Trajectory tracking control and feedforward
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1 Trajectory tracking control and feedforward Aim of this chapter : Jan Swevers May 2013 Learn basic principles of feedforward design for trajectory tracking Focus is on feedforward filter design: comparison of several alternative design approaches 0-0
2 Trajectory tracking control and feedforward 1/39 Outline of this chapter Basic principles of trajectory tracking and assumptions Two alternative schemes to implement feedforward System model inversion for perfect tracking Dealing with nonminimum phase zeros Example
3 Trajectory tracking control and feedforward 2/39 Reference material: E. Gross, M. Tomizuka, W. Messner, Cancellation of discrete time unstable zeros by feedforward control, Transactions of the ASME, Journal of Dynamic Systems, Measurement and Control, March 1994, Vol. 116, pp E. Gross, M. Tomizuka, Experimental flexible beam tim tracking control with a truncated series approximation to uncancelable inverse dynamics, IEEE Transactions on Control Systems Technology, Vol. 2, No. 4, December 1994, pp M. Tomizuka, Zero phase error tracking algorithm for digital control, Transactions of the ASME, Journal of Dynamic Systems, Measurement and Control, March 1987, Vol. 109, pp J.A. Butterworth, L.Y. Pao, D.Y. Abramovitch, The effect of nonminimum-phase zero locations on the performance of feedforward model-inverse control techniques in discrete-time systems, 2008 American Control Conference, Seattle, USA, June 11-13, 2008, pp
4 Trajectory tracking control and feedforward 3/39 Basic principles of trajectory tracking Fast and accurate tracking of time-varying reference signals requires feedforward control. With feedback control only: transfer function between reference y d (k) and output y(k) 1. Feedforward: uses past, present and future information of y d (k) to enhance the tracking performance. Assumptions we make in this chapter: considered system is SISO linear time-invariant (LTI), a perfect discrete-time model of the system is available, no model uncertainty or system variations/distrubances are considered.
5 Trajectory tracking control and feedforward 4/39 Two basic approaches exist to design feedforward signals Filter design: a LTI feedforward filter is designed and filters the given reference signal y d (k) to generate the feedforward signal, feedforward filter is independent of y d (k), real-time computational requirements are low, no system constraints can be imposed, y d (k) must be sufficiently smooth to avoid discontinuities in feedforward signal. Direct design of reference and feedforward signals: use of heuristic rules (for typical motion system dynamics, e.g. double integrator), or optimization based design, different trajectory/feedforward parameterizations (e.g. splines) and objectives (time-, energy-optimal) can be considered, system constraints (kinematic and dynamic) can be explicitly taken into account, if optimization based, computational requirements are higher.
6 Trajectory tracking control and feedforward 5/39 Two alternative schemes to implement feedforward
7 Trajectory tracking control and feedforward 6/39 Scheme (a): independent feedback and feedforward design Without feedforward F (z) = 0: Y (z) = G(z)C(z) (1 + G(z)C(z)) Y d(z). With feedforward F (z) 0: Y (z) = G(z)U(z) = G(z)(U ff (z) + C(z)E(z)), = G(z)(F (z)y d (z) + C(z)(Y d (z) Y (z))), Y (z)(1 + G(z)C(z)) = (G(z)F (z) + G(c)C(z))Y d (z), with H(z) = G(z). Y (z) Y d (z) = G(z)F (z) + G(z)C(z), 1 + G(z)C(z) = 1 if F (z) = H 1 (z),
8 Trajectory tracking control and feedforward 7/39 Scheme (b): dependent feedback and feedforward design Without feedforward F (z) = 1: Y (z) = With feedforward F (z) 1: with Y (z) = Y (z) Y d (z) Y (z) Y d (z) G(z)C(z) (1 + G(z)C(z)) R(z) = G(z)C(z) (1 + G(z)C(z)) Y d(z). = G(z)C(z) 1 + G(z)C(z) R(z), R(z) = F (z)y d(z), G(z)C(z) 1 + G(z)C(z) F (z), = 1 if F (z) = H 1 (z), H(z) = G(z)C(z) 1 + G(z)C(z).
