Robust disturbance rejection methodology for unstable non-minimum phase systems via disturbance observer

ISA Transactions xxx (xxxx) xxx

Contents lists available at ScienceDirect

ISA Transactions journal homepage: www.elsevier.com/locate/isatrans

Research article

Robust disturbance rejection methodology for unstable non-minimum phase systems via disturbance observer Lu Wang, Jianhua Cheng

∗

Marine Navigation Research Institute, College of Automation, Harbin Engineering University, Harbin 150001, PR China

article

info

Article history: Received 30 January 2019 Received in revised form 24 November 2019 Accepted 25 November 2019 Available online xxxx Keywords: Unstable non-minimum phase system Disturbance observer Robust internal stability H∞ control

a b s t r a c t In this work, we propose a disturbance rejection methodology to deal with control problem of unstable non-minimum phase (NMP) systems. Firstly, we propose a two degrees-of-freedom control structure, which consists of an outer loop feedback controller and an inner loop disturbance observer (DOB). Specifically, the controller is designed for desired control performance, whereas the robust DOB is applied to deal with external disturbances and internal uncertainties. By analyzing the robust internal stability, several design requirements on both controller and DOB are presented for the control system. The H2 theory is adopted for controller design. Then, we propose a systematic DOB optimization method for unstable NMP systems. The proposed method synthesizes the relative order, robust internal stability and mixed sensitivity optimization together to formulate the cost function. The standard H∞ method is introduced to acquire the optimal solution that guarantying the design requirements. Simulations show that the proposed method can deal with the control system design for the unstable NMP systems, and it also has better performance comparing with the traditional method. © 2019 Published by Elsevier Ltd on behalf of ISA.

1. Introduction In practical industry applications, the system uncertainties, such as unmodeled dynamics, parameters perturbation, external disturbances, unknown measurement noise will affect system performance, or even lead to instability. As a practical method in engineering, the disturbance rejection methodology has been widely studied in recently years [1,2]. For this methodology, an observer is usually adopted to estimate the system uncertainties online based on the input and output information of controlled object. Among the existing observers [3–9], disturbance observer (DOB) has been widely investigated [7–9]. DOB consists of a nominal plant of the controlled object and a low-pass filter, which is named Q filter. This method was originally proposed by Ohnishi et al. [10], to estimate and compensate the external disturbances. However, the rejection ability against internal uncertainties was not mentioned yet in the original work. Extended state observer (ESO) was originally proposed by Mr. Han [11] for both internal uncertainties and external disturbances. Although these two methodologies are with different form, it is shown in [12] that DOB is an input–output case of ESO when the nominal model is chosen to be a chain of integrator, which means that performance of DOB can be strictly analyzed ∗ Corresponding author. E-mail addresses: [email protected] (L. Wang), [email protected] (J. Cheng).

based on the concept of active disturbance rejection control in state-space with nonlinear and time-varying internal uncertainties. The structure and parameter of Q filter largely determine the performance of a DOB, which determines several key properties, such as robust internal stability, system performance in transient or steady state, etc. In the past few decade, DOB with Chebyshev, Butterworth or binomial coefficient typed Q filters have been widely investigated [13–16]. Notice that the above mentioned low-pass filters were originally developed in the aspect of signal processing, while the purpose of the low-pass filter in DOB is to attenuate disturbance and measurement noise, or even adjust the robustness of the control system. Recently, H∞ control theory has been widely adopted in parameters optimization of the Q filter [17–19]. Linear matrix inequality and Algebraic Riccati Equation are employed in [17,18] respectively to acquire the static gain of a Q filter with fixed structure. In [19], several design requirements are synthesized to establish a cost function, based on which H∞ method is adopted to acquire the Q filter. However, the above methods only deal with the design of DOB for general stable minimum phase systems. The processes that contain right-half of the s-plane (RHP) pole(s), zero(s) or time-delay are regarded as non-minimum phase (NMP) systems according to their special phase characteristics. DOB based control system design for NMP systems have attached more attentions recently [13,20–22]. In [13], the model predictive control (MPC) technique combining with DOB is proposed to deal with the time-delay. However, both RHP poles

https://doi.org/10.1016/j.isatra.2019.11.034 0019-0578/© 2019 Published by Elsevier Ltd on behalf of ISA.

Please cite this article as: L. Wang and J. Cheng, Robust disturbance rejection methodology for unstable non-minimum phase systems via disturbance observer. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.034.

2

L. Wang and J. Cheng / ISA Transactions xxx (xxxx) xxx

and zeros are not taken into account. In [20], the similar results have been extended to deal NMP systems with both time-delay and RHP zeros. In [21], a new filter in parallel with Q filter is proposed, based on which the controlled object is compensated into a minimum phase one. And thus, the traditional DOB design method can be used for the new compensated minimal phase system. In [22], NMP system with both time-delay and RHP zeros is considered, the Q filter is optimized based on H∞ theory, and a prefilter is designed for desired performance. However, the prefilter used in this work has a lot of limitations. Moreover, all the researches mentioned above do not take the unstable poles into account. For an unstable NMP system, it is significantly difficult to design the controller and DOB together for robust internal stability. Several constraints should be considered according to the robust internal stability and system design performance simultaneously. Meanwhile, in the overall closedloop system, feedback controller and DOB will interact with each other. Hence, a reasonable design procedure is very important. In this study, the DOB based control system design and optimization for unstable NMP system is investigated, and a systematic design methodology is proposed. Firstly, system model of unstable NMP system is described, and DOB based control structure is analyzed. The outer loop controller is adopted for desired control performance, whereas the inner loop DOB is used to estimate and compensate the external disturbances as well as internal uncertainties. By analyzing the robust internal stability, several design constraints on feedback controller and Q filter are presented for configuration of the control system. Based on the design constraints and mixed sensitivity design requirements, a cost function is established. Then, standard H∞ theory is adopted for parameters optimization of Q filter. The main contributions of this paper are summarized as: (1) Robust internal stability of 2DOF control structure for unstable NMP system is analyzed, based on which several design constraints are presented. (2) By introducing an equivalent transformation, robust stability constraint is proposed with less conservatism. (3) A systematic DOB design method for unstable NMP system is given in detail for robust internal stability constraints and performance requirements. (4) Design specifications are presented in detail to show the design procedure of the proposed methodology. The rest of this paper is organized as follows: the control problem of unstable NMP system is formulated in Section 2. In Section 3, the internal stability is analyzed, based on which several constraints are proposed, the conservatism and performance of the closed-loop control system are also analyzed. In Section 4, controller and DOB structure design and parameters optimization are introduced. In Section 5, design specifications and simulations on an unstable NMP system are carried out, and Conclusions are summarized in Section 6.

