Evaluate Transfer function Expressions

Description

TF Evaluates a given transfer function expression in the s-domain

Usage

TF(str_expr)

Arguments

Details

TF Evaluates a given transfer function polynomial expression in the s-domain. The evaluation of the expressions are performed similar to symbolic math computations for polynomials. A transfer function model is created as the result of the expression evaluation. Thus, this is an alternative way of creating transfer function models following the natural math expressions found in block diagrams. It also provides an alternative way to perform system interconnections. Only transfer function models are currently supported for system interconnection using this function. System interconnections for other models could be performed using the series, parallel, feedback or connect functions. See the Examples section for further details.

Value

Returns an object of 'tf' class list with a transfer function. Numerator and denominator coefficients could then be retrieved from the object the same way as any other tf object

See Also

tf tf2ss series parallel

Examples

# Example taken from the GitHub page of Julia Control - an electric motor example
J <- 2.0
b <- 0.04
K <- 1.0
R <- 0.08
L <- 1e-4
P <- TF("K/(s*((J*s + b)*(L*s + R) + K^2))")
Cls <- TF("P/(1 + P)") # closed-loop connection

# More examples
TF("s+1")
sys1 <- tf(1, c(1, 2, 5))
sys2 <- tf(2, c(1, 2, 5))
TF("sys1 + sys2") # parallel system interconnection
TF("sys1 * sys2") # series system interconnection
TF("sys1 - sys2")
TF("sys1 - 1")
TF("sys1 + 1")
TF("sys1 - sys2 + sys2")
TF("sys1 / sys2 / sys2")

State-space matrices check.

Description

abcdchk verifies the dimensions of A,B,C,D matrices in its arguments, to ascertain that they are correctly defined.

Usage

abcdchk(a, b, c, d)

Arguments

Details

This is a utility function that is always invoked by other functions to ascertain the dimensions of the arguments a,b,c,d and returns a message if there is an ill-defined entry.

Value

Returns an empty string if matrix dimensions are consistent. Otherwise it returns the associated error message

Examples

A <- rbind(c(0,1), c(-10000,-4))
B <- rbind(0,1)
C <- rbind(c(1,0), c(0,1))
D <- rbind(0,0)
message <-  abcdchk(A,B,C,D)

Pole placement gain selection using Ackermann's formula

Description

Computes the Pole placement gain selection using Ackermann's formula.

Usage

acker(a, b, p)

Arguments

Details

K <- ACKER(A,B,P) calculates the feedback gain matrix K such that the single input system . x <- Ax + Bu

with a feedback law of u <- -Kx has closed loop poles at the values specified in vector P, i.e., P <- eigen(A - B * K).

This method is NOT numerically stable and a warning message is printed if the nonzero closed loop poles are greater than 10 in P.

Examples

F <- rbind(c(0,1),c(0,0))
G <- rbind(0,1)
H <- cbind(1,0);
J <- 0
t <- 1
sys  <-  ss(F,G, H,J)
A  <-  c2d(sys,t);
j <- sqrt(as.complex(-1));
pc  <- rbind(0.78+0.18*j, 0.78-0.18*j)
K  <-  acker(A$A, A$B, pc)

Append the dynamics of a set of systems

Description

append appends the dynamics of a set of n-state-space systems together

Usage

append(...)

Arguments

Details

append(sys1, sys2, sys3,...sysN) first combines the the first two systems and then goes on to combine the resulting state-space system to the next system and so forth. This is achieved by calling the sysgroup(sys1, sys2) at each iteration to group the systems in consecutive pairs until all systems are completely appended to form one system.

sysgroup(sys1, sys2) appends only two systems and is used by append

If a system is not in state-space representation, the function tries to form a state-space representation for such system.

Value

The function returns a state-space model of the formed appended system with A, B, C, D matrices

See Also

series parallel feedback connect

Examples

sys1 <- ss(1,2,3,4)
sys2 <- ss(2,3,4,5)
sys3 <- ss(6,7,8,9)
append(sys1, sys2, sys3)
sys4 <- tf(1, c(1,2,5))
append(sys1, sys2, sys4)

Bode Frequency Response for continuous-time Linear Systems.

Description

bode computes the magnitude and phase of the frequency response of system sys at given frequencies w

General Usage:

bode(sys)

bode(sys, w)

bode(sys, w, iu)

bode(sys, w = seq(0, 100, length = 10000), iu = 1)

bodeplot(sys)

bodeplot(sys, w)

bodeplot(sys, w, subtitle)

Usage

bode(sys, w, iu)

Arguments

Details

bode Compute the magnitude and phase of the frequency response of system sys at given frequencies w. When sys is a transfer function, bode computes the frequency response of the system using the signal package.

bodeplot plots the frequency response computed by bode. For a MIMO state-space system, bodeplot uses selectsys to obtain the bode response for each input-to-output pair and plot them individually. This means that for a 2-input, 2-output system, bodeplot obtains the response for input 1 to output 1, input 1 to output 2, input 2 to output 1 and input 2 to output 2. bodeplot uses the subtitle argument to allow a user assign the plot a sub-title

Value

A list is returned by calling bode containing: w - frequencies

mag - magnitude of the response

phase - phase of the response

A plot is returned by calling bodeplot

See Also

nyquist

Examples

bode(tf(100, c(1,6,100)))
bode(ssdata(tf(100, c(1,6,100))))

bode(tf(4, c(1,1)))
A <- rbind(c(-2, -1), c(1,0)); B <- rbind(1,0);
C <- cbind(0,1); D <- as.matrix(0);
bode(ss(A,B,C,D))

## MIMO plot
A1 <- rbind(c(0,1), c(-25,-4)); B1 <- rbind(c(1,1), c(0,1))
C1 <- rbind(c(1,0), c(0,1)); D1 <- rbind(c(0,0), c(0,0))
sys1 <- ss(A1,B1,C1,D1)
bodeplot(sys1)
# Use:  par(mfrow = c(2,1)); bodeplot(selectsys(sys1,1,2)) to obtain the response for a subsystem
# of sys1 for input 1 and output 2 only
# RESET your plot layout using par(mfrow = c(1,1))

Continuous Time model conversion to Discrete Time model.

