**3. Estimation of multiple time-varying parameters based on the new stripping principle**

The new stripping principle (NSP) in control theory can be effectively to deal with the interactive influence of nodes in complex network systems. Based on this, we use it to strip the unknown disturbance problem in the extended state observer in the time-varying parameter estimation. Therefore, this section proposes an estimation method for multiple time-varying parameters based on the combination of the NSP with the state observer.

### **3.1 The statement of the problem**

Knowing from the definition of *μ*ð Þ*t* that ∣*μ*ð Þ*t* ∣ ≤1, and *μ*\_ðÞ¼� *t γ* ≤ 0, so at a certain moment *t*1, as shown in the **Figure 1**, once *μ*ð Þ*t* reaches the value �*sign*ð Þ *σ*ð Þ*t* ,

*Control Based on PID Framework - The Mutual Promotion of Control and Identification…*

<sup>0</sup> ð Þ� *γ t*<sup>1</sup> � *t*

1 þ *μ t* <sup>0</sup> ð Þ *γ*

, there is *t*>*t*1, there must be *μ*ðÞ�� *t sign*ð Þ *σ*ð Þ*t* . By *σ*ðÞ¼ *t*

∣*e*ð Þ*τ* ∣*dτ*

∣*e*ð Þ*τ* ∣*dτ*

ð*t t*0

∣*e*ð Þ*τ* ∣*dτ*

∣*e*ð Þ*τ* ∣*dτ*

j*e*ð Þj *τ dτ*

ð*t t*0

ð*t t*0

� �

ð*t t*0

> ð*t t*0

<sup>0</sup> ð Þ¼�1 (5)

(6)

(7)

(8)

there is *μ*ðÞ�� *t sign*ð Þ *σ*ð Þ*t* (i.e. *μ*\_ðÞ¼ *t* 0). Otherwise, it contradicts *μ*\_ð Þ*t* ≤0. Here, *t*<sup>1</sup> is found in the following method. Let *t* ¼ *t*1, we have

> *t*<sup>1</sup> ¼ *t* 0 þ

> > ≤ *t* 0 þ 2 *γ*

*σ*ðÞ¼ *t g t*ðÞþ *kμ*ð Þ*t*

¼ *g t*ðÞ� *ksign*ð Þ *σ*ð Þ*t*

ðÞ¼ *t σ*ð Þ*t g t*ðÞ� *ksign*ð Þ *σ*ð Þ*t*

¼ *σ*ð Þ*t g t*ðÞ� *k*∣*σ*ð Þ*t* ∣

≤ ∣*σ*ð Þk *t g t*ð Þ∣ � *k*∣*σ*ð Þ*t* ∣

*μ*ð Þ¼ *t*<sup>1</sup> *μ t*

Then there is

So when *t*>*t*

Ð*t t*0

*g t*ð Þþ *kμ*ð Þ*t*

Then,

**Figure 2.**

**116**

*Overall changes of μ*ð Þ*t over time.*

<sup>0</sup> <sup>þ</sup> <sup>2</sup> *γ*

∣*e*ð Þ*τ* ∣*dτ*, then

*σ*2

The following system with multiple parameters, and the system itself is highly coupled, as shown in the following system:

$$\begin{cases} \dot{\mathbf{x}}\_1 = f\_1(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n, \theta\_1) \\ \dot{\mathbf{x}}\_2 = f\_2(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n, \theta\_2) \\ \vdots \\ \dot{\mathbf{x}}\_n = f\_n(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n, \theta\_n) \\ \mathbf{y} = \mathbf{x} = \begin{bmatrix} \mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n \end{bmatrix}^T \end{cases} \tag{9}$$

where *x*\_ means the derivate function of *x* with respect to time *t*. For this type of coupling problem, the method of NSP was proposed ([7, 9–11]). If *xi* ð Þ *i* ¼ 1, 2, ⋯, *n* is regarded as the interconnected nodes in the network, then the system (9) represents each different network node and the relationship among them. This kind of problems are common in real life, such as WeChat, QQ, Sina Weibo etc. in social networking tools. For example, a complex social system is formed through mutual attention and friendship between people, so here the individual *xi* is one of them, *fi* ð Þ *x*1, *x*2, … , *xn*, *θ<sup>i</sup>* is the interaction (such as relations or research works) among *xi*, there are also other network problems like this.

There are always more or less unknowns in the modeling of such problems, which need to be estimated by using known information, which is the multiparameter estimation problem to be analyzed in this section.

Assuming that *<sup>∂</sup> fi <sup>∂</sup>θ<sup>i</sup>* 6¼ 0, the following subsection uses the method of combining state observer with NSP to study the parameter estimation method. Specific parameter estimation methods, the convergence analysis and simulation research are described in detail in the following subsections respectively.

### **3.2 Parameter estimation method**

This subsection discusses the parameter estimation method based on the combination of the state observer with the new stripping principle. We need to use the state observer to solve the parameter estimation problem, and we consider the time-varying parameters and time-invariant parameters as well.

Firstly, we extend the unknown parameters in the system (9) to states:

*Control Based on PID Framework - The Mutual Promotion of Control and Identification…*

$$\begin{cases}
\dot{\boldsymbol{x}}\_{i} = f\_{1}(\boldsymbol{x}\_{1}, \boldsymbol{x}\_{2}, \dots, \boldsymbol{x}\_{n}, \boldsymbol{x}\_{n+i}) \\
\dot{\boldsymbol{x}}\_{n+i} = \mathbf{g}\_{i}(t), (i = 1, \dots, n)
\end{cases} \tag{10}$$

*ki* >sup *t*≥*t*<sup>0</sup>

*γ*1*<sup>i</sup>* >sup *t*≥*t*<sup>0</sup>

and there is the results as follows:

*DOI: http://dx.doi.org/10.5772/intechopen.97019*

ð*x*1, *x*2, … , *xn*, *xn*þ*<sup>i</sup>*Þ � *fi*

*<sup>x</sup> <sup>j</sup>* <sup>þ</sup> *<sup>∂</sup> fi <sup>∂</sup>xn*þ*<sup>i</sup>*

(19), *xn*þ*<sup>i</sup>* is related to *<sup>x</sup>*\_ *<sup>i</sup>*, And *<sup>∂</sup> fi*

with other function forms of *x*€*i*.

system without unknown *g t*ð Þ:

be obtained that *gi*

**119**

restrain the main part of *<sup>∂</sup> fi*

the following ones:

If *fi*

to P*<sup>n</sup> j*¼1 *∂ fi ∂x <sup>j</sup>* *gi* ð Þ*t*

∣*xi*ð Þ*τ* ∣*dτ*

*Multi-Parameter Estimation of Uncertain Systems Based on the Extended PID Control Method*

*ki*∣*xi*ð Þ*t* ∣ þ *g t* \_ð Þ

∣*xi*ð Þ*τ* ∣*dτ*

Then the system (11) can be used as an observer of the extended system (10),

� � � � �

> � � � � �

lim*t*!<sup>∞</sup> *<sup>x</sup>*^*i*ðÞ¼ *<sup>t</sup> xi*ð Þ*<sup>t</sup>* , *<sup>i</sup>* <sup>¼</sup> 1, … , 2*n:* (18)

*xn*þ*<sup>i</sup>* by Taylor expansion. Select proper *li*ð Þ*t* ð Þ *i* ¼ 1, … , *n* to

*∂ fi <sup>∂</sup>xn*þ*<sup>i</sup>*

At this point, the problem is transformed into a control problem of the system (19).

(i=1,2,...,n). Because the actual value of the parameter *θi*ð Þ *i* ¼ 1, 2, … , *n* is unknown, so *<sup>x</sup>*\_ *<sup>n</sup>*þ*<sup>i</sup>* ð Þ *<sup>i</sup>* <sup>¼</sup> 1, 2, … , *<sup>n</sup>* are also unknown. In order to estimate unknown parameters, it is necessary to find the equivalent or related quantities of *<sup>x</sup>*\_ *<sup>n</sup>*þ*<sup>i</sup>*. From the formula

instead of *<sup>x</sup>*\_ *<sup>n</sup>*þ*<sup>i</sup>*in *<sup>σ</sup>*1*<sup>i</sup>*ð Þ*<sup>t</sup>* , where *kdi* is a design parameter. *<sup>x</sup>*\_ *<sup>n</sup>*þ*<sup>i</sup>* can also be replaced

moment, so the error system (12) is approximately equivalent to the following

*xn*þ*<sup>j</sup>*

*<sup>x</sup>*\_ *<sup>n</sup>*þ*<sup>i</sup>* ¼ �*ln*þ*i*,2ð Þ*<sup>t</sup>* ,ð Þ *<sup>i</sup>* <sup>¼</sup> 1, … , *<sup>n</sup>*

For analyzing the stability of the error systems, the following Lyapunov function

Since *σi*ðÞ¼ *t* E*<sup>i</sup>*1ðÞþ*t ci*E*<sup>i</sup>*2ð Þ*t* (i=1,2,...,n), according to the lemma 1, when the design parameters *b*1*<sup>i</sup>*, *b*2*<sup>i</sup>*, *γ<sup>i</sup>* (i=1,2,...,n) satisfied the conditions that the formula (5.16) and formula (5.17) in Ref. [6], and the time is greater than a certain moment,

According to the lemma 1, when the conditions (16) and (17) are satisfied, it can

ðÞ�*t ln*þ*i*,1ðÞ� *t* 0 ð Þ *i* ¼ 1, … , *n* when the time reaches a certain

*xi* <sup>þ</sup> *cix*\_ *<sup>i</sup>* <sup>¼</sup> <sup>0</sup> ð Þ *<sup>i</sup>* <sup>¼</sup> 1, … , *<sup>n</sup>* (21)

ðÞ�*t ln*þ*<sup>i</sup>*ð Þ*t* , ð Þ *i* ¼ 1, … , *n*

ð Þ *x*^1, *x*^2, … , *x*^*n*, *x*^*<sup>n</sup>*þ*<sup>i</sup>* can be approximately expanded

*x <sup>j</sup>*ð Þ *j* ¼ 1, … , *n* . Then the error system (12) will become

ðÞ�*<sup>t</sup> ln*þ*i*,1ðÞ¼ *<sup>t</sup> <sup>x</sup>*\_ *<sup>n</sup>*þ*<sup>i</sup>* <sup>þ</sup> *ln*þ*i*,2ð Þ*<sup>t</sup>*

is bounded, the observer designed here with *kdi*

*xn*þ*<sup>i</sup>* � *li*ð Þ*t*

*i* ¼ 1, 2, … , *n* (16)

*i* ¼ 1, 2, … , *n* (17)

■

(19)

*x*€*i*

(20)

Ð*t t*0

*ki* Ð*t t*0

� � � � �

Now we prove the theorem 1 according to the lemma 1.

