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LMI-based stability analysis and robust controller design for a class of nonlinear chaotic power systems
Author’s Accepted Manuscript LMI-based Stability Analysis and Robust Controller Design for a Class of Nonlinear Chaotic Power Systems Mokhtar Sha Sadeghi, Navid Mohammad Hassan Khooban
Vafamand, www.elsevier.com/locate/jfranklin
PII: DOI: Reference:
S0016-0032(15)30105-8 http://dx.doi.org/10.1016/j.jfranklin.2016.04.021 FI2616
To appear in: Journal of the Franklin Institute Received date: 2 July 2015 Revised date: 24 November 2015 Accepted date: 20 April 2016 Cite this article as: Mokhtar Sha Sadeghi, Navid Vafamand and Mohammad Hassan Khooban, LMI-based Stability Analysis and Robust Controller Design for a Class of Nonlinear Chaotic Power Systems, Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2016.04.021 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.
LMI-based Stability Analysis and Robust Controller Design for a Class of Nonlinear Chaotic Power Systems
Mokhtar Sha Sadeghi*, Navid Vafamand, Mohammad Hassan Khooban Electronic and Electrical Department, Shiraz University of Technology, Shiraz, Iran Email: [email protected] Email: [email protected] Email: [email protected] * Corresponding author.
Abstract This paper proposes novel linear matrix inequality (LMI) stability analysis and controller design conditions for nonlinear chaotic power systems. The proposed approach is based on the non-quadratic Lyapunov function (NQLF), non-parallel distributed compensation (non-PDC) schematic and Takagi-Sugeno (TS) fuzzy modeling. Utilizing NQLF causes membership functions (MFs) and their time derivative to appear in the design conditions. To solve this problem, an augmented state vector is proposed which results in removing the MFs and their time derivatives from the design conditions. Moreover, structural constraints on Lyapunov matrices are eliminated. The proposed approach provides relaxed stability analysis and controller design conditions due to the framework that is considered during the formulation derivation. Finally, two practical power systems that exhibit chaotic behaviors are considered to evaluate the proposed approach. Simulation results show advantages of the proposed method compared to the recently published works.
Keywords Chaotic Systems, linear matrix inequality (LMI), non-parallel distributed compensation (non-PDC), non-quadratic Lyapunov function (NQLF), Takagi-Sugeno (TS) fuzzy.
1. Introduction Chaotic systems are highly complex nonlinear systems and are extremely sensitive to initial condition. These systems have irregular and complicated behaviors. Dynamic chaos is a very interesting nonlinear effect that has been intensively investigated during the last two decades. Chaotic phenomena exist in many scientific and engineering fields including chemical systems, electronic circuits and power converters [1]. Recently, different techniques have been proposed to stabilize and synchronize the chaotic systems such as: sliding mode controller [1], [2], adaptive backstepping controller [3], fuzzy type-2 controller [4], passivity-based controller [5], finite time control [6], polynomial controller [7], active pining controller [8], Lyapunov-based controller [9] and contraction principalbased controller [10]. Chaos is confirmed to be one of the complicated phenomena in nonlinear systems that can push the power system towards instability [6]. Due to this fact, many researchers have proposed several methods to control chaotic oscillation in electrical systems. One of the first works on chaos control was presented in [11] which studied nonlinear dynamical phenomena in power system networks. An approach for recognizing and classifying chaotic behaviors for certain loading conditions was investigated in [12]. Existence of chaotic behavior in the swing equation of the damped power systems was proved in [13], and the effectiveness of flexible alternating current transmission system (FACTS) devices to damp down the Hopf bifurcation and chaos in these systems was demonstrated in [14]. Recently, in [2], a controller was proposed for a 2-machine interconnected power system based on the variable structure method with robustness to external disturbance inputs. Besides these efforts for controlling chaotic systems, fuzzy model based control has been attracted many researchers to control of nonlinear systems [15]–[18]. Various types of fuzzy model based controllers are proposed to control the nonlinear plants represented by the Takagi-Sugeno (TS) fuzzy model such as static output feedback controller [19] and dynamic one [20], sampled data controller [21] and decentralized controller [22], [23]. In the TS fuzzy model, local dynamics in each consequent part of fuzzy rule is represented by linearizing the original nonlinear system at each operating point. The overall TS model is achieved by fuzzy blending of these local models. The main advantage of TS fuzzy models is that linear control theory can be applied to design controller for nonlinear systems via convex structures. Recently, fuzzy impulsive control is used for stabilization of chaotic systems based on the TS
fuzzy model [24], [25]. However, the main drawback of these approaches is to provide discontinuous control inputs [26]. Parallel distributed compensation (PDC) and non-PDC controllers can handle this problem [27]. The fuzzy control design problem is formulated by using PDC and non-PDC schemes [28]. Several chaotic systems are successfully stabilized by the PDC controller. In [29], LMI stability conditions based on OGY method [11] and the PDC controller is proposed for the discrete time Henon map chaotic system. In [30], H∞ synchronization and state estimation problems are investigated in terms of LMIs for different chaotic systems such as Hopfield and cellular neural networks, Chua’s circuits, unified chaotic systems, Qi systems, and chaotic recurrent multilayer perceptron. The conditions presented in [24], [25], [29]–[31] are derived based on quadratic Lyapunov function (QLF). Utilizing QLF induces conservatism in the controller design conditions. To solve this problem, non-quadratic Lyapunov function (NQLF) is proposed [28]. This function is a fuzzy blending of multiple conventional QLFs. In [32], LMI conditions are derived based on the NQLF and the PDC controller for chaotic systems. Recent literatures indicate that non-PDC controller leads to more relaxed conditions compared to PDC ones [33]. The main advantage of the non-PDC controller is to provide a systematic approach that can cope with stabilization and performance issues based on the Lyapunov theorem. Sufficient conditions for stability and performance analysis of the closedloop TS systems can be restated in terms of LMIs [34] and solved by numerical convex optimization algorithms. By employing NQLF, time derivatives of membership functions (MFs) are appeared in stabilization and performance conditions and their upper bounds must be known to solve these conditions by convex optimization techniques. The time derivatives of MFs are dependent not only on the system states, but also on the control inputs. Therefore, there is a hard task to choose the upper bounds in prior. Few of the literature are concerned with systematic approaches for choosing upper bounds. In [33], [35], global stabilization is reduced to local stabilization to overcome the problem of time derivative of membership function. Hence, an attempt is made to find an estimation of region of attraction. In [35], by means of the Kronecker’s product, the stability conditions are stated by a quadratic structure. The structure quadratically depends on the augmented vector containing the states, time derivatives of the states and also time derivatives of the MFs. In [36], line-integral Lyapunov function (LILF) is proposed in which the time derivatives of the MFs are not appeared. In this paper, novel LMI robust controller design conditions are proposed for the nonlinear systems. This approach is based on the NQLF, non-PDC controller and TS fuzzy model. The proposed stability analysis and controller design
conditions are formulated such that they do not depend on the time derivatives of MFs. The key is to reformulate the problem such that the MFs and their time derivatives will be the arrays of an augmented vector. By utilizing congruence lemma, the augmented vector and the time derivatives of MFs are eliminated. Furthermore, some new slack matrices are introduced based on the properties of MFs. Consequently, restrictions on the Lyapunov matrices structures are not needed compared to the LILF-based conditions. The proposed approach benefits from a less number of stabilization conditions to be satisfied compared to ones derived based on the local stability concept. Finally, to evaluate the performance of the proposed approach, it is applied to chaotic Lorenz system, chaotic permanent magnet synchronous motor (PMSM) and chaotic oscillation damping of power system (CODOPS). Simulation results show the desired performance of the proposed controller. This paper is organized as follows: In Section 2, TS fuzzy model is presented. The robust controller design conditions are proposed in Section 3. Simulation examples are given in Section 4, and finally conclusions are presented in Section 5.
