integration project

SC4050: Integration Project

Control of Inverted PendulumDue on June 12, 2015

Robert Babuska / Gabriel Lopes

Shekhar Gupta(4419677) and Shwetha M R(4420578)

Contents

1 Introduction 2

2 System Identification 3

Lab Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Lagrangian Dynamic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Generalized Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Kinetic and Potential Energy Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Modelling of DC Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Algebraic Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Nonlinear Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 LQR Control 10

Linearised Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Pendulum Up ( = 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

State Space Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

LQG Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Observer Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Continious Time Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Converting the continuous time model to discrete time . . . . . . . . . . . . . . . . . . . . . . 13

Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Comparison of LQG controller performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Model Predictive Control 19

Sample time: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Prediction Horizon: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Control horizon: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Weight tuning: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5 Conclusion 24

Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

LQR control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Model Predictive control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1

Chapter 1

Introduction

The Cart-Inverted Pendulum System (CIPS) is a classical benchmark control problem. Its dynamics resem-

bles with that of many real world systems of interest like missile launchers, pendubots, human walking and

Segway and many more. The control of this system is challenging as it is highly unstable, highly non-linear,

non-minimum phase system and under actuated. Further, the physical constraints on the track position

control voltage etc. also pose complexity in its control design.

For this course our task is to design two controllers that can control the motion of the cart such that the

pendulum is balanced in its upright position with the following requirements.

keeping the pendulum inverted and track the reference signal for the cart position.

The controlled system should have zero steady state error and recover from a small tick against thependulum.

To obtain these objective we are going to follow systematic procedure that are required to obtain controllers

which fulfils these objectives. The first step to do this is by modelling the system. It is very important to have

some information about the plant so that its output can be controlled. Hence most of the control algorithm

rely on the model of the plants that needs to be controlled. We first create a first principle mathematical

model of the Pendulum Cart system and the DC motor. Modelling and Dynamics analysis of Cart pendulum

system is done by writing the Lagrange equations. Then we carry out parametric system identification for

the inverted pendulum using a nonlinear simulation model as we didnt have exact measurement of the input

to force gain km and the damping coefficient b (including viscous friction and the back-emf of the motor).

Once we have the model of the system we are going to design two linear controller for the system. In order

to do that we need to linearize the model which can be done by linearizing the system equation obtained

by Langrange equation at the equilibrium position. After we linearize the system we can derive the state

space model of the system. This continuous time the state space model will be converted into discrete time

state space model. Once we have the discrete time state space model, we can proceed with the designing of

the controller. First we will design a Full State Feedback Controller i.e. Linear Quadratic Gaussian (LQG)

compensator. The compensator aims at providing a proper control input that provides a desired output in

terms of the Pendulum Angle and Cart Position. Once we design the controller first we want to test the

controller in a simulation model and tune the weights, then follow the same procedure on the real setup to

get optimum results.

We then design Model Predictive Control (MPC) using the MATLAB MPC toolbox and simulate the

output by tuning the weights of the controller and then implement the controller on the real-time setup and

further tune the weights.

2

Chapter 2

System Identification

There are two basic approaches to modeling : the First principle modeling and the black box modeling.

Both the approaches have their own advantages and disadvantages. In black box modeling we can use same

algorithm to identify different plants which are complex in nature and are not easy to identify using first

principle due to its complicated mathematical equations. But since the black box modeling is done by

analysing the input and the output data of the plant, the model is generally valid only for the signal it was

calculated from. For example if a step signal was used to identify the plant, this plant should not be used

for the analysis of high frequency signals.

Plants obtained by using first principle are more robust as whole range of input signals can be used to

analyse it. But the drawback of first principle is that there are lot of unknown constants and relations in

the model description. Hence these constants need to obtained by performing real-time experiments on the

plant. In our case we are going to use first principle method to obtain the system model.

Lab Setup

Figure 2.1: The lab setup of The Cart-Inverted Pendulum System (CIPS)

The lab setup is the Linear Servo Base Unit of Quansers consulting. It consists of a cart driven by a DC

motor, via a rack and pinion mechanism, to ensure consistent and continuous traction. The cart is equipped

with a rotary metal shaft to which a free turning pendulum is attached. The Linear Servo Base Unit system

has two encoders: one encoder is used to measure the carts position and the other encoder is used to sense

the position of the pendulum shaft. The Linear Inverted Pendulum is mounted on the cart and is free to fall

along the carts axis of motion.

