ISA Transactions 71 (2017) 530–541
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Control of unstable processes with time delays via ADRC Caifen Fu n, Wen Tan School of Control & Computer Engineering, North China Electric Power University, Beijing 102206, China
art ic l e i nf o
a b s t r a c t
Article history: Received 11 November 2016 Received in revised form 12 June 2017 Accepted 5 September 2017 Available online 14 September 2017
Active disturbance rejection control (ADRC) treats the external disturbance and internal uncertainties as a general disturbance, and uses an extended state observer (ESO) to estimate it in real-time and feeds it back in the control loop, thus can achieve good disturbance rejection performance. However, ADRC is not quite suitable for unstable delayed processes due to its inherent structure. In this paper, a two-degree-offreedom (2DOF) control structure is proposed for unstable time- delayed systems. Set-point tracking and disturbance rejection are separated in this structure and ADRC is solely responsible for disturbance rejection. A method to tune the ADRC parameters using all the information of the system is proposed, and robustness and performance of the proposed method are analyzed. Simulation examples show that 2DOF-ADRC can achieve good tracking and disturbance rejection performance. & 2017 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords: Active disturbance rejection control (ADRC) Unstable delayed processes Two-degree-of-freedom Robustness
TD is used to generate a transition process and extract differential of the input signal. A non-linear TD has the following form [6]:
1. Introduction Disturbances and uncertainties exist in all industrial systems and have adverse effects on the performance of control systems. According to [1], there are a number of disturbance/uncertainty estimation and attenuation techniques, e.g., unknown input observer (UIO) in disturbance accommodation control, perturbation observer, equivalent input disturbance based estimator, extended state observer (ESO), uncertainty and disturbance estimator, disturbance observer (DOB), and generalized proportional integral observer, etc. Among those disturbance estimation approaches, DOB, UIO, and ESO are extensively investigated and applied in practice [2–5]. ESO is generally regarded as a fundamental part of the so-called active disturbance rejection control (ADRC) [6,7]. ADRC generally consists of a tracking differentiator (TD), an ESO, and a state error feedback (SEF) control law. Its structure is shown in Fig. 1. In Fig. 1, the plant is described as
⎧ x (n)(t ) = bu(t ) + f (x(t ), ẋ(t ), ⋯, x (n − 1)(t ), u(t ), d(t )) ⎨ ⎩ y(t ) = x(t ) ⎪
⎪
(1)
where y is the output, u is the input, b is the gain coefficient, x(t ) is the system state, and d is the external disturbance. f (x(t ) , x(̇ t ) , ⋯ , x(n − 1)(t ) , d(t )) is a combination of both the unknown internal dynamics and the external disturbance, and is called the generalized disturbance.
⎧ v1̇ = v2 ⎪ ⎪ v2̇ = v3 ⎪ ⎪⋮ ⎨ ⎪ vṅ − 1 = vn ⎪ ⎛ v v ⎞ ⎪ vṅ = λ nψ ⎜ v1 − r , 2 , ⋯, n ⎟ ⎪ ⎝ λ ⎩ λn− 1 ⎠
(2)
where r is the input, vi (i ¼ 1, 2, …, n) is the output, and λ is the adjustable speed factor. ESO is used to estimate the generalized disturbances, which is the core and essence of ADRC. A non-linear ESO has the following form [6]:
⎧ e = z1 − y ⎪ ⎪ z1̇ = z2 − β1φ1(e) ⎪ ⎪ z2̇ = z3 − β2φ2(e) ⎨ ⎪⋮ ⎪ ż = z n + 1 − βnφn(e) + bu ⎪ n ⎪ ̇ ⎩ z n + 1 = − β φ (e) n+ 1 n+ 1
(3)
where zi(i ¼ 1, 2, …, n þ 1) are the outputs of the ESO, and βi (i ¼ 1, 2, …, n þ 1) are the observer gains. φi(e ) (i ¼ 1, 2, …, n þ 1) are non-linear functions, particularly, defined as
⎧ e / δ1 − α e ≤δ φi(e) = fal(e , α, δ ) = ⎨ α ⎩ e sgn(e) e > δ ⎪
n
Corresponding author. E-mail addresses:
[email protected] (C. Fu),
[email protected] (W. Tan).
http://dx.doi.org/10.1016/j.isatra.2017.09.002 0019-0578/& 2017 ISA. Published by Elsevier Ltd. All rights reserved.
⎪
(4)
C. Fu, W. Tan / ISA Transactions 71 (2017) 530–541
Fig. 1. Structure of ADRC (adapted from [7]).
