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Vibration Control in Flexible Beam

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This example shows how to tune a controller for reducing vibrations in a flexible beam.

Model of Flexible Beam

Figure 1 depicts an active vibration control system for a flexible beam.

Figure 1: Active control of flexible beam

In this setup, the actuator delivering the force $u$ and the velocity sensor are collocated. We can model the transfer function from control input $u$ to the velocity $y$ using finite-element analysis. Keeping only the first six modes, we obtain a plant model of the form

$$ G(s) = \sum_{i = 1}^6 \frac{\alpha_i^2 s}{ s^2 + 2\xi w_i s + w_i^2} $$

with the following parameter values.

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

The resulting beam model for $G(s)$ is given by

% Beam model
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]);

G.InputName = 'uG';  G.OutputName = 'y';

With this sensor/actuator configuration, the beam is a passive system:

isPassive(G)

ans =

  logical

   1

This is confirmed by observing that the Nyquist plot of $G$ is positive real.

nyquist(G)

LQG Controller

LQG control is a natural formulation for active vibration control. The LQG control setup is depicted in Figure 2. The signals $d$ and $n$ are the process and measurement noise, respectively.

Figure 2: LQG control structure

First use lqg to compute the optimal LQG controller for the objective

$$ J = \lim_{T \rightarrow \infty} E \left( \int_0^T (y^2(t) + 0.001 u^2(t)) dt \right) $$

with noise variances:

$$ E(d^2(t)) = 1 ,\quad E(n^2(t)) = 0.01 . $$

[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);

The LQG-optimal controller CLQG is complex with 12 states and several notching zeros.

size(CLQG)

State-space model with 1 outputs, 1 inputs, and 12 states.

bode(G,CLQG,{1e-2,1e3}), grid, legend('G','CLQG')

Use the general-purpose tuner systune to try and simplify this controller. With systune, you are not limited to a full-order controller and can tune controllers of any order. Here for example, let's tune a 2nd-order state-space controller.

C = ltiblock.ss('C',2,1,1);

Build a closed-loop model of the block diagram in Figure 2.

C.InputName = 'yn';  C.OutputName = 'u';
S1 = sumblk('yn = y + n');
S2 = sumblk('uG = u + d');
CL0 = connect(G,C,S1,S2,{'d','n'},{'y','u'},{'yn','u'});

Use the LQG criterion $J$ above as sole tuning goal. The LQG tuning goal lets you directly specify the performance weights and noise covariances.

R1 = TuningGoal.LQG({'d','n'},{'y','u'},diag([1,1e-2]),diag([1 1e-3]));

Now tune the controller C to minimize the LQG objective $J$.

[CL1,J1] = systune(CL0,R1);

Final: Soft = 0.478, Hard = -Inf, Iterations = 40

The optimizer found a 2nd-order controller with $J= 0.478$. Compare with the optimal $J$ value for CLQG:

[~,Jopt] = evalGoal(R1,replaceBlock(CL0,'C',CLQG))

Jopt =

    0.4673

The performance degradation is less than 5%, and we reduced the controller complexity from 12 to 2 states. Further compare the impulse responses from $d$ to $y$ for the two controllers. The two responses are almost identical. You can therefore obtain near-optimal vibration attenuation with a simple second-order controller.

T0 = feedback(G,CLQG,+1);
T1 = getIOTransfer(CL1,'d','y');
impulse(T0,T1,5)
title('Response to impulse disturbance d')
legend('LQG optimal','2nd-order LQG')

Passive LQG Controller

We used an approximate model of the beam to design these two controllers. A priori, there is no guarantee that these controllers will perform well on the real beam. However, we know that the beam is a passive physical system and that the negative feedback interconnection of passive systems is always stable. So if $-C(s)$ is passive, we can be confident that the closed-loop system will be stable.

The optimal LQG controller is not passive. In fact, its relative passive index is infinite because $1-CLQG$ is not even minimum phase.

getPassiveIndex(-CLQG)

ans =

   Inf

This is confirmed by its Nyquist plot.

nyquist(-CLQG)

Using systune, you can re-tune the second-order controller with the additional requirement that $-C(s)$ should be passive. To do this, create a passivity tuning goal for the open-loop transfer function from yn to u (which is $C(s)$). Use the "WeightedPassivity" goal to account for the minus sign.

R2 = TuningGoal.WeightedPassivity({'yn'},{'u'},-1,1);
R2.Openings = 'u';

Now re-tune the closed-loop model CL1 to minimize the LQG objective $J$ subject to $-C(s)$ being passive. Note that the passivity goal R2 is now specified as a hard constraint.

[CL2,J2,g] = systune(CL1,R1,R2);

Final: Soft = 0.478, Hard = 1, Iterations = 37

The tuner achieves the same $J$ value as previously, while enforcing passivity (hard constraint less than 1). Verify that $-C(s)$ is passive.

C2 = getBlockValue(CL2,'C');
passiveplot(-C2)

The improvement over the LQG-optimal controller is most visible in the Nyquist plot.

nyquist(-CLQG,-C2)
legend('LQG optimal','2nd-order passive LQG')

Finally, compare the impulse responses from $d$ to $y$.

T2 = getIOTransfer(CL2,'d','y');
impulse(T0,T2,5)
title('Response to impulse disturbance d')
legend('LQG optimal','2nd-order passive LQG')

Using systune, you designed a second-order passive controller with near-optimal LQG performance.

See Also

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