Performance degradation due to measurement noise in control systems with disturbance observers and saturating actuators

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Journal of the Franklin Institute 356 (2019) 3922–3947 www.elsevier.com/locate/jfranklin

Performance degradation due to measurement noise in control systems with disturbance observers and saturating actuators Gyujin Na a, Nam Hoon Jo b, Yongsoon Eun a,∗ a Department

of Information and Communication Engineering, DGIST, Daegu 42988, Republic of Korea of Electrical Engineering, Soongsil University, Seoul 06978, Republic of Korea

b Department

Received 16 January 2018; received in revised form 31 January 2019; accepted 11 March 2019 Available online 21 March 2019

Abstract Augmenting feedback control systems with disturbance observer (DOB) is a widely used technique in system design to compensate for the effect of exogenous disturbances as well as plant model uncertainties. In practice, actuator saturation should be taken into account in the design of control systems with DOB. In such cases, we have observed performance degradation due to zero mean measurement noise in the form of tracking loss. This phenomenon has never been reported in DOB literature. This paper reports the phenomenon, analyzes the conditions under which the tracking loss occurs, and also presents design guidelines to avoid the tracking loss. Experimental verification is also provided using a BLDC motor drive testbed. © 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.

1. Introduction In control systems design, it is highly desired to achieve robust control performance against model uncertainties and external disturbances. Various robust control tools such as sliding mode control [1,2], backstepping control [3], L1 adaptive control [4], H∞ control [5], etc, were introduced for this purpose. In particular, disturbance observer (DOB), which was first ∗

Corresponding author. E-mail address: [email protected] (Y. Eun).

https://doi.org/10.1016/j.jfranklin.2019.03.001 0016-0032/© 2019 Published by Elsevier Ltd on behalf of The Franklin Institute.

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Fig. 1. Block diagram of DOB control systems with saturating actuators.

proposed in [6], has been widely used as a simple and powerful robust control tool. A large number of industry engineers and academic researchers have utilized DOB in applications such as robotic manipulators [7–12], magnetic disk drives [13], quadrotors [14–16], electric bicycles [17], automatic steering [18], and bank to turn missiles [19]. Actuator saturation is inevitable in all feedback systems due to the physical limitations of actuators, hence, has been well studied for the stability, stabilization, and performance [34,41– 43]. Although linear theory for DOB has matured [20–23], the actual implementation has to be done with saturating actuators. For control systems with DOB, poor transient and even instability may occur if saturating actuators are not properly taken into account in the design [24–26]. This was first recognized in [24] for robust position control and a modified DOB structure that can mitigate the undesired behavior was proposed and analyzed in [24–26]. The modification is simple: feed the saturated control signal to the DOB instead of unsaturated control so that the DOB receives the same amount of actuation as the plant does. All DOB control systems must be implemented in this manner to mitigate the effect of saturation in stability and performance. Throughout this paper, we assume that the DOB control system is implemented in the manner described above in order to mitigate the effect of actuator saturation. Indeed, the modification is adopted in recent applications of DOB [18,24–27]. As it turns out, an unexpected behavior is observed in DOB control systems with saturating actuators that no existing literature provides explanation. Specifically, a zero mean measurement noise degrades control performance in the form of tracking loss: the mean of the output of the system shows constant tracking error only when measurement noise is present, although the measurement noise has zero mean. This phenomenon is similar to Noise Induced Tracking Error (NITE) reported in [29–32] for PI controlled systems with anti-windup. Since the two phenomena are similar, we use the term NITE in the remainder of the paper to refer to noise induced tracking loss in DOB control systems with saturating actuators. It should be pointed out that, due to input saturation, the system of Fig. 1 is nonlinear. For nonlinear systems, parasitic equilibria may be observed [43]. However, tracking errors in the steady state induced by measurement noise are not addressed. In this paper, we analyze DOB control systems with saturating actuators to derive conditions under which the tracking loss occurs, and also quantify the tracking error with respect to system parameters and noise characteristics. The analysis is carried out using stochastic averaging theory [35]. Main contribution of this work is as follows: For the system shown in Fig. 1, 1) we first report that NITE may occur in DOB control systems; 2) we provide the conditions under which NITE occurs and quantify tracking error; and 3) based on the analysis, we propose controller design guidelines that can eliminate or reduce NITE.

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The outline of this paper is as follows: In Section 2, we show NITE phenomenon in DOB control systems with saturating actuators. In Section 3, the DOB control systems with saturating actuators are analyzed and main results are presented. Also, the NITE example mentioned in Section 2 is analyzed in detail. In Section 4, design guidelines are given. In Section 5, BLDC motor experiments are conducted to validate the efficacy of the proposed design guidelines. NITE in unstable systems is additionally analyzed in Section 6 and NITE in MIMO systems is discussed in Section 7. Finally, conclusions are formulated in Section 8. All proofs are included in Appendix. 2. Motivation Consider a single input single output (SISO) DOB control systems with saturating actuators shown in Fig. 1. The transfer function P(s), C(s), Pn (s), Q(s) are plant, outer-loop controller, nominal plant, and low-pass filter, respectively. Signals r, e, uc , u, v, d, y, n, dˆ, z1 , z2 are, respectively, reference, tracking error, controller output, control input, saturated control input, exogenous disturbance, system output, measurement noise, estimated disturbance, and signals associated with DOB. The measurement noise n follows zero mean Gaussian distribution with standard deviation of σ n . The saturating actuator is denoted by satβα (u) with α and β being the lower and upper limits: ⎧ α, u < α ⎪ ⎪ ⎨ β (1) satα (u) = u, α ≤ u ≤ β. ⎪ ⎪ ⎩β, u > β For the design of DOB, Pn (s) is selected to have the same relative degree as that of P(s), and Q(s) is a low pass filter that has the form of Q(s) =

ck (τ s)k + ck−1 (τ s)k−1 + · · · + c1 (τ s) + c0 , (τ s )l + al−1 (τ s )l−1 + · · · + a1 (τ s ) + a0

