DE19830156A1 - Method and arrangement for determining a controlled variable of a technical system, which is described with a given model description in a given room - Google Patents

Abstract

The invention relates to a method and device for determining a controlled variable of a technical system, whereby said is described by a predetermined model description in a predetermined area. According to the inventive method, the model description of the technical system is transformed into a sub-zone of said area wherein a controller model description is determined using a non-linear controller model description on the basis of the transformed model description. The controller model description is transferred back to model description area. The controlled variable is determined using the re-transferred controller model description.

Description

The invention relates to the computer-based determination of a Control variable of a technical system, which with a given model description in a given space is written.

It is known from [1, 2, 3] to use a continuous model to describe a system of a traffic flow. The following state variables are used to describe a state of the system:

• - Traffic flow speed v
• - Vehicle density ρ (ρ = number of vehicles Fz / km)
• - Traffic flow q (q = number of vehicles Fz / h], q = v.ρ).

There is also a means, for example one in a roadway incorporated conductor loop with a counter and a Evaluation unit is known, with which the state quantities (v, ρ, q) of the system of traffic flow measured can be.

The model known from [1, 2, 3] describes, starting from a static relationship between an equilibrium speed V _{eq of} the traffic flow (V _eq = static traffic flow speed in a stationary state of the traffic flow) and the vehicle density ρ, the traffic flow in an equilibrium state.

The following relationship applies:

With:

w _i or ρ _i : freely selectable mapping parameters
l _i or m _i : freely selectable imaging parameters
i: run variable
V _eq or ρ: equilibrium speed or vehicle density.

It is also known from [1, 2, 3] that both the traffic flow velocity v and the vehicle density ρ depend from a location x and from a time t corresponding to the together menhang v = v (x, t) or ρ = ρ (x, t) [x: location variable, t : Time variable] vary.

The model is described by a Continuity equation (equation 2) and an acceleration equation (equation 3) expanded.

The continuity equation (Equation 2), according to the relationship

With:
q: traffic flow
d / dt or d / dx: a partial derivative according to time t or location x
describes the dynamics of the traffic flow under the condition that the traffic flow has a continuous flow without an entry and exit of a vehicle from the system.

The acceleration equation (equation 3) describes the dynamics of the traffic flow outside the equilibrium state given by the static equilibrium speed according to equation 1 by the following relationship:

With:
τ: relaxation time
c ₀ ² : speed variance
η ₀ : viscosity constant
d / dt, d / dx, d ² / dx ² : a partial derivative after the time t or a partial first and a partial second derivative after the location x.

Based on the description of the flow of traffic through a such a model provides a stability analysis of the model characteristic properties of the be written traffic flow.

A local stability analysis of the model shown above by linearization around a stationary working point (v ₀ , ρ ₀ ) shows that the unregulated traffic flow according to the model for a vehicle density ρ in a range [approx. 20 vehicles / km - approx. 50 vehicles / km] exhibits unstable behavior. A disturbance in the flow of traffic increases and leads to conditions which can be observed in real traffic situations, such as a sudden standstill of the flow of traffic (traffic jam) or a "stop-and-go wave".

In the area of the vehicle density ρ [ρ <20 Fz / km] and in the area the system has a sta of the vehicle density ρ [ρ <50 Fz / km] bile behavior.

A distinction is made between the following areas:
ρ <20 vehicles / km: low traffic, high speed, stable behavior
20 vehicles / km <ρ <50 vehicles / km: unstable behavior, small disturbances rock up
ρ <50 vehicles / km: high volume of traffic, slow-moving traffic or traffic jams, stable behavior.

It is also known to use a method of control engineering Traffic flow model to apply a regulated and stable traffic flow in the entire state area of traffic to ensure flow.

From [4] it is known to regulate by a linear zu realizing status feedback. So the Ver traffic flow in a state in which the traffic flow is an in exhibits stable behavior, stabilize and a homogeneous Flow of traffic is ensured.

However, the linear approach from [4] has several disadvantages. Stabilization of the traffic flow is only possible for a small disturbance of the traffic flow or only in a small area (Δv, Δρ) of the state space (v, ρ, q) around the working point (v ₀ , ρ ₀ ) of the linearization. Furthermore, the regulation provides a controlled variable through a linear state feedback, which cannot be applied to the real traffic flow due to the size of its value.

