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Optimization-based feedback control of nonlinear systems subject to input constraints
10:00 am
POB 4.304
In this work, we consider feedback control problems for input constrained nonlinear systems under the influence of uncertainty, which we solve by developing computationally efficient feedback 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 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 (SOS) programs. Given a 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 stabilization can be provingly attained. A class of asymptotically stabilizing constrained control laws based on the 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. For the case of systems subject to unknown, bounded uncertainties that enter the dynamics in an affine way, the aforementioned results are extended to provide robust stabilization under input constraints. With the proposed methods, the min-max conditions typically encountered in Lyapunov control laws with Robust CLFs (CLFs) for such systems are handled in both the (R)CLF analysis and the QP 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. We are leveraging the premise of Vector Lyapunov function 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, 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 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. 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 is based on SOS programming and results in observers with explicit performance guarantees with regards to the behavior of the state determination error.
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