On the Design of Sliding-Mode Static-Output-Feedback Controllers for ...

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 9, SEPTEMBER 2009

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On the Design of Sliding-Mode Static-Output-Feedback Controllers for Systems With State Delay

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X. R. Han, Emilia Fridman, Sarah K. Spurgeon, and Chris Edwards

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Abstract—This paper considers the development of slidingmode-based output-feedback controllers for uncertain systems which are subject to time-varying state delays. A novel method is proposed for design of the switching surface. This method is based on the descriptor approach and leads to a solution in terms of linear matrix inequalities (LMIs). When compared to existing methods (even for systems without delays), the proposed method is efficient and less conservative than other results, giving a feasible solution when the Kimura–Davison conditions are not satisfied. No additional constraints are imposed on the dimensions or structure of the reduced order triple associated with design of the switching surface. The magnitude of the linear gain used to construct the controller is also verified as an appropriate solution to the reachability problem using LMIs. A stability analysis for the full-order time-delay system with discontinuous right-hand side is formulated. This paper facilitates the constructive design of sliding-mode static-output-feedback controllers for a rather general class of time-delay systems. A numerical example from the literature illustrates the efficiency of the proposed method.

24 Index Terms—Linear matrix inequalities (LMIs), sliding-mode 25 control (SMC), static output feedback (SOF), time delay.

I. I NTRODUCTION

26

S

LIDING-MODE control (SMC) [1] is known for its complete robustness to so-called matched uncertainties (which 29 can include time delays that satisfy matching conditions) and 30 disturbances [2]–[4]. The control technique has been applied in 31 many industrial areas [5]–[7]. Many early theoretical develop32 ments in SMC assume that all the system states are accessible. 33 In the case where only a subset of states are measurable, which 34 is relevant to a range of practical applications, either output 35 feedback control or the observer-based method are required. 36 Some work has considered implementation of SMC schemes 37 using observers [8]–[10]. In [11], a sliding-mode observer has 38 been shown to give a significant increase in performance in esti39 mation of the unknown variables of a boost converter compared

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to a traditional current-mode control strategy. A further inter- 40 esting strand considers the fast output sampling method [12], 41 [13]. Recently, in [14], a fast sampling method is employed 42 for a discrete systems in the presence of time-varying delays 43 where a sliding-mode controller is designed using linear matrix 44 inequalities (LMIs) combined with a delta-operator approach. 45 However, all these methods require additional computation. 46 The most straightforward approach is to consider the study of 47 SMC via static output feedback (SOF). 48 One problem of interest in the development of SMC via SOF 49 is the design of the switching surface, which is effectively a 50 reduced order SOF problem for a particular subsystem. Two 51 different methods were proposed to design the sliding surface 52 using eigenvalue assignment and eigenvector techniques in [15] 53 and [16]. A canonical form was provided in [18] via which 54 the SOFSMC design problem is routinely converted to an SOF 55 stabilization problem. As stated in [20], all previous-reported 56 methods for the existence problem are, in fact, equivalent to 57 a particular SOF problem. The solution to the general SOF 58 problem, even for linear time-invariant systems, is still open. 59 LMI methods have been considered within the context of 60 sliding-mode controller design. For example, [21] and [22] 61 presented LMIs methods to design static sliding-mode output- 62 feedback controllers and [23] presented a necessary and suffi- 63 cient condition to solve the existence problem in terms of LMIs 64 for linear uncertain systems. 65 It is important to note that all the work described above 66 does not consider an existence problem involving delay and 67 many practical problems include such effects [24], [25]. In [26], 68 the problem of the development of sliding-mode controllers 69 for operation in the presence of single or multiple, constant 70 or time-varying state delays has been solved. This uses the 71 usual regular form method of solution and the uncertainty is 72 assumed to be matched, where matched describes that uncer- 73 tainty class which is implicit in the range of the input channels 74 and will be rejected by an appropriately designed SMC strategy, 75 although it is important to note that full state availability is 76 assumed. This problem has also been considered in [27] where 77 a class of uncertain time delay systems with multiple fixed 78 delays in the system states is considered. This paper considers 79 unmatched and time-varying parameter uncertainties, together 80 with matched and bounded external disturbances, but again, full 81 state information is assumed to be available to the controller. In 82 [28], Lyapunov functionals were for the first time introduced 83 for the analysis of time-varying delay. In [29], a descriptor 84

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5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Manuscript received October 8, 2008; revised May 13, 2009. This work was supported by the Engineering and Physical Sciences Research Council, U.K., under Grant EP/E 02076311. X. R. Han and S. K. Spurgeon are with the Department of Electronics, University of Kent, Canterbury, CT2 7NZ, U.K. (e-mail: [email protected]; [email protected]). E. Fridman is with the School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel (e-mail: [email protected]). C. Edwards is with the Department of Engineering, University of Leicester, Leicester, LE1 7RH U.K. (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2009.2023635

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0278-0046/$25.00 © 2009 IEEE

