PDE modelling and switch strategy controller design for a flexible detection system with output constraint and actuator failure

A novel approach for modelling a mine detection robot system is presented in this paper using partial differential equations and Hamilton’s principle. The system comprises a rigid arm, a detecting sensor module, and a flexible string. The paper addresses the challenges of boundary deflection constraint and actuator fault. To overcome these challenges, a unique controller is developed using Barrier Lyapunov function and switching strategy. This controller enables the robot system to achieve the desired position and eliminate the vibration of flexible string. The superiority of the designed control law is demonstrated through simulation results. Furthermore, the asymptotic stability of the closed-loop flexible system is established by the use of extending LaSalle’s Invariance principle to an infinite dimensional system. This principle ensures that the system will eventually reach a stable state over time.


Introduction
Flexible systems have extensive applications in various engineering fields, including space exploration, the petrochemical industry, healthcare, and more [1][2][3][4].This paper focuses on the investigation of a mine detection robot system which comprises a rigid arm, a detecting sensor module, and a flexible string, as illustrated in Figure 1.The modeling and control challenges associated with flexible systems have gained significant interest from researchers and practitioners alike.
Flexible systems are commonly represented using ordinary differential equations (ODEs) or partial differential equations (PDEs).ODE models are commonly used when higher modes are neglected, making them convenient for controller design.For instance, in [5], the global exponential stabilization problem is discussed by approximating the dynamics of the Kuramoto-Sivashinsky equation through Galerkin's method.However, the approximated ODE models can exhibit spillover effects.As a result, there has been an increasing focus on the infinite PDE dynamic models [6][7][8].For example, [9] addresses the control design problem for an unstable reaction-diffusion parabolic PDE coupled with a heat relation.In [10], an adaptive boundary iterative learning control law with an infinite PDE model is proposed for a two-link rigid-flexible manipulator.
In practical scenarios, flexible manipulators and strings often exhibit significant deflection and vibration during movement.Various methods have been proposed to address these issues.It's important to note that the dynamics of flexible manipulators and strings differ from each other, requiring tailored solutions for each.For flexible manipulators, in [11], an intelligent network controller is proposed to mitigate the deflection of a flexible manipulator system in the presence of an input dead zone.Similarly, in [12], a boundary controller is developed for a two-link rigid-flexible manipulator，so that prescribed performance and suppress vibration are achieved.Regarding flexible strings, [13] proposes a novel controller with a disturbance observer to suppress vibration in an axially moving belt system.In an oceanic environment, [14] designs a boundary control strategy to restrain transverse vibration in a flexible marine riser, taking into account input saturation.Additionally, [15] presents an adaptive control approach for a thruster-assisted position mooring system, specifically addressing transverse vibrations.
This research focuses on addressing the boundary deflection constraint in the controller design.The motivation for this study stems from the potential consequences of violating boundary constraints, including degraded system performance and possible sensor damage, as depicted in Figure 1.To tackle the issue of output constraints, [16] proposes an integral-barrier Lyapunov function for a nonlinear moving string, ensuring that the desired performance attributes are achieved.Similarly, [17] aims to achieve preset performance targets for outputs in a flexible manipulator system.In addition to the boundary deflection constraint, this paper also considers the actuator faults at the boundary.To handle this problem, a Barrier Lyapunov function and a switching strategy are adopted.Switching strategies have been extensively studied in various contexts.For instance, [18] proposes a novel switching strategy to enhance efficiency and conserve energy in partial load operating conditions.Moreover, [19] presents a novel adaptive fault-tolerant control method based on a switching scheme for a family of nonlinear systems, utilizing healthy actuators as backups.
This paper has three following innovations: 1) A mine detection robot system that is made of a rigid arm, a detecting sensor module, together with a flexible string, and is described by partial differential equations using Hamilton's principle.
2) A novel boundary controller with a switching strategy is proposed, to regulate the discussed system at specific position and suppress the deflection, under boundary deflection constraint and actuator failures 3) It is proven that the closed-loop flexible system is asymptotically stable by LaSalle Invariance Principle which is extended to the infinite dimensional system.

