DE102022209555A1 - Method and control unit for controlling a mechatronic system - Google Patents

Abstract

The approach presented here relates to a method for controlling a mechatronic system (100), the method (400) having a step of reading in (410) of setpoint values (c(x)) for physical variables of the mechatronic system (100), the Setpoint values (c(x)) have been changed over time and, in response to the reading, determining or reading in current values (x) of the physical variables of the mechatronic system (100). Furthermore, the method (400) has a step of integrating (420) an optimization functional (140, J) which is dependent on the setpoint values (c(x)) and the current values (x) over time in order to obtain control values (x*) for to obtain the physical quantities. Finally, the method (400) includes a step of outputting (430) a control signal (150) for controlling the mechatronic system (100) using the determined control values (x*) for the physical variables.

Description

The present invention relates to a method and a control unit for controlling a mechatronic system according to the main claims.

• (i) die Bestimmung von Look-up-Tabellen entweder durch Offline-Berechnungen oder durch Messungen in der realen Welt oder
• (ii) es kann ein nichtlineares Programmierproblem mit Gleichheitsbeschränkungen formuliert werden als:

$$\text{min }J\left(x\right)$$

$$\text{s}\text{.t}\text{. }{c}_1\left(x\right)=\mathrm{0,}$$

$$c_2\left(x\right)=\mathrm{0,}$$

$$c_m\left(x\right)=\mathrm{0,}$$

$$x\in X\subset {ℝ}^2,$$

mit m ≤ n und stetigen Funktionen J, h_i, i = 1, 2, ... , m. Die notwendigen Bedingungen erster Ordnung für die Optimalität lassen sich wie folgt zusammenfassen:Different approaches can be used to optimally control dynamic systems, such as:

• (i) the determination of look-up tables either by offline calculations or by measurements in the real world or
• (ii) a nonlinear programming problem with equality constraints can be formulated as:

$$\text{min}J\left(x\right)$$

$$\text{s}\text{.t}\text{.}{c}_1\left(x\right)=\mathrm{0,}$$

$$c_2\left(x\right)=\mathrm{0,}$$

$$c_m\left(x\right)=\mathrm{0,}$$

$$x\in X\subset {ℝ}^2,$$

with m ≤ n and continuous functions J, h _i , i = 1, 2, ... , m. The necessary first-order conditions for optimality can be summarized as follows:

und auch $$\nabla J\left(x*\right)d=0.$$

Let x* be a local extreme point of J under the condition c(x) = (c ₁ ; c ₂ , ... , c _m ) = 0, and assume that c(x*) = 0, then for all (feasible variations) d ∈ R ⁿ that satisfies the following equations: $$\nabla c\left(x*\right)d=0$$

and also $$\nabla J\left(x*\right)d=0.$$

mit λ ∈ R^m. Es gibt viele Methoden zur Berechnung des optimalen Punktes x* zu berechnen, entweder mit Hilfe der Methode der Lagrange-Multiplikatoren für (4) oder Wurzelfindungsmethoden für (3), wie z. B. die Newton-Iteration.Instead of the latter conditions due to the orthogonal inner products, ∇(x*) is usually considered as a linear combination of the rows of ∇c ^T (x). $$\nabla J\left(x*\right)+{\mathrm{\lambda }}^T\nabla c\left(x*\right)=\mathrm{0,}$$

with λ ∈ R ^m . There are many methods to calculate the optimal point x*, either using the method of Lagrange multipliers for (4) or root finding methods for (3), such as B. the Newton iteration.

The methods in (i) have the disadvantage that the entire operating range cannot be used, a large measurement effort is required, algorithms often require large EC memory or calibration and adjustment is expensive due to wear and aging of the parameters. The mentioned methods in (ii) often suffer from poor interpretability of convergence properties, stability guarantees and physical interpretability when the optimized variables belong to a dynamic real-world system.

• - schlechte Interpretierbarkeit der Konvergenzeigenschaften
• - fehlende Stabilitätsgarantien
• - fehlende physikalische Interpretierbarkeit
• - vollständiger Anwendungsbereich nicht nutzbar
• - großer Messaufwand
• - Algorithmen erfordern oft großen ECU-Speicher
• - Kalibrierung und Anpassung aufgrund von Verschleiß, Alterung der Parameter teuer

Some disadvantages of the prior art can therefore be summarized as follows:

• - poor interpretability of the convergence properties
• - lack of stability guarantees
• - lack of physical interpretability
• - Complete scope of application not usable
• - large measurement effort
• - Algorithms often require large ECU memory
• - Calibration and adjustment expensive due to wear and aging of parameters

Against this background, the present invention creates an improved method and an improved control device for regulating a mechatronic system according to the main claims. Advantageous refinements result from the subclaims and the following description.

• - Einlesen von Sollwerten für physikalischen Größen des mechatronischen Systems, wobei sich die Sollwerte im Zeitablauf geändert hatten und, ansprechend auf das Einlesen, Bestimmen oder Einlesen von aktuellen Werten der physikalischen Größen des mechatronischen Systems;
• - Integrieren eines von den Sollwerten und den aktuellen Werten abhängigen Optimierungsfunktionals über die Zeit, um Regelungswerte für die physikalischen Größen zu erhalten; und
• - Ausgeben eines Regelungssignals zum Regeln des mechatronischen Systems unter Verwendung der ermittelten Regelungswerte für die physikalischen Größen.

According to the approach presented here, a method for controlling a mechatronic system is proposed, the method having the following steps:

• - reading in target values for physical variables of the mechatronic system, the target values having changed over time and, in response to the reading in, determining or reading in current values of the physical variables of the mechatronic system;
• - Integrating an optimization functional over time that is dependent on the setpoint values and the current values in order to obtain control values for the physical variables; and
• - Outputting a control signal for controlling the mechatronic system using the determined control values for the physical variables.

In the present case, a mechatronic system can be understood as a technical system in which electrical variables and/or mechanical variables are converted into one another in order, for example, to control an actuator. For example, such a mechatronic system can be a drive module for a vehicle or generally a machine, a vehicle, a braking and/or steering system or even a power conversion module, depending on which physical variables are to be processed. Without limiting generality, these physical quantities can represent, for example, a current, a voltage, a pressure, a moment, a speed and/or an acceleration. In the present case, an optimization functional can be understood as a formula-like relationship in which input variables, setpoints, reference variables, controlled variables and/or boundary value conditions of the physical variables of the mechatronic system are linked to one another. The current values of the physical variables, which can also be referred to as input variables, are read in specifically when the setpoints for these physical variables have changed. In this way, the control can be optimized specifically if a change in the operating state has taken place in the mechatronic system, i.e. if, for example, a change in the power output of the mechatronic system was controlled (externally) and now the corresponding (current) physical variables are sent to this requested or desired changed power output should be adjusted. In order to be able to carry out an efficient optimization of the values of the individual physical quantities, according to the approach presented here, this optimization functional is integrated over time and not, as in the prior art, solved via lookup tables or complex differentiations. On the one hand, this makes it possible to quickly determine the optimal control values that correspond to the new values for the physical variables (after the control), and it can also be ensured that the new values for the physical variables now obtained also settle quickly and thus meet strict stability criteria .

