Stability analysis of linear/nonlinear switching active disturbance rejection control based MIMO continuous systems

WAN Hui ,QI Xiaohui ,and LI Jie

1.Department of Unmanned Aerial Vehicle Engineering,Army Engineering University of PLA,Shijiazhuang 050003,China;2.The 32nd Research Institute,China Electronics Technology Group Corporation,Shanghai 201800,China

Abstract:In this paper,a linear/nonlinear switching active disturbance rejection control (SADRC) based decoupling control approach is proposed to deal with some difficult control problems in a class of multi-input multi-output (MIMO) systems such as multi-variables,disturbances,and coupling,etc.Firstly,the structure and parameter tuning method of SADRC is introduced into this paper.Followed on this,virtual control variables are adopted into the MIMO systems,making the systems decoupled.Then the SADRC controller is designed for every subsystem.After this,a stability analyzed method via the Lyapunov function is proposed for the whole system.Finally,some simulations are presented to demonstrate the anti-disturbance and robustness of SADRC,and results show SADRC has a potential applications in engineering practice.

Keywords:linear/nonlinear switching active disturbance rejection control (SADRC),multi-input multi-output (MIMO) continuous system,decoupling control,stability analysis.

1.Introduction

After proposing the tracking differentiator (TD) [1,2],non-linear state error feedback (NLSEF) [3] and the extended state observer (ESO) [4],Han formally advanced active disturbance rejection control (ADRC) in 1998 [5].ADRC is an unconventional control technique as it is a significant difference from both modern and classical control theories.It reflects Han’s unique understanding about the control theory,such as“model analysis approach or a direct control approach”[6],“linear and nonlinear of feedback system”[7],and so on.On the one hand,it takes advantage of the proportional-integral-derivative (PID) control law which is almost model free.On the other hand,inspired by the state observer technique of modern control,this theory combines both the internal and external disturbances as well as uncertainties of the plant as“total disturbance”and uses an ESO to estimate and compensate it.It is a big breakthrough of“time invariant and time varying”and“linear and nonlinear”,since the timevarying part and the nonlinear part of the system can be considered as uncertainty.For this reason,ADRC has got increasing attention and a wide range of engineering applications in recent years.

The original ADRC proposed by Han contains nonlinear functions and is mentioned as nonlinear ADRC(NLADRC) in this study.The nonlinear structures of NLADRC make it difficult to perform theoretical research and a retarded period is observed during the preliminary development of ADRC.However,some inspiring results about convergence and stabilization have been achieved recently,laying a good foundation for theoretical studies of NLADRC.For example,the convergence of the NLADRC based closed-loop system is proved for a class of single-input single-output (SISO) systems [8],multi-input multi-put (MIMO) systems [9] and lower triangular systems with uncertainty [10].Meanwhile,stabilization works are mainly around the limit cycle analysis,absolute stability and Lyapunov’s stability. The limit cycle analysis of the ESO including one or two nonlinearities is established via the describing function method[11−13].In some recent published works [14–17],the absolute stability for the nominal satisfying constraint of norm growth is studied separately.Moreover,the local asymptotic stability for the NLADRC based closed-loop system containing multi-nonlinearities is also derived[18].

In 2003,Gao linearized the ADRC,which was called ADRC (LADRC) [19],facilitating the theoretical studies and applications with a splendid advancement of this field.Up to now,the ADRC has successfully been applied in many products [20] and a large number of researches on convergence and stabilization have been achieved.Yoo et al21] and Yang et al22] proved that the estimate error of the linear ESO (LESO) is bounded on the condition that the“total disturbance”is bounded or its derivative is bounded.The works of Shao et al23],Yoo et al24] and Huang et al25] quantitatively discussed the convergence of the discrete-time LESO.Based on the works of Zheng et al26] and Chen et al27],the estimation ability of the LESO and the stability of the closed-loop LADRC based system are demonstrated with and without the plant model.Xue and Huang [28] proved the stability of LADRC for a class of SISO systems (linear or nonlinear,time-varying or time-invariant) with unknown dynamics and disturbance.While Xue et al29] proposed an LADRC controller for a class of MIMO block lower-triangular systems and analyzed the dynamic and steady performance of the closed-loop.Moreover,the performance analysis of ADRC for nonlinear uncertain systems with unknown dynamics and discontinuous disturbances is reported before as well [30].

