New PID designs for sampling control and batch process optimization

Transcription

1 IFAC PID 08, May 9-, 08, Ghent, Belgium New PID designs for sampling control and batch process optimization Tao Liu Institute of Advanced Control Technology Dalian University of Technology Institute of Advanced Control Technology

2 Outline Introduction PID design for sampled control systems 3 Learning PID design for batch optimization 4 Conclusions & Outlook Institute of Advanced Control Technology

3 Introduction Some facts of PID controllers Advantages: Simple structure and low cost; Easily understood and commanded by users; Most widely used and commercialized in industrial applications. Disadvantages [, ]: Generally not optimal in control performance; (often used as an inferior to demonstrate other advanced control designs) Difficult to analyze the robust stability against system uncertainties. (lowly valued for theoretical contribution in top control-relevant journals) [] Karl J. Åström, P.R. Kumar. Control: A perspective. Automatica, 04, 50(), [] Tao Liu, Furong Gao. Industrial Process Identification and Control Design: Step-test and Relay- Experiment-Based Methods. London UK: Springer, 0.

4 Searching results by ISI web of Science More applications for various systems PID Controller: >000 papers in 07 Steadily increasing over the past 30 years! PI Controller: >400 papers in 07 Motivation of this talk Trends:. More digital implementation in computer systems;. Performance improvement for batch/repetitive processes.

5 PID design for sampled control systems Relationship between the PID control and the internal model control (IMC) y sp y - C u d i G G ˆd o G d y d o - e Process model: Frequency domain: Bz ( ) G z Az ( ) G d Bs () e As () s IMC: applicable for stable processes y sp The internal model control (IMC) structure e y - K PID u d The unity feedback control structure i G ˆd o G d d o K C GC Equivalent relationship PID: applicable for stable, integrating, and unstable processes

6 Review of the IMC design in frequency domain Step. Decompose the model into the minimum-phase (MP), non-mp, and all-pass parts. For example: 0 0 where,,. 0 0 G kp( 0s) e ( s)( s) s G mp kp ( s)( s) Step. Specify the desired closed-loop transfer function for set-point tracking. T F FG nmpgap where F and F are two low-pass filters. For a stable process described as above, F F ( s ) s 3 Desired closed-loop transfer function single tuning parameter 0 T ensure ensure G GmpGnmpGap FG mp FG nmp s e 0 3 ( s) 0s strictly proper all-pass, i.e. s G e s G ap nmp 0s 0s s 0 e No overshoot Minimal ISE s 0

7 Discrete-time domain IMC design Step. Decompose the model into the minimum-phase (MP), non-mp, and all-pass parts. For example: a a b b where,,,. G kp( z b )( z b ) z ( z a )( z a ) d G mp kp( z b) ( z a )( z a ) Step. Specify the desired closed-loop transfer function for set-point tracking. T F FG nmpgap where F and F are two low-pass filters. For a stable process described as above, F F ( ) c ( z c ) b ( )( ) b z b Desired closed-loop transfer function T ensure ensure G GmpGnmpGap FG mp FG nmp strictly proper all-pass, i.e. ( ) ( b )( z b ) z ( z ) ( b )( z b ) c c Gap z d G z b nmp ( b )( z b ) ( b )( z b ) d

8 Control Signal Discrete-time domain IMC design Step 3. Determine the IMC controller. IMC ( z) For a stable process described as above, C b 0 C T( z) Gz ( ) ( z a )( z a )( ) ( b ) ( )( ) ( )( ) c IMC( z) kp z b z c b z b In case, it could provoke inter-sample ringing in the control signal! 6 T GC single tuning parameter c Time (sec)

9 Discrete-time domain IMC design Solution: Introduce another filter to remove such a zero for implementation F ( z) 3 z z b b C ( z) F ( z) C ( z) RIMC 3 IMC Control performance assessment For a stable process described by G ( z) K p z z p z d T ( z) d c z c c d 0, k d, Y ( z) z R( z) y( kt z s) kd - k d. c Rz ( ) z z Step response in time domain c, Set-point tracking error IAE ( ) n kd c c c r [ y( kts )] c lim n k d k d c c Disturbance rejection error to a step change IAE d K p y( kts) kd c zp ( )( ) Quantitative tuning

