**6. Adaptive laws**

(37)

10 Advances in Discrete Time Systems

*a*¯ = *h* 2

*c*¯*<sup>n</sup>* = � *h Jn* ∽

<sup>∆</sup>*Qtn* = �� *I*−*h Jtn J* −1 *<sup>n</sup> Kp* � *ηn* �2 −*η* 2 *<sup>n</sup>*+ �� *<sup>I</sup>*−*hM*−<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 new error *f*∆*Q*<sup>2</sup> is

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

+ *M*<sup>−</sup><sup>1</sup>

*f*∆*Q*<sup>2</sup> [*εη<sup>n</sup>* ,*εvn* ] = *h*

(*pn*−*rn*)

(*pn*−*rn*)

2*h*

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

+2

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

��*I* − *hKp*

with **1** a vector with all elements equal one. However, there are some variables and

*n*+1 �2

� *b T*

*<sup>v</sup>*)*v<sup>n</sup>* (38)

*vn*+*hη*˙*rtn* +*hKpηrtn* −*ηrt*

*<sup>n</sup>bn*−4*a*¯*cn* <sup>6</sup> **<sup>1</sup>**

<sup>+</sup>*f*∆*Q*<sup>1</sup> [*εη<sup>n</sup>* ,*εvn* ]+*f*∆*Q*<sup>2</sup> [*εη<sup>n</sup>* ,*εvn* ], (41)

� *b T b*−4*ac*¯ <sup>6</sup> **<sup>1</sup>**

(*pn*−*rn*)

<sup>2</sup>*<sup>h</sup>* <sup>+</sup>

�

+ (39)

. (40)

<sup>+</sup> (*cn* <sup>−</sup> *cn*) <sup>=</sup> (42)

*n*+1 � +

parameters in these coefficients that are not known. So we can approximate them to

*vn*+*hη*˙*rtn* +*hKpηrtn* −*ηrt*

−1 *<sup>n</sup> Kp* ∽ *ηn* � .

� *ηn* �*<sup>T</sup>* � *h Jn* ∽

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

 <sup>−</sup>*b<sup>n</sup>* <sup>2</sup>*a*¯ <sup>±</sup> <sup>1</sup> 2*a*¯

*τ*2*n*=*M*

� *bn*−*b<sup>n</sup>*

2*h*(*I*−*hK*<sup>∗</sup>

*<sup>T</sup>* �

*T*

the matrices *Ui*'s so as to calculate the variables *r<sup>i</sup>* in (13).

� *b T b*−4*ac*¯

�*T* <sup>−</sup>*b<sup>n</sup>* <sup>2</sup>*a*¯ <sup>±</sup> <sup>1</sup> 2*a*¯

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

<sup>6</sup> **<sup>1</sup>**<sup>+</sup> (*cn* <sup>−</sup> *cn*).

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

Introducing the expression of *τn* the functional remains

The adaptation of the control behaviour to the unknown vehicle dynamics occurs by the permanent actualization of the controller matrices *Ui*.

Let the following adaptive law be valid for *i* = 1, ..., 15

$$\mathcal{U}\mathcal{U}\_{i\_{n+1}} \stackrel{\Delta}{=} \mathcal{U}\_{i\_n} - \Gamma\_i \frac{\partial \Delta Q\_{t\_n}}{\partial \mathcal{U}\_i} \,. \tag{43}$$

with a gain matrix Γ*<sup>i</sup>* = Γ *T <sup>i</sup>* <sup>≥</sup> 0 and *<sup>∂</sup>*∆*Qtn ∂Uin* being a gradient matrix for *Uin* .

First we can define an expression for the gradient matrix upon ∆*Qtn* in (30) but considering that *M* is known. This expression is referred to the ideal gradient matrix

$$\frac{\partial \Delta Q\_{t\_n}}{\partial \mathcal{U}\_i} = -2\hbar^2 \boldsymbol{\mathcal{M}}^{-\top} \left( \boldsymbol{\mathcal{M}}^{-1} \boldsymbol{\mathsf{r}}\_{2\_n} \right) \left( \frac{\partial \boldsymbol{r}\_n}{\partial \mathcal{U}\_i} \right)^{\top} - \\ \tag{44}$$

$$-2\hbar^2 \boldsymbol{\mathcal{M}}^{-\top} \boldsymbol{\mathcal{M}}^{-1} (\boldsymbol{\mathcal{p}}\_{\boldsymbol{n}} - \boldsymbol{r}\_{\boldsymbol{n}}) \left( \frac{\partial \boldsymbol{r}\_n}{\partial \mathcal{U}\_i} \right)^{\top} -$$

$$-2\hbar \boldsymbol{\mathcal{M}}^{-\top} (\boldsymbol{I} - \boldsymbol{h} \boldsymbol{K}\_v^{\top}) \boldsymbol{\widehat{\boldsymbol{\mathcal{w}}}}\_{\boldsymbol{n}} \left( \frac{\partial \boldsymbol{r}\_n}{\partial \mathcal{U}\_i} \right)^{\top} .$$

