Mathematical Problems in Engineering

Volume 2016, Article ID 4650380, 9 pages

http://dx.doi.org/10.1155/2016/4650380

## Efficient Multivariable Generalized Predictive Control for Autonomous Underwater Vehicle in Vertical Plane

College of Automation, Harbin Engineering University, Harbin 150001, China

Received 6 July 2016; Accepted 23 October 2016

Academic Editor: Laurent Mevel

Copyright © 2016 Xuliang Yao and Guangyi Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents the design and simulation validation of a multivariable GPC (generalized predictive control) for AUV (autonomous underwater vehicle) in vertical plane. This control approach has been designed in the case of AUV navigating with low speed near water surface, in order to restrain wave disturbance effectively and improve pitch and heave motion stability. The proposed controller guarantees compliance with rudder manipulation, AUV output constraints, and driving energy consumption. Performance index based on pitch stabilizing performance, energy consumption, and system constraints is used to derive the control action applied for each time step. In order to deal with constrained optimization problems, a Hildreth’s QP procedure is adopted. Simulation results of AUV longitudinal control show better stabilizing performance and minimized energy consumption improved by multivariable GPC.

#### 1. Introduction

For AUV (autonomous underwater vehicle), its motion attitude is severely affected by environmental disturbances if it is navigating near water surface. And wave disturbance is the most obvious one. Violent pitch and heave motion often discontinue normal work of AUV [1]. So, it is necessary to design an effective control pattern to the issue of AUV longitudinal motion control. For the AUV that pitch and heave motion are underactuated, external auxiliary device is needed to adjust its attitude. The bow and stern rudders are space-saving and can provide it with antipitch force and torque [2]. Meanwhile, energy consumption for longitudinal motion control is also very large. For small-scale AUV, the carried energy is limited. So, it is necessary to reduce energy consumption for driving rudders, while stabilizing performance is satisfactory.

A variety of methods have been proposed to reduce AUV energy consumption. A dynamic resistance mathematical model of AUV is built in [3], which states the relationship between resistance and velocity, and they applied the terminal SMC to decrease the pitch angle of AUV and optimize the energy consumption. Energy consumption caused by overcoming the resistance and the yaw was analyzed in [4], and the self-tuning PID controller based on the MOGA method is used to optimize the performance index which has the added resistance consumption. Relevant research work based on roll additional resistance is proposed in [5]. Energy consumption used for driving fin stabilizer is added to performance in [6], GPC controller is designed, and recursive least square method is adopted for parameter estimation. An optimal path planning control of AUV based on rolling optimization is proposed in [7], in order to conduct the path of optimal energy consumption.

In the vertical plane, pitch motion is dynamically coupled with surge and heave [8]. So, control system based on AUV model with single degree of freedom usually cannot attain expected pitch and heave stabilizing performance in practical application, and control systems for AUV are often designed with an implicit assumption that vehicle’s surge velocity is constant. Summing up the above, traditional SISO system controller cannot adapt to the coupled AUV dynamic model.

Regarding multivariable generalized predictive control (GPC), it is one of the most popular formulations of MPC methods [9], which has been successfully implemented in many MIMO industry process control. Predictive control can calculate a sequence of future control signals in advance, in order to track a given future reference. Moreover, predictive control allows the integration of system constraints in the controller design and minimizes a cost function defined over a prediction horizon, ensuring near-optimal performance of the system. The predictive control of AUV motion is one of the main research areas in marine control systems field. Several research works in this area were based on using GPC as the main controller for yaw [10], roll [6], depth [11], and propulsion BLDC motor [12].

The implementation of GPC with constraints requires the solution of a QP (quadratic programming) problem, that is, an optimization problem discribed by a quadratic objective function and linear constraints. There are many techniques, such as the ones based on feasible direction methods [13], decreasing ellipsoid volume methods [14], pivoting methods [15], and active set methods [16]. The active set methods belong to primal methods which are commonly used in QP procedure. But the active constraints need to be identified along with the decision variable; the computational work is quite large if there are a few constraints. A dual method can systematically identify the constraints that are not active, and an algorithm called Hildreth’s QP procedure [17] leads to a very simple programming procedure for constrained optimization problems.

