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 Luenberger’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.