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Comments on: “Control of chaotic dynamical systems using support vector machines” [Phys. Lett. A 317 (2003) 429]
Physics Letters A 342 (2005) 278–279 www.elsevier.com/locate/pla
Comments on: “Control of chaotic dynamical systems using support vector machines” [Phys. Lett. A 317 (2003) 429] Huaping Liu ∗ , Fuchun Sun, Zengqi Sun Department of Computer Science and Technology, Tsinghua University, and State Key Laboratory of Intelligent Technology and Systems, Beijing 100084, PR China Received 27 September 2004; accepted 20 April 2005 Available online 27 April 2005 Communicated by A.R. Bishop
Abstract In the above Letter, the authors proposed an interesting approach to design the feedback controller for chaotic systems. However, since the problem how to produce the nonlinear compensation term was not considered, the control scheme present by them is not reasonable and cannot be implemented in general cases. 2005 Elsevier B.V. All rights reserved. Keywords: Support vector machine; Linear feedback; Chaotic oscillators
Very recently, an interesting new approach for controlling chaotic dynamical systems was proposed in Ref. [1], where the dynamic system is decomposed into a sum of linear and nonlinear parts, and the support vector machine (SVM) is used to approximate the nonlinear part. The authors pointed out that the SVM could be used to compensate the nonlinear part and only a linear controller need to be designed. In this Letter, we argue that the control scheme proposed
DOI of original article: 10.1016/j.physleta.2003.09.004. * Corresponding author.
E-mail address: [email protected] (H. Liu). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.04.047
in [1] is not reasonable. For simplicity and avoiding lengthy statements, some symbols and notations with the same meanings as these in [1] are employed in the following derivations. Consider the chaotic system with control effort x˙ = f (x) + Bu = fL (x) + fNL (x) + Bu = Mx + fNL (x) + Bu.
(1)
In [1], a nonlinear function f˜NL (x) presented by SVM is used to approximate the nonlinear part fNL (x). Then the authors obtained the following equation: x˙ = Mx + Bu + fˆNL (x),
(2)
H. Liu et al. / Physics Letters A 342 (2005) 278–279
where fˆNL (x) = fNL (x) − f˜NL (x) is the approximation error. Here, one question arises: How to produce the nonlinear compensation term −f˜NL (x)? Obviously, the chaotic dynamical system cannot compensate itself, and the SVM can only be used to construct f˜NL (x) but cannot introduce it into the dynamical systems. In practice, the compensation term −f˜NL (x) must be produced by an additional external control effort. In this case, the whole control effort should consist two parts: u = u1 + u2 ,
(3)
where u2 is used to compensate the nonlinear term f˜NL (x), and u1 is the linear control law designed in [1] by using the pole-placement technique. Substituting (3) into (1) yields: x˙ = Mx + fNL (x) + Bu1 + Bu2 .
(5)
Substituting (5) into (4) yields: x˙ = Mx + fˆNL (x) + Bu1 ,
exists in general cases (in the examples presented in [1], the authors setB = [1, 1, 1]T , which is even not a square matrix). An alternative approach to obtain u2 is the famous least square approach, which can result in an approximation solution u˜ 2 = (B T B)−1 B T f˜NL (x). However, since the error between u˜ 2 and u2 is significant and cannot be compensated, this approximation will raise some other unpredictable issues. Finally, we point that a similar compensation control approach has been proposed by [2], where the input matrix in square and invertible. In summary, the control scheme presented in Fig. 1 of [1] cannot be implemented because the authors neglected the fact that the nonlinear compensation term must be produced by an external control effort.
Acknowledgements
(4)
Then, to obtain the expected equation (2), we must specify Bu2 = −f˜NL (x).
279
(6)
which is similar to Eq. (8) in [1]. Then we can use the approach presented in [1] to design u1 . After that, the whole control effort can be obtained by (3). Moreover, we must point out a very serious problem in this approach. Though SVM can be used to approximate the nonlinear function f˜NL (x) very well and we can easily obtain Bu2 by (5), what we need in the control design is u2 , but not Bu2 . When the input matrix B is not invertible, no solution to Eq. (5)
This work was jointly supported by the National Excellent Doctoral Dissertation Foundation (Grant No. 200041), the National Key Project for Basic Research of China (Grant No. G2002cb312205), the National Science Foudation of China (Grant Nos. 60474025, 60321002, 60334020, 90405017), and the Basic Research Foundation of Tsinghua University (Grant No. JC2003028).
References [1] A. Kulkarni, V.K. Jayaraman, B.D. Kulkarni, Control of chaotic dynamical systems using support vector machines, Phys. Lett. A 317 (2003) 429–435. [2] Y.C. Chang, A robust tracking control for chaotic Chua’s circuits via fuzzy approach, IEEE Trans. Circuits Systems I Fund. Theory Appl. 48 (7) (2001) 889–895.
