Sistemas com saturação no controle

Transcription

1 1 Sistemas com saturação no controle VIII. Estratégias anti-windup - Introdução SOPHIE TARBOURIECH LAAS - CNRS, Toulouse, France Colaboradores principais : J.-M. Gomes da Silva Jr (UFRGS), G. Garcia (LAAS-CNRS), I. Queinnec (LAAS-CNRS)

2 Outline 2 Introduction to anti-windup strategies 1. Introduction 2. A bit of history 3. Problem statement

3 1. Introduction 3 Non Linear Model + Controller The anti-windup techniques represent an alternative to analysis and synthesis methods previously presented Idea: Deal with the saturations but without modifying the controller in the linear region The anti-windup techniques consist in considering the effects of the nonlinearities like saturations a posteriori Idea: introduce some modifications in the original controller in order to mitigate the undesirable effects of the saturation Successfully used in PID controllers Stability analysis [Teel, Morari & co-authors] external stability global stability results application only to open-loop stable systems

4 1. Introduction 4 To summarize, the principe of anti-windup may be represented through the following scheme: w (disturbance) r (reference) u c unconstrained controller y c u 0 u system z (controlled) y (measured) + v 1 v 2 anti windup controller + +

5 1. Introduction 5 The aims: Study of conditions for guaranteing stability (in local and global contexts) of anti-windup schemes To take into account the amplitude limitations on both the input and the output of the system (limited actuator and sensor) In global context: open-loop properties of stability are required when the open-loop system is unstable : local stability. In local context: design the anti-windup gain in order to enlarge the basin of attraction of the closed-loop system Propose numerically efficient procedures to compute the anti-windup gains Different types of anti-windup compensators

6 1. Introduction 6 Experience with application of anti-windup strategies Aeronautical domains: GARTEUR projects, Airbus cooperation Spatial domain: PIROLA project (robust flying launchers type Ariane 5+) Mechanical domain: flexible beam with piezzo-electrical cells (actuator and sensor) Potential spin-off Reduction of validation costs of control laws Better use of actuator (and/or sensors) capacity Possibility to interact with the engineer at the beginning: for example to choose the actuators and/or sensors (allows to reduce their consummation, size, mass...)

7 2. A bit of history 7 From historical point of view, the anti-windup approach may be decomposed in 6 phases. Phase 1. Discovery ( ) Phase 2. Invention ( ) Phase 3. Criticisms (after 1980) Phase 4. Unification (after beginning 1990) Phase 5. Feasibility (after 1990) Phase 6. Implementation procedure and extensions (after from 2000)

8 2. A bit of history 8 Phase 1 - Discovery It is difficult to find any documents related to engineers experiments in the literature. Since 1940, the literature related to the absolute stability intends to better understand the behaviour of linear systems submitted to input saturations. In 1956 [IRE TAC], Lozier identifies the windup problem and explains why the PI controllers are subject to windup phenomena more than the P controllers. The output begins to oscillate after a saturation of the input The integrator goes on being charged during saturation periods

9 2. A bit of history 9 Phase 2 - Invention Modification of the integral action in case of saturation: Lozier (1956): use of a limiter Fertik, Ross (1967): Ad-hoc method for standard PID controllers Phelan (1977), Krikelis (1980): intelligent integrators More general modifications of the controller: Hanus (1980, 1987): conditioning technique Aström (1983): technique based on observer strategies.

10 2. A bit of history 10 Phase 3 - Criticisms In an ACC paper in 1997, Doyle, Smith and Enns present a critic evaluation of conventional anti-windup techniques: no real formal theory potential links with modern multivariable techniques

11 2. A bit of history 11 Phase 4 - Unification Aström and Rundquist (1989, ACC): they unify several techniques via an observer-based approach Kothare, Campo, Morari, Nett (1989 ACC, 1994 Automatica): except for some cases, the previous techniques may be described through linear and static modifications of the controller.

12 2. A bit of history 12 Phase 5 - Feasibility Stability without performance requirements : Horowitz (1983 IJC) Morari, Zafiriou, Irving (1987) Towards stability with performance requirements: Miyamoto, Vinnicombe (1996 CDC) Teel, Kapoor (1987 ECC) Optimal-based controllers: Gilbert, Kolmanovsky, Tan (1995 IJRNC) Shamma (2000 SCL)

13 2. A bit of history 13 Phase 6 - Implementation procedures and extensions Optimization H : Miyamoto, Vinnicombe (1996 CDC) Edwards, Postlethwaite (1998, 1999 Automatica) Linear matrix inequalities: Mulder, Kothare, Morari (1999 ACC), Gomes da Silva Jr., Tarbouriech (2003 CDC) Grimm et al (2001 ACC) Predictive control: Bemporad, Teel, Zaccarian (2002 ACC) Grimm et al (2003 ACC)

14 3. Problem statement 14 For developing our approach, we need to consider A state-space description of the system and controller A description of the amplitude saturation on actuator and sensor blocks Some anti-windup techniques What are available measures for building the anti-windup loops? Tools: A generalized sector condition is used Quadratic Lyapunov function Finsler s Lemma (use of multipliers = extra degrees of freedom) Convex optimization