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Iterative tuning of modified uncertainty and disturbance estimator for time-delay processes: A data-driven approach
Accepted Manuscript Iterative tuning of modified uncertainty and disturbance estimator for time-delay processes: A data-driven approach Yi Zhang, Li Sun, Jiong Shen, Kwang Y. Lee, Qing-Chang Zhong
PII: DOI: Reference:
S0019-0578(18)30324-0 https://doi.org/10.1016/j.isatra.2018.08.028 ISATRA 2900
To appear in:
ISA Transactions
Received date : 17 December 2017 Revised date : 11 July 2018 Accepted date : 27 August 2018 Please cite this article as: Zhang Y., et al. Iterative tuning of modified uncertainty and disturbance estimator for time-delay processes: A data-driven approach. ISA Transactions (2018), https://doi.org/10.1016/j.isatra.2018.08.028 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Title page showing Author Details
Iterative Tuning of Modified Uncertainty and Disturbance Estimator for Time-Delay Processes: a Data-Driven Approach Yi Zhanga, Li Suna, Jiong Shena*, Kwang Y. Leeb, Qing-Chang Zhongc
a
Yi Zhang, Li Sun and Jiong Shen are with School of Energy and Environment,
Southeast
University,
Sipailou
#2,
Nanjing
210096,
China
(email:
[email protected], [email protected], [email protected]). b
Kwang Y. Lee is with Department of Electrical and Computer Engineering, Baylor
University, Waco, Texas 76798, USA (e-mail: [email protected]). c
Qing-Chang Zhong is with Department of Electrical and Computer Engineering,
Illinois Institute of Technology, Chicago, IL 60616 USA (e-mail: [email protected]).
*Corresponding author: Jiong Shen. Professor. Email address: [email protected] Postal address: School of Energy and Environment, Southeast University #2 Sipailou, Nanjing 210096, P.R.China. Phone:+86 25 83795951
*Highlights (for review)
Highlights:
A normalized analysis leads to a quantitative method to design the MUDE-based control system, balancing the robustness and performance.
The MUDE-based controller is automatically tuned based on the iterative feedback tuning.
The data-driven tuning reduces dependency on the process model, thus improving the system robustness and alleviating the burden of estimator.
*Blinded Manuscript - without Author Details Click here to view linked References
Iterative Tuning of Modified Uncertainty and Disturbance Estimator for Time-Delay Processes: a Data-Driven Approach Abstract: Uncertainty and disturbance widely exist in the process industry, which may deteriorate control performance if not well handled. The uncertainty and disturbance estimator (UDE) emerges as a promising solution by treating the external disturbances and internal uncertainties simultaneously as a lumped term. To overcome its limitation caused by time delay, a modified UDE (MUDE) has been proposed recently. However, its parameter tuning relies heavily on trial-and-error, thus being time-consuming in balancing the robustness and performance. To this end, this paper aims to develop an automatic tuning procedure for the MUDE-based control system. The quantitative relationship between system performance and the scaled parameters is empirically built. Iterative Feedback Tuning (IFT) is utilized to approximate the nominal model towards actual process. Through the empirical formula and optimized model, an automatic design procedure is proposed after taking into account the system robustness and output performance simultaneously. Simulation results show the superiority of the closed-loop performance over the original MUDE controllers. The experimental results validate the feasibility of the method proposed in this paper, depicting a promising prospect in the practical application. Keywords: Modified UDE, FOPDT process, iterative feedback tuning, robustness
1 Introduction Uncertainties and disturbances widely exist in industrial control systems, leading 1
to poor system performance and even instability. Recently, extensive efforts have been devoted to handling various uncertainties and disturbances, including unmodeled dynamics, parameter perturbations and external environmental disturbances. Compared with conventional robust control method, disturbance estimation and rejection techniques provide more flexibilities in dealing with these challenges without requiring a priori knowledge of uncertainties or disturbances [1]. Popular disturbance and uncertainty estimation paradigms includes time delay control (TDC) [2], disturbance observer based control (DOBC) [3, 4] and uncertainty and disturbance estimator (UDE) [5], all of which have received significant attentions in both academia and industry. Among them, UDE is recognized as a promising disturbance estimation technique due to its outstanding robustness performance with simple structure, yet easy to implement [6]. The original UDE algorithm was pioneered in [7] as an improvement of TDC based on the assumption that a continuous signal can be approximated and estimated by using a filter with appropriate bandwidth. Then the estimated signal is compensated in the control law to mitigate the influence of disturbances and uncertainties, and hence provide outstanding robust performance. Due to its distinct simplicity and efficacy, the UDE algorithm has been extensively applied to robust input-output linearization [8, 9], robust control [10, 11] and sliding-mode control [12, 13], and further extended to uncertain linear/nonlinear systems with state delays [14, 15]. The UDE-based control scheme has shown to have a potential for a wide range of applications to many practical systems, such as unmanned aerial vehicles [10], boost 2
converters [11], aero engines [16] and variable-speed wind turbines [17]. Recently, the Modified Uncertainty and Disturbance Estimator (MUDE) based control system has been proposed in [18] to extend its applicability to the systems with time delay. Compared with conventional filtered Smith Predictor (FSP) presented in [19], the proposed MUDE-based control has completely decoupled set-pointing tracking and disturbance rejection performance design and explicit stability analysis. As a model-based control algorithm, the overall performance of MUDE-based control is largely dependent on the accuracy of the model, which is usually obtained from mechanism modeling or system identification. However, with the appearance of increasingly large-scale hybrid systems, mechanism modeling is becoming challenging or even impractical and system identification is usually plagued with various disturbances and noises, therefore the overall performance is usually limited. In addition, the MUDE design parameters are usually selected through trial and error, which is rather time-consuming and sometimes risky. On this occasion, data-driven approach seems to be an ingenious alternative to directly tune the controllers based on the input and output measurement data. This has led to a variety of data-driven control methods, such as the iterative feedback tuning (IFT) [20, 21], virtual reference feedback tuning (VRFT) [22] and the correlation-based tuning (CBT) [23]. Among these methods, only the IFT optimizes the controller parameters by directly minimizing the tracking error between the actual system output and the desired trajectory, thus is the most convenient and has been widely applied in classes of nonlinear systems [24], multivariable systems [25] and time-varying systems [26]. 3
The most distinguishing contribution of IFT is that an unbiased estimate of the performance gradient can be obtained from closed-loop experiments for linear time-invariant (LTI) controllers. The IFT method is particularly appealing to industrial applications because under this scheme, both the model identification and the controller update are iterated without ever opening the control loop. In addition to simple PID controllers, the IFT algorithm is gradually applied to controllers of increasing complexity, like internal controllers [27], smith predictors [28] and DOB-based controllers [29]. Motivated by this trend, this paper aims at applying the IFT method to the MUDE-based control system design. Different from classical feedback control system, the MUDE-based control system includes a nominal model of the process, which poses a knotty challenge for this combination. In this control system, controllers can be regarded as functions of the nominal model weighted by design freedoms that directly determine system performance. To identify these functions, we utilize iterative feedback tuning method to optimize parameters of the nominal model and determine those design freedoms in terms of robustness after taking both output and robust performance into account. Therefore, once the optimal nominal model is obtained, the overall control system is automatically tuned and meanwhile possesses the predefined robustness. The main novelties and contributions of this paper are summarized as follows: 1) Quantitative performance analysis of the MUDE-based control system with respect to design freedoms is performed in a normalized way and systematic design guidelines are provided after balancing both system robustness and output 4
performance; 2) The IFT method is utilized to approximate the nominal model towards actual process, which greatly reduces the dependency of system performance on the plant model and avoids the expense of accurate system modeling; 3) The proposed data-driven tuning method alleviates the burden of MUDE for disturbance rejection in the presence of large disturbance/uncertainties, and hence enhance the overall system performance. The remainder of the paper is organized as follows: Section 2 introduces the MUDE-based control system briefly and Section 3 quantitatively analyzes the relationship between system performance and design parameters. In Section 4, the data-driven iterative tuning method is proposed to optimize the nominal model. Simulation and experimental results shown in Section 5 and 6 demonstrate the feasibility and effectiveness of the proposed iterative tuning methodology. Finally, conclusions are drawn in Section 7.
2. Overview of the MUDE-based control strategy The MUDE-based control strategy is proposed as a modification of the original UDE-based control to extend its applicability to the systems with time delay. Thus it is natural to introduce the original UDE-based control strategy first, followed by necessary explanations of the modifications. Consider an uncertain linear time invariant (LTI) system formulated as x ( A F ) x Bu (t ) d (t )
(1)
where A Rnn is the known state matrix, F R nn is an unknown state matrix, 5
B R nm is the control matrix having full column rank, x ( x1 ,..., xn )T R n is the state, u (t ) [u1 (t ),..., um (t )]T R m is the control input, and d (t ) R n is the external disturbance. The UDE-based control adopts a stable reference model to represent the desired system performance. Assume the reference model is
xm (t ) Am xm (t ) Bmc(t )
(2)
where xm (t ) R n is the reference state vector, c(t ) [c1 (t ),..., cr (t )]T is the desired command to the reference model. The control objective is to obtain a control law u (t ) to drive the state error e(t ) between reference state xm (t ) and actual state x(t ) decay to zero, and the error dynamics satisfy
e(t ) ( Am Ke )e
(3)
where K e R nn is the error feedback gain matrix, which can be determined by common control strategies such as pole placement. Combing (1)-(3) brings
Am x(t ) Bmc(t ) Ax(t ) Fx(t ) Bu (t ) d (t ) K ee(t )
(4)
If systems are described in the canonical form, the control law u (t ) can then be obtained as u (t ) B ( Am x Bm c(t ) Ax K e e Fx d (t ))
(5)
where B ( BT B)1 BT is the pseudoinverse of B. This is the accurate solution of (4) only if the following structural constraint is met: ( I BB ) ( Am x(t ) Bm c(t ) Ax(t ) Fx(t ) K e e(t ) d (t )) 0
(6)
Otherwise, it is the least-squares approximate solution of (4). By applying Laplace transforms to (5), one obtains 6
U( s ) B ( Am X BmC AX K e E ) UDE
(7)
where UDE B [ FX ( s) D( s)] represents the lumped estimation of uncertainty and disturbance. Recalling the primary idea behind the UDE that a continuous signal can be approximated and estimated by a filter with a reasonable bandwidth, the UDE term can be rewritten as
UDE B [( A sI ) X (s) BU (s)]G f (s)
(8)
where G f ( s ) is normally chosen as a low-pass filter
G f ( s)
1 1 Tf s
(9)
where Tf is the filter time constant. Thus, the lumped uncertainty and disturbance can be estimated with the aid of known state and input variables and filter G f ( s ) . As pointed out in [18], the original UDE-based control is incapable of controlling large time delay processes even with quite conservatively tuned parameters, and necessary modifications are made to extend its applicability. To describe a plant with time delay L, the original system (1) is revised as x ( A F ) x Bu (t L) d (t )
(10)
The original control law (7) is composed by three terms [30]: state feedback, error feedback and uncertainty and disturbance estimation, formulated as U ( s ) B ( Am X BmC AX ) B K e E UDE State feedback
Error feedback
(11)
estimation
Modifications are done in the three components respectively to handle the time delay: 1) To synchronize the input signals of UDE, an artificial delay is first introduced into the control signal in (8) as 7
UDEtd B [( A sI ) X (s) BU (s)e s ]G f (s)
(12)
2) Inspired by the concept of the smith predictor, a virtual plant is established to generate feedforward action to obtain fast set-point tracking performance. The virtual plant is assumed as the delay-free part of (10), that is
xv Axv Bu (t )
(13)
Therewith the original state feedback law is modified based on the difference between the virtual plant (13) and the reference model (2) uSF B ( Am xv Bm c(t ) Axv )
(14)
3) Assuming the output of the virtual plant (13) is defined as
yv (t ) xv (t )
(15)
Combing (11) the new error feedback law is derived as uFB B K e ( yv x)
(16)
Therefore the whole MUDE-based control system design is completed, with the final control law given as
u uSF uFB UDEtd
(17)
and the overall structure of the MUDE-based control system is illustrated in Fig. 1. c
B+Bm B+BGf(s)
B+(Am-A)
B+(A-sI)Gf(s) UDEtd
xv
-τs
e
Virtual Plant
uSF
: (A,B)
Artificial Delay u
Controlled Process : (A+F,B)
e-Ls
uFB +
B Ke Artificial Delay e-τs
yv
External Disturbance d _
Fig. 1. The overall structure of the MUDE-based control system.
