AbstractâThe paper establishes the design procedure for the state feedback control of linear discrete-time systems, considerable as an interposed de...

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Volume 11, 2017

Interposed Control Design Conditions for Linear Discrete-time Systems D. Krokavec and A. Filasov´a

Throughout the paper, the notations is narrowly standard in such way that diag[ · ] denotes a block diagonal matrix, xT , X T denotes the transpose of the vector x and matrix X, respectively, for a square matrix X < 0 means that X is a symmetric negative definite matrix, In marks the n-th order unit matrix, IR denotes the set of real numbers and IRn , IRn×r refer to the set of all n-dimensional real vectors and n × r real matrices, respectively.

Abstract—The paper establishes the design procedure for the state feedback control of linear discrete-time systems, considerable as an interposed design criterium in the form of linear matrix inequalities. The goal is to design the feedback control which guarantees bounded H2 performance index for the system transfer function matrix and H∞ norm attenuation for the disturbance transfer function matrix, both combined with D-stable circle region parameters. Analyzing the criteria observance, the task is formulated as a feasible problem subject to integral quadratic constraints included in the Lyapunov discrete-time stability condition. Keywords — H2 /H∞ control strategy, D-stability region, state control, quadratic Lyapunov function, linear matrix inequalities, interposed design criteria.

II. BASIC P RELIMINARIES In this paper, the discrete-time linear MIMO systems are considered, described in the state-space form by the set of equations

I. I NTRODUCTION

q(i+1) = F q(i) + Gu(i) + Ed(i) ,

Tasks relating H2 and H∞ control design have been studied by many authors (see, e.g., [6], [19], [24] and the references therein), where H∞ control design is referred mainly with the system frequency performances while H2 control synthesis sets more suitable achievement on the system transient behavior [10], [21], [27]. Combining H2 and H∞ performance analysis, a mixed H2 /H∞ control problem was formulated in [13] with the goal to optimize H2 norm of the system transfer function matrix subject the constraint on H∞ norm of the disturbance transfer function matrix. To derive the state feedback synthesis conditions, the benefit was substantiated by applying the continuous-time mixed H2 /H∞ performance criterion [2], [4], [7], [20], as well as by formulating the appropriate computational linear matrix inequalities (LMI) technique [8], [16], [22]. To apply LMIs in control law parameter alignment, the H2 /H∞ control design strategies for discrete-time linear systems are analyzed in the paper. In the sense of common practice [11], [25], the approach extends the design method presented in [15] to obtain associated interposed design criteria. Accordingly, exploiting an extended quadratic Lyapunov function, the obtained parameters of the state controller are designed relying on H2 , H∞ constraints and combined with D-stability circle region parameters in the set of LMIs. The outline of this paper is as follows. Section II and Section III. introduce the basic preliminaries in control law parameter design, while in Section IV and Section V. new results in design conditions are established and proven. Section VI. illustrates the properties of the proposed design conditions by a numerical example and in Section VII. some conclusions are established.

y(i) = Cq(i) , n

(2)

r

m

where q(i) ∈ IR , u(i) ∈ IR , and y(i) ∈ IR are vectors of the system, input and output variables, respectively, d(i) ∈ IRp is a bounded unknown disturbance and F ∈ IR n×n , G ∈ IR n×r , C ∈ IR m×n , E ∈ IRn×p . The transfer function matrices to (1), (2) are H(z) = C(zI n − F )−1 G ,

(3)

H d (z) = C(zI n − F )

(4)

−1

E,

where a complex z is the transform variable of the transform Z [17]. Quantifications of the effect of the input onto the output of the system are the so-called H2 and H∞ norms of the transfer function matrix H(z) and H d (z), respectively. Definition 1: [6] The H2 -norm of the transfer functions matrix (3) is defined as π 1 kH(z)k22 = tr ∫ H(ejω )H ∗ (ejω )dω , (5) 2π −π √ where z = ejω , ω is the frequency variable, j := −1 and H ∗ (ejω ) is the adjoint of H(ejω ). Definition 2: [23] The H∞ -norm of the transfer function matrix (4) is defined as kH d (z)k∞ = =

sup

sup

σo (H d (ejω )) =

ω∈h−π,πi

σo (eig(H d (ejω )H ∗d (ejω )) ,

(6)

