Control Objective Research Articles

Constraint‐Driven Multilane Merging Control for Underactuated Connected and Automated Vehicle System With Mismatched Uncertainty

ABSTRACTThis article investigates the merging control for underactuated connected and automated vehicle (CAV) systems with mismatched uncertainty. The objective is to guarantee the safe and prescribed dynamic performance of the system during multilane merging. For CAV multidimensional motions, a strongly coupled vehicle dynamics with time‐varying uncertainties is constructed. For the nominal system, the control objectives are designed as system constraints, with equality constraints guaranteeing merging and time‐varying inequality constraints guaranteeing dynamic performance bounds. Based on the diffeomorphism method and bounded constraint, the system constraints are reconstructed to a unified representation. Combining system constraints and underactuated structure, the nominal controller is derived based on constraint‐following. For the uncertain system, the uncertainty is decomposed into matched and mismatched portions by orthogonal decomposition. The adaptive robust controller is designed based only on matched uncertainty. Through Lyapunov minimax analysis, the control renders the errors uniformly bounded and uniformly ultimately bounded. Simulation results show that merging and platooning are effectively achieved with the proposed control. The system has excellent transient and steady state performance even in the presence of time‐varying uncertainties.

Static actuator-sharing algorithm for concurrent control of multiple plasma properties

Abstract Simultaneous regulation of multiple properties in next-generation tokamaks like ITER and Fusion Pilot Plant (FPP) may require the integration of different plasma control algorithms. Such integration requires the conversion of individual controller commands into physical actuator requests while accounting for the coupling between different plasma properties. This work proposes a tokamak and scenario-agnostic actuator-sharing algorithm (ASA) to perform the above-mentioned command-request conversion and, hence, integrate multiple plasma controllers. The proposed algorithm implicitly solves a quadratic programming (QP) problem formulated to account for the saturation limits and the relation between the controller commands and physical actuator requests. Since the constraints arising in the QP program are linear, the proposed ASA is highly computationally efficient and can be implemented in the tokamak plasma control system (PCS) in real time. Furthermore, the proposed algorithm is designed to handle real-time changes in the control objectives and actuators’ availability. Nonlinear simulations carried out using the Control Oriented Transport SIMulator (COTSIM) illustrate the effectiveness of the proposed algorithm in achieving multiple control objectives simultaneously.

Design and Seismic Performance Study of Multistage Controllable Isolation Bearing for High-Speed Railway Simply Supported Beam

The high-speed railway (HSR) system imposes stringent requirements for track smoothness. However, conventional seismic isolation bearings frequently fail to meet these demands. To address this challenge, a novel seismic isolation bearing was developed based on the principle of functional separation design. This innovative bearing effectively achieves the multistage control objectives, including amplitude limitation to ensure track smoothness during frequent earthquakes, energy dissipation to guarantee train running safety during design earthquakes, and structural integrity maintenance to prevent beam collapse during rare earthquakes. Firstly, an overview of the novel isolation bearing’s structural design and operational principle is provided. Subsequently, a corresponding mechanical model is formulated, with the parameters of the new bearing determined through finite element analysis. The study then compares the seismic performance of the general rubber bearing and the new bearing, using an HSR simply supported bridge as an engineering background. The dynamic response of the bridge under varying seismic waves, pier heights, and bridge spans is meticulously analyzed. The results indicate that the new bearing can achieve multistage control. Compared to general bearings, it reduces bridge displacement vibration by over 46.4% under frequent, design, and rare earthquakes. The bridge deformation under frequent earthquakes remains below 3 mm, thus meeting the track smoothness requirements for normal HSR operations. Additionally, the study reveals that higher pier heights increase the seismic response, peaking at 15 m. The vibration reduction provided by the new bearing varies but remains effective in most earthquake scenarios, with maximum reductions of 92.9% for displacement and 74.17% for bending moment. Furthermore, larger bridge spans also increase the seismic response, with the 24 m span bridge outperforming the 32 m span bridge. In conclusion, the novel seismic isolation bearing significantly enhances the seismic performance of HSR bridges, ensuring train running safety and operational reliability.

