Control system design 2022-2-3
2022-2-3 1 Control system design
Introduction System compensation is the process of designing a controller that will produce an acceptable transient response while maintaining a desired steady-state accuracy. These two design objectives are conflicting in most systems, since small errors imply high gains reduce system stability and may even drive the system unstable Compensation may be thought of as the process of increasing the stability of a system without reducing its accuracy below minimum acceptable standards 2022-2-3 2
2022-2-3 2 Introduction ¨ System compensation is the process of designing a controller that will produce an acceptable transient response while maintaining a desired steady-state accuracy .These two design objectives are conflicting in most systems ,since small errors imply high gains reduce system stability and may even drive the system unstable .Compensation may be thought of as the process of increasing the stability of a system without reducing its accuracy below minimum acceptable standards
Cascade lead compensation ◆ Introduction o The proportional plus derivative compensator has the unfortunate property that its high frequency gain is infinite This means that high frequency effects such as sensor noise and un-modelled high-frequency dynamics, e.g resonance terms, will be amplified with potentially disastrous effects. Of course, a real physical derivative operator cannot be implemented and any implementation will actually have poles that will limit the high-frequency gain. Recognizing this, an alternative to the pure P+D +D(S)=K D +K rop (6.1) 2022-2-3 3
2022-2-3 3 Cascade Lead Compensation ¨ Introduction ¨ The proportional plus derivative compensator has the unfortunate property that its high frequency gain is infinite. This means that high frequency effects, such as sensor noise and un-modelled high-frequency dynamics, e.g. resonance terms, will be amplified with potentially disastrous effects. Of course, a real physical derivative operator cannot be implemented and any implementation will actually have poles that will limit the high-frequency gain. Recognizing this, an alternative to the pure P+D P D prop D (s) K D s K (6.1)
is the So-called "lead compensator Dead(s)=K s+ (6.2) Where Considering the frequency response of (2) D=d(o)=K/1+ (6.3) J0+p0 The low and high -frequency gains are D lead K (64) lea (jo) K 0→0 (65) 2022-2-3 4
2022-2-3 4 is the so-called "lead compensator" 0 0 lead ( ) s p s z D s Kc 0 p z Where o Considering the frequency response of (2) 0 0 lead ( ) j p j z D j Kc (6.2) (6.3) The low and high-frequency gains are: c c D j K p z D j K ( ) ( ) lead 0 0 lead 0 (6.4) (6.5)
so that the ratio of high-to - low frequency gain is D1a(0)二0 The lead compensator is still a high-pass filter but the high frequency gain is limited by the pole at s=-Po. Typically, the ratio of po to zo is kept to below 10 2022-2-3 5
2022-2-3 5 so that the ratio of high-to-low frequency gain is 0 ( 0) ( ) 0 0 lead lead z p D j D j (6.6) The lead compensator is still a high-pass filter but the high frequency gain is limited by the pole at . Typically, the ratio of p0 to z0 is kept to below 10. p0 s
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
