# Lecture Fundamentals of control systems: Chapter 9 - TS. Huỳnh Thái Hoàng

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Lecture "Fundamentals of control systems - Chapter 9: Design of discrete control systems" presentation of content: Introduction, discrete lead - lag compensator and PID controller, design discrete systems in the Z domain,.... Invite you to reference.

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## Nội dung Text: Lecture Fundamentals of control systems: Chapter 9 - TS. Huỳnh Thái Hoàng

1. 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/ www4 hcmut edu vn/ hthoang/ 6 December 2013 © H. T. Hoang - www4.hcmut.edu.vn/~hthoang/ 1
2. Chapter 9 DESIGN OF DISCRETE CONTROL SYSTEMS 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 2
3. Content  Introduction  Discrete lead – lag compensator and PID controller  Design discrete systems in the Z domain  Controllability and observability of discrete systems  Design D i statet t feedback f db k controller t ll using i polel placement  Design state estimator 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 3
4. Discrete lead lag compensators and PID controllers 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 4
5. Control schemes  Serial compensator R(z) Y(z) + GC(z) ZOH G(z) T H( ) H(z)  State feedback control r(k) u(k) x(t) y(k) + x (k  1)  Ad x (k )  Bd u (k ) Cd K 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 5
6. Transfer function of discrete difference term e(k) u(k) D dde(t )  Differential term: u (t )  dt e( kT )  e[( k  1)T ]  Discrete difference: u ( kT )  T  E ( z )  z 1 E ( z ) U ( z)  T  Transfer function of the discrete difference term: 1 z 1 GD ( z )  T z 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 6
7. Transfer function of discrete integral term e(t) u(t) I t Integral l t  Continuous integral:u(t )   e( )d 0 kT ( k 1)T kT  Di Discrete t integral: i t l u (kT )   e( )d   e( )d   e( )d 0 0 ( k 1)T kT u ( kT ) = u[( k - 1)T ] + ò e ( t ) d t = u[( k - 1)T ] + T (e[( k - 1)T ] + e(kT ) ( k -1) T 2  U ( z )  z 1U ( z )  2  T 1 z E( z)  E( z)   TF of discrete integral term: GI ( z )  T z  1 2 z 1 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 7
8. Transfer function of discrete PID controller  C ti Continuous PID controller: t ll K GPID ( s )  K P   K D s s  Discrete PID controller: KIT z  1 KD z  1 GPID ( z )  K P   2 z 1 T z P I D z KD z  1 or GPID ( z )  K P  K I T  z 1 T z P I D 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 8
9. Digital PID controller r(k) (k) e(k) u(k) y(k) + PID D/A G(s)  A/D U ( z) KIT z  1 KD z  1 G PID ( z )   KP   E( z) 2 z 1 T z u( k )  u( k  1)  K P [e( k )  e( k  1)] )  KIT KD [e( k )  e( k  1)]  [e( k )  2e( k  1)  e( k  2)] 2 T 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 9
10. Digital PID control programming float PID_control(float PID control(float setpoint, setpoint float measure) { ek_2 = ek_1; ek_1 = ek; ek = setpoint – measure; uk_1 = uk; uk = uk_1 uk 1 + Kp Kp*(ek (ek-ek_1) ek 1) + Ki Ki*T/2*(ek+ek T/2 (ek+ek_1) 1) +… + Kd/T*(ek – 2ek_1+ek_2); If uk > umax, uk = umax; If uk < umin, uk = umin; return(uk) } Note: Kp, Ki, Kd, uk, uk_1, ek, ek_1, ek_2 must be declared as gglobal variables;; uk_1,, ek_1 and ek_e must be initialized to be zero; umax and umin are constants. 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 10
11. TF of discrete phase lead/lag compensator  Continuous phase lead/lag compensator: sa a  b phase lead GC ( s )  K sb ab p phase lag g  Discretization using trapezoidal integral: (aT  2) z  (aT  2) GC ( z )  K (bT  2) z  (bT  2) ( aT  2) (bT  2)  Denote zC  and pC  ( aT  2) (bT  2) TF TF off discrete di t phase h l d/l compensator lead/lag t z  zC zC  1 zC  pC phase lead GC ( z )  KC z  pC pC  1 zC  pC phase lag 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 11
