Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector...

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Root Locus Unit 7: Part 1: Sketching the Root Locus Engineering 5821: Control Systems I Faculty of Engineering & Applied Science Memorial University of Newfoundland March 5, 2010 ENGI 5821 Unit 7: Root Locus Techniques

Transcript of Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector...

Page 1: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Root Locus

Unit 7: Part 1: Sketching the Root Locus

Engineering 5821:Control Systems I

Faculty of Engineering & Applied ScienceMemorial University of Newfoundland

March 5, 2010

ENGI 5821 Unit 7: Root Locus Techniques

Page 2: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Root Locus

1 Root Locus

Vector Representation of Complex Numbers

Defining the Root Locus

Properties of the Root Locus

Sketching the Root Locus

ENGI 5821 Unit 7: Root Locus Techniques

Page 3: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Root Locus

The Root Locus is a graphical method for depicting location of theclosed-loop poles as a system parameter is varied.

It is applicablefor first and second-order systems, but also to higher ordersystems.

Changes in a parameter, such as the gain K , affects the location ofthe equivalent closed-loop system poles. Root locus allows us todetermine the movement of these poles as K is varied.

Page 4: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Root Locus

The Root Locus is a graphical method for depicting location of theclosed-loop poles as a system parameter is varied. It is applicablefor first and second-order systems, but also to higher ordersystems.

Changes in a parameter, such as the gain K , affects the location ofthe equivalent closed-loop system poles. Root locus allows us todetermine the movement of these poles as K is varied.

Page 5: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Root Locus

The Root Locus is a graphical method for depicting location of theclosed-loop poles as a system parameter is varied. It is applicablefor first and second-order systems, but also to higher ordersystems.

Changes in a parameter, such as the gain K , affects the location ofthe equivalent closed-loop system poles. Root locus allows us todetermine the movement of these poles as K is varied.

Page 6: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Root Locus

The Root Locus is a graphical method for depicting location of theclosed-loop poles as a system parameter is varied. It is applicablefor first and second-order systems, but also to higher ordersystems.

Changes in a parameter, such as the gain K , affects the location ofthe equivalent closed-loop system poles.

Root locus allows us todetermine the movement of these poles as K is varied.

Page 7: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Root Locus

The Root Locus is a graphical method for depicting location of theclosed-loop poles as a system parameter is varied. It is applicablefor first and second-order systems, but also to higher ordersystems.

Changes in a parameter, such as the gain K , affects the location ofthe equivalent closed-loop system poles. Root locus allows us todetermine the movement of these poles as K is varied.

Page 8: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Vector Representation of Complex Numbers

We will need to represent complex numbers and complex functionsF (s) as vectors.

Typically the complex functions we are concernedwith have the following form:

F (s) =(s + z1)(s + z2) · · ·(s + p1)(s + p2) · · ·

Consider the complex number s + q, where q will stand in foreither a zero or a pole. We can represent s + q as a vector in thecomplex plane. The magnitude and angle of this vector are givenby the complex exponential representation,

s + q = re jθ

Where r is the vector length and θ is the angle from the real-axis.

Page 9: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Vector Representation of Complex Numbers

We will need to represent complex numbers and complex functionsF (s) as vectors. Typically the complex functions we are concernedwith have the following form:

F (s) =(s + z1)(s + z2) · · ·(s + p1)(s + p2) · · ·

Consider the complex number s + q, where q will stand in foreither a zero or a pole. We can represent s + q as a vector in thecomplex plane. The magnitude and angle of this vector are givenby the complex exponential representation,

s + q = re jθ

Where r is the vector length and θ is the angle from the real-axis.

Page 10: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Vector Representation of Complex Numbers

We will need to represent complex numbers and complex functionsF (s) as vectors. Typically the complex functions we are concernedwith have the following form:

F (s) =(s + z1)(s + z2) · · ·(s + p1)(s + p2) · · ·

Consider the complex number s + q, where q will stand in foreither a zero or a pole. We can represent s + q as a vector in thecomplex plane. The magnitude and angle of this vector are givenby the complex exponential representation,

s + q = re jθ

Where r is the vector length and θ is the angle from the real-axis.

Page 11: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Vector Representation of Complex Numbers

We will need to represent complex numbers and complex functionsF (s) as vectors. Typically the complex functions we are concernedwith have the following form:

F (s) =(s + z1)(s + z2) · · ·(s + p1)(s + p2) · · ·

Consider the complex number s + q, where q will stand in foreither a zero or a pole.

We can represent s + q as a vector in thecomplex plane. The magnitude and angle of this vector are givenby the complex exponential representation,

s + q = re jθ

Where r is the vector length and θ is the angle from the real-axis.

Page 12: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Vector Representation of Complex Numbers

We will need to represent complex numbers and complex functionsF (s) as vectors. Typically the complex functions we are concernedwith have the following form:

F (s) =(s + z1)(s + z2) · · ·(s + p1)(s + p2) · · ·

Consider the complex number s + q, where q will stand in foreither a zero or a pole. We can represent s + q as a vector in thecomplex plane.

The magnitude and angle of this vector are givenby the complex exponential representation,

s + q = re jθ

Where r is the vector length and θ is the angle from the real-axis.

Page 13: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Vector Representation of Complex Numbers

We will need to represent complex numbers and complex functionsF (s) as vectors. Typically the complex functions we are concernedwith have the following form:

F (s) =(s + z1)(s + z2) · · ·(s + p1)(s + p2) · · ·

Consider the complex number s + q, where q will stand in foreither a zero or a pole. We can represent s + q as a vector in thecomplex plane. The magnitude and angle of this vector are givenby the complex exponential representation,

s + q = re jθ

Where r is the vector length and θ is the angle from the real-axis.

Page 14: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Vector Representation of Complex Numbers

We will need to represent complex numbers and complex functionsF (s) as vectors. Typically the complex functions we are concernedwith have the following form:

F (s) =(s + z1)(s + z2) · · ·(s + p1)(s + p2) · · ·

Consider the complex number s + q, where q will stand in foreither a zero or a pole. We can represent s + q as a vector in thecomplex plane. The magnitude and angle of this vector are givenby the complex exponential representation,

s + q = re jθ

Where r is the vector length and θ is the angle from the real-axis.

Page 15: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Vector Representation of Complex Numbers

We will need to represent complex numbers and complex functionsF (s) as vectors. Typically the complex functions we are concernedwith have the following form:

F (s) =(s + z1)(s + z2) · · ·(s + p1)(s + p2) · · ·

Consider the complex number s + q, where q will stand in foreither a zero or a pole. We can represent s + q as a vector in thecomplex plane. The magnitude and angle of this vector are givenby the complex exponential representation,

s + q = re jθ

Where r is the vector length and θ is the angle from the real-axis.

Page 16: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Normally we would draw s + q as a vector out of the origin.

s q

s + q

s

-q

s + q

Normal: Complex number drawn fromits own zero:

However, we can recognize s = −q as a zero of s + q and draw thesame vector with its tail at −q (see above right).

Page 17: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Normally we would draw s + q as a vector out of the origin.

s q

s + q

s

-q

s + q

Normal: Complex number drawn fromits own zero:

However, we can recognize s = −q as a zero of s + q and draw thesame vector with its tail at −q (see above right).

Page 18: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Normally we would draw s + q as a vector out of the origin.

s q

s + q

s

-q

s + q

Normal: Complex number drawn fromits own zero:

However, we can recognize s = −q as a zero of s + q and draw thesame vector with its tail at −q (see above right).

