**7. Stability analysis**

It is worth noticing that the two first terms in (41) can satisfy *<sup>I</sup>* <sup>−</sup> *h Jtn <sup>J</sup>* −1 *<sup>n</sup> Kp ηn* 2 −*η* 2 *<sup>n</sup>* < <sup>0</sup> by proper selection of *Kp* and the following two terms can fulfill *<sup>I</sup>* <sup>−</sup> *hM*<sup>−</sup><sup>1</sup> *Kv vn* 2 −*v* 2 *<sup>n</sup>* < 0 by proper selection of *Kv* and so the sign of <sup>∆</sup>*Qtn* for any trajectories <sup>∽</sup> *<sup>η</sup>tn* and <sup>∽</sup> *vtn* and initial conditions of them would be depending of error functions *f*∆*Q*<sup>1</sup> and *f*∆*Q*<sup>2</sup> only.

According to (18) and (19), we can argue that *f*∆*Q*<sup>1</sup> in (31) and *f*∆*Q*<sup>2</sup> in (42) are consistent with *εη* and *εv,* it is, in absence of disturbances, they go to zero for *h* tending to zero. On the other hand, by existing disturbances *δηtn* and *δvtn* , and any value of *h*, the maximal global errors are proportional to the disturbances. Clearly, the more extensive the pure dead time *Td*, the larger the magnitude of *f*∆*Q*<sup>1</sup> and *f*∆*Q*<sup>2</sup> , and this dependence is exponential.

To focus the stability problem in more detail, let first the controller matrices *Ui*'s to take the values *U*<sup>∗</sup> *<sup>i</sup>* 's. So, using these constant system matrices in (1), a fixed controller can be designed.

For this particular controller we consider the resulting ∆*Q*<sup>∗</sup> *tn* from (30) accomplishing

$$\begin{split} \Delta Q\_{\rm li}^{\*} &= \left( \left( I - h \boldsymbol{I}\_{\rm li} \boldsymbol{I}\_{\rm u}^{-1} \boldsymbol{K}\_{p} \right) \boldsymbol{\eta}\_{\rm u} \right)^{2} - \boldsymbol{\eta}\_{\rm u}^{2} + \left( \left( I - h \underline{\boldsymbol{M}}^{-1} \boldsymbol{K}\_{\rm v} \right) \boldsymbol{\nu}\_{\rm u} \right)^{2} - \boldsymbol{\upsilon}\_{\rm u}^{2} - \left( \boldsymbol{I}\_{\rm li}^{-1} \boldsymbol{\eta}\_{\rm I} \boldsymbol{\eta}\_{\rm Iu} - \boldsymbol{I}\_{\rm I\_{\rm u}+1}^{-1} \boldsymbol{K}\_{p} \boldsymbol{\tilde{\eta}}\_{\rm Iu} \right)^{2} - \boldsymbol{\eta}\_{\rm I\_{\rm u}}^{2} + \boldsymbol{\upsilon}\_{\rm u}^{2} \\ &+ \boldsymbol{f}\_{\rm A}^{\*} [\boldsymbol{\varepsilon}\_{\rm uq} \boldsymbol{\varepsilon}\_{\rm vq} \boldsymbol{\varepsilon}\_{\rm uq} \boldsymbol{\delta} \boldsymbol{\eta}\_{\rm u} \boldsymbol{\delta} \boldsymbol{\nu}\_{\rm u}, \boldsymbol{\mu}^{-1} \boldsymbol{\underline{M}}]. \end{split} \tag{51}$$

where *f* <sup>∗</sup> <sup>∆</sup>*Qn* is the sum of *f*∆*Q*<sup>1</sup> and *f*∆*Q*<sup>2</sup> obtained from (31) and (42), but with the condition *pn*=*r<sup>n</sup>*

$$f\_{\Delta Q\_{\mathfrak{n}}}^{\*} = f\_{\Delta Q\_{\mathfrak{l}\_{\mathfrak{n}}}}[\mathfrak{p}\_{\mathfrak{n}} = \mathfrak{r}\_{\mathfrak{n}}] + f\_{\Delta Q\_{\mathfrak{2}\_{\mathfrak{n}}}}[\mathfrak{p}\_{\mathfrak{n}} = \mathfrak{r}\_{\mathfrak{n}}].\tag{52}$$

Later, a norm of *f* <sup>∗</sup> <sup>∆</sup>*Qn* will be indicated.

