Method for controlling a dynamic system with jitter effects

Abstract

Method for controlling a dynamic system (12) by means of a control device (10) comprising a sensor (14), a controller (16) and an actuator (18), the method comprising: calculating a controller gain, sampling a state variable (x(t)) of the dynamic system (12) by means of the sensor (14) at predetermined sampling intervals and transmitting sample values ​​(x k ) to the controller (16), calculating by the controller (16) actuator command values ​​(u k ) from the sample values ​​(x k ) using the controller gain and transmitting the actuator command values ​​(u k ) to the actuator (18) and regulating the state variable (x(t)) of the dynamic system (12) by the actuator (18) using the actuator command values ​​(u k), wherein the calculation of the controller gain is based on a stochastic variable which is a function of a signal propagation time (δ) from the sensor (14) via the controller (16) to the actuator (18).

Description

[0001] The present disclosure relates to a method and a control device for controlling a dynamic system.

[0002] In modern control systems, a controller is generally spatially separated from the dynamic system being controlled, so signal propagation times within the control system influence the control behavior. Signal propagation times can vary unpredictably and have a negative impact on the control behavior. The variability of signal propagation times is referred to as jitter.

[0003] Due to jitter, a periodic sample value of a time-varying variable detected by a sensor of the control device for characterizing the dynamic behavior of a dynamic system to be controlled cannot be converted into a correspondingly periodic actuator command value, which impairs the stability of the control device.

[0004] The object of the present invention is therefore to provide a method and a control device for controlling a dynamic system which are robust against jitter.

[0005] This object is achieved according to a first aspect of the present invention by a method for controlling a dynamic system by means of a control device comprising a sensor, a controller and an actuator, the method comprising: - Calculating a controller gain, - Sampling a state variable of the dynamic system using the sensor at specified sampling intervals and transmitting sample values ​​to the controller, - Calculating actuator command values ​​from the sampled values ​​by the controller using the controller gain and transmitting the actuator command values ​​to the actuator and - Regulating the state variable of the dynamic system by the actuator using the actuator command values, wherein the calculation of the controller gain is based on a stochastic variable which is a function of a signal propagation time from the sensor via the controller to the actuator.

[0006] By considering the signal propagation time and thus the jitter—the variability of the signal propagation time from the sensor via the controller to the actuator—as a stochastic variable when calculating the controller gain, the negative effects of jitter on the control behavior of a control system can be compensated for, and stable control can be achieved. The stochastic variable is particularly characterized by its predefined probability distribution, which is a key difference from conventional jitter-incorporating control methods, which only consider a maximum signal propagation time.

[0007] When calculating the controller gain, there is a certain degree of freedom regarding the choice of the stochastic variable, which influences the complexity of the calculation steps. Calculating the controller gain can be performed particularly easily if the stochastic variable has a mean value of zero.

[0008] In order to be able to take into account the variability of the signal propagation time between a maximum signal propagation time and a minimum signal propagation time when calculating the controller gain, a further development of the invention can provide that the stochastic variable is limited by an upper limit value and a lower limit value, wherein the upper limit value and the lower limit value are functions of the minimum and maximum signal propagation time from the sensor via the controller to the actuator.

[0009] In an exemplary embodiment of the present invention, it can be provided that a variance of a distribution of the stochastic variable is adjustable. By adjusting the variance, the probability distribution can be directly influenced, which in turn significantly determines the control behavior. In this case, on the one hand, with increasing variance, the robustness to jitter increases, and on the other hand, the control behavior approaches a classic control behavior in which jitter is not considered as a stochastic variable.

[0010] In a further development of the invention, the calculation of the controller gain can include calculating an actual or de facto sampling interval, which is a function of the stochastic variable with a zero mean, an average signal propagation time from the sensor via the controller to the actuator, the minimum signal propagation time from the sensor via the controller to the actuator, and the specified (nominal) sampling interval. This allows nominal values ​​to be separated from stochastic values, allowing the controller gain to be calculated easily.

[0011] Furthermore, the controller gain calculation can be based on a quadratic cost function with positive-definite weighting matrices. Thus, the calculated controller can stabilize the closed control loop and simultaneously adjust the control performance using the weighting matrices.

[0012] In a further development of the invention, the controller can be designed as a linear-quadratic controller configured to calculate the actuator command values ​​by multiplying the sampled values ​​by the controller gain. This allows the actuator command values ​​to be calculated from the sampled values ​​by matrix multiplication, i.e., in a quick and simple manner.

