**5. Design of the controller**

Let the control system in Fig. 1 be taken as the basic structure for the next development. The guide system therein generates references in time, denoted by *ηr*(*t*), of a geometric path in the 6-DOF with some desired kinematic, termed *vr*(*ηr*(*t*)), over it. Additionally, it is assumed that the disturbances *δηtn* and *δvtn* acting on the samples are uniformly bounded.

We now postulate a functional of the path error energy

$$Q\_{t\_n} = \stackrel{\smile^T}{\hat{\boldsymbol{\eta}}\_{t\_n}} \stackrel{\smile}{\hat{\boldsymbol{\eta}}\_{t\_n}} + \stackrel{\smile}{\hat{\boldsymbol{\sigma}}\_{t\_n}} \stackrel{\smile}{\hat{\boldsymbol{\sigma}}\_{t\_n}} \tag{21}$$

The control goal is to construct the force *τ<sup>n</sup>* so as to minimize *Qtn* asymptotically, with

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

Bearing in mind the presence of disturbances and model uncertainties, the practical goal would be at least that {∆*Qtn* } decrease so as to ultimately remain bounded for *tn* → <sup>∞</sup>. The problem is now to construct the control action *τn* in such a way that this goal be achieved.

∽

*vtn*−*h Jtn εvn*−*h Jtn J*

−1

+*J* −1 *tn*<sup>+</sup>1*Kp* ∽ *ηtn*<sup>+</sup><sup>1</sup> 2 −

*τ<sup>n</sup>* = *τ*1*n*+*τ*2*<sup>n</sup>* , (28)

*<sup>v</sup>* ≥ 0 being another design matrix like *Kp*, but

*n*+1 2 −

−1 *tn*<sup>+</sup>1*η*˙*rtn*<sup>+</sup><sup>1</sup>

+*J* −1 *<sup>n</sup>*+1*Kp* ∽ *ηn*+<sup>1</sup> 

*vtn* . (26)

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

123

*<sup>n</sup> η*˙*rtn* − (27)

(29)

<sup>∆</sup>*Qtn* := *Qtn*<sup>+</sup><sup>1</sup> − *Qtn* < <sup>0</sup>

<sup>∆</sup>*Qtn* <sup>=</sup> <sup>∽</sup> *η T tn*<sup>+</sup><sup>1</sup> ∽ *ηtn*<sup>+</sup><sup>1</sup> +∽ *v T tn*<sup>+</sup><sup>1</sup> ∽ *<sup>v</sup>tn*+1−<sup>∽</sup> *η T tn* ∽ *<sup>η</sup>tn*−<sup>∽</sup> *v T tn* ∽

With (24), (25) and (21) in (26), one gets

*<sup>I</sup>* <sup>−</sup> *h Jtn <sup>J</sup>*

−1

*ηn*+*εηn*−*ηrtn*

*<sup>v</sup>n*+*hM*<sup>−</sup><sup>1</sup>

*vn*+*εvn*−*J*

− *h Jtn J*

− 

+ 

−  −1 *<sup>n</sup> Kp* 

*ηn*+*h Jtn*

2 +

proper modifications attending the particularities of the pure-delay case.

*<sup>τ</sup>*1*n*<sup>=</sup> <sup>−</sup> *Kvvn*−<sup>1</sup>

negative definite term in *<sup>v</sup>n*. Herein *Kv* = *<sup>K</sup><sup>T</sup>*

Analyzing (27) we can conveniently split the control thrust *τn* into two terms as

*h M* −*J* −1 *<sup>n</sup>*+1*η*˙*rtn*<sup>+</sup><sup>1</sup>

is the most convenient to compensate some sign-undefinite terms in (27) and to propitiate a

−1 *tn*<sup>+</sup>1*η*˙*rtn* +*J*

*<sup>n</sup> Kp*(*εηn*−*ηrtn* ) + *εηn*+*εηn*+1−*ηrt*

(*rn*+*τn*) + *εvn*+*εvn*<sup>+</sup>1−*J*

The desired properties of (27) can be conferred through a suitable selection of *τn*. In (Jordán and Bustamante, 2011) a flexible methodology for constructing *τn* was proposed and could serve to support this control objective. We will briefly review it and add the

−1 *tn Kp* ∽ *ηtn* 2 .

<sup>∆</sup>*Qtn* =

We notice that the choice

affecting the kinematic errors.

with (see Jordán and Bustamante, 2011)

$$
\hat{\boldsymbol{\eta}}\_{t\_n} = \boldsymbol{\eta}\_{t\_n} - \boldsymbol{\eta}\_{r\_{l\_n}} = \boldsymbol{\eta}\_n + \boldsymbol{\varepsilon}\_n - \boldsymbol{\eta}\_{r\_{l\_n}} \tag{22}
$$

$$
\stackrel{\frown}{\sigma}\_{t\_n} = \sigma\_{t\_n} - \stackrel{\frown}{f\_{t\_n}} \dot{\eta}\_{t\_n} + \stackrel{\frown}{f\_{t\_n}} \stackrel{\frown}{K\_p} \stackrel{\frown}{\eta}\_{t\_n} = \sigma\_{t\_n} + \varepsilon\_{v\_n} - \stackrel{\frown}{f\_{t\_n}} \dot{\eta}\_{t\_n} + \stackrel{\frown}{f\_{t\_n}} \stackrel{\sm}{K\_p} \stackrel{\sm}{\eta}\_{t\_n \cdot} \tag{23}
$$

where *Kp* = *<sup>K</sup><sup>T</sup> <sup>p</sup>* <sup>≥</sup> 0 is a design gain matrix affecting the geometric path errors. Clearly, if <sup>∽</sup> *ηtn*≡**0**, then by (23) and (2), it yields *vtn*−*vrtn* ≡**0**.

