Identification of uncertain nonlinear systems for robust fuzzy control

ISA Transactions 49 (2010) 27–38

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Identification of uncertain nonlinear systems for robust fuzzy control D. Senthilkumar ∗ , Chitralekha Mahanta Department of Electronics and Communication Engineering, Indian Institute of Technology, Guwahati, Guwahati-781039, Assam, India

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Article history: Received 26 November 2008 Received in revised form 16 July 2009 Accepted 31 July 2009 Available online 14 August 2009 Keywords: Linear programming Parametric uncertainties Interval fuzzy model Takagi–Sugeno (T–S) fuzzy model

abstract In this paper, we consider fuzzy identification of uncertain nonlinear systems in Takagi–Sugeno (T–S) form for the purpose of robust fuzzy control design. The uncertain nonlinear system is represented using a fuzzy function having constant matrices and time varying uncertain matrices that describe the nominal model and the uncertainty in the nonlinear system respectively. The suggested method is based on linear programming approach and it comprises the identification of the nominal model and the bounds of the uncertain matrices and then expressing the uncertain matrices into uncertain norm bounded matrices accompanied by constant matrices. It has been observed that our method yields less conservative results ˘ than the other existing method proposed by Skrjanc et al. (2005) [11,12]. With the obtained fuzzy model, we showed the robust stability condition which provides a basis for different robust fuzzy control design. Finally, different simulation examples are presented for identification and control of uncertain nonlinear systems to illustrate the utility of our proposed identification method for robust fuzzy control. © 2009 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction Fuzzy control is increasingly applied for uncertain and illdefined nonlinear systems and strongly competes with other nonlinear control techniques. Several significant research results were reported in the literature which utilizes an uncertain fuzzy model to deal with robust fuzzy control problems [1–6]. Generally an affine T–S fuzzy model employs a constant term in the consequent part of each rule. But most of aforementioned literature [1–6] utilizes a special type of T–S fuzzy model having a linear rule consequence without the constant term for robust fuzzy control of uncertain nonlinear systems. In these robust fuzzy control methodologies, the controller design is carried out using a class of fuzzy system represented in the Takagi–Sugeno form with uncertainty blocks. Apart from robust fuzzy control, these models are also used for fault identification in uncertain nonlinear systems [7]. In the above cited literature, the uncertain fuzzy model construction incorporates the experience or human knowledge to express the uncertainties. This may be suitable for small systems or systems with sector nonlinearity like a mass–spring system. But most systems have severe nonlinearity and uncertainties which add difficulty to the identification technique. In some of the control design literature [1], the parameters of the nominal fuzzy model are assumed to have certain amount of perturbation and these

∗

Corresponding author. E-mail addresses: [email protected], [email protected] (D. Senthilkumar), [email protected] (C. Mahanta).

are not estimated from the uncertainty of the original system. Therefore, for these applications, the interval fuzzy identification of uncertain nonlinear systems has become an important topic of scientific research. The approximation capability of a fuzzy system is discussed in [8] and it concludes that any nonlinear function can be approximated within an error bound if we use sufficient number of rules. This approximation error will have some influence on the control performance and this band of error is usually added to the uncertainty block of the fuzzy model. Taniguchi et al. [9] presented a systematic method for fuzzy control that includes fuzzy identification, rule reduction and robust compensation for nonlinear systems. Here, the uncertain blocks compensate for modeling error. Lo and Lin [10] presented robust H∞ nonlinear modeling and control via uncertain fuzzy systems. Here, the uncertainties are expressed using non-fuzzy uncertain bounding ˘ matrices. Skrjanc et al. proposed a methodology for interval fuzzy model identification to approximate functions from a finite set of input–output measurements [11–13]. This identification method uses the concepts from linear programming and it provides a lower and upper fuzzy model which enclose the whole band of uncertainties. The approximation capability of this method is explained with a first order (affine) fuzzy model and another singleton fuzzy model. In this paper, we consider fuzzy model identification of uncertain nonlinear systems suitable for robust fuzzy control. We assume that the parameters of the antecedent part are available or the input space is uniformly partitioned for the parameters of the membership functions [6,14]. We consider the identification of the consequent part only. The fuzzy model is expressed with linear terms and uncertain terms, which represent the nominal

0019-0578/$ – see front matter © 2009 ISA. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.isatra.2009.07.005

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D. Senthilkumar, C. Mahanta / ISA Transactions 49 (2010) 27–38

system and parametric uncertainties respectively. We identify the nominal model and the bounds of uncertain terms of the fuzzy ˘ model and it is based on the method proposed by Skrjanc et al. [11]. The model identified in this step cannot be directly employed in robust fuzzy control design and it must be expressed in another special form. The terms of the uncertain matrices are expressed in the form suitable for robust fuzzy control with norm bounded uncertain matrices accompanied by some constant real matrices and we have shown that the obtained model is suitable for robust fuzzy control of uncertain nonlinear systems. This paper is organized as follows. Section 2 describes the interval fuzzy model representation for use in robust fuzzy control problems. In Section 3, the fuzzy model identification method is ˘ presented and it is compared with the method proposed by Skrjanc et al. [11]. In Section 4, we have shown that the obtained model can be used for robust fuzzy control of uncertain nonlinear systems. Illustrative examples for the proposed method are presented in Section 5. Section 6 contains our conclusions. 1.1. Notations

where

The continuous fuzzy model proposed by Takagi and Sugeno [16] represents the dynamics of nonlinear system using fuzzy IF-THEN rules. As in [1–6], rules for the typical fuzzy model of an uncertain nonlinear system employed in robust control design are of the following form: Plant rule i: IF z1 (t ) is Ni1 and z2 (t ) is Ni2 and . . . zp (t ) is Nip THEN x˙ (t ) = (Ai + ∆Ai )x(t ) + (Bui + ∆Bui )u(t ) + (Bwi + ∆Bwi )w (t ) y (t ) = (Ci + ∆Ci )x(t ) + (Dui + ∆Dui )u(t ) i = 1, 2, . . . , r ,

(1)

where z1 (t ), . . . , zp (t ) are premise variables, p is the number of premise variables, Nij (j = 1 . . . p) is the fuzzy set and r is the number of rules. Here, x(t ) = [x1 (t ), . . . , xn (t )]T ∈ Rn is the state vector, u(t ) = [u1 (t ), . . . , umu (t )]T ∈ Rmu is the input vector, y (t ) = [y1 (t ), . . . , yq (t )]T ∈ Rq is the output vector, w (t ) = [w1 (t ), . . . , wmw (t )]T ∈ Rmw is the disturbance input vector. Ai ∈ Rn×n , Bui ∈ Rn×mu , Bwi ∈ Rn×mw , Ci ∈ Rq×n , Dui ∈ Rq×mu , Dwi ∈ Rq×mw are constant real matrices describing the nominal system and ∆Ai , ∆Bui , ∆Bwi , ∆Ci , ∆Dui and ∆Dwi are time varying matrices of appropriate dimensions, which represent parametric uncertainties in the plant and modeling errors. Given a pair of input and output (x(t ), u(t )), the final output of the fuzzy system is inferred as follows [4]: r X

