On-off Control: Audio Applications...On-off Control: Audio Applications Graham C. Goodwin Day 4: Lecture 3 16th September 2004 International Summer School Grenoble, France Centre for

On-off Control: Audio Applications

Graham C. Goodwin

Day 4: Lecture 3

16th September 2004

International Summer SchoolGrenoble, France

Centre for Complex DynamicSystems and Control

1 Background

In this lecture we address the issue of control when the decisionvariables must satisfy a finite set constraint.

Finite alphabet control occurs in many practical situationsincluding: on-off control, relay control, control where quantisationeffects are important (in principle this covers all digital controlsystems and control systems over digital communicationnetworks), and switching control of the type found in powerelectronics.


Exactly the same design methodologies can be applied in otherareas; for example, the following problems can be directlyformulated as finite alphabet control problems:

quantisation of audio signals for compact disc production;

design of filters where the coefficients are restricted to belongto a finite set (it is common in digital signal processing to usecoefficients that are powers of two to facilitate implementationissues);

design of digital-to-analog [D/A] and analog-to-digital [A/D]converters.


2. Finite Alphabet Control

Consider a linear system having a scalar input uk and state vectorxk ∈ Rn described by

xk+1 = Axk + Buk . (1)

A key consideration here is that the input is restricted to belong tothe finite set

U = {s1, s2, . . . , snU}, (2)

where si ∈ R and si < si+1 for i = 1, 2, . . . , nU − 1.


We will formulate the input design problem as a receding horizonquadratic regulator problem with finite set constraints. Thus, giventhe state xk = x, we seek the optimising sequence of present andfuture control inputs:

u(x) , arg minuk∈UN

VN(x,uk ), (3)

where

uk ,

ukuk+1...

uk+N−1

, UN, U × · · · × U. (4)


VN is the finite horizon quadratic objective function

VN(x,uk ) , ‖xk+N‖2P +k+N−1∑

t=k

(‖xt‖2Q + ‖ut‖

2R ), (5)

with Q = Q > 0, P = P > 0, R = R > 0 and where xk = x.


Following the usual receding horizon principle, only the first controlaction, namely

u(x) ,[

1 0 · · · 0]

u(x), (6)

is applied. At the next time instant, the optimisation is repeatedwith a new initial state and the finite horizon window shifted by one.


3. Nearest Neighbour Characterisation of the Solution

Since the constraint set UN is finite, the optimisation problem (3) isnonconvex. Indeed, it is a hard combinatorial optimisation problemwhose solution requires a computation time that is exponential inthe horizon length. Thus, one needs either to use a relatively smallhorizon or to resort to approximate solutions. We will adopt theformer strategy based on the premise that, due to the recedinghorizon technique, the first decision variable is all that is of interest.Moreover, it is a practical observation that this first decisionvariable is often insensitive to increasing the horizon length beyondsome relative modest value.


We vectorise the objective function as follows:Define

xk ,

xk+1xk+2...

xk+N

, Φ ,

B 0 . . . 0 0AB B . . . 0 0...

.... . .

......

AN−1B AN−2B . . . AB B

, Λ ,

AA2...

AN

,

(7)


Given xk = x the predictor xk satisfies

xk = Φuk + Λx. (8)

Hence, the objective function can be re-written as

VN(x,uk ) = V̄N(x) + uk Huk + 2u

k Fx, (9)

where

H , ΦQΦ + R ∈ RN×N , F , ΦQΛ ∈ RN×n,

Q , diag{Q , . . . ,Q ,P} ∈ RNn×Nn, R , diag{R , . . . ,R} ∈ RN×N ,

and V̄N(x) does not depend upon uk .


By direct calculation, it follows that the minimiser, without takinginto account any constraints on uk , is

u

(x) = −H−1Fx. (10)


Definition: Nearest Neighbour Vector Quantiser

Given a countable set of nonequal vectors B = {b1, b2, . . . } ⊂ RnB ,the nearest neighbour quantiser is defined as a mappingqB : RnB → B that assigns to each vector c ∈ RnB the closestelement of B (as measured by the Euclidean norm), that is,qB(c) = bi ∈ B if and only if c belongs to the region

{

c ∈ RnB : ‖c − bi‖2 ≤ ‖c − bj‖

2 for all bj , bi , bj ∈ B}

\{

c ∈ RnB : there exists j < i such that ‖c − bi‖2= ‖c − bj‖

2}

.

(11)


In order to simplify the problem, we introduce the same coordinatetransformation utilised earlier, that is, the one that turns the costcontours into (hper) spheres.

ũk = H1/2uk , (12)

which transforms the constraint set UN into ŨN.


