Method and system for devising an optimum control policy

Abstract

A method for devising an optimum control policy of a controller for controlling a system includes optimizing at least one parameter that characterizes the control policy. A Gaussian process model is used to model expected dynamics of the system. The optimization optimizes a cost function which depends on the control policy and the Gaussian process model with respect to the at least one parameter. The optimization is carried out by evaluating at least one gradient of the cost function with respect to the at least one parameter. For an evaluation of the cost function a temporal evolution of a state of the system is computed using the control policy and the Gaussian process model. The cost function depends on an evaluation of an expectation value of a cost function under a probability density of an augmented state at time steps.

Claims

1. A method for automatically tuning a multivariate PID controller for controlling a system, said method comprising: configuring the multivariate PID controller with a random control policy for controlling the system, said random control policy having at least one parameter that characterizes said random control policy; controlling the system with the multivariate PID controller based on said random control policy; devising an optimum control policy for the multivariate PID controller for controlling the system by optimizing said at least one parameter that characterizes said random control policy; using a Gaussian process model to model expected dynamics of the system, wherein said optimization optimizes a cost function which depends on said random control policy and said Gaussian process model with respect to said at least one parameter; and carrying out said optimization by evaluating at least one gradient of said cost function with respect to said at least one parameter to generate at least one optimized parameter of said optimum control policy, wherein for an evaluation of said cost function a temporal evolution of a state of the system is computed using said random control policy and said Gaussian process model, and wherein said cost function depends on an evaluation of an expectation value of a cost function under a probability density of an augmented state at time steps, tuning the multivariate PID controller by changing said at least one parameter to said at least one optimized parameter; and controlling the system with the tuned multivariate PID controller based on the optimum control policy.

2. The method according to claim 1, wherein said augmented state at a given time step comprises the state at said given time step.

3. The method according to claim 1, wherein said augmented state at a given time step comprises an error between the state and a desired state at a previous time step.

4. The method according to claim 1, wherein said augmented state at a given time step comprises an accumulated error of a previous time step.

5. The method according to claim 3, wherein the augmented state and/or the desired state are Gaussian random variables.

6. The method according to claim 1, further comprising: devising said optimum control policy for the multivariate PID controller for controlling the system by iteratively optimizing said at least one optimized parameter that characterizes said optimum control policy, iteratively updating said Gaussian process model based on a recorded reaction of the system to said optimum control policy, using said updated Gaussian process model to model expected dynamics of the system, wherein said optimization optimizes an updated cost function which depends on said optimized control policy and said updated Gaussian process model with respect to said at least one optimized parameter, and carrying out said optimization by evaluating at least one gradient of the updated cost function with respect to said at least one optimized parameter to generate at least one further optimized parameter of the optimum control policy; and iteratively tuning the multivariate PID controller by changing said at least optimized parameter to said at least one further optimized parameter.

7. The method according claim 1, wherein the system comprises an actuator and/or a robot.

8. The method according to claim 1, wherein a training system for devising said optimum control policy of the multivariate PID controller is configured to carry out the method.

9. The method according to claim 1, wherein a control system for controlling the system is configured to carry out the method.

10. The method according to claim 1, wherein a computer program contains instructions which cause a processor to carry out the method if the computer program is executed by said processor.

11. The method according to claim 10, wherein a machine-readable storage medium is configured to store the computer program.

12. The method according to claim 1, wherein said at least one gradient is determined by application of the chain rule to said cost function.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) The objects, features and advantages of the disclosure will be apparent from the following detailed descriptions of the various aspects of the disclosure in conjunction with reference to the following drawings, where:

(2) FIG. 1 is an illustration of a humanoid robot trained with a system according to the disclosure;

(3) FIG. 2 is a schematic illustration of the mathematics behind the method according to an aspect of the disclosure;

(4) FIG. 3 is a block diagram depicting control structures of a controller to which the disclosure may be applied;

(5) FIG. 4 is a block diagram depicting components of a system according to an aspect of the disclosure;

(6) FIG. 5 is a block diagram depicting components of a system according to another aspect of the disclosure;

(7) FIG. 6 is a flowchart diagram depicting the method according to one aspect of the disclosure.

