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Procedia Computer Science 216 (2023) 213–220
1877-0509 © 2023 The Authors. Published by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0)
Peer-review under responsibility of the scientific committee of the 7th International Conference on Computer Science and Computational
Intelligence 2022
10.1016/j.procs.2022.12.129
10.1016/j.procs.2022.12.129 1877-0509
© 2023 The Authors. Published by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0)
Peer-review under responsibility of the scientic committee of the 7th International Conference on Computer Science and
Computational Intelligence 2022
Available online at www.sciencedirect.com
ScienceDirect
Procedia Computer Science 00 (2022) 000–000
www.elsevier.com/locate/procedia
1877-0509 © 2022 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license
(https://creativecommons.org/licenses/by-nc-nd/4.0)
Peer-review under responsibility of the scientific committee of the 7th International Conference on Computer Science and
Computational Intelligence 2022
7th International Conference on Computer Science and Computational Intelligence 2022
Long short-term memory (LSTM) model-based reinforcement
learning for nonlinear mass spring damper system control
Santo Wijayaa*, Yaya Heryadia, Yulyani Arifina, Wayan Supartab, Lukasc
aComputer Science Department, BINUS Graduate Program – Doctor of Computer Science,Bina Nusantara University, Jakarta 11480, Indonesia.
bDepartment of Electrical Engineering, Faculty of Industrial Technology, Institut Teknologi Nasional Yogyakarta, Yogyakarta 55281, Indonesia.
cCognitive Engineering Research Group (CERG), Universitas Katolik Indonesia Atma Jaya, Jakarta 12930, Indonesia.
Abstract
The Neural Networks (NN) model which is incorporated in the control system design has been studied, and the results show
better performance than the mathematical model approach. However, some studies consider that only offline NN model learning
and does not use the online NN model learning directly on the control system. As a result, the controller's performance decreases
due to changes in the system environment from time to time. The Reinforcement Learning (RL) method has been investigated
intensively, especially Model-based RL (Mb-RL) to predict system dynamics. It has been investigated and performs well in
making the system more robust to environmental changes by enabling online learning. This paper proposes online learning of
local dynamics using the Mb-RL method by utilizing Long Short-Term Memory (LSTM) model. We consider Model Predictive
Control (MPC) scheme as an agent of the Mb-RL method to control the regulatory trajectory objectives with a random shooting
policy to search for the minimum objective function. A nonlinear Mass Spring Damper (NMSD) system with parameter-varying
linear inertia is used to demonstrate the effectiveness of the proposed method. The simulation results show that the system can
effectively control high-oscillating nonlinear systems with good performance.
© 2022 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license
(https://creativecommons.org/licenses/by-nc-nd/4.0)
Peer-review under responsibility of the scientific committee of the 7th International Conference on Computer Science
and Computational Intelligence 2022
Keywords: Model-based Reinforcement Learning; Random-Shooting Policy; LSTM; MPC; Nonlinear System
* Corresponding author. Tel.:+62-812-9906-4921.
E-mail address: santo.wijaya001@binus.ac.id
Available online at www.sciencedirect.com
ScienceDirect
Procedia Computer Science 00 (2022) 000–000
www.elsevier.com/locate/procedia
1877-0509 © 2022 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license
(https://creativecommons.org/licenses/by-nc-nd/4.0)
Peer-review under responsibility of the scientific committee of the 7th International Conference on Computer Science and
Computational Intelligence 2022
7th International Conference on Computer Science and Computational Intelligence 2022
Long short-term memory (LSTM) model-based reinforcement
learning for nonlinear mass spring damper system control
Santo Wijayaa*, Yaya Heryadia, Yulyani Arifina, Wayan Supartab, Lukasc
aComputer Science Department, BINUS Graduate Program – Doctor of Computer Science,Bina Nusantara University, Jakarta 11480, Indonesia.
bDepartment of Electrical Engineering, Faculty of Industrial Technology, Institut Teknologi Nasional Yogyakarta, Yogyakarta 55281, Indonesia.
cCognitive Engineering Research Group (CERG), Universitas Katolik Indonesia Atma Jaya, Jakarta 12930, Indonesia.
