f. Ing. Petr Noskievič, CSc. VŠB –

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FACULTY OF MECHANICAL ENGINEERING

DEPARTMENT OF CONTROL SYSTEMS AND INSTRUMENTATION

Analysis and Control Design of The Hydraulic Test Rig

Student : Jaishree Ananthkumar (ANA0006) Supervisor: Prof. Ing. Petr Noskievič, CSc.

Ostrava 2019

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VŠB – Technical University of Ostrava 4

Annotation

ANANTHKUMAR, Jaishree. Information System Creation: Diploma Thesis. Ostrava:

VŠB-Technical University of Ostrava, Faculty of Mechanical Engineering, Department of Control Systems and Instrumentation, 2019, 66 p. Thesis head: Prof. Ing. Petr Noskievič, CSc

This project includes Identification and Analysis, Creation of Mathematical model, Creation of Simulation model in Simulink environment and eventual control of the Hydraulic test rig with the help of MF634 multifunction I/O card. A large industry employs hundreds of controllers and each one of them must be tuned to be able to produce good control performance. Few of the tuning methods used in industries, in specific, the relay method is discussed. Subsequently, control program for automatic identification of Eigen frequency of the drive on the simulation model and then on the actual test rig is created. Using the critical parameters obtained, the controller parameters are calculated for the design of closed loop control.

Keywords

Hydraulic systems; Parameter identification, Modelling, Fluid flow control, Position control, Fluid dynamics, Control theory, Model Verification, Fluid forces, Closed loop control, Eigen frequency, self-oscillations, Relay, MATLAB-Simulink, PID control, State space representation, State observer

Anotace

ANANTHKUMAR, Jaishree. Tvorba informačního systému: diplomová práce. Ostrava:

VŠB – Technická univerzita Ostrava, Fakulta strojní, Katedra automatizační techniky a řízení, 2019, 66 s. Vedoucí práce: Prof. Ing. Petr Noskievič, CSc

Diplomová práce je zaměřena na analýzu a identifikaci, sestavení matematického modelu a vytvoření simulačního modelu v prostředí Simulink a řízení hydraulického zkušebního pohonu pomocí multifunkční I/O karty MF 634. V průmyslu se používají stovky regulátorů a každý z nich musí být seřízen tak, aby byl schopen zajistit požadovanou kvalitu řízení. Pro identifikaci hydraulického pohonu je použita a rozpracována metoda identifikace pomocí relé ve zpětné vazbě. Byl vytvořen řídicí program umožňující automatickou identifikaci vlastní frekvence hydraulického pohonu pomocí simulačního modelu a také na skutečném zkušebním pohonu. Pomocí změřených veličin byly vypočteny hodnoty kritického zesílení a byl proveden návrh zpětnovazebního řízení, stavové zpětné vazby a pozorovatele.

Klíčová slova

Hydraulické systémy, Identifikace parametrů, Modelování, Řízení průtoku tekutiny, Regulace polohy, Dynamika kapalin, Teorie řízení, Kontrola modelu, Fluidní síly, Řízení uzavřené smyčky, Vlastní kmitočet, oscilace, Relé, MATLAB-Simulink, PID řízení, reprezentace státního prostoru, státní pozorovatel

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Table of Figures

Figure 1: Structure of Hydraulic drive ... 12

Figure 2: Steps involved in this project (Jelali, 2003) ... 13

Figure 3: TP 511 test set ... 20

Figure 4:Test rig connected to Pc ... 22

Figure 5:Simulation model of hydraulic subsystem ... 23

Figure 6:Input to the simulation ... 24

Figure 7:Four-way spool valve (Jelali Mohieddine, 2003) ... 24

Figure 8:Simulation model of servo-valve ... 25

Figure 9:Simulation of Spool position ... 26

Figure 10:Simulation result of spool position ... 26

Figure 11:Simulation of the flow ... 27

Figure 12:Simulation result for flow through valve connected to one side piston rod cylinder ... 27

Figure 13:Simulation model of a cylinder ... 28

Figure 14:QA flow simulation model ... 29

Figure 15:QB flow simulation model ... 29

Figure 16:Graph of piston position in meter ... 30

Figure 17:Graph of Velocity of piston in m/s ... 30

Figure 18:Pressure changes in chamber A(blue line) and chamber B(red line) of the cylinder ... 31

Figure 19:Parameters used ... 31

Figure 20:Relay characteristics ... 32

Figure 21:Block diagram of relay test ... 33

Figure 22:Simulation model for relay testing ... 33

Figure 23: Output oscillations ... 34

Figure 24:Setup to run the test rig ... 35

Figure 25:Data from the system ... 35

Figure 26:Filtered data ... 36

Figure 27:Comparison after processing ... 36

Figure 28:Flow chart of automatic identification ... 37

Figure 29:Application for Automatic identification ... 39

Figure 30:(a) Identified Peaks;(b)Identified period ... 40

Figure 31:Dependence of frequency on position ... 41

Figure 32:Dependence of Damping ratio on position (polynomial order 1) ... 41

Figure 35:Simulation to run the test Rig ... 43

Figure 36:Automatic tuning ... 45

Figure 37:Result using P-controller ... 45

Figure 38:Result using PI controller ... 46

Figure 39:Linear model of system ... 47

Figure 40:Block scheme of state observer (State Space Methods, 2019) ... 48

Figure 41:Plot of the eigen values ... 49

Figure 42:Root locus for poles away from X-axis ... 49

Figure 43:Root locus of poles further left of the plane ... 50

Figure 44:State feedback ... 51

Figure 45:Result of simulation ... 51

Figure 46: Velocity of the model ... 52

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Figure 47: Control of simulation model ... 53

Figure 48: Position output... 54

Figure 49: Error dynamics ... 54

Figure 50: Estimated state variables ... 55

Figure 51:State space control with observer ... 56

Figure 52:State space model ... 56

Figure 53: Model of observer ... 57

Figure 54:Control of the test rig ... 57

Figure 55:State space control pole placement ... 58

Figure 56: (a) Position sensor output, (b) Position output from state space model ... 59

Figure 57:Position estimate ... 60

Figure 58: (a) Estimated Velocity output from the drive, (b) Velocity estimate from the state space model ... 60

Figure 59: (a) Estimated Acceleration output, (b) Acceleration estimate from state space model ... 60

