- Email: [email protected]
Nonlinear Dynamic Ship Positioning
8c-O! 5
Copyright © 1996 IFAC 13th Triennial World Congress, San Francisco, USA
NONLINEAR DYNAMIC SHIP POSmONING
M R Katebi, Yi Zbang and M J Grimble
Industrial Control Centre, Strathclyde University, 50 George Street, Glasgow, GI IQE, UK Tel: +141-5524400 Ext.: 4297 or 2378 Fax: +141-5531232 Email: [email protected]
Abstract: Thruster saturation in high sea states can occur due to lack of thruster power in respond to low frequency or high frequency sea disturbances. This can lead to inaccuracy in ship positions and possible control loop instability. Using the Single Input Describing Function (SIDF) to convert the static nonlinear elements into a quasi-linear system, robust control design techniques such as Ha, and ).L Methods are extended to the nonlinear control design problems to incorporate the thruster nonlinear dynamics. Based on the SIDF approximation, the inverse SIDF compensation is also used to cancel the dominant nonlinearities. The remnant of the nonlinear effects can then be included in the controller robustness requirement. These methods are then compared with the case where the nonlinear elements can be replaced by a unity linear element. Simulation studies for a Ship Dynamic Positioning System are presented to compare the robustness of the proposed methods. Keywords: Marine Systems, Dynamic Systems, Robust Control, Nonlinear Control
1. INTRODUCTION The dynamic positioning control system is required in some ships to keep the ship stationary in response to sea waves, wind and currents. The PID control technique application in the marine systems formed the first generation of the DP systems. This was followed by the implementation of LQG control strategy with the Kalman filters (Grimble, Patton and Wise, 1979), to deal with the conflicting control design requirements, such as, the low thruster modulation and good current disturbance rejection. The Ha, technique is extended here to incorporate the dynamics of the nonlinear elements of thruster system into control design. The dynamic positioning control system has been designed in this paper using Ha, control technique (Katebi, Zhang and Grimble, 1995).
The paper is organised as follows. A brief description of the nonlinear techniques is given in Section 2. The extension of J..l Method is discussed in Section 3. The robustness of the control system is analysed in Section 4. The ship and sea models are given in Section 5. The simulation results are presented in Section 6. Finally the conclusion is drawn in Section 7.
2. NONLINEAR CONTROL DESIGN TECHNIQUE The harmonic linearisation method is simply to replace a nonlinear element using its Describing Function (DF) approximation and consequently convert the nonlinear system into a quasi-linear control problem. Some linear
7999
control design techniques may then be applied to stabilize the closed-loop system. One such method is fl technique which enables the control designer to incorporate a measure of the robust stability and performance into the control design (Katebi and Zhang, 1995 ). For a SISO system, the Single Input Describing Function (SIDF) uses a single sine wave as the input, usually, the main harmonic components, to approximate a nonlinear element by a quasi-linear transfer function. In MIMO system, a number of sine waves should be used as the inputs, that is, the Multi-Input Describing Functions (MIDF) are used to form a quasi-linear system. The basic assumption for the DF approximation is that the general plant can be decomposed into a low pass linear element and a separable nonlinear element. The effect of higher harmonic components can then be sharply attenuated by the linear element. On the other hand, the effect of phase shift on the input of a nonlinear element is usually neglected. The consequence of these approximations is the introduction of a large model uncertainty into the nominal system. It is attempted (Gelbe and Velde, 1968) to reduce the model uncertainty by using the MIDF. However, this approach leads to a complicated model which is not convenient for the closed-loop system analysis and synthesis. The disadvantage of using the DF approximation to formulate the nominal model of a nonlinear element for the control design purpose is that it does not automatically guarantee the stability of an actual control system (Katebi and Zhang, 1995). In fact, the modeling error induced by the DF approximation may results in the poor robustness of closed-loop system and possible limit cycle oscillation. Therefore, the robust analysis against the modeling error induced by the DF approximation and the validity of the DF approximation should be undertaken in the system control analysis and design processes. This is the first method discussed in this paper.
