A Novel Robust Extended Dissipativity State Feedback ...
Transcript of A Novel Robust Extended Dissipativity State Feedback ...
A Novel Robust Extended Dissipativity State Feedback Control system design
for Interval Type-2 Fuzzy Takagi-Sugeno Large-Scale Systems
Mojtaba Asadi Jokar, Iman Zamani, Mohamad Manthouri*, Mohammad Sarbaz
Electrical and Electronic Engineering Department, Shahed University, Tehran, Iran * Corresponding author. Tel.: +98 21 51212029.
E-mail addresses: [email protected], [email protected], [email protected],
Abstract: In this paper, we use the advantage of large-scale systems modeling based on the type-2 fuzzy Takagi
Sugeno model to cover the uncertainties caused by large-scale systems modeling. The advantage of using membership
function information is the reduction of conservatism resulting from stability analysis. Also, this paper uses the
extended dissipativity robust control performance index to reduce the effect of external perturbations on the large-
scale system, which is a generalization of π»β, πΏ2 β πΏβ, passive, and dissipativity performance indexes and control
gains can be achieved through solving linear matrix inequalities (LMIs), so the whole closed-loop fuzzy large-scale
system is asymptotically stable.. Finally, the effectiveness of the proposed method is demonstrated by a numerical
example of a double inverted pendulum system.
Keywords: membership function dependent, type-2 fuzzy, large-scale system, extended dissipativity, state
feedback.
I. Introduction
Since years ago, with the expansion and complexity of industries, a challenge in the engineering has been control
of processes like mechanical engineering, electrical engineering, chemical engineering, or so. Many algorithms and
approaches have proposed in dealing with the instability of systems. But since decades ago, systems have been turned
to large in scope and dynamic. Consequently, control of the process has become an essential task and the main problem
facing such systems is the complexity of mathematical relationships that make it hard to solve in practice. Various
controllers have been designed to encounter the instability of systems in both industry and academic like adaptive
control, fuzzy control, etc. To design an effective and authentic controller, the dynamic of the system should be
identified exactly and it is an important step. In large-scale systems, it is almost impossible to identify the dynamic of
the system accurately. Hence, the well-known method, fuzzy logic is used. Also, it is confirmed that the Takagi-
Sugeno (T-S) model is a powerful tool to approximate any nonlinear systems with arbitrarily high accuracy. The
Takagi-Sugeno fuzzy system shows the dynamic behavior of the nonlinear system with the weighted sum of local
linear systems, which are determined by the membership functions.
Todays, many systems in both industry and academic are large-scale, and have been popular for years [1] and [2].
Besides, many systems specially control systems have become large in scope and complex in computation [3]. These
type of the systems are modeled by a number of independent subsystems which work independently with some
interactions [4]. Since years ago, lots of researches have been done in large-scale systems and gain many attentions
[5], [6], and [7]. Recently, fuzzy systems with IF-THEN rules have become more broad appeal and most of the
nonlinear and complex systems are estimated by fuzzy logic [8]. One of the powerful tools, which can fill the gap
between linear and severe nonlinear systems is Takagi-Sugeno fuzzy model. Many investigations have been done
based on the T-S fuzzy model [9] and [10]. [11] and [12] propose a type of fuzzy inference system popular Takagi-
Sugeno, fuzzy model. In [13], Takagi-Sugeno fuzzy model is selected to represent the dynamic of the unknown
nonlinear system. In [14], the stability of the fuzzy time-varying fuzzy large-scale system based on the piecewise
continuous Lyapunov function investigate. Since systems always have uncertainty or perturbation parameters, several
papers, including [15] and [16] have studied the robust control criterion such as π»β control criterion for large-scale
systems. [17] and [18] design a nonlinear state feedback controller for each subsystem. In [15], the stability and
stabilization conditions of large-scale fuzzy systems obtained through Lyapunov functions and also by using the
continuous piecewise Lyapunov functions, they have performed stability analysis and π»β based controller design for
large-scale fuzzy systems. The author in [19] design an approach for stability analysis of T-S fuzzy systems via
piecewise quadratic stability. The design of robust fuzzy control for exposure to nonlinear time-delay system modeling
error also presented in [20]. Also, in [16], the reference tracking problem using decentralized fuzzy π»β control is
investigated. [5] Designed decentralized linear controllers to stabilize the large-scale system using the Riccati equation
basis, which increases the local state feedback gain if the number of subsystems is large.
Obviously it seems that taking the advantage of modeling systems based on type-2 fuzzy Takagi-Sugeno model to
cover the uncertainties caused by large-scale systems modeling has not been studied yet and several problems remind
unsolved. So the contribution of this paper can be summarized as: In the first step, we analyze the stability of the large-
scale system by using a fuzzy type-2 model based on membership functions, and by using the fuzzy model type-2
decentralized state feedback controller, we stabilize the large-scale system. The advantage of using membership
functions in the stability analysis is the reduction of conservatism in the stability analysis. Also, under the imperfect
premise matching, the type-2 fuzzy controller can choose the premise membership functions and the number of rules
different from the type-2 fuzzy model freely. Next, to reduce the effect of external perturbations on the large-scale
system, we apply the robustness control criterion to the stability analysis, which can guarantee the extended
dissipativity index that describes in definition1 in the appendix. Finally, a large-scale system was given to illustrate
the effectiveness of the proposed method.
The rest of this paper is: Section II formulates the problem. In section III, stability conditions are given. In section
IV, the numerical example presented, and section V, demonstrate the effectiveness of the proposed methods.
II. Problem Formulation
Consider a large-scale nonlinear system with uncertainty parameters that have π subsystem and in a closed-loop
system with a state feedback controller. The mathematical representation of this closed-loop system is based on the
Takagi-Sugeno type-2 fuzzy model. Equation (1) shows a π-rule of the Takagi-Sugeno type-2 fuzzy model for the πth
subsystem in the large-scale system:
π·ππππ ππ πΊππβ πΊπππππ π:
π°π ππ1(π‘)ππ πΉπ1, ππ2(π‘)ππ πΉπ2π πππβ¦πππ πππ(π‘)ππ πΉππ
π
π»π―π¬π΅
{
οΏ½ΜοΏ½π(π‘) =βοΏ½ΜοΏ½ππ(π₯π(π‘))
(
(π΄πππ₯π(π‘) + π΅πππ’π(π‘)) + π·1ππππ(π‘) +β οΏ½Μ οΏ½ππππ₯π(π‘)
π
π=1πβ π
)
π
π=1
(1)
where πΉπππ (π = 1.2β¦ .π) is a fuzzy set, and πππ(π‘) is a measurable variable. π is the number of rules in the subsystem
πth. π₯π(π‘) β πΉπ is the state vector of the πth subsystem. The pairs π΄ππ β πΉ
πΓπ, π΅ππ β πΉπΓπ and π·1ππare matrices of the
πth model of the πth subsystem. π’(π‘) β πΉπdenotes the input vector. οΏ½Μ οΏ½πππis the vector of the interactions between the πth
subsystem and πth at the πth rule, and π₯π(π‘) β πΉπ is the state vector of the πth subsystem. π represents the total
number of subsystems. ππ(π‘) β πΉπ is the disturbance input belonging to π³2[0,β); οΏ½ΜοΏ½ππ(π₯π(π‘)) is a membership
function of the πth rule of the πth subsystem, which is represented by (2).