Figure 60: (a) Error dynamics of the drive and observer, (b) Error dynamics of the state space model and observer ... 61

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List of Tables

Table 1: List of components from TP 511 kit ... 21 Table 2: Ziegler-Nichols table for controller tuning ... 44

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Abbreviations

𝜉_{𝑐𝑟𝑖𝑡} Damping ratio -

𝜉_𝑠𝑣 Damping ratio of servo-valve -

𝐴_𝑜𝑠𝑐 Amplitude of oscillations [V]

a Acceleration of the piston (𝑥̈) [m/s²]

A Chamber A -

A State space Matrix A -

B Chamber B -

B State space Matrix B -

b Viscous damping coefficient [Ns/m]

C Output state space matrix C -

CA Hydraulic capacitance of chamber A [m³/Pa]

CB Hydraulic capacitance of chamber B [m³/Pa]

Cd Flow Coefficient -

D State space matrix D -

e(t) Error over time t -

𝒆̇(𝑡) Error dynamics -

F Force on piston head [N]

FT Coulomb friction force [N]

𝐺_𝐿 Hydraulic conductance [R^-1]

𝐺_𝑅(𝑠) Controller gain -

K Bulk modulus [Pa]

K Feedback gain matrix -

𝐾_𝑀 Cylinder gain [m^-2]

Kp P gain -

Ksv Valve gain [V^-1]

𝐾_𝑄𝑢 Valve gain [m3/s]

𝑳 Observer Gain matrix -

M Magnitude of relay -

m Mass of actuator [kg]

P Pump -

p0 Pressure from pump/source [Pa]

pA Pressure in chamber A [Pa]

pB Pressure in chamber B [Pa]

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pT Pressure at tank [Pa]

QA Flow in chamber A [m³/s]

QB Flow in chamber B [m³/s]

QL, QLA, QLB Internal and external leakage flow [m³/s]

Rcrit Critical gain [V^-1]

SA, SB Surface area of chamber A and chamber B [m²]

SISO Single input single output -

T Tank -

Tcrit Critical time period [s]

Td Derivative time constant [s]

Ti Integral time constant [s]

Tsv Time constant of servo valve [s]

usv Valve control input voltage [V]

𝒖(𝑡) Input -

v Velocity of the piston (ẋ) [m/s]

VA0 Dead volume of piston chamber A [m³]

VB0 Dead volume of piston chamber B [m³]

ωcrit Angular Critical (Eigen) frequency [rad/s]

x Position of piston [m]

xsv Spool position [m]

xsv0 Spool overlap [m]

𝒙(𝑡) Output position -

𝒙̇̂(𝑡) Estimate of 𝑥̇(𝑡) -

𝒙̂(𝑡) Estimate of 𝑥(𝑡) -

𝒙̇(𝑡) First derivative of output -

𝒚(𝑡) Output of the state space model -

𝒚̂(𝑡) Estimate of 𝑦(𝑡) -

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1 Introduction

Hydraulic servo-drives are used in many industrial plants, because they can produce large forces and torques with high speeds. Hydraulic systems are commonly a part of mechatronic systems; they are typically coupled to mechanical systems and automatic control systems (Noskievič, 2013)

A typical hydraulic system is an arrangement of components which are connect to each other to provide a desired form of hydraulic power transfer. The basic structure of a hydraulic system consists of

• Hydraulic power supply,

• Control elements (such as valves, sensors, controller etc.),

• Actuating elements (cylinder and/or motors), and

• Other elements (pipelines, measuring devices, etc.).

Figure 1: Structure of Hydraulic drive

A large industrial process may employ hundreds of these systems with controllers, and these must be tuned exclusively to match the process dynamics in order to provide good and robust control performance. (Jelali, 2003)The tuning procedure, if done manually, is very tedious and time consuming. The resultant system performance mainly depends on the experience and the process knowledge the engineers who helped create it. It is recognized that in practice, many industrial control loops are poorly tuned (Åström, 1984).

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Automatic tuning techniques thus draw more and more attention of the researchers and practicing engineers.

To make progress on the problem as mentioned above, it is first necessary to gain an understanding of how the process operates. This understanding is typically expressed in the form of a mathematical model which describes the steady state and dynamic behavior of the process. Modelling of such systems is required for the design of state observers, filters, and controllers. However, the rather complex structure of such drive systems makes it difficult to develop suitable, preferably low-order models of the dynamic of the plant.

The simulation of the mathematical models of hydraulic systems is done by the control of the fluid flow, control of the fluid reserve in the tanks and by the control of the motion of the mechanisms with hydraulic drives. Realization of the mathematical model can be achieved through MATLAB-Simulink, where with the use of basic blocks the mathematical model can be turned into a simulation model of the systems. The further steps will be to tune the controller using outcome of the Eigen frequency identification and implement this onto the actual test Rig installed in the laboratory.

During the course of this project we employ certain steps to complete the assigned tasks and achieve set goals. The steps depict the flow of the whole process of the project and it somewhat resembles to grey-box testing employed in industries.

Figure 2: Steps involved in this project (Jelali, 2003)

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2 Hydraulic closed loop control drives

Due to increase in computer power and developments in control theory, expectations regarding modelling of the non-linear dynamic behavior of hydraulic servo-systems have increased. More detailed descriptions of dominant non-linear characteristics and relevant dynamics over wider frequency ranges have to be taken into account. The main non- linearities in hydraulic systems arise from the compressibility of the hydraulic fluid, the complex flow properties of the servo-valve and the friction in the hydraulic actuators. They depend on factors, which are difficult to measure or estimate online, such as oil bulk modulus, viscosity and temperature.

2.1 Applications

Fluid power technology which consists of hydraulic actuator, servo-valves etc., to provide large forces and torques with high speeds, were prevalent and developed mainly from the beginning of the 20th century. The first generation of these hydraulic drives consisted of some flow control device driving the hydraulic actuator in an open loop manner. The applications were limited to hydraulic presses, jacks and winches. (C. Goodwin, 2000) The years following World War II saw an increase in the use of servo-control techniques, which paved the way for accurate closed loop motion control. The advancement in the control techniques at that time opened a wide range of applications we still find in industry today.

