1. The document discusses using control theory to analyze sensorimotor systems. It presents the basic control theory framework of inputs, outputs, internal states, disturbances, and noise.
2. Feedback control is described, including how it can potentially stabilize unstable systems and attenuate disturbances. Common control schemes like forward and inverse models are also covered.
3. As an example, the document discusses how control theory has been used to analyze the control system that allows dragonflies to precisely control their flight during foraging.
1. Control-theoretic approach to the
analysis and synthesis of
sensorimotor loops
A few main principles and connections to neuroscience
Neurotheory and Engineering seminar - 05/28/2013
Matteo Mischiati
1
2. โข Control theory framework
- linear time-invariant (LTI) case
โข Properties of feedback
- internal model principle
โข Common control schemes
- forward and inverse models, Smith predictor
- state feedback, observers, optimal control
โข A model of human response in manual
tracking tasks (Kleinman et al. 1970, Gawthrop et al. 2011)
2Matteo Mischiati Control theory primer
3. Control theory framework
๐ข = ๐๐๐๐ข๐ก (๐๐๐ก๐๐)
๐ฆ = ๐๐ข๐ก๐๐ข๐ก ๐ ๐๐๐ ๐๐๐ฆ
๐ = ๐๐๐ก๐๐๐๐๐ ๐ ๐ก๐๐ก๐
๐ = ๐๐๐ ๐ก๐ข๐๐๐๐๐๐
๐ = ๐ ๐๐๐ ๐๐๐ฆ ๐๐๐๐ ๐
Assume you have a system (PLANT) for which you can control certain variables (inputs) and sense
others (outputs). There may be disturbances on your inputs and your sensed outputs may be noisy.
๐ = ๐ ๐, ๐ข
๐ฆ = ๐ ๐, ๐ข
๐ข ๐ฆ
PLANT
๐ข ๐
++
๐
++
๐
๐ฆ ๐
3Matteo Mischiati Control theory primer
4. Control theory framework
๐ข = ๐๐๐๐ข๐ก (๐๐๐ก๐๐)
๐ฆ = ๐๐ข๐ก๐๐ข๐ก ๐ ๐๐๐ ๐๐๐ฆ
๐ = ๐๐๐ก๐๐๐๐๐ ๐ ๐ก๐๐ก๐
๐ = ๐๐๐ ๐ก๐ข๐๐๐๐๐๐
๐ = ๐ ๐๐๐ ๐๐๐ฆ ๐๐๐๐ ๐
SYNTHESIS problem: design a controller that applies the right inputs to the plant, based on the
noisy outputs, to achieve a desired goal while satisfying one or more performance criteria
๐ = ๐ ๐, ๐ข
๐ฆ = ๐ ๐, ๐ข
๐ข ๐ฆ
PLANT
๐ข ๐
++
๐
++
๐
๐ฆ ๐๐ฆ ๐ = โ ๐, ๐ฆ, ๐ฆ๐
๐ข ๐ = ๐ ๐, ๐ฆ, ๐ฆ ๐
CONTROLLER
4Matteo Mischiati Control theory primer
5. Control theory framework
๐ข = ๐๐๐๐ข๐ก (๐๐๐ก๐๐)
๐ฆ = ๐๐ข๐ก๐๐ข๐ก ๐ ๐๐๐ ๐๐๐ฆ
๐ = ๐๐๐ก๐๐๐๐๐ ๐ ๐ก๐๐ก๐
๐ = ๐๐๐ ๐ก๐ข๐๐๐๐๐๐
๐ = ๐ ๐๐๐ ๐๐๐ฆ ๐๐๐๐ ๐
SYNTHESIS problem: design a controller that applies the right inputs to the plant, based on the
noisy outputs, to achieve a desired goal while satisfying one or more performance criteria
Possible Goals:
โข Output regulation (disturbance rejection, homeostasis) : keep ๐ฆ constant despite disturbance
โข Trajectory tracking : keep ๐ฆ ๐ก โ ๐ฆ ๐ก โ ๐ก
Performance criteria:
โข Static performance (at steady state): e.g. lim
๐กโโ
๐ฆ ๐ก โ ๐ฆ ๐ก
โข Dynamic performance: transient time, etcโฆ
โข Stability: not blowing up!
