Robust H ∞ control for nonlinear course system of unmanned surface vessel with polytopic uncertainty based on sum of squares

Introduction

The course control of the unmanned surface vessel (USV) is a crucial part in sailing business. However, the course control system is a complex nonlinear system with uncertainty. In the actual situation, we must consider the various uncertainties caused by the speed, loading conditions, the depth of water and other different navigation conditions that will affect the parameters of the system. What is more, there are randomly external disturbances that can also influence the stability of ship such as wind, wave and current during navigation (Liu, 2017). Therefore, it is of great significance to realize the course control of the USV. In recent years, the robust control about nonlinear course control system with external disturbances and parameteric uncertainty has received much attention. The solutions to this problem mainly include back-stepping, sliding mode control (SMC) and H ∞ control and so on (Wang et al., 2017). As a feasible and typical nonlinear control method, the back-stepping method (Das et al., 2015; Dong et al., 2011) can consider the uncertainty of parameters through the recursive design procedure, and is a powerful way to deal with the uncertainty. In Witkowska et al. (2007), the authors design the nonlinear control rules for course control system by using the back-stepping method. The nonlinear controller configurations are designed for two different course control models and the genetic algorithms is used to tune the parameters of obtained controller. However, the backstepping method is more sensitive to parameter changes. In order to achieve better control effect, it is very important to get the accurate information about the changing parameters, which is very difficult in practical applications. In Yuan et al. (2016), the neural networks control with adaptive updated weight values are introduced to estimate the unknown parameters of the constructed Lyapunov function in backstepping control. This method overcomes the problems faced in Witkowska et al. (2007), and the effectiveness of achieving course tracking of USV is verified via simulation. However, the backstepping technique requires repeated differentiation of the virtual stabilization control function, which makes the back-stepping control very complicated as the number of back-stepping steps increases.

Robust control can handle the uncertain existing in the system and has a wide range of applications such as robotics, aircraft control, and so on. In the field of USV motion control, the robust Proportion Integration Differentiation (PID) control is a very practical and widely used control algorithm. In Yang and Xiaoli Yu (2002), considering the parameter uncertainties and the disturbance uncertainties in the course control of USV, the Lyapunov stability theory is used to design a control algorithm for compensating the uncertainties. The robust PID control law is constituted by combining PID control and compensation control.The results of simulation experiments have shown that the robust control is effective and has a certain degree of robustness to the uncertainties. Robust adaptive control is an important research direction in recent years. It combines robust control with adaptive control methods to target structural and non-structural uncertainties in objects, which makes the control theory closer to reality. In Xing (2017)), a robust adaptive control scheme is proposed based on backstepping technique for a class of nonlinear systems with unknown parameters and unknown modeling errors. The non-triangular structural uncertainties caused by system parameter variations, disturbance torque and modeling errors can be allowed depending on all system states. Simulation studies are used to verify the effectiveness of the proposed scheme and it is shown that the proposed controller can ensure that all signals of closed-loop system are bounded. Coincidentally, an indirect robust adaptive optimal controller for the nonlinear systems with unstructured uncertainties by a unification of a low-level and a high-level controllers in Azimi and Vela (2019). The unknown dynamics of the system are estimated by a joint neural network and concurrent learning adaptation mechanism (NNCL) to inform the adaptive term. The optimal term uses an online quadratic program (QP) formulated to generate the optimal signal while providing system stability via a control Lyapunov function. Simulation results illustrated that tracking and estimation performances were met using the proposed controller even when the system parameters were deviated from their nominal values. In Le-Tien and Albu-Schäffer (2017), a robust adaptive control scheme based on a cascaded structure with a full state feedback controller with integrator terms as inner control loop and computed torque as outer control loop for flexible joint robots. The adaptive control law can enhance position accuracy under uncertainties of the robot model, especially, the high friction caused by harmonic drive with high gear ratio. What is more, the proposed adaptive control approach can simultaneously provide high control performance both in terms of the dynamic behavior and the position accuracy. Finally, global asymptotic convergence of the controllers has been proven.