A few places where Hydraulic servosystems are used are (C. Goodwin, 2000):

• where relatively large forces or torques are required (industrial presses, mobile lifting, digging, material handling equipment, etc.),

• where fast, stiff response of resisting loads is needed (machine tool drives, flight simulators, rolling mills, etc.),

• where accurate control of response is necessary (control surfaces of aircraft, machine tool slides, industrial robots, etc.),

• where manual control of motion involving substantial forces is essential (heavy machinery, aircraft controls, automotive power steering, etc.), and

• as the final actuator subsystem in complex automatically controlled situations (electro- hydraulic flight simulators, industrial robots, fatigue and other programmable test rigs, theatre stage control, etc.).

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2.2 Control systems

However, the rather complex structure of such drive systems makes it difficult to develop suitable, preferably low-order models of the dynamic of the plant. The models are needed for the design of state observers, filters, and controllers. The simulation of the mathematical models of hydraulic systems is done by the control of the fluid flow.

Control of a system deals with finding technically, environmentally and commercially feasible ways of acting on a technological system to control its outputs to desired values at the same time as ensuring a desired level of performance.

The fundamental concept in control engineering is the idea of feedback also known as closed loop control.

The feedback to a system to achieve desired control of the system in question can be generalized by these few steps. (Aström, 1997)

1) Identifying desired state/output of the system

2) Measuring actual system variables of the interested system by the use of sensors 3) Comparing the actual state of the system to the desired state of the system 4) Computing corrective action to bring the actual states to the desired state

5) Applying the computed action with the help of actuators until desired state is reached.

2.3 Types of control

Control of closed loop systems in industries is achieved through myriad control structures.

Few of them will be investigated further in this project

• Classical PID controller

• State space observer-based estimation 2.3.1 PID controller

This specific control structure has become almost universally used in industrial control as a result of its robustness and simplicity. However, simplicity of these controllers is also their downfall, since it limits the range of plants that they can control satisfactorily. Indeed, there exists a set of unstable plants which cannot even be stabilized with any member of the PID family. Nevertheless, the surprising versatility of PID control ensures continued relevance and popularity for this controller. (Ziegler, 1942)

Over time, various tuning methods have been invented to tune PID controller of a system to achieve stability and desired control. (Åström, 1995)

Two of the majorly used tuning methodologies are as follows:

1. Conventional Tuning Methods Ziegler and Nichols proposed in 1942 two different tuning strategies for PID controllers. These methods are successful but

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not extensively used due to its unpredictable execution thus leading to unintended disturbance in the process.

2. Tuning with Relay Method The relay method presented as an alternative to the conventional method of Ziegler-Nichols for closed loop, in the identification of model systems by Åström KJ, Hägglund T (Åström, 1984)in 1984, seems to be the solution. It has the advantage to generate and maintain controlled oscillations.

This method is highly popular due to the simplicity of identification and calibration of a system. Relay method is very efficient in determining the critical gain Rcrit and critical (Eigen) frequency ωcrit.

2.3.2 State-space controller

In a SISO system, a simple representation has sufficient effect on the desired output.

However, when we consider a more complex system with multiple inputs and multiple outputs, it is best to use special model representation. One of the most flexible and useful structures is the state space model. (Víteček, 2013) (C. Goodwin, 2000)

A state space representation is a valuable and frequently used tool for plant modeling. State variables form a set of inner variables which is a complete set, in the sense that, if these variables are known at some time, then any plant output, y(t), can be computed, at all future times, as a function of the state variables and the present and future values of the inputs.

For linear, time invariant systems the state space model is expressed in the following equations:

Continuous time systems:

𝒙̇(𝑡) = 𝑨𝒙(𝑡) + 𝑩𝒖(𝑡) (2.1)

𝒚(𝑡) = 𝑪𝒙(𝑡) + 𝑫𝒖(𝑡) (2.2)

Stability and natural response characteristics of the system can be studied from the eigenvalues of the matrix A. (Luenberger, 1972)

A transfer function can always be derived from a state space model also the vice-versa is possible, i.e., a state space model can be built from a transfer function model. However, only the completely controllable and observable part of the system is described in that state space model. Thus, the transfer function model might be only a partial description of the system.

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The design of this controller is done under the assumption that all system variables are available for control (State space feedback controller) or designing an observer (state space observer) to estimate the values of the variables that are not available for measurement and control as easily. This is discussed in detail in the subsequent chapters of this project.

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3 Modelling and Simulation of Mathematical Models

Owing to increase in understanding of steady state and dynamic behavior of fluid power systems in recent years the creation of computer based mathematical model seems like an attractive possibility for various reasons. By creating a model of the dynamic system, we can describe the dynamic properties of the system. Modelling is not only used to understand the behavior of system but also gives the ability to simulate faults and observe changes in system that would not be possible to detect without investing in expensive components.

The description is brought about by using differential equations representing the properties of the system. Physical laws can be used to describe the dynamic properties and the equations. The model enables us to study system transients and steady state performance.

(Watton, 2007)

Let’s talk about complexity of the models:

• Model accuracy required for control system design is typically simpler than the model accuracy used for system simulation.

• Simpler models are modelled by using a few features such as:

o Overlooking some physical phenomena, o Approximate linearly nonlinear characteristics

• For the design of a control system:

o A preliminary simplified model is constructed for conceptual design of the process.

o A more accurate model is then used for controller design and parameter determination.

After modelling of the system to satisfaction the next step is to simulate it accordingly.

Simulation is usually done using specific numerical method from a plethora of numerical methods available in the market. The results are typically in form of graphs showing the response of the model generated. Nowadays there are simulation programs available in the market that employ these numerical methods, making the process of model solution much easier. The process is pretty simple and straight forward. A simulation model is created from the mathematical model in the simulation program, the condition of the solution are defined and the solution is on track. Subsequently the plots of system variables show the system response.

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3.1 MATLAB – Simulink

MATLAB is a very effective tool for engineering work in a lot of applications in various fields. It is an integrated environment for scientific calculations, modelling and simulation of dynamic systems, analysis of the dynamic properties of the systems, design of controllers, measurement, signal analysis, design of control systems. (Noskievič, 2013) (Jelali, 2003) Simulink is a block diagram environment for multidomain simulation and Model-Based Design. It supports system-level design, simulation, automatic code generation, and continuous test and verification of embedded systems. Simulink provides a graphical editor, customizable block libraries, and solvers for modeling and simulating dynamic systems. It is integrated with MATLAB, enabling you to incorporate MATLAB algorithms into models and export simulation results to MATLAB for further analysis.

