# WUSTL ESE 543 - Lecture 3 and 4 eig cont obs (66 pages)

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## Lecture 3 and 4 eig cont obs

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- Pages:
- 66
- School:
- Washington University in St. Louis
- Course:
- Ese 543 - Control Systems Design by State Space Methods

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Lecture 3 The State Transition Matrix Chapter 9 Time Varying Systems x A t x B t u y C t x D t u Eq 1 x R nx u R nu y R ny x A t x Theorem There exists uniquely for every x 0 a x t that satisfies Eq 1 Theorem Solutions to Eq 1 form a nx dimensional vector space over R Def U nx nx Fundamental matrix if its columns consist of nx linearly independent solutions to Eq 1 U nx nx U1 U nx 2 K A Wise Fundamental Matrix Example 0 0 x x t 0 x1 t x 1 t0 X 10 x 1 t 0 x 2 t tx1 t x2 t 1 2 t 2 X 10 X 20 Since the dimension of the vector space is 2 2 linearly independent initial vectors can be found and each one defines an unique solution X10 0 x t0 X 20 1 0 U1 t 1 X10 2 x t0 0 X 20 2 U 2 t 2 t 0 2 U t 2 1 t 3 K A Wise The State Transition Matrix The solution to the state equation describes the transition of the initial state x t0 state to the state x t at time t x t0 x t Define t t0 U t U 1 t0 as the state transition matrix Let U1 t and U 2 t be two fundamental matrices where U1 U 2 P 1 t t0 U1 t U 1 t0 1 t0 U 2 t P P 1U 2 U 2 t U 2 1 t0 t t0 exhibits the following properties 1 t t I 3 t2 t0 t2 t1 t1 t0 t0 t1 t2 2 1 t t0 t0 t 4 t t0 satisfies x A t x t t0 U t U 1 t0 A t U t U 1 t0 A t t t0 A t U t K A Wise 4 The State Transition Matrix From the previous example 0 2 0 2 1 U t U t0 U t0 2 2 1 t 1 t 0 Now consider the general solution to x Theorem The solution to x 1 2 t02 1 2 1 t t0 0 1 2 2 1 2 t t0 0 1 A t x B t u A t x B t u is t x t t t0 x t0 t B u d t0 Zero Input Response Convolution Integral 5 K A Wise Solution to the State Equation Proof Differentiate the solution to show it satisfies the differential equation t x t t0 x0 t B u d t t 0 t t B u d A t t t0 x0 t t B t u t t0 A t t t A t t t0 x0 t B u d B t u t t 0 x t A t x B t u The output is given by t y t C t t t0 x t0 C t B D t u d t 0 K A Wise Impulse Response Matrix 6 LTI Systems Deriving x Ax Method 1 Derive using a Taylor Series Expand x t about t0 x 2 x t x0 x 0 t t0 0 t t0 2 dx 0 x 0 Ax0 x0 A2 x0 dt k k A t t0 x t x0 t t0 x0 k k 0 Ak t t0 A t t t t0 e 0 k k 0 A t t0 x t e x0 k x 0 d k x t dt k t t0 k K A Wise 7 State Transition Matrix Check to see if it satisfies the properties of the state transition matrix x t e A t t0 x0 A t t e I 1 t t I A t t A t t e 0 e 0 I 2 1 t t0 t0 t A t t A t t A t t e 2 1 e 1 0 e 2 0 3 t2 t0 t2 t1 t1 t0 4 t t0 t0 t1 t2 k 1 Ak t t0 d A t t0 A t t0 e Ae satisfies x A t x k 1 dt k 0 t t0 U t U 1 t0 K A Wise At0 1 e e At A t t e 0 8 Examples 1 Diagonal A matrix Assume real eigenvalues k 1 0 1 k A A 0 2 0 0 k 2 1t e At t e e k k 0 0 kt k 0 2t e 2 Jordan form Assume real eigenvalues A 1 k A 0 e At e t 0 k 0 k k 1 k 1t k k 1t k 1 t te t k k 0 k k 1 k 1 te t t e 9 K A Wise An Easy Way To Compute Assume A matrix has distinct eigenvalues Look at similarity transformation that transforms the A matrix to diagonal form 1 k k T T t 1t At T T 1 A T T T e v1 vnx e k k 0 Examine each term 1 2 1 1 2 1 T T T T T T T T T T T T 1 k 1 T T 1 T k T 1 Factor out T and T 1 on left right k k t At T 1 Te tT 1 e e T k 0 k T T 1t Diagonal matrix with K A Wise e i t on diagonal 10 Examine Transformation To A New Basis LTI x Tx x T 1 x x Tx Substitute in for x 1 x Tx T Ax Bu TAx TBu TAT Ax Bu x TBu A B y Cx Du CT 1x Du Cx Du 1 1 x Tx A TAT B TB C CT D D TAT 1t Ce B CT e TB CT 1 T e At T 1 TB Ce At B At 1 11 K A Wise Definitions Consider the free response no input u Ce At TAT 1t At 1 CT e Ce T x0 Tx0 1 1 Two linear systems are said to be ZERO STATE EQUIVALENT if they have the same impulse response matrix 2 Two linear systems are said to be ZERO INPUT EQUIVALENT if for any initial state in one system there exists a state in the other and vice versa such that the two systems have the same zero input response 3 Two linear time invariant systems are EQUIVALENT IFF A1 B1 C1 D1 A2 B2 C2 D2 1 A2 TA T B2 TB1 1 1 C2 C T D2 D1 1 K A Wise 12 Solution to the State Equation Linear Time Invariant System Model x Ax Bu x t t t 0 x t0 t t Bu d t0 State Transition Matrix t k Ak t 0 e At k k 0 Easiest to form t t0 using eigenvectors eigenvalues P 1 v1 vnx eigenvectors diag 1 nx eigenvalues t 0 e At e P 1 Pt P 1e t P 13 …

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