3

Shekhar Gupta and Shwetha M R Integration Project SC4050

Description

Figure 2.2: Inverted Pendulum on a cart structure

The inverted pendulum setup consists of a cart driven by a DC motor. The motor can steer the cart left and

right on a track approximately one meter long. On the cart, a pendulum is mounted such that it can freely

rotate around an axis that is perpendicular to the direction of motion of the cart. The schematic diagram in

Figure 2.2 shows the construction of the system including all the relevant parameters and variables. Positive

directions of variables are indicated by arrows. This system has one control input u, which is the voltage

driving the cart motor. This input is commanded from the computer and is scaled between 1 (correspondsto the maximal input moving the cart to the left) and +1 (corresponds to the maximal input moving the cart

to the right). There are two measured outputs: x the displacement of the cart from the track center, and

the angle of the pendulum. These measurements are given in their physical units meters and radians,

respectively. The physical parameters of the system are listed in Table 2.1. Most of the values can easily be

determined by measuring the dimensions and masses (such as the pendulum length, mass of the cart, etc.)

Table 2.1: Physical parameters and their valuesSymbol Parameter Value

g acceleration due to gravity 9.81 ms2

l half length of pendulum 0.30 mm mass of pendulum 85 g

J inertia of pendulum ml2

3

M mass of cart 0.49 kg

Lagrangian Dynamic Analysis

A complete theoretical model of the pendulum cart system can be done using Lagrangian Dynamics. First

we choose generalized coordinates, and then derive energy functions, and Lagrangian. Finally, we can use

Lagranges Equation to derive the equations of motion. A schematic of the system is shown in Figure 2.3.

This model and coordinate system will be used in the analysis. For completeness, this derivation involves

inertia about the pendulum center of mass.

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Figure 2.3: Schematic of Pendulum Cart Model includes cart mass M , pendulum of mass m and inertia I

about its center of mass, which is a distance l from the pendulum pivot.

Generalized Coordinate System

The cart pendulum system has two degrees of freedom and can therefore be fully represented using two gen-

eralized coordinates. For this analysis, the generalized coordinates are chosen as the horizontal displacement

of the cart, x, and the rotational displacement of the pendulum,

Kinetic and Potential Energy Functions

The kinetic co-energy function for the cart mass is:

T =Mv1

2

2(2.1)

The velocity of the cart is simply

v1 = x (2.2)

For the pendulum, the co-energy function can be derived from :

T =Mv2

2

2+I2

2(2.3)

The angular velocity of the pendulum is

= (2.4)

Kinetic energy in along both x and and kinetic energy due to rotation

v2 (x) =

x(x l sin ) = x l cos (2.5)

v2 (y) =

x(l cos ) = l sin (2.6)

v22 = v2

2 (x) + v22 () = x2 + l22 2xl cos (2.7)

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Substituting Equation 2.7 into Equation 2.3 and upon simplifying equation 2.1 and 2.3. The total kinetic

co-energy is

T =(M +m)x2

2+ml22

2+I2

2mxl cos (2.8)

Since the cart moves only in the horizontal direction, the potential energy of the system is determined entirely

by the angle of the pendulum, given by

V = mgl cos (2.9)

Lagrangian

From the kinetic co-energy and potential energy functions, the Lagrangian is given by

L = T V (2.10)

Using Equations 2.8 and 2.9, the Lagrangian can be written as

L =(M +m)x2

2+ml22

2+I2

2ml cos (g + x) (2.11)

Applying Euler Lagrange Equation we get,

F bx = d(Lx )

dt Lx

(2.12)

d(L

)

dt L

= 0 (2.13)

Substituting equation 2.11 in equations 2.12 and 2.13 and evaluating the partial derivatives and simplifying

and rearranging, the system state equations are

F = (M +m)x+ bx+ml cos ml 2 sin (2.14)

(I +ml2) +mlx cos +mgl sin = 0 (2.15)

Modelling of DC Motor

Figure 2.4: Schematic of a DC motor

Ldi

dt+Ri = V Vb (2.16)

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Vb = K = Kd

dt(2.17)

Jd2

dt2 bd

dt= T (2.18)

In general, the torque generated by a DC motor is proportional to the armature current and the strength

of the magnetic field. In this example we will assume that the magnetic field is constant and, therefore,

that the motor torque is proportional to only the armature current i by a constant factor k as shown in the

equation below. This is referred to as an armature-controlled motor.

