Time Optimal Hybrid Sliding Mode-PI Control for an Autonomous Underwater Robot

Abstract

This paper presents an underwater robot control system using combination principle among sliding mode control (SMC), Pontryagin maximum principle and linear PI control. The SMC switches according to the Pontryagin's time optimal control principle, in which the solution is obtained by using neural network approach to yield a time optimal response at its reaching phase. PI control is used in place of the SMC at the switching phase to avoid high undesired control activity. Performance of the proposed controller is compared with various classical SMCs and conventional linear control systems. Such comparisons ensure the implementation success and prove it as a real time-optimal controller. The results show the controller's good abilities to deal with plant nonlinearity and parameter uncertainties. The controller yields a time optimal control response without high control chattering.

Keywords

Autonomous Underwater Robot Time optimal control Neural network Hybrid control

1. Introduction

Water has a great influence over human being. It must be investigated and understood so that it can be wisely managed and properly protected. Underwater robots can help us to better understand marine, river, canal and other environmental issues, to protect resources of the earth from pollution and to efficiently utilize them for human welfare. Most commercial unmanned underwater robots are tethered and remotely operated, referred as remotely operated vehicles (ROVs). Extensive use of manned submersibles and ROVs are currently limited to a few applications because of very high operation costs, operator fatigue, and safety issues. As the result, autonomous underwater robots (AURs) have become important tools for underwater operations. They are more and more useful in various kinds of operations such as underwater research, shallow water and deep-water divers support, underwater inspection of pipes and structures, sea floor survey, etc. The demand of AURs is growing. However, it is rather difficult to construct a reliable AUR under the unstructured and hazardous underwater environment due to its highly nonlinear, time-varying dynamic behavior and uncertainties of hydrodynamic coefficients. Therefore, development of an advanced control algorithm for the AURs with the abilities to deal with such environment is needed.

Many advanced underwater robot control systems have been proposed such as sliding mode control (SMC) by Yoerger and Slotine in 1984 (Yoerger, D.R. & Slotine, J.E., 1985), nonlinear control by Nakamura and Savant in 1992 (Nakamura, Y. & Savant, S., 1992), adaptive control by Gianluca et al. in 2001 (Antonelli, G.; Chiaverini, S.; Sarkar, N. & West, M., 2001), neural network control by Lorenz and Yuh in 1996 and Porto and Fogel in 1992 (Lorentz, J. & Yuh, J., 1996)(Porto, V.W. & Fogel D.B., 1990), fuzzy control by Smith et al. (Smith, S.M.; Rae, G.J.S. & Anderson, D.T., 1993), and PID control by Perrier and Canudas-De-wit in 1996 (Perrier, M. & Canudas-De-Wit, C., 1996). Among these control systems, PID control provides the simplest control structure but comes with poor transient performance i.e. overshoot or underdamped response usually appears. SMC provides a satisfactory performance with a simple control structure but comes with undesired high control activity at steady state. By the combination of these two control structures, a satisfactory performance both at transient and steady state can be achieved. To obtain an optimal control response, however, tuning of slope of SMC switching surface has to be done experimentally in practice. By letting the SMC to switch according to Pontryagin's time optimal nonlinear curve, a time optimal control response can be achieved without any manual fine-tuning. Nevertheless, it is very difficult to determine the solution of this nonlinear curve analytically because of two reasons; firstly, it requires an accurate model of the robot, which is highly nonlinear and difficult to achieve in underwater robots, secondly, even if this nonlinear curve can be determined, the procedure usually yields an open loop control, i.e. control signal, u, is obtained as a function of time, u(t) rather than state, u(x). Fuzzy was used by F.Song and S.M.Smith (Song, F. & Smith, S.M., 2000) to solve for this nonlinear curve but there are not any comparisons among the proposed controller and others at the same condition to ensure the controller's time optimal characteristic. In this paper, a neural network approach is employed to solve for the nonlinear curve without any analytical dynamic model of the system. The proposed controller can yield a robust time optimal control response with very good performances both at transient and steady state. The controller's performances are compared with other conventional controller's at the same condition to ensure the implementation success and prove it as a real time optimal controller. In section 2, dynamic model of the underwater robot is described. Section 3 explains the proposed control system and also its simulation results.

