State-dependent Riccati Equation Technique Research Articles

Observer-Based Suboptimal Controller Design for Permanent Magnet Synchronous Motors: State-Dependent Riccati Equation Controller and Impulsive Observer Approaches

Permanent Magnet Synchronous Motors (PMSMs) with high energy efficiency, reliable performance, and a relatively simple structure are widely utilised in various applications. In this paper, a suboptimal controller is proposed for PMSMs without sensors based on the state-dependent Riccati equation (SDRE) technique combined with customised impulsive observers (IOs). Here, the SDRE technique facilitates a pseudo-linearised display of the motor with state-dependent coefficients (SDCs) while preserving all its nonlinear features. Considering the risk of non-available/non-measurable states in the motor due to sensor and instrumentation costs, the SDRE is combined with IOs to estimate the PMSM speed and position states. Customised IOs are proven to be capable of obtaining quality, continuous estimates of the motor states despite the discrete format of the output signals. The simulation results in this work illustrate an accurate state estimation and control mechanism for the speed of the PMSM in the presence of load torque disturbances and reference speed changes. It is clearly shown that the SDRE-IO design is superior compared to the most popular existing regulators in the literature for sensorless speed control.

Cobra aerobatic flight for quadrotors using θ-D nonlinear control

The control design for the Cobra maneuver of quadrotors is presented based on quaternion dynamics. Cobra maneuver commands the quadrotor in ± π / 2 (rad) pitch angle such that the conventional hovering assumption for quadrotor flight is not applicable. The finite-time θ-D method is proposed to control the Cobra maneuver. Based on this technique, the intractable Hamilton–Jacobi–Bellman equation can be approximated by a differential Riccati equation and a series of state-dependent Lyapunov equations, which can be solved recursively and lead to the closed-form state feedback control law . This control law can achieve semi-global asymptomatic stability of the closed-loop system. Numerical simulations verify that the proposed technique is compared favourably with the finite-time state-dependent Riccati equation (SDRE) technique in terms of computation efficiency and control effort, and also shows faster recovery of the quadrotor after the Cobra maneuver compared with conventional dynamics without the derivative of the rotation matrix.

Energy-optimal adaptive artificial potential field method for path planning of free-flying space robots

Aiming at planning a collision-free path for a free-flying space robot (FFSR), an energy-optimal collision avoidance strategy is proposed using the adaptive artificial potential field (AAPF) approach. To address the goals non-reachable with obstacles nearby (GNRON) issues of artificial potential field (APF), an adaptive and secure path planning method is proposed by incorporating a smooth decay function into the repulsive potential field. Meanwhile, a time-varying shaping parameter of attractive potential field is optimized combined with the state-dependent Riccati equation technique to reduce the energy cost of the FFSR. In case the system gets trapped in local minimum, an escape force is elaborately designed to replace artificial potential force based on optimal theory, whose direction vector guides the FFSR to jump out of the local minimum with minimal energy. Finally, simulation results on a planar FFSR demonstrate the effectiveness of the proposed methodology and show improved performance compared with the existing artificial potential field (APF) in terms of escaping local minimum and reducing energy consumption.

Hermitian Solutions of the Quaternion Algebraic Riccati Equations through Zeroing Neural Networks with Application to Quadrotor Control

The stability of nonlinear systems in the control domain has been extensively studied using different versions of the algebraic Riccati equation (ARE). This leads to the focus of this work: the search for the time-varying quaternion ARE (TQARE) Hermitian solution. The zeroing neural network (ZNN) method, which has shown significant success at solving time-varying problems, is used to do this. We present a novel ZNN model called ’ZQ-ARE’ that effectively solves the TQARE by finding only Hermitian solutions. The model works quite effectively, as demonstrated by one application to quadrotor control and three simulation tests. Specifically, in three simulation tests, the ZQ-ARE model finds the TQARE Hermitian solution under various initial conditions, and we also demonstrate that the convergence rate of the solution can be adjusted. Furthermore, we show that adapting the ZQ-ARE solution to the state-dependent Riccati equation (SDRE) technique stabilizes a quadrotor’s flight control system faster than the traditional differential-algebraic Riccati equation solution.