Page 19: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

F (s) =(s + z1)(s + z2) · · ·(s + p1)(s + p2) · · ·

We can replace each term s + zi or s + pi with their correspondingcomplex exponential forms:

F (s) =rz1eθz1 rz2eθz2 · · ·rp1eθp1 rp2eθp2 · · ·

=rz1rz2 · · ·rp1rp2 · · ·

eθz1+θz2+···−θp1−θp2−···

The magnitude and phase of F (s) are as follows:

|F (s)| =rz1rz2 · · ·rp1rp2 · · ·

=|s + z1||s + z2| · · ·|s + p1||s + p2| · · ·

∠F (s) = θz1 + θz2 + · · · − θp1 − θp2 − · · ·= ∠(s + z1) + ∠(s + z2) + · · · − ∠(s + p1)− ∠(s + p2)− · · ·

Page 20: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

F (s) =(s + z1)(s + z2) · · ·(s + p1)(s + p2) · · ·

We can replace each term s + zi or s + pi with their correspondingcomplex exponential forms:

F (s) =rz1eθz1 rz2eθz2 · · ·rp1eθp1 rp2eθp2 · · ·

=rz1rz2 · · ·rp1rp2 · · ·

eθz1+θz2+···−θp1−θp2−···

The magnitude and phase of F (s) are as follows:

|F (s)| =rz1rz2 · · ·rp1rp2 · · ·

=|s + z1||s + z2| · · ·|s + p1||s + p2| · · ·

∠F (s) = θz1 + θz2 + · · · − θp1 − θp2 − · · ·= ∠(s + z1) + ∠(s + z2) + · · · − ∠(s + p1)− ∠(s + p2)− · · ·

Page 21: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

F (s) =(s + z1)(s + z2) · · ·(s + p1)(s + p2) · · ·

We can replace each term s + zi or s + pi with their correspondingcomplex exponential forms:

F (s) =rz1eθz1 rz2eθz2 · · ·rp1eθp1 rp2eθp2 · · ·

=rz1rz2 · · ·rp1rp2 · · ·

eθz1+θz2+···−θp1−θp2−···

The magnitude and phase of F (s) are as follows:

|F (s)| =rz1rz2 · · ·rp1rp2 · · ·

=|s + z1||s + z2| · · ·|s + p1||s + p2| · · ·

∠F (s) = θz1 + θz2 + · · · − θp1 − θp2 − · · ·= ∠(s + z1) + ∠(s + z2) + · · · − ∠(s + p1)− ∠(s + p2)− · · ·

Page 22: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

F (s) =(s + z1)(s + z2) · · ·(s + p1)(s + p2) · · ·

We can replace each term s + zi or s + pi with their correspondingcomplex exponential forms:

F (s) =rz1eθz1 rz2eθz2 · · ·rp1eθp1 rp2eθp2 · · ·

=rz1rz2 · · ·rp1rp2 · · ·

eθz1+θz2+···−θp1−θp2−···

The magnitude and phase of F (s) are as follows:

|F (s)| =rz1rz2 · · ·rp1rp2 · · ·

=|s + z1||s + z2| · · ·|s + p1||s + p2| · · ·

∠F (s) = θz1 + θz2 + · · · − θp1 − θp2 − · · ·= ∠(s + z1) + ∠(s + z2) + · · · − ∠(s + p1)− ∠(s + p2)− · · ·

Page 23: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

F (s) =(s + z1)(s + z2) · · ·(s + p1)(s + p2) · · ·

We can replace each term s + zi or s + pi with their correspondingcomplex exponential forms:

F (s) =rz1eθz1 rz2eθz2 · · ·rp1eθp1 rp2eθp2 · · ·

=rz1rz2 · · ·rp1rp2 · · ·

eθz1+θz2+···−θp1−θp2−···

The magnitude and phase of F (s) are as follows:

|F (s)| =rz1rz2 · · ·rp1rp2 · · ·

=|s + z1||s + z2| · · ·|s + p1||s + p2| · · ·

∠F (s) = θz1 + θz2 + · · · − θp1 − θp2 − · · ·= ∠(s + z1) + ∠(s + z2) + · · · − ∠(s + p1)− ∠(s + p2)− · · ·

Page 24: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

F (s) =(s + z1)(s + z2) · · ·(s + p1)(s + p2) · · ·

We can replace each term s + zi or s + pi with their correspondingcomplex exponential forms:

F (s) =rz1eθz1 rz2eθz2 · · ·rp1eθp1 rp2eθp2 · · ·

=rz1rz2 · · ·rp1rp2 · · ·

eθz1+θz2+···−θp1−θp2−···

The magnitude and phase of F (s) are as follows:

|F (s)| =rz1rz2 · · ·rp1rp2 · · ·

=|s + z1||s + z2| · · ·|s + p1||s + p2| · · ·

∠F (s) = θz1 + θz2 + · · · − θp1 − θp2 − · · ·= ∠(s + z1) + ∠(s + z2) + · · · − ∠(s + p1)− ∠(s + p2)− · · ·

Page 25: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

F (s) =(s + z1)(s + z2) · · ·(s + p1)(s + p2) · · ·

We can replace each term s + zi or s + pi with their correspondingcomplex exponential forms:

F (s) =rz1eθz1 rz2eθz2 · · ·rp1eθp1 rp2eθp2 · · ·

=rz1rz2 · · ·rp1rp2 · · ·

eθz1+θz2+···−θp1−θp2−···

The magnitude and phase of F (s) are as follows:

|F (s)| =rz1rz2 · · ·rp1rp2 · · ·

=|s + z1||s + z2| · · ·|s + p1||s + p2| · · ·

∠F (s) = θz1 + θz2 + · · · − θp1 − θp2 − · · ·

= ∠(s + z1) + ∠(s + z2) + · · · − ∠(s + p1)− ∠(s + p2)− · · ·

Page 26: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

F (s) =(s + z1)(s + z2) · · ·(s + p1)(s + p2) · · ·

We can replace each term s + zi or s + pi with their correspondingcomplex exponential forms:

F (s) =rz1eθz1 rz2eθz2 · · ·rp1eθp1 rp2eθp2 · · ·

=rz1rz2 · · ·rp1rp2 · · ·

eθz1+θz2+···−θp1−θp2−···

The magnitude and phase of F (s) are as follows:

|F (s)| =rz1rz2 · · ·rp1rp2 · · ·

=|s + z1||s + z2| · · ·|s + p1||s + p2| · · ·

∠F (s) = θz1 + θz2 + · · · − θp1 − θp2 − · · ·= ∠(s + z1) + ∠(s + z2) + · · · − ∠(s + p1)− ∠(s + p2)− · · ·

Page 27: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

e.g. Evaluate the following complex function when s = −3 + j4,

F (s) =(s + 1)

s(s + 2)

The magnitude and phase of the zero is:√

20∠116.6o .

The pole at the origin evalutes to 5∠126.9o .

The pole at -2 evaluates to√

17∠104.0o .

|F (s)|∠F (s) =

√20

5√

17∠ [116.9o − 129.9o − 104.0o ]

Page 28: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

e.g. Evaluate the following complex function when s = −3 + j4,

F (s) =(s + 1)

s(s + 2)

The magnitude and phase of the zero is:√

20∠116.6o .

The pole at the origin evalutes to 5∠126.9o .

The pole at -2 evaluates to√

17∠104.0o .

|F (s)|∠F (s) =

√20

5√

17∠ [116.9o − 129.9o − 104.0o ]

Page 29: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

e.g. Evaluate the following complex function when s = −3 + j4,

F (s) =(s + 1)

s(s + 2)

The magnitude and phase of the zero is:√

20∠116.6o .

The pole at the origin evalutes to 5∠126.9o .

The pole at -2 evaluates to√

17∠104.0o .

|F (s)|∠F (s) =

√20

5√

17∠ [116.9o − 129.9o − 104.0o ]

Page 30: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

e.g. Evaluate the following complex function when s = −3 + j4,

F (s) =(s + 1)

s(s + 2)

The magnitude and phase of the zero is:√

20∠116.6o .