2. Dynamic Model of Underwater Robot

Dynamic model of an underwater robot, which is approximately three-plane symmetric, operating at low speed can be decoupled into different degree of freedoms (DOFs) and thus, the robot motion in each DOF can be controlled separately (Indiveri, G., 1998)(Caccia, M.; Indiveri, G. & Veruggio, G., 2000). The model structure of each degree of freedom can be expressed by equations (1) and (2).

\overset{..}{x} = \frac{- k_{x} \dot{x} - k_{\dot{x} ∣ \dot{x}} \dot{x} ∣ \dot{x} ∣ + ɛ}{m} + \frac{τ_{x}}{m}

(1)

\overset{..}{x} = f (\dot{x}, t) + u (t)

(2)

where m is inertia relative to the considered degree of freedom.

x is 1D position (surge, sway, heave, roll, pitch or yaw).

k_x and k_ẋ|ẋ| are linear and quadratic drag coefficients.

τ_x applied force or torque.

ε is disturbance.

$f (\dot{x}, t) = \frac{- k_{\dot{x}} \dot{x} - k_{\dot{x} ∣ \dot{x} ∣} \dot{x} ∣ \dot{x} ∣ + ɛ}{m}$ is dynamics of the system.

u(t) is control input.

An underwater robot yaw model in equation (1), of which the inertia (m) = 85 kg, linear drag coefficient (k_ẋ) = 20.5 Ns/m and quadratic drag coefficient (k_ẋ|ẋ|) = 49.5 Ns²/m² (Indiveri, G., 1998)(Caccia, M.; Indiveri, G. & Veruggio, G., 2000) is employed in all the following simulations of heading control.

3. Control System

3.1 Sliding Mode Control (SMC)

Sliding mode control (SMC) can be considered as a combination of subsystems in which each has a fixed control structure and is effective at particular regions of system behavior. It has come up with two very good characteristics. Firstly, SMC is insensitive to model imprecision caused from unmodelled dynamics, variation in system parameters or the approximation of complex plant behavior by a simplified model. Secondly, SMC allows the n^th-order problems to be represented by the equivalent 1^st-order problems; hence the complexity of the control algorithm is reduced (Slotine, J.E. & Li, W., 1991).

From the characteristics of SMC mentioned earlier, the model uncertainties contained in f(ẋ, t) of equation (2) can be neglected. Double integrator of ẍ = u(t) can, therefore, be considered as a linear approximation of the dynamic model expressed in equation (2) after ignoring the nonlinear term. Hence, a possible variable structure control law of SMC is given by

u (t) = - ρ sgn (s (t))

(3)

u (t) = {\begin{array}{l} - ρ i f s (x, \dot{x}, t) > 0 \\ ρ i f s (x, \dot{x}, t) < 0 \end{array}

(4)

where ρ is sliding gain.

sgn(·) is the signum or the sign function.

The switching function s(t) for the second order underwater robot dynamic as stated in equation (1), is defined by

s (x, \dot{x}, t) = \dot{e} + λ e

(5)

where e is tracking error in the variable = x − xd

xd is desired state.

λ is positive constant and represents slope of the switching surface.

To guarantee that an ideal sliding motion takes place from any initial conditions after the sliding surface is reached, the following inequality must be satisfied (Slotine, J.E. & Li, W., 1991).

s \dot{s} = \frac{1}{2} \frac{d}{d t} s^{2} \leq - η ∣ s ∣

(6)

where η is a strictly positive constant.

