Professor Christopher Kellett

Lyapunov¿s second method is one of the most successful tools for analyzing stability properties of dynamical systems. If a control Lyapunov function is known, then asymptotic stabilizability of an equilibrium of the corresponding dynamical system can be concluded without the knowledge of an explicit solution of the dynamical system. Whereas necessary and sufficient conditions for the existence of nonsmooth control Lyapunov functions are known by now, constructive methods to generate control Lyapunov functions for given dynamical systems are not known up to the same extent. In this paper we build on previous work to compute (control) Lyapunov functions based on linear programming and mixed integer linear programming. In particular, we propose a mixed integer linear program based on a discretization of the state space where a continuous piecewise affine control Lyapunov function can be recovered from the solution of the optimization problem. Different to previous work, we incorporate a semiconcavity condition into the formulation of the optimization problem. Results of the proposed scheme are illustrated on the example of Artstein¿s circles and on a two-dimensional system with two inputs. The underlying optimization problems are solved in Gurobi (2016, http://www.gurobi.com).

In analysis and design of nonlinear dynamical systems, (nonlinear) scaling of Lyapunov functions has been a central idea. This paper proposes a set of tools to make use of such scalings and illustrates their benefits in constructing Lyapunov functions for interconnected nonlinear systems. First, the essence of some scaling techniques used extensively in the literature is reformulated in view of preservation of dissipation inequalities of integral input-to-state stability (iISS) and input-to-state stability (ISS). The iISS small-gain theorem is revisited from this viewpoint. Preservation of ISS dissipation inequalities is shown to not always be necessary, while preserving iISS which is weaker than ISS is convenient. By establishing relationships between the Legendre¿Fenchel transform and the reformulated scaling techniques, this paper proposes a way to construct less complicated Lyapunov functions for interconnected systems.

Hadron synchrotrons accelerate protons and heavy ions to very high energies. The stabilization of the longitudinal beam dynamics in these particle accelerators is a demanding control problemsince typical sample times of the beam-feedback loop are on the order of several ìs to several hundred ns. This work examines if model predictive control is a suitable candidate for longitudinal beam feedback in synchrotrons. We show that, by combining an appropriate problem formulation with efficient numer-ical algorithms, the optimization problem can be solved sufficiently fast on a high-end FPGA. Realistic simulations for the synchrotron SIS18 of the GSI Helmholtz Center for Heavy-Ion Research illustrate that the proposed control scheme can lead to a significant improvement of the control performance.

In this paper we propose a linear program (LP)-based algorithm to schedule battery storage co-located with residential solar photovoltaics (PV), when excess generation is compensated via net metering. While the objective of this LP-based approach is to maximize the operational savings that accrue to customers, an undesirable consequence to the utility is reverse power flow during the peak pricing period. We show in this paper that it is possible to balance the objective of the utility in limiting reverse power flow, with the customer objective of increasing operational savings, in the context of net metering. To balance the specified utility and customer objectives we employ a quadratic program (QP)-based algorithm, which explicitly penalizes reverse power flow. To complete our assessment of net metering, both the LP-based and QP-based scheduling algorithms are applied to measured load and generation data from 145 residential customers located in an Australian distribution network. The results of this case study confirm the absence of reverse power flow when all customers employ a QP-based battery schedule, with the majority of customers exhibiting operational savings.

Several studies have suggested that battery storage co-located with solar photovoltaics (PV) benefits electricity distributors in maintaining system voltages within acceptable limits. However, without careful coordination, these potential benefits might not be realized. In this paper we propose an optimization-based algorithm for the scheduling of residential battery storage co-located with solar PV, in the context of PV incentives such as feed-in tariffs. Our objective is to maximize the daily operational savings that accrue to customers, while penalizing large voltage swings stemming from reverse power flow and peak load. To achieve this objective we present a quadratic program (QP)-based algorithm. To complete our assessment of the customer benefit, the QP-based scheduling algorithm is applied to measured load and generation data from 145 residential customers located in an Australian distribution network. The results of this case study confirm the QP-based scheduling algorithm significantly penalizes reverse power flow and peak loads corresponding to peak time-of-use billing. In the context of feed-in tariffs, the majority of customers exhibited operational savings when QP energy-shifting.

We present a novel method to compute Lyapunov functions for continuous-time systems with multiple local attractors. In the proposed method one first computes an outer approximation of the local attractors using a graphtheoretic approach. Then a candidate Lyapunov function is computed using a Massera-like construction adapted to multiple local attractors. In the final step this candidate Lyapunov function is interpolated over the simplices of a simplicial complex and, by checking certain inequalities at the vertices of the complex, we can identify the region in which the Lyapunov function is decreasing along system trajectories. The resulting Lyapunov function gives information on the qualitative behavior of the dynamics, including lower bounds on the basins of attraction of the individual local attractors. We develop the theory in detail and present numerical examples demonstrating the applicability of our method.

Lyapunov's second or direct method is one of the most widely used techniques for investigating stability properties of dynamical systems. This technique makes use of an auxiliary function, called a Lyapunov function, to ascertain stability properties for a specific system without the need to generate system solutions. An important question is the converse or reversability of Lyapunov's second method; i.e., given a specific stability property does there exist an appropriate Lyapunov function? We survey some of the available answers to this question.