An embodiment of the approach proposed here is particularly favorable in which the optimization functional is used in the integrating step, which represents a cost function to be minimized, in particular where the optimization functional represents a scalar quantity. Such an embodiment offers the advantage of being able to take into account or map a desired target criterion, such as minimizing the energy required for the operation of the mechatronic system, directly through the design or choice of the optimization functional due to the favorable choice of the optimization functional.

According to a further embodiment, an optimization functional based on a quality functional can be used in the integrating step, which represents a boundary condition for which the physical variables of the mechatronic system, in particular wherein the quality functional for the current physical variables and/or for the control values has a value of results in zero and/or where the quality functional is a non-linear functional. Such an embodiment offers the advantage of opening up, by implementing the boundary conditions for the physical variables in the quality functional, a numerically very efficient possibility of being able to take the desired or required boundary conditions for the control of the mechatronic system into account directly in the optimization functional.

Also, according to a further embodiment, the step of integrating can be carried out using a Lie derivative of the quality functional. By using the Lie derivative, a mathematically very compact and numerically efficient processing of the quality functional or the optimization functional formed here can be achieved.

An embodiment of the approach proposed here is also conceivable, in which the step of integrating is carried out using a Lyapunov function of the quality functional after time, in particular where a derivative of the Lyapunov function of the quality functional is used. By using the Lyapunov function of the quality functional in this way, a mathematically very compact and numerically efficient processing of the quality functional or the optimization functional formed here can be achieved.

An embodiment of the approach proposed here is particularly favorable, in which the integrating step involves integrating over a time interval that extends to infinity. By taking all future times into account during integration, such an embodiment offers the smallest possible residual error with reasonable numerical effort, so that the control values obtained can settle reliably and stably.

According to a further embodiment of the approach proposed here, the integrating step may be further carried out using control values obtained in a previous integrating step. Such an embodiment of the approach proposed here offers the advantage of implementing a reliably operating and quickly converting control loop.

An embodiment of the approach proposed here is particularly favorable in which, in the step of reading in, at least one value is read as the physical quantity, which represents a voltage, a current, a pressure, a force and/or a moment. Such an embodiment offers the advantage of being able to edit and/or process central values of physical variables of a mechatronic system with the approach presented here and thus enable rapid and precise control of the mechatronic system in response to a stimulated change in the operating state of the system make possible.

An embodiment of the approach proposed here is also advantageous in which, in the output step, the control signal for controlling a drive system, braking system, vehicle, control system and/or energy conversion system is output. In particular, the examples of mechatronic systems mentioned above can be controlled quickly and efficiently using the approach presented here.

The approach presented here also creates a control unit that is designed to carry out, control or implement the steps of a variant of a method presented here in corresponding devices. This embodiment variant of the invention in the form of a control device can also solve the problem on which the invention is based quickly and efficiently.

A controller unit can be an electrical device that processes electrical signals, for example sensor signals, and outputs control signals depending on them. The controller unit can have one or more suitable interfaces, which can be designed in hardware and/or software. In the case of a hardware design, the interfaces can, for example, be part of an integrated circuit in which functions of the device are implemented. The interfaces can also be their own integrated circuits or at least partially consist of discrete components. In the case of software training, the interfaces can be software modules that are present, for example, on a microcontroller alongside other software modules.

Also advantageous is a computer program product with program code, which can be stored on a machine-readable medium such as a semiconductor memory, a hard drive memory or an optical memory and is used to carry out the method according to one of the embodiments described above if the program is on a computer or a control unit is performed.

• 1 ein eine schematische Darstellung eines Ausführungsbeispiels eines mechatronischen Systems;
• 2 eine Blockbild-Darstellung eines Rückkopplungssystems, wie es in der Integrationseinheit mit dem Optimierungsfunktional;
• 3 eine beispielhafte Darstellung der Funktionsweise des vorstehend genannten Optimierungsfunktionals als eine Tangentialebene an den dreidimensionalen Raum der speziellen Randbedingungswerte;
• 4 ein Ablaufdiagramm eines Ausführungsbeispiels des hier vorgestellten Ansatzes als Verfahren zur Regelung eines mechatronischen Systems; und
• 5 ein Blockschaltbild eines Ausführungsbeispiels einer Implementierungsstruktur der Reglereinheit.

The invention is explained in more detail using the accompanying drawings. Show it:

• 1 a a schematic representation of an exemplary embodiment of a mechatronic system;
• 2 a block diagram representation of a feedback system as it is in the integration unit with the optimization functional;
• 3 an exemplary representation of the operation of the aforementioned optimization functional as a tangential plane to the three-dimensional space of the special boundary condition values;
• 4 a flowchart of an exemplary embodiment of the approach presented here as a method for controlling a mechatronic system; and
• 5 a block diagram of an exemplary embodiment of an implementation structure of the controller unit.

In the following description of preferred exemplary embodiments of the present invention, the same or similar reference numbers are used for the elements shown in the various figures and having a similar effect, with a repeated description of these elements being omitted.

1 shows a schematic representation of an exemplary embodiment of a mechatronic system 100, which is designed here in the form of a vehicle, for example an electric vehicle. In this vehicle as a mechatronic system 100, for example, an accelerator pedal or, more generally, a pedal 105 is provided for adjusting or changing the acceleration or speed of the vehicle as a mechatronic system 100. If, for example, the pedal 105 is now actuated, a corresponding change signal 110 is read in via a read-in interface 115 of a control unit 120, which represents, for example, a change in setpoint values of physical variables of the vehicle as a mechatronic system 100. For example, the position of the pedal 105 can control a corresponding current and/or a corresponding voltage, which is output by the controller unit 120 to an electric motor 125, so that this electric motor 125 wheels 130 of the vehicle as a mechatronic system 100 with a changed torque or can drive at a changed speed. Furthermore, values The values Here, control values x* can be obtained, which are processed in an output unit 145 to form a control signal 150, with which, for example, an actuator such as the electric motor 125 can then be controlled. The control values x* can represent desired values of the physical variables with which, for example, the actuator, here for example the electric motor 130, is operated. Generally speaking, by actuating the pedal 105, a change in the operating variables for operating the electric motor 125 can be achieved, so that by actuating the pedal 105 the electric motor 125 is subjected to a larger or smaller current or a larger or smaller voltage, so that Vehicle accelerates or brakes as a mechatronic system 100. However, it should be noted that the new, desired operating state of the vehicle as a mechatronic system 100 should, on the one hand, settle quickly and stably, but on the other hand should not require energy unnecessarily, so that, for example, an energy storage device is required to provide the energy required for the operation of the actuator or electric motor 125 is operational for as long as possible.

