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A Guide to Design Disturbance Observer

The goal of this paper is to clarify the robustness and performance constraints in the design of control systems based on disturbance observer (DOB). Although the bandwidth constraints of a DOB have long been very well-known by experiences and observations, they have not been formulated and clearly reported yet. In this regard, the Bode and Poisson integral formulas are utilized in the robustness analysis so that the bandwidth constraints of a DOB are derived analytically. In this paper, it is shown that the bandwidth of a DOB has upper and lower bounds to obtain good robustness if the plant has non-minimum phase zero(s) and pole(s), respectively. Besides that the performance of a system can be improved by using a higher-order disturbance observer (HODOB); however, the robustness may deteriorate, and the bandwidth constraints become more severe. New analysis and design methods, which provide good robustness and predefined performance criteria, are proposed for the DOB based robust control systems. The validity of the proposals are verified by simulation results.


INTRODUCTION
1 E. Sariyildiz is with the Department of System Design Engineering, Keio University, Yokohama, 223-8522, Japan (e-mail: emre@sum.sd.keio.ac.jp) 2 Corresponding author information can be added as a footnote.
A DOB, which was proposed by K. Ohnishi et al., is a robust control tool that is used to estimate external disturbances and system uncertainties [1][2][3]. The estimated disturbances, which include system uncertainties, are fed-back by using an inner feedback loop so that the robustness of a system is achieved by using a DOB [2].
Performance goals of a system are achieved by using an outer feed-back loop controller that is designed independently by considering only the nominal plant model, since a DOB can nominalize the inner-loop [4,5]. This control structure is called as two-degreesof-freedom control in the literature [5]. Although a DOB has been widely used in several motion control applications, e.g., robotics, industrial automation and automotive, in the last two decades, it has no systematic analysis and design methods [6][7][8]. Therefore, the performance and robustness of a DOB based control system highly depend on designers' own experiences.
A low pass filter (LPF) and the inverse of a nominal plant model are required to design a DOB. Although the LPF of a DOB is essential to satisfy causality in the innerloop, it is one of the main robustness and performance limitation sources in the control systems based on DOB [9,10]. Besides that the inverse of a nominal plant model causes internal stability problem if the plant has non-minimum phase zero(s); therefore, a special consideration is required when a DOB is implemented to a non-minimum phase plant [11][12][13].
It is a well-known fact that a DOB can estimate disturbances precisely if they stay within the bandwidth of the DOB's LPF [14,15]. Therefore, its bandwidth is desired to set as high as possible to estimate disturbances in a wide frequency range, i.e., to improve the robustness and performance [2]. However, the bandwidth of a DOB is limited by the robustness of a system and noise, so it cannot be shaped freely [2,10].
The noise limitation is directly related to sampling rate and measurement plants and methodology; it puts an upper bound on the bandwidth of a DOB [2]. Several researches have been reported to increase the bandwidth of a DOB by suppressing the noise of measurement [16][17][18]. The robustness of a DOB based control system is directly related to the dynamic characteristics of the DOB's LPF and nominal plant. They also limit the bandwidth of a DOB; however, the relation between the robustness of a system and the dynamic characteristics of the DOB's LPF and nominal plant has not been clearly reported yet [10,19,20]. Recently, it was shown by the authors that if a minimum-phase system has only real parametric uncertainties, then a DOB can guarantee the robustness of the system by increasing its bandwidth, and the stability margin of the system improves as the bandwidth of the DOB is increased [21]. However, it considers only the minimum-phase systems which have real parametric uncertainties when a first order DOB is used.
The main aim of this paper is to clarify the robustness constraints of DOB for a broad range of application area. The Bode integral formula is utilized so that the robustness of minimum-phase and time-delay systems are derived analytically; and the Poisson integral formula is used to derive the robustness constraints of systems with right half plane (RHP) zero(s) and pole(s) [22][23][24][25][26]. It is shown that right half plane (RHP) zero(s) and/or time-delay of a plant limit the bandwidth of a DOB, however RHP pole(s) of a plant put(s) a lower bound on the bandwidth of a DOB to obtain a good robustness.
Besides that increasing the order of a DOB improves the performance of a system by using the bandwidth of a DOB more effectively; however the bandwidth constraints become more severe, and the robustness of a system deteriorates. New analysis and design methods are proposed by using the derived bandwidth constraints. The internal stability problem is solved by using an approximate minimum phase nominal plant model when a plant has non-minimum phase zero(s), and a performance controller is proposed for the conventional two-degrees-of-freedom control structure when a plant has RHP pole(s).
The rest of the paper is organized as follows. In section II, the conventional twodegrees-of-freedom control structure of a DOB based robust control system is presented briefly. In section III, the bandwidth constraints of a DOB are derived analytically by using the Bode and Poisson integral formulas. In section IV, four casestudies are given. The paper ends with conclusion given in the last section. Fig. 1

DISTURBANCE OBSERVER
where L  , S  and T  denote the open-loop, sensitivity and co-sensitivity transfer functions, respectively [10].