DETAILED DESCRIPTION

(8) FIG. 4 shows a block diagram depicting components of a system according to an aspect of the disclosure. Shown is a control system 40, which receives sensor signals S from a sensor 30 via an input unit 50. The sensor senses a state of a physical system 10, e.g. a robot (like e.g. the humanoid robot 1 shown in FIG. 1, or an at least partially self-driving car), or more generally an actuator (like e.g. a throttle valve), in an environment 20. The input unit 50 transforms theses sensor signals S into a signal representing said state x. For example, the input unit 50 may copy the sensor signal S into a predefined signal format. If the sensor signal S is in a suitable format, the input unit 50 may be omitted altogether.

(9) This signal representing state x is then passed on to a controller 60, which may, for example, be given by a PID controller. The controller is parameterized by parameters , which the controller 60 may receive from a parameter storage P. The controller 60 computes a signal representing an input signal u, e.g. via equation (11). This signal is then passed on to an output unit 80, which transforms the signal representing the input signal u into an actuation signal A, which is passed on to the physical system 10, and causes said physical system 10 to act. Again, if the input signal u is in a suitable format, the output unit may be omitted altogether.

(10) The controller 60 may be controlled by software which may be stored on a machine-readable storage medium 45 and executed by a processor 46. For example, said software may be configured to compute the input signal u using the control law given by equation (11).

(11) FIG. 5 shows a block diagram depicting a training system 140, which may be configured to train the control system 40. The training system 140 may comprise an input unit 150 for receiving signals representing an input signal u and a state signal x, which are then passed on to a block 190 which receives present parameters from parameter storage P and computes new parameters . These new parameters are then passed on to parameter storage P to replace present parameters . The block 190 may be operated by software which may be stored on a machine-readable storage medium 210 and executed by a processor 200. For example, block 190 may be configured to execute the steps of the method shown in FIG. 6.

(12) FIG. 6 is a flowchart diagram depicting a method for devising optimum parameters for an optimum control policy of controller 60.

(13) First (1000), a random policy is devised, e.g. by randomly assigning values for parameters and storing them in parameter storage P. The controller 60 then controls physical system 10 by executing its control policy corresponding to these random parameters . The corresponding state signals x are recorded and passed on to block 190.

(14) Next (1010), a GP dynamics model {circumflex over (f)} is trained using the recorded signals x and u to model the temporal evolution of the system state x, x.sub.t+1={circumflex over (f)}(x.sub.t, u.sub.t).

(15) Then (1020), a roll-out of the augmented system state z.sub.t over a horizon H is computed based on the GP dynamics model {circumflex over (f)}, the present parameters and the corresponding control policy () and the gradient of the cost function J w.r.t. to parameters is computed, e.g. by equations (17)-(20).

(16) Based on these gradients, new parameters are computed (1030). These new parameters replace present parameters in parameter storage P.

(17) Next, it is checked whether the parameters have converged sufficiently (1040). If it is decided that they have not, the method iterates back to step 1020. Otherwise, the present parameters are selected as optimum parameters * that minimize the cost function J (1050).

(18) Controller 60 is then executed with a control policy corresponding to these optimum parameters * to control the physical system 10. The input signal u and the state signal x are recorded (1060).

(19) The GP dynamics model {circumflex over (f)} is then updated (1070) using the recorded signals x and u.

(20) Next, it is checked whether the GP dynamics model {circumflex over (f)} has sufficiently converged (1080). This convergence can be checked e.g. by checking the convergence of the log likelihood of the measured data x, t, which is maximized by adjusting the hyperparameters of the GP, e.g. with a gradient-based method. If it is deemed not to have been sufficiently converged, the method branches back to step 1020. Otherwise, the present optimum parameters * are selected as parameters that will be used to parametrize the control policy of controller 60. This concludes the method.

(21) Parts of this disclosure have been published as Model-Based Policy Search for Automatic Tuning of Multivariate PID Controllers, arXiv:1703.02899v1, 2017, Andreas Doerr, Duy Nguyen-Tuong, Alonso Marco, Stefan Schaal, Sebastian Trimpe, which is incorporated herein by reference in its entirety.