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Comprehensive admissibility for descriptor systems
Automatica 66 (2016) 271–275
Contents lists available at ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Technical communique
Comprehensive admissibility for descriptor systems✩ Yu Feng a,1 , Mohamed Yagoubi b a
Information Engineering College, Zhejiang University of Technology, 288 Liuhe Road, Hangzhou, PR China
b
LUNAM Université, Ecole des Mines de Nantes, IRCCyN UMR CNRS 6597, 4 rue Alfred Kastler, Nantes, France
article
info
Article history: Received 28 April 2015 Received in revised form 1 October 2015 Accepted 23 December 2015
Keywords: Descriptor systems Structured controller Unstable and nonproper weights
abstract This paper presents a complete solution to the problem of comprehensive admissibility for descriptor systems with unstable and nonproper weights. In such non-standard circumstances, it is hard to render the closed-loop system admissible due to the uncontrollable and unobservable weights. The necessary and sufficient condition is given and a specifically structured output feedback controller is conducted in terms of the dynamics of the weighs. It is shown that determining suitable controllers for a given descriptor system with the presence of unstable and nonproper weights requires solving an admissibility problem for an augmented system explicitly constructed in this paper. A numerical example is included to illustrate the current result. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Descriptor systems have been widely studied over recent decades owing to their capacity to describe non-dynamic constraints and impulsive behaviors. A number of control issues have been successfully developed for descriptor systems (Darouach, 2014; Duan, 2010; Feng & Yagoubi, 2013; Xu & Lam, 2006; Zou, Ho, & Wang, 2010). Many control problems require a standard framework based on models of the physical plants, disturbances and references together with the control objectives (Chiang & Safonov, 1992; Meinsma, 1995; Stoorvogel, Saberi, & Sannuti, 2000). For instance, control of the longitudinal motion of fighter airplanes has been well addressed under this framework (Chiang & Safonov, 1992; Kwakernaak, 2002). Moreover, in the H∞ control theory, it is often desirable to choose a weight with a pole at the origin since the closed-loop system is finite only if the sensitivity function has a zero at the origin. Moreover, to avoid undesirable high frequency noise sensitivity and limited robustness, it is also often required to select a nonproper weight, with its H∞ norm being large outside the desirable closed-loop bandwidth (Meinsma, 1995). Such
✩ This work was partially supported by the Natural Science Foundation of China under Grant 61203130 and Qianjiang Talent Plan under Grant QJD1302013. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Tingshu Hu under the direction of Editor André L. Tits. E-mail addresses: [email protected] (Y. Feng), [email protected] (M. Yagoubi). 1 Tel.: +86 57185290601.
http://dx.doi.org/10.1016/j.automatica.2016.01.028 0005-1098/© 2016 Elsevier Ltd. All rights reserved.
choices, however, yield non-standard problems for plants containing unstabilizable or undetectable elements. Such problems may be solved in different ways: the perturbation technique (Chiang & Safonov, 1992), the loop transformation (Meinsma, 1995), the polynomial method (Kwakernaak, 1993), or the quasi-stabilization theory (Mita, Xin, & Anderson, 2000). However, all these techniques have their own drawbacks. In fact, the perturbation technique is poorly suited to deal with lightly-damped poles while the loop transformation requires a pretreatment to absorb the weights into the loop. The polynomial method does not apply to cases where the weights have imaginary poles, while the quasistabilization theory does not allow nonproper weights. Recently, in Feng, Yagoubi, and Chevrel (2011), such a nonstandard problem is considered within the framework of descriptor systems. Hence, both unstable and nonproper weights can be naturally handled and an observer-based controller is conducted. Performance control with the presence of weights is further addressed in Feng, Yagoubi, and Chevrel (2012a,b). However, the effects of the weights on the underlying controller’s structure have not yet been clarified. The current paper gives a complete solution to this non-standard problem and the structure of the resulting controller is explicitly exhibited with the dynamics of unstable and nonproper weights. It is observed that the non-standard problem can be transformed into a problem without weights for an augmented system explicitly constructed from the data of the given system. This transformation makes it easy to address additional performance objectives for a larger set of systems. The extended synthesis problem considered in Feng et al. (2012a,b) can be directly solved by applying the current result and unnecessary
272
Y. Feng, M. Yagoubi / Automatica 66 (2016) 271–275
Let us introduce here two partitions of G with regard to Wi and Wo , denoted as GWi and GWo , respectively.
GWi
¯ A11 0 :=
A¯ 12 Ai
Aˆ 12 Aˆ 22
Fig. 1. Comprehensive admissibility control setup.
assumptions concerning the invariant zeros made in Feng et al. (2012a,b) can be removed. The rest of this paper is organized as follows. Section 2 formulates the problem. The main result is presented in Section 3. The concluding remarks are given in Section 4.
GWo :=
Consider the following descriptor system: (1)
where x ∈ Rn , y ∈ Rp and u ∈ Rm . The matrix E ∈ Rn×n may be singular. The descriptor system (1) is said to be regular, impulse-free and stable, if det(sE − A) ̸≡ 0, deg(det(sE − A)) = rank(E ), and all the roots of det(sE − A) = 0 have negative real parts, respectively. If the descriptor system is regular, impulse-free and stable, then it is admissible. In addition, it is said to be finite dynamics stabilizable if F exists such that (E , A + BF ) is regular and stable; it is said to be impulse controllable if F exists such that (E , A + BF ) is regular and impulse-free. Dual notions can be defined for finite dynamics detectability and impulse observability (Xu & Lam, 2006). Let us consider the setup in Fig. 1, where the plant G(s), input weight Wi and output weight Wo are given as follows. Ag − sEg Cg1 Cg2
G :=
Bg1 Dg11 Dg21
Ai − sEi Ci
Wi :=
Bi Di
Bg2 Dg12 0
,
A − sE C1 C2
G :=
B1 D11 D21
Wo :=
B2 D12 0
E=
Eo 0 0
B1 =
0 Eg 0
Ao − sEo Co
Bo Do
,
,
Bo Dg11 Di Bg1 Di , Bi
C1 = Co
Do Cg1
(2)
A=
D11 = Do Dg11 Di ,
B2 = Do Dg11 Ci ,
−s
Eo 0
0 Eˆ
Cˆ 12 Cˆ 22
Co 0
B¯ 12 0
D11 D21
D12 0
Bˆ 11 Bˆ 21
Bˆ 12 Bˆ 22
D11 D21
D12 0
,
(3)
.
