KR19980082593A - Iterative learning control method and device - Google Patents

Abstract

The present invention relates to a method and apparatus for iterative learning control in a nonlinear system. The apparatus according to the present invention is a feedback controller for stabilizing a system at an early stage before learning is performed, an iterative learning controller for converging the system output to a desired output through repetitive learning, and information about a control input of the system obtained after the repeated learning. Includes a front feed neural network controller to store the. According to the method and the apparatus according to the present invention, not only the output of the system can converge to a desired output, but also the information obtained through the iterative learning control is stored so that the learning must be restarted from the beginning whenever the reference input trajectory is changed. In addition, the new trajectory has the advantage of having a generalization capability that allows the output of the system to follow the reference input well.

Description

Iterative learning control method and device

The present invention relates to a method and apparatus for iterative learning control in a nonlinear system.

Iterative learning control is a control technique that ensures that the output of a system that repeatedly performs a given task in a finite time interval exactly follows the desired output. This iterative learning control method is started with the intention to improve the control performance through iterative learning based on the errors of past trials.

The most important problem in the iterative learning control method is to find the optimal control input so that the output of the system is the desired output regardless of the shape of the reference input trajectory of the system. In the iterative learning control, the process of finding the optimal control input for the output of the system to be the desired output is performed through the iterative learning.

In general iterative learning control method, each time the reference input trajectory is changed, the learning must be started again from the beginning. This problem increases the number of learning to find the input trajectory that causes the convergence of the output because the error of the initial state of the iteration is large, especially when the performance of the feedback controller is poor. The neural network can solve the problem of re-learning from the beginning whenever the reference input trajectory changes. As a prior art, a feedback error learning control apparatus using a neural network controller is described.

1 is a block diagram of a feedback error learning control apparatus according to the prior art.

The feedback error learning control device is a system in which the neural network controller 11 becomes the dynamics of the system through feedback learning of the output error in the system with the feedback controller 10. In this way, the feed ahead controller is expressed in the form of linear combination of the known functions, and the weight necessary for the linear combination is updated from the output error. When these basis functions are not known, an approximator such as a multilayer neural network may be used as a feedforward controller. In the feedback error learning control apparatus shown in FIG. 1, control and learning are simultaneously performed, and only the feedback error is required for the learning of the neural network controller, and there is an advantage of not requiring the error back-transfer passing through the plant 12. In addition, since the neural network controller is used, it shows good generalization ability even for untrained trajectories. However, the feedback error learning control scheme illustrated in FIG. 1 has a disadvantage in that logical convergence of the system cannot be guaranteed.

The present invention is to solve the problem of generalization when the desired output trajectory is changed in the general iterative learning control device as described above and the problem of convergence of the system output in the feedback error learning control device as shown in FIG. will be.

In other words, an object of the present invention is to provide an iterative learning control method and apparatus for combining the generalization capability of a feedback error learning control apparatus with the convergence of an iterative learning control apparatus, while ensuring convergence of the output and having a generalization capability of neural networks. There is this.

1 is a block diagram of a feedback error learning control apparatus according to the prior art;

2 is a block diagram of an iterative learning control apparatus according to an embodiment of the present invention;

3 and 4 are the results of applying the iterative learning control method proposed in the present invention to the two-axis robot manipulator compared with the prior art shown in FIG.

Explanation of symbols on the main parts of the drawings

10, 20: feedback controller 11: neural network controller

12, 23: plant 21: iterative learning controller

22: front food new economic flag

Hereinafter, exemplary embodiments of the present invention will be described in detail with reference to the accompanying drawings.

2 is a block diagram of an iterative learning control apparatus according to an embodiment of the present invention. As shown in FIG. 2, the iterative learning control apparatus proposed by the present invention includes a feedback controller (FBC), a short-term memory-based iterative learning controller (ILC), and a long-term memory-based forward feeding neural network. It includes three controllers of the controller (Feedforward Neuro-Controller: FNC). The feedback controller 20 stabilizes the system in an initial state before learning is made. However, when only the feedback controller 20 is used, there is an error between the output of the system and the desired output. The iterative learning controller 21 allows the output of the system to accurately converge to the desired output through the iterative learning. The information about the control input of the system obtained after the repetitive learning is stored in the forward feeding neural network controller 22 through the learning. Information stored in the front feeding neural network controller 22 serves to reduce the output error even for the untrained trajectory.

The control input of the plant 23 is the sum of the control inputs generated by the three controllers described above, and this is expressed by the following equation.

[Equation 1]

In Equation 1, u _fn (t), u _il (t) and u _fb (t) are control inputs by the front feed neural network controller 22, the iteration learning controller 21, and the feedback controller 20, respectively. u (t) is the control input of the plant 23.

