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AN EXTENDED SIMULINK LIBRARY FOR MODELING AND
CONTROL OF INVERTED PENDULA SYSTEMS
Slávka Jadlovská, Ján Sarnovský
Department of Cybernetics and Artificial Intelligence, Faculty of Electrical Engineering and
Informatics, Technical University of Košice, Slovakia
Abstract
The aim of this paper is to present an improved and extended version of the Inverted
Pendula Modeling and Control, a structured thematic Simulink block library which
enhances the capabilities of the MATLAB/Simulink program environment by
providing means for modeling and control of classical and rotary inverted pendula
systems in form of function blocks, demo simulation schemes and applications with
graphical user interface.
1 Introduction
Inverted pendula systems represent a significant class of highly nonlinear mechanical systems
used in classical control education. Although underactuated and extremely unstable by nature, the
systems are controllable and have a number of practical applications: problems such as balancing
a broomstick on a handpalm, stabilization of a walking human or robot, control of a launching rocket
or the vertical movement of a human shoulder or arm can all be simulated by some type of an inverted
pendula system [1][2][3][4][5]. The diversity of modeled systems is reflected in the wide variety of
available inverted pendula models. These may be classified according to several criteria:
• the type of base which actuates the pendula – the system base may either be moving in a linear
manner (classical or linear inverted pendulum system), or in the rotary manner in a horizontal
plane (rotary inverted pendulum system);
• the number of pendulum links attached to the mechanism – every extra pendulum link increases
the number of the system’s degrees of freedom, making control in turn more challenging. For
inverted pendula systems, controllability has been shown for up to 4 links attached to the base [6].
• the distribution of mass along the pendulum rod – the pendulum links can either be homogenous
rods with the mass concentrated in their center of gravity, or the rod may be considered to be
massless, with the mass concentrated in the load at the end.
Fig. 1 shows the schematic depictions of three characteristic representatives of inverted pendula
systems which are most frequently used as testbed systems for nonlinear control design [1][2]: the
classical single (one-link), classical double (two-link) and rotary single inverted pendulum system.
Fig. 1 Typical representatives of inverted pendula systems: a) classical single inverted pendulum system,
b) classical double inverted pendulum system, c) rotary single inverted pendulum system
When the prominent position that inverted pendula systems occupy in nonlinear control theory
was considered by the authors of this paper, the idea of providing integrated program support for the
whole class of systems immediately arose. As a result, a structured thematic Simulink block library,
Inverted Pendula Modeling and Control (IPMaC), has been developed since 2009 as a comprehensive
software framework for the problems of analysis and control of inverted pendula systems. While the
first presented version of the block library [3] was limited to classical inverted pendula systems, this
paper describes an improved and extended 2011 version of the library which deals with two principal
categories of inverted pendula systems (classical and rotary). The importance of the mutual analogy of
the systems is emphasized: if a system of inverted pendula is approached as an instance of
a generalized (n-link) system, crucial procedures related to system modeling and control algorithm
design can be processed into an algorithm and implemented via symbolic software, represented in
MATLAB by the Symbolic Math Toolbox. These considerations can be further extended to control
algorithm design.
2 The Installation and Structure of the IPMaC Block Library
The IPMaC block library was designed in MATLAB/Simulink and it is to be used exclusively
within this program environment. The core of the block library was developed under the configuration
of OS Windows Vista™, MATLAB 7.6 (Release R2008a). After compatibility testing, extensions to
the library were made under OS Windows 7™ Enterprise, MATLAB 7.9 (Release R2009b) as well as
MATLAB 7.12 (R2011a).
The installation process of the IPMaC block library consists of unzipping the provided
InvPend.zip package into a desired directory and calling the included installation script slblocks.m.
Once installed, the library becomes an integral part of the Simulink Library Browser, and the IPMaC
function blocks are fully compatible with the blocks from the rest of Simulink built-in libraries.