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 9, SEPTEMBER 2009

approach to stability and control of linear systems with time86 varying delays, which is based on the Lyapunov–Krasovskii 87 techniques, was combined with results on the SMC of such 88 systems. The systems under consideration were subjected to 89 norm-bounded uncertainties and uncertain bounded delays and 90 the solution given in terms of LMIs. Reference [30] develops 91 a SMC synthesis for a class of uncertain time-delay systems 92 with nonlinear disturbances and unknown delay values whose 93 unperturbed dynamics is linear. The synthesis was based on a 94 new delay-dependent stability criterion. The controller is found 95 to be robust against sufficiently small delay variations and 96 external disturbances. 97 It is important to emphasize that much of the aforementioned 98 literature on SMC of time-delay systems assumes full-state 99 feedback. Reference [31] considered SMC of systems with 100 time-varying delay. This paper considered a solution via LMIs 101 for the existence problem. The current paper extends this contri102 bution to consider the solution of the existence and reachability 103 problems. Specifically, the selection of parameters for stability 104 of the full-order closed-loop system are obtained via LMIs. In 105 this paper, a compensator-based design problem is considered 106 using the proposed SOF approach. Example from the literature 107 illustrates the efficiency of the method. In Section II, the 108 problem formulation is described. The existence problem is 109 formulated in Section III. In Sections IV and V, stability of 110 the full-order closed-loop system is derived via LMIs, and the 111 reachability problem is presented. Compensator-based design 112 is demonstrated in Section VI. 113 114

II. P ROBLEM F ORMULATION Consider an uncertain time-delay system

z(t) ˙ = Az(t) + Ad z (t − τ (t)) + B (u(t) + ξ(t, z, u)) y(t) = Cz(t)

(1)

where z ∈ Rn , u ∈ Rm , and y ∈ Rp with m ≤ p ≤ n. The time-varying delay τ (t) is supposed to be bounded 0 ≤ τ (t) ≤ 117 h, and it may be either slowly varying (i.e., differentiable 118 delay with τ˙ (t) ≤ d < 1) or fast varying (piecewise continuous 119 delay). Assume that the nominal linear system (A, Ad , B, C) 120 is known and that the input and output matrices B and C are 121 both of full rank. The unknown function ξ: R+ × Rn × Rm → 122 Rn , which represents the system nonlinearities plus any model 123 uncertainties, is assumed to satisfy the matching condition and 115 116

ξ(t, z, u) < k1 u + a(t, y)

(2)

for some known function a : R+ × Rp → R+ and positive 125 constant k1 < 1. It can be shown that if rank(CB) = m, there 126 exists a coordinate system in which the system (A, Ad , B, C) 127 has the structure

  A11 A12 Ad11 Ad12 A= Ad = A21 A22 Ad21 Ad22
 0 B= C = [0 T ] (3) B2 124

where B2 ∈ Rm×m is nonsingular and T ∈ Rp×p is orthogo- 128 nal. The system can be represented as 129 z˙1 (t) = A11 z1 (t) + Ad11 z1 (t − τ (t)) + A12 z2 (t) + Ad12 z2 (t − τ (t)) 2 

z˙2 (t) =

(A2i zi (t) + Ad2i zi (t − τ (t)))

i=1

+ B2 (u(t) + ξ(t, z, u)) y(t) = Cz(t).

(4)

Consider the following switching function:

130 AQ1

S = {z(t) ∈ Rn : F Cz(t) = 0}

(5)

for some selected matrix F ∈ Rm×p where by design 131 det(F CB) = 0. Let 132 [F1

F2 ] = F T

(6)

where F1 ∈ Rp−m and F2 ∈ Rm . As a result

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85

F C = [F1 C1

F2 ]

133

(7)

where

 C1 = 0(p−m)×(n−p)

 I(p−m) .

134

(8)

Therefore, F CB = F2 B2 and the square matrix F2 is 135 nonsingular. By assumption, the uncertainty is matched, and 136 therefore the sliding motion is independent of the uncertainty 137 represented by ξ(·). In addition, because the canonical form in 138 (3), where it is necessary that the pair (A11 , A12 ) is controllable 139 and (A11 , C1 ) is observable, can be viewed as a special case 140 of the regular form normally used in sliding-mode controller 141 design, the switching function can also be expressed as 142 s(t) = z2 (t) + KC1 z1 (t)

(9)

143 where K ∈ Rm×(p−m) and is defined as K = F2−1 F1 . Then, a simple SMC law, depending on the output informa- 144 tion F y(t) can be defined by 145

u(t) = −γF y(t) − v(t)

where



v(t) =

F y(t) ρ(t, y) F y(t) , 0,

if F y(t) = 0 otherwise

(10) 146

(11)

where ρ(t, y) is some positive scalar function of the outputs

147

ρ(t, y) = (k1 γ F y(t) + α(t, y) + γ2 ) /(1 − k1 ) where γ and γ2 are positive design scalars [18]. The closed- 148 loops system (4) and (10) can be described by the following 149 equations: 150 z˙1 (t) = (A11 − A12 KC1 )z1 (t) + (Ad11 − Ad12 KC1 )z1 (t − τ (t))

HAN et al.: ON THE DESIGN OF SLIDING-MODE SOF CONTROLLERS FOR SYSTEMS WITH STATE DELAY

s(t) ˙ = (A21 − γB2 KC1 )z1 (t)

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t−τ  (t)

1 z˙1T (s)Rz˙1 (s)ds ≥ h

+ (Ad21 − γB2 KC1 )z1 (t − τ (t)) + (A22 − γB2 )z2 (t) + (Ad22 − γB2 )z2 (t − τ (t))

t−h

t−τ  (t)

t−τ  (t)

z˙1T (s)dsR t−h

z˙1 (s)ds. t−h

(18)

+ B2 (ξ(t, z, u) − v(t)) y(t) = Cz(t).

