Download 19. Systems Concepts (2001)

Michael Arbib: CS564 - Brain Theory and Artificial Intelligence
University of Southern California, Fall 2001
Lecture 19. Systems and Feedback
Reading Assignment:
TMB2 Section 3.1 and 3.2
Note: Self-study on eigenvalues: TMB2, pp.107,108,111-115.
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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A System is Defined by Five Elements
The set of inputs
The set of outputs
(These involve a choice by the modeler)
The set of states: those internal variables of the system — which may
or may not also be output variables — which determine the
relationship between input and output.
state = the system's "internal residue of the past"
The state-transition function : how the state will change when the
system is provided with inputs.
The output function : what output the system will yield with a given
input when in a given state.
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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From Newton to Dynamic Systems
Newton's mechanics describes the behavior
of a system on a continuous time scale.
Rather than use the present state and input to predict the next state,
the present state and input determine
the rate at which the state changes.
Newton's third law says that the force F applied to the system equals the
mass m times the acceleration a.
F = ma
Position x(t)
Velocity v(t) =
Acceleration a(t) =x (tx)
x(t )
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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Newtonian Systems
According to Newton's laws, the state of the system
is given by the position and velocity of the particles
of the system.
We now use
u(t) for the input = force; and
y(t) (equals x(t)) for the output = position.
Note: In general, input, output, and state are more general than in the
following, simple example.
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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State Dynamics
With only one particle, the state is the 2-dimensional vector
 x (t ) 
q(t) = 


x
(
t
)


Then
d
x(t )  x (t ) and
dt
d
x (t ) 
dt
1
u (t ) sin ce u (t )  mx
m
yielding the single vector equation
 x (t ) 0 1 
q (t )  
 

 x(t ) 0 0
 x(t ) 0 
 x (t )  1  u (t )

  m 
The output is given by
1 
y(t) = x(t) =  
0 
 x(t )
 x (t )


The point of the exercise: Think of the state vector as a single point in a
multi-dimensional space.
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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Linear Systems
This is an example of a Linear System:
q = A q + B u
y = Cq
where the state q, input u, and output y are vectors (not necessarily 2dimensional) and A, B, and C are linear operators (i.e., can be
represented as matrices).
Generally a physical system can be expressed by
State Change:
q(t) = f(q(t), u(t))
Output:
y(t) = g(q(t))
where f and g are general (i.e., possibly nonlinear) functions
For a network of leaky integrator neurons:
the state mi(t) = arrays of membrane potentials of neurons,
the output M(t) = s(mi(t)) = the firing rates of output neurons,
obtained by selecting the corresponding membrane potentials and
passing them through the appropriate sigmoid functions.
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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Attractors
For all recurrent networks of interest
(i.e., neural networks comprised of leaky integrator
neurons, and containing loops), given initial state
and fixed input, there are just three possibilities for the asymptotic state:
1. The state vector comes to
rest, i.e. the unit
activations stop changing.
This is called a fixed
point. For given input
data, the region of initial
states which settles into a
fixed point is called its
basin of attraction.
2. The state vector settles
into a periodic motion,
called a limit cycle.
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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Strange attractors
3. Strange attractors describe such
complex paths through the state
space that, although the system is
deterministic, a path which
approaches the strange attractor
gives every appearance of being
random.
Two copies of the system which
initially have nearly identical
states will grow more and more
dissimilar as time passes.
Such a trajectory has become the
accepted mathematical model of
chaos,and is used to describe a
number of physical phenomena
such as the onset of turbulence in
weather.
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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Stability
The study of stability of an equilibrium is concerned with the issue of
whether or not a system will return to the equilibrium in the face of
slight disturbances:
A is an unstable equilibrium
B is a neutral equilibrium
C is a stable equilibrium, since small displacements will tend to
disappear over time.
Note: in a nonlinear system, a large displacement can move the ball
from the basin of attraction of one equilibrium to another.
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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Negative Feedback Controller (Servomechanism)
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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Using Spindles to Tell -Neurons
if a Muscle Needs to Contract
What’s missing in this Scheme?
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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Using -Neurons
to Set the Resting Length of the Muscle



Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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Self-study: the
elegant extension
of this scheme to an
agonist-antagonist
pair of muscles
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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Discrete-Activation Feedforward
“Cortex”
“Spinal Cord”
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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“Ballistic” Correction then Feedback
This long latency reflex was noted by Navas and Stark.
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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The Mass-Spring Model of Muscle
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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Hooke’s Law – The Linear Range of Elasticity
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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The Mass-Spring Model of Muscle
The dynamics of muscle length:
mx = mg - u(t) - k(x - xo) - bx
mg
u(t)
-k(x-xo)
restoring
of
law]
-bx
gravitational force on the mass m
controllable force of muscle contraction
for a constant k > 0 represents the
force of muscle for moderate displacements
length x from resting value xo
[Hooke's
viscosity
Use as state the two-dimensional vector
 x(t) 


q(t) =  .
so that

 x(t) 
0
1   x(t) 

 0
.


b  .
u(t) kxo

q(t) =  k
+


g+
m
m
 m m   x (t) 





Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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An equilibrium state is one which (for fixed values
.
of the input) does not change, i.e., for which q (t) = 0.
Fix the contractile force u(t) to constant u.
..
.
x =x =0
so
..
.
mx = mg - u(t) - k(x - xo) - bx
yields
0 = mg - u - k(x - xo)
Thus, the equilibrium length xe(u) for given force u
satisfies
mg - u
xe(u) = xo +
k
(3,4)
... in the domain of linearity of the spring
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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Linear Systems
.
q (t) = Aq(t) + (t)
where A is a matrix, and (t) is a general input vector.
In a feedback system, (t) will itself be a function of
the state q — so will in general depend on q() for
times < t.
Assume that A is non-singular and thus has inverse
A-1.
Then an equilibrium state qe for fixed (t) =  must
satisfy
0 = Aqe +  so that A-10 = A-1Aqe + A-1i.e.
qe + A-1; and so qe = -A-1,a unique
equilibrium point
Let p = q - qe be the displacement of the system state
from this equilibrium.
.
.
d
p =
(q - qe) = q = Aq + 
dt
= A(p + qe) +  = Ap + Aqe +  = Ap
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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Eigenvectors and Eigenvalues
See TMB2, pp.107,108,111-115 for an exposition
of their use in characterizing the stability of a
linear system.
The key result is:
.
Theorem. The system q = Aq is stable
i.e., q(t)  0 as t  for each initial state q(0)
if both eigenvalues of A have negative real parts.
Moreover, the system will oscillate if the eigenvalues
are complex.
We now consider how that theory is applied
to the mass-spring model.
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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Stability of the Mass-Spring Model
Our mass-spring model
0


 0 1 
.




q (t) =  k b  q(t) +  u(t) kxo 
- g+
m
m 
 m m

has equilibrium state
xe(u)
mg - u

qe(u) = 
where xe(u) = xo +
k
 0 
The eigenvalues are
1 b
1,2 = 2 - +
m
b
 2
 
m
4k 

m
1 

=
2m -b +

b2 - 4km 
which are real if b2  4km.
But since b, k and m are all positive, in this case
0  b2 - 4km < b,
and so both 1 and 2 must be negative.
Again, if 1 and 2 are complex, their real part is b/2m which is always negative.
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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Theorem: The mass-spring system with each b, m
and k positive is always stable and undergoes
damped oscillations just in case b2 < 4km.
Large b will prevent the system oscillating, unless it is
dominated by a large spring term (large k), for fixed
mass m.
For b2 < 4km, the oscillations are quickly damped
either if b is large (high friction or viscosity) or m is
small.
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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For a given m, and a desired resting length x e, the
appropriate force u(xe) must satisfy
mg - u(xe)
xe = x o +
k
kxe = mg - u(xe) + kxo
so that
u(xe) = mg + k(xo-xe)
If the mass is m + m then the length achieved will be
(m+m)g - u(xe)
xo +
= xe + m.g/k
k
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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A simple linear model: the mass-spring model with
-style feedback
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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Substituting u(xe) = mg + k(xo-xe), we obtain
0 
 0 1 

.


q(t) =  (k+h) b  q(t) +  (k+h)xe 
 m m
 m 
Thus at equilibrium,
(k+h)xe
(k+h)x
=
m
m
and so our new model has the expected property that
the equilibrium length is still xe.
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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If the mass changes unexpectedly from m to m+m
while xe and u(xe) remain fixed, then equation (10)
becomes
0
1

  x(t) 
.

 
k
b
q (t) = 
  x. (t) 
m+m 
 m+m
0


u(x
)
+
h(x-x
)
kx
e
e
o 
+ 
g+
m+m
m+m 

However, when we now substitute for u(xe) we
must note that this control value was based on the
original unperturbed value of m, and so we get
0
0
1




.
 (k+h)

m.g+(k+h)x
b
q (t) = 
e
 q(t) + 
m+m 
 m+m
m+m


Thus at equilibrium
m.g+(k+h)xe
(k+h)x
=
m+m
m+m
and so the new equilibrium length is
m.g
x = xe +
k+h
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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The feedback with gain h has served to reduce the
m.g
m.g
perturbation from
to
, so that the larger
k
k+h
is h
the smaller is the effect of the change of load.
Feedback reduces the effect of small
disturbances
to the point where they may be negligible.
Unless h is inordinately large, a large disturbance
will yield a significant perturbation.
In this case, the brain must, essentially, recompute
u(xe) using an estimate of the actual load m+m
rather than the "standard load" m.
This is our explanation of the long latency reflex
noted by Navas and Stark.
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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What is the price we pay for using a high value of h?
We must examine the stability of the new system.
We replace k in the system matrix by k+h, which is
still positive, and possibly perturb the value of m.
Thus the system remains stable.
However, for fixed b, k, and m, we can make
b2 < 4(k + h)m
by making the "feedback gain" h large enough.
This happens as soon as
h > (b2 /4m) - k.
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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Theorem: For the mass-spring model with " "
coactivation with u(t) = u(xe) and feedback term h(x xe) with gain h for desired equilibrium length xe:
(i) the perturbation of the equilibrium length
occasioned by a change in load m will be reduced by
m.g
m.g
the feedback from
to
;
k
k+h
(ii)
the system will be stable; but
(iii) even if the system without feedback exhibits no
oscillations (b2 > 4km), the feedback system will
exhibit oscillations for sufficiently high gain, h > (b2
/4m) - k.
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
30
This model can only be part of the story:
• Our model is only valid for a range of x in which the linear
model of the spring is valid; while the spindle can only
monitor x-xe when this value is positive.
• A complete model would take into account those muscle
spindles which monitor velocity, and the way in which the
Golgi tendon organ monitors force.
• The efficacy of feedback depends crucially on whether or not
the correction signal arrives while it is still valid, and thus
delays in feedback paths can have a great effect on the validity
and stability of a control system.
• Nonlinearities may prove crucial in understanding muscle
tremor in terms of limit cycles rather than the harmonic
oscillations of a linear system.
Michael Arbib CS564 - Brain Theory and Artificial Intelligence, USC, Fall 2001. Lecture 19. Systems Concepts
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