 µi (z (t )) (Ai + ∆Ai )x(t ) + (Bui + ∆Bui )u(t )

i =1

+ (Bwi + ∆Bwi )w (t ) r X  y (t ) = µi (z (t )) (Ci + ∆Ci )x(t ) + (Dui + ∆Dui )u(t ) i =1

+ (Dwi + ∆Dwi )w (t ) ,

p Y

Nij (zj (t )),

(3)

j=1

j =1

and Nij (zj (t )) is the degree of membership of zj (t ) in the fuzzy set N Pr ωi (z (t )) are ωi (z (t )) ≥ 0, Pij .r Some basic properties of µi (z (t )) and ω ( z ( t )) > 0, µ ( z ( t )) ≥ 0 and j i j=1 µj (z (t )) = 1. j =1 In robust control design problems, the uncertain matrices are assumed to be norm bounded and are described by:



∆Ai ∆Ci

∆Bui ∆Dui

   Hxi Fxi (t )(Ex1i Ex2i Ex3i ) ∆Bwi = , ∆Dwi Hyi Fyi (t )(Ey1i Ey2i Ey3i )

(4)

where Hxi , Hyi , Ex1i , Ex2i , Ex3i , Ey1i , Ey2i and Ey3i are known real constant matrices of the appropriate dimension and Fxi (t ), Fyi (t ) are time varying matrix functions with Lebesgue-measurable elements, satisfying FxiT (t )Fxi (t ) ≤ I , and FyiT (t )Fyi (t ) ≤ I .

3.1. Identification with homogenous fuzzy functions Let us consider an uncertain nonlinear function g (v , γ(t )), in which v = [v1 , . . . , vnv ]T ∈ Rnv is the input vector, γ(t ) = [γ1 (t ), . . . , γnγ (t )] ∈ Rnγ is a time varying uncertain vector with known lower and upper bounds γ , γ k of γk (t ) satisfying γ ≤ k k γk (t ) ≤ γ k , k = 1, . . . , nγ . We define the uncertain fuzzy model in the following form

2. Fuzzy model with uncertainty

x˙ (t ) =

ωi (z (t )) =

3. Interval fuzzy model identification

Lower case letters a, b, . . . ∈ R are used to denote real numbers. a and a denote the lower and upper bounds of the real valued closed interval [a, a] respectively. Lower and upper bounds of interval valued functions are denoted by f (x) and f (x) respectively. The simultaneous use of overbars and underbars represents an interval valued quantity, eg. x, f (x) and these are referred from [15]. The symbol ‘∗’ is used to represent blocks that are readily inferred by symmetry.

+ (Dwi + ∆Dwi )w (t ),

ωi (z (t )) , r P ωj (z (t ))

µi (z (t )) =

(2)

ψ(v , t ) =

r X

µi (z (t ))(θ i + ∆θ i (t ))T v ,

(5)

i=1

where θ i = [θi1 , θi2 , . . . , θinv ] is a constant real vector representing the nominal model of the system and ∆θ i (t ) = [∆θi1 (t ), ∆θi2 (t ), . . . , ∆θinv (t )] is an interval vector which represents the parametric uncertainties satisfying |∆θik (t )| ≤ δθik , k = 1, . . . , nv . The upper and lower bounds of the uncertain parameters of the fuzzy model will be δθ Ti |v | and −δθ Ti |v |, where |v | represents the absolute value of v. Now we can define the lower and upper bounds of the fuzzy model as

ψ(v ) =

r X

µi (z )(θ Ti v − δθ Ti |v |)

(6)

µi (z )(θ Ti v + δθ Ti |v |).

(7)

i =1

ψ(v ) =

r X i =1

We now consider the following relation from [11], for identification of interval fuzzy model from a finite set of input–output measurements V = [v1 , . . . , vN ], where vj (j = 1, . . . , N ) is the set of inputs collected from the compact set S:

ψ(vj ) ≤ g (vj , γ(t )) ≤ ψ(vj ),

∀ j.

(8)

Let us define an interval valued function g (vj ), satisfying g (vj ) ≤ g (vj , γ(t )) ≤ g (vj ), where g (vj ) and g (vj ) represent the lower and upper bounds of g (vi ). The interval valued function g (vj ) can be easily constructed by replacing the uncertain quantities with interval variables having bounds equal to that of the uncertain term. If interval valued function g (vj ) satisfies the following inequalities (9) and (10), then the lower and upper fuzzy model will also satisfy the relation given in (8).

ψ(vj ) ≤ g (vj ),

∀j

(9)

g (vj ) ≤ ψ(vj ),

∀ j.

(10)

D. Senthilkumar, C. Mahanta / ISA Transactions 49 (2010) 27–38

If ej and ej are the approximation errors related to the lower and upper fuzzy model then ej = g (vj ) − ψ(vj ),

∀j

(11)

ej = ψ(vj ) − g (vj ),

∀ j.

(12)

If the maximum approximation error related to the fuzzy model and the family of functions is defined as

 λ = max ej + ej ,

(13)

vj ∈V

then ej + ej ≤ λ,

∀ j.

(14)

With (11) and (12), (14) can be written as: g (vj ) − ψ(vj ) + ψ(vj ) − g (vj ) ≤ λ,

∀ j.

(15)

With the inequalities (9), (10) and (14), the optimization problem for identification of the lower and upper fuzzy function can be framed as: min λ

(16)

ψ,ψ

subject to g (vj ) − ψ(vj ) + ψ(vj ) − g (vj ) ≤ λ,

ψ(vj ) − g (vj ) ≤ 0,

j = 1, . . . , N

g (vj ) − ψ(vj ) ≤ 0,

j = 1, . . . , N .

j = 1, . . . , N

With the lower and upper fuzzy model defined in (6) and (7), the above optimization problem can be formulated as following convex linear programming problem min λ

(17)

θj ,δθj

subject to g (vj ) − g (vj ) + 2

r X

µi (z )(δθ Ti |vj |) ≤ λ,

j = 1, . . . , N

i =1

−g (vj ) +

r X

µi (z )(θ

T i vj

− δθ |vj |) ≤ 0,

j = 1, . . . , N

T i

i =1

g (vj ) −

r X

µi (z )(θ Ti vj + δθ Ti |vj |) ≤ 0,

j = 1, . . . , N

i =1

and

−δθik ≤ 0,

i = 1, . . . , r , k = 1, . . . , nv

where λ is a slack variable that represents the maximum approximation error corresponding to the lower and upper fuzzy model. Guidelines for selection of input–output data: In the proposed identification method, the model is built using input–output data to replicate the behavior of a given function. A reasonable amount of input–output data needs to be chosen from the input range. The data can be chosen randomly in the input space or it can be obtained at equidistant points. The more the number of data points, the better the representation of the dynamics of the original nonlinear equation will become. To decide the number of data points, one can start deriving the model with a smaller number of input–output data and again try with a greater number of input–output data until a consistent model is obtained. Remark 1. The interval fuzzy model approximates the dynamics captured by the input–output data and the approximation may be invalid in some regions where the identification data is not present. This limitation is explained in [11] and the same is applicable to the above fuzzy model identification method.

