Nonlinear Dyn DOI 10.1007/s11071-015-2457-7
ORIGINAL PAPER
Robust non-fragile sampled-data control for offshore steel jacket platforms Bao-Lin Zhang · Mao-Mao Meng · Qing-Long Han · Xian-Ming Zhang
Received: 12 February 2015 / Accepted: 10 October 2015 © Springer Science+Business Media Dordrecht 2015
Abstract This paper is concerned with the nonfragile sampled-data control problem for an offshore platform subject to parametric perturbations of the system and admissible gain variations of the controller. By purposefully introducing a time-varying time-delay into control channel, designing a sampled-data controller for the original system is transformed into synthesizing a state feedback controller for a time-varying time-delay system. A sufficient condition on the existence of a robust non-fragile sampled-data controller is derived. Then, a robust non-fragile sampled-data controller is designed and its effectiveness is investigated based on the simulation results. It is demonstrated that (1) the designed non-fragile sampled-data controller is capable of reducing the oscillation amplitudes of the floors of the offshore platform system significantly; and (2) compared with the robust sampled-data controller and the classical robust controller as well as B.-L. Zhang (B)· M.-M. Meng College of Science, China Jiliang University, Hangzhou 310018, Zhejiang, People’s Republic of China e-mail:
[email protected] M.-M. Meng e-mail:
[email protected] Q.-L. Han (B) · X.-M. Zhang Griffith School of Engineering, Griffith University, Gold Coast Campus, QLD 4222, Australia e-mail:
[email protected] X.-M. Zhang e-mail:
[email protected]
the robust delayed controller, the oscillation amplitudes of the floors of the system under the three controllers are almost at the same level, while the control force required by the robust sampled-data controller is less than the one by the continuous-time controllers. Keywords Offshore structures · Active control · Sampled-data control · Non-fragile control
1 Introduction Vibration control for offshore steel jacket platforms have been a hot research topic in the ocean engineering area. Located in marine environment, the offshore platforms are affected by wave, flow, wind, ice, and even earthquake [1–7], which always lead to unavoidable vibration of the offshore platforms. The resultant vibration may result in potential degradation of the safety, integrity, durability of the structure and comfort of the staying. To refrain the vibration to an acceptable level and ensure the performance requirement for the offshore platforms, a considerable amount of research has been conducted [8–11]. Due to some inherent limitation of passive control, active control strategy has a great potential to improve the performance of the offshore platforms [12]. Based on various control techniques, a number of active control schemes have been developed, such as nonlinear control [9], optimal-based control [13,14], H2 control [8,15], sliding mode control [16], adaptive predictive control [17],
123
B.-L. Zhang et al.
H∞ control [15,18], intelligent control [19,20], and network-based control [21]. More recently, by artificially introducing the time-delays into the control channel, several delayed feedback control approaches are utilized to control the offshore platforms. For example, in [22], a delayed dynamic output feedback control scheme is developed to stabilize an offshore steel jacket platform with active tuned mass damper mechanisms. In [23], a sliding mode control scheme by using the mixed current and delayed states is proposed for an uncertain model of the offshore platform. In particular, by considering the variations in the controller gain, a pure delayed non-fragile control scheme is developed in [24]. The aforesaid control approaches are capable of attenuating the wave-induced vibration to an acceptable level and therefore improving the performance of the offshore platforms. However, it is not difficult to see that on the one hand, the above controllers require relatively large force. For instance, to attenuate the vibration of the offshore platform to a satisfactory level, the control force required by the nonlinear controller [9], the dynamic output feedback controller [22] and the sliding mode controller with mixed current and delayed state [23] is more than 105 N. On the other hand, the control schemes mentioned above are involved in designing continuous-time controllers for continuous-time dynamic models of the offshore platform. From the point of view of engineering implementation, when these