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In this paper, a kind of fire new nonlinear integrator and integral action is proposed. Consequently, a conventional Proportional Nonlinear Integral (P_NI) observer and two kinds of added-order P_NI observers are developed to deal with the uncertain nonlinear system. The conditions on the observer gains to ensure the estimated error to be ultimate boundness, which shrinks to zero as the states and control inputs converge to the equilibrium point, are provided. This means that if the observed system is asymptotically stable, the estimated error dynamics is asymptotically stable, too. Moreover, the highlight point of this paper is that the design of nonlinear integral observer is achieved by linear system theory. Simulation results showed that under the normal and perturbed cases, the pure added-order P_NI observer can effectively deal with the uncertain nonlinearities on both the system dynamics and measured outputs.

State observer design plays an essential role in the design of control system. Compared with most type of observers, the Proportional and Integral (PI) observers as an extension of Luenbergerâ€™s observer [

The PI observer was first proposed by [

The observers for nonlinear uncertain systems mainly focus on particular classes of nonlinear systems. For the class of Lipschitz nonlinear system, an observer [

Therefore, in consideration of the recent progress in the integral control domain, the development of integral observer is so far behind. This point is easy to be seen in the literatures [

Motivated by the cognitions above, this paper proposes a conventional P_NI observer and two kinds of added-order P_NI observers along with their design method, respectively. The main contributions are as follows: 1) A kind of fire new nonlinear integrator and integral action is proposed; 2) The gap that there is not nonlinear integral observer is filled by presenting three kinds of nonlinear integral observers; 3) For the system with uncertain nonlinearities that appear on both the system dynamics and measured outputs, two solutions, that is, mixed and pure added-order P_NI observers, are provided; 4) By linear system theory and Lyapunov method, the conditions on the observer gains to ensure the estimated error to be ultimate boundness, which shrinks to zero as the states and control inputs converge to the equilibrium point, are provided. This means that if the observed system is asymptotically stable, the estimated error dynamics is asymptotically stable, too. Moreover, the highlight point of this paper is that the design of nonlinear integral observer is achieved by linear system theory.

Throughout this paper, we use the notation

values, respectively, of a symmetric positive define bounded matrix

The remainder of the paper is organized as follows: Section 2 describes the system under consideration, assumption and definition. Section 3 addresses the design of nonlinear integral observers. Simulations are provided in Section 4. Conclusions are presented in Section 5.

Consider the following observable nonlinear system, ^{ }

where

For convenience, we state all definitions, assumptions and theorems for the case when the equilibrium point is at the origin of

Assumption 1: No loss of generality, suppose that the function

where

For the purpose of this paper, it is convenient to introduce the following definition.

Definition 1:

tial increasing functions [

where

This section proposes three kinds of nonlinear integral observers, respectively. First, a conventional P_NI observer is proposed to deal with the system without uncertainties in measured output; Second, a mixed added- order P_NI observer was developed for the system with the uncertain nonlinearities that appear on both the system dynamics and measured outputs; Finally, a pure added-order P_NI observer is provided to simplify the design of mixed added-order P_NI observer.

For the system (1), a conventional P_NI observer can be designed as follows,

where

mated output;

gain matrices;

Thus, the error dynamics can be obtained by subtracting (3) from (1),

where

and

holds as

Therefore, we ensure that there is a unique solution

Assumption 2: By Assumption 1 and the definitions of

where

Now, the design task is to provide the conditions on the gains

is Hurwitz.

By linear system theory, a quadratic Lyapunov function

solution of Lyapunov equation

We use

where

Now, using (2), (6) and (7), we have,

Substituting (4) into (8), using the inequality (9), Lyapunov equation

where

The first term in the right-hand side of the inequality (10) is negative define when,

Furthermore, if the following inequality,

holds, it can be verified,

where

in finite time. The above argument shows that the error dynamics (4) is ultimate boundness with an ultimate

bound that decreases as

Theorem 1: Under Assumption 1 and 2, if there exist the gain matrices

and the inequality (11) hold, and then the error dynamics (4) is ultimate boundness with an ultimate bound that decreases as

Discussion 1: From the error dynamics (4), it is obvious that the observer (3) is only effective for the system

with

ctively attenuated. However, when

For making up the shortage of conventional P_NI observer and designing an added-order P_NI observer, the system (1) needs to be added order, which is motivated by the design idea presented by [

By the augmented system (15), a mixed added-order P_NI observer can be given as,

where

and the other symbols are the same as these defined in (3).

