Lyapunov method for the stability analysis of uncertain FO systems under input Saturation

Lyapunov method for the stability analysis of uncertain FO systems under input Saturation

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Lyapunov method for the stability analysis of uncertain FO systems under input Saturation Esmat Sadat Alaviyan Shahri, Alireza Alfi, J.A. Tenreiro Machado PII: DOI: Reference:

S0307-904X(20)30013-5 https://doi.org/10.1016/j.apm.2020.01.013 APM 13226

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Applied Mathematical Modelling

Received date: Revised date: Accepted date:

10 April 2019 16 December 2019 8 January 2020

Please cite this article as: Esmat Sadat Alaviyan Shahri, Alireza Alfi, J.A. Tenreiro Machado, Lyapunov method for the stability analysis of uncertain FO systems under input Saturation, Applied Mathematical Modelling (2020), doi: https://doi.org/10.1016/j.apm.2020.01.013

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Highlights • This paper studies the stability analysis of uncertain FO systems using the Lyapunov approach. • Control input saturation is considered for designing the state-feedback control law. • The Lipschitz and the bounded conditions are adopted to derive the stability conditions in terms of LMI.

1

Lyapunov method for the stability analysis of uncertain FO systems under input Saturation Esmat Sadat Alaviyan Shahri a , Alireza Alfi

a∗

, J. A. Tenreiro Machado

b

a

Faculty of Electrical and Robotic Engineering, Shahrood University of Technology, Shahrood 36199 − 95161, Iran b Institute of Engineering, Polytechnic of Porto, Department of Electrical Engineering, Rua Dr. Ant´ onio Bernardino de Almeida 4240-015 Porto, Portugal

Abstract The robust stability and stabilization of a class of Fractional Order systems under input saturation is studied using the Lyapunov method. In the problem formulation the Lipchitz and bounded conditions are adopted. Moreover, sufficient conditions, in the form of a Linear Matrix Inequality, for stabilizing the system via a state feedback control, are derived. Two numerical examples show the effectiveness of the proposed technique. Keywords: Fractional-order system, Lyapunov method, Feedback control, Saturation, Robust stability 1. Introduction Fractional-Order (FO) dynamic systems are capital importance, because many physical phenomena are well characterized by FO differential equations [1, 2, 3, 4, 5]. From the control perspective, the stability and the stabilization of FO systems with different properties is an important topic [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. One of the key challenges from the practical point of view is to control the systems when they are exposed to some kind of input saturation constraint due to physical limitations of the actuators. Therefore, we need to consider the input saturation during the system stability analysis, since this effect leads to performance degradation or even to instability [17, 18, 19, 20]. The stability and stabilization of FO systems including saturation effects have been investigated in the literature [21, 22, 23, 24, 25, 26, 27, 28, 29]. Several studies adopted linear models [21, 25, 26, 27, 28, 29] to analyze the stability of FO systems, and different techniques were proposed to estimate Preprint submitted to

January 15, 2020

the domain of attraction of the closed-loop system. Nevertheless, in practice we have nonlinear effects, resulting in a model mismatch between the real dynamics and the nominal models. This problem motivated the study of FO linear systems with interval uncertainties and attracted considerable attention during the last decades [22, 24, 30, 31]. The robust stability of saturation control was studied in [22]. The stability of a class of uncertain FO systems under input saturation was considered by estimating the solution of a differential equation based on the Lipchitz condition and the Gronwall-Bellman lemma. However, finding the feedback action is a complex task and there is no straightforward procedure to achieve an admissible control law. In general, we find two techniques in the literature for studying the stability of FO systems. The first technique is based on finding or estimating the solution of the differential equations. This approach allows conclusions about the stability, asymptotic stability, and Mittag-Leffler stability, but such techniques reveal some limitations. Hereupon, several research studies applied distinct methods for linear and nonlinear systems with/without saturation control. [14, 21, 22, 23]. The second technique is the FO extension of Lyapunovs technique, namely the direct and the indirect methods. The Lyapunov direct method is an effective tool for the stability analysis of integer and FO systems. For stable FO systems, the decay rate of solution is viewed in the sense of the Mittag-Leffler rather than the exponential function. This prespective motivates the concept of Mittag-Leffler stability of FO systems. In [32] and [33], the Mittag-Leffler stability was firstly proposed for the commensurate and incommensurate cases, respectively. However, the Lyapunov’s second method is often difficult to apply because deriving the Lyapunov functions is more complex for FO models than the integer-order counterpart. One of the reasons for the problem comes from that the computation and estimation of fractional derivatives of the Lyapunov candidate functions. In fact, that task is complex because the Leibniz rule does not hold for such derivatives. Several contributions in the theory of fractional Lyapunov functions can be found in [34, 35, 36, 37]. Tuan and Trinh [34] proposed a rigorous fractional Lyapunov function candidate method to analyze the stability of fractional-order nonlinear systems using a characterization of functions having fractional derivative. An inequality concerning the fractional derivatives of convex Lyapunov functions without the assumption of the existence of the derivative of pseudo-states was proved. Additionally, fractional Lyapunov functions were established for the FO systems without the assumption of 3