Remark 1. NMP plant widely exists in practice. Notice that the inverse of nominal plant should be adopted in the control structure, however, the inverse of the RHP zeros is physically unrealizable according to its non-causal property. Thus, it is a challenging work to design the inner loop DOB as well as outer loop controller to guarantee the robust internal stability. Moreover, the traditional robust stability condition will bring with more conservatism, which will decrease the system performance. In the next section, we first analyze the robust internal stability to give several design requirements of the unstable NMP systems, and the robust stability condition is presented with less conservatism. 3. Internal stability and performance analysis The DOB based 2-DOF control system is described in Fig. 1, u and y denote the input and output of the controlled object, respectively. d and ξ are the external disturbances and measurement noise acting on the system, and dˆ represents the estimation of the DOB. In the DOB based control structure, uncertainties between real and nominal plant are regarded as equivalent disturbances. The DOB structure can estimate the total disturbance online based on the input and output information of controlled object, and compensate the disturbance estimation in the closed-loop for cancelation, as shown in Fig. 1. The low-pass filter Q (s) is used to avoid the implementation of non-proper inverse nominal model and suppress the measurement noise. From Fig. 1, the system transfer function can be expressed as: y(s) = Gyr (s)r(s) + Gyd (s)d(s) − Gyξ (s)ξ (s),

(2)

where

⎧ P(s)Pn (s)C (s) ⎪ ⎪ Gyr = ⎪ ⎪ Q (s)(P(s) − Pn (s)) + Pn (s)(1 + P(s)C (s)) ⎪ ⎪ ⎨ P(s)Pn (s)(1 − Q (s)) Gyd = ⎪ Q (s)(P(s) − Pn (s)) + Pn (s)(1 + P(s)C (s)) ⎪ ⎪ ⎪ P(s)(Q (s) + Pn (s)C (s)) ⎪ ⎪ ⎩ Gyξ = . Q (s)(P(s) − Pn (s)) + Pn (s)(1 + P(s)C (s)) In Eq. (2), Q (s) is a low-pass filter, which follows |Q (jω)| ≈ 1 Pn C and Gyd ≈ 0 at low frequencies. Consequently, Gyr ≈ 1+ Pn C should be satisfied in the low frequency range, which indicates that the uncertain plant will be compensated into a nominal plant. Meanwhile, measurement noise mainly occurs at high frequency such that |ξ (jω)| ≈ 0 at low frequency range. That is, PC Gyξ ≈ 1+ can be obtained at high frequencies, which means PC that the existence of inner loop will not increase the bandwidth from measurement noise to system output. The controller C (s) is then used to stabilize the nominal plant for desired control performance. Thus, the crux of DOB design is the structure design and parameters optimization of Q filter. 3.1. Robust internal stability

2. Problem statement The notations P(s) and Pn (s) represent real plant and nominal plant, respectively. It is assumed that P(s) and Pn (s) are all unstable plants. The nominal model is described as: Pn (s) =

KN− (s)N+ (s) M− (s)M+ (s)

,

Internal stability [23] is a basic requirement for a practical control system. In this section, we analyze the robust internal stability of the closed-loop system. To analyze the internal stability, the transfer function matrix from [r d ξ ]T to [u r0 y¯ ]T is expressed as:

(1)

where K is the static gain of the controlled object. The subscript plus sign + and minus sign − represent the roots in RHP and left-half of the s-plane (LHP), respectively. N+ (j0) = N− (j0) = M+ (j0) = M− (j0) = 1. For a proper controlled object degree{N− (s)} + degree{N+ (s)} ≤ degree{M− (s)} + degree{M+ (s)}.

1 B

[

(QP − QPn + Pn )C Pn C PPn C

(Q − 1)PPn C (Q − 1)Pn (Q − 1)PPn

(Q − 1)Pn C − Q − Pn C (1 − Q )Pn

] ,

(3)

where B ≜ Q (s)(P(s) − Pn (s)) + Pn (s)(1 + P(s)C (s)), y¯ = y + ξ . The closed-loop system depicted in Fig. 1 is robustly internally stable if all the transfer functions in Eq. (3) are stable.

Please cite this article as: L. Wang and J. Cheng, Robust disturbance rejection methodology for unstable non-minimum phase systems via disturbance observer. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.034.

L. Wang and J. Cheng / ISA Transactions xxx (xxxx) xxx

3

Fig. 1. The DOB based controller for the NMP system.

Theorem 1. DOB based control structure in Fig. 1 is robustly internally stable if the following conditions are satisfied: Requirement 1. D(s), Q (s), C (s), Q (s)Pn−1 (s) ∈ RH∞ , where D(s) is defined as D(s) =

1 (1 − Q )(P −1 C −1 + 1) +

Q (P −1 C −1 n

+ 1)

.

(4)

Requirement 2. 1 − Q (s) has no RHP zeros. Proof. It is very hard to analyze the stability of matrix in Eq. (3) directly. Hence, we introduce the notation D(s) expressed in Eq. (4), and then Eq. (3) can be rewritten as the following threeby-three transfer function matrix:

[

A(s) = Ai,j (s)

]

i=1∼3,j=1∼3

,

(5)

definition of D(s), we get:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

P −1 D = P −1 C −1 D =

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

C −1 D =

C (1 − Q ) + (QPn−1 + C )P 1 (1 − Q ) + (QPn−1 + C )P P (1 − Q ) + (QPn−1 + C )P

(6)

.