29

Remark 2. We have assumed that the parameters of the antecedent part is available or the input space is uniformly partitioned [6,14] for the parameters of the antecedent part. A fuzzy model with narrow error band might be possible if the input space is partitioned in an optimal way. By considering the nominal model of the given system, one can identify the antecedent part of the fuzzy model using the approach in [17–19] or by considering the different sectors and applying the method explained in [9,20]. By considering the same antecedent parameters for the uncertain fuzzy model, the consequent part can be obtained by the method described in this section. 3.1.1. Example Similar to the uncertain nonlinear function in [11], we consider the class of G with gnom (v) = cos(v) sin(v) and the uncertainty ∆g (v) = γ sin(8v), 0 ≤ γ ≤ 0.2. This function is similar to the one considered in [11] except the presence of a sine function in the uncertainty part instead of a cosine function and this will make the class of G satisfy g (0) = 0. The functions from the class are defined in the domain S = {v|− 1 ≤ v ≤ 1} and the set of ‘‘measurements’’ is V = {vj |vj = 0.021k, k = −47, −46, . . . 47} ⊂ S. Let us consider the problem of finding the upper and lower fuzzy model in homogenous form for the uncertain nonlinear function g (z , γ (t )). The dimensionality of the input space is 1 and therefore the premise and consequent variables are same as the measurements, i.e., zj = vj , j = 1, . . . , N. Similar to [11], we considered eight and another seven triangular and equidistant membership functions (Set I and II) as shown in Fig. 1. Here, the lower and upper fuzzy model take the following form: R i : IF z is Ni THEN ψ = θi1 v − δθi1 |v|, i = 1, . . . , r, i i

R i : IF z is Ni THEN ψ = θi1 v + δθi1 |v|, i = 1, . . . , r. The fuzzy model is constructed for the two cases of membership function in Fig. 1 and the results are shown in Figs. 2(a) and 4(a), where the dashed line represents the family of functions and the solid line shows the region enclosed by the lower and upper fuzzy model. The approximation errors supg ∈G (ψ(v) − g (v)) and infg ∈G (ψ(v) − g (v)) are presented in Figs. 3(a) and 5(a). For comparison purpose, we consider the method proposed ˘ by Skrjanc et al. [11]. We consider fuzzy model identification using a l1 norm and hence we introduce the slack variables λj ,

λj , j = 1, . . . , N. The interval fuzzy model is identified by the following linear programming problem of minimizing the sum of the absolute value of the estimation errors: N X

λj +

N X

j =1

λj

(18)

j =1

subject to inf(g (vj )) −

r X

µi (z )θ Ti vj ≤ λj ,

j = 1...N

(19)

i =1

inf(g (vj )) −

r X

µi (z )θ Ti vj ≥ 0,

j = 1...N

(20)

i =1

λj ≥ 0,

j = 1...N

(21)

and

− sup(g (vj )) +

r X

T

µi (z )θ i vj ≤ λj ,

j = 1...N

(22)

i=1

sup(g (vj )) −

r X

T

µi (z )θ i vj ≤ 0,

j = 1...N

(23)

i=1

λj ≥ 0,

j = 1 . . . N.

(24)

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D. Senthilkumar, C. Mahanta / ISA Transactions 49 (2010) 27–38

(a) Set – I.

(b) Set – II. Fig. 1. Membership function.

a

b

Fig. 2. Data, lower and upper bound of the fuzzy model using membership function set – I (a) by the proposed method, (b) by the method in [11].

a

b

Fig. 3. Difference between the bounds of the fuzzy function and the actual envelope of the family of functions (Membership function set – I) (a) by the proposed method, (b) by the method in [11].

a

b

Fig. 4. Data, lower and upper bound of the fuzzy model using membership function set – II (a) by the proposed method, (b) by the method in [11].

D. Senthilkumar, C. Mahanta / ISA Transactions 49 (2010) 27–38

a

31

b

Fig. 5. Difference between the bounds of the fuzzy function and the actual envelope of the family of functions (Membership function set – II) (a) by the proposed method, (b) by the method in [11].

In the above minimization problem, we considered inf(g (vj )) and sup(g (vj )) to minimize the difference between the bounds of the family of functions and the fuzzy model. The identification results for the constructed interval fuzzy model are shown in Figs. 2(b) and 4(b). The solid line represents the upper and lower fuzzy model while the dashed set of lines represents the family of functions G for some values of γ . The approximation errors supg ∈G (ψ(v) − g (v)) and infg ∈G (ψ(v) − g (v)) are presented in Figs. 3(b) and 5(b). We can observe that the approximation error is very high in the region near the origin. When compared ˘ to the results of Skrjanc et al. [11], the upper and lower fuzzy model obtained using the proposed linear programming problem encloses the region closely and it is observed that the model approximates the family of functions in a better way (less conservative result). In the above case, we considered an objective function (l1 norm) different from the objective function used in [11]. We checked the results of the fuzzy modeling for the objective function (l∞ -norm) in [11], and here also we observed a very high error in the region around the origin (not shown). The number of linear inequality conditions for solving the problem by our approach is 3N + rnv and for the approach in [11] (with l∞ -norm), is 4N +2. For the above example with membership function set – I and II, the number of linear inequality conditions by our approach is 3 × 95 + 8 = 293 and 3 × 95 + 7 = 292 respectively. With the approach in [11] the number of linear inequality conditions is 4 × 95 + 2 = 382 for both set – I and II. The CPU time for solving the proposed linear programming problem for the membership function shown in Fig. 1 is 3.12 s and 3.04 s for set - I and set - II respectively. For the approach in [11] with l∞ -norm, the CPU time is 3.35 s and 3.32 s for membership function set - I and set - II respectively. 3.2. Lower and upper bounds estimation of interval valued function The linear programming problem explained in the previous subsection for interval fuzzy model identification can be solved only if we know the lower bound g (vj ) and upper bound g (vj ) of the interval valued function g (vj ). Depending upon the uncertain nonlinear system, the interval valued function can be linear or nonlinear. If it is linear, it can be solved easily by inequality constrained linear programming approach. In case of nonlinear interval valued functions, the bounds can be estimated from the methods addressed in the literature [21–24]. Some of the basic interval arithmetic rules involved in interval computation on a and b are listed below: Addition: a + b = [a + b, a + b]

(25)

Subtraction: a − b = [a − b, a − b]

(26)

Multiplication: a b = [inf(ab, ab, ab, ab), sup(ab, ab, ab, ab)]

(27)

Division: a/b = [a, a] [1/b, 1/b].