Finite-Time Suboptimal Control Design for Aerobatic Maneuver of Variable-Pitch Quadrotor

The variable-pitch quadrotors can perform aggressive maneuvers such as aerobatic flip by varying the thrust in both positive and negative vertical directions via the blade pitch, which can provide an additional degree of freedom as opposed to the fixed-pitch quadrotor that only changes the rotational rate of motors. However, an inevitable challenge exists when designing an operative controller for the aerobatic flip due to the resultant singularity of the rotation matrix. In this paper, a finite-time suboptimal control design strategy named finite-time <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\theta -D$</tex-math></inline-formula> algorithm combined with the geometric control technique is proposed for the aerobatic maneuver control of the variable-pitch quadrotor. Based on this scheme, the singularity issue can be circumvented. The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\theta -D$</tex-math></inline-formula> control design provides a closed-form suboptimal control law by offline solving a differential Riccati equation and a set of linear Lyapunov equations analytically, which does not require complex online numerical and iterative solutions compared with the similar finite-time state dependent Riccati equation technique. Simulation results demonstrate that the proposed geometric finite-time <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\theta -D$</tex-math></inline-formula> method provides an effective and efficient control of the variable-pitch quadrotor.

Synthesis of Suboptimal Guidance Law for Anti-Tank Guided Missile with Terminal Impact Angle Constraint Based on the SDRE Technique

In this paper, the homing-phase guidance law is proposed against a stationary target in the planar engagement scenario. In this problem, two technical criteria that the guidance law must achieve are: first, zero terminal miss distance, and second, satisfying the desired impact angle constraint at the final time. This guidance law is synthesized as a nonlinear optimal control problem with an infinite-time horizon and is solved using the state-dependent Riccati equation (SDRE) method. To obtain a guidance law with a finite-time horizon, a state weighting matrix based on the time-to-go is utilized in the SDRE control scheme. The synthesized guidance law is applied to a new generation anti-tank guided missile class, with specific thrust and drag parameters. Nonlinear simulations are conducted to demonstrate the fundamental properties, applicability, and effectiveness of the proposed guidance law.

On the modelling, analysis, and design of a suboptimal controller for a class of wind/PV/battery based DC microgrid

This study deals with the comprehensive modelling, analysis, and control of a DC microgrid (MG) in islanded mode. The proposed DC MG comprises a wind turbine, a photovoltaic (PV) source, battery storage, DC/DC source, and load side converters with DC loads. To this aim, a circuit-oriented modelling of the whole system is developed. The PV source is modelled with a single-diode electrical circuit. Afterward, a mathematical model of the system with state-space representation was derived. A detailed analysis of PV system design is performed because the parameters of PV are appearing in the dynamic model of DC MG. For the purpose of controller development, the dynamic model of the DC MG, which is modelled by a non-linear-non-affine eight-order system, is linearized around an equilibrium point using the Jacobian matrix framework, while stability using eigenvalue is carried out showing that the stability is guaranteed under operating condition. Finally, for the first time, the state-dependent Riccati equation (SDRE) technique is proposed to find the optimal regulation problem for the DC MG with non-linear-non-affine dynamics. The numerical simulation studies first confirm the validity of the performed mathematical, then the effectiveness of the proposed non-linear controller is evaluated under illumination and load change.

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.

Three-dimensional suboptimal nonlinear impact time guidance against non-maneuvering target

In this paper, a nonlinear suboptimal three-dimensional guidance strategy based on a finite-time state-dependent Riccati equation technique is proposed to achieve interception of a non-maneuvering target at a desired impact time. To design the proposed guidance, range error for a spatial engagement between an interceptor and non-maneuvering target is introduced as the difference between the desired range-to-go and range-to-go of the predicted interception point. It is shown that forcing the range error and its rate to the origin are sufficient to achieve impact time constrained interception. Several existing impact time guidance strategies require a time-to-go estimate, to implement their guidance command, which may become erroneous in the presence of measurement noise and affect their performance. However, the proposed guidance does not require any such time-to-go estimate in its design process. Furthermore, guidance for the case of planar engagements is shown to be a special case of the proposed guidance scheme. Finally, numerical simulation studies for multiple engagement geometries, along with a comparison study with an existing strategy, are provided to demonstrate the efficacy of the proposed technique.