Though researchers prefer LADRC due to its simple parameter tuning,clear physical meaning and easy theoretical analysis characteristics,NLADRC is potentially much more effective in tolerance to uncertainties,disturbance and improvement of system dynamics.To combine the advantages of both systems and make NLADRC easier for application,we accomplish the following work:analyzing the stability of NLADRC based nominal continuous systems [13,14,18] and plants with parameter perturbation or disturbance satisfying constraint of norm growth [15],laying a foundation of improving system stability and optimizing control ability;proposing general principles and simple formulas for the parameter tuning of nonlinear ESO (NLESO) and applying them to general cases [31];quantitatively studying the advantages and disadvantages of LADRC and NLADRC,and proposing a new control approach,called linear nonlinear switching ADRC (SADRC) [31,32],providing a new pathway for the development of ADRC.Controller design and stability analysis of the SADRC based on the SISO system is also achieved and corresponding simulations also verify the effectiveness in anti-disturbance and tracking accuracy.

Previous work only focused on the SISO systems,while further investigation should be carried out in MIMO systems which are common in industry and much more challenging due to the multi-variables,multi-disturbances and interactions.Therefore,this paper designs the SADRC based decoupling control approach for a class of MIMO continuous systems and proposes a stability analysis method based on the Lyapunov function which is easy to understand and use in practice via computer calculation.

This paper is structured as follows:Section 2 describes the principle and framework of SADRC,then designs the SADRC based controller for a class of MIMO systems,followed by the stability analysis.Two cases are studied to validate the effectiveness of the proposal by simulations in Section 3.Finally,some conclusions are given in Section 4.

2.Control design and stability analysis

2.1 Framework and algorithm of SADRC

The SADRC aims to combine the advantages of both LADRC and NLADRC.Therefore,we first take a class of SISO continuous systems as an example to illustrate the structure of NLADRC,LADRC and the principle of SADRC.

ADRC generally consists of TD,ESO and state error feedback (SEF).TD is mainly used to extract the derivatives of the reference signal and relatively independent in controller design.ESO is used to estimate the system states and the total disturbance,which is the core and essence of the ADRC.SEF is used to restrain the residual error and achieve the desired control goal.For the sake of facilitating the stability analysis,TD is exempted in the controller design,and we only introduce the structure of ESO and SEF here.

2.1.1 Structure of NLADRC/LADRC

Consider a class ofn-dimensional SISO continuous systems described by

wherex1,···,xnare the system states,f(x1,x2,···,xn,w,t)denotes the“total disturbance”,wrepresents external disturbances,yanduare the output and input of the system,respectively,andbis a constant control coefficient.The objective of NLADRC,LADRC or SADRC is to design a controller to make the outputytrack a reference inputv1,andxitrackvi(i=2,···,n) provided that the latter exists in some sense.

For the above plant,takexn+1=f(x,···,xn,w,t) as the extended state of system (1),then the ESO of system(1) is designed as follows:

wherezi(i=1,2,···,n) is the observed value which provides an estimation of the system state andzn+1provides an estimation of the“total disturbance”;β0i(i=1,2,···,n+1) is the observer gain.

