Adaptive control design and implementation for collective pitch in wind energy conversion systems

Journal Pre-proof Adaptive control design and implementation for collective pitch in wind energy conversion systems Ahmed Lasheen, Mahmoud Elnaggar, Haitham Yassin

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S0019-0578(19)30499-9 https://doi.org/10.1016/j.isatra.2019.11.019 ISATRA 3406

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Received date : 28 October 2018 Revised date : 14 November 2019 Accepted date : 17 November 2019 Please cite this article as: A. Lasheen, M. Elnaggar and H. Yassin, Adaptive control design and implementation for collective pitch in wind energy conversion systems. ISA Transactions (2019), doi: https://doi.org/10.1016/j.isatra.2019.11.019. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Adaptive Control Design and Implementation for Collective Pitch in Wind Energy Conversion Systems Ahmed Lasheen1, Mahmoud Elnaggar1, Haitham Yassin1* Electrical Power Engineering - Faculty of Engineering – Cairo University – Giza – Egypt *

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Adaptive Control Design and Implementation for Collective Pitch in Wind Energy Conversion Systems Abstract: This paper introduces a discrete-time L1 adaptive controller design for collective pitch control for variable speed

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variable pitch wind turbines. In this control algorithm, the blades’ pitch angles are regulated to manipulate the generator

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speed and generated power during high speeds wind airflow. The main merits of the proposed controller are its robustness

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against the wind turbine model uncertainties, guaranteeing the closed loop system stability and performance, and

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applicability during implementation in real life. Commonly used industrial gain-scheduled PI controller is used to be

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compared with the proposed controller. The proposed scheme is simulated using a 5-MW offshore turbine model and the

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obtained results illustrates the superior performance of the proposed controller. Further, a reduced scale wind turbine

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emulator is used to provide hardware in the loop to test the implementation of the proposed controller. The controller is

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implemented using laboratory equipment to demonstrate the feasibility of the controller for real time applications. The

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obtained experimental results validate the closed loop system performance enhancement due to replacing the traditional

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PI controller with the proposed design.

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Keywords: - Collective pitch control, Adaptive control, Wind energy conversion systems.

Wind Energy Conversion System

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WECS

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Nomenclature

FAST

Fatigue, Aero-Dynamics, Structure, And Turbulence

Collective Pitch Control

MRAC

Model Reference Adaptive Control

IPC

Individual Pitch Control

PMSG

Permanent Magnet Synchronous Generator

DEL

Damage Equivalent Load

MSC

Machine Side Converter

DOF

Degree of Freedom

GSC

Machine Side Converter

𝑀, 𝑓

Nonlinear model components

𝑥, 𝑥̂, 𝑥̃

System states, estimated states, and error in

𝑞, 𝑞̇ , 𝑞̈

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CPC

DOF

displacement,

velocity,

state estimation and

𝜃, 𝜃̂, 𝜃̃

acceleration 𝑡, 𝑘

Continuous and discrete time indices

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𝐴𝑐𝑖 , 𝐵𝑐𝑖 , 𝐶𝑐𝑖

Continuous time state space representation matrices at different operating points

System

uncertainties

(uncertainties,

estimated uncertainties, error in estimation) 𝑑, 𝑑̂ , 𝑑̃

Input system disturbance (disturbance, estimated disturbance, error in estimation)

𝐾𝑔

Input manipulation gain to achieve free steady state error tracking

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𝐴𝑖 , 𝐵𝑖 , 𝐶𝑖

Discrete time state space representation

𝐻, 𝐺

Proper stable transfer functions

𝑃, 𝑅

Positive symmetrical matrices

matrices at different operating points 𝐴, 𝐵, 𝐶

Discrete time time-varying state space representation matrices

𝑢0 , 𝑢𝑖𝑝𝑐 , 𝑢𝑐𝑝𝑐

Pitch angle (control signal)

Г𝑑

Adaptation gain

Control signal (operating point look-up

𝐶𝑓

Low pass filter with unity dc gain

𝐾𝑠

State feedback control

table, individual pitch, collective pitch) 𝐴𝑚 ,

State matrix of the required specification

𝑏

Input vector of the required specification

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1. Introduction

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𝑢

𝐾𝑃 , 𝐾𝑖

Proportional and Integral gains

Owing to its environmental advantages, one of the most attractive renewable energy sources is wind energy conversion

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system (WECS). With more than 50 GW wind energy installed in 2018, the total cumulative installed capacity became more

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than 597 GW by the end of 2018 with a yearly growth rate of 9.1% [1]. There are different criteria in classification of WECS

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schemes, fixed speed and variable speed wind turbine according to rotor speed, offshore and onshore systems based on

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implementation site and depending on grid connection, they can be classified as standalone or grid connected WECSs [2]. The

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influence of large wind farms in the power system is high as it is affected by the wind nature [3]. The main objectives of the

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control system for grid connected WECS are maximization of the extracted active power from wind and controlling active and

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reactive power to improve the power quality.

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Varying with the wind speed, the output active power from the wind turbine varies. The relation between this power and

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wind speed could be described by the wind turbine power curve supplied by turbine manufacturers as shown in Figure 1. The

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power curve of the turbine has three regions of operation according to the wind speed. The cut-in speed is the upper limit of

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region 1 which starts from zero wind speed. In this region, there is no sufficient torque to rotate the turbine as the losses in the

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turbine are greater than the wind power so the turbine does not operate. The rated and cut-in wind speeds are the upper and lower

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operating limits of region 2 where the wind power increases rapidly with wind speed. The main control system goal is to extract

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maximum power in region 2. Region 3 falls between the rated and cut-out wind speeds after where the turbine shuts down. In

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this region, the control objective of this region is to produce fixed rated output power from the wind turbine and reduce the flap-

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wise moment on the blades to avoid the damage of drivetrain. If the wind speed becomes higher exceeding the cut-out threshold,

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the turbine stops. This could be achieved by regulating the rotor speed to its rated value using pitch controller of wind turbine

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blades. The two pitch control techniques are individual pitch control (IPC) and collective pitch control (CPC). With reduced

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moment on the blades, the IPC plays role in wind turbine systems. The major target of CPC is to regulate the generator speed

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and power. The main focus of this work is to design and implement the CPC. While PI controller is used to design the IPC as in

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literature. Several literatures addressed pitch controllers design using different approaches. In [4], Individual pitch controller is

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designed using PI controller. To regulate the generator power, variable pitch wind turbine with nonlinear PI control is used [5].

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In [6], the collective pitch fuzzy logic controller considers the generator’s speed and power as the control inputs to control the

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output power at its desired value. A fractional fuzzy PID controller using chaotic optimization algorithms is provided for wind

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turbine pitch control [7]. In [8], adaptive sliding mode back-stepping pitch angle control is proposed for wind turbines to smooth

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turbine output power. CPC controller based on an artificial neural network is proposed [9]. Multi-layer perceptrons with back

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propagation and radial basis function network are used to model the operation. The performance of each technique is investigated

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and it has been achieved that the performance of the second technique has better results comparing with the first one. A wind

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system is controlled by a multiple model predictive control (MPC) to maximize the extracted wind power at region 2 and to

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maintain the desired rated output power by controlling the collective pitch angle [10]. In [11], a fuzzy based MPC is investigated

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to control the collective pitch angle. In [12], model uncertainties are taken into consideration when designing a robust MPC

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based on a continuous-time. The optimization problem is solved offline in order to reduce the on-line computational time using

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explicit MPC. The authors in [13], have controlled the maximum power using adaptive controller. It is assumed that, the

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aerodynamic torque is identified and online approximators are used to estimate the aerodynamic uncertainties. In [14], the pitch

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angle and the generator speed are manipulated by a decentralized control scheme. The proposed controllers are adapted to achieve

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the required transient and steady state performance. A continuous-time L1 adaptive controller is designed to achieve the wind

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turbine control objectives. A robust adaptive pitch controller was recommended in [15]. It provides satisfactory functioning at

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different operating points to satisfy the system constraints.

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Figure 1: Typical wind turbine power output with steady wind speed

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In this work, the design of discrete-time L1 adaptive collective pitch controller is investigated. The controller main target

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is to maintain the turbine power and speed at their rated values by controlling the collective pitch angle. The controller is designed

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and implemented in a complete discrete-time environment. The robustness and stability analyses of the nonlinear wind model 4

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are stated. The proposed controller is implemented to a laboratory emulator of the turbine to validate the proposed controller

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analyses. The main contributions of this work could be pointed out in three points as follows. First, the discrete adaptive controller

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for the collective pitch control is analysed to show the guaranteed closed loop performance. The main objective of controlling

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the collective pitch control is to maintain the generator power and speed to be at their rated values, increase the annual energy

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production, and decrease the damages equivalent loads on the turbine blades. The main advantages of the discrete adaptive

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controller are the ability to stabilize the wind turbine, the ability to reject the disturbance, and it can be implemented in the real

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time. Second, the pitch constraints are taken into consideration in the proposed controller and the stability proof of the discrete

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adaptive controller with saturation is investigated. Finally, complete design, simulation and implementation of the proposed

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controller in a complete discrete-time environment are done. For a chosen sampling interval of the closed loop system, the

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controller grantees the system stability and robustness in real time implementation.