SEF is used to eliminate the residual error and achieve the desired control goal. The control law can be designed as [6]:
u(t ) =
−z^n + 1(t ) + uo(t ) b
(5)
where u0 may employ the form n
u0(t ) =
∑ kifal(vi − zi, α˜i, δ ) i=1
(6)
where ki (i ¼ 1, 2, …, n) are the controller gains. It is obvious that in ADRC the estimated disturbance zn + 1 is combined with the nonlinear state error feedback so that the final control u can reject the disturbance. As can be seen, the structure of a traditional ADRC is complex and its tuning parameters are multiple, which makes it difficult to apply in practice. To overcome the difficulty, the linear version of ADRC is introduced in [8,9] where linear ESO and linear state feedback are used. Furthermore, the number of the parameters for the linear ADRC is reduced to two, which are the controller bandwidth ωc and the observer bandwidth ωo . These two parameters are closely related to the performance of the closed-loop system, thus ADRC is readily applicable in industrial control. ADRC has been successfully applied to various systems [1,10– 14] and recent theoretical analysis can be found in [10,15–19]. What makes ADRC distinctly different from other disturbance estimation and rejection techniques [1] is the new and more general conception of disturbance that includes both internal dynamics and external disturbances, which allows the physical process to be modeled as an ideal cascade integral model and the rest of the dynamics to be treated as disturbance. Though ADRC has made much progress, it has not attracted much attention in process control area, partly due to the fact that ADRC is not quite suitable for processes with large time delay. Several methods were proposed in [6] to deal with time delay in ADRC design. One method is to ignore the time delay and design ADRC for plant without time delay. This leads to limited performance. Another method is to replace the time delay with a Pade approximation and adopts a higher order ADRC design. Other methods try to predict the system output or the control signal based on the Taylor series. However, this has only been successful when the time delay is small. In [20] a predictive active disturbance rejection control method based on the basic idea of Smith Predictor as the mechanism of prediction is proposed, which is able to handle long time delays in the process. However, due to the stability problem, in cannot be directly applied to unstable processes. The research on disturbance rejection for processes with time delay took a new turn recently in [21] where a modified ADRC is obtained by making a small adjustment in the ESO input signal, enabling a significant increase in the achievable observer bandwidth and therefore the performance of ADRC. The result is a promising ADRC based unified solution for time-delay systems for stable, critically stable and unstable plants. However, [22] shows that the modified ADRC proposed in [21] has a drawback within its structure, in that the set-point tracking and the
531
disturbance rejection performances are not completely decoupled, and the controller bandwidth is constrained by the time delay. Open loop unstable processes are comparatively difficult to control than that of stable processes. The control of unstable delayed systems has recently attracted much attention [23–31]. [23] proposed a modified IMC structure for unstable processes with time delays; [24,25] proposed a two-degrees-of-freedom control structure in a modified form of Smith predictor and both the nominal set-point and load disturbance responses are decoupled from each other; To improve the disturbance rejection performance, [26] proposed to use PID controllers cascaded with a lead compensator as the load disturbance rejection controller in the structure proposed in [24]; [27] provided the detailed design of the PID controller cascaded with a lead compensator based on IMC method for unstable process with time delay; [28] proposed a modified smith predictor based cascade control for unstable delayed processes, where two closed-loop controllers are considered in the form of PID controller cascaded with a second order lead compensator; [29] proposed an IMC-PID controller based on pole zero conversion design for a class of processes with time delay, where a first-order lead compensator is cascaded with PID controllers to guarantee the stability of the process; [30,31] designed separately a cascade control system for unstable processes with time delay and an optimal H2-IMC based PID controller for multivariable unstable processes, where the controllers are designed as a PID cascaded with a lead filter based on IMC scheme using optimal H2-minimisation. The main reason for using the PID controller cascaded with a lead compensator is that the lead compensator increases the resonant frequency and thus increases the upper bound of frequency in the low frequency region [32]. However, a cascaded lead compensator would decrease sensor noise rejection performance although it may enhance the disturbance rejection performance of the controlled system, which was not considered in [26–31]. This will be discussed in detail in the following section. Motivated by disturbance rejection performance of ADRC, this paper will investigate ADRC for the control of unstable processes with time delays. Main contributions of the paper are twofold: 1) A two-degree-of-freedom (2DOF) ADRC control structure is proposed where set-point tracking and disturbance rejection are separated in this structure and ADRC is solely responsible for disturbance rejection. In this case ADRC plays its role as its name suggests without having to consider the set-point tracking performance, which will overcome the limitation of one-degree- of-freedom (1DOF) ADRC structure for delayed processes. 2) A method to tune the ADRC parameters for the unstable delayed processes is proposed. By using the full information of the processes the disturbance rejection performance can be improved and parameters are easier to tune. Simulation examples show that the proposed method can get better robustness and load rejection performance than the previous methods. The rest of the paper is arranged as follows. In Section 2, ADRC idea will be reviewed and analyzed, and then in Section 3 a 2DOF ADRC structure is proposed. Robustness and performance of the proposed method is also analyzed. In Section 4, a generalized ADRC method will be used to tune the conventional ADRC. In Section 5, simulation results are provided to demonstrate the effectiveness of the proposed design. Finally, conclusions are given in Section 6.
2. Linear active disturbance rejection control (LADRC) In LADRC design, the controlled plant is assumed to have the following model
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y (n) (t ) = bu(t ) + f (y(t ), u(t ), d(t ))
(7)
where y(t), u(t) and d(t) are the output, input and disturbance of the system, and f(y, u, d) is a combination of the unknown dynamics and the external disturbances of the plant, which is denoted as the generalized disturbance and assumed to be unknown. The parameter n is the order of LADRC and b is the gain of the cascade integral model. In LADRC framework, the central idea is to estimate the unknown generalized disturbance (f(y(t), u(t), d(t))). To do so, an extended state observer (ESO) is used. Let
z1(t ) = y(t ), z2(t ) = ẏ (t ), ⋯, z n(t ) = y (n − 1) (t ), z n + 1(t ) = f (t )
(8)
Assume that f is differentiable and let f ̇ = h. Then (7) can be written as
⎧ z(̇ t ) = A z(t ) + B u(t ) + E h(t ) e e e ⎨ ⎩ y(t ) = Cez(t ) ⎪
⎪
(9)
T where z (t ) = ⎡⎣ z1(t ) z2(t ) ⋯ zn(t ) zn + 1(t )⎤⎦ ,
⎡0 ⎢ ⎢0 Ae = ⎢ ⋮ ⎢0 ⎢⎣ 0
1 0 ⋮ 0 0
0 1 ⋮ 0 0
⋯ ⋯ ⋱ ⋯ ⋯
Fig. 2. Structure of LADRC.
