Control-theoretic approach to sensorimotor loops

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

• Control theory framework
- linear time-invariant (LTI) case
• Properties of feedback
- internal model principle
• Common control schemes
- forward and inverse models, Smith predictor
- state feedback, observers, optimal control
• A model of human response in manual
tracking tasks (Kleinman et al. 1970, Gawthrop et al. 2011)
2Matteo Mischiati Control theory primer

Control theory framework
𝑢 = 𝑖𝑛𝑝𝑢𝑡 (𝑚𝑜𝑡𝑜𝑟)
𝑦 = 𝑜𝑢𝑡𝑝𝑢𝑡 𝑠𝑒𝑛𝑠𝑜𝑟𝑦
𝒙 = 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙 𝑠𝑡𝑎𝑡𝑒
𝑑 = 𝑑𝑖𝑠𝑡𝑢𝑟𝑏𝑎𝑛𝑐𝑒
𝑛 = 𝑠𝑒𝑛𝑠𝑜𝑟𝑦 𝑛𝑜𝑖𝑠𝑒
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.
𝒙 = 𝑓 𝒙, 𝑢
𝑦 = 𝑔 𝒙, 𝑢
𝑢 𝑦
PLANT
𝑢 𝑐
++
𝑛
++
𝑑
𝑦 𝑛

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
𝑢 𝑦
PLANT
𝑢 𝑐
++
𝑛
++
𝑑
𝑦 𝑛𝑦 𝒛 = ℎ 𝒛, 𝑦, 𝑦𝑛
𝑢 𝑐 = 𝑖 𝒛, 𝑦, 𝑦 𝑛
CONTROLLER

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:
• Output regulation (disturbance rejection, homeostasis) : keep 𝑦 constant despite disturbance
• Trajectory tracking : keep 𝑦 𝑡 ≈ 𝑦 𝑡 ∀ 𝑡
Performance criteria:
• Static performance (at steady state): e.g. lim
𝑡→∞
𝑦 𝑡 − 𝑦 𝑡
• Dynamic performance: transient time, etc…
• Stability: not blowing up!
• Robustness: amount of disturbance that can be tolerated
• Limited control effort
𝑢 𝑦
PLANT
𝑢 𝑐
++
𝑛
++
𝑑
𝑦 𝑛𝑦 𝒛 = ℎ 𝒛, 𝑦, 𝑦𝑛
CONTROLLER

ANALYSIS problem: infer the functional structure of the controller, given the observed
performance of the overall system in multiple tasks
Goals:
• Output regulation (disturbance rejection, homeostasis) : keep 𝑦 constant despite disturbance
• Trajectory tracking : keep 𝑦 𝑡 ≈ 𝑦 𝑡 ∀ 𝑡
Performance criteria:
• Static performance (at steady state): e.g. lim
𝑡→∞
𝑦 𝑡 − 𝑦 𝑡
• Dynamic performance: transient time, etc…
• Stability: not blowing up!
• Robustness: amount of disturbance that can be tolerated
• Limited control effort
++
𝑢
𝑛
𝑦
PLANT
++
𝑑
𝑢 𝑐 𝑦 𝑛𝑦 𝒛 = ℎ 𝒛, 𝑦, 𝑦𝑛
CONTROLLER
?

Example of analysis problem:
uncovering the dragonfly control system
𝒗 𝑫𝑭
𝒉𝒆𝒂𝒅
𝒓
• 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).
𝒗 𝑫𝑭
𝒉𝒆𝒂𝒅
𝒇𝒍𝒚
?
𝒓, relative to
dragonfly, in
𝒉𝒆𝒂𝒅 ref. frame
dragonfly accel.
head rotation
dragonfly
head, body &
wing dynamics
dragonfly
visual
system
𝐑𝐄𝐓𝐈𝐍𝐀 ? 𝐓𝐒𝐃𝐍s ?
𝐖𝐈𝐍𝐆 𝐌𝐔𝐒𝐂𝐋𝐄𝐒
𝐍𝐄𝐂𝐊 𝐌𝐔𝐒𝐂𝐋𝐄𝐒
𝒃𝒐𝒅𝒚
𝒃𝒐𝒅𝒚