the global existence of solutions. Chen et al. [35] introduced an inequality for the stability analysis of FO systems by constructing Lyapunov functions. It was shown that a FO system is MittagLeffler stable if there is a convex and positive definite function such that its FO derivative is negative definite. This result generalizes the existing works and gives a useful method for constructing the Lyapunov function in the stability analysis of FO systems. Aguila-Camacho et al. [36] proposed a new lemma in the sense of the Caputo fractional derivatives. The formulation can be applied with the FO Lyapunov direct method for the stability analysis of a general class of FO systems. Li et al. [37] defined the MittagLeffler stability and introduced for the fractional Lyapunov direct method. The principle of fractional comparison was studied and its application was extended from the RiemannLiouville to the Caputo FO systems. In spite of these advances, the stability analysis of FO systems based on the Lyapunov direct method is still an open problem, and a limited number of works is dedicated to this topic. Stemming from the aforementioned observations, this paper focuses on a method for the Lyapunov candidate functions suitable for handling uncertain FO nonlinear system with saturation control. Indeed, finding the feedback action for such systems is a complex task and we do not have a straightforward procedure to achieve an admissible control law. Thereinafter, for approaching the problem, we consider a nonlinear Lipchitz disturbance. In general, the results of this method can be converted to Linear Matrix Inequalities (LMI), and the controller design can be determined in the form of LMI. The main contributions of this paper are: 1) The derivation of sufficient conditions for the robust and asymptotic stability of nonlinear FO systems under control input saturation and nonlinear Lipschitz disturbance using the direct FO Lyapunov method, 2) The development of a linear state-feedback controller for stabilizing the uncertain FO systems, 3) The formulation of the controller design as LMI, leading simpler procedures both in the FO stability analysis and the controller design. The paper is organized as follows. The fundamental definitions and prerequisites are given in Section 2. The main results including the robust stability analysis and the controller design are discussed in Section 3. Simulation results are presented in Section 4. Finally, the main conclusions are drawn in Section 5.

4

2. Fundamental definitions and prerequisites We have several different definitions for the fractional integral and differential operators [1, 2]. In the current work, the Caputo definition is adopted. This fact is due to that the definition of the Caputo derivative is more practical due to the requirements on the initial conditions. Definition 1 [38, 39]: The Caputo derivative of order, v, of a continuous function f is expressed by 1 C v Dt f (t) = Γ(n − v)

Zt 0

f (n) (τ ) dτ , (t − τ )v−n+1

(1)

where n ∈ N, v ∈ R+ , Γ (·) is the Gamma function, f n (·) is nth the derivative of function f (·), and n − 1 < n < n. Theorem 1 [38]: Let us consider the fractional linear system Dtv x = Ax, with x ∈ Rn and A ∈ Rn×n . The linear system is t−v asymptotically stable if and only if there exists a positive definite matrix P ∈ Rn×n such that 1

1

(−(−A) 2−v )T P + P(−(−A) 2−v ) < 0.

(2)

Theorem 2 [36]: Suppose that x = 0 is an equilibrium point of the nonautonomous FO system C v Dt x = f(x, t). (3) If there exists a Lyapunov function V (t, x(t)) and the class-K functions σi (i = 1, 2, 3) satisfies σ1 (kxk) ≤ V (t, x) ≤ σ2 (kxk) , C v Dt V (t, x) ≤ −σ3 (kxk)

v ∈ (0, 1),

(4)

then, the equilibrium point of the system (3) is asymptotically stable. Theorem 3 [36]: Suppose that V = 12 xT Px, x ∈ Rn , P > 0, is a continuous and derivable function. Then, the following inequality is satisfied C

Dtv V ≤ xT PC Dtv x,

t ≥ t0 .