controllers are utilized to control the actual offshore platform systems, it is indispensable to apply the techniques of digital computers to control the continuous-time systems [25]. In fact, to control continuous-time systems by using the digital computers generally involves both continuoustime and discrete-time signals, and the corresponding dynamic systems are known as sampled-data systems. Over the past decades, sampled-data systems have been studied extensively, and a number of important results have been reported, one can see [26–29], and the references therein. Also, the sampled-data control strategies have been applied to control engineering areas, such as vehicle active suspension systems [30], flexible spacecraft [31], mechanical systems [32]. More recently, for a simplified dynamic model of an offshore steel jacket platform with active mass dampers, a fault-tolerant sampled-data H∞ controller has been developed in [33] to reject the wave-induced vibration of the offshore platform. Specifically, in [34], a
123
robust reliable sampled-data controller has been presented to stabilize an offshore steel jacket platform with active tuned mass damper. It is found through simulation results that even if there exist admissible systematic uncertainties as well as actuator failures in the system, the proposed sampled-data controller can stabilize the offshore platform effectively. However, it is not difficult to see that to obtain the satisfactory control performance of the offshore platform, there is still a room for reducing the control cost. Therefore, an interesting question would be whether one can design a sampled-data controller to explore the possibility of improving the performance of the offshore platforms by taking less control cost, which motivates our present study. In this paper, the robust non-fragile sampled-data control problem is considered for an offshore steel jacket platform with active tuned mass damper mechanisms [23]. First, by taking the time-varying gain variation and parametric perturbation into account, a robust non-fragile sampled-data controller is first proposed to stabilize the stability of the offshore platform. Then, based on an input delay approach [29], the resultant closed-loop system of the offshore platform with sampling measurements is transformed into a nonlinear system with time-varying state time-delays. Third, by using the Lyapunov–Krasovskii functional approach, the robust non-fragile sampled-data controller is synthesized by solving a set of linear matrix inequalities. To demonstrate the effectiveness and the advantage of the proposed control schemes, based on the simulation results, some comparisons have been made between the sampled-data controllers and the continuous-time controllers, such as the classic robust non-fragile controller and robust non-fragile delayed controller. In addition, to further demonstrate the advantage of the sampled-data control scheme, if the gain variations of the controllers are not taken into account, the robust sampled-data controller is compared with the robust delayed state feedback controller and the robust sliding mode controller with mixed current and delayed states [23]. It is found that under the designed robust sampled-data controllers and the continuous-time controllers, the oscillation amplitudes of the three floors of the offshore platform are almost in the same level, while the control force required by the former is less than the latter. The rest of the paper is organized as follows. In Sect. 2, an uncertain dynamic model of an offshore
Robust non-fragile sampled-data control
steel jacket platform subject to self-excited wave force is given first. Then by considering the gain variations in the controller, a robust non-fragile sampleddata control problem for the offshore platform system is formulated. In Sect. 3, the design results of a robust non-fragile sampled-data controller is developed. The simulation results are given in Sect. 4 to illustrate the effectiveness and the superiority of the proposed control approach, and conclusions are given in Sect. 5. Throughout this paper, all the matrices are real matrices. The superscript ‘T’ means the transpose of a vector (matrix). For an invertible matrix, the superscript ‘−1’ means the inverse of the matrix. P > 0 (P ≥ 0) means that the matrix P is a real symmetric and positive definite (semi-definite) matrix. I is the identity matrix of appropriate dimensions. For simplicity, the symmetric term in a symmetric matrix is denoted by ∗, X Y X Y e.g., ∗ Z = Y T Z .