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This paper is prepared as follows. In section 2, the modelling of wind turbine and pitch control action are derived. Discrete

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adaptive collective pitch controller is introduced in section 3. In section 4, the WECS with proposed controller is simulated using

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MATLAB/Simulink. A comparison between the simulation results of the gain scheduled PI controller and the proposed controller

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is also provided in section 4. The experimental results of the two controllers are presented and compared in section 5. Section 6

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gives the conclusions.

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2. Model description

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One of software packages developed to simulate the wind turbines operations is the FAST simulator [16]. A realistic

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model of wind turbine with 24 degree of freedom (DOF) can be provided by FAST so it used to emulate the operation and

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performance of a 3-blades, horizontal axis, offshore, 5-MW variable speed variable pitch wind turbine. The modeling of the

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system is performed with a limited enabled number of DOFs, the generator and drivetrain DOFs, to provide a reduced order

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model for designing the controller. The performance of the proposed controller is tested by simulating the system with a full

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order nonlinear model. In section 2.1, the method to obtain the linearized model for design purposes is illustrated. In section 2.2,

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the control action main components are clarified and the constraints on the CPC are formulated.

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2.1. FAST linearization

In FAST, the complete model of nonlinear aero-elastic equations of motion can be expressed as follows [17],

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𝑀(𝑞, 𝑢, 𝑡)𝑞̈ + 𝑓(𝑞, 𝑞̇ , 𝑢, 𝑢𝑑 , 𝑡) = 0

(1)

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Where 𝐹 is the nonlinear vector (forcing function), 𝑀 is the mass matrix. 𝑞̈ , 𝑞̇ , 𝑞 and are the DOF vectors of acceleration,

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velocity and displacement, respectively. 𝑢 and 𝑢𝑑 are the input vectors of the control and wind, respectively. Where t is the

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continuous time index. FAST numerically linearizes the aero-elastic equations of motion at different operating points as [17], 𝑥̇ 𝑖 = 𝐴𝑐𝑖 𝑥𝑖 + 𝐵𝑐𝑖 𝑢𝑐𝑝𝑐 + 𝐵𝑐𝑖 𝑢𝑑

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𝑦 = 𝐶𝑐𝑖 𝑥𝑖

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where 𝑥𝑖 , 𝑢𝑐𝑝𝑐 , y are the perturbations in the system states, CPC action, and generator speed calculated at the 𝑖 𝑡ℎ operating point,

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respectively. The system states are: the rotor speed and the torsional speed and displacement of the drivetrain. 𝐴𝑐𝑖 , 𝐵𝑐𝑖 , 𝐶𝑐𝑖 are constant

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system matrices with proper dimensions. The operating point of the system can be specified by three variables, the generator

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speed, hub-height wind speed and pitch angle. During the operation in region 3, the major purpose of CPC is to regulate the

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generator speed and power to their rated values. So, keeping the generator speed constant at its rated value is necessary during

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the linearization process. Different linearized models are developed at different wind speeds and the nominal pitch angle, with a

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given wind speed, can be obtained at steady state using FAST. From this analysis, it is concluded that the main variable that

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characterizes the linearized model.is the hub height wind speed. Since the variation range of the wind speed in region 3 from

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11.4 m/s to 25 m/s [16], there will be different linearized models to cover all the region. With a step of 2 m/s, seven linearized

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models are obtained to represent all operating points. These models can be written in the discrete-time form as,

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𝑥𝑖 (𝑘 + 1) = 𝐴𝑖 𝑥𝑖 (𝑘) + 𝐵𝑖 𝑢𝑐𝑝𝑐 (𝑘) + 𝐵𝑖 𝑢𝑑 (𝑘) 𝑦(𝑘) = 𝐶𝑖 𝑥𝑖 (𝑘),

(3)

𝑖 = 1,2 … ,7.

where at the 𝑖 𝑡ℎ operating point, discretising equation (1) will obtain matrices 𝐴𝑖 , 𝐵𝑖 . Define the convex set, 𝑆, as

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{𝑆 ∶= (𝐴, 𝐵) ∶= 𝐶𝑜{(𝐴𝑖 , 𝐵𝑖 ), 𝑖 = 1, … ,7}, where 𝐶𝑜 stands for the convex set. Hence, the following form will be the linearized

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model for a certain wind speed, the linearized model takes the form given in (4).

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𝑥(𝑘 + 1) = 𝐴𝑥(𝑘) + 𝐵𝑢𝑐𝑝𝑐 (𝑘) + 𝐵 𝑢𝑑 (𝑘) (4)

𝑦(𝑘) = 𝐶𝑥(𝑘)

2.2. Pitch control action

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There are three components of the pitch control signal. The pitch angle, 𝑢0 , is the first component of this signal. The

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operating point could be represented by this component according to the average wind speed. Zero steady state error is the main

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objective of 𝑢0 . Provided by FAST, a look-up table is usually used to obtain this value. The second component, 𝑢𝐶𝑃𝐶 , is

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responsible for CPC and affects all blades similarly. While, individually concerning to each blade can be done using the last

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component, 𝑢𝑖𝑝𝑐 . PI controllers are usually used for IPC [4]. In this work, collective pitch control with a discrete-time L1 adaptive

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controller is designed. Finally, pitch control command (𝑈𝑃 ) is illustrated in Figure 2 and it can be written as,

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𝑈𝑃 = 𝑢0 + 𝑢𝑖𝑝𝑐 + 𝑢𝑐𝑝𝑐

(5)

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Look up table

+ +

Saturation

Pitch angle

ucpc

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L1 Adaptive Controller

WECS

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uo +

Moment

PI

uipc

Wind Speed

Constra int

Figure 2: Control structure for pitch control

3. L1 adaptive controller for CPC

Wide range variations of altitudes and speeds were the main problem faces the controller design of autopilots for highly

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agile aircrafts. Therefore, since 7 decades ago, research papers have been developed to introduce the design of adaptive

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controllers. Adaptive controllers were introduced to automatically adjust the controller parameters according to changing aircraft

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dynamics [18]- [19]. The first adaptive control test on the flight systems was the X-15 flight [20]. Due to the awful test result of

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this flight, researches on adaptive control is slowed down but it is resurged again in 1970. The research divided into two schemes,

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self-tunning regulator and model reference adaptive control (MRAC). These two schemes were tested and successfully

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implemented for different applications in the literature. The stability, robustness, and convergence analysis of the adaptive control

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was increased in the end of 1970s to 1990s.

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The main concept of the initial adaptive control results was based on the theory of system identification [21]. It combines

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two main parts, the first part is an online parameter estimation and the second part is an automatic control design [22]- [23].

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Regarding the parameter estimation, there are two known structures of MRAC. The first one is based on estimating the process

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parameters, and the controller structure can be obtained using any control algorithm. This structure is called indirect MRAC

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technique. The second structure is called direct MRAC technique. In which the controller parameters is estimated directly by

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adaptation laws. The relationship between the two different structures of the MRAC were stated in [25]. According to [24], in

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both structures, the necessary condition to ensure the identifiability of the system is persistence excitation of the input signal.

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This condition ensures the conversion of the estimated parameters. The research progress of the stability analysis of the adaptive

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control theory have been studied in [26]. In this section, modified indirect MRAC, L1 adaptive controller, is explained and

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discussed as a solution to CPC traditional control problems.

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3.1. Discrete-time L1 adaptive control

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Since 2006, the authors in [27]- [28], proposed a modified structure of the indirect MRAC by adding a low pass filter on

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the control signal, it is added to limit the bandwidth of the control signal. Hence, the input and output of the controlled system 7

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are ensured to be bounded. This modified structure was called L1 adaptive controller [27], for which the robustness and stability

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analysis is proved for different wide range continuous-time systems [27]- [28]. Decoupling between the control loop and the

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adaptation loop is one of the main features of the L1 adaptive controller. The main advantage of this decoupling is that it permits

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very fast adaption without affecting the robustness of the controlled closed loop system. The continuous-time state-feedback and

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output-feedback formulations of the L1 adaptive controller were introduced in [29] and [30], respectively.