Abstract
The Neural Networks (NN) model which is incorporated in the control system design has been studied, and the results show
better performance than the mathematical model approach. However, some studies consider that only offline NN model learning
and does not use the online NN model learning directly on the control system. As a result, the controller's performance decreases
due to changes in the system environment from time to time. The Reinforcement Learning (RL) method has been investigated
intensively, especially Model-based RL (Mb-RL) to predict system dynamics. It has been investigated and performs well in
making the system more robust to environmental changes by enabling online learning. This paper proposes online learning of
local dynamics using the Mb-RL method by utilizing Long Short-Term Memory (LSTM) model. We consider Model Predictive
Control (MPC) scheme as an agent of the Mb-RL method to control the regulatory trajectory objectives with a random shooting
policy to search for the minimum objective function. A nonlinear Mass Spring Damper (NMSD) system with parameter-varying
linear inertia is used to demonstrate the effectiveness of the proposed method. The simulation results show that the system can
effectively control high-oscillating nonlinear systems with good performance.
© 2022 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license
(https://creativecommons.org/licenses/by-nc-nd/4.0)
Peer-review under responsibility of the scientific committee of the 7th International Conference on Computer Science
and Computational Intelligence 2022
Keywords: Model-based Reinforcement Learning; Random-Shooting Policy; LSTM; MPC; Nonlinear System
* Corresponding author. Tel.:+62-812-9906-4921.
E-mail address: santo.wijaya001@binus.ac.id
214 Santo Wijaya et al. / Procedia Computer Science 216 (2023) 213–220
208 Wijaya et al. / Procedia Computer Science 00 (2022) 000–000
1. Introduction
Research into NN has a lengthy history. Psychologist Donald Hebb [1] invented the oldest description of NN and
created the Hebbian Learning scheme based on the brain plasticity process in the 1940s. In 1958, the perceptron,
which is also known as the primary component of neural networks today and constructed as a two -layer neural
network without a training method, was invented by Frank Rosenblatt [2]. In 1975, Paul Werbos created and
published in his Ph.D. thesis of the back-propagation algorithm that is now the most extensively used [3]. In the
following years, several NN architectures emerged, such as feedforward neural networks [4], Recurrent Neural
Networks (RNN) [5], and others. A specific derivative model from RNN to solve the vanishing gradient was
invented by Schmidhuber, which is called Long Short-Term Memory (LSTM) [6]. The LSTM is of great interest in
the control system because they are particularly well-suited for prediction based on time series data, as there may be
unpredictable delays or uncertainties between events in time series.
Pisa et al. [7] proposed the use of LSTM as a model in the Internal Model Control (IMC) scheme applied to a
wastewater treatment plant. The result showed that the LSTM-based IMC approach improves the control metrics
compared to the traditional PI controller. However, the authors used an offline learning approach to train the LSTM
model, which is not robust enough to cope with the uncertainties that may occur in the system, especially
uncertainties in the nonlinear system that might degrade control performance further. Sanaz et al. [8] proposed that
the Mf-RL method be used to identify a dynamic model of a three-phase power converter by using the state-space
Neural Network (ssNN) with Particle Swarm Optimization (PSO) to update the weights under parameter mismatch
and incorporate them into a Model Predictive Control (MPC) scheme for control of a three-phase power converter.
In the paper, the Mf-RL method succeeded in increasing the accuracy of the model in MPC model-based control.
However, this approach did not directly calculate the control actions required in the control system.
The Reinforcement Learning (RL) method is a research topic that has been intensively investigated recently. The
RL may be utilized to learn from the limited labeled data and derive additional essential insights from the unlabeled
data [9]. Nagabandi et al. [10] proposed combining Mb-RL and Mf-RL methods to accomplish various MuJoCo
[11] locomotion tasks. The author first utilizes the Mb-RL method combined with MPC for the initial movement
task of the locomotion model to achieve robustness control with minimum sample data. The Mb -RL approach
enables online learning to update a model based on reward, and the model can be used directly in the control system.
The paper proposed a combined offline and online learning approach utilizing the Mb-RL method. In offline
learning, the LSTM architecture will be determined to have minimum weight parameters based on the dataset
obtained from the controlled object of the system. Simple LSTM architecture is needed to learn faster in the online
learning approach. Then online learning is done for every defined batch size to update the LSTM model online
based on the given control action
() and the measured state response of the system
(). We consider an MPC
scheme with a random-shooting policy to search for the minimum objective function of trajectories error by
calculating a set of random control actions within the given prediction horizon. The first control action of a set of
random control actions that minimize the objective function is applied to the system every time step. We also
consider the LSTM to learn from local dynamics, which is the rate of changes at each time step, to overcome the
difficulties of learning when the difference of the state is too small, especially when the time step difference is small
[10]. The NMSD system is used as an object to be controlled to show the effectiveness of the proposed method. The
paper's primary contribution is two-fold: The online learning of local dynamics using the Mb-RL method utilizing
LSTM as model-based in the MPC scheme with the Random-Shooting policy; Demonstrates the effectiveness of the
proposed method for trajectory control of the NMSD system with parameters varying in the spring inertia and
random disturbance.
