# Boundary Stabilization of Underground Multirobot System with Virtual String Constraints

Authors: Lingling Su; Xianhua Zheng; Yongshi Song; Shang Feng; Lin Zhang
Journal: Journal of Mathematics
 (2022)

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## Abstract

The manipulation of multiple physical connected objects can be described by virtual connected string system. We proposed a Lyapunov-based stabilization control approach for virtual connected string with midway discontinue vertical force. The system is with Riemannian boundary. We use the backstepping method to transform the system into a stable system with Dirichlet boundary. In this way, we get the controller to make sure the closed-loop system converge to the zero point exponentially. Then, we construct a Lyapunov function to analyze the stability for the closed-loop system. We also show the system is well-posedness. Also, we use the active disturbance rejection control (ADRC) to reject the disturbance when disturbance is present. Some numerical stimulations show that the control law is the effective.


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## Body

## 1. Introduction

Multirobot transportation system has been one of the most potential research fields in recent years, and the majority of these studies are focused on multiple robots manipulating a single object, while the manipulated object in this paper is a class of multiple physical connected objects (MPO). The manipulation of MPO has been widely used both in daily life and industrial application. As an example, flood shield is used to obstruct flood disaster. These shields are connected with each other, and each shield is immobilized by fixed oblique support, flexible oblique support, or vertical support, which are represented in Figures1(a)–1(c), respectively. Retaining wall is another typical application of MPO, i.e., retaining mesh, which is supported by multiple linked rods, as shown in Figure 1(d). One more abstracted application, a multiple link model-based human body is held by two-arm service robot, as shown in Figure 1(e). In this application, human beings are abstracted as free-ended multiple connected rigid bodies, and these rigid bodies are manipulated by two arms, respectively. Similar to above applications, the pushing manipulation problem of mobile support robot (MSR, also known as hydraulic support or roof support) on underground fully mechanized coal mining face is a such kind of system as well. Since the manipulated object, i.e., scraper conveyor, is composed of multiple middle troughs (MTs), we simplified scraper conveyor as multiple connected objects with physical connections. As shown in Figures 1(f) and 1(g), both the physical and simulated workplace of multiple MTs manipulated by MSRs on ground is represented, respectively. With simplification, as shown in Figure 1(h), MPO is replaced by external virtual string. Then, the simulated external constraint using virtual connected string is presented, as shown in Figure 1(i).Figure 1
Widely applied scenarios of MPO. (a) Flood shield immobilized by multiple fixed oblique supports. (b) Flood shield immobilized by multiple flexible oblique supports. (c) Flood shield immobilized by multiple vertical supports. (d) Retaining wall supported by multiple bars. (e) Multiple link model-based person held by the two-arm service robot. (f) The simulated workplace of multiple MTs manipulated by MSRs on the ground. The scraper conveyor is composed of many middle troughs (MTs) and each MT is connected with other. MSRs push corresponding MT forward. (g) The diagram of the multirobot system with MPO. (h) Generalized system architecture of the multirobot system, whose manipulating object is replaced with virtual external string. (i) Simulated external constraints through virtual connected string, and both the disturbance and antidamping are presented.Due to loadable characteristics as well as its mobility through pushing and advancing, the manipulation problem of MSRs is one of the most complicated problems on underground fully mechanized coal mining face. Based on our previous research on the underground multirobot system [1, 2], we found the main difficulty is modeling the external constraints. Considering the commonalities of above applications, a reasonable assumption is proposed by replacing MPO with virtual connected string constraints, and the main problem is boundary stabilization of virtual connected string system. Thus, one contribution of this paper is replacing the manipulated MPO with a virtual connected string. Then, the problem is converted to the system stabilization of the connected string system considering pushing disturbance and antidamping.The study for the virtual connected string equations has drawn significant attention in recent years. It can be applied widely in engineering field. For example, aerial cable, traffic problem, and transmission lines can be described by connected string systems [3, 4]. In [5], the stability of strings with various connected feedbacks is studied by spectral analysis. In [2], the control of a connected string converter modeled from electrical power is studied. In [6], the symmetry is considered for the system of strings. Other literature relevant can be found in [7–9] and the references therein.Although there are some works on the stability of connected strings, there is still no result on the Lyapunov stability analysis for connected strings. It needs some mathematical techniques to find the suitable Lyapunov function for connected strings. Moreover, these models are without disturbance. However, disturbance is usually present in practical applications [10–12]. So, we are also concerned with the disturbance for the model. Sliding mode control and active disturbance rejection control (ADRC) are usually used to reject the disturbance [13–19]. ADRC is used in this paper. ADRC shows its advantages because it can not only estimate the disturbance but also counteract the disturbance (see [20–24]).We consider a connected string system, which is connected at pointx=1 in [0, 2]:(1)χttx,t=χxxx,t,x∈0,1∪1,2,t>0,χ1−,t=χ1+,t,t≥0,χx1−,t−χx1+,t=qχt1,t,t≥0,χx0,t=Ut+rt,t≥0,χx2,t=0,t≥0,where χ is the state, Ut is the input, q>0,q≠2, and r is the disturbance; we also assume rt≤M,r.t≤M,∀t>0. The connected term is the discontinuity of vertical force component; it is antistable.We introduce a new variableφx,t=φ1x,t,φ2x,tT, where(2)φ1x,t=χx,t,φ2x,t=χ2−x,t,x∈0,1,t≥0.Then, system (1) becomes(3)φttx,t=φxxx,t,0<x<1,t>0,φ11,t=φ21,t,t≥0,φ1x1,t+φ2x1,t=qφ1t1,t,t≥0,φ1x0,t=Ut+rt,t≥0,φ2x0,t=0,t≥0.It can be seen that the string is unstable because there is an antidamping in the term of discontinuity of the vertical force. Also, the boundary is Riemannian, which gives rise to difficulty to make it converge to zero point. The contribution is compensating the unstable term as well to make sure that it can converge to zero. We used the backstepping transformation (see [25–27]) to deal with the unstable term, and we are the first to give the Lyapunov function for connected strings.The paper is organized as follows. Section2 designs the controller. Section 3 shows MATLAB simulations, and Section 4 gives the conclusion remarks.