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Based on [27], the hardware platform is the only constraints that restrict the increase of the adaptation loop speed domain as the

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real time implementation of the L1 adaptive controller is carried out using a micro-controller. Hence, the robustness and stability

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analysis in discrete-time domain is necessary to take-care off some implementation issues such as the adaptation gain limitations

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and the sampling interval. In [31], the authors introduced a piecewise continuous adaptation law; however, the implementation

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of the control law is still applied in continuous-time form. The authors in [32], have designed an L1 adaptive controller in the

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continuous-time and the effect of the implementation solver is taken into the analysis. It results in deriving a sampling interval

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sufficient stability condition. However, the proposed approach suffers from steady-state error while tracking the reference signal

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due to the elimination of the adaptation law integrator. In [33], L1 adaptive controller was introduced for discrete-time systems

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in a discrete-time platform; design and analysis. Logarithmic Lyapunov function proposed in [34] is used to analyse the system

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stability. However, the introduced analysis is restricted to systems with constant unknown uncertainties with full state

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measurements. In [35], the authors introduced an extensions to the work in [33] in which a wider range of systems are considered.

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Systems with disturbances, time varying uncertainties, unmodeled actuator dynamics and unknown input gains are analysed.

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Stability conditions are derived to handle the time-varying parameters. The analysis in [35] is adopted in this section and used

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to implement a discrete-time controller to regulate the pitch command in WECS.

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Indirect MRAC and L1 adaptive control basic elements are control law, adaptation laws and a state predictor. Consider the 𝑛-

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order discrete-time state-space represented system,

𝑥(𝑘 + 1) = 𝐴𝑚 𝑥(𝑘) + 𝑏(𝑢(𝑘) + 𝜃 𝑇 (𝑘)𝑥(𝑘)) (6) 𝑇

𝑦(𝑘) = 𝑐 𝑥(𝑘)

where 𝑥(𝑘) ∈ ℝ𝑛 is the measured state vector, 𝑘 ≥ 0 is the sampling index, 𝑢(𝑘) ∈ ℝ is the control signal, 𝑦(𝑘) ∈ ℝ is the

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output, 𝐴𝑚 is the state matrix of the tracking model, 𝑏, 𝑐 are known constant vectors satisfying that the pair (𝐴𝑚 , 𝑏) is a

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completely state controllable, and 𝜃(𝑘) represents the system uncertainties. Assuming that, the unknown parameters of the

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system are bounded with known bounds and the changes of the parameters 𝜃, and the disturbance 𝑑, are bounded by arbitrary

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known constants.

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It is required to design a state-feedback discrete-time adaptive controller to maintain the output of the plant and keep

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tracking the discrete-time reference model. The model is subject to a bounded input 𝑟(𝑘) and characterized by the matrix 𝐴𝑚 .

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Let the model system be given as, 𝑥𝑚 (𝑘 + 1) = 𝐴𝑚 𝑥(𝑘) + 𝑏 𝑘𝑔 𝑟(𝑘)

pro of

where 𝑘𝑔 is a reference manipulation gain that can be calculated as 𝑘𝑔 = (𝑐(𝐼 − 𝐴𝑚 )−1 𝑏)−1 to ensure zero steady state

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(7)

error.

Similar to continuous-time L1 adaptive control, the discrete-time controller consists of two main loops. The first loop is

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the adaption loop in which two main targets should be achieved. First, the system dynamics are re-parameterized by using the

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estimated parameters. It can be achieved using a state predictor. Second, a minimum error between the predicted output and the

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system output should be achieved. Regulation of the estimated parameters using the adaptation laws aims to ensure this target

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achievement. The second loop is the control loop, this loop is responsible for producing the control signal which ensures the

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tracking performance between the system output and the reference output with a bounded error.

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Consider the following state predictor,

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𝑥̂(𝑘 + 1) = 𝐴𝑚 𝑥̂(𝑘) + 𝑏 (𝑢(𝑘) + 𝜃̂ 𝑇 (𝑘) 𝑥(𝑘))

(8)

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where 𝜃̂(𝑘) is the estimate of 𝜃 at sample 𝑘. For a simple representation of the adaptation law, define the estimation prediction

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error and the state prediction error as,

𝑥̃(𝑘) = 𝑥̂(𝑘) − 𝑥(𝑘) , 𝜃̃(𝑘) = 𝜃̂ (𝑘) − 𝜃 (𝑘) By subtracting the predictor equation from the system equation, the error dynamics can be rewritten as,

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𝑥̃(𝑘 + 1) = 𝐴𝑚 𝑥̃(𝑘) + 𝑏 𝜃̃ 𝑇 (𝑘) 𝑥(𝑘)

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(9)

(10)

The adaptation law given in (11) is used to update the parameters at each sample as,

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𝑏𝑇 ̂𝑇 𝑇 (𝑥(𝑘 + 1) − 𝐴𝑚 𝑥(𝑘)) − 𝜃 (𝑘)𝑥(𝑘) 𝑏 } 𝜃̂(𝑘 + 1) = 𝑃𝑟 {𝜃̂(𝑘) + Г𝑑 𝑥(𝑘) 𝑏 𝑥 𝑇 (𝑘)𝑥(𝑘)

(11)

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where 0 < Г𝑑 ≤ 1 is the discrete-time adaptation gain, and 𝑃𝑟{∗} is the discrete projection operator used to ensure the

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boundedness of the parameters inside their known convex set [36]. Finally, the following control signal is considered, 𝑢(𝑘) = 𝐶𝑓 (𝑘) ∗ (𝑘𝑔 𝑟(𝑘) − 𝜃̂ 𝑇 (𝑘) 𝑥(𝑘))

(12)

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where (∗) donates the convolution operator, and 𝐶𝑓 (𝑘) is a low-pass filter with 𝐶𝑓 (1) = 1 , and satisfying the following

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condition, ‖𝐺(𝑧)‖1 ‖𝜃‖1 < 1 (13) 𝐻(𝑧) = (𝑧𝐼 − 𝐴𝑚 )−1 𝑏

,

𝐺(𝑧) = 𝐻(𝑧) (1 − 𝐶𝑓 (𝑧))

Lemma 4 in [35] states that for closed loop stability another sufficient stability condition has to be introduced. This

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condition explicitly limits the implementation sampling interval. It states that the system should satisfy the following condition

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to ensure stability, ‖𝑏‖2 ≤

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𝜆𝑚𝑖𝑛 (𝑅)𝜆𝑚𝑖𝑛 (𝑃)

2 ̅̅̅̅ 𝛼 (𝜆𝑚𝑎𝑥 (𝑃)) ∆𝜃

(14)

where 𝑃 and 𝑅 can be obtained by solving Lyapunov-like equation; 𝐴𝑇𝑚 𝑃 𝐴𝑚 − 𝑃 + 𝐼 = −𝑅 , 𝛼 is a known positive constant, and ̅̅̅̅ ∆𝜃 is the upper bound of the uncertainty deviation in 1 sample time.

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For the discrete-time case, the condition in (14) according to [35] is necessary condition for the closed loop stability.

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Assuming that the given system to be controlled is in a continuous-time representation, so the condition in (14) is fulfilled via

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reducing the sampling interval, which reduces the term ‖𝑏‖2 arbitrarily. It also indicates that, in the case of time invariant system,

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no limitation on the sampling interval is needed as ̅̅̅̅ ∆𝜃 = 0. This result matches the result in [33].

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Similar to continuous-time version of L1 adaptive controller, both states and control signal errors are bounded to the reference system. Further analysis is in [35] and omitted in this paper. The minimum tracking error is achieved when the adaptation gain is equal to unity (Г𝑑 = 1). According to [27], the

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tracking error for a continuous L1 adaptive controller is inversely proportional to the adaptation gain. So, the larger the adaptation

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gain, the smaller the error; i.e. an infinite adaptation gain leads to the minimum tracking error. For better understanding the

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condition, it is similar to the case of linear control where the deadbeat performance of the closed-loop system can be achieved

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using a finite gain for a discrete integrator system while in the continuous-time systems this performance can be achieved only

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using an infinite gain. For the system given in (6), assuming that the system is all known, free of uncertainties and suffers from

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input disturbance, then the disturbance can be exactly estimated in one sample using the adaptation law given in (11).

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3.2. L1 adaptive for WECS

This section illustrates the deployment of the previously discussed controller for CPC in WECS. To this end, the CPC control signal is divided into two terms; classical state-feedback term and adaptive term. The control signal is given by, 𝑢𝐶𝑃𝐶 (𝑘) = −𝑘𝑠 𝑥(𝑘) + 𝑢𝑎𝑑 (𝑘)

(15)

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where 𝑘𝑠 is a constant vector with appropriate dimensions. The design is achieved as follows. First, the FAST model is used to provide a linearized model of the system around

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different operating points identified by the wind speed. The wind speed 18m/s is chosen as a midpoint of region 3 operation to

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be the nominal open loop system. A state-feedback controller 𝑘𝑠 is designed for the nominal system to achieve the required

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performance; i.e. 𝐴𝑚 = 𝐴n − 𝑏n 𝑘𝑠 where 𝐴n 𝑎𝑛𝑑 𝑏n describe the nominal system at wind speed 18 m/s.

