State-dependent Differential Riccati Equation Research Articles

Extended State-Dependent Differential Riccati Equation (ESDDRE) Controller Design for a Chemical Plug-Flow Reactor via Time-Delay Partial Differential Equation

Extended State-Dependent Differential Riccati Equation (ESDDRE) Controller Design for a Chemical Plug-Flow Reactor via Time-Delay Partial Differential Equation

Fast SDDRE-Based Maneuvering-Target Interception at Prespecified Orientation

This study considers the 3-D guidance law based on target lead angle information and the state-dependent differential Riccati equation (SDDRE) scheme. In an application-oriented manner, it presents theories to significantly improve critical computational performance and thus aims at a fast implementation for impact-angle-constrained interception of agile maneuvering targets. More specifically, regarding the two major computational burdens using SDDRE, we have replaced the burden in numerical applicability checking by a simple, equivalent, and closed-form condition for the entire state space, which is actually the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dominant</i> burden as supported by complexity analysis and extensive validations. Notably, the proposed analysis not only complements the early findings of applicability guarantee in literature, but also promotes the efficiency of the proposed philosophy when compared to the classic method, where the latter has caused concerns/reservations due to its feasibility and difficulty. On the other hand, we have largely mitigated the second major burden of SDDRE by—after exhaustive trials—selecting the most efficient Riccati-equation solver until the latest benchmarks. Such evaluations are: 1) in favor of a much-less-known achievement, rather than the common QR-based benchmark and 2) subject to both numerical and hardware experiments including, notably, implementations on microcontrollers and field-programmable gate arrays.

Impact Angle Guidance Using Computationally Enhanced State-Dependent Differential Riccati-Equation Scheme

This study considers the latest three-dimensional impact angle guidance based on the state-dependent differential Riccati-equation (SDDRE) scheme, and it presents novel theories that efficiently guarantee the SDDRE’s applicability and largely reduce the computational burden. The unified applicability analysis completely categorizes the state space in terms of a simple equivalent condition, where all the inapplicable cases (leading to implementation breakdowns) are newly discovered and efficiently resolved. The condition almost removes the tedious online checking routine, which accounts for the dominant effort as endorsed by complexity analysis and practical validations. Moving forward to a general scope, we analyze the computational complexity of such an SDDRE controller: first subject to the MATLAB® framework and then the state-of-the-art enhancements, where the latter come from the best performance among extensive trials. Finally, numerical and hardware experiments (notably, microcontroller and field-programmable gate array) strengthen the confidence in the analytical findings, and they enrich the value in robustness and generality that benefit more guidance or control systems.

Sub-optimal controller design for time-delay nonlinear partial differential equation systems: an extended state-dependent differential Riccati equation approach

ABSTRACT In this article, a sub-optimal controller based on Extended State-Dependent Differential Riccati Equation (ESDDRE) is suggested for a class of time-delay nonlinear Partial Differential Equation (PDE) systems. At first, an extended pseudo linearisation presentation for parameterisation form using State-Dependent Coefficients (SDC) is proposed. In this presentation, all the time-delay parts are placed in system matrices as well as in input vectors. By defining a cost function and a Hamiltonian related to the PDE systems, the sub-optimal control law regarded on the ESDDRE is obtained. The stability of the closed-loop system based on the ESDDRE control approach is ensured by using a proper Lyapunov function and also Poincaré inequality. Numerical simulation results for three time-delay nonlinear PDE systems illustrate the appropriate performance of the proposed control approach.

Combination of terminal sliding mode and finite-time state-dependent Riccati equation: Flapping-wing flying robot control

A novel terminal sliding mode control is introduced to control a class of nonlinear uncertain systems in finite time. Having command on the definition of the final time as an input control parameter is the goal of this work. Terminal sliding mode control is naturally a finite-time controller though the time cannot be set as input, and the convergence time is not exactly known to the user before execution of the control loop. The sliding surface of the introduced controller is equipped with a finite-time gain that finishes the control task in the desired predefined time. The gain is found by partitioning the state-dependent differential Riccati equation gain, then arranging the sub-blocks in a symmetric positive-definite structure. The state-dependent differential Riccati equation is a nonlinear optimal controller with a final boundary condition that penalizes the states at the final time. This guides the states to the desired condition by imposing extra force on the input control law. Here the gain is removed from standard state-dependent differential Riccati equation control law (partitioned and made symmetric positive-definite) and inserted into the nonlinear sliding surface to present a novel finite-time terminal sliding mode control. The stability of the proposed terminal sliding mode control is guaranteed by the definition of the adaptive gain of terminal sliding mode control, which is limited by the Lyapunov stability condition. The proposed approach was validated and compared with state-dependent differential Riccati equation and conventional terminal sliding mode control as independent controllers, applied on a van der Pol oscillator. The capability of the proposed approach of controlling complex systems was checked by simulating a flapping-wing flying robot. The flapping-wing flying robot possesses a highly nonlinear model with uncertainty and disturbance caused by flapping. The flight assumptions also limit the input law significantly. The proposed terminal sliding mode control successfully controlled the illustrative example and flapping-wing flying robot model and has been compared with state-dependent differential Riccati equation and conventional terminal sliding mode control.

