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Decoupled control scheme for output tracking of a general industrial nonlinear MIMO system using improved active disturbance rejection scheme
Alexandria Engineering Journal (2019) xxx, xxx–xxx
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
Decoupled control scheme for output tracking of a general industrial nonlinear MIMO system using improved active disturbance rejection scheme Wameedh Riyadh Abdul-Adheem, Ibraheem Kasim Ibraheem * University of Baghdad, College of Engineering, Department of Electrical Engineering, Al-Jadriyah, P.O.B.: 47273, 10001 Baghdad, Iraq Received 17 May 2019; revised 19 September 2019; accepted 19 September 2019
KEYWORDS Active disturbance rejection; Decoupling control; Subsystem couplings; Extended state observer; Generalized disturbance
Abstract Multivariable systems are broadly found in industrial processes with nonlinearities, uncertainties, and both input and state couplings. The basic idea to deal with such problems is to remove the input couplings, then estimating all other issues in an online manner followed be a rejection process. This challenge is presented in this research paper, where a decoupling control scheme is proposed based on the decoupler matrix and the Improved Active Disturbance Rejection Control (IADRC) paradigm. A theoretical analysis based on Hurwitz criterion followed by numerical simulations on a general industrial nonlinear, uncertain and highly coupled Multi-input MultiOutput (MIMO) system proved the validity of the proposed scheme in controlling such kinds of systems. Performance comparisons with the Conventional ADRC (CADRC) based decoupling control scheme are also included. Ó 2019 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction In control discipline, most systems are Multi-Input-MultiOutput (MIMO) in their nature. Undoubtedly, the control theories for MIMO systems will discover straightforward application in a wide assortment of fields (space innovation, electrical machines, and robotic control). The control of MIMO systems is a very intricate task because of the state and input couplings. * Corresponding author. E-mail addresses: [email protected] (W.R. Abdul-Adheem), [email protected] (I.K. Ibraheem). Peer review under responsibility of Faculty of Engineering, Alexandria University.
At the point when the MIMO system is uncertain and nonlinear, the control task will be more sophisticated. In this regard, theoretical outcomes and constructive methodologies for structuring acceptable controllers are incredibly scarce. Recently, researchers have explored the aforementioned difficulties to solve the problem of controlling uncertain nonlinear MIMO systems. In [1], an Internal Model Control (IMC) decoupling controller and an inverted decoupling controller are applied to a coupled-tank liquid level system. These controllers are designed based on identified First Order Plus Dead Time (FOPDT) model of a coupled tank system. While in [2] a decoupling control technique based on the WNN (Wavelet Neural Network) has been used to eliminate the coupling between the temperature control loop and humidity control loop in a Variable-Air-Volume (VAV) Air-Conditioning
https://doi.org/10.1016/j.aej.2019.09.016 1110-0168 Ó 2019 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: W.R. Abdul-Adheem, I.K. Ibraheem, Decoupled control scheme for output tracking of a general industrial nonlinear MIMO system using improved active disturbance rejection scheme, Alexandria Eng. J. (2019), https://doi.org/10.1016/j.aej.2019.09.016
2 system. The work in [3] demonstrated the application of the tensor decomposition for the design of a static decoupling of a MIMO system and only required a Frequency Response Function (FRF) of the system. This technique would permit the MIMO FRF to be written as a linear combination of parallel Single-Input-Single-Output (SISO) FRFs. Authors of [4] applied a linear form of MPC by using input-output State Feedback Linearization with Decoupling (SFLD) to a MIMO nonlinear seeded continuous mixed-suspension mixed-productremoval (MSMPR) crystallization process of acetaminophen in water. An operator-based robust decoupling control system design for multi-input multi-output (MIMO) nonlinear system is considered in [5]. The coupling effects existing in the MIMO nonlinear plants can be decoupled based on a feedback design and robust right coprime factorization approach. Two degrees of freedom controller is designed for multiple inputs, multiple outputs system with multiple time delays in [6]. An H1 norm of closed-loop optimized decoupled smith control method is used to design a feedback controller for disturbance rejection. Near decoupling property of closed-loop control systems through an external constant feedback loop is investigated in [7]. The use of Linear Parameter-Varying (LPV) as decoupling technique for quadrotor helicopter system is demonstrated in [8]. In [9], a decoupler based on a modified adjoint transfer matrix is designed. A novel inverted fuzzy decoupling technique for Multi-Input Multi-Output (MIMO) systems with disturbances has been demonstrated in [10]. Moreover, the work in [11] proposed a novel feedback linearization and feedforward neural network controller design for a twin-rotor multi-input multi-output system (TRMMS), which is a highly nonlinear system with a significant cross-couplings. In [12], the interaction between the inputs and outputs are canceled using inverted decoupling configuration. In [13], a decentralized fuzzy control method for MIMO nonlinear second order systems was developed with application to robot manipulators via a combination of genetic algorithms (GAs) and fuzzy systems. Finally, it must be remembered that other robust control techniques like HInfinity techniques [14–16] can be applied for decoupling MIMO systems, however, this would be on the account of the simplicity of the resulting robust controller, since robust control techniques always consider the wore case scenario for the uncertainties and exogenous disturbances. In this paper, a control scheme is proposed based on the decoupling principle in which the input couplings for the uncertain nonlinear MIMO system is first resolved, converting it into decoupled linear time-invariant SISO systems. Then followed by an application of an IADRC for each of the SISO systems separately. This technique has the advantage of reducing model dependence in its design as compared to aforementioned works. The proposed scheme does not need extensive parameter tuning as with the neural network-based adaptive controlling techniques. Moreover, chattering in control signals, a common phenomenon in the sliding mode control techniques [17–20] is avoided in the proposed scheme. Finally, the proposed control scheme is a direct technique, which means, they estimate/cancel the generalized disturbance in a realtime manner without a need for a decision from an inference engine as with the fuzzy control where a large number of fuzzy rules have to be developed and stored in a database. The contributions of the paper are explained as follows. A decoupled ADRC controller is proposed based on the IADRC
W.R. Abdul-Adheem, I.K. Ibraheem scheme by interfacing the MIMO nonlinear system and the IADRC with a decoupled matrix to cancel the input couplings. Then, the dynamic interactions between different subsystems are refined into the generalized disturbance for later estimation/cancelation from the input channel through a novel Extended State Observer (ESO) based feedback control law. The conventional ADRC (CADRC) approach is improved by two modifications, firstly, adding a nonlinear function to the classical ESO to increase the sensitivity of the observer estimation to the small changes in the estimation errors. Secondly, a more augmented state is added to the dynamics of the classical ESO to estimate the generalized disturbances with higher order derivatives. The new observer is called Nonlinear Higher Order ESO (NHOESO). This paper is structured as given next. Section 2 presents the problem statement followed by a succinct introduction in Section 3 on the Active Disturbance Rejection Control and the formulation of the generalized disturbance. Section 4 introduces the proposed decoupled control scheme, the IADRC, and the closed-loop stability investigation using Hurwitz principle. Section 5 demonstrates the numerical simulations for the proposed decoupled control scheme on a hypothetical highly nonlinear MIMO system. Section 6 concludes the paper results. 2. Problem statement Consider a MIMO nonlinear system given as ( P ðc Þ ni i ¼ fi ðn; g; wÞ þ pq¼1 gi;q ðtÞuq ; yi ¼ ni ; i 2 f1; 2; ::; pg
ð1Þ
The above system belongs to a class of partial exact feedback linearizable system if the relative degree c is less than the order of the MIMO nonlinear system n, the control signal T u is expressed as u ¼ u1 ðtÞ; u2 ðtÞ; up ðtÞ 2 Rp , the measured output y for the MIMO system is defined as T T y ¼ y1 ðtÞ; y2 ðtÞ; yp ðtÞ 2 Rp ; w ¼ w1 ðtÞ; w2 ðtÞ; ; wp ðtÞ 2 Rp is the exogenous disturbance, the state vector of the system given T by (1) is labeled as n ¼ n1 ðtÞ; n2 ðtÞ; ; np ðtÞ 2 Rc , is the total relative degree, c ¼ c1 þ c2 þ þ cp fi 2 C RcðncÞp ; R , i 2 f1; 2; ::; pg is the unknown system function, which includes system uncertainties, exogenous disturbances, and any unwanted dynamics, gi;j 2 CðR; RÞ is the input gain function for i; q 2 f1; 2; ; pg. The internal dynamics of the system (1) can be described as g_ ¼ f0 ðn; g; wÞ, where f0 2 C RcðncÞp ; RðncÞ is an unknown function. Consider the i-th subsystem with state vector denoted as T ðc 1Þ i n ðtÞ ¼ ni ðtÞ; ; ni i ðtÞ 2 Rci ; i 2 f1; 2; ; pg. Let the coefficient bi;q be an approximate estimate for gi;q of the nonlinear system within a rough range of ±50% [21,22], then (1) can be described as, ( P P ðc Þ ni i ¼ fi ðn; g; wÞ þ pq¼1 gi;q ðtÞ bi;q uq þ pq¼1 bi;q uq ð2Þ yi ¼ ni ; i 2 f1; 2; ::; pg The generalized disturbance Fi , i 2 f1; 2; ; pg of the nonlinear MIMO system (1) is expressed as
Please cite this article in press as: W.R. Abdul-Adheem, I.K. Ibraheem, Decoupled control scheme for output tracking of a general industrial nonlinear MIMO system using improved active disturbance rejection scheme, Alexandria Eng. J. (2019), https://doi.org/10.1016/j.aej.2019.09.016
Decoupled control scheme for MIMO systems
3 b n1; ; b n q of the system and the predicted generalized disturbance b n qþ1 .
Fig. 1 Generalized disturbance representation for MIMO nonlinear system.