(5)

Lemma 1 [40]: For any given matrices X and Y with appropriate dimensions, there exists a positive scalar  such that the following relationship holds XT Y ≤ ε XT X + ε−1 YT Y. (6) 5

Remark 1 [25]: The nonlinear function f (·) satisfies the Lipchitz condition if and only if kf (x1 ) − f (x2 )k ≤ λkx1 − x2 k

(7)

where λ > 0 is the Lipschitz constant and xi ∈ Rn , i = 1, 2. Remark 2 [24]: The saturation function denoted by sat(·) : Rm → Rm ,  T sat(u) = sat(u1 ) sat(u2 ) ... sat(um ) , sat(ui ) = sign(ui ) min (kui k , 1) satisfies the Lipschitz condition. Remark 3 [27]: The following inequality can be explored from the special property of saturation function kϕ(x1 ) − ϕ(x2 )k ≤ γ kx1 − x2 k ,

(8)

where γ > 0, ϕ(x) = sat(Kx) − Kx and K ∈ Rm×n is constant matrix. 3. Main results In this section, the theoretical results are provided. First, the robust stability analysis of the uncertain FO systems subject to saturation is established. Then, the controller de- sign and the corresponding optimization problem are given. 3.1. Robust stability analysis Consider the uncertain FO system with 0 < v < 1 described by C

Dtv x = (A + ∆A) x + (B + ∆B) sat(u) + d(x),

(9)

where x ∈ Rn denotes the state vector, A ∈ Rn×n and B ∈ Rm are constant system matrices corresponding to the linear part of the system dynamics and input vector, respectively, u(t) is the control input, d(x) is the disturbance signal that is Lipschitz related to state with Lipschitz constant lw . Moreover, ∆A stands for a time-varying uncertain term regarding to the mismatch model of the linear term, and ∆B represents the input matrix uncertainty satisfying k∆Ak ≤ α, k∆Bk ≤ β, (10)

where α > 0 and β > 0. If we consider a state feedback u=Kx, K ∈ Rm×n , satisfying −u0 ≤ u ≤ u0 , then the closed-loop system can be written as C

Dtv x = (Acl + ∆A) x + Bϕ(x, t) + ∆Bsat(Kx) + d(x), 6

(11)

where Acl = A+BK and ϕ(x, t) = sat(Kx) - Kx. Remark 4: If 0 < ui ≤ 1, then the saturation function works in linear domain, sat(u) = u and the entire closed-loop system is C C

Dtv x = (A + ∆A) x + (B + ∆B) Kx + d(x), Dtv x = (Acl + ∆A + ∆BK) x + d(x).

(12)

In th follow-up, the robust stability of the uncertain closed-loop system (12) is derived with/without considering the disturbance signal d(x). Theorem 4: Consider the closed-loop system (12) with d(x) ≡ 0. If there exists K such that the following relationship is satisfied Acl + αI + β kKk I ≤ 0,

(13)

where I is the identity matrix with appropriate dimension, then the closedloop system is robustly asymptotically stable. Proof : Establish the following Lyapunov function 1 V = xT x. 2

(14)

According to Theorem 3 and substituting the closed-loop (12) with d(x) ≡ 0, we have C v Dt V ≤ xT (Acl + ∆A + ∆BK) x. (15) It is clear that C

Dtv V ≤ xT (Acl + k∆Ak + k∆BKk) x.

(16)

From Eq. (10), we have C

Dtv V ≤ xT (Acl + αI + β kK k I) x.

(17)

Based on Theorem 2, if C Dtv V (t, x) ≤ −σ3 (kxk) is satisfied, then the condition Acl + αI + β kKk I ≤ 0 is obeyed and, therefore, the robust stability is concluded. Theorem 5: Consider the closed-loop system (12). If there exist a positive symmetric definite matrix P, the controller matrix K and a positive scalar ε, such that the following relationship holds P(Acl + αI + βIK) + ε PPT + ε−1 lw2 I < 0, 7

(18)

where lw is the Lipschitz constant of disturbance, then the closed-loop system is robustly asymptotically stable. Proof : Establish the following Lyapunov function 1 V = xT Px, 2

(19)

where P is a symmetric positive-definite matrix. According to Theorem 3 and substituting the closed-loop (12), we have C

Dtv V ≤ xT P ((Acl + ∆A + ∆BK) x + d(x)) .