οΏ½ΜοΏ½ππ(π₯π(π‘)) = πΌππ(π₯π(π‘))π€ππ(π₯π(π‘)) + πΌππ(π₯π(π‘))π€ππ(π₯π(π‘)) (2)
Equation (2) is a type reduction in the type-2 fuzzy structure in which πΌππ and πΌππ are nonlinear functions. As the
nonlinear plant is subject to parameter uncertainties οΏ½ΜοΏ½ππ(π₯π(π‘)) will depend on the parameter uncertainties and thus
leads to the value of πΌππ and πΌππ uncertain. π€ππand π€ππ are lower membership and upper membership degrees,
respectively that characterized by the LMFs and UMFs. Since οΏ½ΜοΏ½ππ(π₯π(π‘)) is a type-2 membership function, it has the
following properties:
βοΏ½ΜοΏ½ππ(π₯π(π‘))
π
π=1
= 1; 0 β€ πΌππ(π₯π(π‘)) β€ 1, 0 β€ πΌππ(π₯π(π‘)) β€ 1, βπ
πΌππ(π₯π(π‘)) + πΌππ(π₯π(π‘)) = 1, βπ
π€ππ(π₯π(π‘)) =βππΉππΌπ (πππΌ(π₯π(π‘)))
π
πΌ=1
; π€ππ(π₯π(π‘)) =βππΉππΌπ (πππΌ(π₯π(π‘)))
π
πΌ=1
ππΉππΌπ (πππΌ(π₯π(π‘))) > ππΉππΌ
π (πππΌ(π₯π(π‘))) β₯ 0; π€ππ(π₯π(π‘)) β₯ π€ππ(π₯π(π‘)) β₯ 0, βπ
ππΉππΌπ and π
πΉππΌπ are the lower membership functions (LMF) and the upper membership functions (UMF), respectively. π
is the number of fuzzy sets of the πth model of the πth subsystem. Thus οΏ½ΜοΏ½ππ(π₯π(π‘)) is a linear combination of π€ππ and
π€ππ denoted by LMFs and UMFs.
Equation (3) is the Takagi-Sugeno type-2 fuzzy representation for state feedback controller. Unlike the PDC control
method, the membership functions and the number of rules of the fuzzy system model and the controller need not be
the same here. Thus, the membership functions and the number of controller rules relative to the plant model can be
freely chosen. For the πth subsystem controller we have:
πͺππππππππ πππ πΊππ β πΊπππππ π:
π°π ππ1(π‘)ππ ππ1π, ππ2(π‘)ππ ππ2
π πππβ¦πππ ππΞ©(π‘)ππ ππΞ©
π
π»π―π¬π΅ π’π(π‘) =βοΏ½ΜοΏ½ππ(π₯π(π‘))πΊπππ₯π(π‘)
π
π=1
(3)
where πππ½π
is the fuzzy set of πth rules of the πth subsystem, corresponding to the function ππ½(π‘) . The state vector is
π₯π(π‘) β πΉπ where π is the number of control rules of the πth subsystem. πΊππ β πΉ
πΓπ is the control gain and οΏ½ΜοΏ½ππ(π₯π(π‘))
is the membership function of πth rules of the πth subsystem with these properties:
οΏ½ΜοΏ½ππ(π₯π(π‘)) =π½ππ(π₯π(π‘))πππ(π₯π(π‘)) + π½ππ(π₯π(π‘))πππ(π₯π(π‘))
β (ππ=1 π½ππ(π₯π(π‘))πππ(π₯π(π‘)) + π½ππ(π₯π(π‘))πππ(π₯π(π‘))
(4)
where
βοΏ½ΜοΏ½ππ(π₯π(π‘))
π
π=1
= 1, 0 β€ π½ππ(π₯π(π‘)) β€ 1, 0 β€ π½ππ(π₯π(π‘)) β€ 1, βπ
π½ππ(π₯π(π‘)) + π½ππ(π₯π(π‘)) = 1, βπ
πππ(π₯π(π‘)) =βππππ½π (πππ½(π₯π(π‘)))
Ξ©
π½=1
, πππ(π₯π(π‘)) =βππππ½π (πππ½(π₯π(π‘)))
Ξ©
π½=1
ππππ½π (πππ½(π₯π(π‘))) > ππππ½
π (πππ½(π₯π(π‘))) β₯ 0, πππ(π₯π(π‘)) > πππ(π₯π(π‘)) β₯ 0
where πππ(π₯π(π‘)) and πππ(π₯π(π‘)) denote the lower and the upper membership degree. π½ππ(π₯π(π‘)) and π½ππ(π₯π(π‘)) are
two nonlinear functions. Relation (4) illustrates the part of the type reduction in type-2 fuzzy structure. Ξ© is the total
number of fuzzy rules for πth controller rules of the πth subsystem. ππππ½π (πππ½(π₯π(π‘)))and π
πππ½π (πππ½(π₯π(π‘))) represent
LMF and UMF, respectively. Finally, the type-2 fuzzy model for πth sub-system will be:
οΏ½ΜοΏ½π(π‘) =βββΜπππ {(π΄ππ +π΅πππΊππ)π₯π(π‘) + π·1ππππ(π‘) +β οΏ½Μ οΏ½ππππ₯π(π‘)
π
π=1πβ π
}
π
π=1
π
π=1
(5)
where βΜπππ(π₯π(π‘)) from (4) and (4) equals the following relation:
βΜπππ(π₯π(π‘)) = οΏ½ΜοΏ½ππ(π₯π(π‘))οΏ½ΜοΏ½ππ(π₯π(π‘)) (6)
has the following properties:
βββΜπππ(π₯π(π‘))
π
π=1
π
π=1
= 1, βπ, π, π
To facilitate the stability analysis of the large-scale type-2 fuzzy control system, we divide the state space π into π
subspace, i.e., state-space equals π = β ππππ=1 . Also, to use the information of type-2 membership functions, LMFs