BANDWIDTH CONSTRAINTS OF DISTURBANCE OBSERVER
The bandwidth constraints of a DOB are derived analytically as follows:

Minimum-phase Plant
Lemma 1: Let us consider the plant model given in (7) and assume that the uncertain plant is minimum-phase and the order of DOB is one. Then, it can be shown that the inner-loop is strictly robust if The equation (11) shows that the Nyquist plot of the inner-loop gets into the unit circle that is shown in Fig. 2 if min max e e  , which contradicts with the error assumption, when 0   . Thus, the first part of the Lemma 1, i.e., strict robustness, is satisfied.
Although the Nyquist plot of the inner-loop gets into the unit circle, i.e., strict roburtness is lost, when 0   , the robust stability can be guaranteed for a wide range of However, if a HODOB is used instead of a DOB, then the robustness cannot be guaranteed for a wide range of DOB's bandwidth even if 0   . It can be easily shown by using a HODOB, e.g., a second order DOB. In that case, the robustness of a system can be guaranteed for a limited bandwidth of a HODOB.
The Fig. 2 Since the relative degree of   i L s is always higher than one when a HODOB is used, (12) can be simplified, and the Bode integral formula is obtained as follows: [22,24]   The robustness of a system depends on the magnitude of sensitivity function peak, which is defined by is not an easy task to determine     sup S jw by using (13) due to the infinite integral range. Although, from a mathematical point of view, (13) can be balanced with a small peak in a wide frequency range, control systems cannot exhibit this response due to uncertainties, digital control implementations, and so on. Besides that the Lemma 1 shows that   i S jw has a peak if a HODOB is used. The Lemma 2 bounds the integral range of (13) when a HODOB is used.
Proof: The equation (14) holds if a HODOB is used. Let us consider the relation given by [30] If (14) is put into (16), then To derive the robustness constraints of a HODOB, the performance and robustness requirements are determined in a predefined frequency range by shaping the sensitivity transfer function. Then, the robustness of a system is analyzed, and the constraints are derived by using the Theorem 1 as follows: Theorem 1: Let us assume that a minimum phase plant is defined by using (7).
Let us also assume that   If a DOB is used, then the system has a good robustness in a wide frequency range, yet its performance is limited by the dynamic characteristics of the DOB. However, if a HODOB is used, then the LPF of a HODOB should satisfy the following inequalities to obtain a good robustness and predefined performance criterion.
Proof: The Lemma 1 proves the robustness of a DOB. Therefore, a HODOB can be considered directly. Let us rewrite (13) by using If the sensitivity constraints given in the Theorem 1 and Lemma 2 are applied into (19), then which can be easily rewritten as follows: where (22) is the function  given in the Theorem 1. If the sensitivity constraint given in the Theorem 1 is applied into (10), then If (22) is applied into (23), then (18) is derived.
As the order of a DOB, which is defined by 1 k  , is increased, the difference between the frequencies w  and w  decreases. Therefore, increasing k causes higher sensitivity peak as derived in (21).
The equation (23) shows that 1   and w  increase as the bandwidth of a HODOB is increased. Therefore, the peak of   i S jw becomes higher with the increasing bandwidth of a DOB as derived in (21).
The equation (18) provides a new design tool to obtain a good robustness and predefined performance criterion, which are determined by and w  . If the LPF of a HODOB satisfies (18), then the robustness and performance goals of a system can be achieved. However, (18) includes conservatism due to sectionally constant sensitivity It can be lessened by using more realistic sensitivity bounds [29,30].

Plant with Time-Delay
The Lemma 3 is used to bound the integral range of (13) when a plant has timedelay.   (25) Proof: Similar to the Lemma 2 [25].
The bandwidth constraints of a DOB due to time-delay are derived by using the Theorem 2 as follows: Theorem 2: Let us assume that a plant is defined by using (7) If the order of a DOB is one, then Proof: Similar to the Theorem 1. The Lemma 3 is used instead of the Lemma 2.
The equation (26) provides a new design tool to obtain a good robustness and predefined performance criterion when a plant has time-delay. It shows that the bandwidth of a DOB is limited due to time-delay. However, the proposed design tool also includes conservatism, since the sensitivity bounds are not realistic.
Some comments are required to determine R . Its smallest value, which satisfies the constraint given in the Theorem 2, should be used to lessen the peak of   i S jw .
The equation (28) As it is expected from the Theorem 1, (29) shows that the peak of   i S jw increases as the order of a DOB is increased.

