Hydraulic closed loop control

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IN THE FIELD OF TECHNOLOGY DEGREE PROJECT

ENGINEERING PHYSICS

AND THE MAIN FIELD OF STUDY ELECTRICAL ENGINEERING, SECOND CYCLE, 30 CREDITS STOCKHOLM SWEDEN 2018,

Hydraulic closed loop control

MARIA ELFVING

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Hydraulic closed loop control

MARIA ELFVING

Master in Systems, Control and Robotics Date: June 13, 2018

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Abstract

The purpose of the thesis work is to investigate methods for closed loop control of hydraulic pressure in transmissions to make them be more precise. This is desirable since it decreases the fuel consumption as well as emissions, and improves the driving performance.

To be able to study the behaviour of the transmission, a Simulink model is designed with the parts relevant to the problem, and from this a linear model is obtained. Three different controllers are designed and implemented in the Simulink model, to compare and analyze different solutions. The controllers implemented are a PI controller, a PID controller and a LQR controller.

The results from the simulation with the different controllers show step responses to be able to evaluate their individual performance. The results show that all of the controllers meet the requirements for a step reponse under better conditions, but under worse ones the LQR controller performs best of the three. The LQR controller is therefore the most suitable of the three controllers for this particular problem.

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Sammanfattning

Syftet med det här examensarbete är att undersöka olika metoder för återkopplad reglering av hydrauliskt tryck i en transmission för att göra det mer exakt. Detta är önskvärt eftersom det minskar bränsleåt- gången och utsläpp, och gör även körupplevelsen bättre.

För att kunna studera transmission tas en Simulink-modell fram in- nehållande de delar som är relevanta för problemet, och från detta kan en linjär modell erhållas. Tre olika regulatorer tas fram och im- plementeras i Simulink-modellen, för att kunna jämföra och analysera de olika lösningarna. De regulatorer som tas fram är PI-regulator, PID- regulator och LQR-regualator.

Resultaten från simuleringen med de olika regulatorerna visar stegsvar under varierande förutsättningar för att kunna utvärdera hur de presterar. Resultaten visar att alla regulatorer uppfyller kraven på ett stegsvar under bättre förhållanden, men LQR-regulatorn presterar bäst under svårare förhållanden. LQR-regulatorn är därför den mest relevanta reglerstrategin för det här problemet av de tre.

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Acknowledgements

I would like to thank Anders Löfgren and Fredrik Wallin from Volvo CE, without whom I never would have gotten anywhere with this project. Thank you for the support and for the hydraulic wisdom!

I would also like to thank Rodrigo Gonzalez and Cristian Rojas from KTH for all their help and their continued support throughout this project.

Also, a special thanks to Elianne Lindmark - without whom I would still be stuck with a faulty Simulink model!

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List of Figures

1.1 Figure showing the requirements for a step response. . . 4

2.1 Torque converter. 1: Pump. 2: Turbine. 3: Turbine shaft. 4: Stator. (Image from: Volvo Service Manual [1]). . . 7

2.2 The transmission pump. (Image from: Volvo Service Man- ual [1]). . . 8

2.3 The planetary gear. A: Sun gear. B: Planet gear. C: Ring gear. D: Planetary carrier. (Image from: Volvo Service Manual [1]). . . 8

3.1 Simulink model. . . 14

3.2 Detrended data . . . 17

3.3 Step response of the Simulink model. . . 18

3.4 AIC values for different Output Error models. . . 19

3.5 Block diagram of the LQR with integral feedback. . . 23

4.1 Simulated response compared to real data. . . 25

4.2 Linear model compared to simulation data. . . 26

4.3 Step response of system with PI controller. . . 27

4.4 PI controller in Simulink simulation. . . 29

4.5 Pressure drop due to low engine speed and opening of the control valve (PI). . . 30

4.6 Varying engine speed with PI controller. . . 31

4.7 Step response of system with PID controller. . . 33

4.8 Output of PID controller in Simulink simulation. . . 35

4.9 Pressure drop due to low engine speed and opening of the control valve (PID). . . 36

4.10 Varying engine speed with PID controller. . . 37

4.11 Step response of system with LQR controller. . . 39

4.12 Output of LQR in Simulink simulation. . . 41

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4.13 Pressure drop due to low engine speed and opening of the control valve (LQR). . . 42 4.14 Varying engine speed with LQR. . . 43

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List of Tables

1.1 Table showing the requirements for a step response. . . . 4 3.1 PI and PID controller gains given from the Ziegler-Nichols

method. . . 21 4.1 Step characteristics of system with PI controller. . . 28 4.2 Step characteristics of system with PI controller in sim-

ulation. . . 28 4.3 Step characteristics of system with PID controller. . . 32 4.4 Step characteristics of system with PID controller in sim-

ulation. . . 34 4.5 Step characteristics of system with LQR. . . 38 4.6 Step characteristics of system with LQR in simulation. . . 40 5.1 Step characteristics of the different controllers in the lin-

ear system. . . 47 5.2 Step characteristics of the different controllers in simu-

lation. . . 47 5.3 Step characteristics of the different controllers in simu-

lation with low engine speed. . . 47

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Chapter 1 Introduction

The basic function of a mechanical transmission is to transfer the power of the engine to the driving wheels of the vehicle. It is what makes it possible for a motor vehicle to drive at different speeds and with different torque, allowing us to use the vehicle in many different en- vironments and in areas with different topographies. A powershift automatic transmission allows us to take this even further and we are able to have heavy machines working in even the most extreme cir- cumstances, such as steep hills and other hard conditions.

