## Abstract

Parameter estimation is an important step in the identification of systems. With the extension of systems, there needs the multi-parameter estimation of systems. The estimation of multi parameters of complex systems based on the extended PID controllers is considered in this chapter. As the related references proved that the integral item of the nonlinear PID controller could deal with the uncertain part of the complex system (which can also be called new stripping principle, simple notes as NSP). Based on this theory, new multi-parameter estimation method is given. Firstly, the unknown parameters are expanded to new states of the system. Two cases, parameters are constant or changing with time, are separately analyzed. In the time-variant case, the unknown parameters are extended to functions which actual forms are uncertain. Secondly the method NSP could be applied to cope with the uncertain part, and then reconstruction state observation to estimate the states. If the states are observed, the unknown parameters are obtained at the same time. Finally the convergence analysis of the error systems and some simulations will be given in this chapter to indicate the effectiveness of the proposed method.

### Keywords

- multi-parameter
- parameter estimation
- complex system
- extended PID control
- convergence analysis

## 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.

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, 2, 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, 6, 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.

## 2. The main idea of NSP

The PID control method applies the error

where

The theory analysis and large applications showed that the PID control

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

where

is satisfied, there will be a finite time

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

Assuming that there is a certain moment

Knowing from the definition of

Here,

Then there is

So when

Then,

From the condition (3), we know that

## 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

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

where

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-parameter estimation problem to be analyzed in this section.

Assuming that

### 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:

That is to say, the parameter

where

Next, we need to design

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

where

where

where

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

■

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

If

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

Known from the conditions that

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

Since

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

It is easy to know that, except for the origin,

From the formula (21),

Known by the condition

In summary, when the time is greater than a certain moment,

** 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

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 time-invariant 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]):

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

According to the system (25), there is

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

We design its observer as follows:

The design of

Firstly, we consider the case where the observation does not contain noise. Set the design parameters in the simulation analysis as

Consider when the observation of the system (25) contains noise, for example, there is noise in the observation that obeys uniformly distributed in

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):

It is assumed here that the true value of

In the simulation analysis, the parameters are time-invariant, so

When there is noise in the observation of the system (25), for example, the observation contains uniformly distributed noise that obeys

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.

#### 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:

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

where

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

Let

The design of

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

When the observation of the system (29) contains noise, for example, the observation contains noise that obeys uniformly distribute in

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

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):

Here we assume that the true value of the unknown parameter does not change with time. Suppose that

For the estimation of time-invariant parameters,

When the observation of the system (29) contains noise, for example, there is noise that obeys uniformly distribute in

In summary, this section analyzes the estimation problem of multiple time-varying 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.

## 4. Conclusions

This chapter studies the state observer method of nonlinear system parameter estimation. When the unknown parameters have explicit expressions, we can use the nonlinear tracking-differentiator-based method to estimate the parameters. The unknown parameters which is relatively non-linear system in nonlinear form or is not easy to express by explicit are main considered in this chapter. According to the different characteristics of the parameters contained in the dynamic process, based on the research of the existing literatures, this chapter proposes a new parameter estimation method based on the state observer and NSP. The parameter estimation method based on the combination of state observer with new stripping principle for dynamic systems containing multiple time-varying parameters. This chapter not only proves the feasibility of the method in theory, but also do the simulations. The simulation results show that the design method can approximate the true value of the parameter within a certain error range. The simulations also consider the presence or absence of observation noise. The simulation results not only show that the parameter estimation method introduced in this chapter is robust to noise, but also show the adaptability of the design parameters. Because it is found in the design parameter adjustment that: adjusting the design parameters within a certain range has little effect on the accuracy of parameter estimation, so in the adjustment of 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 make the estimated error meet the actual demand.

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