*∂x <sup>j</sup>*

8 ><

>:

Known from the conditions that *σ*1*<sup>i</sup>*ðÞ¼ *t gi*

*<sup>x</sup>*\_ *<sup>i</sup>* <sup>¼</sup> <sup>P</sup>*<sup>n</sup> j*¼1

*<sup>x</sup>*\_ *<sup>n</sup>*þ*<sup>i</sup>* <sup>¼</sup> *gi*

*<sup>∂</sup>xn*þ*<sup>i</sup>*

*<sup>x</sup>*\_ *<sup>i</sup>* <sup>¼</sup> *<sup>∂</sup> fi <sup>∂</sup>xn*þ*<sup>j</sup>*

8 ><

>:

there will be *σi*ðÞ¼ *t* E*<sup>i</sup>*1ðÞþ*t ci*E*<sup>i</sup>*2ðÞ¼ *t* 0, namely:

for the system (20) were constructed:

*∂ fi ∂x <sup>j</sup> x <sup>j</sup>* þ

� � � � �

That is to say, the parameter *θ<sup>i</sup>* is extended to the system state *xn*þ*i*, *i* ¼ 1, … , *n*. Then a state observer is built for the extended system (10):

$$\begin{cases} \dot{\hat{x}}\_i = f\_1(\hat{x}\_1, \hat{x}\_2, \dots, \hat{x}\_n, \hat{x}\_{n+i}) + l\_i(t) \\ \dot{\hat{x}}\_{n+i} = l\_{n+i}(t), (i = 1, \dots, n) \end{cases} \tag{11}$$

where *li*ð Þ*t* ð Þ *i* ¼ 1, … , *n* is the function to be designed. Let the error *x* ¼ *x* � *x*^, then the error system of the established state observer is as follows:

$$\begin{cases}
\dot{\overline{\mathbf{x}}}\_i = f\_1(\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_n, \mathbf{x}\_{n+i}) - f\_1(\hat{\mathbf{x}}\_1, \hat{\mathbf{x}}\_2, \dots, \hat{\mathbf{x}}\_n, \hat{\mathbf{x}}\_{n+i}) - l\_i(t) \\
\dot{\overline{\mathbf{x}}}\_{n+i} = \mathbf{g}\_i(t) - l\_{n+i}(t), (i = 1, \dots, n)
\end{cases} \tag{12}$$

Next, we need to design *li*ð Þ*t* to make the error system (12) asymptotically stable. The choice of *li*ð Þ*t* equivalents to the control problem of the uncertain system (12). We design the control items *li*ð Þ*t* ð*i* ¼ 1, … , 2*n*) according to the error of the states, so that the error system (12) is asymptotically stable to zero.

Regarding the relevant conclusions of the estimation problem with multiple time-varying parameters in a nonlinear system, we present it in the form of the following theorem and give a stability analysis.

**Theorem 1** If the system (10) satisfies *fi* (i=1,2,...,n) is differentiable, and *∂ fi <sup>∂</sup>xn*þ*<sup>i</sup>* 6¼ 0 (i=1,2,...,n), take the error feedback *li*ð Þ*<sup>t</sup>* in the following form:

$$\begin{cases} l\_i(t) = \sum\_{j=1}^n k\_{ij} \text{sign}\left(\overline{\mathbf{x}}\_j\right) \\ l\_{n+i}(t) = l\_{n+i,1}(t) + l\_{n+i,2}(t) \\ l\_{n+i,1}(t) = k\_i \mu\_{1i}(t) \int\_{l\_0}^t |\overline{\mathbf{x}}\_i(\tau)| d\tau \\ l\_{n+i,2}(t) = k\_{n+i,i} |u\_i(t)| \overline{\mathbf{x}}\_i \text{sign}\left(\frac{\partial f\_i}{\partial \mathbf{x}\_{n+i}}\right) \\ i = 1, \dots, n \end{cases} \tag{13}$$

where *ui*ðÞ¼ *<sup>t</sup> <sup>μ</sup>i*ð Þ*<sup>t</sup> <sup>b</sup>*1*<sup>i</sup>*j j <sup>E</sup>*<sup>i</sup>*1ð Þ*<sup>t</sup> <sup>α</sup><sup>i</sup>* <sup>þ</sup> *<sup>b</sup>*2*<sup>i</sup>*j j <sup>E</sup>*<sup>i</sup>*2ð Þ*<sup>t</sup> <sup>α</sup><sup>i</sup>* ð Þ, *<sup>μ</sup>*1*<sup>i</sup>*ð Þ*<sup>t</sup>* is determined by binary control as follows:

$$\dot{\mu}\_{\rm 1i}(t) = \begin{cases} -\gamma\_{\rm 1i} \text{sign}(\sigma\_{\rm 1i}(t)), & |\mu\_{\rm 1i}(t)| \le 1, \ |\mu\_{\rm 1i}(t\_0)| \le 1 \\ -\alpha\_{\rm 1i} \mu\_{\rm 1i}(t), & |\mu\_{\rm 1i}(t)| > 1 \end{cases} \tag{14}$$

where *σ*1*<sup>i</sup>*ðÞ¼ *t gi* ðÞ�*<sup>t</sup> ln*þ*i*,1ðÞ¼ *<sup>t</sup> <sup>x</sup>*\_ *<sup>n</sup>*þ*<sup>i</sup>*ð Þþ*<sup>t</sup> ln*þ*i*,2ð Þ*<sup>t</sup>* <sup>≐</sup>*kdi x*€*i*ðÞþ*t ln*þ*i*,2ð Þ*t* . *μi*ð Þ*t* is determined by binary control as follows:

$$\dot{\mu}\_i(t) = \begin{cases} -\gamma\_i \text{sign}(\sigma\_i(t)), & |\mu\_i(t)| \le 1, |\mu\_i(t\_0)| \le 1 \\ -\alpha\_i \mu\_i(t), & |\mu\_i(t)| > 1 \end{cases} \tag{15}$$

where *σi*ðÞ¼ *t* E*<sup>i</sup>*1ðÞþ*t ci*E*<sup>i</sup>*2ð Þ*t* , *k*ð Þ *<sup>n</sup>*þ*<sup>i</sup>* ,*<sup>i</sup>* is greater than 0, so here it is set that the upper bound of *<sup>k</sup>*ð Þ *<sup>n</sup>*þ*<sup>i</sup>* ,*<sup>i</sup>*∣*ui*∣*<sup>=</sup> <sup>∂</sup> fi <sup>∂</sup>xn*þ*<sup>i</sup>* � � is *Km*. *kdi* , *kij*, *ω*1*<sup>i</sup>*, *ωi*, *ci* are all greater than 0, *γi*, *b*1*<sup>i</sup>* and *b*2*<sup>i</sup>* are design parameters in the formula (5.16) and formula (5.17) in Ref. [6], 0<*α<sup>i</sup>* ≤1, *i*, *j* ¼ 1, … , *n*. The design parameters *ki*, *γ*1*<sup>i</sup>* respectively satisfy:

*Multi-Parameter Estimation of Uncertain Systems Based on the Extended PID Control Method DOI: http://dx.doi.org/10.5772/intechopen.97019*

$$k\_i > \sup\_{t \ge t\_0} \left| \frac{\mathcal{g}\_i(t)}{\int\_{t\_0}^t |\overline{\varpi}\_i(\tau)| d\tau} \right| i = 1, 2, \dots, n \tag{16}$$

$$\gamma\_{1i} > \sup\_{t \ge t\_0} \left| \frac{k\_i |\overline{\mathbf{x}}\_i(t)| + \dot{\mathbf{g}}(t)}{k\_i \int\_{t\_0}^t |\overline{\mathbf{x}}\_i(\tau)| d\tau} \right| i = 1, 2, \dots, n \tag{17}$$

Then the system (11) can be used as an observer of the extended system (10), and there is the results as follows:

$$\lim\_{t \to \infty} \dot{\boldsymbol{\pi}}\_i(t) = \boldsymbol{\pi}\_i(t), \ i = 1, \ldots, 2n. \tag{18}$$

■

Now we prove the theorem 1 according to the lemma 1.

If *fi* ð*x*1, *x*2, … , *xn*, *xn*þ*<sup>i</sup>*Þ � *fi* ð Þ *x*^1, *x*^2, … , *x*^*n*, *x*^*<sup>n</sup>*þ*<sup>i</sup>* can be approximately expanded to P*<sup>n</sup> j*¼1 *∂ fi ∂x <sup>j</sup> <sup>x</sup> <sup>j</sup>* <sup>þ</sup> *<sup>∂</sup> fi <sup>∂</sup>xn*þ*<sup>i</sup> xn*þ*<sup>i</sup>* by Taylor expansion. Select proper *li*ð Þ*t* ð Þ *i* ¼ 1, … , *n* to restrain the main part of *<sup>∂</sup> fi ∂x <sup>j</sup> x <sup>j</sup>*ð Þ *j* ¼ 1, … , *n* . Then the error system (12) will become the following ones:

$$\begin{cases} \dot{\overline{\boldsymbol{x}}}\_i = \sum\_{j=1}^n \frac{\partial f\_i}{\partial \mathbf{x}\_j} \overline{\boldsymbol{x}}\_j + \frac{\partial f\_i}{\partial \mathbf{x}\_{n+i}} \overline{\boldsymbol{x}}\_{n+i} - l\_i(t) \\\\ \dot{\overline{\boldsymbol{x}}}\_{n+i} = \mathbf{g}\_i(t) - l\_{n+i}(t), \ (i = 1, \dots, n) \end{cases} \tag{19}$$

At this point, the problem is transformed into a control problem of the system (19).