(12)

Then,

162

V˙ (t) ≤ 2z1T (t)P z˙1T (t) + h2 z˙1T (t)Rz˙1 (t) − (z1 (t) − z1 (t − τ (t)))T R (z1 (t) − z1 (t − τ (t)))

151

III. E XISTENCE P ROBLEM

152 153 154

On the sliding manifold s(t) = 0, it is well known [19] that the reduced-order sliding motion is governed by a free motion with system matrix

× R (z1 (t − τ (t)) − z1 (t − h))

z˙1 (t) = (A11 − A12 KC1 )z1 (t)

− (1 − d)z1T (t − τ (t)) Sz1 (t − τ (t)) .

+(Ad11 − Ad12 KC1 )z1 (t − τ (t)) . (13) 155

Consider a Lyapunov–Krasovskii functional t z1T (s)Ez1 (s)ds

+

t−τ (t) 0 t

z˙1T (s)Rz˙1 (s)dsdθ

+h

Using the descriptor method as in [33] and the free-weighting 163 matrices technique from [34], the right-hand side of the 164 expression 165  0 ≡ 2 z1T (t)P2T + z˙1T (t)P3T +(Ad11 − Ad12 KC1 )z1 (t − τ (t))]

z1T (s)Sz1 (s)ds

+

(14)

V˙ (t) ≤ η T (t)Θη(t) ≤ 0

where the symmetric matrices P > 0 and E, S, R ≥ 0. The condition V˙ (t) < 0 guarantees asymptotic stability of the reduced order system as in [32]. Differentiating V (t) 159 along (13) 156

V˙ (t) = 2z1T (t)P z˙1 (t) + h2 z˙1T (t)Rz˙1 (t) t z˙1T (s)Rz˙1 (s)ds + z1T (t)(E + S)z1 (t) −h t−h z1T (t

− − h)Ez1 (t − h) − (1 − τ˙ (t)) z1T (t − τ (t)) Sz1 (t − τ (t)) . 160

Further using the identity t −h

(15)

θ12 = P − P2T + '(A11 − A12 KC1 )T P2

θ33 = − (E + R)

z˙1T (s)Rz˙1 (s)ds

(16)

t

t z˙1T (s)dsR

t−τ (t)

θ34 = R θ44 = − 2R − (1 − d)S.

and applying Jensen’s inequality

t−τ (t)

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θ24 = 'P2T (Ad11 − Ad12 KC1 )

t−τ (t)

1 ≥ h

(22)

θ22 = − 'P2 − 'P2T + h2 R

z˙1T (s)Rz˙1 (s)ds

−h

z˙1T (s)Rz˙1 (s)ds

θ14 θ24   0, E  ≥ 0, S ≥ 0, R  ≥ 0 and matrices Q22 ∈ 186 variables P 182 183

203 204

It can be shown [18] that there exist a coordinate system in 205 ¯ A¯d , B, ¯ F C) ¯ has the property that 206 which the system triple (A,

  A¯11 A¯12 A¯d11 A¯d12 ¯ ¯ Ad = ¯ A= ¯ A21 A¯22 Ad21 A¯d22
 ¯= 0 B (29) F C¯ = [0 F2 ] B2

¯ = C¯ T F T P¯ B

T δY ] + QT 2 A11

   − [Y M δY ]T AT 12 + E + S − R

−R  θ33 = − E

IV. S TABILITY OF THE F ULL -O RDER C LOSED -L OOP S YSTEM

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(25)

R(p−m)×(p−m) , Q11 ∈ R(n−p)×(n−p) , Q12 ∈ R(n−p)×(p−m) , 187 Y ∈ Rm×(p−m) . If a solution to (24) exists, which may be read- 188 ily obtained from available LMI tools, then the reduced order 189 system (13) is asymptotically stable for all differentiable delays 190 0 ≤ τ (t) ≤ h, τ˙ (t) ≤ d < 1. Moreover, (13) is asymptotically 191 stable for all piecewise-continuous delays 0 ≤ τ (t) ≤ h, if the 192 LMI (24) is feasible with S = 0. 193 Remark: The proposed method is suitable for SOF sliding- 194 mode controller design where Kimura–Davison conditions, 195 written as n ≤ m + p − 1, are not satisfied as in [20]–[22]. No 196 constraints are imposed on the dimensions of the reduced-order 197 triple A11 , A12 , C1 . This represents a constructive and efficient 198 approach to output-feedback-based design for a relatively broad 199 class of systems, which is less conservative than existing results 200 [20]–[22]; an example to illustrate the advantages of the method 201 for systems without time-delay is presented in [17]. 202

