Embed Size (px)
Citation preview
1
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
2
Thanks
Responsible Party: Solmaz Sajjadi-Kia
Collaborators Thanh Nguyen Sharad Sirivastada Emre Kose
Support NSF Grants US D. of Ed GAANN Grants
3
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!
4
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
5
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
||W||2<W2max
6
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.
7
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.
8
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)
9
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.
10
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)
11
System and Controllers
max
212
12111
21
|_| uiu
wDxCy
uDwDxCz
uBwBAxx
Given Nominal Controller
Output Feedback
State Feedback
or
Open loop system
Disturbance attenuation problem (ACC & CDC 07)
Assumption: known wmax (Possibly conservatively)
Requirement: closed loop stability, boundedness (e.g., ISS), acceptable performance
Key: Use of ellipsoids
xKu nom
12
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
13
Analysis
Wmax>Wnom
2
x1
x2
Max βWnom=(1/β)1/2
14
Synthesis
Wmax>Wnom
Nom
1 2 3
Safe1
23
15
Full State-Feedback Control (ACC 07) Synthesis (Wmax>Wnom)
Key condition
MIN gamma or δ FSAFE=XQ-1
16
Safe Switch Condition
Ensures Boundedness
17
Scheduling
Conservatism
1)
2) Elliptic invariant set is conservative
18
Scheduling
Scheduling: Putting Intermediate Controllers
19
Full State-Feedback Control
Scheduling WN=WL<WN-1<…W2<W1=Wmax ; QN=Qnom
Min
For i=1:N-1
Ki =Xi Qi -1 i=1,2,..N
20
Output Feedback (CDC 07)
WLOG Assume
Fact:
Switch Condition
21
A Typical result
22
Full State-Feedback Control
Example
Wnom=2.76
Possible to be exposed to Wmax=15
23
Full State-Feedback Control
24
Full State-Feedback Control
W1=Wmax=15; W2=10; W3=5; W4=Wnom=2.76
25
Full State-Feedback Control
Switch history
Sys. res. in scheduled case vs. the original sys. Res.
26
Output Feedback Example
Analysis: Wnom=1.55
Synthesis: Wmax=5
Given nominal controller in the form
27
Output Feedback Example (One Safe Controller)
i Wi Peak Gain Estimate
3 (nom) 1.55 0.64
2 (safe) 5 16.14
28
Output Feedback Example (Scheduled Safe Controller)
i Wi Peak Gain Estimate
3 (nom) 1.55 0.64
2 3 3.04
1 5 25.29
29
Output Feedback
30
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 …….
31
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
32
Static Anti-windup
P(s)Sat(.)K(s)u ur +
-
y
AW q
+ -
d
33
Static Anti-windup
:)(~sP
Stability and Wellposedness: Small Gain Theorem
1)(~
1)(~
1~~
2~~
2
12
1
sPsPWW
WWuq
iuq
0
12
W
34
Static Anti-windup
Λ=XM-1
Q>0 ,M>0
Performance (stability): L2 Gain
35
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.
36
Example (Static Anti-windup)
System output and input history when anti-windup augmentation applied
37
Over-saturated Anti-windup
P(s)Sat(.)K(s)u ur +
-
y
d
)0(],1,[)(),()()(ˆ
1)(define,0)()(ˆ
1)(
)(ˆ)(0|)(|
1)(
)(ˆ)(|)(|
lim
lim
ggtGtutGtu
tGtutu
tu
tutGutu
tu
tutGutu
38
Over-saturated Anti-windup
Q>0
Performance of saturated system for G(t)є [g,1]
))(max(],1,[)( lim
tu
ugwheregtG
..min tsg
39
Over-saturated Anti-windup
40
Over-saturated Anti-windup
]1,[)(),()()(ˆ
)(
)(ˆ)(|)(|
1)(
)(ˆ)(|)(|
1|)(|0)(
)(
)(ˆ)(|)(ˆ|
1|)(|0)(
lim
lim
lim
limlim
dd
dd
d
dd
dd
ddd
d
gtGtutGtu
ggtu
tutGutuif
tu
tutGutuif
ug
tutq
gtu
tutGutuu
gtutq
41
Over-saturated Anti-windup
Q>0
Λ=XM-1
Performance (stability) of Over-saturated Anti-windup: L2 Gain
42
Over-saturated Anti-windupSystem response: Anti-windup, Over-saturated Anti-Windup, Unconstrained Nominal
Traditional Anti-windup:
Over-saturated Anti-windup:
43
Example (Over-saturated Anti-windup)
inputElevator, limited to ±25 degree
Flapron, limited to ±25 degree
outputPitch angle
Flight path angle
Simulation example of F8 aircraft
Kapasouris, P., Athans, M., and Stein, G., “Design of Feedback Control Systems for Stable Plants with SaturatingActuators”, Proceeding of the 27th IEEE Conf. on Decision and Control, Austin, TX, December 1988.
44
Example (Over-saturated Anti-windup
System response: Unconstrained Nominal, Anti-windup, Unconstrained Nominal
45
Example (Over-saturated Anti-windup)System response: Anti-windup, Over-saturated Anti-Windup, Unconstrained Nominal
46
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!