12. Approaches to design discrete controllers  IIndirect di t design: d i Fi t design First d i a continuous ti controller, then discretize the controller to have a discrete control system. system The performances of the obtained discrete control system are approximate those of the continuous control system y provided p that the sample time is small enough.  Direct design: Directly design discrete controllers in Z domain. Methods: root locus, pole placement, analytical method,, … 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 12
13. D i discrete Design di t controllers t ll in i the th Z domain d i 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 13
14. Procedure for designing discrete lead compensator using the RL z  zC Lead compensator: GC ( z )  KC ( zC  pC ) z  pC *  Step 1: Determine the dominant poles z1, 2 from desired transient response specification: Overshoot (POT)      s1*, 2  n  jn 1   2 Settling g time ts n z  e* Ts*  1, 2  r  z*  e Tn    z*  Tn 1   2 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 14
15. Procedure for designing discrete lead compensator using the RL  Step 2 St 2: Determine D t i th the deficiency d fi i angle l so that th t the th dominant poles z1*, 2 lie on the root locus of the system after compensated: n m  *  180 0   arg( z1*  pi )   arg( z1*  zi ) i 1 i 1 where pi and zi are poles and zeros of G(z) Geometry formula:  *  180 0   angles from poles of G ( z ) to z1*   angles from zer os of G ( z ) to z1* 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 15
16. Procedure for designing discrete lead compensator using the RL  Step 3: Determine the pole & zero of the lead compensator Draw 2 arbitrarily rays starting from the dominant pole z1*such that the angle between the two rays equal to *. The intersection between the two rays and the real axis are the positions of the pole and the zero of the lead compensator. Two methods often used for drawing the rays:  Bisector method  Pole elimination method  Step 4: Calculate the gain KC using the equation: GC ( z )G ( z ) z z*  1 1 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 16
17. Design discrete lead compensator using RL – Example R(s) Y(s) + GC(z) ZOH G(s) T 50 G( s)  T  0.1sec s( s  5)  Design the compensator GC(z) so that the compensated system has dominant poles with  0.707, n  10 (rad/sec) 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 17
18. Design discrete lead compensator using RL – Example (cont’)  Solution:  The open-loop discrete TF: 50 G( s)  1 G( s)  s( s  5)  G ( z )  (1  z )Z    s  1  50   (1  z )Z  2   s ( s  5)  1  z [( 0 . 5  1  e 0 .5 ) z  (1  e 0 .5  0 . 5e  0. 5 )]   10(1  z )  0 .5   5( z  1) ( z  e ) 2  0.21z  0.18a  z(aT  1  e aT ) z  (1  e aT  aTe  aT )  G( z)  Z 2  ( z  1)( z  0s.607 ( s )a )  a ( z  1) 2 ( z  e aT ) 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 18
19. Design discrete lead compensator using RL – Example (cont’)  The desired poles: z1*, 2  re  j where r  e Tn  e 0.10.70710  0.493   Tn 1   2  0.1  10  1  0.707 2  0.707  z1*, 2  0.493e  j 0.707  z1*, 2  0.375  j 0.320 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 19
20. Design discrete lead compensator using RL – Example (cont’)  The deficiency angle Im z +j 0.375+j0.320   180  ( 1   2 )   3 * 1  152.90  2  125.90 P  3  14.60 * 2 1 3 Re z 1 0 A B +1    84 * 0 pc zc j 6 December 2013 © H. T. Hoàng - www4.hcmut.edu.vn/~hthoang/ 20