The pole at the origin evalutes to 5∠126.9o .

The pole at -2 evaluates to√

17∠104.0o .

|F (s)|∠F (s) =

√20

5√

17∠ [116.9o − 129.9o − 104.0o ]

Page 31: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

e.g. Evaluate the following complex function when s = −3 + j4,

F (s) =(s + 1)

s(s + 2)

The magnitude and phase of the zero is:√

20∠116.6o .

The pole at the origin evalutes to 5∠126.9o .

The pole at -2 evaluates to√

17∠104.0o .

|F (s)|∠F (s) =

√20

5√

17∠ [116.9o − 129.9o − 104.0o ]

Page 32: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

e.g. Evaluate the following complex function when s = −3 + j4,

F (s) =(s + 1)

s(s + 2)

The magnitude and phase of the zero is:√

20∠116.6o .

The pole at the origin evalutes to 5∠126.9o .

The pole at -2 evaluates to√

17∠104.0o .

|F (s)|∠F (s) =

√20

5√

17∠ [116.9o − 129.9o − 104.0o ]

Page 33: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

e.g. Evaluate the following complex function when s = −3 + j4,

F (s) =(s + 1)

s(s + 2)

The magnitude and phase of the zero is:√

20∠116.6o .

The pole at the origin evalutes to 5∠126.9o .

The pole at -2 evaluates to√

17∠104.0o .

|F (s)|∠F (s) =

√20

5√

17∠ [116.9o − 129.9o − 104.0o ]

Page 34: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Defining the Root Locus

Consider the following system, designed to track a visual target.

For a unity feedback system such as this we will refer to K1K2s(s+10) as

the open-loop transfer function (if the unity feedback signal werecut the system would be open-loop). Thus, the open-loop polesare at s = 0 and s = −10.

However, depending upon K we will obtain different closed-looppoles, which are the roots of s2 + 10s + K . We can utilize thequadratic formula to obtain these roots for values of K ≥ 0. Wecan then plot the positions of these poles...

Page 35: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Defining the Root Locus

Consider the following system, designed to track a visual target.

For a unity feedback system such as this we will refer to K1K2s(s+10) as

the open-loop transfer function (if the unity feedback signal werecut the system would be open-loop). Thus, the open-loop polesare at s = 0 and s = −10.

However, depending upon K we will obtain different closed-looppoles, which are the roots of s2 + 10s + K . We can utilize thequadratic formula to obtain these roots for values of K ≥ 0. Wecan then plot the positions of these poles...

Page 36: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Defining the Root Locus

Consider the following system, designed to track a visual target.

For a unity feedback system such as this we will refer to K1K2s(s+10) as

the open-loop transfer function (if the unity feedback signal werecut the system would be open-loop).

Thus, the open-loop polesare at s = 0 and s = −10.

However, depending upon K we will obtain different closed-looppoles, which are the roots of s2 + 10s + K . We can utilize thequadratic formula to obtain these roots for values of K ≥ 0. Wecan then plot the positions of these poles...

Page 37: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Defining the Root Locus

Consider the following system, designed to track a visual target.

For a unity feedback system such as this we will refer to K1K2s(s+10) as

the open-loop transfer function (if the unity feedback signal werecut the system would be open-loop). Thus, the open-loop polesare at s = 0 and s = −10.

However, depending upon K we will obtain different closed-looppoles, which are the roots of s2 + 10s + K . We can utilize thequadratic formula to obtain these roots for values of K ≥ 0. Wecan then plot the positions of these poles...

Page 38: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Defining the Root Locus

Consider the following system, designed to track a visual target.

For a unity feedback system such as this we will refer to K1K2s(s+10) as

the open-loop transfer function (if the unity feedback signal werecut the system would be open-loop). Thus, the open-loop polesare at s = 0 and s = −10.

However, depending upon K we will obtain different closed-looppoles, which are the roots of s2 + 10s + K .

We can utilize thequadratic formula to obtain these roots for values of K ≥ 0. Wecan then plot the positions of these poles...

Page 39: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Defining the Root Locus

Consider the following system, designed to track a visual target.

For a unity feedback system such as this we will refer to K1K2s(s+10) as

the open-loop transfer function (if the unity feedback signal werecut the system would be open-loop). Thus, the open-loop polesare at s = 0 and s = −10.

However, depending upon K we will obtain different closed-looppoles, which are the roots of s2 + 10s + K . We can utilize thequadratic formula to obtain these roots for values of K ≥ 0.

Wecan then plot the positions of these poles...

Page 40: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Defining the Root Locus

Consider the following system, designed to track a visual target.

For a unity feedback system such as this we will refer to K1K2s(s+10) as

the open-loop transfer function (if the unity feedback signal werecut the system would be open-loop). Thus, the open-loop polesare at s = 0 and s = −10.

However, depending upon K we will obtain different closed-looppoles, which are the roots of s2 + 10s + K . We can utilize thequadratic formula to obtain these roots for values of K ≥ 0. Wecan then plot the positions of these poles...

Page 41: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus
Page 42: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

The path of the closed-loop poles as K varies is the root locus.

Observations:

for K < 25 the system is overdamped

for K = 25 the system is critically damped

for K > 25 the system is underdamped

Page 43: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

The path of the closed-loop poles as K varies is the root locus.

Observations:

for K < 25 the system is overdamped

for K = 25 the system is critically damped

for K > 25 the system is underdamped

Page 44: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

The path of the closed-loop poles as K varies is the root locus.

Observations:

for K < 25 the system is overdamped

for K = 25 the system is critically damped

for K > 25 the system is underdamped

Page 45: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

The path of the closed-loop poles as K varies is the root locus.

Observations:

for K < 25 the system is

overdamped

for K = 25 the system is critically damped

for K > 25 the system is underdamped

Page 46: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

The path of the closed-loop poles as K varies is the root locus.

Observations:

for K < 25 the system is

overdamped

for K = 25 the system is critically damped

for K > 25 the system is underdamped

Page 47: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

The path of the closed-loop poles as K varies is the root locus.

Observations:

for K < 25 the system is overdamped

for K = 25 the system is

critically damped

for K > 25 the system is underdamped

Page 48: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

The path of the closed-loop poles as K varies is the root locus.

Observations:

for K < 25 the system is overdamped

for K = 25 the system is

critically damped

for K > 25 the system is underdamped

Page 49: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

The path of the closed-loop poles as K varies is the root locus.

Observations:

for K < 25 the system is overdamped

for K = 25 the system is critically damped

for K > 25 the system is

underdamped

Page 50: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

The path of the closed-loop poles as K varies is the root locus.

Observations:

for K < 25 the system is overdamped

for K = 25 the system is critically damped

for K > 25 the system is

underdamped

Page 51: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

The path of the closed-loop poles as K varies is the root locus.

Observations:

for K < 25 the system is overdamped

for K = 25 the system is critically damped

for K > 25 the system is underdamped

Page 52: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

More Observations:

In the underdamped portion the real-part of the pole, σd isconstant. Therefore so is Ts = 4/σd .

The root locus never crosses the jω axis. Therefore thesystem is stable.

Page 53: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

More Observations:

In the underdamped portion the real-part of the pole, σd isconstant. Therefore so is Ts = 4/σd .

The root locus never crosses the jω axis. Therefore thesystem is stable.

Page 54: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

More Observations:

In the underdamped portion the real-part of the pole, σd isconstant. Therefore so is Ts = 4/σd .

The root locus never crosses the jω axis. Therefore thesystem is stable.

Page 55: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

More Observations:

In the underdamped portion the real-part of the pole, σd isconstant. Therefore so is Ts = 4/σd .