Developing PDE-constrained optimal control of multicomponent contamination flows in porous media

This paper develops a robust and efficient PDE-constrained optimal control model for multicomponent pollutions in porous media, which takes into account nonlinear multi-component contamination flows of groundwater. The objective of the pollution optimal control is to identify the optimal injection rates from the top part boundary of domain, which can minimize the least squares error between the concentrations that are simulated and the allowable observed concentrations at observation sites, combining with the affection of costs associated with reducing emissions at injection locations. To discrete the constrained governing system of nonlinear multi-component flows, the splitting improved upwind finite difference scheme is developed for multicomponent PDEs system involving nonlinear chemical reactions of multicomponent pollutants and the pollutant injected rates on the upper part boundary. We employ the differential evolution (DE) optimization algorithm to solve the optimization. We numerically demonstrate the effectiveness of our model by analyzing the flow simulation on a simple geometric aquifer and identifying the optimal injection rates by minimizing the concentration derivation and the abatement costs. We also investigate the simulation of the contamination flow in a more realistic-shaped aquifer, which further validates our model's robustness and efficacy. The developed PDE-constrained control model and algorithm can be applied to applications of groundwater pollution control.

STEWART PLATFORM MULTIDIMENSIONAL TRACKING CONTROL SYSTEM SYNTHESIS

Context. Creating guaranteed competitive motion control systems for complex multidimensional moving objects, including unstable ones, that operate under random controlled and uncontrolled disturbing factors, with minimal design costs, is one of the main requirements for achieving success in this class devices market. Additionally, to meet modern demands for the accuracy of motion control processes along a specified or programmed trajectory, it is essential to synthesize an optimal control system based on experimental data obtained under conditions closely approximating the real operating mode of the test object. Objective. The research presented in this article aims to synthesize an optimal tracking control system for the Stewart platform’s working surface motion, taking into account its multidimensional dynamic model. Method. The article employs a method of a multidimensional tracking control system structural transformation into an equivalent stabilization system for the motion of a multidimensional control object. It also utilizes an algorithm for synthesizing optimal stabilization systems for dynamic objects, whether stable or not, under stationary random external disturbances. The justified algorithm for synthesizing optimal stochastic stabilization systems is constructed using operations such as addition and multiplication of polynomial and fractional-rational matrices, Wiener factorization, Wiener separation of fractional-rational matrices, and the calculation of dispersion integrals. Results. As a result of the conducted research, the problem of defining the concept of analytical design for a Stewart platform’s optimal motion control system has been formalized. The results include the derived transformation equations from the tracking control system to the equivalent stabilization system of the Stewart platform’s working surface motion. Furthermore, the structure and parameters of the main controller transfer function matrix for of this control system have been determined. Conclusions. The justified use of the analytical design concept for the Stewart platform’s working surface optimal motion control system formalizes and significantly simplifies the solution to the problem of synthesizing complex dynamic systems, applying the developed technology presented in [1]. The obtained structure and parameters of the Stewart platform’s working surface motion control system main controller, which is divided into three components W1, W2, and W3, improve the tracking quality of the program signal vector, account for the cross-connections within the Stewart platform, and increase the accuracy of executing the specified trajectory by increasing the degrees of freedom in choosing the controller structure

Robust bilinear tracking control of a parabolic trough solar collector via saturation

This paper investigates the problem of robust tracking control of heat transport in a Parabolic Trough Solar Collector (PTSC), where the output has to track a desired reference trajectory. In this work, the PTSC is modeled by state-space bilinear dynamics. The manipulated variable is the pump volumetric flow rate, and the source term, i.e., solar irradiance, is assumed to be unmeasured. In addition, the actuator’s physical constraints induce saturation bounds on the manipulated variable and need to be considered explicitly in the controller design. To deal with these challenges, we first propose a saturated state-feedback law that meets the control objectives. Then, we reconstruct the unknown time-varying source term using an adaptive estimator. Later, through Lyapunov stability analysis, we prove that the closed-loop system and the output tracking error are uniformly ultimately stable. Numerical simulations attest to the performance of the proposed control strategy.

Observer‐Based Attack Detection and Data‐Driven Resilient Control for Heterogeneous Nonlinear Nonaffine Mass Under FDI Attacks

ABSTRACTThe attack detection and data‐driven resilient control problem for heterogeneous nonlinear nonaffine multi‐agent systems (MASs) in the presence of intermittent false data injection (FDI) attacks is investigated in this article. First, an attack detection and isolation mechanism based on the distributed observer are designed to determine the attack signals and isolate the attacked communication links in time. Then, in order to solve the problem of the unknown system model, the dynamic linearization method is used to convert the nonlinear nonaffine MASs into an equivalent linear data model. Further, based on the linear data model, the data‐driven resilient controller with the attack isolation mechanism is designed to achieve the leader‐following tracking control objective. Finally, the simulation result illustrates the performance of the proposed scheme with comparisons.

