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1、不连续控制系统的未来发展与挑战余星火教授澳大利亚皇家墨尔本理工大学科技发展研究院院长东南大学自动化学院讲座教授2报告提纲报告提纲1. 不连续控制系统的定义不连续控制系统的定义2. 不连续控制系统的类别不连续控制系统的类别3. 不连续控制系统的性质不连续控制系统的性质4. 不连续控制系统中存在的问题不连续控制系统中存在的问题5. 不连续控制系统的未来发展及挑战不连续控制系统的未来发展及挑战1. 不连续控制系统的定义不连续控制系统的定义Consider the following Ordinary Differential Equation (ODE) that describes a typic
2、al control system ? ? ? ? ? ? ? ?where commonly, ? ?,? ?,? ? ?. In here, ? and ? ? are smooth, and the control ? is smooth as well. ANDThe term ? ? ? ? ? ? satisfies the Lipschitz condition to ensure the existence and uniqueness of the solutions. For discontinuous control, it is of the form (simples
3、t form)? ? ? ? ? 0 ? ? ? 0where ? ?.32. 不连续控制系统的类别不连续控制系统的类别 Discontinuous control is everywhere typical types are Sliding mode control Switching control Fuzzy control Optimal control State vector control Impulsive control Control in event-triggered systems4Vector control Vector control of AC Induct
4、ion Motor5Fuzzy control A typical fuzzy control systemFor ? ? ? ? ? ? ? ? ?Rule 1: ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?Rule 2: ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?.Rule m: ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?where ?is fuzzy value 6Optimal control 7Consider the 2ndorder dynamical systemThe time optimal control is Nanjing
5、 Normal University, 27 December 20117Impulsive control 8Nanjing Normal University, 27 December 20118A typical Impulsive control system? ? ? ? ? ? ?,? ? ?,? ? ? ?, ? ? ?, ? ? 0,1,2,The impulsive control occurs at ? ? ?.Control in event-triggered systems 9Nanjing Normal University, 27 December 20119A
6、typical event triggered system For ? ? ? ? ?, ? ?,? ? ? ? ?, ? ?,?The event time ? ? ? is determined by an event-trigger ? ? ? ? ? ? ? . If ?, then ? ? ? ? such that ? ? 0.Switching controlSwitching DecisionsProcessController 4Controller 3Controller 2Controller 1For ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?11
7、Note: This is not the same as which is often wrongly used! 0s sConsider single-input control system1.Define a switching manifold which prescribe the desirable properties s(x)2.Design a discontinuous control u(x)such that Sliding mode control 0lim, 0lim 00 sands ss0)(0)(xsuxsuu? ? ? ? ? ? ? ?12Robust
8、ness in SMC systems Consider a single input single output (SISO) system where x is the state, u is the control and represents uncertainties and disturbance, f and g are smooth functions. When an ideal sliding mode is created, we have There exists a virtual control, called equivalent control, When th
9、e matching condition is satisfied),()()(txuxgxfx0, 0ss ),()()(1 txxfxsxgxsueq)()()(1 xfxsxgxsxgIx No uncertainty nor disturbance is involved!3. 不连续控制系统的性质不连续控制系统的性质 可利用的优势 The benefits of switching are enormous for example, ? ? ? ? ? ? 0system 1? ? ? ? ? ? 0system 2? 0 the systems are both asymptoti
10、cally stable.? 0 the systems are both marginally stable. ? 0 the systems are both unstable.13稳定系统可切换成不稳定系统14不稳定系统可切换成稳定系统15不连续控制类别之间的关系 The relationship between these various discontinuous controls is ambiguous 16)(tuy )(tkyu)(21)(22kyyyV0,0ccVVConsider a double integrator given byConsider the feedb
11、ack controlwhere k0. If we take the Lyapunov functionThen17For 01, there is NO singularity.)(1)()(11txtxtsqp)(11/ 11xxqpxss sVqp 3. 不连续控制系统的性质不连续控制系统的性质 从空间(状态）的角度看 Typical discontinuities are Switching between different dynamics Sliding mode control Fuzzy control Time optimal control Bang-bang cont
12、rolJumping in systems states Impulsive control Control in event-triggered systems313. 不连续控制系统的性质不连续控制系统的性质 从时间（切换频率 ）的角度看 Zeno Phenomena The Zeno phenomenon appears when the execution of the discontinuous control system is such that lim? ? ? ? ? ? where ?(the Zeno time) is a right accumulation point
13、 for the time constants sequence ? ?0 The switching frequency tends to be infinite! Definitions Chattering Zeno: ? ? 0,? ? ?, ? ? 0 Genuinely Zeno: ? ? 0,? ? ?, ? ? 032Bouncing ball Zeno behavior occurs when there are an infinite number of discrete transitions in a finite amount of time.Bouncing Bal
14、l (Liberzon, 2003)PositionVelocityImpact of switching frequency The frequency influences significantly the behaviour of dynamics even methodology may differ significantly due to frequency range Low frequency - many existing methodologies can be used by piecing- together various smooth subsystems in
15、time or in state. Medium frequency same as Low frequency though presents challenges of using Lyapunov theory, e.g. piece-wise Lyapunov function. Various causes: deliberate medium frequency such as switched control systems; time-delay due to digitization, etc. High frequency tends to violate usual smooth dynamics based approaches, may need to use
