**5.2 State estimation and attack reconstruction in linear systems by using super twisting SMO**

Consider the completely observable linearized system Eq. (11) with *C*1ð Þ¼ *x C*1*x*, *C*2ð Þ¼ *x C*2*x*, *B*1ð Þ¼ *x B*, that is,

$$
\dot{\mathbf{x}} = A\mathbf{x} + B\_1 d\_\mathbf{x}(t), \qquad \overline{\mathbf{y}}\_1 = C\_1 \mathbf{x}, \quad \overline{\mathbf{y}}\_2 = C\_2 \mathbf{x} + D\_1 d\_\mathbf{y}(t) \tag{40}
$$

where *B*<sup>1</sup> ∈ R*<sup>n</sup>*�*m*<sup>1</sup> , *C*<sup>1</sup> ∈ R<sup>ð</sup> *<sup>p</sup>*�ð Þ *<sup>m</sup>*�*m*<sup>1</sup> Þ�*<sup>n</sup>*, *C*<sup>2</sup> ∈ Rð Þ� *<sup>m</sup>*�*m*<sup>1</sup> *<sup>n</sup>*.

**Assumption (A5):** The number of uncorrupted/protected measurements is equal or larger than the number of state/plant attack, i.e., *p*<sup>1</sup> ¼ *p* � ð Þ *m* � *m*<sup>1</sup> ≥ *m*1. The system Eq. (40) is assumed to have an input-output vector relative degree *r* ¼ *r*1*;r*2*;* …*;rp*<sup>1</sup> , where *relative degree ri* for *<sup>i</sup>* <sup>¼</sup> <sup>1</sup>*,* <sup>2</sup>*,* …*, p*<sup>1</sup> is defined as follows:

$$\begin{aligned} \mathbf{C}\_{\text{li}}A^{j}B\_{1} &= \mathbf{0} \quad \text{for} \quad \text{all} \quad j \le r\_{i} - \mathbf{1} \\ \mathbf{C}\_{\text{li}}A^{r\_{i}-1}B\_{1} &\neq \mathbf{0} \end{aligned} \tag{41}$$

Without loss of generality, it is assumed that *r*<sup>1</sup> ≤ … ≤*rp*<sup>1</sup> .

#### *5.2.1 Attack observation*

**Assumption (A6):** there exists a full rank matrix.

$$\mathbf{C}\_{a} = \begin{bmatrix} \mathbf{C}\_{1} \\ \vdots \\ \mathbf{C}\_{1}A^{r\_{a\_{1}}-1} \\ \vdots \\ \mathbf{C}\_{p\_{1}} \\ \vdots \\ \mathbf{C}\_{p\_{1}}A^{r\_{a\_{p\_{1}}-1}} \end{bmatrix} \tag{42}$$

The scalar function *Ei* is defined as

*DOI: http://dx.doi.org/10.5772/intechopen.88669*

*Ei* <sup>¼</sup> <sup>1</sup> *if* <sup>~</sup>*yi*þ<sup>1</sup>

� � �

\_

*dx* <sup>¼</sup> ð Þ *CaB TCaB* � ��<sup>1</sup>

> *;* …*;e<sup>r</sup>αi*�<sup>1</sup> *p*1

*e*\_ ¼ *A* � *GlCa*

� � <sup>¼</sup> rank *<sup>B</sup>*<sup>1</sup>

*s*\_ ¼ *Cae*\_ ¼ *Ca A* � *GlCa*

then state/plant attacks are reconstructed as follows:

^

*p*1

h i*<sup>T</sup>*

estimation error *ey* ¼ *Cax* � *y* with

*rαi*�1 <sup>1</sup> *;* …*;e*<sup>1</sup>

<sup>1</sup>*;* …*;e*

then it follows that

*<sup>e</sup>*\_ <sup>¼</sup> *<sup>x</sup>* � \_

Since rank *CaB*<sup>1</sup>

as *e* ! 0; then

obtained as (50).

