Discrete-time static H∞ loop shaping control via LMIs

Author’s Accepted Manuscript Discrete-time static H∞ loop shaping control via LMIs Renan L. Pereira, Karl H. Kienitz, Fernando H.D. Guaracy www.elsevier.com/locate/jfranklin

PII: DOI: Reference:

S0016-0032(17)30020-0 http://dx.doi.org/10.1016/j.jfranklin.2017.01.009 FI2865

To appear in: Journal of the Franklin Institute Received date: 3 June 2016 Revised date: 1 November 2016 Accepted date: 7 January 2017 Cite this article as: Renan L. Pereira, Karl H. Kienitz and Fernando H.D. Guaracy, Discrete-time static H∞ loop shaping control via LMIs, Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2017.01.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Discrete-time static H∞ loop shaping control via LMIs Renan L. Pereiraa,∗, Karl H. Kienitzb , Fernando H. D. Guaracya a UNIFEI

- Campus Itabira, Rua Irm˜ a Ivone Drumond, 200, Distrito Industrial II, 35903-087, Itabira - MG, Brazil b Instituto Tecnol´ ogico de Aeron´ autica, Pra¸ca Marechal Eduardo Gomes, 50, 12228-900, S˜ ao Jos´ e dos Campos - SP, Brazil

Abstract A new synthesis procedure is introduced for static H∞ loop shaping problem in discrete-time case. LMI conditions are derived to obtain a static output feedback controller that ensures the robust stability and performance of the closed-loop system. A numerical example illustrates the use of the proposed control method with application to a VTOL (“vertical take-off landing”) aircraft model. Keywords: Static output feedback; H∞ loop shaping; Discrete-time control; LMIs

1

1. Introduction

2

In the past years, static output feedback controllers have received increasing

3

attention due to the reduced effort for practical controller implementations. The

4

main drawback of this approach is the numerical complexity in the design stage

5

that leads to non-convex problems [1]. To deal with these problems, several

6

methods have been proposed. In [2, 3, 4], iterative algorithms are proposed

7

based on complex matrix equations or LMI optimization problems solved at

8

each step. Typically, such strategies may become a critical issue in large-scale

9

designs. In alternative approaches, the static output feedback controller design

10

is formulated in terms of a single LMI framework, resulting in more simplicity

11

than using iterative algorithms.

12

In particular, one static output feedback H∞ loop shaping controller ap-

13

proach proposed in [5] and generalized in a four-block framework in [6] has

Preprint submitted to Journal of The Franklin Institute

January 16, 2017

14

emerged successfully in recent years and has been applied in various contexts,

15

such as motor control, robot control and flight control. This is motivated mainly

16

by its flexibility to accomplish trade-offs in terms of performance requirements

17

and robustness to uncertainties [7, 8, 9]. The method consists basically in a

18

two-step design. In a first step, a loop shaping procedure is used to shape the

19

singular values of the nominal plant aiming to satisfy desired open-loop require-

20

ments at high and low frequencies [7, 10]. In this step, the engineer can use

21

his experience and knowledge to accomplish trade-offs in terms of requirements

22

such as attenuation of disturbance signals, attenuation of measurement noise,

23

and good reference tracking [10]. In a second step, an H∞ controller is designed

24

which guarantees stabilization with best possible robustness [9]. If the robust

25

stability and performance requirements are not satisfied, then the controller is

26

redesigned.

27

However, so far there has been no reference to a discrete-time version of this

28

approach. Typically, an output continuous controller has been designed and

29

then discretized using some available method (e.g. Tustin) for computer imple-

30

mentation. Nevertheless, it is known that the direct design of a discrete-time

31

controllers ensures better stability and performance requirements for the closed-

32

loop system than an indirect approach that relies on discretization methods [11].

33

Thus, the contribution of this paper, motivated by the results in [8, 9] consists

34

in extending their procedure for a discrete-time static H∞ loop shaping control.

35

Sufficient conditions for the existence of a solution to the static H∞ loop shap-

36

ing problem in discrete-time case are given in an LMI framework, which also

37

provides a Lyapunov matrix ensuring stability and robust performance. The

38

effectiveness of the design method was exemplified on the model of a VTOL

39

aircraft.

