Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
ThAIn4.8
Transfer Functions of Closed-Loop Systems in H2 Optimal Control Hideaki Tanaka, Koji Tsumura, and Masaaki Kanno d
Abstract— This paper is concerned with explicit descriptions of the optimal closed-loop system for H2 control problems in the case of single-input and single-output linear systems. We focus on two typical H2 optimal control problems, the minimal energy problem and optimal tracking control problem, and show that the transfer functions of the optimal closed-loop systems can be described in specific terms of the denominator and numerator polynomials of the plants. Moreover, we derive the positions or the area of poles and zeros of the closed-loop system.
I. I NTRODUCTION The H2 optimization problem is one of main topics in modern control theory and the optimal controllers are given by solving Riccati equations [4] or linear matrix inequalities numerically [3]. The resultant optimal closed-loop systems or their H2 norm can be also given numerically. From the viewpoint of design methodologies, this problem is enough established. On the other hand, clarifying the fundamental property of H2 control systems is a related important issue and several results were reported [1]. In particular, the performance limitation for the H2 optimization problem has been actively investigated in the last decade and it is known that the optimized H2 norm can be described by the properties of the plant such as unstable poles, zeros, and frequency gain [2], [5] or the poles of the plant and those of the resultant closed-loop systems [7], [8]. These results give an effective classification of plants as to whether they are easy or difficult for control and guidelines for designing preferable plants for control. Contrary to this research activity on the attainable H2 norm, explicit descriptions of the transfer functions of the optimal closed-loop systems or their poles/zeros have not been enough investigated except for some limited cases such as state feedback [1], although these information will give much on the properties of the H2 optimal control systems. For example, the entropy, i.e., complexity, of signals in control systems can be described with the unstable zeros of the corresponding transfer functions and, therefore, their information has an important meaning for networked control [9]. To understand the fundamental property of H2 optimal H. Tanaka and K. Tsumura are with the Department of Information Physics and Computing, Graduate School of Information Science and Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan, {hideaki tanaka, koji tsumura}@ipc.i.utokyo.ac.jp M. Kanno is with Institute of Science and Technology, Academic Assembly, Niigata University, 8050 Ikarashi 2-no-cho, Nishi-ku, Niigata-shi 950-2181, Japan,
[email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
r - e eK −6
Fig. 1.
u- ? e -
P
r
y -
Unity feedback system configuration.
control systems or their effect in application, explicit descriptions of the optimal closed-loop systems are of considerable significance. Motivated by the above discussion, in this paper, we deal with some classes of H2 optimal control, the H2 minimal energy problem and H2 optimal tracking control problem, for continuous/discrete time SISO systems. Then, we investigate explicit descriptions of the optimal closed-loop systems and analyze the locations of their poles. Moreover, in the cases where the plants are minimum phase for the H2 minimal energy problem or are stable except for integrators for the H2 optimal tracking control problem, we give the location area of the zeros in the closed-loop systems. We also discuss the significance of the results in several applications of control problems. Notation R, C: set of real numbers and complex numbers, respectively, f ∼ (s) := f (−s), f ∼ (z) := f (z −1 ) for scalar functions f (s) and f (z), respectively, C0 := {s ∈ C : Re(s) = 0}, C− := {s ∈ C : Re(s) < 0}, ∂D := {z ∈ C : |z| = 1}, D := ¯ := {z ∈ C : |z| ≤ 1}, Dc := {z ∈ {z ∈ C : |z| < 1}, D ¯ c := {z ∈ C : |z| > 1}, RH∞ : the set C : |z| ≥ 1}, D of proper and stable real rational functions, H2 : the set of strictly proper and stable real rational functions, H2⊥ : the set of strictly proper and anti-stable real rational functions. II. F ORMULATION A. Preliminary In this paper, we deal with a feedback control system composed of an SISO plant P and a controller K as in Fig. 1, where r, d, u, and y ∈ R are reference, disturbance, control input, and plant output, respectively. Denote the polynomial coprime factorization of P by PN , PD where PD is monic. Define PNs and PNa as the minimum phase factor of PN and the remaining factor of PN containing all the non-minimum phase zeros, respectively, and also
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P =
ThAIn4.8 s a PD and PD as the stable factor and the anti-stable factor of PD , respectively. Also let m, ma , n, and na be the orders a , PN , and PNa , respectively, and ν := m − n be of PD , PD the relative degree of P . The standard description of the stabilizing controllers, the Youla-Kucera parametrization [10], for the control system in Fig. 1 is given as follows: Denote the coprime factorization of P over RH∞ by N (1) P = , D where N, D ∈ RH∞ satisfy B`ezout’s identity:
N X + DY = 1
for the continuous time systems and ( 1, k ≥ 0 r(k) = 0, k < 0 for the discrete time systems, respectively. Then, the control objective is to minimize the H2 norm of the tracking error e := r − y, namely, Z ∞ |e(t)|2 dt Js (P, K) := 0
for the cases of continuous time systems and
(2)
for some X, Y ∈ RH∞ . Then, the set K of all the stabilizing controllers K is given by [10] ½ ¾ X + DQ K= K:K= ; Q ∈ RH∞ , Y − N Q 6= 0 . Y − NQ (3) We show a formula which is frequently used in the proofs of the following theorems:
Jz (P, K) :=
∞ X
|e(k)|2
k=0
for the cases of discrete time systems, respectively, over the set K of the stabilizing controllers. In this case, the quantity Js or Jz is equal to the square of the H2 norm of the sensitivity function: S :=
1 . 1 + PK
Lemma 1 ([6]): If G1 ∈ H2 and G2 ∈ H2⊥ , then kG1 + G2 k22 = kG1 k22 + kG2 k22 .