โข Robustness: amount of disturbance that can be tolerated
โข Limited control effort
๐ = ๐ ๐, ๐ข
๐ฆ = ๐ ๐, ๐ข
๐ข ๐ฆ
PLANT
๐ข ๐
++
๐
++
๐
๐ฆ ๐๐ฆ ๐ = โ ๐, ๐ฆ, ๐ฆ๐
๐ข ๐ = ๐ ๐, ๐ฆ, ๐ฆ ๐
CONTROLLER
5Matteo Mischiati Control theory primer
6. Control theory framework
๐ข = ๐๐๐๐ข๐ก (๐๐๐ก๐๐)
๐ฆ = ๐๐ข๐ก๐๐ข๐ก ๐ ๐๐๐ ๐๐๐ฆ
๐ = ๐๐๐ก๐๐๐๐๐ ๐ ๐ก๐๐ก๐
๐ = ๐๐๐ ๐ก๐ข๐๐๐๐๐๐
๐ = ๐ ๐๐๐ ๐๐๐ฆ ๐๐๐๐ ๐
ANALYSIS problem: infer the functional structure of the controller, given the observed
performance of the overall system in multiple tasks
Goals:
โข Output regulation (disturbance rejection, homeostasis) : keep ๐ฆ constant despite disturbance
โข Trajectory tracking : keep ๐ฆ ๐ก โ ๐ฆ ๐ก โ ๐ก
Performance criteria:
โข Static performance (at steady state): e.g. lim
๐กโโ
๐ฆ ๐ก โ ๐ฆ ๐ก
โข Dynamic performance: transient time, etcโฆ
โข Stability: not blowing up!
โข Robustness: amount of disturbance that can be tolerated
โข Limited control effort
๐ = ๐ ๐, ๐ข
๐ฆ = ๐ ๐, ๐ข
++
๐ข
๐
๐ฆ
PLANT
++
๐
๐ข ๐ ๐ฆ ๐๐ฆ ๐ = โ ๐, ๐ฆ, ๐ฆ๐
๐ข ๐ = ๐ ๐, ๐ฆ, ๐ฆ ๐
CONTROLLER
?
6Matteo Mischiati Control theory primer
7. Example of analysis problem:
uncovering the dragonfly control system
๐ ๐ซ๐ญ
๐๐๐๐
๐
โข We want to precisely characterize the foraging behavior of the dragonfly
(what it does) to gain insight on its neural circuitry (how it does it).
๐ ๐ซ๐ญ
๐๐๐๐
๐๐๐
?
๐, relative to
dragonfly, in
๐๐๐๐ ref. frame
dragonfly accel.
head rotation
dragonfly
head, body &
wing dynamics
dragonfly
visual
system
๐๐๐๐๐๐ ? ๐๐๐๐s ?