and UMFs are described with uncertainty coverage space or briefly FOUs. Now consider dividing FOUs by π + 1
sub-FOU. In the π§th sub-FOU, LMFs and UMFs define as follows:
βππππ§(π₯π(π‘)) =ββ β¦ββππππππ§(π₯π(π‘))πΏππππ1π2β¦ππππ§
π
π=1
2
ππ=1
2
π1=1
π
π=1
,
βππππ§(π₯π(π‘)) =ββ β¦ββππππππ§(π₯π(π‘))πΏππππ1π2β¦ππππ§
π
π=1
2
ππ=1
2
π1=1
π
π=1
(7)
with these properties:
0 β€ πΏππππ1π2β¦ππππ§ β€ πΏππππ1π2β¦ππππ§ β€ 1, 0 β€ βππππ§(π₯π(π‘)) β€ βππππ§(π₯π(π‘)) β€ 1, 0 β€ πππππππ§(π₯π(π‘)) β€ 1
ππ1πππππ§(π₯π(π‘)) + ππ2πππππ§(π₯π(π‘)) = 1
βββ¦ββππππππ§(π₯π(π‘))
π
π=1
2
ππ=1
2
π1=1
π
π=1
= 1, π, π = 1,2β¦ , π; π§ = 1,2β¦ , π + 1; ππ = 1,2;
π₯(π‘)πβ π ; ππ‘βπππ€ππ π , ππππ ππ(π₯π(π‘)) = 0 ;
where πΏππππ1π2β¦ππππ§ and πΏππππ1π2β¦ππππ§ are scalar that must be specified. πππππππ§ are functions that specified by the method
intended to approximate membership functions. Finally, in order to show the sub-FOUs in βΜπππ(π₯π(π‘)) we have:
βΜπππ(π₯π(π‘)) = οΏ½ΜοΏ½ππ(π₯π(π‘))οΏ½ΜοΏ½ππ(π₯π(π‘))
=βπππππ§(π₯π(π‘)) [πΎππππ§(π₯π(π‘))βππππ§(π₯π(π‘)) + πΎππππ§(π₯π(π‘))βππππ§(π₯π(π‘))]
π+1
π§=1
(8)
where for the membership function βΜπππ(π₯π(π‘)) with π, π, π at any one time, among π + 1 sub-FOU is only once
πππππ§(π₯π(π‘)) = 1 and the remainder are zero. πΎππππ§(π₯π(π‘)) and πΎππππ§(π₯π(π‘)) are two functions that have the following
properties:
0 β€ πΎππππ§(π₯π(π‘)) β€ πΎππππ§(π₯π(π‘)) β€ 1, πΎππππ§(π₯π(π‘)) + πΎππππ§(π₯π(π‘)) = 1, βπ, π, π§
III. Main Result
In this section, we will obtain the stability of the closed-loop large-scale system using the type-2 Takagi-Sugeno
model. In [11], the authors introduced a new performance index, referred to extended dissipativity performance index
that holds π»β, πΏ2-πΏβ, passive, and dissipativity performance indexes. This performance indexes describe in definition
1 in the appendix. Therefore, the primary purpose of this section is to design the type-2 Takagi-Sugeno fuzzy state-
feedback controller for the large-scale system such that the closed-loop system is asymptotically stable with the π»β,
πΏ2-πΏβ, passive, and dissipativity performance indexes such that:
1- The closed-loop system with π(π‘) = 0 is asymptotically stable.
2- The closed-loop system holds extended dissipativity performance index.
Theorem1. For given matrices π, π1 , π2 , and π3 satisfying in assumption 1 in the appendix, the system in (5) is
asymptotically stable and satisfies the extended dissipativity performance indexes, if there exist matrices ππ = πππ >
0 , πΎπ = πΎππ > 0, ππ = ππ
π β πΉπΓπ , πππ β πΉπΓπ ,πππππ§ = πππππ§
π β πΉπΓπ , (π = 1,2,β¦ ,π; π = 1,2,β¦ , π; π =
1,2,β¦ , π; π§ = 1,2,β¦ , π + 1)such that the following LMIs hold:
πππππ§ > 0 βπ, π, π, π§ (9)
Ξ©πππ +πππππ§ +ππ > 0 βπ, π, π, π§ (10)
ββ(πΏππππ1π2β¦ππππ§Ξ©πππ β (πΏππππ1π2β¦ππππ§ β πΏππππ1π2β¦ππππ§)πππππ§ + πΏππππ1π2β¦ππππ§ππ) β ππ
π
π=1
π
π=1
< 0 βπ1, π2, β¦ , ππ , π, π, π§ (11)
Ξ2π = [βπΎπ οΏ½ΜοΏ½ππ
ππππ
β βπΌ] < 0
(12)
Ξ1π = [βππ ππβ πΎπ β 2πΌ
] < 0 (13)
where
Ξ©Μπππ = [Ξ©Μ1πππ Ξ©Μ2ππ
β Ξ©Μ3ππ], Ξ©Μ1πππ = π―π(π΄ππππ +π΅πππππ) + π0
β1(π β 1) [β (πποΏ½ΜοΏ½πππ οΏ½ΜοΏ½ππππ)
ππ=1πβ π
] + ππ β οΏ½ΜοΏ½ππππ1ποΏ½ΜοΏ½ππ
Ξ©Μ2ππ = βπ·2πππ π1ππ·2ππ β π»π(π·2πππ2π) β π3π, Ξ©Μ3ππ = π·1ππ β οΏ½ΜοΏ½ππ
ππ1ποΏ½ΜοΏ½ππ β οΏ½ΜοΏ½πππ2π
for all π, π, π; and the feedback gain define as πΊππ = πππππβ1 for all π, π. Remember that π―π(π΄) = π΄ + π΄π. Also
οΏ½ΜοΏ½πππ define as οΏ½ΜοΏ½ππ
π β₯ ββ β βΜπππ(π₯π(π‘))οΏ½Μ οΏ½πππππ=1
ππ=1 β.
