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A simple existence condition of one-degree-of-freedom block decoupling controllers
Automatica 51 (2015) 14–17
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Automatica journal homepage: www.elsevier.com/locate/automatica
Technical communique
A simple existence condition of one-degree-of-freedom block decoupling controllers✩ Kiheon Park 1 School of Electronics and Electrical Engineering, SungKyunKwan University, 2066 Seobu-ro, Jangan-gu, Suwon, Kyunggi-do 440-746, South Korea
article
info
Article history: Received 18 November 2013 Received in revised form 5 August 2014 Accepted 28 September 2014
abstract A block decoupling problem in linear multivariable systems is treated for one-degree-of-freedom controller configuration with unity output feedback. The plant transfer matrix, which may be non-square, is assumed to have unstable simple poles and zeros that may coincide. A simple existence condition of a block decoupling controller is obtained by directional interpolation approaches. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Linear multivariable systems Block decoupling problems Directional interpolations Solvability condition
1. Introduction A number of researchers have been studying the design of diagonal decoupling controllers which amounts to eliminate coupling characteristics between the reference inputs and the plant outputs so that one input affects only a single output. A more general form of decoupling design is the block decoupling problem which includes the diagonal decoupling problem as a special case. The block decoupling problem has been studied in both of the state-space and frequency domains. Wonham and Morse (1970) obtain a solvability condition using geometric approach in the state-space domain. In the frequency domain, Hautus and Heymann (1983) show that the decoupling design and the stability problem can be treated independently by two-degree-of-freedom (2DOF) controller configuration. Lee and Bongirono (1993) show that a diagonal decoupling controller, hence a block decoupling controller, always exists when 2DOF controller configuration is adopted for a plant whose transfer matrix is rectangular with full row rank. Recently, Ku˘cera (2013) treats a 2DOF block decoupling problem in the most general setting in which the measurement output may be different from the output to be decoupled. He obtains the parameterized form of
✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Jie Chen under the direction of Editor André L. Tits. E-mail address: [email protected]. 1 Tel.: +82 31 290 7116; fax: +82 2 2645 9282.
http://dx.doi.org/10.1016/j.automatica.2014.10.072 0005-1098/© 2014 Elsevier Ltd. All rights reserved.
block decoupling controllers and solves an H2 optimal problem as well. As stated previously, 2DOF controller configuration is particularly ideal to decoupling design since stability and decoupling problems are separate issues. In this regard, decoupling design with 1DOF controller configuration is more restrictive than 2DOF configuration and its solvability condition is usually hard to obtain. Howze and Bhattacharyya (1997) point out, however, that the asymptotic property of unity feedback 1DOF controllers is not fragile with respect to the controller parameters, which is not the case of 2DOF controllers. Since this asymptotic tracking property is inherent to the controller configurations, the similar results would be inferred in 1 and 2DOF block decoupling design and hence 1DOF block decoupling design has its own advantage (see also Introduction section of Park, 2012). The existence condition for 1DOF block decoupling controllers is presented by Linnemann and Wang (1993) and Lin and Wu (1998). Linnemann and Wang (1993) show that a block decoupling controller exists if a strict block-adjoint matrix and a stability-factor matrix are externally skew prime. Lin and Wu (1998) derive an existence condition by using partial fraction expansions of the plant transfer matrix and its inverse under the assumption that they have one coincident unstable pole of order of 1 or 2. Lin and Wu (1999) present all achievable input–output block maps as a parameterized form for the case that the plant does not have coincident unstable poles and zeros. The formula for the existence condition in Linnemann and Wang (1993) needs a stability factorization of the plant, which requires tedious derivation of a Smith–McMillan form of the plant. The formula in Lin and Wu
K. Park / Automatica 51 (2015) 14–17
(1998) requires checking of rank equality of two matrices consisting of Kronecker products, which usually causes a dimension inflation problem, of residue matrices. The purpose of this article is to present a simple existence condition of a block decoupling controller when the plant, which may be rectangular, has unstable simple poles and zeros that may be coincident. The only calculation needed is to obtain input and output zero direction vectors from the coprime fractional descriptions of the plant. Throughout the article, we consider only real rational matrices. A rational matrix G(s), not necessarily proper, is said to be stable if it is analytic in Re s ≥ 0. The notations C and C+ denote the complex number plane and the closed right half plane, respectively. The notations GT and σ ∗ denote the transpose of the matrix G and the conjugate transpose of the vector σ , respectively. The degree of a zero or a pole of a rational matrix is defined in the sense of the Smith–McMillan degree. We adopt coprime polynomial matrix fractional descriptions of rational matrices. Though all formulas are described in terms of coprime polynomial fractions, the results derived in this article hold with the coprime stable rational matrix fractions. 2. Block decoupling problem and solvability condition We consider a block decoupling problem for one-degree-offreedom controller configuration with unity output feedback. For a given plant G(s) whose size is n × q, n ≤ q, the problem can be described as finding stabilizing controllers C (s) that make the input–output transfer matrix T (s) block-diagonal and invertible, where T (s) is given by T (s) = G(s)C (s)(I + G(s)C (s))−1 .