Fig. 2 The IPMaC, installed and active in the Simulink Library Browser
Like in any other Simulink blockset, the IPMaC function blocks are thematically arranged into
sublibraries which open when the corresponding item of the IPMaC tree structure in the Simulink
Library Browser is singleclicked. The current version of the IPMaC has kept the core of the sublibrary
structure from the original 2009 version: the Inverted Pendula Models sublibrary contains simulation
models of the selected representatives of inverted pendula systems and the Inverted Pendula Control
sublibrary is made up of function blocks that implement control algorithms. Additional sublibraries
have been designed to incorporate the actuating mechanism block; useful sink and source blocks; or a
heuristic swing-up controller. The sublibraries that the IPMaC is currently structured into are:
• Inverted Pendula Control sublibrary – contains dynamic-masked function blocks which
encapsulate state-feedback control algorithms
• Inverted Pendula Models sublibrary – contains dynamic-masked simulation models of
selected classical and rotary inverted pendula systems, pre-prepared for use in open-loop
analysis as well as in state-feedback control design
• Inverted Pendula Motors sublibrary – contains a block which implements the analytical
model of the most frequently employed actuating mechanism (brushed direct-current motor)
• Inverted Pendula Sinks and Sources sublibrary – concentrates all sink and source blocks
which may find their use in pendula modeling and control, such as an input signal constrained
in time and amplitude or a sink block that displays a signal in degrees rather than radians
• Inverted Pendula Swing-up sublibrary – contains a heuristic controller which swings the
pendulum from the downward to the upright equilibrium and switches to stabilizing control in
the moment the pendulum is sufficiently close to the upright position.
In addition to sublibraries, the root directory of the IPMaC includes two blocks of a special type which
open up into a collection of links that lead to separate MATLAB/Simulink tools:
• Demo Simulations - opens up into a tree structure of links to simulation schemes which
illustrate the ways of interconnecting the IPMaC blocks to solve a variety of problems
• Modeling and Linearization Tools – opens up into a set of links to a pair of applications
which provide a comfortable, user-friendly graphical interface to modeling and linearization.
3 Automatic Approach to Inverted Pendula Mathematical Modeling
The automatic approach to motion equation derivation was promoted for a number of reasons: it
yields a particularly precise approximation of the real system’s dynamics and eliminates any factual or
numeric errors which could arise during manual mathematical modeling. To enable the automatic
derivation of motion equations for inverted pendula systems, a general procedure has to be available
which will output the equations for any given number of pendula [1][3][4]. Analysis of generalized
classical and rotary inverted pendula systems therefore needs to be dealt with.
The generalized system of classical inverted pendula is defined as a set of n>1 rigid,
homogenous rods (pendulum links) which are interconnected in joints and attached to a cart (i.e.
a stable mechanism which allows for movement alongside a single axis). Analogically, the generalized
system of rotary inverted pendula is composed of a set of n>1 interconnected homogenous rods
mounted on a rigid arm which rotates in a horizontal plane, perpendicular to the pendula. Every
system of n inverted pendula is a multi-body composed of the base and the individual pendula, and
thus has
1
n
+
degrees of freedom. It is a typical example of an underactuated system since the number
of actuators is lower than the number of system links: the only input (the force
(
)
F t
acting upon the
cart or the torque
(
)
M t
applied on the rotary arm) is used to control the
n+1 outputs of the system:
cart position
[
]
m
or arm angle
[
]
rad
, and pendula angles
[
]
rad
.
3.1 Derivation of Motion Equations
The Inverted Pendula Model Equation Derivator is a MATLAB GUI application which
generates the motion equations for a user-chosen type of inverted pendula system (classical/rotary,
single/double). Fig. 3 shows an example preview of the Derivator window which contains the
generated model equations for the rotary single inverted pendulum system. The whole derivation
process can be tracked in the command window (see the preview in [3]).
The core of the Derivator tool is represented by MATLAB functions that use the Symbolic Math
Toolbox to implement general procedures that derive the motion equations for a classical or rotary
inverted pendula system. If we view the system’s degrees of freedom as the following vector of
generalized coordinates [2][7]:
( ) ( ) ( ) ( )
(
)
0 1
T
n
t t t t
θ θ θ
= …θ
(1)
then the system can be mathematically described by the
Euler-Lagrange equations of second kind
(one
for each generalized coordinate). The vector form of the equations is:
(
)
( )
(
)
( )
(
)
( )
( )
*
L t L t D t
d
t
dt t
t t
∂ ∂ ∂
− + =
∂
∂ ∂
ɺ ɺ
Q
θ
θ θ
(2)
Fig. 3 Inverted Pendula Model Equation Derivator with the derived rotary single inverted pendulum model
equations
where
(
)
L t
(Lagrange function) is defined as the difference between the system’s kinetic and
potential energy,
(
)
D t
(Rayleigh, dissipation function) describes the viscous (friction) forces and
(
)
*
t
Q
is the vector of generalized external forces acting upon the system. The process of derivation of
the motion equations to describe any kind of inverted pendula system has thus transformed into the
determination of kinetic, potential and dissipation energies related to the base and all pendula:
( )
0
( )
n
n
K Ki
i
E t E t
=
=
∑
,
( )
0
( )
n
n
P Pi
i
E t E t
=
=
∑
,
( )
0
( )
n
i
i
D t D t
=
=
∑
(3)
Using well-known physical formulae and the necessary theoretical assumptions, general
relations that describe the energetic balances of the base and i-th pendulum in a system of n inverted
pendula were derived and can be found in [1] for both classical and rotary inverted pendula systems.