2 Problem formulation Consider an offshore steel jacket platform model with a tuned mass damper (TMD) mechanism showed in Fig. 1 [9,16,23]. The motion equation of the system is given as ⎧ z¨ 1 = − (ω1 +ω1 )2 z 1 −2(ξ1 +ξ1 )(ω1 + ω1 )˙z 1 ⎪ ⎪ ⎪ ⎪ ⎪ − φ1 (K T + K T )(φ1 z 1 + φ2 z 2 ) + f 1 + f 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − φ1 (C T + C T )(φ1 z˙ 1 + φ2 z˙ 2 ) − φ1 u ⎪ ⎪ ⎪ ⎪ ⎪ + φ1 (K T + K T )z T + φ1 (C T + C T )˙z T ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ z¨ 2 = − (ω2 +ω2 )2 z 2 −2(ξ2 +ξ2 )(ω2 + ω2 )˙z 2 − φ2 (K T + K T )(φ1 z 1 + φ2 z 2 ) + f 3 + f 4 ⎪ ⎪ ⎪ ⎪ ⎪ − φ2 (C T + C T )(φ1 z˙ 1 + φ2 z˙ 2 ) − φ2 u ⎪ ⎪ ⎪ ⎪ ⎪ + φ2 (K T + K T )z T + φ2 (C T + C T )˙z T ⎪ ⎪ ⎪ ⎪ u ⎪ 2 ⎪ ⎪ ⎪ z¨ T = − (ωT + ωT ) (z T − φ1 z 1 − φ2 z 2 ) + m ⎪ T ⎪ ⎪ ⎩ − 2(ξT +ξT )(ωT +ωT )(˙z T −φ1 z˙ 1 −φ2 z˙ 2 ) (1) where z 1 and z 2 are generalized coordinates of the first and the second vibration modes, respectively; z T is the horizontal displacement of the TMD; ω1 and ω2 are nominal natural frequencies of the first and the second vibration modes, respectively; ω1 and ω2 rep-
Fig. 1 A steel jacket structure with an active TMD ([9])
resent perturbations with respect to nominal parameters ω1 and ω2 , respectively; ξ1 and ξ2 are nominal damping ratios of the first and the second vibration modes, respectively; ξ1 and ξ2 are perturbations with respect to nominal parameters ξ1 and ξ2 , respectively; φ1 and φ2 are mode shapes vectors of the first and the second vibration modes, respectively; ωT and ξT represent the nominal natural frequency and damping ratio of the TMD, respectively; ωT and ξT are perturbations with respect to nominal parameters ωT and ξT , respectively; C T , m T and K T are the nominal damping, mass and stiffness of the TMD, respectively; C T and K T are perturbations with respect to nominal parameters C T and K T , respectively; f 1 , f 2 , f 3 and f 4 are self-excited wave force terms acting on the offshore platform; u is active control force of the system. Let ⎧ K T (t) = Kˆ T · K˜ T (t) ⎪ ⎪ ⎪ ⎪ ⎨ C (t) = Cˆ · C˜ (t) T T T ⎪ ωi (t) = ωi ωˆ i · ω˜ i (t) + o(ω˜ i (t)) ⎪ ⎪ ⎪ ⎩ ξi (t) = ξˆi · ξ˜i (t), i = 1, 2, T
(2)
where Kˆ T , Cˆ T , ξˆi and ωˆ i are the maximum perturbations with respect to the nominal values of K T , C T , ξi and ωi , respectively; o(·) denotes a higher-order infinitesimal. | K˜ T (t)| ≤ 1, |C˜ T (t)| ≤ 1, |ξ˜i (t)| ≤ 1, and |ω˜ i (t)| ≤ 1, i = 1, 2, T . Then, ignoring the second-order terms ξ˜i (t) · ω˜ i (t) and ω˜ i2 (t) and the higher-order infinitesimal o(ω˜ i (t)), the dynamic equation (1) can be rewritten as
123
B.-L. Zhang et al.
⎧ z¨ 1 = − (ω12 + ω¯ 1 )z 1 − (2ξ1 ω1 + ν¯ 1 )˙z 1 − φ1 u ⎪ ⎪ ⎪ ⎪ ⎪ − φ1 (K T + K T )(φ1 z 1 + φ2 z 2 ) + f 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − φ1 (C T + C T )(φ1 z˙ 1 + φ2 z˙ 2 ) + f 2 ⎪ ⎪ ⎪ ⎪ ⎪ + φ1 (K T + K T )z T + φ1 (C T + C T )˙z T ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ z¨ 2 = − (ω22 + ω¯ 2 )z 2 − (2ξ2 ω2 + ν¯ 2 )˙z 2 − φ2 u
with a21 a23 a25 a41 a43 a45 a61 a63 a65
− φ2 (K T + K T )(φ1 z 1 + φ2 z 2 ) + f 3 ⎪ ⎪ ⎪ ⎪ ⎪ − φ2 (C T + C T )(φ1 z˙ 1 + φ2 z˙ 2 ) + f 4 ⎪ ⎪ ⎪ ⎪ ⎪ + φ2 (K T + K T )z T + φ2 (C T + C T )˙z T ⎪ ⎪ ⎪ ⎪ u ⎪ ⎪ z¨ T = − (ωT2 + ω¯ T )(z T − φ1 z 1 − φ2 z 2 ) + ⎪ ⎪ ⎪ mT ⎪ ⎪ ⎩ − (2ξT ωT + ν¯ T )(˙z T − φ1 z˙ 1 − φ2 z˙ 2 ) (3) where
= −ω12 − K T φ12 , a22 = −K T φ1 φ2 , a24 = φ1 K T , a26 = −K T φ1 φ2 , a42 = −ω22 − K T φ22 , a44 = φ2 K T , a46 = ωT2 φ1 , a62 = ωT2 φ2 , a64 = −ωT2 , a66
= −2ξ1 ω1 − C T φ12 , = −C T φ1 φ2 , = φ1 C T , = −C T φ1 φ2 , = −2ξ2 ω2 − C T φ22 , = φ2 C T , = 2ξT ωT φ1 , = 2ξT ωT φ2 , = −2ξT ωT .
