξ, η) =
ξ η
T
I O O P
ξ η
T
ξ QT31 Q − γ 2wT w ] [Q 31 32 η QT32 T ξ I O Q11 ξ + Q12 η − CBKξ + CBw = η O P Q21 ξ + Q22 η T ξ I O Q11 ξ + Q12 η − CBKξ + CBw + η O P Q21 ξ + Q22 η T T T Q31 Q31 Q31 Q32 ξ ξ + − γ 2 wT w η QT32 Q31 QT32 Q32 η V˙ (ξ, η) +
ξ η
(30)
Θ21 ≡ P Q21 + QT12 + QT32 Q31 , Θ22 ≡ P Q22 + QT22 P + QT32 Q32
(29)
V (ξ, η) is a positive definite function, because of P > 0. To prove that the γ-dissipation inequality V˙ (x(t)) + (z(t)T z(t) − γ 2 w T (t)w(t)) ≤ 0 holds, make the following calculation ;
T
Θ11 ≡ Q11 − CBK + QT11 − K T B T C T 1 T + QT31 Q31 + 2 CBB C T , γ Θ12 ≡ QT21 P + Q12 + QT31 Q32 ,
Furthermore, the closed-loop system (14)∼(17) satisfies condition P1. (Proof ) Define a storage function (which is also a Lyapunov function)
ξ η
Since Θ is a symmetric matrix, it becomes negative definite by Schur complement, if Θ11 is sufficiently small negative definite in consideration of Assumption 1. Here, if we take K as 1 1 T K = (CB)−1 Q11+ (QT31 Q31 + 2 CBB C T )+ Γ (31) 2 γ , then Θ11 becomes −Γ − ΓT , which is negative definite, provided that Γ is a positive definite matrix. Hence, if Γ = diag{γ1 , γ2 , · · · , γm }, γi ≥ γi0 > 0 is sufficiently large, then Θ becomes negative definite under Assumption 1. Accordingly, we have
T T Q31 Q31 QT31 Q32 ξ − γ 2 wT w QT32 Q31 QT32 Q32 η T 1 T T 1 T T <− B C ξ − γw B C ξ − γw ≤ 0 (32) γ γ V˙ (ξ, η) +
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ξ η
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
Consequently, the γ-dissipation inequality holds, and so it follows that we have L2 -gain below γ.
u(t) = −KP Cx(t) + KS ζ(t)
When w = 0, condition P1 has been already proved in Theorem 1. Q.E.D [Corollary 1] Let us consider the case of z = y. Suppose that system (14),(15) has relative degree {1,1,· · ·,1 } (i.e. CB is nonsingular) and is minimum phase. Suppose also that there exists a matrix T such that T B = 0 and T B = 0 hold. Further assume there exists P > 0 satisfying P Q22 + QT22 P + I < 0. And consider the output feedback control (17). Then, there exist constants γi0 > 0, i = 1, 2, · · · , m such that the closed-loop system (14),(15),(17) satisfies condition P2, provided that K is T taken as K = (CB)−1 Q11 + 21 (I + γ12 CBB C T ) + Γ with Γ = diag{γ1 , γ2 , · · · , γm }, γi ≥ γi0 > 0. Namely,
by substituting (34),(39) into (38).
−KD C(Ax(t) + Bu(t) + Bw(t)) Furthermore, arranging this equation, we obtain u(t) = −(Ir + KD CB)−1 (KP C + KD CA)x(t) +(Ir + KD CB)−1 KS ζ(t) −(Ir + KD CB)−1 KD CBw(t) = −KX x(t) + KΞ ζ(t) − KW w(t) where
−1
KX ≡ (Ir + KD CB)
−1
KΞ ≡ (Ir + KD CB)
(40)
(KP C + KD CA)
KS
−1
KW ≡ (Ir + KD CB)
KD CB
By substituting (40) into (33), we obtain the closed-loop system
˙ x(t) = Ax(t) − BKy(t) + Bw(t) possesses L2 -gain less than γ.
˙ x(t) = (A − BKX )x(t) + BKΞ ζ(t) − BKW w(t)
Furthermore, the closed-loop system (14),(15),(17) satisfies condition P1.