If φi(e)(i=1,2,···,n+1) is a nonlinear function,(2) is called the NLESO,a common form of φi(e)(i=1,2,···,n+1) is defined as

whereδrepresents the range of the linearity of φi(e)(i=1,2,···,n+1) which should be pre-determined according to the practical applications;αi(i=1,2,···,n+1) is constant whenαi(i=1,2,···,n+1) is constants,φi(e)(i=1,2,···,n+1) has characteristics of relative small error,big gain,big error,small gain,while whenαi=1(i= 1,2,···,n+1),nonlinear function φi(e)(i=1,2,···,n+1) turns to be linear and (2) is called LESO.

The nonlinear SEF (NLSEF) of system (1) is designed[31] as

A linear one can be described as

If the ESO and the SEF are both linear,the designed controller is called LADRC,otherwise,NLADRC.

2.1.2 SADRC principle

As the key component of ADRC,ESO is used to estimate and eliminate the model uncertainties and elucidate the influence of external disturbances on the system.It turns out that the accuracy of ESO directly affects the control performance.Therefore,in this paper,the linear SEF (LSEF) is chosen here while SADRC represents the switch between LESO and NLESO based on state error,a scheme of SADRC is given as follows:

(i) If there are possible initial state errors between the states of the plant and the ESO,NLESO is adopted during a transition timeT(artificially set) to avoid“peaking phenomenon”;otherwise,this step is skipped and the system directly goes into the following step;

(ii) After step (i),the controller automatically switches between LESO and NLESO according to the errore.Choose a specificδs,whene<δs,NLESO starts to work;otherwise,LESO begins to work.δsis a general boundary of the performance of LESO and NLESO,i.e.,whene<δs,NLESO is superior to LESO,whereas LESO is superior to NLESO.δscan be determined by simulation,experiment or theoretical computation [31].

The above process is also shown inFig.1.In general,this scheme of SADRC unites the merits of LADRC and NLADRC.

Fig.1 Scheme of SADRC

The structure of SADRC will be discussed in detail with the controller design of SADRC based MIMO continuous system in the next section.

2.1.3 MIMO continuous systems

In this paper,we consider a class of MIMO continuous systems composed of coupled subsystems with external disturbances described by

wherei=1,2,···,m,x(t)=(x1T(t),···,xmT(t))T∈Rn×m(g1=g2=···=gm=n),u(t)=(u1(t),···,um(t))T∈Rm,y(t)=(y1T(t),···,ymT(t))T∈Rn×mrepresents the state,the control input and the output of the system (6),respectively;g1=g2=···=gm=nmeans that every subsystem of system (6) isn-dimensional;fi(t,x(t),wi(t)) denotes the“total disturbance”of the subsystemxi(t),andwi(t) represents the corresponding external disturbance;cil(i,l=1,2,···,m) is the control coefficient.

Then the control gain matrixCis described as

Suppose matrixCis invertible or generalized inverse matrices exist,and define virtual control input asuv(t)=(u1v(t),u2v(t),···,umv(t))T∈Rm,system (6) can be rewritten as

wherei=1,2,···,m,uv(t)=Cu(t).

For the subsequent stability analysis,define the following vectorxs(t)=(xs1T(t),···,xsnT(t))T∈Rm×n(h1=h2=···=hn=m),wherexsi(t)=(xsi1(t),···,xsm(t))T∈Rmrepresents the same order states of subsystems,then the system (8) can be rewritten as

2.1.4 Structure of SADRC

Take one of the subsystems as an example,the SESO is designed as follows:

wherezij(i=1,2,···,m;j=1,2,···,si+1) is the estimation ofxij(i=1,2,···,m;j=1,2,···,si+1) inx-subsystem,andzi(si+1)is the estimation of“total disturbance”,si=n;βi0j(i=1,2,···,m;j=1,2,···,si+1) is the NLESO’s gain in SESO,and supposeβi0jL(i=1,2,···,m;j=1,2,···,si+1) the LESO’s gain in SESO,isλij(i=1,2,···,m;j=1,2,···,si+1) multiple of the NLESO’ gain,λijis constant.Then the switching function fsij(e) (i=1,2,···,m;j=1,2,···,si+1) can be described as

whereδisis the critical value for the switching between NLESO and LES;fLij(ei) (i=1,2,···,m;j=1,2,···,si+1) is the nonlinear function of NLESO,defined as

whereαij<1.

