LMIs and LMI Problems This can be generalized to linear differential inclusions (LDi A(t)x where A(t) varies in the convex envelope of a set of Lti models A)eCo41…An}=1∑0A:a120.∑a1=1 a sufficient condition for the asymptotic stability of this ldi is the feasibility of Find p= pf such that a p+pa<0. P>I he random-mean-squares(rMs)gain of a stable LTI system Ax+ Bu the largest input/output gain over all bounded inputs u(t). This gain is the global minimum of the following linear objective minimization problem [ 1, 26, Minimize y over X=X and y such that A X+XA XB C BX -YI D L@G performance. for a stable LTI system Ax+B 17
LMIs and LMI Problems 1-7 This can be generalized to linear differential inclusions (LDI) where A(t) varies in the convex envelope of a set of LTI models: A sufficient condition for the asymptotic stability of this LDI is the feasibility of Find P = PT such that RMS gain . the random-mean-squares (RMS) gain of a stable LTI system is the largest input/output gain over all bounded inputs u(t). This gain is the global minimum of the following linear objective minimization problem [1, 26, 25]. Minimize γ over X = XT and γ such that LQG performance . for a stable LTI system x · = A t( )x A t( ) Co A1,…,An ∈ { } ai Ai : i = 1 n ∑ ai ≥ 0 ai i = 1 N , ∑ = 1 = Ai TP PAi + < 0 , P I > . x · = Ax Bu + y Cx Du = + ATX XA + XB CT BTX –γI DT C D –γI 0 < X 0 > G x · = Ax Bw + y Cx =
where w is a white noise disturbance with unit covariance the lQg or h performance IGll2 is defined by G2: lim e(ty()dt o Gvo)GUo)do It can be shown that G2=inf[ Trace(CPC ) AP+PA + BB <0) Hence G is the global minimum of the LMi problem Minimize Trace(Q)over the symmetric matrices P, Q such that AP+PA+BB<0 Q CP PCT P/>o Again this is a linear objective minimization problem since the objective Trace (Q)is linear in the decision variables(free entries of P, Q). 1-8
1 Introduction 1-8 where w is a white noise disturbance with unit covariance, the LQG or H2 performance ||G||2 is defined by It can be shown that Hence is the global minimum of the LMI problem Minimize Trace (Q) over the symmetric matrices P,Q such that Again this is a linear objective minimization problem since the objective Trace (Q) is linear in the decision variables (free entries of P,Q). G 2 2 : = E T → ∞ lim 1 T --- y T 0 T ∫ ( )t y t( )dt 1 2π ------ GH –∞ ∞ ∫ = ( ) jω G j( ) ω dω G 2 2 inf Trace CPCT { ( ) : AP PAT BBT = + + < 0 } G 2 2 AP PAT BBT + + < 0 Q CP PCT P > 0
Further Mathematical Background Further Mathematical Background Efficient interior-point algorithms are now available to solve the three generic LMI problems(1-3)(1-5 )defined in"The Three Generic LMI Problems"on page 1-5. These algorithms have a polynomial-time complexity That is, the number N(E)of flops needed to compute an E-accurate solution is bounded by N(e)oMN log(vle) where M is the total row size of the LMi system, N is the total number of scalar decision variables, and v is a data-dependent scaling factor. The LMI Control Toolbox implements the Projective Algorithm of Nesterov and Nemirovski [20 19]. In addition to its polynomial-time complexity, this algorithm does not require an initial feasible point for the linear objective minimization problem (1-4)or the generalized eigenvalue minimization problem(1-5) Some LMi problems are formulated in terms of inequalities rather than strict inequalities. For instance, a variant of (1-4)is Minimize c x subject to A(x)o0 While this distinction is immaterial in general, it matters when A(x)can be made negative semi-definite but not negative definite A simple example is Minimize c x subject to|20 (16 Such problems cannot be handled directly by interior-point methods which require strict feasibility of the LMI constraints. A well-posed reformulation of 1-6)would be Minimize cTx subject tox So Keeping this subtlety in mind, we always use strict inequalities in this manual 1-9
