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292  Mathematical Techniques of Fractional Order Systems


            10.3.1 Terminal Sliding Mode Control Law for the Chen
            Chaotic System

            The controller for the system (10.6) with control inputs
                                                       T
                               UðtÞ 5 ½u r1 ; u 1im ; u r2 ; u 2im ; u r3 Š  ð10:17Þ
            is given by:
                  0            12α              ρ                         1
                            k 1 D  ½z r1 1signðz r1 Þjz r1 j Š2a 1 ðz r2 2z r1 Þ2s 1 ðtÞ
                                                ρ
                  B       k 2 D 12α ½z 1im 1signðz 1im Þjz 1im j Š2a 1 ðz 2im 2z 1im Þ2s 2 ðtÞ  C
                  B      12α              ρ                               C
            UðtÞ5  B  k 3 D  ½z r2 1signðz r2 Þjz r2 j Š2ða 3 2a 1 Þz r1 1z r1 z r3 2a 3 z r2 2s 3 ðtÞ  C
                  B    12α                ρ                               C
                  @ k 4 D  ½z 2im 1signðz 2im Þjz 2im j Š2ða 3 2a 1 Þz 1im 1z 1im z r3 2a 3 z 2im 2s 4 ðtÞ A
                          12α               ρ
                       k 5 D  ½z r3 1signðz r3 Þjz r3 j Š1a 2 z r3 2z r1 z r2 2z 1im z 2im 2s 5 ðtÞ
                                                                     ð10:18Þ
            10.3.2 Terminal Sliding Mode Control Law for the Lorenz
            Hyperchaotic System
            The control law for the hyperchaotic Lorenz system shown in (10.8) is given
            in (10.20) with the following control variable:
                                                        T
                              UðtÞ 5 ½u r1 ; u 1im ; u r2 ; u 2im ; u r3 ; u r4 Š ;  ð10:18Þ
            so,

                  0            12α               ρ                        1
                            k 1 D  ½y r1 1 signðy r1 Þjy r1 j Š 2 c 1 ðy r2 2 y r1 Þ 2 s 1
                          12α                  ρ
                  B    k 2 D  ½y 1im 1 signðy 1im Þjy 1im j Š 2 c 1 ðy 2im 2 y 1im Þ 2 y r4 2 s 2  C
                  B         12α               ρ                           C
                  B      k 3 D  ½y r2 1 signðy r2 Þjy r2 j Š 2 c 3 y r1 1 y r2 1 y r1 y r3 2 s 3  C
            UðtÞ 5  B  12α                  ρ                             C
                  B  k 4 D                                                C
                          ½y 2im 1 signðy 2im Þjy 2im j Š 2 c 3 y 1im 1 y 2im 1 y 1im y r3 2 y r4 2 s 4 C
                  B       12α               ρ
                       k 5 D  ½y r3 1 signðy r3 Þjy r3 j Š 2 y r1 y r2 2 y 1im y 2im 1 c 2 y r3 2 s 5
                  @                                                       A
                          12α               ρ
                       k 6 D  ½y r4 1 signðy r4 Þjy r4 j Š 2 y r1 y r2 2 y 1im y 2im 1 c 4 y r4 2 s 6
                                                                     ð10:20Þ
            10.4 ADAPTIVE TERMINAL SLIDING MODE
            SYNCHRONIZATION FOR CHAOTIC AND HYPERCHAOTIC
            SYSTEMS
            In this section, the synchronization of nonidentical systems is presented for
            which the drive system (10.6) is the fractional order complex Chen chaotic
            system and the response system is the fractional order complex Lorenz
            chaotic system. Then, the identical synchronization is done with the frac-
            tional order complex hyperchaotic system with different initial conditions.
               For this purpose consider the following theorem that is valid for the syn-
            chronization of chaotic and hyperchaotic systems (identical and nonidentical):
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