# Output Regulation for a Class of MIMO Uncertain Stochastic Nonlinear Systems by Active Disturbance Rejection Control Approach

Authors: Chunwan Lv; Guoshun Huang; Zhengyong Ouyang; Jian Chen; Lingxin Bao
Journal: Mathematical Problems in Engineering
 (2019)

---

## Abstract

In this paper, the active disturbance rejection control (ADRC) approach is applied to a class of multi-input multioutput (MIMO) uncertain stochastic nonlinear systems. An extended state observer (ESO) is first designed for estimation of both unmeasured states and stochastic total disturbance of each subsystem which represents the total effects of internal unmodeled stochastic dynamics and external stochastic disturbance with unknown statistical property. The ADRC controller based on the states of ESO is further designed to achieve the closed-loop system’s output regulation performance including practical mean square reference signals tracking, disturbance attenuation, and practical mean square stability when the reference signals are zero avoiding solving any partial differential equations in the conventional output regulation theory. Some numerical simulations are presented to demonstrate the effectiveness of the proposed ADRC approach.


---

## Body

## 1. Introduction

During the past couple of years, there have been existing representative control approaches to cope with uncertainties in controlled plants such as the internal model principle [1–3], the robust control [4], and the adaptive control [5]. Nevertheless, a majority of these control methods pay attention to the worst situation so that the design of controller is comparatively conservative. The active disturbance rejection control (ADRC), as a nontraditional control strategy, was first put forward by Han in leading paper [6]. The disturbance coped with by ADRC is much more general which can be the total coupling effects of internal unmodeled system dynamics and external disturbance. The most noteworthy feature of ADRC is that an extended state observer (ESO), as its key part, is designed to estimate the disturbance in real time so that the disturbance can be cancelled in the ESO-based feedback loop. This estimation/cancellation strategy leads to less control energy consumption in the control engineering practice [7].Many industry practitioners have been paying more attention to the ADRC approach as presented in recent survey type paper [8]. Specific applications in different fields emerge in large numbers including synchronous motors [9], DC-DC power converters [10], control system in superconducting RF cavities [11], and flight vehicles control [12]. On the other hand, there have been lots of theoretical researches like the convergence analysis of ADRC for uncertain nonlinear systems [13–16] and the stabilization and output tracking problem for distributed parameter systems by the ADRC approach [17–21].However, tardy theoretical progress has been made about ADRC for stochastic systems. Some progress could be found such as the practical mean square convergence analysis of ESO for the open-loop of a class of MIMO stochastic nonlinear systems [22] and the practical mean square convergence analysis of ADRC for both single-input single-output stochastic nonlinear systems [23] and lower triangular stochastic nonlinear systems [24]. The practical mean square stability of MIMO stochastic nonlinear systems by the ADRC approach is addressed in [25].Although there have been some works on ADRC for stochastic nonlinear systems like [22, 24, 25] where the considered external disturbance is the bounded stochastic noise with unknown statistical characteristics existing widely in practical systems [26–28], the output regulation problem for this class of stochastic nonlinear systems receives less attention. In [22], the authors focus on the convergence of practical mean square estimation errors by ESO for the open-loop of systems without considering the performances of the closed-loop system under the ESO-based feedback. The practical mean square stability of the ADRC’s closed-loop systems is investigated for lower triangular stochastic nonlinear systems in [24] and MIMO stochastic nonlinear ones in [25] where the output-feedback stabilization problem is just a special case of the output regulation one in this paper. As the continuous research of [22–25], the ADRC approach is applied in the output regulation problem for a class of MIMO uncertain stochastic nonlinear systems with large stochastic uncertainties including unknown nonlinear system functions, external stochastic disturbance with unknown statistical property, unknown stochastic inverse dynamics, uncertain nonlinear coupling effects between subsystems, and uncertainties caused by the partially unknown input gains, where the output regulation performance of the resulting closed-loop systems includes practical mean square reference tracking, disturbance attenuation, and practical mean square stability when the reference signals are zero. To be specific, in this paper we consider the output regulation problem for the partial exact feedback linearizable MIMO system [29] subject to vast stochastic uncertainties as follows:(1)dxi=Anixidt+Bnifit,x,ζ,w+∑l=1mbiluldt,dζ=g1t,x,ζ,wdt+g2t,x,ζ,wdW1t,yi=Cnixi,i=1,2,…,m,where x=(x1⊤,…,xm⊤)⊤∈Rn with xi=(xi1,…,xini)⊤ and n=n1+⋯+nm, u=(u1,…,um)⊤∈Rm, and y=(y1,…,ym)⊤∈Rm are the state, control, and measured output of the system, respectively; ζ=(ζ1,…,ζs)⊤∈Rs denotes the state of stochastic inverse dynamics; the functions fi:[0,∞)×Rn+s+1→R, g1:[0,∞)×Rn+s+1→Rs, and g2:[0,∞)×Rn+s+1→Rs×p are unknown; the