International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
State space analysis, eign values and eign vectorsShilpa Shukla
State space analysis concept, state space model to transfer function model in first and second companion forms jordan canonical forms, Concept of eign values eign vector and its physical meaning,characteristic equation derivation is presented from the control system subject area.
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
State space analysis, eign values and eign vectorsShilpa Shukla
State space analysis concept, state space model to transfer function model in first and second companion forms jordan canonical forms, Concept of eign values eign vector and its physical meaning,characteristic equation derivation is presented from the control system subject area.
This presentation explains about the introduction of Nyquist Stability criterion. It clearly shows advantages and disadvantages of Nyquist Stability criterion and also explains importance of Nyquist Stability criterion and steps required to sketch the Nyquist plot. It explains about the steps required to sketch Nyquist plot clearly. It also explains about sketching Nyquist plot and determines the stability by using Nyquist Stability criterion with an example.
state space modeling of electrical systemMirza Baig
Introduction
As systems become more complex, representing them with differential equations or transfer functions becomes cumbersome. This is even more true if the system has multiple inputs and outputs. This document introduces the state space method which largely alleviates this problem. The state space representation of a system replaces an nth order differential equation with a single first order matrix differential equation. The state space representation of a system is given by two equations :
The first equation is called the state equation, the second equation is called the output equation. For an nth order system (i.e., it can be represented by an nth order differential equation) with r inputs and m outputs the size of each of the matrices is as follows:
Several features:The state equation has a single first order derivative of the state vector on the left, and the state vector, q(t), and the input u(t) on the right. There are no derivatives on the right hand side.The output equation has the output on the left, and the state vector, q(t), and the input u(t) on the right. There are no derivatives on the right hand side.
q is nx1 (n rows by 1 column)q is called the state vector, it is a function of timeA is nxn; A is the state matrix, a constantB is nxr; B is the input matrix, a constant u is rx1; u is the input, a function of time C is mxn; C is the output matrix, a constant D is mxr; D is the direct transition matrix, a constant y is mx1; y is the output, a function of time
Derivation of of State Space Model (Electrical)
To develop a state space system for an electrical system, they choosing the voltage across capacitors, and current through inductors as state variables. Recall that
so if we can write equations for the voltage across an inductor, it becomes a state equation when we divide by the inductance (i.e., if we have an equation for einductor and divide by L, it becomes an equation for diinductor/dt which is one of our state variable). Likewise if we can write an equation for the current through the capacitor and divide by the capacitance it becomes a state equation for ecapacitor
There are three energy storage elements, so we expect three state equations. Try choosing i1, i2 and e1 as state variables. Now we want equations for their derivatives. The voltage across the inductor L2 is e1 (which is one of our state variables)so our first state variable equation is
This equation has our input (ia) and two state variable (iL2 and iL1) and the current through the capacitor. So from this we can get our second state equation
Our third, and final, state equation we get by writing an equation for the voltage across L1 (which is e2) in terms of our other state variables
references:
http://lpsa.swarthmore.edu/Representations/SysRepSS.html
https://en.wikipedia.org/wiki/State-space_representation
Root locus is a graphical representation of the closed-loop poles as a system parameter is varied.
It can be used to describe qualitatively the performance of a system as various parameters are changed.
It gives graphic representation of a system’s transient response and also stability.
We can see the range of stability, instability, and the conditions that cause a system to break into oscillation.
State-Space Analysis of Control System: Vector matrix representation of state equation, State transition matrix, Relationship between state equations and high-order differential equations, Relationship between state equations and transfer functions, Block diagram representation of state equations, Decomposition Transfer Function, Kalman’s Test for controllability and observability
A New Approach to Design a Reduced Order ObserverIJERD Editor
In this paper, a new method for designing a reduced order observer for linear time-invariant system is
proposed. The approach is based on matrix inversion with proper dimension. The arbitrariness associated with
the method proposed by O’Reilly is presented here and has been reduced with the help of pole-placement
technique. It also helps reducing the computations regarding the observer design parameters. Illustrative
numerical examples with simulation results are also included.
