Embedded controllers

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Chapter 1

Embedded System: - It is simply the brain of most of the electronics based systems to access, process, stores and controls the data. It is a combination of both software and hardware which creates a dedicated computer system that perform particular, pre-defined tasks those are embedded within the device it controls or if it is part of a larger device like washing machine, mobile phone, car engine control and TV remote control..

Embedded systems can be found in cheap products such as digital watches and also in expensive and complex products like automobiles. Embedded systems control many of the common devices that are currently used in the current world.


I have selected the Netgear ADSL modem/router as an example of Embedded System. Figure (1) shows the various parts of a Netgear DG632 ADSL Modem/router. It acts as a router between an Ethernet port and an ADSL broadband internet connection, and provides typical home router features, such as DHCP.

The labelled parts and their brief operations are described below:

  1. Telephonedecoupling electronics for ADSL: The asymmetrical digital subscriber lines (ADSL) interface is a communication protocol used to transmit high-speed data potentially voice data over the existing two-wire telephone line infrastructure. The data rates are asymmetrical. Data originating at the telephone company's Central Office (CO) is sent 'downstream' to the customer premise equipment (CPE) or remote terminal (RT) end of the line at data rates up to 1.5 Mbits/sec. Data sent from RT to CO in the reverse direction is referred to as 'upstream' data and has a maximum data rate of 150 kBits/sec.
  2. Multicolour LED: Displays Network Status.
  3. Single colour LED: Displaying USB status.
  4. Main processor (A TNETD7300GDU, a member of Texas Instruments AR7 product line): This is the Central Processing Unit (CPU). Its function is to execute a sequence of stored instructions called a program. The program is already stored in the computer memory and is represented by a series of numbers. There are mainly four steps used in an operation fetch, decode, execute and writeback.
  5. JTAG (Joint Test Action Group) test and programming port: It is used with other devices for programming and debugging.
  6. RAM, a single ESMT M12L64164A 8 MB chip: Random access memory (RAM) is a form of computer data storage. With RAM, any piece of data can be accessed and returned in a constant time regardless of its physical location. It allows the access of stored data in any order (randomly).
  7. Flash Memory, obscured by sticker: It is a non-volatile memory that can be electrically erased and reprogrammed. It is non-volatile means no power is needed to maintain the information stored in the chip.
  8. Power supply regulator: It is an electrical regulator designed to maintain a constant output power supply voltage.
  9. Main power supply fuse: It is used to protect the circuit against high currents or input power supply. When there are high currents in the incoming power supply, it melts and breaks down the circuit and thus protecting the entire equipment.
  10. Power connector: It is an electrical connector solely designed to carry a significant amount of electrical power, usually as DC or low-frequency AC.
  11. Reset Button: It resets the device.
  12. Quartz crystal: Quartz crystals have piezoelectric properties, which develop an electric potential upon the application of mechanical stress.
  13. Ethernet Port: It is just simply an Ethernet port to connect.
  14. Ethernet transformer Delta LF8505: It steps down the voltage applied to the requirement of the router.
  15. KS8721B Ethernet PHY transmitter receiver: A transceiver is a device that has both a transmitter and a receiver which is combined and shares common circuitry. So it transmits and receives the signals.
  16. USB port: Universal Serial Bus is used to connect many peripherals using a single standardized interface socket and to improve the plug and play capability by allowing hot swapping, that is, by allowing devices to be connected and disconnected without turning off the device and can also be used for providing power to low-consumption devices without the need for an external power supply and allowing many devices to be used without requiring manufacturer specific.
  17. Telephone (RJ11) port: It translates TCP (Transmission Control Protocol) or UDP (User Control Protocol) communications made between hosts on a private network and hosts on a public network. It allows a single public IP address to be used by many hosts on a private network, which is usually a Local Area Network. It modifies IP packets as they pass through it. The modifications make all the packets which it sends to the public network from the multiple hosts on the private network appear to originate from a single host on the public network.
  18. Telephone connector fuses: These are just the fuses connecting telephone connectors to the outside circuit and used for the protecting purposes of the circuit against any unwanted change in incoming power supply. Embedded systems are totally based on the idea of the microcontroller, a single integrated circuit that contains all the technology required to run an application. Microcontrollers make integrated systems possible by adding different features together into what is effectively a complete computer on a chip, including:

Ø Central Processing Unit

Ø Input/ Output interfaces (such as serial ports)

Ø Peripherals (such as timers)

Ø ROM, EEPROM or Flash memory for program storage

Ø RAM for data storage

Ø Clock generator

Incorporating all of these features into a single chip it is possible to greatly reduce the number of chips and wiring necessary to control an electronic device, dramatically reducing its complexity, size and cost.

