1 Introduction
Autonomous systems often need to operate in complex environments, of which a model is difficult or even impossible to derive from first principles. Learningbased techniques have become a promising paradigm to address these issues (Pillonetto et al., 2014). In particular, Gaussian processes (GPs) have been increasingly employed for system identification and control (Umlauft et al., 2018; Capone and Hirche, 2019; Berkenkamp and Schoellig, 2015; Deisenroth and Rasmussen, 2011). GPs possess very good generalization properties (Rasmussen and Williams, 2006), which can be leveraged to obtain dataefficient learningbased approaches (Deisenroth and Rasmussen, 2011; Kamthe and Deisenroth, 2018)
. By employing a Bayesian framework, GPs provide an automatic tradeoff between model smoothness and data fitness. Moreover, GPs provide an explicit estimate of the model uncertainty that is used to derive probabilistic bounds in control settings
(Capone and Hirche, 2019; Beckers et al., 2019; Umlauft and Hirche, 2020).A crucial performancedetermining factor of datadriven techniques is the quality of the available data. In settings where data is insufficient to achieve accurate predictions, new data needs to be gathered via exploration (Umlauft and Hirche, 2020)
. In classical reinforcement learning settings, exploration is often enforced by randomly selecting a control action with a predetermined probability that tends to zero over time
(Dayan and Sejnowski, 1996). However, this is generally inefficient, as regions of low uncertainty are potentially revisited in multiple iterations. These issues have been addressed by techniques that choose the most informative exploration trajectories (Alpcan and Shames, 2015; Ay et al., 2008; Burgard et al., 2005; Schreiter et al., 2015). The goal of these methods is to obtain a model that is globally accurate. While this is a reasonable aim for systems with a bounded stateaction space, it is unsuited for systems with unbounded ones, particularly if a nonparametric model is used. This is because a potentially infinite number of points is required to achieve a globally accurate model. Furthermore, in practice a model often only needs to be accurate locally, e.g., for stabilization tasks.
In this paper, we propose a model predictive controlbased exploration approach that steers the system towards the most informative points within a bounded subset of the stateaction space. By modeling the system with a Gaussian process, we are able to quantify the information inherent in each data point. Our approach chooses actions by approximating the mutual information of the system trajectory with respect to a discretization of the region of interest. This is achieved by first selecting the single most informative data point within the region of interest, then steering the system towards that point using model predictive control. Through this approximation, the solution approach is rendered computationally tractable.
2 Problem Statement
We consider the problem of exploring the state and control space of a discretetime nonlinear system with Markovian dynamics of the form
(1) 
where , and
are the system’s state vector and control vector at the
th time step, respectively. The system is disturbed by multivariate Gaussian process noise with , . The concatenation , where is employed for simplicity of exposition. The nonlinear function represents the known component of the system dynamics, e.g., a model obtained using first principles, while corresponds to the unknown component of the system dynamics.We aim to obtain an approximation of the function , denoted , which provides an accurate estimate of on a predefined bounded subset of the augmented state space . This is often required in practice, e.g., for local stabilization tasks.
3 Gaussian Processes
In order to faithfully capture the stochastic behavior of (1), we model the system as a Gaussian process (GP), where we employ measurements of the augmented state vector as training inputs, and the differences as training targets.
A GP is a collection of dependent random variables variables, for which any finite subset is jointly normally distributed
(Rasmussen and Williams, 2006). It is specified by a mean function and a positive definite covariance function , also known as kernel. In this paper, we set without loss of generality, as all prior knowledge is already encoded in . The kernel is a similarity measure for evaluations of , and encodes function properties such as smoothness and periodicity.In the case where the state is a scalar, i.e., , given training input samples and training outputs
, the posterior mean and variance of the GP corresponds to a onestep transition model. Starting at a point
, the difference between the subsequent state and the known component is normally distributed, i.e.,(2) 
with mean and variance given by
respectively, where , and the entries of the covariance matrix are computed as , .
In the case where the state is multidimensional, we model dimension of the state transition function using a separate GP. This corresponds to the assumption that the state transition function entries are conditionally independent. For simplicity of exposition, unless stated otherwise, we henceforth assume . However, the methods presented in this paper extend straightforwardly to the multivariate case.
3.1 Performing multistep ahead predictions
The GP model presented in the previous section serves as a onestep predictor given a known test input . However, if only a distribution is available, the successor states’ distribution generally cannot be computed analytically. Hence, the distributions of future states cannot be computed exactly, but only approximated, e.g., using Monte Carlo methods (Candela et al., 2003)
. Alternatively, approximate computations exist that enable to propagate the GP uncertainty over multiple time steps, such as momentmatching and GP linearization
(Deisenroth et al., 2015). In this paper, we employ the GP mean to perform multistep ahead predictions, without propagating uncertainty, i.e., , . However, the proposed method is also applicable using models that propagate uncertainty, e.g., momentmatching or linearizationbased methods (Deisenroth et al., 2015).3.2 Quantifying utility of data
In order to steer the system along informative trajectories, we need to quantify the utility of data points in the augmented state space . To this end, we consider the mutual information between observations at training inputs and evaluations at reference points . Here is a discretization of the bounded subset . Formally, the mutual information between and is given by
(3) 
respectively denote the differential entropy of and the conditional differential entropy of given . In practice, computing (3) for a multistep GP prediction is intractable. However, we can obtain the single most informative data point with respect to by computing the unconstrained minimum of
(4) 
where denotes the determinant of a square matrix. In settings with unconstrained decision spaces, sequentially computing a minimizer of (4) has been shown to yield a solution that corresponds to at least of the optimal value (Krause et al., 2008).
