IST fuzzy-crisp talk

Report
Crisp Control Is Always Better Than
Fuzzy Feedback Control
MICHAEL ATHANS
Professor of Electrical Engineering (Emeritus)
MIT, Cambridge, Mass., USA
and
Visiting Scientist, Instituto de Sistemas e Robotica
Instituto Superior Tecnico, Lisbon, PORTUGAL
[email protected] or [email protected]
EUFIT ´99 DEBATE WITH PROF. L.A. ZADEH
Aachen, Germany, September 1999
1
Debating Points
2
• I like Fuzzy Logic as an alternative to probability theory, especially in
applications involving man-machine interactions
• Fuzzy feedback control methods represent inferior engineering
practice, often by people that never bothered to learn control theory
and design
• Fuzzy feedback control is a vacuous technology for the design of highperformance control systems
• Fuzzy control methods are “parasitic;” they simply implement trivial
interpolations of control strategies obtained by other means
• Theological arguments about “fuzzification”, “defuzzification”,
nonlinear control, and inherent robustness are simply nonsense
• Fuzzy feedback control has failed to capture and utilize alternative
means in dealing with uncertainty using Fuzzy Sets and Fuzzy Logic
• Prof. Zadeh should communicate to his disciples the sorry state of
affairs in fuzzy feedback control and tell them to “shape-up”
Crisp Vs Fuzzy Feedback Control
3
• Crisp control: Normative - prescriptive
• Quantitative models of plant dynamics and disturbances
• Precise definition of performance specifications
• Modeling and environmental uncertainty accounted for
• Rigorous optimization-based design
• Fuzzy control: Empirical - descriptive
• 1st generation (Mamdani). Ad-hoc interpolation of “expert” control
rule-based system
• Vast majority of “fuzzy applications” use this method
• 2nd generation (Takagi-Sugeno). Ad-hoc interpolation of control
strategies derived from crisp feedback control methodologies
• Fuzzy control has failed the noble goal of “fuzzy logic”
in providing alternatives in dealing with uncertainty
The Joy of Feedback
• Measure system response, including effects of disturbances, using
(noisy) sensors
• Compare actual system response to desired system response at
each time
• “Error” signal(s) = (Desired response)-(Actual response)
• Use “error” signals to drive compensator (controller) so as to
generate real-time control corrections so as to keep “errors” small
for all time
• FEEDBACK ESSENTIAL TO GUARANTEE GOOD
PERFORMANCE IN THE PRESENCE OF UNCERTAINTY
4
Why Feedback?
5
• Automatic feedback control systems have been used since the 1930´s
to provide superior performance and higher fidelity than manual
control systems requiring human operators
• The SCIENCE of Feedback Control was developed to allow
engineering designs that deliver this superior performance, NOT to
duplicate poor human control performance
• The performance payoffs are even more dramatic in the case of
coupled multivariable systems, i.e. systems with many sensors and
control inputs
• crisp control theory exploits the tight dynamic coupling
• humans are notorious in lacking ability to develop control rules for
such multivariable systems
• Increased cost of feedback (sensors, actuators, processors,...) is
justified by increased performance capabilities
• sensor/actuator hardware costs greatly exceed processing costs
Fixed Structure Feedback
DISTURBANCES
COMMANDS
CONTROLLER
(COMPENSATOR)
CONTROLS
OUTPUTS
DYNAMIC SYSTEM
(PLANT)
• Compensator structure does not change (no learning)
• No change in digital processor algorithms that approximate the
solution of compensator differential equations and gains
• Design methodologies available for general multivariable case using
(crisp) robust-control theories and algorithms
6
Adaptive Feedback Control
PARAMETER
ADJUSTMENT
LOGIC
REAL-TIME
IDENTIFICATION
DISTURBANCES
CONTROLLER
(COMPENSATOR)
COMMANDS
DYNAMIC SYSTEM
(PLANT)
CONTROLS
OUTPUTS
• Uncertain plant parameters identified in real-time and compensator
parameters are adjusted also in real-time
7
Fault-Tolerant Feedback
REAL-TIME
FAILURE
DETECTION
ISOLATION
CONTROL
RECONFIGURATION
LOGIC
DISTURBANCES
CONTROLLER
(COMPENSATOR)
COMMANDS
DYNAMIC SYSTEM
(PLANT)
CONTROLS
• “Supervisory” level monitors for failures
• Failure isolated and identified
• Compensator structure and algorithms modified
OUTPUTS
8
Crisp Mathematical Control
9
• Based upon analytical description of plant dynamics, model errors,
environment, constraints, and performance objectives
• Optimal Control Theory
• Used to generate “open-loop” preprogrammed control and state
variable trajectories as a function of time
• Feedback Control Theory
• Used to ensure precise command-following and disturbancerejection performance, in the presence of uncertainty, using
feedback of sensed variables
• stability guarantees are essential
• performance guarantees (in the presence of uncertain models)
are desirable
Closed-Loop Stability
10
• “Models have limitations, stupidity does not!”
• Feedback control can result in superior performance
• Careless feedback strategies can cause instabilities