Description

c2d converts a system in continuous-time model to a discrete time model

Usage

c2d(sys, t)

Arguments

Details

c2d converts the continuous-time system: x = Ax + Bu to the discrete-time state-space system: x[n+1] = Phi * x[n] + Gamma * u[n] based on the method of assuming a zero-order hold on the inputs and sample time Transfer function and zero-pole systems are converted to state-space representation before conversion to discrete-time.

Value

Returns the provided system (transfer function, state-space or zero-pole) in an equivalent discrete-time.

See Also

ltitr

Examples

## for TF
c2d(tf(c(1,-1), c(1,4,5)), 0.1)
## for ZPK
sys <- zpkdata( tf(c(1,-1), c(1,4,5)) )
c2d(sys, 0.1)
c2d(zpkdata( tf(c(1,-1), c(1,4,5)) ), 0.1)

Continuous-time Algebraic Riccati Equation solution

Description

Computes the unique solution to the continuous-time Riccati equation:

A'* X + X*A - X * B * R^-1 * B' * X + Q'*Q = 0

Usage

care(A, B, Q, R = 1)

Arguments

Details

X <- care(A, B, Q, R) returns the stablizing solution (if it exists) to the continuous-time Riccati equation.

The care function also returns the gain matrix, G and a vector, L of the closed-loop eigenvalues, where

G = R^-1 B'X*E

L = eig(a-b*g)

Value

Returns the stabilizing matrix, gain and closed-loop eigenvalues in a list.

Note

A, B must be controllable

Examples

a <- matrix(c(-3, 2,1, 1), byrow = TRUE, ncol = 2)
b <- matrix(c(0, 1), nrow = 2)
c <- matrix(c(1, -1), ncol = 2)
q <- t(c)%*%c
r <- 3
care(a, b, q, r)

Closed Feedback Loops

Description

cloop forms a closed feedback loop for a state-space or transfer function system

Usage

cloop(sys, e, f)

Arguments

Details

Other possible usages of cloop:

cloop(sys)

cloop(sys, sgn)

If sys is a state-space model, cloop(sys, SGN) produces a state-space model of the closed-loop system obtained by feeding all the outputs of the system to all the inputs. Positive feedback is used when SGN <- 1 and negative when SGN <- -1

If sys is a transfer function model, cloop(sys, SGN) produces the SISO closed loop system in transfer function form obtained by unity feedback with the sign SGN.

cloop(sys,OUTPUTS,INPUTS) forms the closed loop system obtained by feeding the specific outputs into specific outputs. The vectors OUTPUTS and INPUTS contain indices into the outputs and inputs of the system respectively. Positive feedback is assumed. To form closed loop with negative feedback, negative values are used in the vector INPUTS.

Value

Returns a closed feedback loop system

See Also

feedback

Examples

J <- 2.0; b <- 0.04; K <- 1.0; R <- 0.08; L <- 1e-4
P <- TF("K/(s*((J*s + b)*(L*s + R) + K^2))")
cloop(P)
cloop(ss(1,2,3,4))

Block diagram interconnections of dynamic systems

Description

connect is used to form a state-space model of a system from its block diagram.

Usage

connect(sysapp, q, inputs, outputs)

Arguments

Details

connect This function requires calling the append function to group a set of unconnected dynamics system in one system object. It then uses the q matrix to determine the interconnections between the systems and finally specifies the inputs and outputs for the new aggregate system. This approach helps to realize a block diagram as a single system on which further analysis could be performed. See examples below.

Value

Returns the interconnected system, returned as either a state-space model

See Also

append series parallel feedback

Examples

a1 <- rbind(c(0, 0), c(1,-3))
b1 <- rbind(-2,0)
c1 <- cbind(0,-1)
d <- as.matrix(0)
a2 <- as.matrix(-5)
b2 <- as.matrix(5)
c2 <- as.matrix(1)
d2 <- as.matrix(0)
sysa1 <- ss(a1, b1, c1, d)
sysa2 <- ss(a2, b2, c2, d2)
al <- append(sysa1, sysa2)
connect(al, cbind(2,1,0), cbind(1,2), cbind(1,2))
## OR
connect(append(sysa1, sysa2), cbind(2,1,0), cbind(1), cbind(2))
 ## Not run: 
cbind(2,1,0) means that block 2 has an input from block 1 and block 0 (which doesnt exist)
cbind(1) means that block 1 is the input of the system, and cbind(2) means block 2 is the
output of the system.
if we replace cbind(2) with cbind(1,2), this means that the system has two outputs from
block 1 and 2
i.e. \code{connect(append(sysa1, sysa2), cbind(2,1,0), cbind(1), cbind(1,2))}

## End(Not run)

Form Controllability Matrix

Description

ctrb forms the controllability matrix.

Usage

ctrb(A, B)

Arguments

Details

ctrb ctrb(a, b) returns the controllability matrix, [B AB A^2B ... A^(n-1)B]. If the Controllability matrix has full row rank, the system is controllable.

Value

Returns the controllability matrix.

See Also

obsv

Examples

a1 <- rbind(c(0,0),c(1,-3))
b1 <- rbind(-2,0)
ctrb(a1, b1)

Damping and Natural Frequencies for Continuous Systems

Description

damp computes the natural frequency and damping for continuous systems.

Usage

damp(sys, doPrint = TRUE)

Arguments

Details

A table of the eigenvalues of the matrix a, the associated damping factors, the associated natural frequency (rad/s and Hz.) is displayed by calling the function.

When the continuous system is a state-space model, the eigenvalues of the state matrix are obtained and sorted. If the system is a transfer-function, the poles of the systems are obtained and sorted. For zero-pole systems, the poles are just extracted and sorted. The sorted eigenvalues are printed to output and used to obtain the natural frequencies and damping factors.

Value

Returns the natrural frequencies and damping factors in a list:

omegan = Natural Frequencies (rad/s) zeta = Damping Factors

See Also

esort

Examples

sys1 <- tf(1, c(1,2,5))
damp(sys1)

DC Gain

Description

dcgain Forms the Givens rotation matrix

Usage

dcgain(sys)

Arguments

Details

dcgain(sys) Computes the steady-state gain (or low frequency gain) of a continuous system.

Value

Returns the gain.