4.1 P·SPR·D Control Consider the following general MIMO system: ˙ x(t) = Ax(t) + Bu(t) + Bw(t)
(33)
y(t) = Cx(t)
(34)
z(t) = CP x(t)
(35)
where x(t) ∈ Rn , u(t) ∈ Rr , y(t) ∈ Rm , z(t) ∈ ′ Rm , w(t) ∈ Rl are the state vector, the control input, the measurable output, the controlled output and the disturbance input, respectively. We investigate P·SPR·D control for L2 -gain diturbance attenuation, by making the system be minimum phase and applying the high gain output feedback. In case of regulation problem, PID control is given as
u(t) = −KP y(t) + KI
˙ −y(τ )dτ − KD y(t)
Now we define the SPR gain KS as K S ≡ HS L r×m
where KP , KI , KD ∈ R
Consider the followings.
(36)
(44)
′
(45)
Then, by substituting (34),(39) into (45), we have
In this paper, we propose the following P·SPR·D control ˙ ζ(t) = Dζ(t) − y(t), ζ(0) = 0, D < 0
(37)
˙ u(t) = −KP y(t) + KS ζ(t) − KD y(t)
(38)
where (37) represents a SPR element with negative definite D ∈ Rm×m , and KS ∈ Rr×m denotes the SPR gain.
u(t) = −KP Cx(t) + HS ζ ′ (t) −KD C(Ax(t) + Bu(t) + Bw(t)) Arranging this equation as before, we obtain u(t) = −KX x(t) + KΞ′ ζ ′ (t) − KW w(t)
Since it holds from (33),(34) that
we have
′ ζ˙ (t) = LD′ ζ ′ (t) − Ly(t)
˙ u(t) = −KP y(t) + HS ζ (t) − KD y(t)
are P, I, D gains.
˙ y(t) = C(Ax(t) + Bu(t) + Bw(t))
(43)
m×m
and L ∈ R are called the interwhere HS ∈ R mediate parameter matrix and the adjustable parameter matrix, respectively. The parameterization as (43) is made in order to apply high gain output feedback to the system (49)∼(51) below. Namely, L corresponds to the gain of output feedback and HS is a parameter used to asymptotically stabilize the zero dynamics. The high gain output feedback makes controller parameter tuning easy for improving the control performance.
0
r×m
(41)
Accordingly, by combining (41) and (37), the closed-loop system by the P·SPR·D control becomes ˙ x(t) A − BKX BKΞ x(t) = ˙ ζ(t) −C D ζ(t) −BKW + B + w(t) (42) O
4. L2 -GAIN DISTURBANCE ATTENUATION BY P·SPR·D AND/OR P·SPR·D+I CONTROL
t
+Bw(t)
where (39)
KΞ′
−1
≡ (Ir + KD CB)
(46)
HS
By substituting (46) into (33), we obtain the closed-loop system
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
˙ x(t) = (A − BKX )x(t) + BKΞ′ ζ ′ (t) − BKW w(t) +Bw(t) Accordingly, by combining (47) and (44), we have ˙ x(t) x(t) A − BKX BKΞ′ ′ = −LC LD′ ζ ′ (t) ζ˙ (t) −BKW + B w(t) + O
′
(47)
(48)
(t) ∈ Rm and z (t) ∈ Rm . (t) ∈ RN , v(t) ∈ Rm , y where x
B If system (54),(55) has relative degree {1, 1, · · · , 1} (i.e. C is nonsingular), then the system can be transformed into the normal form. That is, by the nonsingular coordinate transformation C ξ =O = O, TB , TB = x (57) η T
(54)∼(56) can be transformed into the normal form
Next let us consider the following hypothetical system based on the closed-loop system (48) ; ˙ x(t) x(t) A − BKX BKΞ′ ′ = O O ζ ′ (t) ζ˙ (t) O −BKW + B + v(t) + w(t) Im O x(t) + Bv(t) (49) := A + Bw(t)
x(t) x (t) = C −D′ (t) y := C (50) ζ ′ (t) x(t) P x (t) (t) = [ CP O ] ′ (51) z := C ζ (t) ′
(t) ∈ Rn+m , v(t) ∈ Rm , y ∈ Rm , z ∈ Rm where x denote the state vector, the input, the measurable output and the controlled output of the hypothetical system C, B, B, C P }, respectively. And let the input v(t) be {A, given by the output feedback
x(t) ′ v(t) = −L y (t) = −L C −D (52) ζ ′ (t)
Bv +C Bw ξ˙ = Q11 ξ + Q12 η + C
(58)
η˙ = Q21 ξ + Q22 η
(59)
=ξ y
(60)
= Q31 ξ + Q32 η z m
where ξ ∈ R , η ∈ R
N −m
∈R , y
(61) m
∈R and z
m′
.