Known from the conditions that *σ*1*<sup>i</sup>*ðÞ¼ *t gi* ðÞ�*<sup>t</sup> ln*þ*i*,1ðÞ¼ *<sup>t</sup> <sup>x</sup>*\_ *<sup>n</sup>*þ*<sup>i</sup>* <sup>þ</sup> *ln*þ*i*,2ð Þ*<sup>t</sup>* (i=1,2,...,n). Because the actual value of the parameter *θi*ð Þ *i* ¼ 1, 2, … , *n* is unknown, so *<sup>x</sup>*\_ *<sup>n</sup>*þ*<sup>i</sup>* ð Þ *<sup>i</sup>* <sup>¼</sup> 1, 2, … , *<sup>n</sup>* are also unknown. In order to estimate unknown parameters, it is necessary to find the equivalent or related quantities of *<sup>x</sup>*\_ *<sup>n</sup>*þ*<sup>i</sup>*. From the formula (19), *xn*þ*<sup>i</sup>* is related to *<sup>x</sup>*\_ *<sup>i</sup>*, And *<sup>∂</sup> fi <sup>∂</sup>xn*þ*<sup>i</sup>* is bounded, the observer designed here with *kdi x*€*i* instead of *<sup>x</sup>*\_ *<sup>n</sup>*þ*<sup>i</sup>*in *<sup>σ</sup>*1*<sup>i</sup>*ð Þ*<sup>t</sup>* , where *kdi* is a design parameter. *<sup>x</sup>*\_ *<sup>n</sup>*þ*<sup>i</sup>* can also be replaced with other function forms of *x*€*i*.

According to the lemma 1, when the conditions (16) and (17) are satisfied, it can be obtained that *gi* ðÞ�*t ln*þ*i*,1ðÞ� *t* 0 ð Þ *i* ¼ 1, … , *n* when the time reaches a certain moment, so the error system (12) is approximately equivalent to the following system without unknown *g t*ð Þ:

$$\begin{cases} \dot{\overline{\boldsymbol{x}}}\_i = \frac{\partial f\_i}{\partial \mathbf{x}\_{n+j}} \overline{\boldsymbol{x}}\_{n+j} \\\\ \dot{\overline{\boldsymbol{x}}}\_{n+i} = -l\_{n+i,2}(t), (i = 1, \dots, n) \end{cases} \tag{20}$$

Since *σi*ðÞ¼ *t* E*<sup>i</sup>*1ðÞþ*t ci*E*<sup>i</sup>*2ð Þ*t* (i=1,2,...,n), according to the lemma 1, when the design parameters *b*1*<sup>i</sup>*, *b*2*<sup>i</sup>*, *γ<sup>i</sup>* (i=1,2,...,n) satisfied the conditions that the formula (5.16) and formula (5.17) in Ref. [6], and the time is greater than a certain moment, there will be *σi*ðÞ¼ *t* E*<sup>i</sup>*1ðÞþ*t ci*E*<sup>i</sup>*2ðÞ¼ *t* 0, namely:

$$\overline{\mathfrak{X}}\_{i} + c\_{i}\dot{\overline{\mathfrak{X}}}\_{i} = \mathbf{0} \ \ (i = 1, \ldots, n) \tag{21}$$

For analyzing the stability of the error systems, the following Lyapunov function for the system (20) were constructed:

*x*\_ *<sup>i</sup>* ¼ *f* <sup>1</sup>ð Þ *x*1, *x*2, … , *xn*, *xn*þ*<sup>i</sup>*

*Control Based on PID Framework - The Mutual Promotion of Control and Identification…*

That is to say, the parameter *θ<sup>i</sup>* is extended to the system state *xn*þ*i*, *i* ¼ 1, … , *n*.

*x*^*<sup>i</sup>* ¼ *f* <sup>1</sup>ð*x*^1, *x*^2, … , *x*^*n*, *x*^*n*þ*i*Þ þ *li*ð Þ*t*

where *li*ð Þ*t* ð Þ *i* ¼ 1, … , *n* is the function to be designed. Let the error *x* ¼ *x* � *x*^,

Next, we need to design *li*ð Þ*t* to make the error system (12) asymptotically stable. The choice of *li*ð Þ*t* equivalents to the control problem of the uncertain system (12). We design the control items *li*ð Þ*t* ð*i* ¼ 1, … , 2*n*) according to the error of the states,

Regarding the relevant conclusions of the estimation problem with multiple time-varying parameters in a nonlinear system, we present it in the form of the

**Theorem 1** If the system (10) satisfies *fi* (i=1,2,...,n) is differentiable, and

*kijsign x <sup>j</sup>* � �

> Ð*t t*0

∣*xi*ð Þ*τ* ∣*dτ*

*∂ fi <sup>∂</sup>xn*þ*<sup>i</sup>* � �

*ln*þ*<sup>i</sup>*ðÞ¼ *t ln*þ*i*,1ðÞþ*t ln*þ*i*,2ð Þ*t*

*ln*þ*i*,2ðÞ¼ *t kn*þ*i*,*<sup>i</sup>*∣*ui*ð Þ*t* ∣*xisign*

where *ui*ðÞ¼ *<sup>t</sup> <sup>μ</sup>i*ð Þ*<sup>t</sup> <sup>b</sup>*1*<sup>i</sup>*j j <sup>E</sup>*<sup>i</sup>*1ð Þ*<sup>t</sup> <sup>α</sup><sup>i</sup>* <sup>þ</sup> *<sup>b</sup>*2*<sup>i</sup>*j j <sup>E</sup>*<sup>i</sup>*2ð Þ*<sup>t</sup> <sup>α</sup><sup>i</sup>* ð Þ, *<sup>μ</sup>*1*<sup>i</sup>*ð Þ*<sup>t</sup>* is determined by binary

*<sup>μ</sup>*\_ <sup>1</sup>*<sup>i</sup>*ðÞ¼ *<sup>t</sup>* �*γ*1*isign*ð Þ *<sup>σ</sup>*1*<sup>i</sup>*ð Þ*<sup>t</sup>* , <sup>∣</sup>*μ*1*<sup>i</sup>*ð Þ*<sup>t</sup>* ∣ ≤ 1, <sup>∣</sup>*μ*1*<sup>i</sup>*ð Þ *<sup>t</sup>*<sup>0</sup> ∣ ≤<sup>1</sup> �*ω*1*<sup>i</sup>μ*1*<sup>i</sup>*ð Þ*t* , ∣*μ*1*<sup>i</sup>*ð Þ*t* ∣> 1

*<sup>μ</sup>*\_ *<sup>i</sup>*ðÞ¼ *<sup>t</sup>* �*γisign*ð Þ *<sup>σ</sup>i*ð Þ*<sup>t</sup>* , <sup>∣</sup>*μi*ð Þ*<sup>t</sup>* ∣ ≤1, <sup>∣</sup>*μi*ð Þ *<sup>t</sup>*<sup>0</sup> ∣ ≤ <sup>1</sup> �*ωiμi*ð Þ*t* , ∣*μi*ð Þ*t* ∣>1

where *σi*ðÞ¼ *t* E*<sup>i</sup>*1ðÞþ*t ci*E*<sup>i</sup>*2ð Þ*t* , *k*ð Þ *<sup>n</sup>*þ*<sup>i</sup>* ,*<sup>i</sup>* is greater than 0, so here it is set that the

and *b*2*<sup>i</sup>* are design parameters in the formula (5.16) and formula (5.17) in Ref. [6],

is *Km*. *kdi*

0<*α<sup>i</sup>* ≤1, *i*, *j* ¼ 1, … , *n*. The design parameters *ki*, *γ*1*<sup>i</sup>* respectively satisfy:

ðÞ�*<sup>t</sup> ln*þ*i*,1ðÞ¼ *<sup>t</sup> <sup>x</sup>*\_ *<sup>n</sup>*þ*<sup>i</sup>*ð Þþ*<sup>t</sup> ln*þ*i*,2ð Þ*<sup>t</sup>* <sup>≐</sup>*kdi*

*<sup>∂</sup>xn*þ*<sup>i</sup>* 6¼ 0 (i=1,2,...,n), take the error feedback *li*ð Þ*<sup>t</sup>* in the following form:

*j*¼1

*ln*þ*i*,1ðÞ¼ *t kiμ*1*<sup>i</sup>*ð Þ*t*

*x*^*n*þ*<sup>i</sup>* ¼ *ln*þ*i*ð Þ*t* ,ð Þ *i* ¼ 1, … , *n*

*<sup>x</sup>*\_ *<sup>i</sup>* <sup>¼</sup> *<sup>f</sup>* <sup>1</sup>ð*x*1, *<sup>x</sup>*2, … , *xn*, *xn*þ*i*Þ � *<sup>f</sup>* <sup>1</sup>ð*x*^1, *<sup>x</sup>*^2, … , *<sup>x</sup>*^*n*, *<sup>x</sup>*^*n*þ*i*Þ � *li*ð Þ*<sup>t</sup>*

ð Þ*t* ,ð Þ *i* ¼ 1, … , *n*

(10)

(11)

(12)

(13)

(14)

(15)

*x*€*i*ðÞþ*t ln*þ*i*,2ð Þ*t* . *μi*ð Þ*t* is

, *kij*, *ω*1*<sup>i</sup>*, *ωi*, *ci* are all greater than 0, *γi*, *b*1*<sup>i</sup>*

*x*\_ *<sup>n</sup>*þ*<sup>i</sup>* ¼ *gi*

then the error system of the established state observer is as follows:

ðÞ�*t ln*þ*<sup>i</sup>*ð Þ*t* ,ð Þ *i* ¼ 1, … , *n*

so that the error system (12) is asymptotically stable to zero.

