State-dependent Riccati Equation Approach Research Articles

Control Design for Dengue Fever Model with Disturbance

A mathematical model has become a useful tool to predict and control dengue fever dynamics. In reality, the dynamic of dengue fever transmission can be disturbed by uncertainty measurements, so it is needed to consider the disturbance in the model. Then, dengue fever model with disturbance is constructed by using a gain matrix consisting a covariance matrix and random vector. As dengue vaccine has been challenging to reduce the pandemic, a dengue model with vaccination as control is constructed. The aim is to propose a feedback controller that can reduces the infected human (H2 control problem) and the uncertainty measurements (H∞ control problem). The control u denotes the proportion of susceptible humans that one decides to vaccinate at time t. A random mass vaccination with wanning immunity is chosen because vaccine still on development process. A Design of mixed H2 - H∞ control with State-dependent Riccati Equation (SDRE) approach is applied. The SDRE has been an effective method to solve for synthesizing nonlinear feedback controller by transforming the system to an State-dependent coefficient (SDC) form. By comparing the mixed scheme with basic H∞, numerical simulation shows that the control application effectively decreases the number of infected humans and reduces the disturbance.

A PD-Type State-Dependent Riccati Equation With Iterative Learning Augmentation for Mechanical Systems

This work proposes a novel proportional-derivative (PD)-type state-dependent Riccati equation (SDRE) approach with iterative learning control (ILC) augmentation. On the one hand, the PD-type control gains could adopt many useful available criteria and tools of conventional PD controllers. On the other hand, the SDRE adds nonlinear and optimality characteristics to the controller, i.e., increasing the stability margins. These advantages with the ILC correction part deliver a precise control law with the capability of error reduction by learning. The SDRE provides a symmetric-positive-definite distributed nonlinear suboptimal gain <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{K}(\mathrm{x})$</tex> for the control input law <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{u}=-\mathrm{R}^{-1}(\mathrm{x})\mathrm{B}^{T}(\mathrm{x})\mathrm{K}(\mathrm{x})\mathrm{x}$</tex> . The sub-blocks of the overall gain <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{R}^{-1}(\mathrm{x})\mathrm{B}^{T}(\mathrm{x})\mathrm{K}(\mathrm{x})$</tex> , are not necessarily symmetric positive definite. A new design is proposed to transform the optimal gain into two symmetric-positive-definite gains like PD-type controllers as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{u}= -\mathrm{K}_{\mathrm{SP}}(\mathrm{x})\mathrm{e}-\mathrm{K}_{\mathrm{SD}}(\mathrm{x})\dot{\mathrm{e}}$</tex> . The new form allows us to analytically prove the stability of the proposed learning-based controller for mechanical systems; and presents guaranteed uniform boundedness in finite-time between learning loops. The symmetric PD-type controller is also developed for the state-dependent differential Riccati equation (SDDRE) to manipulate the final time. The SDDRE expresses a differential equation with a final boundary condition, which imposes a constraint on time that could be used for finite-time control. So, the availability of PD-type finite-time control is an asset for enhancing the conventional classical linear controllers with this tool. The learning rules benefit from the gradient descent method for both regulation and tracking cases. One of the advantages of this approach is a guaranteed-stability even from the first loop of learning. A mechanical manipulator, as an illustrative example, was simulated for both regulation and tracking problems. Successful experimental validation was done to show the capability of the system in practice by the implementation of the proposed method on a variable-pitch rotor benchmark.

Nonlinear Sub-optimal Control Design for Suppressing HIV Replication

Acquired Immunodeficiency Syndrome is a deadly viral disease caused by the Human Immunodeficiency Virus in vivo, and its purpose is to destroy the immune system of the human body. The disease does not currently have a definitive vaccine or treatment, but treatment with pharmaceutical interventions (antiretroviral therapy, or ART) can slow down the progression of HIV. Daily use of prophylaxis measures may also have serious side effects for the patient, so the dosage and regimen of drugs should be constantly controlled. The dynamic models formulated for HIV infection are nonlinear differential equations. Therefore, nonlinear optimal control methods can be effective in increasing the efficiency of treatment. In this study, a sub-optimal controller based on the state-dependent Riccati equation (SDRE) approach to the dynamic model of HIV is introduced. One of the advantages of the SDRE approach is that the nonlinear properties of the system are preserved in the design control procedure. Furthermore, the specific conditions of infected individuals can be considered via choosing appropriate coefficients in the cost function and limiting the amount of drug administered. In the procedure of control design, all the state variables must be available for feedback in order to use the SDRE controller. In this regard, the Extended Kalman Filter observer is also implemented. The effect of different weighting matrices on these states is examined. In addition, to assess the effectiveness of the proposed control strategy, the well-known performance indicator root mean square error is also considered. Numerical simulations confirm the high efficiency and flexibility of the proposed approach.

