580.691 Learning Theory

Reza Shadmehr

Optimal feedback control

stochastic feedback control with and without additive noise

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Starting at state

Sequence of actions

Observations

Cost to minimize ( ) ( ) ( ) ( ) ( ) ( )

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u u y y

Cost per step

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y y u u

u

x y y x x

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Start at the last time point k=pThe cost at k=p is:

To minimize this cost, set

Under this policy, the value of states are:

( ) ( ) ( ) ( )

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arg min

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p p p

u

p p p T p p p T p p p

p T T p p p T p p p

p T T p p p T p p

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x V x

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p T T p p p T T p p

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x x u u

x x u u

u x

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p p T p T p p

p p T p T p

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2

2

p p p T T p T p p

p T p T p p p T T p p

p T T p T p p

p T p T p T p p p

p T p T T p p

V C T C A W A

L B W B B W A

C T C A W A

G L B W B G

G B W A

x x x

u u u x

x x

x x

x x

We will now show that if we choose the optimal u at step p-1, then cost to go is once again a quadratic function of state x.

Can be simplified to:( 1) ( ) ( 1) ( 1)p T T p p pA W BG x x

( 1) ( ) ( 1) ( 1) ( ) ( ) ( 1) ( 1)

( 1) ( 1) ( ) ( 1) ( 1)

( 1) ( 1) ( 1)

p p p T T p T p T p p p

p T T p T p p p

p T p p

V C T C A W A A W BG

C T C A W A BG

W

x x x

x x

x x

Because it is a scalar, it can be written as:

( 1) ( ) ( 1) ( 1)2 p T T p p pA W BG x x

We just showed that for the last time step, the cost to go is a quadratic function of x:

( ) ( ) ( ) ( )p p T p pV W x x x

The optimal u to at time point p-1 minimizes cost to go J(p-1):

1( 1) ( 1) ( ) ( )

( 1) ( 1) ( 1)

p p T p T p

p p p

G L B W B B W A

G

u x

If at time point p-1 we indeed carry out this optimal policy u, then the cost to go at time p-1 also becomes a linear function of x:

( 1) ( 1) ( 1) ( 1)

( 1) ( 1) ( ) ( 1)

p p T p p

p T p T p p

V W

W C T C A W A BG

x x x

If we now repeat the process and find the optimal u for time point p-2, it will be:

1( 2) ( 2) ( 1) ( 1)

( 2) ( 2) ( 2)

p p T p T p

p p p

G L B W B B W A

G

u x

And if we apply the optimal u at time points p-2 and p-1, then the cost to go at time point p-2 will be a quadratic function of x:

( 2) ( 2) ( 2) ( 2)

( 2) ( 2) ( 1) ( 2)

p p T p p

p T p T p p

J W

W C T C A W A BG

x x

So in general, if for time points t+1, …, p we calculated the optimal policy for u, then the above gives us a recipe to compute the optima policy for time point t.

( 1) ( ) ( )

( 1) ( 1)

1(0) ( 1) ( 1) ( 1) ( ) ( ) ( )

0

k k k

k k

pk T k k k T k k

k

A B

C

J T L

x x u

y x

y y u u

Summary: optimal feedback control

( ) ( ) ( ) ( )

( ) ( )

1( 1) ( 1) ( ) ( )

( 1) ( 1) ( 1)

( 1) ( 1) ( 1) ( 1)

( 1) ( 1) ( ) ( 1)

p p T p p

p T p

p p T p T p

p p p

p p T p p

p T p T p p

V W

W C T C

G L B W B C W A

G

V W

W C T C A W A BG

x x x

u x

x x x

(0)x

(1)y

(0)u

(1)x ( )px

( 1)pu

( )py

Cost to go

1(0) (0) (1) (1)

(0) (0) (0)

(0) (0) (0) (0)

(0) (0) (1) (0)

T T

T

T T

G L B W B B W A

G

J W

W C T C A W A BG

u x

x x

The procedure is to compute the matrices W and G from the last time point to the first time point.

1 2

1 1

2 2

( 1) ( )

3 . / 0.45 . . / 0.3 . /

0 1 0 00

10

1 0

0.01sec

c c

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k N m rad b N m s rad m kg m rad

x x x x

x xk b

x x um m m

A B

y

A B

A I A t

C B t

x x u

x

x x u

Continuous time model of the elbow

Discrete time model of the elbow

Modeling of an elbow movement

Goal: Reach a target at 30 deg in 300 ms time and hold it there for 100 ms.

