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

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Lecture "Fundamentals of control systems - Chapter 7: Mathematical model of discrete time control systems" presentation of content: Introduction to discrete - time system, the Z-transform, transfer function of discrete-time system,... Invite you to reference.

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Nội dung Text: 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 SYSTEMS 6 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 system 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 3
Introduction to discrete- discrete-time systems 6 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 systems r(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-transform 6 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).  The mathematic model of the process of sampling x(t)  X * ( s )   x (kT )e kTs (1) k 0  The Z-transform of the sequence x(k) = x(kT).  X ( z )   x ( k ) z k (2) k 0  Due to z  eTs, the right hand-side of the expression (1) and (2) are identical. So performing Z-transform of a signal is equivalent to discretizing this signal. 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 11
Properties of the Z- Z-transform Given x(k) and y(k) being two sequences which have the Z- transforms: Z x ( k )  X ( z ) Z y ( k )  Y ( z )  Linearity: Z ax ( k )  by ( k )  aX ( z )  bY ( z )  Ti Time shifting: hifti Z x ( k  k0 )  z  k0 X ( z)  Scale in Z-domain: Z a k x (k )  X (a 1 z ) Z kx ( k )   z  Derivative in Z-domain: dX ( z ) dz  Initial-value theorem: x (0)  lim X ( z ) z   Fi l l theorem: Final-value th x ( )  lim(1  z 1 ) X ( z ) z 1 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 12
The Z- Z-transform of basic discrete signals  Dirac impulse: (k) 1 if k  0  (k )   1 0 if k  0 k Z  (k ( k )  1 0  Step p function: u(k) 1 if k  0 1 u(k )   0 if k  0 k 0 Z u( k )  z z 1 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 13
The Z- Z-transform of basic discrete signals (cont’)  Ramp function: r(k) kT if k  0 r(k )   0 if k  0 k 0 Z u( k )  Tz z  12  Exponential p function: x(k) e -akT if k  0 x(k )   0 if k  0 k 0 Z x ( k )  z z  e aT 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 14
Discrete transfer function 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 15
Derive transfer function (TF) from difference equation u(k) y(k) Discrete system  The input-output input output relation ship of a discrete system can be described by the difference equation: a0 y ( k  n )  a1 y ( k  n  1)  ...  an 1 y ( k  1)  an y ( k )  b0u( k  m)  b1u( k  m  1)  ...  bm1u( k  1)  bmu( k ) where n>m, n is the order of the system.  Taking the Z Z-transform transform the two sides of the above equation: a0 z nY ( z )  a1 z n 1Y ( z )  ...  an 1 zY ( z )  anY ( z )  b0 z mU ( z )  b1 z m 1U ( z )  ...  bm1 zU ( z )  bmU ( z ) 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 16
Derive TF from difference equation (con’t (con’t))  Taking the ratio Y(z)/U(z) to obtain the transfer function: Y ( z ) b0 z m  b1 z m 1  ...  bm 1 z  bm G( z)   U ( z ) a0 z n  a1 z n 1  ...  an 1 z  an  The above transfer function can be transformed into the equivalent form:  ( n m ) 1  m 1 m Y ( z) z [b0  b1 z  ...  bm 1 z  bm z ] G( z)   U ( z) a0  a1 z 1  ...  an 1 z n 1  an z n 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 17
Derive TF from difference equation _ Example  Consider a system described by the difference equation. equation Derive its transfer function: y ( k  3)  2 y ( k  2)  5 y ( k  1)  3 y ( k )  2u( k  2)  u( k )  Solution: Taking g the Z-transform the difference equation: q z 3Y ( z )  2 z 2Y ( z )  5zY ( z )  3Y ( z )  2 z 2U ( z )  U ( z ) Y ( z) 2z2  1  G( z)   3 U ( z ) z  2 z 2  5z  3 Y ( z) z 1 ( 2  z 2 )  G( z)   U ( z ) 1  2 z 1  5z 2  3z 3 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 18
Calculate transfer function from block diagram R(s) Y(s) + GC(z) ZOH G(s)  T H(s) Y ( z) GC ( z )G ( z )  The closed-loop TF: Gk ( z )   R ( z ) 1  GC ( z )GH ( z ) where GC (z ) : TF of the controller, derive from difference equation  G ( s )   G ( s ) H ( s )  1 G ( z )  (1  z )Z 1   GH ( z )  (1  z )Z    s   s  6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 19
Calculate TF from block diagram – Example 1  Find the closed-loop closed loop transfer function of the system: R(s) Y(s) + ZOH G(s)  T  0.5 3 G (s)  s2 1  G(s)  1  3  Solution: G ( z )  (1  z )Z    (1  z )Z    s   s ( (ss  2)  20.5 1 3 z (1  e )  (1  z ) 2 ( z  1)( z  e 2 20.5 )  0.948  a  z (1  e  aT ) G( z)  z  0.368 Z   aT  s ( s  a )  ( z  1)( z  e ) 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 20