reconstructed

**15**

*ey* <sup>¼</sup> *<sup>e</sup>*<sup>1</sup>

*<sup>j</sup>* � *<sup>y</sup>i*þ<sup>1</sup> *j*

� �

*Secure State Estimation and Attack Reconstruction in Cyber-Physical Systems: Sliding Mode…*

*<sup>ν</sup>*ðÞ¼ *<sup>s</sup> <sup>ξ</sup>*ðÞþ*<sup>s</sup> <sup>λ</sup>s*j j*<sup>s</sup>* <sup>1</sup>*=*<sup>2</sup> *sign s*ð Þ

*ξ*ðÞ¼ *s βssign s*ð Þ*, λs, β<sup>s</sup>* . 0

**Theorem 3:** Assuming the assumptions (A5) and (A6) hold for system Eq. (40),

Proof: Defining the state estimation error as *e* ¼ *x* � *x*^ and the augmented output

*, y* <sup>¼</sup> *<sup>y</sup>*<sup>1</sup>

By choosing suitable gains *λ<sup>s</sup>* and *β<sup>s</sup>* in the output injections Eq. (49), then.

ð Þ *A; B;Ca* lie in the left half plane, based on the design methodologies in [35], It follows that *e* ¼ 0 is an asymptotically stable equilibrium point of Eq. (52) and dynamics are independent of *dx*ð Þ*t* once a sliding motion on the sliding manifold

where ð Þ *υ<sup>c</sup> eq* is the equivalent output error injection required to maintain the system on the sliding manifold. Since *CaB*<sup>1</sup> is full rank, the attack reconstruction is

According to (A1), *D*<sup>1</sup> is full rank; then sensor attacks in Eq. (40) are

for all *t* . *T* [33]. Then, the error dynamics Eq. (52) is rewritten as

*s* ¼ *Cae* ¼ 0 has been attained. During the sliding mode *s*\_ ¼ *s* ¼ 0, it is

<sup>1</sup>*;* …*; y*

*<sup>x</sup>*^ <sup>¼</sup> *Ae* <sup>þ</sup> *<sup>B</sup>*1*dx*ðÞ�*<sup>t</sup> Gl ya* � *Cax*^ � � � *Gnυ<sup>c</sup> ya* � *Cax*^ � � (52)

*rαi*�1 <sup>1</sup> *;* …*; y*<sup>1</sup>

� �*<sup>e</sup>* <sup>þ</sup> *<sup>B</sup>*1*dx*ðÞ�*<sup>t</sup> Gnυc*ð Þ *Cae* (54)

� �*<sup>e</sup>* <sup>þ</sup> *CaB*1*dx*ðÞ�*<sup>t</sup> CaGnυc*ð Þ¼ *Cae* 0 (55)

*CaGn*ð Þ *υ<sup>c</sup> eq* ! *CaB*1*dx*ð Þ*t* (56)

� � and by assumption the invariant zeros of the triple

and the continuous injection term *ν*ð Þ*:* is given by the super twisting algorithm [34]:

�≤*<sup>ε</sup> for all j*≤*i, else Ei* <sup>¼</sup> 0 (48)

ð Þ *CaB TCaGn*ð Þ *<sup>υ</sup><sup>c</sup> eq* (50)

*p*1

h i*<sup>T</sup>*

*ya* ¼ *Cax* (53)

*;* …*; y<sup>r</sup>αi*�<sup>1</sup> *p*1

(49)

(51)

where integers 1≤*r<sup>α</sup><sup>i</sup>* ≤*ri* are such that rankð Þ¼ *CaB* rankð Þ *B* and *r<sup>α</sup><sup>i</sup>* are chosen such that P*p*<sup>1</sup> *<sup>i</sup>*¼<sup>1</sup> *<sup>r</sup><sup>α</sup><sup>i</sup>* is minimal.