For the subsequent stability analysis,define

Let

The SESO (10) can be rewritten as

wherei=1,2,···,m;si=n.

The LESF of SADRC can be described as

whereviq(q=1,2,···,n) is the reference input;kiq(q=1,2,···,n) is the controller gain.

2.1.5 Decoupling controller design based on SADRC

Consider the engineering application,make the following assumptions for the MIMO systems.

Assumption 1The reference inputs and their all-order derivatives are assumed to be bounded.

Assumption 2The changing rate of disturbances are assumed to be bounded.

Defining the desired values of every state:xd(t) =(x1dT(t),···,xndT(t))T∈Rm×n(h1=h2=···=hn=m),error vector:δ(t)=(δ1T(t),···,δnT(t))T∈Rm×n(r1=r2=···=rn=m),whereδi(t)=(xi1(t) −xdi1(t),···,xim(t) −xdim(t))∈Rm,substitutexd(t) andδ(t) into (9),then we obtain

Then the objective of the controller will find LESF to makeδ(t) converge asymptotically to zero.

SESO is used to estimate the“total disturbance”of the system,takef(t,x(t),w(t)) as the extended state of system (9),define ς(t)=(ς1T(t),···,ςnT(t),ςn+1T(t))T=(xs1T(t),···,xsnT(t),f(t,x(t),w(t)))Tthen the state-space model of system (9) with the extended state can be described as

whereh(t)=(t,x(t),w(t)),and assumeh(t) is bounded;

The SESO for system (18) can be designed as

The LESF of the system is defined as

whereK=andKg=diag(kg1,···,kgm)(g=1,2,···,n) is the control gain;(t,x(t),w(t)) is the estimation off(t,x(t),w(t)).

Substituting (20) into (19),we have

whereAH=A−BK.If parameters ofKare chosen to makeAHHurwitz,system (21) is globally asymptotically stable.

2.2 Stability analysis

Then the LESF of (20) can be rewritten as

Substituting (24) into (17),we get

Combining (23) and (25),we have

2.2.1 Observer error system stability analysis

Theorem1 Consider the observer error dynamics of(21) under Assumption 2,if the observer gain vectorLis chosen such that (A0−LQ) is stable,thenM(t) exponentially converges to the bounded ballBr1=M(t)∈Rm(n+1)×1,||ς˜(t)||≤2λmax(P0)hmax,whereλmax(P0) is the maximum eigenvalue ofP0,P0is the solution of the equation (A0−LQ)TP0+P0(A0−LQ)=−Im(n+1)×m(n+1),andhmaxis the absolute maximum value ofh(t) [33].

ProofDefine the Lyapunov functionVas follows:

Taking the derivative ofV,we get

ThusV˙(t)<0,whenever ||M(t)||>2λmax(P0)hmax.M(t)exponentially converges to bounded ballBr1=M(t)∈Rm(n+1)×1,≤2λmax(P0)hmax.

Remark 1From (19),we can conclude that,ifQconverges to zero,will converge to zero,thus→ς.

2.2.2 Closed-loop system stability analysis

Consider the closed-loop system (26). Define δcl(t)=[δ(t),M(t)]T,the the system (26) can be rewritten as

Theorem 2Consider the closed-loop system (27) under Assumption 2,if the gain vectorKand the observer gain vectorLare chosen such thatAclis Hurwitz,thenδcl(t) exponentially converges to the bounded ballBr2=δcl(t)∈R(mn+m(n+1))×1,||δcl(t)||≤2λmax(Pcl)hmax,whereλmax(Pcl) is the maximum eigenvalue ofPcl,Pclis the solution of equationAclTPcl+PclAcl=−I(mn+m(n+1))×(mn+m(n+1)),hmaxis the absolute maximum value ofh(t).