By choosing ρ in equation (3) large enough to have sufficient control energy to reach the sliding surface and maintain a sliding motion, it can guarantee that equation (6) is verified (Slotine, J.E. & Li, W., 1991). In the other word, ρ must be greater than the entire modelled and unmodelled system uncertainties. Normally, ρ is set to the maximum value of the output. Simulation results, which are obtained from using the control law in equation (4), switching function in (5) with ρ = 150 m/s² and λ = 1 are illustrated in Fig. 1.

Fig. 1.

Simulation results of SMC

It can be seen that the SMC performance to track the desired set point is very good. The sliding mode takes place after 13.38 sec and maintains on the sliding line when high frequency switching occurs. The system is forced to behave as first-order system, so no overshoot occurs when attempting to track to the set point. The nonlinear term, which is neglected in the control law design may be construed as a disturbance or uncertainty in the nominal double integrator system and has been completely rejected. Such performance, however, is obtained at the price of extreme high control activity, which is called control chattering. In general, chattering is highly undesirable, since it may excite high-frequency dynamics neglected in the course of modeling, and furthermore cause wear and tear on the actuators. In the literature, this can be eliminated by smoothing out the control discontinuity in a thin boundary layer neighboring the switching surface but in the expense of less tracking precision.

3.2 Proportional-Integral (PI) Control

PI control is a conventional control, which is used widely in control applications. It is common practice to use it for steady state regulation or tracking problem while the transient response control is less important. A continuous PI control law is shown in equation (7).

G (s) = \frac{u (s)}{e (s)} = K_{P} + \frac{K_{I}}{s}

(7)

Simulation results of PI control with proportional gain (K_P) = 17, integral gain (K_I) = 0.005 are depicted in Fig. 2. It is found from the results that PI control yield an overdamp response and give a very good steady state response and smooth control action.

Fig. 2.

Simulation results of PI Control (K_P = 17, K_I = 0.005)

3.3 Hybrid Sliding Mode-PI Control

Consider the simulation results from the two previous sections, SMC yields good response and robustness to model uncertainties with high undesired control activity while PI control yields good steady state response but sluggish transient response. The good properties of each controller can be combined as a hybrid system to give a better performance. In general, the operation of SMC consists of two phases, reaching phase and switching phase. Reaching phase is the operation period until the controller brings system states to reach switching line whereas switching phase is the operation period during the controller brings system states along the switching line toward set points. In the hybrid sliding mode-PI control, the controller is in operation to bring the system states near to the set point in the reaching phase and some parts of switching phase of SMC. Then, PI controller takes over to bring the system states to set points at steady state. Consequently, a good response both in transient and steady state can be achieved. A switching condition is essential to change the controller from SMC to PI control to guarantee the stability of the switched system. This stability usually be ensured by keeping each controller in the loop for a long enough time to allow the transient effects to dissipate (Liberzon, D. & Morse, A.S., 1999). The control block diagram of hybrid sliding mode-PI control is shown in Fig. 3. The supervisor is a state-dependent switcher. It selects a controller between SMC and PI to be active by regarding the sum of absolute of error and error rate to be less than a given threshold that is small enough to bring the system state to the steady state region of PI controller. When the set point changes that cause the sum of absolute of error and error rate to exceed the threshold, the supervisor switches the active controller back to SMC. All the disturbances, which occur during the switching back period, can be totally dealt with by the SMC as may be seen later in Fig. 16; ladder input tracking performances of the proposed controller.

Fig. 3.

Block diagram of hybrid sliding mode-PI control

Fig. 4.

Simulation results of hybrid sliding mode-PI control

Fig. 5.

Phase trajectories when the controller is switching between maximum and minimum values

Fig. 6.

Open loop control block diagram

Fig. 7.

Open loop step response for maximum and minimum control input

Fig. 8.

2-6-2 neural network served as a nonlinear time optimal curve approximator

Fig. 9.

Open loop step response and training sets

Fig. 10.

Open loop step responses after training (a) good training (b) bad training

Fig. 11.

Block diagram of neural network based-time optimal hybrid sliding mode for heading control with smooth control action

Fig. 12.

