Bayesian Safe Learning and Control with Sum-of-Squares Analysis and Polynomial Kernels

This paper proposes a control strategy for input constrained second order systems with one unstable mode. For this class of systems, the largest region of attraction under bounded control is bounded in one direction whilst infinite in the other. Standard MPC type control with guaranteed stability either requires large control horizons or typically yields a much smaller region of attraction. We show that by extending MPC concepts a control strategy with a horizon of one is developed that provides asymptotic stability with the largest possible region of attraction under bounded control.

This article describes a method for modelling non-linear dynamic systems from measurement data. The method merges the linear local model blending approach in the velocity-based linearisation form with Bayesian Gaussian process (GP) modelling. The new Fixed-Structure GP (FSGP) model has a predetermined linear model structure with varying and probabilistic parameters represented by GP models. These models have several advantages for the modelling of local model parameters as they give us adequate results, even with small data sets. Furthermore, they provide a measure of the confidence in the prediction of the varying parameters and information about the dependence of the parameters on individual inputs. The FSGP model can be applied for the extended local linear equivalence class of non-linear systems. The obtained non-linear system model can be, for example, used for control-system design. The proposed modelling method is illustrated with a simple example of non-linear system modelling for control design.

This contribution presents a new development in the design of control system based on evolving Gaussian process (GP) models . GP models provide a probabilistic, nonparametric modelling approach for black-box identification of nonlinear dynamic systems. They can highlight areas of the input space where prediction quality is poor, due to the lack of data or its complexity, by indicating the higher variance around the predicted mean. GP models contain noticeably less coefficients to be optimised than commonly used parametric models. While GP models are Bayesian models, their output is normal distribution, expressed in terms of mean and variance. Latter can be interpreted as a confidence in prediction and used in many fields, especially in control system. Evolving GP model is the concept approach within which various ways of model adaptations can be used. Successful control system needs as much as possible data about process to be controlled . If the prior knowledge about the system to be controlled is scarce or the system varies with time or operating region, this control problem can be solved with an iterative method which adapts model with information obtained with streaming data and concurrently optimises hyperparameter values. This contribution provides: a survey of adaptive control algorithms for dynamic systems described in publications where GP models have been used for control design, a novel and improved closed-loop controller with evolving GP models and an example for the illustration of proposed control algorithm.

For the problem of stabilizing an inverted pendulum with restricted travel, a saturating control law is developed that satisfies the amplitude constraint on the cart and has a large region of attraction. The explicit expression of the region of attraction is also obtained. The analysis and design are performed based on the linearized model of the system, as in the study of Lin et al. The control law has two design parameters: T and k, which are both positive. As T approaches 0, the region of attraction approaches the maximal one. These parameters are chosen to optimize the transient response of the closed-loop system considering the input constraint, observation noise, etc. The effectiveness of the control law is demonstrated by simulations and experiments.

Earth observation from satellite sensory data pose challenging problems, where machine learning is currently a key player. In recent years, Gaussian Process (GP) regression and other kernel methods have excelled in biophysical parameter estimation tasks from space. GP regression is based on solid Bayesian statistics, and generally yield efficient and accurate parameter estimates. However, GPs are typically used for inverse modeling based on concurrent observations and in situ measurements only. Very often a forward model encoding the well-understood physical relations is available though. In this work, we review three GP models that respect and learn the physics of the underlying processes in the context of inverse modeling. First, we will introduce a Joint GP (JGP) model that combines in situ measurements and simulated data in a single GP model. Second, we present a latent force model (LFM) for GP modeling that encodes ordinary differential equations to blend data-driven modeling and physical models of the system. The LFM performs multi-output regression, adapts to the signal characteristics, is able to cope with missing data in the time series, and provides explicit latent functions that allow system analysis and evaluation. Finally, we present an Automatic Gaussian Process Emulator (AGAPE) that approximates the forward physical model via interpolation, reducing the number of necessary nodes. Empirical evidence of the performance of these models will be presented through illustrative examples of vegetation monitoring and atmospheric modeling.

Quantitative prediction of subjective pain intensity from whole-brain fMRI data using Gaussian processes

