Inverted Pendulum

Introduction

Proportional (Derivative) Control Determined through LQR Analysis

This experiment will demonstrate the control of a system using full state feedback, where the control system is designed using the principles of Linear Quadratic Regulator (LQR) theory. This is an analysis technique that would be seen in more detail during advanced studies. You are not required to completely understand the derivation of the method itself. However, you should understand how to formulate the required matrices (Q and R) and why the overall goal (to minimize the cost, J) leads to the requirements on those matrices, as well as the reasoning behind the formulation of the cost equation, which serves as the purpose behind the method’s derivation.

Linear Quadratic Theory (LQR)

Consider the system

where is a state vector and is a control input vector. Therefore, and are matrices of dimensions and , respectively. Let us suppose that we wish to design a state-feedback controller

such that the closed-loop system

is stable. Here is a matrix of gains of dimensions . In achieving closed-loop stability, we wish to use as little control effort as possible. We also wish to avoid large overshoot of the trajectories during the transients. One way to achieve both of these objectives is to find a control law which minimizes the following quadratic cost

where and are weighing matrices of appropriate dimensions. In order for the previous problem to make sense we must ensure that the two terms in the integral are positive. The requirement that

for all vectors is satisfied if the matrix is positive semi-definite (all its eigenvalues are greater than or equal to zero). The requirement that

for all vectors is satisfied if the matrix is positive definite (all its eigenvalues are greater than zero). This optimization problem has a closed-form solution. The optimal gain matrix is given by

where the matrix is the positive definite solution of the following Algebraic Riccati Equation (ARE):

Example of a Simple Mass/Spring/Damper System

We will demonstrate the previous LQR design using a simple mass-spring-damper system as shown in Fig. 7.2. The equation of motion is

or in state-space form

where and . Let and . The system matrices are

Choose the weighting matrices as

The Riccati equation gives

Expanding, we obtain

Solving these equations, we get

or that

Notice that the solution matrix P is positive definite. Its eigenvalues are .

The optimal gain matrix is given by

The closed-loop system matrix is

and the eigenvalues are

Control of an Inverted Pendulum

Objective

The purpose of this experiment is to design a balance control using the principles of linear quadratic regulator (LQR) theory for an inverted pendulum mechanism.

Equipment Required

• Inverted Pendulum mechanism

• Q8-USB interface board

• MATLAB, SIMULINK and QUARC software.

Introduction

In this lab, we demonstrate a control design using modern “state-space” methods. The plant consists of an inverted pendulum on a cart. The plant has two distinct equilibrium points of which one is stable and the other is unstable. In this experiment, we will show how the plant in its unstable configuration can be controlled using a suitable controller. The methodology used to design the control law is based on the linear quadratic regulator (LQR) theory. The inverted pendulum mechanism to be used in this lab is shown in Fig. 7.3.

Mathematical Model of the Inverted Pendulum Mechanism

Consider an inverted pendulum of mass and length connected on a sliding cart of mass as shown in Fig. 7.4.

The angle is measured from the up-right position. We can apply a force to the cart through the cartwheel motors. We wish to design a controller to balance the pendulum to the upright position . Note that the problem of balancing a pendulum is akin to the problem of stabilizing a rocket during ascent.

The equations of motion for the system can be easily derived using the free-body diagrams shown in Fig. 7.6 as follows (here we assume that the track on which the cart moves without sliding):

where is the moment of inertia about the pendulum's mass center given by .

This system has two equilibria which are and .

The first equilibrium corresponds to the down position (stable) and the second equilibrium corresponds to the upright position (unstable) of the pendulum. Linearizing the previous equations about the equilibrium of interest we obtain the following equations:

The input force in the above equations can be written in terms of the input voltage applied to the motor of the cart and the cart velocity as shown below.

First, we write the motor equation as

or

where

• : voltage applied to motor (Volts)

• : current in motor (Amp)

• : back EMF constant (Volt/(rad/sec))

The torque generated at the output of the motor is given by

where is the motor torque constant given by

where is the conversion factor from volt-amp to N-m given by

Thus, the force transmitted to the cart via the pinion is

and on substituting the values of

we get

Letting and , the input force F can be written as

Substitution of the above expression for F in the linearized equations results in

The above set of linear differential equations is of the form , where . We seek a linear control law of the form , where is a vector of gains.

Linear Quadratic Regulator (LQR) Design

The LQR formulation provides a linear control law

which minimizes the quadratic cost given by

for some user-defined matrices and . It is assumed that is positive semi-definite (all the eigenvalues are non-negative) and is positive definite (all the eigenvalues are positive). Here is the input voltage, given by

is the state vector, and given by

is the controller gain vector. Note that the input voltage is scalar and therefore, is also scalar. In practice, the weighting matrices and are chosen to be diagonal. A simple way for choosing and is the following (Bryson's rule):

Assuming the time-domain design specifications are:

one can choose

Further tuning of and may be required for improving the controller performance. Note that increasing means increasing the penalty on the control input (using less control effort). Similarly, increasing means increasing the penalty on the states (states reaching equilibrium position quickly and with less overshoot).

The block diagram of the experimental setup in the lab is shown in Fig. 7.7.

(Note: Conventional error signal shown, but )

where is position of the cart, is velocity of the cart, is angular position of the rod from the vertical, is angular velocity of the rod, and is a vector of feedback gains .

Controller Design Procedure

1. Obtain the A and B matrices of the system using , and .

2. Choose and based on the following:

   Assume that:

Controller Implementation and Evaluation

1. Turn the power on.

2. Open the MATLAB and locate the file InvPendulum_LQR.mdl under the path C:\AE4610_Controls_Lab\Inverted_Pendulum and open it. This is the block diagram for this part of the experiment.

Analysis

• Individual: Include the results from the controller design procedure Steps 1-5.

• Group: Build a SIMULINK block diagram using Fig. 7.7 and obtain the simulated response to the same command input used in the experiment. Compare the simulated response with the experimental response and explain any differences between them.