Sensor Equations

The p — 1 nonzero elements of the regressor vector / j for each variable i are computed by using PLS algorithm where the predicted variable block Y contains the measurements of the fth sensor taken from X, and the predictor block denoted by X contains the observations from the remaining p —1 sensors. Equations 8.1 and 8.2 are defined such that the nxp matrix X which contains the measurements from all the p sensors is utilized directly. These PLS regressions are repeated for i = 1, iP- For the fth sensor, the p — 1 elements of the regressor vector /3j are [329]... 

In the case of actively controlled structural dynamics, sensors may measure a variety of signals, such as accelerations (accelerometers), displacements (Hall sensors, capacitive sensors, laser interferometers, etc.), forces (force transducers), or - typically for adaptronic structures - strain or strain velocities (strain gauges, piezoelectric sensors, etc.). Most of these cases can be represented by the following sensor equation ... 

If the electrical excitation and the mechanical load are limited to amplitude ranges where the dependence of the characteristic of the electrical transfer path and the actuator transfer path on the mechanical load as well as the dependence of the characteristic of the sensor transfer path and the mechanical transfer path on the electrical excitation can be neglected, then the vectorial operators in sensor equation (6.55) and in actuator equation (6.56) can be simplified to a hnear superposition of scalar operators ... 

If the mappings T in the sensor equation (6.75) and the actuator equation (6.76) are purely hysteretic they can be modeled by a Prandtl-Ishhnskii operator H, a modified Prandtl-Ishlinskii operator M or a Preisach hysteresis operator R depending on the degree of symmetry of the branching behaviour. The calculation of these hysteresis operators and the corresponding compensators from the measured output-input characteristic requires special computer-aided synthesis procedures which is based on system identification methods. Due to a lack of space, this article cannot further comment on these synthesis methods. However, a detailed description of both the synthesis method and the mathematical basics can be found in the literature [332,341,350-352,356]. 

Here D is the vector of the dielectric displacement (size 3x1, unit C/m ), S is the strain (size 6x1, dimension 1), E is a vector of the electric field strength (size 3x1, unit V/m) and T is a vector of the mechanical tension (size 6x1, unit N/m ). As the piezoelectric constants depend on the direction in space they are described as tensors e- is the permittivity constant also called dielectric permittivity at constant mechanical tension T (size 3x3, unit F/m) and 5 , is the elastic compliance matrix (size 6x6, unit m /N). The piezoelectric charge coefficient df " (size 6x3, unit C/N) defines the dielectric displacement per mechanical tension at constant electrical field and (size 3x6, unit m/V) defines the strain per eiectric fieid at constant mechanical tension [84], The first equation describes the direct piezo effect (sensor equation) and the second the inverse piezo effect (actuator equation). 

The boundary condition at the air space between the test sensor and insulating block is given by equation 12.29. The initial condition is T(x) = Ti(x), where 7)(x) is the initial temperature distribution in the fabric, air gap and test sensor. Equations 12.31 and 12.32 are solved subject to equations 12.29, 12.33 and 12.34, and the initial condition. 

The design of smart materials and adaptive stmctures has required the development of constitutive equations that describe the temperature, stress, strain, and percentage of martensite volume transformation of a shape-memory alloy. These equations can be integrated with similar constitutive equations for composite materials to make possible the quantitative design of stmctures having embedded sensors and actuators for vibration control. The constitutive equations for one-dimensional systems as well as a three-dimensional representation have been developed (7). 

The doped Zr02 stmctures are used as electrochemical sensors, as, for example, when used to detect oxygen in automotive exhaust (see Exhaust CONTROL, automotive). The sensor voltage is governed by the Nemst equation (eq. 17) where the activities are replaced by oxygen partial pressures and the air inside the chamber is used as reference. 

In order for a solution for the systems of equations expressed in equation 11 to exist, the number of sensors must be at least equal to the number of analytes. To proceed, the analyst must first determine the sensitivity factors using external standards, ie, solve equation 11 for Kusing known C and R. Because concentration C is generally not a square data matrix, equation 11 is solved by the generalized inverse method. K is given by... 

Given that one sample is to be analy2ed for r analytes usings sensors (p > r) by making a series of n standard additions in > r) to the sample and recording the sensor responses after each addition, the equation becomes... 

Most sensor volumes, whether in LC (e.g., a UV absorption cell) or in GC (e.g., a katharometer cell), are cylindrical in shape, are relatively short in length and have a small length-to-diameter ratio. The small length-to-diameter ratio is in conflict with the premises adopted in the development of the Golay equation for dispersion in an open tube and, consequently, its conclusions are not pertinent to detector sensors. Atwood and Golay [12] extended the theory of dispersion in open tubes to tubes of small length-to-diameter ratio. The theory developed is not pertinent here as it will be seen that, with correctly designed cells, that dispersion from viscous sources can be... 

The time constant in the previous equations is, in principle, the product of the sensor capacity and the resistance of the flow into the sensor,... 

Air is normally the reference gas used in the exhaust gas sensor. If the oxygen partial pressure in the engine exhaust gas is known as a function of the engine air/fuel ratio, the theoretical galvanic potential of the sensor is easily determined by the Nernst equation. 

NADH, 121, 122, 180 Nafion coating, 118, 123, 124, 126 Nanometer electrodes, 116, 128 Nernst equation, 3, 15, 80 Nernstian behavior, 143 Nernst Planck equation, 5 Neuronal sensors, 188 Neurotransmitters, 40, 116, 124 Neutral carrier electrodes, 154 Nickel, 123... 

Radical ions, 33, 44 Raman spectroelectrochemistry, 45 Randles-Sevcik equation, 31 Rate constant, 12, 18 Rate determining step, 4, 14 Reaction mechanism, 33, 36, 113 Reaction pathway, 4, 33 Reaction rate, 12 Receptor-based sensors, 186 Redox recycling, 135... 

For a circular aperture a typical set of basis functions are the Zemike polynomials (Noll, 1976), but for other geometries alternative basis functions may be more appropriate. The objective of most wavefront sensors is to produce a set of measurements, m, that can be related to the wavefront by a set of linear equations... 

The sensor is taken to be of the linear type, i.e., it transduces the incoming chemical information into electrical output according to the equation el.signal = constant + slope (chemical signal) without loss of clarity, the constant can be set to zero and the slope to 1.00. 

---