Fi ¼ fi ðn; g; wÞ þ
p X gi;q ðtÞ bi;q uq ; i 2 f1; 2; ::; pg
ð3Þ
q¼1
Fig. 1 illustrates the system given in (2) considering the generalized disturbance of (3). The objectives are to design a MIMO controller for the MIMO system of (1) which provides the following: disassociates the state couplings, disassociates the input couplings, rejects the effect of the generalized disturbance Fi , i 2 f1; 2; ::; pgon the outputs, and maintains acceptable performance during both the transient and steady-state. 3. Active Disturbance Rejection Control Active Disturbance Rejection Control is an innovative strategy, its fundamental principle lies in augmenting the statespace model of the system with a supplementary fictitious first-order differential equation. This fictitious equation defines all the undesirable system uncertainties, dynamical properties, and external disturbance, denoted as the ‘‘generalized disturbance” [23]. This fictitious state, as well as the states of the dynamical plant, is estimated in a real-time manner by the ESO which is the central part of the active disturbance rejection control (ADRC) [24]. It achieves immediate and dynamic estimation and elimination to the generalized disturbance by returning the predicted generalized disturbance to the control port with basic calculations using control law with output feedback. With ADRC, a controlling a sophisticated uncertain nonlinear plant is transformed approximately into a simple linear time-invariant system [25]. The supremacy that makes it such an effective, robust control instrument, is that it is a model-free method, instead of a model-based technique. Essentially, ADRC comprises of a Tracking Differentiator (TD) which is used to generate the transient profile of the reference input (i.e., the noise-free signal itself together with its n1 derivatives, where n is the TD order, a Nonlinear State Error feedback (NLSEF), and an ESO as shown in Fig. 2., where T r 2 R is the desired signal, ð r1 r2 rq Þ 2 Rn is the transient profile, q is the relative degree, v 2 R is the control T 2 Rnþ1 is the expanded estiinput, b n2 b n1 b n qþ1 mated vector which, it includes the predicted states
Numerous engineering control challenges have been adequately resolved over the past 20 years through the successful application of ADRC. These include the tension and velocity regulations in dredger cutter motor [26], high-performance motion control [27], chemical processes [28], vibrational MEMS gyroscopes [29], hysteresis compensation [30], noncircular turning processes [31], high pointing accuracy and rotation speed [32], omnidirectional mobile robot control [33], the control of model-scale helicopters [34], flexible-joint manipulator control [35], and speed control of PMDC motor [25]. More recently, Manufacturers that have successfully adopted ADRC technology include Texas Instruments Inc., which utilized ADRC in its motion control chips, i.e. TMS320F28052M, TMS320F28054M, TMS320F28069M, (InstaSPINTM-Motion) [36]. Parker-Hannifin Corporation recently adopted ADRC over its production lines at an extrusion plant and halved its energy usage [37]. Finally, the National Superconducting Cyclotron Laboratory in the United States deployed ADRC in some high-energy particle accelerators for the regulation of electromagnetic fields [38]. 4. Main results In this section, we present the proposed IADRC-based decoupled scheme. Then, the IADRC is aimed to serve the intended functions of the disturbance and uncertainty cancelation for the MIMO nonlinear uncertain systems. Finally, the proposed control schemes are validated theoretically by demonstrating the closed-loop system stability investigation. 4.1. The offered IADRC-based decoupled scheme In this control scheme, the controller is applied after adding a decoupler element between the IADRC and the nonlinear MIMO system. Recall equations (2) and (3), the simplified model is expressed as, ( P ðc Þ ni i ¼ Fi þ pq¼1 bi;q uq ð4Þ yi ¼ ni ; i 2 f1; 2; ::; pg The matrix form of (4) is given as, ðc Þ 1 0 0 1 1 n1 i u1 F1 B ðc2 Þ C B B u2 C B n2 C B F2 C B C C B C B . C C þ B ð5Þ B . C¼B .. C B . C B . C B @ @ A . . A @ . A ð cp Þ Fp up np 1 0 b11 b12 b1p B b21 b22 b2p C C B where B ¼ B . .. .. C is a non-singular input .. @ .. . . A . 0
bp1 bp2 bpp gain matrix. Its inverse is the matrix B 0 1 0 b11 b1p b11 B . C B . .. . . 1 B C B .. .. A ¼ @ .. B ¼ @ .. . bp1 bp1 bpp
expressed as 1 b1p 1 .. C C . A bpp
ð6Þ
Please cite this article in press as: W.R. Abdul-Adheem, I.K. Ibraheem, Decoupled control scheme for output tracking of a general industrial nonlinear MIMO system using improved active disturbance rejection scheme, Alexandria Eng. J. (2019), https://doi.org/10.1016/j.aej.2019.09.016
4
W.R. Abdul-Adheem, I.K. Ibraheem ui ¼ vi b n i;ci þ1 ; i 2 f1; 2; ::; pg
ð11Þ
where, b n i;ci þ1 is the estimation of the generalized disturbance Fi of (3) produced by an ESO designed for this purpose. Now, the uncertain nonlinear MIMO system of (4) is converted into p uncertain nonlinear systems expressed by (10). The nominal control signal vi is a nonlinear combination of the tracking error and can be described as follows vi ¼ Wðe~i;l Þ; i 2 f1; 2; ::; pg; l 2 f1; 2; ci g
ð12Þ
where W : R ! R is any nonlinear combination function with sector bounded feature and satisfies W(0) = 0 and is the tracking error vector, defined e~i;l 2 Rci T as~ ei;l ¼ e~i;1 ; e~i;2 ; ; e~i;ci : The tracking error is expressed as n i;l , for i 2 f1; 2; ::; pg and l 2 f1; 2; c g, where e~i;l ¼ ri;l b ci
Fig. 2 Structure of conventional SISO ADRC configuration, q is the relative degree of the nonlinear system.
The proposed decoupler element used in this scheme is given as, 0 1 0 1 u1 u1 B C B u2 C u2 C B B C C B . C ¼ B1 B ð7Þ B C B . C B ... C @ . A @ A up
up
By substituting model (5), we get, 0 ðci Þ 1 0 n1 F1 B ðc2 Þ C B B n2 C B F2 B C B . C¼B .. B . C B @ . A @ . ð cp Þ Fp np 0 B B B B B @
(
the decoupler (7) element in the system 1
0
B C B C C þ BB1 B B C B A @
u1 u2 .. . up
1 C C C C C A
This leads to the decoupled model, ðc Þ 1 0 1 0 u 1 n1 i F1 1 C C ðc Þ B F2 C B B u2 C n2 2 C B C B C B C ¼ .. C .. C .. C CþB B C B C @ A . . A @ . A c ð pÞ up Fp np
ð8Þ
This model can be expressed as, ðc Þ
ni i ¼ Fi þ ui ; yi ¼ ni ; i 2 f1; 2; ::; pg:
ð9Þ
ðl1Þ
Let ni;l ¼ ni , for i 2 f1; 2; ::; pg; l 2 f1; 2; ::; ci g. Also, let ni;ci þ1 ¼ Fi ) n_ i;ci þ1 ¼ F_ i . The subsystem (9) can be expressed as, 8 > n_ ¼ ni;2 ; > > i;1 > > > n_ ¼ ni;3 ; > > < i;2 .. ð10Þ . > > > > n_ ¼ F þ u ; i 2 f1; 2; ::; pg > i i;ci > i > > :_ ni;c þ1 ¼ F_ i i
As shown in Fig. 3, the virtual control signal, ui is generated according to the following formula,
i
ðl1Þ
ri;l ¼ ri is the ðl 1Þth derivative of the reference signal ri , b and n i;l is an estimation of the state ni;l produced by a suitable designed ESO. 4.2. Improved ADRC (IADRC) The structure of the IADRC that will be used in the proposed decoupled control scheme for MIMO nonlinear systems is the same as that of the Conventional ADRC (CADRC) with the same units except the conventional fal-based ESO, where it is replaced by a novel Nonlinear Higher Order ESO (NHOESO). The dynamical structure of the TD is given as [22], 8 2 f1; 2; ; ci 1g < r_i;l ¼ ri;lþ1 ; l ð13Þ r jri;2 j : r_i;ci ¼ Ri sign ri;1 ri þ i;22R ; i 2 f1; ::; pg i where Ri , i 2 f1; ::; pg, is an application-dependent design parameter, and its value controls the convergence speed of the differentiator output. The NLSEF has the following nonlinear error function [22], ( e~i;l e~i;l di;l 1a d i;l ð14Þ fali;l ðe~i;l ; ai;l ; di;l Þ ¼ ai;l e~i;l signðe~i;l Þ e~i;l di;l with l 2 f1; 2; ; ci g, i 2 f1; ; pg, ai;l ; di;l are design parameters, usually, di;l is a small number and ai;l 2 ð0; 1Þ. With proper selection of these parameters, the error e~i;l approaches zero in a very short time. The novel NHOESO is proposed as follows, 8 _ > l b b b > n ¼ n þ a i;l xo;i gi yi n i;1 ; l 2 f1; 2: ; ci 1g; i;l i;lþ1 > > > > > > _ > ci þ1 b b b > ¼ n þ a x g y n ; n > i;c þ1 i;c þ1 i;c þ2 i;1 o;i i i > i i i > > > > _ > ci þ2 :b gi yi b n i;1 i 2 f1; ::; pg n i;ci þ2 ¼ ai;ci þ2 xo;i ð15Þ T where the vector b n i;1 ; :; b n i;ci , i 2 f1; ::; pg contains the observed model states and b n i;ci þ1 ; i 2 f1; ::; pgis the estimated generalized disturbance, xo;i is the bandwidth of the NHOESO for the i-th subsystem to be tuned, and ai;s , s 2 f1; ::; ci þ 2g is
Please cite this article in press as: W.R. Abdul-Adheem, I.K. Ibraheem, Decoupled control scheme for output tracking of a general industrial nonlinear MIMO system using improved active disturbance rejection scheme, Alexandria Eng. J. (2019), https://doi.org/10.1016/j.aej.2019.09.016
Decoupled control scheme for MIMO systems
5 In what follows, the closed-loop stability of the MIMO nonlinear system is investigated for the decoupled control scheme with a generalized disturbance denoted as Fi . Theorem 1 (Stability of the Closed–Loop System). Consider a nonlinear n-dimensional uncertain MIMO system of (10). If the augmented system of (10) is governed by a Linearization Control Law (LCL) ui , n i;ci þ1 ; i 2 f1; ::; pg ui ¼ vi b
ð21Þ
where vi is given as Fig. 3
The proposed IADRC-based decoupled control scheme.