This presentation explains about the introduction of Nyquist Stability criterion. It clearly shows advantages and disadvantages of Nyquist Stability criterion and also explains importance of Nyquist Stability criterion and steps required to sketch the Nyquist plot. It explains about the steps required to sketch Nyquist plot clearly. It also explains about sketching Nyquist plot and determines the stability by using Nyquist Stability criterion with an example.
state space modeling of electrical systemMirza Baig
Introduction
As systems become more complex, representing them with differential equations or transfer functions becomes cumbersome. This is even more true if the system has multiple inputs and outputs. This document introduces the state space method which largely alleviates this problem. The state space representation of a system replaces an nth order differential equation with a single first order matrix differential equation. The state space representation of a system is given by two equations :
The first equation is called the state equation, the second equation is called the output equation. For an nth order system (i.e., it can be represented by an nth order differential equation) with r inputs and m outputs the size of each of the matrices is as follows:
Several features:The state equation has a single first order derivative of the state vector on the left, and the state vector, q(t), and the input u(t) on the right. There are no derivatives on the right hand side.The output equation has the output on the left, and the state vector, q(t), and the input u(t) on the right. There are no derivatives on the right hand side.
q is nx1 (n rows by 1 column)q is called the state vector, it is a function of timeA is nxn; A is the state matrix, a constantB is nxr; B is the input matrix, a constant u is rx1; u is the input, a function of time C is mxn; C is the output matrix, a constant D is mxr; D is the direct transition matrix, a constant y is mx1; y is the output, a function of time
Derivation of of State Space Model (Electrical)
To develop a state space system for an electrical system, they choosing the voltage across capacitors, and current through inductors as state variables. Recall that
so if we can write equations for the voltage across an inductor, it becomes a state equation when we divide by the inductance (i.e., if we have an equation for einductor and divide by L, it becomes an equation for diinductor/dt which is one of our state variable). Likewise if we can write an equation for the current through the capacitor and divide by the capacitance it becomes a state equation for ecapacitor
There are three energy storage elements, so we expect three state equations. Try choosing i1, i2 and e1 as state variables. Now we want equations for their derivatives. The voltage across the inductor L2 is e1 (which is one of our state variables)so our first state variable equation is
This equation has our input (ia) and two state variable (iL2 and iL1) and the current through the capacitor. So from this we can get our second state equation
Our third, and final, state equation we get by writing an equation for the voltage across L1 (which is e2) in terms of our other state variables
references:
http://lpsa.swarthmore.edu/Representations/SysRepSS.html
https://en.wikipedia.org/wiki/State-space_representation
Root locus is a graphical representation of the closed-loop poles as a system parameter is varied.
It can be used to describe qualitatively the performance of a system as various parameters are changed.
It gives graphic representation of a system’s transient response and also stability.
We can see the range of stability, instability, and the conditions that cause a system to break into oscillation.
State-Space Analysis of Control System: Vector matrix representation of state equation, State transition matrix, Relationship between state equations and high-order differential equations, Relationship between state equations and transfer functions, Block diagram representation of state equations, Decomposition Transfer Function, Kalman’s Test for controllability and observability
A New Approach to Design a Reduced Order ObserverIJERD Editor
In this paper, a new method for designing a reduced order observer for linear time-invariant system is
proposed. The approach is based on matrix inversion with proper dimension. The arbitrariness associated with
the method proposed by O’Reilly is presented here and has been reduced with the help of pole-placement
technique. It also helps reducing the computations regarding the observer design parameters. Illustrative
numerical examples with simulation results are also included.
This project was developed for an Embedded systems class: we implemented a PID controller for a mechanical inverted pendulum. It was very interesting to experiment in practice with a simple control plant.
