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Robust stabilization of uncertain nonlinear slowly-varying systems: Application in a time-varying inertia pendulum T. Binazadeh, M.H. Shaﬁei n School of Electrical and Electronic Engineering, Shiraz University of Technology, Modares Boulevard, Shiraz, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 28 July 2013 Received in revised form 27 November 2013 Accepted 7 December 2013 Available online 1 January 2014 This paper was recommended for publication by Dr. Qing-Guo Wang

This paper considers the problem of robust stabilization of nonlinear slowly-varying systems, in the presence of model uncertainties and external disturbances. The main contribution of this paper is an extension of the Slowly-Varying Control Lyapunov Function (SVCLF) technique to design a robust stabilizing controller for nonlinear slowly-varying systems with matched uncertainties. In the proposed strategy, the Lyapunov redesign method is utilized to design a robust control law. This method, originally, leads to a discontinuous controller which suffers from chattering. In this paper, this problem is removed by using a saturation function with a high slope, as an approximation of the signum function. Since, using the saturation function leads to loss of asymptotic stability and, instead, guarantees only the boundedness of the system's states; therefore, some sufﬁcient conditions are proposed to guarantee the asymptotic stability of the closed-loop uncertain nonlinear slowly-varying system (without chattering). Also, in order to show the applicability of the proposed method, it is applied to a time-varying inertia pendulum. The efﬁciency of the designed controller is demonstrated through analysis and simulations. & 2013 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Nonlinear uncertain slowly-varying system Slowly-Varying Control Lyapunov Function (SVCLF) Robust stabilizing controller Boundedness

1. Introduction Among the nonlinear time-varying systems, an important category is the nonlinear systems with slowly-varying parameters. Such systems are called slowly-varying systems [1] and may be considered between time-invariant and time-varying systems. Slowly-varying systems have many applications in physics and control engineering [2–5]. For these systems, the algorithms developed for time-invariant systems, may cause instability. On the other hand, the general time-varying based methods may be too conservative and complicated due to control law. A considerable amount of scientiﬁc works have been done in the area of analysis of nonlinear slowly-varying systems, for instance see [6–8]. However, there are a few works presenting a framework to design a nonlinear stabilizing controller for nonlinear slowly-varying systems and most of the existing papers in this ﬁeld, are concentrated on linear slowly-varying systems [9– 11]. Authors of [12–14] have proposed stabilizing control laws for nonlinear slowly-varying systems, based on the Control Lyapunov Function (CLF) method. The CLF method has been, originally, introduced by Sontag to stabilize the nonlinear time-invariant systems [15]. The CLF-based stabilization method for nonlinear n

Corresponding author. E-mail addresses: [email protected] (T. Binazadeh), shaﬁ[email protected] (M.H. Shaﬁei).

time-varying systems leads to a controller with a complicated time-varying structure [16]. The proposed controllers in [13,14], which were called SVCLF controller, are structurally as simple as the CLF controller for nonlinear time-invariant systems. In [13] the SVCLF controller has been designed for nonlinear slowly-varying systems with a scalar input and a scalar slowly-varying parameter while the authors of [14] have extended this method to nonlinear multi input time-varying systems with slowly-varying parameters However, in spite of the advantages of the proposed methods in [12–14], model uncertainties and external disturbances, which exist in many of practical systems [17–20], have not been considered in the model of the nonlinear slowly-varying systems. The main result of this paper is an extension of the SVCLF technique to design a robust stabilizing controller for uncertain nonlinear slowly-varying systems with matched uncertainties. The proposed strategy utilizes the Lyapunov redesign method to conquer the uncertainties. In the design procedure, ﬁrst, the nominal controller is designed for the nominal system (the slowlyvarying system without any uncertainties and external disturbances), and then, using the Lyapunov redesign method, an additional term is added to the control law to guarantee the robust asymptotic stabilization in the presence of uncertainties and external disturbances. The Lyapunov redesign method, originally, leads to a discontinuous controller with signum function. Since, discontinuous controllers suffer from chattering, to alleviate this problem; the signum function is replaced with the saturation

0019-0578/$ - see front matter & 2013 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2013.12.009

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T. Binazadeh, M.H. Shaﬁei / ISA Transactions 53 (2014) 373–379

function with a high slope. However, this causes to loss of the asymptotic stability and, instead, only the boundedness of the system's states could be guaranteed. In this paper, in addition to designing a robust controller for nonlinear slowly-varying systems, some sufﬁcient conditions are also presented which guarantee the asymptotic stability of the closed-loop system (without chattering). In order to show the applicability of the proposed method, it is applied to the time-varying inertia pendulum, which is one of the famous benchmarks among the nonlinear time-varying systems. Computer simulations show the efﬁciency of the proposed method in robust asymptotic stabilization of the nonlinear time-varying inertia pendulum in the presence of model uncertainties.