ω∈h−π,πi

where σo means the largest singular value of the matrix H d (ejω ). Definition 3: [12] A square matrix F is stable if every eigenvalue of F lies in the unit circle in the plain of the complex variable z. If F is stable, then the dynamical system (1), (2) has the stable transfer function matrix (3), i.e., the poles of all elements of H(z) lies in the unit circle in the plain of the complex variable z.

The work presented in this paper was supported by VEGA, the Grant Agency of the Ministry of Education and the Academy of Science of Slovak Republic under Grant No. 1/0608/17. This support is very gratefully acknowledged. D. Krokavec and A. Filasov´a are with Department of Cybernetics and Artificial Intelligence, Faculty of Electrical Engineering and Informatics, Technical University of Koˇsice, Letn´a 9/B, 042 00 Koˇsice, Slovakia

[email protected]tuke.sk; [email protected] ISSN: 1998-0159

(1)

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Proof: Since a solution of (1), (2) is

Proposition 1: [18] (Lyapunov inequality (LI)) The linear discrete-time system (1), (2) with a bounded disturbance is stable if and only if there exist a symmetric positive definite matrix X ∈ IR n×n such that X = XT > 0 , # " −X ∗ < 0. F X −X

q(n) = F n q(0) +

A(l) = F l G , (8)

W (n) =

=

F l GGTF T l .

(22)

F W (n)F T = n P F l+1 GGT (F T ) l+1 = F l GGT F T l

(23)

l=1

and, subtracting (22) from (23), it yields ∗

F W (n)F T − W (n) = F n GGT (F T ) n − GGT . (24)

(11)

Thus, considering that (20) for l = n insert the input variable value u(−1) which is identically equal zero, and defining a stationary solution W (n) = Wo , then (24) implies

XF T − aX −̺X

#

<0

(12)

F Wo F T − Wo + GGT = 0 .

X = XT > 0 ,

(13)

MD (F , X) < 0 .

(14)

Proposition 3: [5] (quadratic performance) If the matrix F of system (1), (2) is stable and d(i) is bounded then 2 T (y T(i)y(i) − γ∞ d (i)d(i)) > 0 ,

X = XT > 0 ,

(26)

F XF T − X + GGT < 0 ,

(27)

tr(CXC T ) > γ22 .

(28)

Proof: Let (27) yields for a symmetric positive definite matrix X. Then subtracting (19) from (27) leads to the strict inequality

(15)

where γ∞ ∈ IR is the H∞ norm of the discrete-time disturbance transfer function matrix (4). Proposition 4: [6], [14] (bounded real lemma (BRL)) The discrete-time linear system (1), (2) with a bounded disturbance is stable if there exist a symmetric positive definite matrix X ∈ IR n×n and a positive scalar γ∞ ∈ IR such that X = XT > 0 , γ∞ > 0 , (16) −X ∗ ∗ ∗ F X −X ∗ ∗ < 0. (17) CX 0 −γ∞ I m ∗ 0 ET 0 −γ∞ I p

F (X − Wo )F T − (X − Wo ) < 0

(29)

and with X > Wo the Lyapunov property implies that (29) is negative definite if and only if F is stable. Moreover, the relation X > Wo gives tr(CXC T ) > tr(CWo C T ) = γ22

(30)

and so (30) implies (28). This concludes the proof. III. S TATE F EEDBACK D ESIGN By applying the controllable system (1), (2) and the control law u(i) = −Kq(i) , (31)

Lemma 1: If the matrix F of the system (1), (2) is stable, then γ22 = tr(CWo C T ) , (18)

where K ∈ IR r×n , then the closed-loop system description takes the form

where F Wo F T − Wo + GGT = 0 ,

(25)

