14 - Gaussian Stochastic Processes

Transcription

1 14-1 Gaussian Stochastic Processes S. Lall, Stanford Gaussian Stochastic Processes Linear systems driven by IID noise Evolution of mean and covariance Example: mass-spring system Steady-state behavior Stochastic processes Stationary processes State-space formulae Autocovariance Example: low-pass filter Systems on bi-infinite time Fourier series and the Z-transform The power spectral density

2 14-2 Gaussian Stochastic Processes S. Lall, Stanford Linear systems driven by IID noise consider the linear dynamical system x(t + 1) = Ax(t) + Bu(t) + v(t) with the input u(), u(1),... is not random the disturbance v(), v(1), v(2),... is white Gaussian noise E v(t) = µ v (t) cov v(t) = Σ v the initial state is random x() N(µ x (), Σ x ()), independent of v(t) for all t view this as stochastic simulation of the system what are the statistical properties (mean and covariance) of x(t)?

3 14-3 Gaussian Stochastic Processes S. Lall, Stanford Evolution of Mean and Covariance we have x(t + 1) = Ax(t) + Bu(t) + v(t) taking the expectation of both sides, we have, as before µ x (t + 1) = Aµ x (t) + Bu(t) + µ v (t) taking the covariance of both sides, we have Σ x (t + 1) = AΣ x (t)a T + Σ v i.e, the state covariance Σ x (t) = cov(x(t)) obeys a Lyapunov recursion

4 14-4 Gaussian Stochastic Processes S. Lall, Stanford State Covariance The solution to the Lyapunov recursion is Σ x (t) = A t Σ x ()(A t ) T + t k= A k Σ v (A k ) T Because the covariance of the state Σ x (t) = cov x(t) is Σ x (t) = A t Σ x ()(A t ) T + [ A t... A I ] Σ v Σ v (A t ) T.... A T Σv I = A t Σ x ()(A t ) T + t k= A k Σ v (A k ) T

5 14-5 Gaussian Stochastic Processes S. Lall, Stanford Example: Mass-Spring System k 1 k 2 k 3 m 1 m 2 m 3 b 1 b 2 b 3 masses m i = 1, springs k i = 2, dampers b i = 3 ẋ(t) = A c x(t) + B c1 w(t) + B c2 u(t) where A c = B c1 = B c2 = 1 u(t) is deterministic force applied to mass 3 w(t) R 3 is random forcing w(t) N(,.2I) applied to all masses

6 14-6 Gaussian Stochastic Processes S. Lall, Stanford Example: Mass-Spring System discretization x(t + 1) = Ax(t) + B 1 w(t) + B 2 u(t) let v(t) = B 1 w(t), so E v(t) = cov v(t) = B 1 Σ w B T 1 and we have the inputs are 1 x(t + 1) = Ax(t) + B 2 u(t) + v(t) u t w t

7 14-7 Gaussian Stochastic Processes S. Lall, Stanford Example: Mass-Spring System simulate three things the evolution of the mean µ x (t + 1) = Aµ x (t) + Bu(t) + µ v (t) the evolution of the covariance Σ x (t + 1) = AΣ x (t)a T + Σ v the state trajectory for a particular realization of the random process at each time t plot actual state in this particular run x(t) mean state µ xi (t) 9% confidence interval [µ xi (t) h(t), µ xi (t) + h(t)], where h(t) is as usual h(t) = ( (Σx (t) ) )1 ii F 2 (.9) χ 2 1

8 14-8 Gaussian Stochastic Processes S. Lall, Stanford Example: Stochastic Simulation of Mass-Spring System position and velocity of mass 1 2 mean of state 9% confidence interval for realization of state 1 mean of state x5 9% confidence interval for x5 realization of state x x 1 x t t

9 14-9 Gaussian Stochastic Processes S. Lall, Stanford Example: Ellipsoids.5 time = 1.5 time = 6.5 time = 11.5 time = time = time = time = time = time = time = time = time =

10 14-1 Gaussian Stochastic Processes S. Lall, Stanford Steady-State Behavior the Lyapunov equation is the same as the one we used for controllability analysis if A is stable, then the limit is Σ xss = lim t Σ x (t) = A k Σ v (A k ) T k= the steady-state covariance as in controllability, this is the unique solution to the Lyapunov equation Σ xss AΣ xss A T = Σ v if Σ v = BB T then Σ xss is the controllability Gramian

11 14-11 Gaussian Stochastic Processes S. Lall, Stanford Stochastic processes A stochastic process is an infinitely long random vector. It has mean and covariance x() E x(1) x(2) =. µ x () µ x (1) µ x (2). x() cov x(1) x(2) =. Σ x (, ) Σ x (, 1)... Σ x (1, ) Σ x (1, 1) Σ x (2, ) Σ x (2, 1). For each w Ω, the random variable x returns of the entire sequence x(), x(1),... If x(), x(1),... are Gaussian and IID, then x is called white Gaussian noise (WGN) In this case, Σ x (i, j) = if i j

