Subsections

• 5.1 Separation of finite dimensional and infinite dimensional components in DDE
• 5.2 Conditional representation of DDE
• 5.3 Description of DDE by a finite number of functions and integrals
• 5.4 Delay operators. Description of DDE in terms of delay operators
• 5.5 Initial condition for DDE

5 Representation of systems in the Toolbox

5.1 Separation of finite dimensional and infinite dimensional components in DDE

Systems with delays

$$\dot{x} \tau \le$$ (5.1)

are infinite dimensional systems because of the presence of the functional component { x(t + s),  - $\tau$$\le$s < 0 } which characterizes delay. So the right-hand side of system (5.1) is a mapping

$$\theta \bf R^{n}_{} \tau \to \bf R^{n}_{}$$ (5.2)

Remark 5.1. The central concept that we use for describing and modeling of systems with delays consists of separation (distinguishing) of finite dimensional and infinite dimensional components in the structure of systems. For this reason we use for DDE the notation (5.1) and consider the state of DDE as the pair x_t = {x(t); x(t + s),  - $\tau$$\le$s < 0} $\in$ H = $\bf R^{n}_{}$×Q[- $\tau$, 0). Note, one of the conventional notations for DDE is

$$\dot{x}$$(t) = $$\cal {F}$$[t, x(t + $$\hat{s}$$)] ,  - $$\tau$$$$\le$$$$\hat{s}$$$$\le$$0 ,

however for our aims it is more convenient to use (5.1). $\Box$

The right-hand side of system

$$\dot{x} \tau_{1}^{} \int\limits_{-\tau_2}^{0} \gamma$$ (5.3)

is the mapping

$$\equiv \tau_{1}^{} \int\limits_{-\tau_2}^{0} \gamma$$ (5.4)

here F[ ^. , ^. , ^. , ^. ] : $\bf R$×$\bf R^{n}_{}$×$\bf R^{n}_{}$×$\bf R^{m}_{}$$\to$$\bf R^{n}_{}$; $\gamma$( ^. , ^. ) is m×n matrix with continuous elements on $\bf R$×[- $\tau$, 0]; $\tau_{1}^{}$ and $\tau_{2}^{}$ are positive constants.

The right-hand side of system

$$\dot{x} \tau_{1}^{} \int\limits_{-\tau_2}^{0} \phi$$ (5.5)

is the mapping

$$\equiv \tau_{1}^{} \int\limits_{-\tau_2}^{0} \phi$$ (5.6)

here F[ ^. , ^. , ^. , ^. ] : $\bf R$×$\bf R^{n}_{}$×$\bf R^{n}_{}$×$\bf R^{m}_{}$$\to$$\bf R^{n}_{}$; $\phi$( ^. ) : $\bf R^{n}_{}$$\to$$\bf R^{m}_{}$; $\tau_{1}^{}$ and $\tau_{2}^{}$ are positive constants.

5.2 Conditional representation of DDE

For structural presentation of time-delay systems it is convenient to use the following conditional representation

$$\dot{x}$$ (5.7)

i.e. just write in the right-hand side of the equations the mapping f (t, x, y( ^. )) (5.2) without indication of solutions.

Remark 5.2. Remember that for ODE

$$\dot{x}$$(t) = g(t, x(t))$$\left(\vphantom{ g(t,x): {\bf R} \times {\bf R}^n \to {\bf R}^n }\right.$$ g(t, x) : $$\bf R$$×$$\bf R^{n}_{}$$$$\to$$$$\bf R^{n}_{}$$ $$\left.\vphantom{ g(t,x): {\bf R} \times {\bf R}^n \to {\bf R}^n }\right)$$

the conditional representation is

$$\dot{x}$$ (5.8)

i.e. the argument t is not pointed out in state variable x(t), and we just write the mapping g(t, x) (without indication of solutions) in the right part of the equation. $\Box$

The conditional representation of system (5.3) is

$$\dot{x} \tau_{1}^{} \int\limits_{-\tau_2}^{0} \gamma$$ (5.9)

and the conditional representation of system (5.5) is

$$\dot{x} \tau_{1}^{} \int\limits_{-\tau_2}^{0} \phi$$ (5.10)

5.3 Description of DDE by a finite number of functions and integrals

System (5.7) is the infinite dimensional system, and the problem of its simulating consists of describing f (t, x, y( ^. )) by finite number of parameters, because for computer simulation finite algorithms with finite number of input parameters can be used.