constants bil(i,l=1,2,…,m) are the partially unknown control coefficients with some known nominal values bil∗ satisfying Assumption (A3); {W1(t)}t≥0 is a p-dimensional standard Wiener process defined on a complete probability space (Ω,F,{Ft}t≥0,P) with Ω being a sample space, F a σ-field, {Ft}t≥0 a filtration, and P the probability measure; w(t)≜ψ(t,W2(t))∈R is the external stochastic disturbance where ψ(·):[0,∞)×Rq→R is an unknown bounded function satisfying Assumption (A1) and {W2(t)}t≥0 is a q-dimensional standard Wiener process defined on (Ω,F,{Ft}t≥0,P) as well and is mutually independent with {W1(t)}t≥0; in addition, we denote(2)Ani=0Ini-100ni×ni,Bni=0,…,0,1ni×1⊤,Cni=1,0,…,01×ni.For each1≤i≤m, the stochastic total disturbance of each i-subsystem is defined as follows:(3)xini+1≜fit,x,ζ,w+∑l=1mbil-bil∗ul,which represents the total effects of unknown nonlinear system functions, external stochastic disturbance with unknown statistical property, unknown stochastic inverse dynamics, uncertain nonlinear coupling effects between subsystems, and uncertainties caused by the partially unknown input gains. So the stochastic uncertainties in the considered system are very complex.For given, bounded, deterministic reference signalsvi(t)(i=1,2,…,m) whose derivatives v˙i(t),v¨i(t),…,vi(ni+1)(t) are assumed to be bounded, the control objective is to design an output-feedback control such that for any initial states the output yi converges practically to vi(t) in mean square sense and at the same time xij converge practically to vi(j-1)(t) in mean square sense for all j=2,…,ni. The output-feedback stabilization problem for the class of MIMO uncertain stochastic nonlinear systems is covered by letting vi(t)≡0 for all t≥0.On the basis of a partial exact feedback linearizable MIMO system [29] widely investigated in control theory and the fact that stochastic uncertainties are ubiquitous in practical control engineering and often cause disadvantageous effects on control performance, the ADRC approach is addressed for the output regulation problem of system (1) including practical mean square reference signals tracking, disturbance attenuation, and practical mean square stability when the reference signals are zero in this paper. It should be noticed that system (1) representing a partial exact feedback linearizable MIMO system [29] with vast stochastic uncertainties is quite general and has physical and engineering background. Firstly, the SISO nonlinear systems and MIMO nonlinear ones widely addressed by the ADRC approach in available literatures [6, 7, 13, 14, 21, 30] are covered as special cases of system (1) when ψ(·) is the function of time variable t only: w(t)≜ψ(t) and g2(·)≡0. Secondly, the external stochastic disturbance w(t) is quite general in the sense that it is not required to know its statistical characteristics since the function ψ(·) can be unknown and the bounded stochastic noise investigated in [26–28] in many practical systems is also covered as its special case. Finally, system (1) covers some stochastic systems considered in the aforementioned literatures like SISO stochastic nonlinear systems in [31] when m=1, ζ(·)≡0, and b11=1.The main contributions of this paper can be summarized as follows: (a) The ADRC approach is systematically proposed to solve the output regulation problem for a class of MIMO uncertain stochastic nonlinear systems without difficulty in solving any partial differential equations compared with the conventional output regulation theory. (b) The stochastic uncertainties dealt with by ADRC in this paper are very general including unknown nonlinear system functions, external stochastic disturbance with unknown statistical property, unknown stochastic inverse dynamics, uncertain nonlinear coupling effects between subsystems, and uncertainties caused by the partially unknown input gains. (c) Most available output-feedback controls for stochastic nonlinear systems are designed to guarantee the global asymptotic stability in probability provided that the noise vector field vanishes at the origin [32] or only the noise-to-state (or input-to-state) stability in probability [33] otherwise. In this paper, however, the practical mean square convergence is obtained by the ADRC approach without assuming that the noise vector field should be vanishing at the origin.The rest of this paper is presented as follows. In Section2, both ESO and ESO-based feedback control are designed for the x-subsystem of (1), the assumptions of the main result are stated, and the output regulation performance of the closed-loop is summarized as Theorem 2. In Section 3, a rigorous proof of Theorem 2 is given. Finally, in Section 4, some numerical simulations are presented to illustrate the effectiveness of the proposed ADRC approach.The following notations are used throughout this paper.Rn denotes the n-dimensional Euclidean space and Rn×m represents the space of all real n×m-matrices; for a vector or matrix K, K⊤ denotes its transpose; for a square matrix K, Tr(K) denotes its trace; In×n denotes the n×n unit matrix; λmin(K) and λmax(K) denote the minimal and maximal eigenvalues of the symmetric real matrix K, respectively; K denotes the Euclidean norm of the vector K and the corresponding induced norm when K is a matrix; (a(ij))m×n denotes an m×n matrix with entries a(ij); for a differentiable function f:Rn→R, ∂f/∂z≜(∂f/∂z1,…,∂f/∂zn)⊤ for z=(z1,…,zn)⊤; for a twice differentiable function f:Rn→R, ∂2f/∂z2≜(∂2f/∂zizj)n×n(i,j=1,2,…,n) for z=(z1,…,zn)⊤; for a matrix valued function f:Rn→Rm×s, f≜(f(ij))m×s(i=1,2,…,m,j=1,2,…,s).