Applications of Embedded Systems

There are infinite uses for embedded systems in consumer products with as the inclusion of microcontrollers to replace general purpose microprocessors. They have decreased the unit manufacturing costs significantly and end user prices dramatically, resulting in increased sales and an improved profit margin.

The use of embedded system for example a modern automobile may contain dozens of embedded systems to control a wide range of processes within the car, ranging from brake balance control to air conditioning to the ignition system. Microcontrollers are designed to deliver maximum performance for minimum size and weight. They are designed to perform repeated functions for long periods of time without failing or requiring service.


Chapter 2

Fourier Series: - Any finite power periodic signal f(t) with a period of T seconds is expressed as a sum of sine waves and cosine waves which is known as trigonometric Fourier series.

The fundamental frequency of f(t) is

radian per second or 1/T Hz.

The signal as a whole contains upto n harmonics, a second harmonic at 2/T Hz, a third harmonic at 3/T Hz, etc. Also, we can express the above equation in terms of angle in radian. If ƒ(x) denotes a function of the real variable x and is periodic, of period 2π, and ƒ(x+2π) = ƒ(x), for all real numbers x. We can write such a function as an infinite sum, or series of simpler functions. We start with an infinite sum of sine and cosine functions on the interval [−π,π], which is given below.

The coefficients an and bn are calculated directly from the signal using the following two equations:

A function f is said to be even if

f(-x) = f(x) for all x Ñ” R

and odd

if f(-x) = -f(x) for all x Ñ” R

Recall the product of two even functions is even, the product of two odd functions is even and the product of an even and an odd function is odd.

Compare the multiplication of even and odd functions to the addition of even and

odd integers.

If f(x) is an odd function then

While if f is an even function, then

So any arbitrary equation can be solved out using Fourier series.

Fourier Transform:

Fourier transform moves the data sample from the time domain to the frequency domain; which allows a seemingly random sample to be broken into its major frequencies. It is used in all kinds of signal processing. Fourier series is an infinite sum of sine and cosine functions that can be used to characterize any periodic function. The individual sine and cosine functions will have different frequencies. This is what allows the transformation to the frequency domain.

Most often, the unqualified term Fourier transform refers to the transform of functions of a continuous real argument, such as time (t). In this case the Fourier transform describes a function Æ’(t) in terms of basic complex exponentials of various frequencies.The discussion of Fourier Series above dealt with functions periodic on the interval

This can be generalised to functions periodic on any interval.

Functions of arbitrary periodicity

Functions with a periodicity of

can be decomposed into contributions from

Which are periodic over a period of 2L.

The Fourier series may then be written as:

or in exponential form can be written as


This allows a function of arbitrary period to be analysed.

Applications of Fourier Transform:

The Fourier transform has many applications, in fact any field of physical science that uses sinusoidal signals, such as engineering, physics, applied mathematics, and chemistry, will make use of Fourier series and Fourier transforms. Here are some examples from physics, engineering, and signal processing.

1) Communications:

According communications theory the signal is usually a voltage, and Fourier theory is essential to understanding how a signal behaves when it passes through filters, amplifiers and communications channels. Even discrete digital communications which use 0's or 1's to send information still have frequency contents. This is perhaps easiest to grasp in the case of trying to send a single square pulse down a channel.

The field of communications spans a range of applications from high-level network management down to sending individual bits down a channel. The Fourier transform is usually associated with these low level aspects of communications.

Let's take simple digital pulse that is to be sent down a telephone line, it will ideally look like this:

If we take the Fourier transform of this to show what frequencies make up this signal we get something like:

This means that the square pulse is a sum of infinite frequencies. However if the telephone line only has a bandwidth of 10MHz then only the frequencies below 10MHz will get through the channel. This will cause the digital pulse to be distorted e.g.

This fact has to be considered when trying to send large amounts of data down a channel, because if too much is sent then the data will be corrupted by channel and will be unusable.