Today, a number of improvements are continuously being made to make transmissions more efficient, decrease the fuel consumption as well as emissions, and make the driving performance better. One such improvement is to have more precise control over the hydraulic pressure in the transmission, which decreases fuel consumption since we use exactly what is needed and no more, and also extends the life- time of the system since it is not subjected to unnecessary wear and tear.

1.1 Problem statement

The purpose of the thesis work is to investigate methods for closed loop control of hydraulic pressure in transmissions for off road ma- chinery. Right now the hydraulic pressure is controlled through open

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sion needs to be increased in order to achieve maximum allowed pressure without risking fatiguing clutches with too high pressure.

The main expected outcome of this project is to formulate and estimate a mathematical model for control purposes of the involved parts, i.e.

valves and pump, implemented in Simulink, as well as one or more controller(s) implemented and evaluated in a simulation study.

1.2 Literature overview

To provide background and theoretical knowledge before the start of the practical part of this project, a literature study has been conducted.

It basically involved two parts: the first part was simply to get a better understanding of the problem and gain insight about the system, i.e. the transmission. The second part was to see what other people have done and try to see what controllers could be suitable for this case.

Material from Volvo was used to attain better knowledge of the system. This included specifications and presentations about the system and the parts relevant to my project. For the theoretical background regarding the transmission, the book Motor Vehicle [2] as well as the aforementioned material from Volvo have been used as refer- ences.

To get a clear view of how other people have resolved problems similar to the control part of this project, several papers that presented different ideas of how to do it have been studied.

Two different controllers have been decided upon: a PID controller, presented in Section 2.2.1, and a linear quadratic regulator (LQR), presented in Section 2.2.2.

For the PID controller the course book in the basic control course at KTH [3] will be used for reference. This controller was chosen because it is very commonly used and is good to have as a benchmark.

The ideas presented in Pourebrahim et al. 2016 [4] and Rodrigues et al. 2011 [5] as well as Chapter 9 in Control Theory: Multi- variable and Nonlinear Methods [6] and Chapter 11 in Computer-Controlled Systems [7] will be used for reference for the LQR. This controller was chosen

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because it is easy to trade-off between response time and and control effort [6].

1.3 State of the art

Currently, automatic transmissions are commonly used in heavy duty vehicles. To actuate the different clutches and brakes in a heavy duty vehicle, a control system is needed. However, in the past there has only existed a need to keep the pressure in the hydraulic system over a certain threshold, and the upper limit has not been as important. In many cases, open loop control has been used with a lower limit and that has been satisfactory, and that has been the case in the system considered in this project as well.

To keep fuel consumption down, it is becoming more and more important to give just the right amount of pressure and not exceed this limit by too much. As an effect of that, many studies similar to this one have been conducted to try to solve that problem by implementing more ad- vanced control strategies. There have been several different solutions put forth, including simple PID controllers (Meng et al. 2016 [8]), fault- tolerant controllers (Wang and Wang 2013 [9]) and predictive control strategies (Balau et al. 2011 [10]).

The article regarding a PID controller [8] explains modelling of valves similar to the ones in this case, but instead of regulating the main pressure it is clutch actuation that is considered. The article describing a fault-tolerant controller [9] is more mathematical and does not have as much physical modelling, and the paper is more focused on describing different kinds of faults. The article describing a predictive control strategy [10] is very similar to this case and could be a possibility to test in the future.

1.4 Requirement specification

The requirements for the step responses are presented in Table 1.1 and in Figure 1.1. The requirements are given by Volvo and are chosen to

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The maximum rise time for the step, defined as the time from 10 % to 90 % of the step height, is 50 ms. The maximum overshoot for the step response is 2 bar and the tolerance, meaning the acceptable difference between the desired and the actual value, is 2 %. The maximum settling time for the step, defined as time to reach and stay within the tolerance of the final value, is 200 ms.

Var. Definition Description Req.

Tr Rise time Time from 10 % to 90 % of the

step height 50 ms

M Max. overshoot Maximum peak of the signal ex- ceeding its target

2 bar Tol Tolerance Acceptable difference between

desired and actual value after reaching steady state

2 %

Ts Settling time Time to reach and stay within tolerance of the final value

200 ms

Table 1.1: Table showing the requirements for a step response.

Figure 1.1: Figure showing the requirements for a step response.

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1.5 Thesis outline

Chapter 2 consists of the theoretical background. In Section 2.1 the theory about the automatic transmission is presented and in Section 2.2 theory about the controllers to be used is introduced.

Chapter 3 covers modelling of the system. It also covers the design of the controllers and how they are implemented.

Chapter 4 presents the evaluation of both the Simulink model and the mathematical model, as well as the results from the simulation.

Chapter 5 encloses the discussion about the results from the simulation.

Chapter 6 contains the conclusion of the report and what future work could be done after this study to further improve the proposed solutions.

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Chapter 2 Background

In this chapter the theoretical background will be presented, both for the automatic transmission and for the controllers that shall be used.

2.1 The automatic transmission

At the center of this thesis work is the automatic transmission, and the goal is to improve its performance. The purpose here is to give a quick overview of the different parts in an automatic transmission to get a clear picture of the entire system.