*li*ðÞ¼ *<sup>t</sup>* <sup>P</sup>*<sup>n</sup>*

*i* ¼ 1, … , *n*

following theorem and give a stability analysis.

8

>>>>>>>>>><

>>>>>>>>>>:

�

�

*<sup>∂</sup>xn*þ*<sup>i</sup>* � �

determined by binary control as follows:

�

Then a state observer is built for the extended system (10):

\_

(

*<sup>x</sup>*\_ *<sup>n</sup>*þ*<sup>i</sup>* <sup>¼</sup> *gi*

(

control as follows:

where *σ*1*<sup>i</sup>*ðÞ¼ *t gi*

upper bound of *<sup>k</sup>*ð Þ *<sup>n</sup>*þ*<sup>i</sup>* ,*<sup>i</sup>*∣*ui*∣*<sup>=</sup> <sup>∂</sup> fi*

**118**

*∂ fi*

\_

*Control Based on PID Framework - The Mutual Promotion of Control and Identification…*

$$V\_i = \frac{1}{2} \left( K\_m \overline{\boldsymbol{\pi}}\_i^2 + \overline{\boldsymbol{\pi}}\_{n+i}^2 \right) \ (i = 1, \dots, n) \tag{22}$$

According to the system (25), there is *<sup>∂</sup><sup>f</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.97019*

�

We design its observer as follows:

\_

8 ><

>:

\_

*<sup>x</sup>*^<sup>2</sup> <sup>¼</sup> *<sup>k</sup>*1*μ*11Ð*<sup>t</sup>*

and parameters can be controlled within 10�<sup>2</sup>

*x*\_ <sup>2</sup> ¼ *g*1ð Þ*t*

established.

again.

**Figure 3.**

**121**

*and estimation errors based on NSP.*

described as below:

are both 0 almost never exists, so it can be considered that the condition *<sup>∂</sup><sup>f</sup>*

*Multi-Parameter Estimation of Uncertain Systems Based on the Extended PID Control Method*

*x*^<sup>1</sup> ¼ *f*ð Þþ *x*^1, *x*^2, *t k*11*sign y*ð Þ � *x*^<sup>1</sup>

*t*0

Next, the extended state system based on the parameter estimation method is

*x*\_ <sup>1</sup> ¼ �∣*x*1∣*x*<sup>2</sup> þ *x*<sup>1</sup> þ *cosx*<sup>2</sup> ¼ *f x*ð Þ 1, *x*2, *t*

<sup>∣</sup>*x*1ð Þ*<sup>τ</sup>* <sup>∣</sup>*d<sup>τ</sup>* <sup>þ</sup> *<sup>k</sup>*21∣*u*1∣*x*1*sign <sup>∂</sup><sup>f</sup>*

.

The design of *μ*<sup>11</sup> and *u*<sup>1</sup> is shown in the theorem 1, here we will not repeat them

Firstly, we consider the case where the observation does not contain noise. Set the design parameters in the simulation analysis as *k*<sup>1</sup> ¼ 0*:*1, *kd*<sup>1</sup> ¼ 0*:*1, *k*<sup>11</sup> ¼ 40, *k*<sup>21</sup> ¼ 10, *b*<sup>11</sup> ¼ 1, *b*<sup>12</sup> ¼ 10, *α*<sup>1</sup> ¼ 0*:*5, *c*<sup>1</sup> ¼ 1, *ω*<sup>11</sup> ¼ *ω*<sup>1</sup> ¼ 3, *γ*<sup>11</sup> ¼ *γ*<sup>1</sup> ¼ 10, *μ*11ð Þ¼ 0 *μ*1ð Þ¼ 0 0. Suppose that the initial state of the state observer is 0, 0 ð Þ. We get the estimation of the states and parameters and the estimation errors are shown in the **Figure 3**, the estimation results are satisfied. And after a certain period of time (for example, this simulation is about 7 seconds), the estimated errors of the states

*The case with single parameter and the observation without noise: states, time-varying parameters estimation*

*<sup>∂</sup><sup>θ</sup>* ¼ �∣*x*∣ � *sinθ*. The situation that *x* and *θ*

*∂x*<sup>2</sup> � � � *x*^1,*x*^<sup>2</sup> � � *<sup>∂</sup><sup>θ</sup>* 6¼ 0 is

(26)

(27)

It is easy to know that, except for the origin, *Vi* > 0 ð Þ *i* ¼ 1, … , *n* . Let us analyze the derivative function of *Vi* with respect to time,

$$\begin{split} \dot{V}\_{i} &= K\_{m} \overline{\mathbf{x}}\_{i} \overline{\dot{\mathbf{x}}}\_{i} + \overline{\mathbf{x}}\_{n+i} \overline{\dot{\mathbf{x}}}\_{n+i} \\ &= K\_{m} \overline{\mathbf{x}}\_{i} \overline{\dot{\mathbf{x}}}\_{i} + k\_{n+i,i} |u\_{i}| \overline{\mathbf{x}}\_{i} \text{sign} \left( \frac{\partial f\_{i}}{\partial \mathbf{x}\_{n+i}} \right) \overline{\mathbf{x}}\_{n+i} \\ &= K\_{m} \overline{\mathbf{x}}\_{i} \dot{\overline{\mathbf{x}}}\_{i} + k\_{n+i,i} |u\_{i}| \overline{\mathbf{x}}\_{i} \dot{\overline{\mathbf{x}}}\_{i} / \left( |\partial f\_{i}/\partial \mathbf{x}\_{n+i}| \right) \\ &= \left[ K\_{m} - k\_{n+i,i} |u\_{i}| / \left( |\partial f\_{i}/\partial \mathbf{x}\_{n+i}| \right) \right] \overline{\mathbf{x}}\_{i} \dot{\overline{\mathbf{x}}}\_{i} \\ &\quad (i = 1, \dots, n) \end{split} \tag{23}$$

From the formula (21),

$$\dot{V}\_i = -c\_i \left[ K\_m - k\_{n+i,i} |u\_i| / \left| \frac{\partial f\_i}{\partial \mathbf{x}\_{n+i}} \right| \right] \dot{\tilde{\mathbf{x}}}\_i^2 \ (i = 1, \dots, n) \tag{24}$$

Known by the condition *kn*þ*i*,*<sup>i</sup>*∣*ui*∣*<sup>=</sup> <sup>∂</sup> fi <sup>∂</sup>xn*þ*<sup>i</sup>* has an upper bound *Km*, then *<sup>V</sup>*\_ *<sup>i</sup>* <sup>&</sup>lt;<sup>0</sup> ð Þ *<sup>i</sup>* <sup>¼</sup> 1, … , *<sup>n</sup>* .

In summary, when the time is greater than a certain moment, *x*^*i*, *x*^*<sup>n</sup>*þ*<sup>i</sup>* can be used as the estimation of *xi*, *θ<sup>i</sup>* ð Þ *i* ¼ 1, … , *n* respectively. During this progress, there is nothing to do with the specific form of *g t*ð Þ. ■

**Remark 1** When the parameter is a time-invariant parameter, it is easy to prove that the theorem 1 still works. Because at this time the expanded states *gi* ðÞ¼ *t* 0ð Þ *i* ¼ 1, … , *n* in the (10), then we can take *ln*þ*i*,1ðÞ¼ *t* 0 ð Þ *i* ¼ 1, … , *n* in our control law.■

The subsection focuses on the estimation problem of multiple time-varying parameters in general nonlinear systems. A parameter estimation method based on the combination of the state observer with the new stripping principle is given. Stability analysis is also carried out. The following simulation studies further verify the effectiveness of the parameter estimation method proposed in this subsection.

#### **3.3 Simulation analysis**

This subsection simulates the parameter estimation method proposed in the previous subsection. We have studied the estimation of a single time-varying and time-invariant parameter, and the estimation of multiple time-varying and timeinvariant parameters in a dynamic system respectively. We also consider whether the observation contains observation noise or not. Further verify the robustness of the parameter estimation method.

#### *3.3.1 Single parameter estimation simulation analysis*

**Example 1** We choose the nonlinear system as follow (that is, example 2 in Ref. [5]):

$$
\dot{\mathfrak{x}} = -|\mathfrak{x}|\theta + \mathfrak{x} + \cos \theta \tag{25}
$$

Here, we assume that the true value of the unknown parameter changes with time *θ* ¼ 1 þ *sin* ð Þ 2*t* , and the initial state of the system is *x*ð Þ¼ 0 2.

*Multi-Parameter Estimation of Uncertain Systems Based on the Extended PID Control Method DOI: http://dx.doi.org/10.5772/intechopen.97019*

According to the system (25), there is *<sup>∂</sup><sup>f</sup> <sup>∂</sup><sup>θ</sup>* ¼ �∣*x*∣ � *sinθ*. The situation that *x* and *θ* are both 0 almost never exists, so it can be considered that the condition *<sup>∂</sup><sup>f</sup> <sup>∂</sup><sup>θ</sup>* 6¼ 0 is established.