PARP1 is one of six enzymes required for the highly error-prone DNA repair pathway microhomology-mediated end joining (MMEJ) and needs to be inhibited when over-expressed. In order to study the PARP1 inhibitory effect of fused tetracyclic or pentacyclic dihydrodiazepinoindolone derivatives (FTPDDs) by quantitative structure-activity relationship technique, six models were established by four kinds of methods, heuristic method, gene expression programming, random forester, and support vector regression with single, double, and triple kernel function respectively. The single, double, and triple kernel functions were RBF kernel function, the integration of RBF and polynomial kernel functions, and the integration of RBF, polynomial, and linear kernel functions respectively. The problem of multi-parameter optimization introduced in the support vector regression model was solved by the particle swarm optimization algorithm. Among the models, the model established by support vector regression with triple kernel function, in which the optimal R2 and RMSE of training set and test set were 0.9353, 0.9348 and 0.0157, 0.0288, and R2cv of training set and test set were 0.9090 and 0.8971, shows the strongest prediction ability and robustness. The method of support vector regression with triple kernel function is a great promotion in the field of quantitative structure-activity relationship, which will contribute a lot to designing and screening new drug molecules. The information contained in the model can provide important factors that guide drug design. Based on these factors, six new FTPDDs have been designed. Using molecular docking experiments to determine the properties of new derivatives, the new drug was ultimately successfully designed.

The excellent generalisation ability of the Support Vector Machine (SVM) algorithm has made it one of the most popular statistical learning theories in supervised machine learning. The classification accuracy and effectiveness of SVM is highly sensitive to the kernel function used during the training process. This paper compares linear, polynomial, and Gaussian kernel functions for evaluating their contribution to SVM for accurately and effectively classifying healthy and faulty status of rotating machinery. A three-phase induction motor and a four-stroke diesel engine were considered as the machinery for this study. Acoustic signals coming from these machines were collected using microphones and Fast Fourier Transform (FFT) was used to extract the magnitudes of the dominant frequency components of the signals. The extracted ominant frequency components are considered as acoustic signatures and their variations are taken as condition monitoring parameters. The results show that with the second-order polynomial kernel function, SVM achieved an accuracy of at least 2.4% greater than the other kernel functions with 1.2% less training time. Furthermore, the third-order polynomial kernel function found to be the second best choice.

This article considers the problems of robust stability and stabilization for a class of fractional-order linear time-invariant systems via 0<α<1 subject to input saturation with convex polytopic uncertainties, by a kind of [Formula: see text]–proportional–integral–derivative control synthesis. A new saturation-based stability condition is suggested to estimate the stable region ([Formula: see text]) and region of attraction (ε) using the ellipsoid approach. The main concept of the presented strategy is based on transforming the proportional–integral–derivative controller design problem into the static output feedback control problem. Gronwall–Bellman lemma and sector bounded condition of the saturation function is used for stability investigation and stabilization of such systems. The static output feedback control laws are obtained with the aid of a non-iterative linear matrix inequality implication. From the static output feedback controller, the proportional–integral–derivative controller is then restored. Stability problem is addressed as an optimization problem to obtain the controller gains and the stable region, as well as the largest region of attraction. Some theorems and numerical examples are provided to illustrate the results of proposed mechanism.

In this work, we discuss the modeling, control, and implementation of a rimless wheel with torso. We derive and compare two control methodologies: a discrete-time controller (DT) that updates the controls once-per-step and a continuous-time controller (CT) that updates gains continuously. For the discrete controller, we use least-squares estimation method to approximate the Poincaré map on a certain section and use discrete-linear-quadratic-regulator (DQLR) to stabilize a (closed-form) linearization of this map. For the continuous controller, we introduce moving Poincaré sections and stabilize the transverse dynamics along these moving sections. For both controllers, we estimate the region of attraction of the closed-loop system using sum-of-squares methods. Analysis of the impact map yields a refinement of the controller that stabilizes a steady-state walking gait with minimal energy loss. We present both simulation and experimental results that support the validity of the proposed approaches. We find that the CT controller has a larger region of attraction and smoother stabilization as compared with the DT controller.

For the challenging task of modeling actual complex motions, we propose a new class of Gaussian process (GP) models that are data-driven and also take into account prior knowledge of the motion intention. As a theoretical basis, we show that the GP regression is mathematically equivalent to regularized least-squares estimation for random functions with known prior means. Compared with the popular GP models in machine learning literature, the proposed GP motion model priors with conditional kernels have at least two advantages: 1) they are nonstationary and more applicable to represent complex motions by integrating the basic kinematic principles; 2) conditional kernels are further devised by incorporating the motion intent so that the resultant GP models are more versatile and would expectedly entail more accurate trajectory prediction. A superior property of the GP models with conditional kernels is found to improve the computational efficiency. Finally, illustrative examples are provided to show the superiority of the proposed motion models and to verify the theoretical results given in the paper.