¯ (ξ(t, yt ) − vy (t)) . z(t) ˙ = A¯0 z(t) + A¯d z(t − τ ) + B

216

(34)

For large enough γ > 0, these conditions are delay independent 217 with respect to the delay in z2 .} However, for derivation of 218 this condition using Lyapunov–Krasovskii techniques, it is 219 necessary to consider the case where τ˙ ≤ d < 1. A stability 220

HAN et al.: ON THE DESIGN OF SLIDING-MODE SOF CONTROLLERS FOR SYSTEMS WITH STATE DELAY

condition for the full-order closed-loop system can be derived 222 using the following Lyapunov–Krasovskii functional: 221

t V (t) = z (t)P¯ z(t) +

¯ z T (s)Ez(s)ds

T

t−h

t

0 t ¯ z (s)Sz(s)ds +h T

+

¯ z(s)dsdθ z˙ T (s)R ˙

(35)

−h t+θ

t−τ (t)




 ¯ ¯ S¯ ≥ 0 and R ¯ = R1 0 ≥ 0 as it is 223 where the matrix E, 0 0 224 desired to determine a stability condition for the time delay 225 system which is delay independent with respect to delay in 226 z2 (t). Differentiating V (t) along the closed-loop trajectories ¯ z(t) V˙ (t) ≤ 2z T (t)P¯ z˙ T (t) + h2 z˙ T (t)R ˙ T ¯ − (z(t) − z (t − τ (t))) R (z(t) − (t − τ (t)))

¯ z(t) is satisfied if ς T (t)Φh ς(t) + h2 z˙ T (t)R ˙ 0, E such that LMI (41) holds. Given that the design parameters k1 , α(t, y), γ2 , and P¯2 have been selected so that (44) holds, the closed-loop system (33) is asymptotically stable for all differentiable delays 0 ≤ τ (t) ≤ h, τ˙ (t) ≤ d ≤ 1. V. F INITE -T IME R EACHABILITY TO THE S LIDING M ANIFOLD

248 249

(44)

244 245 246 247

Corollary: An ideal sliding motion takes place on the sur- 250 face S if 251   −1 L   −1 L B A¯0 z(t) + B A¯d z(t − τ ) < γ2 − η (45) 2 2

 ¯ (ξ(t, z, u) − v(t)) × A¯0 z + A¯d z(t − τ ) + B  ¯T  A0
  0n  I(n−m) 2 ¯ T = 1 (t)  ¯ T  h R1 Ad 0 T ¯ B T

However, by definition

V˙ (t) < −2γ2 F y(t) < 0 if z(t) = 0

T ¯ ¯ R + ξ(t, z, u) − v(t)T B 

I(n−qm) 0

(42)

(38)

¯ z(t) h2 z˙ T (t)R ˙  T = h2 z T (t)A¯0 + z T (t − τ )A¯d

×

< −2 F y(t) (ρ(t, y) − k1 u(t) − α(t, y)) .

From (37), if (41) is valid, then from (42) and (43),

Setting 1(t) = col{z(t), z(t − h), z(t − τ ), ξ(t, z, u) − v(t)}




= 2y T F T (ξ(t, z, u) − v(t))

ρ(t, y) = k1 ρ(t, y) + k1 γ F y(t) + α(t, y) + γ2

¯ (ξ(t, z, u) − v(t)) < 0 (37) + 2z T P¯ B

232

¯ (ξ(t, z, u) − v(t)) 2z T P¯ B

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Substitute the right-hand side of (34) into (36). Setting ς(t) = col{z(t), z(t − h), z(t − τ (t))}, it follows that ¯ z(t) V˙ (t) ≤ ς(t)T Φh ς(t) + h2 z˙ T (t)R ˙

231

¯ ≥ 235 Inequality (41) is an LMI in the decision variables P¯1 > 0, E ¯ ¯ 0, S ≥ 0 and R1 ≥ 0. Equation (37) is valid if (41) is satisfied 236 and given 237

= −2ρ(t, y) F y(t) + 2 F y(t) ξ(t, z, u)

¯ + S)z(t) ¯ ¯ − z T (t − h)Ez(t − h) + z T (t)(E T ¯ (t − τ (t)) . (36) − (1 − d)z (t − τ (t)) Sz

229 230

¯ z(t) ˙ < 0 233 Using the Schur complement, ξ T (t)Φh ξ(t)+h2 z˙ T (t)R holds if 234
   I(n−m) ¯ R1 hA¯T 0 0     0
(n,n−m)   Φh  (41)   < 0. I T  (n−m) ¯1  R hA¯d   0 ¯1 ∗∗∗ −R

≤ −2y T F T v(t) + 2 F y(t) ξ(t, z, u)

− (z (t − τ (t)) − z(t − h))T ¯ (z (t − τ (t)) − z(t − h)) ×R

227 228

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¯L where the matrices A¯L 0 and Ad represent the last m rows of 252 ¯ ¯ A0 and Ad , respectively, and η is a small scalar satisfying 253 0 < η < γ2 . 254 Proof: 255 (40)

s(t) ˙ = F C¯ A¯0 z¯(t) + F C¯ A¯d z(t−τ )+F2 B2 (ξ(t, z, u)−v(t)) . (46)

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 9, SEPTEMBER 2009

Let Vc : Rm → R be defined by T  Vc (s) = sT (t) F2−1 P¯2 F2−1 s(t).