(2)

where τ > 0 is the filter time constant, and k and l are non-negative integers with l − k greater than or equal to the relative degree of Pn (s) so that Q(s)Pn−1 (s) becomes proper. It is also assumed that a0 = c0 so that the dc-gain becomes unity. For details of DOB design, see [20–22,25,26,36–39] and the references therein. Now, consider the system of Fig. 1 with 2 1 , Pn (s) = , C(s) = 4, s(s + 3) s(s + 4) 1 Q(s) = , α = −1, β = 1. (0.1s + 1)2 P (s) =

(3)

The response of the system of Fig. 1 with the parameter defined in Eq. (3) is shown in Fig. 2(a) for r being the unit step, d being constant of 0.4 applied at t = 50 s, and noise n being zero. As expected, the system output asymptotically tracks the reference and rejects the disturbance shortly after t = 50 s. Fig. 2(b) and (c) shows the corresponding control input and saturated control input, respectively. The output response from simulating the dynamics of Fig. 1 with Eq. (3) and measurement noise is shown in Fig. 3(a). The measurement noise n is Gaussian distributed with zero mean

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and σn = 0.04. The output begins to lose tracking at t = 50 s. The only difference in operating conditions of Figs. 2 and 3 is the existence of zero mean measurement noise. Yet, the response in Fig. 3 shows a loss of tracking. This phenomenon is referred to as NITE. While [29– 32] report and analyze NITE phenomenon in systems with PI controller and anti-windup, no literature on systems with DOB reports this phenomenon. For the characteristics of noise n to be other than Gaussian, a similar phenomenon occurs. For instance, Fig. 4 shows the step response and control signal of the same system with zero mean uniformly distributed noise whose standard deviation is 0.04. Indeed, we can see that NITE occurs. The amplitude of NITE is greater than that for the Gaussian case. One may consider adding a low pass filter in the feedback loop for measurement noise reduction. However, adding a filter changes the closed loop system dynamics from the original design and may even result in destabilization of the system. Reference [37] indeed shows detailed analysis and illustration of DOB based control system’s destabilization by adding a noise reduction filter. Analysis of NITE in DOB control systems with actuator saturation will be carried out in the subsequent section for a Gaussian distributed noise.

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(c) Saturated control signal Fig. 3. Step response, control signal, and saturated control signal of Fig. 1 with Eq. (3) when the characteristics of n is zero mean Gaussian and σn = 0.04.

3. Noise induced tracking error 3.1. Approach Analyzing system of Fig. 1 with random measurement noise is a difficult task which involves Fokker–Planck equation [34]. Therefore, we use stochastic averaging [35] to approximate the behavior of Fig. 1 by a system with only deterministic signal. This approximation is with high accuracy when the bandwidth of the measurement noise is high compared to that of the control systems. The brief review of stochastic averaging theory is given in Appendix. Applying the stochastic averaging to the system of Fig. 1 with respect to Gaussian measurement noise yields the system of Fig. 5. The details of the averaging process are similar to those in [29], hence, omitted here. For the system of Fig. 5, all signals are denoted by the same symbols as in Fig. 1, but with a bar to denote that they are the results of averaging. The transfer function W(s) denotes Q(s)Pn−1 (s), and is introduced for the notational simplicity in the analysis given later. Notice that the measurement noise n does not appear in the system of Fig. 5, and the effect of n is averaged into the function hαβ (u¯; κ ) that replaces the saturating

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(c) Saturated control signal Fig. 4. Step response, control signal, and saturated control signal of Fig. 1 with Eq. (3) when the characteristics of n is zero mean uniform and σn = 0.04.

Fig. 5. Averaged version of the feedback system of Fig. 1.

actuator satβα (u) in the original system. The function hαβ (u¯; κ ) is defined as hαβ (u¯; κ ) = En [satβα (u¯ − (C∞ + W∞ )n)]   u¯ − α α+β u¯ − α = + erf √ 2 2 2κ

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¯ 2 (u¯; κ ) for several values of κ. Fig. 6. The function sat2−2 (u¯ ) and h−2

    u¯ − β (u¯ − α)2 κ u¯ − β + √ exp − erf √ 2 2κ 2 2π 2κ   (u¯ − β )2 κ − √ exp − 2κ 2 2π

−

(4)

where En [ · ] is the expectation with respect to the distribution of n and the error function erf(ξ ) is defined as  ξ 2 erf(ξ ) = √ exp(−t 2 ) dt. (5) π 0 The parameter κ in hαβ (u¯; κ ) is defined as κ = (C∞ + W∞ )σn where C∞ = lims→∞ C(s) and W∞ = lims→∞ Q(s)Pn−1 (s). The shape of hαβ (u¯; κ ) is depicted for several values of κ in Fig. 6. Notice that hαβ (u¯; κ ) has the same saturation limits as satβα (u¯ ) and it converges to the saturation nonlinearity as κ tends to zero. The function hαβ (u¯; κ ) is a “smoothed” saturation and the degree of smoothing is determined by κ. The derivation of hαβ (u¯; κ ), again, is explained in [29]. We point out that the actual saturating actuator is an intrinsic property of the physical plant. Thus, the saturation is physically fixed. The theory of stochastic averaging states that the effect of the sensor noise manifests itself as if satβα (u) were replaced with hαβ (u¯; κ ) in the system. Clearly, the physical limitations are respected in the analysis because the lower and upper limits of hαβ (u¯; κ ) are identical to those of satβα (u) showing that no physics is violated. We also emphasize that hαβ (u¯; κ ) is not what we introduce to the system for any design purpose. It is a result of applying stochastic averaging that appears in the course of analysis. The actual system will operate with satβα (u), not hαβ (u¯; κ ). We now illustrate the accuracy of the averaging approach. Responses of the system in Fig. 5 with parameters defined in Eq. (3) with σn = 0.04 and σn = 0.06, respectively, are shown in Fig. 7(a). For comparison, in Fig. 7(b), we show again the responses of the system in Fig. 1 with zero mean Gaussian noise. For both cases of σn = 0.04 and σn = 0.06, the

