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Workshop
Future Directions in Systems and Control Theory Tuesday, June 22
Viewgraphs - Volume 1
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DIRECTIONS IN SYSTEMS AND CONTROL THEORY
MICHAEL ATHANS
Visiting Scientist, Institute de Sistemas e Robötica Institute Superior Tecnico, Lisbon, Portugal
and Professor of Electrical Engineering (emeritus)
Massachusetts Institute of Technology Cambridge, Mass., USA
[email protected] or [email protected]
June 1999
Control Is Vital Technology
Half a century of significant advances in theory, methodology, and software Mushrooming applications Exciting opportunities for future research More and more people discover the value and benefits of feedback control • The "Joy of Feedback"
DISTRIBUTION STATEMENT A Approved for Public Release
Distribution Unlimited
flfjQ QUALITY IHS?HCTED 1
20000118 124
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Define a research agenda for the next decade and beyond Survey research needs for theories, methodologies, algorithms in • System theoretic issues for deterministic and stochastic systems
• Linear and nonlinear robust control • System identification • Robust adaptive control • Fault tolerant and reconfigurable systems
• Hybrid control • Hierarchical, decentralized, and distributed large scale systems
• Intelligent systems • Chaotic and complex systems (a la Santa Fe Institute ...)
• Novel concepts
Optimal Control Theory
Maximum Principle of Pontryagin and Bellman's Dynamic Programming are the foundations of optimal control theory • why no new powerful theorems in the last 35 years?
Neurodynamic programming uses iterative methods to approximate cost-to-go function and store it in a neural network One could also use repetitive numerical solutions of the two-point- boundary-value problem to train neural network for approximate implementation of optimal nonlinear feedback control law Combination of genetic algorithms with classical algorithms to handle issues of locally optimal solutions Efficient and reliable numerical methods for approximating solution of Hamilton-Jacobi-Bellman equation would lead to major advances in several domains ,f -, Are there new advances in dynamic team and dynamic game theory and algorithms? " ,
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I I I I I I I I I I I I I I I I I I I I I
Classical Kaiman filters and (suboptimal) extended Kaiman filters remain the workhorses of applied estimation Blending of hypothesis-testing concepts and filtering algorithms can lead to significant performance payoffs • driven by multi-sensor multi-object tracking problems that include
changes in maneuvers, temporary occlusion of target, motion constraints etc.
• performance vs. combinatorial complexity tradeoffs are required Distributed/decentralized estimation problems are much more difficult than their centralized counterparts • example: decentralized team-theoretic hypotheses-testing
problems
Robust Multivariable Control
DISTURBANCES
Feedback imperative to • stabilize open-loop unstable plants and/or • deliver superior performance in the presence of plant and exogenous
signal uncertainties Robust control issues: in the presence of parametric and unmodeled dynamic errors, guarantee • stability-robustness • performance-robustness
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Structured errors: known bounds on uncertain parameters Unstructured errors: unmodeled dynamics (possibly infinite-dimensional) bounded in the frequency domain For all legal structured and unstructured uncertainty must guarantee
• stability-robustness • performance-robustness to posed specifications (command-following,
disturbance-rejection, insensitiv'rty to sensor noise)
Mixed jd -synthesis Setup
Real Fa En
rameter rors BEfl
IBA2MM
B4sLs»l
A Unmodeled Dynamics
Errors
w z
Uncertainty block Perfor Spl
^■•S'v^:^p(s)t.-Ä;'4:üi ■-; -.Tftvi:. d e
u y Generalized plant
Dynamic compensator
-1 i^l H«-
1 1 I I 1 1 I 1 I 1 I I I I I I I I I I
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The DK or DGK Iteration
0. Solve for nominal H«, compensator K(s) - '
1. Evaluate stability and performance robustness using - upper bound (tub for u. or mixed \l
-Determine frequency-dependent D or DG scales
* Hub(°>) < 1 a|lffi?
jf NO ~ YES. Design is robust.
M. Fit stable, min.phase rational transfer functions $ to D or DG scales
I <■ .-S3. Redefine plant, including scales, for new H- problem lj||
" r—^^.
j4. Solve new H- problem. Determine new H— |§! I compensator K^s} \
H-oo Problem in DK and DGK Iteration
* <Dy»amk) From D-ttention
77*1 ̂ •■»wj»
^^^5 HtDLf'l—*■
t
• y ^ '
—HSS)S^-
Scales (Dynamic) from DG-Ucration ' T
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Directions in Linear Robust Controi
Mixed ^-synthesis methodology well established • computationally intensive for several uncertain parameters,
leading to compensators of increasing dimensionality Integration with adaptive control to reduce parameter uncertainty bounds? Alternatives for robust synthesis (must be valid for both structured and unstructured uncertainty) • view parameters as random variables?
• other?
Nonlinear Feedback Control Methods
DISTURBANCES
COMMANDS
Research goal is to design for stability and performance robustness
Common present methodologies • Lyapunov-based designs • Gain-scheduling • State-feedback linearization • Sliding-mode controllers • Backstepping methods
Do we need a new paradigm?
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Gain-Scheduling
Family of linear dynamic models Family of linear dynamic compensators
Stability-robustness requirements place additional constraints on how gain scheduling is implemented
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Nonlinear Robust Stability with Feedback Linearization
Model error
Nominal nonlinear system
S(0=amGit))+Bm<&<t))u(t)
x(t)=f(x(t))+G(x(t))u(t) -K(s)
To obtain non-conservative stability-robustness results, one needs to routinely solve Hamilton-Jacobi-Bellman partial differential equation inequalities
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System Identification
Open-loop vs. closed-loop • role of excitation signals
Classical methods have difficulty when there are several poles and zeroes near the jco-axis, especially in MIMO settings Identification for robust control Identification in behavioral modeling setting
Robust Adaptive Control
jDYNAMIC SYSTEM! (PLANp !
Classical adaptive MIMO designs ackward. Stability-robustness to unmodeled dynamics still a big problemldentification for robust control Sequential identification/robust-controller resynthesis? Robustified Multiple-Model Adaptive Control (MMAC) methods
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Classical Adaptive Control
Traditional methods (MRAC, STURE,...) developed for plant parameter estimation; do not take into account unstructured uncertainty • focus on command-following, not disturbance-rejection
• difficult to generalize to MIMO systems • stability-robustness not guaranteed, ad-hoc fixes still do not lead
to global stability guarantees • unmeasurable plant disturbances and sensor noise can create
stability and performance problems Benefits of adaptive control should be quantified • must quantify tradeoffs of using persistent excitation that increase
output disturbances vs. longer-term performance improvements due to learning
Operation in closed-loop not well understood
Sequential Identification/Compensator Redesign
Continuous readjustment of compensator parameters seems to be wrong way to go Sequential closed-loop identification/compensator redesign may be superior engineering approach • advances in identification for robust control • needs rethinking of proper persistent excitation signals • should take advantage of major advances in the design of robust
linear controllers (mixed (i-synthesis,...) • open closed-loop bandwidth slowly, monitor for instabilities so as
to revert to prior stabilizing compensator • should improve performance degradation caused by "detuning" of
fixed compensators because of large parameter errors
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Multiple Model Adaptive Estimation (MMAE)
u(t- 1),
IKFTHI *i(t/t)
|KF#2
r,(t)
hm
|KF#N
r2(t)
^N(t/t)
Si(t)
S2ÖI ML
-*K@>
Pi(t) Posterior
! Probability \ i Evaluator j
Residual covariances (off-line)
ÜÖL PNCO
S(t/t)
Posterior hypotheses probabilities
If true system is included in bank of Kaiman filters, it is identified with probability one. Otherwise, we identify nearest "probabilistic neighbor"
Multiple Model Adaptive Control (MMAC) Measurements; z(t) Controls; u(t)
*l(t) mm
|KF#2f
ri(t)
*2(t)
|KF#N¥ ^N(t)
fm ▼ ▼▼
i-G2p
*> -GN
Posterior Probability Evaluator
PlW P?.(t)
U(t)
?N(t)
Posterior hypotheses probabilities
Applied control is the probabilistic average of "matched LQG controls" Other variants possible, e.g. apply control with the largest current posterior probability (switching...)