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Second, consider the system in (4), the system is represented as time-varying system. It can be reconsidered as a nominal

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system with time-varying uncertainties to compensate for the state matrix alteration and time-varying disturbance to compensate

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for both the input-vector change and the linearization disturbance term 𝐵𝑖 𝑑 𝑢𝑑 (𝑘). Hence, the system can be reformulated as, 𝑥(𝑘 + 1) = 𝐴𝑚 𝑥(𝑘) + 𝑏(𝑢𝑎𝑑 (𝑘) + 𝜃 𝑇 (𝑘)𝑥(𝑘) + 𝑑(𝑘))

(16)

where 𝑥(𝑘) is the measured states, 𝜃 is the system uncertainties, 𝑑 is the system total input disturbance, and 𝑏 is a known

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input vector. By comparing the system in (4) and the above reformulation, the uncertainties term is added to reformulate the

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time-varying state matrix of the system as 𝐴𝑛 = 𝐴𝑚 + 𝑏𝜃 𝑇 (𝑘), while the disturbance term is added to compensate any changes

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in the input vector in addition to the linearization disturbance component as 𝐵𝑢(𝑘) + 𝐵𝑢𝑑 (𝑘) = 𝑏(𝑢𝑎𝑑 (𝑘) + 𝑑(𝑘)). As the

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wind turbine operating region is bounded, hence the time-varying parameters in the system in (4) is bounded which implies that

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both uncertainties 𝜃 𝑇 (𝑘) and disturbance 𝑑(𝑘) are bounded. The sampling interval and the low-pass filter are selected to fulfil

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the sufficient stability conditions in (13) and (14).

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Stability proof of closed loop systems using L1 adaptive control in case of input saturation is discussed in Appendix A.

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The stability proof is an extension of the work in [37] in discrete frame work. The system in Appendix A is considered as an

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uncertain system with input saturation. This proof can be directly used in CPC for WECS by reformulating the added disturbance

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and the uncertainties to update the adaptation law. Proofing the closed loop stability of the system while taking the pitch angle

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constraint into consideration is one of the main contributions of this paper as stated previously.

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The new adaptation law is analysed and proven to be stable in [35]. The adaptation law and the state predictor are updated as follows. Let,

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𝜃̂ (𝑘) 𝑥(𝑘) 𝜗̂(𝑘) = [ ] , 𝜑(𝑘) = [ ] 1 𝑑̂ (𝑘)

(17)

Following [35] discussed in the previous subsection, the controller is given as predictor, adaptation law, and control law, 𝑥̂(𝑘 + 1) = 𝐴𝑚 𝑥̂(𝑘) + 𝑏(𝑢𝑎𝑑 (𝑘) + 𝜃̂ 𝑇 (𝑘)𝑥(𝑘) + 𝑑̂ (𝑘))

(18)

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𝑏𝑇 ̂𝑇 ̂ (𝑘 + 1) 𝑇 𝑏 (𝑥(𝑘 + 1) − 𝐴𝑚 𝑥(𝑘)) − 𝜗 (𝑘)𝜑(𝑘) 𝜃 𝑏 ̂ ̂ (𝑘 (𝑘) } 𝜗 + 1) = [ ] = 𝑃𝑟 {𝜗 + Г𝑑 𝑥(𝑘) 𝜑 𝑇 (𝑘)𝜑(𝑘) 𝑑̂ (𝑘 + 1)

𝑢𝐶𝑃𝐶 (𝑘) = −𝑘𝑠 𝑥(𝑘) + 𝐶𝑓 (𝑘) ∗ (𝑘𝑔 𝑟(𝑘) − 𝜗̂ 𝑇 (𝑘) 𝜑(𝑘))

(19)

(20)

The controller structure is illustrated in Figure 3. The control signal is considered of two components: the state feedback which

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regulates the system to attain the essential transient performance and the adaptive component which deals with the system

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uncertainties and disturbances.

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Control Signal

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ucpc

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Adaption Laws

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Figure 3: L1 adaptive control for CPC

4. Simulation results

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This section is further divided into three main subsections. These subsections objective is to demonstrate the simulation

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results showing an improved performance of the proposed adaptive controller in comparison with the traditionally used controller;

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PI with gain schedules. In the first subsection, the parameters of the FAST turbine used in the simulation are illustrated, and the

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wind speed profile at hub height is given. Second, PI controller design via gain schedule is explained. Finally, a comparison is

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given to discuss the simulation results using both proposed and gain schedule PI controllers.

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FAST model of 5 MW turbine is used in the simulation. The main parameters regarding the used turbine during the simulation are listed in Table 1. More specifications and details of this wind turbine, the reader can refer to [16]. Table 1: Wind Turbine General Specifications

Rated Power

5 𝑀𝑊

Cut-in, Rated, Cut-out Wind speed

3 𝑚/𝑠, 11.4 𝑚/𝑠, 25 𝑚/𝑠

Cut-in, Rated Rotor Speed

6.9 𝑟𝑝𝑚, 12.1 𝑟𝑝𝑚

Turbine

Rotor, Hub Diameter

126 𝑚, 3 𝑚

Data

Hub Height

90 𝑚

Overhang, Shaft Tilt, Percone

5 𝑚, 5 𝑑𝑒𝑔., 2.5 𝑑𝑒𝑔.

Rotor Mass

110,000 𝑘𝑔

Nacelle Mass

240,000 𝑘𝑔

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Generator Data

347,460 𝑘𝑔

Gearbox ratio

97:1

Rated speed

1173.7 𝑟𝑝𝑚

Electrical Efficiency

94.4 %

Inertia about high speed shaft

534.116 𝑘𝑔. 𝑚2

4.1. Simulated FAST model

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Tower Mass

One of the main advantages of FAST is its ability to simulate the actual conditions that found in the field. To achieve

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these conditions, two major features of FAST model are considered. The first feature is that a reduced order model is obtained

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by the FAST software and it is used in the design of the proposed controller as discussed in section 2, so the 24 DOF supported

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by FAST are enabled during simulation to obtain the results. The second one is that the system is tested with a real life wind

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profile generated using TurbSim software package as stochastic wind profile [23]. This software package helps in generating a

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multi-dimensional wind profile covering the whole blades swept area and the tower. There are different wind models available

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in TurbSim. The model labeled “Great Plains Low Level Jet” is used here since it reflects many aerodynamic phenomena such

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as the coherent structure. Figure 4 depicts the wind profile generated by TurbSim which is applied at the hub height of the turbine.

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Figure 4: Stochastic wind speed profile

4.2. PI controller with Gain – Scheduled Parameters In this work, the performance proposed controller is compared with the gain-scheduled PI controller’s performance. Two

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steps frequency response analysis is used to design the PI controller as described in [38], [39]. First, a linearized model of the

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system at certain wind speed is developed. Then this model is used to design the PI controller achieving optimum cross over

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frequency and phase margin of 60o [16], [39]. In this work, the wind speed 12 m/sec is chosen as the nominal wind speed to get

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the PI parameters in this step, and the PI gains are labelled 𝑘𝑝𝑛 and 𝑘𝑖𝑛 for the proportional and integral terms, respectively.

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However, for the full wind speed range, the linearized model changes at each wind speed which implies that it is not

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accurate the using of the same obtained PI controller gains. Therefore, in the second step, the pitch effectiveness with the wind

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speed [16] is used to change the PI parameters for compensating the wind speed variations. So, the controller parameters are

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modified as, 13

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𝑘𝑝 =

𝑘𝑝𝑛 1 + 9.1𝑈𝑃

,

𝑘𝑖 =

𝑘𝑖𝑛 1 + 9.1𝑈𝑃

(21)

where 𝑈𝑃 is the pitch control action. Open loop frequency response of the FAST turbine model cascaded by the designed PI

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controller at different wind speeds is plotted in Figure 5. It shows that the ability of the gain-scheduled PI controller to compensate

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the variation of the wind speed. So, with time varying model, the gain-scheduled PI controller achieves the required performance.

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A comparison between the system’s phase margin with and without the gain-scheduling is drawn in Figure 6 to illustrate that the

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phase margin becomes approximately fixed at 600 after gain scheduling.

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4.3. Performance Discussion

In this section, the proposed controller performance is discussed in comparison with the performance of the later discussed

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PI controller with gain scheduling. The FAST model discussed and the wind speed profile shown in Figure 4 are used to test the

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controllers’ performance. Figure 7 illustrated the results obtained. Three variables are captured and plotted; the generated power,

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the generator speed, and the turbine blades’ flapwise moments. Table 2 shows the collected data statistics to illustrate the

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difference in performance between the two controllers.

