# Optimization-based feedback control of nonlinear systems subject to input constraints

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In this work, we are studying and solving feedback control problems for input constrained nonlinear systems under the influence of uncertainty. Our results are developed by fusing fundamental Lyapunov stability concepts with tools and techniques from the field of convex optimization that enable the derivation of computationally efficient control laws accompanied by robust stabilization guarantees. When a nonlinear control system is subject to input constraints, a critical aspect of the stabilization problem with simple control laws based on a particular Control Lyapunov Function (CLF) is to characterize a subset of the state space starting from where stabilization to the origin is guaranteed. We consider polynomial systems which are affine in a control input constrained in a convex and compact polytope. We propose two alternative analysis methods that ultimately yield sufficient conditions for asymptotic stabilization under such input constraints and provide an estimate of the stabilization set for the system and the given CLF. Both methods relax the problem to the solution of Sum-of-Squares programs, which nominally can be cast as Semidefinite Programs that are solvable with interior point algorithms. Given a particular CLF, it is also possible to sequentially optimize over its coefficients to the end of reshaping or enlarging the stabilization set, and thus, favorably altering the set of initial conditions from where the control objectives can be attained. A class of constrained control laws based on a particular CLF is shown to attain values equal to the minimizer of a Quadratic Program (QP), which is guaranteed to remain feasible along any closed loop trajectory emanating from the stabilization set. The input constraints are always respected and the closed loop system is rendered asymptotically stable. Additionally, such a QP is of a rather low dimension and can be solved efficiently, enabling the embedded implementation of the proposed control laws even on resource-constrained computational platforms. For the case of systems subject to unknown, bounded uncertainties that enter the dynamics in an affine way, the aforedescribed results are extended to provide robust stabilization subject to input constraints. With the proposed methods, the min-max conditions typically encountered in Lyapunov methods with Robust CLFs (RCLFs) for such systems are handled in both the (R)CLF analysis and the feedback control problem. Therefore, one can estimate a subset of the robust stabilization set with SOS programming and, subsequently, calculate - online - the stabilizing control inputs using state feedback to render the system robustly practically stable. An often encountered challenge in nonlinear control design and implementation is the large dimension of the underlying system, often resulting from the interconnection of multiple subsystems which interact with each other. The concept of Vector (Control) Lyapunov functions allows studying or warranting the applicable stability notion by focusing at the subsystem level and the respective subsystem-to-subsystem interactions. We are leveraging the premise of VCLF methods with our results on the robust stabilization problem to enable the solution of the input constrained robust stabilization problem for large scale systems, either in a distributed or a decentralized way (or in a combination of both), depending on whether state information is exchanged between interacting subsystems or not. Lastly, we examine how uncertainty in the measurements of the system can affect the stabilization problem under input constraints. We propose a control framework with which one can steer a system to a neighborhood of the origin using only imperfect state feedback. The latter is achieved by enforcing a causality relationship between stabilizing the system from the point of view of an imperfect feedback control law and stabilizing the actual system. Ultimately, we use control laws based, again, on the minimizer of simple QPs, to provingly achieve the robust stabilization objective in a subset of the measurement space which is characterized by solving a sequence of SOS programming problems. For the case where only imperfect measurements either of a subset of the state vector of the system or of a linear combination of state vector components are available, we propose an extension of Lyapunov-based nonlinear observer design results from the literature to account for uncertainty in the dynamics and the measurement equation. The robust observer synthesis process takes place through SOS programming and produces observers with explicit performance guarantees with regards to the behavior of the state determination error. The factors considered in this work are relevant to contemporary safety-critical control applications; nonlinearity, input constraints, uncertainty, and the need for embeddability and low footprint implementation are ubiquitous in control problems across fields ranging from robotics to industrial engineering, space exploration and cyber-physical systems. The proposed methods aim to collectively provide a theoretically sound, algorithmically implementable and practically useful framework to study and tackle challenging control problems