8
x
3. Performance analysis of the MUDE-based control system with respect to three design freedoms To facilitate the subsequent analysis of the MUDE-based control system, the simplified equivalent structure is obtained after a series of dreary transformations as shown in Fig. 2. d c
Gc
u
r'
_
Independent MUDE Gp
y
+ dˆ
+ _
Gfe-τs Pn-1Gf
Q
Fig. 2. The simplified equivalent structure of the MUDE-based control system.
In Fig. 2, c, d, are the set-point, external disturbance and noise, respectively,
r ' , dˆ , u and y are the input of the independent UDE control structure (excluding controllers Gc and Q), the estimated uncertainty and disturbance, the control signal and the system output, respectively. The actual plant is described as Gp, while the nominal model of the plant is described as Gp0
k e s Pn e s Ts 1
(18)
where k is steady-state gain, T is the time constant, is the estimate of delay time L. A classical first-order-plus-dead-time (FOPDT) model is utilized here since it can represent most industrial processes. Pn is the stable delay-free part of the nominal model (18). In correspondence with the state-space expressions in Fig. 1, the feedback 9
controller Q in Fig. 2 is described as Q
T Ke k
(19)
The feedforward controller Gc is described as Gc Gm ( Pn1 Qe s )
(20)
where Gm is the transfer function of the reference model (2), 1 Tm s 1
(21)
1 1 , Bm Tm Tm
(22)
Gm
with the time-domain parameters set as Am
where Tm is the tracking time constant. Thus (20) can be rewritten as
Gc
1 Ts 1 T Ke e s Tm s 1 k k
(23)
Note that both controllers Q and Gc are functions of the nominal model Gp0, namely Q f q (G p 0 )
(24)
Gc f c (G p 0 )
(25)
where tracking time constant Tm and error feedback gain Ke are two design parameters to be determined in the functions. The inherent two-degree-of-freedom structure of the UDE-based control assures that the set-point response and the disturbance response design can be completely decoupled [31]: the set-point response is determined by the reference model Gm and
10
the disturbance response is determined by the error feedback gain Ke and the filter Gf. Although some brief guidelines have been given in the literatures for parameter tuning of the UDE-based control system, the quantitative relation between system performance and those design freedoms have never been analyzed, which gives main motivation for the work in this section. In this work, we aim to analyze the relationship between system output/robust performance and design freedoms both qualitatively and quantitatively and obtain an empirical correlation among them, which will be used for data-driven tuning in the next section. 3.1. Set-point tracking performance and Disturbance rejection performance Set-point tracking performance and disturbance rejection performance are two main considerations in a control system design. In this paper we use the integral absolute error to evaluate the output performance, which is defined as
IAE 0 y(t ) c(t ) dt 0 e(t ) dt
(26)
The IAE value under a step change in the set-point IAEsp is a numerical indicator of set-point tracking performance while the IAE value under a step change in the disturbance IAEld indicates the disturbance rejection performance. In the nominal case, the system output can be represented as
k T f s 1 e s 1 s y e c e s d s Tm s 1 Tf s 1 1 Ts KeTe
(27)
To make the deductions more general, the transformations are conducted as follows to realize the normalization of the transfer functions:
s' s 11
(28)
T f
(29)
T
(30)
Ke
=
T
, (0,1)
(31)
where s ' is the normalized parameter, and represent the normalized filter time constant and the normalized error feedback gain respectively,
is the
normalized time delay. Thus (27) can be rewritten as
y
1 Tm
e c s'
s ' 1
k s ' 1 e s ' 1
s ' 1 1
s ' e
e s ' d
(32)
s'
We can infer from (32) that the normalized IAEsp, IAEsp / only depends on the normalized tracking time constant Tm , and the normalized IAEld , IAEld / k / is related to the normalized filter time constant , the normalized error feedback gain
and the normalized time delay . Generally, a larger Tm leads to enhanced system robustness, but the response speed will be slower, and vice versa. To balance the robustness and system response speed, the tracking time constant Tm is normally chosen equal to the estimated time delay .
3.2. Robustness The sensitivity function S(s) and the complementary sensitivity function T(s) are two common numerical robustness indicators in robust control theory. In this work, the robustness index is chosen as
M st max S ( jw) , T ( jw) , w w
Similarly, the normalized sensitivity function
S ( s ')
sensitivity function can be obtained from Fig. 2 as follows:
12
(33) and complementary
1 s ' 1 s ' e s ' 1 1 S ( s ') 1 OP( s ') 1 s ' e s ' s ' 1 1
(34)
1 1 s ' s ' OP( s ') e s ' T ( s ') 1 1 OP( s ') s ' e s ' s ' 1 1
(35)
Combing (33), (34) and (35), we can find that the system robustness index Mst is dependent on the normalized filter time constant , the normalized error feedback gain and the normalized time delay . Fig. 3 shows the values of IAEld / k / and IAEsp / for three typical FOPDT processes under different robustness indices: a) lag dominant process ( 0.1 ); b) balanced process ( 0.5 ); c) delay dominant process ( 0.9 ). This figure clearly shows the relationship between the robustness/output performance and normalized design parameters: 1) A larger leads to a larger IAEld / k / with the same robustness index, which means that when the robustness index Mst is fixed, a larger leads to a poorer disturbance rejection performance. 2) For a fixed , a larger Mst corresponds with a smaller IAEld / k / , which means that the disturbance rejection performance is better. This also indicates that when the normalized filter time constant is fixed, the explicit tradeoff between robustness and disturbance rejection performance can be achieved by choosing an appropriate Mst.
13
Fig. 3 (a). The values of IAEld / k / and IAEsp / :
0.1 .
Fig. 3 (b). The values of IAEld / k / and IAEsp / :
0.5 .
14
Fig. 3 (c). The values of IAEld / k / and IAEsp / :
Fig. 4.
in terms of and Mst. 15
0.9 .
Generally, the closed-loop bandwidth 1 / T f is required to be smaller than 1/L . Meanwhile, the disturbance rejection ability is supposed to be strong as much as possible. Taking both of the two considerations into account, the normalized filter time constant is set as 1. In order to quantitatively describe the relationship among
, and Mst, abundant data pairs ( , , M st ) are produced through enumeration (see Fig. 4) and then the curve-fitting technique is utilized to acquire the empirical correlation of in terms of and Mst. The empirical correlation is as follows:
a1 M st a2 M st b1
(36)
a1 0.3989 1.967 2.0 M st 2.5 . where a2 0.7589 1.777 b 0.7601 1.056 2.671 1
3.3. Determination of the normalized design parameters On the basis of the aforementioned findings, the normalized design parameters can be determined in the following way: 1) Set the normalized filter time constant as 1. 2) After the optimal nominal model is obtained, choose estimated time delay as the tracking time constant Tm of the reference model. 3) Calculate the normalized error feedback gain according to (36) based on the obtained nominal model Gp0 and predefined robustness index Mst.
4. Iterative tuning of the nominal model in the MUDE-based control system Consider the independent MUDE control structure shown in Fig. 2, the transfer
16
functions are given as
y 1 u 1 G e s P 1G G f n f p dˆ
Gp G p (1 G f e s ) 1 G f e s r ' 1 Pn1G f G p Pn1G f d G f (e s Pn1G p ) Pn1G f G p Pn1G f
(37)
In the nominal case, namely Gp Gp 0 Pn e s , (37) can be rewritten as
y * G p 0 * u 1 dˆ * 0
G p 0 (1 G f e s ) 1 G f e s r ' G f e s Pn1G f d G f e s Pn1G f
(38)
* * where y * , u , dˆ are the output, the control input and the estimated uncertainty
and disturbance in the nominal case. Suppose = k T , then the goal is to find T
an optimal parameter vector * to approximate Gp with the aid of IFT method. To achieve this goal, when the independent MUDE control structure is only excited by r ' and , the following quadratic objective function is considered: 1 J E 2N
y (t ) yr (t ) N
2
t 1
1 2N
N
dˆ (t ) dˆr (t ) t 1
2
(39)
denotes expectation with respect to the weakly stationary noise , yr is In (39), E the desired output of the independent MUDE structure, dˆr (t ) is the desired estimated uncertainty and disturbance, N is the sampling number in each experiment. Note that the response of the estimated disturbance is also considered here. The optimal parameter vector * need to make y and dˆ behave as close to y * and
dˆ * defined in (38) as possible, therefore the definition of yr and dˆr are defined respectively as follows:
yr G p 0 r ' ˆ d r 0 The optimal parameter vector * is defined by 17
(40)
* arg min J ( )
(41)
Obviously, the optimal parameter vector * corresponds with the solution satisfying the following equation
0
1 N y (t ) yr (t ) 1 N ˆ J dˆ (t ) E y (t ) yr (t ) d (t ) i i N t 1 i i N t 1
(42)
where yr (t ) / i is defined as yr (t ) G p 0 r ' i i t
(43)
4.1. Stochastic approximation algorithm employed in the IFT In the case when the gradient J i could be computed, the solution to (42) can be obtained by the following iterative algorithm
i 1 i i R i 1
J i
(44)
where the subscript i represents the iteration number, i is a positive real scalar that determines the step size. The sequence i must obey some constraints for the algorithm to converge to a local minimum of the objective function [32]. R i is some appropriate positive definite matrix, typically chosen as a Gauss-Newton approximation of the Hessian of J: y (t ) y (t ) y (t ) y (t ) T dˆ (t ) dˆ (t ) T (45) r r i i i i i t 1 i The calculation of y (t ) / i and dˆ (t ) / i are rather intractable since it involves 1 Ri N
N
unknown noise . However, this problem can be fixed by referring to the stochastic approximation algorithm [33] that the both of the two gradients can be replaced by an unbiased estimate. In addition, it can be inferred from the IFT method that the unbiased estimate can be obtained by performing specified experiments on the 18
closed-loop system. Therefore, the next step is to design IFT experiments to obtain useful data for the calculation of the unbiased estimates of the gradients.