Linear time-invariant systems
𝑃 𝑠 =
𝑠−𝑧1 𝑠−𝑧2 …(𝑠−𝑧 𝑚)
𝑠−𝑝1 𝑠−𝑝2 …(𝑠−𝑝 𝑛)
, 𝑧𝑖∈ ℂ = 𝑧𝑒𝑟𝑜𝑠, 𝑝𝑖 ∈ ℂ = 𝑝𝑜𝑙𝑒𝑠
• Stability (of a system) ⇔ 𝑝1, … , 𝑝 𝑛 have negative real part
• Performance (of a system): depends on location of poles and zeros
• Related to transfer function in frequency domain:
𝑢 𝑡 = sin 𝜔𝑡 ⇒ 𝑦 𝑡 = 𝑃 𝑗𝜔 ⋅ sin(𝜔𝑡 + Φ 𝑃 𝑗𝜔 )
𝒙 = 𝐴𝒙 + 𝐵𝑢
𝑦 = 𝐶𝒙 + 𝐷𝑢
++
𝑢
𝑛
𝑦
PLANT
++
𝑑
𝑢 𝑐 𝑦 𝑛𝑦 𝒛 = 𝐻𝒛 + 𝐺 𝑦
𝑢 𝑐 = 𝐼𝒛 + 𝐿 𝑦
CONTROLLER
𝑃(𝑠)
++
𝑈(𝑠)
𝑁(𝑠)
𝑌(𝑠)
PLANT
++
𝐷(𝑠)
𝑈 𝐶(𝑠) 𝑌𝑛(𝑠)𝑌(𝑠) 𝐶(𝑠)
CONTROLLER
Laplace transform
Y 𝑠 = 𝑃 𝑠 ∙ 𝑈(𝑠)

Linear time-invariant systems
Laplace transform for signals:
• Final value theorem : lim
𝑡→∞
𝑦 𝑡 = lim
𝑠→0
𝑠 ∙ 𝑌 𝑠 (if limit exists)
• If u 𝑡 ⟷ 𝑈 𝑠 , then u 𝑡 ⟷
𝑈 𝑠
𝑠
Typical reference/disturbance signals:
- Step 𝑌 𝑠 = 𝑎
1
𝑠
- Ramp 𝑌 𝑠 = 𝑎
1
𝑠2
- Sinusoid 𝑌 𝑠 =
𝜔
𝑠2+𝜔2
𝑃(𝑠)
++
𝑈(𝑠)
𝑁(𝑠)
𝑌(𝑠)
PLANT
++
𝐷(𝑠)
𝑈 𝐶(𝑠) 𝑌𝑛(𝑠)𝑌(𝑠) 𝐶(𝑠)
CONTROLLER
Y 𝑠 = 𝑃 𝑠 ∙ 𝑈(𝑠)
𝑡
𝑦(𝑡)
𝑎
𝑡
𝑦(𝑡)
𝑎𝑡
𝑡
𝑦(𝑡)
sin(𝜔𝑡)

Feedforward (inverse model)
Stability
Depends on poles (and zeros!) of 𝑃(𝑠)
Performance (static and dynamic)
Arbitrarily good if 𝑃 𝑠 ≈ 𝑃 𝑠 and its inverse exists and is stable: Y 𝑠 ≈
𝑃 𝑠 ∙ 𝑃−1 𝑠 ⋅ 𝑌(𝑠) ≈ 𝑌(𝑠)
Robustness to disturbance (disturbance rejection)
None : Y 𝑠 = 𝑃 𝑠 ∙ ( 𝑃−1 𝑠 ⋅ 𝑌 𝑠 + 𝐷(𝑠)) ≈ 𝑌(𝑠) + 𝑃(𝑠) ⋅ 𝐷(𝑠)
𝑃(𝑠)𝑈(𝑠) 𝑌(𝑠)
PLANT
++
𝐷(𝑠)
𝑈 𝐶(𝑠)𝑌(𝑠) 𝑃−1
(𝑠)
CONTROLLER
Y 𝑠 = 𝑃 𝑠 ∙ 𝑈(𝑠)Uc 𝑠 = 𝑃−1
𝑠 ∙ 𝑌(𝑠)

