Optimal Disturbance Observer Design for High Tracking Performance in Motion Control Systems

: In this paper, a stability-driven optimal disturbance observer (DO) is proposed. The proposed method does not require any plant inverse dynamics to detect introduced disturbances or a stabilizing Q ﬁlter. It does not require additional compensators to resolve causality problems, due to the relative degree, or ﬁlters to solve instability problems of non-minimum phase plants. Using this method enables wideband and narrowband disturbances to be attenuated by simply multiplying the corresponding peak ﬁlters by the baseline weight function. Furthermore, the proposed DO guarantees the stability of closed-loop systems because the already designed outer-loop systems are considered as a target plant to be stabilized and because of the Lyapunov stability-based H ∞ control. In the application example, it was conﬁrmed that the proposed method is effective, and the position error signals were improved by 20.9% in commercial hard disk drives and 36.6% in optical image stabilization systems.


Introduction
One of the main causes of tracking performance degradation in the motion control systems is a disturbance. The disturbances act as additional input signals and generate unwanted output signals. In the steady-state, the unwanted output signals are added to a carefully designed output signal, resulting in distorting of the output signal. To reduce the effects of the disturbances, a disturbance observer (DO) has been used in many industrial applications [1].
Many motor control systems have employed DOs. In permanent-magnet synchronous motors, DOs have been used to remove generalized multiple disturbances, such as cogging torques, load torques, friction torques, measurement error effects, dead-time effects, and parameter perturbations [2]. To remove narrowband high-frequency disturbances, a discrete-time DO based on an infinite-impulse-response filter was proposed and applied to a permanent-magnet synchronous motor [3]. A robust high-order DO for the state-dependent Riccati-equation-based suboptimal speed controller of an interior permanent-magnet synchronous motor drive was proposed [4]. Unlike a conventional DO, in that study, the proposed high-order DO guarantees fast convergence of the estimated error. To reduce both noise and uncertainty, a modified DO was proposed, and the method was applied to brushless DC motor drive systems [5]. To obtain design flexibility, an algebraic-calculation-based DO was also proposed [6]. In two-axis control systems, DOs have also been used to enhance performance. A DO was applied to a double-gimbaled variable-speed control moment gyroscope. In the study, a DO was used to decouple the inner-and outer-gimbal system and to reject disturbances [7]. Star tracker systems also use DOs. A star tracker system based the stability to all types of models, but the method still required additional filters to solve the causality and instability problems.
In this study, a new unified design for the DO is proposed to remove the above three-step design procedure with guaranteed stability. For stability, the proposed design method uses a linear matrix inequality (LMI)-based H ∞ control because the LMI-based control guarantees the stability of the closed loop systems. However, it is not simple to apply the DO design to the standard LMI framework because the closed loop systems are mixed with a plant, a conventional controller, and a DO. To apply the DO design to the LMI framework, in this study, the closed loop system with the DO is reinterpreted as a parallel system that consists of a plant, a conventional controller (C) forming outer-loop systems, and a DO (C conv DO ) forming inner-loop systems. Because the DO can be written by , the closed loop system with the DO can be considered as a plant controlled by the two parallel controllers, C and C conv DO . From the DO point of view, the DO controls a fictitious plant written by a function of the plant and the outer-loop controller C, that is, f (P, C). In this study, an f (P, C) is considered as a control target, which is a new design framework for DOs. Considering the function of the DO, the DO must be able to estimate the incoming disturbances well and remove the disturbance from the total control input. To reach the goal, a cost function to be minimized is defined by z = |d − u DO |, where d and u DO are the incoming and estimated disturbances. Now, only one challenge of how and how much to eliminate the disturbance remains. In this study, the problem is solved by applying a weighting function that controls the capability of the disturbance rejection. With the proposed design framework and the suggested cost function, the generalized systems are augmented and finally the optimal DO is designed. Unlike previous studies, the type of the model needs not be considered in the proposed method. Therefore, additional filters and compensators to solve causality and instability problems caused by the model types are not required. The proposed method requires only one design specification-i.e., the frequency characteristics of the disturbances to be attenuated-and guarantees closed-loop system stability. This work makes the following contributions.
• The proposed method guarantees the stability of the closed-loop system because it provides optimal control based on the Lyapunov stability. • It does not require plant inverse dynamics.
• Stabilizing Q filters are not required.
• The relative degree does not need to be considered. Therefore, additional filters to compensate for the relative degree are not necessary. • Compensators based on all-pass filters are not required for non-minimum phase plants.
• Both wide and narrow band disturbances can be attenuated.
• Because this method provides more stability margins, it can reduce a sensitivity peak.
The rest of the paper is organized as follows. Section 2 contains the main research results. In this section, the conventional DO is discussed, and a new framework for optimal DO design is suggested. Section 3 includes the plant identification and controller design for stabilizing the outer loop. In addition, illustrative examples, including time and frequency responses, are presented. The conclusion follows.

Optimal Disturbance Observer Design
This section contains the main research results. The conventional DO is analyzed, and an optimal design framework is proposed. The formulation for the suggested method is also described.

