I Introduction
Networked control systems (NCSs) are rapidly gaining in popularity, both in academia and industry. Advancements in control strategies and network technologies enable the systems to closely interact with their environment and share data. Treating communication as a shared resource, as suggested in [1], is an important step to scale NCSs to problems involving many agents.
In this paper, we consider NCSs with multiple spatially distributed agents, whose dynamics are independent, but that are coupled through a joint control objective and communicate via a shared network. Figure 1 depicts two agents representative for one communication link in such an NCS. While communication between agents is beneficial or even necessary for coordination (e.g., formation control [2], or multiagent balancing [3]), the network constitutes a shared and scarce resource and, hence, its usage shall be limited.
Eventtriggered state estimation (ETSE) [4, 5, 6, 7, 8] has been proposed to reliably exchange sensor or state data between agents, but with limited interagent communication. Many ETSE methods utilize dynamics models to predict other agents’ states or measurements (see Fig. 1), in order to anticipate their behavior without the need for continuous data transmissions. Event triggering rules then ensure that an update is sent whenever the prediction is not good enough. Hence, the capability of the dynamics model in making accurate predictions critically determines how much communication can be saved.
In order to improve prediction accuracy, we propose to combine ETSE with model learning. We develop a framework, where an agent uses its inputoutput data to update the model of its dynamics, and communicates this model to other agents. With improved models, it is possible to achieve superior prediction accuracy and thus lower communication even further compared to standard ETSE, especially when facing changing dynamics. Since learning and communicating models can be resourceintensive operations themselves, we develop triggering rules that quantify the model accuracy and decide when to learn. The result is an eventtriggered learning scheme, which sits on top of the standard ETSE framework (cf. Fig. 1).
Contributions
In detail, this paper makes the following contributions:

introducing the novel idea of eventtriggered learning;

derivation of a datadriven learning trigger based on statistical properties of intercommunication times;

probabilistic guarantees ensuring the effectiveness of the proposed learning triggers;

concrete implementation of eventtriggered learning for linear Gaussian dynamic systems; and

demonstration of improved communication behavior in numerical simulations and hardware experiments on a cartpole system.
Related work
Various eventtriggering approaches have been proposed for improving resource usage in state estimation; for recent overviews see, e.g., [9, 10, 11, 12] and references therein. Approaches differ, among others, in the type of models that are used for predictions. The sendondelta protocol [13] assumes a trivial constant model by triggering when the difference between the current and lastly communicated value passes a threshold. This protocol is extended to linear predictions in [14], which are obtained from approximating the signal derivative from data. More elaborate protocols use dynamics models of the observed process, which typically leads to more effective triggering [15, 10]. The vast majority of ETSE methods (e.g., [4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 15]) employ linear dynamics models, which we also consider herein. Nonlinear prediction models are used in [17, 18], for example. None of these works considers model learning to improve prediction accuracy, as we do herein. While we implement eventtriggered learning for linear Gaussian problems, the general idea also applies to other types of estimation problems and dynamics models.
In order to obtain effective learning triggers, we take a probabilistic view on intercommunication times (i.e., the time between two communication events) and trigger learning experiments whenever the expected communication differs from the empirical. A similar interpretation of intercommunication times is considered in [19], where it is used to infer stability results. We extend this idea with statistical convergence results to design the event trigger for learning.
Deciding if learning is necessary is related to the problem of change detection, which seeks to identify times when the distribution governing a random process changes. Change detection has received considerable attention in statistical literature; see [20] for an overview. While one main application of this work is fault detection, we use the proposed trigger to initiate learning experiments to capture the changed distribution. Furthermore, we do not analyze the stochastic process directly, but instead target intercommunication times, which are a natural onedimensional feature in the ETSE framework and also amenable to theoretical analysis, which is incorporated in the trigger design. Applications of change detection in the context of NCSs focus on fault detection in the presence of delays [21], or network design and performance in general [22]. Iterative learning control has been proposed for multiagent problems in [23] to improve eventtriggered control for repetitive tasks, however, the idea of triggering learning when there is significant change is new to the best of the authors’ knowledge.
Ii Eventtriggered Learning
In this section, we explain the main idea of eventtriggered learning using the schematic in Fig. 1. The figure depicts a canonical problem, where one agent (‘Sending agent’ on the left) has information that is relevant for another agent at a different location (‘Receiving agent’). This setting is representative for remote monitoring scenarios or two agents of a multiagent network, for instance. For resourceefficient communication, a standard ETSE architecture is used (shown in black). The main contribution of this work is to incorporate learning into the ETSE architecture. By designing an event trigger also for model learning (in green), learning tasks are performed only when necessary. We next explain the core components of the proposed framework.
