TâS fuzzy model with nonlinear consequence and ... - Semantic Scholar
Applied Soft Computing 7 (2007) 772–782 www.elsevier.com/locate/asoc
T–S fuzzy model with nonlinear consequence and PDC controller for a class of nonlinear control systems R. Rajesh *, M.R. Kaimal Department of Computer Science, University of Kerala, Kariavattom Campus, University of Kerala, Thiruvananthapuram-695581, India Received 30 December 2003; received in revised form 7 September 2005; accepted 16 January 2006 Available online 18 April 2006
Abstract In this paper a new Takagi–Sugeno (T–S) fuzzy model with nonlinear consequence (TSFMNC) is presented which can approximate a class of smooth nonlinear systems, nonlinear dynamical systems and nonlinear control systems. It is also proved that Takagi–Sugeno fuzzy controller with nonlinear consequence (TSFCNC) can be used to approximate a class of nonlinear state-feedback controllers using the so-called parallel distributed compensation (PDC) method. The inverted pendulum problem has been simulated with TSFCNC and compared with Takagi–Sugeno fuzzy controller with linear consequence (TSFCLC) and the results show that TSFCNC performs better than TSFCLC. A real-life example of dynamic positioning of ship is simulated and the results also show that TSFCNC performs better than TSFCLC. # 2006 Elsevier B.V. All rights reserved. Keywords: T–S fuzzy model; T–S fuzzy model with nonlinear consequence; Parallel distributed compensation; Inverted pendulum; Fuzzy control; Feedback control
1. Introduction The essential element for the study of a nonlinear control problem is to get a tractable model of a dynamical system for use in control system design. The design should be simple enough to work with, but must retain the essential features of the system. Various schemes using fuzzy logic models [1– 4,9,12,13,15,18] have been developed in the last few decades since the Zadeh’s seminal paper on fuzzy sets [19], one of which is the Takagi–Sugeno fuzzy model [15]. Wang et. al. proposed the so-called PDC [17] as a design frame work and also modified Tanaka’s stability theorem to include a control algorithm. Recently in [16] it has been proven that T–S fuzzy model with linear rule consequence and PDC controller as a universal framework for nonlinear systems. This paper presents Takagi–Sugeno fuzzy model with nonlinear consequence (TSFMNC). It is proved that TSFMNC can approximate a class of nonlinear systems, nonlinear dynamic systems and nonlinear control systems. Takagi–Sugeno fuzzy controller with nonlinear consequence (TSFCNC) is designed
* Corresponding author at: Department of Computer Science and Engineering, Bharathiar University, Coimbatore-641046, India. E-mail addresses: [email protected] (R. Rajesh), [email protected] (M.R. Kaimal). 1568-4946/$ – see front matter # 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2006.01.014
using parallel distributed compensation (PDC) method which can better approximate a class of nonlinear state-feedback controllers. A famous benchmarking problem, namely, the inverted pendulum problem has been simulated using both TSFCNC and TSFCLC, and the results show better performance with TSFCNC. A real-life example of dynamic positioning of ship is simulated and the results also show that TSFCNC performs better than TSFCLC. This paper is organized as follows. Section 2 deals with the approximation of nonlinear functions using Takagi–Sugeno fuzzy models with nonlinear consequence. Section 3 deals with the applications to modeling and control of nonlinear dynamic systems. Section 4 deals with a real-life example of dynamic positioning of ships. Section 5 concludes the paper. 2. Approximation of nonlinear functions using Takagi– Sugeno fuzzy models with nonlinear consequence 2.1. Takagi–Sugeno fuzzy models with nonlinear consequence Suppose that the nonlinear function f ðxÞ : Rn ! R is defined over a compact region D Rn with the following assumptions: A1. f ð0Þ ¼ 0.
R. Rajesh, M.R. Kaimal / Applied Soft Computing 7 (2007) 772–782
A2. f 2 C12 . Therefore, f ; @@xf and fore bounded over D.
@2 f @x2
are continuous and there-
A3. f ðxÞ is expressable in the form of f ðxÞ ¼ aðxÞcðxÞ and can be approximated as f0 ðxÞ ¼ a0 cðxÞ in the region D0 ¼ fx=jxi j < e0 g and f j1 j2 ;...; jn ðxÞ ¼ a j1 j2 ;...; jn cðxÞ in the region D j1 j2 ;...; jn ¼ fx=x 2 D; ji e xi ð ji þ 1Þe 8 ig where ji are integers, e0 and e are small positive numbers. x¼ ½x1 ; x2 ; . . . ; xn T . aðxÞ is continuous and a0 ¼ aðx00...0 Þ, a j1 j2 ; . . . ; jn ¼ aðx j1 j2 ;...; jn Þ 2 R1n . cðxÞ : Rn ! Rn is defined over compact region D Rn . cðxÞ ¼ ½c1 ðx1 Þ; c2 ðx2 Þ; . . . ; cn ðxn ÞT and ci ðxi Þ is either equal to sin ðxi Þ or equal to xi . Then f ðxÞ can be approximated by TSFMNC. The rules of TSFMNC is of the following form: ˆ ¼ Rule 0: IF x1 is about 0 . . . and xn is about 0 THEN fðxÞ a0 cðxÞ. Rule j1 j2 ; . . . ; jn : IF x1 is about j1 e. . . and xn is about jn e THEN fˆ ¼ a j1 j2 ;...; jn cðxÞ. For Rule 0, choose the possibility of firing h0 ðxÞ as, 1 inside D0 and, 0 outside. The possibility of firing for the j1 j2 ; . . . ; jn th rule is given by the product of all membership functions associated Q with the j1 j2 ; . . . ; jn th rule and is h j1 j2 ;...; jn ðxðtÞÞ ¼ ni¼1 M ji ðxi ðtÞÞ where the membership function for xi is given by (1). It is assumed that h j1 j2 ;...; Pjn ðxÞ have already been normalized, i.e. h j1 j2 ;...; jn ðxÞ 0 and j1 j2 ;...; jn h j1 j2 ;...; jn ðxÞ ¼ 1. ( jxi ji ej jxi ji ej < e M ji ðxi Þ ¼ 1 (1) e 0 else where Then by using center of gravity method for defuzzification, the TSFMNC can be represented as: X ˆ ¼ h0 ðxÞa0 cðxÞ þ h j1 j2 ;...; jn ðxÞa j1 j2 ;...; jn cðxÞ y ¼ fðxÞ j1 j2 ;...; jn
(2) 2.2. Analysis of approximation In this subsection, we will prove the fact that any smooth nonlinear function satisfying the Assumptions A1–A3 can be approximated, to any degree of accuracy, using T–S fuzzy model with nonlinear consequence. In the following discussions, we will only concentrate on one of such regions D j1 j2 ;...; jn by assuming that x 2 D j1 j2 ;...; jn . In the following, for simplicity, f is substituted instead of j1 j2 ; . . . ; jn . Consider eðxÞ, the approximation error between f ðxÞ and ˆ fðxÞ: X keðxÞk ¼ f ðxÞ hf ðxÞaf cðxÞ f X ¼ f ðxÞ hf ðxÞaf cðxf Þ f X hf ðxÞaf ðcðxÞ cðxf ÞÞ f
773
X X hf ðxÞ f ðxf Þ hf ðxÞaf ðcðxÞ ¼ f ðxÞ f f X hf ðxÞk f ðxÞ f ðxf Þk cðxf ÞÞ f X þ hf ðxÞkaf ðcðxÞ cðxf ÞÞk max k f ðxÞ f
f
f ðxf Þk þ max kaf ðcðxÞ cðxf ÞÞk f
Since x 2 D j1 j2 ;...; jn , the distance pffiffiffi between x and any vertex pffiffiffi point of D j1 j2 ;...; jn is less than ne, i.e. jx x j1 j2 ;...; jn j ne, and since cðxÞ is continuous over x, i.e. jcðxÞ cðx j1 j2 ;...; jn Þj < d where d is an arbitrarily small number, it is possible to make eðxÞ arbitrarily small by just reducing e. Now consider the approximation of de dx. Before finding the approximation of @e @x, consider the following two facts: Fact 1. [16] Define " # @h j1 j2 ;...; jn @h j1 j2 ;...; jn @h j1 j2 ;...; jn @h j1 j2 ;...; jn ¼ ; ; @x x @x x1 @x x2 @x xn where it exists, then X @h j j ;...; j 1 2 n ¼0 @x x j1 j2 ;...; jn
(3)
Fact 2. [16] @h j1 j2 ;...; jn ðx x j1 j2 ;...; jn Þ ¼ I @x x j1 j2 ;...; jn X
(4)
Now consider the approximation of @e @x, the difference ˆ between @@xf and @@xf. In the following, for simplicity, f is substituted instead of j1 j2 ; . . . ; jn . @ P h ðxÞa cðxÞ f f @e @ f f ¼ @x @x @x x @ f X @hf ðxÞ X @cðxÞ ¼ af cðxÞ hf ðxÞaf @x x @x x @x x f f @ f X @hf ðxÞ X @hf ðxÞ ¼ af cðxf Þ þ a cðx Þ f f @x x @x x @x x f f X @hf ðxÞ X @cðxÞ af cðxÞ hf ðxÞaf @x @x x x f f @ f X @hf ðxÞ X ¼ f ðxf Þ af ðcðxÞ @x x @x x f f @hf ðxÞ X @cðxÞ hf ðxÞaf cðxf ÞÞ @x x @x x f
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@ f X @ f @hf ðxÞ ¼ ð f ðxÞ þ ðxf xÞ þ Oðe2 ÞÞ @x x @x x @x x f X @hf ðxÞ X @cðxÞ af ðcðxÞ cðxf ÞÞ hf ðxÞaf @x @x x x f f @ f X@ f X ðxf xÞ@hf ðxÞ ¼ af ðcðxÞ @x x @x x @x x f f @hf ðxÞ X @cðxÞ þ OðeÞ hf ðxÞaf cðxf ÞÞ @x x @x x f ðFrom Fact1Þ X @hf ðxÞ ¼ af ðcðxÞ cðxf ÞÞ @x x f X @cðxÞ þ OðeÞ hf ðxÞaf @x x f ðFrom Fact2Þ X @hf ðxÞ @cðxÞ ¼ a ðcðxÞ cðxf ÞÞ þaðxÞ f f @x x @x x X @cðxÞ X @cðxÞ þ hf ðxÞaf hf ðxÞaðxÞ @x @x x x f f X þ OðeÞ ða aðxÞÞðcðxÞ f f @hf ðxÞ @cðxÞ cðxf ÞÞ @x x @x x X @cðxÞ þ h ðxÞðaf aðxÞÞ þ OðeÞ f f @x x ðFrom Fact2Þ Since x 2 D j1 j2 ;...; jn , the distance pffiffiffi between x and any vertex pffiffiffi point of D j1 j2 ;...; jn is less than ne, i.e. jx x j1 j2 ;...; jn j ne, and since cðxÞ and aðxÞ are continuous over x, i.e. jcðxÞ cðx j1 j2 ;...; jn Þj < d and jaðxÞ aðx j1 j2 ;...; jn Þj < d1 where d and d1 are arbitrarily small numbers, it is possible to make eðxÞ arbitrarily small by just reducing e. Next consider region D0 . In region D0 , it is known from Taylor series that eðxÞ and de dx can also be made arbitrarily small by reducing e0 . Therefore, the following theorem is obtained by summarizing the results above:
Fig. 1. Plot of f ðx1 ; x2 Þ ¼ cos 2 ðx1 Þ sin ðx1 Þ þ x1 x2 .