The following SMO [33] is used to estimate the states of system Eq. (40):

$$
\dot{\hat{\mathbf{x}}} = A\hat{\mathbf{x}} + G\_l \left( \mathbf{y}\_a - \mathbf{C}\_a \hat{\mathbf{x}} \right) + G\_n \nu\_c \left( \mathbf{y}\_a - \mathbf{C}\_a \hat{\mathbf{x}} \right) \tag{43}
$$

where the matrices of appropriate dimensions *Gl* and *Gn* are to be designed, and *υc*ð Þ*:* is an injection vector

$$\nu\_c \left( y\_a - \mathbf{C}\_a \hat{\mathbf{x}} \right) = \begin{cases} -\rho \frac{P \left( y\_a - \mathbf{C}\_a \hat{\mathbf{x}} \right)}{||P \left( y\_a - \mathbf{C}\_a \hat{\mathbf{x}} \right)||} & \text{if } \left( y\_a - \mathbf{C}\_a \hat{\mathbf{x}} \right) \neq \mathbf{0} \\\ \mathbf{0} & \text{otherwise} \end{cases} \tag{44}$$

where *ρ* . 0 is larger than the upper bound of unknown input *d t*ð Þ.

The definition of the symmetric positive definite matrix *P* can be found in [33]. The auxiliary output *ya* is defined by

$$\mathbf{y}\_a = \begin{bmatrix} \mathbf{y}\_1 \\ \nu (\mathbf{y}\_1 - \mathbf{y}^{r\_1}) \\ \vdots \\ \nu (\bar{\mathbf{y}}\_1^{r\_1 - 1} - \bar{\mathbf{y}}\_1^{r\_1 - 1}) \\ \vdots \\ \mathbf{y}\_{p\_1} \\ \vdots \\ \nu (\bar{\mathbf{y}}\_{p\_1}^{r\_{p\_1} - 1} - \bar{\mathbf{y}}\_{p\_1}^{r\_{p\_1} - 1}) \end{bmatrix} \tag{45}$$

where the constituent signals in Eq. (45) are given from the continuous secondorder sliding mode observer as

$$\begin{aligned} \dot{\boldsymbol{y}}\_i^1 &= \nu \left( \boldsymbol{y}\_i - \boldsymbol{y}\_i^1 \right) \\ \dot{\boldsymbol{y}}\_i^2 &= \mathbf{E}\_1 \nu \left( \ddot{\boldsymbol{y}}\_i^2 - \boldsymbol{y}\_i^2 \right) \\ &\vdots \\ \dot{\boldsymbol{y}}\_i^{r\_{ai}-1} &= \mathbf{E}\_{r\_{ai}-2} \nu \left( \ddot{\boldsymbol{y}}\_i^{r\_{ai}-1} - \boldsymbol{y}\_i^{r\_{ai}-1} \right) \end{aligned} \tag{46}$$

for 1≤*i*≤ *p*1, with

$$\tilde{\mathcal{Y}}\_i^1 = \mathcal{Y}\_i,\ \tilde{\mathcal{Y}}\_i^j = \nu \left( \tilde{\mathcal{Y}}\_i^{j-1} - \mathcal{Y}\_i^{j-1} \right), \quad 2 \le j \le r\_{a\_i} - 1 \tag{47}$$

*Secure State Estimation and Attack Reconstruction in Cyber-Physical Systems: Sliding Mode… DOI: http://dx.doi.org/10.5772/intechopen.88669*

The scalar function *Ei* is defined as

$$E\_i = \mathbf{1} \quad \text{if} \quad \left| \bar{\mathbf{y}}\_j^{i+1} - \mathbf{y}\_j^{i+1} \right| \le \epsilon for \text{ all } \ j \le i, \text{ else } E\_i = \mathbf{0} \tag{48}$$

and the continuous injection term *ν*ð Þ*:* is given by the super twisting algorithm [34]:

$$\begin{aligned} \nu(\mathfrak{s}) &= \xi(\mathfrak{s}) + \lambda\_{\mathfrak{s}} |\mathfrak{s}|^{1/2} \text{sign}(\mathfrak{s}) \\ \dot{\xi}(\mathfrak{s}) &= \beta\_{\mathfrak{s}} \text{sign}(\mathfrak{s}), \quad \lambda\_{\mathfrak{s}} \beta\_{\mathfrak{s}} \succeq \mathbf{0} \end{aligned} \tag{49}$$