Next, the extended state system based on the parameter estimation method is described as below:

$$\begin{cases}
\dot{\mathbf{x}}\_1 = -|\mathbf{x}\_1|\mathbf{x}\_2 + \mathbf{x}\_1 + \cos \mathbf{x}\_2 = f(\mathbf{x}\_1, \mathbf{x}\_2, t) \\
\dot{\mathbf{x}}\_2 = \mathbf{g}\_1(t)
\end{cases} \tag{26}$$

We design its observer as follows:

*Vi* <sup>¼</sup> <sup>1</sup> 2

the derivative function of *Vi* with respect to time,

From the formula (21),

*<sup>V</sup>*\_ *<sup>i</sup>* <sup>&</sup>lt;<sup>0</sup> ð Þ *<sup>i</sup>* <sup>¼</sup> 1, … , *<sup>n</sup>* .

**3.3 Simulation analysis**

in Ref. [5]):

**120**

the parameter estimation method.

*3.3.1 Single parameter estimation simulation analysis*

law.■

Known by the condition *kn*þ*i*,*<sup>i</sup>*∣*ui*∣*<sup>=</sup> <sup>∂</sup> fi*

is nothing to do with the specific form of *g t*ð Þ. ■

*Kmx*<sup>2</sup> *<sup>i</sup>* <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> *n*þ*i*

*Control Based on PID Framework - The Mutual Promotion of Control and Identification…*

*<sup>V</sup>*\_ *<sup>i</sup>* <sup>¼</sup> *Kmxix*\_ *<sup>i</sup>* <sup>þ</sup> *xn*þ*ix*\_ *<sup>n</sup>*þ*<sup>i</sup>*

<sup>¼</sup> *Kmxix*\_ *<sup>i</sup>* <sup>þ</sup> *kn*þ*i*,*<sup>i</sup>*∣*ui*∣*xisign*

<sup>¼</sup> *Km* � *kn*þ*i*,*i*j*ui*j*<sup>=</sup>* <sup>j</sup>*<sup>∂</sup> fi*

ð Þ *i* ¼ 1, … , *n*

*<sup>V</sup>*\_ *<sup>i</sup>* ¼ �*ci Km* � *kn*þ*i*,*<sup>i</sup>*j*ui*j*=*<sup>j</sup> *<sup>∂</sup> fi*

<sup>¼</sup> *Kmxix*\_ *<sup>i</sup>* <sup>þ</sup> *kn*þ*i*,*<sup>i</sup>*∣*ui*∣*xix*\_ *<sup>i</sup><sup>=</sup>* <sup>j</sup>*<sup>∂</sup> fi*

that the theorem 1 still works. Because at this time the expanded states *gi*

In summary, when the time is greater than a certain moment, *x*^*i*, *x*^*<sup>n</sup>*þ*<sup>i</sup>* can be used as the estimation of *xi*, *θ<sup>i</sup>* ð Þ *i* ¼ 1, … , *n* respectively. During this progress, there

0ð Þ *i* ¼ 1, … , *n* in the (10), then we can take *ln*þ*i*,1ðÞ¼ *t* 0 ð Þ *i* ¼ 1, … , *n* in our control

The subsection focuses on the estimation problem of multiple time-varying parameters in general nonlinear systems. A parameter estimation method based on the combination of the state observer with the new stripping principle is given. Stability analysis is also carried out. The following simulation studies further verify the effectiveness of the parameter estimation method proposed in this subsection.

This subsection simulates the parameter estimation method proposed in the previous subsection. We have studied the estimation of a single time-varying and time-invariant parameter, and the estimation of multiple time-varying and timeinvariant parameters in a dynamic system respectively. We also consider whether the observation contains observation noise or not. Further verify the robustness of

**Example 1** We choose the nonlinear system as follow (that is, example 2

Here, we assume that the true value of the unknown parameter changes with

time *θ* ¼ 1 þ *sin* ð Þ 2*t* , and the initial state of the system is *x*ð Þ¼ 0 2.

*x*\_ ¼ �∣*x*∣*θ* þ *x* þ *cosθ* (25)

**Remark 1** When the parameter is a time-invariant parameter, it is easy to prove

It is easy to know that, except for the origin, *Vi* > 0 ð Þ *i* ¼ 1, … , *n* . Let us analyze

*<sup>=</sup>∂xn*þ*i*<sup>j</sup> *xix*\_ *<sup>i</sup>*

*<sup>∂</sup>xn*þ*<sup>i</sup>* j *x*\_ 2

*<sup>∂</sup>xn*þ*<sup>i</sup>* has an upper bound *Km*, then

ð Þ *<sup>i</sup>* <sup>¼</sup> 1, … , *<sup>n</sup>* (22)

*∂ fi <sup>∂</sup>xn*þ*<sup>i</sup>* 

*<sup>=</sup>∂xn*þ*i*<sup>j</sup>

*xn*þ*<sup>i</sup>*

*<sup>i</sup>* ð Þ *i* ¼ 1, … , *n* (24)

(23)

ðÞ¼ *t*

$$\begin{cases} \dot{\hat{\boldsymbol{\alpha}}}\_{1} = f(\hat{\boldsymbol{\alpha}}\_{1}, \hat{\boldsymbol{\alpha}}\_{2}, t) + k\_{11} \text{sign}(\boldsymbol{y} - \hat{\boldsymbol{\alpha}}\_{1}) \\ \dot{\hat{\boldsymbol{\alpha}}}\_{2} = k\_{1} \mu\_{11} \int\_{t\_{0}}^{t} |\overline{\mathbf{x}}\_{1}(\tau)| d\tau + k\_{21} |\mu\_{1}| \overline{\mathbf{x}}\_{1} \text{sign}\left(\frac{\partial f}{\partial \mathbf{x}\_{2}}\Big|\_{\hat{\boldsymbol{\alpha}}\_{1}, \hat{\boldsymbol{\alpha}}\_{2}}\right) \end{cases} \tag{27}$$

The design of *μ*<sup>11</sup> and *u*<sup>1</sup> is shown in the theorem 1, here we will not repeat them again.

Firstly, we consider the case where the observation does not contain noise. Set the design parameters in the simulation analysis as *k*<sup>1</sup> ¼ 0*:*1, *kd*<sup>1</sup> ¼ 0*:*1, *k*<sup>11</sup> ¼ 40, *k*<sup>21</sup> ¼ 10, *b*<sup>11</sup> ¼ 1, *b*<sup>12</sup> ¼ 10, *α*<sup>1</sup> ¼ 0*:*5, *c*<sup>1</sup> ¼ 1, *ω*<sup>11</sup> ¼ *ω*<sup>1</sup> ¼ 3, *γ*<sup>11</sup> ¼ *γ*<sup>1</sup> ¼ 10, *μ*11ð Þ¼ 0 *μ*1ð Þ¼ 0 0. Suppose that the initial state of the state observer is 0, 0 ð Þ. We get the estimation of the states and parameters and the estimation errors are shown in the **Figure 3**, the estimation results are satisfied. And after a certain period of time (for example, this simulation is about 7 seconds), the estimated errors of the states and parameters can be controlled within 10�<sup>2</sup> .

#### **Figure 3.**

*The case with single parameter and the observation without noise: states, time-varying parameters estimation and estimation errors based on NSP.*

Consider when the observation of the system (25) contains noise, for example, there is noise in the observation that obeys uniformly distributed in ½ � �0*:*001, 0*:*001 , that is, *y t*ðÞ¼ *x t*ð Þþ Eð Þ*t* , where EðÞ� *t U*½ � �0*:*001, 0*:*001 . Under these circumstances, design parameters are still taken as *k*<sup>1</sup> ¼ 0*:*001, *kd*<sup>1</sup> ¼ 0*:*01, *k*<sup>11</sup> ¼ 40, *k*<sup>21</sup> ¼ 10, *b*<sup>11</sup> ¼ 1, *b*<sup>12</sup> ¼ 10, *α*<sup>1</sup> ¼ 0*:*5, *c*<sup>1</sup> ¼ 1, *ω*<sup>11</sup> ¼ *ω*<sup>1</sup> ¼ 3, *γ*<sup>11</sup> ¼ *γ*<sup>1</sup> ¼ 10, *μ*11ð Þ¼ 0 *μ*1ð Þ¼ 0 0, and suppose that the initial state of the state observer is 0, 0 ð Þ. The estimation errors of the state and parameter are shown in the **Figure 4**. Where the estimation error of the state is 10�<sup>3</sup> , which is larger than the estimation error without noise. The parameter estimation error controlled within 2 � <sup>10</sup>�<sup>2</sup> is larger than the parameter estimation error without noise as well.

Previously, we studied the estimation problem based on the principle of disturbance stripping for the estimation of a single time-varying parameter, and then we will analyze the situation that the unknown parameter does not change with time.

**Example 2** This example is still focusing on the nonlinear system of the system (25):

$$
\dot{\mathbf{x}} = -|\mathbf{x}|\theta + \mathbf{x} + \cos \theta \tag{28}
$$

The simulation shows that the state and parameters have close to the true value within 1 second, the estimation error can be controlled within 10�<sup>2</sup> within 5 seconds, and the state estimation converges to the true state value faster due to the

*The case with single parameter and the observation without noise: states, time-varying parameters estimation*

*Multi-Parameter Estimation of Uncertain Systems Based on the Extended PID Control Method*

*DOI: http://dx.doi.org/10.5772/intechopen.97019*

When there is noise in the observation of the system (25), for example, the observation contains uniformly distributed noise that obeys ½ � �0*:*001, 0*:*001 . That is, *y t*ðÞ¼ *x t*ðÞþ Eð Þ*t* , where EðÞ� *t U*½ � �0*:*001, 0*:*001 . Under these circumstances, the design parameters in simulation analysis are still taken as *k*<sup>1</sup> ¼ 0, *kd*<sup>1</sup> ¼ 0, *k*<sup>11</sup> ¼ 40, *k*<sup>21</sup> ¼ 10, *b*<sup>11</sup> ¼ 1, *b*<sup>12</sup> ¼ 10, *α*<sup>1</sup> ¼ 0*:*5, *c*<sup>1</sup> ¼ 1, *ω*<sup>11</sup> ¼ *ω*<sup>1</sup> ¼ 3, *γ*<sup>11</sup> ¼ *γ*<sup>1</sup> ¼ 10, *μ*11ð Þ¼ 0 *μ*1ð Þ¼ 0 0. Suppose the initial state of the state observer is 0, 0 ð Þ. The estimated errors of the parameters and states are shown in **Figure 6**, the estimation

However, the parameter estimation error is larger than the parameter estimation without noise, but the estimation error can still be controlled within 2 � <sup>10</sup>�<sup>2</sup>

In summary, this subsection studies the application of parameter estimation methods based on the combination of NSP with state observer in the estimation of single parameters of nonlinear systems. This subsection not only analyzed the two cases of time-invariant and time-varying parameters through simulation, but also analyzed the situation that the observations of the system include observation noise. In these simulation studies, based on the preliminary adjusted design parameters, when analyzing the time-varying and time-invariant parameters, and the presence or absence of observation noise, the design parameters were basically not changed, but the simulation results show that the state and parameters in the observer (27) can asymptotically converge to the true value. These studies show the feasibility and robustness of the combination of the state observer with the stripping principle in

the single parameter estimation of nonlinear systems.

