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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.

% Parametersxi = 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 modelG = 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$G$是正的真实。

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 uselqgto 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);

LQG-optimal控制器CLQGis 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 tunersystuneto try and simplify this controller. Withsystune, 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 controllerCto 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 forCLQG:

[~,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)

Usingsystune, 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 fromyntou(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 modelCL1to minimize the LQG objective$J$subject to$-C(s)$being passive. Note that the passivity goalR2is now specified as a hard constraint.

[CL2,J2,g] = systune(CL1,R1,R2);
Final: Soft = 0.478, Hard = 1, Iterations = 51

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')

最后,比较的脉冲响应$d$to$y$.

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

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

See Also

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