Control Systems on R Views

Control Systems Toolbox – System Interconnection

Thu, 24 Aug 2017 00:00:00 +0000

Introduction

Dynamic systems are usually represented by a model before they can be analyzed computationally. These dynamic systems are systems that change, evolve or have their states altered or varied with time based on a set of defined rules. Dynamic systems could be mechanical, electrical, electronic, biological, sociological, and so on. Many such systems are usually defined by a set rules that are represented as a set of nonlinear differential equations. A generic first-order differential form is as shown below: \[ \frac{dx}{dt} = f ( x(t), u(t), t ) ........... (1) \]

where x(t) is the state vector representing the configuration (or state) of the system at the time, t, u(t) represents the external control input at time, t, and f is a function that gives the rate of change of the state vector for a specific input, state and time.

The system of (1) could be said to be time invariant if we assume that f does not explicitly depend on time. This implies that the relation of (1) becomes:

\[ \frac{dx}{dt} = f (x, u) ........... (2) \]

For such time-invariant systems, the control input may remain time-variant, but the coefficients of the function f are constant since the underlying physical laws themselves are not usually time-dependent. Over a sufficiently small operating range, the dynamics of most systems are approximately linear, and the following relation entails a linear time-invariant (LTI) system:

\[ \frac{dx}{dt} = Ax + Bu ........... (3) \]

Several engineering challenges till today are being solved using LTI techniques, while most results obtained from control theory are based on the described assumptions associated with LTI systems.

Laplace Transform

For those systems that are described by differential equations, it is possible to convert the system’s time-domain model into a frequency-domain (s-domain) representation. Two striking benefits for doing this are:

the differential equations that define the system are transformed into algebraic equations, which are often easier to analyze
The frequency response of the system can be obtained

This frequency-domain output / input representation is called the transfer function. Many control system models are available in transfer function representation, and this model is mostly used in the block diagram representation of control systems.

Figure 1 shows the block diagram for the pitch-control system of an Unmanned Free Swimming Submersible (UFSS) vehicle. This system and its block diagram shall be detailed in the Case Study section of this article.

Transfer Function Expressions

A transfer function expression is an algebraic expression in the frequency-domain. It is usually the ratio of the output to the input of a control system. Evaluating such expressions while retaining their mathematical form (as in symbolic mathematics) should be possible using a scientific computing language. An example of such expression is shown below:

\[ G(s) = \frac{Output}{Input} = \frac{s^2 + 3s + 4}{2s^3 + 4s^2 + 5s + 1} \]

Models of physical systems often appear as transfer functions with the coefficients of state as electrical, mechanical, chemical, or other physical quantities. Users would find it convenient to simply input these transfer functions as-is into a computer, and quickly analyze the system.

In the following mechanical example of a mass-spring-damper system of [1], the transfer function contains symbols m, b and k, which represent mass, damping constant, and spring constant variables respectively, as shown below:

Figure 2: Mass-Spring-Damper diagram \[ \frac{X(s)}{F(s)} = \frac{1}{ms^2+bs+k} \]

where m = 1; k = 1; b = 0.2

In this Google Summer of Code (GSoC) 2017 project, we have tried to include such capability within the control library, making it possible to create a transfer function model from the evaluation of a rational expression (the transfer function). A function named TF() was created to accept transfer function expressions, and return a transfer function model after evaluation. Therefore, to create this mass-spring-damper model within R, use the control library as shown in the snippet below:

 m <- 1;  k <- 1;  b <- 0.2
TF("1/(m*s^2 + b *s + k)")

## 
## y1:
##                 1 
##   - - - - - - - - - - -
##      s^2 + 0.2 s + 1 
## 
## 
## Transfer Function: Continuous time model

Electric Motor Example

Consider the following example from a University of Michigan course page on Control Systems using MATLAB – an example of an electric motor [2]

Figure 3: showing electric motor physical model [2]

Wikipedia: A DC motor is any class of rotary electric machines that converts direct current electrical energy into mechanical energy. It can provide translational motion when coupled with wheel, drums and cables. DC motors are used in propulsion of electric vehicles, elevator and hoists, or in drives for steel rolling mills. Several applications are: air compressors, vacuum cleaner, hair drier, sewing machine, and even toys or any system that requires variable speed.