๐๐๐๐ ๐๐๐๐๐๐๐
๐๐๐๐ ๐๐๐๐๐๐๐
๐๐๐ ๐
๐๐๐ ๐
7Matteo Mischiati Control theory primer
8. Linear time-invariant systems
๐ข = ๐๐๐๐ข๐ก (๐๐๐ก๐๐)
๐ฆ = ๐๐ข๐ก๐๐ข๐ก ๐ ๐๐๐ ๐๐๐ฆ
๐ = ๐๐๐ก๐๐๐๐๐ ๐ ๐ก๐๐ก๐
๐ = ๐๐๐ ๐ก๐ข๐๐๐๐๐๐
๐ = ๐ ๐๐๐ ๐๐๐ฆ ๐๐๐๐ ๐
๐ ๐ =
๐ โ๐ง1 ๐ โ๐ง2 โฆ(๐ โ๐ง ๐)
๐ โ๐1 ๐ โ๐2 โฆ(๐ โ๐ ๐)
, ๐ง๐โ โ = ๐ง๐๐๐๐ , ๐๐ โ โ = ๐๐๐๐๐
โข Stability (of a system) โ ๐1, โฆ , ๐ ๐ have negative real part
โข Performance (of a system): depends on location of poles and zeros
โข Related to transfer function in frequency domain:
๐ข ๐ก = sin ๐๐ก โ ๐ฆ ๐ก = ๐ ๐๐ โ sin(๐๐ก + ฮฆ ๐ ๐๐ )
๐ = ๐ด๐ + ๐ต๐ข
๐ฆ = ๐ถ๐ + ๐ท๐ข
++
๐ข
๐
๐ฆ
PLANT
++
๐
๐ข ๐ ๐ฆ ๐๐ฆ ๐ = ๐ป๐ + ๐บ ๐ฆ
๐ข ๐ = ๐ผ๐ + ๐ฟ ๐ฆ
CONTROLLER
๐(๐ )
++
๐(๐ )
๐(๐ )
๐(๐ )
PLANT
++
๐ท(๐ )
๐ ๐ถ(๐ ) ๐๐(๐ )๐(๐ ) ๐ถ(๐ )
CONTROLLER
Laplace transform
Y ๐ = ๐ ๐ โ ๐(๐ )
8Matteo Mischiati Control theory primer
9. Linear time-invariant systems
Laplace transform for signals:
โข Final value theorem : lim
๐กโโ
๐ฆ ๐ก = lim
๐ โ0
๐ โ ๐ ๐ (if limit exists)
โข If u ๐ก โท ๐ ๐ , then u ๐ก โท
๐ ๐
๐
Typical reference/disturbance signals:
- Step ๐ ๐ = ๐
1
๐
- Ramp ๐ ๐ = ๐
1
๐ 2
- Sinusoid ๐ ๐ =
๐
๐ 2+๐2
๐(๐ )
++
๐(๐ )
๐(๐ )
๐(๐ )
PLANT
++
๐ท(๐ )
๐ ๐ถ(๐ ) ๐๐(๐ )๐(๐ ) ๐ถ(๐ )
CONTROLLER
Y ๐ = ๐ ๐ โ ๐(๐ )
๐ก
๐ฆ(๐ก)
๐
๐ก
๐ฆ(๐ก)
๐๐ก
๐ก
๐ฆ(๐ก)
sin(๐๐ก)
9Matteo Mischiati Control theory primer
10. โข Control theory framework
- linear time-invariant (LTI) case
โข Properties of feedback
- internal model principle
โข Common control schemes
- forward and inverse models, Smith predictor
- state feedback, observers, optimal control
โข A model of human response in manual
tracking tasks (Kleinman et al. 1970, Gawthrop et al. 2011)
10Matteo Mischiati Control theory primer
11. Feedforward (inverse model)
Stability
Depends on poles (and zeros!) of ๐(๐ )
Performance (static and dynamic)
Arbitrarily good if ๐ ๐ โ ๐ ๐ and its inverse exists and is stable: Y ๐ โ
๐ ๐ โ ๐โ1 ๐ โ ๐(๐ ) โ ๐(๐ )
Robustness to disturbance (disturbance rejection)
None : Y ๐ = ๐ ๐ โ ( ๐โ1 ๐ โ ๐ ๐ + ๐ท(๐ )) โ ๐(๐ ) + ๐(๐ ) โ ๐ท(๐ )
๐(๐ )๐(๐ ) ๐(๐ )
PLANT
++
๐ท(๐ )
๐ ๐ถ(๐ )๐(๐ ) ๐โ1
(๐ )
CONTROLLER
Y ๐ = ๐ ๐ โ ๐(๐ )Uc ๐ = ๐โ1
๐ โ ๐(๐ )
11Matteo Mischiati Control theory primer
17. Internal model principle
To perfectly track: We need:
Step
Ramp
Sinusoid
Internal model principle: To achieve asymptotical tracking of a reference signal
(rejection of a disturbance signal) via feedback, the controller (or the plant) must
contain an โinternal modelโ of the signal.