[0013] The object defined above is achieved in a further aspect by a control device for controlling a dynamic system, wherein the control device comprises: a sensor, a controller, an actuator, and a memory in which a computer-readable code is stored, which, when executed on a data processing device, operates the control device according to a method described above. The data processing device can be, for example, a computer or a microcontroller.

[0014] The control device may further comprise a network, wherein the controller communicates with the sensor or actuator, or with the sensor and actuator, via the network. This provides a high degree of design freedom, as the relative spatial arrangement of the controller with respect to the dynamic system can be essentially freely selected. For example, the controller may be configured as a cloud-based controller.

[0015] In a further aspect, the problem defined above is solved by a control system comprising a control device as described above and a dynamic system to be controlled. The dynamic system can be several vehicles whose relative distance is to be controlled. The vehicles can be autonomous vehicles, for example.

[0016] The invention is described in detail below with reference to the accompanying drawings, in which: Fig. 1 a schematic diagram showing the architecture of a networked control system, Fig. 2 a schematic representation of a flow chart of a control in the Fig. 1 shown control system, Fig. 3 a schematic representation of a dynamic system comprising several vehicles, Fig. 4A is a graphical representation showing a distance error of a first vehicle of the Fig. 3 illustrated vehicles shows, Fig. 4B is a graphical representation showing a distance error of a second vehicle of the Fig. 3 illustrated vehicles shows, Fig. 5A is a graphical representation showing the mean value of an exemplary control cost function, Fig. 5B is a graphical representation showing the variance of an exemplary control cost function, Fig. 6 Simulation results of a control of another dynamic system.

[0017] Fig. Figure 1 is a schematic diagram showing the architecture of a control system 100. The control system 100 includes a control device 10 and a dynamic system 12 to be controlled by the control device 10.

[0018] The control device 10 comprises: a sensor 14 which is configured to sample a time-dependent state variable x(t) of the dynamic system 12 at predetermined sampling intervals and to record discrete sample values ​​x k to the controller 16. The controller 16 is configured to transmit discrete actuator command values ​​u k from the sample values ​​x k and the calculated actuator command values ​​u kto the actuator 18. The actuator 18 is configured to determine the state variable x(t) of the dynamic system 12 using the actuator command values ​​u k to regulate, for example by means of a continuous actuator output signal u(t) transmitted to the dynamic system 12. In Fig. 1, t indicates the time.

[0019] The controller 16 may be in signal communication with one or both of the sensor 14 and the actuator 18 via a network 20. The network 20 may be wireless or wired.

[0020] The controller 16 can be designed as any data processing device, such as a computer or microcontroller, capable of executing a computer-readable program code that carries out the method described below. The program code can be stored in a memory 16a that is in data communication with the controller 16. The memory 16a can, as in Fig. 1, be integrated into the controller 16. The memory 16a can be designed, for example, as a read-only memory (ROM), random access memory (RAM), hard disk, and the like, or a combination thereof.

[0021] The data processing device may include one or more data processing units. A data processing unit may refer to any type of entity that enables processing of data or signals. A data processing unit may be embodied as an analog circuit, digital circuit, logic circuit, microcontroller, central processing unit (CPU), graphics processing unit (GPU), digital signal processor (DSP), field programmable gate array (FPGA), or a combination thereof, or may include one or more of these units.

[0022] Fig. 2 is a schematic representation of a flow chart of a control in the Fig. 1 shown control system 100. The aspects relevant to the control process are briefly described below. I: The sensor 14 samples at predetermined time (nominal) sampling intervals h ∈ ℝ >0 a state variable of the dynamic system 12, ie at given times t k = kh, where k is a natural number, ie k ∈ ℕ. By sampling the state variable x(t), discrete samples x k which are transmitted to the controller 16 via the network 20. II: The network 20 causes a delay in the signal propagation time from the sensor 14 to the controller 16 and / or from the controller 16 to the actuator 18. Further delays are caused by computational processes. The total signal propagation time from the sensor 14 via the controller 16 to the actuator 18 (hereinafter referred to as the "end-to-end signal propagation time") is Fig. 2 by δk and is smaller than the specified sampling interval h. III: The controller 16 receives the discrete sample values ​​x k and calculates discrete actuator command values ​​u k which are transmitted to the actuator 18. A signal delay may also occur during this transmission, as indicated under II. IV: The end-to-end signal propagation time varies stochastically between a minimum end-to-end signal propagation time δ min and a maximum end-to-end signal propagation time δ max This variation is, as already mentioned, called jitter and is in Fig. 2 marked with j. Due to the signal delay δ k the actuator command values ​​u reach k the actuator 18 at time t k +δ k , whereby this time point can be unpredictably in the range between t k + δ min and t k + δ max varies.