, which is more than the estimation error without noise.

.

effect of error feedback.

*and estimation errors based on NSP.*

**Figure 5.**

error of the state is 10�<sup>3</sup>

**123**

It is assumed here that the true value of *θ* is a constant *θ* ¼ 1 that does not change with time, and the initial state value is *x*ð Þ¼ 0 2.

In the simulation analysis, the parameters are time-invariant, so *g*1ðÞ¼ *t* 0, the feedback item *l*<sup>21</sup> can be ignored, and the design parameters *k*<sup>1</sup> ¼ 0, *kd*<sup>1</sup> ¼ 0, *k*<sup>11</sup> ¼ 40, *k*<sup>21</sup> ¼ 10, *b*<sup>11</sup> ¼ 1, *b*<sup>12</sup> ¼ 10, *α*<sup>1</sup> ¼ 0*:*5, *c*<sup>1</sup> ¼ 1, *ω*<sup>11</sup> ¼ *ω*<sup>1</sup> ¼ 3, *γ*<sup>11</sup> ¼ *γ*<sup>1</sup> ¼ 10, *μ*11ð Þ¼ 0 *μ*1ð Þ¼ 0 0. Suppose that the initial state of the state observer is 0, 0 ð Þ. The state and parameter estimation and estimation errors are shown in the **Figure 5**.

#### **Figure 4.**

*The case with single parameter and the observation with noise: states, time-varying parameters estimation and estimation errors based on NSP.*

*Multi-Parameter Estimation of Uncertain Systems Based on the Extended PID Control Method DOI: http://dx.doi.org/10.5772/intechopen.97019*

**Figure 5.**

Consider when the observation of the system (25) contains noise, for example, there is noise in the observation that obeys uniformly distributed in ½ � �0*:*001, 0*:*001 , that is, *y t*ðÞ¼ *x t*ð Þþ Eð Þ*t* , where EðÞ� *t U*½ � �0*:*001, 0*:*001 . Under these circumstances, design parameters are still taken as *k*<sup>1</sup> ¼ 0*:*001, *kd*<sup>1</sup> ¼ 0*:*01, *k*<sup>11</sup> ¼ 40, *k*<sup>21</sup> ¼ 10, *b*<sup>11</sup> ¼ 1, *b*<sup>12</sup> ¼ 10, *α*<sup>1</sup> ¼ 0*:*5, *c*<sup>1</sup> ¼ 1, *ω*<sup>11</sup> ¼ *ω*<sup>1</sup> ¼ 3, *γ*<sup>11</sup> ¼ *γ*<sup>1</sup> ¼ 10, *μ*11ð Þ¼ 0 *μ*1ð Þ¼ 0 0, and suppose that the initial state of the state observer is 0, 0 ð Þ. The estimation errors of the state and parameter are shown in the **Figure 4**. Where the estimation error of the

*Control Based on PID Framework - The Mutual Promotion of Control and Identification…*

estimation error controlled within 2 � <sup>10</sup>�<sup>2</sup> is larger than the parameter estimation

Previously, we studied the estimation problem based on the principle of disturbance stripping for the estimation of a single time-varying parameter, and then we will analyze the situation that the unknown parameter does not change with time. **Example 2** This example is still focusing on the nonlinear system of the

It is assumed here that the true value of *θ* is a constant *θ* ¼ 1 that does not change

In the simulation analysis, the parameters are time-invariant, so *g*1ðÞ¼ *t* 0, the feedback item *l*<sup>21</sup> can be ignored, and the design parameters *k*<sup>1</sup> ¼ 0, *kd*<sup>1</sup> ¼ 0, *k*<sup>11</sup> ¼ 40, *k*<sup>21</sup> ¼ 10, *b*<sup>11</sup> ¼ 1, *b*<sup>12</sup> ¼ 10, *α*<sup>1</sup> ¼ 0*:*5, *c*<sup>1</sup> ¼ 1, *ω*<sup>11</sup> ¼ *ω*<sup>1</sup> ¼ 3, *γ*<sup>11</sup> ¼ *γ*<sup>1</sup> ¼ 10, *μ*11ð Þ¼ 0 *μ*1ð Þ¼ 0 0. Suppose that the initial state of the state observer is 0, 0 ð Þ. The state and parameter estimation and estimation errors are shown in the **Figure 5**.

*The case with single parameter and the observation with noise: states, time-varying parameters estimation and*

, which is larger than the estimation error without noise. The parameter

*x*\_ ¼ �∣*x*∣*θ* þ *x* þ *cosθ* (28)

state is 10�<sup>3</sup>

system (25):

**Figure 4.**

**122**

*estimation errors based on NSP.*

error without noise as well.

with time, and the initial state value is *x*ð Þ¼ 0 2.

*The case with single parameter and the observation without noise: states, time-varying parameters estimation and estimation errors based on NSP.*

The simulation shows that the state and parameters have close to the true value within 1 second, the estimation error can be controlled within 10�<sup>2</sup> within 5 seconds, and the state estimation converges to the true state value faster due to the effect of error feedback.

When there is noise in the observation of the system (25), for example, the observation contains uniformly distributed noise that obeys ½ � �0*:*001, 0*:*001 . That is, *y t*ðÞ¼ *x t*ðÞþ Eð Þ*t* , where EðÞ� *t U*½ � �0*:*001, 0*:*001 . Under these circumstances, the design parameters in simulation analysis are still taken as *k*<sup>1</sup> ¼ 0, *kd*<sup>1</sup> ¼ 0, *k*<sup>11</sup> ¼ 40, *k*<sup>21</sup> ¼ 10, *b*<sup>11</sup> ¼ 1, *b*<sup>12</sup> ¼ 10, *α*<sup>1</sup> ¼ 0*:*5, *c*<sup>1</sup> ¼ 1, *ω*<sup>11</sup> ¼ *ω*<sup>1</sup> ¼ 3, *γ*<sup>11</sup> ¼ *γ*<sup>1</sup> ¼ 10, *μ*11ð Þ¼ 0 *μ*1ð Þ¼ 0 0. Suppose the initial state of the state observer is 0, 0 ð Þ. The estimated errors of the parameters and states are shown in **Figure 6**, the estimation error of the state is 10�<sup>3</sup> , which is more than the estimation error without noise. However, the parameter estimation error is larger than the parameter estimation without noise, but the estimation error can still be controlled within 2 � <sup>10</sup>�<sup>2</sup> .

In summary, this subsection studies the application of parameter estimation methods based on the combination of NSP with state observer in the estimation of single parameters of nonlinear systems. This subsection not only analyzed the two cases of time-invariant and time-varying parameters through simulation, but also analyzed the situation that the observations of the system include observation noise. In these simulation studies, based on the preliminary adjusted design parameters, when analyzing the time-varying and time-invariant parameters, and the presence or absence of observation noise, the design parameters were basically not changed, but the simulation results show that the state and parameters in the observer (27) can asymptotically converge to the true value. These studies show the feasibility and robustness of the combination of the state observer with the stripping principle in the single parameter estimation of nonlinear systems.

\_

8 >>>>><

*DOI: http://dx.doi.org/10.5772/intechopen.97019*

>>>>>:

8

>>>>>>>>>>>>>>>><

>>>>>>>>>>>>>>>>:

**Figure 7**.

**125**

parameters are shown in **Figure 8**.

accurate estimation value.

\_

\_

\_

Let *x*<sup>1</sup> ¼ *y*<sup>1</sup> � *x*^1, *x*<sup>2</sup> ¼ *y*<sup>2</sup> � *x*^2, where *li* is set as follows:

*<sup>l</sup>*<sup>31</sup> <sup>¼</sup> *<sup>k</sup>*1*μ*11Ð*<sup>t</sup>*

*<sup>l</sup>*<sup>41</sup> <sup>¼</sup> *<sup>k</sup>*2*μ*12Ð*<sup>t</sup>*

The design of *μ*11, *μ*12, *u*<sup>1</sup> and *u*<sup>2</sup> is shown in the theorem 1.