Procedures which generate the equations of motion for any given system type and number of pendula
were subsequently constructed and implemented as symbolic MATLAB functions (invpenderiv.m for
classical, rotinvpenderiv.m for rotary) which produce the equations in the simplified and rearranged
form, equivalent to the most likely form obtained by manual derivation.
The mathematical models generated by the Derivator will hereafter be referred to as
force/torque models, to distinguish them from voltage models, which will be derived in the following
section.
3.2 Simulation Models of Classical and Rotary Inverted Pendulum Systems
The Inverted Pendula Models sublibrary of the IPMaC block library contains library blocks
which represent the simulation models of a classical single, classical double and rotary single inverted
pendulum system: the Classical Single Inverted Pendulum, the Classical Double Inverted Pendulum
and the Rotary Single Inverted Pendulum block. The structure of all blocks is composed of logically
designed subsystem blocks which are interconnected with respect to their mutual physical relations
(Fig. 4); each subsystem block implements a nonlinear differential equation that is part of the system’s
Lagrange mathematical model, obtained from the Inverted Pendula Model Equation Derivator
application. Every model is composed of n+1 (2 for single, 3 for double) nonlinear second-order
differential equations that describe the base and each pendulum. The motion equations that make up
the respective mathematical models are specified in [1][2][3].
4
dfi1dt pendulum angular velocity
3
dfi0dt arm angular velocity
2
fi1 pendulum angle
1
fi0 arm angle
M external torque
fi0 arm angle
fi1 pendulum angle
dfi0dt arm angular velocity
dfi1dt pendulum angular velocity
Rotary Single Inverted Pendulum
dfi0/dt
d2fi0/dt2
fi1
dfi1/dt
d2fi1/dt2
Pendulum
M
fi1
dfi1/dt
d2fi/dt2
fi0
dfi0/dt
d2fi0/dt2
Arm
1
M external torque
3
d2fi0/dt2
2
dfi0/dt
1
fi0
-C-
init(3)
-C-
init(1)
1
s
x
o
Integrator1
1
s
x
o
Integrator
f(u)
Fcn
4
d2fi/dt2
3
dfi1/dt
2
fi1
1
M
d2fi0dt
d2fi0dt
dfi0dt
dfi0dt
fi0
3
d2fi1/dt2
2
dfi1/dt
1
fi1
-C-
init(4)
-C-
init(2)
1
s
x
o
Integrator2
1
s
x
o
Integrator1
f(u)
Fcn
2
d2fi0/dt2
1
dfi0/dt
fi1
fi1
d2fi1dt
dfi1dt
dfi1dt
Fig. 4 The structure of the Rotary Single Inverted Pendulum library block
Each included system is equipped with a dynamic block mask (Edit – Mask Subsystem) which
enables the user to edit the physical parameters and initial conditions, to enable or disable the input
(force/torque) port and to flexibly adjust the number of the block’s output ports, which is equivalent to
fitting a real model with sensors. This was implemented by creating scripts which add the
Input/Output Port block into the scheme whenever the input or one of the outputs is marked as
required; and replace it by the Ground/Terminator block if this is not the case.
3.3 Simulation Models of Direct-Current Motors
The practical use of force/torque models generated by the Derivator and included in the
Inverted Pendula Models sublibrary is limited purely to the simulational environment: we are unable
to manually generate a force or torque which, if applied to the base, would stabilize the pendula in
a chosen position. As a result, electric motors need to be coupled with inverted pendula systems to act
as mechanisms which actually produce the force or torque which actuates both the base and the
attached pendula.