Nonlinear State Dependent Riccati Equation Controller for 3-DOF Airfoil with Cubic Structural Nonlinearity

In this paper, Limit Cycle Oscillations (LCOs) of a Two-Dimensional (2-D) airfoil with cubic structural nonlinearity in the plunge Degree of Freedom (DOF) are investigated. The aerodynamic loads are modelled using the unsteady Theodorsen’s theory. Wagner function and Jones’ approximation are used to transform the unsteady aerodynamic loads from frequency domain into time domain. The aeroelastic differential equations are solved using the routine ODE45 in MATLAB to get the system response. Both subcritical and supercritical LCOs are observed in the 3-DOF airfoil with structural nonlinearity in the plunge DOF. Such LCOs are undesirable phenomena and should be suppressed within the flight envelope. A nonlinear state feedback controller is designed to minimize the amplitude of LCOs presented in the considered nonlinear aeroelastic system. The State-Dependent Riccati Equation (SDRE) approach in combination with Kalman filter technique is used to design a controller for the 2-D airfoil with trailing edge control surface (flap). The forces and moments produced by this flap’s action are used to stabilize the system and suppress the existence of LCOs. The efficiency of the proposed nonlinear controller by the SDRE and Kalman filter in suppressing the existence of subcritical and supercritical LCOs are verified by the numerical simulation results.

Symbolic Regulator Sets for a Weakly Nonlinear Discrete Control System with a Small Step

For a class of discrete weakly nonlinear state-dependent coefficient (SDC) control systems, a suboptimal synthesis is constructed over a finite interval with a large number of steps. A one-point matrix Padé approximation (PA) of the solution of the initial problem for the discrete matrix Riccati equation is constructed based on the state-dependent Riccati equation (SDRE) approach and the asymptotics by the small-step of the boundary layer functions method. The symmetric gain coefficients matrix for Padé control synthesis is constructed based on the one-point PA. As a result, the parametric closed-loop control is obtained. The results of numerical experiments illustrate, in particular, the improved extrapolation properties of the constructed regulator, which makes the algorithm applicable in control systems for a wider range of parameter variation.

Control of active suspension system in the presence of nonlinear spring and damper

Two methods which are computationally simple and easy to apply are developed by using state dependent Riccati equation (SDRE) approach and approximating sequence of Riccati equation (ASRE) approach to control active suspension system in the presence of nonlinear spring and damper. Additionally, effectivenesses of the both control methods developed by utilizing two recently introduced SDRE and ASRE techniques are compared. First, methodologies of both approaches are presented. After that, nonlinear dynamics of the vehicle suspension system is described in terms of conveniently selected state variables for better control performance. Then, a cost function is written by using suspension and tire deflections, sprung mass velocity and acceleration, and unsprung mass velocity variables to improve ride quality, suspension deflection, and tire deflection. Additionally, a convenient representation of this cost function in terms of state variables is obtained to realize better control. A bump expressed as sinusoidal function and roughness of the road expressed as white noise are taken into consideration as the disturbances from the road. Finally, quarter vehicle suspension system equivalent model of Ford Fiesta Mk2 is used as an example and simulations obtained by using the developed control methods are checked against the performance requirements and corresponding passive suspension system.

Nonlinear observer-based optimal control of an active transfemoral prosthesis

This paper designs a joint controller/observer framework using a state dependent Riccati equation (SDRE) approach for an active transfemoral prosthesis system. An integral state control technique is utilized to design a tracking controller for a robot/prosthesis system. This framework promises a systematic flexible design using which multiple design specifications such as robustness, state estimation, and control optimality are achieved without the need for model linearization. Performance of the proposed approach is demonstrated through simulation studies, which show improvements versus a robust adaptive impedance controller and an extended Kalman filter-based state estimation method. Numerical results confirm the benefits of our method over the above-mentioned approaches with regard to control optimality and state estimation.

Symbolic Padé representation of stabilizing regulators for a class of nonlinear control systems with a parameter

An approach for the construction of a parametric set of stabilizing controls for a class of pseudo-linear control systems with a parameter that varies on a certain interval of the positive semi-axis is described. The algorithm is based on a combination of the State-dependent Riccati equation approach (SDRE) for feedback control design, matrix Padé approximations and intellectual choice theory. Firstly, asymptotic expansions for the solutions of certain matrix algebraic state-dependent Riccati equations in the vicinity of the boundary points of the interval of the parameter variation are constructed. The obtained asymptotic expansions are matched into one symbolic construction using two-point diagonal Padé approximations and as a result a parametric set of solutions for the entire domain of parameter variation is obtained. The resulting controllers have the form of the Kalman regulators and can be used for systems with different control gains coefficients. The symbolic and parametric representation of the regulators allows to increase the efficiency of calculations and to propose algorithms for the intellectual selection of the parameters taking into account the changes in the characteristics of motion in real time.