III. H2 M INIMAL E NERGY P ROBLEM
In this paper, we consider the H2 minimum energy problem and H2 optimal tracking control problem for the above plants and the controllers as explained in the following subsections.
In this section, we derive a description of the optimized T for the H2 minimal energy problem and show a result on the locations of poles and zeros of T .
B. H2 Minimal Energy Problem
Firstly, we give the following assumptions on the plant P (s): Assumption 1: The relative degree ν of P (s) is less than or equal to 1. Assumption 2: P (s) does not have poles and zeros in C0 . Moreover, we define polynomials A(s) and B(s) that satisfy
In the H2 minimal energy problem, we attempt to minimize the necessary energy of the control input u, i.e., H2 norm of u, for the stabilization under the conditions that the reference r is zero and the disturbance d is a unit impulse. Therefore, the problem is to minimize Z ∞ |u(t)|2 dt Es (P, K) :=
A. Continuous Time Case
a ∼ a ) (−1)na (PNa )∼ , APNa + BPD = (−1)ma (PD
0
deg(A(s)) < ma , deg(B(s)) = na .
for the cases of continuous time systems and Ez (P, K) :=
∞ X
|u(k)|2
k=0
for discrete time systems, respectively, over the set K of the stabilizing controller. The quantity Es or Ez is equal to the square of the H2 norm of the complementary sensitivity function: PK T := . 1 + PK C. H2 Optimal Tracking Control Problem In the H2 optimal tracking control problem, we suppose that the disturbance d is zero and the reference r is a unit step defined by ( 1, t ≥ 0 r(t) = 0, t < 0
(4)
Note that A(s) and B(s) always exist and are unique from the condition on their degrees. By employing A(s) and B(s), we can derive the following theorem on the H2 minimal energy problem: Theorem 1: Suppose that P (s) satisfies Assumptions 1 and 2, and that A(s) and B(s) satisfy (4). Then the optimal controller Kopt (s) for the H2 minimal energy problem is given by s APD Kopt = , BPNs and the optimized transfer function Topt (s) from d to u is given by Topt =
APNa . a m (−1) a (PD )∼ (−1)na (PNa )∼
Moreover, let pai (i = 1, . . . , ma ) and zia (i = 1, . . . , na ) be the unstable poles and the non-minimum phase zeros of the
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ThAIn4.8 plant, respectively. Then the poles of the closed-loop system are given by −pai (i = 1, . . . , ma ) and −zia (i = 1, . . . , na ). Proof: From the standard manipulation with the coprime factorization (1) of P (s), B`ezout’s identity (2) and the parametrization (3) of all the stabilizing controllers, we get T = N (X + DQ). As possible N , D, X, and Y , we define PN PD N= , D= , F F AF , X= a ∼ m a (−1) (PD ) (−1)na (PNa )∼ PNs BF Y = a )∼ (−1)na (P a )∼ P s , (−1)ma (PD N D
from Lemma 1. It is now clear that Es is minimum at Q = 0. Hence, the optimal controller Kopt is given by Kopt =
and the optimized transfer function Topt (s) from d to u is given by Topt = N X =
(5)
