This paper proposed the H∞ state feedback and H∞ output feedback design methods for unstable plants, which improved the original H∞ state feedback and H∞ output feedback. For the H∞ state feedback design of unstable plants, it presents the complete robustness constraint which is based on solving Riccati equation and Bode integral. For the H∞ output feedback design of unstable plants, the medium-frequency band should be considered in particular. Besides, this paper presents the method to select weight function or coefficients in the H∞ design, which employs Bode integral to optimize the H∞ design. It takes a magnetic levitation system as an example. The simulation results demonstrate that the optimal performance of perturbation suppression is obtained with the design of robustness constraint. The presented method is of benefit to the general H∞ design.
Some constraints are often ignored in theory design so that the designed system could not been achieved [
There are two types of H∞ design, which are cycle formation based on coprime factorization, H∞ state feedback and H∞ output feedback such as DGKF [
The key to achieve the H∞ control design is up to the weight function. The weight function is considered particularly for an unstable plant in H∞ control design. There are two different types of unstable plants. The first type is that the frequency band of mathematical model is 10 times larger than unstable mode, for example, in designing autopilot, the unstable mode is less than 1 rad/sec but the bandwidth is larger than 40 rad/sec [
In terms of control theory, there may be instability in control design for an unstable plant. Feedback characteristics must be considered in the design of the feedback system. Feedback systems have some performance such as robustness, sensitivity, and disturbance rejection, which can be changed only by feedback. The low sensitivity and disturbance rejection are the reasons why a system needs feedback control, but the robustness is essential performance in the feedback system. Therefore, the purpose of feedback control system design is to achieve low sensitivity and disturbance rejection.
It is known that the sensitivity function describes the performance of the control system. The schematic of the feedback control system is shown in Figure
Feedback control system.
Define the sensitivity function
Taking derivative of (
Figure
Nyquist curve of a system.
Define the maximum peak of sensitivity as
It is known that the response curve for open-loop systems is closer to the point (−1,
As is shown in Figure
The relationship between phase margin
The inequation can be obtained by
Equation (
Suppression of output disturbance
As shown in Figure
Then, the sensitivity reflects the ability of the system to track the input signal and smaller response to little error.
The sensitivity demonstrates that the effect of the system and the robustness could be reflected by its peak. The performance of the system is often reflected by the sensitivity function, which should be decreased in the design. Therefore, the purpose of this paper is to obtain the optimal peak of sensitivity function.
In this section, the characteristic of the system in combined design and the limitation of the characteristic in an unstable plant are discussed.
Assuming that the open-loop transfer function
If the controlled plant is stable, the integral is zero,
Sensitivity reduction at low frequency unavoidably leads to sensitivity increase at higher frequencies.
There, Bode integral is not a real limitation and the negative area is limited between some frequency bands while the positive area is distributed over others small in average. It means
This is the real constraint with the integral interval
From (
The ideal sensitivity function.
There is an unstable pole 6 rad/s and a bandwidth of 40 rad/sec about the fighter jet X-29 in [
Although this is theoretical analysis without the practical system and the constraint of Bode integral is independent of design methods, it can be used to assist the design. The H∞ control design for the unstable magnetic levitation system is detailed in the following.
Figure
Model of the electromagnetic levitation system.
State feedback is the basic control method, and H∞ state feedback design is the simplest control method in H∞ control. But state feedback is not a standard problem in H∞ [
The weighted output is first set in H∞ design and define the output
Assume that the transfer function from input
The solution of H∞ state feedback is the central controller in the full information problem, and the Riccati equation is
Assuming that
This theory is a proof result [
From Theory
So,
Theoretically, it is possible to minimize
It is considerable that H∞ state feedback is to suppress disturbance signal
Together with weighted output (
Define the open-loop transfer function of the state feedback system as
The crossover frequency can be approximated by [
Substituting (
The weight coefficients is set as
And the crossover frequency is
The bandwidth is controlled by weight coefficients using (
Signal flow diagram of the system with state feedback.
The plant is the part after the input current of electromagnetic coils, and the controlled input is the current
The corresponding transfer function is as follows:
And the transfer function of the plant is given by
From (
It should be noted that gamma is just a design parameter in designing H∞ state feedback and the purpose is to achieve the minimum peak
When
When
Notice that the H∞ standard problem is an output feedback problem [
Let us discuss with the most commonly used mixed sensitivity problem in H∞ control. The following H∞ optimization problem is alluded to as mixed sensitivity problem:
In the formula,
The rectangular characteristic shown in Figure
In the Nyquist diagram shown in Figure
Nyquist locus of
Figure
If the designed control law includes integral law,
If the unstable pole of the magnetic levitation system is 70 rad/sec and the bandwidth is designed in the same order of 70 rad/sec, the mathematical model in (
The sensitivity design in conventional
The weight function
Figure
Black diagram of the
It is noted that the
Figure
The Bode plot of
The purpose of H∞ state feedback is to suppress disturbance so that the norm from
The disturbance attenuation performance of the system.
Because
The sensitivity function
Figure
Singular value characteristic of the system.
The crossover frequency
The above is designed according to the weight function
Sensitivity function with integral control.
Of course, this kind of sensitivity characteristic with integral control can also be obtained by specifying the weight function in the H∞ design. For example, when the performance weighting function
The first kind of unstable plants with a small unstable mode will not be discussed because the difference between
This paper presents H∞ state feedback control and H∞ output feedback control with respect to the unstable plant, of which the key is the design of weight function or coefficients. In this paper, the weight function or coefficients are obtained subject to the Bode integral constraint, avoiding the repeated attempt. The deficiency in usual designs has been modified, which adds robustness constraint under Bode integral law into H∞ state feedback control design and points out that the purpose of H∞ design is to achieve optimal performance by adjusting
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was partly supported by the Youth Foundation of Hebei Educational Committee (no. ZD2016203), the Doctoral Foundation of Liaoning Province (no. 20170520333), and the Natural Science Foundation of Hebei Province (no. F2017501088).