Extending the example of the telephone line, whenever you dial a number on a " touch-tone " phone you hear a series of different tones. Each of these tones is composed of two different frequencies that add together to produce the sound you hear.

The Fourier transform is an ideal method to illustrate this, as it shows these two frequencies e.g.

2) Optics:

According to electromagnetic theory, the intensity of light is proportional to the square of the oscillating electric field which exists at any point in space. The Fourier transform of this signal is the equivalent of breaking the light into its component parts of the spectrum, a mathematical spectrometer.

One simple example application of the fourier transform in optics is the diffraction of light when it passes through narrow slits. The ideas described here can be equally applied to acoustic, x-ray, and microwave diffraction, or any other form of wave diffraction.

Suppose we have the experiment shown in the picture below where we have a slit in an opaque screen.

If the gap is of width W then the transmission function will be

This square transmission function is equivalent to a filter response such as an ideal band-pass filter. Light is only transmitted between the edges of the slit.

The Fourier transform of this gives us its frequency content,

Which is used to calculate the diffraction intensity at various angles from the normal. The resultant intensity is given by the square of the fourier transform:

In other words, on the screen there is a bright region based around the slit which rapidly falls off to either side of the slit. Roughly what you would expect if you shone a light through a hole in a wall.

If the screen is altered to give us two parallel slits, with width W and distance D apart, then the transmission function is

If we the use this, and the Fourier transform to calculate the diffracted intensity we get
This represents closely spaced maxima and minima (light and dark bands) that slowly decrease the further from the center of the two slits you go.

It is the Fourier transform used to calculate the result of a Young's double slit experiment.



Chapter 4

Differential Equations:-

A differential equation is an algebraic equality which involves either differentials or derivatives. Consider the class of differential equations of the form:

Where coefficients ai and bi are constants, u=u(t) the input is a known time function and y=y(t) the output is the unknown solution of the equation. Generally m≤n, and n is the order of the differential equation. The set of initial conditions is

Let us consider a differential equation

With initial conditions

So the solution of above differential equation is the sum of the free response and the forces response where the first term (free response) result from initial conditions and second term (forces response) result from the input sequence.

The free response is given by

By putting the initial conditions, we get

And forces response calculated with convolution and is given by

By solving, we get

So the total response is given by

Difference Equations:

A difference equation is an algebraic equality which involves more than one value of the dependent variable(s) corresponding to more than one value of the independent variable(s). the dependent variables do not involve either differentials or derivatives. Consider the class of difference equations

Where k is the integer-valued discrete-time variable, the coefficients ai and bi are constants, a0 and an are nonzero, the input u(k) is a known time sequence, and output y(k) is the unknown sequence solution of the equation. Since y(k+n) is an explicit function of y(k), y(k+1), ......, y(k+n-1), then n is the order of the difference equation. The time sequence over which a solution is desired , and a set of n initial conditions for y(k) must be specified to obtain a unique solution for y(k). Let k=0, 1, 2.... Then the set of initial conditions is y(0), y(1), ......,y(n-1). For example let us consider a differential equation

With initial conditions x(0)=0 and x(1)=1. By applying z transformation we get

Thus the z-transformation X(z) of the solution sequence x(k) is

Where the first term results from initial conditions and second term results from the input sequence. Their inverse gives free response and forced response respectively. Free response is given by

And forces response is given by

And the total response is given by

Chapter 5

Let's assume that the system got one integrator, first order system and a second order system linked with each other in an open loop.   

Now before we draw the time response of each system, we have to get familiar with some of the terms as explained below

1. Percentage Overshoot:

The overshoot is an indication of the largest error between the reference input and output. It is usually given as a percentage.

Percentage overshoot= OS%=

2. Settling Time: The settling time of any output device is the time elapsed from the application of an ideal instantaneous step input to the time at which the output has entered and remained within a specified error band, usually symmetrical about the final value. Here we consider the error band as 2%.

3. Rise time: The rise time is the time required for system to change from 10% to 100% of its final value.

4. D.C Gain:  It is the final value or the final output of the system of device.

Now in order to draw the time response of the chosen system, we draw the time domain of each element of the system with help of Matlab, and analyse each element of the system.

Integrator :

The Matlab commands for time response a integrator element are given below.

num=[1]; % numerator

den=[1 0]; % denominator

g=tf(num,den) % transfer function

step(g) % plot step response

Here we see that the response of the integrator element is quite fast and keep on increasing with time and the DC gain is infinite.