T = Ki (2.19)

since torque is force times the radius we have

T = Fr (2.20)

where r is the radius of the pinion

i =V

R(2.21)

F =KV

rR(2.22)

F = kmu (2.23)

where km is equal toKrR and V = u

Algebraic Loops

When analysing or simulating a system described by a block diagram, it is necessary to form the differential

equations that describe the complete system. In many cases the equations can be obtained by combining

the differential equations that describe each subsystem and substituting variables. This simple procedure

cannot be used when there are closed loops of subsystems that all have a direct connection between inputs

and outputs, known as an algebraic loop. When algebraic loops are present, it is necessary to solve algebraic

equations to obtain the differential equations for the complete system.

Hence for our simulation model well further simplify these equation so that we can incorporate these

equation in Matlab simulink. Equation 2.14 and 2.15 can be written as

x =1

M(kmuN bx)

=1

I(Nl cos + Pl sin )

(2.24)

where

P = m(l sin l cos ) + gN = m(x l2 sin + l cos )

(2.25)

and F is replaced by kmu as derived in the DC motor modeling.

Nonlinear Simulation Model

As described above we used equation 2.24 and 2.25 to create a non linear simulink model. Figure 2.5 shows

the simulink model.

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Figure 2.5: Non-linear simulink model of Cart inverted pendulum system

We used this simulink model for the parameter estimation.

Parameter Estimation

As discussed before the drawback of first principle modelling is that there are lot of unknown constants and

relations in the model description. Since we modelled our system using first principles we have to obtain

these unknown parameters of the model by performing real-time experiments on the Cart inverted pendulum

system. We are neglecting rotational viscous friction, and translational Coulomb friction in our model. Hence

there will be always some difference between the model and the real setup. For example the non-presence of

oscillations in real setup is caused by presence of dry (Coulomb) friction but in our model simulation we can

find oscillation. Hence the only constants that we have to determine are the input to force gain km and the

damping coefficient b (including viscous friction and the back-emf of the motor). The advantage of modelling

by first principle is the model is valid for all sets of input output data and not just on the data set that was

used for parameter estimation. Due to this we do not need to worry about which kind of input signal to

take. A signal as simple as impulse should be enough in our case. Since our model is highly non-linear due

the presence of various trigonometric term as well as due the presence of friction, to have better estimate

of the parameters, we want to use Non-linear least square approach. The non-linear simulink model (refer

to Fig 2.5) that was derived earlier is used for this estimation. We used Matlab function lsqnonlin for our

identification. To do this identification we had to provide the initial values to the parameters. We used

various input signal for this estimation with each data we got different values for km and b. Even for same

data set if we chose different initial values for the parameters we got different values. When we used input to

be a multi-input sine wave we got curve fitting with VAF (variance accounted for) of 96%. Figure 2.6 shows

the curve fitting with the identification data. The values obtained for the input to force gain was km = 4.45

and the value for damping coefficient b was found out to be 11.54.

Model Validation

Once we had obtained the parameter values, we validated these values against different set of input output

data. We took a different set of input data and compared the output of our model with the calculated

parameters against the actual setups output. The VAF value for the data validation was found to be 86%.

Figure 2.7 shows the graph of validation data.

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Figure 2.6: Model output with identification data

Figure 2.7: Model output comparison with actual setup output for validation data

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Chapter 3

LQR Control

Linearised Model

Before proceeding with further analysis, we must linearize the state equations. The mathematical model of

the system is nonlinear, it has to be linearized around some appropriate working point. Since the analysis

and control design techniques we will be employing apply only to linear systems, this set of equations needs

to be linearized. In case of availability of mathematical model of the system, linearization is performed by

calculating first two members of corresponding Tylor series (function value and first derivation). Specifically,

we will linearize the equations about the vertically upward equilibrium position, = 0, and will assume that

the system stays within a small neighbourhood of this equilibrium. Focusing on small variations of about

the equilibrium point 0:

= 0 + (3.1)

= (3.2)

From a Taylor Series expansion, a first order approximation of any function of is

f() f() + dfd

(3.3)

Also, because higher order terms are neglected

2 0 (3.4)

Hence

2 0 (3.5)

Pendulum Up ( = 0)

For = 0, Equation 2.26 yields to first order

cos cos (0) + ( sin 0) = 1sin sin (0) + (cos (0)) = 0

(3.6)

