US 11,308,180 B1Granted

Fractional order dynamic inversion controller for non-linear systems

AssigneeKing Abdulaziz University

InventorsIbrahim M. Mehedi, Ubaid M. Al-Saggaf, Md. Sarowar Alam, Mohd Heidir bin Mohd Shah, Md. Masud Rana

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Abstract

A device, method, and non-transitory computer readable medium for fractional order dynamic inversion control of nonlinear systems under the influence of a disturbance. The method for fractional order dynamic inversion control includes receiving a mathematical model of a controlled device, the mathematical model including a set of movement trajectories, wherein the movement trajectories include control objectives, formulating a first set of linear, time-varying, fractional order differential equations from the mathematical model, the first set including dynamic constraints on the control objectives, transforming the first set into linear equations by differentiating a norm measure function along the movement trajectories, thus generating a second set of fractional order differential equations, and determining a control signal for controlling the movement trajectories in the presence of disturbances by dynamically inverting the second set of fractional order differential equations.

Claims19

Dependent claims

A method for fractional order dynamic inversion control, comprising:

receiving a mathematical model of a controlled device, the mathematical model including a set of movement trajectories, wherein the movement trajectories include control objectives;
formulating a first set of linear, time-varying, fractional order differential equations from the mathematical model, the first set including dynamic constraints on the control objectives;
transforming the first set into linear equations by differentiating a norm measure function along the movement trajectories, thus generating a second set of fractional order differential equations;
formulating the first set of linear, time-varying, fractional order differential equations based on:
d
α
dt
α
⁢
ξ
+
c
⁡
(
t
)
⁢
ξ
=
0
,
where 0<α≤1 and
d
α
⁢
ξ
dt
α
=
2
⁢
ne
⁢
⁢
(
F
⁡
(
x
)
+
G
⁡
(
x
)
⁢
u
-
d
α
dt
α
⁢
y
meas
)
,
where ξ is an error function, c(t) is a constraint at time t, n is a positive real valued constant, e is an error in a position of the controlled device, F(x) is a force applied to move the controlled device along the movement trajectories, G(x) is a frictional force, u is an input vector, α is a fractional order of a derivative
d
α
dt
α
,
and ymeas is an output vector;
wherein uc is the dynamic constraints on the control objectives of a linear servo cart given by:
u
c
=
G
+
⁡
(
F
⁡
(
x
)
+
c
⁡
(
t
)
⁢
e
-
d
α
dt
a
⁢
y
meas
)
,
where G+ is a real part of the frictional force;
determining a control signal for controlling the movement trajectories in the presence of disturbances by dynamically inverting the second set of fractional order differential equations;
and
controlling the movement trajectories of the cart with the control signal.

97 more elements

A system for fractional order dynamic inversion control of a controlled device, comprising:

a controlled device configured to move along a set of movement trajectories based on control objectives;
a motor configured to generate a force which moves the controlled device along the movement trajectories;
a computing device connected to the motor, the computing device having circuitry and program instructions, which when executed by at least one processor, are configured to:
receive a mathematical model of the controlled device, the mathematical model including the set of movement trajectories, wherein the set of movement trajectories include the control objectives;
formulate a first set of linear, time-varying, fractional order differential equations from the mathematical model, the first set including dynamic constraints on the control objectives;
transform the first set into linear equations by differentiating a norm measure function along the movement trajectories, thus generating a second set of fractional order differential equations;
formulate the first set of linear, time-varying, fractional order differential equations based on:
d
α
dt
α
⁢
ξ
+
c
⁡
(
t
)
⁢
ξ
=
0
,
where 0<α≤1 and
d
α
⁢
ξ
dt
α
=
2
⁢
ne
⁡
(
F
⁡
(
x
)
+
G
⁡
(
x
)
⁢
u
-
d
α
dt
α
⁢
y
meas
)
,
where ξ is an error function, c(t) is a constraint at time t, n is a positive real valued constant, e is an error in a position of the controlled device, F(x) is a force applied to move the controlled device along the movement trajectories, G(x) is a frictional force, u is an input vector, α is a fractional order of a derivative
d
α
dt
α
,
and ymeas is an output vector;
wherein uc is the dynamic constraints on the control objectives of a linear servo cart given by:
u
c
=
G
+
⁡
(
F
⁡
(
x
)
+
c
⁡
(
t
)
⁢
e
-
d
α
dt
α
⁢
y
meas
)
,
where G+ is a real part of the frictional force;
and
determine a control signal for controlling the movement trajectories in the presence of disturbances by dynamically inverting the second set of fractional order differential equations.

98 more elements

A non-transitory computer readable medium having program instructions stored therein that, when executed by one or more processors, cause the one or more processors to perform a method for fractional order dynamic inversion control, comprising:

receiving a mathematical model of a controlled device, the mathematical model including a set of movement trajectories, wherein the movement trajectories include control objectives;
formulating a first set of linear, time-varying, fractional order differential equations from the mathematical model, the first set including dynamic constraints on the control objectives;
transforming the first set into linear equations by differentiating a norm measure function along the movement trajectories, thus generating a second set of fractional order differential equations;
formulating the first set of linear, time-varying, fractional order differential equations based on:
d
α
dt
α
⁢
ξ
+
c
⁡
(
t
)
⁢
ξ
=
0
,
where 0<α≤1 and
d
α
⁢
ξ
dt
α
=
2
⁢
ne
⁡
(
F
⁡
(
x
)
+
G
⁡
(
x
)
⁢
u
-
d
α
dt
α
⁢
y
meas
)
,
where ξ is an error function, c(t) is a constraint at time t, n is a positive real valued constant, e is an error in a position of the controlled device, F(x) is a force applied to move the controlled device along the movement trajectories, G(x) is a frictional force, u is an input vector, α is a fractional order of a derivative
d
α
dt
α
,
and ymeas is an output vector;
wherein uc is the dynamic constraints on the control objectives of a linear servo cart given by:
u
c
=
G
+
⁡
(
F
⁡
(
x
)
+
c
⁡
(
t
)
⁢
e
-
d
α
dt
α
⁢
y
meas
)
,
where G+ is a real part of the frictional force;
and
determining a control signal for controlling the movement trajectories in the presence of disturbances by dynamically inverting the second set of fractional order differential equations;
controlling the movement trajectories of the cart with the control signal.

96 more elements

Dependencies are inferred from the claim text.