On-off Control: Audio Applications Graham C. Goodwin Day 4: Lecture - - PowerPoint PPT Presentation

On-off Control: Audio Applications

Graham C. Goodwin Day 4: Lecture 3 16th September 2004 International Summer School Grenoble, France

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1 Background

In this lecture we address the issue of control when the decision variables must satisfy a finite set constraint. Finite alphabet control occurs in many practical situations including: on-off control, relay control, control where quantisation effects are important (in principle this covers all digital control systems and control systems over digital communication networks), and switching control of the type found in power electronics.

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Exactly the same design methodologies can be applied in other areas; for example, the following problems can be directly formulated as finite alphabet control problems: quantisation of audio signals for compact disc production; design of filters where the coefficients are restricted to belong to a finite set (it is common in digital signal processing to use coefficients that are powers of two to facilitate implementation issues); design of digital-to-analog [D/A] and analog-to-digital [A/D] converters.

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• 2. Finite Alphabet Control

Consider a linear system having a scalar input uk and state vector xk ∈ Rn described by xk+1 = Axk + Buk. (1) A key consideration here is that the input is restricted to belong to the finite set

U = {s1, s2, . . . , snU},

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

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We will formulate the input design problem as a receding horizon quadratic regulator problem with finite set constraints. Thus, given the state xk = x, we seek the optimising sequence of present and future control inputs: u(x) arg min

uk∈UN VN(x, uk),

(3) where uk

                

uk uk+1

. . .

uk+N−1

                 , UN U × · · · × U.

(4)

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VN is the finite horizon quadratic objective function VN(x, uk) xk+N2

P + k+N−1

• t=k

(xt2

Q + ut2 R),

(5) with Q = Q > 0, P = P > 0, R = R > 0 and where xk = x.

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Following the usual receding horizon principle, only the first control action, namely u(x)

• 1

· · ·

• u(x),

(6) is applied. At the next time instant, the optimisation is repeated with a new initial state and the finite horizon window shifted by one.

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• 3. Nearest Neighbour Characterisation of the Solution

Since the constraint set UN is finite, the optimisation problem (3) is

• nonconvex. Indeed, it is a hard combinatorial optimisation problem

whose solution requires a computation time that is exponential in the horizon length. Thus, one needs either to use a relatively small horizon or to resort to approximate solutions. We will adopt the former strategy based on the premise that, due to the receding horizon technique, the first decision variable is all that is of interest. Moreover, it is a practical observation that this first decision variable is often insensitive to increasing the horizon length beyond some relative modest value.

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We vectorise the objective function as follows: Define xk

                

xk+1 xk+2

. . .

xk+N

                 , Φ                 

B

. . .

AB B

. . . . . . . . . ... . . . . . .

AN−1B AN−2B

. . .

AB B

                 , Λ                 

A A2

. . .

AN

                 ,

(7)

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Given xk = x the predictor xk satisfies xk = Φuk + Λx. (8) Hence, the objective function can be re-written as VN(x, uk) = ¯ VN(x) + u

kHuk + 2u kFx,

(9) where H ΦQΦ + R ∈ RN×N, F ΦQΛ ∈ RN×n, Q diag{Q, . . . , Q, P} ∈ RNn×Nn, R diag{R, . . . , R} ∈ RN×N, and ¯ VN(x) does not depend upon uk.

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By direct calculation, it follows that the minimiser, without taking into account any constraints on uk, is u

 (x) = −H−1Fx.

(10)

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Definition: Nearest Neighbour Vector Quantiser

Given a countable set of nonequal vectors B = {b1, b2, . . . } ⊂ RnB, the nearest neighbour quantiser is defined as a mapping qB : RnB → B that assigns to each vector c ∈ RnB the closest element 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 − bi2 ≤ c − bj2 for all bj bi, bj ∈ B
• \
• c ∈ RnB : there exists j < i such that c − bi2 = c − bj2

.

