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The complex structured singular value. (English) Zbl 0772.93023
Summary: A tutorial introduction to the complex structured singular value $$(\mu)$$ is presented, with an emphasis on the mathematical aspects of $$\mu$$. The $$\mu$$-based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems. Several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared, and a simple synthesis problem is studied. Uncertain systems are represented using Linear Fractional Transformations (LFTs) which naturally unify the frequency-domain and state space methods.

##### MSC:
 93B35 Sensitivity (robustness) 93C05 Linear systems in control theory 93C35 Multivariable systems, multidimensional control systems 93C73 Perturbations in control/observation systems
##### Keywords:
linear fractional transformations
##### Software:
Mu Analysis and Synthesis Toolbox
Full Text:
##### References:
 [1] Anderson, B.; Agathoklis, P.; Jury, E.; Mansour, M., Stability and the matrix Lyapunov equation for discrete 2-dimensional systems, IEEE trans. on circuits and sys., 33, 261-267, (1986) · Zbl 0588.93052 [2] Bamieh, B.; Dahleh, M., On robust stability with structured time-invariant perturbations, (), CCEC-92-0331 [3] Balakrishnan, V.; Boyd, S.; Balemi, S., Branch and bound algorithm for computing the minimum stability degree of parameter-dependent linear systems, () · Zbl 0759.93036 [4] Balas, G.; Doyle, J.; Glover, K.; Packard, A.; Smith, R., The μ analysis and synthesis toolbox, Math works and MUSYN, (1991) [5] Bartlett, A.C.; Hollot, C.V.; Lin, H., Root locations of an entire polytope of polynomials: it suffices to check the edges, Mathematics of control, signals, and systems, 1, 1, (1988) · Zbl 0652.93048 [6] Barmish, B.; Khargonekar, P.; Shi, Z.; Tempo, R., Robustness margin need not be a continuous function of the problem data, Systems and control letters, 15, 91-98, (1989) · Zbl 0703.93051 [7] Beck, C., Computational issues in solving lmis, (), 1259-1260 [8] Boyd, S.; Desoer, C., Subharmonic functions and performance bounds on linear time invariant feedback systems, IMA J. of mathematical control and information, 2, 153-170, (1985) [9] Boyd, S.; El Ghaoui, L., Method of centers for minimizing generalized eigenvalues, Linear algebra and its applications, (1993), to appear · Zbl 0781.65051 [10] Boyd, S.; Yang, Q., Structured and simultaneous Lyapunov functions for system stability problems, Int. J. control, 49, 2215-2240, (1989) · Zbl 0683.93057 [11] Chen, M.J.; Desoer, C.A., Necessary and sufficient condition for robust stability of linear distributed feedback systems, Int. J. control, 35, 255-267, (1982) · Zbl 0489.93041 [12] Chen, M.J.; Fan, K.H.; Nett, C.N., The structured singular value and stability of uncertain polynomials: A missing link, () · Zbl 0826.93055 [13] Dahleh, M.A.; Khammash, M.H., Controller design for plants with structured uncertainty, Automatica, 29, 37-56, (1992) · Zbl 0772.93028 [14] Daniel, R.W.; Kouvaritakis, B.; Latchman, H., Principal direction alignment: a geometric framework for the complete solution to the μ problem, (), 45-56 · Zbl 0588.93036 [15] Davis, C.; Kahan, W.; Weinberger, H., Norm preserving dilations and their applications to optimal error bounds, SIAM J. on numerical analysis, 19, 445-469, (1982) · Zbl 0491.47003 [16] de Gaston, R.; Safonov, M., Exact calculation of the multiloop stability margin, IEEE trans. aut. control, 33, 156-171, (1988) · Zbl 0674.93036 [17] Demmel, J., The componentwise distance to the nearest singular matrix, SIAM J. on matrix analysis and applications, 13, 10-19, (1992) · Zbl 0749.65031 [18] Desoer, C.A.; Vidyasagar, M., () [19] Doyle, J.C., Analysis of feedback systems with structured uncertainties, (), 242-250 [20] Doyle, J.C., Matrix interpolation theory and $$H∞$$ performance bounds, () [21] Doyle, J.C., Structured uncertainty in control system design, () [22] Doyle, J.; Glover, K.; Khargonekar, P.; Francis, B., State space solutions to $$H2$$ and $$H∞$$ control problems, IEEE trans. aut. control, 34, 831-847, (1989) · Zbl 0698.93031 [23] Doyle, J. C. and A. Packard. Uncertain