12 Advances in Discrete Time Systems

 *M*<sup>−</sup>*<sup>T</sup> M*<sup>−</sup><sup>1</sup> − *M*<sup>−</sup>*<sup>T</sup>*

<sup>∆</sup>*Qtn* (*U*′

This feature will be useful in the next analysis.

*<sup>∂</sup>Ui* .

It is worth noticing that the two first terms in (41) can satisfy

For this particular controller we consider the resulting ∆*Q*<sup>∗</sup>

by proper selection of *Kp* and the following two terms can fulfill

0 by proper selection of *Kv* and so the sign of <sup>∆</sup>*Qtn* for any trajectories <sup>∽</sup>

conditions of them would be depending of error functions *f*∆*Q*<sup>1</sup> and *f*∆*Q*<sup>2</sup> only.

larger the magnitude of *f*∆*Q*<sup>1</sup> and *f*∆*Q*<sup>2</sup> , and this dependence is exponential.

*M*<sup>−</sup><sup>1</sup> 

*<sup>i</sup>*) <sup>−</sup> <sup>∆</sup>*Qtn* (*U*′′

≥ 0 and *δM*−<sup>1</sup> =

Moreover, with (46) in mind we can conclude for any pair of values of *Ui*, say *<sup>U</sup>*′

*<sup>i</sup>* ) ≤

≤

In summary, the practical laws which conform the digital adaptive controller are

*Uin*<sup>+</sup><sup>1</sup> ∆ = *Uin* − <sup>Γ</sup>*<sup>i</sup>*

sampled state functions obtained from (44) after extracting of the common factors *δM*−<sup>2</sup> and

It is worth noticing that <sup>∆</sup>*Qtn* and <sup>∆</sup>*Qtn* , satisfy convexity properties in the space of elements

*<sup>∂</sup>*∆*Qtn* (*U*′′

*∂Ui*

*<sup>∂</sup>*∆*Qtn* (*U*′′

*∂Ui*

Finally, it is seen from (45) that also here the noisy measures *ηδtn* and *vδtn* will propagate into

According to (18) and (19), we can argue that *f*∆*Q*<sup>1</sup> in (31) and *f*∆*Q*<sup>2</sup> in (42) are consistent with *εη* and *εv,* it is, in absence of disturbances, they go to zero for *h* tending to zero. On the other hand, by existing disturbances *δηtn* and *δvtn* , and any value of *h*, the maximal global errors are proportional to the disturbances. Clearly, the more extensive the pure dead time *Td*, the

To focus the stability problem in more detail, let first the controller matrices *Ui*'s to take

*<sup>i</sup>* 's. So, using these constant system matrices in (1), a fixed controller can be

*i* )

*i* )

*<sup>∂</sup>*∆*Qtn ∂Ui*

*I* − *h Jtn J*

−1 *<sup>n</sup> Kp ηn* 2 −*η* 2 *<sup>n</sup>* < <sup>0</sup>

> *Kv vn* 2 −*v* 2 *<sup>n</sup>* <

> > *vtn* and initial

*<sup>η</sup>tn* and <sup>∽</sup>

*I* − *hM*<sup>−</sup><sup>1</sup>

*tn* from (30) accomplishing

 *U*′ *<sup>i</sup>* <sup>−</sup> *<sup>U</sup>*′′ *i* 

 *U*′ *<sup>i</sup>* <sup>−</sup> *<sup>U</sup>*′′ *i* 

 *M*<sup>−</sup><sup>1</sup> − *M*<sup>−</sup><sup>1</sup> 

≥ 0. Here *Ain* and *Bin* are

≤ (48)

. (49)

. (50)

*<sup>i</sup>* and *<sup>U</sup>*′′

*<sup>i</sup>* , it

and *δM*−<sup>2</sup> =

of the *Ui*'s.

is valid

*δM*−<sup>1</sup> , respectively.

the adaptive laws *<sup>∂</sup>*∆*Qtn*

**7. Stability analysis**

the values *U*<sup>∗</sup>

designed.