**Theorem 3:** Assuming the assumptions (A5) and (A6) hold for system Eq. (40), then state/plant attacks are reconstructed as follows:

$$\hat{d}\_{\mathbf{x}} = \left( (\mathbf{C}\_{a}\mathbf{B})^{T}\mathbf{C}\_{a}\mathbf{B} \right)^{-1} (\mathbf{C}\_{a}\mathbf{B})^{T}\mathbf{C}\_{a}\mathbf{G}\_{n} (\boldsymbol{\nu}\_{c})\_{eq} \tag{50}$$

Proof: Defining the state estimation error as *e* ¼ *x* � *x*^ and the augmented output estimation error *ey* ¼ *Cax* � *y* with

$$\boldsymbol{e}\_{\mathcal{V}} = \begin{bmatrix} \boldsymbol{e}\_1^1, \dots, \boldsymbol{e}\_1^{r\_{\tilde{a}}-1}, \dots, \boldsymbol{e}\_{p\_1}^1, \dots, \boldsymbol{e}\_{p\_1}^{r\_{\tilde{a}}-1} \end{bmatrix}^T, \quad \boldsymbol{y} = \begin{bmatrix} \boldsymbol{y}\_1^1, \dots, \boldsymbol{y}\_1^{r\_{\tilde{a}}-1}, \dots, \boldsymbol{y}\_{p\_1}^1, \dots, \boldsymbol{y}\_{p\_1}^{r\_{\tilde{a}}-1} \end{bmatrix}^T \tag{51}$$

then it follows that

*Ca* ¼

such that P*p*<sup>1</sup>

*υc*ð Þ*:* is an injection vector

*Control Theory in Engineering*

*<sup>i</sup>*¼<sup>1</sup> *<sup>r</sup><sup>α</sup><sup>i</sup>* is minimal.

\_

8 ><

>:

*<sup>υ</sup><sup>c</sup> ya* � *Cax*^ � � <sup>¼</sup> �*<sup>ρ</sup>*

The auxiliary output *ya* is defined by

order sliding mode observer as

for 1≤*i*≤ *p*1, with

**14**

~*y* 1 *<sup>i</sup>* ¼ *yi ,* ~*y j*

*C*1 ⋮ *C*1*Arα*1�<sup>1</sup> ⋮ *Cp*<sup>1</sup> ⋮

(42)

(44)

(45)

(46)

*Cp*<sup>1</sup>

The following SMO [33] is used to estimate the states of system Eq. (40):

*P ya* � *Cax*^ � � *P ya* � *Cax*^ � � � � �

where *ρ* . 0 is larger than the upper bound of unknown input *d t*ð Þ.

*ya* ¼

*y*\_ 1

*y*\_ 2

⋮ *y*\_ *rαi*�1

where integers 1≤*r<sup>α</sup><sup>i</sup>* ≤*ri* are such that rankð Þ¼ *CaB* rankð Þ *B* and *r<sup>α</sup><sup>i</sup>* are chosen

where the matrices of appropriate dimensions *Gl* and *Gn* are to be designed, and

�

0 *otherwise*

The definition of the symmetric positive definite matrix *P* can be found in [33].