Sums-of-Squares optimization represents an important tool for the direct computation of a local Lyapunov function for a nonlinear dynamic system. Specifically, it can certify a sub-levelset of the cost-to-go from an optimal feedback controller like the Linear Quadratic Regulator (LQR), geometrically an ellipsoid in the state space, as Region of Attraction (ROA) of the closed-loop system. More complex robotic tasks however require switching control to first take the system into the ROA before invoking the LQR stabilizer. In this paper, we propose computationally efficient measures of the ROA distance based on quadratic and conic optimization to effectively supervise such a trajectory as it approaches the ROA. As a one-dimensional condensate of the multi-dimensional state trajectory, monitoring the ROA distance evolution allows us to early detect deviations, e.g. due to input saturation or time delay, in order to quickly take corrective action such as replanning. Importantly, computing the ROA distance adds only a small overhead on top of the ROA calculation itself and can be done concurrently.

Support Vector Machine (SVM) in dealing with a classification problem, separates classes using decision boundaries with the primary objective of establishing a large margin between support vectors of the respective class groups; it utilizes kernels to achieve non-linear decision boundaries. This current work examines the performance of four SVM kernels (Sigmoid, Linear, Radial Basis Function (RBF) and Polynomial kernel functions) in addressing classification problems using two datasets from two domains. The two datasets are the Knowledge Discovery in Dataset (KDD) and a set of features extracted from voiced and unvoiced frames. The Polynomial kernel function had the best classification performance on the KDD dataset with accuracy and precision of 99.77% and 99.8% respectively but recorded the worst performance against the voice-feature dataset with an accuracy of 74.96%. Inductively, the polynomial kernel can be best suited for some classification datasets but can return the worst classification performance on another classification dataset. The RBF shows consistent high performance across the two data domains with accuracies of 96.04% and 99.77% and can be considered a general-purpose kernel guaranteed to yield satisfactory classification performance regardless of the dataset type or data domains. The performance of polynomial kernels over the two separate datasets supports the “No Free Launch Theorem”, which when applied to machine learning, means that if an algorithm performs well over a class of problem, it may have worse performance on other class of problem. This implies that there might not be one specific machine learning algorithm that gives the best possible performance for a set of problems, it is therefore important for researchers to try out various algorithms before concluding on the best possible result on any dataset.

Elevated temperatures limit the peak performance of systems because of frequent interventions by thermal throttling. Non-uniform thermal states across system nodes also cause performance variation within seemingly equivalent nodes leading to significant degradation of overall performance. In this paper we present a framework for creating a lightweight thermal prediction system suitable for run-time management decisions. We pursue two avenues to explore optimized lightweight thermal predictors. First, we use feature selection algorithms to improve the performance of previously designed machine learning methods. Second, we develop alternative methods using neural network and linear regression-based methods to perform a comprehensive comparative study of prediction methods. We show that our optimized models achieve improved performance with better prediction accuracy and lower overhead as compared with the Gaussian process model proposed previously. Specifically we present a reduced version of the Gaussian process model, a neural network–based model, and a linear regression–based model. Using the optimization methods, we are able to reduce the average prediction errors in the Gaussian process from $4.2^\circ$ C to $2.9^\circ$ C. We also show that the newly developed models using neural network and Lasso linear regression have average prediction errors of $2.9^\circ$ C and $3.8^\circ$ C respectively. The prediction overheads are 0.22, 0.097, and 0.026 ms per prediction for reduced Gaussian process, neural network, and Lasso linear regression models, respectively, compared with 0.57 ms per prediction for the previous Gaussian process model. We have implemented our proposed thermal prediction models on a two-node system configuration to help identify the optimal task placement. The task placement identified by the models reduces the average system temperature by up to $11.9^\circ$ C without any performance degradation. Furthermore, these models respectively achieve 75, 82.5, and 74.17 percent success rates in correctly pointing to those task placements with better thermal response, compared with 72.5 percent success for the original model in achieving the same objective. Finally, we extended our analysis to a 16-node system and we were able to train models and execute them in real time to guide task migration and achieve on average 17 percent reduction in the overall system cooling power.

This paper introduces an approach to classification of RNA-seq read counts using grey relational analysis (GRA) and Bayesian Gaussian process (GP) models. Read counts are transformed to microarray-like data to facilitate normal-based statistical methods. GRA is designed to select differentially expressed genes by integrating outcomes of five individual feature selection methods including two-sample t-test, entropy test, Bhattacharyya distance, Wilcoxon test and receiver operating characteristic curve. GRA performs as an aggregate filter method through combining advantages of the individual methods to produce significant feature subsets that are then fed into a nonparametric GP model for classification. The proposed approach is verified by using two benchmark real datasets and the five-fold cross-validation method. Experimental results show the performance dominance of the GRA-based feature selection method as well as GP classifier against their competing methods. Moreover, the results demonstrate that GRA-GP considerably dominates the sparse Poisson linear discriminant analysis classifiers, which were introduced specifically for read counts, on different number of features. The proposed approach therefore can be implemented effectively in real practice for read count data analysis, which is useful in many applications including understanding disease pathogenesis, diagnosis and treatment monitoring at the molecular level.