Although there has recently been a rise in the percentage of brushless (EC, BLDC) direct-
current motors, three-phase synchronous motors and asynchronous (induction) motors [1][7][10], the
great majority of authors dealing with modeling and control of inverted pendula systems still use the
brushed direct-current (DC) motor as a high-performance drive to actuate their system. Hence, it was
decided that the IPMaC/Inverted Pendula Motors sublibrary would presently contain a single block:
the DC Motor for Inverted Pendula Systems, which implements the mathematical model of a brushed
direct-current (DC) motor in form of a voltage-to-force or a voltage-to-torque conversion relationship,
derived in [1]. Appending the DC motor model to an inverted pendula system yields a voltage model
of the system.
Fig. 5 The dynamically changing dialog box of the DC Motor for Inverted Pendula Systems block
The appearance of the block and the dynamic block mask can be adjusted to correspond with
the type of inverted pendula system (classical/rotary) it is associated with (Fig. 5).
4 Implementation of State-Feedback Control Algorithms
It is well-known from experience that, once released and independently of the initial state, an
inverted pendula system will always stabilize with all pendula in their natural hanging position, i.e. in
their stable equilibrium. The principal control objective for all considered inverted pendula systems
was therefore to stabilize all pendulum links in the vertical upright (inverted) position, which
represents the unstable equilibrium of the system. Several additional problems (subobjectives) were
approached and solved:
• stabilizing the system following the pendulum’s initial deflection (nonzero initial conditions)
• compensation of a time-constrained (impulse) or permanent (step) disturbance input signal,
• tracking a desired position (reference trajectory) of the base.
Linear state-feedback control was emphasized as the principal technique for the control of the inverted
pendula systems since control of several degrees of freedom at once can only be ensured if the whole
state vector is taken into consideration.
4.1 Linear Approximation of Inverted Pendula Systems
To be able to employ linear control techniques, the linear approximation of the nonlinear
inverted pendulum system needs to be obtained. The motion equations generated by the Derivator are
first rewritten into the standard (minimal ODE – ordinary differential equation) form [8]:
( )
(
)
( ) ( ) ( )
(
)
( ) ( )
(
)
( )
,
t t t t t t t
+ + =
ɺɺ ɺ ɺ
M
θ θ N θ θ θ P θ V
(4)
which provides the only way to express this kind of system in the nonlinear state-space form of
(
)
(
)
(
)
( ) ( )
( )
( ) , ,
( ) , ,
t t u t t
t t u t t
=
=
ɺ
x f x
y g x
(5)
by defining the state vector as
( ) ( ) ( )
(
)
T
t t t
=
ɺ
x θ θ
and isolating the second derivative
(
)
t
ɺɺ
θ
from (4).
All inverted pendula systems included in the IPMaC were modeled in a way which defines the
“all upright” equilibrium as
(
)
T
S
t
= =
x x 0
. If the input
(
)
0
S
u t u
= =
, then the state-space description
of the continuous linearized system is given as
(
)
(
)
(
)
( ) ( ) ( )
t t u t
y t t du t
= +
= +
ɺ
x Ax b
Cx
, (6)
and is obtained by expanding (5) into the Taylor series around a given equilibrium point and
neglecting the terms of the order greater than 1.
Fig. 6 Inverted Pendula Model Linearizator & Discretizer with the linearized and discretized state-space
matrices of the classical double inverted pendulum system
The process of transformation of the Lagrange mathematical model of an inverted pendulum
system into a linearized state-space matrix form can be considerably sped up with help of another GUI
application designed as part of the IPMaC: the Inverted Pendula Model Linearizator & Discretizer
(Fig. 6). In case the type of inverted pendula system (classical/rotary), model parameters and operating
(equilibrium) point have been provided by the user, the application generates the numeric form of the
A
,
b
,
C
, d continuous state-space matrices (see (6)) of the system in a selected operating point. In case
a discrete-time model is required, the Linearizator & Discretizer returns the discretized state-space
matrices
F
,
g
,
C
, d of the chosen inverted pendula system provided the sampling period constant T
vz
was entered and the continuous-time matrices have already been successfully computed.