ProofDefine the Lyapunov functionV1as follows:

Taking the derivative ofV1(t),we obtain

3.Case study

In this section,two case studies are used to test the antidisturbance and robustness of SADRC,LADRC,and NLADRC.For a fair comparison,the parameters of most controllers are transported from the public literatures.

3.1 Binary distillation column system

3.1.1 Binary distillation column modeling

The binary distillation column system without time-delay is described as follows [34]:

wherey1andy2are the outputs of the system;ud1andud2represent the inputs;K11=12.8,K12=−18.9,K21=6.6,K22=−19.4,T11=16.7,T12=21,T21=10.9 andT22=14.4 are model parameters,where parameter perturbation is ubiquitous.

Carrying out the Laplace transform to (32),we obtain

Design LADRC,NLADRC and SADRC controllers fory1channel andy2channel respectively in system (34).The parameters of these controllers are shown inTable 1.

Table 1 Parameters of LADRC,NLADRC and SADRC controllers

InTable 1,wcrepresents the controller bandwidth of LADRC ;woandwoNrepresent the observers of LESO and NLESO,respectively.

In the SADRC based controller for the binary distillation column system,the observer gainLis described as

According to (15),the value ofλ0i(i=1,2,3) is correlated withe.Make curves ofλ0iaboute,the characteristics are shown inFig.2.

Fig.2 Curves of λ0i (i=1,2,3) about e

FromFig.2,we can conclude that although the value ofλ0i(i=1,2,3) varies withe,it is wobbling within a range.In the entire interval,set 0.002 as the step size,solve Lyapunov functions via Matlab:(A0−LQ)TP0+P0(A0−LQ)=I6×6,AclTPcl+PclTAcl=I10×10.We obtain that there always exist solutions forP0andPcl.According to Theorem 1 and Theorem 2,the observer error system and the closed-loop system of the SADRC based binary distillation column system is stable independently.

3.1.2 Simulation results

(i) Anti-disturbance simulation

Set the initial value of two channels:y1(0)=0,y2(0)=0,particularlyy1channel has initial state error,letz11=0.2.Set the target outputs:yd1=1,yd2=1 att=0 s,and add the disturbances of the magnitude 0.001 and 0.3,respectively,into the two outputs att=25 s.The integrated absolute error (IAE) for each channel is calculated in the whole 50 s.The results are shown inFig.3,Fig.4andTable 2,where IAE=+

FromFig.3,Fig.4andTable 2,it can be concluded that:i) All three control methods can realize the decoupling control for the binary distillation column system;ii) LADRC is sensitive to the initial state error,and the performance may deteriorate due to the initial error,while NLADRC and SADRC are insensitive to this;iii) When the disturbance is relative small,the performances of LADRC,NLADRC and SADRC are nearly the same,which means the observer gains of three are all properly chosen and can estimate the disturbance;iv) When the disturbance turns big,the performance of LADRC and SADRC are nearly the same and both superior to NLADRC.Overall,SADRC may be superior to both NLADRC and LADRC in anti-disturbance and can effectively deal with complex circumstances.

Fig.3 Tracking performance for the binary distillation column system

Fig.4 Observed“total disturbance”for the binary distillation column system

In fact,the switching thresholdδsand the linearity thresholdδcan also affect the anti-disturbance performance of the control system,as both of them have influences on the observer gains of SADRC.As a footnote,on the condition that the relationship between the bandwidth of LESO and the one of NLESO remains the same,letδs=0.003,0.005,0.008,δ=0.0005,0.002,respectively,repeat the above anti-disturbance simulation experiments,the performances of the SADRC controllers with differentδsandδare shown inFig.5andFig.6,summarized inTable 3andTable 4.