function f ðx1 ; x2 Þ ¼ cos 2 ðx1 Þ sin ðx1 Þ þ x1 x2 , where x1 ; x2 ¼ ½p=2; p=2. The plot of the function is shown in Fig. 1. TSFMNC is constructed where j1 j2 th rule is of the form: Rule j1 j2 : IF x1 is j1 e and x2 is j2 e THEN sin ðx1 Þ 2 f ðx1 ; x2 Þ ¼ ½cos ð j1 eÞ; j2 e x2 TSFMLC is constructed where j1 j2 th rule is of the form: Rule j1 j2 : IF x1 is j1 e and x2 is j2 e THEN 2 cos ð j1 eÞ sin ð j1 eÞ x1 ; j2 e f ðx1 ; x2 Þ ¼ x2 j2 e In both the cases e ¼ 10p=180 and ji ¼ 9; 8; . . . ; 1; 0; 1; . . . ; 8; 9 for i ¼ 1; 2. Fig. 2 shows the plot of TSFMNC and Fig. 3 shows the plot of TSFMLC. Fig. 4 shows the difference of the function and TSFMNC and Fig. 5 shows the difference of the function and TSFMLC. Fig. 6 shows the difference of TSFMNC and TSFMLC. 1024 data points (32 for x1 32 for x2 ) are taken for simulation and found the
Theorem 1. For a smooth nonlinear function f ðxÞ : Rn ! R satisfying Assumptions A1, A2 and A3can be approximated, to any degree of accuracy, by a T–S fuzzy model with nonlinear consequence. Furthermore, its derivative can be approximated to any degree of accuracy, except for a finite number of points. 2.3. Example An example is given in this subsection for illustration. Consider the approximation of two-dimensional nonlinear
Fig. 2. Plot of TSFMNC.
R. Rajesh, M.R. Kaimal / Applied Soft Computing 7 (2007) 772–782
775
Fig. 6. Difference of TSFMNC and TSFMLC.
Fig. 3. Plot of TSFMLC.
The errors show that TSFMNC better approximates the function than TSFMLC. 3. Applications to modeling and control of nonlinear dynamic systems 3.1. Approximation of nonlinear dynamic systems using Takagi–Sugeno fuzzy models with nonlinear consequence The TSFMNC is used to describe dynamic systems. It is of the following form: Rule 0: IF x1 is about 0 . . . and xn is about 0 THEN x˙ ¼ A0 cðxÞ. Rule j1 j2 ; . . . ; jn : IF x1 is about j1 e . . . and xn is about jn e THEN x˙ ¼ A j1 j2 ;...; jn cðxÞ.
Fig. 4. Difference of f ðx1 ; x2 Þ and TSFMNC.
following error in modeling: XX ð f ðx1 ; x2 Þ TSFMLCðx1 ; x2 ÞÞ2 ¼ 0:0088 x1
x2
x1
x2
x1
x2
XX ð f ðx1 ; x2 Þ TSFMNCðx1 ; x2 ÞÞ2 ¼ 0:0079 XX
(5) (6)
ðTSFMLCðx1 ; x2 Þ TSFMNCðx1 ; x2 ÞÞ2
¼ 1:0078 104
where x ¼ ½x1 ; x2 ; . . . ; xn T are the system states and cðxÞ ¼ ½c1 ðx1 Þ; c2 ðx2 Þ; . . . ; cn ðxn ÞT . By using center of gravity method for defuzzification, TSFMNC can be represented as Eq. (8) where h j1 j2 ;...; jn ðxÞ is the possibility for the j1 j2 ; . . . ; jn th rule to fire: X h j1 j2 ;...; jn ðxÞA j1 j2 ;...; jn cðxÞ (8) x˙ ¼ h0 ðxÞA0 cðxÞ þ j1 j2 ;...; jn
(7)
Consider the nonlinear system (9) where f ðxÞ is a vector field defined over compact region D Rn satisfying Assumptions A1, A3 and the following Assumption A4: 2
A4. f 2 Cn2 . Therefore, f, @@xf and @@x2f are continuous and therefore bounded over D: x˙ ¼ f ðxÞ
(9) T
Fig. 5. Difference of f ðx1 ; x2 Þ and TSFMLC.
Suppose f ðxÞ can be written as ½ f1 ðxÞ; . . . ; fn ðxÞ . What we ˆ ¼ mean by approximation is finding a TSFMNC given by fðxÞ ˆ ½ fˆ1 ðxÞ; . . . ; fˆn ðxÞT such that k f ðxÞ fðxÞk is small. Since ˆ k f ðxÞ fðxÞk is small iff each of its components (which are nonlinear functions) is small, then by applying Theorem 1 proven in the previous section, the following corollary is obtained:
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Corollary 2. Any smooth nonlinear system(9) satisfying the Assumptions A1, A3 and A4can be approximated, to any degree of accuracy, by a TSFMNC(8). Similarly, smooth nonlinear control system x˙ ¼ f ðxÞ þ gðxÞu can also be approximated using a TSFMNC given by X x˙ ¼ h0 ðxÞðA0 cðxÞ þ B0 uÞ þ h j1 j2 ;...; jn ðxÞ j1 j2 ;...; jn
ðA j1 j2 ;...; jn cðxÞ þ B j1 j2 ;...; jn uÞ
where x1 denotes the angle (in radians) of the pendulum from the vertical, x2 ¼ x˙ 1 the angular velocity, g the gravity constant, m the mass of the pendulum, M the mass of the cart, 2l is the length of the pendulum, u the force applied to the cart and a ¼ 1=ðm þ MÞ. The values used in the simulation are, g ¼ 9:8 m/s2, m ¼ 2 kg, M ¼ 8 kg, a ¼ 0:1, 2l ¼ 1 m. By neglecting the higher order terms Eq. (11) can be written as x˙ 2 ¼
3.2. Approximation of nonlinear state-feedback controllers using Takagi–Sugeno fuzzy controller with nonlinear consequence Takagi–Sugeno fuzzy controller with nonlinear consequence (TSFCNC) is designed using parallel distributed compensation (PDC) method. The j1 j2 ; . . . ; jn th rule of the TSFCNC is given by
The system (12) is approximated by the following two-rule Takagi–Sugeno fuzzy model with linear consequence: R1 : If x1 is about 0 then # " # "0 1 x1 0 x˙ 1 g a u ¼ 0 x2 þ x˙ 2 4l=3 aml 4l=3 aml R2 : If x1 is about p=2 then # "0 1 x x˙ 1 1 2g ¼ 0 x2 x˙ 2 2
pð4l=3 aml cos ð88 ÞÞ " # 0
a cos ð88 Þ þ u 4l=3 aml cos 2 ð88 Þ
(10)
j1 j2 ;...; jn
Following similar argument as in the above sections, the following theorem is obtained:
(12)
3.4. Modeling of TSFMLC and design of TSFCLC
Rule 0: IF x1 is about 0 . . . and xn is about 0 THEN u ¼ K0 cðxÞ. Rule j1 j2 ; . . . ; jn : IF x1 is about j1 e . . . and xn is about jn e THEN u ¼ K j1 j2 ;...; jn cðxÞ. The output of the TSFCNC is X u ¼ h0 K0 cðxÞ h j1 j2 ;...; jn ðxÞK j1 j2 ;...; jn cðxÞ
g sin ðx1 Þ a cos ðx1 Þu 2 4l=3 aml cos ðx1 Þ 4l=3 aml cos 2 ðx1 Þ
By utilizing the concept of parallel distributed compensation (PDC), the following two rules are designed for the controller:
Theorem 3. For smooth nonlinear state feedback controller u ¼ KcðxÞ where x is defined over a compact region can be approximated, to any degree of accuracy, by TSFCNC (10).
R1 : If x1 is about 0 then u1 ¼ K1 x. R2 : If x1 is about p=2 then u2 ¼ K2 x. 3.5. Modeling of TSFMNC and design of TSFCNC
Remark 4. Most of the real-life control system applications like ball and beam system, truck and trailer docking system, rotational pendulum, etc., are having some of its state variables in terms of sin function and hence can be better approximated using this TSFCNC. 3.3. Inverted pendulum — an application A famous benchmark problem namely the inverted pendulum control problem is chosen for study. The nonlinear and non-stable behavior of the inverted pendulum problem renders the use of conventional method very difficult. The control objective is to balance the inverted pendulum for the approximate range of vertical angle, namely x1 2 ðp=2; p=2Þ. The equations of motion of the pendulum [17] are x˙ 2 ¼
g sin ðx1 Þ amlx22 sin ð2x2 Þ=2 a cos ðx1 Þu 4l=3 aml cos 2 ðx1 Þ
(11)
Eq. (12) can also be written as
x˙ 1 x˙ 2
"
¼
0
# 1 sin ðx1 Þ 0 x2 #
g 4l=3 aml cos 2 ðx1 Þ " 0 a cos ðx1 Þ þ u 4l=3 aml cos 2 ðx1 Þ
(13)
It is clear from the Eq. (13) that Eq. (12) is expressable in the form of x˙ ¼ AðxÞcðxÞ þ BðxÞu. Now the system (13) is approximated by the following two-rule Takagi–Sugeno fuzzy model with nonlinear consequence: R1 : If x1 is about 0 then # " # "0 1 sin ðx1 Þ 0 x˙ 1 g a u ¼ þ 0 x2 x˙ 2 4l=3 aml 4l=3 aml
R. Rajesh, M.R. Kaimal / Applied Soft Computing 7 (2007) 772–782
777
Table 1 Performence of TSFCLC and TSFCNC Performance index
TSFCLC
TSFCNC
J
2.9634e+005
2.9455e+005
R2 : If x1 is about p=2 then # "0 1 sin ðx1 Þ x˙ 1 g ¼ 0 x˙ 2 x2 4l=3 aml cos 2 ð88 Þ " # 0
a cos ð88 Þ þ u 4l=3 aml cos 2 ð88 Þ By utilizing the concept of parallel distributed compensation (PDC), the following two rules are designed for the controller where cðxÞ ¼ ½sin ðx1 Þ; x2 T : R1 : If x1 is about 0 then u1 ¼ K1 cðxÞ. R2 : If x1 is about p=2 then u2 ¼ K2 cðxÞ.
