Acquire the ability to adjust the parameters of the control system in state feedback for a given nonlinear inverted pendulum

The aim of the exercise is to acquire the ability to adjust the parameters of the control system in

state feedback for a given nonlinear inverted pendulum model z

exercises 5. The aim of steering is to stabilize the pendulum at the point of equilibrium for

non-zero starting point.

Please: Define a control criterion with the constraints 𝑐 (𝑥) ≤ 0 and illustrate

graphically.

Tuning criteria to be used in the exercise:

A. Limitations as above hard and the objective function constant, e.g. zero

B. Limitations as above hard and objective function = rise time

C. Limitations as above soft and constant objective function (only minimization of overrun)

D. Limitations as above soft and objective function = rise time (minimize tn +

overrun)

E. Limitations as above but with the minimization of time t2 (dynamic restriction displacement)

only t2 criterion

F. Limitations as above but with the minimization of time t2 (dynamic restriction displacement)

criterion t2 + tn

I'm really fresh new in this case. I did optimalization and calling program. I did also constraints program, but i'm exactly sure that I did it wrong. If you could please tell me how to manage with point A

Simulink:

[Sam Chak]

Hi @Michal Tchorzewski

In an ideal environment, a simple PD controller like , where and can drive the non-zero initial angular position to the equilibrium very quickly, approximately within 1.2 seconds, and without oscillatory transient. However, the model provided in the Simulink doesn't look like an inverted pendulum. Please correct me if I'm wrong.

[Michal Tchorzewski]

This is inverted pendulum described by a mathematical formula:

this formula is the lower balance position beacuse:

[Sam Chak]

Hi @Michal Tchorzewski

The pendulum equation can be used if you want to balance the inverted position at the unstable equilibrium or . However, the reference position is not shown in your Simulink model. Futhermore, I'm unsure about the input term, .

Anyhow, to swing up the pendulum from any position and balance it at the inverted position , the following controller can be used:

, where and .

But swinging up from the lowest point requires a relatively high torque. If the reference pendulum position is modified to become

, where rad, then the initial swing-up and the swing-down actions from will begin from the same amount of torque. In other words,

if , then the pendulum will swing upward to ;

if , then the pendulum will swing downward to .

Answer 1

[Michal Tchorzewski]

Ok, my new question is how should I do hard limitation? I'm not sure how to write it in Matlab. I know that hard limitation is this limits that are like phisical constraints. Could you please help?

Answer 2

[Sam Chak]

Hi @Michal Tchorzewski

To assign the hard limiter that emulates the actuator output restriction by mechanical means, you can probably follow this example:

x = linspace(-pi, pi, 301);

y1 = sin(x); % original signal

y2 = max(-0.5, min(y1, 0.5)); % actuator with amplitude constraint ±0.5

plot(x, y1, x, y2)