Performance Limitations
It is a well-known fact that RHP zero(s) and pole(

The Poisson's Integral Formulas
The Poisson's integral formulas are used to derive the bandwidth constraints of a DOB analytically when a plant has RHP pole(s) and/or zero(s).  [29,30]. The integral ranges of (32) and (33) are bounded by     2 2 x pi

Plant with RHP Zero(s)
An approximate nominal plant model is used to solve the internal stability problem when a plant has RHP zero(s). It is defined by The bandwidth constraints of a DOB due to RHP zero(s) are derived by using the Theorem 3 as follows: Theorem 3: Let us assume that a plant, which has a RHP zero at RHP z , is defined by using (7) where (41) and (42)

Plant with RHP Pole(s)
Against the conventional design of DOB based robust control systems, the outer loop controller should be designed as a stabilizing controller instead of a performance one, since the inner-loop is always unstable when a plant has RHP pole(s). Therefore, a new controller   P C s , shown in Fig. 3, is used to achieve performance requirements. The bandwidth constraint due to RHP pole(s) is derived by using the Theorem 4 as follows: Theorem 4: Let us assume that a plant, which has a RHP pole at RHP p , is defined by using (7) Proof: Similar to the Theorem 3.

The equation (45) provides a new design tool when a plant has RHP pole(s). It
shows that the bandwidth of a DOB has a lower bound to obtain a good robustness when the plant is unstable; however, it also suffers from the conservatism.
Consequently, the design constraints of a DOB are summarized in Table 1.

SIMULATION RESULTS
In this section, four case studies are carried out to verify the proposals.

Minimum-Phase Plant:
Let us consider nominal and uncertain plant models by using rad to obtain robust stability. Fig. 4 shows that as the order of DOB is increased, the bandwidth of DOB is used more effectively and noise suppression is Higher Order DOB

Plant with
Time Delay improved. However, the robustness deteriorates, and the bandwidth constraints of DOB become more severe, as expected from the Theorem 1. The robustness and performance of the system can be improved by using the Theorem 1 as follows: The design parameters k , and where W B denotes the bandwidth of DOB.

Plant with Time-Delay
Let us consider the time-delay constraints by using the following plant model.  Fig. 6 shows the frequency responses of inner-loop sensitivity and co-sensitivity transfer functions. The real bandwidth constraint, which is significantly different from the derived one due to conservatism, is obtained as . The sensitivity weighting function is same as given above.

Plant with a RHP Zero
Let us consider the RHP zero constraints by using the following plant model.  [31]. It is assumed that

Plant with a RHP Pole:
Let us consider the RHP pole constraints by using the following plant model.    The source of conservatism, i.e., the approximate sensitivity/co-sensitivity bounds, can be seen in Fig. 10. In this figure, the grey areas are determined by the sectionally constant sensitivity bounds, and the black areas denote the errors which cannot be considered in the robustness analysis. It clearly shows that, there is a significant difference between the areas, which are bounded by the real sensitivity function and its approximate bound.
Let us again consider the plant with time-delay example, which has the most severe conservative result, to obtain more accurate bandwidth constraint by decreasing the conservatism. If we consider the nominal plant model with time-delay, then the sensitivity function frequency response is derived as follows: where g is the bandwidth of the first order DOB; and  is delay time. The frequencies 1 w and 2 w , given in the Fig. 10, can be obtained approximately as follows: Step Response under disturbances Fig. 9.
Step responses of the unstable plant when it is controlled by using the proposed robust controller The conservatism can be decreased by using 1 w and 2 w instead of R . If we use the sectionally constant sensitivity bound with 1 w and 2 w , then the bandwidth limitation of DOB, which is obtained as 70 rad/s in the second example, is derived as 95 rad/s. Hence, the conservatism can be lessened by considering more realistic sensitivity bounds.

CONCLUSION
This paper has been concerned with the problem of robustness and performance constraints in the design of DOB based control systems. The bandwidth constraints of DOB are derived analytically by using the Bode and Poisson integral formulas, and new analysis and design tools are proposed. The proposed tools include conservatism; however, it can be lessened by using a more realistic sensitivity/co-sensitivity bound, which increases the complexity of analysis, as shown in the paper. The experiences with the proposed tools showed us that the conservatism is not a severe problem, since the proposed tools give a deep insight into the design constraints of DOB based control systems. Therefore, the analysis and design tools are very useful, and they can be easily implemented into many different DOB based robust control problems. Table 1 Summary of the design constraints of a DOB     Step responses of the unstable plant when it is controlled by using the proposed robust controller   Step responses of the unstable plant when it is controlled by using the proposed robust controller