Simulation results of neural network based-time optimal hybrid sliding mode-PI control

Fig. 13.

Controller performance comparison among PI, hybrid sliding mode-PI and neural network based-time optimal hybrid sliding mode-PI control

Fig. 14.

Controller performance comparison among hybrid sliding mode-PI with (1) λ = 0.25 (2) λ = 1.0 (3) λ = 10 and (4) neural network based-time optimal hybrid sliding mode-PI control

Fig. 15.

Error phase portrait comparison among hybrid sliding mode-PI with (1) λ = 0.25 (2) λ = 1.0 (3) λ = 10 and (4) neural network based-time optimal hybrid sliding mode-PI control

Fig. 16.

Ladder input tracking performance of neural network based-time optimal hybrid sliding mode-PI controller.

Fig. 4 depicts the results of hybrid sliding mode-PI control. The results show good response in both transient and steady state without control chattering at steady state.

3.4 Neural Network Based-Time Optimal Hybrid Sliding Mode-PI Control

To improve response of the system, a time optimal control using Pontryagin's maximum principle is employed to bring the system states as quickly as possible to set points. The principle minimizes a performance index expressed in equation (8).

S = \int_{o}^{T} 1 d t = T

(8)

where T is time to bring system states to set points. The optimal control input u(t) that minimizes S will maximize the scalar Hamiltonian function, H in equation (9).

H (t) = \sum_{i = 1}^{n} p_{i} (t) f_{i} (x (t), u (t)) - 1

(9)

where $\begin{array}{l} {\dot{p}}_{i} (t) = - & \frac{\partial H (t)}{\partial x_{i} (t)}, i = 1, 2, ..., n .; \end{array}$ ; n is system order.

\begin{array}{l} u (t) \leq ∣ max . o u t p u t ∣ \end{array}

The solution of equation (9) in a nonlinear system is difficult to obtain analytically. Even if this equation was solved, the optimal control u would still be open loop, u(t), instead of closed loop, u(x). It's noteworthy that the solution of optimal u is in the form of switching function of a bang-bang controller. In general, the phase trajectories for u with maximum and minimum values of a bang-bang control are shown in Fig. 5 (Pontryagin, L.S.; Boltyanskii, V.G.; Gamkrelidze, R.V. & Mishchenko, E.F., 1962).

The figure shows an optimal nonlinear switching curve, AOB, which divides the control action into two parts; maximum and minimum values, and brings the system states to set points in a minimum time. As mentioned earlier that getting this curve analytically is very difficult, however, this curve can be obtained by an open loop control system shown in Fig. 6 (Song, F. & Smith, S.M., 2000) for heading control.

Under the maximum control command, the system output is saturated after a period of time as shown in Fig. 7.

This open loop step response is the maximum actuator capability of the autonomous underwater robot that obtains the fastest response. This curve can be used as the switching curve in the time optimal controller design [8]. In order to obtain this highly nonlinear curve, a basic curve fitting may be used, however it's not good enough. It needs extremely high order equation. Moreover, the graph has to be divided into several sections to yield a satisfactory result. For the neural network, that is a universal approximator, the whole highly nonlinear curve can be well approximated at once. Fig. 8 shows a 2–6–2 neural network, which is applied to serve this purpose.

The input to the network consists of two system state variables, yaw (or heading) and yaw rate for heading control, while the outputs are slope, λ_i(t), and intersection, ci(t), which are parameters of straight line at each system states. Training sets are obtained by analyzing data from open loop response experiment. The data are heading; x, yaw rate; ẋ, slope; λ and intersection; c at each system state. Fig. 9 and Table 1 show the training set from open loop step response.

Table 1.

