Improving Quality of Regulation of a Nonlinear MIMO Dynamic Plant

The paper presents a robust two-degrees of freedom control system for a multi-input, multi-output (MIMO) nonlinear dynamic process. There are discussed the advantages of the model following control (MFC) structure for MIMO systems, its stability and robustness properties and a method for synthesis of a multi-controller structure of the MFC model loop. The problems under study are exemplified by synthesis of a position and yaw angle control system for a 3DOF nonlinear mathematical model of a drillship, where the MFC structure is used to reduce effects of the impact of the sea current and wind forces on the ship's hull. perturbations, as well as great disturbances dumping at the input and at the output of the plant to be controlled (10)- (12). In the paper we fully utilize this theory and adopt the MFC advantages to synthesize a robust control system of improved quality for a class of nonlinear MIMO systems. The problems under study are exemplified by synthesis of a position and yaw angle control system for a drillship described by a 3DOF nonlinear mathematical model of low- frequency motions over the drilling point. The effectiveness of the proposed MFC structure is shown by its use to compensate the environmental disturbances, the impact of


I. INTRODUCTION
The nonlinear plants are commonly encountered in many different areas of science and technology.However, despite the great progress in analysis and synthesis of nonlinear control made over the past years the majority of proposed solutions is still based on linear controllers.Such an approach consists in designing either robust [1], [2] or adaptive controllers with varying parameters systematically tuned up with changing plant operating conditions [3], [4].These may be supplemented by different control structures [5]- [7] and auxiliary algorithms utilizing e.g.recalculations of the control systems after system change or failure [4], [8], [9].
In the paper an adaptive modal MIMO controller with (stepwise) varying parameters in the process of operation is studied.The modal controllers making up the considered adaptive control system are designed for all possible operating points of the nonlinear MIMO plant.The appropriate set of parameter values of the tuned controller is selected during system operation on the basis of auxiliary measured signals, on which the operating points of the nonlinear plant are dependent.
To make the robustness of the controller higher and the compensation of disturbances more effective we suggest incorporating it into the Model Following Control (MFC) structure.The robust MFC system is known for its outstanding robustness to plant parameter and/or structure perturbations, as well as great disturbances dumping at the input and at the output of the plant to be controlled [10]- [12].In the paper we fully utilize this theory and adopt the MFC advantages to synthesize a robust control system of improved quality for a class of nonlinear MIMO systems.
The problems under study are exemplified by synthesis of a position and yaw angle control system for a drillship described by a 3DOF nonlinear mathematical model of lowfrequency motions over the drilling point.The effectiveness of the proposed MFC structure is shown by its use to compensate the environmental disturbances, the impact of the sea current and wind forces on the ship's hull.
The paper is organized as follows.A mathematical description of the adopted nonlinear control plant is brought in Section 2. Sections 3 and 4 present the structure of the proposed control system, its properties and a method of synthesis.An example, which demonstrates the disturbance damping property, is presented in Section 5. Finally the paper is concluded with Section 6.

II. NONLINEAR MODEL OF A DRILL SHIP
A nonlinear mathematical model of ship's low-frequency motions in 3DOF has been developed on the basis of tests carried out on a physical 1/20-scale model of the "Wimpey Sealab" drilling vessel [13].It may be presented in the form of nonlinear state-space equations:  The yaw angle and the ship's position are defined in an Earth-based reference system the origin of which is located over the drilling point on the seabed.In contrast, force and speed components with respect to water are determined in a moving system related with the ship's body and the axes directed to the front and the starboard of the ship with the origin placed in its gravity center.These are shown in Fig. 1.  associated with the moving ship.The detailed description of the wind disturbances modeled for the "Wimpey Sealab" model may be found in [15].These forces are converted into dimensionless values and introduced into (1).Furthermore, it is assumed that the wind above the water surface w V is turbulent and consists of two components: the slowlyvarying component w V (average wind speed) and the high- frequency component In the Model Following Control structure (Fig. 3), described for the first time in [10], the basic control task is performed by the main controller matched in the most optimal way to the process model.The corrective control signal, generated by the auxiliary controller, depends on the difference between the outputs of the adopted model and the actual process.Thus, the effect produced by the processmodel mismatch, caused e.g. by disturbances and by possible process perturbations, can be neutralized.1).When the classic control loop, with the controller synthesized by the pole-placement method, is used, the non-measurable forces and moment derived from the wind cause the steady-state positioning errors (Fig. 5).The use of a MIMO MFC control structure brought the ship to the drilling point and assumed preset course angle without steady-state errors (Fig. 6).The chosen here auxiliary controller ( ) P s R contained the same modal controller as in the model loop together with three PI controllers with parameters chosen as following: 1 k = , and 0.05 i T = . The integration blocks bring the steady state errors to zero, thus the control goal has been met and the proposed MFC control structure has proved its ability to dump disturbances.

VI. CONCLUSIONS
In the paper an application of the MFC control structure to control of a nonlinear MIMO plant has been discussed.A method for synthesis of an adaptive gain scheduling modal controller in the model loop as well as conditions the auxiliary controller has to satisfy have been given.The presented example of a positioning control system for a drilling vessel, with the wind acting on the ship, shows efficiency of the method and the appropriateness of its use to control strongly nonlinear MIMO plants under the influence of non-measurable disturbances.The method makes it also possible to implement easily such a control system in typical off-the-shelf controllers.

Fig. 1 .CFig. 2 .
Fig. 1.Ship's co-ordinate systems.Wind disturbances are one of the most important environmental disturbances occurring in the dynamic positioning of vessels.The wind forces and the moment for each degree of freedom are usually defined as follows:( ) ( ) ( ) the value of the direction of the relative wind ' p γ converted to a reference system {X o , Y o } FOLLOWING CONTROL FOR MIMO PROCESSES

Fig. 3 .Fig. 4 .
Fig. 3.The MFC structure.Let the system components be described by continuous transfer function matrices of appropriate dimensions: model ( ) s M , process ( ) s P , main ( ) M s R and auxiliary ( ) P s R controller.For the MFC system with a perturbed process the disturbance sensitivity function is defined in the frequency domain s jω = , for (0, ) ω ∈ ∞ as