2. Methods
The research method conducted in this paper consists of three phases: the initial phase, the design phase, and the
simulation phase, as shown in Fig. 1. In the initial phase, we build the dynamic simulator using an NMSD system,
then run it to obtain the dynamics data of systems input and outputs. The data will be divided into training, testing,
and validation dataset to determine the minimum nodes of LSTM architecture with offline training. After that, in the
design phase, the MPC is designed to use the LSTM model as a predictor to calculate the optimized action to
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Wijaya et al. / Procedia Computer Science 00 (2022) 000–000 209
minimize the objective function in the pre-determined horizon window. In addition, the MPC acts as an agent in the
RL framework designed with a random-shooting policy. Lastly, in the simulation phase, we show the effectiveness
of the proposed method in controlling the NMSD system to accomplish a regulatory objective.
2.1. Nonlinear Mass Spring Damper System
The NMSD system is an oscillation system consisting of a mass spring and damper elements that store kinetic
and potential energy. This model can be used in many modeling approaches to model the system's dynamics, such as
Flexible Bodies [12] and Mechanical Systems [13]. The NMSD system is particularly a mechanical system studied
intensively in the control theory. However, it is still unavailable in the benchmark environment, such as MuJoco.
In the paper, the arbitrary free-body diagram of the NMSD system satisfying the motion of Newton’s second law
is shown in Fig. 2. It is assumed that the NMSD system consists of three masses or inertia of (kg), in this
case identical masses considered where with initial positions . Damper
damping constant also assumed to be identical with respectively. Since the nonlinear
spring is considered, the stiffness function is given by with parameter varying linear
inertia of and nonlinear inertia of . The control inputs of force to the NMSD
system are and . Then, is a disturbance that disrupts the motion of the NMSD system; in this case, it is
considered a random value between to . It is assumed that there is no measurement data for the
position , and the control actions are bounded between to .
Equation (1), (2), and (3) shows the NMSD system nonlinear differential equation function after
derivation with Newton's law. SciPy Library will be used in the paper to obtain the dynamics of the NMSD
differential equations using the scipy.integrate.odeint function with simulation time step seconds,
absolute error = and relative error = .
(1)
(2)
(3)
2.2. LSTM Neural Network Model
The LSTM model based on RNN is well-suited for learning time series or sequence data. Previous works [14, 15]
comprehensively describe the LSTM network. This paper's LSTM model is constructed with Tensorflow and Keras
Library [16] in the Google Colaboratory environment. Three layers are constructed, the Lambda layer, the LSTM
layer and the Dense layer. The LSTM parameters are setup as follows, #return_sequence = True,
#activation_function = Leaky ReLU ( ), #recurrent_activation = Sigmoid, #look_back = , #loss_function =
Mean Square Error, #optimizer = Adam (learning rate = ), #epochs = , #batch_size = .
Fig. 1. Overall research method of the paper
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Parameterization of an LSTM dynamics function is defined as
, where denotes the vector of
network weights. The input to the model is the control action , the measured disturbance () and the
measured state of the NMSD system at every time step . The output of the model is the predicted state
response at next time step The paper also employed learning a dynamics function based on
the rate of change in measurement data, , proposed by Nagabandi et al. [10], for better prediction
results in the case of the small-time step . In this case, the predicted next state becomes .
We considered the learned dynamics function is defined as
. The additional input to
the model is the rate of change , and . Following that, the output of the
model is changed to the rate of change of the predicted state response at next time step .
Thus, totally the LSTM model require 9 input neurons in the Lambda layer and 2 output neurons in the Dense layer.
The paper also preprocessed the input dataset based on the Z-score normalization method [17] to standardize
mean = and standard deviation = for fast training of the LSTM model. It is formulated as , where
denotes the normalize data, denotes the source date, denotes the dataset mean, and denotes the dataset
standard deviation. The Z-score normalization is chosen because the dataset feature distribution obtained from the
NMSD system does not contain extreme outliers. Fig. 3 shows the overall configuration of the LSTM neural
network model with input-output parameters and data preprocessing.