Here we assume : [Assumption 2] There exists P > 0 such that P Q22 + QT22 P + QT32 Q32 < 0
Applying Theorem 2 to system (54)∼(56) we have : C, B, B, C P } [Propositon 1] Suppose that system {A, given by (54)∼(56) has relative degree {1, 1, · · · , 1} (i.e. B is nonsingular) and is minimum phase. Suppose also C =0 = 0 and TB that there exists a matrix T such that TB hold. Further assume Assumption 2 holds. And consider an output feedback control
where L ∈ Rm×m is the output feedback gain. Then, the closed-loop system of the hypothetical system becomes ˙ x(t) x(t) A − BKX BKΞ′ ′ = −LC LD′ ζ ′ (t) ζ˙ (t) −BKW + B w(t) (53) + O which is the same as (48).
. Then, there exist constants with a gain matrix L ∈ R γi0 > 0, i = 1, 2, · · · , m such that the closed-loop system (54)∼(56),(62) satisfies condition P2, provided that L is B TC B T + B) −1 Q11 + 1 QT Q31 + 12 C taken as L = (C 31 2 γ Γ with Γ = diag(γ1 , γ2 , · · · , γm ), γi ≥ γi0 > 0. Namely,
Now if we put D′ = DL−1 and ζ ′ = Lζ, then the lower column of (53) becomes equal to (37). Therefore, if we set KS = HS L, D = D′ L, (53) coincides with (42).
x(t) − BL y (t) + Bw(t) ˙ (t) = A x
Consequently, (53) becomes equal to the closed-loop error system with the P·SPR·D control (33),(34),(37),(38), provided that the output feedback gain L is taken equal to the adjustable parameter of (43).
v(t) = −L y (t)
(62)
m×m
possesses L2 -gain below γ.
Furthermore, the closed-loop system (54)∼(56),(62) satisfies condition P1. (Proof ) Obvious from Theorem 2.
Q.E.D
4.2 Design of P·SPR·D Controller for L2 Gain Disturbance Attenuation
Now let us consider to apply Proposition 1 to the hypothetical system (49)∼(51) to design the P·SPR·D controller. So check first the relative degree with respect to v. Since
Let us apply Theorem 2 to design the P·SPR·D controller for L2 gain disturbance attenuation. Consider the following general MIMO system ;
O ˙ (t) ∂y B = C −D′ =C = −D′ Im ∂v(t)
x(t) + Bv(t) ˙ (t) = A x + Bw(t) x (t) = C (t) y P x (t) = C (t) z
(54) (55) (56)
, this matrix has to be nonsingular in order to apply Proposition 1. So let D′ be a nonsingular matrix. Accordingly, we can transform system (49)∼(51) into the normal form to obtain its zero dynamics. Consider the following transfomation from (57) ;
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
ξ η
=
x C C −D′ x = Tn O ζ′ T
(63)
= O, = O, TB T = [ Tn O ] , TB
where
and so η ∈ Rn . Note We know that ξ ∈ Rm from ξ = y that the inverse matrix of (63) becomes
C −D′ Tn O
−1
=
O Tn−1 ′ ′ −1 −1 −D D CTn−1
Transform the hypothetical system (49)∼(51) by (63) to get A − BKX BKΞ′ ξ˙ = C −D′ × Tn O O O η˙ O Tn−1 ξ ′ ′ −1 −1 −1 η −D D CTn O C −D′ + v Im Tn O C −D′ −BKW + B + w (64) Tn O O Then, we can obtain the normal form of the hypothetical system (49)∼(51) ;
4.3 Determination of Controller Parameters An important task in our method is to satisfy Assumption 3, that is, to determine KP , KD and HS such that the zero dynamics (69), i.e. η˙ = Tn A − B(Ir + KD CB)−1 (KP C + KD CA ′ −HS D −1 C) Tn−1 η (70) is asymptotically stable.
Since eigenvalues of zero dynamics (70) are the same with those of the matrix ′
A − B(Ir + KD CB)−1 (KP C + KD CA − HS D −1 C)(71) we consider a method of finding KP , KD , HS such that this matrix be asymptotically stable. We first change the matrix of (71) as follows. ′
A − B(Ir + KD CB)−1(KP C + KD CA − HS D −1 C)
C ′ =A − B(Ir + KD CB)−1 KP − HS D −1 KD (72) CA
ξ˙ = −CBKΞ′ D′−1 ξ ′ +C (A − BKX )Tn−1 + BKΞ′ D −1 CTn−1 η − D′ v +C(−BKW + B)w
Then, the closed-loop system (33)∼(35),(37),(38) satisfies the conditions P1 and P2. Also, from the property of high gain output feedback, we can improve the convergence speed by adjusting the high gain L to some extent.