Further Mathematical Background 1-9 Further Mathematical Background Efficient interior-point algorithms are now available to solve the three generic LMI problems (1-3)–(1-5) defined in “The Three Generic LMI Problems” on page 1-5. These algorithms have a polynomial-time complexity. That is, the number N(ε) of flops needed to compute an ε-accurate solution is bounded by N(ε) ð M N3 log(V/ε) where M is the total row size of the LMI system, N is the total number of scalar decision variables, and V is a data-dependent scaling factor. The LMI Control Toolbox implements the Projective Algorithm of Nesterov and Nemirovski [20, 19]. In addition to its polynomial-time complexity, this algorithm does not require an initial feasible point for the linear objective minimization problem (1-4) or the generalized eigenvalue minimization problem (1-5). Some LMI problems are formulated in terms of inequalities rather than strict inequalities. For instance, a variant of (1-4) is Minimize cTx subject to A(x) ð 0. While this distinction is immaterial in general, it matters when A(x) can be made negative semi-definite but not negative definite. A simple example is (1-6) Such problems cannot be handled directly by interior-point methods which require strict feasibility of the LMI constraints. A well-posed reformulation of (1-6) would be Minimize cTx subject to x Š 0. Keeping this subtlety in mind, we always use strict inequalities in this manual. Minimize c T x subject to x x x x ≥ 0
References [1] Anderson, B DO, and S. Vongpanitlerd, Network Analysis, Prentice-Hall, Englewood Cliffs, 1973 [2] Apkarian, P, P Gahinet, and G. Becker, "Self-Scheduled Hoo Control of Linear Parameter-Varying Systems, Proc. Amer. Contr. Conf, 1994, pp 856-860 3] Bambang, R, E. Shimemura, and K. Uchida, "Mixed H2/Ho Control wit. Pole Placement, State-Feedback Case, Proc. Amer. Contr. Conf, 1993, pp 2777-2779 IEEE Trans. Aut Contr., AC-28(1983), pp 848-850 s via Linear Control [4]Barmish, B.R., "Stabilization of Uncertain System 5] Becker, G, Packard, P, "Robust Performance of Linear-Parametrically Varying Systems Using Parametrically-Dependent Linear Feedback, " Systems and Control Letters, 23(1994), pp. 205-215 [6] Bendsoe, M P, A Ben-Tal, and J Zowe, "Optimization Methods for Truss Geometry and Topology Design, "to appear in Structural Optimization [7 Ben-Tal, A, and A Nemirovski, "Potential Reduction Polynomial-Time Method for Truss Topology Design, to appear in SIAM .. Contr. Opt [8] Boyd, S, and Q. Yang, "Structured and Simultaneous lyapunov functions for System Stability Problems, "Int. Contr, 49(1989), pp. 2215-2240 [9]Boyd, S, L El Ghaoui, E Feron, v. Balakrishnan, Linear Matrix Inequalities in systems and Control Theory, SIAM books, Philadelphia, 1994 [10] Chilali, M, and P Gahinet, "Ho Design with Pole Placement Constraints an LMI Approach, to appear in IEEE Trans. Aut Contr. Also in Proc. Conf. Dec.Conr.,1994,pp.553-558 [11] Gahinet, P, and P. Apkarian, "A Linear Matrix Inequality approach to h Control, Int. Robust and Nonlinear Contr., 4(1994), pp. 421-440 [12] Gahinet, P, P. Apkarian, and M. chilali, "Affine Parameter-Dependent Lyapunov Functions for Real Parametric Uncertainty, " Proc. Conf. Dec Contr. 1994,pp.2026-2031 1-10
1 Introduction 1-10 References [1] Anderson, B.D.O, and S. Vongpanitlerd, Network Analysis, Prentice-Hall, Englewood Cliffs, 1973. [2] Apkarian, P., P. Gahinet, and G. Becker, “Self-Scheduled H∞ Control of Linear Parameter-Varying Systems,” Proc. Amer. Contr. Conf., 1994, pp. 856-860. [3] Bambang, R., E. Shimemura, and K. Uchida, “Mixed H2 /H∞ Control with Pole Placement,” State-Feedback Case,“ Proc. Amer. Contr. Conf., 1993, pp. 2777-2779. [4] Barmish, B.R., “Stabilization of Uncertain Systems via Linear Control,“ IEEE Trans. Aut. Contr., AC–28 (1983), pp. 848-850. [5] Becker, G., Packard, P., “Robust Performance of Linear-Parametrically Varying Systems Using Parametrically-Dependent Linear Feedback,” Systems and Control Letters, 23 (1994), pp. 205-215. [6] Bendsoe, M.P., A. Ben-Tal, and J. Zowe, “Optimization Methods for Truss Geometry and Topology Design,” to appear in Structural Optimization. [7] Ben-Tal, A., and A. Nemirovski, “Potential Reduction Polynomial-Time Method for Truss Topology Design,” to appear in SIAM J. Contr. Opt. [8] Boyd, S., and Q. Yang, “Structured and Simultaneous Lyapunov Functions for System Stability Problems,” Int. J. Contr., 49 (1989), pp. 2215-2240. [9] Boyd, S., L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, SIAM books, Philadelphia, 1994. [10] Chilali, M., and P. Gahinet, “H∞ Design with Pole Placement Constraints: an LMI Approach,” to appear in IEEE Trans. Aut. Contr. Also in Proc. Conf. Dec. Contr., 1994, pp. 553-558. [11] Gahinet, P., and P. Apkarian, “A Linear Matrix Inequality Approach to H∞ Control,” Int. J. Robust and Nonlinear Contr., 4 (1994), pp. 421-448. [12] Gahinet, P., P. Apkarian, and M. Chilali, “Affine Parameter-Dependent Lyapunov Functions for Real Parametric Uncertainty,” Proc. Conf. Dec. Contr., 1994, pp. 2026-2031