*l*<sup>32</sup> ¼ *k*31∣*u*1∣*x*1*sign*

*l*<sup>42</sup> ¼ *k*42∣*u*2∣*x*2*sign*

*x*^<sup>1</sup> ¼ 5*x*^3*x*^<sup>2</sup> þ *l*<sup>1</sup>

*Multi-Parameter Estimation of Uncertain Systems Based on the Extended PID Control Method*

*<sup>x</sup>*^<sup>2</sup> ¼ �*x*^<sup>4</sup> *cos <sup>x</sup>*^<sup>2</sup>

*x*^<sup>3</sup> ¼ *l*<sup>3</sup> ¼ *l*<sup>31</sup> þ *l*<sup>32</sup>

*x*^<sup>4</sup> ¼ *l*<sup>4</sup> ¼ *l*<sup>41</sup> þ *l*<sup>42</sup>

*l*<sup>1</sup> ¼ *k*11*sign*ð Þþ *x*<sup>1</sup> *k*12*sign*ð Þ *x*<sup>2</sup> *l*<sup>2</sup> ¼ *k*21*sign*ð Þþ *x*<sup>1</sup> *k*22*sign*ð Þ *x*<sup>2</sup>

<sup>0</sup>∣*x*1ð Þ*τ* ∣*dτ*

<sup>0</sup>∣*x*2ð Þ*τ* ∣*dτ*

We consider the case that the observation does not contain noise first. By using the design of the aforementioned observer (31), design parameters in simulation analysis are as *k*<sup>1</sup> ¼ *k*<sup>2</sup> ¼ 0*:*01, *kd*<sup>1</sup> ¼ *kd*<sup>2</sup> ¼ 0*:*1, *ω*<sup>11</sup> ¼ *ω*<sup>12</sup> ¼ 3, *γ*<sup>11</sup> ¼ *γ*<sup>12</sup> ¼ 10, *k*<sup>11</sup> ¼ 15, *k*<sup>21</sup> ¼ 0*:*1, *k*<sup>12</sup> ¼ 0*:*1, *k*<sup>22</sup> ¼ 1, *k*<sup>31</sup> ¼ 50, *k*<sup>42</sup> ¼ 50, *b*<sup>11</sup> ¼ *b*<sup>21</sup> ¼ 15, *b*<sup>12</sup> ¼ *b*<sup>22</sup> ¼ 25, *α*<sup>1</sup> ¼ *α*<sup>2</sup> ¼ 0*:*5, *c*<sup>1</sup> ¼ *c*<sup>2</sup> ¼ 5, *ω*<sup>1</sup> ¼ 5, *ω*<sup>2</sup> ¼ 0*:*55, *γ*<sup>1</sup> ¼ 10, *γ*<sup>2</sup> ¼ 150, *μ*1ð Þ¼ 0 *μ*2ð Þ¼ 0 0. Suppose the initial state of the state observer is 0, 0 ð Þ. We obtain the following states, parameters estimation and estimation errors as shown in

When the observation of the system (29) contains noise, for example, the observation contains noise that obeys uniformly distribute in ½ � �0*:*001, 0*:*001 , namely *y t*ðÞ¼ *x t*ðÞþ Eð Þ*t* , where EðÞ� *t U*½ � �0*:*001, 0*:*001 . In this case, the design parameters are the same as the above, and the estimated error of the states and

Simulation results in **Figures 7** and **8** show that the observer designed in this section is applicable to the estimation of time-varying parameters and it has certain robustness to noise. The estimation error of the state is similar either with or without observation noise. For parameter estimation, when there is no noise in the

there is noise in the observation, the parameter estimation effect of *θ*<sup>2</sup> is not ideal, and the design parameters need to be adjusted appropriately to obtain the more

We have studied the estimation of multiple time-varying parameters based on the principle of disturbance stripping above. The following will analyze the situa-

<sup>2</sup> � 3*x*<sup>1</sup>

observation, the parameter estimation error is controlled within 5 � <sup>10</sup>�<sup>2</sup>

*x*\_ <sup>1</sup> ¼ 5*θ*1*x*<sup>2</sup> *<sup>x</sup>*\_ <sup>2</sup> ¼ �*θ*2*cosx*<sup>2</sup>

*y*<sup>1</sup> ¼ *x*<sup>1</sup> *y*<sup>2</sup> ¼ *x*<sup>2</sup>

tion where the unknown parameters do not change with time. **Example 4** This example is still researching the system (29):

> 8 >>>><

> >>>>:

*∂ f* 1 *∂x*<sup>3</sup> � �

*∂ f* 2 *∂x*<sup>4</sup> � �

<sup>2</sup> � 3*x*^<sup>1</sup> þ *l*<sup>2</sup>

(31)

(32)

, but when

(33)

**Figure 6.** *The observation contains noise: Statse, time-invariant parameters estimation and estimation errors based on NSP.*

### *3.3.2 Multiple parameter estimation simulation analysis*

This subsection will study the simulation results with multiple parameter estimates in the dynamic process.

**Example 3** Consider the following nonlinear system with two unknown parameters:

$$\begin{cases} \dot{\mathbf{x}}\_1 = \mathbf{5}\theta\_1 \mathbf{x}\_2 \\ \dot{\mathbf{x}}\_2 = -\theta\_2 \cos \mathbf{x}\_2^2 - \mathbf{3} \mathbf{x}\_1 \\ \mathbf{y}\_1 = \mathbf{x}\_1 \\ \mathbf{y}\_2 = \mathbf{x}\_2 \end{cases} \tag{29}$$

Here, we assume that the true value of the unknown parameter changes with time *θ*<sup>1</sup> ¼ *sin* ð Þ 2*t* , *θ*<sup>2</sup> ¼ *cos*ð Þ 2*t* , and *θ*1ð Þ¼ 0 1, *θ*2ð Þ¼ 0 1, the initial state of the system is *x*ð Þ¼ 0 ð Þ 5*:*4, �1*:*4 . In this example, *θ*1, *θ*<sup>2</sup> are unknown parameters. According to the parameter estimation method, the unknown parameters are extended to the state, and we obtain the following extended state system:

$$\begin{cases} \dot{\mathbf{x}}\_1 = \mathbf{5x}\_3 \mathbf{x}\_2 \\ \dot{\mathbf{x}}\_2 = -\mathbf{x}\_4 \cos \mathbf{x}\_2^2 - \mathbf{3} \mathbf{x}\_1 \\ \dot{\mathbf{x}}\_3 = \mathbf{g}\_1(t) \\ \dot{\mathbf{x}}\_4 = \mathbf{g}\_2(t) \end{cases} \tag{30}$$

where *g*1ð Þ*t* , *g*2ð Þ*t* are unknown functions of time *t*.

The state observer of the above system is established as follows:

*Multi-Parameter Estimation of Uncertain Systems Based on the Extended PID Control Method DOI: http://dx.doi.org/10.5772/intechopen.97019*

$$\begin{cases} \dot{\hat{\mathbf{x}}}\_1 = \mathbf{5}\hat{\mathbf{x}}\_3\hat{\mathbf{x}}\_2 + l\_1\\ \dot{\hat{\mathbf{x}}}\_2 = -\hat{\mathbf{x}}\_4\cos\hat{\mathbf{x}}\_2^2 - 3\hat{\mathbf{x}}\_1 + l\_2\\ \dot{\hat{\mathbf{x}}}\_3 = l\_3 = l\_{31} + l\_{32}\\ \dot{\hat{\mathbf{x}}}\_4 = l\_4 = l\_{41} + l\_{42} \end{cases} \tag{31}$$

Let *x*<sup>1</sup> ¼ *y*<sup>1</sup> � *x*^1, *x*<sup>2</sup> ¼ *y*<sup>2</sup> � *x*^2, where *li* is set as follows:

$$\begin{cases} l\_1 = k\_{11} \text{sign}(\overline{\mathbf{x}\_1}) + k\_{12} \text{sign}(\overline{\mathbf{x}\_2}) \\ l\_2 = k\_{21} \text{sign}(\overline{\mathbf{x}\_1}) + k\_{22} \text{sign}(\overline{\mathbf{x}\_2}) \\ l\_{31} = k\_{1} \mu\_{11} \int\_0^t |\overline{\mathbf{x}\_1}(\tau)| d\tau \\\\ l\_{32} = k\_{31} |\mu\_1| \overline{\mathbf{x}\_1} \text{sign}\left(\frac{\partial f\_1}{\partial \mathbf{x}\_3}\right) \\ \vdots \\ l\_{41} = k\_{2} \mu\_{12} \int\_0^t |\overline{\mathbf{x}\_2}(\tau)| d\tau \\ l\_{42} = k\_{42} |\mu\_2| \overline{\mathbf{x}\_2} \text{sign}\left(\frac{\partial f\_2}{\partial \mathbf{x}\_4}\right) \end{cases} \tag{32}$$

The design of *μ*11, *μ*12, *u*<sup>1</sup> and *u*<sup>2</sup> is shown in the theorem 1.

We consider the case that the observation does not contain noise first. By using the design of the aforementioned observer (31), design parameters in simulation analysis are as *k*<sup>1</sup> ¼ *k*<sup>2</sup> ¼ 0*:*01, *kd*<sup>1</sup> ¼ *kd*<sup>2</sup> ¼ 0*:*1, *ω*<sup>11</sup> ¼ *ω*<sup>12</sup> ¼ 3, *γ*<sup>11</sup> ¼ *γ*<sup>12</sup> ¼ 10, *k*<sup>11</sup> ¼ 15, *k*<sup>21</sup> ¼ 0*:*1, *k*<sup>12</sup> ¼ 0*:*1, *k*<sup>22</sup> ¼ 1, *k*<sup>31</sup> ¼ 50, *k*<sup>42</sup> ¼ 50, *b*<sup>11</sup> ¼ *b*<sup>21</sup> ¼ 15, *b*<sup>12</sup> ¼ *b*<sup>22</sup> ¼ 25, *α*<sup>1</sup> ¼ *α*<sup>2</sup> ¼ 0*:*5, *c*<sup>1</sup> ¼ *c*<sup>2</sup> ¼ 5, *ω*<sup>1</sup> ¼ 5, *ω*<sup>2</sup> ¼ 0*:*55, *γ*<sup>1</sup> ¼ 10, *γ*<sup>2</sup> ¼ 150, *μ*1ð Þ¼ 0 *μ*2ð Þ¼ 0 0. Suppose the initial state of the state observer is 0, 0 ð Þ. We obtain the following states, parameters estimation and estimation errors as shown in **Figure 7**.

