Analysis of SDC matrices for successfully implementing the SDRE scheme

The state-dependent Riccati equation (SDRE) approach for stabilization of nonlinear affine systems was recently reported to be effective in many practical applications; however, there is no guideline on the construction of state-dependent coefficient (SDC) matrix when the SDRE solvability condition is violated, which may result in the SDRE scheme being terminated. In this study, we present several easy checking conditions so that the SDRE scheme can be successfully implemented. Additionally, when the presented checking conditions are satisfied, the sets of all feasible SDC matrices and their structures are explicitly depicted for the planar system.     

Exponential nonlinear observer based on the differential state-dependent Riccati equation

This paper presents a novel nonlinear continuous-time observer based on the differential state-dependent Riccati equation(SDRE) filter with guaranteed exponential stability.Although impressive results have rapidly emerged from the use of SDRE designs for observers and filters,the underlying theory is yet scant and there remain many unanswered questions such as stability and convergence.In this paper,Lyapunov stability analysis is utilized in order to obtain the required conditions for exponential stability of the estimation error dynamics.We prove that under specific conditions,the proposed observer is at least locally exponentially stable.Moreover,a new definition of a detectable state-dependent factorization is introduced,and a close relation between the uniform detectability of the nonlinear system and the boundedness property of the state-dependent differential Riccati equation is established.Furthermore,through a simulation study of a second order nonlinear model,which satisfies the stability conditions,the promising performance of the proposed observer is demonstrated.Finally,in order to examine the effectiveness of the proposed method,it is applied to the highly nonlinear flux and angular velocity estimation problem for induction machines.The simulation results verify how effectively this modification can increase the region of attraction and the observer error decay rate.     

Nonlinear feedback control of slewing beams

This paper is concerned with feedback control of slewing beams. It allows for the hub to have finite rotation which leads to a fully nonlinear system. It uses the method of state-dependent Riccati equation (SDRE) to obtain feedback control laws. The model assumes finite rigid body rotations and infinitesimal (small) elastic displacements. It considers both the classical Euler–Bermoulli model and the Timoshenko model for the beam. It uses finite element method to obtain approximate equations that can be used to construct feedback control laws.     

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.     

Guaranteed cost control of affine nonlinear systems via partition of unity method

We consider the problem of guaranteed cost control (GCC) of affine nonlinear systems in this paper. Firstly, the general affine nonlinear system with the origin being its equilibrium point is represented as a linear-like structure with state-dependent coefficient matrices. Secondly, partition of unity method is used to approximate the coefficient matrices, as a result of which the original affine nonlinear system is equivalently converted into a linear-like system with modeling error. A GCC law is then synthesized based on the equivalent model in the presence of modeling error under certain error condition. The control law ensures that the system under control is asymptotically stable as well as that a given cost function is upper-bounded. A suboptimal GCC law can be obtained via solving an optimization problem in terms of linear matrix inequality (LMI), in stead of state-dependent Riccati equation (SDRE) or Hamilton–Jacobi equations that are usually required in solving nonlinear optimal control problems. Finally, a numerical example is provided to illustrate the validity of the proposed method.     

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

In 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%.     

自由漂浮空间机器人避奇异规划—跟踪算法

针对路径相关空间内自由漂浮空间机器人无法进行有效跟踪控制的问题,设计了一种避奇异轨迹规划—跟踪算法,用于完成路径相关空间机械臂末端轨迹跟踪控制的任务．首先,分析奇异条件并设定安全边界曲线,求解回避奇异的基座姿态角阈值,从而得到避奇异参考轨迹及初始状态值．接着,利用自由漂浮空间机器人非线性动力学模型具有状态依赖参数的类线性结构特点,基于状态依赖Riccati方程设计跟踪控制器对末端速度进行跟踪,保证闭环系统的局部渐近稳定性．所提方法克服了传统方法将工作空间约束在路径无关空间的缺点．仿真结果表明,该算法具有比比例微分（proportional derivative,PD）控制更高的跟踪精度．同时,在存在输入干扰的情况下仍然能够实现有效跟踪．     