40

This text is organized as follows. Section 2 presents the discrete-time static

41

output feedback H∞ control problem. In Section 3 a solvability condition for

42

existence of a discrete-time static H∞ loop shaping controller is established and

43

a numerical example is presented. Finally, Section 4 contains the conclusion.

44

The notation used is standard: Rn×m denotes the set of real n × m matrices 2

47

and M > 0 (or M < 0) means and positive (or negative) ⎡ M is symmetric ⎤ B A ⎦ is used to denote a realization of definite. The notation G := ⎣ C D system transfer matrix G.

48

2. Discrete-time static output feedback H∞ control problem

45

46

Consider a strictly proper discrete-time linear time invariant system which maps exogenous inputs w and control inputs u to controlled outputs z and measured outputs y, ⎡ ⎢ ⎢ ⎢ ⎣

x(k + 1) z(k) y(k)

⎤

⎡

⎥ ⎥ ⎥=P ⎦

⎢ ⎥ ⎢ ⎥ ⎢ w(k) ⎥ ⎣ ⎦ u(k)

where the generalized plant P is given by ⎡ A B1 ⎢ ⎢ P := ⎢ C1 D11 ⎣ C2 D21

x(k)

B2

⎤ (1)

⎤

⎥ ⎥ D12 ⎥ ⎦ 0

(2)

49

with matrices A ∈ Rn×n , B2 ∈ Rn×m , C2 ∈ Rp×n . Moreover, it is assumed that

50

(A, B2 , C2 ) is stabilizable and detectable in order to ensure the sufficient con-

51

ditions that allow to stabilize the system by dynamic or static output feedback

52

control [12, 5, 6]. The H∞ performance for this system P is defined as follows. Definition 1 (H∞ performance). Suppose that the system P is stable. Then, its H∞ performance is defined as P ∞ =

z(k)2 with w(k)2 =0 w(k)2 sup

w(k) ∈ 2 and z(k) ∈ 2 .

(3)

53

Based on the Bounded Real Lemma [12, 13], an upper bound for the H∞

54

performance of the system P can be computed using an LMI framework, as

55

shown in the following lemma for discrete-time case.

3

Lemma 1. [12, 13] The system P has H∞ performance γ if and only if there exists an unique matrix X > 0 ⎛ −X −1 ⎜ ⎜ ⎜ A ⎜ ⎜ ⎜ C1 ⎝ 0

such that ⎞

AT

C1T

0

−X

0

B1

0

−γI

D11

B1T

T D11

−γI

⎟ ⎟ ⎟ ⎟ < 0. ⎟ ⎟ ⎠

(4)

56

The discrete-time static output feedback H∞ control problem consists in

57

determining a gain matrix K such that the output feedback law u(k) = Ky(k)

58

yields a closed-loop transfer function from w to z (denoted by Tzw ) with an H∞

59

norm smaller than γ (Figure 1). w

z

P y

u

K Figure 1: H∞ control problem.

60

For this class of discrete-time linear systems, the following lemma provides

61

necessary and sufficient conditions for the existence of a discrete-time static

62

output feedback H∞ controller. Lemma 2. [12] [13] Consider the generalized plant (2). There exists a discretetime static output feedback controller K such that Tzw ∞ ≤ γ, if and only if there exist R > 0 and S > 0 satisfying the following conditions: ⎛ ⎞ ⎛ ⎛ ⎞T ARAT − R ARC1T B1 ⎜ ⎟ 0 NR NR ⎟ ⎝ ⎠ ⎜ ⎜ C1 RAT −γI + C1 RC1T D11 ⎟ ⎝ ⎝ ⎠ 0 I 0 T B1T D11 −γI

⎞ 0 I

⎠<0 (5)

4

⎛ ⎝

NS 0

⎞T

⎛

⎜ ⎠ ⎜ ⎜ ⎝ I

0

AT SA − S

AT SB1

C1T

B1T SA

−γI + B1T SB1

T D11

C1

D11

−γI

⎞

⎛ ⎟ ⎟ ⎝ NS ⎟ ⎠ 0

⎞ 0 I

⎠<0 (6)

R = S −1 ,

(7)

63

where NR and NS denote bases of the null spaces of [B2T

T D12 ] and [C2

64

respectively.

