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

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.

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.

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.

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 2̂

(F) force applied to the cart

(x) cart position coordinate

(theta) pendulum angle from vertical (down)

The dynamics of the system are given by

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:

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,

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

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

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*CTC (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

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

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.

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,

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

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,

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)

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

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

Plot of the steady state error

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

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.

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.

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