Various methods of non-linear control are from [5] lung technology known. It is also shown in [5] that on due to its robustness with regard to a fault a structurally variable controller is used. To determine the parameters of the structure riablen controller in [5] the method of equivalent Re gelation used.

It is also known that a regulated traffic flow model can be used to control the real traffic flow. State variables of a real traffic situation are measured for this purpose. These state variables are applied to the control system, the control system determining a controlled variable, such as the traffic flow velocity v _, for example. Using a display means, such as an alternating traffic sign of a traffic control system, this controlled variable is given to the traffic flow, in accordance with the example above, a target speed.

The invention is based on the problem of a computer-aided method for determining a controlled variable of a technical to specify the system in which the regulated technical cal system the technical system is stabilized and at which the controlled variable is applicable to the technical system.

The problem is solved by the method according to claim 1 and solved by the arrangement according to claim 13.

In the method according to claim 1, a controlled variable a technical system, which with a given Mo dell description is described in a given space, certainly. To do this, the model description is placed in a subspace of space transformed. In this subspace, the transformed model description using a nonlinear controller model be a controller model description Right. This controller model description is in the original transformed space of the model description. Under Use of the back-transformed controller model description the controlled variable is determined.

The arrangement according to claim 13 for determining a controlled variable of a technical system, which is written with a given model description in a given space, comprises a processor which is set up such that the following steps can be carried out:

• - Transformation of the model description into a subspace of space;  
• - Determination of a controller model description from the transfor model description using a specifiable non-linear controller model;
• - Back transformation of the controller model description into the room the model description;
• - Determination of the controlled variable using the reverse trans formed controller model description.

The method and the arrangement ensure that a Controlled variable of a technical system is determined, the regulated technical system stabilizes a fault, and that the controlled variable has such a value that the Control variable on the basis of the technical system real system is applicable.

Advantageous developments of the invention result from the dependent claims.

In a further development, it is advantageous to use the invention Use regulation of the technical system. With that a Disturbance of the technical system can be stabilized so that the technical system in the entire state space (v, ρ, q) has stable behavior.

In a further embodiment of the invention, this is techni system a traffic flow. This makes it possible for Ver regulate the flow of traffic so that it is homogeneous and trouble-free State of traffic flow is reached.

In a development of the invention, it is advantageous to represent the flow of traffic using the following relationship:

With:

w _i or ρ _i : freely selectable mapping parameters
l _i or m _i : freely selectable imaging parameters
i: run variable
V _eq or ρ: equilibrium speed or vehicle density.

The relationship shown above is a suitable model of the real system of homogeneous traffic flow and is suitable thus especially to regulate the system.

To the location and / or time dependence of the state variables ei It must be considered as a further development of the traffic flow tion of the invention advantageous, the flow of traffic through a Continuity equation and / or an acceleration equation to describe.

In a further development of the invention, it is advantageous to use the following relationship to determine the continuity equation

With:
q: traffic flow
d / dt or d / dx: a partial derivative according to time t or location x,
and / or represent the acceleration equation by the following relationship:

With:
τ: relaxation time
c ₀ ² : speed variance
η ₀ : viscosity constant
d / dt, d / dx, d ² / dx ² : a partial derivative after the time t or a partial first and a partial second derivative after the location x.

The relationships shown above are a good model for the location and time dependency of the state variables of the real system of traffic flow and are therefore suitable especially to regulate the system.

A particularly simple method results in a white Training of the invention when the transformation in Un terraum of the room is carried out in that several Di dimensions of the space of the space of the technical system on egg ne dimension of the subspace can be reduced.

It is particularly advantageous in an embodiment of the Er finding, the non-linear controller model by a non-linear describe near structure-variable controllers. This will the robustness against a disturbance increases and a good one Control behavior guaranteed.

Because of the simple method, it is preferred to use a Development of the invention a method of an equivalent Scheme for the design of the non-linear structure variable Controller used.

It is particularly advantageous in the context of a Use traffic management system, because it is a homogeneous and stable traffic flow of the real system can be achieved can. The controlled variable can be done with the help of a display and / or a variable that can be determined from the controlled variable turn participants are displayed.

Embodiments of the invention are shown in FIGS. 1 to 3 and are explained in more detail below.