• - x = (x₁, x₂, ..., x_n)^T einen Zustand eines dynamischen Systems. Jede Komponente x₁, x₂, ..., x_n des Zustands x entspricht einer physikalischen Größe. Zum Beispiel, kann dies ein Strom, eine Spannung, eine Geschwindigkeit, eine Position, ein Druck usw. sein, d. h. der Zustand kann sich aus elektrischen und mechanischen Größen zusammensetzen.
• - u = (u₁, u₂, ... , u_m)^T einen Eingang oder Steuereingang eines dynamischen Systems.
• - c(x) = (c₁(x), c₂(x), ..., c_m(x))^T einen Vektor mit Gleichheitsbeschränkungen für den Zustand x. Die Lösungen eines dynamischen Systems sollen die Nebenbedingung erfüllen. (Häufig ist eine solche Nebenbedingung eine Kraftgleichung, die von von physikalischen Größen wie Strom oder Druck abhängt, wobei die gewünschte Kraft gegeben ist.)
• - c* = c(x) einen gewünschten Wert für die Gleichheitsbedingung.
• - J(x) die Ziel- oder Leistungsfunktion bzw. das hier als solches bezeichnete Optimierungsfunktional, welches beispielsweise eine skalare Gleichung abbildet, die vom Zustand x abhängt. J(x) kann z. B. verwendet werden, um den Energieverlust zu beschreiben, der minimiert werden soll.
• - $L_{f}h$
  
  eine Lie-Ableitung der Form $L_{f}h:=\frac{\partial h}{\partial x}f.$
  

The central focus for such a functionality of stable settling or efficient calculation of the control values x* or the control signal 150 determined therefrom is the design of the integration unit 135 with the optimization functional 140, which is also referred to below as the variable J for the sake of simplicity. In order to provide a basic understanding of the problem discussed here, a definition of the notations used in the following description is first given. This refers in particular to the variable

• - x = (x ₁ , x ₂ , ..., x _n ) ^T a state of a dynamic system. Each component x ₁ , x ₂ , ..., x _n of the state x corresponds to a physical quantity. For example, this can be a current, a voltage, a speed, a position, a pressure, etc., ie the state can be made up of electrical and mechanical quantities.
• - u = (u ₁ , u ₂ , ... , u _m ) ^T an input or control input of a dynamic system.
• - c(x) = (c ₁ (x), c ₂ (x), ..., c _m (x)) ^T a vector with equality constraints for the state x. The solutions of a dynamic system should fulfill the secondary condition. (This is often the case Secondary condition is a force equation that depends on physical quantities such as current or pressure, where the desired force is given.)
• - c* = c(x) a desired value for the equality condition.
• - J(x) is the target or performance function or the optimization functional referred to here as such, which, for example, represents a scalar equation that depends on the state x. J(x) can e.g. B. can be used to describe the energy loss that should be minimized.
• - $L_{f}H$
  
  a Lie derivation of the form $L_{f}H:=\frac{\partial H}{\partial x}f.$
  

For the detailed description of an exemplary embodiment, the case can be considered that the vector x in the non-linear programming problem according to equation (1) refers to the state of a dynamic system $$\dot{x}=F\left(x,u\right),x\left(t_0\right)=x_0.$$

In this context, it is proposed to use the knowledge of the dynamic system to solve the problem of nonlinear programming. For this purpose, taking into account the optimality condition according to equation (3), h(x) := ∇J(x*)d, which is referred to here as the quality functional, together with a feedback u = ϕ(x; h(x)) in equation (5), a feedback system is constructed,

implementiert ist. 2 shows a block diagram representation of a feedback system, as in the integration unit 135 with the optimization functional 140 $$\dot{x}=\tilde{F}\left(x\right),x\left(t_0\right)=x_0.$$

is implemented.

3 shows an exemplary representation of the functionality of the aforementioned optimization functional 140 as a tangential plane 300 to the three-dimensional space of the special boundary condition values.

In order to solve equation (1), the feedback system (6) should have a unique solution x(t) such that for t →∞, h(x(t)) = 0 and c(x(t)) = 0 apply , where the latter means that x(t) corresponds to a regular point as t →∞, see 3 for an illustration of a regular point x* for two constraints c(x).

A detailed description of the procedure presented here will be described below.

mit $$x\in D\subset {\text{R}}^n,x_0\in D,u={\left(u_1,u_2,\cdots ,u_m\right)}^T\in U\subset {\text{R}}^m$$

The following class of multi-input affine nonlinear dynamical systems is considered: $$\dot{x}=F\left(x,u\right)=f\left(x\right)+{\displaystyle \sum _{k=1}^N{G}_k\left(x\right)u_k,x\left(t_0\right)}=x_0,$$

with $$x\in D\subset {\text{R}}^n,x_0\in D,u={\left(u_1,u_2,\cdots ,u_m\right)}^T\in U\subset {\text{R}}^m$$

und für jedes t∈ [t₀, t₁), das Anfangswertproblem eine eindeutige Lösung $x:[t_0,t_1)\to D,t_0<t\le t_1<\infty$

erzeugt. Dabei zu beachten ist das Optimierungsproblem $$\underset{x}{\text{min}}J\left(x\right)$$

$$\text{s}\text{.t}\text{. }h\left(x\right)=\mathrm{0,}$$

mit der Kostenfunktion J : ℝⁿ →ℝ₊ und der nicht-linearen Nebenbedingung (bzw. des Gütefunktionals) h : ℝⁿ → ℝIt is assumed that for each $x\left(t_0\right)\in D$

and for every t∈ [t ₀ , t ₁ ), the initial value problem has a unique solution $x:[{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DE102022209555A1\_0092.png"\; state="\{\{state\}\}">t</figure-callout>}}_0,t_1)\to D,{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DE102022209555A1\_0092.png"\; state="\{\{state\}\}">t</figure-callout>}}_0<t\le t_1<\infty$

generated. The optimization problem must be taken into account $$\underset{x}{\text{min}}J\left(x\right)$$

$$\text{s}\text{.t}\text{.}H\left(x\right)=\mathrm{0,}$$

with the cost function J : ℝ ⁿ →ℝ ₊ and the non-linear constraint (or the quality functional) h : ℝ ⁿ → ℝ

was bedeutet, dass x* = x, so dass h(x*) = 0 impliziert, J(x*) < J(x)|_x≠x*.It can be assumed that the cost function J(x) together with the constraint h(x) can be reformulated into a condition: $$H\left(x\right)=\mathrm{0,}$$

which means that x* = x, so h(x*) = 0 implies J(x*) < J(x)| _x≠x* .