k1(r (t ) − z^1(t )) + ⋯ + k n(r (n − 1)(t ) − z^n(t )) − z^n + 1(t ) b ^ ^ = Ko(r (t ) − z (t ))
u(t ) =
(17)
where r^(t ) is an extended reference signal composed of the reference signal r(t) and its derivatives with the order up to n 1, T r^(t ) = ⎡⎣ r (t ) r (̇ t ) ⋯ r (n − 1)(t ) 0⎤⎦
(18)
and
⎡ 0⎤ ⎡ 0⎤ 0⎤ ⎢ ⎥ ⎥ ⎢ ⎥ 0 0⎥ ⎢ ⎥ ⎢ 0⎥ , Be = ⎢ ⋮ ⎥ , Ee = ⎢ ⋮ ⎥ , ⋮⎥ ⎢ b⎥ ⎢ 0⎥ 1⎥ ⎢ ⎥ ⎥ ⎢⎣ ⎥⎦ ⎣ 0 ⎦(n + 1)× 1 0 ⎦(n + 1)×(n + 1) 1 (n + 1)× 1
Ko = ⎡⎣ k1 k2 ⋯ k n 1⎤⎦/b
(10)
Ce = ⎡⎣ 1 0 0 ⋯ 0⎤⎦ . 1 ×(n + 1)
(19)
LADRC structure is shown in Fig. 2. It can be noted that though ADRC can handle disturbance rejection and set-point tracking simultaneously, the disturbance rejection and set-point tracking performance are not completely independent.
A full-order Luenberger state-observer can be designed as 3. Two-degree-of-freedom (2DOF) ADRC
⎧ ^̇ ⎪ z (t ) = Ae z^(t ) + Beu(t ) + L o(y(t ) − y^ (t )) ⎨ ⎪ ⎩ y^ (t ) = Cez^(t )
(11)
where Lo is the observer gain vector: T L o = ⎡⎣ β1 β2 ⋯ βn βn + 1⎤⎦
(12)
To improve the tracking performance of the conventional ADRC, the following two-degree-of-freedom (2DOF) structure is adopted in this paper. The structure is shown in Fig. 3, where the controller C is used to track the reference signal, and N is the desired closed-loop transfer function.
When Ae LoCe is asymptotically stable, z^1(t ) , ⋯z^n(t ) approximate y (t) and its derivatives (up to the order of n 1), and z^ (t ) ap-
3.1. Performance
proximates the generalized disturbance f(y(t), u(t), d(t)). The estimated generalized disturbance can be used in control effort so as to be rejected. We choose the control law as
Set-point tracking and disturbance rejection are considered for the 2DOF-ADRC scheme. Denote the ADRC controller as K. It can be shown that the inputs r, d1, d2 and the outputs u, y in the scheme are related by
n+1
u(t ) =
−z^n + 1(t ) + uo(t ) b
(13)
where uo(t) is to be determined afterward. Then the conventional plant (7) becomes
y (t ) = f (y , u, d) − z^n + 1(t ) + uo(t ) (n)
u=
(C + KN )r − KPd1 − Kd2 1 + KP
(20)
y=
(PC + PKN )r + Pd1 + d2 1 + KP
(21)
(14)
If the ESO is properly designed, i.e., z^n + 1(t ) ≈ f (y, u, d ), the conventional plant will be reduced to a nth-order integral system
y (n) (t ) ≈ uo(t )
(15)
The final system can be effectively controlled with the following state-feedback law
uo(t ) = k1(r (t ) − y(t )) + k2(r (̇ t ) − ẏ (t )) + ⋯ + k n(r (n − 1)(t ) − y (n − 1) (t ))
(16)
where r(t) is the reference signal. Since z^1(t ) , ⋯ , z^n(t ) approximate
y(t ) , ⋯ , y(n − 1) (t ), the final control law can be approximated as
Fig. 3. Two-degree-of-freedom (2DOF) ADRC.
C. Fu, W. Tan / ISA Transactions 71 (2017) 530–541
To have a desired closed-loop N from r to y, it is necessary that
PC + PKN =N 1 + PK
(22)
It is clear that if PC ¼ N, then the condition is always valid. So the controller C can be chosen as
C = P −1N
(23)
If P is of non-minimum phase, then according to IMC design approach P can be decomposed as
P = PAPM
(24)
where PM is the minimum phase part, and PA is the all-pass part. In this case the controller C can be chosen as −1 C = PM F
(25)
where
F=
1 (λs + 1)k
(27)
In this case we always have PC¼ N, thus the closed-loop from r to y has the desired tracking performance. So the set-point response of 2DOF scheme is only related to C. ADRC is only used in the feedback loop and is solely responsible for stability and disturbance rejection. The set-point response and the disturbance response are completely decoupled. Remark 1. The tuning of the set-point tracking parameter λ is the same as in IMC design: when λ is decreased, the nominal set-point tracking performance becomes faster while the output energy of the set-point tracking controller increases, which consequently results in more aggressive action in the presence of process uncertainty. In contrast, increasing λ requires lower output energy of the set-point tracking controller, which in turn results in degraded set-point tracking performance and less aggressive dynamic behavior of the set-point response in the presence of process uncertainty.
with Δ1, Δ2 ∈ H∞.
