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Novel observer-based output feedback control synthesis of discrete-time nonlinear control systems via a fuzzy approach
Author’s Accepted Manuscript Novel Observer-Based Output Feedback Control Synthesis of Discrete-Time Nonlinear Control Systems Via A Fuzzy Approach Guotao Hui, Xiangpeng Xie www.elsevier.com/locate/neucom
PII: DOI: Reference:
S0925-2312(16)30371-X http://dx.doi.org/10.1016/j.neucom.2016.05.033 NEUCOM17049
To appear in: Neurocomputing Received date: 22 September 2015 Revised date: 5 May 2016 Accepted date: 10 May 2016 Cite this article as: Guotao Hui and Xiangpeng Xie, Novel Observer-Based Output Feedback Control Synthesis of Discrete-Time Nonlinear Control Systems Via A Fuzzy Approach, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.05.033 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.
Novel Observer-Based Output Feedback Control Synthesis of Discrete-Time Nonlinear Control Systems Via A Fuzzy Approach
Guotao Hui a , Xiangpeng Xie b,c a College
of Information Science and Engineering, Northeastern University, Shenyang, 110004, China.
b Institute
of Advanced Technology, Nanjing University of Posts and Telecommunications, Nanjing 210003, China.
c School
of Automation, Huazhong University of Science and Technology, Wuhan 430074, China.
Abstract The paper is focus on less conservative design of observer-based output feedback control synthesis of discrete-time Takagi-Sugeno fuzzy systems on account of an improved homogenous polynomial approach. Compared with those previous methods in the cited literature, a novel fuzzy observer-based output feedback law is constructed in virtue of utilizing one promising solution to homogenous polynomials that are parameter-dependent on both the current-time and those multi-steps past-time normalized fuzzy weighting functions with a pair of prescribed degrees. It is rather remarkable that more information contained in the normalized fuzzy weighting functions is integrated to build the proposed control law, and thus the conservatism left behind previous results can be released in this work. Moreover, a robust version of the underlying result is also proposed. Finally, the effectiveness of
Preprint submitted to Elsevier
14 May 2016
the given approach is demonstrated via some illustrative experiments. Key words: Fuzzy model, Signal estimation, Output feedback, Observer design, Nonlinear control.
1
Introduction
For the past few years, stability analysis and control synthesis by the aid of Takagi-Sugeno (T-S) models have acted as one of the most popular topics in data-based control fields [1,2]. Indeed, over the past decade tremendous investigations have been devoted to the study of T-S control systems, such as those important results: stabilization of fuzzy-model-based systems [3–7], distributed filtering for fuzzy-model-based systems [8–10], fuzzy fault detection for nonlinear systems that are with sensor fault or limited communication capacity [11], adaptive sliding-mode control for fuzzy-model-based systems [12–14], adaptive fuzzy identification and control of uncertain fuzzy systems [15–18], applications to various actual models [19,20], and so on. It is noteworthy that depending on the choice of different Lyapunov functions, the derived conditions belong to different levels of conservatism [21]. In spite of the low-complexity of the obtained results, the simple and common Lyapunov functions may lead to a bank of conservative design conditions. The main reason for this deficiency is that the requirement of some monotonically decreasing conditions for one prescribed common Lyapunov matrix is sometimes too strict to get any set of feasible solutions. For the purpose of getting Corresponding author X. Xie at: Institute of Advanced Technology, Nanjing University of Posts and Telecommunications, Nanjing 210003, China. Email address: [email protected] (Xiangpeng Xie).
2
the major deficiency over, several kinds of featured Lyapunov functions were proposed in the past decade. Particularly, the effectiveness of higher-degree parameter-dependent Lyapunov functions has been well illustrated in [22,23], respectively. This class of Lyapunov functions seems to be one of the best alternatives to seeking more efficient results, especially for the case of discrete-time T-S fuzzy systems, e.g., [24] and the cited references therein.
Also, it is rather remarkable that most of existing research literature are derived with the assumption that all the system states could be obtainable, but it isn’t usually satisfied in real-life operations [25]. By way o estimating the system states, a series of important work were gained successively in the past years, for example, nonfragile H∞ filtering of continuous-time fuzzy systems [26], filter design and model reduction for interval Type-2 fuzzy systems with D stability constraints [27,28], output feedback control of fuzzy stochastic systems [29], robust H∞ filter design for affine fuzzy systems [30], more efficient fuzzy observer designs [31], dynamic output feedback control of T–S fuzzy systems [32], fuzzy observer-based output feedback control designs [33]. Acting as a landmark result, the obtained design conditions in [33] are given for ensuring the existence of fuzzy observer-based controllers in terms of bilinear matrix inequalities, which are not convex and very difficult to be solved. Accompanied by a compromise, a two-step procedure is utilized but a great deal of conservatism is inevitably introduced. Recently, a single-step result has been offered for achieving the linear matrix inequality(LMI)-based observer-based controller design by the authors in [34]. It has been illustrated in [34] that both the controller and the observer’s parameters can be captured by solving a bank of strict LMIs which are numerically feasible with the usage of the famous Matlab toolbox. More recently, another observer-based output feedback 3
law that relies on two or more steps normalized fuzzy weighting functions have been proposed in [35] and [36], respectively.
Based on the above technical discussion, the main purpose of the work is to pursue more relaxed fuzzy observer-based output feedback controller designs by developing an improved homogenous polynomial approach. Compared with those existing results which are merely linearly parameter-dependent on the current-time/past-time normalized fuzzy weighting functions with a lower degree, a novel fuzzy observer-based output feedback controller is developed on account of an improved homogenous polynomials approach. Owing to the reason that added information contained in the normalized fuzzy weighting functions is integrated into the proposed observer-based control law, the conservatism induced by the existing results can be released in this work. Moreover, a robust version of the underlying result is also proposed in this study.
2
Background and Preliminaries
In this work, an asterisk (∗) that is located inside some matrix represents the transpose of its symmetric term; Z+ stands for the set of all positive integers; R denotes the set of all real numbers; p! represents the factorial, e.g., z! = z(z − 1) · · · (1) in case of z ∈ Z+ and 0! = 1; If M represents a matrix, then He(M ) means He(M ) = M + M T . And, the left-hand side of a relation can be defined as Left(·).
Based on the powerful T-S model, a family of discrete-time nonlinear systems 4
are represented like the one used in [34]: ⎧ r ⎪ ⎪ ⎪ x(t + 1) = hi (z(t))(Ai x(t) + Bi u(t)) ⎪ ⎨ i=1
r ⎪ ⎪ ⎪ ⎪ hi (z(t))Ci x(t) ⎩ y(t) =
(1)
i=1
and x(t) ∈ Rn1 denotes the system state vector and is unmeasurable; u(t) ∈ Rn2 represents the control input; y(t) ∈ Rn3 denotes the measured output vector; Ai , Bi and Ci are a bank of known system matrices; hi (z(t)) represents the i-th normalized fuzzy weighting function.
A batch of vital definitions about homogeneous matrix polynomials are shown in this section, which are consistent with the cited references, such as [22–24] and [36], and so on. The set Δr is described as {α ∈ Rr ;
r i=1
αi = 1; α ≥ 0}. And here α1k1 α2k2 · · · αrkr
means the monomials while one has α ∈ r , ki ∈ Z+ , i ∈ {1, · · · , r} and k = k1 k2 · · · kr . K(g) represents the set of r-tuples induced by collecting all possible combinations of nonnegative integers ki , i ∈ {1, · · · , r}, satisfying that k1 + k2 + · · · + kr = g. Pk ∈ Rn×n , ∀k ∈ K(g) represent matrix-valued coefficients. As an illustrative example, if some homogeneous polynomials of degree g = 3 are with r = 3 variables, then the possible values of the partial degrees are K(3) = {300, 030, 003, 210, 201, 120, 021, 102, 012, 111}. As those prescribed definitions that are utilized in [24], for r-tuples k and k , one denotes k ≥ k if ki ≥ ki , (i = 1, ..., r). The usual operations of summation, k +k , and subtraction, k −k (whenever k ≥ k ), are operated componentwise. Similarly, another two key definitions about the r-tuple χi ∈ K(1) and the coefficient π(k), ∀k ∈ K(g) are given in the following form: 5
χi =
0
···0
1−th element
0···
1
i−th element
0
,
r−th element
π(k) = (k1 !) × (k2 !) × · · · × (kr !).
(2)
For another illustrative example, once the parameter of involved homogeneous polynomials is chosen as g = 2 and r = 2, we get χ1 = 1 0, χ2 = 0 1, π(2 0) = (2!) × (0!) = 2, π(1 1) = (1!) × (1!) = 1 and π(0 2) = (0!) × (2!) = 2. Given one homogeneous polynomial with r variables (i.e., k = k1 k2 · · · kr ) and j ∈ {1, · · · , m} with m ∈ Z+ , related shortenings are selected like those in [35]: ⎧ ⎪ ⎪ ⎪ hi = hi (z(t)), hi (t + 1) = hi (z(t + 1)), hi (t − j) = hi (z(t − j)), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ h(t − j) = [h (t − j), · · · , h (t − j)]T , h = [h , · · · , h ]T , 1
1
r
r
⎪ ⎪ ⎪ ⎪ h(t + 1) = [h1 (t + 1), · · · , hr (t + 1)]T , hk = hk11 hk22 · · · hkr r , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ h(t − j)k = h1 (t − j)k1 h2 (t − j)k2 · · · hr (t − j)kr .