This concludes the proof. Lemma 2: The matrix F of the system (1), (2) is stable and kH(z)k2 < γ2 if there exists a symmetric positive definite matrix X ∈ IRn×n such that

and the matrix F of the discrete-time linear system (1), (2) is D-stable if and only if there exists a symmetric positive definite matrix X ∈ IRn×n such that

q(i+1) = (F −GK)q(i)+Ed(i) = Fc q(i)+Ed(i) , (32)

(19)

while Wo ∈ IR n×n is a positive definite symmetric matrix and γ2 ∈ IR is H2 norm of the discrete-time system transfer function matrix H(z). ISSN: 1998-0159

n−1 P l=0

it yields

i=0

n−1 X

Pre-multiplying the left side of (22) by F and postmultiplying the right side by F T results in

(10)

(X, F X, XF ) ↔ (1, z, z )

∞ X

(21)

l=0

is the LMI region characteristic function. Related by the substitution

MD (F , X) =

(20)

then as an explicit test for linear independence of A(l) can be used its Gramian [1]

(9)

where a, ̺ ∈ IR, ̺ ≤ a, 0 < a < 1 and # " −̺ z ∗ − a fD (z) = z−a −̺

−̺X F X − aX

A(l)u(n − 1 − l) ,

where

(7)

D = {z ∈ C : fD (z) < 0} ,

"

n−1 X l=0

Hereafter, ∗ labels the symmetric item in a symmetric matrix. Proposition 2: [3] (LMI region) A subset D of the complex plane Z is called a stable circle LMI region if

T

Volume 11, 2017

y(i) = Cq(i) ,

(33)

Fc = F − GK .

(34)

where

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Rewriting that Fc X = F X − GY ,

Y = KX ,

and premultiplying the left side of (46) and postmultiplying the right side by the matrix X = P −1 leads to

(35)

XFcT P Fc X − X +

the standard LMI design conditions are given by the following theorems (some of the proofs are omitted since evidently imply from the formulas stated in Section II). Theorem 1: (LI synthesis) The control (31) to the system (1), (2) exists if there exist a symmetric positive definite matrix X ∈ IR n×n and a matrix Y ∈ IR r×n such that X = XT > 0 , "

−X F X − GY

∗ −X

< 0.

(1−̺)2 −a2 X 1−̺

= = (1−̺)X − (−a)X(1−̺)−1 X −1 (−a)X .

(37)

−X F X − GY CX 0

∗ −X 0 ET

γ∞ > 0 ,

∗ ∗ −γ∞ I m 0

Thus, exploiting (35), it is evident that (50) is identical with (42) which implies that D-circle stability region in the design condition means a quadratic constraint on the Lyapunov function, acceptable in the sense of the LyapunovKrasovskii theorem [9]. Theorem 4: (H2 control synthesis) The control (31) to the system (1), (2) exists and kH(z)k2 < γ2 if there exist symmetric positive definite matrices X ∈ IR n×n , Z ∈ IR m×m and a matrix Y ∈ IR r×n such that

(39)

∗ ∗ ∗ −γ∞ I p

< 0 . (40)

When the above conditions hold, the control law gain is given by (38). Theorem 3: (D-stable LI synthesis) The control (31) to the system (1), (2) exists and the closed-loop eigenvalues are clustered in the D-stable circle region if for given a, ̺ ∈ IR, ̺ ≤ a, 0 < a < 1 there exist a symmetric positive definite matrix X ∈ IR n×n and a matrix Y ∈ IR r×n such that T

X = X > 0, "

−̺X F X − GY − aX

X = XT > 0 ,

Z = ZT > 0 , −X F X − GY G ∗ −X 0 < 0, ∗ ∗ −I r X XC T > 0. ∗ Z

(41)

∗ −̺X

#

< 0.

