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