Properties of Feedback
𝑌 𝑠 = 𝑃 𝑠 𝐷 𝑠 + 𝐶 𝑠 𝐸 𝑠 = 𝑃 𝑠 𝐷 𝑠 + 𝐶 𝑠 ( 𝑌 𝑠 − 𝑌 𝑠 ) ⇒
𝑌 𝑠 =
𝑃 𝑠 𝐶 𝑠
1 + 𝑃 𝑠 𝐶 𝑆
𝑌 𝑠 +
𝑃 𝑠
1 + 𝑃 𝑠 𝐶 𝑆
𝐷(𝑠)
Stability
Depends on 1 + 𝑃 𝑠 𝐶 𝑆 . Can potentially stabilize unstable plants.
Disturbance rejection
Depends on 1 + 𝑃 𝑠 𝐶 𝑆 . Can potentially attenuate/cancel effect of 𝐷 𝑠 .
PLANT
++
𝐷(𝑠)
𝑈 𝐶(𝑠)𝐸(𝑠) 𝐶(𝑠)
CONTROLLER
𝑌(𝑠)
Y 𝑠 = 𝑃 𝑠 ∙ 𝑈(𝑠)Uc 𝑠 = 𝐶 𝑠 ∙ 𝐸(𝑠)
+
-

Static performance
lim
𝑡→∞
𝑒 𝑡 = lim
𝑠→0
𝑠 ∙ 𝐸 𝑠 , 𝐸 𝑠 =
1
1 + 𝑃 𝑠 𝐶 𝑆
𝑌 𝑠 −
𝑃 𝑠
1 + 𝑃 𝑠 𝐶 𝑆
𝐷(𝑠)
Let’s see how different controllers perform:
PLANT
++
𝐷(𝑠)
CONTROLLER
𝑌(𝑠)
+
-
𝑒. 𝑔.
1
1 + 𝜏𝑠
𝑘

Static performance
lim
𝑡→∞
𝑒 𝑡 = lim
𝑠→0
𝑠 ∙ 𝐸 𝑠 , 𝐸 𝑠 =
1
1 + 𝑃 𝑠 𝐶 𝑆
𝑌 𝑠 −
𝑃 𝑠
1 + 𝑃 𝑠 𝐶 𝑆
𝐷(𝑠)
Proportional controller: 𝐶 𝑠 = 𝑘
lim
𝑡→∞
𝑒 𝑡 = lim
𝑠→0
𝑠 ∙
1
1+𝑘 𝑃 𝑠
𝑎
𝑠
=
𝑎
1+𝑘
≠ 0 errors in tracking a step
(but small if 𝑘 is large)
lim
𝑡→∞
𝑒 𝑡 = lim
𝑠→0
𝑠 ∙
1
1+𝑘 𝑃 𝑠
𝑎
𝑠2 → ∞ cannot track a ramp at all
PLANT
++
𝐷(𝑠)
CONTROLLER
𝑌(𝑠)
+
-
𝑎
𝑠
𝑎
𝑠2
𝑒. 𝑔.
1
1 + 𝜏𝑠
𝑘

Static performance
lim
𝑡→∞
𝑒 𝑡 = lim
𝑠→0
𝑠 ∙ 𝐸 𝑠 , 𝐸 𝑠 =
1
1 + 𝑃 𝑠 𝐶 𝑆
𝑌 𝑠 −
𝑃 𝑠
1 + 𝑃 𝑠 𝐶 𝑆
𝐷(𝑠)
Proportional+Integral (PI) controller: 𝐶 𝑠 = 𝑘 𝑝 +
𝑘 𝑖
𝑠
=
𝑘 𝑝 𝑠+𝑘 𝑖
𝑠
lim
𝑡→∞
𝑒 𝑡 = lim
𝑠→0
𝑠 ∙
1
1+
𝑠
𝑃 𝑠
𝑎
𝑠
= 0 perfect in tracking a step
lim
𝑡→∞
𝑒 𝑡 = lim
𝑠→0
𝑠 ∙
1
1+
𝑠
𝑃 𝑠
𝑎
𝑠2 =
𝑎
𝑘 𝑖
≠ 0 errors in tracking a ramp
PLANT
++
𝐷(𝑠)
CONTROLLER
𝑌(𝑠)
+
-
𝑎
𝑠
𝑎
𝑠2
1
1 + 𝜏𝑠
𝑘 𝑝 +
𝑘𝑖
𝑠
(but small if 𝑘𝑖 is large)

