Control Systems with Actuator Saturation:

Transcription

1 Control Systems with Actuator Saturation: Anti-windup Mechanism and Its Activation Xiongjun Wu Shanghai Jiao Tong University Zongli Lin Shanghai Jiao Tong University and University of Virginia The 5 th Swedish-Chinese Conference on Control

2 1 Actuator Saturation and Integrator Windup 2 Anti-windup Mechanism and Its Activation 3 L 2 Gain Anti-windup Design 4 Anti-windup Design for Large Region of Stability 5 Multiple Activations of Anti-windup Mechanism 6 Questions and Comments 2

3 1 Actuator Saturation and Integrator Windup k P R k I G (s) Y Actuator saturation is a common phenomenon Integrators are commonly present in a controller 3

4 1 Actuator Saturation and Integrator Windup k P =1 R =1(t ) k I = s 2 Y(0)=0 Y 4

5 1 Actuator Saturation and Integrator Windup Role of integrator: elimination of steady state error = reset of reference k P =1 R 1 =1(t ) R 2 =3x1(t ) - k I = s 2 Y(0)=0 Y R 1 =1(t ) R 2 =3x1(t ) 5

6 1 Actuator Saturation and Integrator Windup k P =1 R =1(t ) - k I = s 2 Y(0)=0 Y 6

7 1 Actuator Saturation and Integrator Windup k P =2 R =1(t ) - k I = s Y(0)=0 Y With saturation Without saturation 7

8 1 Actuator Saturation and Integrator Windup k P =2 R =1(t ) - - k I = s Y(0)=0 Y k a - K a =0 K a =1 K a =10 8

9 2 Anti-windup Mechanism and Its Activation R - - C (s) 1-1 G (s) Y E c - Systematic design of E c that guarantees a large size of stability region (domain of attraction) good closed-loop performances such as a small L 2 gain from the disturbance to the output 9

10 2 Anti-windup Mechanism and Its Activation The traditional approach to anti-windup design is to activate the anti-windup mechanism as soon as the saturation occurs. Immediate Activation 10

11 2 Anti-windup Mechanism and Its Activation A Recent Innovation [Sajjadi-Kia & Jabbari, CDC 09]: To delay the activation of the anti-windup mechanism until the degree of saturation reaches to a certain level. h / g, g 1 d d h Delayed Activation Result: Improved transient performance. 11

12 2 Anti-windup Mechanism and Its Activation Our Proposal [Wu & Lin, CDC 10]: To activate the anti-windup mechanism in anticipation of the occurrence of saturation. h / g, g 1 a a h Anticipatory Activation 12

13 2 Anti-windup Mechanism and Its Activation Problem: To examine the effectiveness of anticipatory activation of anti-windup mechanism in L 2 1. gain anti-windup design 2. anti-windup design for large region of stability 13

14 3 L 2 Gain Anti-windup Design Consider the following system xp Ap xp Bww B2sat h( u) y C2xp D21w D22sat h( u) z C1xp D11w D12sat h( u) Let a linear dynamic controller be given in the form of xc Ac xc Bcy y Bcww u Ccxp Dcy y 14

15 3 L 2 Gain Anti-windup Design Under the immediate activation scheme, a static anti-windup design is to add to the controller a correction term proportional to q u sat h ( u) as soon as saturation occurs: xc Ac xc Bcy y Bcww 1 u Ccxp Dcy y 2 where, This leads to the following closed-loop system: 1 1 q q 2 2 x Ax 1 Bww ( Bq Bη ) W q z 1 Czx Dzww ( Dzq Dzη) W q u 1 WCux WDuww W ( Duq Duη) W q q 1 W WCuu where, q u u u x x p sat( ), x c 15

16 3 L 2 Gain Anti-windup Design Theorem [Sajjadi-Kia & Jabbari, CDC 09]: The closed-loop system is stable and the L 2 gain from w to z is less than if there exist a diagonal matrix M > 0, a matrix Q > 0 and a vector X such that T AQ QA * * * * T T T MBq X Bη M * 0 * CuQ DuqM Duη X M * 0 T T Bw 0 Duw I * CzQ DzqM Dzη X 0 Dzw I If this LMI is feasible, then the anti-windup gain can be obtained as XM