65

3. Static H∞ loop shaping control for discrete-time systems

66

3.1. Problem statement

D21 ],

Consider a strictly proper discrete-time system G, having m inputs and p outputs with the following state-space realization ⎡ ⎤ Bn An ⎦ G := ⎣ Cn 0 and a post- and pre-compensator, respectively, given by ⎡ ⎡ ⎤ Bw2 Aw2 Aw1 ⎦ W1 := ⎣ W2 := ⎣ Cw2 Dw2 Cw1

(8)

Bw1 Dw1

⎤ ⎦

(9)

where An ∈ Rn×n , Aw1 ∈ Rnw1 ×nw1 , Aw2 ∈ Rnw2 ×nw2 and y, u are, respectively, the output and the input of the plant. Now, designate the state vectors of the post-, pre-compensator and nominal plant by xw2 , xw1 and x. Then, a minimal state-space representation of the shaped plant Gs = W2 GW1 is ⎤ ⎡ ⎤⎡ ⎡ An x (k) x (k + 1) 0 Bn Cw1 Bn Dw1 ⎥ ⎢ ⎥⎢ ⎢ ⎥ ⎢ ⎥⎢ ⎢ ⎥ ⎢ xw2 (k) ⎢ xw2 (k + 1) ⎥ ⎢ Bw2 Cn Aw2 0 0 ⎥=⎢ ⎥⎢ ⎢ ⎥ ⎢ ⎥⎢ ⎢ ⎢ xw1 (k + 1) ⎥ ⎢ Bw1 ⎥ ⎢ xw1 (k) 0 0 Aw1 ⎦ ⎣ ⎦⎣ ⎣ y(k) Dw2 Cn Cw2 0 0 u(k)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (10)

5

which will be represented in short notation by ⎡ ⎤ ⎡ ⎤⎡ ⎤ B xs (k + 1) A xs (k) ⎣ ⎦=⎣ ⎦⎣ ⎦ y (k) C 0 u (k) 

T

(11)

and A, B and C are the state-space matrices of

67

where xs =

68

the shaped plant. The choice of the pre- and post-compensators W1 and W2 has

69

been investigated in recent years [14, 15, 16, 17]. A naive approach is to ensure

70

that W1 guarantees high gain at low frequencies, roll-off rates of approximately

71

20 dB/decade at the desired bandwidth and choose W2 as a constant reflecting

72

the importance of the outputs to be controlled [18].

xT xTw2 xTw1

In the loop-shaping design procedure framework, the discrete-time shaped plant (which may include pre- and post-compensators) can be represented as   ˜ −1 N ˜ if and only if there exists V˜ , U ˜ ∈ H + (where H + denotes the Gs = M ∞ ∞ space of functions with all poles in the open unit disc of the z- plane) such that [8, 9] ˜ V˜ + N ˜U ˜ = I, M

(12)

˜ and N ˜ are the normalized left coprime factors of Gs given by where M ⎤ ⎡   A + HC B H ⎦. ˜ M ˜ := ⎣ N E −1 C 0 E −1

(13)

Matrix H is an observer gain given by H = −AQC T (I + CQC T )−1 and E is a symmetric matrix such that E T E = I + CQC T , where Q is the stabilizing solution of the discrete-time algebraic Riccati equation (DARE)[8, 9] AQAT − Q − AQC T (E T E)−1 CQAT + BB T = 0.

(14)

Taking into account these definitions, the discrete-time generalized shaped plant Ps can be written as [8, 9] ⎡ ⎢ ⎢ Ps = ⎢ ⎣

0

I

˜ −1 M

Gs

˜ −1 M

Gs

⎤

⎡

A

⎢ ⎢ ⎥ ⎢ 0 ⎥ ⎢ := ⎥ ⎢ ⎦ ⎢ C ⎣ C 6

−HE 0 E E

B

⎤

⎥ ⎥ I ⎥ ⎥. ⎥ 0 ⎥ ⎦ 0

(15)

