The loop shaping paradigm. Lecture 7. Loop analysis of feedback systems (2) Essential specifications (2)


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1 Lecture 7. Loop analysis of feedback systems (2). Loop shaping 2. Performance limitations The loop shaping paradigm. Estimate performance and robustness of the feedback system from the loop transfer L(jω) 2. Choose C(jω) to shape L(jω) = C(jω)G(jω) Graphical convenience of Bode plots: process and controller curves are additive Syst3 lecture 7 Syst3 lecture 7 2 Essential specifications () Choose a large loop gain L(jω) at low frequencies for good tracking of reference signals and good rejection of load disturbances. Example: integral action causes L(j) = +. Perfect tracking of constant references and perfect rejection of constant disturbances. Essential specifications (2) Choose a loop gain L(jω) that has good rolloff, that is, rapid attenuation of the gain at high frequencies. This makes the feedback system less sensitive to measurement noise and to unmodelled dynamics. Example: the asymptotic rolloff of C(jω) depends on its relative degree, that is, the difference between the number of poles and the number of zeros. Integral action is good for the rolloff. Derivative action is bad for the rolloff. Syst3 lecture 7 3 Syst3 lecture 7 4
2 Essential specifications (3) A desirable loop gain L = CP Maximize the phase margin φ m and the gain margin G m around the crossover frequencies. This makes the feedback system robustly stable, that is stable in spite of process uncertainties. Bode s relationship approximates to L(jω ) π 2 d d 2 log L(jω ) L(jω) BF : S(s) slope PM HF : S(s) ω C Consequence: the phase margin depends on the slope of log L in the vicinity of ω c Example: A small phase margin causes oscillations in the step response. Integral action is bad for the phase margin. Derivative action is good for the phase margin. The crossover frequency ω c approximates the bandwidth of the feedback system. Pushing LF or HF specifications inevitably deteriorates the stability margins. The tradeoff will be worse in the presence of RHP poles or zeros. Syst3 lecture 7 5 Syst3 lecture 7 6 Bode s ideal loop shaping L(jω) = ( jω ω c ) n Gain curve has constant slope n. Constant phase lag nπ/2. (n does not need to be an integer, e.g. n =.5 to have a phase margin of 45 deg). BUT controller gain does not come with no cost Choosing C(jω) to shape L(jω) Advantage of Bode plots: curves add up! log C(jω)P (jω) = log C(jω) + log P (jω) arg C(jω)P (jω) = arg(jω) + arg P (jω) Proportional control: Amplitude curve is vertically shifted by 2 log K and phase curve is unchanged. Dynamic controllers localize (in frequency) the loop shaping. RHP poles and zeros cannot be cancelled Syst3 lecture 7 7 Syst3 lecture 7 8
3 PI controller : + st i T i G 9 Good for static performance but phase lag choose T i ω B PD controller : + st d T d G +9 Phase lead is beneficial to stability margins but the closedloop bandwidth is increased limit T d Phaselag compensation K T is+, > T i s+ = K T is + T i s + = K (s + z), z p s + p 69 ωt i ωt i Static gain is increased by factor. Phase lag will not affect the phase margin if T i ω C. Synthesis guidelines: adjust K to assign ω C with sufficient phase margin. Evaluate the necessary reduction of static error to choose ; Maximize T i without degrading the phase margin. Syst3 lecture 7 9 Syst3 lecture 7 Phaselead compensation C(s) = K(T d s + ) T d s+, < = PD control Lowpass filter = K s+z s+p, p z K K +9 ωt d φ max Limitation of derivative action at high frequency. ωt d Phaselead compensation Useful formulas: max = 2 φ max = arcsin ω max = z p ( log ( )) + log T d T d ( ) + Synthesis guidelines: Choose ω c to assign closedloop bandwidth. Evaluate the necessary phase lead at ω c (not more than 6 = 6) and adjust T d to place the maximal phase lead at ω C. Choose K to have L(jω c ) =. Syst3 lecture 7 Syst3 lecture 7 2
