Gary J. Balas Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN USA

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1 μ-synthesis Gary J. Balas Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN USA Keywords: Robust control, ultivariable control, linear fractional transforation (LFT), odel uncertainty, structured singular value μ, μ synthesis, ultivariable stability argin K, D K iteration, H control. Contents. Introduction 2. Control Design via D K Iteration 2.. Linear Fractional Transforations, LFTs 2.2. Robust Control Proble Forulation 2.3. D K Iteration for Coplex Uncertainty Two-Step Procedure for Scalar entries d of D Two-Step Procedure for Full D 2.4. ( DG, ) K Iteration for Real and Coplex Uncertainty 3. Control Design Using Fixed-Order Scalings 4. Conclusion Acknowledgeents Glossary Bibliography Biographical Sketch Suary This chapter presents a perspective on the design of robust controllers via μ -synthesis techniques. μ -synthesis is a ultivariable feedback controller design process that accounts for robustness to structured uncertain variations in the open-loop plant dynaics during the synthesis process.. Introduction μ -synthesis concerns the design of ultivariable feedback controller that is robust to structured uncertain variations in the open-loop plant odel. μ refers to the structured singular value which is the reciprocal of the ultivariable stability argin, denoted K. The goal is to autoate the synthesis of ultivariable feedback controllers that achieve desired perforance objectives and are insensitive to odeled plant uncertainty. Stability argins for ultivariable systes can be traced back to Sandberg and Zaes in the 960 s. Their input-output results, which did not include odel uncertainty, were based on conic-sectors, positivity and loop-gain. They for the basis of the structured singular value and ultivariable gain argin. Their sall-gain nonlinear stability Encyclopedia of Life Support Systes (EOLSS)

2 results were in fact based on singular values which now play a significant role in ultivariable syste theory and analysis. The notation of the ultivariable stability argin, K, was introduced by Safonov and Athans. The structured singular value terinology, μ, was introduced by Doyle. The calculation of these analysis etrics required solution of a scaled singular value proble. Optiizing the singular value of the atrix over a set of diagonal scaling reduced the conservatis of these etrics. Initial algoriths for optially scaled singular value probles focused on coplex valued uncertainty. Algoriths for optial scaling techniques and their generalization to real valued uncertainty followed. Through out the reainder of this chapter, the concept of the structured singular value μ and the ultivariable stability argin K, the reciprocal of μ, will be denoted as μ to iprove the readability of the text. The structured singular value is the appropriate tool for analyzing the robustness (both stability and perforance) of a linear, tie-invariant syste with structured uncertainty. Hence, a ultivariable controller synthesis technique which directly seeks to iniize μ would be advantageous. The concept of μ synthesis was introduced in the 980 s, cobining H control design and the diagonal scaling techniques fro the structured singular value. The structured uncertainty was odeled as H -nor bounded uncertain gains in one or ore syste inputs and outputs. The algoriths, denoted as D K iteration, involves iteratively optiizing a set of diagonal scaling frequency response atrices D( jω ) for a fixed controller Ks. ( ) Each of these optiizations are known to be convex individually, though the cobined proble is not convex. Thus the D K iteration algorith for μ -synthesis cannot be guaranteed to be globally optial. It practice, this algoriths works very well and are coputational efficient and fast. Each D Kiteration iproves the bound on perforance and robustness of the controller until subsequent iterations show no iproveent. These algoriths are available in coercial software packages and have been applied successfully to nuerous real applications over the past 5 years. The D K iteration algorith for coplex odel uncertainty, described in the previous paragraph, has been extended to include real paraeter uncertainty. This algorith is denoted as ( DG, ) K iteration and has been applied to several real applications. An alternative approach to μ -synthesis replaces the D -scale state-space realization step in the algorith with optiization of a fixed order diagonal scaling atrix Ds ( ). This approach is also discussed in this section. 2. Control Design via D K Iteration The structured singular value, μ, is a tool for analyzing the stability and perforance robustness of a syste subjected to structured, linear fractional perturbations. In this section, the echanics of a controller design ethodology based on structured singular value objectives are presented. These ethods rely heavily on the upper bound for μ. 2.. Linear Fractional Transforations, LFTs Encyclopedia of Life Support Systes (EOLSS)

3 Linear Fractional Transforations (LFT) are a powerful and flexible approach to represent uncertainty in atrices and systes. Consider first a coplex atrix M as in Figure, relating vectors r and v, Figure : Matrix M relating r and y If r and v are partitioned into a top part and botto part, then we can draw the relationship in ore detail, explicitly showing the partitioned atrix M in Figure 2. Figure 2: Partitioned atrix M relating r and y Suppose a atrix Δ relates v 2 to r 2, Figure 3, as Figure 3: Matrix Δ relating v 2 and r 2 The linear fractional transforation of M by Δ interconnects these two eleents, as shown in Figure 4. Encyclopedia of Life Support Systes (EOLSS)