A Hybrid Multiobjective Control Strategy Based on Combination of Modulated Model Predictive Control and Phase‐Shifted Modulation for a Unidirectional Five‐Level Rectifier

ABSTRACTThe unidirectional multilevel rectifier has immense potential in high‐power medium‐voltage industrial applications. This paper proposes a multiobjective control method based on the combination of optimal‐vector‐pair modulated model predictive control (OVP‐M2PC) and a phase‐shifted modulation strategy for a unidirectional five‐level rectifier. This strategy takes both of the inductor current tracking and capacitor voltage balancing as the control objectives. First, an optimal vector pair is determined based on the AC‐side current variation rate, and then, through combining the calculated action duration of OVP with carrier phase‐shifted modulation, inductor current tracking is achieved. Second, capacitor voltage balancing control is carried out by compensating for the duty cycles which also decouples the input current tracking and output capacitor voltage balance control. Compared with traditional FCS‐MPC, under the proposed control strategy, complex cost function calculation and weighting factor tuning are not required, which much reduced computational burden. Multiobjective control is achieved with better performance in both steady‐state and dynamic conditions. The superiority of the proposed strategy over traditional FCS‐MPC are validated through simulations and hardware experiments.

Development of a control algorithm for a boost converter with the possibility of dynamic correction of control system parameters

RELEVANCE of the study lies in the development of an algorithm for controlling a DC voltage converter capable of providing a high-quality transient process with a wide change in the input parameters of the control object, affecting its nonlinear properties. purpose. THE PURPOSE. To consider methods for the development of continuous and discrete control systems for a step-up DC voltage converter and to obtain an analytical dependence of accounting for the nonlinear properties of the converter that affect the quality of regulation of the output voltage of the voltage stabilizer METHODS. The method of the average geometric root was chosen as a method for calculating the coefficients of the DC voltage converter control system, and the current circuit is adjusted using the technical optimum method, and the voltage circuit using the symmetric optimum method. The obtained analytical solutions of mathematical models and simulation models were compared in the SimInTech software environment. results. RESULTS. The simulation results show that the analytical solution of the mathematical model of the voltage stabilization system has a shorter transition time than the simulation results. This is due to the fact that in the mathematical model according to which the continuous control algorithm was synthesized, the presence of a power input filter, as well as the degree of their precharge, is not taken into account. Because of this, the transition time on simulation models is slower, applied 4 times than in the mathematical model. conclusion. CONCLUSION. When comparing the algorithms with each other, it can be seen that the algorithm obtained by converting a continuous algorithm by the Tusten method has the shortest transition time than all other algorithms. The continuous algorithm and the algorithm obtained by the inverse Euler transformation method have a speed close to each other, and the algorithm obtained by the direct Euler transformation turns out to be the slowest.

Identification of the electric motor mathematical model based on a data sample with feature engineering

The object of this study is a mathematical model of a synchronous electric motor, obtained on the basis of experimental data, which takes into account the temperature mode and uses artificial features to increase the accuracy of its operation. A characteristic feature of this work is that the model takes into account the temperature mode as a component of the technical-operational state of the object. The resulting mathematical model could make it possible to synthesize an optimal automatic control system in terms of the operational state of the object. The problem addressed was to increase the accuracy of the identified mathematical models by applying the approach of feature engineering. The results showed that the identification of mathematical models by the initial data leads to a low level of accuracy of the obtained models, namely 65–70 % for the first output channel, 80–85 % for the second, and 75–80 % for the third, fourth, and fifth output channels. Accordingly, building models with a higher threshold of accuracy requires the use of other, more significant data for identification. This paper reports a method for reformatting the original data into artificial features and provides results of their effectiveness in relation to the original channels. The resulting artificial features and the original features were used for further identification; the resulting mathematical model has on average higher accuracy thresholds, namely 82 %, 93 %, 88 %, 85 % for the corresponding output channels. The results prove the effectiveness of applying the principle of feature engineering since the accuracy of the resulting model is 5–10 % higher compared to the baseline. The scope of practical application of the results includes the synthesis of automatic control systems based on mathematical models of control objects obtained as a result of identification.

