# Linear time-invariant system

As a linear time invariant system , also known as LZI system and LTI system ( English linear time-invariant system ), a system referred to when its behavior of both the property of linearity has, as is also independent of time shifts. This independence from time shifts is called time invariance .

The importance of these systems lies in the fact that they have particularly simple transformation equations and are therefore easily accessible for system analysis. Many technical systems, such as those in communications or control technology, have these properties, at least to a good approximation. In this context, a system can be, for example, a transmission system . Some LZI systems can be described by linear ordinary differential equations (or difference equations ) with constant coefficients.

## properties

### Linearity

Overlay principle

A system is called linear if every sum of any number of input signals leads to a sum of output signals proportional to it . The superposition principle , also known as the superposition principle, must apply. Mathematically, this is described by a transformation , which represents the transfer function of the system, between the input and output signals: ${\ displaystyle s_ {i} (t)}$${\ displaystyle g_ {i} (t)}$${\ displaystyle {\ mathcal {T}}}$

${\ displaystyle \ sum _ {i} a_ {i} g_ {i} (t) = \ sum _ {i} a_ {i} {\ mathcal {T}} \ left \ {s_ {i} (t) \ right \} = {\ mathcal {T}} \ left \ {\ sum _ {i} a_ {i} s_ {i} (t) \ right \}}$

The constant coefficients represent the individual proportionality factors. ${\ displaystyle a_ {i}}$

A signal is clearly applied to the input of the system and the reaction is observed. The reaction to a second signal is then examined independently of this. When applying an input signal that is the sum of the two previously examined signals, it can be determined that the reaction at the output corresponds to the addition of the two individual answers if the system is linear.

### Time invariance

Displacement principle

A system is called time-invariant if the following applies for any time shift by t 0 :

${\ displaystyle {\ mathcal {T}} \ left \ {s (t-t_ {0}) \ right \} = g (t-t_ {0})}$

For the time invariance, the output signal must maintain the time reference to the input signal and react identically. This principle is also called the displacement principle.

### Connection with convolution integral

The arbitrarily running input signal can be approximated by applying the superposition theorem and time invariance using a time sequence of individual square-wave pulses . At the limit for a square-wave pulse with a duration approaching 0, the output signal approaches a shape that only depends on the transfer function of the system, but no longer on the course of the input signal. ${\ displaystyle s (t)}$

Mathematically, these square-wave impulses , which tend towards zero, are described by Dirac impulses and the sums in the transformation equation become integrals . The input signal can be expressed as a convolution integral or with the symbol for the convolution operation as: ${\ displaystyle \ delta (t)}$${\ displaystyle s (t)}$${\ displaystyle *}$

${\ displaystyle s (t) = \ int _ {- \ infty} ^ {\ infty} s (\ tau) \ delta (t- \ tau) \ mathrm {d} \ tau = (s * \ delta) (t )}$

The output signal is via the convolution integral ${\ displaystyle g (t)}$

${\ displaystyle g (t) = \ int _ {- \ infty} ^ {\ infty} s (\ tau) h (t- \ tau) \ mathrm {d} \ tau = (s * h) (t)}$

linked to the input signal , representing the impulse response of the system. If the input signal is a Dirac impulse, the impulse response is. The transfer function of the system is the Laplace transform of the impulse response. ${\ displaystyle s (t)}$${\ displaystyle h (t)}$${\ displaystyle g (t) = h (t)}$

## Solution of linear time-invariant differential equations

An explicit linear system of differential equations is given in the form

{\ displaystyle {\ begin {aligned} {\ frac {dx (t)} {dt}} = A \, x (t) + b \, u (t), \, x (t = 0) = x_ { 0} \ end {aligned}}}

with the state vector , the system matrix , the input , the input vector and the initial condition . The solution consists of a homogeneous and a particulate part. ${\ displaystyle x (t) \ in \ mathbb {R} ^ {n}}$${\ displaystyle A \ in \ mathbb {R} ^ {nxn}}$${\ displaystyle u (t) \ in \ mathbb {R}}$${\ displaystyle b \ in \ mathbb {R} ^ {n}}$${\ displaystyle x_ {0} \ in \ mathbb {R} ^ {n}}$

### Homogeneous solution

The homogeneous differential equation is obtained by setting the input equal to zero.

