**2. The main idea of NSP**

The PID control method applies the error Eð Þ*t* between the reference input and observation. The PID control is the linear combination of the error, its differential and its integration. That is

$$u(t) = k\_P \epsilon(t) + k\_I \int\_{t\_0}^t \epsilon(\tau)d\tau + k\_D \dot{\epsilon}(t) \tag{1}$$

where *kP*, *kI*, *kD* are design parameters, Eð Þ*t* is the error, E\_ð Þ*t* is the differential of the error, Ð*<sup>t</sup> t*0 Eð Þ*τ dτ* is the integration of the error, *t*<sup>0</sup> is the intial time.

The theory analysis and large applications showed that the PID control *u* often had the conflict in fast and overshoot. Luckily, the nonlinear PID could solve this problem [8], which used the nonliner function such as sat funtion, fal function. It was the nonlinear combination of the error, its difference and its integration. At the same time, it also applied the nonlinear tracking-differentiator to filter the noise *Multi-Parameter Estimation of Uncertain Systems Based on the Extended PID Control Method DOI: http://dx.doi.org/10.5772/intechopen.97019*

of the observation, and got the differential of the signal which may be not differentiable. The detail of this nonlinear PID controller can be seen in [8].

Basd on the idea of the extended PID controller, the NSP thought was proposed [6, 10, 12, 13]. They found that the integration of the error in the extended PID controller could stripping the unknown item in the complex systems. So we could use the NSP to deal with the system with unknown parts. The basic conclusion to be used in the following analysis, which is the most important thought in NSP involved in [6, 10, 12, 13]. The core idea will be simplified here, given in the form of a lemma, and with detailed proof.

**Lemma 1** If the dynamic process *μ*ð Þ*t* takes the following form:

$$\dot{\mu}(t) = \begin{cases} -\gamma \text{sign}(\sigma(t)), & |\mu(t)| \le 1, |\mu(t\_0)| \le 1 \\ -a\mu(t), & |\mu(t)| > 1 \end{cases} \tag{2}$$

where *σ*ðÞ¼ *t g t*ðÞþ *kμ*ð Þ*t* Ð*t t*0 ∣*e*ð Þ*τ* ∣*dτ*, *e t*ð Þ is the difference between the state observer system and the original system (which is *x* as mentioned below), *g t*ð Þ is the unknown quantity with the known variation range, *γ* >0, *ω* >0 is the undetermined constant. When the condition

$$k > \sup\_{t \ge t\_0} \left| \frac{g(t)}{\int\_{t\_0}^t |e(\tau)| d\tau} \right| \tag{3}$$

is satisfied, there will be a finite time *t* 0 , if *t*>*t* 0 , then *<sup>σ</sup>*ðÞ� *<sup>t</sup>* 0. ■

**Proof:** Let us prove it by contradiction method. It is supposed that when *t*>*t* 0 , *σ*ð Þ*t* is not always 0.

Assuming that there is a certain moment *σ*ð Þ*t* 6¼ 0, we might set *σ*ð Þ*t* >0 by the local scope. From the formula (2), there is *μ*\_ðÞ¼� *t γ*. The integral on both sides about time *t* is calculated, and we obtained:

$$\begin{aligned} \label{eq:1} \int\_{t'}^{t} \hat{\mu}(\tau)d\tau &= \int\_{t'}^{t} (-\gamma)d\tau\\ \mu(t) - \mu(t') &= -\gamma(t - t')\\ \mu(t) &= \mu(t') - \gamma(t - t') \end{aligned} \tag{4}$$

$$\begin{aligned} \label{eq:1} \mathbf{1} &\quad \xrightarrow{\mu} \mathbf{1} \\ \begin{array}{c} \big[ \begin{array}{c} \mathbf{1} \\ \mathbf{0} \\ \mathbf{1} \end{array} \Big] \\ \cline{2-3} \end{aligned} \qquad \begin{aligned} \label{eq:1} \mathbf{1} &\quad \xrightarrow{\mu} \mathbf{1} \\ \mathbf{1} &\quad \xrightarrow{\mu} \mathbf{1} \\ \mathbf{1} &\quad \xrightarrow{\mu} \mathbf{1} \\ \mathbf{1} &\quad \xrightarrow{\mu} \mathbf{1} \end{aligned}$$

**Figure 1.** *Local changes of μ*ð Þ*t over time.*

dynamic process model. So this chapter will study a method for estimating multiple

The reference [1] proposed a general form of establishing the state observer of the nonlinear system, and gave a direct method to deal with the nonlinear control system [2]. On the basis of the references [1–3], several specific state observers was provided to realize the estimation of a single time invariant parameter, and appropriate design parameters were selected according to the relevant results in the book [4]. By analyzing the stability the error system, a design method that made the error system asymptotically converge to zero was obtained. The simulation results showed that this method can estimate the parameters

For the estimation of time-varying parameters, the article [5] analyzed a system with one time-varying parameter. The design of the state observer in this article used the combination of binary control with PID control, which can handle the unknown items in the extended states. Although there was no rigorous theoretical proof in this article, the effect of parameter estimation did have excellent characteristics of fast convergence with less chatter. The reference [6] gave a method of combining binary control with nonlinear PID controller, and conducted a rigorous theoretical proof. Then it was extended to the regulation of high-level systems, and the principle of disturbance stripping [7] for the regulation of complex network systems. This laid the foundation for the theoretical analysis of the estimation methods of multiple time-varying parameters below. So this chapter is based on [5–7] and other references. The method of estimating a time-varying parameter in the nonlinear system in [5] is extended to the estimation of multiple time-varying parameters in a dynamic system by using the principle of disturbance stripping in the article [7]. The simulation studies showed that this method was also suitable for