Since a quasi-linear transfer function, SIDF or MIDF, is a function of the frequency and amplitudes of all nonlinear element input signals, the definition of the Structured Singular Value (SSV) employed in the fl methods should be modified for the quasi-linear system. This requires the introduction of qao -norm, which is defined and used to indicate the norm size of a quasi-linear transfer function. To simplify the quasi-linear system design problem and controller calculation, some compensation techniques have been investigated on the basis of the DFs for the certain classes of separable nonlinear systems (Zhang, 1994). The inverse DF pre-compensation may be used to cancel the dominant non-linearity by cascading the inverse of DFs with the system (Zhang, Katebi and Grimble, 1994). Some
Figure 1 The total model uncertainties. of separable nonlinear control systems can be designed using the linear compensation technique, in which the nonlinear dynamic can be replaced using a linear unity element when the remnant non-linearity is qao -norm bounded. The applications of these methods lead to the development of a linear nominal system with a quasilinear model perturbation. Thus, the controller calculation is simplified to one for the linear control systems.
4. ROBUSTNESS ISSUES To define the SSVs for the quasi-linear system, assume that the nominal closed loop system is stable and the inverse of the describing function N(A,oo) exists. The real model of the nonlinear system may now be represented by (Figure I):
3. EXTENSION OF fl METHODS With the introduction of the DF approximation to replace the nonlinear elements, the fl analysis and synthesis can be applied to the resulting quasi-linear system to deal with the nonlinear control system analysis and design ( Katebi and Zhang, 1995). On the other hand, the fl methods allows the modeling errors induced by the DF approximation and even the high frequency perturbation effects on the linear components to be included in the control design.
1
The model used for control design is GN. The term in bracket is the input multiplicative uncertainty due to the SIDF approximation. Using small gain theorem, the following transfer functions are derived:
A.
8000
= W'A,;
T
=[1 +MGNtMGN
2
The transfer function, T, is the complementary sensitivity function which will be used to test the stability robustness of the closed-loop system. The small gain theorem gives the following sufficient condition for stability concerning the input multiplicative uncertainty:
1 n < (j(T)
-;;;.r /). ) V\
O(/).) :5 (j(/).n) + 0(/).1) + (j(/).//).n) :5 O(/).n) + 0(/)./) + (j(/).I )O(/).n)
7
Using Il criteria to ensure robustness and absence of limit cycle oscillation:
1
= a[(I + MGN)-l MGN]
3
8
Since qOOnorm of /).n and T is defined as a function of
Rearranging this inequality gives an upper bound on the linear element model uncertainties for the stability robustness and the absence of limit cycle oscillation:
the frequency 0) and the control-input amplitude A, their SVs are also dependent on 0) and A. In order to obtain a tighter bound on the singular values of T for stability robustness, Doyle, et al, 1990 defines the structured singular value Il as: _1_ = inf ({cr(~.) s. t. det(I - ~.T) = o}} p(T)
4
A."A
5. SYSTEM DESCRIPTION
where/).n is a block diagonal matrix and ~ is a set of all ~ matrices with a given, fixed block diagonal structure. If there is no /). n to satisfy the above relationship, then Il(T) is defined as zero. This gives the following robustness test for the nonlinear system using the SSVs and SIDF approximation: -;;;.r/). ) V\ n
1
< p(T)
5
The model uncertainty in the linear element may also force the system to limit cycle oscillation. The robustness of the closed loop system can therefore be guaranteed if the total effect of model uncertainties in the linear and nonlinear elements is taken into account in the control design as shown in Figure 2. Let /). n and A represent the model uncertainties of the nonlinear and linear elements, respectively as shown Fig. 5. Define /)./ = N- 1/)., N. The input multiplicative uncertainties are now at the input of the nominal model, GN. A total uncertainty transfer function may now be defined as; 6 Find the maximum singular values of both sides of this equation to give:
The H 00 control design technique described here uses the linear state space description of the ship, the sea disturbances and the modelling error. The general form of the state space model is as follows: x(t)
= Ax(t) + Bu(t);y(t) = Cx(t) + Du(t)
10
where x(t) is the system state, u(t) is the thruster inputs, y(t) is the position measurements and A,B,C and D are system matrices with appropriate dimensions. In the following section, the non-linear dynamics of the ship and the disturbances are briefly discussed. A linear state description of these models is then developed. The non-linear ship dynamics may be described as:
(m- X,,}u-(m- I:}rv = X H+ X .. + X,.. + X .. (m-I:}v+(m-X,,)ru = YH+~ +Y,..+Y.. (ID -N,}r:;:; NH +N.. +N,.. +N.. where m = displacement; Izz
11
= Yaw inertia; Xo, Y0 and ~)
= added masses and inertia; u, v,
r = Surge, sway speeds and Yaw rate (rad/s); ( hi = Hydrodynamic forces and moment; ( )A = Aerodynamic forces and moment; ( >w = Wave forces and moment; ( = Thruster forces and moment.