The optimiser u(x) can be defined in terms of this auxiliaryvariable as

u(x) = H−1/2 arg minũk∈Ũ

NJN(x, ũk ), (13)

whereJN(x, ũk ) , ũk ũk + 2ũ

k H−/2Fx. (14)


The level sets of JN are spheres in RN, centred at

ũ

(x) , −H−/2Fx. (15)


Hence, the constrained optimiser (3) is given by the nearestneighbour to ũ

(x), namely

arg minũk∈Ũ

NJN(x, ũk ) = q

ŨN (−H−/2Fx). (16)


Summary Theorem: Closed Form Solution

Let UN = {v1, v2, . . . , vr }, where r = (nU)N. Then the optimiser

u(x) in (3) is given by

u(x) = H−1/2qŨ

N (−H−/2Fx), (17)

where the nearest neighbour quantiser qŨ

N (·) maps RN to ŨN,

defined as

ŨN, {ṽ1, ṽ2, . . . , ṽr }, ṽi = H

1/2vi , vi ∈ UN . (18)


The receding horizon controller satisfies

u(x) =[

1 0 · · · 0]

H−1/2qŨ

N (−H−/2Fx). (19)

This solution can be illustrated as the composition of the followingtransformations:

x ∈ Rn−H−

2 F−−−−−−−→ ũ

∈ RN

H−12 qŨ

N (·)−−−−−−−−−−→ u ∈ UN

[1 0 · · · 0]−−−−−−−−−−→ u ∈ U .

(20)


4. State Space Partition

The optimal expression partitions the domain of the quantiser intopolyhedra, called a Voronoi partition.Since the constrained optimiser u(x) is defined in terms ofqŨ

N (·), an equivalent partition of the state space can be derived.


Theorem

The constrained optimising sequence u(x) can be characterisedas

u(x) = vi ⇐⇒ x ∈ Ri ,

where

Ri ,{

z ∈ Rn : 2(vi − vj)Fz ≤ ‖vj‖

2H − ‖vi‖

2H for all vj , vi , vj ∈ U

N}

\{

z ∈ Rn : there exists j < i such that 2(vi − vj)Fz = ‖vj‖

2H − ‖vi‖

2H

}

.

(21)


5. Examples: 5.1 Open Loop Stable Plant

Consider an open loop stable plant described by

xk+1 =

[

0.1 20 0.8

]

xk +

[

0.10.1

]

uk , (22)

and the binary constraint set U = {−1, 1}. The receding horizoncontrol law with R = 0 and

P = Q =

[

1 00 1

]

, (23)

partitions the state space into the regions depicted in the nextfigure, for constraint horizons N = 2 and N = 3.


−80 −60 −40 −20 0 20 40 60 80

−0.5

0

0.5

1

2

3

4

−80 −60 −40 −20 0 20 40 60 80

−0.5

0

0.5

1

23

4

5

6

7

8

x1k

x2k

x2k

N = 2

N = 3

R

RR

R

R

RR

R

R

R

R

R

Figure: State space partition for the plant (22).


The receding horizon control law is

u(x) =

−1 if x ∈ X1,

1 if x ∈ X2,

where

X1 =⋃

i=2N−1+1,2N−1+2,...,2N

Ri , X2 =⋃

i=1,2,...,2N−1

Ri .


5.2 Open Loop Unstable Plant

Consider

xk+1 =

[

1.02 20 1.05

]

xk +

[

0.10.1

]

uk , (24)

controlled with a receding horizon controller with parameters U, P,Q and R as above. The constraint horizon is chosen to be N = 2.


The following figure illustrates the induced state space partitionand a closed loop trajectory, which starts at x = [−10 0]. As canbe seen, due to the limited control action available, the trajectorybecomes unbounded.


−40 −20 0 20 40 60 80−5

−4

−3

−2

−1

0

1

2

1

2

3

4

x1k

x2k

R

R

R

R

Figure: State trajectories of the controlled plant (24) with initial conditionx = [−10 0].


The situation is entirely different when the initial condition ischosen as x = [0.7 0.2]. As depicted in the following figure, theclosed loop trajectory now converges to a bounded region, whichcontains the origin in its interior. Within that region, the behaviouris not periodic, but appears to be random, despite the fact that thesystem is deterministic. Neighbouring trajectories diverge due tothe action of the unstable poles of the plant. However, the controllaw manifests itself by maintaining the plant state ultimatelybounded.


−1 −0.5 0 0.5 1

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

1

2

3

4

x1k

x2k

R

R

R

R

Figure: State trajectories of the controlled plant (24) with initial conditionx = [0.7 0.2].