˜ and ΔM ˜ in the coprime factors Assuming the additive uncertainties ΔN     ˜ ΔM ˜  ≤ 1/γ, the perturbed discrete-time of the plant such that  ΔN ∞

shaped plant model can be defined as [7, 19, 18, 9]   −1  ˜ + ΔN ˜ , ˜ + ΔM ˜ Gs (Δ) = M N

(16)

˜ and ΔM ˜ are stable and unknown transfer matrices. The discretewhere ΔN time static H∞ loop shaping control problem (Figure 2) consists in finding a controller K that stabilizes the closed-loop system satisfying Tzw ∞ ≤ γ, i.e., ⎡  ⎤    K  −1 ⎣ ˜ −1  < 1 = γ ⎦ (I − Gs K) M (17)   ε  I  ∞

73 74 75

where ε yields an upper bound for nonparametric uncertainties and γ is the T  H∞ norm from φ to uT y T [9]. The solution for this problem proposed herein is presented in the following section.

Figure 2: H∞ loop shaping control problem.

76

3.2. LMI conditions

77

Taking into account the generalized shaped plant Ps presented in (15), LMI

78

conditions are now provided for the static H∞ loop shaping problem in the

79

discrete-time case. Theorem 1. There exists a discrete-time static H∞ loop shaping controller K that satisfies (17), if γ > 1 and if there exists a solution R > 0 that satisfies the 7

following inequalities: ⎛ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎞

T

R

R(A + HC)

(A + HC) R

⎠>0

R

R + γBB T

AR

0

HE

RAT

R

RC T

0

0

CR

γI

ET H T

0

−E T

(18)

⎞

⎟ ⎟ ⎟ ⎟>0 ⎟ −E ⎟ ⎠ γI

(19)

Proof. Comparing equation (2)⎡ with (15) ⎤and choosing the base of the null   I ⎦, one obtains from LMI condition space of C2 , D21 as NS = ⎣ −E −1 C (6), the equivalent inequality  −1 C < 0. AT SA + AT SHC + C T H T SA + C T H T SHC − S − γC T EE T (20) Consider the condition (18), since S

−1

= R (static condition (7)) [12, 5, 6],

this condition can be rewritten as, ⎛ S −1 ⎝ (A + HC) S −1

S −1 (A + HC)

which is equivalent to ⎞⎛ ⎛ 0 S S −1 ⎠⎝ ⎝ −1 0 S S(A + HC)

(A + HC) S

T

S

−1

T

S

⎞ ⎠>0

⎞⎛ ⎠⎝

(21)

⎞

S −1

0

0

S −1

⎠>0. (22)

Using the Schur complement, (22) results in AT SA + AT SHC + C T H T SA + C T H T SHC − S < 0.

(23)

Thus (20) is indeed satisfied, concluding the first part of the proof. ⎡ Analogously, ⎤ I 0 ⎢ ⎥   ⎢ ⎥ T as NR = ⎢ −B 0 ⎥ and choosing the base of the null space of B2T , D12 ⎣ ⎦ 0 I 8

using LMI condition (5), ⎛ ARAT − R − γBB T ⎜ ⎜ ⎜ CRAT ⎝ −E T H T

ARC T

−HE

−γI + CRC T

E

ET

−γI

⎞ ⎟ ⎟ ⎟<0. ⎠

(24)

80

Now, using the Schur complement and a few additional mathematical ma-

81

nipulations, (24) can be rewritten as (19), concluding the proof of Theorem 1.

82

The condition expressed in the form of (19) can be advantageous over the use of

83

(24) for a class of linear systems with parametric uncertainties when scheduling

84

control is intended [20, 21, 22, 10].



86

Remark 1. In Theorem 1, the LMI conditions are only sufficient due to elimi −1 nation of the quadratic term −γC T EE T C. Thus, the sufficient conditions

87

presented for synthesis of a discrete-time static H∞ loop shaping controller may

88

be conservative.

