**4. Spatial‐based adaptive iterative learning control of nonlinear rotary systems with spatially periodic parametric variation**

Consider an NTI system described by

$$\dot{\mathbf{x}}(t) = f\_t\left(\mathbf{x}, \,\boldsymbol{\varphi}\_f(\boldsymbol{\theta})\right) + \mathcal{B}\_c \mathbf{g}\_t\left(\mathbf{x}, \,\boldsymbol{\varphi}\_\varepsilon(\boldsymbol{\theta})\right) \mathbf{u}(t), \,\mathbf{y}(t) = \Psi \mathbf{x}(t) \tag{47}$$

where

update law denoted by *PR*(.) is employed to stop the parametric vector from leaving the set.

a

£- - ç ÷ +

2 1

*<sup>i</sup> jj j j zj j j*

æ ö ¶


*j*

æ ö ¶

¶ è ø

ˆ 2

*y d*

&

( )

<sup>1</sup> <sup>ˆ</sup> <sup>4</sup> *zz z <sup>r</sup>*

22 2

ˆˆ ˆ 11 13 1

L

^ *R*

zw

2 2 2 2

+ + <sup>=</sup>P + + % % % % %

are damping ratios satisfying 0<*ξ<sup>i</sup>* <*ζ<sup>i</sup>* <1. The gain of *R*

xw

*i i ni ni*

where *k* is the number of periodic frequencies, *ωni* is the *i*th disturbance frequency in rad/rev,

Consider the control law of (41) and (45) employed to a nonlinear system with unmodeled dynamics, parametric uncertainty, and output disturbance given by (30). Suppose that

+ ++ D+D

2

*j j si y*

*g z dd y g*

*<sup>y</sup>* | 2 to (42), we have

e

1

2

12

ˆ

2

( )

*T T*

ˆ

*r R*

*z u*

^ (*s*˜) are related by

(*s*˜) as a low‐order and attenuated‐type internal model filter, i.e.,

 and *ζ<sup>i</sup>* .

*r*

*h* , *L <sup>g</sup>L <sup>f</sup>*

(*s*˜) is specified to stabilized the feedback system. Then, the

*r*−1

^) is bounded away from zero. Moreover,

 w

*i ni ni*

 w

^ *<sup>y</sup>*, *d* ^˙ *<sup>y</sup>*, ⋯, *d* ^ *y*

 e

<sup>ˆ</sup> % %% % = - <sup>1</sup> ˆ ˆ <sup>ˆ</sup> () () () () *U s RsCsZ s <sup>R</sup>* (45)

*s s*(46)

^

(*s*˜) at those periodic

(*r*) are known and bounded,

*h* are Lipschitz contin‐

(44)

e

 e

1

*dd P P*

1 ˆ ˆ

*j*=1 *<sup>r</sup>* 1 4*gj* |*d* ^ *si*1 + *d* ^˙

a


*j*

*V cz d z*

*j j r r*

1


ˆ 2

&

1 1 ˆ ˆ 4 4

*g d*

= =

1 1

å å

*r r*

=

1

å

*r*

=

The tracking error *Z*1(*s*˜) and the control input *U*

^

frequencies can be varied by adjusting the values of *ξ<sup>i</sup>*

uous functions, and at least one column of *W*¯(*<sup>y</sup>*

where we have chosen *R*

and *ξ<sup>i</sup>*

*y* ^ *<sup>m</sup>*, *y* ^˙ *<sup>m</sup>*, ⋯, *y* ^ *m* (*r*)

*d* ^ *si*1 (*r*−1) , *d* ^ *si*2 (*r*−2)

**Theorem 3.1**

and *ζ<sup>i</sup>*

, ⋯, *d* ^˙ *sir* <sup>−</sup><sup>1</sup>

Proof: Refer to [36].

suppose that a loop‐shaping filter *C*

1

*j j*

*d*

å

*r*

ee

=

(where *r* is the relative degree) and *d*

^

parametric update law given by (43) yields the bounded tracking error.