Sort Complex Continuous Eigenvalues in Descending Order

Description

esort sorts the complex continuous eigenvalues in descending order

Usage

esort(p)

Arguments

Details

esort sorts the complex eigenvalues based on their real part. The unstable eigenvalues (positive real part) are first shown.

This function is used to sort eigenvalues and system poles in damp

Value

Returns the sorted eigenvalues and the cooresponding indices in a list:

s = sorted eigenvalues idx = index

See Also

damp

Feedback Connection of LTI systems

Description

feedback forms a feedback connection for two LTI state-space or transfer function systems

Usage

feedback(sys1, sys2, in1, out1)

Arguments

Details

When sys1 and sys2 are transfer functions feedback(sys1, sys2, SIGN) produces the SISO closed loop system in transfer function form obtained by connecting the two SISO transfer function systems in feedback with the sign SIGN.

feedback(sys1, sys2, SIGN) produces an aggregate state-space system consisting of the feedback connection of the two systems 1 and 2. If SIGN = 1 then positive feedback is used. If SIGN = -1 then negative feedback is used. In all cases, the resulting system has the same inputs and outputs as system 1.

feedback(sys1, sys2, inputs, outputs) produces the feedback system formed by feeding all the outputs of system2 into the inputs of system 1 specified by INPUTS1 and by feeding the outputs of system 2 specified by OUTPUTS1 into all the inputs of system 2. Positive feedback is assumed. To connect with negative feedback, use negative values in the vector INPUTS1.

feedback() calls fdbcksys() to perform the feedback connection for two systems. Unity feedback calls are possile, for example, feedback(sys1, 1), feedback(1, sys1)

Value

Returns the feedback system in tf or ss model

See Also

cloop parallel series

Examples

C <- pid(350,300,50)
P <- TF(" 1/(s^2 + 10* s + 20)")
feedback(C,P)
feedback(P,P,1)
feedback(P,P,-1)
feedback(P,P)
feedback(P,1)
feedback(TF("C*P"))
## Not run:  On Octave: feedback(C*P)

Low level frequency response function

Description

This function obtains the low level frequency response of a system.

Usage

freqresp(sys, w = seq(0, 100, length = 10000), iu = 1)

Arguments

Value

freqresp(sys, w) returns a vector of frequencies for sys in complex form

See Also

bode nyquist

Examples

H <- freqresp(ssdata(tf(c(1,1), c(1,2,1))), (seq(0, 100, length = 10000)))
H <- freqresp(tf(c(1,1), c(1,2,1)), seq(0, 100, length = 10000))

Generate periodic signal

Description

gensig generates a periodic signal. More useful when used in combination with lsim

Usage

gensig(signal, tau, tfinal, tsam)

Arguments

Details

gensig generates a periodic signal of the following types: square, sin, cos, pulse

Possible usage: gensig(signal)

Value

Returns a list of two single column matrices, u and t

u is the vector of signal values

t is the time vector of the signal

See Also

lsim

Examples

## Not run:  A square wave signal 
sig <-  gensig('square', 4, 10, 0.1)
plot(sig$t, sig$u, type = "l", col = "blue")
grid(5,5, col = "lightgray")

## Not run:  A sine wave signal 

sig <-  gensig('sin')
plot(sig$t, sig$u, type = "l", col = "blue")
grid(5,5, col = "lightgray")

Complex Givens Rotation

Description

givens_rot Forms the Givens rotation matrix

Usage

givens_rot(a, b)

Arguments

Details

givens_rot(a, b) returns the complex Givens rotation matrix This function is called by ordschur

Value

Returns the complex Givens rotation matrix.

See Also

ordschur

Impulse Response for Linear Systems

Description

impulse obtains the impulse response of the linear system:

dx/dt = Ax + Bu

y = Cx + Du

to an impulse applied to the input

Usage

impulse(sys, t, input)
impulseplot(sys, t, input)

Arguments

Details

impulse produces the impulse response of linear systems using lsim

impulseplot produces the impulse response as a plot against time.

These functions can handle both SISO and MIMO (state-space) models.

Other possible calls using impulse and impulseplot are:

impulse(sys)

impulse(sys, t)

impulseplot(sys)

impulseplot(sys, t)

Value

A list is returned by calling impulse containing:

t Time vector

x Individual response of each x variable

y Response of the system

The matrix y has as many rows as there are outputs, and columns of the same size of length(t). The matrix x has as many rows as there are states. If the time vector is not specified, then the automatically set time vector is returned as t

A plot of y vs t is returned by calling impulseplot

See Also

initial step ramp

Examples

res <- impulse(tf(1, c(1,2,1)))
res$y
res$t
impulse(tf(1, c(1,2,1)), seq(0, 10, 0.1))
impulseplot(tf(1, c(1,2,1)))
impulseplot(tf(1, c(1,2,1)), seq(0, 10, 0.1))

## Not run:  State-space MIMO systems 
A <- rbind(c(0,1), c(-25,-4)); B <- rbind(c(1,1), c(0,1));
C <- rbind(c(1,0), c(0,1)); D <- rbind(c(0,0), c(0,0))
res1 <- impulse(ss(A,B,C,D), input = 1)
res2 <- impulse(ss(A,B,C,D), input = 2)
res1$y # has two rows, i.e. for two outputs
res2$y # has two rows, i.e. for two outputs
impulseplot(ss(A,B,C,D), input = 1:2) # OR
impulseplot(ss(A,B,C,D), input = 1:ncol(D))
impulseplot(ss(A,B,C,D), seq(0,3,0.01), 1:2)

Initial Condition Response for Linear Systems

Description

initial obtains the time response of the linear system:

dx/dt = Ax + Bu

y = Cx + Du

to an initial condition.

Usage

initial(sys, x0, t)
initialplot(sys, x0, t)

Arguments

Details

initial produces the time response of linear systems to initial conditions using lsim

initialplot produces the time response to initial conditions as a plot againts time.

The functions can handle both SISO and MIMO (state-space) models.