## 2. The Control Design

An invertible backstepping transformation is introduced by(4)θ1x,t=φ1x,t+qq+c4−q2φ1x,t+φ2x,t−qq+c4−q2φ11,t+φ21,t+2q+c4−q2∫x1φ1ty,t+φ2ty,tdy,(5)θ2x,t=φ2x,t,where q>0,q≠2 and c>0,c≠2; its inverse transformation is(6)φ1x,t=θ1x,t+qq+c4−c2θ1x,t+θ2x,t−qq+c4−c2θ11,t+θ21,t+2q+c4−c2∫x1θ1ty,t+θ2ty,tdy,φ2x,t=θ2x,t.By transformation (4) and (5), system (3) is going to be(7)θttx,t=θxxx,t,0<x<1,t>0,θ11,t=θ21,t,t≥0,θ1x1,t+θ2x1,t=−cθ1t1,t,t≥0,θ1x0,t=4−c24+qcUt+rt−2q+c4+qcθ1t0,t+θ2t0,t,t≥0,θ2x0,t=0,t≥0.By the transformation, the connected antidamping is moved to the boundary. We give a controller:(8)U1t=2q+c4−c2θ1t0,t+θ2t0,t+k4+qc4−c2θ10,t,k>0,to deal with the antidamping term at point x=0. Under the control of(9)Ut=U0t+U1t,where U0t is a new control to be designed to contract the disturbance. System (7) becomes(10)θttx,t=θxxx,t,0<x<1,t>0,θ11,t=θ21,t,t≥0,θ1x1,t+θ2x1,t=−cθ1t1,t,t≥0,θ1x0,t=4−c24+qcU0t+rt+kθ10,t,t≥0,θ2x0,t=0,t≥0.We consider system (10) in the Hilbert space(11)ℋ=u1,v1,u2,v2∈H10,1×L20,12|u11=u21.With the inner product induced norm, for∀u1,v1,u2,v2∈ℋ,(12)u1,v1,u2,v2=∫01u1′x2+v1x2+u2′x2+v2x2dx+ku102.Introduce an unbounded linear operatorA:DA⊂ℋ⟶ℋ:(13)AZ=v1,u1″,v2,u2″,∀Z=u1,v1,u2,v2∈DA,DA=Z∈H∩H20,1×H10,12|u1′1+u2′1=−cv11,u1′0=ku10,u2′0=0,v11=v21,and then, system (10) can be written in the evolution equation form in ℋ:(14)ddtZt=Aθ1⋅,t,θ1t⋅,t,θ2⋅,t,θ2t⋅,t−4−c24+qcBU0t+rtZ⋅,0=θ1⋅,0,θ1t⋅,0,θ2⋅,0,θ2t⋅,0,where Zt=θ1⋅,t,θ1t⋅,t,θ2⋅,t,θ2t⋅,t and ℬ=0,δx,0,0. We get the following two lemmas.Lemma 1.
A generates an exponentially stable C0-semigroup.Proof.
We will first demonstrate thatA generates a C0-semigroup. A direct computation gives(15)ReAu1,v1,u2,v2T,u1,v1,u2,v2T=∫01v1′x,tu1″x,t+u1″x,tv1x,t+v2′x,tu2″x,t+u2″x,tv2x,tdx+kv10u10=−cv112≤0.
So,A is dissipative.
For any givenϕ1,φ1,ϕ2,φ2∈ℋ, solve Au1,v1,u2,v2=ϕ1,φ1,ϕ2,φ2, to obtain(16)v1x=ϕ1x,v2x=ϕ2x,which gives(17)u1x=kx+1u10+∫0xφ1ηx−ηdη,u2x=u20+∫0xφ2ηx−ηdη,u10=−1k∫01φ1η+φ2ηdη+cv11,u20=k+1u10+∫01φ1η1−ηdη−∫01φ2η1−ηdη.