(4)
BK DK
,
(5)
The problem under consideration is to find a controller K such that the interconnection in Fig. 1 is comprehensively admissible. Before ending this section, we make the following assumptions: (A1) Wi and Wo only possess unstable and impulsive modes; (A2) (E¯ , A¯ 11 , B¯ 12 ) is finite dynamics stabilizable and impulse controllable; (A3) (Eˆ , Aˆ 22 , Cˆ 22 ) is finite dynamics detectable and impulse observable.
Theorem 2. Consider the partitions (3) and (4). The interconnection in Fig. 1 is comprehensively admissible, if and only if Xi ∈ R(ng +no )×ni , Yi ∈ R(ng +no )×ni , Πi ∈ Rm×ni , Xo ∈ Rno ×(ng +ni ) , Yo ∈ Rno ×(ng +ni ) and Πo ∈ Rno ×p exist such that
B¯ 12 Πi D Π 12¯ i EYi Πo Cˆ 22 ΠD o 21ˆ Xo E
Ao 0 0
Bo Cg1 Ag 0
AK =
D12 = Do Dg12 ,
Dg21 Ci ,
D21 = Dg21 Di .
(7)
BK D1k1
Ai + D2k1 Γi Bk1 Γi
,
EK =
Ck2 Ak −Γo Ck
Ωo Γi
Cg2
= Ao Yo − Aˆ 12 − Xo Aˆ 22 , = −Xo Bˆ 21 − Bˆ 11 , = Eo Y o .
(8)
with
Bo Dg11 Ci Bg1 Ci , Ai
Bo Dg12 Bg2 , 0
(6)
AK − sEK CK
C2 = 0
A¯ 11 Yi − A¯ 12 − Xi Ai , C¯ 11 Yi − C¯ 12 , Xi E i .
= = =
Moreover, an admissible controller is given by
C¯ 12 C¯ 22
B¯ 11 Bi
Definition 1 (Comprehensive Admissibility Feng et al., 2012a,b). The feedback system in Fig. 1 is said to be comprehensively admissible if Fl ( G, K ), where Fl (·, ·) is the lower linear fractional transformation, is internally stable and the closed-loop system defined as Tz w = Fl (G, K ) is admissible.
K :=
0 0 , Ei
where EK ∈ Rnk ×nk , AK ∈ Rnk ×nk , BK ∈ Rnk ×p , CK ∈ Rm×nk and DK ∈ Rm×p . EK may be singular.
where
0 Ei
3. Main result
where e ∈ Rq , y ∈ Rp , v ∈ Rl and u ∈ Rm are the controlled output, measurement, disturbance and control input, respectively; Ai ∈ Rni ×ni , Ao ∈ Rno ×no , Bi ∈ Rni ×mi , Bo ∈ Rno ×q , Ci ∈ Rl×ni , Co ∈ Rpo ×no , Di ∈ Rl×mi and Do ∈ Rpo ×q are known constant matrices. Moreover, Eg , Ei and Eo may be singular. The direct feedthrough matrix from u to y is left out to make the arguments simpler and can be handled via standard methods. Then, the resulting weighted system G is written as
Ao 0
E¯ 0
,
C¯ 11 C¯ 21
AK − sEK CK
K := x˙ (t ) = Ax(t ) + Bu(t ), y(t ) = Cx(t ) + Du(t ),
−s
The system data of the partitions can be obtained directly from (2); see Feng et al. (2012a) for more details. Consider an output feedback controller K as
2. Problem formulation
Ei 0 0
0 Ek 0
Γi = C¯ 21 Yi − C¯ 22 ,
Ao − Γo Dk2
0 0 , Eo
D2k2 Bk2
D2k1 Bk1 ,
BK =
Ωo
Γo = Xo B˜ 22 + B˜ 12 ,
,
ΩiT T CKT = Ck1 , 1 T Dk2
Y. Feng, M. Yagoubi / Automatica 66 (2016) 271–275
Ωi = Πi + D1k1 Γi , Ωo = Πo − Γo Dk1 , Xi2 B˜ 12 = Bˆ 12 Xi1 , B˜ 22 = Bˆ 22 , −I n i Xi1 = Ino 0 Xi , Xi2 = 0 Ing Xi , 1 1 1 Dk1 Dk2 Ck Dk1 = , Dk2 = , Ck = 2 2 2 , Dk1
Ck
Cc = C¯ 11 + D12 D1k1 C¯ 21
Bk1
Bk2
Ck1 Ck2
D1k1 D2k1
D1k2 D2k2
K :=
(9)
A − sE C1 C2
−I n o
B2
D11
D12
D21 0
0 0
0
0 0
.
C¯ k2 ¯Ak − sE¯ k C¯ k1
D2k1 B¯ k , D1k1
Then, the closed-loop system is given by Ac − sEc Cc
Bc Dc
0 0 I 0
0 0 , 0 I
B¯ c Dc
,
(13)
B¯ 12 Ωi Ai + D2k1 Γi B¯ k Γi
B¯ 12 C¯ k1 C¯ k2 , A¯ k
B¯ 11 + B¯ 12 D1k1 D21 + Xi Bi , Bc = D2k1 D21 − Bi B¯ k D21
0 0, E¯ k
D12 C¯ k1 .
