State-dependent Riccati Equation Research Articles

A benchmark mechatronics platform to assess the inspection around pipes with variable pitch quadrotor for industrial sites

This paper investigates the inspection-of-pipe topic in a new framework, by rotation around a pipe, peculiar to industrial sites and refineries. The evolution of the ultimate system requires prototype design and preliminary tests. A new benchmark has been designed and built to mimic the rotation around a pipe, with the main purpose of assessing the different types of rotors and control systems. The benchmark control system presents a mechatronics package including mechanical design and machining, electronics and motor drive, motor-blade installation, computer programming, and control implementation. The benchmark is also modular, working with two modes of one- and two-degree-of-freedom (DoF), easily interchangeable. To cover a full rotation, conventional fixed-pitch drones fail to provide negative thrusts; nonetheless, variable-pitch (VP) rotor quadcopters can produce that in both directions. A closed-loop nonlinear optimal method is chosen as a controller, so-called, “the state-dependent Riccati equation (SDRE)” approach. Optimal control policies are challenging for experimentation though it has been successfully done in this report. The advantage of the VP is also illustrated in a rotation plus radial motion in comparison with fixed-pitch rotors while a wind gust disturbs the inspection task. The proposed VP system compensated the disturbance while the fixed pitch was pushed away by the wind gust. The solution methods to the SDRE were mixed, a closed-form exact solution for the one-DoF system, and a numerical one for the two-DoF. Solving the Riccati online in each time step is a critical issue that was effectively solved by the implementation approach, through online communication with MATLAB software. Both simulations and experiments have been performed along with a discussion to prove the application of VP systems in rotary-motion pipe inspection.

A novel frequency tracker for sinusoidal signal based on state dependent Riccati Equation filter

This paper presents a new approach for frequency estimation and tracking based on State Dependent Riccati Equation Filter (SDREF). The SDREF is an alternative concept to the well-known Extended Kalman Filter (EKF) to design a robust nonlinear filter, with low computational costs and better performances. The aim is to estimate and track the time-varying fundamental frequency of harmonic sinusoidal signals in general and more particularly the fundamental electric frequency. Through this study, the performance of the proposed approach is compared to the widely adopted the EKF, taking into account several aspects both theoretically and real-life applications. It has been shown that the proposed estimator performs better than the usual EKF estimator in terms of noise robustness, estimation accuracy, and tracking ability both in numerical simulations and experimental study.

A framework for 3D tracking of frontal dynamic objects in autonomous cars

Both recognition and 3D tracking of frontal dynamic objects are crucial problems in an autonomous vehicle, while depth estimation as an essential issue becomes a challenging problem using a monocular camera. Since both camera and objects are moving, the issue can be formed as a structure from motion (SFM) problem. In this paper, to elicit features from an image, the YOLOv3 approach is utilized beside an OpenCV tracker. Subsequently, to obtain the lateral and longitudinal distances, a nonlinear SFM model is considered alongside a state-dependent Riccati equation (SDRE) filter and a newly developed observation model. Additionally, a switching method in the form of switching estimation error covariance is proposed to enhance the robust performance of the SDRE filter. The stability analysis of the presented filter is conducted on a class of discrete nonlinear systems. Furthermore, the ultimate bound of estimation error caused by model uncertainties is analytically obtained to investigate the switching significance. Simulations are reported to validate the performance of the switched SDRE filter. Finally, real-time experiments are performed through a multi-thread framework implemented on a Jetson TX2 board, while radar data is used for the evaluation.