The root locus never crosses the jω axis. Therefore thesystem is stable.

Page 56: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Properties of the Root Locus

The transfer function for a general closed-loop system is,

T (s) =KG (s)

1 + KG (s)H(s)

We are concerned with the poles of T (s). We will have a polewhenever KG (s)H(s) = −1. The root locus is the locus of pointsin the s-plane for which this is true. We can express this equalityas follows,

|KG (s)H(s)|∠KG (s)H(s) = 1∠(2k+1)180o k = 0,±1,±2,±3, . . .

A particular s is on the root locus if its magnitude is unity and itsangle is an odd multiple of 180◦. Satisfying both of theserequirements means that s is a closed-loop pole of the system.

Page 57: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Properties of the Root Locus

The transfer function for a general closed-loop system is,

T (s) =KG (s)

1 + KG (s)H(s)

We are concerned with the poles of T (s). We will have a polewhenever KG (s)H(s) = −1. The root locus is the locus of pointsin the s-plane for which this is true. We can express this equalityas follows,

|KG (s)H(s)|∠KG (s)H(s) = 1∠(2k+1)180o k = 0,±1,±2,±3, . . .

A particular s is on the root locus if its magnitude is unity and itsangle is an odd multiple of 180◦. Satisfying both of theserequirements means that s is a closed-loop pole of the system.

Page 58: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Properties of the Root Locus

The transfer function for a general closed-loop system is,

T (s) =KG (s)

1 + KG (s)H(s)

We are concerned with the poles of T (s).

We will have a polewhenever KG (s)H(s) = −1. The root locus is the locus of pointsin the s-plane for which this is true. We can express this equalityas follows,

|KG (s)H(s)|∠KG (s)H(s) = 1∠(2k+1)180o k = 0,±1,±2,±3, . . .

A particular s is on the root locus if its magnitude is unity and itsangle is an odd multiple of 180◦. Satisfying both of theserequirements means that s is a closed-loop pole of the system.

Page 59: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Properties of the Root Locus

The transfer function for a general closed-loop system is,

T (s) =KG (s)

1 + KG (s)H(s)

We are concerned with the poles of T (s). We will have a polewhenever KG (s)H(s) = −1.

The root locus is the locus of pointsin the s-plane for which this is true. We can express this equalityas follows,

|KG (s)H(s)|∠KG (s)H(s) = 1∠(2k+1)180o k = 0,±1,±2,±3, . . .

A particular s is on the root locus if its magnitude is unity and itsangle is an odd multiple of 180◦. Satisfying both of theserequirements means that s is a closed-loop pole of the system.

Page 60: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Properties of the Root Locus

The transfer function for a general closed-loop system is,

T (s) =KG (s)

1 + KG (s)H(s)

We are concerned with the poles of T (s). We will have a polewhenever KG (s)H(s) = −1. The root locus is the locus of pointsin the s-plane for which this is true.

We can express this equalityas follows,

|KG (s)H(s)|∠KG (s)H(s) = 1∠(2k+1)180o k = 0,±1,±2,±3, . . .

A particular s is on the root locus if its magnitude is unity and itsangle is an odd multiple of 180◦. Satisfying both of theserequirements means that s is a closed-loop pole of the system.

Page 61: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Properties of the Root Locus

The transfer function for a general closed-loop system is,

T (s) =KG (s)

1 + KG (s)H(s)

We are concerned with the poles of T (s). We will have a polewhenever KG (s)H(s) = −1. The root locus is the locus of pointsin the s-plane for which this is true. We can express this equalityas follows,

|KG (s)H(s)|∠KG (s)H(s) = 1∠(2k+1)180o k = 0,±1,±2,±3, . . .

A particular s is on the root locus if its magnitude is unity and itsangle is an odd multiple of 180◦. Satisfying both of theserequirements means that s is a closed-loop pole of the system.

Page 62: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Properties of the Root Locus

The transfer function for a general closed-loop system is,

T (s) =KG (s)

1 + KG (s)H(s)

We are concerned with the poles of T (s). We will have a polewhenever KG (s)H(s) = −1. The root locus is the locus of pointsin the s-plane for which this is true. We can express this equalityas follows,

|KG (s)H(s)|∠KG (s)H(s) = 1∠(2k+1)180o k = 0,±1,±2,±3, . . .

A particular s is on the root locus if its magnitude is unity and itsangle is an odd multiple of 180◦. Satisfying both of theserequirements means that s is a closed-loop pole of the system.

Page 63: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Properties of the Root Locus

The transfer function for a general closed-loop system is,

T (s) =KG (s)

1 + KG (s)H(s)

We are concerned with the poles of T (s). We will have a polewhenever KG (s)H(s) = −1. The root locus is the locus of pointsin the s-plane for which this is true. We can express this equalityas follows,

|KG (s)H(s)|∠KG (s)H(s) = 1∠(2k+1)180o k = 0,±1,±2,±3, . . .

A particular s is on the root locus if its magnitude is unity and itsangle is an odd multiple of 180◦.

Satisfying both of theserequirements means that s is a closed-loop pole of the system.

Page 64: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Properties of the Root Locus

The transfer function for a general closed-loop system is,

T (s) =KG (s)

1 + KG (s)H(s)

We are concerned with the poles of T (s). We will have a polewhenever KG (s)H(s) = −1. The root locus is the locus of pointsin the s-plane for which this is true. We can express this equalityas follows,

|KG (s)H(s)|∠KG (s)H(s) = 1∠(2k+1)180o k = 0,±1,±2,±3, . . .

A particular s is on the root locus if its magnitude is unity and itsangle is an odd multiple of 180◦. Satisfying both of theserequirements means that s is a closed-loop pole of the system.

Page 65: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

e.g. Consider the following system,

We will test a couple of points to see if they are closed-loop poles.We evaluate KG (s)H(s) graphically for s = −2 + j3,

θ1 + θ2 − θ3 − θ4 = 56.31◦ + 71.57◦ − 90◦ − 108.43◦ = −70.55◦

Therefore s = −s + j3 is not on the root locus.

Page 66: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

e.g. Consider the following system,

We will test a couple of points to see if they are closed-loop poles.We evaluate KG (s)H(s) graphically for s = −2 + j3,

θ1 + θ2 − θ3 − θ4 = 56.31◦ + 71.57◦ − 90◦ − 108.43◦ = −70.55◦

Therefore s = −s + j3 is not on the root locus.

Page 67: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

e.g. Consider the following system,

We will test a couple of points to see if they are closed-loop poles.

We evaluate KG (s)H(s) graphically for s = −2 + j3,

θ1 + θ2 − θ3 − θ4 = 56.31◦ + 71.57◦ − 90◦ − 108.43◦ = −70.55◦

Therefore s = −s + j3 is not on the root locus.

Page 68: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

e.g. Consider the following system,

We will test a couple of points to see if they are closed-loop poles.We evaluate KG (s)H(s) graphically for s = −2 + j3,

θ1 + θ2 − θ3 − θ4 = 56.31◦ + 71.57◦ − 90◦ − 108.43◦ = −70.55◦

Therefore s = −s + j3 is not on the root locus.

Page 69: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

e.g. Consider the following system,

We will test a couple of points to see if they are closed-loop poles.We evaluate KG (s)H(s) graphically for s = −2 + j3,

θ1 + θ2 − θ3 − θ4 = 56.31◦ + 71.57◦ − 90◦ − 108.43◦ = −70.55◦

Therefore s = −s + j3 is not on the root locus.

Page 70: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

e.g. Consider the following system,

We will test a couple of points to see if they are closed-loop poles.We evaluate KG (s)H(s) graphically for s = −2 + j3,

θ1 + θ2 − θ3 − θ4

= 56.31◦ + 71.57◦ − 90◦ − 108.43◦ = −70.55◦

Therefore s = −s + j3 is not on the root locus.