(1)
Here we assume that the plant G(s) is free of unstable hidden poles. For ease of presentation, we consider the case that the block diagonal T (s) has two blocks of T1 (s) and T2 (s) whose sizes are n1 × n1 and n2 × n2 , respectively, with n1 + n2 = n. In this case, the input–output transfer matrix T (s) is said to be blockdecoupled with the partition (n1 , n2 ). Since T (s) is required to be block-diagonal and invertible, G(s) is assumed to have full row 1 rank. Let G(s) = A−1 (s)B(s) = B1 (s)A− 1 (s) denote coprime polynomial matrix fractional descriptions. There always exist polynomial matrices X (s), Y (s), X1 (s) and Y1 (s) such that
X1 −B
Y1 A
A1 B1
−Y X
A = 1 B1
−Y X
X1 −B
Y1 A
= I.
(2)
Lemma 2 (Park, 2009). Suppose that G(s) is a square stable rational matrix with full normal rank and it has distinct simple zeros zi ∈ C+ , i = 1 → m. Let σi∗ be an output zero direction vector of G(zi ) so that σi∗ G(zi ) = 0 and G−1 (s) be denoted by the partial fractional expression G−1 (s) =
m Mi + F (s), s − zi i=1
diag {T1 (s), T2 (s)} = B1 (s)Y1 (s) + B1 (s)K (s)A(s),
(3)
and a block decoupling controller exists if there exists a stable K (s) that makes B1 Y1 + B1 KA block-diagonal. The purpose of this article is to present a simple existence condition of a block decoupling controller when the plant G(s) has simple poles and zeros in C+ . The following lemmas are useful to prove Theorem 1. Lemma 1 is a special case of Lemma 3.3 in Lin and Wu (1998). (It is noted here that Lemma 3.3 in Lin and Wu (1998) is valid without the inequality condition p ≥ q.)
(5)
where Mi is the residue matrix at zi and F (s) is a stable matrix. Then the jth row of Mi is either zero or proportional to σi∗ . That is, the jth row of Mi is of the form kij σi∗ , kij ∈ C . In this article we assume that the plant G(s) has simple zeros and poles in C+ . It is well known that when the plant has no coincident poles and zeros in C+ , there exists a diagonal decoupling controller. So it is assumed here that the plant has common poles and zeros in C+ . Assumption 1. B1 (s) has the simple zeros vi ∈ C+ , i = 1 → m1 and A(s) has the simple zeros wj ∈ C+ , j = 1 → m2 , with vi ̸= wj for any i and j. B1 (s) and A(s) have common simple zeros zk ∈ C+ , k = 1 → m3 . Since vi , wj , and zk are simple zeros, there exist nonzero vectors βi , αj , σk and µk such that
βi∗ B1 (vi ) = 0,
A(wj )αj = 0
(6)
σk B1 (zk ) = 0 and A(zk )µk = 0, ∗
(7)
for i = 1 → m1 , j = 1 → m2 and k = 1 → m3 . For a given vector x = [x1 x2 · · · xn ]T , let us define its two sub-vectors xa and xb as xa = [x1 x2 · · · xn1 ]T ,
xb = [xn1 +1 xn1 +2 · · · xn1 +n2 ]T .
(8)
When there exist stable rational matrices T1 (s) and T2 (s) satisfying the equality in (3), it is suggestive to observe that the matrices T1 and T2 satisfy some directional interpolation conditions. In fact, ∗ ∗ pre-multiplying βi∗ = [βia∗ βib∗ ] and σk∗ = [σka σkb ] to (3) with s = vi T and s = zk , respectively, and post-multiplying αj = [αja αjbT ]T and
µk = [µTka µTkb ]T to (3) with s = wj and s = zk , respectively, yields, with the aid of the equality B1 (s)Y1 (s) + X (s)A(s) = I which is obtained from (2), that
βia∗ T1 (vi ) = 0,
T1 (wj )αja = αja ,
σka T1 (zk ) = 0,
T1 (zk )µka = µka ,
(10)
βib T2 (vi ) = 0,
T2 (wj )αjb = αjb ,
(11)
σkb T2 (zk ) = 0,
T2 (zk )µkb = µkb ,
(12)
∗
It is known that the class of all stabilizing controllers is given by C (s) = (Y + A1 K )(X − B1 K )−1 , where K (s) is a stable rational matrix. Therefore, the transfer matrix T (s) of block-decoupled systems with internal stability must be of the form
15
∗
∗
(9)
for i = 1 → m1 , j = 1 → m2 and k = 1 → m3 . Also differentiating (3) yields that diag {T1′ (s), T2′ (s)} = B′1 Y1 + B1 Y1′ + B′1 KA + B1 K ′ A + B1 KA′ .
(13)
Now pre-multiplying σk∗ and post-multiplying µk to this equation with s = zk , we obtain an interpolation condition ∗ ′ ∗ ′ σka T1 (zk )µka + σkb T2 (zk )µkb = ρk ,
k = 1 → m3 ,
(14)
where ρk = σk∗ B′1 (zk )Y1 (zk )µk . Now we present the main result.