Numerical Methods for Monitoring Rare Events in Nonlinear Stochastic Systems

In this article, we consider the development of numerical methods of large deviations analysis for rare events in nonlinear stochastic systems. The large deviations of the controlled process from a certain stable state are the basis for predicting the occurrenceof a critical situation (a rare event). The rare event forecasting problem is reduced to the Lagrange-Pontryagin optimal control problem.The presented approach for solving the Lagrange-Pontryagin problem differs from the approach used earlier for linear systems in that it uses feedback control. In the nonlinear case, approximate methods based on the representation of the system model in the state-space form with state-dependent coefficients (SDC) matrixes are used: the state-dependent Riccati equation (SDRE) and the asymptotic sequence of Riccati equations (ASRE). The considered optimal control problem allow us to obtain a numerical-analytical solutionthat is convenient for real-time implementation. Based on the developed methods of large deviations analysis, algorithms for estimating the probability of occurrence of a rare event in a dynamical systemare presented. The numerical applicability of the developed methods is shown by the example of the FitzHugh-Nagumo model for the analysis of switching between excitable modes. The simulation results revealed an additional problem related to the so-called parameterization problem of the SDC matrices. Since the use of different representations for SDC matrices gives different results in terms of the system trajectory, the choice of matrices is proposed to be carried out at each algorithm iteration so as to provide conditions for the solvability of the Lagrange-Pontryagin problem.

Time-varying sliding mode control of missile based on suboptimal method

This paper proposes a time-varying sliding mode control method to address nonlinear missile body kinematics based on the suboptimal control theory. The analytical solution of suboptimal time-varying sliding surface and the corresponding suboptimal control law are obtained by solving the state-dependent Riccati equation analytically. Then, the Lyapunov method is used to analyze the motion trend in sliding surface and the asymptotic stability of the closed-loop system is validated. The suboptimal control law is transformed to the form of pseudo-angle-of-attack feedback. The simulation results indicate that the satisfactory performance can be obtained and the control law can overcome the influence of parameter errors.

SDRE and LQR Controls Comparison Applied in High-Performance Aircraft in a Longitudinal Flight

This paper presents the design of the LQR (Linear Quadratic Regulator) and SDRE (State-Dependent Riccati Equation) controllers for the flight control of the F-8 Crusader aircraft considering the nonlinear model of longitudinal movement of the aircraft. Numerical results and analysis demonstrate that the designed controllers can lead to significant improvements in the aircraft's performance, ensuring stability in a large range of attack angle situations. When applied in flight conditions with an angle of attack above the stall situation and influenced by the gust model, it was demonstrated that the LQR and SDRE controllers were able to smooth the flight response maintaining conditions in balance for an angle of attack up to 56% above stall angle. However, for even more difficult situations, with angles of attack up to 76% above the stall angle, only the SDRE controller proved to be efficient and reliable in recovering the aircraft to its stable flight configuration.

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.

Suboptimal sliding manifold For nonlinear supply chain with time delay

In this paper, a basic dynamic model of the supply chain system is constructed in which the oscillation problems caused by the time delay in remanufacturing, ordering, the disturbance of the system parameters are considered along with the customers’ demand estimation. Time delay in production systems affects the efficiency of supply chain and lead to the bullwhip effect. This paper proposes a new method according to the model structure for controlling the bullwhip effect based on state dependent Riccati equation (ESDRE) and designing suboptimal sliding manifolds for a nonlinear supply chain in the presence of input and state delays. A switching control scheme is obtained based on the designed suboptimal sliding manifold. It is proved that this control scheme can guarantee that the nonlinear supply chain system is asymptotically stable and understand soft switching among subsystems of the nonlinear supply chain to mitigate fluctuations in the system variables. The efficiency of suboptimal sliding manifold method was examined by comparing LQR method in the presence of various time delays. Also, simulation investigation in real supply chain proved this efficiency.

Biological control of the chaotic sugarcane borer-parasitoid agroecosystem

As a raw material for ethanol, the production of sugarcane in Brazil is a very important and profitable agribusiness sector. In the last decades, with the prohibition of burning cane fields and the expansion of plantations, there has been a significant increase in the population of pests, the most important of which is a sugarcane borer (Diatraea saccharalis). It affects the stalks of plants, causing losses and decreasing the industrial yield in the sugar and alcohol production process. Thus, the purpose of this work is to study an interaction between the sugarcane borer and its larval parasite (Cotesia flavipes). The interaction between these populations is modeled by a host-parasitoid system, for which the study is done considering an influence of seasonal variation on the dynamics of the system. It has been shown that this variation generates chaotic dynamics in the system. In order to keep the pest population below economic damage, a biological control strategy has been proposed, which consists of introducing into the environment the amount of pest parasitoids determined by the State Dependent Riccati Equation (SDRE) method.