First Order:-

num=[1]; % numerator

den=[1 2]; % denominator

g=tf(num,den) % transfer function

step(g) % plot step response of first order equation

Here we see the time response of the first order element with the help of Matlab graph and values of the various parameters.

Second Order :

num=[100]; % numerator

den=[1 4 100]; % denominator

g=tf(num,den) % transfer function

step(g) % plotting step response of second order equation gives time reponse

In the above figure, time domain analysis for second order element is shown with the help of Matlab graph and values of the various parameters. We see that response of the second order can be divided into two parts.

1. Steady state performance: The steady state response corresponds to the final behaviour of the system when all the transients have almost died out or the transient behaviour have finished.

2. Transient performance: The transient behaviour corresponds to the initial behaviour of the system before it has begun to settle down into its steady state behaviour.

Chapter 6

The figures below show the time and frequency response of the chosen system.

The figure below shows the frequency and the time response of the first order element as chosen in previous chapter with corresponding Matlab commands.

Frequency response for Integrator (LAG):

num=[1]; % numerator

den=[1 0]; % denominator

g=tf(num,den) % transfer function

bode(g) % bode plot of an Integrator(LAG)

Time response of Integrator:-

num=[1]; % numerator

den=[1 0]; % denominator

g=tf(num,den) % transfer function

step(g) % step response plot time response of an Integrator

Frequency response for first order :

num=[1]; % numerator

den=[1 2]; % denominator

g=tf(num,den) % transfer function

bode(g) % bode plot of first order equation.

Time response for first order :

num=[1]; % numerator

den=[1 2]; % denominator

g=tf(num,den) % transfer function

step(g) % plot step response of first order equation

Frequency response of Second order :

Time response of second order :

The difference between the two responses of a second order can be summarized as below

1) Different matlab commands are used to plot the two different graphs as shown below

num=[100]; % numerator

den=[1 4 100]; % denominator

g=tf(num,den) % transfer function

step(g) % plotting step response of second order equation gives time response

num=[100]; % numerator

den=[1 4 100]; % denominator

g=tf(num,den) % transfer function

bode(g) % plot bode diagram

2) The time response graph plotted shows the behaviour of the system with time at different time intervals, however the bode plot or frequency response shows the behaviour of the system over a range of frequencies.

3) The time response is plotted on the Cartesian coordinate axis, magnitude and time axis, however the bode plot is plotted on semilog Cartesian coordinate axis system whereas the horizontal axis represents a log scale and the vertical axis represents a linear magnitude axis. The bode plot is also comprises of two plots, magnitude and phase plot.

Now let's examine the feature of gain response over the different range of frequencies in bode plot;

Gain in dB


So the equation 6.1 can be written as

4) So we can say that for a second order underdamped system, the gain at natural frequency is inversely proportional to the damping ratio, ξ. As the damping decreases and system becomes more oscillatory, the gain at the natural frequency increases which is also same in time domain analysis but the methods of calculations are totally different.

5) In time domain analysis, the stability is dependent upon the value of damping ratio, however, in frequency domain, it is expressed in terms of gain.

6)In the chosen system, in time domain analysis we can see that the system shows its initial conditions and then settle down, however it do not show about the stability of the system at different frequencies. However, frequency response is showing the value of the gain at the different frequencies and so the stability.

7) In the time response, we can calculate the steady state error( E(s)=GR(s)) to a specified input, the specifications like percentage overshoot, the settling time and the rise time, however no such information is available in frequency response.

8) In time response, the response is divided into two regions called the transient and steady state portions of the response, however in frequency response, it shows the response of the system over a specified range of frequencies.

Chapter 7

Both feedforward plus feedback control can significantly improve performance over simple feedback control whenever there is a major disturbance that can be measured before it affects the process output. In the most ideal situation, feedforward control can entirely eliminate the effect of the measured disturbance on the process output. Even when there are modelling errors, feedforward control can often reduce the effect of the measured disturbance on the output better than that achievable by feedback control alone. However, the decision as to whether or not to use feedforward control depends on whether the degree of improvement in the response to the measured disturbance justifies the added costs of implementation and maintenance. The economic benefits of feedforward control can come from lower operating costs and/or increased stability of the product due to its more consistent quality.