4.2. IFT experiments design When the dependent MUDE control structure is stimulated by external input r ' and noise , the output is obtained as
y
Gp 1 G f e s Pn1G f Gp
1 G f e s
r '
1 G f e s Pn1G f G p
(46)
Then the gradient y can be derived as 2 2 s 2 G pG f r ' G 2f Pn1G p y G p G f Pn r ' (1 G f e )G f G p Pn Pn e s 2 1 G f e s Pn1G f G p 2 1 G f e s Pn1G f G p
(47)
Combing with (37) brings Gp Gp Pn y e s G f Pn2 y G u f 1 G f e s Pn1G f G p 1 G f e s Pn1G f G p
(48)
Similarly, the estimated uncertainty and disturbance can be obtained as
dˆ
G f (e s Pn1Gp ) 1 G f e s Pn1G f Gp
r '
Pn1G f 1 G f e s Pn1G f G p
(49)
Then the gradient dˆ can be derived as 2 s 2 G f r ' Pn1G 2f dˆ G f G p Pn r ' G f (1 G f e ) Pn Pn e s 2 1 G f e s Pn1G f G p 2 1 G f e s Pn1G f G p
(50)
Combing with (37) brings
Pn dˆ 1 1 e s 2 =G f Pn y Gf u 1 G f e s Pn1G f G p 1 G f e s Pn1G f G p It is observed
that
Gp 1 Gf e
s
P G f Gp 1 n
and
1 Gf e
s
1 Pn1G f Gp
(51)
are the
transfer functions from r ' to y and from r ' to u, respectively. Inferring from this insight, the unbiased estimate of y / and dˆ / can be obtained through three 19
closed-loop experiments with specific reference signals. Let rij ' , yij , u ij and ij (j=1,2,3) denote the input signal, output signal, control signal and noise signal of the dependent MUDE control structure in the j-th experiment of i-th iteraton respectively. In each iteration, three IFT experiments are designed as follows: 1) Use the predefined exciting signal as the input of the dependent MUDE control structure ri1 ' , collect output data yi1 and the input data ui1. 2) Use yi1 as the input of the dependent MUDE control structure, namely ri 2 ' yi1 . In this experiment, we obtain
Gp
1 G f e s
i 2
(52)
Pn1G f 1 ui 2 yi1 i 2 1 G f e s Pn1G f Gp 1 G f e s Pn1G f Gp
(53)
yi 2
1 G f e s Pn1G f Gp
yi1
1 G f e s Pn1G f G p
3) Use ui1 as the input of the dependent MUDE control structure, namely ri 3 ' ui1 . In this experiment, we obtain
yi 3 ui 3
Gp
1 G f e s
i 3
(54)
Pn1G f 1 u i 3 i1 1 G f e s Pn1G f Gp 1 G f e s Pn1G f G p
(55)
1 G f e s Pn1G f Gp
ui1
1 G f e s Pn1G f Gp
Therefore, in the i-th iteration, y / i and dˆ / i can be respectively estimated as
yi1 (e s ) 2 ( Pn ) est yi 2 G f yi 3 G f Pn i i i
(56)
dˆ ( Pn ) (e s ) est i1 G f Pn2 ui 2 G f ui 3 i i i
(57)
Recalling (42), the estimation of the objective function gradient J / i can be
20
described as
ˆ yi1 yr 1 N J 1 N ˆ (t ) est di1 (t ) est y ( t ) y ( t ) est d i1 r i i N t 1 i1 i N t 1 t t i t (58)
4.3. Unbiasedness of the objective function gradient estimation In order to guarantee convergence of the iteration algorithm, the objection function gradient estimation has to be unbiased, namely J E est i
J i
(59)
To prove the unbiasedness, some assumptions are made in advance. Assumption 1: Noise ij in each experiment are random bounded with zero mean value. Assumption 2: Noise ij from different experiments are mutually independent. Proof: The estimation of the objective function gradient can be written as J E est i 1 E N
y y 1 yi1 (t ) yr (t ) i1 r t 1 i t i t N
1 E N
yi1 (t ) yr (t ) Fi 2i 2 t
N
N
t 1
dˆi1 ˆ d ( t ) i1 t 1 i t 1 N Fi 3i 3 t E dˆ (t ) H i 2i 2 t H i 3i 3 t N t 1 N
(60) where Fij (j=2,3) and Hij(j=2,3) are filters which can be calculated from (52)-(57). According to Assumption 1 and 2, we can obtain that
E y (t ) y (t ) F 0 i1 r ij ij t E dˆi1 (t ) H ijij t 0
21
(j=2,3)
(61)
Then (60) can be rewritten as
J 1 E est E N i
y y 1 yi1 (t ) yri (t ) i1 ri t 1 i t i t N N
dˆi1 ˆ d ( t ) i1 t 1 i t N
J i (62)
Therefore the unbiasedness of the objective function gradient estimation has been proved. Remark 1: The convergence of the proposed data-driven tuning method is briefly discussed here by referring to [20, 21, 29] . The necessary condition is that the closed-loop should be stable with the updated parameters in each iteration. Assume the following assumptions on the noise, the controller, the closed loop system and the step sizes of the algorithm. Let D be a convex compact subset of R 3 . Assumption 1: There exist a neighbourhood O of D such that Pn , are twice continuously differentiable with in O . Assumption 2: All elements of the gradient and the Hessian matrix of Pn and e
s
,
are stable in each iteration. Assumption 3: The independent MUDE structure is stable for D in each iteration. Consider the convergence of the stochastic algorithm, i converges to a suboptimal solution if the following conditions hold: 1): i D in each iteration. 2): E est[J i ] = J i . 22
3): The elements of the sequence i
satisfy i 0 ,
and
i 1 i
2 i 1 i
.
4): R i is a symmetric matrix and satisfies 1 I R i I , I I for some
0. The above four conditions are the conditions of the stochastic approximation algorithm. The convergence rate of the IFT algorithm strongly depends on i and
R i , the iteration step size i can be typically chosen as 0 i , where 0 is the initial step size. A line search algorithm is utilized in [34] to determine the iteration step size in order to improve parameter iteration. However, the convergence of the proposed data-driven tuning method not only depends on the convergence of the IFT algorithm, but also depends on the stability of the closed loop system in each iteration. Assumption 1-3 guarantee the system stability in each iteration. Remark 2: Although the data-driven tuning method proposed in this section is executed on a dependent MUDE control structure, it can also be applied to the MUDE-based control system. The advantages of conducting IFT experiments on the dependent MUDE control structure are twofold: 1) the robustness of the independent MUDE control structure is better than that of the whole MUDE system. This helps to maintain the stability of the closed-loop system in each iteration. 2) It is simpler to calculate the gradient of J based on the independent MUDE control structure.
4.4. Robust stability In this section, the structured singular value (SSV) is used to perform quantitative robustness analysis for the independent MUDE structure in each iteration. Suppose a 23
multiplicative perturbation structure in Fig. 5, and the uncertainty in the model is parameterized by a multiplicative uncertainty with the weight wI ( s )
s 0.2 0.5s 1
(63)
It represents a relative uncertainty of 20% at low frequencies and 200% at high frequencies, which is able to describe various unmodeled dynamics including parameter uncertainty, nonlinearity and even model order uncertainty. Assume the input and output of the perturbation are z and v, respectively, then the closed-loop control structure in Fig. 5 can be rearranged as the standard M feedback structure in Fig. 6. wI
r'
u
z
y
Gp0
+ dˆ + _
v
Gfe-τs Pn-1Gf
Fig. 5. Multiplicative perturbation structure.
z
v M
Fig. 6. M structure.
24
Fig. 7. Robust stability comparison for different
(dotted line: 1/ wI ( jw) ; solid line:
G f e s ).
It can be obtained from Fig. 5 that
z wI G f e s v
(64)
which means that M wI G f e s . Then ‘generalized small-gain theorem’ [35] is employed to yield
G f e s
1 wI
w
(65)
where w is the frequency. Fig. 7 compares the robust stability for the independent MUDE control structure with different . This picture explicitly demonstrates that a large value of is beneficial for maintaining the stability of the system in each iteration. 25
4.5. Summary of the iterative tuning procedure In this section, the implementation procedure of the proposed iterative tuning method is summarized as follows: 1) Choose the normalized filter time constant and set the iteration number i=0. 2) Select the initial parameter vector 0 k0 T0 0 T . 3) Execute three IFT experiments with the predefined input signal ri1 ' on the independent MUDE structure and collect corresponding control output and input data denoted as (yi1, ui1), (yi2, ui2), (yi3, ui3). 4) Calculate the estimates of gradients and positive definite matrix R i and update
i . 5) Judge whether i 1 i
is smaller than a predefined threshold value, if it is, turn
to step 6), otherwise turn to step 7). 6) Output i 1 as the optimal parameter vector * , set the tracking time constant Tm equal to the estimated time delay , and calculate the normalized error feedback gain with a given Mst according to (36). End the procedure. 7) Set i 1 0 , then go to step 2). To further explain the proposed iterative tuning procedure, an algorithm flow chart is presented in Fig. 8.
26
Start
choose λ
set initial parameter vector T 0 k0 T0 0
Execute three IFT experiments, and collect control input and output data (yi1, ui1), (yi2, ui2), (yi3, ui3)
dˆ y (t ) y J calculate est i1 , est i1 , r , est , Ri i i i i
i 1 0
update parameter vetor J i 1 i i R i 1 i
if i 1 i
No
Yes output *
Calculate Tm, χ
End
Fig. 8. Flowchart of the iterative tuning procedure.