(20)

Eq. (20) can be also rewritten as C

Dtv V ≤ xT P (Acl + k∆Ak + k∆BKk) x + xT P d(x).

(21)

Based on Eq. (10) and applying Lemma 1, it yields C

Dtv V ≤ xT P (Acl + α I + β kKk I) x + εxT PPT x + ε−1 dT (x)d(x) (22)

and, using Lipschitz condition of d(x), we have C

Dtv V ≤ xT P (Acl + αI + β kKk I) x + εxT PPT x + ε−1 lw2 xT x.

(23)

Based on Theorem 2, if C Dtv V (t, x) ≤ −σ3 (kxk) satisfies, then it must be P(Acl + αI + β kK k I) + εPPT + ε−1 lw2 I < 0 and, therefore, the robust stability is concluded. If 1 < ui ≤ M < ∞, then the robust stability analysis can be addressed as follows Theorem 6: Consider the closed-loop system (11) with d(x) ≡ 0. If there exist a positive symmetric definite matrix P and three positive scalars, θ1 , θ2 and ε2 , such that the following relationship is satisfied PAcl + θ1 PPT + ε2 PBBT PT + θ2 I < 0,

(24)

then the closed-loop system is robustly asymptotically stable. Proof : Let us use the Lyapunov function with Eq. (19). According to Theorem 3 and substituting the closed-loop (11) with d(x) ≡ 0, we have C

Dtv V ≤ xT P ((Acl + ∆A) x + Bϕ(x, t) + ∆Bsat(Kx)) . 8

(25)

Eq. (25) can be rewritten as C

Dtv V ≤ xT P (Acl + ∆A) x + xT PBϕ(x, t) + xT P∆Bsat(Kx).

(26)

Applying Lemma 1, it yields C

Dtv V ≤ xT PAcl x + xT P∆Ax + xT PBϕ(x, t) + xT P∆Bsat(Kx) T T T T T ≤ xT PAcl x + ε1 xT PPT x + ε−1 1 x ∆A ∆Ax + ε2 x PBB P x T T −1 T T T +ε−1 2 ϕ(x, t) ϕ(x, t) + ε3 x P∆B∆B P x + ε3 sat(Kx) sat(Kx). (27) Using Eqs. (7)-(8) and (10), the following inequality is obtained. T 2 Dtv V ≤ xT PAcl x + ε1 xT PPT x + ε−1 1 x α x −1 T 2 T 2 T 2 T +ε2 xT PBBT PT x + ε−1 2 x γ x + ε3 x Pβ P x + ε3 x λ x

−1 2 −1 2 2 = xT PAcl + (ε1 + ε3 β 2 ) PPT + ε2 PBBT PT + ε−1 I x. 1 α + ε2 γ + ε3 λ (28) Based on Theorem 2, if C Dtv V (t, x) ≤ −σ3 (kxk) is satisfied, then
it must −1 2 −1 2 2 be PAcl + (ε1 + ε3 β 2 ) PPT + ε2 PBBT PT + ε−1 I ≤ 0 1 α + ε2 γ + ε3 λ and, therefore, the robust stability is concluded. Furthermore, by defining two variables θ1 and θ2 as C

θ1 = ε1 + ε3 β 2 , −1 2 −1 2 2 θ2 = ε−1 1 α + ε2 γ + ε3 λ ,

(29)

the inequality (28) can be expressed in the form of Eq. (24). This completes the proof. Remark 5: Considering Q = P−1 and multiplying by Q both sides of Eq. (24), the stability condition becomes Acl Q + θ1 I + ε2 BBT + Qθ2 Q < 0.