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Fault-Tolerant Control
," CONTROL RECONFIGURATION s';: LOGIC
<*" REAL-TIME
FAILURE DETECTION ISOLATION
nnA
. DISTURBANCES.
TITTTT
Detect soft/hard failures in sensors and actuators, abrupt dynamic changes Impact of parametric and dynamic errors on FDI algorithms, e.g. GLR .. Controller reconfiguration algorithms Role of hybrid system theory (mode switching...) Role of "intelligent" supervisory methods Robust stability and performance guarantees
Hybrid Control Discrete-state system
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 for feedback system based upon feedback of behavior of continuous variables
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Issues in Hybrid Control
Deals with switched dynamical sytems, software enabled control, and multimodal control Impact on several disciplines and applications, e.g. • fault-tolerant reconfigurable control • hierarchical systems
Formal modeling leads to some qualitative properties (reachability,...) • difficult to establish existence, uniqueness and sensitivity of
solutions • some verification software is available
Design-directed theories, methodologies, software are at their infancy. They seem to require advances in • optimal control theory
• dynamic team theory • dynamic game theory
Complexity and combinatorics are main obstacles
Decentralized Control
t T T T T DISTURBANCES
Weak dynamic coupling Multiple time-scales Stability and performance robustness guarantees?
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Issues in Decentralized Control
Non-classical information patterns "fight" with classical optimization
• signaling strategies • second-guessing strategies • cost of communications and communications delays • "easy" convex centralized problems transform into "nasty"
nonconvex combinatorial ones in decentralized setting
WHO should know WHAT, WHEN, WHERE and WHY • Performance impact of enhanced information patterns in teams
Cooperative multiperson dynamic game theory can be exploited
• Nash strategies • Stackelberg (leader-follower) strategies
Can we isolate superior alternative hierarchical and spatial decompositions and architectures? Is there a decentralized "maximum principle?"
25
Intelligent Systems
Fuzzy systems • fuzzy feedback control • fuzzy supervisory layers (fuzzy hybrid?)
• uncertainty modeling? Neural networks • training and learning • specific means of approximating nonlinear static maps
Genetic/evolutionary algorithms • minimal assumptions on problem structure • used for bypassing local optima • require extensive number of iterations for convergence
• hard convergence and rate of convergence proofs All three have attracted a lot of attention in computer science community. Normative vs. empirical approaches.
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Chaotic systems: consequence of solution properties for classes of systems of ordinary differential equations • analysis by simulation studies
Complex systems: macroscopic behavior of systems that contain a very large number of interacting simple agents with specific survival, reproductive, etc. characteristics • artificial life: studies involving large-scale simulations providing
trends of species survival, migration,... What is role of systems and control theory (apart from simulations with sexy graphics?) Recent research on "Highly Optimized Tolerance" by J.C. Doyle
27
Concluding Remarks
Our heritage is fifty years of exciting and important developments in theory, design methodologies and algorithms There exist numerous opportunities for relevant basic theoretical and methodological research • adaptive, multimodal, complex systems • my favorite: software that routinely solves Hamilton-Jacobi-
Bellman partial differential equations Expanding applications that benefit from theoretical developments Should always be on lookout for significant engineering/scientific/ socioeconomic applications that suggest novel theory development and extensions Expect continuation of "clashes" between ad-hoc approaches from the computer science community and the quantitative model-based methodologies of our discipline • we shall overcome!
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Fundamental Problems
In Adaptive Control
Workshop on Future Directions in Systems and Control Theory
Cascais, Portugal 22 June 1999
Brian D O Anderson Australian National University
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I
OUTLINE
THREE FUNDAMENTAL PROBLEMS OF ADAPTIVE CONTROL
» Changing experimental conditions accurate but inexact models
»Impractical control objectives » Transient instability
VINNICOMBE METRIC
ITERATIVE CONTROL AND IDENTIFICATION
ADDRESSING THE FUNDAMENTAL PROBLEMS
MULTIPLE MODEL ADAPTIVE CONTROL AND VINNICOMBE METRIC
CONCLUSIONS
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A Result with Practical Difficulties
Let P be unknown linear system.
Let P have known degree and be minimum phase.
[Statements regarding operation of an adaptive controller.]
Then as? —» °° all signals in the adaptive loop remain bounded, the controller converges and the performance index is minimized (i.e. asymptotically, all is right with the world).
This result sidesteps three problems which arise in adaptive control, including adaptive control of nonlinear, distributed etc systems.
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Problem Of Changing Experimental Conditions with Accurate but Inexact Models
Arbitrarily small open-loop modelling errors can lead to arbitrarily bad closed- loop performance, when a nice controller is designed for the model and connected to the real plant.
Large open-loop modelling errors may give small closed-loop errors.
Example: P1=:P0+ high frequency resonance, C stabilizes P0 with closed- loop bandwidth excluding the resonance.
Problem is that closeness of models is evaluated with particular experimental conditions.
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Problem Of Changing Experimental Conditions with Accurate but Inexact Models
Insertion, removal or change of a controller is a change of experimental conditions.
Qualitative conclusion: controller changes should be small.
Question: How can we measure smallness?
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Problem of Impractical Control Objective
Some non-adaptive control problems, while theoretically soluble, have impractical solutions. Example: LQG problem yielding e
degrees phase margin for the loop.
1
If one tried an adaptive control solution, not knowing the plant initially, there would be trouble (large transient signals, failure to converge etc.)
Question: How can we arrange that adaptive control algorithms flag over- ambitious control objectives, and back off in seeking to achieve them?
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I I I I I I! I I I l! I I lj
lj
I lj
I I I I I
Problem of Transient Instability
"All signals in the adaptive closed-loop remain bounded" permits 1012 amps motor current.
A property of one famous adaptive control convergence result is the following:
Let £0 denote the vector of initial conditions and
Z(t), the vector of all interesting signals at time t.
For arbitrary N > 0, there exists £0 with ||£0| < 1
and tx > 0 such that \\z(ti)\ > N
Suppose that at t = 0, we know how to connect a stabilizing controller. How can we adapt it - not knowing the plant - to retain frozen closed-loop stability?
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OUTLINE BHH BEB HBB SBH SP gHH2XSEKHBBES8BOBBBEBiB30BBBBBB^BB^BKBHBt8HBIBBSBBB^
THREE FUNDAMENTAL PROBLEMS OF ADAPTIVE CONTROL VINNICOMBE METRIC
» Generalized stability margins » Vinnicombe Metric and link to
stability margin » Consequences of the
metric/margin connection
ITERATIVE CONTROL AND IDENTIFICATION
ADDRESSING THE FUNDAMENTAL PROBLEMS
MULTIPLE MODEL ADAPTIVE CONTROL AND VINNICOMBE METRIC
CONCLUSIONS
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Generalized Stability Margin
r.
* c + 6
[uj = T(P, C)
T(P,C) = p{i+cpylc p{i+cpyr
_{I+CP)A
C {i+cpy1
*y
p i
{i+cpyl[c i]
Define generalized stability margin by
bP,c = \\T(P,C)l! if (P,C) is stable
=0 otherwise
- small if close to instability,
(1+PC close to zero)
- small if closed loop bandwidth
exceeds open loop [C(1 + PC)"1 large]
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Generalized Stability Marqin
I I
Measure of difficulty of controlling a
Sup particular plant P is bopt(P)= bPC
- Nice formula available, showing that right
half plane poles or zeros make bopt(P) smaller,
i.e. V^IL b'99er
- nonlinear version available
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Vinnicombe Metric and Link to Stability Margin
Introduce distance measure between two plants
For scalar plants,
max 7 n sv(PhP2)= °>
(i+|p2l2)2a+N2)2
(provided a winding number condition is satisfied), and
5v(p1,p2) = i otherwise.
