Lyapunov Matrix-Based Guaranteed Cost Dynamic Positioning Control for Unmanned Marine Vehicles With Time Delay

Authors: st xin yang
ying  hao
tieshan  li
yang  xiao
Venue: IEEE_org_ICIST_2024_Conference
Published: 2024-08-15 15:13:20

Abstract: This paper presents the design of a Lyapunov matrix-based guaranteed cost dynamic positioning controller for unmanned marine vehicles (UMVs) under time delays. Compared to traditional time delays compensation methods. This article first introduces a novel Lyapunov matrix-based complete type Lyapunov--Krasovskii functional (LKF), incorporating more comprehensive information about time delays and system states. The strategy employs the linear matrix inequality (LMI) method and Jensen's inequality to derive sufficient conditions for the existence of the controller under the constructed complete LKF framework. The proposed controller ensures that the state errors of UMVs asymptotically converge to zero, with the adaptive $H_{\infty}$ performance index not exceeding $\gamma_{0}$, and that the cost function has an upper bound. Finally, the effectiveness of the proposed approach is verified through simulation results.

Main paper: 
 i. introduction: 
 Unmanned Marine Vehicles (UMVs) play a pivotal role in enhancing maritime safety and security by performing highrisk operations effectively without compromising human lives, thereby revolutionizing search and rescue missions and coastal surveillance [1] - [3] . Compared to traditional anchor mooring, dynamic positioning (DP) offers a more versatile, precise, and environmentally friendly method for positioning vessels, making it particularly suitable for use in complex or dynamic marine environments [4] . Over the years, numerous control strategies have been proposed to ensure robust DP control in UMVs. For instance, [5] introduces a dynamic output feedback control method, specifically tailored for DP ships to counter denial of service attacks. In [6] , the design of an adaptive sliding mode fault-tolerant compensation mechanism is presented, targeting the maintenance of DP control in UMVs despite thruster faults and unknown ocean disturbances. It is crucial to recognize that time delays are typically inevitable [7] - [9] . Consequently, there is an urgent need to develop a strategy to compensate for these time delays.In DP systems for UMVs, time delays due to networkmediated signal and control command transmission represents a significant challenge that often compromises system stability and performance [10] , [11] . This issue has led to the development of various advanced time delays compensation methods [12] - [14] . Among these methods, enhanced time delays compensation approaches for autonomous underwater vehicles have shown promise [12] . In [13] , model-free proportionalderivative controllers are innovatively incorporated into the Lyapunov-Krasovskii functional (LKF) framework to effectively counteract the impacts of delays. Advanced strategies utilizing Lyapunov matrix-based LKF methods have proven particularly effective. These approaches leverage comprehensive information about time delays and system states, providing control strategy that efficiently accommodates time delays systems. The primary motivation of this paper is to develop a complete LKF based on the Lyapunov matrix to mitigate the effects of time delays on UMVs.On another research front, guaranteed cost control (GCC) has been extensively studied [15] - [17] . This strategy offers the advantage of setting an upper limit on a specified performance index, ensuring that any system performance degradation remains below this predefined cost threshold. As vessels often navigate in complex and varied ocean environments, the impact of wind and wave disturbances becomes significant [17] . In response, [18] investigated a robust H ∞ guaranteed cost controller aimed at enhancing path-following performance. The GCC method presented in [19] offers a way to reduce energy consumption for surface vessels in DP, thereby increasing its practical applicability. These results have inspired our research into GCC theory, particularly its application to DP ships. Thus, how to propose a guaranteed cost controller based on the Lyapunov matrix to achieve effective DP control for UMVs is the second research motivation of this paper.The primary objective of this paper is to design a Lyapunov matrix-based guaranteed cost dynamic positioning controller, utilizing the LMI method to ensure stability. The paper's main contributions are evaluated in comparison to recent advancements in the field.1) We propose a novel time delays compensation method for UMVs that incorporates more detailed time delays and state information by employing a Lyapunov matrixbased complete-type LKF, which reduces conservatism compared to conventional time delays compensation techniques. 