2. TS FUZZY MODEL A TS fuzzy system is described by a set of fuzzy IF-THEN rules that represent local linear input-output relations of a nonlinear system. The overall fuzzy model is obtained by fuzzy blending of the linear system models. The -th rule of a general -rules TS fuzzy model can be written as follows: ( )
Plant rule i: ̇( ) { ( )
( ) ( )
where ( ) ( ) and
( )
[ ( )
( )
, (1)
( )
( )]
is the premise variable vector that its elements are a function of states,
is the state vector, ( )
is the input vector,
( )
is the disturbance input vector
is the -th fuzzy membership function related to the -th premise variable. The overall TS fuzzy system with
rule (1), singleton fuzzifier and center average defuzzifier, is of the form: ̇( )
( )
∑
( ( )){ ∑
∑
( ( )){ ∑
( ( ))
( )
( )
( )}
( ( )) ( )}
∑
( ( )){
∑
( )}
( ( )){
( )
( )
( )}
(2)
(3)
( ( ))
(4)
( ( )) is the normalized MF in relation with the -th rule such that:
where ∑
( ( ))
∏
( ( ))
(5)
The problem is to derive stabilization analysis and robust controller design conditions in term of LMIs such that the L2-L2 performance level is guaranteed. For brevity, in the following,
, , ̇,
and
denote
( ( )), ( ), ̇ ( ),
( ) and ( ), respectively. Also, a star (*) in a symmetric matrix denotes the transpose of its symmetric element.
3. MAIN RESULTS To obtain stability conditions, NQLF candidate and non-PDC controller are defined as: (6) ∑
(7)
∑
(8)
The constraint
for
are sufficient conditions that assure that (6) will be a Lyapunov function
candidate. Non-PDC controller, NQLF and TS fuzzy model share the same fuzzy sets in the premise part. By substituting (7) into (2), the closed-loop TS system is obtained as: ̇
∑∑
{(
)
}
∑∑
{(
)
}
∑∑
{(
)
}
(9)
Lemma 1 (Congruence complement) [34]: For arbitrary square matrix appropriate dimensions:
⇔
.
and non-singular matrix
with
Lemma 2 [37]: For arbitrary matrices
,
and
with appropriate dimensions, the inequality
holds. Lemma 3 [37]: For an arbitrary vector
and positive scalar
one has:
⇔
.
3.1 Introducing the slack matrices To obtain stabilization conditions, two slack matrices will be used in the forthcoming theorem. From the property of the normalized MFs (5), and their time derivatives, one has: ∑ ̇
(10)
Using (10), the following four null product matrices are obtained: ̂ ∑∑ ̇
̂ ̂
̂ ∑∑ ̇ ̇ (̂
(11)
̂ )̂
(12)
∑∑ ̇
∑∑ ̇ ̇ ( where ̂
[
(13)
) ] and ̂ ̂ ̂
(14)
and
are matrices with appropriate dimensions.
3.2 FUZZY CONTROL DESIGN AND DISTURBANCE REJECTION In this sub-section, we are interested in designing a stabilizing controller in the presence of disturbance in order to reduce the effect of disturbance on output. To do this, L2-L2 controller design is proposed and is tried to maintain the supreme of ratio of output energy to disturbance energy less than a pre-determinate threshold: ‖ ‖ ‖ ‖
(15)
Lemma 4 [34]: For zero initial conditions, inequality (15) is implied by: ̇
(16)
The next Theorem gives sufficient conditions to guarantee L2-L2 level performance of the closed-loop system (9).
subject to | ̇ |
Theorem 1: For positive scalars
for
, the TS fuzzy system (9) is guaranteed to be
asymptotically stable with a guaranteed L2-L2 performance level , if there exist symmetric matrices , ̂
and matrices
̂ and ̂(
)(
̂(
)
) for
)(
, ̂, ̂, ̂,
with appropriate dimensions such that the
following LMIs are satisfied. (17) [ [
[ [
[̂ ]
[
] ]
[
]
] [̂ ] ]
̂
̂(
̂
̂
̂
̂(
(18)
]
[̂ ]
[
[
[ )(
]
(19)
] (20)
)
̂( )
̂
]
)(
)
̂(
)
̂(
)(
(21)
)
(22)
̂
̂
(
)
̂
(
)
̂
̂
(
)
̂
(
)
̂(
)(
̂(
)(
)
̂(
[
)(
(23) )
)
]
where ̂
[
̂
̂
̂
̂
̂
] [
(24)
] ̂
Proof: Consider Lemma 4 and the NQLF (6). One has: ̇ ̇
̇
̇
̇
̇
̇
⇒
Since ̇
̇
̇
(25) ̇ ̇
⇒
̇
̇
, therefore, (25) is equal to: (26)
Substituting the closed-loop TS system (9) into (26), results: {∑ ∑
{(
)
}}
{∑ ∑
{
(
)
}}
∑ ̇
∑∑
(
)
∑ ̇
∑∑
∑∑
{
(
)
∑∑
∑∑ with ̂
̇ {
}
}
̂ [
[
]̂
̇ ̂ [
]̂
̇ ̂ [
]̂
∑∑
(27)
]. By adding the null terms (11) and (12) to (27), (27) leads to:
∑∑
̂ [
]̂
̂ ∑∑ ̇ ̇ (̂
∑∑
∑
∑∑
̂ ̂ ̂
̂ ( ̂ )̂
∑
∑∑
̂ (̂
̂ )̂
̇ ̂ ̂ ̂
̂ )̂
̂ ∑∑ ̇
∑∑
̂ ̂
∑∑ ̇ ̇ ̂ ̂ ̂
∑∑
̇ ̂ ̂ ̂
∑ ̇ ̂ ̂ ̂ (28)
∑ ̇ ̇ ̂ (̂
where
̂ )̂
̂
̂
[
] (29)
̂
̂
̂
[
̂ ]
(30)
If ̂
̂(
̂
̂
̂
̂(
̂
̂
̂
̂
)(
)
̂( )
)(
)
̂(
)
̂(
)(
(31)
)
(32) ̂
hold, then
̂
(33)
, where
equals to:
(̂ )
̂ {∑
(̂
∑
∑ ̇ ( ̂(
̂ )
)(
))
(34) ∑ ̇ ̇ ( ̂(
̂
[ ̇
)
̂(
))
)(
∑∑
̇ (̂(
)
̂(
) )}
̂
̂
̂ ̇
)(
̂
̂
̂ ̇ [ ̇
̂]
(35)
̂ ̂]
with ̂ ̂
̂
̂
(
)
̂
(
)
̂
̂
(
)
̂
(
)
̂(
)(
̂(
)(
)
̂(
[ By employing Lemma 1, ̂
)(
) )
]
will be negative if: (36)
Considering (29) and (31), LMIs (20), (21) and (22) are obtained. In the following, it is trying to prove (18) and (19). From (30) and (32), one concludes
, where
[
equals to: ̂
]
[
] ̂
[
](
)[
̂
[
]
]
(37)
where is the identity matrix with appropriate dimensions and ̂
[
]
̂
(38)
Utilizing Shure complement as follows: ̂ [
]
(39)
provides (21). Using the same procedure as applied on (32) to derive (18), (33) leads to (19). The proof is ■
completed.