The blocks are structured in model libraries and cover the need for modelling of dynamic systems independent on their physical domain. The standard library can be extended by the use of different toolboxes which are system identification, signal processing etc., to name a few. The models in these libraries are based on the physical modelling, and their structure is based on the energy flow through the system.

3.2 Simulation

Simulation is the process of constructing a model of a system which contains a problem and conducting experiments with the model on a computer for a specific purpose of experimentation to solve the problem. (Watton, 2007)

Credibility of simulation results not only depends on model correctness, but also is significantly influenced by accurate formulation of the problem. Therefore, validation, verification, and testing (VV&T) techniques must be employed throughout the life cycle of a simulation study starting with problem formulation and concluding with presentation of simulation results.

A model is a representation of a system, concept, problem, or phenomena. It can have inputs, parameters, and outputs.

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4 Analysis of The Structure of The Hydraulic Drive of The Test Rig

The hydraulic test rig that is to be used during this project is illustrated in the image below.

It is a hydraulic training equipment TP 511: Basic closed loop hydraulic training from Festo.

Figure 3: TP 511 test set

It has a PC dedicated to running this test rig assembly with MATLAB. Hydraulic closed- loop control circuits are normally operated with continuous valves which provides the fluid flow to the hydraulic actuator. The control of the actuator can be achieved using a feedback from a system variable like position, velocity, force etc.

The components of the training kit from Festo didactic catalog 2015/2016 are as follows:

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VŠB – Technical University of Ostrava 21 Table 1: List of components from TP 511 kit

1x PID controller 1x State controller 2x Pressure sensor 1x Pressure gauge 1x Hydraulic motor 1x Flow sensor 1x Pressure filter 1x Flow control valve 1x Shut-off valve

2x 4-way distributor with pressure gauge.

2x T-distributor

1x 4/3-way regulating valve 1x Linear drive

2x Weight, 5 kg, for linear drive

In this setup, the components are connected in a feedback form to control the position of the actuator. The position value is obtained from the position sensor and the input is the control voltage provided to the valve for the flow fluid power.

The control of the system can be achieved by using the PID controller that is available in the kit and even by using MATLAB Simulink to give command to the system. This is achieved by installing a Multifunction I/O card for interfacing the real-world signals to the PC.

The control program, which is a MATLAB script, is stored in the PC. The card is installed by connecting it to PCI Express expansion slot available in the PC. The card has 8-channel fast 14-bit A/D converter and 8 independent 14-bit D/A converters for processing of digital signals from PC and analog signals back and forth from the system. The input and output ranges are ±10V.

These signals are connected via 8-bit digital input port and 8-bit digital output port which are TTL compatible. The card is designed for standard data acquisition and control applications and optimized for use with Real-Time Windows Target for Simulink.

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VŠB – Technical University of Ostrava 22 Figure 4:Test rig connected to Pc

The figure perfectly sums up the functions of the MF 634 card. It is the medium between the control signals from the PC which are digital signals and the signals to and from the system which are analog signal. The signals from the system are from the sensors for accurate controlling of the system and the input to the system is via the control signal which is analog voltage signal to the valve.

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5 Simulation Modeling of hydraulic drive

A mathematical model is constructed from basic physical laws, such as Newton's laws and equilibrium equations. The result of this is generally a non-linear dynamic (simulation) model of the hydraulic servosystem, which includes components like actuator, servo-valve, pipelines, and power supply.

This model is used to perform various simulations, with realistic physical parameters, so that structural insight into the relevant dynamics and non-linearities of the system can be gained.

5.1 Simulation of Hydraulic Subsystems

In order to reduce the complexity of the modelling of the system, it is useful to create a few subsystems:

1. Servo-valve: which transfers the control input into the oil flows, depending on the actuator pressures. Its actual behavior is generally not ideal.

2. Hydraulic actuator including load mass: with the driving oil flows QA and QB from the valve and the external force F as inputs, and correspondingly the actuator pressures pA and pB, and position x and velocity (ẋ) v as outputs.

The complete hydraulic system can be modelled using these two main subsystems which are working with inputs such as u, pressures pA, pB, p0, pT

Figure 5:Simulation model of hydraulic subsystem

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VŠB – Technical University of Ostrava 24 Figure 6:Input to the simulation

The input to the simulation is given as 0V to 5V and then back to 0V.

5.2 Mathematical model of a Servo-Valve

Flows through valve orifices are described by the orifice Equation, which takes the direction of the pressure drop (flow direction) into account (Noskievič, 2013)

𝑄_𝑖 = 𝐶_𝑑. 𝑎𝑏𝑠(𝑥_𝑠𝑣± 𝑥_𝑠𝑣0). √𝑎𝑏𝑠(∆𝑝_𝑖). 𝑠𝑔𝑛(∆𝑝_𝑖) for i= PA, AT, PB, BT. (5.1)

Figure 7:Four-way spool valve (Jelali Mohieddine, 2003)

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The decision of flow of fluid to chambers is determined by using the value of (𝑥_𝑠𝑣 ± 𝑥_𝑠𝑣0) component. If this value is equal to or over 0, the fluid flow is from pump to chamber A and then flows back to tank from chamber B. If the value of the term is less than 0, the fluid from is from pump to chamber B and then flows back to tank from chamber A.

5.3 Simulation model of servo-valve

Using the mathematical model obtained from various published literature we can model a non-linear model of a servo valve in MATLAB Simulink environment. We make use of various blocks available in the library to model the structure of valve. The following Figure 8 illustrates the simulation model of the valve.

Figure 8:Simulation model of servo-valve

The model can be described in two parts:

First part describes the spool position 𝑥_𝑆𝑉of the valve, given by mathematical model

𝑇_𝑆𝑉²𝑥̈_𝑆𝑉+ 2𝜉_𝑆𝑉𝑇_𝑆𝑉𝑥̇_𝑆𝑉+ 𝑥_𝑆𝑉 = 𝐾_𝑆𝑉𝑢_𝑆𝑉 (5.2)

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VŠB – Technical University of Ostrava 26 Figure 9:Simulation of Spool position

The result of this part of simulation is shown in the Figure 10 below:

Figure 10:Simulation result of spool position

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The spool is opened from -1 to 1 at which the valve position is at the absolute end. The value 1 indicates that the valve is completely open. The sign indicates the direction of the movement of spool. Value 0 indicates that the valve is completely closed.