Two multi-instant homogeneous matrix polynomials with a pair of prescribed parameters (g1 , g2 ) are represented as follows:
Pzg1 (t−m)zg2 (t) =
m
1 ),i∈{1··· ,m}; (a(i)∈K(g ) k ∈K(g2 )
Pzg1 (t−m+1)zg2 (t+1) =
h(t − i)
a(i)
hk Pa(m)···a(1)k ,
i=1
1 ),i∈{1··· ,m}; (a(i)∈K(g ) k ∈K(g2 )
m
h(t − i + 1)
a(i)
h(t + 1)k Pa(m)···a(1)k ,
i=1
while m, g1 , g2 ∈ Z+ , and Pa(m)···a(1)k is a matrix-valued coefficient. Given an illustrative example for r = 2, m = 1 and g1 = g2 = 2, one gets Pzg1 (t−m)zg2 (t) = h(t − 1)20 h20 P2020 + h(t − 1)20 h11 P2011 + h(t − 1)20 h02 P2002 + h(t − 1)11 h20 P1120 + h(t − 1)11 h11 P1111 + h(t − 1)11 h02 P1102 + h(t − 1)02 h20 P0220 + h(t − 1)02 h11 P0211 + h(t − 1)02 h02 P0202 . 6
3
Main results
In this work, a new fuzzy observer-based control law is constructed by the following form: ⎧ ⎪ ⎪ ⎪ xˆ(t + 1) = Az(t) xˆ(t) + Bz(t) u(t) + G−1 ˆ(t)), ⎪ zg3 (t−m)zg4 (t) Kzg1 (t−m)zg2 (t) (y(t) − y ⎪ ⎪ ⎪ ⎨
yˆ(t) = Cz(t) xˆ(t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u(t) = α−1
ˆ(t), zg5 (t−m)zg6 (t) Fzg1 (t−m)zg2 (t) x (3)
with m ∈ Z+ , Kzg1 (t−m)zg2 (t) , Gzg3 (t−m)zg4 (t) , Fzg1 (t−m)zg2 (t) and αzg5 (t−m)zg6 (t) are multi-instant homogenous matrix polynomials of degree (g1 , g2 ) and (g3 , g4 ), respectively (g1 , g2 , g3 , g4 ∈ Z+ ), and one has
Kzg1 (t−m)zg2 (t) = Gzg3 (t−m)zg4 (t) = Fzg1 (t−m)zg2 (t) = αzg5 (t−m)zg6 (t) =
m
),i∈{1,2,...,m}; (a(i)∈K(gk1 ∈K(g ) 2)
),i∈{1,2,...,m}; (a(i)∈K(gk3 ∈K(g ) 4)
),i∈{1,2,...,m}; (a(i)∈K(gk1 ∈K(g ) 2)
),i∈{1,2,...,m}; (a(i)∈K(gk5 ∈K(g ) 6)
h(t − i)
a(i)
hk Ka(m)···a(1)k ,
i=1
m
h(t − i)
a(i)
hk Ga(m)···a(1)k ,
i=1
m
h(t − i)
a(i)
hk Fa(m)···a(1)k ,
i=1
m
h(t − i)
a(i)
h(t)k αa(m)···a(1)k ,
i=1
In addition, Ka(m)···a(1)k ∈ Rn1 ×n3 , Ga(m)···a(1)k ∈ Rn1 ×n1 , and Fa(m)···a(1)k ∈ Rn2 ×n1 are a set of matrices to be resolved, αa(m)···a(1)k are scalars to be resolved. Remark 1 : Particularly, the above fuzzy observer-based control law (3) may be reducible to the previous one given in [36] if one set g1 = g2 = g3 = g4 = g5 = g6 = 1. In a word, the previous one given in [36] can be thought to be a special case of this work. Compared with those previous methods in the cited 7
literature, (3) is constructed in virtue of utilizing one promising solution to homogenous polynomials that are parameter-dependent on both the currenttime and those multi-steps past-time normalized fuzzy weighting functions with a pair of prescribed degrees. It is rather remarkable that more information contained in the normalized fuzzy weighting functions is integrated to build the proposed control law, and thus the conservatism left behind previous results can be released in this work. Embedding the above observer-based control law (3) to the origin system, the closed-loop system turns into another form: ⎛
⎞
⎜ ⎜ Υ(1, 1) ⎜ x˜(t + 1) = ⎜ ⎜ ⎜ ⎝
0
−αz−1 g (t−m)zg 5
Az(t) − G−1 zg (t−m)zg
⎛ ⎜ ⎜ ⎜ where x˜(t) = ⎜ ⎜ ⎜ ⎝
6 (t)
3
Bz(t) Fzg1 (t−m)zg2 (t)
4 (t)
Kzg1 (t−m)zg2 (t) Cz(t)
⎟ ⎟ ⎟ ⎟x ⎟ ˜(t), ⎟ ⎠
(4)
⎞
x(t) x(t) − xˆ(t)
⎟ ⎟ ⎟ ⎟ and Υ(1, 1) = Az(t) +α−1 ⎟ zg5 (t−m)zg6 (t) Bz(t) Fzg1 (t−m)zg2 (t) . ⎟ ⎠
For the sake of achieving the task of observer-based control design, a multiinstant Lyapunov function candidate is proposed with the help of multi-instant homogeneous matrix polynomials: ⎛
⎞
⎜ ⎜ P1zg (t−m)zg (t) 5 6 ⎜ T V (˜ x(t)) = x˜ (t) ⎜ ⎜ ⎜ ⎝
0
among (5), P1zg5 (t−m)zg6 (t) = P2zg5 (t−m)zg6 (t) = m, g5 , g6 ∈
0 P2zg5 (t−m)zg6 (t)
a(i)∈K(g5
),i∈{1,··· ,m};k ∈K(g
m i=1
(5)
h(t − i)a(i) hk P1a(m)···a(1)k ,
h(t − i)a(i) hk P2a(m)···a(1)k , and
k(i)∈K(g5 ),i∈{1,··· ,m};k ∈K(g6 ) Z+ , P1k(m)···k(1)k ∈ Rn1 ×n1 , P2k(m)···k(1)k
8
6)
m i=1
⎟ ⎟ ⎟ ⎟x ⎟ ˜(t), ⎟ ⎠
∈ Rn1 ×n1 .
Theorem 1. The closed-loop system (4) is called to be asymptotically stable, if there exist matrices Fa(m)···a(1)k ∈ Rn2 ×n1 and Ka(m)···a(1)k ∈ Rn1 ×n3 (a(j) ∈ K(g1 ), j ∈ {1, 2, ..., m}, k ∈ K(g2 )); matrices Ga(m)···a(1)k ∈ Rn1 ×n1 (a(j) ∈ K(g3 ), j ∈ {1, 2, ..., m}, k ∈ K(g4 )); symmetric matrices P1a(m)···a(1)k ∈ Rn1 ×n1 and P2a(m)···a(1)k ∈ Rn1 ×n1 , αa(m)···a(1)k , (a(j) ∈ K(g5 ), j ∈ {1, 2, ..., m}, k ∈ K(g6 )); such that all the LMIs in terms of (6) are simultaneously satisfied: ⎡
⎤
⎢ 11 ∗ ∗ ⎢ Γp(m)···p(1)qk ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 0 Γ22 ∗ p(m)···p(1)qk ⎢ ⎢ Γp(m)···p(1)qk = ⎢ ⎢ ⎢ 31 32 33 ⎢Γ ⎢ p(m)···p(1)qk Γp(m)···p(1)qk Γp(m)···p(1)qk ⎢ ⎢ ⎢ ⎣ 42
0
Γp(m)···p(1)qk
0
∗ ∗ ∗ Γ44 p(m)···p(1)qk
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ > 0, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(6)
∀ p(j) ∈ K(g7 + d1 ); j = 1, 2, ..., m; q ∈ K(g6 + d2 ); k ∈ K(g8 + d3 ); with g7 = max{g1 , g3 , g5 }, g8 = max{g2 + 1, g4 + 1, g5 , g6 + 1},
Γ11 p(m)···p(1)qk =
5 ),j∈{1,2,...,m}; (a(j)∈K(g p(j)≥a(j);k ∈K(g6 );k≥k )
⎧⎛ ⎞ m ⎨ (g7 + d1 − g5 )! ⎠ (g8 + d3 − g6 )! ⎝ ⎩ π(k − k ) j=1 π(p(j) − a(j))
(g6 + d2 )! P1a(m)···a(1)k , × π(q) Γ22 p(m)···p(1)qk =
5 ),j∈{1,2,...,m}; (a(j)∈K(g p(j)≥a(j);k ∈K(g6 );k≥k )
⎧⎛ ⎞ m ⎨ (g + d − g )! (g + d3 − g6 )! 7 1 5 ⎠ 8 ⎝ ⎩ π(k − k ) j=1 π(p(j) − a(j))
(g6 + d2 )! P2a(m)···a(1)k , × π(q) Γ33 p(m)···p(1)qk =
5 ),j∈{1,2,...,m}; (a(j)∈K(g p(j)≥a(j);k ∈K(g6 );k≥k )
⎧⎛ ⎞ m ⎨ (g + d − g )! (g + d3 − g6 )! 7 1 5 ⎠ 8 ⎝ ⎩ π(k − k ) j=1 π(p(j) − a(j))
×
(g6 + d2 )! 2αa(m)···a(1)k I − π(q)
5 ),j∈{2,...,m};p(j)≥a(j+1); (a(j)∈K(g a(1)∈K(g5 );k≥a(1),k ∈K(g6 ),q≥k )
9
⎧⎛ ⎞ ⎨ m−1
(g + d − g )! 7 1 5 ⎠ ⎝ ⎩ π(p(j) − a(j + 1)) j=1
d2 ! (g7 + d1 )! (g8 + d3 − g5 )! P1a(m)···a(1)k , × π(p(m)) π(k − a(1)) π(q − k )
⎧⎛ ⎞ m ⎨ (g + d − g )! 7 1 3 44 ⎠ ⎝ Γp(m)···p(1)qk = ⎩ π(p(j) − a(j)) j=1 a(j)∈K(g3 ),j∈{1,2,...,m};p(j)≥a(j);k ∈K(g4 );k≥k (g + d − g )! (g + d )!