3.3 FUZZY CONTROL DESIGN AND DECAY RATE In most practical applications, it is desired to increase the closed-loop system response. . The convergence rate of the systems is related to decay rate. This subsection deals with the decay rate performance of non-PDC controller design. Definition 1: If Lyapunov function satisfies the following inequality ̇
(40)
where
is the decay rate, then Lyapunov function exponentially converges to zero and the system is
exponentially stable. The next Theorem guarantees the exponential stability of the closed-loop system (9).
Theorem 2: For given positive scalars
and
exponentially stable if there exist symmetric matrices (
)(
) for
subject to | ̇ |
for ,
,
,
, and matrices
, the TS fuzzy system (9) is ,
and
with appropriate dimensions such that the following LMIs are satisfied;
(
)(
)
(41) (42) (43) (
)(
(44)
) (
(
)
)(
)
(
)
(
)(
(45)
)
(46) (
)
(
)
(
)
(
)
(
)(
(
)(
)
[
(
)(
(47)
) )
]
where
Proof: Assume that
. By recalling (40) and the NQLF (6), one has:
̇ ̇ ̇
̇
̇
̇
̇
∑∑
(
)
∑∑
∑∑
[
]
∑∑
̇ {
̇
}
[ ]
Consider the null terms (13) and (14). Therefore, (48) is equal to: ∑∑
[
]
∑∑ ̇ ̇ (
)
∑∑
∑∑ ̇
̇
[ ]
(48)
∑∑
∑
∑∑
(
)
∑
̇
∑∑ ̇ ̇
(
)
̇
∑∑
(49) ∑ ̇
(
)
∑ ̇ ̇
(
)
where
(50)
By using the same procedure as discussed in proof procedure of the Theorem 1 (i.e. (31)-(36)), the LMIs (41)-(47) ■
will be obtained. The proof is completed.
Remark 1: As it is stated before, the number of stabilization conditions is few compared to local stabilization ones. The number of LMI conditions in Theorem 2 is where
is the number of fuzzy rules,
and the number of LMI conditions in [33] is
is the number of states and
is the number of premise variables.
Remark 2: The conditions derive based on QLF and LILF leads to conservative results. The reason lies in the natural structure of QLFs and LILFs. In LMI conditions derived by LILF, although time derivative of MFs does not emerge. However, some structural restrictions must be imposed on LILF matrices. In this regard, the off-diagonal elements of the LILF matrices must be the same. Therefore, in LILF
number of Lyapunov matrices is considered
in which, all the off-diagonal elements are the same and only the diagonal ones can be different. Contrary to LILF, in NQLF, the entire elements of Lyapunov matrices can change. This is the main advantage of utilizing NQLF compare to LILF. Remark 3: Utilizing NQLF has a main drawback. Since the NQLF is a fuzzy blending of common QLFs, therefore, the time derivatives of MFs are appeared in the deriving the stability and stabilization conditions. In order to formulate the conditions in term of LMIs, these time derivatives must be handled. A well-known approach is to consider lower and/or upper bounds [28]. Since we cannot pre-determine these bounds, the LMI conditions are validated for a specific region of states in where the assumption of time derivative of MFs bounds is held. This
assumption leads to a local analysis [33], [35]. In Theorems 1 and 2, it was assumed that the bounds of ̇ (i.e. | ̇ |
) are given. Considering the chain-time derivatives of MFs, one has:
̇ ̇
(51)
Therefore, finding a bound for ̇ depends on finding an upper bound for ̇ . Form (9), it is clear that ̇ depends on the values of
and
which are LMI decision variables. Thus, the given upper bounds
in the LMIs depend on
the values of the LMI variables, which make the problem much more complex. In Theorem 3, we propose new design conditions to overcome the aforementioned problem and to derive the region of attraction. { |
Theorem 3: For the given region
derivatives of membership functions | ̇ | matrices
and ̃
for
(
) (
for
)
}, the constraints on the time
hold, if there exist symmetric matrices
̃ for
and
with appropriate dimensions such that the following
LMIs are satisfied: (52) ̃ {̃
̃ ̃
̃
̃
(53)
̃
̃
[
̃
]
(54)
where ̃
(
[
)
]
Proof: Substituting the non-perturbed closed-loop TS system (9) into (51), results in: |∑ ∑
{
(
)
}|
|∑ ∑
Based on Lemma 2 and defining the slack matrix
{
( ∑
) (
, (55) is satisfied if:
) }|
(55)
[
]
∑∑
(
)
[
∑∑ ]
(
[
)
[
]
(56)
]
hold. Based on Lemma 3, one has:
∑∑
(
[
)
[
][
]
∑∑
] ,(
Suppose that [
) ][
(
) (
(
[ )
(57)
) ]
. Therefore:
]
(58)
Substituting (58) into (57) and applying Lemma 3, one has:
∑∑
(
)
∑∑
[
][
Pre- and post-multiplying (59) by
(
(
(59)
) ]
) with
∑
, and considering Schur complement provide:
∑
(60)
and ∑∑
̃
(61)
where ̃
(
[
)
]
Inequality (60) is implied by (52). Furthermore, (61) can be written as: ∑
If
̃
∑
{̃
̃ }
(62)
̃ {̃
̃ ̃
hold, then
̃
(63)
̃ , where
is equal to:
[ ] ̃ [ ]
(64)
with ̃
̃
[
̃ ̃
]
By employing Lemma 1,
will be positive if:
̃
(65)
Considering (63) and (65), the LMIs (53) and (54) are obtained. The proof is completed.
■
4. Simulation Results In this section, three chaotic systems are presented. First, Lorenz system is considered and results of the proposed approach is compared with recently sampled-data controller [38]. Second, PMSM system with exhibits a chaotic behavior is considered and comparison results of non-PDC controller designed by this paper and PDC method are discussed. Third, CODOPS system [6] is considered. For this system, initially, the nonlinear dynamic of the system is presented. Then, the equivalent TS fuzzy model is proposed and Theorems 1-3 are utilized to design robust nonPDC controller for the CODOPS system.
4.1. Chaotic Lorenz system Consider the chaotic Lorenz equation with an input term: ̇ { ̇
(66) ̇
where
.
4.1.1. Equivalent TS fuzzy model of Lorenz system An equivalent two-rule TS fuzzy model is obtained with the following state space matrices [38]: [
]
[
]
[ ]
The MFs are defined as: ( )
(
)
( )
( )
4.1.2. Lorenz system simulation results For simulation, the system parameters are chosen based on [38] with decay rate ( )
∑
( )
,
,
and
. The sampled data controller
, is designed as:
( )
(67) [
where the sampling instants belong to
] and
[ Set the decay rate
]
and the upper bounds
in the Theorem (2). Non-PDC feedback gains and
Lyapunov matrices are calculated as: [
]
⌊ The initial condition is chosen as
⌋ ( )
[
[ ⌊ ] and a sampling constant
] ⌋ for sampled-data
controller. Figs. 1 and 2 indicate the Lorenz system states and control effort, respectively. Furthermore, table 1 demonstrates the control input signal norm derived by the proposed approach and [38].
10
x
1
5
0
-5
0
0.5
1 Time[sec]
1.5
2
(a) 15
x
2
10 5 0 -5
0
0.5
1 Time[sec] (b)
1.5
2
0
0.5
1 Time[sec] (c)
1.5
2
15
x
3
10
5
0
Fig. 1. Lorenz system states (Theorem 1 “” and ref. [38] “···”). (a) The state x1. (b) The state x2. (c) The state x3.
100
u
0 -100 -200 -300
0.5
1 1.5 2 Time[sec] Fig. 2. Lorenz system control input (Theorem 1 “” and ref. [38] “···”).