The second part of the simulation model is the flow through the valve which can be modelled as in the Figure 11.

Figure 11:Simulation of the flow

The result of this part of simulation is described in terms of graphical view.

Figure 12:Simulation result for flow through valve connected to one side piston rod cylinder

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The blue line represents QA, that is flow through the control edge P-A into the chamber.

The red line represents QB, that is flow through the control edge B-T from the chamber.

This is made into a subsystem to as to model the whole hydraulic system and then fed to another subsystem that is the simulation model of a hydraulic cylinder.

5.4 Simulation model of cylinder

The equation of piston motion governing the load motion arises by applying Newton's second law to the forces on the piston. The resulting force equation is (Noskievič, 2013) 𝑚𝑥̈ + 𝑏𝑥̇ = 𝑆_𝐴𝑃_𝐴− 𝑆_𝐵𝑃_𝐵− 𝐹 − 𝐹_𝑇𝑠𝑔𝑛(𝑥̇) (5.3) The simulation model is created using this equation and various blocks available for mathematical description in Simulink environment.

Figure 13:Simulation model of a cylinder

Pressures can be described using the following differential equations:

𝑝_𝐴̇ = ¹

𝐶𝐴(𝑄_𝐴− 𝑄_𝐿𝐴− 𝑆_𝐴𝑣 − 𝑄_𝐿) (5.4) 𝑝_𝐵̇ = ¹

𝐶𝐵(𝑆_𝐵𝑣 − 𝑄_𝐵− 𝑄_𝐿𝐴− 𝑄_𝐿) (5.5) Also, leakage can be given by

𝑄_𝐿 = 𝐺_𝐿∗ Δ𝑝 (5.6)

Calculation of the pressures is done employing initial conditions pA0 and pB0 into the simulation model. This is to help start off simulation from a set point rather from 0 i.e., similar to what we observe in real systems.

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VŠB – Technical University of Ostrava 29 Figure 14:QA flow simulation model

Figure 15:QB flow simulation model

The hydraulic capacities can be described using the equations:

𝐶_𝐴 =^𝑉^𝐴0^+𝑆𝑥

𝐾 (5.7)

𝐶_𝐵= ^𝑉^𝐵0^{+𝑆(ℎ−𝑥)}

𝐾 (5.8)

5.5 Result of simulation of cylinder

The simulation output of this cylinder model is in terms of piston position x (in Meter), velocity of piston v (in m/s) and the pressures pA and pB (in Pascal).

The response of the system is slower and different when the piston must retract due to the asymmetric characteristic of the one-sided piston rod cylinder. The input to the system is also shown in graphical form in Figure 6. All the outputs are read in form of graphical representation. The obtained graph is as illustrated in the following figures:

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VŠB – Technical University of Ostrava 30 Figure 16:Graph of piston position in meter

Figure 17:Graph of Velocity of piston in m/s

The beginning error is due to the initial parameters that are set.

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VŠB – Technical University of Ostrava 31 Figure 18:Pressure changes in chamber A (blue line) and chamber B (red line) of the cylinder

The parameters used are all in SI units and are selected as shown in the figure below of MATLAB script.

Figure 19:Parameters used

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6 Identification method of Eigen Frequency of the Hydraulic Cylinder

The natural frequency of the hydraulic cylinder varies with the piston position and influences the controller tuning. Their progression in dependence on the piston position can be useful for the design of the controller of the position closed loop-controlled system. We describe the identification of the natural frequency of the hydraulic cylinder using the self-excited oscillations by the arranging a nonlinear element in the feedback or in place of the controller.

6.1 Implementation of eigen frequency identification

By placing a non-linear element in the feedback or in place of the controller we can start off with observing sustained oscillations in the process value. The non-linear element must have relay characteristics described in the Figure 20.

Figure 20:Relay characteristics

Arrangement of the test is as shown in the block diagram (Figure 21) with the relay placed in the place of the controller. The test can also be run by using the relay in the feedback loop and switching off the controller. The simulation model of the hydraulic system obtained in the previous chapters is used and the controller is removed, and a relay of magnitude M is placed in the feedback loop.

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VŠB – Technical University of Ostrava 33 Figure 21:Block diagram of relay test

The simulation model is as shown as in Figure 22

Figure 22:Simulation model for relay testing

The test on the simulation model can be carried out by following the steps described below:

1. Bring the system to steady state.

2. Switch on the relay with magnitude M= 2 for example, with switch on and switch off point set close to zero (if relay is used in place of controller) or set to the value of the control input (if relay is placed in the feedback)

3. As the output crosses the Set Point (control input value), the manipulated input is switched to the opposite value (-2).

4. Repeat the steps until sustained oscillation is obtained. Read the critical period Tcrit and amplitude of the oscillations Aosc from the obtained output and compute Kcrit from Equation 6.1.

𝑅

_{𝑐𝑟𝑖𝑡}

=

^4𝑀

𝜋𝐴_𝑜𝑠𝑐

(6.1)

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The resulting oscillations are represented in a graphical form and is as shown in Figure 23 when the set point(position) is given as 0.2m.

Figure 23: Output oscillations

6.2 Setup of the test Rig

For actual implementation of the control programme on hydraulic test rig, there are a few steps to be accomplished

• First the simulation model is to be replaced by the actual system.

• The program in MATLAB Simulink needs to be in communication with the hydraulic rig for the controlling of the system. This is accomplished by using the MF 634 multifunction I/O card, which is used for connecting PC compatible computers to real world signals.

First the I/O card is installed in the PC to run using MATLAB Simulink. It converts real world signals i.e., Voltage to equivalent digital signal. The converted signal is then used to implement eigen frequency identification setup in Simulink with a digital relay placed in the feedback loop and the controller is turned off.

The Simulink setup is as illustrated below:

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VŠB – Technical University of Ostrava 35 Figure 24:Setup to run the test rig

6.3 Data logging

The position sensor values (from the block Analog input-position sensor) are stored to the base workspace of MATLAB for the set desired position. The obtained data are then used to identify the critical parameters.

Figure 25:Data from the system

As we can see from the figure above that the data obtained from the sensor is extremely noisy making the identification of the critical parameters from the oscillations nearly impossible without any postprocessing. The noise in the signal is expected as the laboratory is surrounded by heavy machinery causing errant magnetic fields which leads noisy measurement.