where F is a stable polynomial of degree m. Hereafter, we will show that they satisfy (2) and as a first step, we claim that X and Y are in RH∞ . The degrees of the denominator and numerator polynomials of X and Y can be calculated with the degree conditions in (4) as deg(AF ) < ma + m, a ∼ ) (−1)na (PNa )∼ PNs ) = ma + n, deg((−1)ma (PD deg(BF ) = na + m, s a ∼ ) = na + m. ) (−1)na (PNa )∼ PD deg((−1)ma (PD Hence, we get νX ≥ (ma + n) − (ma + m − 1) = 1 − ν and νY = 0, where νX and νY are the relative degrees of X and Y , respectively. Therefore, from Assumption 1, X and Y are in RH∞ . With this fact and (4), we can show that N , D, X and Y satisfy (2) by direct calculations. By using (5), Es can be calculated as follows: Es = kT k22 = kN (X + DQ)k22 ° a s µ ° PN PN AF =° a )∼ (−1)na (P a )∼ P s ° F (−1)ma (PD N N ¶°2 a s ° PD PD + Q ° ° F 2 °2 ° a s a s a ° PN PN PD PD ° APN ° Q + =° ° ° (−1)ma (P a )∼ (−1)na (P a )∼ F2 D N 2 ° µ a a ° A P P N D =° ° (−1)ma (P a )∼ (−1)na (P a )∼ P a N D D ¶°2 s s a ∼ ° PN ) (−1)na (PNa )∼ PD (−1)ma (PD ° . (6) + Q ° F2 2
is obtained. Moreover, from the fact that (−1)
ma +na
a ∼ s a ∼ s ) PN ) PD (PN (PD F2
A a PD
Assumption 3: P (s) is minimum phase. Then, we can show the following theorem: Theorem 2: Suppose that P (s) satisfies Assumptions 1, 2 and 3. Then the optimized transfer function Topt (s) from d to u for the H2 minimal energy problem is given by Topt =
D
2
a ∼ a (−1)ma (PD ) − PD , a ∼ m a (−1) (PD )
and all the zeros of Topt exist in C0 . Proof: From Assumption 3, we have PNa = 1 and na = 0. Substitute them into (4) to get a ∼ a ) , deg(A) < ma , deg(B) = 0. A + BPD = (−1)ma (PD (8) From (8) we have a ∼ a ) − PD , A = (−1)ma (PD
B = 1.
Hence, Topt is given as Topt =
a ∼ a ) − PD (−1)ma (PD a m ∼ a (−1) (PD )
from Theorem 1. Next, we show the locations of the zeros. The zeros of Topt are the roots of a ∼ a ) − PD = 0. (−1)ma (PD
∈ H2⊥ and
Q ∈ H2 , we can derive °2 ° °2 ° s s a ∼ ° A ° ° (−1)ma (PD PN ° ) (−1)na (PNa )∼ PD ° ° ° Q° (7) = ° a ° + ° ° P F2
APNa a )∼ (−1)na (P a )∼ . (−1)ma (PD N
In the H2 norm minimization problem of the control input u by state feedback, it is well known that the stable poles psi and the mirror image −pai of the unstable poles pai of the plant become the poles of the resultant closed-loop system [1]. Theorem 1 is the corresponding result for the output feedback case. In this case, the mirror images of the unstable poles of the plant −pai are still the poles of the resultant closed-loop system, but the other poles are the mirror images of the unstable zeros of the plant −zia rather than the stable poles psi . Although Theorem 1 is applicable for the non-minimum phase plants, the locations of the zeros of the resultant transfer function Topt may not be given in an explicit form. However, in the case that the plant is minimum phase, we can derive a stronger result. Make the following assumption:
PaPa
By noting that (−1)ma +naD(PNa )∼ (P a )∼ is an all-pass function, D N ° °2 s s a ∼ m ° A PN ° (−1) a (PD ) (−1)na (PNa )∼ PD ° (6) = ° a + Q° ° PD F2 2 (7)
s APD X = , Y BPNs
(9)
Let pai be the unstable poles of the plant, then,
2
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a PD =
ma Y
i=1
(s − pai ),
a ∼ (PD ) =
ma Y
i=1
(−s − pai ).