Proof. Consider the quadratic Lyapunov function as follows:
π(π‘) = β π₯ππ(π‘)πππ₯π(π‘)
ππ=1 , 0 < ππ = ππ
π β πΉπΓπ , βπ (14)
The main objective is to develop a condition guaranteeing that π(π‘) > 0and οΏ½ΜοΏ½(π‘) < 0 for all π₯π(π‘) β 0, the type-2
fuzzy large-scale control system is guaranteed to be asymptotically stable, implying that π₯π(π‘) β 0 as π‘ β β. To
ensure that οΏ½ΜοΏ½(π‘) < 0 for all π₯π(π‘) β 0 we have:
οΏ½ΜοΏ½(π‘) =β{οΏ½ΜοΏ½ππ(π‘)πππ₯π(π‘) + π₯π
π(π‘)πποΏ½ΜοΏ½π(π‘)}
π
π=1
=β2{(βββΜπππ{(π΄ππ + π΅πππΊππ)π₯π(π‘) + π·1ππππ(π‘)}
π
π=1
π
π=1
)
π
πππ₯π(π‘)}
π
π=1
+β2{βββΜπππ {βοΏ½Μ οΏ½ππππ₯π(π‘)
π
π=1πβ π
}
π
π=1
π
π=1
}πππ₯π(π‘)
π
π=1
(15)
Same as [21] for interconnections terms by using Lemma1 in the appendix and noting that οΏ½ΜοΏ½ππ β₯ββ β βΜπππ(π₯π(π‘))οΏ½Μ οΏ½πππ
ππ=1
ππ=1 β we have
β{[βοΏ½ΜοΏ½πππ₯π(π‘)
π
π=1πβ π
]
π
[βοΏ½ΜοΏ½πππ₯π(π‘)
π
π=1πβ π
]}
π
π=1
β€β{[βοΏ½ΜοΏ½πππ₯π(π‘)
π
π=1πβ π
]
π
[βοΏ½ΜοΏ½πππ₯π(π‘)
π
π=1πβ π
]}
π
π=1
β€β{(π β 1) [βπ₯ππ(π‘)οΏ½ΜοΏ½ππ
π οΏ½ΜοΏ½πππ₯π(π‘)
π
π=1πβ π
]}
π
π=1
(16)
and by using Lemma2 in the appendix we have and by considering 0 < π0 < ππππwe have
οΏ½ΜοΏ½(π‘)
β€β2{(βββΜπππ{(π΄ππ + π΅πππΊππ)π₯π(π‘) + π·1ππππ(π‘)}
π
π=1
π
π=1
)
π
πππ₯π(π‘)}
π
π=1
+β{π0β1(π β 1) [βπ₯π
π(π‘)οΏ½ΜοΏ½πππ οΏ½ΜοΏ½πππ₯π(π‘)
π
π=1πβ π
]}
π
π=1
+β(βββΜπππππ
π
π=1
π
π=1
(π₯π(π‘)ππππππ₯π(π‘)))
π
π=1
(17)
let ππ = ππβ1, ππ(π‘) = ππ
β1π₯π(π‘) , πππ = πΊππππ, οΏ½ΜοΏ½ππ = πΆππππ, then we have
οΏ½ΜοΏ½(π‘)
β€β
{
βββΜπππ
(
πππ(π‘)(πππ΄ππ
π + πππππ΅ππ
π + π΄ππππ +π΅πππππ)ππ(π‘)
π
π=1
π
π=1
π
π=1
+ π0β1(π β 1) [βππ
π(π‘)(πποΏ½ΜοΏ½πππ οΏ½ΜοΏ½ππππ)ππ(π‘)
π
π=1πβ π
] + ππ(πππ(π‘)ππ(π‘))
)
}
(18)
π§π(π‘) =βββΜπππ{οΏ½ΜοΏ½ππππ(π‘) + π·2ππππ(π‘)}
π
π=π
π
π=π
(19)
now by consider the following performance index we have
οΏ½ΜοΏ½(π‘) β π½(π‘) β€βπππ {βββΜπππΞ©Μπππ
π
π=1
π
π=1
}ππ
π
π=1
(20)
π½(π‘) =β(π§πππ1ππ§π + 2π§π
ππ2πππ(π‘) + πππ(π‘)π3πππ(π‘))
π
π=1
=β(βββΜπππ{οΏ½ΜοΏ½ππππ(π‘) + π·2ππππ(π‘)}π
π
π=π
π
π=π
π1π{οΏ½ΜοΏ½ππππ(π‘) + π·2ππππ(π‘)}
π
π=1
+ 2{οΏ½ΜοΏ½ππππ(π‘) + π·2ππππ(π‘)}ππ2πππ(π‘) + ππ
π(π‘)π3πππ(π‘) (21)
where
ππ(π‘) = [ππ(π‘)ππ(π‘)
], Ξ©Μπππ = [Ξ©Μ1πππ Ξ©Μ2ππ
β Ξ©Μ3ππ], Ξ©Μ1πππ = π―π(π΄ππππ + π΅πππππ) + π0
β1(π β 1) [β (πποΏ½ΜοΏ½πππ οΏ½ΜοΏ½ππππ)
ππ=1πβ π
] + ππ β
οΏ½ΜοΏ½ππππ1ποΏ½ΜοΏ½ππ, Ξ©Μ2ππ = βπ·2ππ
π π1ππ·2ππ βπ»π(π·2πππ2π) β π3π, Ξ©Μ3ππ = π·1ππ β οΏ½ΜοΏ½ππππ1ποΏ½ΜοΏ½ππ β οΏ½ΜοΏ½πππ2π.
by using Schur complement we have
Ξ©πππ = [Ξ©11πππ Ξ©12ππ Ξ©13ππβ Ξ©22ππ Ξ©23ππβ β βI
], Ξ©11πππ = π―π(π΄ππππ +π΅πππππ) + π0β1(π β 1) [β (πποΏ½ΜοΏ½ππ
π οΏ½ΜοΏ½ππππ)ππ=1πβ π
] + ππ, Ξ©12ππ =
π·1ππ β οΏ½ΜοΏ½πππ2π, Ξ©13ππ = οΏ½ΜοΏ½ππππ1π
π , Ξ©22ππ = βπ»π(π·2πππ2π) β π3π, Ξ©23ππ = π·2πππ π1π
π .
if we can prove β β βΜπππΞ©πππππ=1
ππ=1 < 0 then we have:
οΏ½ΜοΏ½(π‘) β π½(π‘) β€βπππ {βββΜπππΞ©Μπππ
π
π=1
π
π=1
}ππ
π
π=1
< 0 (22)
now by using (8) considering the information of the sub-FOUs is brought to the stability analysis with the introduction
of some slack matrices through the following inequalities using the S-procedure:
let ππ = πππis an arbitrary matrix with appropriate dimensions. Then
{βββπππππ§(π₯π(π‘)) [(πΎππππ§(π₯π(π‘))βππππ§(π₯π(π‘)) + πΎππππ§(π₯π(π‘))βππππ§(π₯π(π‘))) β 1]ππ
π+1
π§=1
π
π=1
π
π=1
} = 0 (23)
also, consider 0 β€ πππππ§ =πππππ§π
βββ(1β πΎππππ§(π₯π(π‘)))
π
π=1
π
π=1
(βππππ§(π₯π(π‘)) β βππππ§(π₯π(π‘)))πππππ§ β₯ 0 (24)
by using (23) and (24) for β β βΜπππΞ©πππππ=1
ππ=1 < 0 we have
β{βββ(πππππ§(π₯π(π‘)) [πΎππππ§(π₯π(π‘))βππππ§(π₯π(π‘)) + (1 β πΎππππ§(π₯π(π‘)))βππππ§(π₯π(π‘))])
π+1
π§=1
Ξ©πππ
π
π=1
π
π=1
}
π
π=1
βββββπππππ§(π₯π(π‘))
π+1
π§=1
(1 β πΎππππ§(π₯π(π‘)))
π
π=1
π
π=1
(βππππ§(π₯π(π‘)) β βππππ§(π₯π(π‘)))πππππ§
π
π=1
+β{βββπππππ§(π₯π(π‘)) [(πΎππππ§(π₯π(π‘))βππππ§(π₯π(π‘)) + (1 β πΎππππ§(π₯π(π‘)))βππππ§(π₯π(π‘)))
π+1
π§=1
π
π=1
π
π=1
π
π=1
β 1]ππ}
=β{[βββπππππ§(π₯π(π‘)) (βππππ§(π₯π(π‘))Ξ©πππ β (βππππ§(π₯π(π‘)) β βππππ§(π₯π(π‘)))πππππ§
π+1
π§=1
π
π=1
π
π=1
π
π=1
+ βππππ§(π₯π(π‘))ππ)β ππ]}