(65)
By defining the following matrices
′
′
η˙ = −Tn BKΞ′ D −1 ξ ′ + Tn (A − BKX + BKΞ′ D −1 C)Tn−1 η =ξ y
= [CP z
ξ O Tn−1 O] = CP Tn−1 η −D′−1 D′−1 CTn−1 η
Fη1 = (Ir + KD CB)−1 (KP − HS D −1 )
η˙ = Tn
′ A − B(KX − KΞ′ D −1 C) Tn−1 η
Fη2 = (Ir + KD CB)
(74)
(67)
Fη = [ Fη1
(75)
(68)
(69)
In order to satisfy the minimum phase requirement, the zero dynamics (69) has to be asymptotically stable. Thus we assume : [Assumption 3] There exist parameter matrices KP ,HS , KD such that the zero dynamics (69) is asymptotically stable. Corresponding to Assumption 2, we assume also : [Assumption 4] There exists P > 0 such that
where
Q22 ≡ Tn (A − B(KX − KΞ′ D
−1
KD C Fη2 ] , Cη = CA
(72) can be expressed as A − BFη Cη (76) This can be regarded as a matrix of the closed-loop system due to the output feedback −Fη Cη η for system {A, B, Cη }. Accordingly, in order to get matrices KP , KD , HS asymptotically stabilizing the zero dynamics, we can apply any existing method of output feedback stabilization (e.g. Tamura and Shimizu (2006)) to the system {A, B, Cη }. After determining the output feedback gain Fη stabilizing (76) , we can obtain KP , KD , HS stabilizing (71) from the relations (73) and (74). Besides, we can make it by trial and error, checking eigenvalues of matrix (71) directly. From Proposition 1 the high gain L can be attained :
P Q22 + QT22 P + QT32 Q32 < 0 ′
(73)
(66)
Accordingly, from (66) the zero dynamics is expressed as
−1
C)Tn−1 ,
Q32 ≡ CP Tn−1 Consequently, we can apply Proposition 1 under Assumptions 3 and 4 in order to obtain L of the output feedback (52) which solves the L2 -gain disturbance attenuation problem for (49)∼(51). Accordingly, by setting SPR parameter matrices as KS = HS L, D = D′ L, we obtain the P·SPR·D controller (37),(38) for the system (33)∼(35).
B TC B) −1 Q11 + 1 QT Q31 + 1 C T + Γ B L = (C 31 2 γ2 1 1 C(−BKW + B) = −D′−1 − CBKΞ′ D′−1 + 2 γ2 ×(−BKW + B)T C T + Γ = −D′−1 − CB(Ir + KD CB)−1 HS D′−1 1 1 + C(−B(Ir + KD CB)−1 KD CB + B) 2 γ2
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
×(−B(Ir + KD CB)−1 KD CB + B)T C T + Γ
(77)
1.4 1.2
controlled output z = θ
3
A design procedure is arranged as below. [Algorithm 1] ′ [Step 1] Give a nonsingular matrix D ∈ Rm×m . [Step 2] Detemine gain matrices KP , HS , KD ∈ Rr×m which make the zero dynamics matrix (71) be asymptotically stable and satisfy Assumption 4. [Step 3] Choose L as (77), based on Proposition 1, and determine KS = HS L and D = D′ L. [Step 4] The P·SPR·D controller is given by (37),(38).
1 0.8 0.6 0.4 0.2 0 0
0.5
1 time t
1.5
2
Fig.1 Step Desired Output Response for z ∗ = 1 0.16
As a practical design policy, it may be effective to decide Γ i.e. L subjectively, observing the output responses concretely, since D, KP , KS , KD can be computed immediately given L in the proposed method.
0.14
controlled output z = θ3
0.12
4.4 P·SPR·D+I Coontrol
0.1 0.08 0.06 0.04 0.02
The P·SPR·D control certainly satisfies condition P1 (i.e. the closed-loop system converges to the origin x = 0 when w = 0). But an off-set arises when w = 0. Hence we necessitate to add I action to P·SPR·D control to get rid of it. Moreovere, in order to cope with servo problem, the I action is also needed. Our method can handle the set-point servo problem easily by adding the I action.