Instead of using a root-finding method for Equation (9), the goal here is to exploit the dynamical system to determine x* = x such that h(x*) =0.

$$\text{s}\text{.t }\dot{x}=F\left(x,u\right),x\left(t_0\right)=x_0,$$

wobei die Endkosten Null sind und die Endzeit frei ist, d. h. es die Form eines eines Optimierungsproblems mit unendlichem Horizont hat. Hier hat $$L=\dot{h}=\frac{\partial h}{\partial x}\dot{x},$$

die Mayer-Form, die ergibt $$J\left(x,u\right)={\displaystyle \int {}_{t_0}^{\infty }}\frac{\partial h}{\partial x}F\left(x,u\right)\text{d}t.$$

A useful approach is that of an infinite horizon optimization problem: $$\underset{x,u(\cdot )}{\text{min}}J\left(x,u\right)={\displaystyle \int {}_{{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DE102022209555A1\_0092.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}^{\infty }}L\left(x\left(t\right),u\left(t\right),t\right)\text{d}t$$

$$\text{s}\text{.t}\dot{x}=F\left(x,u\right),x\left(t_0\right)=x_0,$$

where the final cost is zero and the final time is free, i.e. it has the form of an optimization problem with an infinite horizon. Here has $$L=\dot{H}=\frac{\partial H}{\partial x}\dot{x},$$

the Mayer form that results $$J\left(x,u\right)={\displaystyle \int {}_{{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DE102022209555A1\_0092.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}^{\infty }}\frac{\partial H}{\partial x}F\left(x,u\right)\text{d}t.$$

• Theorem 3.1. Es wird das dynamische System $$\dot{h}=\frac{\partial h}{\partial x}F\left(x,\varphi \left(x\right)\right),h\left(x\left(t_0\right)\right)=h_0,t\ge \mathrm{0,}$$
  
  mit dem Leistungsfunktional (12) Optimierungsfunktional 140 betrachtet. Es existiert eine kontinuierlich differenzielle Funktion V ◯ h(x) = V(h) : ℝ→ ℝ und ein Steuergesetz $\varphi \left(x\right):D\to U$
  
  derart, dass $$V\left(0\right)=0$$
  
  $$V\circ h\left(x\right)>\mathrm{0,}h\in ℝ,h\ne \mathrm{0,}$$
  
  $$\frac{\partial V}{\partial x}\left(x\right)F\left(x,\varphi \left(x\right)\right)<\mathrm{0,}h\in ℝ,h\ne \mathrm{0,}$$
  
  $$H\left(x,\varphi \left(x\right)\right)=\mathrm{0,}h\in ℝ,$$
  
  $$H\left(x,u\right)\ge \mathrm{0,}h\in ℝ,u\in U,$$
  

wobei $$H\left(x,u\right)=L\left(x\right)+\frac{\partial V}{\partial x}\left(x\right)F\left(x,u\right).$$

For further analysis, the originally described Lyapunov-based setup is used, considered and adapted:

• Theorem 3.1. It becomes the dynamic system $$\dot{H}=\frac{\partial H}{\partial x}F\left(x,\varphi \left(x\right)\right),H\left(x\left(t_0\right)\right)=H_0,t\ge \mathrm{0,}$$
  
  with the performance functional (12) optimization functional 140 considered. There is a continuous differential function V ◯ h(x) = V(h) : ℝ→ ℝ and a control law $\varphi \left(x\right):D\to U$
  
  such that $$v\left(0\right)=0$$
  
  $$v\circ H\left(x\right)>\mathrm{0,}H\in ℝ,H\ne \mathrm{0,}$$
  
  $$\frac{\partial v}{\partial x}\left(x\right)F\left(x,\varphi \left(x\right)\right)<\mathrm{0,}H\in ℝ,H\ne \mathrm{0,}$$
  
  $$H\left(x,\varphi \left(x\right)\right)=\mathrm{0,}H\in ℝ,$$
  
  $$H\left(x,u\right)\ge \mathrm{0,}H\in ℝ,u\in U,$$
  

where $$H\left(x,u\right)=L\left(x\right)+\frac{\partial v}{\partial x}\left(x\right)F\left(x,u\right).$$

asymptotisch stabil und es existiert eine Umgebung des Ursprungs $D_0\subset D$

derart, dass $$J\left(x_0,\varphi \left(x(\cdot );h\left(x(\cdot )\right)\right)\right)=V\left(x_0\right),x\in D_0.$$

$$J\left(x_0,\varphi \left(x(\cdot );h\left(x(\cdot )\right)\right)\right)=\underset{u\in S\left(x_0\right)}{\text{min}}J\left(x_0,u(\cdot )\right),$$

With the feedback u = ϕ(x;h(x)) the solution h ≡ 0 of the control loop system results $$\dot{H}-\psi \left(H\right),$$

asymptotically stable and a neighborhood of the origin exists ${\mathrm{<figure-callout\; id="0"\; label="D"\; filenames="DE102022209555A1\_0092.png"\; state="\{\{state\}\}">D</figure-callout>}}_0\subset D$

such that $$J\left(x_0,\varphi \left(x(\cdot );H\left(x(\cdot )\right)\right)\right)=v\left(x_0\right),x\in {\mathrm{<figure-callout\; id="0"\; label="D"\; filenames="DE102022209555A1\_0092.png"\; state="\{\{state\}\}">D</figure-callout>}}_0.$$

$$J\left(x_0,\varphi \left(x(\cdot );H\left(x(\cdot )\right)\right)\right)=\underset{u\in S\left(x_0\right)}{\text{min}}J\left(x_0,u(\cdot )\right),$$

Proof. From (14c) it follows that $$\dot{v}\left(H\left(x\right)\right)=\frac{\partial v}{\partial x}\left(x\right)F\left(x,\varphi \left(x\right)\right)<\mathrm{0,}t\ge \mathrm{0,}H\ne 0.$$

ergibt sich $$L\left(x\left(t\right),u\left(t\right)\right)=-\dot{V}\left(h\left(x\left(t\right)\right)\right)+L\left(x\left(t\right),u\left(t\right)\right)+\frac{\partial V}{\partial x}\left(x\left(t\right)\right)F\left(x\left(t\right),u\left(t\right)\right)$$

$$=-\dot{V}\left(h\left(x\left(t\right)\right)\right)+H\left(x\left(t\right),u\left(t\right)\right).$$