Table 1. Norms of Lorenz system control effort. Approach H∞ Theorem 1 183.8955 [38] 234.0773
H2 573.4377 1.046×103
4.2. Chaotic permanent magnet synchronous motor system A PMSM can exhibit bifurcation, limit cycle or chaotic behavior when motor parameters lie in a certain area [39], [40]. In chaos, the motor torque will change randomly and the motor speed will oscillate [41]. The nonlinear dynamic of PMSM can be described by: ̇ ̇
(68) (
)
{ 4.2.1. Equivalent TS model of PMSM system [
Consider the PMSM system (68) with the parameters [42]: [
],
[
] and
[
],
[ ],
,
]. By employing the procedure discussed in
[40], the following TS model will be derived: ̇
∑
where
( ){
[
} ]
(69)
[
] and
[
]
[
]
[ ]
with the normalized MFs ( )
(
( )
)
(
)
Based on [40], the following stabilizing PDC controller is designed ∑
( )
[
Utilizing theorem 2 and considering decay rate [
and the upper bounds ]
⌊
]
, one has:
[ ⌋
The simulation was done with the initial condition ( )
] ⌊
⌋
[
] and a step size of
[40]. The
state evolution and control effort signal are demonstrated in Figs. 3 and 4. In addition, norms of the control inputs are presented in table 2. 15
x
1
10
5
0
0
0.5
1 Time[sec] (a)
1.5
2
10
x
2
5
0
-5
0
0.5
1 Time[sec] (b)
1.5
2
0
0.5
1 Time[sec] (c)
1.5
2
2
x
3
0 -2 -4 -6
Fig. 3. PMSM system states (Theorem 1 “” and ref. [40] “···”). (a) The state x1. (b) The state x2. (c) The state x3. 600
u
400 200 0 -200
0
0.2
0.4 0.6 Time[sec]
0.8
1
Fig. 4. PMSM system control input (Theorem 1 “” and ref. [40] “···”).
Table 2. Norms of PMSM system control effort. Approach Theorem 1 [40]
H∞ 80.9534 435.892
H2 2.6558×103 4.1860×103
From Fig. 3 one concludes that, the closed-loop systems constructed by the PDC controller [40] and non-PDC controller in this paper, have roughly the same system responses. However, as can be seen in Fig. 4 and Table 2, the norms of control input signal derived by the non-PDC controller are significantly smaller than the ones derived by [40].
4.3. Chaotic oscillation damping of power system In this sub-section, an interconnected power system will be introduced and the dynamic model will be derived. The simple interconnected power system is shown in Fig. 5.
3
4 6
𝑮𝟏
7
6
𝑮𝟐 5
Fig. 5. A general scheme power system model In this figure, G1 and G2 are the equivalent generators and 3 and 4 are the main transformers of the generators 1 and 2, respectively. Then, 5, 6 and 7 are considered as the load, circuit breakers and tie line, respectively. This system exhibits some important aspects of behavior of multi-machine systems [6]. Different forms of chaotic systems with linear or nonlinear damping specifications have been represented in literatures [4]. In the state space form, the CODOPS equation (called swing equations) can be described as [2], [4], [6]: ̇ {
where
( ) ̇
(
is the relative angle between voltages of generators 1 and 2,
from generator 1 to 2,
and
power of generator 1,
( ) is the input signal and
)
( )
( )
(70)
is the maximum electric power transmitting
are inertia and damping coefficient of generator, respectively,
is the machinery
( ) is an external disturbance with unknown but bounded
amplitude. This chaotic power system exhibits complex dynamics for constant values of and
|
and
system without controller for
|
and initial states of
[6]. Fig. 6 shows irregular motion of this
( )
( )
and
.
10
X2
5
0
-5
-10 -0.5
0
0.5
1
1.5
2
X1
Fig. 6. x1–x2 plane time response of the chaotic power system.
4.3.1. Constructing the TS fuzzy modeling The system (70) is reformulated as: ̇ {
where ( )
(
( ) ̇
)
( )
(71)
( )
( ) is the new input. There is one nonlinear term in the model (71). Based
on the sector nonlinearity concept [34], one has: ( )
{
( ) ( )
( )
( ) ( )
(72)
( )
(73)
The MFs are derived by solving the equations (73) as following: ( )
( )
( )
( )
(74)
Therefore, the equivalent two rule TS fuzzy model is obtained as follows: ̇ where
∑
(
)
(75)
[
]
[
]
[ ]
4.3.2. Scenario 1: The white noise as the external disturbance Set
,
and {
Consequently,
{ | |
in Theorems 1, 2. Furthermore, consider the local region } and
. In Theorem 3, the upper bounds of
}.
for
are needed. Recall the membership functions (74), one has: ( )
| (
Therefore,
|
|
( )
( )|
(76)
) . By employing Theorems 1-3, the following feedback gain matrices and Lyapunov
matrices are obtained: [
]
[
[
]
]
[
]
Fig. 7 illustrates the disturbance input and the states evolution. The states of chaotic system based on the non-PDC controller designed by the proposed approach converge to equilibrium point. It should be noted that the effect of the external disturbance on the system output
is significantly attenuated. Moreover, the region of attraction and the
outermost Lyapunov function contour that lay in this region is plotted in Fig. 8. The figure demonstrates the regions { | |
},
{
} and
{
}.
4.3.3. Scenario 2: External disturbance with unknown but bounded amplitude In order to challenge our proposed method to study the robustness of the proposed controller, we also introduce (
)
(
) as a disturbance signal. In Theorems 1-3, set { | |
and a compact set
} which provides
,
,
,
. The feedback gain matrices
and Lyapunov matrices are obtained as: [ [
] ]
[
] [
]
Fig. 9 shows the states evolution in presence of the new disturbance. Fig. 9.a illustrates the disturbance signal. The states of chaotic system based on the non-PDC controller designed by the proposed approach converge to
equilibrium point. In addition, in Fig. 10, absolute of time derivative of membership functions is plotted. As it can be seen, the maximum value is smaller than
, which satisfies the constraint on the time derivatives of
membership function.
Disturbance signal
60 40 20 0 -20 -40
0
0.5
1 Time [sec]
1.5
2
(a)
1
x
1
0.5
0
-0.5
0
0.5
1
1.5 Time [sec]
2
2.5
3
2
2.5
3
(b)
0.5 0
x
2
-0.5 -1 -1.5 -2
0
0.5
1
1.5 Time [sec]
(c) Fig. 7. The scenario 1 simulation. (a) The disturbance signal. (b) The system state x1. (c) The system state x2.
5 4 3 2
2
1
x
0 -1 -2 -3 -4 -5 -5
0 x1
5
Fig. 8. Region of attraction for the scenario 1. 1
x
1
0.5
0
-0.5
0
0.5
1 Time [sec]
1.5
2
1.5
2
(a) 10
x
2
0
-10
-20
0
0.5
1 Time [sec]
(b) Fig. 9. The scenario 2 simulation. (a) The system state x1. (b) The system state x2.
i
Absolute of time derivative of h
3
2
1
0
0
0.2
0.4 0.6 Time [sec]
0.8
1
Fig. 10. Absolute of time derivative of membership functions evolution for the scenario 2.
5. Conclusions In this paper, based on the NQLF, novel local LMI stability analysis and non-PDC controller design conditions were proposed for the nonlinear chaotic power systems described by TS fuzzy model. The proposed approach had three main superiorities. Frist, time derivatives of the MFs did not emerge in the LMI conditions. Second, Lyapunov matrices structural restrictions were eliminated. Third, the proposed approach provided a less number of stabilization conditions compared to the ones derived based on the local stability concept. Finally, in order to demonstrate the feasibility of this algorithm, it was applied to the Lorenz system and practical chaotic PMSM system and power chaotic system. The comparison results verified the effectiveness and merits of the proposed algorithm for stability analysis and robust controller synthesis in the presence of external disturbance inputs.