Lemma 1. Let σ and µ be given n × 1 column vectors. If σ ∗ µ = 0, then there exists an n × n constant matrix P satisfying the equalities
Theorem 1. Under Assumption 1, a block decoupling controller with the partition (n1 , n2 ) exists if and only if
σ ∗ P = 0 and P µ = µ.
∗ ∗ σka µka = 0 and σkb µkb = 0,
(4)
k = 1 → m3 .
(15)
16
K. Park / Automatica 51 (2015) 14–17
∗ Proof. (Necessity) Post-multiplying µka to σka T1 (zk ) = 0 in (10) ∗ ∗ yields σka T1 (zk )µka = σka µka = 0 and the similar approach to ∗ T2 (zk ) in (12) yields σkb µkb = 0. (Sufficiency) We will show that if the conditions in (15) are satisfied then there exists a stable rational matrix K (s) that makes B1 Y1 + B1 KA block-diagonal. First notice that, since B1 (s) has full row rank, there exists a unimodular matrix U (s) such that B1 (s)U (s) = [B1o (s) 0] with B1o (s) square and invertible. Then Eq. (3) becomes
diag {T1 , T2 } = B1 Y1 + [B1o 0]U −1 KA
= B1 Y1 + B1o K˜ 1 A,
(16)
where K˜ 1 is the upper part of the matrix K˜ = U −1 K = [K˜ 1T K˜ 2T ]T . Let us define
Φ (s) = diag {Tp1 (s), Tp2 (s)} − B1 (s)Y1 (s).
(17)
If we can find stable matrices Tp1 (s) and Tp2 (s) that make the matrix −1 K˜ 1 = B1o Φ (s)A−1 stable, then the stable matrix K = U K˜ , where K˜ 2 being an arbitrary stable matrix, makes B1 Y1 +B1 KA block-diagonal. In the next we will show that any stable rational matrices T1 (s) and T2 (s) satisfying the directional interpolation conditions in (9)–(12) 1 −1 and (14) are the ones making K˜ 1 = B− stable. 10 Φ (s)A ∗ If σka µka = 0, then there exists a square matrix Pka , by Lemma 1, such that ∗ σka Pka = 0,
Pka µka = µka .
(18)
Next let Wka and Wkb be the constant matrices satisfying the equation ∗ ∗ Wkb µkb = ρk , σka Wka µka + σkb
(19)
where ρk = σk∗ B′1 (zk )Y1 (zk )µk . It is trivial to show that such matrices always exist. (We first choose Wkb arbitrarily and then find Wka satisfying (19), or vice versa.) Now we find a stable rational matrix satisfying the direct interpolation conditions, Tp1 (vi ) = 0, Tp1 (zk ) = Pka ,
Tp1 (wj ) = I ,
(20)
′ Tp1 (zk ) = Wka ,
(21)
for i = 1 → m1 , j = 1 → m2 and k = 1 → m3 . It should be noticed that this Tp1 (s) is a particular solution for T1 (s) that satisfies the directional interpolation conditions in (9), (10) and (14). A particular solution Tp2 (s) for T2 (s) can be obtained by the same way to satisfy the directional interpolation conditions in (11), (12) and (14). That is, we find a stable Tp2 (s) satisfying Tp2 (vi ) = 0, Tp2 (zk ) = Pkb ,
Tp2 (wj ) = I ,
(22)
′ Tp2 (zk ) = Wkb ,
1 −1 thing is to show that the matrix K˜ 1 = B− is stable. Since 1o Φ (s)A the Smith forms of B1 (s) and B1o (s) are identical, B1o (s) has simple zeros in C+ at vi , i = 1 → m1 and zk , k = 1 → m3 . Furthermore, it is easy to confirm that the output zero direction vectors of B1o (s) at vi and zk are βi∗ and σk∗ , respectively. Since vi , zk are simple zeros of B1o (s) and wj , zk are simple zeros of A(s), they are simple poles 1 −1 of B− (s), respectively. Consider their partial fractional 1o (s) and A expressions m3 m1 Mk Mv i + , s − v s − zk i k=1 i =1
Ωb =
A−1 = Ωa (s) + F2 (s),
m3 m2 Nwj Nk Ωa = + , s − wj s − zk j=1 k=1
K˜ 1 = Ωb (s)Φ (s)F2 (s) + F1 (s)Φ (s)Ωa (s)
+ Ωb (s)Φ (s)Ωa (s) + F1 (s)Φ (s)F2 (s).