Validating effectiveness of crown-of-thorns starfish control thresholds to limit coral loss throughout the Great Barrier Reef

AbstractPopulation irruptions of the Pacific crown-of-thorns starfish (CoTS, Acanthaster cf. solaris) are a key source of coral loss on Australia’s Great Barrier Reef (GBR). CoTS management currently involves their manual control (culling) to threshold densities below which net coral decline theoretically ceases based on analysis of a validated population dynamics model. Spatial variability in coral growth and community composition, and their predicted changes under continuing global warming, necessitate further consideration of current coral representation in CoTS models. Here, we consider the sensitivity of equilibrium coral-CoTS thresholds to coral growth rates and consider how the demographic composition of CoTS at a site may relate to culling thresholds. We found thresholds should be location-specific if the objective of CoTS control is coral recovery, but location-specific thresholds may not be needed if the objective is to limit coral cover loss based on coral growth and CoTS demography. The consequence of using a higher CPUE threshold than the analytical equilibrium coral-CoTS thresholds in terms of coral cover loss is suggested to be limited at coral cover < 40%, and varying control thresholds thereof may make little difference. Introducing a 0.06 CoTS.min−1 threshold for 40–60% coral cover may reduce coral loss at ~ 40% where it is likely greatest. With regional GBR coral cover averaging < 40%, this study validates the 0.04–0.08 CoTS.min−1 tiered threshold system for CoTS control and suggests methods developed from localised studies can be more broadly applicable and well-defined objectives (e.g. avoiding coral cover decline at a site) can help guide what thresholds are used and the sensitivity around these.

Optimal Determining Air Supply Humidity for Multi-Location Demands Under Different Objectives in an Indoor Moisture Environment: A Comprehensive Method and Case Study

Within high-precision indoor environments, such as semiconductor fabrication or textile plants, humidity control is paramount for preserving product integrity and reducing energy expenditure. The non-uniform indoor air environment poses a significant challenge in achieving humidity regulation that meets the distinct requirements of various locations. Traditional feedback control mechanisms may lead to instability, overshooting, and oscillation in indoor parameters. This paper proposes a comprehensive method to address humidity assurance issues in high-precision indoor environments by establishing analytical expressions that link the demand parameters at different locations with air supply parameters. Using a case study, this paper examines several typical operational scenarios with diverse control objectives, including minimizing dehumidification energy consumption, minimizing air supply humidity adjustment values, and constraints on adjustable air supply inlets. This method enables rapid calculation of air supply humidity and regulation of humidity parameters at multiple locations within the indoor environment. It considers various locations, requirements, optimization targets, and precision, demonstrating that it can quickly determine the optimal air supply parameters based on the objective function. This method facilitates rapid adjustment and high-precision assurance of different humidity requirements at multiple locations, making it suitable for high-precision design and control of indoor humidity environments.

Learning‐based tracking control of AUV: Mixed policy improvement and game‐based disturbance rejection

AbstractA mixed adaptive dynamic programming (ADP) scheme based on zero‐sum game theory is developed to address optimal control problems of autonomous underwater vehicle (AUV) systems subject to disturbances and safe constraints. By combining prior dynamic knowledge and actual sampled data, the proposed approach effectively mitigates the defect caused by the inaccurate dynamic model and significantly improves the training speed of the ADP algorithm. Initially, the dataset is enriched with sufficient reference data collected based on a nominal model without considering modelling bias. Also, the control object interacts with the real environment and continuously gathers adequate sampled data in the dataset. To comprehensively leverage the advantages of model‐based and model‐free methods during training, an adaptive tuning factor is introduced based on the dataset that possesses model‐referenced information and conforms to the distribution of the real‐world environment, which balances the influence of model‐based control law and data‐driven policy gradient on the direction of policy improvement. As a result, the proposed approach accelerates the learning speed compared to data‐driven methods, concurrently also enhancing the tracking performance in comparison to model‐based control methods. Moreover, the optimal control problem under disturbances is formulated as a zero‐sum game, and the actor‐critic‐disturbance framework is introduced to approximate the optimal control input, cost function, and disturbance policy, respectively. Furthermore, the convergence property of the proposed algorithm based on the value iteration method is analysed. Finally, an example of AUV path following based on the improved line‐of‐sight guidance is presented to demonstrate the effectiveness of the proposed method.