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Vafamand, www.elsevier.com/locate/jfranklin
PII: DOI: Reference:
S0016-0032(15)30105-8 http://dx.doi.org/10.1016/j.jfranklin.2016.04.021 FI2616
To appear in: Journal of the Franklin Institute Received date: 2 July 2015 Revised date: 24 November 2015 Accepted date: 20 April 2016 Cite this article as: Mokhtar Sha Sadeghi, Navid Vafamand and Mohammad Hassan Khooban, LMI-based Stability Analysis and Robust Controller Design for a Class of Nonlinear Chaotic Power Systems, Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2016.04.021 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.
LMI-based Stability Analysis and Robust Controller Design for a Class of Nonlinear Chaotic Power Systems
Mokhtar Sha Sadeghi*, Navid Vafamand, Mohammad Hassan Khooban Electronic and Electrical Department, Shiraz University of Technology, Shiraz, Iran Email: [email protected] Email: [email protected] Email: [email protected] * Corresponding author.
Abstract This paper proposes novel linear matrix inequality (LMI) stability analysis and controller design conditions for nonlinear chaotic power systems. The proposed approach is based on the non-quadratic Lyapunov function (NQLF), non-parallel distributed compensation (non-PDC) schematic and Takagi-Sugeno (TS) fuzzy modeling. Utilizing NQLF causes membership functions (MFs) and their time derivative to appear in the design conditions. To solve this problem, an augmented state vector is proposed which results in removing the MFs and their time derivatives from the design conditions. Moreover, structural constraints on Lyapunov matrices are eliminated. The proposed approach provides relaxed stability analysis and controller design conditions due to the framework that is considered during the formulation derivation. Finally, two practical power systems that exhibit chaotic behaviors are considered to evaluate the proposed approach. Simulation results show advantages of the proposed method compared to the recently published works.
Keywords Chaotic Systems, linear matrix inequality (LMI), non-parallel distributed compensation (non-PDC), non-quadratic Lyapunov function (NQLF), Takagi-Sugeno (TS) fuzzy.
1. Introduction Chaotic systems are highly complex nonlinear systems and are extremely sensitive to initial condition. These systems have irregular and complicated behaviors. Dynamic chaos is a very interesting nonlinear effect that has been intensively investigated during the last two decades. Chaotic phenomena exist in many scientific and engineering fields including chemical systems, electronic circuits and power converters [1]. Recently, different techniques have been proposed to stabilize and synchronize the chaotic systems such as: sliding mode controller [1], [2], adaptive backstepping controller [3], fuzzy type-2 controller [4], passivity-based controller [5], finite time control [6], polynomial controller [7], active pining controller [8], Lyapunov-based controller [9] and contraction principalbased controller [10]. Chaos is confirmed to be one of the complicated phenomena in nonlinear systems that can push the power system towards instability [6]. Due to this fact, many researchers have proposed several methods to control chaotic oscillation in electrical systems. One of the first works on chaos control was presented in [11] which studied nonlinear dynamical phenomena in power system networks. An approach for recognizing and classifying chaotic behaviors for certain loading conditions was investigated in [12]. Existence of chaotic behavior in the swing equation of the damped power systems was proved in [13], and the effectiveness of flexible alternating current transmission system (FACTS) devices to damp down the Hopf bifurcation and chaos in these systems was demonstrated in [14]. Recently, in [2], a controller was proposed for a 2-machine interconnected power system based on the variable structure method with robustness to external disturbance inputs. Besides these efforts for controlling chaotic systems, fuzzy model based control has been attracted many researchers to control of nonlinear systems [15]–[18]. Various types of fuzzy model based controllers are proposed to control the nonlinear plants represented by the Takagi-Sugeno (TS) fuzzy model such as static output feedback controller [19] and dynamic one [20], sampled data controller [21] and decentralized controller [22], [23]. In the TS fuzzy model, local dynamics in each consequent part of fuzzy rule is represented by linearizing the original nonlinear system at each operating point. The overall TS model is achieved by fuzzy blending of these local models. The main advantage of TS fuzzy models is that linear control theory can be applied to design controller for nonlinear systems via convex structures. Recently, fuzzy impulsive control is used for stabilization of chaotic systems based on the TS
fuzzy model [24], [25]. However, the main drawback of these approaches is to provide discontinuous control inputs [26]. Parallel distributed compensation (PDC) and non-PDC controllers can handle this problem [27]. The fuzzy control design problem is formulated by using PDC and non-PDC schemes [28]. Several chaotic systems are successfully stabilized by the PDC controller. In [29], LMI stability conditions based on OGY method [11] and the PDC controller is proposed for the discrete time Henon map chaotic system. In [30], H∞ synchronization and state estimation problems are investigated in terms of LMIs for different chaotic systems such as Hopfield and cellular neural networks, Chua’s circuits, unified chaotic systems, Qi systems, and chaotic recurrent multilayer perceptron. The conditions presented in [24], [25], [29]–[31] are derived based on quadratic Lyapunov function (QLF). Utilizing QLF induces conservatism in the controller design conditions. To solve this problem, non-quadratic Lyapunov function (NQLF) is proposed [28]. This function is a fuzzy blending of multiple conventional QLFs. In [32], LMI conditions are derived based on the NQLF and the PDC controller for chaotic systems. Recent literatures indicate that non-PDC controller leads to more relaxed conditions compared to PDC ones [33]. The main advantage of the non-PDC controller is to provide a systematic approach that can cope with stabilization and performance issues based on the Lyapunov theorem. Sufficient conditions for stability and performance analysis of the closedloop TS systems can be restated in terms of LMIs [34] and solved by numerical convex optimization algorithms. By employing NQLF, time derivatives of membership functions (MFs) are appeared in stabilization and performance conditions and their upper bounds must be known to solve these conditions by convex optimization techniques. The time derivatives of MFs are dependent not only on the system states, but also on the control inputs. Therefore, there is a hard task to choose the upper bounds in prior. Few of the literature are concerned with systematic approaches for choosing upper bounds. In [33], [35], global stabilization is reduced to local stabilization to overcome the problem of time derivative of membership function. Hence, an attempt is made to find an estimation of region of attraction. In [35], by means of the Kronecker’s product, the stability conditions are stated by a quadratic structure. The structure quadratically depends on the augmented vector containing the states, time derivatives of the states and also time derivatives of the MFs. In [36], line-integral Lyapunov function (LILF) is proposed in which the time derivatives of the MFs are not appeared. In this paper, novel LMI robust controller design conditions are proposed for the nonlinear systems. This approach is based on the NQLF, non-PDC controller and TS fuzzy model. The proposed stability analysis and controller design
conditions are formulated such that they do not depend on the time derivatives of MFs. The key is to reformulate the problem such that the MFs and their time derivatives will be the arrays of an augmented vector. By utilizing congruence lemma, the augmented vector and the time derivatives of MFs are eliminated. Furthermore, some new slack matrices are introduced based on the properties of MFs. Consequently, restrictions on the Lyapunov matrices structures are not needed compared to the LILF-based conditions. The proposed approach benefits from a less number of stabilization conditions to be satisfied compared to ones derived based on the local stability concept. Finally, to evaluate the performance of the proposed approach, it is applied to chaotic Lorenz system, chaotic permanent magnet synchronous motor (PMSM) and chaotic oscillation damping of power system (CODOPS). Simulation results show the desired performance of the proposed controller. This paper is organized as follows: In Section 2, TS fuzzy model is presented. The robust controller design conditions are proposed in Section 3. Simulation examples are given in Section 4, and finally conclusions are presented in Section 5.
