1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
|
#' Estimate Impulse Response Coefficients
#'
#' \code{impulseest} is used to estimate impulse response coefficients from
#' the data
#'
#' @param data an object of class \code{idframe}
#' @param M Order of the FIR Model (Default:\code{30})
#' @param K Transport delay in the estimated impulse response
#' (Default:\code{0})
#' @param regul Parameter indicating whether regularization should be
#' used. (Default:\code{FALSE})
#' @param lambda The value of the regularization parameter. Valid only if
#' \code{regul=TRUE}. (Default:\code{1})
#'
#' @details
#' The IR Coefficients are estimated using linear least squares. Future
#' Versions will provide support for multivariate data and regularized
#' regression
#'
#' @references
#' Arun K. Tangirala (2015), \emph{Principles of System Identification:
#' Theory and Practice}, CRC Press, Boca Raton. Sections 17.4.11 and 20.2
#'
#' @seealso \code{\link{step}}
#'
#' @examples
#' uk <- rnorm(1000,1)
#' yk <- filter (uk,c(0.9,-0.4),method="recursive") + rnorm(1000,1)
#' data <- idframe(output=data.frame(yk),input=data.frame(uk))
#' fit <- impulseest(data)
#' plot(fit)
#'
#' @export
impulseest <- function(data,M=30,K=0,regul=F,lambda=1){
N <- dim(data$output)[1]
ind <- (M+K+1):N
z_reg <- function(i) data$input[(i-K):(i-M-K),]
Z <- t(sapply(ind,z_reg))
Y <- data$output[ind,]
# Dealing with Regularization
if(regul==F){
lambda = 0
}
# Fit Linear Model and find standard errors
fit <- lm(Y~Z-1)
df <- nrow(Z)-ncol(Z);sigma2 <- sum(resid(fit)^2)/df
vcov <- sigma2 * solve(t(Z)%*%Z)
se <- sqrt(diag(vcov))
out <- list(coefficients=coef(fit),residuals=resid(fit),lags=K:(M+K),
x=colnames(data$input),y=colnames(data$output),se = se)
class(out) <- "impulseest"
return(out)
}
#' Impulse Response Plots
#'
#' Plots the estimated IR coefficients along with the significance limits
#' at each lag.
#'
#' @param model an object of class \code{impulseest}
#' @param sig Significance Limits (Default: \code{0.975})
#'
#' @seealso \code{\link{impulseest}},\code{\link{step}}
#' @export
plot.impulseest <- function(model,sig=0.975){
lim <- model$se*qnorm(0.975)
ylim <- c(min(coef(model)),max(coef(model)))
title <- paste("Impulse Response \n From",model$x,"to",model$y)
plot(model$lags,coef(model),type="h",xlab="Lag",ylab= model$y,
main = title)
abline(h=0);points(x=model$lags,y=lim,col="blue",lty=2,type="l")
points(x=model$lags,y=-lim,col="blue",lty=2,type="l")
}
#' Step Response Plots
#'
#' Plots the step response of a system, given the IR model
#'
#' @param model an object of class \code{impulseest}
#'
#' @seealso \code{\link{impulseest}}
#'
#' @examples
#' uk <- rnorm(1000,1)
#' yk <- filter (uk,c(0.9,-0.4),method="recursive") + rnorm(1000,1)
#' data <- idframe(output=data.frame(yk),input=data.frame(uk))
#' fit <- impulseest(data)
#' step(fit)
#'
#' @export
step <- function(model){
title <- paste("Step Response \n From",model$x,"to",model$y)
stepResp <- cumsum(coef(model))
plot(model$lags,stepResp,type="s",xlab="Lag",ylab= model$y,
main = title)
abline(h=0)
}
#' Estimate frequency response
#'
#' Estimates Frequency Response with fixed frequency resolution using
#' spectral analysis
#'
#' @param data an \code{idframe} object
#' @param npad an integer representing the total length of each time series
#' to analyze after padding with zeros. This argument allows the user to
#' control the spectral resolution of the SDF estimates: the normalized
#' frequency interval is deltaf=1/npad. (Default:)
#'
#' @export
spa <- function(data,npad=255){
require(sapa)
temp <- cbind(data$output,data$input)
# Non-parametric Estimation of Spectral Densities -
# WOSA and Hanning window
gamma <- SDF(temp,method="wosa",sampling.interval = data$Ts,npad=npad)
freq <- seq(from=1,to=ceiling(npad/2),by=1)/ceiling(npad/2)*pi/data$Ts
out <- idfrd(response = gamma[,2]/gamma[,3],freq=freq,Ts= data$Ts)
return(out)
}
#' Estimate empirical transfer function
#'
#' Estimates the emperical transfer function
#'
#' @export
etfe <- function(data){
temp <- cbind(as.ts(data$output),as.ts(data$input))
tempfft <- mvfft(temp)/dim(temp)[1]
freq <- seq(from=1,to=ceiling(dim(tempfft)[1]/2),
by=1)/ceiling(dim(tempfft)[1]/2)*pi/data$Ts
resp <- as.complex(tempfft[,1]/tempfft[,2])
out <- idfrd(response=resp[1:ceiling(length(resp)/2)],freq=freq,
Ts=data$Ts)
return(out)
}
|