So from (14a), (14b) and (19) it follows that V(·) is a Lyapunov function, and therefore from the direct Lyapunov method it follows that the zero solution h = 0 of the dynamic system (13) is locally asymptotically stable. There $$0=-\dot{v}\left(H\left(x\right)\right)+\frac{\partial v}{\partial x}\left(x\right)F\left(x,u\right)<\mathrm{0,}t\ge \mathrm{0,}H\ne 0$$

surrendered $$L\left(x\left(t\right),u\left(t\right)\right)=-\dot{v}\left(H\left(x\left(t\right)\right)\right)+L\left(x\left(t\right),u\left(t\right)\right)+\frac{\partial v}{\partial x}\left(x\left(t\right)\right)F\left(x\left(t\right),u\left(t\right)\right)$$

$$=-\dot{v}\left(H\left(x\left(t\right)\right)\right)+H\left(x\left(t\right),u\left(t\right)\right).$$

$$=-\underset{t\to \infty }{\text{lim}}V\left(h\left(x\left(t\right)\right)\right)+V\left(\left(h\left(x_0\right)\right)\right)+{\displaystyle \int {}_{t_0}^{\infty }}H\left(x\left(t\right),u\left(t\right)\right)dt$$

$$=V\left(\left(h\left(x_0\right)\right)\right)+{\displaystyle \int {}_{t_0}^{\infty }}H\left(x\left(t\right),u\left(t\right)\right)dt$$

$$\ge V\left(\left(h\left(x_0\right)\right)\right)\ge J\left(x_0,\mathrm{\Phi }\left(x;h\left(x\right)\right)\right),$$

was zu Gleichung (18) führt.Due to (14e), and u = ϕ(x), $$J\left(x_0,u(\cdot )\right)={\displaystyle \int {}_{{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DE102022209555A1\_0092.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}^{\infty }}-\dot{v}\left(H\left(x\left(t\right)\right)\right)+H\left(x\left(t\right),u\left(t\right)\right)dt$$

$$=-\underset{t\to \infty }{\text{lim}}v\left(H\left(x\left(t\right)\right)\right)+v\left(\left(H\left(x_0\right)\right)\right)+{\displaystyle \int {}_{{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DE102022209555A1\_0092.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}^{\infty }}H\left(x\left(t\right),u\left(t\right)\right)dt$$

$$=v\left(\left(H\left(x_0\right)\right)\right)+{\displaystyle \int {}_{{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="DE102022209555A1\_0092.png"\; state="\{\{state\}\}">t</figure-callout>}}_0}^{\infty }}H\left(x\left(t\right),u\left(t\right)\right)dt$$

$$\ge v\left(\left(H\left(x_0\right)\right)\right)\ge J\left(x_0,\mathrm{\Phi }\left(x;H\left(x\right)\right)\right),$$

which leads to equation (18).

The feedback can be designed to be particularly advantageous. The following therefore proposes how, firstly, the feedback u = Φ(x) can be derived, which is based on an exact feedback linearization, and secondly, how the optimal points x* can be determined under the condition h(x*) = 0.

näher berücksichtigt, wobei der Einfachheit halber die Richtungsableitungen ausgedrückt werden durch die Lie-Ableitung $L_{f}h=\frac{\partial h}{\partial x}f$

einer Funktion h nach f.Equation (15) with the condition from equation (14d) is considered, and this and the non-linear dynamic system from equation (7): $$\frac{d}{dt}H\left(x\right)=\frac{\partial H}{\partial x}\dot{x}=L_{f}H+L_{G1}H{u}_1+L_{G2}H{u}_2+\cdots +L_{Gm}H{u}_m,$$

considered in more detail, whereby for simplicity the directional derivatives are expressed by the Lie derivative $L_{f}H=\frac{\partial H}{\partial x}f$

a function h to f.

mit einem neuen Eingang v linearisiert das System aus Gleichung (27) genau, d. h., $$\frac{d}{dt}h=v.$$

The feedback u = (u ₁ , u ₂ , ... , u _M ), with u _k , k = 1, 2, ... , m is defined as $$u_k=\left(v-L_{f}H\right)=\frac{L_{Gk}H}{{\Vert \left(L_{G1}H,\cdots ,L_{Gm}H\right)\Vert }^2},$$

with a new input v the system from equation (27) linearizes exactly, i.e. $$\frac{d}{dt}H=v.$$

Now consider a Lyapunov function candidate as V = 1/2h ² with its time derivative $$H\frac{\text{d}}{\text{d}t}H=Hv.$$

garantiert, dass für jede Anfangsbedingung h₀ = h(x(t₀)), h(t) → 0 als t→∞.v is chosen so that it applies to the scalar system $H\frac{\text{d}}{\text{d}t}H<\mathrm{0,}H\ne 0$

guarantees that for any initial condition h ₀ = h(x(t ₀ )), h(t) → 0 as t→∞.

für die Gleichung (29) folgendes ergibt $$\dot{h}=-\mathrm{\lambda }h.$$

A possible choice can be $$v=-\mathrm{\lambda }H,\mathrm{\lambda }>\mathrm{0,}$$

for equation (29) the following results $$\dot{H}=-\mathrm{\lambda }H.$$

The feedback from Equation (28) can be applied to the dynamical system from Equation (7) to determine the solutions x(t) that solve h(x) = 0. It has the following form: $$\dot{x}=f\left(x\right)+{\displaystyle \sum _{k=1}^N{G}_k\left(x\right)\left(v-L_{f}H\right)\frac{L_{G_k}H}{{\Vert \left(L_{G_1}H,\cdots ,L_{G_n}H\right)\Vert }^2},x_0=x\left(t_0\right).}$$

Note that the derived closed-loop control system corresponds to equation (6).

die Terme auf der rechten Seite entsprechend der vorstehenden Beschreibung wie folgt geschrieben $$\frac{G{G}^T{\left(\frac{\partial h}{\partial x}\right)}^T\frac{\partial h}{\partial x}}{\frac{\partial h}{\partial x}G{G}^T{\left(\frac{\partial h}{\partial x}\right)}^T}f\left(x\right)={\displaystyle \sum _{k=1}^N{g}_k\left(x\right)L_{f}h\frac{L_{g_k}h}{{\Vert \left(L_{g_1}h,\cdots ,L_{g_n}h\right)\Vert }^2},}$$

$$v\frac{G{G}^T{\left(\frac{\partial h}{\partial x}\right)}^T}{\frac{\partial h}{\partial x}G{G}^T{\left(\frac{\partial h}{\partial x}\right)}^T}={\displaystyle \sum _{k=1}^N{g}_k\left(x\right)v\frac{L_{g_k}h}{{\Vert \left(L_{g_1}h,\cdots ,L_{g_n}h\right)\Vert }^2}.}$$