×
8
3
4
π(k − k )
6
2
Ga(m)···a(1)k + GTa(m)···a(1)k
π(q)
⎧⎛ ⎞ ⎨ m−1
(g + d − g )! 7 1 5 ⎠ ⎝ − ⎩ π(p(j) − a(j + 1)) j=1 a(j)∈K(g5 ),j∈{2,...,m};p(j)≥a(j+1);a(1)∈K(g5 );k≥a(1),k ∈K(g6 ),q≥k
×
d2 ! (g7 + d1 )! (g8 + d3 − g5 )! P2a(m)···a(1)k , π(p(m)) π(k − a(1)) π(q − k )
Γ31 p(m)···p(1)qk =
5 ),j∈{1,2,...,m};p(j)≥a(j); (a(j)∈K(g i∈{1,2,...,r},k ∈K(g6 ),k−k −χi ≥0 )
×
(g6 + d2 )! αa(m)···a(1)k Ai + π(q)
⎧⎛ ⎞ m ⎨ (g + d − g )! (g + d3 − g6 − 1)! 7 1 5 ⎠ 8 ⎝ ⎩ π(k − k − χi ) j=1 π(p(j) − a(j))
1 ),j∈{1,2,...,m};p(j)≥a(j); (a(j)∈K(g i∈{1,2,...,r},k ∈K(g2 ),k−k −χi ≥0 )
⎧⎛ ⎞ m ⎨ (g + d − g )! 7 1 1 ⎠ ⎝ ⎩ π(p(j) − a(j)) j=1
×
(g8 + d3 − g2 − 1)! (g6 + d2 )! Bi Fa(m)···a(1)k , π(k − k − χi ) π(q)
Γ32 p(m)···p(1)qk = −
1 ),j∈{1,2,...,m};p(j)≥a(j); (a(j)∈K(g i∈{1,2,...,r},k ∈K(g2 ),k−k −χi ≥0 )
⎧⎛ ⎞ m ⎨ (g + d − g )! (g + d3 − g2 − 1)! 7 1 1 ⎠ 8 ⎝ ⎩ π(k − k − χi ) j=1 π(p(j) − a(j))
(g6 + d2 )! Bi Fa(m)···a(1)k , × π(q) Γ42 p(m)···p(1)qk =
3 ),j∈{1,2,...,m};p(j)≥a(j); (a(j)∈K(g i∈{1,2,...,r},k ∈K(g4 ),k−k −χi ≥0 )
(g6 + d2 )! Ga(m)···a(1)k Ai − × π(q)
⎧⎛ ⎞ m ⎨ (g + d − g )! (g + d3 − g4 − 1)! 7 1 3 ⎠ 8 ⎝ ⎩ π(k − k − χi ) j=1 π(p(j) − a(j))
1 ),j∈{1,2,...,m};p(j)≥a(j); (a(j)∈K(g i∈{1,2,...,r},k ∈K(g2 ),k−k −χi ≥0 )
⎧⎛ ⎞ m ⎨ (g + d − g )! 7 1 1 ⎠ ⎝ ⎩ π(p(j) − a(j)) j=1
×
(g8 + d3 − g2 − 1)! (g6 + d2 )! Ka(m)···a(1)k Ci . π(k − k − χi ) π(q)
PROOF. Taking into account the Lyapunov function candidate in term of (5), its one-step difference (i.e., ΔV (˜ x(t))) along the solution to the estimation error (4) can be written as the following equation: 10
⎛
⎛
⎞⎞
⎜ ⎜ ⎜ ⎜ P1zg (t−m)zg (t) 5 6 ⎜ ⎜ T T ⎜ ΔV (˜ x(t)) = x˜ (t) ⎜Υ ΞΥ − ⎜ ⎜ ⎜ ⎜ ⎝ ⎝
0
⎛
0 P2zg5 (t−m)zg6 (t)
⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ x ⎟⎟ ˜(t), (7) ⎟⎟ ⎠⎠
⎞
⎜ ⎜ P1zg (t−m+1)zg (t+1) 5 6 ⎜ with Ξ = ⎜ ⎜ ⎜ ⎝
0
⎛
0 P2zg5 (t−m+1)zg6 (t+1)
⎜ ⎜ Az(t) + αz−1 (t−m)z (t) Bz(t) Fzg (t−m)zg (t) g5 g6 1 2 ⎜ Υ=⎜ ⎜ ⎜ ⎝
0
⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠ ⎞
−αz−1 g (t−m)zg 5
Bz(t) Fzg1 (t−m)zg2 (t) 6 (t)
Az(t) − G−1 zg (t−m)zg 3
4 (t)
Kzg1 (t−m)zg2 (t) Cz(t)
Thus, the estimation error (4) can be called to be globally asymptotically stable if the following inequality is ensured: ⎛
⎞
⎜ ⎜ P1zg (t−m)zg (t) 5 6 ⎜ ΥT ΞΥ − ⎜ ⎜ ⎜ ⎝
0
0 P2zg5 (t−m)zg6 (t)
⎟ ⎟ ⎟ ⎟ < 0. ⎟ ⎟ ⎠
(8)
Therefore, it is rather remarkable that (8) can be satisfied if the matrix inequality (9) holds in true: ⎡
⎤
⎢ ∗ ⎢ P1zg5 (t−m)zg6 (t) ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 0 P2zg5 (t−m)zg6 (t) ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Ω31 −Bz(t) Fzg1 (t−m)zg2 (t) ⎢ ⎢ ⎢ ⎢ ⎣
0
Ω42
∗
∗ ⎥ ⎥
∗
∗
Ω33
∗
0
Ω44
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ > 0, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(9)
with Ω31 = αzg5 (t−m)zg6 (t) Az(t) + Bz(t) Fzg1 (t−m)zg2 (t) , Ω33 = 2αzg5 (t−m)zg6 (t) I − 11
⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠
P1zg5 (t−m+1)zg6 (t+1) , Ω42 = Gzg3 (t−m)zg4 (t) Az(t) −Kzg1 (t−m)zg2 (t) Cz(t) , Ω44 = He(Gzg3 (t−m)zg4 (t) )− P2zg5 (t−m+1)zg6 (t+1) . Thereupon, a key equality is utilized by representing the expression of Left(9): ⎡
⎤
⎢ ∗ ⎢ φ11 P1zg5 (t−m)zg6 (t) ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 0 φ22 P2zg5 (t−m)zg6 (t) ⎢ Left(9) = ⎢ ⎢ ⎢ ⎢ ˜ 31 ⎢ Ω −φ32 Bz(t) Fzg1 (t−m)zg2 (t) ⎢ ⎢ ⎢ ⎢ ⎣
˜ 42 Ω
0
m
= (
p(j)∈K(g7 +d1 ),j∈{1,2,...,m}; q∈K(g6 +d2 );k∈K(g8 +d3 )
∗
∗ ⎥ ⎥
∗
∗
˜ 33 Ω
∗
0
˜ 44 Ω
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
h(t − l)
p(l)
q
k
h h(t + 1) Γp(m)···p(1)qk ,
l=1
)
(10)
and, φ11 = φ22 =
φ32 =
m
r
l=1
i=1
m
r
l=1
i=1 !g7 +d1 −g1
hi (t − l)
hi (t − l)
!g7 +d1 −g5 r
r
!g8 +d3 −g6
hi
i=1 !g8 +d3 −g2 −1
i=1 r
hi
i=1
φ33 = ˜ 31 = Ω
i=1 r
!g7 +d1 m−1
hi (t − m) !−1
hi
i=1
l=1
r i=1
φ42 =
m
r
l=1
i=1
hi (t − l)
!g7 +d1 −g5
r
φ44 =
m
r
l=1
i=1
,
!g8 +d3 −g5
r
hi
i=1
!d 2
hi (t + 1)
i=1
φ11 αzg5 (t−m)zg6 (t) Az(t) + φ32 Bz(t) Fzg1 (t−m)zg2 (t) ,
hi (t − l)
!g7 +d1 −g3
r
!g8 +d3 −g4 −1
r
hi
i=1
!g6 +d2
hi (t + 1)
hi (t − l)
!g7 +d1 −g3
r
,
i=1
˜ 44 = φ44 He(Gzg (t−m)zg (t) ) − φ33 P2zg (t−m+1)zg (t+1) , Ω 3 4 5 6
,
!g6 +d2
hi (t + 1)
˜ 42 = φ42 Gzg (t−m)zg (t) Az(t) − φ32 Kzg (t−m)zg (t) Cz(t) , Ω 3 4 1 2
!g6 +d2
hi (t + 1)
i=1
˜ 33 = 2φ11 αzg (t−m)zg (t) I − φ33 P1zg (t−m+1)zg (t+1) , Ω 5 6 5 6 r
r
!g8 +d3 −g4
hi
i=1
r
!g6 +d2
hi (t + 1)
,
i=1
and g7 = max{g1 , g3 , g5 }, g8 = max{g2 +1, g4 +1, g5 , g6 +1}. More importantly, Γp(m)···p(1)qk is given in the equation (6). On the basis of the above equality (10), the inequality (9) can be evidently 12
,
ensured once if one gets Γp(m)···p(1)qk > 0 for all possible combinations of p(m) · · · p(1)qk. As a consequence, it is concluded that the closed-loop system (4) can be called to be asymptotically stable if all the LMIs in terms of (6) hold on. This completes the proof.
Remark 2 : It is rather remarkable that the proposed result via T-S fuzzy models are on account of an implicit assumption that the controller will be implemented exactly. Anyway, the uncertainties do occur in the implementation of a designed controller in real-life systems. Such uncertainties can be due to, among different reasons, roundoff errors in numerical computation during the controller implementation and the need to offer practicing engineers with safe-tuning margins. In what follows, similar to previous results of [10,26], the result given in Theorem 1 of this paper can be easily extended to robust control designs. In this work, the polytopic-type representation (i.e., Ai = w s=1
w s=1
αs Asi , Bi =
w s=1
αs Bis and Ci =
w s=1
αs Cis while
αi = 1, αs ≥ 0 and w ∈ Z+ ) is applied for modeling uncertainties, and
the following result is obtained. Corollary 1. While one has Ai = w s=1
w s=1
αs Asi , Bi =
w s=1
αs Bis and Ci =
αs Cis , the closed-loop system (4) is called to be robust asymptotically sta-
ble, if there exist matrices Fa(m)···a(1)k ∈ Rn2 ×n1 and Ka(m)···a(1)k ∈ Rn1 ×n3 (a(j) ∈ K(g1 ), j ∈ {1, 2, ..., m}, k ∈ K(g2 )); matrices Ga(m)···a(1)k ∈ Rn1 ×n1 (a(j) ∈ K(g3 ), j ∈ {1, 2, ..., m}, k ∈ K(g4 )); symmetric matrices P1a(m)···a(1)k ∈ Rn1 ×n1 and P2a(m)···a(1)k ∈ Rn1 ×n1 , αa(m)···a(1)k , (a(j) ∈ K(g5 ), j ∈ {1, 2, ..., m}, k ∈ K(g6 )); such that all the LMIs in terms of (11) are simultaneously satisfied: 13
⎡
⎤
⎢ 11 ∗ ∗ ⎢ Γp(m)···p(1)qk ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 0 Γ22 ∗ p(m)···p(1)qk ⎢ s ⎢ Γp(m)···p(1)qk = ⎢ ⎢ ⎢ 31s 32s 33 ⎢Γ ⎢ p(m)···p(1)qk Γp(m)···p(1)qk Γp(m)···p(1)qk ⎢ ⎢ ⎢ ⎣ 42s
0
Γp(m)···p(1)qk
0
∗ ∗ ∗ Γ44 p(m)···p(1)qk
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ > 0, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(11)
∀ p(j) ∈ K(g7 + d1 ); j = 1, 2, ..., m; q ∈ K(g6 + d2 ); k ∈ K(g8 + d3 ); s ∈ {1, · · · , w};
22 33 with g7 = max{g1 , g3 , g5 }, g8 = max{g2 +1, g4 +1, g5 , g6 +1}, Γ11 p(m)···p(1)qk , Γp(m)···p(1)qk , Γp(m)···p(1)qk
and Γ44 p(m)···p(1)qk are the same as those in Theorem 1, and
Γ31s p(m)···p(1)qk =
5 ),j∈{1,2,...,m};p(j)≥a(j); (a(j)∈K(g i∈{1,2,...,r},k ∈K(g6 ),k−k −χi ≥0 )
(g6 + d2 )! αa(m)···a(1)k Asi + × π(q)
⎧⎛ ⎞ m ⎨ (g + d − g )! 7 1 5 ⎠ (g8 + d3 − g6 − 1)! ⎝ ⎩ π(p(j) − a(j)) π(k − k − χi ) j=1
1 ),j∈{1,2,...,m};p(j)≥a(j); (a(j)∈K(g i∈{1,2,...,r},k ∈K(g2 ),k−k −χi ≥0 )
⎧⎛ ⎞ m ⎨ (g + d − g )! 7 1 1 ⎠ ⎝ ⎩ π(p(j) − a(j)) j=1
(g8 + d3 − g2 − 1)! (g6 + d2 )! s Bi Fa(m)···a(1)k , × π(k − k − χi ) π(q)
Γ32s p(m)···p(1)qk = −
