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Research Papers

# A Framework for Discrete-Time $H2$ Preview Control

[+] Author and Article Information
A. J. Hazell

Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UKandrew_hazell@fmail.co.uk

D. J. N. Limebeer1

Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UKdavid.limebeer@eng.ox.ac.uk

1

Corresponding author.

J. Dyn. Sys., Meas., Control 132(3), 031005 (Apr 21, 2010) (14 pages) doi:10.1115/1.4000810 History: Received August 22, 2007; Revised June 18, 2009; Published April 21, 2010; Online April 21, 2010

## Abstract

The purpose of this paper is to provide a set of synthesis and design tools for a wide class of $H2$ preview control systems. A generic preview design problem, which features both previewable and nonpreviewable disturbances, is embedded in a standard generalized regulator framework. Preview regulation is accomplished by a two-degrees-of-freedom output-feedback controller. A number of theoretical issues are studied, including the efficient solution of the standard $H2$ full-information Riccati equation and the efficient evaluation of the full-information preview gain matrices. The full-information problem is then extended to include the efficient implementation of the output-feedback controller. The synthesis of feedforward controllers with preview is analyzed as a special case—this problem is of interest to designers who wish to introduce preview as a separate part of a system design. The way in which preview reduces the $H2$-norm of the closed-loop system is analyzed in detail. Closed-loop norm reduction formulas provide a systematic way of establishing how much preview is required to solve a particular problem, and determine when extending the preview horizon will not produce worthwhile benefits. The paper concludes with a summary of the main features of preview control, as well as some controller design insights. New application examples are introduced by reference.

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## Figures

Figure 1

A generalized regulator problem with both previewable and nonpreviewable disturbances. The transfer function G is the system to be controlled, K is the controller to be synthesized, and Φ=IZ−N is an N-step delay line (where Z is the Z-transform variable). The disturbance w is not previewable, the control and measurement signals are u and y, respectively, r̂ is the previewable disturbance, and r is the future value of r̂. The filter Wr is used to model the expected frequency content of r.

Figure 2

A simple single-input-single-output (SISO) open-loop preview-tracking problem. The transfer function Φ=Z−N is an N-step delay, G is the plant to be controlled, and K is a feedforward controller. The signal r is the future value of the reference, and e is the tracking error.

Figure 3

Pole-zero plot of the H2-optimal K(Z) for the case where cz=1.05, G(Z)=(Z−cz)/(Z−0.5) and N=20. Crosses represent the poles and circles represent the zeros.

Figure 4

Structure of the H2-optimal preview controller. The signal u is the control, the measurement is y, and r is the future value of the previewable disturbance. The preview length is N, lr is the dimension of r, nr is the order of Wr, and ng is the order of G.

Figure 5

Two equivalent representations of the previewable disturbance rejection problem. These representations are equivalent in the sense that the transfer functions from η and w to z and y are identical. Recall that Φ=Z−NI, which commutes with Wr under multiplication.

Figure 6

A feedforward controller design problem. The notation follows that of Fig. 1.

Figure 7

The structure of the H2-optimal discrete-time preview controller. The signal u(k) is the control, the measurement is y(k), and r(k) is the futuremost value of the previewable disturbance.

Figure 8

A simple preview-tracking problem. The feedback signal is derived from the states of G, Wr, We, and Φ, together with η. The signal u is the control, r is the previewed reference, and z=[z1′z2′]′ is the output to be minimized.

Figure 9

Closed-loop response of the system described in Eq. 50 and Fig. 8 with Wr=1 and We=1000. The plotted output is the signal zg in Fig. 8, and shows the relative nonresponsiveness of the low-preview-horizon system.

Figure 10

Closed-loop response of the system described in Eq. 50 and Fig. 8; the reference weight is given by Wr=Z/(Z−0.99), with We=1000. The improved step response (of zg) for short preview horizons is clearly visible. Note the high-amplitude control in the N=30 case.

Figure 11

Closed-loop response of the example system described in Eq. 50 and Fig. 8. The preview horizon is fixed at N=50 and αi is used to achieve similar closed-loop rise times. While the closed-loop responses (zg) are similar, the control signals are quite different; especially near the beginning of the preview horizon.

Figure 12

Closed-loop response of the example system described in Eq. 50 and Fig. 9; the weighting functions are Wr=1 and We=100/(1−Z). The plotted output is the signal zg in Fig. 8, and is relatively insensitive to the preview horizon. The control signal becomes “spread out,” and lower in amplitude, as the preview horizon is increased.

Figure 13

Bode plots of the closed-loop transfer functions Tr→e and Tr→u, which result from the application of the H∞-optimal controls. The unweighted plant is considered in (a), and a low-pass We(We=1/(Z−1)) is employed in (b).

Figure 14

Preview tracking with integral action. The signal z=[z1′z2′z3′]′ is the output of the closed-loop transfer function whose H2-norm is to be minimized; y=[y1′y2′]′ is the measurement signal. The transfer functions We1, We2, and Wn are shaping filters. The other notation follows that of Fig. 1.

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