10 IMC-based PID design in frequency domain Step 4. Determine the equivalent controller in a unity feedback control structure. For G kp( 0s) e ( s)( s) K C GC M lim Ks ( ), let Ks () s0 s M (0) K( s) [ M (0) M (0) s s ] s! Ds K() s kc s s I s F k I ( s)( s) [( ) ( ) ( ) ] K k 3 s p s 0 s 0 s e Step 5. Determine the PID controller by the Taylor approximation []. Since C D M(0) / M (0) M (0) / Important advantage: single tuning parameter F (0.0 0.) [] Lee, Y., Park, S., Lee, M., & Brosilow, C. PID controller tuning for desired closed-loop responses for SI/SO systems. AIChE Journal, 44, 06-5, 998. D

11 IMC-based PID design in frequency domain Alternatively, the delay term in the process model or the IMC controller may be rationally approximated to derive a PID, e.g., The first-order Taylor approximation []: Pade approximation [3]: Model reduction [4, 5]: Frequency response fitting [6]: s e s s e ( s) / ( s) ( s)( s) m i ( ) s [] Rivera, D. E., Morari, M., Skogestad, S. Internal model control. 4. PID controller design. Ind. Eng. Chem. Res., 5, 5-65, 986. [3] Fruehauf P. S., Chien I.-L., Lauritsen M. D. Simplified IMC-PID tuning rules. ISA Transactions, 994, 33, [4] Skogestad, S. Simple analytical rules for model reduction and PID controller tuning. Journal of Process Control, 003, 3, [5] Lee, J. Cho, W. Edgar, T. F. Simple analytic PID controller tuning rules revisited, Industrial & Engineering Chemistry Research, 53(3), , 04. [6] Wang, Q. G., Yang, X. P. Single-loop controller design via IMC principles. Automatica, 37, , 00. PID e s min K (j ) K(j ) i i Half rule (0., ) i cb

12 IMC-based PID design in discrete-time domain For sampling control implementation, the first-order differentiation is generally used to discretize the above frequency domain design. For direct PID design in discrete-time domain, let the equivalent controller in a unity feedback control structure be lim K( z) owing to. z M( z) z A PID controller is determined by the Taylor approximation [, ]. M () K z M M z z z! K C GC ( ) () ()( ) ( )... D( z ) KPID( z) kc ( D) I( z ) z D single tuning parameter K( z) (0.0, 0.) kc M() I M() / M() D M() / M() [] Wang, D., Liu T.*, Sun X., Zhong Chongquan. Discrete-time domain two-degree-of-freedom control for integrating and unstable processes with time delay. ISA Transactions, 63, -3, 06. [] Cui J., Chen Y., Liu T.*. Discrete-time domain IMC-based PID control design for industrial processes with time delay. The 35th Chinese Control Conference (CCC), Chengdu, China, , 06.

13 IMC-based PID design in discrete-time domain Closed-loop system robust stability analysis Small gain theorem + T The T + structure Magnitude/Phase margin, Maximal sensitivity index For a stable process described by k () p d z G z z T ( z) ( ) T( z) m z G ( z) C ( z) p PID G p z C PID z ( ) ( ) ( z) [ G( z) G ( z)] G ( z) m m m M s No analytical solution! kp d Td ( z) z K ( Td ) z Ti( z ) z Td k Td ( z) m() z z T z z Td p d z K ( Td ) i( )

14 DP IMC-based PID design in discrete-time domain Disturbance rejection performance G( z) ( z 0.9) ( z0.8669)( z0.9048) z d Disturbance response peak d c 0.8 Tuning guideline: The single adjustable parameter can be monotonically increased or decreased in a range of to meet a good trade-off c (0.8, ) between the closed-loop control performance and its robust stability. c

15 Control Signal Output IMC-based PID design in discrete-time domain C An illustrative example: G() s e s 0.5s Sampling period: Ts 0.0( s) ( z ) G( z) z ( z)( z0.9608) C PID IMC ( z) ( z 0.99)( c ) ( z ) 5 ( z) ( z ).4956( z ) z c Proposed Ref.[0] Time(sec) Proposed 6 Ref.[0] Avoid differential kick! Time(sec) [0] Panda R.C. Synthesis of PID tuning rule using the desired closed-loop response, Industrial & Engineering Chemistry Research, 47(), , 008.