Page 71: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

e.g. Consider the following system,

We will test a couple of points to see if they are closed-loop poles.We evaluate KG (s)H(s) graphically for s = −2 + j3,

θ1 + θ2 − θ3 − θ4 = 56.31◦ + 71.57◦ − 90◦ − 108.43◦

= −70.55◦

Therefore s = −s + j3 is not on the root locus.

Page 72: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

e.g. Consider the following system,

We will test a couple of points to see if they are closed-loop poles.We evaluate KG (s)H(s) graphically for s = −2 + j3,

θ1 + θ2 − θ3 − θ4 = 56.31◦ + 71.57◦ − 90◦ − 108.43◦ = −70.55◦

Therefore s = −s + j3 is not on the root locus.

Page 73: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

If we test a point and find it is on the RL (e.g. s = −2 + j√

2/2)we may then want to determine the corresponding value of K .

Since |KG (s)H(s)| = 1 on the RL,

K =1

|G (s)H(s)|=

∏pole lengths∏zero lengths

Page 74: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

If we test a point and find it is on the RL (e.g. s = −2 + j√

2/2)we may then want to determine the corresponding value of K .Since |KG (s)H(s)| = 1 on the RL,

K =1

|G (s)H(s)|=

∏pole lengths∏zero lengths

Page 75: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

If we test a point and find it is on the RL (e.g. s = −2 + j√

2/2)we may then want to determine the corresponding value of K .Since |KG (s)H(s)| = 1 on the RL,

K =1

|G (s)H(s)|

=

∏pole lengths∏zero lengths

Page 76: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

If we test a point and find it is on the RL (e.g. s = −2 + j√

2/2)we may then want to determine the corresponding value of K .Since |KG (s)H(s)| = 1 on the RL,

K =1

|G (s)H(s)|=

∏pole lengths∏zero lengths

Page 77: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

If we test a point and find it is on the RL (e.g. s = −2 + j√

2/2)we may then want to determine the corresponding value of K .Since |KG (s)H(s)| = 1 on the RL,

K =1

|G (s)H(s)|=

∏pole lengths∏zero lengths

Page 78: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Sketching the Root Locus

One possibility is to sweep through a sampling of points in thes-plane and test each one for inclusion in the RL.

It is much more preferable to utilize some insights about the RL toidentify its major characteristics and therefore obtain a roughsketch. The following rules will help achieve this...

1. Number of branches:

Consider a branch to be the path that one pole traverses.

There will be one branch for every closed-loop pole.

e.g. Two branches for the previous quadratic example.

2. Symmetry:

The poles for real physical systems either lie on the real-axis orcome in conjugate pairs. Hence...

The root locus is symmetrical about the real axis.

Page 79: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Sketching the Root Locus

One possibility is to sweep through a sampling of points in thes-plane and test each one for inclusion in the RL.

It is much more preferable to utilize some insights about the RL toidentify its major characteristics and therefore obtain a roughsketch.

The following rules will help achieve this...

1. Number of branches:

Consider a branch to be the path that one pole traverses.

There will be one branch for every closed-loop pole.

e.g. Two branches for the previous quadratic example.

2. Symmetry:

The poles for real physical systems either lie on the real-axis orcome in conjugate pairs. Hence...

The root locus is symmetrical about the real axis.

Page 80: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Sketching the Root Locus

One possibility is to sweep through a sampling of points in thes-plane and test each one for inclusion in the RL.

It is much more preferable to utilize some insights about the RL toidentify its major characteristics and therefore obtain a roughsketch. The following rules will help achieve this...

1. Number of branches:

Consider a branch to be the path that one pole traverses.

There will be one branch for every closed-loop pole.

e.g. Two branches for the previous quadratic example.

2. Symmetry:

The poles for real physical systems either lie on the real-axis orcome in conjugate pairs. Hence...

The root locus is symmetrical about the real axis.

Page 81: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Sketching the Root Locus

One possibility is to sweep through a sampling of points in thes-plane and test each one for inclusion in the RL.

It is much more preferable to utilize some insights about the RL toidentify its major characteristics and therefore obtain a roughsketch. The following rules will help achieve this...

1. Number of branches:

Consider a branch to be the path that one pole traverses.

There will be one branch for every closed-loop pole.

e.g. Two branches for the previous quadratic example.

2. Symmetry:

The poles for real physical systems either lie on the real-axis orcome in conjugate pairs. Hence...

The root locus is symmetrical about the real axis.

Page 82: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Sketching the Root Locus

One possibility is to sweep through a sampling of points in thes-plane and test each one for inclusion in the RL.

It is much more preferable to utilize some insights about the RL toidentify its major characteristics and therefore obtain a roughsketch. The following rules will help achieve this...

1. Number of branches:

Consider a branch to be the path that one pole traverses.

There will be one branch for every closed-loop pole.

e.g. Two branches for the previous quadratic example.

2. Symmetry:

The poles for real physical systems either lie on the real-axis orcome in conjugate pairs. Hence...

The root locus is symmetrical about the real axis.

Page 83: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Sketching the Root Locus

One possibility is to sweep through a sampling of points in thes-plane and test each one for inclusion in the RL.

It is much more preferable to utilize some insights about the RL toidentify its major characteristics and therefore obtain a roughsketch. The following rules will help achieve this...

1. Number of branches:

Consider a branch to be the path that one pole traverses.

There will be one branch for every closed-loop pole.

e.g. Two branches for the previous quadratic example.

2. Symmetry:

The poles for real physical systems either lie on the real-axis orcome in conjugate pairs. Hence...

The root locus is symmetrical about the real axis.

Page 84: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Sketching the Root Locus

One possibility is to sweep through a sampling of points in thes-plane and test each one for inclusion in the RL.

It is much more preferable to utilize some insights about the RL toidentify its major characteristics and therefore obtain a roughsketch. The following rules will help achieve this...

1. Number of branches:

Consider a branch to be the path that one pole traverses.

There will be one branch for every closed-loop pole.

e.g. Two branches for the previous quadratic example.

2. Symmetry:

The poles for real physical systems either lie on the real-axis orcome in conjugate pairs. Hence...

The root locus is symmetrical about the real axis.

Page 85: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

Sketching the Root Locus

One possibility is to sweep through a sampling of points in thes-plane and test each one for inclusion in the RL.

It is much more preferable to utilize some insights about the RL toidentify its major characteristics and therefore obtain a roughsketch. The following rules will help achieve this...

1. Number of branches:

Consider a branch to be the path that one pole traverses.

There will be one branch for every closed-loop pole.

e.g. Two branches for the previous quadratic example.

2. Symmetry:

The poles for real physical systems either lie on the real-axis orcome in conjugate pairs. Hence...

The root locus is symmetrical about the real axis.

Page 86: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

3. Real-axis segments:

Consider evaluating the anglular contribution of the open-looppoles and zeros at points P1, P2, P3, and P4 below:

The angular contribution of a pair of complex open-loop poles(or zeros) is zero

The contribution of poles or zeros to the left of the point iszero

Using only the real-axis poles or zeros to the right of the point,we find that the angle sum alternates between 0o and 180o .

On the real axis, for K > 0 the root locus exists to theleft of an odd number of real-axis, finite open-loop polesand/or finite open-loop zeros.

Page 87: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

3. Real-axis segments:

Consider evaluating the anglular contribution of the open-looppoles and zeros at points P1, P2, P3, and P4 below:

The angular contribution of a pair of complex open-loop poles(or zeros) is zero

The contribution of poles or zeros to the left of the point iszero

Using only the real-axis poles or zeros to the right of the point,we find that the angle sum alternates between 0o and 180o .

