Solving the Control Synthesis Problem Through Supervised Machine Learning of Symbolic Regression

A new two-step approach for solving a control system synthesis problem by symbolic regression methods

In this thesis, a number of search techniques are proposed as a solution for program and discrete controller synthesis (DCS). Classic synthesis techniques facilitate exhaus- tive search, while genetic programming has recently proven the potential of generic search techniques. But is genetic programming the right search technique for the synthesis prob- lem? In this thesis we challenge this belief and argue in favor of simulated annealing, a different class of general search techniques. We show that, in hindsight, the success of genetic programming has drawn from what is arguably a hybrid between simulated annealing and genetic programming, and compare the fitness of classic genetic program- ming, the hybrid form, and pure simulated annealing. Our experimental evaluation suggests that pure simulated annealing offers better results for automated programming than techniques based on genetic programming. Discrete Controller Synthesis (DCS) and Program Synthesis have similar goals: they are automated techniques to infer a control strategy and an implementation, respectively, that is correct by construction. We also investigate the application of the search tech- niques that we have been used for program synthesis for the computation of deterministic strategies solving symbolic Discrete Controller Synthesis (DCS) problems, where a model of the system under control is given along with desired objective behaviours. We experi- mentally confirm that relative performance results are similar to program synthesis, and give a complexity analysis of our simulated annealing algorithm for symbolic DCS. From the performance results we obtain, we draw the conclusion that simulated annealing, when combined with efficient model-checking techniques, is worth further investigating to solve symbolic DCS problems. A tool is designed to explore the parameter space of different synthesis techniques. Besides using it to synthesise a discrete control strategies for reactive systems (controller synthesis) and for protocol adapters for the coordination of different threads (software synthesis), we can also use it to study the influence of turning various screws in the syn- thesis process. For simulated annealing, PranCS allows the user to define the behaviour of the cooling schedule. For genetic programming, the user can select the population size.

The problem of optimal control synthesis of complex nonlinear and multidimensional systems oriented on the building of integrated control systems with the adaptation and self-organization properties is of much current interest. The development of optimization methods employed in control problems demonstrated their effectiveness and has led to the construction of algorithms of approximate solution of optimization problems that significantly extends the class of control problems that can be set. In this work we consider a synthesis of optimal control for affine nonlinear system, in which a special emphasis is placed on the degenerate problems of optimal control. Our approach is based on the extension of the method of minimization of generalized work function by properly transforming the initial nondegenerate problem of optimal control into degenerate problem of optimal control synthesis. The search for the solution is based on the suboptimal strategy that includes the construction of minimizing sequences that converge interval-wise to the optimal solution of the initial problem. We obtain some optimality conditions as well as the necessary and sufficient conditions of existence of optimal solution.

<p>Symbolic regression (SR) is a function identification process, the task of which is to identify and express the relationship between the input and output variables in mathematical models. SR is named to emphasise its ability to find the structure and coefficients of the model simultaneously. Genetic Programming (GP) is an attractive and powerful technique for SR, since it does not require any predefined model and has a flexible representation. However, GP based SR generally has a poor generalisation ability which degrades its reliability and hampers its applications to science and real-world modeling. Therefore, this thesis aims to develop new GP approaches to SR that evolve/learn models exhibiting good generalisation ability. This thesis develops a novel feature selection method in GP for high-dimensional SR. Feature selection can potentially contribute not only to improving the efficiency of learning algorithms but also to enhancing the generalisation ability. However, feature selection is seldom considered in GP for high-dimensional SR. The proposed new feature selection method utilises GP’s built-in feature selection ability and relies on permutation to detect the truly relevant features and discard irrelevant/noisy features. The results confirm the superiority of the proposed method over the other examined feature selection methods including random forests and decision trees on identifying the truly relevant features. Further analysis indicates that the models evolved by GP with the proposed feature selection method are more likely to contain only the truly relevant features and have better interpretability. To address the overfitting issue of GP when learning from a relatively small number of instances, this thesis proposes a new GP approach by incorporating structural risk minimisation (SRM), which is a framework to estimate the generalisation performance of models, into GP. The effectiveness of SRM highly depends on the accuracy of the Vapnik-Chervonenkis (VC) dimension measuring model complexity. This thesis significantly extends an experimental method (instead of theoretical estimation) to measure the VC-dimension of a mixture of linear and nonlinear regression models in GP for the first time. The experimental method has been conducted using uniform and non-uniform settings and provides reliable VC-dimension values. The results show that our methods have an impressively better generalisation gain and evolve more compact model, which have a much smaller behavioural difference from the target models than standard GP and GP with bootstrap, The proposed method using the optimised non-uniform setting further improves the one using the uniform setting. This thesis employs geometric semantic GP (GSGP) to tackle the unsatisfied generalisation performance of GP for SR when no overfitting occurs. It proposes three new angle-awareness driven geometric semantic operators (GSO) including selection, crossover and mutation to further explore the geometry of the semantic space to gain a greater generalisation improvement in GP for SR. The angle-awareness brings new geometric properties to these geometric operators, which are expected to provide a greater leverage for approximating the target semantics in each operation, and more importantly, to be resistant to overfitting. The results show that compared with two kinds of state-of-the-art GSOs, the proposed new GSOs not only drive the evolutionary process fitting the target semantics more efficiently but also significantly improve the generalisation performance. A further comparison on the evolved models shows that the new method generally produces simpler models with a much smaller size and containing important building blocks of the target models.</p>

