GAIN-SCHEDULED CONTROL FOR UNMODELED SUBSYSTEM DYNAMICS. Stephen J. Fedigan 1 Carl R. Knospe 2

Transcription

1 GAIN-SCHEDULED CONTROL FOR UNMODELED SUBSYSTEM DYNAMICS Stephen J. Fedigan 1 Carl R. Knospe 2 1 DSP Solutions R&D Center, Control Systems Branch, Texas Instruments, Inc. M/S 8368, P.O. Box , Dallas, TX Phone: ( / Fax: ( / fedigan@ti.com 2 Center for Magnetic Bearings, Dept. of Mechanical, Aerospace, and Nuclear Engineering University of Virginia, Charlottesville, VA Phone: ( / Fax: ( / crk4y@virginia.edu Abstract: A method is developed, based on an extended D K iteration procedure, for synthesizing controllers that are gain-scheduled on a priori unmodeled dynamics. A controller is designed for an example problem, and its performance is a significant improvement over a controller which is robust to the same set of unmodeled dynamics. In addition, a weighting filter synthesis method is presented which enables time-invariant parametric gain-scheduling problems to be translated into dynamic gain-scheduling problems, which reduces design complexity at the cost of controller performance IFAC. Keywords: Adaptive control, uncertain dynamic systems, automation. 1. INTRODUCTION In many plants, the dynamics of certain components cannot be completely characterized at design time and are only known after the control system has been installed in its operating environment. This scenario commonly occurs in vibration control problems, where the sensors and actuators are mounted on a larger substructure whose dynamics are difficult to predict. For example, in rotating machinery systems with active magnetic bearings, the dynamics of the machine will depend on the machine s mounting configuration, its support structure, and even the building s foundation properties. Consequently, the dynamics of the substructure will vary from one installation to the next. These dynamics must be considered in the controller design, since ignoring them can lead to serious performance degradation or even instability (Lantto, The most direct solution to this problem is to identify the subsystem dynamics in situ, and re-tune or redesign the controller for the fully-identified plant. While this solution is practical for a one-of-a-kind control system or one in limited production, it is much less attractive for a control system produced in larger quantities, as it usually involves fielding trained personnel to re-tune or re-design the controller, a timeconsuming and expensive proposition. Another potential solution is to treat these unknown subsystem dynamics as dynamical uncertainties about a nominal plant, which includes a model of the nominal system and the nominal subsystem. This is a reasonable approach, if the uncertainty levels are low, or the performance requirements are not too stringent. It is also attractive because automated procedures such as D K iteration (Stein and Doyle, 1991 exist to synthesize controllers that guarantee performance in the presence of these uncertainties. However, the controller performance can be disappointing because the uncertainty levels are often quite high. To improve the performance, an on-line adaptive approach (Aström and Wittenmark, 1995 could be adopted, and the adaptive system could learn the unmodeled dynamics. This is viable if the subsystem dynamics are known to be low order. If they are