2. TS FUZZY MODEL A TS fuzzy system is described by a set of fuzzy IF-THEN rules that represent local linear input-output relations of a nonlinear system. The overall fuzzy model is obtained by fuzzy blending of the linear system models. The -th rule of a general -rules TS fuzzy model can be written as follows: ( )
Plant rule i: ̇( ) { ( )
( ) ( )
where ( ) ( ) and
( )
[ ( )
( )
, (1)
( )
( )]
is the premise variable vector that its elements are a function of states,
is the state vector, ( )
is the input vector,
( )
is the disturbance input vector
is the -th fuzzy membership function related to the -th premise variable. The overall TS fuzzy system with
rule (1), singleton fuzzifier and center average defuzzifier, is of the form: ̇( )
( )
∑
( ( )){ ∑
∑
( ( )){ ∑
( ( ))
( )
( )
( )}
( ( )) ( )}
∑
( ( )){
∑
( )}
( ( )){
( )
( )
( )}
(2)
(3)
( ( ))
(4)
( ( )) is the normalized MF in relation with the -th rule such that:
where ∑
( ( ))
∏
( ( ))
(5)
The problem is to derive stabilization analysis and robust controller design conditions in term of LMIs such that the L2-L2 performance level is guaranteed. For brevity, in the following,
, , ̇,
and
denote
( ( )), ( ), ̇ ( ),
( ) and ( ), respectively. Also, a star (*) in a symmetric matrix denotes the transpose of its symmetric element.
3. MAIN RESULTS To obtain stability conditions, NQLF candidate and non-PDC controller are defined as: (6) ∑
(7)
∑
(8)
The constraint
for
are sufficient conditions that assure that (6) will be a Lyapunov function
candidate. Non-PDC controller, NQLF and TS fuzzy model share the same fuzzy sets in the premise part. By substituting (7) into (2), the closed-loop TS system is obtained as: ̇
∑∑
{(
)
}
∑∑
{(
)
}
∑∑
{(
)
}
(9)
Lemma 1 (Congruence complement) [34]: For arbitrary square matrix appropriate dimensions:
⇔
.
and non-singular matrix
with
Lemma 2 [37]: For arbitrary matrices
,
and
with appropriate dimensions, the inequality
holds. Lemma 3 [37]: For an arbitrary vector
and positive scalar
one has:
⇔
.
3.1 Introducing the slack matrices To obtain stabilization conditions, two slack matrices will be used in the forthcoming theorem. From the property of the normalized MFs (5), and their time derivatives, one has: ∑ ̇
(10)
Using (10), the following four null product matrices are obtained: ̂ ∑∑ ̇
̂ ̂
̂ ∑∑ ̇ ̇ (̂
(11)
̂ )̂
(12)
∑∑ ̇
∑∑ ̇ ̇ ( where ̂
[
(13)
) ] and ̂ ̂ ̂
(14)
and
are matrices with appropriate dimensions.
3.2 FUZZY CONTROL DESIGN AND DISTURBANCE REJECTION In this sub-section, we are interested in designing a stabilizing controller in the presence of disturbance in order to reduce the effect of disturbance on output. To do this, L2-L2 controller design is proposed and is tried to maintain the supreme of ratio of output energy to disturbance energy less than a pre-determinate threshold: ‖ ‖ ‖ ‖
(15)
Lemma 4 [34]: For zero initial conditions, inequality (15) is implied by: ̇
(16)
The next Theorem gives sufficient conditions to guarantee L2-L2 level performance of the closed-loop system (9).
subject to | ̇ |
Theorem 1: For positive scalars
for
, the TS fuzzy system (9) is guaranteed to be
asymptotically stable with a guaranteed L2-L2 performance level , if there exist symmetric matrices , ̂
and matrices
̂ and ̂(
)(
̂(
)
) for
)(
, ̂, ̂, ̂,
with appropriate dimensions such that the
following LMIs are satisfied. (17) [ [
[ [
[̂ ]
[
] ]
[
]
] [̂ ] ]
̂
̂(
̂
̂
̂
̂(
(18)
]
[̂ ]
[
[
[ )(
]
(19)
] (20)
)
̂( )
̂
]
)(
)
̂(
)
̂(
)(
(21)
)
(22)
̂
̂
(
)
̂
(
)
̂
̂
(
)
̂
(
)
̂(
)(
̂(
)(
)
̂(
[
)(
(23) )
)
]
where ̂
[
̂
̂
̂
̂
̂
] [
(24)
] ̂
Proof: Consider Lemma 4 and the NQLF (6). One has: ̇ ̇
̇
̇
̇
̇
̇
⇒
Since ̇
̇
̇
(25) ̇ ̇
⇒
̇
̇
, therefore, (25) is equal to: (26)
Substituting the closed-loop TS system (9) into (26), results: {∑ ∑
{(
)
}}
{∑ ∑
{
(
)
}}
∑ ̇
∑∑
(
)
∑ ̇
∑∑
∑∑
{
(
)
∑∑
∑∑ with ̂
̇ {
}
}
̂ [
[
]̂
̇ ̂ [
]̂
̇ ̂ [
]̂
∑∑
(27)
]. By adding the null terms (11) and (12) to (27), (27) leads to:
∑∑
̂ [
]̂
̂ ∑∑ ̇ ̇ (̂
∑∑
∑
∑∑
̂ ̂ ̂
̂ ( ̂ )̂
∑
∑∑
̂ (̂
̂ )̂
̇ ̂ ̂ ̂
̂ )̂
̂ ∑∑ ̇
∑∑
̂ ̂
∑∑ ̇ ̇ ̂ ̂ ̂
∑∑
̇ ̂ ̂ ̂
∑ ̇ ̂ ̂ ̂ (28)
∑ ̇ ̇ ̂ (̂
where
̂ )̂
̂
̂
[
] (29)
̂
̂
̂
[
̂ ]
(30)
If ̂
̂(
̂
̂
̂
̂(
̂
̂
̂
̂
)(
)
̂( )
)(
)
̂(
)
̂(
)(
(31)
)
(32) ̂
hold, then
̂
(33)
, where
equals to:
(̂ )
̂ {∑
(̂
∑
∑ ̇ ( ̂(
̂ )
)(
))
(34) ∑ ̇ ̇ ( ̂(
̂
[ ̇
)
̂(
))
)(
∑∑
̇ (̂(
)
̂(
) )}
̂
̂
̂ ̇
)(
̂
̂
̂ ̇ [ ̇
̂]
(35)
̂ ̂]
with ̂ ̂
̂
̂
(
)
̂
(
)
̂
̂
(
)
̂
(
)
̂(
)(
̂(
)(
)
̂(
[ By employing Lemma 1, ̂
)(
) )
]
will be negative if: (36)
Considering (29) and (31), LMIs (20), (21) and (22) are obtained. In the following, it is trying to prove (18) and (19). From (30) and (32), one concludes
, where
[
equals to: ̂
]
[
] ̂
[
](
)[
̂
[
]
]
(37)
where is the identity matrix with appropriate dimensions and ̂
[
]
̂
(38)
Utilizing Shure complement as follows: ̂ [
]
(39)
provides (21). Using the same procedure as applied on (32) to derive (18), (33) leads to (19). The proof is ■
completed.
3.3 FUZZY CONTROL DESIGN AND DECAY RATE In most practical applications, it is desired to increase the closed-loop system response. . The convergence rate of the systems is related to decay rate. This subsection deals with the decay rate performance of non-PDC controller design. Definition 1: If Lyapunov function satisfies the following inequality ̇
(40)
where
is the decay rate, then Lyapunov function exponentially converges to zero and the system is
exponentially stable. The next Theorem guarantees the exponential stability of the closed-loop system (9).