5. Simulation results In order to demonstrate the feasibility and effectiveness of the iterative tuning 27
methodology of the MUDE-based control system proposed in this paper, we now present simulation results for three different processes.
5.1. Example 1: First order plus time delay process Consider a simple FOPDT process described as 1 s e (66) s 1 The excitation signal r ' is given in Fig. 9. Suppose the system is affected by G p (s)
Gaussian white noise N (0,0.052 ) . The initial parameter vector is set as
0 0.8 1.2 1.2 . The normalized filter coefficient is chosen as 1. The T
obtained optimal parameter vector is * = 1.0029 1.0068 0.9943 , which is very T
close to the real value r [1 1 1]T , hence the tracking time constant Tm is chosen equal to the estimated time delay 0.9943. The robustness index Mst is given as 2.0 and
is 0.4969 for * . According to empirical correlation, the normalized error feedback gain is calculated as 0.5737. Figs. 10 and 11 show the system actual output and the desired output in the first iteration and in the 10th iteration, respectively, and Fig. 12 compares the estimated uncertainty and disturbance in the first and the 10th iteration. It can be observed that the system actual output has approximated the desired output quite closely after 10 iterations, and the estimated uncertainty and disturbance is gradually approaching 0. The convergence of the parameters is clearly shown in Fig. 13. Since the optimal nominal model is very close to the actual process, the obtained MUDE based control system satisfies the predefined robustness index. This simulation clearly shows that the proposed iterative tuning method can amend the nominal model to approach the actual process by utilizing the closed-loop experiments data. 28
Fig. 9. Excitation signal of the experiments in Example 1.
Fig. 10. System output after the first iteration in Example 1.
29
Fig. 11. System output after 10 iterations in Example 1.
Fig. 12. Estimated uncertainty and disturbance in Example 1.
30
Fig.13. Parameter convergence diagram in Example 1.
To further verify the influence of the initial parameter vector 0 on the final optimal vector * , three additional experiments are carried out. Table 1 shows different initial parameter vectors and corresponding optimal parameter vectors, where m is the iteration number. The distance between * and r , defined as
* - r
2
, is used to estimate the performance of IFT algorithm in different
experiments. It is apparent that the distance between resulting and actual parameter vector will be smaller if the initial parameter vector is closer to the actual parameter vector. These experiments indicate that an appropriate initial nominal model is beneficial to improving the performance of the proposed tuning method.
31
Table 1 Experiments with different initial parameter vector in Example 1 Case
0
1
0.8 1.2 1.2
1.0029 1.0068 0.9943
2
0.7 1.3 1.3
3
1.5 2 1.4
4
0.5 1.5 1.5
r
m
* - r
1 1 1
10
0.0093
1.0013 0.9733 1.0152
1 1 1
10
0.0308
1.0012 0.9623 1.0243
1 1 1
10
0.0449
0.9938 0.8743 1.0989
1 1 1
10
0.1601
* T
T
T
T
T
T
T
T
T
T
T
T
2
5.2. Example 2: High-order process 1 Consider a lag-dominated third-order process described by
G p ( s)
1 ( s 1)(0.6s 1)(0.3s 1)
(67)
The simulation settings including the excitation signal r ' and noise are the same as in Example 1. The initial parameter vector is set as 0 = 1 1.3 0.3 . The T
normalized filter coefficient is chosen as 1. After 10 iterations, the parameter vector converges to * [0.9884 1.2360 0.8076]T . To demonstrate the performance improvement of the proposed iterative tuning method, both the original and the proposed MUDE based control system are compared. For the original MUDE controller, parameters are tuned based on the initial nominal model G p 0 , the tracking time constant Tm is chosen equal to the estimated time delay 0.3, and the normalized error feedback gain is calculated as 1.7486 when the robustness index Mst is 2.0. For the proposed MUDE controller, parameters are tuned based on the optimal nominal model G *p , the tracking time constant Tm is chosen equal to the estimated time delay 0.8076 and the normalized error feedback gain is calculated as 0.7407 under the same 32
robustness index. The system outputs and control signals with the original and proposed MUDE controller under a unity step set-point change together with input disturbance of magnitude 1 introduced at time 50s are given in Fig. 14. Oscillations in the original MUDE-based system in both outputs and control actions are quite severe and frequent, while after optimizing the nominal model through the proposed tuning method, both output and control signal are smooth and fast with no oscillations or overshoots. This example clearly indicates that the performance of the MUDE-based control system relies heavily on the accuracy of the process model, and the proposed iterative tuning method can provide guaranteed performance without exact process model.
Fig. 14(a). System outputs in Example 2.
33
Fig. 14(b). System control signals in Example 2
5.3. Example 3: High-order process 2 Consider a benchmark forth-order process with four equal poles described by
G p (s)
1 ( s 1) 4
(68)
The initial parameter vector is set as 0 2 2 2 . The normalized filter T
coefficient is chosen as 1. After 10 iterations, the parameter vector converges to
* [1.0149 2.4279 1.8652]T , namely
G*p 1.0149e1.8652 s (2.4279s 1) .
For
comparison purpose, three reduced models of (68) presented in [36] are utilized to compare the accuracy of the reduced-order model. Those three models are obtained from model reduction methods of AMIGO, SIMC and DRO, respectively, shown as follows: G p1 ( s)
1 e 1.42 s 2.9 s 1
34
(69)
1 e2.5 s 1.5s 1 1 G p3 (s) e1.9 s 2.1s 1 G p 2 (s)
(70) (71)
Fig. 15 shows the Nyquist diagrams of the actual process and different reduced-order models. It shows among those obtained approximated FOPTD models, the optimal nominal model obtained by the proposed method is the closest to the actual process. Example 2 and 3 indicate that although the proposed data-driven tuning algorithm are derived based on the FOPTD model, it can also be applied to some high-order systems that have analogical dynamic characteristics with FOPTD model.
Fig. 15. Nyquist diagrams of G p , G *p , G p1 , G p 2 , G p 3 in Example 3.
35
6. Real-time experiments To demonstrate the feasibility of the proposed iterative tuning method in industrial applications, real-time experiments are conducted on a Coupled Tanks setup shown in Fig. 16, which represents a chemical plant fragment. The set-up has four water tanks coupled through connecting pipes and allows for the design of different controllers and tests in real-time using Matlab and Simulink environment. Tank 2 is chosen as the experimental subject, where the controlled variable is the level of tank 2, and the manipulated variable is the voltage of the pump that pumps water into tank 2. To obtain the initial plant model, an open loop test is conducted by increasing the pump voltage from 1.75V to 2.35V, and the resulting open loop response data is shown in Fig. 17. The identified initial model is obtained as Gp 0 (s)
19.207 1.7452 s e 88.24s 1
Controller
(72)
Tank 1
Tank 3
Tank 2
Tank 4
Monitor
Fig. 16 Experimental setup
36
Fig.17 Open loop response of water level in tank 2
Fig. 18 Parameter convergence diagram in the real-time experiment
37
Fig.19 System outputs and control signals before tuning and after tuning
Table 2 Parameters setting in real-time experiments IAEsp1
IAEld
IAEsp2
(101~400s)
(401~700s)
(701~900s)
21.2586
37.3377
131.3811
29.7762
43.9854
28.6513
24.6914
23.6268
Method
Tm
Original MUDE
1.7452
1
Proposed MUDE
1.0001
1
Accordingly, the initial parameter vector is set as 0 = 19.207 88.24 1.7452 . T
Similarly, the normalized filter coefficient is selected as 1. The tracking time constant Tm is chosen equal to the estimated time delay 1.7452. The robustness index Mst is given as 2.0 and is 0.0194 for 0 . According to the empirical correlation, the normalized error feedback gain is calculated as 21.2586. After 10 iterations, the parameter vector converges to * [21.4984 102.8139 1.0001]T (shown in Fig. 18). The convergence of the parameters shows the effectiveness of the proposed iterative 38
tuning method. Then the obtained optimized model can be described as
(21.4984 /102.8139s 1)e1.0001s . The output response of both initial model and optimized model under the same test input are also shown in Fig. 17 (dashed line and dotted line respectively). It is obvious that after 10 iterations, the optimized model output response is almost the same as the actual experimental output. For the proposed MUDE-based control system, the normalized filter coefficient is selected as 1. The tracking time constant Tm is chosen equal to the estimated time delay 1.0001 and the normalized error feedback gain is calculated as 43.9854. The parameters of the original MUDE and proposed MUDE are summarized in Table 2. Fig. 19 shows experimental outputs and control signals of the MUDE-based control system before and after tuning respectively, with set-point change of 3 and -3 at t=100s and at t=700s respectively and an input disturbance added to the system at t=400s. It is clear that the set-point tracking performance after tuning is much faster than that of before tuning and hence overshoots and actuator windup are observed in the output and control signal respectively. Moreover, the disturbance rejection performance is also significantly improved with less overshoots and settling time. The indices IAEsp and IAEld during each time period are also calculated in Table 2 and an apparent decrease can be seen in both IAEsp and especially IAEld after applying the proposed iterative tuning method. The comparative results indicate that under the same robustness index, the proposed iterative tuning method provides an obviously better disturbance rejection performance while the set-point tracking performance is also improved. The real-time experiments demonstrate the effectiveness of proposed 39
methodology and its application prospect to industrial cases.
7. Conclusions In this paper, we propose a data-driven iterative tuning method for the MUDE-based control system with the aid of IFT algorithm. This paper aims to realize automatic parameter tuning of the MUDE-based control system even in the case of inaccurate plant model while satisfying a predefined robustness index. To achieve this, quantitative performance analysis with respect to design freedoms is first conducted and an empirical correlation of system robustness index is derived to describe the relationship among them. Then the IFT algorithm is used to approximate nominal model towards actual process through conducting specific closed-loop experiments iteratively. Based on the obtained optimized nominal model and empirical correlation, system design parameters are determined. The proposed data-driven iterative tuning method provides a systematic automatic design procedure for the MUDE-based control system after balancing both system output performance and robustness. It largely reduces the dependency of system performance on the plant model and avoids the expense of accurate system modeling. Numerical examples have demonstrated the improved set-point tracking and disturbance rejection performance after optimization and the real-time experiments have proved its feasibility and prospect in industrial applications.
40
Acknowledgements This work is supported by National Nature Science Foundation of China under Grant 51576041, the Natural Science Foundation of Jiangsu Province, China under Grant BK20170686, and the open funding of the state key lab for power systems, Tsinghua University under Grant SKLD17MK11. The authors would like to give our sincere appreciation to the anonymous reviewers for their careful review and valuable suggestion. Special thanks goes to Dr. Donghai Li at Tsinghua University for his support in the experimental validation.