16 Improved IMC-based PID design for disturbance rejection G p For a slow process described by with a large time constant, ke s s p kp d Discrete-time domain model: G( z) z with a pole z close to the unit circle z p z p p Load disturbance transfer function (optimal GC T by IMC) y G( GC) d y Gd ( GC) dˆ i o G G The slow pole of or affects the disturbance rejection performance! d Solution: Introduce an asymptotic constraint to remove the effect of slow dynamics. Modify the desired transfer function: p f p [ ( ) e ] Discrete-time domain: p lim( T ) 0 z d T lim ( T ) 0 s/ RIMC ( f s ) p z/ z p ( s) e RIMC lim ( T ) 0 d s T d ( z) nd ( ) ( z) c 0 nd z c ( ) nd nd ( z ) ( ) p c c nd ( zp )( c) 0

17 Improved IMC-based PID design for disturbance rejection An illustrative example: G s e (0s)(s) long tail Slow pole: s 0 Comparison with the standard IMC and SIMC by Skogestad (JPC, 003) [] Tao Liu, Furong Gao. New insight into internal model control filter design for load disturbance rejection. IET Control Theory & Application, 4 (3), , 00.

18 Discrete-time PID design for integrating and unstable processes Problem: The classical PID control structure could not suppress large overshoot for set-point tracking! There exists severe water-bed effect! Solution: A two-degree-of-freedom (DOF) control structure r u c - u u f + + e y r + - y Advantage: the set-point tracking is decoupled from load disturbance rejection. Set-point tracking: open-loop control IMC design: T r GC Disturbance rejection: closed-loop control using a PID controller, s C f The desired transfer function for set-point tracking is the same as the standard IMC design The desired transfer function for disturbance rejection : ( ) T z z z z m m f d( ) ( 0 m ) ( f ) m z d

19 Discrete-time PID design for integrating and unstable processes Integrating process: G ( z) I k ( z z ) p 0 ( z )( z z ) p z d Unstable process: kp( z z0) GU ( z) z ( z z )( z z ) u p d The following asymptotic tracking constraints must be satisfied, lim( T ) 0 z d lim( T ) 0 zu z d or z p (close to the unit circle) T GC f d GC f Derive the closed-loop controller: d lim ( Td ) 0 z dz C f Td T d G Taylor approximation PID

20 Control Signal Output Discrete-time PID design for integrating and unstable processes Integrating process: 5s 0.e Gs () s(5s). 0.8 Frequency domain design: s(5s) C () s s 0.( cs ) 5s e Tr () s ( cs ) Td () s Cf () s G( s) T ( s) s d() 4 ( f s ) T s 4 5 d s e 5s f 4 5 5(0. f ) e Proposed 0. Ref.[3] Discretized ref.[3] Ref.[5] Time (sec) Proposed Ref.[3] 5 Discretized ref.[3] Ref.[5] Time (sec) [] Wang, D., Liu T.*, Sun X., Zhong Chongquan. Discrete-time domain two-degree-of-freedom control for integrating and unstable processes with time delay. ISA Transactions, 63, -3, 06. [3] Liu T.*, Gao, F. Enhanced IMC design of load disturbance rejection for integrating and unstable processes with slow dynamics. ISA Transactions, 50 (), 39-48, 0.