Theorem 2: For given positive scalars
and
exponentially stable if there exist symmetric matrices (
)(
) for
subject to | ̇ |
for ,
,
,
, and matrices
, the TS fuzzy system (9) is ,
and
with appropriate dimensions such that the following LMIs are satisfied;
(
)(
)
(41) (42) (43) (
)(
(44)
) (
(
)
)(
)
(
)
(
)(
(45)
)
(46) (
)
(
)
(
)
(
)
(
)(
(
)(
)
[
(
)(
(47)
) )
]
where
Proof: Assume that
. By recalling (40) and the NQLF (6), one has:
̇ ̇ ̇
̇
̇
̇
̇
∑∑
(
)
∑∑
∑∑
[
]
∑∑
̇ {
̇
}
[ ]
Consider the null terms (13) and (14). Therefore, (48) is equal to: ∑∑
[
]
∑∑ ̇ ̇ (
)
∑∑
∑∑ ̇
̇
[ ]
(48)
∑∑
∑
∑∑
(
)
∑
̇
∑∑ ̇ ̇
(
)
̇
∑∑
(49) ∑ ̇
(
)
∑ ̇ ̇
(
)
where
(50)
By using the same procedure as discussed in proof procedure of the Theorem 1 (i.e. (31)-(36)), the LMIs (41)-(47) ■
will be obtained. The proof is completed.
Remark 1: As it is stated before, the number of stabilization conditions is few compared to local stabilization ones. The number of LMI conditions in Theorem 2 is where
is the number of fuzzy rules,
and the number of LMI conditions in [33] is
is the number of states and
is the number of premise variables.
Remark 2: The conditions derive based on QLF and LILF leads to conservative results. The reason lies in the natural structure of QLFs and LILFs. In LMI conditions derived by LILF, although time derivative of MFs does not emerge. However, some structural restrictions must be imposed on LILF matrices. In this regard, the off-diagonal elements of the LILF matrices must be the same. Therefore, in LILF
number of Lyapunov matrices is considered
in which, all the off-diagonal elements are the same and only the diagonal ones can be different. Contrary to LILF, in NQLF, the entire elements of Lyapunov matrices can change. This is the main advantage of utilizing NQLF compare to LILF. Remark 3: Utilizing NQLF has a main drawback. Since the NQLF is a fuzzy blending of common QLFs, therefore, the time derivatives of MFs are appeared in the deriving the stability and stabilization conditions. In order to formulate the conditions in term of LMIs, these time derivatives must be handled. A well-known approach is to consider lower and/or upper bounds [28]. Since we cannot pre-determine these bounds, the LMI conditions are validated for a specific region of states in where the assumption of time derivative of MFs bounds is held. This
assumption leads to a local analysis [33], [35]. In Theorems 1 and 2, it was assumed that the bounds of ̇ (i.e. | ̇ |
) are given. Considering the chain-time derivatives of MFs, one has:
̇ ̇
(51)
Therefore, finding a bound for ̇ depends on finding an upper bound for ̇ . Form (9), it is clear that ̇ depends on the values of
and
which are LMI decision variables. Thus, the given upper bounds
in the LMIs depend on
the values of the LMI variables, which make the problem much more complex. In Theorem 3, we propose new design conditions to overcome the aforementioned problem and to derive the region of attraction. { |
Theorem 3: For the given region
derivatives of membership functions | ̇ | matrices
and ̃
for
(
) (
for
)
}, the constraints on the time
hold, if there exist symmetric matrices
̃ for
and
with appropriate dimensions such that the following
LMIs are satisfied: (52) ̃ {̃
̃ ̃
̃
̃
(53)
̃
̃
[
̃
]
(54)
where ̃
(
[
)
]
Proof: Substituting the non-perturbed closed-loop TS system (9) into (51), results in: |∑ ∑
{
(
)
}|
|∑ ∑
Based on Lemma 2 and defining the slack matrix
{
( ∑
) (
, (55) is satisfied if:
) }|
(55)
[
]
∑∑
(
)
[
∑∑ ]
(
[
)
[
]
(56)
]
hold. Based on Lemma 3, one has:
∑∑
(
[
)
[
][
]
∑∑
] ,(
Suppose that [
) ][
(
) (
(
[ )
(57)
) ]
. Therefore:
]
(58)
Substituting (58) into (57) and applying Lemma 3, one has:
∑∑
(
)
∑∑
[
][
Pre- and post-multiplying (59) by
(
(
(59)
) ]
) with
∑
, and considering Schur complement provide:
∑
(60)
and ∑∑
̃
(61)
where ̃
(
[
)
]
Inequality (60) is implied by (52). Furthermore, (61) can be written as: ∑
If
̃
∑
{̃
̃ }
(62)
̃ {̃
̃ ̃
hold, then
̃
(63)
̃ , where
is equal to:
[ ] ̃ [ ]
(64)
with ̃
̃
[
̃ ̃
]
By employing Lemma 1,
will be positive if:
̃
(65)
Considering (63) and (65), the LMIs (53) and (54) are obtained. The proof is completed.
■
4. Simulation Results In this section, three chaotic systems are presented. First, Lorenz system is considered and results of the proposed approach is compared with recently sampled-data controller [38]. Second, PMSM system with exhibits a chaotic behavior is considered and comparison results of non-PDC controller designed by this paper and PDC method are discussed. Third, CODOPS system [6] is considered. For this system, initially, the nonlinear dynamic of the system is presented. Then, the equivalent TS fuzzy model is proposed and Theorems 1-3 are utilized to design robust nonPDC controller for the CODOPS system.
4.1. Chaotic Lorenz system Consider the chaotic Lorenz equation with an input term: ̇ { ̇
(66) ̇
where
.
4.1.1. Equivalent TS fuzzy model of Lorenz system An equivalent two-rule TS fuzzy model is obtained with the following state space matrices [38]: [
]
[
]
[ ]
The MFs are defined as: ( )
(
)
( )
( )
4.1.2. Lorenz system simulation results For simulation, the system parameters are chosen based on [38] with decay rate ( )
∑
( )
,
,
and
. The sampled data controller
, is designed as:
( )
(67) [
where the sampling instants belong to
] and
[ Set the decay rate
]
and the upper bounds
in the Theorem (2). Non-PDC feedback gains and
Lyapunov matrices are calculated as: [
]
⌊ The initial condition is chosen as
⌋ ( )
[
[ ⌊ ] and a sampling constant
] ⌋ for sampled-data
controller. Figs. 1 and 2 indicate the Lorenz system states and control effort, respectively. Furthermore, table 1 demonstrates the control input signal norm derived by the proposed approach and [38].
10
x
1
5
0
-5
0
0.5
1 Time[sec]
1.5
2
(a) 15
x
2
10 5 0 -5
0
0.5
1 Time[sec] (b)
1.5
2
0
0.5
1 Time[sec] (c)
1.5
2
15
x
3
10
5
0
Fig. 1. Lorenz system states (Theorem 1 “” and ref. [38] “···”). (a) The state x1. (b) The state x2. (c) The state x3.
100
u
0 -100 -200 -300
0.5
1 1.5 2 Time[sec] Fig. 2. Lorenz system control input (Theorem 1 “” and ref. [38] “···”).