Comments on: “Control of chaotic dynamical systems using support vector machines” [Phys. Lett. A 317 (2003) 429] Huaping Liu ∗ , Fuchun Sun, Zengqi Sun Department of Computer Science and Technology, Tsinghua University, and State Key Laboratory of Intelligent Technology and Systems, Beijing 100084, PR China Received 27 September 2004; accepted 20 April 2005 Available online 27 April 2005 Communicated by A.R. Bishop
Abstract In the above Letter, the authors proposed an interesting approach to design the feedback controller for chaotic systems. However, since the problem how to produce the nonlinear compensation term was not considered, the control scheme present by them is not reasonable and cannot be implemented in general cases. 2005 Elsevier B.V. All rights reserved. Keywords: Support vector machine; Linear feedback; Chaotic oscillators
Very recently, an interesting new approach for controlling chaotic dynamical systems was proposed in Ref. [1], where the dynamic system is decomposed into a sum of linear and nonlinear parts, and the support vector machine (SVM) is used to approximate the nonlinear part. The authors pointed out that the SVM could be used to compensate the nonlinear part and only a linear controller need to be designed. In this Letter, we argue that the control scheme proposed
DOI of original article: 10.1016/j.physleta.2003.09.004. * Corresponding author.
E-mail address: [email protected] (H. Liu). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.04.047
in [1] is not reasonable. For simplicity and avoiding lengthy statements, some symbols and notations with the same meanings as these in [1] are employed in the following derivations. Consider the chaotic system with control effort x˙ = f (x) + Bu = fL (x) + fNL (x) + Bu = Mx + fNL (x) + Bu.
(1)
In [1], a nonlinear function f˜NL (x) presented by SVM is used to approximate the nonlinear part fNL (x). Then the authors obtained the following equation: x˙ = Mx + Bu + fˆNL (x),
(2)
H. Liu et al. / Physics Letters A 342 (2005) 278–279
where fˆNL (x) = fNL (x) − f˜NL (x) is the approximation error. Here, one question arises: How to produce the nonlinear compensation term −f˜NL (x)? Obviously, the chaotic dynamical system cannot compensate itself, and the SVM can only be used to construct f˜NL (x) but cannot introduce it into the dynamical systems. In practice, the compensation term −f˜NL (x) must be produced by an additional external control effort. In this case, the whole control effort should consist two parts: u = u1 + u2 ,
(3)
where u2 is used to compensate the nonlinear term f˜NL (x), and u1 is the linear control law designed in [1] by using the pole-placement technique. Substituting (3) into (1) yields: x˙ = Mx + fNL (x) + Bu1 + Bu2 .
(5)
Substituting (5) into (4) yields: x˙ = Mx + fˆNL (x) + Bu1 ,
exists in general cases (in the examples presented in [1], the authors setB = [1, 1, 1]T , which is even not a square matrix). An alternative approach to obtain u2 is the famous least square approach, which can result in an approximation solution u˜ 2 = (B T B)−1 B T f˜NL (x). However, since the error between u˜ 2 and u2 is significant and cannot be compensated, this approximation will raise some other unpredictable issues. Finally, we point that a similar compensation control approach has been proposed by [2], where the input matrix in square and invertible. In summary, the control scheme presented in Fig. 1 of [1] cannot be implemented because the authors neglected the fact that the nonlinear compensation term must be produced by an external control effort.
Acknowledgements
(4)
Then, to obtain the expected equation (2), we must specify Bu2 = −f˜NL (x).
279
(6)
which is similar to Eq. (8) in [1]. Then we can use the approach presented in [1] to design u1 . After that, the whole control effort can be obtained by (3). Moreover, we must point out a very serious problem in this approach. Though SVM can be used to approximate the nonlinear function f˜NL (x) very well and we can easily obtain Bu2 by (5), what we need in the control design is u2 , but not Bu2 . When the input matrix B is not invertible, no solution to Eq. (5)
This work was jointly supported by the National Excellent Doctoral Dissertation Foundation (Grant No. 200041), the National Key Project for Basic Research of China (Grant No. G2002cb312205), the National Science Foudation of China (Grant Nos. 60474025, 60321002, 60334020, 90405017), and the Basic Research Foundation of Tsinghua University (Grant No. JC2003028).
References [1] A. Kulkarni, V.K. Jayaraman, B.D. Kulkarni, Control of chaotic dynamical systems using support vector machines, Phys. Lett. A 317 (2003) 429–435. [2] Y.C. Chang, A robust tracking control for chaotic Chua’s circuits via fuzzy approach, IEEE Trans. Circuits Systems I Fund. Theory Appl. 48 (7) (2001) 889–895.