The basic parts of the automatic transmission that is considered in this project are: a torque converter, transmission pump, planetary gear sets, clutches and brakes and last but not least: valves. The system achieves shifting by locking and unlocking a complex system of planetary gears using pressurized hydraulic fluid.

2.1.1 Torque converter

The function of the torque converter, shown in Figure 2.1, is to transfer the power from the engine to the transmission. The torque converter mainly consists of three parts: a pump, stator and turbine.

The pump is connected to and is driven by the engine and translates the power of the engine to hydrodynamic power. The turbine converts

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that hydrodynamic power, or oil flow, to torque into the transmission.

Between the pump and the turbine is the stator.

The pump and the turbine rotate separately. When there is a big difference in rotation speed between the pump and the turbine, the stator redirects the flow of the oil from the pump to the turbine and gives torque amplification. With this, the torque can become two to three times bigger.

Figure 2.1: Torque converter. 1: Pump. 2: Turbine. 3: Turbine shaft. 4:

Stator. (Image from: Volvo Service Manual [1]).

2.1.2 Transmission pump

The transmission pump, shown in Figure 2.2, is driven by the engine, via the pump rotor in the torque converter and thereby has the same speed as the engine. The pump provides pressurized oil to two cir- cuits, one oil circuit that controls gearshifting (called the main pressure circuit, the one this project concerns) and one oil circuit for the torque converter and for lubrication. The flow from the pump is proportional to the engine speed.

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Figure 2.2: The transmission pump. (Image from: Volvo Service Manual [1]).

2.1.3 Planetary gear sets

The planetary gear sets, shown in Figure 2.3, allows for the same set of gears to produce all of the different gear ratios needed.

A planetary gear consists of four parts: the sun gear, the planet gears, the ring gear and the carrier. All the gears rotate around a common axle, which goes through the center of the sun gear and the ring gear.

The planet gear not only rotates around this axle, but also around their own axles.

All parts of the planetary gear can be the input, the output or kept still and thereby give different gear ratios, and we can also combine several planetary gears coaxially to get higher or more ratios.

Figure 2.3: The planetary gear. A: Sun gear. B: Planet gear. C: Ring gear. D: Planetary carrier. (Image from: Volvo Service Manual [1]).

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2.1.4 Clutches and brakes

In this case there are three clutches and five brakes that can hold parts of the planetary gear set still and connect different shafts to each other.

Both the clutches and the brakes are actuated by pressurized hydraulic fluid that enters a piston inside the clutch or brake. In both there are also springs that make sure that the clutch or brake is released when the pressure is reduced.

2.1.5 Valves

The main pressure valve is used to regulate the supplied pressure to the clutch control valves – this pressure is called the main pressure.

To control this main pressure, two valves are used: the main stage and the pilot stage. The pilot stage is an electro-proportional valve, meaning that it can regulate the pilot pressure using an electric control signal. The pilot pressure is used to control the pressure setting of the main stage.

The pilot stage is designed to handle a relatively small flow, compared to the much larger flow delivered by the pump. The main stage is designed to handle the complete flow from the pump.

If no current is supplied to the pilot stage’s solenoid, the spring forces will determine the main pressure. The force from the solenoid is acting to open the pilot stage, hence the pilot pressure will be reduced when current is supplied, which in turn reduces the main pressure.

2.2 Controllers

The controllers that have been chosen to be implemented in this project are one PID controller and one linear-quadratic regulator (LQR). The theory behind will be presented in this section. A more detailed description about the design of the controllers are given in Section 3.2.

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2.2.1 PID controller

The first controller to be implemented is a PID controller. The PID controller is presented in discrete-time form in Equation (2.1):

u(k) = Kpe(k) + Ki

Xn k=1

e(k) + Kd(e(k) e(k 1)), (2.1)

where u is the control input to the system, k is the discrete time and the error e(k) is defined as: e(k) = r(k) y(k). Kp, Ki and Kdare the proportional, integral and derivative gains respectively.

It is very common to use a PI controller instead of a PID controller or a PD controller since measurement noise can have a big negative im- pact on controllers with derivative action. A PI controller is achieved simply by setting the derivative gain, Kd, equal to zero.

2.2.2 Linear-quadratic regulator (LQR)

We consider a discrete-time system in state space description as the one shown in Equation (2.2):

xt+1 = Axt+ But, x0 = x^init. (2.2) The problem we often face here is that we simultaneously want x1, x2,... to be “small” to get good regulation and u0, u1,... to be “small” to use as little input effort as possible. These two objectives often inter- fere with each other, since a large u can drive x to zero fast. It is this problem that linear-quadratic regulator theory tackles.

We define a quadratic cost function, shown in Equation (2.3):

J(U ) =

N 1X

⌧ =0

(x^T_⌧Qx⌧ + u^T_⌧Ru⌧) + x^T_NQfxN, (2.3)

where U = (u0, ..., u_{N 1})and

Q = Q^T 0, Qf = Q^T_f 0, R = R^T > 0

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are given state cost, final state cost and input cost matrices, respectively.

The LQR problem is basically: find u^lqr₀ , ..., u^lqr_{N 1}that minimizes J(U).

To give some further explanation of the cost function we can start by saying that N is called the time horizon, that decides during which time interval we want the regulator to act.

The first term in the cost function measures state deviation, the second term measures input size and the last term measures final state deviation.

It is very common for Q and R to take the following form:

R = ⇢I, Q = Qf = C^TC.

The cost function then becomes:

J(U ) = XN

⌧ =0

||y^⌧||²+ ⇢

N 1X

⌧ =0

||u^⌧||²,

where y = Cx.