Feedforward control is always used along with feedback control because a feedback control system is required to track setpoint changes and to suppress unmeasured disturbances those are always present in any real process. However here we are going to study feedforward and feedbackward separately and compare their result. Feedforward is a useful complement to feedback. Some of its properties are:

  1. Reduce effects of disturbances that can be measured
  2. Improve response to reference signals
  3. No risk for instability
  4. Design of feedforward is simple but it requires good models
  5. Beneficial to combine with feedback
Response of Feedforward System: Let us consider a feedforward system as shown below where a discrete feedforward G is applied to process H where H=1 and G is given by

Solving the above equation using z-transform, we get

Then the feedforward transfer function is given by

num=[1 0.8 0.5];

den=[1 0.5 0.5];


Response of Feedbackward System :

Here the same discrete function G is applied as a feedbackward to a process H. Then the transfer function is given by

num=[1 0.5 0.5];

den=[1 0.8 0.5];


From the above two figures and looking at values of various parameters like overshoot, system D.C gain and responses of both feedback and feedforward, we can summarized as



Closed loop

Open loop

Acts only when there are deviations

Acts before deviations shows up

Market driven


Robust to model errors for some w

No robust to model errors for all w

Risk of instability

No risk of instability

Feedforward is a nice complement to feedback. Properly used it can improve a control system substantially. Its use is increasing. It requires good models and should always be used together with feedbackrward.

Chapter 8

Positive definite functions:

A function   is positive definite (PD) if

Global Lyapunov stability theorems

Consider a function V(x): Rn → R such that

* with equality if and only if x = 0 (positive definite)

* (negative definite)

Then V(x) is called a Lyapunov function candidate and the system is asymptotically stable in the sense of Lyapunov (i.s.L.). (Note that V(0) = 0 is required; otherwise V(x) = 1 / (1 + | x | ) would "prove" that is locally stable. An additional condition called "properness" or "radial unboundedness" is required in order to conclude global asymptotic stability.)

Stability in terms of Lyapunov stability Theorem:-

Consider an equation, where compared to the Van der Pol oscillator equation the friction term is changed:


so that the corresponding system is

Let us assume

Which is clearly Lyapunov positive definite function. Its derivative is

If the parameter ε is positive, stability is asymptotic for

LaSalle's theorem:

LaSalle's theorem (1960) allows us to conclude G.A.S. (globally asymptotic stability) of a system with only along with an observability type condition

We consider

Suppose there is a function such that

Then the system

Here will apply the Krasovskii-LaSalle principle to establish the local asymptotic stability of a simple system, the pendulum with friction. The differential equation is given by

Where θ is the angle the pendulum makes with the vertical normal, m is the mass of the pendulum, l is the length of the pendulum, k is the friction coefficient, and g is acceleration due to gravity. Let us assume

Using Krasovskii-LaSalle principle, we can show that all trajectories which begin in a ball of certain size around the origin x1 = x2 = 0 asymptotically converge to the origin. We define V(x1, x2) as

This V(x1, x2) is simply the scaled energy of the system. Clearly, V(x1,x2) is positive definite in an open ball of radius π around the origin. Computing the derivative,

Observe that , we could conclude that every trajectory approaches the origin by Lyapunov's second theorem. Unfortunately, and is only negative semidefinite. However, the set

This is simply the set of

S = {(x1, x2) | x2 = 0}

Which does not contain any trajectory of the system, except the trivial trajectory x = 0. Indeed, if at some time t, x2(t) = 0, then because x1 must be less π away from the origin, . As a result, the trajectory will not stay in the set S.

All the conditions of the local Krasovskii-LaSalle principle are satisfied, and we can conclude that every trajectory that begins in some neighborhood of the origin will converge to the origin as .

Central limit theorem :

The central limit theorem is also known as the second fundamental theorem of probability.

Let X1, X2, X3 ... Xn be a sequence of n independent and identically distributed random variables each having finite values of expectation µ and variance σ2 > 0. The central limit theorem states that as the sample size n increases, the distribution of the sample average of these random variables approaches the normal distribution with a mean µ and variance σ2 / n irrespective of the shape of the original distribution.