(24)
(25)
(26)
By Lemma 2, each row of Mv i (Mk ) is zero or proportional to βi∗ (σk∗ ). Since Tp1 (s) and Tp2 (s) are chosen to satisfy the interpolation conditions in (9)–(12), it follows that Mv i Φ (vi ) = 0 (Mk Φ (zk ) = 0). This implies that Mv i Φ (s) (Mk Φ (s)) has the factor s − vi (s − zk ), and this assures that Ωb (s)Φ (s) is stable. Similarly, each column of Nwj (Nk ) is zero or proportional to αj (µk ) by the dual property of Lemma 2 and this leads that Φ (wj )Nwj = 0 (Φ (zk )Nk = 0) so that Φ (s)Nwj (Φ (s)Nk ) has the factor s − wj (s − zk ), and this assures that Φ (s)Ωa (s) is stable. It is also true that Mv i Φ (vi )Nwj = Mv i Φ (wj )Nwj = 0, which implies that Mv i Φ (s)Nwj has the factor (s − vi )(s − wj ). By the same arguments, we can show that Mv i Φ (s)Nk , Mk Φ (s)Nwj and Mk Φ (s)Nr , k ̸= r, have the factors (s − vi )(s − zk ), (s − zk )(s − wj ), and (s − zk )(s − zr ), respectively. Next, define H (s) = Mk Φ (s)Nk . Then we can show that H (zk ) = 0 and H ′ (zk ) = 0 using the fact that Tp1 (s) and are chosen to satisfy the interpolation conditions in (14). This implies that H (s) has the factor (s − zk )2 and this assures that Ωb (s)Φ (s)Ωa (s) and, hence K˜ 1 , is stable and this completes the proof. Remark 1. Though the existence condition in Theorem 1 is stated for the two-block case, extension to the general case should be obvious. We note that the sufficiency part of the proof provides construction procedures to find a block decoupled solution Tp (s) = diag {Tp1 (s), Tp2 (s)}. When the plant G(s) is square, the corresponding controller C (s) associated with Tp (s) can be calculated by the equation C = G−1 (I − Tp )−1 Tp = G−1 Tp (I − Tp )−1 .
(27)
Since the solution rational matrices Tp1 (s) and Tp2 (s) can be obtained with any arbitrary orders, we can always find a proper block-decoupling controller C (s) by managing the orders of Tp1 (s) and Tp2 (s). When the plant G(s) is rectangular with full row rank, the corresponding controller C (s) is more involved than the one in (27) and not stated here. 3. An example Consider the plant with the coprime polynomial matrix fractional descriptions of
(23)
for i = 1 → m1 , j = 1 → m2 and k = 1 → m3 , where Pkb is the ∗ matrix satisfying σkb Pkb = 0 and Pkb µkb = µkb . Now the remaining
1 B− 1o = Ωb (s) + F1 (s),
where F1 and F2 are stable rational matrices. Inserting (24) and (25) 1 −1 into K˜ 1 (s) = B− yields 1o Φ (s)A
−3 s−1
s−1
s + 3 s − 1 G(s) = s + 3 4 s+3
s−1
s
s+3 s−1
s−1 −1
s+3 5s + 11
s−1
(s + 2)(s + 3)
= A B = B1 A1 , −1
−1
(28)
with
s
3
A = 1 − s −1
s−1 0
0
s−1 B= 0 3
s+3 −1
0 0 , s+2
s−1 0 , 4
(29)
K. Park / Automatica 51 (2015) 14–17
s−1 s−1 4
−3
s+3 0 0
0
B1 =
A1 =
s −1 s−1 0
0 0 , 1
−(s + 2) 0 s+2
(30)
.
3
0],
µ1 = [3
−1
the unstable common poles and zeros of the plant. Possible future researches include extending the results of this article to multiple zero cases and characterizing all achievable input–output maps. References
The zeros of the matrices B1 (s) and A(s) in C+ locate only at s = 1. An output zero direction vector σ1∗ satisfying σ1∗ B1 (1) = 0 and an input zero direction vector µ1 satisfying A(1)µ1 = 0 are given by
σ1∗ = [1
17
1]T .
(31)
It is readily observed that the only possible block partition is (2, 1). ∗ ∗ ∗ ∗ In fact, σ1a µ1a = 0 and σ1b µ1b = 0 with σ1a = [1 3], σ1b = 0, T µ1a = [3 −1] and µ1b = 1. We can find a particular solution diag {Tp1 (s), Tp2 (s)} by following the procedures in the sufficiency part of the proof of Theorem 1. 4. Conclusion A new approach using directional interpolations to find an existence condition of a block decoupling controller is suggested. The existence condition is described by orthogonality of the subvectors of output and input zero direction vectors associated with
Hautus, M. L. J., & Heymann, M. (1983). Linear feedback decoupling-Transfer function analysis. IEEE Transactions on Automatic Control, 28(8), 823–832. Howze, J. W., & Bhattacharyya, S. P. (1997). Robust tracking, error feedback, and two-degree-of-freedom controllers. IEEE Transactions on Automatic Control, 42(7), 980–983. Ku˘cera, V. (2013). Optimal decoupling controllers revisited. Control and Cybernetics, 42, 139–154. Lee, H. P., & Bongirono, J. J., Jr. (1993). Wiener–Hopf design of optimal decoupling controllers for plants with non-square transfer matrices. International Journal of Control, 58(6), 1227–1246. Lin, C. A., & Wu, C. M. (1998). Block-decoupling linear multivariable systems: Necessary and sufficient conditions. Automatica, 34(2), 237–243. Lin, C. A., & Wu, C. M. (1999). Block decoupling control of linear multivariable systems. Asian Journal of Control, 1(3), 146–152. Linnemann, A., & Wang, Q. G. (1993). Block decoupling with stability by unity output feedback—solution and performance limitations. Automatica, 29(3), 735–744. Park, K. (2009). H2 design of decoupling controllers based on directional interpolations. In Proc. of joint 48th IEEE CDC and 28th chinese control conference (pp. 5333–5338). Park, K. (2012). Parameterization of decoupling controllers in the generalized plant model. IEEE Transactions on Automatic Control, 57(4), 1067–1070. Wonham, W. M., & Morse, A. S. (1970). Decoupling and pole assignment in linear multivariable systems: A geometric approach. SIAM Journal on Control, 8, 1–18.