the associated design parameter, it is selected such that the following matrix is Hurwitz. 2 3 ai;1 1 0 0 6 a 0 7 1 0 6 7 i;2 6 7 7 E¼6 ð16Þ 6 7 .. 6 7 1 0 0 4 ai;ci þ1 5 . ai;ci þ2 0 0 0 ðci þ2Þðci þ2Þ The nonlinear function gi : R ! R; i 2 f1; ::; pgis designed as in [39], gi ðei Þ ¼ Ki;a jei jai signðei Þ þ Ki;b jei jbi ei
ð17Þ
where Ki;a ; Ki;b ; ai and bi are positive design parameters, and n i;1 . ei is defined as ei ¼ y b i
The stability of the closed-loop system with the proposed IADRC-based decoupled control scheme is demonstrated next. Some assumptions are needed and adopted in the stability investigation of the closed-loop system; they are given as follows. Assumption (A1). The plant states ni;l , i 2 f1; ::; pg; l 2 f1; :; ci g and the generalized disturbance Fi of the MIMO plant of the system (10) are observed using a stable NHOESO given by (15) which produces the predicted system states b n i;l ; i 2 f1; ::; pg and l 2 f1; :; ci g and the generalized disturbance b n i;c þ1 as t ! 1 respectively, i.e., i
t!1
ð18Þ
and n i;ci þ1 ¼ 0 lim Fi b
t!1
ð19Þ
t!1
i 2 f1; ::; pg; l 2 f1; 2; ; ci g:
t!1
Proof. The closed-loop tracking error e~i;l between the output n i;l of the nonlinear sysof the TD ri;l and the estimated states b tem can be described as, n i;l ; i 2 f1; ::; pg; l 2 f1; 2; ; ci g e~i;l ¼ ri;l b
ð23Þ
With assumptions A1 and A2 hold, the tracking error ~ei;j can be expressed as, ð24Þ
For the system of (10), the states ni;l are defined in terms of the plant output, ðl1Þ
ni;l ¼ yi
; i 2 f1; ::; pg; l 2 f1; 2; ; ci g
ð25Þ
Substitute (25) in (24), and e~i;l is expressed as, ðl1Þ ðl1Þ e~i;l ¼ ri yi ; i 2 f1; ::; pg; l 2 f1; 2; ; ci g
ð26Þ
Differentiating (26) w.r.t t, gives ðlÞ ð lÞ e~_ i;l ¼ ri yi ¼ e~i;lþ1 ; i 2 f1; ::; pg; l 2 f1; 2; ; ci g
ð27Þ
then, e~i;l ; i 2 f1; ::; pg; l 2 f1; 2; ; ci g are given below 8 > e~_ ¼ e~i;2 ; > > i;1 > > < e~_ i;2 ¼ e~i;3 ; > ... > > > > :_ ðc Þ ðc Þ ðc Þ e~i;ci ¼ ri i yi i ¼ ri i n_ i;ci ; i 2 f1; ::; pg
ð28Þ
This together with (9) gives,
Assumption (A2). A TD generates a transient profile signal ri;l ; i 2 f1; 2; ; pg; l 2 f1; :; ci g with a minimum mean square error. The signal ri;l approaches a reference signal ðl1Þ with l 2 f1; :; ci g as t ! 1, i.e., ri ðl1Þ ri;l ¼ 0; i 2 f1; ::; pg; l 2 f1; 2; ; ci g lim ri
ð22Þ
With i 2 f1; ::; pg; l 2 f1; 2; ; ci g, where ki;l : R ! Rþ is an n i;l ; i 2 f1; ::; pg; even gain nonlinear function, e~i;l ¼ ri;l b l 2 f1; 2; ; ci g are the closed-loop errors. Furthermore, if assumption A1 and assumption A2 are valid, then, the closed loop system is asymptotically stable, i.e., lim e~i;l ¼ 0;
ðl1Þ e~i;l ¼ ri ni;l ; i 2 f1; ::; pg; l 2 f1; 2; ; ci g
4.3. Stability analysis of the closed-loop system
n i;l ¼ 0; i 2 f1; 2; ; pg; l 2 f1; :; ci g lim ni;l b
ei;1 þ þ ki;l ðe~i;l Þ~ ei;l þ þ ki;ci e~i;ci e~i;ci vi ¼ ki;1 ðe~i;1 Þ~
ð20Þ
8 > e~_ i;1 ¼ e~i;2 ; > > > > < e~_ i;2 ¼ e~i;3 ; .. > > . > > > :_ ðc Þ e~i;ci ¼ ri i Fi þ ui ; i 2 f1; ::; pg
ð29Þ
Please cite this article in press as: W.R. Abdul-Adheem, I.K. Ibraheem, Decoupled control scheme for output tracking of a general industrial nonlinear MIMO system using improved active disturbance rejection scheme, Alexandria Eng. J. (2019), https://doi.org/10.1016/j.aej.2019.09.016
6
W.R. Abdul-Adheem, I.K. Ibraheem of (34) lie in the left-half plane, i.e., it is Hurwitz stable matrix. h
From (21), we get 8 e~_ i;1 ¼ e~i;2 ; > > > > > < e~_ i;2 ¼ e~i;3 ;
ð30Þ
.. > > . > > > : _ ðc Þ e~i;ci ¼ ri i vi þ b n i;ci þ1 Fi ; i 2 f1; ::; pg
To show the performance of the proposed IADRC-based decoupled control scheme for MIMO systems, consider the following general industrial uncertain nonlinear MIMO system,
It follows from Assumption A1 that, 8 > e~_ ¼ e~i;2 ; > > i;1 > > < e~_ i;2 ¼ e~i;3 ;
ð31Þ
.. > > . > > > :_ ðc Þ e~i;ci ¼ ri i vi ; i 2 f1; ::; pg
The dynamics of the tracking errors of (31) with the control signal vi of (22) yields the dynamics for the closed-loop errors given as, 8 > e~_ ¼ e~i;2 ; > > i;1 > > < e~_ i;2 ¼ e~i;3 ; > ... > > > > :_ e~i;ci ¼ ki;1 ðe~i;1 Þ~ ei;1 ki;2 ðe~i;2 Þ~ ei;2 ki;ci e~i;ci e~i;ci ; i 2 f1; ::; pg ð32Þ
The dynamics given in (32) can be represented in matrix form as, e~_ i ¼ Ai e~i ; i 2 f1; ::; pg
ð33Þ
Where 0 B B B B B Ai ¼ B B B B @
0
1
0
0
0
0 .. .
0
1
0 .. .
0 .. .
1
0 0 0 1 0 0 0 0 0 1 ki;1 ðe~i;1 Þ ki;2 ðe~i;2 Þ ki;3 ðe~i;3 Þ ki;ci 1 e~i;ci 1 ki;ci e~i;ci
C C C C C C C C C A
ð34Þ
T and e~i ¼ e~i;1 ; e~i;2 ; ; e~i;ci . The characteristic polynomial of Ai is given by jkI Ai j ¼ kci þ ki;ci e~i;ci kci 1 þ ki;ci 1 e~i;ci 1 kci 2 þ þ ki;1 ðe~i;1 Þ
5. Numerical simulations
88 _ n1;1 ¼ n1;2 ; > > > > < > > > > > n_ 1;2 ¼ f1 ðn; g; w1 Þ þ g1;1 ðtÞu1 þ g1;2 ðtÞu2 ; > > > > > > : > > > y1 ¼ n1;1 ; > < 8 n_ 2;1 ¼ n2;2 ; > >> < > > > > _ > > > n2;2 ¼ f2 ðn; g; w2 Þ þ g2;1 ðtÞu1 þ g2;2 ðtÞu2 ; > > > : > > > y2 ¼ n2;1 ; > > > : g_ ¼ n1;2 þ n2;1 þ sinðgÞ þ sinðtÞ
where y1 , y2 are the outputs, u1 ,u2 are inputs,
n ¼ n1;1 ; n1;2 ; n2;1 ; n2;2 2 R4 is the external state vector, g 2 R is the internal state of (38). The variables y1 , y2 , u1 ,u2 ,w1 ,w2 2 R and the unknown functions are given as, f1 ¼ n1;1 þ n2;1 þ g þ sin n1;2 þ n2;2 w1 ; f2 ¼ n1;2 þ n2;2 þ gþ sin n1;1 þ n2;1 Þw2 ; g1;1 ðtÞ ¼ 1 þ 101 sinðtÞ; g1;2 ðtÞ ¼ 1 þ 101 cosðtÞ, g2;1 ðtÞ ¼ 1 þ 101 2t ; g2;2 ðtÞ ¼ 1. Suppose that the exogenous disturbancesw1 ; w2 and the reference signals r1 ; r2 are as follows: w1 ¼ 1 þ sin ðtÞ; w2 ¼ 2t cos ðtÞ; r1 ¼ sin ðtÞ; r2 ¼ cos ðtÞ: The initial values of the model are taken as follows: n1;1 ;n1;2 ;n2;1 ;n2;2 ; g ¼ ð0:5; 0:5;1; 1; 0Þ. Two ADRC Configurations will be used in the simulations for the proposed schemes. The difference between them is the type of ESO used for estimation. The first configuration is the Conventional ADRC (CADRC), which consists of: 1. Conventional TD described by [22], 8 < r_i;1 ¼ ri;2 ;
r jri;2 j : r_i;2 ¼ Ri sign ri;1 ri þ i;22R ; i 2 f1; 2g i where Ri , i 2 f1; 2g is a design parameter.
ei;l fali;l ðe~i;l ; a; dÞ ¼ ki;l ðe~i;l ; ai;l ; di;l Þ~
(
ð36Þ
with l 2 f1; 2; ; ci g, i 2 f1; ::; pg, where ki ðe~i;l ; ai;l ; di;l Þ ¼
1 d
ð39Þ
ð35Þ
The NLSEF controller adopted in this paper is utilizing the fali;l ðÞ expressed by (14) and is redrafted as a function of ki;l ðÞ as follows,
(
ð38Þ
1ai;l
ai;l 1 e~i;l
e~i;l di;l e~i;l di;l
2. A fal-based control law given as, n 1;3 ; u1 ¼ k1;1 falðe~1;1 ; a1;1 ; d1;1 Þ þ k1;2 falðe~1;2 ; a1;2 ; d1;2 Þ b b u ¼ k2;1 falðe~2;1 ; a2;1 ; d2;1 Þ þ k2;2 falðe~2;2 ; a2;2 ; d2;2 Þ n 2;3 : 2
ð40Þ ð37Þ
which is an even positive function. The design parameters (ai;l ; di;l Þ of (37) are chosen to guarantee that the eigenvalues
n i;l fori; l 2 f1; 2g is the tracking error, where e~i;l ¼ ri;l b ki;l ; ai;l ; di;l i; l 2 f1; 2g are design parameters of the fal-based control law.
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Decoupled control scheme for MIMO systems
7
3. fal-based ESO given as follows, 8 _ > b > n i;2 þ 3xo;i fal yi b n i;1 ; ai ; di ; n i;1 ¼ b > > > < _ b n i;3 þ ui þ 3x2o;i fal yi b n i;1 ; ai ; di ; n i;2 ¼ b > > > > _ > :b n i;1 ; ai ; di ; i 2 f1; 2g n i;3 ¼ x3o;i fal yi b
ð41Þ
8 _ > > n i;1 ¼ b n i;2 þ ai;1 xo;i gi yi b n i;1 ; >b > > > > > _ > b > n i;4 þ ai;3 x3o;i gi yi b n i;1 ; n i;3 ¼ b > > > > > > _ > :b n i;1 ; i 2 f1; 2g n i;4 ¼ ai;4 x4o;i gi yi b
ð42Þ
T where the vectors b n i;1 ; b n i;2 , i 2 f1; 2g are the observed model
T n i;2 , i 2 f1; 2g are the observed model where the vectors b n i;1 ; b
states and b n i;3 ; i 2 f1; 2g are the estimated generalized disturbance, xo;i , i 2 f1; 2g is the bandwidth of the NHOESO of the i-th subsystem. The second configuration is the Improved ADRC (IADRC), which consists of:
states and b n i;3 ; i 2 f1; 2g are the estimated generalized disturbance, ai;s , i 2 f1; 2gand s 2 f1; 2; 3; 4g, are design parameters, and xo;i is the bandwidth of the NHOESO for the i-th subsystem. The nonlinear function gi : R ! R is designed as in [27],
1. Conventional TD given by (39). 2. fal-based control law given by (40). 3. A novel NHOESO proposed as,
Table 1
ð43Þ
with i 2 f1; 2g;where Ki;a ; Ki;b ; ai and bi are positive design parameters.