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The digital filters available in MATLAB is not ideal for processing of the data to remove noise using filters dynamically. This dynamic filtering of the signal leads to change in the characteristics of the signal rendering the identification process and subsequent calculations unusable. When we do the identification process, it also includes the dynamics of the filter that was used to the remove noise. Thus, compromising the results we need to obtain.

Solution to this is to log the data from the position sensor to the base workspace and then process it to remove noise, without causing changes to the signal characteristics. A simple digital lowpass filter is used to achieve this. This filter is able to produce mostly noise free signal output.

The lowpass filter used has a passband frequency of 100Hz with a sampling frequency of 10kHz.

Let us compare the signal before filtering and after filtering. The figure below illustrates the differences plainly.

Figure 26:Filtered data

Also, to help visualize the difference between the two, the signals are plotted on the same graph as seen in the figure

Figure 27:Comparison after processing

The blue signal is the signal from the sensor and the red is the same filtered signal of the sensor.

As we infer, this filter is very satisfactory for filtering out the noise and thus integral for the automatic identification of Eigen frequency of the system.

This filter is unavailable in the MATLAB version that is installed in the Laboratory as it was introduced in the signal processing toolbox in 2018a version; thus, the identification was done on the data that has been saved after running the test Rig with the relay.

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7 Design of control Algorithm for Automatic Eigen frequency Identification

The process of identification of the eigen frequency and other critical parameters required for the calculation of controller tuning parameters is to be automated to make the process faster and less tedious. The output obtained from the test rig is very noisy thus demanding filtration process and sensitive identification of the peaks and the zero-crossing of the oscillation signal.

The steps taken in the control algorithm is as illustrated below.

Figure 28:Flow chart of automatic identification

The flow chart indicates the most significant steps taken to identify the amplitude and the time period of the oscillations obtained from the system.

The output of this program is ultimately the Damping ratio, Critical gain and Eigen Frequency.

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7.1 Calculations

These are calculated using the following formulae (Noskievič, 2005):

Critical gain:

𝑅_{𝑐𝑟𝑖𝑡} = ^4𝑀

𝜋𝐴_𝑜𝑠𝑐 (7.1)

Where M and Aosc can be obtained by measurement of the damped oscillations attained previously.

Critical Frequency:

𝜔_{𝑐𝑟𝑖𝑡} = ^2𝜋

𝑇_{𝑐𝑟𝑖𝑡}√1− 𝜉𝑐𝑟𝑖𝑡2 (7.2)

This expression can be calculated using the damping ratio value which is given by the equation below. Where Tcrit is also obtained from the damped oscillations attained during the course of identification. The 𝜔_{𝑐𝑟𝑖𝑡} is in terms of rad/s.

Damping ratio:

𝜉_{𝑐𝑟𝑖𝑡} = ^𝑅^{𝑐𝑟𝑖𝑡}^𝑇^{𝑐𝑟𝑖𝑡}

√16𝜋²+𝑅_{𝑐𝑟𝑖𝑡}²𝑇_{𝑐𝑟𝑖𝑡}²

(7.3) The damping ratio value can be very low as to 0.05-0.6 range

These are then used to tune and design the closed loop controllers for the system.

7.2 Application for identification

The identification of eigen frequency can be done using the simulation model created using the data sheets available for the test rig, the test rig in the lab via the MF card interface with the computer and lastly we can save the oscillatory data onto a file and use this data for identification and to find controller parameters.

To make this choice easier to handle, an application is created in the application designer in MATLAB.

The application looks like the figure below, consisting of three buttons for the type of identification model to use and a plot figure to show the dependence of frequency with respect to change is position of the actuator.

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VŠB – Technical University of Ostrava 39 Figure 29:Application for Automatic identification

7.1.1 Use simulation model

The Simulink model described in the previous chapters is used to obtain oscillations for 10 position values between 0.01m to 0.2m which is the whole stroke length of the model. The peaks are detected, and the critical parameters are calculated after each value of position.

A function called findpeaks which is available under digital signal processing library in MATLAB is used to find the peak values of the oscillations and the time of occurrence of these peaks in positive and negative direction.

[pospks,postimes] = findpeaks(x, tout) [negpks,negtimes] = findpeaks(-x, tout)

Where pospks and negpks contain the values of the positive and negative peak values respectively. The times of these peaks are saved in postimes and negtimes.

7.1.2 Data obtained from the test rig

The data are obtained from the implementation of this tuning method for every position from 1-9 which is at 2cm intervals of total stroke of the actuator, which is 20cm.

The system is first brought to a steady state and then the controller is switched off and the ideal relay is switched on, to obtain sustained damped oscillations. We discussed about the handling of the noisy signal obtained from the system.

Since we need to identify the peaks and the time period, we use signal processing toolbox in MATLAB to accomplish this. The red and blue triangles are the positive peaks and negative

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peaks of the obtained oscillating signal. The green stars are the threshold crossing for finding time period of the signal.

The identified peaks are illustrated as in figure

Figure 30:(a) Identified Peaks; (b) Identified period

The natural frequency of the drive which limits the controller tuning depends on the piston position. The course of the frequency with respect to the changes in the piston position can be seen in the figure below. The mean amplitude value is 0.3436 V and the time period mean value is 0.0550s.

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VŠB – Technical University of Ostrava 41 Figure 31:Dependence of frequency on position

The frequency change follows a trace of the second order polynomial with a standard deviation of 0.1098.

The dependence of Damping ratio on the Piston position can also be seen in the following figures. The approximation was done for polynomial of order 1, 2 and 3 to see which is the best possible fit.

Figure 32:Dependence of Damping ratio on position (polynomial order 1)

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VŠB – Technical University of Ostrava 42 Figure 33:Dependence of Damping ratio on position (polynomial order 2)

Figure 34:Dependence of Damping ratio on position (polynomial order 3)

The change in damping ratio follows a trace of the third order polynomial with a standard deviation of 2.444.

7.1.3 Use the test Rig

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This button enables the use of the test rig for automatic identification of the Eigen frequency of the rig for 8 piston position. Since the program to identify the peaks and hence the frequency calculation involves usage of filters and functions that are not available in the MATLAB version that is installed on the Lab PC, this option is not used.