4 The LocAL algorithm
The system dynamics (1) considerably limit the decision space at every time step . Furthermore, after a data point is collected, both the GP model and mutual information change. Hence, we employ a model predictive control (MPC)based approach to steer the system towards areas of high information. Ideally, at every MPCstep , we would like to minimize (3) with respect to a series of inputs . However, this is generally infeasible, limiting its applicability in an MPC setting. Hence, we consider an approximate solution approach that sequentially computes the most informative data point by minimizing (4) separately from the MPC optimization. This is achieved as follows. At every time step , an unconstrained minimizer of (4) is computed. Afterwards, the MPC computes the approximate optimal inputs by minimizing a constrained optimization problem that penalizes the weighted distance to the reference point
(5) 
The ensuing state is then measured, the GP model is updated, and the procedure is repeated. These steps yield the Localized Active Learning (LocAL) algorithm, which is presented in Algorithm 1.
The square weight matrix should be chosen such that the MPC steers the system as close to as possible. This represents a systemdependent task. Alternatively, can be chosen such that the MPC cost function corresponds to a quadratic approximation of the mutual information, e.g., such that holds for .
The computational complexity of the overall algorithm can be adjusted in various manners. For example, a new input can be computed only after a predefined number of time steps, as opposed to every time step. This is a commonly employed technique in MPC (Camacho, 2013). Furthermore, the discretization can be made coarse to facilitate the solution of the first optimization step.
4.1 Sensitivity analysis
We now provide a sensitivity analysis of (4) for a single time step. Let be the difference between the augmented state and the most informative data point at time step . Moreover, let denote the corresponding difference in mutual information, and assume the kernel is upper bounded by the scalar . Then, there exists a constant , such that
(6) 
holds, where .
Proof Sketch
Inhalt…
s.t. 
5 Numerical Experiments
In this section, we apply the proposed algorithm to four different dynamical systems. We begin with a toy example, with which we can easily illustrate the explored portions of the state space. Afterwards, we apply the proposed approach to a pendulum, a cartpole, and a synthetic model that generalizes the mountain car problem. The exploration is repeated times for each system using different starting points sampled from a normal distribution. To quantify the performance of each approach, we compute the root mean square model error (RMSE) on
points sampled from a uniform distribution on the region of interest
.We employ a squaredexponential kernel in all examples, and train the hyperparemters online using gradientbased log likelihood maximization (Rasmussen and Williams, 2006). We employ an MPC horizon of , and choose weight matrix for the MPC optimization step as
where
denotes the standard deviation of the GP kernel corresponding to the
th dimension, and denote the corresponding lengthscales. In order to ease the computational burden, we apply the first inputs computed by the LocAL algorithm before computing a new solution.We additionally explore each system using a onestep greedy entropybased cost function, as suggested in Koller et al. (2018) and Schreiter et al. (2015), and compare the results. In all three cases, the LocAL algorithm yields a better model in the regions of interest than the entropybased algorithm.
5.1 Toy Problem
Consider the continuoustime nonlinear dynamical system
(7) 
with state space and input space . We are interested in obtaining an accurate dynamical model within the region . To obtain a discretetime system in the form of (1), we discretize (7) with a discretization step of and set the prior model to . The results are displayed in Figure 1.
The LocAL algorithm yields a substantial improvement in model accuracy in every run. This is because the system stays close to the region of interest during the whole simulation. By contrast, the greedy entropybased method covers a considerably more extensive portion of the state space. This comes at the cost of a poorer model on , as indicated by the respective RMSE.
5.2 Surface exploration
We apply the LocAL algorithm to the dynamical system given by
(8) 
This setting can be seen as a surface exploration problem, i.e., an agent navigates a surface to learn its curvature. We aim to obtain an accurate model of the dynamics within
To run the LocAL algorithm, we employ a discretization step of and set the prior model to . The results are shown in Figure 2.
The LocAL algorithm manages to significantly improve its model after time steps, while the entropybased strategy does not yield any improvement. This is because every variable of the state space is unbounded, i.e., the state space can be explored for a potentially infinite amount of time without ever reaching the region of interest .
5.3 Pendulum
We now consider a twodimensional pendulum, whose state is given by the angle and angular velocity . The input torque is constrained to the interval . Our goal is to obtain a suitable model within the region given by
Obtaining a precise model around this region is particularly useful for the commonly considered task of stabilizing the pendulum around the upward position . The results are depicted in Figure 3.
The RMSE indicates that the LocAL algorithm yields a similar model improvement in every run. The model obtained with the entropybased strategy, by contrast, exhibits a significantly stronger variance.
5.4 Cartpole
We apply the LocAL algorithm to the cartpole system (Barto et al., 1983). In this example, the state space is given by , where is the cart velocity, is the pendulum angle, and is the angular velocity of the pendulum. Here we ignore the cart position without loss of generality, as it has no influence on the system dynamics. The region of interest is
Similarly to the pendulum case, obtaining an accurate model on this region is useful to address the balancing task. The discretization step is set to , the prior model is . The results are shown in Figure 4.
Similarly to the pendulum case, the model obtained with the LocAL algorithm exhibits low variance compared to the one obtained with the entropybased approach. This is because the region of interest is explored more thoroughly with our approach.
6 Conclusion
A technique for efficiently exploring bounded subsets of the stateaction space of a system has been presented. The proposed technique aims to minimize the mutual information of the system trajectories with respect to a discretization of the region of interest. It employs Gaussian processes both to model the unknown system dynamics and to quantify the informativeness of potentially collected data points. In numerical simulations of four different dynamical systems, we have demonstrated that the proposed approach yields a better model after a limited amount of time steps than a greedy entropybased approach.
Acknowledgements
This work was supported by the European Research Council (ERC) Consolidator Grant ”Safe datadriven control for humancentric systems (COMAN)” under grant agreement number 864686 .
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