+ββββπππππ§(π₯π(π‘))
π+1
π§=1
π
π=1
π
π=1
π
π=1
πΎππππ§(π₯π(π‘)) (βππππ§(π₯π(π‘)) β βππππ§(π₯π(π‘))) (Ξ©πππ +πππππ§ +ππ) < 0 (25)
also, the following equation must be checked
[βββπππππ§(π₯π(π‘)) (βππππ§(π₯π(π‘))Ξ©πππ β (βππππ§(π₯π(π‘)) β βππππ§(π₯π(π‘)))πππππ§ + βππππ§(π₯π(π‘))ππ)
π+1
π§=1
π
π=1
π
π=1
βππ] < 0 βπ (26)
and Ξ©Μπππ +πππππ§ +ππ > 0 for all π, π, π, π§ due to (βππππ§(π₯π(π‘)) β βππππ§(π₯π(π‘))) β€ 0. Recalling that only one
πππππ§(π₯π(π‘)) = 1 for each fixed value of π, π, π at any time instant such that β πππππ§(π₯π(π‘))π+1π§=1 = 1, the first set of
inequality is satisfied by
[ββ(βππππ§(π₯π(π‘))Ξ©πππ β (βππππ§(π₯π(π‘)) β βππππ§(π₯π(π‘)))πππππ§ + βππππ§(π₯π(π‘))ππ) βππ
π
π=1
π
π=1
]
< 0 βπ, π, π, π§ (27)
Expressing βππππ§(π₯π(π‘)) and βππππ§(π₯π(π‘))with (7) and recalling that
β β β¦β β πππππππ§(π₯π(π‘))ππ=1
2ππ=1
2π1=1
ππ=1 = 1, for all z and πππππππ§ β₯ 0 for all π, ππ , π, π πππ π§ the first set of
inequalities will be satisfied if the following inequalities hold
[ββ β¦ββππππππ§(π₯π(π‘))
π
π=1
2
ππ=1
2
π1=1
π
π=1
ββ(πΏππππ1π2β¦ππππ§Ξ©πππ β (πΏππππ1π2β¦ππππ§ β πΏππππ1π2β¦ππππ§)πππππ§
π
π=1
π
π=1
+ πΏππππ1π2β¦ππππ§ππ) βππ] < 0 βπ1, π2, β¦ , ππ , π, π, π§ (28)
consequently (27) can be guaranteed by
ββ(πΏππππ1π2β¦ππππ§Ξ©πππ β (πΏππππ1π2β¦ππππ§ β πΏππππ1π2β¦ππππ§)πππππ§ + πΏππππ1π2β¦ππππ§ππ) β ππ
π
π=1
π
π=1
< 0 βπ1, π2, β¦ , ππ , π, π, π§ (29)
therefore, there is always a sufficiently small scalar π > 0 such that Ξ©Μπππ β€ βππΌ. This means that
οΏ½ΜοΏ½(π‘) β π½(π‘) β€ βπ |βππ
π
π=1
|
2
(30)
thus π½(π‘) β₯ οΏ½ΜοΏ½(t) hold for any π‘ β₯ 0, which means
β« π½(π )π‘
0
ππ β₯ π(π₯(π‘)) β π(π₯(0)) (31)
then by considering π = βπ(π₯(0))in (31) we have:
β« π½(π )π‘
0
ππ β₯ π(π₯(π‘)) + π, βπ‘ β₯ 0 (32)
According to Definition 1 in the appendix, if we want to design a controller with a robust π»β performance, then we
must set the π value to zero. For substitution π(π₯(π‘)) in (32) considering πΎπ > 0 by Characteristic (πΎπ β πΌ)πΎπβ1(πΎπ β
πΌ) β₯ 0 where βπΎπβ1 β€ πΎπ β 2πΌ then we have:
Ξ1π = [βππ ππβ πΎπ β 2πΌ
] < 0 (33)
Finally ππ > πΎπ and (14) proved if :
π(π₯(π‘)) =βπ₯ππ(π‘)πππ₯π(π‘)
π
π=1
β₯βπ₯ππ(π‘)πΎππ₯π(π‘)
π
π=1
β₯ 0 (34)
Also
π(π₯(π‘)) =βπ₯ππ(π‘)πππ₯π(π‘)
π
π=1
β₯βπ₯ππ(π‘)πΎππ₯π(π‘)
π
π=1
β₯ 0 (35)
According to Definition 1, we need to prove that the following inequality holds for any matrices Οπ , π1π ,π2ππππ π3π satisfying assumption 1 in the appendix:
β« π½(π‘)π‘
0
ππ‘ β π§π(π‘)ππ§(π‘) β₯ π (36)
To this end, we consider the two cases of βπβ = 0 and βπβ β 0, respectively. Firstly, we consider the case
whenβπβ = 0. Also in this case by considering π1 = βπΌ , π2 = 0 , π3 = πΎ2πΌ and π = 0 the π»β performance
index will be hold.
β« π½(π )π‘
0
ππ =βπ₯ππ(π‘)πΎππ₯π(π‘)
π
π=1
+ π β₯ π, βπ‘ β₯ 0 (37)
By using (37) and considering π§π(π‘)ππ§(π‘) β‘ 0 the (36) hold. Secondly, we consider the case of βπβ β 0. In this case,
it is required under Assumption1 in the appendix that βπ1β + βπ2β = 0 and βπ·2ππβ = 0, which implies that π1 =0, π2 = 0 πππ π3 > 0. Then:
π½(π ) =βπππ(π )π3πππ
π(π )
π
π=1
β₯ 0
now by considering οΏ½ΜοΏ½ππππποΏ½ΜοΏ½ππ β€ πΎπ due to
Ξ2π = [βπΎπ οΏ½ΜοΏ½ππ
ππππ
β βπΌ] < 0 (38)
and βπ·2ππβ = 0 satisfy in assumption1 for any π‘ β₯ 0, the following inequalities hold
β« π½(π )π‘
0
ππ β π§π(π‘)ππ§(π‘)
β₯ β« π½(π )π‘
0
ππ ββ(βββΜπππ{πΆπππ₯π(π‘) + π·2ππππ(π‘)}π»
π
π=π
π
π=π
ππ{πΆπππ₯π(π‘) + π·2ππππ(π‘)}
π
π=1
= β« π½(π )π‘
0
ππ ββ(βββΜπππ
π
π=π
π
π=π
(πππ(π‘)οΏ½ΜοΏ½ππ
ππποΏ½ΜοΏ½ππππ(π‘))
π
π=1
β₯ β« π½(π )π‘
0
ππ ββ(βββΜπππ
π
π=π
π
π=π
(π₯ππ(π‘)πΎππ₯π(π‘))
π
π=1
β₯ π (39)
Finally by Ο(π‘) β‘ 0 we have:
οΏ½ΜοΏ½(π‘) β€ π§π(π‘)(ββββΜπππ
π
π=π
π
π=π
(π1ππ)
π
π=1
)π§(π‘) β π |βππ
π
π=1
|
2
(40)
According to assumption1 in the appendix π1ππ < 0 for any π, π, then we have
οΏ½ΜοΏ½(π‘) β€ βπ |βππ
π
π=1
|
2
(41)
thus the closed-loop system asymptotically stable by Ο(π‘) β‘ 0. This completes the proof.