Since *<sup>ε</sup>ηn*+<sup>1</sup> *, <sup>ε</sup>vn*<sup>+</sup><sup>1</sup> , *<sup>δ</sup>η*, *<sup>δ</sup><sup>v</sup>* <sup>∈</sup> *<sup>l</sup>*∞. Additionally, *<sup>M</sup>*<sup>−</sup><sup>1</sup> *M* is bounded. Thus, one concludes *f* <sup>∗</sup> <sup>∆</sup>*Qn* ∈ *l*∞ as well.

So, it is noticing that ∆*Q*<sup>∗</sup> *tn* <sup>&</sup>lt; 0, at least in an attraction domain equal to

$$\mathcal{B} = \left\{ \stackrel{\frown}{\eta}\_{t\_{\mathfrak{t}}} \stackrel{\frown}{\mathfrak{v}}\_{t\_{\mathfrak{t}}} \in \mathcal{R}^{\mathfrak{G}} \cap \mathcal{B}\_{0}^{\*} \right\},\tag{53}$$

with B<sup>∗</sup> <sup>0</sup> a residual set around zero

$$\mathcal{B}\_0^\* = \left\{ \stackrel{\frown}{\eta}\_{t\_n} \stackrel{\smile}{\mathcal{D}}\_{t\_n} \in \mathcal{R}^6 / \Delta \mathcal{Q}\_{t\_n}^\* - f\_{\Delta Q\_n}^\* \le 0 \right\} \tag{54}$$

and with the design matrices satisfying the conditions

$$2\frac{J\_n f\_{t\_n}^{-1}}{h} > K\_p > 0\tag{55}$$

$$\frac{2}{h}I > K\_v^\* \ge 0,\tag{56}$$

which is equivalent to

$$
\frac{2}{h}M \ge \frac{2}{h}\underline{M} > K\_v \ge 0.\tag{57}
$$

The residual set B<sup>∗</sup> <sup>0</sup> depends not only on *εηn*+<sup>1</sup> and *εvn*<sup>+</sup><sup>1</sup> and the measure noises *δηtn* and *δvtn* , but also on *M*<sup>−</sup><sup>1</sup> *M*. In consequence, B<sup>∗</sup> <sup>0</sup> becomes the null point at the limit when *h* → 0, *δηtn* , *δvtn* → 0 and *M* = *M*.

#### **7.1. Stability proof**

The problem of stability of the adaptive control system is addressed in the sequel. Let a Lyapunov function be

$$V\_{t\_n} = Q\_{t\_n} + \frac{1}{2} \sum\_{i=1}^{15} \sum\_{j=1}^{6} \left( \stackrel{\sim}{\tilde{u}\_j}\_j \right)\_{i\_{n+1}} \Gamma\_i^{-1} \left( \stackrel{\sim}{\tilde{u}\_j}\_j \right)\_{i\_{n+1}} - \tag{58}$$
 
$$- \frac{1}{2} \sum\_{i=1}^{15} \sum\_{j=1}^{6} \left( \stackrel{\sim}{\tilde{u}\_j}\_j \right)\_{i\_n} \Gamma\_i^{-1} \left( \stackrel{\sim}{\tilde{u}\_j}\_j \right)\_{i\_n} \ . \tag{59}$$

If on the contrary 4*a*¯*c*¯*<sup>n</sup>* > *b*

and yields a new functional ∆*Q*∗∗

that the positive quantity

<sup>0</sup> results larger than <sup>B</sup><sup>∗</sup>

also on the magnitude of

**7.2. Variable boundness**

where ∆*Q*<sup>∗</sup>

B∗∗

*f* ∗

<sup>∆</sup>*Qn* in (52).

Qualitatively speaking, when

*<sup>I</sup>*−*h Jtn <sup>J</sup>*

−1 *<sup>n</sup> Kp ηn* 2 +*η* 2 *<sup>n</sup>*−

*f* ∗ <sup>∆</sup>*Qn* <sup>&</sup>lt;<sup>−</sup> *T*

*<sup>τ</sup>*2*n*=−<sup>1</sup> 2*a M*<sup>−</sup><sup>1</sup>

<sup>∆</sup>*Vtn* <sup>≤</sup> <sup>∆</sup>*Q*∗∗

*tn* is (51) with a real root of (40) and <sup>B</sup>∗∗

*c*¯*<sup>n</sup>* − <sup>1</sup> <sup>4</sup>*h*<sup>2</sup> *b T nbn* 

*c*¯*<sup>n</sup>* − <sup>1</sup> <sup>4</sup>*h*<sup>2</sup> *b T nbn* .