Show it:

Fig. 1 Schematic representation of a real system of a traffic flow

Fig. 2 Schematic representation of the development of a non-linear control system for the traffic flow system

Fig. 3 control of a real traffic flow system.

In Fig. 1, a real system of a traffic flow is shown schematically.

On an observed route section 101 of a route, vehicles 102 are moved in their direction of travel 106 by their respective drivers 103 .

State variables of the system are measured at a predetermined location, a measuring point 104 , within the observed route section 101 .

For this purpose, a conductor loop 105 is worked into a carriageway 109 , the number i _{Fz of} the vehicles 102 that cross the measuring point 104 within a predetermined time period Δt, and the respective speed v _{iFz of} the vehicle 102 that crosses the measuring point 104 , measures.

The measured values (i _Fz , v _iFz ) are transmitted to an evaluation unit 107 coupled to the conductor _loops 105 . The evaluation unit 107 determines a _target speed v _Soll 108 as a function of the transmitted variables, which is displayed to the traffic participants using a traffic control system 110 which is coupled to the evaluation unit 107 .

In Fig. 2, the development of a non-linear Regelsy system for the traffic flow system is shown schematically.

1. Model description of the traffic flow system in the state space (step 201 )

The model description (step 201 ) of the traffic flow system in the state space is carried out by:

With:
w _i or ρ _i : freely selectable mapping parameters
l _i or m _i : freely selectable imaging parameters
i: run variable
V _eq or ρ: equilibrium speed or vehicle density
in which:
w ₁ = 100 km / h or w ₂ = 10 km / h
ρ ₁ = 100 vehicles / km or ρ ₂ = 160 vehicles / km
l ₁ = 3.2 or l ₂ = 2
m ₁ = 0.9 or m ₂ = 0 is set.

For a free speed v _free in the limit (ρ → 0) the following applies: v _free = w ₁ + w ₂ = 110 km / h.

For ρ <ρ ₁ , w ₁ = 0 applies to prevent an increase in the V _eq (ρ) relationship.

The location and time dependence (x, t) of the state variable speed v = v (x, t) and the state variable ρ = ρ (x, t) are taken into account by the continuity equation (equation 2) and the acceleration equation (equation 3)

With:
q: traffic flow
d / dt or d / dx: a partial derivative according to time t or location x,

With:
τ: relaxation time
c ₀ ² : speed variance
η ₀ : viscosity constant
d / dt, d / dx, d ² / dx ² : a partial derivative after the time t or a partial first and a partial second derivative after the location x,
in which:
τ ₀ = 6 s or c ₀ = 13.31 m / s or η ₀ = 59.33 m / s.

The effect of a speed restriction on the traffic flow is described by scaling Equation 1:

V _eq (ρ, u) = (1 + u) V _eq (ρ) (Equation 4)

With:
u: controller output variable
uV _eq (ρ): controlled variable
vfree (1 + u): maximum speed displayed

2. Transformation of the model description into the subspace (step 202 )

For the transformation of the model description into the subspace, a collective coordinate z (equation 5) with:

z = x - v _s .t (equation 5)

introduced, where v _s indicates the speed of a solitary wave. This solitary wave is an asymptotic solution to model equations 1, 2 and 3, which waves have a constant profile and propagate at a constant speed v _s .

The transformed model description (step 203 ) (equation 6) for a solitary wave results in:

With:

: a partial derivative of the first or second Order of traffic flow speed the collective coordinate z

The transformed continuity equation (equation 7) provides the constant flow q ₀ as a secondary condition (equation 8):

ρ (vv _s ) = q ₀ = const. (Equation 8).

3. Determination of the controller model description using a nonlinear structure-variable controller (step 204 )

To regulate the transformed model description is on due to the control properties, a non-linear structural variant only controller used [5].

For this purpose, the transformed model description (equation 6) is shown taking equation 4 into account as follows:

With:
f (v, dv / dz) or b (v, dv / dz): mapping rules.

The design of the nonlinear structure variable controller he follows using the equivalent control method [5].

The rule law (equation 12) is:
u = u _e + u _n (Equation 12)
With:
u: controller output variable
u _e , u _n : equivalent or non-continuous part of the controller output variable.