_ where α1 ð:Þ and α2 ð:Þ are class K functions. If ‖θðtÞ‖ (for every t A ½0; 1Þ) satisﬁes the condition (10), then the SVCLF controller (11) leads to V_ r αγð‖x‖Þ in the trajectories of the closed-loop system (where α A ð0; 1Þ is chosen such that the inﬁmum in (10) is positive and qðx; θÞ is given in the cost function (3)). qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ T a2 ðx; θÞ þ qðx; θÞbðx; θÞb ðx; θÞ αγð‖x‖Þ _ r inf sup ‖θ‖ ð10Þ θ A Ω;x A D ‖∂V=∂θ‖ t

2. Problem formulation and preliminaries

Therefore, the control law (11) guarantees the asymptotic stabilization of the nominal system (4).

u ¼ kðx; θðtÞÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8 T T < b ðx;θðtÞÞ aðx;θðtÞÞ þ a2 ðx;θðtÞÞ þ qðx;θðtÞÞbðx;θðtÞÞb ðx;θðtÞÞ ; T bðx;θðtÞÞb ðx;θðtÞÞ ¼ : 0;

where

ba0

where

b¼0 ð11Þ

Consider the following uncertain nonlinear slowly-varying system: x_ ¼ f ðx; θðtÞÞ þ gðx; θðtÞÞ½u þdðx; u; θðtÞÞ

ð1Þ

where x A D ℜn (D contains the origin) and u A ℜp are the state and input vectors, respectively. θðtÞ A Ω ℜq is a vector of time-varying parameters, whose variations are bounded and small enough (the slowly-varying parameters) and the perturbation term dðx; u; θðtÞÞ is resulted from modeling error or uncertainties and external disturbances, which exists in any practical problem. In a typical situation, dðx; u; θðtÞÞ is unknown, but some information about it, like an upper bound on ‖dðx; u; θðtÞÞ‖ is usually known. In this paper it is assumed that ‖dðx; θðtÞÞ‖1 r ρðxÞ;

8 ðx; θÞ A D Ω

ð2Þ

where ρðxÞ is a known positive function. The task is to design a robust controller such that the closedloop system (1) is asymptotically stable in the presence of dðx; u; θðtÞÞ. Additionally, in order to make the performance quantitative, it is desirable to minimize the following cost function: Z 1 J¼ ðqðxðτÞ; θðτÞÞ þ uðτÞT uðτÞÞdτ ð3Þ

Proof. See [14].

□

Remark 1. One of the important issues in the design the proposed control law is its link with optimality. Consider the nonlinear system (4) with the cost function (3). Finding the optimal controller for this system leads to solving the following timevarying partial differential equation (which is called time-varying Hamilton–Jacobi–Bellman (HJB) equation): 1 V nt þ q þ V nx f V nx gg T V xnT ¼ 0 4 The optimizing control action is un ¼ 0:5g T V xnT which may be evaluated after solving HJB equation and ﬁnding V nx [21]. Unfortunately, the HJB equation is not analytically solvable for most practical systems. Therefore, approximate techniques have been used to solve this equation and estimate V nx which leads to suboptimal controllers. It has been shown in [14] that the SVCLF controller is also a suboptimal controller and the response of the system (4) with this controller may be very close to its optimal solution.