In this paper, the main contributions are as follows. First, coupling effect of surge and heave on pitch is considered and made as a premise of issue discussion; then an augmented state space model is transformed to a CARIMA model which is suitable for GPC. In order to be more similar to the actual situation, wave disturbance is considered in vertical plane of AUV motion. Second, the energy consumption of driving rudders is calculated and added to the performance index, which expends the application range of GPC used for energy-saving and attitude control of AUV. Third, by using the novel performance index, Hildreth’s QP is adopted to deal with the constraints and obtain the optimal solution, which can be considered as the compromise between energy-saving and attitude stabilization. The rest of the paper is organized as follows. Section 2 described the AUV vertical plane dynamics. GPC design is proposed in Section 3. Section 4 gives the simulation results of an AUV to verify the effectiveness of proposed approach.

#### 2. AUV Vertical Plane Dynamics

The vertical plane model of AUV is linearized for predefined operating conditions to extract the linear model. First, the matrix equations of motion can be given as follows:which can be simply expressed as discretized state space form

An augmented state space model can be used if the model input is , and . is a zero-mean white noise sequence.

Notice that, from (2), the following difference equation can be given as and then subtracting (3) from (2) leads to

In order to relate the output to the state variable , we deduce that where .

Choosing a new state variable vector , we have the following:where is the identity matrix with dimensions , which is the number of outputs, and is a zero matrix. In (6), , , , and have dimensions , , , and , respectively.

For notational simplicity, we denote (6) bywhere , , , and are matrices corresponding to the forms given in (7). In the following, the dimensionality of the augmented state space equation is taken to be .

*Remark 1. *The disturbance due to surface waves is the main disturbance type which is considered in this paper and written as . Wave height is affected by wind, and the wave profile is affected by water depth. Sea state is used to classify the sea conditions and expressed by wind speed and wave height. In surface waves modeling, we assumed the long crested and unidirectional sea state, which means all the waves travel in a primary direction. In order to complete an irregular long crested wave sea state simulation, the superposition principle is used here. The single parameter Pierson-Moskowitz spectrum [18] is used in this model and its spectrum density is defined by the following equation:where is the significant wave height (m), is wave frequency. Assuming the AUV body is small compared with the incoming wave and the wave speed is large enough in relation to the body diameter . The transient force and moment are found by the use of Morison’s equation [19] and integration along the length of the body at each step as follows:where is fluid density, is the drag coefficient, and is the added mass coefficient.

In (9), where is the equal interval frequency waveband and chosen in this thesis to discretize the wave spectrum, is encountering frequency, , is wave number, and is random phase shift () in relation to each frequency.

#### 3. Multivariable GPC Controller Design

##### 3.1. Multivariable GPC

It is known that the GPC algorithm is based on using a CARIMA modelwhere the unmeasurable disturbance is given by a white noise colored by .

We consider the most usual case when , because the coloring polynomials are very difficult to estimate with sufficient accuracy in practice, especially in the multivariable case.

Multiply (11) by

With zero initial conditions, it can be seen that both (7) and (11) are equivalent if

By making , we can get the matrix and .

The vector output prediction equation can be calculated and expressed in condensed form which predicts the future dynamic behavior of the AUV longitudinal motion along the horizon :where , and other procedures are detailed in [9].

Notice that matrix is diagonal; the problem is reduced to the recursion of scalar Diophantine equations, which become much simpler to program and require less computation. The computation of and is also considerably simplified.

Conventional GPC performance index can be written as follows:where is a future reference sequence and and are positive definite weighting matrices.

*Remark 2. *CARIMA (controlled auto regressive integrated moving average) model [20] has been long used in GPC. The main reason we use the I-O model (CARIMA model) rather than state space model (7) is that we can directly use the input and output signals of the plant to avoid the use of a state observer, and the complexity of the controller’s front stage design is reduced. However, system identification technology is usually used in GPC to obtain the required I-O model (GPC can be considered as an adaptive control when the system identification is carried out online). Apparently, it will increase the difficulty of controller design by using system identification. State space model of AUV is widely used in many classic literatures [21, 22] and can be modeled from AUV dynamics equations. Therefore, it is wise and simple to obtain an I-O model by equivalent transformation of state space model.

##### 3.2. Improved Performance Index

The energy consumption used for driving bow and stern rudders can be expressed as where is selected according to energy consumption used for pitch stabilizing and depth tracking and is variation of rudder angle. is the driving moment of rudder.

Then the Lagrange dynamics equation is chosen to analyze flat wing-type rudder with thin foil condition as shown in Figure 1. Rudder length is , the distance from midpoint to shaft is , and cross-section minor axis length is .