Adaptive Sliding Mode RBF Neural Network Control
Sliding mode control is an effective approach for the robust control of a class of nonlinear systems with uncertainties defined in compact sets. The direction of the control action at any moment is determined by a switching condition to force the system t
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Adaptive Sliding Mode RBF Neural Network Control
Sliding mode control is an effective approach for the robust control of a class of nonlinear systems with uncertainties defined in compact sets. The direction of the control action at any moment is determined by a switching condition to force the system to evolve on the sliding surface so that the closed-loop system behaves like a lower order linear system. For the method to be applicable, a so-called matching condition should be satisfied, which requires that the uncertainties be in the range space of the control input to ensure an invariance property of the system behavior during the sliding mode. Sliding mode control is frequently used for the control of nonlinear systems incorporated with neural network. Stability, reaching condition, and chattering phenomena are known important difficulties. For mathematically known models, such a control is used directly to track the reference signals. However, for uncertain systems with disturbance, to eliminate chattering phenomena, there is the need to design neural network compensator and then the sliding mode control law is used to generate the control input.
9.1
Typical Sliding Mode Controller Design
Sliding mode control (SMC) was first proposed and elaborated in the early 1950s in the Soviet Union by Emelyanov and several co-researchers such as Utkins and Itkis. During the last decades, significant interest on SMC has been generated in the control research community. For linear system x_ ¼ Ax þ bu; x 2 Rn ; u 2 R
© Tsinghua University Press, Beijing and Springer Nature Singapore Pte Ltd. 2018 J. Liu, Intelligent Control Design and MATLAB Simulation, https://doi.org/10.1007/978-981-10-5263-7_9
ð9:1Þ
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9 Adaptive Sliding Mode RBF Neural Network Control
A sliding variable can be designed as sðxÞ ¼ cT x ¼
n X
ci xi ¼
n1 X
i¼1
ci xi þ xn
ð9:2Þ
i¼1
where x is state vector, c ¼ ½ c1 cn1 1 T . In sliding mode control, parameters c1 ; c2 ; ; cn1 should be selected so that the polynomial pn1 þ cn1 pn2 þ c2 p þ c1 is Hurwitz, where p is Laplace operator. For example, n ¼ 2; sðxÞ ¼ c1 x1 þ x2 , to guarantee the polynomial p þ c1 Hurwitz, the eigenvalue of p þ c1 ¼ 0 should has negative real part, i.e., c1 [ 0; e.g., if we set c1 ¼ 10, then sðxÞ ¼ 10x1 þ x2 . For another example, n ¼ 3; sðxÞ ¼ c1 x1 þ c2 x2 þ x3 , to guarantee the polynomial p2 þ c2 p þ c1 Hurwitz, the eigenvalue of p2 þ c2 p þ c1 ¼ 0 should has negative real part. For example, we can design k [ 0 in ðp þ kÞ2 ¼ 0, and then, we can get p2 þ 2kp þ k2 ¼ 0. Therefore, we have c2 ¼ 2k; c1 ¼ k2 ; e.g., if we set k ¼ 5, we can get c1 ¼ 25; c2 ¼ 10 and then sðxÞ ¼ 25x1 þ 10x2 þ x3 . Now, we consider a second-order system and there are two steps in the SMC design. The first step is to design a sliding surface so that the plant restricted to the sliding surface has a desired system response. The second step is to construct a controller to drive the plant’s state trajectory to the sliding surface. These constructions are built on the
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