This tutorial illustrates how to use MotionView and MotionSolve to design a control system that stabilizes an inverted pendulum.  The goal is to design a regulator using the pole placement method.  The inverted pendulum MDL model file is supplied.

The tutorial steps include:

Figure 1 shows the classic inverted pendulum on a slider.  The system has two degrees of freedom leading to four state variables.  The vertically upright position of the pendulum is unstable.  The goal is to design a regulator to stabilize this configuration.

Figure 1: Inverted pendulum model

We want to find a full-state feedback control law to achieve our goal.  The control input is a force applied to the slider along the global X-axis.  Plant output is the pendulum angle of rotation about the global Y-axis.

Start by loading the file inv_pendu.mdl, located in the mbd_modeling\motionsolve folder, into MotionView.  Upon examination of the model topology, you will notice that everything needed for this exercise is already included in the model.  However, depending on which task you are performing, you will need to activate or deactivate certain entities.

Step 1: Determine the stability of the open loop model

Compute the eigenvalues to determine the stability of the Inverted pendulum.

Compute Eigenvalues

Table - Open Loop Eigenvalues

There is one eigenvalue with a positive real part, indicating that the system is unstable in the current configuration.

The result animation H3D will be loaded in the adjacent window.

Step 2: Obtaining a Linearized Model

Usually, the first step in a control system design is to obtain a linearized model of the system in the state space form,

where A,B,C,and D are the state matrices, x is the state vector, u is the input vector, and y is the output vector.  The A,B,C,and D matrices depend on the choice of states, inputs, and outputs.  The states are chosen automatically by MotionSolve and the chosen states are reported in one of the output files.  We need to define only the inputs and outputs.

Linear tab in Simulation Settings dialog for specifying the MATLAB matrix files output

You should get six new files with base name inv_pendu_state_matrices and extensions .a, .b, .c, .d, .pi, .po. The .pi and .po files contain information about the input and output variables.

The states chosen by the MotionSolve solver are:

Step 3: Control System Design in MATLAB

A detailed discussion of control system design is beyond the scope of this document.  However, the steps to design a regulator using pole placement [1] to stabilize the inverted pendulum are described briefly.  For details, refer to the standard controls text and the MATLAB documentation.

It can be easily verified using MATLAB that the system is completely state controllable [1, 2].  We employ a full-state feedback control law
u = -k*x, where u is the control input, k is the gain vector, and x is the state vector.  Then, assuming the desired pole locations are stored in vector P, you may use the pole placement method to compute k.  For desired poles at [-20 –20 –20 –20] (rad/s), the acker function in MATLAB yields
k=1e3[-2.4186  -0.0163 -0.070 -0.0033].

Step 4: Implementing the Control Force in MotionView

The control force is simply u=-k*x.  The model contains a solver variable called Control Force Variable - CL. It is defined using the expression:

Notice that it is simply the dot product between the gain vector (k) and the state vector (x) elements.  This solver variable is used to define a force named Control Force - CL.

Activate the force Control Force - CL if it is deactivated.

Step 5: Check the Stability of a Closed Loop System

Table - Closed Loop Eigenvalues

They all have negative real parts, hence the system is stabilized.  Note that the negative real parts are close to the desired poles (-20 rad/s = -3.038 Hz).

Step 6: Add Disturbance Force and Run the Simulation

Fx= `step(TIME,.1,0,.5,50) + step(TIME,1,0,1.5,-50)`

Fy=0

Fz=0

Run a dynamic simulation with MotionSolve.

Follow these steps.

 

The plots of disturbance force, control force, slider x displacement, and pendulum angular displacement are shown below.

Figure 2: Plots of disturbance and control forces as well as slider translational and pendulum angular displacements.

References

Feedback Control of Dynamic Systems, G. G. Franklin, J. D. Powell, and A. Emami-Naeini, Third Edition, Addison Wesley.

See also

MATLAB Documentation, www.mathworks.com.