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Fig. 8. An equivalent system of Fig. 1 including a deadzone nonlinearity.

responses of Fig. 1 and Fig. 5 are almost identical, which validates the approach of the analysis. 3.2. NITE Analysis Consider the system of Fig. 1. While [23] provides a thorough stability analysis for linear DOB systems, the system of Fig. 1 is not linear due to saturating actuator. Therefore, the stability analysis is more involved than a linear case. Following the analysis in [28], we can represent the system of Fig. 1 in the form given in Fig. 8. Here, signals u and v are the same as in Fig. 1 and H(s) is given by H (s) =

C(s)P (s) + W (s)P (s) − Q(s) . 1 + C(s)P (s) + W (s)P (s) − Q(s)

(6)

Since we investigate internal stability, the external signals r, n, and d of Fig. 1 are set to zero. Then, the global asymptotic stability of Fig. 8 (equivalently that of Fig. 1) can be verified using the LMI condition given in the following assumption.

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Assumption 1. There exist a square matrix M > 0 with an appropriate dimension and a scalar N > 0 such that T

MB + C T N A M + MA <0 (7) BT M + N C N D + DT N − 2N where A, B, C, and D are a realization of H(s) in Eq. (6). The LMI in Eq. (7) guarantees the global asymptotic stability of the system of Fig. 1 with Lyapunov function V (x) = x T Mx where x is the state of H(s) in the realization using A, B, C, and D. We assume that the system in Fig. 1 satisfies Assumption 1. It must be noted that A, B, C, and D are not a realization of the plant, P(s). They are a realization of H(s), which includes the parameters for the controller and DOB, i.e., C(s), Q(s), and W(s), and is a representation of the closed loop dynamics. If P (s) = Pn (s), i.e., W (s) = Q(s)P−1 (s), then, H(s) reduces to H (s) =

C(s)P (s) , 1 + C(s)P (s)

(8)

which is a well known closed loop dynamics from r to y. Assumption 2. The reference r and the exogenous disturbance d are constants and satisfy α < lim

s→0

C(s) r − d < β. C(s)P (s) + W (s)P (s)

(9)

Assumption 2 implies that the steady state value of the control input, denoted by uss , satisfies α < uss < β. This ensures that the control input is in the linear region of the actuator given in Eq. (1). The derivation of uss is provided in Appendix. For the analysis of NITE, we consider the system of Fig. 5 instead of Fig. 1 as the approach is explained in Section 3.1. For the system of Fig. 5, we have the following results. Theorem 1. Let Assumption 1 hold. Then, the system of Fig. 5 is globally asymptotically stable. For constant r and d satisfying Assumption 2, the steady state error e¯ss of Fig. 5 is given by e¯ss =

r u¯ss − hαβ (u¯ss ; κ ) + 1 + C0 Pn0 C0 + W0

(10)

where u¯ss is the control input in the steady state, and C0 , Pn0 , W0 are, respectively, the dc-gains of C(s), Pn (s), and W(s). Also, u¯ss satisfies the following equation: u¯ss = C0 r + (1 − C0 P0 − W0 P0 )hαβ (u¯ss ; κ ) − (C0 P0 + W0 P0 )d.

(11)

Here, P0 , κ are, respectively, the dc-gain of P(s) and κ = (C∞ + W∞ )σn . Proof. See Appendix.  The interpretations of Theorem 1 are as follows: First, we consider the case when σ n approaches to zero, i.e., no measurement noise exists. Then, κ goes to zero and hαβ (u¯; κ ) approaches to satβα (u¯ ). This renders the second term in (Eq. 10) go to zero and e¯ss be given by 1+Cr0 Pn0 which is identical to the standard result in linear DOB systems [20,36]. Note also that the error is independent of d because the DOB has compensated for it. To be more specific, e¯ss coincides with that of the linear DOB case only for those r and d that satisfy Assumption 2. This is because the actuator of the system of Fig. 1 is saturating. It

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has been known that reference tracking and disturbance rejection capabilities are limited due to actuator saturation [33]. While Assumption 2 limits r and d such that the operating control input belongs to the linear region of the actuator in the steady state, Assumption 1 guarantees the global asymptotic stability of the system of Fig. 1. Next, we consider the case with measurement noise, i.e., σ n > 0. Then, the tracking error β α (u¯ss ;κ ) yields an additional term u¯ss −h due to measurement noise. This term is attributed to the C0 +W0 β

α (u¯ss ;κ ) zero mean Gaussian measurement noise and quantifies NITE. In order to evaluate u¯ss −h , C0 +W0 the value of u¯ss has to be obtained from (Eq. 11). The solution u¯ss is a function of system parameters, but in particular, dependent on κ (hence σ n ) and d. This contrasts with the case of no measurement noise where error is independent of d thanks to the action of DOB. Therefore, with measurement noise DOB capability turns out to be limited. β α (u¯ss ;κ ) Finally, the NITE term u¯ss −h depends on u¯ss − hαβ (u¯ss ; κ ). This term becomes large if C0 +W0 u¯ss is near or exceeds the saturation limits, which can be seen in Fig. 6. Moreover, the NITE term also increases if σ n increases, because hαβ (u¯ss ; κ ) deviates further from u¯ss .