Compact symbolic expressions have been shown to be more efficient than neural network (NN) models in terms of resource consumption and inference speed when implemented on custom hardware such as field-programmable gate arrays (FPGAs), while maintaining comparable accuracy (Tsoi et al 2024 EPJ Web Conf. 295 09036). These capabilities are highly valuable in environments with stringent computational resource constraints, such as high-energy physics experiments at the CERN Large Hadron Collider. However, finding compact expressions for high-dimensional datasets remains challenging due to the inherent limitations of genetic programming (GP), the search algorithm of most symbolic regression (SR) methods. Contrary to GP, the NN approach to SR offers scalability to high-dimensional inputs and leverages gradient methods for faster equation searching. Common ways of constraining expression complexity often involve multistage pruning with fine-tuning, which can result in significant performance loss. In this work, we propose S y m b o l N e t , a NN approach to SR specifically designed as a model compression technique, aimed at enabling low-latency inference for high-dimensional inputs on custom hardware such as FPGAs. This framework allows dynamic pruning of model weights, input features, and mathematical operators in a single training process, where both training loss and expression complexity are optimized simultaneously. We introduce a sparsity regularization term for each pruning type, which can adaptively adjust its strength, leading to convergence at a target sparsity ratio. Unlike most existing SR methods that struggle with datasets containing more than O ( 10 ) inputs, we demonstrate the effectiveness of our model on the LHC jet tagging task (16 inputs), MNIST (784 inputs), and SVHN (3072 inputs).

This paper considers evolutionary methods of symbolic regression for the creation of artificial intelligence of robotic systems. Methods of symbolic regression are reviewed and the features of their application to the solution of the problem of synthesis of control of robotic systems are indicated. The measure of the complexity of artificial intelligence is determined and the advantage of using the principle of small variations of the basic solution is shown, while creating intelligent control systems. A method of variational genetic programming is described and an example of its use for the synthesis of intellectual control is given.

Medical decision-making makes frequent use of algorithms that combine risk equations with rules, providing clear and standardized treatment pathways. Symbolic regression (SR) traditionally limits its search space to continuous function forms and their parameters, making it difficult to model this decision-making. However, owing to its ability to derive data-driven, interpretable models, SR holds promise for developing data-driven clinical risk scores. To that end, we introduce Brush, an SR algorithm that combines decision-tree-like splitting algorithms with nonlinear constant optimization, allowing for seamless integration of rule-based logic into SR and classification models. Brush achieves Pareto-optimal performance on SRBench and was applied to recapitulate two widely used clinical scoring systems, achieving high accuracy and interpretable models. Compared with decision trees (DTs), random forests (RFs) and other SR methods, Brush achieves comparable or superior predictive performance while producing simpler models. This article is part of the discussion meeting issue 'Symbolic regression in the physical sciences'.