The time-varying perturbation matrix A(t) is a 3× 3 block matrix A(t) = Ai j (t) 3×3
ν¯ i = 2ωi [ξi ωˆ i · ω˜ i (t) + ξˆi · ξ˜i (t)]
where
ω¯ i = 2ωi2 ωˆ i · ω˜ i (t), i = 1, 2, T
A11 (t) = −φ12 · A0 (t) − A1 (t), A12 (t) = −φ1 φ2 · A0 (t), A22 (t) = −φ22 · A0 (t) − A2 (t), A13 (t) = φ1 · A0 (t), A21 (t) = A12 (t), A23 (t) = φ2 · A0 (t), A31 (t) = φ1 · A T (t), A32 (t) = φ2 · A T (t), A33 (t) = −A T (t).
Denote T x(t) = x1 (t) x2 (t) x3 (t) x4 (t) x5 (t) x6 (t) where x1 (t) = z 1 (t), x2 (t) = z˙ 1 (t), x3 (t) = z 2 (t) x4 (t) = z˙ 2 (t), x5 (t) = z T (t), x6 (t) = z˙ T (t) Then the dynamic equation (3) can be expressed as
x(t) ˙ = (A + A(t))x(t) + Bu(t) + D f (x, t) x(0) = x0
(4)
where ⎡ ⎧ 0 1 0 0 ⎪ ⎪ ⎪ ⎢ a21 a22 a23 a24 ⎪ ⎪ ⎢ ⎪ ⎪ ⎢ 0 ⎪ 0 0 1 ⎪ ⎢ ⎪ A = ⎪ ⎢ a41 a42 a43 a44 ⎪ ⎪ ⎢ ⎪ ⎪ ⎣ 0 ⎨ 0 0 0 a a a a 61 62 63 64 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ B = 0 −φ1 0 −φ2 0 ⎪ ⎪ ⎪ ⎪ T ⎪ ⎪ ⎪ 0 1 0 0 0 0 ⎪ ⎩D = 0 0 0 1 0 0
123
⎤ 0 0 a25 a26 ⎥ ⎥ 0 0 ⎥ ⎥ a45 a46 ⎥ ⎥ 0 1 ⎦ a65 a66 T 1 mT
with ⎧ 0 0 ⎪ ⎪ A (t) = 0 ⎨ K T C T ⎪ 0 0 ⎪ ⎩ Ai (t) = , i = 1, 2, T ω¯ i (t) ν¯ i (t)
(6)
(7)
x0 is the initial value, f (x, t) is the nonlinear selfexcited wave force as f 1 (x1 , x3 , t) + f 2 (x1 , x3 , t) (8) f (x, t) = f 3 (x1 , x3 , t) + f 4 (x1 , x3 , t) which is uniformly bounded and satisfies the following cone-bounding constraint [9] f (x, t) ≤ μ x (t)
(5)
where μ is a positive scalar. Denote ⎧ E a = E˜ E˜ 0 ⎪ ⎪ ⎨ T Ha = H˜ T H˜ 0T ⎪ ⎪ ⎩ ˜ Fa (t) = diag{ F(t), F˜0 (t)}
(9)
(10)
Robust non-fragile sampled-data control
˜ E˜ 0 , H˜ , H˜ 0 , F(t) ˜ where the matrices E, and F˜0 (t) are defined as those in [23]. Then similar to [23], matrix A(t) can be simplified as
Assumption 1 The state variables of the system are measured at times instants tk , . . . , tk+1 , . . ., and only x(tk ) are available for interval tk ≤ t < tk+1 .