Training sets from the open loop step response

x	x_dot	lambda	c
−3.4885	−1.5384	0.0099	−1.504
−2.8748	−1.5289	0.0224	−1.4646
−2.2669	−1.5076	0.0507	−1.3927
−1.6721	−1.4604	0.1151	−1.2679
−1.1057	−1.3584	0.2633	−1.0672
−0.5995	−1.1492	0.6171	−0.7793
−0.2101	−0.7631	1.6061	−0.4257
−0.0173	−0.1700	9.8549	0
0.0173	0.1700	9.8549	0
0.2101	0.7631	1.6061	0.4257
0.5995	1.1492	0.6171	0.7793
1.1057	1.3584	0.2633	1.0672
1.6721	1.4604	0.1151	1.2679
2.2669	1.5076	0.0507	1.3927
2.8748	1.5289	0.0224	1.4646
3.4885	1.5384	0.0099	1.504

The network is trained by using Levenberg-Marquardt Backpropagation algorithm. A set of weights and biases is selected among the converged weights and biases. Examples of open loop step responses after training are shown in Fig. 10.

The weights and biases after training are implemented into the control system to automatically approximate the switching line of sliding mode control at each system states.

Simulation block diagram of the neural network based-time optimal sliding mode for heading control is shown in Fig. 11.

Simulation results of the above control system are shown in Fig. 12.

The simulation results show a very good performance both in transient and steady state without control chattering at steady state.

A comparison of the controllers in accordance with the conditions set up below is shown in Fig. 13.

(1)

PI control with K_P = 75, K_I = 0.005.

(2)

Hybrid sliding mode-PI control with λ = 0.5, ρ = 150, K_P = 75, K_I = 0.005.

(3)

Neural network based-time optimal hybrid sliding mode-PI control with ρ = 150, K_P = 75, K_I = 0.005.

In parameters selection, K_P, K_I, and λ are selected by trial and error just to see the different control response between when it is standalone and hybrid with sliding mode and neural network based-time optimal sliding mode. The effect of the other values λ of shall be investigated later in Fig. 14. ρ is the maximum of the control output.

It can be seen from the results that PI control gives a large overshoot underdamped response, hybrid sliding mode-PI control gives a sluggish overdamped reponse and neural network based-time optimal hybrid sliding mode control yields a time optimal critical damped response. A comparison between hybrid sliding mode-PI control with λ = 0.25, 1, 10 and neural network based-time optimal hybrid sliding mode-PI control is shown in Fig. 14–15.

From Fig. 14, it can be seen that neural network based-time optimal hybrid sliding mode-PI control gives the optimal response. Fig. 15 shows an error phase portrait of the four controllers. The hybrid sliding mode-PI controllers have the straight sliding line while the neural network based-time optimal hybrid sliding mode-PI control has a nonlinear time optimal sliding line. In case of λ = 10, the controller gives an overshoot response and the ideal sliding motion is not obtained, which can be seen from Fig. 15 that the switching can not be maintained on the sliding line toward directly to the origin but it goes forth and back to the origin. This happens because the sliding line is too steep. When the phase portrait trajectory reaches the sliding line, there is not enough energy to maintain an ideal sliding motion.

Fig. 17.

Ramp input tracking performance of the neural network based-time optimal hybrid sliding mode-PI controller

Consider the scalar λ of the sliding mode controller; tuning of this parameter in practice has to be done experimentally. By neural network based-time optimal hybrid sliding mode control algorithm, this parameter is automatically tuned at every time of system states changing to yield a robust time optimal control with very good performance both in transient and steady state.

Fig. 16 and 17 illustrate ladder and ramp input tracking performance respectively.

4. Conclusions

The proposed control system, neural network based-time optimal hybrid sliding mode-PI, is a robust nonlinear time optimal controller. Its performance, which can be seen from the simulations, is better than PI and conventional sliding mode controller. It inherits the advantages of sliding mode controller in transient response and PI control at steady state. Furthermore, it has the capabilities to tune the sliding surface automatically at different system states. As the result, it yields the time optimal response without control chattering. The controller is robust to model uncertainties, unmodelled dynamics of the underwater robot. Since the controller is not model based, the complexity of mathematical modeling process can be avoided.

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