State-dependent Riccati Equation Research Articles

Heterogeneity-Aware Graph Partitioning for Distributed Deployment of Multiagent Systems.

In this work, we examine the distributed coverage control problem for deploying a team of heterogeneous robots with nonlinear dynamics in a partially known environment modeled as a weighted mixed graph. By defining an optimal tracking control problem, using a discounted cost function and state-dependent Riccati equation (SDRE) approach, a new partitioning algorithm is proposed to capture the heterogeneity in robots dynamics. The considered partitioning cost, which is a state-dependent proximity metric, penalizes both the tracking error and the control input energy that occurs during the movement of a robot, on a straight line, to an arbitrary node of the graph in a predefined finite time. We show that the size of the subgraph associated with each robot depends on its resources and capabilities in comparison to its neighbors. Also, a distributed deployment strategy is proposed to optimally distribute robots aiming at persistently monitoring specified regions of interest. Finally, a series of simulations and experimental studies is carried out to demonstrate the viability and efficacy of the proposed methodology in deploying heterogeneous multiagent systems.

Numerical Exploratory Analysis of Dynamics and Control of an Atomic Force Microscopy in Tapping Mode with Fractional Order

In this paper, we investigate the mechanism of atomic force microscopy in tapping mode (AFM-TM) under the Casimir and van der Waals (VdW) forces. The dynamic behavior of the system is analyzed through a nonlinear dimensionless mathematical model. Numerical tools as Poincaré maps, Lyapunov exponents, and bifurcation diagrams are accounted for the analysis of the system. With that, the regions in which the system presents chaotic and periodic behaviors are obtained and investigated. Moreover, the fractional calculus is introduced into the mathematical model, employing the Riemann-Liouville kernel discretization in the viscoelastic term of the system. The 0-1 test is implemented to analyze the new dynamics of the system, allowing the identification of the chaotic and periodic regimes of the AFM system. The dynamic results of the conventional (integer derivative) and fractional models reveal the need for the application of control techniques such as Optimum Linear Feedback Control (OLFC), State-Dependent Riccati Equations (SDRE) by using feedback control, and the Time-Delayed Feedback Control. The results of the control techniques are efficient with and without the fractional-order derivative.

Optimized Intermittent Pharmaceutical Treatment of Cancer Using Non-Linear Optimal Control Techniques

Cancer remains one of the most important diseases and causes of death. In this study, a non-linear mathematical model of tumor growth with immune response, under the effects of chemotherapeutic treatment is studied. Two cost-efficient optimal control approaches are presented based on direct collocation and state dependent Riccati equation methods in order to optimize the pharmaceutical treatment-dosage to the patients. Finally, the numerical results from each method are presented, providing an overall better regimen, when compared to similar previous studies, by successfully eradicating the tumor and minimizing the side-effects of chemotherapy

Nonlinear Suboptimal Guidance Law With Impact Angle Constraint: An SDRE-Based Approach

In this article, a guidance law with impact angle constraint is developed using the state-dependent Riccati equation (SDRE) technique. First, the nonlinear guidance dynamics consisting of line-of-sight (LOS) angle rate and error is formulated under a planar engagement scenario. After the state-dependent coefficient parameterization, a nonlinear regulator problem is formed to concurrently zero LOS angle rate and error. Second, a preliminary guidance law is directly developed under the framework of the SDRE technique. By applying the Lyapunov stability analysis, a nonlinear suboptimal guidance law is obtained through further parameter modifications. Then, the proposed guidance law is extended to attack maneuvering targets. Finally, the effectiveness of the proposed guidance law is demonstrated through numerical simulations.

Numerical Solution of SDRE Control Problem – Comparison of the Selected Methods

Abstract Methods for solving non-linear control systems are still being developed. For many industrial devices and systems, quick and accurate regulators are investigated and required. The most effective and promising for nonlinear systems control is a State-Dependent Riccati Equation method (SDRE). In SDRE, the problem consists of finding the suboptimal solution for a given objective function considering nonlinear constraints. For this purpose, SDRE methods need improvement. In this paper, various numerical methods for solving the SDRE problem, i.e. algebraic Riccati equation, are discussed and tested. The time of computation and computational effort is presented and compared considering selected nonlinear control plants.