Static performance
lim
𝑡→∞
𝑒 𝑡 = lim
𝑠→0
𝑠 ∙ 𝐸 𝑠 , 𝐸 𝑠 =
1
1 + 𝑃 𝑠 𝐶 𝑆
𝑌 𝑠 −
𝑃 𝑠
1 + 𝑃 𝑠 𝐶 𝑆
𝐷(𝑠)
To perfectly track: We need:
Step
Ramp
How about sinusoid? We need:
𝑎
𝑠
𝑎
𝑠2
𝜔
𝑠2 + 𝜔2
𝑃(𝑠)𝐶 𝑠 =
1
𝑠
⋅ (𝑃 𝑠 𝐶 𝑠 )′
1
𝑠2 ⋅ (𝑃 𝑠 𝐶 𝑠 )′
𝑦 𝑡 = sin 𝜔𝑡
⇓
𝑒 𝑡 = 𝐺 𝑗𝜔 ⋅ sin(𝜔𝑡 + Φ 𝐺 𝑗𝜔 )
𝐺 𝑠
1
𝑠2 + 𝜔2
⋅ (𝑃 𝑠 𝐶 𝑠 )′
⇓
𝐺 𝑗𝜔 = 1 + 𝑃 𝑗𝜔 𝐶 𝑗𝜔 −1
→ 0

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 “internal model” of the signal.
It is a necessary condition, not a sufficient condition (need also stability)
𝑎
𝑠
𝑎
𝑠2
𝜔
𝑠2 + 𝜔2
1
𝑠
⋅ (𝑃 𝑠 𝐶 𝑠 )′
1
𝑠2
⋅ (𝑃 𝑠 𝐶 𝑠 )′
1
𝑠2 + 𝜔2
⋅ (𝑃 𝑠 𝐶 𝑠 )′

Feedback vs. Feedforward
Feedback
• is needed if plant is unstable or for disturbance rejection
• does not require full knowledge of the plant
• incorporating the knowledge of possible reference and disturbance
signals is very useful (internal model principle)
Feedforward
• if plant is known, and no disturbance, its performance can’t be beat
• no sensory delays
𝑌 𝑠 =
𝑃 𝑠 𝐶 𝑠
1 + 𝑃 𝑠 𝐶 𝑆
𝑌 𝑠 +
𝑃 𝑠
1 + 𝑃 𝑠 𝐶 𝑆
𝐷(𝑠) 𝑌 𝑠 = 𝑃 𝑠 ∙ ( 𝑃−1
𝑠 ⋅ 𝑌 𝑠 + 𝐷(𝑠))
≈ 𝑌(𝑠) + 𝑃(𝑠) ⋅ 𝐷(𝑠)

Feedback + Feedforward
The feedback controller kicks in only if inverse model is incorrect.
PLANT
++
𝑈 𝐹𝐵(𝑠)
𝐸(𝑠) 𝐶(𝑠)𝑌(𝑠)
+
-
𝑃−1
(𝑠)
INVERSE MODEL
FEEDBACK
𝑈 𝐹𝐹(𝑠)

The corrective command from the feedback path can be also used as a
learning/adaptation signal by the inverse model.
PLANT
++
𝑈 𝐹𝐵(𝑠)
+
-
𝑃−1
(𝑠)
INVERSE MODEL
FEEDBACK
𝑈 𝐹𝐹(𝑠)

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.
PLANT
++
𝑈 𝐹𝐵(𝑠)
+
-
𝑃−1
(𝑠)
INVERSE MODEL
FEEDBACK
𝑈 𝐹𝐹(𝑠)
𝑒−𝑠𝜏

Forward model
The control signal is sent through a model of the plant (“forward model”) to
predict the sensory output.
𝑃(𝑠) 𝑌(𝑠)
PLANT
𝑈 𝐶(𝑠)𝐸(𝑠) 𝐶(𝑠)𝑌(𝑠)
+
-
𝑃(𝑠)
FORWARD MODEL
CONTROLLER
predicted sensory output

Forward model
The control signal is sent through a model of the plant (“forward model”) 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.
𝑃(𝑠) 𝑌(𝑠)
PLANT
𝑈 𝐶(𝑠)𝐸(𝑠) 𝐶(𝑠)𝑌(𝑠)
+
-
𝑃(𝑠)
FORWARD MODEL
𝑒−𝑠𝜏
CONTROLLER