10


State Space Representation

For linear state control of the inverted pendulum, it is necessary to convert the state equations to state space

representation in the form:

x = Ax+Bu

To get the state-space form, equation 2.14 and 2.15 are linearized using eqn. 3.5 and 3.6, then they are

rearranged into a series of first order differential equations. Since the equations are linear, they can then be

put into the standard matrix form shown below.

x

x

=

0 1 0 0

0 (I+ml2)b

I(M+m)+Mml2m2gl2

I(M+m)+Mml2 0

0 0 0 1

0 mlbI(M+m)+Mml2mgl(M+m)

I(M+m)+Mml2 0

x

x

+

0(I+ml2)km

I(M+m)+Mml2

0mlkm

I(M+m)+Mml2

u

y =

[1 0 0 0

0 0 1 0

]x

x

+ [ 00]u

(3.7)

LQG Control

The linear quadratic regulator (LQR) is a well-known design technique that provides practical feedback

gains. A synthesis problem is formulated to minimize a criterion, which is a quadratic function of the states

and the control signals. The problem is called the Linear Quadratic (LQ) control problem. The resulting

optimal controller is linear. The LQ-controller can also be interpreted as a pole-placement controller.

The solution of the LQ-control problem, if all the states are available, is the result of an algebraic Riccati

equation, where the properties of LQ-controllers are also discussed. But since we cannot estimate all the

states, we use a observer based LQR controller.

Implementation

We first linearise the system and then find out a state feedback gain that stabilizes the pendulum in the

equilibrium point. In order to do this we check the rank of Controllability matrix.

C = [B AB A2B A3B] (3.8)

Rank of C should be 4 for the system to be controllable.This can be checked with matlab command ctrb.

LQ- Controller can be implemented in matlab using lqr command.

The LQ-Controller needs as input the weighted matrices: Q for the states and R for the inputs. Changing

the relative magnitude between the elements in the weighting matrices means a compromise between the

speed of the recovery and the magnitudes of the control signals.

Looking at the function to minimize:

J =

Nk=1

x(k)TQx(k) + u(k)

TRu(k) (3.9)

Q and R are symmetric weighting matrices that are positive semi-definite and positive definite respectively.

Page 11 of 26


By increasing the weighting matrices relative to one another we can put emphasis on the state and control

trajectories. We can obtain the value of K by using matlab function lqr giving different values of Q and

R. The R matrix influences the control action. Keeping Q equals to the identical matrix, and amounting R

unit by unit, the step response will become slower, less effort will be put in the control action. Leaving the

R matrix equals to the unit, the control action increases if the Q is increased. The Q matrix acts on the

states of the system. Q(1, 1) amounts to the cart displacement, Q(2, 2) corresponds to the cart displacement,

Q(3, 3) to the pendulum angle and Q(4, 4) to the pendulum angular displacement. Since we are not bothered

about the cart velocity and angular velocity of the cart we can keep Q(2, 2) and Q(4, 4) as zero. Since the

constraint on cart position is difficult to meet, we choose Q(1, 1) Q(3, 3).We start the tuning with the value of Q = CTC and R = [1] and check the response. We will increase the

weight matrix Q such that Q(1, 1) is always greater then Q(3, 3). The final value of Q and R after tuning

for which the system was stable and was able to track the reference signal is given below:

Q =

450 0 0 0

0 0 0 0

0 0 150 0

0 0 0 0

, R = [5] (3.10)

Observer Design

To design the observer, we first check the rank of observability matrix.

W =

C

CA

CA2

CA3

(3.11)We observe that the rank of W is 4 and therefore we can design an observer to measure the states which are

not currently measurable. i.e. the speed of the cart displacement and the angular velocity of the pendulum.

We calculate the eigenvalues of the closed loop system and place the observer pole at 10 times the slowest

pole. We found that the slowest closed loop pole was at -4.09 [before tuning], so we places the observer poles

at

P = [40,41,42,43] (3.12)We implement the placement of observer using the matlab command place. The final state space matrix of

the closed loop plant is

Ac = [ABK]Bc = B

Cc = C

Dc = D

(3.13)

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Figure 3.1: Simulink Model of LQR continuous time Observer based Model

Observer state space matrix is

Ahat = [A LC]Bhat = [B L]

Chat =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

Dhat =

0 0 0

0 0 0

0 0 0

0 0 0

(3.14)

Where K is the solution of the riccati equation and L is the output of matlab command place used for

observer pole placement.

