US 12,517,480 B2Granted

Method and apparatus for M-level control and digital-to-analog conversion

AssigneeETH Zurich

InventorsErik Hampus Niklas Malmberg, Raphael Urs Keusch, Hans-Andrea Loeliger

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Abstract

A method is disclosed for steering a physical analog system (e.g., an electric motor) using a discrete-level (e.g., binary) control signal. The discrete-level control signal is computed by an iterative scheme that can handle a long planning horizon. A preference for infrequent level switches can be taken into account. The quality of the fit to the target trajectory can be expressed not only by the quadratic error, but also by other norms. The method can be used also for digital-to-analog conversion.

Claims18

Dependent claims

A method for controlling an analog physical system with input signal u=(u1, . . . , uK) to follow a given target trajectory y̆=(y̆1, . . . , y̆K), wherein at least one component ( . . . , ) of said input signal is discrete-valued with M levels, M∈{2, 3, . . . }, wherein

uk=(uk,1, . . . , uk,L)∈L is the input signal of the analog physical system at time k∈{1, . . . , K},
K∈ is a planning horizon,
L is the number of input signal components,
(with 1«≤L) is the index of said discrete-valued component of u,
y̆k=(k̆k,1, . . . , k̆k,H)∈H is the given target trajectory at time k,
H is the number of components of the target trajectory,
wherein said method uses an extended system model with output signal {tilde over (y)}=({tilde over (y)}1, . . . , {tilde over (y)}K), {tilde over (y)}k∈{tilde over (H)} for some {tilde over (H)}≥H, and with input signal ũ=(ũ1, . . . , ũK), ũk∈{tilde over (L)} for some {tilde over (L)}≥L, such that
uk=Sũk (81)
for some matrix S and all k∈{1, . . . , K}, and with time-dependent parameters θ=(θ1, . . . , θK) and optional additional parameters ζ=(ζ1, . . . , ζK), wherein ũ=(ũ1, . . . , ũk) is determined by iteratively executing, in iterations i=1, 2, 3, . . . ,
wherein said method comprises the steps, performed by a control unit, of
(a) executing a maximization step in said extended system model with parameters θ fixed to θ(i−1) and ζ fixed to ζ(i−1), thereby obtaining a candidate input signal (ũ1(i), . . . , ũK(i)) with optional precision information (VŨ1(i), . . . , VŨK(i)), and an output signal ({tilde over (y)}1(i), . . . , {tilde over (y)}K(i)), with optional precision information (V{tilde over (Y)}1(i), . . . , V{tilde over (Y)}K(i)), and
(b) determining new parameters θ(i)=(θ1(i), . . . , θK(i)) and optionally ζ(i)=(ζ1(i), . . . , ζK(i)), where ζk(i) is a function of ũk(i) and VŨk(i), and ζk(i) is a function of {tilde over (y)}k(i) and V{tilde over (Y)}k(i), as computed in Step (a),
and wherein said extended system model can be expressed as
p(ũ,ẙ,{tilde over (x)}0,{tilde over (e)}|θ,ζ)∝p(ẙ|ũ,{tilde over (x)}0,{tilde over (e)},ζ)ρ(ũ,θ)p({tilde over (x)}0)p({tilde over (e)}) (82)
wherein
p(ũ,ẙ,{tilde over (x)}0,{tilde over (e)}|0,ζ) is a probability density function in ũ, ẙ, {tilde over (x)}0 and {tilde over (e)} with parameters θ and ζ,
ẙ=(ẙ1, . . . , ẙK), ẙk∈{tilde over (H)}, is the target trajectory of said extended system model comprising y̆ and optionally additional trajectories,
“∝” denotes equality up to a scale factor,
said analog physical system is modeled by a discrete-time linear state space model with recursions
xk=Akxk−1+Bkuk+Dkek (83)
yk=Ckxk (84)
for k∈{1, . . . , K}, and with state vector xk∈N, initial state x0, noise ek∈G, model output yk∈H, and matrices Ak, Bk, Ck, Dk,
said method uses an extended version of model of eq. 83 and eq. 84 which is given by the state space recursions
{tilde over (x)}k=Ãk{tilde over (x)}k−1+{tilde over (B)}kSũk+{tilde over (D)}k{tilde over (e)}k (85)
{tilde over (y)}k={tilde over (C)}k{tilde over (x)}k (86)
for k∈{1, . . . , K}, and with state vector xk∈Ñ with Ñ≥N, initial state {tilde over (x)}0, noise {tilde over (e)}k ∈{tilde over (G)} with {tilde over (G)}≥G, and wherein the matrices Ãk, {tilde over (B)}k, {tilde over (C)}k and {tilde over (D)}k are derived from Ak, Bk, Ck and Dk,
p(ẙ|ũ,{tilde over (x)}0,{tilde over (e)},ζ) is given by
p
⁡
(
y
∘
❘
u
~
,
x
~
0
,
e
~
,
ζ
)
∝
exp
⁡
(
-
1
2
⁢
∑
k
=
1
K
w
k
(
y
∘
k
-
y
~
k
)
T
⁢
W
k
(
y
∘
k
-
y
~
k
)
)
⁢
ψ
⁡
(
y
~
,
ζ
)
(
87
)
where Wk is a real non-negative weight matrix and where {tilde over (y)}k is the extended system model output signal determined by eq. 85 and eq. 86,
ψ({tilde over (y)},ζ) is an optional factor in eq. 87, which means it is either 1, or if present, it may take the form
ψ
⁡
(
y
~
,
ζ
)
=
∏
k
=
1
K
ψ
k
(
y
~
k
,
ζ
k
)
(
88
)
such that, for fixed ζk, ψk({tilde over (y)}k,ζk) is a Gaussian probability density function in {tilde over (y)}k, up to a scale factor,
p({tilde over (x)}0) is a Gaussian probability density,
p({tilde over (e)}) factors as
p
⁡
(
e
~
)
=
∏
k
=
1
K
p
⁡
(
e
~
k
)
(
89
)
where each p({tilde over (e)}k) is a Gaussian probability density,
ρ(ũ,θ) factors as
ρ
⁡
(
u
~
,
θ
)
=
∏
k
=
1
K
ρ
k
(
u
~
k
,
θ
k
)
(
90
)
such that,
for fixed θk, ρk(ũk,θk) is a Gaussian probability density function in ũk, up to a scale factor, and
for at least one index {tilde over (l)}∈{1, . . . , {tilde over (L)}}, ρk(ũk,θk) contains as a factor the product
(ũk,{tilde over (l)};m1,σk,12) . . . (ũk,{tilde over (l)};mJ,σk,J2) (91)
of J≥2 Gaussian probability density functions with different means m1, . . . , mJ and with variances σk,12, . . . , σk,J2 determined by θk.