- nonlinear and multivariable version available.
Suppose (PhC) is stable. Then
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Vinnicombe Metric and Link to Stability Margin
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Consequences ofthe Metric/Margin Connection
Px hard to control <=» bopt(Px) is small
<^> b(P\,C) is even smaller for some
stabilizing C
<^> little scope to change C without
risking instability
<=> little scope for modelling errors
in Ph or satisfactory behaviour
with C should Px drift
Small changes in C (= safe changes in C) require
knowledge of bPC for their determination.
Suppose q stabilizes P{ and C2 replaces q with
<5v(q,c2)<&P)Cl
C2 may give less robust design than q :
sin-1 fep5c2 - s*n~ ^P,Ci -$v(chc2)
and equality is possible.
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Consequences ofthe Metric/Margin Connection J
Svic^lTiPAhTiPAh* bP,c{ h,c2
\r(p,Cif <sv(cl5c2)
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OUTLINE
1 • THREE FUNDAMENTAL PROBLEMS 1 OF ADAPTIVE CONTROL
| • VINNINICOMBE METRIC
I . ITERATIVE CONTROL AND ] IDENTIFICATION , » Algorithm framework ' » Acceptable cautious controller
adjustments i » Desirable cautious controller
adjustments
• ADDRESSING THE FUNDAMENTAL PROBLEMS
• MULTIPLE MODEL ADAPTIVE CONTROL AND VINNICOMBE METRIC
• CONCLUSIONS
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Iterative Control and Identification Algorithm Framework
1 Assume controller C, is connected to real plant P and
gives stable closed loop.
2 Identify a model Pj of P using closed - loop data.
3 Using Pj redesign controller to obtain Ci+1 - ensuring Ci+1
is not too different to Cj.
4 Check that (Pj,Ci+1) behaves like (P,Ci+1). if so, update i
and go back to step 3. If not, go to step 5.
5 Reidentify P to obtain a new model Pj+1, connected to a
controller Cj+1.
• Repeat cycle until desired performance is attained or is
seen to be unattainable.
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Iterative Control and Identification Algorithm Framework
For plants with only unstable poles at origin, and with
Cj obtained by IMC design (with iterative closed-loop
bandwidth broadening), windsurfer approach to
adaptive control results.
17
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Cautious Contoller Adjustment: What is Acceptable?
Assumption: the statement that "Pj is a good model of P
using closed - loop data with controller Q"
means
r(p,q) = r(i>-,c-)
Hence
bp,Ci = bPj >q T(prci) -1
oo
Hence we should select Q+1 such that
5w(Q,Ci+1) < kbp c *e(0,l)
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Cautious Contoller Adjustment: What is Acceptable?
Assumption: for unknown P, the design goal is to obtain
stabilizing Cto minimize J(P,C).
Let P have right coprime description MT1 and Q right
coprime description UV~l.
C={C(Q):C(Q) = (U-DQ)(V + NQT1}
for Q stable delineates set of stabilizing compensators
Assumption: j(P^C{Q))ioxCeC depends on Q in a
convex manner.
I In Ho andfl^, dependence is convex ]
Q
* arg mm , ^ / x
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Cautious Contoller Adjustment: What is Acceptable?
— ^* If Sv(citdl+l)^kbp c., choose Q+1-Q+i
Else look at set for a e[0,l]
c[<xQ*) = (u-aDQ*)(v + aNQ*)~
a = 0:C=Q a = l: C=C*+l
Choose a e (0,1) with
Sy(Citc(aQ*)) = IApCi
Ci+l = c(aQ*)
and
j(Pj,Q+i)<j(Pj,Ci)
(Safety and performance improvement)
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OUTLINE
THREE FUNDAMENTAL PROBLEMS OF ADAPTIVE CONTROL
VINNICOMBE METRIC
ITERATIVE CONTROL AND IDENTIFICATION
ADDRESSING THE FUNDAMENTAL PROBLEMS
MULTIPLE MODEL ADAPTIVE CONTROL AND VINNICOMBE METRIC
CONCLUSIONS
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Addressing the Fundamental Problems
Changing experimental conditions given accurate but
inexact models
- Limit <Sv(Cj,Ci+1) using |T(PJ,C^
-Relyon|T(Piq)LsT(Pj>Ci
Impractical control objectives
Possibility 1: Identification may not be sufficiently
reliable to conclude T(PJ,CJ) = T(P,C|)
"Y-4^0 u r2 = 0
Suppose closed loop bandwidth is a>0
Suppose estimate PC(1 + PC)~1 at 10ö)0, SNR = 1
P(l + PC)"1 at 106)o estimate will have SNR = 1
If P has resonance at \0co0 not captured in P,, pXP.CJL
could be due to P(1 + PC)"1 at 10ö)0 and misestimated. 22
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Addressing the Fundamental Problems
Possibility 2: T(PjfCj) may be very large, so scope
for adjusting Cj is negligible.
Transient instability
Assume stabilizing compensator is known for the unknown plant.
Then transient instability is avoided (so long as controllers are not switched too fast).
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THREE FUNDAMENTAL PROBLEMS OF ADAPTIVE CONTROL
VINNICOMBE METRIC
ITERATIVE CONTROL AND IDENTIFICATION
ADDRESSING THE FUNDAMENTAL PROBLEMS
MULTIPLE MODEL ADAPTIVE CONTROL AND VINNICOMBE METRIC
CONCLUSIONS
24,
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Multiple Model Adaptive Control and Vinnicombe Metric
Consider a set of plants
P(S) = X S'X ■ 0.3 < X < 3
or an even bigger set
n = {P: P = Px(s)[l + A(5)]wiih|iA(7ö))L < ö}
Multiple model adaptive control assumes
there exist N and Xx < X2 <... < A^such that
- a nice controller Cu exists for Pu
- any plant in II is controlled nicely by
some Cu
Multiple model control provides an algorithm
to switch among the C^. Safety is an issue.
24a
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Multiple Model Adaptive Control and Vinnicombe Metric
I I
m
Answer for Px(s) «- A; —>
0.3 X, /-i A,.. A, 7+1 A, 7+2
All plants in A,- obey
8v(Px,Pu)nPu,Cul<0.l
This ensures
7(PA„cA,.)-r(P,,c,)L<o.i|r(P,,c, 24b
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OUTLINE BOBBI
THREE FUNDAMENTAL PROBLEMS OF ADAPTIVE CONTROL
VINNICOMBE METRIC
ITERATIVE CONTROL AND IDENTIFICATION
ADDRESSING THE FUNDAMENTAL PROBLEMS
MULTIPLE MODEL ADAPTIVE CONTROL AND VINNICOMBE METRIC
CONCLUSIONS
» Main messages
» Shortcomings and future possibilities
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Main Messages
•
Adaptive control should have modest steps only in the controller.
The Vinnicombe metric plus closed-loop measurements indicate what is modest.
• A desired control objective may be impractical; good algorithms need to flag this and adjust their goal. Some do this.
• Transient instability should be avoided, and can be avoided.
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Shortcomings and Future Possibilities
•
The Vinnicombe metric is a single number. It would be better to look at the whole imaginary axis
There is no protection against step changes in the plant (Hypothesis testing rather than parameter estimates should be used if possible to identify change)
There is a lack of methodology for optimally using closed-loop data obtained for one plant and more than one controller in identifying that plant
The ideas look fruitful for nonlinear problems. Much has already been done on the Vinnicombe metric / robust stability question.