For the transfer function P −1 D, its unstable poles can only come from the unstable zeros from P(s), because D(s) ∈ RH∞ . According to Requirement 1 and 2, we introduce the following notations as 1−Q ≜

N1 (s) M1 (s)

N2 (s)

, QPn−1 + C ≜

M2 (s)

,P ≜

N3 (s) M3 (s)

,

the denominator polynomial of P −1 D is

where

N1 (s)M2 (s)M3 (s) + M1 (s)N2 (s)N3 (s).

A11 (s) = (1 − D)C A12 (s) = (Q − 1)D A13 (s) = (Q − 1)P −1 D A21 (s) = P −1 D A22 (s) = (Q − 1)P −1 C −1 D A23 (s) = −(1 + QPn−1 C −1 )P −1 D A31 (s) = D A32 (s) = (Q − 1)C −1 D A33 (s) = (1 − Q )P −1 C −1 D. According to Requirement 1, if D(s), Q (s), C (s) ∈ RH∞ is satisfied, it can be concluded that the components of A11 (s), A12 (s) and A31 (s) are stable transfer function. Since 1 − Q (s) ∈ RH∞ , A13 (s) is stable if and only if A21 (s) is stable, A32 (s) is stable if and only if C −1 D is stable. Notice that A22 (s) = −A33 (s), that is, only transfer functions A21 (s), A22 (s), A23 (s) and A32 (s) should be analyzed. For transfer function A22 (s), it is stable if P −1 C −1 D is stable. For transfer function A23 (s), we have A23 (s) = −P −1 D − QPn−1 C −1 P −1 D −1 −1 −1

= −A21 (s) − QPn P −1

C

D.

Since Q (s)Pn (s) ∈ RH∞ according to Requirement 1, A23 (s) is stable if A21 (s) and P −1 C −1 D are stable. In summary, if transfer functions P −1 D, P −1 C −1 D and C −1 D are in RH∞ , all the components of A(s) are stable. According to the

(7)

If the denominator polynomial in Eq. (7) has the same RHP roots as P −1 , the component N1 (s)M2 (s)M3 (s) should have same RHP roots with P −1 . According to Requirement 1 and 2, only polynomials N2 (s), N3 (s) and M3 (s) may have RHP roots, while N1 (s), M1 (s) and M2 (s) do not have RHP roots. Notice that N3 (s) will not have the same RHP roots with M3 (s), thus, the polynomial in Eq. (7) will not have the RHP roots, P −1 D is stable. For the transfer function P −1 C −1 D, its unstable poles can only come from the unstable zeros from P(s) and C (s), since D(s) ∈ RH∞ . For the transfer function C −1 D, its unstable poles can only come from the unstable zeros from C (s). According to the system uncertainty definition, we have

⎧ ⎪ −1 −1 ⎪ ⎨P C D = ⎪ ⎪ ⎩

C −1 D =

1 1 + Q ∆ + PC P 1 + Q ∆ + PC

(8)

.

By introducing the notations Q∆ ≜

N4 (s) M4 (s)

, PC ≜

N5 (s) M5 (s)

,

the denominator polynomial of P −1 C −1 D is M4 (s)M5 (s) + N4 (s)M5 (s) + M4 (s)N5 (s).

(9)

If the denominator polynomial in Eq. (9) has the same RHP roots as P −1 C −1 , the component M4 (s)M5 (s) should have same RHP roots with P −1 C −1 . Since Q , ∆ ∈ RH∞ , and N5 (s) will not have the same RHP roots with M5 (s), thus, the polynomial in Eq. (9)

Please cite this article as: L. Wang and J. Cheng, Robust disturbance rejection methodology for unstable non-minimum phase systems via disturbance observer. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.034.

4

L. Wang and J. Cheng / ISA Transactions xxx (xxxx) xxx

will not have the RHP roots, P −1 C −1 D is stable. Similarly, it can be concluded C −1 P is stable. Thus, the DOB based control system described in Fig. 1 is robustly internally stable if Requirement 1 and 2 are satisfied. □

Since |Pn (jω)C (jω) + Q (jω)| < |Pn (jω)C (jω)| + |Q (jω)|, the robust stability constraint of the close-loop system will be satisfied if Q filter meets the following inequality

|Q (jω)| < H(ω), ∀ω,

(18)

H(ω) = |∆−1 (jω)(1 + Pn (jω)C (jω))| − |Pn (jω)C (jω)|.

(19)

Theorem 2. By introducing the transformation

ˆ ≜ P(s)

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

, Pˆ n (s) ≜

Pn (s) Pn (s)C (s) + 1

,

transfer function D(s) is stable when

ˆ (jω)) ≤ |W∆ (jω)|, ∀ω, ∥W∆ (s) · Q (s)∥∞ < 1, σ¯ (∆

(10)

ˆ ˆ (s)), weighting function W∆ (s) reflects where P(s) = Pˆ n (s)(1 + ∆ the frequency responses of internal uncertainties, and the selection of W∆ (s) is expressed in Section 5 in detail. Proof. By applying the equivalent transformation, the transfer function in Eq. (2) can be rewritten as y(s) = Gyr (s)r(s) + Gyd (s)d(s) − Gyξ (s)ξ (s), ′

′

′

(11)

G′yd (s) = G′yξ (s) =

ˆ Pˆ n (s)P(s)C (s) ˆ − Pˆ n (s))Q + Pˆ n (s) (P(s) ˆ Pˆ n (s)P(s)(1 − Q (s)) ˆ − Pˆ n (s))Q + Pˆ n (s) (P(s)

, ,

ˆ P(s)Q (s) ˆ − Pˆ n (s))Q + Pˆ n (s) (P(s)

·

ˆ −1 (jw)| > H(ω), ∀ω. |∆

(20)

ˆ (s) and triangle inequality, we According to the definition of ∆ get

ˆ −1 (jω)| |∆ = |∆−1 (jω)(1 + Pn (jω)C (jω)) + (1 + Pn (jω)C (jω))|

(21)

≥ |∆−1 (jω)(1 + Pn (jω)C (jω))| − |(1 + Pn (jω)C (jω))|. Then, we can finally come to a conclusion that by using the proposed transformation, the proposed Q (s) optimization method can obtain higher bandwidth for better disturbance rejection performance.

where G′yr (s) =

According to the above inequality, it is clearly that the conservatism of proposed robust stability is no more than that of directly method without any transformation if