89

3.3. Discrete-time static H∞ loop shaping controller synthesis

85

In Theorem 1, the solvability condition is defined in terms of the existence of a positive-definite matrix R > 0. If there exists a feasible solution for (18)-(19), the discrete-time H∞ control law u(k) = −Ky(k) is then calculated following the method described in [5, 6] with the closed-loop given by, ⎡ ⎡ ⎤ A − BKC −HE − BKE ⎢ Bcl Acl ⎢ ⎦=⎢ Tzw = ⎣ −KC −KE ⎣ Ccl Dcl C E

⎤ ⎥ ⎥ ⎥ . ⎦

(25)

90

Following the cited method and using (4), which ensures the internal stability

91

and the H∞ norm constraint, one may determine a discrete-time static H∞

92

controller solving the following LMI with matrix R > 0 obtained from (18) and

93

(19). Rewriting (4) accordingly, one gets ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

⎞

−R−1

ATcl

T Ccl

0

Acl

−R

0

Bcl

Ccl

0

−γI

Dcl

0

T Bcl

T Dcl

−γI

9

⎟ ⎟ ⎟ ⎟ < 0. ⎟ ⎟ ⎠

(26)

Equivalently [12, 5, 6] one may solve the following LMI for K, Ω − ΞK T Ψ − ΨT KΞT < 0 where matrices Ω, Ξ and Ψ are defined as ⎛ −R−1 AT 0 ⎜ ⎜ ⎜ A −R 0 ⎜ ⎜ Ω=⎜ 0 0 −γI ⎜ ⎜ ⎜ C 0 0 ⎝ 0 0 −E T H T ΞT =

Ψ= 94



 0

C

C

(27)

⎞

T

0

⎟ ⎟ −HE ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ E ⎟ ⎠ −γI

0 0 −γI ET

0

0

0

E

BT

I

0

0





.

(28a)

(28b)

(28c)

3.4. Design procedure

95

Finally, a control design procedure to obtain discrete-time static H∞ loop

96

shaping controller with guaranteed robustness properties can be summarized as

97

follows:

98

1. Initially select W1 and W2 to define the shaped plant Gs = W2 GW1 ; then

99

determine Q > 0 and calculate E to obtain the observer gain H from (14).

100

2. Solve LMIs (18) and (19) to obtain the solution R > 0 with the smallest

101

possible value of γ < 4 (resulting in maximum robust stability margin

102

[18]). If (18) and (19) are not feasible, redesign the dynamic weighting

103

matrix W1 until achieving feasibility. Methods for selecting W1 can be

104

found in [14] and [16], among others.

105 106

3. Use the resulting matrix R to find the discrete-time static output feedback H∞ controller K defined in (27).

10

4. Finally, the output feedback controller for implementation Ks is then given by Ks = W1 KW2 [7, 19, 18, 9] and can be written as ⎡ ⎤ ⎡ A xw1 (k + 1) BKCw2 Bw1 KDw2 ⎢ ⎥ ⎢ w1 ⎢ ⎥ ⎢ ⎢ xw2 (k + 1) ⎥ = ⎢ 0 Aw2 Bw2 ⎣ ⎦ ⎣ u(k) Cw1 Dw1 KCw2 Dw1 KDw2

⎤⎡

xw1 (k)

⎥⎢ ⎥⎢ ⎥ ⎢ xw2 (k) ⎦⎣ y(k)

⎤ ⎥ ⎥ ⎥ ⎦ (29)

or, equivalently, in short notation as ⎡ ⎤ ⎡ xST (k + 1) A ⎣ ⎦ = ⎣ KST u(k) CKST 107

BKST DKST

⎤⎡ ⎦⎣

xST (k) y(k)

⎤ ⎦ .

(30)

3.5. Numerical Example - VTOL Aircraft

108

To illustrate the effectiveness of the proposed method, consider the example

109

system discussed in [23] where a robust controller was designed in order to keep

110

the closed-loop system stable when subject to airspeed changes. Herein, the

111

model is slightly modified to address the present problem adopting a sampling

112

time Ts = 150[ms], resulting in the following discrete-time state-space matrices ⎡

