Probability-Dependent Gain-Scheduled Control for Discrete-Time Stochastic Systems with Randomly Occurring Sensor Saturations Wangyan Li, Guoliang Wei, Fei Han Department of Control Science and Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China E-mail:
[email protected]
Abstract: This paper is devoted to tackling the control problem for a class of discrete-time stochastic systems with randomly occurring sensor saturations by utilizing gain-scheduled method, the sensor saturation phenomenon is assumed to occur in a randomly way based on time-varying Bernoulli distribution with measurable probability in real time. The aim of the paper is to design a gain-scheduled controller with probability-dependent gain which can be achieved by solving a convex optimization problem via semi-definite programme method. Subsequently, a new kind functional, probability-dependent Lyapunov functional is proposed to make the theory sound. Finally, an illustration example will demonstrate the effectiveness of the procedures we design. Key Words: Randomly Occurring Sensor Saturations, Time-Varying Bernoulli Distribution, Probability-Dependent Lyapunov Functional, Gain-Scheduled Controller
1
INTRODUCTION
Over the past decades, the gain-scheduling method is one of the most popular methods of controller designing and has been extensively studied both from theoretical and practical viewpoint, see e.g., [1–5, 15]. The gain-scheduling method, whose main idea is to design controller gains as functions of the scheduling parameters, which are supposed to be available in real time, and it then updates the controller with a predetermined set of tuning parameters designed to optimize the closed-loop performance. However, the gain-scheduling controller we design has not only the constant part but also timevarying part of the system which can be scheduled on-line according to the corresponding time-varying parameters, which will naturally lead to less conservative than the conventional ones with fix gains only, see e.g., [4,5,15]. Meanwhile, instead of using the information of system states, static output feedback (SOF) control directly makes use of system outputs to design controllers, which is much simple and easy to implement, and has been extensively used in various kinds of engineering fields [6–9, 16]. However, in most existing literatures, few attentions have focused on the SOF control problem for nonlinear stochastic systems and the randomly occurring phenomenon. The randomly occurring phenomenon is a newly emerged research topic which has drawn many researchers’ attentions, see e.g., [4, 5, 10–12, 14, 15]. It refers to those phenomenon appears in a random way based on a certain kind of probabilistic law including the randomly occurring nonlinearities, the missing measurements, randomly This work was supported in part by the National Natural Science Foundation of China under Grants 61074016, the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, the Program for New Century Excellent Talents in University under Grant NCET-11-1051, the Shanghai Natural Science Foundation of China under Grant 10ZR1421200, the Leverhulme Trust of the U.K., the Alexander von Humboldt Foundation of Germany, and the Innovation Fund Project for Graduate Student of Shanghai under Grant JWCXSL1202.
c 978-1-4673-5534-6/13/$31.00 2013 IEEE
occurring actuator faults, the randomly varying sensor delays and randomly occurring sensor saturations and so on, we refer paper [12] for more detail. If not handled appropriately, this phenomenon could cause a reduction of performance and/or launch a threat to the safety and reliability of the plant. As we all know, sensor saturation phenomenon is very common in practical engineering. It refers to that sensors cannot provide signals of unlimited amplitude, due mainly to the physical or technological constraints. In recent years, networked control systems (NCSs) have been used in a variety of engineering areas. Because of random occurrence of networked induced phenomena in NCSs, such as random sensor failures leading to intermittent saturation, sensor aging resulting in changeable saturation level, changes in the interconnections of subsystems, sudden environment changes, etc, sensor saturation may occur in a random way. We consider this phenomenon as randomly occurring sensor saturation, which has received increasing attention. Paper [10] discussed 𝐻∞ state estimation for discrete-time complex networks with randomly occurring sensor saturations and randomly varying sensor delays, while paper [11] turned to design 𝐻∞ filter for system with randomly occurring sensor saturations and missing measurements. However, to the best of authors’ knowledge, rare published literatures have dealt with randomly occurring sensor saturations, therefore, this paper attempts to flourish the research on this phenomenon by using the gain-scheduled method. The main contributions of this paper are summarized as follows: 1) a new SOF control problem is addressed for a class of discrete-time nonlinear stochastic systems with randomly occurring phenomenon by utilizing gainscheduled method; 2) a sequence of stochastic variables satisfying Bernoulli distribution is introduced to describe the time-varying features of the randomly occurring sensor saturations phenomenon; 3) a time-varying Lyapunov functional dependent on the saturating probability is pro4728
posed and applied to improve the performance of system; and 4) the gain-scheduled controller parameters can be adjusted online according to the saturating probability estimated through statistical tests. Notation. In this paper, ℝ𝑛 , ℝ𝑛×𝑚 , 𝕀+ , denote, respectively, the 𝑛-dimensional Euclidean space, the set of all 𝑛 × 𝑚 real matrices, the set of all positive integers. ∣ ⋅ ∣ refers to the Euclidean norm in ℝ𝑛 . 𝐼 denotes the identity matrix of compatible dimension. The notation 𝑋 ≥ 𝑌 (respectively, 𝑋 > 𝑌 ), where 𝑋 and 𝑌 are symmetric matrices, means that 𝑋 − 𝑌 is positive semi-definite (rxespectively, positive definite). For a matrix 𝑀 , 𝑀 𝑇 and 𝑀 −1 represent its transpose and inverse, respectively. The shorthand diag{𝑀1 , 𝑀2 , . . . , 𝑀𝑛 } denotes a block diagonal matrix with diagonal blocks being the matrices 𝑀1 , 𝑀2 , . . . , 𝑀𝑛 . In symmetric block matrices, the symbol ∗ is used as an ellipsis for terms induced by symmetry. Matrices, if they are not explicitly stated, are assumed to have compatible dimensions. In addition, 𝔼{𝑥} and Prob{𝑦} will, respectively, mean expectation of 𝑥 and probability of 𝑦.