(11)

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In order to simplify the problem, we introduce the same coordinate transformation utilised earlier, that is, the one that turns the cost contours into (hper) spheres.

˜

uk = H1/2uk, (12) which transforms the constraint set UN into ˜

U

N.

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The optimiser u(x) can be defined in terms of this auxiliary variable as u(x) = H−1/2 arg min

˜ uk∈ ˜ UN JN(x, ˜

uk), (13) where JN(x, ˜ uk) ˜ u

k ˜

uk + 2˜ u

kH−/2Fx.

(14)

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

˜

u

 (x) −H−/2Fx.

(15)

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Hence, the constrained optimiser (3) is given by the nearest neighbour to ˜ u

 (x), namely

arg min

˜ uk∈ ˜ UN JN(x, ˜

uk) = q ˜

UN(−H−/2Fx).

(16)

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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−1/2q ˜

UN(−H−/2Fx),

(17) where the nearest neighbour quantiser q ˜

UN(·) maps RN to ˜

U

N,

defined as

˜ U

N {˜

v1, ˜ v2, . . . , ˜ vr},

˜

vi = H1/2vi, vi ∈ UN. (18)

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The receding horizon controller satisfies u(x) =

• 1

· · ·

• H−1/2q ˜

UN(−H−/2Fx).

(19) This solution can be illustrated as the composition of the following transformations: x ∈ Rn −H− 

2 F

− − − − − − − → ˜

u

 ∈ RN H− 1

2 q ˜

UN(·)

− − − − − − − − − − → u ∈ UN [1 0 · · · 0] − − − − − − − − − − → u ∈ U .

(20)

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• 4. State Space Partition

The optimal expression partitions the domain of the quantiser into polyhedra, called a Voronoi partition. Since the constrained optimiser u(x) is defined in terms of q ˜

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

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Theorem

The constrained optimising sequence u(x) can be characterised as u(x) = vi ⇐⇒ x ∈ Ri, where

Ri

• z ∈ Rn : 2(vi − vj)Fz ≤ vj2

H − vi2 H for all vj vi, vj ∈ UN

\

• z ∈ Rn : there exists j < i such that 2(vi − vj)Fz = vj2

H − vi2 H

• .

(21)

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• 5. Examples: 5.1 Open Loop Stable Plant

Consider an open loop stable plant described by xk+1 =

• 0.1

2 0.8

• xk +
• 0.1

0.1

• uk,

(22) and the binary constraint set U = {−1, 1}. The receding horizon control law with R = 0 and P = Q =

• 1

1

• ,

(23) partitions the state space into the regions depicted in the next figure, for constraint horizons N = 2 and N = 3.

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−80 −60 −40 −20 20 40 60 80 −0.5 0.5

1 2 3 4

−80 −60 −40 −20 20 40 60 80 −0.5 0.5

1 2 3 4 5 6 7 8

x1

k

x2

k

x2

k

N = 2 N = 3 R R R R R R R R R R R R

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

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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.

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5.2 Open Loop Unstable Plant

Consider xk+1 =

• 1.02

2 1.05

• xk +
• 0.1

0.1

• 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.

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The following figure illustrates the induced state space partition and a closed loop trajectory, which starts at x = [−10 0]. As can be seen, due to the limited control action available, the trajectory becomes unbounded.

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−40 −20 20 40 60 80 −5 −4 −3 −2 −1 1 2

x1

k

x2

k

R R R R

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

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The situation is entirely different when the initial condition is chosen as x = [0.7 0.2]. As depicted in the following figure, the closed loop trajectory now converges to a bounded region, which contains the origin in its interior. Within that region, the behaviour is not periodic, but appears to be random, despite the fact that the system is deterministic. Neighbouring trajectories diverge due to the action of the unstable poles of the plant. However, the control law manifests itself by maintaining the plant state ultimately bounded.

