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This paper concerns the verification of continuous-time polynomial spline trajectories against linear temporal logic specifications (LTL without 'next'). Each atomic proposition is assumed to represent a state space region described by a multivariate polynomial inequality. The proposed approach samples a trajectory strategically, to capture every one of its region transitions. This yields a discrete word called a trace, which is amenable to established formal methods for path checking. The original continuous-time trajectory is shown to satisfy the specification if and only if its trace does. General topological conditions on the sample points are derived that ensure a trace is recorded for arbitrary continuous paths, given arbitrary region descriptions. Using techniques from computer algebra, a trace generation algorithm is developed to satisfy these conditions when the path and region boundaries are defined by polynomials. The proposed PolyTrace algorithm has polynomial complexity in the number of atomic propositions, and is guaranteed to produce a trace of any polynomial path. Its performance is demonstrated via numerical examples and a case study from robotics. [ABSTRACT FROM AUTHOR]

Constrained finite-horizon linear-quadratic optimal control problems are studied within the context of discrete-time dynamics that arise from the series interconnection of subsystems. A structured algorithm is devised for computing the Newton-like steps of primal-dual interior-point methods for solving a particular re-formulation of the problem as a quadratic program. This algorithm has the following properties: (i) the computation cost scales linearly in the number of subsystems along the cascade; and (ii) the computations can be distributed across a linear processor network, with localised problem data dependencies between the processor nodes and low communication overhead. The computation cost of the approach, which is based on a fixed permutation of the primal and dual variables, scales cubically in the time horizon of the original optimal control problem. Limitations in these terms are explored as part of a numerical example. This example involves application of the main results to model data for the cascade dynamics of an automated irrigation channel in particular. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A recently proposed time-varying generalisation of Vinnicombe's v-gap is shown to have the properties of a metric on a class of causal linear continuous-time systems that admit normalised strong graph representations. An alternative time-varying gap metric is then defined in more general terms and a corresponding robust stability result is derived for linear feedback interconnections. Finally, it is shown that the time-varying gap and v-gap metrics induce the same topology, this being the coarsest topology under which feedback stability and closed-loop sensitivity are both robust properties. [ABSTRACT FROM AUTHOR]

Feedback interconnections of causal linear systems are studied in a continuous-time setting. The developments include a linear time-varying (LTV) generalization of Vinnicombe's v-gap metric and an integral-quadratic-constraint-based robust L2-stability theorem for uncertain feedback interconnections of potentially open-loop unstable systems. These main results are established in terms of Toeplitz--Wiener--Hopf and Hankel operators, and the Fredholm index, for a class of causal linear systems with the following attributes: (i) a system graph (i.e., subspace of L2 input-output pairs) admits normalized strong right (i.e., image) and left (i.e., kernel) representations, and (ii) the corresponding Hankel operators are compact. These properties are first verified for stabilizable and detectable LTV state-space models to initially motivate the abstract formulation, and subsequently verified for frequency-domain multiplication by constantly proper Callier--Desoer transfer functions in analysis that confirms consistency of the developments with the time-invariant theory. To conclude, the aforementioned robust stability theorem is applied in an illustrative example concerning the feedback interconnection of distributed-parameter systems over a network with time-varying gains. [ABSTRACT FROM AUTHOR]

Abstract: This paper investigates several aspects of linear–quadratic (LQ) optimal control for Roesser models over a two-dimensional (2D) signal-index set of finite extent. First, we consider the characterisation and computation of open-loop control laws when constraints on the system semi-states are imposed at both the south-west and north-east boundaries of the frame (i.e. signal-index set) of interest; by virtue of the quarter-plane causal structure of the Roesser model, the south-west and north-east boundary conditions are analogous to initial conditions and terminal constraints, respectively. A necessary and sufficient characterisation of optimality is obtained and explicitly computable formulae are derived to characterise the corresponding control inputs and performance index under reasonable assumptions on the problem data. In the second part of the paper, the problem of optimal LQ control via semi-state feedback is considered. A 2D Riccati-like difference equation is introduced to characterise, in a sufficient sense, a solution to this problem. [Copyright &y& Elsevier]

Abstract: A method to compute the gain is developed for the class of linear periodic continuous-time systems that admit a finite-dimensional state-space realisation. A bisection search for the smallest upper bound on the gain is employed, where at each step an equivalent discrete-time problem is considered via the well-known technique of time-domain lifting. The equivalent problem involves testing a bound on the gain of a linear shift-invariant discrete-time system, with the same state dimension as the periodic continuous-time system. It is shown that a state-space realisation of the discrete-time system can be constructed from point solutions to a linear differential equation and two differential Riccati equations, all subject to only single-point boundary conditions. These are well behaved over the corresponding one period intervals of integration, and as such, the required point solutions can be computed via standard methods for ordinary differential equations. A numerical example is presented and comparisons made with alternative techniques. [Copyright &y& Elsevier]