2
Consider the following discrete-time nonlinear stochastic systems: =
𝑥(𝑘)
=
𝐴𝑥(𝑘) + 𝐷𝑥(𝑘 − 𝑑) + 𝐵𝑢(𝑘) +𝑁 𝑓 (𝑧(𝑘)) + 𝐸𝑥(𝑘)𝑤(𝑘), 𝜌(𝑘), 𝑘 = −𝑑, −𝑑 + 1, ..., 0,
(1) (2)
𝑛
where 𝑥(𝑘) ∈ ℝ is the state, 𝑑 is a constant delay and 𝑧(𝑘) := 𝐺𝑥(𝑘) + 𝐺𝑑 𝑥(𝑘 − 𝑑), 𝜔(𝑘) is an one-dimensional Gaussian white noise sequence satisfying 𝔼{𝜔(𝑘)} = 0 and 𝔼{𝜔 2 (𝑘)} = 𝜎 2 , 𝜌(𝑘) is the initial state of the system. 𝐴,𝐷, 𝐵, 𝐸, 𝑁 , 𝐺 and 𝐺𝑑 are constant real matrices of appropriate dimensions and 𝐵 is of full-column. The nonlinear function 𝑓 (⋅) with (𝑓 (0) = 0) is assumed as nonlinear disturbances and satisfies the following sector-bounded condition: [𝑓 (𝑧(𝑘)) − 𝐹1 𝑧(𝑘)]𝑇 [𝑓 (𝑧(𝑘)) − 𝐹2 𝑧(𝑘)] ≤ 0,
(3)
where 𝑓 (⋅) belonged to the sector [𝐹1 , 𝐹2 ] and 𝐹1 and 𝐹2 are given constant real matrices. Remark 1 For the technique convenience, the nonlinear function 𝑓 (𝑧(𝑘)) can be decomposed into a linear and a nonlinear part as 𝑓 (𝑧(𝑘)) = 𝑓𝑠 (𝑧(𝑘))+𝐹1 𝑧(𝑘), then, from (3), we have 𝑓𝑠𝑇 (𝑧(𝑘))(𝑓𝑠 (𝑧(𝑘))−𝐹 𝑧(𝑘)) ≤ 0, where 𝐹 = 𝐹2 − 𝐹1 > 0. The measurement output with sensor saturation is described as: 𝑦(𝑘) = 𝜉(𝑘)𝜚(𝐶𝑥(𝑘)) + (1 − 𝜉(𝑘))𝐶𝑥(𝑘),
The variable 𝜉(𝑘) ∈ ℝ is a random white sequence characterizing the probabilistic sensor saturation, which obeys the following time-varying Bernoulli distribution: Prob{𝜉(𝑘) = 1} Prob{𝜉(𝑘) = 0}
= =
𝔼{𝜉(𝑘)} = 𝑝(𝑘) 1 − 𝔼{𝜉(𝑘)} = 1 − 𝑝(𝑘), (5)
where 𝑝(𝑘) is a time-varying positive scalar sequence and belongs to [𝑝1 𝑝2 ] ⊆ [0 1] with 𝑝1 and 𝑝2 being the lower and upper bounds of 𝑝(𝑘), respectively. Throughout the paper, for simplicity, we assume that 𝜉(𝑘), 𝜔(𝑘) and 𝜌(𝑘) are uncorrelated. In this paper, we are interested in designing the following gain-scheduled controller 𝑢(𝑘) = 𝐾(𝑝)𝑦(𝑘),
(6)
where 𝐾(𝑝) is the controller gain sequence to be designed and assumed as the following structure
PROBLEM FORMULATION
𝑥(𝑘 + 1)
[𝜚(𝑥) − 𝑎𝑥][𝜚(𝑥) − 𝑥] ≤ 0 and ∣𝑥∣ ≤ 𝑎−1 , where 𝑎 is a positive scalar satisfying 0 < 𝑎 < 1; So the nonlinear function 𝜚(𝐶𝑥(𝑘)) satisfies [𝜚(𝐶𝑥(𝑘)) − 𝑎𝐶𝑥(𝑘)]𝑇 [𝜚(𝐶𝑥(𝑘)) − 𝐶𝑥(𝑘)] ≤ 0, while ∣𝑎𝐶𝑥(𝑘)∣ ≤ 1 and 𝑎 satisfies 0 < 𝑎 < 1.