are sufficiently smooth, *f* , *g* , *h* , *L <sup>f</sup>*

1 <sup>2</sup> ˆ( ) <sup>2</sup> *k*

*s s R s*


= =

1 1

*si y j i j j*

å å

1

With (43), add and subtract terms ∑

118 Robust Control - Theoretical Models and Case Studies

&

$$\mathbf{x}(t) = \begin{bmatrix} \mathbf{x}\_1(t) & \cdots & \mathbf{x}\_n(t) \end{bmatrix}^T, \quad \Psi = \begin{bmatrix} 0 & \cdots & 0 & 1 \end{bmatrix}, \ B\_c = \begin{bmatrix} 0 & \cdots & 0 & 1 \end{bmatrix}^T$$

*<sup>y</sup>*(*t*) is the system output, *u*(*t*) is the control input, and *φ<sup>f</sup>* (*θ*)= *<sup>φ</sup>*<sup>1</sup> (*θ*) <sup>⋯</sup> *<sup>φ</sup>p*(*θ*) and *φg*(*θ*) are system parameters that are periodic with respect to angular position *θ* (i.e., spatially periodic). Using the aforementioned change of coordinate, we may transform (47) in the time domain into

$$
\hat{\rho}(\boldsymbol{\theta})\dot{\hat{\mathbf{x}}}(\boldsymbol{\theta}) = f\_i(\hat{\mathbf{x}}, \boldsymbol{\varphi}\_f(\boldsymbol{\theta})) + B\_i \mathbf{g}\_i(\hat{\mathbf{x}}, \boldsymbol{\varphi}\_\mathbf{g}(\boldsymbol{\theta})) \hat{\boldsymbol{\mu}}(\boldsymbol{\theta}), \ \hat{\mathbf{y}}(\boldsymbol{\theta}) = \Psi \hat{\mathbf{x}}(\boldsymbol{\theta}) \tag{48}
$$

in the *θ*‐domain. If *ω* ^ (*θ*) equals one of the state variables, (48) is an NPI system in the *θ*‐domain.

**Remark 4.1.** As mentioned previously, uncertainties for rotary systems may be treated as periodic disturbances or periodic parameters. Periodic parametric variation is, in fact, a sensible and practical assumption.

### **4.1 Definitions and assumptions**

In this section, we list and present the definitions and assumptions to be used in the subsequent sections.

**Definition 4.1**. (Lie derivative) The Lie derivative is defined as

$$L\_f^0 h(\mathbf{x}) = h(\mathbf{x}),\\ L\_f h(\mathbf{x}) = \frac{\partial h}{\partial \mathbf{x}} f(\mathbf{x}),\\ L\_f^2 h(\mathbf{x}) = L\_f L\_f h(\mathbf{x}) = \frac{\partial \left( L\_f h \right)}{\partial \mathbf{x}} f(\mathbf{x}),\\ L\_x L\_f h(\mathbf{x}) = \frac{\partial \left( L\_f h \right)}{\partial \mathbf{x}} g(\mathbf{x}), \dots$$

**Definition 4.2**. (Diffeomorphism) A diffeomorphism is considered as a mapping *T* (.):*D* ⊂*R <sup>n</sup>* →*R <sup>n</sup>* being continuously differentiable on *D* and has a continuously differentia‐ ble inverse *T* <sup>−</sup>1(.).

**Definition 4.3**. (Adaptation rate) Instead of constant adaptation rate in regular adaptive control, a varying adaptation rate will be used. Consider a matrix *Γ*(*θ*, *φc*) defined by

$$\Gamma\left(\theta,\,\varphi\_{\,\,c}\right) = \begin{cases} 0, & \theta = 0\\ a\left(\theta\right), & 0 < \theta < \varphi\_{\,\,c} \\ \beta, & \varphi\_{\,\,c} \le \theta \end{cases} \tag{49}$$

where *φc* is the lowest common multiple of the parametric periods, *<sup>β</sup>* <sup>=</sup>*diag*{*β*<sup>1</sup> <sup>⋯</sup> *<sup>β</sup>*ℓ} with nonzero positive constant *β<sup>i</sup>* , and *α*(*θ*)=*diag*{*α*<sup>1</sup> (*θ*) <sup>⋯</sup> *<sup>α</sup>*ℓ(*θ*)} with *α<sup>i</sup>* (*θ*) a strictly increasing function, *α<sup>i</sup>* (0)=0, and *α<sup>i</sup>* (*φc*) =*β<sup>i</sup>* .

**Assumption 4.1**. The desired trajectory (or reference command signal) *ym* is sufficiently smooth or *ym* (*n*) , *ym* (*n*−1) , ⋯, *y*˙ *<sup>m</sup>* exists.