Now we implement the linearised closed loop plant with observer on the simulation using the above matrices,

as shown in Figure 3.1

Continious Time Simulation Results

Given below the graph of Closed loop system response.

We observe that the system stabilizes within 2 seconds in the simulation model. Now we proceed to convert

the model into discrete time.

Converting the continuous time model to discrete time

The system must be converted to a discrete time system. To do so, it was needed the sampling frequency.

Using the bode plot of the module of the open loop transfer function, it is possible to select the sampling

frequency. The selection of the sampling frequency is important to not incur in aliasing problems. For

Page 13 of 26


Figure 3.2: Continuous time LQR simulation model system response

discrete-time systems, the Shannons sampling theorem, based on the Nyquist frequency, is used to avoid

aliasing. A continuous time signal, that does not contain frequency components greater than cross over

frequency (c), is uniquely determined if the sampling frequency s is higher than 2c.

One way to measure the sampling time is the closed loop bandwidth. A good rule of thumb is to use sampling

frequency to be in between 10 bandwidth to 30 bandwidth which can be determined by closed loop bodeplot. Another approach is to use sampling time to be in between risetime/4 to risetime/10.

When we give a step input to the system, the rise time of the system is found to be 0.2 seconds. So we take

the sampling period to be 0.01 seconds.

We use the matlab command c2d to convert the state matrix in continuous time to discrete time. Use the

same values of Q and R as continuous time and implement the LQ controller in discrete time using matlab

command dlqr. Now we convert the observer poles in continuous time to discrete time using the conversion,

z = ehs (3.15)

where z is discrete time pole,

s is continuous time pole and

h is sampling frequency

So the discrete time observer poles are

P = [0.7, 0.72 , 0.73 , 0.69] (3.16)

We repeat the same procedure as continuous time by using the place command to place the observer and

create the discrete time LQcontroller with observer The discrete time closed loop state space model is given

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Figure 3.3: Discrete time LQR simulation model

by.

Phic = [PhiGammaK]Gammac = Gamma

Cdc = Cd

Ddc = Dd

(3.17)

The observed state space model is given by

Phihat = [Phi LCd]Gammahat = [Gamma,L]

Cdhat =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

Ddhat =

0 0 0

0 0 0

0 0 0

0 0 0

(3.18)

Given below is the simulation of Discrete time LQG Controller.

Challenges

The discrete time controller with the same Q and R values gave good response on the simulation model

where as when tested on the pendulum setup didnt stabilise the system. This happened because of high

dynamics of the observer pole placed at 0.4. It created turbulent oscillation and destabilised the system.

When we placed the observer poles near 0.7, the system oscillation reduced and it gave a smooth response.

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Figure 3.4: Comparison of Disturbance rejection in the Simulation Model to Pendulum setup

Comparison of LQG controller performance

We compare the simulation output of LQG controller which is given an initial disturbance of 0.01 in angle

with the disturbance rejection of the real setup. As seen in the figure 3.4 we observe that the disturbance

rejection in simulation model is fast whereas in the setup it takes more time to settle down. This is because

the simulation model does not consider the viscous force of the pendulum and the static friction in the cart.

Now we test the reference tracking in the simulation model by changing the reference from 0 to 0.1

for cart displacement. The output of the simulation is shown in figure 3.5 Figure 3.6 shows the reference

tracking on the real setup. We observe there is a very small error of 0.02m during reference tracking. This

error in the reference tracking is so less that the control input generated by it is not enough to overcome the

static friction, hence unavoidable. Figure 3.7 and 3.6 show the control input and the observer output during

reference tracking, respectively.

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Figure 3.5: LQR Reference tracking on Simulation

Figure 3.6: LQR Reference tracking on the cart pendulum setup

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Figure 3.7: Control Output LQR Reference tracking on the cart pendulum setup

Figure 3.8: Observer Output LQR Reference tracking on the cart pendulum setup

Page 18 of 26

Chapter 4

Model Predictive Control

The design of model predictive controller for inverted pendulum system is based on the mathematical model

of the system and its linearized version. Generally, the computation of control signal for model predictive

controller (MPC) is based on minimization of quadratic criterion mentioned in the equation 4.1. Model

Predictive Control takes into account of the following three ideas for its functioning:

1. Explicit use of a model to predict the process output along a future time horizon.

2. Calculation of a control sequence to optimize an objective function.

3. A receding horizon strategy, so that at each instant the horizon is moved towards the future, which

involves the application of the first control signal of the sequence calculated at each step.