Vibration suppression and attitude control for the formation flight of flexible satellites by optimally tuned on-off state-dependent Riccati equation approach

In this paper, vibration suppression and attitude control for the formation flight of flexible satellites using optimally tuned on-off SDRE (state-dependent Riccati equation) approach is discussed. A formation consisting of flexible satellites has highly nonlinear dynamics and the corresponding satellites are subject to vibrations as well as uncertainties due to the practical conditions. Vibrations that are mainly caused by flexible modes of the satellites disorganize the coordination and hinder the formation stability as well as decreasing its performance and lifetime. Hence, flexibility should be considered in formation model and the coordination control needs to address such challenges. Owing to capabilities of SDRE approach for nonlinear systems, it is used as the coordination control. Satellites are assumed to be equipped with thrusters as their actuators which requires the control to be applied as on-off pulses. To this end, an algorithm is suggested to efficiently convert SDRE control into on-off pulses. For optimal tuning of the controller, the particle swarm optimization (PSO) algorithm is employed. Stability of the system has also been analyzed via a Lyapunov-based approach utilizing the region of attraction concept. The proposed on-off SDRE approach has shown to effectively suppress the vibrations in the presence of uncertainties leading to the accurate coordination of the whole formation while consuming less energy. Simulation results show the capability, efficiency, robustness and stability of the suggested approach.

Optimal controller design for DC microgrid based on state-dependent Riccati Equation (SDRE) approach

ABSTRACT In the present paper, an optimal control method based on State-Dependent Riccati Equation (SDRE) is used to design a controller for DC microgrid system. In the proposed method, optimal control law is proposed for stabilisation and tracking of online reference paths using pseudo-linearisation and by preserving all the nonlinear characteristics of the system. The proposed method not only traces reference paths, but also is able to minimise control effort and energy consumption. Simulation results indicate efficient performance of the proposed method with regard to tracking reference paths despite the changes in operating point, external load disturbance and parametric uncertainty in comparison with linear quadratic regulators.

State-dependent indirect pseudospectral method for nonlinear optimal control problems

The state-dependent Riccati equation (SDRE) method is an efficient approach to solve nonlinear optimal control problems (OCPs), but nonlinear necessary conditions for the first-order optimality are seldom met in the SDRE approach. In this paper, a state-dependent indirect pseudospectral (SDIP) technique is developed to design nonlinear optimal controllers. To preserve the nonlinearity of the system and reduce the computational cost as well, the state-dependent coefficient (SDC) parameterization technique is employed. Then the optimality conditions are derived under input and state constraints, and spectral methods are used to discretize the optimality conditions into a series of mixed linear complementarity problems (MLCPs). The developed SDIP method is able to handle the finite and infinite-horizon nonlinear OCPs in a unified framework. Numerical comparisons also verify the performance of the developed SDIP method.

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.

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.

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.

An update algorithm design using moving Region of Attraction for SDRE based control law

State Dependent Riccati Equation (SDRE) methods have the considerable advantages over other nonlinear control methods. However, stability issues can be arisen in SDRE based control system due to the lack of the global asymptotic stability property. Therefore, the previous studies have usually shown that local asymptotic stability can be ensured by estimating a Region of Attraction (ROA) around the equilibrium point. These estimated regions for stability may become narrow or the condition to keep the states in this region may be very conservative. To resolve these issues, this paper proposes a novel SDRE method employing an update algorithm to re-estimate the ROA when the states tend to move out of the stable region. The tendency is checked using a condition which is developed based on a new theorem. The theorem proves that it is possible to redesign the previous ROA with respect to the current states lying close to its boundary for ensuring the “non-local” stability along the trajectory without the need of solving SDRE at each time instant, unlike the standard SDRE approach. Therefore, the new theorem is now able to enhance the stability of the SDRE based closed-loop control system. The feasibility of the proposed SDRE control method is tested in both simulations and experiments. A validated 3-DOF laboratory helicopter is used for experiments and the control objective for the helicopter is to realise a preplanned movement in both elevation and travel axes. The results reveal that the proposed SDRE approach enables the controlled plant to track the desired trajectory as satisfactorily as the standard SDRE approach, while only solving SDRE when needed. The proposed SDRE method reduces the computational load for practical implementation of the control algorithm whilst ensuring the stability over the operational region.