The physical parameters for this example are [3]:

Moment of inertia of the rotor, J = 3.2284E-6 kg.m^2

Motor viscous friction constant, b = 3.5077E-6 N.m.s

Electromotive force constant, Ke = 0.0274 V/rad/sec

Motor torque constant, Kt = 0.0274 N.m/Amp

Electric resistance, R = 4 Ohm

Electric inductance, L = 2.75E-6 H

where, Kt = Ke = K

The input of the system is the voltage source (V) applied to the motor’s armature, while the output is the rotational speed of the shaft.

Open-Loop Response of the electric motor

The open-loop transfer function for the DC motor described above in the s-domain is:

\[ P(s) = \frac{K}{(s(Js + b)(Ls + R) + K^2)} \]

The step function is usually used to analyze the system’s step response, so it is applied here to analyze the open-loop response of the system. Using the control library, this model in the s-domain could be created and analyzed in R as follows:

 J <- 3.2284E-6;
b <- 3.5077E-6;
K <- 0.0274;
R <- 4;
L <- 2.75E-6;
P_motor <- TF("K/(s*((J*s + b)*(L*s + R) + K^2))")
time <- seq(0,0.2,0.001)
stepplot(P_motor, t = time)

Figure 1: Figure 4: Electric Motor Open-loop Step Response

From the above plot showing the system’s step response, it is clear that when a step input (of 1 volt) is applied to the system, the electric motor’s position grows unbounded - system instability.

The stability of the system can also be determined from the poles of the transfer function using the pole function:

pole(P_motor)

##               [,1]
## [1,]  0.000000e+00
## [2,] -1.454487e+06
## [3,] -5.922604e+01

As shown above, one of the poles of the open-loop transfer function is on the imaginary axis while the other two poles are in the left half of the complex s-plane. A pole on the imaginary axis indicates that the free response of the system will not only grow unbounded, but also will not decay to zero [3].

Closed-Loop Response of the electric motor

We can add feedback the motor model with a controller set to 1, and then investigate the system’s closed-loop step response for stability. This can be obtained using the following R commands:

P_motor2 <- feedback(P_motor, 1)
print(P_motor2)

## 
## y1:
##                                                     0.0274 
##   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
##     8.8781e-12 s^3 + 1.291361e-05 s^2 + 0.0007647908 s + 0.0274 
## 
## 
## Transfer Function: Continuous time model

# OR

P_motor2 <- cloop(P_motor, -1)
print(P_motor2)

## 
## y1:
##                                                     0.0274 
##   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
##     8.8781e-12 s^3 + 1.291361e-05 s^2 + 0.0007647908 s + 0.0274 
## 
## 
## Transfer Function: Continuous time model

stepplot(P_motor2, t = time)

Figure 2: Figure 5: Electric Motor Closed-loop Step Response

The above closed-loop step response shows that the addition of feedback has stabilized the system.

The damping and natural frequencies associated with the poles of this closed-loop system could be obtained using the following damp function:

damp(P_motor2)

## 
##      Eigenvalue            Damping     Freq. (rad/s)    Freq. (Hz). 
##      ----------            -------     -------------    -----------
## -29.61     +     35.28j   0.64           46.06           0.46      
## -29.61     -     35.28j   0.64           46.06           0.46      
## -1454487.32 +      0.00j   1.00           1454487.32      0.00

## $omega
##              [,1]
## [1,] 4.606386e+01
## [2,] 4.606386e+01
## [3,] 1.454487e+06
## 
## $zeta
## [1] 0.642853 0.642853 1.000000

Furthermore, using the TF() function, transfer function models could be summed, subtracted, multiplied, and divided. This is exemplified in the following section.

System Interconnections

In many cases, systems or subsystems have to be interconnected to obtain a particular controlled or regulated activity. Interconnecting models of components allows you to easily construct large control system models.

Analyzing systems that are modeled in a block diagram should be easily done using a scientific computing tool like R. The time response or frequency response of a system (such as shown in Figure 1), can be obtained after the block diagram is simplified. Simplification means that the several components (blocks described by different transfer functions) that make up the complete system have to be algebraically reduced to one transfer function to represent the system. This mathematical reduction is done based on the rules of control theory.

