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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

III. Design of Controller for SG ♦ Loop Shaping 20 (s+0.09226) W prf = ------------- s (s+25.41) W pof = 1 15

III. Design of Controller for SG Ak = 0 18.0988 0 0 0 0 0 0 -24.2661 0.0033 -0.0214 0.0005 0.0004 0.0222 0 0 0.0051 -0.4717 0.4703 0.4804 0.4939 0 0 -0.6075 -5.3975 7.2676 8.7923 5.3422 0 0 -5.4296 -49.9288 61.4308 84.0272 49.3102 0 0 5.5183 51.0049 -63.5168 -86.4916 -50.3826 0 0 -0.5947 -5.0709 6.9288 8.3876 5.0143 Bk = 1.0e+003 * 0 0 0.0084 0.1816 0 1.9374 0 -0.4474 0 -0.4021 0 -0.5761 0 -1.1627 Ck = 0.6884 38.9507 0 0 0 0 0 Dk = 0 0 16

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

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

III. Design of Controller for SG ♦ Loop Shaping ♦ Controller (not provieded) 40 (s+0.1075) W prf = ------------- s (s+2.715) W pof = 5 19

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

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

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

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

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