Stabilization of systems with asynchronous sensors and controllers

We study the stabilization of networked control systems with asynchronous sensors and controllers. Offsets between the sensor and controller clocks are unknown and modeled as parametric uncertainty. First we consider multi-input linear systems and provide a sufficient condition for the existence of linear time-invariant controllers that are capable of stabilizing the closed-loop system for every clock offset in a given range of admissible values. For first-order systems, we next obtain the maximum length of the offset range for which the system can be stabilized by a single controller. Finally, this bound is compared with the offset bounds that would be allowed if we restricted our attention to static output feedback controllers.Comment: 32 pages, 6 figures. This paper was partially presented at the 2015 American Control Conference, July 1-3, 2015, the US

Robustness of controllers designed using Galerkin type approximations

One of the difficulties in designing controllers for infinite-dimensional systems arises from attempting to calculate a state for the system. It is shown that Galerkin type approximations can be used to design controllers which will perform as designed when implemented on the original infinite-dimensional system. No assumptions, other than those typically employed in numerical analysis, are made on the approximating scheme

Optimal Control with Information Pattern Constraints

Despite the abundance of available literature that starts with the seminal paper of Wang and Davison almost forty years ago, when dealing with the problem of decentralized control for linear dynamical systems, one faces a surprising lack of general design methods, implementable via computationally tractable algorithms. This is mainly due to the fact that for decentralized control configurations, the classical control theoretical framework falls short in providing a systematic analysis of the stabilization problem, let alone cope with additional optimality criteria. Recently, a significant leap occurred through the theoretical machinery developed in Rotkowitz and Lall, IEEE-TAC, vol. 51, 2006, pp. 274-286 which unifies and consolidates many previous results, pinpoints certain tractable decentralized control structures, and outlines the most general known class of convex problems in decentralized control. The decentralized setting is modeled via the structured sparsity constraints paradigm, which proves to be a simple and effective way to formalize many decentralized configurations where the controller feature a given sparsity pattern. Rotkowitz and Lall propose a computationally tractable algorithm for the design of H2 optimal, decentralized controllers for linear and time invariant systems, provided that the plant is strongly stabilizable. The method is built on the assumption that the sparsity constraints imposed on the controller satisfy a certain condition (named quadratic invariance) with respect to the plant and that some decentralized, strongly stablizable, stabilizing controller is available beforehand. For this class of decentralized feedback configurations modeled via sparsity constraints, so called quadratically invariant, we provided complete solutions to several open problems. Firstly, the strong stabilizability assumption was removed via the so called coordinate free parametrization of all, sparsity constrained controllers. Next we have addressed the unsolved problem of stabilizability/stabilization via sparse controllers, using a particular form of the celebrated Youla parametrization. Finally, a new result related to the optimal disturbance attenuation problem in the presence of stable plant perturbations is presented. This result is also valid for quadratically invariant, decentralized feedback configurations. Each result provides a computational, numerically tractable algorithm which is meaningful in the synthesis of sparsity constrained optimal controllers

Structural decomposition of general singular linear systems and its applications

Ph.DDOCTOR OF PHILOSOPH

Distributed Cooperative Communications and Wireless Power Transfer

In telecommunications, distributed cooperative communications refer to techniques which allow different users in a wireless network to share or combine their information in order to increase diversity gain or power gain. Unlike conventional point-to-point communications maximizing the performance of the individual link, distributed cooperative communications enable multiple users to collaborate with each other to achieve an overall improvement in performance, e.g., improved range and data rates. The first part of this dissertation focuses the problem of jointly decoding binary messages from a single distant transmitter to a cooperative receive cluster. The outage probability of distributed reception with binary hard decision exchanges is compared with the outage probability of ideal receive beamforming with unquantized observation exchanges. Low- dimensional analysis and numerical results show, via two simple but surprisingly good approximations, that the outage probability performance of distributed reception with hard decision exchanges is well-predicted by the SNR of ideal receive beamforming after subtracting a hard decision penalty of slightly less than 2 dB. These results, developed in non-asymptotic regimes, are consistent with prior asymptotic results (for a large number of nodes and low per-node SNR) on hard decisions in binary communication systems. We next consider the problem of estimating and tracking channels in a distributed transmission system with multiple transmitters and multiple receivers. In order to track and predict the effective channel between each transmit node and each receive node to facilitate coherent transmission, a linear time-invariant state- space model is developed and is shown to be observable but nonstabilizable. To quantify the steady-state performance of a Kalman filter channel tracker, two methods are developed to efficiently compute the steady-state prediction covariance. An asymptotic analysis is also presented for the homogenous oscillator case for systems with a large number of transmit and receive nodes with closed-form results for all of the elements in the asymptotic prediction covariance as a function of the carrier frequency, oscillator parameters, and channel measurement period. Numeric results confirm the analysis and demonstrate the effect of the oscillator parameters on the ability of the distributed transmission system to achieve coherent transmission. In recent years, the development of efficient radio frequency (RF) radiation wireless power transfer (WPT) systems has become an active research area, motivated by the widespread use of low-power devices that can be charged wirelessly. In this dissertation, we next consider a time division multiple access scenario where a wireless access point transmits to a group of users which harvest the energy and then use this energy to transmit back to the access point. Past approaches have found the optimal time allocation to maximize sum throughput under the assumption that the users must use all of their harvested power in each block of the harvest-then-transmit protocol. This dissertation considers optimal time and energy allocation to maximize the sum throughput for the case when the nodes can save energy for later blocks. To maximize the sum throughput over a finite horizon, the initial optimization problem is separated into two sub-problems and finally can be formulated into a standard box- constrained optimization problem, which can be solved efficiently. A tight upper bound is derived by relaxing the energy harvesting causality. A disadvantage of RF-radiation based WPT is that path loss effects can significantly reduce the amount of power received by energy harvesting devices. To overcome this problem, recent investigations have considered the use of distributed transmit beamforming (DTB) in wireless communication systems where two or more individual transmit nodes pool their antenna resources to emulate a virtual antenna array. In order to take the advantages of the DTB in the WPT, in this dissertation, we study the optimization of the feedback rate to maximize the energy efficiency in the WPT system. Since periodic feedback improves the beamforming gain but requires the receivers to expend energy, there is a fundamental tradeoff between the feedback period and the efficiency of the WPT system. We develop a new model to combine WPT and DTB and explicitly account for independent oscillator dynamics and the cost of feedback energy from the receive nodes. We then formulate a Normalized Weighted Mean Energy Harvesting Rate (NWMEHR) maximization problem to select the feedback period to maximize the weighted averaged amount of net energy harvested by the receive nodes per unit of time as a function of the oscillator parameters. We develop an explicit method to numerically calculate the globally optimal feedback period