Nonlinear SDRE based adaptive fuzzy control approach for age-specific drug delivery in mixed chemotherapy and immunotherapy

Determination of a near-optimal personalized drug delivery scenario in mixed chemotherapy and immunotherapy based on the age of cancer patients was presented in this paper. For this purpose, a mathematical model for cancer dynamics is considered in the form of ordinary differential equations (ODE) which contains cancer cells, tumor cells, and chemotherapy drug intervention. This model is modified by adding immunotherapy intervention effects to alter cancer dynamics. We consider two patients with known and unknown model parameters which are mentioned as a reference and unknown patients respectively. The nonlinear composite adaptive controller is introduced by compounding State Dependent Riccati Equation (SDRE) and Model Reference Adaptive Control (MRAC) techniques for achieving the drug delivery of an unknown patient. The drug delivery scenario for a reference patient with a known mathematical model and parameters is determined via the SDRE technique and then for any unknown patient, the personalized mixed therapy protocol is achieved using the treatment regimen of the reference patient as a reference model in MRAC. To reduce the side effects of chemotherapy drugs, we determine drug maximum dose limits using fuzzy control by considering the age of cancer patients. We regarded patients in four age groups of child, young, middle-aged, and old. In the proposed methodology, unknown patients are considered as a black-box simulator and the mathematical model parameters of the patient are not essential for the design of drug administration protocol and we only need patients’ age. After chemotherapy, the treatment is completed by immunotherapy to avoid cancer relapse. The effect of reference and unknown patient ages in obtaining drug delivery protocol is deeply surveyed. Numerical simulations confirm the effectiveness, robustness, and flexibility of the proposed strategy for patients of different ages. The Proposed algorithm can be an effective tool to aid oncologists in prescribing age-specific optimal treatment protocols for cancer patients.

A novel controller for nonlinear uncertain systems using a combination of SDRE and function approximation technique: Regulation and tracking of flexible-joint manipulators

This paper presents a novel combined State Dependent Riccati Equation (SDRE) / Function Approximation Technique (FAT)-based control design for nonlinear uncertain systems. The SDRE is employed to construct an optimal controller and the function approximation technique is utilized to estimate time-varying disturbances and uncertainties. Moreover, a robust term in the proposed control law compensates for the truncation error. The closed-loop stability and boundedness of the tracking error and FAT weights approximation error are proved in the sense of Lyapunov, with consideration of truncation error. Due to the great importance of the adequate performance of transient response from practical point of view, performance evaluation has been accomplished. The proposed scheme is computationally simple due to utilizing the FAT to represent uncertainties and disturbances as a function of time. Compared with the SDRE based SMC, the proposed controller is superior in terms of capability to track a fast and highly complicated trajectory and no need to determine time-varying disturbances and uncertainties bounds. The case study is a Selective Compliant Articulated Robot for Assembly (SCARA) flexible joint manipulator as a representative of highly nonlinear, coupled, large robotic systems. Simulation results easily verify the effectiveness and superiority of the proposed controller.

Optimal Factorization of the State-Dependent Riccati Equation Technique in a Satellite Attitude and Orbit Control System

The satellite attitude and orbit control system (AOCS) can be designed with success by linear control theory if the satellite has slow angular motions and small attitude maneuver. However, for large and fast maneuvers, the linearized models are not able to represent all the perturbations due to the effects of the nonlinear terms present in the dynamics and in the actuators (e.g., saturation). Therefore, in such cases, it is expected that nonlinear control techniques yield better performance than the linear control techniques. One candidate technique for the design of AOCS control law under a large maneuver is the State-Dependent Riccati Equation (SDRE). SDRE entails factorization (that is, parameterization) of the nonlinear dynamics into the state vector and the product of a matrix-valued function that depends on the state itself. In doing so, SDRE brings the nonlinear system to a (nonunique) linear structure having state-dependent coefficient (SDC) matrices and then it minimizes a nonlinear performance index having a quadratic-like structure. The nonuniqueness of the SDC matrices creates extra degrees of freedom, which can be used to enhance controller performance, however, it poses challenges since not all SDC matrices fulfill the SDRE requirements. Moreover, regarding the satellite's kinematics, there is a plethora of options, e.g., Euler angles, Gibbs vector, modified Rodrigues parameters (MRPs), quaternions, etc. Once again, some kinematics formulation of the AOCS do not fulfill the SDRE requirements. In this paper, we evaluate the factorization options (SDC matrices) for the AOCS exploring the requirements of the SDRE technique. Considering a Brazilian National Institute for Space Research (INPE) typical mission, in which the AOCS must stabilize a satellite in three-axis, the application of the SDRE technique equipped with the optimal SDC matrices can yield gains in the missions. The initial results show that MRPs for kinematics provides an optimal SDC matrix.

