Michael Cantoni

Standard formulations of prescribed worst-case disturbance energy-gain control policies for linear time-varying systems depend on all forward model data. In discrete time, this dependence arises through a backward Riccati recursion. This article is about the infinite-horizon $\ell _{2}$ gain performance of state feedback policies with only finite receding-horizon preview of the model parameters. The proposed synthesis of controllers subject to such a constraint leverages the strict contraction of lifted Riccati operators under uniform controllability and observability. The main approximation result is a sufficient number of preview steps for the incurred performance loss to remain below any set tolerance, relative to the baseline gain bound of the associated infinite-preview controller. Aspects of the result are explored in a numerical example.

A feedback control system is proposed for balancing the deviations of water levels from set points along open channels subject to uncertain supply–demand mismatch that exceeds individual pool capacity. Decentralized controllers adjust the gate flows between pools to regulate potentially weighted differences between neighboring water-level errors to zero in a steady state. A sequential SISO loop-shaping procedure is developed for the design of each local flow controller based on distributed parameter transfer function models of the channel dynamics. Recursive feasibility of the procedure for relevant performance specifications and stability of the resulting MIMO closed loop are verified by supporting analysis. Both numerical simulations and field trial results are presented.

A feedback control system is proposed for balancing the deviations of water levels from set-points along open channels subject to uncertain supply-demand mismatch that exceeds individual pool capacity. Decentralized controllers adjust the gate flows between pools to regulate potentially weighted differences between neighbouring water-level errors to zero in steady state. A sequential SISO loop-shaping procedure is developed for the design of each local flow controller based on distributed parameter transfer function models of the channel dynamics. Recursive feasibility of the procedure for relevant performance specifications, and stability of the resulting MIMO closed-loop, are verified by supporting analysis. Both numerical simulations and field trial results are presented.

An input-output model for networks with link uncertainty is developed. The main result presents a set of integral quadratic constraints (IQCs) that collectively imply robust stability of the uncertain network dynamics. The model dependency of each IQC is localized according to an edge-based partition of the network graph. The class of admissible network partitions affords scope for trading-off scalability against conservativeness. This is illustrated by numerical example.

This paper investigates the stereographic projection of points along the Nyquist plots of single input single output (SISO) linear time invariant (LTI) systems subject to probabilistic uncertainty. At each frequency, there corresponds a complex-valued random variable with given probability distribution in the complex plane. The chordal distance between the stereographic projections of this complex value and the corresponding value for a nominal model, as per the well-known Nu-Gap metric of Vinnicombe, is also a random quantity. The main result provides the cumulative density function (CDF) of the chordal distance at a given frequency. Such a stochastic distance framework opens up a fresh and a fertile research direction on probabilistic robust control theory.

The robustness of instability is investigated for feedback interconnections of rational transfer functions, motivated by the relevance of such in the study of oscillatory dynamics. An analysis is presented in terms of normalized coprime factorizations and the $\mathscr {L}_{2}$ -gap metric. The main result is a necessary and sufficient condition for feedback instability to persist across a particular uncertainty set for one open-loop component of the interconnection, with the other component fixed. This uncertainty set is a subset of an ${\mathscr {L}}_{2}$ -gap ball. It includes the associated $\nu$ -gap metric ball, and other winding-number based partitions that enable necessity to be established for sufficiently large ball radius.

The closed-loop stability and infinite-horizon performance of receding-horizon approximations are studied for non-stationary linear-quadratic regulator (LQR) problems. The approach is based on a lifted reformulation of the optimal control problem, under assumed uniform controllability and observability, leading to a strict contraction property of the corresponding Riccati operator. Leveraging this contraction property, a stabilizing linear time-varying state-feedback approximation of the infinite-horizon optimal control policy is constructed to meet a performance-loss specification. Its synthesis involves only finite preview of the time-varying problem data at each time step, over a sufficiently long prediction horizon.

Contraction properties of the Riccati operator are studied within the context of non-stationary linear-quadratic optimal control. A lifting approach is used to obtain a bound on the rate of strict contraction, with respect to the Riemannian metric, across a sufficient number of iterations. This number of iterations is related to an assumed uniform controllability and observability property of the dynamics and stage-cost in the original formulation of the problem.

A supremum-of-quadratics representation for a class of extended real valued barrier functions is developed and applied in the context of solving a continuous time linear regulator problem subject to a single state constraint of bounded norm. It is shown that this very simple state constrained regulator problem can be equivalently formulated as an unconstrained two-player game. By demonstrating equivalence of the upper and lower values, and exploiting existence and uniqueness of the optimal actions for both players, state feedback characterizations for the corresponding optimal policies for both players are developed. These characterizations are illustrated by a simple example.

We present an event-triggered boundary control scheme for hyperbolic systems. The trigger condition is based on predictions of the state on determinate sets, and the control input is updated only when the predictions deviate from the reference by a given margin. Nominal closed-loop stability, the absence of Zeno behaviour, and robustness to uncertainty and disturbances, are all established analytically. For the special case of linear systems, the trigger condition can be expressed in closed-form as an $L_{2}$ -scalar product of kernels with the distributed state. The presented controller can also be combined with existing observers to solve the event-triggered output-feedback control problem. A numerical simulation demonstrates the effectiveness of the proposed approach.