Smith predictor
Assuming 𝑃 𝑠 ≈ 𝑃 𝑠 and 𝜏 ≈ 𝜏:
𝑌 𝑠 = 𝑃 𝑠 𝑈𝑐 𝑠 =
𝑃 𝑠 𝐶 𝑠
1 + 𝑃 𝑠 𝐶 𝑆
𝑌 𝑠
Delay has been moved outside the control loop.
PLANT
𝑃(𝑠) 𝑌(𝑠)𝑈 𝐶(𝑠)𝐸(𝑠) 𝐶(𝑠)𝑌(𝑠)
+
- -
𝑃(𝑠)
𝑒−𝑠𝜏
𝑒−𝑠 𝜏
+ -
delay model
plant model
error in sensory output prediction
CONTROLLER

Models of the cerebellum
1. Cerebellum as an inverse
model in a feedback+feedforward
motor control scheme
Wolpert, Miall & Kawato, 1998 “Internal models in the cerebellum”
Not in the sense of my presentation !

Models of the cerebellum
2. Cerebellum as a
forward model in a
Smith predictor
control scheme
Wolpert, Miall & Kawato, 1998 “Internal models in the cerebellum”

State feedback
𝑦
PLANT
𝑢𝑦 𝒛 = ℎ 𝒛, 𝑦, 𝒙
𝑢 = 𝑖 𝒛, 𝑦, 𝒙
CONTROLLER
𝒙
𝑦
PLANT
𝑢𝑦
CONTROLLER
𝒙
𝒛 = 𝐻𝒛 + 𝐺 𝑦 + 𝑀𝒙
𝑢 = 𝐼𝒛 + 𝐿 𝑦 + 𝑁𝒙
Linear time-invariant case:

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 = −𝐾𝒙 + 𝐾𝑟 𝑦
𝑦
PLANT
𝑢𝑦 𝒛 = ℎ 𝒛, 𝑦, 𝒙
𝑢 = 𝑖 𝒛, 𝑦, 𝒙
CONTROLLER
𝒙
𝑦
PLANT
𝑢𝑦
CONTROLLER
𝒙
Linear time-invariant case:
𝐾
𝐾𝑟 +
-

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
𝑡→∞
𝒙 𝑡 = 𝒙 (with right 𝐿)
𝑦
PLANT
𝑢
𝒙
𝑦 = 𝐶𝒙
𝒙 = 𝐴 𝒙 + 𝐵𝑢 + 𝐿(y − C 𝒙)
OBSERVER

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
𝑡→∞
𝒙 𝑡 = 𝒙 (with right 𝐿)
Separation principle: if the plant is reachable & observable, can replace 𝒙
with 𝒙 and design 𝐾 independently of 𝐿 (use observed state just as real one)
𝑦
PLANT
𝑢
𝒙
𝑦 = 𝐶𝒙
𝒙 = 𝐴 𝒙 + 𝐵𝑢 + 𝐿(y − C 𝒙)
𝑦
𝐾
𝐾𝑟 +-
OBSERVER

Optimal control
Linear Quadratic Gaussian (LQG) optimal output regulation: linear plant,
additive Gaussian white noise on state (with covariance Σ 𝑑) and output (𝜎 𝑛);
minimize quadratic cost :
𝔼 lim
𝑇→∞
1
𝑇 0
𝑇
𝒙 𝑇 𝑡 𝑄𝒙 𝑡 + 𝑢 𝑡 2 𝑑𝑡
Solution is linear observer (Kalman filter) with linear memoryless controller:
𝐿 = 𝑃𝐶 𝑇 𝜎 𝑛
−1, 𝐴𝑃 + 𝑃𝐴 𝑇 + Σ 𝑑 − 𝑃𝐶 𝑇 𝜎 𝑛
−1 𝐶𝑃 𝑇 = 0
𝐾 = 𝐵 𝑇 𝑆, 𝐴 𝑇 𝑆 + 𝑆𝐴 + 𝑄 − 𝑆𝐵𝐵 𝑇 𝑆 = 0
𝑦
PLANT
𝑢 𝒙 = 𝐴𝒙 + 𝐵𝑢 + 𝒅
𝑦 = 𝐶𝒙
𝒙
𝒙 = 𝐴 𝒙 + 𝐵𝑢 + 𝐿(yn − C 𝒙)
𝑦 = 0
𝐾
+-
OBSERVER (KALMAN FILTER)
𝒅
++
𝑛
𝑦 𝑛