0 −0.02 0
1.5
2
1.4
controlled output z = θ
3
1.2 1 0.8 0.6 0.4 0.2 0 0
(78)
0.5
1 time t
1.5
2
∗
Fig.3 Step Desired Output Response for z = 1 0.09 0.08 0.07
controlled output z = θ3
˙ u(t) = −KP y(t) + KS ζ(t) − KD y(t) t +KI (z ∗1 − z 1 (τ ))dτ
1 time t
Fig.2 Step Disturbance Response Against τd1 = 1
For that purpose let us assume that among controlled output variables z, variables z 1 to track the desired value z ∗1 is a partial vector of z (i.e. z 1 ⊂ z) and can be measured to use for integral control. In such a case consider the following P·SPR·D+I control adding the integral action. ˙ ζ(t) = Dζ(t) − y(t), ζ(0) = 0, D < 0
0.5
(79)
0
Since stability and L2 -gain disturbance attenuation are guaranteed sufficiently by the high gain feedback, we devise here only a countermove for a steady state error.
0.06 0.05 0.04 0.03 0.02 0.01 0 0
0.5
1 time t
1.5
2
Fig.4 Step Disturbance Response Against τd1 = 1
Incidentally, in a special case where the controlled output z equals to the measurable output y, i.e., y ≡ z, the prososed methods can be simplified to a great extent.
˙ ζ(t) = Dζ(t) + (y ∗ − y(t)), ζ(0) = 0, D < 0
(80)
∗
˙ u(t) = KP (y − y(t)) + KS ζ(t) − KD y(t) t + KI (y ∗ − y(τ ))dτ
Meanwhile, when w = 0, at an equilibrium state xe holding the controlled output z at z ∗ must satisfy the following relation;
(81)
0
∗
0 = Axe + Bu, z = CP xe Since this relation consists of (n + m′ ) equations and (n + r) variables, when r ≥ m′ , (r − m′ ) state variables xeN can be set arbitrary value x∗eN , but the remained dependently. state variables xeB and u are determined ∗ x eN Putting such an equilibrium as x∗ = and xeB (x∗eN , z ∗ ) u∗ = u(x∗eN , z ∗ ), we have
5. SIMULATION RESULTS Let us consider a position control problem of 3-inertia system (Hara et al. (1995)). Letting θi , i = 1, 2, 3, be the rotating angle of each rotating mass and τ the manipulated torque and τdi , i = 1, 2, 3, the disturbance torque, an equation of motion of the system is represented as
0 = Ax∗ + Bu∗ , z ∗ = CP x∗ Then, the measurable output y should be y ∗ = Cx∗ . Thus we can consider the following P·SPR·D+I control : 7509
J1 θ¨1 = d1 θ˙1 − ka (θ1 − θ2 ) − da (θ˙1 − θ˙2 ) + τ + τ d1 J2 θ¨2 = ka (θ1 − θ2 ) + da (θ˙1 − θ˙2 ) − d2 θ˙2 − kb (θ2 − θ3 ) −db (θ˙2 − θ˙3 ) + τ d2
18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
J2 θ¨3 = kb (θ2 − θ3 ) + db (θ˙2 − θ˙3 ) − d3 θ˙3 + τ d3 where Ji , i = 1, 2, 3, denotes the inertia moment, di , i = 1, 2, 3, a, b, the coefficient of viscosity friction and ki , i = a, b, the spring constant. Let x = (θ1 , θ2 , θ3 , θ˙1 , θ˙2 , θ˙3 )T , u = τ and w = (τd1 , τd2 , τd3 )T , then a state space representation of the system is given as x˙ = Ax + Bu + Bw
that convergence speed for the set-point servo problem got improved very much compared to (78),(79), though step disturbance response became a little slower. It can be said that satisfactory control performances were obtained for both cases. Note also that even if gains KP , KS , KD , KI change around the above values to some degree, the closed-loop system maintains to be asymptotically stable.
