LQ Optimal Control Research Articles

LQ optimal control of uncertain fractional differential systems

LQ optimal control of uncertain fractional differential systems

An Inversion-Free Iterative Algorithm With a Scalar Tuning Parameter for Coupled Riccati Matrix Equations Arising in LQ Optimal Control of Markov Jump Systems

An Inversion-Free Iterative Algorithm With a Scalar Tuning Parameter for Coupled Riccati Matrix Equations Arising in LQ Optimal Control of Markov Jump Systems

Stochastic LQ Optimal Control With Initial and Terminal Constraints

Stochastic LQ Optimal Control With Initial and Terminal Constraints

The Equivalence Conditions of Optimal Feedback Control-Strategy Operators for Zero-Sum Linear Quadratic Stochastic Differential Game with Random Coefficients

From the previous work, when solving the LQ optimal control problem with random coefficients (SLQ, for short), it is remarkably shown that the solution of the backward stochastic Riccati equations is not regular enough to guarantee the robustness of the feedback control. As a generalization of SLQ, interesting questions are, “how about the situation in the differential game?”, “will the same phenomenon appear in SLQ?”. This paper will provide the answers. In this paper, we consider a closed-loop two-person zero-sum LQ stochastic differential game with random coefficients (SDG, for short) and generalize the results of Lü–Wang–Zhang into the stochastic differential game case. Under some regularity assumptions, we establish the equivalence between the existence of the robust optimal feedback control strategy operators and the solvability of the corresponding backward stochastic Riccati equations, which leads to the existence of the closed-loop saddle points. On the other hand, the problem is not closed-loop solvable if the solution of the corresponding backward stochastic Riccati equations does not have the needed regularity.

Bifurcation analysis of a wheel‐slip control loop under low wheel‐slip regulation considering time‐delay

AbstractSafety and efficiency are key aspects of research and development in the automotive industry, especially in the field of safety‐critical subsystems, such as brake systems. The effect of time delay arise in brake systems, this adverse effect can cause performance degradations or make the brake system unstable. In this study, the bifurcation analysis of a wheel slip control feedback loop is presented. The control system consists of a wheel and a tyre, delayed actuator dynamics, and an LQR (linear quadratic regulator) that is designed to operate under relatively low wheel slip, during service braking. As a result, it is shown, that if time delay is neglected, the steady‐state solution of the model is globally stable using the LQ optimal controller gains. However, considering nonzero time‐delay may cause a subcritical Hopf‐bifurcation, and an unstable limit‐cycle exists around the steady‐state solution in a large domain of time‐delay parameters. Critical bifurcation maps and simulation results with more detailed system models are generated and shown in the study.

Limitations of energy SNR in control over channels with and without feedback

This paper presents limitations of energy signal-to-noise ratio (SNR) in control of discrete-time linear systems over noisy channels with and without feedback. The minimum SNR required for stabilization is shown to be the modulus of the product of unstable poles of the plants. Two types of optimal control laws that achieve this minimum are constructed based on the gains for expensive control, that is, LQ optimal control without input weight. It is observed that channel feedback does not improve the SNR limitation, which means feedback for control is efficient enough for communication.

Solvability of general fully coupled forward–backward stochastic difference equations with delay and applications

AbstractA class of fully coupled forward–backward stochastic difference equations with delay (FBSDDEs) over infinite horizon are considered in this article. By establishing a non‐homogeneous explicit relation between the forward and backward equations in terms of Riccati‐like difference equations, we derive the unique solution to the FBSDDEs under certain conditions. Then, we deduce that the FBSDDEs are solvable if and only if the corresponding stochastic delayed system is ‐degree open‐loop mean‐square exponentially stabilizable. Finally, as an application, the FBSDDEs are employed to demonstrate the maximum principle of the stochastic LQ optimal control problem.

Indefinite Backward Stochastic Linear-Quadratic Optimal Control Problems

This paper is concerned with a backward stochastic linear-quadratic (LQ, for short) optimal control problem with deterministic coefficients. The weighting matrices are allowed to be indefinite, and cross-product terms in the control and state processes are presented in the cost functional. Based on a Hilbert space method, necessary and sufficient conditions are derived for the solvability of the problem, and a general approach for constructing optimal controls is developed. The crucial step in this construction is to establish the solvability of a Riccati-type equation, which is accomplished under a fairly weak condition by investigating the connection with forward stochastic LQ optimal control problems.