257

Then, using the fact that F2T = P¯2 B2 it follows that 

T



T

F2−1

258

(47)

F2−1

P¯2 F2−1 F C¯ A¯0 = B2−1 A¯L 0 P¯2 F2−1 F C¯ A¯d = B2−1 A¯L d.

(48)

Then, it can be verified that T −1 ¯L V˙ c = 2sT (t)B2−1 A¯L 0 z(t) + 2s (t)B2 Ad z(t − τ )

+ 2sT (t) (ξ(t, z, u) − v(t))    ≤ 2 s(t) B2−1 A¯L 0 z(t) + 2 s(t)

Fig. 1.

Output against time.

< − 2η s(t)

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   × B2−1 A¯L d z(t − τ ) − 2γ2 s(t) (49)

if z(t) and z(t − τ ) ∈ Ω. It follows that there exists a t0 such that z(t) and z(t − τ ) ∈ Ω for all t > t0 . Consequently, (49) 261 holds for all t > t0 . A sliding motion will thus be attained in 262 finite time. 263 Example 1: The following model of a liquid monopropel264 lant rocket motor has been considered in [35]. It is assumed 265 that the variable κ = 0.8 in this case, where Ad (1, 1) = −κ 266 and A(1, 1) = κ − 1. The outputs have been chosen to be the 267 second and fourth states so that in (1)

259 260



−0.2  0 A= −1 0   0 1 B=  0 0

0 0 0 0 0 −1 1 1
C=

 0 −1   1 0



−0.8  0 Ad =  0 0

0 1 0 0 0 0 0 0

 0 0  0 0



0 1 0 0 . 0 0 0 1

(50)

Clearly, the Kimura–Davison conditions are not met. Here, the rate at which the delay varies with time has been examined with d = 0, a slow varying delay. The gain from the LMI tool ¯2 = 1, γ = 2.9, and 271 solver with δ = 50, ' = 0.5, h = 0.45s, P 272 M = [5 0.2], yields F = [−1 − 2.0754]. The poles of the 273 sliding-mode dynamics are {−2.67, −0.4, −0.2}. A simu274 lation was performed with the initial state values [1 1 1 1]. 275 As shown, the LMI solver gave a feasible result for stability 276 for h ≤ 0.45s, but in simulation, the closed-loop system only 277 became unstable for h ≥ 1s (Fig. 1); this is due to the conserv278 ativeness of the method. The LMI solver gave feasible closed279 loop stability results for controller gain γ ≥ 2.9 while a choice 280 of γ ≥ 1 in simulation was able to stabilize the system with the 281 compensation of longer settling time. The switching function 282 for h = 0.45s, γ = 2.9 is shown in Fig. 2. 268 269 270

Fig. 2.

Switching function.

VI. C OMPENSATOR -BASED E XISTENCE P ROBLEM

283

For certain system triples (A11 , A12 , C1 ), LMI (24) is known 284 to be infeasible. In this case, consider a dynamic compensator 285 similar to that of El-Khazali and DeCarlo [36] 286 z˙c (t) = Hzc (t) + Dy(t)

(51)

where the matrices H ∈ Rq×q and D ∈ Rq×p are to be deter- 287 mined. Define a new hyperplane in the augmented state space, 288 formed from the plant and compensator state spaces, as 289   Sc = (z(t), zc (t)) ∈ Rn+q : Fc zc (t) + F Cz(t) = 0 (52)

where Fc ∈ Rm×q and F ∈ Rm×p . Define D1 ∈ Rq×(p−m) 290 and D2 ∈ Rq×m as 291 [D1

D2 ] = DT

(53)

then the compensator can be written as z˙c (t) = Hzc (t) + D1 C1 z1 (t) + D2 z2 (t)

292

(54)

HAN et al.: ON THE DESIGN OF SLIDING-MODE SOF CONTROLLERS FOR SYSTEMS WITH STATE DELAY

293 294

7

where C1 is defined in (8). The sliding motion, obtained by eliminating the coordinates z2 (t), can be written as z˙1 (t) = (A11 − A12 KC1 )z1 (t) − A12 Kc zc (t) + (Ad11 − Ad12 KC1 )z1 (t − τ ) − Ad12 Kc zc (t − τ ) z˙c (t) = (D1 − D2 K)C1 z1 (t) + (H − D2 Kc )zc (t)

(55)

where K = F2−1 F1 and Kc = F2−1 Fc , then similar to [37], the 296 design problem becomes one of selecting a compensator, re297 presented by the matrices D1 , D2 , and H, and a hyperplane, 298 represented by the matrices K and Kc , so that the system


 z˙1 (t) z1 (t) A11 − A12 KC1 −A12 Kc = (D1 − D2 K)C1 H − D2 Kc z˙c (t) zc (t)    295