Integrated nonlinear suboptimal control‐and‐estimator based on the state‐dependent differential Riccati equation approach

AbstractThis paper is devoted to the theory and application of a combined nonlinear suboptimal control and estimator based on the state‐dependent differential Riccati equation (SDDRE), aiming finite‐time tracking of manipulators that possess flexible joints. Though measuring the angular positions of links is possible for flexible joint manipulators (FJMs), it is not feasible to measure the changes in the position angles of actuators. In addition, the existence of measurement noise and the influence of unknown workspace disturbance are always unavoidable. Therefore, combining an estimator with a controller could result in full‐state feedback for the system. Designing the proposed integrated estimator algorithm includes two parts: The first part is the theoretical extension of the SDDRE approach expressed with nonlinear state variables and inputs of the system in a finite‐time horizon. In the second part, enhancing and upgrading software and hardware interface, the proposed procedure for FJMs is modeled and simulated. Then, the experimental implementation is applied in the online closed‐loop form.

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.

Cooperative Control of Underwater Vehicle-Manipulator Systems Based on the SDC Method.

The paper considers the problem of cooperative control synthesis for a complex of underwater vehicle–manipulator systems (UVMS) to perform the work of moving a cargo along a given trajectory. Here, we used the approach based on the representation of nonlinear dynamics models in the form of state space with state-dependent coefficients (SDC-form). That allowed us to apply methods of suboptimal control with feedback based on the state-dependent differential Riccati equation (SDDRE) solution at a finite time interval, providing the change in control intensity with the transient effect of the system matrices in SDC form. The paper reveals two approaches to system implementation: a general controller for the whole system and a set of independent subcontrollers for UVMSs. The results of both approaches are similar; however, for the systems with a small number of manipulators, the common structure is recommended, and for the systems with a large number of manipulators, the approach with independent subcontrollers may be more acceptable. The proposed method of cooperative control was tested on the task of cooperative control for two UVMSs with six-link manipulators Orion 7R. The simulation results are presented in the article and show the effectiveness of the proposed method.

Quaternion-based state-dependent differential Riccati equation for quadrotor drones: Regulation control problem in aerobatic flight

AbstractThe quaternion is a powerful and common tool to avoid singularity in rotational dynamics in three-dimensional (3D) space. Here it has been particularly used as an alternative to Euler angles and rotation matrix. The application of the quaternion is exercised in quadrotor modeling and control. It changes the dynamics and represents a singularity-free attitude model. Here for the first time (for the best knowledge of authors), the state-dependent differential Riccati equation (SDDRE) control has been implemented on the quaternion-based model of a quadcopter. The proposed control structure is capable of aerobatic flight, and the Pugachev’s Cobra maneuver is chosen to assess the capability of the quaternion-based SDDRE approach. The introduced control simulator is validated by comparison with conventional dynamics based on Euler angles, controlled using a proportional-derivative (PD) controller on a normal regulation flight. The simulator successfully performed the Cobra maneuver and also validated the proposed structure. The more precision in regulation along with lower energy consumption demonstrated the superiority of the introduced approach.

Nonlinear Optimal Feedback Control of the Two-Level Open Non-Markovian Stochastic Quantum System

In this paper, a nonlinear optimal feedback control based on the nonlinear state estimator for a two-level open non-Markovian stochastic quantum system is proposed. The proposed nonlinear state estimator is designed by using the state-dependent differential Riccati equation and constructed to optimally estimate the state. The estimated state is updated by the output data of continuous weak measurement of the controlled quantum system and used to design the nonlinear state feedback controller to track the output of the reference model. The output state of reference model is chosen as the desired performance. The numerical simulation results verify the achievability of the proposed state feedback control strategy, and the capability to steer the state of system from any arbitrary initial state to the final reference target state with high state transfer success rate.