1 ),j∈{1,2,...,m};p(j)≥a(j); (a(j)∈K(g i∈{1,2,...,r},k ∈K(g2 ),k−k −χi ≥0 )
⎧⎛ ⎞ m ⎨ (g + d − g )! 7 1 1 ⎠ (g8 + d3 − g2 − 1)! ⎝ ⎩ π(p(j) − a(j)) π(k − k − χi ) j=1
(g6 + d2 )! s Bi Fa(m)···a(1)k , × π(q) Γ42s p(m)···p(1)qk =
3 ),j∈{1,2,...,m};p(j)≥a(j); (a(j)∈K(g i∈{1,2,...,r},k ∈K(g4 ),k−k −χi ≥0 )
(g6 + d2 )! Ga(m)···a(1)k Asi − × π(q)
⎧⎛ ⎞ m ⎨ (g + d − g )! 7 1 3 ⎠ (g8 + d3 − g4 − 1)! ⎝ ⎩ π(p(j) − a(j)) π(k − k − χi ) j=1
1 ),j∈{1,2,...,m};p(j)≥a(j); (a(j)∈K(g i∈{1,2,...,r},k ∈K(g2 ),k−k −χi ≥0 )
(g8 + d3 − g2 − 1)! (g6 + d2 )! Ka(m)···a(1)k Cis . × π(k − k − χi ) π(q)
14
⎧⎛ ⎞ m ⎨ (g + d − g )! 7 1 1 ⎠ ⎝ ⎩ π(p(j) − a(j)) j=1
4
Numerical Experiments
Example. In this section, the same nonlinear model as the one in [35] is adopted for needful comparisons. ⎧ ⎪ ⎪ ⎪ x1 (t + 1) = x1 (t) − x1 (t)x2 (t) + (5 + x1 (t))u(t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
x2 (t + 1) = −x1 (t) − 0.5x2 (t) + 2x1 (t)u(t),
⎪ ⎪ ⎪ ⎪ ⎪ β − 1 β + 1 ⎪ ⎪ ⎪ + x1 (t) x1 (t) + x2 (t), ⎩ y(t) =
2
(12)
2β
where x1 (t) is assumed to be measurable and x2 (t) is assumed to be unmeasurable. β is a varying variable with β > 0. The following discrete-time T-S fuzzy model is employed for representing the involved nonlinear systems: Rule 1: If x1 (m) is β, then ⎛
⎞
⎡
⎜ ⎟ ⎢ ⎜ x1 (t + 1) ⎟ ⎢ 1 ⎜ ⎟ ⎢ ⎜ ⎟=⎢ ⎜ ⎟ ⎢ ⎜ ⎟ ⎢ ⎝ ⎠ ⎣
⎞
⎥
⎟
⎡
⎤
⎢ ⎥ ⎢5 + β ⎥ ⎢ ⎥ ⎢ ⎥ u(t), ⎟+⎢ ⎥ ⎟ ⎢ ⎥ ⎠ ⎣ ⎦
⎜ ⎜ x (t) ⎟ −β ⎥ ⎥⎜ 1 ⎟ ⎥ ⎟
−1 −0.5
x2 (t + 1)
⎤⎛ ⎜ ⎥⎜ ⎥⎜ ⎦⎝
x2 (t)
2β
⎞
⎛
⎟ ⎤⎜ ⎜ x1 (t) ⎟ ⎟ ⎜ ⎟, y(t) = ⎣ β 1 ⎦ ⎜ ⎟ ⎜ ⎟ ⎜ ⎠ ⎝ ⎡
x2 (t)
Rule 2: If x1 (t) is −β, then ⎛
⎞
⎡
⎤
⎜ ⎟ ⎢ ⎜ x1 (t + 1) ⎟ ⎢ 1 ⎜ ⎟ ⎢ ⎜ ⎟=⎢ ⎜ ⎟ ⎢ ⎜ ⎟ ⎢ ⎝ ⎠ ⎣
x2 (t + 1)
⎡
⎤
⎥
⎢ ⎥ ⎢5 − β ⎥ ⎢ ⎥ ⎥ x(t) + ⎢ ⎥ u(t). ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎦ ⎣ ⎦
β ⎥ ⎥
−1 −0.5
−2β
15
⎞
⎛
⎟ ⎤⎜ ⎜ x1 (t) ⎟ ⎟ ⎜ ⎟. y(t) = ⎣ 1 1 ⎦ ⎜ ⎟ ⎜ ⎟ ⎜ ⎠ ⎝ ⎡
x2 (t)
Borrowed from that in [35], two involved normalized fuzzy weighting functions are chosen as: h1 (z(t)) = (β + x1 (t))/2β, h2 (z(t)) = (β − x1 (t))/2β. For the purpose of giving clear comparisons, both the previous results ([35] and [36]) and the proposed result act on the underlying nonlinear systems. As a matter of convenience, some shortenings are adopted as follows: Case 1 of [35] means Theorem 1 of [35] with d1 = d2 = d3 = 1; Case 2 of [35] means Theorem 1 of [35] with d1 = d2 = d3 = 2; Case 1 of [36] means Theorem 1 of [36] with d1 = 1, d2 = 4; Case 2 of [36] means Theorem 1 of [36] with d1 = 1, d2 = 5; Case 1 of this paper means Theorem 1 of this paper with g1 = g3 = g5 = 2, g2 = g4 = g6 = 2 and d1 = d2 = d3 = 1; Case 2 of this paper means Theorem 1 of this paper with g1 = g3 = g5 = 1, g2 = g4 = g6 = 2 and d1 = d2 = d3 = 2.
Every method has its feasible stabilization region, i.e., (0, β]. It is plain to us that the stabilization region becomes more relaxed as its maximum of feasible β is bigger. The maximum of feasible β are calculated through the above six methods, respectively. The calculated results about β are listed in Table I. From Table I, it is easy to observe that the proposed methods in this work offer more bigger values of β than the results given in [34] and [35]. This reality shows that the relaxation quality has been improved on account of using the approach in this paper. In the case of β = 1.639, it is unfortunate that the results provided by [34] 16
Table 1 Maximum of feasible β via previous methods and the proposed method Methods
Case 1 of [35]
Case 2 of [35]
β
1.507
1.542
Methods
Case 1 of [36]
Case 2 of [36]
β
1.610
1.616
Methods
Case 1 of this paper
Case 2 of this paper
β
1.619
1.639
0.5
e1(t)
0 −0.5 −1 −1.5 0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
t 1.5
2
e (t)
1 0.5 0 −0.5 0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
t
Fig. 1. The responses of the estimation errors (e1 (t) and e2 (t))
and [35] haven’t give any feasible solution. As it may, employing Theorem 1 of this paper with g1 = g3 = g5 = 1, g2 = g4 = g6 = 2 and d1 = d2 = d3 = 2, a set of feasible solutions are achieved by solving the LMIs given by (6): α1020 = 6.8638 × 10−13 , α1011 = 1.4303 × 10−12 , α1002 = 8.0400 × 10−13 , α0120 = 6.8558 × 10−13 , α0111 = 1.2105 × 10−12 , α0102 = 7.7432 × 10−13 , ⎡
⎤
⎡
⎤
F1020 = 10−12 × ⎣ −0.0772 0.1636 ⎦ , F1011 = 10−12 × ⎣ −0.1977 0.1340 ⎦ , ⎡
⎤
⎡
⎤
F1002 = 10−12 × ⎣ −0.1626 −0.2950 ⎦ , F0120 = 10−12 × ⎣ −0.0926 0.1299 ⎦ ,
17
0.2
0.1
u(t)
0
−0.1
−0.2
−0.3
−0.4
1
2
3
4
5
6
7
8
9
10
11 12
13 14
15 16
17 18
19
20
t
Fig. 2. The response of the control input (u(t)) 2
x1(t)
1 0 −1 0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
t 2
2
x (t)
1 0 −1 −2 0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
t
Fig. 3. The responses of the system states (x1 (t) and x2 (t)) ⎡
⎤
⎡
⎤
F0111 = 10−12 × ⎣ −0.2259 0.1511 ⎦ , F0102 = 10−12 × ⎣ −0.0853 −0.2906 ⎦ , ⎡
⎤T
⎡
⎤T
K1020 = 10−10 × ⎣ 0.1449 −0.4043 ⎦ , K1011 = 10−10 × ⎣ 0.4114 −0.4528 ⎦ , ⎡
⎤T
⎡
⎤T
K1002 = 10−10 × ⎣ 0.5849 −0.4265 ⎦ , K0120 = 10−10 × ⎣ 0.0705 −0.4205 ⎦ , ⎡
⎤T
⎡
⎤T
K0111 = 10−10 × ⎣ 0.3907 −0.4536 ⎦ , K0102 = 10−10 × ⎣ 0.6291 −0.2995 ⎦ , ⎡
⎤
⎡
⎤
⎢ ⎥ ⎢ ⎥ ⎢ 0.1047 −0.1105 ⎥ ⎢ 0.3340 −0.4603 ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ , G1011 = 10−10 × ⎢ ⎥, G1020 = 10−10 × ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦
0.0391 0.6969
18
−0.0066 0.5632
⎡
⎤
⎡
⎤
⎢ ⎥ ⎢ ⎥ ⎢ 0.3145 −0.2657 ⎥ ⎢ 0.1065 −0.0949 ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ , G0120 = 10−10 × ⎢ ⎥, G1002 = 10−10 × ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦
0.1202 0.6307
⎡
0.0163 0.7169
⎤
⎡
⎤
⎢ ⎥ ⎢ ⎥ ⎢ 0.3485 −0.5361 ⎥ ⎢ 0.3078 −0.3133 ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ , G0102 = 10−10 × ⎢ ⎥. G0111 = 10−10 × ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦
0.1045 0.5230
0.1487 0.6414
Presetting the initial conditions x(0) = (1, − 1)T and xˆ(0) = (3, − 3)T of the nonlinear system (12), the evolutions of the estimation errors (e1 (t) = x1 (t) − xˆ1 (t) and e2 (t) = x2 (t) − xˆ2 (t)), the control input (u(t)), and the system states (x1 (t) and x2 (t)) are shown in Fig. 1–Fig. 3 respectively. It is clear to observe from Fig. 1–Fig. 3 that the closed-loop system belongs to be asymptotically stable, in other words, the converging property of the underlying nonlinear system with the initial conditions x(0) = (1, − 1)T and xˆ(0) = (3, − 3)T has been successfully illustrated.
5
Conclusion
This study presents fuzzy observer-based output feedback control designs of discrete-time nonlinear systems on account of an improved homogenous polynomial approach. In virtue of the fact that added information contained in the normalized fuzzy weighting functions is integrated into the proposed observerbased control law, the conservatism induced by the existing results has been released in this work. Moreover, a robust version of the underlying result has also been proposed. Illustrative experiments have been utilized to show the effectiveness of the developed approach over the available ones. In the future, 19
finding a way to extend the proposed technique to either nonlinear networked control theory or robust control theory may be potential studies.
Acknowledgements
The work described in this paper was supported by the China Postdoctoral Science Foundation Funded Project (2014M551110), National Natural Science Foundation of China (61403073), and the Fundamental Research Funds for the Central Universities (N140404018). .