Control of slowly-varying linear systems

State feedback control of slowly varying linear continuous-time and discrete-time systems with bounded coefficient matrices is studied in terms of the frozen-time approach. This study centers on pointwise stabilizable systems. These are systems for which there exists a state feedback gain matrix placing the frozen-time closed-loop eigenvalues to the left of a line Re s=-γ<0 in the complex plane (or within a disk of radius ρ<1 in the discrete-time case). It is shown that if the entries of a pointwise stabilizing feedback gain matrix ar continuously differentiable functions of the entries of the system coefficient matrices, then the closed-loop system is uniformly asymptotically stable if the rate of time variation of the system coefficient matrices is sufficiently small. It is also shown that for pointwise stabilizable systems with a sufficiently slow rate of time variation in the system coefficients, a stabilizing feedback gain matrix can be computed from the positive definite solution of a frozen-time algebraic Riccati equation     

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 SDRE control of mobile base cooperative manipulators: Collision free path planning and moving obstacle avoidance

This work proposes application of a state-dependent Riccati equation (SDRE) controller for wheeled mobile cooperative manipulators. Implementation of the SDRE on a wheeled mobile manipulator (WMM) considering holonomic and non-holonomic constraints is difficult and leads to instability of the system. The present study introduces a method of controlling the WMMs including: a general formulation, state-dependent coefficient parameterization, and control structure of the SDRE. Overcoming the problem of instability of the WMM resulted in control design for a system of cooperative manipulators mounted on a wheeled mobile platform. Optimal load distribution (OLD) was employed to distribute the load between the cooperative arms. The presence of obstacles and the probability of a collision between multiple robots in a workspace are the motivations behind employment of the artificial potential field (APF) approach. Two cooperative manipulators mounted on a mobile platform retrieved from Scout robot were modeled and simulated for situations such as controlling multiple mobile bases (collision avoidance), a cooperative system of manipulators, and moving obstacle avoidance. The OLD improved the load capacity, precision, and stability in motion of the cooperative system. Compatibility of the APF within the structure of the SDRE controller is another promising aspect of this research.     

基于非线性状态依赖Riccati方程的直线倒立摆一致性控制

直线倒立摆作为一种典型的非线性系统,是一种经典的控制理论研究对象.本文将状态依赖的Riccati方程(SDRE)方法与极点配置方法结合,进行倒立摆非线性控制的研究.该方法与SDRE相比,不再需要实时计算Riccati方程,同时克服了线性最优控制(LQR),线性鲁棒(H∞)控制等控制域不足的问题,可实现几乎任意初始摆角的稳定控制,而且在稳定点附近保持与某期望的线性控制方法完全相同.实验表明了该控制方法的有效性和对扰动的鲁棒性.最后讨论了SDRE进行一致性起摆控制的硬件可行性,以及系统对于传感器零点漂移的鲁棒性.     

Controller design for a class of nonlinear systems with input saturation using convex optimization

In this paper, we propose a new design strategy for nonlinear systems with input saturation. The resulting nonlinear controllers are locally asymptotically stabilizing the origin. The proposed methodology is based on exact feedback linearization which is used to reformulate the nonlinear system as a linear system having state-dependent input saturation. Linear saturating state feedback controllers and soft variable-structure controllers are developed based on this system formulation. The resulting convex optimization problems can be written in terms of linear matrix inequalities and sum of squares conditions for which efficient solvers exist. Polynomial approximation based on Legendre polynomials is used to extend the methodology to a more general class of nonlinear systems. To demonstrate the benefit of this design method, a stabilizing controller for a single link manipulator with flexible joint is developed.     

Global -gain design for a class of nonlinear systems

It is known that the so-called control problem of a nonlinear system is locally solvable if the corresponding problem for the linearized system can be solved by linear feedback. In this paper we prove that this condition suffices to solve also a global control problem, for a fairly large class of nonlinear systems, if one is free to choose a state-dependent weight of the control input. Using a two-way (backward and forward) recursive induction argument, we simultaneously construct, starting from a solution of the Riccati algebraic equation, a global solution of the Hamilton–Jacobi–Isaacs partial differential equation arising in the nonlinear control, as well as a state feedback control law that achieves global disturbance attenuation with internal stability for the nonlinear systems.     