The control algorithm by the iterative learning control device shown in FIG. 2 is as follows.

First stage

Design a feedback controller 20 to stabilize the system. The performance of the feedback controller 20 is not a big problem at this stage, and the main purpose of the feedback controller 20 is to stabilize the system.

2nd step

Set the desired output y ^d (t) in the finite time interval t∈ [0, T].

3rd step

The control inputs u _il (t) and t ∈ [0., T] by the iterative learning controller 21 are obtained so that the output of the system converges to the desired output. The detailed process of this step is as follows.

(a) In the first trial, the control input is set to '0'. That is, u _il ⁰ (t) = 0.

(b) Apply the input u ^k (t) (Equation 2) to the system at the kth iteration.

[Equation 2]

(c) The control input u _il ^{k + 1} (t) by the iterative learning controller at the next trial is updated using the following equation (3).

[Equation 3]

In Equation 3, y ^d (t) is a desired output, and l (·) is an input update law. This input update rule is described below.

(d) Repeat (b) and (c) above until the output error falls within the specified error range at all times t∈ [0, T].

4th step

For the n th desired output trajectory, the neural network is trained until the output u _fn ⁿ (t) of the front feed neural network controller 22 satisfies the following expression (4).

[Equation 4]

In Equation 4, u _fn ^n-1 (t) is the output of the front feed neural network controller 22 learned by the trajectories before the n th learning trajectory, and u _il ⁿ (t) is repeated for the n th learning trajectory. The locus obtained by the learning controller 21, epsilon, indicates the maximum allowable value of the error, respectively.

5th step

The error value that is not learned by the neural network controller 22 that feeds the input u _il ⁿ (t) by the iterative learning controller 21 u _il ⁿ (t) -u _fn ⁿ (t) + u _fn ^n-1 ( t) The two processes of the fourth step and the fifth step may be regarded as the movement of information from the short term memory to the long term memory.

6th step

Steps 3 to 5 are repeated for the other trajectories.

In the third step, the iterative learning controller 21 can obtain a control input for converging the output of the system to a desired output. However, this control input does not help any new output trajectory when the desired output trajectory is changed. In order for the information learned by the iterative learning controller 21 to be useful information for other trajectories, it is necessary to leave the learned information in the feeding neural network controller 22. The fourth step and the fifth step are processes for this, and in this process, the short-term memory-based information obtained by using the iterative learning controller 21 is transferred to the long-term memory-based front feeding neural network controller 22. By this front feeding neural network controller 22, the iterative learning control device according to the present invention can show good control performance even for other types of trajectories that have not been learned. The control method proposed by the present invention is similar to the memory characteristics of a person. In the case of human beings, they have general information through learning, and if they are in a similar situation, they perform a given task based on the previously learned information, and the errors occurring in the execution of the task are reduced through repetitive trials. Similarly, the front feeding neural network controller 22 of the iterative learning control device according to the present invention has information learned in the past and takes appropriate action based on the previously learned information even in a new situation. It is eliminated by repetitive learning by the repetitive learning controller 21. As the front feed neural network controller 22, a general neural network can be used. Here, an example of using a piecewise linearly trained network (PLTN) as the front feed neural network controller 22 will be described.

The iterative learning control method according to the present invention described above is applied to a nonlinear system represented by Equation 5 below to examine the convergence of the output.

[Equation 5]

In Equation 5, u = [u ₁ ,... , u _m ] ^T ∈ R ^m is the system input, y = [y ₁ ,.. , y _m } ^T ∈ ℃ ^m is the system output, x = [x ₁ ,… , x _n ] ^T ∈ R ⁿ is a system state variable. In addition, f: R ⁿ → ° C ⁿ , G: R ^{n × m} → ° C ⁿ , and h: R ⁿ → ° C ^m are all functions capable of differentiating infinitely. In this specification, the case where the input order and the output order are the same will be described.

If the relative order of the system defined by Equation 5 is x = x ⁰ , then {r ₁ ,... , r _m } From the definition of relative order, the system satisfies the following two conditions.

First condition

The following equation 6 holds for all x in the vicinity of x ⁰ .

[Equation 6]

Second condition

The m × m non-interfering matrix of Equation 7 below is a nonsingular matrix at x = x ⁰ .

[Equation 7]

For the system defined as above, there is the following theorem regarding the input update law that allows the output of the system to converge to the desired output during iterative learning control.

theorem

In the iterative learning control of the pseudo nonlinear system defined as in Equation 5, the input update law is set as in Equation 8.