Let the sum of n random variables be Sn, given by

Sn = X1 + ... + Xn. Then, defining a new random variable

The distribution of Zn converges towards the standard normal distribution N(0,1) as n approaches∞ This is often written as


Is the simple mean. The diagram below shows the application of central limit theorem.

Pictures of a distribution being "smoothed out" by summation (showing original density of distribution and three subsequent summations).(Wikipedia 2009b)

Converse Lyapunov theorems :

A typical converse Lyapunov theorem has the form

* if the trajectories of system satisfy some property

* then there exists a Lyapunov function that proves it

A sharper converse Lyapunov theorem is more specific about the form of the Lyapunov function

Lyapunov functions:- Lyapunov functions are functions which can be used to prove the stability of a certain fixed point in a dynamical system or autonomous differential equations.

* there are many different types of Lyapunov theorems, the key in all cases is to find a Lyapunov function and verify that it has the required properties

* There are several approaches to finding Lyapunov functions and verifying the properties.

One common approach:

* decide form of Lyapunov function (e.g., quadratic), parameterized by some parameters (called a Lyapunov function candidate)

* try to find values of parameters so that the required hypotheses hold

Lyapunov fractal with the sequence AAAABBB ( Wikipedia 2009)

Algebraic stability Criterion: The algebraic stability criterion is of five times. Let's discuss all five one by one for the chosen system.

The Routh Criteria: All roots of the characteristics equation have negative real parts if and only if the elements of the first column of the Routh table have the same sign. Otherwise, the number of roots with positive real parts is equal to the number of change of sign.

Applying this criterion to the assumed characteristic equation

S2              1              100

S              4                0


So we see there is no change of sign in the first column of the Routh table so the chosen characteristic equation is has got real parts and the system as a whole is stable.

Hurwitz Criteria:This criterion is applied using determinants formed from the coefficients formed from the roots of the characteristics equation. All roots of the characteristic equation have negative real parts if and only if ∆i >0, for i=1, 2,...., n. The given characteristics equation is

∆2 =   4            0     

1        100 

∆2 =400

∆1 =  4

Since each determinant is positive, the system is stable.

Nyquist Stability Criterion :

Let's assume that

Po= number of poles (≥) of GH in the RHP (right hand plane) for continuous systems or outside the unit circle for discrete-time systems.

N=total number of CW (clockwise encirclements of (-1, 0) point

Then depending upon Nyquist stability criterion, the closed loop control system whose open loop transfer function is GH is stable if and only if

N= - Po ≤ 0

If N >0, the total number of zeros Zo of 1+GH in the right hand plane for continuous system or outside the unit circle for discrete system, is determined by

Zo=N+ Po

If N≤ 0, the (-1, 0) point is not enclosed by the Nyquist stability plot. Therefore N≤ 0, if the region to the right of the contour in the prescribed direction does not include the (-1, 0) point. In the chosen system N≤ 0 as obvious from the Nyquist graph as plotted below.

If N≤ 0 and Po=0, then the system is absolutely stable if and only if N=0; that is, if and only if the (-1, 0) point does not lie in the shaded region. Also Po=0 because the roots of the characteristics equation of the chosen system are complex conjugate and lie in the left hand plane. So the chosen system is stable.

Nyquist Plot :

The Nyquist plot of the chosen transfer function is plotted with the help of the following commands.

num=[100]; % numerator

den=[1 4 100]; % denominator

nyquist(num,den) % Nyquist plot

So as clear from the plot

N ≤ 0 and Po=0 because the roots of the characteristics equation of the chosen system are complex conjugate and lie in the left hand plane. So the chosen system is stable.

Bode plot:   The bode plot of the given transfer function is plotted with the help of matlab with the use of commands as shown below.

num=[100]; % numerator

den=[1 4 100]; % denominator

g=tf(num,den) % transfer function

bode(g) % plot bode diagram

As the stability of bode plot is given in terms of the gain margin and phase margin as defined below.

Gain Margin :

The number of decibels that |GH (w )| is below 0

db at the phase crossover frequency wp (arg GH(wp ) = - 180°).

The factor by which the gain can be increased before the system becomes marginally stable and is given by infinity in the above graph.

Phase Margin :

The number of degrees arg GH (w ) is above

-180 ° at the gain crossover frequency w 1 (|GH (w 1) = 1).

The additional phase lag that will make the system marginally stable which in the above case is around 90o.

Thus we can say that the system is stable.

Chapter 9