2) A novel guaranteed cost DP control strategy is designed, which ensuring the stability of DP systems for UMVs while providing an upper bound on a prespecified cost function. The remainder of this paper is structured as follows: Section II describes the UMVs model with time delays. Section 3 reviews basic concepts and preliminary results, which serve as the theoretical basis for the proposed LKF method based on the Lyapunov matrix. A complete-type LKF based on the Lyapunov matrix is presented in Section 4. Section 5 introduces guaranteed cost dynamic positioning controller. Finally, Section 6 presents simulations to illustrate the validity of the theoretical results.The UMVs model typically employs a three degrees of freedom motion equation to describe its dynamic behavior in the marine environment. These three degrees of freedom include yaw, surge, and sway. Therefore, the dynamic equations of the UMVs are often simplified and expressed in the following form [20] :EQUATIONEQUATIONwhere matrix ξ represents the inertia matrix, and the veloc-ity vector v(t) = v 1 (t), v 2 (t), v 3 (t)T describes the ship's motion in different directions, where v 1 (t) represents the surge velocity, v 2 (t) indicates the sway velocity, and v 3 (t) corresponds to the yaw rate. The position vectorλ(t) = x o (t), y o (t), θ(t)T is used to describe the ship's position and orientation on the water surface, where x o (t) and y o (t) represent the coordinates of the ship in the horizontal plane, and θ(t) denotes the ship's heading angle. The matrix C is the damping matrix. The matrix D represents the mooring moment matrix, which models external disturbances such as wind, waves, and ocean currents acting on the UMVs. The matrix G is the thrust allocation matrix, responsible for distributing thrust to the ship's propellers. Additionally, the rotation matrix S(θ(t)) is given by:S(θ(t)) =   cos(θ(t)) -sin(θ(t)) 0 sin(θ(t)) cos(θ(t)) 0 0 0 I   ,For the control of UMVs in the northern region, where the yaw angle θ(t) is small, the matrix S(θ(t)) can be approximated by the identity matrix I. We define the following matricesA 1 = -ξ -1 C, B = ξ -1 G, and F = -ξ -1 D. let x(t) = λ T (t), v T (t)T . Thus, the dynamic equation of UMVs can be written as follows:EQUATIONwhereA = 0 I 0 A 1 , B 1 = 0 B , F = 0 F . ϖ(t) ∈ L 2 [0, ∞) represents disturbance. Defined reference signal x ref = λ ref v ref, the error vector is defined as e(t) = x(t) -x ref .The error dynamics of the UMVs can be expressed as follows:EQUATIONlet e(t) ∈ R n denote the state vector, u ∈ R p represent the control input vector. The term B 2 ω(t) is defined asAx ref + ϖ(t), where ω(t) = x ref ϖ(t), and B 2 = A I .Considering the unavoidable time delay during signal transmission, it follows from equation (4) that:EQUATIONwhere d > 0 represents the time delay, and g : R n ×R n → R m is assumed to satisfy the following inequality. Assumption 1: Let matrices N > 0 and Y > 0, where N ∈ R m×m and Y ∈ R 2n×2n . The nonlinear function g(•) satisfies the following inequality:g T (e(t), e(t -d))N -1 g(e(t), e(t -d)) ≤ [e T (t) e T (t -d)]Y[e T (t) e T (t -d)] T .Remark 1: Assumption 1 ensures that the function g(t) is bounded. When e(t) = 0 or e(t -d) = 0, Assumption 1 in this article is the general form of Assumption 1 in reference [17] .To bring both linear and angular velocities to zero and minimize the impact of external disturbances such as wind, waves, and currents, the output Z(t), can be formulated as follows:EQUATIONDefinition 1: [21] The system is described byEQUATIONGiven a constantγ 0 > 0, ω(t) ∈ L 2 [0, ∞), if for any ϵ > 0, the following condition ∞ 0 Z T (t)Z(t)dt ≤ γ 2 0 ∞ 0 ω T (t)ω(t)dt + ϵ,is satisfied, then the system ( 7) is said to achieve an adaptive H ∞ performance index that does not exceed γ 0 .Definition 2: The cost function related to system ( 5) is described as