The Dynamic Inversion embedded SDRE control law combines dynamic inversion and ... Region of Attraction around the origin of the closed loop system.
Stability Domain Estimation for Dynamic Inversion Embedded SDRE Flight Controller Praveen Shankar∗ and Rama K. Yedavalli† Department of Aerospace Engineering and Aviation The Ohio State University Columbus,Ohio-43210, USA
David B. Doman‡ Air Force Research Laboratory WPAFB, Ohio, USA The Dynamic Inversion embedded SDRE control law combines dynamic inversion and state dependent riccati equation techniques such that the desired dynamics required for dynamic inversion is calculated by solving an SDRE. The resulting closed loop system is similar to a pure SDRE controller. This controller has been previously applied to the control of the rotational dynamics of a flight vehicle and simulations have shown that the origin of the closed loop system is a locally asymptotically stable equilibrium point. The stability of SDRE controlled systems have been previously studied in literature, however, it is still difficult to show global stability properties for this controller. The analytical solution of the state dependent riccati equation is possible for systems of order less than or equal to 2 by choosing the state dependent system matrix carefully. The global stability properties can then be determined by studying the analytical expression of the closed loop system. The stability of SDRE controlled systems of order greater than 2 such as the rotational dynamics of a flight vehicle are determined by calculating the Domain or Region Of Attraction (ROA) around the equilibrium point. A method for estimating the ROA of a nonlinear system by defining an overvaluing matrix using vector norms has been extended in past literature to calculate the stability domains for SDRE controlled systems. In this paper, we use this method of vector norms to analyze the stability properties of the DISDRE controlled rotational dynamics of a flight vehicle and illustrate our results with a Region of Attraction around the origin of the closed loop system.
1 of 11 American Institute of Aeronautics and Astronautics
Nomenclature B = Control Effectiveness Matrix f = Nonlinear Dynamics h = Vector Norm Jx = Moment of Inertia about x-axis Jy = Moment of Inertia about y-axis Jz = Moment of Inertia about z-axis Jxz = Moment of Inertia about xz-plane L = Rolling Moment M = Pitching Moment Mo = Overvaluing Matrix N = Yawing Moment p = Roll Angular Velocity q = Pitch Angular Velocity r = Yaw Angular Velocity u = Control Vector um = Importance Vector x = State Vector
I.
Introduction
Shankar et.al present a combined Dynamic Inversion and SDRE control law and apply it to the rotational dynamics of a flight vehicle.1 While the resulting closed loop system is locally asymptotically stable, however, the global stability properties cannot be determined because the closed loop system is not known in closed form. A global stability analysis for second order systems under SDRE control was done by Erdem et.al , where the matrix A(x) is parameterized in such a way so as to get an analytical solution P(x) to the State Dependent Riccati Equation.2 However, for systems with state dimension greater than 2, it is difficult to get an analytical solution to the State Dependent Riccati Equation. Alleyne et.al describe a method to determine the stability properties of a SDRE controlled system by calculating Region of Attraction (ROA) and provide some examples.3 This method is based on the definition of an overvaluing comparison system for the original one using vector norms that was proposed by Borne et.al. to calculate the Region of Attraction (ROA) for a nonlinear system.4 This paper attempts to characterize the stability properties by determining the ROA of the Dynamic Inversion embedded SDRE controlled system developed for the rotational dynamics of flight vehicles using the method of vector norms. The rest of the paper is organized as follows: Section II gives an overview of some of the concepts used in this paper. Section III discusses the method to calculate the stability domain of a nonlinear system. Sections IV and V describe the application of the Dynamic Inversion embedded SDRE control to flight vehicles. Section VI explains the calculation of ROA for a flight vehicle. Some results and conclusions are given in Sections VII and VIII.
II. A.
Overview
State Dependent Riccati Equation Technique
SDRE stabilization refers to the use of State Dependent Riccati Equations to construct nonlinear feedback control laws for nonlinear systems.5 The main idea is to represent the nonlinear system x˙ = f (x) + B(x)u
(1)
x˙ = A(x)x + B(x)u
(2)
u = −R−1 (x)B T (x)P (x)x
(3)
in the form and to use the feedback
2 of 11 American Institute of Aeronautics and Astronautics
where P(x) is obtained from the SDRE P (x)A(x) + AT (x)P (x) − P (x)B(x)R−1 (x)B T (x)P (x) = −Q(x)
(4)
and Q(.) and R(.) are design parameters that satisfy the point wise definiteness condition Q(x) > 0, R(x) > 0. The resulting closed loop dynamics have a linear-like structure given by x˙ = ACL (x)x where ACL (x) is the closed loop system. B.
Dynamic Inversion
Dynamic Inversion is a design technique used to synthesize flight controllers whereby the set of existing or undesirable dynamics are cancelled out and replaced by a designer selected set of desired dynamics. Consider a nonlinear system (5) x˙ = f (x) + g(x)u where x is the state and u is the control. Then, we can design a control law given by u = g(x)−1 (x˙ des − f (x))
(6)
where x˙ des is the designer selected set of desired dynamics. Dynamic inversion is similar to model following control, in that both methodologies invert dynamical equations of the plant. This cancellation and replacement is accomplished by careful algebraic selection of the feedback function. It is for this reason that this methodology is also called feedback linearization. It applies to both SISO and MIMO systems, provided that the control effectiveness function (in the SISO case) or the control influence matrix (in the MIMO case) is invertible. The method works for both full-state feedback (input-state feedback linearization), and output feedback (input-output feedback linearization). C.
Dynamic Inversion Embedded SDRE Control
Consider the nonlinear system of the form x˙ = f (x) + Bu = A(x)x + Bu