2.3. Model Predictive Control
The MPC scheme is well known in the control theory and has been intensively investigated in combination with
the Machine Learning method and Neural Network model [18, 19]. The MPC scheme utilizes a model to make the
system dynamics prediction in the pre-determined prediction horizon () and calculate a sequence of control action
inside the prediction window to minimize the objective function () by optimization, apply the first control action
from the calculated sequence at each time step, and then re-do the algorithm at the next time step. The input to the
MPC controller () is the vector of from the NMSD system.
Fig. 4 shows the design of the LSTM model-based MPC scheme. The learned dynamics function of the LSTM
model
received vector input () and calculate the predicted state response rate of
change inside the prediction window () where the length of the horizon is every time step (). Then the
MPC scheme calculate the objective function () of error between reference trajectory () and the predicted state
response (). The minimization problem of is solved with optimization solver to calculate a set of optimized
control actions within the prediction horizon (). But, to get the solution using optimization directly is difficult
considering the model's nonlinearity. In this case, we employed a Random-Shooting policy [20] where set of
control action vector () with length are generated, recalculate to obtain a set of the objective function (),
then find the minimum objective function (Ω) within the set, and get the first control action () of in which
has the minimum objective function, finally apply the to the system for each time step.
Fig. 3. LSTM Neural Network Model with Input – Output Parameters
Fig. 2. Free-body diagram of the mass spring damper model
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2.4. Control Performance Evaluation
There are three control performance parameters in the paper. The trajectory deviation performance is
calculated as a Root Mean Square Error
, where denotes the simulation time
steps, denotes the reference values, denotes the predicted output values. The total impulse is formulated as
and the action smoothness is formulated as
, where denotes the
control action.
2.5. Model-based Reinforcement Learning Framework
In the paper, the agent is the MPC controller, the model is the LSTM model, and the control inputs represent the
action to the environment. The maximum reward is awarded if the system follows the reference trajectory or is
considered as a minimization of RMSE. A framework of the Mb-RL in the paper, as shown in Fig. 5 and Algorithm
1. By exploration with random actions applied to the system and obtaining the exploration dataset of the first
to be normalized with Z-score, which can be used to train
of the LSTM model, re-train
for every size.
At , the MPC controller start to calculate optimized actions, then applied them to the system and obtained
the exploitation dataset, re-train
is continue using the exploitation dataset. Retraining use 1 epochs, and use the
weights of
from the previous iteration. The algorithm stopped at maximum simulation time.
Algorithm 1 Mb-RL Framework
01:
Initialize
based on offline learning.
16:
Generate random sets of random action sequences vector
02:
Set simulation time (), normalization time (),
17:
with length , .
03:
batch size (), control time ().
18:
For to
04:
Generate random action data points ().
19:
Applied
to predict the next time step state.
05:
For to
20:
Calculate objective func.
.
06:
If isInteger() then
21:
Find minimum objective function .
07:
Do normalization to obtain new and .
22:
End For
08:
End If
23:
Get the first control action from the minimum .
09:
If ( ) AND isInteger() then
24:
Set
10:
Retrain
and clear
25:
End If
11:
End If
26:
Simulate the NMSD system with
12:
If ( ) then
27:
ODEsolver seconds.
13:
Set
28:
Populate dataset (), batch .
14:
Elseif ( ) then
29:
Reset objective function
15:
Get the current dataset ().
30:
End For
Fig. 4. Design of MPC Scheme with Random-Shooting Policy
Fig. 5. The Model-based Reinforcement Learning Framework
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3. Results and Discussions
3.1. Offline Learning
At this phase, the LSTM minimum configuration will be determined. Then, random action data points are
generated to analyze the NMSD system and collect training, validation, and testing datasets through the system
dynamic simulator. Fig. 6 shows the response of the NMSD system using 20.000 data points for training data. Fig. 8
shows the response of the NMSD system using 100.000 data points for training data which can be generated again
for testing data. Finally, the normalization using Z-score is done using 20.000 data points before applying the
dataset to the model for training, and will be calculated using the training dataset as shown in Fig. 7.