D12 Ωi
¯ := G
A¯ 11 − sE¯ 0
B¯ 12 Πi Ai − sEi
B¯ 11 + Xi Bi −Bi
C¯ 11 C¯ 21
D12 Πi
D11 D21
Γi
A¯ k − sE¯ k K¯ := C¯ k1 C¯ k2
B¯ 12 0 D12 0
0 Ini , 0 0
B¯ k D1k1 . D2k1
With Eq. (6) and two transformation matrices
M2 =
In g + n o 0
Xi , −Ini
In g + n o 0
N2 =
Yi , −Ini
ˆ := G
,
Aˆ 12 Aˆ 22
Ao 0
−s
Eo 0
0 Eˆ
Cˆ 12 Cˆ 22
Co 0
Bˆ 11 Bˆ 21
B˜ 12 B˜ 22
D11 D21
˜ 12 D 0
,
˜ 12 = D12 0 , and the other where B˜ 12 and B˜ 22 are given in (8), D data are given in the partition (4). Proceeding with the controller K¯ specified in (11) and (12) gives
M2 A¯ c N2 − sM2 E¯ c N2 C¯ c N2
M2 B¯ c Dc
,
where
A11 + B¯ 12 D1k1 C¯ 21 0 Ac = D2k1 C¯ 21 B¯ k C¯ 21
¯
E 0 Ec = 0 0
Yi −I −I 0
Therefore, the uncontrollable dynamics of Wi are cancelled in the closed-loop system. Second, we prove that the unobservable dynamics of Wo are also cancelled in the closed-loop system. To this end, let us consider further the closed-loop system (13), which ¯ , K¯ ), where is nothing other than Fl (G
Fl (Gˆ , K¯ ) :=
where
¯
A¯ c − sE¯ c C¯ c
0 Ei 0
(12)
Fl (G, K ) :=
I 0 N1 = 0 0
¯ is represented alternatively by Gˆ as the system G
Ak Bk2 ¯Ek = Ek 0 , ¯Ak = , (11) 0 Eo −Γo Ck Ao − Γo Dk2 1 T 2 T Bk1 , (Ck ) (Ck ) B¯ k = , (C¯ k1 )T = , (C¯ k2 )T = . Ωo (D1k2 )T (D2k2 )T
where
0 0 , 0 I
C¯ c = C¯ 11 + D12 D1k1 C¯ 21
By setting Πi = DK C¯ 22 − DK C¯ 21 Yi − CK YiK , Eq. (6) holds. Eq. (7) can be conducted by following the same line for the partition in (4). Sufficiency: We prove that under (6) and (7), the controller (8), together with the free parameters specified in K (9) that internally stabilizes the augmented system G (10), achieves comprehensive admissibility. First, we show that the uncontrollable dynamics of Wi are cancelled in the closed-loop system. For simplicity, we denote the controller (8) as
Ωi
Fl (G, K ) :=
E¯ E¯ c = 0 0
0 = D12 (DK C¯ 22 − DK C¯ 21 Yi − CK YiK ) − C¯ 11 Yi + C¯ 12 , Xi Ai = A¯ 11 Yi + B¯ 12 (DK C¯ 21 Yi − DK C¯ 22 + CK YiK ) − A¯ 12 , ¯ i. Xi Ei = EY
Ai + D2k1 Γi − sEi B¯ k Γi
0 0 I 0
together with (6), there holds
Proof. Necessity: We prove that if the closed-loop system in Fig. 1 is comprehensively admissible, then (6) and (7) hold. First, let us consider the partition (3). According to Feng et al. (2011), matrices Xi ∈ R(ng +no )×ni , Yi ∈ R(ng +no )×ni and YiK ∈ Rnk ×ni exist such that
K :=
Xi −I −I 0
A¯ 11 + B¯ 12 D1k1 C¯ 21 ¯Ac = D2k1 C¯ 21 B¯ k C¯ 21
(10)
I 0 M1 = 0 0
Xi In i
B1
−Y o
Φ = A¯ 12 + B¯ 12 D1k1 C¯ 22 .
where
internally stabilizing the augmented system G of the form
G :=
By the two transformation matrices
Dk2
Ak − sEk
D12 C¯ k1 ,
D12 Ωi
C¯ 12 + D12 D1k1 C¯ 22
Dc = D11 + D12 D1k1 D21 ,
where the matrices Ek , Ak , Bk1 , Bk2 , Ck1 , Ck2 , D1k1 , D2k1 , D1k2 and D2k2 are parameters of the controller K given by
273
0 Ei 0 0
0 0 Ei 0
0 0 , 0 E¯ k
Ao 0 M2 A¯ c N2 = 0 0
Aˆ 12 + B˜ 12 Dk1 Cˆ 22 Aˆ 22 + B˜ 22 Dk1 Cˆ 22 Bk1 Cˆ 22 Ωo Cˆ 22
B11 + B¯ 12 D1k1 D21 Bi Bc = , D2k1 D21 B¯ k D21
Eo 0 M2 E¯ c N2 = 0 0
0 Eˆ 0 0
Φ Ai D2k1 C¯ 22 B¯ k C¯ 22
B¯ 12 Ωi 0 Ai + D2k1 Γi B¯ k Γi
¯
B¯ 12 C¯ k1 0 , C¯ k2 A¯ k
0 0 Ek 0
0 0 , 0 Eo
B˜ 12 Ck B˜ 22 Ck Ak −Γo Ck
B˜ 12 Dk2 B˜ 22 Dk2 , Bk2 Ao − Γo Dk2
ˆ
B11 + B˜ 12 Dk1 D21 Bˆ 21 + B˜ 22 Dk1 D21 M2 Bc = , Bk1 D21 Ωo D21
274
Y. Feng, M. Yagoubi / Automatica 66 (2016) 271–275
Fig. 3. Weighted system.
comprehensive admissibility. Note that the weighted system can be represented as:
Fig. 2. Structured controller.