Ac




Ad11 − Ad12 KC1 + 0  

−Ad12 Kc 0




z1 (t − τ ) zc (t − τ )



It follows that (56)

is stable. To obtain the compensator gains this problem can be shown to be a new output-feedback problem with

 

A11 0 A12 K Kc C1 0 0 Ac = − D1 H 0 0 D2 −Iq 0 Iq           
Acd =

Aq




Bq

Kq

Aqd




Bqd




Cq

 Ad11 0 Ad12 0 K Kc C1 0 − . D1 H 0 0 0 0 0 Iq            

Kq

Cq

(57)

The existence problem represented by system (56), where Ac and Acd are partitioned as in (57) and D2 is a tuning parameter, 303 can be solved as for the noncompensated case (13). Similarly 304 to (27), 
  Q11 Q12 Kq Cq Q2 = Kq 0(p−m+q)×(n−p) Ip−m+q Q22 M δQ22

301 302

= [ Kq Q22 M = [Y M

δY ]

δKq Q22 ]

(58)

where Y = Kq Q22 , M ∈ R(p−m+q)×(n−p) is a tuning matrix. 306 Example 2: Consider the delay system     0 25 −1 0.1 0.1 0 A= 1 0 0  Ad =  0 0.3 −0.1  −5 0 1 0 0.2 0  
 0 0 1 0 B = 0 C = (59) 0 0 1 1

305

307

from which
A11 =

0 1

25 0

308

 λ(A11 − A12 KC1 ) = ± (25 + K 2 ) and so (24) is infeasible. Now, consider designing a first-order 309 compensator. Choosing D2 = 1, it follows that 310     0 25 0 0.1 0.1 0 Aq =  1 0 0  Aqd =  0 0.3 0  0 0 0 0 0 0     −1 0 −0.1 0 Bq =  0 0  Bqd =  0 0 1 −1 0 0
 0 1 0 Cq = . 0 0 1

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Acd

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Fig. 3. Compensator-based controller design h = 2.5s.




A12 =

−1 0

Choosing δ = 50, ' = 5, M = [10 4] , and d = 0 (slowly 311 varying delay) with the maximum allowable delay h = 0.25s, 312 the LMITOOL solver returns 313
 −25.38 −0.37 Kq = . −25.32 −5.03

The augmented system with compensator given by

 H DC 0 0 Aa = Ada = 0 A 0 Ad

 0 I 0 Ba = Ca = q B 0 C

314

is asymptotically stablized by the controller

315

[Fc

F ] = [−0.369

− 25.38

Taking the controller in the form of (10) and (11) where γ = 10, 316 ρ = 10. Simulation results with the switching gain and initial 317 conditions [0.5, 0, 0], as shown in Figs. 3 and 4. 318 VII. C ONCLUSION

 C1 = [1

0].

1].

319

A descriptor Lyapunov–Krasovskii functional method has 320 been introduced for SOF switching function design for systems 321

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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 56, NO. 9, SEPTEMBER 2009

Fig. 4. Compensator-based controller design h = 2.5s.

with state time-varying delays. The delay is assumed bounded with a known upper bound, either slowly or fast varying. In addition, a novel stability analysis of the full-order closed325 loop discontinuous time-delay system has been performed via 326 the Krasovskii method, which is delay independent in z2 (t) 327 (and thus the delay is restricted to be slowly varying) and 328 delay dependent in z1 (t), i.e., in the state of the reduced-order 329 system. The proposed SOF design approach also applies to 330 compensator-based design. Examples show the effectiveness of 331 the method. For future work, the results can be extended to the 332 interval delay case, where the lower bound on the delay is taken 333 into account. The Razumikin approach can be employed for the 334 stability analysis of the full-order closed-loop system with fast 335 varying delay.