n
Each term is now briefly described. The mathematical equations to calculate the hydrodynamic forces are:
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Copyright © 1996 IFAC 13th Triennial World Congress, San Francisco, USA
NONLINEAR DYNAMIC SHIP POSmONING
M R Katebi, Yi Zbang and M J Grimble
Industrial Control Centre, Strathclyde University, 50 George Street, Glasgow, GI IQE, UK Tel: +141-5524400 Ext.: 4297 or 2378 Fax: +141-5531232 Email: [email protected]
Abstract: Thruster saturation in high sea states can occur due to lack of thruster power in respond to low frequency or high frequency sea disturbances. This can lead to inaccuracy in ship positions and possible control loop instability. Using the Single Input Describing Function (SIDF) to convert the static nonlinear elements into a quasi-linear system, robust control design techniques such as Ha, and ).L Methods are extended to the nonlinear control design problems to incorporate the thruster nonlinear dynamics. Based on the SIDF approximation, the inverse SIDF compensation is also used to cancel the dominant nonlinearities. The remnant of the nonlinear effects can then be included in the controller robustness requirement. These methods are then compared with the case where the nonlinear elements can be replaced by a unity linear element. Simulation studies for a Ship Dynamic Positioning System are presented to compare the robustness of the proposed methods. Keywords: Marine Systems, Dynamic Systems, Robust Control, Nonlinear Control
1. INTRODUCTION The dynamic positioning control system is required in some ships to keep the ship stationary in response to sea waves, wind and currents. The PID control technique application in the marine systems formed the first generation of the DP systems. This was followed by the implementation of LQG control strategy with the Kalman filters (Grimble, Patton and Wise, 1979), to deal with the conflicting control design requirements, such as, the low thruster modulation and good current disturbance rejection. The Ha, technique is extended here to incorporate the dynamics of the nonlinear elements of thruster system into control design. The dynamic positioning control system has been designed in this paper using Ha, control technique (Katebi, Zhang and Grimble, 1995).
The paper is organised as follows. A brief description of the nonlinear techniques is given in Section 2. The extension of J..l Method is discussed in Section 3. The robustness of the control system is analysed in Section 4. The ship and sea models are given in Section 5. The simulation results are presented in Section 6. Finally the conclusion is drawn in Section 7.
2. NONLINEAR CONTROL DESIGN TECHNIQUE The harmonic linearisation method is simply to replace a nonlinear element using its Describing Function (DF) approximation and consequently convert the nonlinear system into a quasi-linear control problem. Some linear
7999
control design techniques may then be applied to stabilize the closed-loop system. One such method is fl technique which enables the control designer to incorporate a measure of the robust stability and performance into the control design (Katebi and Zhang, 1995 ). For a SISO system, the Single Input Describing Function (SIDF) uses a single sine wave as the input, usually, the main harmonic components, to approximate a nonlinear element by a quasi-linear transfer function. In MIMO system, a number of sine waves should be used as the inputs, that is, the Multi-Input Describing Functions (MIDF) are used to form a quasi-linear system. The basic assumption for the DF approximation is that the general plant can be decomposed into a low pass linear element and a separable nonlinear element. The effect of higher harmonic components can then be sharply attenuated by the linear element. On the other hand, the effect of phase shift on the input of a nonlinear element is usually neglected. The consequence of these approximations is the introduction of a large model uncertainty into the nominal system. It is attempted (Gelbe and Velde, 1968) to reduce the model uncertainty by using the MIDF. However, this approach leads to a complicated model which is not convenient for the closed-loop system analysis and synthesis. The disadvantage of using the DF approximation to formulate the nominal model of a nonlinear element for the control design purpose is that it does not automatically guarantee the stability of an actual control system (Katebi and Zhang, 1995). In fact, the modeling error induced by the DF approximation may results in the poor robustness of closed-loop system and possible limit cycle oscillation. Therefore, the robust analysis against the modeling error induced by the DF approximation and the validity of the DF approximation should be undertaken in the system control analysis and design processes. This is the first method discussed in this paper.
Since a quasi-linear transfer function, SIDF or MIDF, is a function of the frequency and amplitudes of all nonlinear element input signals, the definition of the Structured Singular Value (SSV) employed in the fl methods should be modified for the quasi-linear system. This requires the introduction of qao -norm, which is defined and used to indicate the norm size of a quasi-linear transfer function. To simplify the quasi-linear system design problem and controller calculation, some compensation techniques have been investigated on the basis of the DFs for the certain classes of separable nonlinear systems (Zhang, 1994). The inverse DF pre-compensation may be used to cancel the dominant non-linearity by cascading the inverse of DFs with the system (Zhang, Katebi and Grimble, 1994). Some
Figure 1 The total model uncertainties. of separable nonlinear control systems can be designed using the linear compensation technique, in which the nonlinear dynamic can be replaced using a linear unity element when the remnant non-linearity is qao -norm bounded. The applications of these methods lead to the development of a linear nominal system with a quasilinear model perturbation. Thus, the controller calculation is simplified to one for the linear control systems.