A(t) = E a Fa (t)Ha
Assumption 2 The interval between any two sampling instants is bounded by h > 0, i.e., the following inequality holds
(11)
where Fa (t) represents the time-varying parameter perturbation satisfying FaT (t)Fa (t)
≤ I, ∀t ≥ 0
(12)
To stabilize the offshore steel jacket platform, in this paper, a robust non-fragile sampled-data controller is designed as u(t) = (K + K (t))x(tk )
(14)
with E b and Hb are known matrices of appropriate dimensions, and the time-varying matrix Fb (t) is an unknown continuous function satisfying FbT (t)Fb (t) ≤ I, ∀t ≥ 0
(16)
Lemma 1 [35] Let x(t) ∈ Rn be a vector-valued function with first-order continuous-derivative entries. Then, the following integral inequality holds for any matrices X, M1 , M2 ∈ Rn×n and Z ∈ R2n×2n , and a scalar function h := h(t) ≥ 0:
(13)
where K is a 1 × 6 gain matrix to be designed, x(tk ) is the sampled value of the state variable x(t) at time instant tk , tk ≤ t < tk+1 . The time-varying matrix K (t) represents the gain variation satisfying K (t) = E b Fb (t)Hb
tk+1 − tk ≤ h
−
t
x˙ T (s)X x(s)ds ˙ ≤ ξ T (t)ϒξ(t) + hξ T (t)Z ξ(t)
t−h
(17) where
M1T + M1 −M1T + M2 ∗ −M2T − M2 X Y x(t) ≥0 , ξ(t) := ∗ Z x(t − h) ϒ :=
with Y := M1 M2
(15) 3 Non-fragile sampled-data controller
Remark 1 It is clear from (13) that the feedback control signal is based on the sampled-data value x(tk ) of the continuous-time state variable at time instant tk , which means that the active control of the offshore platform is implemented via sampled-data system frameworks. Compared with the nonlinear and robust control [9], integral sliding mode control [16,18,23] and dynamic output feedback control [22], the proposed sampleddata control scheme is a more convenient way to realize active control via digital computers. In addition, during the design of the sampled-data controller (13), the controller gain variations are also considered. It indicates that compared with the controllers mentioned above, if there exist time-varying perturbations in the controller gain, the proposed controller can still stabilize the offshore platform and improve its control performance. In what follows, the following assumptions and lemmas are necessary to obtain the main results.
In this section, some sufficient conditions on the existence of the controller (13) will be derived. Let d(t) = t − tk , ∀k ≥ 0
(18)
It is clear that d(t) ≤ h, and ˙ = 1, t = tk d(t)
(19)
Then, the robust non-fragile sampled-data controller (13) can be written as u(t) = (K + K (t))x(t − d(t)), tk ≤ t < tk+1 (20) Substituting (20) into (4), one yields the closed-loop system as ˜ ˜ x(t) ˙ = A(t)x(t) + B(t)x(t − d(t)) + D f (x, t) (21)
123
B.-L. Zhang et al.