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PII: DOI: Reference:
S0019-0578(18)30324-0 https://doi.org/10.1016/j.isatra.2018.08.028 ISATRA 2900
To appear in:
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Received date : 17 December 2017 Revised date : 11 July 2018 Accepted date : 27 August 2018 Please cite this article as: Zhang Y., et al. Iterative tuning of modified uncertainty and disturbance estimator for time-delay processes: A data-driven approach. ISA Transactions (2018), https://doi.org/10.1016/j.isatra.2018.08.028 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Title page showing Author Details
Iterative Tuning of Modified Uncertainty and Disturbance Estimator for Time-Delay Processes: a Data-Driven Approach Yi Zhanga, Li Suna, Jiong Shena*, Kwang Y. Leeb, Qing-Chang Zhongc
a
Yi Zhang, Li Sun and Jiong Shen are with School of Energy and Environment,
Southeast
University,
Sipailou
#2,
Nanjing
210096,
China
(email:
[email protected], [email protected], [email protected]). b
Kwang Y. Lee is with Department of Electrical and Computer Engineering, Baylor
University, Waco, Texas 76798, USA (e-mail: [email protected]). c
Qing-Chang Zhong is with Department of Electrical and Computer Engineering,
Illinois Institute of Technology, Chicago, IL 60616 USA (e-mail: [email protected]).
*Corresponding author: Jiong Shen. Professor. Email address: [email protected] Postal address: School of Energy and Environment, Southeast University #2 Sipailou, Nanjing 210096, P.R.China. Phone:+86 25 83795951
*Highlights (for review)
Highlights:
A normalized analysis leads to a quantitative method to design the MUDE-based control system, balancing the robustness and performance.
The MUDE-based controller is automatically tuned based on the iterative feedback tuning.
The data-driven tuning reduces dependency on the process model, thus improving the system robustness and alleviating the burden of estimator.
*Blinded Manuscript - without Author Details Click here to view linked References
Iterative Tuning of Modified Uncertainty and Disturbance Estimator for Time-Delay Processes: a Data-Driven Approach Abstract: Uncertainty and disturbance widely exist in the process industry, which may deteriorate control performance if not well handled. The uncertainty and disturbance estimator (UDE) emerges as a promising solution by treating the external disturbances and internal uncertainties simultaneously as a lumped term. To overcome its limitation caused by time delay, a modified UDE (MUDE) has been proposed recently. However, its parameter tuning relies heavily on trial-and-error, thus being time-consuming in balancing the robustness and performance. To this end, this paper aims to develop an automatic tuning procedure for the MUDE-based control system. The quantitative relationship between system performance and the scaled parameters is empirically built. Iterative Feedback Tuning (IFT) is utilized to approximate the nominal model towards actual process. Through the empirical formula and optimized model, an automatic design procedure is proposed after taking into account the system robustness and output performance simultaneously. Simulation results show the superiority of the closed-loop performance over the original MUDE controllers. The experimental results validate the feasibility of the method proposed in this paper, depicting a promising prospect in the practical application. Keywords: Modified UDE, FOPDT process, iterative feedback tuning, robustness
1 Introduction Uncertainties and disturbances widely exist in industrial control systems, leading 1
to poor system performance and even instability. Recently, extensive efforts have been devoted to handling various uncertainties and disturbances, including unmodeled dynamics, parameter perturbations and external environmental disturbances. Compared with conventional robust control method, disturbance estimation and rejection techniques provide more flexibilities in dealing with these challenges without requiring a priori knowledge of uncertainties or disturbances [1]. Popular disturbance and uncertainty estimation paradigms includes time delay control (TDC) [2], disturbance observer based control (DOBC) [3, 4] and uncertainty and disturbance estimator (UDE) [5], all of which have received significant attentions in both academia and industry. Among them, UDE is recognized as a promising disturbance estimation technique due to its outstanding robustness performance with simple structure, yet easy to implement [6]. The original UDE algorithm was pioneered in [7] as an improvement of TDC based on the assumption that a continuous signal can be approximated and estimated by using a filter with appropriate bandwidth. Then the estimated signal is compensated in the control law to mitigate the influence of disturbances and uncertainties, and hence provide outstanding robust performance. Due to its distinct simplicity and efficacy, the UDE algorithm has been extensively applied to robust input-output linearization [8, 9], robust control [10, 11] and sliding-mode control [12, 13], and further extended to uncertain linear/nonlinear systems with state delays [14, 15]. The UDE-based control scheme has shown to have a potential for a wide range of applications to many practical systems, such as unmanned aerial vehicles [10], boost 2
converters [11], aero engines [16] and variable-speed wind turbines [17]. Recently, the Modified Uncertainty and Disturbance Estimator (MUDE) based control system has been proposed in [18] to extend its applicability to the systems with time delay. Compared with conventional filtered Smith Predictor (FSP) presented in [19], the proposed MUDE-based control has completely decoupled set-pointing tracking and disturbance rejection performance design and explicit stability analysis. As a model-based control algorithm, the overall performance of MUDE-based control is largely dependent on the accuracy of the model, which is usually obtained from mechanism modeling or system identification. However, with the appearance of increasingly large-scale hybrid systems, mechanism modeling is becoming challenging or even impractical and system identification is usually plagued with various disturbances and noises, therefore the overall performance is usually limited. In addition, the MUDE design parameters are usually selected through trial and error, which is rather time-consuming and sometimes risky. On this occasion, data-driven approach seems to be an ingenious alternative to directly tune the controllers based on the input and output measurement data. This has led to a variety of data-driven control methods, such as the iterative feedback tuning (IFT) [20, 21], virtual reference feedback tuning (VRFT) [22] and the correlation-based tuning (CBT) [23]. Among these methods, only the IFT optimizes the controller parameters by directly minimizing the tracking error between the actual system output and the desired trajectory, thus is the most convenient and has been widely applied in classes of nonlinear systems [24], multivariable systems [25] and time-varying systems [26]. 3
The most distinguishing contribution of IFT is that an unbiased estimate of the performance gradient can be obtained from closed-loop experiments for linear time-invariant (LTI) controllers. The IFT method is particularly appealing to industrial applications because under this scheme, both the model identification and the controller update are iterated without ever opening the control loop. In addition to simple PID controllers, the IFT algorithm is gradually applied to controllers of increasing complexity, like internal controllers [27], smith predictors [28] and DOB-based controllers [29]. Motivated by this trend, this paper aims at applying the IFT method to the MUDE-based control system design. Different from classical feedback control system, the MUDE-based control system includes a nominal model of the process, which poses a knotty challenge for this combination. In this control system, controllers can be regarded as functions of the nominal model weighted by design freedoms that directly determine system performance. To identify these functions, we utilize iterative feedback tuning method to optimize parameters of the nominal model and determine those design freedoms in terms of robustness after taking both output and robust performance into account. Therefore, once the optimal nominal model is obtained, the overall control system is automatically tuned and meanwhile possesses the predefined robustness. The main novelties and contributions of this paper are summarized as follows: 1) Quantitative performance analysis of the MUDE-based control system with respect to design freedoms is performed in a normalized way and systematic design guidelines are provided after balancing both system robustness and output 4
performance; 2) The IFT method is utilized to approximate the nominal model towards actual process, which greatly reduces the dependency of system performance on the plant model and avoids the expense of accurate system modeling; 3) The proposed data-driven tuning method alleviates the burden of MUDE for disturbance rejection in the presence of large disturbance/uncertainties, and hence enhance the overall system performance. The remainder of the paper is organized as follows: Section 2 introduces the MUDE-based control system briefly and Section 3 quantitatively analyzes the relationship between system performance and design parameters. In Section 4, the data-driven iterative tuning method is proposed to optimize the nominal model. Simulation and experimental results shown in Section 5 and 6 demonstrate the feasibility and effectiveness of the proposed iterative tuning methodology. Finally, conclusions are drawn in Section 7.
2. Overview of the MUDE-based control strategy The MUDE-based control strategy is proposed as a modification of the original UDE-based control to extend its applicability to the systems with time delay. Thus it is natural to introduce the original UDE-based control strategy first, followed by necessary explanations of the modifications. Consider an uncertain linear time invariant (LTI) system formulated as x ( A F ) x Bu (t ) d (t )
(1)
where A Rnn is the known state matrix, F R nn is an unknown state matrix, 5
B R nm is the control matrix having full column rank, x ( x1 ,..., xn )T R n is the state, u (t ) [u1 (t ),..., um (t )]T R m is the control input, and d (t ) R n is the external disturbance. The UDE-based control adopts a stable reference model to represent the desired system performance. Assume the reference model is
xm (t ) Am xm (t ) Bmc(t )
(2)
where xm (t ) R n is the reference state vector, c(t ) [c1 (t ),..., cr (t )]T is the desired command to the reference model. The control objective is to obtain a control law u (t ) to drive the state error e(t ) between reference state xm (t ) and actual state x(t ) decay to zero, and the error dynamics satisfy
e(t ) ( Am Ke )e
(3)
where K e R nn is the error feedback gain matrix, which can be determined by common control strategies such as pole placement. Combing (1)-(3) brings
Am x(t ) Bmc(t ) Ax(t ) Fx(t ) Bu (t ) d (t ) K ee(t )
(4)
If systems are described in the canonical form, the control law u (t ) can then be obtained as u (t ) B ( Am x Bm c(t ) Ax K e e Fx d (t ))
(5)
where B ( BT B)1 BT is the pseudoinverse of B. This is the accurate solution of (4) only if the following structural constraint is met: ( I BB ) ( Am x(t ) Bm c(t ) Ax(t ) Fx(t ) K e e(t ) d (t )) 0
(6)
Otherwise, it is the least-squares approximate solution of (4). By applying Laplace transforms to (5), one obtains 6
U( s ) B ( Am X BmC AX K e E ) UDE
(7)
where UDE B [ FX ( s) D( s)] represents the lumped estimation of uncertainty and disturbance. Recalling the primary idea behind the UDE that a continuous signal can be approximated and estimated by a filter with a reasonable bandwidth, the UDE term can be rewritten as
UDE B [( A sI ) X (s) BU (s)]G f (s)
(8)
where G f ( s ) is normally chosen as a low-pass filter
G f ( s)
1 1 Tf s
(9)
where Tf is the filter time constant. Thus, the lumped uncertainty and disturbance can be estimated with the aid of known state and input variables and filter G f ( s ) . As pointed out in [18], the original UDE-based control is incapable of controlling large time delay processes even with quite conservatively tuned parameters, and necessary modifications are made to extend its applicability. To describe a plant with time delay L, the original system (1) is revised as x ( A F ) x Bu (t L) d (t )
(10)
The original control law (7) is composed by three terms [30]: state feedback, error feedback and uncertainty and disturbance estimation, formulated as U ( s ) B ( Am X BmC AX ) B K e E UDE State feedback
Error feedback
(11)
estimation
Modifications are done in the three components respectively to handle the time delay: 1) To synchronize the input signals of UDE, an artificial delay is first introduced into the control signal in (8) as 7
UDEtd B [( A sI ) X (s) BU (s)e s ]G f (s)
(12)
2) Inspired by the concept of the smith predictor, a virtual plant is established to generate feedforward action to obtain fast set-point tracking performance. The virtual plant is assumed as the delay-free part of (10), that is
xv Axv Bu (t )
(13)
Therewith the original state feedback law is modified based on the difference between the virtual plant (13) and the reference model (2) uSF B ( Am xv Bm c(t ) Axv )
(14)
3) Assuming the output of the virtual plant (13) is defined as
yv (t ) xv (t )
(15)
Combing (11) the new error feedback law is derived as uFB B K e ( yv x)
(16)
Therefore the whole MUDE-based control system design is completed, with the final control law given as
u uSF uFB UDEtd
(17)
and the overall structure of the MUDE-based control system is illustrated in Fig. 1. c
B+Bm B+BGf(s)
B+(Am-A)
B+(A-sI)Gf(s) UDEtd
xv
-τs
e
Virtual Plant
uSF
: (A,B)
Artificial Delay u
Controlled Process : (A+F,B)
e-Ls
uFB +
B Ke Artificial Delay e-τs
yv
External Disturbance d _
Fig. 1. The overall structure of the MUDE-based control system.