## 2. Main Results

By analogy with [30], the one-parameter tuning linear ESO for the x-subsystem of (1) is designed as follows:(4)x^˙i1=x^i2+ai1ryi-x^i1,x^˙i2=x^i3+ai2r2yi-x^i1,⋮x^˙ini=x^ini+1+ainirniyi-x^i1+∑l=1mbil∗ul,x^˙ini+1=aini+1rni+1yi-x^i1,i=1,2,…,m,where aij(i=1,2,…,m,j=1,2,…,ni+1) are designed parameters such that the following matrix is Hurwitz:(5)Ei=-ai110⋯0-ai201⋯0⋯⋯⋯⋯⋯-aini00⋱1-aini+100⋯ni+1×ni+1,r>0 is the gain parameter to be tuned, and bil∗ is the nominal value of bil satisfying Assumption (A3). ESO (4) is designed to estimate both stochastic total disturbance and unmeasured states by choosing suitable parameters aij and tuning the gain parameter r. Specially, x^i(ni+1) is the estimate of the stochastic total disturbance xi(ni+1) defined by (3). It should be noted that we only need to tune one parameter r in ESO (4) based on the estimation accuracy and the variation of the stochastic total disturbance. Generally, the higher the estimation accuracy is needed and the faster the stochastic total disturbance varies, the larger the parameter r needs to be set. Here and throughout the paper, we always drop r for the solution of (4) by abuse of notation without confusion.For all1≤i≤m, let(6)vi1,vi2,…,vini+1=vi,v˙i,…,viniand(7)ϱizi1,…,zini=ki1zi1+ki2zi2+⋯+kinizini,∀zi1,…,zini⊤∈Rni.ESO (4) based output-feedback control is designed as(8)ui=∑l=1mb^il∗ϱlx^l1-vl1t,x^l2-vl2t,…,x^lnl-vlnlt-x^lnl+1+vlnl+1t,1≤i≤m,where b^il∗ are defined in (13) and the feedback gain parameters kij(i=1,2,…,m,j=1,2,…,ni) are chosen such that the following matrix is Hurwitz:(9)Fi=010⋯0001⋯0⋯⋯⋯⋯⋯000⋱1ki1ki2⋯kini-1kinini×ni.To obtain practical mean square convergence of the closed-loop of thex-subsystem of (1) under ESO (4) based output-feedback control (8) including ESO’s estimation of unmeasured states and stochastic total disturbance, practical mean square reference signals tracking, disturbance attenuation, and practical mean square stability, the following assumptions are needed.Assumption (A1) is about the unknown functionψ(·) defining the external stochastic disturbance.Assumption (A1). ψ(t,θ):[0,∞)×Rq→R is continuously differentiable and twice continuously differentiable with respect to t and θ, respectively, and there exists a known constant D1>0 such that, for all θ=(θ1,…,θq)⊤∈Rq,(10)ψt,θ+∂ψt,θ∂t+∂ψt,θ∂θ+∑i=1q∂2ψt,θ∂θi2≤D1.Remark 1.