This paper presents a systematic method for generating optimal helicopter flight trajectories in real time for cockpit display. Helicopter flights in the event of a single engine failure are critical, and display of optimal flight paths on-board is desirable. Due to the large number of states and constraints, however, numerical solutions on a mainframe computer can take a long tune. In this paper, optimal solutions are first obtained off-line using a ground computer. For a given set of initial helicopter states and flight parameters, optimal solutions are fitted by a Fourier series. The coefficients of the Fourier approximations are then interpolated as functions of initial flight states and parameters. The approximate optimal trajectories can be quickly generated on-board using the proposed method. Introduction Engine failure is a major safety issue for all helicopters, especially during takeoff and landing operations. Due to the complexity of the situation and the dependence of safety on many parameters, optimal flight trajectories can provide useful guidance for pilot operation. In particular, on-board display of optimal trajectories is desirable. This paper presents a method for generating optimal flight trajectories in real-tune that can be used for on-board display. Depending on its capability and performance during an engine failure, a transport helicopter can be certified as Category A or Category B. The Federal Aviation Administration (FAA) grants Category B certification to a single or multi-engine helicopter with gross weight less than 20,000 Ib and capable of landing safely in case of one or all engine failures. Category A certification, on the other hand, is granted to multi-engine helicopters with independent engine systems. Category A helicopters must be able to either land safely or continue flight in case of One Engine Inoperation (OEI). This makes Category A helicopters uniquely suited for remote places where emergency landing sites are not available. I Ph.D. Candidate, Center for Control Sciences, Member AIAA * Assistant Professor, Dept. of Aerospace Eng. and Mechanics, Member AIAA + Civil Rotorcraft Group Leader, Rotorcraft and Powered Lift Branch The decision to continue the flight or to land represents an energy management problem. It has been shown''' that this decision depends on initial conditions at the time of failure as well as flight parameters. Due to the numerous initial variables involved in this decision, a method to communicate all this information to the pilot instantaneously, at engine failure, is needed. This method must keep the emergency procedure simple and manageable. In particular, on-board cockpit display of optimal flight trajectories is useful. This requires a reliable real-tune solution of optimal trajectories, on board, with limited time, limited computing power, and limited memory available. If possible, an analytical solution of optimal trajectories is desirable. Unfortunately, this can only be achieved for systems of low order and where the equations of motion and the cost function are simple. Therefore, for a complex system, an open-loop numerical procedure becomes necessary. Research efforts were made to solve optimal control problems in real-time. Schultz et al. used calculus of variation and energy state method to generate optimal trajectories on-board for the National Aerospace Plane (NASP). In his study, the equations of motion of the NASP are less complex than those of the helicopter, and an interval of time of few minutes is allowed. Bryson used neighboring control theory to generate near-optimal solutions that could be used in real-time. This method is elegant but requires a large amount of memory to store the needed perturbation matrices. Also, it is valid on a relatively small neighborhood of the nominal trajectory. Perturbation methods and table lookups are often used if only few parameters are involved. In the current problem, the equations of motion of the helicopter are complex and too many path equality and inequality constraints are present. Also, limitations on tune, computing power, and memory available make all the above methods unsuitable for this problem. This paper proposes a technique of real-time optimal trajectory generation based on approximation and

Symbolic regression is a process to find a mathematical expression that represents the relationship between a set of explanatory variables and a measured variable. It has become a best-known problem for GP (genetic programming), as GP can use the tree representation to represent solutions as expression trees. Since the success of memetic algorithms (MAs (Memetic algorithms (MAs) can be regarded as a class of methods that combine population-based global search and local search [6, 30])) has proved the importance of local search in augmenting the global search ability of GP, GP with local search is investigated to solve symbolic regression tasks in this work. An important design issue of MAs is the balance between the global exploration of GP and the local exploitation, which has a great influence on the performance and efficiency of MAs. This work proposes a GP-based memetic algorithm for symbolic regression, termed as aMeGP (a daptive Me metic GP), which can balance global exploration and local exploitation adaptively. Compared with GP, two improvements are made in aMeGP to invoke and stop local search adaptively during evolution. The proposed aMeGP is compared with GP-based and nonGP-based symbolic regression methods on both benchmark test functions and real-world applications. The results show that aMeGP is generally better than both GP-based and nonGP-based reference methods with its evolved solutions achieving lower root mean square error (RMSE) for most test cases. Moreover, aMeGP outperforms the reference GP-based methods in the convergence ability, which can converge to lower RMSE values with faster or similar speeds.