Remark 1. We point out that the result of (Eq. 10) in Theorem 1 comes from the stability of Fig. 5. Assumption 1 guarantees the stability of Fig. 1 and Fig. 5 simultaneously as detailed in the proof. If the stability of Fig. 5 is guaranteed by other means, not necessarily by Assumption 1, the result of Theorem 1 still holds. This can be seen in the proof that derivation of e¯ss does not depend on M and N. Remark 2. Assumption 1 gives global stability, which can not be obtained for P(s) having poles with positive real parts [43]. For such unstable plant, NITE appears as defined in Theorem 1, as long as the initial conditions of the plant, the controller, and the DOB lie within the domain of attraction. Estimating the domain of attraction can be done using convex hull method [41,42] or others. Corollary 1. Let Assumptions 1 and 2 hold. Assume that the plant P(s) has a pole at the origin. Then, the system of Fig. 5 is globally asymptotically stable and the steady state error e¯ss is given by e¯ss =

r u¯ss + d + 1 + C0 Pn0 C0 + W0

(12)

and steady state input u¯ss is the unique solution of hαβ (u¯ss ; κ ) = −d

(13)

where C0 , Pn0 , W0 are, respectively, the dc-gains of C(s), Pn (s) and W(s). Proof. See Appendix.  As an illustration, consider again the system of Fig. 5 with Eq. (3). It can be shown that Assumption 1 is satisfied with appropriate M and N. Fig. 9 shows e¯ss of the system for −1 < d < 1 (satisfying Assumption 2) and σn = 0.01, σn = 0.04, and σn = 0.06. Notice that NITE becomes severe as |d| increases. In particular, when d = 0.4, NITE is −0.4506 which is almost identical to the observed tracking error from Fig. 3. Fig. 10 shows NITE over a range of σ n for d = 0.1, d = 0.4, and d = 0.9. As explained, larger amplitude of the noise yields more severe NITE.

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4. Design guidelines for NITE mitigation Based on the analysis in the previous section, we provide two design guidelines for NITE mitigation in the following subsections.

4.1. Include integral control in C(s) From (Eq. 10), the steady state error is quantified by e¯ss =

r u¯ss − hαβ (u¯ss ; κ ) + . 1 + C0 Pn0 C0 + W0

(14)

If the outer-loop controller C(s) has a pole at the origin (integral control), C0 becomes infinite in (Eq. 14), resulting e¯ss = 0. Therefore, including integral control in C(s) eliminates NITE. Fig. 11 shows the step response of the system of Fig. 1 with P(s), Pn (s), Q(s) given as in Eq. (3) and C(s) given to be 4s+1 instead of C(s) = 4. Indeed, by including integral control in s C(s), NITE is eliminated as seen in Fig. 11(a). The corresponding control input and saturated control input are, respectively, shown in Fig. 11(b) and (c). Control system designers tend not to use integral control in the outer-loop controller C(s) because DOB rejects the disturbance. This guideline, however, shows that the integral control may still be needed. Stability of the closed loop system with an added integral controller shall be verified using Assumption 1 or the other known methods [40].

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4.2. Select Q(s) such that Q(s)Pn−1 (s) is strictly proper For the case where adding integral control is not preferred because of poor transient response or some other reason, the following guideline can be used. Consider Eq. (14) again. Clearly, NITE is proportional to u¯ss − hαβ (u¯ss ; κ ). From Eqs. (4) and (5), and also from Fig. 6, it is straightforward to show that |u¯ss − hαβ (u¯ss ; κ1 )| > |u¯ss − hαβ (u¯ss ; κ2 )|

(15)

for κ 1 > κ 2 . Therefore, smaller κ is preferred for mitigating NITE. Recall that κ is given by κ = (C∞ + W∞ )σn , where W∞ is the proportional gain of Q(s)Pn−1 (s). Therefore, Q(s)Pn−1 (s) being strictly proper sets W∞ to zero, resulting in smaller κ, and consequently smaller NITE. Fig. 12(a) shows the step response of the system of Fig. 1 with P(s), Pn (s), C(s) given 1 1 as in Eq. (3) but Q(s) given to be (0.1s+1) 3 instead of (0.1s+1)2 . Indeed, NITE is substantially reduced as can be seen in Fig. 12. The corresponding control input and saturated control input are, respectively, shown in Fig. 12(b) and (c). Note that this method does not completely eliminate NITE. However, it effectively reduces NITE for the following reason. When Q(s)Pn−1 (s) is of relative degree 0, W∞ increases as

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the bandwidth of Q(s) increases. Typically, system designers choose parameters of Q(s) to give a high bandwidth for Q(s) in order to achieve a fast disturbance rejection response. So, W∞ tends to be a very large number dominating the value of κ = (C∞ + W∞ )σn . Indeed, in our illustration, κ reduces to 4σ n from 104σ n . Notice, however, that the efficacy of this method depends on r and d. Although κ is reduced, u¯ss − hαβ (u¯ss ; κ ) may still be substantial depending on r and d. Indeed, Fig. 10 illustrates that the level of reduction of e¯ss by the reduction of κ is dependent on d. If the relative degree of Pn (s) is high, the relative degree of Q(s) also becomes high, which may introduce difficulty in system stabilization. For such cases, it is recommended to use the method in Section 4.1, i.e., including integral control in the C(s). One may set up H2 and/or H∞ performance metric to design C(s) and Q(s) in the system of Fig. 1. Then, the resulting κ and C0 will determine the level of NITE. Remark 3. Convex hull method [41,42] is well known for designing a state feedback controller for the system with saturated inputs. Instead of C(s) and DOB in Fig. 1, convex hull method may be used to design a closed loop system that can deal with disturbance. Analyzing the performance of such closed loop system with measurement noise is a future work.