Conventional Disturbance Observer
In the presence of disturbances, DOs have been utilized intensively to reduce the disturbances. A DO monitor introduces disturbances and subtracts the disturbances from the total control effort. A typical DO is shown in Figure 1. In the figure, P is a plant to be controlled; P n is a nominal plant model; d is an external disturbance; C is a controller when there is no disturbance; Q is a low pass filter to stabilize DO when the disturbance is introduced. If it is assumed that the output of the Q filter is not connected to the feedback loop, then the input of the Q filter, i Q , can be calculated as In the equation, if P and P n are identical, then i Q = d. The role of the Q filter including additional compensators is to stabilize the DO, and to solve causality problems by relative degree and stability problems of the non-minimum phase plants. Finally, the estimated disturbanced is applied to subtract d from the overall control input. In Figure 1, the closed loop system can be rewritten by where QP −1 n /(1 − Q) is a transfer function of the DO, C conv BO . Therefore, the closed loop system is a parallel system controlled by two controllers C and C conv BO . From the DO point of view, the DO controls a composite plant defined by a function of P and C, that is, f (P, C). Thus, the DO design problem is equivalent to design the DO to f (P, C). In this study, because f (P, C) is a pre-designed fictitious model in the absence of external disturbances, the optimal DO is designed to f (P, C), which is a new DO design method.

Design Framework for Optimal Disturbance Observer
With a given outer-loop controller, a DO should estimate introduced disturbances and subtract the disturbance from the control energy. To estimate the disturbance accurately, the output of the DO should be as close to d as possible in the frequency range of interest, which can be obtained by With this motivation, a new framework for optimal DO design is proposed, as shown in Figure 2.
Using the block diagram, augmented systems are derived, and a linear-matrix-inequality-based DO is designed. In the block diagram, W is a weight function to control the performance of the DO; w is an external disturbance; z is an output to be minimized; C DO is an optimal controller that minimizes a transfer function from w to z. In addition, y DO is a measurement, which is an input of the C DO . u DO is an output of the C DO , which is designed to be equivalent to w in the frequency range of interest. As for disturbances, there are wideband disturbances and narrowband disturbances. To attenuate each disturbance, we propose separated weighting functions, W base and W peak , which are used to attenuate wideband disturbances and narrowband disturbances, respectively. W base for reducing wideband disturbance is represented by where M, A, ω * B and n are the upper bound, the lower bound, the bandwidth, and an order of W base (s), respectively [45]. Here, using W base , wideband disturbances can be reduced. To reduce narrowband disturbances, a peak-filter-based weight function is used as where ζ i determines the width of the peak; ω i is a peak frequency; g i is the magnitude of the peak response; N is the number of the narrow band disturbances [46]. In Equation (5), if we would like to reduce one narrow disturbance, then one second order filter should be designed. If we would like to reduce N narrow band disturbances, then 2 × N ordered filters should be designed. Therefore, the single second order filter reduces one narrow band disturbance. Finally, to reduce both wide and narrow band disturbances, two weight functions are combined as and their typical frequency responses are illustrated in Figure 3. In the figure, the gain of W −1 base W − peak 1 is −14 dB at 1 Hz, which means that if an external disturbance (w) with the 1 Hz frequency component is introduced, the designed DO generates u DO that satisfies |w − u DO | = 10 −14/20 . This means that wide band disturbances that exist in the region below 10 Hz are reduced by approximately 10 −14/20 . At 50 Hz, the gain of W −1 base W − peak 1 is −20 dB, which implies that the introduced disturbance can be attenuated by −20 dB. In this sample weight, even though a single narrowband disturbance is considered, multiple narrowband disturbances could be considered with N > 1 in Equation (5). However, if excessive high-order filters are used to reduce the multiple narrow band disturbances, the proposed design framework decreases the performance of the disturbance rejection to maintain stability. Therefore, critical disturbances to the closed loop systems must firstly be attenuated.

Optimal Disturbance Observer Design
To realize the proposed framework, all transfer functions should be described by state space representation. Here, P is the identified model and can be written by state space representation.
P :ẋ P (t) = A P x P (t) + B P u P (t) y P (t) = C P x P (t) In addition, W and outer-loop controller C are described by state space representation, as in (8) and (9).
Considering (20), the system matrix of C is as follows.
Using Equation (7), a state equation of P is calculated bẏ A state equation of C is represented bẏ Moreover, a state equation of W can be calculated bẏ z to be minimized is obtained by . (14) and the input of the DO is captured as Using Equations (11)-(15), the augmented systems are written aṡ where In addition, a state space representation of C DO is defined by Then, the closed loop systems, T zw , can be obtained by Theorem 1. The following two statements are equivalent.
where Q, F, L, R, X, and Y are variables. X > 0 means that X is positive definite and * denotes an ellipsis for terms induced by a symmetric matrix. Then, an optimal controller C * DO is given by where U and V are any nonsingular matrices satisfying YX + VU = I.
Theorem 2. C * DO stabilizes the closed loop systems.
Proof of Theorem 2. Because C * DO is a Lyapunov-stability-based H ∞ dynamic output feedback controller, C * DO always stabilizes the closed-loop systems [37][38][39].