Table 3 Comparison of IAE for the binary distillation column system in anti-disturbance simulation (δ=0.002)

Table 4 Comparison of IAE for the binary distillation column system in anti-disturbance simulation (δ=0.000 5)

Fig.5 Tracking performance for the binary distillation column system with a different δs and δ

Table 2 Comparison of IAE for the binary distillation column system in anti-disturbance simulation

From the above results,it can be concluded that:differentδsandδhave little influence on the performance of the condition that the relationship between the bandwidth of LESO and the one of NLESO is fixed.This is because thatδscan be determined by theoretical compution [31],when the relationship between the bandwidth of LESO and the one of NLESO is fixed,so is the range ofδs.The performances with differentδsof the same order of magnitude are similar.δaims to suppress high frequency oscillations at zero and though it has influence on the observer gain of NLESO,asδ<δs,whenδsis fixed,δin this paper is too small to have a large effect on the performance of the whole control system.

(ii) Robustness simulation

Set the initial value of two channels:y1(0)=0,y2(0)=0,and either channel exists state error;set the target outputs:yd1=1,yd2=1 att=0 s.Add random perturbation within a range of ±10% to all the parameters in the system (33) before the simulation starts and repeat the simulation by 200 times.Every time the IAE for each channel is calculated in the whole 50 s.The records of overshootσand IAE for every experiment are shown inFig.7,and summarized inTable 5,whereσ=σy1+σy2;IAE=IAEy1+IAEy2.

Fig.7 Robustness performance for the binary distillation column system

Table 5 Comparison of IAE for the binary distillation column system in robustness simulation

FromFig.7andTable 5we can conclude that:i) The intensity of the points reveals that all three control methods have decent robustness; ii) The robustness of LADRC is a little better among all three control methods,and SADRC and NLADRC are nearly the same.That is caused by the small tracking error,which means that NLADRC mainly works in the whole simulation in the SADRC controller,so the performances of the SADRC controller and the NLADRC one are similar,and the observer gain of LADRC is bigger than that of NLADRC,leading to that the tracking error converges to zero more quickly,as a result,the IAE of LADRC is thus smaller.

3.2 Attitude control for 3-DOF Hover

3.2.1 3-DOF Hover model

The attitude model of the 3-DOF Hover system can be described [36] as

where ϕ,θandψrepresent the pitch angle,roll angle and yaw angle of the system,respectively;l=0.197 m represents the distance between the propeller motor and the pivot on the axis;Vi(i=f,b,l,r) represent the front motor voltage,the back motor voltage,the left motor voltage and the right motor voltage;Kf=0.1188 N/V represents the thrust-force constant;KM= 0.0036 N·m/V is the thurst-torque constant;Kafx=Kafy=0.008 N·m/rad/s,Kafz=0.009 N·m/rad/s are the drag constants about thex,y,zaxes separately;Jx=Jz=0.0552 kg·m2,Jy=0.11 kg·m2represent the moment of inertia about thex,y,zaxes respectively.

Introducing virtual control inputUi(i=1,2,3),taking the coupling interactions among subsystems as“internal disturbance”of the system and considering the external disturbancewi(i= 1,2,3) of every subsystem,let η=ω=η˙,then the system (36) can be described as

Fdisrepresents the“total disturbance”,and can be described as

Design LADRC,NLADRC and SADRC controllers for ϕ channel,θchannel andψchannel respectively in system (37). The parameters of these controllers are shown inTable 6.

Table 6 Parameters of LADRC,NLADRC,SADRC controllers

In the SADRC based controller for 3-DOF Hover system,the observer gainLis described as

Within the entire intervals ofλ01,λ02andλ03,set 0.002 as the step size,solve Lyapunov functions via Matlab;(A0−LQ)TP0+P0(A0−LQ)=I9×9,AclTPcl+PclTAcl=I15×15,and we obtain that there always exist solutions forP0andPcl.According to Theorem 1 and Theorem 2,the observer error system and the closed-loop system of the SADRC based 3-DOF Hover system is stable separately.