**Assumption 4.2.** For a *θ*‐domain NPI system described by

$$\dot{\hat{\mathbf{x}}}(\theta) = f\left(\hat{\mathbf{x}}(\theta), \,\boldsymbol{\varphi}\_{\uparrow}(\theta)\right) + B\_{\text{c}}\mathbf{g}\left(\hat{\mathbf{x}}(\theta), \,\boldsymbol{\varphi}\_{\downarrow}(\theta)\right) \\ \hat{\boldsymbol{\mu}}(\theta), \,\hat{\mathbf{y}}(\theta) = \Psi \\ \hat{\mathbf{x}}(\theta)$$

the nonlinear functions *f* (*x* ^(*θ*)) and *g*(*<sup>x</sup>* ^(*θ*)) are assumed to linearly relate to the system parameters *φf* and *φg*, i.e.,

$$f\left(\hat{\mathfrak{x}}(\boldsymbol{\theta}),\boldsymbol{\varphi}\_{\boldsymbol{\upbeta}}(\boldsymbol{\theta})\right) = \sum\_{i=1}^{p} \boldsymbol{\uprho}\_{i}(\boldsymbol{\theta}) f\_{i}\left(\hat{\mathfrak{x}}(\boldsymbol{\theta})\right), \\ \boldsymbol{g}\left(\hat{\mathfrak{x}}(\boldsymbol{\theta}),\boldsymbol{\uprho}\_{\boldsymbol{\upbeta}}(\boldsymbol{\theta})\right) = \boldsymbol{\uprho}\_{\boldsymbol{\upbeta}}(\boldsymbol{\theta}) g\left(\hat{\mathfrak{x}}(\boldsymbol{\theta})\right)$$

**Remark 4.2**. Assumption 1 may be satisfied by considering a reference trajectory without sudden change of slope. Assumption 2 may be satisfied by many systems, e.g., LTI and NTI systems.

### **4.2 Spatial‐based adaptive iterative learning control**

For tidy presentation, the *θ* notation will be dropped from most of the equations in the sequel. Rewrite (48) as

$$
\dot{\hat{\mathbf{x}}} = f\left(\hat{\mathbf{x}}, \boldsymbol{\varphi}\_{\uparrow}\right) + B\_c \mathbf{g}\left(\hat{\mathbf{x}}, \boldsymbol{\varphi}\_{\mathbf{g}}\right) \\
\hat{\boldsymbol{\mu}}\_{\prime} \cdot \hat{\mathbf{y}} = \hat{a} \hat{\mathbf{y}} = \Psi \hat{\mathbf{x}} \tag{50}
$$

where the output *y* ^ is equal to the angular velocity *<sup>ω</sup>* ^ , which is set to be the first state of the system. Also note that

$$f\left(\hat{\mathfrak{x}},\boldsymbol{\varphi}\_{\boldsymbol{f}}\right) = f\_t\left(\hat{\mathfrak{x}},\boldsymbol{\varphi}\_{\boldsymbol{f}}\right) \Big/ \hat{\mathfrak{x}}\_1 \text{ and } \operatorname{g}\left(\hat{\mathfrak{x}},\boldsymbol{\varphi}\_{\boldsymbol{g}}\right) = \operatorname{g}\_t\left(\hat{\mathfrak{x}},\boldsymbol{\varphi}\_{\boldsymbol{g}}\right) \Big/ \hat{\mathfrak{x}}\_1$$

The system (50) is valid within the set *D*<sup>0</sup> ={*x* ^ <sup>∈</sup>*<sup>R</sup>* <sup>|</sup> *<sup>x</sup>* ^ <sup>1</sup> ≠0}. Within this set, a diffeomorphism *T* (*x* ^):*<sup>D</sup>*0⊂*D* (as defined previously) exists and may be described by

$$\hat{\mathbf{z}} = T\left(\hat{\mathbf{x}}\right) = \left[L\_f^0 h\left(\hat{\mathbf{x}}\right) \quad L\_f h\left(\hat{\mathbf{x}}\right) \quad \cdots \quad L\_f^{n-1} h\left(\hat{\mathbf{x}}\right)\right]^\dagger \tag{51}$$

where *z* ^ <sup>=</sup> *<sup>z</sup>* ^ <sup>1</sup> ⋯ *z* ^ *<sup>n</sup> <sup>T</sup>* . Using (51), we may transform (50) into

$$\dot{\hat{z}} = A\_c \hat{z} + B\_c \Big[ L\_f^n h\left(\hat{\mathbf{x}}\right) + L\_{\hat{\mathbf{x}}} L\_f^{n-1} h\left(\hat{\mathbf{x}}\right) \hat{\boldsymbol{\mu}} \Big]\_{\hat{\mathbf{x}} = \boldsymbol{T}^{-1}\left(\hat{\mathbf{z}}\right)}, \quad \hat{\mathbf{y}} = \hat{\mathbf{z}}\_1 \tag{52}$$

where

( ) ( )