Distributed Autonomous Economic Control Strategy for the AC/DC Interconnected Microgrid Considering the Regulation Boundary of the Consensus Variable

ABSTRACTThe autonomous economic control for the islanded AC/DC interconnected microgrid has various modes due to the restriction of the feasible regions of inner and inter‐microgrid resources. To properly handle the cooperative relationship between resources with different regulation characteristics and realize the frequency, voltage stability, and economic operation of the AC/DC interconnected microgrid, a distributed control strategy considering the regulation boundary of the consensus variable is proposed. Firstly, the influence of the variable feasible regions on the consensus control objective is analyzed, and the fundamental principle of the consensus control algorithm considering the variable regulation boundary is expounded. Secondly, the trilevel control strategy consisting of the local equipment control and inner and inter‐microgrid cooperative control is constructed. The local equipment control adopts droop control, and the cooperative control adopts consensus control to realize the elimination of the frequency and voltage deviation and the economic dispatch of active power within and between microgrids. Finally, the AC/DC interconnected microgrid model is built based on PSCAD/EMTDC, and the simulation results verify the effectiveness of the proposed control strategy.

Optimization of Potato Cultivation Through the Use of Biostimulator Supporter

Seed potato treatment is vital for plant protection, yield enhancement, and product quality. In the conducted research, the plant biostimulator Supporter was applied to evaluate its impact on potato yields and its structure. Supporter contains both synthetic and SL amino acids, which promote plant growth by enhancing nutrient utilization and fostering the development of a more effective root system. Such a formulation allows to maintain better resistance to environmental stresses, which may include drought or nutrient deficiency, among others. The field study was conducted in 2015–2017 in four towns located in different regions of Poland (Barankowo, Głubczyce, Kędrzyno, and Ryn) using a randomized complete block design with a split-plot design. Varieties (‘Innovator’, ‘Lilly’, ‘Lady Claire’, and ‘Verdi’) were tested. The experiment compared the cultivation technology using Supporter biostimulator with which seed potatoes were treated compared to conventional cultivation (control object) by soaking the tubers in distilled water before planting. The total yield of potato tubers after Supporter application was higher by 13.3%, while the commercial yield increased by 21.1% compared to the traditional cultivation method. The most productive, regardless of cultivation technology and years of research, in terms of total tuber yield was the ‘Lilly’ variety with an average yield of 47.95 t∙ha−1, while the least productive variety was the ‘Innovator’ variety with an average yield of 29.93 t∙ha−1. The ‘Lady Claire’ variety had the highest commercial tuber yield, while the ‘Innovator’ variety had the lowest.

Validation of a Model Predictive Control Strategy on a High Fidelity Building Emulator

In recent years, advanced controllers, including Model Predictive Control (MPC), have emerged as promising solutions to improve the efficiency of building energy systems. This paper explores the capabilities of MPC in handling multiple control objectives and constraints. A first MPC controller focuses on the task of ensuring thermal comfort in a residential house served by a heat pump while minimizing the operating costs when subject to different pricing schedules. A second MPC controller working on the same system tests the ability of MPC to deal with demand response events by enforcing a time-varying maximum power usage limitation signal from the electric grid. Furthermore, multiple combinations of the control parameters are tested in order to assess their influence on the controller performance. The controllers are tested on the BOPTEST framework, which offers standardized test cases in high-fidelity emulation models, and pre-defined baseline control strategies to allow fair comparisons also across different studies. Results show that MPC is able to handle multi-objective optimal control problems, reducing thermal comfort violations by between 66.9% and 82% and operational costs between 15.8% up to 20.1%, depending on the specific scenario analyzed. Moreover, MPC proves its capability to exploit the building thermal mass to shift heating power consumption, allowing the latter to adapt its time profile to time-varying constraints. The proposed methodology is based on technologically feasible steps that are intended to be easily transferred to large scale, in-field applications.

Adaptive output feedback control of a bidirectional AC/DC totem‐pole PFC converter for wireless charging of electric vehicles