The stability of the above-mentioned closed control loop will also be discussed in more detail. First, according to $$G=\left(G_1(\cdot ),G_2(\cdot ),\dots ,G_m(\cdot )\right),$$

the terms on the right are written as follows according to the description above $$\frac{G{G}^T{\left(\frac{\partial H}{\partial x}\right)}^T\frac{\partial H}{\partial x}}{\frac{\partial H}{\partial x}G{G}^T{\left(\frac{\partial H}{\partial x}\right)}^T}f\left(x\right)={\displaystyle \sum _{k=1}^N{G}_k\left(x\right)L_{f}H\frac{L_{G_k}H}{{\Vert \left(L_{G_1}H,\cdots ,L_{G_n}H\right)\Vert }^2},}$$

$$v\frac{G{G}^T{\left(\frac{\partial H}{\partial x}\right)}^T}{\frac{\partial H}{\partial x}G{G}^T{\left(\frac{\partial H}{\partial x}\right)}^T}={\displaystyle \sum _{k=1}^N{G}_k\left(x\right)v\frac{L_{G_k}H}{{\Vert \left(L_{G_1}H,\cdots ,L_{G_n}H\right)\Vert }^2}.}$$

und $$C\left(x\right):=\frac{G{G}^T{\left(\frac{\partial h}{\partial x}\right)}^T}{\frac{\partial h}{\partial x}G{G}^T{\left(\frac{\partial h}{\partial x}\right)}^T}$$

nimmt das dynamische genannte System (3.3.1) die kompakte Form an: $$\dot{x}=f\left(x\right)-B\left(x\right)f\left(x\right)+vC\left(x\right),x_0=x\left(t_0\right).$$

With $$b\left(x\right):=\frac{G{G}^T{\left(\frac{\partial H}{\partial x}\right)}^T\frac{\partial H}{\partial x}}{\frac{\partial H}{\partial x}G{G}^T{\left(\frac{\partial H}{\partial x}\right)}^T}$$

and $$C\left(x\right):=\frac{G{G}^T{\left(\frac{\partial H}{\partial x}\right)}^T}{\frac{\partial H}{\partial x}G{G}^T{\left(\frac{\partial H}{\partial x}\right)}^T}$$

the dynamic system called (3.3.1) takes the compact form: $$\dot{x}=f\left(x\right)-b\left(x\right)f\left(x\right)+vC\left(x\right),x_0=x\left(t_0\right).$$

Theorem 3.2. For the matrix defined in equation (37) applies $$0\le x^{T}bx\le x^{T}x,fOrallx\in {ℝ}^n.$$

durch ein äußeres Produkt gebildet wird, hat sie den Rang 1. Somit hat auch $N:=G{G}^T{\left(\frac{\partial h}{\partial x}\right)}^T\frac{\partial h}{\partial x}$

den Rang 1, was bedeutet, dass nur ein Eigenwert von N ungleich Null ist. Damit gilt: $$N=N^T,\text{ tr}\left(N\right)={\displaystyle \sum {}_{i=1}^n\mathrm{\lambda }\left(N\right).}$$

Proof: There the matrix ${\left(\frac{\partial H}{\partial x}\right)}^T\frac{\partial H}{\partial x},$

is formed by an external product, it has rank 1. Therefore also has $N:=G{G}^T{\left(\frac{\partial H}{\partial x}\right)}^T\frac{\partial H}{\partial x}$

has rank 1, which means that only one eigenvalue of N is non-zero. This means: $$N=N^T,\text{tr}\left(N\right)={\displaystyle \sum {}_{i=1}^n\mathrm{\lambda }\left(N\right).}$$

also λ_max(B) = 1 und λ_min(B) = 0. Die Ungleichung (40) folgt aus dem Min-Max-Theorem.Due to the cyclic nature of the track $\text{tr}\left(N\right)=\frac{\partial H}{\partial x}G{G}^T{\left(\frac{\partial H}{\partial x}\right)}^T,$

so λ _max (B) = 1 and λ _min (B) = 0. Inequality (40) follows from the min-max theorem.

sodass angenommen wird, dass $$\Vert C\left(x\right)\Vert \le a{\Vert x\Vert }^k,k>\mathrm{0,}a>\mathrm{0,}\Vert \frac{\partial h}{\partial x}G\Vert \ne 0.$$

Conjecture 3.3. Without loss of generality, for the matrix defined in equation (38), $$\Vert C\left(x\right)\Vert =\Vert \frac{G{G}^T{\left(\frac{\partial H}{\partial x}\right)}^T}{\frac{\partial H}{\partial x}G{G}^T{\left(\frac{\partial H}{\partial x}\right)}^T}\Vert =\frac{\Vert G\Vert }{\Vert \frac{\partial H}{\partial x}G\Vert },$$

so that it is assumed that $$\Vert C\left(x\right)\Vert \le a{\Vert x\Vert }^k,k>\mathrm{0,}a>\mathrm{0,}\Vert \frac{\partial H}{\partial x}G\Vert \ne 0.$$

betrachtet und es angenommen wird, dass der folgende Zusammenhang existiert $$V=\frac{1}{2}{x}^{T}x,$$

derart, dass $$\dot{V}=\frac{\partial V}{\partial x}f\left(x\right)\le 0.$$

gilt.Theorem 3.4. Becomes the dynamic system without input $$\dot{x}=f\left(x\right),$$

considered and it is assumed that the following relationship exists $$v=\frac{1}{2}{x}^{T}x,$$

such that $$\dot{v}=\frac{\partial v}{\partial x}f\left(x\right)\le 0.$$

applies.

begrenzt.Then the trajectories of are $$\dot{x}=f\left(x\right)-b\left(x\right)f\left(x\right)-\mathrm{\lambda }H\left(x\right)C\left(x\right),x_0=x\left(t_0\right),$$

limited.

mit der zeitlichen Ableitung $$\dot{V}=x^T\dot{x}$$

$$=x^{T}f\left(x\right)-x^{T}Bf\left(x\right)-x^T\mathrm{\lambda }hC\left(x\right)$$

$$=x^T\left(I-B\right)f\left(x\right)-x^T\mathrm{\lambda }hC\left(x\right).$$

The following can be given as evidence. Becomes the quadratic Lyapunov function candidate $$v=\frac{1}{2}{x}^{T}x,$$

with the time derivative $$\dot{v}=x^T\dot{x}$$

$$=x^{T}f\left(x\right)-x^{T}bf\left(x\right)-x^T\mathrm{\lambda }HC\left(x\right)$$

$$=x^T\left(I-b\right)f\left(x\right)-x^T\mathrm{\lambda }HC\left(x\right).$$

gilt $$\dot{V}\le -x^T\mathrm{\lambda }h\left(x\right)C\left(x\right)$$

$$\le \mathrm{\lambda }\left|h\left(x\right)\right|{\Vert x\Vert }^{k}a.$$

Since it is assumed that x ^T f(x) ≤ 0, and due to $$0\le I-b\le I,$$

applies $$\dot{v}\le -x^T\mathrm{\lambda }H\left(x\right)C\left(x\right)$$

$$\le \mathrm{\lambda }\left|H\left(x\right)\right|{\Vert x\Vert }^{k}a.$$

Since ḣ = -λ h, h ₀ = (t ₀ ), h(t) = h ₀ e ^{-λ(tt 0 )} , follows, $$\dot{v}\le \mathrm{\lambda }a\left|{\mathrm{<figure-callout\; id="0"\; label="H"\; filenames="DE102022209555A1\_0092.png"\; state="\{\{state\}\}">H</figure-callout>}}_0\right|e^{-\mathrm{\lambda }\left(t-t_0\right)}{\Vert x\Vert }^k.$$