The parameters of the first configuration (CADRC).
Unit
First channel parameters
TD fal-based ESO
fal-based Control law
Table 2
gi ðei Þ ¼ Ki;a jei jai signðei Þ þ Ki;b jei jbi ei
Second channel parameters
Parameter
Value
Parameter
Value
R1 xo;1 a1 d1 d1;1 d1;2 a1;1 a1;2 k1;1 k1;2 d1
92.2713 68.3308 0.2630 0.7537 0.0010 0.2834 0.1629 0.7946 12.8015 11.2999 40
R2 xo;2 a2 d2 d2;1 d2;2 a2;1 a2;2 k2;1 k2;2 d2
88.4424 53.1690 0.4020 0.8687 0.14456 0.73456 0.02730 0.93745 18.3095 19.52670 40
The parameters of the second configuration (IADRC).
Unit
TD NHOESO
fal-based Control law
First channel parameters
Second channel parameters
Parameter
Value
Parameter
Value
R1 xo;1 a1;1 a1;2 a1;3 a1;4 K1;a a1 K1;b b1 d1;1 d1;2 a1;1 a1;2 k1;1 k1;2 d1
192.7715 135.6086 2.31423 4.5361 2.0465 0.1658 0.9000 0.9000 0.1000 0.0100 0.0341 0.6008 0.0207 0.3372 18.3186 8.9993 40
R2 xo;2 a2;1 a2;2 a2;3 a2;4 K2;a a2 K2;b b2 d2;1 d2;2 a2;1 a2;2 k2;1 k2;2 d2
148.9279 22.8802 3.3264 4.66885 1.48218 0.04076 0.9000 0.9000 0.1000 0.0100 0.0082 0.8162 0.0120 0.7222 6.84822 6.5260 40
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8
W.R. Abdul-Adheem, I.K. Ibraheem
1 1 , then, 1 1
Since the input gain matrix B ¼ 1 1 2 B1 ¼ 21 1 , the proposed decoupler element used in this
u1 u2
¼
1 2 1 2
1 2 1 2
!
u1 u2
¼
u1 2 u1 2
þ
u2 2 u2 2
! ð44Þ
scheme given in (7) is found as,
Then, the suggested feedback control laws u1 and u2 are formulated as
Fig. 4 The output response of (38) using CADRC-based decoupled control scheme, (a) Output y1 , (b) Output y2 , (c) Control signals u1 and u2 , (d) Estimated generalized disturbances b n 2;3 . n 1;3 and b
Fig. 5 The output response of (38) using IADRC-base decoupled control scheme, (a) Output y1 , (b) Output y2 , (c) Control n 1;3 and signals u1 and u2 , (d) Estimated generalized disturbances b b n 2;3 .
2
2
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Decoupled control scheme for MIMO systems Table 3
9
Performance of the decoupled control scheme.
Performance index
CADRC
IADRC
%Reduction
ITAE1 ITAE2 ISU1 ISU2
0.1628 0.3536 312.9118 291.5853
0.1210 0.0937 308.4248 225.7019
25.7% 73.5% 1.4% 22.6%
8 u u > < u1 ¼ Sat 21 þ 22 ; d1 > : u2 ¼ Sat u21 þ u22 ; d2
ð45Þ
where di ,i 2 f1; 2g is a design parameter, and the function Satðu; dÞ is defined as, 8 >
> :
u d < u < d d u d
ð46Þ
The virtual control signals ui ; i 2 f1; 2g in (45) and indicated in Fig. 3 is derived from the fal-based control law given in (40). The desired transient trajectoriesðr1;1 ; r1;2 ÞT , and ðr2;1 ; r2;2 ÞT are generated from the reference signalsr1 ; r2 via
the tracking differentiators described by (39). The parameters of the first and second configurations for the two channels are tabulated in Tables 1 and 2, respectively. For the two configurations, the output responses of the numerical simulations for the proposed IADRC-based decoupled scheme are shown in Figs. 4 and 5, respectively. The performance indices for the two cases are listed in Table 3, where Rt ITAE ¼ 0 f tjy rjdt is the time absolute error integration, R tf 2 ISU ¼ 0 v dt is the integrated squared control signal v, and tf is the period of the simulation time. As illustrated in Table 3, the reduction was very evident in the values of the ITAE and ISU indices of the two channels for the second configuration, except for ISU1 where it slightly reduced from its value in the CADRC. This has been reflected in the control efforts u1 and u2 shown in Fig. 4 (c) and 5(c),
Fig. 6 The output tracking of (38) due to reference inputs r1 and r2 using CADRC-based decoupled control scheme, (a) ðr1 ; r2 Þ ¼ ðsin ðtÞ; 0Þ, (b) ðr1 ; r2 Þ ¼ ð0; cos ðtÞÞ.
Please cite this article in press as: W.R. Abdul-Adheem, I.K. Ibraheem, Decoupled control scheme for output tracking of a general industrial nonlinear MIMO system using improved active disturbance rejection scheme, Alexandria Eng. J. (2019), https://doi.org/10.1016/j.aej.2019.09.016
10
W.R. Abdul-Adheem, I.K. Ibraheem
Fig. 7 The output tracking of (38) due to reference inputs r1 and r2 using IADRC-based decoupled control scheme, (a) ðr1 ; r2 Þ ¼ ðsin ðtÞ; 0Þ, (b) ðr1 ; r2 Þ ¼ ð0; cos ðtÞÞ.
where u1 and u2 for the IADRC witnessed less activity than the CADRC. The tracking output response for the IADRC is better than in the CADRC, specifically during the transient period, where both configurations have entirely attenuated the effect of the exogenous disturbances w1 and w2 , the state couplings for each subsystem, and the time-varying input gains b1;1 , b1;2 , b2;1 , and b2;2 on the output response of the two channels. The transient response of the outputs y1 and y2 in response to reference inputs r1 and r2 sequentially applied to the nonlinear system (38) are illustrated in Figs. 6 and 7. As can be seen from these figures, the decoupling requirement, stated in the problem statement, is very well satisfied, with a smooth response on each output channel. The proposed scheme converted the nonlinear system of (1) into two noninteracted SISO subsystems. 5.1. Discussion For controlling nonlinear MIMO systems, the nonlinear couplings between different subsystems are considered as the most significant difficulty. Therefore, it is necessary to adopt a control technique that is both simple and robust. A decoupled control scheme was suggested in this paper, it makes use of the IADRC because of its robustness and model-independent features. The nonlinear coupling, along with other uncertain-
ties, were considered as a part of the generalized disturbances n1;3 andn2;3 that needed to be observed and canceled by an ESO-based control law, e.g., ADRC configuration. In the proposed IADRC-based decoupled control scheme, the decoupling process for the input couplings is accomplished through the decoupler unit, where the control input for each channel is composed by gathering several control signals from each ADRC controller. For this reason, the energy consumed by each input is anticipated to be substantial. The chattering in the control signals u1 andu2 in the decoupled control scheme using IADRC lasted for a shorter amount of time than in the case of CADRC because the proposed NHOESO needs more time for its output to become settled. Finally, looking back at Figs. 6 and 7, the diagonalization in the output response of both configurations is due to the offdiagonal terms cancelation (input cross-couplings) of the nonlinear system using B-matrix in the decoupled control scheme. The other reason for the diagonal output response is because of the cancellation of the state couplings between different individual channels within the ADRC configuration. 6. Conclusion In this paper, the ADRC paradigm has been utilized for controlling nonlinear MIMO systems. The suggested control
Please cite this article in press as: W.R. Abdul-Adheem, I.K. Ibraheem, Decoupled control scheme for output tracking of a general industrial nonlinear MIMO system using improved active disturbance rejection scheme, Alexandria Eng. J. (2019), https://doi.org/10.1016/j.aej.2019.09.016
Decoupled control scheme for MIMO systems scheme, namely, the decoupled control used the IADRC due to its aforementioned supreme features. The investigation into the MIMO systems indicated potential issues concerning the computation of the inverse of the input gain matrix. The proposed control scheme efficiently eliminated the input couplings via the input gain matrix, while the state couplings, exogenous disturbances, and system uncertainties have been removed excellently via the NHOESO which is the central part of the IADRC configuration. The proposed IADRC-based decoupled control scheme converted the uncertain nonlinear MIMO system into a distinct multiple SISO linear time-invariant systems with suitable state feedback control law. It can be concluded that the performance of the proposed IADRC-based decoupled control scheme is significantly higher than its counterpart, namely, the CADRC-based decoupling control scheme, regarding output tracking, control energy, and chattering as indicated by Table 3. A possible future work is to implement the proposed decoupled control scheme on a real nonlinear MIMO platform and to compare the practical results with that of the simulations presented in this paper. Declaration of Competing Interest The authors declare that they have no conflict of interest. References [1] S.R. Mahapatro, B. Subudhi, S. Ghosh, P. Dworak, A comparative study of two decoupling control strategies for a coupled tank system, IEEE Reg. 10 Annu. Int. Conf. Proc./ TENCON 3447–3451 (2017), https://doi.org/10.1109/tencon. 2016.7848695. [2] X. Ji, J. Li, T. Gao, Research on wavelet neural network decoupling control of variable-air-volume air-conditioning system, Proc. 30th Chin. Control Decis. Conf. CCDC 2018 (2018) 1848–1853, https://doi.org/10.1109/ccdc.2018.8407427. [3] J. Stoev, J. Ertveldt, T. Oomen, J. Schoukens, Tensor methods for MIMO decoupling and control design using frequency response functions, Mechatronics 45 (2017) 71–81, https://doi. org/10.1016/j.mechatronics.2017.05.009. [4] R. Parekh, B. Benyahia, C.D. Rielly, A global state feedback linearization and decoupling MPC of a MIMO continuous MSMPR cooling crystallization process, 28th European Symposium on Computer Aided Process Engineering, vol. 43, 2018, pp. 1607–1612, https://doi.org/10.1016/b978-0-444-642356.50280-1. [5] S. Bi, Y. Xiao, X. Fan, Operator-based robust decoupling control for MIMO nonlinear systems, in: 11th World Congr Intell. Control Autom., 2014, pp. 2602–2606, https://doi.org/ 10.1109/WCICA.2014.7053135. [6] A. Ahuja, S. Narayan, J. Kumar, Optimal two degrees of freedom decoupling smith control for MIMO systems with multiple time delays, in: 2016 IEEE 1st International Conference on Power Electronics, Intelligent Control and Energy Systems (ICPEICES), 2016, pp. 1–5, https://doi.org/10.1109/ icpeices.2016.7853279. [7] L. Jetto, V. Orsini, Enhancing the near decoupling property of closed-loop control systems through external constant feedback loop, J. Process Control 69 (2018) 70–78, https://doi.org/ 10.1016/j.jprocont.2018.06.007. [8] T.T.R. van de Wiel, R. To´th, V.I. Kiriouchine, Comparison of parameter-varying decoupling based control schemes for a quadrotor, IFAC-PapersOnLine 51 (26) (2018) 55–61, https:// doi.org/10.1016/j.ifacol.2018.11.168.