Roughly speaking, Assumption (A1) indicates that both the external stochastic disturbance and its “variation” should be bounded which is reasonable since it is as a part of the stochastic total disturbance estimated by ESO.Assumption (A2) is a prior assumption about the unknown functionsfi(·), g1(·), and g2(·) in system (1).Assumption (A2). fi(·)(i=1,…,m) are continuously differentiable and twice continuously differentiable with respect to t and other arguments, respectively, and gi(·)(i=1,2) are locally Lipschitz continuous in (x,ζ,w) uniformly in t∈[0,∞). There exist known constants Di≥0(i=2,3,4) and a nonnegative continuous function ς∈C(R;R) such that, for all t≥0, x∈Rn, ζ∈Rs, w∈R, i=1,2,…,m,(11)∂fit,x,ζ,w∂t+g1t,x,ζ,w≤D2+D3x+ςw,(12)∂fit,x,ζ,w∂x+∂fit,x,ζ,w∂ζ+∑j,l=1s∂2fit,x,ζ,w∂ζj∂ζl+∂fit,x,ζ,w∂w+∂2fit,x,ζ,w∂w2+∑j=1p∑i=1sg2ijt,x,ζ,w≤D4+ςw.Assumption (A3) is about the prior estimatesbil∗(i,l=1,2,…,m) for the unknown control parameters bil(i,l=1,2,…,m) in system (1).Assumption (A3). The matrix with entries bil∗(i,l=1,2,…,m) in (4) is invertible with the inverse matrix given by(13)b11∗b12∗⋯b1m∗b21∗b22∗⋯b2m∗⋮⋮⋱⋮bm1∗bm2∗⋯bmm∗-1=b^11∗b^12∗⋯b^1m∗b^21∗b^22∗⋯b^2m∗⋮⋮⋱⋮b^m1∗b^m2∗⋯b^mm∗,and the nominal values bil∗ of bil satisfy(14)Ξ≜1-∑i,l,d=1m2λmaxQibil-bil∗b^ld∗adnd+1>0,where Qi are the positive definite matrix solution satisfying QiEi+Ei⊤Qi=-I(ni+1)×(ni+1).The main result on practical mean square convergence of the closed-loop of thex-subsystem of (1), (4), and (8), which includes ESO’s practical mean square estimation of both unmeasured states and stochastic total disturbance and output regulation performance, is summarized in Theorem 2.Theorem 2.
Under Assumptions (A1)–(A3) and supposing thatsupt≥0(vi(t),v˙i(t),…,vi(n+1)(t))≤M for some constant M≥0, then the closed-loop of the x-subsystem of (1), (4), and (8) is practically mean square convergent in the sense that there are a constant r∗>0 (specified by (32) later) and a r-dependent constant tr∗>0 with r>r∗ such that, for any initial values and all t≥tr∗,(15)Exijt-x^ijt2≤Γr2ni+3-2j,1≤i≤m,1≤j≤ni+1,and(16)Exijt-vijt2≤Γr,1≤i≤m,1≤j≤ni,where Γ>0 is a r-independent constant.Remark 3.
It should be noticed that the practical mean square convergence addressed in this paper includes ESO’s practical mean square estimation of both unmeasured states and stochastic total disturbance and output regulation performance for the closed-loop system, but the practical mean square convergence addressed in [22] only refers to ESO’s practical mean square estimation of both unmeasured states and stochastic total disturbance for the open-loop system. In addition, there does not exist an essential difference between the upper bound of the estimation error of ESO in practical mean square sense in this paper and the one in [22] since the high gain parameter in ESO is denoted by r in this paper and by 1/ε in [22], respectively.