(28)

The bounds (with some overestimation) of the interval valued function can be calculated using the above interval arithmetic rules. Closer bounds for the interval function can be found using a branch and bound technique [21–24]. The bounds of the interval valued function can be found using the interval arithmetic rules combined with a branch and bound technique which splits boxes adaptively until the overestimation becomes insignificant. The main advantage of this technique is that the bounds obtained are global, but this technique suffers from more computation time and hence the suitability of this approach is restricted to small problems. It is also possible to find the bounds for the interval function involving nonlinear operations on interval a like, sin(a), cos(a), etc. The readers are referred to the literature [21–24] for more details on interval analysis using a branch and bound algorithm. 4. Application to robust fuzzy control The interval fuzzy model obtained using the linear programming method explained in the previous section is not suitable for controller design for robust fuzzy control. The uncertain fuzzy model employed in robust fuzzy control must be expressed in the form explained using (2) and (4). Consider the following nonlinear system described by x˙ (t ) = gx (x(t ), u(t ), w (t ), γ(t )) y (t ) = gy (x(t ), u(t ), w (t ), γ(t )),

(29)

where x(t ) = [x1 (t ), . . . , xn (t )] ∈ R is the state vector, u(t ) = [u1 (t ), . . . , umu (t )]T ∈ Rm is the input vector, y (t ) = [y1 (t ), . . . , yq (t )]T ∈ Rq is the output vector, w (t ) = [w1 (t ), . . . , wmw (t )]T ∈ Rmw is the disturbance input vector, γ(t ) = [γ1 (t ), . . . , γnγ (t )]T ∈ Rnγ is the uncertain vector with known lower bound γ and upper k bound γ k satisfying γ ≤ γk (t ) ≤ γk , (k = 1, . . . , nγ ). T

n

k

With v (t ) = [xT (t ), uT (t ), w T (t )]T , the functions in (29) can be expressed in the following form g (v (t ), γ(t )) =



gx (v (t ), γ(t )) . gy (v (t ), γ(t ))



(30)

Suppose g (v (t ), γ(t )) is written as [g1 (v (t ), γ(t )), . . . , g(n+q) (v (t ), γ(t ))]T and satisfying the condition gl (0, γ(t )) = 0,

l = 1, . . . , n + q

(31)

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D. Senthilkumar, C. Mahanta / ISA Transactions 49 (2010) 27–38

then according to the concept presented in [25], gl (v (t ), γ(t )), l = 1, . . . , n + q can be approximated by an interval fuzzy model of the form

ψl (v , t ) =

r X

µi (z (t ))(θ li + ∆θ li (t ))T v ,

l = 1, . . . , n + q. (32)

i=1

where ∆θlijk (t ) = hlijk flijk (t )elijk . Let us assume that hlijk and elijk take only positive values; then the lower and upper bounds of hlijk flijk (t )elijk are given by −hlijk elijk and hlijk elijk . If hlijk and elijk satisfy the relation in (39), then the band of ∆θlijk (t ) can always be found between the lower and upper bounds of hlijk flijk (t )elijk :

With (30) and (32), the overall fuzzy model for the nonlinear system (29) can be written as follows:

− hlijk elijk ≤ ∆θlijk (t ) ≤ hlijk elijk .

x˙ (t ) = [ψ1 (v , t ), . . . , ψn (v , t )]

With the above inequality, the relation between the lower and upper bounds of ∆θlijk (t ) and hlijk flijk (t )elijk is given by

y (t ) = [ψn+1 (v , t ), . . . , ψn+q (v , t )].

(33)

(39)

The terms θ li and ∆θ li (t ) corresponding to (32) and (33) represent the system matrices and uncertain matrices of the fuzzy model (2) and are given by

max ∆θlijk (t ) = δθlijk = hlijk elijk ,

 T θ xi = θ T1i , . . . , θ Tni = [Ai Bui Bwi ] , T  θ yi = θ T(n+1)i , . . . , θ T(n+q)i = [Ci Dui Dwi ] ,  T ∆θ xi (t ) = ∆θ T1i (t ), . . . , ∆θ Tni (t ) = [∆Ai (t ) ∆Bui (t ) ∆Bwi (t )] ,  T ∆θ yi (t ) = ∆θ T(n+1)i (t ), . . . , ∆θ T(n+q)i (t ) = [∆Ci (t ) ∆Dui (t ) ∆Dwi (t )] . (34) The parameters θ xi , θ yi and the bounds δθ xi , δθ yi of ∆θ xi (t ), ∆θ yi (t ) can be found from the linear programming problem

Here δθlijk ≥ 0. To satisfy the above condition in (40), we assume p hlijk = elijk = δθlijk . Let us consider the uncertain terms in the results of robust fuzzy control [1,2,27,3–6,28]. The terms involving the uncertainties are expressed in any one of the following form

explained in the previous section. But the uncertain term needs to be expressed in a special form shown in (4) for application in robust fuzzy control. Let us assume that the matrices in the right side of the Eq. (4) take the following form hxi1  0



Hxi =   ..

0 hxi2

.

0 hyi1  0



Hyi =   .. .

0 0 hyi2

...

0 0 



..  , .

..

. ...

hxin

...

(35)

0 0 



..  , .

..

. . . . hyiq Fxi (t ) = diag(fxi11 (t ), fxi12 (t ), . . . , fxijk (t ), . . .), j = 1, . . . , n, k = 1, . . . , (n + mu + mw ), Fyi (t ) = diag(fyi11 (t ), fyi12 (t ), . . . , fyijk (t ), . . .), j = 1, . . . , q, 0

0

k = 1, . . . , (n + mu + mw ),

(36)

Exi = [Ex1i Ex2i Ex3i ] = [exi1 exi2 . . . exin ]T , Eyi = Ey1i Ey2i Ey3i = eyi1 eyi2 . . . eyin







T

,

(37)

where hxij = [hxij1 hxij2 . . . hxij(n+mu +mw ) ] and hyij = [hyij1 hyij2 . . . hyij(n+mu +mw ) ]. The vector 0 has a length n + mu + mw with all zero entries. In (37), exij = diag(exij1 , exij2 , . . . , exijn ) and eyij = diag(eyij1 , eyij2 , . . . , eyijn ). The entries of matrices in (36) satisfy the condition |fxijk (t )| ≤ 1, and |fyijk (t )| ≤ 1. The above form of expressing the matrices of the uncertain term in the fuzzy model is inspired from [26]. With (35), (36), (37) and defining l ∈ {x, y}, the matrices ∆θ xi (t ) and ∆θ yi (t ) can be expressed as ∆θ li (t ) = Hli Fli (t )Eli and

∆θ (t ) ∆θ (t ) . . . li11 li12 ∆θli21 (t ) ∆θli22 (t ) . . .  .. .. ..  . . . ∆θ li (t ) =   ∆θ (t ) ∆θ (t ) lij2  lij1 .. .. . . k = 1, . . . , (n + mu + mw ), l = x, j = 1, . . . , n for ∆θxi (t ), l = y, j = 1, . . . , q for ∆θyi (t ),

∆θli1k (t ) ∆θli2k (t ) ∆θlijk (t )

. . .