4.2 Inverted Pendula Control
The expanded Inverted Pendula Control sublibrary of the IPMaC chiefly provides software
support for the state-feedback methods of controller design for inverted pendula systems. Most
importantly, the
State-Feedback Controller with Feedforward Gain
block implements the standard
state-feedback control law calculated either from the continuous-time state-space description
(
)
(
)
(
)
(
)
(
)
(
)
(
)
R ff u ff u
u t u t u t d t t k w t d t
= + + = − + +kx
,
(7)
or from the discrete-time linear state-space description
(
)
(
)
(
)
(
)
(
)
(
)
(
)
R ff u D ffD u
u i u i u i d i x i k w i d i
= + + =− + +k
,
(8)
where k (
D
k
) is the feedback gain which brings the state vector
(
)
t
x
(
(
)
i
x
) into the origin of the state
space, k
ff
is the feedforward (setpoint) gain which makes the output track the reference command and
(
)
u
d t
(
(
)
u
d i
) is the unmeasured disturbance [1][2][3][9][10]. The preferred state-space description is
selected in the dynamic block mask (Fig. 7) which also allows the user to choose the method to
determine the feedback gain
k
(
D
k
) from between the pole-placement algorithm and the linear
quadratic regulation (LQR) optimal control method. To match a particular control subobjective, the
block’s appearance may be adjusted by optional enabling or disabling of the reference command input
(
)
w t
(
(
)
w i
) and/or the disturbance input
(
)
u
d t
(
(
)
u
d i
).
Fig. 7 The dialog box of the State-Feedback Controller with Feedforward Gain block, dynamically changing
according to the chosen method
The structure of the State-Feedback Controller with Summator block implements a summator
(
)
v i
of all past error values [11], which ensures that the system output will track the changes in the
reference command and eliminate the influence of permanent disturbances. The control law is given as
(
)
(
)
(
)
(
)
(
)
(
)
(
)
1 2R ff S
u i u i u i u i i k w i v i
= + + = − + +k x
,
(9)
The layout of the block mask once again allows the user to adjust the number of block input ports to
match a control subobjective and to compute the state-feedback vector
1
k
using a preferred method
and state-space description.
The
Luenberger Estimator
block provides a complete, reconstructed state vector by evaluating
a model of the original discrete-time system in the structure:
(
)
(
)
(
)
(
)
(
)
(
)
ˆ ˆ
1
i i u i i i
+ = + + −x Fx g L y Cx
(10)
where
L
is the estimator gain matrix and
(
)
ˆ
i
x
is the reconstructed state vector.
5 Inverted Pendula Demo Simulations Section – Overview
The Inverted Pendula Demo Simulations section, accessible through the root directory of the
IPMaC, displays a structured set of links to simulation schemes which demonstrate the functionality of
the designed library blocks. Each demo scheme can be accessed by doubleclicking a block that briefly
describes the issue it solves. Making a classic masked block to act as a link to another file involves
creating an OpenFcn callback function within the Block Properties.
Compared to the Demo Simulations section of the 2009 version of the library, the collection of
links has now been notably expanded to include a wider variety of addressed problems. The most
significant changes to the Demo Simulations section of the IPMaC were the following:
Fig. 8 Structure of the root directory of the Demo Simulations section
• the rotary single inverted pendulum model was introduced into the library
• a DC motor block was implemented into a separate sublibrary; both force/torque models and
voltage models of inverted pendula were used in simulations
• the control section was expanded to include more state-feedback controller blocks and control
schemes; control algorithms were verified using every implemented simulation model
• the tree structure of the Demo Simulations section was completely revised (Fig. 8).
5.1 Open-loop Dynamic Analysis
Evaluatory open-loop responses to an impulse signal were performed for all analyzed nonlinear
simulation models both without the motor (force/torque model) and with the motor (voltage model).
The respective simulation schemes can be located in the Nonlinear Analysis and Nonlinear Analysis
with DC Motor sections. Examples are shown in Fig. 9 (force model of a classical double inverted
pendulum) and Fig. 10 (voltage model of a classical single inverted pendulum).