(4)
where 𝐶 is a constant real matrix of appropriate dimensions and 𝜚(𝑥) = sign(𝑥)min{1, ∣𝑥∣}. Here, the notation of “sign” means the signum function, and we use the notation 𝜚 to denote saturation functions. Noting that, without loss of generality, the saturation level is taken as unity. Remark 2 According to the definition of the saturation function, we can get that the nonlinear function 𝜚 satisfies
𝐾(𝑝) = 𝐾0 + 𝑝(𝑘)𝐾𝑢 ,
(7)
for every time step 𝑘, 𝑝(𝑘) is the time-varying parameter of the controller gain, 𝐾0 , 𝐾𝑢 are the constant parameters of the controller gain to be designed. The closed-loop systems of the static output feedback gainscheduled controller is 𝑥(𝑘 + 1)
=
𝐴𝑥(𝑘) + 𝐷𝑥(𝑘 − 𝑑) + 𝐵𝐾(𝑝) ×[𝜉(𝑘)𝜚(𝐶𝑥(𝑘)) + (1 − 𝜉(𝑘))𝐶𝑥(𝑘)] +𝑁 𝑓 (𝑧(𝑘)) + 𝐸𝑥(𝑘)𝑤(𝑘). (8)
Before formulating the problem to be investigated, we first introduce the following stability concepts: Definition 1 The closed-loop system (8) is said to be exponentially mean-square stable if, with 𝑤𝑘 = 0, there exist constant 𝛼 > 0 and 𝜏 ∈ (0, 1) such that } } { { 𝔼 ∥𝑥𝑘 ∥2 ≤ 𝛼𝜏 𝑘 sup 𝔼 ∥𝑥𝑖 ∥2 , 𝑘 ∈ 𝕀+ . (9) −𝑑≤𝑖≤0
In this paper, our purpose is to design a probabilitydependent gain-scheduled controller of the form (6) for the system (1) by exploiting a probability-dependent Lyapunov functional and LMI method such that, for all admissible sensor saturations and exogenous stochastic noises, the closed-loop system (8) is exponentially meansquare stable.
3
MAIN RESULTS
The following lemmas will be used in the proofs of our main results in this paper. Lemma 1 (Schur Complement) [13] Given constant ma𝑇 trices Σ1 , Σ2 , Σ3 where Σ1 = Σ𝑇 1 and 0 < Σ2 = Σ2 . 𝑇 −1 Then Σ1 + Σ3 Σ2 Σ3 ≥ 0 if and only if [ ] [ ] −Σ2 Σ3 Σ1 Σ𝑇 3 ≥ 0 or ≥ 0. Σ3 −Σ2 Σ𝑇 Σ1 3
2013 25th Chinese Control and Decision Conference (CCDC)
4729
+𝐸𝑥(𝑘)𝑤(𝑘)] − 𝑥𝑇 (𝑘)𝑄(𝑝(𝑘))𝑥(𝑘)
Lemma 2 [14] Let the matrix 𝐵 ∈ 𝑅𝑛×𝑚 be of fullcolumn rank. There always exist two orthogonal matrices [ ] 𝑇 𝑈 ∈ 𝑅𝑛×𝑛 and 𝑉 ∈ 𝑅𝑛×𝑛 such that 𝐵 = 𝑈 Σ 0 𝑉 , Σ = diag{𝜎1[, 𝜎2 , ⋅ ⋅ ⋅ ], 𝜎m }. If matrix 𝑆 has such struc𝑇 𝑛×(𝑛−𝑚) 12 ture: 𝑆 = 𝑈 𝑆011 𝑆 𝑆22 𝑈 , where 𝑆11 , 𝑆12 ∈ 𝑅 and 𝑆22 ∈ 𝑅(𝑛−𝑚)×(𝑛−𝑚) , then there exists a nonsingular matrix 𝑅 ∈ 𝑅𝑚×𝑚 such that 𝑆𝐵 = 𝐵𝑅.
≤
+𝑥𝑇 (𝑘)𝑄𝜏 𝑥(𝑘) − 𝑥𝑇 (𝑘 − 𝑑)𝑄𝜏 𝑥(𝑘 − 𝑑) { ¯ + (1 − 𝑝(𝑘))𝐵𝐾(𝑝)𝐶)𝑥(𝑘) 𝔼 [(𝐴
¯ +𝑝(𝑘)𝐵𝐾(𝑝)𝜚(𝐶𝑥(𝑘)) + 𝐷𝑥(𝑘 − 𝑑) 𝑇 ¯ + (1 − 𝑝(𝑘)) +𝑁 𝑓𝑠 (𝑧(𝑘))] 𝑄(𝑝(𝑘 + 1))[(𝐴 ×𝐵𝐾(𝑝)𝐶)𝑥(𝑘) + 𝑝(𝑘)𝐵𝐾(𝑝)𝜚(𝐶𝑥(𝑘)) ¯ +𝐷𝑥(𝑘 − 𝑑) + 𝑁 𝑓𝑠 (𝑧(𝑘))] + 𝑝(𝑘)(1 − (𝑝(𝑘))
Theorem 1 Consider the discrete-time nonlinear stochastic systems (8). If there exist positive-difinite matrices 𝑄(𝑝(𝑘)) and 𝑄𝜏 , slack matrix 𝑆 and nonsingular matrices 𝑌 (𝑝) and 𝑅, such that the following LMIs hold:(10), where ¯ Λ Δ𝑝 (𝑘) ¯ 𝐴
= =
𝑆𝑇 𝐵
= =
𝐾(𝑝) 𝑌 (𝑝)
= =
×[𝐵𝐾(𝑝)(𝜚(𝐶𝑥(𝑘)) − 𝐶𝑥(𝑘))]𝑇 𝑄(𝑝(𝑘 + 1)) ×𝐵𝐾(𝑝)[𝜚(𝐶𝑥(𝑘)) − 𝐶𝑥(𝑘)] +𝜎 2 𝑥𝑇 (𝑘)𝐸 𝑇 𝑄(𝑝(𝑘 + 1))𝐸𝑥(𝑘) −𝑥𝑇 (𝑘)𝑄(𝑝(𝑘))𝑥(𝑘) − 𝑥𝑇 (𝑘 − 𝑑)𝑄𝜏 𝑥(𝑘 − 𝑑)
−𝑄(𝑝(𝑘 + 1)) + 𝑆 + 𝑆 𝑇 , 𝑝(𝑘)(1 − 𝑝(𝑘)), ¯ = 𝐷 + 𝑁 𝐹1 𝐺𝑑 , 𝐴 + 𝑁 𝐹1 𝐺, 𝐷 𝐵𝑅,
+𝑥𝑇 (𝑘)𝑄𝜏 𝑥(𝑘) + 2𝑓𝑠𝑇 (𝑧(𝑘))𝐹 𝐺𝑥(𝑘)
+2𝑓𝑠𝑇 (𝑧(𝑘))𝐹 𝐺𝑑 𝑥(𝑘 − 𝑑) − 2𝑓𝑠𝑇 (𝑧(𝑘))𝑓𝑠 (𝑧(𝑘))
−2𝜚𝑇 (𝐶𝑥(𝑘))𝜚(𝐶𝑥(𝑘)) + (2 + 2𝑎)𝜚𝑇 (𝐶𝑥(𝑘)) } (14) ×𝐶𝑥(𝑘) − 2𝑎(𝐶𝑥(𝑘))𝑇 𝐶𝑥(𝑘) .
