User manual MATLAB ROBUST CONTROL TOOLBOX 3

[. . . ] Robust Control ToolboxTM 3 User's Guide Gary Balas Richard Chiang Andy Packard Michael Safonov How to Contact The MathWorks Web Newsgroup www. mathworks. com/contact_TS. html Technical Support www. mathworks. com comp. soft-sys. matlab suggest@mathworks. com bugs@mathworks. com doc@mathworks. com service@mathworks. com info@mathworks. com Product enhancement suggestions Bug reports Documentation error reports Order status, license renewals, passcodes Sales, pricing, and general information 508-647-7000 (Phone) 508-647-7001 (Fax) The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 For contact information about worldwide offices, see the MathWorks Web site. Robust Control ToolboxTM User's Guide © COPYRIGHT 2005­2010 by The MathWorks, Inc. The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. [. . . ] Further applications of LMIs arise in estimation, identification, optimal design, structural design [6], [7], matrix scaling problems, and so on. The main strength of LMI formulations is the ability to combine various design constraints or objectives in a numerically tractable manner. A nonexhaustive list of problems addressed by LMI techniques includes the following: · Robust stability of systems with LTI uncertainty (µ-analysis) ([24], [21], [27]) · Robust stability in the face of sector-bounded nonlinearities (Popov criterion) ([22], [28], [13], [16]) · Quadratic stability of differential inclusions ([15], [8]) · Lyapunov stability of parameter-dependent systems ([12]) · Input/state/output properties of LTI systems (invariant ellipsoids, decay rate, etc. ) ([9]) · Multi-model/multi-objective state feedback design ([4], [17], [3], [9], [10]) · Robust pole placement · Optimal LQG control ([9]) · Robust H control ([11], [14]) · Multi-objective H synthesis ([18], [23], [10], [18]) · Design of robust gain-scheduled controllers ([5], [2]) 3-6 LMIs and LMI Problems · Control of stochastic systems ([9]) · Weighted interpolation problems ([9]) To hint at the principles underlying LMI design, let's review the LMI formulations of a few typical design objectives. Stability The stability of the dynamic system is equivalent to the feasibility of Find P = PT such that AT P + P A < 0, P > I. 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 3-7 3 Introduction to Linear Matrix Inequalities 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], [25], [26]. Minimize over X = XT and such that LQG Performance For a stable LTI system 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 3-8 LMIs and LMI Problems Again this is a linear objective minimization problem since the objective Trace (Q) is linear in the decision variables (free entries of P, Q). 3-9 3 Introduction to Linear Matrix Inequalities Further Mathematical Background Efficient interior-point algorithms are now available to solve the three generic LMI problems Equation 3-2­Equation 3-4 defined in "Three Generic LMI Problems" on page 3-5. These algorithms have a polynomial-time complexity. That is, the number N() of flops needed to compute an -accurate solution is bounded by M N3 log(V/) 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. Robust Control Toolbox software 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 Equation 3-3or the generalized eigenvalue minimization problem Equation 3-4. Some LMI problems are formulated in terms of inequalities rather than strict inequalities. For instance, a variant of Equation 3-3 is Minimize cTx subject to A(x) < 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 (3-6) Such problems cannot be handled directly by interior-point methods which require strict feasibility of the LMI constraints. A well-posed reformulation of Equation 3-5 would be Minimize cTx subject to x 0. Keeping this subtlety in mind, we always use strict inequalities in this manual. 3-10 Bibliography Bibliography [1] Anderson, B. D. O. , and S. Vongpanitlerd, Network Analysis, Prentice-Hall, Englewood Cliffs, 1973. Becker, "Self-Scheduled H Control of Linear Parameter-Varying Systems, " Proc. Uchida, "Mixed H2 /H Control with Pole Placement, " State-Feedback Case, Proc. [4] Barmish, B. R. , "Stabilization of Uncertain Systems via Linear Control, " IEEE Trans. [5] Becker, G. , and Packard, P. , "Robust Performance of Linear-Parametrically Varying Systems Using Parametrically-Dependent Linear Feedback, " Systems and Control Letters, 23 (1994), pp. [. . . ] This check box defaults to MATLAB Expression in the drop-down menu. This option lets you specify the block to linearize to an uncertain variable using a MATLAB expression containing Robust Control Toolbox functions. To learn more about the options, see "Configuring the Linearization for Specific Blocks and Subsystems" in the Simulink Control Design documentation. 5-22 Computing Uncertain State-Space Models from Simulink® Models c In the Enter an expression to specify the linearization of the Simulink block field, enter an expression, which must evaluate to an uncertain variable or uncertain model, such as ureal, umat, ultidyn or uss. d Click OK to save the changes. Note You can also specify a block to linearize to an uncertain variable at the command line. For an example, see "Example - Specifying a Block to Linearize To an Uncertain Variable at the Command Line" on page 5-23. 4 Run the linearize command to compute an uncertain linearization. [. . . ]