8
x
3. Performance analysis of the MUDE-based control system with respect to three design freedoms To facilitate the subsequent analysis of the MUDE-based control system, the simplified equivalent structure is obtained after a series of dreary transformations as shown in Fig. 2. d c
Gc
u
r'
_
Independent MUDE Gp
y
+ dˆ
+ _
Gfe-τs Pn-1Gf
Q
Fig. 2. The simplified equivalent structure of the MUDE-based control system.
In Fig. 2, c, d, are the set-point, external disturbance and noise, respectively,
r ' , dˆ , u and y are the input of the independent UDE control structure (excluding controllers Gc and Q), the estimated uncertainty and disturbance, the control signal and the system output, respectively. The actual plant is described as Gp, while the nominal model of the plant is described as Gp0
k e s Pn e s Ts 1
(18)
where k is steady-state gain, T is the time constant, is the estimate of delay time L. A classical first-order-plus-dead-time (FOPDT) model is utilized here since it can represent most industrial processes. Pn is the stable delay-free part of the nominal model (18). In correspondence with the state-space expressions in Fig. 1, the feedback 9
controller Q in Fig. 2 is described as Q
T Ke k
(19)
The feedforward controller Gc is described as Gc Gm ( Pn1 Qe s )
(20)
where Gm is the transfer function of the reference model (2), 1 Tm s 1
(21)
1 1 , Bm Tm Tm
(22)
Gm
with the time-domain parameters set as Am
where Tm is the tracking time constant. Thus (20) can be rewritten as
Gc
1 Ts 1 T Ke e s Tm s 1 k k
(23)
Note that both controllers Q and Gc are functions of the nominal model Gp0, namely Q f q (G p 0 )
(24)
Gc f c (G p 0 )
(25)
where tracking time constant Tm and error feedback gain Ke are two design parameters to be determined in the functions. The inherent two-degree-of-freedom structure of the UDE-based control assures that the set-point response and the disturbance response design can be completely decoupled [31]: the set-point response is determined by the reference model Gm and
10
the disturbance response is determined by the error feedback gain Ke and the filter Gf. Although some brief guidelines have been given in the literatures for parameter tuning of the UDE-based control system, the quantitative relation between system performance and those design freedoms have never been analyzed, which gives main motivation for the work in this section. In this work, we aim to analyze the relationship between system output/robust performance and design freedoms both qualitatively and quantitatively and obtain an empirical correlation among them, which will be used for data-driven tuning in the next section. 3.1. Set-point tracking performance and Disturbance rejection performance Set-point tracking performance and disturbance rejection performance are two main considerations in a control system design. In this paper we use the integral absolute error to evaluate the output performance, which is defined as
IAE 0 y(t ) c(t ) dt 0 e(t ) dt
(26)
The IAE value under a step change in the set-point IAEsp is a numerical indicator of set-point tracking performance while the IAE value under a step change in the disturbance IAEld indicates the disturbance rejection performance. In the nominal case, the system output can be represented as
k T f s 1 e s 1 s y e c e s d s Tm s 1 Tf s 1 1 Ts KeTe
(27)
To make the deductions more general, the transformations are conducted as follows to realize the normalization of the transfer functions:
s' s 11
(28)
T f
(29)
T
(30)
Ke
=
T
, (0,1)
(31)
where s ' is the normalized parameter, and represent the normalized filter time constant and the normalized error feedback gain respectively,
is the
normalized time delay. Thus (27) can be rewritten as
y
1 Tm
e c s'
s ' 1
k s ' 1 e s ' 1
s ' 1 1
s ' e
e s ' d
(32)
s'
We can infer from (32) that the normalized IAEsp, IAEsp / only depends on the normalized tracking time constant Tm , and the normalized IAEld , IAEld / k / is related to the normalized filter time constant , the normalized error feedback gain
and the normalized time delay . Generally, a larger Tm leads to enhanced system robustness, but the response speed will be slower, and vice versa. To balance the robustness and system response speed, the tracking time constant Tm is normally chosen equal to the estimated time delay .
3.2. Robustness The sensitivity function S(s) and the complementary sensitivity function T(s) are two common numerical robustness indicators in robust control theory. In this work, the robustness index is chosen as
M st max S ( jw) , T ( jw) , w w
Similarly, the normalized sensitivity function
S ( s ')
sensitivity function can be obtained from Fig. 2 as follows:
12
(33) and complementary
1 s ' 1 s ' e s ' 1 1 S ( s ') 1 OP( s ') 1 s ' e s ' s ' 1 1
(34)
1 1 s ' s ' OP( s ') e s ' T ( s ') 1 1 OP( s ') s ' e s ' s ' 1 1
(35)
Combing (33), (34) and (35), we can find that the system robustness index Mst is dependent on the normalized filter time constant , the normalized error feedback gain and the normalized time delay . Fig. 3 shows the values of IAEld / k / and IAEsp / for three typical FOPDT processes under different robustness indices: a) lag dominant process ( 0.1 ); b) balanced process ( 0.5 ); c) delay dominant process ( 0.9 ). This figure clearly shows the relationship between the robustness/output performance and normalized design parameters: 1) A larger leads to a larger IAEld / k / with the same robustness index, which means that when the robustness index Mst is fixed, a larger leads to a poorer disturbance rejection performance. 2) For a fixed , a larger Mst corresponds with a smaller IAEld / k / , which means that the disturbance rejection performance is better. This also indicates that when the normalized filter time constant is fixed, the explicit tradeoff between robustness and disturbance rejection performance can be achieved by choosing an appropriate Mst.
13
Fig. 3 (a). The values of IAEld / k / and IAEsp / :
0.1 .
Fig. 3 (b). The values of IAEld / k / and IAEsp / :
0.5 .
14
Fig. 3 (c). The values of IAEld / k / and IAEsp / :
Fig. 4.
in terms of and Mst. 15
0.9 .
Generally, the closed-loop bandwidth 1 / T f is required to be smaller than 1/L . Meanwhile, the disturbance rejection ability is supposed to be strong as much as possible. Taking both of the two considerations into account, the normalized filter time constant is set as 1. In order to quantitatively describe the relationship among
, and Mst, abundant data pairs ( , , M st ) are produced through enumeration (see Fig. 4) and then the curve-fitting technique is utilized to acquire the empirical correlation of in terms of and Mst. The empirical correlation is as follows:
a1 M st a2 M st b1
(36)
a1 0.3989 1.967 2.0 M st 2.5 . where a2 0.7589 1.777 b 0.7601 1.056 2.671 1
3.3. Determination of the normalized design parameters On the basis of the aforementioned findings, the normalized design parameters can be determined in the following way: 1) Set the normalized filter time constant as 1. 2) After the optimal nominal model is obtained, choose estimated time delay as the tracking time constant Tm of the reference model. 3) Calculate the normalized error feedback gain according to (36) based on the obtained nominal model Gp0 and predefined robustness index Mst.
4. Iterative tuning of the nominal model in the MUDE-based control system Consider the independent MUDE control structure shown in Fig. 2, the transfer
16
functions are given as
y 1 u 1 G e s P 1G G f n f p dˆ
Gp G p (1 G f e s ) 1 G f e s r ' 1 Pn1G f G p Pn1G f d G f (e s Pn1G p ) Pn1G f G p Pn1G f
(37)
In the nominal case, namely Gp Gp 0 Pn e s , (37) can be rewritten as
y * G p 0 * u 1 dˆ * 0
G p 0 (1 G f e s ) 1 G f e s r ' G f e s Pn1G f d G f e s Pn1G f
(38)
* * where y * , u , dˆ are the output, the control input and the estimated uncertainty
and disturbance in the nominal case. Suppose = k T , then the goal is to find T
an optimal parameter vector * to approximate Gp with the aid of IFT method. To achieve this goal, when the independent MUDE control structure is only excited by r ' and , the following quadratic objective function is considered: 1 J E 2N
y (t ) yr (t ) N
2
t 1
1 2N
N
dˆ (t ) dˆr (t ) t 1
2
(39)
denotes expectation with respect to the weakly stationary noise , yr is In (39), E the desired output of the independent MUDE structure, dˆr (t ) is the desired estimated uncertainty and disturbance, N is the sampling number in each experiment. Note that the response of the estimated disturbance is also considered here. The optimal parameter vector * need to make y and dˆ behave as close to y * and
dˆ * defined in (38) as possible, therefore the definition of yr and dˆr are defined respectively as follows:
yr G p 0 r ' ˆ d r 0 The optimal parameter vector * is defined by 17
(40)
* arg min J ( )
(41)
Obviously, the optimal parameter vector * corresponds with the solution satisfying the following equation
0
1 N y (t ) yr (t ) 1 N ˆ J dˆ (t ) E y (t ) yr (t ) d (t ) i i N t 1 i i N t 1
(42)
where yr (t ) / i is defined as yr (t ) G p 0 r ' i i t
(43)
4.1. Stochastic approximation algorithm employed in the IFT In the case when the gradient J i could be computed, the solution to (42) can be obtained by the following iterative algorithm
i 1 i i R i 1
J i
(44)
where the subscript i represents the iteration number, i is a positive real scalar that determines the step size. The sequence i must obey some constraints for the algorithm to converge to a local minimum of the objective function [32]. R i is some appropriate positive definite matrix, typically chosen as a Gauss-Newton approximation of the Hessian of J: y (t ) y (t ) y (t ) y (t ) T dˆ (t ) dˆ (t ) T (45) r r i i i i i t 1 i The calculation of y (t ) / i and dˆ (t ) / i are rather intractable since it involves 1 Ri N
N
unknown noise . However, this problem can be fixed by referring to the stochastic approximation algorithm [33] that the both of the two gradients can be replaced by an unbiased estimate. In addition, it can be inferred from the IFT method that the unbiased estimate can be obtained by performing specified experiments on the 18
closed-loop system. Therefore, the next step is to design IFT experiments to obtain useful data for the calculation of the unbiased estimates of the gradients.
