A design strategy of optimal output regulators for dual-rate discrete-time systems, whose output sampling period is an integer multiple of the input updating period, is proposed. At first, by using the discrete lifting technique, the dual-rate discrete-time system is converted to a single-rate augmented system in form and the lifted state-space model is constructed. Correspondingly, the performance index of the original system is modified to the performance index of the single-rate augmented system. And the original problem is transformed into an output regulation problem for the augmented system. Then, according to the optimal regulator theory, an optimal output regulator for the dual-rate discrete-time system is derived. In the meantime, the existence conditions of the optimal output regulator are discussed. Finally, a numerical example is included to illustrate the effectiveness of the proposed method.
Sampling systems are obtained by discretization of a continuous signal for an actual system. For sampling systems, multirate systems arise when the components of the same system have several different sampling rates [
An optimal output regulator is designed by using linear quadratic regulator theory. The output regulation problem aims to find the optimal control law that can minimize the sum of the dynamic deviation of output and the dissipated energy of the control variables with the given weight [
In this regard, preview control based on the optimal regulation theory for discrete-time multirate systems has produced very good research results recently [
On the basis of [
Consider the following linear discrete-time system:
For system ( The state vector The input vector
If (A1) and (A2) hold, the system is dual-rate sampled. That is, the state vector Assume Suppose the system has zero-order-hold, that is
(A5) guarantees the existence of the state feedback during the design of the optimal output regulator.
We introduce the quadratic performance index function for system (
We would like to design an optimal output regulator for system (
In this section, the optimal output regulator for system (
Based on the multirate study methods, the dual-rate discrete-time system is converted into a single-rate augmented system in form by using the lifting technique. Lifting technique is a typical approach to multirate control. By using this technique, a fast-rate signal can be mapped to a slow-rate signal with increased dimensionality. While this operation maps a fast-rate signal to a slow-rate signal, the inverse operation maps a lifted signal to a fast-rate signal in [
Next, a lifted state-space model is constructed.
First, the lifting technique is applied to the input vectors. According to (A2), (A3), and (A4), the input vector
Second, we apply the lifting technique to the state vectors and output vectors. According to (A1), the state vectors can only be measured at
Using the first group of equations of (
We introduce the vectors as follows:
Introduce the vectors as follows:
Combine (
Now we have successfully transformed the discrete-time system (
Notice that the performance index (
Now the problem becomes an optimal regulator problem of system (
Substituting
Further, by means of contract transformation in the matrix, we can eliminate the cross term of
Now the problem becomes an optimal control problem of the augmented system (
By using the optimal regulator theory in [
If the discrete-time system (
To prove Theorem
If (A5) holds,
First, we prove that
Noticing that elementary transformation of the matrix cannot change the rank of the matrix, and applying the PBH test, we have
Next, we prove that if (A5) holds,
Noticing that
Obviously, we can derive
If
One has the following:
Set
So
Because
is detectable if and only if
And notice
In Section
According to Lemmas
According to (
And due to (A4),
In this section, a numerical example is included to illustrate the effectiveness of the proposed method.
Consider a linear discrete-time system:
Suppose
Through verifying, all of the conditions of Theorem
Next we perform MATLAB simulation results. The output response of the discrete-time system with dual-rate setting is shown in Figure
The output response of the dual-rate discrete-time system.
The control input of the dual-rate discrete-time system.
We notice that the output response in Figure
In this paper, we studied the optimal output regulator for a class of linear discrete-time systems with dual-rate setting. By using the discrete lifting technique, the dual-rate discrete-time system is converted to a single-rate augmented system in form, and the lifted state-space model is constructed. Then we can use the method for single-rate systems to study the optimal regulator problem of the lifted system. By using optimal regulator theory, an optimal output regulator for the dual-rate discrete-time system is finally derived. This approach is also a guideline for future study of the optimal preview control problem for dual-rate systems. Furthermore, when assumption (A3) does not hold, for example, in the cases where
The authors declare that there are no competing interests regarding the publication of this paper.
This work is supported by the National Natural Science Foundation of China (no. 61174209).