The Inverted Pendulum, Spring Mass and Integrated Spring Mass Approach to Tre...Dr. James Stoxen DC
The Inverted Pendulum, Spring-Mass and Integrated Spring Mass Approach to Treating Plantar Fasciitis
A presentation by Dr. James Stoxen DC
The Malaysia World Congress in Sports and Exercise Medicine
August 26-30, 2014
Royale Chulan in Kula Lumpur, Malaysia
Design of observers for nonlinear systems using the Frobenius theorem. Presentation for the defense of my MSc Thesis at the School of Applied Mathematics, NTU Athens.
MODELLING AND SIMULATION OF INVERTED PENDULUM USING INTERNAL MODEL CONTROLJournal For Research
The internal model control (IMC) philosophy relies on the internal model principle, which states that control can be achieved only if the control system encapsulates, either implicitly or explicitly, some representation of the process to be controlled. In particular, if the control scheme is developed based on an exact model of the process, then perfect control is theoretically possible. Transfer function of Inverted Pendulum is selected as the base of design, which examines IMC controller. Matlab/simulink is used to simulate the procedures and validate the performance. The results shows robustness of the IMC and got graded responses when compared with PID. Furthermore, a comparison between the PID and IMC was shows that IMC gives better response specifications.
Sometimes companies do not realize why the goals of its employees are not aligned with the organization's goals. This presentation identifies the potential problems and possible solutions.
Mathematical model analysis and control algorithms design based on state feed...hunypink
XZ-Ⅱtype rotary inverted pendulum is a typical mechatronic system; it completes real-time motion control using DSP motion controller and motor torque. In this paper, we recognize XZ-Ⅱrotational inverted pendulum and learn system composition, working principle, using method, precautions and software platform. We master how to build mathematical model and state feedback control method (pole assignment algorithm) of the one order rotational inverted pendulum system and finish simulation study of system using Mat lab. In the end we grasp debugging method of the actual system, and finish online control of the one order rotational inverted pendulum system as well.
Experimental verification of SMC with moving switching lines applied to hoisti...ISA Interchange
In this paper we propose sliding mode control strategies for the point-to-point motion control of a hoisting crane. The strategies employ time-varying switching lines (characterized by a constant angle of inclination) which move either with a constant deceleration or a constant velocity to the origin of the error state space. An appropriate design of these switching lines results in non-oscillatory convergence of the regulation error in the closed-loop system. Parameters of the lines are selected optimally in the sense of two criteria, i.e. integral absolute error (IAE) and integral of the time multiplied by the absolute error (ITAE). Furthermore, the velocity and acceleration constraints are explicitly taken into account in the optimization process. Theoretical considerations are verified by experimental tests conducted on a laboratory scale hoisting crane.
International Journal of Engineering Inventions (IJEI) provides a multidisciplinary passage for researchers, managers, professionals, practitioners and students around the globe to publish high quality, peer-reviewed articles on all theoretical and empirical aspects of Engineering and Science.
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
Linear quadratic regulator and pole placement for stabilizing a cart inverted...journalBEEI
The system of a cart inverted pendulum has many problems such as nonlinearity, complexity, unstable, and underactuated system. It makes this system be a benchmark for testing many control algorithm. This paper presents a comparison between 2 conventional control methods consist of a linear quadratic regulator (LQR) and pole placement. The comparison indicated by the most optimal steps and results in the system performance that obtained from each method for stabilizing a cart inverted pendulum system. A mathematical model of DC motor and mechanical transmission are included in a mathematical model to minimize the realtime implementation problem. From the simulation, the obtained system performance shows that each method has its advantages, and the desired pendulum angle and cart position reached.
Navigation of Mobile Inverted Pendulum via Wireless control using LQR TechniqueIJMTST Journal
Mobile Inverted Pendulum (MIP) is a non-linear robotic system. Basically it is a Self-balancing robot
working on the principle of Inverted pendulum, which is a two wheel vehicle, balances itself up in the vertical
position with reference to the ground. It has four configuration variables (Cart position, Cart Velocity,
Pendulum angle, Pendulum angular velocity) to be controlled using only two control inputs. Hence it is an
Under-actuated system. This paper focuses on control of translational acceleration and deceleration of the
MIP in a dynamically reasonable manner using LQR technique. The body angle and MIP displacement are
controlled to maintain reference states where the MIP is statically unstable but dynamically stable which
leads to a constant translational acceleration due to instability of the vehicle. In this proposal, the
implementation of self balancing robot with LQR control strategy and the implementation of navigation
control of the bot using a wireless module is done. The simulation results were compared between PID control
and LQR control strategies.