## 3. Proof of the Main Result

Proof of Theorem2.
For all1≤i≤m,1≤j≤ni+1, we set(17)ηij=rni+1-jxij-x^ij,ηi=ηi1,…,ηini+1⊤,η=η1⊤,…,ηm⊤⊤,κij=xij-vijt,κi=κi1,…,κini⊤,κ=κ1,…,κm⊤,(18)Δi=ϱix^i1-vi1t,x^i2-vi2t,…,x^ini-vinit-ϱiκi,(19)Φixi=xi2,xi3,⋯,ϱix^i1-vi1t,x^i2-vi2t,…,x^ini-vinit+xini+1-x^ini+1⊤,Φx=Φ1x1⊤,Φ2x2⊤,…,Φmxm⊤⊤.By Itô’s formula, we can obtain that(20)dfit,x,ζ,walong1=∂fit,x,ζ,w∂t+∂fit,x,ζ,w∂x⊤Φx+∂fit,x,ζ,w∂ζ⊤g1t,x,ζ,w+12Trg2⊤t,x,ζ,w∂2fit,x,ζ,w∂ζ2g2t,x,ζ,w+∂fit,x,ζ,w∂w∂ψt,W2t∂t+12∑i=1q∂2ψt,W2t∂θi2+12∂2fit,x,ζ,w∂w2∑i=1q∂ψt,W2t∂θi2dt+∂fit,x,ζ,w∂ζ⊤g2t,x,ζ,wdW1t+∂fit,x,ζ,w∂w∂ψt,W2t∂θ⊤dW2t≜ϑi1dt+ϑi2dW1t+ϑi3dW2t,where we set(21)ϑi2=ϑi2,j1×p,ϑi2,j=∑l=1s∂fit,x,ζ,w∂ζlg2ljt,x,ζ,w,ϑi3=ϑi3,j1×q,ϑi3,j=∂fit,x,ζ,w∂w∂ψt,W2t∂θj.
By Assumptions (A1)-(A2), we can easily conclude that there existr-independent positive constants δi1,δi2,δi3,δi4 such that(22)ϑi1≤δi1+δi2η+δi3κ,ϑi22+ϑi32≤δi4.
Moreover,(23)ddtalong4∑l=1mbil-bil∗ul=ddtalong4∑l,d=1mbil-bil∗b^ld∗ϱdx^d1-vd1t,…,x^dnd-vdndt-x^dnd+1+vdnd+1t=∑l,d=1mbil-bil∗b^ld∗∑j=1nd-1kdjx^dj+1-vdj+1t+adjrnd-jηd1+∑l,d=1mbil-bil∗b^ld∗kdndx^dnd+1-vdnd+1t+adndηd1+∑k=1mbdk∗uk+∑l,d=1mbil-bil∗b^ld∗-radnd+1ηd1+∑l,d=1mbil-bil∗b^ld∗v˙dnd+1t≜ϑi4.Suppose that r>1. It follows that there exist r-independent positive constants δi5,δi6,ξi such that(24)ϑi4≤δi5η+δi6κ+rξiη,where(25)ξi=∑l,d=1mbil-bil∗b^ld∗adnd+1.
Thus, it follows that the closed-loop of thex-subsystem of (1), (4), and (8) is equivalent to(26)dκi=Aniκidt+Bniϱiκi+Δi+ηini+1dt,dηi=rAni+1ηidt-rai1ηi1⋯aini+1ηi1dt+Bni+1ϑi1+ϑi4dt+Bni+1ϑi2dW1t+Bni+1ϑi3dW2t,1≤i≤m,where Δi is defined as that in (18). It follows from the definition of Δi in (18) that(27)Δi2≤max1≤j≤nikij2xi1-x^i12+⋯+xini-x^ini2=max1≤j≤nikij21r2niηi12+⋯+1r2ηini2≤max1≤j≤nikij2ηi2,i=1,2,…,m.The remaining proof is arranged in the following three steps.
Step  1. We prove that the solution (κ,η) of system (26) is practically mean square bounded.