SMC (Hu and Xiao, 2017) is one of the most efficient techniques to cope with external disturbance and parametric uncertainty. The literature (Perera and Soares, 2012) proposed a sliding mode method based on pre-filter to overcome the rudder angle and rudder rate limitation in the ship course control system. The introduced pre-filter can allow the large instantaneous course error and reduce the requirements of larger rudder angle and rudder rate. Furthermore, the pre-filter based sliding mode controller is robust to nonlinear course system with uncertain parameters. In Renhai et al. (2017), the algorithm of ship course control is proposed based on differential flatness and sliding mode technique. The states and control input are firstly represented by the flat output and its derivatives, so that the trajectory of the state variable and control input can be known through flat output without integrating the function. The flatness can also considerably simplifies the design process of the sliding mode controller by reducing the problem to control a linearized system. However, a fatal drawback to SMC is existence of the chattering when the controller of the system is switched during the operation. How to reduce the effect of chattering is an important and challenging topic.

The main idea of H ∞ control is to design a controller that makes the internal stabilization of the system and minimizes the infinity norm of the closed transfer function from the disturbances to the regulated output. In Hu et al. (2003), the robust control of the nonlinear ship course-keeping system with uncertainty is investigated. Firstly, an approximated linear reference model of the original nonlinear system is obtained in input/output (I/O) behaviour via the H ∞ I/O linearization formulation. Then, the uncertain nonlinear system of the ship can be formulated as linear reference model plus an uncertainty block, which was addressed by linear H ∞ control and μ -synthesis technique. But, the design of the H ∞ controller depends on the solution of the complex Hamilton-Jacobi-Isaacs (HJI) equation, which will increase the difficulty of solving the control rules. In Rodrigues et al. (2018), the H ∞ state feedback controller was designed for linear time-invariant systems with polytopic uncertainty. By using parameter-dependent lyapunov function and Elimination Lemma, the solution of the controller is formulated as some linear matrix inequalities (LMIs). In Dong and Yang (2013), the robust static output feedback controllers were proposed to solve the problem of linear systems with polytopic uncertainties by searching a new LMI condition. Compared with the existing methods, the new one is applicable for the situation where the uncertain output matrices are not required to be full row rank. Both Rodrigues et al. (2018) and Dong and Yang (2013) use the LMI approaches to solve the robust H ∞ control law, which simplifies the computation compared with the method in Hu et al. (2003) to a certain degree, but the method applies only to linear systems. However, in recent years, the Sum of squares (SOS) technique has provided a new framework for solving many difficult problems in the control of nonlinear systems. Specifically, SOS generalizes the LMI algorithm tool in linear robust control theory. In Ma and Yang (2008), the fault-tolerant H ∞ control problem in a polynomial nonlinear system was transformed into an SOS optimization problem, avoiding the direct solution of the HJI inequality. Narimani and Lam (2010) proposed a new polynomial Lyapunov function, combined with SOS technology to give the solvability condition of stability analysis of polynomial fuzzy control system. In Jiang and Jiang (2015), a new global adaptive dynamic programming method was proposed in which SOS technology is used to optimize the controller online, which avoid the complicated calculation of the neural network fitting method.

In fact, H ∞ control offers effective ways to analyse both linear and nonlinear systems with uncertainties and external disturbances, and has formed relatively mature theory. However, the computation is still a difficult problem in H ∞ control for nonlinear systems such as Hu et al. (2003). To simplify computing and design of H ∞ controller, some constraint conditions are made in nonlinear systems such as Zemzemi et al. (2016) and Hu et al. (2016). In Zemzemi et al. (2016), a scheme of designing robust observer is proposed for a class of nonlinear systems with time-varying uncertainties and whose nonlinear function is assumed to satisfy the Lipschitz condition. The parameters of proposed observer are determined by using LMIs techniques. In this reference, the authors assume that the Lipschitz constant of the system is known, however, it is difficult to determine the value of the Lipschitz constant in fact. In Hu et al. (2016), the robust reliable H ∞ control problem for uncertain nonlinear systems is settled in terms of LMI. The uncertain of nonlinear system is described by the matrices multiplication and a new method of annihilating uncertain matrix is proposed. However, both in Zemzemi et al. (2016) and Hu et al. (2016), the nonlinear uncertainty must be assumed to satisfy a matching condition and the nonlinearity of the system is subject to certain assumptions.