0.9945 0.0037 −0.0023 −0.0694

⎤

⎥ ⎢ ⎥ ⎢ ⎢ 0.0065 0.8587 −0.0413 −0.5628 ⎥ ⎥ ⎢ A=⎢ ⎥ ⎢ 0.0145 0.0488 0.9136 0.1873 ⎥ ⎦ ⎣ 0.0011 0.0038 0.1431 1.0147 ⎡ ⎤T 0.0680 0.7834 −0.7680 −0.0590 ⎦ B=⎣ 0.0239 −1.0669 0.6851 0.0529   C= 0 1 0 0 . 113 114 115 116 117 118

(31)

Moreover, no pre- and post-compensators are used (i.e. W1 = W2 = I). Solving LMI conditions (18), (19) and (27), the discrete-time static H∞ loop  T shaping controller obtained was K = 0.5380 −0.6537 with a robustness margin γ = 2.0959. This indicates that closed-loop stability is ensured   ˜ ΔM ˜ as the normalized coprime factors uncertainty satisfies  ΔN 0.4771.

11

as long    ≤ ∞

Continuous−time Discrete−time

1.2 1.2 1.1

1

Amplitude

Amplitude

1.1

0.9

1

0.9

0.8 0.8 0.7 0.7 0.6

0

20

40 60 Samples (a)

80

0

100

1000

2000 3000 Samples (b)

4000

5000

Figure 3: a) Step response for the discrete-time controller - Ts = 150[ms]. b)Step response for both discrete- and continuous-time controllers - Ts = 2[ms].

119

As the proposed method consists basically in using a static gain, such ap-

120

proach can be compared with the procedure proposed in [5] for the continuous-

121 122

time case. Thus, solving the LMI conditions provided in [5] the static H∞ loop  T shaping controller is K = 1.1137 −1.3326 ensuring a robustness margin

123

γ = 2.0041. The use of this gain in a discrete-time loop with large sample period,

124

e.g. 150[ms], results in an unstable system (as one would possibly expect), while

125

the discrete-time solution obtained with the method proposed herein yields the

126

step response shown in Fig. 3a. When the sample period is reduced, e.g. to

127

Ts = 2[ms], the calculated discrete-time static gain approaches the continuous-

128

time gain and both step responses become close, as one would expect and as

129

shown in Fig. 3b. The example illustrates the use of the method while serving

130

as a basic check and comparison with the continuous-time counterpart method.

131

4. Conclusion

132

In this paper a novel method for static H∞ loop shaping control problem

133

for discrete-time systems has been presented. The method is based on a new 12

134

sufficient condition which may be efficiently solved using an LMI solver. In

135

practical terms, some adjustment of the dynamic weighting matrix W1 may be

136

necessary to achieve good performance and avoid inappropriate control input

137

amplitudes. In future work, we intend to expand this method to the design

138

of static gain-scheduling controllers for discrete-time systems. Also, we intend

139

to investigate the possibility of expanding the current results for systems with

140

time-delays or fuzzy model descriptions, thus generalizing results such as those

141

given in [24] and [25].

142

Acknowledgement

143

The authors acknowledge the support provided by CAPES (grant 88887.092490

144

/2015-00) and FAPESP (grant 2011/17610-0) and thank the anonymous review-

145

ers, whose suggestions helped improving this paper.

146

References

147 148

[1] V. L. Syrmos, C. T. Abdallah, P. Dorato, K. Grigoriadis, Static output feedback - a survey, Automatica Vol. 33, No. 02 (1997) 125 – 137.

149

[2] D. Moerder, A. Calise, Convergence of a numerical algorithm for calculating

150

optimal output feedback gains, IEEE Transactions on Automatic Control

151

Vol. 30, No. 09 (1985) 900–903.

152 153

[3] Y. Cao, J. Lam, Y. Sun, Static output feedback stabilization: an ILMI approach, Automatica Vol. 34, No. 12 (1998) 1641–1645.

154

[4] J. Gadewadikar, F. Lewis, L. Xie, V. Kucera, M.Abu-Khalaf, Parameter-

155

ization of all stabilizing H∞ static state-feedback gains: application to

156

output-feedback design, Automatica Vol. 43, No. 09 (2007) 1597–1604.

157

[5] E. Prempain, I. Postlethwaite, Static H∞ loop shaping control of a fly-by-

158

wire helicopter, Automatica 41 (2005) 1517–1528.