21 Control Signal Output Discrete-time PID design for integrating and unstable processes Sampling period: G( z) C ( z) ( z ) z ( z)( z0.9608) s ( z c ) 5 ( z)( z )( c) ( z z )( ) 4 0 f d( ) 4 ( z f ) T z 4 d C Ts 0.( s) z 5 f d ( p f ) p 4( f ) ( p) ( d d p )( f ) ( ) ( ) f p ( ) ( z z )( z )( z ) 4 f 0 p f ( z) kpz z0 z f 0 z z f z ( )[( ) ( )( ) ] Perturbation: the process gain and time constant are actually 0% larger Proposed 0. Ref.[3] Discretized ref.[3] Ref.[5] Time (sec) Time (sec) Proposed Ref.[3] Discretized ref.[3] Ref.[5] [5] Torrico B.C., Cavalcante M.U., Braga A.P.S., Normey-Rico J.E., Albuquerque A.A.M. Simple tuning rules for dead-time compensation of stable, integrative, and unstable first-order dead-time processes. Ind. Eng. Chem. Res., 5, , 03.

22 Discrete-time PID design for long time delay processes Predictor based control structure Set-point filter PID Predictor Delay-free output prediction [] Tao Liu*, Pedro García, Yueling Chen, Xuhui Ren, Pedro Albertos, Ricardo Sanz. New Predictor and DOF Control Scheme for Industrial Processes with Long Time Delay. IEEE Transactions on Industrial Electronics, 65 (5), , 08. [] Yueling Chen, Tao Liu*, Pedro García, Pedro Albertos. Analytical design of a generalized predictorbased control scheme for low-order integrating and unstable systems with long time delay. IET Control Theory & Application, 0(8), , 06.

23 Discrete-time PID design for long time delay processes The nominal system transfer function K( z) G ( z) d y z Kf z z r z K( z) Gˆ ( z) p p ( ) ( ) ( ) Gp () z K( z) Gˆ ( z) dp K( z) F ( z) z w( z) Closed-loop transfer function for disturbance rejection uz () Td () z K( z) G( z) wz () Kz ( ) G( z) The desired closed-loop transfer function (free of time delay) ( ) nd l c i Td ( z) iz ( z c ) i0 n d i0 l i K( z) Td ( z) T ( z) G ( z) d p Taylor approximation PID

24 Control Signal Output Discrete-time PID design for long time delay processes Example: Gs () 7.5s e 5.5s. 0.8 Sampling period: Ts 0.5( s) PI controller: K G() z Kz () z z ( ) ( z ) c ( z ) 0 ( zp c) ( c) / ( zp )( c) ( z ) ( ) c f f () z ( c ) ( 0 z)( z f ) z Proposed 0. Ref.[6] Ref.[4] Ref.[9] 3.5 Time (s) Proposed 3 Ref.[6].5 Ref.[4] Ref.[9] Time (s) [6] Kirtania, K.M. Choudhury, A.A.S. A novel dead time compensator for stable processes with long dead times. J. Process Control, (3), 6-65, 0. [4] Zhang, W., Rieber, J.M., Gu, D. Optimal dead-time compensator design for stable and integrating processes with time delay. J. Process Control, 8(5), , 008. [9] Normey-Rico J.E., Camacho E. F. Unified approach for robust dead-time compensator design. J Process Control, 9: 38-47, 009.

25 Temperature Discrete-time PID design for long time delay processes A temperature control system of a 4-liter jacketed reactor for pharmaceutical crystallization Sampling period: Ts 3( s) Step response identification: Integrating type process model: G() s G() z e s(760.40s) ( z ) ( z)( z0.996) 00.5s z 34 Pt00 Jacketed reactor Heater PLC Closed-loop controller: 4 ( f ) ( 0 z z ) K( z) z( z )( z ) 4 / ( ) K 0 f 4 (0.996 f ) 4 4 (0.996) ( f) ( 0.996)( f) (0.996 ) Set-point filter: zz ( ) ( ) 4 3 f s f ( z) 4 3 ( f ) ( 0 z z )( z s ) Process Model Time (s)

26 Control signal Temperature response Discrete-time PID design for long time delay processes Heat up the aqueous solution from the room temperature (5 o C) to 45 o C. Load disturbance arises from feeding 00(ml) solvent of distilled water. 0 u 00 Input constraint: regulate the heating power Compared with the filtered PID method [] based on delayed output feedback, more than 30 minutes are saved for recovering the solution temperature to the operation zone of (45±0.) o C against the load disturbance Proposed Ref.[8] Ref.[] Time (min) Time (min) Proposed Ref.[8] Ref.[] [8] Normey-Rico J.E, Camacho E.F. Unified approach for robust dead-time compensator design. J. Process Control, 9, 38-47, 009. [] Jin, Q.B., Liu, Q. Analytical IMC-PID design in terms of performance/robustness tradeoff for integrating processes: From -Dof to -Dof. J. Process Control, 4(3), -3, 04.