Remark1: It can be seen from (8) that if more sub-FOUs are considered the more information about the FOU is
contained in the local LMFs and UMFs. Thus, using the information of membership functions into the stability
condition is resulting in a more relaxed stability analysis result.
Remark2: From (28), the advantage of using the type-2 fuzzy system in the form of (5) can be seen that local LMFs
and UMFs determine the stability condition.
Remark3: By expressing βππππ§(π₯π(π‘)) and βππππ§(π₯π(π‘)) in the form of (7), they are characterized by the constant
scalersπΏππππ1π2β¦ππππ§ and πΏππππ1π2β¦ππππ§. Also, noting that the cross terms β ππππππ§(π₯π(π‘))ππ=1 are independent of π and π.
By these favorable properties we need only to check (28) at some discrete points (πΏππππ1π2β¦ππππ§ and πΏππππ1π2β¦ππππ§) instead
of every single point of the local LMFs and UMFs.
Remark4: Under the imperfect premise matching, the type-2 fuzzy controller can choose the premise membership
functions and the number of rules different from the type-2 fuzzy model freely.
Corollary: In the particular case, if we do not consider disturbance, then we have the following result. First, we
consider a large-scale nonlinear system that is composed of π nonlinear subsystems with interconnections. A p-rule
type-2 fuzzy T-S model is employed to describe the dynamics of the πth nonlinear subsystem as follows:
π·ππππ πΉπππ π: π°π ππ1(π‘)ππ πΉπ1, ππ2(π‘)ππ πΉπ2
π πππβ¦ πππ πππ(π‘)ππ πΉπππ
π»π―π¬π΅
οΏ½ΜοΏ½π(π‘) =βοΏ½ΜοΏ½ππ(π₯π(π‘))
(
(π΄πππ₯π(π‘) + π΅πππ’π(π‘)) +β οΏ½Μ οΏ½ππππ₯π(π‘)
π
π=1πβ π
)
π
π=1
(42)
where πΉππΌπ is a type-2 fuzzy set of rule π corresponding to the function πππΌ(π‘),π = 1,2, β¦ ,π; πΌ = 1,2,β¦ , π; π =
1,2,β¦ , π; π is a positive integer; π₯π(π‘) β πΉπ is the πth subsystem state vector; the π΄ππ β πΉ
πΓπ and π΅ππ β πΉπΓπare the
known system and input matrices, respectively; π’π β πΉπ is the input vector. οΏ½Μ οΏ½πππ denotes the interconnection matrix
between the πth and πth subsystems; (π΄ππ , π΅ππ) are the πth local model; The firing strength of the πth rule of πth
subsystem is of the form (2). Like controller in (3) the membership functions and the number of rules of the fuzzy
system model and the controller need not be the same here. Thus, the membership functions and the number of
controller rules relative to the plant model can be freely chosen. For the πth subsystem controller we have:
πͺπππππππππ πΉπππ π:
π°π π1(π₯(π‘))ππ ππ1π, ππ2(π₯(π‘))ππ ππ2
π πππβ¦πππ ππΞ©(π₯(π‘)) ππ ππΞ©
π
π»π―π¬π΅ π’π(π‘) =βοΏ½ΜοΏ½ππ(π₯π(π‘))πΊπππ₯π(π‘)
π
π=1
(43)
where ππΞ²π
is a type-2 fuzzy set of rule πth corresponding to the functionπππ½(π₯(π‘)), π½ = 1,2,β¦ ,Ξ©; π = 1,2, β¦ , π; Ξ© is
a positive integer; πΊπ β πΉπΓπ are the constant feedback gains to be determined. The firing strength of the πth rule is
the form of (4). Finally, we have the following type-2 fuzzy T-S large-scale control system:
οΏ½ΜοΏ½π(π‘) =ββοΏ½ΜοΏ½πποΏ½ΜοΏ½ππ {(π΄ππ +π΅πππΊπ)π₯π(π‘) +β οΏ½Μ οΏ½ππππ₯π(π‘)
π
π=1πβ π
}
π
π=1
π
π=1
(44)
Now, decentralized state feedback type-2 fuzzy T-S controller design presented for the continuous-time large-scale
type-2 fuzzy T-S model system in (55).
Theorem2. Consider a large-scale type-2 fuzzy T-S system model in (42). Decentralized state feedback type-2
fuzzy controller in the form of (43) exist, and can guarantee the asymptotic stability of the closed-loop type-2 fuzzy
control system (44) if there exist ππ = πππ > 0, πΊπ = πΊπ
π > 0, ππ = πππ β πΉπΓπ, πππ β πΉ
πΓπ , πππππ§ = πππππ§π β
πΉπΓπ ,(π = 1,2,β¦ , π; π = 1,2,β¦ , π; π = 1,2,β¦ , π; π§ = 1,2,β¦ , π + 1) such that the following LMIs hold:
πππππ§ > 0 βπ, π, π, π§ (45)
(
(
(πππ΄ππ
π +πππππ΅ππ
π +π΄ππππ + π΅πππππ) + π0β1(π β 1) [β(πποΏ½ΜοΏ½ππ
π οΏ½ΜοΏ½ππππ)
π
π=1πβ π
] + πππΌ
)
+πππππ§ +ππ
)
> 0 βπ, π, π, π§ (46)
ββ
(
πΏππππ1π2β¦ππππ§
(
(πππ΄ππ
π +πππππ΅ππ
π +π΄ππππ +π΅πππππ) + π0β1(π β 1) [β(πποΏ½ΜοΏ½ππ
π οΏ½ΜοΏ½ππππ)
π
π=1πβ π
]
π
π=1
π
π=1
+ πππΌ
)
β (πΏππππ1π2β¦ππππ§ β πΏππππ1π2β¦ππππ§)πππππ§ + πΏππππ1π2β¦ππππ§ππ
)
βππ < 0 βπ1, π2, β¦ , ππ , π, π, π§
(47)
where πΏππππ1π2β¦ππππ§ and πΏππππ1π2β¦ππππ§, π = 1,2, β¦ ,π; π = 1,2,β¦ , π; π = 1,2,β¦ , π; π§ = 1,2,β¦ , π + 1; ππ = 1,2; π =
1,2,β¦ , π are predefine constant scalers satisfying (7).