13

159

[6] S. Patra, S. Sen, G. Ray, Design of static H∞ loop shaping controller

160

in four-block framework using LMI approach, Automatica Vol. 44 (2008)

161

2214–2220.

162

[7] D. McFarlane, K. Glover, A loop shaping design procedure using H∞ syn-

163

thesis, IEEE Transactions on Automatic Control Vol. 37, No. 6 (1992)

164

759–769.

165

[8] D.-W. Gu, P. Petkov, M. Konstantinov, Formulae for discrete H∞ loop

166

shaping design procedure controllers, in: 15th Triennial World Congress,

167

2002.

168 169

[9] D.-W. Gu, P. H. Petkov, M. M. Konstantinov, Robust control design with MATLAB, Springer, 2005.

170

[10] R. Pereira, K. Kienitz, Design of gain-scheduled controllers based on para-

171

metric H∞ loop shaping, International Journal of Modelling, Identification

172

and Control Vol. 23 (2015) 77–84.

173

[11] F. H. Guaracy, L. H. Ferreira, C. A. Pinheiro, The discrete-time controller

174

for H∞ /LTR problem with mixed-sensitivity properties, Automatica Vol.

175

58 (2015) 28–31.

176

[12] P. Gahinet, P. Apkarian, A linear matrix inequality approach to H∞ con-

177

trol, International Journal of Robust and Nonlinear Control Vol. 4 (1994)

178

421–448.

179 180

[13] P. Gahinet, Explicit controller formulas for LMI based H∞ synthesis, Automatica Vol. 32 (1996) 1007–1014.

181

[14] A. Almeida, A. Filho, Algorithmic design for a robust control benchmark

182

problem, in: Proc. 21 st Brazilian Congress of Mechanical Engineering,

183

2011.

184

[15] M. Osinuga, S. Patra, A. Lanzon, Smooth weight optimization in H∞ loop-

185

shaping design, Systems & Control Letters Vol. 59, No. 11 (2010) 663–670. 14

186

[16] S. Patra, S. Sen, G. Ray, Pre-compensator selection for H∞ loop-shaping

187

control, International Journal of Control, Automation and Systems Vol. 08,

188

No. 1 (2010) 45–51.

189 190

191 192

[17] A. Lanzon, Weight optimisation in H∞ loop-shaping, Automatica Vol. 41, No. 7 (2005) 1201–1208. [18] Skogestad, I. Postlewaite, Multivariable feedback control: Analysis and Design, New York: Wiley, 2005.

193

[19] R. A. Hyde, K. Glover, The application of scheduled H∞ controllers to a

194

VSTOL aircraft, IEEE Transactions on Automatic Control Vol. 38, No. 07

195

(1993) 1021–1039.

196

[20] P. Apkarian, P. Gahinet, A convex characterization of gain-scheduled H∞

197

controllers, IEEE Transactions on Automatic Control Vol. 40, No. 5 (1995)

198

853–864.

199

[21] P. Apkarian, P. Gahinet, G. Becker, Self-scheduled H∞ control of linear

200

parameter-varying systems: a design example, Automatica Vol. 31, No. 09

201

(1995) 1251–1261.

202

[22] R. Apkarian, P; Adams, Advanced gain-scheduling techniques for uncertain

203

systems, IEEE Transactions on Control Systems Technology Vol. 6, No. 1

204

(1998) 21–32.

205

[23] L. H. Keel, S. P. Bhattacharyya, J. W. Howze, Robust control with struc-

206

ture perturbations, IEEE Transactions on Automatic Control Vol. 33, No.

207

01 (1988) 68–78.

208

[24] J. Qiu, Y. Wei, H. R. Karimi, New approach to delay-dependent H∞ control

209

for continuous-time Markovian jump systems with time-varying delay and

210

deficient transition descriptions, Journal of the Franklin Institute Vol. 352

211

(2015) 189–215.

15

212

[25] J. Qiu, S. X. Ding, H. Gao, S. Yin, Fuzzy-model-based reliable static output

213

feedback H∞ control of nonlinear hyperbolic PDE systems, IEEE Trans-

214

actions on Fuzzy Systems Vol. 24 (2016) 388–400.

16