C¯ c N2 = Co
ˆ 12 Dk1 Cˆ 22 Cˆ 12 + D
1 0 E = 0 0 0
ˆ 12 Dk2 . D
ˆ 12 Ck D
By the two transformation matrices
I 0 M3 = 0 0
Xo I I 0
0 0 I 0
0 0 , 0 I
I 0 N3 = 0 0
−Yi
−I
0 0 I 0
I 0 0
0 , 0 I
Aˆ c − sEˆ c
Bˆ c
Cˆ c
Dc
,
(14)
Aˆ 22 + B˜ 22 Dk1 Cˆ 22 Aˆ c = Bk1 Cˆ 22 Ωo Cˆ 22
Eˆ Eˆ c = 0 0
0 Ek 0
B˜ 22 Ck Ak −Γo Ck
D11
B˜ 22 Dk2 , Bk2 Ao − Γo Dk2
ˆ 12 Dk1 Cˆ 22 − Co Yo Cˆ c = Cˆ 12 + D
Aˆ 22 − sEˆ Πo Cˆ 22
G¯ :=
Ao − sEo
ˆ 12 − Co Yo C Cˆ 22
0
− Co
D11 D21 0
0 In o
B˜ 22
− Γo ˆD12 0 0
Ing +ni −X o
0 , −I n o
N4 =
Ing +ni −Yo
0
0
0
Πi = −0.8,
.
Using Eq. (7) and two transformation matrices M4 =
0
XiT = 0.16
ˆ 12 Dk2 − Co . D
Bˆ 21 Πo D21
0
0 1 A = 0 0 0
1 0 0 0 0
0 0 0 0 0
0 0 1 1 0
0 0 −1 , 0 1
1
T
0
,
0.5 0
0 0
C1 =
0 1
0 0
0 , 0
1
−1 ,
D12 =
−0.1 0
,
D21 = 0.25.
Solving the two generalized Sylvester equations yields
Therefore, the unobservable dynamics of Wo are cancelled in the closed-loop system. Finally, it suffices to show that the closed-loop system is admissible. It is easy to see that the closed-loop system (14) is nothing other than Fl (G¯ , K ), where K is given in (9) and
0 0 0 , 0 5
0
ˆ 12 Ck D
−1 0 = ,
C2 = 0
Bˆ 21 + B˜ 22 Dk1 D21 , Bˆ c = Bk1 D21 Ωo D21
0 0, Eo
0 0 0 1 0
BT2 = 0
where
0 0 1 0 0
0 0 B1 = 0.25 , 0 −0.5
together with (7), there holds
Fl (Gˆ , K¯ ) :=
0 0 0 0 0
0 , −I n o
the system G¯ is represented alternatively by G in (10). Under Assumptions (A2) and (A3), it is observed that G is both finite dynamics stabilizable and impulse controllable, and finite dynamics detectable and impulse observable. Therefore, a stabilizing controller K (9) always exists for G. Remark 3. Fig. 2 exhibits explicitly the structure of the controller (8), where the free parameters can be used for additional performance objective. In addition, it has been shown (Feng et al., 2012a) (Remark 3) that (6) and (7) can be solved as a linear problem. Example 4. Consider Fig. 3 that was used in Feng et al. (2012b) for the mixed sensitivity problem. Here, we use the present result for
ΠoT = 0
0
T −0.2 , 0 XoT = YoT = 0 0
YiT = 0.8
0.16
0 0
0 0
T
0 0
0
T −1 ,
,
T −1 .
Hence, the comprehensive admissibility problem for the system in Fig. 2 is transformed into finding a controller K in (9) to internally stabilize the augmented system G in (10). Hence, the resulting controller K can be constructed by (8). 4. Conclusion Admissibility control with unstable and nonproper weights is considered for continuous-time descriptor systems. The problem is transformed into a problem without weights for an augmented system explicitly constructed from the data of the given system. The structure of the underlying controller is clearly exhibited with the dynamics of the weights, which paves a way to additional performance objective control with weights. References Chiang, R. Y., & Safonov, M. G. (1992). Robust control toolbox user’s guide. South Natick, MA: The Mathworks. Darouach, M. (2014). Observers and observer-based control for descriptor systems revisited. IEEE Transactions on Automatic Control, 59(5), 1367–1373. Duan, G. R. (2010). Analysis and design of descriptor linear systems. New York: Springer-Verlag. Feng, Y., & Yagoubi, M. (2013). On state feedback H∞ control for discrete-time singular systems. IEEE Transactions on Automatic Control, 58(10), 2674–2679. Feng, Y., Yagoubi, M., & Chevrel, P. (2011). Parametrization of extended stabilizing controllers for continuous-time descriptor systems. Journal of The Franklin Institute, 348(9), 2633–2646. Feng, Y., Yagoubi, M., & Chevrel, P. (2012a). Extended H2 controller synthesis for continuous descriptor systems. IEEE Transactions on Automatic Control, 57(6), 1559–1564.
Y. Feng, M. Yagoubi / Automatica 66 (2016) 271–275 Feng, Y., Yagoubi, M., & Chevrel, P. (2012b). H∞ control with unstable and nonproper weights for descriptor systems. Automatica, 48(5), 991–994. Kwakernaak, H. (1993). Robust control and H∞ optimization—tutorial paper. Automatica, 29, 255–273. Kwakernaak, H. (2002). Mixed sensitivity design: An aerospace case study. In 15th Triennial world congress, Barcelona, Spain (pp. 49–53). Meinsma, G. (1995). Unstable and nonproper weights in H∞ control. Automatica, 31(11), 1655–1658.