9 Trajectory tracking control and feedforward 8/39 System model inversion for perfect tracking In both cases: Independent design: F (z) = H 1 (z). H(z) = G(z). Dependent design: H(z) = G(z)C(z) 1 + G(z)C(z). Remark that with the independent design and feedforward configuration, in case of perfect model inversion, a perfect system model and no disturbances, the feedback controller is out of work. This is not the case in the dependent configuration.
10 Trajectory tracking control and feedforward 9/39 General form: H(z) = B(z) A(z) = B s(z)b u (z), A(z) with B(z) and A(z) the numerator and denominator of the stable open-loop or closed loop system transfer function, respectively. The numerator B(z) is partitioned in two parts: B s (z) contains all stable (invertible) zeros (on and inside the unit circle), and B u (z) contains all unstable (non-invertible) zeros (outside the unit circle).
11 Trajectory tracking control and feedforward 10/39 A(z) = z n + a 1 z n a n, B s (z) = b s0 z m + b s1 z m b sm, B u (z) = b u0 z r + b u1 z r b ur, with n m + r, d = n m r is the relative degree of the system, equal to the number of system delays. Inverting H(z) entirely F (z) = H 1 (z) = A(z) B s (z)b u (z), yields an unstable feedforward filter, which is undesirable. Different substitutions schemes exist to replace B u (z) in the filter by a stable approximation. We will discuss the following three schemes: (i) ZPET: the Zero Phase Error Tracking scheme, (ii) ZMET: the Zero Magnitude Error Tracking scheme, and (iii) TSA: the Truncated Series Approximation scheme.
12 Trajectory tracking control and feedforward 11/39 ZPET: Zero Phase Error Tracking scheme The feedforward filter obtained using the ZPET scheme equals: F (z) = H 1 (z) = A(z) B s (z)b u:zp(z) with B u:zp(z) = B u(z) 2 z=1 z r B f u(z) = B u(1) 2 z r B f u(z) B f u(z) is obtained by flipping the coefficients of B u (z): F (z) has a preview of d + r samples. B f u(z) = b u0 + b u1 z b ur z r..
13 Trajectory tracking control and feedforward 12/39 The relation between Y d (z) and Y (z) equals: Y d (z) Y (z) = H(z)F (z) = T (z), = B s(z)b u (z) A(z) = B u(z)b f u(z) B u (1) 2 z r. A(z) B s (z)b u:zp(z), The phase of T (z) is zero, hence the name Zero Phase Error Tracking (ZPET). This will be illustrated later.
14 Trajectory tracking control and feedforward 13/39 ZMET: Zero Magnitude Error Tracking scheme The feedforward filter obtained using the ZMET scheme equals: F (z) = H 1 (z) = A(z) B s (z)b u:zm(z), with B u:zm(z) = B f u(z). B f u(z) is obtained by flipping the coefficients of B u (z): F (z) has a preview of d samples. B f u(z) = b u0 + b u1 z b ur z r.
15 Trajectory tracking control and feedforward 14/39 The relation between Y d (z) and Y (z) equals: Y d (z) Y (z) = H(z)F (z) = T (z), = B s(z)b u (z) A(z) = B u(z) B f u(z). A(z) B s (z)b u:zm(z), The magnitude of T (z) is one, hence the name Zero Magnitude Error Tracking (ZMET). This will be illustrated later.
16 Trajectory tracking control and feedforward 15/39 TSA: Truncated Series Approximation scheme The feedforward filter obtained using the TSA scheme equals: F (z) = H 1 (z) = A(z) B s (z)b u:ta(z) with 1/B u:ta(z) a finite Taylor series expansion about the origin of length q of 1/B u (z), that is convergent in the region that includes the unit circle. 1 B u:ta(z) = α 0 + α 1 z 1 + α 2 z α q 1 z q 1. The procedure to calculate this series expansion is discussed below. F (z) has a preview of d + r + q 1 samples.