0

where qðx; θÞ is a positive-deﬁnite function. Since, choosing the cost function is effective on the characteristics of the transient responses of system's states, qðx; θÞ should be selected such that the system's performance to be acceptable in terms of settling time, overshoot and etc. In the following theorem, the nominal controller is designed (according to SVCLF method) to asymptotically stabilize the following nonlinear slowly-varying system (called the nominal system): x_ ¼ f ðx; θðtÞÞ þ gðx; θðtÞÞu

ð4Þ

For this purpose, considering the parametric Lyapunov function Vðx; θðtÞÞ : D Ω-R (which is called the slowly-varying control Lyapunov function), the functions aðx; θðtÞÞ and bðx; θðtÞÞ are deﬁned as follows (note that V x ¼ ∂V=∂x is assumed as a row vector): aðx; θðtÞÞ ¼ V x f ðx; θðtÞÞ;

ð5Þ

bðx; θðtÞÞ ¼ V x gðx; θðtÞÞ:

ð6Þ

Theorem 1. Considering the nonlinear slowly-varying system (4), suppose that γð‖x‖Þ is a class K function and there exists a parametric Lyapunov function V ðx; θðtÞÞ : D Ω-R such that α1 ð‖x‖Þ r Vðx; θðtÞÞ r α2 ð‖x‖Þ; aðx; θðtÞÞ r γð‖x‖Þ; ∂V o 1; ∂θ

8 ðx; θðtÞÞ A D Ω

8 ðx; θðtÞÞ A D Ω

8 ðx; θðtÞÞ A D Ω

where

ð7Þ

3. Robust stabilization of nonlinear slowly-varying systems In this section, an additional term (v) is designed (based on the Lyapunov redesign method), such that the overall feedback law (u ¼ kðx; θðtÞÞ þ v) guarantees robust asymptotic stabilization of the nonlinear system (1) in the presence of the unknown vector function dðx; u; θðtÞÞ. Theorem 2. Consider the system (1). Take nonlinear slowly-varying r 4 0, such that Br ¼ x A Rn : ‖x‖ rr D and suppose that the elements of v are as follows: ( ηðxÞsgnðbi Þ; ηðxÞj bi j Z ε for i ¼ 1; ⋯; p ð12Þ vi ¼ ηðxÞj bi j o ε η2 ðxÞbεi ; where bi is the ith element of the row vector bðx; θðtÞÞ, deﬁned in (6), and ηðxÞ Z ρðxÞ is a positive continuous function of the state variables. Also, ε o 4ð1 βÞαγ½α2 1 ðα1 ðrÞÞ=p where β is an arbitrary value belonging to ð0; 1Þ. Also, α1 ð:Þ; α1 ð:Þ; γð:Þ and α have been introduced in Theorem 1. The feedback law u ¼ kðx; θÞ þv (where kðx; θÞ is the nominal controller (11)) guarantees that for any ‖xðt 0 Þ‖o α2 1 ðα1 ðrÞÞ, there exists a ﬁnite time T such that ‖xðtÞ‖ r α1 1 ðα2 ðμÞÞ; where μ ¼ γ

bðx; θðtÞÞ ¼ 0

1

8 t Z t 0 þT

ð13Þ

ðpε=½4ð1 βÞαÞ.

ð8Þ

Proof. According to the result of Theorem 1, there exists a Lyapunov function Vðx; θðtÞÞ, for the nominal system (4), such that

ð9Þ

∂V ∂V þ ½f ðx; θðtÞÞ þ gðx; θðtÞÞkðx; θðtÞÞ r αγð‖x‖Þ; V_ ¼ ∂t ∂x

T. Binazadeh, M.H. Shaﬁei / ISA Transactions 53 (2014) 373–379

8 ðx; θÞ A D Ω

ð14Þ

Now, if the control signal, u is chosen as u ¼ kðx; θÞ þ v

ð15Þ

Then, V_ in the direction of the perturbed closed-loop system (1), is as follows: ∂V ∂V ∂V þ ½f ðx; θðtÞÞ þ gðx; θðtÞÞkðx; θðtÞÞ þ gðx; θðtÞÞ½v þdðx; θðtÞÞ V_ ¼ ∂t ∂x ∂x ð16Þ Considering (6) and (14), V_ satisﬁes the following inequality: V_ r αγð‖x‖Þ þ bðx; θðtÞÞðv þ dÞ;