3.6. Comparison By chossing Q ¼ ½100 0; 0 0 and R ¼ 1 and by using linear quadratic regulator design method, the gain values of TSFCLC are obtained as K1 ¼ ½196:5089 47:1922; K2 ¼ ½3575:4 1168:6 and the gain values of TSFCNC are obtained as K1 ¼ ½196:5089 47:1922; K2 ¼ ½5616:1 1464:7. The performance index is evaluated by Z timestep t T J¼ x ðtÞQxðtÞ þ uT ðtÞRuðtÞ (14) t 0 Table 1 shows the performance index of the TSFCLC and 2
0 60 6 60 AðXÞ ¼ 6 6 h111 x1 6 4 h121 x1 h131 x1 TSFCNC for a simulation of the pendulum for 3 s, with integration timestep ¼ 0:01, starting with initial angle, x1 ¼ 85 . Fig. 7 shows the plot of u for both TSFCLC and TSFCNC. The results shows better performance for the new TSFCNC. 4. Dynamic positioning of ships — a real-life example The model of dynamic positioning of ships (DPS) [6– 8,10,11,14] exhibits nonlinear interaction in three degrees of freedom (surge, sway, and yaw). The DPS control problem is
Fig. 7. Performance comparision of pendulum angle u for TSFCLC and TSFCNC.
usually solved under the assumption that the kinematic equations can be linearized about a constant yaw angle such that linear theory and gain scheduling techniques can be applied. Later, it was solved by globally exponentially stable (GES) nonlinear control law in [7] and solved using Takagi– Sugeno fuzzy controller with linear consequence in [5]. The state space model of the normalized DPS system [5,7] is as given below: X˙ ¼ AðXÞ þ Bu where 2
(15)
3 2 0 x˙ 1 6 x˙ 2 7 60 6 7 6 6 x˙ 3 7 60 7 6 X¼6 7 B¼6 6 h311 x ˙ 6 47 6 4 x˙5 5 4 h321 x˙6 h331
0 0 0 h112 x2 h122 x2 h132 x2
0 0 0 h113 x3 h123 x3 h133 x3
0 0 0 h312 h322 h332
cos ðx3 Þx4 sin ðx3 Þx4 0 h211 x4 h221 x4 h231 x4
3 0 0 7 7 0 7 7 h313 7 7 h323 5 h333
3 u1 u ¼ 4 u2 5 u3
(16)
3 0 7 0 7 7 1 7 h213 x6 7 7 h223 x6 5 h233 x6
(17)
sin ðx3 Þx5 cos ðx3 Þx5 0 h212 x5 h222 x5 h232 x5
2
where x ¼ ½x1 x2 x3 x4 x5 x6 T ¼ ½x y f u v rT . ðx1 ¼ x; x2 ¼ yÞ is the earth-fixed position, x3 ¼ f is the yaw angle, x4 ¼ u is the surge velocity, x5 ¼ v is the sway velocity, and x6 ¼ r is the yaw velocity. The control thrust in the surge, sway and yaw modes are respectively, t 1 ; t2 , and t3 where u ¼ ½u1 u2 u3 T ¼ ½t1 t2 t3 T . In this section, Takagi–Sugeno fuzzy controller with nonlinear consequence (TSFCNC) is used to design fuzzy controller for the dynamic positioning of ships and is compared with Takagi–Sugeno fuzzy controller with linear consequence (TSFCLC).
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4.1. TSFCLC for DPS
Parallel distributed compensation is used to design fuzzy controller rules and are as given below:
The Takagi–Sugeno fuzzy controller with linear consequence with three rules obtained by approximating AðXÞ at yaw angle 0; p=2 and p=2 are as given below: R1 : IF x3 is about 0 THEN x˙ ¼ A1 x þ B1 u R2 : IF x3 is about p=2 THEN x˙ ¼ A2 x þ B2 u R3 : IF x3 is about p=2 THEN x˙ ¼ A3 x þ B3 u
4.2. TSFCNC for DPS
2 60 6 6 60 6 6 60 6 6 AðXÞ ¼ 6 h111 6 6 6 6h 6 121 6 6 4 h131 where a ¼ sin ð2 Þ; b ¼ cos ð88 Þ and 2 0 0 0 1 a 6 0 0 0 a 1 6 6 60 0 0 0 0 6 6 A1 ¼ 6 h111 h112 h113 h211 h212 6 6 h121 h122 h123 h221 h222 6 6 4 h131 h132 h133 h231 h232 2
0
b
1
0
0 0
0 0
1 0
b 0
0 1
h112 h122
h113 h123
h211 h221
h212 h222
h132
h133
h231
h232
7 7 7 7 7 7 h213 7 7 h223 7 7 7 h233 5
0
0
b
1
0
0
0
1
b
0
0 h112
0 h113
0 h211
0 h212
h122 h123 h131 h132 h133 2 0 60 6 6 60 B1 ¼ B2 ¼ B3 ¼ 6 6h 6 311 6 4 h321
h221 h231
h222 h232 h233 3
2
0
60 6 6 60 A3 ¼ 6 6h 6 111 6 4 h121
h331
0
0
0
7 7 7 0 7 7 h313 7 7 7 h323 5
0 h312 h322 h332
0 h112 h122 h132
and
Ri : IF x3 is about j1;i and IF x4 is about j2;i and IF x5 is about j3;i THEN x˙ ¼ Ai c þ Bi u i ¼ 1; 2; . . . ; 12
(18)
3
7 7 7 1 7 7 h213 7 7 7 h223 5
and
where c ¼ ½x1 ; x2 ; sin ðx3 Þ; x4 ; x5 ; x6 T ; j1;1 ¼ j1;2 ¼ j1;3 ¼ j1;4 ¼ p=2; j1;5 ¼ j1;6 ¼ j1;7 ¼ j1;8 ¼ 0; j1;9 ¼ j1;10 ¼ j1;11 ¼ j1;12 ¼ p=2; j2;1 ¼ j2;2 ¼ j2;5 ¼ j2;6 ¼ j2;9 ¼ j2;10 ¼ 10; j2;3 ¼ j2;4 ¼ j2;7 ¼ j2;8 ¼ j2;11 ¼ j2;12 ¼ 10; j3;1 ¼ j3;3 ¼ j3;5 ¼ j3;7 ¼ j3;9 ¼ j3;11 ¼ 6j3;2 ¼ j3;4 ¼ j3;6 ¼ j3;8 ¼ j3;10 ¼ j3;12 ¼ 6, and 3 2 M2;i M3;i 0 0 0 M1;i 7 6 0 M4;i M5;i M6;i 0 7 60 7 6 60 0 0 0 0 1 7 7 6 7 6 Ai ¼ 6 h111 h112 di h113 h211 h212 h213 7 and 7 6 6 h121 h122 di h123 h221 h222 h223 7 7 6 7 6 4 h131 h132 di h133 h231 h232 h233 5 2
0
h333
The Eq. (17) representing AðXÞ in the state space model (15) can also be written as 3 1 1 cos ðx3 Þ sin ðx3 Þ 0 7 x5 2 2 72 3 7 x1 1 1 x4 sin ðx3 Þ cos ðx3 Þ 0 7 76 7 2 2 76 x2 7 0 0 0 1 7 7 76 6 sin ðx Þ 3 7 76 h113 x3 7 (20) 7 h211 h212 h213 76 7 x4 sin ðx3 Þ 7 6 76 7 74 x 5 h123 x3 5 h221 h222 h223 7 7 sin ðx3 Þ 7 x6 7 5 h133 x3 h231 h232 h233 sin ðx3 Þ
which gives importance to the variables x5 and x4 by providing it in the 3rd column of 1st and 2nd, respectively, and leads to nonlinear consequent. It is clear that the approximation of the above AðXÞ requires approximations in x3 ; x4 and x5 . By assuming the range of yaw angle (x3 ), the surge velocity (x4 ) and the sway velocity (x5 ), respectively, as ½p=2; p=2; ½10; 10 and ½6; 6, the 12 rules of Takagi–Sugeno fuzzy controller with nonlinear consequence are obtained at different regions and are as given below:
3
0
6 60 6 60 6 6 A2 ¼ 6 h111 6 6 h121 6 6 4 h131
0
0 1
0
0
0
3
7 7 7 7 7 7 h213 7 7 h223 7 7 7 h233 5
R1 : IF x3 is about 0 THEN u ¼ F1 x R2 : IF x3 is about p=2 THEN u ¼ F2 x R3 : IF x3 is about p=2 THEN u ¼ F3 x
(19)
0
0
0
0 0
3
60 6 6 60 Bi ¼ 6 6h 6 311 6 4 h321
h312 h322
7 7 7 7 7 h313 7 7 7 h323 5
h331
h332
h333
0 0
(21)
R. Rajesh, M.R. Kaimal / Applied Soft Computing 7 (2007) 772–782
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Table 2 Gain values of TSFCLC Gain F1
F2
F3
Value 2 0:9613 4 0:0346 0:0041 2 0:0014 4 0:9937 0:1125 2 0:0014 6 0:9937 6 4 0:1125
0:0001 0:1144 0:9934
0:0348 0:9665 0:1143 0:9996 0:0110 0:0001
1:3614 0:0000 0:0002
0:0002 1:8917 0:4727
0:0043 0:1124 0:9937
1:3890 0:0011 0:0059
0:0042 1:9188 0:4721
0:0043 0:1124 0:9937
1:3890 0:0011 0:0059
0:0042 1:9188 0:4721
0:9996 0:0110 0:0001
3 0:0001 0:3115 5 0:5725 3 0:0016 0:3168 5 0:5725 3 0:0016 0:3168 7 7 0:5725 5
and di ¼ p=2 for i ¼ 14; 912 and di ¼ 1 for i ¼ 58 and 6 M1;1 ¼ M1;3 ¼ M1;5 ¼ M1;7 ¼ M1;9 ¼ M1;11 ¼ 2 6 M1;2 ¼ M1;4 ¼ M1;6 ¼ M1;8 ¼ M1;10 ¼ M1;12 ¼ 2 M2;1 ¼ M2;2 ¼ M2;3 ¼ M2;4 ¼ M2;9 ¼ M2;10 ¼ M2;11 ¼ M2;12 ¼ b M2;5 ¼ M2;6 ¼ M2;7 ¼ M2;8 ¼ 1 1 M3;1 ¼ M3;2 ¼ M3;3 ¼ M3;4 ¼ 2 a M3;5 ¼ M3;6 ¼ M3;7 ¼ M3;8 ¼ 2 1 M3;9 ¼ M3;10 ¼ M3;11 ¼ M3;12 ¼ 2 10 M4;1 ¼ M4;2 ¼ M4;5 ¼ M4;6 ¼ M4;9 ¼ M4;10 ¼ 2 10 M4;3 ¼ M4;4 ¼ M4;7 ¼ M4;8 ¼ M4;11 ¼ M4;12 ¼ 2 1 M5;1 ¼ M5;2 ¼ M5;3 ¼ M5;4 ¼ 2 a M5;5 ¼ M5;6 ¼ M5;7 ¼ M5;8 ¼ 2 1 M5;9 ¼ M5;10 ¼ M5;11 ¼ M5;12 ¼ 2 M6;1 ¼ M6;2 ¼ M6;3 ¼ M6;4 ¼ M6;9 ¼ M6;10 ¼ M6;11 ¼ M6;12 ¼ b M6;5 ¼ M6;6 ¼ M6;7 ¼ M6;8 ¼ 1
4.3. Comparison
Parallel distributed compensation is used to design fuzzy controller rules and are as given below:
For the following values of Q and R, linear quadratic regulator techniques are applied to find out the gain values
Fig. 8. Performance comparison of x3 for TSFCLC and TSFCNC.
