The ultimate goal of a cloud service provider is the maximization of its profit. In addition to reduce the number of SLA violations (and hence to reduce the amount of monetary penalty to possibly pay to service consumer), one way to increase the profit is to reduce the Total Cost of Ownership (TCO), which comprises capital and
It has been argued [60] that energy costs are among the most important factors impacting on TCO, and that this influence will grow in the near future due to the increase of electricity costs. Therefore, the reduction of operating costs is usually pursued through the reduction of the amount of energy absorbed by the physical resources of the data center. Various techniques already exist that aim at reducing the amount of electrical energy consumed by the physical infrastructure underlying the IaaS substrate, ranging from energy-efficient hardware and energy- aware design strategies, to server consolidation, whereby multiple virtual machines
run on the same physical resource [166]. Moreover, VM live migration can be used
to dynamically consolidate VMs residing on multiple under-utilized servers onto a single server, so that the remaining servers can be set to an energy-saving state. Unfortunately, these techniques alone are not enough to guarantee application performance requirements because of the complexity of cloud computing systems, where (1) system resources have to be dynamically and unpredictably shared among several independent applications, (2) the requirements of each application must be met in order to avoid economical penalties, (3) the workload of each application generally changes over time, (4) applications may span multiple computing nodes, and (5) system resources may be possibly distributed world-wide.
Linear Systems Theory
Linear System Theory[146] investigates the behavior and the response of a linear
system to arbitrary input signals. In this thesis, we use linear system theory as a mean to characterize the behavior of computing systems. Specifically, the class of dynamical systems we are interested in are discrete-time linear time-invariant causal systems, which are the ones that are usually employed to describe current
computing systems from the control-theoretic point-of-view [84].
In this thesis, we focus on discrete-time linear time-invariant casual sys- tems.
In the rest of this chapter, we present an overview of some of the key concepts and results related to discrete-time linear systems, especially from the perspective of identification and control (topics that will be covered in the subsequent chapters).
First, in Section 3.1 we provide a characterization of dynamical systems, with
a particular emphasis on linear systems. Then, in Section3.2, we discuss some
technique to approximate nonlinear systems with linear ones. In Section3.3, we
present main representations of linear systems. In Section3.4, we introduce the
realization theory, which provides a bridge between the state-space and the input-
output representations. Finally, in Section3.5, we discuss about the stability of
linear systems.
3.1
Characterization of Dynamical Systems
A system is a collection of interacting components. An electric motor, an airplane, and a biological unit such as the human arm are examples of systems. A dynamical
systemis a system consisting of a set of possible states, together with a rule that
determines the present state in terms of past and future states [18]. A more formal
definition is provided by [94]:
Definition 3.1.1 (Dynamical System). A dynamical system is composed of three parts:
• the state of a system, which is a representation of all the information about the system at some particular value of an independent variable (e.g., the time);
• the state-space of a system, which is a set that contains all of the possible states to which a system can be assigned;
• the state-transition function that is used to update and change the state from one moment to another.
Usually the independent variable denotes the time; however, it can represent other type of information, as is the case of image processing, where the independent variable denotes the space.
Dynamical systems can be characterized by the interaction of different type of signals, which are quantities that change their value as a function of an independent variable (which, in this thesis, is the time).
Definition 3.1.2 (Signal). A signal g is a uniquely defined mathematical function (single-valued function) of an independent variable k. The set for which the independent variable k is defined, is called the domain of the signal.
To denote the “whole signal”, that is the sequence of values g(k), for every k, we use the notation g(·) or g; while, to denote the value of the signal at a specific k, we use the notation g(k). A vector-valued signal g(·) is a signal whose values are vectors. If the domain of the signal represents discrete time, then g(·) defines a
discrete-time signal.1 In Fig.3.1is shown an example of discrete-time signal.
Figure 3.1: A discrete signal.
In this thesis, we assume that the independent variable represents the time and that it takes values on the set N; thus, we are only concerned to discrete- time signals.
Signals interacting in a dynamical system can be classified into:
• Inputs (or exogenous variables) u(·), which consist of variables from the environment that influence the system. They are the only variables that can be manipulated by the user in order to have effect on the system.
• Outputs (or endogenous variables) y(·), that quantify effects of inputs on the system.
• States x(·), which represent the internal state of the system and provide a way to characterize the effects of the input on the produced output and on the future state of the system.
When the input signal u(·) takes scalar values, the system is called single-input (SI); otherwise, it is called multiple-input (MI). When the output signal y(·) takes scalar values the system is called single-output (SO); otherwise, it is called multiple-output (MO). Thus, a SISO system is a single-input single-output system, a MISO system is a multiple-input single-output system, and a MIMO system is a multiple-input multiple-output system.
Dynamical systems can evolve either in continuous-time or in discrete-time; in the former, the independent variable is defined over some continuous time interval (e.g., the set R), while in the latter, is defined over a discrete interval (e.g., the set N). A dynamical system is causal (or nonanticipatory) if its current output depends on past and current inputs but not on future input; conversely, a noncausal (or anticipatory) system can predict or anticipate what will be applied in the future. Physical systems are usually considered causal systems; noncausal systems can be found in off-line post-processing systems, like image or audio processing. We say that a dynamical system is memoryless (or without memory) if its output, for each value of the independent variable, is dependent only on the input evaluated at the same value of the independent variable. A system that is not memoryless is said to have memory. All memoryless systems are also causal, since they depend only on its current input; the converse, in general, is not true. A system is said to be time-invariant if a time shift in the input causes a corresponding time shift in the output. If a system is not time-invariant, it is said to be time-varying.
A system is linear if it obeys to the principle of superposition, whereby the response to a weighted sum of any two inputs is the (same) weighted sum of the responses to each individual input. More formally, the system y = T (u) is linear, if
T is a linear operator that is, for any valid inputs u1and u2, and for any α, β ∈ R:
y3= T (αu1+ β u2)
= αT (u1) + β T (u2)
= αy1+ β y2
(3.1)
A system that does not obey to the superposition principle is called nonlinear.
In this thesis, we are only concerned to casual discrete-time linear systems.