8 ðx; θÞ A D Ω

ð17Þ

375

also satisﬁed with μ ¼ γ 1 ðpε=½4ð1 βÞαÞ and ε o 4ð1 βÞαγ½α2 1 ðα1 ðrÞÞ=p). Thus, for any ‖xðt 0 Þ‖ o α2 1 ðα1 ðrÞÞ, there is a ﬁnite time T such that ‖xðtÞ‖ r α1 1 ðα2 ðμÞÞ;

8 t Zt 0 þ T

□

ð25Þ

Remark 2. Theorem 2, just, proves the boundedness of the state variables in the closed-loop uncertain system (1). Although, choosing smaller values of μ makes the state variables closer to the origin, it is not equivalent to asymptotical stability of the closed-loop uncertain system (1). The following theorem determines the additional conditions which result in the robust asymptotic stabilization of the nonlinear uncertain system (1).

According to the upper bound of d (deﬁned in (2)), the inequality (17) can be rewritten as

Theorem 3. Consider the nonlinear system (1) with the conditions stated in Theorem 2. Also, suppose that there exist a positive deﬁnite function ϕðxÞ : D-R, and two positive constants η0 and ρ1 satisfying the following inequalities:

V_ r αγð‖x‖Þ þ bv þ ‖b‖1 ‖d‖1 r αγð‖x‖Þ þ bv þ ‖b‖1 ρ

γð‖x‖Þ Zϕ2 ðxÞ

ð18Þ

Since, the proposed v in (12), is element-wise, the vector quantities in (18) may be replaced as follows: p

p

i¼1

i¼1

V_ r αγð‖x‖Þ þ ∑ ½bi vi þρj bi j r αγð‖x‖Þ þ ∑ ½bi vi þ ηj bi j

ð19Þ

Inserting v from (12) into (19), leads to p jb j2 V_ r αγð‖x‖Þ þ ∑ η2 i þ ηj bi j þ ∑ ½ ηbi sgnðbi Þ þ ηj bi j ε iAI i¼1

ð26Þ

where functions γð:Þ, ηð:Þ and ρð:Þ have been introduced, previously. If ε o 4αð1 βÞη20 =ðpρ21 Þ, then, the feedback law u ¼ kðx; θÞ þ v (where kðx; θÞ is the nominal controller (11) and v is given in (12)) guarantees the asymptotic stabilization of the nonlinear system (1) in the presence of the unknown term dðx; u; θðtÞ. Proof. Consider the proof of Theorem 2, the ﬁrst inequality in (19) is restated here p

i2 =I

ð20Þ where I is the set of indexes of elements of the vector b which satisfy ηðxÞjbi jo ε (i.e., when ηðxÞjbi j o ε, then i A I). Since, bi sgnðbi Þ ¼ jbi j, the second sigma in (20) is equal to zero. Thus jb j2 V_ r αγð‖x‖Þ þ ∑ η2 i þ ηj bi j ð21Þ ε iAI

V_ r αγð‖x‖Þ þ ∑ ½bi vi þ ρj bi j

ð27Þ

i¼1

Substituting the control signal v, from (12) into (27), results in p jb j2 V_ r αγð‖x‖Þ þ ∑ η2 i þ ρj bi j þ ∑ ½ ηbi sgnðbi Þ þ ρj bi j ε iAI i¼1 i2 =I

ð28Þ

2

The term ( η ðjbi j =εÞ þ ηjbi j) has a quadratic form in ηjbi j and

reaches to its maximum value, ε=4, at η bi ¼ ε=2. Also, the worst _ occurs, when the set I consists of all case (less negative value for V) indexes i ¼ 1; 2; :::; p (i.e., the number of terms in the sigma in (21) is equal to p). Therefore, always one has ε ð22Þ V_ r αγð‖x‖Þ þ p 4 2

ηðxÞ Z η0 4 0 ρðxÞ r ρ1 ϕðxÞ

It is worth noting that, in the other cases, when some (or all) elements of v follow the ﬁrst relation of (12) (i.e., ηjbi j Z ε; for some or all values of i), the value of V_ is more negative than the right-hand side of (22). Therefore, the inequality (22) is satisﬁed for all cases. Now, this inequality may be rewritten as follows: ε ð23Þ V_ r βαγð‖x‖Þ ð1 βÞαγð‖x‖Þ þ p 4