4.2. IFT experiments design When the dependent MUDE control structure is stimulated by external input r ' and noise , the output is obtained as
y
Gp 1 G f e s Pn1G f Gp
1 G f e s
r '
1 G f e s Pn1G f G p
(46)
Then the gradient y can be derived as 2 2 s 2 G pG f r ' G 2f Pn1G p y G p G f Pn r ' (1 G f e )G f G p Pn Pn e s 2 1 G f e s Pn1G f G p 2 1 G f e s Pn1G f G p
(47)
Combing with (37) brings Gp Gp Pn y e s G f Pn2 y G u f 1 G f e s Pn1G f G p 1 G f e s Pn1G f G p
(48)
Similarly, the estimated uncertainty and disturbance can be obtained as
dˆ
G f (e s Pn1Gp ) 1 G f e s Pn1G f Gp
r '
Pn1G f 1 G f e s Pn1G f G p
(49)
Then the gradient dˆ can be derived as 2 s 2 G f r ' Pn1G 2f dˆ G f G p Pn r ' G f (1 G f e ) Pn Pn e s 2 1 G f e s Pn1G f G p 2 1 G f e s Pn1G f G p
(50)
Combing with (37) brings
Pn dˆ 1 1 e s 2 =G f Pn y Gf u 1 G f e s Pn1G f G p 1 G f e s Pn1G f G p It is observed
that
Gp 1 Gf e
s
P G f Gp 1 n
and
1 Gf e
s
1 Pn1G f Gp
(51)
are the
transfer functions from r ' to y and from r ' to u, respectively. Inferring from this insight, the unbiased estimate of y / and dˆ / can be obtained through three 19
closed-loop experiments with specific reference signals. Let rij ' , yij , u ij and ij (j=1,2,3) denote the input signal, output signal, control signal and noise signal of the dependent MUDE control structure in the j-th experiment of i-th iteraton respectively. In each iteration, three IFT experiments are designed as follows: 1) Use the predefined exciting signal as the input of the dependent MUDE control structure ri1 ' , collect output data yi1 and the input data ui1. 2) Use yi1 as the input of the dependent MUDE control structure, namely ri 2 ' yi1 . In this experiment, we obtain
Gp
1 G f e s
i 2
(52)
Pn1G f 1 ui 2 yi1 i 2 1 G f e s Pn1G f Gp 1 G f e s Pn1G f Gp
(53)
yi 2
1 G f e s Pn1G f Gp
yi1
1 G f e s Pn1G f G p
3) Use ui1 as the input of the dependent MUDE control structure, namely ri 3 ' ui1 . In this experiment, we obtain
yi 3 ui 3
Gp
1 G f e s
i 3
(54)
Pn1G f 1 u i 3 i1 1 G f e s Pn1G f Gp 1 G f e s Pn1G f G p
(55)
1 G f e s Pn1G f Gp
ui1
1 G f e s Pn1G f Gp
Therefore, in the i-th iteration, y / i and dˆ / i can be respectively estimated as
yi1 (e s ) 2 ( Pn ) est yi 2 G f yi 3 G f Pn i i i
(56)
dˆ ( Pn ) (e s ) est i1 G f Pn2 ui 2 G f ui 3 i i i
(57)
Recalling (42), the estimation of the objective function gradient J / i can be
20
described as
ˆ yi1 yr 1 N J 1 N ˆ (t ) est di1 (t ) est y ( t ) y ( t ) est d i1 r i i N t 1 i1 i N t 1 t t i t (58)
4.3. Unbiasedness of the objective function gradient estimation In order to guarantee convergence of the iteration algorithm, the objection function gradient estimation has to be unbiased, namely J E est i
J i
(59)
To prove the unbiasedness, some assumptions are made in advance. Assumption 1: Noise ij in each experiment are random bounded with zero mean value. Assumption 2: Noise ij from different experiments are mutually independent. Proof: The estimation of the objective function gradient can be written as J E est i 1 E N
y y 1 yi1 (t ) yr (t ) i1 r t 1 i t i t N
1 E N
yi1 (t ) yr (t ) Fi 2i 2 t
N
N
t 1
dˆi1 ˆ d ( t ) i1 t 1 i t 1 N Fi 3i 3 t E dˆ (t ) H i 2i 2 t H i 3i 3 t N t 1 N
(60) where Fij (j=2,3) and Hij(j=2,3) are filters which can be calculated from (52)-(57). According to Assumption 1 and 2, we can obtain that
E y (t ) y (t ) F 0 i1 r ij ij t E dˆi1 (t ) H ijij t 0
21
(j=2,3)
(61)
Then (60) can be rewritten as
J 1 E est E N i
y y 1 yi1 (t ) yri (t ) i1 ri t 1 i t i t N N
dˆi1 ˆ d ( t ) i1 t 1 i t N
J i (62)
Therefore the unbiasedness of the objective function gradient estimation has been proved. Remark 1: The convergence of the proposed data-driven tuning method is briefly discussed here by referring to [20, 21, 29] . The necessary condition is that the closed-loop should be stable with the updated parameters in each iteration. Assume the following assumptions on the noise, the controller, the closed loop system and the step sizes of the algorithm. Let D be a convex compact subset of R 3 . Assumption 1: There exist a neighbourhood O of D such that Pn , are twice continuously differentiable with in O . Assumption 2: All elements of the gradient and the Hessian matrix of Pn and e
s
,
are stable in each iteration. Assumption 3: The independent MUDE structure is stable for D in each iteration. Consider the convergence of the stochastic algorithm, i converges to a suboptimal solution if the following conditions hold: 1): i D in each iteration. 2): E est[J i ] = J i . 22
3): The elements of the sequence i
satisfy i 0 ,
and
i 1 i
2 i 1 i
.
4): R i is a symmetric matrix and satisfies 1 I R i I , I I for some
0. The above four conditions are the conditions of the stochastic approximation algorithm. The convergence rate of the IFT algorithm strongly depends on i and
R i , the iteration step size i can be typically chosen as 0 i , where 0 is the initial step size. A line search algorithm is utilized in [34] to determine the iteration step size in order to improve parameter iteration. However, the convergence of the proposed data-driven tuning method not only depends on the convergence of the IFT algorithm, but also depends on the stability of the closed loop system in each iteration. Assumption 1-3 guarantee the system stability in each iteration. Remark 2: Although the data-driven tuning method proposed in this section is executed on a dependent MUDE control structure, it can also be applied to the MUDE-based control system. The advantages of conducting IFT experiments on the dependent MUDE control structure are twofold: 1) the robustness of the independent MUDE control structure is better than that of the whole MUDE system. This helps to maintain the stability of the closed-loop system in each iteration. 2) It is simpler to calculate the gradient of J based on the independent MUDE control structure.
4.4. Robust stability In this section, the structured singular value (SSV) is used to perform quantitative robustness analysis for the independent MUDE structure in each iteration. Suppose a 23
multiplicative perturbation structure in Fig. 5, and the uncertainty in the model is parameterized by a multiplicative uncertainty with the weight wI ( s )
s 0.2 0.5s 1
(63)
It represents a relative uncertainty of 20% at low frequencies and 200% at high frequencies, which is able to describe various unmodeled dynamics including parameter uncertainty, nonlinearity and even model order uncertainty. Assume the input and output of the perturbation are z and v, respectively, then the closed-loop control structure in Fig. 5 can be rearranged as the standard M feedback structure in Fig. 6. wI
r'
u
z
y
Gp0
+ dˆ + _
v
Gfe-τs Pn-1Gf
Fig. 5. Multiplicative perturbation structure.
z
v M
Fig. 6. M structure.
24
Fig. 7. Robust stability comparison for different
(dotted line: 1/ wI ( jw) ; solid line:
G f e s ).
It can be obtained from Fig. 5 that
z wI G f e s v
(64)
which means that M wI G f e s . Then ‘generalized small-gain theorem’ [35] is employed to yield
G f e s
1 wI
w
(65)
where w is the frequency. Fig. 7 compares the robust stability for the independent MUDE control structure with different . This picture explicitly demonstrates that a large value of is beneficial for maintaining the stability of the system in each iteration. 25
4.5. Summary of the iterative tuning procedure In this section, the implementation procedure of the proposed iterative tuning method is summarized as follows: 1) Choose the normalized filter time constant and set the iteration number i=0. 2) Select the initial parameter vector 0 k0 T0 0 T . 3) Execute three IFT experiments with the predefined input signal ri1 ' on the independent MUDE structure and collect corresponding control output and input data denoted as (yi1, ui1), (yi2, ui2), (yi3, ui3). 4) Calculate the estimates of gradients and positive definite matrix R i and update
i . 5) Judge whether i 1 i
is smaller than a predefined threshold value, if it is, turn
to step 6), otherwise turn to step 7). 6) Output i 1 as the optimal parameter vector * , set the tracking time constant Tm equal to the estimated time delay , and calculate the normalized error feedback gain with a given Mst according to (36). End the procedure. 7) Set i 1 0 , then go to step 2). To further explain the proposed iterative tuning procedure, an algorithm flow chart is presented in Fig. 8.
26
Start
choose λ
set initial parameter vector T 0 k0 T0 0
Execute three IFT experiments, and collect control input and output data (yi1, ui1), (yi2, ui2), (yi3, ui3)
dˆ y (t ) y J calculate est i1 , est i1 , r , est , Ri i i i i
i 1 0
update parameter vetor J i 1 i i R i 1 i
if i 1 i
No
Yes output *
Calculate Tm, χ
End
Fig. 8. Flowchart of the iterative tuning procedure.