Application: Quantization of Audio Signals

Modern music recording equipment use digital recording – typically 16 bit:

Naïve idea:

Round toQuantized

LevelsAnalogue Audio


CD Mastering Stations


Audio in Quantizer Quantized Output

Error Feedback

Noise Shaping Quantizer

More Conventional Form (after block diagram manipulation)


Reformulation as Novel Optimization Problem

H(ρ)

Incorporation of a perception filter


Design Criterion: Finite Horizon Constrained Optimization

1 2 ( ).k N

Nt k

V e t+ −

== ∑

Perception Filter:

1

1( ) 1 ,i

iH hρ ρ

∞−

== + ∑

then the overall perceived error is given by:

( )( ) ( ) ( ) ( ) .e t H a t u tρ= −


Block Optimization

( ) ( ) ( 1) ... ( 1) .T

u k u k u k u k N⎡ ⎤= + + −⎣ ⎦r

( ) ( )21

( ) ( ) ( ) ( ) .k N

Nt k

V u k H a t u tρ+ −

=

⎛ ⎞= −⎜ ⎟

⎝ ⎠∑

r

( )*( )

( ) arg min ( ) .NNu k Uu k V u k

∈=

r

r r

Finite Alphabet

Define the future quantized audio signals as a vector

Recall cost function:

Optimal control (actually the quantized audio)


Recall the Geometry of the Constrained Optimization Problem

Geometric interpretation of quadratic programming


After a Simple Transformation

Geometry of finite alphabet optimization as aminimum Euclidean distance problem


Feedback form of the Solution to Finite Alphabet Control Problem

Convert to State Space

1( ) 1 ( ) .H C I A Bρ ρ −= + −

( )( )

( 1) ( ) ( ) ( )

( ) ( ) ( ) ( )

x t Ax t B a t u t

e t Cx t a t u t

+ = + −

= + −


Theorem

Suppose UN = {v1, v2, …,vr}, where r = nUN and H(ρ) has realization as above, then the optimizing sequence is given by:

where:

*( )u kr

( )* 1( ) ( ) ( )NUu k q a k x k−= Ψ Ψ +Γ%r r

01 0

11 1 0

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

( 1) N N

hCa kCA h ha ka k

a k N h h hCA − −

⎡ ⎤⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥+= Γ = Ψ = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥+ − ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

Kr O M

MM M O OK


Moving Horizon Optimization

Moving horizonPrinciple, N = 5


Final MHOQ for Audio Quantization

( )1( ) [10 0] ( ) ( )NUu k q a k x k−= Ψ Ψ +Γ%r

K

Closed form – Vector Quantizer

MHOQ: Moving Horizon Optimal Quantizer


Special Case: Horizon = 1

Consider a unitary prediction horizon, i.e. N = 1. With N = 1, H(ρ) reduces to its first element which according to the definitions given above satisfies

i.e. it is exactly the

11 ( )H ρ′+

111 ( ) 1 ( ) ( )C I A B Hρ ρ ρ

−′+ = + − =H

Perception Filter


MHOQ with horizon N = 1

Horizon 1 mpc solution to optimal audio quantization

Does this appear familiar?


Horizon 1 mpc solution

The standard noise shaping filter solution (in conventional feedback form)


Key Observation ☺

Optimization based Audio Quantizer

Standard Noise Shaping Quantizer for N = 1

Thus standard noise shaping quantizer is special case of MHOQ.

( ) 1( )( )

HHFρρρ−=


Example

Psycho-acoustic studies:

12.245 0.66411 21 1.335 0.644

( ) 1H ρρ ρ

ρ ρ−−−

− −− += +Perception Filter:

12.245 0.664111 0.91

( )F ρρ

ρ ρ−−−

−+=Noise Shaping Filter:


Frequency responses of H and F


Music Quantization = MPC



Effect of Increasing Horizon

Mean SquareQuantization

Error

Optimization Horizon


Question: Just how well can we do?

It is interesting to plot the spectrum of the errors due to naïve quantization and the errors arising from the MHOQ (See next figure).


Spectrum of Errors due to Quantization


Observations

• MHOQ has reduced quantization noise energy in low frequency band.

• This has resulted in an increase in quantization noise energy at high frequencies.

• Actually this is in accord with (approximate) Bode integral

0 1log log

npjw

ii

S e dw pπ π=

⎛ ⎞ =⎜ ⎟⎝ ⎠

∑∫

(pi – unstable poles of H i.e. unstable zeros of 1-Fsince ).11 FH −=

GCG_Day4_3.pdfApplication: Quantization of Audio SignalsCD Mastering StationsReformulation as Novel Optimization ProblemDesign Criterion: Finite Horizon Constrained OptimizationBlock OptimizationRecall the Geometry of the Constrained Optimization ProblemAfter a Simple TransformationFeedback form of the Solution to Finite Alphabet Control ProblemTheoremMoving Horizon OptimizationFinal MHOQ for Audio QuantizationSpecial Case: Horizon = 1MHOQ with horizon N = 1Key Observation ExampleMusic Quantization = MPCEffect of Increasing HorizonQuestion: Just how well can we do?Spectrum of Errors due to QuantizationObservations

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On-off Control: Audio Applications...On-off Control: Audio Applications Graham C. Goodwin Day 4: Lecture 3 16th September 2004 International Summer School Grenoble, France Centre for