Now, consider the second sigma in (28), since, bi sgnðbi Þ ¼ jbi j and ρðxÞ r ηðxÞ (refer to Theorem 2), this sigma is less than or equal to zero, therefore jb j2 ð29Þ V_ r αγð‖x‖Þ þ ∑ η2 i þ ρj bi j ε iAI substituting the conditions (26), into the inequality (29), leads to jb j2 V_ r αϕ2 ðxÞ þ ∑ η20 i þ ρ1 ϕðxÞj bi j ð30Þ ε iAI

Now, two cases are considerable Case 1. The set I is null and there is no member belonging to it. Therefore, there is not any summation in (30) and consequently

where β is an arbitrary real value belonging to ð0; 1Þ. If ð1 βÞαγð‖x‖Þ Z p4ε (or equivalently, ‖x‖ Zγ 1 ðpε=½4ð1 βÞα 9 μÞ) then

V_ r αϕ2 ðxÞ

V_ r βαγð‖x‖Þ

Case 2. The set I has m (1 r m r p) member(s). In this case, the inequality (30) may be rewritten as

8 ‖x‖ Zμ

ð24Þ

This leads to boundedness of state variables of the closed-loop system (see Theorem 4 in (Section 6) where condition (A3) is satisﬁed for W 3 ðxÞ ¼ βαγð‖x‖Þ and the condition μo α2 1 ðα1 ðrÞÞ is

8 x A D:

αð1 βÞ 2 jb j2 ϕ ðxÞ η20 i þ ρ1 ϕðxÞj bi j V_ r αβϕ2 ðxÞ þ ∑ m ε iAI

ð31Þ

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T. Binazadeh, M.H. Shaﬁei / ISA Transactions 53 (2014) 373–379

the quadratic term, inside the sigma, may be converted to a matrix form, i.e., 3" " #T 2 αð1 βÞ # ρ1 =2 ϕðxÞ ϕðxÞ m 2 _ 4 5 ð32Þ V r αβϕ ðxÞ þ ∑ η20 j bi j j bi j ρ1 =2 iAI ε

ε o 4αð1 βÞη20 =ðmρ21 Þ,

the matrix in (32) will be positive If deﬁnite and consequently, the summation in (32) will contain a number of terms which are less than or equal to zero. Therefore, in the second case, also, one has, V_ r αβϕ2 ðxÞ

4. Application of the proposed method in robust stabilization of a time-varying inertia pendulum In order to show the applicability of the proposed method, and also to clarify the design procedure, it is applied to one of the famous benchmarks of nonlinear control (inertia pendulum). Since, the proposed method is related to nonlinear time-varying systems, a time-varying pendulum is chosen. This pendulum which includes a plate, a beam and a traveling mass (which moves along the beam) is mechanically constructed [22]. Fig. 1 shows a schematic diagram of this time-varying pendulum. Also, its mathematical model is as follows [22]: € þ ðcv þ 2mm sðtÞs_ ðtÞÞθðtÞ _ þ ðP 0 þmm gsðtÞÞ sin θðtÞ ¼ τðtÞ ðI 0 þ mm s2 ðtÞÞθðtÞ ð33Þ

Component

Mass (kg)

Plate

mp ¼ 0:0713

Beam Travelling mass

Sizes (m)

ap ¼ 0:044 bp ¼ 0:063 Lb ¼ 1 mb ¼ 0:29 mm ¼ 0:5025 r m ¼ 0:05

Centre of mass distance from O (m) dp ¼ 0:01 db ¼ 0:5

8xAD

Thus, both cases result in the asymptotic stability of the uncertain closed-loop system (1) for all x A D. Although, in the inequality ε o 4αð1 βÞη20 =ðmρ21 Þ, there is an unknown integer m, if ε o 4αð1 βÞη20 =ðpρ21 Þ, the above mentioned inequality will also be satisﬁed for every value of m A ½1; p. □

where

Table 1 The values of physical parameters of the inertia pendulum [22].