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H O S T E D BY
Alexandria University
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ORIGINAL ARTICLE
Decoupled control scheme for output tracking of a general industrial nonlinear MIMO system using improved active disturbance rejection scheme Wameedh Riyadh Abdul-Adheem, Ibraheem Kasim Ibraheem * University of Baghdad, College of Engineering, Department of Electrical Engineering, Al-Jadriyah, P.O.B.: 47273, 10001 Baghdad, Iraq Received 17 May 2019; revised 19 September 2019; accepted 19 September 2019
KEYWORDS Active disturbance rejection; Decoupling control; Subsystem couplings; Extended state observer; Generalized disturbance
Abstract Multivariable systems are broadly found in industrial processes with nonlinearities, uncertainties, and both input and state couplings. The basic idea to deal with such problems is to remove the input couplings, then estimating all other issues in an online manner followed be a rejection process. This challenge is presented in this research paper, where a decoupling control scheme is proposed based on the decoupler matrix and the Improved Active Disturbance Rejection Control (IADRC) paradigm. A theoretical analysis based on Hurwitz criterion followed by numerical simulations on a general industrial nonlinear, uncertain and highly coupled Multi-input MultiOutput (MIMO) system proved the validity of the proposed scheme in controlling such kinds of systems. Performance comparisons with the Conventional ADRC (CADRC) based decoupling control scheme are also included. Ó 2019 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction In control discipline, most systems are Multi-Input-MultiOutput (MIMO) in their nature. Undoubtedly, the control theories for MIMO systems will discover straightforward application in a wide assortment of fields (space innovation, electrical machines, and robotic control). The control of MIMO systems is a very intricate task because of the state and input couplings. * Corresponding author. E-mail addresses: [email protected] (W.R. Abdul-Adheem), [email protected] (I.K. Ibraheem). Peer review under responsibility of Faculty of Engineering, Alexandria University.
At the point when the MIMO system is uncertain and nonlinear, the control task will be more sophisticated. In this regard, theoretical outcomes and constructive methodologies for structuring acceptable controllers are incredibly scarce. Recently, researchers have explored the aforementioned difficulties to solve the problem of controlling uncertain nonlinear MIMO systems. In [1], an Internal Model Control (IMC) decoupling controller and an inverted decoupling controller are applied to a coupled-tank liquid level system. These controllers are designed based on identified First Order Plus Dead Time (FOPDT) model of a coupled tank system. While in [2] a decoupling control technique based on the WNN (Wavelet Neural Network) has been used to eliminate the coupling between the temperature control loop and humidity control loop in a Variable-Air-Volume (VAV) Air-Conditioning
https://doi.org/10.1016/j.aej.2019.09.016 1110-0168 Ó 2019 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: W.R. Abdul-Adheem, I.K. Ibraheem, Decoupled control scheme for output tracking of a general industrial nonlinear MIMO system using improved active disturbance rejection scheme, Alexandria Eng. J. (2019), https://doi.org/10.1016/j.aej.2019.09.016
2 system. The work in [3] demonstrated the application of the tensor decomposition for the design of a static decoupling of a MIMO system and only required a Frequency Response Function (FRF) of the system. This technique would permit the MIMO FRF to be written as a linear combination of parallel Single-Input-Single-Output (SISO) FRFs. Authors of [4] applied a linear form of MPC by using input-output State Feedback Linearization with Decoupling (SFLD) to a MIMO nonlinear seeded continuous mixed-suspension mixed-productremoval (MSMPR) crystallization process of acetaminophen in water. An operator-based robust decoupling control system design for multi-input multi-output (MIMO) nonlinear system is considered in [5]. The coupling effects existing in the MIMO nonlinear plants can be decoupled based on a feedback design and robust right coprime factorization approach. Two degrees of freedom controller is designed for multiple inputs, multiple outputs system with multiple time delays in [6]. An H1 norm of closed-loop optimized decoupled smith control method is used to design a feedback controller for disturbance rejection. Near decoupling property of closed-loop control systems through an external constant feedback loop is investigated in [7]. The use of Linear Parameter-Varying (LPV) as decoupling technique for quadrotor helicopter system is demonstrated in [8]. In [9], a decoupler based on a modified adjoint transfer matrix is designed. A novel inverted fuzzy decoupling technique for Multi-Input Multi-Output (MIMO) systems with disturbances has been demonstrated in [10]. Moreover, the work in [11] proposed a novel feedback linearization and feedforward neural network controller design for a twin-rotor multi-input multi-output system (TRMMS), which is a highly nonlinear system with a significant cross-couplings. In [12], the interaction between the inputs and outputs are canceled using inverted decoupling configuration. In [13], a decentralized fuzzy control method for MIMO nonlinear second order systems was developed with application to robot manipulators via a combination of genetic algorithms (GAs) and fuzzy systems. Finally, it must be remembered that other robust control techniques like HInfinity techniques [14–16] can be applied for decoupling MIMO systems, however, this would be on the account of the simplicity of the resulting robust controller, since robust control techniques always consider the wore case scenario for the uncertainties and exogenous disturbances. In this paper, a control scheme is proposed based on the decoupling principle in which the input couplings for the uncertain nonlinear MIMO system is first resolved, converting it into decoupled linear time-invariant SISO systems. Then followed by an application of an IADRC for each of the SISO systems separately. This technique has the advantage of reducing model dependence in its design as compared to aforementioned works. The proposed scheme does not need extensive parameter tuning as with the neural network-based adaptive controlling techniques. Moreover, chattering in control signals, a common phenomenon in the sliding mode control techniques [17–20] is avoided in the proposed scheme. Finally, the proposed control scheme is a direct technique, which means, they estimate/cancel the generalized disturbance in a realtime manner without a need for a decision from an inference engine as with the fuzzy control where a large number of fuzzy rules have to be developed and stored in a database. The contributions of the paper are explained as follows. A decoupled ADRC controller is proposed based on the IADRC
W.R. Abdul-Adheem, I.K. Ibraheem scheme by interfacing the MIMO nonlinear system and the IADRC with a decoupled matrix to cancel the input couplings. Then, the dynamic interactions between different subsystems are refined into the generalized disturbance for later estimation/cancelation from the input channel through a novel Extended State Observer (ESO) based feedback control law. The conventional ADRC (CADRC) approach is improved by two modifications, firstly, adding a nonlinear function to the classical ESO to increase the sensitivity of the observer estimation to the small changes in the estimation errors. Secondly, a more augmented state is added to the dynamics of the classical ESO to estimate the generalized disturbances with higher order derivatives. The new observer is called Nonlinear Higher Order ESO (NHOESO). This paper is structured as given next. Section 2 presents the problem statement followed by a succinct introduction in Section 3 on the Active Disturbance Rejection Control and the formulation of the generalized disturbance. Section 4 introduces the proposed decoupled control scheme, the IADRC, and the closed-loop stability investigation using Hurwitz principle. Section 5 demonstrates the numerical simulations for the proposed decoupled control scheme on a hypothetical highly nonlinear MIMO system. Section 6 concludes the paper results. 2. Problem statement Consider a MIMO nonlinear system given as ( P ðc Þ ni i ¼ fi ðn; g; wÞ þ pq¼1 gi;q ðtÞuq ; yi ¼ ni ; i 2 f1; 2; ::; pg
ð1Þ
The above system belongs to a class of partial exact feedback linearizable system if the relative degree c is less than the order of the MIMO nonlinear system n, the control signal T u is expressed as u ¼ u1 ðtÞ; u2 ðtÞ; up ðtÞ 2 Rp , the measured output y for the MIMO system is defined as T T y ¼ y1 ðtÞ; y2 ðtÞ; yp ðtÞ 2 Rp ; w ¼ w1 ðtÞ; w2 ðtÞ; ; wp ðtÞ 2 Rp is the exogenous disturbance, the state vector of the system given T by (1) is labeled as n ¼ n1 ðtÞ; n2 ðtÞ; ; np ðtÞ 2 Rc , is the total relative degree, c ¼ c1 þ c2 þ þ cp fi 2 C RcðncÞp ; R , i 2 f1; 2; ::; pg is the unknown system function, which includes system uncertainties, exogenous disturbances, and any unwanted dynamics, gi;j 2 CðR; RÞ is the input gain function for i; q 2 f1; 2; ; pg. The internal dynamics of the system (1) can be described as g_ ¼ f0 ðn; g; wÞ, where f0 2 C RcðncÞp ; RðncÞ is an unknown function. Consider the i-th subsystem with state vector denoted as T ðc 1Þ i n ðtÞ ¼ ni ðtÞ; ; ni i ðtÞ 2 Rci ; i 2 f1; 2; ; pg. Let the coefficient bi;q be an approximate estimate for gi;q of the nonlinear system within a rough range of ±50% [21,22], then (1) can be described as, ( P P ðc Þ ni i ¼ fi ðn; g; wÞ þ pq¼1 gi;q ðtÞ bi;q uq þ pq¼1 bi;q uq ð2Þ yi ¼ ni ; i 2 f1; 2; ::; pg The generalized disturbance Fi , i 2 f1; 2; ; pg of the nonlinear MIMO system (1) is expressed as
Please cite this article in press as: W.R. Abdul-Adheem, I.K. Ibraheem, Decoupled control scheme for output tracking of a general industrial nonlinear MIMO system using improved active disturbance rejection scheme, Alexandria Eng. J. (2019), https://doi.org/10.1016/j.aej.2019.09.016
Decoupled control scheme for MIMO systems
3 b n1; ; b n q of the system and the predicted generalized disturbance b n qþ1 .
Fig. 1 Generalized disturbance representation for MIMO nonlinear system.