So,u1,v1,u2,v2∈ℋ is unique. Hence, A−1 exists, and from the Sobolev embedding theorem, it is also compact on ℋ. Using Lumer–Phillips theorem (see [28]), A generates a C0-semigroup of contractions eAt.
Next, we show thatA is exponentially stable. We consider the evolution equation:(18)ddtZt=AZt,Zt=θ1⋅,t,θ1t⋅,t,θ2⋅,t,θ2t⋅,t,which equals to the following PDE:(19)θttx,t=θxxx,t,0<x<1,t>0,θ11,t=θ21,t,t≥0,θ1x1,t+θ2x1,t=−cθ1t1,tt≥0,θ1x0,t=kθ10,t,t≥0,θ2x0,t=0t≥0.
LetVt be a function given by(20)Vt=12∫01θ1x2x,t+θ1t2x,t+θ2x2x,t+θ2t2x,tdx+12kθ120,t+δ∫01θ1xx,tθ1tx,t+θ2xx,tθ2tx,tdx+δ∫01θ1tx,tθ2xx,t+θ2tx,tθ1xx,tdx,where δ>0 and δ is sufficiently small.
From Cauchy–Schwartz and Young’s inequalities, it is found that, forδ≤min1/3,k/3, there exists m1,m2>0 so that m1Zt2≤Vt≤m2Zt2, where(21)m1=min1−3δ2,k−3δ2,m2=max1+3δ2,k+3δ2,Zt2=θ1x2+θ1t2+θ102+θ2x2+θ2t2.
Hence,Vt is a proper Lyapunov function for (23). By taking the deviation of Vt in t, we have(22)V˙t=∫01θ1xx,tθ1xtx,t+θ1tx,tθ1ttx,t+θ2xx,tθ2xtx,t+θ2tx,tθ2ttx,tdx+kθ10,tθ1t0,t+δ∫01θ1xtx,tθ1tx,t+θ1xx,tθ1ttx,t+θ2xtx,tθ2tx,t+θ2xx,tθ2ttx,tdx+δ∫01θ1ttx,tθ2xx,t+θ1tx,tθ2xtx,t+θ2ttx,tθ1xx,t+θ2tx,tθ1xtx,tdx,=∫01θ1xx,tθ1xtx,t+θ1tx,tθ1xxx,t+θ2xx,tθ2xtx,t+θ2tx,tθ2xxx,tdx+kθ10,tθ1t0,t+δ∫01θ1xtx,tθ1tx,t+θ1xx,tθ1xxx,t+θ2xtx,tθ2tx,t+θ2xx,tθ2xxx,tdx+δ∫01θ1xxx,tθ2xx,t+θ1tx,tθ2xtx,t+θ2xxx,tθ1xx,t+θ2tx,tθ1xtx,tdx,=θ1t1,tθ1x1,t−θ1t0,tθ1x0,t+θ2t1,tθ2x1,t−θ2t0,tθ2x0,t+kθ10,tθ1t0,t−δ2∫01θ1x2x,t+θ1t2x,t+θ2x2x,t+θ2t2x,tdx+δ2θ1t21,t+θ1x21,t+θ2t21,t+θ2x21,t−δ2θ1x20,t+θ1t20,t+θ2x20,t+θ2t20,t+δθ1x1,tθ2x1,t−θ1x0,tθ2x0,t+θ1t1,tθ2t1,t−θ1t0,tθ2t0,t,=2+c22δ−cθ1t21,t−δ2∫01θ1x2x,t+θ1t2x,t+θ2x2x,t+θ2t2x,tdx−δ2k2θ120,t+θ1t0,t+θ2t0,t2.
Letδ<min1/3,k/34c/4+c2 and α=max1,k2δ/2m1, and we get V˙t≤−αVt, which gives the result.Lemma 2.
ℬ is admissible to eAt.Proof.
By direct computation, we obtain(23)A∗u1,v1,u2,v2=−v1,−u1″,−v2,−u2″,DA∗=u1,v1,u2,v2∈H∩ℋ20,1×ℋ10,12|u1′1+u2′1=cv11,u1′0=ku10,u2′0=0,v1=v2.
The dual system of (10) is(24)θtt∗x,t=θxx∗x,t,0<x1,t0,θ1∗1,t=θ2∗1,t,t≥0,θ1x∗1,t+θ2x∗1,t=cθ1t∗1,t,t≥0,θ1x∗0,t=kθ1∗0,t,t≥0,θ2x∗0,t=0,t≥0,yt=−4−c24+qcθ1t∗0,t.