(42)

i−1 (1−̺)2 −a2 X T q (l)P q(l) , (43) 1−̺ l=0

2

(44)

2

Z > CXC T = CXX −1 XC T ,

Using the disturbance free part of (32) then (44) implies (1−̺)2 − a2 P q(i) < 0 . 1−̺

ISSN: 1998-0159

(1−̺)2 − a2 P <0 1−̺

(56)

with Z ∈ IRm×m being symmetric and positive definite, and applying appropriate the Schur complement property, then (56) implies (53). This concludes the proof. Corollary 2: It is evident that

(45)

The negativeness of (45) demands to be valid FcT P Fc − P +

(53)

Prom this it follows easily (52). By H2 control nomination the inequality (28) could be minimized, but this form cannot be directly included into the set of LMIs. Introducing the inequality

−a + (1−̺) q T (i)P q(i) < 0 . 1−̺

q T (i) FcT P Fc − P +

(52)

Supplanting F in (54) by (35) modifies the LMI (54) as −X F X − GY G XF T − Y T GT −X 0 < 0 . (55) T G 0 −I r

where a, ̺ ∈ IR, ̺ ≤ a, 0 < a < 1 and a positive definite matrix P ∈ IRn×n , then it yields for the first forward difference of the Lyapunov function (43) ∆v(q(i)) = v(q(i + 1)) − v(q(i)) = = q T(i+1)P q(i+1) − q T (i)P q(i)+

(51)

When the above conditions hold, the control law gain is given by (38). Proof: Rearranging the inequality (27) by using the Schur complement property it yields G −X F X T XF −X 0 < 0. (54) T G 0 −I r

When the above conditions hold, the control law gain is given by (38). Corollary 1: Considering the extended Lyapunov function v(q(i)) = q T(i)P q(i)+

(49)

Thus, using the Schur complement property, either it yields " # (1−̺)X −aX −X XFcT + < 0 (50) −aX (1−̺)X Fc X −X

Theorem 2: (BRL synthesis) The control (31) to the system (1), (2) exists and kH d (z)k∞ < γ∞ if there exist a symmetric positive definite matrix X ∈ IR n×n , a matrix Y ∈ IR r×m and a positive scalar γ∞ ∈ IR such that X = XT > 0 ,

(47)

while, moreover,

When the above conditions hold, the control law gain is given as K = Y X −1 . (38)

(1−̺)2 − a2 X < 0. 1−̺

Subsequently, the Schur complement property implies a reformulated form of (47) as " # 2 −a2 −X + (1−̺) X XFcT 1−̺ < 0, (48) Fc X −X

(36) #

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η = tr(Z) > tr(CXC T ) > γ22 .

(46)

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(57)

INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTERS IN SIMULATION

IV. M ULTI - OBJECTIVE D ESIGN An integration of the above presented approaches can be formulated by the multi-objective principle. Theorem 5: (D-stable BRL synthesis) The control (31) to the system (1), (2) exists, kH d (z)k∞ < γ∞ and the closedloop system matrix eigenvalues are clustered in the D-stable circle region if for given a, ̺ ∈ IR, ̺ ≤ a, 0 < a < 1 there exist a symmetric positive definite matrix X ∈ IR n×n a matrix Y ∈ IR r×n and a positive scalar γ∞ ∈ IR such that X = X T > 0 , γ∞ > 0 , (58) −̺X ∗ ∗ ∗ F X −GY −aX −̺X ∗ ∗ < 0 . (59) CX 0 −γ∞ I m ∗ 0 ET 0 −γ∞ I p

V. I NTERPOSED D ESIGN C RITERIA Combining the algorithms for H2 and H∞ control design, as well as the D-stable circle constraints, the following theorems can be introduced. Theorem 6: (mixed H2 /H∞ synthesis) The state feedback control (31) to the system (1), (2) exists and kH(z)k2 < γ2 as well as kH d (z)k∞ < γ∞ if there exist symmetric positive definite matrices X ∈ IR n×n , Z ∈ IR m×m , a matrix Y ∈ IR r×n and a positive scalar γ∞ ∈ IR such that X = XT > 0 ,

−1 +γ∞

i−1 P l=0

(1−̺)2 −a2 1−̺

i−1 P

(60)