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PII: DOI: Reference:
S0925-2312(16)30371-X http://dx.doi.org/10.1016/j.neucom.2016.05.033 NEUCOM17049
To appear in: Neurocomputing Received date: 22 September 2015 Revised date: 5 May 2016 Accepted date: 10 May 2016 Cite this article as: Guotao Hui and Xiangpeng Xie, Novel Observer-Based Output Feedback Control Synthesis of Discrete-Time Nonlinear Control Systems Via A Fuzzy Approach, Neurocomputing, http://dx.doi.org/10.1016/j.neucom.2016.05.033 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.
Novel Observer-Based Output Feedback Control Synthesis of Discrete-Time Nonlinear Control Systems Via A Fuzzy Approach
Guotao Hui a , Xiangpeng Xie b,c a College
of Information Science and Engineering, Northeastern University, Shenyang, 110004, China.
b Institute
of Advanced Technology, Nanjing University of Posts and Telecommunications, Nanjing 210003, China.
c School
of Automation, Huazhong University of Science and Technology, Wuhan 430074, China.
Abstract The paper is focus on less conservative design of observer-based output feedback control synthesis of discrete-time Takagi-Sugeno fuzzy systems on account of an improved homogenous polynomial approach. Compared with those previous methods in the cited literature, a novel fuzzy observer-based output feedback law is constructed in virtue of utilizing one promising solution to homogenous polynomials that are parameter-dependent on both the current-time and those multi-steps past-time normalized fuzzy weighting functions with a pair of prescribed degrees. It is rather remarkable that more information contained in the normalized fuzzy weighting functions is integrated to build the proposed control law, and thus the conservatism left behind previous results can be released in this work. Moreover, a robust version of the underlying result is also proposed. Finally, the effectiveness of
Preprint submitted to Elsevier
14 May 2016
the given approach is demonstrated via some illustrative experiments. Key words: Fuzzy model, Signal estimation, Output feedback, Observer design, Nonlinear control.
1
Introduction
For the past few years, stability analysis and control synthesis by the aid of Takagi-Sugeno (T-S) models have acted as one of the most popular topics in data-based control fields [1,2]. Indeed, over the past decade tremendous investigations have been devoted to the study of T-S control systems, such as those important results: stabilization of fuzzy-model-based systems [3–7], distributed filtering for fuzzy-model-based systems [8–10], fuzzy fault detection for nonlinear systems that are with sensor fault or limited communication capacity [11], adaptive sliding-mode control for fuzzy-model-based systems [12–14], adaptive fuzzy identification and control of uncertain fuzzy systems [15–18], applications to various actual models [19,20], and so on. It is noteworthy that depending on the choice of different Lyapunov functions, the derived conditions belong to different levels of conservatism [21]. In spite of the low-complexity of the obtained results, the simple and common Lyapunov functions may lead to a bank of conservative design conditions. The main reason for this deficiency is that the requirement of some monotonically decreasing conditions for one prescribed common Lyapunov matrix is sometimes too strict to get any set of feasible solutions. For the purpose of getting Corresponding author X. Xie at: Institute of Advanced Technology, Nanjing University of Posts and Telecommunications, Nanjing 210003, China. Email address: [email protected] (Xiangpeng Xie).
2
the major deficiency over, several kinds of featured Lyapunov functions were proposed in the past decade. Particularly, the effectiveness of higher-degree parameter-dependent Lyapunov functions has been well illustrated in [22,23], respectively. This class of Lyapunov functions seems to be one of the best alternatives to seeking more efficient results, especially for the case of discrete-time T-S fuzzy systems, e.g., [24] and the cited references therein.
Also, it is rather remarkable that most of existing research literature are derived with the assumption that all the system states could be obtainable, but it isn’t usually satisfied in real-life operations [25]. By way o estimating the system states, a series of important work were gained successively in the past years, for example, nonfragile H∞ filtering of continuous-time fuzzy systems [26], filter design and model reduction for interval Type-2 fuzzy systems with D stability constraints [27,28], output feedback control of fuzzy stochastic systems [29], robust H∞ filter design for affine fuzzy systems [30], more efficient fuzzy observer designs [31], dynamic output feedback control of T–S fuzzy systems [32], fuzzy observer-based output feedback control designs [33]. Acting as a landmark result, the obtained design conditions in [33] are given for ensuring the existence of fuzzy observer-based controllers in terms of bilinear matrix inequalities, which are not convex and very difficult to be solved. Accompanied by a compromise, a two-step procedure is utilized but a great deal of conservatism is inevitably introduced. Recently, a single-step result has been offered for achieving the linear matrix inequality(LMI)-based observer-based controller design by the authors in [34]. It has been illustrated in [34] that both the controller and the observer’s parameters can be captured by solving a bank of strict LMIs which are numerically feasible with the usage of the famous Matlab toolbox. More recently, another observer-based output feedback 3
law that relies on two or more steps normalized fuzzy weighting functions have been proposed in [35] and [36], respectively.
Based on the above technical discussion, the main purpose of the work is to pursue more relaxed fuzzy observer-based output feedback controller designs by developing an improved homogenous polynomial approach. Compared with those existing results which are merely linearly parameter-dependent on the current-time/past-time normalized fuzzy weighting functions with a lower degree, a novel fuzzy observer-based output feedback controller is developed on account of an improved homogenous polynomials approach. Owing to the reason that added information contained in the normalized fuzzy weighting functions is integrated into the proposed observer-based control law, the conservatism induced by the existing results can be released in this work. Moreover, a robust version of the underlying result is also proposed in this study.
2
Background and Preliminaries
In this work, an asterisk (∗) that is located inside some matrix represents the transpose of its symmetric term; Z+ stands for the set of all positive integers; R denotes the set of all real numbers; p! represents the factorial, e.g., z! = z(z − 1) · · · (1) in case of z ∈ Z+ and 0! = 1; If M represents a matrix, then He(M ) means He(M ) = M + M T . And, the left-hand side of a relation can be defined as Left(·).
Based on the powerful T-S model, a family of discrete-time nonlinear systems 4
are represented like the one used in [34]: ⎧ r ⎪ ⎪ ⎪ x(t + 1) = hi (z(t))(Ai x(t) + Bi u(t)) ⎪ ⎨ i=1
r ⎪ ⎪ ⎪ ⎪ hi (z(t))Ci x(t) ⎩ y(t) =
(1)
i=1
and x(t) ∈ Rn1 denotes the system state vector and is unmeasurable; u(t) ∈ Rn2 represents the control input; y(t) ∈ Rn3 denotes the measured output vector; Ai , Bi and Ci are a bank of known system matrices; hi (z(t)) represents the i-th normalized fuzzy weighting function.
A batch of vital definitions about homogeneous matrix polynomials are shown in this section, which are consistent with the cited references, such as [22–24] and [36], and so on. The set Δr is described as {α ∈ Rr ;
r i=1
αi = 1; α ≥ 0}. And here α1k1 α2k2 · · · αrkr
means the monomials while one has α ∈ r , ki ∈ Z+ , i ∈ {1, · · · , r} and k = k1 k2 · · · kr . K(g) represents the set of r-tuples induced by collecting all possible combinations of nonnegative integers ki , i ∈ {1, · · · , r}, satisfying that k1 + k2 + · · · + kr = g. Pk ∈ Rn×n , ∀k ∈ K(g) represent matrix-valued coefficients. As an illustrative example, if some homogeneous polynomials of degree g = 3 are with r = 3 variables, then the possible values of the partial degrees are K(3) = {300, 030, 003, 210, 201, 120, 021, 102, 012, 111}. As those prescribed definitions that are utilized in [24], for r-tuples k and k , one denotes k ≥ k if ki ≥ ki , (i = 1, ..., r). The usual operations of summation, k +k , and subtraction, k −k (whenever k ≥ k ), are operated componentwise. Similarly, another two key definitions about the r-tuple χi ∈ K(1) and the coefficient π(k), ∀k ∈ K(g) are given in the following form: 5
χi =
0
···0
1−th element
0···
1
i−th element
0
,
r−th element
π(k) = (k1 !) × (k2 !) × · · · × (kr !).
(2)
For another illustrative example, once the parameter of involved homogeneous polynomials is chosen as g = 2 and r = 2, we get χ1 = 1 0, χ2 = 0 1, π(2 0) = (2!) × (0!) = 2, π(1 1) = (1!) × (1!) = 1 and π(0 2) = (0!) × (2!) = 2. Given one homogeneous polynomial with r variables (i.e., k = k1 k2 · · · kr ) and j ∈ {1, · · · , m} with m ∈ Z+ , related shortenings are selected like those in [35]: ⎧ ⎪ ⎪ ⎪ hi = hi (z(t)), hi (t + 1) = hi (z(t + 1)), hi (t − j) = hi (z(t − j)), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ h(t − j) = [h (t − j), · · · , h (t − j)]T , h = [h , · · · , h ]T , 1
1
r
r
⎪ ⎪ ⎪ ⎪ h(t + 1) = [h1 (t + 1), · · · , hr (t + 1)]T , hk = hk11 hk22 · · · hkr r , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ h(t − j)k = h1 (t − j)k1 h2 (t − j)k2 · · · hr (t − j)kr .