4 Practice these design guidelines with Matlab It is good to use different design paradigms in parallel (e.g. rltool to have simultaneously Bode plots, Nyquist curve, and step response)) Never concentrate on a single transfer function (at least Y R and Y D but preferably also U R and U D ) Avoid pushing the specs and interpret physically the different actions of the controller Non minimum phase systems For systems with no RHP pole/zero and no delay, Bode s relationship between magnitude and phase implies d log L(jω) arg L(jω) π/2 d Systems with RHP pole(s) and/or zero(s) and/or timedelays have a larger phase lag. They are called nonminimum phase systems Syst3 lecture 7 3 Syst3 lecture 7 4 Additional bandwidth limitations for non minimum phase systems Factor P (s) as P mp (s)p nmp (s) with P nmp (jω) = and arg P nmp (jω) < Suppose (after loop shaping) that arg P mp (jω c ) = nπ/2. (This means that the slope of the amplitude curve of P mp (jω) is n around ω c ). A phase margin φ m for the overall system imposes arg P nmp (jω c ) π + φ m nπ/2 := 2 (For n = 2 and φ m = 45 deg, = π/4) This constraint imposes restrictions on the location of the crossover frequency. A RHP zero This means G nmp (s) = z s z + s arg G nmp (jω) = 2 arctan ω z The bandwidth constraint becomes ω c z < tan A zero in the RHP near the imaginary axis severely limits the closedloop bandwidth! Syst3 lecture 7 5 Syst3 lecture 7 6
5 A RHP pole This means G nmp (s) = s + p s p A time delay This means G nmp (s) = e st arg G nmp (jω) = 2 arctan p ω The bandwidth constraint becomes ω c > p tan arg G nmp (jω) = ωt The bandwidth constraint becomes ω c T 2 p large corresponds to fast unstable dynamics. Control of fast unstable dynamics requires high closedloop bandwidth! Syst3 lecture 7 7 Syst3 lecture 7 8 Difficult or illposed control problems Problems for which the nonminimum phase part imposes conflicting requirements on the closedloop bandwidth. For instance: a RHP pole + a timedelay, a RHP pole in a noisy environment, a (fast) RHP pole and a (slow) RHP zero. Popular examples: cartpole system; fight aircrafts; unrideable bicycles The X 29 aircraft Specification: 45 deg phase margin for all flight conditions. At one flight condition, the system has one pole at s = 6 and one zero at s = 26. The bandwidth constraint for a system with one RHP pole and one RHP zero is ω c z + p ω c ( p z ) tan Consequence: the achievable phase margin is φ m = 32 deg for n =.5 and φ m = 2.6 deg for n =. The X 29 aircraft never flew... Syst3 lecture 7 9 Syst3 lecture 7 2
6 Klein s unridable bicycle Bicycle with rear wheel steering. steering angle to tilt angle is P (s) = K Transfer function from V as s 2 mgl/j The system has a RHP zero at s = V a. V is the forward velocity and a is the horizontal distance from rear contact point to center of mass. With front steering, the zero would be at s = V a. Pole balancing Inverted pendulum has a RHP pole at s = g l. Assuming that the neural lag of a human is.7s, the bandwidth constraint yields l >.45 with =. Consequence: difficult to balance a pole whose length is less than.5m. The system has one unstable pole at p = ω = mgl/j. With numerical values V = 5m/s, l =.2m, a =.7, m = 7kg, one has p = 2.6 and z = 7.4. The achievable phase margin is then φ m = deg with n =.5. How to make the rear wheel steering bicycle ridable? Syst3 lecture 7 2 Syst3 lecture 7 22 Summary The loop transfer L(jω) provides a very compact information about the performance and robustness of the feedback system. Loop shaping design consists in defining the controller C(jω) to shape the loop transfer L = P C. Simple considerations about the phase margin provide an excellent qualitative understanding of the bandwidth constraints that limit the performance of any real life control system. Syst3 lecture 7 23
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