Subsections

• 6.1 Linear time-varying systems
• 6.2 Linear time-invariant systems
• 6.3 Linear time-invariant systems with exponential distributed delays

6 Linear systems with delays

6.1 Linear time-varying systems

To simulate the linear system with delays

$$\dot{x} \sum_{i=1}^{m} \tau_{i}^{} \int\limits_{-\tau (t)}^{0}$$ (6.1)

(here A₀(t), A_i(t) (i = 1,..., m) are n×n matrices with piece-wise continuous elements, G(t, s) is n×n matrix with piece-wise continuous elements, B(t) is an n-dimensional vector with piece-wise continuous elements) one can use the function dde45lin.

The conditional representation of (6.1) is

$$\dot{x} \sum_{i=1}^{m} \tau_{i}^{} \int\limits_{-\tau (t)}^{0}$$ (6.2)

so, to simulate the system it is necessary to define the following finite number of parameters:

- matrices A₀(t), A_i(t) (i = 1,..., m), B(t),

- a matrix-function G(t, s),

- an initial time-moment t₀,

- an initial point x⁰,

- an initial prehistory y⁰( ^. ) = {y⁰(s), - $\tau$ $\leq$ s < 0}.

6.2 Linear time-invariant systems

To simulate the linear time-invariant system with delay

$$\dot{x} \tau \tau \int\limits_{-\tau}^{0}$$ (6.3)

with respect to the basic initial functions one can use the function test.

Here A₀, A_$\tau$ are constant n×n matrices, G(s) is n×n matrix with piece-wise continuous on [- $\tau$, 0] elements.

The conditional representation of (6.3) is

$$\dot{x} \tau \tau \int\limits_{-\tau}^{0}$$ (6.4)

and to simulate the system it is necessary to define the following finite number of parameters:

- matrices A₀, A_$\tau$,

- a matrix-function G(s),

- an initial point x⁰,

- an initial prehistory y⁰( ^. ) = {y⁰(s), - $\tau$ $\leq$ s < 0}.

Remark 6.1. For time-invariant system (6.3) we consider the initial time-moment t₀ equals to zero, i.e. t₀ = 0. $\Box$

Remark 6.2. The matrix-function G(s) can be described using M-file. $\Box$

6.3 Linear time-invariant systems with exponential distributed delays

In many cases linear time-invariant systems with delays have the form

$$\dot{x} \tau \tau \int\limits_{-\tau}^{0}$$ (6.5)

where A₀, A_$\tau$, D₀, D₁ and D₂ are constant n×n matrices.

The conditional representation of (6.5) is

$$\dot{x} \tau \tau \int\limits_{-\tau}^{0}$$ (6.6)

One can simulate such systems with respect to the basic initial functions using teste.