2 P aug z f z ( s 0 0 s ( w f w To setup the plant for synthesis, an uncertainty description must be chosen for the substructure dynamics, and the known and the unknown elements of the plant must be separated. In this problem, a multiplicative uncertainty description is chosen, with the form: z := y y f Fig. 1: Setup for dynamic gain-scheduled synthesis. potentially high order, a large number of adaptive parameters will be required, and convergence time will become an issue. For systems produced in large quantities, with highly uncertain subsystem dynamics and high performance requirements, an off-line adaptive approach is recommended herein for solving this problem. This approach, which will be referred to as a posteriori gain-scheduling (Fedigan, 1998; Fedigan et al., 1998a; Fedigan et al., 1998b, involves synthesizing a controller, which is gain-scheduled on the a priori unknown subsystem dynamics. Before activating the control system, the subsystem dynamics are identified by an automated procedure and entered into the gainscheduled controller. Because the gain-scheduled controller has a knowledge of the a priori unmodeled dynamics, its performance will be superior to a controller which is robust to the same dynamics. The approach, which will be called dynamic (as opposed to the usual parametric gain-scheduling, detailed in (Fedigan, 1998, is based on established methods (Apkarian and Gahinet, 1995; Helmersson, 1995; Packard, 1994 for synthesizing gain-scheduled controllers. 2. SYNTHESIS METHOD To synthesize a dynamic gain-scheduled controller, an extended D-K iteration method based on the work of (Lu and Balas, 1995; Helmersson, 1995 will be used. Although it was developed originally for synthesizing parametric gain-scheduled controllers, this method can be used without modification for synthesizing dynamic gain-scheduled controllers. As in traditional D-K iteration, the plant P is organized into three sets of inputs (uncertainty, performance, control and three sets of outputs (uncertainty, performance, sensor as shown in Figure 1. P K u u f diag( 1, 2,, nc w Ss ( = Ws ( s, (1 where s ( is a diagonal matrix whose elements have unit magnitude and unknown order and Ws ( is a diagonal matrix of stable, minimum phase weighting filters which modulate the size of the uncertainties. Next, the substructure dynamics, which are embedded in the plant, are pulled out, and the plant is written as a linear fractional transformation (LFT of s (, which represents the dynamic uncertainty, and the nominal plant Ps (. To complete the setup, the -block is copied and supplied to the controller by feedthru connections as shown in Figure 1. The original plant and the wires form an augmented plant P aug. This casts the problem as a µ -synthesis design problem with repeated uncertainty blocks. Unlike standard D-K iteration, one cannot select unity D-scales to start out the iteration due to the special structure of the augmented plant. If this is done, in the following K-step, the controller will sever the feedthru connections in order to reject the disturbance caused by the second -block. This causes the procedure to synthesize a controller which is robust to rather than scheduled on the unmodeled substructure dynamics. This difficulty can be avoided (Lu and Balas, 1995 by either (1 starting out with coupled D-scales for the repeated uncertainty or (2 starting out with a seed controller. The first option is not attractive, since there is not a obvious, systematic way to generate coupled D-scales. However, the second option is feasible, since an initial gain-scheduled controller can be designed using the LPV-LFT methods of (Packard, Although the seed controller is conservative because it uses constant D-scales, it serves as an excellent starting point for further D-K iteration. When the iteration is complete, a controller is designed which is linear fractionally scheduled on a normbounded dynamic system ( s. This resulting controller guarantees a certain -norm level of disturbance rejection, provided the substructure dynamics stay within a design envelope specified by the weighting filters. Unlike traditional D-K iteration, where the controller has a fixed order, the order of the dynamic gain-scheduled controller equals the plant order plus the scale order plus the -block order. Thus, the order of the controller grows with the complexity of the unmodeled substructure dynamics. 3. EXAMPLE PROBLEM To flesh out this concept, an example problem will be considered. The plant is a collection of masses,

3 f c z f c f d k p m p m p K S R K S Controlled System Controller Substructure Fig. 2: Physical schematic of the example problem. springs, and dampers, whose configuration is shown in Figure 2. The controlled system dynamics, modeled by a two-mass system, exhibit both a rigid body mode and a flexible mode. The controlled system is attached to ground through a flexible substructure, whose dynamics are modeled as Ss ( = Ws ( s, (2 where s ( is a stable dynamic system of unit magnitude and Ws ( is a fourth order bandpass filter of the form Ws ( c p Ss ( = Ws ( s z r1 z r2 g bp ( 1+ s z bp1 2 ( 1 + s z bp2 2 = , (3 ( 1 + s p bp1 2 ( 1 + s p bp2 2 z s Amplitude Dynamic Perturbation Envelope of Admissible Perturbations Frequency (rads/sec Fig. 3: Magnitude plot of the weighting filter and the example substructure. 4. SYNTHESIS RESULTS & DISCUSSION Using the plant model described in the previous section, both robust and gain-scheduled controllers were designed for the example problem. The robust controller was designed for benchmarking purposes by standard D-K iteration. The procedure yielded a 26 th order controller with an -norm of With the robust controller in hand for the sake of comparison, the gain-scheduled controller was synthesized. Due to the unstructured performance block, the seed controller could be designed by the method of (Apkarian and Gahinet, 1995, which casts the controller synthesis as a convex optimization problem. This starting controller s performance was improved upon by further D-K iterations, since dynamic rather than constant D -scales were used. The resulting 10 th order dynamic gain-scheduled controller had an - norm of 0.25, a factor of 5 improvement over its robust counterpart whose magnitude response is plotted in Figure 3. The filter coefficients have been selected to yield a gain of 0.10 in the 1.0 to rads/sec passband and a gain of 0.01 in the stopband. The filter limits the substructure s amplification factor to 10.0 but imposes no phase restrictions. The controlled system and the substructure interact thru a controller which measures the relative displacement between the lower controlled system mass and the substructure displacement and applies equal but opposite control forces. An external disturbance force of unit magnitude acts upon the upper controlled system mass, and the control objective is to find a stabilizing controller which minimizes the -norm between disturbance force input, and mass displacements and actuator travel. To further investigate this gain-scheduled controller, a 4 th order example substructure was devised, with the following form: s 2 2ζ a ω a s ω2 ( + + a Ss ( = g (4 ( s 2 + 2ζ r1 ω r1 s+ ω2 r1 ( s 2 + 2ζ r2 ω r2 s+ ω2 r2 g ω 2 r1 ω2 r2 = kω2 a which has one anti-resonance at 8.00 rads/sec and two resonances at 4.00 and 30.0 rads/sec with amplification factors of 10.0 and 20.0 respectively. To prove that this substructure falls within the design envelope, its magnitude response has been plotted along with the magnitude response of the weighting filter in Figure 3. To satisfy the requirements of the small gain theorem, ( s must be stable. Fortunately, this will be the case, since any physical structural (5

