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Discrete Dynamics in Nature and Society
Volume 2010, Article ID 569850, 16 pages
http://dx.doi.org/10.1155/2010/569850
Research Article

Robust Controller Design Using the Nevanlinna-Pick Interpolation in Gyro Stabilized Pod

1Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin 150001, China
2School of Electric Information Engineering, Northeast Petroleum University, Daqing 163318, China
3College of Earth Science, Northeast Petroleum University, Daqing 163318, China

Received 22 September 2010; Accepted 13 November 2010

Academic Editor: Li Xian Zhang

Copyright © 2010 Bin Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The sensitivity minimization of feedback system is solved based on the theory of Nevanlinna-Pick interpolation with degree constraint without using weighting functions. More details of the dynamic characteristic of second-order system investigated, which is determined by the location of spectral zeroes, the upper bound of , the length of the spectral radius and the additional interpolation constraints. And the guidelines on how to tune the design parameters are provided. Gyro stabilized pod as a typical tracking system is studied, which is based on the typical structure of two-axis and four-frame. The robust controller is designed based on Nevanlinna-Pick interpolation with degree constraint. When both friction of LuGre model and disturbance exist, the closed-loop system has stronger disturbance rejection ability and high tracking precision. Numerical examples illustrate the potential of the method in designing robust controllers with relatively low degrees.

1. Introduction

Gyro stabilized pod can be used to isolate line of sight (LOS) from the movement and vibration of carrier and guarantee pointing and tracking for target in electro-optical tracking system, so in modern weapon systems it has been widely applied [1, 2]. The carrier vibration in the azimuth, pitch and roll direction induces the LOS of image sensor to rotate and causes the image blur and affects miss distance of the target, leading to the tracking performance degradation. In order to overcome these problems, we must establish a stable servosystem to isolate LOS from the movement and vibration of carrier, so as to guarantee system reliability and image quality [3]. The servosystem must not only have high precision and good dynamic quality, but also have disturbance rejection ability and large scale of adaptation range [4]. Along with improving control precision of gyro stabilized system, conventional methods have been limited, therefore, new design methods need to be found to improve system performance.

Up to now, PID control is still the most popular method applied in gyro stabilized pod. With the development of control technology in inertial system, a variety of control methods are produced, such as neural network control [5], Fuzzy control [6], control [7], and optimal control [8], and they have gradually been applied to control the gyro stabilized platform. Switching control method has received much more attention recently, but still remaining at the theory analysis [911].

Weighting functions are adopted in conventional robust controller design, which are chosen to reflect the design objectives and the knowledge of the disturbances and sensor noise. In the optimal control design including and control, a key step is the selection of weighting functions [12]. In many occasions, as in the scale case, the weights are chosen purely as a design parameter without any physical bases. What is worse is that the high order of weighting function will lead to high order of controller. Gahinet and Apkarian [13], and Skelton et al. [14] have introduced a technique for feedback design that allows such a constraint on the degree of the controller, in which performance and robustness are expressed by linear matrix inequalities. The relation between Nevanlinna-Pick (shortly denoted by NP) interpolation with degree constraint and sensitivity reduction is studied by Georgiou and Lindquist, which provides an alternative handle on McMillan degrees in feedback design [15]. The main difference between NP interpolation method and the existing design methods is that the frequency weighting functions are not used to shape the frequency response of S. Another advantage of the method based on the NP interpolation theory is that controllers with relatively low degree can be obtained directly, without any model and controller reduction [16, 17].

Sensitivity minimization is one of the most fundamental and important issues in designing feedback system controllers. Although the idea of the application of NP theory with degree constraint to sensitivity, minimization was presented in [18]. Nagamune firstly gave the computation method for NP interpolation with degree constraint [19].

More details of the dynamic characteristic of second-order system is investigated in this paper, which is determined by the location of spectral zeroes, the upper bound of , the length of the spectral radius, and the additional interpolation constraints. And the guidelines on how to tune the design parameters are provided.