Proof. We consider the following quadratic Lyapunov function candidate to investigate the stability of the type-
2 fuzzy T-S large-scale control system
π(π‘) =βπ₯ππ(π‘)πππ₯π(π‘)
π
π=1
(48)
where 0 < ππ = πππ β πΉπΓπ.
The main objective is to develop a condition guaranteeing that π(π‘) > 0and οΏ½ΜοΏ½(π‘) < 0 for all π₯π(π‘) β 0, the type-2
fuzzy T-S large-scale control system is guaranteed to be asymptotically stable, implying that π₯π(π‘) β 0 as π‘ β β. We
have:
οΏ½ΜοΏ½(π‘) =β{οΏ½ΜοΏ½ππ(π‘)πππ₯π(π‘) + π₯π
π(π‘)πποΏ½ΜοΏ½π(π‘)}
π
π=1
=β
{
(ββοΏ½ΜοΏ½πποΏ½ΜοΏ½ππ {(π΄ππ +π΅πππΊππ)π₯π(π‘) +βοΏ½Μ οΏ½πππ₯π(π‘)
π
π=1πβ π
}
π
π=1
π
π=1
)
π
πππ₯π(π‘)
π
π=1
+ π₯ππ(π‘)ππ (ββοΏ½ΜοΏ½πποΏ½ΜοΏ½ππ {(π΄ππ +π΅πππΊππ)π₯π(π‘) +β οΏ½Μ οΏ½πππ₯π(π‘)
π
π=1πβ π
}
π
π=1
π
π=1
)
}
(49)
Such as proof of Theorem1 for interconnection term by considering οΏ½ΜοΏ½ππ β₯ ββ β βΜππποΏ½Μ οΏ½πππππ=1
ππ=1 β. we have:
οΏ½ΜοΏ½(π‘)
β€β2{(βββΜπππ{(π΄ππ + π΅πππΊππ)π₯π(π‘)}
π
π=1
π
π=1
)
π
πππ₯π(π‘)}
π
π=1
+β{π0β1(π β 1) [βπ₯π
π(π‘)οΏ½ΜοΏ½πππ οΏ½ΜοΏ½πππ₯π(π‘)
π
π=1πβ π
]}
π
π=1
+β(βββΜπππππ
π
π=1
π
π=1
(π₯π(π‘)ππππππ₯π(π‘)))
π
π=1
(50)
let ππ = ππβ1, ππ(π‘) = ππ
β1π₯π(π‘) , πππ = πΊππππ, then we have
οΏ½ΜοΏ½(t)
=β
{
βββΜπππ(π₯π(π‘))
(
πππ(π‘)(πππ΄ππ
π + πππππ΅ππ
π +π΄ππππ +π΅πππππ)ππ(π‘)
π
π=1
π
π=1
π
π=1
+ π0β1(π β 1) [βππ
π(π‘)(πποΏ½ΜοΏ½πππ οΏ½ΜοΏ½ππππ)ππ(π‘)
π
π=1πβ π
] + ππ(πππ(π‘)ππ(π‘))
)
}
(51)
we then express the type-2 membership function in the form of (8) and by considering the information of the sub-
FOUs brought to stability analysis with the introduction of some slack matrices as in (23) and (24). Then we have
οΏ½ΜοΏ½(π‘) < 0for all π₯π(π‘) β 0 from:
[
βββπππππ§(π₯π(π‘))
(
βππππ§(π₯π(π‘))
(
(πππ΄ππ
π +πππππ΅ππ
π +π΄ππππ +π΅πππππ)
π+1
π§=1
π
π=1
π
π=1
+ π0β1(π β 1) [β(πποΏ½Μ οΏ½ππππ
ποΏ½Μ οΏ½ππππππ)
π
π=1πβ π
] + πππππΌ
)
β (βππππ§(π₯π(π‘)) β βππππ§(π₯π(π‘)))πππππ§
+ βππππ§(π₯π(π‘))ππ
)
βππ
]
< 0 βπ (52)
(52) satisfied if the following inequality hold:
[
ββ
(
βππππ§(π₯π(π‘))
(
(πππ΄ππ
π +πππππ΅ππ
π +π΄ππππ +π΅πππππ) + π0β1(π β 1) [β(πποΏ½Μ οΏ½ππππ
ποΏ½Μ οΏ½ππππππ)
π
π=1πβ π
]
π
π=1
π
π=1
+ πππππΌ
)
β (βππππ§(π₯π(π‘)) β βππππ§(π₯π(π‘)))πππππ§ + βππππ§(π₯π(π‘))ππ
)
βππ
]
< 0 βπ, π, π, π§
(53)
also, the second set of inequalities will be satisfied if the following inequalities hold:
ββ
(
πΏππππ1π2β¦ππππ§
(
(πππ΄ππ
π + πππππ΅ππ
π + π΄ππππ +π΅πππππ) + π0β1(π β 1) [β(πποΏ½Μ οΏ½ππππ
ποΏ½Μ οΏ½ππππππ)
π
π=1πβ π
]
π
π=1
π
π=1
+ πππππΌ
)
β (πΏππππ1π2β¦ππππ§ β πΏππππ1π2β¦ππππ§)πππππ§ + πΏππππ1π2β¦ππππ§ππ
)
βππ < 0 βπ1, π2, β¦ , ππ , π, π, π§
(54)
This completes the proof.
IV. Simulation
Consider a double-inverted pendulum system connected by a spring, the modified equations of the motion for the
interconnected pendulum are given by [21].
{
οΏ½ΜοΏ½π1 = π₯π2
οΏ½ΜοΏ½π2 = βππ2
4π½ππ₯π1 +
ππ2
4π½πsin(π₯π1) π₯π2 +
2
π½ππ₯π2 +
1
π½ππ’π +β
ππ2
8π½ππ₯π1
2
π=1πβ π
, π = {1,2} (55)
where π₯π1denotes the angle of the πth pendulum from the vertical; π₯π2is the angular velocity of the πth pendulum. The
objective here is to design robust decentralized state feedback π»β fuzzy type-2 controller for the T-S fuzzy type-2
large-scale in the form of such that the resulting closed-loop system is asymptotically stable with an π»β disturbance
attenuation level πΎ. A concise framework on the decentralized state feedback control shown in Fig. 1. A concise
framework on the decentralized State Feedback control.
Fig. 1. A concise framework on the decentralized State Feedback control.
In this simulation, the masses of two pendulums chosen as π1 = 2ππ and π2 = 2.5ππ; the moments of inertia areπ½1 =2ππ.π2 and π½2 = 2.5ππ.π
2; the constant of the connecting torsional spring is π = 8π/π; the length of the pendulum
is π = 1π; the gravity constant is π = 9.8π/π 2 . We choose two local models, i.e., by linearizing the interconnected
pendulum around the origin and π₯π1 = (Β±88Β°, 0), respectively, each pendulum can be represented by the following
IT2 T-S fuzzy model with two fuzzy rules.