This can be seen as the output and input cost, which are competing objectives and we ideally want them both to be small.

Integral feedback can be used in the LQR to eliminate steady state error. This is done by adding another state, z, in the controller that is the integral of the error, where the error is defined as the difference between the reference and the output.

The addition of the constant reference means that the output is driven to that, since the LQR controller drives the states to zero.

This augmented system is shown in Equation (2.4):

x(k + 1) z(k + 1) =

Ax(k) + Bu(k) y(k) r(k) =

Ax(k) + Bu(k)

Cx(k) r(k) . (2.4) All of this together gives a controller on the following form:

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u = Kx Kiz + ud

where [KKi] is the state feedback for the augmented system with the added state and where udis the reference input.

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Chapter 3 Modelling and Control

In this chapter the method used to model the system and designing the controllers is presented.

3.1 Modelling

This chapter describes how the physical model of the system was developed in Simulink, how that model was linearized and how a state space model was produced to be used for the control part of this project.

3.1.1 Physical modelling in Simulink

There is an existing model that has been developed at Volvo that is very complex. It is used to give a good representation of the physical system and to simulate many different scenarios. In this case, a Simulink model of only the involved parts will be developed with the existing model for reference.

The involved components are presented in Chapter 2.1, but will be simplified in the Simulink model. In the Simulink model there will be four parts: a pump, a pilot valve, a main valve and a on/off valve.

The on/off valve is a simplified model of a clutch control valve. The on/off valve is used to simulate the filling of a clutch. The Simulink

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Figure 3.1: Simulink model.

Modelling of the pump

The only purpose of the model of the pump in this project is to give an input to the system that somewhat resembles real life. The pump is not a part of the control problem, except as a disturbance or a restriction.

The flow and therefore pressure that the pump provides sets a limit to what we can achieve, and therefore a good model is needed.

The inputs are engine rotation speed and the main pressure from the system. There are also some constants relevant to the physical pump, i.e. pump displacement and dynamic viscosity of the hydraulic fluid.

The engine speed and the pump displacement are used to calculate an ideal flow and the other inputs are used to determine pump efficiency, which together gives us the effective flow as shown in Equation (3.1)

Qpump = veng· D · ⌘ (3.1) where Qpump is the flow, veng is the engine speed, D is the pump displacement and ⌘ is the pump efficiency. The pump efficiency is approximated with the help of a look-up table provided by the pump manufacturer.

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Modelling of the pilot valve

The inputs to the pilot valve are the current, the main flow and the main pressure.

The current is used to calculate the pilot pressure, Pp,I, through a look- up table with values measured from the actual valve. To find what the maximum pressure that can actually be achieved, Equation (3.2) is used:

P_p,Q = B V

Z T_{f inal} 0

Q_mdt. (3.2)

where Pp,Q is the pilot pressure, Qm is the main flow, B is the bulk modulus and V is the volume of the hydraulic fluid. The minimum of the pilot pressure calculated from the current, Pp,I and the pilot pressure calculated from the main flow, Pp,Q, is then the actual pilot pressure.

The difference between the main pressure and the pilot pressure and the damping orifice area is used in Equation (3.3) to calculate the pilot flow:

Qp = Cd· Ad· r2

⇢ · (Pm Pp). (3.3)

where Qpis the flow, Cdis the discharge coefficient, Adis the damping orifice area, ⇢ is the fluid density, Pmis the main pressure and Ppis the pilot pressure.

Modelling of the main valve

The main valve contains a spool and a spring. The difference between the main pressure and the pilot pressure is used to calculate the force against the spool in the valve. The spring force and the damping force is also calculated to work against the force on the spool from the hydraulic fluid. The net sum of the forces is used and the distance the spool moves is deduced through integration. This is shown in Equa-

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X

f orces

= Ad· (P^m Pp) Fspring Fvd,

dspool =

TZZf inal

0

P

f orces

m dt².

(3.4)

where Pmis the main pressure, Ppis the pilot pressure, Adis the orifice area, Fspring is the spring force, Fvd is the viscous damping, dspoolis the spool displacement and m is the mass of the spool.

The spool displacement is then used to calculate the area of the opened orifice of the valve. This is then used together with the main pressure to calculate the flow through Equation (3.3).

Modelling of the control valve

The control valve and the clutch is approximated by an on/off valve and an orifice. This on/off valve is controlled by a signal and the flow is then calculated by Equation (3.5):

Q_c = C_d· Ad· r2

⇢ · Pm. (3.5)

where Qc is the flow, Cdis the discharge coefficient, ⇢ is the fluid density, Pmis the main pressure and Adis the orifice area, calculated from the assupmtion that a main pressure of 20 bars should give a flow of 30 liters per minute.

3.1.2 Mathematical model

After obtaining a model that represents the reality in a way that is considered good enough, the data from the inputs and outputs of that model can be used to create a mathematical model. Since the control techniques that are to be used require a linear model and the Simulink model is not linear, there is a need to linearize it around a working point. This linear, mathematical model will be used for control purposes.

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The non-linearities in the Simulink model mainly consists of the calcu- lation of flow from pressure, shown in Equation (3.3) and present in all of the valves, and the pilot pressure obtained from the current through a look-up table.