multivariable systems from a state space perspective. Proc. of American Control Conf., pp. 2147-2152. [24] Doyle, J.; Packard, A.; Zhou, K., Review of LFTs, LMIs and μ, (), 1227-1232 [25] Doyle, J.C.; Stein, G., Multivariable feedback design: concepts for a classical/modern synthesis, IEEE trans. aut. control, AC-26, 4-16, (1981) · Zbl 0462.93027 [26] Doyle, J.C.; Wall, J.; Stein, G., Performance and robustness analysis for structured uncertainty, (), 629-636 [27] Dunford, N.; Schwartz, J.T., () [28] Fan, M.K.H.; Tits, A.L., Characterization and efficient computation of the structured singular value, IEEE trans. aut. control, AC-31, 734-743, (1986) · Zbl 0607.93019 [29] Fan, M.; Tits, A.; Doyle, J., Robustness in the presence of joint parametric uncertainty and unmodeled dynamics, IEEE trans. aut. control, 36, 25-38, (1991) · Zbl 0722.93055 [30] Foo, Y.K.; Postlethwaite, I., Extensions of the small-μ test for robust stability, IEEE trans. aut. control, 33, 172-176, (1988) · Zbl 0639.93047 [31] Freudenberg, J.S.; Looze, D.P.; Cruz, J.B., Robustness analysis using singular value sensitivities, Int. J. control, 35, 95-116, (1982) · Zbl 0473.93031 [32] Helton, W., A numerical method for computing the structured singular value, Systems and control letters, 21-26, (1988) · Zbl 0653.65050 [33] Special issue on linear multivariable control systems. IEEE Trans. Aut. Control, {\bf26,} 621-623. [34] Jarre, F., An interior-point method for minimizing the maximum eigenvalue of a linear combination of matrices, Technical report SOL 91-8, Stanford, CA, (1991) [35] Kato, T., () [36] Kharitonov, V.L., Asymptotic stability of an equilibrium position of a family of systems of linear differential equations, Differential equations, 14, 1483-1485, (1979) · Zbl 0409.34043 [37] Khargonekar, P.; Kaminer, I., Robust stability analysis with structure norm bounded unstable uncertainty, (), 2700-2701, Boston, MA [38] Khammash, M.; Pearson, J.B., Performance robustness of discrete-time systems with structured uncertainty, Ieee tac, AC-36, 398-412, (1991) · Zbl 0754.93063 [39] Khargonekar, P.; Petersen, I.; Zhou, K., Robust stabilization of uncertain linear systems: quadratic stability and $$H∞$$ control theory, IEEE trans. aut. control, 35, 356-361, (1990) · Zbl 0707.93060 [40] Lu, W.M.; Zhou, K.; Doyle, J., Stability and stabilization of LSI multidimensional systems, (), 1239-1244 [41] Luenberger, D., () [42] Morton, B. and J. Doyle (1985). Private communication. [43] Morton, B.; McAfoos, R., A mu-test for real-parameter variations, (), 135-138, Boston [44] Nesterov, Yu.E.; Nemirovsky, A.S., () [45] Nesterov, Yu.E.; Nemirovsky, A.S., () [46] Newlin, M.; Smith, R., Model validation and a generalization of μ, (), 1257-1258 [47] Newlin, M.P.; Young, P.M.; Doyle, J.C., Improving the bounds for mixed μ problems via branch and bound techniques, (1993), In preparation [48] Osborne, E.E., On preconditioning of matrices, J. assoc. comp. Mach., 7, 338-345, (1960) · Zbl 0106.31604 [49] Overton, M., Large-scale optimization of eigenvalues, NYU computer science department report, (1990) [50] Packard, A., What’s new with μ, () [51] Packard, A.; Whitney Balsamo, J., A maximum modulus theorem for linear fractional transformations, Systems and control letters, 11, 365-367, (1988) · Zbl 0673.93009 [52] Packard, A.; Doyle, J., Quadratic stability with real and complex perturbations, IEEE trans. aut. control, 35, 198-201, (1990) · Zbl 0705.93060 [53] Packard, A.; Fan, M.; Doyle, J., A power method for the structured singular value, (), 2132-2137, Austin, TX [54] Packard, A.; Pandey, P., Continuity of the real/complex structured singular value, (), 875-884, to appear, March 1993 [55] Packard, A.; Teng, J., Robust stability with time-varying perturbations, (), 765-767 [56] Packard, A.; Zhou, K., Improved upper bounds for the structured singular value, (), 934-935 [57] Packard, A.; Zhou, K.; Pandey, P.; Leonhardsen, J.; Balas, G., Optimal I/O similarity scaling for full-information and state-feedback problems, Systems and control letters, 19, 271-280, (1991) [58] Poolla, K. (1991). Personal communication. [59] Popov, V.M., Absolute