180 more elements

Dependencies are inferred from the claim text.

Facts

Application: 17/928,593Filed 26 Jan 2021
Priority: WO PCT/EP2020/06621811 Jun 2020
Published: US 2023/0205149 A129 Jun 2023
Granted: 6 Jan 2026
Term ends: Not estimatedThe source records don’t support an estimate.

Classification5 codes

G05B 13/048
Involving the use of models or simulators: using a predictor
G05B 17/02
Systems involving the use of models or simulators of said systems: electric
H03M 1/661
Digital/analogue converters: improving the reconstruction of the analogue output signal beyond the resolution of the digital input signal, e.g. by…
G05B
Control or regulating systems in general

Family

US 2023/0205149 A1Pre-grant publication · published 29 Jun 2023
US 12,517,480 B2This patent

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Abstract

uk=(uk,1, . . . , uk,L)∈L is the input signal of the analog physical system at time k∈{1, . . . , K},
K∈ is a planning horizon,
L is the number of input signal components,
(with 1«≤L) is the index of said discrete-valued component of u,
y̆k=(k̆k,1, . . . , k̆k,H)∈H is the given target trajectory at time k,
H is the number of components of the target trajectory,
wherein said method uses an extended system model with output signal {tilde over (y)}=({tilde over (y)}1, . . . , {tilde over (y)}K), {tilde over (y)}k∈{tilde over (H)} for some {tilde over (H)}≥H, and with input signal ũ=(ũ1, . . . , ũK), ũk∈{tilde over (L)} for some {tilde over (L)}≥L, such that
uk=Sũk (81)
for some matrix S and all k∈{1, . . . , K}, and with time-dependent parameters θ=(θ1, . . . , θK) and optional additional parameters ζ=(ζ1, . . . , ζK), wherein ũ=(ũ1, . . . , ũk) is determined by iteratively executing, in iterations i=1, 2, 3, . . . ,
wherein said method comprises the steps, performed by a control unit, of
(a) executing a maximization step in said extended system model with parameters θ fixed to θ(i−1) and ζ fixed to ζ(i−1), thereby obtaining a candidate input signal (ũ1(i), . . . , ũK(i)) with optional precision information (VŨ1(i), . . . , VŨK(i)), and an output signal ({tilde over (y)}1(i), . . . , {tilde over (y)}K(i)), with optional precision information (V{tilde over (Y)}1(i), . . . , V{tilde over (Y)}K(i)), and
(b) determining new parameters θ(i)=(θ1(i), . . . , θK(i)) and optionally ζ(i)=(ζ1(i), . . . , ζK(i)), where ζk(i) is a function of ũk(i) and VŨk(i), and ζk(i) is a function of {tilde over (y)}k(i) and V{tilde over (Y)}k(i), as computed in Step (a),
and wherein said extended system model can be expressed as
p(ũ,ẙ,{tilde over (x)}0,{tilde over (e)}|θ,ζ)∝p(ẙ|ũ,{tilde over (x)}0,{tilde over (e)},ζ)ρ(ũ,θ)p({tilde over (x)}0)p({tilde over (e)}) (82)
wherein
p(ũ,ẙ,{tilde over (x)}0,{tilde over (e)}|0,ζ) is a probability density function in ũ, ẙ, {tilde over (x)}0 and {tilde over (e)} with parameters θ and ζ,