Memoryless stabilization of uncertain linear systems includingtime-varying state delays

The problem of stabilizing a class of uncertain time-delay systems via memoryless linear feedback is examined. The systems under consideration are linear systems with time-varying state delays. They also contain uncertain parameters (possibly time-varying) whose values are known only to within a prescribed compact bounding set. The main contribution given is to enlarge the class of time-delay systems for which one can construct a stabilizing memoryless linear feedback controller. Within this framework, a novel notion of robust memoryless stabilizability is first introduced via the method of Lyapunov functionals. Then a sufficient condition for the stabilizability is proposed. It is shown that solvability of a parameterized Riccati equation can be used to determine whether the time-delay system satisfies the sufficient condition. If there exists a positive definite symmetric solution satisfying the Riccati equation, a suitable memoryless linear feedback law can be derived     

Suboptimal control of feedback-linearizable nonlinear plant

An optimal control problem is formulated for a class of nonlinear systems for which there exists a coordinate representation (diffeomorphism) transforming the original system into a system with a linear main part and a nonlinear feedback. In this case the coordinate transformation significantly changes the form of original quadratic functional. The penalty matrices become dependent on the system state. The linearity of the structure of the transformed system and the quadratic functional make it possible to pass over from the Hamilton-Jacoby-Bellman equation to the Riccati-type equation with state-dependent parameters upon the control synthesis. Note that it is impossible to solve the obtained form of Riccati equation analytically in the general case. It is necessary to approximate the solution; this approximation is realized by numerical methods using symbolic computer packages or interpolation methods. In the latter case, it is possible to obtain the suboptimal control. The presented example illustrates the application of the proposed control method for the feedback-linearizable nonlinear system.     

Robust stabilization of uncertain systems with time-varyingmultistate delay

The authors deal with the problem of stabilizing a class of uncertain linear systems with time-varying multi-state delay and subject to norm-bounded parameter uncertainty via memoryless linear state feedback. Some sufficient conditions for the robust stabilizability are derived fur this class of uncertain systems. If there exists a positive-definite symmetric solution satisfying the algebraic Riccati equation (or inequality), a suitable memoryless state feedback law can be derived also. Moreover, all such parametric algebraic Riccati inequalities have been transformed into some linear matrix inequality problems, so there is no tuning of the parameters to gain a stabilizing solution     

On design of time-varying feedback control for a class of second-order nonlinear systems

This paper solves the tracking problem by designing the time-varying control for a class of nonlinear pendulum system. First, by analyzing the pendulum dynamics, we get a second-order nonlinear strict-feedback system, and the tracking problem is converted into a stabilizing problem of this nonlinear system. Then, we consider a candidate output feedback control with time-varying parameters, and analyze the system dynamics to get the stabilizing condition through utilizing the backstepping approach. Although this condition is nonlinear, which is not easy to be solved, under some special conditions, the parameters can be calculated. Finally, the simulation on the tracking problem for the pendulum system is presented, which verifies our results.     

Sampled-data controller design for uncertain systems

Presents a sampled-data controller design methodology for uncertain systems. A continuous system with bounded time-varying uncertainty is sampled at intervals of length T. The controller is designed using a Riccati equation approach by neglecting O(T²) uncertainty terms in the discretized system. Stability is verified for this choice of T with the U(T²) terms included. If there exists a stabilizing continuous controller, then there also exists a stabilizing sampled-data controller for a sufficiently small choice of T     

Robust output feedback control of polynomial growth nonlinear systems with measurement uncertainty

Even in the presence of uncertainty in both state and output equations, we prove that global asymptotic stabilization is still possible by output feedback for a family of uncertain nonlinear systems dominated by a triangular system with a polynomial output‐dependent growth rate. In contrast to the linear growth requirement in the recent work the nonlinear perturbations in this paper are allowed to satisfy a linear growth condition with a polynomial output‐dependent rate. To handle simultaneously the polynomial nonlinearities and unknown parameter in the system output, we propose a high‐gain estimator with a dynamic gain that is updated online through a Riccati‐type dynamic equation. Then, an estimator‐based controller is designed by a recursive algorithm that makes it possible to assign the controller gains step by step. The globally stabilizing output‐feedback controller developed in this paper is robust with respect to uncertainties in the system dynamics and output equations.     

Control of nonlinear plants with state-dependent coefficients

The theoretical fundamentals for solving the linear quadratic problems may be sometimes used to design the control actions for the nonlinear systems. The method relying on the Riccati equation with state-dependent coefficients is one of the promising and rapidly developing tools for design of the nonlinear controllers. The set of possible suboptimal solutions is generated by the ambiguous representation of the nonlinear system as a linearly structured system with state-dependent coefficients and the lack of sufficiently universal algorithms to solve the Riccati equation also having state-dependent coefficients. The paper proposed a method to design a guaranteed control for the uncertain nonlinear plant with state-dependent parameters. An example of designing the controller for an uncertain nonlinear system was presented.