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Mita, T., Xin, X., & Anderson, D. O. (2000). Extended H∞ control—H∞ control with unstable weights. Automatica, 36, 735–741. Stoorvogel, A. A., Saberi, A., & Sannuti, P. (2000). Performance with regulation constraints. Automatica, 36, 1443–1456. Xu, S., & Lam, J. (2006). Robust control and filtering of singular systems. Berlin: Springer-Verlag. Zou, Z., Ho, D., & Wang, Y. (2010). Fault tolerant control for singular systems with actuator saturation and nonlinear perturbation. Automatica, 46, 569–576.
Contents lists available at ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Technical communique
Comprehensive admissibility for descriptor systems✩ Yu Feng a,1 , Mohamed Yagoubi b a
Information Engineering College, Zhejiang University of Technology, 288 Liuhe Road, Hangzhou, PR China
b
LUNAM Université, Ecole des Mines de Nantes, IRCCyN UMR CNRS 6597, 4 rue Alfred Kastler, Nantes, France
article
info
Article history: Received 28 April 2015 Received in revised form 1 October 2015 Accepted 23 December 2015
Keywords: Descriptor systems Structured controller Unstable and nonproper weights
abstract This paper presents a complete solution to the problem of comprehensive admissibility for descriptor systems with unstable and nonproper weights. In such non-standard circumstances, it is hard to render the closed-loop system admissible due to the uncontrollable and unobservable weights. The necessary and sufficient condition is given and a specifically structured output feedback controller is conducted in terms of the dynamics of the weighs. It is shown that determining suitable controllers for a given descriptor system with the presence of unstable and nonproper weights requires solving an admissibility problem for an augmented system explicitly constructed in this paper. A numerical example is included to illustrate the current result. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Descriptor systems have been widely studied over recent decades owing to their capacity to describe non-dynamic constraints and impulsive behaviors. A number of control issues have been successfully developed for descriptor systems (Darouach, 2014; Duan, 2010; Feng & Yagoubi, 2013; Xu & Lam, 2006; Zou, Ho, & Wang, 2010). Many control problems require a standard framework based on models of the physical plants, disturbances and references together with the control objectives (Chiang & Safonov, 1992; Meinsma, 1995; Stoorvogel, Saberi, & Sannuti, 2000). For instance, control of the longitudinal motion of fighter airplanes has been well addressed under this framework (Chiang & Safonov, 1992; Kwakernaak, 2002). Moreover, in the H∞ control theory, it is often desirable to choose a weight with a pole at the origin since the closed-loop system is finite only if the sensitivity function has a zero at the origin. Moreover, to avoid undesirable high frequency noise sensitivity and limited robustness, it is also often required to select a nonproper weight, with its H∞ norm being large outside the desirable closed-loop bandwidth (Meinsma, 1995). Such
✩ This work was partially supported by the Natural Science Foundation of China under Grant 61203130 and Qianjiang Talent Plan under Grant QJD1302013. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Tingshu Hu under the direction of Editor André L. Tits. E-mail addresses: [email protected] (Y. Feng), [email protected] (M. Yagoubi). 1 Tel.: +86 57185290601.
http://dx.doi.org/10.1016/j.automatica.2016.01.028 0005-1098/© 2016 Elsevier Ltd. All rights reserved.
choices, however, yield non-standard problems for plants containing unstabilizable or undetectable elements. Such problems may be solved in different ways: the perturbation technique (Chiang & Safonov, 1992), the loop transformation (Meinsma, 1995), the polynomial method (Kwakernaak, 1993), or the quasi-stabilization theory (Mita, Xin, & Anderson, 2000). However, all these techniques have their own drawbacks. In fact, the perturbation technique is poorly suited to deal with lightly-damped poles while the loop transformation requires a pretreatment to absorb the weights into the loop. The polynomial method does not apply to cases where the weights have imaginary poles, while the quasistabilization theory does not allow nonproper weights. Recently, in Feng, Yagoubi, and Chevrel (2011), such a nonstandard problem is considered within the framework of descriptor systems. Hence, both unstable and nonproper weights can be naturally handled and an observer-based controller is conducted. Performance control with the presence of weights is further addressed in Feng, Yagoubi, and Chevrel (2012a,b). However, the effects of the weights on the underlying controller’s structure have not yet been clarified. The current paper gives a complete solution to this non-standard problem and the structure of the resulting controller is explicitly exhibited with the dynamics of unstable and nonproper weights. It is observed that the non-standard problem can be transformed into a problem without weights for an augmented system explicitly constructed from the data of the given system. This transformation makes it easy to address additional performance objectives for a larger set of systems. The extended synthesis problem considered in Feng et al. (2012a,b) can be directly solved by applying the current result and unnecessary
272
Y. Feng, M. Yagoubi / Automatica 66 (2016) 271–275
Let us introduce here two partitions of G with regard to Wi and Wo , denoted as GWi and GWo , respectively.
GWi
¯ A11 0 :=
A¯ 12 Ai
Aˆ 12 Aˆ 22
Fig. 1. Comprehensive admissibility control setup.
assumptions concerning the invariant zeros made in Feng et al. (2012a,b) can be removed. The rest of this paper is organized as follows. Section 2 formulates the problem. The main result is presented in Section 3. The concluding remarks are given in Section 4.