This paper is organized as follows. In Section 2, the sensitivity reduction problem is revisited and it is reviewed how the problem is reduced to the NP interpolation problem. In Section 3, the NP interpolation theory with degree constraint is reviewed, and the dynamic characteristic of second-order system is investigated which is determined by the interpolation conditions. In Section 4, gyro stabilized pod as a typical tracking system is studied, which is based on the typical structure of two-axis and four-frame. The robust controller is designed based on NP interpolation with degree constraint.

2. Sensitivity Reduction of Tracking System and NP Interpolation

Considering a typical one degree of freedom stabilization system simplified to be suited to other applications such as stabilizing a sensor for electro-optical data gathering systems, the control configuration is shown in Figure 1. It consists of the interconnected and the controller forced by command , sensor noise , plant input disturbance , and plant output disturbance .

569850.fig.001
Figure 1: Configuration of feedback system.

Without loss of generality, is assumed to be real rational and proper, and all the unstable poles and zeroes of are assumed to be simple and strictly outside the unit circle. Otherwise, [2022] are referred for the reduction of cases with multiple and boundary zeroes and poles to this case.

Suppose that has poles and zeroes in the closed right half plane (including point at infinity) let us denote them and . If , is unstable, which is needed to be stabilized by feedback. Therefore, a controller is designed so that the closed-loop system fulfills certain design specifications, as depicted in Figure 1. It is well known that the transfer function from to that is called sensitivity function, which is also the transfer function from disturbance to the output, plays an important role in designing feedback controllers, that is because the frequency response of the sensitivity function determines the ability of the closed-loop system for disturbance rejection, tracking, robustness against noise and uncertainties, and so on.

The first object is to stabilize the tracking system in such a way that the transfer functions between any two arbitrary points in the closed-loop system is stable that is called internal stability. It is well known that internal stability can be guaranteed by the following Theorem 2.1.

Theorem 2.1 (see [23]). Internal stability of tracking system is achieved if and only if the sensitivity function is stable and satisfies the interpolation conditions:

Secondly, a reduction of the -norm of implies an attenuation of the effect of disturbances on the output since . For robustness and tracking performance it is also desirable to put a specified uniform bound on the absolute value of the sensitivity function. For example, we need to bound , since . Therefore we require that for some prescribed .

With this bound, the suboptimal solution set to a scalar control problem is equivalent to the solution set to the classical NP interpolation problem By Pick’s Theorem, this problem has a solution if and only if the Pick matrix is positive definite [24].

3. NP Interpolation with Degree Constraint

3.1. Problem Formulation

For simplicity, we assume that the interpolation points,  ,  , are distinct and the interpolation values, ,  , are self-conjugate. Without loss of generality, we can also assume that and that is real. The NP interpolation with degree constraint relating to the sensitivity function can be described as follows.

Given the sets of interpolation points and interpolation values , , satisfying a function from a subset of is desired, where .

The merit of bounding the degree of a closed-loop transfer function is attributed to the following Theorem 3.1. And one of the main design parameters of is stated in Theorem 3.2, which are called the spectral zeros.

Theorem 3.1 (see [25]). Suppose is strictly proper and that satisfies the interpolation conditions (2.2). Then the controller satisfies the degree bound where and are the number of unstable zeros and poles of , respectively.

Theorem 3.2 (see [16]). For each monic real stable polynomial: there exists a unique pair of real polynomials: which satisfy(1) is strictly bounded real,(2), , (3).

In other words, for each , there exists a unique solution to the NP problem with degree constraint. In addition, the relation between the set of self-conjugate zeros of inside the unit circle in the complex plane and the set of the solution pairs of the NP problem with degree constraint is bijective. In particular, zeros of is called the spectral zeros of the function , and the distance between the spectral zero and the origin is called the spectral radius. In [16], it is shown how to determine the unique numerically from the set of spectral zeros of the function by transforming the infinite dimensional entropy maximization problem into a dual finite dimensional strictly convex minimization problem.

If it is impossible to find an meeting by using the spectral zeros, we must introduce extra interpolation constraints: where . That is the degree bound of will be raised. Then the NP interpolation with degree constraint is redefined as The additional interpolation sets can be chosen freely except with conditions that they do not violating the positivity of the corresponding Pick matrix and that the total interpolation data set forms self-conjugate pairs, if they are not real. It is also important to note that additional interpolation constraints can influence both gain and phase of . Next, the relation between the location of the spectral zeros/additional constrains and the shape of the corresponding .