17 Trajectory tracking control and feedforward 16/39 The relation between Y d (z) and Y (z) equals: Y d (z) Y (z) = H(z)F (z) = T (z) = B s(z)b u (z) A(z) A(z) B s (z)b u:ta(z) = B u (z)(α 0 + α 1 z 1 + α 2 z α q 1 z q 1 ).
18 Trajectory tracking control and feedforward 17/39 Example: Consider one nonminimum phase zero α: 1 B u (z) = 1 (z α). Two possible Taylor series expansions about the origin: 1 z α = z 1 + αz 2 + α 2 z z > α, 1 z α = 1 α 1 α 2 z1 1 α 3 z2... z < α. Since α > 1, the second infinite sequence is convergent in the region that includes the unit circle.
19 Trajectory tracking control and feedforward 18/39 The product of the q term truncated series and the system zero (z α) equals: ( 1 α 1 α 2 z1 1 α 3 z α q zq 1 )(z α) = 1 zq α q. This is the resulting relation between Y d (z) and Y (z), that is T (z), after applying this approach. The gain of T (z) lies within a band 1 ± 1 α q, and the maximum value of the phase lead and lag, θ max, is given by: θ max = arcsin( 1 α q ). The gain and phase cycling from their maximum to their minimum values occurs q/2 times from DC to the Nyquist frequency.
20 Trajectory tracking control and feedforward 19/39 So, by choosing q such that α q is large, we can get arbitrarily close to 1. The farther α is from the unit circle, the smaller q needs to be to get a satisfactory frequency response. Remark that the DC gain of the resulting T (z) is 1 1 α 1. This can be q corrected by scaling the series expansion with α q /(α q 1).
21 Trajectory tracking control and feedforward 20/39 Systematic approach to calculate the Truncated Series Approximation Consider 1 B u (z) = 1 b u0 z r + b u1 z r 1, b u(r 1) z + b ur and we wish to obtain the noncausal response or the expansion of it in positive powers of z. Step 1: Note that all roots of 1/B u (z) lie outside the unit circle, and therefore we substitute z by z 1 such that this transfer function is transformed into a stable transfer function. 1 B u (z 1 ) = = 1 b u0 z r + b u1 z r b u(r 1) z 1 + b ur, z r b ur z r + b u(r 1) z r b u1 z + b u0.
22 Trajectory tracking control and feedforward 21/39 Step 2: Calculate q terms of the impulse response of this stable transfer function (e.g. using Matlab), yielding: 1 B u (z 1 ) α 0 + α 1 z 1 + α 2 z α q 1 z q+1. Step 3: replace z 1 by z, yielding 1 B u:ta(z) = α 0 + α 1 z 1 + α 2 z α q 1 z q 1. Step 4: scale the series expansion such that the DC-gain of T (z) is equal to 1.
23 Trajectory tracking control and feedforward 22/39 This q-term series expansion is the solution of the following least squares problem: 2π q 1 1 min α i B u (e jθ ) α i e ijθ 2 dθ 0 If it is known that the desired trajectory has dominant frequency components in a particular band, a frequency weighting can be applied to emphasize this region: i=0 min α i 2π 0 q 1 1 B u (e jθ ) α i e ijθ 2 W (e jθ ) 2 dθ i=0 This solution can be calculated analytically (easy for low order polynomials) or numerically by gridding the frequency axis.