(40)

min ∆θlijk (t ) = −δθlijk = −hlijk elijk .




T ∆AT (µ)P + P ∆A(µ) = ε E1x (µ)E1x (µ) +

1

ε

PHx (µ)HxT (µ)P ,

T or ∆AT (µ)P + P ∆A(µ) = E1x (µ)E1x (µ) + PHx (µ)HxT (µ)P (41)

where P is the Lyapunov variable and ε is some positive scalar. Other uncertain terms (∆Bui , ∆Bwi , ∆Ci , ∆Dui , ∆Dwi ) are also transformed in a similar way. The Lyapunov variable P is a control design variable and usually it is not available p at the time of modeling. The assumption hlijk = elijk = δθlijk may give a conservative result while designing the controller. Hence, another variable i is introduced and the robust stability condition is shown in the next subsection. 4.1. Robust stability condition Let us consider a nonlinear system which can be described by the T–S fuzzy model given below: Plant rule i: IF z1 (t ) is Ni1 and . . . zp (t ) is Nip , THEN x˙ (t ) = (Ai + ∆Ai )x(t ) + (Bi + ∆Bi )u(t ) + w (t ) y (t ) = Ci x(t ),

i = 1, 2, . . . , r ,

(42)

where Nij is the fuzzy set, x(t ) ∈ R is the state vector, u(t ) ∈ Rmu is the control input, w (t ) ∈ Rmw is the unknown but bounded disturbance input and y (t ) ∈ Rq is the output vector. Here, Ai ∈ Rn×n , Bi ∈ Rn×mu and Ci ∈ Rq×n , ∆Ai and ∆Bi are time-varying matrices with appropriate dimensions and z1 (t ), z2 (t ), . . . , zp (t ) are premise variables. The parameters of the uncertain matrices are assumed to be expressed in the form given in (35)–(37). Given a pair of (x(t ), u(t )), the final output of the fuzzy system is inferred as follows: n

x˙ (t ) =

r X

µi (z (t ))[(Ai + ∆Ai )x(t ) + (Bi + ∆Bi )u(t ) + w (t )]

i=1

y (t ) =

r X

µi (z (t ))Ci x(t ),

(43)

i =1

. . .      .. .

where

µi (z (t )) =

ωi (z (t )) , r P ωi (z (t ))

ωi (z (t )) =

p Y

Nij (zj (t )),

j =1

i=1

(38)

and Nij (zj (t )) is the degree of the membership Pr of zj (t ) in Nij . Therefore, µi (z (t )) ≥ 0, for i = 1, 2, . . . , r and i=1 µi (z (t )) > 0 for all t.

D. Senthilkumar, C. Mahanta / ISA Transactions 49 (2010) 27–38

Suppose the following fuzzy control rule is employed to stabilize the system represented by (43). Control rule i: IF z1 (t ) is Ni1 and . . . zp (t ) is Nip , THEN u(t ) = Ki x(t ), i = 1, 2, . . . , r . Then the overall fuzzy control law is represented by u(t ) =

r X

µi (z (t ))Ki x(t )

(44)

i=1

where Ki is the local control gain. The stability condition for the robust fuzzy state feedback control design is presented in the following Theorem: Theorem 1. Consider the fuzzy model (43) with the T–S state feedback control law (44). If there exists a symmetric and positive definite matrix P, diagonal matrices i , some matrices Kj , (j = 1, 2, . . . , r ) such that the following parameterized matrix inequality is satisfied: r X r X

 µi µj (Ai + Bi Kj )T P + P (Ai + Bi Kj )

i=1 j=1

1 + PHxi i HxiT P + (Ex1i + Ex2i Kj )T − i (Ex1i + Ex2i Kj ) ≤ 0

(45)

then the uncertain nonlinear system represented by (43) is globally stable. Proof. The proof is given in Appendix.



The above theorem gives the basic stability condition for an uncertain fuzzy system. Several approaches were discussed in the literature for solving these parametric inequality based problems with less conservative results [29–32]. Based on the concepts presented, the following algorithm is proposed for identification and robust fuzzy control design of uncertain nonlinear systems. Algorithm 1. Given the nonlinear equation of the form (29) representing the dynamics of the nonlinear system and the details about the membership functions, premise and antecedent variable, Step 1: Construct the vector V = {v1 , v2 , . . . , vN } that represents the input-measurement data. Find the upper and lower bounds of the nonlinear equations expressed in the form (29) with the concepts presented in the Section 3.2. Step 2: With the input-measurement data V and the bounds of the nonlinear equation obtained in the previous step, solve the linear programming problem (17) to find the parameters θ 1i , . . . , θ (n+q)i and δθ 1i , . . . , δθ (n+q)i corresponding to the nonlinear equations g1 , . . . , gn+q . If the nonlinear programming problem is not feasible, go to Step-5. Step 3: Arrange the elements of θ 1i , . . . , θ (n+q)i and δθ 1i , . . . , δθ (n+q)i in the form shown in (34) to get the system matrices and the bounds of the uncertain matrices. p Step 4: With hlijk = elijk = δθlijk , construct the matrices Hxi , Hyi , Exi and Eyi in the form shown in (35) and (37). Design the robust fuzzy controller using the concepts presented in Section 4.1. If the LMIs for the robust fuzzy control design are feasible, Stop. Step 5: Adjust the membership functions, rules and the premise and antecedent variables and again try to solve the problem using Algorithm or Stop. Remark 3. The vector g (v (t ), γ(t )) contains n + q individual equations and the nonlinearity may not be present in all individual equations. Some equations may be represented as linear combinations of the antecedent variables, e.g., x˙ 2 (t ) = x1 (t ) + x2 (t ). The sub-models corresponding to these equations can be directly written without solving the linear programming problems. Hence we can eliminate these simple linear equations and consider only the nonlinear equations while identifying the consequent part.