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1 (0.003 +1)

Fig. 13. The existence of NITE in the BLDC motor testbed.

5. Experimental results of BLDC motor The phenomenon of NITE as well as the validity of the design guidelines are illustrated in a brushless DC (BLDC) motor drive testbed. The testbed consists of a DSP evaluation board and a 26.2W eight-pole BLDC motor drive. The DSP evaluation board is equipped with a Texas Instruments TMS320F28335 floating-point DSP and a 50W three phase full-bridge inverter. Controllers are implemented using discretization with Tustin approximation method and operate at a sampling rate of 30 kHz. The control objective is to regulate the angular velocity of the motor in the presence of disturbance. The overall control system is represented as in Fig. 1 with Pn (s) =

22800 1 [rpm/V], Q(s) = . s(0.03s + 1) (0.003s + 1)2

(16)

The voltage command is constrained between α = 0V and β = 12V. The speed of BLDC motor is measured by a built-in Hall-effect sensor. In fact, the motor speed is obtained by counting the number of timer interrupts between two sequential interrupts from the Hall sensors. Thus, the speed feedback signal contains a quantization error, which can be regarded as a high frequency measurement noise. Exogenous disturbance is emulated by applying constant disturbance to voltage input of BLDC. Fig. 13 shows the experimental result of DOB controller with C(s) = 0.015 and Eq. (16), where the speed reference is 2000 rpm and disturbances of 7V and 8.5V are injected at t = 5 s. The Hall sensor measurement shown in gray is quite noisy, so a low pass filtered angular velocity is shown together in red. The experimental responses confirm the existence of NITE, which has not been reported anywhere before. Also, the size of the disturbance affects that of NITE, as pointed out in the interpretations of Section 3.

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Speed(rpm)

2060 2040 2020 2000 1980 1960

0

1

(a)

2

= 7,

3

( )=

4

5

0.012( +4)

6

7

, and

8

( )=

9

10

(sec)

10

(sec)

1 (0.003 +1)

Speed(rpm)

2060 2040 2020 2000 1980 1960

0

(b)

1

2

= 8.5,

3

( )=

4

5

0.012( +4)

6

7

, and

8

( )=

9

1 (0.003 +1)

Fig. 14. Applying the first design guideline to the BLDC motor control.

Speed(rpm)

2060 2040 2020 2000 1980 1960

0

1

(a)

2

= 7,

3

4

5

6

( ) = 0.015, and

7

( )=

8

9

10

(sec)

10

(sec)

1 (0.003 +1)

Speed(rpm)

2060 2040 2020 2000 1980 1960

0

(b)

1

2

= 8.5,

3

4

5

( ) = 0.015, and

6

7

( )=

8

9

1 (0.003 +1)

Fig. 15. Applying the second design guideline to the BLDC motor control.

To illustrate the validity of the first guideline, we have changed the outer loop controller C(s) from P controller to a PI controller 0.012(s+4) . The responses are shown in Fig. 14. s As predicted by Section 4.1, NITE disappears and the mean value of the angular velocity is regulated at 2000 rpm without any bias. Clearly, asymptotic tracking is achieved before t = 5 s, and the disturbance is rejected with some transients. This validates the first design guideline.

G. Na, N.H. Jo and Y. Eun / Journal of the Franklin Institute 356 (2019) 3922–3947

3937

Amplitude

2 1.5 1

0.5 0

0

50

100

(sec)

150

(a) Output and reference

Amplitude

10 5 0 -5

-10

0

50

100

(sec)

150

(b) Control signal

Amplitude

10 5 0 -5

-10

0

50

100

(sec)

150

(c) Saturated control signal Fig. 16. Step response, control signal, and saturated control signal of Fig. 1 with (Eq. 17) when measurement noise does not exist.

To illustrate the validity of the second guideline, we have changed Q(s) from (0.0031s+1)2 to (0.0031s+1)3 . The relative degree of Q(s) = (0.0031s+1)3 is increased by 1. Such selection of Q(s) renders Q(s)Pn−1 (s) be strictly proper and yields W∞ = 0. The responses are shown in Fig. 15. As predicted by Section 4.2, NITE is reduced although not completely eliminated. For Fig. 15(a) where d = 7, NITE is almost eliminated, but for the case of d = 8.5 (Fig. 15(b)), it still remains. However, comparing Figs. 13 and 15 shows reduction of NITE. This validates the second design guideline. 6. NITE in unstable systems In this section, as argued in Remark 2, we show that even if the global stability is not guaranteed due to unstable poles of P(s), NITE is locally analyzed and Theorem 1 is useful. The following example is the case where P(s) has unstable pole.

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Amplitude

2 1.5 1

0.5 0

0

50

100

(sec)

150

Amplitude

(a) Output and reference

20 0

-20 0

50

100

(sec)

150

(b) Control signal

Amplitude

10 5 0 -5

-10

0

50

100

(sec)

150

(c) Saturated control signal Fig. 17. Step response, control signal, and saturated control signal of Fig. 1 with (Eq. 17) when measurement noise is zero mean Gaussian and σn = 0.1.

