Hernan Haimovich

A moving horizon approach to Networked Control system design

This paper presents a control system design strategy for multivariable plants where the controller, sensors and actuators are connected via a digital, data-rate limited, communications channel. In order to minimize bandwidth utilization, a communication constraint is imposed which restricts all transmitted data to belong to a finite set and only permits one plant to be addressed at a time. We emphasize implementation issues and employ moving horizon techniques to deal with both control and measurement quantization issues. We illustrate the methodology by simulations and a laboratory-based pilot-scale study.

A systematic method to obtain ultimate bounds for perturbed systems

In this paper, we develop a systematic method to obtain ultimate bounds for both continuous- and discrete-time perturbed systems. The method is based on a componentwise analysis of the system in modal coordinates and thus exploits the system geometry as well as the perturbation structure without requiring calculation of a Lyapunov function for the system. The method is introduced for linear systems having constant componentwise perturbation bounds and is then extended to the case of state-dependent perturbation bounds. This extension enables the method to be applied to non-linear systems by treating the perturbed non-linear system as a linear system with a perturbation bounded by a non-linear function of the state. Examples are provided where the proposed systematic method yields bounds that are tighter or at least not worse than those obtained via standard Lyapunov analysis. We also show how our method can be combined with Lyapunov analysis to improve on the bounds provided by either approach.

Brief paper: Componentwise ultimate bound and invariant set computation for switched linear systems

We present a novel ultimate bound and invariant set computation method for continuous-time switched linear systems with disturbances and arbitrary switching. The proposed method relies on the existence of a transformation that takes all matrices of the switched linear system into a convenient form satisfying certain properties. The method provides ultimate bounds and invariant sets in the form of polyhedral and/or mixed ellipsoidal/polyhedral sets, is completely systematic once the aforementioned transformation is obtained, and provides a new sufficient condition for practical stability. We show that the transformation required by our method can easily be found in the well-known case where the subsystem matrices generate a solvable Lie algebra, and we provide an algorithm to seek such transformation in the general case. An example comparing the bounds obtained by the proposed method with those obtained from a common quadratic Lyapunov function computed via linear matrix inequalities shows a clear advantage of the proposed method in some cases.

Brief paper: Control design with guaranteed ultimate bound for perturbed systems

We present a new control design method for perturbed multiple-input systems, which guarantees any desired componentwise ultimate bound on the system state. The method involves eigenvalue/eigenvector assignment by state feedback and utilises a componentwise bound computation procedure. This procedure directly takes into account both the system and perturbation structures by performing componentwise analysis, thus avoiding the need for bounds on the norm of the perturbation. The perturbation description adopted can accommodate numerous types of uncertainties, including uncertain time-delays in the feedback loop. We apply the method to an example taken from the literature to illustrate its simplicity and generality.

Brief paper: Systematic ultimate bound computation for sampled-data systems with quantization

We present a novel systematic method to obtain componentwise ultimate bounds in perturbed sampled-data systems, especially when the perturbations arise due to quantization. The proposed method exploits the system geometry as well as the perturbation structure, and takes intersample behavior into account. The main features of the method are its systematic nature, whereby it can be readily computer coded, without requiring adjustment of parameters for its application, and its suitability for dealing with highly structured perturbation schemes, whereby the information on the perturbation structure is directly taken into account. The latter feature distinguishes the method from other approaches that require a bound on the norm of the perturbation and thus disregard information on the perturbation structure. We apply the method to a numerical example taken from the literature to illustrate its simplicity and potential.

Bounds and invariant sets for a class of switching systems with delayed-state-dependent perturbations

We present a novel method to compute componentwise transient bounds, componentwise ultimate bounds, and invariant regions for a class of switching continuous-time linear systems with perturbation bounds that may depend nonlinearly on a delayed state. The main advantage of the method is its componentwise nature, i.e. the fact that it allows each component of the perturbation vector to have an independent bound and that the bounds and sets obtained are also given componentwise. This componentwise method does not employ a norm for bounding either the perturbation or state vectors, avoids the need for scaling the different state vector components in order to obtain useful results, and may also reduce conservativeness in some cases. The present paper builds upon and extends to switching systems with delayed-state-dependent perturbations previous results by the authors. In this sense, the contribution is three-fold: the derivation of the aforementioned extension; the elucidation of the precise relationship between the class of switching linear systems to which the proposed method can be applied and those that admit a common quadratic Lyapunov function (a question that was left open in our previous work); and the derivation of a technique to compute a common quadratic Lyapunov function for switching linear systems with perturbations bounded componentwise by affine functions of the absolute value of the state vector components. In this latter case, we also show how our componentwise method can be combined with standard techniques in order to derive bounds possibly tighter than those corresponding to either method applied individually.

Feedback Stabilization of Switching Discrete-Time Systems via Lie-Algebraic Techniques

This technical note addresses the stabilization of switching discrete-time linear systems with control inputs under arbitrary switching. A sufficient condition for the uniform global exponential stability (UGES) of such systems is the existence of a common quadratic Lyapunov function (CQLF) for the component subsystems, which is ensured when the closed-loop component subsystem matrices are stable and generate a solvable Lie algebra. The present work develops an iterative algorithm that seeks the feedback maps required for stabilization based on the previous Lie-algebraic condition. The main theoretical contribution of the technical note is to show that this algorithm will find the required feedback maps if and only if the Lie-algebraic problem has a solution. The core of the proposed algorithm is a common eigenvector assignment procedure, which is executed at every iteration. We also show how the latter procedure can be numerically implemented and provide a key structural condition which, if satisfied, greatly simplifies the required computations.

Bounds and invariant sets for a class of discrete-time switching systems with perturbations

We present a novel method to compute componentwise ultimate bounds and invariant regions for a class of switching discrete-time linear systems with perturbation bounds that may depend nonlinearly on a delayed state. The method has the important advantage that it allows each component of the perturbation vector to have an independent bound and that the bounds and sets obtained are also given componentwise. This componentwise method does not employ a standard norm for bounding either the perturbation or state vectors, and thus may avoid conservativeness due to different perturbation or state vector components having substantially different bounds. We also establish the relationship between the class of switching linear systems to which the proposed method can be applied and those that admit a common quadratic Lyapunov function. We illustrate the application of our method via numerical examples, including the fault tolerance analysis of the feedback control of a winding machine.

Analysis and Improvements of a Systematic Componentwise Ultimate-bound Computation Method

Abstract We perform in-depth analysis and provide improvements of a systematic componentwise ultimate-bound computation method recently introduced in the literature. This method was shown to have many advantages over traditional ultimate-bound computation methods based on the use of quadratic Lyapunov functions. The analysis performed enhances our understanding of the componentwise methodology, and simplifies the search for improvements. The improvements provided aim at reducing the conservatism of the componentwise ultimate-bound computation methods even further, hence leading to tighter bounds. These improvements do not alter the systematic nature of the method.

Geometric characterization of multivariable quadratically stabilizing quantizers

In this paper, we present an explicit geometric characterization of quadratically stabilizing state feedback laws that are based on the use of multivariable quantizers of minimum dimension. This characterization consists of a set of necessary and sufficient conditions for a quantized static state feedback to render a given quadratic function a Lyapunov function for the closed-loop system. These necessary and sufficient conditions provide a means to analyse and design such quantized feedback laws and are derived from set inclusion conditions that are necessarily satisfied by the quantization regions and values of a quadratically stabilizing quantizer.