Other possible calls using initial and initialplot are:

initial(sys)

initial(sys, x0)

initialplot(sys)

initialplot(sys, x0)

Value

A list is returned by calling initial containing:

x Individual response of each x variable

y Response of the system

t Time vector

The matrix y has as many rows as there are outputs, and columns of the same size of length(t). The matrix X has as many rows as there are states. If the time vector is not specified, then the automatically set time vector is returned as t

A plot of y vs t is returned by calling initialplot

See Also

step impulse ramp

Examples

res <- initial(tf(1, c(1,2,1)))
res$y
res$t
A <- rbind(c(-2, -1), c(1,0)); B <- rbind(1,0);
C <- cbind(0,1); D <- as.matrix(0);
x0 <- matrix(c( 0.51297, 0.98127))
initialplot(ss(A,B,C,D), x0)
initialplot(tf(1, c(1,2,1)), t = seq(0, 10, 0.1))

## Not run:  State-space MIMO systems 
A <- rbind(c(0,1), c(-25,-4)); B <- rbind(c(1,1), c(0,1));
C <- rbind(c(1,0), c(0,1)); D <- rbind(c(0,0), c(0,0))
res <- initial(ss(A,B,C,D))
res$y # has two rows, i.e. for two outputs
initialplot(ss(A,B,C,D))

SISO / MIMO Check

Description

issiso checks if state-space system is a single-input single-output system ismimo checks if state-space system is a multiple-input multiple-output system

Usage

issiso(sys)

Arguments

Value

Returns TRUE or FALSE

Time response of a Linear system

Description

lsim Computes the time response of a Linear system described by:

x = Ax + Bu

y = Cx + Du

to the input time history u.

Usage

lsim(sys, u, t, x0)

Arguments

Details

lsim(sys, u, t) provides the time history of the linear system with zero-initial conditions.

lsim(sys, u, t, x0) provides the time history of the linear system with initial conditions. If the linear system is represented as a model of tf or zpk it is first converted to state-space before linear simulation is performed. This function depends on c2d and ltitr

Value

Returns a list of two matrices, x and y. The x values are returned from ltitr call.

See Also

ltitr lsimplot

Examples

signal <- gensig('square',4,10,0.1)
H <- tf(c(2, 5, 1),c(1, 2, 3))
response <- lsim(H, signal$u, signal$t)
plot(signal$t, response$y, type = "l", main = "Linear Simulation Response", col = "blue")
lines(signal$t, signal$u, type = "l", col = "grey")
grid(5,5, col = "lightgray")
## Not run:  based on example at: https://www.mathworks.com/help/ident/ref/lsim.html 

## Not run:  MIMO system response 
A <- rbind(c(0,1), c(-25,-4)); B <- rbind(c(1,1), c(0,1))
C <- rbind(c(1,0), c(0,1)); D <- rbind(c(0,0), c(0,0))
response <- lsim(ss(A,B,C,D), cbind(signal$u, signal$u), signal$t)
plot(signal$t, response$y[1,], type = "l",
 main = "Linear Simulation Response", col = "blue"); grid(7,7)
plot(signal$t, response$y[2,], type = "l",
main = "Linear Simulation Response", col = "blue"); grid(7,7)

Plot time response of an LTI system

Description

lsimplot Plots the time response of a Linear system described by:

x = Ax + Bu

y = Cx + Du

to the input time history u.

Usage

lsimplot(sys, u, t, x0)

Arguments

Details

lsimplot(sys, u, t) plots the time history of the linear system with zero-initial conditions.

lsimplot(sys, u, t, x0) plots the time history of the linear system with given initial conditions.

If the linear system is represented as a model of tf or zpk it is first converted to state-space before linear simulation is performed. This function depends on c2d and ltitr

Value

Returns a plot for the response of the system

See Also

lsim stepplot rampplot

Examples

signal <- gensig('square',4,10,0.1)
H <- tf(c(2, 5, 1),c(1, 2, 3))
lsimplot(H, signal$u, signal$t)

## Not run:  MIMO system response 
A <- rbind(c(0,1), c(-25,-4)); B <- rbind(c(1,1), c(0,1))
C <- rbind(c(1,0), c(0,1)); D <- rbind(c(0,0), c(0,0))
lsimplot(ss(A,B,C,D), cbind(signal$u, signal$u), signal$t)

LTI frequency response kernel

Description

This function computes the frequency response of the following system:

g(w) = (wI-A) \ B

for the complex frequencies contained in the vector W. The column vector B must have as many rows as the matrix A.

Usage

ltifr(A, B, w)

Arguments

Value

Returns the frequency response in vector. freqresp utilizes this function for state-space systems.

See Also

freqresp

Examples

## use \code{\link{freqresp}}

Time response of a Linear Time-Invariant system

Description

ltitr Computes the time response of a Linear Time-Invariant system

Usage

ltitr(a, b, u, x0)

Arguments

Details

ltitr computes the time response of a Linear Time-Invariant system in state-space representation of the form: x[n+1] = Ax[n] + Bu[n] to an input, U

ltitr(a, b, u) computes the time response with zero-initial conditions since x0 is not supplied.

Value

Returns a matrix X which has as many rows as there are outputs y (and with max(dim(U)) columns).

See Also

lsim gensig

Examples

A <- diag(1, 2)
B <- rbind(1, 1)
x0 <- rbind(-1, -2)
u <- cbind(1, 2, 3, 4, 5)
X <- ltitr(A, B, u)
X <- ltitr(A, B, u, x0)

A <- replicate(6, abs(rnorm(6)))
B <- replicate(3, abs(rnorm(6)))
U <- replicate(100, rnorm(3))
x0 <-  rnorm(6)
X <- ltitr(A, B, U)
X <- ltitr(A, B, U, x0)

Nyquist Frequency Response for continuous-time Linear Systems.