Contents lists available at ScienceDirect
Automatica journal homepage: www.elsevier.com/locate/automatica
Technical communique
A simple existence condition of one-degree-of-freedom block decoupling controllers✩ Kiheon Park 1 School of Electronics and Electrical Engineering, SungKyunKwan University, 2066 Seobu-ro, Jangan-gu, Suwon, Kyunggi-do 440-746, South Korea
article
info
Article history: Received 18 November 2013 Received in revised form 5 August 2014 Accepted 28 September 2014
abstract A block decoupling problem in linear multivariable systems is treated for one-degree-of-freedom controller configuration with unity output feedback. The plant transfer matrix, which may be non-square, is assumed to have unstable simple poles and zeros that may coincide. A simple existence condition of a block decoupling controller is obtained by directional interpolation approaches. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Linear multivariable systems Block decoupling problems Directional interpolations Solvability condition
1. Introduction A number of researchers have been studying the design of diagonal decoupling controllers which amounts to eliminate coupling characteristics between the reference inputs and the plant outputs so that one input affects only a single output. A more general form of decoupling design is the block decoupling problem which includes the diagonal decoupling problem as a special case. The block decoupling problem has been studied in both of the state-space and frequency domains. Wonham and Morse (1970) obtain a solvability condition using geometric approach in the state-space domain. In the frequency domain, Hautus and Heymann (1983) show that the decoupling design and the stability problem can be treated independently by two-degree-of-freedom (2DOF) controller configuration. Lee and Bongirono (1993) show that a diagonal decoupling controller, hence a block decoupling controller, always exists when 2DOF controller configuration is adopted for a plant whose transfer matrix is rectangular with full row rank. Recently, Ku˘cera (2013) treats a 2DOF block decoupling problem in the most general setting in which the measurement output may be different from the output to be decoupled. He obtains the parameterized form of
✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Jie Chen under the direction of Editor André L. Tits. E-mail address: [email protected]. 1 Tel.: +82 31 290 7116; fax: +82 2 2645 9282.
http://dx.doi.org/10.1016/j.automatica.2014.10.072 0005-1098/© 2014 Elsevier Ltd. All rights reserved.
block decoupling controllers and solves an H2 optimal problem as well. As stated previously, 2DOF controller configuration is particularly ideal to decoupling design since stability and decoupling problems are separate issues. In this regard, decoupling design with 1DOF controller configuration is more restrictive than 2DOF configuration and its solvability condition is usually hard to obtain. Howze and Bhattacharyya (1997) point out, however, that the asymptotic property of unity feedback 1DOF controllers is not fragile with respect to the controller parameters, which is not the case of 2DOF controllers. Since this asymptotic tracking property is inherent to the controller configurations, the similar results would be inferred in 1 and 2DOF block decoupling design and hence 1DOF block decoupling design has its own advantage (see also Introduction section of Park, 2012). The existence condition for 1DOF block decoupling controllers is presented by Linnemann and Wang (1993) and Lin and Wu (1998). Linnemann and Wang (1993) show that a block decoupling controller exists if a strict block-adjoint matrix and a stability-factor matrix are externally skew prime. Lin and Wu (1998) derive an existence condition by using partial fraction expansions of the plant transfer matrix and its inverse under the assumption that they have one coincident unstable pole of order of 1 or 2. Lin and Wu (1999) present all achievable input–output block maps as a parameterized form for the case that the plant does not have coincident unstable poles and zeros. The formula for the existence condition in Linnemann and Wang (1993) needs a stability factorization of the plant, which requires tedious derivation of a Smith–McMillan form of the plant. The formula in Lin and Wu
K. Park / Automatica 51 (2015) 14–17
(1998) requires checking of rank equality of two matrices consisting of Kronecker products, which usually causes a dimension inflation problem, of residue matrices. The purpose of this article is to present a simple existence condition of a block decoupling controller when the plant, which may be rectangular, has unstable simple poles and zeros that may be coincident. The only calculation needed is to obtain input and output zero direction vectors from the coprime fractional descriptions of the plant. Throughout the article, we consider only real rational matrices. A rational matrix G(s), not necessarily proper, is said to be stable if it is analytic in Re s ≥ 0. The notations C and C+ denote the complex number plane and the closed right half plane, respectively. The notations GT and σ ∗ denote the transpose of the matrix G and the conjugate transpose of the vector σ , respectively. The degree of a zero or a pole of a rational matrix is defined in the sense of the Smith–McMillan degree. We adopt coprime polynomial matrix fractional descriptions of rational matrices. Though all formulas are described in terms of coprime polynomial fractions, the results derived in this article hold with the coprime stable rational matrix fractions. 2. Block decoupling problem and solvability condition We consider a block decoupling problem for one-degree-offreedom controller configuration with unity output feedback. For a given plant G(s) whose size is n × q, n ≤ q, the problem can be described as finding stabilizing controllers C (s) that make the input–output transfer matrix T (s) block-diagonal and invertible, where T (s) is given by T (s) = G(s)C (s)(I + G(s)C (s))−1 .
