# Solution via Laplace transform and matrix exponential

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1 EE263 Autumn 2015 S. Boyd and S. Lall Solution via Laplace transform and matrix exponential Laplace transform solving ẋ = Ax via Laplace transform state transition matrix matrix exponential qualitative behavior and stability 1

2 Laplace transform of matrix valued function suppose z : R + R p q Laplace transform: Z = L(z), where Z : D C C p q is defined by Z(s) = integral of matrix is done term-by-term 0 e st z(t) dt convention: upper case denotes Laplace transform D is the domain or region of convergence of Z D includes at least {s Rs > a}, where a satisfies z ij(t) αe at for t 0, i = 1,..., p, j = 1,..., q 2

3 Derivative property L(ż) = sz(s) z(0) to derive, integrate by parts: L(ż)(s) = 0 e st ż(t) dt = e st z(t) t t=0 = sz(s) z(0) + s 0 e st z(t) dt 3

4 Laplace transform solution of ẋ = Ax consider continuous-time time-invariant (TI) LDS ẋ = Ax for t 0, where x(t) R n take Laplace transform: sx(s) x(0) = AX(s) rewrite as (si A)X(s) = x(0) hence X(s) = (si A) 1 x(0) take inverse transform x(t) = L 1 ( (si A) 1) x(0) 4

5 Resolvent and state transition matrix (si A) 1 is called the resolvent of A resolvent defined for s C except eigenvalues of A, i.e., s such that det(si A) = 0 Φ(t) = L 1 ( (si A) 1) is called the state-transition matrix; it maps the initial state to the state at time t: x(t) = Φ(t)x(0) (in particular, state x(t) is a linear function of initial state x(0)) 5

6 Example 1: Harmonic oscillator ẋ = [ ] 0 1 x

7 Example 1: Harmonic oscillator si A = [ ] s 1, so resolvent is 1 s (si A) 1 = [ s s s s 2 +1 s s 2 +1 ] (eigenvalues are ±i) state transition matrix is Φ(t) = L 1 ([ s s s s 2 +1 s s 2 +1 ]) [ ] cos t sin t = sin t cos t a rotation matrix ( t radians) [ cos t sin t so we have x(t) = sin t cos t ] x(0) 7

8 Example 2: Double integrator ẋ = [ ] 0 1 x

9 Example 2: Double integrator si A = [ ] s 1, so resolvent is 0 s (eigenvalues are 0, 0) state transition matrix is so we have x(t) = (si A) 1 = ([ 1 Φ(t) = L 1 s 0 [ ] 1 t x(0) 0 1 [ 1 s 1 s 1 2 s 0 1 s 1 2 s ]) = ] [ ] 1 t 0 1 9

10 Characteristic polynomial X (s) = det(si A) is called the characteristic polynomial of A X (s) is a polynomial of degree n, with leading (i.e., s n ) coefficient one roots of X are the eigenvalues of A X has real coefficients, so eigenvalues are either real or occur in conjugate pairs there are n eigenvalues (if we count multiplicity as roots of X ) 10

11 Eigenvalues of A and poles of resolvent i, j entry of resolvent can be expressed via Cramer s rule as ( 1) i+j det ij det(si A) where ij is si A with jth row and ith column deleted det ij is a polynomial of degree less than n, so i, j entry of resolvent has form f ij(s)/x (s) where f ij is polynomial with degree less than n poles of entries of resolvent must be eigenvalues of A but not all eigenvalues of A show up as poles of each entry (when there are cancellations between det ij and X (s)) 11

12 Matrix exponential define matrix exponential as e M = I + M + M 2 + 2! converges for all M R n n looks like ordinary power series e at = 1 + ta + (ta)2 2! with square matrices instead of scalars

13 Matrix exponential (I C) 1 = I + C + C 2 + C 3 + (if series converges) series expansion of resolvent: (si A) 1 = (1/s)(I A/s) 1 = I s + A s 2 + A2 s 3 + (valid for s large enough) so Φ(t) = L 1 ( (si A) 1) = I + ta + (ta)2 2! with this definition, state-transition matrix is Φ(t) = L 1 ( (si A) 1) = e ta + 13

14 Matrix exponential solution of autonomous LDS solution of ẋ = Ax, with A R n n and constant, is x(t) = e ta x(0) generalizes scalar case: solution of ẋ = ax, with a R and constant, is x(t) = e ta x(0) 14

