_{∞}Optimal Performance Design of an Unstable Plant under Bode Integral Constraint

^{1}

^{2}

^{1}

^{1}

^{1}

^{1}

^{2}

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 [_{∞} state feedback and H_{∞} output feedback design together with the magnetic levitation system, which is applicable to unstable plants.

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 [_{∞} design in general. The second type is that the unstable mode and the bandwidth of the closed-loop system are approximate, for example, in magnetic levitation systems in [_{∞} control design. This paper mainly discusses the second type of unstable plants, and the analysis results will benefit the explanation of the design of the first type.

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_{∞} [_{∞} output feedback design, which will be discussed in this section and next section. The relation between the solution of state feedback and H_{∞} norm

The weighted output is first set in H_{∞} design and define the output

Assume that the transfer function from input _{∞} design is

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 _{∞} design. But the norm _{∞} state feedback should not be set directly to

So,

Theoretically, it is possible to minimize _{∞} state feedback design is just to solve Riccati equation (

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 [_{∞} controller is theoretically possible to design. However, if considered from the point of view of engineering practice, the system should add current feedback to suppress various disturbances in the current loop to improve the response characteristic of the current (i.e., electromagnetic force) [

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, _{∞} controller is designed to ensure the stability of the system, while the performance and robustness correspond to the low-frequency and high-frequency characteristics of the system, so the weight function of

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 _{∞} design, the solution to (_{∞} optimal solution is all-pass characteristics and if demonstrated by

Figure _{∞} controller to be designed. From Figure

Black diagram of the

It is noted that the _{∞} optimization problem. We use MATLAB function hinfsyn() to get _{∞} controller (after neglecting the high-frequency term

Figure _{∞} state feedback design, and its logarithmic integral equals to 220 in coincident with (

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 _{∞} output feedback design, and it is rectangular in practical application, which decreases after the bandwidth. It shows that the area of the rectangle is equivalent in Figure _{∞} output feedback design under the controller (

Figure _{∞} controller (

Singular value characteristic of the system.

The crossover frequency _{∞} optimal design.

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 _{∞} mixed sensitivity design, the approximate equation

The first kind of unstable plants with a small unstable mode will not be discussed because the difference between _{∞} design could be applied in terms of the flat curve in Figure _{∞} design.

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 _{∞} output feedback design, and together with the magnetic levitation system, it is pointed that the sensitivity function is an adjustable weight function to obtain all-pass and weight sensitivity to control sensitivity peak and bandwidth. Simulation results demonstrate that the H_{∞} design of state feedback control and output feedback control subject to the Bode integral constraint could achieve optimal performance.

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).

_{∞}state feedback control of an electromagnetic suspension system

_{∞}control of linear discrete-time stochastic uncertain systems

_{∞}non-fragile synchronous guaranteed control of uncertainty complex dynamic network with time-varying delay

_{2}and H

_{∞}control problems

_{∞}theory—a design example

_{2}and H

_{∞}control for maglev vehicles

_{∞}controllers for electromagnetic suspension systems

_{2}/H

_{∞}control of nonminimum phase-switching converters

_{2}-gain analysis and control synthesis for a class of uncertain switched nonlinear systems

_{2}-gain analysis of nonlinear systems and nonlinear state feedback H

_{∞}control