Slow Decay and Turnpike for Infinite-Horizon Hyperbolic Linear Quadratic Problems

This paper is devoted to analyzing the explicit slow decay rate and turnpike in infinite-horizon linear quadratic optimal control problems for hyperbolic systems. Under suitable weak observability or controllability conditions, lower and upper bounds of the corresponding algebraic Riccati operator are proved. Then based on these two bounds, the explicit slow decay rate of the closed-loop system with Riccati-based optimal feedback control is obtained. The averaged turnpike property for this problem is also further discussed. We then apply these results to LQ optimal control problems constrained to networks of one-dimensional wave equations and also some multidimensional ones with local controls which lack a geometric control condition.

Necessary optimality conditions of fractional-order discrete uncertain optimal control problems

This paper is primarily dedicated to the necessary conditions of optimal control for uncertain discrete-time systems of fractional-order governed by left Riemann-Liouville-like fractional-order nabla difference equations. First, an existence result of solutions for backward right Riemann-Liouville-like fractional-order nabla difference equations is derived from the Banach contraction mapping theorem. Besides, necessary optimality conditions of these fractional-order uncertain optimal control problems are proposed by the variational method and Lagrange multiplier technique. Finally, a special LQ optimal control problem for uncertain discrete-time systems of fractional-order is solved by necessary conditions of optimality derived.

열간압연 코일러 시스템의 코일 장력에 대한 최적 서보 제어

This paper designs an optimal servo controller for the coiler system in hot rolling mill in order to suppress the tension fluctuation of the tail-end part after a steel plate leaves the final stand of hot rolling mill. The state space model of the coiler system is firstly modeled. Next, the optimal servo controller combined with a disturbance observer is designed using the inverse LQ optimal control method and the full state observer algorithm. Finally, it is shown by a computer simulation that the proposed control system is very effective to control tension fluctuations between the pinch roll and the mandrel under disturbances.

Optimal control of SDEs with expected path constraints and related constrained FBSDEs

In this paper, we consider optimal control of stochastic differential equations subject to an expected path constraint. The stochastic maximum principle is given for a general optimal stochastic control in terms of constrained FBSDEs. In particular, the compensated process in our adjoint equation is deterministic, which seems to be new in the literature. For the typical case of linear stochastic systems and quadratic cost functionals (i.e., the so-called LQ optimal stochastic control), a verification theorem is established, and the existence and uniqueness of the constrained reflected FBSDEs are also given.

Backlash Fault Suppression Using LQ Optimal Control Based RST Controllers in Wind Turbine Systems Using Bond Graphs and Matlab/Simulink

Abstract The presence of backlash in wind turbines is a source of limitations as it introduces nonlinearities that reduce their efficiency in speed/torque control which affect the performance of the power quality. Because of production tolerances during rotation, the teeth contact is lost for a small angle; until it is re-established, it produces a backlash phenomenon. The desire to eliminate this phenomenon is often hard to realise due to the nonlinear dynamic behaviour, which arises with the presence of backlash fault in a system. Therefore, the goal of this study is to develop an LQ optimal control structure in a form of an R-S-T controller in order to reduce the disturbing torque transmitted inside the dead zone of a gearbox in the wind turbine system. The actual system is also developed to be used as a demonstration model at lectures or presentations. The efficacy of the proposed control is illustrated via simulations.

Constrained Dynamic Mean-Variance Portfolio Selection in Continuous-Time

This paper revisits the dynamic MV portfolio selection problem with cone constraints in continuous-time. We first reformulate our constrained MV portfolio selection model into a special constrained LQ optimal control model and develop the optimal portfolio policy of our model. In addition, we provide an alternative method to resolve this dynamic MV portfolio selection problem with cone constraints. More specifically, instead of solving the correspondent HJB equation directly, we develop the optimal solution for this problem by using the special properties of value function induced from its model structure, such as the monotonicity and convexity of value function. Finally, we provide an example to illustrate how to use our solution in real application. The illustrative example demonstrates that our dynamic MV portfolio policy dominates the static MV portfolio policy.