GWo :=
Consider the following descriptor system: (1)
where x ∈ Rn , y ∈ Rp and u ∈ Rm . The matrix E ∈ Rn×n may be singular. The descriptor system (1) is said to be regular, impulse-free and stable, if det(sE − A) ̸≡ 0, deg(det(sE − A)) = rank(E ), and all the roots of det(sE − A) = 0 have negative real parts, respectively. If the descriptor system is regular, impulse-free and stable, then it is admissible. In addition, it is said to be finite dynamics stabilizable if F exists such that (E , A + BF ) is regular and stable; it is said to be impulse controllable if F exists such that (E , A + BF ) is regular and impulse-free. Dual notions can be defined for finite dynamics detectability and impulse observability (Xu & Lam, 2006). Let us consider the setup in Fig. 1, where the plant G(s), input weight Wi and output weight Wo are given as follows. Ag − sEg Cg1 Cg2
G :=
Bg1 Dg11 Dg21
Ai − sEi Ci
Wi :=
Bi Di
Bg2 Dg12 0
,
A − sE C1 C2
G :=
B1 D11 D21
Wo :=
B2 D12 0
E=
Eo 0 0
B1 =
0 Eg 0
Ao − sEo Co
Bo Do
,
,
Bo Dg11 Di Bg1 Di , Bi
C1 = Co
Do Cg1
(2)
A=
D11 = Do Dg11 Di ,
B2 = Do Dg11 Ci ,
−s
Eo 0
0 Eˆ
Cˆ 12 Cˆ 22
Co 0
B¯ 12 0
D11 D21
D12 0
Bˆ 11 Bˆ 21
Bˆ 12 Bˆ 22
D11 D21
D12 0
,
(3)
.
(4)
BK DK
,
(5)
The problem under consideration is to find a controller K such that the interconnection in Fig. 1 is comprehensively admissible. Before ending this section, we make the following assumptions: (A1) Wi and Wo only possess unstable and impulsive modes; (A2) (E¯ , A¯ 11 , B¯ 12 ) is finite dynamics stabilizable and impulse controllable; (A3) (Eˆ , Aˆ 22 , Cˆ 22 ) is finite dynamics detectable and impulse observable.
Theorem 2. Consider the partitions (3) and (4). The interconnection in Fig. 1 is comprehensively admissible, if and only if Xi ∈ R(ng +no )×ni , Yi ∈ R(ng +no )×ni , Πi ∈ Rm×ni , Xo ∈ Rno ×(ng +ni ) , Yo ∈ Rno ×(ng +ni ) and Πo ∈ Rno ×p exist such that
B¯ 12 Πi D Π 12¯ i EYi Πo Cˆ 22 ΠD o 21ˆ Xo E
Ao 0 0
Bo Cg1 Ag 0
AK =
D12 = Do Dg12 ,
Dg21 Ci ,
D21 = Dg21 Di .
(7)
BK D1k1
Ai + D2k1 Γi Bk1 Γi
,
EK =
Ck2 Ak −Γo Ck
Ωo Γi
Cg2
= Ao Yo − Aˆ 12 − Xo Aˆ 22 , = −Xo Bˆ 21 − Bˆ 11 , = Eo Y o .
(8)
with
Bo Dg11 Ci Bg1 Ci , Ai
Bo Dg12 Bg2 , 0
(6)
AK − sEK CK
C2 = 0
A¯ 11 Yi − A¯ 12 − Xi Ai , C¯ 11 Yi − C¯ 12 , Xi E i .
= = =
Moreover, an admissible controller is given by
C¯ 12 C¯ 22
B¯ 11 Bi
Definition 1 (Comprehensive Admissibility Feng et al., 2012a,b). The feedback system in Fig. 1 is said to be comprehensively admissible if Fl ( G, K ), where Fl (·, ·) is the lower linear fractional transformation, is internally stable and the closed-loop system defined as Tz w = Fl (G, K ) is admissible.
K :=
0 0 , Ei
where EK ∈ Rnk ×nk , AK ∈ Rnk ×nk , BK ∈ Rnk ×p , CK ∈ Rm×nk and DK ∈ Rm×p . EK may be singular.
where
0 Ei
3. Main result
where e ∈ Rq , y ∈ Rp , v ∈ Rl and u ∈ Rm are the controlled output, measurement, disturbance and control input, respectively; Ai ∈ Rni ×ni , Ao ∈ Rno ×no , Bi ∈ Rni ×mi , Bo ∈ Rno ×q , Ci ∈ Rl×ni , Co ∈ Rpo ×no , Di ∈ Rl×mi and Do ∈ Rpo ×q are known constant matrices. Moreover, Eg , Ei and Eo may be singular. The direct feedthrough matrix from u to y is left out to make the arguments simpler and can be handled via standard methods. Then, the resulting weighted system G is written as
Ao 0
E¯ 0
,
C¯ 11 C¯ 21
AK − sEK CK
K := x˙ (t ) = Ax(t ) + Bu(t ), y(t ) = Cx(t ) + Du(t ),
−s
The system data of the partitions can be obtained directly from (2); see Feng et al. (2012a) for more details. Consider an output feedback controller K as
2. Problem formulation
Ei 0 0
0 Ek 0
Γi = C¯ 21 Yi − C¯ 22 ,
Ao − Γo Dk2
0 0 , Eo
D2k2 Bk2
D2k1 Bk1 ,
BK =
Ωo
Γo = Xo B˜ 22 + B˜ 12 ,
,
ΩiT T CKT = Ck1 , 1 T Dk2
Y. Feng, M. Yagoubi / Automatica 66 (2016) 271–275
Ωi = Πi + D1k1 Γi , Ωo = Πo − Γo Dk1 , Xi2 B˜ 12 = Bˆ 12 Xi1 , B˜ 22 = Bˆ 22 , −I n i Xi1 = Ino 0 Xi , Xi2 = 0 Ing Xi , 1 1 1 Dk1 Dk2 Ck Dk1 = , Dk2 = , Ck = 2 2 2 , Dk1
Ck
Cc = C¯ 11 + D12 D1k1 C¯ 21
Bk1
Bk2
Ck1 Ck2
D1k1 D2k1
D1k2 D2k2
K :=
(9)
A − sE C1 C2
−I n o
B2
D11
D12
D21 0
0 0
0
0 0
.