The following also applies:
s = λv + dv / dz (Equation 13)
V _L (s) = (1/2) s ² (Equation 14)
With:
s: switching variable
λ: system parameters, λ <0
V _L : Lyapunov-like function
V _L (s): mapping rule.

The switching variables s are selected such that the system is stable for s = 0 (sliding state).

The controller output variable u is determined so that the derivative of the Lyapunow-like function V _L is negative according to the collective coordinate z:

dV _L / dz <0. (Equation 15)

The sliding state s = 0 is equivalent to ds / dz = 0 be wrote.

Taking into account the scaling (equation 4) and the transformed model description (equation 6), the equivalent part of the controller output variable u _{e is represented} as follows:

The non-continuous part of the controller output variable u _n is shown as follows:

With:
K: system parameters, K <0.

This provides a controlled system in the subspace (step 205 ).

4. Transform the controller model description back into the state of the system (step 206 )

For the inverse transformation (step 206 ), the non-continuous portion of the controller output variable u _{n is} neglected.

The back transformation results in:

Neglecting the acceleration term dv / dt, which is usually not measured in practice, results in:

The regulated traffic flow system in the original space of the technical system (step 207 ) is thus described by the following relationships (equations 20, 2 and 21):

V _eq (ρ, u) = (1 + u _e ) V _eq (ρ) = v, (Equation 20)

A local stability analysis of the controlled system in the original room shows the following properties of the controlled system:
In the entire state space of the technical system, the controlled system exhibits stable behavior with regard to any faults.

The homogeneous and stable state of the controlled system (ρ _hom , q _hom , v _hom ), which results from the non-linear and structure-variable control, corresponds to the spatially averaged initial conditions of the system variables (ρ, q, v).

The controlled variable supplies maximum values (maximum control intervention approx. 25 km / h) which apply to the real traffic flow system are cash.

In Fig. 3 is shown schematically how using the regulated model of the traffic flow system, the real Sy stem traffic flow is homogenized.

At a predetermined location 301 of an observed traffic flow 302 , the state variables (ρ, q, v) of the traffic flow 302 are measured at predetermined time intervals Δt. The measurement is started at a predeterminable time t = 0 s.

The measured output state variables of the real system are ρ _start , q _start , v _start .

The measured state variables (ρ, q, v) are applied to the regulated model of the system. If the real system malfunctions, the measured state variables change (ρ _Stör , q _Stör , v _Stör )

The controlled model determines the controlled variable v _{target as a} function of the current state variables of the system (ρ _Stör , q _Stör , v _Stör ) and the output state variables (ρ _Start , q _Start , v _Start ).

This is displayed to a traffic participant 304 with the aid of a traffic control system 303 . At a point in time t ₁ , the real system again reaches the stable initial state (ρ _Start , q _Start , v _Start ).

Some alternatives to the invention are set out below ben:

An alternative approach to speed balance is:

The acceleration equation can also be replaced by another Approach to be replaced provided the characteristic eigen like instability in the medium density range and that Occurrence of a solitary wave as an asymptotic solution is guaranteed.  

The following publications have been cited in this document:
[1]: Kerner, BS, et al., "Structure and parameters of clusters in traffic flowl", Phys. Rev. E50 (1), pp. 54-83, 1994.
[2]: Kuehne, R., Pal, SK, "Influencing Road Traffic and Physics of Phase Transitions", Physik in heute, 15th year, No. 3, pp. 84-92, 1984.
[3]: Zackor, H., et al., "Investigations of the course of traffic in the area of performance and in the event of an unstable river", Research Road Construction and Road Traffic Technology, Issue 524, 1988.
[4]: Cremer, M., et al., "Use of control technology to improve the flow of traffic on expressways", Research Road Construction and Traffic Engineering, Issue 307, 1980.
[5]: Lenz, H., Berstecher, R., Lang, M., "Adaptive Sliding-Mode Control of the Absolute Gain", IFAC Nonlinear Oontrol Systems Design Symposium, Enschede, Netherlands, 1998.

Claims (30)

1. A method for determining at least one controlled variable of a technical system, which is described with a given model description in a given room, comprising the following steps:

• - transformation of the model description into a subspace of the room;
• - Determination of a controller model description from the transformed model description using a predefinable non-linear controller model;
• - Back transformation of the controller model description into the space of the model description;
• - Determination of the controlled variable using the back-transformed controller model description.