(28)
It represents simultaneous input multiplicative and inverse output multiplicative uncertainty. [34–36] showed that the measure is effective for comparing robustness of single-loop/multivariable feedback control structures, and can be used as a guideline for robust tuning. To analyze the robustness of the closed-loop system with the plant uncertainty described in (32), we can pull out the uncertainties Δ1 and Δ2, and put the system into an equivalent M − Δ structure as shown in Fig. 4(b), where
⎡Δ 0⎤ 1 ⎥ Δ: = ⎢ ⎣ 0 Δ2 ⎦
(29)
and M is the transfer matrix from signals d1, d2 to u, y. By the small μ theorem [37], the closed-loop system in Fig. 4 (b) is robustly stable for all ‖Δ‖∞ ≤ γ if and only if
μΔ(M ) < 1/γ . (26)
is a filter of order k and the desired closed-loop N can be chosen as
N = PAF
PΔ = (1 − Δ1)−1P (1 − Δ2 ),
533
(30)
Thus μΔ(M ) is a measure of system robustness. The smaller it is, the more robust the closed-loop system is. For the 2DOF-ADRC scheme (Fig. 3), we have
⎡ PK K ⎤ − ⎥ ⎢− 1 + PK 1 + PK ⎥ M=⎢ ⎥ ⎢ P 1 ⎢⎣ 1 + PK 1 + PK ⎥⎦
(31)
and we can compute μΔ(M ) to get a robustness measure for the 2DOFADRC scheme. Since uncertainties are different in different frequency range, it is necessary to compare it in a specific frequency range. Parameter variation is not the only possible uncertainty. Sensor noise and high frequency uncertainty is also inevitable in practical control. As the robustness measure plot mainly reflects the robust performance of the controlled system at low and mid frequency range, in the examples we will use the complementary sensitivity function (T) to show the performance against sensor noise or high frequency uncertainty for the 2DOF-ADRC scheme.
3.2. Robust stability
4. Tuning of ADRC
In this paper, the structured singular value (SSV) approach [33] is used to perform the robustness analysis for the 2DOF-ADRC scheme. It is an effective method for robustness analysis, however, it is very complex and problem-specific. To alleviate the burden, a robustness measure is proposed to evaluate the robustness of a feedback control system with a general uncertainty structure in [34]. The uncertainty structure is shown in Fig. 4(a) and described as:
It is obvious that an ADRC has two sets of gains to tune: Lo, the observer gain for ESO, and Ko, the controller gain for the nth-order integral plant. For practical reason, the tuning of these two gains are reduced to two tuning parameters as suggested in [8]: ωc , the controller bandwidth, and ωo , the observer bandwidth. This method simplifies the tuning procedure, however, since only partial information (order n and high-frequency gain b) is used in the tuning, and the performances are limited for unstable and
Fig. 4. Robustness analysis with general uncertainty structure.
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C. Fu, W. Tan / ISA Transactions 71 (2017) 530–541
delayed systems. In this section, we will present a method to tune ADRC using all the information of the process model. We consider a general single-input-single-output system with the following state-space realization
⎧ ẋ(t ) = Ax(t ) + Bu(t ) P (s ) = ⎨ ⎩ y(t ) = Cx(t )
(32)
The dimension of the states of the system is assumed to be n. Assume that the generalized disturbance f (due to external disturbances and model uncertainties) affects the system in the following form:
⎧ ẋ(t ) = Ax(t ) + Bu(t ) + B f (t ) d ⎨ ⎩ y(t ) = Cx(t ) ⎪
(33)
For such a system, following the ADRC idea, we can define the extended plant as
⎧ ̇ t ) = A˜ e z(t ) + B˜eu(t ) + E˜eh(t ) ⎪ z( ⎨ ⎪ ⎩ y(t ) = C˜ ez(t )
(34)
where
⎡ x(t ) ⎤ z(t ) = ⎢ ⎥, h(t ) = f ̇ (t ) ⎣ f (t )⎦
(35)
and
⎡A B ⎤ ⎡ B⎤ ⎡ 0⎤ d A˜ e = ⎢ ⎥, B˜e = ⎢ ⎥, C˜ e = ⎡⎣ C 0⎤⎦, E˜e = ⎢ ⎥ ⎣ 0⎦ ⎣ 1⎦ ⎣0 0⎦
(36)
For the extended plant (34), similar to the original ADRC, a fullorder Luenberger state-observer can be designed
⎧ ^̇ ⎪ z (t ) = A˜ e z^(t ) + B˜eu(t ) + L˜ o(y(t ) − y^ (t )) ⎨ ⎪ y^ (t ) = C˜ z^(t ) ⎩ e
⋯
k˜1 k˜ 0 ⎤⎦ = : ⎡⎣ K¯ o k˜ 0 ⎤⎦
(38)
(39)
(40)
K¯ o is the first n components of K˜ o . Substitute this control law into (34), the closed-loop system becomes
⎧ ẋ(t ) ≈ Ax(t ) + B(r^(t ) − K¯ x(t ) − k˜ f (t )) + B f (t ) o 0 d ⎪ ⎪ ⎨ = (A − BK¯ o)x(t ) + Br^(t ) + (Bd − Bk˜ 0)f (t ) ⎪ ⎪ ⎩ y(t ) = Cx(t )
K (s ) = Ko(sI − Ae + BeKo + L oCe )−1L o en + 1s n + 1 + ens n + ⋯ + e1s
⎡k ⎡ cn ⎤ ⎢ 1 ⎢c ⎥ ⎢ n − 1⎥ = 1 ⎢ 0 ⎢ ⋮ ⎥ b⎢ ⋮ ⎢ ⎢⎣ c0 ⎥⎦ ⎣0
(44)
k2 ⋯ 1 ⎤⎡ β1 ⎤ ⎥ ⎥⎢ k1 ⋯ k n ⎥⎢ β2 ⎥ ⋮ ⋱ ⋮ ⎥⎢ ⋮ ⎥ ⎥ ⎥⎢ 0 ⋯ k1 ⎦⎣ βn + 1⎦
(45)
and
0 ⎡ en + 1⎤ ⎡ 1 0 ⎢ e ⎥ ⎢β 1 0 ⎢ n ⎥=⎢ 1 ⋮ ⋮ ⎢ ⋮ ⎥ ⎢⋮ ⎢⎣ e1 ⎥⎦ ⎢ β β ⎣ n n − 1 βn − 2
⋯ 0⎤⎡ 1 ⎤ ⎥⎢ ⎥ ⋯ 0⎥⎢ k n ⎥ ⋱ ⋮ ⎥⎢ ⋮ ⎥ ⎥⎢ ⎥ ⋯ 1⎦⎣ k1 ⎦
(46)
So to find an equivalent conventional ADRC, we should first find the numerator and denominator of the transfer function (44), then solving Eqs. (45) and (46) and get k1, ⋯ , k n and β1, ⋯ , βn + 1 for the conventional ADRC.