Description

nyquist computes the real and imaginary parts of the frequency response of system sys at given frequencies w

Usage

nyquist(sys, w, iu)

Arguments

Details

nyquist Compute the real and imaginary parts of the frequency response of system sys at given frequencies w. When sys is a transfer function, nyquist computes the frequency response of the system using the signal package.

nyquistplot plots the frequency response computed by nyquist. For a MIMO state-space system, nyquistplot uses selectsys to obtain the nyquist response for each input-to-output pair and plot them individually. This means that for a 2-input, 2-output system, nyquistplot obtains the response for input 1 to output 1, input 1 to output 2, input 2 to output 1 and input 2 to output 2. nyquistplot uses the subtitle argument to allow a user assign the plot a sub-title

Other possible calls using nyquist and nyquistplot are:

nyquist(sys) nyquist(sys, w) nyquist(sys, w = seq(0, 100, length = 10000), iu = 1) nyquistplot(sys) nyquistplot(sys, w) nyquistplot(sys, w, subtitle)

Value

A list is returned by calling nyquist containing: h.real - real part of the frequency response

h.imag - imaginary part of the frequency response

A plot is returned by calling nyquistplot

See Also

bode

Examples

nyquist(tf(100, c(1,6,100)))
nyquist(ssdata(tf(100, c(1,6,100))))

## Not run:  MIMO plot 
A1 <- rbind(c(0,1), c(-25,-4)); B1 <- rbind(c(1,1), c(0,1))
C1 <- rbind(c(1,0), c(0,1)); D1 <- rbind(c(0,0), c(0,0))
sys1 <- ss(A1,B1,C1,D1)
nyquistplot(sys1)
## Not run:  Use nyquistplot(selectsys(sys1,1,2)) to obtain the response for a subsystem
 of sys1 for input 1 and output 2 only.

 RESET your plot layout using par(mfrow = c(1,1)
   
## End(Not run)

Observability Matrix

Description

This function creates the observability matrix.

Usage

obsv(A, C)

Arguments

Value

obsv(A, C) returns the observability matrix, obsvm. where obsvm = | C CA CA^2 ... CA^(n-1) |

See Also

ctrb

Examples

A <- rbind(c(0,1), c(-25,-4))
C <- rbind(c(1,0), c(0,1))
 obsv(A, C)

Ordered schur decomposition

Description

ordschur Orders a schur decomposition

Usage

ordschur(Ui, Si, idx)

Arguments

Details

ordschur finds an orthogonal matrix, U so that the eigenvalues appearing on the diagonal of Si are ordered according to the increasing values of the array index where the i-th element of index corresponds to the eigenvalue appearing as the element Si[i,i].

ordschur could also be used in this syntax: ordschur(Si, idx)

Value

Returns a list of ordered (U, S)

Parallel Connection of two systems

Description

parallel connects two systems in the parallel block form below

|–>[System1]–| u–>+ 0—>y |<–[System2]–|

Usage

parallel(sys1, sys2, in1, in2, out1, out2)

Arguments

Details

psys <- parallel(sys1, sys2) produces a state- space system consisting of the parallel connection of sys1 and sys2 that connects all the inputs together and sums all the outputs of the two systems.

The parallel connection is performed by appending the two systems, summing the specified inputs and outputs, and removing the, now redundant, inputs and outputs of system 2.

If sys1 and sys2 are transfer functions, then parallel(sys1, sys2) produces a parallel connection of the two transfer function systems.

parallel(sys1, sys2,IN1,IN2,OUT1,OUT2) connects the two systems in parallel by connecting the inputs specified by IN1 and IN2 and by summing the outputs specified by OUT1 and OUT2. The vector IN1 contains indexes into the input vectors of sys1 while, IN2 contains indexes for sys2, . Vectors OUT1 and OUT2 contain indexes for the outputs of the sys1 and sys2 respectively.

Value

The function returns a state-space model of the parallel-connected system with A, B, C, D matrices

See Also

series feedback connect

Examples

sys2 = ss(1,2,3,4)
sys3 = ss(6,7,8,9)
parallel(sys2, sys3)
parallel(tf(1, c(1,2,3)), ss(1,2,3,4))
parallel(tf(1, c(1,2,3)),tf(2, c(3,2,3)))

Proportional-Integral-Derivative (PID) Controller

Description

pid Parallel form of the model of a PID controller

Usage

pid(p, i, d)

Arguments

Details

pid creates the transfer function model for a PID, PI, PD, and P-controller.

Value

Returns a transfer function model for the PID, PI, PD or P-controller.

Examples

C <- pid(350,300,50) # PID-control
P <- TF(" 1/(s^2 + 10* s + 20)")
T <- feedback(TF("C*P"), 1)
stepplot(T, seq(0,2,0.01))

C <- pid(300,0,0) # P-control
T <- feedback(TF("C*P"), 1)
stepplot(T, seq(0,2,0.01))

C <- pid(30,70,0) # PI-control
T <- feedback(TF("C*P"), 1)
stepplot(T, seq(0,2,0.01))

C <- pid(300,0,10) # PD-control
T <- feedback(TF("C*P"), 1)
stepplot(T, seq(0,2,0.01))

Pole placement gain selection

Description

Computes the Pole placement gain selection using Ackermann's formula.

Usage

place(a, b, p)

Arguments

Details

K <- place(A,B,P) calculates the feedback gain matrix K such that the single input system . x <- Ax + Bu

with a feedback law of u <- -Kx has closed loop poles at the values specified in vector P, i.e., P <- eigen(A - B * K). This function is just a wrapper for the acker function.

This method is NOT numerically stable and a warning message is printed if the nonzero closed loop poles are greater than 10 in P.

Examples

F <- rbind(c(0,1),c(0,0))
G <- rbind(0,1)
H <- cbind(1,0);
J <- 0
t <- 1
sys  <-  ss(F,G, H,J)
A  <-  c2d(sys,t);
j <- sqrt(as.complex(-1));
pc  <- rbind(0.78+0.18*j, 0.78-0.18*j)
K  <-  place(A$A, A$B, pc)

Obtain Poles for a System

Description

This function obtains the poles for a given system

Usage

pole(sys)

Arguments

Details

pole returns the poles for a given system either a transfer function, state-space or zero-pole models. If sys is a transfer function, it computes the roots of the denominator If sys is a state-space object, it computes the eigenvalues of the A matrix. If sys is a zpk object, it retrieves the poles from the object.

Value

The function returns a column matrix containing the poles for the given system

Examples

H1 <- tf(c(2, 5, 1),c(1, 3, 5))
pole(zpk(NULL, c(-1,-1), 1))
pole(ssdata(tf(1, c(1,2,1))))

Print Polynomial

Description

Print polynomial as a character string.