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.

Nonlinear control strategies for a UAV carrying a load with swing attenuation

Two novel nonlinear control schemes for Unmanned Aerial Vehicles (UAV) carrying a load and their comparative results are presented in this paper. The goal is to carry the load to a desired position, with oscillation attenuation along the trajectory. The proposed control structures are hierarchical schemes consisting of nonlinear controllers to stabilize the vehicle translational movements and the payload swing together with a well-known state-dependent differential Riccati equation controller to stabilize the rotational dynamics. We present new methodologies where two nonlinear controllers are proposed to obtain precise aerial vehicle positioning and efficient load oscillation reduction by exploiting the natural coupling between the horizontal quadcopter movement and the payload oscillation. It is shown that asymptotic stability can be guaranteed by the use of the Lyapunov approach and La Salle’s invariance principle. Numerical experiments were carried out to validate the nonlinear control behaviors where the results show improvements with respect to a strategy from the literature.

Nonlinear sliding sector design for multi‐input systems with application to helicopter control

SummaryThe ability of helicopters to hover and land vertically has spurred an interesting field of research on the development of autonomous flight for these rotatory wing aircrafts. Linear control theory with gain scheduling, which is based on linearizing the system at the equilibrium points, dominated the helicopter autopilot design. Unlike the linear cascaded autopilot structure used in the existing literature, this paper uses state‐dependent linear like structure, including rate‐limited actuator dynamics, with cascaded autopilot topology. This approach allows nonlinear control laws to be implemented throughout the entire flight envelope, providing satisfactory robustness and stability over the various parameter uncertainties and time delays. The cascaded autopilot topology with nonlinear dynamical equations contains a new sliding sector control (SSC) mechanism which is derived for multi‐input nonlinear dynamical systems. The proposed SSC structure for multi‐input nonlinear systems is used in the inner loop of the cascaded autopilot system where the fastest dynamics are required to be controlled for rapid changes in the helicopter dynamical characteristics which enables one to stabilize the helicopter over a wide range of flight conditions. The proposed cascaded autopilot topology with the new SSC mechanism is tested in simulations to assess its robustness and stability properties. To establish its feasibility, the proposed control method is replaced with a suboptimal control method, namely state‐dependent differential Riccati equation (SDDRE) method, for the inner loop and the results of the proposed control architecture are compared with those of SDDRE method.

Observer-based feedback control of two-level open stochastic quantum system

The observer-based feedback control for the two-level bilinear open stochastic quantum system is proposed in this paper. The state of open stochastic quantum system (OSQS) is described in the Cartesian coordinate system. The proposed state observer is designed by using state-dependent differential Riccati equation (SDDRE) and constructed for optimally estimating the state of OSQS from measurement output of the system. The state of observer is continuously updated by the output data of continuous weak measurement (CWM). A Lyapunov Feedback control is designed based on estimated state of the observer for the state transfer of OSQS. An exponential Lyapunov function is chosen to ensure the stability of the system. The observer-based Lyapunov feedback control (OLFC) strategy is developed according to the stochastic Lyapunov stability theorem. The numerical simulation results verify the achievability of the proposed OLFC strategy in terms of state estimation and state transfer of OSQS. Numerical simulations demonstrate that the observer tracks the state of system asymptotically with minimum error of ± 3%. The proposed OLFC has the ability to move the state of OSQS from arbitrary initial state to the final target eigenstate with high fidelity ≥ 90%.

Extreme learning machine-based non-linear observer for fault detection and isolation of wind turbine

ABSTRACTThis paper presents a robust fault detection and isolation (FDI) scheme for a variable speed wind turbine. The proposed scheme (extreme learning machine–state-dependent differential Riccati equation (ELM-SDDRE)) is an observer model-based approach, especially, a non-linear observer using SDDRE based on an improved model of the wind turbine by using the ELM. The standard SDDRE can be used for small model uncertainties. However, when the uncertainties are large, the SDDRE cannot detect and isolate the faults. The main objective of the ELM is the prediction of unknown nominal model dynamics to construct a new improved nominal model used by the observer for FDI. This makes the effect of uncertainties weak and consequently allows better faults detection. The faults considered in this paper are sensor faults in the rotating speeds of the rotor and generator outputs. The effectiveness of the proposed approach is illustrated through simulation.

