- Do i need symbolic math toolbox for heaviside function full#
- Do i need symbolic math toolbox for heaviside function code#
Do i need symbolic math toolbox for heaviside function code#
The MATLAB code will be structureX = 6X = lmivar(1,structureX) structureY = Y = lmivar(1,structureY) lmiterm(,C,C) % term CXC lmiterm(,B,A) % term BYA lmiterm(,1,F) % term XF lmiterm(,F,1) % term FX lmiterm(,1,1) % term Y typeLMI2 = lmiterm(typeLMI2,D,D) % term DXD In this example, it is easily observed that the second and third term of the first parameter of lmiterm define the locations of the corresponding term in the LMI.
Do i need symbolic math toolbox for heaviside function full#
Let Y be a full symmetric matrix of dimension 4. Consider the following set LMIs: T TT0 + DXD (13) In this case, we have two decision matrices, X and Y and two LMIs. It can be shown that 2T2inf Tr = G CPC subject to T T0 + + CXC, we simply use typeLMI1 = lmiterm(typeLMI1,C,C) Lets consider now a more involved example. (10) Here ( ) j G represents the transfer function of the system. (9) where is a white noise disturbance with unit covariance, the LQG or 2H performance 2G is defined by 4 22: HTr j j d =G G G. LQG performance for a stable LTI system = + `= )x Ax BGy Cx This condition can also be stated as follows: The system = x Ax is stable (and A is Hurwitz) if the LMI T0 + + AS SA Q, or its strict version T0 + + S S. Applications Finding a solution x to the LMI system ( ) 0 ` Q Q, there exists a unique solution T0 = > S S for the Lyapunov equation given by T+ = AS SA Q (or T+ = A S SA Q). For a proof, see Dullerud and Paganini (2002). (ii) The matrix inequality x y 3 * > M R0R Q.
(6) The Schur complement formula: The following are equivalent (i) The matrix inequalities > Q 0 and 1 * > M RQ R 0 both hold. The nonstrict version is, 0 1( ) 01 n nx x = + + + F x F F. CONVEX NOT CONVEX LMI (1) is a strict inequality. Convexity: A set nC R is convex if 1 2(1 ) C u u + x x for any 1 2, C x x and any u. This is an important property since powerful numerical solution techniques are available for the problems involving convex solution sets. A set of LMIs can always be converted to one LMI, i.e., suppose we have the two LMIs 0 0 0( ) F is a convex set. Affinity: A function ( ) g x is an affine function of x if it can be written as ( ) ( ) g x f x a = + where ( ) f x is a linear function and a is a constant. Linearity: A function ( ) f x is a linear function of x if 1 2 1 2( ) ( ) ( ) f x x f x f x + = + for all scalar and. , )n n F x F F F (4) Note that left hand side of (1) is affine and left hand side of (2) is linear in x. Different Representations In most applications, LMIs do not naturally arise in the canonical form (1), but rather in the form 1 1(. ,nF F F are real symmetric matrices, and x is a vector of decision variables. F (1) is called a linear matrix inequality where ( ) F x is an affine function of the real vector | |T1 2.nx x x = x, 0 1. INTRODUCTION TO LMIS Definition The matrix inequality 0 1( ) 01 n nx x = + + + > F x F F. The second section introduces MATLAB LMI Toolbox and explains some of the useful functions by example codes. First section gives the definition and various representations of the LMIs and common problems involving LMIs. Lee2 () ABSTRACT This document gives a brief introduction to linear matrix inequalities (LMIs). LINEAR MATRIX INEQUALITIES AND MATLAB LMI TOOLBOX B.