2

2

2

1 1 I 0 ¼ 12 mp a2p þbp þmp dp þ 12 mb L2b þ mb db þ 12mm r 2m

P 0 ¼ mp gdp þ mb gdb

ð34Þ

and τðtÞ is the applied torque to the pendulum. Also, the parameter cv is the viscous damping coefﬁcient, g=9.8 is the acceleration due to gravity and mp ; mb and mm are masses of the plate, the beam and the travelling mass, respectively. The parameter sðtÞ is the distance of the travelling mass from the point O (see Fig. 1), which

is the slowly-varying parameter and may change slowly in the range of 0:1 r sðtÞ r1 (due to physical constraints of the mechanical system). The other parameters are given in Table 1. The goal is to regulate θðtÞ-π. Thus, choosing state variables as _ x1 ¼ θ πðradÞ, x2 ¼ θðrad=sÞ and the control input u ¼ τ, the statespace equations of (33) are as follows: # " # " # " x2 0 x_ 1 u ð35Þ ¼ þ ξ1 ðsÞ sin ðx1 þ πÞ ξ2 ðs; s_ Þx2 ξ3 ðsÞ x_ 2 where ξ1 ; ξ2 and ξ3 are as follows: ξ1 ðsÞ ¼

P 0 þ mm gsðtÞ I 0 þ mm s2 ðtÞ

cv þ 2mm sðtÞs_ ðtÞ I 0 þ mm s2 ðtÞ 1 ξ3 ðsÞ ¼ I 0 þ mm s2 ðtÞ ξ2 ðs; s_ Þ ¼

ð36Þ

Suppose that there are uncertainties in the ξ2 due to the unknown parameter cv and inaccurate measurement of the s_ ðtÞ. Also, ξ2 can be rewritten as ξ2 ðs; s_ Þ ¼ ξ3 ðsÞðcv þ2mm ss_ Þ ¼ ξ^ 2 ðs; s_ Þ þ ξ3 ðsÞδ

ð37Þ

where ξ^ 2 is the nominal part of ξ2 which is known precisely; while, δ is an unknown function that lumps together the uncertain term due to parameter uncertainty. By considering 0 r cv r 1 and the maximum error 0:5ðrad=sÞ in the estimation of s_ , one can obtain jδj r 1. Thus, Eq. (35) can be rewritten as follows (which has the structure of the system (1) with θðtÞ ¼ sðtÞ): # " # " # " x2 0 x_ 1 ðu þ dÞ ð38Þ ¼ þ ξ1 ðsÞ sin ðx1 Þ ξ^ 2 ðs; s_ Þx2 ξ3 ðsÞ x_ 2 where, dðx; sÞ ¼ δx2

ð39Þ

therefore j dj r j x2 j

ð40Þ

Now, the task is to design a robust asymptotic stabilizing control law for the system (38) over the set D ¼ fx A R2 : jx1 j r π & jx2 jr 1g by considering the following cost function: Z 1 J¼ ð10ðx21 þ x22 Þ þ u2 Þdt ð41Þ |ﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄ} 0 q

4.1. Design of the nominal controller Considering the state-space Eq. (38) with d ¼ 0 as the nominal system, a Lyapunov function may be chosen as follows: Fig. 1. Schematic diagram of the time-varying pendulum [22].

1 1 1 V ¼ xT Px ¼ p11 x21 þ p22 x22 þ p12 x1 x2 2 2 2

ð42Þ

T. Binazadeh, M.H. Shaﬁei / ISA Transactions 53 (2014) 373–379

where pij s are the elements of the symmetric positive deﬁnite matrix P. Thus, condition (7) is satisﬁed for, α1 ð‖x‖Þ ¼ λmin ðPÞ‖x‖2 ; α2 ð‖x‖Þ ¼ λmax ðPÞ‖x‖2

ð43Þ

Moreover, the Lyapunov function (42) is independent from the slowly-varying parameter (θðtÞ ¼ sðtÞ), thus, ∂V=∂θ ¼ 0, and conditions (9) is also satisﬁed. Also, according to deﬁnitions (5) and (6), aðx; θðtÞÞ ¼ V x f ¼ p11 p12 ξ^ 2 x1 x2 þ p12 p22 ξ^ 2 x22 þp22 ξ1 x2 sin x1 þ p12 ξ1 x1 sin x1 bðx; θðtÞÞ ¼ V x g ¼ ξ3 ðp22 x2 þ p12 x1 Þ