On global stability of planar systems with state‐dependent Riccati equation control

AbstractThis paper is concerned with the global asymptotic stability (GAS) of a class of planar systems with a state‐dependent Riccati equation (SDRE) control. Based on the philosophy of feedback domination along with the depiction of the influence arising from system nonlinearities, a sufficient condition indicating the allowable intervals of control gains is established by which the GAS of the system with SDRE control can be guaranteed successfully. The presented condition is delicately equipped with a tunable parameter skillfully enhancing the feasibility while providing a new insight and perspective in ensuring global stability of the SDRE control. A numerical example is provided to verify the effectiveness of the proposed result.

Modeling and Nonlinear Control of a Two-Wheeled Self Balancing Human Transporter

The two-wheeled self-balancing human transporters (TW-SBHT) are being widely used in transportation nowadays owing to their advantages such as energy saving, environmental protection, simple structure, and flexible operation. The modelling and control of TW-SBHT have emerged as one of the trending research areas in the field of control system design of mobile robots. Being a complex and nonlinear system, the control problem of TW-SBHT is a challenging task and needs to be effectively tackled to achieve the control objectives of maintaining uniform speed and dynamic stability. Though linear control strategies for TW-SBHT have been already proposed in the literature, they cannot offer an effective control for large external disturbances. Therefore, in this work, a non-linear control i.e. State-Dependent Riccati Equation (SDRE) has been implemented for effective control of TW-SBHT and its performance is compared with the linear controls including Proportional-Integral-Derivative (PID) and Linear-Quadratic Regulator (LQR) techniques. Initially, the more accurate model of TW-SBHT has been derived by applying the suitable modifications in existing models. Then, the application of SDRE for control of TW-SBHT has been presented and its performance is compared with linear control strategies.

Nonlinear optimal attitude control of spacecraft using novel state-dependent coefficient parameterizations

This paper addresses the attitude stabilization problem with the optimization objective regarding system performance and energy consumption for rigid spacecraft. Novel state-dependent coefficient (SDC) parameterizations are developed to reduce the optimization problem to solving the state-dependent Riccati equation (SDRE). To be explicit, a proper SDC parameterization is selected for the stabilization mission among infinite feasible SDC representations by utilizing an improved guideline. Subsequently, a novel analytical SDC reconstruction is evoked alternatively at the mission breakdown region in which the derived SDC parameterization fails to operate. Behind the scenes, the anti-unwinding behavior is supported by the explicit sampling SDC forms derived for the scalar part of the unit quaternion. The novelty of two developed SDC parameterizations lies in that they not only render the maximal pointwise controllable space to facilitate the control mission, but also cover the whole maneuvering domain without any breakdowns. Moreover, the computational burdens for online SDRE solvability checking and numerical SDC reconstruction are alleviated significantly. Finally, comparative numerical simulations are implemented to highlight the superior performance of the presented scheme.

Fast real-time SDRE controllers using neural networks

This paper describes the implementation of fast state-dependent Riccati equation (SDRE) control algorithms through the use of shallow and deep artificial neural networks (ANN). Several ANNs are trained to replicate an SDRE controller developed for a satellite attitude dynamics simulator (SADS) to display the technique’s efficacy. The neural controllers have reduced computational complexity compared with the original SDRE controller, allowing its execution at a significantly higher rate. One of the neural controllers was validated using the SADS in a practical experiment. The experimental results indicate that the training error is sufficiently small for the neural controller to perform equivalently to the original SDRE controller.

Geometric control using the state-dependent Riccati equation: application to aerial-acrobatic maneuvers

Acrobatic flip is one of the most challenging representatives of aggressive maneuvers to test the performance of an aerial system’s capability or a controller. A variable-pitch rotor quadcopter generates thrust in both vertical directions for the special design of the rotor’s actuation mechanism. This research proposes two possible solutions for the flip: a regulation solution based on the geometric control approach; and tracking a predefined optimal smooth trajectory covering a turnover. The first solution uses a geometric control approach that is immune to singular points since the rotation matrix is integrated on the manifold on . The second solution proposes an optimal trajectory generation for flip maneuver using open-loop optimal control, two-point boundary value problem (TPBVP) approach. Since generated open-loop state information is not applicable without a controller, the state-dependent differential Riccati equation (SDDRE) is chosen for trajectory tracking.