**Figure 1.** Digital adaptive control system of an UUV with sample disturbances and pure time delay in the control communication link

Then, replacing (6) and (7) in (22) for *tn*+<sup>1</sup> one gets

$$
\hat{\boldsymbol{\eta}}\_{\boldsymbol{t}\_{n+1}} = \boldsymbol{\eta}\_{\boldsymbol{n}+1} + \boldsymbol{\varepsilon}\_{\boldsymbol{n}+1} - \boldsymbol{\eta}\_{\boldsymbol{r}\_{\boldsymbol{t}\_{n+1}}} = \boldsymbol{\eta}\_{\boldsymbol{n}} + h\boldsymbol{\eta}\_{\boldsymbol{n}} + \boldsymbol{\varepsilon}\_{\boldsymbol{n}} + \boldsymbol{\varepsilon}\_{\boldsymbol{n}+1} - \boldsymbol{\eta}\_{\boldsymbol{r}\_{\boldsymbol{t}\_{n+1}}}.\tag{24}
$$

Similarly, (6) and (7) in (23) for *tn*+<sup>1</sup> one obtains

$$\begin{split} \overset{\circ}{\boldsymbol{\mathfrak{v}}}\_{t\_{n+1}} &= \boldsymbol{\mathfrak{v}}\_{t\_{n+1}} - \boldsymbol{J}\_{t\_{n+1}}^{-1} \boldsymbol{\dot{\eta}}\_{\boldsymbol{r}\_{n+1}} + \boldsymbol{J}\_{t\_{n+1}}^{-1} \boldsymbol{K}\_{p} \overset{\circ}{\boldsymbol{\mathfrak{v}}}\_{t\_{n+1}} = \\ &= \boldsymbol{\mathfrak{v}}\_{n} + \boldsymbol{h} \underline{\boldsymbol{M}}^{-1} \left( \boldsymbol{\mathfrak{r}}\_{\delta\_{n-d}} + \boldsymbol{\mathfrak{r}}\_{n-d} \right) + \boldsymbol{\mathfrak{e}}\_{\boldsymbol{\mathfrak{v}}\_{n}} + \boldsymbol{\mathfrak{e}}\_{\boldsymbol{\mathfrak{v}}\_{n+1}} - \boldsymbol{J}\_{t\_{n+1}}^{-1} \boldsymbol{\dot{\eta}}\_{\boldsymbol{r}\_{n+1}} + \boldsymbol{J}\_{t\_{n}+1}^{-1} \boldsymbol{K}\_{p} \overset{\circ}{\boldsymbol{\mathfrak{v}}}\_{t\_{n+1}}. \end{split} \tag{25}$$

The control goal is to construct the force *τ<sup>n</sup>* so as to minimize *Qtn* asymptotically, with <sup>∆</sup>*Qtn* := *Qtn*<sup>+</sup><sup>1</sup> − *Qtn* < <sup>0</sup>

$$
\Delta Q\_{t\_n} = \stackrel{\smile^{\tau}}{\eta}\_{t\_{n+1}} \stackrel{\smile}{\eta}\_{t\_{n+1}} + \stackrel{\smile^{\tau}}{\sigma}\_{t\_{n+1}} \stackrel{\smile}{\sigma}\_{t\_{n+1}} - \stackrel{\smile^{\tau}}{\eta}\_{t\_n} \stackrel{\smile}{\eta}\_{t\_n} - \stackrel{\smile^{\tau}}{\sigma}\_{t\_n} \stackrel{\smile}{\sigma}\_{t\_n}.\tag{26}$$

Bearing in mind the presence of disturbances and model uncertainties, the practical goal would be at least that {∆*Qtn* } decrease so as to ultimately remain bounded for *tn* → <sup>∞</sup>. The problem is now to construct the control action *τn* in such a way that this goal be achieved.

With (24), (25) and (21) in (26), one gets

6 Advances in Discrete Time Systems

**5. Design of the controller**

with (see Jordán and Bustamante, 2011)

∽

∽

where *Kp* = *<sup>K</sup><sup>T</sup>*

link

*vtn* = *vtn*−*J*

*ηtn*≡**0**, then by (23) and (2), it yields *vtn*−*vrtn* ≡**0**.

Then, replacing (6) and (7) in (22) for *tn*+<sup>1</sup> one gets

Similarly, (6) and (7) in (23) for *tn*+<sup>1</sup> one obtains

<sup>=</sup> *<sup>v</sup>n*+*hM*<sup>−</sup><sup>1</sup>

−1 *tn*<sup>+</sup>1*η*˙*rtn*<sup>+</sup><sup>1</sup>

=*ηn*+1+*εn*+1−*ηrt*

+*J* −1 *tn*<sup>+</sup>1*Kp* ∽ *ηtn*<sup>+</sup><sup>1</sup>

*rδn*−*d*+*τn*−*<sup>d</sup>*

∽ *ηtn*<sup>+</sup><sup>1</sup>

*vtn*<sup>+</sup><sup>1</sup> = *vtn*<sup>+</sup>1−*J*

∽

−1 *tn η*˙*rtn* +*J*

Let the control system in Fig. 1 be taken as the basic structure for the next development. The guide system therein generates references in time, denoted by *ηr*(*t*), of a geometric path in the 6-DOF with some desired kinematic, termed *vr*(*ηr*(*t*)), over it. Additionally, it is assumed

*ηtn* = *ηtn*−*ηrtn* = *η<sup>n</sup>* + *εn*−*ηrtn* (22)

−1 *tn η*˙*rtn* +*J*

−1 *tn Kp* ∽ *ηtn*

*ηtn*=*vtn*+*εvn* −*J*

*<sup>p</sup>* <sup>≥</sup> 0 is a design gain matrix affecting the geometric path errors. Clearly, if <sup>∽</sup>

**Figure 1.** Digital adaptive control system of an UUV with sample disturbances and pure time delay in the control communication

=*ηn*+*hqn*+*εn*+*εn*+1−*ηrt*

−1 *tn*<sup>+</sup>1*η*˙*rtn*<sup>+</sup><sup>1</sup> *n*+1

= (25)

+*J* −1 *tn*+1*Kp* ∽ *ηtn*<sup>+</sup><sup>1</sup> .