[Equation 8]

In Equation 8, u _ff ^{k + 1} (t) is a feed-in control input in the k-th iteration learning, and S ^k : [0, T] → ° C ^{m × m} is a bounded function to obtain learning gain. E ^k (t) = [e ₁ ^k (t),... , e _m ^k (t)] The i th term of ^T ∈ ° C ^m is e _i ^k (t) = y _i ^d (t) -y _i ^k (t). Here, e _i ^{k (j)} (t) is the derivative of e _i ^k (t) j times.

If S ^k (t) in Equation 8 satisfies the following inequality as in Equation 9, the output y ^k (t) of the system is uniformly equal to the desired output y ^d (t) by the iterative learning control. Converge (Equation 10).

[Equation 9]

[Equation 10]

In order to use the above theorem, the iterative learning control apparatus according to the present invention as shown in FIG. 2 has an input update rule as shown in Equation 11 below.

[Equation 11]

In Equation 11, S ^k : [0, T] → ° C. ^{m × m} represents a learning gain as a bounded function, e _i ^k (t) = [e ₁ ^k (t),... , e _m ^k (t)] The i th term of ^T ∈ ° C ^m is e _i ^k (t) = y _i ^d (t) -y _i ^k (t). Here, e _i ^{k (j)} (t) is the derivative of e _i ^k (t) j times.

If S ^k (t), which is the learning gain of Equation 11, satisfies the inequality as in Equation 9 below, the output y ^k (t) of the system is equalized to the desired output y ^d (t) by the iterative learning control. (uniformly) converge. Proof of this is as follows.

proof

During the iterative learning process, the output of the front feed neural network controller 22 does not change. That is, u _fn ^{k + 1} (t) = u _fn ^k (t). When u _fn ^{k + 1} (t) is added to the left side of Equation 11 and u _fn ^k (t) is added to the right side of Equation 11, Equation 11 can be expressed as Equation 12 below.

[Equation 12]

In the iterative learning control system according to the invention shown in Figure ^{_{2, u il k (t)}} + u fn k (t) By defining the as feed forward inputs u _ff ^k (t), the equation (2) and the equation In Equation 12, Equation 11 may be expressed as Equation 13 below.

[Equation 13]

Equation 13 has the same shape as Equation 8, and according to the theorem described above, if S ^k (t), the learning gain in the apparatus according to the present invention, satisfies the inequality as in Equation 9 below, The output y ^k (t) of is uniformly converged to the desired output y ^d (t) by the iterative learning control.

In order for the system's output to converge exactly to the desired output, the initial conditions of the system and controller must be the same for each iteration.

Example

The iterative learning control method proposed in the present invention was applied to a two-axis robot manipulator.

The dynamics of the robot manipulator with m-degree of freedom can be represented by the Lagrange-Euler equation as shown in Equation 14 below.

[Equation 14]

In Equation 14, Are the joint angle, inertia matrix, Coriolis and centripetal force, force by gravity, and control input of robot manipulator, respectively. State variable When the output of the system is set to y (t) = θ (t), Equation 14 representing robot dynamics can be expressed as shown in Equation 5 above. When Equation 14 is expressed in the form of Equation 5, f (x), g (x), and h (x) are represented by Equation 15 below.

[Equation 15]

By differentiating the output y (t) = θ (t) twice, we get

[Equation 16]

The relative orders of the system {r ₁ ,... , r _m } = {2,… 2}, and the non-interfering matrix is J (x (t)) = M ⁻¹ (θ (t)).

When the PD controller having the proportional gain K _p and the derivative gain K _d is used as the feedback controller 20, the control input u (t) can be expressed by Equation 17 below.

[Equation 17]

In Equation 17, θ ^d (t) = y ^d (t) is a desired trajectory of each joint of the robot.

With reference to Equation 11, which is the input update law proposed in the present invention, the input update law of the present embodiment can be represented by the following equation (18).

Equation 18

In Equation 18, Is an estimate of the inertia matrix and α is a constant.

Therefore, the condition for convergence of the output trajectory to the desired trajectory can be expressed by Equation 19 below.

[Equation 19]

The mass of each axis of the horizontal motion two-axis robot menu plate used in this embodiment is m ₁ = m ₂ = 2.0 kg, and the length is l ₁ = l ₂ = 0.5m. The gain of the feedback controller 20 was selected as in Equation 20 below.

[Equation 20]

The desired trajectories for both axes are given by Equation 21 below.

[Equation 21]

The learning rate α was set at 0.8. The input of the front feeding neural network controller 22 is a derivative value of the reference input and the reference input, that is, to be. For learning the neural network, the derivatives of the output are approximated by Equation 22 below.