C¯ k2 ¯Ak − sE¯ k C¯ k1
D2k1 B¯ k , D1k1
Then, the closed-loop system is given by Ac − sEc Cc
Bc Dc
0 0 I 0
0 0 , 0 I
B¯ c Dc
,
(13)
B¯ 12 Ωi Ai + D2k1 Γi B¯ k Γi
B¯ 12 C¯ k1 C¯ k2 , A¯ k
B¯ 11 + B¯ 12 D1k1 D21 + Xi Bi , Bc = D2k1 D21 − Bi B¯ k D21
0 0, E¯ k
D12 C¯ k1 .
D12 Ωi
¯ := G
A¯ 11 − sE¯ 0
B¯ 12 Πi Ai − sEi
B¯ 11 + Xi Bi −Bi
C¯ 11 C¯ 21
D12 Πi
D11 D21
Γi
A¯ k − sE¯ k K¯ := C¯ k1 C¯ k2
B¯ 12 0 D12 0
0 Ini , 0 0
B¯ k D1k1 . D2k1
With Eq. (6) and two transformation matrices
M2 =
In g + n o 0
Xi , −Ini
In g + n o 0
N2 =
Yi , −Ini
ˆ := G
,
Aˆ 12 Aˆ 22
Ao 0
−s
Eo 0
0 Eˆ
Cˆ 12 Cˆ 22
Co 0
Bˆ 11 Bˆ 21
B˜ 12 B˜ 22
D11 D21
˜ 12 D 0
,
˜ 12 = D12 0 , and the other where B˜ 12 and B˜ 22 are given in (8), D data are given in the partition (4). Proceeding with the controller K¯ specified in (11) and (12) gives
M2 A¯ c N2 − sM2 E¯ c N2 C¯ c N2
M2 B¯ c Dc
,
where
A11 + B¯ 12 D1k1 C¯ 21 0 Ac = D2k1 C¯ 21 B¯ k C¯ 21
¯
E 0 Ec = 0 0
Yi −I −I 0
Therefore, the uncontrollable dynamics of Wi are cancelled in the closed-loop system. Second, we prove that the unobservable dynamics of Wo are also cancelled in the closed-loop system. To this end, let us consider further the closed-loop system (13), which ¯ , K¯ ), where is nothing other than Fl (G
Fl (Gˆ , K¯ ) :=
where
¯
A¯ c − sE¯ c C¯ c
0 Ei 0
(12)
Fl (G, K ) :=
I 0 N1 = 0 0
¯ is represented alternatively by Gˆ as the system G
Ak Bk2 ¯Ek = Ek 0 , ¯Ak = , (11) 0 Eo −Γo Ck Ao − Γo Dk2 1 T 2 T Bk1 , (Ck ) (Ck ) B¯ k = , (C¯ k1 )T = , (C¯ k2 )T = . Ωo (D1k2 )T (D2k2 )T
where
0 0 , 0 I
C¯ c = C¯ 11 + D12 D1k1 C¯ 21
By setting Πi = DK C¯ 22 − DK C¯ 21 Yi − CK YiK , Eq. (6) holds. Eq. (7) can be conducted by following the same line for the partition in (4). Sufficiency: We prove that under (6) and (7), the controller (8), together with the free parameters specified in K (9) that internally stabilizes the augmented system G (10), achieves comprehensive admissibility. First, we show that the uncontrollable dynamics of Wi are cancelled in the closed-loop system. For simplicity, we denote the controller (8) as
Ωi
Fl (G, K ) :=
E¯ E¯ c = 0 0
0 = D12 (DK C¯ 22 − DK C¯ 21 Yi − CK YiK ) − C¯ 11 Yi + C¯ 12 , Xi Ai = A¯ 11 Yi + B¯ 12 (DK C¯ 21 Yi − DK C¯ 22 + CK YiK ) − A¯ 12 , ¯ i. Xi Ei = EY
Ai + D2k1 Γi − sEi B¯ k Γi
0 0 I 0
together with (6), there holds
Proof. Necessity: We prove that if the closed-loop system in Fig. 1 is comprehensively admissible, then (6) and (7) hold. First, let us consider the partition (3). According to Feng et al. (2011), matrices Xi ∈ R(ng +no )×ni , Yi ∈ R(ng +no )×ni and YiK ∈ Rnk ×ni exist such that
K :=
Xi −I −I 0
A¯ 11 + B¯ 12 D1k1 C¯ 21 ¯Ac = D2k1 C¯ 21 B¯ k C¯ 21
(10)
I 0 M1 = 0 0
Xi In i
B1
−Y o
Φ = A¯ 12 + B¯ 12 D1k1 C¯ 22 .
where
internally stabilizing the augmented system G of the form
G :=
By the two transformation matrices
Dk2
Ak − sEk
D12 C¯ k1 ,
D12 Ωi
C¯ 12 + D12 D1k1 C¯ 22
Dc = D11 + D12 D1k1 D21 ,
where the matrices Ek , Ak , Bk1 , Bk2 , Ck1 , Ck2 , D1k1 , D2k1 , D1k2 and D2k2 are parameters of the controller K given by
273
0 Ei 0 0
0 0 Ei 0
0 0 , 0 E¯ k
Ao 0 M2 A¯ c N2 = 0 0
Aˆ 12 + B˜ 12 Dk1 Cˆ 22 Aˆ 22 + B˜ 22 Dk1 Cˆ 22 Bk1 Cˆ 22 Ωo Cˆ 22
B11 + B¯ 12 D1k1 D21 Bi Bc = , D2k1 D21 B¯ k D21
Eo 0 M2 E¯ c N2 = 0 0
0 Eˆ 0 0
Φ Ai D2k1 C¯ 22 B¯ k C¯ 22
B¯ 12 Ωi 0 Ai + D2k1 Γi B¯ k Γi
¯
B¯ 12 C¯ k1 0 , C¯ k2 A¯ k
0 0 Ek 0
0 0 , 0 Eo
B˜ 12 Ck B˜ 22 Ck Ak −Γo Ck
B˜ 12 Dk2 B˜ 22 Dk2 , Bk2 Ao − Γo Dk2
ˆ
B11 + B˜ 12 Dk1 D21 Bˆ 21 + B˜ 22 Dk1 D21 M2 Bc = , Bk1 D21 Ωo D21
274
Y. Feng, M. Yagoubi / Automatica 66 (2016) 271–275
Fig. 3. Weighted system.
comprehensive admissibility. Note that the weighted system can be represented as:
Fig. 2. Structured controller.