3.2.2 Simulation results

(i) Anti-disturbance simulation

Set the initial value of three channels:ϕ(0)=0°,θ(0)=0°,ψ(0)=0°,andϕchannel andθchannel have initial state error separately,letz1ϕ=0.04°,z1θ=0.4°.Set the target outputs:ϕd=3°,θd=3°,ψd=3° att=0 s.Add the disturbances of the magnitude 0.04°,0.6°,and 1° into the three outputs att=25 s for 3 s,respectively.The results are shown inFig.8,Fig.9andTable 7.

Fig.8 Tracking performance for the 3-DOF Hover system

Fig.9 Observed“total disturbance”for the 3-DOF Hover system

Table 7 Comparison of IAE for the 3-DOF Hover system in antidisturbance simulation

The performances of SADRC controllers with differentδsandδalso have been taken into consideration in this part.On the condition that the relationship between the bandwidth of LESO and the one of NLESO remains the same,letδs=0.003,0.005,0.008,δ=0.0005,0.002,respectively,repeat the above anti-disturbance simulation experiments in Subsection 3.2.2,the performances of the SADRC controllers with differentδsandδare shown inFig.10andFig.11and summarized inTable 8andTable 9.

Table 8 Comparison of IAE for the 3-DOF Hover system in antidisturbance simulation (δ=0.002)

Table 9 Comparison of IAE for the 3-DOF Hover system in antidisturbance simulation (δ=0.000 5)

Fig.10 Tracking performance for the 3-DOF Hover system with a different δs and δ

Fig.11 Observed“total disturbance”for the 3-DOF Hover system of each channel with a different δs and δ

The results once again prove that differentδsandδwill influence the performance,but the influences are not obvious under the current experimental condition.

(ii) Robustness simulation

Set the initial value of three channels:ϕ(0)=0°,θ(0)=0°,ψ(0)=0°,and none channel exists has error.Set the target outputs:ϕd=3°,θd=3°,ψd=3° att=0 s.Add random perturbation within a range of ±10% to all the parameters in the system (37) before the simulation starts and repeat the simulation by 200 times.Every time the IAE for each channel is calculated in the whole 50 s.The records of overshootσand IAE for every experiment are shown inFig.12,and summarized inTable 10,whereσ=σϕ+σθ+σψ;IAE=IAEϕ+IAEθ+IAEψ.

Table 10 Comparison of IAE for the 3-DOF Hover system

Fig.12 Robustness performance for the 3-DOF Hover system

From the above results,we can conclude that:i) The proved decoupling control approach can also be applied to the systems that the number of inputs and outputs is not equal;ii) The performance of LADRC is sensitive to the initial state error,when the initial state error is relatively big,its performance may deteriorate due to the“peaking phenomenon”,while the initial state error has little influence on the performance of NLADRC and SADRC;iii) The NLADRC is more suitable to deal with relative small disturbance while LADRC is suitable to deal with the relative big one.In general,SADRC may be superior to both NLADRC and LADRC in terms of the anti-disturbance and robustness in occasions where the amplitude of the disturbance is unsure or unstable,as it combines the advantages of both LADRC and NLADRC.

4.Conclusions

In this paper,the SADRC based decoupling control approach is proposed for a class of MIMO continuous systems,which can be applied to systems with equal or unequal number of inputs or outputs.Stability of the closedloop system is also proved.The stability analysis method is based on the Lyapunov function and does not require an accurate model,only the number of subsystems,the order of the subsystem and the parameters of controllers are needed.It is convenient for engineering applications and can provide references for parameter tuning.Two cases also verify that the SADRC may be superior to both NLADRC and LADRC in the anti-disturbance and robustness in some occasions as it combines the advantages of both systems.However,the proposed stability analysis method is achieved by means of computer calculation and is not rigorous enough in theory to some extent.Other approaches will be characterized to give strict theoretical proofs for stability analysis in the near future.