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Scheduling in Anti-windup Controllers: State and Output Feedback Cases. Faryar Jabbari Mechanical an Aerospace Engineering Department University of California, Irvine (UCI) November 13, 2007. Thanks. Responsible Party: Solmaz Sajjadi-Kia Collaborators Thanh Nguyen Sharad Sirivastada

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scheduling in anti windup controllers state and output feedback cases

Scheduling in Anti-windup Controllers: State and Output Feedback Cases

Faryar Jabbari

Mechanical an Aerospace Engineering Department

University of California, Irvine (UCI)

November 13, 2007

thanks
Thanks
  • Responsible Party:
      • Solmaz Sajjadi-Kia
  • Collaborators
      • Thanh Nguyen
      • Sharad Sirivastada
      • Emre Kose
  • Support
      • NSF Grants
      • US D. of Ed GAANN Grants
surveys
Surveys
  • IJRNC: Michele and Bernstein, eds. (1995)
  • IJRNC: Saberi and Stoorvogel, eds. (1999)
  • Franco Blanchini\'s review article(TAC, 2000)
  • Tarbouriech, et al., Springer, (1999)
  • Kapila and Grigoriadis, Marcel Dekker (2003)
  • IJRNC: Saberi and Stoorvogel, eds. (2004)
  • Much more!
motivation
Motivation
  • Old Problem: actuator limitation is ubiquitous
  • `Safe\' (Low gain) LTI controllers are often excessively conservative
  • Broad approaches:
      • Oldest: Anti-windup
        • Nominal high performance controller (linear design)
        • Anti-windup augmentation
      • Relatively new: Explicit account of saturation nonlinearity
        • Nonlinear design or low gain designs
current techniques to deal with saturation
Direct Approach

Considers the controllers limitation at the very beginning of the design

Anti-windup

Augmentation on top of the nominal controller designed without considering controller bound

Current Techniques to Deal with Saturation

||W||2<W2max

anti windup
Anti-windup
  • Starting in 60\'s (Sandberg, among many)
  • Huge body or work, at times intuitive or even ad-hoc
  • Many attempts at unifying, interpreting of all techniques
  • New rigorous stability and performance results
    • Morari group
    • Teel group
    • Many others (literally too numerous to review!)
    • Positivity, small gain, LMI\'s, etc.
anti windup continued
Anti windup (continued)
  • High performance when no saturation
  • Ideal for `occasional\' saturation
  • Relatively weak performance when in saturation
  • Typically open loop performance -- so open-loop stability `often\' needed (exceptions: Tell, et al. ACC-05, and a few references there)
  • A single controller/augmentations for all saturation levels (even almost zero?), disturbances, tracking signals, etc.
explicit direct approach
Explicit – direct – approach
  • Low-high gain (Saberi and Lin, 199x)
  • Early LPV : Nguyen and Jabbari (1999, 2000), Scorletti, et al (2001)
  • Scheduling: Older work (full state):
    • Gutman and Hagander (1985)
    • Wredenhagen and Belanger (1994)
    • Megretski (1996 IFAC)
  • Scheduling: Recent work}
    • Lin (1997), a little bit of observer
    • Teel (1995), Tarboriech, et al (1999, 2000) - state feedback
    • Wu, Packard and Grigoriadis (2000) - pure LPV
    • Stoustrup (2005-07)
    • Kose and Jabbari (2002, 2003)
direct approach
Direct Approach
  • Stability and performance guarantees
  • Performance not strong in small signal operation

`Some\' have nice properties:

  • A family of controllers (rather than one)
  • Computationally tractable (e.g., a convex search)
  • High actuator utilization
  • Performance guarantees dependent on actuator size and disturbance estimate
  • Approach flexible to incorporate different design approaches, actuator rate limits, state constraints, tracking, etc.
basic idea 1 combining with scheduling
Basic Idea 1: Combining with Scheduling
  • Start with a nominal controller (from somewhere!)
  • Keep it as long as possible
  • Once saturated, switch to a new (family of) of controller (s) that can avoid saturation but can provide guaranteed stability and performance
  • Make sure there are no `cracks\' or escape routs!

Assumptions:

  • Full state or full order controllers (relaxed later)
  • Disturbance attenuation problem (for now)
  • Information of worst case disturbance (e.g. energy or peak)
  • A small number of controllers (for now -- technical detail)
system and controllers
System and Controllers

Disturbance attenuation problem (ACC & CDC 07)

Open loop system

Assumption: known wmax (Possibly conservatively)

Requirement: closed loop stability, boundedness (e.g., ISS), acceptable performance

Key: Use of ellipsoids

Given Nominal Controller

State Feedback

or

Output Feedback

a simple safe controller
A simple `safe’ controller
  • Objective:

-Use Knom(s) as long as possible,

-Once Knom(s) saturates, implement Ksafe(s) that ensure reasonable

behavior

  • Steps:

- Analysis:

What is the largest disturbance the system can tolerate?

