1. 루프-쉐이핑 H 제어기 설계 (Design of Loop-Shaping H Controller)
Jan. 15, 2007
Sohn, Jong Joo

2. Table of Contents Recall of the Previous Work
Loop-Shaping H Design Procedure
Design of Controller for SG
Summary
Future Works 1

3. I. Recall of the Previous Work
Phase I : Model Identification
1. Thermal-Hydraulic Model : RELAP
2. Plant Experimentation
3. Model Structure Selection
4. Model(Parameter) Estimation
5. Validation
Phase II : Design of Controller
1. Loop-Shaping H Design Procedure
2. Implementation of overall closed loop system
2. Simulation of various operating situations
4. Performance Evaluation & validation 2

4. 1+Tz*s Gs(s) = K * -----------------
I. Recall
 Identified System Models of 4 Major System Inputs
1+Tz*s
Gf (s) = K * * exp(-Td*s)
s(1+Tp1*s)(1+Tp2*s)
Feedwater Flowrate :
1+Tz*s Gs(s) = K *
s(1+Tp1*s)
Steam Flowrate :
1+Tz*s
Gtf (s) = K * * exp(-Td*s)
s(1+Tp1*s)(1+Tp2*s)
Feedwater temperature :
1+Tz*s
Gp(s) = K * * exp(-Td*s)
s(1+Tp1*s)(1+Tp2*s)
Reactor Power : 3

5. I. Recall  Model Validation
♦ Identified Models were validated comparing with Thermal-Hydraulic Model 4

6. II. Loop-Shaping H Design Procedure
 Concept
 Design Procedure 5

7. II. Loop-Shaping H Design Procedure
 Concept
H Loop - Shaping Problem
- A design procedure that is based on H robust stabilization combined with classical loop shaping, proposed by McFarlane and Glover(1992)
- incorporating loop shaping : to obtain
performance/robust stability trade-offs
- H optimization problem : to guarantee closed-loop stability and a level of stability at all frequency 6

8. II. Loop-Shaping H Design Procedure
 Concept - Standard H Optimal Control Theory
General Presentation of H Optimal Control
where
Controller
♦ To find stabilized controller K(s) such that minimize 7

9. II. Loop-Shaping H Design Procedure
The controller design consists essentially three stages as follows :
♦ Stage 1 : Open-loop shaping
♦ Stage 2 : Robust stabilization of the shaped plant with respect to uncertainty using H optimization.
♦ Stage 3 : Construction of Final Controller 8

10. II. Loop-Shaping H Design Procedure
Stage 1 : Open-loop shaping
♦ The nominal plant G and shaping functions are combined to form
the shaped plant,
Gs = Wpof G Wprf W
prf
pof G s
- Select of a controller which achieves sufficiently high (low) open- loop gain at low (high) frequency,
can guarantee concerning closed - loop performance (robust stability) can be made at these frequencies. 9

11. II. Loop-Shaping H Design Procedure
Stage 2 : Robust stabilization
- The normalized left coprime factorization of a shaped plant model, Gs, can be expressed as
Gs = M -1 N
- A set of perturbed plant models, Gp, can be characterized by
Gp = ( M + △M ) -1 ( N + △N )
where △M and △N are stable unknown transfer functions which represent the uncertainty in the nominal shaped plant(Gs) △N N u y
△ M
M -1  Ks _ + GP P
Uncertainty Model
y : measured variables
u : control input 10

12. II. Loop-Shaping H Design Procedure
- A feedback controller Ks to maximize the stability margins of the closed-loop system under the uncertain perturbations. 
where P is the augmented open - loop plant as Gs,
 is the H∞ norm from  to ,
 is the stability margin that defines a stable family of the
perturbed plant
y : measured variables
u : control input 11

13. II. Loop-Shaping H Design Procedure
Stage 3 : Final controller K
- Once the Ks for the shaped model Gs is obtained,
the final feedback controller K for the nominal plant G is then constructed by combining the H∞ controller Ks with the shaping functions Wprf and Wpof
such that K = Wprf Ks Wpof
Wprf G Ks
Wpof K 12

14. III. Design of Controller for SG
 Controller for TF from Feedwater to Level (At 5% of Rx. Power )
 Controller for TF from Feedwater to Level (At 80% of Rx. Power ) 13

15. III. Design of Controller for SG
 Controller for TF from Feedwater to Level (At 5% of Rx. Power )
♦ System Gain 14

16. III. Design of Controller for SG
♦ Loop Shaping
20 (s )
W prf =
s (s+25.41)
W pof = 1 15

17. III. Design of Controller for SG
Ak =
Bk = 1.0e+003 *
Ck =
Dk = 16

18. III. Design of Controller for SG
♦ Unit Step Response of Feedback System 17

19. III. Design of Controller for SG
 Controller for TF from Feedwater to Level (At 80% of Rx. Power )
♦ System Gain 18

20. III. Design of Controller for SG
♦ Loop Shaping
♦ Controller
(not provieded)
40 (s )
W prf =
s (s+2.715)
W pof = 5 19

21. III. Design of Controller for SG
♦ Unit Step Response of Feedback System 20

22. IV. Summary
Controllers for Feedwater- to- Level Model are under development.
- Developed controllers for 5 & 80% showed good control performance.
9 sets of controller will be developed for each of major system inputs. 21

23. V. Future Works - Reduction of controller system order.
1. Development of Controllers for each design point of each transfer function.
- Reduction of controller system order.
- Interpolation of the sets of controller using appropriate interpolation method between design points.
2. Implementation of overall control System and Evaluation
- Simulation of various operating situations
- Performance & Robustness Evaluation 22

24. Design Considerations
Survey on Control Theory
Design Considerations
PID Control
LQG Control
H Control
1.Applicability
- SISO
- Frequency Domain
- MIMO
- Time Domain
- Time & Frequency
Domain
2. Robustness Design
- Not Guaranteed
- No Guaranteed
Stability Margins
- Guaranteed Stability
Margins
3. Implementation of
Design Spec.
- Trial-and-Error
Method
- Trade-off Between
Conflicting Objectives
in Various Frequency
Ranges 23

25. Advantages of Loop-Shaping Design Techniques
- Pole-zero cancellations phenomena can be avoided
- Wide range of perturbations can be covered
by adoption of coprime factor uncertainty model
- Explicit solution can be obtained because coprime
factorization is normalized 24