Finite-Time Feedback Linearization (FTFL) Controller Considering Optimal Gains on Mobile Mechanical Manipulators

The aim of this paper is to implement a novel finite-time feedback linearization (FTFL) controller considering optimal gains on mobile manipulators with differential wheels theoretically and experimentally. To achieve the finite time constraint, optimal gains are obtained from solving the state dependent differential Riccati equation in a time varying dynamic system. Yet, unlike other works, using input-output linearization, the outcome of state dependent coefficient (SDC) parametrization is a couple of linear matrices. This helps the microcontroller to control the process in real time and very quick. Conceiving holonomic and non-holonomic constraints, mobile manipulator dynamic is derived. In addition, due to the fact that it is shown the proposed dynamic is not input-state linearizable, both the FTFL and state dependent Riccati equation (SDRE) are applied on the dynamic system using output feedback strategy and stability analyses are done based on lyapunov principle. Moreover, point-to-point motion and trajectory tracking simulations depict advantages of applying the FTFL method over other nonlinear schemes including sliding mode control (SMC), conventional feedback linearization, SDRE, and finite time SDRE on both the mobile manipulator and wheeled base systems. In addition, certain analysis regarding the look ahead positions and dynamic parameters are carried out to show the superiority of FTFL. Finally, the obtained results are validated by experimental setup of a mobile manipulator (Scout robot).

Sliding mode control design based on the state-dependent Riccati equation: theoretical and experimental implementation

ABSTRACTIn this paper, a suboptimal sliding mode control method is derived from combination of the sliding mode control (SMC) and the state-dependent Riccati equation (SDRE) technique, applied for a class of nonlinear closed-loop systems. One of the distinguished features of this control method is its robustness towards uncertainty. Due to lack of optimality in SMC method, in this paper, a robust and suboptimal method is presented by considering the SDRE in design of the sliding surface in two types of: algebraic and integral sliding surfaces. In addition, due to the use of the state-dependent differential Riccati equation in the integral form of sliding surface, proposed method is able to provide a robust attitude with desired finite-time control option. The sensitivity of various percentage of uncertainty in the physical structure of the system is studied and control strategies for general manipulators are provided. The proposed control structure was implemented on Scout robot theoretically and practically by the LabVIEW software; and the results were compared by considering the uncertainty in its structure. In comparison with conventional SMC, the proposed method reduced the required time to reach the sliding surface almost 50%.

Finite time SDRE control design for mobile robots with differential wheels

The State-dependent differential Riccati equation (SDDRE) control design for mobile robots with differential wheels is introduced for the first time, considering holonomic and non-holonomic constraints. Differential constraints of wheels put the SDDRE or even conventional State-dependent Riccati equation (SDRE) controller into uncontrollability condition. Since the actuators of wheels are not directly related to coordinates of the base, a proper kinematic transformation in design provided that relation; as a result, controllability of the controller over mobile robot was obtained. Moreover, finite time structure of the SDDRE resulted in terminal time control. Simulation data confirmed the accuracy and capability of the proposed design in nonlinear optimal control domain for both regulation and trajectory tracking cases. Time analysis of the controller was presented to assess the effectiveness of the SDDRE design for the final boundary condition. The SDRE was compared with output feedback linearization control to verify the proposed structure and to show the performance of design.

Finite time sliding sector guidance law with acceleration saturation constraint

A novel guidance law utilising variable structure control with finite time sliding sector is proposed. In contrast with the normally used notion of asymptotic stability in conventional sliding sector, a finite time sliding sector is defined as a subset of state space in which the Lyapunov function candidate satisfies the finite time stability condition. For the guidance system, the analytical solution of a state-dependent differential Riccati equation is achieved to determine the finite time sliding sector. Then, based on the finite time sliding sector, a sliding sector guidance law is designed by considering the target acceleration as a bounded uncertainty. Furthermore, the finite time sliding sector guidance law is even extended to the case where missile acceleration saturation constraint is considered. Simulation results show that the new guidance law is highly effective.

Analytical representation of the state-dependent coefficients in the SDRE/SDDRE scheme for multivariable systems

Recently, the easy-to-implement state-dependent Riccati equation (SDRE) strategy has been shown effective for numerous practical applications. Since being similar to SDRE, the newly emerged state-dependent differential Riccati equation (SDDRE) approach shares most of the benefits of SDRE, and exhibits interesting potential from both the analytical and practical viewpoints. By generalizing previous works to the general-order nonlinear time-variant systems, we try to resolve several problems related to the design flexibility (the infinitely many candidates of the state-dependent coefficient matrix), since they appear at the beginning of the implementation of both schemes. Finally, we demonstrate the proposed scheme via examples.