AbstractThis paper deals with the control problem of a single‐phase bidirectional alternating current/direct current (AC/DC) totem‐pole power factor correction (PFC) converter intended to be used in wireless power transfer (WPT) chargers for electric vehicles (EVs). The converter is operating as a bridgeless boost rectifier with the PFC functionality during grid to vehicle (G2V) mode and as a PFC inverter during vehicle to grid (V2G) mode. Firstly, the operating principle is presented. Secondly, an overall mathematical model governing all the operating modes of the converter is elaborated. Thirdly, due to the nonlinearity of the studied system, a nonlinear controller–based Lyapunov technique is designed to achieve the following control objectives: (i) unity power factor (UPF) during both power transfer modes G2V and V2G, (ii) regulation of the DC‐link voltage, and (iii) asymptotic stability of the closed‐loop system. Further, as the developed control law requires the measurement of all voltages and currents, an adaptive observer–based Kalman‐like technique is designed to estimate all the required signals and parameters. The performances of the proposed output feedback control technique are validated through several simulation scenarios showing that all control objectives are achieved. It should be noted that the use of the Kalman observer reduces the complexity of the systems as it requires only two measurements (grid voltage and current) making it an advantageous solution over existing solutions where three measurements are generally used. Furthermore, due to the low‐pass filtering behavior of the observer and its susceptibility to estimate the converter's parameters, the robustness of the controller and its insensitivity to the parameter uncertainties are greatly improved.

Integrated optimization control strategy for vehicle longitudinal and lateral assisted driving

In order to coordinate the control objectives of the longitudinal and lateral auxiliary driving system of the vehicle and the vehicle dynamics control requirements, this paper takes the four-wheel independent drive electric vehicle as the control object, and combines deep learning and model predictive control to establish the vehicle generalized control force prediction-dynamic optimization distribution control strategy. The upper-level algorithm calculates the generalized control force that meets the vehicle control requirements according to the reference objectives of the longitudinal and lateral auxiliary driving strategies; the middle-level module includes the vehicle expected generalized force neural network prediction model and the active set numerical solution algorithm; the lower level calculates the tire slip rate and sideslip angle according to the expected tire force, realizes the tire force control requirements by controlling the motor torque and the braking system, and adjusts the vehicle movement direction through the steering system. This paper uses simulation tests and hardware-in-the-loop tests to verify the algorithm. The results show that the established algorithm can simultaneously meet the longitudinal and lateral control requirements of the vehicle, with a lateral path following error of 0.0117 m and a vehicle speed error of 0.59 km/h.

Control of a Directional Downhole Drilling System Using a State Barrier Avoidance Based Method

Abstract Vibrations such as bit-bouncing, stick–slip, and whirl motion can significantly reduce downhole drilling's performance and efficiency. These phenomena are likely to arise once the dynamical states of drilling fall into undesired operating regimes, as verified by both theoretical analyses and experimental studies. To effectively avoid such undesired operating conditions of a downhole drilling process, this paper introduces an advanced nonlinear state-constrained control method to formulate a setpoint tracking problem of high-order directional drilling dynamics under state constraints. These constraints are carefully defined using the field test results, and their shapes are in complex state-space regions, causing additional complexity to the control design. Moreover, unlike vertical drilling, the directional drilling system requires modeling of coupled axial and torsional dynamics with a higher degree-of-freedom (DOF). In this study, the proposed method will tackle these challenges by converting the original high-order constrained control problem into a standard nonlinear control problem. Leveraging on the linear parameter varying (LPV) method that is applied to the converted system, we can actively prevent drilling from falling into the undesired regimes, to achieve the constrained control objective.

Advanced Model with a Trajectory Tracking Stabilisation System and Feasible Solution of the Optimal Control Problem

In this study, we consider the extended optimal control problem and search for a control function in the class of feasible functions for a real control object. Unlike the classical optimal control problem, the control function should depend on the state, not time. Therefore, the control synthesis problem for the initial-state domain should be solved, instead of the optimal control problem with one initial state. Alternatively, an optimal trajectory motion stabilisation system may be constructed. Both approaches—control and trajectory motion stabilisation system syntheses—cannot be applied to real-time control, as the task is too complex. The minimum threshold of quality criteria is searched for in the space of mathematical expression codes. Among other problems, the search space is difficult to define and the gradient is hard to determine. Therefore, the advanced control object model is used to obtain a feasible control function. The advanced model is firstly obtained before solving the optimal control problem and it already includes a trajectory motion stabilisation system; in particular, this stabilisation system is synthesised in advance at the control system design stage. When the optimal control problem appears, it is solved in real time in the classical statement, and a control function is searched for as a function of time. The advanced control object model also uses the reference model to generate the optimal trajectory. The search for the optimal control function is performed in real time and considers the synthesised stabilisation system of motion along a determined trajectory. Machine learning control via symbolic regression, namely, the network operator method, is used to directly solve the control synthesis problem. An example solution of the optimal control problem, with an advanced model moving in the environment with obstacles for a group of two mobile robots, is presented. The obtained solution is a control function for a reference model that generates a trajectory from a class of trajectories stabilised with the object’s control system.