Abstract: It is shown how noisy closed-loop frequency-response measurements can be used to obtain pointwise in frequency bounds on the possible difference between the actual closed-loop system and the closed-loop comprising a nominal model of the plant and the stabilising controller. To this end, Vinnicombe''s gap metric framework for robustness analysis plays a central role. Indeed, an optimisation problem and corresponding algorithm are proposed for estimating the chordal distance between the frequency responses of the nominal plant model and a plant that is consistent with the closed-loop data and a priori information, when projected onto the Riemann sphere. [Copyright &y& Elsevier]

Abstract: A new, computationally tractable, bound is derived for the level of closed-loop performance achieved by a given finite-dimensional feedback compensator with a plant for which a finite number of frequency response samples are computable. The bound involves quantities reflecting the performance of the controller with a finite-dimensional, nominal model of the plant, quantities that can be determined from the finite number of frequency response samples of the true plant, and quantities related to the complexity (in the sense of Vinnicombe) of all systems involved. This bound can be used to ‘validate’ closed-loop performance in the case that the true plant frequency response samples are of a plant which is not completely known or, to measure the performance of a finite-dimensional controller with a computationally intractable (e.g.infinite-dimensional) model of the true plant. [Copyright &y& Elsevier]

A new approach to robust performance problems is proposed in this paper. The approach involves the optimisation of the so-called performance weights subject to a constraint, formulated in terms of the structured singular value, which ensures the existence of a stabilising feedback compensator that achieves robust performance with respect to the optimised performance weights and an uncertain plant set. Optimising over the performance weights in this way gives rise to an algorithm for systematically trading-off desired performance against specified plant uncertainty and performance limitations due to plant dynamics. The algorithm also yields a robust controller. The designer is only required to specify the uncertain plant set and an optimisation directionality, which is used to reflect desired closed-loop performance over frequency in terms of a corresponding cost function. Design of this directionality appears to be simpler than designing the performance weights directly. [Copyright &y& Elsevier]

Quantitative robust stability results are established in this paper for feedback systems that evolve in continuous-time and exhibit linear, periodically time-varying (LPTV) behavior. The results presented are analogous to results known to hold for linear, time-invariant (LTI) systems, although they do not follow directly from these. System uncertainty is measured using the gap metric, which quantifies the distance between systems in terms of the aperture between their graphs (the subspaces corresponding to all input-output pairs for each system). The main robustness result characterizes the largest gap-ball of LPTV plants stabilized by a nominal LPTV feedback controller known to stabilize a nominal LPTV plant at the center of this ball. A key step in the proof of this result makes use of a formula derived for the directed gap between LPTV systems. This formula is essentially a generalization of Georgiou's for LTI systems. Importantly, all of the results presented apply to a class of sampled-data control systems, as a special case. [ABSTRACT FROM AUTHOR]

The graph representation of a system (the set of all input-output pairs) has gained considerable attention in the control literature in view of its usefulness for the analysis of feedback systems. In this paper it is shown that the graph of any stabilizable, linear, periodically time-varying (LPTV), continuous-time system can be expressed as the range and kernel of bounded, causal, LPTV systems that are, respectively, left and right invertible by bounded, causal, LPTV systems. These so-called strong-right and strong-left representations are closely related to the perhaps more common notion of coprime factor representations. As an example of their usefulness, a neat characterization of closed-loop stability is obtained in terms of strong-right and strong-left representations of the plant and controller graphs. This in turn leads to a Youla-style parametrization of stabilizing controllers. All of the results obtained accommodate possibly infinite-dimensional input and output spaces and apply, as a special case, to sampled-data control-systems. Furthermore, they are particularly useful for robustness analysis in terms of the gap metric. [ABSTRACT FROM AUTHOR]

Market and network integration of distributed energy resources can be facilitated by their coordination within a virtual power plant (VPP). However, VPP operation subject to network limits and different market and physical uncertainties is a challenging task. This paper introduces a framework that co-optimizes the VPP provision of multiple market (e.g., energy, reserve), system (e.g., fast frequency response, inertia, upstream reactive power), and local network (e.g., voltage support) services with the aim of maximizing its revenue. To ensure problem tractability, while accommodating the uncertain nature of market prices, local demand, and renewable output and while operating within local network constraints, the framework is broken down into three sequentially coordinated optimization problems. Specifically, a scenario-based robust optimization for day-ahead resource scheduling, with linearized power flows, and two receding horizon optimizations for close-to-real-time dispatch, with a more accurate second-order cone relaxation of the power flows. The results from a real Australian case study demonstrate how the framework enables effective deployment of VPP flexibility to maximize its multi-service value stack, within an uncertain operating environment, and within technical limits. [ABSTRACT FROM AUTHOR]

A central planner wishes to engage a collection of sensors to measure a quantity. Each sensor seeks to trade-off the effort it invests to obtain and report a measurement, against contracted reward. Assuming that measurement quality improves as a sensor increases the effort it invests, the problem of the reward contract design is investigated. To this end, a game is formulated between the central planner and the sensors. Using this game, it is established that the central planner can enhance the quality of the estimate by rewarding each sensor based on the distance between the average of the received measurements and the measurement provided by the sensor. Optimal contracts are designed from the perspective of the budget required to achieve a specified level of error performance. [ABSTRACT FROM AUTHOR]