27 Our recent publications on discrete-time domain PID design Tao Liu*, Pedro García, Yueling Chen, Xuhui Ren, Pedro Albertos, Ricardo Sanz. New Predictor and DOF Control Scheme for Industrial Processes with Long Time Delay. IEEE Transactions on Industrial Electronics, 08, 65 (5), Yueling Chen, Tao Liu*, Pedro García, Pedro Albertos. Analytical design of a generalized predictor-based control scheme for low-order integrating and unstable systems with long time delay. IET Control Theory & Application, 06, 0(8), Dong Wang, Tao Liu*, Ximing Sun, Chongquan Zhong. Discrete-time domain twodegree-of-freedom control for integrating and unstable processes with time delay. ISA Transactions, 06, 63, -3. Tao Liu *, Furong Gao. Enhanced IMC design of load disturbance rejection for integrating and unstable processes with slow dynamics. ISA Transactions, 0, 50 (), Tao Liu, Furong Gao *. New insight into internal model control filter design for load disturbance rejection. IET Control Theory & Application, 00, 4 (3), Jiyao Cui, Yueling Chen, Tao Liu *. Discrete-time domain IMC-based PID control design for industrial processes with time delay. The 35th Chinese Control Conference (CCC), Chengdu, China, July 7-9, 06, Hongyu Tian, Xudong Sun, Dong Wang, Tao Liu *. Predictor based two-degree-offreedom control design for industrial stable processes with long input delay. The 35th Chinese Control Conference (CCC), Chengdu, China, July 7-9, 06,

28 PID design for batch process optimization Industrial batch processes, e.g., injection molding machines, chemical reactors Features:. Repetitive operation for production;. Historical cycle information for progressively improving system performance; 3. Time or batch varying uncertainties.

29 PID design for batch process optimization Y k- Y k m e s + - Memory Ŷ k IMC-based iterative learning control (ILC) for perfect tracking using historical cycle information R V k + - C UC Ŷ k- U k- + U k G mo G e m s + D G d - + Y Advantage: batch control optimization Tao Liu, Furong Gao, Youqing Wang. IMC-based iterative learning control for batch processes with uncertain time delay. Journal of Process Control, 0 (), 73-80, 00.

30 PID design for batch process optimization Type I: PID-based ILC system (modify the input error for batch operation) Set-point Type II: ILC-based PID control system (controller retuning for batch operation) Set-point Wang, Y., Gao, F., Doyle F.III., Survey on iterative learning control, repetitive control, and run-to-run control. J. Process Control, 9(0), , 009.

31 Youqing Wang, Tao Liu, Zhong Zhao. Advanced PI control with simple learning set-point design: Application on batch processes and robust stability analysis. Chemical Engineering Science, 7(), 53-65, 0. PID design for batch process optimization Indirect-type ILC based on the PI control structure Advantage: No need to modify the closed-loop PI controller for ILC design, i.e., The PI controller and ILC updating law can be separately designed.

32 PID design for batch process optimization Indirect-type ILC design based on the PI control structure and current output error MEMORY y s(t,k) L 3 L L e(t+,k-) δe (t,k) s y s(t,k) - z e s(t,k) MEMORY e (t, k) s e s(t,k) K I e(t,k) K p MEMORY u(t,k) Plant Y r y(t,k) Shoulin Hao, Tao Liu*, Qing-Guo Wang. PI based indirect-type iterative learning control for batch processes with time-varying uncertainties A D FM model based approach. J. Process Control, submitted, 08.

33 PID design for batch process optimization Indirect-type ILC based on the PID control structure plus feedforward control ILC scheme Memory PID P : x( t, k) [ Am A( t, k)] x( t, k) [ Bm B( t, k)] u( t, k) ( t, k) y( t, k) Cx( t, k), 0 t Tp; x(0, k) x(0), k=,,. t : k : T p : time index batch index batch period Tao Liu, Xue Z. Wang, Junghui Chen. Robust PID based indirect-type iterative learning control for batch processes with time-varying uncertainties. Journal of Process Control, 4 (), 95-06, 04.