2 T (y T(l)y(l) − γ∞ d (l)d(l)) ,

n×n

where P ∈ IR is a positive definite symmetric matrix and γ∞ ∈ IR is the H∞ norm of the disturbance transfer function matrix, then using (2) it yields ∆v(q(i)) = q T(i+1)P q(i+1) − q T (i)P q(i)+ −1 T q (i)C T Cq(i) − γ∞ dT(i)d(i)+ +γ∞ 2 −a2 T + (1−̺) q (i)P q(i) < 0 . 1−̺ Now let, with (32) and the notation q Tc (i) = q T (i) uT (i)

(61)

X = XT > 0 ,

the inequality (61) is written as

"

=

FcTP Fc −P

Pc −1 T +γ∞ C C T

(1−̺)2 −a2 P 1−̺

E P

(n+p)×(n+p)

while Pc < 0, Pc ∈ IR matrix T = diag X

(63)

= +

Ip

,

# (64) PE , −γ∞ I p

. Defining the transform X = P −1

(65)

(71)

(72)

(73)

(74)

(75)

When the above conditions hold, the control law gain is given by (38). Proof: This is an immediate consequence of the above LMI conditions.

and premultiplying the left side and postmultiplying the right side of (64) by the matrix T then " # 2 −a2 −1 XFcTP Fc X −X +γ∞ XC TCX + (1−̺) 1−̺ X E <0 . ET −γ∞Ip (66) As explained above, an equivalent form of (66) is " # −̺X XFcT − aX + F c X − aX −̺X " # (67) −1 γ∞ XC TCX 0 + <0 −1 T 0 γ∞ E E

VI. I LLUSTRATIVE EXAMPLE To illustrate the proposed method, a system whose dynamics is described by equations (1), (2) is considered with the sampling period ts = 0.01 s, the disturbance noise variance σd2 = 0.028 and the matrix parameters 1.0142 −0.0018 0.0651 −0.0546 −0.0057 0.9582 −0.0001 0.0067 , F = 0.0103 0.0417 0.9363 0.0563 0.0004 0.0417 0.0129 0.9797

Thus, using (35) and applying the Schur complement property then (67) implies (59). This concludes the proof. ISSN: 1998-0159

Z = ZT > 0 ,

γ∞ > 0 , −̺X ∗ ∗ ∗ F X −GY −aX −̺X ∗ ∗ < 0. CX 0 −γ∞ I m ∗ 0 ET 0 −γ∞ I p −X F X − GY G ∗ −X 0 < 0, ∗ ∗ −I r X XC T > 0. ∗ Z

(62)

∆v(q c (i)) = q Tc (i)Pc q c (i) < 0 ,

γ∞ > 0 , (68) ∗ ∗ < 0 , (69) ∗ −γ∞ I p G 0 < 0, (70) −I r

When the above conditions hold, the control law gain is given by (38). Proof: Setting down a unique solution of K within the above conditions then (39), (40), (51)-(53) imply (68)-(71). This concludes the proof. Theorem 7: (interposed H2 /H∞ synthesis) The control (31) to the system (1), (2) exists, kH d (z)k∞ < γ∞ , kH d (z)k∞ < γ∞ and the closed-loop system matrix eigenvalues are clustered in the D-stable circle region if for given a, ̺ ∈ IR, ̺ ≤ a, 0 < a < 1 there exist symmetric positive definite matrices X ∈ IR n×n , Z ∈ IR m×m , a matrix Y ∈ IR r×n and a positive scalar γ∞ ∈ IR such that

q T(l)P q(l)+

l=0

Z = ZT > 0 ,

−X ∗ ∗ F X − GY −X ∗ CX 0 −γ∞ I m 0 ET 0 −X F X − GY ∗ −X ∗ ∗ X XC T > 0. ∗ Z

When the above conditions hold, the control law gain is given by (38) Proof: The proof of this theorem is a modification of the argument given in the Corollary 1. Considering the extended Lyapunov function v(q(i)) = q T(i)P q(i) +