33

5. Illustrative examples 5.1. Inverted pendulum on a cart Consider the nonlinear equation with parameter uncertainties representing the equation of motion of an inverted pendulum on a cart given in [1,33]. x˙1 (t ) = x2 (t ) x˙2 (t ) =

gr sin (x1 (t )) − amlx22 (t ) sin (2x1 (t )) /2 − a cos (x1 (t )) u(t ) 4l/3 − aml cos2 (x1 (t ))

y(t ) = x1 (t ).

(46)

Here x1 (t ) and x2 (t ) represent the angular displacement about the vertical axis (in rad) and the angular velocity (in rad/sec) respectively, gr = 9.8 m/s2 is the acceleration due to gravity, a = 1/(m + M ), m ∈ [mmin mmax ] = [2 3] kg is the mass of the pendulum, M ∈ [Mmin Mmax ] = [8 10] kg is the mass of the cart, 2l = 1 m is the length of the pendulum and u is the force applied on the cart (in Newton). We consider the operating domain x1 (t ) ∈ [−π /3 π /3], x2 (t ) ∈ [−4 4] and the input u(t ) ∈ [−1000 1000]. We choose the membership functions and the rules as given below. Plant rule i: IF x1 is about Ni THEN x˙ (t ) = (Ai + ∆Ai )x(t ) + (Bi + ∆Bi )u(t ) y(t ) = Ci x(t ), where Ni is the triangular fuzzy set of x1 about 0, ±π /12, ±π /6, ±π /4, ±π /3 for i = 1, 2, . . . , 5 respectively. In the equation of motion of the nonlinear system, x˙1 (t ) and y(t ) are linear equations and the consequent part corresponding to these functions can be written directly. Only the consequent part corresponding to x˙ 2 (t ) needs to be identified. The matrices Ai and ∆Ai take the following form:

 Ai =

∆Ai =

0 ai21





1 , ai22



0 ∆ai21

Bi =



1 , ∆ai22



0 , bi21

∆Bi =





0 , ∆ai21



Ci = 1

0 .



With the uncertainty in m and M, the interval equation for estimating the bounds of x˙ 2 (t ) can be obtained as shown below: g =

gr sin (x1 ) − (m/(m + M ))lx22 sin (2x1 ) /2 − (1/(m + M )) cos (x1 ) u 4l/3 − (m/(m + M ))l cos2 (x1 )

.

(47) With the guidelines given in Section 3, we obtain the input data as V = {x1i |x1i = 0.0698l1 , l1 = −15, −14, . . . , 15, x2j |x2j = 2l2 , l2 = −2, −1, . . . , 2, uk |uk = 500l3 , l3 = −2, −1, . . . , 2} ⊂ S and the output data is estimated from the interval valued function (47). The linear programming problems was solved using SeDuMi [34] with YALMIP [35] interface. The uncertain fuzzy model is obtained by the algorithm presented in Section 4 and the parameters of the matrices Ai , Bi , δ Ai and δ Bi are given in Table 1. All the computations were performed on a Pentium IV 3.4 GHz processor with 1 GB RAM. The computation time for finding the bounds of the function was 69.29 s and the linear programming problem for finding the nominal model and the bounds of the fuzzy model was solved in 33.25 s. We consider the robust H∞ tracking control problem from [1]. We consider the same reference model and the reference input given in [1]:



x˙ r1 x˙ r2



 =

0 −4

1 −3









xr1 0 + . xr2 sin(t )

(48)

34

D. Senthilkumar, C. Mahanta / ISA Transactions 49 (2010) 27–38 1

0

0.8 –0.5 0.6 –1

0.4

–1.5

0.2 0

–2

–0.2 –2.5 0

5

10

15

0

20

(a) x1 (t ) and xr1 (t ).

5

10

15

20

(b) x2 (t ) and xr2 (t ).

Fig. 6. Trajectories of state variables x(t ) (dashed line: m = 2 kg and M = 8 kg, solid line: m = 3 kg and M = 10 kg) and the reference trajectories xr (t ) (dotted line) for an inverted pendulum. Table 1 Parameters of matrices Ai , Bi , δ Ai and δ Bi — Inverted pendulum on a cart. Rule i

a21

a22

b21

δ a21

δ a22

δ b21

1 2 3 4 5

16.2703 15.5785 14.9298 13.5178 12.1137

0 0 0 0 0

−0.1590 −0.1516 −0.1313 −0.1022 −0.0692

5.6613 3.3232 1.7417 1.2361 0.811

0.0062 0.0076 0.0204 0.0090 0.0040

0.0195 0.0179 0.0158 0.0127 0.0087

Table 2 Parameters of the feedback gain matrices Ki — Inverted pendulum.

Ki1 Ki2

i=1

i=2

i=3

i=4

i=5

780.0632 244.1326

800.652 250.8751

799.3351 247.0909

861.6303 266.3371

986.8387 304.1544

The H∞ tracking controller is designed with the concept presented in [1] with the disturbance input w = [0 ∆g sin(x1 )/(4l/3− aml cos2 (x1 ))]T , where ∆g = g (6370 × 103 /6370 × 103 + H ), H ∈ [0, 100]. The LMI optimization problem is solved using the LMI relaxations shown in Lemma 1. The parameters of the feedback gain matrices Ki = [Ki1 Ki2 ] obtained are given in Table 2. The simulation results for different values of m and M with the initial condition x(0) = [π /3 0]T and xr (0) = [0 0]T are shown in Fig. 6. 5.1.1. Comparison In the example presented in Section 3.1.1, the proposed method is compared with the method in [11]. In this subsection, another comparison is made using the above nonlinear system (Inverted pendulum on a cart). The parameters obtained by the proposed method for the homogenous fuzzy model are shown in Table 1. But in case of [11], the linear programming problem provides an infeasible solution for this example (inverted pendulum on a cart). Next, an affine fuzzy modeling problem is considered for building a fuzzy model of the same nonlinear system (inverted pendulum on a cart) with plant rules in the following form: Plant rule i: IF x1 is about Ni THEN x˙ (t ) = (a0i + ∆a0i ) + (Ai + ∆Ai )x(t ) + (Bi + ∆Bi )u(t ) y(t ) = Ci x(t ). Here a0i = [0 a20i ]T and ∆a0i = [0 ∆a20i ]T are the affine terms of the fuzzy model and the remaining terms are same as the homogenous fuzzy model. The parameters of the fuzzy model obtained by the proposed method method are shown in Table 3. By the method in [11], the lower fuzzy model (with affine terms) takes the following form: Plant rule i: IF x1 is about Ni THEN x˙ (t ) = a0i + Ai x(t ) + Bi u(t ) y(t ) = Ci x(t ).

Fig. 7. Equivalent circuit of a buck converter.

