State-dependent Coefficient Research Articles

A Novel Robust Flight Controller Design for an Air-Breathing Hypersonic Vehicle

A novel robust flight control design method is proposed for a generic air-breathing hypersonic vehicle based on a state-dependent Riccati equation technique and a nonlinear disturbance observer. The highly nonlinear dynamics of the hypersonic vehicle are firstly brought to a linear structure having a state-dependent coefficient form. And then a state-dependent Riccati equation is solved at each sampling moment to obtain a nonlinear feedback optimal control law. In order to enhance robustness of the closed-loop system, a nonlinear disturbance observer is introduced to estimate the uncertainty caused by parametric variations and external disturbances. The resulting composite controller achieves not only promising robustness and disturbance rejection performance but also flexible adjustment in the response time. Compared to a Kriging controller, the proposed controller has great advantages in the system response time and robustness. The feasibility of the proposed method is validated by simulation results.

Feedback linearization of singularly perturbed systems based on canonical similarity transformations

This paper discusses the problem of feedback linearization of a singularly perturbed system in a state-dependent coefficient form. The result is based on the introduction of a canonical similarity transformation. The transformation matrix is constructed from separate blocks for fast and slow part of an original singularly perturbed system. The transformed singular perturbed system has a linear canonical form that significantly simplifies a control design problem. Proposed similarity transformation allows accomplishing linearization of the system without considering the virtual output (as it is needed for normal form method), a technique of a transition from phase coordinates of the transformed system to state variables of the original system is simpler. The application of the proposed approach is illustrated through example.

Research on High-Precision Measurement Systems of Modern Aircraft

High-precision measurement systems of modern aircraft are studied in this paper. A measurement system with correction in the structure of inertial navigation system is introduced for highprecision aircraft. In the correction algorithms, a linear error model of the navigation system is usually used. With the aim of increasing the accuracy of the navigation system, we propose a nonlinear correction algorithm based on the state dependent coefficient representation of the nonlinear model.

Decentralised non‐linear SDRE observer–controller scheme for large‐scale power systems

An efficient controller design is essential to damp out the rotor speed oscillations for real-life power systems during nominal and faulty conditions. This study presents a decentralised non-linear state observer and controller scheme for power system stabilisers (PSS). Conventional PSS designs are based on linearised models of power system, which is then followed by gain scheduling to cover the complete operating region. Here, the authors propose a derivative-free state-dependent Riccati equation (SDRE) controller and observer scheme for the PSS. Unlike the conventional methods, the proposed scheme is directly based on non-linear models and does not require linearised models. For the proposed SDRE scheme, the non-linear power system model is first converted in state-dependent coefficient form; then the SDRE controller is designed to achieve the desired performance. The SDRE observer is designed to estimate the states of the power systems based on output information; finally, the estimated states are fed back to the SDRE controller. The proposed combined SDRE observer–controller scheme is tested on a 68-bus benchmark system and is compared with the conventional control scheme. Extensive simulations are performed to show the efficacy of the proposed scheme under nominal and faulty conditions.

Trajectory control of aggressive maneuver by agile autonomous helicopter

In this paper, a new state-dependent coefficient parameterization of an agile helicopter dynamics is derived to deal effectively with the optimal trajectory control of aggressive maneuver such as the infinity maneuver with agility. The angular velocity of the main rotor and engine throttle as state and input, respectively, are involved in the dynamic model to improve the maneuvering capability of helicopter. Since the presented six degrees-of-freedom helicopter model is highly nonlinear and nonaffine, particularly nonlinear in actuator, the state-dependent Riccati equation is implemented in the presence of saturation bounds for actuators to achieve precision trajectory control and conquer the challenge of aggressive maneuver tracking control. In addition, new helicopter dynamic representation results in the conventional state-dependent Riccati equation to be applied without prevalent simplifications within dynamic and actuators and also certain augmentations on controllers such as trim or feedforward compensators. Indeed, the proposed controller structure is verified by simulations for both regulation and trajectory tracking problem.