4 Figure 4: Lower linear fractional transforation of M and Δ Eliinate v 2 and r 2, leaving the relationship between r and v ( ) = + 2Δ 22Δ 2 v M M I M M r FL L ( M,Δ) ( ) = F M,Δ r The notation F L indicates that the lower loop of M is closed with Δ. If the upper loop of M is closed with Ω, then we have and upper linear fractional transforation as shown in Figure 5 Figure 5: Upper linear fractional transforation of M and Ω where U ( ) ( ) F M,Ω := M + M Ω I M Ω M. Encyclopedia of Life Support Systes (EOLSS)

5 - - - TO ACCESS ALL THE 5 PAGES OF THIS CHAPTER, Click here Bibliography Balas G., Doyle J., Glover K., Packard A., Sith R. (993). μ-analysis and Synthesis Toolbox (μ - Tools)}. Natick, MA: MathWorks. [A Matlab toolbox for ultivariable control] Chiang R., Safonov M. (992). Robust Control Toolbox. Natick, MA: MathWorks. [A Matlab toolbox for ultivariable control]. Doyle J. (982). Structured uncertainty in control syste design. IEE Proceedings, 29-D(6), [Introduction of structured singular value and terinology]. Doyle J. (993). Synthesis of robust controllers and filters with structured plant uncertainty. In Proceedings of the IEEE Conference on Decision and Control, New York, NY. [Introduction to the D-K iteration approach to μ -synthesis.]. J.C. Doyle, Glover, K., Khargonekar, P. and Francis, B. (988). State space solutions to standard H 2 and H 8 control probles. IEEE Trans. on Autoatic Control, AC-24 (8), [State-space solutions to H 8 control proble]. Doyle J., Stein G. (98). Multivariable feedback design: Concepts for a classical/odern synthesis. IEEE Trans. on Autoatic Control, AC-26 (), [Excellent introduction to robust control design concepts]. Doyle J., Lenz, K., and Packard, A. (987). Design Exaples using μ synthesis: Space Shuttle Lateral Axis FCS during reentry. NATO ASI Series, vol F34, Modelling, Robustness and Sensitivity Reduction in Control Systes. R.F. Curtain Editor, 28-54, Springer-Verlag. [First application of μ -synthesis to a flight control exaple]. Safonov M. (982). Stability argins of diagonally perturbed ultivariable feedback systes. IEE Proceedings, 29-D (6), [Algorith to copute ultivariable stability argin]. Safonov M., Athans M. (98). A ulti-loop generalization of the circle criterion for stability argin analysis. IEEE Trans. on Autoatic Control, AC-26(2), [Introduction to the ultivariable stability argin]. Safonov M., Chiang R. (993). Real/coplex k -synthesis without curve fitting. In Control and Dynaic Systes, Vol. 56 (Part 2), , Acadeic Press. [An excellent overview of μ -synthesis, or k - synthesis and introduction to direct, fixed-order D-scale synthesis. The introduction section draws on this overview.]. Sandberg L. (964). On the $l_2$-boundedness of solutions of nonlinear functional equations. Bell Syste Technical Journal, 43 (4), [Stability argins for ultivariable systes]. Stein G., Doyle J. (99). Beyond singular values and loopshapes. AIAA Journal of Guidance, Control and Dynaics, 4 (), [Overview and application of μ -synthesis to an exaple.]. Young P. (996). Controller design with real paraetric uncertainty. International Journal of Control, 65, [Introduction to ixed μ -synthesis]. Zaes G. (966). On the input-output stability of tie-varying nonlinear feedback systes-- parts I and II. IEEE Trans. on Autoatic Control, AC-5 (2,3), , [Stability argins for Encyclopedia of Life Support Systes (EOLSS)

6 ultivariable systes]. Biographical Sketch Gary J. Balas received the BS and MS degree in civil and electrical engineering fro UC Irvine and the PhD degree in Aeronautics fro the California Institute of Technology in 990. He is a Professor in the Departent of Aerospace Engineering and Mechanics at the University of Minnesota. He is a coorganizer and developer of the MUSYN Robust Control Short Course and the μ-analysis and Synthesis Toolbox used with MATLAB and the president of MUSYN Inc. He is an Associate Fellow of the AIAA and a Senior Meber of the IEEE. Encyclopedia of Life Support Systes (EOLSS)