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Figure 7 shows that the proposed discrete-time L1 adaptive controller suppresses the fluctuations in all the variables. This

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is clearly shown in Table 2 by the decrease in the standard deviations’ values of the proposed controller. Taking the gain-

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scheduled PI controller as a comparison reference, L1 adaptive controller reduces the standard deviations by 75%, 71.3%, and

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7.5% in speed, power, and flapwise moments, respectively. Moreover, it reduces the maximum absolute error by 69.3%, 60.2%,

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and 16.2%, respectively. Regarding the generator performance, on the first hand, , both controllers have approximately the same

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average generator speed value; on the other hand, the average generator power is increased by 2.8% while using the proposed

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controller. Based on the above discussion, it is fair to claim that the adaptive controller, in comparison to the traditionally used

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controller, improves the regulation of the generator speed around its rated value, increases the generated power, and meanwhile

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increases the blades’ life time.

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Bode Diagram 50

Model at 12 m/sec Model at 14 m/sec Model at 16m/sec Model at 18 m/sec Model at 20 m/sec Model at 22 m/sec

-100 -150 -200

Phase (deg)

-250 360 270 180 90 0 -2

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10

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Magnitude (dB)

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10

3

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4

10

Frequency (rad/s)

Figure 5: Bode plot of the system controlled by gain-scheduled PI controller

In region 3, the wind turbine operates normally at its rated power, and the controller’s objectives is to maintain this

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power close to its rated value with minimum damage equivalent load (DEL) on the mechanical structure of the turbine’s elements;

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the blades and the tower. DEL is used as a comparison index represents the fatigue load applied on the wind turbine [40].

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Assuming a 5M turbine is operating at a capacity factor of 0.35 [41], Table 2 shows that replacing the traditional PI controller

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by the proposed adaptive controller increases the annual energy production by 0.42 GWh. Moreover, using the standard deviation

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of the generated power data as a comparison index, the proposed technique smoothes the fluctuations in the generated power by

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71.3%. Last of all, Table 2 shows that it reduces the DEL by 29.8%, 17.6%, 18.24%, and 14.68% for the moment on the tower

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fore-aft and the moment on the three blades, respectively. The above discussion illustrates that the proposed adaptive controller

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achieves the required objectives of WECS control. These objectives achievement is the main goal of this work, as mentioned

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previously in the introduction section as the first paper contribution.

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Figure 6. Phase margin using PI controller with fixed parameters versus gain-scheduling parameters 15

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Figure 7: Discrete-time L1 adaptive controller vs Gain-scheduled PI controller: (a) generator speed (b) generated power (c) flapwise moment on the first blade.

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Table 2. Data analysis of the results obtained using the gain scheduled PI controller versus using the proposed controller Performance Indices

Gain scheduled PI

L1 Adaptive

Error Standard Deviation

11.159

2.779

Mean

1173.83

1173.34

Maximum Absolute Error

42.928

13.15

Error Standard Deviation

301.8236

86.5314

4801.581

4938.821

1311.696

522.034

2189.278

2025.084

4588.218

4701.816

1311.696

522.034

1420

1170

1370

1120

1430

1220

3020

2120

14.72

15.14

Generator speed

Electric power (kW)

Mean Maximum Absolute Error Standard Deviation

Flap wise moment Mean Value (kNm) Maximum Absolute Error 1st blade 2nd blade moment (kNm) 3rd blade DEL for tower fore-aft (kNm)

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5. Experimental results

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A small-scale experimental setup, is used to obtain experimental results which validate the real time implementation of

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this work. The setup is based on 0.75 kW separately excited dc motor coupled to 0.3 kW Permanent magnet synchronous

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generator (PMSG). A 3-phase fully controlled converter is used to control the torque of the DC motor. The motor emulates a

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wind turbine whose aerodynamic power fluctuates with the wind turbine. A PMSG is connected to the grid through two stage

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converters, a machine side converter (MSC) back-to-back with a grid-side converter (GSC) with DC link connecting the two

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converters. A dSPACE DS1104 R&D controller is used to control the torque set-point of the dc motor and the control action of

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the MSC. The control of grid side converter (GSC) is implemented through a Data Acquisition card. The hardware setup

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schematic and a photo of the hardware setup are shown in Figure 8.

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MSC

GSC

PMSG DC motor

( I abc ) grid

Interface Board & Gate Drive Circuit

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(Vabc ) grid

Signals

Interface Board & Gate Drive Circuit

Gating

Vdc

 gen

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Signals

Tm

Gating

Current Control

(Vabc ) gen

( I abc ) gen

3-phase supply

National Instruments Data acquisition Board

Host PC 2

Host PC 1

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Grid

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Figure 8. The experimental setup Emulator (a) Hardware Setup Schematic Diagram, (b) Hardware Setup in the Laboratory

The detailed parameters of the DC motor and PMSG used in this work are listed in Table 3.In the hardware setup, a

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capacitor of 450 µ𝐹 capacitance connects the two converters, and the grid voltage is set to 42 𝑉𝑟𝑚𝑠 /50 Hz with a transformer

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which steps the voltage up to 380 V. The converters’ frequency is set to 4.8 𝑘𝐻𝑧. The test is applied by using 7 min of stochastic

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wind speed followed by constant wind speed stairs for almost 100 sec. The wind speed data with the collected generator data are

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plotted in Figure 9. Figure 9.b shows the improvement of the proposed controller in regulating the generator speed around its

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nominal value which reflects the generated power fluctuations in Figure 9.c.

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Table 3. Parameters of the machines used in the hardware work PMSG Rated Power (kW)

0.3

Rated Torque (Nm)

2.86

Rated Speed (rpm)

1000 5

Number. of PaerPoles Armature Resistance (Ω) Armature Inductance (mH) Voltage Constant (mV/rpm)

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DC motor

0.75

Rated Power (kW) Rated Armature Voltage (V) Rated Armature Current (A)

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Rated Speed (rpm)

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The obtained data is summarized in Table 4 with the same comparison indices using in the simulation results over the

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first 300 sec. The table confirms the obtained conclusion via simulation by showing a significant improvement in the regulated

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generator speed and the generated power while implementation. These hardware results in combined with the previous simulation

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results illustrate the capability of implementing the proposed design technique for different types of systems, which highlights

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the third contribution mentioned previously in the introduction section.

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Table 4. Experimental Results Summary

Performance Indices

Generator speed (PU)

0.3546

0.1489

Mean

0.9984

1.0000

Error Standard Deviation

0.0709

0.0114

Maximum Absolute Error

213.9095

71.5771

Mean

287.1165

295.4204

44.1523

10.0537

Error Standard Deviation

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L1 adaptive controller

Maximum Absolute Error

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Gain scheduled PI

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Figure 9: Experimental results comparison of different system variables when using a gain- scheduled PI controller and the proposed controller: (a) stochastic wind speed profile (b) generator speed (c) generator power

6. Conclusion

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The paper presents a design of discrete-time L1 adaptive controller for a variable speed variable pitch wind turbine. The

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proposed controller main target is to manipulate the generator speed and its output power at constant value during over rated

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wind speed. The proposed adaptive controller compensates for the uncertainties of the system due to non-linearities and

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disturbances. The robustness and stability analyses of the closed loop system, the wind turbine controlled by the adaptive

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controller, are introduced. The analyses show that the controller is able to handle the bounded uncertainties and disturbances due

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to wind speed rapid changes. It forces the turbine to follow the rated speed and power during transient and steady state with

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controllable upper bounded error.

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A 5-MW wind turbine FAST model is used via simulation to compare the proposed controller’s performance with

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traditionally-used gain-scheduled PI controllers. Simulation results show the superior performance of the proposed controller in

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regulating the turbine speed, generator power, and minimizing the fatigue on the blades. Using an experimental setup with a 20

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reduced power rating, the proposed controller performance is compared to the gain-scheduled PI controller. The experimental

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results demonstrate that the adaptive controller is not only easily implementable in discrete-time environment, but also achieves

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significant improvement in regulating the generator speed and power.