Constrained LQR Control of Dual Induction Motor Single Inverter Drive

Control of dual induction motor fed by a single voltage source inverter is a challenging task especially during unbalanced load conditions on each motor. Such drive configurations are common, for example, in traction. Conventional control algorithms solve the control of each motor in a separate control loop and the overall control action is designed as a weighted contribution of each control loop. In such a case, the control performance strongly relies on the specific design of weighting coefficients, which should be state dependent and time varying value in majority of cases. In this contribution, we formulate the control problem as an optimization task and investigate the use of state-dependent Riccati equation and predictive control concepts. We show that it is sufficient to consider a simple linear quadratic regulator (LQR) with rule-based predictive controller for current limitation. The proposed control algorithm naturally weights contribution of tracking error of each drive. Moreover, due to high level of symmetry in the system, only two parameters are tuned manually. Experimental testing is used to validate the control algorithm on a laboratory prototype of dual induction motor drive with a rated power of 2 × 4.5 kW.

Trajectory tracking of an uncertain wheeled mobile robotic manipulator with a hybrid control approach

In this paper, a novel hybrid control approach is used for trajectory-tracking control of wheeled mobile robotic manipulators (WMRMs) in the presence of model uncertainties and external disturbances. Due to the existence of nonholonomic constrains in wheeled mobile robots, the proposed controller should be able to tackle the challenge of underactuation in such systems. To attain this objective, the dynamic equations of motion for a WMRM are derived in closed form using the Gibbs–Appell (G-A) formulation, which is an efficient method for obtaining the dynamic motion equations of nonholonomic systems. Then, a novel control scheme is developed which is both optimal and robust and which enjoys updated gains under an adaptive law. In this regard, to benefit from the advantages of two distinct methods, a high-order sliding mode control (HOSMC) is employed as a robust controller with the aim of handling system uncertainties and a state-dependent Riccati equation (SDRE) is used as a nonlinear optimal controller. In order to systematically deal with system uncertainties and to avoid the manual calculation of the upper bound of uncertainties, the gain of HOSMC is updated via an adaptive law. The most important advantage of this novel approach over the existing methods is that not only is it robustly stable against external disturbances when controlling an uncertain nonlinear system but also optimal for a predefined quadratic cost function. Lyapunov stability theory is employed to verify the stability of the proposed controller. Finally, to demonstrate the superiority of this novel controller, computer simulation results for a WMRM are compared with those of an adaptive sliding mode controller. It can be observed that the proposed hybrid control approach is capable of optimizing control inputs while achieving stability for a WMRM in the presence of model uncertainties.

Computational Enhancement of the SDRE Scheme: General Theory and Robotic Control System

This article presents a new efficient variant of the state-dependent Riccati equation (SDRE) scheme in a general scope, by analytically alleviating the computational burden in solving the pointwise algebraic Riccati equations (AREs). The novel contributions include a more efficient construction of feasible state-dependent coefficients (named alternative SDRE), which is critical at the early design stage, and more efficient solvability check for the classical SDRE scheme. For the selected robotic application—balance control of a two-wheeled robot, the contribution lies in the system-specific analysis that further enhances the computational performance toward an agile mobility. This novelty is with respect to a state-of-the-art ARE solver. An offline/ a priori analytical formulation replaces the very first stage of the solving process, which more efficiently integrates the ARE solver into the SDRE design framework. Notably, all the results not only benefit the classical SDRE scheme but, more significantly , favor the proposed alternative SDRE owing to its much better computational efficiency—mainly in terms of time while promisingly for memory saving. In addition, simulations reveal more potential advantages using the variants—such as the control effort efficiency or required regulation time—within and beyond the scope of SDRE.