3.2. The Relation between Spectral Zeros and

If the spectral zero locates near the unit circle , it will lifts up to the level near the upper bound at the frequency , as depicted in Figure 2. This is because if is a spectral zero close to the unit circle , and hence . Then, by the water-bed effect, the sensitivity will often be lowered in other parts of the spectrum [9].

569850.fig.002
Figure 2: Influences of a spectral zero on .
3.3. The Relation between Extra Interpolation Constraints and

Introducing an extra interpolation condition , where is close to , fixes the modulus of the sensitivity at a level close to and the phase of close to that of at the frequency , as depicted in Figure 3. Thus, can control the magnitude and the phase of the sensitivity. However, the more extra interpolation constraints, the larger bound of . From Theorem 3.1, the controller degree bound will increase by the number of extra interpolation constraints. Thus, we should try to use additional constraints as little as possible.

569850.fig.003
Figure 3: Influences of an extra interpolation constraint on .
3.4. The Relation between NP Interpolation Conditions and the Dynamic Characteristic

In this section, we will propose general strategies to tune the design parameters by investigating the relation between NP interpolation conditions and the dynamic characteristic of second-order system.

Suppose the continuous-time plant described by and the norm bound constraint

First, by Möbius transform: , which conformally maps the right half plane into the complement of the unit disk: the set of corresponding to is obtained in the discrete-time setting

Note that the condition in (3.13) is a boundary interpolation constraint, which makes the NP interpolation problem complicated. This difficulty can be circumvented by introducing another variable . Define a new function as follows: the NP interpolation with degree constraint becomes

If , the set (3.15) will be

Especially, we will introduce an extra interpolation condition to investigate the influence to dynamic characteristic of second-order system. We choose as the adding constraint. The NP interpolation problem with degree constraint is redefined as

Next, we will study how the following four factors influence the dynamic characteristic of second-order system: the location of the spectral zeros, the upper bound of , the length of spectral radius, and the additional constrains.

(a) The Influence of the Location of the Spectral Zeros.

When we locate 5 groups spectral zeros: , , , , , which are self-conjugate, and of the same length of spectral radius, the corresponding step response of closed-loop system is depicted in Figure 4, denoted by . The closer to positive real semiaxis the spectral zeros, the larger the overshoot and the longer the settling time . When the spectral zeros are located in left half plane, there is no overshoot, and the settling time becomes shorter. For the described plant (3.10), when the located spectral zeros are near the positive real semiaxis, the settling time is about 180 seconds. When they are near the negative real semiaxis, the settling time is about 5 seconds, but the undershoot becomes larger. In the following (b), (c), (d), the spectral zeros are chosen as .

569850.fig.004
Figure 4: The influence of the location of the spectral zeros.

(b) The Influence of the upper bound of .

The upper bound of is specified as 1.1, 2, 5, 8, 9. The corresponding step responses are depicted in Figure 5, denoted by , , , , . When is close to 1, the closed-loop system becomes unstable. Increasing the value of , the overshoot becomes larger. The rise times are almost the same. For the given plant (3.10), is appropriate.

569850.fig.005
Figure 5: The influence of the upper bound of .

(c) The Influence of the length of spectral radius.

When we choose the spectral zeros: , , , they have the same phase lying in imaginary axis. However, they have different distance from to origin. Three step response curve are depicted in Figure 6, denoted by , , . As we can see, the rise time becomes longer with the spectral radius smaller. When the spectral radius is less than 0.5, there is no overshoot.

569850.fig.006
Figure 6: The influence of the length of spectral radius.

(d) The Influence of the additional constrains.

The extra interpolation conditions are chosen as 1.5, 1.2, 1.1, 1.05, 1.01, 1.005, 1.002, 1.0005. Their step responses are depicted in Figure 7, denoted by , . The settling time is almost unchanged. But, the steady error becomes larger with the value of the additional interpolation points increasing.

569850.fig.007
Figure 7: The influence of the additional constrains.