𝑅𝐾(𝑝) = 𝑌 (𝑝),
𝑅−1 𝑌 (𝑝) 𝑌0 + 𝑝(𝑘)𝑌𝑢 ,
(11)
Denote the following matrix variables:
in this case, the constant gains of the desired controller can be obtained as follows: 𝐾0 = 𝑅−1 𝑌0 ,
𝐾𝑢 = 𝑅−1 𝑌𝑢 ,
𝜂(𝑘) = [𝑥𝑇 (𝑘) 𝑥𝑇 (𝑘 − 𝑑) 𝜚𝑇 (𝐶𝑥(𝑘)) 𝑓𝑠𝑇 (𝑧(𝑘))]𝑇 , (15) { } 𝔼 {Δ𝑉 (𝑘)} ≤ 𝔼 𝜂 𝑇 (𝑘)Ω𝜂(𝑘) ,
(12)
and the closed-system (8) is then exponentially meansquare stable for all 𝑝(𝑘) ∈ [𝑝1 𝑝2 ].
⎡ Ω
Proof : Define the Lyapunov functional 𝑉 (𝑘) :
=
𝑥𝑇 (𝑘)𝑄(𝑝(𝑘))𝑥(𝑘) +
𝑘−1 ∑
𝑥𝑇 (𝑠)𝑄𝜏 𝑥(𝑠),
(13)
𝑠=𝑘−𝑑
=
𝔼 {Δ𝑉 (𝑘)} { 𝔼 𝑥𝑇 (𝑘 + 1)𝑄(𝑝(𝑘 + 1))𝑥(𝑘 + 1) −𝑥𝑇 (𝑘)(𝑄(𝑝(𝑘)) − 𝑄𝜏 )𝑥(𝑘) } −𝑥𝑇 (𝑘 − 𝑑)𝑄𝜏 𝑥(𝑘 − 𝑑) 𝔼 {[𝐴𝑥(𝑘) + 𝑝(𝑘)𝐵𝐾(𝑝)[𝜚(𝐶𝑥(𝑘)) − 𝐶𝑥(𝑘)] +(𝜉(𝑘) − 𝑝(𝑘))𝐵𝐾(𝑝)[𝜚(𝐶𝑥(𝑘)) − 𝐶𝑥(𝑘)] +𝐵𝐾(𝑝)𝐶𝑥(𝑘) + 𝑁 𝑓 (𝑧(𝑘)) + 𝐷𝑥(𝑘 − 𝑑) +𝐸𝑥(𝑘)𝑤(𝑘)]𝑇 𝑄(𝑝(𝑘 + 1))[𝐴𝑥(𝑘) +𝑝(𝑘)𝐵𝐾(𝑝)[𝜚(𝐶𝑥(𝑘)) − 𝐶𝑥(𝑘)] +(𝜉(𝑘) − 𝑝(𝑘))𝐵𝐾(𝑝)[𝜚(𝐶𝑥(𝑘)) − 𝐶𝑥(𝑘)] +𝐵𝐾(𝑝)𝐶𝑥(𝑘) + 𝐷𝑥(𝑘 − 𝑑) + 𝑁 𝑓 (𝑧(𝑘)) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
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=
Ω1 ⎢ Ω2 ⎣ Ω 4 Ω7
∗ Ω3 Ω5 Ω8
∗ ∗ Ω6 Ω9
⎤ ∗ ∗ ⎥ , ∗ ⎦ Ω10
(16)
(17)
¯ + (1 − 𝑝(𝑘))𝐵𝐾(𝑝)𝐶]𝑇 𝑄(𝑝(𝑘 + 1))[𝐴 ¯ Ω1 =[𝐴 + (1 − 𝑝(𝑘))𝐵𝐾(𝑝)𝐶] + 𝜎 2 𝐸 𝑇 𝑄(𝑝(𝑘 + 1))𝐸
noting 𝔼{𝜉(𝑘) − 𝑝(𝑘)} = 0 , 𝔼{𝜔(𝑘)} = 0 and 𝔼{[𝜉(𝑘) − 𝑝(𝑘)]2 } = 𝑝(𝑘)(1 − 𝑝(𝑘)), we can get =
}
𝑄𝜏 − 𝑄(𝑝(𝑘)) − 2𝑎𝐶 𝑇 𝐶 0 (𝑎 + 1)𝐶 𝐹𝐺 ¯ + (1 − 𝑝(𝑘))𝐵𝑌 (𝑝)𝐶 𝑆𝑇 𝐴 𝜎2 𝑆 𝑇 𝐸 Δ𝑝 (𝑘)𝐵𝑌 (𝑝)𝐶
∗ −𝑄𝜏 0 𝐹 𝐺𝑑 ¯ 𝑆𝑇 𝐷 0 0
∗ ∗ −2𝐼 0 𝑝(𝑘)𝐵𝑌 (𝑝) 0 Δ𝑝 (𝑘)𝐵𝑌 (𝑝)
+ Δ𝑝 (𝑘)(𝐵𝐾(𝑝)𝐶)𝑇 𝑄(𝑝(𝑘 + 1)𝐵𝐾(𝑝)𝐶 + 𝑄𝜏 − 𝑄(𝑝(𝑘)) − 2𝑎𝐶 𝑇 𝐶, ¯ 𝑇 𝑄(𝑝(𝑘 + 1))[𝐴 ¯ + (1 − 𝑝(𝑘))𝐵𝐾(𝑝)𝐶], Ω2 =𝐷 ¯ 𝑇 𝑄(𝑝(𝑘 + 1))𝐷 ¯ − 𝑄𝜏 , Ω3 =𝐷 ¯ Ω4 =𝑝(𝑘)(𝐵𝐾(𝑝))𝑇 𝑄(𝑝(𝑘 + 1))[𝐴 + (1 − 𝑝(𝑘))𝐵𝐾(𝑝)𝐶] + Δ𝑝 (𝑘)(𝐵𝐾(𝑝))𝑇 × 𝑄(𝑝(𝑘 + 1)𝐵𝐾(𝑝)𝐶 + (𝑎 + 1)𝐶, ¯ Ω5 =𝑝(𝑘)(𝐵𝐾(𝑝))𝑇 𝑄(𝑝(𝑘 + 1))𝐷, Ω6 =Δ𝑝 (𝑘)[𝐵𝐾(𝑝)]𝑇 𝑄(𝑝(𝑘 + 1))𝐵𝐾(𝑝) + 𝑝2 (𝑘)[𝐵𝐾(𝑝)]𝑇 𝑄(𝑝(𝑘 + 1))[𝐵𝐾(𝑝)] − 2𝐼, ¯ + (1 − 𝑝(𝑘))𝐵𝐾(𝑝)𝐶] + 𝐹 𝐺, Ω7 =𝑁 𝑇 𝑄(𝑝(𝑘 + 1))[𝐴 𝑇 ¯ + 𝐹 𝐺𝑑 , Ω8 =𝑁 𝑄(𝑝(𝑘 + 1))𝐷
∗ ∗ ∗ −2𝐼 𝑆𝑇 𝑁 0 0
∗ ∗ ∗ ∗ ¯ −Λ 0 0
∗ ∗ ∗ ∗ ∗ ¯ −𝜎 2 Λ 0
∗ ∗ ∗ ∗ ∗ ∗ ¯ −Δ𝑝 (𝑘)Λ
2013 25th Chinese Control and Decision Conference (CCDC)
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎦
(10)
Ω9 =𝑝(𝑘)𝑁 𝑇 𝑄(𝑝(𝑘 + 1))𝐵𝐾(𝑝), 𝑇
Ω10 =𝑁 𝑄(𝑝(𝑘 + 1))𝑁 − 2𝐼.
(18)
If Ω ≤ 0, we can conclude the following matrix by Schur Complement:(19) , where Λ = 𝑄−1 (𝑝(𝑘 + 1)),
(20)
by preforming the congruence transformation diag{𝐼, 𝐼, 𝐼, 𝐼, 𝑆, 𝜎 2 𝑆, Δ𝑝 (𝑘)𝑆} to (19), we have(21), where ˆ = 𝑆 𝑇 𝑄−1 (𝑝(𝑘 + 1))𝑆, Λ
(22)
from inequality
Remark 3 In the above theorem, a static output feedback controller has been designed based on a set of LMIs. However, the LMIs are actually infinite owing to the timevarying parameter 𝑝(𝑘) ∈ [𝑝1 𝑝2 ]. In this case, the desired controller cannot be obtained directly from Theorem 1 due to the infinite number of LMIs. To handle such a problem, in the next theorem, we have to convert this problem to a computationally accessible one by assigning a specific form to 𝑝(𝑘). Let us set 𝑄(𝑝(𝑘)) = 𝑄0 + 𝑝(𝑘)𝑄𝑢 . Theorem 2 Consider the discrete-time nonlinear stochastic system with infinite-distributed delays and missing measurements (8). If there exist positive-difinite matrices 𝑄0 , 𝑄𝑢 and 𝑄𝜏 , slack matrix 𝑆 and nonsingular matrices 𝑌 (𝑝) and 𝑅, such that the following LMIs hold:(27), where
¯ (23) 𝑆 𝑇 𝑄−1 (𝑝(𝑘 + 1))𝑆 ≥ 𝑆 𝑇 + 𝑆 − 𝑄(𝑝(𝑘 + 1)) ≜ Λ,
¯ = −𝑄0 − 𝑝𝑙 𝑄𝑢 + 𝑆 + 𝑆 𝑇 , Λ ¯ = 𝐴 + 𝑁 𝐹1 𝐺, 𝐷 ¯ = 𝐷 + 𝑁 𝐹1 𝐺𝑑 , 𝐴
we can get(24), by using Lemma 2, we have 𝑆 𝑇 𝐵 = 𝐵𝑅, denoting 𝑅𝐾(𝑝) = 𝑌 (𝑝), then (24) can be written as(10), furthermore, we can know from Lemma 1 that Ω < 0 and, subsequently, 𝔼 {Δ𝑉 (𝑘)} < −𝜆min (−Ω)𝔼∣𝜂(𝑘)∣2 ,
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
𝑄𝜏 − 𝑄(𝑝(𝑘)) − 2𝑎𝐶 𝑇 𝐶 0 (𝑎 + 1)𝐶 𝐹𝐺 ¯ + (1 − 𝑝(𝑘))𝐵𝐾(𝑝)𝐶 𝐴 𝐸 𝐵𝐾(𝑝)𝐶
∗ −𝑄𝜏 0 𝐹 𝐺𝑑 ¯ 𝐷 0 0
∗ ∗ −2𝐼 0 𝑝(𝑘)𝐵𝐾(𝑝) 0 𝐵𝐾(𝑝)
𝑅𝐾(𝑝) = 𝑌 (𝑝),
−1
𝐾(𝑝) = 𝑅 𝑌 (𝑝), Δ𝑖𝑗 = 𝑝𝑖 (1 − 𝑝𝑗 ), 𝑌 𝑚 (𝑝) = 𝑌0 + 𝑝𝑚 𝑌𝑢 , 𝑄𝑖 (𝑝(𝑘)) = 𝑄0 + 𝑝𝑖 𝑄𝑢 ,
(26)
where 𝜆min (−Ω) is the minimum eigenvalue of (−Ω). Finally, we can confirm that the closed-loop system is exponentially mean-square stable, then the proof of this theorem is complete. ⎡