[Equation 22]

The results of this example are shown in FIGS. 3 and 4. (A) of each figure shows the locus of axis 1, and (b) of each figure shows the locus of axis 2. As shown to FIG.

FIG. 3 is a case where only the feedback controller 20 and the repetition learning controller 21 are used, and FIG. 4 is a case where the repeat control apparatus according to the present invention including the front feeding neural network controller 22 is used. This is the result of applying to the system of the present embodiment after learning. 3 and 4, k denotes the number of repetitions.

In the case of FIG. 4, the initial trajectory error is much smaller than in the case of FIG. 3. This means that by using the front feed neural network controller 22, it is possible to reduce the initial trajectory error even before repetitive learning, and shows the generalization capability of the front feed neural network controller 22. According to the iterative learning control method proposed in the present invention, since the initial trajectory error is small, the number of times the output trajectory is repeated to converge to the desired trajectory can be reduced. In addition, as the front feeding neural network controller 22 learns more trajectories, the performance of the iterative learning control device according to the present invention is improved.

As described above, the method and apparatus according to the present invention can not only allow the output of the system to converge to a desired output, but also store information obtained through the iterative learning control so that learning can be performed whenever the reference input trajectory is changed. Compensating the disadvantage of having to start again from the beginning, the new trajectory has the advantage that the output of the system to follow the reference input well.

Claims (7)

In the iterative learning control device to repeat the output of the system repeatedly performing a given task in a finite time interval to the desired output through repetitive learning based on the error of the past trial, A feedback controller that stabilizes the system in its initial state before learning is made; A short-term memory-based iterative learning controller for accurately converging the output of the system to the desired output through iterative learning; And And a forward feed neural network controller for storing information about a control input of the system obtained after the repetitive learning. The repetition learning control apparatus according to claim 1, wherein the front feeding neural network controller uses a general neural network as a neural network. The apparatus of claim 2, wherein the front feeding neural network controller uses a Piecewise Linearly Trained Network (PLTN) as a neural network. In the iterative learning control method to repeat the output of the system repeatedly performing a given task in a finite time interval to the desired output through repetitive learning based on the error of the past trial, Designing a feedback controller to stabilize the system; Setting a desired output y ^d (t) in a finite time interval t 시간 [0, T]; A third step of obtaining a control input u _il (t), t ∈ [0., T] by the iterative learning controller so that the output of the system converges to a desired output; a fourth step of training the neural network until the output u _fn ⁿ (t) of the front feeding neural network controller satisfies the following equation for the n th desired output trajectory; Where u _fn ^n-1 (t) is the output of the front feed neural network controller 22 learned by the trajectories before the n th learning trajectory, and u _il ⁿ (t) is the iterative learning controller 21 for the n th learning trajectory. The trajectory obtained by), ε respectively indicate the maximum allowable value of the error; A fifth value of changing the input u _il ⁿ (t) by the iterative learning controller to an error value not learned by the front feeding neural network controller u _il ⁿ (t) -u _fn ⁿ (t) + u _fn ^n-1 (t) step; And And a sixth step of repeating the third to fifth steps with respect to another trajectory. The method of claim 4, wherein the third step, (a) setting the control input to '0' in the first trial, ie u _il ⁰ (t) = 0; (b) applying an input u ^k (t) to the system during the k th iteration, as in the following equation; (c) updating the control input u _il ^{k + 1} (t) by the iterative learning controller at the next execution using the following equation; And Where y ^d (t) is the desired output and l (·) is the input update law and (d) repeating steps (b) and (c) until the output error falls within a predetermined error range at all times t∈ [0, T]. 6. The iterative learning control method according to claim 5, wherein the input update rule of step (c) is as follows. Where S ^k : [0, T] → ° C ^{m × m} represents the learning gain as a bounded function, e _i ^k (t) = [e ₁ ^k (t),... , e _m ^k (t)] The i th term of ^T ∈ R ^m is e _i ^k (t) = y _i ^d (t) -y _i ^k (t), and e _i ^{k (j)} (t) is e _i ^k (t) is the derivative of j times. The method of claim 6, wherein the learning gain S ^k (t) satisfies an inequality such as the following equation. Where I is an identity matrix and J (x ^k (t)) is when the system is represented by In the above equation, u = [u ₁ ,... , u _m ] ^T ∈ R ^m is the system input, y = [y ₁ ,.. , _m } ^T ∈ R ^m is the system output, x = [x ₁ ,… , x _n ] ^T ∈ R ⁿ is a system state variable. In addition, f: R ⁿ → ° C ⁿ , G: R ^{n × m} → R ⁿ , and h: R ⁿ → ° C ^m are all infinitely different functions. M × m non-interfering matrix, such as the following equation.