Fig. 10. Performance comparison of x5 for TSFCLC and TSFCNC.
Fig. 9. Performance comparison of x4 for TSFCLC and TSFCNC.
Ri : IF x3 is about j1;i and IF x4 is about j2;i and IF x5 is about j3;i THEN u ¼ Ki c i ¼ 1; 2; . . . ; 12. where c ¼ ½x1 ; x2 ; sin ðx3 Þ; x4 ; x5 ; x6 T .
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of Fi s in TSFCLC and Ki s in TSFCNC and are given in Tables 2 and 3, respectively: 3 2 1 0 0 0 0 0 60 1 0 0 0 07 2 3 7 6 1 0 0 60 0 1 0 0 07 7 4 5 Q¼6 6 0 0 0 0 0 0 7 and R ¼ 0 1 0 7 6 0 0 1 40 0 0 0 0 05 0 0 0 0 0 0
The performance index J1 and J2 given by the following equations are shown in Table 4. It shows that TSFCNC performs better than TSFCLC: X
J1 ¼
x1 ðtÞ þ x2 ðtÞ þ x3 ðtÞ þ x4 ðtÞ þ x5 ðtÞ þ x6 ðtÞ
t¼0:01:0:01:2
(22)
Table 3 Gain values of TSFCNC Gain K1
Value 2 0:5723 6 0:7362 6 4 0:4183
K2
2
K3
2
K4
2
K5
2
K6
2
K7
2
K8
2
K9
2
K10
2
K11
2
K12
2
0:6258 6 0:4106 6 4 0:6254 0:6358 6 0:6103 6 4 0:4170 0:5597 6 0:5257 6 4 0:6737 0:5723 6 0:7362 6 4 0:4183 0:6258 6 0:4106 6 4 0:6254 0:6358 6 0:6103 6 4 0:4170 0:5597 6 0:5257 6 4 0:6737 0:5723 6 0:7362 6 4 0:4183 0:6258 6 0:4106 6 4 0:6254 0:6358 6 0:6103 6 4 0:4170 0:5597 6 0:5257 6 4 0:6737
0:4171 0:1749 0:8866
0:1642 0:7306 3:3640
0:3910 0:3188 0:2950
0:4692 0:4491 0:7759
0:1573 0:6245 3:4878
0:5234 0:4113 0:1514
0:4658 0:1138 0:8810
0:1733 0:7175 3:3548
0:5218 0:3208 0:2969
0:4176 0:5047 0:7375
0:1494 0:6091 3:4764
0:4171 0:1749 0:8866
0:1642 0:7306 3:3640
0:3910 0:3188 0:2950
0:4692 0:4491 0:7759
0:1573 0:6245 3:4878
0:5234 0:4113 0:1514
0:4658 0:1138 0:8810
0:1733 0:7175 3:3548
0:3918 0:4084 0:1507
0:5218 0:3208 0:2969
0:4176 0:5047 0:7375
0:1494 0:6091 3:4764
0:4171 0:1749 0:8866
0:1642 0:7306 3:3640
0:3910 0:3188 0:2950
0:4692 0:4491 0:7759
0:1573 0:6245 3:4878
0:5234 0:4113 0:1514
0:4658 0:1138 0:8810 0:4176 0:5047 0:7375
0:1733 0:7175 3:3548 0:1494 0:6091 3:4764
0:3918 0:4084 0:1507
0:5218 0:3208 0:2969 0:3918 0:4084 0:1507
0:7157 0:8662 0:0094
3 0:1786 0:0060 7 7 0:9011 5
0:7228 0:5848 0:3387
3 0:1249 0:0425 7 7 0:9773 5
0:7200 0:6521 0:0162
3 0:1797 0:0501 7 7 0:8959 5
0:7177 0:7924 0:3959 0:7157 0:8662 0:0094
3 0:1239 0:0024 7 7 0:9891 5 3 0:1786 0:0060 7 7 0:9011 5
0:7228 0:5848 0:3387
3 0:1249 0:0425 7 7 0:9773 5
0:7200 0:6521 0:0162
3 0:1797 0:0501 7 7 0:8959 5
0:7177 0:7924 0:3959 0:7157 0:8662 0:0094
3 0:1239 0:0024 7 7 0:9891 5 3 0:1786 0:0060 7 7 0:9011 5
0:7228 0:5848 0:3387
3 0:1249 0:0425 7 7 0:9773 5
0:7200 0:6521 0:0162
3 0:1797 0:0501 7 7 0:8959 5
0:7177 0:7924 0:3959
3 0:1239 0:0024 7 7 0:9891 5
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5. Conclusion This paper presents T–S fuzzy model with nonlinear consequence. It is proved that it can approximate a particular class of nonlinear functions, nonlinear dynamic systems and nonlinear control systems. Moreover, TSFCNC can approximate state feedback controllers by using parallel distributed compensation method. A benchmarking control problem, namely, inverted pendulum is simulated and the results show that TSFCNC performs better than TSFCLC. A real-life example of dynamic positioning of ship is simulated and the results also show that TSFCNC performs better than TSFCLC. Fig. 11. Performance comparison of x6 for TSFCLC and TSFCNC.
Acknowledgements The authors are very thankful to Indian Space Research Organization for financial support through RESPOND project. The authors are also thankful to the reviewers for their valuable help in improving the quality of the paper. References
Fig. 12. Performance comparison of x1 y1 for TSFCLC and TSFCNC.
J2 ¼
X
u1 ðtÞ þ u2 ðtÞ þ u3 ðtÞ
(23)
t¼0:01:0:01:2
The dotted lines in Figs. 8–11 show the performance of TSFCLC and the solid lines show the performance of TSFCNC. The figures show that TSFCNL performs better than TSFCLC. The plot of x1 versus y1 , shown in Fig. 12, shows the path of the ship from the earth fixed position ð10; 10Þ to the position (0, 0). Since the dotted line representing TSFCLC and solid line representing TSFCNL concides, it seems to be a straight solid line. But at the finishing point, after zooming it is clear that TSFCNC performs better than TSFCLC.
Table 4 Performance of TSFCLC and TSFCNC Performance index
TSFCLC
TSFCNC
J1 J2
200.6146 189.0391
200.4305 54.5321
[1] J.J. Buckley, Universal fuzzy controllers, Automatica 28 (1992) 1245– 1248. [2] O. Castillo, P. Melin, Special issue on soft computing for control of nonlinear dynamical systems, Appl. Soft Comput. 3 (2003) 303–304. [3] O. Castillo, P. Melin, Intelligent adaptive model-based control of robotic dynamic systems with a hybrid fuzzy-neural approach, Appl. Soft Comput. 3 (2003) 363–378. [4] J.L. Castro, Fuzzy logic controllers are universal approximators, IEEE Trans. Syst. Man Cybern. 25 (1998) 629–635. [5] W.-J. Chang, G.-J. Chen, Y.-L. Yeh, Fuzzy control of dynamic positioning of ships, J. Mar. Sci. Technol. 10 (2002) 47–53. [6] T.I. Fossen, Guidance and Control of Ocean Vehicles, Wiley, New York, 1994. [7] T.I. Fossen, A. Grovlen, Nonlinear output feedback control of dynamically positioned ships using vectorial observer backstepping, IEEE Trans. Control Syst. Technol. 6 (1998) 121–128. [8] J.M. Godhavn, T.I. Fossen, S.P. Berge, Nonlinear and adaptive backstepping designs for tracking control os ships, Int. J. Adapt. Control Signal Process. 12 (1998) 649–670. [9] M. Golob, Decomposed fuzzy proportional-integral-derivative controllers, Appl. Soft Comput. 1 (2001) 201–214. [10] M.-H. Kim, Nonlinear control and robust observer design for marine vehicles, Ph.D. thesis, Virgina Polytechnic Institute and State University, 2000. [11] A. Loria, T.I. Fossen, E. Panteley, A separation principle for dynamic positioning of ships: theoretical and experimental results, IEEE Trans. Control Syst. Technol. 8 (2000) 332–343. [12] E.H. Mamdani, Application of fuzzy algorithms for control of simple dynamic plant, Proc. Inst. Electr. Eng. 121 (1974) 1585–1588. [13] P. Melin, O. Castillo, Adaptive intelligent control of aircraft systems with a hybrid approach combining neural networks, fuzzy logic and fractal theory, Appl. Soft Comput. 3 (2003) 353–362. [14] K.Y. Pettersen, H. Nijmeijer, Global practical stabilization and tracking for underactuated ship — a combined averaging and backstepping approach, in: Proceedings of the IFAC Conference on System Structure and Control, Nantes, France, 1998, pp. 59–64. [15] T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Syst. Man Cybern. 15 (1985) 116– 132.
