Linearization Around An Operating Point Computer Science Essay

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Control system is a huge area that requires a lot of knowledge and study in order to have a full understanding of the dynamics of any required system and the recommended ways to control it. The area of control systems has changed drastically during the last decades where different approaches have been developed to solve different control problems. Through undergraduate study, the area of linear control system has been the main focus where many methods and techniques are introduced to solve Linear Time Invariant (LTI) systems such as Root Locus, Bode Plot and Nyquist Criteria. However, in the real world there are many practical systems that behave in a nonlinear pattern. Thus, understanding the area of nonlinear control systems is strongly needed. Nonlinear control systems deal with the design and analysis of nonlinear systems, time variant or both. This report will go through the definition of nonlinear control, the principles and theorems, the history and development and this area and some simple and simulation examples to illustrate the ideas.

2.0 Definition

The nonlinearity of a system comes from different sources. Some physical quantities such as vehicle's velocity or electrical signals have lower and upper limits. When these limits are reached, linearity is lost. There are also some systems that are nonlinear by nature such as Thermal, Fluidic, automotive systems, aerospace systems and many others. Moreover, many mechanical systems are subject to many nonlinear elements and factors such as relays and frictions. In addition, nonlinearity might occur when wide operating points are taken into consideration during controlling a specific system.

There are many factors that enable specialists to distinguish between linear and nonlinear control systems. Some of the main differences between linear and nonlinear control systems are:

A Linear control system is defined by Matrix representation ẋ= Ax (A is a matrix) while a nonlinear control system is defined by nonlinear differential equations, ẋ = f(x).

The principle of homogeneous and superposition must be satisfied in a system in order to be a linear system, otherwise it is a nonlinear system.

The stability of a nonlinear system depends on several factors like the initial conditions and limit cycle [1] ; unlike stability of linear systems that depend on the location of the poles of the system.

If the input is bounded, the output definitely will be bounded in a linear system. Also if the input is sinusoidal, the output will also be a sinusoidal with the same frequency in a linear system. This situation is not valid for a nonlinear system

Nonlinear control systems require more precise mathematical analysis than linear control systems in the process of analysis and design.

3.0 Underlying principles and Theorems

There are several well-developed techniques for analyzing nonlinear feedback systems as well as several control design approaches which either linearize the system to enable the usage of common linear design techniques or introduce an auxiliary nonlinear feedback that transforms the system into an equivalent linear system. This section will discuss some of the common methods used in analysis and design.

3.1 Linearization around an operating point

This method is used effectively when the nonlinearity portion is weak around the operating point or for small range of operation. Linearization method is commonly used because of its simplicity. Linear approximation can be determined either analytically or numerically. There are certain procedures that need to be followed in order to linearize a nonlinear system around an operating point:

Select a range of operating points for a nonlinear system

Find a value that relates the change of input to the change of output

Generate a linear equation for the system and use it in the analysis whether by Laplace Transform or other methods.

The following figure shows an example of linearization between the pressure drop and flow around an operating range. nonlinear.png

Fig3.1 Linearization around an operating point

Moreover, it is possible to linearize a system by using the Taylor series expansion.

The final expression of the linearized model is

This can be rewritten in the state space form

Where A and B are the Jacobian matrices and X0, U0 are the operating points

Example: Given the system

We want to linearize the system at the equilibrium point (when the derivatives of X1 and X2 equal to 0)

Then the equilibrium point for the input u=0 becomes

There are two possible values, if we choose X0 = , the linearized system is

This method has an advantage of utilizing the linear techniques in analysis and design. However, One of the drawbacks of the Linearization method is that it does not insure the stability when a nonlinear system has unstable zero dynamics.

3.2 Input to output linearization

This approach relies on a basic mathematics technique "the change of variables" in order to transform a nonlinear system into an equivalent linear system from the input to the output. This is achieved by obtaining a suitable control input and then designing a stabilizing outer state feedback loop.

Example: Given the nonlinear system

If we choose a variable Ï… such that

This yield

This is linear from the artificial input Ï… to the output y.

This approach resembles the "pole-zero cancellation" except for nonlinear systems and it is suitable for simple cases. However, it has some downsides like sensitivity to parameters as all state variables should be measureable to insure they stay stable or bounded. In addition, this method is only applicable for stable processes

3.3 Lyapunov Redesign Method

It is a method that depends on Lyapunov function in order to design a stabilized nonlinear feedback control system. Lyapunov function is a scalar function that is used normally to insure the stability of ordinary differential equation. There is no general rule to find Lyapunov function but it varies according to the situation. This method focuses mainly on designing a control law for the nonlinear system such that the derivative of the chosen Lyapunov function is negative for all possible initial conditions, disturbances and other uncertainties. The resulting control law is usually a nonlinear state feedback law. The stabilizing feedback control law U(x) is given by Sontag's formula

With the assumption of having a nonlinear system linear in the input

Example: consider the following nonlinear system of the form

The Lyapunov function for this example is given by

Applying Sontag's formula, The assumption f(0) = 0 is satisfied, also

This gives

Now if we substitute in U(x), the control signal becomes

The advantages of this method are that it insures the stability of the closed loop system and provides robustness for the system. On the other hand, obtaining a Lyapunov function for the system is quite difficult.