• Closed-loop stability must be guaranteed for family of plants (stabilityrobustness)
• stability guarantees for “nominal” plant and nominal plant
simulations are not enough
• control engineers must be paranoid about closed-loop stability
Crisp Feedback Theory Status
11
• Start with global nonlinear dynamic model of plant (nonlinear
differential or difference equations)
• Using “linearization” establish a collection of linear models in vicinity of
operating conditions
• Generate linear multivariable dynamic compensator with guaranteed
stability-robustness and performance-robustness properties for each
linear model
• Use “gain-scheduling” of the parameters of the linear compensator
collection to derive a single global nonlinear dynamic compensator for
the global nonlinear plant
Linearization, Gain-Scheduling
Global
Nonlinear
Dynamic
Compensator
Global
Nonlinear
Plant
LM:3
LM:4
LM:2
LM:1
LC:3
LM:k
Family of linear dynamic models
LC:4
LC:2
LC:1
LC:k
Family of linear dynamic
compensators
12
Robust Feedback Control Design
13
• Start with nominal state-space model of linear MIMO dynamic system
• Define bounds on model errors (class of “legal” errors)
• parametric uncertainty; upper and lower bounds for key
coefficients
• unstructured uncertainty; worst size of dynamic errors as a function
of frequency (bending modes, torsional modes, actuator/sensor
errors, ....)
• Model exogenous signals (a key requirement for superior
performance)
• power spectral densities of commands, disturbances and sensor
noise
• Quantify robust-performance specifications in the frequency domain
Design is meaningless unless performance specs are quantified
Robust MIMO Feedback Design
14
• LQG or H2 method
• performance goal: minimize RMS errors of stochastic performance
variables
• H•method
• performance goal: minimize maximum errors assuming worst-case
exogenous disturbances
• Robust feedback design is done via mixed- (structured singular
value) synthesis. Iterative generation of H•dynamic compensators (of
increasing complexity) to guarantee stability-robustness and
performance-robustness
• Generate linear multivariable dynamic compensator with guaranteed
stability-robustness and performance-robustness properties for each
linear model (model errors are explicitly accounted for)
• Use “gain scheduling” of the parameters of the linear compensator
collection to derive a single global nonlinear dynamic compensator for
the global nonlinear plant
• controller involves real-time solution of coupled nonlinear
differential equations
Fuzzy Feedback Systems (Mamdani)
15
• 1st generation fuzzy feedback control systems
• start with set of “expert” discrete-valued control rules (if-then...),
often obtained from human operators
• interpolate between discrete control rules using “membership
functions” from fuzzy set theory
• No explicit quantitative statement of performance specifications
• No quantitative modeling of plant dynamics, disturbance and sensor
noise characteristics
• No stability-robustness or performance-robustness guarantees
• Lots of “theology”, hand-waiving and scientifically unfounded claims
• Simulation based results (where does model used for simulation come
from?)
Fuzzy Control (Mamdani)
de\e
NL
NS
ZE
PS
PL
NL
PL
PL
PL
PS
ZE
NS
PL
PS
PS
ZE
NS
ZE
PS
PS
ZE
NS
NS
PS
ZE
ZE
NS
NS
NL
PL
NS
NS
NL
NL
NL
16
Weakness of Mamdani-Type Fuzzy
Control Philosophy
• Attempt to emulate or duplicate human control behavior
• Basic problem
• premise: Human is good controller
• fallacy: Human is very poor controller for complex, multivariable,
marginally stable dynamic plants
• Fuzzy feedback controllers “work” for very simple SISO dynamic
systems where high precision is not required
• mostly PI controllers (a few PID with a crisp channel)
• no guarantees of closed-loop stability, stability-robustness and of
performance in presence of uncertainty
• hard to extrapolate designs to new applications
99% of fuzzy feedback control applications deal with essentially
1st or 2nd-order, overdamped, SISO systems
17
Michio Sugeno Says....
18
• “Stability has been one of the central issues since Mamdani´s
pioneering work. Most of the critical comments to fuzzy control are
due to the lack of a general method for its stability analysis. We are
still seeking an appropriate tool for the stability analysis of fuzzy
control systems, though this situation is now improved......The success
of fuzzy control, however, does not imply that we do not need a
stability theory for it. Perhaps the main drawback of the lack of stability
analysis would be that we cannot take a model-based approach to
fuzzy controller design.”
• Reference: M. Sugeno, “On Stability of Fuzzy Systems Expressed by
Fuzzy Rules with Singleton Consequences,” IEEE Trans. on Fuzzy
Systems, Vol. 7, April 1999
From Jenkins and Passino...
•
•
•
19
Reference: D.F. Jenkins and K.M. Pasino, “An Introduction to Nonlinear
Analysis of Fuzzy Control Systems,” J. Intelligent and Fuzzy Systems, Vol. 7,
1999
“The fuzzy controller design methodology primarily involves distilling human