34 Robust PID tuning for indirect-type ILC A PID control law in discrete-time domain u ( t) k e( t) k e( t) k [ e( t ) e( t)] PID P I D e( t, k) Y r( t) y( t, k) e( t) e( t ) approximate [ e( t) e( t ) e( t )] / State-space closed-loop PID system description x( t ) A B[ kˆ ˆ ˆ ( ) I D( CI Am ) kpc] Bk x t I () t e( t) C I e( t ) 0 xt () y() t C 0 et ( ) where A Am A() t B Bm B() t kˆ (I k CB ) ( k k ) P D m P I kˆ (I k CB ) k I D m I kˆ (I k CB ) k D D m D

35 Robust PID tuning for indirect-type ILC The H infinity control objective for closed-loop system robust stability e( t) ( t) PID where PID denotes the robust performance level. Theorem : The PID control system is guaranteed robustly stable if there exist P P 0 0 matrices P,,, and positive scalars,, such that R R T T P A A B B Dg T T T T * P 0 PH C P A P B * * PIDI * * * PIDI 0 0 * * * * I 0 * * * * * I where Dg [I 0] T T T H [I 0] A [ A, 0] A [ A P, A P] [ B, 0] T B [ B R, B R ] B T P P * P P A P B R A P B R m m m m T CP P CP P

36 Robust PID tuning for indirect-type ILC Correspondingly, the PID controller is determined by [ kˆ ( CI A ) kˆ C kˆ ] [ R R ] P D m P I k k ˆ (I k CB ) I I D m k k ˆ (I k CB ) k P P D m I k kd 0 where is user specified for implementation. If, it is a PI controller. D An optimal program for tuning PID to accommodate for the uncertainty bounds, Min A( t), B( t) PID Guideline: A smaller value of leads to faster output response with a more aggressive control action, and vice versa. PID Tao Liu, Xue Z. Wang, Junghui Chen. Robust PID based indirect-type iterative learning control for batch processes with time-varying uncertainties. Journal of Process Control, 4 (), 95-06, 04.

37 Robust PI tuning for indirect-type ILC Another robust tuning of PI controller by assigning the closed-loop system poles to a prescribed circular region, D(, r) centered at (,0) with radius r and, i.e., r ( A) D(, r) while the closed-loop transfer function satisfies Theorem : The PI control system is guaranteed robustly D-stable if there exist T matrices P, P, P P, R, R, and positive scalar, such that T T 0 PCˆ 0 PFˆ ˆ T ˆ T * PII D 0 D 0 * * * * * I 0 0 * * * * P 0 * * * * * I ( ) T T ˆ ˆ ˆ ˆ PA AP ( r ) P EE T T T ˆ T P ˆ ˆ ˆ T PF A R FB PA EE PF ˆ AP BR AP BR T T T T AP T ˆ ˆ T P P FA R FB CP P CP P 3 P EE where P P P 3 H( z) ˆ H ( z) C( zi A) D PI ˆ 3 T

38 Robust PI tuning for indirect-type ILC Correspondingly, the PI controller is determined by ( K K ) C K R R P P I I K I Kˆ I K Kˆ C ( CC ) Kˆ T T P P I To optimize the robust H infinity control performance, the PI controller can be determined by solving the following optimization program, Min A( t), B() t PI ( R R ) P ( Kˆ Kˆ ) Guideline: A smaller value of leads to faster output response with a more aggressive control action, and vice versa. PI P I Shoulin Hao, Tao Liu*, Qing-Guo Wang. PI based indirect-type iterative learning control for batch processes with time-varying uncertainties A D FM model based approach. J. Process Control, submitted, 08.