{\ displaystyle {\ begin {aligned} {\ frac {dx (t)} {dt}} = A \, x (t), \, x (t = 0) = x_ {0} \ end {aligned}} }

This solution can now be described by a Taylor series representation:

{\ displaystyle {\ begin {aligned} x (t) = \ phi (t) x_ {0} = (E + \ phi _ {1} t + \ phi _ {2} t ^ {2} + ... + \ phi _ {n} t ^ {n} + ...) x_ {0} \ end {aligned}}}

where is the identity matrix. If one uses this solution of the equation above, one obtains: ${\ displaystyle E}$

{\ displaystyle {\ begin {aligned} {\ frac {d} {dt}} (\ phi (t) x_ {0}) & = A \, \ phi (t) \, x_ {0} \\ (\ phi _ {1} +2 \, \ phi _ {2} \, t + ... + n \, \ phi _ {n} \, t ^ {n-1} + ...) x_ {0} & = (A + A \ phi _ {1} t + A \ phi _ {2} t ^ {2} + ... + A \ phi _ {n} t ^ {n} + ...) \, x_ {0} \ end {aligned}}}

The unknown matrices can now be determined by comparing coefficients : ${\ displaystyle \ phi _ {n}}$

{\ displaystyle {\ begin {aligned} \ phi _ {1} & = A \\\ phi _ {2} & = {\ frac {1} {2}} A \, \ phi _ {1} = {\ frac {1} {2!}} A ^ {^ {2}} \\\ phi _ {3} & = {\ frac {1} {3}} A \, \ phi _ {2} = {\ frac {1} {3!}} A ^ {3} \\ & ... \\\ phi _ {n} & = {\ frac {1} {n!}} A ^ {n}. \ End {aligned }}}

The following notation is widely used for the fundamental matrix: ${\ displaystyle \ phi _ {n}}$

{\ displaystyle {\ begin {aligned} \ phi (t) = e ^ {At} = E + At + {\ frac {1} {2!}} A ^ {2} t ^ {2} + {\ frac { 1} {3!}} A ^ {3} t ^ {3} + ... + {\ frac {1} {n!}} A ^ {n} t ^ {n} + ... \ end { aligned}}}

### Particulate solution

Based on and follows: ${\ displaystyle u (t) \ neq 0}$${\ displaystyle x_ {0} = 0}$

{\ displaystyle {\ begin {aligned} {\ frac {d} {dt}} x (t) = A \, x (t) + b \, u (t) \ end {aligned}}}

The particular solution is sought in the form:

{\ displaystyle {\ begin {aligned} x_ {p} (t) = \ phi (t) \ xi (t) = e ^ {At} \ xi (t), \ end {aligned}}}

where is an unknown function vector with . From the two equations above it follows: ${\ displaystyle \ xi (t)}$${\ displaystyle \ xi (0) = 0}$

{\ displaystyle {\ begin {aligned} {\ frac {d} {dt}} x_ {p} (t) & = A \, x_ {p} (t) + b \, u (t) \\\ xi (t) {\ frac {d} {dt}} \ phi (t) + \ phi (t) {\ frac {d} {dt}} \ xi (t) & = A \, x_ {p} (t ) + b \, u (t) \\ A \, \ phi (t) \ xi (t) + \ phi (t) {\ frac {d} {dt}} \ xi (t) & = A \, x_ {p} (t) + b \, u (t) \\ A \, x_ {p} (t) + \ phi (t) {\ frac {d} {dt}} \ xi (t) & = A \, x_ {p} (t) + b \, u (t) \\\ end {aligned}}}

This can be used to determine: ${\ displaystyle \ xi (t)}$

{\ displaystyle {\ begin {aligned} {\ frac {d} {dt}} \ xi (t) = \ phi ^ {- 1} (t) bu (t) \\\ end {aligned}}}

By integration with the help of the properties of the fundamental matrix, one obtains:

{\ displaystyle {\ begin {aligned} \ phi (t) \ xi (t) & = \ phi (t) \ int _ {0} ^ {t} \ phi ^ {- 1} (\ tau) bu (\ tau) d \ tau \\ x_ {p} (t) & = \ int _ {0} ^ {t} \ phi (t- \ tau) bu (\ tau) d \ tau \\ x_ {p} (t ) & = \ int _ {0} ^ {t} e ^ {A (t- \ tau)} bu (\ tau) d \ tau \\\ end {aligned}}}

The solution to a linear time-invariant differential equation is:

{\ displaystyle {\ begin {aligned} x (t) = e ^ {At} x_ {0} (t) + \ int _ {0} ^ {t} e ^ {A (t- \ tau)} bu ( \ tau) d \ tau \\\ end {aligned}}}

## LZI systems in various forms of representation

The following part is limited to systems with a finite number of internal degrees of freedom.