(10)
ThAIn4.8 By substituting (10) into (9), we get Qma (s − pai ) Qi=1 = 1. ma a i=1 (s + pi )
a a For PD to be a polynomial with real coefficients, PD should a a have the root p¯i if it has the root pi such that Im(pai ) 6= 0. Now define a function:
R(s, pai ) :=
|s − pai |2 . |s + p¯ai |2
The difference of the denominator and numerator of R(s, pai ) can be calculated as |s + p¯ai |2 − |s − pai |2 s − p¯ai ) s + pai ) − (s − pai )(¯ = (s + p¯ai )(¯ = (|s|2 + pai s + p¯ai s¯ + |pai |2 ) − (|s|2 − pai s¯ − p¯ai s + |pai |2 ) = pai s + p¯ai s¯ + pai s¯ + p¯ai s
Fig. 2. Locations of poles and zeros of the continuous time H2 minimal energy problem
= (pai + p¯ai )(s + s¯) = 4Re(pai )Re(s), where Re(pai ) > 0 because pai is an unstable C+ , Re(pai )Re(s) > 0 and this implies
pole. For s ∈
R(s, pai ) < 1.
a a ∼ na APNa + BPD = z ma (PD ) z (PNa )∼ ,
R(s, pai ) > 1.
deg(A) < ma , deg(B) = na .
(11)
Also for s ∈ C− , Re(pai )Re(s) < 0 and this implies (12)
From (11) or (12), ¯ Q ma ¯ ¯ i=1 (s − pai ) ¯ ¯ Qm ¯ 6= 1 for s ∈ C+ or s ∈ C− . a ¯ a ¯ i=1 (s + pi )
Hence, the zeros of Topt do not exist in C− or C+ and this concludes that they should exist in C0 . Example 1: For verifying Theorems 1 and 2, we calculate the poles and the zeros from our results and compare them with numerical solutions. Define a minimum phase plant P (s) by P (s) =
Assumption 5: P (z) does not have poles and zeros in ∂D. Moreover, define polynomials A(z) and B(z) that satisfy
(s + 3)(s + 2)(s + 1) . (s2 − 2s + 3)(s2 − 5s + 10)
Then, the theorems give the poles and the zeros of the resultant closed-loop system for the H2 minimum energy problem, which are plotted in Fig. 2. Fig. 2 also contains the corresponding numerical solutions of the optimized closedloop system computed by using Robust Control Toolbox in Matlab. We can observe that they coincide and, more importantly, the zeros are located on the imaginary axis. B. Discrete time Case In the previous subsection, we give descriptions of the optimal Topt for continuous time systems. In this subsection, we derive the corresponding results for the case of discrete time systems. Firstly, assume the following for the plant P (z): Assumption 4: The relative degree ν of P (z) is less than or equal to 1.
(14)
Note that the polynomials A(z) and B(z) always exist and are unique from the conditions on their degrees. By employing A(z) and B(z), we can derive the following theorem on the H2 minimal energy problem in the discrete time case: Theorem 3: Suppose that P (z) satisfies Assumptions 4 and 5, and that A(z) and B(z) satisfy (14). Then the optimal controller Kopt (z) for the H2 minimal energy problem is given by s APD Kopt = , BPNs and the optimized transfer function Topt (z) from d to u is given by AP a Topt = ma a ∼ Nna a ∼ . z (PD ) z (PN ) Moreover, let λai (i = 1, . . . , ma ) and ηia (i = 1, . . . , na ) be the unstable poles and the non-minimum phase zeros of the plant, respectively. Then, the poles of the closed-loop system are given by 1/λai (i = 1, . . . , ma ) and 1/ηia (i = 1, . . . , na ). The proof is similar to that of Theorem 1 and is omitted due to the limitation of pages. As in the case of continuous time systems, we can derive a stronger result on the locations of zeros of the resultant Topt when P (z) satisfies: Assumption 6: P (z) is minimum phase. Then, we can derive the following theorem: Theorem 4: Suppose that P (z) satisfies Assumptions 4, 5 and 6. Then the optimized transfer function Topt (z) from d to u is given by Q ma a a ∼ (−λai )PD ) − i=1 z ma (PD , Topt = a )∼ z ma (PD
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ThAIn4.8 IV. H2 O PTIMAL T RACKING C ONTROL P ROBLEM In the previous section, we give descriptions of the resultant T for the H2 minimal energy problem. From this, we expect that the similar results can be derived for other cases. Motivated by this, in this section, we give descriptions of the optimized transfer function S from r to e for the H2 optimal tracking control problem and show a result on the locations of the poles and the zeros.