[0023] The temporal fluctuation impairs the stability of the control in the control system 100. This can be counteracted by a control method in which the signal propagation time is taken into account as a stochastic variable and the gain K of the controller 16 is calculated taking this stochastic variable into account.

[0024] The starting point of the procedure is the following equation of state (1) of the dynamic system 12: x˙(t)=Acx(t)+Bcu(t) ∀t∈ℝ≥0, where x ∈ ℝ n is the state variable of the dynamic system 12, ẋ(t) is the time derivative of the state variable and u(t) ∈ ℝ m is the actuator output signal.

[0025] This equation of state can be solved using the Euler forward method with a sampling interval h̃ ∈ ℝ >0 be discretized, which leads to the following equation of state (2): xk+1=(I+Ach˜)xk+Bch˜uk, k ∈ ℕ represents a discrete time. A c and B c represent coefficient matrices and I an identity matrix. The jitter is now treated as a random variable, which is accounted for by a stochastic variable δ that follows a certain probability distribution. For δ, the following applies: 0≤δmin≤δ≤δmax≤h.

[0026] To simplify the subsequent calculation steps, it is advisable to work with a stochastic quantity that has a mean value of zero. For this purpose, a limited stochastic quantity θ=E(δ)−δ∈[−δmax−δmin2,δmax−δmin2] with variance σ ∈ ℝ >0 introduced. E(δ) is the expected value of δ. The variance σ is adjustable in this method. By adjusting the variance, the probability distribution can be directly influenced, which in turn can directly influence the control behavior. On the one hand, with increasing variance, the robustness to jitter increases, and on the other hand, the control behavior approaches classical control behavior, in which jitter is not considered as a stochastic variable.

[0027] This transformation yields a factual or actual sampling interval h̃, which is a function of the stochastic quantity θ with mean zero, a mean end-to-end signal propagation time δ = 0.5 (δ max -δ min ), the minimum end-to-end signal propagation time δ min and the specified (nominal) sampling interval h is: h˜=h+δmin+δ¯+θ.

[0028] Thus, the discretized equation of state (2) can be rewritten as follows: xk+1=(I+Ac[h+δmin+δ˜])︸A0xk+Bc[h+δmin+δ˜]︸B0uk+(Ac︸A1xk+Bc︸B1uk)θk.

[0029] The first two terms in this equation represent nominal terms, while the last term is a stochastic term. This separation of variables simplifies the subsequent calculation of the controller gain.

[0030] The goal is to create a stochastic linear-quadratic controller (LQR) of the form u k = Kx k with controller gain K, which stabilizes the dynamics of the equation of state (2) in the sense of the quadratic mean, ie E(xk)→0 and E(xkxkT)→0 for k → ∞. For this purpose, a quadratic cost function of the form l(x,u)=‖x‖Q2+‖u‖R2 = with positive definite weighting matrices Q, R > 0 with the aim of obtaining a quadratic cost function V(x)=‖x‖P2(with P≻0) with infinite horizon, so that E(V(xk+1))≤V(xk)−l(xk,Kxk).

[0031] This problem can be solved using the following matrix inequality (3): [E(A0E+B0Y)T(A1E+B1Y)TETQ1 / 2YTR1 / 2(A0E+B0Y)E000(A1E+B1Y)0σ−1E00Q1 / 2E00I0R1 / 2Y000I]≥0.

[0032] The aim is to determine the volume of the ellipsoid {x T Px ≤ 1}, i.e., minimize log det(E). Note that Y and E are matrix variables written as E = P -1 and Y = KE. From these matrix variables, the controller gain is given as K = YE -1 .

[0033] In summary, the following optimization problem (4): minE,Y−log det(E) is to be solved under the constraint (matrix inequality) (3) and the constraint E ≽ 0.

[0034] This optimization problem yields E and Y and thus the controller gain K, which is based on the stochastic quantity θ(δ).

[0035] With the controller gain determined in this way, the controller 16 can calculate the sample values ​​x k by matrix multiplication with the controller gain K the actuator command values ​​u k calculate, i.e. in a simple and resource-saving way.