Two multi-instant homogeneous matrix polynomials with a pair of prescribed parameters (g1 , g2 ) are represented as follows:
Pzg1 (t−m)zg2 (t) =
m
1 ),i∈{1··· ,m}; (a(i)∈K(g ) k ∈K(g2 )
Pzg1 (t−m+1)zg2 (t+1) =
h(t − i)
a(i)
hk Pa(m)···a(1)k ,
i=1
1 ),i∈{1··· ,m}; (a(i)∈K(g ) k ∈K(g2 )
m
h(t − i + 1)
a(i)
h(t + 1)k Pa(m)···a(1)k ,
i=1
while m, g1 , g2 ∈ Z+ , and Pa(m)···a(1)k is a matrix-valued coefficient. Given an illustrative example for r = 2, m = 1 and g1 = g2 = 2, one gets Pzg1 (t−m)zg2 (t) = h(t − 1)20 h20 P2020 + h(t − 1)20 h11 P2011 + h(t − 1)20 h02 P2002 + h(t − 1)11 h20 P1120 + h(t − 1)11 h11 P1111 + h(t − 1)11 h02 P1102 + h(t − 1)02 h20 P0220 + h(t − 1)02 h11 P0211 + h(t − 1)02 h02 P0202 . 6
3
Main results
In this work, a new fuzzy observer-based control law is constructed by the following form: ⎧ ⎪ ⎪ ⎪ xˆ(t + 1) = Az(t) xˆ(t) + Bz(t) u(t) + G−1 ˆ(t)), ⎪ zg3 (t−m)zg4 (t) Kzg1 (t−m)zg2 (t) (y(t) − y ⎪ ⎪ ⎪ ⎨
yˆ(t) = Cz(t) xˆ(t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ u(t) = α−1
ˆ(t), zg5 (t−m)zg6 (t) Fzg1 (t−m)zg2 (t) x (3)
with m ∈ Z+ , Kzg1 (t−m)zg2 (t) , Gzg3 (t−m)zg4 (t) , Fzg1 (t−m)zg2 (t) and αzg5 (t−m)zg6 (t) are multi-instant homogenous matrix polynomials of degree (g1 , g2 ) and (g3 , g4 ), respectively (g1 , g2 , g3 , g4 ∈ Z+ ), and one has
Kzg1 (t−m)zg2 (t) = Gzg3 (t−m)zg4 (t) = Fzg1 (t−m)zg2 (t) = αzg5 (t−m)zg6 (t) =
m
),i∈{1,2,...,m}; (a(i)∈K(gk1 ∈K(g ) 2)
),i∈{1,2,...,m}; (a(i)∈K(gk3 ∈K(g ) 4)
),i∈{1,2,...,m}; (a(i)∈K(gk1 ∈K(g ) 2)
),i∈{1,2,...,m}; (a(i)∈K(gk5 ∈K(g ) 6)
h(t − i)
a(i)
hk Ka(m)···a(1)k ,
i=1
m
h(t − i)
a(i)
hk Ga(m)···a(1)k ,
i=1
m
h(t − i)
a(i)
hk Fa(m)···a(1)k ,
i=1
m
h(t − i)
a(i)
h(t)k αa(m)···a(1)k ,
i=1
In addition, Ka(m)···a(1)k ∈ Rn1 ×n3 , Ga(m)···a(1)k ∈ Rn1 ×n1 , and Fa(m)···a(1)k ∈ Rn2 ×n1 are a set of matrices to be resolved, αa(m)···a(1)k are scalars to be resolved. Remark 1 : Particularly, the above fuzzy observer-based control law (3) may be reducible to the previous one given in [36] if one set g1 = g2 = g3 = g4 = g5 = g6 = 1. In a word, the previous one given in [36] can be thought to be a special case of this work. Compared with those previous methods in the cited 7
literature, (3) is constructed in virtue of utilizing one promising solution to homogenous polynomials that are parameter-dependent on both the currenttime and those multi-steps past-time normalized fuzzy weighting functions with a pair of prescribed degrees. It is rather remarkable that more information contained in the normalized fuzzy weighting functions is integrated to build the proposed control law, and thus the conservatism left behind previous results can be released in this work. Embedding the above observer-based control law (3) to the origin system, the closed-loop system turns into another form: ⎛
⎞
⎜ ⎜ Υ(1, 1) ⎜ x˜(t + 1) = ⎜ ⎜ ⎜ ⎝
0
−αz−1 g (t−m)zg 5
Az(t) − G−1 zg (t−m)zg
⎛ ⎜ ⎜ ⎜ where x˜(t) = ⎜ ⎜ ⎜ ⎝
6 (t)
3
Bz(t) Fzg1 (t−m)zg2 (t)
4 (t)
Kzg1 (t−m)zg2 (t) Cz(t)
⎟ ⎟ ⎟ ⎟x ⎟ ˜(t), ⎟ ⎠
(4)
⎞
x(t) x(t) − xˆ(t)
⎟ ⎟ ⎟ ⎟ and Υ(1, 1) = Az(t) +α−1 ⎟ zg5 (t−m)zg6 (t) Bz(t) Fzg1 (t−m)zg2 (t) . ⎟ ⎠
For the sake of achieving the task of observer-based control design, a multiinstant Lyapunov function candidate is proposed with the help of multi-instant homogeneous matrix polynomials: ⎛
⎞
⎜ ⎜ P1zg (t−m)zg (t) 5 6 ⎜ T V (˜ x(t)) = x˜ (t) ⎜ ⎜ ⎜ ⎝
0
among (5), P1zg5 (t−m)zg6 (t) = P2zg5 (t−m)zg6 (t) = m, g5 , g6 ∈
0 P2zg5 (t−m)zg6 (t)
a(i)∈K(g5
),i∈{1,··· ,m};k ∈K(g
m i=1
(5)
h(t − i)a(i) hk P1a(m)···a(1)k ,
h(t − i)a(i) hk P2a(m)···a(1)k , and
k(i)∈K(g5 ),i∈{1,··· ,m};k ∈K(g6 ) Z+ , P1k(m)···k(1)k ∈ Rn1 ×n1 , P2k(m)···k(1)k
8
6)
m i=1
⎟ ⎟ ⎟ ⎟x ⎟ ˜(t), ⎟ ⎠
∈ Rn1 ×n1 .
Theorem 1. The closed-loop system (4) is called to be asymptotically stable, if there exist matrices Fa(m)···a(1)k ∈ Rn2 ×n1 and Ka(m)···a(1)k ∈ Rn1 ×n3 (a(j) ∈ K(g1 ), j ∈ {1, 2, ..., m}, k ∈ K(g2 )); matrices Ga(m)···a(1)k ∈ Rn1 ×n1 (a(j) ∈ K(g3 ), j ∈ {1, 2, ..., m}, k ∈ K(g4 )); symmetric matrices P1a(m)···a(1)k ∈ Rn1 ×n1 and P2a(m)···a(1)k ∈ Rn1 ×n1 , αa(m)···a(1)k , (a(j) ∈ K(g5 ), j ∈ {1, 2, ..., m}, k ∈ K(g6 )); such that all the LMIs in terms of (6) are simultaneously satisfied: ⎡
⎤
⎢ 11 ∗ ∗ ⎢ Γp(m)···p(1)qk ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 0 Γ22 ∗ p(m)···p(1)qk ⎢ ⎢ Γp(m)···p(1)qk = ⎢ ⎢ ⎢ 31 32 33 ⎢Γ ⎢ p(m)···p(1)qk Γp(m)···p(1)qk Γp(m)···p(1)qk ⎢ ⎢ ⎢ ⎣ 42
0
Γp(m)···p(1)qk
0
∗ ∗ ∗ Γ44 p(m)···p(1)qk
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ > 0, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(6)
∀ p(j) ∈ K(g7 + d1 ); j = 1, 2, ..., m; q ∈ K(g6 + d2 ); k ∈ K(g8 + d3 ); with g7 = max{g1 , g3 , g5 }, g8 = max{g2 + 1, g4 + 1, g5 , g6 + 1},
Γ11 p(m)···p(1)qk =
5 ),j∈{1,2,...,m}; (a(j)∈K(g p(j)≥a(j);k ∈K(g6 );k≥k )
⎧⎛ ⎞ m ⎨ (g7 + d1 − g5 )! ⎠ (g8 + d3 − g6 )! ⎝ ⎩ π(k − k ) j=1 π(p(j) − a(j))
(g6 + d2 )! P1a(m)···a(1)k , × π(q) Γ22 p(m)···p(1)qk =
5 ),j∈{1,2,...,m}; (a(j)∈K(g p(j)≥a(j);k ∈K(g6 );k≥k )
⎧⎛ ⎞ m ⎨ (g + d − g )! (g + d3 − g6 )! 7 1 5 ⎠ 8 ⎝ ⎩ π(k − k ) j=1 π(p(j) − a(j))
(g6 + d2 )! P2a(m)···a(1)k , × π(q) Γ33 p(m)···p(1)qk =
5 ),j∈{1,2,...,m}; (a(j)∈K(g p(j)≥a(j);k ∈K(g6 );k≥k )
⎧⎛ ⎞ m ⎨ (g + d − g )! (g + d3 − g6 )! 7 1 5 ⎠ 8 ⎝ ⎩ π(k − k ) j=1 π(p(j) − a(j))
×
(g6 + d2 )! 2αa(m)···a(1)k I − π(q)
5 ),j∈{2,...,m};p(j)≥a(j+1); (a(j)∈K(g a(1)∈K(g5 );k≥a(1),k ∈K(g6 ),q≥k )
9
⎧⎛ ⎞ ⎨ m−1
(g + d − g )! 7 1 5 ⎠ ⎝ ⎩ π(p(j) − a(j + 1)) j=1
d2 ! (g7 + d1 )! (g8 + d3 − g5 )! P1a(m)···a(1)k , × π(p(m)) π(k − a(1)) π(q − k )
⎧⎛ ⎞ m ⎨ (g + d − g )! 7 1 3 44 ⎠ ⎝ Γp(m)···p(1)qk = ⎩ π(p(j) − a(j)) j=1 a(j)∈K(g3 ),j∈{1,2,...,m};p(j)≥a(j);k ∈K(g4 );k≥k (g + d − g )! (g + d )!
