LTI system theory

As a linear time invariant system, as LTI system LTI system (English linear time- invariant system) is a system referred to when it has both the characteristic of linearity is also independent of time shifts. This independence of temporal shifts is referred to as time-invariance.

The importance of these systems is that they have a particularly simple transformation equations of the system analysis are thus easily accessible. Many technical systems such as in communication or control systems have, at least to a good approximation, these properties. A system can be, for example, a transmission system in this context. Some LTI systems can be described by linear ordinary differential equations with constant coefficients.

  • 2.1 Time range
  • 2.2 image area
  • 3.1 Example of mechanics

Properties

Linearity

A system is then called linear if every sum of any number of input signals leads to a proportional thereto sum ​​of output signals. It must allow the superposition principle, also called the superposition principle, apply. Mathematically, this is described by a transformation which is the transfer function of the system, between the input and output signals:

The constant coefficients represent the individual proportionality

Clearly, a signal is applied to the input of the system and the reaction is observed. Thereafter, the response to a second signal of which is analyzed independently. When applying an input signal, which is the sum of the two previously refereed signals, it can be established that the response at the output of the addition of the two individual responses corresponding to when the system is linear.

Time invariance

A system is called time-invariant then, if for any time shift of t0:

For the time invariance of the output signal must be maintained for the time reference input signal and respond identically. This principle is also referred to as the displacement principle.

Connection with convolution integral

The any path input signal s ( t) can be approximated by applying the superposition theorem and the invariance by a temporal sequence of single square pulses. The border crossing for a rectangular pulse whose duration goes to 0, the output signal of a shape approximates that depends only on the transfer function of the system, but no more. From the waveform of the input signal

Mathematically describe these against the duration of zero aspiring rectangular pulses by Dirac impulses δ ( t) and the buzz in the transformation equation shall become integrals. The input signal s (t) can be expressed as equivalent convolution integral or with the symbol for the convolution operation as:

The output signal g (t) is the convolution integral

Associated with the input signal s (t). The function h (t) is also referred to as an impulse response, with which the convolution integral is one for LTI systems generally applicable transformation.

LTI systems in various forms of representation

The following section is limited to systems with finitely many internal degrees of freedom.

Time domain

The most common representation in the time domain system, the state space representation, having the general form

Herein are the vectors u input vector, state vector x and output vector y. Are the matrices A system matrix, input matrix B, C and D output matrix penetration matrix constant, the system is linear and time-invariant. For the addition and multiplication of vectors and matrices see Matrix (mathematics).

Image area

For simpler systems, in particular SISO systems ( Single Input, Single Output ) systems with only one each input and output size is often selected the description by a transfer function ( " image area " or " frequency range " )

Here, Z is the numerator polynomial in s, and N is the denominator polynomial in s Are all the coefficients of the two polynomials constant, the system is time-invariant.

The transfer function is useful for graphical display as a locus or Bode diagram.

Examples

  • Electrical Engineering: filter circuits or amplifier
  • Mechanics: Gear
  • Thermodynamics: central heating, engine cooling
  • Converter between the types of systems mentioned above: Electric motor ( electricity power ), temperature sensor (temperature - current)
  • Mathematics (digital simulation): all kinds regulator eg PID controller

Example from mechanics

The free fall without friction is described by the differential equation

Path Z, the acceleration at the surface g and the mass of the falling object m. Transferred to the state space representation and shorten out of m one obtains the state differential equation

Where g is considered (usually a constant ) outside influence, and thus forms an ( the only one) member of the input vector. One is interested in an obvious way for the current position p and velocity v, is the initial equation

1 with a matrix as an output matrix and a zero matrix as penetration matrix, since the outputs are identical to the conditions. In this view, it is an LTI system, since all the matrices of the linear differential equation system constant, that is time-invariant, as is.

Considering, however, that the acceleration due to gravity is g depending on the distance of the centers of mass

With the mass of the Earth and the Earth's radius, so the system is not linearly dependent on the state z, so no LTI system.

If the gravitational acceleration g due to a usually much smaller height z with respect to the radius of the earth continue to be regarded as constant

But the friction between the mass and considered one of the air as much influential as a linear function of linear taken into account ( see also Free fall with Stokes friction), we obtain the state equation

With the coefficient of friction. If considered as a form constant of the falling object, it is still a LTI system.

Linear system Image (mathematics)
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