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Identification of Complex Systems
Munther A. Dahleh
Department of Electrical Engineering and Computer Science Laboratory for Information and Decision Systems
MIT
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Motivation
• Robust Control Theory: Major development in the past 20 years.
• Design tools
- Start with the nominal process, a description of the unmodeled dy-
namics and exogenous inputs.
- Provide performance tradeoffs.
• An example of an I/O description of a process:
y = Gu + 6Gu + WAu + w.
• System identification has not had a parallel development.
• Existing theory did not address control-oriented modeling.
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Objective
• Develop a paradigm for identification of complex systems using limited-
complexity models.
• Motivation: consistency with existing robust control paradigms.
• Main considerations
- Explicit under modeling.
- Error measures. - Noise (deterministic or stochastic).
• Standard issues
- Experiment Design.
- Algorithms and Computations.
- Error bounds.
- Complexity.
• Reference: S.R. Venkatesh. Identification for complex systems. Ph.D. Thesis # LIDS-TH-2394, MIT, Cambridge, MA.
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Bahnst Control
• Interconnection:
- G: Nominal system.
- A: Describes plant uncertainty.
- K stabilizing feedback controller.
- w: exogenous input.
- z: regulated output.
• Uncertainty:
- Uncertainty description is linear fractional.
- Example: Additive or multiplicative uncertainty.
- A is an unknown system, with bounded "induced" norm (Hoc, ^i)-
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I I I (MPE) System Identification
I ■ I • Minimum Prediction Error Paradigm:
" y = fe{u,w) see
■! - w is a stochastic Process (generally white noise).
| - A parametrized set of models.
- Actual process is in the parametrization. Ii - Estimate the parameters from Data by minimizing the "prediction
error .
m\ • Positive results.
■ 1 _ Consistency, if actual process is in the model parametrization.
- Asymptotic distribution of the parameters.
11 - Sample complexity issues.
• Limitations
- Stationarity assumptions on the noise.
| - Results are asymptotic. - Under modeling is not accounted for a priori. Equivalently, noise is
I the only source of mismatch. - Difficult to integrate with existing control paradigms.
I I I I I
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Set-]\4^rTiVi^rshTp^yster^dentification
• Prior Information
- The Process: y = G{Oo)u + &{u)+w
- w:= output noise, w eW.
- A:= Plant uncertainty, A E A. - model parametrization M(9) = {G(9) + A, A E A} C X, X:--
model set.
- 9 £ 0 <E ]Rn
- u:= Input, «eW (||w||oc < !)•
• Positive issues
- Consistent with robust control.
- Explicit under modeling.
• Negative issues
- No clear separation between the finite models and the unmodeled.
- W = Bioc is a conservative model for noise.
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Problem Description
• Process class, T, contains stable LTI systems,
- Set is complex; not approximable by a finite dimensional set.
- equipped with a metric.
- Real process is in this set.
• Model parametrization Q
- Finite dimensional set (subspace).
-GcT
• Noise
- Stochastic or deterministic.
- Deterministic: set is not too rich or too constrained.
- Set description of white noise.
• Data: finite observations of the input and output.
y = TQU + w
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proH^m Formulation
• Output equation:
y = T0u + w T0eT,weW
• Define Go as Go = argmm||To-G||
• Prior: There exists a 7 such that dist(T0, Q) < 1-
• Problem: Does there exists an input u, and an algorithm Gn so that:
1. limn_^oo supr0)W ||Go - Gn\\ - 0, 2. Time complexity of convergence is polynomial.
To 'G
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Comments
} • Choice of T
- Operator spaces (#oo, ^i)- 1 - The space l<i more convenient mathematically.
- Hardy-Sobolev space U\ Hubert space with inner product
' <f,9>n=<f,9> + <f''9'>
I - The following inequality holds:
j IITHoo < ||T||i < c\\T\\H i
- Robust control can deal with U.
1 • Prior structure is additive. Need more complicated structures.
j • Outcome of identification:
| — An estimate Gn. ■' - A (parametric) estimate of the error ||G0 - Gh\\, Dn.
} - An estimate % of the unmodeled part % < 7.
- Additional structure on the unmodeled part, A = Q .
- The model
{Gn + A I \\G - Gn\\ <Dn,&£ A, || A|| < %}
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Description of the Unmodeled
• Go is the best approximation of T0 in T = U.
• Express To as T0 = Go+'Ao
Ao G ^.
• Example: Q is the set of FIR of order M, then
g± = {xM+1f{\)} f arbitrary}.
• Example: Q = {6/1 - a\ 0 G R}, then OO
£x = {A | £ (1 + k2)a% = 0.} fc=0
• If Q is a subspace, then A is nice set!
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Diameter of Uncertainty
• Set of unfalsified models:
Sn(y,u) = {Ge Q\y{n) = Gu(n)+&u(n)+w(n), A e A, ||A|| <^weW}
• Diameter of uncertainty at time n:
Ai(T,£, W,-u) = sup sup diam(5n) TeTweW
• Asymptotic diameter of uncertainty D = limn_>oc Dn.
. Problem: Does Dn converge to zero for some input u in polynomial time?
A.
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Results
• Exact modeling
- Bounded-but-unknown noise.
- Low correlated noise models. - Bridge the gap between stochastic and deterministic formulations.
• Under modeling
- FIR model structures - General finite dimensional subspaces of finite-dimensional systems.
• Identification in practice.
• Connection to robust control.
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Correlation Methods: Exact Modeling
• Problem Set-up:
- y = GQU + w - Go e Q is the subspace of FIR models of order M.
• The estimate is computed as follows:
r?u(T) = GNr»(T), r<M
• When does G.v converge to G0?
• Input is persistently exciting of order M. ■
• Noise:
- quasi-stationary, uncorrelated with u (e.g., white noise).
- Set-valued with Correlation Constraints: (%v < ^)
wN = {denN\±JL\r»(T)\<clN}
- Set-valued with Periodogram Constraints:
<W{w), we [0,2TT] WS = {de IR*1 l £ dkJwk
k=0
•
- Both W.v, W& contain white noise with high probability.
Convergence is uniform with polynomial sample complexity.
Does not. converge if noise is only "bounded-but-unknown".
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Under Modeling: Basic Idea
a
T
Gc
A
Go
■w
O
w
o-
^
y
G'n r,v
• Two-step algorithm
1. Filter the data to eliminate the mimodeled part.
2. Use previous correlation method.
• T is constructed from the annihilators of A.
• What about the input?
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Tnput Design: Motivation
• Q is the subspace of FIR models of order 0 (constants).
• A = (0,/i,/2, .-.)■
• y(k) = g0u(k) + Ej=i fju{k - j) + w{k)
• Tradeoff
- If w = 0, then the best u is the unit pulse.
- If w ^ 0, then u needs to be longer.
• Periodic inputs will not work:
y(k) = (go + // + . • .)«(*) + (/i + /m + • ■ -M* -!) + ■•■ + W*)
• Obtain first order correlations with u
rJi=SorJ"(0) + X:/i^(*)+^(0) ^ *=i
• Conditions on the input (Robust Input)
1. SUPI<KA- |${§| -> ^ero.
O ÜWHVQ) —^. ypTQ l- r*(0) ^ -e'a-
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Input Design
Not robust inputs
— Periodic - Standard Chirp u{t) = exp(iorf2), a G B.
max \<i<N
rfW *f(0)
> e for any e and some N
- PBRS: For large period m, correlations up to m can be made small.
Not sufficient.