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[13] H. Saaj, C. M. Bandyopadhyay, and B. Unbehauen, “A minor correction to ’a new algorithm for discrete-time sliding mode control using fast output sampling feedback’,” IEEE Trans. Ind. Electron., vol. 51, no. 1, pp. 244–247, Feb. 2004. [14] Y. Q Xia, M. Y Fu, H. J. Yang, and C. P. Liu, “Robust sliding mode control for uncertain time-delay systems based on delta operator,” IEEE Trans. Ind. Electron., vol. 56, 2009, to be published. [15] S. H. Zak and S. Hui, “On variable structure output feedback controllers for uncertain dynamic systems,” IEEE Trans. Autom. Control, vol. 38, no. 10, pp. 1509–1512, Oct. 1993. [16] R. El-Khazali and R. A. Decarlo, “Output feedback variable structure controllers,” IEEE Trans. Autom. Control, vol. 31, pp. 805–816, 1995. [17] X. R. Han, E. Fridman, S. K. Spurgeon, and C. Edwards, “Sliding mode controllers using output information: An LMI approach,” in Proc. UKACC, 2008. [18] C. Edwards and S. K. Spurgeon, “Sliding mode stabilization of uncertain systems using only output information,” Int. J. Control, vol. 62, no. 5, pp. 1129–1144, 1995. [19] A. S. I. Zinober, “An introduction to sliding mode variable structure control,” in Variable Structure and Lyapunov Control. London, U.K.: Springer-Verlag, 1994. [20] C. Edwards, S. K. Spurgeon, and R. G. Hebden, “On the design of sliding mode output feedback controllers,” Int. J. Control, vol. 76, no. 9/10, pp. 893–905, 2003. [21] C. Edwards, S. K. Spurgeon, and A. Akoachere, “A sliding mode static output feedback controller based on linear matrix inequalities applied to an aircraft system,” Trans. ASME, J. Dyn. Syst. Meas. Control, vol. 122, no. 4, pp. 656–662, Dec. 2000. [22] C. Edwards, A. Akoachere, and S. K. Spurgeon, “Sliding-mode output feedback controller design using linear matrix inequalities,” IEEE Trans. Autom. Control, vol. 46, no. 1, pp. 15–19, Jan. 2001. [23] H. H. Choi, “Variable structure output feedback control for a class of uncertain dynamic systems,” Automatica, vol. 38, no. 2, pp. 335–341, Feb. 2002. [24] K. Natori and K. Ohnishi, “A design method of communication disturbance observer for time-delay compensation, taking the dynamic property of network disturbance into account,” IEEE Trans. Ind. Electron., vol. 55, no. 5, pp. 2152–2168, May 2008. [25] R. C. Luo and L.-Y. Chung, “Stabilization for linear uncertain system with time latency,” IEEE Trans. Ind. Electron., vol. 49, no. 4, pp. 905–910, Aug. 2002. [26] F. Gouaisbaut, M. Dambrine, and J. P. Richard, “Robust control of delay systems: A sliding mode control design via LMI,” Syst. Control Lett., vol. 46, no. 4, pp. 219–230, 2002. [27] X. Li and R. A. DeCarlo, “Robust sliding mode control of uncertain time delay systems,” Int. J. Control, vol. 76, no. 13, pp. 1296–1305, 2003. [28] E. Fridman and U. Shaked, “An improved stabilization method for linear time-delay systems,” IEEE Trans. Autom. Control, vol. 47, no. 11, pp. 1931–1937, Nov. 2002. [29] E. Fridman, F. Gouaisbaut, M. Dambrine, and J.-P. Richard, “Sliding mode control of systems with time-varying delays via descriptor approach,” Int. J. Syst. Sci., vol. 34, no. 8/9, pp. 553–559, 2003. [30] Y. Orlov, W. Perruquetti, and J. P. Richard, “Sliding mode control synthesis of uncertain time-delay systems,” Asian J. Control, vol. 5, no. 4, pp. 568–577, Dec. 2003. [31] X. R. Han, E. Fridman, S. K. Spurgeon, and C. Edwards, “On the design of sliding mode static output feedback controllers for systems with time-varying delay,” in Proc. Variable Structure Syst., 2008. [32] J. Hale and S. Lunel, Introduction to Functional Differential Equations. New York: Springer-Verlag, 1993. [33] E. Fridman, “New Lyapunov–Krasovskii functionals for stability of linear retarded and neutral type systems,” Syst. Control Lett., vol. 43, no. 4, pp. 233–240, 2001. [34] H. Y. Wang, Q.-G. Lin, and C. M. Wu, “Delay-range-dependent stability for systems with time-varying delay,” Automatica, vol. 43, no. 2, pp. 371–376, Feb. 2007. [35] Z. Feng, C. Mian, and G. Weibing, “Variable structure control of timedelay systems with a simulation study on stabilizing combustion in liquid propellant rocket motors,” Automatica, vol. 31, no. 7, pp. 1031–1037, Jul. 1995. [36] R. El-Khazali and R. A. DeCarlo, “Variable structure output feedback control,” in Proc. Amer. Control Conf., Chicago, IL, 1992, pp. 871–875. [37] C. Edwards and S. K. Spurgeon, “Linear matrix inequality methods for designing sliding mode output feedback controllers,” Proc. Inst. Elect. Eng., vol. 150, no. 5, pp. 539–545, Sep. 2003.