Fi ¼ fi ðn; g; wÞ þ
p X gi;q ðtÞ bi;q uq ; i 2 f1; 2; ::; pg
ð3Þ
q¼1
Fig. 1 illustrates the system given in (2) considering the generalized disturbance of (3). The objectives are to design a MIMO controller for the MIMO system of (1) which provides the following: disassociates the state couplings, disassociates the input couplings, rejects the effect of the generalized disturbance Fi , i 2 f1; 2; ::; pgon the outputs, and maintains acceptable performance during both the transient and steady-state. 3. Active Disturbance Rejection Control Active Disturbance Rejection Control is an innovative strategy, its fundamental principle lies in augmenting the statespace model of the system with a supplementary fictitious first-order differential equation. This fictitious equation defines all the undesirable system uncertainties, dynamical properties, and external disturbance, denoted as the ‘‘generalized disturbance” [23]. This fictitious state, as well as the states of the dynamical plant, is estimated in a real-time manner by the ESO which is the central part of the active disturbance rejection control (ADRC) [24]. It achieves immediate and dynamic estimation and elimination to the generalized disturbance by returning the predicted generalized disturbance to the control port with basic calculations using control law with output feedback. With ADRC, a controlling a sophisticated uncertain nonlinear plant is transformed approximately into a simple linear time-invariant system [25]. The supremacy that makes it such an effective, robust control instrument, is that it is a model-free method, instead of a model-based technique. Essentially, ADRC comprises of a Tracking Differentiator (TD) which is used to generate the transient profile of the reference input (i.e., the noise-free signal itself together with its n1 derivatives, where n is the TD order, a Nonlinear State Error feedback (NLSEF), and an ESO as shown in Fig. 2., where T r 2 R is the desired signal, ð r1 r2 rq Þ 2 Rn is the transient profile, q is the relative degree, v 2 R is the control T 2 Rnþ1 is the expanded estiinput, b n2 b n1 b n qþ1 mated vector which, it includes the predicted states
Numerous engineering control challenges have been adequately resolved over the past 20 years through the successful application of ADRC. These include the tension and velocity regulations in dredger cutter motor [26], high-performance motion control [27], chemical processes [28], vibrational MEMS gyroscopes [29], hysteresis compensation [30], noncircular turning processes [31], high pointing accuracy and rotation speed [32], omnidirectional mobile robot control [33], the control of model-scale helicopters [34], flexible-joint manipulator control [35], and speed control of PMDC motor [25]. More recently, Manufacturers that have successfully adopted ADRC technology include Texas Instruments Inc., which utilized ADRC in its motion control chips, i.e. TMS320F28052M, TMS320F28054M, TMS320F28069M, (InstaSPINTM-Motion) [36]. Parker-Hannifin Corporation recently adopted ADRC over its production lines at an extrusion plant and halved its energy usage [37]. Finally, the National Superconducting Cyclotron Laboratory in the United States deployed ADRC in some high-energy particle accelerators for the regulation of electromagnetic fields [38]. 4. Main results In this section, we present the proposed IADRC-based decoupled scheme. Then, the IADRC is aimed to serve the intended functions of the disturbance and uncertainty cancelation for the MIMO nonlinear uncertain systems. Finally, the proposed control schemes are validated theoretically by demonstrating the closed-loop system stability investigation. 4.1. The offered IADRC-based decoupled scheme In this control scheme, the controller is applied after adding a decoupler element between the IADRC and the nonlinear MIMO system. Recall equations (2) and (3), the simplified model is expressed as, ( P ðc Þ ni i ¼ Fi þ pq¼1 bi;q uq ð4Þ yi ¼ ni ; i 2 f1; 2; ::; pg The matrix form of (4) is given as, ðc Þ 1 0 0 1 1 n1 i u1 F1 B ðc2 Þ C B B u2 C B n2 C B F2 C B C C B C B . C C þ B ð5Þ B . C¼B .. C B . C B . C B @ @ A . . A @ . A ð cp Þ Fp up np 1 0 b11 b12 b1p B b21 b22 b2p C C B where B ¼ B . .. .. C is a non-singular input .. @ .. . . A . 0
bp1 bp2 bpp gain matrix. Its inverse is the matrix B 0 1 0 b11 b1p b11 B . C B . .. . . 1 B C B .. .. A ¼ @ .. B ¼ @ .. . bp1 bp1 bpp
expressed as 1 b1p 1 .. C C . A bpp
ð6Þ
Please cite this article in press as: W.R. Abdul-Adheem, I.K. Ibraheem, Decoupled control scheme for output tracking of a general industrial nonlinear MIMO system using improved active disturbance rejection scheme, Alexandria Eng. J. (2019), https://doi.org/10.1016/j.aej.2019.09.016
4
W.R. Abdul-Adheem, I.K. Ibraheem ui ¼ vi b n i;ci þ1 ; i 2 f1; 2; ::; pg
ð11Þ
where, b n i;ci þ1 is the estimation of the generalized disturbance Fi of (3) produced by an ESO designed for this purpose. Now, the uncertain nonlinear MIMO system of (4) is converted into p uncertain nonlinear systems expressed by (10). The nominal control signal vi is a nonlinear combination of the tracking error and can be described as follows vi ¼ Wðe~i;l Þ; i 2 f1; 2; ::; pg; l 2 f1; 2; ci g
ð12Þ
where W : R ! R is any nonlinear combination function with sector bounded feature and satisfies W(0) = 0 and is the tracking error vector, defined e~i;l 2 Rci T as~ ei;l ¼ e~i;1 ; e~i;2 ; ; e~i;ci : The tracking error is expressed as n i;l , for i 2 f1; 2; ::; pg and l 2 f1; 2; c g, where e~i;l ¼ ri;l b ci
Fig. 2 Structure of conventional SISO ADRC configuration, q is the relative degree of the nonlinear system.
The proposed decoupler element used in this scheme is given as, 0 1 0 1 u1 u1 B C B u2 C u2 C B B C C B . C ¼ B1 B ð7Þ B C B . C B ... C @ . A @ A up
up
By substituting model (5), we get, 0 ðci Þ 1 0 n1 F1 B ðc2 Þ C B B n2 C B F2 B C B . C¼B .. B . C B @ . A @ . ð cp Þ Fp np 0 B B B B B @
(
the decoupler (7) element in the system 1
0
B C B C C þ BB1 B B C B A @
u1 u2 .. . up
1 C C C C C A
This leads to the decoupled model, ðc Þ 1 0 1 0 u 1 n1 i F1 1 C C ðc Þ B F2 C B B u2 C n2 2 C B C B C B C ¼ .. C .. C .. C CþB B C B C @ A . . A @ . A c ð pÞ up Fp np
ð8Þ
This model can be expressed as, ðc Þ
ni i ¼ Fi þ ui ; yi ¼ ni ; i 2 f1; 2; ::; pg:
ð9Þ
ðl1Þ
Let ni;l ¼ ni , for i 2 f1; 2; ::; pg; l 2 f1; 2; ::; ci g. Also, let ni;ci þ1 ¼ Fi ) n_ i;ci þ1 ¼ F_ i . The subsystem (9) can be expressed as, 8 > n_ ¼ ni;2 ; > > i;1 > > > n_ ¼ ni;3 ; > > < i;2 .. ð10Þ . > > > > n_ ¼ F þ u ; i 2 f1; 2; ::; pg > i i;ci > i > > :_ ni;c þ1 ¼ F_ i i
As shown in Fig. 3, the virtual control signal, ui is generated according to the following formula,
i
ðl1Þ
ri;l ¼ ri is the ðl 1Þth derivative of the reference signal ri , b and n i;l is an estimation of the state ni;l produced by a suitable designed ESO. 4.2. Improved ADRC (IADRC) The structure of the IADRC that will be used in the proposed decoupled control scheme for MIMO nonlinear systems is the same as that of the Conventional ADRC (CADRC) with the same units except the conventional fal-based ESO, where it is replaced by a novel Nonlinear Higher Order ESO (NHOESO). The dynamical structure of the TD is given as [22], 8 2 f1; 2; ; ci 1g < r_i;l ¼ ri;lþ1 ; l ð13Þ r jri;2 j : r_i;ci ¼ Ri sign ri;1 ri þ i;22R ; i 2 f1; ::; pg i where Ri , i 2 f1; ::; pg, is an application-dependent design parameter, and its value controls the convergence speed of the differentiator output. The NLSEF has the following nonlinear error function [22], ( e~i;l e~i;l di;l 1a d i;l ð14Þ fali;l ðe~i;l ; ai;l ; di;l Þ ¼ ai;l e~i;l signðe~i;l Þ e~i;l di;l with l 2 f1; 2; ; ci g, i 2 f1; ; pg, ai;l ; di;l are design parameters, usually, di;l is a small number and ai;l 2 ð0; 1Þ. With proper selection of these parameters, the error e~i;l approaches zero in a very short time. The novel NHOESO is proposed as follows, 8 _ > l b b b > n ¼ n þ a i;l xo;i gi yi n i;1 ; l 2 f1; 2: ; ci 1g; i;l i;lþ1 > > > > > > _ > ci þ1 b b b > ¼ n þ a x g y n ; n > i;c þ1 i;c þ1 i;c þ2 i;1 o;i i i > i i i > > > > _ > ci þ2 :b gi yi b n i;1 i 2 f1; ::; pg n i;ci þ2 ¼ ai;ci þ2 xo;i ð15Þ T where the vector b n i;1 ; :; b n i;ci , i 2 f1; ::; pg contains the observed model states and b n i;ci þ1 ; i 2 f1; ::; pgis the estimated generalized disturbance, xo;i is the bandwidth of the NHOESO for the i-th subsystem to be tuned, and ai;s , s 2 f1; ::; ci þ 2g is
Please cite this article in press as: W.R. Abdul-Adheem, I.K. Ibraheem, Decoupled control scheme for output tracking of a general industrial nonlinear MIMO system using improved active disturbance rejection scheme, Alexandria Eng. J. (2019), https://doi.org/10.1016/j.aej.2019.09.016
Decoupled control scheme for MIMO systems
5 In what follows, the closed-loop stability of the MIMO nonlinear system is investigated for the decoupled control scheme with a generalized disturbance denoted as Fi . Theorem 1 (Stability of the Closed–Loop System). Consider a nonlinear n-dimensional uncertain MIMO system of (10). If the augmented system of (10) is governed by a Linearization Control Law (LCL) ui , n i;ci þ1 ; i 2 f1; ::; pg ui ¼ vi b
ð21Þ
where vi is given as Fig. 3
The proposed IADRC-based decoupled control scheme.