(1)
Here we assume that the plant G(s) is free of unstable hidden poles. For ease of presentation, we consider the case that the block diagonal T (s) has two blocks of T1 (s) and T2 (s) whose sizes are n1 × n1 and n2 × n2 , respectively, with n1 + n2 = n. In this case, the input–output transfer matrix T (s) is said to be blockdecoupled with the partition (n1 , n2 ). Since T (s) is required to be block-diagonal and invertible, G(s) is assumed to have full row 1 rank. Let G(s) = A−1 (s)B(s) = B1 (s)A− 1 (s) denote coprime polynomial matrix fractional descriptions. There always exist polynomial matrices X (s), Y (s), X1 (s) and Y1 (s) such that
X1 −B
Y1 A
A1 B1
−Y X
A = 1 B1
−Y X
X1 −B
Y1 A
= I.
(2)
Lemma 2 (Park, 2009). Suppose that G(s) is a square stable rational matrix with full normal rank and it has distinct simple zeros zi ∈ C+ , i = 1 → m. Let σi∗ be an output zero direction vector of G(zi ) so that σi∗ G(zi ) = 0 and G−1 (s) be denoted by the partial fractional expression G−1 (s) =
m Mi + F (s), s − zi i=1
diag {T1 (s), T2 (s)} = B1 (s)Y1 (s) + B1 (s)K (s)A(s),
(3)
and a block decoupling controller exists if there exists a stable K (s) that makes B1 Y1 + B1 KA block-diagonal. The purpose of this article is to present a simple existence condition of a block decoupling controller when the plant G(s) has simple poles and zeros in C+ . The following lemmas are useful to prove Theorem 1. Lemma 1 is a special case of Lemma 3.3 in Lin and Wu (1998). (It is noted here that Lemma 3.3 in Lin and Wu (1998) is valid without the inequality condition p ≥ q.)
(5)
where Mi is the residue matrix at zi and F (s) is a stable matrix. Then the jth row of Mi is either zero or proportional to σi∗ . That is, the jth row of Mi is of the form kij σi∗ , kij ∈ C . In this article we assume that the plant G(s) has simple zeros and poles in C+ . It is well known that when the plant has no coincident poles and zeros in C+ , there exists a diagonal decoupling controller. So it is assumed here that the plant has common poles and zeros in C+ . Assumption 1. B1 (s) has the simple zeros vi ∈ C+ , i = 1 → m1 and A(s) has the simple zeros wj ∈ C+ , j = 1 → m2 , with vi ̸= wj for any i and j. B1 (s) and A(s) have common simple zeros zk ∈ C+ , k = 1 → m3 . Since vi , wj , and zk are simple zeros, there exist nonzero vectors βi , αj , σk and µk such that
βi∗ B1 (vi ) = 0,
A(wj )αj = 0
(6)
σk B1 (zk ) = 0 and A(zk )µk = 0, ∗
(7)
for i = 1 → m1 , j = 1 → m2 and k = 1 → m3 . For a given vector x = [x1 x2 · · · xn ]T , let us define its two sub-vectors xa and xb as xa = [x1 x2 · · · xn1 ]T ,
xb = [xn1 +1 xn1 +2 · · · xn1 +n2 ]T .
(8)
When there exist stable rational matrices T1 (s) and T2 (s) satisfying the equality in (3), it is suggestive to observe that the matrices T1 and T2 satisfy some directional interpolation conditions. In fact, ∗ ∗ pre-multiplying βi∗ = [βia∗ βib∗ ] and σk∗ = [σka σkb ] to (3) with s = vi T and s = zk , respectively, and post-multiplying αj = [αja αjbT ]T and
µk = [µTka µTkb ]T to (3) with s = wj and s = zk , respectively, yields, with the aid of the equality B1 (s)Y1 (s) + X (s)A(s) = I which is obtained from (2), that
βia∗ T1 (vi ) = 0,
T1 (wj )αja = αja ,
σka T1 (zk ) = 0,
T1 (zk )µka = µka ,
(10)
βib T2 (vi ) = 0,
T2 (wj )αjb = αjb ,
(11)
σkb T2 (zk ) = 0,
T2 (zk )µkb = µkb ,
(12)
∗
It is known that the class of all stabilizing controllers is given by C (s) = (Y + A1 K )(X − B1 K )−1 , where K (s) is a stable rational matrix. Therefore, the transfer matrix T (s) of block-decoupled systems with internal stability must be of the form
15
∗
∗
(9)
for i = 1 → m1 , j = 1 → m2 and k = 1 → m3 . Also differentiating (3) yields that diag {T1′ (s), T2′ (s)} = B′1 Y1 + B1 Y1′ + B′1 KA + B1 K ′ A + B1 KA′ .