[0036] The method thus has two essential phases. During a first phase, which can also be referred to as the offline phase, the controller gain K is determined based on a stochastic variable. During a second phase, which can be referred to as the online phase, the controller 16 is operated in a closed control loop in which sample values ​​x k provided by the sensor 14, these are transmitted to the controller 16, the controller 16 actuator command values ​​u k by matrix multiplication according to u k= Kx k calculated, these are transmitted to the actuator 18 and the actuator 18 based on the actuator command values ​​u k regulates the state variable of the dynamic system 12.

[0037] The method described above can also be used for multiple jitter sources, for example, considering q ∈ ℕ matrix pairs (A i , B i ) (where i = 1, ..., q) with q independent stochastic quantities θ i .

[0038] Hereinafter, by reference to Fig. 3 describes an exemplary control system in which the dynamic system 12 comprises two vehicles 12-1 and 12-2. Other possible applications include, for example, the control / coordination of vehicles in a restricted area, such as a logistics center or a production facility, or the control of robots, especially robot arms.

[0039] In the Fig. In the example shown in Figure 3, the vehicles 12-1 and 12-2 are connected to the controller 16 via a network 20 in data exchange connection, whereby the signal propagation times between the vehicles 12-1, 12-2 and the controller 16 exhibit fluctuations. Initially, a joint approach of the three in Fig. The vehicles 12-0, 12-1, and 12-2 shown in Figure 3 are assumed, with the leading vehicle 12-0 not being controlled and traveling at a constant speed v0 = 1 m / s. The other two vehicles 12-1 and 12-2 follow the leading vehicle 12-0 at speeds v1 and v2, respectively. The goal is to control the distance error e1 = d(v1) - d and e2 = d(v2) - d to the respective origin (indicated by the respective dashed vertical lines), which indicates a safety distance ds to the immediately preceding vehicle. The controlled vehicles 12-1 and 12-2 are controlled via their respective accelerations a1 and a2. The state vector of the dynamic system 12 is thus given by x = (e1, v0 - v1, e2, v1 - v2) T and the actuator command vector by u = (a1, a2) TAs sensors for detecting the state variable, ie the state vector, of the dynamic system 12, for example, distance measuring sensors such as RADAR or LiDAR sensors can be used. Any electrical or electromechanical devices capable of adjusting the amount of energy supplied to a drive device of the respective vehicles (e.g., an internal combustion engine or an electric motor) can be used as actuators. The sensors and actuators of the vehicles 12-1 and 12-2 are in Fig. 3 not shown.

[0040] The nominal sampling interval h is set to 0.5 seconds. The end-to-end signal propagation time δ is set between a minimum signal propagation time δ min of 0.1 seconds and a maximum signal propagation time δ maxof 0.5 seconds is assumed to be uniformly distributed. Q = diag(10,1,10,1) and R = diag(1,1) are used as weighting matrices for the quadratic cost function. The variance σ is set to 0.4.

[0041] In Fig. 4A, the time course of the distance error e1(t) of the vehicle 12-1 during control by a method according to the present disclosure is indicated by the curve SLQR. For comparison, the time course of the distance error during control by a method in which only the maximum end-to-end signal propagation time δ max is taken into account (curve with reference symbol RLQR). This method is also called robust LOR method. Furthermore, Fig. 4A also shows the time course of the distance error when controlled using a method that does not take into account fluctuations in the signal propagation time (curve with reference symbol LQR). This method is also referred to as the nominal LQR method. The corresponding confidence intervals, which correspond to one standard deviation, are shown in Fig. 4A highlighted by the respective filled areas and marked with reference symbol STD SLQR , HOURS RLQR or STD LQR designated.

[0042] Fig. 4B is an analogous representation to Fig. 4A of the distance error e2(t) of the vehicle 12-2. The results in the Fig. 4A and Fig. 4B were obtained by Monte Carlo simulation.

[0043] As can be seen from the Fig. 4A and Fig. As can clearly be seen in Figure 4B, stabilization fails when using a nominal LOR method. In contrast, the distance error according to the present disclosure and the robust LOR method converges quadratically, i.e., in the root mean square sense. For a more detailed analysis, a closed-loop simulation was performed for k = 0, ..., 25 time steps, and the running costs (as given in the following equation) were added: Jcl=∑k=025‖x(k)‖Q2+‖Kx(k)‖R2.

[0044] To make statistical statements about J cl To be able to make the best decisions (mean µ and variance var), a Monte Carlo simulation of the closed control loop was performed (each over k = 0, ..., 25 time steps). This was repeated for different values ​​of the variance σ to illustrate the effect of this setting parameter on the cost function. The results are shown in the Fig. 5A and Fig. 5B. The curve labeled SLQR refers to a method according to the present disclosure. The curve labeled RLQR refers to a robust control method.