*y*1

*<sup>ν</sup> <sup>y</sup>*<sup>1</sup> � *<sup>y</sup>*<sup>1</sup>

<sup>1</sup> � <sup>~</sup>*y<sup>r</sup>*1�<sup>1</sup> 1 � �

> ⋮ *yp*1 ⋮

where the constituent signals in Eq. (45) are given from the continuous second-

*i* � �

> *<sup>i</sup>* � *<sup>y</sup>*<sup>2</sup> *i* � �

> > *<sup>i</sup>* � *y*

*j*�1 *i* � �

*<sup>i</sup>* <sup>¼</sup> *Er<sup>α</sup>i*�<sup>2</sup>*<sup>ν</sup>* <sup>~</sup>*y<sup>r</sup>αi*�<sup>1</sup>

*ν* ~*y<sup>r</sup>*1�<sup>1</sup>

*ν* ~*y rp*1�1 *<sup>p</sup>*<sup>1</sup> � *y*

*<sup>i</sup>* <sup>¼</sup> *<sup>ν</sup> yi* � *<sup>y</sup>*<sup>1</sup>

*<sup>i</sup>* <sup>¼</sup> *<sup>ν</sup>* <sup>~</sup>*<sup>y</sup> <sup>j</sup>*�<sup>1</sup>

*<sup>i</sup>* <sup>¼</sup> *<sup>E</sup>*1*<sup>ν</sup>* <sup>~</sup>*y*<sup>2</sup>

1 � � ⋮

*rp*1�1 *p*1 � �

*<sup>i</sup>* � *y*

*rαi*�1 *i* � �

*,* 2 ≤*j*≤ *r<sup>α</sup><sup>i</sup>* � 1 (47)

*Ar<sup>α</sup>p*<sup>1</sup> �<sup>1</sup>

*<sup>x</sup>*^ <sup>¼</sup> *Ax*^ <sup>þ</sup> *Gl ya* � *Cax*^ � � <sup>þ</sup> *Gnυ<sup>c</sup> ya* � *Cax*^ � � (43)

*if ya* � *Cax*^ � � 6¼ <sup>0</sup>

$$\dot{\varepsilon} = \varkappa - \dot{\hat{\mathbf{x}}} = A\varepsilon + B\_1 d\_\mathbf{x}(t) - G\_l \left( \mathcal{Y}\_a - \mathcal{C}\_a \hat{\mathbf{x}} \right) - G\_n \nu\_\varepsilon \left( \mathcal{Y}\_a - \mathcal{C}\_a \hat{\mathbf{x}} \right) \tag{52}$$

By choosing suitable gains *λ<sup>s</sup>* and *β<sup>s</sup>* in the output injections Eq. (49), then.

$$\mathcal{Y}\_a = \mathbf{C}\_a \mathbf{x} \tag{53}$$

for all *t* . *T* [33]. Then, the error dynamics Eq. (52) is rewritten as

$$
\dot{e} = (\overline{A} - \mathcal{G}\_l \mathcal{C}\_a)e + \overline{B}\_1 d\_\mathbf{x}(t) - \mathcal{G}\_n \nu\_c(\mathcal{C}\_a e) \tag{54}
$$

Since rank *CaB*<sup>1</sup> � � <sup>¼</sup> rank *<sup>B</sup>*<sup>1</sup> � � and by assumption the invariant zeros of the triple ð Þ *A; B;Ca* lie in the left half plane, based on the design methodologies in [35], It follows that *e* ¼ 0 is an asymptotically stable equilibrium point of Eq. (52) and dynamics are independent of *dx*ð Þ*t* once a sliding motion on the sliding manifold *s* ¼ *Cae* ¼ 0 has been attained. During the sliding mode *s*\_ ¼ *s* ¼ 0, it is

$$\dot{\varepsilon} = \mathbf{C}\_{a}\dot{e} = \mathbf{C}\_{a}(\overline{A} - \mathbf{G}\_{l}\mathbf{C}\_{a})e + \mathbf{C}\_{a}\overline{B}\_{1}d\_{\mathbf{x}}(t) - \mathbf{C}\_{a}\mathbf{G}\_{n}\nu\_{\varepsilon}(\mathbf{C}\_{a}\mathbf{e}) = \mathbf{0} \tag{55}$$

as *e* ! 0; then

$$\mathbf{C}\_{a}\mathbf{G}\_{n}(\boldsymbol{\nu}\_{c})\_{eq} \to \mathbf{C}\_{a}\overline{\mathbf{B}}\_{1}d\_{\boldsymbol{x}}(t) \tag{56}$$

where ð Þ *υ<sup>c</sup> eq* is the equivalent output error injection required to maintain the system on the sliding manifold. Since *CaB*<sup>1</sup> is full rank, the attack reconstruction is obtained as (50).