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7. References

[7] [8] [9] [10] [11] [12] [13]

[14]

[15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

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The World Wind Energy Association. New Record in Worldwide Wind Installations, http://www.wwindea.org.. Y. Amirat, M. Benbouzid, B. Bensaker and et al, “The State of the Art of Generators for Wind Energy Conversion Systems,” Electromotion, vol. 14, no. 4, pp. 163–172, 2007. I. Poultangari, R. Shahnazi and M. Sheikhan, “RBF neural network based PI pitch controller for a class of 5-MW wind turbines using particle swarm optimization algorithm,” ISA Transactions, vol. 51, pp. 641–648, 2012. H. Liu, Q. Tang, Y. Chi, Z. Zhang and X. Yuan, “Vibration reduction strategy for wind turbine based on individual pitch control and torque damping control,” International Transactions on Electrical Energy System, 2016. Y. Ren, L. Li, J. Brindley and L. Jiang, “Nonlinear PI control for variable pitch wind turbine,” Control Engineering Practice, vol. Vol. 50, p. pp. 84–94, 2016. T. L. Van, T. H. Nguyen and D. C. Lee, “Advanced Pitch Angle Control Based on Fuzzy Logic for Variable-Speed Wind Turbine Systems,” IEEE Transactions on Energy Conversion, vol. 30, no. 2, pp. 578-587, 2015. A. Asgharnia, R. Shahnazi and A. Jamali, “Performance and robustness of optimal fractional fuzzy PID controllers for pitch control of a wind turbine using chaotic optimization algorithms,” ISA Transactions, vol. 79, pp. 27-44, 2018. X. Yin, Y. Lin, W. Li, H. Liu and Y. Gu, “Adaptive sliding mode back-stepping pitch angle control of avariabledisplacement pump controlled pitch system for wind turbines,” ISA Transactions, vol. 58, pp. 629-634, 2015. A. S. Yilmaz and Z. Özer, “Pitch angle control in wind turbines above the rated wind speed by multi-layer perceptron and radial basis function neural networks,” Expert System with Applications, vol. 36, p. 9767–9775, 2009. M. Soliman, O. P. Malik and D. T. Westwick, “Multiple Model Predictive Control for Wind Turbines With Doubly Fed Induction Generators,” IEEE Trans. on Sustainable Energy, vol. 2, no. 3, pp. 215-225, 2011. A. lasheen and A. L. Elshafei, “Wind-turbine collective-pitch control via a fuzzy predictive algorithm,” Renewable Energy, vol. 87, pp. 298 - 306, 2016. A. Lasheen, M. Saad, H. Emara and A. L. Elshafei, “Continuous-time tube-based explicit model predictive control for collective pitching of wind turbines,” Energy, vol. 1, pp. 1222-1233, 2017. W. Meng, Q. Yang, Y. Ying and etc, “Adaptive power capture control of variable-speed wind energy conversion systems with guaranteed transient and steady-state performance,” IEEE Trans. Energy Convers, vol. 28, no. 3, pp. 716– 825, 2013. M. Elnaggar, M. Saad, H. Abdel Fattah and A. L. Elshafei, “L1 adaptive fuzzy control of wind energy conversion systems via variable structure adaptation for all wind speed regions,” IET Renewable Power Generation, vol. 12, no. 1, pp. 18-27, 2018. P. Bagheri and Q. Sun, “Adaptive robust control of a class of non-affine variable-speed variable-pitch wind turbines with unmodeled dynamics,” ISA Transactions, vol. 63, pp. 233-241, 2016. J. M. Jonkman, S. Butterfield, W. Musial and et al:, “Definition of a 5-MW reference wind turbine for offshore system development,” USA, 2007. J. Jonkman and M. Buhl, “FAST User's Guide,” National Renewable Energy Labratory, Technical Report NREL/EL500-38230, 2005. P. C. Gregory, “Air research and development command plans and programs,” in in Proceedings of the Self-Adaptive Flight Control Symposium, Ohio, pp. 8-15, 1959. E. Mishkin and L. Braun, Adaptive Control Systems, New York: McGraw-Hill, 1961. L. W. Taylor and E. J. Adkins, “Adaptive control and the X-15,” in Princeton University Conference on Aircraft Flying Qualities, Princeton, NJ, USA, June 1965. L. Ljung, System Identification—Theory for the User, Englewood Cliffs, NJ, USA: Prentice-Hall, 1987. K. J. Åström and B. Wittenmark, “On self-tuning regulators,” Automatica, vol. 9, no. 2, pp. 185-199, 1973. R. Kalman, “Design of self-optimizing control systems,” Transactions on ASME, vol. 80, p. 468–478, 1958. K. J. Åström and T. Bohlin, “Numerical Identification of Linear Dynamic Systems from Normal Operating Records,” in Theory of Self-Adaptive Control Systems, New York, USA, Percival H. Hammond, ed., Springer , pp. 96-111, 1966. B. Egardt, Stability of Adaptive Controllers, Berlin, Germany: Springer-Verlag, 1979.

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[26] Y. D. Landau, Adaptive Control: The Model Reference Approach, New York, NY, USA: Marcel Dekker, 1979. [27] C. Cao and N. Hovakimyan, L1 Adaptive Control Theory, Guaranteed Robbusness with Fast Adaptation, Philadelphia, PA: SIAM, 2010. [28] C. Cao and N. Hovakimyan, “Stability margins of L1 adaptive control architecture,” IEEE Trans. Automatic Control, vol. 55, no. 2, pp. 480 - 487, Feb. 2010. [29] C. Cao and N. Hovakimyan, “Design and Analysis of a Novel L1 Adaptive Control Architecture With Guaranteed Transient Performance,” IEEE Trans. on Automatic Control, vol. 53, no. 2, pp. 586-591, 2008. [30] C. Cao and N. Hovakimyan, “L1 Adaptive Output Feedback Controller for Systems of Unknown Dimension,” IEEE Trans. on Automatic Control, vol. 53, no. 3, pp. 815-821, Apr. 2008. [31] C. Cao and N. Hovakimyan, “Design, L1 Adaptive Output-Feedback Controller for Non-Strictly-Positive-Real Reference Systems: Missile Longitudinal Autopilot,” Journal of Guidance, Control, and Dynamics., vol. 32, no. 3, pp. 717-726, May-June 2009. [32] H. Jafarnejadsani, H. Lee and N. Hovakimyan, “An L1 Adaptive Control Design for Output-Feedback Sampled-Data Systems,” in American Control Conference, Seattle, USA, May 2017. [33] H. Jafarnejadsani and N. Hovakimyan, “Optimal Filter Design for Discrete-Time Formulation of L1 adaptive Control,” in AIAA SciTech, Florida, Jan. 2015. [34] S. Akhtar and D. S. Bernstein, “Lyapunov-Stable Discete-Time Model Reference Adaptive Control,” in American Control Conf., Portland, USA, June 2005. [35] M. Elnaggar, M. Saad, H. A. Fattah and A. L. Elshafei, “Discrete Time L1 Adaptive Control for Systems with Timevarying Parameters and Disturbances,” in IEEE Conf. on Desicion and Control, LV. USA, Dec. 2016. [36] P. Ioannou and B. Fidan, Adaptive Control Tutorial, SIAM, 2006. [37] D. Li, N. Hovakimyan and C. Cao, “Positive invariant set estimation of L1 adaptive controller in the presence of input saturation,” Internationl Journal of Adaptive Control and Signal Processing, vol. 27, no. 11, pp. 1012-1030, 2013. [38] D. J. Leith and . E. Leithead, “Appropriate realization of gain‐ scheduled controllers with application to wind turbine regulation ,” International Journal of Control, vol. 65, no. 2, pp. 223-248, 2000. [39] N. Yoonsu, Control System Design, Wind Turbines, 2011. [40] G. J. Hayman and M. Buhl, “MLife User’s Guide for Version 1.00,” NREL, 2012. [41] L. Y. Pao and K. E. Johnson, “A tutorial on the dynamic and control of wind turbines and wind farms,” in American Control Conference, St. Louis, Mo, USA, 2009. [42] M. Elnaggar, M. S. Saad, H. A. AbdelFattah and A. L. Elshafei, “Further extensions to L1 adaptive control theory : Analysis and application,” Ph.D. Cairo University, 2017. [43] C. Cao and N. Hovakimyan, L1 Adaptive Control Theory, Guaranteed Robbusness with Fast Adaptation, Philadelphia, PA: SIAM, 2010.

22

Journal Pre-proof

364 365

Appendix A First, consider an uncertain Linear Time-Invariant (LTI) system subject to an input disturbance in discrete-time as, 𝑥(𝑘 + 1) = 𝐴 𝑥(𝑘) + 𝑏 𝑢(𝑘) + 𝑏𝑑 𝑑(𝑘)

(22)

where 𝐴 ∈ 𝛺 is the state matrix describing the nominal system with a perturbation defined within the known convex set 𝛺

367

where 𝛺 = 𝐶𝑜{𝐴1 , 𝐴2 , … , 𝐴𝑗 } and 𝐶𝑜 denotes the convex hull with 𝑗 elements. 𝑏 and 𝑏𝑑 are vectors with proper dimensions,

368

and 𝑑(𝑘) represents a system’s bounded input disturbance. Assume a known constant 𝑑̅ such that, ‖𝑑(𝑘)‖ ≤ 𝑑̅ . Assume

369

(𝐴 , 𝑏) is a completely state controllable pair, the state-feedback controller is described as, 𝑢𝑠 (𝑘) = 𝑘𝑠 𝑇 𝑥(𝑘)

,

pro of

366

𝑢𝑠 (𝑘) 𝑢(𝑘) = 𝑠𝑎𝑡(𝑢𝑠 (𝑘)) = { 𝑢𝑚𝑎𝑥 − 𝑢𝑚𝑎𝑥

|𝑢𝑠 (𝑘)| ≤ 𝑢𝑚𝑎𝑥 𝑢𝑠 (𝑘) > 𝑢𝑚𝑎𝑥 } 𝑢𝑠 (𝑘) < −𝑢𝑚𝑎𝑥

(23)

where 𝑘𝑠 ∈ ℝ𝑛 is a state-feedback vector, 𝑠𝑎𝑡(∗) denotes the saturation function with a domain 𝑢𝑚𝑎𝑥 which is the actuator

