Backstepping PDE Design: A Convex Optimization Approach

Abstract

Backstepping design for boundary linear PDE is formulated as a convex optimization problem. Some classes of parabolic PDEs and a first-order hyperbolic PDE are studied, with particular attention to non-strict feedback structures. Based on the compactness of the Volterra and Fredholm-type operators involved, their Kernels are approximated via polynomial functions. The resulting Kernel-PDEs are optimized using Sumof-Squares (SOS) decomposition and solved via semidefinite programming, with sufficient precision to guarantee the stability of the system in the L 2-norm. This formulation allows optimizing extra degrees of freedom where the Kernel-PDEs are included as constraints. Uniqueness and invertibility of the Fredholm-type transformation are proved for polynomial Kernels in the space of continuous functions. The effectiveness and limitations of the approach proposed are illustrated by numerical solutions of some Kernel-PDEs.

Application of quasilinear and CGA models for designing significantly nonlinear control systems

“…The System (11) has a unique solution only if the QLM (1) of the object is completely controllable [10], that is, when the following controllability condition is met: (9) and (11) are the main calculated relations of the method for the synthesis of nonlinear ACS using quasilinear MM. They allow for a nonlinear plant specified by a quasilinear model (1), when natural condition (12) is satisfied, to find the sought quasilinear control law (2).…”

“…The study of the structure and properties of CGA-models showed that they provide the possibility of analytical synthesis of ACS. At present, many analytical methods for the synthesis of nonlinear ACS have been developed, such as the method of linearization by feedback [9], the method of point transformations [1,4,10], the method of inverse step [11], the method of passivation [12], output control [13], method of quasilinear models [10]. However, application of these methods is possible after conversion of mathematical model of control object into one of many special forms of equation representation.…”

Trajectory Tracking and Stabilization of Nonholonomic Wheeled Mobile Robot Using Recursive Integral Backstepping Control

“…In the literature, various nonlinear control techniques have been suggested for posture stabilization that can be categorized into exact and dynamic feedback linearization [6,8], continuous time-varying feedback controller [9], and discontinuous feedback controller [10]. Similarly, for the trajectory tracking problem, various approaches have been presented, such as static and dynamic-based input-output feedback linearization [2,5,6,[11][12][13][14][15][16][17][18][19], backstepping control [20][21][22][23][24][25][26], and sliding mode control [27].…”

Backstepping-Based Estimation of Thermoacoustic Oscillations in a Rijke Tube With Experimental Validation

“…Nevertheless, due to the limitations of the standard Volterra transformation, most of the results in literature consider only strict-feedback connection with the ODE subsystem. Recently, some applications have appeared for very specific classes of systems with non-strict-feedback components [6], [23], [24], [25].…”