39 PI based set-point learning design for indirect-type ILC Memory ILC scheme PID Based on the PI or PID closed-loop, a learning set-point command is designed as where y ( t, k) y ( t, k ) L e( t, k ) L e ( t, k) L e ( t, k) s s s 3 s y ( t, k) s is the set-point command in the previous cycle, e ( t, k) e ( t, k) e ( t, k ) es ( t, k) es ( t, k) es( t, k) s s s Like PI

40 PI based set-point learning design for indirect-type ILC Memory ILC scheme PID Feedforward controllers are used to adjust the process input u( t, k) u ( t, k) F e ( t, k) F e ( t, k) F e ( t, k) PID s s 3 s Note: The setpoint tracking errors at the current moment, one-step ahead moment, and the error integral in the current cycle are used to construct the feedforward control.

41 PI based set-point learning design for indirect-type ILC Two-dimensional (D) system description of the indirect ILC scheme where ( t, k) e( t, k ) G [ I] T Dw [ I 0 0 C ] x( t, k) x( t, k) es( t, k) es( t, k) Dw () t es( t, k) es( t, k) e( t, k) e( t, k ) x( t, k) es ( t, k) ( t, k) G es ( t, k) e( t, k ) T A B( kp ki kd F ) C B[( kp ki kd F ) L F kd ] C L C L CA CB kp ki kd F C CB kp ki kd F L F kd ( ) [( ) ] B[( k k k F ) L F k ] B( k k k F ) L p i d 3 3 i p i d L I L 3 3 p i d 3 3 i I CB( kp ki kd F ) L CB[( k k k F ) L F k ] L L

42 PI based set-point learning design for indirect-type ILC The control objectives for robust tracking from batch to batch NTp N BP ILC ILC t0 k0 J ( ( t, k ) ( t, k ) ) 0 D Roesser s system stability []: h h x ( i, j) A A A A x ( i, j) (, ) v i j v (, ) A A A A x i j x ( i, j) h x ( i, j) y( i, j) C C v x ( i, j) i, j=0,,,. Robust stability condition []: T A PA P 0 A A A A A A A A A P diag{ P, P } [] Rogers, E., Galkowski, K. & Owens, D. H. Control Systems Theory and Applications for Linear Repetitive Processes, Berlin: Springer, 007.

43 PI based set-point learning design for indirect-type ILC Define x( t, k) h x ( t, k) es ( t, k) v x ( t, k) e( t, k) es ( t, k) Lyapunov-Krasovskii function used for analyzing D asymptotic stability h h x ( t, k) x ( t, k) V VQ V v Q v x ( t, k) x ( t, k ) The objective function of robust batch operation for minimization N Tp N N Tp N BP ILC ILC t0 k0 t0 k0 J ( ( t, k ) ( t, k ) V ) V 0 x(0, 0) x(0,) x(, 0) 0; es (0, 0) es (0,) es (, 0) 0; es (0, 0) es (0,) es (, 0) 0; e(0, 0) e(0,) e(, 0) 0. N T N p V t0 k0 0

44 PI based set-point learning design for indirect-type ILC Theorem 3: The D control system is guaranteed robustly stable with a H infinity control performance level, ILC, if there exist Q, Q 0 0, Q3 0, Q4 0, matrices ˆF, ˆF,, ˆL, ˆL, and positive scalars,, such that 3 T T Q AA B B Dw T T T * Q 0 QG PA PB * * ILCI * * * ILCI 0 0 * * * * I 0 * * * * * I where Q diag{ Q Dg [I 0] T H [I 0], Q, Q3, Q4} [ A T, 0, 0, A T C T ] T A [ A, 0, 0, 0] B [ B, 0, 0, B C ] A 3 ˆL B B( kp ki kd F ) C, B [( k k k F ) Lˆ Fˆ k ], p i d d B [( k k k F ) Lˆ Fˆ k ] p i d 3 3 i B( k ˆ p ki kd F ) L T T T T AmQ Bm( kp ki kd F ) CQ Bm( kp ki kd F ) Lˆ B ˆ mf BmkdQ CQ ˆ L ˆ CQ L ( ) ( ) ˆ ˆ CA Q CB k k k F CQ CB k k k F L CB F CB k Q B ˆ ˆ ˆ m( kp ki kd F ) L3 Bm F3 BmkiQ3 Bm( kp ki kd F ) L Lˆ ˆ 3 L ˆ ˆ Q3 L3 L CBm ( kp k ˆ ˆ ˆ i kd F ) L3 CBm F3 CBmkiQ3 Q4 CBm ( kp ki kd F ) L m m p i d m p i d m m d