371

saturation bound. Let ρ > 0 be an arbitrary positive number, 𝑃 ∈ ℝ𝑛 𝑥 𝑛 be a positive definite matrix, and let 𝑉(𝑥(𝑘)) =

372

𝑥 𝑇 (𝑘) 𝑃 𝑥(𝑘). Then the ellipsoid 𝜀(𝑃, 𝜌) defined as, 𝜀(𝑃, 𝜌) = {𝑥 ∈ ℝ𝑛 ∶ 𝑥 𝑇 𝑃 𝑥 ≤ 𝜌 } is said to be invariant by definition if

373

for all 𝑥(𝑘) ∈ 𝜀(𝑃, 𝜌) the following inequality is satisfied, 𝑉(𝑥(𝑘 + 1)) = 𝑥 𝑇 (𝑘 + 1) 𝑃 𝑥(𝑘 + 1) ≤ 𝜌 . Also, let the

374

vector 𝑘𝑎 ∈ ℝ𝑛 , such that 𝒪(𝑘𝑎 ) be a subset defined as 𝒪(𝑘𝑎 ) = {𝑥 ∈ ℝ𝑛 ∶ |𝑘𝑎 𝑇 𝑥 | ≤ 𝑢𝑚𝑎𝑥 }. This definition ensures that if

375

the vector 𝑘𝑎 is used as the state-feedback controller, then for any 𝑥 that lies inside the subset 𝒪(𝑘𝑎 ), the control signal will

376

satisfy the actuator saturation.

377

Lemma A-1 [Lemma 5-1 in [42]] : For a given ellipsoid 𝜀(𝑃, 𝜌), if there exist some 𝑘𝑎 ∈ ℝ𝑛 and a positive number 𝜂 that

378

satisfy that 𝜀(𝑃, 𝜌) ⊂ 𝒪(𝑘𝑎 ) and satisfy the following inequalities,

lP

re-

370

𝑇

1+𝜂 ) 𝜆𝑚𝑎𝑥 (𝑏𝑑𝑇 𝑃 𝑏𝑑 ) − 1 ) 𝑃 ≤ 0 𝜌𝜂

∀ 𝑖 = [1 … . 𝑗]

1+𝜂 ) 𝜆𝑚𝑎𝑥 (𝑏𝑑𝑇 𝑃 𝑏𝑑 ) − 1 ) 𝑃 ≤ 0 𝜌𝜂

∀ 𝑖 = [1 … . 𝑗]

urn a

(1 + 𝜂)(𝐴𝑖 + 𝑏 𝑘𝑠 𝑇 ) 𝑃(𝐴𝑖 + 𝑏 𝑘𝑠 𝑇 ) + (𝑑̅ 2 ( 𝑇

(1 + 𝜂)(𝐴𝑖 + 𝑏 𝑘𝑎 𝑇 ) 𝑃(𝐴𝑖 + 𝑏 𝑘𝑎 𝑇 ) + (𝑑̅ 2 (

(24)

(25)

379

Then the ellipsoid 𝜀(𝑃, 𝜌) is an invariant set for the system defined in (22) with the control action defined in (23). The lemma

380

proof is an extension of the proof in [37].

382

Second, this lemma is used to proof the stability of the closed loop system with L1 adaptive control as following. Consider the system,

Jo

381

𝑥(𝑘 + 1) = 𝐴𝑛 𝑥(𝑘) + 𝑏 ( 𝑢(𝑘) + 𝜃 𝑇 𝑥(𝑘))

(26)

383

where 𝐴𝑛 is a known state matrix describing the nominal system, ( 𝐴𝑛 , 𝑏) is a completely state controllable pair, and 𝜃 ∈

384

ℝ𝑛 represents the system matched uncertainty which is assumed to be bounded; i.e. 𝜃𝑖 ≤ 𝜃𝑖 ≤ 𝜃𝑖

385

𝜃𝑖 𝑎𝑛𝑑 𝜃𝑖 are known constants. The control action can be divided into two terms. The first term is a linear state-feedback to

𝑖 = [1 , 2 , … , 𝑛], where

23

Journal Pre-proof

386

transform the nominal system to the model system. The second term is the adaptive control term which attempts to cancel the

387

system uncertainties and follow the reference signal. Consider, 𝑢(𝑘) = 𝑠𝑎𝑡 (𝑢𝑠 (𝑘) + 𝑢𝑎𝑑 (𝑘)) 𝑛

𝑘𝑠𝑇

,

𝑢𝑠 (𝑘) = − 𝑘𝑠𝑇 𝑥(𝑘)

(27)

. On the other hand, 𝑢𝑎𝑑 (𝑘) is designed as an L1 adaptive controller. Defining

388

where 𝑘𝑠 ∈ ℝ satisfies that 𝐴𝑚 = 𝐴𝑛 − 𝑏

389

the deficiency in the control action, 𝑢𝑑𝑒 (𝑘) , due to the actuator saturation as,

390

pro of

𝑢𝑑𝑒 (𝑘) = 𝑢(𝑘) − (𝑢𝑠 (𝑘) + 𝑢𝑎𝑑 (𝑘)) Substituting (28) into (26) results in,

(28)

𝑥(𝑘 + 1) = 𝐴𝑚 𝑥(𝑘) + 𝑏 ( 𝑢𝑎𝑑 (𝑘) + 𝑢𝑑𝑒 (𝑘) + 𝜃 𝑇 𝑥(𝑘))

(29)

391

The adaptive control loops are presented; the adaptation loop and the control loop. The predictor is modified to include the

392

deficiency effect presented due to the actuator saturation limits. For the adaptation loop, consider the following state predictor

393

which re-parametrizes the system in (26) and taking the control deficiency into consideration,

𝜃̂ (𝑘) is

re-

𝑥̂(𝑘 + 1) = 𝐴𝑚 𝑥̂(𝑘) + 𝑏 (𝑢𝑎𝑑 (𝑘) + 𝑢𝑑𝑒 (𝑘) + 𝜃̂ 𝑇 (𝑘) 𝑥(𝑘))

(30)

394

where

395

x̃(k) = x̂(k) − x(k) , θ̃(k) = θ̂ (k) − θ . Subtracting (29) from (30) leads to, x̃(k + 1) = Am x̃(k) + b θ̃T (k)x(k). The

396

parameters are updated each sample using the adaptation law,

the

estimate

of

𝜃

at

sample

𝑘 .

Define

the

prediction

and

estimation

lP

𝑏𝑇 ̂𝑇 𝑇 (𝑥(𝑘 + 1) − 𝐴𝑚 𝑥(𝑘)) − 𝑢𝑎𝑑 (𝑘) − 𝑢𝑑𝑒 (𝑘) − 𝜃 (𝑘)𝑥(𝑘) 𝑏 } 𝜃̂(𝑘 + 1) = 𝑃 𝑟 {𝜃̂(𝑘) + 𝑥(𝑘) 𝑏 𝑥 𝑇 (𝑘)𝑥(𝑘)

errors

as,

(31)

where 𝑃𝑟{∗} is the discrete projection operator used to ensure the boundedness of the parameters inside their known convex set.

398

The control signal is defined as in (12) and satisfies (13).

399

Lemma A-2 : The prediction error is uniformly bounded as,

urn a

397

‖𝑥̃(𝑘)‖∞ ≤ √(𝑒

𝜃𝑚𝑎𝑥 𝑎

− 1) /(𝜇 𝜆𝑚𝑖𝑛 (𝑃𝐿 ))

(32)

where 𝑃𝐿 > 0 is a positive definite matrix that satisfies a Lyapunov-like equation 𝐴𝑚 𝑇 𝑃𝐿 𝐴𝑚 − 𝑃𝐿 + 𝐼 = −𝑅 with 𝑅 > 0, and 𝐼

401

is the identity matrix with proper dimension. 𝑎 and 𝜇 are positive finite constants and 𝜃𝑚𝑎𝑥 = 4 𝑚𝑎𝑥‖𝜃‖.

402

Proof: Same results from Lemma 4 in [35]. Accordingly, the constant used in (32) is given by,

Jo

400

𝑎=

1 𝛾1 𝑏 𝑇 𝑃𝐿 𝑏

𝜃∈

(33)

403

with 𝛾1 is a positive constant and 𝑏 is the discrete input vector in (26) obtained from discretizing the continuous-time system

404

model. This analysis shows that the error ‖𝑥̃‖∞ can be arbitrarily decreased by the choice of the sampling interval. Decreasing

405

the sampling interval arbitrarily decreases the discrete-time input vector 𝑏.