It is a necessary condition, not a sufficient condition (need also stability)
๐
๐
๐
๐ 2
๐
๐ 2 + ๐2
๐(๐ )๐ถ ๐ =
1
๐
โ (๐ ๐ ๐ถ ๐ )โฒ
๐(๐ )๐ถ ๐ =
1
๐ 2
โ (๐ ๐ ๐ถ ๐ )โฒ
๐(๐ )๐ถ ๐ =
1
๐ 2 + ๐2
โ (๐ ๐ ๐ถ ๐ )โฒ
17Matteo Mischiati Control theory primer
18. Feedback vs. Feedforward
Feedback
โข is needed if plant is unstable or for disturbance rejection
โข does not require full knowledge of the plant
โข incorporating the knowledge of possible reference and disturbance
signals is very useful (internal model principle)
Feedforward
โข if plant is known, and no disturbance, its performance canโt be beat
โข no sensory delays
๐ ๐ =
๐ ๐ ๐ถ ๐
1 + ๐ ๐ ๐ถ ๐
๐ ๐ +
๐ ๐
1 + ๐ ๐ ๐ถ ๐
๐ท(๐ ) ๐ ๐ = ๐ ๐ โ ( ๐โ1
๐ โ ๐ ๐ + ๐ท(๐ ))
โ ๐(๐ ) + ๐(๐ ) โ ๐ท(๐ )
18Matteo Mischiati Control theory primer
19. โข Control theory framework
- linear time-invariant (LTI) case
โข Properties of feedback
- internal model principle
โข Common control schemes
- forward and inverse models, Smith predictor
- state feedback, observers, optimal control
โข A model of human response in manual
tracking tasks (Kleinman et al. 1970, Gawthrop et al. 2011)
19Matteo Mischiati Control theory primer
20. Feedback + Feedforward
The feedback controller kicks in only if inverse model is incorrect.
๐(๐ )๐(๐ ) ๐(๐ )
PLANT
++
๐ ๐น๐ต(๐ )
๐ธ(๐ ) ๐ถ(๐ )๐(๐ )
+
-
๐โ1
(๐ )
INVERSE MODEL
FEEDBACK
๐ ๐น๐น(๐ )
20Matteo Mischiati Control theory primer
21. Feedback + Feedforward
The feedback controller kicks in only if inverse model is incorrect.
The corrective command from the feedback path can be also used as a
learning/adaptation signal by the inverse model.
๐(๐ )๐(๐ ) ๐(๐ )
PLANT
++
๐ ๐น๐ต(๐ )
๐ธ(๐ ) ๐ถ(๐ )๐(๐ )
+
-
๐โ1
(๐ )
INVERSE MODEL
FEEDBACK
๐ ๐น๐น(๐ )
21Matteo Mischiati Control theory primer
22. Feedback + Feedforward
The feedback controller kicks in only if inverse model is incorrect.
The corrective command from the feedback path can be also used as a
learning/adaptation signal by the inverse model.
Significant sensory delays are still a problem.
๐(๐ )๐(๐ ) ๐(๐ )
PLANT
++
๐ ๐น๐ต(๐ )
๐ธ(๐ ) ๐ถ(๐ )๐(๐ )
+
-
๐โ1
(๐ )
INVERSE MODEL
FEEDBACK
๐ ๐น๐น(๐ )
๐โ๐ ๐
22Matteo Mischiati Control theory primer
23. Forward model
The control signal is sent through a model of the plant (โforward modelโ) to
predict the sensory output.