The sending agent in Fig. 1 monitors the state of a dynamic process (either directly measured or obtained via state estimation) and can transmit this state information to the remote agent via a network link. In order to save network resources, an eventbased protocol is used. The receiving agent makes modelbased predictions (‘Model’ and ’Prediction’) to anticipate the state at times of no communication. The sending agent implements a copy of the same prediction and compares it to the current state in the ‘Event Trigger’, which triggers a communication whenever the prediction deviates too much from the actual state. This general scheme is standard in ETSE literature (see [9, 10, 11, 12] and references therein). The effectiveness of this scheme in reducing communication will generally depend on the accuracy of the prediction, and thus the quality of the model.
The key idea of this work is to trigger learning experiments and learn improved models from data when prediction performance of the current model is poor. The updated model is then shared with the remote agent to improve its predictions. Because performing a learning task is costly itself (e.g., involving computation and communication resources, as well as possibly causing deviation from the control objective), we propose eventtriggering rules also for model learning.
While the idea of using event triggering to save communication in estimation or control is quite common by now, this work proposes event triggering also on a higher level. Triggering of learning tasks yields improved prediction models, which are the basis for ETSE at the lower level.
The general idea of eventtriggered learning applies to diverse scenarios. In the following, we make the idea concrete for ETSE of linear Gaussian systems (introduced in Sec. III). For this case, model learning can be solved with standard leastsquares estimation (Sec. IV). We propose a trigger design that is based on probabilistic analysis of intercommunication times (Sec. V and VI). By means of numerical examples and physical experiments on a cartpole system (Sec. VII and VIII), we show that communication can effectively be reduced through eventtriggered learning.
Iii Linear Gaussian Problem
In this section, we make the problem of eventtriggered learning precise for linear Gaussian dynamic systems. For this, we introduce all standard elements of the ETSE architecture shown in Fig. 1 (black).
Iiia Process
We assume the linear dynamics (‘Process’ in Fig. 1)
(1) 
with discretetime index , state , control input , system matrices ,
, and independent identically distributed (i.i.d.) Gaussian random variables
. For simplicity, we assume the state can be measured, but the framework readily extends to the case where the state is locally estimated.The system is assumed stabilizable. Hence, stable closed loop dynamics can be achieved through state feedback
(2) 
where is a known reference. The reference can be used for tracking problems or to excite the system, which is particularly important during learning experiments and will be further discussed later. The closed loop dynamics then reads
(3) 
with stable matrix .
IiiB Predictions
The ‘Prediction’ block, which is implemented on the sending and the receiving agent (cf. Fig. 1), utilizes a model to predict the true process. After initializing , both agents iterate
(4) 
The prediction (4) is deterministic and deviates from the true value (3) over time. We bound this error through an event trigger: whenever a certain error threshold is reached, the actual state is transmitted and the prediction reset
(5) 
where the binary variable
denotes positive () or negative triggering decisions.IiiC Event Trigger
The ‘Event Trigger’ on the sending agent has access to both, prediction and true state. It can thus track the error
(6) 
and trigger a communication when it becomes too large:
(7) 
This way, communication is reduced to the necessary instants given by a significant prediction error.
IiiD Problem Formulation
Precise models are key to reduce communication rates. In this work, we address mismatches between model and true process , which may stem from imprecise initial models or changing dynamics, for example. The development of a model learning framework to improve the effectiveness of ETSE in reducing communication is the main objective of this paper. This includes, in particular, a decision rule (the learning trigger ) for deciding when a new model ought to be learned.
Iv Model Learning
This section addresses learning of the dynamics model (4) from inputoutput data of the system (3) (’Model Learning’ in Fig. 1). In the numerical and physical experiments of Sec. VII and VIII, the data is obtained from dedicated learning experiments. In other settings, data could be recorded during normal operation. We here present standard least squares to estimate the system matrices, but any other technique for linear system identification [24] could be used.