5. Simulation results
Example 1. (Process with One Unstable Pole and High-order) Consider a high-order unstable process previously studied in [23] and [26].
(41)
To attenuate the generalized disturbance, the following 'matching condition' must be met:
Bd = Bk˜ o
cns n + cn − 1s n − 1 + ⋯ + c0
where
Here r^(t ) is an extended reference signal determined by the tracking differentiator (TD), and the controller gain is defined as
K˜ o = ⎡⎣ k˜ n
(43)
Since the generalized plant (A˜ e , B˜e, C˜ e ) contains all the information of the plant, the observer gain L˜ o and control gain K˜ o can be tuned simply via the bandwidth idea. i.e., choose K¯ o such that the eigenvalues of A − BK¯ o will be all placed at −ωc , and choose L˜ o such that the eigenvalues of A˜ e − L˜ oC˜ e will be all placed at −ωo. It will be shown that this method can achieve desired closed-loop performance. However, the generalized ADRC will be more complex in implementation than the original ADRC. To utilize the simple structure of the original ADRC, we can find an equivalent conventional ADRC controller for the tuned generalized ADRC controller [39]. It is noted that the feedback transfer function of a nth-order conventional LADRC (Fig. 2) is
(37)
T Now all the states z (t ) = ⎣⎡ x(t ) f (t )⎦⎤ are estimated by ESO if L˜ o is properly designed. As in the original ADRC, the generalized disturbance f can be used as a state for feeding back in order to reject it quickly. Now the states x are different from those in the original ADRC, thus the state-feedback law will be somehow different. With a properly designed ESO, the final state-feedback law we propose has the following form:
u(t ) = r^(t ) − K˜ oz^(t ) ≈ r^(t ) − K¯ ox(t ) − k˜ 0f (t )
⎧ ^̇ ⎪ z (t ) = (A˜ e − L˜ oC˜ e )z^(t ) + B˜eu(t ) + L˜ oy(t ) ⎨ ⎪ ⎩ u(t ) = r^(t ) − K˜ oz^(t )
=
where the observer gain is T L˜ o = ⎡⎣ β˜1 β˜2 ⋯ β˜n β˜n + 1⎤⎦
disturbance f is 'matching' with the control input u (so called 'matching condition' in the literature on robust control of nonlinear uncertain systems [38]). In the original ADRC, from the state- space data given in (10), Bd = B /b, so the condition (42) is always met. It is shown that ADRC works under the assumption of matching condition. Whether it is applicable to systems with unmatching condition will be further investigated. Nevertheless, ADRC is a combination of a novel control structure and a novel design philosophy. The existence of the general disturbance is assumed, the real disturbance is unknown, so just like PID control, there may be drawbacks, but it has potential applications in industrial control. In summary, for a general linear system (32), the generalized ADRC controller has the following state-space form
(42)
The condition is amount to saying that the generalized
P=
1 e−0.5s (5s − 1)(2s + 1)(0.5s + 1)
(47)
As studied in [23], the process can be approximated with a
C. Fu, W. Tan / ISA Transactions 71 (2017) 530–541
model with one unstable pole and time delay:
P¯ =
1 e−0.939s (5s − 1)(2.07s + 1)
(48)
Tan et al. [23] have proposed a modified IMC (MIMC) structure for controlling the process with three controllers:
K0 = 2(2.07s + 1), K1 =
5s + 1 , K2 = 3.584(2.4s + 1) 0.2s + 1
(49)
Rao et al. [26] have proposed a PID with a lead filter as the disturbance estimator in a two-degrees-of-freedom control scheme for controlling this process with three controllers:
1.7747 , Gc2 = 4.997 + 14.32s , s 0.98 0.47s + 1 Gc 3 = (6.20 + + 8.72s ) s 0.0198s + 1 Gc1 = 5.583 +
(50)
In order to obtain the same set-point rising speed as that of in Tan's [23] and Rao's [26] methods for comparison, the desired tracking performance is chosen as
N=
1 e−0.939s (s + 1)2
(51)
and the tracking controller is chosen as
C=
(5s − 1)(2.07s + 1) (s + 1)2
(52)
for the 2DOF-ADRC scheme. To design the disturbance rejection controller, 1st-order Pade approximation is used to approximate the delay and the full model for the generalized ADRC design is
P1 =
−s + 2.13 10.35s 3 + 24.97s 2 + 5.241s − 2.13
(53)
The resulting system is then of 3rd -order, thus a 3rd-order ADRC is used. The control and observer gains for the generalized ADRC are tuned with ωc = 20, ωo = 0.8, and the gains for the conventional ADRC can be obtained by solving (45) and (46) with b ¼ 1.