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R EFERENCES

337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369

[1] V. I. Utkin, “Variable structure systems with sliding modes,” IEEE Trans. Autom. Control, vol. AC-22, no. 2, pp. 212–222, Apr. 1977. [2] V. I. Utkin, Sliding Modes in Control and Optimization. New York: Springer-Verlag, 1992. [3] C. Edwards and S. K. Spurgeon, Sliding Mode Control—Theory and Applications. New York: Taylor & Francis, 1998. [4] J. Y. Hung, W. B. Gao, and J. C. Hung, “Variable structure control: A survey,” IEEE Trans. Ind. Electron., vol. 40, no. 1, pp. 2–22, Feb. 1993. [5] Y. Shtessel, S. Baev, and H. Biglari, “Unity power factor control in three-phase AC/DC boost converter using sliding modes,” IEEE Trans. Ind. Electron., vol. 55, no. 11, pp. 3874–3882, Nov. 2008. [6] Y. Pan, Ü. Özgüner, and O. H. Daˇgci, “Variable-structure control of electronic throttle valve,” IEEE Trans. Ind. Electron., vol. 55, no. 11, pp. 3899–3907, Nov. 2008. [7] M. Defoort, T. Floquet, A. Kökösy, and W. Perruquetti, “Sliding-mode formation control for cooperative autonomous mobile robots,” IEEE Trans. Ind. Electron., vol. 55, no. 11, pp. 3944–3953, Nov. 2008. [8] C. Edwards and S. K. Spurgeon, “Sliding mode observers for fault detection,” Automatica, vol. 36, no. 4, pp. 541–553, Apr. 2000. [9] K. C. Veluvolu and Y. C. Soh, “Discrete-time sliding mode state and unknown inputs estimations for nonlinear systems,” IEEE Trans. Ind. Electron., vol. 56, 2009, to be published. [10] S. M. N. Hasan and I. Husain, “A Luenberger-sliding mode observer for online parameter estimation and adaptation in high-performance induction motor drives,” IEEE Trans. Ind. Electron., vol. 45, no. 2, pp. 772–781, 2009. [11] M. Oettmeier, J. Neely, S. Pekarek, R. Decarlo, and K. Uthaichana, “MPC of switching in a boost converter using a hybrid state model with a sliding mode observer,” IEEE Trans. Ind. Electron., vol. 56, 2009, to be published. [12] S. Janardhanan and B. Bandyopadhyay, “Output feedback sliding-mode control for uncertain systems using fast output sampling technique,” IEEE Trans. Ind. Electron., vol. 53, no. 5, pp. 1677–1682, Oct. 2006.

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X. R. Han received the degree in electrical engineering from Fushun Mining College, Fushun, China, in 2002, and the M.Sc. degree in electrical and electronic engineering from the University of Leicester, Leicester, U.K., in 2006. He is currently working toward the Ph.D. degree at the University of Kent, Canterbury, U.K. His research interests include sliding-mode control and time-delay systems.

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Emilia Fridman received the M.Sc. degree in mathematics from Kuibyshev State University, Kuibyshev, U.S.S.R., in 1981, and the Ph.D. degree in mathematics from Voroneg State University, Voroneg, U.S.S.R., in 1986. From 1986 to 1992, she was an Assistant and then an Associate Professor in the Department of Mathematics, Kuibyshev Institute of Railway Engineers, U.S.S.R. Since 1993, she has been with Tel Aviv University, Tel Aviv, Israel, where she is currently Professor of electrical engineering—systems. Her research interests include time-delay systems, distributed parameter systems, H∞ control, singular perturbations, nonlinear control, and asymptotic methods. She has published about 80 articles in international scientific journals. She serves as an Associate Editor for Automatica and the IMA Journal of Mathematical Control and Information.

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Sarah K. Spurgeon received the B.Sc. and D.Phil. degrees from the University of York, York, U.K., in 1985 and 1988, respectively. She is currently a Professor of control engineering and the Director of the Electronics Department, University of Kent, Canterbury, U.K. Her research interests are in the area of robust nonlinear control and estimation, particularly via sliding-mode techniques in which area she has published in excess of 200 refereed papers. Dr. Spurgeon is a Fellow of the Institution of Engineering and Technology, a Fellow of the Institute for Mathematics and Applications, a Fellow of the Institute of Measurement and Control, and was elected a Fellow of the Royal Academy of Engineering in 2008. She received the IEEE Millennium Medal in 2000.

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Chris Edwards was born in Swansea, U.K. He received the B.Sc. degree in mathematics from the University of Warwick, Coventry, U.K., in 1987, and the Ph.D. degree from Leicester University, Leicester, U.K., in 1995. From 1987 to 1991, he was a Research Officer for British Steel Technical, Port Talbot, U.K., where he was involved with mathematical modeling of rolling and finishing processes. Since 1991, he has been with Leicester University, where he was first a Ph.D. student supported by a British Gas Research Scholarship, appointed as a Lecturer with the Control Systems Research Group in 1996, and promoted to Senior Lecturer in 2004 and Reader in 2007. He is a coauthor of over 200 refereed papers and two books on sliding-mode control.

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HAN et al.: ON THE DESIGN OF SLIDING-MODE SOF CONTROLLERS FOR SYSTEMS WITH STATE DELAY

AUTHOR QUERIES AUTHOR PLEASE ANSWER ALL QUERIES AQ1 = “Assume that” was deleted. Please check if correct. AQ2 = Please provide publication update in Ref. [9]. AQ3 = Please provide month of publication in Ref. [10]. AQ4 = Please provide publication update in Ref. [11]. AQ5 = Please provide publication update in Ref. [14]. AQ6 = Please provide issue number and month of publication in Ref. [16]. AQ7 = Please provide page range in Ref. [17]. AQ8 = Please provide page range in Ref. [31]. AQ9 = Please specify the degree received by Mr. Han from Fushun Mining College, Fushun, China, in 2002.

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