ð44Þ

The necessary condition in the SVCLF strategy is that in the points where bðx; θðtÞÞ ¼ 0 (or x2 ¼ p12 x1 =p22 ), the term aðx; θðtÞÞ should satisfy condition (8). In such points, aðx; θðtÞÞ is as follows: aðx; θðtÞÞj x2 ¼

p12 x1 =p22

¼

p12 ðp22 Þ2

½p11 p22 ðp12 Þ2 x21

ð45Þ

aðx; θðtÞÞj x2 ¼

p12 x1 =p22

r ζx21

ð46Þ λðx21 þx22 Þ; λ 4 0,

Now, choosing the class K function γð‖x‖Þ ¼ then, in the points where bðx; θðtÞÞ ¼ 0, the function γð‖x‖Þ will be simpliﬁed as follows: 2 ! p γð‖x‖Þj x2 ¼ p12 x1 =p22 ¼ λ 1 þ 12 ð47Þ x21 p22 Consequently, in order to satisfy condition (8), the inequality ζx21 r λð1 þ ðp12 =p22 Þ2 Þx21 should be satisﬁed, which determine the acceptable range for λ (i.e., λ r ζ=ð1 þðp12 =p22 Þ2 Þ). Also, it is worth noting that if the chosen Lyapunov function to be multiplied by a positive scalar constant, the resulting controller will be unchanged. This is because that when the Lyapunov function is multiplied by a scalar constant, both a and b functions will be scaled with that constant. Therefore, considering the structure of the control law (11), both nominator and denominator of the control law will be multiplied with a same coefﬁcient. Consequently, one of the parameters of the Lyapunov function (42) may be selected without loss of generality. Assume that p22 ¼ 1 is selected to simplify the above mentioned relations. For other free parameters, we have the following constraints:

p11 should be a positive value. p12 should be a positive value smaller than pﬃﬃﬃﬃﬃﬃﬃ p11 (in order to 2 p11 p22 p12

The last parameter which should be determined in nominal phase design, is α A ð0; 1Þ. This parameter should be chosen such that the inﬁmum in (10) is positive. It may be investigated by using numerical methods that for λ ¼ 2:5 and x A D this inﬁmum is positive for α r 0:52.

4.2. Design of the robust controller Considering the state space Eq. (38) with the uncertain term d a 0. The additional feedback term (v), will be designed to suppress this uncertainty. According to (2) and (40) the nonnegative function, ρðxÞ, may be chosen as, ρðxÞ ¼ jx2 j. Moreover, by choosing, ηðxÞ ¼ η0 ¼ supjx2 j ¼ 1, the condition, ηðxÞ Z ρðxÞ, is also satisﬁed for all x A D. Therefore, the additional part v, is as follows (according to (12)): (

Since, the matrix P is positive deﬁnite and symmetric, the coefﬁcients p11 ; p22 and p11 p22 ðp12 Þ2 are positive. In the Eq. (45), it is evident that if p12 be also a positive value, the coefﬁcient of ð x21 Þ is positive. Therefore, there is a positive constant ζ, such that,

377

v¼

sgnðbÞ; bε ;

j bj Z ε j bj o ε

ð48Þ

where b ¼ ξ3 ðp22 x2 þ p12 x1 Þ. Finally, in order to guarantee the asymptotic stability of the closed-loop system, the necessary condition on ε may be evaluated according to Theorem 3. Refer to the function γð:Þ which was chosen, previously ( ¼ 2:5ðx21 þ x22 Þ), by considering ϕ2 ðxÞ ¼ ðx21 þ x22 Þ and ρ1 ¼ 1, the inequalities (26) are satisﬁed. Also, set α ¼ β ¼ 0:5, and p ¼ 1 then ε o 4αð1 βÞη20 =ðpρ21 Þ ¼ 4 0:5 0:5 1=1 ¼ 1. Thus according to Theorem 3, by choosing ε o1, the feedback law u ¼ kðx; θÞ þ v guarantees the asymptotic stabilization of the uncertain nonlinear system (38).