A generates a C0-semigroup, so A∗ generates a C0-semigroup.
The energy of system (25) is(25)Et=12∫01θ1x∗x,t2+θ1t∗x,t2+θ2x∗x,t2+θ2t∗x,t2dx+kθ1∗0,t2.
Then,Et≤E0. Let ρt=∫01x−1θ1x∗x,tθ1t∗x,t+xθ2x∗x,tθ2t∗x,tdx. Differentiate ρt with respect to t to yiel(26)ρ˙t=12θ1x∗20,t+θ1t∗20,t+θ2x∗21,t+θ2t∗21,tdx−Et+12kθ1∗20,t.
We, thus, have(27)∫0Tyt2dt=4−c24+qc2∫0Tθ1t∗0,t2dt≤2T4−c24+qc2E0.
Similar to the computation ofA−1(28)A∗−1u1v1u2v2=1+kxa−∫0xv1ηx−ηdη−u1xb−∫1xv2ηx−ηdη−u2x,∀u1v1u2v2∈ℋ,where(29)a=1k∫01φ1η+φ2ηdη−cv11,b=k+1a−∫01φ1η1−ηdη+∫01φ2η1−ηdη,ℬ∗A∗−1u1v1u2v2=−u10.
Therefore, onℋℬ∗A∗−1 is bounded. Combined with (26), we show that ℬ is admissible to eAt (see [29, 30]).
From the two lemmas above, the evolution equation (14) has a unique weak solution in ℋ. Therefore, for ∀θ1.,t,θ1t.,t,θ2.,t,θ2t.,tT∈ℋ,U0∈Lloc20,∞, system (10) has a unique weak solution θ1x,t,θ1tx,t,θ2x,t,θ2tx,tT∈ℋ, ∀u1,v1,u2,v2T∈DA∗,(30)ddtθ1θ1tθ2θ2t,u1v1u2v2=θ1θ1tθ2θ2t,A∗u1v1u2v2−g104−c24+qcU0t+rt,that is,(31)ddt∫01θ1xx,tu1′x+θ1tx,tv1x+θ2xx,tu2′x+θ2tx,tv2xdx=∫01−θ1xx,tv1′x−θ1tx,tu12x−θ2xx,tv2′x−θ2tx,tu22xdx−v104−c24+qcU0t+rt.
Next, we will reject the disturbanced. An observer is chosen as(32)y1t=∫01θ1tx,t+12cxθ1xx,t+θ2tx,t+12cxθ2xx,tdx,t≥0,y2t=∫01θ1tx,t+θ2tx,tdx,t≥0.
From equality (31), we have(33)y˙1t=−c2y2t−4−c24+qcU0+rt.
We choose the high gain estimator to be (see [23])(34)y^˙t=−4−c24+qcU0t+r^t−c2y2t−1εy^t−y1t,r^˙t=1ε24+qc4−c2y^t−y1t,where ε>0 is the design small parameter and r^ is the approximation of disturbance r. Let(35)y˜t=y^t−y1t,r˜t=r^t−rt,be the errors. Then, y∼andr∼ satisfy(36)ddty˜td˜t=−1ε−4−c24+qc)1ε24+qc4−c20y˜td˜t+0−1d˙t=Ay˜td˜t+Bd˙t.
The collocated feedback controller of (10) is designed to be(37)U0t=−satr^t,where satx=M,x≥M+1,x,x∈−M−1,M+1,−M,x≤−M−1. with controller U0t=U0t+U1t; the closed-loop system of (10) is(38)θttx,t=θxxx,t,0<x<1,t>0,θ11,t=θ21,t,t≥0,θ1x1,t+θ2x1,t=cθ1t1,t,t≥0,θ1x0,t=4−c24+qc−satr˜t+rt+rt+kθ10,t,t≥0,θ2x0,t=0,t≥0,y^t=−4−c24+qc−satr˜t+r^t−c2y2t−1εy^t−y1t,t≥0,r^t=1ε24−qc4+c2y^t−yt,t≥0.