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

$$\frac{\partial \overline{\partial \mathbf{Q}}\_{t\_n}}{\partial \mathbf{I}\_l} = -2\hbar^2 \underline{\underline{\mathbf{M}}}^{-\top} \left(\underline{\underline{\mathbf{M}}}^{-1} \mathbf{r}\_{2\_{\text{s}}}\right) \left(\frac{\partial \mathbf{r}\_n}{\partial \mathbf{U}\_l}\right)^{\top} - \tag{45}$$

$$-2\hbar^2 \underline{\underline{\mathbf{M}}}^{-\top} \underline{\underline{\mathbf{M}}}^{-\top} (\mathbf{p}\_n - \mathbf{r}\_n) \left(\frac{\partial \mathbf{r}\_n}{\partial \mathbf{U}\_l}\right)^{\top} -$$

$$-2\hbar \underline{\underline{\mathbf{M}}}^{-\top} (I - \hbar \mathbf{K}\_v^{\top}) \hat{\underline{\mathbf{v}}}\_n \left(\frac{\partial \mathbf{r}\_n}{\partial \mathbf{U}\_l}\right)^{\top},$$

and the property

$$\frac{\partial \overline{\Delta Q}\_{t\_n}}{\partial U\_l} = \frac{\partial \Delta Q\_{t\_n}}{\partial U\_l} + \Delta\_{U\_{l\_n}} \tag{46}$$

where

$$
\Delta\_{U\_{\rm li}} = \delta\_{M^{-2}} A\_{\dot{\imath}\_{\rm n}} + \delta\_{M^{-1}} B\_{\dot{\imath}\_{\rm n}\prime} \tag{47}
$$

and *δM*−<sup>2</sup> = *M*<sup>−</sup>*<sup>T</sup> M*<sup>−</sup><sup>1</sup> − *M*<sup>−</sup>*<sup>T</sup> M*<sup>−</sup><sup>1</sup> ≥ 0 and *δM*−<sup>1</sup> = *M*<sup>−</sup><sup>1</sup> − *M*<sup>−</sup><sup>1</sup> ≥ 0. Here *Ain* and *Bin* are sampled state functions obtained from (44) after extracting of the common factors *δM*−<sup>2</sup> and *δM*−<sup>1</sup> , respectively.

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

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

$$
\Delta Q\_{t\_n}(\boldsymbol{\mathcal{U}\_i}) - \Delta Q\_{t\_n}(\boldsymbol{\mathcal{U}\_i}'') \le \frac{\partial \Delta Q\_{t\_n}(\boldsymbol{\mathcal{U}\_i}'')}{\partial \boldsymbol{\mathcal{U}\_i}} \left(\boldsymbol{\mathcal{U}\_i} - \boldsymbol{\mathcal{U}\_i}''\right) \le \tag{48}
$$

$$\leq \frac{\partial \overline{\Delta Q}\_{t\_n}(\mathcal{U}\_i^{''})}{\partial \mathcal{U}\_i} \left(\mathcal{U}\_i^{'} - \mathcal{U}\_i^{''}\right). \tag{49}$$

∆*Q*<sup>∗</sup> *tn* =

+*f* <sup>∗</sup>

Later, a norm of *f* <sup>∗</sup>

So, it is noticing that ∆*Q*<sup>∗</sup>

which is equivalent to

The residual set B<sup>∗</sup>

*δvtn* → 0 and *M* = *M*.

but also on *M*<sup>−</sup><sup>1</sup>

where *f* <sup>∗</sup>

*l*∞ as well.

with B<sup>∗</sup>

*pn*=*r<sup>n</sup>*

*<sup>I</sup>*−*h Jtn <sup>J</sup>*

<sup>∆</sup>*Q*[*εη<sup>n</sup>* ,*εvn <sup>δ</sup>ηtn* ,*δvtn* ,*M*−<sup>1</sup>

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

*f* ∗

<sup>∆</sup>*Qn* will be indicated.

B∗ <sup>0</sup>= ∽ *ηtn* , ∽

and with the design matrices satisfying the conditions

*M*. In consequence, B<sup>∗</sup>

B = ∽ *ηtn* , ∽

> 2 *Jn J* −1 *tn*

2 *<sup>h</sup> <sup>M</sup>* <sup>≥</sup>

2 *<sup>h</sup> <sup>I</sup>* <sup>&</sup>gt; *<sup>K</sup>*<sup>∗</sup>

2

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>

<sup>0</sup> a residual set around zero

*<sup>I</sup>*−*hM*−<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>

<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

*tn* <sup>&</sup>lt; 0, at least in an attraction domain equal to

*<sup>v</sup>tn* ∈ R<sup>6</sup> ∩ B<sup>∗</sup>

*<sup>v</sup>tn*∈R6/∆*Q*<sup>∗</sup>

0 

*tn* <sup>−</sup> *<sup>f</sup>* <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* ,

<sup>∆</sup>*Qn* ≤ 0

*<sup>h</sup>* <sup>&</sup>gt; *Kp* <sup>&</sup>gt; <sup>0</sup> (55)

*<sup>h</sup> <sup>M</sup>* <sup>&</sup>gt; *Kv* <sup>≥</sup> 0. (57)

<sup>0</sup> becomes the null point at the limit when *h* → 0, *δηtn* ,

*<sup>v</sup>* ≥ 0, (56)

*M*], (51)

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

<sup>∆</sup>*Qn* = *f*∆*Q*1*<sup>n</sup>* [*pn*=*rn*] + *f*∆*Q*2*<sup>n</sup>* [*pn*=*rn*]. (52)

*M* is bounded. Thus, one concludes *f* <sup>∗</sup>

, (53)

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

129

<sup>∆</sup>*Qn* ∈

(54)

This feature will be useful in the next analysis.

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

$$\mathcal{U}\mathcal{U}\_{\bar{i}\_{n+1}} \stackrel{\Delta}{=} \mathcal{U}\_{\bar{i}\_n} - \Gamma\_i \frac{\partial \overline{\Delta \mathcal{Q}}\_{\bar{t}\_n}}{\partial \mathcal{U}\_{\bar{i}}}.\tag{50}$$

Finally, it is seen from (45) that also here the noisy measures *ηδtn* and *vδtn* will propagate into the adaptive laws *<sup>∂</sup>*∆*Qtn <sup>∂</sup>Ui* .