Figure 6: Block Diagram Interconnection

For an example, Figure 6 shows the interconnection of dynamic system models of a plant P(s), a controller C(s), sensor dynamics S(s), and a filter F(s). To analyze the system requires constructing a single (equivalent) model that represents the entire closed-loop control system.

In the second coding phase of the GSoC 2017, the following model interconnection functions have been produced as shown in Figure 7 below:

Figure 7: System Model Interconnections

Append Connection using the `append` function

A number of systems, whether in transfer-function, state-space or zero-pole model, could be appended into one state-space model using the append function. The block diagram arrangement is found in Figure 7 (e).

The syntax is: append(sys1, sys2, …, sysN). The resulting model is in state-space form.

For example:

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

Series Connection using the `series` function

This function connects two systems in series, in transfer-function, state-space or zero-pole model, into one model. The block diagram arrangement is found in Figure 7 (a).

Syntax: series(sys1, sys2)

Arithmetically, this could be achieved using the TF() function in the following manner:

TF(“sys1*sys2”)

For now, TF(“sys1*sys2”) is only possible when sys1 and sys2 are transfer-function models

Parallel Connection using the `parallel` function

The parallel function connects two systems in parallel, in transfer-function, state-space or zero-pole model, into one model. The block diagram arrangement is found in Figure 7 (c).

Syntax: parallel(sys1, sys2)

Arithmetically, this could be achieved using the TF() function in the following manner:

TF(“sys1 + sys2”)

For now, TF(“sys1+ sys2”) is only possible when sys1 and sys2 are transfer-function models

Feedback Connection using the `feedback` function

This function sets up a closed-loop transfer function from output to input based on the standard configuration shown in Figure 7 (b).

Syntax: feedback(sys1, sys2)

Arithmetically, this could be achieved using the TF() function in the following manner:

TF(“sys1 / (1 + sys2*sys1)”), but this is not recommended in terms of accuracy. feedback(sys1, sys2) is preferred.

General block diagram building using `connect` function

Using the connect function, more complicated (especially multiple-input multiple-output (MIMO) ) block diagrams could be set up. For this case, all systems that are to be connected are first appended, then connected using their inputs and outputs.

Syntax: connect(append(sys1, sys2, sys3), connections, inputs, outputs)

Case Study – Unmanned Free Swimming Submersible (UFSS) Vehicle

The Unmanned Free-Swimming Submersible (UFSS) vehicle was one of the first autonomous underwater vehicles (AUVs). It was developed in the late 1970s as a test vehicle to demonstrate the potential of autonomous vehicles over long distances, and to study laminar flow [5]. A picture showing an Unmanned Free-Swimming Submersible (UFSS) vehicle is shown in Figure 8 below.

Figure 8: Unmanned Free Swimming Submersible Vehicle (UFSS). Source: [5]

Nise [4] relates the following concerning the UFSS:

The depth of the vehicle is controlled as follows. During forward motion, an elevator surface on the vehicle is deflected by a selected amount. This deflection causes the vehicle to rotate about the pitch axis. The pitch of the vehicle creates a vertical force that causes the vehicle to submerge or rise.

A detailed block diagram for the pitch control system is shown in Figure 9 below.

Figure 9: Block diagram for the pitch control system of UFSS

Rationale:

The subsystems of the UFSS vehicle are developed independently, and studied using computer simulation before integration into the vehicle [5]. Thus, for this reason, we are examining the pitch control system independently.

For this case study, we shall use the developing control library in R to:

Obtain the closed-loop UFSS pitch control system by block diagram reduction of the system in Figure 9 above, using series, parallel and feedback.
Obtain the step response of the pitch control system simplified by series, parallel and feedback
Obtain the closed-loop UFSS pitch control system by block diagram reduction of the system in Figure 9 above, using the algebraic operations provided by TF().
Obtain the step response of the pitch control system from the method of algebraic operations.