4 system will always be stable and the inverse realization of the weighting filter will be stable, due to the weighting filter s minimum phase property. Next, this particular substructure was pre-multiplied by the inverse realization of the weighting filter to determine ( s and the controller was evaluated on ( s by K eval ( s = F L { Ks ( s, } where F L (. is the lower linear fractional transformation (LFT of Ks ( on s ( (Zhou, The amplitude response of K eval ( s and ( s are plotted together, and it now becomes clear how the gain-scheduled controller takes advantage of its knowledge of the substructure s dynamics. As shown in Figure 4, the controller s response is similar to a classical proportional-derivative (P.D. controller, with notches placed at the perturbation s resonance frequencies to gain stabilize the substructure. 5. WEIGHTING FILTER SYNTHESIS FOR PARAMETRIC GAIN- SCHEDULING PROBLEMS While this synthesis technique is valuable for plants with dynamics which are unknown at the time of design, it can also prove useful for plants which have many time-invariant parameters that are unknown at design time. For example, suppose that (4 is nominal substructure, and further suppose that the two resonant frequencies and damping factors vary 30 % about their nominal values. While this problem could be tackled by LPV methods of (Becker, 1996, four parameters would lead to an impractical number of LMI constraints. While these parameters could also be pulled out of the substructure in a feedback interconnection and LFT gain-scheduling methods applied, the block sizes of the parameters would be quite large, making the DG, -scale fitting step in DG, K iteration (Young, 1993 impractical. Designing the seed controller remains practical using the methods of (Apkarian and Gahinet, 1995; Packard, 1994, since it is a convex optimization problem, but the initial controller would be very conservative since it would admit infinitely-fast time-varying parameters. One way to make this problem tractable is to convert this large parametric gain-scheduling problem to a small dynamic gain-scheduling one. To accomplish this, a systematic way will be set forth to translate the parameter perturbations about the nominal model into additive dynamical perturbations, whose magnitude is specified by a frequency-dependent weighting filter, i.e. Ss ( = S 0 ( s + Ws ( s = F U 0Ws ( 1S 0 ( s (6. (7, s ( Transfer Function Amplitude Evaluated Controller Dynamic Perturbation Frequency Response (rads/sec Fig. 4: Illustration of how the gain-scheduled controller takes advantage of its knowledge of the perturbation dynamics by placing notches at the resonance frequencies of the perturbation. To perform this conversion, the parameter values are written as multiplicative perturbations about the nominal values. In other words, the parameters were expressed as ω r1 = ω r1 ( 1+ w ωr1 δ ωr1 ζ r1 = ζ r1 ( 1 + w ζr1 δ ζr1 ω r1 = 4.0 ω r2 = 30.0 ω r2 = ω r2 ( 1+ w ωr2 δ ωr2 ζ r2 = ζ r2 ( 1 + w ζr2 δ ζr2 (8 ζ r1 = ζ r2 = where the overbars are the nominal parameter values, the w s are the relative weights, equal to 0.30, and the δ s are the normalized perturbations, which range on the interval ( 1, 1. Next, these expressions were substituted into (4, the δs were pulled out of the substructure transfer function in an LFT manner. This can be done because the transfer function coefficients are multi-linear in the δs (Zhou, With an LFT description of the substructure, the worst case magnitude deviation of the substructure frequency response from nominal over the range of parameter values was determined at each frequency. This was done by subtracting the LFT description of the substructure from the nominal model as illustrated in Figure 5. Using a mixed- µ analysis (Young, 1993, an upper bound on this worst case deviation (and then the weighting filter magnitude could be established at each frequency. Next, the complex Ceptstrum technique (Oppenheim and Schafer, 1989 was applied on this magnitude data to obtain phase data for the corresponding minimum phase system. The resulting magnitude and phase data were submitted to a transfer function fitting procedure to approximate the weighting filter data. An sixth order fit of the weighting data is shown in Figure 6, which exhibits two peaks centered about the nominal substructure resonance frequencies of 4.0 and 30.0 rads/sec.