the associated design parameter, it is selected such that the following matrix is Hurwitz. 2 3 ai;1 1 0 0 6 a 0 7 1 0 6 7 i;2 6 7 7 E¼6 ð16Þ 6 7 .. 6 7 1 0 0 4 ai;ci þ1 5 . ai;ci þ2 0 0 0 ðci þ2Þðci þ2Þ The nonlinear function gi : R ! R; i 2 f1; ::; pgis designed as in [39], gi ðei Þ ¼ Ki;a jei jai signðei Þ þ Ki;b jei jbi ei
ð17Þ
where Ki;a ; Ki;b ; ai and bi are positive design parameters, and n i;1 . ei is defined as ei ¼ y b i
The stability of the closed-loop system with the proposed IADRC-based decoupled control scheme is demonstrated next. Some assumptions are needed and adopted in the stability investigation of the closed-loop system; they are given as follows. Assumption (A1). The plant states ni;l , i 2 f1; ::; pg; l 2 f1; :; ci g and the generalized disturbance Fi of the MIMO plant of the system (10) are observed using a stable NHOESO given by (15) which produces the predicted system states b n i;l ; i 2 f1; ::; pg and l 2 f1; :; ci g and the generalized disturbance b n i;c þ1 as t ! 1 respectively, i.e., i
t!1
ð18Þ
and n i;ci þ1 ¼ 0 lim Fi b
t!1
ð19Þ
t!1
i 2 f1; ::; pg; l 2 f1; 2; ; ci g:
t!1
Proof. The closed-loop tracking error e~i;l between the output n i;l of the nonlinear sysof the TD ri;l and the estimated states b tem can be described as, n i;l ; i 2 f1; ::; pg; l 2 f1; 2; ; ci g e~i;l ¼ ri;l b
ð23Þ
With assumptions A1 and A2 hold, the tracking error ~ei;j can be expressed as, ð24Þ
For the system of (10), the states ni;l are defined in terms of the plant output, ðl1Þ
ni;l ¼ yi
; i 2 f1; ::; pg; l 2 f1; 2; ; ci g
ð25Þ
Substitute (25) in (24), and e~i;l is expressed as, ðl1Þ ðl1Þ e~i;l ¼ ri yi ; i 2 f1; ::; pg; l 2 f1; 2; ; ci g
ð26Þ
Differentiating (26) w.r.t t, gives ðlÞ ð lÞ e~_ i;l ¼ ri yi ¼ e~i;lþ1 ; i 2 f1; ::; pg; l 2 f1; 2; ; ci g
ð27Þ
then, e~i;l ; i 2 f1; ::; pg; l 2 f1; 2; ; ci g are given below 8 > e~_ ¼ e~i;2 ; > > i;1 > > < e~_ i;2 ¼ e~i;3 ; > ... > > > > :_ ðc Þ ðc Þ ðc Þ e~i;ci ¼ ri i yi i ¼ ri i n_ i;ci ; i 2 f1; ::; pg
ð28Þ
This together with (9) gives,
Assumption (A2). A TD generates a transient profile signal ri;l ; i 2 f1; 2; ; pg; l 2 f1; :; ci g with a minimum mean square error. The signal ri;l approaches a reference signal ðl1Þ with l 2 f1; :; ci g as t ! 1, i.e., ri ðl1Þ ri;l ¼ 0; i 2 f1; ::; pg; l 2 f1; 2; ; ci g lim ri
ð22Þ
With i 2 f1; ::; pg; l 2 f1; 2; ; ci g, where ki;l : R ! Rþ is an n i;l ; i 2 f1; ::; pg; even gain nonlinear function, e~i;l ¼ ri;l b l 2 f1; 2; ; ci g are the closed-loop errors. Furthermore, if assumption A1 and assumption A2 are valid, then, the closed loop system is asymptotically stable, i.e., lim e~i;l ¼ 0;
ðl1Þ e~i;l ¼ ri ni;l ; i 2 f1; ::; pg; l 2 f1; 2; ; ci g
4.3. Stability analysis of the closed-loop system
n i;l ¼ 0; i 2 f1; 2; ; pg; l 2 f1; :; ci g lim ni;l b
ei;1 þ þ ki;l ðe~i;l Þ~ ei;l þ þ ki;ci e~i;ci e~i;ci vi ¼ ki;1 ðe~i;1 Þ~
ð20Þ
8 > e~_ i;1 ¼ e~i;2 ; > > > > < e~_ i;2 ¼ e~i;3 ; .. > > . > > > :_ ðc Þ e~i;ci ¼ ri i Fi þ ui ; i 2 f1; ::; pg
ð29Þ
Please cite this article in press as: W.R. Abdul-Adheem, I.K. Ibraheem, Decoupled control scheme for output tracking of a general industrial nonlinear MIMO system using improved active disturbance rejection scheme, Alexandria Eng. J. (2019), https://doi.org/10.1016/j.aej.2019.09.016
6
W.R. Abdul-Adheem, I.K. Ibraheem of (34) lie in the left-half plane, i.e., it is Hurwitz stable matrix. h
From (21), we get 8 e~_ i;1 ¼ e~i;2 ; > > > > > < e~_ i;2 ¼ e~i;3 ;
ð30Þ
.. > > . > > > : _ ðc Þ e~i;ci ¼ ri i vi þ b n i;ci þ1 Fi ; i 2 f1; ::; pg
To show the performance of the proposed IADRC-based decoupled control scheme for MIMO systems, consider the following general industrial uncertain nonlinear MIMO system,
It follows from Assumption A1 that, 8 > e~_ ¼ e~i;2 ; > > i;1 > > < e~_ i;2 ¼ e~i;3 ;
ð31Þ
.. > > . > > > :_ ðc Þ e~i;ci ¼ ri i vi ; i 2 f1; ::; pg
The dynamics of the tracking errors of (31) with the control signal vi of (22) yields the dynamics for the closed-loop errors given as, 8 > e~_ ¼ e~i;2 ; > > i;1 > > < e~_ i;2 ¼ e~i;3 ; > ... > > > > :_ e~i;ci ¼ ki;1 ðe~i;1 Þ~ ei;1 ki;2 ðe~i;2 Þ~ ei;2 ki;ci e~i;ci e~i;ci ; i 2 f1; ::; pg ð32Þ
The dynamics given in (32) can be represented in matrix form as, e~_ i ¼ Ai e~i ; i 2 f1; ::; pg
ð33Þ
Where 0 B B B B B Ai ¼ B B B B @
0
1
0
0
0
0 .. .
0
1
0 .. .
0 .. .
1
0 0 0 1 0 0 0 0 0 1 ki;1 ðe~i;1 Þ ki;2 ðe~i;2 Þ ki;3 ðe~i;3 Þ ki;ci 1 e~i;ci 1 ki;ci e~i;ci
C C C C C C C C C A
ð34Þ
T and e~i ¼ e~i;1 ; e~i;2 ; ; e~i;ci . The characteristic polynomial of Ai is given by jkI Ai j ¼ kci þ ki;ci e~i;ci kci 1 þ ki;ci 1 e~i;ci 1 kci 2 þ þ ki;1 ðe~i;1 Þ
5. Numerical simulations
88 _ n1;1 ¼ n1;2 ; > > > > < > > > > > n_ 1;2 ¼ f1 ðn; g; w1 Þ þ g1;1 ðtÞu1 þ g1;2 ðtÞu2 ; > > > > > > : > > > y1 ¼ n1;1 ; > < 8 n_ 2;1 ¼ n2;2 ; > >> < > > > > _ > > > n2;2 ¼ f2 ðn; g; w2 Þ þ g2;1 ðtÞu1 þ g2;2 ðtÞu2 ; > > > : > > > y2 ¼ n2;1 ; > > > : g_ ¼ n1;2 þ n2;1 þ sinðgÞ þ sinðtÞ
where y1 , y2 are the outputs, u1 ,u2 are inputs,
n ¼ n1;1 ; n1;2 ; n2;1 ; n2;2 2 R4 is the external state vector, g 2 R is the internal state of (38). The variables y1 , y2 , u1 ,u2 ,w1 ,w2 2 R and the unknown functions are given as, f1 ¼ n1;1 þ n2;1 þ g þ sin n1;2 þ n2;2 w1 ; f2 ¼ n1;2 þ n2;2 þ gþ sin n1;1 þ n2;1 Þw2 ; g1;1 ðtÞ ¼ 1 þ 101 sinðtÞ; g1;2 ðtÞ ¼ 1 þ 101 cosðtÞ, g2;1 ðtÞ ¼ 1 þ 101 2t ; g2;2 ðtÞ ¼ 1. Suppose that the exogenous disturbancesw1 ; w2 and the reference signals r1 ; r2 are as follows: w1 ¼ 1 þ sin ðtÞ; w2 ¼ 2t cos ðtÞ; r1 ¼ sin ðtÞ; r2 ¼ cos ðtÞ: The initial values of the model are taken as follows: n1;1 ;n1;2 ;n2;1 ;n2;2 ; g ¼ ð0:5; 0:5;1; 1; 0Þ. Two ADRC Configurations will be used in the simulations for the proposed schemes. The difference between them is the type of ESO used for estimation. The first configuration is the Conventional ADRC (CADRC), which consists of: 1. Conventional TD described by [22], 8 < r_i;1 ¼ ri;2 ;
r jri;2 j : r_i;2 ¼ Ri sign ri;1 ri þ i;22R ; i 2 f1; 2g i where Ri , i 2 f1; 2g is a design parameter.
ei;l fali;l ðe~i;l ; a; dÞ ¼ ki;l ðe~i;l ; ai;l ; di;l Þ~
(
ð36Þ
with l 2 f1; 2; ; ci g, i 2 f1; ::; pg, where ki ðe~i;l ; ai;l ; di;l Þ ¼
1 d
ð39Þ
ð35Þ
The NLSEF controller adopted in this paper is utilizing the fali;l ðÞ expressed by (14) and is redrafted as a function of ki;l ðÞ as follows,
(
ð38Þ
1ai;l
ai;l 1 e~i;l
e~i;l di;l e~i;l di;l
2. A fal-based control law given as, n 1;3 ; u1 ¼ k1;1 falðe~1;1 ; a1;1 ; d1;1 Þ þ k1;2 falðe~1;2 ; a1;2 ; d1;2 Þ b b u ¼ k2;1 falðe~2;1 ; a2;1 ; d2;1 Þ þ k2;2 falðe~2;2 ; a2;2 ; d2;2 Þ n 2;3 : 2
ð40Þ ð37Þ
which is an even positive function. The design parameters (ai;l ; di;l Þ of (37) are chosen to guarantee that the eigenvalues
n i;l fori; l 2 f1; 2g is the tracking error, where e~i;l ¼ ri;l b ki;l ; ai;l ; di;l i; l 2 f1; 2g are design parameters of the fal-based control law.
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Decoupled control scheme for MIMO systems
7
3. fal-based ESO given as follows, 8 _ > b > n i;2 þ 3xo;i fal yi b n i;1 ; ai ; di ; n i;1 ¼ b > > > < _ b n i;3 þ ui þ 3x2o;i fal yi b n i;1 ; ai ; di ; n i;2 ¼ b > > > > _ > :b n i;1 ; ai ; di ; i 2 f1; 2g n i;3 ¼ x3o;i fal yi b
ð41Þ
8 _ > > n i;1 ¼ b n i;2 þ ai;1 xo;i gi yi b n i;1 ; >b > > > > > _ > b > n i;4 þ ai;3 x3o;i gi yi b n i;1 ; n i;3 ¼ b > > > > > > _ > :b n i;1 ; i 2 f1; 2g n i;4 ¼ ai;4 x4o;i gi yi b
ð42Þ
T where the vectors b n i;1 ; b n i;2 , i 2 f1; 2g are the observed model
T n i;2 , i 2 f1; 2g are the observed model where the vectors b n i;1 ; b
states and b n i;3 ; i 2 f1; 2g are the estimated generalized disturbance, xo;i , i 2 f1; 2g is the bandwidth of the NHOESO of the i-th subsystem. The second configuration is the Improved ADRC (IADRC), which consists of:
states and b n i;3 ; i 2 f1; 2g are the estimated generalized disturbance, ai;s , i 2 f1; 2gand s 2 f1; 2; 3; 4g, are design parameters, and xo;i is the bandwidth of the NHOESO for the i-th subsystem. The nonlinear function gi : R ! R is designed as in [27],
1. Conventional TD given by (39). 2. fal-based control law given by (40). 3. A novel NHOESO proposed as,
Table 1
ð43Þ
with i 2 f1; 2g;where Ki;a ; Ki;b ; ai and bi are positive design parameters.