expert knowledge about how to control a system into a set of rules. While a
significant amount of attention has been given to the advantages of the
heuristic fuzzy control design methodology .... relatively little attention has
been given to its potential disadvantages. For example, the following
questions are cause for concern
• will the behaviors observed by a human expert include all possible
unforseen situations that can occur due to disturbances, noise, or plant
parameter variations?
• can the human expert realisticaly and reliably foresee problems that could
arise from closed-loop system instabilities or limit cycles
• will the expert really know how to incorporate stability criteria and
performance objectives into a rule-base to ensure that reliable operation
can be obtained?
Authors advocate the use of Tagaki-Sugeno models with crisp stability criteria
Shortcomings of Fuzzy Controller
Methodology
e1
e2
en
•
•
•
•
•
MIMO
Fuzzy
Controller
u1
u1  h1 (e1 , e2 ,..., en )
u2
u2  h2 (e1 , e2 ,..., en )
um
.............
um  hm (e1 , e2 ,..., en )
Fuzzy rules just generate nonlinear static functions
Impossible to generate multidimensional “if-then” rule tables
Cannot generate “differential equation” controller rules
It is not easy to differentiate noisy sensor signals by finite differencing, as it is
almost always done in fuzzy applications
• no utilization of dynamic (e.g. Kalman) filtering of sensor noise
I have never seen a multiple-input multiple-output (MIMO) fuzzy control
application using Mamdani-type methods
• combinatorial complexity for high-order and multivariable applications
20
Challenge to Fuzzy Control Experts
•
M~m
•
m
•
f(t)
M
xo
x1
x(t)
•
21
Observe only noisy position x(t)
• with broadband sensor noise
Find force f(t) to relocate cart
• not just balance stick
No static fuzzy rule-based system
can solve this problem
• human cannot stabilize system
with knowledge only of x(t)
To change cart position and for
inverted pendulum stabilization, the
controller must be dynamic, i.e. it
must implement “differential
equations” from x(t) to f(t)
Why is Fuzzy Control Popular with the
Masses
LEARNING FUZZY CONTROL
•
•
Working pragmatic knowledge of
fuzzy sets and membership
functions ..... 1 week
Working pragmatic knowledge of
Mamdani method ..... 1 week
LEARNING CRISP CONTROL
•
•
•
•
•
•
Differential equations ... 8 weeks
Linear algebra ... 10 weeks
SISO servos .... 14 weeks
State space methods/stability
theory ... 14 weeks
Optimal control .... 8 weeks
Multivariable robust control ... 14
weeks
22
Takagi-Sugeno Fuzzy Control
• Approach developed to overcome criticism regarding closed-loop
stability guarantees
• Approximate global nonlinear dynamics by “interpolating” linear statespace models with membership functions
• Design full-state feedback controllers for each linear model (using
crisp control methods, e.g. LQR, H2, H•, etc.) and “interpolate” using
membership functions
• technique is inferior to that of “gain-scheduling”
• It is possible to use quadratic Lyapunov functions to obtain sufficient
conditions for nominal stability
• results are disappointing; at best applicable to low performance
systems
• Current methodology does not address stability-robustness and
performance-robustness issues
• Current methodology does not address output feedback requiring
dynamic compensator designs
23
Recent References on Fuzzy Stability
24
• M. Sugeno, “On Stability of Fuzzy Systems Expressed by Fuzzy Rules
with Singleton Consequences,” IEEE Trans. on Fuzzy Systems, Vol. 7,
April 1999
• S.H. Zak, “Stabilizing Fuzzy System Models Using Linear Controllers,”
IEEE Trans. on Fuzzy Systems, Vol. 7, April 1999
• M. Margaliot and G. Langholz, “Fuzzy Lyapunov-based Approach to
the Design of Fuzzy Controllers,” Fuzzy Sets and Systems, Vol. 106,
August 1999
• D.F. Jenkins and K.M. Pasino, “An Introduction to Nonlinear Analysis
of Fuzzy Control Systems,” J. Intelligent and Fuzzy Systems, Vol. 7,
1999
• A. Kandel, Y. Luo,and Y.Q. Zhang, “Stability Analysis of Fuzzy Control
Systems,” Fuzzy Sets and Systems, Vol. 105, July 1999
• Y. Tang, N. Zhang and Y. Li, “Stable Fuzzy Adaptive Control for a
Class of Nonlinear Systems,” Fuzzy Sets and Systems, Vol. 104, June
1999
Trends in Fuzzy Stability Studies
• Must have a (linear, nonlinear, multi-model,...) state-space model
• Classical crisp stability theory results are applied
• Popov criterion
• Circle criterion
• Lyapunov stability theory
• Linear Matrix Inequalities (LMI)
• Bounded-input bounded-output (L2) stability theory
BIG QUESTI ON
If a tsate spa c emodel i s a il
v a bel why not us e
supe ror
i cri sp d esi gn te chnique stha t gua rante e
stabi l iy,
t sta b lii t y-robustne s s a, nd pe rformanc e
robustne s s?
25
Takagi-Sugeno Models
26
• Start with R linear s tate- s pace m odels,each valid in a s pecific regionSk of Rn
xÝ(t)  Ak x(t )  Bk u(t); k  1, 2,..., R; x(t) Sk
• Define R scalar valued m embership functions, k ( x(t)), 0   k (x(t))  1,s uch that
1 if x(t) S k