Design and Simulation of Aircraft Autopilot to Control the Pitch AngleEhab Al hamayel
In this paper, we are going to design an aircraft autopilot to control the pitch angle by apply the state-space controller design technique. In particular, we will attempt to place the closed-loop poles of the system by designing a controller that calculates its control based on the state of the system. Because the dynamic equations covering the motion of the motion of the aircraft are a very complicated set of six nonlinear coupled differential equations. We will use a linearized longitudinal model equation under certain assumption to build the aircraft pitch controller also we will verify the design and check the response using MatLab&Simulink.
Design and Simulation of Different Controllers for Stabilizing Inverted Pendu...IJERA Editor
The Inverted Pendulum system has been identified for implementing controllers as it is an inherently unstable system having nonlinear dynamics. The system has fewer control inputs than degrees of freedom which makes it fall under the class of under-actuated systems. It makes the control task more challenging making the inverted pendulum system a classical benchmark for the design, testing, evaluating and comparing. The inverted pendulum to be discussed in this paper is an inverted pendulum mounted on a motor driven cart. The aim is to stabilize the system such that the position of the cart on the track is controlled quickly and accurately so that the pendulum is always erected in its vertical position. In this paper the linearized model was obtained by Jacobian matrix method. The Matlab-Simulink models have been developed for simulation for optimal control design of nonlinear inverted pendulum-cart dynamic system using different control methods. The methods discussed in this paper are a double Proportional-Integral-Derivative (PID) control method, a modern Linear Quadratic Regulator (LQR) control method and a combination of PID and Linear Quadratic Regulator (LQR) control methods. The dynamic and steady state performance are investigated and compared for the above controllers.
Position Control of Satellite In Geo-Stationary Orbit Using Sliding Mode Con...ijsrd.com
In this paper, sliding mode control, or SMC, in control theory is a form of variable structure control (VSC). It is a nonlinear control method that alters the dynamics of a nonlinear system by application of a high-frequency switching control. It switches from one continuous structure to another based on the current position in the state space. The state feedback control law is not a continuous function of time. Hence, sliding mode control is a variable structure control method. The multiple control structures are designed so that trajectories always move toward a switching condition, and so the ultimate trajectory will not exist entirely within one control structure. Instead, the ultimate trajectory will slide along the boundaries of the control structures. The motion of the system as it slides along these boundaries is called a sliding mode and the geometrical locus consisting of the boundaries is called the sliding (hyper) surface. Using this law we can control the Satellite's position in Geostationary Orbit.
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Troubleshooting and Enhancement of Inverted Pendulum System Controlled by DSP...Thomas Templin
An inverted pendulum is a pendulum that has its center of mass above its pivot point. It is often implemented with the pivot point mounted on a cart that can move horizontally and may be called a cart-and-pole system. A normal pendulum is always stable since the pendulum hangs downward, whereas the inverted pendulum is inherently unstable and trivially underactuated (because the number of actuators is less than the degrees of freedom). For these reasons, the inverted pendulum has become one of the most important classical problems of control engineering. Since the 1950s, the inverted-pendulum benchmark, especially the cart version, has been used for the teaching and understanding of the use of linear-feedback control theory to stabilize an open-loop unstable system.
The objectives of this project are to:
• Focus on hardware and software troubleshooting and enhancement of an inverted-pendulum system controlled by a DSP28355 microprocessor and CCSv7.1 software.