The parameters of the first configuration (CADRC).
Unit
First channel parameters
TD fal-based ESO
fal-based Control law
Table 2
gi ðei Þ ¼ Ki;a jei jai signðei Þ þ Ki;b jei jbi ei
Second channel parameters
Parameter
Value
Parameter
Value
R1 xo;1 a1 d1 d1;1 d1;2 a1;1 a1;2 k1;1 k1;2 d1
92.2713 68.3308 0.2630 0.7537 0.0010 0.2834 0.1629 0.7946 12.8015 11.2999 40
R2 xo;2 a2 d2 d2;1 d2;2 a2;1 a2;2 k2;1 k2;2 d2
88.4424 53.1690 0.4020 0.8687 0.14456 0.73456 0.02730 0.93745 18.3095 19.52670 40
The parameters of the second configuration (IADRC).
Unit
TD NHOESO
fal-based Control law
First channel parameters
Second channel parameters
Parameter
Value
Parameter
Value
R1 xo;1 a1;1 a1;2 a1;3 a1;4 K1;a a1 K1;b b1 d1;1 d1;2 a1;1 a1;2 k1;1 k1;2 d1
192.7715 135.6086 2.31423 4.5361 2.0465 0.1658 0.9000 0.9000 0.1000 0.0100 0.0341 0.6008 0.0207 0.3372 18.3186 8.9993 40
R2 xo;2 a2;1 a2;2 a2;3 a2;4 K2;a a2 K2;b b2 d2;1 d2;2 a2;1 a2;2 k2;1 k2;2 d2
148.9279 22.8802 3.3264 4.66885 1.48218 0.04076 0.9000 0.9000 0.1000 0.0100 0.0082 0.8162 0.0120 0.7222 6.84822 6.5260 40
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8
W.R. Abdul-Adheem, I.K. Ibraheem
1 1 , then, 1 1
Since the input gain matrix B ¼ 1 1 2 B1 ¼ 21 1 , the proposed decoupler element used in this
u1 u2
¼
1 2 1 2
1 2 1 2
!
u1 u2
¼
u1 2 u1 2
þ
u2 2 u2 2
! ð44Þ
scheme given in (7) is found as,
Then, the suggested feedback control laws u1 and u2 are formulated as
Fig. 4 The output response of (38) using CADRC-based decoupled control scheme, (a) Output y1 , (b) Output y2 , (c) Control signals u1 and u2 , (d) Estimated generalized disturbances b n 2;3 . n 1;3 and b
Fig. 5 The output response of (38) using IADRC-base decoupled control scheme, (a) Output y1 , (b) Output y2 , (c) Control n 1;3 and signals u1 and u2 , (d) Estimated generalized disturbances b b n 2;3 .
2
2
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Decoupled control scheme for MIMO systems Table 3
9
Performance of the decoupled control scheme.
Performance index
CADRC
IADRC
%Reduction
ITAE1 ITAE2 ISU1 ISU2
0.1628 0.3536 312.9118 291.5853
0.1210 0.0937 308.4248 225.7019
25.7% 73.5% 1.4% 22.6%
8 u u > < u1 ¼ Sat 21 þ 22 ; d1 > : u2 ¼ Sat u21 þ u22 ; d2
ð45Þ
where di ,i 2 f1; 2g is a design parameter, and the function Satðu; dÞ is defined as, 8 >
> :
u d < u < d d u d
ð46Þ
The virtual control signals ui ; i 2 f1; 2g in (45) and indicated in Fig. 3 is derived from the fal-based control law given in (40). The desired transient trajectoriesðr1;1 ; r1;2 ÞT , and ðr2;1 ; r2;2 ÞT are generated from the reference signalsr1 ; r2 via
the tracking differentiators described by (39). The parameters of the first and second configurations for the two channels are tabulated in Tables 1 and 2, respectively. For the two configurations, the output responses of the numerical simulations for the proposed IADRC-based decoupled scheme are shown in Figs. 4 and 5, respectively. The performance indices for the two cases are listed in Table 3, where Rt ITAE ¼ 0 f tjy rjdt is the time absolute error integration, R tf 2 ISU ¼ 0 v dt is the integrated squared control signal v, and tf is the period of the simulation time. As illustrated in Table 3, the reduction was very evident in the values of the ITAE and ISU indices of the two channels for the second configuration, except for ISU1 where it slightly reduced from its value in the CADRC. This has been reflected in the control efforts u1 and u2 shown in Fig. 4 (c) and 5(c),
Fig. 6 The output tracking of (38) due to reference inputs r1 and r2 using CADRC-based decoupled control scheme, (a) ðr1 ; r2 Þ ¼ ðsin ðtÞ; 0Þ, (b) ðr1 ; r2 Þ ¼ ð0; cos ðtÞÞ.
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10
W.R. Abdul-Adheem, I.K. Ibraheem
Fig. 7 The output tracking of (38) due to reference inputs r1 and r2 using IADRC-based decoupled control scheme, (a) ðr1 ; r2 Þ ¼ ðsin ðtÞ; 0Þ, (b) ðr1 ; r2 Þ ¼ ð0; cos ðtÞÞ.
where u1 and u2 for the IADRC witnessed less activity than the CADRC. The tracking output response for the IADRC is better than in the CADRC, specifically during the transient period, where both configurations have entirely attenuated the effect of the exogenous disturbances w1 and w2 , the state couplings for each subsystem, and the time-varying input gains b1;1 , b1;2 , b2;1 , and b2;2 on the output response of the two channels. The transient response of the outputs y1 and y2 in response to reference inputs r1 and r2 sequentially applied to the nonlinear system (38) are illustrated in Figs. 6 and 7. As can be seen from these figures, the decoupling requirement, stated in the problem statement, is very well satisfied, with a smooth response on each output channel. The proposed scheme converted the nonlinear system of (1) into two noninteracted SISO subsystems. 5.1. Discussion For controlling nonlinear MIMO systems, the nonlinear couplings between different subsystems are considered as the most significant difficulty. Therefore, it is necessary to adopt a control technique that is both simple and robust. A decoupled control scheme was suggested in this paper, it makes use of the IADRC because of its robustness and model-independent features. The nonlinear coupling, along with other uncertain-
ties, were considered as a part of the generalized disturbances n1;3 andn2;3 that needed to be observed and canceled by an ESO-based control law, e.g., ADRC configuration. In the proposed IADRC-based decoupled control scheme, the decoupling process for the input couplings is accomplished through the decoupler unit, where the control input for each channel is composed by gathering several control signals from each ADRC controller. For this reason, the energy consumed by each input is anticipated to be substantial. The chattering in the control signals u1 andu2 in the decoupled control scheme using IADRC lasted for a shorter amount of time than in the case of CADRC because the proposed NHOESO needs more time for its output to become settled. Finally, looking back at Figs. 6 and 7, the diagonalization in the output response of both configurations is due to the offdiagonal terms cancelation (input cross-couplings) of the nonlinear system using B-matrix in the decoupled control scheme. The other reason for the diagonal output response is because of the cancellation of the state couplings between different individual channels within the ADRC configuration. 6. Conclusion In this paper, the ADRC paradigm has been utilized for controlling nonlinear MIMO systems. The suggested control
Please cite this article in press as: W.R. Abdul-Adheem, I.K. Ibraheem, Decoupled control scheme for output tracking of a general industrial nonlinear MIMO system using improved active disturbance rejection scheme, Alexandria Eng. J. (2019), https://doi.org/10.1016/j.aej.2019.09.016
Decoupled control scheme for MIMO systems scheme, namely, the decoupled control used the IADRC due to its aforementioned supreme features. The investigation into the MIMO systems indicated potential issues concerning the computation of the inverse of the input gain matrix. The proposed control scheme efficiently eliminated the input couplings via the input gain matrix, while the state couplings, exogenous disturbances, and system uncertainties have been removed excellently via the NHOESO which is the central part of the IADRC configuration. The proposed IADRC-based decoupled control scheme converted the uncertain nonlinear MIMO system into a distinct multiple SISO linear time-invariant systems with suitable state feedback control law. It can be concluded that the performance of the proposed IADRC-based decoupled control scheme is significantly higher than its counterpart, namely, the CADRC-based decoupling control scheme, regarding output tracking, control energy, and chattering as indicated by Table 3. A possible future work is to implement the proposed decoupled control scheme on a real nonlinear MIMO platform and to compare the practical results with that of the simulations presented in this paper. Declaration of Competing Interest The authors declare that they have no conflict of interest. References [1] S.R. Mahapatro, B. Subudhi, S. Ghosh, P. Dworak, A comparative study of two decoupling control strategies for a coupled tank system, IEEE Reg. 10 Annu. Int. Conf. Proc./ TENCON 3447–3451 (2017), https://doi.org/10.1109/tencon. 2016.7848695. [2] X. Ji, J. Li, T. Gao, Research on wavelet neural network decoupling control of variable-air-volume air-conditioning system, Proc. 30th Chin. Control Decis. Conf. CCDC 2018 (2018) 1848–1853, https://doi.org/10.1109/ccdc.2018.8407427. [3] J. Stoev, J. Ertveldt, T. Oomen, J. Schoukens, Tensor methods for MIMO decoupling and control design using frequency response functions, Mechatronics 45 (2017) 71–81, https://doi. org/10.1016/j.mechatronics.2017.05.009. [4] R. Parekh, B. Benyahia, C.D. Rielly, A global state feedback linearization and decoupling MPC of a MIMO continuous MSMPR cooling crystallization process, 28th European Symposium on Computer Aided Process Engineering, vol. 43, 2018, pp. 1607–1612, https://doi.org/10.1016/b978-0-444-642356.50280-1. [5] S. Bi, Y. Xiao, X. Fan, Operator-based robust decoupling control for MIMO nonlinear systems, in: 11th World Congr Intell. Control Autom., 2014, pp. 2602–2606, https://doi.org/ 10.1109/WCICA.2014.7053135. [6] A. Ahuja, S. Narayan, J. Kumar, Optimal two degrees of freedom decoupling smith control for MIMO systems with multiple time delays, in: 2016 IEEE 1st International Conference on Power Electronics, Intelligent Control and Energy Systems (ICPEICES), 2016, pp. 1–5, https://doi.org/10.1109/ icpeices.2016.7853279. [7] L. Jetto, V. Orsini, Enhancing the near decoupling property of closed-loop control systems through external constant feedback loop, J. Process Control 69 (2018) 70–78, https://doi.org/ 10.1016/j.jprocont.2018.06.007. [8] T.T.R. van de Wiel, R. To´th, V.I. Kiriouchine, Comparison of parameter-varying decoupling based control schemes for a quadrotor, IFAC-PapersOnLine 51 (26) (2018) 55–61, https:// doi.org/10.1016/j.ifacol.2018.11.168.
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