(30)

Theorem 7: Consider the closed-loop system (11). If there exist a positive symmetric definite matrix P and three positive scalars, θ3 , θ4 and ε2 , such that the following relationship is satisfied PAcl + θ3 PPT + ε2 PBBT PT + θ4 I ≤ 0,

(31)

then, the closed-loop system is robustly asymptotically stable. Proof : Using Lyapunov function with Eq. (19) and according to Theorem 3 and substituting the closed-loop (11), we have C

Dtv V ≤ xT P ((Acl + ∆A) x + Bϕ(x, t) + ∆Bsat(Kx) + d(x)) . 9

(32)

Eq. (32) can be rewritten as C

Dtv V ≤ xT P (Acl + ∆A) x + xT PBϕ(x, t) + xT P∆Bsat(Kx) + xT Pd(x). (33) Applying Lemma 1, it yields C

Dtv V ≤ xT PAcl x + xT P∆Ax + xT PBϕ(x, t) + xT P∆Bsat(Kx) + xT Pd(x) T T T T T ≤ xT PAcl x + ε1 xT PPT x + ε−1 1 x ∆A ∆Ax + ε2 x PBB P x T T −1 T T T +ε−1 2 ϕ(x, t) ϕ(x, t) + ε3 x P∆B∆B P x + ε3 sat(Kx) sat(Kx) T +ε4 xT PPx + ε−1 4 d (x)d(x). (34) Using Eqs. (7)-(8) and (10), the following inequality is obtained T T −1 T 2 T 2 T Dtv V ≤ xT PAcl x + ε1 xT PPT x + ε−1 1 x α x + ε2 x PBB P x + ε2 x γ x T −1 2 T T 2 T +ε3 xT Pβ 2 PT x + ε−1 3 x λ x + ε4 x PP x + ε4 lw x x T T 2 = x (PAcl + (ε1 + ε3 β + ε4 ) PP
+ ε2 PBBT PT −1 2 −1 2 −1 2 2 + ε−1 1 α + ε2 γ + ε3 λ + ε4 lw I)x, (35) where lw is the Lipschitz constant of the disturbance. Based on Theorem 2, if C Dtv V (t, x) ≤ −σ3 (kx k) is satisfied, then it must be
−1 2 −1 2 −1 2 2 PAcl +(ε1 + ε3 β 2 + ε4 ) PPT +ε2 PBBT PT + ε−1 1 α + ε2 γ + ε3 λ + ε4 lw I ≤ 0 and, therefore, the robust stability is concluded. Furthermore, by defining two variables θ3 and θ4 as C

θ3 = ε1 + ε3 β 2 + ε4 , −1 2 −1 2 −1 2 2 θ4 = ε−1 1 α + ε2 γ + ε3 λ + ε4 lw ,

(36)

the inequality (35) can be expressed in the form of Eq. (31). This completes the proof. Remark 6: Considering Q = P−1 and multiplying Q both sides of Eq. (31), the stability condition becomes Acl Q + θ3 I + ε2 BBT + Qθ4 QT < 0.

(37)

Remark 7: Since there are some conditions where the system states are not fully available that motivates the use of the output feedback control technique.

10

3.2. Controller design The design procedure for the feedback control law based on LMI is given in the follow-up. Consider the open-loop system (9) with d ≡ 0. As mentioned previously, if the constant matrix, K, exists, such that Eq. (24) holds, then the closed-loop system (11) is robustly stable. The procedure for designing the controller can be summarized as follows. First, assume that Q as a determinate symmetric positive-definite matrix (e.g., µI, µ > 0), and second, solve Eq. (30). For d 6= 0, we have a situation similar to the case of d ≡ 0. This means that Q is a specified symmetric positive-definite matrix and the respective LMI given in (37) must be solved. 4. Simulation results In this section, the efficiency of the proposed method of analysis is applied to two examples. Example 1 Consider a system with model:  C v         D t x1 0 1 0 0 x1 1 0.05  C Dtv x2   0       0 1 0   C v =   x2  +  1  sat (u)+  0  e−3x1  Dt x3   0  0  0 0 1   x3   1  C v Dt x4 −γ −β −α 0 x4 1 0 (38) where α = 0.8, β = −1.4, γ = 0.1 and v = 0.7. Additionally, we consider     1 0 0 0 1  0 1 0 0   1     ∆A = 0.035 ·   0 0 1 0  · sin (0.5t) , ∆B = 0.02 ·  1  · cos (t) . (39) 0 0 0 1 1 The open-loop system is unstable and the proposed algorithm is applied for finding the state-feedback controller. Choosing and applying the controller design strategy, we obtain   K = −1.55 −2.4 −2 −2 , θ3 = 0.2, (40) θ4 = 0.35, ε2 = 0.3. The time responses of the open system for initial conditions   x0 = 2 .91 −5.89 5.04 1.79 , is depicted in Fig. 1. Furthermore, Figs. 11

2 and 3 present time responses of the closed-loop system in case of absent and present of the disturbance. The results demonstrate good performance of the proposed controller in the presence of disturbance.