• Use the swing-up strategy to move the pendulum into the unstable upward position (‘saddle’). The cart/pole system employs linear bearings for back-and-forward motion. The motor shaft has a pinion gear that rides on a track permitting the cart to move in a linear fashion. Both rack and pinion are made of hardened steel and mesh with a tight tolerance. The rack-and-pinion mechanism eliminates undesirable effects found in belt-driven and free-wheel systems, such as slippage or belt stretching, ensuring consistent and continuous traction.
• The motor shaft is coupled to a high-resolution optical encoder that accurately measures the position of the cart. The angle of the pendulum is also measured by an optical encoder, and the system employs an LQR controller to stabilize the pendulum rod at the unstable-equilibrium position.
• Addition of real-time status reporting and visualization of the system.
For the project, the Quanser High Frequency Linear Cart (HFLC) was used. The HFLC system consists of a precisely machined solid aluminum cart driven by a high-power 3-phase brushless DC motor. The cart slides along two high-precision, ground-hardened stainless steel guide rails, allowing for multiple turns and continuous measurement over the entire range of motion.
Our team implemented a control strategy that consists of a linear stabilizing LQR controller, proportional-integral swing-up control, and a supervisory coordinator that determines the control strategy (LQR or swing-up) to be used at any given time. The function of the linear stabilizer is to stabilize the system when it is in the vicinity of the unstable equilibrium. When the pendulum is in its natural state (straight-down stable-equilibrium node), the swing-up controller provides the cart/pendulum system with adequate energy to move the pendulum to the unstable equilibrium inside the “region of attraction” in which the linearized LQR controller is functional.
Troubleshooting and Enhancement of Inverted Pendulum System Controlled by DSP...
Inverted Pendulum Control System
1. Robust Control of Inverted
Pendulum using LQR with
FeedForward control and Steady
State error tracking
EL7253: State Space Design for Linear Control Systems
by
Aniket Govindaraju
0507565
Department of Electrical and Computer Engineering
Polytechnic Institute of NYU
under
Dr. Prashanth Krishnamurthy
Department of Electrical and Computer Engineering
2. Abstract
The inverted pendulum is an unstable non-minimum phase plant, H-infinity
output feedback is far less robust. The objective is to determine the control
strategy that delivers better performance with respect to pendulum’s angle
and cart’s position. A Linear Quadratic Regulator with feedforward control
and steady state error tracking technique for controlling the linearized system
of the inverted pendulum model is presented. The result is compared against
previous H-infinity results obtained. Simulation study in MATLAB environment
shows that the LQR technique is capable of controlling the multi output
system successfully. A robust LQR Tracking controller is realized and results
show that good performance can be achieved and the uncertainities can be
compensated using the proposed controller.
3. Introduction
An Inverted Pendulum System is one of the most well known equipment in the
field of control systems theory. It is a non-linear problem which is linearized
in the control schemes. In general, the control of this system by classical
methods is a difficult task. This is mainly because the non-linear system has
two degrees of freedom (i.e, angle of the inverted pendulum and the position
of the cart) and only one control input.
The control strategies such as LQR and robust control is used to overcome
the Inverted Pendulum problem. Robust control scheme used in this paper is
a Linear Quadratic Regulator Tracking Controller with a FeedForward
controller. Performance of the pendulum’s angle and cart’s position is
assessed and presented.
Informally, a controller designed for a particular set of parameters is said to
be robust if it would also work well under a different set of assumptions.
High-gain feedback is a simple example of a robust control method; with
sufficiently high gain, the effect of any parameter variations will be negligible.
But high gains mean high control costs and performance costs. To avoid
that, a Linear Quadratic Controller is used.
4. System Modelling
The system in this example consists of an inverted pendulum mounted on a
motorized cart. The Inverted Pendulum system is unstable without control, i.e,
the cart needs to be controlled to balanced pendulum. Additionally the
dynamics of the system are non linear. The objective of the control system is
to balance the pendulum by applying force to the cart. A real world example
relates directly to this system is the attitude control of a booster rocket at
take-off.
In this case, we consider a two dimensional problem where the pendulum is
constrained to move in the vertical plane and the control input is the force F
that moves the cart horizontally and the outputs are the angle of the pendulum
and position of the cart.