Table 1. Norms of Lorenz system control effort. Approach H∞ Theorem 1 183.8955 [38] 234.0773
H2 573.4377 1.046×103
4.2. Chaotic permanent magnet synchronous motor system A PMSM can exhibit bifurcation, limit cycle or chaotic behavior when motor parameters lie in a certain area [39], [40]. In chaos, the motor torque will change randomly and the motor speed will oscillate [41]. The nonlinear dynamic of PMSM can be described by: ̇ ̇
(68) (
)
{ 4.2.1. Equivalent TS model of PMSM system [
Consider the PMSM system (68) with the parameters [42]: [
],
[
] and
[
],
[ ],
,
]. By employing the procedure discussed in
[40], the following TS model will be derived: ̇
∑
where
( ){
[
} ]
(69)
[
] and
[
]
[
]
[ ]
with the normalized MFs ( )
(
( )
)
(
)
Based on [40], the following stabilizing PDC controller is designed ∑
( )
[
Utilizing theorem 2 and considering decay rate [
and the upper bounds ]
⌊
]
, one has:
[ ⌋
The simulation was done with the initial condition ( )
] ⌊
⌋
[
] and a step size of
[40]. The
state evolution and control effort signal are demonstrated in Figs. 3 and 4. In addition, norms of the control inputs are presented in table 2. 15
x
1
10
5
0
0
0.5
1 Time[sec] (a)
1.5
2
10
x
2
5
0
-5
0
0.5
1 Time[sec] (b)
1.5
2
0
0.5
1 Time[sec] (c)
1.5
2
2
x
3
0 -2 -4 -6
Fig. 3. PMSM system states (Theorem 1 “” and ref. [40] “···”). (a) The state x1. (b) The state x2. (c) The state x3. 600
u
400 200 0 -200
0
0.2
0.4 0.6 Time[sec]
0.8
1
Fig. 4. PMSM system control input (Theorem 1 “” and ref. [40] “···”).
Table 2. Norms of PMSM system control effort. Approach Theorem 1 [40]
H∞ 80.9534 435.892
H2 2.6558×103 4.1860×103
From Fig. 3 one concludes that, the closed-loop systems constructed by the PDC controller [40] and non-PDC controller in this paper, have roughly the same system responses. However, as can be seen in Fig. 4 and Table 2, the norms of control input signal derived by the non-PDC controller are significantly smaller than the ones derived by [40].
4.3. Chaotic oscillation damping of power system In this sub-section, an interconnected power system will be introduced and the dynamic model will be derived. The simple interconnected power system is shown in Fig. 5.
3
4 6
𝑮𝟏
7
6
𝑮𝟐 5
Fig. 5. A general scheme power system model In this figure, G1 and G2 are the equivalent generators and 3 and 4 are the main transformers of the generators 1 and 2, respectively. Then, 5, 6 and 7 are considered as the load, circuit breakers and tie line, respectively. This system exhibits some important aspects of behavior of multi-machine systems [6]. Different forms of chaotic systems with linear or nonlinear damping specifications have been represented in literatures [4]. In the state space form, the CODOPS equation (called swing equations) can be described as [2], [4], [6]: ̇ {
where
( ) ̇
(
is the relative angle between voltages of generators 1 and 2,
from generator 1 to 2,
and
power of generator 1,
( ) is the input signal and
)
( )
( )
(70)
is the maximum electric power transmitting
are inertia and damping coefficient of generator, respectively,
is the machinery
( ) is an external disturbance with unknown but bounded
amplitude. This chaotic power system exhibits complex dynamics for constant values of and
|
and
system without controller for
|
and initial states of
[6]. Fig. 6 shows irregular motion of this
( )
( )
and
.
10
X2
5
0
-5
-10 -0.5
0
0.5
1
1.5
2
X1
Fig. 6. x1–x2 plane time response of the chaotic power system.
4.3.1. Constructing the TS fuzzy modeling The system (70) is reformulated as: ̇ {
where ( )
(
( ) ̇
)
( )
(71)
( )
( ) is the new input. There is one nonlinear term in the model (71). Based
on the sector nonlinearity concept [34], one has: ( )
{
( ) ( )
( )
( ) ( )
(72)
( )
(73)
The MFs are derived by solving the equations (73) as following: ( )
( )
( )
( )
(74)
Therefore, the equivalent two rule TS fuzzy model is obtained as follows: ̇ where
∑
(
)
(75)
[
]
[
]
[ ]
4.3.2. Scenario 1: The white noise as the external disturbance Set
,
and {
Consequently,
{ | |
in Theorems 1, 2. Furthermore, consider the local region } and
. In Theorem 3, the upper bounds of
}.
for
are needed. Recall the membership functions (74), one has: ( )
| (
Therefore,
|
|
( )
( )|
(76)
) . By employing Theorems 1-3, the following feedback gain matrices and Lyapunov
matrices are obtained: [
]
[
[
]
]
[
]
Fig. 7 illustrates the disturbance input and the states evolution. The states of chaotic system based on the non-PDC controller designed by the proposed approach converge to equilibrium point. It should be noted that the effect of the external disturbance on the system output
is significantly attenuated. Moreover, the region of attraction and the
outermost Lyapunov function contour that lay in this region is plotted in Fig. 8. The figure demonstrates the regions { | |
},
{
} and
{
}.
4.3.3. Scenario 2: External disturbance with unknown but bounded amplitude In order to challenge our proposed method to study the robustness of the proposed controller, we also introduce (
)
(
) as a disturbance signal. In Theorems 1-3, set { | |
and a compact set
} which provides
,
,
,
. The feedback gain matrices
and Lyapunov matrices are obtained as: [ [
] ]
[
] [
]
Fig. 9 shows the states evolution in presence of the new disturbance. Fig. 9.a illustrates the disturbance signal. The states of chaotic system based on the non-PDC controller designed by the proposed approach converge to
equilibrium point. In addition, in Fig. 10, absolute of time derivative of membership functions is plotted. As it can be seen, the maximum value is smaller than
, which satisfies the constraint on the time derivatives of
membership function.
Disturbance signal
60 40 20 0 -20 -40
0
0.5
1 Time [sec]
1.5
2
(a)
1
x
1
0.5
0
-0.5
0
0.5
1
1.5 Time [sec]
2
2.5
3
2
2.5
3
(b)
0.5 0
x
2
-0.5 -1 -1.5 -2
0
0.5
1
1.5 Time [sec]
(c) Fig. 7. The scenario 1 simulation. (a) The disturbance signal. (b) The system state x1. (c) The system state x2.
5 4 3 2
2
1
x
0 -1 -2 -3 -4 -5 -5
0 x1
5
Fig. 8. Region of attraction for the scenario 1. 1
x
1
0.5
0
-0.5
0
0.5
1 Time [sec]
1.5
2
1.5
2
(a) 10
x
2
0
-10
-20
0
0.5
1 Time [sec]
(b) Fig. 9. The scenario 2 simulation. (a) The system state x1. (b) The system state x2.
i
Absolute of time derivative of h
3
2
1
0
0
0.2
0.4 0.6 Time [sec]
0.8
1
Fig. 10. Absolute of time derivative of membership functions evolution for the scenario 2.
5. Conclusions In this paper, based on the NQLF, novel local LMI stability analysis and non-PDC controller design conditions were proposed for the nonlinear chaotic power systems described by TS fuzzy model. The proposed approach had three main superiorities. Frist, time derivatives of the MFs did not emerge in the LMI conditions. Second, Lyapunov matrices structural restrictions were eliminated. Third, the proposed approach provided a less number of stabilization conditions compared to the ones derived based on the local stability concept. Finally, in order to demonstrate the feasibility of this algorithm, it was applied to the Lorenz system and practical chaotic PMSM system and power chaotic system. The comparison results verified the effectiveness and merits of the proposed algorithm for stability analysis and robust controller synthesis in the presence of external disturbance inputs.
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