๐(๐ ) ๐(๐ )
PLANT
๐ ๐ถ(๐ )๐ธ(๐ ) ๐ถ(๐ )๐(๐ )
+
-
๐(๐ )
FORWARD MODEL
CONTROLLER
predicted sensory output
23Matteo Mischiati Control theory primer
24. Forward model
The control signal is sent through a model of the plant (โforward modelโ) to
predict the sensory output.
The (delayed) sensory output can be used as a learning/adaptation signal for
the forward model.
Direct use of the delayed sensory output in the control is problematic
because of time mismatch.
๐(๐ ) ๐(๐ )
PLANT
๐ ๐ถ(๐ )๐ธ(๐ ) ๐ถ(๐ )๐(๐ )
+
-
๐(๐ )
FORWARD MODEL
๐โ๐ ๐
CONTROLLER
predicted sensory output
24Matteo Mischiati Control theory primer
25. Smith predictor
Assuming ๐ ๐ โ ๐ ๐ and ๐ โ ๐:
๐ ๐ = ๐ ๐ ๐๐ ๐ =
๐ ๐ ๐ถ ๐
1 + ๐ ๐ ๐ถ ๐
๐ ๐
Delay has been moved outside the control loop.
PLANT
๐(๐ ) ๐(๐ )๐ ๐ถ(๐ )๐ธ(๐ ) ๐ถ(๐ )๐(๐ )
+
- -
๐(๐ )
๐โ๐ ๐
๐โ๐ ๐
+ -
delay model
plant model
predicted sensory output
error in sensory output prediction
CONTROLLER
25Matteo Mischiati Control theory primer
26. Models of the cerebellum
1. Cerebellum as an inverse
model in a feedback+feedforward
motor control scheme
Wolpert, Miall & Kawato, 1998 โInternal models in the cerebellumโ
Not in the sense of my presentation !
26Matteo Mischiati Control theory primer
27. Models of the cerebellum
2. Cerebellum as a
forward model in a
Smith predictor
control scheme
Wolpert, Miall & Kawato, 1998 โInternal models in the cerebellumโ
27Matteo Mischiati Control theory primer
29. State feedback
If the plant is reachable, it is possible to achieve any arbitrary choice of
closed-loop poles with an appropriate linear and memoryless controller:
u = โ๐พ๐ + ๐พ๐ ๐ฆ
๐ = ๐ ๐, ๐ข
๐ฆ = ๐ ๐, ๐ข
๐ฆ
PLANT
๐ข๐ฆ ๐ = โ ๐, ๐ฆ, ๐
๐ข = ๐ ๐, ๐ฆ, ๐
CONTROLLER
๐
๐ฆ
PLANT
๐ข๐ฆ
CONTROLLER
๐
๐ = ๐ด๐ + ๐ต๐ข
๐ฆ = ๐ถ๐ + ๐ท๐ข
Linear time-invariant case:
๐พ
๐พ๐ +
-
29Matteo Mischiati Control theory primer
30. Observers
Observer: dynamical system designed to estimate the full state (when not
fully available)
If the plant is observable, it is possible to achieve lim
๐กโโ
๐ ๐ก = ๐ (with right ๐ฟ)
๐ฆ
PLANT
๐ข
๐
๐ = ๐ด๐ + ๐ต๐ข
๐ฆ = ๐ถ๐
๐ = ๐ด ๐ + ๐ต๐ข + ๐ฟ(y โ C ๐)
OBSERVER
30Matteo Mischiati Control theory primer
31. Observers
Observer: dynamical system designed to estimate the full state (when not
fully available)
If the plant is observable, it is possible to achieve lim
๐กโโ