. (24)

*n*+1

+*εvn* +*εvn*<sup>+</sup>1−*J*

*vtn* , (21)

, (23)

that the disturbances *δηtn* and *δvtn* acting on the samples are uniformly bounded.

*Qtn* <sup>=</sup> <sup>∽</sup> *η T tn* ∽ *<sup>η</sup>tn*+<sup>∽</sup> *v T tn* ∽

> −1 *tn Kp* ∽

We now postulate a functional of the path error energy

<sup>∆</sup>*Qtn* = *<sup>I</sup>* <sup>−</sup> *h Jtn <sup>J</sup>* −1 *<sup>n</sup> Kp ηn*+*h Jtn* ∽ *vtn*−*h Jtn εvn*−*h Jtn J* −1 *<sup>n</sup> η*˙*rtn* − (27) − *h Jtn J* −1 *<sup>n</sup> Kp*(*εηn*−*ηrtn* ) + *εηn*+*εηn*+1−*ηrt n*+1 2 − − *ηn*+*εηn*−*ηrtn* 2 + + *<sup>v</sup>n*+*hM*<sup>−</sup><sup>1</sup> (*rn*+*τn*) + *εvn*+*εvn*<sup>+</sup>1−*J* −1 *tn*<sup>+</sup>1*η*˙*rtn*<sup>+</sup><sup>1</sup> +*J* −1 *tn*<sup>+</sup>1*Kp* ∽ *ηtn*<sup>+</sup><sup>1</sup> 2 − − *vn*+*εvn*−*J* −1 *tn*<sup>+</sup>1*η*˙*rtn* +*J* −1 *tn Kp* ∽ *ηtn* 2 .

The desired properties of (27) can be conferred through a suitable selection of *τn*. In (Jordán and Bustamante, 2011) a flexible methodology for constructing *τn* was proposed and could serve to support this control objective. We will briefly review it and add the proper modifications attending the particularities of the pure-delay case.

Analyzing (27) we can conveniently split the control thrust *τn* into two terms as

$$
\pi\_{\mathfrak{l}} = \mathfrak{T}\_{\mathfrak{l}\_{\mathfrak{u}}} + \mathfrak{T}\_{\mathfrak{u}\_{\mathfrak{u}}}.\tag{28}
$$

We notice that the choice

$$\boldsymbol{\tau}\_{1\_{n}} = -\boldsymbol{K}\_{\boldsymbol{\upsilon}} \boldsymbol{\upsilon}\_{n} - \frac{1}{h} \underline{\boldsymbol{M}} \left( -\boldsymbol{J}\_{n+1}^{-1} \boldsymbol{\dot{\eta}}\_{r\_{n+1}} + \boldsymbol{J}\_{n+1}^{-1} \boldsymbol{K}\_{p} \boldsymbol{\dot{\eta}}\_{n+1} \right) \tag{29}$$

is the most convenient to compensate some sign-undefinite terms in (27) and to propitiate a negative definite term in *<sup>v</sup>n*. Herein *Kv* = *<sup>K</sup><sup>T</sup> <sup>v</sup>* ≥ 0 being another design matrix like *Kp*, but affecting the kinematic errors.