Robust Optimal Control for Precision Improvement of Guided Gliding Vehicle Positioning

In this paper, a new method for controlling and guidance of guided gliding vehicle (GGV) is provided. In this regard, a hybrid structure of nonlinear optimal control has been proposed to minimize control effort. The inner loop has a regulating function that guarantees the stability of motion and rotation equations and reduces the effect of external disturbances. The outer loop provides optimal tracking with a straight line scroll criterion for the GGV. The use of state dependent Riccati equation control involves determining the appropriate state dependent coefficient (SDC) form, which requires a complete understanding of the dynamics of the system. One of the methods for determining the relative dynamics of a nonlinear time-varying system and calculating SDC is the online identification method. The advantage of this method in determination of the SDC is more evident when it is not possible to fully understand the dynamics of the system or external factors and disturbance affect the system. This method also eliminates the problem of uncertainty and accurate measurement of parameters. In other words, it provides an adaptive model for different conditions. In this paper, the ANFIS network has been used for the first time for continuous and online identification of the system dynamics and calculating SDC. In order to ensure the stability of identification, the identifier will be trained first with the particle swarm optimization and the online back-propagation learning algorithm. The stability of the closed loop system with a proposed hybrid structure is investigated with the help of the Lyapunov function and the concept of passivity. The optimal and robust performance of the proposed framework has been investigated in terms of the ability to simultaneously pursue the target in the optimal path despite disturbance and optimize the control effort with multiple simulations.

Construction of Parametric Regulators for Nonlinear Control Systems Based on the Pade Approximations of the Matrix Riccati Equation Solution

An approach to the construction of state controllers for nonlinear control systems with a parameter represented in a pseudo-linear form based on the Padé approximations of the solution of a matrix algebraic Riccati equation with state-dependent coefficients is demonstrated. Two classes of problems are considered. The first class of problems represents the systems in which the small value of the parameter on the right-hand side of the system equations leads to a weakly controlled system, and in case of a large value of the same parameter we have a large gain control. The second class of problems is devoted to quasilinear regularly perturbed systems, where the parameter on the right-hand side can take arbitrary positive values.

Control for a Class of Second-Order Systems via a State-Dependent Riccati Equation Approach

This study investigates the state-dependent Riccati equation (SDRE) controller for a class of second-order nonlinear systems. By fully exploiting the design degree of freedom (DOF) arising from the nonunique state-dependent coefficient (SDC) matrices, we explicitly calculate the ranges of control input via the SDRE scheme. Moreover, when a permissible control input is determined, we also explicitly parameterize the SDC matrices that result in the designated control value, in terms of system states and parameters, so the engineer can easily implement the scheme. Notably, this is the first analytical result that explores the range of control input using the design DOF of SDC matrices. In addition, by applying the analytical results, it is shown that the second-order systems are always globally stabilizable, without any supplementary assumptions on weighting matrices (as extended from existing SDRE global results), and the corresponding stabilizing SDC matrices are also explicitly presented. Finally, illustr...

Active disturbance rejected predictive functional control for space vehicles with RCS

Reaction control system (RCS) is a powerful and efficient actuator for space vehicles attitude control, which is typically characterized as a pulsed unilateral effector only with two states (off/on). Along with inevitable internal uncertainties and external disturbances in practice, this inherent nonlinear character always hinders space vehicles autopilot from pursuing precise tracking performance. Compared to most of pre-existing methodologies that passively suppress the uncertainties and disturbances, a design based on predictive functional control (PFC) and generalized extended state observer (GESO) is firstly proposed for three-axis RCS control system to actively reject that with no requirement for additional fuel consumption. To obtain a high fidelity predictive model on which the performance of PFC greatly depends, the nonlinear coupling multiple-input multiple-output (MIMO) flight dynamics model is parameterized as a state-dependent coefficient form. And based on that, a MIMO PFC algorithm in state space domain for a plant of arbitrary orders is deduced in this paper. The internal uncertainties and external disturbances are lumped as a total disturbance, which is estimated and cancelled timely to further enhance the robustness. The continuous control command synthesised by above controller-rejector tandem is finally modulated by pulse width pulse frequency modulator (PWPF) to on-off signals to meet RCS requirement. The robustness and feasibility of the proposed design are validated by a series of performance comparison simulations with some prominent methods in the presence of significant perturbations and disturbances, as well as measurement noise.