Scope
rad2deg
Scope
Impulse
F external force on the cart
fi0 cart position
fi1 lower pendulum angle
fi2 upper pendulum angle
Classical Double Inverted Pendulum
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Simulation time [s]
Cart Position [m]
Cart Position Analysis - Double Inverted Pendulum
Cart Position
0 2 4 6 8 10 12 14 16 18 20
-350
-300
-250
-200
-150
-100
-50
0
50
Simulation time [s]
Pendulum Angle [deg]
Pendulum Angle (Lower & Upper) Analysis -
Double Inverted Pendulum
Pendulum Angle (Lower)
Pendulum Angle (Upper)
Fig. 9 Classical double inverted pendulum system – simulation scheme and time behavior results (cart
position and pendula angles)
Scope2
Scope1
Scope
rad2deg1
Scope
rad2deg
Scope
Impulse
V motor voltage
dfi0dt cart velocity
F force induced on the cart
DC Motor for Inverted Pendula Systems
F external force on the cart
fi0 cart position
fi1 pendulum angle
dfi0dt cart velocity
dfi1dt pendulum angular velocity
Classical Single Inverted Pendulum
0 2 4 6 8 10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Cart Position Analysis
Classical Single Inverted Pendulum with DC Motor
Simulation time [s]
Cart Position [m]
0 2 4 6 8 10 12 14 16 18 20
-300
-250
-200
-150
-100
-50
0
Pendulum Angle Analysis -
Classical Single Inverted Pendulum with DC Motor
Simulation time [s]
Pendulum Angle [deg]
Fig. 10 Classical single inverted pendulum system with a DC motor – simulation scheme and time
behavior results (cart position and pendulum angle)
Reasonable behavior of the open-loop responses of the simulation models (damped oscillatory
transient state, system reaching the stable equilibrium point with all pendula pointing downward,
visible backward impact of the pendulum/pendula on the base, motor-driven classical inverted
pendulum response being much more rigid than that of its rotary equivalent) means the models can be
considered accurate enough to serve as a reliable testbed for control algorithms.
5.2 Verifications of Inverted Pendula State-Feedback Control
The schemes in the Demo Simulations section document a number of simulation experiments
which were performed to prove that the implemented controllers are able to meet the required control
objectives for all available simulation models of inverted pendula systems.
Each demo simulation scheme is designed as an independent unit. All computations required for
the simulation to run (model parameter definition, state-space matrices of the linear approximation
necessary for control algorithm design) are programmed to occur during the initialization of the
scheme, which eliminates the need for additional m-files. Moreover, if the parameters of the
simulation model are changed by the user in an open scheme, the Linearizator & Discretizer tool can
be called from a provided link to obtain modified state-space matrices (Fig. 11, Fig. 12).
The example schemes below were picked from the State-Feedback Control and Summator
Control sections to illustrate the way of interconnecting the blocks to control the voltage model of a
rotary single inverted pendulum system. Fig. 11 depicts the simulation results given that the control
objective was to maintain the desired arm angle value while keeping the pendulum upright. An
estimator block was included to simulate measurement limitations.
Model already linearized and discretized using default parameters.
Use Linearizator to obtain modified matrices if a change in parameters occurs. --->>>
x
w
u
State-Feedback Controller
x
u
State-Feedback Controller with Feedforward Gain1
Signal 1
Signal Builder1
Scope
rad2deg1
Scope
rad2deg
M external torque
fi0 arm angle
fi1 pendulum angle
dfi0dt arm angular velocity
Rotary Single Inverted Pendulum
xe(i+1)=Fxe(i)+Gu(i)+L(y(i)-ye(i))
u
y
xe
Luenberger Estimator
Linearizator
Inverted Pendula Model
Linearizator and Discretizer
[ref]
Goto3
[pend1]
Goto2
[arm1]
Goto1
-K-
Gain
[pend1]
From5
[arm1]
From1
[ref]
From
V motor voltage
dfi0dt arm angular velocity
M torque induced on the cart
DC Motor for Inverted Pendula Systems
0
Constant
0 2 4 6 8 10 12 14 16 18 20
-50
0
50
100
150
200
Arm Angle - Reference Trajectory
Rotary Inverted Pendulum
Simulation time [s]
Arm Angle [deg]
Arm Angle Setpoint [deg]
Arm Angle [deg]
0 2 4 6 8 10 12 14 16 18 20
-10
-5
0
5
10
15
Pendulum Angle Stabilization
Rotary Inverted Pendulum
Simulation time [s]
Pendulum Angle [deg]
Pendulum Angle Setpoint [deg]
Pendulum Angle [deg]
Fig. 11 Rotary single inverted pendulum: simulation results for LQR control with estimator included;
note that the arm is supposed to rotate for exactly a half-circle before returning to its initial position
Finally, it is shown in Fig. 12 that applying state-feedback control with feedforward gain on
a system is unsufficient if the steady-state effect of a permanent disturbance input needs to be
eliminated. The compensation of permanent disturbances is successfully executed by a LQR algorithm
with a summator included in the control structure.
Model already linearized and discretized using default parameters.