Robust Inputs
- High order chirp u(t) = exp(zatf3), a = TT(1 + \/5):
max Ki<N
CM tf(0)
<C log(iV)
JV
- i.i.d. Bernoulli process with zero mean and variance 1
P | maxrf («) > a\ < nexp(-n/(a<))
- For both signals rSu{0) >f(0)
—> zero
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TTTR Models
• Assumptions:
- T is £1 or ft. - 0 is the set of FIR systems of order M.
- Noise is in W.
- Prior X: dist(T0,£) is bounded by 7.
- Input: High order chirp.
• Results
- The least squares estimator converges to Go:
sup sup \\Gn - Go\\ < (C'i7 + C2M)l^0-
- The estimator is untuned
- Complexity is polynomial.
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Example: One-Pole Model
• T = k (for simplicity).
. A = {(A - a)/(A), / e k}-
• Equivalent output equation:
y{k) = 9u(k) + (A - a)f(X)u + w, u = 1-aX
■u
• Comments
- There is no direct separation between the model and the unmodeled
dynamics. - Least squares estimator will not converge. The unmodeled part af-
fects rf(0). - Need to exploit the structure of the unmodeled.
• A two-step approach
- Pre-filter the data using the non-causal filter a~k. This corresponds
to multiplying the data by
/1 a or .. a
a \
v ... i ; - Apply a least squares estimator to the filtered Data.
- Can interpret as a weighted LS.
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GeneraLResult
• Assumptions
-r = n. -Q A finite dimensional subspace of stable systems, with dimension
M. - Noise is in W- - Prior X: dist(T0,£) is bounded by 7.
- Input: High order chirp.
• Results
- There exists a filter T such that the two-step algorithm converges to
Go sup sup \\Gn - Goll < (C1.7 - L-2M)—y=- Teiw€W" v
- The estimator is untuned
- Complexity is polynomial.
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MPEjaevisited
. If w is WN, then it can be shown that for a fixed T0 G T, the LS estimator
satisfies: _ ^ i.. . lim GN = G = min||(To - G)$ä||2 ™-P-l
JY-J-OO " Gey
where $u is the spectral density function of u.
• Comments:
- Convergence is only pointwise (w.p.l).
- There is no uniform convergence rate.
- Results are asymptotic. - Results require quasi-stationarity.
- G^Go unless the input is robust.
- No robust inputs are constructed.
- Theory addresses i2 error bounds.
• Contrast with previous results:
- Convergence is uniform (w.p.l if w is WN).
- A uniform convergence rate is derived.
- Estimates are based on finite data.
- No stationarity assumptions.
- Robust inputs are constructed.
- Theory addresses more general error bounds.
I
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jjjpntifi^atmn Tn Practice
• The set of unfalsified models, SN is convex.
- Define zw(t) = y(t) - Gu(t) - w{t) for t = 0, ... N
- Let zw be the vector constructed from zw(t).
- The set SN can be written as:
SN = {G\zw = US1 <g\A>=0,i = l,..-M,weW}
U is the Toeplits matrix constructed from u and gx's are the annihi-
lators of A.
- Eliminating A
SN = {G | zwTVzw < 7, «> e W}
V can be computed from the resulting QP.
• Let DN{-y) = diam(Sjv)-
• Estimate JN as follows: min 7
Av(7)<(Ci7 + C2M)- 5V
N
Estimates capture tradeoff between parametric and non parametric er-
rors
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•
Example
Process: -j_ TM = irä8Ä+A(A)-
The unmodeled: 200
A(A) = £ 5(k)Xk
/200 o , \~1/2 &i{k) 5(k) = [j:m(k2+i)j ^
where Si{k) = (A - a)v(k) and v, a vector of length 200, was selected
using a random number generator.
. Noise: N(0, .3)
Input: High order chirp.
I
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0.5 1.5 2 frequency
2.5 3.5
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Example
Comparisons:
Process T\
Algorithm
MUD LS
WLS
.97
.67
.60
% Error
33 40
Actual Error
.03
.33
Estimated-Error
.06 .0002 .0002
unmodeled-erro i 1.0 1.9 =1 2.2
Process To
Algorithm 9 % Error Actual Error I Estimated-Error unmodeled error
1.01 1.95 2.3
Estimating the unmodeled:
I
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2.5
2 Noisy Estimate Norm=200
200 300 400 900 1000
Estimate Unmodeled Dynamics
100 200 300 400 500 600 700 800 900 1000 frequency
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Robust Control
c
A
Standard robustness results extend for perturbations in U.
Stability if and only if for all 9 G [0, 2TT) and A G A
C l + (Ar A^J) (e^O
Idea is to compute the range of A at each frequency: oc
s<H(OT:£(i+rtBsl fc=0
Through the singular value decomposition, we can find a continuous
function in 9 W{9), such that
5(ö) = {w(ö)Q);|h.e||2<l
Stability if and only if for all 9 e [0, 2TT) and ||fa f]||2 < 1
C J6\ _L l + fo fl^T Uc ) (e )7"°
Above is a standard one-rank problem. Both analysis and synthesis are
equivalent to convex problems.
A theory for the general robustness problem (including parametric un- certainty) has been developed with such error bounds.
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■ I I I Conclusions
I ■ • A new paradigm for system identification
- Bridges the gap between stochastic and deterministic formulations.
| _ Under modeling is explicit in the formulation.
m - Based on sample-path analysis.
I _ Captures tradeoffs in error estimates.
I« Formulation compatible with robust control
- Derive error bounds in terms of || • \\n-
Ij - Derive robustness results for this norm.
m - Exploit the Hubert space structure.
I I I I I I I I I I
Future directions.
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ISSUES IN CONTROL DESIGN :
THE IMPORTANCE OF SYSTEM IDENTIFICATION IN CLOSED LOOP
I.D. LANDAU Laboratoire d'Automatique de Grenoble
ENSIEG BP46 38402 Saint Martin d'Heres, France E-mail: [email protected]
June 1999
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CONTROLLER DESIGN AND VALIDATION
Performance specs. Robustness' specs.
DESIGN ÄTHO> t^i^ y- yaw i?^
MODEL(S)
T rui-
L
Ref.
1) Identification of the dynamic model(s)
2) Performance and robustness specifications
3) Compatible controller design method
4) Controller implementation
5) Real-time controller validation (and on site re-tuning)
6) Controller maintenance (same as 5)
(5) and (6) require closed-loop identification
i n
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REAL-TIME CONTROLLER VALIDATION ^
äS§ifti??f«i
m
noise
PLANT
V MODEL
i
design system
Comparison of "achieved" and "desired" performances
A useful interpretation :
Check to what extent the model used for design allows achievement of:
- desired nominal performance - desired robustness specs, (sensitivity fct.)
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mm
mim
REAL-TIME CONTROLLER VALIDATION iV-:
Time domain tests (too inaccurate in some cases )
Closeness tests of achieved and desired poles and/or sensitivity functions
by identification of the closed loop
PLANT
I
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CLOSED LOOP PLANT IDENTIFICATION
Why?
There are systems where open loop operation is not suitable (instability, drift,..)
A controller may already exist (ex .: PID )
Re-tuning of the controller a) to improve achieved performances b) controller maintenance
Iterative identification and controller redesign
Cannot be dissociated from the controller and robustness issues
DUAL PROBLEM CONTROLLER REDUCTION
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-Introduction
- Background in robust digital control
- Identification in Closed loop - Some facts - Closed loop output error (CLOE) algorithms
- Validation of models identified in closed loop
- Iterative identification in closed loop and controller re-design
- Experimental results
- Controller reduction by identification in closed loop
- Experimental results
- Identification in closed loop of nonlinear plants
- Conclusions and Perspectives
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Background in Robust Digital Control
sä
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I
I
I I
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1
I
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ROBUST STABILITY
ImG
uncertainty disk CO
CO = 7C ReG
GCe-i®)
Family of Plant Models : G' e F(G,8, Wx)
G - nominal model; SCz"1)^ < 1 Wx(z"1) - size (and type) of uncertainity
Robust Stability Condition :
a related sensitivity . .a type of uncertainity function *^ A
\\^\l < 1 >g, defines an upper template
/for the modulus of the S I < IW I"*! sensitivity function
defines the size of the tolerated uncertainty
There are also lower templates (because of the relationship between various sensitivity fct)
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I 1
I 1
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j^^äw^s*«
'IM
> ■■ <^> S ■■. PO,* ^^*" f,v '-.-Xi «»V 1 v> * ^ ji y,v| 1, w S ,«,ts .
