**1. Introduction**

Dynamic process model is the basis to study the uncertain systems. Generally speaking, the establishment of dynamic process model for the research object is the first step to solve the problem, and the parameter estimation of the established dynamic process model is the next key procedure to solve the problem. So the identification of dynamic processes is of great significance.

The design of the state observer in the control theory is to construct a dynamic system artificially, to make it approximate the real state of the dynamic system by selecting a certain form of the observer. The criterion for designing the state observer is to make the error system asymptotically converge to the origin, that is to say, as time goes by, the error will asymptotically converge to zero. It is based on this design idea we use in the parameter estimation problem. In the identification of the model, the dynamic process model is often accompanied with unknown disturbances. In the analysis of the estimation of multiple time-varying parameters, when the parameters are expanded to the states, there are also unknown parts in the

dynamic process model. So this chapter will study a method for estimating multiple time-varying parameters based on the combination of disturbance stripping principle with state observer.

of the observation, and got the differential of the signal which may be not differen-

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

*<sup>μ</sup>*\_ðÞ¼ *<sup>t</sup>* �*γsign*ð Þ *<sup>σ</sup>*ð Þ*<sup>t</sup>* , <sup>∣</sup>*μ*ð Þ*<sup>t</sup>* ∣ ≤1, <sup>∣</sup>*μ*ð Þ *<sup>t</sup>*<sup>0</sup> ∣ ≤<sup>1</sup> �*ωμ*ð Þ*t* , ∣*μ*ð Þ*t* ∣>1

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

> � � � � �

> > 0 , if *t*>*t* 0

**Proof:** Let us prove it by contradiction method. It is supposed that when *t*>*t*

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

> ð*t t*0 ð Þ �*γ dτ*

<sup>0</sup> ð Þ¼�*γ t* � *t*

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

<sup>0</sup> ð Þ

*μ τ* \_ð Þ*dτ* ¼

*g t*ð Þ Ð*t t*0 ∣*e*ð Þ*τ* ∣*dτ*

� � � � �

∣*e*ð Þ*τ* ∣*dτ*, *e t*ð Þ is the difference between the state

, then *<sup>σ</sup>*ðÞ� *<sup>t</sup>* 0. ■

(2)

(3)

0 ,

(4)

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

tiable. The detail of this nonlinear PID controller can be seen in [8].

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

*k*>sup *t*≥*t*<sup>0</sup>

ð*t t*0

*μ*ðÞ�*t μ t*

*μ*ðÞ¼ *t μ t*

�

Ð*t t*0

lemma, and with detailed proof.

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

where *σ*ðÞ¼ *t g t*ðÞþ *kμ*ð Þ*t*

constant. When the condition

*σ*ð Þ*t* is not always 0.

**Figure 1.**

**115**

*Local changes of μ*ð Þ*t over time.*

is satisfied, there will be a finite time *t*

about time *t* is calculated, and we obtained:

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 effectively [3].

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 estimation of time-invariant parameters.

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 research methods and results presented in this chapter.