To design model predictive controller we used Matlabs Model predictive GUI based toolbox. First we importthe discretized state space model of the setup and define the manipulated variable as the u and and x as

the measurable output. When we get to the tree structure on the design tool under the controller node we

select the controller which contains multiple tabs such as Horizon, constraint and weight tuning. We then

tune these parameters one by one. The tuning of these parameters has been described below.

Sample time:

As discussed before the selection of sample time is decided based on the cut-off frequency or rise time of

the system. If we decrease the value of sampling time rejection of unknown disturbance usually improves.

Usually very low value of sampling time increases the computational power significantly. In our case when we

tried using sampling time of 0.01 second we did not encounter any computation problem and we continued

with 0.01 sampling time.

Prediction Horizon:

The prediction horizon, Np, is the number of future control intervals the MPC controller must evaluate by

prediction when optimizing its manipulated variables at control interval Ts. Generally value of Np is selected

based on how fast you want your controller to respond that is we want our response time to be equal to the

prediction horizon times the sampling time. As we have already designed LQG controller we have seen that

achieving 2 seconds of settling time is possible. Hence we want our closed loop system to have a settling

time similar to LQG, in range of 2 seconds. To achieve this we select our prediction horizon to be 200 as

our sampling time is 0.01s. By going through various literature we realise that a prediction interval of 2 to 3

second is good enough for a good controller. It is also know that low value of prediction interval can create

an internally unstable controller hence we do not decrease this value.

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Control horizon:

Control horizon is interval after which a controller forces the control signal to become constant. In general Ncshould be very small compared to prediction horizon. An important effect of control horizon is the smoothing

of the control signal and the control signal is forced towards steady state value which is important for steady

state property. Another effect of decreasing Nc is, it decreases the computational effort drastically. The

control horizon value should be more if there is dead time in the process. Since in our setup we do not find

any dead time we chose Control horizon to be 30 (30*0.01=0.3s).

Weight tuning:

Model predictive controller (MPC) is based on minimization of quadratic criterion.

J(k) = eTQ(k)e+ uTRu (4.1)

where e is a vector of predicted control errors, u is a vector of future control signal samples and square matrices

Q and R allows to set weighting of individual vector elements. MPC uses receding horizon principle: only

a finite number of future values is used in the criterion and only the first element of the obtained control

sequence is applied to the controlled system. Future outputs of the controlled system, and consequently

control errors, are computed on base of the linear model. Since the system is described by state space model,

future system outputs depend only on current states of the system and future control signals:

y(k) = Cx(k) +Du (4.2)

As the matrices Q and R are diagonal the predictive control criterion can be rewritten into the following

form:

J(k) = 1

Nj=1

[e1(k + j)]2

+ 2

Nj=1

[e2(k + j)]2

+ 3

Ncj=1

[u(k + j 1)]2 (4.3)

where

e1(i) = wr(i) r(i)e2(i) = 0 (i) = (i)

Matlab control toolbox optimizes this objective function by converting it into Quadratic programming

problem and gives the optimized control signal that is used to track the output signal. Since we had already

designed the LQR controller we started tuning weights here by giving the values we had obtained in LQR

controller i.e. weight on x as 450, weight on as 150 and weight on input as 1. When we simulated this

scenario we were able to track the signal, but the control signal was very high. Hence we scaled all the

weights and tuned it till control input reduced to value in between -1 and 1 which corresponded to value

when all weights were scaled by 10. Hence the new weights were 4500 weight on x, 1500 weight on and

150 weight on input u. The simulation result with these parameters are shown in the figure 4.1

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Figure 4.1: Reference tracking in simulation after weight tuning of input signal

It can be seen from the above figure that when the weight on input was low the control input was very

high. Once we increased the weight on input the control input is well within the limit. Even though now the

controller was able to do reference tracking, even for a small disturbance we were not able to do disturbance

rejection and the system was becoming unstable. To overcome this we tried increasing the weight on and

decreasing the weight on x. Hence with the weight of 1500 on x and 4500 on the model was able to do

disturbance rejection.