, and *α*(*θ*)=*diag*{*α*<sup>1</sup>

 aq

, ,0 , *c c*

<sup>ì</sup> <sup>=</sup> <sup>ï</sup> G = << <sup>í</sup>

b

qj

nonzero positive constant *β<sup>i</sup>*

(0)=0, and *α<sup>i</sup>*

120 Robust Control - Theoretical Models and Case Studies

, ⋯, *y*˙ *<sup>m</sup>* exists.

*x fx* &

( () () q j q

q

the nonlinear functions *f* (*x*

parameters *φf* and *φg*, i.e.,

(*φc*) =*β<sup>i</sup>* .

**Assumption 4.2.** For a *θ*‐domain NPI system described by

ˆ ˆ () () ()

 q jq

^(*θ*)) and *g*(*<sup>x</sup>*

 jq

=

*i*

**4.2 Spatial‐based adaptive iterative learning control**

&

j

( )( ) jj

^ is equal to the angular velocity *<sup>ω</sup>*

*p*

function, *α<sup>i</sup>*

or *ym* (*n*) , *ym* (*n*−1)

systems.

Rewrite (48) as

where the output *y*

system. Also note that

q

0, 0

 q j (49)

(*θ*) a strictly increasing

*c*

(*θ*) <sup>⋯</sup> *<sup>α</sup>*ℓ(*θ*)} with *α<sup>i</sup>*

 q  q

^(*θ*)) are assumed to linearly relate to the system

 q

> q

 jq

<sup>ï</sup> £ <sup>î</sup>

where *φc* is the lowest common multiple of the parametric periods, *<sup>β</sup>* <sup>=</sup>*diag*{*β*<sup>1</sup> <sup>⋯</sup> *<sup>β</sup>*ℓ} with

**Assumption 4.1**. The desired trajectory (or reference command signal) *ym* is sufficiently smooth

 q jq

 q

*f x f x gx g x*

*f i i g g*

= = å<sup>1</sup> ˆ , ˆ ˆ , , ˆ

**Remark 4.2**. Assumption 1 may be satisfied by considering a reference trajectory without sudden change of slope. Assumption 2 may be satisfied by many systems, e.g., LTI and NTI

For tidy presentation, the *θ* notation will be dropped from most of the equations in the sequel.

*x f x Bg x u y x*

ˆ ˆ ˆ ˆˆ ˆ ˆ =+ = () () , , ,

= = 1 1 ( )( ) ˆ ˆˆ ˆ ˆˆ , , and , , *ft f gt g f x f x x gx g x x*

jw

jj

*fc g* = Y (50)

^ , which is set to be the first state of the

 q j q j q

) () () ( ) ( () ()) () () ( )

= ( , *fc g* ) + = *Bg x u y x* ( ˆ ˆˆ ˆ () () , ) () () () , Y

$$A\_\circ = \begin{bmatrix} \mathbf{0} & I\_{\left(\mathfrak{u}-1\right)\times\left(\mathfrak{u}-1\right)} \\ \mathbf{0} & \mathbf{0}\_{1\times\left(\mathfrak{u}-1\right)} \end{bmatrix}$$

According to Assumption 3.2, we may rewrite (52) as

$$\dot{\hat{z}} = A\_c \hat{z} + B\_c \left[ \rho \left( \hat{z} \right) + \Theta^T \mathcal{W}\_f \left( \hat{z} \right) + \varphi\_{\hat{g}} \mathcal{W}\_{\hat{g}} \left( \hat{z} \right) \hat{\mu} \right], \tag{53}$$

where <sup>Θ</sup> <sup>=</sup> *<sup>φ</sup>*<sup>1</sup> <sup>⋯</sup> *<sup>φ</sup><sup>p</sup> <sup>φ</sup><sup>g</sup>* <sup>⋯</sup> *<sup>T</sup>* is the actual parametric vector, *φg* is a parameter mapped via the diffeomorphism, *Wf* (*z* ^) is a vector of nonlinear terms, and *ρ*(*<sup>z</sup>* ^) and *Wg*(*<sup>z</sup>* ^) are two nonlinear functions.

