This thesis develops Lyapunov-based control techniques for nonlinear process systems subject to input constraints and stochastic uncertainty. The problems considered include those which focus on the null-controllable region (NCR) for unstable systems. The NCR is the set of states in the state-space from where controllability to desired equilibrium point is possible. For unstable systems, the presence of input constraints induces bounds on the NCR and thereby limits the ability of any controller to steer the system at will. Common approaches for applying control to such systems utilize Control Lyapunov Functions (CLFs) . Such functions can be used for both designing controllers and also preforming closed--loop stability analysis. Existing CLF-based controllers result in closed--loop stability regions that are subsets of the NCR and do not guarantee closed--loop stability from the entire NCR. In effort to mitigate this shortcoming, we introduce a special type of CLF known as a Constrained Control Lyapunov Function (CCLF) which accounts for the presence of input constraints in its definition. CCLFs result in closed--loop stability regions which correspond to the NCR. We demonstrate how CCLFs can be constructed using a function defined by the NCR boundary trajectories for varying values of the available control capacity. We first consider linear systems and utilize available explicit characterization of the NCR to construct CCLFs. We then develop a Model Predictive Control (MPC) design which utilizes this CCLF to achieve stability from the entire NCR for linear anti-stable systems. We then consider the problem of nonlinear systems where explicit characterizations of the NCR boundary are not available. To do so, the problem of boundary construction is considered and an algorithm which is computationally tractable is developed and results in the construction of the boundary trajectories. This algorithm utilizes properties of the boundary pertaining to control equilibrium points to initialize the controllability minimum principle. We then turn to the problem of closed--loop stabilization from the entire NCR for nonlinear systems. Following a similar development as the CCLF construction for linear systems, we establish the validity of the use of the NCR as a CCLF for nonlinear systems. This development involves relaxing the conditions which define a classical CLF and results in CCLF-based control achieving stability to an to an equilibrium manifold. To achieve stabilization from the entire NCR, the CCLF-based control design is coupled with a classical CLF-based controller in a hybrid control framework. In the final part of this thesis, we consider nonlinear systems subject to stochastic uncertainty. Here we design a Lyapunov-based model predictive controller (LMPC) which provides an explicitly characterized region from where stability can be probabilistically obtained. The design exploits the constraint-handling ability of model predictive controllers in order to inherent the stabilization in probability characterization of a Lyapunov-based feedback controller. All the proposed control designs along with the NCR boundary computation are illustrated using simulation results. / Thesis / Doctor of Philosophy (PhD)

In the thesis, two different model predictive control (MPC) strategies are investigated for linear systems with uncertainty in the presence of constraints: namely robust MPC and stochastic MPC. Firstly, a Youla Parameter is integrated into an efficient robust MPC algorithm. It is demonstrated that even in the constrained cases, the use of the Youla Parameter can desensitize the costs to the effect of uncertainty while not affecting the nominal performance, and hence it strengthens the robustness of the MPC strategy. Since the controller u = K x + c can offer many advantages and is used across the thesis, the work provides two solutions to the problem when the unconstrained nominal LQ-optimal feedback K cannot stabilise the whole class of system models. The work develops two stochastic tube approaches to account for probabilistic constraints. By using a semi closed-loop paradigm, the nominal and the error dynamics are analyzed separately, and this makes it possible to compute the tube scalings offline. First, ellipsoidal tubes are considered. The evolution for the tube scalings is simplified to be affine and using Markov Chain model, the probabilistic tube scalings can be calculated to tighten the constraints on the nominal. The online algorithm can be formulated into a quadratic programming (QP) problem and the MPC strategy is closed-loop stable. Following that, a direct way to compute the tube scalings is studied. It makes use of the information on the distribution of the uncertainty explicitly. The tubes do not take a particular shape but are defined implicitly by tightened constraints. This stochastic MPC strategy leads to a non-conservative performance in the sense that the probability of constraint violation can be as large as is allowed. It also ensures the recursive feasibility and closed-loop stability, and is extended to the output feedback case.

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