 k (x(t))   0 if x(t) S j for k  j
; let

linear int erpolations otherwise
 Global nonlinear model
 R

 R

xÝ(t )   Ak  k (x(t))x(t)   Bk  k (x(t))u(t)
k 1

k1

 Alm os t im pos sible to define the
m em bers hip functions k (x(t)) for
high- dim ensional problems
 1 
 (x(t ))   

 

R
R

1
 1 (x(t )) 
1


 ( x(t )) 
 ( x(t)) 

k
R
Takagi-Sugeno Feedback Law
27
• For each linear plant, design full- s tate feedback gain m atrices ,typically by crisp
feedback m ethods (eigenstructure-ass ignm ent,LQR, H , etc.) of the form
Kk x(t),
k  1,2,..., R.
• Generate global nonlinear feedback by interpolating with the same m embers hip
functions
 R

u(t)  
 K j  j (x(t))
x(t)
j1

• Global closed- loop sys tem
 R

 R
 R
xÝ(t )  
 K j  j (x(t))

 Ak  k (x(t))   Bk k ( x(t ))
x(t)
k 1
j1

k 1
• Quadratic Lyapunov functions provide sufficient conditions for stability.
Find P  0 so
P(Ak  Bk K j )  (Ak  Bk K j ) T P  0
for all j,k  1,2,..., R
i.e. all m is matched linear plant/linear gain combinations mus t be s table!
!! This
s eldom happens in high-perform ance designs.
Set-Point Vs. Task-Based Control
• Prof. Zadeh asserts
• crisp control theory only deals with set-point control; it cannot
handle task-based control
• Fact
• hybrid control systems do provide the methodology for integrating
task-based and set-point control
28
Hybrid Control
Discrete-state system
CONTROLLER
(COMPENSATOR)
DYNAMIC SYSTEM
(PLANT)
Continuous-time system
•
•
Architectures involving interactions between a finite-state event-driven
system and a continuous-state continuous-time system
Discrete level can establish different modes of operation (tasks) for feedback
system
29
Car Parking
•
•
Prof. Zadeh asserts that control theory cannot solve parallel parking problem
Fact: Time-optimal solution using simplified dynamics is shown
• optimal control theory using more complex nonholonomic car dynamic
model can also be used using arbitrary initial car location and orientation
• automated crisp solution can be implemented if customer is willing to pay
the price
30
Highway Driving
31
• Prof. Zadeh asserts that it will never be possible to construct an
automated automobile driving system using conventional control
theory
• FACT: Such a prototype system has been already been demonstrated
by PATH on the I-5 freeway in San Diego including
• longitudinal control with minimal inter-car spacing to triple freeway
lane capacity
• lateral control (lane changing and lane-centerline following)
• automated merging and demerging capabilities
• using hybrid control methodologies
• by some of Prof. Zadeh´s colleagues (Varayia, Sastry, Hedrick, ...)
at UC-Berkeley, among others
• Most certainly the fatality rate of such automated highway systems will
be far less that those involving human drivers
• Similar efforts are ongoing by Daimler-Benz in Europe
Barriers to “Computing With Words”
32
• Prof. Zadeh advocates computing with words using fuzzy logic
concepts
• noble task; provides a foundation for a computational theory of
perceptions
• What is not usually stressed is that such computations require the
solution of exceedingly complex equations in real-time
• in June 1997 talk at the Portuguese Academy of Sciences, Prof.
Zadeh showed an example which illustrated that even simple
“word computations” require solution of systems of complex
nonlinear integro-differential equations
• such real-time computations are beyond capabilities of current and
projected computers
• must wait for completely new computers with novel architectures
and software
Fuzzy Dynamical Systems
33
• Appropriate framework for capturing system uncertainty
• References
• P.E. Kloeden, “Fuzzy Dynamical Systems,” Fuzzy Sets and
Systems, Vol. 7, 1982
• Y. Friedman and U. Sandler, “Evolution of Systems under Fuzzy
Dynamics Laws,” Fuzzy Sets and Systems, Vol. 84, 1996
• Y. Friedman and U. Sandler, “Fuzzy Dynamics as Alternative to