Pn (s)C (s) + Q (s) (1 + Pn (s)C (s))Q (s)

,

which is described as the structure in Fig. 2 equivalently. According to the definition of D(s), it can be easily verified that D(s) = G′yr (s). Thus, we only need to analysis whether G′yr (s) is stable. The DOB based control system can be transformed equivalently into the form shown in Fig. 3 for robust stability analysis. The robust stability condition can be rewritten as follows:

ˆ = Pˆ n (s)(∆ ˆ (jω) · Q (jω)) < 1, ∀ω, P(s) ˆ (s) + 1), σ¯ (∆

(12)

where σ¯ (·) represents the maximum singular value. By choosing a stable transfer function W2 (s) as the weighting function:

ˆ (jω)) ≤ |W∆ (jω)|, ∀ω, σ¯ (∆

(13)

the sufficient robust stability condition for the control system can be expressed as follows to formulate the optimization function

|Q (jω)| < |W∆−1 (jω)|, ∀ω ⇔ ∥W∆ (s) · Q (s)∥∞ < 1.

Remark 2. 1 − Q (s) should be minimized to attenuate external disturbance d. According to the transfer function Gyξ , Q (s) should also be minimized to reject measurement noise ξ . The above two conditions are obviously conflicting. Notice that in practice, external disturbance usually exists in the low frequency range, while system uncertainties and measurement noise mainly happens in the high frequency range. This reveals that the DOB can be designed by weighted frequency optimization [15]. 3.3. Performance analysis The rejection performance of disturbance and measurement noise is significant in the control system, thus, system performance is analyzed in this subsection. The disturbance contains two parts, the external disturbance and equivalent disturbance caused by the mismatch between the nominal and real plant. By considering the equivalent structure in Fig. 2, the disturbance estimation can be expressed as

ˆ = d(s)

(14)

ˆ − Pˆ n (s))Q (s)C (s) (P(s) ˆ − Pˆ n (s))Q (s) + Pˆ n (s) (P(s) +

3.2. Conservatism analysis

ˆ P(s)Q (s)

as:

ˆ (s)∥∞ , ∥Q (s)∥∞ ≤ ∥∆ −1

(15)

ˆ (s) is defined in Eq. (12). where ∆ If this equivalent transformation is not applied, the robust stability constraint is expressed as follows

    ∆(s) · Q (s) + Pn (s)C (s)  < 1.  Pn (s)C (s) + 1 ∞

(22)

d(s)

ˆ − Pˆ n (s))Q (s) + Pˆ n (s) (P(s) ˆ ˆ − Pˆ n (s))C (s)r(s) P(s)d(s) + (P(s)

= Q (s) From Theorem 2, the robust stability constraint is expressed

r(s)

ˆ − Pˆ n (s))Q (s) + Pˆ n (s) (P(s)

.

According to the principle of DOB, it can be expressed in the open-loop form as follows y(s) = Pˆ n (s)(C (s)r(s) + d′ (s) − dˆ ′ (s)),

(23)

ˆ′

′

where d (s) is the estimation of the DOB, d (s) represents the equivalent total disturbance with both external disturbances and system uncertainties, which can be expressed as:

(16) d′ (s) =

ˆ − Pˆ n (s))C (s)r(s) + P(s)d(s) ˆ (P(s) ˆ − Pˆ n (s))Q (s) + Pˆ n (s) (P(s)

.

Since Q (s) exists in the above Eq. implicitly, it cannot be optimized straightforward. By introducing the transformation of sufficient condition of robust stability, Eq. (16) can be rewritten as

For a DOB based control system, by considering the above equations, we get

|Pn (jω)C (jω) + Q (jω)| < |∆−1 (jω)(1 + Pn (jω)C (jω))|, ∀ω,

y(s) =

(17)

Pn (s)C (s) Pn (s)C (s) + 1

r(s) +

Pn (s) Pn (s)C (s) + 1

˜ , d(s)

(24)

Please cite this article as: L. Wang and J. Cheng, Robust disturbance rejection methodology for unstable non-minimum phase systems via disturbance observer. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.034.

L. Wang and J. Cheng / ISA Transactions xxx (xxxx) xxx

5

Fig. 2. Equivalent structure of closed-loop system.

Fig. 3. Equivalent transformation for robust stability analysis.

˜ ≜ d′ (s) − dˆ ′ (s). The system tracking error e = r − y is where d(s) expressed as

⎧ Pn (s) 1 ⎪ ˜ ⎪ ⎪ e(s) = P (s)C (s) + 1 r(s) − P (s)C (s) + 1 d(s) ⎪ ⎪ n n ⎪ ⎨ ˜ = (P(s) − Pn (s))C (s)r(s) (1 − Q (s))+ d(s) ⎪ B(s) ⎪ ⎪ ⎪ P(s)(1 + Pn (s)C (s))d(s) ⎪ ⎪ ⎩ (1 − Q (s)).

(25)

B(s)

where B(s) is defined in Eq. (3). The first component of e(s) in Eq. (25) can be eliminated completely by using feedforward compensation, hence, the second component in Eq. (25) determines output error of closed-loop ˜ system. For the estimation error d(s), the first component is caused by system uncertainties, while the second component is caused by external disturbance. Final Value Theorem is adopted for analysis of the steady-state performance, the disturbance estimation error of steady-state is expressed as:

˜ ∞) = lim d(t) ˜ = lim sd(s) ˜ . d( t →∞

(26)

s→0

Notice that B(s)

P(s)(1 + Pn (s)C (s)) ⎪ ⎪ ⎩ lim = c2 , s→0

The method proposed in [24] is applied for controller design. All the linear time-invariant (LTI) controllers which satisfying Requirement 1 in Theorem 1 and have zero steady-state error can be parameterized as C (s) =

G(s) 1 − G(s)Pn (s)

lim S(s) = lim 1 −

where c1 and c2 are constants. Then we have

s→0

˜ ∞) = c1 lim sr(s) lim(1 − Q (s)) + c2 lim sd(s)(1 − Q (s)). d( s→0

4.1. Controller design

[

B(s)

s→0

4. Control system design of unstable NMP system

, G(s) =

[1 + sg(s)]M+ (s) K

,

(28)

where a stable transfer function g(s) is designed to make G(s) proper. The sensitivity transfer function from reference input signal to output error is S(s) = 1 − G(s)Pn (s). It can be verified that