12 14-12 Gaussian Stochastic Processes S. Lall, Stanford Generating stochastic processes Suppose v(), v(1),... is white Gaussian noise, with covariance cov(v(t)) = I, and x(t + 1) = Ax(t) + Bv(t) x() = y(t) = Cx(t) We have y = Tv, where T is the Toeplitz matrix y() v() y(1) y(2) y(3) = H(1) v(1) H(2) H(1) v(2) H(3) H(2) H(1) v(3) and H(), H(1),... is the impulse response { CA t B if t > H(t) = otherwise

13 14-13 Gaussian Stochastic Processes S. Lall, Stanford Example Let W = cov(y) = TT T. For example, suppose x(t + 1) =.95x(t) + v(t) x() = y(t) = x(t) An image plot of W(i, j), is below. Notice that it becomes constant along diagonals. Also plotted is R(i) = W(i + 1, i) R(i) i

14 14-14 Gaussian Stochastic Processes S. Lall, Stanford The Output Covariance We have the covariance of the output W = cov(y) satisfies W(i, j) = T ik (T jk ) T k= = H(i k) ( H(j k) ) T k= since T(i, j) = H(i j). Therefore, evaluating W along the j th diagonal W(i + j, i) = H(i + j k) ( H(i k) ) T k= = H(j p)h T ( p) p= i and so lim W(i + j, i) = i p= H(j p)h T ( p)

15 14-15 Gaussian Stochastic Processes S. Lall, Stanford The Output Covariance Let R(j) = p= H(j p)h T ( p) Then we have lim cov i y(i) y(i + 1) y(i + 2) y(i + 3). = R() R() R( 2)... R(1) R() R() R( 2)... R(2) R(1) R() R() R(3) R(2) R(1) R().... We must have R(i) = R( i) T As i becomes large, the output covariance W becomes Toeplitz

16 14-16 Gaussian Stochastic Processes S. Lall, Stanford Asymptotic Stationarity This means that the pdf of y(i) y(i + 1) y(i + 2). y(i + N) tends to a limit as i becomes large. The output process is therefore called asymptotically stationary.

17 14-17 Gaussian Stochastic Processes S. Lall, Stanford Stationary Stochastic Processes A stochastic process y(), y(1),... is called stationary if for every i and every N > the pdf of y(i) y(i + 1) y(i + 2). y(i + N) is independent of i. If y(), y(1),... is stationary, then W(i, j) = cov ( y(i), y(j) ) = R(i j) for some sequence of matrices R(), R(1),... Any segment of the signal y(i),..., y(i + N) has the same statistical properties as any other R is called the autocovariance of the process

18 14-18 Gaussian Stochastic Processes S. Lall, Stanford State-Space Formulae Suppose v(), v(1),... is white Gaussian noise, with covariance cov(v(t)) = I, and x(t + 1) = Ax(t) + v(t) x() N(, Σ x ()) y(t) = Cx(t) We have x() v() x(1) x(2) x(3) = I v(1) A I v(2) A 2 A I v(3) = P v() v(1) + Jx(). I A A 2 A 3. x()

19 14-19 Gaussian Stochastic Processes S. Lall, Stanford State-Space Formulae Therefore x(), x(1), x(2),..., is a stochastic process with covariance cov x() Σ v x(1) = P Σ v... P T + JΣ x ()J T. Σv Hence for i j cov ( x(i), x(j) ) j = A i k Σ v (A j k ) T + A i Σ x ()(A j ) T k=1 ( j ) = A i j A j k Σ v (A j k ) T + A j Σ x ()(A j ) T k=1 = A i j Σ x (j) similarly, for i j we have cov ( x(i), x(j) ) = Σ x (j)(a j i ) T.

20 14-2 Gaussian Stochastic Processes S. Lall, Stanford State Covariance Hence we have the state covariance x() Σ x () Σ x (1)A T Σ x (2)(A 2 ) T Σ x (3)(A 3 ) T x(1) cov x(2) = AΣ x () Σ x (1) Σ x (2)A T Σ x (3)(A 2 ) T A 2 Σ x () AΣ x (1) Σ x (2) Σ x (3)A T As t we have Σ x (t) Σ xss, so x(t) Σ xss Σ xss A T Σ xss A T 2 Σ xss A T 3 x(t + 1) lim cov AΣ xss Σ xss Σ xss A T Σ xss A T 2 t x(t + 2) = A 2 Σ xss AΣ xss Σ xss Σ xss A T x(t + 3) A 3 Σ xss A 2 Σ xss AΣ xss Σ xss

21 14-21 Gaussian Stochastic Processes S. Lall, Stanford The Autocovariance The autocovariance of the output y is R(i) = CA i Σ xss C T for i where Σ xss is the unique solution to the Lyapunov equation Σ xss AΣ xss A T = I

22 14-22 Gaussian Stochastic Processes S. Lall, Stanford Example: Low-Pass Filter Let s look at the low-pass filter where λ =.1. Ĝ(z) = c (z e λ ) 3 The breakpoint frequency is 2π rad s when sampling period h = 1. The constant c is chosen such that Ĝ(1) =

23 14-23 Gaussian Stochastic Processes S. Lall, Stanford Example: Low-Pass Filter The input, output, autocorrelation, and breakpoint frequency are below