We first define the positive definite functionsV1i(·):Rni→R, V2i(·):Rni+1→R, and V(·):R2n+m→R as follows:(28)V1iκi=κi⊤Hiκi,V2iηi=ηi⊤Qiηi,Vκ,η=∑i=1mV1iκi+∑i=1mV2iηi,where Hi and Qi are the positive definite matrix solutions satisfying the Lyapunov equations HiFi+Fi⊤Hi=-Ini×ni and QiEi+Ei⊤Qi=-I(ni+1)×(ni+1), respectively.
It is easy to obtain that(29)λminHiκi2≤V1iκi≤λmaxHiκi2,∂V1iκi∂κini≤2λmaxHiκi,∂2V1iκi∂κni2≤2λmaxHi,λminQiηi2≤V2iηi≤λmaxQiηi2,∂V2iηi∂ηini+1≤2λmaxQiηi,∂2V2iηi∂ηini2≤2λmaxQi,∂2V2iηi∂ηini+12≤2λmaxQi,∂2V2iηi∂ηini∂ηini+1≤2λmaxQi.
Apply Itô’s formula toV(κ,η) with respect to t along the solution (κ,η) of system (26) to obtain(30)dVκ,η=∑i=1m∑j=1ni-1∂V1iκi∂κijκij+1+∂V1iκi∂κiniϱiκi+∂V1iκi∂κiniΔi+∂V1iκi∂κiniηini+1dt+r∑i=1m∑j=1ni∂V2iηi∂ηijηij+1-aijηi1-∂V2iηi∂ηini+1aini+1ηi1dt+∑i=1m∂V2iηi∂ηini+1ϑi1+ϑi4dt+∑i=1m12∂2V2iηi∂ηini+12ϑi22+ϑi32dt+∑i=1m∂V2iηi∂ηini+1ϑi2dW1t+∑i=1m∂V2iηi∂ηini+1ϑi3dW2t.
It is easy to conclude that there exist0<μ<1 and r1>0 such that(31)-r1Ξ2+4μmax1≤i≤mλmax2Himax1≤i≤mmax1≤j≤nikij2+4μmax1≤i≤mλmax2Hi+1+∑i=1m2λmaxQiδi2+4max1≤i≤m⁡λmax2Qiδi32μ+∑i=1m2λmaxQiδi5+4max1≤i≤m⁡λmax2Qiδi62μ<0,where Ξ is given in (14). Now we suppose that(32)r>r∗≜max1,r1,21-μΞ.Set(33)β1=maxmax1≤i≤mλmaxHi,max1≤i≤mλmaxQi.
It follows from (22), (24), (27), (31), and Young’s inequality that(34)dEVκ,ηdt≤-Eκ2+∑i=1m2λmaxHimax1≤j≤nikij·Eκi·ηi+∑i=1m2λmaxHiEκi·ηini+1-rEη2+∑i=1m2λmaxQiEηi·δi1+δi2η+δi3κ+δi5η+δi6κ+rξiη+∑i=1mλmaxQiδi4≤-Eκ2+μ4Eκ2+4μmax1≤i≤mλmax2Hi·max1≤i≤mmax1≤j≤nikij2Eη2+μ4Eκ2+4μmax1≤i≤mλmax2HiEη2-rEη2+Eη2+∑i=1mλmax2Qiδi12+∑i=1m2λmaxQiδi2Eη2+μ4Eκ2+4max1≤i≤m⁡λmax2Qiδi32μEη2+∑i=1m2λmaxQiδi5Eη2+μ4Eκ2+4max1≤i≤m⁡λmax2Qiδi62μEη2+r∑i=1m2λmaxQiξiEη2+∑i=1mλmaxQiδi4≤-1-μEκ2-rΞ2Eη2+β2≤-1-μβ1EVκ,η+β2,where β2≜∑i=1mλmax2(Qi)δi12+∑i=1mλmax(Qi)δi4 and Ξ is given in (14). It is easy to conclude that EV(κ(0),η(0))≤M1r2max1≤i≤mni for some r-independent constant M1. Hence for any r>r∗ and any ϵ>0, there exists tr≜rϵ such that, for all t≥tr, we have(35)EVκ,η≤e-1-μ/β1tEVκ0,η0+β2∫0te-1-μ/β1t-sds≤e-1-μ/β1trEVκ0,η0+β1β21-μ≤M2,for some r-independent constant M2>0.