⎧ (P(s) − Pn (s))C (s) ⎪ ⎪ = c1 ⎨ lim s→0

Since lims→0 sr(s) is a constant, that is 1 − Q (s) should have at least one zero on imaginary axis of s-plane to avoid the influence caused by system uncertainties. However, for high order or periodic external disturbance, 1 − Q (s) should eliminate all the unstable RHP poles of d(s). For the high frequency measurement noise, it can be concluded from Gyξ (s) that the DOB will enlarge the influence caused by measurement noise. Thus, |Q (jω)| should be designed as small as possible to decrease the gain from measurement noise to system output.

s→0

(27)

s→0

KN+ (s)N− (s) (1 + sg(s))M+ (s) M+ (s)M− (s)

K

]

= 0.

(29)

In other words, closed-loop system with controller in Eq. (28) has zero steady-state error.

Please cite this article as: L. Wang and J. Cheng, Robust disturbance rejection methodology for unstable non-minimum phase systems via disturbance observer. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.034.

6

L. Wang and J. Cheng / ISA Transactions xxx (xxxx) xxx

Lemma 1 ([25]). Assume that L2 represents all the strictly proper functions. Let H2 represents the subset of L2 and analytic in Res > 0, H2⊥ analytic in Res ≤ 0. For F1 ∈ H2 , F2 ∈ H2⊥ , the following equation is satisfied

∥F1 + F2 ∥22 = ∥F1 ∥22 + ∥F2 ∥22 .

(30)

To design the controller, the optimal setpoint tracking performance is considered as follows min ∥W (s)S(s)∥2 , W (s) = 1/s.

(31)

According to Eq. (28) and Lemma 1, the optimization function can be separated into two parts in H2 and H2⊥

∥W (s)S(s)∥22 = ∥SH2 ∥22 + ∥SH⊥ ∥22

(32)

2

where

⎧ N+ (−s) − N+ (s) ⎪ ⎪ ⎨ SH2 = sN+ (s)

M− (s) − N+ (−s)N− (s) N+ (−s)N− (s) ⎪ ⎪ ⎩ SH⊥ = − g(s). sM− (s)

2

(33)

s.t.

M− (s) − N+ (−s)N− (s) sN+ (−s)N− (s)

(34)

and the controller is shown as [1 + sgopt (s)]M+ (s) M− (s)M+ (s) C (s) = f (s) = f (s), K KN+ (−s)N− (s)

min

Q (s)∈Ω Q (s)∈RH∞

(35)

(36)

Q (s) =

α0 = h(0), α1 = β1 h(0) + h (0) 3h′′ (0)

The crux of DOB configuration is the structure design and parameters optimization of the Q filter. According to the structural constraints proposed in Section 3, the set of Q (s) is defined as: N(s) M(s)

, M(s) =

m ∑

ai si , N(s) =

i=0

n ∑ , bj sj , j=0

(37)

am ̸ = 0, bn ̸ = 0, a0 = b0 , m − n ≥ k}, where M(s) and N(s) are coprime polynomials, k is the relative order of nominal plant. The optimized Q filter should not only satisfy the proposed design constrains, but also attenuate the total disturbance and measurement noise as far as possible. According to Remark 2, to achieve better disturbance and measurement noise rejection performance, both Q (s) and 1−Q (s) should be minimized as much as possible at corresponding frequency range. Thus the following optimization functions can be obtained:

⎧ ⎨ min |1 − Q (jω)| · |W1 (jω)| Q (s)∈Ω

⎩ min |Q (jω)| · |Wξ (jω)|, Q (s)∈Ω

˜ K˜ (s) 1 + P(s)

.

(41)

(42)

where Po (s) is a stable transfer function, while Pl (s) is an allpass portion with all the RHP zeros of nominal plant. Without loss of generality, Po (s) is usually chosen as the following form:

αn , (s + α )n

where n is the relative order of W2 (s). The specifications will be described in Section 5.

.

4.2. Robust DOB design

Ω = {Q (s)|Q (s) =

˜ K˜ (s) P(s)

˜ = P˜ o (s)P˜ l (s), P(s)

Po (s) =

′

h′′′ (0)

(40)

For an unstable NMP system, the optimized Q filter should contain the RHP zeros of Pn (s) as its RHP zeros to make Q (s)Pn−1 (s) ˜ should have the stable. Thus, the virtually controlled object P(s) corresponding RHP zeros, which is defined as:

Let h(s) = sC (s), it follows that:

α2 = β1 h′ (0) + h′′ (0)/2, β1 = −

[ ]  λW1 (s)(P˜ K˜ + 1)−1     W2 (s)P˜ K˜ (P˜ K˜ + 1)−1  < 1, ∞

where Q (s) can be obtained as:

where a low-pass filter f (s) is employed to make C (s) proper. The proposed controller is physically realizable, however, the implementation of such a high-order transfer function in practice is impossible. In this paper, Maclaurin expansion series is introduced to acquire a parameterized PID controller, which is expressed as follows

α2 s2 + α1 s + α0 . s(β1 s + 1)

(39)

where |W2 (jω)| is chosen such that |W2 (jω)| = max{|∆(jω)|, |Wξ (jω)|}, ∀ω. And also, W2−1 (s)Pn−1 (s) should be a proper transfer function. Then we present the method to get the optimized solution. It is difficult to optimize Q (s) directly according to optimization function in Eq. (39). The H∞ method is used to solve the proposed optimization function [26]. By defining the virtually controlled ˜ object and controller as P(s) and K˜ (s), respectively, the Q filter design problem can be rewritten into a standard H∞ problem as:

s.t.