2. The method according to claim 1, wherein several control variables be be true. 3. The method according to claim 1 or 2, used for control of the technical system. 4. The method according to any one of claims 1 to 3, in which the technical system describes a traffic flow. 5. The method of claim 4, wherein the flow of traffic is described by the following relationship:
With:
w _i or ρ _i : freely selectable mapping parameters
l _i or m _i : freely selectable imaging parameters
i: run variable
V _eq or ρ: equilibrium velocity or Vehicle density. 6. The method according to claim 4 or 5, wherein the traffic flow is described by a continuity equation. 7. The method of claim 6, wherein the continuity equation has the following relationship:
With:
q: traffic flow
d / dt or d / dx: a partial derivative according to time t or location x. 8. The method according to any one of claims 4 to 7, wherein the Traffic flow described by an acceleration equation becomes. 9. The method of claim 8, wherein the acceleration equation has the following relationship:
With:
τ: relaxation time
c ₀ ² : speed variance
η ₀ : viscosity constant
d / dt, d / dx, d ² / dx ² : a partial derivative after the time t or a partial first and a partial second derivative after the location x. 10. The method according to any one of claims 1 to 9, wherein the Transformation is carried out in that dimensions of the Space of the technical system to a dimension of the sub of the room. 11. The method according to any one of claims 1 to 10, wherein the nonlinear controller model a nonlinear structural variant blen controller describes. 12. The method according to claim 11, in which to design the nonlinear structure variable controller a method of a equivalent regulation is used. 13. The method according to claim 1 to 12, in which a traffic participants the controlled variable and / or one of the controlled variable determinable size is displayed. 14. Arrangement for determining at least one controlled variable of a technical system which is described with a predefined model description in a predefined space, which arrangement comprises a processor which is set up in such a way that the following steps can be carried out:

• - transformation of the model description into a subspace;
• - Determination of a controller model description from the transformed model description using a predefinable non-linear controller model;
• - Back transformation of the controller model description into the space of the model description;
• - Determination of the controlled variable using the back-transformed controller model description.

15. An arrangement according to claim 14, wherein the processor is such is set up that several controlled variables can be determined. 16. The arrangement of claim 14 or 15, wherein the processor is set up to regulate the technical system.   17. The arrangement according to one of claims 14 to 16, wherein the Processor is set up so that the technical system describes a traffic flow. 18. The arrangement according to claim 17, wherein the processor is set up such that the traffic flow is described by the following relationship:
With:
w _i or ρ _i : freely selectable mapping parameters
l _i or m _i : freely selectable imaging parameters
i: run variable
V _eq or ρ: equilibrium speed or vehicle density. 19. The arrangement of claim 17 or 18, wherein the processor is set up so that the flow of traffic through a con continuity equation is described. 20. The arrangement according to claim 19, wherein the processor is set up in such a way that the continuity equation has the following relationship:
With:
q: traffic flow
d / dt or d / dx: a partial derivative according to time t or location x. 21. Arrangement according to one of claims 17 to 20, wherein the Processor is set up so that the flow of traffic through an acceleration equation is described.   22. The arrangement according to claim 21, wherein the processor is set up such that the acceleration equation has the following relationship:
With:
τ: relaxation time
c ₀ ² : speed variance
η ₀ : viscosity constant
d / dt, d / dx, d ² / dx ² : a partial derivative after the time t or a partial first and a partial second derivative after the location x. 23. Arrangement according to one of claims 14 to 22, wherein the Processor is set up so that the transformation there is feasible that dimensions of the space of the techni system towards a dimension of the subspace of the room be returned. 24. The arrangement according to one of claims 14 to 23, wherein the Processor is set up so that the non-linear Reg a non-linear structure-variable controller writes. 25. The arrangement of claim 24, wherein the processor is such is set up to design the nonlinear structure variable controller is a method of equivalent control is used. 26. Arrangement according to one of claims 14 to 24, with a Display means.   27. The arrangement according to one of claims 14 to 26, wherein the Processor is set up so that a traffic participant always determine the controlled variable and / or one from the controlled variable bar size is displayed. 28. A set of multiple arrangements according to claim 26 or 27 used in a traffic control system.