Usage

poly2str(p, svar = "x", smul = "*", d = options("digits")$digits)

Arguments

Details

Modified from package *pracma*. Modification: To hide any coefficient and power that is equal to 1 So that instead of '1s^3' we have 's^3' and instead of 's^1', we have 's'

Value

Returns the usual string representing a polynomial in mathematics.

Examples

poly2str(c(2, -3, 1, 20, -11))

Subtracting Polynomials

Description

Subtract two polynomials given as vectors

Usage

polysub(a, b)

Arguments

Details

Simply calls polyadd from pracma package in the following manner: pracma::polyadd(a, -b)

Value

Returns a Vector representing the resulting polynomial.

Examples

polysub(c(1, 1, 1), 1)
polysub(c(1, 1, 1), c(0, 0, 1))

Ramp Response for Linear Time-Invariant Systems

Description

ramp obtains the ramp response of the linear system:

dx/dt = Ax + Bu

y = Cx + Du

Usage

ramp(sys, t, input)
rampplot(sys, t, input)

Arguments

Details

ramp produces the ramp response of linear systems using lsim

rampplot produces the ramp response as a plot against time.

These functions can handle both SISO and MIMO (state-space) models.

#' Other possible calls using ramp and rampplot are:

ramp(sys)

ranp(sys, t)

rampplot(sys)

rampplot(sys, t)

Value

A list is returned by calling ramp containing:

t Time vector

x Individual response of each x variable

y Response of the system

The matrix y has as many rows as there are outputs, and columns of the same size of length(t). The matrix x has as many rows as there are states. If the time vector is not specified, then the automatically set time vector is returned as t

A plot of y vs t is returned by calling rampplot

See Also

initial step impulse

Examples

res <- ramp(tf(1, c(1,2,1)))
res$y
res$t
ramp(tf(1, c(1,2,1)), seq(0, 6, 0.1))
rampplot(tf(1, c(1,2,1)))
rampplot(tf(1, c(1,2,1)), seq(0, 6, 0.1))

## Not run:  State-space MIMO systems 
A <- rbind(c(0,1), c(-25,-4)); B <- rbind(c(1,1), c(0,1));
C <- rbind(c(1,0), c(0,1)); D <- rbind(c(0,0), c(0,0))
res1 <- ramp(ss(A,B,C,D), input = 1)
res2 <- ramp(ss(A,B,C,D), input = 2)
res1$y # has two rows, i.e. for two outputs
res2$y # has two rows, i.e. for two outputs
rampplot(ss(A,B,C,D), input = 1:2) # OR
rampplot(ss(A,B,C,D), input = 1:ncol(D))
rampplot(ss(A,B,C,D), seq(0,3,0.01), 1:2)

Select/Remove Subsystem in State-space Model

Description

selectsys extracts a subsystem from a larger state-space system. removesys removes specified inputs, outputs, and state from a state-space system.

Usage

selectsys(statesys, inputs, outputs, states)
       removesys(statesys, inputs, outputs, states)

Arguments

Details

subsys <- selectsys(statesys,inputs,outputs) will extract a state space subsystem with the specified inputs and outputs.

subsys <- selectsys(statesys, inputs,outputs,states) will return the state space subsystem with the specified inputs, outputs, and states.

subsys <- removesys(statesys, inputs, outputs) will remove the specified inputs and outputs from the system.

subsys <- removesys(statesys, inputs, outputs, states) will also return a state-space model with the specified inputs, outputs, and states removed from the system.

Value

Returns a subsystem of the state-space model

See Also

append

Examples

A <- rbind(c(33,2,5), c(23,200,2), c(9,2,45))
B <- rbind(c(4,5), c(12,5), c(82,1))
C <- rbind(c(34,56,2), c(6,2,112))
D <- rbind(c(2,0), c(0,19))
sys1 <- ss(A, B, C, D)
 selectsys(sys1, 1, 1) # extract subsystem for only input 1 and output 1
 selectsys(sys1, 2,2) # extract subsystem for only input 2 and output 2
 selectsys(sys1, 2, 1:2) # extract subsystem for only input 1 and output 1 to 2
 selectsys(sys1, 1:2, 2) # extract subsystem for only input 1 to 2 and output 2 to 2
 selectsys(sys1, 2, 2, 1:2) # extract subsystem for only input 2 and output 2 but states 1 to 2
 removesys(sys1, 1,2) # removes input 1 and output 2

Series Connection of two systems

Description

series connects two systems in the series block form below

u —>[System1]—>[System2]—-> y

Usage

series(sys1, sys2, outputs, inputs)

Arguments

Details

seriessys <- series(sys1, sys2) connects the two state-space systems in series such that the outputs of sys1 specified are connected to the inputs of sys2 specified by input2. If sys1 and sys2 are both transfer functions, series(systf1, systf2) produces the SISO system in transfer function form obtained by connecting the two SISO transfer function systems in series. If a system is not in state-space representation, the function tries to form a state-space representation for such system.

Value

The function returns a state-space model of the aggregate system with A, B, C, D matrices

See Also

parallel feedback connect

Examples

series(tf(1, c(1,2,3)), tf(2, c(2,3,5)))
sys2 = ss(1,2,3,4)
sys3 = ss(6,7,8,9)
series(sys2, sys3)
series(tf(1, c(1,2,3)), ss(1,2,3,4))

Create State-space Model.

Description

ss creates the model for a system represented in state-space form

Usage

ss(A, B, C, D, Ts = NULL)

Arguments

Details

ss creates a model object for state-space systems.

Value

Returns a list object of 'ss' class.

See Also

tf zpk

Examples

A <- rbind(c(-2, -1), c(1,0))
B <- rbind(1,0)
C <- cbind(0,1)
D <- 0;
sys <- ss(A,B,C,D)

## Not run:   OR 

sys <- ss(c(-2,-1,1,0), c(1,0), c(0,1), 0)

## Not run:   Access individual state-space sys elements as 

sys$A
sys$B
sys$C
sys$D

State-space model conversion to Transfer function model.

Description

ss2tf converts the model for a state-space system to transfer function representation

Usage

ss2tf(a, b, c, d, iu)

Arguments

Details

ss2tf converts a model object in state-space form to transfer function model by calculating the transfer function of the system: . x = Ax + Bu y = Cx + Du

#' Other possible usages for ss2tf are: ss2tf(a,b,c,d) ss2tf(sys) ss2tf(sys, iu)

where sys is an object of state-space class

Value

Returns an object of 'tf' class containing num and den. The numerator coefficients are returned in matrix num with as many rows as outputs y.