Thus

<sup>∆</sup>*Qtn* = *<sup>I</sup>* <sup>−</sup> *h Jtn <sup>J</sup>* −1 *<sup>n</sup> Kp ηn* 2 −*η*<sup>2</sup> *n*+ *I* − *hM*<sup>−</sup><sup>1</sup> *Kv vn* 2 −*v* 2 *<sup>n</sup>*− (30) − *J* −1 *tn η*˙*rtn* +*J* −1 *tn Kp* ∽ *ηtn* 2 − *η* 2 *rtn* + −*h Jtn εvn*−*h Jtn J* −1 *<sup>n</sup> Kpεηn*+*εηn*+*εηn*+<sup>1</sup> 2 + +2*ε T <sup>η</sup><sup>n</sup> ηrtn* −*ε* 2 *<sup>η</sup><sup>n</sup>* + *εvn* +*εvn*<sup>+</sup><sup>1</sup> −∆*J* −1 *<sup>n</sup>*+1*η*˙*rtn*<sup>+</sup><sup>1</sup> +∆*J* −1 *<sup>n</sup>*+1*Kp* ∽ *ηtn*<sup>+</sup><sup>1</sup> 2 −*ε*<sup>2</sup> *vn* + +2 *<sup>I</sup>* <sup>−</sup> *h Jtn <sup>J</sup>* −1 *<sup>n</sup> Kp ηn <sup>T</sup>* −*h Jtn εvn*−*h Jtn J* −1 *<sup>n</sup> Kpεηn*+ +*εηn*+*εηn*+<sup>1</sup> − 2*η T n εηn*−*ηrtn* +2 *I* − *hM*<sup>−</sup><sup>1</sup> *Kv vn T εvn* +*εvn*<sup>+</sup><sup>1</sup> −∆*J* −1 *<sup>n</sup>*+1*η*˙*rtn*<sup>+</sup><sup>1</sup> +∆*J* −1 *<sup>n</sup>*+1*Kp* ∽ *ηtn*<sup>+</sup><sup>1</sup> + + *hM*<sup>−</sup><sup>1</sup> (*pn*−*rn*) *<sup>T</sup> εvn* +*εvn*<sup>+</sup><sup>1</sup> −∆*J* −1 *<sup>n</sup>*+1*η*˙*rtn*<sup>+</sup><sup>1</sup> +∆*J* −1 *<sup>n</sup>*+1*Kp* ∽ *ηtn*<sup>+</sup><sup>1</sup> − −*v T <sup>n</sup> εvn* + *h Jtn* ∽ *vtn*+*h Jtn J* −1 *<sup>n</sup> η*˙*rtn* +*h Jtn J* −1 *<sup>n</sup> Kpηrtn* −*ηrt n*+1 2 + +2 *<sup>I</sup>* <sup>−</sup> *h Jtn <sup>J</sup>* −1 *<sup>n</sup> Kp ηn <sup>T</sup> h Jtn* ∽ *vtn*+*h Jtn J* −1 *<sup>n</sup> η*˙*rtn* +*h Jtn J* −1 *<sup>n</sup> Kpηrtn* −*ηrt n*+1 + + *hM*<sup>−</sup><sup>1</sup> (*pn*−*rn*) 2 +2 *I* − *hM*<sup>−</sup><sup>1</sup> *Kv vn T hM*<sup>−</sup><sup>1</sup> (*pn*−*r<sup>n</sup>* ) − −2*v<sup>T</sup> n* −*J* −1 *tn η*˙*rtn* +*J* −1 *tn Kp* ∽ *ηtn* + *hM*<sup>−</sup><sup>1</sup> *τn*<sup>2</sup> 2 + +2 *<sup>v</sup>n*+*hM*<sup>−</sup><sup>1</sup> (*pn*−*rn*) <sup>+</sup>*εvn* <sup>+</sup>*εvn*<sup>+</sup>1−∆*<sup>J</sup>* −1 *<sup>n</sup>*+1*η*˙*rtn*<sup>+</sup><sup>1</sup> +∆*J* −1 *<sup>n</sup>*+1*Kp* ∽ *ηtn*<sup>+</sup><sup>1</sup> *T hM*<sup>−</sup><sup>1</sup> *τn*<sup>2</sup>

*f*∆*Q*<sup>1</sup>

[*εη<sup>n</sup>* ,*εvn* ] = �

+ �

+2

�

+ � *hM*<sup>−</sup><sup>1</sup>

−*v T <sup>n</sup> εvn*+2

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

� *hM*−<sup>1</sup> *τn*<sup>2</sup> �2 +*b T n* � *hM*−<sup>1</sup> *τn*<sup>2</sup> �

*v*)*vn*+2*hM*<sup>−</sup><sup>1</sup>

−1

−1 *<sup>n</sup> Kp* � *ηn* �*<sup>T</sup>* � *h Jtn* ∽ *vtn*+*h Jtn J*

(*pn*−*rn*)

*<sup>v</sup>* being an auxiliary matrix equal to *<sup>K</sup>*<sup>∗</sup>

*<sup>n</sup> η*˙*rtn* +*h Jtn J*

�2 +2

−1 *tn Kp* ∽ *ηtn* � ,

*τ*2*n*=*M*

 <sup>−</sup>*b<sup>n</sup>* 2*a*

which is consistent with *εη<sup>n</sup>* , *εvn* . So we have

��*I*−*h Jtn <sup>J</sup>*

+*f*∆*Q*<sup>1</sup> [*εη<sup>n</sup>* ,*εvn* ]+*a*

<sup>∆</sup>*Qtn* =

*a* = *h* 2

*cn* = � *h Jtn* ∽ *vtn*+*h Jtn J*

with *K*<sup>∗</sup>

it is

*<sup>b</sup><sup>n</sup>* <sup>=</sup> <sup>2</sup>*h*(*I*−*hK*<sup>∗</sup>

+2

+ � *hM*<sup>−</sup><sup>1</sup>

−2*v T n* � −*J* −1 *tn η*˙*rtn* +*J*

��*<sup>I</sup>* <sup>−</sup> *h Jtn <sup>J</sup>*

−*h Jtn εvn*−*h Jtn J*

��*<sup>I</sup>* <sup>−</sup> *h Jtn <sup>J</sup>*

*<sup>ε</sup>vn*+*εvn*<sup>+</sup>1−∆*<sup>J</sup>*

+*εηn*+*εηn*+<sup>1</sup>

*<sup>ε</sup>vn*+*εvn*<sup>+</sup>1−∆*<sup>J</sup>*

� +2

(*pn*−*rn*)

�

−1

−1 *<sup>n</sup>*+1*η*˙*rtn*<sup>+</sup><sup>1</sup>

��*I* − *hM*<sup>−</sup><sup>1</sup>

�*<sup>T</sup>* �

*<sup>ε</sup>vn*+*εvn*<sup>+</sup>1−∆*<sup>J</sup>*

��*I*−*hM*−<sup>1</sup>

−1

��*I* − *hM*<sup>−</sup><sup>1</sup>

−1 *<sup>n</sup> Kp* � *ηn* �*<sup>T</sup>* �

−1 *<sup>n</sup>*+1*η*˙*rtn*<sup>+</sup><sup>1</sup>

*<sup>n</sup> Kpεηn*+*εηn*+*εηn*+<sup>1</sup>

+∆*J* −1 *<sup>n</sup>*+1*Kp* ∽ *ηtn*<sup>+</sup><sup>1</sup> �2 −*ε* 2 *vn*+

*Kv* � *vn* �*T*

*<sup>ε</sup>vn*+*εvn*<sup>+</sup>1−∆*<sup>J</sup>*

−1 *<sup>n</sup>*+1*η*˙*rtn*<sup>+</sup><sup>1</sup>

*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 *rtn* <sup>+</sup>

where the last three terms conform a complete quadratic polynomial in *τn*<sup>2</sup> with coefficients

