Robust anti-windup control for marine cyber-physical systems

. In this paper the robust output control with anti-windup compensation and its implementation to the robotic boat are addressed. The detailed control design and stability analysis of the closed-loop systems are provided in the work. Extensive experimental verification of the dynamic positioning system based on various modifications of the basic controller is carried out by means of robotic boat. The corresponding experimental results are presented and analysed.


Introduction
Development of cyber-physical systems (CPS) is a brand new ongoing scientific trend in applied control theory. It assumes design of sophisticated systems comprised of multiple smart devices interconnected between each other and able to make decisions on their own and adapt to external changes without human intervention. There is a wide range of applications of such systems including for instance manufacturing and transportation.
This research is focused on development of robust output control algorithm with anti-windup compensation for MIMO plants with uncertain parameters and bounded inputs. In particular a solution suggested in this paper can be used in dynamic positioning (DP) problem to control a surface vessel being a component of marine CPS.
If an integral term being a part of the internal model is included in the control system, input saturation might cause various adverse effects when implementing regulator to real plants. Anti-windup modification first mentioned in [1] can be used to resolve this issue. There is a number of scientific works developing anti-windup approaches. Overview of modern anti-windup design is consider in [2]. Application of the toolkit of linear matrix inequalities (LMI) to synthesis is presented in [3][4][5][6][7][8]. The anti-windup is combined with internal control mode (ICM) in [9][10][11]. In the application to discrete-time systems, the anti-windup scheme is considered in [12]. The consecutive compensator approach described in [13,14] and based on the pacification approach [15] is used as a basis for control design for marine CPS. The relevant problem of output regulation for MIMO systems with unknown disturbance compensation and application of the suggested control approach to surface vessels to control its velocities is addressed in [16]. Detailed theoretical background on ship control is provided in [17]. Some results on saturated control design being applied to the research setup of the robotic boat is presented [18,19] and to the quadcopters in [20][21][22].
The consecutive compensator approach is augmented with an auxiliary integral loop and anti-windup scheme. Such solution is needed, on the one hand, to eliminate the steady state error and, on the other hand, to compensate integral windup.
This paper is organized as follows. The problem of this research is formulated in Section 2. Control design is given in Section 3. Stability analysis of the closed-loop system is provided in Section 4. Experimental verification and the corresponding test results are described and discussed in Section 5. The paper is summarized in Conclusions.

Problem formulation
Consider the plant where � � � � is the state, � � � � � is the disturbance, which derivative and initial conditions are � � � � is the measurable output, � � � � is the input satisfying where � ��� and � ��� are the input saturation limits, � is the control signal generated by the nominal linear controller, �, �, �, � are matrices and vectors of corresponding dimensions.
Assumption 4: The disturbance is bounded � � � � and the nominal control signal � � needed for its compensation at steady state satisfies The objective is to asymptotically stabilize the output under conditions of external disturbance effects and presence of the integral windup.

Robust control design
The plant model (1), (2) can be represented as where �(�) is the Hurwitz polynomial due to Assumption 1. Rewrite the plant model (5) as and express (6) as two interconnectyed subsystems as which is equivalent to the state-space model where A 11 is the Hurwitz matrix due to Assumption 1, if the matrix A 22 is chosen in companion form, then where �(�) is the memoryless nonlinearity, � > 0, � > 0, � > 0 , A q , b q , c q are given as where σ > 0 and � � � (� = 1, �) are chosen for the system (10) to be Hurwitz, � � (� = 1, �) are coefficients of an arbitrary polynomial of the degree ρ -1.
Perform the change of coordinates � = � � − � and compute the derivative This change of coordinates puts the system (13) into the form (14).
Perform the second change of variables � = �� � � � � � �� � � � � � + � and compute the derivative where I is the identity matrix of the corresponding dimension.
For the sake of simplicity denote This change of coordinates taking into accound (15) puts the system (14) into the form (16) : � As noted above, the matrix A 11 is Hurwitz due to Assumption 1. The matrix is Hurwitz due to the choice of the parameter � � 0.
There exists a number � � such that for �� � � � the matrix is Hurwitz, since only the element (3, 3) depends on the parameter �, which can be set to be sufficiently large.
The overall state matrix of the closed-loop system (16) can be forced to be Hurwitz due to the structure of the matrix � � and sufficiently large value of the parameter �, which is included only in the block (4,4). Statement 1: Consider the plant model given at statespace (7), (8) and the control law (9)- (12). Hurwitzness of the state matrix of the resultant closed-loop system (16) follows from Hurwitzness of all its diagonal blocks, which is forced by the parameters �, �, �.
The transfer function � � (�) can be derived as follows where Represent the matrix ℱ in the block form Remark 1: Note that, Hurwitzness of the numerator and denominator of the transfer function (22) can be achieved by choosing matrix A q and vectors b q and c q .
In order to determine steady state behavior use the Sylvester equation applied to the model (13) and focus on the fourth row which shows that at steady state Find the relation between � � � Π � and � � � Π � . Consider the auxiliary variable z 0 taking into account (24), the steady state value of which is zero. From (10) follows where �(0) is a vector of initial conditions. Rewrite (25) find that the steady-state error of y as well as � � � � converges to zero.
The practical verification of this research has been carried out using the robotic boat shown in Figure 1. It is intended to test DP systems of marince CPS under laboratory conditions. The boat is equipped with the main engine, bow and stern tunnel thrusters and heading servo drive (Fig. 2). Dimensions of the boat are (0.432×0.096×0.052) m. Its workspace is represented by the specific basin made of plywood sheets with dimensions (1.50×1.10×0.1) m. The computer vision system emulating satelite navigation systems is based on the RGB video signal provided by the digital camera placed above the basin. The control signals transmitted via radio channel saturate at the rates [−127; 127] due to the pulse-width modulation technique. The particular form of the control law is (9)- (12) applied to the boat model with the relative degree being equal to 2 is Three experiments have been carried out. The first one focuses on the simple robust output control with the internal model (11) deactivated (set � � 0). The second one is the robuts output control with the integral term and excluding the anti-windup scheme (12) (set �� � 0� � � 0). The third one is robuts output control with the integral term and anti-windup scheme, which is the suggested algorithm (set � � 0� � � �).      The experimental results are given in Figures 3-8. Cyan curves illustrate the control signals generated by the nominal controller without saturation. Blue curves are assigned to the simplest controller with the integral term deactivated. Green curves correspond to the controller augmeted with the integral term. Red curves refer to the controller augmented with the itegral term and anti-windup scheme.
The experimental data shows that the first controller leads to the slight steady-state error. It is compensated by means of the integral term which accumulates the error value and increase the control signal. In this case integral windup occurs and leads to the overshoot increase. This effect can be fixed using the anti-windup approach.

Conclusions
This research focuses on the design the robust output control algorithm with anti-windup compensation for DP systems. The theoretical background is based on the consecutive compensator approach, which uses the highgain principle to suppress plant uncertainties. The paper provides control design strategy, strict stability analysis and experimental verification by means of the robotic boat setup.