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 “internal model” 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.
𝑦
PLANT
𝒖𝑦
𝑦 = 𝐶𝒙
𝜼 = 𝑆𝜼 + 𝐺𝑒𝒛 = 𝑆 𝒛
𝑦 = 𝑇 𝒛
+
-
𝑒
STABILIZING
CONTROLLER
𝜼INT.MODELEXOSYSTEM

Human response model
D. L. Kleinman, S. Baron and W. H. Levison, “An Optimal Control Model of
Human Response. Part I: Theory and Validation”, 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:
𝑚𝑜𝑛𝑖𝑡𝑜𝑟 𝑌(𝑠)
++
𝐷(𝑠) (computer)
𝑈(𝑠)
𝑗𝑜𝑦𝑠𝑡𝑖𝑐𝑘 𝑃𝐽𝑀 𝑠 ∈ 𝑘,
𝑘
𝑠
,
𝑘
𝑠2
𝑠𝑒𝑛𝑠𝑜𝑟𝑦
𝑓𝑒𝑒𝑑𝑏𝑎𝑐𝑘
“Human controller”:
𝑛𝑒𝑢𝑟𝑜𝑚𝑜𝑡𝑜𝑟
dynamics
𝑈(𝑠)
++
𝑚𝑜𝑡𝑜𝑟 𝑛𝑜𝑖𝑠𝑒
𝑛𝑒𝑢𝑟𝑎𝑙
computation
𝑃 𝑁 𝑠𝐶 𝑠
𝑒−𝑠𝜏
𝑃 𝑁 𝑠 ≈
1
𝜏 𝑛 𝑠+1
𝜏 𝑛 ≈ 100𝑚𝑠
𝜏 ≈ 150 − 250𝑚𝑠

Task: Output regulation/disturbance rejection with linear time-invariant
plant and significant delay on any potential feedback line
𝑒−𝑠𝜏
𝑃(𝑠)
++
𝑈(𝑠)
𝑁(𝑠)
𝑌(𝑠)
++
𝐷(𝑠)
𝑈 𝐶(𝑠) 𝑌𝑛(𝑠)𝐶(𝑠)
CONTROLLER
𝑃𝐽𝑀 𝑠 ∙ 𝑃 𝑁(𝑠)
ANALYSIS problem: infer a model of the neural controller 𝐶(𝑠) from the
observed performance of the subjects tested.

Task: Output regulation/disturbance rejection with linear time-invariant
plant and significant delay on any potential feedback line
𝑒−𝑠𝜏
𝑃(𝑠)
++
𝑈(𝑠)
𝑁(𝑠)
𝑌(𝑠)
++
𝐷(𝑠)
𝑈 𝐶(𝑠) 𝑌𝑛(𝑠)𝐶(𝑠)
CONTROLLER
𝑃𝐽𝑀 𝑠 ∙ 𝑃 𝑁(𝑠)
ANALYSIS problem: infer a model of the neural controller 𝐶(𝑠) from the
observed performance of the subjects tested.
So what are the performances?
• Very good and robust to disturbances up to 2Hz (sum of sinusoids), for all
three types of joystick dynamics
• Apparently delay-free Must be some kind of
FEEDBACK + FORWARD model !

Hypothesis: optimal control to minimize average error & control effort
𝔼 lim
𝑇→∞
1
𝑇 0
𝑇
𝒙 𝑡 2
+ 𝛼 𝑢 𝑡 2
+ 𝛽 𝑢 𝑡 2
𝑑𝑡
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)
𝑦
PLANT
𝑢
𝒙(𝑡 − 𝜏)
𝒙 = 𝐴𝒙 + 𝐵𝑢 + 𝒅
𝑦 = 𝐶𝒙
KALMAN FILTER
𝑦 = 0
𝐾
+-
𝒅
++
𝑛
𝑦 𝑛
* D. Kleinman, “Optimal control of linear systems with time-delay and observation noise”, IEEE Trans. Autom. Control, 1969
𝑒−𝑠𝜏PREDICTOR
𝒙(𝑡) 𝑦𝑛(𝑡 − 𝜏)

Controller freq. response with plant
𝑘
𝑠
Controller freq. response with plant
𝑘
𝑠2
Matteo Mischiati Control theory primer 39

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., “Intermittent control: a computational theory of human control”, Biol. Cybern., 2011
Actual response to sinusoid Response without int.model

• Crash course in control theory (for LTI systems)
- many concepts can be extended to more general settings
• 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 !

Control-theoretic approach to sensorimotor loops

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