406

Using the definition of 𝜃̃, the adaptive control action can be written as, 24

Journal Pre-proof

𝑢𝑎𝑑 (𝑘) = 𝐶𝑓 (𝑘) ∗ (𝑘𝑔 𝑟(𝑘) − 𝜃 𝑇 (𝑘) 𝑥(𝑘) − 𝜃̃ 𝑇 (𝑘) 𝑥(𝑘))

(34)

407

The control signal, 𝑢𝑎𝑑 (𝑘), can be further divided into two terms. The first term includes the reference and the unknown

408

parameters 𝜃 and the second term includes only the error in the estimation 𝜃̃ (𝑘) . To represent the first term in the state-space

409

formulation, consider the following system,

pro of

𝑥𝑓 (𝑘 + 1) = 𝐴𝑓 𝑥𝑓 (𝑘) + 𝑏𝑓 (𝑘𝑔 𝑟(𝑘) − 𝜃 𝑇 (𝑘) 𝑥(𝑘)) 𝑦𝑓 (𝑘) = 𝑐𝑓 𝑥(𝑘)

(35)

410

where (𝐴𝑓 , 𝑏𝑓 , 𝑐𝑓 ) is a minimum state-space realization of the filter 𝐶𝑓 (𝑧) with order 𝑛𝑓 . The second term follows from

411

representing x̃(k) in z-domain as, 𝑥̃(𝑧) = 𝐻(𝑧)𝜂̃(𝑧)

(36)

where 𝜂̃(𝑧) is the z-transform of the term 𝜃̃ 𝑇 (𝑘) 𝑥(𝑘). As the pair (𝐴𝑚 , 𝑏) is controllable, then there exists an arbitrary vector

413

𝑐̅ that ensures that the system 𝑐̅𝑇 (𝑧𝐼 − 𝐴𝑚 )−1 𝑏 is a minimum phase system with relative degree 1 (Lemma A.12.1 [43]). It

414

follows that,

re-

412

̆ (𝑧) 𝑥̃(𝑧) = 𝐶𝑓 (𝑧)𝜂̃(𝑧) 𝐻 1

(37)

415

̆ (𝑧) = 𝐶𝑓 (𝑧) where 𝐻

416

To summarize the previous analysis, the L1 adaptive control action is reformulated using (28),(34),(35), and (37) as,

𝑐̅𝑇 is proper stable.

lP

𝑐̅𝑇 𝐻(𝑧)

𝑢(𝑘) = 𝑠𝑎𝑡(− 𝑘𝑠𝑇 𝑥(𝑘) + 𝑦𝑓 (𝑘) + 𝑑𝑢 (𝑘)) = − 𝑘𝑠𝑇 𝑥(𝑘) + 𝑦𝑓 (𝑘) + 𝑑𝑢 (𝑘) + 𝑢𝑑𝑒 (𝑘)

417

(38)

where 𝑑𝑢 (𝑘) is the deviation due to the error in the parameter estimation and it is upper bounded as follows, 𝜃𝑚𝑎𝑥 1 𝑐̅𝑇 ‖ √(𝑒 𝑎 − 1) /(𝜇 𝜆𝑚𝑖𝑛 (𝑃)) 𝑐̅𝑇 𝐻(𝑧) 1

urn a

𝑑𝑢 (𝑘) ≤ ‖𝐶𝑓 (𝑧)

(39)

418

The closed-loop system is expressed in an augmented state-space model using the model in (26) and the control action in (38)

419

as, 𝐴𝑛 𝑥𝑇 (𝑘 + 1) = [ 0

420

0 𝑏 [ 𝑇 (𝑘) + [ −𝑏𝑓 ] 𝜃 𝐴𝑓 ] 𝑥𝑇

0 𝑏 𝑇 0] 𝑥𝑇 (𝑘) + [𝑏 ] 𝑘𝑔 𝑟(𝑘) + [ ] 𝑠𝑎𝑡 (𝑑𝑢 (𝑘) + [− 𝑘𝑠 0 𝑓

𝑐𝑓 ] 𝑥𝑇 (𝑘))

Recalling the saturation relation (Lemma 1 in [37]),

Jo

𝑠𝑎𝑡( 𝑢1 + 𝑢2 ) = 𝑠𝑎𝑡(𝑢1 ) + 𝑢 ̃2

421

(41)

for some 𝑢 ̃2 ≤ |𝑢2 |. Hence, the sampling interval can be chosen to ensure that ‖𝑑𝑢 ‖ < 𝑢𝑚𝑎𝑥 , which implies that, 𝐴𝑛 𝑥𝑇 (𝑘 + 1) = [ 0

422

(40)

0 𝑏 [ 𝑇 (𝑘) + [ −𝑏𝑓 ] 𝜃 𝐴𝑓 ] 𝑥𝑇

0 0] 𝑥𝑇 (𝑘) + [𝑏

𝑓

𝑏 𝑘𝑔 𝑟(𝑘) 𝑏 ] + [ ] 𝑠𝑎𝑡 ([− 𝑘𝑠𝑇 0] [ 𝑑 ̃𝑢 0

𝑐𝑓 ] 𝑥𝑇 (𝑘))

(42)

̃𝑢 ≤ ‖𝑑𝑢 ‖∞ . Finally, define the following convex hull elements, where 𝑑 𝐴 = 𝑐𝑜{𝐴1 , 𝐴2 , 𝐴3 , … 𝐴𝑗 }

(43) 25

Journal Pre-proof

𝐴𝑛 𝐴1 = [ 0

0 𝑏 𝜃 𝐴𝑓 ] + [−𝑏𝑓 ] [ 1

𝜃2

… 𝜃𝑛

0] , 𝐴2 = [𝐴𝑛 0

…

,

0 𝑏 𝜃 𝐴𝑓 ] + [−𝑏𝑓 ] [ 1

𝐴𝑛 𝐴𝑗 = [ 0

𝜃2

0 𝑏 𝐴𝑓 ] + [−𝑏𝑓 ] [𝜃1

𝜃2

… 𝜃𝑛−1

… 𝜃𝑛

𝜃𝑛

0]

0]

where 𝑗 = 2𝑛 to cover all the uncertainty 𝜃 convex set axis.

424

Theorem A-1 [Therorem 5-1 in [42]]: For a given ellipsoid 𝜀(𝑃, 𝜌), assume there exist some 𝑘𝑎 ∈ ℝ𝑛+𝑛𝑓 and a positive number

425

𝜂 that satisfy that 𝜀(𝑃, 𝜌) ⊂ 𝒪(𝑘𝑎 ) and satisfy the following inequalities, (1 + 𝜂) (𝐴𝑖 + [𝑏] [− 𝑘𝑠𝑇 0

𝑇

𝑐𝑓 ]) 𝑃 (𝐴𝑖 + [𝑏] [− 𝑘𝑠𝑇 0

pro of

423

2

𝑘𝑔 𝑟(𝑘) 1+𝜂 0 𝑐𝑓 ]) + (‖ ‖ ( ) 𝜆𝑚𝑎𝑥 ([𝑏 𝜌𝜂 𝑓 𝑑̃𝑢 2

𝑇 𝑘𝑔 𝑟(𝑘) 1+𝜂 0 (1 + 𝜂) (𝐴𝑖 + [𝑏] 𝑘𝑎 𝑇 ) 𝑃 (𝐴𝑖 + [𝑏] 𝑘𝑎 𝑇 ) + (‖ ‖ ( ) 𝜆𝑚𝑎𝑥 ([𝑏 ̃𝑢 0 0 𝜌𝜂 𝑓 𝑑

𝑏 𝑇 0 0] 𝑃 [𝑏𝑓

𝑏 𝑇 0 0] 𝑃 [𝑏𝑓

𝑏 0]) − 1 ) 𝑃 ≤ 0

𝑏 0]) − 1 ) 𝑃 ≤ 0

(44)

∀ 𝑖 = [1 … . 𝑗]

Then the ellipsoid 𝜀(𝑃, 𝜌) is an invariant set for the system defined in (26) with the control signal defined in (27).

427

Proof : The proof is a direct application of the conditions in Lemma 1. The conditions in (24) and (25) are reformulated to handle

428

the state-space given in (42)

re-

426

Jo

urn a

lP

429

26

Journal Pre-proof

Highlights

431 •

A proposed discrete adaptive controller for the CPC is analyzed and the pitch constraints are taken into consideration

432

in the proposed controller.

433 •

Complete design, simulation and implementation of the proposed controller in a complete discrete-time environment

434

are done.

435 •

Simulation and experimental results of the proposed controller are compared with gain scheduled PI controller. For a

436

chosen sampling interval of the closed loop system, the proposed controller grantees the system stability and robustness.

pro of

430

Jo

urn a

lP

re-

437

27

Journal Pre-proof

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

pro of

438 439 440 441 442 443 444 445

Jo

urn a

lP

re-

446 447 448 449 450

28