Optimal Sliding Mode Controller for an Active Transfemoral Prosthesis Using State-Dependent Riccati Equation Approach

In this paper, a nonlinear robust optimal controller is designed for an active transfemoral prosthesis using sliding mode control and state-dependent Riccati equation control. In this method, unlike most nonlinear control methods, the Jacobian approach is not used for model linearization and system's state space equations are used directly. This is a reason for the accuracy and flexibility of the proposed controller in design compared to other conventional methods. Since the proposed approach is energy-driven, the main objective of this research is to optimize the energy consumption of the robot/prosthesis system, reduce the effects of disturbances, noise, and parametric uncertainties, and desirable trajectory tracking of the vertical displacement in hip and thigh and knee angles. To this aim, the integral state control technique and the boundary layer are used for reconciliation between tracking performance and control signal chattering. In this study, the performance of the controller is assessed for nominal system and uncertain system with ± 30% parameters’ uncertainty by considering the saturation bounds of control signals. The simulation results demonstrate a decrease in the control effort, good robustness in the presence of uncertainty, and desirable tracking performance compared to a robust adaptive impedance control approach.

Discrete Time-Coupled State-Dependent Riccati Equation Control of Nonlinear Mechatronic Systems

Abstract A discrete-time-coupled state-dependent Riccati equation (CSDRE) control strategy is structured in this paper for synthesizing state feedback controllers satisfying the combined nonlinear quadratic regulator (NLQR) and H∞ robust control performance objectives. Under smoothness assumptions, the nonlinear plant dynamics can be formulated into state-dependent coefficient form through direct parameterization. By solving a pair of coupled state-dependent Riccati equations, the optimal stabilizing solutions can achieve inherent stability, nonlinear quadratic optimality, and H∞ disturbance attenuation performance. The established two-player Nash's game theory is utilized for developing both of the finite and infinite time optimal control laws. Furuta swing-up pendulum, a representative nonholonomic underactuated nonlinear system, is stabilized in real-time for validating the robustness and potential of proposed approach in mechatronics applications.

A nonlinear optimal control based on the SDRE technique for the two-wheeled self-balancing robot

ABSTRACT In this article, for the first time, a nonlinear optimal controller based on the state-dependent Riccati equation (SDRE) technique is proposed for the self-balancing two-wheeled Robot. The proposed optimal control is similar to linear quadratic Gaussian (LQG) control. The LQG controller is based on linearisation, but the proposed nonlinear optimal controller is based on parameterisation. For this purpose, at first, a three-degree-of-freedom (3-DOF) model of the two-wheeled robot including, longitudinal displacement, yaw, and pitch motion of the chassis are obtained using Kane’s method. Then, using the parameterisation method, the state-dependent coefficient (SDC) matrices are derived for the design of the nonlinear quadratic Gaussian (NLQG) controller. Consequently, the 3-DOF model of the two-wheeled robot and the NLQG controller algorithm are simulated in the MATLAB software environment. Also, using the proposed controller and the open-loop simulations the performance of the two-wheeled Robot is discussed. The simulation results demonstrate that the optimal controller based on the SDRE approach performs remarkably well when the system has an extremely nonlinear behaviour. Abbreviations: KF: Kalman filter; CG: Centre of gravity; DOF: degree of freedom; LQG: linear quadratic Gaussian; NLQG: nonlinear quadratic Gaussian; SDRE: state dependent Riccati equation; LQR: linear quadratic regulator; RBF: radial basis function; SDC: state-dependent coefficient.