When we apply the controller with these parameter values in the real time setup we found out that it had

lot of oscillation and the cart never became stationary, even the pendulum had oscillation. We realised that

this was this was due to fact that there was high weight on the error in , because of which even for small

error in the control input was very high and it was making the cart move rapidly. To overcome this we

increased the weight on x till the oscillation stopped and the cart was able to track the reference. Hence the

final weight that we obtained was 2500 on x, 4500 on and 150 on input u. Figure 4.3 shows the reference

tracking for the setup and figure 4.2 shows the reference tracking in the simulation model. We observe that

there is a very small steady state error in the reference tracking of the real-time setup and this is because

the error is so small that the control signal generated by it cannot overcome static friction.

Figure 4.4 shows the comparison between the disturbance rejection in the simulation to the pendulum setup.

The simulation is given an initial disturbance of 0.01 radians whereas the pendulum setup is given distur-

bance throughout the experiment.

We observe that the disturbance in the pendulum setup takes more time to settle in comparison with the

simulation. This same behaviour was observed in the LQG controller and is due to the model mismatch for

the unaccounted viscosity of the pendulum and static friction of the cart.

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Figure 4.2: Shows reference tracking in simulation after weight tunning

Figure 4.3: Shows reference tracking in real setup, after tunning the MPC controller

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Figure 4.4: Disturbance rejection in simulation and in real-time setup with MPC controller

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Chapter 5

Conclusion

Modeling

As discussed before there are two approaches to model a system: The First principle model and the black

box model. We use the first principle modelling instead of the black box model this is because it is usually

more reliable in comparison with the black box modelling and also the pendulum cart system can be easily

modelled mathematically. The drawback of black box modelling is that it is generally valid only for the signal

it was calculated from, which could be avoided by taking an input which persistently excites the system,

but finding such a input is difficult. When modelling the system we neglect the rotational viscous friction,

translational Coulomb friction and stiction for easy calculation. We observe that this approach gives us a

good model with VAF of 96% for identification data and a VAF of 86% for validation data. This establishes

that the model is reliable. For further improvement in the accuracy of the model we can model the coulomb

friction as:

Fxc = Fc.sign(x,)

where Fc is coulomb friction constant. The negative sign is because friction acts against the movement of

cart.

LQR control

Even though the implementation of a state feedback controller was simple in comparison with the MPC

controller, the LQR controller yields really good results. The downside of this controller is that it requires

really good model of the system. Hence system modelling should be reliable for this to work but if the system

is really complex then modelling is not easy and it can become tedious. The model we used was accurate, so

once we had the controller working in simulation, the same controller worked well on the real setup without

much tuning. One problem we encountered in the real setup was the rigorous oscillations of the cart. This

was because of an aggressive observer pole used for the state estimation was responding to noise. Once we

made this observer poles less aggressive it was able to estimate the state and worked smoothly.

Model Predictive control

Implementing model predictive control was little tricky. Tuning the parameters to achieve the objective was

the toughest part as most of the time it yielded infeasible solution. We began the tuning first by treating the

system as SISO system with the constraint that should to be equal to zero to meet the controller objective.

When we simulated the system treating this constraint as hard constraint, it yielded infeasible solution. This

is due to the fact that the cart cannot move without disturbing the pendulum. So we modelled the system as

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SIMO system with constraint on the input signal to be between -1 and 1. This resulted in a feasible solution

which was able to track the reference but was unstable for even small disturbance. Finally we removed this

constraint and we increased the weight till the input signal was under the recommended limit of -1 to 1

and then used a saturation block to be on safer side. Another problem we faced was that even though the

controller was working in simulation it was not able to control the real setup. We re-tuned the controller to

stabilise the system.

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Bibliography

[1] Karl Johan Astrom and Richard M. Murray. Feedback Systems : An Introduction for Scientists and

Engineers.

[2] Aarshita Joshi and Nimmy Paulose Discrete Time Model Predictive Control Approach for Inverted Pen-

dulum System with Input Constraints

[3] Petr Chalupa and Vladimir Bobal

Modelling and Predictive Control of Inverted Pendulum

[4] Control Tutorials for Matlab and Simulink

http://ctms.engin.umich.edu/CTMS/index.php?aux=Home

[5] Mathworks Tutorial for MPC Toolbox

http://uk.mathworks.com/products/mpc/

26

integration project

Documents

controlled system

pendulum cart system

control design

statespace model

continuous time model

model validation83 lqr

nonlinear simulation

control horizon