Use Linearizator to obtain modified matrices if a change in parameters occurs. --->>>
Step1
Step
x
w
y
du
u
State-Feedback Controller with Summator
x
u
State-Feedback Controller with a Summator
x
w
du
u
State-Feedback Controller
x
u
State-Feedback Controller with Feedforward Gain1
Signal 1
Signal Builder1
Scope
rad2deg1
Scope
rad2deg
M external torque
fi0 arm angle
fi1 pendulum angle
dfi0dt arm angular velocity
Rotary Single Inverted Pendulum1
M external torque
fi0 arm angle
fi1 pendulum angle
dfi0dt arm angular velocity
Rotary Single Inverted Pendulum
xe(i+1)=Fxe(i)+Gu(i)+L(y(i)-ye(i))
u
y
xe
Luenberger Estimator1
xe(i+1)=Fxe(i)+Gu(i)+L(y(i)-ye(i))
u
y
xe
Luenberger Estimator
Linearizator
Inverted Pendula Model
Linearizator and Discretizer
[pend2]
Goto5
[arm2]
Goto4
[ref]
Goto3
[pend1]
Goto2
[arm1]
Goto1
-K-
Gain
[pend1]
From5
[pend2]
From3
[arm2]
From2
[arm1]
From1
[ref]
From
V motor voltage
dfi0dt arm angular velocity
M torque induced on the cart
DC Motor for Inverted Pendula Systems1
V motor voltage
dfi0dt arm angular velocity
M torque induced on the cart
DC Motor for Inverted Pendula Systems
0
Constant
0 5 10 15 20
-50
0
50
100
150
200
Simulation time [s]
Arm angle [deg]
Arm Angle (Permanent Disturbance Compensation)
Rotary Single Inverted Pendulum
Arm angle reference
LQR with summator
LQR without summator
0 5 10 15 20
-30
-20
-10
0
10
20
30
Simulation time [s]
Pendulum angle [deg]
Pendulum Angle (Permanent Disturbance Compensation)
Rotary Single Inverted Pendulum
Pendulum angle reference
LQR with summator
LQR without summator
Fig. 12 Rotary single inverted pendulum – simulation results for LQR control with a summator
(integral action)
5.3 Verifications of Inverted Pendula Swing-up
To illustrate the problem of pendulum swing-up, the Swing-up Controller block was included in
the IPMaC/Inverted Pendula Swing-up sublibrary. The block implements three heuristic, energy-based
control laws (cosine value controller, zero speed controller and absolute value controller) that swing
the pendulum up from the downward to the upright equilibrium [12]. A transition mechanism switches
between swing-up and stabilizing control provided the pendulum is sufficiently close to the upright
position. The block mask allows the user to select a swing-up method, the input voltage magnitude
and the size of the balancing region.
0 2 4 6 8 10 12 14 16 18 20
-100
-50
0
50
100
150
Simulation time [s]
Arm angle [deg]
Arm Angle - Swing-up and Stabilization
Rotary Single Inverted Pendulum
Absolute value controller
Zero speed controller
Cosine value controller
0 2 4 6 8 10 12 14 16 18
20
-350
-300
-250
-200
-150
-100
-50
0
50
Pendulum angle [deg]
Pendulum Angle - Swing-up and Stabilization
Rotary Single Inverted Pendulum
Simulation time [s]
Absolute value controller
Zero speed controller
Cosine value controller
Fig. 13 Swing-up and stabilization of the rotary single inverted pendulum system – comparison of methods
The effectiveness of the three swing-up methods was compared in a simulation experiment
involving the rotary single inverted pendulum model (Fig. 13).
6 Conclusion
The purpose of this paper was to present a comprehensive and unifying approach to the problem
of modeling and control of classical and rotary inverted pendula dynamical systems. A structured,
thematic Simulink block library, Inverted Pendula Modeling and Control (IPMaC), was developed to
serve as a software framework for all solved issues. The library is composed of a set of custom
function blocks which centers around the simulation models of the classical single, classical double
and rotary single inverted pendulum system. An integral part of the library is the Demo Simulations
section, which is basically a collection of links to simulation schemes which illustrate the ways of
interconnecting the blocks to solve various problems related to analysis and control of inverted
pendula systems.
Practical importance of symbolic mathematical software, represented in MATLAB by Symbolic
Math Toolbox, was demonstrated: general symbolic procedures were implemented that either yield the
motion equations of a mathematical model for a user-chosen classical or rotary inverted pendulum
system, or perform the symbolic linear transformation of a specified system in a chosen equilibrium
point. Applications with graphical user interface were developed to provide a comfortable and user-
friendly access to both procedures.