The data provided about the relationship between input current and the pilot pressure suggested that a linear region is from 0.15 [A] to 0.25 [A], meaning that a good working point is 0.20 [A]. To obtain good data for mathematical modelling a pseudorandom noise signal for the current was used as input. The pseudorandom noise signal was created using the Signal Builder block in Simulink. This pseudorandom noise and the output after it has been detrended can be seen in Figure 3.2.

Figure 3.2: Detrended data

The simulation was run for 10 seconds with a sample time of 1 ms. The first second was then removed to only get good data points, resulting in 9002 data points. The input and output data is then used to create a time-domain data object. This data is then detrended by removing the means and the data is divided equally into evaluation and validation

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The sample time of 1 ms was chosen after looking at the step response of the Simulink model, shown in Figure 3.3. The rise time of the system is 17 ms and as a rule of thumb there should be around 10 samples per rise time, giving the sample time of 1 ms.

Figure 3.3: Step response of the Simulink model.

Four different models with many different polynomial orders are evaluated. For control purposes, an order higher than three is not desired and therefore not considered. The models used are ARX, AR- MAX, Box-Jenkins (BJ) and Output-Error (OE). The Akaike’s Informa- tion Criterion (AIC) for the different models is used to compare the different models.

The Akaike’s Information Criterion (AIC) [11] is a way to compare the relative quality of statistical models. It compares how much information is lost when a model is used to represent the original pro- cess.

This comparison of different models using AIC yielded the result that an ARMAX model was the best fit. However, when looking at the zeros and poles it was evident that one of the zeros was outside the unit circle, giving a non-minimum phase system. It can be argued that the

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non-linear system in the Simulink model is not non-minimum phase since the step response does not have an initial negative response, so modelling it with a non-minimum phase linear system is not a good approximation. After making this observation, it was decided to try the OE model with the lowest AIC since it might be better to try and not model the noise, which in this case proved to be a reasonable as- sumption.

The OE model is represented on the form shown in Equation (3.6):

y(t) = B(q)

F (q)u(t nk) + e(t) (3.6)

where y(t) is the output, u(t) is the input and e(t) is the error. There are three design parameters: nb, nf and nk. nb and nf are the orders of the polynomials B(q) and F (q), respectively. nk is the input delay in number of samples.

After evaluating many different OE models, the AIC values for the best ones are shown in Figure 3.4.

Figure 3.4: AIC values for different Output Error models.

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It can be seen that between a model of order 2 and of order 3 the improvement is quite large, while the difference between a model of order 3 and 4 and bigger is quite small. From this the conclusion can be drawn that the model of order 3 is the best to use since not much is gained from using a higher order model, and for control purposes it can be beneficial to use a model of lower order. The resulting OE model is shown in Equation (3.7), with model orders nb = 3, nf = 3 and nk = 1.

y(t) = B(q)

F (q)u(t) + e(t) B(z) = 2.92q 1.694q² 1.077q³ F (z) = 1 2.766q + 2.592q² 0.8231q³

(3.7)

A comparison of the validation data output and the output given by the linear model is shown in Section 4.2.

3.2 Controller design

In this section the design of the controllers will be described.

3.2.1 PID controller

After obtaining the mathematical model the first and easiest step to- ward a PID controller is using the Ziegler-Nichols method for obtaining values for Kp, Kiand Kd. This tuning method was chosen because it is a straightforward way to tune a PID controller. It is an heuristic method and does not guarantee good results, but in this case it proved successful.

The basic idea is to first implement a PID controller with only the P part active and increase that until we get steady oscillations. The gain used to achieve this is denoted K0 and the oscillation period of those steady oscillations is denoted T0.

The values used to achieve a PI and PID controller are presented in Table 3.1.

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Control type Kp Ki Kd

PI 0.45· K⁰ T0/1.2 - PID 0.6· K0 T0/2 T0/8

Table 3.1: PI and PID controller gains given from the Ziegler-Nichols method.

To achieve a stable system that still meets the requirements, a fraction of the calculated values is used in the PI/PID controller. For the PI controller this fraction is 0.1, and for the PID controller this fraction is 0.045. These values were chosen to give the fastest systems possible while still meeting the requirements, mainly to keep the overshoot small enough. The resulting PI controller is shown in Equation (3.8) and the resulting PID controller is shown in Equation (3.9).

Kp+ Ki

T_s

z 1

Kp = 0.00739, Ki = 2.96, T s = 0.001

(3.8)

Kp+ Ki

Ts

z 1 + Kd

z 1

T_s Kp = 0.00333, Ki = 1.33, Kd= 2.08e 06, Ts= 0.001

(3.9)

3.2.2 Linear quadratic regulator (LQR)

After obtaining the mathematical model the first step towards a LQR controller was to create a state space representation of that model:

x(t + 1) =Ax(t) + Bu(t)

y =Cx(t). (3.10)

To add the integral feedback an integral state is added and the matrices A, B and C are augmented. This is done by adding zeros as shown in

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then be used to obtain the control matrices Q and R for the design of the LQR controller.

Aaugmented = 2 66 4

0

A 0

0 0 0 0 0

3 77

5 , B^augmented = 2 66 4 B

0 3 77 5

Caugmented =⇥

C 0 ⇤

(3.11)

The matrix Q should have the same size as the augmented matrix A in the state space representation and is chosen to be C^TC with an added weight for the integral state. The design C^TC is chosen since it is the output y we want to bring to the reference value. Since C is the re- lationship between the states x and the output y this gives us a good output weighting. The weight for the integral state is varied to get a good response. The resulting Q is shown in Equation (3.12):

Q = 2 66 4

2.1318 0.6185 0.7859 0 0.6185 0.1794 0.2280 0 0.7859 0.2280 0.2897 0

0 0 0 0.7

3 77

5 (3.12)

The matrix R should be square and have the same number of columns as the augmented matrix B in the state space representation, which in this case is 1, and is chosen to be R = ⇢. The parameter ⇢ is then varied to get a good response, resulting in R = 5.