The upper fuzzy model takes a similar form as the lower fuzzy model. The linear programming problem from [11] is used to identify the parameters of the lower and upper fuzzy model and the results are shown in Table 4. In the robust fuzzy control literature [1–6] the fuzzy controller is designed with the homogenous fuzzy model. In this example, the method in [11] provides an infeasible solution for homogenous fuzzy function identification. Hence a homogenous fuzzy model cannot be derived using the method in [11]. But the method proposed in this paper provides a feasible solution for fuzzy model identification in homogenous as well as affine forms. Hence the proposed method has better approximation capability than the method in [11]. 5.2. Regulation of a PWM buck converter We next consider the regulation problem of a PWM buck converter. Consider the equivalent circuit of a PWM buck converter shown in Fig. 7. By the averaged modeling method, the dynamic equation is obtained as [36,37]:

˙iL (t ) = − 1 vC (t ) − 1 (RM iL (t ) − Vin − VD )du − VD

L L L 1 1 v˙ C (t ) = iL (t ) − vC (t ) C RC where iL is the inductor current, vC is the capacitor voltage, RM = 0.27  is the static drain to source resistance of the power MOSFET, VD = 0.82 V is the forward voltage of the power diode and du is the duty ratio of PWM buck converter. Other parameters are L = 0.09858 mH, C = 0.2025 mF and R = 6 . We consider some uncertainty in the input voltage Vin ∈ [Vin min Vin max ] = [27 33]V. With x(t ) = [x1 (t ) x2 (t )] = [iL (t ) vC (t )] and u(t ) = du , the uncertain nonlinear system can be represented as follows:



  x˙ 1 (t ) 0 = 1 x˙ 2 (t ) C

−1



L  x1 (t ) −1  x2 (t )





RC

# " # VD − ( RM x1 (t ) − Vin − VD ) − + u(t ) + L . (49) L "

1

0

0

D. Senthilkumar, C. Mahanta / ISA Transactions 49 (2010) 27–38

35

Table 3 Parameters of matrices a0i , δ a0i , Ai , Bi , δ Ai and δ Bi by the proposed method — Inverted pendulum on a cart (affine fuzzy model). Rule i

a20i

a21

a22

b21

δ a20i

δ a21

δ a22

δ b21

1 2 3 4 5

0 0 0 0 0

16.1512 15.6940 14.8059 13.5023 12.1046

0 0 0 0 0

−0.159 −0.1516 −0.1313 −0.1022 −0.0692

0.9193 0.8973 0.7528 0.5162 0.4747

2.2547 0.5043 0.6499 0.7425 0.5381

0.00795 0.00119 0.00320 0.00173 0.01352

0.01851 0.01780 0.01577 0.01257 0.00874

Table 4 Parameters of matrices of lower and upper fuzzy model by the method in [11] — Inverted pendulum on a cart (affine fuzzy model). Rule i

a20i

a21

a22

b21

a20i

a21

a22

b21

1 2 3 4 5

−19.279 −18.9947 −18.6561 −16.4212 −14.6017

15.7790 15.4305 14.9071 13.4453 12.1521

0 0 0 0 0

−0.1588 −0.1520 −0.1292 −0.1031 −0.0689

19.2785 18.9953 18.6558 16.4215 14.6018

15.7798 15.4289 14.9082 13.4449 12.1523

0 0 0 0 0

−0.1588 −0.1520 −0.1292 −0.1031 −0.0689

2

1

M 0 0

0.005

0.01

0.015

u

0.02

15

θ

m

10 5 0 0

0.005

0.01

0.015

0.02

Fig. 9. TORA system. Fig. 8. Responses of x1 (t ) and x2 (t ) for regulation of PWM converter.

5.3. TORA

As given in [37], with x1 (t ) as the premise variable, the membership functions are defined as N1 = (x1 − x1 )/(x1 − x1 ) and N2 = 1 − N1 . x1 and x1 represents the lower and upper bounds of x1 (t ). Similar to the previous example, the uncertain fuzzy model is obtained by solving the linear programming problem presented in Section 3 and the parameters of the fuzzy model are



0 A1 = 10 × 4.938 3

A2 = 103 ×



0 4.938

 −10.14 , −0.823  −10.14 , −0.823

We next consider the identification of translational oscillator with an eccentric rotational proof mass actuator (TORA) [38,39,20] depicted in Fig. 9. The objective of the problem is to damp the base translational oscillations by applying a suitable input. Let x1 and x2 denote the translational position and velocity of the cart with x2 = x˙ 1 . Let x3 = θ and x4 = x˙ 3 denote the angular position and velocity of the rotational proof mass. Then the system dynamics can be described by the following equation x˙ = f (x) + g (x)u, where u is the torque applied to the eccentric mass and

δ A1 = 0,

δ A = 0,  2    301.68 312.64 B1 = 103 × , B2 = 103 × , 0 0     211.5 209.9 δ B1 = , δ B2 = . 0

   1 − ε 2 cos2 x3 f (x) =  x4   ε cos x (x − ε x2 sin x 3

+





x1 − x1d x − x1d − µ2 K2 1 x2 − x2d x2 − x2d

x2d + VD Vin + VD + RM x1

1

4



3

   ,  )

1 − ε 2 cos2 x3

With the above fuzzy model, we now consider the controller design presented in [37]. We consider the state feedback case and by the method given in [37], the control law for for set-point tracking of PWM converter is



x2 −x1 + ε x24 sin x3



0

u(t ) = −µ1 K1

(51)



 0 −ε cos x 3    1 − ε 2 cos2 x  3  , g (x) =   0   1 

1 − ε 2 cos2 x3 (50)

where K1 = [0.0541 − 0.0298] and K2 = [0.0878 − 0.0339] are the local control gains obtained by solving the inequalities given in Theorem 1 from [37]. The responses of x1 (t ) and x2 (t ) are given in Fig. 8.

with ε = 0.1 being a constant which depends upon the system parameters. Let us define the new state variables z1 = x1 + ε sin x3 , z2 = x2 + ε x4 cos x3 , y1 = x3 , y2 = x4 and employ the feedback transformation

v=

1 1−ε

2

cos2

 y1


 ε cos y1 z1 − (1 + y22 )ε sin y1 + u

(52)

36

D. Senthilkumar, C. Mahanta / ISA Transactions 49 (2010) 27–38

to bring the system into the following simpler form:

6. Conclusion

z˙1 = z2 ,

In this paper, we proposed a method for identification of an uncertain fuzzy model in a special form meant for robust fuzzy control. The bounds of the given dynamic equation of the uncertain nonlinear system are estimated using an interval analysis technique combined with the branch and bound technique which is available in the literature. With the estimated bounds, the nominal model and the uncertainty bounds in the fuzzy model are found using a linear programming approach. The uncertain matrices are expressed in a special form using the bounds obtained from the linear programming and we have shown that the obtained model can be used for robust fuzzy control of uncertain nonlinear systems. The number of linear inequality conditions in the linear programming problem is reduced by the proposed method and the computation time is also less with better approximation than the method proposed by ˘ Skrjanc et al. [11]. The limitation of the proposed method lies in determining the bounds of the uncertain equation by a branch and bound technique. The computational complexity of the problem will increase with an increase in dimension of the branching space and the proposed method is not suitable for systems with a great number of uncertain terms. The proposed modeling technique is illustrated with different simulation examples. Comparison results show that the proposed method has better performance than the other method given in [11].

z˙2 = −z1 + ε sin y1 , y˙ 1 = y2 , y˙ 2 = v.