Solution using `series`, `parallel` and `feedback`

K1 <- 1
K2  <- 1
G1 <- tf(-K1,1)
G2 <- tf(c(0,2), c(1,2))

numg3 <- -0.125*c(1, 0.435)
deng3 <- pracma::polymul(c(1, 1.23), c(1, 0.226, 0.0169))
G3 <- tf(numg3, deng3)

H1 <- tf(c(-K2, 0), c(0, 1))

G4 <- series(G2, G3)
G5 <- feedback(G4, H1)
Ge <- series(G1, G5)
T1 <- feedback(Ge, 1)
stepplot(T1, t = seq(0,40,0.1))

Figure 3: Figure 10: R plot showing UFSS vehicle system response - Block diagram reduction by series, parallel and feedback

Solution via algebraic operations

G4 <- TF("G3*G2")
G5 <- TF("G4/(1+ G4*H1)")
Ge <- TF("G5*G1")
T2 <- TF("Ge/(1+Ge)")
stepplot(T2, t = seq(0,40,0.1))

Figure 4: Figure 11: R plot showing UFSS vehicle system response - Block diagram reduction by Algebraic Operations

Clearly, the step responses (Figure 10 and Figure 11) of the pitch control system using the two methods are identical to each other - a justification to the framework of the control systems toolbox.

Feedback

Interested users are welcome to install the development version of this package using devtools::install_github("benubah/control"), and feel free to get in touch for any related issues.

Conclusion

In this article, we have discussed on our progress with the Control Systems Toolbox in R as a GSoC 2017 project. We have highlighted the use of this toolbox to carry out the following tasks:

Create Transfer function models by evaluating transfer function expressions in the s-domain
Obtain the poles and their associated damping and natural frequencies of a system
Open-loop and closed-loop responses
System Model Interconnections using series, parallel and feedback and algebraic functions in the UFSS vehicle case study.

Other functions that have been developed in this GSoC phase are: givens_rot, ordschur, care, ctrb, acker, place.

You may want to read about what was achieved in the first GSoC coding phase here.

References

http://ctms.engin.umich.edu/CTMS/index.php?example=Introduction&section=SystemModeling
http://ctms.engin.umich.edu/CTMS/index.php?example=MotorSpeed&section=SystemModeling
http://ctms.engin.umich.edu/CTMS/index.php?example=MotorPosition&section=SystemAnalysis
Norman, S. Nise, Control Systems Engineering, 6th. Ed, 2011.
http://www.navalunderseamuseum.org/ufss/

Control Systems Toolbox in R - a GSoC 2017 Project

Thu, 06 Jul 2017 00:00:00 +0000

Introduction

Control theory is an interdisciplinary branch of mathematics and engineering that has the objective of controlling physical systems. A control system is a device or a collection of devices that manage, command, direct or regulate the behavior of other devices or systems. Control systems engineering is a major application of control theory, and involves the study, design/modeling of automatic control systems. Control theory has also been applied to several other fields such as finance, sociology, psychology, physiology and other sciences that analyze dynamic systems.

In recent times, control systems have played a major role in the development and advancement of modern technology and civilization. Control systems are essential in any field of engineering and science, as they are an important and integral part of systems such as space-vehicle systems, robotic systems, modern manufacturing systems, nuclear reactors, navigation systems and industrial operations involving control of temperature, pressure, humidity, flow, speed/velocity, etc.

Example of a Control System

Imagine the operation of an Automatic Electric Iron where its heating elements are controlled by the temperature output of the iron. As long as there is power supply to the electric iron, and a desired heat-level is selected, the electric iron automatically controls at what time to turn-on and switch-off heating so as to maintain that temperature. This type of control system is called a feedback control system: a system in which the output is fed back to the input as a control signal.

Figure 1 shows a simple block diagram depicting a sample feedback control system for a heating system.

Figure 1: Block Diagram Model for a Heating System

Another example is the control system for an aircraft or a space shuttle. Truly, such control system would be complex as there are several sub-systems that are interconnected in such transport systems. Therefore, it is very necessary to simulate how these systems could behave using a computer before implementing them physically. This is the reason why computer simulation of control systems have been proliferated.

Control Systems R-GSoC Project

For this Google Summer of Code (GSoC) 2017 project, we are working to develop a control system toolbox for the R language. The rationale is that there is a control toolbox for most commercial and open-source technical computing platforms out there, and while the R language is experiencing tremendous growth in its capabilities and user base, it seems necessary to extend the language to serve other scientific and engineering communities. For example, in Python, the development of NumPy and the general SciPy stack, added some dimensions and attracted several scientific domains to the use of Python. This GSoC project hopes to make basic design, analysis and simulation of control systems possible within the R ecosystem.