[0045] In this example, a minimum mean value µ results at a variance σ of approximately 0.4, and a minimum variance var results at a variance σ of approximately 0.8. In the range 0.161 ≤ σ ≤ 0.894, the control method according to the present disclosure is superior to the robust LOR method with regard to the mean value.

[0046] It should be noted that the superiority of the control method according to the present disclosure over a robust LQR method depends on the stability of the dynamic system when it is uncontrolled. Fig. The dynamic system shown in Figure 3 was a stable, uncontrolled system. Therefore, to demonstrate the performance of the control method according to the present disclosure, an unstable, uncontrolled dynamic system is considered below, which is described by the following state differential equation: x˙=[2300,1]x(t)+

[01] u(t).

[0047] This is discretized with a nominal sampling interval h = 0.25. Furthermore, a uniformly distributed jitter in the range δ ∈ [0.01; 0.25] is assumed. For the weights of the LQR cost function, Q = I (identity matrix) and R = 1 are chosen, and the variance σ is set to 0.1.

[0048] Fig. Figure 6 shows the convergence behavior of the control method according to the present disclosure and the robust control method. The curve labeled SLQR represents the mean value over time for the control method according to the present disclosure. The curve labeled STD SLQR The area labeled RLQR represents the confidence interval corresponding to one standard deviation. The curve labeled RLQR represents the mean over time for the robust control procedure. The curve labeled STD RLQR The area denoted is the confidence interval, which corresponds to one standard deviation.

[0049] Out of Fig. 6 clearly shows the improved stability that can be achieved with a control method according to the present disclosure compared to a conventional robust control method. For the expected values ​​of the cost function, one obtains E(Jcl)=1.92⋅104 for the robust control procedure and E(Jcl)=2.22⋅103 for the control method according to the present disclosure.

Claims

[1] Method for controlling a dynamic system (12) by means of a control device (10) comprising a sensor (14), a controller (16) and an actuator (18), wherein the method comprises: - Calculating a controller gain, - Sampling a state variable (x(t)) of the dynamic system (12) using the sensor (14) at predetermined sampling intervals (h) and transmitting sampled values ​​(x k ) to the controller (16), - Calculation by the controller (16) of actuator command values ​​(u k ) from the sampled values ​​(x k ) using the controller gain and transmitting the actuator command values ​​(u k ) to the actuator (18) and - Regulating the state variable (x(t)) of the dynamic system (12) by the actuator (18) using the actuator command values ​​(u k), where the calculation of the controller gain is based on a stochastic quantity which is a function of a signal propagation time (δ) from the sensor (14) via the controller (16) to the actuator (18). [2] Method according to claim 1, wherein the stochastic quantity has a mean value of zero. [3] Method according to claim 1 or 2, wherein the stochastic quantity is limited by an upper limit and a lower limit, wherein the upper limit and the lower limit are functions of a minimum signal propagation time (δ min ) and a maximum signal propagation time (δ max ) from the sensor (14) via the controller (16) to the actuator (18). [4] Method according to one of claims 1 to 3, wherein a variance (σ) of a distribution of the stochastic quantity is adjustable. [5] Method according to claim 2 or according to claim 2 and one of claims 3 or 4, wherein the calculation of the controller gain includes the calculation of an actual sampling interval which is a function of the stochastic quantity with mean value zero, a mean signal propagation time from the sensor (14) via the controller (16) to the actuator (18), the minimum signal propagation time (δ min ) from the sensor (14) via the controller (16) to the actuator (18) and the specified sampling interval (h). [6] Method according to any one of claims 1 to 5, wherein the calculation of the controller gain is based on a quadratic cost function with positive definite weighting matrices. [7] Method according to any one of claims 1 to 6, wherein the controller (16) is configured as a linear-quadratic controller which is set up to control the actuator command values ​​(u k ) by multiplying the sample values ​​(x k ) to calculate with the controller gain. [8] Control device (10) for controlling a dynamic system (12), wherein the control device (10) comprises: a sensor (14), a controller (16), an actuator (18) and a memory (16a) in which a computer-readable code is stored which, when executed on a data processing device, operates the control device (10) according to a method according to any one of claims 1 to 7. [9] Control device (10) according to claim 8, which further comprises a network (20), wherein the controller (16) is in signal exchange connection with the sensor (14) or the actuator (18) or the sensor (14) and the actuator (18) via the network (20). [10] Control system (100) comprising a control device (10) according to claim 8 or 9 and a dynamic system (12) to be controlled.

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