www.gusucode.com > control_featured 案例源码程序 matlab代码 > control_featured/BeamVibrationControlExample.m
%% Vibration Control in Flexible Beam % This example shows how to tune a controller for reducing vibrations in a % flexible beam. % Copyright 2015 The MathWorks, Inc. %% Model of Flexible Beam % Figure 1 depicts an active vibration control system for a flexible beam. % % <<../beam.png>> % % *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) %% % 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. % % <<../positive.png>> % % *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) %% 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); %% % The optimizer found a 2nd-order controller with $J= 0.478$. % Compare with the optimal $J$ value for |CLQG|: [~,Jopt] = evalSpec(R1,replaceBlock(CL0,'C',CLQG)) %% % 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) %% % 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); %% % 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.