5 δ ωr1 I δ ζr δ ωr2 I Weighting Filter Magnitude Response δ ζr S( jω Magnitude W t ( jω 10 3 S 0 ( jω Fig. 5: Illustration of how the worst case magnitude deviation of a system from its nominal model can be cast as a mixed-performance analysis problem. Having synthesized the weighting filter, the new substructure was written as an LFT on ( s, and was interconnected with the rest of the plant, which is itself an LFT on ( s. To complete the design setup, this plant was augmented with feedthru connections, and a dynamic gain-scheduled controller was synthesized by the previously discussed extended D K iteration procedure. The iteration was seeded with an LFT gainscheduled controller employing constant D-scales using the convex optimization of (Apkarian and Gahinet, 1995, and further D K iterations were performed with dynamic D-scales to improve the controller performance. In this case, the dynamic scales did not improve upon the original LFT gainscheduled controller, which had a -norm performance level of The performance of this controller compared very favorably to the corresponding robust controller, synthesized by D K iteration, which had a performance level of CONCLUSIONS A novel way of gain-scheduling controllers on transfer functions has been presented, which is based on established D K iteration methods. These gainscheduled controllers achieve higher performance than controllers which are robust to the same set of dynamics. A weighting filter synthesis method has been presented which enables time-invariant parametric gain-scheduling problems to be translated into dynamic gain-scheduling problems, which can reduce design complexity at the cost of controller performance. This paper has covered controller synthesis for unmodeled subsystem dynamics while previous papers (Fedigan et al., 1998a; Fedigan et al., 1998b Frequency (rads/sec Fig. 6: Weighting filter which approximates worst case magnitude deviation data. have covered controller synthesis for unknown subsystem parameters. Future papers will outline a hybrid gain-scheduling concept (Fedigan, 1998, to design controllers gain-scheduled on both parameters and unmodeled subsystem dynamics. This will be achieved by combining the LFT and LPV gainscheduling methods. 7. REFERENCES P. Apkarian and P. Gahinet. A Convex Characterization of Gain-Scheduled H Controllers. IEEE Transactions on Automatic Control, 40(5: , May, K. Åström and B. Wittenmark. Adaptive Control, Addison-Wesley Publishing, New York, G. Becker. Additional Results on Parameter- Dependent Controllers for LPV Systems. Proceedings of the 1996 International Federation of Automatic Control (IFAC, 13th Triennial World Congress, pages , San Francisco, California, June-July, S. Fedigan. A Posteriori Gain-Scheduled Control for Flexible Subsystem Dynamics. Ph.D. Dissertation, University of Virginia, S. Fedigan, C. Knospe, and R. Williams. Gain Scheduling for Substructure Properties. To appear in Proceedings of the 1998 American Control Conference (ACC, Philadelphia, June, S. Fedigan, C. Knospe, and R. Williams. Gain Scheduling for Substructure Properties in AMBs. To appear in Proceedings of the Sixth International Symposium on Magnetic Bearings, Boston, August 1998.

6 A. Helmersson. Methods for Robust Gain- Scheduling. Ph.D. Dissertation, Linchöping University, E. Lantto, J. Väänänen, and M. Antila. Effect of Foundation Stiffness on AMB Suspension. Proceedings of the Fifth International Symposium on Magnetic Bearings, pages 37-42, Kenazawa, Japan, August, W. Lu and G. Balas. Robust, Gain-Scheduled Control Design of a Lightly Damped Plant. Proceedings of the 34th Conference on Decision and Control (CDC, New Orleans, Louisiana, December, A. Oppenheim and R. Schafer. Discrete-Time Signal Processing. Prentice-Hall, Englewood Cliffs, New Jersey, A. Packard. Gain Scheduling via Linear Fractional Transformations. Systems & Control Letters, 22: 79-92, G. Stein and J. Doyle. Beyond Singular Values and Loopshapes. AIAA Journal of Guidance, Control, and Dynamics, 14(1: 5-16, January, P. Young. Robustness with Parametric and Dynamic Uncertainty. Ph.D. Dissertation, California Institute of Technology, K. Zhou, J. Doyle, and K. Glover. Robust and Optimal Control, Prentice-Hall, Upper Saddle River, New Jersey, 1996.