State-dependent Riccati Equation Method Research Articles

A suboptimal control scheme based on integrated real-time nonlinear estimation-identification approach for developed flexible joint manipulators structure

As the safety of human-robot interaction is of great interest in various applications, the flexibility of machine joints becomes increasingly important. In this paper, a general model is developed for electrically driven flexible-joint robots (ED-FJR), in which the coupled phenomena of stiffness-damping and friction in the joints are modeled as nonlinear and time-varying parameters. Because the output torque of the actuator mounted on the link cannot be measured, its state variables are estimated by the suboptimal estimator. Furthermore, the unknown parameters of joint flexibility are identified by the identification algorithm based on time-varying nonlinear least-squares error (TV-NLS), which are then used in the proposed control algorithm to compensate for negative effects and errors. The state-dependent Riccati equation (SDRE) method is developed in this paper using the combined estimator-identifier approach. The novel proposed method for a 3-degree-of-freedom (3-DOF) mechanical arm with electrically driven flexible joints is implemented and validated. The obtained results show that the system's performance has improved by 83% in terms of modeling accuracy. Furthermore, when compared to other methods, the proposed algorithm reduces tracking error by 59%, making it easier to reach the target point.

Guidance strategy of motion camouflage for spacecraft pursuit-evasion game

This work is inspired by a stealth pursuit behavior called motion camouflage whereby a pursuer approaches an evader while the pursuer camouflages itself against a predetermined background. We formulate the spacecraft pursuit-evasion problem as a stealth pursuit strategy of motion camouflage, in which the pursuer tries to minimize a motion camouflage index defined in this paper. The Euler-Hill reference frame whose origin is set on the circular reference orbit is used to describe the dynamics. Based on the rule of motion camouflage, a guidance strategy in open-loop form to achieve motion camouflage index is derived in which the pursuer lies on the camouflage constraint line connecting the central spacecraft and evader. In order to dispose of the dependence on the evader acceleration in the open-loop guidance strategy, we further consider the motion camouflage pursuit problem within an infinite-horizon nonlinear quadratic differential game. The saddle point solution to the game is derived by using the state-dependent Riccati equation method, and the resulting closed-loop guidance strategy is effective in achieving motion camouflage. Simulations are performed to demonstrate the capabilities of the proposed guidance strategies for the pursuit–evasion game scenario.

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.

Statistical Investigation about the SDRE Optimality for Satellites with Nonlinearities

The Attitude Control System (ACS) of a Satellite can be designed with success by linear control theory if the satellite has slow angular motions. However, for fast maneuvers, the linearized models are not able to represent the effects of the nonlinear terms. One candidate technique to design the ACS of the satellite under fast maneuvers is the State-Dependent Riccati Equation (SDRE), which provides an effective algorithm for synthesizing nonlinear feedback control taking into account the nonlinearities. Nonetheless, much criticism has been leveled against the SDRE method since it does not assure global asymptotic stability (GAS), because there are situations in which global asymptotic stability cannot be achieved (e.g., systems with multiple equilibrium points). One way to study the GAS is by estimating the region of attraction (ROA) which in turn is fundamental to investigate the performance of the controller designed by the SDRE method The Brazilian National Institute for Space Research (INPE) is responsible to build a Remote-Sensing Satellites, called Amazonia-1. The ACS of the Amazonia-1 satellite must be stabilized in three axes so that the optical payload can point to the desired target. In this paper, one investigates the performance of the Amazonia-1 ACS using a statistical test (unpaired t-test) that compares the optimality inside the ROA controlled by LQR and SDRE. By several simulations running a full Monte Carlo perturbation satellite model one observes a significant difference between the optimality of the two controllers in favor of SDRE when nonlinearities are accounted for.

State-dependent Riccati equation feedback stabilization for nonlinear PDEs

The synthesis of suboptimal feedback laws for controlling nonlinear dynamics arising from semi-discretized PDEs is studied. An approach based on the State-dependent Riccati Equation (SDRE) is presented for 2 and ∞ control problems. Depending on the nonlinearity and the dimension of the resulting problem, offline, online, and hybrid offline-online alternatives to the SDRE synthesis are proposed. The hybrid offline-online SDRE method reduces to the sequential solution of Lyapunov equations, effectively enabling the computation of suboptimal feedback controls for two-dimensional PDEs. Numerical tests for the Sine-Gordon, degenerate Zeldovich, and viscous Burgers’ PDEs are presented, providing a thorough experimental assessment of the proposed methodology.