5. For this example, let's assume the following quantities:
(M) mass of the cart 1 kg
(m) mass of the pendulum 0.4 kg
(b) coefficient of friction for cart 0.2 N/m/sec
(l) length to pendulum center of mass 0.6 m
(I) mass moment of inertia of the pendulum 0.012 kg.m^2
(F) force applied to the cart
(x) cart position coordinate
(theta) pendulum angle from vertical (down)
The dynamics of the system are given by
6. Summing the forces in the free-body diagram of the cart in the horizontal
direction, the following equation of motion is obtained
Summing the forces in the free-body diagram of the pendulum in the
horizontal direction, the following expression for the reaction force N is
obtained
Solving the two equations, we get
The sum of the forces perpendicular to the pendulum is,
To get rid of P and N, sum the moments about the centroid of the pendulum
to get the following equation,
Combining these last two expressions, you get the second governing
equation,
Let represent the deviation of the pendulum's position from equilibrium, that
is, = + . Again presuming a small deviation ( ) from equilibrium, we can
use the following small angle approximations of the nonlinear functions in our
system equations:
7. Hence, our governing equations become,
where F = u
Transfer Function
To obtain the transfer function of the linearized system equations, we take
laplace transform of the system equations,
And further solving the equations, we get
where
From the transfer function, we can cancel out the pole and zero at the origin
and that leads to,
8. State Space Model
By substituting the values in the model above, we get
System Performance
Using MATLAB, we can see that the poles of the system are at 0, -4.5517,
-0.1428 and 4.5005. As we can see, there is an unstable pole at 4.5005.
Impulse Response of Open loop system
9. As we can see from the plot, the system response is entirely unsatisfactory
and is unstable
Hence, we have to design a controller to stabilize the system and achieve
desirable performance of the system.
Controller Design
We use the LQR Tracking Controller to stabilize the system and also reduce
the control and performance cost. This controller also reduces the steady
state error of the system by employing a Steady State Error Tracking by
using a FeedForward loop to asymptotically tracking the error.
To use LQR, we first have to make sure the system is controllable and
observable. In this case, the Inverted Pendulum system satisfies that
condition.
The cost function is defined as
10. LQR minimizes the performance cost of the system using wieghting
parameters, Q and R for the states and input respectively. For this system,
we choose
R = 1 and Q = p*CT
C (where p is varied to obtain optimal cost)
In this case,
The (1,1) and (3,3) elements of the Q matrix provide weight to the state
variables corresponding to the position of the cart and angle of the pendulum
respectively. They can be increased for better performance but that would
increase the control cost. Hence, these values were selected to balance the
control cost and the performance cost.
The uncertainities of the system were treated as random disturbances to the
input u and sensor reading at the output y. To incorporate these
uncertainities in our design, randomly generated variable disturbances were
added to the system
11. The controller using LQR has the form
where P is found by solving the Algebraic Riccati Equation
and the optimal control law is given by
Now, the system obtained has good dynamic properties and has stabilized
using the optimal control law. But the steady state error has to be addressed.
Designing an Asymptotic Error Tracking
controller to reduce Steady State Error
12. where K is the LQR controller and the Error Tracking controller is
Now, the input u changes from -Kx to r - Kx, where is the
precompensator.
To find ,
The transfer system becomes,
And using K from the LQR, we have
The C matrix is modified to reflect the fact that the reference is a command
only on the cart position, i.e, only the first row of actual C matrix.
13. Designing an Observer
As we know, not all states can be accurately known at all times. Hence we
design an observer-based controller.
Now we can estimate our system state by the following equation,
The Complete System is now formed resulting in the following state space
equations,
14. Robustness of the system is verified in the results and the desired
performance is achieved at a low cost. Hence, the Robust Control of an
inverted Pendulum is obtained using LQR Error Tracking Controller.
Simulation Results:
The initial Eigen Values of the System
Gains calculated at 20 instances of time
15. As it can be seen, the gain matrix K adjusts its value at every instance to
maintain the desired system performance under uncertainity.