Step  2. We prove the convergence of the estimation errors of ESO for both unmeasured states and stochastic total disturbance in practical mean square sense.
Similar to the operations used in (34), it follows from (35) that, for all t≥tr, we have(36)dEV2ηdt≤-rΞ2Eη2+μ2Eκ2+β2≤-rΞ2max1≤i≤m⁡λmaxQiEV2η+μM22min1≤i≤m⁡λminHi+β2.Set(37)β3=Ξ2max1≤i≤m⁡λmaxQi,β4=μM22min1≤i≤m⁡λminHi+β2.Then(38)EV2η≤e-rβ3t-trEV2ηtr+β4∫trte-rβ3t-sds.We can see from (35) that the first term of the right-hand side of (38) is bounded by e-rβ3 multiplied by a r-independent constant and the second term is bounded by 1/r multiplied by a r-independent constant when t≥tr+1; thus there exists a r-independent constant β5>0 such that, for all t≥tr+1,(39)EV2η≤β5r,and thus(40)Eηi2≤EV2ηλminQi≤β5rλminQi.Therefore, for all t≥tr+1, 1≤i≤m,1≤j≤ni+1, we have(41)Exij-x^ij2=1r2ni+2-2jEηij2≤1r2ni+2-2jEηi2≤β5λminQir2ni+3-2j.
Step  3. We prove the convergence of the reference signals tracking errors in practical mean square.
For anyr>r∗ and all t≥tr+1, it follows from (34) and (40) that(42)dEV1κdt≤-1-μ2Eκ2+4μmax1≤i≤mλmax2Hi·max1≤i≤mmax1≤j≤nikij2+4μmax1≤i≤mλmax2HiEη2≤-1-μ2Eκ2+4μmax1≤i≤mλmax2Hi·max1≤i≤mmax1≤j≤nikij2+4μmax1≤i≤mλmax2Hi·mβ5rmin1≤i≤m⁡λminQi≤-β6EV1κ+β7r,where we set(43)β6=1-μ/2max1≤i≤m⁡λmaxHiand(44)β7=4μmax1≤i≤mλmax2Himax1≤i≤mmax1≤j≤nikij2+4μmax1≤i≤mλmax2Himβ5min1≤i≤m⁡λminQi.Therefore, for any r>r∗ and all t≥tr+1, we have(45)Eκ2≤1min1≤i≤m⁡λminHie-β6t-tr-1EV1κtr+1+β7rmin1≤i≤m⁡λminHi∫tr+1te-β6t-sds.It follows from (35) that the first term of the right-hand side of (45) is bounded by e-β6rϵ multiplied by a r-independent constant and the second term is bounded by 1/r multiplied by a r-independent constant when for all t≥2tr+1 so that there exist tr∗≥2tr+1 and Γ≥β5/min1≤i≤m⁡λminQi>0 such that, for all 1≤i≤m,1≤j≤ni, we have(46)Exij-vij2=Eκi2≤Eκ2≤Γr,∀t≥tr∗.This completes the proof of Theorem 2.


