In this section we give some numerical examples to illustrate our results in this chapter. Our conditions for BIBO stability depend on the product aτ, where a is the feedback gain and τ is a positive constat time delay. In the following examples we fix τ = 1 and investigate the boundedness of the solutions for cases when we changea.
Throughout this section we use the approximation method described in Section 2.7with dis-cretization parameter h = 0.01. In most of the cases below we plotted the numerical solutions on the interval [0,100]. Increasing the time interval gives similar figures, so these numerical simulations indicate that the solutions are bounded or unbounded on [0,∞).
In the following examples we assume that the output of the system is the whole state x(t), t≥0. The next example shows that if our conditions of Theorem4.2.3hold, then the solution is BIBO stable.
Example 4.4.1. We consider the scalar linear equation
˙
z(t) =−az(t−1) +bz(t) +r(t), t≥0 (4.4.1) with initial condition
z(t) =ϕ(t) −1≤t≤0,
wherea, b∈R,r ∈L1∞ is the reference input and ϕ(t) is a continuous function on [−1,0].
We consider (4.4.1) in the form
˙
z(t) =bz(t) +u(t), t≥0,
whereu(t) =−a z(t−1)+r(t) is the control law. The solution of the uncontrolled equation ˙z(t) = bz(t) is unbounded for b >0, so we investigate control laws which guarantee the boundedness of the solutions.
In the first 5 figures we fix the initial functionϕ(t) = cos 3t. First we takea= 1.58,b= 0.001 and a constant reference inputr(t) = 10. In this case condition (4.2.14) is not satisfied, and we can observe in Figure4.1that the solution of (4.4.1) is unbounded. In the case of Figure 4.2the condition a < π2 is satisfied, so Theorem 4.2.3implies the boundedness of the solution of (4.4.1) for sufficiently smallb. If condition (4.2.13) holds, then Theorem 4.2.3implies the boundedness of the solution. In [47] the following estimate was given to approximate $:
$:=
Z ∞ 0
|w(t)|dt≤$1 := α2o+βo2
aα2o , (4.4.2)
Chapter4. BIBO Stability of Controlled Nonlinear Systems with Time Delay 40
Figure 4.1: a= 1.58,b= 0.001,r(t) = 10, ϕ(t) = cos 3t
Figure 4.2: a= 1.57,b= 0.001,r(t) = 10, ϕ(t) = cos 3t
Figure 4.3: a= 1.57,b= 0.01, r(t) = 10, ϕ(t) = cos 3t
Figure 4.4: a = 1.3, b = 0.1, r(t) = 10, ϕ(t) = cos 3t
whereλo =αo+iβo is the dominant characteristic root of ˙z(t) =−az(t−τ), i.e., the solution of
λ=−ae−λ, λ∈C (4.4.3)
with greatest real part andαo <0 andβo∈[0,2τπ ). Also, in [51] the following improved estimate was given
$:=
Z ∞ 0
|w(t)|dt≤$2:= 1 a
1− 2βoeαoπ2βo αo
1−eαoπβo ,
, (4.4.4)
where αo <0 and βo ∈ (0,2τπ). Using Maple we computed the dominant root of (4.4.3) corre-sponding toa= 1.57. The numerical solution isλo =−0.00036083807 +i1.570566577, using this root we get from (4.4.2) the estimate $1 = 12066695.85 and from (4.4.4), $2 = 7681900.002.
Therefore Theorem4.2.3yields the boundedness of the solution if kbk∞< 1
$1 = 0.8287272775×10−7 or kbk∞< 1
$2 = 0.1301761283×10−6.
Chapter4. BIBO Stability of Controlled Nonlinear Systems with Time Delay 41
Figure 4.5: a= 1.3,b= 0.1,r(t) = 100t1+t + 10e−t,ϕ(t) = cos 3t
Figure 4.6: a= 1.3,b= 0.1,r(t) = 100t1+t+ 10e−t,ϕ(t) = 10−t−11
Figure 4.7: a= 0.3, b= 0.25,r(t) =100t1+t+ 10e−t,ϕ(t) = 10−t−11
Figure 4.8: a = 0.3, b = 0.25, r(t) = t, ϕ(t) = 10−t−11
We emphasize that they are only sufficient conditions. Our numerical investigation shows that forb= 0.001 the solution is bounded (see Figure4.2), but forb= 0.01 we lose the boundedness, as Figure 4.3demonstrates.
Second, we fix a = 1.3 and b = 0.1 such that conditions (4.2.13) and (4.2.14) are satisfied.
Figures 4.4 and 4.5 show the case when we change the reference input r(t) = 10 to r(t) =
100t
1+t+10e−t. Both cases the solutions converge to a limit, the qualitative behaviour of the solution is the same using different reference input, but the value of the limit depends on the selection of r. Also for the change of the initial condition the solution is still bounded and convergent as Figure4.6 shows.
Finally, if b < a < 1/e, for example a = 0.3 and b = 2.5, then the solution converges monotonically to a limit for anyϕand for all constantr(t) see Figure (4.7), but Figures4.8and 4.9illustrate that if limt→∞r(t) =∞ ( for exampler(t) =t) orb≥a, then the solution of (4.4.1)
is unbounded.
Chapter4. BIBO Stability of Controlled Nonlinear Systems with Time Delay 42
Figure 4.9: a = 0.3, b = 0.3, r(t) = 100t1+t + 10e−t, ϕ(t) = 10−t−11
Figure 4.10: a = 1.58, b = 0.001, r(t) = 10,ϕ(t) = cos 3t
Figure 4.11: a= 1.57, b = 0.001, r(t) = 10,ϕ(t) = cos 3t
The following example illustrates our result of Theorem4.2.1.
