A robust suspension-control strategy is presented for electric vehicles equipped with in-wheel motors and dynamic vibration absorbers (DVAs). A quarter-car model incorporating an in-wheel motor and an adjustable DVA is developed to mitigate excess unsprung mass. A finite-horizon stochastic linear quadratic regulator (FHS-LQR) that explicitly accounts for control delays and parameter uncertainties is coupled with an NSGA-II multi-objective optimizer to tune both DVA mechanics and controller weights. Compared with passive and conventional LQR baselines, the proposed framework reduces the root-mean-square values of sprung-mass acceleration, suspension deflection, tire dynamic load, and motor dynamic force by up to 40% under ISO 8608 road profiles and maintains performance for input delays up to 100 ms, demonstrating real-time feasibility and improved ride comfort.

Optimal control of stochastic linear systems is fundamental in control theory, with applications in robotics, finance, and engineering. The Stochastic Linear Quadratic Regulator (SLQR) derives optimal feedback laws via the Riccati equation but requires numerical discretization of the resulting stochastic dynamics. Despite extensive studies on numerical methods for stochastic differential equations, their performance within the SLQR framework remains insufficiently explored. This study compares two predictor–corrector schemes of different orders: the Order 1.0 Predictor-Corrector (PC) method and the Order 2.0 Weak PC method. A one-dimensional linear quadratic problem with a closed-form solution enables precise error evaluation against the analytical trajectory. Convergence analysis across varying time step sizes shows that the Order 2.0 Weak PC method achieves consistently higher accuracy, particularly for fine discretizations, underscoring the importance of higher-order schemes in stochastic optimal control.

A Value Iteration Algorithm for Stochastic Linear Quadratic Regulator

This brief studies the discounted stochastic linear quadratic regulator (LQR) problem for systems suffering from additive noise of unknown mean. A completely model-free (MF) value iteration (VI) algorithm is developed to learn the optimal control policy using off-line system trajectories. The generated control policies are proven to converge to a small neighborhood of the optimal ones with high probability. In addition, an MF algorithm is proposed to learn a feasible discount factor. The proposed MF algorithms are illustrated through several examples.

In this article, we analyze the stability properties of stochastic linear systems in closed loop with an optimal policy that minimizes a discounted quadratic cost in expectation. In particular, the linear system is perturbed by both additive and multiplicative stochastic disturbances. We provide conditions under which mean-square boundedness, mean-square stability, and recurrence properties hold for the closed-loop system. We distinguish two cases, when these properties are verified for any value of the discount factor sufficiently close to 1, or when they hold for a fixed value of the discount factor in which case tighter conditions are derived, as illustrated in an example. The analysis exploits properties of the optimal value function, as well as a detectability property of the system with respect to the stage cost, to construct a Lyapunov function for the stochastic linear quadratic regulator problem.

Solutions to increase the accuracy of radio systems are analyzed. It is shown that invariance in a single-loop automatic control system affects its stability. Article [13] offers to apply a method for the synthesis of continuous high precision double-loop automatic control systems which are equivalent to combined systems in conditions when some values (entry useful action) can't be measured for development of tracking systems (in particular, radio systems, where the entry useful action can't be measured, and therefore combined control is impossible). The article considers extension of the method for the synthesis of double-loop automatic control systems to discrete systems equivalent to combined systems in an environment with simultaneous entry useful (preset) action and external disturbances and interferences. The developed method makes it possible to synthesize discrete high-precision automatic control tracking systems equivalent to combined ones in an environment with a controlled variable (entry useful action) which cannot be measured. A discrete transfer function from an error in double-loop system (5), an invariance condition (6), and a characteristic equation (8) are obtained. In this case, the numerator polynomial of the transfer function from error must have a difference of polynomials. The article demonstrates that invariance conditions (improved accuracy) can be achieved without any destabilization in the first loop. In this discrete tracking system equivalence to combined systems is achieved by two control loops and not by three loops as in the differential coupling method. A double-loop discrete automatic control system equivalent to a combined system was synthesized. A stochastic regulator was calculated and made, and the influence of this regulator on the astatism of the system (that is, on its accuracy) was analyzed. The proposed method can be used to develop discrete tracking systems (especially radio systems, where entry useful action cannot be measured due to interference), and it can be applied for laser radar tracking systems, and control systems for aircraft of various purposes.