Thus, the ordinary least squares (OLS) estimator for the model matrices yields
Due to the autoregressive structure of the system (3), we can ensure identifiability for certain types of input signals . Sinusoidal input signals with increasing frequencies (chirp) are known to yield invertible matrices (see condition of persistent excitation, e.g., [24]). We thus use chirp signals as reference
to excite the system when a learning experiment shall be performed. Since the estimator is designed to achieve minimal variance it is straightforward to obtain
by averaging over the squared residuals.The specific learning method only affects the ’Model Learning’ block in Fig. 1 and has no influence on the decision whether learning is necessary (’Event Trigger’). Hence, the general eventtriggered learning approach is agnostic to what learning or identification method is used.
V Foundations in Stochastic Analysis
In this section, we briefly discuss theoretical results from stochastic analysis as background for deriving the event trigger for model learning in the next section. More detailed treatments of stochastic processes and stochastic differential equations (SDEs) can be found in [25] and [26], for example.
Intercommunication times (i.e., the time between two triggering events (7)) will play a key role in the proposed triggering scheme for model learning. In the following, we express intercommunication times as random variables (stopping times) and deduce statistical properties directly from the model (in Sec. VB). To be amenable to stopping time analysis, we shall first describe the eventtriggered estimation scheme in terms of continuoustime SDEs (Sec. VA).
The analysis in this section is done under the assumption that the prediction model (4) is perfect (i.e., , , and ), which will be motivated in the next section.
Va OrnsteinUhlenbeck processes
Consider (3) with , i.e.,
(11) 
Because of the perfect model assumption, the reference signal cancels out in the later analysis step (15) and is hence irrelevant. Transforming system (11) to continuous time yields an OrnsteinUhlenbeck (OU) process
(12) 
with , , and a multidimensional Wiener process. The matrices , can be computed from , the sampling time of the discretetime system (11), and the covariance (refer to [26, 27] for details). Continuous and discretetime processes coincide in the sense that they have the same distribution for any given sampling time.
The OU process (12) can be written explicitly in integrated form as
(13) 
With the perfect model assumption, the discretetime predictions (4) transform to
(14) 
in continuous time. Using (13) and (14), we thus obtain for the continuoustime prediction error
(15) 
By comparing to (13), it follows that is an OU process starting in zero. We will further analyze the behavior of and make a connection between stopping times of this process and communication behavior.
VB Stopping Times
Due to the existence and uniqueness result for SDEs [25]
, continuity of sample paths is assured for OU processes. We will leverage this property to pinpoint the exact moment the process crosses a given threshold. Exactly like in the discretetime setting, state information is communicated whenever
, which resets to . Accordingly, the stopping time is defined as the first time when the stochastic process exits an dimensional sphere with radius :(16) 
The random variable hence corresponds to the intercommunication time, the time between communication events. Because they coincide in the setting herein, ‘intercommunication time’ and ‘stopping time’ will be used synonymously hereafter. In Sec. VI, learning triggers will be proposed that are based on a comparison of empirical and expected intercommunication times. The computation of expected intercommunication times is discussed next.
For certain classes of SDEs, it is possible to compute as the solution
to specific partial differential equations (PDEs). The following lemma states this result for the OU process.
Lemma 1
Assume the boundary value problem
(17)  
with gradient , Hessian , and an dimensional sphere with radius , has a unique bounded solution . Then, .
The result is obtained by using the more general AndronovVittPontryagin formula from [26] and adapting it to the OU process (as was also done in [19]).
The onedimensional case can be solved analytically (see Example 4.2 on page 111 in [26]) and gives interesting insights. From the solution, dependencies between and the parameters and can be seen. The magnitude of stochastic effects is pushing the process out of the domain, while the stable part drives it back to zero. Hence, more stable leads to longer stopping times, while larger leads to shorter.
For general dimension , one could attempt solving the PDE (17), which is, however, challenging because it is nonlinear and possibly high dimensional. Typically, one has to resort to numerical solutions, which usually yield the whole function and are computationally intensive. Since we actually only require , an alternative is to approximate through Monte Carlo simulations, which we use in the experiments herein. Because the error is always reset to at triggering instants for the scenario herein, we shall omit the dependence on the initial value and write instead of in the following.
Vi Event Trigger for Model Learning
In this section, we design the event trigger for model learning (green ’Event Trigger’ block in Fig. 1). We first discuss the general idea of how to trigger learning experiments, and then state two concrete instances.
Via General idea
The learning trigger must reliably detect when the prediction accuracy of the current model is poor, and thus learning a new model from data promises improved predictions. We cannot directly compare the model to the true process because it is unknown. Owing to the stochasticity of the process, it is also not straightforward to quantify model accuracy from data. Since we ultimately care about the communication behavior that results from the models, we propose to utilize intercommunication times to trigger learning in the following way.