*<sup>n</sup> Kpηrtn* −*ηrtn*<sup>+</sup><sup>1</sup>

*Kv* � *vn* �*T hM*<sup>−</sup><sup>1</sup>

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

Clearly, for eliminating these terms we need to implement one of the roots of the polynomial,

� *b T <sup>n</sup> bn*−4*acn* <sup>6</sup> **<sup>1</sup>**

<sup>±</sup> <sup>1</sup> 2*a*

+∆*J* −1 *<sup>n</sup>*+1*Kp* ∽ *ηtn*<sup>+</sup><sup>1</sup> � +

�2 +2*ε T <sup>η</sup><sup>n</sup> ηrtn* −*ε*

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

−*h Jtn εvn*−*h Jtn J*

−1 *<sup>n</sup>*+1*η*˙*rtn*<sup>+</sup><sup>1</sup>

+∆*J* −1 *<sup>n</sup>*+1*Kp* ∽ *ηtn*<sup>+</sup><sup>1</sup> �*T hM*<sup>−</sup><sup>1</sup> *τn*<sup>2</sup>

−1 *<sup>n</sup> Kpεηn*+

> +∆*J* −1 *<sup>n</sup>*+1*Kp* ∽ *ηtn*<sup>+</sup><sup>1</sup> � −

+*cn*, (32)

+ (35)

*n*+1 � +

, (36)

*<sup>n</sup> Kpηrtn* −*ηrt*

(*pn*−*rn*) −

(*pn*−*rn*) (34)

−1

�2

−1

*Kv*.

*<sup>n</sup> η*˙*rtn* +*h Jtn J*

2

*<sup>η</sup>n*+ (31)

125

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

(33)

where *p<sup>n</sup>* is defined as in *pδtn* but with predictions *v<sup>n</sup>* and *η<sup>n</sup>* instead of *vtn* and *ηtn* , respectively. Similarly *Jn* stays for *J*[*ηn*]. Additionally ∆*J* −1 *<sup>n</sup>*+<sup>1</sup> = *J* −1 *tn*<sup>+</sup><sup>1</sup> − *J* −1 *<sup>n</sup>*+1.

Finally, as seen in (30), the unique remaining design variable is *τn*<sup>2</sup> .

A glance into (30) let us identify sign-undefinite terms which can not be compensated with *τn*<sup>2</sup> , precisely because they are functions of unknown global prediction errors *εη*, *εv*, and ∆*J*. We can group them into the function

$$f\_{\mathbf{A}\mathbf{Q}\_{1}}\left[\boldsymbol{\varepsilon}\_{\eta\_{n}}\boldsymbol{\varepsilon}\_{\sigma\_{n}} - h\boldsymbol{I}\_{\operatorname{I}}\boldsymbol{I}\_{\operatorname{I}}\boldsymbol{\varepsilon}\_{\eta\_{n}}^{\operatorname{I}} - \boldsymbol{K}\_{p}\boldsymbol{\varepsilon}\_{\eta\_{n}} + \boldsymbol{\varepsilon}\_{\eta\_{n}} + \boldsymbol{\varepsilon}\_{\eta\_{n+1}}\right]^{2} + 2\boldsymbol{\varepsilon}\_{\eta\_{n}}^{\operatorname{T}}\boldsymbol{\eta}\_{\operatorname{I}\iota\_{n}} - \boldsymbol{\varepsilon}\_{\eta\_{n}}^{2} + \\ \quad + \left(\boldsymbol{\varepsilon}\_{\eta\_{n}} + \boldsymbol{\varepsilon}\_{\upsilon\_{n+1}} - \Delta\boldsymbol{I}\_{\operatorname{I}}^{-1}\cdot\boldsymbol{\tilde{\eta}}\_{\operatorname{I}\iota\_{n+1}} + \Delta\boldsymbol{\varepsilon}\_{\eta\_{n}}^{-1} + \boldsymbol{K}\_{p}\boldsymbol{\tilde{\eta}}\_{\operatorname{I}\iota\_{n+1}}\right)^{2} - \boldsymbol{\varepsilon}\_{\upsilon\_{n}}^{2} + \\ \quad + 2\left(\left(I - h\boldsymbol{I}\_{\operatorname{I}}\boldsymbol{I}\_{\operatorname{I}}\boldsymbol{I}\_{\operatorname{I}}^{-1}\boldsymbol{K}\_{p}\right)\boldsymbol{\eta}\_{n}\right)^{\operatorname{T}} \left(-h\boldsymbol{I}\_{\operatorname{I}}\boldsymbol{\varepsilon}\_{\operatorname{I}}\boldsymbol{\varepsilon}\_{\operatorname{I}} - h\boldsymbol{I}\_{\operatorname{I}}\boldsymbol{I}\_{\operatorname{I}}\boldsymbol{I}\_{\operatorname{I}}^{-1}\boldsymbol{K}\_{p}\boldsymbol{\varepsilon}\_{\operatorname{I}\eta\_{n}} + \\ \quad + \boldsymbol{\varepsilon}\_{\eta\_{n}} + \boldsymbol{\varepsilon}\_{\eta\_{n+1}}\right) + 2\left(\left(I - h\boldsymbol{M}\underline{\boldsymbol{M}}^{-1}\boldsymbol{K}\_{\operatorname{U}}\right)\boldsymbol{\varepsilon}\_{\eta$$