4.3. Computer simulations Computer simulations are presented to show the performance of the designed controller in robust asymptotic stabilization with acceptable characteristics of the transient responses of the state variables and the control signal. Simulation results are illustrated in Figs. 2–5. The time response of state variables for the closedloop system with d ¼ 0; u ¼ kðx; θÞ and d a 0; u ¼ kðx; θÞ þ v are shown in Figs. 2 and 4, respectively. Also, Figs. 3 and 5 demonstrate the time response of the nominal controller (i.e., u ¼ kðx; θÞ) and the robust controller (i.e., u ¼ kðx; θÞ þ v). The cost value is 11.4 where d ¼ 0; u ¼ kðx; θÞ and 12.3 where d a 0; u ¼ kðx; θÞ þ v. Since, conquering an unknown uncertainty in the system needs additional energy consuming, thus adding the term v to the control law, although, makes the controller robust; the cost function will be increased with respect to the nominal system.

be positive).

In order to ﬁnd the best values for p11 and p12 (with considering the cost function (41)), two nested loops were used (in the written computer program) where the outer loop varies for positive value pﬃﬃﬃﬃﬃﬃﬃ of p11 and in the inner loop p12 varies from zero to p11 . The minimum cost function was achieved for p11 ¼ 11:9 and p12 ¼ 0:3. Now by substituting p11 ¼ 11:9, p12 ¼ 0:3 and p22 ¼ 1 in (44), the functions aðx; θðtÞÞ and bðx; θðtÞÞ are determined and also the nominal controller u ¼ kðx; θðtÞÞ can be constructed with substituting aðx; θðtÞÞ and bðx; θðtÞÞ from (44) and q from (41) in (11). Also, for these values of pij 's, the functions aðx; θðtÞÞ and γð‖x‖Þ, in the points where bðx; θðtÞÞ ¼ 0, are easily evaluated as aðx; θðtÞÞ

jx2 ¼ 0:3x1 ¼ 3:543x21 and γð‖x‖Þ x ¼ 0:3x ¼ 1:09λx21 . Therefore, in 2

1

order to satisfy condition (46), the positive constant ζ may be chosen as ζ ¼ 3:543, and consequently the acceptable range for λ is λ r ζ=1:09 r3:25.

Fig. 2. Time response of state variables for the closed-loop system with d ¼ 0; u ¼ kðx; θÞ.

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simulations showed the performance of the designed controller in robust asymptotic stabilization with acceptable characteristics of the transient responses of the state variables and the control signal.

Appendix Consider the following nonlinear time-varying system: x_ ¼ f ðx; tÞ

ðA1Þ

Theorem 4. Suppose that V : D 0 1 -R be a continuously differentiable function such that for every ðx; t Þ A D 0; 1 , Fig. 3. Time response of the nominal controller u ¼ kðx; θÞ.

α1 ð‖x‖Þ r Vðx; tÞ r α2 ð‖x‖Þ ∂V ∂V þ f ðx; tÞ r W 3 ðxÞ ∂t ∂x

ðA2Þ 8 ‖x‖ Z μ 4 0

ðA3Þ

where functions α1 ð:Þ and α2 ð:Þ are class K functions and W 3 ðxÞ is a continuous positive deﬁnite function. Take r 4 0, such that Br ¼ fx A Rn ; ‖x‖ r rg D and suppose that ‖xðt 0 Þ‖ rα2 1 ðα1 ðrÞÞ. Then, there exist a class KL function βð:; :Þ and T Z 0 (T is dependent on xðt 0 Þ and μ where μo α2 1 ðα1 ðrÞÞ ), such that the solution of (A1) satisﬁes ‖xðtÞ‖r βð‖xðt 0 Þ‖; t t 0 Þ;

8 t 0 rt r t 0 þT

‖xðtÞ‖r α2 1 ðα1 ðμÞÞ;

8 t Z t0 þ T

Proof. See [1].

ðA4Þ

□

References

Fig. 4. Time response of state variables for the closed-loop system with d a 0; u ¼ kðx; θÞþ v.

Fig. 5. Time response of the robust controller u ¼ kðx; θÞ þ v.

5. Conclusion In this paper, based on the SVCLF method, a robust stabilizing control law for uncertain nonlinear slowly-varying systems was proposed. Using the idea of Lyapunov redesign method, an additional term was added to the SVCLF controller such that the overall feedback law guaranteed the asymptotic stabilization of the uncertain nonlinear slowly-varying systems. Moreover, some sufﬁcient conditions were proposed to warranty the asymptotic stabilization without chattering. Finally, in order to show the efﬁciency and applicability of the proposed method, it was applied to the nonlinear time-varying inertia pendulum. Computer

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