Consider a DOB based system given by 3s + 2 4s + 2 , Pn (s) = , C(s) = 8.4, s(s − 1) s(s − 1) 1 Q(s) = , α = −2, β = 2, r = 1, d = 1.5, 0.01s + 1 P (s) =

(17)

where d is injected after 50 s. The results of Fig. 1 simulated with (Eq. 17) and σn = 0 are illustrated in Fig. 16. The system output, corresponding control input, and saturated control input are shown in Fig. 16(a), (b), and (c), respectively. The DOB compensates the disturbance after 50 s and the system output tracks the reference. The results of Fig. 1 simulated with (Eq. 17) and σn = 0.1 are illustrated in Fig. 17. After 50 s, the system output has performance degradation as the occurrence of NITE. Now, we consider the averaged system of Fig. 5 with (Eq. 17) and σn = 0.1. The system output, corresponding control input, and saturated control input are shown in Fig. 18(a), (b), and (c), respectively. The system output in Fig. 18 has very similar shape with Fig. 17, which

G. Na, N.H. Jo and Y. Eun / Journal of the Franklin Institute 356 (2019) 3922–3947

3939

Amplitude

2 1.5 1

0.5 0

0

50

(sec)

100

150

(a) Output and reference

Amplitude

10 5 0 -5

-10

0

50

(sec)

100

150

(b) Control signal

Amplitude

10 5 0 -5

-10

0

50

(sec)

100

150

(c) Saturated control signal Fig. 18. Step response, control signal, and saturated control signal in the averaged control system of Fig. 5 with (Eq. 17) when measurement noise is zero mean Gaussian and σn = 0.1.

2 1 0 -1 -2 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Fig. 19. Steady state errors of the system of Fig. 5 with (Eq. 17) for −2 < d < 2 and σn = 0.1.

shows that the proposed method can be applied to the system where the local stability is guaranteed. Now, we calculate NITE based on Eq. (10) of Theorem 1. The steady state errors for −2 < d < 2 and σn = 0.1 are shown in Fig. 19 and the errors over a range of σ n and d = 1.5 are shown in Fig. 20. When d = 1.5 and σn = 0.1, NITE has −0.3059, which is

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G. Na, N.H. Jo and Y. Eun / Journal of the Franklin Institute 356 (2019) 3922–3947 0.2 0 -0.2 -0.4 -0.6 -0.8

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Fig. 20. Steady state errors of the system of Fig. 5 with (Eq. 17) for a range of σ n and d = 1.5.

very similar with the actual tracking error shown in Fig. 17. Clearly, even if Assumption 1 is not satisfied, NITE is analyzed. 7. NITE in MIMO systems In this section, we show that NITE occurs in MIMO systems with DOB and also show that the mitigation guidelines given in Section 4 are valid for MIMO cases. Consider a coupled MIMO dynamical system given by y¨1 + y˙1 = sat2−2 (u1 ) + d1 , .5 y¨2 + 3y˙2 − y˙1 = sat1−1 .5 (u2 ) + d2 ,

(18)

where y1 and y2 are system outputs, u1 and u2 are system inputs, and d1 and d2 are external disturbances. Now, we design a MIMO DOB based feedback control system following [44,45]. For design convenience, we change Eq. (18) into a state space form. Define x1 = y1 , x2 = x˙1 , x3 = y2 , and x4 = x˙3 . Then, Eq. (18) is represented by ⎡ ⎤ ⎡ ⎤ 0 1 0 0 0 0 ⎢ 0 −1 0 ⎢ ⎥ 0 ⎥ ⎢ ⎥x + ⎢ 1 0 ⎥(v + d ), x˙ = ⎣ 0 ⎣ 0 0 ⎦ 0 0 1 ⎦ 0 1 0 −3 0 1

1 0 0 0 x, y = 0 0 1 0 y˜ = y + n, (19) .5 T 2 where x = [x1 , x2 , x3 , x4 ]T ∈ R4 , v = [v1 , v2 ]T = [sat2−2 (u1 ), sat1−1 .5 (u2 )] ∈ R , d = T 2 T 2 T 2 T 2 [d1 , d2 ] ∈ R , y = [y1 , y2 ] ∈ R , y˜ = [y˜1 , y˜2 ] ∈ R , and n = [n1 , n2 ] ∈ R are noises with standard deviations of σ n,1 and σ n,2 . The control purpose of Eq. (18) is to track references denoted by r1 and r2 . The outer-loop controllers for uc,1 and uc,2 are designed as

uc,1 = C1 (s)(r1 − y˜1 ) = 2(r1 − y˜1 ), uc,2 = C2 (s)(r2 − y˜2 ) = 4(r2 − y˜2 ). For compensating d, we design two DOBs. First DOB for rejecting d1 is given by



1 0 0 p + v1 , p˙1 = −a0 /(τ 2 ) −a1 /(τ ) 1 a0 /(τ 2 )

(20)

G. Na, N.H. Jo and Y. Eun / Journal of the Franklin Institute 356 (2019) 3922–3947

3941

Amplitude

2 1.5 1

0.5 0

0

50

100

(sec)

150

(a) Output and reference

Amplitude

2 1.5 1

0.5 0

0

50

100

(sec)

150

(b) Output and reference Fig. 21. Step responses of the MIMO DOB control system (19)–(23) when measurement noises do not exist.





