State-dependent Riccati Equation Approach Research Articles

On Factorization of the Nonlinear Drift Term for SDRE Approach

AbstractThis study presents necessary and sufficient conditions for the existence of a constant matrix A that satisfies Ax = f, (A, B) is stabilizable and (A, C) is observable (resp., detectable), where x and f are two constant vectors with x ≠ 0, while B and C are two constant matrices with approximate dimension. The conditions presented are very easy to check and a set of such feasible matrices A are also provided explicitly when the existence conditions are satisfied. In practical applications, this study may cooperate with the state-dependent Riccati equation (SDRE) implementation at those states where the SDRE scheme fails to operate, yet the presented existence conditions hold there.

Numerical State-Dependent Riccati Equation Approach for Missile Integrated Guidance Control

A x , B x = state-dependent system matrices of size n n and n m Q x , R x = state-dependent weighting matrices of sizes n n and m m r, _ r = range and range rate of the target with respect to the missile S = solution to the Riccati equation T = rocket motor thrust per unit mass acting along the longitudinal axis of the missile u = control vector of size m 1 upert = control perturbation vector of size m 1 x = state vector of size n 1 xpert = state perturbation vector of size n 1 X, Y, Z = relative position components of the target with respect to the missile y, z = position of the moving masses along the pitch and yaw axes with respect to the body yc, zc = moving-mass position commands , = pitch and yaw Euler angles of the missile y, z = line-of-sight angles

Optimal Sliding Mode Controllers for Spacecraft Attitude Manoeuvres

This paper studies optimal sliding mode controller designs using integral sliding mode control (ISM) to deal with some spacecraft attitude control problems. Integral sliding mode control combining the first order sliding mode and optimal control is applied to quaternion-based attitude control for rest-to-rest manoeuvres with external disturbances. For the optimal control part, the state dependent Riccati equation (SDRE) approach and control Lyapunov function (CLF) are used to solve the infinite-time nonlinear optimal problem. The second method of Lyapunov is used to show that asymptotic stability is achieved. An example of rest-to-rest attitude manoeuvres is presented and simulation results are included to verify and compare the usefulness of the various controllers.

Position control of flexible manipulator using non-linear H∞ with state-dependent Riccati equation

The paper is concerned with the control of the tip position of a single-link flexible manipulator. The non-linear model of the manipulator is derived and tested, assuming the number of model shape functions to be two. It is known that the assumed modes method introduces uncertainty to the model by neglecting higher-order dynamics. There are other sources of uncertainty, such as friction. In addition, the model is non-linear. Therefore, for the next task, which is the controller design, the H∞ approach is proposed to deal efficiently with uncertainties, and the non-linear nature of the problem is addressed by the use of the state-dependent Riccati equation (SDRE) technique. Following the SDRE approach, the state-feedback non-linear control law is derived, which minimizes a quadratic cost function. This solution is then mapped into the H∞ optimization problem. The resulting control law has been tested with the simulation model of the flexible manipulator and the results are discussed in the paper.

Nonlinear feedback controllers and compensators: a state-dependent Riccati equation approach

State-dependent Riccati equation (SDRE) techniques are rapidly emerging as general design and synthesis methods of nonlinear feedback controllers and estimators for a broad class of nonlinear regulator problems. In essence, the SDRE approach involves mimicking standard linear quadratic regulator (LQR) formulation for linear systems. In particular, the technique consists of using direct parameterization to bring the nonlinear system to a linear structure having state-dependent coefficient matrices. Theoretical advances have been made regarding the nonlinear regulator problem and the asymptotic stability properties of the system with full state feedback. However, there have not been any attempts at the theory regarding the asymptotic convergence of the estimator and the compensated system. This paper addresses these two issues as well as discussing numerical methods for approximating the solution to the SDRE. The Taylor series numerical methods works only for a certain class of systems, namely with constant control coefficient matrices, and only in small regions. The interpolation numerical method can be applied globally to a much larger class of systems. Examples will be provided to illustrate the effectiveness and potential of the SDRE technique for the design of nonlinear compensator-based feedback controllers.

State-dependent Riccati equation approach for optimal control of a tandem cold metal rolling process

The tandem cold rolling of metal strip is a large multiinput-multioutput process that presents a difficult challenge to the control designer because of the following factors: 1) the complex interactions between the process variables; 2) the nonlinearities; and 3) the interstand time delays that change significantly with the mill speed. Control systems using the present technology have produced an acceptable product but are limited in their capability for improvement in performance and robustness, and therefore, there is a need for a better approach. It is considered that the state-dependent algebraic Riccati equation technique for the control of nonlinear systems that has been quite successful in the aerospace industry might fulfil this need. This paper presents the results of an initial work performed to investigate the theoretical and applied aspects of this technique for the control of the tandem cold rolling mill. In this paper, nonlinear state space equations are derived from a mathematical model of the mill, and a controller using the state-dependent Riccati equation approach (with trims) is developed. By simulation of typical operating conditions, the controller, when coupled to the model, is shown to be effective in reducing the effects of disturbances in entry strip thickness and hardness.

Existence of SDRE stabilizing feedback

The state-dependent Riccati equation (SDRE) approach to nonlinear system stabilization relies on representing a nonlinear system's dynamics in a manner to resemble linear dynamics, but with state-dependent coefficient matrices that can then be inserted into state-dependent Riccati equations to generate a feedback law. Although stability of the resulting closed-loop system need not be guaranteed a priori, simulation studies have shown that the method can often lead to suitable control laws. In this note, we consider the nonuniqueness of state-dependent representations. In particular, we show that if there exists any stabilizing feedback leading to a Lyapunov function with star-convex level sets, then there always exists a representation of the dynamics such that the SDRE approach is stabilizing. The main tool in the proof is a novel application of the S-procedure for quadratic forms.

A Stability Result With Application to Nonlinear Regulation1

This work considers a class of nonlinear systems whose feedback controller is generated via the solution of a State Dependent Riccati Equation (SDRE) as proposed in Banks and Manha and Cloutier. A pseudo-linear representation of the class of nonlinear systems is described and a stability analysis is performed. This analysis leads to sufficiency conditions under which local asymptotic stability is present. These conditions allow for the computation of a Region of Attraction estimate for system stability. These results are then applied to study stability and convergence properties of closed loop systems that arise when the SDRE technique is used. Many of the benefits of Linear Quadratic (LQ) Optimal Control, such as a tradeoff between state regulation and input effort, are readily transparent in the nonlinear scheme. The tradeoff ability is the major advantage of the SDRE over several other nonlinear control schemes. The computed Region of Attraction, while sufficient, is demonstrated to also be quite conservative. An example is used to examine the SDRE approach.