y = Cx
6. CONCLUDING REMARKS
z = CP x where A= 0
0 0 −ka J1 ka J2 0
0 0 0 ka 1 J −ka − kb J2 kb J3
0 0 0 B= , 1/J1 0 0
0 0 0
1 0 0 −d1 − da 0 J1 kb da J2 J2 −kb 0 J3 0 0 0 0 0 0 1 B = J1 0 0 1 J2 0 0
C = [1 0 0 0 0 0]
0 0 1 0 0 1 da 0 J1 −da − d2 − db db J2 J2 −db − d3 db J3 J3 0 0 0 0 0 1 J3
CP = [ 0 0 1 0 0 0 ] where plant parameters are given as J1 = J2 = 0.001, J3 = 0.002, d1 = 0.05, d2 = 0.001, d3 = 0.007, da = db = 0.001, ka = 920, kb = 80. Using Algorithm 1, we can solve L2 -gain disturbance attenuation problem. We set γ = 1. We first determine KP , KD , HS asymptotically stabilizing the zero dynamics (71) and then set controller parameters of P·SPR·D+I control (78),(79) as follows. D′ = −1, HS = 1, KP = 4, KD = 1, KI = 10 Γ = 5, L = 5.5, KS = 5.5 The simulation results of set-point servo problem for z ∗ = 1 and x(0) = 0, w = 0 is shown in Fig.1. The simulatioin results against a step disturbance w = (τd1 , τd2 , τd3 )T = ( 1 0 0 )T is shown in Fig.2, when z ∗ = 0 and x(0) = 0. Next, we consider P·SPR·D+I control (80),(81), setting controller parameters as follows. D′ = −1, HS = 1, KP = 10, KD = 0.5, KI = 10 Γ = 5, L = 5.5, KS = 5.5 The simulation results of set-point servo problem for z ∗ = 1 and x(0) = 0, w = 0 is shown in Fig.3. In this case we set y ∗ = 1 corresponding to z ∗ = 1. The simulatioin results against a step disturbance w = [ 1 0 0 ]T is shown in Fig.4, when we set z ∗ = 0, y ∗ = 0 and x(0) = 0. We see
We investigated the P·SPR·D and/or P·SPR·D+I control for the L2 -gain disturbance attenuation problem, based on the idea that one makes a system have relative degree {1, 1, · · · , 1} and be minimum phase by use of the SPR element such that high gain output feedback can be applied. The use of SPR element contributes powerfully to stabilizing the closed-loop system. The P·SPR·D or P·SPR·D+I control is very practical because of its simple structure. Compared to H∞ control, our method can handle the set-point servo prblem easily by using I action. Besides, we do not have to solve the Riccati inequality or LMI for designing the controller. Although we studied only to stabilize the system and to attain L2 -gain below γ against disturbances, how to determine KP , KS , KD , D giving better control performance is considered a future topic. REFERENCES T.Hara,Y.Chida,M.Saeki,K.Nonami, Bench Mark Problem for Robust Control (I), SICE Jjournal, Vol.34, No.5, pp.403-409, 1995 (in Japanese) A.Isidori, Nonlinear Control Systems, Third Edition, Springer-Verlag, 1995 A.Isidori, Global Almost Disturbance Decoupling with Stability for Nonminimum Phase Systems, Syst. Contr. Lett., Vol.27, pp.191-194, 1996 A.Isidori, Nonlinear Control Systems II, Chap. 13, Disturbance Attenuation, Springer-Verlag, 1999 H.K. Khalil, Nonlinear Systems, 3rd ed., Prentice Hall, 2002 J.LaSalle and S.Lefschetz, Stability by Liapunov’s Direct Method, Academic Press, 1961 R.Marino, W.Rosenblock, A. van der Schaft, P.Tomei, Nonlinear H∞ Almost Disturbance Decoupling, Syst. Contr. Lett., Vol.23, pp.159-168, 1994 A. van der Schaft, L2 -Gain and Passivity Techniques in Nonlinear Control, 2nd ed., Springer-Verlag, 2000 T.Shen, Robust Control of Nonlinear System Based on Dissipativity, SICE, 2002 (in Japanese) R.Sepulcure, M.Jankovic and P.V.Kokotovic, Constructive Nonlinear Control, Springer-Verlag, 1997 K.Shimizu, Generalization and Proof of High Gain output Feedback Theorem, Trans. SICE, Vol.40, No.12, pp.1247-1249, 2004 (in Japanese) K.Shimizu, P·SPR·D and P·SPR·D·I Control for Linear Multi-Variable Systems-Stabilization Based on High gain Output Feedback-, Proceedings of 2009 American Control Conference, pp.4666-4672, St. Louis, 2009 K.Tamura and K.Shimizu, Eigenvalue-Assignment Method by PID Control for MIMO Systems, Systems, Control and Informatioin, Vol.19, No. 5, pp.193-202, 2006 (in Japanese) K.Tamura and K.Shimizu, A Control Method for MIMO Systems(P+quasi·I+D Control)–Stabilization Based on High Gain Output Feedback–, Trans.SICE, Vol.44,No.5, pp.434-443, 2008 (in Japanese)
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