Abstract:  This work proposes an adaptive output feedback model predictive control (MPC) framework for uncertain systems subject to external disturbances. In the absence of exact knowledge about the plant parameters and complete state measurements, the MPC optimization problem is reformulated in terms of their estimates derived from a suitably designed robust adaptive observer. The MPC routine returns a homothetic tube for the state estimate trajectory. Sets that characterize the state estimation errors are then added to the homothetic tube sections, resulting in a larger tube containing the true state trajectory. The two-tier tube architecture provides robustness to uncertainties due to imperfect parameter knowledge, external disturbances, and incomplete state information. Additionally, recursive feasibility and robust exponential stability are guaranteed and validated using a numerical example. 

Abstract:  This article presents a homothetic tube-based adaptive model predictive control strategy to handle discrete-time linear time-invariant (LTI) systems with parametric uncertainties and hard constraints imposed on the states and the control inputs. The proposed solution systematically fuses a gradient descent-based adaptive parameter identification strategy with a suitably designed tube-based model predictive controller (MPC). An estimated model is utilized in the MPC for the purpose of state predictions. The parameters of the estimated plant model are updated at every time instant through an adaptive update law by utilizing the measured states and inputs from the uncertain plant. The task of satisfying the hard constraints in the presence of errors in state predictions, arising due to model mismatch between the estimated model and the uncertain plant, is accounted for by suitably tightening the constraints within the MPC optimization routine. The proposed tube-based adaptive MPC is analytically proved to be recursively feasible if initially feasible, and the closed-loop states are guaranteed to be bounded and asymptotically converging to the origin. The claimed properties are further validated through a simulation example.

Abstract:  In this paper, we address the problem of computing the maximal admissible robust positive invariant (MARPI) set for discrete-time linear time-varying systems with parametric uncertainties and additive disturbances. The system state and input are subjected to hard constraints, and the system parameters and the exogenous disturbance are assumed to belong to known convex polytopes. We provide necessary and sufficient conditions for the existence of the non-empty MARPI set, and explore relevant features of the set that lead to an efficient finite-time converging algorithm with a suitable stopping criterion. The analysis hinges on backward reachable sets defined using recursively computed halfspaces and the minimal RPI set. A numerical example is used to validate the theoretical development. 

Abstract: In this paper, an adaptive observer is proposed for multi-input multi-output (MIMO) discrete-time linear time-invariant (LTI) systems. Unlike existing MIMO adaptive observer designs, the proposed approach is applicable to LTI systems in their general form. Further, the proposed method uses recursive least square (RLS) with covariance resetting for adaptation that is shown to guarantee that the estimates are bounded, irrespective of any excitation condition, even in the presence of a vanishing perturbation term in the error used for updation in RLS. Detailed analysis for convergence and boundedness has been provided along with simulation results for illustrating the performance of the developed theory. 

Abstract: Model predictive control (MPC) for uncertain systems in the presence of hard constraints on state and input is a non-trivial problem, and the challenge is increased manyfold in the absence of state measurements. In this letter, we propose an adaptive output feedback MPC technique, based on a novel combination of an adaptive observer and robust MPC, for single-input single-output discrete-time linear time-invariant systems. At each time instant, the adaptive observer provides estimates of the states and the system parameters that are then leveraged in the MPC optimization routine while robustly accounting for the estimation errors. The solution to the optimization problem results in a homothetic tube where the state estimate trajectory lies. The true state evolves inside a larger outer tube obtained by augmenting a set, invariant to the state estimation error, around the homothetic tube sections. The proof for recursive feasibility for the proposed ‘homothetic and invariant’ two-tube approach is provided, along with simulation results on an academic system.