Suppression of ground resonance in rotorcraft using active landing gear

This article examines the fundamental aspects of controlling ground resonance in rotorcraft equipped with actively controlled landing gear. Ground resonance is a mechanical instability affecting rotorcraft on the ground. It occurs at certain rotor speeds, where the lead–lag motion of the rotor couples with the motion of fuselage creating a self-excited oscillation. Typically, passive or semi-active lag dampers are used to avoid instability; however, these are undesirable from a design and maintenance perspective. Innovations in active landing gear for rotorcraft, such as articulated robotic legs, have provided an alternate approach to avoid the instability, eliminating the need for lag dampers with respect to ground resonance. This article extends classic ground resonance to include movable landing gear and identifies key physical parameters affecting dynamic behavior. Applying LQ optimal control to this model, it is shown that ground resonance instability can be eliminated using active landing gear as the control mechanism, even when there is no lag damping present in the rotor. In addition, while superior performance is achieved when landing gear movement can occur both longitudinally and laterally, it is still possible to stabilize ground resonance with inputs in a single direction, albeit with reduced performance.

LQ optimal control of fractional-order discrete-time uncertain systems

This paper primarily focuses on LQ optimal control problem of fractional-order discrete-time systems based on uncertainty theory that is a significant tool to model belief degree. First, an equivalent LQ problem, which is integer-order one, is presented by expanding the state variables with the unchanging control variables. This equivalent problem is then solved by dynamic programming approach. Accordingly, the solution of the proposed LQ optimal control problem reduces to solve a system of backward matrix difference equations. Finally, a numerical example is provided to show how to solve the proposed LQ optimal control problem. As an application, an LQ optimal control problem of macroeconomic system is discussed by the achieved results.

Development of driver braking control model based on ride comfort index

We propose a new driver braking control model. The driver braking control model is developed based on a ride comfort index. First, a brake model of the experimental vehicle is shown. Next, a driver braking control model is developed by using LQ optimal control. The driver braking control model is designed so that a ride comfort index can become small and ride comfort performance can be improved. Finally, numerical simulations are carried out to show effectiveness of the driver braking control model.

Open-loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems of Markovian regime switching system

This paper investigates the stochastic linear quadratic (LQ, for short) optimal control problem of Markovian regime switching system. The representation of the cost functional for the stochastic LQ optimal control problem of Markovian regime switching system is derived by the technique of Itô’s formula with jumps. For the stochastic LQ optimal control problem of Markovian regime switching system, we establish the equivalence between the open-loop (closed-loop, resp.) solvability and the existence of an adapted solution to the corresponding forward-backward stochastic differential equation with constraint. (i.e., the existence of a regular solution to Riccati equations). Also, we analyze the interrelationship between the strongly regular solvability of Riccati equations and the uniform convexity of the cost functional. Finally, we present an example which is open-loop solvable but not closed-loop solvable.

A Simulation Study on Turnpikes in Stochastic LQ Optimal Control

Abstract This paper presents a simulation study on turnpike phenomena in stochastic optimal control problems. We employ the framework of Polynomial Chaos Expansions (PCE) to investigate the presence of turnpikes in stochastic LQ problems. Our findings indicate that turnpikes can be observed in the evolution of PCE coefficients as well as in the evolution of statistical moments. Moreover, the turnpike phenomenon can be observed for optimal realization trajectories and with respect to the optimal stationary distribution. Finally, while adding variance penalization to the objective alters the turnpike, it does not destroy the phenomenon.

On the stochastic linear quadratic control problem with piecewise constant admissible controls

A linear quadratic optimal control problem for a system described by Itô differential equations with state and control dependent white noise under the assumption that the set of admissible controls consists of a class of piecewise constant stochastic processes is considered. The considered LQ optimal control problem is converted into a LQ optimization problem for a stochastic controlled system with finite jumps and multiplicative white noise perturbations. One of the original contribution of this work is the proof of the equivalence between the solvability of the considered optimal control problem and the solvability of the problem with given terminal values associated to a matrix linear differential equation (MLDE) with finite jumps and constraints. Another original contribution consists in the proof of the global existence of the solution of the problem with given terminal value of the MLDE if the cost weights matrices are positive semidefinite. The results obtained in the case of a LQ optimal control for systems with finite jumps are then applied to derive explicit formulae of the optimal controls for the optimization problem under piecewise constant controls.