Wnom

- Synthesis

Constructing the safe controller

analysis

x2

x1

Analysis

2

Wmax>Wnom

Max β

Wnom=(1/β)1/2

synthesis

2

3

Safe

1

Nom

1

2

3

Synthesis

Wmax>Wnom

full state feedback control acc 07
Full State-Feedback Control (ACC 07)
  • Synthesis (Wmax>Wnom)

Key condition

MIN gamma or δ

FSAFE=XQ-1

safe switch condition
Safe Switch Condition

Ensures Boundedness

scheduling
Scheduling
  • Conservatism

1)

2) Elliptic invariant set is conservative

scheduling1
Scheduling
  • Scheduling: Putting Intermediate Controllers
full state feedback control
Full State-Feedback Control
  • SchedulingWN=WL<WN-1<…W2<W1=Wmax ; QN=Qnom

For i=1:N-1

Min

Ki =Xi Qi-1 i=1,2,..N

output feedback cdc 07
Output Feedback (CDC 07)

WLOG Assume

Fact:

Switch Condition

full state feedback control1
Full State-Feedback Control
  • Example

Wnom=2.76

Possible to be exposed to Wmax=15

full state feedback control3
Full State-Feedback Control

W1=Wmax=15; W2=10; W3=5; W4=Wnom=2.76

full state feedback control4
Full State-Feedback Control

Sys. res. in scheduled case vs. the original sys. Res.

Switch history

output feedback example
Output Feedback Example

Given nominal controller in the form

Analysis: Wnom=1.55

Synthesis: Wmax=5

future work
Future Work
  • Continuous (e.g., spline based) family of controllers: messy but straight forward (will place a bound on how fast the gain can be increased)
  • Mismatch in order of controller and plant: augment the order of the controller
  • Tracking
  • Non-ellipsoidal sets
  • Adding scheduling to the traditional anti-windup scheme …….
slide31

Going the other way around:

  • Start with a basic Ant-windup set up
  • Use Different anti-windups for different levels of saturation
  • Shouldn’t small saturation leave to better performance guarantee than a sever saturation? (Ans: yes!)
  • But first: Something interesting shows up!!
  • Let us review the basic `Static’ anti-windup set up
slide32

Static Anti-windup

d

y

r

u

+

Sat(.)

K(s)

P(s)

-

+

-

AW

q

slide33

Static Anti-windup

Stability and Wellposedness: Small Gain Theorem

slide34

Static Anti-windup

Performance (stability): L2 Gain

Q>0 ,

M>0

Λ=XM-1

slide35

Example (Static Anti-windup)

Grimm, G., Teel, A.R., and Zaccarian, L., “Results on Linear LMI-Based External Anti-windup Design”, IEEE Trans.

on Automatic Control, Vol. 48, No. 9, Sep. 2003.

slide36

Example (Static Anti-windup)

System output and input history when anti-windup augmentation applied

slide37

Over-saturated Anti-windup

d

y

r

u

+

Sat(.)

K(s)

P(s)

-

slide38

Over-saturated Anti-windup

Performance of saturated system for G(t)є [g,1]

Q>0

slide41

Over-saturated Anti-windup

Performance (stability) of Over-saturated Anti-windup: L2 Gain

Q>0

Λ=XM-1

slide42

Over-saturated Anti-windup

System response: Anti-windup, Over-saturated Anti-Windup, Unconstrained Nominal

Traditional Anti-windup:

Over-saturated Anti-windup:

slide43

Example (Over-saturated Anti-windup)

Simulation example of F8 aircraft

Elevator, limited to ±25 degree

Flapron, limited to ±25 degree

input

Pitch angle

Flight path angle

output

Kapasouris, P., Athans, M., and Stein, G., “Design of Feedback Control Systems for Stable Plants with Saturating

Actuators”, Proceeding of the 27th IEEE Conf. on Decision and Control, Austin, TX, December 1988.

slide44

Example (Over-saturated Anti-windup

System response: Unconstrained Nominal, Anti-windup, Unconstrained Nominal

slide45

Example (Over-saturated Anti-windup)

System response: Anti-windup, Over-saturated Anti-Windup, Unconstrained Nominal

slide46

Summary

  • Tradeoff between `matched uncertainty’ vs better performance guarantee
  • Dynamic Anti-windup case: Reasonably straight forward: the uncertainly is of the LPV (self-scheduled) variety – constant Lyapunov functions suffice
  • Combine `over saturation’ and scheduling is next!
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