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−1 −0.5 0.5 1 −0.3 −0.2 −0.1 0.1 0.2 0.3

x1

k

x2

k

R R R R

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

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Application: Quantization of Audio Signals

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

Round to Quantized Levels Analogue Audio

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CD Mastering Stations

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Audio in Quantizer Quantized Output Error Feedback

Noise Shaping Quantizer More Conventional Form (after block diagram manipulation)

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Reformulation as Novel Optimization Problem

H(ρ) Incorporation of a perception filter

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Design Criterion: Finite Horizon Constrained Optimization

1 2( ). k N N t k

V e t

+ − =

=

∑

Perception Filter:

1 1

( ) 1 ,

i i

H h ρ ρ

∞ − =

= + ∑

then the overall perceived error is given by:

( )

( ) ( ) ( ) ( ) . e t H a t u t ρ = −

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Block Optimization

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

T

u k u k u k u k N ⎡ ⎤ = + + − ⎣ ⎦ r

( ) ( )

2 1

( ) ( ) ( ) ( ) .

k N N t k

V u k H a t u t ρ

+ − =

⎛ ⎞ = − ⎜ ⎟ ⎝ ⎠

∑

r

( )

* ( )

( ) arg min ( ) .

N N u k U

u 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)

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Recall the Geometry of the Constrained Optimization Problem

Geometric interpretation of quadratic programming

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After a Simple Transformation

Geometry of finite alphabet optimization as a minimum Euclidean distance problem

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Feedback form of the Solution to Finite Alphabet Control Problem

Convert to State Space

1

( ) 1 ( ) . H C I A B ρ ρ

−

= + −

( ) ( )

( 1) ( ) ( ) ( ) ( ) ( ) ( ) ( ) x t Ax t B a t u t e t Cx t a t u t + = + − = + −

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Theorem

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

• ptimizing sequence is given by:

where:

*( )

u k r

( )

* 1

( ) ( ) ( )

N U

u k q a k x k

−

= Ψ Ψ + Γ

%

r r

1 1 1 1

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

N N

h C a k CA h h a k a k a k N h h h CA

− −

⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ + = Γ = Ψ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ + − ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ K r O M M M M O O K

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Moving Horizon Optimization

Moving horizon Principle, N = 5

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Final MHOQ for Audio Quantization

( )

1

( ) [10 0] ( ) ( )

N U

u k q a k x k

−

= Ψ Ψ + Γ

%

r K Closed form – Vector Quantizer

MHOQ: Moving Horizon Optimal Quantizer

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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

1

1 ( ) H ρ ′ +

1 1

1 ( ) 1 ( ) ( ) C I A B H ρ ρ ρ

−

′ + = + − = H

Perception Filter

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MHOQ with horizon N = 1

Horizon 1 mpc solution to optimal audio quantization

Does this appear familiar?

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Horizon 1 mpc solution The standard noise shaping filter solution (in conventional feedback form)

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Key Observation ☺

Optimization based Audio Quantizer Standard Noise Shaping Quantizer for N = 1 Thus standard noise shaping quantizer is special case of MHOQ.

( ) 1 ( )

( )

H H

F

ρ ρ

ρ

−

=

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Example

Psycho-acoustic studies:

1 2.245 0.664 1 1 2 1 1.335 0.644

( ) 1 H

ρ ρ ρ

ρ ρ

− − − − − − +

= +

Perception Filter:

1 2.245 0.664 1 1 1 0.91

( ) F

ρ ρ

ρ ρ

− − − − +

=

Noise Shaping Filter:

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Frequency responses of H and F

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Music Quantization = MPC

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Effect of Increasing Horizon

Mean Square Quantization Error Optimization Horizon

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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).

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Spectrum of Errors due to Quantization

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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

1

log log

np jw i i

S e dw p

π

π

=

⎛ ⎞ = ⎜ ⎟ ⎝ ⎠

∑ ∫

(pi – unstable poles of H i.e. unstable zeros of 1-F since ).

1 1 F

H

−

=