According to (A1), *D*<sup>1</sup> is full rank; then sensor attacks in Eq. (40) are reconstructed

$$
\hat{d}\_{\mathcal{Y}}(t) = \overline{D}\_1^{-1} \left( \overline{y}\_2 - C\_2 \hat{\mathfrak{x}} \right) \tag{57}
$$

**Assumption (A10):** The norm-bounded solution of the internal dynamics

*Secure State Estimation and Attack Reconstruction in Cyber-Physical Systems: Sliding Mode…*

ð Þ *x ;* …*; ηm*11ð Þ *x ;* …*; ηm*1*rm*<sup>1</sup>

If assumption (A9) is satisfied, then it is always possible to find *n* � *r* functions

<sup>∈</sup> <sup>R</sup>*<sup>n</sup>* <sup>n</sup>

In order to estimate the derivatives *δij*ð Þ*t* ∀*i* ¼ 1*,* …*, m*1*,* ∀*j* ¼ 1*,* …*, ri* of the output. *yi* in finite time, higher-order sliding mode differentiators [28] are used here

> <sup>1</sup>*, z*\_ *i* <sup>1</sup> <sup>¼</sup> *vi* 1

*ri*�<sup>1</sup> � *vi ri*�2 � � <sup>þ</sup> *<sup>z</sup><sup>i</sup>*

is a local diffeomorphism in a neighborhood of any point *x*∈ Ω ⊂ Ω ⊂ R*n*, i.e.,

*<sup>x</sup>* <sup>¼</sup> <sup>Ψ</sup>�<sup>1</sup>

<sup>0</sup> � *yi* ð Þ*<sup>t</sup>* � � <sup>þ</sup> *<sup>z</sup><sup>i</sup>*

ð Þ <sup>1</sup>*=*<sup>2</sup> *sign z<sup>i</sup>*

*<sup>r</sup>*1�<sup>1</sup>*,* \_ *δ* ^1 *<sup>r</sup>*<sup>1</sup> <sup>¼</sup> \_ *η*^1 *r*1

ð Þ¼ *x z m*<sup>1</sup> *rm*1�<sup>1</sup>*,* \_ *δ* ^ *m*<sup>1</sup> *<sup>r</sup>*<sup>1</sup> <sup>¼</sup> \_ *η*^ *m*<sup>1</sup> *rm*1

Next, integrate Eq. (60) with *δ* replaced by ^*δ*; estimate of internal dynamics is

ð Þ *<sup>x</sup>*^ � �*<sup>T</sup>*

∈ R*rt*

*η*^*<sup>r</sup>*þ<sup>1</sup>ð Þ *x*^ *η*^*<sup>r</sup>*þ<sup>2</sup>ð Þ *x*^ ⋮ *η*^*n*ð Þ *x*^

1

CCCA

0

BBB@

ð Þ *<sup>x</sup> ; <sup>η</sup>r*þ1ð Þ *<sup>x</sup> ;* …*; <sup>η</sup>n*ð Þ *<sup>x</sup>* �

*ri , z*\_ *i ri* ¼ �*λ<sup>i</sup> ri sign z<sup>i</sup>*

> ð Þ¼ *<sup>x</sup> <sup>z</sup>*<sup>1</sup> *rm*1

∈ R*ri*

^*;* ^*γ* � � (68)

ð Þ *x*^ *; η*^*<sup>r</sup>*þ<sup>1</sup>ð Þ *x*^ *;* …*; η*^*n*ð Þ *x*^

ð Þ¼ *<sup>x</sup> <sup>z</sup>*<sup>1</sup> *r*1

ð Þ *δ; γ* (64)

(63)

*ri* � *vi ri*�1 � �

(65)

(66)

(67)

(69)

(70)

*γ*\_ ¼ *g*ð Þ *δ; γ* is assumed to be locally asymptotically stable [29].