By the same way as Subsection 3.1, the error dynamics can be obtained by subtracting (16) from (15),

where

Now, the design task is to provide the conditions on the gain matrices

such that

is Hurwitz.

Now, using (2), (6) and (7), we have,

where

By the same way as Subsection 3.1, we can obtain a quadratic Lyapunov function

then using (18), if the following inequality,

holds, we have,

and then the time derivative of

where

and

Thus, the trajectory of the error dynamics (17) reaches the set,

in finite time. As shown in Subsection 3.1, the following theorem can be established.

Theorem 2: Under Assumption 1 and 2, if there exist the gain matrices

and the inequality (19) hold, and then the error dynamics (17) is ultimate boundness with an ultimate bound that

decreases as

Remark 1: It is obvious that the order of the system (15) and observer (16) are all added. This is why our observer is called the added-order observer. In addition, the observer (16) is designed by using the estimated errors

Discussion 2: From the error dynamics (17), it is easy to see that: 1) By increasing

Obviously, the design method above is too complicated such that some sort of compromise is needed in practice. Therefore, a simplified observer will be proposed in the next subsection.

Based on Discussion 2, it is obvious that only the actions of

where

are the same as these defined in (3).

By the same way as Subsection 3.2, the error dynamics can be obtained by subtracting (23) from (15),

and then by letting

Theorem 3: Under Assumption 1 and 2, if there exist the gain matrices

hold, and then the error dynamics (24) is ultimate boundness with an ultimate bound that decreases as

Remark 2: It is easy to see that the observer (23) is designed only by the estimated error

Discussion 3: From the error dynamics (24) and demonstration above, it is obvious that: 1)

simplified; 3) By increasing

and

tions on system (1) but also the stability of the error dynamics is easier to be achieved. Moreover, since the integral action can attenuate measurement noise, the observer (23) can be suitable for handling measurement noise, too.

Discussion 4: Although the works of [

Discussion 5: Compared with the integrators and integral actions proposed by [

tively; 3) The integral actions: here not only include bounded integral actions, such as

so contains the unbounded one; however, they are all bounded in [

Remark 3: From the stability analysis of Subsections 3.1 - 3.3, it is obvious that: Just the integrator is taken

as the product of estimated error and reciprocal of derivative

can be transformed into the linear form on the estimated error. Just with this ingenious mathematical transformation [

Consider the pendulum system [

where

and

measured output

linear action can be written as,

By the design method proposed here, the augmented system can be given as,

and then, the pure added-order P_NI observer can be given as,

By the design method proposed here, we can take

is Hurwitz.

Therefore, the control input can be taken as,

For demonstrating the performance of the pure added-order observer, the simulations are implemented under normal and perturbed parameter cases, respectively.

Normal case: The initial states are

Perturbed case: The initial states are

only verified the justification of Theorem 3 but also shows that the observer (23) has strong robustness and can effectively deal with the uncertain nonlinearities on both the system dynamics and measured outputs.

This paper proposed a conventional P_NI observer and two kinds of added-order P_NI observers along with their design method. The main contributions are as follows: 1) A kind of fire new nonlinear integrator and integral action is proposed; 2) The gap that there is not nonlinear integral observer is filled by presenting three kinds of nonlinear integral observers; 3) For the system with uncertain nonlinearities that appear on both the system dynamics and measured outputs, two solutions, that is, mixed and pure added-order P_NI observers, are provided; 4) The conditions on the observer gains to ensure the estimated error to be ultimate boundness, which shrinks to zero as the states and control inputs converge to the equilibrium point, are provided. This means that if the observed system is asymptotically stable, the estimated error dynamics is asymptotically stable, too. In addition, the highlight point of this paper is that the design of nonlinear integral observer was achieved by linear system theory.