(13)
Now pre-multiplying σk∗ and post-multiplying µk to this equation with s = zk , we obtain an interpolation condition ∗ ′ ∗ ′ σka T1 (zk )µka + σkb T2 (zk )µkb = ρk ,
k = 1 → m3 ,
(14)
where ρk = σk∗ B′1 (zk )Y1 (zk )µk . Now we present the main result.
Lemma 1. Let σ and µ be given n × 1 column vectors. If σ ∗ µ = 0, then there exists an n × n constant matrix P satisfying the equalities
Theorem 1. Under Assumption 1, a block decoupling controller with the partition (n1 , n2 ) exists if and only if
σ ∗ P = 0 and P µ = µ.
∗ ∗ σka µka = 0 and σkb µkb = 0,
(4)
k = 1 → m3 .
(15)
16
K. Park / Automatica 51 (2015) 14–17
∗ Proof. (Necessity) Post-multiplying µka to σka T1 (zk ) = 0 in (10) ∗ ∗ yields σka T1 (zk )µka = σka µka = 0 and the similar approach to ∗ T2 (zk ) in (12) yields σkb µkb = 0. (Sufficiency) We will show that if the conditions in (15) are satisfied then there exists a stable rational matrix K (s) that makes B1 Y1 + B1 KA block-diagonal. First notice that, since B1 (s) has full row rank, there exists a unimodular matrix U (s) such that B1 (s)U (s) = [B1o (s) 0] with B1o (s) square and invertible. Then Eq. (3) becomes
diag {T1 , T2 } = B1 Y1 + [B1o 0]U −1 KA
= B1 Y1 + B1o K˜ 1 A,
(16)
where K˜ 1 is the upper part of the matrix K˜ = U −1 K = [K˜ 1T K˜ 2T ]T . Let us define
Φ (s) = diag {Tp1 (s), Tp2 (s)} − B1 (s)Y1 (s).
(17)
If we can find stable matrices Tp1 (s) and Tp2 (s) that make the matrix −1 K˜ 1 = B1o Φ (s)A−1 stable, then the stable matrix K = U K˜ , where K˜ 2 being an arbitrary stable matrix, makes B1 Y1 +B1 KA block-diagonal. In the next we will show that any stable rational matrices T1 (s) and T2 (s) satisfying the directional interpolation conditions in (9)–(12) 1 −1 and (14) are the ones making K˜ 1 = B− stable. 10 Φ (s)A ∗ If σka µka = 0, then there exists a square matrix Pka , by Lemma 1, such that ∗ σka Pka = 0,
Pka µka = µka .
(18)
Next let Wka and Wkb be the constant matrices satisfying the equation ∗ ∗ Wkb µkb = ρk , σka Wka µka + σkb
(19)
where ρk = σk∗ B′1 (zk )Y1 (zk )µk . It is trivial to show that such matrices always exist. (We first choose Wkb arbitrarily and then find Wka satisfying (19), or vice versa.) Now we find a stable rational matrix satisfying the direct interpolation conditions, Tp1 (vi ) = 0, Tp1 (zk ) = Pka ,
Tp1 (wj ) = I ,
(20)
′ Tp1 (zk ) = Wka ,
(21)
for i = 1 → m1 , j = 1 → m2 and k = 1 → m3 . It should be noticed that this Tp1 (s) is a particular solution for T1 (s) that satisfies the directional interpolation conditions in (9), (10) and (14). A particular solution Tp2 (s) for T2 (s) can be obtained by the same way to satisfy the directional interpolation conditions in (11), (12) and (14). That is, we find a stable Tp2 (s) satisfying Tp2 (vi ) = 0, Tp2 (zk ) = Pkb ,
Tp2 (wj ) = I ,
(22)
′ Tp2 (zk ) = Wkb ,
1 −1 thing is to show that the matrix K˜ 1 = B− is stable. Since 1o Φ (s)A the Smith forms of B1 (s) and B1o (s) are identical, B1o (s) has simple zeros in C+ at vi , i = 1 → m1 and zk , k = 1 → m3 . Furthermore, it is easy to confirm that the output zero direction vectors of B1o (s) at vi and zk are βi∗ and σk∗ , respectively. Since vi , zk are simple zeros of B1o (s) and wj , zk are simple zeros of A(s), they are simple poles 1 −1 of B− (s), respectively. Consider their partial fractional 1o (s) and A expressions m3 m1 Mk Mv i + , s − v s − zk i k=1 i =1
Ωb =
A−1 = Ωa (s) + F2 (s),
m3 m2 Nwj Nk Ωa = + , s − wj s − zk j=1 k=1
K˜ 1 = Ωb (s)Φ (s)F2 (s) + F1 (s)Φ (s)Ωa (s)
+ Ωb (s)Φ (s)Ωa (s) + F1 (s)Φ (s)F2 (s).