Fixed-time near-optimal control for repointing maneuvers of a spacecraft with nonlinear terminal constraints

Repointing maneuvers of a spacecraft in staring mode are investigated where the optical axis is required to align with the target orientation. Different from traditional three-axis reorientation maneuvers, the rotation about the optical axis is free of constraints for repointing maneuvers. Both static target observation and moving target detection constraints are considered. The problem is then formulated as a finite-time horizon optimal control problem with nonlinear terminal constraints. A simple and efficient state-dependent Riccati equation(SDRE) based dynamic programming approach is applied to tackle this nonlinear optimal control problem. The convergence of the attitude from initial conditions to the desired terminal constraint is rigorously proved for the first time. Considering the inability of the SDRE method to deal with the problem of large angle maneuvers, an improved SDRE approach combined with a waypoint is proposed to enhance control performance. Finally, numerical investigations are conducted and compared with the real optimal solutions obtained by using the optimization software.

SDRE Preview Control for a LPV Modelled Autonomous Vehicle

This paper aims to present a Preview Controller by taking State-Dependent Riccati Equation (SDRE) approach to the standard Preview Control solution based on Linear Quadratic (LQ) framework. A Matlab/Simulink simulation study for performing a lane change task for a LPV modelled autonomous vehicle has been conducted. The control objective is to achieve reference tracking under parameter variation. It is assumed that the system has access to the future reference information for Np preview steps. The results show that the SDRE Preview Controller demonstrates good transient behaviour and achieve reference tracking objective.

Development of a novel nonlinear estimator based on state-dependent Riccati equation technique for articulated vehicles

In this article, for the first time, a novel nonlinear estimator based on state-dependent Riccati equation filter technique is developed for state estimation of the articulated heavy vehicles. The state-dependent Riccati equation filter approach has a structure similar to the Kalman filter, and compared to the Kalman filter, which is based on linearization, the state-dependent Riccati equation filter is based on parameterization. Also, the state-dependent Riccati equation approach due to the nonuniqueness of the state-dependent coefficient matrix prevented singularity (uncontrollability or unobservability), and this advantage can be used to improve the efficiency of the system. For this purpose, using the Newton's method, a ten-degrees-of-freedom nonlinear model of the articulated heavy vehicle including the longitudinal, lateral and yaw motion of the tractor, the articulation angle, and rotational motion of each wheel is developed. Then, the developed model of articulated heavy vehicle is verified using nonlinear TruckSim model in high-speed lane change maneuver. The vehicle model validation results indicated that the development model is near to the actual articulated vehicle nonlinear model and can be utilized in the nonlinear estimator design. Then, using the state-dependent Riccati equation filter approach, the state variable estimation algorithm based on parametrization is designed and the articulated vehicle states are estimated online. In order to assess the performance and the efficiency of the developed estimator, the simulations with two standard maneuvers including low-speed 90 ° turn maneuver and high-speed lane change maneuver in the slippery road are performed. The simulation results indicate the remarkable impacts of the developed estimator based on state-dependent Riccati equation filter technique on the state estimation of the articulated heavy vehicles.

Real-time obstacle avoidance of mobile robots using state-dependent Riccati equation approach

In this paper, state-dependent Riccati equation (SDRE) method-based optimal control technique is applied to a robot. In recent years, issues associated with the robotics have become one of the developing fields of research. Accordingly, intelligent robots have been embraced greatly; however, control and navigation of those robots are not easy tasks as collision avoidance of stationary obstacles to doing a safe routing has to be taken care of. A moving robot in a certain time has to reach the specified goals. The robot in each time step needs to identify criteria such as velocity, safety, environment, and distance in respect to defined goals and then calculate the proper control strategy. Moreover, getting information associated with the environment to avoid obstacles, do the optimal routing, and identify the environment is necessary. The robot must intelligently perceive and act using adequate algorithms to manage required control and navigation issues. In this paper, smart navigation of a mobile robot in an environment with certain stationary obstacles (known to the robot) and optimal routing through Riccati equation depending on SDRE is considered. This approach enables the robot to do the optimal path planning in static environments. In the end, the answer SDRE controller with the answer linear quadratic controller will be compared. The results show that the proposed SDRE strategy leads to an efficient control law by which the robot avoids obstacles and moves from an arbitrary initial point × 0 to a target point. The robust performance of SDRE method for a robot to avoid obstacles and reach the target is demonstrated via simulations and experiments. Simulations are done using MATLAB software.

Coupled State-Dependent Riccati Equation Control for Continuous Time Nonlinear Mechatronics Systems

This paper considers a novel coupled state-dependent Riccati equation (SDRE) approach for systematically designing nonlinear quadratic regulator (NLQR) and H∞ control of mechatronics systems. The state-dependent feedback control solutions can be obtained by solving a pair of coupled SDREs, guaranteeing nonlinear quadratic optimality with inherent stability property in combination with robust L2 type of disturbance reduction. The derivation of this control strategy is based on Nash's game theory. Both finite and infinite horizon control problems are discussed. An under-actuated robotic system, Furuta rotary pendulum, is used to examine the effectiveness and robustness of this novel nonlinear control approach.