๐ ๐ก = ๐ (with right ๐ฟ)
Separation principle: if the plant is reachable & observable, can replace ๐
with ๐ and design ๐พ independently of ๐ฟ (use observed state just as real one)
๐ฆ
PLANT
๐ข
๐
๐ = ๐ด๐ + ๐ต๐ข
๐ฆ = ๐ถ๐
๐ = ๐ด ๐ + ๐ต๐ข + ๐ฟ(y โ C ๐)
๐ฆ
๐พ
๐พ๐ +-
OBSERVER
31Matteo Mischiati Control theory primer
32. Optimal control
Linear Quadratic Gaussian (LQG) optimal output regulation: linear plant,
additive Gaussian white noise on state (with covariance ฮฃ ๐) and output (๐ ๐);
minimize quadratic cost :
๐ผ lim
๐โโ
1
๐ 0
๐
๐ ๐ ๐ก ๐๐ ๐ก + ๐ข ๐ก 2 ๐๐ก
Solution is linear observer (Kalman filter) with linear memoryless controller:
๐ฟ = ๐๐ถ ๐ ๐ ๐
โ1, ๐ด๐ + ๐๐ด ๐ + ฮฃ ๐ โ ๐๐ถ ๐ ๐ ๐
โ1 ๐ถ๐ ๐ = 0
๐พ = ๐ต ๐ ๐, ๐ด ๐ ๐ + ๐๐ด + ๐ โ ๐๐ต๐ต ๐ ๐ = 0
๐ฆ
PLANT
๐ข ๐ = ๐ด๐ + ๐ต๐ข + ๐
๐ฆ = ๐ถ๐
๐
๐ = ๐ด ๐ + ๐ต๐ข + ๐ฟ(yn โ C ๐)
๐ฆ = 0
๐พ
+-
OBSERVER (KALMAN FILTER)
๐
++
๐
๐ฆ ๐
32Matteo Mischiati Control theory primer
33. Internal model principle
Internal model principle (state space): to achieve asymptotical tracking of a
reference signal (rejection of a disturbance signal) produced by an exosystem,
the controller must contain an โinternal modelโ of the exosystem.
(Francis & Wonham, Automatica, 1970)
It is a necessary condition, not a sufficient condition (need also stability).
General principle with extensions to nonlinear systems.
๐ฆ
PLANT
๐๐ฆ
๐ = ๐ด๐ + ๐ต๐ข
๐ฆ = ๐ถ๐
๐ผ = ๐๐ผ + ๐บ๐๐ = ๐ ๐
๐ฆ = ๐ ๐
+
-
๐
STABILIZING
CONTROLLER
๐ผINT.MODELEXOSYSTEM
33Matteo Mischiati Control theory primer
34. โข Control theory framework
- linear time-invariant (LTI) case
โข Properties of feedback
- internal model principle
โข Common control schemes
- forward and inverse models, Smith predictor
- state feedback, observers, optimal control
โข A model of human response in manual
tracking tasks (Kleinman et al. 1970, Gawthrop et al. 2011)
34Matteo Mischiati Control theory primer
35. Human response model
D. L. Kleinman, S. Baron and W. H. Levison, โAn Optimal Control Model of
Human Response. Part I: Theory and Validationโ, Automatica, 1970
(revisited, more recently, by Gawthrop et al. 2011)
Task: by controlling a joystick (position, velocity or acceleration control),
subject is asked to keep a cursor on the screen as close as possible to a
target location, while unknown disturbances are applied by the computer.