Ko = [9406.895 998.3894 56.6806 1],
(54)
T L o = ⎡⎣ 4.1063 17.5144 10.875 1.6927⎤⎦
(55)
The responses of the designed control system for a step
535
reference at t ¼ 0 s and a step input disturbance of at t ¼ 40 s are shown in Fig. 5 for 2DOF-ADRC and for modified ADRC [21] ( ωc = 0.7, ωo = 15). For comparison, the responses of the control scheme proposed in Tan's [23] and Rao's [26] are also shown in Fig. 5. It is noted that ADRC achieves a close disturbance rejection performance as the control scheme [26]. The tracking performance for modified ADRC [21] is the worst though it achieves the best nominal disturbance rejection performance. The tracking performance of 2DOF-ADRC achieves the desired. The integral absolute errors (IAE) are 3.875, 5.263, 4.451 and 4.258 respectively for the 2DOF-ADRC, modified ADRC [21], Tan's [23] and Rao's [26]. The robustness measures of the 2DOF-ADRC, modified ADRC [21], Tan's [23] and Rao's [26] are shown in Fig. 6. The robustness measure of the proposed 2DOF-ADRC is smaller at low and mid frequency ranges, which means that the proposed setting is more insensitive to low and mid frequencies uncertainty. To verify the observations from the robustness measure plot, suppose that the process time delay is actually 20% larger, the gain 20% larger, and the unstable time constant 20% smaller, which is the worst case in parameter variations. The responses for the perturbed system are shown in Fig. 7. It is clear that the proposed method has good robust stability. The integral absolute errors (IAE) are 4.021, 4.692, 5.761 and 4.23 respectively for the 2DOFADRC, modified ADRC [21], Tan's [23] and Rao's [26]. In order to show the sensor noise rejection in high-frequency range, the complementary sensitivity function T of the 2DOFADRC, modified ADRC [21], Tan's [23] and Rao's [26] are provided in Fig. 8(a). T for the method by Rao's [26] is the largest at high frequency thus the setting is very sensitive to sensor noise. The reason lies in the PID cascaded a lead structure that is adopted as the disturbance estimator in [26]. It makes T larger at high frequencies. In contrast, T for the proposed 2DOF-ADRC is smaller at high frequencies thus it will be more insensitive to sensor noise. At the same time, it can be seen that T for modified ADRC by [21] is larger at high frequency and is expected to be more sensitive to sensor noise than the proposed 2DOF-ADRC. The reason is that there is a limit on the controller bandwidth for the modified ADRC [21], and to achieve good disturbance rejection performance the observer bandwidth ωo has to be chosen large enough. To verify the observations from the complementary sensitivity plot, suppose a white sensor noise with the noise power 0.0001, and sampling period 0.01 s is introduced into the process output. Time response for the above four schemes are shown in Fig. 8(b) for the nominal parameters. It is clear that the proposed 2DOF-ADRC has good sensor noise rejection performance. Rao's [26] and modified
Fig. 5. Responses of Example 1.
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KC = 3.5671,
τI = 1.491,
τD = 1.3364,
α = 0.15, and β = 0.0058.
(59)
For the 2DOF-ADRC tuning, the desired tracking performance is chosen as follows to obtain the same set-point responses with [24,27]
N=
1 e−0.3s (0.51s + 1)2
(60)
and the tracking controller is chosen as
C=
(3s − 1)(s − 1) 2(0.51s + 1)2
(61)
To design the disturbance rejection controller, 1st-order Pade approximation is used to approximate the delay and the full model for generalized ADRC is Fig. 6. Robustness measures for Example 1.
P1 =
ADRC [21] are sensitive to sensor noise. The integral absolute errors (IAE) are 4.324, 6.153, 4.902 and 8.359 respectively for the 2DOFADRC, modified ADRC [21], Tan's [23] and Rao's [26]. Example 2.. (Process with Two Unstable Poles) Consider the process with two unstable poles studied by Liu et al. [24] and Shamsuzzoha et al. [27].
2 P= e−0.3s (3s − 1)(s − 1)
(56)
According to Liu et al.’s [24] method, the set-point tracking controller is designed as kd = 3, λc = 0.51. The tracking controller is
C=
1.5s 2 + s + 0.5 , (0.51s + 1)2
(57)
and the disturbance estimator is tuned as
F3/3 =
32.82s 3 + 439.41s 2 + 232.64s + 129.79 0.56s 3 + 0.8s 2 + 100s
.
(58)
Shamsuzzoha et al. [27] have proposed an enhanced disturbance rejection for open-loop unstable processes with time delay. The disturbance estimator by the [27] is in the form of PID with a lead compensator given by:
−2s + 13.33 3s 3 + 16s 2 − 25.67s + 6.667
(62)
The resulting system is then of 3rd-order, thus a 3rd-order ADRC is used. The control gain and observer gain for the generalized ADRC are tuned with ωc = 30, ωo = 2, and the gains for the conventional ADRC can be obtained by solving (45) and (46) with b ¼1.
