22 research outputs found
Phase limitations of Zames-Falb multipliers
Phase limitations of both continuous-time and discrete-time Zames-Falb
multipliers and their relation with the Kalman conjecture are analysed. A phase
limitation for continuous-time multipliers given by Megretski is generalised
and its applicability is clarified; its relation to the Kalman conjecture is
illustrated with a classical example from the literature. It is demonstrated
that there exist fourth-order plants where the existence of a suitable
Zames-Falb multiplier can be discarded and for which simulations show unstable
behavior. A novel phase-limitation for discrete-time Zames-Falb multipliers is
developed. Its application is demonstrated with a second-order counterexample
to the Kalman conjecture. Finally, the discrete-time limitation is used to show
that there can be no direct counterpart of the off-axis circle criterion in the
discrete-time domain
Convex searches for discrete-time Zames-Falb multipliers
In this paper we develop and analyse convex searches for Zames--Falb
multipliers. We present two different approaches: Infinite Impulse Response
(IIR) and Finite Impulse Response (FIR) multipliers. The set of FIR multipliers
is complete in that any IIR multipliers can be phase-substituted by an
arbitrarily large order FIR multiplier. We show that searches in discrete-time
for FIR multipliers are effective even for large orders. As expected, the
numerical results provide the best -stability results in the
literature for slope-restricted nonlinearities. Finally, we demonstrate that
the discrete-time search can provide an effective method to find suitable
continuous-time multipliers.Comment: 12 page
On the exponential convergence of input-output signals of nonlinear feedback systems
We show that the integral-constraint-based robust feedback stability theorem
for certain Lurye systems exhibits the property that the endogenous
input-output signals enjoy an exponential convergence rate for all initial
conditions of the linear time-invariant subsystem. More generally, we provide
conditions under which a feedback interconnection of possibly open-loop
unbounded subsystems to admit such an exponential convergence property, using
perturbation analysis and a combination of tools including integral quadratic
constraints, directed gap measure, and exponential weightings. As an
application, we apply the result to first-order convex optimisation methods. In
particular, by making use of the Zames-Falb multipliers, we state conditions
for these methods to converge exponentially when applied to strongly convex
functions with Lipschitz gradients.Comment: This paper has been submitted to Automatic
Robust Scale-Free Synthesis for Frequency Control in Power Systems
The AC frequency in electrical power systems is conventionally regulated by
synchronous machines. The gradual replacement of these machines by asynchronous
renewable-based generation, which provides little or no frequency control,
increases system uncertainty and the risk of instability. This imposes hard
limits on the proportion of renewables that can be integrated into the system.
In this paper we address this issue by developing a framework for performing
frequency control in power systems with arbitrary mixes of conventional and
renewable generation. Our approach is based on a robust stability criterion
that can be used to guarantee the stability of a full power system model on the
basis of a set of decentralised tests, one for each component in the system. It
can be applied even when using detailed heterogeneous component models, and can
be verified using several standard frequency response, state-space, and circuit
theoretic analysis tools. Furthermore the stability guarantees hold
independently of the operating point, and remain valid even as components are
added to and removed from the grid. By designing decentralised controllers for
individual components to meet these decentralised tests, every component can
contribute to the regulation of the system frequency in a simple and provable
manner. Notably, our framework certifies the stability of several existing
(non-passive) power system control schemes and models, and allows for the study
of robustness with respect to delays.Comment: 10 pages, submitte
Robust control design with real parameter uncertainty using absolute stability theory
The purpose of this thesis is to investigate an extension of mu theory for robust control design by considering systems with linear and nonlinear real parameter uncertainties. In the process, explicit connections are made between mixed mu and absolute stability theory. In particular, it is shown that the upper bounds for mixed mu are a generalization of results from absolute stability theory. Both state space and frequency domain criteria are developed for several nonlinearities and stability multipliers using the wealth of literature on absolute stability theory and the concepts of supply rates and storage functions. The state space conditions are expressed in terms of Riccati equations and parameter-dependent Lyapunov functions. For controller synthesis, these stability conditions are used to form an overbound of the H2 performance objective. A geometric interpretation of the equivalent frequency domain criteria in terms of off-axis circles clarifies the important role of the multiplier and shows that both the magnitude and phase of the uncertainty are considered. A numerical algorithm is developed to design robust controllers that minimize the bound on an H2 cost functional and satisfy an analysis test based on the Popov stability multiplier. The controller and multiplier coefficients are optimized simultaneously, which avoids the iteration and curve-fitting procedures required by the D-K procedure of mu synthesis. Several benchmark problems and experiments on the Middeck Active Control Experiment at M.I.T. demonstrate that these controllers achieve good robust performance and guaranteed stability bounds
Network flow optimization and distributed control algorithms
This thesis concerns the problem of designing distributed algorithms for achieving efficient and fair bandwidth allocations in a resource constrained network. This problem is fundamental to the design of transmission protocols for communication networks, since the fluid models of popular protocols such as TCP and Proportional Fair Controller can be viewed as distributed algorithms which solve the network flow optimization problems corresponding to some fairness criteria. Because of the convexity of the optimization problem as well as its decoupling structure, there exist classical dual algorithm and primal/dual algorithm which are both distributed. However, the main difficulty is the possible instability of the dynamics of these algorithms caused by transmission delays. We use customized Lyapunov-Krasovskii functionals to obtain the stability conditions for these algorithms in networks with heterogeneous time-varying delays. There are two main features of our results. The first is that these stability conditions can be enforced by a small amount of information exchange among relevant users and links. The second is that these stability conditions only depend on the upper bound of delays, not on the rate of delay variations. We further our discussion on scalable algorithms with minimum information to maintain stability. We present a design methodology for such algorithms and prove the global stability of our scalable controllers by the use of Zames-Falb multipliers. Next we extend this method to design the first scalable and globally stable algorithm for the joint multipath routing and flow optimization problem. We achieve this by adding additional delays to different paths for all users. Lastly we discuss the joint single path routing and flow optimization problem, which is a NP hard problem. We show bounded price of anarchy for combined flow and routing game for simple networks and show for many-user networks, simple Nash algorithm leads to approximate optimum of the problem
Physics-guided neural networks for feedforward control with input-to-state stability guarantees
Currently, there is an increasing interest in merging physics-based methods
and artificial intelligence to push performance of feedforward controllers for
high-precision mechatronics beyond what is achievable with linear feedforward
control. In this paper, we develop a systematic design procedure for
feedforward control using physics-guided neural networks (PGNNs) that can
handle nonlinear and unknown dynamics. PGNNs effectively merge physics-based
and NN-based models, and thereby result in nonlinear feedforward controllers
with higher performance and the same reliability as classical, linear
feedforward controllers. In particular, conditions are presented to validate
(after training) and impose (before training) input-to-state stability (ISS) of
PGNN feedforward controllers. The developed PGNN feedforward control framework
is validated on a real-life, high-precision industrial linear motor used in
lithography machines, where it reaches a factor 2 improvement with respect to
conventional mass-friction feedforward