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[16] H.O. Wang, J. Li, K. Tanaka, T–S Fuzzy model with linear rule consequence and PDC controllers: a universal framework for nonlinear control systems, Int. J. Fuzzy Syst. (2003) 106–113. [17] H.O. Wang, K. Tanaka, M. Griffin, Parallel distributed compensation of nonlinear systems by Takagi–Sugeno Fuzzy Model, in: Proc. Fuzz-IEEE/ IFES’95 (1995) 531–538.
[18] Y. Yang, C. Zhou, J. Ren, Model reference adaptive robust fuzzy control for ship steering autopilot with uncertain nonlinear systems, Appl. Soft Comput. 3 (2003) 305–316. [19] L.A. Zadeh, Fuzzy Sets, Inform. Control 8 (1965) 338–353.
T–S fuzzy model with nonlinear consequence and PDC controller for a class of nonlinear control systems R. Rajesh *, M.R. Kaimal Department of Computer Science, University of Kerala, Kariavattom Campus, University of Kerala, Thiruvananthapuram-695581, India Received 30 December 2003; received in revised form 7 September 2005; accepted 16 January 2006 Available online 18 April 2006
Abstract In this paper a new Takagi–Sugeno (T–S) fuzzy model with nonlinear consequence (TSFMNC) is presented which can approximate a class of smooth nonlinear systems, nonlinear dynamical systems and nonlinear control systems. It is also proved that Takagi–Sugeno fuzzy controller with nonlinear consequence (TSFCNC) can be used to approximate a class of nonlinear state-feedback controllers using the so-called parallel distributed compensation (PDC) method. The inverted pendulum problem has been simulated with TSFCNC and compared with Takagi–Sugeno fuzzy controller with linear consequence (TSFCLC) and the results show that TSFCNC performs better than TSFCLC. A real-life example of dynamic positioning of ship is simulated and the results also show that TSFCNC performs better than TSFCLC. # 2006 Elsevier B.V. All rights reserved. Keywords: T–S fuzzy model; T–S fuzzy model with nonlinear consequence; Parallel distributed compensation; Inverted pendulum; Fuzzy control; Feedback control
1. Introduction The essential element for the study of a nonlinear control problem is to get a tractable model of a dynamical system for use in control system design. The design should be simple enough to work with, but must retain the essential features of the system. Various schemes using fuzzy logic models [1– 4,9,12,13,15,18] have been developed in the last few decades since the Zadeh’s seminal paper on fuzzy sets [19], one of which is the Takagi–Sugeno fuzzy model [15]. Wang et. al. proposed the so-called PDC [17] as a design frame work and also modified Tanaka’s stability theorem to include a control algorithm. Recently in [16] it has been proven that T–S fuzzy model with linear rule consequence and PDC controller as a universal framework for nonlinear systems. This paper presents Takagi–Sugeno fuzzy model with nonlinear consequence (TSFMNC). It is proved that TSFMNC can approximate a class of nonlinear systems, nonlinear dynamic systems and nonlinear control systems. Takagi–Sugeno fuzzy controller with nonlinear consequence (TSFCNC) is designed
* Corresponding author at: Department of Computer Science and Engineering, Bharathiar University, Coimbatore-641046, India. E-mail addresses: [email protected] (R. Rajesh), [email protected] (M.R. Kaimal). 1568-4946/$ – see front matter # 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2006.01.014
using parallel distributed compensation (PDC) method which can better approximate a class of nonlinear state-feedback controllers. A famous benchmarking problem, namely, the inverted pendulum problem has been simulated using both TSFCNC and TSFCLC, and the results show better performance with TSFCNC. A real-life example of dynamic positioning of ship is simulated and the results also show that TSFCNC performs better than TSFCLC. This paper is organized as follows. Section 2 deals with the approximation of nonlinear functions using Takagi–Sugeno fuzzy models with nonlinear consequence. Section 3 deals with the applications to modeling and control of nonlinear dynamic systems. Section 4 deals with a real-life example of dynamic positioning of ships. Section 5 concludes the paper. 2. Approximation of nonlinear functions using Takagi– Sugeno fuzzy models with nonlinear consequence 2.1. Takagi–Sugeno fuzzy models with nonlinear consequence Suppose that the nonlinear function f ðxÞ : Rn ! R is defined over a compact region D Rn with the following assumptions: A1. f ð0Þ ¼ 0.
R. Rajesh, M.R. Kaimal / Applied Soft Computing 7 (2007) 772–782
A2. f 2 C12 . Therefore, f ; @@xf and fore bounded over D.
@2 f @x2
are continuous and there-
A3. f ðxÞ is expressable in the form of f ðxÞ ¼ aðxÞcðxÞ and can be approximated as f0 ðxÞ ¼ a0 cðxÞ in the region D0 ¼ fx=jxi j < e0 g and f j1 j2 ;...; jn ðxÞ ¼ a j1 j2 ;...; jn cðxÞ in the region D j1 j2 ;...; jn ¼ fx=x 2 D; ji e xi ð ji þ 1Þe 8 ig where ji are integers, e0 and e are small positive numbers. x¼ ½x1 ; x2 ; . . . ; xn T . aðxÞ is continuous and a0 ¼ aðx00...0 Þ, a j1 j2 ; . . . ; jn ¼ aðx j1 j2 ;...; jn Þ 2 R1n . cðxÞ : Rn ! Rn is defined over compact region D Rn . cðxÞ ¼ ½c1 ðx1 Þ; c2 ðx2 Þ; . . . ; cn ðxn ÞT and ci ðxi Þ is either equal to sin ðxi Þ or equal to xi . Then f ðxÞ can be approximated by TSFMNC. The rules of TSFMNC is of the following form: ˆ ¼ Rule 0: IF x1 is about 0 . . . and xn is about 0 THEN fðxÞ a0 cðxÞ. Rule j1 j2 ; . . . ; jn : IF x1 is about j1 e. . . and xn is about jn e THEN fˆ ¼ a j1 j2 ;...; jn cðxÞ. For Rule 0, choose the possibility of firing h0 ðxÞ as, 1 inside D0 and, 0 outside. The possibility of firing for the j1 j2 ; . . . ; jn th rule is given by the product of all membership functions associated Q with the j1 j2 ; . . . ; jn th rule and is h j1 j2 ;...; jn ðxðtÞÞ ¼ ni¼1 M ji ðxi ðtÞÞ where the membership function for xi is given by (1). It is assumed that h j1 j2 ;...; Pjn ðxÞ have already been normalized, i.e. h j1 j2 ;...; jn ðxÞ 0 and j1 j2 ;...; jn h j1 j2 ;...; jn ðxÞ ¼ 1. ( jxi ji ej jxi ji ej < e M ji ðxi Þ ¼ 1 (1) e 0 else where Then by using center of gravity method for defuzzification, the TSFMNC can be represented as: X ˆ ¼ h0 ðxÞa0 cðxÞ þ h j1 j2 ;...; jn ðxÞa j1 j2 ;...; jn cðxÞ y ¼ fðxÞ j1 j2 ;...; jn
(2) 2.2. Analysis of approximation In this subsection, we will prove the fact that any smooth nonlinear function satisfying the Assumptions A1–A3 can be approximated, to any degree of accuracy, using T–S fuzzy model with nonlinear consequence. In the following discussions, we will only concentrate on one of such regions D j1 j2 ;...; jn by assuming that x 2 D j1 j2 ;...; jn . In the following, for simplicity, f is substituted instead of j1 j2 ; . . . ; jn . Consider eðxÞ, the approximation error between f ðxÞ and ˆ fðxÞ: X keðxÞk ¼ f ðxÞ hf ðxÞaf cðxÞ f X ¼ f ðxÞ hf ðxÞaf cðxf Þ f X hf ðxÞaf ðcðxÞ cðxf ÞÞ f
773
X X hf ðxÞ f ðxf Þ hf ðxÞaf ðcðxÞ ¼ f ðxÞ f f X hf ðxÞk f ðxÞ f ðxf Þk cðxf ÞÞ f X þ hf ðxÞkaf ðcðxÞ cðxf ÞÞk max k f ðxÞ f
f
f ðxf Þk þ max kaf ðcðxÞ cðxf ÞÞk f
Since x 2 D j1 j2 ;...; jn , the distance pffiffiffi between x and any vertex pffiffiffi point of D j1 j2 ;...; jn is less than ne, i.e. jx x j1 j2 ;...; jn j ne, and since cðxÞ is continuous over x, i.e. jcðxÞ cðx j1 j2 ;...; jn Þj < d where d is an arbitrarily small number, it is possible to make eðxÞ arbitrarily small by just reducing e. Now consider the approximation of de dx. Before finding the approximation of @e @x, consider the following two facts: Fact 1. [16] Define " # @h j1 j2 ;...; jn @h j1 j2 ;...; jn @h j1 j2 ;...; jn @h j1 j2 ;...; jn ¼ ; ; @x x @x x1 @x x2 @x xn where it exists, then X @h j j ;...; j 1 2 n ¼0 @x x j1 j2 ;...; jn
(3)
Fact 2. [16] @h j1 j2 ;...; jn ðx x j1 j2 ;...; jn Þ ¼ I @x x j1 j2 ;...; jn X
(4)
Now consider the approximation of @e @x, the difference ˆ between @@xf and @@xf. In the following, for simplicity, f is substituted instead of j1 j2 ; . . . ; jn . @ P h ðxÞa cðxÞ f f @e @ f f ¼ @x @x @x x @ f X @hf ðxÞ X @cðxÞ ¼ af cðxÞ hf ðxÞaf @x x @x x @x x f f @ f X @hf ðxÞ X @hf ðxÞ ¼ af cðxf Þ þ a cðx Þ f f @x x @x x @x x f f X @hf ðxÞ X @cðxÞ af cðxÞ hf ðxÞaf @x @x x x f f @ f X @hf ðxÞ X ¼ f ðxf Þ af ðcðxÞ @x x @x x f f @hf ðxÞ X @cðxÞ hf ðxÞaf cðxf ÞÞ @x x @x x f
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@ f X @ f @hf ðxÞ ¼ ð f ðxÞ þ ðxf xÞ þ Oðe2 ÞÞ @x x @x x @x x f X @hf ðxÞ X @cðxÞ af ðcðxÞ cðxf ÞÞ hf ðxÞaf @x @x x x f f @ f X@ f X ðxf xÞ@hf ðxÞ ¼ af ðcðxÞ @x x @x x @x x f f @hf ðxÞ X @cðxÞ þ OðeÞ hf ðxÞaf cðxf ÞÞ @x x @x x f ðFrom Fact1Þ X @hf ðxÞ ¼ af ðcðxÞ cðxf ÞÞ @x x f X @cðxÞ þ OðeÞ hf ðxÞaf @x x f ðFrom Fact2Þ X @hf ðxÞ @cðxÞ ¼ a ðcðxÞ cðxf ÞÞ þaðxÞ f f @x x @x x X @cðxÞ X @cðxÞ þ hf ðxÞaf hf ðxÞaðxÞ @x @x x x f f X þ OðeÞ ða aðxÞÞðcðxÞ f f @hf ðxÞ @cðxÞ cðxf ÞÞ @x x @x x X @cðxÞ þ h ðxÞðaf aðxÞÞ þ OðeÞ f f @x x ðFrom Fact2Þ Since x 2 D j1 j2 ;...; jn , the distance pffiffiffi between x and any vertex pffiffiffi point of D j1 j2 ;...; jn is less than ne, i.e. jx x j1 j2 ;...; jn j ne, and since cðxÞ and aðxÞ are continuous over x, i.e. jcðxÞ cðx j1 j2 ;...; jn Þj < d and jaðxÞ aðx j1 j2 ;...; jn Þj < d1 where d and d1 are arbitrarily small numbers, it is possible to make eðxÞ arbitrarily small by just reducing e. Next consider region D0 . In region D0 , it is known from Taylor series that eðxÞ and de dx can also be made arbitrarily small by reducing e0 . Therefore, the following theorem is obtained by summarizing the results above:
Fig. 1. Plot of f ðx1 ; x2 Þ ¼ cos 2 ðx1 Þ sin ðx1 Þ þ x1 x2 .