Application Example
In this section, we applied the proposed method to two motion control systems-that is, hard disk drives (HDDs) and a pinhole camera model.

Case 1
Firstly, the proposed design method was applied to HDDs as a motion control system, as shown in Figure 4. The dynamics of a plant model were measured by a laser Doppler vibrometer (LDV) and a dynamic signal analyzer (DSA). The mathematical models are identified by where K t is the torque constant; J m is the inertia; B d is the damping coefficient, and K s is the damping coefficient; K s is the spring coefficient that determines low-frequency poles. Here, ω n and ζ n describe a baseline of high-frequency characteristics, and the remaining terms represent detailed high-frequency features over the baseline. For the high-frequency dynamics, m = 1 is chosen. The values of the parameters are listed in Table 1.  Figure 4 shows the identified model and the nominal model. For outer-loop design, a second-order nominal model is used. It is represented by the first two terms of the right side of Equation (18). Its state space model is expressed as In this study, an estimator-based state feedback controller is designed to stabilize the plant as follows.
C :˙x where L and K are the estimator and state feedback control gains of C, respectively. To design the state feedback controller, a pole placement method is used [47]. In the outer-loop design, 914 Hz is chosen as the 0-dB crossover frequency, as stability margins, 12.4 dB and 52.6 • are used. For outer-loop design, estimator-based state feedback control was used. In the presence of disturbances, the proposed method was applied to remove the disturbances. Because only wide band disturbances were considered in this study, W base was used as a weight function. For weight function design, M = 10 0/20 , A = 10 −14/20 , ω * B = 2π100 and n = 1 were used. The designed weight function is represented by a red solid line in Figure 3. The designed open-loop transfer functions are shown in Figure 5.    In the magnitude plot in Figure 5, the loop gain designed by C * DO is added to the gain of the combined open-loop transfer function, and the additional gain results in an increasing 0-dB crossover frequency. Although it appears that C * DO just adds gain, C * DO also adds phase in the phase plot. This additional phase maintains or even increases the relative stability; in general, this advantage cannot be obtained by simple high-gain control. Figure 6 shows Nyquist plots of the outer loop and the combined loop. In the plot, C * DO pushes the inner-loop transfer function to the right side, which indicates more stability margins, resulting in a lower peak in the sensitivity function. The maximum peak of the sensitivity function, S, is defined as where L denotes an open-loop transfer function. Moreover, 1 + L(jω) indicates the distance between (−1, 0) and L(jω). The distance is represented by the radius of the circle tangent to L(jω) and is located at (−1, 0). Furthermore, the radius is larger in more stable systems with more stability margins. In Figure 6, r 2 > r 1 , which implies that C * DO offers more stability margins to the closed loop systems. Therefore, C * DO decreases the peak of the sensitivity function. The compared sensitivity functions are shown in Figure 7. In Figure 6, r 1 = 0.67 and r 2 = 0.825. Therefore, the corresponding sensitivity peaks are 20 log 10 (1/0.67) = 3.48 dB and 20 log 10 (1/0.825) = 1.67 dB, respectively, as shown in Figure 7. As the sensitivity function is improved, the torque disturbance function is also is improved. The torque disturbance function is defined by a transfer function from plant input to plant output.
With C * DO , the peak of the torque disturbance function is reduced by 4.8 dB, as shown in Figure 8. The weight function W can control the performance of the disturbance rejection and frequency range of interest. The lower bound of W −1 controls the performance of the disturbance rejection, and a cutoff frequency of W −1 determines the frequency range of interest. To measure the entire performance in all frequency ranges, random disturbances were applied to a zero-order hold equivalent discretized model with a sample frequency of 10.68 kHz. The measured power spectral density and accumulated position error signal (PES) are compared in Figure 9. With C * DO , the performance was improved by 20.9%. For readability, the detailed results of the frequency responses are summarized in Table 2.

Case 2
As for the second example, an optical image stabilization (OIS) system was used [41]. Because the identified model is a lightly damped system, a notch filter was firstly designed to suppress the oscillated responses of the plant. To design the outer loop controller C, an estimator-based state feedback controller was designed, as shown in Equation (20). In addition, a weight function was design, as shown in Figure 10, because hand shake disturbances exist in the frequency range below 100 Hz. Using the weighting function, designed sensitivity functions were compared, as shown in Figure 11. In the figure, the sensitivity peak is reduced, which implies that additional stabilities increase. In addition, the rejection performance to the plant output disturbances is also improved in the low frequency range. As the sensitivity function is improved, the torque disturbance function is improved, as shown in Figure 12. In the figure, it is shown that the disturbance rejection performance is improved in the frequency range below 100 Hz. In particular, it shows a disturbance reduction effect of −15 dB at 1 Hz. This results from the weight function, as in Figure 10. As in the Section 3.1, in this Section 3.1, a random disturbance experiment was conducted. Figure 13 shows the compared power spectral densities. In OIS, error is defined by image pixel deviations because the parameter is crucial to image qualities.