The MATLAB function lqi is used to compute the optimal state feedback K. Since we are operating in discrete time the integrator output xi is computed using the forward Euler formula, xi[k + 1] = xi[k] + T_s(r[k] e[k]), where Tsis the sample time of the system. The resulting K is shown in Equation (3.13):

K =⇥

0.7018 0.5154 0.3414 0.1543 ⇤

(3.13) The block diagram for the LQR with the added integral is shown in Figure 3.5.

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Integrator -K u System

r e

xi

u y

x

Figure 3.5: Block diagram of the LQR with integral feedback.

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Chapter 4 Evaluation and Simulation results

In this chapter the evaluation and simulation results will be presented.

4.1 Evaluation of Simulink model

To evaluate the model, a experimental measurement from the test bench was used. In that case a step from 12.8 bar to 21.8 bar was made, so the same thing was simulated using the obtained model. The result is shown in Figure 4.1.

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Figure 4.1: Simulated response compared to real data.

To evaluate this result the error was calculated, according to Equation (4.1), where Ym is the output from the measurement and Ysis the output from Simulink. It can be seen that the error is 1.85 %.

Error = 1 N

XN k=1

|Y^m,k Ys,k|

|Ym,k| = 0.0185. (4.1)

4.2 Evaluation of mathematical model

To evaluate the mathematical model we compared it to the validation data. The fit was found to be 63.84 % and the result can be seen in Figure 4.2.

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Figure 4.2: Linear model compared to simulation data.

4.3 Simulation results: PI controller

In this section the results of using the PI controller that was designed in the previous chapter will be presented, both from the linear model and from simulation in Simulink.

4.3.1 Linear model

In Figure 4.3 the step response of the linear system with the PI controller is shown. The step response characteristics are shown in Table 4.1 where the tolerance is 2 % and the rise time is calculated between 10 % and 90 % of the steady state response.

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(a) Step response.

(b) Step input.

Figure 4.3: Step response of system with PI controller.

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Rise time 9 ms Settling time 123 ms

Overshoot 4.7 %

Steady state error 0 %

Table 4.1: Step characteristics of system with PI controller.

4.3.2 Simulation

In Figure 4.4 the performance of the simulation in Simulink with the PI controller applied is shown. A constant current is first applied that gives a pressure of 11.7 bar. After one second the controller is applied with a reference of 20.15 bar and the clutch control valve is opened at time 2 seconds and closed at time 2.25 seconds. In Figure 4.4a the pressure is shown and in Figure 4.4b the output current from the PI controller is shown.

Rise time 9 ms

Settling time 142 ms Overshoot 1.8 bar Steady state error 0 %

Table 4.2: Step characteristics of system with PI controller in simulation.

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(a) Performance of simulation with PI controller.

(b) Output current from the PI controller.

Figure 4.4: PI controller in Simulink simulation.

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In Figure 4.5 the performance of the system is shown when the engine speed is kept at 800 rpm (compared to 2000 rpm used in the other simulations) and we do the same step from 11.7 bar på 20.15 bar and then the clutch control valve is opened at time 2 seconds and closed again at 2.25 seconds. It can be seen that the overshoot is very large.

Figure 4.5: Pressure drop due to low engine speed and opening of the control valve (PI).

In Figure 4.6a the response of the system is shown when the engine speed is being varied and the PI controller is being used. The varia- tion of the engine speed is shown in Figure 4.6b. The control valve is opened for 250 ms after 4 seconds, when the engine speed is at its high- est, and then again after 7 seconds, when the engine speed is low.

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(a) Response of the system with PI controller and varying engine speed.

(b) The varying engine speed.

Figure 4.6: Varying engine speed with PI controller.

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4.4 Simulation results: PID controller

In this section the results of using the PID controller that was designed in the previous chapter will be presented, both from the linear model and from simulation in Simulink.

4.4.1 Linear model

In Figure 4.7 the step response of the linear system with the PID controller applied is shown. The step response characteristics are shown in Table 4.3 where the tolerance is 2 % and the rise time is calculated between 10 % and 90 % of the steady state response.

Rise time 9 ms

Settling time 111 ms

Overshoot 5 %

Table 4.3: Step characteristics of system with PID controller.

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(a) Step response.

(b) Step input.

Figure 4.7: Step response of system with PID controller.

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4.4.2 Simulation

In Figure 4.8 the performance of the simulation in Simulink with a PID controller applied is shown. A constant current is first applied that gives a pressure of 11.7 bar. After one second the controller is applied with a reference of 20.15 bar and the clutch control valve is opened at time 2 seconds and closed at time 2.25 seconds. In Figure 4.8a the pressure is shown and in Figure 4.8b the output current from the PID controller is shown. It can be seen that the rise time is 9.5 ms and that the settling time is 157 ms.

Rise time 9.5 ms

Table 4.4: Step characteristics of system with PID controller in simulation.

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(a) Performance of simulation with PID controller.

(b) Output current from the PID controller.

Figure 4.8: Output of PID controller in Simulink simulation.