which is consistent with *εη<sup>n</sup>* , *εvn* . So we have

8 Advances in Discrete Time Systems

<sup>∆</sup>*Qtn* =

− *J* −1 *tn η*˙*rtn* +*J*

+2*ε T <sup>η</sup><sup>n</sup> ηrtn* −*ε*

+2 

+ *hM*<sup>−</sup><sup>1</sup>

−*v T <sup>n</sup> εvn* + *h Jtn* ∽ *vtn*+*h Jtn J*

+2 

+ *hM*<sup>−</sup><sup>1</sup>

−2*v<sup>T</sup> n* −*J* −1 *tn η*˙*rtn* +*J*

+2 

+*εηn*+*εηn*+<sup>1</sup>

*εvn* +*εvn*<sup>+</sup><sup>1</sup>

*I* − *h Jtn J*

−1 *tn Kp* ∽ *ηtn* 2 − *η* 2 *rtn* + 

2 *<sup>η</sup><sup>n</sup>* + 

> −1 *<sup>n</sup> Kp ηn <sup>T</sup>*

*I* − *h Jtn J*

 − 2*η T n* 

−∆*J* −1 *<sup>n</sup>*+1*η*˙*rtn*<sup>+</sup><sup>1</sup>

(*pn*−*rn*)

−1 *<sup>n</sup> Kp ηn <sup>T</sup> h Jtn* ∽ *vtn*+*h Jtn J*

> 2 +2

−1 *tn Kp* ∽ *ηtn* + *hM*<sup>−</sup><sup>1</sup> *τn*<sup>2</sup> 2 +

respectively. Similarly *Jn* stays for *J*[*ηn*]. Additionally ∆*J*

Finally, as seen in (30), the unique remaining design variable is *τn*<sup>2</sup> .

(*pn*−*rn*)

*I* − *h Jtn J*

*<sup>v</sup>n*+*hM*<sup>−</sup><sup>1</sup>

We can group them into the function

*<sup>T</sup>*

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

*εvn* +*εvn*<sup>+</sup><sup>1</sup>

*εηn*−*ηrtn*

+∆*J* −1 *<sup>n</sup>*+1*Kp* ∽ *ηtn*<sup>+</sup><sup>1</sup> +

*εvn* +*εvn*<sup>+</sup><sup>1</sup>

*<sup>n</sup> η*˙*rtn* +*h Jtn J*

*I* − *hM*<sup>−</sup><sup>1</sup>

(*pn*−*rn*) <sup>+</sup>*εvn* <sup>+</sup>*εvn*<sup>+</sup>1−∆*<sup>J</sup>*

−1

−∆*J* −1 *<sup>n</sup>*+1*η*˙*rtn*<sup>+</sup><sup>1</sup>

 +2 

−*h Jtn εvn*−*h Jtn J*

−∆*J* −1 *<sup>n</sup>*+1*η*˙*rtn*<sup>+</sup><sup>1</sup>

−1

*Kv vn T hM*<sup>−</sup><sup>1</sup>

where *p<sup>n</sup>* is defined as in *pδtn* but with predictions *v<sup>n</sup>* and *η<sup>n</sup>* instead of *vtn* and *ηtn* ,

A glance into (30) let us identify sign-undefinite terms which can not be compensated with *τn*<sup>2</sup> , precisely because they are functions of unknown global prediction errors *εη*, *εv*, and ∆*J*.

−1 *<sup>n</sup>*+1*η*˙*rtn*<sup>+</sup><sup>1</sup>

*I* − *hM*<sup>−</sup><sup>1</sup>

+∆*J* −1 *<sup>n</sup>*+1*Kp* ∽ *ηtn*<sup>+</sup><sup>1</sup> 2 −*ε*<sup>2</sup> *vn* +

−1 *<sup>n</sup> Kpεηn*+

> *Kv vn T*

+∆*J* −1 *<sup>n</sup>*+1*Kp* ∽ *ηtn*<sup>+</sup><sup>1</sup> −

*<sup>n</sup> η*˙*rtn* +*h Jtn J*

*n*+1 2 +

+∆*J* −1 *<sup>n</sup>*+1*Kp* ∽ *ηtn*<sup>+</sup><sup>1</sup> *T hM*<sup>−</sup><sup>1</sup> *τn*<sup>2</sup>

−1 *<sup>n</sup>*+<sup>1</sup> = *J*

−1

*<sup>n</sup> Kpηrtn* −*ηrt*

(*pn*−*r<sup>n</sup>* ) −

−1 *tn*<sup>+</sup><sup>1</sup> − *J* −1 *<sup>n</sup>*+1.

*n*+1 +

*I* − *hM*<sup>−</sup><sup>1</sup>

*<sup>n</sup> Kpηrtn* −*ηrt*

−1

−*h Jtn εvn*−*h Jtn J*

*Kv vn* 2 −*v* 2

−1

*<sup>n</sup> Kpεηn*+*εηn*+*εηn*+<sup>1</sup>

*<sup>n</sup>*− (30)