π π’ππ π: πΌπΉ π΄πππ₯π(π‘)ππ πΉππ ππ»πΈπ
οΏ½ΜοΏ½π(π‘) =ββοΏ½ΜοΏ½πποΏ½ΜοΏ½ππ {(π΄ππ +π΅πππΊππ)π₯π(π‘) + π·1ππππ(π‘) +β οΏ½Μ οΏ½πππ₯π(π‘)
π
π=1πβ π
}
π
π=1
π
π=1
π§π(π‘) =ββοΏ½ΜοΏ½πποΏ½ΜοΏ½ππ{πΆπππ₯π(π‘) + π·2ππππ(π‘)}
π
π=1
π
π=1
(56)
where
π΄11 = [0 18.81 0
] , π΄12 = [0 15.38 0
] , οΏ½Μ οΏ½12 = [00.25
] , π΅1π = [00.5] , π·1π = [
00.5] , πΆ1π = [1 1]
(57)
for the first subsystem, and
π΄21 = [0 19.01 0
] , π΄22 = [0 15.58 0
] , οΏ½Μ οΏ½21 = [00.20
] , π΅2π = [00.5] , π·2π = [
00.5] , πΆ2π = [1 1]
(58)
for the second subsystem.
The normalized type-2 membership functions shown in Fig. 2 where ππ = 88Β°.
Fig. 2. IT2 Membership function in Example.
Given the initial conditions π₯1(0) = [1.2,0]π , π₯2(0) = [0.8,0]
π Fig. 3 shows that the double-inverted pendulum
system is not stable in the open-loop case.
Fig. 3 State responses for open-loop double-inverted pendulums system.
Taken the controller gains solved by Theorem1, Fig. 4 shows the state responses for the closed-loop large-scale system
by considering the external disturbances π1(π‘) = 0.8πβ0.2π‘ sin(0.2π‘) and π2(π‘) = 0.6π
β0.2π‘ sin(0.2π‘)it can be seen
that minimum of π»β disturbance attenuation level πΎπππ = 0.333 and the desired controller gains obtained in Table
1.
Fig. 4. State responses for closed-loop double-inverted pendulums system based on Theorem1.
0
0.2
0.4
0.6
0.8
1
1.2
- 8 8 0 8 8
DEG
REE
OF
MEM
BER
SHIP
X(TIME)
Rule1(LMF) Rule1(UMF) Rule2(LMF) Rule2(UMF)
Table 1
πΎ [πΊ11 πΊ12 πΊ21 πΊ22] 0.333 [
β34.3381 β60.0235 β174.0191 β485.0611β16.2743 β31.7905 β93.5045 β268.4191
]
V. Conclusion
In this paper, the robust decentralized state feedback π»β type-2 fuzzy controller design has been investigated for
continuous-time large-scale type-2 Takagi-Sugeno fuzzy systems. Through some linear matrix inequality techniques,
it has been shown that the state fuzzy controller gain can be calculated by solving a set of LMIs. Then the resulting
closed-loop fuzzy control system is asymptotically stable under extended dissipativity performance indexes.
Uncertainty in the model of large-scale systems is the result of the use of the type-1 fuzzy Takagi-Sugeno model.
Therefore, in this paper, the type-2 fuzzy model is used to cover modeling uncertainty for large scale systems. We
also stabilize the large-scale system by using the type-2 fuzzy state feedback controller model with imperfect premise
membership functions. The advantage of using membership function information in sustainability analysis is to reduce
the conservatism of the obtained conditions. Then, in order to reduce the effect of external perturbations on the large-
scale system, we applied the robustness control criterion name as extended dissipativity performance indexes to
stability analysis, which was able to guarantee the π»β criterion, the πΏ2 β πΏβ, passive, and dissipativity performances.
Finally, a numerical example of a double-inverted pendulum system has been concerned to verify the effectiveness of
the developed methods. The result of this simulation is to improve the control characteristics and make the conditions
relax, as well as more complete coverage of the uncertainties in the system. An interesting problem for future research
is to deal with the robust decentralized static output feedback π»β type-2 fuzzy control design for large-scale systems.
VI. Appendix
Assumption 1. ([22]) Let π,π1, π2and π3be matrices such that the following conditions hold:
(1) π = ππ, π1 = π1π πππ π3 = π3
π;
(2) π β₯ 0 and π1 β€ 0;
(3) βπ·2πβ. βπβ = 0;
(4) (β π1β + β π2β). βπβ = 0;
(5) π·2πππ1π·2π + π·2π
ππ2 +π2ππ·2π + π3 > 0.
Definition 1. ([22]) For given matrices π, π1 , π2and π3 satisfying Assumption 1, system (5) is said to be extended
dissipative if there exists a scalar π such that the following inequality holds for any π‘ > 0 and all π(π‘) β β2[0,β):
β« π½(π )π‘
0
ππ‘ β π§π(π‘)ππ§(π‘) β₯ π, (59)
where π½(π‘) = π§π(π‘)π1π§(π‘) + 2π§π(π‘)π2π€(π‘) + π€
π(π‘)π3π€(π‘).
It can be seen from Definition 1 that the following performance indexes hold.
(1) Choosing π = 0,π1 = βπ°,π2 = 0,π3 = πΎ2π° and π = 0 the inequality (59) reduces to the π»β performance
[13].
(2) Let π = π°,π1 = 0,π2 = 0, π3 = πΎ2π° and π = 0 the inequality (59) becomes the πΏ2 β πΏβ(energy-to-peak)
performance [14].
(3) If the dimension of output π§(π‘) is the same as that of disturbance π€(π‘), then the inequality (59) with π =
0, π1 = 0,π2 = π°,π3 = πΎπ° and π = 0 becomes the passivity performance [15].
(4) Let π = 0,π1 = βππ°,π2 = π°,π3 = βππ° with π > 0 and π > 0, inequality (59) becomes the very-strict
passivity performance [16].
(5) Let π = 0,π1 = π,π2 = π, π3 = π β πΌπ° and π = 0, inequality (59) reduces to the strict (π, π, π )-
dissipativity [17].
Lemma 1. [23] (Jensenβs inequality) For any constant positive semidefinite symmetric matrix W β ππ§Γπ§, WT = W β₯
0 two positive integers d2 and d1satisfy d2 β₯ d1 β₯ 1 then the following inequality holds
(β π₯(π)
π2
π=π1
)
π
π(β π₯(π)
π2
π=π1
) β€ οΏ½Μ οΏ½ β π₯π(π)ππ₯(π)
π2
π=π1
where οΏ½Μ οΏ½ = π2 β π1 + 1.
Lemma 2. for given matrices xΜ β Rn, yΜ and scaler ΞΊ > 0 we have
2οΏ½Μ οΏ½ποΏ½Μ οΏ½ β€ π β1οΏ½Μ οΏ½ποΏ½Μ οΏ½ + π οΏ½Μ οΏ½ποΏ½Μ οΏ½
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