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In Figure 4.9 the performance of the system is shown when the engine speed is kept at 800 rpm (compared to 2000 rpm used in the other simulations) and we do the same step from 11.7 bar på 20.15 bar and then the clutch control valve is opened at time 2 seconds and closed again at 2.25 seconds. It can be seen that the overshoot is very large.

Figure 4.9: Pressure drop due to low engine speed and opening of the control valve (PID).

In Figure 4.10a the response of the system is shown when the engine speed is being varied and the PID controller is being used. The varia- tion of the engine speed is shown in Figure 4.10b. The control valve is opened for 250 ms after 4 seconds, when the engine speed is at its high- est, and then again after 7 seconds, when the engine speed is low.

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(a) Response of the system with PID controller and varying engine speed.

Figure 4.10: Varying engine speed with PID controller.

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4.5 Simulation results: Linear quadratic reg- ulator (LQR)

In this section the results of using a LQR that was designed in the previous chapter will be presented, both from the linear model and from simulation in Simulink.

4.5.1 Linear model

In Figure 4.11 the step response of the linear system with the LQR controller applied is shown. The step response characteristics are shown in Table 4.5 where the tolerance is 2 % and the rise time is calculated between 10 % and 90 % of the steady state response.

Rise time 3 ms

Settling time 5.2 ms

Overshoot 0 %

Table 4.5: Step characteristics of system with LQR.

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(a) Step response.

(b) Step input.

Figure 4.11: Step response of system with LQR controller.

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4.5.2 Simulation

In Figure 4.12 the performance of the simulation in Simulink with the LQR controller applied is shown. A constant current is first applied that gives a pressure of 11.7 bar. After one second the controller is applied with a reference of 20.15 bar and the clutch control valve is opened at time 2 seconds and closed at time 2.25 seconds. In Figure 4.12a the pressure is shown and in 4.12b the output current from the LQR controller is shown. It can be seen that the rise time is 9 ms and that the settling time is 74 ms.

Rise time 9 ms

Table 4.6: Step characteristics of system with LQR in simulation.

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(a) Performance of simulation with LQR.

(b) Output current from the LQR controller.

Figure 4.12: Output of LQR in Simulink simulation.

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In Figure 4.13 the performance of the system is shown when the engine speed is kept at 800 rpm (compared to 2000 rpm used in the other simulations) and we do the same step from 11.7 bar på 20.15 bar and then the clutch control valve is opened at time 2 seconds and closed again at 2.25 seconds. It can be seen that the overshoot is within the required bounds.

Figure 4.13: Pressure drop due to low engine speed and opening of the control valve (LQR).

In Figure 4.14a the response of the system is shown when the engine speed is being varied and the LQR controller is being used. The varia- tion of the engine speed is shown in Figure 4.14b. The control valve is opened for 250 ms after 4 seconds, when the engine speed is at its high- est, and then again after 7 seconds, when the engine speed is low.

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(a) Response of the system with LQR and varying engine speed.

Figure 4.14: Varying engine speed with LQR.

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Chapter 5 Discussion

In this chapter the results will be discussed and compared. The conclusion will also be presented as well as the ideas for potential future work regarding this problem.

5.1 Modelling

5.1.1 Simulink model

In Figure 4.1 the performance of the Simulink model compared to data from the real system is shown. In Equation (4.1) the error is shown to be 1.85 %. These results are satisfactory and shows that reality is approximated in a relatively good way.

Not all disturbances and noise from the real system are represented in the model, but that is hard to achieve without having a too complex model.

5.1.2 Mathematical model

The fit of the linear model is good, as can be seen in Figure 4.2. It might seem low with a fit about 60 %, but this is mainly due to the linear model not modelling the noise that is introduced to the system

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with the steps in the input and the fact that the nonlinear Simulink model is approximated with a linear model.

The low fit percentage could also be a sign that a model of higher order is needed, but as shown in Figure 3.4 not much is gained from using a model of higher order. Also, not having a too high model order is desirable for control purposes.

The limitations of using a linear model is of course that the Simulink model is not linear, nor is the actual system, and in some ways this means that the model is not true. The idea is however that this should be reasonably good, but this could of course be a problem.

5.2 Controllers

5.2.1 PI and PID controller

Linear model

It was relatively easy to find both a PI controller and a PID controller that satisfied the requirements of the step response. It can be seen in Tables 4.1 and 4.3 that the PI and PID controllers have compara- ble step characteristics, and both meet the requirements for a step response.

Simulink

With the PI and PID controllers implemented in the Simulink model, it can be seen in Figures 4.4 and 4.8 that the PI controller performed slightly better than the PID controller, with both a shorter rise time and settling time and a smaller overshoot.

When subjected to the case of a slower engine speed and opening of the control valve it can be seen in Figures 4.5 and 4.9 that neither of them do a satisfactory job. Both the step response and the response when the clutch control valve is opened has an overshoot that is too large.

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In Figures 4.6 and 4.10 we can also see the same results when the engine speed is being varied.

5.2.2 Linear quadratic regulator (LQR)

Linear model

It took some trial and error, but a satisfactory LQR controller was found. As can be seen in Figure 4.11 the controller is very fast with no overshoot. As can be seen in Table 4.5, this controller meets the requirements for a step response.

Simulink

With the LQR controller implemented in the Simulink it can be seen in Figure 4.12 that it is satisfactory. The overshoot is not too big, the rise time is small and even though the added noise and disturbances are present here the settling time is still very short.

