Properties of the Cascade lead compensator As, the angle contributed by the compensator to some arbitrary point s, at on the s-plane is illustrated in Figure 1 Im s-plane Re 2022-2-3 6
2022-2-3 6 Properties of the Cascade Lead Compensator ¨ As , the angle contributed by the compensator to some arbitrary point s1 at on the s-plane is illustrated in Figure 1. s-plane z 0 s 1 p0 p z Im Re
The net contribution is e=6.-0n>0 So that the lead compensator al ways makes a positive contribution to the angle criterion. This has the effect of allowing the closed-loop poles to move to the left in the s- lane The problem is then how to choose the relative location of the pole and the zero We reproduce the advice of D'AzZo and houris Method 1 Use the zero to cancel a low frequency real pole. This can simplify the root locus and reduce the complexity of the problem. The compensator pole is then placed such that S1 becomes a point on the desired root-locus. For a type-1 system, the real pole (excluding the pole at zero that is closest to the origin should be cancelled. For a type 0 system the second closest pole to the origin should be cancelled 2022-2-3
2022-2-3 7 The net contribution is 0 c z p So that the lead compensator always makes a positive contribution to the angle criterion. This has the effect of allowing the closed-loop poles to move to the left in the s- plane. The problem is then how to choose the relative location of the pole and the zero. We reproduce the advice of D'Azzo and Houpis. Method 1 Use the zero to cancel a low frequency real pole. This can simplify the root locus and reduce the complexity of the problem. The compensator pole is then placed such that s1 becomes a point on the desired root-locus. For a type-1 system, the real pole (excluding the pole at zero) that is closest to the origin should be cancelled. For a type 0 system, the second closest pole to the origin should be cancelled
Example 1 The following Matlab code illustrates these principles for the system with open-loop transfer function G s(s+ (67 Define the plant G1=tf(1,conv([1,01,[1,1]));H=1; root-locus locus(GI*H) 2022-2-3 8
2022-2-3 8 Example 1 The following Matlab code illustrates these principles for the system with open-loop transfer function ( 1 ) 1 1 s s G Define the plant G1 = tf(1,conv([1, 0],[1, 1])); H=1; root-locus rlocus(G1*H) (6.7)
0XmE -15 15 -0.5 Real Axis 2022-2-3 9
2022-2-3 9 -2 -1.5 -1 -0.5 0 0.5 1 -1.5-1 -0.50 0.51 1.5 Real Axis Imag Axis
Clearly, we cannot achieve a closed-loop pole at without some dynamic compensation. However, if we use the zero of a cascade lead compensator to cancel the pole at and place the pole at we get D1=zpk([-1],[-4],1); Go1=D1*G1★H; locus(Go1) which will have a closed-loop pole at the desired location when the gain is Kc =rlocfind(Gol,-2+23) KC 2022-2-3 10
2022-2-3 10 Clearly, we cannot achieve a closed-loop pole at without some dynamic compensation. However, if we use the zero of a cascade lead compensator to cancel the pole at and place the pole at we get: D1 = zpk([-1],[-4],1); Go1 = D1*G1*H; rlocus(Go1) which will have a closed-loop pole at the desired location when the gain is Kc =rlocfind(Go1,-2+2j) Kc = 8