The presented Simulink block library offers solid program support for both modeling and
control of inverted pendula systems, based on the modern object-oriented approach to problem
solving. Since the library was designed as an open system with the classical and rotary inverted
pendula simulation models at its core, it will be possible to add a wider variety of controller blocks
and control schemes to the already included state-feedback and optimal control algorithms. Model
predictive control algorithms and exact linearization are considered among other candidate structures.
Acknowledgements
This work has been supported by the Scientific Grant Agency of Slovak
Republic under project Vega No.1/0286/11 Dynamic Hybrid Architectures of
the Multiagent Network Control Systems (50%). This work is also the result
of the project implementation Development of the Center of Information and
Communication Technologies for Knowledge Systems (project number:
26220120030) supported by the Research & Development Operational Program
funded by the ERDF (50%).
References
[1] S. Jadlovská. Modeling and Optimal Control of Inverted Pendula Systems. Diploma (Master)
Thesis. Supervisor: prof. Ing. Ján Sarnovský, PhD., Technical University of Košice, Faculty of
Electrical Engineering a Informatics, Košice, 2011.
[2] S. Jadlovská, A. Jadlovská. A Simulink Library for Inverted Pendula Modeling and Simulation.
17th Annual Conference Proceedings of the International Scientific Conference - Technical
Computing Prague 2009, Congress Center of Czech Technical University in Prague, November
19, 2009, ISBN 978-80-7080-733-0.
[3] S. Jadlovská, A. Jadlovská. Inverted Pendula Simulation and Modeling – a Generalized
Approach. Proceedings of the 9th International Scientific – Technical Conference on Process
Control ‘10, Říp 2010, June 7-10, University of Pardubice, Czech Republic, ISBN 978-80-
7399-951-3.
[4] S. Jadlovská, J. Sarnovský. Matlab-Based Tools for Analysis and Control of Inverted Pendula
Systems. Electrical Engineering and Informatics – Proceeding of the Faculty of Electrical
Engineering and Informatics of the Technical University of Košice, 2011, ISBN 978-80-553-
0460-1
[5] K. Sultan. Inverted Pendulum, Analysis, Design and Implementation. From:
http://www.mathworks.com/matlabcentral/fileexchange/3790, accessed on 1-8-2009
[6] H. X. Li, J. Y. Wang, Y. D. Gu, Y. B. Feng. Hardware Implementation of the Quadruple
Inverted Pendulum with Single Motor. Progress in Natural Science, Vol. 14, Issue 9, 2002.
[7] M. Schlegel, J. Mešťánek. Limitations on the Inverted Pendula Stabilizability According to
Sensor Placement. 16th International Conference Process Control : Proceedings 2007 (June 11-
14, 2007, Štrbské Pleso, Slovakia).
[8] A. Bogdanov. Optimal Control of a Double Inverted Pendulum on a Cart. Technical Report
CSE-04-006, OGI School of Science and Engineering, OHSU, 2004
[9] M. Demirci. Design of Feedback Controllers for a Linear System with Applications to Control
of a Double Inverted Pendulum. International Journal of Computational Cognition, Vol. 2, No.
1, p. 65-84, 2004
[10] J. Lam. Control of an Inverted Pendulum. From:
https://research.cc.gatech.edu/humanoids/sites/edu.humanoids/files/Control%20of%20inverted
%20pendulum.pdf, accessed on 10-5-2011
[11] O. Modrlák. Teorie automatického řízení I. - Základy analýzy a syntézy ve stavovém prostoru
(Studijní materiály). Technical University Liberec, 2002.
[12] M. Korda. Rotary and Linear Pendulum Control. Bachelor Thesis. Faculty of Electrical
Engineering, Prague, 2009.
Ing. Slávka Jadlovská
Department of Cybernetics and Artificial Intelligence
Faculty of Electrical Engineering and Informatics
Technical University of Košice
Letná 9, 042 00 Košice, Slovak Republic
e-mail: slavka.jadlovska@tuke.sk
prof. Ing. Ján Sarnovský, CSc.
Department of Cybernetics and Artificial Intelligence
Faculty of Electrical Engineering and Informatics
Technical University of Košice
Letná 9, 042 00 Košice, Slovak Republic
e-mail: jan.sarnovsky@tuke.sk