**7.3. Instability for large sampling time and pure delay**

This closes the stability and convergence proof.

small (*<sup>h</sup>* << 1), and so, the kinematic error <sup>∽</sup>

consult for instance (Ioannou and Sun, 1996).

rise a turn off dominant in the error function *f* <sup>∗</sup>

*tn* in

*tn* <sup>=</sup> <sup>∆</sup>*Q*<sup>∗</sup>

complex roots in (40). So a suboptimal control action is employed instead equal to

*<sup>b</sup>n*=−*M*<sup>−</sup><sup>1</sup>

*tn* <sup>+</sup> *<sup>c</sup>*¯*<sup>n</sup>* <sup>−</sup> <sup>1</sup>

With respect to the boundness of the adaptive matrices *Ui*´s it is seen from (45) that the gradients are bounded. Also the third term is more dominant than the remainder ones for *h*

more significantly than the others. From (43) one concludes than the increasing of |*Ui*| may not be avoided long term, however some robust modification techniques like a projection zone can be employed to achieve boundness. This is not developed here. The author can

Broadly speaking, the influence of the analyzed parameters will play a role in the instability when the chosen *h* is something large, even smaller than one, because the quadratic terms

The study of this phenomenon is rather complex. It involves the function ∆*Q*<sup>∗</sup>

*<sup>I</sup>*−*hM*−<sup>1</sup>

∆*Qn* .

> *Kv vn* 2 +*v* 2 *<sup>n</sup>*+ *J* −1 *tn <sup>η</sup>*˙*rtn* −*<sup>J</sup>* −1 *tn*+1*Kp* ∽ *ηtn* 2 +*η* 2

the path trajectories may not be bounded into a residual set because the domain for the initial conditions in this situation is partially repulsive. So, depending on the particular

*<sup>h</sup>* (*<sup>I</sup>* <sup>−</sup> *hK*<sup>∗</sup>

Stability Analysis of Nonlinear Discrete-Time Adaptive Control Systems with Large Dead-Times

<sup>4</sup>*h*<sup>2</sup> *<sup>b</sup> T*

*<sup>n</sup> b<sup>n</sup>* occurs at some time *tn*, one chooses the real part of the

*v* ) ∽

*<sup>n</sup>bn*<0 in B∩B∗∗

<sup>0</sup> in (59), since its dimension depends not only on *εηn*+<sup>1</sup> and *εvn*<sup>+</sup><sup>1</sup> but

<sup>0</sup> is a new residual set. It is worth noticing

can be reduced by choosing *h* small. Nevertheless,

*<sup>v</sup>tn* influences the intensity and sign of *<sup>∂</sup>*∆*Qtn*/*∂Ui*

*vtn* , (60)

http://dx.doi.org/10.5772/52224

131

<sup>0</sup> , (61)

*tn* in (51) and

*rtn* , (62)

with <sup>∼</sup> *uj in* = *uj*−*u* ∗ *j in* , where *u<sup>j</sup>* and *u* ∗ *<sup>j</sup>* are vectors corresponding to the column *j* of the adaptive controller matrix *Ui* and its corresponding one *<sup>U</sup>*<sup>∗</sup> *<sup>i</sup>* in the fixed controller, respectively. Then the differences <sup>∆</sup>*Vtn* = *Vtn*<sup>+</sup><sup>1</sup> − *Vtn* can be bounded as follows

$$
\Delta V\_{t\_n} = \Delta Q\_{t\_n} + \frac{1}{2} \sum\_{i=1}^{15} \sum\_{j=1}^{6} \left( \Delta \mathbf{u}\_j^T \right)\_{i\_n} \Gamma\_i^{-1} \left( \left( \widetilde{\mathbf{u}}\_j \right)\_{i\_{n+1}} + \left( \widetilde{\mathbf{u}}\_j \right)\_{i\_n} \right) \tag{59}
$$

$$
= \Delta Q\_{t\_n} + \sum\_{i=1}^{15} \sum\_{j=1}^{6} \left( \Delta \mathbf{u}\_j^T \right)\_{i\_n} \Gamma\_i^{-1} \left( \widetilde{\mathbf{u}}\_j \right)\_{i\_n} - 
$$