Figure 35:Simulation to run the test Rig

The switches are used to switch the controller with the relay placed in the feedback as the simulation time reaches 1 second and the piston reaches steady state. When the relay is switched ON, the oscillations can be observed on the output position. The desired value is then set to 0 at 3s to stop the oscillations. A for loop is created in MATLAB script to run this simulation for 9 iterations with increments of 1 to the control input staring from value 1 to 9.

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8 Design of Closed Loop Control

As discussed in the beginning, the critical parameters obtained by the Eigen frequency identification can be used to design a controller for closed loop control.

The controller that are used:

1. Classical PID controller

2. State space feedback with observer

8.1 Parameters calculation of PID controller

The gain and the time-period estimated previously is used to get the proportional gain Kp, Integral time-constant Ti and derivative time-constant Td for the tuning of the controller. The controller action can be described using the formula (Víteček, 2013):

𝑢(𝑡) = 𝐾_𝑝𝑒(𝑡) +^𝐾^𝑝

𝑇𝑖∫ 𝑒(𝑡)𝑑𝑡₀^𝑡 + 𝐾_𝑝𝑇_𝑑 ^𝑑

𝑑𝑡𝑒(𝑡) (8.1)

The table to calculate these values are given by Ziegler–Nichols method:

Table 2: Ziegler-Nichols table for controller tuning

Type of controller Kp Ti Td

P 0.5Rcrit ∞ 0

PI 0.45Rcrit Tc/1.2 0

PD 0.4Rcrit ∞ 0.05Tc

PID 0.6Rcrit 0.5Tc 0.125Tc

An application to find PID parameters for each type of controller and for desired position was constructed for the ease of implementation. It can be done so as to automate control as well as identification.

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VŠB – Technical University of Ostrava 45 Figure 36:Automatic tuning

The control button is pressed to apply the calculated values of the controller chosen.

Result of the calculation can be seen in the plots below for desired position=3

Figure 37:Result using P-controller

The desired position is reached within 0.5s, which is good response time with no overshoot.

The same response is garnered from both P and PD controller.

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The signal obtained is overly noisy as the sensor measurement is affected by the magnetic field present in the lab due to heavy machinery around.

Figure 38:Result using PI controller

Since the system is an integrating system, the introduction of the integrating part to the controller causes overshoot. The response is the same in both PI and PID controller.

8.2 State space representation

As we discussed in the beginning that, state space controller can be used to control the states of the system, of which there can be several. It is developed to control complicated and complex non-linear systems.

The system to be represented using state space representation has to be mathematically modelled accurately and described in the form of matrices. The hydraulic drive can be represented as state space model after linearizing the whole system.

v

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VŠB – Technical University of Ostrava 47 Figure 39:Linear model of system

The valve dynamics have been neglected here due to the faster dynamics of the cylinder. It can be modelled as a proportional system using the flow gain KQu. The obtained eigen frequency is used to describe the cylinder.

The open loop connection of the control valve and hydraulic cylinder can be represented as:

[ 𝑥̇

𝑣̇

𝑎̇

] = [

0 1 0

0 0 1

0 ⁻¹

𝑇_𝑀²

−2𝜉𝑀 𝑇𝑀

] [ 𝑥 𝑣 𝑎

] + [ 0 0

𝐾_𝑄𝑢𝐾_𝑀 𝑇_𝑀²

] 𝑢 (8.2)

𝑦 = [1 0 0]𝑥 (8.3)

As we can see we have reduced our system to a state space representation which can be fulfilled using the parameters that was estimated in the previous chapters.

8.2.1 State space observer

Practically in most of the real-life applications, measurement or status of all the state variable of the system are not available. In this case, it is general practice to make use of state observer A state observer is used to estimate the values of an unmeasured states from other states, which are measurable and that are interrelated to it. If a system is observable, it is possible to fully reconstruct the system state from its output measurements using the state observer.

The observer of the system acts as a virtual sensor to measure all the unmeasured variables of the system. We concentrate on designing a Luenberger observer of our system which estimates the real system variables using the form (Víteček, 2013)

𝒙̇̂(𝑡) = 𝑨𝒙̂(𝑡) + 𝑩𝒖(𝑡) + 𝑳(𝒚(𝑡) − 𝒚̂(𝑡)) (8.4) 𝒚

̂(𝑡) = 𝑪𝒙̂(𝑡) (8.5)

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The error dynamics are expressed as poles of A-LC

𝒆̇(𝑡) = 𝒙̇(𝑡) − 𝒙̇ ̂ (𝑡) (8.6)

The matrix L is the gain matrix of the observer that has values which are multiples of the real valued part of the poles selected. The higher the L matrix values are the faster is the observer behavior, which is essential for fast estimation of the real state variables. These poles are placed on the left side of the plane as much as possible.

Care should be taken as to not give an unrealistic values of gain matrix. Typically, the values are three or five times the pole value.

Output of the observer system is fed back to the input of both the observer and the plant through the gains matrix K. The figure below describes the connection of the observer to the system which is represented using state space model.

Figure 40:Block scheme of state observer (State Space Methods, 2019)

8.2.2 State space feedback

Having done representing the system in the form of state space model, we can find the eigen values or poles of the system. For our system, the poles are placed are seen in the figure below:

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VŠB – Technical University of Ostrava 49 Figure 41:Plot of the eigen values

Placement of poles further to the left as possible stabilizes the system. The figures below describe this phenomenon in an illustrative manner:

Figure 42:Root locus for poles away from X-axis

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VŠB – Technical University of Ostrava 50 Figure 43:Root locus of poles further left of the plane

Now that the poles are placed at desired locations, it is possible to calculate a set of constant gains, one for each state variable, such that the feedback to the system from the states, through the gains results in a closed loop with poles in the desired locations.

In practicality, one rarely has access to all the system variables at all times. Thus, making way to the concept of employing an observer.

8.3 Implementing state feedback

Since we have access to all the state variables in the simulation model created during the course of this project, we will design a state feedback controller.

The poles of the system are identified and are placed on the left plane. The Gain Matrix K of state feedback is computed and each of the gain value used with one state variable.

The system is described by using the following matrices.

𝐴 = [

0 1 0

0 0 1

0 −3664.1 −0018.4 ]

𝐵 = [ 0 0 6378.4

]

𝐶 = [1 0 0]

𝐷 = 0

The equation of the system is given by

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Where s is the gain, w is the set point, u is the control input, K is the controller feedback gain and x is the output.