5. Simulation results In order to demonstrate the feasibility and effectiveness of the iterative tuning 27
methodology of the MUDE-based control system proposed in this paper, we now present simulation results for three different processes.
5.1. Example 1: First order plus time delay process Consider a simple FOPDT process described as 1 s e (66) s 1 The excitation signal r ' is given in Fig. 9. Suppose the system is affected by G p (s)
Gaussian white noise N (0,0.052 ) . The initial parameter vector is set as
0 0.8 1.2 1.2 . The normalized filter coefficient is chosen as 1. The T
obtained optimal parameter vector is * = 1.0029 1.0068 0.9943 , which is very T
close to the real value r [1 1 1]T , hence the tracking time constant Tm is chosen equal to the estimated time delay 0.9943. The robustness index Mst is given as 2.0 and
is 0.4969 for * . According to empirical correlation, the normalized error feedback gain is calculated as 0.5737. Figs. 10 and 11 show the system actual output and the desired output in the first iteration and in the 10th iteration, respectively, and Fig. 12 compares the estimated uncertainty and disturbance in the first and the 10th iteration. It can be observed that the system actual output has approximated the desired output quite closely after 10 iterations, and the estimated uncertainty and disturbance is gradually approaching 0. The convergence of the parameters is clearly shown in Fig. 13. Since the optimal nominal model is very close to the actual process, the obtained MUDE based control system satisfies the predefined robustness index. This simulation clearly shows that the proposed iterative tuning method can amend the nominal model to approach the actual process by utilizing the closed-loop experiments data. 28
Fig. 9. Excitation signal of the experiments in Example 1.
Fig. 10. System output after the first iteration in Example 1.
29
Fig. 11. System output after 10 iterations in Example 1.
Fig. 12. Estimated uncertainty and disturbance in Example 1.
30
Fig.13. Parameter convergence diagram in Example 1.
To further verify the influence of the initial parameter vector 0 on the final optimal vector * , three additional experiments are carried out. Table 1 shows different initial parameter vectors and corresponding optimal parameter vectors, where m is the iteration number. The distance between * and r , defined as
* - r
2
, is used to estimate the performance of IFT algorithm in different
experiments. It is apparent that the distance between resulting and actual parameter vector will be smaller if the initial parameter vector is closer to the actual parameter vector. These experiments indicate that an appropriate initial nominal model is beneficial to improving the performance of the proposed tuning method.
31
Table 1 Experiments with different initial parameter vector in Example 1 Case
0
1
0.8 1.2 1.2
1.0029 1.0068 0.9943
2
0.7 1.3 1.3
3
1.5 2 1.4
4
0.5 1.5 1.5
r
m
* - r
1 1 1
10
0.0093
1.0013 0.9733 1.0152
1 1 1
10
0.0308
1.0012 0.9623 1.0243
1 1 1
10
0.0449
0.9938 0.8743 1.0989
1 1 1
10
0.1601
* T
T
T
T
T
T
T
T
T
T
T
T
2
5.2. Example 2: High-order process 1 Consider a lag-dominated third-order process described by
G p ( s)
1 ( s 1)(0.6s 1)(0.3s 1)
(67)
The simulation settings including the excitation signal r ' and noise are the same as in Example 1. The initial parameter vector is set as 0 = 1 1.3 0.3 . The T
normalized filter coefficient is chosen as 1. After 10 iterations, the parameter vector converges to * [0.9884 1.2360 0.8076]T . To demonstrate the performance improvement of the proposed iterative tuning method, both the original and the proposed MUDE based control system are compared. For the original MUDE controller, parameters are tuned based on the initial nominal model G p 0 , the tracking time constant Tm is chosen equal to the estimated time delay 0.3, and the normalized error feedback gain is calculated as 1.7486 when the robustness index Mst is 2.0. For the proposed MUDE controller, parameters are tuned based on the optimal nominal model G *p , the tracking time constant Tm is chosen equal to the estimated time delay 0.8076 and the normalized error feedback gain is calculated as 0.7407 under the same 32
robustness index. The system outputs and control signals with the original and proposed MUDE controller under a unity step set-point change together with input disturbance of magnitude 1 introduced at time 50s are given in Fig. 14. Oscillations in the original MUDE-based system in both outputs and control actions are quite severe and frequent, while after optimizing the nominal model through the proposed tuning method, both output and control signal are smooth and fast with no oscillations or overshoots. This example clearly indicates that the performance of the MUDE-based control system relies heavily on the accuracy of the process model, and the proposed iterative tuning method can provide guaranteed performance without exact process model.
Fig. 14(a). System outputs in Example 2.
33
Fig. 14(b). System control signals in Example 2
5.3. Example 3: High-order process 2 Consider a benchmark forth-order process with four equal poles described by
G p (s)
1 ( s 1) 4
(68)
The initial parameter vector is set as 0 2 2 2 . The normalized filter T
coefficient is chosen as 1. After 10 iterations, the parameter vector converges to
* [1.0149 2.4279 1.8652]T , namely
G*p 1.0149e1.8652 s (2.4279s 1) .
For
comparison purpose, three reduced models of (68) presented in [36] are utilized to compare the accuracy of the reduced-order model. Those three models are obtained from model reduction methods of AMIGO, SIMC and DRO, respectively, shown as follows: G p1 ( s)
1 e 1.42 s 2.9 s 1
34
(69)
1 e2.5 s 1.5s 1 1 G p3 (s) e1.9 s 2.1s 1 G p 2 (s)
(70) (71)
Fig. 15 shows the Nyquist diagrams of the actual process and different reduced-order models. It shows among those obtained approximated FOPTD models, the optimal nominal model obtained by the proposed method is the closest to the actual process. Example 2 and 3 indicate that although the proposed data-driven tuning algorithm are derived based on the FOPTD model, it can also be applied to some high-order systems that have analogical dynamic characteristics with FOPTD model.
Fig. 15. Nyquist diagrams of G p , G *p , G p1 , G p 2 , G p 3 in Example 3.
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6. Real-time experiments To demonstrate the feasibility of the proposed iterative tuning method in industrial applications, real-time experiments are conducted on a Coupled Tanks setup shown in Fig. 16, which represents a chemical plant fragment. The set-up has four water tanks coupled through connecting pipes and allows for the design of different controllers and tests in real-time using Matlab and Simulink environment. Tank 2 is chosen as the experimental subject, where the controlled variable is the level of tank 2, and the manipulated variable is the voltage of the pump that pumps water into tank 2. To obtain the initial plant model, an open loop test is conducted by increasing the pump voltage from 1.75V to 2.35V, and the resulting open loop response data is shown in Fig. 17. The identified initial model is obtained as Gp 0 (s)
19.207 1.7452 s e 88.24s 1
Controller
(72)
Tank 1
Tank 3
Tank 2
Tank 4
Monitor
Fig. 16 Experimental setup
36
Fig.17 Open loop response of water level in tank 2
Fig. 18 Parameter convergence diagram in the real-time experiment
37
Fig.19 System outputs and control signals before tuning and after tuning
Table 2 Parameters setting in real-time experiments IAEsp1
IAEld
IAEsp2
(101~400s)
(401~700s)
(701~900s)
21.2586
37.3377
131.3811
29.7762
43.9854
28.6513
24.6914
23.6268
Method
Tm
Original MUDE
1.7452
1
Proposed MUDE
1.0001
1
Accordingly, the initial parameter vector is set as 0 = 19.207 88.24 1.7452 . T
Similarly, the normalized filter coefficient is selected as 1. The tracking time constant Tm is chosen equal to the estimated time delay 1.7452. The robustness index Mst is given as 2.0 and is 0.0194 for 0 . According to the empirical correlation, the normalized error feedback gain is calculated as 21.2586. After 10 iterations, the parameter vector converges to * [21.4984 102.8139 1.0001]T (shown in Fig. 18). The convergence of the parameters shows the effectiveness of the proposed iterative 38
tuning method. Then the obtained optimized model can be described as
(21.4984 /102.8139s 1)e1.0001s . The output response of both initial model and optimized model under the same test input are also shown in Fig. 17 (dashed line and dotted line respectively). It is obvious that after 10 iterations, the optimized model output response is almost the same as the actual experimental output. For the proposed MUDE-based control system, the normalized filter coefficient is selected as 1. The tracking time constant Tm is chosen equal to the estimated time delay 1.0001 and the normalized error feedback gain is calculated as 43.9854. The parameters of the original MUDE and proposed MUDE are summarized in Table 2. Fig. 19 shows experimental outputs and control signals of the MUDE-based control system before and after tuning respectively, with set-point change of 3 and -3 at t=100s and at t=700s respectively and an input disturbance added to the system at t=400s. It is clear that the set-point tracking performance after tuning is much faster than that of before tuning and hence overshoots and actuator windup are observed in the output and control signal respectively. Moreover, the disturbance rejection performance is also significantly improved with less overshoots and settling time. The indices IAEsp and IAEld during each time period are also calculated in Table 2 and an apparent decrease can be seen in both IAEsp and especially IAEld after applying the proposed iterative tuning method. The comparative results indicate that under the same robustness index, the proposed iterative tuning method provides an obviously better disturbance rejection performance while the set-point tracking performance is also improved. The real-time experiments demonstrate the effectiveness of proposed 39
methodology and its application prospect to industrial cases.
7. Conclusions In this paper, we propose a data-driven iterative tuning method for the MUDE-based control system with the aid of IFT algorithm. This paper aims to realize automatic parameter tuning of the MUDE-based control system even in the case of inaccurate plant model while satisfying a predefined robustness index. To achieve this, quantitative performance analysis with respect to design freedoms is first conducted and an empirical correlation of system robustness index is derived to describe the relationship among them. Then the IFT algorithm is used to approximate nominal model towards actual process through conducting specific closed-loop experiments iteratively. Based on the obtained optimized nominal model and empirical correlation, system design parameters are determined. The proposed data-driven iterative tuning method provides a systematic automatic design procedure for the MUDE-based control system after balancing both system output performance and robustness. It largely reduces the dependency of system performance on the plant model and avoids the expense of accurate system modeling. Numerical examples have demonstrated the improved set-point tracking and disturbance rejection performance after optimization and the real-time experiments have proved its feasibility and prospect in industrial applications.
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Acknowledgements This work is supported by National Nature Science Foundation of China under Grant 51576041, the Natural Science Foundation of Jiangsu Province, China under Grant BK20170686, and the open funding of the state key lab for power systems, Tsinghua University under Grant SKLD17MK11. The authors would like to give our sincere appreciation to the anonymous reviewers for their careful review and valuable suggestion. Special thanks goes to Dr. Donghai Li at Tsinghua University for his support in the experimental validation.
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