^fSlÄIÄJwipiä^
. *^^{S*3S ROBUST CONTROLLER DESIGN
^Cftbi',i V&V
Combining nominal performance design with shaping of the sensitivity functions
Pole placement with sensitivity function shaping using convex optimization
uses convex parametrization w. r. to auxiliary poles and controller parameters. m
Kra ?^U^--i
f««l WlPfH £S^»:«?WM
Pi
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14
I I
t
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1
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IDENTIFICATION IN CLOSED LOOP )
SÄ
Jim. ru
*■'. • K - ■
Ü 0t
öS»';
i iftti
Obj.: Development of Algorithms ^hich :
- take advantage of the "improved" input spectrum
- are insensitive to noise in closed loop operation
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CLOSED LOOP PLANT IDENTIFICATION )
FILTERED OPEN LOOP ALGORITHMS
JULTL noise
MR
" /~* iigsimmi
i AT i ^ I •V'.V -!
I/O data filters : S/P or AS/P or ....
(may be related to the control objective)
m Pip; «If'''
!^;!' »I
mm m
- biased estimates
- require (theoretically) time varying filters
- FOL can be seen as approx. of CLOE alg.
- are used in standard indirect adaptive control
18
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CLOSED LOOP OUTPUT ERROR ALGORITHMS
JUTL noise
CLOE
9(t+l) = 9(t) + F(t)<D(t)8CL(t+l)
<D(t) = (|)(t) = [-y(t),..,ü(t),.J (CLOE) or
O(t) = |<l)(t) (F-CLOE)
;.-.■■■■■;. iM?*^.;.;
- Unbiased parameter estimation (asympt.)
- Mild sufficient positive real condition for convergence ( S/P or P/P)
- Insensitive to plant disturbances
19
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Objective of the Identification in Closed Loop
(Identification for Control)
Find the "plant model" which minimizes the discrepancy between the "real"closed loop system and the "simulated" closed loop system
W
-
R A/lodel
Parametric | Adaptation
Algorithm
CLOE Algorithms (Closed Loop Output Error)
I
fi
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Jim Excitation on the reference
w M i 1 +
£CL
±C » 1/S u q-dß
A y r
r V
N/lrvH^I
R
u = -|y + Xr ~ R - T
Excitation added to the controller output
w
Jim ru
r+Q^ B/A -+&-, + X ■ ■ +
R/S ^
y
+
6—► £CL
■5 ^u. ^ 9 1 ^\J ^ D/M
1 R/S < J
u = 's y+ru - R - u = -g-y+ru
21
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I Closed loop system: | (1) y(t+l) = -A*y(t) + B*u(0 = eT(p(0 |
0T = [ai,.., anA, bi,.., bnB] ■ q>T(t) = [-y(t),..,-y(t-nA+l), u(t),.., u(t-nB+l)] I
u(t) = -hit) + ru 1 Closed loop adjustable predictor: »
(2') y°(t+l) = -A*(t)y(t) + B*(t)u(t) = 9T(t)(|)(t) I
(2") y(t+l) = -A*(t+l)y(t) + B*(t+l)u(t) =9 (t+l)<|)(t) |
0 =[äi,.., änA, bi,.., bnßj
(j)T(t) = [-y(t),..,-y(t-nA+l), ü(t),.., u(t-nB+l)]
I I I
u(t) =-|y(t) + ru -
Closed loop prediction( output) error I e£L(t+l) = y(t+l) - y°(t+l) a priori I
£cL(t+1) = y(t+1) - y(t+1) a posteriori
22
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Adjustable Predictor:
y°(t+l) = eT(t)<t>(t)
y(t+l) = eT(t+l)(^(t)
9T(t) =[äi(t),.., anA(t), bi(t),.., b„B(t)]
^T(t) = [-y(t),..,-y(t-nA+l), u(t),.., u(t-nB+l)]
The Algorithm
9(t+l) = 9(t) + F(t)(|)(t)8cL(t+l)
FCt+l)"1 = ?H(t)F(t) + 9i2(t)(()(t)(j)T(t)
0 < ^i(t) < 1 ; 0 < ?i2(t) < 2
ecL(t+1) = «0**1) 1 + (|)T(t)F(t)(t)(t)
ecL°(t+D =y(t+l) - eT(t)<Kt)
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(the "plant model" is in the "model set" )
eCL(t+i) = |[e -e(t+i)7<|)(t) + ^w(t+i)
Deterministic environment (w=0)
limt—ecL(t+l) = limt^EcL°(t+l) = 0
(J)(t) <oo;Vt
Stochastic Environment (w * 0 )
Prob{limt-^9(t) e Dc} = 1
where : Dc = {o : <j>T(t,9)[e - e] = o]
//••
H(Z )_P(z-l) 2-^-K'
where : maxtA,2 < A, <2 24
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(Adaptive) Filtered Cloded Loop Output Error
IT, ecL(t+i) = ff[e-e(t+i)]|(t)(t) =£[e-e(t+i)J <t>f(t)
F-CLOE:
AF-CLOE:
P = AS + BR; (J)f(t) = 4 <|)(t)
_ S P(t,q-l) = A(t,q-l)S + B(t,q-l)R «MO = -^y 4>(t)
The Algorithm
9(t+l) = 0(t) + F(t)<l)f(t)ecL(t+l)
FCt+l)"1 = ^i(t)F(t) + X2(t)(t>f(t)<l)fT(t)
0 < ?ii(t) < 1; 0 < ^2(t) < 2
, is £CL°(t+l) / N ecL(t+l) = ^— (approx.) 1 + <t>f
T(t)F(t)ct>f(t)
EcL°(t+D =y(t+D - eT(t)<t>(t)
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- Deterministic Stability Condition :
- Stochastic Convergence Condition (O.D.E.):
?^i.^=S.P.R. P(z-!) 2
maxtX,2 < A, <2
26
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Asymptotic BIAS - Frequency Distribution ED ("plant model" not necessarily in the "model set")
CLOE/F-CLOE/AF-CLOE algorithms
6 = argmin0
r
J -K
Sypp[|G - Gp ISypp^C©)
(J)w(co)]dco
(excitation added to the controller output)
excitation on the reference :
Sypp(k(co) -> |TurPWco); Tur = TÄ/P
Syp = Sensitivity of the closed loop predictor
- noise does not affect parameter estimation (as.)
- precision of estimated model enhanced in the critical frequency regions for design
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li
I
Validation of Models Identified in
Closed Loop
28
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Validation of Models Identified in Closed Loop
-&
Controller dependent validation !
- Statistical Validation
- Pole Closeness Validation
1) Identify the closed loop poles (identification of the closed loop system)
2) Compute the closed loop poles based on the identified model
3) Compare (visually and using v-gap)
- Time Domain Validation
1) Compare real time results and simulation results
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Identification in Closed Loop j
STATISTICAL MODEL VALIDATION
|RN(i)|<^; i>l
-/ ^
normalized number of data crosscorrelatjon.