Virtual strategies in the kinematic and dynamical models applied to fault-tolerant strategy of underwater vehicles by using state-dependent Riccati equations

A new kind of kinematic underwater vehicle modelling, which uses the screw theory and the Davis method, is proposed. This latter one makes possible to treat the direct and inverse kinematic in the same way, using virtual kinematic chains. By a specific way, the kinematic model allows simplifying the Jacobian matrix, since it is not restricted to the base frame. A transformation from Euler angles to Octonions avoids possible singularities. In addition, a new linear quadratic regulator control that uses the design of variables gains based on low-pass filter and norm. Global stability of the State-Dependent Riccati Equation is performed by using Lyapunov analysis with La'Salle's Principle. The allocation of faults in this strategy by using virtual fuzzy thrusters is addressed, which minimises the fault impact expressed as an abrupt discontinuity in the thruster system, as observed in the simulation results.

Guaranteed Continuity and Computational Improvement in SDRE Controllers for Cancer Treatment Analysis

Abstract This study provides a novel analysis and design of the state-dependent Riccati equation (SDRE) control in cancer treatment application. The key assumption to ensure continuous SDRE controllers—in terms of the solvability of pointwise Riccati equations—is replaced by a simplified equivalent condition, which largely alleviates the computational burden. At the discontinuities, an alternative solution is novelly suggested, because the conventional/empirical α−parameterization technique to seek a continuous SDRE implementation without breakdowns is analyzed to be ineffective, which is the first counterexample in literature. Representatively, among discontinuities, an objective conflict against tumor eradication is discovered. Another value of the proposed analysis is supported by the generality demonstrations, in various fields beyond biomedical systems. Finally, the robustness of SDRE scheme to parameter variations is established via simulations, which more promotes the alternative solution as applied throughout the treatment course.

SDRE based optimal finite-time tracking control of a multi-motor driving system

This paper presents a nonlinear optimal finite-time tracking controller based on a state-dependent Riccati equation (SDRE) for the multi-motor driving system (MDS). Conventional fast terminal sliding mode controller (FTSMC) can get the finite-time convergence of the tracking error, but how to achieve the optimality simultaneously is not an easy task. To address this optimal control problem, a SDRE based optimal controller is developed via a fast terminal sliding surface to design the nonlinear optimal feedback of the FTSMC. To further increase the robustness of the SDRE based method, an extended state observer (ESO) and a robust controller are incorporated into the optimal controller design to deal with the system uncertainties and external disturbances. Consequently, the proposed controller can not only guarantee the finite-time stability of the tracking error, but also achieve the optimal control performance under the system unknown dynamics. Experimental results demonstrate the effectiveness of the proposed scheme.

Digital implementation of a continuous-time nonlinear optimal controller: An experimental study with real-time computations

In this paper, the digital implementation of a continuous-time robust nonlinear optimal controller is presented as an experimental study with real-time computations. Complicated computations, solutions, and algorithms of nonlinear optimal policies were always reported as limits to experimental implementations. This work uses a combination of integral sliding mode control (ISMC) and the state-dependent Riccati equation (SDRE) approach for controlling an experimental setup, a rotary inverted pendulum (RIP) with nonlinear dynamics. Designing in the continuous-time domain and performing an experiment using digital computers are common and that leads to extra tuning in practice. Digital components are considered in simulations to provide a more real output and omit extra tuning. Analysis of sampling time effect on instability of a stable controller was done and the obtained bound of sampling time was verified in the experiment. The experimental study showed that the computations of the proposed controller were able to be programmed into the platform interface with time-varying sampling time which was bounded to the generated sampling time in the simulation. Successful swinging up and stabilization of the RIP demonstrated the effectiveness of the ISMC plus SDRE approach. The comparison of the proposed controller with the solo SDRE controller validated the results and showed the performance of the design.

A Modified SDRE-based Sub-optimal Hypersurface Design in SMC

Sliding Mode Control (SMC) plays a prominent role in dealing with matched uncertainties. In classical SMC design, the sliding surface (SS) is crucial to the guarantee for the stability and desired performance, especially if the system is nonlinear. A possible way to fulfill these desired performances for nonlinear systems is to use State Dependent Riccati Equation (SDRE) method, enabling SS to be designed even optimally. However, SDRE may suffer an inherent stability problem as well as a computational burden. To overcome these issues, in a recent study, a new SDRE method has been proposed. Therefore, this study takes advantages of the advanced SDRE method in designing a sub-optimal SS and also provides some comparative results with the conventional one to establish the feasibility of the proposed SDRE-based SMC control architecture experimentally. Experiments are conducted by using a 3-DOF helicopter platform and the results reveal that the proposed SDRE-based SMC is able to produce smoother SS than the conventional counterpart.