Fig. 3. Locations of poles and zeros of the discrete time H2 minimum energy problem
¯c and all the zeros of Topt exist in D The proof is similar to that of Theorem 2 and is omitted due to the limitation of pages. It is interesting to point out that the locations of the zeros in this discrete time case is slightly different from Theorem 2 for continuous time systems. In the networked control, the entropy of signals in the systems represents their complexity and give the necessary channel capacity for signal transmission. In the case of linear time-invariant systems, the entropy of signals can be represented by the product of the non-minimum phase zeros of the corresponding transfer function [9]. Theorem 4 tells that all the zeros of the resultant closed-loop system are unstable for the H2 minimal energy problem in the case of a minimum phase plant, and suggests that we can calculate the entropy when we know the product of such zeros. On this point, we show the following corollary: Corollary 1: Suppose that P (z) satisfies Assumptions 4, 5 and 6. Moreover, let λai (i = 1, . . . , ma ) be the unstable poles of the plant. Then, the product of non-minimum phase zeros of Topt is given by Qma a Qma 1 i=1 λa i=1 λi − Pma a Pma 1i . i=1 λa i=1 λi − i The proof is omitted due to the limitation of pages. Example 2: For verifying Theorem 4, we calculate the poles and the zeros, and compare them with numerical solutions. Define a minimum phase plant P (z) by P (z) =
(z 2
(z − 13 )(z 2 − 12 )(2z 2 − 51 z + 51 ) . + 2z − 4)(z 2 + z + 5)(z 2 − 3z + 8)
Then, the theorem gives the poles and the zeros of the resultant closed-loop system for the H2 minimal energy problem, and they are plotted in Fig. 3. Fig. 3 also contains the corresponding numerical solutions by Matlab. We can observe that they coincide and also that the zeros are located outside the unit disc.
A. Continuous Time Systems 1) Case of Plants Containing an Integrator: Firstly, we consider the following cases: Assumption 7: P (s) can be described by P (s) = P0s(s) . Assumption 8: P0 (s) does not have poles and zeros in C0 . Assumption 9: The relative degree ν of P (s) is 0 Assumptions 7 and 8 imply that P has one integrator. We a a s a then decompose PD as PD = sPD PD where PD is the antis stable factor and PD is the stable factor. Denote the degree a of PD by ma − 1 and define polynomials A(s) and B(s) that satisfy a a ∼ APNa + sBPD = (−1)ma −1 (PD ) (−1)na (PNa )∼ , deg(A) = ma − 1, deg(B) < na .
(16)
Note that A(s) and B(s) always exist and are unique from their degree condition. By employing A(s) and B(s), we can derive the following theorem for the H2 optimal tracking control problem: Theorem 5: Suppose that P (s) satisfies Assumptions 7, 8 and 9, and that A(s) and B(s) satisfy (16). Then the optimal controller Kopt (s) of the H2 tracking problem is given by s APD Kopt = , BPNs and the optimized transfer function Sopt (s) from r to e is given by a sBPD Sopt = a )∼ (−1)na (P a )∼ . (−1)ma −1 (PD N Moreover, let pai (i = 1, . . . , ma − 1) and zia (i = 1, . . . , na ) be the unstable poles and non-minimum phase zeros of the plant, respectively. Then the poles of the closed-loop system are given by −pai (i = 1, . . . , ma − 1) and −zia (i = 1, . . . , na ). The proof is omitted due to the limitation of pages. 2) Case of Stable Plants except for an Integrator: We next consider a special case that plant P (s) further satisfies the following assumption: Assumption 10: P0 (s) is stable. Then, from Theorem 5, we can get the following results: Theorem 6: Suppose that P (s) satisfies Assumptions 7, 8, 9 and 10. Then the optimized transfer function Sopt (s) from r to e is given by (P a )∼ − P a Sopt = N a ∼ N , (PN ) and all the zeros of Sopt exist in C0 . The proof is omitted due to the limitation of pages.
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ThAIn4.8 V. C ONCLUSION
B. Discrete Time Systems In this subsection, we derive corresponding results for the case of discrete time systems. 1) Case of Plants Containing an Integrator: Firstly, we consider the following cases: Assumption 11: P (z) can be described by P (z) = P0 (z)/(z − 1). Assumption 12: P0 (z) does not have poles and zeros in ∂D. Assumption 13: The relative degree ν of P (z) is 0. Assumptions 11 and 12 imply that P includes only one a s integrator. Now, we decompose PD as PD = (z − 1)PD PD a where PD is the the anti-stable factor except for z − 1 of s the denominator and PD is the stable factor. Also denote the a degree of PD by ma − 1 and define polynomials A(z) and B(z) satisfying a a ∼ na APNa + (z − 1)BPD = z ma −1 (PD ) z (PNa )∼ , deg(A) = ma − 1, deg(B) < na .