Now the comparison principle is exploited, so Ilxll = (2 V) ^½ is inserted into the right-hand side of equation (53) to give $$\dot{v}\le \mathrm{\lambda }\left|{\mathrm{<figure-callout\; id="0"\; label="H"\; filenames="DE102022209555A1\_0092.png"\; state="\{\{state\}\}">H</figure-callout>}}_0\right|e^{-\mathrm{\lambda }\left(t-t_0\right)}a{2}^{\frac{k}{2}}{v}^{\frac{k}{2}}.$$

If the upper bound of the differential inequality (54) is taken, the separable equation follows $$\frac{\text{d}v}{\text{d}t}=\mathrm{\lambda }\left|H\left(x\right)\right|a{2}^{\frac{k}{2}}{v}^{\frac{k}{2}}.$$

$$\frac{2}{2-k}{V}^{\frac{2-k}{2}}-\frac{2}{2-k}{V}_0{}^{\frac{2-k}{2}}=-a\left|h_0\right|e^{-\mathrm{\lambda }\left(t-t_0\right)}{2}^{\frac{k}{2}}+a\left|h_0\right|2^{\frac{k}{2}},$$

$$\frac{2}{2-k}{V}^{\frac{2-k}{2}}=-a\left|h_0\right|e^{-\mathrm{\lambda }\left(t-t_0\right)}{2}^{\frac{k}{2}}+C$$

mit $C:a\left|h_0\right|2^{\frac{k}{2}}+\frac{2}{2-k}{V}_0{}^{\frac{2-k}{2}}$

Daraus folgt, $$V\left(t\right)={\left(\frac{k-2}{2}\left(a\left|h_0\right|2^{\frac{k}{2}}{e}^{-\mathrm{\lambda }\left(t-t_0\right)}-C\right)\right)}^{\frac{2}{2-k}}.$$

Integrating the separate equation gives: $${\displaystyle \int {}_{v_0}^v{v}^{-\frac{k}{2}}\text{dv}={\displaystyle \int {}_{t_0}^t}\mathrm{\lambda }a\left|H_0\right|e^{-\mathrm{\lambda }\left(t-t_0\right)}}{2}^{\frac{k}{2}}\text{d}t,$$

$$\frac{2}{2-k}{v}^{\frac{2-k}{2}}-\frac{2}{2-k}{v}_0{}^{\frac{2-k}{2}}=-a\left|H_0\right|e^{-\mathrm{\lambda }\left(t-t_0\right)}{2}^{\frac{k}{2}}+a\left|H_0\right|2^{\frac{k}{2}},$$

$$\frac{2}{2-k}{v}^{\frac{2-k}{2}}=-a\left|{\mathrm{<figure-callout\; id="0"\; label="H"\; filenames="DE102022209555A1\_0092.png"\; state="\{\{state\}\}">H</figure-callout>}}_0\right|e^{-\mathrm{\lambda }\left(t-t_0\right)}{2}^{\frac{k}{2}}+C$$

with $C:a\left|{\mathrm{<figure-callout\; id="0"\; label="H"\; filenames="DE102022209555A1\_0092.png"\; state="\{\{state\}\}">H</figure-callout>}}_0\right|2^{\frac{k}{2}}+\frac{2}{2-\mathrm{<figure-callout\; id="0"\; label="k\; v"\; filenames="DE102022209555A1\_0092.png"\; state="\{\{state\}\}">k</figure-callout>}}{v}_{}{}_0{}^{\frac{2-k}{2}}$

It follows, $$v\left(t\right)={\left(\frac{k-2}{2}\left(a\left|{\mathrm{<figure-callout\; id="0"\; label="H"\; filenames="DE102022209555A1\_0092.png"\; state="\{\{state\}\}">H</figure-callout>}}_0\right|2^{\frac{k}{2}}{e}^{-\mathrm{\lambda }\left(t-t_0\right)}-C\right)\right)}^{\frac{2}{2-k}}.$$

Daraus folgt, für t →∞, $V\le {\left(\frac{2-k}{2}C\right)}^{\frac{2}{2-k}}$

$$

und $\Vert x\Vert \le \sqrt{2}{\left(\frac{2-k}{2}C\right)}^{\frac{1}{2-k}},$

was zeigt, dass Ilxll begrenzt ist. □At t →∞ is the upper one $v^{+}\to {\left(\frac{2-k}{2}C\right)}^{\frac{2}{2-k}}$

It follows, for t →∞, $v\le {\left(\frac{2-k}{2}C\right)}^{\frac{2}{2-k}}$

$$

and $\Vert x\Vert \le \sqrt{2}{\left(\frac{2-k}{2}C\right)}^{\frac{1}{2-k}},$

which shows that Ilxll is limited. □

angewendet werden, um die Bedingung c(x*) = 0 zu erfüllen. Alternativ kann die Optimalitätsbedingung h auch um einen Strafterm erweitert werden $$\tilde{h}=h+\mathrm{\gamma }\tilde{\mathrm{\Gamma }}c\left(x\right),\tilde{\mathrm{\Gamma }}\in {ℝ}^{1\times m},\mathrm{\gamma }\ne 0.$$

For a general nonlinear constraint c(x) = 0, an injection into the feedback system of the form $$\dot{x}=f\left(x\right)-b\left(x\right)f\left(x\right)-\mathrm{\lambda }H\left(x\right)C\left(x\right)+\mathrm{\gamma }{c}^T\left(x\right)x,\mathrm{\gamma \Gamma }{c}^T\left(x\right)x\le 0$$

be applied to satisfy the condition c(x*) = 0. Alternatively, the optimality condition h can also be extended to include a penalty term $$\tilde{H}=H+\mathrm{\gamma }\tilde{\mathrm{\Gamma }}c\left(x\right),\tilde{\mathrm{\Gamma }}\in {ℝ}^{1\times m},\mathrm{\gamma }\ne 0.$$

• - Bestimmung von optimalen Sollwerten für die optimale Regelung dynamischer Systeme (z. B. Fahrzeugsysteme, mechatronische Systeme).
• - Bestimmung optimaler Sollwerte für die optimale Regelung dynamischer Systeme (z. B. Kraftfahrzeugsysteme, mechatronische Systeme) in Echtzeit während des Betriebs.
• - Berücksichtigung möglicher nichtlinearer Eigenschaften bei der Bestimmung der der optimalen Sollwerte.
• - Berücksichtigung der expliziten und exakten Dynamik für die Bestimmung der optimalen Sollwerte.