A near-optimal decentralized control approach for polynomial nonlinear interconnected systems

This article tackles the decentralized near-optimal control problem for the class of nonlinear polynomial interconnected system based on a shifted Legendre polynomials direct approach. The proposed method converts the interconnected optimal control problems into a nonlinear programming one with multiple constraints. In light of the formulated NLP optimization, state and control coefficients are used to design a nonlinear decentralized state feedback controller. Overall closed-loop system stability sufficient conditions are investigated with the help of Grönwall lemma. The triple inverted pendulum case is considered for simulation. Satisfactory results are obtained in both open-loop and closed-loop schemes with comparison to collocation and state-dependent Riccati equation techniques.

High-voltage ride through strategy for DFIG-based wind farms under bipolar blocking of HVDC system

Excessive reactive power generated by the filter and compensation devices of rectifier station will flow into the AC system when DC bipolar block occurs, which brings a high voltage threat to doubly-fed induction generator (DFIG)-based wind farms at the sending-end. This paper proposes a nonlinear robust tracking control (RTC) strategy to enhance the high-voltage ride through (HVRT) capability of DFIG-based wind farms. In the proposed strategy, the nonlinear characteristics of wind farms and the disturbance attenuation requirements during DC bipolar block is fully considered. The RTC problem is formulated with control objective of absorbing the reactive power injected into the AC system, and state-dependent Riccati equation (SDRE) technique is adopted to obtain an approximate solution. Case studies are performed on MATLAB/Simulink, the results illustrate the proposed HVRT control strategy has a better transient performance of suppressing overvoltage under Bipolar Blocking of HVDC System.

A nonlinear robust optimal controller for an active transfemoral prosthesis: An estimator-based state-dependent Riccati equation approach

This article presents an estimator-based nonlinear robust optimal controller for an active prosthetic leg for transfemoral amputees. The proposed controller is derived from a combination of the state-dependent Riccati equation technique to optimize the energy consumption of the robot/prosthesis system and the sliding mode control to reduce the effects of the model parametric uncertainties and ground reaction forces as nonparametric uncertainties. In addition, the integral state control technique is employed to improve tracking; also, to have a compromise between tracking performance and control signal chattering, the boundary layer is then used. In this study, the performance of both the controller and estimator in the presence of noise and disturbance is assessed for nominal system while ±40% parametric uncertainty with respect to saturation bounds of control signals is considered. The results of the simulation in this research with ±40% parametric uncertainty compared to a robust adaptive impedance control approach with the only variation of ±30% on the system parameters, show high performance of the proposed controllers in reducing energy consumption, good robustness, improved position tracking performance, and good performance in estimating state variables, even in the presence of large initial errors compared to the extended Kalman filter.

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.

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.

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.

Simulation and Validation of Satellite Attitude Control Algorithms in a Spherical Air Bearing

This work presents the application and comparison of two satellite attitude control methods, using the satellite simulation platform developed by ITA, the MuSat, as a test bench. One of the algorithms under test applies feedback linearization, a method that was already validated using this test bed. The second control algorithm, developed during this research, applies control by state-dependent Riccati equation (SDRE) technique. A numerical simulation using a computational model of the MuSat, followed by an experiment using the physical platforms, is used to compare the control performance of both algorithms. The system in which the control is applied consists in a frictionless spherical air bearing that is able to rotate in three axes, allowing for validating attitude control algorithms in laboratory. The experimental results obtained indicate that the system had a better control performance under the SDRE algorithm than the algorithm based on feedback linearization.

Path control in limit handling and drifting conditions using State Dependent Riccati Equation technique

Tyres operated at or close to their friction limits show a highly nonlinear motion to force response. This state is called limit handling. The objective of this research is to minimize lateral path tracking error while the tyres operate in limit handling. The State Dependent Riccati Equation technique is employed to develop a feedback-feedforward steering controller. It gives a systematic approach to take into account model nonlinearities such as combined slip tyre characteristics. Furthermore, the controller is implemented in a test vehicle and tested on a low friction skid pad. The controller shows robust path tracking performance even when the rear wheels are operated beyond their friction limits, and large body sideslip prevails.