,

min

Q (s)∈Ω Q (s)∈RH∞

[ ]  λW1 (s)(1 − Q (s))    < 1,   W2 (s)Q (s) ∞

max λ,

2

C (s) =

max λ,

M− (s)

To minimize SH⊥ , we have gopt (s) =

where Wξ and W1 are the weighting functions reflecting the frequency range of the measurement noise and external disturbances, respectively. Considering the robust stability condition, we can finally acquire the optimization function as:

(38)

Theorem 3. For the proposed methodology, the following conclusions can be reached: 1. Q (s)Pn (s) should be a proper transfer function. 2. Q (s) and Q (s)Pn−1 (s) are both stable transfer function without RHP poles. 3. 1 − Q (s) has no RHP zeros. Proof. 1. According to loop shaping theory, it is obtained ∥1 − Q (s)∥∞ < ∥ λW1 (s) ∥∞ and ∥Q (s)∥∞ < ∥ W 1(s) ∥∞ from the cost 1 2 function in Eq. (39). Then we have the following expression as: lim |W2 (jω)Q (jω)| < 1,

ω→∞

(43)

which means that W2 (s)Q (s) is a proper transfer function. Then we have Q (s)Pn−1 (s) = W2 (s)Q (s) · W2−1 (s)Pn−1 (s). Since W2 (s) is selected such that W2−1 (s)Pn−1 (s) is proper, Q (s)Pn−1 (s) is a proper transfer function. 2. According to H∞ optimal control theory, the control system of the virtual H∞ closed-loop structure is internally stable, thus Q (s) is a stable transfer function without RHP poles.

Please cite this article as: L. Wang and J. Cheng, Robust disturbance rejection methodology for unstable non-minimum phase systems via disturbance observer. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.034.

L. Wang and J. Cheng / ISA Transactions xxx (xxxx) xxx

According to H∞ control theory, there is no RHP zeros-poles cancelation between the virtually controlled object and the optimized H∞ controller. From the definition of Eq. (42), virtually controlled object have the corresponding RHP zeros of Pn (s). Thus, ˜ = P(s) ˜ K˜ (s) will contains the RHP zeros of Pn (s). Since L(s) ˜ has L(s) no RHP poles, Q (s) will contains all the RHP zeros of the nominal plant. In other words, Q (s)Pn−1 (s) is a stable transfer function. 3. According to the definition of Eq. (41), 1 − Q (s) is obtained as: 1 − Q (s) =

1

˜ K˜ (s) + 1 P(s)

.

(44)

˜ is a stable plant, and the Since the virtual controlled object P(s) virtual controller K˜ (s) optimized by H∞ theorem is also stable. Thus, 1 − Q (s) has no RHP zeros. □ Theorem 3 indicates that if the weighting functions and virtually controlled object are well-defined, the optimization results will satisfy the relative order constraint, and Q (s)Pn−1 (s) is a proper stable transfer function. Meanwhile, both Q (s) and 1 − Q (s) are in RH∞ . The design procedure reveals that the proposed methodology can successfully satisfy the required constraints. 5. Simulations 5.1. Case 1 In this subsection, the design procedure of the proposed method is presented specifically. An unstable NMP system is considered as follows: P(s) =

k(a1 s + 1)(a2 s + 1) (b1 s + 1)(b2 s + 1)(b3 s + 1)

·

s2

ωn2 , + 2ξ ωn s + ωn2

(45)

where k = 3, a1 ∈ [−0.225, −0.27], a2 ∈ [2.7, 3.3], b1 ∈ [−1.1, −0.9], b2 ∈ [1.8, 2.2], b3 ∈ [0.27, 0.33], ωn = 30rad, ξ = 0.1 and the nominal value is given as a1 = −0.25, a2 = 3.0, b1 = −1.0, b2 = 2.0, b3 = 0.3. Thus the nominal plant is given as 3.0(−0.25s + 1)(3.0s + 1) . (46) Pn (s) = (−1.0s + 1)(2.0s + 1)(0.3s + 1)

7

˜ Step 3. Selection of virtually controlled object P(s) ∈ Σp should consider the relative order of Q (s), and the RHP zeros of Pn (s) should also be considered. Then, optimized K˜ opt (s) can be optimized according to H∞ framework in Section 4.2. From Eq. (42), the virtually controlled object is chosen as ˜ = P˜ o (s)P˜ l (s) = P(s)

1 (s +

1)2

·

−0.25s + 1 . 0.25s + 1

(51)

The virtually optimal controller is obtained while λ = 1.2140 as K˜ opt =

899.9544(0.25s + 1)(s + 1)2 (s + 0.3925) s2 (s2 + 16.99s + 458.2)

.

Step 4. Obtain the optimal Q filter according to Eq. (41). From K˜ opt (s), Q filter is given as Q (s) =

899.9544(−0.25s + 1)(s + 0.3925) s4

+ 16.99s3 + 233.2114s2 + 811.6464s + 353.2321

It is obvious that the optimized Q (s) can eliminate the RHP pole of Pn−1 (s), which fulfills the Requirement 1 of Theory 1. The relative order of Q filter is 2, which is same as that of W2 (s). From the frequency responses in Fig. 5, Q (s) ≤ W2 (s) around all the frequency axis, which fulfill the stability design requirement. The whole DOB based control structure has been successfully constructed. The method proposed in [20] is also used for comparison, the following Q filter has same relative order as Q (s):

−0.25s + 1 . (54) (τ + 2τ s + 1 0.25s + 1 where τ = 1.0 is adjusting parameter to guarantee robust Q (s) =

1

s)2

·

stability. The all-pass portion in Q ′ (s) is to make Q ′ (s)Pn−1 (s) stable. In simulation, constantly external disturbance is considered. Fig. 6 shows the step responses, while Fig. 7 shows the estimation effect of total disturbance. Notice that the factor 1 − Q (s) reflect the disturbance rejection performance. For the Q (s) acquired from the proposed methodology, we have 1 − Q (s) =

s4 + 16.99s3 + 458.2s2 s4 + 16.99s3 + 233.2114s2 + 811.6464s + 353.2321

and for Q ′ (s) acquired from tradition method, we have

−28.8s4 − 82.2s3 + 60.7s2 + 49.3s + 1 C (s) = . 81s4 + 414s3 + 96s2 − 11s + 2

1 − Q ′ (s) =

The parameterized PID controller is given in a unified form as

C (s) =

−0.2116s2 − 20.9029s − 0.4442 . 0.4764s2 + 47.6427s

(48)

