Lecture Fundamentals of control systems: Chapter 7 - TS. Huỳnh Thái Hoàng

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Lecture Fundamentals of control systems: Chapter 7 - TS. Huỳnh Thái Hoàng Lecture Notes Fundamentals of Control Systems Instructor: Assoc. Prof. Dr. Huynh Thai Hoang Department of Automatic Control Faculty of Electrical & Electronics Engineering Ho Chi Minh City University of Technology Email: hthoang@hcmut.edu.vn huynhthaihoang@yahoo.com Homepage: www4.hcmut.edu.vn/~hthoang/6 December 2013 © H. T. Hoang - www4.hcmut.edu.vn/~hthoang/ 1 Chapter 7 MATHEMATICAL MODEL OF DISCRETE TIME CONTROL SYSTEMS6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 2 Content Introduction to discrete-time discrete time system The Z-transform Transfer function of discrete-time system State-space equation of discrete-time system6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 3 Introduction to discrete- discrete-time systems6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 4 Digital control systems r(kT) u(kT) (kT) uR(t) y(t) Computer D/A Plant yfb(kT) A/D Sensor “Computer” = computational equipments based on microprocessor technology (microprocessor, (microprocessor microcontroller microcontroller, PC, DSP,…). Advantages of digital control system:  Flexibility  Easy to implement complex control algorithms  Computer can control many plants at the same time. 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 5 Discrete control systemsr(kT) Discrete u(kT) uR(t) y(t) Hold Plant processing yfb(kT) Sampling Sensor Discrete control systems are control systems which have signals at several points being discrete signal.6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 6 Sampling Sampling is the reduction of a continuous signal to a discrete signal. x(t) x*(t) Mathematical expression T describing the sampling process: x(t)  X * ( s )   x ( kT )e kTs k t k 0 0 x*(t) Shannon’s Theorem: 1 t f   2 fc T 0 If quantization error is negligible, then A/D converters are approximate the ideal samplers. 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 7 Sampled--data hold SampledSampled data hold is the reconstruction of discrete signal to Sampled-data a continuous signal. x*(t) xR (t) ZOH Zero-order hold (ZOH): keep x*(t) signal unchanged between two consecutive sampling instants instants. t 0 Transfer function of the ZOH. ZOH xR(t) 1  e Ts GZOH ( s )  t s 0 If quantization error is negligible, then D/A converters are approximate the zero-order hold. 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 8 The Z- Z-transform6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 9 Definition of the Z- Z-transform Consider x(k), x(k) kk=0 0,1,2,… 12 being a discrete signal signal. The Z- Z transform of x(k) is defined as:  X ( z )  Z x (k )   x ( k ) z k k   where:  z  eTs (s is the Laplace variable, T is the sampling period)  X(z) : Z-transform of x(k). Z Notation: x ( k )  X ( z )  If x(k) = 0,  k < 0 then X ( z )  Z x ( k )   x ( k ) z k k 0 Region Of Convergence (ROC): set of z such that X(z) is finite.6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 10 An interpretation of the Z- Z-transform Suppose x(t) being a continuous signal, sample x(t) at the sampling periode T , we have a discrete signal x(k) = x(kT). Th ...