(24)
(25)
(26)
By Lemma 2, each row of Mv i (Mk ) is zero or proportional to βi∗ (σk∗ ). Since Tp1 (s) and Tp2 (s) are chosen to satisfy the interpolation conditions in (9)–(12), it follows that Mv i Φ (vi ) = 0 (Mk Φ (zk ) = 0). This implies that Mv i Φ (s) (Mk Φ (s)) has the factor s − vi (s − zk ), and this assures that Ωb (s)Φ (s) is stable. Similarly, each column of Nwj (Nk ) is zero or proportional to αj (µk ) by the dual property of Lemma 2 and this leads that Φ (wj )Nwj = 0 (Φ (zk )Nk = 0) so that Φ (s)Nwj (Φ (s)Nk ) has the factor s − wj (s − zk ), and this assures that Φ (s)Ωa (s) is stable. It is also true that Mv i Φ (vi )Nwj = Mv i Φ (wj )Nwj = 0, which implies that Mv i Φ (s)Nwj has the factor (s − vi )(s − wj ). By the same arguments, we can show that Mv i Φ (s)Nk , Mk Φ (s)Nwj and Mk Φ (s)Nr , k ̸= r, have the factors (s − vi )(s − zk ), (s − zk )(s − wj ), and (s − zk )(s − zr ), respectively. Next, define H (s) = Mk Φ (s)Nk . Then we can show that H (zk ) = 0 and H ′ (zk ) = 0 using the fact that Tp1 (s) and are chosen to satisfy the interpolation conditions in (14). This implies that H (s) has the factor (s − zk )2 and this assures that Ωb (s)Φ (s)Ωa (s) and, hence K˜ 1 , is stable and this completes the proof. Remark 1. Though the existence condition in Theorem 1 is stated for the two-block case, extension to the general case should be obvious. We note that the sufficiency part of the proof provides construction procedures to find a block decoupled solution Tp (s) = diag {Tp1 (s), Tp2 (s)}. When the plant G(s) is square, the corresponding controller C (s) associated with Tp (s) can be calculated by the equation C = G−1 (I − Tp )−1 Tp = G−1 Tp (I − Tp )−1 .
(27)
Since the solution rational matrices Tp1 (s) and Tp2 (s) can be obtained with any arbitrary orders, we can always find a proper block-decoupling controller C (s) by managing the orders of Tp1 (s) and Tp2 (s). When the plant G(s) is rectangular with full row rank, the corresponding controller C (s) is more involved than the one in (27) and not stated here. 3. An example Consider the plant with the coprime polynomial matrix fractional descriptions of
(23)
for i = 1 → m1 , j = 1 → m2 and k = 1 → m3 , where Pkb is the ∗ matrix satisfying σkb Pkb = 0 and Pkb µkb = µkb . Now the remaining
1 B− 1o = Ωb (s) + F1 (s),
where F1 and F2 are stable rational matrices. Inserting (24) and (25) 1 −1 into K˜ 1 (s) = B− yields 1o Φ (s)A
−3 s−1
s−1
s + 3 s − 1 G(s) = s + 3 4 s+3
s−1
s
s+3 s−1
s−1 −1
s+3 5s + 11
s−1
(s + 2)(s + 3)
= A B = B1 A1 , −1
−1
(28)
with
s
3
A = 1 − s −1
s−1 0
0
s−1 B= 0 3
s+3 −1
0 0 , s+2
s−1 0 , 4
(29)
K. Park / Automatica 51 (2015) 14–17
s−1 s−1 4
−3
s+3 0 0
0
B1 =
A1 =
s −1 s−1 0
0 0 , 1
−(s + 2) 0 s+2
(30)
.
3
0],
µ1 = [3
−1
the unstable common poles and zeros of the plant. Possible future researches include extending the results of this article to multiple zero cases and characterizing all achievable input–output maps. References
The zeros of the matrices B1 (s) and A(s) in C+ locate only at s = 1. An output zero direction vector σ1∗ satisfying σ1∗ B1 (1) = 0 and an input zero direction vector µ1 satisfying A(1)µ1 = 0 are given by
σ1∗ = [1
17
1]T .
(31)
It is readily observed that the only possible block partition is (2, 1). ∗ ∗ ∗ ∗ In fact, σ1a µ1a = 0 and σ1b µ1b = 0 with σ1a = [1 3], σ1b = 0, T µ1a = [3 −1] and µ1b = 1. We can find a particular solution diag {Tp1 (s), Tp2 (s)} by following the procedures in the sufficiency part of the proof of Theorem 1. 4. Conclusion A new approach using directional interpolations to find an existence condition of a block decoupling controller is suggested. The existence condition is described by orthogonality of the subvectors of output and input zero direction vectors associated with
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