17 3 L 2 Gain Anti-windup Design Delayed activation scheme: Theorem [Sajjadi-Kia & Jabbari, 09]: The closed-loop system is stable with an L 2 gain from w to z less than if there exist a diagonal matrix M 0, a matrix Q 0 and a vector X such that 1( g) * * * * T Bw ( g) I 0 * * 2( g) 0 M * 0 0, g gd, Cz( g) Q Dzw( g) 3( g) I 0 CuQ Duw 4( g) 0 M where ( g) A( g) Q QA ( g), ( g) MB ( g) X B ( g), T T T T 1 2 q ( g) D ( g) M D ( g) X, ( g) D ( g) M D ( g) X 3 zq z 4 uq u T A( g) Q QA ( g) * * Bw ( g) I * 0, g 1 Cz ( g) Q Dzw( g) I The anti-windup gain can then be obtained as: XM 1. 17

18 3 L 2 Gain Anti-windup Design Anticipatory activation scheme: Theorem: The closed-loop system is stable and the L 2 gain from w to z is less than if there exist a diagonal matrix M 0, a matrix Q 0 and a vector X such that T A( g) Q QA ( g) * * Bw ( g) I * 0, g 1, Cz ( g) Q Dzw ( g) I 1( g) * * * * T Bw ( g) I 0 * * 2( g) 0 M * 0 0, g {1, ga}. Cz ( g) Q Dzw ( g) 3( g) I 0 CuQ Duw 4( g) 0 M The anti-windup gain can then be obtained as: XM 1. 18

19 3 L 2 Gain Anti-windup Design Simulation Results Consider plant and controller with A B B A B B p 2 1 c cy cw C2 D D21, Dc Dcy D cw C D11 D and select g d , g Immediate activation scheme: Delayed activation scheme: Anticipatory activation scheme: a h 1,

20 3 L 2 Gain Anti-windup Design Transient performance under different activation schemes 20

21 4 Anti-windup Design for Large Region of Stability Consider a linear system with actuator saturation xp Ap xp Bpsat h( u) y Cpxp Assume that a linear dynamic compensator has been designed as xc Ac xc Bc y u Ccxc Dcy y Objective: To examine how delayed and anticipatory activation of the anti-windup mechanism will affect the size of the domain of attraction of the resulting closed-loop system as compared to the immediate activation scheme. 21

22 4 Anti-windup Design for Large Region of Stability With the immediate activation scheme: The controller law with anti-windup compensation is given by xc Ac xc Bc y Ec(sat h( u) u) u Ccxc Dcy y The closed-loop system can be written as where x ( A BF) x Bsat ( Fx) x Ap BpDcC BpCc x, A, x B C A c p c c Bp B, F DcCp Cc. E c h 22

23 4 Anti-windup Design for Large Region of Stability The design of E c can be formulated into the following optimization problem with BMI constraints max P0, E, H s.t. a) X ( P) R c b) Binlinear matrix nequalities c) ( P) L( H ) n where X R co{ x1, x2,, xl}, for some a priori given points xi R, is referred to as a shape reference set. An iterative LMI based algorithm was developed in [Cao, Lin & Ward, TAC 02] to solve the above BMI optimization problem. Similar algorithms can be developed under the framework of delayed activation and anticipatory activation. 23

24 4 Anti-windup Design for Large Region of Stability Simulation examples Consider a stable plant x1 0.1x1 0.5sat( u1) 0.4sat( u2) x2 0.1x2 0.4sat( u1) 0.3sat( u2) or an unstable plant with the first equation replaced by Let a PI controller be given by xc1= x1, xc1= x2 u1 = 10 x1 xc1, u2=10 x1 x E x 0.1x 0.2x 0.5sat( u ) 0.4sat( u ) Let gd , and ga , we can obtain c delayed c , E c anticpatory

25 4 Anti-windup Design for Large Region of Stability Stable plant Unstable plant 25

26 5 Multiple Activations of Anti-windup Mechanism Wu & Lin, CCC 11: Multiple Anti-windup Loops for Multiple Activations 26

27 5 Multiple Activations of Anti-windup Mechanism L 2 gain anti-windup design: A B B A B B p 2 1 c cy cw C2 D D21, Dc Dcy D cw C D11 D ˆ 1 = , e10,

28 3 L 2 Gain Anti-windup Design: Multiple Case large input small input Transient performance under different activation schemes 28

29 3 L 2 Gain Anti-windup Design: Multiple Case large input small input Transient performance under different activation schemes 29

30 6 Questions and Comments 30