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0.0000 −0.0010 0.0556 0.0000 G= 0.0125 −0.0304 , E = 0.0125 −0.0002 1 0 0 1 C= . 0 0 1 0

0.0063 0.0216 , 0.0131 0.0044

0.45

0.4

0.35

0.3

0.25

0.2

0.15

Solving (39)–(40) using Self-Dual-Minimization (SeDuMi) package, the H∞ control design problem is feasible while 0.0755 0.0645 −0.0733 −0.0054 0.0645 1.0991 −0.0586 −0.1332 , X= −0.0733 −0.0586 0.4652 0.0062 −0.0054 −0.1332 0.0062 0.2011 −0.0073 1.3335 0.3439 1.0031 Y = , −0.4794 0.0571 −2.3094 0.3805

0.1

0.05

y (t) 1

y2(t)

0

0

0.2

0.4

0.6

0.8

1 t [s]

1.2

1.4

1.6

1.8

2

Fig. 1. Closed-loop system output response - H∞

0.45

0.4

which results the control loop structure parameters −0.6037 2.0568 0.8192 6.3077 K1 = , −13.6687 0.7487 −7.0524 2.2354 ρ(F c ) = 0.8320 ± 0.0842 i 0.8952 ± 0.0599 i , γ∞ < 1.7586 ,

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y(t)

0.35

0.3

y(t)

0.25

0.2

0.15

0.1

tr (CXC T ) = 0.7309 > γ22 ,

0.05

y1(t) y (t) 2

0

Frobenius norm of K 1 = 16.9464 . Solving (68)-(71) then 0.1415 0.1030 −0.1383 −0.0049 0.1030 1.3972 −0.1322 −0.1814 , X= −0.1383 −0.1322 0.6909 0.0183 −0.0049 −0.1814 0.0183 0.3182 −0.0732 3.0467 0.5694 1.5381 Y = , −0.7805 0.0778 −3.8454 0.6599 1.3909 −0.0854 Z= , γ∞ < 3.1650 , −0.0854 1.5655 −1.7621 3.2559 0.9198 6.6108 K2 = , −14.0272 0.6547 −8.3200 2.7112 ρ(F c ) = 0.7671 ± 0.0499 i 0.9058 ± 0.0382 i ,

0.4

0.6

0.8

1 t [s]

1.2

1.4

1.6

1.8

2

while the common solution of (68)–(71) gives the following result 0.0411 0.0437 −0.0969 0.0207 0.0437 2.0370 −0.1092 −0.0801 , Q= −0.0969 −0.1092 0.6430 −0.0142 0.0207 −0.0801 −0.0142 0.0958 0.0982 4.7108 −0.0011 0.9448 Y = , −0.0353 0.0164 −5.2944 0.5093 1.2588 −0.0725 Z= , γ∞ < 3.0524 , −0.0725 1.5552 −9.4462 3.0486 −0.5891 14.3629 K4 = , −40.7424 0.6257 −13.9865 12.5492 ρ(F c ) = 0.7210 ± 0.0856 i 0.8196 ± 0.0616 i ,

Frobenius norm of K 2 = 44.2207 . On the other side, applying the D-circle parameters a = 0.5, ρ = 0.41 the LMI-based conditions (58), (59) are solvable with 0.0243 0.0265 −0.0579 0.0120 0.0265 1.4924 −0.0659 −0.0588 , X= −0.0579 −0.0659 0.3791 −0.0061 0.0120 −0.0588 −0.0061 0.0623 0.0643 2.9413 −0.0268 0.5633 Y = , 0.0127 −0.0821 −3.0039 0.3096

tr Z = 2.8140 > tr (CXC T ) = 0.8213 > γ22 . Frobenius norm of K 4 = 48.1519 .