ẙ=(ẙ1, . . . , ẙK), ẙk∈{tilde over (H)}, is the target trajectory of said extended system model comprising y̆ and optionally additional trajectories,
“∝” denotes equality up to a scale factor,
said analog physical system is modeled by a discrete-time linear state space model with recursions
xk=Akxk−1+Bkuk+Dkek (83)
yk=Ckxk (84)
for k∈{1, . . . , K}, and with state vector xk∈N, initial state x0, noise ek∈G, model output yk∈H, and matrices Ak, Bk, Ck, Dk,
said method uses an extended version of model of eq. 83 and eq. 84 which is given by the state space recursions
{tilde over (x)}k=Ãk{tilde over (x)}k−1+{tilde over (B)}kSũk+{tilde over (D)}k{tilde over (e)}k (85)
{tilde over (y)}k={tilde over (C)}k{tilde over (x)}k (86)
for k∈{1, . . . , K}, and with state vector xk∈Ñ with Ñ≥N, initial state {tilde over (x)}0, noise {tilde over (e)}k ∈{tilde over (G)} with {tilde over (G)}≥G, and wherein the matrices Ãk, {tilde over (B)}k, {tilde over (C)}k and {tilde over (D)}k are derived from Ak, Bk, Ck and Dk,
p(ẙ|ũ,{tilde over (x)}0,{tilde over (e)},ζ) is given by
p
⁡
(
y
∘
❘
u
~
,
x
~
0
,
e
~
,
ζ
)
∝
exp
⁡
(
-
1
2
⁢
∑
k
=
1
K
w
k
(
y
∘
k
-
y
~
k
)
T
⁢
W
k
(
y
∘
k
-
y
~
k
)
)
⁢
ψ
⁡
(
y
~
,
ζ
)
(
87
)
where Wk is a real non-negative weight matrix and where {tilde over (y)}k is the extended system model output signal determined by eq. 85 and eq. 86,
ψ({tilde over (y)},ζ) is an optional factor in eq. 87, which means it is either 1, or if present, it may take the form
ψ
⁡
(
y
~
,
ζ
)
=
∏
k
=
1
K
ψ
k
(
y
~
k
,
ζ
k
)
(
88
)
such that, for fixed ζk, ψk({tilde over (y)}k,ζk) is a Gaussian probability density function in {tilde over (y)}k, up to a scale factor,
p({tilde over (x)}0) is a Gaussian probability density,
p({tilde over (e)}) factors as
p
⁡
(
e
~
)
=
∏
k
=
1
K
p
⁡
(
e
~
k
)
(
89
)
where each p({tilde over (e)}k) is a Gaussian probability density,
ρ(ũ,θ) factors as
ρ
⁡
(
u
~
,
θ
)
=
∏
k
=
1
K
ρ
k
(
u
~
k
,
θ
k
)
(
90
)
such that,
for fixed θk, ρk(ũk,θk) is a Gaussian probability density function in ũk, up to a scale factor, and
for at least one index {tilde over (l)}∈{1, . . . , {tilde over (L)}}, ρk(ũk,θk) contains as a factor the product
(ũk,{tilde over (l)};m1,σk,12) . . . (ũk,{tilde over (l)};mJ,σk,J2) (91)
of J≥2 Gaussian probability density functions with different means m1, . . . , mJ and with variances σk,12, . . . , σk,J2 determined by θk.

180 more elements

Facts

Application

17/928,593Filed 26 Jan 2021

Priority

WO PCT/EP2020/06621811 Jun 2020

Published

US 2023/0205149 A129 Jun 2023

Granted

6 Jan 2026

Term ends

Not estimatedThe source records don’t support an estimate.

Classification5 codes

G05B 13/048
Involving the use of models or simulators: using a predictor
G05B 17/02
Systems involving the use of models or simulators of said systems: electric
H03M 1/661
Digital/analogue converters: improving the reconstruction of the analogue output signal beyond the resolution of the digital input signal, e.g. by…
G05B
Control or regulating systems in general