Ko = [61341.2837 2914.1104 92.2411 1],
(63)
T L o = ⎡⎣ 0.4256 4.9673 2.9762 1.5846⎤⎦
(64)
The responses of the designed control system for a step reference at t ¼ 0 s and a step input disturbance at t ¼ 20 s are shown in Fig. 9 for 2DOF-ADRC and for modified ADRC [21] ( ωc = 0.7, ωo = 40). Again, there is a limit on the controller bandwidth for the modified ADRC [21], so ωo has to be chosen large enough to achieve acceptable disturbance rejection performance. For comparison, the responses of the control scheme proposed in Liu's [24] and Shamsuzzoha's [27] are also shown in Fig. 9. It is noted that Shamsuzzoha's [27] PID with a lead compensator achieves best disturbance rejection performance. The proposed 2DOF-ADRC achieves better disturbance rejection performance than Liu's [24] control scheme. The tracking performance for modified ADRC [21] is the worst. The integral absolute errors (IAE) are 1.969, 3.63, 2.092 and 1.748 respectively for the 2DOF-ADRC, modified ADRC [21], Liu's [24] and Shamsuzzoha's [27]. The robustness measures of the 2DOF-ADRC, modified ADRC [21], Liu's [24] and Shamsuzzoha's [27] are shown in Fig. 10. The
Fig. 7. Responses of Example 1 with model uncertainties.
C. Fu, W. Tan / ISA Transactions 71 (2017) 530–541
537
Fig. 8. Analysis and simulation for Example 1 with sensor noise.
Fig. 9. Responses of Example 2.
measure for the PID with a lead compensator scheme by [27] does not diminish at the high frequency thus the setting is very sensitive to high-frequency uncertainty. The robustness measure of the proposed 2DOF-ADRC is smaller at high frequency but larger at mid-frequency, which means that the proposed setting is more insensitive to high-frequency uncertainty but more sensitive to low-frequency uncertainty than Shamsuzzoha's [27]. The robust performance of the controller is tested for model mismatch by assuming simultaneous perturbation of 5% in each parameter towards the worst direction. The actual process is assumes as P˜ = 2.1e−0.315s/(2.85s − 1)(0.95s − 1). Fig. 11 shows the responses of the perturbed system for the proposed 2DOF-ADRC, modified ADRC [21], Liu's [24] and Shamsuzzoha's [27] controllers. It is clear that the proposed method has good robustness in the presence of the process uncertainty. The integral absolute errors (IAE) are 2.067, 3.618, 2.213 and 1.805 respectively for the 2DOF-ADRC, modified ADRC [21], Liu's [24] and Shamsuzzoha's [27]. In order to show the sensor noise rejection in high-frequency range, the complementary sensitivity function T of the 2DOFADRC, modified ADRC [21], Liu's [24] and Shamsuzzoha's [27] are provided in Fig. 12(a). T for the method by Shamsuzzoha's [27] is the highest at high frequencies thus the setting is very sensitive to sensor noise. T for the proposed 2DOF-ADRC is small enough at high frequencies thus it will be insensitive to sensor noise. At the
Fig. 10. Robustness measures for Example 2.
same time, it can be seen that T for modified ADRC by [21] is higher at high frequencies and is expected to be more sensitive to sensor noise than the proposed 2DOF-ADRC. To verify the observations from complementary sensitivity plot, suppose a white sensor noise with the noise power 0.0001,
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Fig. 11. Responses of Example 2 with model uncertainties.
Fig. 12. Analysis and simulation for Example 2 with sensor noise.
Fig. 13. Responses of Example 3.
and sampling period 0.01 s is introduced into the process output. The responses for the above four schemes are shown in Fig. 12(b) for the nominal parameters. It is clear that the proposed 2DOF-ADRC has good sensor noise rejection performance.
The Shamsuzzoha's [27] is very sensitive to sensor noise. The integral absolute errors (IAE) are 3.371, 4.954, 3.536 and 5.278 respectively for the 2DOF-ADRC, modified ADRC [21], Liu's [24] and Shamsuzzoha's [27].
C. Fu, W. Tan / ISA Transactions 71 (2017) 530–541
N=
539
1 e−1.2s (s + 1)2
(68)
and the tracking controller is chosen as
C=
(s − 1)(0.5s + 1) (s + 1)2
(69)
To design the disturbance rejection controller, 2nd-order Pade approximation is used to approximate the delay and the full model for generalized ADRC is
P1 =
Fig. 14. Robustness measures for Example 3.
Example 3.. (Unstable Process with Large Time Delay) The following unstable process with comparatively large time delay is studied by Liu et al. [24] and Shamsuzzoha et al. [27].
P=
(65)
In Liu et al.’s [24] structure, the set-point tracking controller is designed as kc = 2, λc = 1,and C = (0.5s2 + 0.5s + 1) /(s + 1)2 and a 5th-order disturbance estimator is given below: 36.12s 5 + 337.86s 4 + 1168.12s 3 + 1331.04s 2 + 120.05s + 1 s(2.08s 4 + 122.32s 3 + 9.84s 2 + 1331.54s + 100)
.
(66)
Shamsuzzoha et al.’s [27] used the same structure as [24] but designed the disturbance estimator as a PID with a lead compensator is given by:
KC = 1.1165, τI = 61.3412, τD = 0.4983, α = 0.6, and β = 0.0145.