$$
\quad - \frac{1}{2} \sum\_{i=1}^{15} \sum\_{j=1}^{6} \left( \Delta \mathbf{u}\_j^T \right)\_{i\_n} \Gamma\_i^{-1} \left( \Delta \mathbf{u}\_j \right)\_{i\_n} 
$$

$$
\leq \Delta Q\_{t\_n} - \sum\_{i=1}^{15} \sum\_{j=1}^{6} \left( \frac{\partial \Delta Q\_{t\_n}}{\partial \mathbf{u}\_j} \right)^{\top} \left( \widetilde{\mathbf{u}}\_j \right)\_{i\_n} 
$$

$$
\leq \Delta Q\_{t\_n} - \sum\_{i=1}^{15} \sum\_{j=1}^{6} \left( \frac{\partial \Delta \overline{Q}\_{t\_n}}{\partial \mathbf{u}\_j} \right)^{\top} \left( \widetilde{\mathbf{u}}\_j \right)\_{i\_n} 
$$

$$
\leq \Delta Q\_{t\_n}^{\diamond} < 0 \text{ or } \mathbf{B} \odot \mathbf{B}\_{t\_n}^{\diamond} 
$$

with ∆*u<sup>j</sup> in* a column vector of *Uin*<sup>+</sup>1−*Uin* .

The column vector ∆*u<sup>j</sup> in* at the first inequality was replaced by the column vector −Γ*<sup>i</sup> <sup>∂</sup>*∆*Qtn ∂u<sup>j</sup>* and then by −Γ*<sup>i</sup> <sup>∂</sup>*∆*Qtn ∂u<sup>j</sup>* in the right member according to (46) and (48)-(49). So in the second and third inequality, the convexity property of <sup>∆</sup>*Qtn* in (48) was applied for any pair *<sup>U</sup>*′ <sup>=</sup> *Uin* , *<sup>U</sup>*′′ <sup>=</sup> *<sup>U</sup>*<sup>∗</sup> *i* .

This analysis has proved convergence of the error paths when real square root exist from *b T <sup>n</sup>bn*−4*a*¯*c*¯*<sup>n</sup>* of (40).

If on the contrary 4*a*¯*c*¯*<sup>n</sup>* > *b T <sup>n</sup> b<sup>n</sup>* occurs at some time *tn*, one chooses the real part of the complex roots in (40). So a suboptimal control action is employed instead equal to

$$
\sigma\_{\Sigma\_{\boldsymbol{\nu}}} = \frac{-1}{2\overline{\boldsymbol{\pi}}} \underline{\boldsymbol{M}}^{-1} \overline{\boldsymbol{b}}\_{\boldsymbol{\nu}} = \frac{-\underline{\boldsymbol{M}}^{-1}}{\underline{\boldsymbol{h}}} (\boldsymbol{I} - \boldsymbol{h} \boldsymbol{K}\_{\boldsymbol{\nu}}^{\*}) \overset{\rightsquigarrow}{\overline{\boldsymbol{\sigma}}}\_{\boldsymbol{t}\_{\boldsymbol{\nu}}} \tag{60}
$$

and yields a new functional ∆*Q*∗∗ *tn* in

14 Advances in Discrete Time Systems

The problem of stability of the adaptive control system is addressed in the sequel. Let a

∼ *u T j in*<sup>+</sup><sup>1</sup>

Γ−1 *i* ∼ *uj in*<sup>+</sup><sup>1</sup>

<sup>∼</sup> *uj in*<sup>+</sup><sup>1</sup> + ∼ *uj in* 

at the first inequality was replaced by the column vector

in the right member according to (46) and (48)-(49).

*<sup>j</sup>* are vectors corresponding to the column *j* of

− (58)

*<sup>i</sup>* in the fixed controller,

(59)

*Vtn* = *Qtn* +

15 ∑ *i*=1

6 ∑ *j*=1

, where *u<sup>j</sup>* and *u*

the adaptive controller matrix *Ui* and its corresponding one *<sup>U</sup>*<sup>∗</sup>

1 2

6 ∑ *j*=1 ∆*u T j in* Γ −1 *i* ∼ *uj in* −

15 ∑ *i*=1

−1 2

<sup>∆</sup>*Vtn* = <sup>∆</sup>*Qtn* +

15 ∑ *i*=1

15 ∑ *i*=1

15 ∑ *i*=1

6 ∑ *j*=1

6 ∑ *j*=1

*tn* <sup>&</sup>lt; 0 in B∩B<sup>∗</sup>

 *<sup>∂</sup>*∆*Qtn ∂u<sup>j</sup>* 

*i* .  *<sup>∂</sup>*∆*Qtn ∂u<sup>j</sup>*

 *<sup>∂</sup>*∆*Qtn ∂u<sup>j</sup>*

0 ,

*Uin*<sup>+</sup>1−*Uin*

*T* ∼ *uj in*

*T* ∼ *uj in*

 .