Combining with the above analysis and the characteristics of the ship course control system, in this paper, the nonlinear robust H ∞ control will be investigated for ship course control systems in present of parameter uncertainties and external disturbances. Firstly, we formulate the uncertainties of parameters as the form of convex polytopic according to the analysis of the causes, characteristics of uncertainties in parameters. The mathematical model of ship course control system with polytopic uncertain will be obtained subsequently. Secondly, by using the positive polynomial theory and parameter-dependent Lyapunov function approach, the sufficient condition for stability with H ∞ performance of nonlinear ship systems is derived. Finally, the robust H ∞ state feedback controller is designed and the corresponding solvability condition is given. It is worth noting that the solutions to both analysis and synthesis problems presented in the paper are formulated as terms of state dependent LMIs, which can be verified by the sum of squares technique easily. So that the computation problem meeting in nonlinear H ∞ control can be overcame.

Aiming at the uncertainty of parameters and the interference of wind and wave, the nonlinear H ∞ control method based on SOS is proposed for course control. The main contributions of the paper are as follows:

The remainder of the paper is organized as follows. Section 2 shows the related preliminaries and the problem formulation. Section 3 is devoted to the robust consensus design without and with disturbance attenuation. In Section 4, simulations are provided to validate the proposed approach. Some concluding remarks are given in Section 5.

Mathematical model and problem formulation

The model of ship course control system

The motion of the ship is shown in Figure 1. The control system discussed in this paper is the course control of the ship during navigation. In the ship course control system, the controlled parameter is the course of ship ψ ( t ) , and the control input is the rudder angle δ ( t ) . The simplest mathematical model describing the steering dynamics of a ship is the Nomoto model (Nomoto et al., 1956), which is a linear equation and derived from Newton’s laws of dynamics. However, the nonlinear characteristics of the ship will appear if the effect of the hydrodynamic factors is considered. Based on the Nomoto mode, the Norrbin model (Norrbin, 1996) in which the nonlinear characteristics were considered could improve precision of the ship course control model. Considering the influence of wind, wave and other external disturbances, the Norrbin model can be expressed as

T ψ ·· + H N ( ψ · ) = K ( δ + ω )

(1)

H N ( ψ · ) = n 3 ψ · 3 + n 2 ψ · 2 + n 1 ψ · + n 0

(2)

where ω ∈ L 2 [ 0 , ∞ ] is external disturbance, H N ( ψ · ) is a nonlinear function with respect to ψ · , n i ( i = 0 , ⋯ , 3 ) is the Norrbin coefficient, which can be determined by a spiral experiment. The n 0 = n 2 = 0 when the port and starboard of ship are symmetric (Xia and Luan, 2015). The n 1 = 1 for a course-stable ship while n 1 = − 1 corresponds to a course-unstable ship, and n 3 is a positive number. The parameters T and K are defined as

T = T 0 L U , K = K 0 U L

where T 0 and K 0 are the parameters of the system model, U and L are the speed and length of the ship, respectively. Let x 1 = ψ , x 2 = ψ · , then the system in (1) can be expressed as a state space equation as follows

{ x · 1 = x 2 x · 2 = − a 1 x 2 − a 2 x 2 3 + a 3 u + a 3 ω

(3)

where a 1 = n 1 U T 0 L , a 2 = n 3 U T 0 L , a 3 = K 0 U 2 T 0 L 2 , u = δ is the control input.

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Figure 1.

Ship motion coordinate system.

The ship is susceptible to disturbances such as wind, current, the depth of water and the load of ship during the voyage, which will not only influence the course of the ship, but also some disturbances, such as the depth of water and the load of ship, will make system parameters change during some certain ranges (Van Amerongen, 1984; Van Amerongen and Udink, 1973). For example, changes in the load of ship will affect the value of H N ( ψ · ) . Assume that the parameter a = ( a 1 , a 2 , a 3 ) T in the system (3) is a time-invariant uncertain parameter, which satisfies a ¯ i ⩽ a i ⩽ a ¯ i . The state space expression of Norbrin model under consideration of disturbances and parameters uncertainty can be modeled

x · = A ( x , ξ ) x + B 2 ( x , ξ ) u + B 1 ( x , ξ ) ω

(4)

where x = col ( x 1 , x 2 ) , [ A ( x , ξ ) , B 1 ( x , ξ ) , B 2 ( x , ξ ) ] ∈ co { Ω Σ } , ξ ∈ Θ q = card ( Ω Σ ) with a given set