Performance analysis of the vectorized high order expansions method in the accurate landing problem of reusable boosters

This paper addresses the design of the terminal phase maneuver of the booster landing in the presence of initial deviations and uncertainties. The primary purpose of this study is to evaluate the effectiveness of the vectorized high order expansions method to extract commands and near-optimal trajectory, while the accurate satisfaction of the terminal conditions is of great importance. After reviewing high order expansions methods, the Vectorized High Order Expansions method is briefly discussed. Then, the solution for the booster landing problem is extracted up to the third order. To study the solution quality in the presence of uncertainties, mass, density, and gravity are assumed to be constant in extracting the 3rd order solution. The assumptions however are eliminated in the simulations, and the effectiveness of high order expansions solution is studied subjectively and numerically in the presence of initial deviations and uncertainties. After implementing simulations and considering different assumptions, it will be shown that the 3rd order solution holds a desirable quality in accurately satisfying final conditions in presence of uncertainties and initial deviations and provides far better results than the 1st order solution. Moreover, in this paper, a novel method is proposed for online updating, and the performance and effect of this modification are studied using simulations. Reviewing the results of the 3rd order solution combined with the online update strategy, it becomes clear that the implementation of the online update significantly reduces landing point errors and improves the performance of the high order method. Furthermore, the study compares the performance of this strategy with the well-known state-dependent Riccati equation method to better evaluate the effectiveness of the proposed approach.

An Asymptotic Method for Predicting Risks in Problems of Stochastic Monitoring and Control

To ensure the stabilization of the equilibrium state in a nonlinear system in the presence of noise, it is not enough to solve the local stabilization problem, it is also necessary to ensure continuous monitoring of a possible transition to a critical state leading to system failure. To organize such monitoring, we use the large deviations principle applied to dynamical systems with small perturbations. For the purposes of monitoring, the optimal path that we call the A-profile is important. We use the A-profile to build a situational forecast in the risk control problem for a multi-agent system. In addition to the nonlinear mechanism of internal stabilization of the level h for each of the agents, there are forces of mean field interaction between the agents. The weak limit in this model with the number of agents tending to infinity is described by the FokerPlanck-Kolmogorov equation, but the use of approximation up to O(h2) leads to a finite-dimensional Wentzel-Freidlin scheme. According to the scheme, we obtain an explicit A-profile as a solution of the degenerate Abel equation of the second kind. At the same time, the approximation in h makes it possible to develop a method of successive approximations for the A-profile. In this paper, the A-profile is synthesized as a solution of the optimal control problem, where the state-dependent Riccati equation method and the method of the approximating sequence of Riccati equations are used. In the article, these methods are applied and compared within the framework of the risk control problem.

Optimal consensus control of multi-agent systems based on the state-dependent riccati equation

This paper is concerned with optimal consensus control of a multi-agent system, leaderless or leader-follower control. The hierarchical state-dependent Riccati equation (SDRE) controller was the main framework. The Lyapunov-based method is used for solving time-varying systems. Not only positive definite solution of the related algebraic Riccati equation is realized by the SDRE method based on Lyapunov, but also the negative definite solution is realized. Furthermore, the leader-follower tracking structure is developed based on two nonlinear differential equations; the feedforward differential equation and the state-dependent Riccati differential equation, which consider the error in the cost function through the co-sate equation. The application of the proposed controller structure in controlling the lifting system of multiple 3D cranes is studied. The theoretical results have been verified by simulating the leaderless and leader-follower consensus control problem.

Modified Infinite-Time State-Dependent Riccati Equation Method for Nonlinear Affine Systems: Quadrotor Control

This paper presents modeling and infinite-time suboptimal control of a quadcopter device using the state-dependent Riccati equation (SDRE) method. It establishes a solution to the control problem using SDRE and proposes a new procedure for solving the problem. As a new contribution, the paper proposes a modified SDRE-based suboptimal control technique for affine nonlinear systems. The method uses a pseudolinearization of the closed-loop system employing Moore–Penrose pseudoinverse. Then, the algebraic Riccati equation (ARE), related to the feedback compensator gain, is reduced to state-independent form, and the solution can be computed only once in the whole control process. The ARE equation is applied to the problem reported in this study that provides general formulation and stability analysis. The effectiveness of the proposed control technique is demonstrated through the use of simulation results for a quadrotor device.

DC microgrid fault‐tolerant control using state‐dependent Riccati equation techniques

AbstractAn attempt is made to design a fault‐tolerant control system using a nonlinear technique, called the state‐dependent Riccati equation (SDRE) method. The proposed mechanism consists of a master controller, an observer‐based fault detection and isolation system, and three emergency controllers, designed using the SDRE methods. The master controller, as a suboptimal nonlinear regulator, is designed to return DC microgrid (MG) to its desired equilibrium point in normal operation condition. On the other hand, an SDRE observer is designed to establish a unified framework for fault detection and isolation in the DC MG. Indeed, in the case of unexpected fault occurrence, the proposed mechanism can isolate the faulty DG, and then, the emergency controllers are triggered immediately to stabilize the MG. Simulation results indicate that the proposed method is so useful to achieve the control objectives and online protection of the considered MG.

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.

