# lec2 - ME6401 Nonlinear Control Systems Outline Nonlinear...

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1 Outline ± Nonlinear System Classifications & stems Representation ± Examples of Nonlinear Systems ± Solution of Nonlinear State Differential Equations ME6401, Nonlinear Control Sys ± Some Common Behavior of Nonlinear Systems What are Nonlinear Systems? ± Nonlinear Systems are dynamical systems that may contain one or more u y System nonlinear components. ± They do not necessarily satisfy the principal of superposition : x α ⎯→ α α α i i i i i i i i i i i i i i i i y x y x y u y u system system system system ) 0 ( ) 0 ( : Response Free : Response Forced

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2 Nonlinearity Classifications ± Structural ± Inherent (natural): Naturally associated with stems Inherent (natural): Naturally associated with system's hardware and motion, e.g., centripetal acc., friction, saturation, etc. ± Intentional (artificial): Artificially introduced into the controller, e.g., switching (on-off), adaptive and sliding-mode controllers. ME6401, Nonlinear Control Sys ± Mathematical ± Linearizable or ‘soft’ nonlinearities ± Nonlinearizable or ‘hard’ nonlinearities: e.g., friction, backlash, hysteresis, etc State-Space Representation ) u x h y u x x & ± Continuous-Time Systems : ) , , ( ), , , ( t t f = = ) ), ( ), ( [ ) ( ) ), ( ), ( [ ) 1 ( k k k k k k k k u x h y u x f x = = + ± Discrete-Time Systems : ± Linear system u D x C u x h u B x A u x f ) ( ) ( ) , , ( ) ( ) ( ) , , ( t t t t t t + = + = ± where A, B, C, D are matrices of appropriate dimensions
3 Exercise Derive a set of state equations for the revolving pendulum example in the last lecture stems pendulum example in the last lecture. ME6401, Nonlinear Control Sys Examples of Nonlinear Systems ± Mechanical Systems ± Quas Static nonlinearities e g friction Quasi-Static nonlinearities, e.g., friction, backlash ± Rigid Body in Space, e.g., satellite, aircraft, etc. ± Multi-link bodies, e.g., robotic manipulator ± Electromechanical Examples ± Brushless dc motors ± Induction motors

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4 Brushless DC Motor stems m q m e d m q q d q m d d K d v L L K i p i L R dt di v L i p i L R dt di ω + ω ω = + ω + = 1
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lec2 - ME6401 Nonlinear Control Systems Outline Nonlinear...

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