C¯ c N2 = Co
ˆ 12 Dk1 Cˆ 22 Cˆ 12 + D
1 0 E = 0 0 0
ˆ 12 Dk2 . D
ˆ 12 Ck D
By the two transformation matrices
I 0 M3 = 0 0
Xo I I 0
0 0 I 0
0 0 , 0 I
I 0 N3 = 0 0
−Yi
−I
0 0 I 0
I 0 0
0 , 0 I
Aˆ c − sEˆ c
Bˆ c
Cˆ c
Dc
,
(14)
Aˆ 22 + B˜ 22 Dk1 Cˆ 22 Aˆ c = Bk1 Cˆ 22 Ωo Cˆ 22
Eˆ Eˆ c = 0 0
0 Ek 0
B˜ 22 Ck Ak −Γo Ck
D11
B˜ 22 Dk2 , Bk2 Ao − Γo Dk2
ˆ 12 Dk1 Cˆ 22 − Co Yo Cˆ c = Cˆ 12 + D
Aˆ 22 − sEˆ Πo Cˆ 22
G¯ :=
Ao − sEo
ˆ 12 − Co Yo C Cˆ 22
0
− Co
D11 D21 0
0 In o
B˜ 22
− Γo ˆD12 0 0
Ing +ni −X o
0 , −I n o
N4 =
Ing +ni −Yo
0
0
0
Πi = −0.8,
.
Using Eq. (7) and two transformation matrices M4 =
0
XiT = 0.16
ˆ 12 Dk2 − Co . D
Bˆ 21 Πo D21
0
0 1 A = 0 0 0
1 0 0 0 0
0 0 0 0 0
0 0 1 1 0
0 0 −1 , 0 1
1
T
0
,
0.5 0
0 0
C1 =
0 1
0 0
0 , 0
1
−1 ,
D12 =
−0.1 0
,
D21 = 0.25.
Solving the two generalized Sylvester equations yields
Therefore, the unobservable dynamics of Wo are cancelled in the closed-loop system. Finally, it suffices to show that the closed-loop system is admissible. It is easy to see that the closed-loop system (14) is nothing other than Fl (G¯ , K ), where K is given in (9) and
0 0 0 , 0 5
0
ˆ 12 Ck D
−1 0 = ,
C2 = 0
Bˆ 21 + B˜ 22 Dk1 D21 , Bˆ c = Bk1 D21 Ωo D21
0 0, Eo
0 0 0 1 0
BT2 = 0
where
0 0 1 0 0
0 0 B1 = 0.25 , 0 −0.5
together with (7), there holds
Fl (Gˆ , K¯ ) :=
0 0 0 0 0
0 , −I n o
the system G¯ is represented alternatively by G in (10). Under Assumptions (A2) and (A3), it is observed that G is both finite dynamics stabilizable and impulse controllable, and finite dynamics detectable and impulse observable. Therefore, a stabilizing controller K (9) always exists for G. Remark 3. Fig. 2 exhibits explicitly the structure of the controller (8), where the free parameters can be used for additional performance objective. In addition, it has been shown (Feng et al., 2012a) (Remark 3) that (6) and (7) can be solved as a linear problem. Example 4. Consider Fig. 3 that was used in Feng et al. (2012b) for the mixed sensitivity problem. Here, we use the present result for
ΠoT = 0
0
T −0.2 , 0 XoT = YoT = 0 0
YiT = 0.8
0.16
0 0
0 0
T
0 0
0
T −1 ,
,
T −1 .
Hence, the comprehensive admissibility problem for the system in Fig. 2 is transformed into finding a controller K in (9) to internally stabilize the augmented system G in (10). Hence, the resulting controller K can be constructed by (8). 4. Conclusion Admissibility control with unstable and nonproper weights is considered for continuous-time descriptor systems. The problem is transformed into a problem without weights for an augmented system explicitly constructed from the data of the given system. The structure of the underlying controller is clearly exhibited with the dynamics of the weights, which paves a way to additional performance objective control with weights. References Chiang, R. Y., & Safonov, M. G. (1992). Robust control toolbox user’s guide. South Natick, MA: The Mathworks. Darouach, M. (2014). Observers and observer-based control for descriptor systems revisited. IEEE Transactions on Automatic Control, 59(5), 1367–1373. Duan, G. R. (2010). Analysis and design of descriptor linear systems. New York: Springer-Verlag. Feng, Y., & Yagoubi, M. (2013). On state feedback H∞ control for discrete-time singular systems. IEEE Transactions on Automatic Control, 58(10), 2674–2679. Feng, Y., Yagoubi, M., & Chevrel, P. (2011). Parametrization of extended stabilizing controllers for continuous-time descriptor systems. Journal of The Franklin Institute, 348(9), 2633–2646. Feng, Y., Yagoubi, M., & Chevrel, P. (2012a). Extended H2 controller synthesis for continuous descriptor systems. IEEE Transactions on Automatic Control, 57(6), 1559–1564.
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