State Dependent Riccati Equation Based Rotor-Side Converter Control for Doubly Fed Wind Generator

This paper proposes a rotor-side converter (RSC) control of doubly fed induction generator (DFIG) through state-dependent Riccati equation (SDRE) approach to enhance the low voltage ride through (LVRT) ability. In order to guarantee the stability of the DFIG-based wind farm under external disturbances, a robust input-to-state stability (ISS)-based controller is presented in cooperation with the SDRE approach. The robust ISS controller is proved to be able to stabilize the wind generation system, while the system dynamics are not guaranteed to be of high performance. By parameterizing the robust ISS controller, the SDRE approach is used to provide guidance for the selection of ISS control parameters. Due to design flexibility, the SDRE control has the ability to enhance dynamic performances of wind generation system through optimizing the state-dependent coefficients. Time-domain simulations are carried out to verify the effectiveness of the proposed control approach. Compared with the conventional PI controller, exact linearization-based nonlinear controller, and robust ISS controller, the proposed SDRE-based RSC control has the ability to optimize the dynamics of the wind generation system during and after faults, and has better LVRT performance.

System Parameters’ Identification and Optimal Tracking Control for Nonlinear Systems

This paper proposes a parameters’ identification methodology via the continuous-time least-squares algorithm for those nonlinear systems which are linear with respect to their parameters. The parameter identification method can be used for easily determining all the parameters in real plants from only input-output data measurements, such that a posteriori, a model-based control strategy can be synthesized. It is worth mentioning that when a control engineer wants to design a modern and sophisticated controller, usually the system model is used for such purposes, which depends on its parameters, however, in general the system parameters are not easy to determine. Also, this paper uses the parameters’ identification methodology for the design of an optimal tracking controller for state-dependent coefficient factorized (SDCF) nonlinear systems. Both, the parameters’ identification scheme and the optimal control strategy are applied via simulations for the control of a Permanent Magnet Synchronous Motor (PMSM), a three-phase nonlinear machine.

Nonlinear Optimal Control Design for Underactuated Two-Wheeled Inverted Pendulum Mobile Platform

In terms of the state-dependent Riccati equation (SDRE) control framework, a nonlinear motion control is investigated for the two-wheeled inverted pendulum (TWIP) mobile robot platform. As a critical design issue, the state dependent coefficient matrix is established based on the sound understanding of dynamic characteristics of the TWIP robot. The developed SDRE control solution has the merit of robust posture stabilization when the inverted pendulum robot experiences strong nonlinear behaviors due to abrupt external disturbances.

Nonlinear Control for Spacecraft Pursuit-Evasion Game Using the State-Dependent Riccati Equation Method

Spacecraft pursuit-evasion game is formulated as a two-player zero-sum differential game. The Hill reference frame is used to describe the dynamics of the game. The goal is to derive feedback control law which forms a saddle point solution to the game. Both spacecraft use continuous-thrust engines. The linear quadratic differential game theory is applied to derive a linear control law. This theory is extended to derive a nonlinear control law using the state-dependent Riccati equation method. A state-dependent coefficient matrix is developed for this purpose. Various scenarios are considered for comparing the performance of the linear and nonlinear laws. In all the scenarios, the efficacy of the nonlinear control law is found to be superior to that of the linear control law with computation cost similar to the linear law.

Solving Constrained Optimal Control Problems Using State-Dependent Factorization and Chebyshev Polynomials

The present work introduces a method to solve constrained nonlinear optimal control problems using state-dependent coefficient factorization and Chebyshev polynomials. A recursive approximation technique known as approximating sequence of Riccati equations is used to replace the nonlinear problem by a sequence of linear-quadratic and time-varying approximating problems. The state variables are approximated and expanded in Chebyshev polynomials. Then, the control variables are written as a function of state variables and their derivatives. The constrained nonlinear optimal control problem is then converted to quadratic programming problem, and a constrained optimization problem is solved. Different final state conditions (unspecified, partly specified, and fully specified) are handled, and the effectiveness of the proposed method is demonstrated by solving sample problems.

Controller design of cooperative manipulators using state-dependent Riccati equation

SUMMARYThis study examined the use of a state-dependent Riccati equation (SDRE) for controller design and analysis of cooperative manipulators. The connection of end-effectors when holding an object imports constraint and complexity into the problem. Optimal load distribution (OLD) was used to divide the load between arms using a desired rate and omitting Lagrange multipliers. General dynamic structure, OLD formulation, and controller design are presented for an arbitrary number of manipulators. State-dependent coefficient parameterizations for rigid and flexible joint manipulators assuming friction for joints of them were investigated by two methods: controlling each robot independently and an entire system of robots uniformly. The effectiveness of the method, a decrease in errors, and increased stability in motion were also observed. The increase in the number of manipulators greatly expanded the state vector of the system. The SDRE was able to address this by simulation of four arms, each one possessing seven degrees of freedom (DoF). Analyses of a practical model (Scout robot) consisting of two arms with three DoF were presented and the results for connected arms and free arms were compared. The experimental data validated the simulation results and indicated that cooperation definitely improves load-carrying capacity and precision of trajectory tracking.