The Complete System State Space Model is given by,
16. And the eigen values of the final system are,
It can be seen that the system is stable as all the poles are in the left hand
side of the plane.
So far, we have seen that the design stabilized the system. Now we check its
performance and robustness.
All the graphs correspond to the system response with uncertainites (20
instances)
17. Impulse Response
As we can see from the graph, the settling time when the system is affected
by an Impulse is around 4 seconds for the angle of the pendulum (impulse
response of the second output) and the rise time and overshoot are also
minimal.
Step Response
18. Again, as observed in the plots, position(blue) has a small steady state error
(i.e, when the stystem reaches steady state, the position is at about 0.1) but
the angle(green) has negligible steady state error.
Bode Plot
From the above Bode Plot, the gain margin of the position and angle are
between 4 and 10 and the phase margin of both is infinite which indicates the
system is robust and has minimal overshoot. The problem with infinite phase
margin is that the system will have trouble tracking the signals. But this was
solved by adding the Steady State Error Tracking Controller. Hence, we have
a Robust Control System.
The Steady State error of the angle during the 20 instances
20. As we can see, the angle of the pendulum has minimal steady state error
(i.e, almost 0)
Trajectory of the Position of the cart
Trajectory of the Angle of the Pendulum
21. As can be seen, the angle of the pendulum varies between -2.3*10^-3 and
10^-2. This shows that the controller has achieved its objective.
Comparison with existing H-infinity results
As referred from IEEE journals, both H-infinity and LQR Tracking control
systems are robust but performance varies. Comparison of transient
properties for one instance is
for position of cart and angle of pendulum respectively.
As we can see, LQR Tracking Control has better transient response.
22. Conclusion
In this paper, an LQR Tracking Controller was successfully designed for te
Inverted Pendulum system. Based on the results, it is concluded that the
control method is capable of controlling the pendulum’s angle and cart’s
position of the linearized system. The controller has good transient
responses, optimized performance and control cost and is Robust. By
varying the values of the weight matrices, better performance can be
achieved at higher costs.
Since the Tracking controller was included in the design, the infinite phase
margin (which poses a tracking problem) is dealt with and the Steady State
error tracking improved the performance of the system. Since all the states
can not be observed, a full-order observer was also designed.
The resultant design was a Robust Control System for an Inverted Pendulum
System with desired performance and optimal cost.
23. References
1. Lee, S.S., and Lee, J.M, "Robust control of the inverted pendulum and mobile
robot," Assembly and Manufacturing, 2009. ISAM 2009. IEEE International
Symposium on , vol., no., pp.398-401, 17-20 Nov. 2009.
2. Zhang Jing, and Xu Lin, "Robust control of the inverted pendulum," Systems and
Control in Aerospace and Astronautics, 2008. ISSCAA 2008. 2nd International
Symposium on , vol., no., pp.1-3, 10-12 Dec. 2008.
3. Sek Un Cheang, and Wei Ji Chen, "Stabilizing control of an inverted pendulum
system based on a, loop shaping design procedure," Intelligent Control and
Automation, 2000. Proceedings of the 3rd World Congress on , vol.5, no.,
pp.3385-3388 vol.5, 2000.
4. J. Li and T.-C. Tsao, “Robust performance repetitive control systems,” ASME J.
Dynamic Systems, Measurement, and Control, vol. 123, no. 3, pp. 330-337, 2001.
5. T.-Y. Doh and M. J. Chung, “Repetitive control design for linear systems with
time-varying uncertainties,” IEE Proc. - Control Theory and Applications, vol. 150,
no. 4, pp. 427-432, 2003.
6. K. Zhou and J. C. Doyle, Essentials of Robust Control, Prentice-Hall, Inc., 1998.
7. M.-C. Tsai and W.-S. Yao, “Analysis and estimation of tracking errors of plug-in
type repetitive control system,” IEEE Trans. on Automatic Control, vol. 50, no. 8,
pp. 1190-1195, 2005.