Step 2. Design the weighting functions according to the robust stability and system performance. Firstly, we consider the ˆ (s) = multiplicative system uncertainties, which is obtained as ∆ ˆ − Pˆ n (s))/Pˆ n (s). And also, W2 (s) should suppress the measure(P(s) ment noise of −30 dB at frequency of 100 rad/s. Thus, we have: 0.08s2 + 1.0s + 14.0

. (49) 18.0 Frequency responses of W2 (s) are shown in Fig. 4, it is shown that the selection of W2 (s) fulfill the requirement in Eq. (13). For all the possible parameter perturbation, it satisfies W2 (s) ≤ ∆−1 (s). For disturbance rejection performance, W1 (s) is simply chosen as 1 W1 (s) = 2 . (50) s W2 (s) =

. (53)

Step 1. Controller design based on nominal plant. According to Section 4.1, the controller is designed as (47)

(52)

0.25s3 + 1.5s2 + 2.5s 0.25s3 + 1.5s2 + 2.25s + 1.0

,

.

Notice that both 1 − Q (s) and 1 − Q ′ (s) have at least a zero on imaginary axis of s-plane, that is to say, these DOBs can eliminate the influence caused by constantly external disturbances completely. For the traditional method, the frequency responses of Q has a high magnitude at medium frequency range, thus the disturbance estimation and control output with traditional Q (s) have serious overshot, while the DOB with proposed Q (s) will converge within 40 s. If the robust stability condition shown in Eq. (10) cannot be satisfied, the proposed DOB will lead instable. The Q filter is defined as follows: −449.9683s + 1799.8732 Q ′ (s) = 3 . (55) s + 19.48s2 + 468.8317s + 1799.8732 This Q ′ (s) fulfill all the design requirements expect robust stability condition in Eq. (10). The frequency responses of these two Q filters are shown in Fig. 8. It can be seen that Q ′ (s) do not fulfill robust stability condition when the frequency is larger than 12.14 rad/s. It is shown in Fig. 9 that the DOB based control structure with Q ′ (s) is instable and the system output will diverge quickly in 10 s, while the Q filter that fulfill the robust stability condition can make the control system work well.

Please cite this article as: L. Wang and J. Cheng, Robust disturbance rejection methodology for unstable non-minimum phase systems via disturbance observer. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.034.

8

L. Wang and J. Cheng / ISA Transactions xxx (xxxx) xxx

Fig. 4. Weighting function of W2 (s).

Fig. 5. Frequency responses of optimized Q (s).

Fig. 6. Control effect with constant external disturbance.

Please cite this article as: L. Wang and J. Cheng, Robust disturbance rejection methodology for unstable non-minimum phase systems via disturbance observer. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.034.

L. Wang and J. Cheng / ISA Transactions xxx (xxxx) xxx

9

Fig. 7. Disturbance estimation effect with constant external disturbance.

Fig. 8. Frequency responses comparison of Q filters.

Fig. 9. Control effect with proposed DOB and Q ′ (s).

Please cite this article as: L. Wang and J. Cheng, Robust disturbance rejection methodology for unstable non-minimum phase systems via disturbance observer. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.034.

10

L. Wang and J. Cheng / ISA Transactions xxx (xxxx) xxx

Fig. 10. Frequency responses of optimized Q (s).

Fig. 11. Control effect with slope external disturbances.

Fig. 12. Estimation error with slope external disturbances.

Please cite this article as: L. Wang and J. Cheng, Robust disturbance rejection methodology for unstable non-minimum phase systems via disturbance observer. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.034.

L. Wang and J. Cheng / ISA Transactions xxx (xxxx) xxx

5.2. Case 2 The depth control system of a torpedo can be expressed as the following form [27]: P(s) =

k(a1 s + 1)(a2 s + 1) s2 (b1 s + 1)(b2 s + 1)

,

(56)

where k = 138.67, a1 = 0.4, a2 = −0.02, b1 = 1/1.15, b2 = 1/8.33, and the nominal values are chosen as k0 = 150, a1 = 0.5, a2 = −0.03, b1 = 1, b2 = 1/8. The controlled object has two poles on imaginary axis, which is regarded as a critical stable system. Step 1. Controller design based on nominal plant. According to the controller design method in Section 4.1, The parameterized PID controller is given as follows: C (s) =

0.1752s2 − 0.0423s − 0.0024 0.5453s2 + s

.

(57)

Step 2. Similar to the descriptions in Section 5.1, we choose the follows weighting functions: W1 (s) =

1 s2

, W2 (s) =

0.05s2 + 0.7s + 10.0 12.0

.

(58)

˜ Step 3. Select the virtual controlled object P(s) ∈ Σp should consider the relative order of Q (s), and the RHP zeros of Pn (s) should also be considered. Then, optimized K˜ opt (s) can be obtained according to the H∞ framework in Section 4.2. From Eq. (42), the virtually controlled object is selected as follows: ˜ = P˜ o (s)P˜ l (s) = P(s)

1 (s + 1)2

·

−0.03s + 1 . 0.03s + 1

(59)

The virtually optimal controller is obtained while λ = 5.75 as: K˜ opt =

240.04(s + 33.33)(s + 1)2 (s + 1.273) s2 (s2 + 48.86s + 979.2)

.

(60)

Step 4. From K˜ opt (s), Q filter is shown as: Q (s) =

8000.71(−0.03s + 1)(s + 1.273) s4 + 48.86s3 + 739.16s2 + 7694.96s + 10184.9

.