Ω Σ ≜ { Σ = [ A ( x ) = [ 0 0 0 − a 1 − a 2 x 2 2 ] , B 1 ( x ) = [ 0 a 3 ] , B 2 ( x ) = [ 0 a 3 ] ] : a i = a _ i o r a ¯ i , i = 1 , 2 , 3 }

To analyze the problem of disturbance suppression, define the regulated output of system as z ( t ) = x 1 ( t ) , then system (4) can be expressed as

{ x · = A ( x , ξ ) x + B 2 ( x , ξ ) u + B 1 ( x , ξ ) ω z = Cx

(5)

where C = [ 10 ] .

Remark 1: In (5), the ξ ∈ Θ q = card ( Ω Σ ) refers to a convex polyhedron containing q nodes. The convex polyhedron contains all possible values of uncertain parameter vectors. When the uncertain parameter vector takes different values, the corresponding vector ξ also takes different values, so the uncertainty of the original model parameters is converted into the uncertainty of the vector ξ , and each component is a positive number, which provides an effective processing method for the elimination of uncertainties.

Main results

In this section, the state feedback controller will be designed to achieve robust H ∞ control for unmanned ship course control system. Firstly, suppose u ( t ) = 0 . The criterion is obtained that guarantees the system (5) satisfying the asymptotic stability with the L 2 -gain performance γ . The method of designing the H ∞ controller is given subsequently.

Theorem 1: Consider the ship system (5) with u ( t ) = 0 . For a given scalars γ > 0 and a small enough positive scalar α , if there exist polynomial matrices P i ( x 1 ) and constant matrices ϒ 1 , ϒ 2 , scalars ε 1 i > 0 , ε 2 i > 0 , i = 1 , 2 , ⋯ , q such that

where M ( i ) ( x ) = [ M 11 ( i ) ( x ) * * * − B 1 i T ( x ) ϒ 1 T − γ 2 I * * C 0 − I 0 α ( M 14 ( i ) ( x ) ) T − α ϒ 2 B 1 i ( x ) 0 α 2 he ( ϒ 2 ) ] ,

then the system is internally stable and has H ∞ performance γ .

Proof: For the system (5), we define the parameter dependent Lyapunov function as follows

Let Φ ( x ) = V · ( x , ξ ) + z T z − γ 2 ω T ω , we have

where Q ( x , ξ ) = [ Q 11 ( x , ξ ) * B 1 T ( x , ξ ) P ( x 1 , ξ ) − γ 2 I ] , Q 11 ( x , ξ ) = P ( x 1 , ξ ) A ( x , ξ ) + A T ( x , ξ ) P ( x 1 , ξ ) + ∂ P ( x 1 , ξ ) ∂ x 1 A i 1 ( x ) x + C T C , ν = col ( x , ω ) .

Obviously, if there is a polynomial matrix P ( x 1 , ξ ) with appropriate dimension such that Q ( x , ξ ) < 0 , then Φ ( x ) < 0 can be obtained. It is concluded that:

Consider the following formula

where E ( x ) = P ( x 1 , ξ ) + ϒ 1 + A T ( x , ξ ) ϒ 2 . It is clear that he ( ϒ 2 ) < 0 if (10) holds. Thus, Q ( x , ξ ) < 0 if Z ( x , ξ ) < 0 , and Z ( x , ξ ) < 0 can be equivalent to H ( x , ξ ) < 0 by using the Schur complement where

In fact, Q ( x , ξ ) is a sub-matrix of H ( x , ξ ) . If H ( x , ξ ) < 0 has a feasible solution, the solution is feasible to Q ( x , ξ ) < 0 . On the other hand, since α can take a sufficiently small positive number, if P ( x 1 , ξ ) is a feasible solution to Q ( x , ξ ) < 0 and it is also a solution to H ( x , ξ ) < 0 . Furthermore, pre-multiply G ( x , ξ ) by Γ T ( x , ξ ) and post-multiply by Γ ( x , ξ ) , we can obtain

where

By (16), R ( x , ξ ) < 0 is equivalent to G ( x , ξ ) < 0 , and we further have

Obviously if α can take a small scalar such that α ⩽ 1

Therefore, if G ( x , ξ ) < 0 , we can get Q ( x , ξ ) < 0 . Notice that ∑ i = 1 8 ξ i M ( i ) ( x ) < 0 is equivalent to G ( x , ξ ) < 0 , which implies G ( x , ξ ) < 0 if (10) holds. Moreover x T P i ( x 1 ) x > 0 is guaranteed by (11).