𝑢 = 𝑠𝑤 − 𝐾𝑥 (8.7) The state feedback gain K is given by:

K= [2.8723 0.0144 0.0008]

The other feedback gain is the scaling block to convert value of output in meter to volts.

Which is split as K1, K2 and K3 for individual gains of the system variables.

Figure 44:State feedback

For the set point s=0.1

The feedback gain was used to control the piston position when they are multiplied to respective state variables and then added.

Figure 45:Result of simulation

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The required position is reached before 1s but it is pretty slow response. The output here corresponds to the piston position.

Figure 46: Velocity of the model

8.4 Implementing state observer

First it is necessary to check the controllability and the observability of the described system to design the observer. A system is controllable if and only if the system is observable.

The code to check the controllability of the system in MATLAB is as follows:

P=rank(ctrb(A,B)) Q=rank(obsv(A,C))

The system is controllable and observable if the result of the above code is full rank value.

In our case, if the value returned is 3.

The result of the above code was indeed 3, thus enabling us to move forward with the design of the observer.

The eigen values obtained by the code eig(A)is used as starting point for placement of the poles. The poles are further moved towards the left to obtain better control, stability and faster response. The feedback gain matrix K is calculated for the poles chosen.

The observer is constructed using the equation described previously and the poles of the observer is taken as 5 times the dominant poles of the state space feedback.

Now with all the components and parameters calculated, the state space model and the observer are built in Simulink.

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8.4.1 On Simulation model

The observer was designed using the steps explained in the previous segments. It was connected to the simulation model that was created in the beginning. The Simulink model of the whole setup can be seen in the figure below:

The matrices defined previously is used and the observer matrix L can be defined as:

𝐿 = [

250 16500

−482490 ]

Figure 47: Control of simulation model

The model in figure has three parts, the first one is the simulation model of the hydraulic system, the next is the state space model of the simulation model and the last is the state space observer.

The switches are provided to switch the control between state space feedback control or state space observer control for the simulation model or the state space model.

The input was set at 5V control input. The output of the model is as shown below:

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VŠB – Technical University of Ostrava 54 Figure 48: Position output

The desired position is reached within 1s without any overshoot.

The error estimate of the plant model and the observer estimates is as follows:

Figure 49: Error dynamics

The error reaches 0 value quickly and stays at that value during the steady state.

Now let’s take a look are the Velocity and acceleration estimates of the system:

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VŠB – Technical University of Ostrava 55 Figure 50: Estimated state variables

Since the range of the variables are not comparable to each other, individual variables are not seen precisely.

8.4.2 Observer for the Test rig

A state space model of the test rig is constructed in Simulink using simple mathematical blocks according to the expression below.

𝒙̇(𝑡) = 𝑨𝒙(𝑡) + 𝑩𝒖(𝑡) (8.8)

𝒚(𝑡) = 𝑪𝒙(𝑡) + 𝑫𝒖(𝑡)

(8.9)

Using the following matrices to describe the system 𝐴 = [

0 1 0

0 0 1

0 −13726 −00016 ]

𝐵 = [ 0 0 23893

]

𝐶 = [1 0 0]

𝐷 = 0

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The observer is designed for this model as the state space model is assumed to be similar to the dynamics of the test rig in the lab. This observer is tuned in Simulink and then used to control the laboratory drive. The state space control with observer is setup in Simulink as illustrated in the figures below

Figure 51:State space control with observer

Here the state feedback gain is represented as K that is calculated by placement of poles which will be discussed in detail in next segment. The state space model is constructed as shown below

Figure 52:State space model

The observer is modelled as below. The observer gain L is calculated for fast response to track the system variables.

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VŠB – Technical University of Ostrava 57 Figure 53: Model of observer

Once the observer is tuned to satisfaction it is used with the laboratory test rig and the control is evaluated.

When working with the actual drive, the input is set in volts so, input to the observer must be scaled to meter as it was designed to receive input in meter. This is calculated by dividing the maximum stroke length of the cylinder by the maximum voltage value.

The Simulink model used to control the drive is as below:

Figure 54:Control of the test rig

Here a controller switch is placed to switch between P controller or the state space observer for control.

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8.4.3 Pole placement

By placing the poles in desired positions, we can obtain the desired response. The poles placed further to the left plane are more stable thus leading to better control dynamics.

The eigen values of the system was used as the starting point for the pole placement and was moved to the left to obtain faster response and stability.

The poles chosen are:

p1 = -15;

p2 = -116.88+ 008.08i;

p3 = -116.88 - 008.08i;

The plot of the poles placed are be seen in the figure below:

Figure 55:State space control pole placement

The feedback gain matrix K is calculated for the poles chosen. The matrix is defined as:

K =[ 8.6172 0.1468 0.0097]

The observer gains were set almost 5 times the dominant pole of the system op1 = -300;

op2 = -301;

op3 = -302;

The observer matrix L is given by:

f. Ing. Petr Noskievič, CSc. VŠB –

FACULTY OF MECHANICAL ENGINEERING

DEPARTMENT OF CONTROL SYSTEMS AND INSTRUMENTATION

Analysis and Control Design of The Hydraulic Test Rig

Student : Jaishree Ananthkumar (ANA0006) Supervisor: Prof. Ing. Petr Noskievič, CSc.

Ostrava 2019

Annotation

Keywords

Anotace

Klíčová slova

Table of Contents

Table of Figures

List of Tables

Abbreviations

1 Introduction

2 Hydraulic closed loop control drives

2.1 Applications

2.2 Control systems

2.3 Types of control

3 Modelling and Simulation of Mathematical Models

3.1 MATLAB – Simulink

3.2 Simulation

4 Analysis of The Structure of The Hydraulic Drive of The Test Rig

5 Simulation Modeling of hydraulic drive

5.1 Simulation of Hydraulic Subsystems

5.2 Mathematical model of a Servo-Valve

5.3 Simulation model of servo-valve

5.4 Simulation model of cylinder

5.5 Result of simulation of cylinder

6 Identification method of Eigen Frequency of the Hydraulic Cylinder

6.1 Implementation of eigen frequency identification

𝑅

=

6.2 Setup of the test Rig

6.3 Data logging

7 Design of control Algorithm for Automatic Eigen frequency Identification

7.1 Calculations

7.2 Application for identification

8 Design of Closed Loop Control

8.1 Parameters calculation of PID controller

8.2 State space representation

8.3 Implementing state feedback

8.4 Implementing state observer