This article investigates the output regulation problem of multi-input–multi-output (MIMO) linear stochastic systems and unstable linear exogenous systems. First, stochastic output regulator equations are constructed to characterize the solvability of the output regulation problem. Then, a necessary and sufficient condition on the solvability of the output regulator equation is established. Under this condition, not only the relationship between the output regulation of linear stochastic systems and that of linear deterministic systems is established, but also the solvability in some special cases is discussed. In the case where the control input is not invoked in the tracking error, there almost always exists a bounded and deterministic solution if the stochastic output regulator equation has a bounded solution, which provides a new approach without estimating the Brownian motion. This approach can deal with the crucial issue that the stochastic output regulator equation may not be solved in a practical sense, since the solution depends on the Brownian motion which may not be measured in most real scenes. Several numerical examples are presented to illustrate the main results.

This paper presents a comprehensive framework addressing optimal nonlinear analysis and feedback control synthesis for nonlinear stochastic dynamical systems. The focus lies on establishing connections between stochastic Lyapunov theory and stochastic Hamilton–Jacobi–Bellman theory within a unified perspective. We demonstrate that the closed-loop nonlinear system’s asymptotic stability in probability is ensured through a Lyapunov function, identified as the solution to the steady-state form of the stochastic Hamilton–Jacobi–Bellman equation. This dual assurance guarantees both stochastic stability and optimality. Additionally, optimal feedback controllers for affine nonlinear systems are developed using an inverse optimality framework tailored to the stochastic stabilization problem. Furthermore, the paper derives stability margins for optimal and inverse optimal stochastic feedback regulators. Gain, sector, and disk margin guarantees are established for nonlinear stochastic dynamical systems controlled by nonlinear optimal and inverse optimal Hamilton–Jacobi–Bellman controllers.

We consider anindefiniteoptimal regulator problem for a class of nonlinear stochastic systems with asquare-rootnonlinearity, random and possibly unbounded coefficients, and the quadratic-linear criterion with weight matrices being indefinite in general. This represents a certain nonlinear generalisation of the indefinite stochastic linear-quadratic control with random coefficients. All solutions to this problem are obtained in an explicit closed-form as an affinestate-feedbackcontrol the coefficients of which are determined by the solution pairs of certain backward stochastic differential equations with algebraic equality and inequality constraints.

In this paper, we prove both necessary and sufficient maximum principles for infinite horizon discounted control problems of stochastic Volterra integral equations with finite delay and a convex control domain. The corresponding adjoint equation is a novel class of infinite horizon anticipated backward stochastic Volterra integral equations. Our results can be applied to discounted control problems of stochastic delay differential equations and fractional stochastic delay differential equations. As an example, we consider a stochastic linear-quadratic regulator problem for a delayed fractional system. Based on the maximum principle, we prove the existence and uniqueness of the optimal control for this concrete example and obtain a new type of explicit Gaussian state-feedback representation formula for the optimal control.

Providing end-to-end network delay guarantees in packet-switched networks such as the Internet is highly desirable for mission-critical and delay-sensitive data transmission, yet it remains a challenging open problem. Since deterministic bounds are based on the worst-case traffic behavior, various frameworks for stochastic network calculus have been proposed to provide less conservative, probabilistic bounds on network delay, at least in theory. However, little attention has been devoted to the problem of regulating traffic according to stochastic burstiness bounds, which is necessary in order to guarantee the delay bounds in practice. We design and analyze a stochastic traffic regulator that can be used in conjunction with results from stochastic network calculus to provide probabilistic guarantees on end-to-end network delay. Two alternative implementations of the stochastic regulator are developed and compared. Numerical results are provided to demonstrate the performance of the proposed stochastic traffic regulator.

Economic MPC of Markov Decision Processes: Dissipativity in undiscounted infinite-horizon optimal control

Three new algorithms for synthesizing control for the model of an electrohydraulic disk brake system are presented, which are based on the synergetic control theory. The first algorithm is developed relying on the classical method of analytical design of aggregated regulators in an assumption of a completely defined object. The second algorithm represents an algorithm of nonlinear adaptation on a target manifold and is designed for an object with a nonrandom disturbance in the control channel. The third algorithm takes account of the random disturbance in the discrete description of this object and rests on the strategies minimizing the dispersion of the output macrovariable. The results of a comparative numerical simulation of the three control algorithms are presented and the recommendations concerning the selection of the regulator parameters are formulated depending on the level of systematic disturbances and random noise.