function f ðx1 ; x2 Þ ¼ cos 2 ðx1 Þ sin ðx1 Þ þ x1 x2 , where x1 ; x2 ¼ ½p=2; p=2. The plot of the function is shown in Fig. 1. TSFMNC is constructed where j1 j2 th rule is of the form: Rule j1 j2 : IF x1 is j1 e and x2 is j2 e THEN sin ðx1 Þ 2 f ðx1 ; x2 Þ ¼ ½cos ð j1 eÞ; j2 e x2 TSFMLC is constructed where j1 j2 th rule is of the form: Rule j1 j2 : IF x1 is j1 e and x2 is j2 e THEN 2 cos ð j1 eÞ sin ð j1 eÞ x1 ; j2 e f ðx1 ; x2 Þ ¼ x2 j2 e In both the cases e ¼ 10p=180 and ji ¼ 9; 8; . . . ; 1; 0; 1; . . . ; 8; 9 for i ¼ 1; 2. Fig. 2 shows the plot of TSFMNC and Fig. 3 shows the plot of TSFMLC. Fig. 4 shows the difference of the function and TSFMNC and Fig. 5 shows the difference of the function and TSFMLC. Fig. 6 shows the difference of TSFMNC and TSFMLC. 1024 data points (32 for x1 32 for x2 ) are taken for simulation and found the
Theorem 1. For a smooth nonlinear function f ðxÞ : Rn ! R satisfying Assumptions A1, A2 and A3can be approximated, to any degree of accuracy, by a T–S fuzzy model with nonlinear consequence. Furthermore, its derivative can be approximated to any degree of accuracy, except for a finite number of points. 2.3. Example An example is given in this subsection for illustration. Consider the approximation of two-dimensional nonlinear
Fig. 2. Plot of TSFMNC.
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775
Fig. 6. Difference of TSFMNC and TSFMLC.
Fig. 3. Plot of TSFMLC.
The errors show that TSFMNC better approximates the function than TSFMLC. 3. Applications to modeling and control of nonlinear dynamic systems 3.1. Approximation of nonlinear dynamic systems using Takagi–Sugeno fuzzy models with nonlinear consequence The TSFMNC is used to describe dynamic systems. It is of the following form: Rule 0: IF x1 is about 0 . . . and xn is about 0 THEN x˙ ¼ A0 cðxÞ. Rule j1 j2 ; . . . ; jn : IF x1 is about j1 e . . . and xn is about jn e THEN x˙ ¼ A j1 j2 ;...; jn cðxÞ.
Fig. 4. Difference of f ðx1 ; x2 Þ and TSFMNC.
following error in modeling: XX ð f ðx1 ; x2 Þ TSFMLCðx1 ; x2 ÞÞ2 ¼ 0:0088 x1
x2
x1
x2
x1
x2
XX ð f ðx1 ; x2 Þ TSFMNCðx1 ; x2 ÞÞ2 ¼ 0:0079 XX
(5) (6)
ðTSFMLCðx1 ; x2 Þ TSFMNCðx1 ; x2 ÞÞ2
¼ 1:0078 104
where x ¼ ½x1 ; x2 ; . . . ; xn T are the system states and cðxÞ ¼ ½c1 ðx1 Þ; c2 ðx2 Þ; . . . ; cn ðxn ÞT . By using center of gravity method for defuzzification, TSFMNC can be represented as Eq. (8) where h j1 j2 ;...; jn ðxÞ is the possibility for the j1 j2 ; . . . ; jn th rule to fire: X h j1 j2 ;...; jn ðxÞA j1 j2 ;...; jn cðxÞ (8) x˙ ¼ h0 ðxÞA0 cðxÞ þ j1 j2 ;...; jn
(7)
Consider the nonlinear system (9) where f ðxÞ is a vector field defined over compact region D Rn satisfying Assumptions A1, A3 and the following Assumption A4: 2
A4. f 2 Cn2 . Therefore, f, @@xf and @@x2f are continuous and therefore bounded over D: x˙ ¼ f ðxÞ
(9) T
Fig. 5. Difference of f ðx1 ; x2 Þ and TSFMLC.
Suppose f ðxÞ can be written as ½ f1 ðxÞ; . . . ; fn ðxÞ . What we ˆ ¼ mean by approximation is finding a TSFMNC given by fðxÞ ˆ ½ fˆ1 ðxÞ; . . . ; fˆn ðxÞT such that k f ðxÞ fðxÞk is small. Since ˆ k f ðxÞ fðxÞk is small iff each of its components (which are nonlinear functions) is small, then by applying Theorem 1 proven in the previous section, the following corollary is obtained:
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Corollary 2. Any smooth nonlinear system(9) satisfying the Assumptions A1, A3 and A4can be approximated, to any degree of accuracy, by a TSFMNC(8). Similarly, smooth nonlinear control system x˙ ¼ f ðxÞ þ gðxÞu can also be approximated using a TSFMNC given by X x˙ ¼ h0 ðxÞðA0 cðxÞ þ B0 uÞ þ h j1 j2 ;...; jn ðxÞ j1 j2 ;...; jn
ðA j1 j2 ;...; jn cðxÞ þ B j1 j2 ;...; jn uÞ
where x1 denotes the angle (in radians) of the pendulum from the vertical, x2 ¼ x˙ 1 the angular velocity, g the gravity constant, m the mass of the pendulum, M the mass of the cart, 2l is the length of the pendulum, u the force applied to the cart and a ¼ 1=ðm þ MÞ. The values used in the simulation are, g ¼ 9:8 m/s2, m ¼ 2 kg, M ¼ 8 kg, a ¼ 0:1, 2l ¼ 1 m. By neglecting the higher order terms Eq. (11) can be written as x˙ 2 ¼
3.2. Approximation of nonlinear state-feedback controllers using Takagi–Sugeno fuzzy controller with nonlinear consequence Takagi–Sugeno fuzzy controller with nonlinear consequence (TSFCNC) is designed using parallel distributed compensation (PDC) method. The j1 j2 ; . . . ; jn th rule of the TSFCNC is given by
The system (12) is approximated by the following two-rule Takagi–Sugeno fuzzy model with linear consequence: R1 : If x1 is about 0 then # " # "0 1 x1 0 x˙ 1 g a u ¼ 0 x2 þ x˙ 2 4l=3 aml 4l=3 aml R2 : If x1 is about p=2 then # "0 1 x x˙ 1 1 2g ¼ 0 x2 x˙ 2 2
pð4l=3 aml cos ð88 ÞÞ " # 0
a cos ð88 Þ þ u 4l=3 aml cos 2 ð88 Þ
(10)
j1 j2 ;...; jn
Following similar argument as in the above sections, the following theorem is obtained:
(12)
3.4. Modeling of TSFMLC and design of TSFCLC
Rule 0: IF x1 is about 0 . . . and xn is about 0 THEN u ¼ K0 cðxÞ. Rule j1 j2 ; . . . ; jn : IF x1 is about j1 e . . . and xn is about jn e THEN u ¼ K j1 j2 ;...; jn cðxÞ. The output of the TSFCNC is X u ¼ h0 K0 cðxÞ h j1 j2 ;...; jn ðxÞK j1 j2 ;...; jn cðxÞ
g sin ðx1 Þ a cos ðx1 Þu 2 4l=3 aml cos ðx1 Þ 4l=3 aml cos 2 ðx1 Þ
By utilizing the concept of parallel distributed compensation (PDC), the following two rules are designed for the controller:
Theorem 3. For smooth nonlinear state feedback controller u ¼ KcðxÞ where x is defined over a compact region can be approximated, to any degree of accuracy, by TSFCNC (10).