If an accurate model is used, average intercommunication times are expected to be equal to (as computed according to Sec. VB under the assumption ). If, however, observed intercommunication times deviate on average from , we conclude an inaccurate model and trigger a learning experiment (). Hence, the proposed learning trigger compares empirically observed intercommunication times with the expected intercommunication time computed from the current model, and triggers learning when the two are significantly apart.
ViB Exact learning trigger
Based on the foregoing discussion, we propose the following learning trigger:
(18) 
where indicates that a new model shall be learned; is computed according to Sec. VB assuming a perfect model; and are the last empirically observed intercommunication times ( the duration between two triggers (7)). The moving horizon is chosen to yield robust triggers. The threshold parameter quantifies the error we are willing to tolerate. We denote (18) as the exact learning trigger because it involves the exact expected value , as opposed to the trigger derived in the next subsection.
Even though the trigger (18) is meant to detect inaccurate models, there is always a chance that the trigger fires not due to an inaccurate model, but instead due to the randomness of the process (and thus randomness of intercommunication times ). Even for a perfect model, (18) may trigger at some point. This is inevitable due to the stochastic nature of the problem. Therefore, we propose to chose to yield effective triggering with a userdefined confidence level. For this, we make use of Hoeffding’s inequality:
Lemma 2 (Hoeffding’s inequality [28])
Let be i.i.d. bounded random variables with for all . Then
(19) 
The application of Hoeffding’s inequality requires intercommunication times to be upper bounded by some . This is easily enforced in practice by triggering at the latest when
is reached. Moreover, we specify a maximal probability
that we are willing to tolerate for the difference being caused through randomness. We then have the following result for the trigger (18):Theorem 1 (Exact learning trigger)
Let the parameters , , and be given, and assume a perfect model . For
(20) 
we have
(21) 
Substituting (20) for into the right hand side of Hoeffding’s inequality yields the desired result.
The theorem quantifies the expected convergence rate of the empirical mean to the expected value for a perfect model. This result can be used as follows: the user specifies the desired confidence level , the number of intercommunication times considered in the empirical average, and the maximum intercommunication time . By choosing as in (20), the exact learning trigger (18) is guaranteed to make incorrect triggering decisions (false positives) with a probability of less than .
ViC Approximated learning trigger
As discussed in Sec. VB, obtaining can be difficult and computationally expensive. Instead of solving the PDE (17), we propose to approximate by sampling . For this, we simulate the OU process (15) until it reaches a sphere with radius for times, and average the obtained stopping times . Hence,
(22) 
This yields the approximated learning trigger
(23) 
This approximation leads to a different choice of .
Theorem 2 (Approximated learning trigger)
Let the parameters , , , and be given, and assume . For
(24) 
we have
(25) 
We insert , use the triangle inequality, and additivity of probability measures:
where the last step follows with and Hoeffding’s inequality for and . We avoid solving the PDE (17), but the obtained trigger (23) is less efficient than (18), i.e., the required sample size for equal accuracy is higher.
Remark 1 (practical implementation)
In the experiments reported in the next sections, we use trigger (23), which is easier to compute than (18). In order to improve the trigger’s robustness, we trigger learning experiments only when (23) holds for a certain duration rather than instantaneous. This way, the trigger is less prone to unmodeled shortterm effects, which also decreases the probability of false positives.
Vii Simulation
This section illustrates the proposed eventtriggered learning scheme with a numerical example. For this, we consider two agents as in Fig. 1 and a onedimensional process
(26) 
with sample time , reference , and . To simulate imperfect model information, we assume that only a slightly perturbed model is available for making predictions,
(27) 
We implement all main components of the proposed eventtriggered learning scheme as shown in Fig. 1; that is, the ETSE components as described in Sec. III with , model learning as in Sec. IV, and the approximated learning trigger (23). As trigger parameters, we chose a confidence level , as well as , , and . Theorem 2 then yields . In order to obtain a robust learning trigger, in the empirical intercommunication time average, , is reset after a learning experiment, and (23) is only checked after having observed at least stopping times .