We get the following theorem concerning the stability when disturbance in involved.Theorem 1.
Ifr and r˙ are bounded measurable, then, for any initial value θ⋅,t∈ℋ, there exists a unique solution θ,θt∈C0,∞;ℋ for the closed-loop system (39) of (7):(39)θttx,t=θxxx,t,0<x<1,t>0,θ11,t=θ21,t,t≥0,θ1x1,t+θ2x1,t=cθ1t1,t,t≥0,θ1x0,t=4−c24+qc−satr˜t+rt+rt+kθ10,t,t≥0,θ2x0,t=0,t≥0,y˜t=−1εy˜t−4−c24+qcr˜t,t≥0,r˜t=1ε24−qc4+c2y˜t−rt,t≥0,y1t=∫01θ1tx,t+12cxθ1xx,t+θ2tx,t+12cxθ2xx,tdx,t≥0,y2t=∫01θ1tx,t+θ2tx,tdx,t≥0.Moreover, the solution of (39) converges to any arbitrary given neighborhood of zero when t⟶∞andε⟶0.For the proof of Theorem1, we omit the proof, since it can be easily proved followed by [19, 24].


## 3. Numerical Simulations

In this section, we compute the state for the closed-loop system (39) by the finite difference method [19]. Here, q=5,c=1.5,k=5,andd=3sin2t. Figure 2 shows that the displacement of θ1andθ2 converges to zero. Figure 3 shows the velocity of θ1tandθ2t tend to zero. Figure 4 shows the estimation of r^ fits well with the value of r. Thus, the control law is effective.Figure 2
(a) The displacement ofθ1; (b) the displacement of θ2.
(a)(b)Figure 3
(a) The velocity ofθ1t; (b) the velocity of θ2t.
(a)(b)Figure 4
The displacement ofr, r^, and r˜.


## 4. Conclusion

A virtual connected string system with midway discontinue vertical force model is proposed for the underground multirobot system, and the Lyapunov-based stabilization control approach for virtual connected string is studied in the paper. The system is with Riemannian boundary. The backstepping is used to move the antistable term to the boundary thus to give the controller and Lyapunov approach is used to analyze the stability for the closed-loop system when the disturbance is absent. Then, ADRC is used to estimate and reject the disturbance. We can study then stings with discontinue vertical force in the further research.



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