Remark 3: In Theorem 1, (10) and (11) are expressed in the form of SOS matrix. However, the non-negative polynomial cannot be deduced as SOS. While the SOS polynomial is stricter, the SOS is much more computationally tractable than nonnegativity. At the same time, it is easier to get the exact solution by replacing nonnegativity with the SOS property in many cases. The equivalence, proven by Hilbert (Reznick, 2000), between nonnegativity and SOS in the cases of univariate polynomials, any (even) degree, quadratic polynomials, in any number of variables, and quartic polynomials in two variables.

An important advantage of Theorem 1 is the elimination of the product term of P ( x 1 , ξ ) from the system matrices A ( x , ξ ) and B 1 ( x , ξ ) by introducing ϒ 1 and ϒ 2 . Thus, we can use parameter-dependent Lyapunov function to solve the robust control problems with polytopic uncertainties, which may potentially lead to smaller conservativeness than quadratic stability conditions.

Theorem 1 focuses on the robust analysis problem of the course control of the unmanned ship with bounded parameter uncertainties, which can also be extended to design robust control synthesis problem of the uncertain ship system. For system (5), we design a nonlinear state-feedback controller as follows

Thus, the closed loop system of (5) can be respented as

where A cl ( x , ξ ) = A ( x , ξ ) + B 2 ( x , ξ ) K ( x ) .

Theorem 2: Consider the closed loop system (20) of the ship (5). For given scalars γ > 0 , λ > 0 , 0 < α ⩽ 1 , if there exist polynomial matrices X i ( x 1 ) , Y ( x ) , constant matrices S of the appropriate dimension and scalars ε 1 i > 0 , ε 2 i > 0 , i = 1 , 2 , ⋯ , 2 3 such that the following conditions hold

where Ψ ( i ) ( x ) = [ Ψ 11 ( i ) ( x ) * * * − B 1 i T ( x ) − γ 2 I * * C S T 0 − I * α ( Ψ 14 ( i ) ( x ) ) T − α λ B 1 i ( x ) 0 α 2 λ ( S + S T ) ] ,

then the uncertain course control system of the unmanned ship is internally stable and has H ∞ performance γ . Moreover the control gain matrix is given by K ( x ) = Y ( x ) S − T .

Proof: For system (20), choose a Lyapunov function candidate to be

It is follow a similar process of Theorem 1, by replacing A i ( x ) in M ( i ) ( x ) with A cli ( x , ξ ) , and letting ϒ 1 ( x ) = ϒ , ϒ 2 ( x ) = λ ϒ . It is worth noting that S is reversible if (21) holds. So, we can let ϒ − 1 = S and multiply M ( i ) ( x ) on the left by diag ( ϒ − 1 , I , I , ϒ − 1 ) , and on the right by diag ( ϒ − T , I , I , ϒ − T ) we can obtain

where J 11 ( i ) ( x ) = − ϒ − 1 A i T ( x ) − A i ( x ) ϒ − T − B 2 i K ( x ) ϒ − T − ϒ − 1 K T ( x ) B 2 i T ( x ) + ϒ − 1 d P i ( x 1 ) d x 1 A i 1 ( x ) x ϒ − T ,

Let ϒ − 1 = S , ϒ − 1 P i ( x ) ϒ − T = X i ( x ) , K ( x ) ϒ − T = Y ( x ) , we can get Ψ ( i ) ( x ) and K ( x ) = Y ( x ) S − T . □