1 0 0 q + q˙1 = y˜ , −a0 /(τ 2 ) −a1 /(τ ) 1 a0 /(τ 2 ) 1 a0 a1 dˆ1 = − 2 (q1,1 − y˜1 ) − q1,2 + q1,2 − p1,1 , τ τ u1 = uc,1 − dˆ1 ,

(21)

where p1 = [ p1,1 , p1,2 ]T ∈ R2 , q1 = [q1,1 , q1,2 ]T ∈ R2 , and dˆ1 ∈ R is the first DOB output (estimated disturbance). Second DOB for rejecting d2 is given by



1 0 0 p + v , p˙2 = −a0 /(τ 2 ) −a1 /(τ ) 2 a0 /(τ 2 ) 2



1 0 0 q + q˙2 = y˜ , −a0 /(τ 2 ) −a1 /(τ ) 2 a0 /(τ 2 ) 2 a0 a1 dˆ2 = − 2 (q2,1 − y˜2 ) − q2,2 − q1,2 + 3q2,2 − p2,1 , τ τ u2 = uc,2 − dˆ2 , (22) where p2 = [ p2,1 , p2,2 ]T ∈ R2 , q2 = [q2,1 , q2,2 ]T ∈ R2 , and dˆ2 ∈ R is the second DOB output. Next, we show simulation results. The simulation parameters are chosen as a0 = 1, a1 = 2, τ = 0.1, r1 = 1, r2 = 1, d1 = 0.4, d2 = 0.6.

(23)

The system response y1 is shown in Fig. 21(a) for r1 being unit step, d1 being constant of 0.4 applied at 50 s, and noise n1 being zero. The system response y2 is also shown in Fig. 21(b) for r2 being unit step, d2 being constant of 0.6 applied at 50 s, and noise n2 being zero. The system outputs asymptotically track r1 and r2 , although external disturbances exist.

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Amplitude

2 1.5 1

0.5 0

0

50

100

(sec)

150

(a) Output and reference

Amplitude

2 1.5 1

0.5 0

0

50

100

(sec)

150

(b) Output and reference Fig. 22. Step responses of the MIMO DOB control system (19)–(23) when measurement noises exist.

Amplitude

2 1.5 1

0.5 0

0

50

100

(sec)

150

(a) Output and reference

Amplitude

2 1.5 1

0.5 0

0

50

100

(sec)

150

(b) Output and reference Fig. 23. Step responses of the MIMO DOB control system (19)–(23) when a design guideline is used.

The output responses simulated with zero mean Gaussian measurement noises n1 and n2 of σn,1 = 0.04 and σn,2 = 0.04 are shown in Fig. 22. All of system outputs lose tracking at 50 s. The simulation results illustrate the occurrence of NITE in MIMO DOB systems. Now, we apply first design guideline into the control system. The outer-loop controllers are changed into C1 (s) = 2 + 0.s5 and C2 (s) = 4 + 1s , respectively. As shown in Fig. 23, NITE is clearly eliminated, which proves that the proposed design guideline is still effective in the MIMO case. Comprehensive analysis for MIMO case is future work.

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3943

8. Conclusions Noise induced tracking error phenomenon was first reported in this paper for feedback control systems with saturating actuators and disturbance observers. Mathematical analysis of the system is carried out and the tracking error is quantified in terms of system parameters and noise standard deviation. Experimental results on a BLDC motor testbed confirm that zero mean measurement noise indeed induces tracking error in such systems. Design guidelines to avoid or reduce NITE have also been proposed and the effectiveness is validated on the BLDC motor testbed. We expect this results would benefit practicing control system design engineers in the field of industrial electronics. Acknowledgments This work was partly supported by Institute for Information and Communications Technology Promotion (IITP) grant funded by the Korea government (MSIP) (2014-0-00065, Resilient Cyber-Physical Systems Research), partially supported by GLR Program (NRF-2013K1A1A2A02078326) and Basic Science Research Program (No. 2017R1D1A1B03032236) both through the National Research Foundation of Korea. This work was also supported by the DGIST R&D Programs of the Ministry of Science and ICT (18-ST-02 and 18-EE-01). Appendix A. Stochastic averaging theory A brief review of stochastic averaging theory [35] is presented here. Consider the system x˙ = f (x, n (t ))

(A.1)

where x ∈ Rk , f: Rk × R → Rk , and n (t) is zero mean wide sense stationary random process with standard deviation of σ n and bandwidth of ωn . The subscript is intended to parameterize the noise in such a manner that ωn → ∞ as → 0. Then, for sufficiently small, the solution x(t) of Eq. (A.1) is well approximated by the solution of the averaged equation x˙¯ = f¯(x¯ )

(A.2)

where x¯ ∈ Rk , f¯: Rk → Rk , and f¯ is the conditional expected value of f with respect to the distribution of n (t), i.e., f¯(x¯ ) = En (t ) [ f¯(x¯, n (t ))].

(A.3)

Since the bandwidth of the measurement noise is typically much larger than that of the closed-loop system, the behavior of Eq. (A.1) can be studied using the averaged system of Eq. (A.2). Appendix B. Derivation of Eq. (9) Consider Fig. 1. We seek a necessary condition on r and d such that, without measurement noise, the control in the steady state remains in the linear region of the actuator. Setting n = 0

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and ignoring saturation (local behavior only), one can obtain u=

C(s) C(s)P (s) + W (s)P (s) r− d. 1 − Q(s) + C(s)P (s) + W (s)P (s) 1 − Q(s) + C(s)P (s) + W (s)P (s)

(A.4)

By using the fact that the dc-gain of Q(s) is 1, the steady state value of the control signal, denoted by uss , is obtained as uss = lim

s→0

C(s) r − d. C(s)P (s) + W (s)P (s)

(A.5)

Consequently, Eqs. (A.5) and (1) combine to give Eq. (9). Proof of Theorem 1. First, we show that the system of Fig. 5 is globally asymptotically stable. The control signal u¯ is given by u¯ = (−C(s)P (s) − W (s)P (s) + Q(s))hαβ (u¯; κ ). Define q¯ = u¯ − H (s) =

hαβ (u¯; κ ).