Modelling and Control of an Annular Momentum Control Device

The results of a modelling and control study for an advanced momentum storage device supported on magnetic bearings are documented. The control challenge posed by this device lies in its dynamics being such a strong function of flywheel rotational speed. At high rotational speed, this can lead to open loop instabilities, resulting in requirements for minimum and maximum control bandwidths and gains for the stabilizing controllers. Using recently developed analysis tools for systems described by complex coefficient differential equations, the closed properties of the controllers were analyzed and stability properties established. Various feedback controllers are investigated and discussed. Both translational and angular dynamics compensators are developed, and measures of system stability and robustness to plant and operational speed variations are presented

Finite settling time stabilization for linear multivariable time-invariant discrete-time systems: An algebraic approach

The problem of Total Finite Settling Time Stabilization of linear time-invariant discrete-time systems is investigated in this thesis. This problem falls within the same area of the well-known deadbeat (time-optimal) control and in particular, constitutes a generalization of this problem. That is, instead of seeking time-optimum performance, it is required that all internal and external variables (signals) of the closed-loop system settle to a new steady state after a finite time from the application of a step change to any of its inputs and for every initial condition. The state/output deadbeat control is a special case of the Total FSTS problem. Using a mathematical and system theory framework based on sequences and the polynomial equation (algebraic) approach, we are able to tackle the FSTS problem in a unifying manner. The one-parameter (unity) feedback configuration is mainly used for the solution of the FSTS problem and FSTS related control strategies. The whole problem is reduced to the solution of a polynomial matrix Diophantine equation which guarantees not only internal stability but also internal FSTS and is further reduced to the solution of a linear algebra problem over R. This approach enables the parametrizat ion of the family of all FSTS controllers, as well as those which are causal, in a Youla-Bongiorno-Kucera type parametrization. The minimal McMillan degree FSTS problem is completely solved for vector plants and a parametrization of the FSTS controllers according to their McMillan degree is obtained. In the MIMO case bounds of the minimum McMillan degree controllers are derived and families of FSTS controllers with given lower/upper McMillan degree bounds are provided in parametric form. Having parametrized the family of all FSTS controllers, the state deadbeat regulation is treated as a special case of FSTS and complete parametrization of all the deadbeat regulators is presented. In addition, further performance criteria, or design constraints are imposed such as, FSTS tracking and/or disturbance rejection, partial assignment of controller dynamics, l1-, l∞-optimization and robustness to plant parameter variations. Finally, the Simultaneous-FSTS problem is formulated, and necessary as well as sufficient conditions for its solution are derived. Also, a two-parameter control scheme is introduced to alleviate some of the drawbacks of the one-parameter control. A parametrization of the family of FSTS controllers as well as the FSTS controllers for tracking and/or disturbance rejection is given as an illustration of the particular advantages of the two-parameter FSTS controllers

Contributions to Passivity Theory and Dissipative Control Synthesis

This thesis contains contributions to some relevant problems in the field of control theory and controller design technology, namely to the areas of passivity analysis and dissipative control synthesis for linear and nonlinear dynamical systems. The first of our contributions consists in presenting a solution to a problem which had been unsolved for many years: the problem of the equivalence between the notions of strict positive realness and strict passivity of linear systems. Both properties imply the asymptotic stability of a linear system, although the former is a frequency-domain concept and the latter is a time-domain concept. Subsequently, we approach the equally classical topic of static output feedback stabilization of linear systems, a problem to which a definite solution remains to be given. We present a new necessary and sufficient LMI condition for stabilization based on the notion of strict dissipativity, and we propose a new noniterative strategy for controller design which consists in solving a single convex optimization problem. In addition, we also introduce a new dissipativity-based strategy for feedback stabilization of nonlinear systems using the notion of linear annihilators and the celebrated Finsler’s Lemma. This approach allows for analysing the dissipativity properties of rational nonlinear plants in terms of a polytopic LMI condition. A new stabilizability condition that would not be feasible in the case of a passive representation of the system is presented as well, making it possible to derive a closed-form expresion for the controller’s feedthrough term as a direct consequence of the local dissipativity analysis of the plant. This feature simplifies the remaing steps of the controller design procedure considerably, both in the case of a static or a dynamic output feedback