2 +

Thus

$$\Delta Q\_{li} = \left( \left( I - hI\_{li} \right)^{-1} \mathbf{K}\_p \right) \boldsymbol{\upmu}\_n \right)^2 - \boldsymbol{\upmu}\_n^2 + \left( \left( I - h\underline{\mathbf{M}}^{-1} \ \mathbf{K}\_p \right) \boldsymbol{\upmu}\_n \right)^2 - \boldsymbol{\upnu}\_n^2 - \left( I\_{lu}^{-1} \boldsymbol{\upmu}\_{l\_{li}} - I\_{l\_{n+1}}^{-1} \ \mathbf{K}\_p \boldsymbol{\upmu}\_{l\_{li}} \right)^2 - \boldsymbol{\upmu}\_{l\_{l\_{li}}} + \left( I\_{l\_{n+1}}^{-1} \ \mathbf{K}\_p \boldsymbol{\upmu}\_{l\_{n+1}} \right)^2 \right) \tag{32}$$
 
$$+ f \boldsymbol{\upmu}\_{l\_1} \left[ \boldsymbol{\upmu}\_{l\_0} \ \boldsymbol{\upmu}\_{l\_1} \right] + a \left( h \underline{\mathbf{M}}^{-1} \ \boldsymbol{\upmu}\_{l\_2} \right)^2 + b\_n^{\top} \left( h \underline{\mathbf{M}}^{-1} \ \boldsymbol{\upmu}\_{l\_2} \right) + c\_{n\star} \tag{33}$$

where the last three terms conform a complete quadratic polynomial in *τn*<sup>2</sup> with coefficients

$$a = h^2\tag{33}$$

$$\mathbf{b}\_{n} = \mathfrak{L}h(\mathrm{I} - h\mathrm{K}\_{\upsilon}^{\ast})\mathbf{z}\_{n} + \mathfrak{L}h\underline{\underline{M}}^{-1}(\mathfrak{p}\_{n} - \mathfrak{r}\_{n})\tag{34}$$

*cn* = � *h Jtn* ∽ *vtn*+*h Jtn J* −1 *<sup>n</sup> η*˙*rtn* +*h Jtn J* −1 *<sup>n</sup> Kpηrtn* −*ηrtn*<sup>+</sup><sup>1</sup> �2 + (35) +2 ��*<sup>I</sup>* <sup>−</sup> *h Jtn <sup>J</sup>* −1 *<sup>n</sup> Kp* � *ηn* �*<sup>T</sup>* � *h Jtn* ∽ *vtn*+*h Jtn J* −1 *<sup>n</sup> η*˙*rtn* +*h Jtn J* −1 *<sup>n</sup> Kpηrtn* −*ηrt n*+1 � + + � *hM*<sup>−</sup><sup>1</sup> (*pn*−*rn*) �2 +2 ��*I* − *hM*<sup>−</sup><sup>1</sup> *Kv* � *vn* �*T hM*<sup>−</sup><sup>1</sup> (*pn*−*rn*) − −2*v T n* � −*J* −1 *tn η*˙*rtn* +*J* −1 *tn Kp* ∽ *ηtn* � ,

with *K*<sup>∗</sup> *<sup>v</sup>* being an auxiliary matrix equal to *<sup>K</sup>*<sup>∗</sup> *<sup>v</sup>* <sup>=</sup> *<sup>M</sup>*<sup>−</sup><sup>1</sup> *Kv*.

Clearly, for eliminating these terms we need to implement one of the roots of the polynomial, it is

$$\mathbf{r}\_{2\_{\pi}} = M \left( \frac{-\mathbf{b}\_{n}}{2a} \pm \frac{1}{2a} \sqrt{\frac{\mathbf{b}\_{n}^{\top}\mathbf{b}\_{n} - 4ac\_{n}}{6}} \mathbf{1} \right),\tag{36}$$

with **1** a vector with all elements equal one. However, there are some variables and parameters in these coefficients that are not known. So we can approximate them to

$$
\vec{a} = \vec{h}^2\tag{37}
$$

**6. Adaptive laws**

with a gain matrix Γ*<sup>i</sup>* = Γ

matrices which will denote by *<sup>∂</sup>*∆*Qtn*

and the property

where

permanent actualization of the controller matrices *Ui*. Let the following adaptive law be valid for *i* = 1, ..., 15

*T*

*∂*∆*Qtn ∂Ui*

*∂*∆*Qtn ∂Ui*

The adaptation of the control behaviour to the unknown vehicle dynamics occurs by the

First we can define an expression for the gradient matrix upon ∆*Qtn* in (30) but considering

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

(*I* − *hK*<sup>∗</sup> *v*) ∽ *vn ∂r<sup>n</sup> ∂Ui*

Now, in order to be able to implement adaptive laws like (43) we have to replace the unknown *M* in (44) by its lower bound *M*. In this way, we can generate implementable gradient

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

(*I* − *hK*<sup>∗</sup> *v*) ∽ *vn ∂r<sup>n</sup> ∂Ui*

<sup>=</sup> *<sup>∂</sup>*∆*Qtn ∂Ui*

*∂*∆*Qtn ∂Ui*

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

being a gradient matrix for *Uin* .

 *∂r<sup>n</sup> ∂Ui*

 *∂r<sup>n</sup> ∂Ui*

(*pn*−*rn*)

*T*

*T* −

*T* ,

∆*Uin* = *δM*−<sup>2</sup> *Ain* + *δM*−1*Bin* , (47)

 *∂r<sup>n</sup> ∂Ui*

(*pn*−*rn*)

*T*

*T* −

*T* .

 *∂r<sup>n</sup> ∂Ui*

, (43)

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

127

− (44)

− (45)

+ ∆*Uin* , (46)