The location of spectral zeros and the upper bound of have more influence to the overshoot and the settling time of feedback system. The spectral zeros should be located in the left half plane, and the less , the better, if the closed-loop system is stable. The spectral radius affects the dynamic characteristic little except only some influence to the rise time. The additional interpolation conditions have more impact on the steady error, and the extra interpolation points of less value should be chosen from the simulation. We will mainly utilize these four strategies for controller design, instead of using weighting functions.

4. Example Illustrating the New Design Method

4.1. Mathematical Model of Gyro Stabilized Pod

The gyro stabilized pod investigated in this paper is based on the typical structure of two-axis and four-frame. The stabilization loop of each axis is almost the same, including stabilization controller, motor drive circuits, DC torque motor, and rate gyro. Take the pitch framework for example, the block diagram of stabilization loop is as in Figure 8, where is the transfer function of correction link. represents friction torque disturbance. represents measurement noise. The approximate open-loop transfer function of the plant:

569850.fig.008
Figure 8: Block diagram of stabilization loop.

Design requirements:(R1) overshoot: ;(R2) settling time:  sec;(R3) oscillatory times: ;(R4) steady-state error of system approaches zero.

In this subsection, the robust control problem is solved by the proposed method. The plant treated is transformed from  (4.1) which has a bound unstable zero . The upper bound of the can be chosen 1.7. The set of functions meeting the interpolation constraints is represented as

As described in Section 3.4, we introduce the variable , and defined another set of functions:

It can be shown that the set has only one element, namely, , which does not satisfy design requirements. Hence, it is necessary to add some interpolation constraints for the achievement of the specification. Taking into account the low gain requirement at low frequencies, we introduce one additional interpolation constraint , and redefine the set by

Now we have one spectral zero of the function as the design parameter. When the spectral zero is located at , the corresponding sensitivity function is calculated as

Due to the reverse variable transform, the sensitivity function corresponding to (4.6) becomes And, the controller calculated by (3.3) is

4.2. Tracking Performance and Disturbances Rejection

As can be seen in Figure 9, design requirements (R1)–(R4) are all met. Next, we will check the tracking performance and the disturbance rejection ability using sinusoidal signal of three different frequencies: 0.1 Hz, 1 Hz, 10 Hz. Both the friction and the pulse jamming are considered simultaneously. The friction torque is described by LuGre model where is static friction force, is Coulomb friction force, is Stribeck velocity, is viscous friction coefficient, and is angular velocity. Obviously, the friction force will be different when tracking different frequency sinusoidal signal, as depicted in Figures 1012. The pulse jamming is inserted into the output of the plant at  sec, 0.5 sec, 0.05 sec, and the lasting time is 5 sec, 0.5 sec, and 0.05 sec.

569850.fig.009
Figure 9: Step response of tracking system.
fig10
Figure 10: Reference signal is sinusoidal signal of 0.1 Hz.
fig11
Figure 11: Reference signal is sinusoidal signal of 1 Hz.
fig12
Figure 12: Reference signal is sinusoidal signal of 10 Hz.

As can be seen in Figures 1012, the proposed method has advantages of short regulating time, high precision, and high disturbance rejection ability.

4.3. Influence of Model Uncertainty to Tracking Performance

Suppose that the perturbed plant is described by

Sinusoidal signals of different frequency, 1 Hz, 10 Hz, are adopted as the reference signal. The disturbances are the same as in Section 4.2. The main difference between Figures 11(b), 12(b) and 13 is that the later has longer settling time when the pulse jamming is added to the output of the plant. The Oscillatory times become more and more when the plant has uncertainty.

fig13
Figure 13: Output curves of different frequency.

5. Conclusions

In this paper, we have proposed a new approach to the robust controller design based on the NP interpolation theory with degree constraint. There are two kinds of design parameters: the spectral zeros and the extra interpolation constrains. We have investigated how the following four factors influence the dynamic characteristic of second-order system: the location of the spectral zeros, the upper bound of , the length of spectral radius, and the additional constrains. And we also provide the guidelines on how to tune the design parameters. Gyro stabilized pod as a typical tracking system is studied. Simulations show that the proposed method has advantages of short regulating time, high precision, strong disturbance rejection ability, and good robustness.

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