## 4. Numerical Simulations

In this section, we aim to verify the validity of the ADRC approach by considering the following MIMO uncertain stochastic system:(47)dx11=x12dt,dx12=c1x12+c2x21+c3ζ+c4sin⁡c5t+c6W2t+u1+u2dt,dx21=x22dt,dx22=c7x21+c8x12+c9ζ+c10sin⁡c5t+c6W2t+u1-u2dtdζ=c11sin⁡ζ·x11+c12cos⁡ζ·x22dt+c13sin⁡ζ·sinc5t+c6W2tdW1t,y1=x11,y2=x21,where ci(i=1,2,…,13) are unknown parameters satisfying |ci|≤M(i=1,2,…,13) for a given known constant M>0. w(t)≜sin⁡c5t+c6W2(t) is a bounded nonwhite noise that exists in many practical dynamical systems like the motion of oscillators [26, 27], where c5 and c62 denote the central frequency and strength of frequency disturbance, respectively. In this case, m=2,n1=n2=2,n=n1+n2=4,s=1,p=q=1,b11=b12=b21=1,b22=-1. In addition, the constants Di(i=1,2,3,4) and the function ς(w) in Assumptions (A1)-(A2) can be specified as D1=max{1,c5,c6,c62}, D2=0, D3=max{c11,c12}, D4=max{c1,c2,c3,c4,c7,c8,c9,c10}, and ς(w)=c13|w|. We can easily check that Assumptions (A1)-(A2) hold in this case.The stochastic total disturbance(x13,x23)⊤ is defined by(48)x13=c1x12+c2x21+c3ζ+c4sin⁡c5t+c6W2t,x23=c7x21+c8x12+c9ζ+c10sin⁡c5t+c6W2t.An ESO (49) is designed for system (47) as follows:(49)x^˙11=x^12+3ry1-x^11,x^˙12=x^13+3r2y1-x^11+u1+u2,x^˙13=r3y1-x^11,x^˙21=x^22+3ry2-x^21,x^˙22=x^23+3r2y2-x^21+u1-u2,x^˙23=r3y2-x^21.In this case the corresponding matrices in (5) become(50)E1=E2=-310-301-100,where all eigenvalues are -1 and thus these two matrices are Hurwitz. Choose ϱi:R2→R(i=1,2) in (7) as follows:(51)ϱ1z11,z12=-2z11-3z12,ϱ2z21,z22=-2z21-2z22with the corresponding matrices in (9)(52)F1=01-2-3,F2=01-2-2being Hurwitz. ESO (49) based feedback control is designed as(53)u1=12ϱ1x^11-v11,x^12-v12-x^13+v13+12ϱ2x^21-v21,x^22-v22-x^23+v23,u2=12ϱ1x^11-v11,x^12-v12-x^13+v13-12ϱ2x^21-v21,x^22-v22-x^23+v23,where in this case b^il∗(i,l=1,2) in (8) are specified as b^11∗=b^12∗=b^21∗=1/2,b^22∗=-1/2. Both system (47) and system (49) are discretized by the Milstein approximation method proposed in [34]. In Figures 1 and 2, the initial values are(54)x0=1,-1,1,-1,ζ0=0,x^0=0,0,0,0,0,0,the time discrete step is taken as(55)Δt=0.001,and the gain r in (49) is taken as(56)r=100.The uncertain parameters and reference signals in Figures 1 and 2 are taken as(57)c1=1,c2=2,c3=1,c4=13,ci=1i=5,6,…,13,v11t=sin⁡t+1,v21t=cost+1,and(58)c1=1,c2=2,c3=2,c4=1,c5=1,c6=2,c7=1,c8=1.5,ci=2i=9,10,11,12,13,v11t=sin⁡⁡2t+1,v21t=cos2t+1,respectively.Figure 1
Estimation of unmeasured states and stochastic total disturbance of the ADRC’s closed-loop system and the closed-loop reference signal tracking with the uncertain parameters and the reference signal given in (57).
(a) (b) (c) (d) (e) (f)Figure 2
Estimation of unmeasured states and stochastic total disturbance of the ADRC’s closed-loop system and the closed-loop reference signal tracking with the uncertain parameters and the reference signal given in (58).
(a) (b) (c) (d) (e) (f)It is observed from both Figures1 and 2 that ESO (49) is very valid to estimate both the state x=(x11,x12,x21,x22)⊤ and the stochastic total disturbance x=(x13,x23)⊤ defined in (48), despite varying system parameters and reference signals between (57) and (58). In addition, it is seen from Figures 1 and 2 that the tracking effects of xi1(i=1,2) to vi1(t)(i=1,2) and xi2(i=1,2) to vi2(t)(i=1,2) are very satisfactory. Figure 2 shows that the estimation effects of ESO and the performance of reference signals tracking are still very effective regardless of the larger parameters than the one given in (57), which validates the good robust performance of ADRC.


## 5. Concluding Remarks

In this paper, the active disturbance rejection control (ADRC) approach is applied to the output regulation problem for a class of multi-input multioutput (MIMO) uncertain stochastic nonlinear systems subject to vast stochastic uncertainties. The stochastic uncertainties of each subsystem including unknown nonlinear system functions, external stochastic disturbance with unknown statistical property, unknown stochastic inverse dynamics, uncertain nonlinear coupling effects between subsystems, and uncertainties caused by the partially unknown input gains are first regarded as the stochastic total disturbance of each subsystem. The stochastic total disturbance is then estimated by a linear ESO in real time and cancelled in the ESO-based output feedback loop. The output regulation performance of the resulting ADRC’s closed-loop system is obtained with rigorous theoretical proof including practical mean square reference signals tracking, disturbance attenuation, and practical mean square stability when the reference signals are zero. Finally, the validity of the estimation performance of ESO, efficient reference tracking of ADRC, and good robust performance of ADRC are shown by some numerical simulations.



---

*Source: 5674212-2019-01-22.xml*