(20)
Note that A(z) and B(z) always exist and are unique from their degree conditions. By employing A(z) and B(z), we can derive the following result: Theorem 7: Suppose that P (z) satisfies Assumptions 11, 12 and 13, and that A(z) and B(z) satisfy (20). Then the optimal controller Kopt (z) of the H2 tracking problem is given by s APD Kopt = , BPNs and the optimized transfer function Sopt (z) from r to e is given by (z − 1)BP a Sopt = ma −1 a ∼ naD a ∼ . z (PD ) z (PN ) Moreover, let λai (i = 1, . . . , ma − 1) and ηia (i = 1, . . . , na ) be the unstable poles and non-minimum phase zeros of the plant, respectively. Then the poles of the closed-loop system are given by 1/λai (i = 1, . . . , ma − 1) and 1/ηia (i = 1, . . . , na ). The proof is omitted due to the limitation of pages. 2) Case of Stable Plants except for an Integrator: We next consider a special case that plant P (z) further satisfies the following assumption: Assumption 14: P0 (z) is stable. Then, from Theorem 7, we can get the following result: Theorem 8: Suppose that P (z) satisfies Assumptions 11, 12, 13 and 14. Then the optimized transfer function Sopt (z) from r to e is given by Sopt =
In this paper, we have given the numerators and the denominators of the transfer functions of the optimal controllers and the optimized closed-loop systems for continuous/discrete time H2 minimal energy problem and H2 optimal tracking control problem. From these results, the locations of the poles are completely clarified. On the other hand, in general cases, the locations of the zeros are still not given in a completely analytic form because the numerator of the transfer functions is described by a polynomial, which should satisfy a condition. However, in special cases, the location areas of the zeros are known. The results on the locations of the zeros in the discrete time case imply that even if the plant is minimum phase, H2 optimal control gives the anti-stable numerator of the resultant closed-loop systems under some conditions, therefore, such optimized control is not necessarily preferable from the viewpoint of decreasing the signal complexity. These results are restricted to some cases such that a plant is minimum phase or stable and, therefore, the general case is left for the future work. Moreover, the problems in this paper, the H2 minimum energy problem and the H2 optimal tracking control problem, are also special cases because only one of the pair of the outputs of the plant and the controller is evaluated in the optimization function. The case of the general evaluation function in the H2 optimization problem is also left for the future work. R EFERENCES [1] B.D.O. Anderson and J.B. Moore, Optimal Control: Linear Quadratic Methods. Prentice Hall, Englewood Cliffs, New Jersey, 1990. [2] T. Bakhtiar, “H2 control performance limitations for SIMO feedback control systems,” Ph.D. dissertation, The University of Tokyo, 2006. [3] S. Boyd, L. EL Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, 1994. [4] R. W. Brockett, Finite Dimensional Linear Systems, Series on decision and control, John Wiley & Sons, Inc., New York, 1970. [5] J. Chen, S. Hara, and G. Chen, “Best Tracking and Regulation Performance Under Control Energy Constraint,” IEEE Transactions on Automatic Control, vol. 48, no. 8, pp. 1320–1336, 2003. [6] J. Doyle, B. Francis, and A. Tannenbaum Feedback Control Theory. Macmillan, New York, 1992. [7] M. Kanno, S. Hara, H. Anai, and K. Yokoyama, “Sum of roots, polynominal spectral factorization, and control performance limitations,” in Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, LA, USA, 2007, pp. 2968–2973. [8] H. Tanaka, M. Kanno, and K. Tsumura, “Characterization of discretetime h2 control performance limitation based on poles and zeros,” in Proceedings of the 47th IEEE Conference on Decision and Control, 2008, pp. 3700 – 3705. [9] K. Tsumura and J. Maciejowski, “Stabilizability of SISO Control Systems under Constraints of Channel Capacities,” in Proceedings of SICE 3rd Annual Conference on Control Systems, 2003, pp. 457–462. [10] K. Zhou, K. Glover, and J. Doyle, Robust and Optimal Control. Prentice Hall, Upper Saddle River, New Jersey, 1996.
z ma (PNa )∼ − PNa , z na (PNa )∼
and all the zeros of Sopt exist in ∂D. The proof is omitted due to the limitation of pages.
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