24 Trajectory tracking control and feedforward 23/39 Example to illustrate ZPET, ZMET and TSA schemes Consider the following NMP system: with p = 0.5 and a > 1. B s (z) = K c and B u (z) = z a. H(z) = K c z a z p K c is selected such that the DC-gain of H(z) is equal to 1: H(1) = 1. ZPET: Preview is 1 sample. F (z) = 1 K c (z p)( az + 1) ( a + 1) 2 z T (z) = (z a)( az + 1) ( a + 1) 2 z
25 Trajectory tracking control and feedforward 24/39 ZMET: No preview. TSA: Preview is q samples. F (z) = 1 K c (z p) ( az + 1) T (z) = (z a) ( az + 1) F (z) = 1 K c (z p)(α 0 + α 1 z α q z q 1 ) 1 T (z) = (z a)(α 0 + α 1 z α q z q 1 ) 1
26 Trajectory tracking control and feedforward 25/39 Reference output is a smooth step: 9 th order polynomial with 100 leading zeros. Reference was generated using polytraj.m
27 Trajectory tracking control and feedforward 26/39
28 Trajectory tracking control and feedforward 27/39 Case 1: a = 1.1 a low frequency NMP zero Comparison of ZPET, ZMET and TSA with q = 90: FRF of total transfer function T (z). ZPET: Blue-DashDot, ZMET: Green-Dashed, TSA: Red-Solid.
29 Trajectory tracking control and feedforward 28/39 Tracking error without feedforward.
30 Trajectory tracking control and feedforward 29/39 Comparison of ZPET, ZMET and TSA with q = 90: Tracking error y d (k) y(k). ZPET: Blue-DashDot, ZMET: Green-Dashed, TSA: Red-Solid.
31 Trajectory tracking control and feedforward 30/39 Comparison of ZPET, ZMET and TSA with q = 90: Feedforward input u ff (k) or r(k). ZPET: Blue-DashDot, ZMET: Green-Dashed, TSA: Red-Solid.
32 Trajectory tracking control and feedforward 31/39 Case 2: a = -1.1 a high frequency NMP zero Comparison of ZPET, ZMET and TSA with q = 90: FRF of total transfer function T (z). ZPET: Blue-DashDot, ZMET: Green-Dashed, TSA: Red-Solid.
33 Trajectory tracking control and feedforward 32/39 Tracking error without feedforward.
34 Trajectory tracking control and feedforward 33/39 Comparison of ZPET, ZMET and TSA with q = 90: Tracking error y d (k) y(k). ZPET: Blue-DashDot, ZMET: Green-Dashed, TSA: Red-Solid.
35 Trajectory tracking control and feedforward 34/39 Comparison of ZPET, ZMET and TSA with q = 90: Feedforward input u ff (k) or r(k). ZPET: Blue-DashDot, ZMET: Green-Dashed, TSA: Red-Solid.
36 Trajectory tracking control and feedforward 35/39 Example revisited The TSA scheme yields the best results but requires a very long preview. If we restrict the preview to a horizon comparable to that of the ZPET approach, by e.g. taking q = 3, the performance of the TSA scheme is worse. Case 1: a = 1.1 a low frequency NMP zero
37 Trajectory tracking control and feedforward 36/39 Comparison of ZPET, ZMET and TSA with q = 3: Tracking error y d (k) y(k). ZPET: Blue-DashDot, ZMET: Green-Dashed, TSA: Red-Solid.
38 Trajectory tracking control and feedforward 37/39 Case 1: a = -1.1 a high frequency NMP zero ZPET: Blue-DashDot, ZMET: Green-Dashed, TSA: Red-Solid.
39 Trajectory tracking control and feedforward 38/39 Comparison of ZPET, ZMET and TSA with q = 3: Tracking error y d (k) y(k). ZPET: Blue-DashDot, ZMET: Green-Dashed, TSA: Red-Solid.
40 Trajectory tracking control and feedforward 39/39 Summary The goal of feedforward control is to compensate the overall closed-loop system dynamics. We focus on feedforward filter design. The design of the feedforward filter can be either dependent or independent of the feedback controller. Perfect tracking is achieved for minimum phase systems by using a feedforward filter which is the inverse system transfer function. For non-minimum phase systems: substitute the NMP zeros using the ZPET, ZMET or the TSA scheme. We assume a perfect system model and no disturbances! Other (also robust) approaches exist!
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