The content of this chapter is arranged as follows: The Section 2 simply introduces the main idea of NSP and gives detail proof of it. The Section 3 puts forward an estimation method that contains multiple time-varying parameters in a nonlinear system. It describes the applicable objects of this kind of parameter estimation method, and gives a design of a specific state observer. Theoretical analysis and simulation research verifies the feasibility of the method. Section 4 summarizes the

The PID control method applies the error Eð Þ*t* between the reference input and observation. The PID control is the linear combination of the error, its differential

> ð*t t*0

where *kP*, *kI*, *kD* are design parameters, Eð Þ*t* is the error, E\_ð Þ*t* is the differential of

Eð Þ*τ dτ* is the integration of the error, *t*<sup>0</sup> is the intial time. The theory analysis and large applications showed that the PID control *u* often had the conflict in fast and overshoot. Luckily, the nonlinear PID could solve this problem [8], which used the nonliner function such as sat funtion, fal function. It was the nonlinear combination of the error, its difference and its integration. At the same time, it also applied the nonlinear tracking-differentiator to filter the noise

Eð Þ*τ dτ* þ *kD*E\_ð Þ*t* (1)

time-varying parameters based on the combination of disturbance stripping

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

principle with state observer.

the estimation of time-invariant parameters.

**2. The main idea of NSP**

and its integration. That is

*t*0

the error, Ð*<sup>t</sup>*

**114**

research methods and results presented in this chapter.

*u t*ðÞ¼ *kP*EðÞþ*t kI*

effectively [3].

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* , 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

$$
\mu(t\_1) = \mu(t') - \gamma(t\_1 - t') = -\mathbf{1} \tag{5}
$$

Then there is

$$\begin{aligned} t\_1 &= t' + \frac{1 + \mu(t')}{\mathcal{Y}} \\ &\le t' + \frac{2}{\mathcal{Y}} \end{aligned} \tag{6}$$

From the condition (3), we know that *<sup>σ</sup>*ð Þ*<sup>t</sup>* <sup>2</sup> <sup>&</sup>lt; 0, it leads a contradictory. When *σ*ð Þ*t* <0, the contradiction can be derived in the same way, and the overall change of

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

*<sup>μ</sup>*ð Þ*<sup>t</sup>* will be shown in **Figure 2**. In summary, the conclusion is established. ■

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

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

The following system with multiple parameters, and the system itself is highly

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

*x*\_ *<sup>n</sup>* ¼ *f <sup>n</sup>*ð Þ *x*1, *x*2, … , *xn*, *θ<sup>n</sup> <sup>y</sup>* <sup>¼</sup> *<sup>x</sup>* <sup>¼</sup> ½ � *<sup>x</sup>*1, *<sup>x</sup>*2, … , *xn <sup>T</sup>*

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,

ð Þ *x*1, *x*2, … , *xn*, *θ<sup>i</sup>* is the interaction (such as relations or research works) among *xi*,

*<sup>∂</sup>θ<sup>i</sup>* 6¼ 0, the following subsection uses the method of combining

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 multi-

state observer with NSP to study the parameter estimation method. Specific parameter estimation methods, the convergence analysis and simulation research

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

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

(9)

⋮

8 >>>>>><

>>>>>>:

there are also other network problems like this.

parameter estimation problem to be analyzed in this section.

are described in detail in the following subsections respectively.

the time-varying parameters and time-invariant parameters as well.

**stripping principle**

of the NSP with the state observer.

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

**3.1 The statement of the problem**

*fi*

**117**

Assuming that *<sup>∂</sup> fi*

**3.2 Parameter estimation method**

coupled, as shown in the following system:

So when *t*>*t* <sup>0</sup> <sup>þ</sup> <sup>2</sup> *γ* , there is *t*>*t*1, there must be *μ*ðÞ�� *t sign*ð Þ *σ*ð Þ*t* . By *σ*ðÞ¼ *t g t*ð Þþ *kμ*ð Þ*t* Ð*t t*0 ∣*e*ð Þ*τ* ∣*dτ*, then

$$\begin{aligned} \sigma(t) &= \mathbf{g}(t) + k\mu(t) \int\_{t\_0}^t |e(\tau)| d\tau \\ &= \mathbf{g}(t) - k\text{sign}(\sigma(t)) \int\_{t\_0}^t |e(\tau)| d\tau \end{aligned} \tag{7}$$

Then,

$$
\sigma^2(t) = \sigma(t) \left[ \mathbf{g}(t) - k \text{sign}(\sigma(t)) \int\_{t\_0}^t |e(\tau)| d\tau \right]
$$

$$
= \sigma(t)\mathbf{g}(t) - k|\sigma(t)| \int\_{t\_0}^t |e(\tau)| d\tau \tag{8}
$$

$$
\leq |\sigma(t)| |\mathbf{g}(t)| - k|\sigma(t)| \int\_{t\_0}^t |e(\tau)| d\tau
$$

**Figure 2.** *Overall changes of μ*ð Þ*t over time.*

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

From the condition (3), we know that *<sup>σ</sup>*ð Þ*<sup>t</sup>* <sup>2</sup> <sup>&</sup>lt; 0, it leads a contradictory. When *σ*ð Þ*t* <0, the contradiction can be derived in the same way, and the overall change of *<sup>μ</sup>*ð Þ*<sup>t</sup>* will be shown in **Figure 2**. In summary, the conclusion is established. ■