See Also

tf2ss ss2zp

Examples

sys2 <- tf2ss(tf(1, c(1,2,1)))
ss2tf(sys2)

## Not run:   OR 

ss2tf(sys2$A,sys2$B,sys2$C,sys2$D)

# a single input multiple output system
A <- rbind(c(0,1), c(-10000,-4)); B <- rbind(0,1); C <- rbind(c(1,0), c(0,1));
D <- rbind(0,0)
ss2tf(A, B, C, D)

# a MIMO system
A = rbind(c(0,1), c(-25,-4)); B = rbind(c(1,1), c(0,1));
C = rbind(c(1,0), c(0,1)); D = rbind(c(0,0), c(0,0))
ss2tf(A,B,C,D,1) # to obtain output for input 1
ss2tf(A,B,C,D,2) # to obtain output for input 2

## OR

systems <- vector("list", ncol(D))
for(i in 1:ncol(D)){ systems[[i]] <- ss2tf(A,B,C,D,i) }
systems
systems[[1]]
systems[[2]]

State-space representation to zero-pole-gain representation

Description

ss2zp converts a system represented in state-space form to zero-pole-gain model

Usage

ss2zp(a,b,c,d,iu)

Arguments

Details

ss2zp converts a system represented in zero-pole form to state-space by converting from zero-pole to transfer function and from transfer functon to state-space The vector P contains the pole locations of the denominator of the transfer function.

Other possible usages for ss2zp are:

ss2zp(a,b,c,d)

ss2zp(sys)

ss2zp(sys, iu)

where sys is an object of state-space class

Value

Returns a list object of 'zpk' class, consisting of z, p and k. The numerator zeros are returned in the columns of matrix Z with number of columns equal to number of outputs. The gains for each numerator transfer function are returned in column vector K. P, a column vector contains the pole locations of the denominator of the transfer function.

See Also

zp2ss ss2tf

Examples

A <- rbind(c(-2, -1), c(1,0)); B <- rbind(1,0);
C <- cbind(0,1); D <- 0;
sys2 <- ss(A,B,C,D)
ss2zp(sys2$A,sys2$B,sys2$C,sys2$D)
ss2zp( zp2ss ( tf2zp( c(1,1,1), c(1,2,1) ) ) )

## Not run:   A MIMO system 
A = rbind(c(0,1), c(-25,-4)); B = rbind(c(1,1), c(0,1));
C = rbind(c(1,0), c(0,1)); D = rbind(c(0,0), c(0,0))
ss2tf(A,B,C,D,1) # to obtain output for input 1
ss2tf(A,B,C,D,2) # to obtain output for input 2

## Not run:   OR 

systems <- vector("list", ncol(D))
for(i in 1:ncol(D)){ systems[[i]] <- ss2zp(A,B,C,D,i) }
systems
systems[[1]]
systems[[2]]

Retrieve State-space data

Description

ssdata retrieves the model for a state-space system from a sys object

Usage

ssdata(sys1)

Arguments

Details

ssdata retrieves a model object for a state-space system, from a sys object of tf, ss and zpk classes

Value

Returns a list object of ss class containing A, B, C and D matrices

See Also

ss tfdata zpkdata

Examples

sys1 <- tf(c(1), c(1,2,1))
 ssdata(sys1)
 A <- rbind(c(-2, -1), c(1,0)); B <- rbind(1,0);
 C <- cbind(0,1); D <- 0;
 sys2 <- ss(A,B,C,D)
 ssdata(sys2)
 ss2zp(ssdata(zpk(NULL, c(-1,-1), 1)))

Step Response for Linear Systems

Description

step obtains the time response of the linear system:

dx/dt = Ax + Bu

y = Cx + Du

Usage

step(sys, t, input)
stepplot(sys, t, input)

Arguments

Details

step produces the step response of linear systems using lsim

stepplot produces the step response as a plot againts time.

The functions can handle both SISO and MIMO (state-space) models.

Other possible calls using step and stepplot are:

step(sys)

step(sys, t)

stepplot(sys)

stepplot(sys, t)

Value

A list is returned by calling step containing:

x Individual response of each x variable

y Response of the system

t Time vector

The matrix y has as many rows as there are outputs, and columns of the same size of length(t). The matrix X has as many rows as there are states. If the time vector is not specified, then the automatically set time vector is returned as t

A plot of y vs t is returned by calling stepplot

See Also

initial impulse ramp

Examples

res <- step(tf(1, c(1,2,1)))
res$y
res$t
step(tf(1, c(1,2,1)), seq(0, 10, 0.1))
stepplot(tf(1, c(1,2,1)))
stepplot(tf(1, c(1,2,1)), seq(0, 10, 0.1))

## Not run:   State-space MIMO systems 

A <- rbind(c(0,1), c(-25,-4)); B <- rbind(c(1,1), c(0,1));
C <- rbind(c(1,0), c(0,1)); D <- rbind(c(0,0), c(0,0))
res1 <- step(ss(A,B,C,D), input = 1)
res2 <- step(ss(A,B,C,D), input = 2)
res1$y # has two rows, i.e. for two outputs
res2$y # has two rows, i.e. for two outputs
stepplot(ss(A,B,C,D), input = 1:2) # OR
stepplot(ss(A,B,C,D), input = 1:ncol(D))

Create Transfer function Model.

Description

tf creates the model for a transfer function

Usage

tf(num, den, Ts = NULL)

Arguments

Details

tf creates a model object for a transfer function, Where num is the numerator and den is the denominator of the transfer function.

Value

Returns an object of 'tf' class list with a proper transfer function or with warnings when not proper.

See Also

ss zpk TF tf2ss tf2zp

Examples

tf(1, c(1,2,1))
sys1 <- tf(1, c(1,2,1))
sys1$num
sys1$den

## Not run:   for single-input multi-output systems (SIMO) each numerator row for one output 
num = rbind(c(0,1,1), c(1,0,1))
den = rbind(c(1,3,2))
tf(num, den)

Transfer function model conversion to State-space model.