Three-Dimensional Finite-time Suboptimal Nonlinear Impact Angle Guidance

Interceptor guidance strategies are crucial in efficiently achieving desired terminal objectives. In this paper, a guidance strategy is proposed for interception of targets at pre-specified impact angle within finite time, by an interceptor in three dimensional space. The impact angle is defined as the azimuth and elevation of line-of-sight in the inertia reference frame, at interception. A suboptimal nonlinear guidance law is derived using the finite-time state dependent Riccati equation technique, for coupled engagement dynamics. The performance of the proposed guidance is tested for stationary and constant velocity targets using numerical simulations.

Nonlinear Business Cycle and Optimal Policy: A VSTAR Perspective

This paper studies optimal macroeconomic policy when nonlinearity in the business cycle is described by a vector smooth transition autoregression (VSTAR). A structural identification of the VSTAR that yields a low-dimension and certainty-equivalent nonlinear quadratic regulator (NLQR) problem is derived. Optimal rules are calculated by adapting from the engineering theory the approach of State Dependent Riccati Equation, which allows standard dynamic programming techniques to solve NLQR problems. The methodology is employed to study optimal conventional and quantitative easing (QE) monetary policy using a VSTAR model esti-mated on data for the United States during 1979-2018. The model allows for regime changes during periods of economic slack and when interest rates are near the zero lower bound. The results highlight the quantitative significance of nonlinearity in the analysis of optimal monetary policy and how the size, timing and composition of QE can influence macroeconomic dynamics.

Discrete-Time State-Dependent Proportional-Integral Control for Torque Tracking of Hydrostatic Transmissions

This paper presents both design and implementation of a discrete-time proportional-integral (PI) tracking control for the desired output torque of a nonlinear hydrostatic transmission system affected by disturbances and uncertainties. The control is conceived in a decentralized form, in which the bent-axis angle and the torque of the hydraulic motor are controlled separately. The motor bent-axis angle is adjusted by a pure feedforward control law, whereas the torque of the hydraulic motor is controlled using a PI state feedback from a online-solution of the state-dependent Riccati equation. State variables and external disturbances are reconstructed by a discrete-time nonlinear observer. The stability of the closed-loop system as well as the observer are investigated by linear matrix inequalities. The achieved tracking performance indicates the robustness of the overall control structure in the presence of system disturbances and uncertainties. The proposed control is evaluated by means of simulations and experiments using the dedicated test rig at the Chair of Mechatronics, University of Rostock.

Synthesis of Control and State Observer for Weakly Nonlinear Systems Based on the Pseudo-Linearization Technique

In this paper an approach for the construction of nonlinear output tracking control on a finite time interval for a class of weakly nonlinear systems with state-dependent coefficients is considered. The proposed method of control synthesis consists of two main stages. On the first stage, a nonlinear state feedback regulator is constructed using a previously proposed control algorithm based on the State-Dependent Riccati Equation (SDRE). On the second stage, the problem of full-order observer construction is formulated and then is reduced to the differential game problem. The form of its solution is obtained with the help of the guaranteed (minimax) control principle, which allows to find the best observer coefficients with respect to a given functional considering the worst-case uncertainty realization. The form of the obtained equations made it possible to use the algorithm from the first stage to determine the observer matrix. The proposed approach is characterized by the nonapplicability of the estimation and control separation principle used for linear systems, since the matrix of observer coefficients turned out to be dependent on the feedback coefficients matrix. The use of numerical-analytical procedures for determination of observer and feedback coefficients matrices significantly reduces the computational complexity of the control algorithm.