×
8
3
4
π(k − k )
6
2
Ga(m)···a(1)k + GTa(m)···a(1)k
π(q)
⎧⎛ ⎞ ⎨ m−1
(g + d − g )! 7 1 5 ⎠ ⎝ − ⎩ π(p(j) − a(j + 1)) j=1 a(j)∈K(g5 ),j∈{2,...,m};p(j)≥a(j+1);a(1)∈K(g5 );k≥a(1),k ∈K(g6 ),q≥k
×
d2 ! (g7 + d1 )! (g8 + d3 − g5 )! P2a(m)···a(1)k , π(p(m)) π(k − a(1)) π(q − k )
Γ31 p(m)···p(1)qk =
5 ),j∈{1,2,...,m};p(j)≥a(j); (a(j)∈K(g i∈{1,2,...,r},k ∈K(g6 ),k−k −χi ≥0 )
×
(g6 + d2 )! αa(m)···a(1)k Ai + π(q)
⎧⎛ ⎞ m ⎨ (g + d − g )! (g + d3 − g6 − 1)! 7 1 5 ⎠ 8 ⎝ ⎩ π(k − k − χi ) j=1 π(p(j) − a(j))
1 ),j∈{1,2,...,m};p(j)≥a(j); (a(j)∈K(g i∈{1,2,...,r},k ∈K(g2 ),k−k −χi ≥0 )
⎧⎛ ⎞ m ⎨ (g + d − g )! 7 1 1 ⎠ ⎝ ⎩ π(p(j) − a(j)) j=1
×
(g8 + d3 − g2 − 1)! (g6 + d2 )! Bi Fa(m)···a(1)k , π(k − k − χi ) π(q)
Γ32 p(m)···p(1)qk = −
1 ),j∈{1,2,...,m};p(j)≥a(j); (a(j)∈K(g i∈{1,2,...,r},k ∈K(g2 ),k−k −χi ≥0 )
⎧⎛ ⎞ m ⎨ (g + d − g )! (g + d3 − g2 − 1)! 7 1 1 ⎠ 8 ⎝ ⎩ π(k − k − χi ) j=1 π(p(j) − a(j))
(g6 + d2 )! Bi Fa(m)···a(1)k , × π(q) Γ42 p(m)···p(1)qk =
3 ),j∈{1,2,...,m};p(j)≥a(j); (a(j)∈K(g i∈{1,2,...,r},k ∈K(g4 ),k−k −χi ≥0 )
(g6 + d2 )! Ga(m)···a(1)k Ai − × π(q)
⎧⎛ ⎞ m ⎨ (g + d − g )! (g + d3 − g4 − 1)! 7 1 3 ⎠ 8 ⎝ ⎩ π(k − k − χi ) j=1 π(p(j) − a(j))
1 ),j∈{1,2,...,m};p(j)≥a(j); (a(j)∈K(g i∈{1,2,...,r},k ∈K(g2 ),k−k −χi ≥0 )
⎧⎛ ⎞ m ⎨ (g + d − g )! 7 1 1 ⎠ ⎝ ⎩ π(p(j) − a(j)) j=1
×
(g8 + d3 − g2 − 1)! (g6 + d2 )! Ka(m)···a(1)k Ci . π(k − k − χi ) π(q)
PROOF. Taking into account the Lyapunov function candidate in term of (5), its one-step difference (i.e., ΔV (˜ x(t))) along the solution to the estimation error (4) can be written as the following equation: 10
⎛
⎛
⎞⎞
⎜ ⎜ ⎜ ⎜ P1zg (t−m)zg (t) 5 6 ⎜ ⎜ T T ⎜ ΔV (˜ x(t)) = x˜ (t) ⎜Υ ΞΥ − ⎜ ⎜ ⎜ ⎜ ⎝ ⎝
0
⎛
0 P2zg5 (t−m)zg6 (t)
⎟⎟ ⎟⎟ ⎟⎟ ⎟⎟ x ⎟⎟ ˜(t), (7) ⎟⎟ ⎠⎠
⎞
⎜ ⎜ P1zg (t−m+1)zg (t+1) 5 6 ⎜ with Ξ = ⎜ ⎜ ⎜ ⎝
0
⎛
0 P2zg5 (t−m+1)zg6 (t+1)
⎜ ⎜ Az(t) + αz−1 (t−m)z (t) Bz(t) Fzg (t−m)zg (t) g5 g6 1 2 ⎜ Υ=⎜ ⎜ ⎜ ⎝
0
⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠ ⎞
−αz−1 g (t−m)zg 5
Bz(t) Fzg1 (t−m)zg2 (t) 6 (t)
Az(t) − G−1 zg (t−m)zg 3
4 (t)
Kzg1 (t−m)zg2 (t) Cz(t)
Thus, the estimation error (4) can be called to be globally asymptotically stable if the following inequality is ensured: ⎛
⎞
⎜ ⎜ P1zg (t−m)zg (t) 5 6 ⎜ ΥT ΞΥ − ⎜ ⎜ ⎜ ⎝
0
0 P2zg5 (t−m)zg6 (t)
⎟ ⎟ ⎟ ⎟ < 0. ⎟ ⎟ ⎠
(8)
Therefore, it is rather remarkable that (8) can be satisfied if the matrix inequality (9) holds in true: ⎡
⎤
⎢ ∗ ⎢ P1zg5 (t−m)zg6 (t) ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 0 P2zg5 (t−m)zg6 (t) ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ Ω31 −Bz(t) Fzg1 (t−m)zg2 (t) ⎢ ⎢ ⎢ ⎢ ⎣
0
Ω42
∗
∗ ⎥ ⎥
∗
∗
Ω33
∗
0
Ω44
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ > 0, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(9)
with Ω31 = αzg5 (t−m)zg6 (t) Az(t) + Bz(t) Fzg1 (t−m)zg2 (t) , Ω33 = 2αzg5 (t−m)zg6 (t) I − 11
⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠
P1zg5 (t−m+1)zg6 (t+1) , Ω42 = Gzg3 (t−m)zg4 (t) Az(t) −Kzg1 (t−m)zg2 (t) Cz(t) , Ω44 = He(Gzg3 (t−m)zg4 (t) )− P2zg5 (t−m+1)zg6 (t+1) . Thereupon, a key equality is utilized by representing the expression of Left(9): ⎡
⎤
⎢ ∗ ⎢ φ11 P1zg5 (t−m)zg6 (t) ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 0 φ22 P2zg5 (t−m)zg6 (t) ⎢ Left(9) = ⎢ ⎢ ⎢ ⎢ ˜ 31 ⎢ Ω −φ32 Bz(t) Fzg1 (t−m)zg2 (t) ⎢ ⎢ ⎢ ⎢ ⎣
˜ 42 Ω
0
m
= (
p(j)∈K(g7 +d1 ),j∈{1,2,...,m}; q∈K(g6 +d2 );k∈K(g8 +d3 )
∗
∗ ⎥ ⎥
∗
∗
˜ 33 Ω
∗
0
˜ 44 Ω
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
h(t − l)
p(l)
q
k
h h(t + 1) Γp(m)···p(1)qk ,
l=1
)
(10)
and, φ11 = φ22 =
φ32 =
m
r
l=1
i=1
m
r
l=1
i=1 !g7 +d1 −g1
hi (t − l)
hi (t − l)
!g7 +d1 −g5 r
r
!g8 +d3 −g6
hi
i=1 !g8 +d3 −g2 −1
i=1 r
hi
i=1
φ33 = ˜ 31 = Ω
i=1 r
!g7 +d1 m−1
hi (t − m) !−1
hi
i=1
l=1
r i=1
φ42 =
m
r
l=1
i=1
hi (t − l)
!g7 +d1 −g5
r
φ44 =
m
r
l=1
i=1
,
!g8 +d3 −g5
r
hi
i=1
!d 2
hi (t + 1)
i=1
φ11 αzg5 (t−m)zg6 (t) Az(t) + φ32 Bz(t) Fzg1 (t−m)zg2 (t) ,
hi (t − l)
!g7 +d1 −g3
r
!g8 +d3 −g4 −1
r
hi
i=1
!g6 +d2
hi (t + 1)
hi (t − l)
!g7 +d1 −g3
r
,
i=1
˜ 44 = φ44 He(Gzg (t−m)zg (t) ) − φ33 P2zg (t−m+1)zg (t+1) , Ω 3 4 5 6
,
!g6 +d2
hi (t + 1)
˜ 42 = φ42 Gzg (t−m)zg (t) Az(t) − φ32 Kzg (t−m)zg (t) Cz(t) , Ω 3 4 1 2
!g6 +d2
hi (t + 1)
i=1
˜ 33 = 2φ11 αzg (t−m)zg (t) I − φ33 P1zg (t−m+1)zg (t+1) , Ω 5 6 5 6 r
r
!g8 +d3 −g4
hi
i=1
r
!g6 +d2
hi (t + 1)
,
i=1
and g7 = max{g1 , g3 , g5 }, g8 = max{g2 +1, g4 +1, g5 , g6 +1}. More importantly, Γp(m)···p(1)qk is given in the equation (6). On the basis of the above equality (10), the inequality (9) can be evidently 12
,
ensured once if one gets Γp(m)···p(1)qk > 0 for all possible combinations of p(m) · · · p(1)qk. As a consequence, it is concluded that the closed-loop system (4) can be called to be asymptotically stable if all the LMIs in terms of (6) hold on. This completes the proof.
Remark 2 : It is rather remarkable that the proposed result via T-S fuzzy models are on account of an implicit assumption that the controller will be implemented exactly. Anyway, the uncertainties do occur in the implementation of a designed controller in real-life systems. Such uncertainties can be due to, among different reasons, roundoff errors in numerical computation during the controller implementation and the need to offer practicing engineers with safe-tuning margins. In what follows, similar to previous results of [10,26], the result given in Theorem 1 of this paper can be easily extended to robust control designs. In this work, the polytopic-type representation (i.e., Ai = w s=1
w s=1
αs Asi , Bi =
w s=1
αs Bis and Ci =
w s=1
αs Cis while
αi = 1, αs ≥ 0 and w ∈ Z+ ) is applied for modeling uncertainties, and
the following result is obtained. Corollary 1. While one has Ai = w s=1
w s=1
αs Asi , Bi =
w s=1
αs Bis and Ci =
αs Cis , the closed-loop system (4) is called to be robust asymptotically sta-
ble, if there exist matrices Fa(m)···a(1)k ∈ Rn2 ×n1 and Ka(m)···a(1)k ∈ Rn1 ×n3 (a(j) ∈ K(g1 ), j ∈ {1, 2, ..., m}, k ∈ K(g2 )); matrices Ga(m)···a(1)k ∈ Rn1 ×n1 (a(j) ∈ K(g3 ), j ∈ {1, 2, ..., m}, k ∈ K(g4 )); symmetric matrices P1a(m)···a(1)k ∈ Rn1 ×n1 and P2a(m)···a(1)k ∈ Rn1 ×n1 , αa(m)···a(1)k , (a(j) ∈ K(g5 ), j ∈ {1, 2, ..., m}, k ∈ K(g6 )); such that all the LMIs in terms of (11) are simultaneously satisfied: 13
⎡
⎤
⎢ 11 ∗ ∗ ⎢ Γp(m)···p(1)qk ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 0 Γ22 ∗ p(m)···p(1)qk ⎢ s ⎢ Γp(m)···p(1)qk = ⎢ ⎢ ⎢ 31s 32s 33 ⎢Γ ⎢ p(m)···p(1)qk Γp(m)···p(1)qk Γp(m)···p(1)qk ⎢ ⎢ ⎢ ⎣ 42s
0
Γp(m)···p(1)qk
0
∗ ∗ ∗ Γ44 p(m)···p(1)qk
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ > 0, ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(11)
∀ p(j) ∈ K(g7 + d1 ); j = 1, 2, ..., m; q ∈ K(g6 + d2 ); k ∈ K(g8 + d3 ); s ∈ {1, · · · , w};
22 33 with g7 = max{g1 , g3 , g5 }, g8 = max{g2 +1, g4 +1, g5 , g6 +1}, Γ11 p(m)···p(1)qk , Γp(m)···p(1)qk , Γp(m)···p(1)qk
and Γ44 p(m)···p(1)qk are the same as those in Theorem 1, and
Γ31s p(m)···p(1)qk =
5 ),j∈{1,2,...,m};p(j)≥a(j); (a(j)∈K(g i∈{1,2,...,r},k ∈K(g6 ),k−k −χi ≥0 )
(g6 + d2 )! αa(m)···a(1)k Asi + × π(q)
⎧⎛ ⎞ m ⎨ (g + d − g )! 7 1 5 ⎠ (g8 + d3 − g6 − 1)! ⎝ ⎩ π(p(j) − a(j)) π(k − k − χi ) j=1
1 ),j∈{1,2,...,m};p(j)≥a(j); (a(j)∈K(g i∈{1,2,...,r},k ∈K(g2 ),k−k −χi ≥0 )
⎧⎛ ⎞ m ⎨ (g + d − g )! 7 1 1 ⎠ ⎝ ⎩ π(p(j) − a(j)) j=1
(g8 + d3 − g2 − 1)! (g6 + d2 )! s Bi Fa(m)···a(1)k , × π(k − k − χi ) π(q)
Γ32s p(m)···p(1)qk = −
1 ),j∈{1,2,...,m};p(j)≥a(j); (a(j)∈K(g i∈{1,2,...,r},k ∈K(g2 ),k−k −χi ≥0 )