follows:EQUATION)where Ω T = Ω ≥ 0 and R T q = R q ≥ 0. A stabilization controller u(t) for system ( 5) is called a guaranteed cost controller if it ensures that J ≤ J * , where J * is a positive scalar. The value J * is known as the guaranteed cost.For UMVs (5) affected by time delays, this paper proposes a guaranteed cost DP controller based on the Lyapunov matrix. The controller is designed to drive the state error of the UMVs asymptotically converges to zero, while also satisfying the specified H ∞ performance criteria and guaranteeing an upper limit on the predefined cost function.We will construct a complete-type LKF for UMVs (5) based on Lyapunov matrix. In the following section, we begin by defining the Lyapunov matrix.We will now present relevant concepts related to linear timedelay systems as follows [22] :EQUATIONwhere e(t) ∈ R n represents the state vector, d > 0 is the time delay. A, A 1 ∈ R n×n are system matrices. Definition 3: [22] Given a matrix P > 0, if the matrix Q : [-d, d] → R n×n meets the following conditions:EQUATION) Definition 4: [22] If the system (9) is asymptotically stable, there exists a Lyapunov matrix Q(•) associated with matrix P for system (9) . Lemma 1: Suppose there are matrices H = H T > 0 and K 11 ∈ R p×n , and for any U > 0, the following LMI condition is satisfied:EQUATIONwhereΛ 2 = AX -B 1 Y 1 + (AX -B 1 Y 1 ) T + U , X = H -1 , Y 1 = K 11 H -1, and U = H -1 LH -1 , then there exists a controller u 1 (t) = -K 11 e(t) that guarantees system ( 9) is asymptotically stable. Proof 1: Select the Lyapunov function:V c (e(t)) = e T (t)He(t) + t t-de T (θ)Le(θ)dθ.We can derive:dV c (e(t)) dt (9) = Λ T 0 Ω 1 Λ 0 where Λ 0 = [e T (t), e T (t -d)] T , Ω 1 = Λ 2 A 1 X (A 1 X) T -U , Λ 2 = AX -B 1 Y 1 + (AX -B 1 Y 1 ) T + U, X = H -1 , Y 1 = K 11 H -1 , U = H -1 LH -1 .Using Lyapunov stability theory, the controller u 1 (t) = -K 11 e(t) guarantees the asymptotic stability of system (9).We construct a LKF V(•):EQUATIONwhereEQUATIONwhereΓ 1 (e(t)) = 0 -d Q(-d -τ )A 1 e(t + τ )dτ and matrices R, Q 1 satisfying the R T = R > 0, Q T 1 = Q 1 > 0.In this section, we will provide a detailed explanation of the controller design process and conduct a systematic analysis of its stability.We propose the following guaranteed cost DP controller for UMVs in (5) :EQUATIONwhere K 11 , K 21 , K 22 are feedback gain matrices. K 11 is already determined in Lemma 1, while K 21 and K 22 will be provided in Theorem 1.Theorem 1: Consider the UMVs ( 5) under Assumption 1. The guaranteed cost DP controller is defined by (14) . For the given positive definite matricesN ∈ R m×m , Y := Y 11 Y 12 Y T 12 Y 22∈ R 2n×2n , P ∈ R n×n , and a positive constant γ 0 , if there exist positive definite matrices R, Q 1 ∈ R n×n , and matrices K 21 ∈ R p×p , K 22 ∈ R p×n such that P -Q 1 -P 1 > 0 and the following inequality holds,EQUATIONwhereE 1 = 1 2 Q(0)B 1 (K 21 + K T 21 )B T 1 Q(0) -Y 11 -C T z C z -γ -2 0 Q(0)B 2 B T 2 Q(0) -Q(0)F NF T Q(0), E 2 = 1 2 Q(0)B 1 K 22 + Y 12 , E 3 = Q(0)B 1 K 21 B T 1 + Q(0)F NF T + γ -2 0 Q(0)B 2 B T 2 , E 4 = - R d + B 1 K 21 B T 1 + F NF T + γ -2 0 B 2 B T 2 ,then, the state of the UMVs in system (5) asymptotically converge to zero, while maintaining an H ∞ norm bound of γ 0 . Proof 2: The time derivative of V(e(t)) along the trajectory of the UMVs (5) can be calculated as follows:dV(e(t)) dtEQUATIONwhereU 0 (e) = e T (t)(P -Q 1 -P 1 )e(t) + e T (t -d)Q 1 e(t -d) + 0 -d e T (t + τ )A T 1 Q T (-d-τ )RQ(-d-τ )A 1 e(t + τ )dτ. P 1 = 0 -d A T 1 Q T (-d -τ )RQ(-d -τ )A 1 dτ.Substituting ( 14) into ( 16), we have dV(e(t)) dt(5)EQUATIONwhereΓ(t) = [e T (t) e T (t -d) Γ T 1 (e(t))] T , E :=   P +Q 1 +P 1 -E 1 E 2 E 3 E T 2 -Q 1 + Y 22 1 2 K T 22 B T 1 E T 3 1 2 B 1 K 22 E 4   ,whereE 1 = 1 2 Q(0)B 1 (K 21 + K T 21 )B T 1 Q(0) -Y 11 -C T z C z -γ -2 0 Q(0)B 2 B T 2 Q(0) -Q(0)F NF T Q(0), E 2 = 1 2 Q(0)B 1 K 22 + Y 12 , E 3 = Q(0)B 1 K 21 B T 1 + Q(0)F NF T + γ -2 0 Q(0)B 2 B T 2 , E 4 = - R d + B 1 K 21 B T 1 + F NF T + γ -2 0 B 2 B T 2 ,For E < 0, it implies that dV(EQUATIONIf Theorem 1 holds, then the t t0 Γ T (τ )EΓ(τ )dτ < 0 is satisfied:EQUATIONClearlyEQUATIONWe obtainEQUATIONBy integrating equation ( 18)) from 0 to ∞, we obtainEQUATIONWhen the disturbance ω(t) is absent, combining (8), (14) , and (18) yields:dV(e(t)) dt+ e T (t)Ωe(t) + u T (t)R q u(t)EQUATIONwhereO = -(Y + K 21 )B T 1 Q(0) K 