Integrated Control and Protection Architecture for Islanded PV-Battery DC Microgrids: Design, Analysis and Experimental Verification

Direct current (dc) microgrids have gained significant interest in research due to dc generation/storage technologies—such as photovoltaics (PV) and batteries—increasing performance and reducing in cost. However, proper protection and control systems are critical in order to make dc microgrids feasible. This paper aims to propose a novel integrated control and protection scheme by using the state-dependent Riccati equation (SDRE) method for PV-battery based islanded dc microgrids. The dc microgrid under study consists of photovoltaic (PV) generation, a battery energy storage system (BESS), a capacitor bank and a dc load. The aims of this study are fast fault detection and voltage control of the dc load bus. To do so, the SDRE observer-controller—a nonlinear mathematical model—is employed to model the operation of the dc microgrid. Simulation results show that the proposed SDRE method is effective for fault detection and robust against external disturbances, resulting in it being capable of controlling the dc load bus voltage during disturbances. Finally, the dc microgrid and its proposed protection scheme are implemented in an experimental testbed prototype to verify the fault detection algorithm feasibility. The experimental results indicate that the SDRE scheme can effectively detect faults in a few milliseconds.

State Dependent Riccati Equation (SDRE) controller design for moving obstacle avoidance in mobile robot

In this paper, a new navigating method for a non-holonomic wheeled moving robot in a dynamic environment with the moving obstacles is proposed. This method is indeed the combination of the nonlinear optimal controller based on State-Dependent Riccati Equation (SDRE) and an obstacle avoidance algorithm named artificial potential field (APF). The corresponding cost function of the SDRE is obtained of APF algorithm. APF algorithm forces the robot to approach the target as an attracting (low-potential) point and to get away from the obstacle as a repulsing (high-potential) point. This method calculates the best path from origin to destination which also implicitly guaranties the stability. The obtained path is the best according to the both amount of traveled distance and input energy. Moreover, this approach not only avoid both fixed and moving obstacle, but also create a smooth path in presence of them. Keeping the nonlinear structure of the system instead of eliminating them during the linearization process is the advantage of SDRE method. Here, the robot navigation is done in the presence of the three different movements of obstacle: (1) fixed speed, (2) fixed acceleration and (3) non-uniform circular. The represented simulation results indicate a suitable performance of the proposed algorithm.

Suboptimal closed‐form feedback control of input‐affine non‐linear systems

A closed-form feedback solution to finite-time optimal control problems for input-affine non-linear systems is presented here. Based on the state-dependent Riccati equation (SDRE) framework, this suboptimal solution, called balanced-cost SDRE (BCSDRE), is proposed to minimise the overall control effort required for driving the system from an arbitrary initial condition to an acceptable neighbourhood around origin, within a pre-specified final time. The state error is dynamically penalised to balance the overall state error and the control cost, resulting in a modified SDRE. Unlike the existing finite-horizon SDRE methods, the proposed technique does not require to solve algebraic Riccati and Lyapunov equations at every time step, which makes this approach computationally efficient. Based on the Lyapunov theory, the origin of the BCSDRE controlled system is shown to be locally exponentially stable. Furthermore, the state profile is also shown to be locally predefined time ultimately bounded, under appropriate tuning of the proposed controller. The proposed technique is implemented to design a controller for the Van der Pol oscillator and impact angle guidance design problem, and its performance is compared with the existing finite-horizon SDRE-based techniques.

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.

Vehicle tip-over prevention using gain-scheduled State-dependent Riccati Equation–based anti-rollover controller

The most hazardous kind of vehicle crash among all road accidents is vehicle rollover. Present-day rollover prevention systems in commercial vehicles mitigate rollover by preventing any wheel lift-off from the ground. These systems make use of actuators such as differential brakes and demand all the wheels on the ground for satisfactory operation. Such systems are not effective in recovering a vehicle from intense rollover scenarios where the wheels on one side are lifted off the ground, and the vehicle is about to rollover to the other side after reaching the tip-over point. A few studies have investigated the possibility of reinstating a vehicle at the tip-over point with the wheels on a side lifted off. The high complexity and computation time of the optimal control strategies such as nonlinear model predictive controller make it unsuitable for real-time implementations. This study proposes a novel gain-scheduled State-dependent Riccati Equation–based optimal anti-rollover controller for reinstating a vehicle from the tip-over point. An inverted double pendulum on a cart vehicle model is used as the plant model. The anti-rollover controller is found to be presentable as a two-dimensional gain-scheduled lookup table with specific state dependencies in existence. It eliminates the necessity of solving the nonlinear performance index minimization problem online. State-dependent Riccati Equation method adequately accounts for the nonlinearities involved, yet possesses a small computational time per sample. The anti-rollover controller is evaluated with a 10 degrees of freedom full vehicle model with a nonlinear pure slip tyre model that incorporates the dynamical effects neglected in the controller formulation. Finally, the anti-rollover controller is evaluated in real-life initial conditions using a sophisticated pick-up truck model obtained from TruckSim® software through a co-simulation with the anti-rollover controller setup in MATLAB®/Simulink® environment. The State-dependent Riccati Equation controller was found to be effective in reinstating the higher-order models from the tip-over point in all the case studies conducted.

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

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.

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.

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.