Simulation results

Consider the course control system of unmanned ship in the form of (3), the parameters of the ship course control system can be determined according to the literature (Fossen and Paulsen, 1992) where n 1 = 1 , n 3 = 0 . 4 , U = 5 , L = 45 , T 0 = 3 . 44 , K 0 = 4 . 5 . In this example, assume that γ = 0 . 8 , λ = 2 × 10 − 5 , α = 1 . The uncertain parameters n 3 , T 0 and K 0 have 20% perturbation due to the change of the external environment. The disturbance ω ( t ) shown in Figure 2 is a uniformly distributed random signal with a duration of 10 seconds. According to Theorem 2, the controller will be obtained by utilizing the SOSTOOLS as follows

Figure 3 plots the state response of the closed loop system without disturbance. It is clear that the closed loop system is asymptotically stable. Figure 4 shows the state response with the disturbance ω ( t ) . Obviously, in the first 10 seconds, the state trajectories are influenced by the the disturbance ω ( t ) , but the system is asymptotically stable when ω ( t ) turn to zero after 10 seconds later. Moreover, H ∞ performance of the closed loop system is shown in Figure 5. Figure 6 and Figure 7 show, respectively, the state response of the system under different model parameters ( a 1 , a 2 , a 3 ) without and with disturbance. It can be seen that when the parameters change, the course response curve of the system hardly changes, which shows that the system has better robustness to the parameter changes.

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Figure 2.

External disturbance.

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Figure 3.

State response without disturbance.

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Figure 4.

State response with disturbance.

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Figure 5.

The H ∞ performance γ 2 .

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Figure 6.

State response under different model parameters without disturbance

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Figure 7.

State response under different model parameters with disturbance.

Figure 8 to Figure 9 show the comparison about the course response curve of the USV between the proposed method and the robust PID. method in Yang and Xiaoli Yu (2002). Figure 8 shows the comparison curve without interference in the system. It can be seen that the robust PID method has a faster response speed but its overshoot is almost over 13 % . The large overshoot of the course will make the USV deviate from the original route. However, the method proposed in this paper has almost no overshoot.Although the response time of the proposed method takes about 5 seconds, in reality it takes about 7 seconds for robust PID method to return USV to its original route due to the excessive overshoot. Figure 9 shows the comparison curve with interference in the system. It shows that the robust PID method has a larger overshoot almost 17 % and a steady state error occurs when there is external interference. On the contrary, the method proposed in this article only has a small amount of overshoot, which means a strong ability to suppress the interference. Figure 10 and Figure 11 show, respectively, the course response of robust PID method and proposed method under different model parameters ( a 1 , a 2 , a 3 ) without disturbance. It can be seen that when the parameters change, the course response curve of robust PID method changes significantly. The overshoot and steady state values change with the model parameters. The course response curve of the proposed method hardly changes, which shows the better robustness to the parameter changes. What is more, the comparison effect of the two methods about the robustness are shown as Figure 12 and Figure 13, when there is external interference in system.

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Figure 8.

Course contrast curve without disturbance.

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Figure 9.

Course contrast curve with disturbance.

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Figure 10.

Course response of robust PID method under different model parameters without disturbance.

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Figure 11.

Course response of proposed method under different model parameters without disturbance.

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Figure 12.

Course response of robust PID method under different model parameters with disturbance.

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Figure 13.

Course response of robust proposed method under different model parameters with disturbance.

Conclusion

In this paper, we focus on the robust H ∞ control of nonlinear unmanned ship course control systems with uncertain parameters and external disturbance. By analyzing the factors that causing the uncertainty of the parameters of the ship, the course control system is modeled as a polynomial system with polytopic uncertainty. Sufficient conditions in terms of state dependent LMI are given for analysis and synthesis of H ∞ control of the ship course control system via the parameter-dependent Lyapunov function method. These conditions can be verified using semidefinite programming, such as sum of squares technique, which essentially solves a LMI feasibility problem. Finally, a numerical example is given to illustrate the effectiveness of the proposed approach. Further research work can also be focused on the following two aspects. First, due to the limited execution ability of the steering gear, it is not possible to provide too large a rudder angle output, so it is necessary to consider the nonlinear robust H ∞ control problem when the rudder angle input is restricted. Second, the design of a nonlinear robust H ∞ controller is based on the solution of multiple SOS matrices in this paper, but when the number of unknown parameters increase, the number of SOS matrices will grow exponentially. Therefore, how to effectively reduce the number of SOS matrices in the obtained sufficient conditions requires further study.