Figure 2 illustrates the behavior of the eventtriggered estimation and learning system for this numerical example. The actual communication is captured by the empirical intercommunication time, which is the inverse of the communication rate and computed as the moving average , where is equal to or the number of stopping times observed since the start or last learning experiment. Since the initial model (27) is inaccurate, we observe a significant deviation between empirical intercommunication time and modelbased one (22). At around , 2000 intercommunication times have been observed, and the trigger (23) is checked. Because the triggering condition (righthand side of (23)) is clearly true, a learning experiment is performed (until about ). Directly after learning, the moving average of the empirical intercommunication time is reset, hence its fluctuations. With the updated model, empirical and modelbased stopping time coincide well; hence, no further learning experiment is triggered thereafter.
Viii Physical Experiment
We demonstrate the proposed eventtriggered learning approach in experiments on a cartpole system (see Fig. 3), which is a common benchmark in control [29]. We consider balancing about the upright equilibrium, whose dynamics are approximately linear (1) with position , velocity , angle , angular velocity as its states, and motor voltage as input . The dynamics are stabilized with a standard controller (2), and we consider remote state estimation of the stabilized process (3) as per the architecture in Fig. 1.
For making predictions in ETSE, we initially use a linear model supplied by the manufacturer [30], and we take as triggering threshold. As the learning trigger, we use (23), where the empirical mean stopping time is computed over a fixed duration , which is an alternative to including a fixed number of samples in the average. While the physical system does not exactly match the assumption in Sec. VA (e.g., no strictly linear Gaussian dynamics), the theoretical analysis of Sec. VI is still helpful in guiding our choice of a triggering threshold . We choose in the trigger (23), which is slightly lower then what we would obtain through Theorem 2. Since we additionally enforce the triggering condition to be true for , we reduce the risk of false positives.
Figure 4 shows the results of the experiment. While the model used for predictions is not perfect, there is still a good match between the empirical and modelbased stopping time: the empirical stopping time mostly remains within the gray area and thus no learning is triggered initially. At s, we add weights on top of the pole (cf. Fig. 4), thus changing the system dynamics. The empirical intercommunication time drops (i.e., indicating more communication) and clearly captures the change in dynamics. After the triggering condition being true (the black graph being outside the gray area) for , a learning experiment is triggered. A new model is then learned according to Sec. IV (an openloop model is identified instead of a closedloop one, which makes no difference here). After updating the model, the empirical and modelbased stopping times coincide very well. The eventtriggered learning thus successfully detected and compensated for the changed dynamics, and reduced communication thereafter.
Ix Conclusion
Eventtriggered learning is proposed in this paper as a novel concept to trigger model learning when needed. The concept is applied to eventtriggered state estimation (ETSE) and shown to lead to reduced communication in simulation and physical experiments. For this setting, we obtained (provably) effective learning triggers by means of statistical stopping time analysis.
This paper is the first to develop the concept of eventtriggered learning. Extending the method to nonlinear dynamic systems is an obvious next step we plan for future work. While eventtriggered learning has been motivated as an extension to ETSE, the concept generally addresses the fundamental question of when to learn, and thus potentially has much wider relevance.
Acknowledgment
The authors thank Felix Grimminger for his support with the cartpole system, and Alonso Marco for his help with speeding up the learning experiments.
References
 [1] J. P. Hespanha, P. Naghshtabrizi, and Y. Xu, “A survey of recent results in networked control systems,” Proc. IEEE Special Issue on Technology of Networked Control Systems, vol. 95, no. 1, pp. 138–162, Jan. 2007.
 [2] Y. Cao, W. Yu, W. Ren, and G. Chen, “An overview of recent progress in the study of distributed multiagent coordination,” IEEE Trans. on Industrial informatics, vol. 9, no. 1, pp. 427–438, 2013.
 [3] S. Trimpe and R. D’Andrea, “The Balancing Cube: A dynamic sculpture as test bed for distributed estimation and control,” IEEE Control Systems Magazine, vol. 32, no. 6, pp. 48–75, Dec. 2012.
 [4] J. K. Yook, D. M. Tilbury, and N. R. Soparkar, “Trading computation for bandwidth: reducing communication in distributed control systems using state estimators,” IEEE Transactions on Control Systems Technology, vol. 10, no. 4, pp. 503–518, Jul. 2002.
 [5] S. Trimpe and R. D’Andrea, “An experimental demonstration of a distributed and eventbased state estimation algorithm,” in 18th IFAC World Congress, 2011, pp. 8811–8818.
 [6] J. Sijs and M. Lazar, “Event based state estimation with time synchronous updates,” IEEE Transactions on Automatic Control, vol. 57, no. 10, pp. 2650–2655, 2012.