97%
N = 256 —►|RN(i)| < 0.136
|RN(i)|<0.15
Controller dependent validation! 30
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Identification in Closed Loop ] MODEL VALIDATION by POLE CLOSENESS
1) Identification of achieved poles and/or sensitivity functions by identification of the closed loop
JTUU +
^fer.s&sixs.;
BA A
Adaptation Algorithm
t> = estimation of achieved closed loop poles
The same signals are used for the the identification of the "plant model" in closed loop and for the identification of the "closed loop"
2) Computation of the closed loop poles and/or the sensitivity functions based on the identified model
3) Comparison (visual or using v -gap)
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Iterative Identification in Closed Loop and Controller Re-Design
■'IM i .I t\nmmmmmmiimiinmmS^Si^SmiimmMarfmmiim»mm**mtm
T +
1/S
R
*s v^- 1/S u q-dß
A
y
{*■ i ™ Mr\rlol
R ■^-
Step 1 : Identification in Closed Loop
- Keep controller constant - Identify a new model such that: £CL - Validate the new model
Step 2 : Controller Re - Design
- Compute a new controller such that :^CL"
Repeat 1, 2, 1, 2 ,....
i
£CL
Rem.: the first iteration gives the most significant improvement
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How to use OL and CL Identification s. D
OPEN LOOP IDENTIFICATION
I OPEN LOOP
MODEL VALIDATION
OPEN LOOP BASED CONTROLLER (OLBC)
I PLANT IDENTIFICATION
IN CLOSED LOOP(with OLBC)
IDENTIFICATION OF THE CLOSEPliiOP
I COMPARATIVE
CLOSED LOOP VALIDATION
i ~ BEST PLANT MODEL
I CLOSED LOOP BASED CONTROLLER (CLBC)
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Experimental Results: Identification in Closed Loop
and Controller Re-design wmmmmmmmmmmm jj*f^-^---^M^ämmiinmmmi' ' riiirr»————""""^
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Identification in Closed Loop +
Controller Re-design
The Flexible Transmission
motor axis
DC axis
position Position sensor motor
' L i I * D A C
PC D C
-1 cE
fVurtrnllor *o*- V '
ref
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ADAPTECH WinPIM
dB
20 1——
~ 'modele idetitif 16 •en boucle fermee
- l - ■ ■ - ■ ■!--.-■
0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
Frequentielles f/fe Reponses
Frequency response of the flexible transmission (identified models)
ADAPTECH WirxREG
6 .20 r
5.80
5.60
5.40
5 .20
Sim_BO =t=R
Sim_BO : r^ponse en simulation avec le module identifie en Boucle Ouverte
Reel : reponse en temps reel de la boucle fermee
45 50 5.5 60 65 70 75 80 85 90
Reponses Boucle Fermee
Step responbse of the closed loop system (model identified in open loop(BO).)
I
36
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I I I I I I I I li I I I I I I I I I I I I
ADAPTECH WinREG
6.20
5.80
5.60
5.40
5.20
Sim BF
=i=n
H Sim_BF : r6ponse en simulation avec le modele identifie en Boucle Fermee
pi Reel : reponse en temps reel de la boucle fermee
45
Reponses
50 55 60
Boucle
65 70 75
Fermee
80 85 90
Step response of the closed loop system (model identified in closed loop ( B.F.))
ADAPTECH WinREG
0.50
-0.50
o : poles identifies.
x : p61es calcules avec le modele iden ifie en BO
-0.50 0 0.50
Poles_BF
Poles of the Closed Loop (model identified in open loop (B.O.))
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0.50
-0.50
ADAPTECH
poles identifie
WinREG
x : poles calcules avec le modele ident^ifie en BF
-1 -0.50 0.50
P61es_BF
Poles of the Cloed Loop (model identified in closed loop (B.F.))
ADAPTECH WinREG
5.80
5.60
5 .40
5 .20
"eir-r Reel Siitvu
Reel : r^ponse en temps reel avec le r^gulateur base svur le modele idenfcifie en BF
Sirma : reponse en simulation
_i t _i L.
45
Repons(
50 55 60 65 70 75 80 85 90
Boucle Fermee
Step response of the closed loop system, (controller based on the model identified in closed lop)
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CONTROLLER REDUCTION USING
IDENTIFICATION IN CLOSED LOOP
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It is an important issue in many applications
Several approaches can be considered
The most important: controller reduction should be done with the aim to preserve the closed loop properties
Identification in closed loop offers two useful approaches for controller reduction :
1) (indirect) Identify a reduced order model in closed loop operation
(will capture the essentials characteristics of the model in the critical frequency regions for design)
2)(direct) Reduced order controller identification in closed loop
(will preserve the closed loop characteristics in the critical frequency regions)
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Reduced Order Controller Identification in Closed Loop
nominal simulated closed loop
rum +
R/S ^
q-d§/A
+
i
reduced order controller
X. -
R/S u
/-
q-dg/A < 1 P.A.A.
£CL
Algorithms similar to those used for plant model identification in closed loop
The algorithms will try to minimize the discrepancy between the two closed loops
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Asymptotic BIAS - Frequency Distribution
K = ß- (nominal controller)
K = E, (estimated reduced order controller) S
.—rp
9 =[fo,fl,...,fng,Si,S2»--->Sn§J
"K
9*= argmine SypF [ |K - Kp |Syp|2^r(cö) -7t
+ <|>p(aO] dco
- noise does not affect parameter estimation (as.)
- precision of estimated model enhanced in the critical frequency regions for design
- precision can be further "tuned" by the choice of r(t)
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- Comparison of the sensitivity functions
- Comparison of the controller frequency characteristics
Rem.: The v-gap seems very useful
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Use of Real Data for Controller Reduction Si
true closed loop system
J1U1 rr R/S &
PLANT i J
R/S u
q-dB/A
/ reduced order
controller !ar(t mode, identified in closed loop
P.A.A.
£CL
Validation of the estimated reduced controller using similar tests as for plant model validation in closed loop
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Experimental results: Controller Reduction using
Identification in Closed Loop wsäM
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ACTIVE SUSPENSION
primary acceleration (disturbance)
elastomere cone
piston residual acceleration
1 !gd)rirCJlii? 1
nertia chamber
Active suspension identified model T
0.05 0.5
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Model:
nA = 6, nB = 8 , d= 0 ; (order of the system : 8)
Fixed part of the controller :
R= R'HR HR = 1 + q-1 (opening the loop at 0.5fs)
Design : Pole placement with sensitivity functions shaping (see templates next viewgraph)
Resulting controller (nominal) :nR=ll,nS = 13
Theoretical dimension of pole placement controller
nR=5, nS= 7
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Syp - Output sensivity function
nominal (nR = 11, nS = 13)
= 7. nS = 9
nR = 5, nS = 7
0.35 0.5
T r Sup - Input sensivity function 1 1 1
0.5
48
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Dual Vinnicombe stability criterion :
8(Kn,Ki) < b(Kn) v-gap ' ^ stability margin
Controller 5(Kn,Ki) 8(Sypn,Sypi) 5(Supn,SupF b(K)
Kn nR=11,nS=13
0 0 0 0.079
K1 nR=7,nS=9
0.039 0.122 0.041 0.069
K2 nR=5,nS=7
0.187 0.119 0.135 0.072
-5
-20
-30 -
Controller frequency response -i r 1—: r n r
nR = 5, nS = 7 -
0.05 0.1 0.45 0.5
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Identification of Continuous Time Nonlinear Plant Models in Closed Loop Operation
jgs&jssaasm
U
ru + NL Plant
NL Controller i
U
+
* y NL Model
s
NL Controller <—
/—
P.A.A.