Unperturbed movement Arm held at start for 200ms Force pulse to the arm for 50ms

0 0.1 0.2 0.3 0.4sec

0

0.1

0.2

0.3

0.4

0.5

noitisoP

0.05 0.1 0.15 0.2 0.25 0.3 0.35sec

-75

-50

-25

0

25

50

75

rotoMdnammoc

0 0.1 0.2 0.3 0.4sec

0

0.1

0.2

0.3

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0.05 0.1 0.15 0.2 0.25 0.3 0.35sec

-5

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0 0.1 0.2 0.3 0.4sec

0

0.1

0.2

0.3

0.4

0.5

noitisoP0.05 0.1 0.15 0.2 0.25 0.3 0.35

sec

-30

-20

-10

0

10

rotoMdnammoc

0 0.1 0.2 0.3 0.4sec

1

1.25

1.5

1.75

2

2.25

2.5

L

L

0 0.1 0.2 0.3 0.4sec

0

500000

1́ 106

1.5́ 106

2́ 106

soPtsoc T

0 0.1 0.2 0.3 0.4sec

0

5000

10000

15000

20000

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0 0.1 0.2 0.3 0.4 0.5 0.6sec

0

500000

1´106

1.5́ 106

2´106

soPtsoc

Movement with a via point: we set the cost to be high at the time when we are supposed to be at the via points.

0 0.1 0.2 0.3 0.4 0.5 0.6sec

-10

0

10

20

30

rotoMdnammoc

0 0.1 0.2 0.3 0.4 0.5 0.6sec

0

0.2

0.4

0.6

0.8

noitisoP

0 0.1 0.2 0.3 0.4 0.5 0.6sec

0

200

400

600

800

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G

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pk T k k k T k k

k

A C N Q

B N R

L T

J L T

x x u ε ε

y x ε ε

u u y y

u u y y

Stochastic optimal feedback control

Biological processes have noise. For example, neurons fire stochastically in response to a constant input, and muscles produce a stochastic force in response to constant stimulation. Here we will see how to solve the optimal control problem with additive Gaussian noise.

Cost to minimize

Because there is noise, we are no longer able to observe x directly. Rather, the best we can do is to estimate it. As we saw before, for a linear system with additive noise the best estimate of state is through the Kalman filter. So our goal is to determine the best command u for the current estimate of x so that we can minimize the global cost function.

Approach: as before, at the last time point p the cost is a quadratic function of x. We will find the optimal motor command for time point p-1 so that it minimizes the expected cost to go. If we perform the optimal motor command at p-1, then we will see that the cost to go at p-1 is again a quadratic function of x.

Preliminaries: Expected value of a squared random variable. In the following example, we assume that x is the random variable.

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x y y x ε x ε

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tr T R

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arg min ,

2

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p p T p p p T p p p

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x x x u

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1( 1) ( 1) ( ) ( ) ( 1)

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p T py y

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p p T p T p p

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p p p

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dL B W B B W A

d

L B W B B W A

G L B W B B W A

G

ε ε

u xu

u x

u x

So we see that if our system has additive state or measurement noises, the optimal motor command remains the same as if the system had no noises at all. When we use the optimal policy at time point p-1, we see that, as before, the cost-to-go at p-1 is a quadratic function of x. The matrix W at p-1 remains the same as when the system had no noise.

The problem is that we do not have x. The best that we can do is to estimate x via the Kalman filter. We do this in the next slide.

1 21 1

1 2 2 2 2

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ˆ ˆ

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ˆ ˆ

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p pp p

p p p p p

p p p p p pp p

p p p p p pp p p

p pp

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G

A B

K C

A B AK C

A B AK C

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u x

x x u

x x y x

x x u y x

x x

x x u y x

u x

On trial p-1, our best estimate of x is the prior.

We compute the prior for the current trial from the posterior of the last trial.

The posterior estimate.

Our short-hand way to note the prior estimate of x on trial p-1.

Although the noises in the system do not affect the gain G, the estimate of x is of course affected by the noises because the Kalman gain is influenced by them.

Kalman gain

( 1) ( ) ( )

( 1) ( 1)

1(0) ( ) ( ) ( ) ( 1) ( 1) ( 1)

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0,

0,

k k kx x

k ky y

pk T k k k T k k

k

A B N Q

C N R

J L T

x x u ε ε

y x ε ε

u u y y

Summary of stochastic optimal control for a linear system with additive Gaussian noise and quadratic cost

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ˆ

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p p T p T p

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p p T p p p

p T p T p p

p p T p py y

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W C T C

w T

G L B W B B W A

G

J W w

W C T C A W A BG

w tr W Q T E w

x x

ε ε

u x

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Cost to go at the start

Cost to go at the end

( )ptr T R