(53)

We identify the T–S fuzzy model for the TORA system from (53). Unlike the previous examples, only the approximation error is included in the uncertain blocks of the fuzzy model. Let us consider the fuzzy rules and premise parameters defined in [20]. With x(t ) = [z1 (t ) z2 (t ) y1 (t ) y2 (t )]T , the uncertain fuzzy model takes the following form: Rule 1: IF y1 (t ) is ‘‘about −π or π rad’’ THEN x˙ (t ) = (A1 + ∆A1 )x(t ) + (B1 + ∆B1 )u(t ). Rule 2: IF y1 (t ) is ‘‘about −π /2 or π /2 rad’’ THEN x˙ (t ) = (A2 + ∆A2 )x(t ) + (B2 + ∆B2 )u(t ). Rule 3: IF y1 (t ) is ‘‘about 0 rad’’ and y2 (t ) is ‘‘about 0’’ THEN

Appendix

x˙ (t ) = (A3 + ∆A3 )x(t ) + (B3 + ∆B3 )u(t ).

Proof of Theorem 1. Consider the Lyapunov function candidate V (t ) = xT (t )P x(t ). Then,

Rule 4: IF y1 (t ) is ‘‘about 0 rad’’ and y2 (t ) is ‘‘about −a or a’’ THEN

V˙ (t ) = x˙ T (t )Px(t ) + xT (t )P x˙ (t )

=

x˙ (t ) = (A4 + ∆A4 )x(t ) + (B4 + ∆B4 )u(t ).

0

 −1 Ai =  0 a41i

1 0 0 0



0  0 ∆Ai =  0 ∆a41i

0 a23i 0 a43i 1 0 0 0





0 0 , 1 0 0 ∆a23i 0 ∆a43i

 0  , Bi =  0 

i = 1, . . . , 4 (54)

b41i



0 0  , 1  ∆a44i

 µi µj xT (t ) (Ai + Bi Kj )T P

o
 T  + (∆Ai + ∆Bi Kj )T P + ∗ x(t ) .

(57)

Since Fxi (t ) in (36) and i in (45) are diagonal matrices, = Fxi (t ) √1 . With this condition and the inequality X T Y +

√1 Fxi (t ) i

i

Y T X ≤ X T X + YY T given in [1], we obtain



0

(56)

n

i=1 j=1

The uncertain fuzzy model is identified by the procedure explained in Section 3 and we obtained the following system matrices with the parameters and uncertainty bounds given in Table 5.



r X r X





0  0  ∆Bi =  , 0  ∆b41i

i = 1, . . . , 4.

(55)

We now consider the state regulation problem with [0 0 0 0]T as the desired equilibrium point and design the fuzzy PDC controller discussed in [20]. With the control input in the form Pas r u(t ) = i=1 µi Ki x(t ), the LMIs are solved and the parameters of the feedback gains are obtained as K1 = [17.1021 − 8.3725 − 6.1768 − 10.4373] K2 = [15.8944 − 7.6050 − 5.7597 − 9.7674] K3 = [14.5570 − 6.8362 − 5.2915 − 9.0124] K4 = [15.1720 − 7.2377 − 5.5140 − 9.3890]. Simulation results with the initial condition x(0) = [1 0 0 0]T is shown in Fig. 10.

   T  T x (t ) (∆Ai + ∆Bi Kj ) P + ∗ x(t )    T  x(t ) = xT (t ) (Hxi Fxi (t )(Ex1i + Ex2i Kj ))T P + ∗    T  √ 1 = xT (t ) (Hxi i √ Fxi (t )(Ex1i + Ex2i Kj ))T P + ∗ x(t ) i    T  √ 1 = xT (t ) (Hxi i Fxi (t ) √ (Ex1i + Ex2i Kj ))T P + ∗ x(t ) i  1 ≤ xT (t ) (Ex1i + Ex2i Kj )T √ FxiT (t )Fxi (t ) i  1 T × √ (Ex1i + Ex2i Kj ) + PHxi i Hxi P x(t ) i
1 T ≤ x (t ) (Ex1i + Ex2i Kj )T − (Ex1i + Ex2i Kj ) + PHxi i HxiT P x(t ). i T

(58) Substituting (58) into (57) yields: V˙ (t ) ≤

r X r X

n  µi µj x¯ T (t ) ((Ai + Bi Kj )T P ) + (∗)T

i=1 j=1

 o 1 T ¯ + (Ex1i + Ex2i Kj )T − i (Ex1i + Ex2i Kj ) + PHxi i Hxi P x(t ) .

(59)

D. Senthilkumar, C. Mahanta / ISA Transactions 49 (2010) 27–38

37

Table 5 Parameters of matrices Ai , Bi , δ Ai and δ Bi – TORA. Rule i

a23i

a41i

a43i

b41i

δ a23i

δ a41i

δ a43i

δ a44i

δ b41i

1 2 3 4

0.0047 0.0663 0.1116 0.0982

−0.1140

0.0012 0.0061 −0.0127 −0.0627

1.0095 0.9993 1.0112 1.0076

0.0006 0.0007 0.0048 0.0001

0 0.0038 0.0105 0.0004

0.0055 0.0064 0.0185 0.0040

0 0.0015 0.0110 0.0006

0 0.0007 0.0027 0.0006

0.0041 0.1132 0.1114 1

0.5 0.5

0 0

–0.5

–0.5

–1

–1 0

20

40

60

80

100

1.5

1.5

1

1

0.5

0.5

0

0

–0.5

–0.5

–1

–1

0

20

40

60

80

100

0

20

40

60

80

100

–1.5

–1.5 0

20

40

60

80

100

Fig. 10. Control response for TORA system.

Hence proved.



Lemma 1 ([29]). The parameterized linear matrix inequality, r X r X

µi µj Mij < 0

(60)

i=1 j=1

is fulfilled, if the following condition holds:

Mii < 0, 1 r −1

i = 1, 2, . . . , r

Mii +

1 2

(Mij + Mji ) < 0,

(61) 1 ≤ i 6= j ≤ r .

(62)

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