Plant:
๐๐๐๐๐ก๐๐ ๐(๐ )
++
๐ท(๐ ) (computer)
๐(๐ )
๐๐๐ฆ๐ ๐ก๐๐๐ ๐๐ฝ๐ ๐ โ ๐,
๐
๐
,
๐
๐ 2
๐ ๐๐๐ ๐๐๐ฆ
๐๐๐๐๐๐๐๐
โHuman controllerโ:
๐๐๐ข๐๐๐๐๐ก๐๐
dynamics
๐(๐ )
++
๐๐๐ก๐๐ ๐๐๐๐ ๐
๐๐๐ข๐๐๐
computation
๐ ๐ ๐ ๐ถ ๐
๐โ๐ ๐
๐ ๐ ๐ โ
1
๐ ๐ ๐ +1
๐ ๐ โ 100๐๐
๐ โ 150 โ 250๐๐
35Matteo Mischiati Control theory primer
36. Human response model
Task: Output regulation/disturbance rejection with linear time-invariant
plant and significant delay on any potential feedback line
๐โ๐ ๐
๐(๐ )
++
๐(๐ )
๐(๐ )
๐(๐ )
++
๐ท(๐ )
๐ ๐ถ(๐ ) ๐๐(๐ )๐ถ(๐ )
CONTROLLER
๐๐ฝ๐ ๐ โ ๐ ๐(๐ )
ANALYSIS problem: infer a model of the neural controller ๐ถ(๐ ) from the
observed performance of the subjects tested.
36Matteo Mischiati Control theory primer
37. Human response model
Task: Output regulation/disturbance rejection with linear time-invariant
plant and significant delay on any potential feedback line
๐โ๐ ๐
๐(๐ )
++
๐(๐ )
๐(๐ )
๐(๐ )
++
๐ท(๐ )
๐ ๐ถ(๐ ) ๐๐(๐ )๐ถ(๐ )
CONTROLLER
๐๐ฝ๐ ๐ โ ๐ ๐(๐ )
ANALYSIS problem: infer a model of the neural controller ๐ถ(๐ ) from the
observed performance of the subjects tested.
So what are the performances?
โข Very good and robust to disturbances up to 2Hz (sum of sinusoids), for all
three types of joystick dynamics
โข Apparently delay-free Must be some kind of
FEEDBACK + FORWARD model !
37Matteo Mischiati Control theory primer
38. Human response model
Hypothesis: optimal control to minimize average error & control effort
๐ผ lim
๐โโ
1
๐ 0
๐
๐ ๐ก 2
+ ๐ผ ๐ข ๐ก 2
+ ๐ฝ ๐ข ๐ก 2
๐๐ก
Theoretical solution * (with assumptions similar to LQG problem):
- Optimal observer (Kalman filter) to estimate delayed state (as in LQG)
- Optimal least mean-squared predictor to predict current state
- Optimal linear memoryless controller (as in LQG)
๐ฆ
PLANT
๐ข
๐(๐ก โ ๐)
๐ = ๐ด๐ + ๐ต๐ข + ๐
๐ฆ = ๐ถ๐
KALMAN FILTER
๐ฆ = 0
๐พ
+-
๐
++
๐
๐ฆ ๐
* D. Kleinman, โOptimal control of linear systems with time-delay and observation noiseโ, IEEE Trans. Autom. Control, 1969
๐โ๐ ๐PREDICTOR
๐(๐ก) ๐ฆ๐(๐ก โ ๐)
38Matteo Mischiati Control theory primer
39. Controller freq. response with plant
๐
๐
Controller freq. response with plant
๐
๐ 2
Matteo Mischiati Control theory primer 39
40. Human response model
Gawthrop et al. * (2011):
- Introduced, in both estimator and predictor, a copy of the exosystem
generating sinusoidal disturbances (internal model principle!)
- Show that intermittent control is also compatible with results
* P. Gawthrop et al., โIntermittent control: a computational theory of human controlโ, Biol. Cybern., 2011
Actual response to sinusoid Response without int.model
40Matteo Mischiati Control theory primer
41. โข Crash course in control theory (for LTI systems)
- many concepts can be extended to more general settings
โข An example of control-theoretic approach to
modeling sensorimotor loops
- need to iterate between modeling/experiments to discern
among alternatives and improve understanding of the system
Conclusions
THANK YOU FOR YOUR ATTENTION !
41Matteo Mischiati Control theory primer
Editor's Notes
Fast feedback loop involving the forward model can be seen as playing the role of an inverse model.
Fast feedback loop involving the forward model can be seen as playing the role of an inverse model.