R1 : If x1 is about 0 then u1 ¼ K1 x. R2 : If x1 is about p=2 then u2 ¼ K2 x. 3.5. Modeling of TSFMNC and design of TSFCNC
Remark 4. Most of the real-life control system applications like ball and beam system, truck and trailer docking system, rotational pendulum, etc., are having some of its state variables in terms of sin function and hence can be better approximated using this TSFCNC. 3.3. Inverted pendulum — an application A famous benchmark problem namely the inverted pendulum control problem is chosen for study. The nonlinear and non-stable behavior of the inverted pendulum problem renders the use of conventional method very difficult. The control objective is to balance the inverted pendulum for the approximate range of vertical angle, namely x1 2 ðp=2; p=2Þ. The equations of motion of the pendulum [17] are x˙ 2 ¼
g sin ðx1 Þ amlx22 sin ð2x2 Þ=2 a cos ðx1 Þu 4l=3 aml cos 2 ðx1 Þ
(11)
Eq. (12) can also be written as
x˙ 1 x˙ 2
"
¼
0
# 1 sin ðx1 Þ 0 x2 #
g 4l=3 aml cos 2 ðx1 Þ " 0 a cos ðx1 Þ þ u 4l=3 aml cos 2 ðx1 Þ
(13)
It is clear from the Eq. (13) that Eq. (12) is expressable in the form of x˙ ¼ AðxÞcðxÞ þ BðxÞu. Now the system (13) is approximated by the following two-rule Takagi–Sugeno fuzzy model with nonlinear consequence: R1 : If x1 is about 0 then # " # "0 1 sin ðx1 Þ 0 x˙ 1 g a u ¼ þ 0 x2 x˙ 2 4l=3 aml 4l=3 aml
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Table 1 Performence of TSFCLC and TSFCNC Performance index
TSFCLC
TSFCNC
J
2.9634e+005
2.9455e+005
R2 : If x1 is about p=2 then # "0 1 sin ðx1 Þ x˙ 1 g ¼ 0 x˙ 2 x2 4l=3 aml cos 2 ð88 Þ " # 0
a cos ð88 Þ þ u 4l=3 aml cos 2 ð88 Þ By utilizing the concept of parallel distributed compensation (PDC), the following two rules are designed for the controller where cðxÞ ¼ ½sin ðx1 Þ; x2 T : R1 : If x1 is about 0 then u1 ¼ K1 cðxÞ. R2 : If x1 is about p=2 then u2 ¼ K2 cðxÞ.
3.6. Comparison By chossing Q ¼ ½100 0; 0 0 and R ¼ 1 and by using linear quadratic regulator design method, the gain values of TSFCLC are obtained as K1 ¼ ½196:5089 47:1922; K2 ¼ ½3575:4 1168:6 and the gain values of TSFCNC are obtained as K1 ¼ ½196:5089 47:1922; K2 ¼ ½5616:1 1464:7. The performance index is evaluated by Z timestep t T J¼ x ðtÞQxðtÞ þ uT ðtÞRuðtÞ (14) t 0 Table 1 shows the performance index of the TSFCLC and 2
0 60 6 60 AðXÞ ¼ 6 6 h111 x1 6 4 h121 x1 h131 x1 TSFCNC for a simulation of the pendulum for 3 s, with integration timestep ¼ 0:01, starting with initial angle, x1 ¼ 85 . Fig. 7 shows the plot of u for both TSFCLC and TSFCNC. The results shows better performance for the new TSFCNC. 4. Dynamic positioning of ships — a real-life example The model of dynamic positioning of ships (DPS) [6– 8,10,11,14] exhibits nonlinear interaction in three degrees of freedom (surge, sway, and yaw). The DPS control problem is
Fig. 7. Performance comparision of pendulum angle u for TSFCLC and TSFCNC.
usually solved under the assumption that the kinematic equations can be linearized about a constant yaw angle such that linear theory and gain scheduling techniques can be applied. Later, it was solved by globally exponentially stable (GES) nonlinear control law in [7] and solved using Takagi– Sugeno fuzzy controller with linear consequence in [5]. The state space model of the normalized DPS system [5,7] is as given below: X˙ ¼ AðXÞ þ Bu where 2
(15)
3 2 0 x˙ 1 6 x˙ 2 7 60 6 7 6 6 x˙ 3 7 60 7 6 X¼6 7 B¼6 6 h311 x ˙ 6 47 6 4 x˙5 5 4 h321 x˙6 h331
0 0 0 h112 x2 h122 x2 h132 x2
0 0 0 h113 x3 h123 x3 h133 x3
0 0 0 h312 h322 h332
cos ðx3 Þx4 sin ðx3 Þx4 0 h211 x4 h221 x4 h231 x4
3 0 0 7 7 0 7 7 h313 7 7 h323 5 h333
3 u1 u ¼ 4 u2 5 u3
(16)
3 0 7 0 7 7 1 7 h213 x6 7 7 h223 x6 5 h233 x6
(17)
sin ðx3 Þx5 cos ðx3 Þx5 0 h212 x5 h222 x5 h232 x5
2
where x ¼ ½x1 x2 x3 x4 x5 x6 T ¼ ½x y f u v rT . ðx1 ¼ x; x2 ¼ yÞ is the earth-fixed position, x3 ¼ f is the yaw angle, x4 ¼ u is the surge velocity, x5 ¼ v is the sway velocity, and x6 ¼ r is the yaw velocity. The control thrust in the surge, sway and yaw modes are respectively, t 1 ; t2 , and t3 where u ¼ ½u1 u2 u3 T ¼ ½t1 t2 t3 T . In this section, Takagi–Sugeno fuzzy controller with nonlinear consequence (TSFCNC) is used to design fuzzy controller for the dynamic positioning of ships and is compared with Takagi–Sugeno fuzzy controller with linear consequence (TSFCLC).
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4.1. TSFCLC for DPS
Parallel distributed compensation is used to design fuzzy controller rules and are as given below:
The Takagi–Sugeno fuzzy controller with linear consequence with three rules obtained by approximating AðXÞ at yaw angle 0; p=2 and p=2 are as given below: R1 : IF x3 is about 0 THEN x˙ ¼ A1 x þ B1 u R2 : IF x3 is about p=2 THEN x˙ ¼ A2 x þ B2 u R3 : IF x3 is about p=2 THEN x˙ ¼ A3 x þ B3 u
4.2. TSFCNC for DPS
2 60 6 6 60 6 6 60 6 6 AðXÞ ¼ 6 h111 6 6 6 6h 6 121 6 6 4 h131 where a ¼ sin ð2 Þ; b ¼ cos ð88 Þ and 2 0 0 0 1 a 6 0 0 0 a 1 6 6 60 0 0 0 0 6 6 A1 ¼ 6 h111 h112 h113 h211 h212 6 6 h121 h122 h123 h221 h222 6 6 4 h131 h132 h133 h231 h232 2
0
b
1
0
0 0
0 0
1 0
b 0
0 1
h112 h122
h113 h123
h211 h221
h212 h222
h132
h133
h231
h232
7 7 7 7 7 7 h213 7 7 h223 7 7 7 h233 5
0
0
b
1
0
0
0
1
b
0
0 h112
0 h113
0 h211
0 h212
h122 h123 h131 h132 h133 2 0 60 6 6 60 B1 ¼ B2 ¼ B3 ¼ 6 6h 6 311 6 4 h321
h221 h231
h222 h232 h233 3
2
0
60 6 6 60 A3 ¼ 6 6h 6 111 6 4 h121
h331
0
0
0
7 7 7 0 7 7 h313 7 7 7 h323 5
0 h312 h322 h332
0 h112 h122 h132
and
Ri : IF x3 is about j1;i and IF x4 is about j2;i and IF x5 is about j3;i THEN x˙ ¼ Ai c þ Bi u i ¼ 1; 2; . . . ; 12
(18)
3
7 7 7 1 7 7 h213 7 7 7 h223 5
and
where c ¼ ½x1 ; x2 ; sin ðx3 Þ; x4 ; x5 ; x6 T ; j1;1 ¼ j1;2 ¼ j1;3 ¼ j1;4 ¼ p=2; j1;5 ¼ j1;6 ¼ j1;7 ¼ j1;8 ¼ 0; j1;9 ¼ j1;10 ¼ j1;11 ¼ j1;12 ¼ p=2; j2;1 ¼ j2;2 ¼ j2;5 ¼ j2;6 ¼ j2;9 ¼ j2;10 ¼ 10; j2;3 ¼ j2;4 ¼ j2;7 ¼ j2;8 ¼ j2;11 ¼ j2;12 ¼ 10; j3;1 ¼ j3;3 ¼ j3;5 ¼ j3;7 ¼ j3;9 ¼ j3;11 ¼ 6j3;2 ¼ j3;4 ¼ j3;6 ¼ j3;8 ¼ j3;10 ¼ j3;12 ¼ 6, and 3 2 M2;i M3;i 0 0 0 M1;i 7 6 0 M4;i M5;i M6;i 0 7 60 7 6 60 0 0 0 0 1 7 7 6 7 6 Ai ¼ 6 h111 h112 di h113 h211 h212 h213 7 and 7 6 6 h121 h122 di h123 h221 h222 h223 7 7 6 7 6 4 h131 h132 di h133 h231 h232 h233 5 2
0
h333
The Eq. (17) representing AðXÞ in the state space model (15) can also be written as 3 1 1 cos ðx3 Þ sin ðx3 Þ 0 7 x5 2 2 72 3 7 x1 1 1 x4 sin ðx3 Þ cos ðx3 Þ 0 7 76 7 2 2 76 x2 7 0 0 0 1 7 7 76 6 sin ðx Þ 3 7 76 h113 x3 7 (20) 7 h211 h212 h213 76 7 x4 sin ðx3 Þ 7 6 76 7 74 x 5 h123 x3 5 h221 h222 h223 7 7 sin ðx3 Þ 7 x6 7 5 h133 x3 h231 h232 h233 sin ðx3 Þ