22 -(Y + K 21 )B T 1 . We have E + diag(Ω, 0, 0) O T O -4R -1 q < 0 Hence, ∞ 0 e T (t)Ωe(t) + u T (t)R q u(t) dt ≤ J * .where J * = V(e(t)), with V(e(t)) defined in (12) .VI. SIMULATION EXAMPLE The proposed control method's effectiveness is demonstrated through a standard floating production vessel model, as referenced in [23] . The matrices ξ, C, and D are specified in [23] , and the thruster configuration matrix G is derived from [24] .The initial condition is given as ϕ(s) = [0 0 0 0 0 0.2] T , with the reference signal set to x ref = [0.01 -0.01 0.05 0.01 0.04 0.01] T . The time delay is d = 1, and the H ∞ performance index γ 0 = 2.The controller gain matrix K 11 is obtained by solving the LMI (11) from Lemma 1, as follows: We set the matrix Q = I. The (i, j)-th element of the matrix Q(θ), denoted as Q ij (θ), is determined using the method proposed in [22] . Figures 1 2 show the values ofK 11 =         3.Q ij (θ) for θ ∈ [0, 1].Finally, by solving LMI (15) as described in Theorem 1, the controller gain matrices K 21 and K 22 are computed as: (5) . Figure 5 shows the control inputs produced by the controller as defined in (14) . In Figure 3 , it is clear that the error curves under the proposed control initially exhibit small fluctuations before gradually converging to zero. This demonstrates the effectiveness of the proposed control strategy. Figure 5 illustrates the response curves of the guaranteed cost DP controller u(t).K 21 = 1 × 10 4         0.In this paper, we have addressed the guaranteed cost dynamic positioning control problem for UMVs with time delays. First, we propose a complete-type LKF for UMVs with time delays, which leads to less conservativeness. Furthermore, a novel approach for designing guaranteed cost dynamic positioning controller for DP systems is proposed. The specific form of this controller is derived from feasible solutions of LMIs. The proposed method was validated through simulation, demonstrating its effectiveness. Future work will focus on extending the control strategy to systems with time-varying delays, further enhancing the robustness of DP control for UMVs. 
 
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Reviews: review: title: title: new insight and opinion into the development of  UMVs with time delays
main_review: main_review: In this manuscript  titled "Lyapunov Matrix-Based Guaranteed Cost Dynamic Positioning Control for Unmanned Marine Vehicles With Time Delay", the authors summarized and discussed a Lyapunov matrix-based dynamic positioning controller for unmanned marine vehicles (UMVs) that accounts for time delays using a novel Lyapunov–Krasovskii functional and ensures state errors converge to zero with bounded performance through linear matrix inequalities and Jensen’s inequality.This work provides new insight and opinion into the development of  UMVs with time delays,a novel approach for designing guaranteed cost dynamic positioning controller for DP systems is proposed. The manuscript is well-organized and clearly stated. I would suggest accepting it after the following concerns are addressed: Analysis and complete interpretation of all data
title: title: The paper is written clearly, exceptionally excellent.
main_review: main_review: This paper excels in terms of quality, clarity, originality, and significance, but I would still like to offer some suggestions.
1. Can the method proposed in this paper be applied to actual systems?
2. Discuss the rationality of the assumptions made in the paper.
title: title: Accept
main_review: main_review: The paper presents a well-structured and rigorous approach to designing a guaranteed cost dynamic positioning controller for UMVs under time delays.  The proposed method is based on solid theoretical foundations, utilizing the LKF framework, LMI technique, and Jensen's inequality to derive sufficient conditions for the controller's existence.  The simulation results demonstrate the effectiveness of the approach, providing evidence of its practical value.The following comments should be considered to improve the paper’s quality further.1.Compared to conventional time delays compensation techniques, what are the specific advantages of the time delays compensation method proposed in this paper?2. In guaranteed cost analysis, the disturbance is considered to be absent, what is the purpose of this process?