𝑆 𝑇 𝐵 = 𝐵𝑅,
(28)
in this case, the constant gains of the desired controller can be obtained as follows: 𝐾0 = 𝑅−1 𝑌0 ,
∗ ∗ ∗ −2𝐼 𝑁 0 0
∗ ∗ ∗ ∗ −Λ 0 0
∗ ∗ ∗ ∗ ∗ −𝜎 −2 Λ 0
𝐾𝑢 = 𝑅−1 𝑌𝑢
∗ ∗ ∗ ∗ ∗ ∗
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎦
∗ −𝑄𝜏 0 𝐹 𝐺𝑑 ¯ 𝑆𝑇 𝐷 0 0
∗ ∗ −2𝐼 0 𝑝(𝑘)𝑆 𝑇 𝐵𝐾(𝑝) 0 Δ𝑝 (𝑘)𝑆 𝑇 𝐵𝐾(𝑝)
∗ ∗ ∗ −2𝐼 𝑆𝑇 𝑁 0 0
∗ ∗ ∗ ∗ ˆ −Λ 0 0
∗ ∗ ∗ ∗ ∗ ˆ −𝜎 2 Λ 0
∗ ∗ ∗ ∗ ∗ ∗ ˆ −Δ𝑝 (𝑘)Λ
𝑄𝜏 − 𝑄(𝑝(𝑘)) − 2𝑎𝐶 𝑇 𝐶 0 (𝑎 + 1)𝐶 𝐹𝐺 ¯ + (1 − 𝑝(𝑘))𝑆 𝑇 𝐵𝐾(𝑝)𝐶 𝑆𝑇 𝐴 𝜎2 𝑆 𝑇 𝐸 Δ𝑝 (𝑘)𝑆 𝑇 𝐵𝐾(𝑝)𝐶
∗ −𝑄𝜏 0 𝐹 𝐺𝑑 ¯ 𝑆𝑇 𝐷 0 0
∗ ∗ −2𝐼 0 𝑝(𝑘)𝑆 𝑇 𝐵𝐾(𝑝) 0 Δ𝑝 (𝑘)𝑆 𝑇 𝐵𝐾(𝑝)
∗ ∗ ∗ −2𝐼 𝑆𝑇 𝑁 0 0
∗ ∗ ∗ ∗ ¯ −Λ 0 0
∗ ∗ ∗ ∗ ∗ ¯ −𝜎 2 Λ 0
∗ ∗ ∗ ∗ ∗ ∗ ¯ −Δ𝑝 (𝑘)Λ
𝑀 𝑖𝑗𝑙𝑚
⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣
𝑄𝜏 − 𝑄𝑖 (𝑝(𝑘)) − 2𝑎𝐶 𝑇 𝐶 0 (𝑎 + 1)𝐶 𝐹𝐺 ¯ + (1 − 𝑝𝑖 )𝐵𝑌 𝑚 (𝑝)𝐶 𝑆𝑇 𝐴 𝜎2 𝑆 𝑇 𝐸 Δ𝑖𝑗 𝐵𝑌 𝑚 (𝑝)𝐶
∗ −𝑄𝜏 0 𝐹 𝐺𝑑 ¯ 𝑆𝑇 𝐷 0 0
∗ ∗ −2𝐼 0 𝑝𝑖 𝐵𝑌 𝑚 (𝑝) 0 Δ𝑖𝑗 𝐵𝑌 𝑚 (𝑝)
∗ ∗ ∗ −2𝐼 𝑆𝑇 𝑁 0 0
∗ ∗ ∗ ∗ ¯𝑙 −Λ 0 0
(19)
−Δ−1 𝑝 (𝑘)Λ
𝑄𝜏 − 𝑄(𝑝(𝑘)) − 2𝑎𝐶 𝑇 𝐶 0 (𝑎 + 1)𝐶 𝐹𝐺 ¯ + (1 − 𝑝(𝑘))𝑆 𝑇 𝐵𝐾(𝑝)𝐶 𝑆𝑇 𝐴 𝜎2 𝑆 𝑇 𝐸 Δ𝑝 (𝑘)𝑆 𝑇 𝐵𝐾(𝑝)𝐶
⎡
(29)
∗ ∗ ∗ ∗ ∗ ¯𝑙 −𝜎 2 Λ 0
2013 25th Chinese Control and Decision Conference (CCDC)
∗ ∗ ∗ ∗ ∗ ∗ ¯𝑙 −Δ𝑖𝑗 Λ
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎦
(21)
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎦
(24)
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ < 0, ⎥ ⎥ ⎦
(27)
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and the closed-system (8) is then exponentially meansquare stable for all 𝑝(𝑘) ∈ [𝑝1 𝑝2 ]. Proof : Firstly, set 𝛼1 (𝑘) =
𝑝2 − 𝑝(𝑘) , 𝑝2 − 𝑝1
𝛼2 (𝑘) =
𝑝(𝑘) − 𝑝1 , 𝑝2 − 𝑝1
(31)
with 𝛼𝑖 (𝑘) ≥ 0 (𝑖 = 1, 2) and 𝛼1 (𝑘) + 𝛼2 (𝑘) = 1. Similarly, let 𝛽1 (𝑘) =
𝑝2 − 𝑝(𝑘 + 1) , 𝑝2 − 𝑝1
𝑝(𝑘 + 1) − 𝑝1 , 𝑝2 − 𝑝1 (32)
𝛽2 (𝑘) =
and we have 𝑝(𝑘 + 1) = 𝛽1 (𝑘)𝑝1 + 𝛽2 (𝑘)𝑝2 ,
(33)
with 𝛽𝑖 (𝑘) ≥ 0 (𝑖 = 1, 2), 𝛽1 (𝑘) + 𝛽2 (𝑘) = 1. From the above transformation, we can easily get 𝑄(𝑝(𝑘))
=
2 ∑
𝛼𝑖 (𝑘)𝑄𝑖 (𝑝(𝑘)),
¯ = Λ
𝑖=1
𝑌 (𝑝(𝑘))
=
2 ∑
2 ∑
Set the time-varying Bernoulli distribution sequences as 𝑝(𝑘) = 𝑝1 + (𝑝2 − 𝑝1 )∣ sin(𝑘)∣ and the sector nonlinear function 𝑓 (𝑢) is taken by