Example 4.4.2. We consider the sub-linear equation
˙
z(t) =−az(t−1) +b|z(t)|12 +r(t), t≥0 (4.4.5) with initial condition
z(t) =ϕ(t) −1≤t≤0,
where a, b ∈ R, r ∈ L1∞ is the reference input and ϕ(t) is a continuous function on [−1,0]. In Theorem4.2.1there is no condition onbto get BIBO stability of the solutions. We assume that the condition (4.2.1) is not satisfied and if we take in this casea= 1.58, b= 0.001,r(t) = 10 and ϕ(t) = cos 3t, then the solution of (4.4.5) is unbounded as we can see in Figure 4.10. But if we decrease a such that a < π/2, i.e., the condition (4.2.1) is satisfied, then the solution becomes
Chapter4. BIBO Stability of Controlled Nonlinear Systems with Time Delay 43
Figure 4.12: a = 1.3, b = 3, r(t) = 10, ϕ(t) = cos 3t
Figure 4.13: a = 1.3, b = 3, r(t) = 10, ϕ(t) = 10 +et
Figure 4.14: a = 1.3, b = 3, r(t) = 5 + 10e−t,ϕ(t) = 10 +et
Figure 4.15: a = 1.3, b = 3, r(t) = 10, ϕ(t) = 10 +et
bounded for allbsee Figures4.11and4.12. Also if we change the initial condition or/and reference input, then the solution is oscillatory and it is still bounded as Figures 4.13 and 4.14 show. If a≤ 1e then the solution of (4.4.5) is convergent for allb, bounded functionr(t) and for any initial
condition (see Figure4.15).
The following example illustrates our result of Theorem4.2.6.
Example 4.4.3. We consider the super-linear equation
˙
z(t) =−az(t−1) +bz3(t) +r(t), t≥0 (4.4.6) with initial condition
z(t) =ϕ(t) −1≤t≤0,
where a, b ∈ R, r ∈ L1∞ is the reference input and ϕ(t) is a continuous function on [−1,0].
Here in this example to get the boundedness or locally BIBO stability of the solutions of (4.4.6),
Chapter4. BIBO Stability of Controlled Nonlinear Systems with Time Delay 44 it is not enough that the condition (4.2.14) holds, but we need extra conditions on the initial condition ϕ(t) and the reference input r(t) should be small enough (see Theorem 4.2.6). If we take a= 1.57, b= 0.0001, r(t) = 1 and ϕ(t) = cos 3t, then the solution is bounded similarly to the results in Examples 4.4.1 and 4.4.3 (see Figure 4.16). Moreover if we fix a, b and increase the initial condition or/and input reference we lose the boundedness as Figures 4.17 and 4.18 illustrate. In the case whena= 1.3,b= 0.01,ϕ(t) = cos 3tandr(t) = 3.65 the solution of (4.4.6) is bounded as in Figure 4.19, and if we increase r(t) we lose the boundedness (see Figure 4.20 where the critical constant value is used for r(t) where the boundedness is lost). Figures 4.21 and 4.22illustrate the case whenb < a < 1e we lose the boundedness for solution of (4.4.6) if we
change input reference or/and the initial condition.
Figure 4.16: a= 1.57, b= 0.0001, r(t) = 1,ϕ(t) = cos 3t
Figure 4.17: a= 1.57,b= 0.0001, r(t) = 14 + 10e−t,ϕ(t) = cos 3t
Figure 4.18: a= 1.57, b= 0.0001, r(t) = 1,ϕ(t) = 10
Figure 4.19: a = 1.3, b = 0.01, r(t) = 3.65,ϕ(t) = cos 3t
Chapter4. BIBO Stability of Controlled Nonlinear Systems with Time Delay 45
Figure 4.20: a = 1.3, b = 0.01, r(t) = 3.65184021809,ϕ(t) = cos 3t
Figure 4.21: a= 0.3,b= 0.001,r(t) = 2, ϕ(t) = cos 3t
Figure 4.22: a= 0.3,b= 0.001,r(t) = 2.015,ϕ(t) = cos 3t
Chapter 5
Boundedness of Solutions of Nonlinear Volterra Difference Equations with Constant Delay
In this chapter we study the boundedness of the solutions of nonlinear VDEs with constant delay.
Moreover we give some special cases and examples to illustrate our results on the boundedness.
5.1 Nonlinear Volterra difference equations
In this section we obtain sufficient conditions for the boundedness of the solutions of nonlinear VDEs.
We consider the nonlinear VDE x(n+ 1) =
n
X
j=0
f(n, j, x(j−σ(j))) +h(n), n≥0, (5.1.1)
with the initial condition
x(n) =ψ(n), −τ ≤n≤0, (5.1.2)
whereτ is a positive integer constant and the following conditions are satisfied.
(B1) The function f(n, j,·) :Rd→Rd is a given mapping for any fixed 0≤j≤n.
(B2) For any 0≤j≤n, there exists an ai(n, j)∈R+ such that fori= 1, . . . , d
|fi(n, j, x)| ≤ai(n, j)φ(kxk), x∈Rd (5.1.3) hold with a monotone non-decreasing mappingφ:R+→R+.
46
Chapter5. Boundedness of Solutions of Nonlinear VDEs with Constant Delay 47 (B3) h(n) = (h1(n), . . . , hd(n))T ∈Rd and 0≤σ(n)≤τ, n≥0.
(B4) ψ∈S [−τ,0],Rd .
For some recent literature on the boundedness of the solutions of linear VDEs, we refer the readers to [55,71,72]. We give some applications of our main result for sub-linear, linear, and super-linear VDEs. We study the boundedness of solutions of convolution cases and we get a result parallel to the corresponding result of Lipovan [87] for integral equation. Also we give some examples to illustrate our main results.