When the observation of the system (29) contains noise, for example, the observation contains noise that obeys uniformly distribute in ½ � �0*:*001, 0*:*001 , namely *y t*ðÞ¼ *x t*ðÞþ Eð Þ*t* , where EðÞ� *t U*½ � �0*:*001, 0*:*001 . In this case, the design parameters are the same as the above, and the estimated error of the states and parameters are shown in **Figure 8**.

Simulation results in **Figures 7** and **8** show that the observer designed in this section is applicable to the estimation of time-varying parameters and it has certain robustness to noise. The estimation error of the state is similar either with or without observation noise. For parameter estimation, when there is no noise in the observation, the parameter estimation error is controlled within 5 � <sup>10</sup>�<sup>2</sup> , but when there is noise in the observation, the parameter estimation effect of *θ*<sup>2</sup> is not ideal, and the design parameters need to be adjusted appropriately to obtain the more accurate estimation value.

We have studied the estimation of multiple time-varying parameters based on the principle of disturbance stripping above. The following will analyze the situation where the unknown parameters do not change with time.

**Example 4** This example is still researching the system (29):

$$\begin{cases} \dot{\mathbf{x}}\_1 = \mathbf{5}\theta\_1 \mathbf{x}\_2 \\\\ \dot{\mathbf{x}}\_2 = -\theta\_2 \cos \mathbf{x}\_2^2 - \mathbf{3} \mathbf{x}\_1 \\\\ \mathbf{y}\_1 = \mathbf{x}\_1 \\\\ \mathbf{y}\_2 = \mathbf{x}\_2 \end{cases} \tag{33}$$

*3.3.2 Multiple parameter estimation simulation analysis*

estimates in the dynamic process.

parameters:

**124**

**Figure 6.**

*NSP.*

This subsection will study the simulation results with multiple parameter

*The observation contains noise: Statse, time-invariant parameters estimation and estimation errors based on*

*Control Based on PID Framework - The Mutual Promotion of Control and Identification…*

**Example 3** Consider the following nonlinear system with two unknown

*x*\_ <sup>1</sup> ¼ 5*θ*1*x*<sup>2</sup> *<sup>x</sup>*\_ <sup>2</sup> ¼ �*θ*2*cosx*<sup>2</sup>

*x*\_ <sup>1</sup> ¼ 5*x*3*x*<sup>2</sup> *<sup>x</sup>*\_ <sup>2</sup> ¼ �*x*4*cosx*<sup>2</sup>

*x*\_ <sup>3</sup> ¼ *g*1ð Þ*t x*\_ <sup>4</sup> ¼ *g*2ð Þ*t*

The state observer of the above system is established as follows:

Here, we assume that the true value of the unknown parameter changes with time *θ*<sup>1</sup> ¼ *sin* ð Þ 2*t* , *θ*<sup>2</sup> ¼ *cos*ð Þ 2*t* , and *θ*1ð Þ¼ 0 1, *θ*2ð Þ¼ 0 1, the initial state of the system is *x*ð Þ¼ 0 ð Þ 5*:*4, �1*:*4 . In this example, *θ*1, *θ*<sup>2</sup> are unknown parameters. According to the parameter estimation method, the unknown parameters are extended to the state, and we obtain the following extended state system:

*y*<sup>1</sup> ¼ *x*<sup>1</sup> *y*<sup>2</sup> ¼ *x*<sup>2</sup>

8 >>><

>>>:

8 >>><

>>>:

where *g*1ð Þ*t* , *g*2ð Þ*t* are unknown functions of time *t*.

<sup>2</sup> � 3*x*<sup>1</sup>

<sup>2</sup> � 3*x*<sup>1</sup>

(29)

(30)

**Figure 7.**

*The case with two parameters and the observation without noise: states, time-varying parameters estimation and estimation errors based on NSP.*

**Figure 9.**

**Figure 10.**

*NSP.*

**127**

*estimation errors based on NSP.*

*The case with two parameters and the observation with noise: states, time-varying parameters estimation and*

*Multi-Parameter Estimation of Uncertain Systems Based on the Extended PID Control Method*

*DOI: http://dx.doi.org/10.5772/intechopen.97019*

*The observation with noise: The states, time-invariant parameters estimation and estimation errors based on*

#### **Figure 8.**

*The case with two parameters and the observation with noise: states, time-varying parameters estimation and estimation errors based on NSP.*

Here we assume that the true value of the unknown parameter does not change with time. Suppose that *θ*<sup>1</sup> ¼ 1*:*4, *θ*<sup>2</sup> ¼ �1*:*4, the initial state of the system is *x*ð Þ¼ 0 ð Þ 5*:*4, �1*:*4 .

*Multi-Parameter Estimation of Uncertain Systems Based on the Extended PID Control Method DOI: http://dx.doi.org/10.5772/intechopen.97019*

**Figure 9.**

*The case with two parameters and the observation with noise: states, time-varying parameters estimation and estimation errors based on NSP.*

**Figure 10.** *The observation with noise: The states, time-invariant parameters estimation and estimation errors based on NSP.*

Here we assume that the true value of the unknown parameter does not change with time. Suppose that *θ*<sup>1</sup> ¼ 1*:*4, *θ*<sup>2</sup> ¼ �1*:*4, the initial state of the system is *x*ð Þ¼ 0

*The case with two parameters and the observation with noise: states, time-varying parameters estimation and*

*The case with two parameters and the observation without noise: states, time-varying parameters estimation*

*Control Based on PID Framework - The Mutual Promotion of Control and Identification…*

ð Þ 5*:*4, �1*:*4 .

**126**

*estimation errors based on NSP.*

**Figure 8.**

**Figure 7.**

*and estimation errors based on NSP.*

For the estimation of time-invariant parameters, *g*1ðÞ¼ *t g*2ðÞ¼ *t* 0, so take *l*<sup>31</sup> ¼ *l*<sup>41</sup> ¼ 0. Take design parameters in simulation analysis *k*<sup>1</sup> ¼ *k*<sup>2</sup> ¼ 0, *kd*<sup>1</sup> ¼ *kd*<sup>1</sup> ¼ 0, *ω*<sup>11</sup> ¼ *ω*<sup>12</sup> ¼ 0*:*35, *γ*<sup>11</sup> ¼ *γ*<sup>12</sup> ¼ 10, *k*<sup>11</sup> ¼ *k*<sup>21</sup> ¼ 10, *k*<sup>12</sup> ¼ *k*<sup>22</sup> ¼ 5, *k*<sup>31</sup> ¼ *k*<sup>32</sup> ¼ 15, *k*<sup>41</sup> ¼ *k*<sup>42</sup> ¼ 10, *b*<sup>11</sup> ¼ *b*<sup>21</sup> ¼ 1, *b*<sup>12</sup> ¼ *b*<sup>22</sup> ¼ 25, *α*<sup>1</sup> ¼ *α*<sup>2</sup> ¼ 0*:*1, *c*<sup>1</sup> ¼ *c*<sup>2</sup> ¼ 6, *ω*<sup>1</sup> ¼ *ω*<sup>2</sup> ¼ 0*:*35, *γ*<sup>1</sup> ¼ 10, *γ*<sup>2</sup> ¼ 15, *μ*1ð Þ¼ 0 *mu*2ð Þ¼ 0 0. Suppose the initial state of the state observer is 0, 0 ð Þ. The state and parameter estimation, and estimation error obtained by simulation are shown in **Figure 9**. The simulation result in **Figure 9** shows that the above parameter method is still applicable to the estimation of time-invariant parameters. We can see from the simulation results that the estimation error of the state is controlled within 5 � <sup>10</sup>�<sup>3</sup> , and the estimation error of the parameter estimation is controlled within 2 � <sup>10</sup>�<sup>2</sup> , or even better (see **Figure 9** *<sup>θ</sup>*<sup>1</sup> � ^*θ*1). If we tune the design parameters properly, we can get a more accurate estimate.

design parameters, according to the characteristics of the error system, the thought and method of control system design can be used to give an approximate value to make the state and the parameter converge, and it can also make fine adjustments to

*Multi-Parameter Estimation of Uncertain Systems Based on the Extended PID Control Method*

make the estimated error meet the actual demand.

*DOI: http://dx.doi.org/10.5772/intechopen.97019*

**Author details**

\*† and Wei Wang2†

† These authors contributed equally.

provided the original work is properly cited.

\*Address all correspondence to: fjinping@henu.edu.cn

1 School of Mathematics and Statistics, Henan University, KaiFeng, China

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

2 School of Mathematics, Renmin University of China, Beijing, China

Jinping Feng<sup>1</sup>

**129**

When the observation of the system (29) contains noise, for example, there is noise that obeys uniformly distribute in ½ � �0*:*001, 0*:*001 , that is, *y t*ðÞ¼ *x t*ðÞþ Eð Þ*t* , where EðÞ� *t U*½ � �0*:*001, 0*:*001 . In this case, the design parameters are the same as above, and the estimated errors of the states and parameters are shown in **Figure 10**, where the estimated error of the states are controlled within 5 � <sup>10</sup>�<sup>3</sup> . The parameter estimation error is larger than the parameter estimation error without noise, but the parameter estimation error can still be controlled within 5 � <sup>10</sup>�<sup>2</sup> .

In summary, this section analyzes the estimation problem of multiple timevarying parameters in nonlinear systems based on the parameter estimation method combined the observer with the new stripping principle. Simulation research shows that the parameter estimation method proposed this chapter can estimate multiple time-varying parameters (this section only considers the estimation of two parameters), and the time-invariant and time-varying conditions of the parameters in the analysis both illustrate the applicability of the parameter estimation method. In addition, the simulation research on whether there is observation noise in the observations verifies the robustness and feasibility of the parameter estimation method proposed in this section.