(30)
therefore, we have 𝑝(𝑘) = 𝛼1 (𝑘)𝑝1 + 𝛼2 (𝑘)𝑝2 ,
𝑝1 = 0.19, 𝑝2 = 0.51, 𝜎 2 = 1, 𝑎 = 0.411.
𝑓 (𝑢) =
which satisfies (3). Also, select the initial state 𝜌 = [2 − 2]𝑇 . According to Theorem 2, the constant controller parameters 𝐾0 , 𝐾𝑢 can be obtained as follows: ] [ 5.0383 −24.7648 , 𝐾0 = −154.3846 758.7742 ] [ 0.0023 −0.0002 . 𝐾𝑢 = −0.0693 0.0063 Then, according to the measured time-varying probability parameters 𝑝(𝑘), the gain-scheduled controller gain 𝐾(𝑝) and parameter-dependent Lyapunov matrix 𝑄(𝑝(𝑘)) can be calculated at every time step 𝑘 as in Table 1 and Table 2, separately.
¯ 𝑙, 𝛽𝑙 (𝑘)Λ
Table 1: Computing results
𝑙=1
𝛼𝑚 (𝑘)𝑌 𝑚 (𝑝),
𝐹1 + 𝐹 2 𝐹2 − 𝐹 1 𝑢+ sin(𝑢), 2 2
(34)
𝑘
𝑝(𝑘)
0
0.4593
1
0.4810
2
0.4322
.. .
.. .
𝑚=1
Meanwhile, it is easy to find that 2 ∑
𝑖𝑗𝑙𝑚
𝛼𝑖 (𝑘)𝛼𝑗 (𝑘)𝛼𝑚 (𝑘)𝛽𝑙 (𝑘)𝕄
< 0.
(35)
𝑖,𝑗,𝑙,𝑚=1
From (30)-(35), we can have that (10) in Theorem 1 is true, then the proof is now complete.
Table 2: Computing results
Remark 4 By using the methods proposed in the proof of theorem 2, we choose 4 variables, then, it’s easy to calculate the computation complexity as 24 depend on the upper and lower bound of 𝑝(𝑘).
4
AN ILLUSTRATIVE EXAMPLE
In this section, the gain-scheduled static output feedback controller is designed for the discrete-time nonlinear stochastic systems with randomly occurring sensor saturations. The system parameters are given as follows: ] ] [ [ 0.13 0.21 0.97 0 , , 𝑁 = 𝐴= 0.28 0.33 0 0.21 ] ] [ [ 0 0.199 0.01 0 , , 𝐶= 𝐵= 0.6 2.209 86.1 2.81 ] [ ] [ 0.023 0.14 0.06 0 , 𝐹1 = 𝐷= , 0.15 0.18 0 0.01 ] ] [ [ 0.08 0.12 0.1 0 , , 𝐺= 𝐹2 = 0.08 0.02 0 0.01 ] ] [ [ 0.03 0.19 0.011 0.09 , 𝐺𝑑 = , 𝐸= 0.21 0.02 0.18 0.09
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𝐾(𝑝) ] 5.0394 −24.7649 [−154.4164 758.7771 ] 5.0394 −24.7649 −154.4179 758.7772 ] [ 5.0389 −24.7648 −154.4009 758.7757 .. . [
𝑘
𝑝(𝑘)
0
0.4593
1
0.4810
2
0.4322
.. .