On the real axis, for K > 0 the root locus exists to theleft of an odd number of real-axis, finite open-loop polesand/or finite open-loop zeros.

Page 88: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

3. Real-axis segments:

Consider evaluating the anglular contribution of the open-looppoles and zeros at points P1, P2, P3, and P4 below:

The angular contribution of a pair of complex open-loop poles(or zeros) is zero

The contribution of poles or zeros to the left of the point iszero

Using only the real-axis poles or zeros to the right of the point,we find that the angle sum alternates between 0o and 180o .

On the real axis, for K > 0 the root locus exists to theleft of an odd number of real-axis, finite open-loop polesand/or finite open-loop zeros.

Page 89: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

3. Real-axis segments:

Consider evaluating the anglular contribution of the open-looppoles and zeros at points P1, P2, P3, and P4 below:

The angular contribution of a pair of complex open-loop poles(or zeros) is zero

The contribution of poles or zeros to the left of the point iszero

Using only the real-axis poles or zeros to the right of the point,we find that the angle sum alternates between 0o and 180o .

On the real axis, for K > 0 the root locus exists to theleft of an odd number of real-axis, finite open-loop polesand/or finite open-loop zeros.

Page 90: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

3. Real-axis segments:

Consider evaluating the anglular contribution of the open-looppoles and zeros at points P1, P2, P3, and P4 below:

The angular contribution of a pair of complex open-loop poles(or zeros) is zero

The contribution of poles or zeros to the left of the point iszero

Using only the real-axis poles or zeros to the right of the point,we find that the angle sum alternates between 0o and 180o .

On the real axis, for K > 0 the root locus exists to theleft of an odd number of real-axis, finite open-loop polesand/or finite open-loop zeros.

Page 91: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

3. Real-axis segments:

Consider evaluating the anglular contribution of the open-looppoles and zeros at points P1, P2, P3, and P4 below:

The angular contribution of a pair of complex open-loop poles(or zeros) is zero

The contribution of poles or zeros to the left of the point iszero

Using only the real-axis poles or zeros to the right of the point,we find that the angle sum alternates between 0o and 180o .

On the real axis, for K > 0 the root locus exists to theleft of an odd number of real-axis, finite open-loop polesand/or finite open-loop zeros.

Page 92: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

e.g. G (s) = K(s+3)(s+4)(s+1)(s+2) , H(s) = 1

Page 93: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

e.g. G (s) = K(s+3)(s+4)(s+1)(s+2) , H(s) = 1

Page 94: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

4. Starting and ending points:

Where does the RL begin and end?

It begins at K = 0 and ends atK =∞. Consider the closed-loop transfer function:

T (s) =KG (s)H(s)

1 + KG (s)H(s)

=K NG (s)

DG (s)NH(s)DH(s)

1 + K NG (s)DG (s)

NH(s)DH(s)

=KNG (s)NH(s)

DG (s)DH(s) + KNG (s)NH(s)

If we let K → 0 the poles of T (s) approach the combined poles ofG (s) and H(s). If K →∞ the poles of T (s) approach thecombined zeros of G (s) and H(s).

Thus, the RL begins at the open-loop poles of G (s)H(s)and ends at the zeros of G (s)H(s).

Page 95: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

4. Starting and ending points:

Where does the RL begin and end? It begins at K = 0 and ends atK =∞.

Consider the closed-loop transfer function:

T (s) =KG (s)H(s)

1 + KG (s)H(s)

=K NG (s)

DG (s)NH(s)DH(s)

1 + K NG (s)DG (s)

NH(s)DH(s)

=KNG (s)NH(s)

DG (s)DH(s) + KNG (s)NH(s)

If we let K → 0 the poles of T (s) approach the combined poles ofG (s) and H(s). If K →∞ the poles of T (s) approach thecombined zeros of G (s) and H(s).

Thus, the RL begins at the open-loop poles of G (s)H(s)and ends at the zeros of G (s)H(s).

Page 96: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

4. Starting and ending points:

Where does the RL begin and end? It begins at K = 0 and ends atK =∞. Consider the closed-loop transfer function:

T (s) =KG (s)H(s)

1 + KG (s)H(s)

=K NG (s)

DG (s)NH(s)DH(s)

1 + K NG (s)DG (s)

NH(s)DH(s)

=KNG (s)NH(s)

DG (s)DH(s) + KNG (s)NH(s)

If we let K → 0 the poles of T (s) approach the combined poles ofG (s) and H(s). If K →∞ the poles of T (s) approach thecombined zeros of G (s) and H(s).

Thus, the RL begins at the open-loop poles of G (s)H(s)and ends at the zeros of G (s)H(s).

Page 97: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

4. Starting and ending points:

Where does the RL begin and end? It begins at K = 0 and ends atK =∞. Consider the closed-loop transfer function:

T (s) =KG (s)H(s)

1 + KG (s)H(s)

=K NG (s)

DG (s)NH(s)DH(s)

1 + K NG (s)DG (s)

NH(s)DH(s)

=KNG (s)NH(s)

DG (s)DH(s) + KNG (s)NH(s)

If we let K → 0 the poles of T (s) approach the combined poles ofG (s) and H(s). If K →∞ the poles of T (s) approach thecombined zeros of G (s) and H(s).

Thus, the RL begins at the open-loop poles of G (s)H(s)and ends at the zeros of G (s)H(s).

Page 98: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

4. Starting and ending points:

Where does the RL begin and end? It begins at K = 0 and ends atK =∞. Consider the closed-loop transfer function:

T (s) =KG (s)H(s)

1 + KG (s)H(s)

=K NG (s)

DG (s)NH(s)DH(s)

1 + K NG (s)DG (s)

NH(s)DH(s)

=KNG (s)NH(s)

DG (s)DH(s) + KNG (s)NH(s)

If we let K → 0 the poles of T (s) approach the combined poles ofG (s) and H(s). If K →∞ the poles of T (s) approach thecombined zeros of G (s) and H(s).

Thus, the RL begins at the open-loop poles of G (s)H(s)and ends at the zeros of G (s)H(s).

Page 99: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

4. Starting and ending points:

Where does the RL begin and end? It begins at K = 0 and ends atK =∞. Consider the closed-loop transfer function:

T (s) =KG (s)H(s)

1 + KG (s)H(s)

=K NG (s)

DG (s)NH(s)DH(s)

1 + K NG (s)DG (s)

NH(s)DH(s)

=KNG (s)NH(s)

DG (s)DH(s) + KNG (s)NH(s)

If we let K → 0 the poles of T (s) approach the combined poles ofG (s) and H(s). If K →∞ the poles of T (s) approach thecombined zeros of G (s) and H(s).

Thus, the RL begins at the open-loop poles of G (s)H(s)and ends at the zeros of G (s)H(s).

Page 100: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

4. Starting and ending points:

Where does the RL begin and end? It begins at K = 0 and ends atK =∞. Consider the closed-loop transfer function:

T (s) =KG (s)H(s)

1 + KG (s)H(s)

=K NG (s)

DG (s)NH(s)DH(s)

1 + K NG (s)DG (s)

NH(s)DH(s)

=KNG (s)NH(s)

DG (s)DH(s) + KNG (s)NH(s)

If we let K → 0 the poles of T (s) approach the combined poles ofG (s) and H(s).

If K →∞ the poles of T (s) approach thecombined zeros of G (s) and H(s).

Thus, the RL begins at the open-loop poles of G (s)H(s)and ends at the zeros of G (s)H(s).