3.4 Gain Scheduling

This method depends basically on using a group of linear controllers. Each of these linear controllers provides control and stability for the system at different operating conditions. The choice of a proper linear controller for a specific range of operating conditions depends on variables that are measured such as controller output, system output and state variables [xx]. gain scheduling.png

Fig3.2 Gain Scheduling block diagram

Gain scheduling is commonly used in the aerospace industry. The behavior of an airplane changes relative to its speed, so the choice of linear controllers depends on a function of plane's speed. Since this method deals with linear controllers, linear techniques are applicable. However, this method requires an engineering process to find a relation between the measured variables and the controller's parameters.

3.5 Fuzzy Control

It is a process that operates by changing inputs and outputs of a nonlinear control system in order to have sufficient control on the required system. There are specific rules that organize the change of inputs to produce the desired output. These rules are generated by experts or operators. There are three main stages for this operation. The first stage is "fuzzification". In this stage, the inputs and measurements are linked to a specific category with an indication of how strongly they link to this category. According to the rules, the category of outputs of the system that corresponds to the selected inputs will be chosen. This process is called "inference". The next process is "defuzzification". In this process, a value will be given according to the category of the output. The following graph represents the mechanism of fuzzy control:


Fig3.3 Fuzzy control process


It is required to control the speed of a motor by changing the input voltage. If for some reason the motor runs faster or slower than a defined set point we need either to slow it down or speed it up to go back to the defined set point If we define

The inputs: Too slow, Just Right, Too fast

The output action: Less voltage (Slow down), No change, More voltage (Speed up)

The rule base:

If the motor is running too slow, then more voltage

If the motor speed is about right, then no change

If the motor speed is too fast, then less voltage

Now, we define the membership functions for inputs and output variable as shown in the figure below

Fig3.4 Membership functions

Now, suppose the speed increases from a set point of 2420 to 2437.4 rpm

Fig3.5 Motor speed above set point

From fig3.5, we see that this intersects the triangles of rule 2 and 3 at 0.4and 0.3. So we change the height of the triangles of the input voltage

Fig3.6 Motor voltage

Now, area of "Not much change" triangle is 0.008 and area of "Slow down" triangle is 0.012. Then the output is found by calculating the point at which a pivot would balance the two triangles.

By solving,

0.008D1 = 0.012D2 ….. (1)

D1 + D2 = 0.04 ……. (2)

Solving (1) and (2) we get

D1 = 0.024 , D2 = 0.016

Thus the voltage required is 2.40-0.024 = 2.376 V

An advantage of this method is that there is no specific design for it as it depends on the experience and intelligence of who will provide the rules [xx]. However, this method does not insure stability of the system. Moreover it requires many tuning parameters and it depends on trial and error for optimization [xx].

4.0 History and major contributors

In the Early 1990's, nonlinear control systems were being successfully applied, for example the Tirrill regulator and the Centrifugal governor but without a major theoretical understanding behind it. However, some physicists and mathematicians had interest in developing techniques of solving nonlinear differential equations that occurred in modeling problems in such areas as mechanical vibration and electronic oscillators.

The main milestones of the history of nonlinear control development from 1883 until the early 1960's and the major contributors are summarized in the following timeline:


Lindstedt and Poincare examined solutions for second-order nonlinear differential equation limit cycle.


The phase plane One of the most useful early methods was introduced by Poincare which was used to study the second-order nonlinear differential equations. This approach proved its value for control system design from the late 1930s onward [2].

Following Poincare, the phase plane approach received many contributions like the conditions of having limit cycles [2].

Lyapunov original work on nonlinear systems stability theory was published in 1892 in Russia however it has been neglected until the end of 1950's [2].


Bush and his colleagues of MIT carried out a major research regarding the effect of nonlinearities in control systems. Moreover, they studied nonlinear differential equations considering differential analyzers [2].

The war between the European powers motivated the development of servomechanism to design accurate fire control systems. The main methods used were the phase plane and the describing function with relay systems being studied by both methods [2].


Following Duffing's seminal work, oscillations of the second order nonlinear differential equations were studied using several methods of harmonic balance techniques [2].