⎧⎛ ⎞ m ⎨ (g + d − g )! 7 1 1 ⎠ (g8 + d3 − g2 − 1)! ⎝ ⎩ π(p(j) − a(j)) π(k − k − χi ) j=1
(g6 + d2 )! s Bi Fa(m)···a(1)k , × π(q) Γ42s p(m)···p(1)qk =
3 ),j∈{1,2,...,m};p(j)≥a(j); (a(j)∈K(g i∈{1,2,...,r},k ∈K(g4 ),k−k −χi ≥0 )
(g6 + d2 )! Ga(m)···a(1)k Asi − × π(q)
⎧⎛ ⎞ m ⎨ (g + d − g )! 7 1 3 ⎠ (g8 + d3 − g4 − 1)! ⎝ ⎩ π(p(j) − a(j)) π(k − k − χi ) j=1
1 ),j∈{1,2,...,m};p(j)≥a(j); (a(j)∈K(g i∈{1,2,...,r},k ∈K(g2 ),k−k −χi ≥0 )
(g8 + d3 − g2 − 1)! (g6 + d2 )! Ka(m)···a(1)k Cis . × π(k − k − χi ) π(q)
14
⎧⎛ ⎞ m ⎨ (g + d − g )! 7 1 1 ⎠ ⎝ ⎩ π(p(j) − a(j)) j=1
4
Numerical Experiments
Example. In this section, the same nonlinear model as the one in [35] is adopted for needful comparisons. ⎧ ⎪ ⎪ ⎪ x1 (t + 1) = x1 (t) − x1 (t)x2 (t) + (5 + x1 (t))u(t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
x2 (t + 1) = −x1 (t) − 0.5x2 (t) + 2x1 (t)u(t),
⎪ ⎪ ⎪ ⎪ ⎪ β − 1 β + 1 ⎪ ⎪ ⎪ + x1 (t) x1 (t) + x2 (t), ⎩ y(t) =
2
(12)
2β
where x1 (t) is assumed to be measurable and x2 (t) is assumed to be unmeasurable. β is a varying variable with β > 0. The following discrete-time T-S fuzzy model is employed for representing the involved nonlinear systems: Rule 1: If x1 (m) is β, then ⎛
⎞
⎡
⎜ ⎟ ⎢ ⎜ x1 (t + 1) ⎟ ⎢ 1 ⎜ ⎟ ⎢ ⎜ ⎟=⎢ ⎜ ⎟ ⎢ ⎜ ⎟ ⎢ ⎝ ⎠ ⎣
⎞
⎥
⎟
⎡
⎤
⎢ ⎥ ⎢5 + β ⎥ ⎢ ⎥ ⎢ ⎥ u(t), ⎟+⎢ ⎥ ⎟ ⎢ ⎥ ⎠ ⎣ ⎦
⎜ ⎜ x (t) ⎟ −β ⎥ ⎥⎜ 1 ⎟ ⎥ ⎟
−1 −0.5
x2 (t + 1)
⎤⎛ ⎜ ⎥⎜ ⎥⎜ ⎦⎝
x2 (t)
2β
⎞
⎛
⎟ ⎤⎜ ⎜ x1 (t) ⎟ ⎟ ⎜ ⎟, y(t) = ⎣ β 1 ⎦ ⎜ ⎟ ⎜ ⎟ ⎜ ⎠ ⎝ ⎡
x2 (t)
Rule 2: If x1 (t) is −β, then ⎛
⎞
⎡
⎤
⎜ ⎟ ⎢ ⎜ x1 (t + 1) ⎟ ⎢ 1 ⎜ ⎟ ⎢ ⎜ ⎟=⎢ ⎜ ⎟ ⎢ ⎜ ⎟ ⎢ ⎝ ⎠ ⎣
x2 (t + 1)
⎡
⎤
⎥
⎢ ⎥ ⎢5 − β ⎥ ⎢ ⎥ ⎥ x(t) + ⎢ ⎥ u(t). ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎦ ⎣ ⎦
β ⎥ ⎥
−1 −0.5
−2β
15
⎞
⎛
⎟ ⎤⎜ ⎜ x1 (t) ⎟ ⎟ ⎜ ⎟. y(t) = ⎣ 1 1 ⎦ ⎜ ⎟ ⎜ ⎟ ⎜ ⎠ ⎝ ⎡
x2 (t)
Borrowed from that in [35], two involved normalized fuzzy weighting functions are chosen as: h1 (z(t)) = (β + x1 (t))/2β, h2 (z(t)) = (β − x1 (t))/2β. For the purpose of giving clear comparisons, both the previous results ([35] and [36]) and the proposed result act on the underlying nonlinear systems. As a matter of convenience, some shortenings are adopted as follows: Case 1 of [35] means Theorem 1 of [35] with d1 = d2 = d3 = 1; Case 2 of [35] means Theorem 1 of [35] with d1 = d2 = d3 = 2; Case 1 of [36] means Theorem 1 of [36] with d1 = 1, d2 = 4; Case 2 of [36] means Theorem 1 of [36] with d1 = 1, d2 = 5; Case 1 of this paper means Theorem 1 of this paper with g1 = g3 = g5 = 2, g2 = g4 = g6 = 2 and d1 = d2 = d3 = 1; Case 2 of this paper means Theorem 1 of this paper with g1 = g3 = g5 = 1, g2 = g4 = g6 = 2 and d1 = d2 = d3 = 2.
Every method has its feasible stabilization region, i.e., (0, β]. It is plain to us that the stabilization region becomes more relaxed as its maximum of feasible β is bigger. The maximum of feasible β are calculated through the above six methods, respectively. The calculated results about β are listed in Table I. From Table I, it is easy to observe that the proposed methods in this work offer more bigger values of β than the results given in [34] and [35]. This reality shows that the relaxation quality has been improved on account of using the approach in this paper. In the case of β = 1.639, it is unfortunate that the results provided by [34] 16
Table 1 Maximum of feasible β via previous methods and the proposed method Methods
Case 1 of [35]
Case 2 of [35]
β
1.507
1.542
Methods
Case 1 of [36]
Case 2 of [36]
β
1.610
1.616
Methods
Case 1 of this paper
Case 2 of this paper
β
1.619
1.639
0.5
e1(t)
0 −0.5 −1 −1.5 0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
t 1.5
2
e (t)
1 0.5 0 −0.5 0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
t
Fig. 1. The responses of the estimation errors (e1 (t) and e2 (t))
and [35] haven’t give any feasible solution. As it may, employing Theorem 1 of this paper with g1 = g3 = g5 = 1, g2 = g4 = g6 = 2 and d1 = d2 = d3 = 2, a set of feasible solutions are achieved by solving the LMIs given by (6): α1020 = 6.8638 × 10−13 , α1011 = 1.4303 × 10−12 , α1002 = 8.0400 × 10−13 , α0120 = 6.8558 × 10−13 , α0111 = 1.2105 × 10−12 , α0102 = 7.7432 × 10−13 , ⎡
⎤
⎡
⎤
F1020 = 10−12 × ⎣ −0.0772 0.1636 ⎦ , F1011 = 10−12 × ⎣ −0.1977 0.1340 ⎦ , ⎡
⎤
⎡
⎤
F1002 = 10−12 × ⎣ −0.1626 −0.2950 ⎦ , F0120 = 10−12 × ⎣ −0.0926 0.1299 ⎦ ,
17
0.2
0.1
u(t)
0
−0.1
−0.2
−0.3
−0.4
1
2
3
4
5
6
7
8
9
10
11 12
13 14
15 16
17 18
19
20
t
Fig. 2. The response of the control input (u(t)) 2
x1(t)
1 0 −1 0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
t 2
2
x (t)
1 0 −1 −2 0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20
t
Fig. 3. The responses of the system states (x1 (t) and x2 (t)) ⎡
⎤
⎡
⎤
F0111 = 10−12 × ⎣ −0.2259 0.1511 ⎦ , F0102 = 10−12 × ⎣ −0.0853 −0.2906 ⎦ , ⎡
⎤T
⎡
⎤T
K1020 = 10−10 × ⎣ 0.1449 −0.4043 ⎦ , K1011 = 10−10 × ⎣ 0.4114 −0.4528 ⎦ , ⎡
⎤T
⎡
⎤T
K1002 = 10−10 × ⎣ 0.5849 −0.4265 ⎦ , K0120 = 10−10 × ⎣ 0.0705 −0.4205 ⎦ , ⎡
⎤T
⎡
⎤T
K0111 = 10−10 × ⎣ 0.3907 −0.4536 ⎦ , K0102 = 10−10 × ⎣ 0.6291 −0.2995 ⎦ , ⎡
⎤
⎡
⎤
⎢ ⎥ ⎢ ⎥ ⎢ 0.1047 −0.1105 ⎥ ⎢ 0.3340 −0.4603 ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ , G1011 = 10−10 × ⎢ ⎥, G1020 = 10−10 × ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦
0.0391 0.6969
18
−0.0066 0.5632
⎡
⎤
⎡
⎤
⎢ ⎥ ⎢ ⎥ ⎢ 0.3145 −0.2657 ⎥ ⎢ 0.1065 −0.0949 ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ , G0120 = 10−10 × ⎢ ⎥, G1002 = 10−10 × ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦
0.1202 0.6307
⎡
0.0163 0.7169
⎤
⎡
⎤
⎢ ⎥ ⎢ ⎥ ⎢ 0.3485 −0.5361 ⎥ ⎢ 0.3078 −0.3133 ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ , G0102 = 10−10 × ⎢ ⎥. G0111 = 10−10 × ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦
0.1045 0.5230
0.1487 0.6414
Presetting the initial conditions x(0) = (1, − 1)T and xˆ(0) = (3, − 3)T of the nonlinear system (12), the evolutions of the estimation errors (e1 (t) = x1 (t) − xˆ1 (t) and e2 (t) = x2 (t) − xˆ2 (t)), the control input (u(t)), and the system states (x1 (t) and x2 (t)) are shown in Fig. 1–Fig. 3 respectively. It is clear to observe from Fig. 1–Fig. 3 that the closed-loop system belongs to be asymptotically stable, in other words, the converging property of the underlying nonlinear system with the initial conditions x(0) = (1, − 1)T and xˆ(0) = (3, − 3)T has been successfully illustrated.
5
Conclusion
This study presents fuzzy observer-based output feedback control designs of discrete-time nonlinear systems on account of an improved homogenous polynomial approach. In virtue of the fact that added information contained in the normalized fuzzy weighting functions is integrated into the proposed observerbased control law, the conservatism induced by the existing results has been released in this work. Moreover, a robust version of the underlying result has also been proposed. Illustrative experiments have been utilized to show the effectiveness of the developed approach over the available ones. In the future, 19
finding a way to extend the proposed technique to either nonlinear networked control theory or robust control theory may be potential studies.
Acknowledgements
The work described in this paper was supported by the China Postdoctoral Science Foundation Funded Project (2014M551110), National Natural Science Foundation of China (61403073), and the Fundamental Research Funds for the Central Universities (N140404018). .
References
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