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

*∂Uin*

that *M* is known. This expression is referred to the ideal gradient matrix

= −2*h* 2 *M*<sup>−</sup>*<sup>T</sup>*

> −2*h* 2 *M*<sup>−</sup>*<sup>T</sup> M*<sup>−</sup><sup>1</sup>

*<sup>∂</sup>Ui* with

= −2*h* 2 *M*<sup>−</sup>*<sup>T</sup>*

> −2*h* 2 *M*<sup>−</sup>*<sup>T</sup> M*<sup>−</sup><sup>1</sup>

<sup>−</sup>2*hM*<sup>−</sup>*<sup>T</sup>*

*∂*∆*Qtn ∂Ui*

<sup>−</sup>2*hM*<sup>−</sup>*<sup>T</sup>*

*<sup>i</sup>* <sup>≥</sup> 0 and *<sup>∂</sup>*∆*Qtn*

$$\overline{\mathfrak{B}}\_n = 2h(\boldsymbol{I} - h\boldsymbol{K}\_v^\circ)\mathfrak{v}\_n\tag{38}$$

$$\bar{\mathfrak{c}}\_{n} = \left( h \mathfrak{l}\_{n} \hat{\mathfrak{v}}\_{n} + h \mathfrak{h}\_{\mathfrak{l}\_{\text{fr}}} + h \mathbf{K}\_{p} \mathfrak{y}\_{\text{l}\_{\text{n}}} - \mathfrak{y}\_{\text{l}\_{\text{n}+1}} \right)^{2} + \underbrace{\tag{39}}\_{\text{c}} \tag{30}$$

$$\begin{split} &+2\left(\left(I-hK\_{P}\right)\eta\_{n}\right)^{\top}\left(hf\_{n}\stackrel{\smile}{\mathfrak{D}}\_{n}+h\dot{\eta}\_{\mathbb{F}\_{\mathbb{H}}}+hK\_{P}\eta\_{\mathbb{F}\_{\mathbb{H}}}-\eta\_{\mathbb{F}\_{\mathbb{H}}}\right)+\\ &-2\boldsymbol{\sigma}\_{n}^{\top}\left(-f\_{n}^{-1}\dot{\eta}\_{\mathbb{F}\_{\mathbb{H}}}+f\_{n}^{-1}K\_{P}\stackrel{\smile}{\mathfrak{H}}\_{n}\right). \end{split}$$

Finally we get the second component of *τn* in an implementable way

$$\mathbf{r}\_{2\_{\text{ul}}} = \underline{\mathbf{M}} \left( \frac{-\overline{\mathbf{b}}\_{n}}{2\overline{a}} \pm \frac{1}{2\overline{a}} \sqrt{\frac{\overline{\mathbf{b}}\_{n}^{\mathrm{r}}\overline{\mathbf{b}}\_{n} - 4\overline{a}\overline{c\_{n}}}{6}} \mathbf{1} \right). \tag{40}$$

Introducing the expression of *τn* the functional remains

$$\begin{split} \Delta \mathbf{Q}\_{li} &= \left( \left( I - h \mathbf{I}\_{li} \mathbf{I}\_{n}^{-1} \mathbf{K}\_{p} \right) \boldsymbol{\mathfrak{y}}\_{n} \right)^{2} - \boldsymbol{\mathfrak{y}}\_{n}^{2} + \left( \left( I - h \underline{\mathbf{M}}^{-1} \, \mathbf{K}\_{v} \right) \boldsymbol{\mathfrak{y}}\_{n} \right)^{2} - \boldsymbol{\mathfrak{y}}\_{n}^{2} - \left( I\_{l\_{\rm li}}^{-1} \, \boldsymbol{\mathfrak{y}}\_{l\_{\rm I}} - I\_{l\_{\rm I+1}}^{-1} \, \mathbf{K}\_{p} \tilde{\boldsymbol{\mathfrak{y}}}\_{l\_{\rm I}} \right)^{2} - \boldsymbol{\mathfrak{y}}\_{r\_{l\_{\rm I}}}^{2} + \\ &+ f\_{\rm \Delta Q\_{1}} \, [\boldsymbol{\mathfrak{e}}\_{\eta\_{\rm u}} \, \boldsymbol{\mathfrak{e}}\_{\nu\_{\rm u}}] + f\_{\rm \Delta Q\_{2}} \, [\boldsymbol{\mathfrak{e}}\_{\eta\_{\rm u}} \, \boldsymbol{\mathfrak{e}}\_{\nu\_{\rm u}}], \end{split} \tag{41}$$

where the new error *f*∆*Q*<sup>2</sup> is

$$\begin{split} f\_{\Delta Q\_{2}}[\boldsymbol{\varepsilon}\_{\eta\_{n}}, \boldsymbol{\varepsilon}\_{\eta\_{n}}] &= h \left( \mathbf{b}\_{n} - \overline{\mathbf{b}}\_{n} \right)^{\top} \left( \frac{-\overline{\mathbf{b}}\_{n}}{2\overline{a}} \pm \frac{1}{2\overline{a}} \sqrt{\frac{\overline{\mathbf{b}}^{\top} \overline{\mathbf{b}} - 4\overline{a}\overline{c}}}{6} \mathbf{1} \right) + \left( c\_{n} - \overline{c}\_{n} \right) = \\ &= \frac{\underline{M}^{-1} \left( \boldsymbol{\mu}\_{n} - \boldsymbol{r}\_{n} \right)^{\top} \left( 2h \left( I - h \overline{\mathbf{b}}\_{v}^{\top} \right) \boldsymbol{\uppi}\_{n} + 2h \underline{\mathbf{M}}^{-1} \left( \boldsymbol{\upmu}\_{n} - \boldsymbol{r}\_{n} \right) \right)}{2h} + \\ &+ \frac{\underline{M}^{-1} \left( \boldsymbol{\upmu}\_{n} - \boldsymbol{r}\_{n} \right)^{\top} \sqrt{\frac{\overline{\mathbf{b}}^{\top} \overline{\mathbf{b}} - 4\overline{a}\overline{c}}} \mathbf{1} + \left( c\_{n} - \overline{c}\_{n} \right). \end{split} \tag{42}$$

The properties of the error functions *f*∆*Q*<sup>1</sup> and *f*∆*Q*<sup>2</sup> and their influence in the stability of the control system is analyzed later. Previous to this task, we will illustrate the way we generate the matrices *Ui*'s so as to calculate the variables *r<sup>i</sup>* in (13).