${H} _\infty $ Based Estimation of Nonlinear Systems

This letter presents a ${H} _{\infty }$ criterion-based recursive filter of nonlinear systems with nonlinear dynamics and measurement. The filter is based on the deterministic-statistical approach to estimation. The solution is based on three components: 1) the mean square error criterion; 2) the state dependent coefficients form; and 3) calculus of variations. The computation of the estimator’s gains needs the solution of a non-symmetric differential Riccati matrix equation. A simulation example is presented.

Model predictive control of hypersonic vehicles accommodating constraints

A novel tracking control strategy is proposed for a hypersonic vehicle based on model predictive control and the extended linearisation method. Unlike the general linearisation techniques, the state-dependent coefficient parameterisation method is used to establish a linear structure of non-linear dynamics. Based on the acquired linear system, two predictive controllers are designed for the velocity and altitude tracking in the presence of constraints. Compared with the conventional model predictive control which consumes a large amount of computer resources due to the online constrained optimisation, the predictive functional control adopts the heuristic approach to deal with constraints hence suiting fast systems more. Simulation results show the designed controllers not only meet constraint requirements but also ensure the fast response and tracking performance, and the resulting closed-loop system is robust with respect to the parameter variations and external disturbances.

A Nonlinear Dynamics-Based Estimator for Functional Electrical Stimulation: Preliminary Results From Lower-Leg Extension Experiments

Miniature inertial measurement units (IMUs) are wearable sensors that measure limb segment or joint angles during dynamic movements. However, IMUs are generally prone to drift, external magnetic interference, and measurement noise. This paper presents a new class of nonlinear state estimation technique called state-dependent coefficient (SDC) estimation to accurately predict joint angles from IMU measurements. The SDC estimation method uses limb dynamics, instead of limb kinematics, to estimate the limb state. Importantly, the nonlinear limb dynamic model is formulated into state-dependent matrices that facilitate the estimator design without performing a Jacobian linearization. The estimation method is experimentally demonstrated to predict knee joint angle measurements during functional electrical stimulation of the quadriceps muscle. The nonlinear knee musculoskeletal model was identified through a series of experiments. The SDC estimator was then compared with an extended kalman filter (EKF), which uses a Jacobian linearization and a rotation matrix method, which uses a kinematic model instead of the dynamic model. Each estimator's performance was evaluated against the true value of the joint angle, which was measured through a rotary encoder. The experimental results showed that the SDC estimator, the rotation matrix method, and EKF had root mean square errors of 2.70°, 2.86°, and 4.42°, respectively. Our preliminary experimental results show the new estimator's advantage over the EKF method but a slight advantage over the rotation matrix method. However, the information from the dynamic model allows the SDC method to use only one IMU to measure the knee angle compared with the rotation matrix method that uses two IMUs to estimate the angle.

Interplay of fast and slow dynamics in rare transition pathways: The disk-to-slab transition in the 2d Ising model.

Rare transitions between long-lived stable states are often analyzed in terms of free energy landscapes computed as functions of a few collective variables. Here, using transitions between geometric phases as example, we demonstrate that the effective dynamics of a system along these variables are an essential ingredient in the description of rare events and that the static perspective provided by the free energy alone may be misleading. In particular, we investigate the disk-to-slab transition in the two-dimensional Ising model starting with a calculation of a two-dimensional free energy landscape and the distribution of committor probabilities. While at first sight it appears that the committor is incompatible with the free energy, they can be reconciled with each other using a two-dimensional Smoluchowski equation that combines the free energy landscape with state dependent diffusion coefficients. These results illustrate that dynamical information is not only required to calculate rate constants but that neglecting dynamics may also lead to an inaccurate understanding of the mechanism of a given process.

On Stabilizing Control of Gaussian Processes for Unknown Nonlinear Systems

This paper proposes a novel controller design to stabilize Gaussian process (GP) dynamical models for partially unknown nonlinear systems. The unknown system is identified as the GP model. Because existing GP models are difficult to analyze in terms of controller design, a novel GP identification method is proposed that obtains the state dependent coefficient matrix of the system, making the controller design more efficient. The GP model is represented as a piecewise linear system with a bounded uncertainty that is derived based on the characteristics of the GP. Extending quadratic stability theory derives a novel stabilizing controller for such systems.