3.5 Design Procedure

　

 The obstacle avoidance is expected to realize in longitudinal direction only when an effective set-point tracking can be realized. In the sequel, we discuss the design for this tracking system. The synthesis methodology is parallel to that of linear time-invariant robust control [1] [2]. The interconnection structure for control system design is shown in Fig. 1. The plant model includes the low-pass filter modeled by two first order delay components and denoted by Gf. Also the depth and Euler angles are measured. In this figure. r, ym, z1 and z2 refer to the reference, measurement signal and controlled outputs, respectively.

　

　

 The weighted plant P is

　

　

where BwI (U ) and BwD. are the state-space data correspond to WI and WD respectively. Cm is the output matrix and Cs is the selection matrix of output. Low-pass filter model Gf is as follows,

　

　

 Once the parameterized model including weight is obtained, the controller design becomes similar to the linear time-invariant control design.

 For comparison, two control system designs are carried out. The first is a linear time-invariant robust controller and is denoted as LTI. The weight functions are chosen as

　

　

 Then, the design process is simply iteration on improving the weights to get a satisfactory controller such that,

　

　

where S the plant output sensitivity function. The robust controller is designed for the equilibrium point:

　

　

 The second design results in LPV control. We considered the bounded range and rate of the schedule variable in controller synthesis. That is, the control system works well even if the speed ranges in 0.5 to 1.1 [m/s] and the acceleration changes in -0.15 to 0.15 [m/s²]. During controller synthesis computation, for tractability either symmetric matrix X or Y in (7) is fixed. Thus different results of optimization with γ = 0.4620 or 0.4242 are obtained. They correspond to two LPV controllers. Note also different weight functions are used for each parameter vertex [4][7].

　

 Several test missions are considered: 2m, 4m diving. These missions correspond to the advance speed varying from 55% to 122% of its nominal value. Linear time-invariant robust control provides good tracking performance for 2m submerge test. The change of speed U may be referred as uncertainty of the model. But with the increase of depth, the speed variation increases as well under required performance requirement. The result of linear time-invariant robust control becomes degenerate for 4m submerge test. We see that LPV control surpasses the LTI control in set-point tracking performance. Furthermore, the LPV control obtained with X matrix fixed shows better time response than that of LPV control with Y matrix fixed.

　

　

　

 WorldUP^TM is a popular virtual reality tool for real-time simulation. Using WorldUP^TM, a 3-dimensional underwater space for collision avoidance simulations is constructed as shown in Fig. 3. Also the underwater vehicle is modeled as shown in Fig.4. The underwater vehicle that is denoted as a nonlinear simulation block is controlled by the LPV controller block, see Fig.5. This can be referred as the integration of LPV control and virtual reality simulations, Fig.6.

　

 Virtual reality techniques bring about several advantages in the present control system simulation. One merit is that it provides us a real sense on the motion of the vehicle. More important merit is that it helps us to determine: which one is critical to the specific case among many kinds of control performance such as settling time, overshoot etc. For example, the LPV controller whose responses are shown in Fig. 2(left) exhibits short settling time, but it is inferior to the LPV controller in Fig. 2(right) in large depth diving. Thus the former LPV controller is likely to be used in short-distance, small-obstacle avoidance. In contrast, the later one is suitable for long-distance, large-obstacle avoidance. For another instance, virtual reality simulation shows usefulness when we wish to investigate the effect of the overshoot to the controlled system. We learned that the overshoot is catastrophic when the vehicle is near the seabed and conduct the downward obstacle avoidance.

　

 It may be argued that control system with intelligence will settle the above problems. This is our future research subject. And virtual reality is expected to play an important role as well in developing an intelligent control system.

　

 An effective obstacle avoidance control scheme for underwater vehicle is presented with the help of virtual reality techniques in simulation. For the real-time implementation of the LPV controller, we produce the C codes using the Real-Time Workshop.

　

 As for obstacle avoidance, speed dependent controllers are designed. This is realized taking advantage of the polytopic LPV model and parameter dependent Lyapunov function. Nonlinear simulation using Simulink^TM on MATLAB^TM showed that LPV control overpasses the linear robust control in carrying out obstacle avoidance. Furthermore, with the help of virtual reality techniques, we testify our results instead of the practical experiment in water tank. This seamless design environment used in the present study has been proved effective in that the development cycle is greatly shortened and the experiment cost is reduced.

　

[1] Apkarian, P, Gahinet, P, and Becker, G. "Self-Scheduled H〓 Control of Linear Parameter-Varying Systems: A Design Example," Automatica, Vol 31, No 9, pp 1251-1261, 1995.

[2] Apkarian, P, Becker, G, Gahinet, P, and Kajiwara, H "LMI Techniques in Control Engineering from Theory to Practice," Workshop Notes CDC, Kobe, Japan, Dec, 1996.

[3] Apkarian, P, Adams RJ. "Advanced Gain-Scheduling Techniques for Uncertain Systems", IEEE Trans. Automat. Contr Tech. Vol 6, No 1, pp2l-32, 1998.

[4] Bendotti, P, and Bodenheimer, AB. "Linear Parameter-Varying versus Linear Time-Invariant Control Design for a Pressured Water Reactor," Int. J. Robust and Nonlinear Control, Vol 9, pp 969-995, 1999.

[5] Fossen TI "Guidance and Control of Ocean Vehicles," John Wiely and Sons, Chichester, 1994.

[6] Gahinet, P, Nemirovski, A, Laub AJ, and Chilali, M. "LMI Control Toolbox," MathWorks Inc. , Natick, MA, 1995.

[7] Gao R, Kondo E, Kajiwara H, Koterayama W, Nakamura M "Depth Control of an Underwater Vehicle Using Linear Parameter-varying Techniques, " Int. J. Offshore and Pole Engineering, Vol 13, No 1, pp 52-59, 2003.

[8] Gao R "Linear Parameter-Varying Control of an Underwater Vehicle, " PhD thesis, Kyushu University 2003.

[9] Nakamura, M, Kajiwara, H, and Koterayama., W. "Development of an ROV Operated Both as Towed and Self-Propulsive Vehicle." Ocean Engineering, Vol 28, pp 1-43, 2000.

[10] Wu, F, Yang, XH, Packard, A, and Beker, G "Induced L2 Norm Control for LPV System with Bounded Parameter Variation Rates," Proc American Control Conf, Seattle, Washington, pp 2379-238, 1995.