stability of nonlinear systems of automatic control, Automat. remote control, 22, 857-875, (1962) · Zbl 0107.29601 [60] Power, S.C., () [61] Redheffer, R., Inequalities for a matrix Riccati equation, J. of mathematics and mechanics, 8, (1989) [62] Redheffer, R., On a certain linear fractional transformation, J. math. phys., 39, 269-286, (1960) · Zbl 0102.10402 [63] Rohn, J.; Poljak, S., Radius of nonsingularity, Mathematics of control, signals and systems, (1992), to appear, 1993 [64] Rotea, M.A.; Corless, M.; Da, D.; Petersen, I., Systems with structured uncertainty: relations between quadratic and robust stability, (), 885-894, to appear · Zbl 0785.93076 [65] Rotea, M.A.; Khargonekar, P.P., Stabilization of uncertain systems with norm bounded uncertainty—A control Lyapunov function approach, SIAM J. control optim., 27, 1462-1476, (1989) · Zbl 0682.93050 [66] Safonov, M.G., Tight bounds on the response of multivariable systems with component uncertainty, (), 451-460 [67] Safonov, M.G., () [68] Safonov, M.G., Stability margins of diagonally perturbed multivariable feedback systems, (), 251-256 [69] Safonov, M.G., Stability of interconnected systems having slope bounded nonlinearities, () · Zbl 0547.93056 [70] Safonov, M.G., Optimal diagonal scaling for infinity norm optimization, Systems and control letters, 7, 257-260, (1986) [71] Safonov, M.G.; Doyle, J.C., Minimizing conservativeness of robustness singular values, () [72] Safonov, M.G.; Le, V.X., An alternative solution to the $$H∞$$ optimal control problem, Systems and control letters, 10, 155-158, (1988) · Zbl 0652.93013 [73] Sezginer, R.; Overton, M., The largest singular value of e^{x}ae−x is convex on convex sets of commuting matrices, IEEE trans. on aut. control, 35, 229-230, (1990) · Zbl 0704.93023 [74] Sideris, A.; de Gaston, R.R.E., Multivariable stability margin calculation with uncertain correlated parameters, (), 766-773 [75] Sideris, A.; Sanchez Peña, R., Robustness margin calculations with dynamic and real parametric uncertainty, IEEE trans. aut. control, 35, 970-974, (1990) · Zbl 0723.93056 [76] Sideris, A.; Sánchez Peña, R.S., Fast computation of the multivariable stability margin for real interrelated uncertain parameters, IEEE trans. aut. control, 34, 1272-1276, (1989) · Zbl 0689.93050 [77] Skogestad, S. (1987) Personal communication. [78] Skogestad, S.; Morari, M.; Doyle, J., Robust control of ill-conditioned plants: high-purity distillation, IEEE trans. aut. control, 33, 1092-1105, (1988) · Zbl 0669.93055 [79] Smith, R.; Doyle, J., Model validation: A connection between robust control and identification, IEEE trans. aut. control, 37, 942-952, (1992) · Zbl 0767.93020 [80] Stein, G.; Doyle, J., Beyond singular values and loopshapes, AIAA J. of guidance and control, 14, 5-16, (1991) · Zbl 0751.93031 [81] Wang, W.; Doyle, J.; Beck, C.; Glover, K., Model reduction of LFT systems, (), 1233-1238 [82] Willems, J.C., Least squares stationary optimal control and the algebraic Riccati equation, IEEE trans. aut. control, 16, 621-634, (1971) [83] Willems, J.C., () [84] Willems, J.L., The circle criterion and quadratic Lyapunov functions for stability analysis, IEEE trans. aut. control, 18, 184, (1973) · Zbl 0265.93022 [85] Young, P. (1992) Personal communication. [86] Young, P.M.; Doyle, J.C., Computation of μ with real and complex uncertainties, (), 1230-1235 [87] Young, P.M.; Doyle, J.C., Properties of the mixed μ problem and its bounds, (1993), In preparation [88] Young, P.M.; Newlin, M.P.; Doyle, J.C., Μ analysis with real parametric uncertainty, (), 1251-1256 [89] Young, P.M.; Newlin, M.P.; Doyle, J.C., Practical computation of the mixed μ problem, (), to appear · Zbl 0831.93020 [90] Zames, G.; Zames, G., On the input-output stability of nonlinear time-varying feedback systems, part II, IEEE trans. aut. control, IEEE trans. aut. control, 11, 465-476, (1966) [91] Zhou, K.; Khargonekar, P., Stability robustness bounds for linear state-space models with structured uncertainty, IEEE trans. aut. control, 31, 621-623, (1987) · Zbl 0616.93060
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