Statistical Mechanics,” Fuzzy Sets and Systems, Vol. 106, 1999
• Must propagate the Possibility Density Function using ChapmanKolmogorov integral equations
• to solve these requires enormous computational power
• feedback control system design using such Chapman-Kolmogorov
equations is extremely complex and its real-time computational
requirements are astronomical
Linear-Quadratic-Fuzzy (LQF) Optimal
Control
• Form ulation of s tandard LQ problem using
fuzzy m em bership functions for proces s
and measurem ent noise
x(t  1)  Ax(t)  Bu(t)  Lw(t)
y(t  1)  Cx(t  1)  v(t  1)
1
J  lim
T 2T
T
 x T (t)Qx(t)  u T (t )Ru(t )
kT
• Technical difficulties
(1). The conditional state m em bers hip function,given pas t obs ervations,
involves the solution of nonlinear partial differential equations
(2). Min/max fuzzy arithmetic further complicates life
(3). Comm on membership functions are nondifferentiable
34
The Numbers Game: So What?
• Prof. Zadeh claims that from 1981 to 1996 there are 15,631 INSPEC
and 5,660 Math Reviews citations with “fuzzy”, and 2,997 INSPEC
citations with “fuzzy control”
• There are at least 250,000 citations on Kalman filtering alone, and
there must be several million citations on other aspects of “crisp”
modern control theory
• Note that Modern Control Theory started in about 1959 and Zadeh´s
seminal paper on Fuzzy sets was written in 1965
35
The Numbers Game: Comparisons
• Prof. Zadeh credits Japanese with innovative insight to popularize
fuzzy control applications and bring “fuzzy” commercial products into
the marketplace
• oriental vs western philosophy
• Numerical facts
• in December 1989 the Nikkei 225 was at 39,000
• in December 1989 the Dow Jones was at 2,700
• on August 18, 1999 the Nikkei 225 was at 17,879
• on August 18, 1999 the Dow Jones was at 10,991
36
Fuzzy Applications
37
• Lot´s of “hoopla” about commercial applications (air-conditioners,
washing machines, camcorders, ...)
• The innovation is adding special sensors/actuators and feedback to
previously open-loop systems
• even better performance would be obtained for the same
sensor/actuator architectures if engineers used crisp control
methods
• Example: Phillips design for Mercedes CD player rejecting fuzzy
control design in favor of H-based
one
•
It is time we moved from " voodoo engineering" into solid and respectable
science and technology, and
Prof. Zadeh should take a leadership role in this transition
Crisp and Fuzzy Control Complement?
• Prof. Zadeh´s asserts: Fuzzy controls do not replace crisp
controls, but they can complement each other
• Basic engineering problem: How does an engineer integrate a
crisp and a fuzzy control design (and why???)
d(t) Disturbance
Command
Error
r(t)
e(t)
-
Fuzzy
controller
Control
u(t)
Dynamic
system
Output
y(t)
38
My Dillema
39
• Without stability guarantees, Mamdani fuzzy controllers cannot be
used for 3rd or higher order systems
• To obtain stability guarantees, even fuzzy control afficionados admit
that they must use some nominal state space model for system
dynamics for fuzzy control designs (Sugeno et al)
• plus, lots of crisp tools (Lyapunov theory, circle criterion, Popov
criterion, linear quadratic regulators, pole placement, ...)
• they still have to worry about unmodeled dynamics and uncertain
parameters
• Given that a state space model is necessary, why bother to introduce
fuzzy ideas when conventional crisp control methods can deal with the
design problem directly???
• and, at the same time, address explicitly and directly disturbances,
sensor noise, model errors, performance specifications, nominal
stability, robust stablity, and performance-robustness
Optimal Control
• Used for determining best way of adjusting controls, as functions of
time, such that system response is “optimal” (in well-defined sense)
from any initial state
 State Dynamics (continuous - time) :
xÝ(t)  f  x(t), u(t); x(to )  x o
 Cost Functional : J(u)  K(x(t f )) 