15 Properties of matrix exponential matrix exponential is meant to look like scalar exponential some things you d guess hold for the matrix exponential (by analogy with the scalar exponential) do in fact hold but many things you d guess are wrong example: you might guess that e A+B = e A e B, but it s false (in general) [ ] [ ] A =, B = [ ] [ ] e A =, e B 1 1 = [ ] [ ] e A+B = e A e B =

16 Properties of matrix exponential e A+B = e A e B if AB = BA i.e., product rule holds when A and B commute thus for t, s R, e (ta+sa) = e ta e sa with s = t we get e ta e ta = e ta ta = e 0 = I so e ta is nonsingular, with inverse ( e ta) 1 = e ta 16

17 Example: matrix exponential let s find e A, where A = we already found [ ] so, plugging in t = 1, we get e A = let s check power series: since A 2 = A 3 = = 0 e ta = L 1 (si A) 1 = [ ] e A = I + A + A2 2! [ ] 1 t = I + A 17

18 Time transfer property for ẋ = Ax we know x(t) = Φ(t)x(0) = e ta x(0) interpretation: the matrix e ta propagates initial condition into state at time t more generally we have, for any t and τ, x(τ + t) = e ta x(τ) (to see this, apply result above to z(t) = x(t + τ)) interpretation: the matrix e ta propagates state t seconds forward in time (backward if t < 0) 18

19 Time transfer property recall first order (forward Euler) approximate state update, for small t: x(τ + t) x(τ) + tẋ(τ) = (I + ta)x(τ) exact solution is x(τ + t) = e ta x(τ) = (I + ta + (ta) 2 /2! + )x(τ) forward Euler is just first two terms in series 19

20 Sampling a continuous-time system suppose ẋ = Ax sample x at times t 1 t 2 : define z(k) = x(t k ) then z(k + 1) = e (t k+1 t k )A z(k) for uniform sampling t k+1 t k = h, so z(k + 1) = e ha z(k), a discrete-time LDS (called discretized version of continuous-time system) 20

21 Piecewise constant system consider time-varying LDS ẋ = A(t)x, with A(t) = A 0 0 t < t 1 A 1 t 1 t < t 2 where 0 < t 1 < t 2 < (sometimes called jump linear system) for t [t i, t i+1] we have. x(t) = e (t t i)ai e (t 3 t 2 )A 2 e (t 2 t 1 )A 1 e t 1A 0 x(0) (matrix on righthand side is called state transition matrix for system, and denoted Φ(t)) 21

22 Qualitative behavior of x(t) suppose ẋ = Ax, x(t) R n then x(t) = e ta x(0); X(s) = (si A) 1 x(0) ith component X i(s) has form where a i is a polynomial of degree < n X i(s) = ai(s) X (s) thus the poles of X i are all eigenvalues of A (but not necessarily the other way around) 22

23 Qualitative behavior of x(t) first assume eigenvalues λ i are distinct, so X i(s) cannot have repeated poles then x i(t) has form x i(t) = where β ij depend on x(0) (linearly) n j=1 β ije λ j t eigenvalues determine (possible) qualitative behavior of x: eigenvalues give exponents that can occur in exponentials real eigenvalue λ corresponds to an exponentially decaying or growing term e λt in solution complex eigenvalue λ = σ + iω corresponds to decaying or growing sinusoidal term e σt cos(ωt + φ) in solution 23

24 Qualitative behavior of x(t) Rλ j gives exponential growth rate (if > 0), or exponential decay rate (if < 0) of term Iλ j gives frequency of oscillatory term (if 0) eigenvalues Is Rs 24

25 Repeated eigenvalues now suppose A has repeated eigenvalues, so X i can have repeated poles express eigenvalues as λ 1,..., λ r (distinct) with multiplicities n 1,..., n r, respectively (n n r = n) then x i(t) has form x i(t) = r p ij(t)e λ j t where p ij(t) is a polynomial of degree < n j (that depends linearly on x(0)) j=1 25

26 Stability we say system ẋ = Ax is stable if e ta 0 as t meaning: state x(t) converges to 0, as t, no matter what x(0) is all trajectories of ẋ = Ax converge to 0 as t fact: ẋ = Ax is stable if and only if all eigenvalues of A have negative real part: Rλ i < 0, i = 1,..., n 26

27 Stability the if part is clear since for any polynomial, if Rλ < 0 we ll see the only if part next lecture lim t p(t)eλt = 0 more generally, max i Rλ i determines the maximum asymptotic logarithmic growth rate of x(t) (or decay, if < 0) 27

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