Non-Model-Based Frequency Control Design

Steve Rogers, srogers@isr.us, Brian Stolarik, bstolark@isr.us Institute for Scientific Research, 119 Roush Circle, Box 27220, Fairmont, WV 26555-2720

Abstract Many legacy baseline simulations exist in various languages written over the past few decades. These occasionally must be upgraded and new controllers must be designed or the existing ones redesigned to accommodate changes. Many control designs rely on the ability to obtain linear models at various operating conditions. In many cases it is difficult or impossible to linearize the baseline legacy simulations. Lack of linearization capability limits the use of control designs utilizing the rich possibilities of modern control theory. H-infinity and H2 theory have combined the use of classical control criteria with modern norm-based control theory. Even in the absence of linearized models we can take advantage of H∞ ideas by using only simulation outputs. In this paper we review some of the classical control criteria used in H∞ approaches and apply them to the outputs of a simple helicopter simulation output. Keywords : H infinity, frequency control

Introduction H∞ was developed in the 80’s and has been very popular since1-4. In nearly all of the development there is a reliance on the ability to obtain linear models of the plant to affect the design. Frequency spectrum data from the linearized models is then compared with a desired spectrum to arrive at an optimal or sub optimal controller. A rich body of designs has been developed in this fashion. Unfortunately, the necessary linear models require a significant effort to develop, validate, and maintain. In addition, the controller gains generally must be adjusted when transitioning from less accurate linear mo dels to more accurate nonlinear models or hardware-in-the-loop environments. Linear models are used for proof-of concepts. However, with the many legacy designs and decades of research, there is little need to investigate new controller structures. If designs could be developed directly from simulation outputs of the nonlinear models, control development could proceed much faster. The advantages resulting from designs directly from more accurate nonlinear models include: 1) less reliance and therefore, less effort required for linear model development, 2) by proceeding directly to nonlinear models more development resources are available for model validation, and 3) overall control development may be significantly reduced. Considering today’s high-powered computers, comparison studies may be performed quickly with even complex simulations. The design process may be automated in a straightforward manner. The paper is organized as follows: 1) Introduction, 2)

Classical control using H∞ ideas, and 3) Simulation on unstable Helicopter Model.

Classical Control Using H∞ Ideas In reference 1, performance and stability criteria are generated based on available linear models. Figure 1 shows the typical assumed architecture of the system S.

C P

u y

F i g u r e 1 T h e C l o s e d L o o p S y s t e m S

In Figure 1 u is the input vector, y is the output vector, C is the compensator, and P is the given plant rational function. Key functions include1:

• Closed loop function T = PC/(1+PC) • Sensitivity function S = (1+PC)-1 • Tracking error for input u, Su • Closed loop compensator Q = CS • Closed loop plant PS • Open loop transfer function L = PC

Figure 2 shows a model reference approach to frequency control design.

Model Reference Frequency Design Architecture

uC P

yM

M is a low pass filter with a desired frequency response. U(w) and Y(w) are frequency responses used for controller C design.

fft fft

U(w) Y(w)

Figure 2 Model Reference Frequency Design A frequency domain performance requirement is an inequality that the designable transfer function T must satisfy, and an interval in ω for which the inequality is required to hold. The frequency domain performance requirements may be written in the form of a disk inequality |K(jω) – T(jω)| <= R(jω), ωa <= ω <= ωb, which T must satisfy. Various classical measures of performance include: phase/gain margin, tracking error, bandwidth, closed loop roll-off, and disturbance rejection. The above

may be applied at different frequencies to develop a composite performance requirement. Table 1 shows an example with five composite requirements. Other classical time domain requirements such as overshoot and settling time may be formulated as in Table 11. The frequency ranges used in Table 1 are normalized to the sampling frequency – which is 1. In this approach great care must be taken to ensure consistency between multiple requirements. The disk inequalities vary greatly depending on the requirement suite being considered1. Constraint Disk

inequality Freq Band K(j

ω) R(jω)

Plant |T(jω)-1| ≤ 1/|P(jω)|

0 ≤ ω ≤ .03 1 1/|P(jω)|

Tracking |T(jω)-1| ≤ m1 0 ≤ ω ≤ .05 1 m1 Gain-phase

|T(jω)-1| ≤ m2 .05 ≤ ω ≤ .07 1 m2

Bandwidth |T(jω)| ≤ .707 .07 ≤ ω ≤ .1 0 .707 Roll-off |T(jω)| ≤

m3|P(jω)| .1 ≤ ω ≤ 1 0 m3|P(jω)

| Table 1 Design Example with 5 constraints

The difficulty with the above and similar approaches 1,2 is the necessity of beginning with a plant model P. In order to automate and/or improve overall design efficiency one should consider putting simulation outputs to better use. Consider the relation |Ysp(jω) - Y(jω)| = |Ysp(jω)| - |Y(jω)|, 0 <= ω <= 1, where Ysp is the closed loop frequency span from 0 to the sampling frequency. If the difference of the above equation is minimized the controller that regulates the output Y will force it to conform to the desired Ysp frequency characteristics throughout the frequency spectrum. Fast fourier transforms (fft) may be run on simulation outputs for comparison purposes and to validate the design. The simplified design procedure is minimize {|fft(Ysp)| - |fft(Y)|} by manipulating the controller C.

Simulation on unstable Helicopter Model The above output fft comparison was tested on a linear helicopter model simulation5.

Figure 3 Roll, yaw, vert. velocity, and forward velocity The model is unstable & interactive. There are four inputs so four outputs are regulated. In our case we are designing

an autopilot to regulate roll rate, yaw rate, forward velocity, and vertical velocity. The tracking performance is shown in Figure 3. The input setpoints were passed through a low pass filter with a 0.2 second time constant. The frequency response fft’s are shown in Figure 4. The tracking portion of the frequency span is very small and was swamped by the remainder of the frequency spectrum; consequently, we had to add an extra term to ensure tracking. In addition, if magnitudes are used, another term is necessary to give the controller the proper signs, otherwise . The minimization criteria to ensure tracking is minimize {[|fft(Ysp)| - |fft(Y)|] + α|Ysp – Y|}, where Ysp and Y are vectors for SISO systems and column vectors for MIMO systems and α is a scalar weight for SISO systems and a diagonal matrix for MIMO systems. Another option is to use system identification methods to estimate a closed loop model and force its spectrum to equal the frequency spectrum of a desired low pass filter. In this study the fft approach was used.

Figure 4 FFT magnitude responses for heli simulation

The results show that this approach may be used for frequency domain control by passing the setpoint through a low pass filter with desired frequency response characteristics.

References 1. Helton, J., and Merino, O., Classical Control Using

H Methods – An Introduction to Design, Society for Industrial and Applied Mathematics, 1998, ISBN 0-89871-424-9

2. Glad, T. & Ljung, L. Control Theory – Multivariable and Nonlinear Methods , Taylor & Francis, 1997, ISBN 0-7484-0878-9

3. Francis, B., First Course in H∞ – Control, Springer-Verlag, New York, 1987

4. Skogestad, S. etal, Multivariable Feedback Control , 1996, John Wiley, New York

5. Bryson, A., Applied Linear Optimal Control , Cambridge University Press, 2002