The major objective of this year’s GSoC project is to try to:

Design and develop a framework for the package that defines data structures and creates flexibility of usage and extension for future users and interested contributors to easily adopt the package. The framework is also expected to mimic the I/O style of existing control systems toolkits.
To port a set of functions from the RLabPlus control-toolbox and adapt them to the defined data structures and general framework.
To test the functions, create documentation, usage examples and build them for distribution.

Benefits of a Control Systems Toolbox in R

Some benefits of having a control-systems-toolbox in R are:

A control package in R could be useful to engineers and students who are looking to migrate to an open-source language like R that has better visualization options and interfaces to several technologies. Models for several dynamic systems could then be analyzed using R.
The package could assist in further developments in areas like optimal control theory which has several applications to economics and management sciences.

Current State of the Project

At the time of writing this article, the first phase of coding for GSoC has elapsed, and we are delighted to meet our target of creating 18 functions that define the general framework and structures. Consequently, a user can, at this stage, perform the following:

Create LTI system models of state-space, transfer function and zero-pole classes
Conversions from one system model to other system models
Convert continuous-time system models to discrete-time models
Obtain time response for LTI systems
Install the development version of the package using: devtools::install_github(“benubah/control”)

Practical Example

Analyzing the Time Response of a Bandpass RLC Filter Circuit

Wikipedia: “A bandpass filter is a device that passes frequencies within a certain range and rejects (attenuates) frequencies outside that range.” Bandpass filters are commonly used in wireless transmitters and receivers to prevent transmitters from interfering with other stations and to allow signals within selected frequency ranges to be decoded, while preventing signals at unwanted frequencies from passing through to the receiver. In astronomy, bandpass filters are used to allow only certain portion of light spectrum pass through to an instrument. They have several other applications in neuroscience, digital signal processing, atmospheric sciences, etc.

The following example is taken from a Mathworks web-page and it shows how to analyze the time response of a common RLC bandpass filter circuit using the current available functions of the Control Systems Toolbox in R. The attenuation properties of the filter circuit, G2 can be ascertained by simulating how this filter transforms sine waves with frequency 0.9, 1, and 1.1 rad/s.

Figure 2: Bandpass RLC Filter Circuit.

For this example, we will try to show how the following could be achieved using the developing control library:

Creating a transfer function model
Converting a transfer function to state-space representation
Converting a continuous-time model to discrete-time model
Performing Linear simulation using lsim
Obtaining step response using lsim

The following values are given R = 1, L = 1, C = 1, R2 = 20

The transfer function for this circuit is:

\[ G(s) = \frac{V_output}{V_input} = \frac{s/(RC)}{s^2 + s/(RC) + 1/(LC)} \]

On the R Console (or in an R script), after installing the control package, the following code obtains the time response of the circuit:

library(control)

## 
## Attaching package: 'control'

## The following object is masked from 'package:base':
## 
##     append

R <- C <- L <- 1
R2  <-  20
t  <-  seq(0,250,0.05)
G1 <-  tf(c(1/(R*C), 0), c(1, 1/(R*C), 1/(L*C) ) )
G2  <-  tf(c(1/(R2*C), 0), c(1, 1/(R2*C), 1/(L*C) ) )
print(G1)

## 
## y1:
##              s 
##   - - - - - - - - -
##      s^2 +  s + 1 
## 
## 
## Transfer Function: Continuous time model

tf2ss(G1) # example: convert the transfer function to state-space representation

## 
##  sys.A = 
## 
##      x1   x2
## x1   -1   -1
## x2    1    0
##    
## sys.B = 
## 
##      u1
## x1    1
## x2    0
##    
## sys.C = 
## 
##      x1   x2
## y1    1    0
##    
## sys.D = 
## 
##      u1
## y1    0
##    
## 
##  State-Space system: Continuous time model

c2d(G1, 0.1) # example: convert the system G1 from continuous-time to discrete-time model