tf
to
L x(t), u(t)dt
 State Dynamics (discrete - time) :
x(t  1)  f  x(t), u(t); x(0)  xo
 Cost Function : J(u)  K  x(T ) 
T1
 L x(t  1), u(t)
t0
40
41
An Example
• Old-fashioned F-4 aircraft
• Objective: Reach operational altitude in minimum time
• Shown is expected flight path
Altitude
60,000 ft
Conventional
T=720s
Range
42
Optimal Control Theory
• Pontryagin maximum principle (1957) main theoretical tool for
analyzing and solving optimal control problems
• Extension of Kuhn-Tucker conditions in Nonlinear Programming
problems to dynamic case
• Maximum Principle leads to numerical solution of Two-PointBoundary-Value (TPBV) problem to calculate
• optimal controls vs. time
• resulting optimal dynamic state trajectories and responses
• Several algorithms exist for solving TPBV problems
43
Linearization, Gain-Scheduling
Global
Nonlinear
Dynamic
Compensator
Global
Nonlinear
Plant
LM:3
LM:4
LM:2
LM:1
LC:3
LM:k
Family of linear dynamic models
LC:4
LC:2
LC:1
LC:k
Family of linear dynamic
compensators
44
MIMO Linear Feedback
45
• Must design MIMO compensator to ensure stability and satisfaction of
performance specifications
• Digital approximation of MIMO compensator solves in real-time highorder LTI differential equations
Concluding Remarks
46
• Crisp control theory offers a powerful methodology for designing SISO
and MIMO optimal and high-performance feedback control systems
• extensive knowledge of theoretical developments required
• quantitative modeling of plant, disturbances, specs. is essential
• systematic prescriptive/normative approach to control design
• leads to high-performance (high-gain, high-bandwidth) designs
• Fuzzy feedback control methods (Mamdani) are suitable for trivial
control problems requiring low accuracy (minimal performance)
• no training in control theory necessary
• no models, no specifications, no guarantees
• impossible to guarantee stability
• empirical ad-hoc approach to design
• leads to low-performance (low-gain, low-bandwidth) designs
Fuzzy control is a “parasitic” technology

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