Input-to-State Stability (ISS) and the ISS-Lyapunov function are useful tools for the analysis and design of nonlinear systems. Motivated by the fact that many feedback control laws, such as model predictive or event-based control, lead to discontinuous discrete-time dynamics, we investigate ISS-Lyapunov functions for such systems. ISS-Lyapunov functions were originally introduced in a so-called implication-form and, in many cases, this has been shown to be equivalent to an ISS-Lyapunov function of dissipative-form. However, for discontinuous dynamics, we demonstrate via an example that this equivalence no longer holds. We therefore propose a stronger implication-form ISS-Lyapunov function and provide a complete characterization of ISS-Lyapunov functions for discrete-time systems with discontinuous dynamics.

The use of comparison functions has become standard in systems and control theory, particularly for the purposes of studying stability properties. The use of these functions typically allows elegant and succinct statements of stability properties such as asymptotic stability and input-to-state stability and its several variants. Furthermore, over the last 20 years several inequalities involving these comparison functions have been developed that simplify their manipulation in the service of proving more significant results. Many of these inequalities have appeared in the body of proofs or in appendices of various papers. Our goal herein is to collect these inequalities in one place. © 2014 The Author(s).

Input-to-state stability (ISS) with respect to two measurement functions subsumes many ISS-type properties such as input-to-output stability (IOS) and state-independent input-to-output stability (SI-IOS). It also subsumes a version of incremental ISS. In this paper, we develop the notion of ISS with respect to two measurement functions for discrete-time systems and highlight the importance of commensurability of the two measurement functions and compactness of the zero set of the lower measurement function. Of particular interest, we show that the obvious extension of an implication-form ISS-Lyapunov function does not imply ISS with respect to two measurement functions.

In this paper the design of nonlinear model predictive control (NMPC) for multi-mega watt wind turbines is addressed. In an attempt to minimize energy production costs, we propose an NMPC scheme to maximize the wind energy harvest while mitigating structural loads. The proposed controller relies on noisy look-ahead wind field information obtained via LIDAR measurements. By means of simulations we demonstrate the advantages of the proposed NMPC approach.

In this paper, we study the notion of ISS with respect to two measurement functions for discrete-time systems. An example in previous work shows that, in the two measurement function case, the existence of an implication-form ISS-Lyapunov function does not imply ISS. Here, we define a "strong" ISS-Lyapunov function and demonstrate that its existence is equivalent to ISS with respect to two measurement functions.

Recently it has been shown that general bilinear systems satisfy a nonlinear L2-gain property, which is a qualitatively equivalent special case of the more general integral input-to-state stability (iISS) property. By exploiting a small-gain theorem that attends this nonlinear L2-gain property, a nontrivial example of a feedback interconnection for general bilinear systems is constructed that preserves this nonlinear L2-gain property, and hence iISS, in closed loop.

Scaling of Lyapunov functions has been a central tool in analysis and design of nonlinear systems. In this paper, the essence of some scaling techniques used extensively in the literature is reformulated in view of preservation of dissipation inequalities of integral input-to-state stability (iISS) and input-to-state stability (ISS). The iISS small-gain theorem is revisited from this viewpoint. Preservation of ISS dissipation inequalities is shown to not always be necessary, while preserving iISS which is weaker than ISS is convenient. By establishing relationships between the Legendre-Fenchel transform and the reformulated scaling techniques, this paper proposes a way to construct less complicated Lyapunov functions for interconnected systems.

Several qualitative equivalences are demonstrated between various robust stability prop-erties for discrete-time nonlinear systems. In particu-lar, two different summation-to-summation estimates are shown to be equivalent to the standard defini-tions of Input-to-State Stability (ISS) and integral ISS (iISS), respectively. Furthermore, ISS and iISS are shown to be qualitatively equivalent, via a nonlinear change of coordinates, to linear and nonlinear l2-gain properties, respectively. These equivalences, together with a previous result on the equivalence of 0-input global asymptotic stability and iISS provide interesting relationships between discrete-time robust stability properties that do not hold in continuous-time.

This paper considers the finite-field multi-way relay channel with pairwise message sharing, where multiple users exchange messages through a single relay and where the users may share parts of their source messages (meaning that some message parts are known/common to more than one user). In this paper, we design an optimal functional-decode-forward coding scheme that takes the shared messages into account. More specifically, we design an optimal function for the relay to decode (from the users on the uplink) and forward (back to the users on the downlink). We then show that this proposed function-decode-forward coding scheme can achieve the capacity region of the finite-field multi-way relay channel with pairwise message sharing. This paper generalizes our previous result for the case of three users to any number of users. © 2014 IEEE.

We present a design of a predictive control scheme for longitudinal beam dynamics in heavy ion synchrotrons. Specifically, we consider a linear-quadratic model predictive control (MPC) approach, whereby the quadratic program is solved via a fast gradient method. Furthermore, we investigate whether the fast gradient method allows for real-time feasible implementation of the proposed approach on a field programmable gate array (FPGA). Our results indicate that sampling rates in the order of 1MHz are achievable.

Stability and performance of networked control systems has been a recent area of interest in the control literature. The inclusion of a shared communication network between plant and controller inevitably leads to occasional random data loss. We provide novel results relating to the probability of such a system being globally asymptotically stable or input to state stable. Copyright © 2005 IFAC.

We consider the problem of orbital transfer under propulsion constraints. The solution is arrived at by appealing to several recent nonlinear control design methods including backstepping, forwarding, and a modified form of Jurdjevic-Quinn control.