£CL
NL-CLOE
I
i
- Applicable to nonlinear plant models which are nonlinear in the parameters
- P.A.A. with a similar structure as for the linear case
- O contains computable gradient expressions (functions of ü and Ay)
- The noise problems are for the moment an open issue
50
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-Identification in closed loop can provide better plant models for design
- Identification in closed loop can not be dissociated from the controller and robustness issues
- Specific closed loop identification algorithms with convenient properties have been developped
- Identification in closed loop has started to make its way in practice
- Algorithms for identification in closed loop have been successfully applied for controller reduction
- Algorithms for identification of nonlinear plant models in closed loop operation have been developped
- Comparative evaluation of the controller reduction techniques (to be done)
- The nonlinear case raise numerous and interesting problems (to be explored)
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I.D. Landau, R.Lozano, M. M'Saad "Adaptive Control" Springer Verlag, London, 1997, (Chap. 5, 9,13)
I.D. Landau, A. Karimi "Recursive Algorithms for Identification in Closed Loop: A Unified approach and Evaluation", Automatica, voll 33, no 8,1997
A. Karimi, I.D. Landau "Comparison of the closed loop identification methods in terms of the bias distribution" Syst and Control letters, 1998
M. Gevers "Towards a joint design of identification and control?" in "Essays in Control: Perspectives in the Theory and'its Applications"Birkhauser, Boston 1993
L. Ljung "System Identification: Theory for the user", Prentice Hall, 1998
B.D.O Anderson, M. Gevers "Fundamental problems in adaptive control" \nnPerspectives in Control"(D.Normand -Cyrot,ed) Springer, 1998
P. Bendotti,B. Codrons, CM. Falinower,M. Gevers "Control oriented low order modelling of a complex PWR plant: a comparison between open loop and closed loop mathods" 37 IEEE-CDC , Tampa, Dec. 1998
F. De Bruyne, B.D.O. Anderson, I.D. Landau"Recursive iterative feedback tuning " submitted to 38 IEEE-CDC, 1999
I.D. Landau, B.D.O. Anderson, F. De Bruyne "Closed- loop output error identification algorithms for nonlinear plants" submitted to 38 IEEE-CDC, 1999
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I ■ftp Hai Vision Based Navigation
and Control of Robotic Systems
Joäo Sentieiro and Jos6 Santos-Victor Institute Superior T6cnico
Institute de Sistemas e Robötica Lisbon, PORTUGAL
¥ VisLab - Computer Vision Lab
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Vision Based Control: First Steps
• Shirai and Inoue (1973) : visual feedback for precise motion control of a manipulator.
• Hill and Park (1979): "continuous use" of image information; term Visual Servoing first used
• Weiss, (1980): dynamic visual control of robots, stabilizing initial concepts.
VisLab - Computer Vision Lab
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• .'. ' , 1
mm Initial Difficulties Mmwmm -ft N _ >
Limited Computing Power Expensive and complex image digitizers and processors Slow sampling frequencies and dynamics
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Why vision based control of robots
Many robots used in (constrained /changed) industrial environments
Need for dealing with more general environments Vision allows non-contact measurements (perception) of the environment (as humans)
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Challenges for Control Science
Noisy image measurements time consuming (still!) image processing
large amount of data to process non-linearites of camera-lens system difficulty of inferring 3D information from 2D images
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Challenges for Control Science
Large delays in the feedback loop Unreliable sensor data Inadequately modeled or difficult to model plants or systems Poorly conditioned transformations
fc Vis Lab - Computer Vision Lab
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MM &4$ Image-based look and move structure
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Position-Based Visual Servoing
World Coordinates
o Cartesian control law
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Image Processing
Image coordinates
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fr
Image-Based Visual Servoing
World Coordinates
Image based control law
Image coordinates
Vis Lab - Computer Vision Lab
K
Talk Outline (example cases)
Visual Navigation: Learning from bees - Problem - Motivation
Visual Docking Remote control of Cellular Robots
Binocular Tracking with a robot Head
Conclusions
Vis Lab - Computer Vision Lab
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Visual Navigation
(Srinivasan 91)
VisLab - Computer Vision Lab
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I I I I I
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Right Flow
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Optical Flow computation
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I I I I I I I I
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Rigid Body Motion : T, W
Perspective Projection
Optical Flow
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v(x,y)=fy
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Jy J
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Z(x,y) = i-yxx/fx-vyy/fy
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Motion Model
Global Parametric Model
u{x,y)=M0 + uxx + u,y + u^xy + u^x2
v(x,y) = v0 + vxx + vTy + v^xy+v„y*
Affine Model
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v(x,y) = v0+vxx + vyy
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RESULTS
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Visual Control of Cellular Robots
Vis Lab - Computer Vision Lab
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Robot-Eye Coordination
m=M m'
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Image and plane coordinates of 4 points (minimum) required
Vis Lab - Computer Vision Lab
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«ISTITUTA
Control Strategy Goal Point
•Decoupled angular and linear
velocities dynamic systems
•Separate Heading and speed
control using the angular error
and distance to target point.
fr
Angular error
Heading direction
Vis Lab - Computer Vision Lab
Control Problem structure
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Controller Zero Order
Hold Ge(s)
Camera unknown Non-Linear mapping - M
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•Measure angular error between robot heading and direction to target
•Generate differential voltages to control the robot heading direction
•Similar loop can be used to control the linear speed as a function of the distance to goal
Vis Lab - Computer Vision Lab I
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Binocular Visual Tracking
Descartes 1664
h VisLab - Computer Vision Lab
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JHSWVTÄ
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fr
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> Robot head pose
>* Target pose
*" Image features
Head/Object Motion
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Kinematic controller
8q = -j;l5f + j;1Jp 8 p
p = K(q,f) Forward kinem.
Vis Lab - Computer Vision Lab
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St
k
Fixation Constraint
Small errors:
iMiSHi
/-/C
Introduces a linkage between the camera and target poses
P° ~ K(q, 7°)
This constraint can be used for simplify the system kinematics (Jacobians) and dynamics (decoupling various DOFs) and one can easily introduce motion models for the target.
Vis Lab - Computer Vision Lab
I mm
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Conclusions
"Opportunistic" use of visual information for control purposes.
Challenging image geometries
Ill-conditioned (perspective) observations
Many issues of stability and performance analysis remain to be addressed.
fr Vis Lab - Computer Vision Lab
REFERENCES (Alexandre Bernardino, Cesar Silva, Etienne Grossmann)
fr
Binocular Tracking,: Integrating perception and Control Alexandre Bernardino, Jose Santos-Victor, accepted for publication IEEE Trans. Robotics and Automation, 1999
Topological Maps for Visual Navigation, Josd Santos-Victor, Raquel Vassallo, Hans-Jorg Schneebeli, VisLab-TR 01/99 - 1st International Conference on Computer Vision Systems, Las Palmas, Canarias, January, 1999
Visual behaviors for Binocular Tracking, Alexandre Bernardino, Jose Santos-Victor Robotics and Autonomous Systems, Elsevier, 1998
Vision based Control of Cellular Robots Jose Santos-Victor, Robotics and Autonomous Systems, Elsevier 23(4), July 1998.
Egomotion Estimation Using Log-Polar Images, Cesar Silva, Jose Santos-Victor, International Conference of Computer Vision ICCV - Bombay, India, January 1998
Visual Behaviors for Docking, Jos« Santos-Victor, G. Sandini, Computer Vision and Image Understanding - CVIU, Vol 67(3), September 1997
Robust Egomotion Estimation from the Normal Flow using Search Subspaces, Cesar Silva, Jose Santos-Victor, IEEE Transactions on PAMI, Vol 19(9), September 1997
Performance Evaluation of Optical Flow Estimators: Assessment of a New Affine Flow Method, Etienne Grossmann, Jose Santos-Victor, - Robotics and Autonomous Systems, Elsevier, 21(1) July 1997
Embedded Visual Behaviors for Navigation, Jose Santos-Victor, Giulio Sandini, Robotics and Autonomous Systems, Elsevier, 1997,19(3-4), March 1997
VisLab - Computer Vision Lab