All simulations are done in the forced mode, where u(i) = −K j q(i) + Wj wo , j ∈ h1, 4i , the set of gain matrices is given above and the associated signal gain matrices Wj are computed by using the static decoupling principle [26] as

which results the control loop structure parameters −8.2100 2.5999 −0.6640 13.0181 K3 = , −37.4652 0.4664 −13.3803 11.3358 ρ(F c ) = 0.7421 ± 0.0785 i 0.8306 ± 0.0817 i ,

Wj = (C(I n − (F − GK j ))−1 G)−1 . Therefore, the signal gain matrices are given as 0.9608 7.6607 −3.8384 W1 = , W2 = −9.7879 7.6909 −27.2241 0.1562 9.3619 −4.5478 W3 = , W4 = −9.9902 7.1958 −29.6214

tr (CXC T ) = 0.4897 > γ22 ,

Frobenius norm of K 3 = 18.2214 . ISSN: 1998-0159

0.2

Fig. 2. Closed-loop system output response - H2 /H∞

tr Z = 2.9563 > tr (CXC T ) = 1.1409 > γ22 ,

γ∞ < 1.8527 ,

0

49

19.8243 , 32.4430 22.4230 . 36.1152

INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTERS IN SIMULATION

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design strategy is easy implementable, making it an eligible method to factual applications. The numerical example is given to show the feasibility and advantage of the criteria.

0.5

0.4

0.3

R EFERENCES

y(t)

0.2

[1] P. J. Antsaklis and A. N. Michel, A Linear Systems Primer, Boston, MA, USA: Birkh¨auser, 2007. [2] R. Bambang, E. Shimemura and K. Uchjida, ”Mixed H2 /H∞ control with pole placement. State feedback case,” Proc. 2009 American Control Conf., San Francisco, CA, USA, pp. 2777-2779, 1993. [3] M. Chilali and P. Gahinet, ”H∞ design with pole placement constraints. An LMI Approch”, IEEE Tran. Automatic Control, vol. 41, no. 3, pp. 358-361, 1996. [4] J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, ”Statespace solutions to standard H2 and H∞ control problems”, IEEE Tran. Automatic Control, vol. 34, no. 8, pp. 831-847, 1989. [5] A. Filasov´a and D. Krokavec, ”H∞ control of discrete-time linear systems constrained in state by equality constraints,” Int. J. Applied Mathematics and Computer Science, vol. 22, no. 3, pp. 551-560, 2012. [6] B. A. Francis, A Course in H∞ Control Theory, Berlin, Germany: Springer-Verlag, 1987. [7] J. C. Geromel, P. L. D. Peres and S. R. Souza, ”Mixed H2 /H∞ control for continuous-time linear systems,” Proc. 31st Conf. Decision and Control, Tuscon, AZ, USA, pp. 3717-3722, 1992. [8] E. N. Goncalves, R. M. Palhares and R. H. C. Takahashi, ”Multiobjective optimization applied to robust H2 /H∞ state-feedback control synthesis,” Proc. 2004 American Control Conference, Boston, MA, USA, pp. 4619-4624, 2004. [9] W. M. Haddad and V. Chellaboina, Nonlinear Dynamical Systems and Control. A Lyapunov-Based Approach, Princeton, NJ, USA: Princeton University Press, 2008. [10] M. Hou and R. J. Patton, ”An LMI approach to H /H∞ fault detection observers,” Proc. UKACC Int. Conf. Control CONTROL’96, Exeter, England, pp. 305-10, 1996. [11] M. Kchaou, M. Souissi, and A. Toumi, ”Robust H∞ output feedback control with pole placement constraints for uncertain discrete-time fuzzy systems,” Soft Computing, vol. 16, no. 4, 13p, 2012. [12] H. K. Khalil, Nonlinear Systems, Englewood Cliffs, NJ, USA: Prentice Hall, 2002. [13] P. P. Khargonekar and M. A. Rotea, ”Mixed H2 /H∞ control. A Convex optimization approach,” IEEE Tran. Automatic Control, vol. 36, no. 7, pp. 824-831, 1991. [14] D. Krokavec and A. Filasov´a, ”On pole placement LMI constraints in control design for linear discrete-time systems,” Proc. 19th Int. Conf. ˇ Process Control PC’2013, Strbsk´ e Pleso, Slovakia, pp. 69-74, 2013. [15] D. Krokavec and A. Filasov´a, ”Discrete-time state feedback control design for linear systems,” Proc. 20th Int. Conf. Process Control ˇ PC’2015, Strbsk´ e Pleso, Slovakia, pp. 7-12, 2015. [16] M. Meisami-Azad, J. Mohammadpour, and K. M. Grigoriadis, ”Upper bound mixed H2 /H∞ control and integrated design for collocated structural systems,” Proc. 2009 American Control Conf., St. Louis, MO, USA, pp. 4563-4568, 2009. [17] K. Ogata, Discrete-Time Control Systems, Upper Saddle River, NJ, USA: Prentice-Hall, 1995. [18] M. C. Oliveira de, J. Bernussou and J. C. Geromel, ”A new discretetime robust stability condition,” Systems & Control Letters, vol. 37, no. 4, pp. 261-265, 1999. [19] Y. V. Orlov and L. T. Aguilar, Advanced H∞ Control. Towards Nonsmooth Theory and Applications, New York, NY, USA: Springer Science, 2014. [20] M. A. Rotea and P. P. Khargonekar ”H2 -optimal control with an H∞ constraint. The state feedback case,” Automatica, vol. 27, no. 2, pp. 307-316, 1991. [21] A. Schaft van der, L2 -Gain and Passivity Techniques in Nonlinear Control, London, UK: Springer-Verlag, 2000. [22] C. Scherer, ”Mixed H2 /H∞ control,” Trends in Control. A European Perspective, A. Isidori Ed. Berlin, Germany: Springer-Verlag, pp. 173216, 1995. [23] R. E. Skelton, T. Iwasaki, and K. Grigoriadis, A Unified Algebraic Approach to Linear Control Design, London, UK: Taylor & Francis, 1998. [24] A. A. Stoorvogel, The H∞ Control Problem. State-Space Approach, Englewood Cliffs, N.J., USA: Prentice-Hall, 1992. [25] J. G. Van Antwerp and R. D. Braatz, ”A tutorial on linear and bilinear matrix inequalities,” J. Process Control, vol. 10, pp. 363-385, 2000. [26] Q. G. Wang, Decoupling Control, Berlin, Germany: Springer-Verlag, 2003. [27] A. G. Wu and G. R. Duan, ”Enhanced LMI representations for H2 performance of polytopic uncertain systems. Continuous-time case,” Int. J. Automation and Computing, vol. 3, no. 3, pp. 304-308, 2006.