Analyzing the structure of DDE one can see that in concrete cases right-hand sides of such equations are combinations of finite dimensional functions and integrals.

The right-hand sides of systems (5.3), (5.5) can be described just by finite number of functions. For example, to describe system (5.3) (or (5.5)) we should set the mappings F[ ^. , ^. , ^. , ^. ], $\tau_{1}^{}$( ^. ), $\tau_{2}^{}$( ^. ) and $\gamma$( ^. , ^. ) (or $\phi$( ^. , ^. )).

For presentation DDE in Toolbox it is convenient to use their structural presentation in terms of delay operators.

5.4 Delay operators. Description of DDE in terms of delay operators

One can see that in many DDE delays are described by the following mappings:

a)

linear operator of pure delay

$$\cal {P} \tau \tau \bf R \to \tau$$ (5.11)

b)

linear operator of distributed delay

$$\cal {P} \int\limits_{-\tau (t)}^{0} \lambda$$ (5.12)

c)

nonlinear operator of distributed delay

$$\cal {P} \int\limits_{-\tau (t)}^{0} \phi$$ (5.13)

here $\lambda$(t, s) is an n×n matrix with elements continuous on $\bf R$×[- $\tau$, 0]; $\phi$( ^. ) : $\bf R^{n}_{}$$\to$$\bf R$.

Mappings (5.11), (5.12) and (5.13) are called delay operators [18]. Note, to define the delay operator (5.12) it is necessary to set two functions: $\lambda$(t, s) and $\tau$(t). To define the delay operator (5.13) it is necessary to set two functions: $\phi$( ^. ) and $\tau$(t).

In specific cases delay differential equations can be presented, as a rule, as combinations of finite dimensional functions and delay operators. For example, the system

$$\dot{x}$$(t) = G[t, x(t), x(t - $$\tau$$(t))]

can be written as

$$\dot{x} \left[\vphantom{t,x,{\cal P}[t,y(\cdot)]}\right. \cal {P} \left.\vphantom{t,x,{\cal P}[t,y(\cdot)]}\right]$$ (5.14)

where $\cal {P}$ is the pure delay operator (5.11) defined on $\bf R$×Q(- $\tau$, 0]. The system

$$\dot{x}$$(t) = G[t, x(t), x(t - $$\tau$$(t)),$$\int\limits_{-\tau}^{0}$$$$\phi$$(x(t + s)) ds]

can be presented in the form

$$\dot{x}$$ = G$$\left[\vphantom{t,x,{\cal P}[t,y(\cdot)], {\cal D}[y(\cdot)]}\right.$$t, x,$$\cal {P}$$[t, y( ^. )],$$\cal {D}$$[y( ^. )]$$\left.\vphantom{t,x,{\cal P}[t,y(\cdot)], {\cal D}[y(\cdot)]}\right]$$ ,

where $\cal {P}$ is a pure delay operator (5.11) and $\cal {D}$ is a distributed delay operator of the form (5.13).

Actually one can use different types of delay operators. In the present version of Toolbox we realized numerical algorithms for simulating nonlinear time-delay systems with delay operators of the forms (5.11), (5.12) and (5.13).

5.5 Initial condition for DDE

We consider initial conditions of systems with delays as the pair h⁰ = {x⁰, y⁰( ^. )} $\in$ H of a vector x⁰ $\in$ $\bf R^{n}_{}$ and a function y⁰( ^. ) $\in$ Q[- $\tau$, 0), i.e. at an initial time t₀ a solution x(t) satisfies the conditions

x(t₀) = x₀ , (5.15)

$$\tau \leq$$ (5.16)

So in order to simulate systems with delays using functions of Toolbox it is necessary, as a rule, to set into M-files an initial data as the pair {x⁰, y⁰( ^. )}.

The numerical algorithms which are realized in Toolbox guarantee accurate results for continuous initial functions {x⁰, y⁰( ^. )} $\in$ C[- $\tau$, 0], i.e. when

x⁰ = $$\lim_{s \to 0-}^{}$$y⁰(s) .

Nevertheless using functions of Toolbox one can simulate DDE with respect to piece-wise continuous initial data {x⁰, y⁰( ^. )} $\in$ $\bf R^{n}_{}$×Q[- $\tau$, 0). However in this case caution should be exercised.

Remark 5.3. In the present version of Toolbox the initial function y⁰( ^. ) should be an elementary function and cannot be described using M-file. $\Box$