1 A Survey on Computational Optimal Control.- Issues in the Direct Transcription of Optimal Control Problems to Sparse Nonlinear Programs.- Optimization in Control of Robots.- Large-scale SQP Methods and their Application in Trajectory Optimization.- Solving Optimal Control and Pursuit-Evasion Game Problems of High Complexity.- 2 Theoretical Aspects of Optimal Control and Nonlinear Programming.- Continuation Methods In Boundary Value Problems.- Second Order Optimality Conditions for Singular Extremals.- Synthesis of Adaptive Optimal Controls for Linear Dynamic Systems.- Control Applications of Reduced SQP Methods.- Time Optimal Control of Mechanical Systems.- 3 Algorithms for Optimal Control Calculations.- Second Order Algorithm for Time Optimal Control of a Linear System.- An SQP-type Solution Method for Constrained Discrete-Time Optimal Control Problems.- Numerical Methods for Solving Differential Games, Prospective Applications to Technical Problems.- Construction of the Optimal Feedback Controller for Constrained Optimal Control Problems with Unknown Disturbances.- Repetitive Optimization for Predictive Control of Dynamic Systems under Uncertainty.- Optimal Control of Multistage Systems Described by High-Index Differential-Algebraic Equations.- A New Class of a High Order Interior Point Method for the Solution of Convex Semiinfinite Optimization Problems.- A Structured Interior Point SQP Method for Nonlinear Optimal Control Problems.- 4 Software for Optimal Control Calculations.- Automated Approach for Optimizing Dynamic Systems.- ANDECS: A Computation Environment for Control Applications of Optimization.- Application of Automatic Differentiation to Optimal Control Problems.- OCCAL: A mixed symbolic-numeric Optimal Control CALculator.- 5 Applications of Optimal Control.- A Robotic Satellite with Simplified Design.- Nonlinear Control under Constraints of a Biological System.- An Object-Oriented Approach to Optimally Describe and Specify a SCADA System Applied to a Power Network.- Near-Optimal Flight Trajectories Generated by Neural Networks.- Performance of a Feedback Method with Respect to Changes in the Air-Density during the Ascent of a Two-Stage-To-Orbit Vehicle.- Linear Optimal Control for Reentry Flight.- Steady-State Modelling of Turbine Engine with Controllers.- Shortest Paths for Satellite Mounted Robot Manipulators.- Optimal Control of the Industrial Robot Manutec r3.

Recent work has argued that many machine learning techniques exhibit a 'double descent' in model risk, where increasing model complexity beyond an interpolation zone can overcome the bias-variance tradeoff to produce large, over-parameterised models that generalise well to unseen data. While the double descent characteristic has been identified in many learning methods, it has not been explored within symbolic regression research. This paper presents an initial exploration into the presence of double descent behaviour in symbolic regression over a range of parameter settings. Results suggest that symbolic regression via genetic programming does not exhibit a clear double descent risk curve relative to model size or function set. Unlike other methods, models evolved through symbolic regression do not appear to strongly interpolate training data, which promotes a degree of robustness towards noise in training data. However, models evolved by symbolic regression can still be large and do not present a strong overfitting characteristic. Given that a prime motivation for symbolic regression is to produce compact interpretable models, these results suggest that methods aimed at regularising evolved models should be a key feature of all symbolic regression methods.

Genetic Programming (GP) for symbolic regression is often prone to overfitting the training data, causing poor performance on unseen data. A number of recent works in the field have been devoted to regulating this problem by investigating both the structural and functional complexity of GP individuals during the evolutionary process. This work uses the Rademacher complexity and incorporates it into the fitness function of GP, utilising it as a means of controlling the functional complexity of GP individuals. The experiment results confirm that the new GP method has a notable generalization gain compared to the standard GP and Support Vector Regression (SVR) in most of the considered problems. Further investigations also show that the new GP method generates symbolic regression models that could not only release the overfitting trend in standard GP but also are significantly smaller in size compared to their counterparts in standard GP.

Symbolic regression for defect interactions in 2D materials

The paper is devoted to application of Cartesian Genetic Programming (CGP) for generating optimal trajectories of a mobile robots group. The problem of a control system synthesis for a mobile robots group is solved. The proposed algorithm uses numerical approach from the class of symbolic regression methods to which Cartesian Genetic Programming belonging. It allows to receive a control function in the form of a mathematical expression. We consider several stages to get optimal trajectories for mobile robots group moving along which the robots wouldn’t collide with each other and obstacles. Initially, we solve the problem of synthesis for each robot in order to get the stabilized robot control system relative some point in the state space. At the second stage, spatial trajectories are found along which robots move from the current state to the obtained equilibrium points without collisions. It was proposed to improve an initial algorithm by using the principal of small variation of basic solution. There is considered a group of three robots and the control system for them with phase constraints in the paper.

An identification control is a control of an object which mathematical model is unknown. For such objects initially the problem of identification is solved, and then for the resulting object model the problem of control synthesis is solved. As a result of control synthesis we obtain a multi-dimensional function that describes the dependence of control on the object state. After the implementation of this function in the control unit of the real object, it achieves the goal of control with optimal value of the quality criteria. To solve the identification and synthesis problems we use the method of network operator, which with the help of evolutionary algorithms finds the structure and parameters of the functions describing the model and functional dependence of control from the object state. In this paper, an example of using a network operator to solve the problem of identification control synthesis for a multilink robot is given.