Stability Evaluation of the SDRE Technique based on Java in a CubeSat Attitude and Orbit Control Subsystem

In 2013, the STRaND (University of Surrey and Surrey Satellite Technology Ltd) and the PhoneSat (NASA) programs attracted the attention of the aerospace community applying commercial off-the-shelf smartphones in CubeSats. Both programs deployed CubeSats using smartphones based on Google's Android, in which application development is mainly based on Java programming language. Some of these CubeSats had actuators, e.g., STRaND-1 had three reaction wheels mounted in an orthogonal configuration to provide three-axis control, whereas PhoneSat 2.0 beta had magnetorquers to de-tumble the spacecraft. Taking into account a CubeSat that runs Android operating system (based on a smartphone), it is natural to evaluate the attitude and orbit control subsystem (AOCS) based on Java. Elsewhere, we shown State-Dependent Riccati Equation (SDRE) is a feasible non-linear control technique that can be applied in such CubeSats using Java. Moreover, we shown, through simulation using a Monte Carlo perturbation model, SDRE provides better performance than the PID controller, a linear control technique. In this paper, we tackle the next fundamental problem: stability. We evaluate stability from two perspectives: (1) parametric uncertainty of the inertia tensor and (2) a Monte Carlo perturbation model based on a uniform attitude probability distribution. Through the combination of these two perspectives, we grasp the stability properties of SDRE in a broader sense. In order to handle the uncertainty appropriately, we combine SDRE with H∞. The Nanosatellite Constellation for Environmental Data Collection (CONASAT), a CubeSat from the Brazilian National Institute for Space Research (INPE), provided the nominal parameters for the simulations. The initial results of the simulations shown that the SDRE controller is stable to ± 20% uncertainty in the inertia tensor for attitudes uniformly distributed and angular velocity up to 0.15 radians/second.

Optimal control methods for drug delivery in cancerous tumour by anti-angiogenic therapy and chemotherapy.

There are numerous mathematical models simulating the behaviour of cancer by considering variety of states in different treatment strategies, such as chemotherapy. Among the models, one is developed which is able to consider the blood vessel‐production (angiogenesis) in the vicinity of the tumour and the effect of anti‐angiogenic therapy. In the mentioned‐model, normal cells, cancer cells, endothelial cells, chemotherapy and anti‐angiogenic agents are taking into account as state variables, and the rate of injection of the last two are considered as control inputs. Since controlling the cancerous tumour growth is a challenging matter for patient's life, the time schedule design of drug injection is very significant. Two optimal control strategies, an open‐loop (calculus of variations) and a closed‐loop (state‐dependent Riccati equation), are applied on the system in order to find an optimal time scheduling for each drug injection. By defining a proper cost function, an optimal control signal is designed for each one. Both obtained control inputs have reasonable answers, and the system is controlled eventually, but by comparing them, it is concluded that both methods have their own benefits which will be discussed in details in the conclusion section.

Neural Stochastic Contraction Metrics for Learning-Based Control and Estimation

We present Neural Stochastic Contraction Metrics (NSCM), a new design framework for provably-stable robust control and estimation for a class of stochastic nonlinear systems. It uses a spectrally-normalized deep neural network to construct a contraction metric, sampled via simplified convex optimization in the stochastic setting. Spectral normalization constrains the state-derivatives of the metric to be Lipschitz continuous, thereby ensuring exponential boundedness of the mean squared distance of system trajectories under stochastic disturbances. The NSCM framework allows autonomous agents to approximate optimal stable control and estimation policies in real-time, and outperforms existing nonlinear control and estimation techniques including the state-dependent Riccati equation, iterative LQR, EKF, and the deterministic neural contraction metric, as illustrated in simulation results.