which gives importance to the variables x5 and x4 by providing it in the 3rd column of 1st and 2nd, respectively, and leads to nonlinear consequent. It is clear that the approximation of the above AðXÞ requires approximations in x3 ; x4 and x5 . By assuming the range of yaw angle (x3 ), the surge velocity (x4 ) and the sway velocity (x5 ), respectively, as ½p=2; p=2; ½10; 10 and ½6; 6, the 12 rules of Takagi–Sugeno fuzzy controller with nonlinear consequence are obtained at different regions and are as given below:
3
0
6 60 6 60 6 6 A2 ¼ 6 h111 6 6 h121 6 6 4 h131
0
0 1
0
0
0
3
7 7 7 7 7 7 h213 7 7 h223 7 7 7 h233 5
R1 : IF x3 is about 0 THEN u ¼ F1 x R2 : IF x3 is about p=2 THEN u ¼ F2 x R3 : IF x3 is about p=2 THEN u ¼ F3 x
(19)
0
0
0
0 0
3
60 6 6 60 Bi ¼ 6 6h 6 311 6 4 h321
h312 h322
7 7 7 7 7 h313 7 7 7 h323 5
h331
h332
h333
0 0
(21)
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Table 2 Gain values of TSFCLC Gain F1
F2
F3
Value 2 0:9613 4 0:0346 0:0041 2 0:0014 4 0:9937 0:1125 2 0:0014 6 0:9937 6 4 0:1125
0:0001 0:1144 0:9934
0:0348 0:9665 0:1143 0:9996 0:0110 0:0001
1:3614 0:0000 0:0002
0:0002 1:8917 0:4727
0:0043 0:1124 0:9937
1:3890 0:0011 0:0059
0:0042 1:9188 0:4721
0:0043 0:1124 0:9937
1:3890 0:0011 0:0059
0:0042 1:9188 0:4721
0:9996 0:0110 0:0001
3 0:0001 0:3115 5 0:5725 3 0:0016 0:3168 5 0:5725 3 0:0016 0:3168 7 7 0:5725 5
and di ¼ p=2 for i ¼ 14; 912 and di ¼ 1 for i ¼ 58 and 6 M1;1 ¼ M1;3 ¼ M1;5 ¼ M1;7 ¼ M1;9 ¼ M1;11 ¼ 2 6 M1;2 ¼ M1;4 ¼ M1;6 ¼ M1;8 ¼ M1;10 ¼ M1;12 ¼ 2 M2;1 ¼ M2;2 ¼ M2;3 ¼ M2;4 ¼ M2;9 ¼ M2;10 ¼ M2;11 ¼ M2;12 ¼ b M2;5 ¼ M2;6 ¼ M2;7 ¼ M2;8 ¼ 1 1 M3;1 ¼ M3;2 ¼ M3;3 ¼ M3;4 ¼ 2 a M3;5 ¼ M3;6 ¼ M3;7 ¼ M3;8 ¼ 2 1 M3;9 ¼ M3;10 ¼ M3;11 ¼ M3;12 ¼ 2 10 M4;1 ¼ M4;2 ¼ M4;5 ¼ M4;6 ¼ M4;9 ¼ M4;10 ¼ 2 10 M4;3 ¼ M4;4 ¼ M4;7 ¼ M4;8 ¼ M4;11 ¼ M4;12 ¼ 2 1 M5;1 ¼ M5;2 ¼ M5;3 ¼ M5;4 ¼ 2 a M5;5 ¼ M5;6 ¼ M5;7 ¼ M5;8 ¼ 2 1 M5;9 ¼ M5;10 ¼ M5;11 ¼ M5;12 ¼ 2 M6;1 ¼ M6;2 ¼ M6;3 ¼ M6;4 ¼ M6;9 ¼ M6;10 ¼ M6;11 ¼ M6;12 ¼ b M6;5 ¼ M6;6 ¼ M6;7 ¼ M6;8 ¼ 1
4.3. Comparison
Parallel distributed compensation is used to design fuzzy controller rules and are as given below:
For the following values of Q and R, linear quadratic regulator techniques are applied to find out the gain values
Fig. 8. Performance comparison of x3 for TSFCLC and TSFCNC.
Fig. 10. Performance comparison of x5 for TSFCLC and TSFCNC.
Fig. 9. Performance comparison of x4 for TSFCLC and TSFCNC.
Ri : IF x3 is about j1;i and IF x4 is about j2;i and IF x5 is about j3;i THEN u ¼ Ki c i ¼ 1; 2; . . . ; 12. where c ¼ ½x1 ; x2 ; sin ðx3 Þ; x4 ; x5 ; x6 T .
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of Fi s in TSFCLC and Ki s in TSFCNC and are given in Tables 2 and 3, respectively: 3 2 1 0 0 0 0 0 60 1 0 0 0 07 2 3 7 6 1 0 0 60 0 1 0 0 07 7 4 5 Q¼6 6 0 0 0 0 0 0 7 and R ¼ 0 1 0 7 6 0 0 1 40 0 0 0 0 05 0 0 0 0 0 0
The performance index J1 and J2 given by the following equations are shown in Table 4. It shows that TSFCNC performs better than TSFCLC: X
J1 ¼
x1 ðtÞ þ x2 ðtÞ þ x3 ðtÞ þ x4 ðtÞ þ x5 ðtÞ þ x6 ðtÞ
t¼0:01:0:01:2
(22)
Table 3 Gain values of TSFCNC Gain K1
Value 2 0:5723 6 0:7362 6 4 0:4183
K2
2
K3
2
K4
2
K5
2
K6
2
K7
2
K8
2
K9
2
K10
2
K11
2
K12
2
0:6258 6 0:4106 6 4 0:6254 0:6358 6 0:6103 6 4 0:4170 0:5597 6 0:5257 6 4 0:6737 0:5723 6 0:7362 6 4 0:4183 0:6258 6 0:4106 6 4 0:6254 0:6358 6 0:6103 6 4 0:4170 0:5597 6 0:5257 6 4 0:6737 0:5723 6 0:7362 6 4 0:4183 0:6258 6 0:4106 6 4 0:6254 0:6358 6 0:6103 6 4 0:4170 0:5597 6 0:5257 6 4 0:6737
0:4171 0:1749 0:8866
0:1642 0:7306 3:3640
0:3910 0:3188 0:2950
0:4692 0:4491 0:7759
0:1573 0:6245 3:4878
0:5234 0:4113 0:1514
0:4658 0:1138 0:8810
0:1733 0:7175 3:3548
0:5218 0:3208 0:2969
0:4176 0:5047 0:7375
0:1494 0:6091 3:4764
0:4171 0:1749 0:8866
0:1642 0:7306 3:3640
0:3910 0:3188 0:2950
0:4692 0:4491 0:7759
0:1573 0:6245 3:4878
0:5234 0:4113 0:1514
0:4658 0:1138 0:8810
0:1733 0:7175 3:3548
0:3918 0:4084 0:1507
0:5218 0:3208 0:2969
0:4176 0:5047 0:7375
0:1494 0:6091 3:4764
0:4171 0:1749 0:8866
0:1642 0:7306 3:3640
0:3910 0:3188 0:2950
0:4692 0:4491 0:7759
0:1573 0:6245 3:4878
0:5234 0:4113 0:1514
0:4658 0:1138 0:8810 0:4176 0:5047 0:7375
0:1733 0:7175 3:3548 0:1494 0:6091 3:4764
0:3918 0:4084 0:1507
0:5218 0:3208 0:2969 0:3918 0:4084 0:1507
0:7157 0:8662 0:0094
3 0:1786 0:0060 7 7 0:9011 5
0:7228 0:5848 0:3387
3 0:1249 0:0425 7 7 0:9773 5
0:7200 0:6521 0:0162
3 0:1797 0:0501 7 7 0:8959 5
0:7177 0:7924 0:3959 0:7157 0:8662 0:0094
3 0:1239 0:0024 7 7 0:9891 5 3 0:1786 0:0060 7 7 0:9011 5
0:7228 0:5848 0:3387
3 0:1249 0:0425 7 7 0:9773 5
0:7200 0:6521 0:0162
3 0:1797 0:0501 7 7 0:8959 5
0:7177 0:7924 0:3959 0:7157 0:8662 0:0094
3 0:1239 0:0024 7 7 0:9891 5 3 0:1786 0:0060 7 7 0:9011 5
0:7228 0:5848 0:3387
3 0:1249 0:0425 7 7 0:9773 5
0:7200 0:6521 0:0162
3 0:1797 0:0501 7 7 0:8959 5
0:7177 0:7924 0:3959
3 0:1239 0:0024 7 7 0:9891 5
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5. Conclusion This paper presents T–S fuzzy model with nonlinear consequence. It is proved that it can approximate a particular class of nonlinear functions, nonlinear dynamic systems and nonlinear control systems. Moreover, TSFCNC can approximate state feedback controllers by using parallel distributed compensation method. A benchmarking control problem, namely, inverted pendulum is simulated and the results show that TSFCNC performs better than TSFCLC. A real-life example of dynamic positioning of ship is simulated and the results also show that TSFCNC performs better than TSFCLC. Fig. 11. Performance comparison of x6 for TSFCLC and TSFCNC.
Acknowledgements The authors are very thankful to Indian Space Research Organization for financial support through RESPOND project. The authors are also thankful to the reviewers for their valuable help in improving the quality of the paper. References
Fig. 12. Performance comparison of x1 y1 for TSFCLC and TSFCNC.
J2 ¼
X
u1 ðtÞ þ u2 ðtÞ þ u3 ðtÞ
(23)
t¼0:01:0:01:2
The dotted lines in Figs. 8–11 show the performance of TSFCLC and the solid lines show the performance of TSFCNC. The figures show that TSFCNL performs better than TSFCLC. The plot of x1 versus y1 , shown in Fig. 12, shows the path of the ship from the earth fixed position ð10; 10Þ to the position (0, 0). Since the dotted line representing TSFCLC and solid line representing TSFCNL concides, it seems to be a straight solid line. But at the finishing point, after zooming it is clear that TSFCNC performs better than TSFCLC.
Table 4 Performance of TSFCLC and TSFCNC Performance index
TSFCLC
TSFCNC
J1 J2
200.6146 189.0391
200.4305 54.5321
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