The practical stabilization problem is investigated for a class of linear systems with actuator saturation and input additive disturbances. Firstly, the case of the input additive disturbance being a bounded constant and a variety of different situations of system matrices are studied for the three-dimensional linear system with actuator saturation, respectively. By applying the Riccati equation approach and designing the linear state feedback control law, sufficient conditions are established to guarantee the semiglobal practical stabilization or oscillation for the addressed system. Secondly, for the case of the input additive disturbances being time-varying functions, a more general class of systems with actuator saturation is investigated. By employing the Riccati equation approach, a low-and-high-gain linear state feedback control law is designed to guarantee the global or semiglobal practical stabilization for the closed-loop systems.
Actuator saturation (control saturation), as a common and typical nonlinear constraint for control systems, is often encountered in various industrial systems, especially in many physical-controlled systems with magnitude limitation in the input. In general, linear systems can be completely controlled by using the linear state feedback, and the semiglobal (or local) stabilization can be achieved [
When the dimension of the integrator is greater than or equal to 3, the linear systems with input saturation cannot be stabilized by using the linear state feedback control law, so the global asymptotic stabilization of the system cannot be attained. However, if all eigenvalues of the open-loop system have negative real parts, the global asymptotic stabilization for the addressed system can be guaranteed by designing a globally stable boundary control law. Accordingly, the exponentially semiglobal stabilization problems have been widely studied in [
On the other hand, the study of the linear systems with actuator saturation also covers the practical stabilization problem, that is, a controller is designed such that the trajectories of closed-loop system can enter into an arbitrarily small prescribed neighborhood of the origin in finite time and remain thereafter. The practical stabilization problems have been gaining an increasing research interest, and many important results have been reported, see, for example, [
Motivated by the above discussion, we aim to investigate the practical stabilization problem for three-dimensional (multidimensional) system with actuator saturation and input additive disturbances. By employing the Riccati equation approach, the linear state feedback control law is designed to guarantee the practical stabilization for the system with actuator saturation and time-invariant input additive disturbances. Moreover, the practical stabilization of a general multidimensional system with actuator saturation and time-varying input additive disturbances is studied, and the low-and-high-gain linear state feedback control law is synthesized by using the Riccati equation approach. The main contributions of this paper can be highlighted as follows: (1) the practical stabilization problem of three-dimensional linear systems is investigated for the first time, which covers actuator saturation as well as input additive disturbances; (2) the low-and-high-gain linear state feedback control law is designed for multidimensional system with actuator saturation and time-varying input additive disturbances.
The notations in this paper are quite standard except where otherwise stated. The superscript “
Consider the following three-dimensional system with actuator saturation and input additive disturbance:
The saturation function
Before proceeding further, we make the following assumptions.
The matrix pair
The input disturbance is bounded, that is,
Based on Assumption
These forms of the matrix pair
Define an ellipsoid
For system (
Letting
Consider the system ( If If If
Firstly, let us prove (i).
(i1) Consider the following system
Let
Define
We choose the Lyapunov function
(i2) Assuming that matrix
When
Secondly, let us prove (ii).
(ii1) Consider the following system:
In this case, we choose the matrix
By noting
Define
Similarly, when
(ii2) Consider the following system:
In this case, we choose the matrix
Define
(
In this case, we choose the matrix
Set
Finally, let us prove (iii).
Assuming that matrix
Based on (i)–(iii), the proof of this theorem is now complete.
It should be pointed out that, because of the mathematical complexity and computational difficulty, almost all papers concerning the actuator saturation and input additive disturbances have considered the two-dimensional systems. In this paper, we make the first attempt to investigate the practical stabilization problem for three-dimensional system with actuator saturation and input additive disturbances. To the best of our knowledge, the research topic addressed in this paper is new and meaningful. The above attempts distinguish our research results from the existing ones.
The sufficient conditions are established in Theorem
In this section, we investigate a more general class of systems with actuator saturation and time-varying disturbance input. A low-and-high-gain is designed to guarantee the global or semiglobal practical stabilization for the closed-loop systems.
Consider the following linear system with actuator saturation and time-varying disturbance input
Before proceeding further, we make the following assumptions.
The matrix pair
The uncertain element
For the system (
If If
As discussed in [
For Problem
Let
Let Assumption
Now, we construct the low-gain state feedback law as follows:
Based on Lemma
We design the high-gain state feedback control law as follows:
Taking the low-gain and high-gain state feedback control law into account, the low-and-high-gain state feedback control law is constructed as follows:
Let Assumption
To begin with, the cases of saturation and unsaturation are discussed respectively. For convenience, set
Firstly, we choose the Lyapunov function
When
It follows from (
When
(2) The actuator is not saturated. When
If
Specifically, it follows from
In this section, the practical stabilization problem is investigated for a general multidimensional system. From a practical point of view, it is more significant to consider the high-dimensional systems. It is worth mentioning that the system under consideration is comprehensive that includes the actuator saturation and the time-varying input disturbances. By using the Riccati equation approach, the low-and-high-gain state feedback control law is designed such that the global or semiglobal practical stabilization for the multidimensional system can be guaranteed. On the other hand, we are now researching into a method for the system with uncertainties, time-delay, and/or input disturbance in more general cases. The corresponding results will appear in the near future.
In this paper, we have made an attempt to investigate the practical stabilization problem for a class of system with actuator saturation and input additive disturbances. For the case of the input additive disturbance being bounded constant, the three-dimensional system has been studied where the system matrices satisfy a class of the controllability canonical form. Eight different forms of the system matrices have been discussed. Subsequently, by applying the Riccati equation approach and designing the linear state feedback control law, the sufficient conditions of the semiglobal practical stabilization or oscillation for the addressed systems have been established. For the case when the input additive disturbances are time-varying functions, by using of the Riccati equation approach as well as combining the low-gain linear state feedback and high-gain linear state feedback, a low-high-gain linear state feedback control law has been designed such that the global or semiglobal practical stabilization for a general multidimensional system with actuator saturation can be guaranteed. One of the future research topics would be the extension of the main results obtained in this paper to networked control systems [
This work was supported by the National Natural Science Foundation of China under Grant 10771047.