.. .
𝑄(𝑝(𝑘)) ] 6.1127 −2.5731 6.4562 ] [−2.5731 6.1294 −2.5608 6.5132 ] [−2.5608 5.9408 −2.7008 −2.7008 5.8675 .. . [
Fig. 1 gives the response curves of state 𝑥(𝑘) of uncontrolled systems. Fig. 2 depicts the simulation results of state 𝑥(𝑘) of the controlled systems. The simulation results have illustrated our theoretical analysis.
5
CONCLUSIONS
In this paper, the gain-scheduled control problem for a class of discrete stochastic systems with randomly occurring sensor saturations has been studied, the sensor saturation phenomenon is assumed to occur in a randomly way based on time-varying Bernoulli distribution with measurable probability in real time. By employing probability-dependent Lyapunov functional, we design a gain-scheduled controller with the gain including both constant part and time-varying parameters such that,
2013 25th Chinese Control and Decision Conference (CCDC)
7 x (k) 1
6
x (k) 2
5 4 3 2 1 0 −1 −2
0
20
40
60
80
100
k
Figure 1: State evolution 𝑥(𝑘) of uncontrolled systems 2 x1(k) x2(k)
1.5 1 0.5 0 −0.5 −1 −1.5 −2
0
20
40
60
80
100
k
Figure 2: State evolution 𝑥(𝑘) of controlled systems
[5] G. Wei, Z. Wang and B. Shen, Probability-dependent gainscheduled control for discrete stochastic delayed systems with randomly occurring nonlinearities, International Journal of Robust and Nonlinear Control, Mar. 2012. [6] Y. Y. Cao, J. Lam and Y. X. Sun, Static output feedback stabiliztion: An LMI approach, Automatica, Vol.34, No.12, 16411645, 1998. [7] J. C. Geromel, R. Souza and R. E. Skelton, Static output feedback controllers: Stability and convexity, IEEE Trans. Automat. Contr., Vol. 43, No. 1, 120-125, 1998. [8] I. N. Kar, Design of static output feedback controller for uncertain systems, Automatica, Vol. 35, 169-175, 1999. [9] V. L. Syrmos, C. T. Abdallah, P. Dorato and K. Grigoriadis, Static output feedback a survey, Automatica, Vol. 33, 125137, 1997. [10] D. Ding, Z. Wang, B. Shen and H. Shu, 𝐻∞ state estimation for discrete-time complex networks with randomly occurring sensor saturations and randomly varying sensor delays, IEEE Transactions on Neural Networks and Learning Systems, Vol. 23, No. 5, 725-736, May. 2012. [11] Z. Wang, B. Shen, X. Liu, 𝐻∞ filtering with randomly occurring sensor saturations and missing measurements, Automatica, Vol. 48, No. 3, 556-562, Mar. 2012. [12] B. Shen, Z. Wang, J. Liang, and Y. Liu, Recent advances on filtering and control for nonlinear stochastic complex systems with incomplete information: a survey, Mathematical Problems in Engineering, Vol. 2012, Sep. 2011. [13] S. Boyd, L. EI Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, SIAM, Philadelphia, PA, 1994. [14] F. Yang, Z. Wang, Y. S. Hung and M. Gani, 𝐻∞ control for networked systems with random communication delays, IEEE Trans on Automatic Control, Vol. 61, No. 3, 511-518, 2006. [15] W. Li, G. Wei, and L. Wang, Probability-dependent static output feedback control for discrete-time nonlinear stochastic systems with missing measurements, Mathematical Problems in Engineering, Vol. 2012, Aug. 2012. [16] J. Qiu, G. Feng, and H. Gao, Fuzzy-model-based piecewise 𝐻∞ static-output-feedback controller design for networked nonlinear systems fuzzy systems, IEEE Transactions on, vol. 18, no. 5, 919-934, 2010.
for the admissible sensor saturations, time-delays and noise disturbances, the closed-loop system, which is still exponentially mean-square stable. Furthermore, we can extend the main results to more complex and realistic systems, for instance, complex networks system or system with norm-bounded or polytopic uncertainties. Meanwhile, we can also consider corresponding dynamic output feedback control and filtering problem as well as the relevant applications in networked control systems or sensor networks.
REFERENCES [1] P. Apkarian and R. J. Adams, Advanced gain-scheduling techniques for uncertain systems, IEEE Transactions On Control Systems Technology, Vol.6, No.1, 21-32, 1998. [2] Y. Y. Cao, Z. Lin and Y. Shamash, Set invariance analysis and gain-scheduling control for LPV systems subject to actuator saturation, Systems and Control Letters, Vol.46, 137151, 2002. [3] W. J. Rugh and J. S. Shamma, Research on gain scheduling, Automatica, Vol.36, 1401-1425, 2000. [4] G. Wei, Z. Wang and B.Shen, Probability-dependent gainscheduled filtering for stochastic systems with missing measurements, IEEE Transactions on Circuits and Systems II: Express Briefs, Vol. 58, No. 11, 753-757, 2011.
2013 25th Chinese Control and Decision Conference (CCDC)
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