Description

tf2ss converts the model for a transfer function to state-space representation

Usage

tf2ss(num, den)

Arguments

Details

tf2ss converts a model object for a transfer function to a state-space model, Where num is the numerator and den is the denominator of the transfer function and sys is a transfer function object

Another possible call is tf2ss(sys) where sys is object of transfer-function model.

Value

Returns an object of 'ss' class.

See Also

ss2tf tf2zp

Examples

tf2ss(tf(1, c(1,2,1)))

## Not run:   OR 

sys <- tf(1, c(1,2,1))
tf2ss(sys)

## Not run:   OR 

sys2 <- tf2ss(1, c(1,2,1))

Transfer function model conversion to Zero-Pole-Gain model.

Description

tf2zp converts the model for a transfer function to zero-pole-gain representation

Usage

tf2zp(num, den)

Arguments

Details

tf2zp converts a model object for a transfer function to a zero-pole model, Where num is the numerator and den is the denominator of the transfer function and sys is a transfer function object

Another possible call is: tf2zp(sys)

where sys is an object of transfer-function model.

Value

Returns a list object of 'zpk' class.

See Also

tf2ss zp2tf

Examples

syszp1 <- tf2zp(c(1,1), c(1,2,1))
syszp1
syszp2 <- tf2zp(c(2,2,1), c(1,2,1))
syszp2
unclass(syszp2) # to see list of the zeros,poles and gain as vectors
tf2zp(zp2tf(c(-1,-1), c(-1,-2), 5))

Transfer function check.

Description

tfchk verifies the structure of a transfer function

Usage

tfchk(num, den)

Arguments

Details

This is a utility function that is always invoked by other functions to verify the structure of num, den. Where num is the numerator and den is the denominator of the transfer function. If the transfer function is not proper, it returns a list with length(num) = length(den).

Value

Returns a list with a proper transfer function or with warnings when not proper.

Examples

tf1 <- tfchk(1, c(1,2,1))

Retrieve Transfer function data

Description

tfdata retrieves the model for a transfer function from a sys object

Usage

tfdata(sys1)

Arguments

Details

tfdata retrieves a model object for a transfer function, from a sys object of tf, ss and zpk classes

Value

Returns a list object of tf class containing numerator and denominator coefficients in desecending values of s. For multiple-input multiple-output systems (MIMO) a list containing tf sys objects for as many outputs is returned

See Also

tf ssdata zpkdata

Examples

sys1 <- zpk(NULL, c(-1,-1), 1)
tfdata(sys1)
A <- rbind(c(-2, -1), c(1,0)); B <- rbind(1,0);
C <- cbind(0,1); D <- 0
tfdata( ss(A, B, C, D) )
tfdata(ss2zp( A,B,C,D))
tfdata(tf(c(1), c(1,2,1)))

 ## Not run:   MIMO system 
A = rbind(c(0,1), c(-25,-4)); B = rbind(c(1,1), c(0,1));
C = rbind(c(1,0), c(0,1)); D = rbind(c(0,0), c(0,0))
tfdata(ss(A,B,C,D))

Convert Zero-Pole-Gain Model to State-Space Model

Description

zp2ss converts a system represented in zero-pole form to state-space

Usage

zp2ss(z,p,k)

Arguments

Details

zp2ss converts a system represented in zero-pole form to state-space by converting from zero-pole to transfer function and from transfer functon to state-space

Another possible usage is: zp2ss(sys)

where sys is an object of zero-pole-gain model.

Value

Returns a list object of 'ss' class.

See Also

ss2zp zp2tf

Examples

zp2ss(NULL, c(-1,-1), 1)
zp2ss(tf2zp(c(1,1,1), c(1,2,1)))

Zero-pole-gain model conversion to Transfer function model

Description

zp2tf converts the model for a zero-pole-gain system to transfer function representation

Usage

zp2tf(z, p, k)

Arguments

Details

zp2tf converts a model object for a zero-pole-gain system to a transfer function model

Another possible usage is: zp2tf(sys)

where sys is an object of zero-pole-gain model.

Value

Returns a list object of 'tf' class.

See Also

zp2ss tf2zp

Examples

systf <- zp2tf(zpk(NULL, c(-1,-1), 1))
zp2tf(tf2zp(c(2,2,1), c(1,2,1)))

Create Zero-Pole-Gain Model.

Description

zpk creates the model for a system represented in zero-pole form

Usage

zpk(zero, pole, gain, Ts = NULL)

Arguments

Details

zpk creates a model object for zero-pole systems.

Value

Returns a list object of 'zpk' class.

See Also

ss tf

Examples

sys <- zpk(NULL, c(-1,-1), 1)
sys <- zpk(c(1,2), c(3,4), 5)
sys <- zpk(c(1,2), c(3+1i,4+2i), 5)

## Not run:   Access individual sys elements as 
sys$z
sys$p
sys$k

Retrieve zero-pole data from LTI system object

Description

zpkdata retrieves the model for a zero-pole-gain system from a sys object

Usage

zpkdata(sys1)

Arguments

Details

zpkdata retrieves a model object for a zero-pole-gain system, from a sys object of tf, ss and zpk classes

Value

Returns a list object of zpk class containing zero, pole and gain matrices. For multivariable systems, the zeros of each system is listed as a column in the zeros matrix, the poles are listed as a column-vector as well as the gain

See Also

zpk tfdata ssdata

Examples

sys1 <- zpk(NULL, c(-1,-1), 1)
zpkdata(sys1)
sys3 <- tf(c(1), c(1,2,1))
zpkdata(sys3)

## Not run:   MIMO system of 2-inputs and 2-outputs 
A = rbind(c(0,1), c(-25,-4)); B = rbind(c(1,1), c(0,1));
C = rbind(c(1,0), c(0,1)); D = rbind(c(0,0), c(0,0))
zpkdata(ss(A,B,C,D))

## OR
syszp <- zpkdata(ss(A,B,C,D))
syszp[[1]]
syszp[[2]]
syszp[[1]]$z # retrieve zeros of system 1 - Input 1 to Outputs 1 and 2
syszp[[2]]$z # retrieve zeros of system 2 - Input 2 to Outputs 1 and 2