Problem formulation
The geometry of a mine detection manipulator is depicted in Figure 1.The system under investigation comprises three main components: a rigid arm, a detecting sensor module and a flexible string, located at the end of the system.The state variables of the detection manipulator are defined in two coordinate systems.XOY represents the inertial coordinate system while xOy is the rotating coordinate system associated with the Remark 1: For convenience, the notation t in the control inputs ( ) t  , F(t) is omitted in the following of this paper.

Dynamic modelling
In order to describe the two-link detection manipulator, a position vector P is given as follows: The principle of Hamilton is used to obtain the dynamic equations of the system consist of arm and string [20,21] shown in (2) where t is the operating interval between the time constants t 1 and t 2 ; δ, W(t), E p (t)and E k (t) represent the variational operator, non-conservative work, the potential energy and kinetic energy, respectively. (2) The Equation ( 3) determines E k (t) of the considered flexible system where J, M, l 2 , ρ, and m are the inertia moment for the rigid arm, the arm's mass, the string's length, the string's mass per unit length, the detecting sensor module's mass, respectively.
The kinetic energy can be obtained from ( 6) considering gravitational and strain energy, where T is the string tension.If the system inputs are τ and F, the virtue work can be calculated by (7).

 
Remark 3: The discussed system with two-link arm-string manipulator is mathematically described by Equations ( 8)-( 9) of partial differential equations and Equation ( 10) of an ordinary differential equation, subject to boundary conditions (Equations ( 12) and ( 13)).It is important that the system exhibits coupling between the joint angle and elastic deflection, which introduces additional complexities in designing the controller.

Control design
Boundary control laws for τ and u are designed in this section to achieve the following twofold objectives: a) regulate the manipulator to the desired position; b) suppress the deflection and vibration.That is to say, the twofold control objectives are

Boundary controller without output constraint
Theorem 1: The asymptotic stability of the discussed system defined by ( 8)- (10) with boundary conditions ( 12) and ( 13) is guaranteed by the proposed boundary controller represented as ( 14) and ( 15) can ensure.
In ( 14) and ( 15), the tracking error is clarified as and i K stand for the controller gains which are both greater than 0. Proof: The following Lyapunov function candidate is considered in this paper: 1 2 0 0 0.5 0.5 0.5 0.5 ' , 0.5 0.5 , The energy objective and control objective are included in Lyapunov function candidate (16).
Taking the derivative of Equation ( 16), it is obtained that , cos ,

V J t t L t h x t t t h x t t t h x t t dx G h l t h l t Ll t t h x t h x t dx mL t t h l t h l t Th x t h x t K e t e t mL t h l t t t h l t t m
Substituting governing Equations ( 8)-( 10) by the use of the conditions (11) and (12), and applying ( 14) and (15) as control laws, (18) is obtained.
Since K 2 and G 2 are positive constants, the time derivative (18) in ( 16) is negative semidefinite.It's not enough to prove the asymptotic stability of the discussed system by standard Lyapunov arguments.
Next, the extended LaSalle's Invariance Principle is applied.Due to the LaSalle's Invariance Principle extended to infinite dimensional system [19], we first define

y y y y y y h l t h l t h x t h x t
The closed-system described by ( 8) -( 10) can be compactly written as In (20), the infinite dimensional operator  is defined as where The spaces mentioned in ( 21) is described as 1 2 0 0 0.5 0.5 0.5 0.5 ' , 0.5 0.5 , From ( 18) we know, 0, It can be seen that the operator  is dissipative.Based on Lax-Milgram theory [22], the equation   , 0, I A y q q H        has a unique solution.Hence, the operator   I A   ranged onto H .The operator  generates a C 0 -semi-group of contractions T(t) on  due to the Lumer-Phillips theorem [23].
Then, we will show that the solution trajectories of system (22) are recompact in for t > 0.
We assume is given.Consider the equation where We can obtain (27) by solving (26) where β i is defined by the boundary conditions. ( According to the above analysis, the Equation (26) exists a unique solution , which means that .Hence, there exists that maps into ( ). is obtained to be a compact operator for the compact embedding of the latter space into [24].The spectrum of comprises solely of distinct eigenvalues as a result of a compact operator of .
is also proved to be compact for any in the resolvent set of .Therefore, the solution trajectories (20) are recompact in when [24].Making , we can get from Equation ( 18) when K2 and G2 are positive constants.31) can be treated using separation of variables, for it is a constant-coefficient linear system [23].Then,   , h x t can be presented as follows:

 
) where X(x) and Y(t) stand for the unknown functions respect to space and time, respectively.On the basis of (31), we have T q q q q q q q    Gy G y mL y y y y Ty m q y q L y y y y T y q L y y y y y q y y l y q y L y y y y T T q d d x y q Therefore, based on Equation (37), the following equations can be derived: (37) Then, we have c1=0, c2=0 Therefore,   0 X x  is derived, implying h(x,t)=0, ( , ) 0 h x t   .Substituting control scheme ( 13)-( 14) into (30) to (32) yields So, the Theorem 1 is proven.

With boundary state constraint
In this section, boundary state constraint and actuator fault are considered in controller design.Considering actuator failure, the objective is to keep the boundary state stay at the constrained space, which means  

, h l t B
 , where B denotes the constraint.
The controller with boundary state constraint is designed as We define the following Barrier Lyapunov function: Substituting the control laws (41) and (42), by a lengthy calculation we can obtain (44) After similar proofs, we can get Next, a monitoring function is presented to guarantee the boundary state   2 , w l t to stay at the constraint B.
Then, we have Therefore, we can conclude

53)
Assumption 1: There area total of m actuator switchings on   0, because of actuator failures.
Assumption2: There is no switching at t 0 .
The monitoring function is designed by   (56) The current actuator will switch to the next one when the monitoring function (54) is not met at the time instant t k+1 .

Results of simulation
Two simulation cases are provided in this part for validation of the proposed controller's effectiveness in Section 3. The simulation objective is ( 8)- (10) with boundary conditions ( 12)- (13).The final objective is .Parameters of the discussed system can see Table 1.14)-( 15) and the controller gain is shown in Table 2. Figure 2 -Figure 5 give the simulation result.In Figure 2, the desired position can be achieved.Figure 4 shows that we can eliminate the deflection of the flexible string.Figure 3 and Figure 5 give the corresponding control inputs.From simulation results we can see, the final objective d    and   , 0 h x t  can be fulfilled.We use the control law (41)-( 42) and monitoring function (54) to achieve the final aim.Table 3 shows the controller gains.Figure 6 -Figure 10 show the simulation results.Assuming that a actuator is locked dead at , the boundary control input is (See Figure 9).From Figure 10 we know, the actuator switching happens at the time of 1.2456s.Similarly, Figure 6 and Figure 8 show that the final objective and can be achieved.at the end of the flexible string with boundary state constraint.

Conclusions
Through Hamilton's principle, the PDE model of a mine detection robot system has been derived which consist of a detecting sensor module, a rigid arm, and a flexible string.In the presence of boundary deflection constraint and actuator fault, a novel controller is designed with Barrier Lyapunov function and switching strategy.With the designed boundary control method, it can eliminate the flexible string's deflection, and the desired position also can be reached.Via LaSalle's Invariance principle extended to infinite system, the asymptotic stability for the closed-loop system is proven.The designed controller's effectiveness has been indicated by the study.

Figure 1 .
Figure 1.The geometry of a mine detection manipulator.

Figure 2 .
Figure 2. Joint position tracking without boundary state constraint.Figure 3. Torque control input without boundary state constraint.

Figure 3 .
Figure 2. Joint position tracking without boundary state constraint.Figure 3. Torque control input without boundary state constraint.

8 Figure 4 .
Figure 4. Deflection of the flexible string without boundary state constraint.

Figure 5 .
Figure 5. Boundary control input F without boundary state constraint.

Figure 6 .
Figure 6.Joint position tracking with boundary state constraint.Figure 7. Torque control input with boundary state constraint.

Figure 7 . 9 Figure 8 .
Figure 6.Joint position tracking with boundary state constraint.Figure 7. Torque control input with boundary state constraint.

Figure 9 .
Figure 9. Boundary control input F with boundary state constraint.

Figure 10 .
Figure 10.at the end of the flexible string with boundary state constraint.
are the control torque at the rigid arm's top and the control force attached at the flexible string's end for vibration suppression τ.L 1 is the length of rigid arm.

Table 1 .
Parameters in the detection arm-string system.