(A.6)

From Eq. (A.6), the transfer function from q¯ to u¯ is given by

C(s)P (s) + W (s)P (s) − Q(s) . 1 + C(s)P (s) + W (s)P (s) − Q(s)

(A.7)

Let a state space realization of H(s) to be x˙¯ = Ax¯ + Bq¯, u¯ = C x¯ + Dq¯,

(A.8)

where x¯ is the state of the realization. Now, consider a quadratic function V (x¯ ) = x¯T M x¯ with a positive definite matrix M given in Assumption 1. Taking the time derivative of V (x¯ ) yields V˙ (x¯ ) = (Ax¯ + Bq¯ )T M x¯ + x¯T M(Ax¯ + Bq¯ ) = x¯T AT M x¯ + q¯T BT M x¯ + x¯T MAx¯ + x¯T MBq¯.

(A.9)

We now show that q¯T N (u¯ − q¯ ) is always greater than 0 for q¯ = 0 and N > 0. From the property of hαβ (u¯; κ ) in Eq. (4), we obtain (u¯ − hαβ (u¯; κ ))hαβ (u¯; κ ) > 0 for u¯ = α+2 β and this gives q¯T N (u¯ − q¯ ) > 0 for any N > 0. Thus, we obtain the following inequality for all x¯, q¯ = 0 V˙ (x¯ ) < x¯T AT M x¯ + q¯T BT M x¯ + x¯T MAx¯ + x¯T MBq¯ + 2q¯T N (C x¯ + Dq¯ − q¯ ). The right hand side of Eq. (A.10) can be written in LMI:



 T  AT M + MA MB + C T N x¯ q¯T x¯ . BT M + N C N D + DT N − 2N q¯

(A.10)

(A.11)

Then, Assumption 1 guarantees V˙ (x¯ ) < 0 and establishes the global asymptotic stability of the system of Fig. 5. We note that the rest of the proof remains valid if the stability of the system of Fig. 5 is guaranteed, not by Assumption 1, but by other means, e.g., Circle criterion or Popov criterion [34,40]. Next, we find the steady state error e¯ss and steady state control u¯ss of Fig. 5 for constant r and d satisfying Assumption 2. In Fig. 5, the control signal u¯ is given by u¯ = C(s)(r − y¯) + Q(s)hαβ (u¯; κ ) − W (s)y¯.

(A.12)

G. Na, N.H. Jo and Y. Eun / Journal of the Franklin Institute 356 (2019) 3922–3947

3945

This can be rewritten as u¯ = C(s)e¯ + Q(s)hαβ (u¯; κ ) − W (s)(r − e¯),

(A.13)

from which we obtain e¯ =

u¯ − Q(s)hαβ (u¯; κ ) + W (s)r . C(s) + W (s)

(A.14)

Then, the steady state error is given by e¯ss =

u¯ss − hαβ (u¯ss ; κ ) + W0 r , C0 + W0

(A.15)

where C0 , W0 are dc-gains of C(s) and W(s), respectively. In order to evaluate e¯ss , the value of u¯ss must be obtained. Combining y¯ = P (s)(hαβ (u¯; κ ) + d ) with Eq. (A.12), we have u¯ = C(s)r + Q(s)hαβ (u¯; κ ) − (C(s) + W (s))P (s)(hαβ (u¯; κ ) + d ).

(A.16)

Then, the steady state input u¯ss must satisfy u¯ss = C0 r + (1 − C0 P0 − W0 P0 )hαβ (u¯ss ; κ ) − (C0 P0 + W0 P0 )d,

(A.17)

where P0 is the dc-gain of P(s). In order to obtain u¯ss , one has to find the steady state of the system in Fig. 5 for given r and d. Fortunately, if C0 P0 + W0 P0 > 0, it can be shown that Eq. (A.17) has an unique solution u¯ss , hence, one can obtain u¯ss from Eq. (A.17) instead of finding the steady state of the system in Fig. 5. Now, we show that Eq. (A.17) has an unique solution u¯ss under Assumption 2 and C0 P0 + W0 P0 > 0. Dividing Eq. (A.17) by C0 P0 + W0 P0 yields u¯ss + (C0 P0 + W0 P0 − 1)hαβ (u¯ss ; κ ) C0 = r − d. C0 P0 + W0 P0 C0 P0 + W0 P0

(A.18)

The right hand side term is identical to uss in Eq. (A.5), hence, by Assumption 2, the right hand side is a value between α and β. Then, we can rewrite Eq. (A.18) as u¯ss − hαβ (u¯ss ; κ ) + hαβ (u¯ss ; κ ) = uss . C0 P0 + W0 P0

(A.19)

The left hand side of Eq. (A.19) is a strictly increasing function of u¯ss , the range of which include the open interval (α, β). Therefore, there exists an unique intersection that defines u¯ss .  Proof of Corollary 1. Since P(s) has a pole at the origin, for an equilibrium to exist, the input to the plant P(s) must be 0 in the steady state, i.e., d + hαβ (u¯ss ; κ ) = 0.

(A.20)

Therefore, by using Eq. (A.15), we can obtain e¯ss =

u¯ss + d + W0 r , C0 + W0

where u¯ss is the solution of Eq. (A.20). 

(A.21)

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