So in the second and third inequality, the convexity property of <sup>∆</sup>*Qtn* in (48) was applied for

This analysis has proved convergence of the error paths when real square root exist from

= <sup>∆</sup>*Qtn* +

15 ∑ *i*=1

≤ <sup>∆</sup>*Qtn* −

≤ <sup>∆</sup>*Qtn* −

a column vector of

≤ ∆*Q*<sup>∗</sup>

 ∆*u<sup>j</sup> in*

and then by −Γ*<sup>i</sup>*

*<sup>U</sup>*′ <sup>=</sup> *Uin* , *<sup>U</sup>*′′ <sup>=</sup> *<sup>U</sup>*<sup>∗</sup>

6 ∑ *j*=1 ∆*u T j in* Γ −1 *i* ∆*u<sup>j</sup> in*

−1 2 1 2

∼ *u T j in* Γ−1 *i* ∼ *uj in* ,

15 ∑ *i*=1

6 ∑ *j*=1

∗

respectively. Then the differences <sup>∆</sup>*Vtn* = *Vtn*<sup>+</sup><sup>1</sup> − *Vtn* can be bounded as follows

6 ∑ *j*=1 ∆*u<sup>T</sup> j in* Γ −1 *i*

**7.1. Stability proof**

Lyapunov function be

with

with ∆*u<sup>j</sup> in*

−Γ*<sup>i</sup>*

 *b T*

The column vector

*<sup>n</sup>bn*−4*a*¯*c*¯*<sup>n</sup>* of (40).

 *<sup>∂</sup>*∆*Qtn ∂u<sup>j</sup>* 

any pair

∼ *uj in* = *uj*−*u* ∗ *j in*

$$
\Delta V\_{t\_n} \le \Delta Q\_{t\_n}^{\ast \ast} = \Delta Q\_{t\_n}^{\ast} + \overline{c}\_n - \frac{1}{4\hbar^2} \overline{b}\_n^{\tau} \overline{b}\_n < 0 \text{ in } \mathcal{B} \cap \mathcal{B}\_0^{\ast \ast} \tag{61}
$$

where ∆*Q*<sup>∗</sup> *tn* is (51) with a real root of (40) and <sup>B</sup>∗∗ <sup>0</sup> is a new residual set. It is worth noticing that the positive quantity *c*¯*<sup>n</sup>* − <sup>1</sup> <sup>4</sup>*h*<sup>2</sup> *b T nbn* can be reduced by choosing *h* small. Nevertheless, B∗∗ <sup>0</sup> results larger than <sup>B</sup><sup>∗</sup> <sup>0</sup> in (59), since its dimension depends not only on *εηn*+<sup>1</sup> and *εvn*<sup>+</sup><sup>1</sup> but also on the magnitude of *c*¯*<sup>n</sup>* − <sup>1</sup> <sup>4</sup>*h*<sup>2</sup> *b T nbn* .

This closes the stability and convergence proof.

#### **7.2. Variable boundness**

With respect to the boundness of the adaptive matrices *Ui*´s it is seen from (45) that the gradients are bounded. Also the third term is more dominant than the remainder ones for *h* small (*<sup>h</sup>* << 1), and so, the kinematic error <sup>∽</sup> *<sup>v</sup>tn* influences the intensity and sign of *<sup>∂</sup>*∆*Qtn*/*∂Ui* more significantly than the others. From (43) one concludes than the increasing of |*Ui*| may not be avoided long term, however some robust modification techniques like a projection zone can be employed to achieve boundness. This is not developed here. The author can consult for instance (Ioannou and Sun, 1996).

### **7.3. Instability for large sampling time and pure delay**

Broadly speaking, the influence of the analyzed parameters will play a role in the instability when the chosen *h* is something large, even smaller than one, because the quadratic terms rise a turn off dominant in the error function *f* <sup>∗</sup> ∆*Qn* .