Page 101: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

4. Starting and ending points:

Where does the RL begin and end? It begins at K = 0 and ends atK =∞. Consider the closed-loop transfer function:

T (s) =KG (s)H(s)

1 + KG (s)H(s)

=K NG (s)

DG (s)NH(s)DH(s)

1 + K NG (s)DG (s)

NH(s)DH(s)

=KNG (s)NH(s)

DG (s)DH(s) + KNG (s)NH(s)

If we let K → 0 the poles of T (s) approach the combined poles ofG (s) and H(s). If K →∞ the poles of T (s) approach thecombined zeros of G (s) and H(s).

Thus, the RL begins at the open-loop poles of G (s)H(s)and ends at the zeros of G (s)H(s).

Page 102: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

4. Starting and ending points:

Where does the RL begin and end? It begins at K = 0 and ends atK =∞. Consider the closed-loop transfer function:

T (s) =KG (s)H(s)

1 + KG (s)H(s)

=K NG (s)

DG (s)NH(s)DH(s)

1 + K NG (s)DG (s)

NH(s)DH(s)

=KNG (s)NH(s)

DG (s)DH(s) + KNG (s)NH(s)

If we let K → 0 the poles of T (s) approach the combined poles ofG (s) and H(s). If K →∞ the poles of T (s) approach thecombined zeros of G (s) and H(s).

Thus, the RL begins at the open-loop poles of G (s)H(s)and ends at the zeros of G (s)H(s).

Page 103: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

e.g. G (s) = K(s+3)(s+4)(s+1)(s+2) , H(s) = 1

Note that we don’t know yet what the exact trajectory of the rootlocus will be.

Page 104: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

e.g. G (s) = K(s+3)(s+4)(s+1)(s+2) , H(s) = 1

Note that we don’t know yet what the exact trajectory of the rootlocus will be.

Page 105: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

e.g. G (s) = K(s+3)(s+4)(s+1)(s+2) , H(s) = 1

Note that we don’t know yet what the exact trajectory of the rootlocus will be.

Page 106: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

What if the number of open-loop poles and zeros is mismatched?e.g.

F (s) =K

s(s + 1)(s + 2)

A function can have both poles and zeros at infinity. For example,as s →∞.

F (s) ≈ K

s · s · sWe therefore consider F (s) to have three zeros at infinity. If weinclude both finite and infinite poles and zeros every function hasan equal number of poles and zeros.

The root locus for F (s) (i.e. KG (s)H(s) = F (s)) would start atthe three finite poles and go towards the zeros at infinity. Yet howdo we get to these zeros at infinity?

Page 107: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

What if the number of open-loop poles and zeros is mismatched?e.g.

F (s) =K

s(s + 1)(s + 2)

A function can have both poles and zeros at infinity. For example,as s →∞.

F (s) ≈ K

s · s · sWe therefore consider F (s) to have three zeros at infinity. If weinclude both finite and infinite poles and zeros every function hasan equal number of poles and zeros.

The root locus for F (s) (i.e. KG (s)H(s) = F (s)) would start atthe three finite poles and go towards the zeros at infinity. Yet howdo we get to these zeros at infinity?

Page 108: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

What if the number of open-loop poles and zeros is mismatched?e.g.

F (s) =K

s(s + 1)(s + 2)

A function can have both poles and zeros at infinity.

For example,as s →∞.

F (s) ≈ K

s · s · sWe therefore consider F (s) to have three zeros at infinity. If weinclude both finite and infinite poles and zeros every function hasan equal number of poles and zeros.

The root locus for F (s) (i.e. KG (s)H(s) = F (s)) would start atthe three finite poles and go towards the zeros at infinity. Yet howdo we get to these zeros at infinity?

Page 109: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

What if the number of open-loop poles and zeros is mismatched?e.g.

F (s) =K

s(s + 1)(s + 2)

A function can have both poles and zeros at infinity. For example,as s →∞.

F (s) ≈ K

s · s · sWe therefore consider F (s) to have three zeros at infinity. If weinclude both finite and infinite poles and zeros every function hasan equal number of poles and zeros.

The root locus for F (s) (i.e. KG (s)H(s) = F (s)) would start atthe three finite poles and go towards the zeros at infinity. Yet howdo we get to these zeros at infinity?

Page 110: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

What if the number of open-loop poles and zeros is mismatched?e.g.

F (s) =K

s(s + 1)(s + 2)

A function can have both poles and zeros at infinity. For example,as s →∞.

F (s) ≈ K

s · s · s

We therefore consider F (s) to have three zeros at infinity. If weinclude both finite and infinite poles and zeros every function hasan equal number of poles and zeros.

The root locus for F (s) (i.e. KG (s)H(s) = F (s)) would start atthe three finite poles and go towards the zeros at infinity. Yet howdo we get to these zeros at infinity?

Page 111: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

What if the number of open-loop poles and zeros is mismatched?e.g.

F (s) =K

s(s + 1)(s + 2)

A function can have both poles and zeros at infinity. For example,as s →∞.

F (s) ≈ K

s · s · sWe therefore consider F (s) to have three zeros at infinity.

If weinclude both finite and infinite poles and zeros every function hasan equal number of poles and zeros.

The root locus for F (s) (i.e. KG (s)H(s) = F (s)) would start atthe three finite poles and go towards the zeros at infinity. Yet howdo we get to these zeros at infinity?

Page 112: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

What if the number of open-loop poles and zeros is mismatched?e.g.

F (s) =K

s(s + 1)(s + 2)

A function can have both poles and zeros at infinity. For example,as s →∞.

F (s) ≈ K

s · s · sWe therefore consider F (s) to have three zeros at infinity. If weinclude both finite and infinite poles and zeros every function hasan equal number of poles and zeros.

The root locus for F (s) (i.e. KG (s)H(s) = F (s)) would start atthe three finite poles and go towards the zeros at infinity. Yet howdo we get to these zeros at infinity?

Page 113: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

What if the number of open-loop poles and zeros is mismatched?e.g.

F (s) =K

s(s + 1)(s + 2)

A function can have both poles and zeros at infinity. For example,as s →∞.

F (s) ≈ K

s · s · sWe therefore consider F (s) to have three zeros at infinity. If weinclude both finite and infinite poles and zeros every function hasan equal number of poles and zeros.

The root locus for F (s) (i.e. KG (s)H(s) = F (s)) would start atthe three finite poles and go towards the zeros at infinity.

Yet howdo we get to these zeros at infinity?

Page 114: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

What if the number of open-loop poles and zeros is mismatched?e.g.

F (s) =K

s(s + 1)(s + 2)

A function can have both poles and zeros at infinity. For example,as s →∞.

F (s) ≈ K

s · s · sWe therefore consider F (s) to have three zeros at infinity. If weinclude both finite and infinite poles and zeros every function hasan equal number of poles and zeros.

The root locus for F (s) (i.e. KG (s)H(s) = F (s)) would start atthe three finite poles and go towards the zeros at infinity. Yet howdo we get to these zeros at infinity?

Page 115: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

5. Behaviour at infinity:

The root locus approaches straight lines as asymptotesfor zeros at infinity.

These asymptotes are defined aslines with real-axis intercept σa and angle θa.

σa =

∑finite poles−

∑finite zeros

#finite poles−#finite zeros

θa =(2k + 1)π

#finite poles−#finite zeros

where k is an integer and θa is the angle (in radians) tothe positive real-axis. We get as many asymptotes asthere are branches corresponding to zeros at infinity.

The derivation for these formulae can be found atwww.wiley.com/college/nise under Appendix L.1.

Page 116: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

5. Behaviour at infinity:

The root locus approaches straight lines as asymptotesfor zeros at infinity. These asymptotes are defined aslines with real-axis intercept σa and angle θa.

σa =

∑finite poles−

∑finite zeros

#finite poles−#finite zeros

θa =(2k + 1)π

#finite poles−#finite zeros

where k is an integer and θa is the angle (in radians) tothe positive real-axis. We get as many asymptotes asthere are branches corresponding to zeros at infinity.