Once the dataset is obtained, offline training of the LSTM model is conducted. Fig. 9 shows the results of
training and validation of the LSTM model with hidden node layers: 3, 5, and 10. Several activation functions have
been experimented with (tanh, sigmoid, ReLU, Leaky ReLU with =0.3); the best activation function is Leaky
ReLU with = 0.5, notably, the model has a reasonable convergence rate and good validation result with only five
layers of hidden nodes. However, the result with activation function Leaky ReLU with = 0.5 only is shown due to
the space limitation.
Employing online learning means that a test of model accuracy only needed within the length of the prediction
horizon of MPC compared to the testing dataset. In this case, we consider using =10, the model's accuracy is
based on the predicted ten-step-ahead state output. Even with a short horizon, it has been proven that an MPC
controller can successfully control a robot's movement in a MuJoco environment [10]. Fig. 10 shows the test result
of the ten-step-ahead prediction output and the RMSE = 1.62×10−4. Thus, we selected the LSTM with five layers
of hidden nodes. The weight parameters obtained in the offline training will warm-start the model in the simulation
phase.
Fig. 7. Z-score normalization result of and
Fig. 6. Response of the NMSD system for training data
Fig. 8. Response of the NMSD system for validation data
Fig. 9. Training and validation result of the LSTM model
Fig. 10. Test result of 10 step-ahead prediction using LSTM 5 nodes
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3.2. Online Learning
In the simulation phase, we setup timesteps, timesteps, data
points, timesteps, horizons. It is also assumed that the actuators arbitrarily have a limit of
before control action is started and after control action is
started. Random disturbance of . Lastly, we also applied randomly distributed Gaussian noise to the
measurement data of the NMSD system output, and there is a random parameter varying on the inertia of the spring
N/m. Then, there are two scenarios are considered for the random-shooting policy to
generate sets of random actions, , randomly. Case A, to generate a set of randomly between
and case B to generate a set of randomly between . The control scenario is a regulatory
objective to follow the pre-determined reference trajectory. There are three reference trajectories set up after control
time () is reached, (1) IF THEN AND , (2) IF
THEN AND , (3) IF THEN
AND .
As shown in Fig. 11, the simulation result indicated that the proposed method effectively controls the NMSD
system according to the objective control scenario. Table 1 shows the control performance result based on Random-
Shooting Policy (RSP) cases A and B. Notably, the RMSE and total Impulse performance in case A were better than
in case B. In contrast, the performance of action smoothness in case B was better than in case A. The other
important parameter is simulation time; case A, which has more extensive sets of to search minimum objective
function, produces better control performance but with a longer simulation time. We also compare the proposed
LSTM model with a Multi-Layer Perceptron (MLP) model for better insights. In this comparison, case A of the
Random-Shooting policy is selected. Furthermore, we selected the MLP model with 30 hidden nodes ( ) as
it contains similar weight parameters as the LSTM model with five hidden nodes ( ). Table 2 shows the
control performance comparison; notably, the LSTM model performs better in RMSE and action smoothness but
with a longer simulation time.
Table 1. Control Performance of RSP Cases A and B
Case
Sim_Time
A
B
Table 2. Control Performance of RSP Case A of LSTM and MLP
Model
Sim_Time
LSTM
5 nodes
MLP
5 nodes
Fig. 11. Simulation result of Random Shooting Case A
220 Santo Wijaya et al. / Procedia Computer Science 216 (2023) 213–220
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4. Conclusions
Several studies considered only offline learning of the NN model and did not use the NN model directly in the
control system. As a result, the environmental changes of the system over time might not be incorporated into the
model; hence it may degrade the controller performance. The paper proposed online learning of local dynamics
using the Mb-RL method utilizing the LSTM model to make the system more robust to environmental changes by
enabling online learning. The MPC as an agent, provides a reward function calculated based on a random-shooting
policy, which generates random control action and finds the minimum objective function through a closed-loop
strategy. The online learning scheme retrains the LSTM model and minimizes model mismatch with the system. The
NMSD system with varying parameters is employed to show the effectiveness of the pro posed method. The
simulation results showed that the proposed method effectively controls the system with good performance.
Extending the proposed method to a real-world mechanical system and investigating how the framework works
on the real-time control system are considered for future research. The proposed approach requires considerable
computing power to shorten the calculation time. This issue motivates the adoption of cloud-based control
structures, where the controller is operated remotely as a cloud-based service, enabling a real-time control system.
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