80

x1 x2 x3 x4

60 40

States

20 0 −20 −40 −60 −80 −100 0

5

time (Sec)

10

15

Figure 1: State trajectories versus time of the open-loop system (38) for Example 1

Remark 8: It is worth mentioning that the stability condition in [22] were obtained by estimating the solution of a differential equation based on the Gronwall-Bellman lemma, while the presented theoretical results are derived by the Lyapunov Theorem. Based on the results reported in [22], the closed-loop system (9) with d(x) ≡ 0 is robustly asymptotically stable if , where spec(Acl ) stands the spectrum of the matrix |arg(spec(Acl ))| ≥ vπ 2 Acl . Remark 9: In comparison with the controller designed in [22], where the   state-feedback control law K = −12.2117 −12.5632 −11.9487 −12.2583 is adopted based on Remark 7, it can be inferred that the control effort obtained in the proposed technique is lower than using the approach of [22].

12

6

x1 x2

4

x3 x4

States

2

0

-2

-4

-6

0

50

100

150

200

250

300

350

400

time(Sec)

Figure 2: State trajectories versus time of the closed-loop system derived using Eq. (40) when d(x) = 0 for Example 1

Example 2 To illustrate the effectiveness of the proposed control rithm, the FO Wien-bridge oscillator is used [41] with model      C v   −2 −1 x1 1 Dt x1 = + sat (kx1 ) , C v −1 −1 x2 1 Dt x2   where v = 0.5 and K = k 0 . In addition, it is assumed that     1 0 1 ∆A = 0.3 · · sin (0.05t) , ∆B = 0.2 · · cos (t) . 0 1 1   1 0 Choosing µI = , we obtain 0 1   K = 0.8 0 , θ1 = θ2 = 0.1, ε2 = 0.1.

algo-

(41)

(42)

(43)

The time response of the closed-loop system for v = 0.5 with initial con T ditions x (0) = 3 −5 is shown in Fig. 4. Moreover, Fig. 5 displays the comparison results for the case of using Eq. (43) and adopting the controller proposed in [22]. Comparing the results of Eq. (43) with those of [22], 13

6

x1 x2 x3

4

x4

States

2

0

-2

-4

-6

0

500

1000

1500

time(Sec)

Figure 3: State trajectories versus time of the closed-loop system derived using Eq. (40) when d(x) 6= 0 for Example 1

we verify that time response of the proposed controller is better than the result obtained with [22], because corresponds to a solution of the optimization problem. In summary, the results demonstrate that the system states converge to zero in the presence of disturbance by applying the appropriate controller, which indicates the validity of the main results.

14

4

x1 x2

3 2

States

1 0 -1 -2 -3 -4 -5

0

50

100

150

200

250

300

350

400

450

500

time(Sec)

Figure 4: State trajectories versus time of the closed-loop system derived using Eq. (43) for Example 2

5. Conclusion This paper addressed the stability and the stabilization of uncertain FO systems using the Lyapunov approach, while including the effects of input saturation and model uncertainties. The Lipschitz and the bounded conditions were considered for determining the state-feedback control law with the help of LMI. This strategy was proved to be capable of stabilizing such type of systems. Simulation results demonstrated the validity of the proposed control method. References [1] C A Monje, Y Q Chen, B M Vinagre, D Xue, V Feliu-Batlle, Fractionalorder systems and controls: fundamentals and applications, Springer Science and Business Media, 2010. [2] J Sabatier, O P Agrawal, J A Tenreiro Machado, Advances in fractional calculus, Netherlands: Springer, 2007. 15

3.5 [22] Proposed method

3

States

2.5

2

1.5

1

0.5

0

0

50

100

150

200

250

300

350

400

450

500

time(Sec)

Figure 5: Comparison of the state trajectory x1 (t) derived using (43) and [22] for Example 2

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