(61)

From the frequency responses in Fig. 10, it is verified that designed Q filter satisfies the robust stability condition. The slope external disturbance with frequency 40 s and amplitude interval form 0 to 1 is considered in the simulation. The traditional Q filter with same structure as Eq. (54) is also adopted for comparison. Control performance comparison is shown in Fig. 11, while disturbance estimation error comparison is shown in Fig. 12. The simulation results indicate that better disturbance rejection performance can be obtained by using the proposed method. We can also find the settling time of the proposed methodology is less than 10 s, while that of the traditional method is more than 20 s. And the overshot of the proposed method is about 6%, whereas the overshot of the traditional method is more than 40%. These results reveal that the proposed methodology can provide the system with better transient state performance. 6. Conclusions In this work, we propose a robust disturbance rejection methodology for unstable NMP systems. Internal stability is first analyzed, based on which several constraints are proposed to make sure the designed control system is robustly stable. Then, it is shown that the proposed constraints will bring the system with less conservatism and better performance. Several design

11

requirements are synthesized together to establish the cost function and thus optimize the Q filter of DOB. At last, the design specification is presented to show that all the constraints can be satisfied. Simulations show that the proposed methodology can be adopted for unstable NMP system successfully. Comparing with the existing method, the proposed methodology has better disturbance rejection performance. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported part by Key Project of National Natural Science Foundation of China under Grant 61633008, Preeminent Youth Fund of Heilongjiang Province under Grant JJ2018JQ0059, China Postdoctoral Science Foundation under Grant 2018M641806, Basic Scientific Research Fund under Grant HEUCFG201821, Class A Project of Young Talents of Science and Technology Innovation Talents of Harbin Science and Technology Bureau under Grant 2015RAQXJ010. References [1] Chen W, Yang J, Guo L, Li S. Disturbance-observer-based control and related methods-an overview. IEEE Trans Ind Electron 2016;63(2):1083–95. [2] Sariyildiz E, Oboe R, Ohnishi K. Disturbance observer-based robust control and its applicatiosn: 35th anniversary overview. IEEE Trans Ind Electron 2020;67(3):2042–53. [3] Du J, Liu Z, Wang Y, Wen C. An adaptive sliding mode observer for lithium-ion battery state of charge and state of health estimation in electric vehicles. Control Eng Pract 2016;54(8):81–90. [4] Lin S, Yen J, Chen M, Chang S. An adaptive unknown periodic input observer for discrete-time LTI SISO systems. IEEE Trans Automat Control 2017;62(8):4073–9. [5] Wang C, Zou Z, Qi Z, Ding Z. Predictor-based extended-state-observer design for consensus of MASs with delays and disturbances. IEEE Trans Cybern 2018. [6] Sun L, Zheng Z. Disturbance observer-based robust saturated control for spacecraft proximity maneuvers. IEEE Trans Control Syst Technol 2018;26(2):684–92. [7] Zhang W, Tomizuka M, Wu P, Wei Y. A double disturbance observer design for compensation of unknown time delay in a wireless motion control system. IEEE Trans Control Syst Techonol 2018;26(2):675–84. [8] Jo N, Jeon C, Shim H. Noise reduction disturbance observer for disturbance attenuation and noise suppression. IEEE Trans Ind Electron 2017;64(2):1381–91. [9] Wang L, Su J, Xiang G. Robust motion control system design with scheduled disturbance observer. IEEE Trans Ind Electron 2016;63(10):6519–29. [10] Ohnishi K, Shibata M, Murakami T. Motion control for advanced mechatronics. IEEE/ASME Trans Mechatronics 1996;1(1):56–67. [11] Han J. From PID to active disturbance rejection control. IEEE Trans Ind Electron 2009;56(3):900–6. [12] Xue W, Huang Y. Comparison of the DOB based control, a special kind of PID control and ADRC. In: Proceeding American control conference, San Francisco, June 29–July 1. IEEE; 2011, p. 4373–9. [13] Chen X, Yang J, Li S, Li Q. Disturbance observer based multi-variable control of ball mill grinding circuits. J Process Control 2009;19(7):1205–13. [14] Xing D, Su J, Liu Y. Robust approach for humanoid joint control based on a disturbance observer. IET Control Theory Appl 2011;5(14):1630–6. [15] Umeno T, Kaneko T, Y. H. Robust servosystem design with two degrees of freedom and its application to novel motion control of robot manipulators. IEEE Trans Ind Electron 1993;40(5):473–85. [16] Kempf C, Kobayashi S. Disturbance observer and feedforward design for a high-speed direct-drive positioning table. IEEE Trans Control Syst Technol 1999;7(5):513–26. [17] Wang C, Tomizuka M. Design of robustly stable disturbance observers based on closed loop consideration using H∞ optimization and its applications to motion control system. In: Proceeding American control conference, Boston, June 30–July 2, 2004. IEEE; 2004, p. 3764–9.

Please cite this article as: L. Wang and J. Cheng, Robust disturbance rejection methodology for unstable non-minimum phase systems via disturbance observer. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.034.

12

L. Wang and J. Cheng / ISA Transactions xxx (xxxx) xxx

[18] Thum K, Du C, Lewis M, Ong H. H∞ disturbance observer design for high precision track following in hard disk drives. IET Control Theory Appl 2009;3(12):1591–8. [19] Yun N, Su J. Design of a disturbance observer for a two-link manipulator with flexible joints. IEEE Trans Control Syst Technol 2014;22(2):809–15. [20] Zhou P, Chai T, Zhao J. Stability and robustness of disturbanceobserver-based motion control systems. IEEE Trans Circuits Syst II 2012;59(8):525–9. [21] Jo N, Shim H, Son Y. Disturbance observer for non-minimum phase linear systems. Int J Control Autom Syst 2010;8(5):994–1002. [22] Wang L, Su J. Disturbance rejection control for non-minimum phase systems with optimizal disturbance observer. ISA Trans 2015;57(1):1–9.

[23] Morari M. Robust process control. Englewood Cliffs, New Jersey: Prentice-Hall; 1987. [24] Ye F, Sun B, Ou L, Zhang W. Disturbance observer-based control for consensus tracking of multi-agent systems with input delays from a frequency domain perspective. Systems Control Lett 2018;114(10):66–75. [25] Zhou K, Doyle JC. Essentials of robust control. Upper Saddle River New Jersey: Prentice-Hall; 1998. [26] Su J, Wang L, Yun N. A design of disturbance observer in standard H∞ control framework. Internat J Robust Nonlinear Control 2015;25(16):2894–910. [27] Liu X, Li D, Jiang X, Hu X. Simulation study on auto-disturbance-rejection control for unstable systems and non-minimum phase systems. Control Decis. 2001;16(4):420–9.

Please cite this article as: L. Wang and J. Cheng, Robust disturbance rejection methodology for unstable non-minimum phase systems via disturbance observer. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.11.034.