4. ROBUSTNESS ISSUES To define the SSVs for the quasi-linear system, assume that the nominal closed loop system is stable and the inverse of the describing function N(A,oo) exists. The real model of the nonlinear system may now be represented by (Figure I):
3. EXTENSION OF fl METHODS With the introduction of the DF approximation to replace the nonlinear elements, the fl analysis and synthesis can be applied to the resulting quasi-linear system to deal with the nonlinear control system analysis and design ( Katebi and Zhang, 1995). On the other hand, the fl methods allows the modeling errors induced by the DF approximation and even the high frequency perturbation effects on the linear components to be included in the control design.
1
The model used for control design is GN. The term in bracket is the input multiplicative uncertainty due to the SIDF approximation. Using small gain theorem, the following transfer functions are derived:
A.
8000
= W'A,;
T
=[1 +MGNtMGN
2
The transfer function, T, is the complementary sensitivity function which will be used to test the stability robustness of the closed-loop system. The small gain theorem gives the following sufficient condition for stability concerning the input multiplicative uncertainty:
1 n < (j(T)
-;;;.r /). ) V\
O(/).) :5 (j(/).n) + 0(/).1) + (j(/).//).n) :5 O(/).n) + 0(/)./) + (j(/).I )O(/).n)
7
Using Il criteria to ensure robustness and absence of limit cycle oscillation:
1
= a[(I + MGN)-l MGN]
3
8
Since qOOnorm of /).n and T is defined as a function of
Rearranging this inequality gives an upper bound on the linear element model uncertainties for the stability robustness and the absence of limit cycle oscillation:
the frequency 0) and the control-input amplitude A, their SVs are also dependent on 0) and A. In order to obtain a tighter bound on the singular values of T for stability robustness, Doyle, et al, 1990 defines the structured singular value Il as: _1_ = inf ({cr(~.) s. t. det(I - ~.T) = o}} p(T)
4
A."A
5. SYSTEM DESCRIPTION
where/).n is a block diagonal matrix and ~ is a set of all ~ matrices with a given, fixed block diagonal structure. If there is no /). n to satisfy the above relationship, then Il(T) is defined as zero. This gives the following robustness test for the nonlinear system using the SSVs and SIDF approximation: -;;;.r/). ) V\ n
1
< p(T)
5
The model uncertainty in the linear element may also force the system to limit cycle oscillation. The robustness of the closed loop system can therefore be guaranteed if the total effect of model uncertainties in the linear and nonlinear elements is taken into account in the control design as shown in Figure 2. Let /). n and A represent the model uncertainties of the nonlinear and linear elements, respectively as shown Fig. 5. Define /)./ = N- 1/)., N. The input multiplicative uncertainties are now at the input of the nominal model, GN. A total uncertainty transfer function may now be defined as; 6 Find the maximum singular values of both sides of this equation to give:
The H 00 control design technique described here uses the linear state space description of the ship, the sea disturbances and the modelling error. The general form of the state space model is as follows: x(t)
= Ax(t) + Bu(t);y(t) = Cx(t) + Du(t)
10
where x(t) is the system state, u(t) is the thruster inputs, y(t) is the position measurements and A,B,C and D are system matrices with appropriate dimensions. In the following section, the non-linear dynamics of the ship and the disturbances are briefly discussed. A linear state description of these models is then developed. The non-linear ship dynamics may be described as:
(m- X,,}u-(m- I:}rv = X H+ X .. + X,.. + X .. (m-I:}v+(m-X,,)ru = YH+~ +Y,..+Y.. (ID -N,}r:;:; NH +N.. +N,.. +N.. where m = displacement; Izz
11
= Yaw inertia; Xo, Y0 and ~)
= added masses and inertia; u, v,
r = Surge, sway speeds and Yaw rate (rad/s); ( hi = Hydrodynamic forces and moment; ( )A = Aerodynamic forces and moment; ( >w = Wave forces and moment; ( = Thruster forces and moment.
n
Each term is now briefly described. The mathematical equations to calculate the hydrodynamic forces are:
8001
8002
8003
8004