The concept of state

The state of a system may be defined as The set of variables (called the state variables) which at some initial time Iq, together with the input variables completely determine the behaviour of the system for time t to - 

The state variables are the smallest number of states that are required to describe the dynamic nature of the system, and it is not a necessary constraint that they are measurable. The manner in which the state variables change as a function of time may be thought of as a trajectory in n dimensional space, called the state-space. Two-dimensional state-space is sometimes referred to as the phase-plane when one state is the derivative of the other. 

State-space methods for control system design 233 

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The concept of equilibrium is central in thermodynamics, for associated with the condition of internal eqmlibrium is the concept of. state. A system has an identifiable, reproducible state when 1 its propei ties, such as temperature T, pressure P, and molar volume are fixed. The concepts oi state a.ndpropeity are again coupled. One can equally well say that the properties of a system are fixed by its state. Although the properties T, P, and V may be detected with measuring instruments, the existence of the primitive thermodynamic properties (see Postulates I and 3 following) is recognized much more indirectly. The number of properties for wdiich values must be specified in order to fix the state of a system depends on the nature of the system and is ultimately determined from experience. 

EQUATION OF STATE. Alsu called characterisin ci/tniinni. a relation, empirical or derived, between thermodynamic properties of a substance or system. The equation of stale must be single-valued in terms of its variables This is a direct consequence >r the concept of state. 

To rescue the concepts of state of consciousness and altered state of consciousness for more precise scientific use, I introduce the terms and abbreviation discrete state of consciousness (d-SoC) and discrete state of consciousness (d-ASC). I discuss in Chapter 2 the basic theoretical concepts for defining these crucial terms. Here, I first describe certain kinds of experiential data that led to the concepts of discrete states and then go on to a formal definition of d-SoC and d-ASC. 

Thus bits of knowledge that are specific for a d-ASC for one individual may be part of ordinary consciousness for another. Arguments over the usefulness of the concept of states of consciousness may reflect differences in the structure of the ordinary d-SoC of various investigators, as we discussed in Chapter 9. 

All the information that can be known about a system in a given stationary state is contained in the state function 5 (r), particularly molecular electron densities. The use of electron density to interpret atoms in molecules, bonds and structures constitutes a bridge between the concept of state function and the physical model of matter in real space. 

The projects undertaken by Italy in the late 1990s to produce CRMs for environmental research in Antarctica have also prompted similar actions to achieve full traceability to the SI of the certified properties. This was the case of the CRM MURST ISS A1 Antarctic Sediment prepared by ISS and the Institute for Reference Materials and Measurements (IRMM) of the Joint Research Centre (JRC), EC, for which an approach was proposed to obtain traceability for the Cu concentration (45). To this end, after microwave digestion of sediment and separation of the analyte by Ion-Exchange Chromatography (lEC), ID quadrupole ICP MS was employed. As traceability also implies the concept of stated uncertainties (see definitions in Table 1.3), the complete uncertainty budget was estimated. Thanks to this and to the fact that ID is a primary method of measurement, a clear mathematical relationship could be established between Cu concentration and isotope ratios. 

The first law of thermodynamics also states that t/ is a state function. State functions are very important in thermodynamics they depend only on the present state of a system and not on its past history. Neither q nor iv are state functions. An understanding of the concept of state function is furthered by considering the example of one s taking a trip from San Diego, Cahfomia, to Denver, Colorado. The change in altitude that one experiences during this trip does not depend on the route taken and, thus, is similar to a state function. In comparison, the distance traveled between the two cities does depend on the route one follows similarly, q and iv are path-dependent quantities. 

In thermodynamics the concept of state is important because state functions have the special characteristics of exactness. In this section we show how to test whether a quantity is exact, hence, whether it is a state function. Consider a differential equation in two variables, x and y,... 

To obtain information about heats of reaction indirectly, we use what is known as Hess s law the enthalpy change for any process is independent of the particular way the process is carried out. The underlying concept upon which this idea is built is that enthalpy is a state function. A state function is a variable whose value depends only on the state of the system and not on its history. When you drive your car, your location is a state function, but the distance that you have traveled to get there is not. For a chemical reaction, the concept of state functions can be very important. We rarely if ever know the microscopic details of how reactant molecules are actually converted into product molecules. But it is relatively easy to determine what the reactants and the products were. Because enthalpy is a state function, if we can find a way to determine the change in enthalpy for ar particular path that leads from the reactants to the desired products, we will know that the result will apply for whatever actual path the reaction may take. The block diagram in Figure 9.10 illustrates this concept. 

The system boundary separates the system or subsystem from its environment. The environment of a system is that part of the rest of the universe that has some form of relation with the system. The environment can influence the behavior of the system, but not its dynamic characteristics. The system boundary is the boundary between a system and its environment that can be defined on the basis of a boundary criterion. Common boundary criteria are based on the concept of state that will be discussed first. 

In contrast to the concept of state controllability in control theory which refers to the structural property of a system that the state vector can be driven to the origin in any desired period of time by suitable inputs, we are here concerned with I/0-controllabiIity of a plant which characterizes the potential to control the outputs of the system by the available inputs. 

The concept of state-of-component and state-of-system faults is worth discussing briefly here. If a state-of-component exists—in other words, the fault occurs because of a component failure—then OR gates are used. The use of OR gates connotes that any of the listed fault inputs can cause the event. If a state-of-system fault occurs, that means that something in the syston failed that caused the event to occur and thus connotes an AND gate—all the fault inputs must occur for the event to occm. 

Del Piero G, DeseriL (1997) On the concepts of state and free energy in linear viscoelasticity. Arch Rational Mech Anal 138 1-35... 

A simple example of the concept of state variables is encountered in the gas law pV = nRT this law describes the state of an ideal gas by specifying three state variables pressure p, volume V and absolute temperature T. If two of these state variables are known, the system state has been unambiguously determined. Later in this section we shall learn that a number of important energy quantities can be defined in such a way that they become state quantities. 

Figure 2.25. The concept of state variable can be illustrated by an example. Let the states (1) and (2) denote two points in a terrain. When we move from (1) to (2) the change of elevation is independent of the route AH = H2 — Hi. The elevation H is a state variable. The distance Si,2 is clearly dependent on the chosen route, i.e. S is not a state variable (see Mathematical appendix, subchapter A6).

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