45 PI based set-point learning design for indirect-type ILC Correspondingly, the PI type ILC controller is determined by L L L Lˆ Q 4 Lˆ Q Lˆ Q The feedforeward controller is determined by F F Fˆ Q Fˆ Q To optimize the set-point tracking performance, the PI type ILC controller can be determined by solving the following optimization program, Min A( t), B() t ILC Guideline: A smaller value of leads to faster output response with a more aggressive control action, and vice versa. ILC Shoulin Hao, Tao Liu*, Qing-Guo Wang. PI based indirect-type iterative learning control for batch processes with time-varying uncertainties A D FM model based approach. J. Process Control, submitted, 08.

46 PI based indirect-type ILC for batch injection molding injection molding machine The nozzle pressure response to the hydraulic valve input was modeled [] by.39( 5%) z 0.98( 5%) z y( t, k ) u( t, k ) ( t, k ).607( 5%) z ( 5%) z Shi, J., Gao, F., Wu, T.-J. Integrated design and structure analysis of robust iterative learning control system based on a two-dimensional model. Ind. Eng. Chem. Res., 44, , 005.

47 PI based indirect-type ILC for batch process opimtization Equivalently, the process model is rewritten in a state-space form x( t, k ) ( A) x( t, k ) ( B) u( t, k ) ( t, k ) y( t, k ), 0 x( t, k ) time-varying uncertainties ( t) 0 0 ( t) At () ( t) ( t) ( t) 0 ( t) Bt () ( t) 0 0 ( t) ( t) Robustly tuned PI controllers: kp.889 ki ILC controllers: F [.3, 0.] F 0 F L L 0 L [ A B ( k k k F ) C] m m p i d

48 PI based indirect-type ILC for batch process opimtization Case : time-invariant uncertainties Desired output profile Case : repetitive disturbance 00, 0 t 00; Yr 00+5( t 00), 00 t 0; 300, 0 t Tp 00.

49 PI based indirect-type ILC for batch process opimtization Case 3: Time-varying uncertainties ( t) 0. ( tk, ) sin( t ( k)) ( k) [0, ] ATE( k) T p t e( t, k) / T p Plot of the output error for batch operation [] Tao Liu, Xue Z. Wang, Junghui Chen. Robust PID based indirect-type iterative learning control for batch processes with time-varying uncertainties. Journal of Process Control, 4 (), 95-06, 04. [] Y. Wang, Tao Liu, Z. Zhao. Advanced PI control with simple learning set-point design: Application on batch processes and robust stability analysis. Chemical Engineering Science, 7 (), 53-65, 0. [3] Shi, J., Gao, F., Wu, T.-J. Integrated design and structure analysis of robust iterative learning control system based on a two-dimensional model. Ind. Eng. Chem. Res., 44, , 005.

50 Conclusions & Outlook Main Results: Analytical PID design in discrete-time domain for sampled control systems DOF control structure based PID design for improving disturbance rejection Predictor-based PID design for long time delay systems Robust PID tuning methods with respect to the system uncertainty bounds PI based indirect type ILC design for batch process optimization Outlook: Data-driven PID tuning for sampled control systems Fractional-order PID design & PID scheduling for nonlinear systems PID +Memory for learning/intelligent control of industrial batch processes, repetitive systems, and robots etc. Institute of Advanced Control Technology 50

51 Acknowledgement Thanks to my collaborators and students for PID design Pedro Albertos Furong Gao Pedro García Wojciech Paszke Youqing Wang My group photo in 07 Institute of Advanced Control Technology 5

52 Institute of Advanced Control Technology Research interests: Advanced control system design & system optimization; On-line monitoring, control design and control optimization of chemical production batch process; Real-time model predictive control and optimization of crystallization & drying processes; Design of in-situ measurement and control devices; PBM and CFD modeling of crystallization processes. Institute of Advanced Control Technology 5

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