ABSTRACT In this paper, we derive stability margins for optimal and inverse optimal stochastic feedback regulators. Specifically, gain, sector, and disk margin guarantees are obtained for discrete-time nonlinear stochastic dynamical systems controlled by nonlinear optimal and inverse optimal controllers that minimise a nonlinear-nonquadratic performance criterion. Furthermore, using the newly developed notion of stochastic dissipativity we derive a return difference inequality to provide connections between stochastic dissipativity and optimality of nonlinear controllers for discrete-time stochastic dynamical systems. In particular, using extended Kalman-Yakubovich-Popov conditions characterising stochastic dissipativity we show that our optimal feedback control law satisfies a return difference inequality predicated on a difference operator of a controlled Markov dispersion process and is stochastically dissipative with respect to a specific quadratic supply rate.

Robust Reinforcement Learning for Stochastic Linear Quadratic Control with Multiplicative Noise

The paper presents a problem statement for a robust stochastic regulator synthesis based on the principles of control on manifolds, a solution algorithm of a new information system and its structure maintaining the synthesis algorithm for a stochastic discrete object. The object under study is presented as a system of stochastic, difference nonlinear equations. The mathematical tools of the classical method of analytical design of aggregated regulators are used, which were developed earlier for a deterministic nonlinear object with a complete description. The robust stochastic nonlinear regulator ensures the following characteristics of the control system: a) minimum deviation of the output variable; b) minimum dispersion of the target invariant; c) minimum average value of the control quality functional.

As the penetration of renewable energy, generally, uncertainties, increase in microgrids, a more dynamic and complex system is emerging that makes the load frequency control (LFC) of islanded microgrids more challenging due to their stochastic dynamic nature. This stochastic nature can create oscillatory frequency response, which eventually leads to deteriorating control function and instability even under primary and traditional secondary controllers. Due to the uncertainties, the measurements and microgrid states are continually perturbed; hence, they cannot be directly utilized for feedback control. Therefore, it is required to utilize an observer-based stochastic feedback control that correctly recognizes these uncertainties and sieves incorrect states, frequency deviation here, and can properly maintain the balance between the energy sources and the demand. This article develops an optimal observer-based secondary control for LFC, framed in two layers that are the stochastic unknown input observer (SUIO) and linear quadratic regulator. The proposed SUIO not only can address the uncertainties, e.g., renewable energy, load, and measurement noise, with efficient control effort but also performs robust against parameter changes of the microgrid. The simulation results confirm the efficacy of this proposed control as compared to the traditional secondary control method.

A Stochastic LQR Model for Child Order Placement in Algorithmic Trading

In this paper, the air-fuel ratio regulation problem of compressed natural gas (CNG) engines considering stochastic L2 disturbance attenuation is researched. A state observer is designed to overcome the unmeasurability of the total air mass and total fuel mass in the cylinder, since the residual air and residual fuel that are included in the residual gas are unmeasured and the residual gas reflects stochasticity. With the proposed state observer, a stochastic robust air-fuel ratio regulator is proposed by using a CNG engine dynamic model to attenuate the uncertain cyclic fluctuation of the fresh air, and the augmented closed-loop system is mean-square stable. A validation of the proposed stochastic robust air-fuel ratio regulator is carried out by the numerical simulation of two working conditions. The accuracy control of the air-fuel ratio is realized by the proposed stochastic robust air-fuel ratio regulator, which in turn leads to an improvement in fuel economy and emission performance of the CNG engines.

In this paper, the air-fuel ratio regulation problem of compressed natural gas (CNG) engines is considered by employing stochastic model predictive control (MPC) technology. A stochastic model predictive regulator based on a discrete-time dynamic model of CNG engines is proposed, taking into account the residual gas, and the closed-loop system is deduced to be stochastically stable. A numerical simulation is performed to demonstrate the effectiveness of the proposed control scheme under two working conditions. The simulation results show that the performance of the proposed stochastic model predictive regulator is better than that of the open-loop controller.