Consider a reference trajectory *ym*(*t*) satisfying Assumption 3.1, which may be transformed into its counterpart in the *θ*‐domain, i.e., *y* ^ *<sup>m</sup>*(*θ*)= *ym*(*<sup>λ</sup>* <sup>−</sup>1(*θ*)) <sup>=</sup> *ym*(*t*). Define another state or coordinate transformation:

$$
\hat{z}\_{1r} = \hat{y}\_{\; \; \; \; \prime} (\theta)\_{\prime} \,\, \hat{z}\_{2r} = \dot{\hat{y}}\_{\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \hat{z}\_{nr} = \hat{y}\_{\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \;$$

We may form a state space model, which produces the reference trajectory, as

$$
\dot{\hat{z}}\_r = A\_c \hat{z}\_r + B\_c \hat{y}\_m^{(\iota)} \tag{54}
$$

where *z* ^ *<sup>r</sup>* <sup>=</sup> *<sup>z</sup>* ^ <sup>1</sup>*<sup>r</sup>* ⋯ *z* ^ *nr <sup>T</sup>* . Define the tracking error as *e* ^= *<sup>z</sup>* ^ <sup>−</sup> *<sup>z</sup>* ^ *<sup>r</sup>*. Then, the error dynamics can be obtained using the first equation of (53) and (54), i.e.,

$$\dot{\hat{e}} = A\hat{e} + B\_{\text{c}} \left[ \Theta^{\top} \mathcal{W}\_{f} + \rho + \sigma - \hat{y}\_{\text{w}}^{(\text{v})} + \varphi\_{\text{g}} \mathcal{W}\_{\text{g}} \hat{\mu} \right] \tag{55}$$

where σ = ce with c =[ G

$$A = \begin{bmatrix} \mathbf{0}\_{\left(\mathfrak{u}-1\right)\times 1} & I\_{\left(\mathfrak{u}-1\right)\circ \left(\mathfrak{u}-1\right)} \\ & c \end{bmatrix}$$

Next, specify an LKF as

$$V = \sigma^2 \Big/ 2\rho\_\wp + 1/2 \int\_{\theta - \phi\_\ddagger}^{\theta} \Phi^\top(\tau) \,\beta^{-1} \Phi(\tau) d\tau \tag{56}$$

where ®=Θ =Θ and Θ is a vector of parameters (to be defined later). Θ is the estimate of ಲ . The objective for the following steps is to establish a suitable control input and parametric update law rendering the derivative of the LKF negative semidefinite. Calculating the derivative of V , we obtain

$$\dot{V} = V\_1 + V\_2 \text{ \text{ } } \ V\_1 = \frac{\sigma}{\rho\_\varepsilon} c \dot{\hat{e}} - \frac{\sigma^2}{2\rho\_\varepsilon^2} \dot{\rho}\_{\xi'} \text{ \text{ } } V\_2 = 1/2 \left[ \Phi^\top(\theta) \beta^{-1} \Phi(\theta) - \Phi^\top(\theta - \phi\_\varepsilon) \beta^{-1} \Phi(\theta - \phi\_\varepsilon) \right]. \tag{57}$$

Substituting the error dynamics (55) into V1 and recalling that o =ce, we have

$$V\_1 = \sigma \left[ \frac{1}{\rho\_\mathcal{g}} \left( \overleftrightarrow{\alpha} + \boldsymbol{\rho} - \hat{\boldsymbol{y}}\_m^{(n)} \right) + \frac{\Theta^T}{\rho\_\mathcal{g}} \boldsymbol{W}\_f - \frac{\dot{\boldsymbol{\rho}}\_\mathcal{g}}{\rho\_\mathcal{g}} \frac{\sigma^2}{2} + \boldsymbol{W}\_\mathcal{g} \hat{\boldsymbol{\mu}} \right]. \tag{58}$$