### Time range

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

${\ displaystyle {\ begin {matrix} {\ dot {x}} (t) = Ax (t) + Bu (t) \\ y (t) = Cx (t) + Du (t) \ end {matrix} }}$

Here the vectors are input vector , state vector and output vector. If the matrices system matrix, input matrix, output matrix and pass-through matrix are constant, the system is linear and time-invariant. For the addition and multiplication of vectors and matrices see Matrix (Mathematics) . ${\ displaystyle u}$${\ displaystyle x}$${\ displaystyle y}$${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle C}$${\ displaystyle D}$

### Image area

For simpler continuous systems, in particular SISO systems (single input, single output systems) with only one input and one output variable, the description using a Laplace transfer function (in the Laplace " image range " or " frequency range ") is often chosen:

${\ displaystyle G (s) = {\ frac {Z (s)} {N (s)}}}$

Here the numerator polynomial is in , and the denominator polynomial is in . If all coefficients of both polynomials are constant, the system is time-invariant. The transfer function is suitable for stability analysis and for graphical representation as a locus or Bode diagram . ${\ displaystyle Z}$${\ displaystyle s}$${\ displaystyle N}$${\ displaystyle s}$

For discrete systems, a corresponding description is made using the z- transfer function (with the complex z-plane as the image area)

## Examples

• Electrical engineering: filter circuits or amplifiers
• Mechanics: transmission
• Thermodynamics : central heating, engine cooling
• Converter between the aforementioned system types: electric motor (current-power), temperature sensor (temperature-current)
• Mathematical (digital simulation): controllers of all kinds, e.g. B. PID controller

### Example from mechanics

The free fall without rubbing is described by the equation

${\ displaystyle m {\ ddot {z}} = mg}$

with the path , the acceleration on the surface of the earth and the mass of the falling object . Transferred to the state space representation and truncating from , the state differential equation is obtained ${\ displaystyle z}$${\ displaystyle g}$${\ displaystyle m}$${\ displaystyle m}$

${\ displaystyle {\ begin {bmatrix} {\ ddot {z}} \\ {\ dot {z}} \ end {bmatrix}} = {\ begin {bmatrix} 0 & 0 \\ 1 & 0 \ end {bmatrix}} {\ begin {bmatrix} {\ dot {z}} \\ z \ end {bmatrix}} + {\ begin {bmatrix} 1 \\ 0 \ end {bmatrix}} {\ begin {bmatrix} g \ end {bmatrix}} }$

where it is regarded as a (usually constant) external influence, and thus forms one (the only) term of the input vector. If one is naturally interested in the current position and speed , the starting equation is ${\ displaystyle g}$${\ displaystyle p}$${\ displaystyle v}$

${\ displaystyle {\ begin {bmatrix} v \\ p \ end {bmatrix}} = {\ begin {bmatrix} 1 & 0 \\ 0 & 1 \ end {bmatrix}} {\ begin {bmatrix} {\ dot {z}} \ \ z \ end {bmatrix}} + {\ begin {bmatrix} 0 \\ 0 \ end {bmatrix}} {\ begin {bmatrix} g \ end {bmatrix}}}$

with a 1 matrix as the output matrix and a zero matrix as the pass-through matrix, since the outputs are identical to the states. In this consideration it is a LZI system, since all matrices of the linear differential equation system are constant, i.e. time-invariant.

However, if you take into account that the acceleration due to gravity g depends on the distance between the centers of gravity

${\ displaystyle g = G \, {\ frac {m _ {\ mathrm {E}}} {(r _ {\ mathrm {E}} + z) ^ {2}}} = G \, {\ frac {m_ { \ mathrm {E}}} {r _ {\ mathrm {E}} ^ {2} + 2r _ {\ mathrm {E}} z + z ^ {2}}}}$

with the earth's mass and the earth's radius , the system is non-linearly dependent on the state z, i.e. no LZI system. ${\ displaystyle m _ {\ mathrm {E}}}$${\ displaystyle r _ {\ mathrm {E}}}$

The acceleration due to gravity is still regarded as constant due to a mostly much smaller height compared to the earth's radius${\ displaystyle g}$${\ displaystyle z}$${\ displaystyle z \ ll r _ {\ mathrm {E}}}$

${\ displaystyle g \ approx G \, {\ frac {m _ {\ mathrm {E}}} {r _ {\ mathrm {E}} ^ {2}}}}$

but the friction between the considered mass and air is considered to be much more influential in linear dependence on linear (see also case with air resistance # case with Stokes friction ), one obtains the state differential equation ${\ displaystyle {\ dot {z}}}$

${\ displaystyle {\ begin {bmatrix} {\ ddot {z}} \\ {\ dot {z}} \ end {bmatrix}} = {\ begin {bmatrix} - \ beta & 0 \\ 1 & 0 \ end {bmatrix} } {\ begin {bmatrix} {\ dot {z}} \\ z \ end {bmatrix}} + {\ begin {bmatrix} 1 \\ 0 \ end {bmatrix}} {\ begin {bmatrix} g \ end { bmatrix}}}$

with the coefficient of friction . If considered as the shape constant of the falling object, it is still an LZI system. ${\ displaystyle \ beta}$${\ displaystyle \ beta}$