[13] Haddad, W.M. and D.S. Berstein, "Parameter-Dependent Lyapunoy Functions, Constant Real Parameter Uncertainty, Popov Criterion Robust analysis and Synthesis: Part 1 and 2, Proc. Conf. Dec. Contr 1991 [14] Iwasaki, T, and R.E. Skelton, "All Controllers for the General H Control Problem: LMI Existence Conditions and State-Space Formulas, Automatica 30(1994),pp.1307-1317 [15] Horisberger H P, and P. R Belanger, "Regulators for Linear Time- Va Plants with Uncertain Parameters " IEEE Trans. Aut Contr. [16] How, J P, and S R. Hall, "Connection between the Popov Stability Criterion and bounds for real Parameter Uncertainty Proc. Amer. Contr Conf,1993,pp.1084-1089 [17] Khargonekar, PP, and M.A. Rotea, "Mixed H2/H Control: a Convex Optimization Approach, "IEEE Trans. Aut Contr., 39(1991), pp. 824-837 [18] Masubuchi, I, A Ohara, and N. Suda, "LMI-Based Controller Synthesis A Unified Formulation and solution submitted to Int, robust and Nonlinear contr. 1994 [19 Nemirovski, A, and P Gahinet, "The Projective Method for Solving Linear Matrix Inequalities, " Proc. Amer. Contr. Conf, 1994, pp. 840-844 [20] Nesterov, Yu, and A Nemirovski, Interior Point Polynomial Methods in Convex Programming: Theory and Applications, SIAM Books, Philadelphia 1994. [21] Packard, A, and J.C. Doyle, "The Complex Structured Singular value, Automatica, 29(1994), pp. 71-109 [22] Popov, VM "Absolute Stability of Nonlinear Systems of Automatic Control, Automation and Remote Control, 22(1962), pp. 857-875. [23] Scherer, C, "Mixed H2 H. Control, to appear in Trends in Control: a European Perspective, volume of the special contributions to the ECC 1995 24] Stein, G and J.C. Doyle, "Beyond Singular Values and Loop Shapes, " J Guidance,14(1991),pp.5-16. [25] Vidyasagar M, Nonlinear System Analysis, Prentice-Hall, Englewood Cliffs. 1992 1-11
References 1-11 [13] Haddad, W.M. and D.S. Berstein,“Parameter-Dependent Lyapunov Functions, Constant Real Parameter Uncertainty, and the Popov Criterion in Robust Analysis and Synthesis: Part 1 and 2,” Proc. Conf. Dec. Contr., 1991, pp. 2274-2279 and 2632-2633. [14] Iwasaki, T., and R.E. Skelton, “All Controllers for the General H∞ Control Problem: LMI Existence Conditions and State-Space Formulas,” Automatica, 30 (1994), pp. 1307-1317. [15] Horisberger, H.P., and P.R. Belanger, “Regulators for Linear Time-Varying Plants with Uncertain Parameters,” IEEE Trans. Aut. Contr., AC–21 (1976), pp. 705-708. [16] How, J.P., and S.R. Hall, “Connection between the Popov Stability Criterion and Bounds for Real Parameter Uncertainty,” Proc. Amer. Contr. Conf., 1993, pp. 1084-1089. [17] Khargonekar, P.P., and M.A. Rotea,“Mixed H2 /H∞ Control: a Convex Optimization Approach,” IEEE Trans. Aut. Contr., 39 (1991), pp. 824-837. [18] Masubuchi, I., A. Ohara, and N. Suda, “LMI-Based Controller Synthesis: A Unified Formulation and Solution,” submitted to Int. J. Robust and Nonlinear Contr., 1994. [19] Nemirovski, A., and P. Gahinet, “The Projective Method for Solving Linear Matrix Inequalities,” Proc. Amer. Contr. Conf., 1994, pp. 840-844. [20] Nesterov, Yu, and A. Nemirovski, Interior Point Polynomial Methods in Convex Programming: Theory and Applications, SIAM Books, Philadelphia, 1994. [21] Packard, A., and J.C. Doyle, “The Complex Structured Singular Value,” Automatica, 29 (1994), pp. 71-109. [22] Popov, V.M., “Absolute Stability of Nonlinear Systems of Automatic Control,” Automation and Remote Control, 22 (1962), pp. 857-875. [23] Scherer, C., “Mixed H2 H∞ Control,” to appear in Trends in Control: A European Perspective, volume of the special contributions to the ECC 1995. [24] Stein, G. and J.C. Doyle, “Beyond Singular Values and Loop Shapes,” J. Guidance, 14 (1991), pp. 5-16. [25] Vidyasagar, M., Nonlinear System Analysis, Prentice-Hall, Englewood Cliffs, 1992