where c=[0 G1 ···· cm-1]. Hence, we may specify û as

$$\hat{\boldsymbol{\mu}} = -\mathbf{1} / \mathcal{W}\_{\boldsymbol{\mathcal{S}}} \left( k \boldsymbol{\sigma} + \tilde{\boldsymbol{\Theta}}^{\top} \mathcal{W} \right) \tag{59}$$

where k is a positive variable, ⊙ is the corresponding estimate of

$$\overline{\Theta} = \left[ \frac{1}{\overline{\rho}\_{\mathcal{S}}} \quad \frac{\Theta^{T}}{\overline{\rho}\_{\mathcal{S}}} \quad \frac{\dot{\overline{\rho}}\_{\mathcal{S}}}{\overline{\rho}\_{\mathcal{S}}} \right]^{T}, \text{ and } \mathcal{W} = \left[ \left( \overline{c}\overline{e} + \rho - \widehat{y}\_{m}^{(\boldsymbol{\imath})} \right) \quad \mathcal{W}\_{\boldsymbol{f}} \quad -\frac{\sigma^{2}}{2} \right]^{T}.$$

Robust Adaptive Repetitive and Iterative Learning Control for Rotary Systems Subject to Spatially Periodic 123 Uncertainties http://dx.doi.org/10.5772/63082

This will simplify V1, i.e.,

$$V\_1 = -k\sigma^2 + \sigma \Phi^\dagger W\tag{60}$$

Using the periodicity of @ (0)=0 (0-02), we may rewrite V, as

$$V\_2 = 1/2\left[\left(\overline{\Theta} - \overline{\tilde{\Theta}}\right)^T \beta^{-1} \left(\overline{\Theta} - \overline{\tilde{\Theta}}\right) - \left(\overline{\Theta} - \overline{\tilde{\Theta}}(\theta - \varphi\_c)\right)^T \beta^{-1} \left(\overline{\Theta} - \overline{\tilde{\Theta}}(\theta - \varphi\_c)\right)\right].\tag{61}$$

According to the following algebraic relationship,

$$\left(\left(a-b\right)^{T}\beta\left(a-b\right)-\left(a-c\right)^{T}\beta\left(a-c\right)-1/2\left(c-b\right)^{T}\beta\left[2\left(a-b\right)+\left(b-c\right)\right]\right)$$

where a, b, and c are vectors, (61) implies that

$$V\_2 = 1/2\left(\tilde{\overline{\Theta}}\left(\theta - \varphi\_c\right) - \tilde{\overline{\Theta}}\right)^\top \beta^{-1} \left[2\left(\tilde{\Theta} - \tilde{\overline{\Theta}}\right) + \left(\tilde{\overline{\Theta}} - \tilde{\overline{\Theta}}\left(\theta - \varphi\_c\right)\right)\right] \tag{62}$$

Therefore, we may specify a periodic parametric update law as

$$\tilde{\vec{\Theta}}(\theta) = \tilde{\vec{\Theta}}(\theta - \varphi\_{\varepsilon}) + \Gamma(\theta, \varphi\_{\varepsilon}) \mathcal{W} \sigma; \ \tilde{\vec{\Theta}}(\theta) = 0 \text{ if } -\varphi\_{\varepsilon} \le \theta \le 0 \tag{63}$$

Recall that I (0, q.) is the adaptation rate as defined in (49). For q. ≤0, V2 becomes

$$
\Delta V\_{\gamma} = -\sigma \Phi^T \mathbf{V} \mathbf{V} - \mathbf{1} / 2 \,\sigma^T \mathbf{V} \mathbf{V}^T \beta \mathbf{W} \sigma \tag{64}
$$

With (60) and (64), we conclude that

$$\dot{V} = -k\sigma^2 - 1/2\,\sigma^T V V^\dagger \beta V \sigma \le -k\sigma^2\tag{65}$$

The objective is achieved. The main results are summarized in the following theorem.

Theorem 4.1 Consider a spatial-based nonlinear system (50) with spatially periodic parameters satisfying Assumption 3.2. The error dynamics described by (55) exists under Assumption 3.1. Assume that the control input is determined by (59) along with the periodic parametric adaptation law (63). Then, the tracking error e will converge to 0 with the performance characteristics described by

$$\int\_{\theta-\varphi\_{\mathbb{C}}}^{\theta} \left\| \hat{e} \right\|^{2} d\tau \to 0, \text{ as } \theta \to \infty$$

**Proof:** Refer to [37].