3
0.5s + 3s + 5.667s 2 − 0.8333s − 8.333
(70)
The resulting system is then of 4th-order, thus a 4th-order ADRC is used. It is noted that if 1st-order Pade approximation is used we can get a 3rd-order ADRC that achieves similar performance as the 4thorder disturbance estimator in Liu's [24]. For simplicity, we compare with the 5th-order disturbance estimator of [24] since it has the better disturbance rejection performance, so we choose to show the ADRC design with 2nd-order Pade approximation. The control gain and observer gain for the generalized ADRC are tuned with ωc = 30, ωo = 0.68, and the gains for the conventional ADRC can be obtained by solving (45) and (46) with b ¼ 1.
Ko = [106.3492 212.5218 164.8019 14.6933 1],
1 e−1.2s (s − 1)(0.5s + 1)
F5/5 =
s 2 − 5s + 8.333 4
(67)
For the 2DOF-ADRC tuning, the desired tracking performance is chosen as followed to obtain the same set-point responses as in [24,27]
(71) T
Lo = [ 102.7067 3422.9873 −4854.4554 5294.1912 66.4426 ]
(72)
The responses of the designed control system for a step reference at t ¼ 0 s and a step change of load disturbance with magnitude of 0.05 to the process input at t ¼ 50 s are shown in Fig. 13 for 2DOF-ADRC. We cannot find the settings for modified ADRC [21] to control this unstable process with large time delay so it is not shown here. For comparison, the responses of the control scheme proposed in Liu's [24] and Shamsuzzoha's [27] are also shown in Fig. 13. It is noted that Shamsuzzoha's [27] PID with a lead compensator achieves the best disturbance rejection performance. The proposed 2DOF-ADRC achieves better disturbance rejection performance than Liu's [24] control scheme. The integral absolute errors (IAE) are 6.891, 8.215 and 6.396 respectively for the 2DOF-ADRC, Liu's [24] and Shamsuzzoha's [27]. The robustness measures of the 2DOF-ADRC, Liu's [24] and Shamsuzzoha's [27] are shown in Fig. 14. The measure for Liu's
Fig. 15. Responses of Example 3 with model uncertainties.
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Fig. 16. Analysis and simulation for Example 3 with sensor noise.
[24] method is smaller at high frequency but larger at mid-frequency, which means that the setting is more insensitive to highfrequency uncertainty but more sensitive to low-frequency uncertainty. The measure for Shamsuzzoha et al. [27] does not diminish at the high frequency thus the setting is very sensitive to high-frequency uncertainty. The robustness measure of the proposed 2DOF-ADRC is smaller than Liu's [24] method at mid-frequency, which means that the proposed setting is more insensitive to low-frequency uncertainty than Liu's [24]. The robust performances of the controllers are tested for model mismatch by assuming simultaneous perturbation of 5% in each parameter towards the worst direction. The actual process is assumed as P˜ = 1.05e−1.26s /(0.95s − 1)(0.475s + 1). Fig. 15 shows the responses of the perturbed system for the proposed 2DOF-ADRC, Liu's [24] and Shamsuzzoha's [27] controllers. Liu et al.'s [24] fifthorder disturbance estimator gives an unstable oscillatory response. The proposed 2DOF-ADRC method shows good performance in the perturbed case. The integral absolute errors (IAE) are 16.4 and 10.37 respectively for the 2DOF-ADRC and Shamsuzzoha's [27]. In order to show the sensor noise rejection in high-frequency range, the complementary sensitivity function T of the 2DOFADRC, Liu's [24] and Shamsuzzoha's [27] are provided in Fig. 16(a). T for the method by Shamsuzzoha's [27] is the largest at high frequencies thus the setting is very sensitive to sensor noise. T for Liu's [24] method and the proposed 2DOF-ADRC is small enough at high frequencies thus they are insensitive to sensor noise. To verify the observations from complementary sensitivity plot, suppose a white sensor noise with the noise power 0.0001, and sampling period 0.01 s is introduced into the process output. The response for the above three schemes are shown in Fig. 16(b) for the nominal parameters. It is clear that Liu's [24] method and the proposed 2DOF-ADRC has good sensor noise rejection performance. Shamsuzzoha's [27] is very sensitive to sensor noise. The integral absolute errors (IAE) are 15.27, 17.77 and 23.23 respectively for the 2DOF-ADRC, Liu's [24] and Shamsuzzoha's [27].
6. Conclusions A two-degree-of-freedom (2DOF) active disturbance rejection control (ADRC) structure was proposed for unstable time-delayed systems in this paper. Set-point tracking and disturbance rejection are separated in the structure and ADRC is solely responsible for disturbance rejection. Robustness and performance of the proposed method were analyzed. A method was proposed to tune
ADRC parameters using the full information of the processes. Simulation results showed that the proposed method can achieve good tracking and disturbance rejection performance. Though the proposed ADRC tuning method can relieve some burden in parameter tuning for time-delayed systems, it is still not quite convenient if delays are large. In this case the modified ADRC (in which the control signal is delayed by the same amount before it enters into the ESO) may be used in the proposed 2DOF-ADRC configuration. This will be investigated in the future research. Experiment tests of real systems via 2DOF-ADRC will also be investigated in the future work.
Acknowledgements This work is supported by National Nature Science Foundation of China under Grant 61573138, 61403137 and the Fundamental Research Funds for the Central Universities.
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