ð Þ *ri=*ð Þ *ri*þ<sup>1</sup> *sign zi*

*ri*�<sup>1</sup> � *vi ri*�2

� �

�

ð Þ¼ *<sup>x</sup> <sup>z</sup>*<sup>1</sup>

Therefore, the following exact estimates are available in finite time:

\_ ^*γ* ¼ *g δ*

^*γ* 1 ^*γ* 2 ⋮ ^*γ<sup>n</sup>*�*<sup>r</sup>*

0

BBB@

and with some initial condition from the stability domain of the internal

1

CCCA ¼

Therefore, the asymptotic estimate for the mapping (63) is identified as

ð Þ *x*^ *;* …*; η*^*<sup>m</sup>*11ð Þ *x*^ *;* …*; η*^*<sup>m</sup>*1*rm*<sup>1</sup>

n o

� �*<sup>T</sup>* <sup>¼</sup> *<sup>η</sup>*^*<sup>i</sup>*1ð Þ *<sup>x</sup>*^ *; <sup>η</sup>*^*<sup>i</sup>*2ð Þ *<sup>x</sup>*^ *;* …*; <sup>η</sup>*^*ir*<sup>1</sup>

*ηr*þ1ð Þ *x ,* …*, ηn*ð Þ *x* such that

*DOI: http://dx.doi.org/10.5772/intechopen.88669*

Ψð Þ¼ *x col η*11ð Þ *x ;* …*; η*1*r*<sup>1</sup>

*ri*�<sup>1</sup>*, vi*

*δ* ^1 <sup>1</sup> <sup>¼</sup> *<sup>η</sup>*^<sup>1</sup>

⋮ *δ* ^*<sup>m</sup>*<sup>1</sup> <sup>1</sup> ¼ *η*^ *m*<sup>1</sup>

*ri*�<sup>1</sup> ¼ �*λ<sup>i</sup>*

for *i* ¼ 1*,* …*, m*1. By construction,

<sup>1</sup> ð Þ¼ *<sup>x</sup> <sup>z</sup><sup>m</sup>*<sup>1</sup>

^*<sup>i</sup>*1*;* ^*δ<sup>i</sup>*2*;* …*; δ*

∀*i* ¼ 1*,* …*, m*1*, δ*

Ψð Þ¼ *x*^ *col η*^11ð Þ *x*^ *;* …*; η*^1*r*<sup>1</sup>

**17**

<sup>1</sup>ð Þ¼ *<sup>x</sup> <sup>z</sup>*<sup>1</sup>

^*δ<sup>i</sup>* <sup>¼</sup> *<sup>δ</sup>*

*ri*�<sup>1</sup> *<sup>z</sup><sup>i</sup>*

<sup>0</sup>*,* …*, δ* ^1 <sup>1</sup> <sup>¼</sup> *<sup>η</sup>*^<sup>1</sup> *r*1

> <sup>0</sup> *,* …*,* ^*δ<sup>m</sup>*<sup>1</sup> *rm*1 ¼ *η*^ *m*<sup>1</sup> *rm*1

> > ^*ir*1

^ <sup>¼</sup> ^*<sup>δ</sup>* 1 *; δ* ^2 *;* …*; δ* ^*<sup>m</sup>*<sup>1</sup> � �*<sup>T</sup>*

dynamics, a asymptotic estimate ^*γ* can be obtained locally

^*γ* ¼

�

*z*\_ *i* <sup>0</sup> <sup>¼</sup> *vi* 0*, v<sup>i</sup>* <sup>0</sup> ¼ �*λ<sup>i</sup>* <sup>0</sup> *z<sup>i</sup>* <sup>0</sup> � *yi* ð Þ*<sup>t</sup>* � � � �

⋮ *z*\_ *i ri*�<sup>1</sup> <sup>¼</sup> *<sup>v</sup><sup>i</sup>*