In summary, the advantages of the new approach presented here can be mentioned. The invention enables:

• - Determination of optimal setpoints for the optimal control of dynamic systems (e.g. vehicle systems, mechatronic systems).
• - Determination of optimal setpoints for the optimal control of dynamic systems (e.g. automotive systems, mechatronic systems) in real time during operation.
• - Consideration of possible non-linear properties when determining the optimal setpoints.
• - Consideration of the explicit and exact dynamics for determining the optimal setpoints.

1. 1. Warten auf eine Änderung einer gewünschten oder vorgegebenen Zwangsbedingung c* = c(x) und Messung oder Schätzung der Komponenten des Zustands x. (Die gewünschte Nebenbedingung c* = c(x) kann von einer übergeordneten Steuereinheit vorgegeben werden. Beispielsweise: Bei einem elektrischen Antrieb gibt das Motorsteuergerät ein gewünschtes Motordrehmoment T* = c* aufgrund der Messung einer Gaspedal-Aktivität vor.)
2. 2. Numerische Integration des vorstehend beschriebenen geregelten dynamischen Systems in Bezug auf die Zeit im Steuergerät bzw. der Reglereinheit.
3. 3. Ausgabe der ermittelten/integrierten optimalen Punkte x* an das überlagerte Regelsystem.
4. 4. Regelung des Zustandes x des mechatronischen dynamischen Systems

The control unit or regulator unit 120 can carry out the following steps cyclically (or for each sample) or event-oriented:

1. 1. Wait for a change in a desired or given constraint c* = c(x) and measure or estimate the components of state x. (The desired secondary condition c* = c(x) can be specified by a higher-level control unit. For example: In an electric drive, the engine control unit specifies a desired engine torque T* = c* based on the measurement of accelerator pedal activity.)
2. 2. Numerical integration of the controlled dynamic system described above with respect to the time in the control device or the regulator unit.
3. 3. Output of the determined/integrated optimal points x* to the superimposed control system.
4. 4. Control of the state x of the mechatronic dynamic system

• - Einsatz für alle Varianten dynamischer Systeme oder mechatronischer Systeme, z. B.
• - Stellantriebe
• - Traktionsantriebe
• - synchrone Antriebe
• - Automotive Systeme
• - Fahrzeuge
• - Fahrräder
• - Bremssysteme
• - Lenksysteme
• - Stromumrichter
• - Windenergiesysteme

The approach presented here is suitable for several application scenarios, such as:

• - Use for all variants of dynamic systems or mechatronic systems, e.g. b.
• - Actuators
• - Traction drives
• - synchronous drives
• - Automotive systems
• - Vehicles
• - Cycles
• - Braking systems
• - Steering systems
• - Power converter
• - Wind energy systems

4 shows a flowchart of an exemplary embodiment of the approach presented here as a method 400 for controlling a mechatronic system. The method 400 includes a step 410 of reading in setpoints for physical quantities of the mechatronic system, the setpoints having changed over time and, in response to the reading, determining or reading in current values of the physical quantities of the mechatronic system. Furthermore, the method 400 includes a step 420 of integrating an optimization functional that is dependent on the setpoint values and the current values over time in order to obtain control values for the physical variables. Finally, the method 400 includes a step 430 of outputting a control signal for controlling the mechatronic system using the determined control values for the physical variables

5 shows a block diagram of an exemplary embodiment of an implementation structure of the controller unit 120. First, the boundary conditions c*=c(x) are read into the read-in interface. This is followed by a numerical integration of the optimization functional over time in the integrating unit 135, whereby this optimization functional is generally referred to as ẋ = F (x), x(t ₀ ) = x ₀ can be reproduced. In a subsequent output unit 145, the optimal control values x* are then first determined in a determination unit 500, before the relevant control signals 150 are determined from these control values x* in a control unit 510, which then in turn determine the state values x for the mechatronic system 100 form a subsequent state space.

The exemplary embodiments described and shown in the figures are only chosen as examples. Different exemplary embodiments can be combined with one another completely or with regard to individual features. An exemplary embodiment can also be supplemented by features of a further exemplary embodiment.

Furthermore, method steps according to the invention can be repeated and carried out in an order other than that described.

If an exemplary embodiment includes an “and/or” link between a first feature and a second feature, this can be read as meaning that, according to one embodiment, the exemplary embodiment has both the first feature and the second feature and, according to a further embodiment, either only the first Feature or only the second feature.

Claims (12)

Method for controlling a mechatronic system (100), the method (400) having the following steps: - Reading in (410) of setpoint values (c(x)) for physical quantities of the mechatronic system (100), the setpoint values (c(x)) being changed over time and, in response to the reading in, determining or reading in current values ( x) the physical variables of the mechatronic system (100); - Integrating (420) an optimization functional (140, J) dependent on the setpoint values (c(x)) and the current values (x) over time in order to obtain control values (x*) for the physical variables; and - Outputting (430) a control signal (150) for controlling the mechatronic system (100) using the determined control values (x*) for the physical variables. Procedure (400) according to Claim 1 , characterized in that in step (420) of integrating, the optimization functional (140, J) is used, which represents a cost function (J) to be minimized, in particular wherein the optimization functional (140, J) represents a scalar quantity. Method (400) according to one of the preceding claims, characterized in that in the step (420) of integrating, an optimization functional (140, J) is used based on a quality functional (h (x)) which provides a boundary condition (h) for the represents the physical variables of the mechatronic system (100), in particular wherein the quality functional (h(x)) for the current values (x) of the physical variables and/or for the control values (x*) results in a value of zero and/or where the quality functional (h(x)) is a non-linear functional. Procedure (400) according to Claim 3 , characterized in that the step (420) of integrating is carried out using a Lie derivative (L) of the quality functional (h (x)). Procedure (400) according to Claim 3 or 4 , characterized in that the step (420) of integrating is carried out using a Lyapunov function of the quality functional (h (x)) in time, in particular wherein a derivative (v) of the Lyapunov function of the quality functional (h (x) ) is used. Method (400) according to one of the preceding claims, characterized in that in the step (420) of integrating, integration is carried out over a time interval extending to infinity. Method (400) according to one of the preceding claims, characterized in that the step (420) of integrating is further carried out using control values (x*) obtained in a previous step (420) of integrating. Method (400) according to one of the preceding claims, characterized in that in the step (410) of reading in at least one value (x) is read in as physical variables, which is a voltage, a current, a pressure, a force and / or a moment represented. Method (400) according to one of the preceding claims, characterized in that in the step (430) of outputting, the control signal (150) for controlling a drive system (125), braking system, vehicle, control system and/or energy conversion system is output. Controller unit (120), which is set up to carry out and/or control the steps (410, 420, 430) of the method (400) according to one of the preceding claims in corresponding units (115, 135, 145). Computer program that is set up to execute and/or control the steps (410, 420, 430) of the method (400) according to one of the preceding claims when the computer program is executed on a control unit (120) or a device. Machine-readable storage medium on which the computer program can be written Claim 11 is stored.

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