Minorsky made it clear in a published paper that Lyapunov's method might be viable to solve nonlinear control problems [2].


Hamel and Tsypkin developed techniques for precise measurement of the limit cycle of relay based systems after realizing the independent relation between the relay output and input once it is switched [2].


Further studies were carried on Different nonlinear effects of second order systems such as: the effects of torque saturation [2].

Simulation played a major role in the study of nonlinear systems however it took a lot of time for set up, it was slow and the equipment used were bulky, noisy and expensive [2].


The describing function theory was extended to examine more complex behavior of nonlinear systems such as determining the forced harmonic response of the system and the stability of a possible limit cycle[2] 


A new method for obtaining solutions for second order nonlinear differential equations was introduced by Van der Pol, Krylov and Bogoliubov using averaging methods [2]. 

5.0 Simulation examples using MATLAB

5.1: Linearization around an operating point

If we consider the system given in example section 3.1

This system can be represented in Simulink as follows

Fig5.1 Simulink model of example1

Then the equilibrium point and the linearized model can be found through a MATLAB program as follows

clear all


x0=[1;1]; %initial values of states

u0=0; %initial value of input

file = 'DCS_EX1'; %Simulink model

[x,u]=trim(file,x0,u0); %find the equilibrium points

[a,b,c,d]=linmod(file,x,u) %find the SS linearized model

Then the result is similar to the answer obtained previously which confirms the linearization

5.2: Fuzzy control

Consider the example of control of motor speed in section 3.7. Using the fuzzy control toolbox functions we can define the membership functions (input/output) and the rule base as follows

%Motor speed Fuzzy control

clear all


a=newfis('Motorspeed'); %New fuzzy inference system

%Define the input

a=addvar(a,'input','MotorSpeed',[2304 2540]);

a=addmf(a,'input',1,'TooSlow','trimf',[2304 2362 2420]);

a=addmf(a,'input',1,'AboutRight','trimf',[2390 2420 2450]);

a=addmf(a,'input',1,'TooFast','trimf',[2420 2480 2540]);

%Define the output

a=addvar(a,'output','Voltage',[2.32 2.48]);

a=addmf(a,'output',1,'LessVoltage','trimf',[2.32 2.36 2.4]);

a=addmf(a,'output',1,'Nochange','trimf',[2.38 2.4 2.42]);

a=addmf(a,'output',1,'MoreVoltage','trimf',[2.4 2.44 2.48]);

%Define the rules

ruleList=[ ...

1 3 1 1

2 2 1 1

3 1 1 1 ];



fis = readfis('Motorspeed');

disp('Desired voltage')

disp(' ')

disp(evalfis(2437.4,fis))%Evaluating the required voltage

Which gives the same result as found previously

6.0 Recent development

In recent years, following the several developments and advances in computer-control hardware and software capabilities, there has been a renewed interest in developing improved control and identification strategies for non-linear systems. One of the main issues that held back fast development in the past is the lack of tools to perform some complex mathematical computations involved which later were overcame by the developed computers and software. Moreover, there are other factors that played a role in driving this development such as:

Advances in non-linear systems theory

The development of efficient identification methods for empirical non-linear models, and their widespread availability in commercial software packages

Some of the nonlinear control design methods that have received considerable attention includes: non-linear model predictive control, Fuzzy control and Sliding mode control.

Model Predictive Control currently enjoying quite much popularity in the industry and especially in the field of process control. The success of this approach with linear systems motivated researchers to extend it to solve nonlinear control problems. The nonlinear MPC is similar to the linear MPC in terms of control problem formulation however for nonlinear MPC a nonlinear dynamic model is used to predict future process behavior. The control algorithm is obtained by solving a constrained control problem at each sampling instant. Hence, when looking to use the nonlinear MPC one should take into consideration if the benefits of this approach will justify the great computational power needed in comparison with other techniques.

7.0 Conclusion

In conclusion, this report provided a brief picture about the topic of nonlinear control systems. As undergraduate students, many information and knowledge were learnt from looking up through this topic. Digital Control System (DCS) course was a great help in understanding some points in nonlinear control system. Many techniques learnt in DCS course were used in this research topic such as state space representation, state variables and simulation diagram. DCS course covers only the control of linear discrete time systems. After doing this research on nonlinear control, it was found that there are also nonlinear discrete time systems that need to be controlled. However, it is not an easy task to obtain linear discrete time systems for nonlinear continuous systems since there is no direct relation between the input and output of nonlinear systems. Also, designing a nonlinear discrete controller requires using advanced techniques. Thus, the control of nonlinear discrete-time systems requires having a combined knowledge of nonlinear and digital control theory.