www.gusucode.com > control_featured 案例源码程序 matlab代码 > control_featured/NetworkedControlSystemExample.m
%% Passive Control with Communication Delays % This example shows how to mitigate communication delays in a passive % control system. % Copyright 2015 The MathWorks, Inc. %% Passivity-Based Control % By the Passivity Theorem, the negative-feedback interconnection of % two strictly passive systems $H_1$ and $H_2$ is always stable. % % <<../passivitythm.png>> % % When the physical plant is passive, it is therefore advantageous to % use a passive controller for robustness and safety reasons. In networked % control systems, however, communication delays can undo the benefits of % passivity-based control and lead to instability. % To illustrate this point, we use the plant and 2nd-order passive % controller from the "Vibration Control in Flexible Beam" example. % See this example for background on the underlying control problem. % Load the plant model $G$ and passive controller $C$ (note that $C$ % corresponds to $-C$ in the other example). load BeamControl G C bode(G,C,{1e-2,1e4}) legend('G','C') %% % The control configuration is shown below as well as the impulse response % from $d$ to $y$. % % <<../negativefb.png>> % impulse(feedback(G,C)) %% Destabilizing Effect of Communication Delays % Now suppose there are substantial communication delays between the sensor % and the controller, and between the controller and the actuator. This % situation is modeled in Simulink as follows. open_system('DelayedFeedback') %% % The communication delays are set to T1 = 1; T2 = 2; %% % Simulating this model shows that the communication delays destabilize % the feedback loop. % % <<../noscatter.png>> % %% Scattering Transformation % To mitigate the delay effects, you can use a simple linear transformation of % the signals exchanged between the plant and controller over the network. % % <<../networkedcontrol.png>> % % *Figure 1: Networked Control System* % % This is called the "scattering transformation" and given by the formulas % % $$ \left(\begin{array}{c}u_l\\v_l\end{array}\right) = % \left(\begin{array}{cc}1 & b \\ 1 & -b \end{array}\right) % \left(\begin{array}{c}u_G\\y_G\end{array}\right) , \quad % \left(\begin{array}{c}u_r\\v_r\end{array}\right) = % \left(\begin{array}{cc}1 & b \\ 1 & -b \end{array}\right) % \left(\begin{array}{c}y_C\\u_C\end{array}\right) , $$ % % or equivalently % % $$ \left(\begin{array}{c}u_l\\u_G\end{array}\right) = S % \left(\begin{array}{c}v_l\\y_G\end{array}\right) , \quad % \left(\begin{array}{c}v_r\\u_C\end{array}\right) = S^{-1} % \left(\begin{array}{c}u_r\\y_C\end{array}\right) , \quad % S = \left(\begin{array}{cc}1 & 2b \\ 1 & b \end{array}\right) $$ % % with $b>0$. Note that in the absence of delays, the two scattering % transformations cancel each other and the block diagram in Figure 1 is % equivalent to the negative feedback interconnection of $G$ and $C$. % % When delays are present, however, $(u_l,v_l)$ is no longer equal to % $(u_r,v_r)$ and this scattering transformation alters the properties of % the closed-loop system. In fact, observing that % % $$ u_l = (1-bC(s))/(1+bC(s)) v_l , \quad % v_r = (G(s)/b-1)/(G(s)/b+1) u_r $$ % % and that $bC$ and $G/b$ strictly passive ensures that % % $$ \| (1-bC)/(1+bC) \|_\infty < 1 , \quad \| (G/b-1)/(G/b+1) \|_\infty < % 1 , $$ % % the Small Gain Theorem guarantees % that the feedback interconnection of Figure 1 is always stable no matter % how large the delays. Confirm this by building a Simulink model of % the block diagram in Figure 1 for the value $b=1$. b = 1; open_system('ScatteringTransformation') %% % Simulate the impulse response of the closed-loop system as done before. % The response is now stable and similar to the delay-free response in % spite of the large delays. % % <<../scatter.png>> % % For more details on the scattering transformation, see T. Matiakis, % S. Hirche, and M. Buss, "Independent-of-Delay Stability of Nonlinear % Networked Control Systems by Scattering Transformation," Proceedings of % the 2006 American Control Conference, 2006, pp. 2801-2806.