Domain of attraction and guaranteed cost control for non‐linear quadratic systems. Part 2: controller design

Abstract

This study addresses the problem of designing a guaranteed cost controller for non-linear discrete quadratic systems. More specifically, the main contribution concerns a sufficient condition for the existence of a state-feedback controller achieving a certain guaranteed cost, whenever the state trajectories start from a prescribed set of initial conditions. The controller design requires the solution of a convex optimisation problem involving linear matrix inequalities, which can be efficiently solved via available optimisation algorithms. Finally, the proposed design methodology is illustrated via an example concerning the attitude control of an unmanned aerial vehicle.

Guaranteed Cost Control of Uncertain Discrete Time Systems Subjected to Actuator Saturation via Homogenous Polynomial Lyapunov Function

“…where Q 2 R n n and R 2 R n n are given positive semi-de nite symmetric matrices and J is called the Guaranteed Cost. [A + BK] T P [A + BK] P + Q + K T RK < 0; (10) where P is called the guaranteed cost matrix. Now, consider the discrete time system with norm bounded uncertainty and saturation:…”

“…In the theory of optimal control, Lyapunov functions [6][7][8] are the scalar functions that are used widely to prove the stability of a dynamical system. For instance, quadratic Lyapunov function, which has drawn valuable attention, serves for the systems with state variables involving the existence of linear matrix inequalities [9,10]. More general classes of Lyapunov functions have been considered, i.e.…”

Static Output-Feedback Control for Vehicle Suspensions: A Single-Step Linear Matrix Inequality Approach

“…where the vector ∈ R ×1 collects the free entries of the matrices distinct from and , together with the remaining LMI variables, and is a given affine map that makes the matrix inequality an LMI. This kind of LMI formulation is very common in practice and appears in a large number of state-feedback control problems [22]; some recent works can be found in [23][24][25][26][27][28]. More precisely, the LMI formulation (1) arises naturally in a wide variety of state-feedback controller designs, where the state control gain matrix ∈ R × is explicitly given by…”

Guaranteed cost control of rigid‐body attitude systems under control saturation

“…Furthermore, when taking more precise nonlinear models, system nonlinearities hamper the possibility of directly applying linear GCC methods to attitude control. In this context, to the best of the authors' knowledge, there exists nearly no literature to develop guaranteed cost controller based on nonlinear attitude models except 18 . Nevertheless, it focuses only on the attitude dynamics without considering the kinematics, 18 and extra requirements on system states are included so that the system stability can be ensured only in a small region.…”

“…In this context, to the best of the authors' knowledge, there exists nearly no literature to develop guaranteed cost controller based on nonlinear attitude models except 18 . Nevertheless, it focuses only on the attitude dynamics without considering the kinematics, 18 and extra requirements on system states are included so that the system stability can be ensured only in a small region. This is the first motivation of the presented work in which we develop a new GCAC method based on full nonlinear attitude dynamics without modeling errors.…”