where
Note from (16) and (18) that
˜ = A + E a Fa (t)Ha A(t) ˜ B(t) = B K + B E b Fb (t)Hb
(22)
Proposition 1 For given scalars μ > 0 and h > 0, the system (21) is robustly stable if there exist 6 × 6 matrices P¯ > 0, R¯ > 0, M¯ 1 , M¯ 2 , Z¯ 1 ≥ 0, Z¯ 2 , Z¯ 3 ≥ 0, a 1 × 6 matrix K¯ , and a scalar ε > 0 such that
Λ11 Λ12 D h P¯ AT μ P¯ εE a ⎢ ∗ Λ22 0 h K¯ T B T 0 0 ⎢ ⎢ ∗ ∗ −I h D T 0 0 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗
−h R¯ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗
0 −I ∗ ∗ ∗ ∗
ε B E b P¯ HaT
⎤
0 0 0 P¯ HbT ⎥ ⎥ 0 0 0 ⎥ ⎥ εh E a εh B E b 0 0 ⎥ ⎥ 0 0 0 0 ⎥ ⎥ −ε I 0 0 0 ⎥ ⎥ ∗ −ε I 0 0 ⎥ ⎥ ∗ ∗ −ε I 0 ⎦ ∗ ∗ ∗ −ε I
0 , R > 0 . Taking the derivative of V (xt ) along the state trajectory of the closed-loop system (21) yields
+ 2x (t)P D f (x, t) + h x˙ (t)R x(t) ˙ t x˙ T (s)R x(s)ds ˙ −
123
t
x˙ T (s)R x(s)ds ˙
t−d(t)
(28) By Lemma 1, for any 6 × 6 matrices R > 0, M1 , M2 , Z 1 ≥ 0, Z 2 and Z 3 ≥ 0 satisfying ⎤ R M1 M2 ⎣ ∗ Z1 Z2 ⎦ ≥ 0 ∗ ∗ Z3 ⎡
(29)
the following inequality holds: −
t
x˙ T (s)R x(s)ds ˙
t−d(t) T
≤ x (t)(M1T + M1 + h Z 1 )x(t) − 2x T (t)(M1T − M2 − h Z 2 )x(t − d(t)) − x T (t − d(t))(M2T + M2 − h Z 3 )x(t − d(t)) (30)
⎤ x(t) ξ(t) = ⎣ x(t − d(t)) ⎦ , f (x, t) ⎡ ⎤ Φ11 (t) Φ12 (t) P D ∇(t) = ⎣ ∗ Φ22 (t) 0 ⎦ ∗ ∗ −I ˜ ˜
(t) = A(t) B(t) D where ˜ + μ2 I + h Z 1 + M1T + M1 Φ11 (t) = A˜ T (t)P + P A(t) ˜ − M1T + M2 + h Z 2 Φ12 (t) = P B(t) Φ22 (t) = − M2T − M2 + h Z 3 Then from (27), (28) and (30), we have
˜ + A˜ T (t)P]x(t) V˙ (xt ) =x T (t)[P A(t) ˜ − d(t)) + 2x T (t)P B(t)x(t
t−h
x˙ (s)R x(s)ds ˙ ≤−
⎡
Proof Choose the Lyapunov–Krasovskii functional as 0 t V (xt ) = x T (t)P x(t) + x˙ T (α)R x(α)dαdβ ˙
T
−
T
Let
Moreover, the gain matrix in (20) is given by K = K¯ P¯ −1 .
−h
t t−h
The following proposition provides a sufficient condition on the existence of the controller (20).
⎡
V˙ (xt ) ≤ ξ T (t)[∇(t) + T (t)(h R) (t)]ξ(t)
(31)
T
(27)
A sufficient condition for robust stability of system (21) is that there exist matrices P > 0, R > 0, K , M1 , M2 , Z 1 ≥ 0, Z 2 and Z 3 ≥ 0 such that
Robust non-fragile sampled-data control
V˙ (xt ) ≤ νx T (t)x(t) < 0, ∀t ≥ 0
(32)
⎡
where ν > 0. To guarantee (32), we require that ∇(t) + T (t)(h R) (t) < 0, ∀t ≥ 0
(33)
In what follows, we only need to prove that the linear matrix inequality (33) is equivalent to (23). For this, denote F(t) = ⎡
0 Fa (t) 0 Fb (t)
ϒ12 ϒ22 ∗ ∗ ∗ ∗ ∗ ∗ ∗
PD 0 −I ∗ ∗ ∗ ∗ ∗ ∗
ϒ16 = ε P E a ,
⎤ μI ϒ16 ϒ17 HaT 0 T 0 0 0 0 Hb ⎥ ⎥ 0 0 0 0 0 ⎥ ⎥ 0 ⎥ 0 ϒ46 ϒ47 0 ⎥ −I 0 0 0 0 ⎥ ⎥ 0, if there exist 6 × 6 matrices P¯ > 0, R¯ > 0, M¯ 1 , M¯ 2 ,
123
B.-L. Zhang et al.