The study of this phenomenon is rather complex. It involves the function ∆*Q*<sup>∗</sup> *tn* in (51) and *f* ∗ <sup>∆</sup>*Qn* in (52).

Qualitatively speaking, when

$$f\_{\Lambda Q\_{\rm in}} < -\left(\left(I - \hbar I\_{\rm in} f\_{\rm in}^{-1} K\_{\rm p}\right) \mathfrak{q}\_{\rm in}\right)^2 + \mathfrak{q}\_{\rm in}^2 - \left(\left(I - \hbar \underline{\mathbf{M}}^{-1} K\_{\rm s}\right) \mathfrak{q}\_{\rm in}\right)^2 + \mathfrak{v}\_{\rm in}^2 + \left(I\_{\rm in}^{-1} \mathfrak{q}\_{\rm I\_{\rm in}} - I\_{\rm I\_{\rm in + 1}}^{-1} K\_{\rm p} \overset{\smile}{\mathfrak{q}\_{\rm I\_{\rm in}}}\right)^2 + \mathfrak{q}\_{\rm I\_{\rm in}}^2,\tag{62}$$

the path trajectories may not be bounded into a residual set because the domain for the initial conditions in this situation is partially repulsive. So, depending on the particular initial conditions and for *h* >> 0, or similarly for *Td* large, the adaptive control system may turn unstable.

c) Proof of boundness of the adaptive controller matrices *Ui*'s and the way to ensure this

Stability Analysis of Nonlinear Discrete-Time Adaptive Control Systems with Large Dead-Times

http://dx.doi.org/10.5772/52224

133

d) Analysis of a stability condition involving both huge sampling times and a large pure

With the end of illustrating the features of our control system approach, we simulate a path-tracking problem in 6 DOF´s for an underwater vehicle in a planar motion with some

A continuous-time model of a fully-maneuverable underwater vehicle is employed for the numerical simulations and a pure dead time in the control transmission was included. Details

We present the simulation of the adaptive control algorithm summarized in the previous section for an immersion in the depth (motion in the heave modus *z*) with translations aside

Fig. 2 illustrates the evolution of the six controlled modes of the UUV in the descending. The initial conditions of the UUV at *t*<sup>0</sup> were *η*(0)=[1.1; 1.1; 1.1; 0.1; 0.1; 0.1] and *v*(0)=[0.1; 0.1; 0.1; 0.1; 0.1; 0.1]. The design matrices were set in *Kp* = *Kv* = *I*. The sampling

It is seen that after a short transient, which is not longer than 7 s, the UUV is positioned at the coordinates to the start point. Then it begins to the maneuver of descending. In the meantime the matrices *Ui*'s are adapted and the path tracking result asymptotically convergent with unappreciated stationary errors. The predictors are sufficiently accurate during the evolution

In the simulation, between 20 s and 30 s, it can be seen a high interaction between the traslational modes which are tracked (namely: *x*, *y*, *z*) and the rotational modes which are regulated about zero only (namely: *ϕ*, *θ*, *ψ*). In the last ones, it one observes a significant variation of these magnitudes. However, this is reasonable considering the dynamics with

Often in complex digital control systems, for instance digital adaptive controllers in conjugation with complex dynamics with high degree of state interaction, the role played by the sampling time *h* is the algid point in the stability and control performance analysis. This is so because of the potential instability that may occur by improperly selected large *h*. This phenomenon is commonly magnified when a dead time is present. In any case, the appearance of perturbations and delays together in the dynamics makes the problem difficult

An important example which meets these particularities is found in the control problem represented by the guidance of UUV´s in 6 DOF´s, which was taken here as case study.

(Section 7.2),

delays (Section 7.3).

sporadic immersions to the floor.

period was selected *h* = 0.1 s.

large dead-times we are dealing with.

of the states.

**10. Conclusions**

to seize and comprehend.

of this dynamics are given in (Jordán & Bustamante, 2009a).

in the modes surge *x* and sway *y* simultaneously.

**9. Case study**

In conclusion, when comparing two digital controllers, the sensitivity of the stability to *h* and indirectly the presence of large pure delays in the dynamics, is fundamental to draw out robust properties and finally to range them.