When subjected to the case of a slower engine speed it can be seen in Figure 4.13 that the result for the step response is still satisfactory.

However, when the clutch control valve is opened the overshoot is too big.

When the engine speed is being varied and the clutch control valve is being opened we can see the same results as in the other test cases in Figure 4.14.

5.2.3 Comparison of controller results

In Tables 5.1, 5.2 and 5.3 the step characteristics of the different controllers both in the linear system and Simulink is shown. For Simulink it is shown with both high and low engine speed.

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PI PID LQR Rise time 9 ms 9 ms 3 ms Settling time 123 ms 111 5.2 ms Overshoot 4.7 % 5 % 0 %

Table 5.1: Step characteristics of the different controllers in the linear system.

PI PID LQR

Rise time 9 ms 9.5 ms 9 ms Settling time 142 ms 157 74 ms Overshoot 1.8 bar 1.83 bar 1.73 bar

Table 5.2: Step characteristics of the different controllers in simulation.

PI PID LQR

Overshoot Too large Too large Acceptable

Table 5.3: Step characteristics of the different controllers in simulation with low engine speed.

These results show that for this particular problem the LQR is more suitable than a PI or PID controller. This is mainly because of the LQR controllers robustness against slower engine speed, where the PI and PID controllers did not meet the requirements.

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Chapter 6 Conclusion and Future work

6.1 Conclusion

The goal of this study was to investigate methods for closed loop control of hydraulic pressure in a transmission. To achieve this the sub- goals were to create a Simulink model of the system, a mathematical model and to design one or more controllers.

During the course of this project, a Simulink model has been created as well as a linearized mathematical model. In addition to this, three controllers have been designed: a PI controller, a PID controller and a LQR controller. The controllers’ performance in the modelled system have been simulated and compared.

It has been shown that all of the controllers could regulate the hydraulic pressure in the transmission with satisfactory results when the engine speed is high enough, but both the PI and PID controller could not handle lower engine speed. The LQR controller did much better than the other two controllers with a slower engine speed, but still did not do well enough when the clutch control valve is opened during periods of low engine speed. This is supported by the information presented in Section 5.2.

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6.2 Future work

As future work, other control strategies suitable for this project could be tried, such as self-tuning PIDs or a Model Predictive Controller.

Another thing that could be done further is to implement the controllers in an actual transmission, perhaps on a test rig. After doing this, further analysis on the performance of the controllers could be done and be used to decide what needs to be improved or what other controller to try out next.

When it comes to the Simulink model, the complexity and accuracy of it could be improved. For example, the modelling of the different parts could be done much more physical and not rely on measurements and look-up tables.

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Bibliography

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[3] T. Glad and L. Ljung, Reglerteknik: grundläggande teori. Studentlit- teratur AB, 2006.

[4] M. Pourebrahim, A. S. Ghafari, and M. Pourebrahim, “Designing a lqr controller for an electro-hydraulic-actuated-clutch model,”

in 2016 2nd International Conference on Control Science and Systems Engineering (ICCSSE), pp. 82–87, 2016.

[5] C. Rodrigues, R. Kuiava, and R. Ramos, “Design of a linear quadratic regulator for nonlinear systems modeled via norm- bounded linear differential inclusions,” IFAC Proceedings Volumes, vol. 44, no. 1, pp. 7352–7357, 2011. 18th IFAC World Congress.

[6] T. Glad and L. Ljung, Control Theory: Multi-variable and Nonlinear Methods. Studentlitteratur AB, 2000.

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Theory and Design. Dover Publications, 2011.

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TRITA TRITA-EECS-EX-2018:174 ISSN 1653-5146

Hydraulic closed loop control

Hydraulic closed loop control

MARIA ELFVING

Hydraulic closed loop control

MARIA ELFVING

Abstract

Sammanfattning

Acknowledgements

Contents

List of Figures

List of Tables

Chapter 1 Introduction

1.1 Problem statement

1.2 Literature overview

1.3 State of the art

1.4 Requirement specification

1.5 Thesis outline

Chapter 2 Background

2.1 The automatic transmission

2.1.1 Torque converter

2.1.2 Transmission pump

2.1.3 Planetary gear sets

2.1.4 Clutches and brakes

2.1.5 Valves

2.2 Controllers

2.2.1 PID controller

2.2.2 Linear-quadratic regulator (LQR)

Chapter 3

Modelling and Control

3.1 Modelling

3.1.1 Physical modelling in Simulink

3.1.2 Mathematical model

3.2 Controller design

3.2.1 PID controller

3.2.2 Linear quadratic regulator (LQR)

Chapter 4

Evaluation and Simulation results

4.1 Evaluation of Simulink model

4.2 Evaluation of mathematical model

4.3 Simulation results: PI controller

4.3.1 Linear model

4.3.2 Simulation

4.4 Simulation results: PID controller

4.4.1 Linear model

4.4.2 Simulation

4.5 Simulation results: Linear quadratic reg- ulator (LQR)

4.5.1 Linear model

4.5.2 Simulation

Chapter 5 Discussion

5.1 Modelling

5.1.1 Simulink model

5.1.2 Mathematical model

5.2 Controllers

5.2.1 PI and PID controller

5.2.2 Linear quadratic regulator (LQR)

5.2.3 Comparison of controller results

Chapter 6

Conclusion and Future work

6.1 Conclusion

6.2 Future work

Bibliography