Z¯ 1 ≥ 0, Z¯ 2 , Z¯ 3 ≥ 0, a 1 × 6 matrix K¯ , and a scalar ε > 0 such that (24) and ⎡
⎤ Λ11 Λ12 D h P¯ AT μ P¯ εE a P¯ HaT ⎢ ∗ Λ22 0 h K¯ T B T 0 0 0 ⎥ ⎢ ⎥ T ⎢ ∗ 0 0 0 ⎥ ∗ −I h D ⎢ ⎥ ⎢ ∗ 0 ⎥ ∗ ∗ −h R¯ 0 εh E a ⎢ ⎥ 0 and h > 0, if there exist 6 × 6 matrices P¯ > 0, R¯ > 0, M¯ 1 , M¯ 2 , Z¯ 1 ≥ 0, Z¯ 2 , Z¯ 3 ≥ 0, a 1 × 6 matrix K¯ such that (24) and ⎡ ⎤ Λ11 Λ12 D h P¯ AT μ P¯ ε B E b 0 ⎢ ∗ Λ22 0 h K¯ T B T 0 0 P¯ HbT ⎥ ⎢ ⎥ T ⎢ ∗ ∗ −I hD 0 0 0 ⎥ ⎢ ⎥ ⎢ ∗ 0 ⎥ ∗ ∗ −h R¯ 0 εh B E b ⎢ ⎥ ⎢ ∗ ∗ ∗ ∗ −I 0 0 ⎥ ⎢ ⎥ ⎣ ∗ ∗ ∗ ∗ ∗ −ε I 0 ⎦ ∗ 0 is an artificially introduced constant timedelay, K (t) is given by (14). A sufficient condition on the existence of the control law (42) is presented by the following proposition. Proposition 2 For given scalars μ > 0 and τ > 0, if there exist 6 × 6 matrices P¯ > 0, R¯ > 0, Q¯ > 0, M¯ 1 , M¯ 2 , Z¯ 1 ≥ 0, Z¯ 2 , Z¯ 3 ≥ 0, a 1 × 6 matrix K¯ , and a scalar ε > 0 such that (24) and ⎡
Θ11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗
Θ12 Θ22 ∗ ∗ ∗ ∗ ∗ ∗ ∗
D τ P¯ AT 0 τ K¯ T B T −I τ D T ∗ −τ R¯ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
⎤ μ P¯ εE a ε B E b P¯ HaT 0 T 0 0 0 0 P¯ Hb ⎥ ⎥ 0 0 0 0 0 ⎥ ⎥ 0 ετ E a ετ B E b 0 0 ⎥ ⎥ −I 0 0 0 0 ⎥ ⎥ ∗ −ε I 0 0 0 ⎥ ⎥ ∗ ∗ −ε I 0 0 ⎥ ⎥ ∗ ∗ ∗ −ε I 0 ⎦ ∗ ∗ ∗ ∗ −ε I
0 and τ > 0, if there exist 6 × 6 matrices P¯ > 0, R¯ > 0, Q¯ > 0, M¯ 1 , M¯ 2 , Z¯ 1 ≥ 0, Z¯ 2 , Z¯ 3 ≥ 0, a 1 × 6 matrix K¯ , and a scalar ε > 0 such that (24) and ⎡
Θ11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗
Θ12 Θ22 ∗ ∗ ∗ ∗ ∗
D τ P¯ AT 0 τ K¯ T B T −I τ D T ∗ −τ R¯ ∗ ∗ ∗ ∗ ∗ ∗
⎤ μ P¯ εE a P¯ HaT 0 0 0 ⎥ ⎥ 0 0 0 ⎥ ⎥ 0 ετ E a 0 ⎥ ⎥ < 0 (46) −I 0 0 ⎥ ⎥ ∗ −ε I 0 ⎦ ∗ ∗ −ε I
Robust non-fragile sampled-data control
then, under the robust delayed state feedback control law (45), the system (4) is robustly stable. Moreover, the gain matrix in (45) is given by K = K¯ P¯ −1 . In (42), let τ = 0. Then, one yields a robust nonfragile control law as u(t) = [K + K (t)]x(t)
(47)
which can be designed by a corollary stated as follows. Corollary 4 For a given scalar μ > 0, if there exist a 6 × 6 matrix P¯ > 0, a 1 × 6 matrix K¯ , and a scalar ε > 0 such that ⎡
⎤ Ω D μ P¯ εE a ε B E b P¯ HaT P¯ HbT ⎢ ∗ −I 0 0 0 0 0 ⎥ ⎢ ⎥ ⎢ ∗ ∗ −I 0 0 0 0 ⎥ ⎢ ⎥ ⎢∗ ∗ ∗ −ε I 0 0 0 ⎥ ⎢ ⎥