We present a control design for semilinear and quasilinear 2 × 2 hyperbolic partial differential equations with the control input at one boundary and a nonlinear ordinary differential equation coupled to the other. The controller can be designed to asymptotically stabilize the system at an equilibrium or relative to a reference signal. Two related but different controllers for semilinear and general quasilinear systems are presented and the additional challenges in quasilinear systems are discussed. Moreover, we present an observer that estimates the distributed PDE state and the unmeasured ODE state from measurements at the actuated boundary only, which can be used to also solve the output feedback control problem. [ABSTRACT FROM AUTHOR]

A model-based approach to control system design is developed for regulating the discharge flows at the outlets of a pipeline network supplied by an irrigation channel. The open channel is also controlled automatically to regulate the supply-point water level. The hydraulic pressure at the source of the network is therefore dynamic when flow load varies. Regulation of the piped discharge flows is achieved by adjusting outlet control valves on the basis of the specified flow demand and sensor measurements. A blend of feedforward and feedback control is proposed. The steady-state behaviour of a nonlinear distributed-parameter model of the network is used to determine the feedforward control action. The feedback control action is used to compensate for modeling error. The design of the feedback controller is based on a frequency-domain transfer function model of the system dynamics, and classical loop-shaping ideas. This model is obtained via the admittance matrix of linearized network equations. The synthesis of both centralized and decentralized feedback controller configurations is considered. Simulations that involve a nonlinear distributed-parameter model of the network dynamics are used to illustrate controller performance. [ABSTRACT FROM AUTHOR]

We solve the problem of stabilizing a general class of 1-d semilinear hyperbolic systems with an arbitrary number of states convecting in each direction and with the actuation and sensing restricted to one boundary. The control design is based on the dynamics on the characteristic lines along which the inputs propagate through the domain and the predictability of states in the interior of the domain up to the time they are affected by the inputs. In the context of broad solutions, the state-feedback controller drives systems with globally Lipschitz nonlinearities from an arbitrary initial condition to the origin in minimum time. Alternatively, it is possible to satisfy a tracking objective at the uncontrolled boundary or, for systems with C 1 -coefficients and initial conditions, to design the control inputs to obtain classical C 1 -solutions that also reach the origin in finite time. Further, we design an observer that estimates the distributed state from boundary measurements only. The observer combined with the state-feedback controller solves the output-feedback control problem. [ABSTRACT FROM AUTHOR]

This paper is about an encryption based approach to the secure implementation of feedback controllers for physical systems. Specifically, Paillier's homomorphic encryption is used to digitally implement a class of linear dynamic controllers, which includes the commonplace static gain and PID type feedback control laws as special cases. The developed implementation is amenable to Field Programmable Gate Array (FPGA) realization. Experimental results, including timing analysis and resource usage characteristics for different encryption key lengths, are presented for the realization of an inverted pendulum controller; as this is an unstable plant, the control is necessarily fast. [ABSTRACT FROM AUTHOR]

Abstract: Actuator saturation can have a significant impact on control system performance. In particular, when actuators saturate because of large initial condition mismatch at startup, or because of large disturbances, the controller can suffer from so-called integrator windup. This paper describes the application of recent anti-windup and bumpless transfer (AWBT) compensation techniques to the problem of open water channel control. This is nontrivial in light of the (marginally) unstable nature of uncontrolled open water channels, which for the purposes of controller design, can modelled as a concatenation of integrators, linked by saturation-prone gates used to regulate the flow of water. AWBT compensator design is considered within the context of both continuous- and discrete-time controllers and models. All simulation studies are carried out using an experimentally validated, high-fidelity model of the Haughton Main Channel in Queensland, Australia. The AWBT compensation schemes considered achieve excellent performance. [Copyright &y& Elsevier]

Abstract: In Australia, gravity fed irrigation systems are critical infrastructure essential to agricultural production and export. By supplementing these large scale civil engineering systems with an appropriate information infrastructure, sensors, actuators and a communication network it is feasible to use systems engineering ideas to improve the exploitation of the irrigation system. This paper reports how classical ideas from system identification and control can be used to automate irrigation systems to deliver a near on-demand water supply with vastly improved overall distribution efficiency. [Copyright &y& Elsevier]

When operated independently, distributed energy resources (DER) cannot readily participate in electricity markets or contribute toward network stability and security. These issues can be addressed by aggregated control of DER within a virtual power plant (VPP) framework. This work presents a scenario-based optimization approach to operating a multi-energy VPP under uncertainty. This enables optimized participation in multiple energy and ancillary service markets, while alleviating local network constraints and providing reactive power support. The case study deploys an electricity-hydrogen VPP demonstrating the efficacy of the proposed framework. The results highlight the tight interaction between local network support and system-level market participation, and how provision of local voltage and reactive power services can enhance the VPP's flexibility to maximize its market revenues. This is particularly relevant in the context of distribution system operation and emerging DER marketplaces that are constrained by local network limits. [ABSTRACT FROM AUTHOR]