 [7] J. Wu, Q.S. Jia, K. H. Johansson, and L. Shi, “Eventbased sensor data scheduling: Tradeoff between communication rate and estimation quality,” IEEE Transactions on Automatic Control, vol. 58, no. 4, pp. 1041–1046, 2013.
 [8] S. Trimpe and R. D’Andrea, “Eventbased state estimation with variancebased triggering,” IEEE Transactions on Automatic Control, vol. 59, no. 12, pp. 3266–3281, 2014.
 [9] D. Shi, L. Shi, and T. Chen, EventBased State Estimation: A Stochastic Perspective. Springer, 2015.
 [10] S. Trimpe and M. C. Campi, “On the choice of the event trigger in eventbased estimation,” in Int. Conf. on Eventbased Control, Communication, and Signal Processing, 2015, pp. 1–8.
 [11] J. Sijs, B. Noack, M. Lazar, and U. D. Hanebeck, “Timeperiodic state estimation with eventbased measurement updates,” in EventBased Control and Signal Processing. CRC Press, 2016.
 [12] S. Trimpe, “Eventbased state estimation: An emulationbased approach,” IET Control Theory & Applications, vol. 11, no. 11, pp. 1684–1693, Jul. 2017.
 [13] M. Miskowicz, “Sendondelta concept: an eventbased data reporting strategy,” Sensors, vol. 6, no. 1, pp. 49–63, 2006.
 [14] Y. S. Suh, “Sendondelta sensor data transmission with a linear predictor,” Sensors, vol. 7, no. 4, pp. 537–547, 2007.
 [15] J. Sijs, L. Kester, and B. Noack, “A study on event triggering criteria for estimation,” in 17th International Conference on Information Fusion, July 2014, pp. 1–8.
 [16] M. Muehlebach and S. Trimpe, “Distributed eventbased state estimation for networked systems: An lmi approach,” IEEE Transactions on Automatic Control, vol. 63, no. 1, pp. 269–276, Jan. 2018.
 [17] S. Trimpe and J. Buchli, “Eventbased estimation and control for remote robot operation with reduced communication,” in IEEE Int. Conf. on Robotics and Automation, 2015, pp. 5018–5025.
 [18] M. MartínezRey, F. Espinosa, A. Gardel, and C. Santos, “Onboard eventbased state estimation for trajectory approaching and tracking of a vehicle,” Sensors, vol. 15, no. 6, pp. 14 569–14 590, 2015.
 [19] Y. Xu and J. P. Hespanha, “Communication logics for networked control systems,” in American Control Conference, 2004, pp. 572–577.
 [20] T. L. Lai, “Sequential changepoint detection in quality control and dynamical systems,” Journal of the Royal Statistical Society. Series B (Methodological), pp. 613–658, 1995.
 [21] Y. Wang, S. X. Ding, H. Ye, and G. Wang, “A new fault detection scheme for networked control systems subject to uncertain timevarying delay,” IEEE Transactions on Signal Processing, vol. 56, no. 10, pp. 5258–5268, 2008.
 [22] M. Haghighi and C. J. Musselle, “Dynamic collaborative change point detection in wireless sensor networks,” in Int. Conf. on CyberEnabled Distributed Computing and Knowledge Discovery, 2013, pp. 332–339.
 [23] T. Zhang and J. Li, “Eventtriggered iterative learning control for multiagent systems with quantization,” Asian Journal of Control, 2016.
 [24] L. Ljung, System identification: Theory for the User. Prentice Hall, New Jersey, 1999.
 [25] B. Øksendal, “Stochastic differential equations,” in Stochastic differential equations. Springer, 2003, pp. 65–84.
 [26] Z. Schuss, Theory and Applications of Stochastic Processes: An Analytical Approach, ser. Applied Mathematical Sciences. Springer New York, 2009.
 [27] K. J. Åström, Introduction to stochastic control theory. Dover Publications, 2006.
 [28] U. von Luxburg and B. Schölkopf, Statistical Learning Theory: Models, Concepts, and Results. Amsterdam, Netherlands: Elsevier North Holland, May 2011, vol. 10, pp. 651–706.
 [29] O. Boubaker, “The inverted pendulum benchmark in nonlinear control theory: A survey,” International Journal of Advanced Robotic Systems, vol. 10, no. 5, p. 233, 2013.
 [30] Quanser Inc., “IP02  selferecting single inverted pendulum (SESIP)  linear experiment #6: PV and LQR control  instructor manual.”
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