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Fig. 3. Closed-loop system output response - D/H∞

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Fig. 4. Closed-loop system output response - H2 /D/H∞

The trajectories of the output of this system are drawn for the system state initial vector q(0) = 0 and the desired steady state vector of the output variables wTo = [ 0.2 0.4 ]. The simulation results for H∞ and H2 /H∞ methodology as well as D/H∞ and H2 /D/H∞ principle are presented with respect to the closed-loop systems responses in the Fig. 1 Fig. 4. From the numerical results above it can see that by adding an additional H2 constraint, these new conditions can provide comparable H∞ -norm of the closed-loop disturbance transfer function but substantially decrease the value of H2 -norm of the closed-loop system transfer function. Consequently, the feasible solutions can be obtained in the same manner. It is natural that in terms of closed loop system dynamics, it is essential to define the pole cluster of the closed loop characteristic polynomial by using the D-stability region, also because it is linked to a common matrix of the Ljapunov function verifying the closed-loop stability. VII. C ONCLUDING REMARKS This paper modifies the use of a control design approach, destined for MIMO linear systems with disturbance attenuations by using γ2 and γ∞ norms of the closedloop discrete-time transfer function matrices, solving the interposed H2 /D/H∞ control design task. Using a originally constructed extended BRL with D-stability region parameters, the problems of H∞ and H2 /H∞ control are redefined and proven, documenting that the D-circle stability region means a new quadratic constraint on the Lyapunov function in the discrete system stability condition. The proposed control design method is linear and established as a set of LMIs utilizing quadratic constraints. This ISSN: 1998-0159

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