## 
## y1:
##   0.09500408 z^1 - 0.09500408 
##   - - - - - - - - - - - - - - - - -
##      z^2 - 1.895329 z + 0.9048374 
## 
## 
## Sample Time = 0.1 
## Transfer function: Discrete time model

response <- lsim(G2,sin(t),t)
par(mfrow = c(3,1))
plot(t, sin(t), type = "l", xlab = "Time, secs", ylab = "Amplitude", main = "Linear Simulation Response - w = 1", col = "lightgray")
lines(t, response$y, type = "l", col = "blue", lwd = 2)
grid(5,5, col = "lightgray")

response2 <- lsim(G2, sin(0.9*t), t)
plot(t, sin(t), type = "l", xlab = "Time, secs", ylab = "Amplitude", main = "Linear Simulation Response - w = 0.9", col = "lightgray")
lines(t, response2$y, type = "l", col = "blue", lwd = 2)
grid(5,5, col = "lightgray")

response3 <- lsim(G2, sin(1.1*t), t)
plot(t, sin(t), type = "l", xlab = "Time, secs", ylab = "Amplitude", main = "Linear Simulation Response - w = 1.1", col = "lightgray")
lines(t, response3$y, type = "l", col = "blue", lwd = 2)
grid(5,5, col = "lightgray")

Figure 1: Figure 3: R Plot for Linear Simulation Response of RLC bandpass circuit

The output of the simulation displayed in Figure 3 shows a pure sine wave (in light-gray color) behind the wave response from our linear simulation (in blue). At w = 1 rad/s, after the transients have died off, the wave remains a pure sine wave – no attenuation observed. However, the waves at 0.9 and 1.1 rad/s are significantly attenuated.

Please get the code here: https://github.com/benubah/controldev/blob/master/R/lsimExample.R

To obtain a step response for the RLC filter circuit using lsim as shown in Figure 4, the following code is used:

dims <- dim(as.matrix(t))
   u <- matrix(rep(1,dims[1]), dims[1], dims[2])
   stepresponse <- lsim(G2, u, t)
plot(t, stepresponse$y, type = "l", xlab = "Time, secs", ylab = "Amplitude", main = "Step Response - G2", col = "red", lwd = 2)
grid(7,7, col = "lightgray", lwd = 2)

Figure 2: Figure 4: R Plot for Step Response of the RLC bandpass circuit

Next Coding Phase

In the next coding stage, we plan to attempt creating functions that would perform the following:

Evaluate transfer function expressions in the s-domain within R as part of the framework.
Control system block diagram algebra functions – system interconnections

Challenges

The expected challenge for this project is to accurately set up a framework that defines data structures and how functions operate within the framework in such a manner that would be attractive (due to simplicity, reusability and extensibility) to prospective users and contributors.

To overcome this challenge, through the assistance of my mentors, we have examined the data architectures and plotting layout of other open-source scientific languages, and from these, designed a plan and framework that would integrate reasonably with the architecture of the R language while remaining open to improvements.

Dependencies

The Control Systems Toolbox for the R language is expected to depend on R packages like, pracma, signal and expm.

Feedback

We would be glad to hear from the R community – feel free to get in touch with any suggestions, opinions or comments.

Conclusion

In this article, we have introduced the Control Systems Toolbox GSoC 2017 project, its current state, benefits and practical examples showing some functions that are currently available. We have also listed the future coding plans for the next GSoC coding phase. We hope to be able to update the R community as things develop.

Control Systems on R Views

Control Systems Toolbox – System Interconnection

Introduction

Laplace Transform

Transfer Function Expressions

Electric Motor Example

Open-Loop Response of the electric motor

Closed-Loop Response of the electric motor

System Interconnections

Append Connection using the append function

Series Connection using the series function

Parallel Connection using the parallel function

Feedback Connection using the feedback function

General block diagram building using connect function

Case Study – Unmanned Free Swimming Submersible (UFSS) Vehicle

Rationale:

Solution using series, parallel and feedback

Solution via algebraic operations

Feedback

Conclusion

References

Control Systems Toolbox in R - a GSoC 2017 Project

Introduction

Example of a Control System

Control Systems R-GSoC Project

Benefits of a Control Systems Toolbox in R

Current State of the Project

Practical Example

Analyzing the Time Response of a Bandpass RLC Filter Circuit

Next Coding Phase

Challenges

Dependencies

Feedback

Conclusion

Append Connection using the `append` function

Series Connection using the `series` function

Parallel Connection using the `parallel` function

Feedback Connection using the `feedback` function

General block diagram building using `connect` function

Solution using `series`, `parallel` and `feedback`