Enforce Passivity Constraint for Flexible Beam

This example shows how to enforce passivity constraint for vibration control in a flexible beam using the Passivity Enforcement block.

Flexible Beam Model

The following figure depicts an active vibration control system for a flexible beam.

In this setup, the actuator delivering the force $\mathit{u}\left(\mathit{t}\right)$ and the velocity sensor are collocated. You can model the transfer function from control input $\mathit{u}$ to the velocity $\mathit{y}$ using finite-element analysis. Keeping only the first six modes, you obtain a plant model of the following form.

$\mathit{G}\left(\mathit{s}\right)={\sum }_{\mathit{i}=1}^6\frac{{\alpha }_{\mathit{i}}^2\mathit{s}}{{\mathit{s}}^2+2\xi {\mathit{w}}_{\mathit{i}}\mathit{s}+{\mathit{w}}_{\mathit{i}}^2}$.

Configure the model parameters.

xi = 0.05;
alpha = [0.09877, -0.309, -0.891, 0.5878, 0.7071, -0.8091];
w = [1, 4, 9, 16, 25, 36];

Construct the resulting beam model $\mathit{G}\left(\mathit{s}\right)$.

G = tf(alpha(1)^2*[1,0],[1, 2*xi*w(1), w(1)^2]) + ...
    tf(alpha(2)^2*[1,0],[1, 2*xi*w(2), w(2)^2]) + ...
    tf(alpha(3)^2*[1,0],[1, 2*xi*w(3), w(3)^2]) + ...
    tf(alpha(4)^2*[1,0],[1, 2*xi*w(4), w(4)^2]) + ...
    tf(alpha(5)^2*[1,0],[1, 2*xi*w(5), w(5)^2]) + ...
    tf(alpha(6)^2*[1,0],[1, 2*xi*w(6), w(6)^2]);

Check whether the model is passive.

isPassive(G)

ans = logical
   1

With this sensor and actuator configuration, the beam is a passive system.

Open the Simulink® model and disable passivity constraint enforcement.

mdl = 'passivityBeam';
open_system(mdl)
constrained = 0;

Design LQG Controller

Before you apply the constraints, design an LQG controller for the beam model. LQG control is a natural formulation for active vibration control. The LQG control setup is depicted in this figure.

The signals $\mathit{d}$ and $\mathit{n}$ are the process and measurement noise, respectively.

The LQG objective is to minimize

$\mathit{J}=\underset{\mathit{T}\to \infty }{\lim }\mathit{E}\left({\int }_0^{\mathit{T}}\left({\mathit{y}}^2\left(\mathit{t}\right)+0.001{\mathit{u}}^2\left(\mathit{t}\right)\right)\mathit{dt}\right)$,

with noise variance

$\mathit{E}\left({\mathit{d}}^2\left(\mathit{t}\right)\right)=1$, $\mathit{E}\left({\mathit{n}}^2\left(\mathit{t}\right)\right)=0.01$.

Design the LQG controller.

[a,b,c,d] = ssdata(G);
M = [c d;zeros(1,12) 1];  % [y;u] = M * [x;u]
QWV = blkdiag(b*b',1e-2);
QXU = M'*diag([1 1e-3])*M;
CLQG = lqg(ss(G),QXU,QWV);

Simulate the LQG controller and plot its performance.

% Simulate the model.
out = sim(mdl);

% Extract trajectories.
logsout = out.logsout;

% Plot trajectories of y.
y = logsout.getElement('y');       
y_vector = y.Values.Data(:,:)';
plot(y.Values.time,y_vector)
grid on
title('y')

Passivity Constraint

The beam model is passive from control input $\mathit{u}$ to the velocity $\mathit{y}$.

The Passivity Enforcement block accepts passivity constraint in the form ${\mathit{u}}_{\mathit{p}}={\mathit{f}}_{\mathit{p}}\left(\mathit{x}\right)+{\mathit{g}}_{\mathit{p}}\left(\mathit{x}\right)\mathit{u}$ and ${\mathit{y}}_{\mathit{p}}={\mathit{h}}_{\mathit{p}}\left(\mathit{x}\right)$. In this application, ${\mathit{f}}_{\mathit{p}}\left(\mathit{x}\right)=0$, ${\mathit{g}}_{\mathit{p}}\left(\mathit{x}\right)=1$, and ${\mathit{h}}_{\mathit{p}}\left(\mathit{x}\right)=\mathit{y}$.

Simulate Controller with Passivity Constraint

To view the constraint implementation, open the Constraint > Constrained subsystem.

Enable the passivity constraint enforcement.

constrained = 1;

Run the model and plot the simulation results.

% Simulate the model.
out = sim(mdl);

% Extract trajectories.
logsout = out.logsout;

% Plot trajectories of y.
y = logsout.getElement('y');       
y_vector = y.Values.Data(:,:)';
plot(y.Values.time,y_vector)
grid on
title('y')

When you enforce the passivity constraint, the vibration in flexible beam is greatly reduced.

Close the model.

bdclose(mdl)