The derivation for these formulae can be found atwww.wiley.com/college/nise under Appendix L.1.

Page 117: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

5. Behaviour at infinity:

The root locus approaches straight lines as asymptotesfor zeros at infinity. These asymptotes are defined aslines with real-axis intercept σa and angle θa.

σa =

∑finite poles−

∑finite zeros

#finite poles−#finite zeros

θa =(2k + 1)π

#finite poles−#finite zeros

where k is an integer and θa is the angle (in radians) tothe positive real-axis. We get as many asymptotes asthere are branches corresponding to zeros at infinity.

The derivation for these formulae can be found atwww.wiley.com/college/nise under Appendix L.1.

Page 118: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

5. Behaviour at infinity:

The root locus approaches straight lines as asymptotesfor zeros at infinity. These asymptotes are defined aslines with real-axis intercept σa and angle θa.

σa =

∑finite poles−

∑finite zeros

#finite poles−#finite zeros

θa =(2k + 1)π

#finite poles−#finite zeros

where k is an integer and θa is the angle (in radians) tothe positive real-axis. We get as many asymptotes asthere are branches corresponding to zeros at infinity.

The derivation for these formulae can be found atwww.wiley.com/college/nise under Appendix L.1.

Page 119: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

5. Behaviour at infinity:

The root locus approaches straight lines as asymptotesfor zeros at infinity. These asymptotes are defined aslines with real-axis intercept σa and angle θa.

σa =

∑finite poles−

∑finite zeros

#finite poles−#finite zeros

θa =(2k + 1)π

#finite poles−#finite zeros

where k is an integer and θa is the angle (in radians) tothe positive real-axis.

We get as many asymptotes asthere are branches corresponding to zeros at infinity.

The derivation for these formulae can be found atwww.wiley.com/college/nise under Appendix L.1.

Page 120: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

5. Behaviour at infinity:

The root locus approaches straight lines as asymptotesfor zeros at infinity. These asymptotes are defined aslines with real-axis intercept σa and angle θa.

σa =

∑finite poles−

∑finite zeros

#finite poles−#finite zeros

θa =(2k + 1)π

#finite poles−#finite zeros

where k is an integer and θa is the angle (in radians) tothe positive real-axis. We get as many asymptotes asthere are branches corresponding to zeros at infinity.

The derivation for these formulae can be found atwww.wiley.com/college/nise under Appendix L.1.

Page 121: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

5. Behaviour at infinity:

The root locus approaches straight lines as asymptotesfor zeros at infinity. These asymptotes are defined aslines with real-axis intercept σa and angle θa.

σa =

∑finite poles−

∑finite zeros

#finite poles−#finite zeros

θa =(2k + 1)π

#finite poles−#finite zeros

where k is an integer and θa is the angle (in radians) tothe positive real-axis. We get as many asymptotes asthere are branches corresponding to zeros at infinity.

The derivation for these formulae can be found atwww.wiley.com/college/nise under Appendix L.1.

Page 122: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

e.g. Sketch the RL for the following system:

We first apply rules 1-4:

Page 123: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

e.g. Sketch the RL for the following system:

We first apply rules 1-4:

Page 124: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

e.g. Sketch the RL for the following system:

We first apply rules 1-4:

Page 125: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

We now compute the real-axis intercept and the angles of allasymptotes:

σa =

∑finite poles−

∑finite zeros

#finite poles−#finite zeros

=(0− 1− 2− 4)− (−3)

4− 1= −4/3

θa =(2k + 1)π

#finite poles−#finite zeros

= π/3 for k = 0

= π for k = 1

= 5π/3 for k = 2

Notice that we have one real-axis intercept but multiple angles.There are three asymptotes—one for each infinite zero. We obtainthree unique values for θa before the angles start to repeat.

Page 126: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

We now compute the real-axis intercept and the angles of allasymptotes:

σa =

∑finite poles−

∑finite zeros

#finite poles−#finite zeros

=(0− 1− 2− 4)− (−3)

4− 1= −4/3

θa =(2k + 1)π

#finite poles−#finite zeros

= π/3 for k = 0

= π for k = 1

= 5π/3 for k = 2

Notice that we have one real-axis intercept but multiple angles.There are three asymptotes—one for each infinite zero. We obtainthree unique values for θa before the angles start to repeat.

Page 127: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

We now compute the real-axis intercept and the angles of allasymptotes:

σa =

∑finite poles−

∑finite zeros

#finite poles−#finite zeros

=(0− 1− 2− 4)− (−3)

4− 1

= −4/3

θa =(2k + 1)π

#finite poles−#finite zeros

= π/3 for k = 0

= π for k = 1

= 5π/3 for k = 2

Notice that we have one real-axis intercept but multiple angles.There are three asymptotes—one for each infinite zero. We obtainthree unique values for θa before the angles start to repeat.

Page 128: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

We now compute the real-axis intercept and the angles of allasymptotes:

σa =

∑finite poles−

∑finite zeros

#finite poles−#finite zeros

=(0− 1− 2− 4)− (−3)

4− 1= −4/3

θa =(2k + 1)π

#finite poles−#finite zeros

= π/3 for k = 0

= π for k = 1

= 5π/3 for k = 2

Notice that we have one real-axis intercept but multiple angles.There are three asymptotes—one for each infinite zero. We obtainthree unique values for θa before the angles start to repeat.

Page 129: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

We now compute the real-axis intercept and the angles of allasymptotes:

σa =

∑finite poles−

∑finite zeros

#finite poles−#finite zeros

=(0− 1− 2− 4)− (−3)

4− 1= −4/3

θa =(2k + 1)π

#finite poles−#finite zeros

= π/3 for k = 0

= π for k = 1

= 5π/3 for k = 2

Notice that we have one real-axis intercept but multiple angles.There are three asymptotes—one for each infinite zero. We obtainthree unique values for θa before the angles start to repeat.

Page 130: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

We now compute the real-axis intercept and the angles of allasymptotes:

σa =

∑finite poles−

∑finite zeros

#finite poles−#finite zeros

=(0− 1− 2− 4)− (−3)

4− 1= −4/3

θa =(2k + 1)π

#finite poles−#finite zeros

= π/3 for k = 0

= π for k = 1

= 5π/3 for k = 2

Notice that we have one real-axis intercept but multiple angles.There are three asymptotes—one for each infinite zero. We obtainthree unique values for θa before the angles start to repeat.

Page 131: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

We now compute the real-axis intercept and the angles of allasymptotes:

σa =

∑finite poles−

∑finite zeros

#finite poles−#finite zeros

=(0− 1− 2− 4)− (−3)

4− 1= −4/3

θa =(2k + 1)π

#finite poles−#finite zeros

= π/3 for k = 0

= π for k = 1

= 5π/3 for k = 2

Notice that we have one real-axis intercept but multiple angles.There are three asymptotes—one for each infinite zero. We obtainthree unique values for θa before the angles start to repeat.

Page 132: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

We now compute the real-axis intercept and the angles of allasymptotes:

σa =

∑finite poles−

∑finite zeros

#finite poles−#finite zeros

=(0− 1− 2− 4)− (−3)

4− 1= −4/3

θa =(2k + 1)π

#finite poles−#finite zeros

= π/3 for k = 0

= π for k = 1

= 5π/3 for k = 2

Notice that we have one real-axis intercept but multiple angles.There are three asymptotes—one for each infinite zero. We obtainthree unique values for θa before the angles start to repeat.

Page 133: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

We now compute the real-axis intercept and the angles of allasymptotes:

σa =

∑finite poles−

∑finite zeros

#finite poles−#finite zeros

=(0− 1− 2− 4)− (−3)

4− 1= −4/3

θa =(2k + 1)π

#finite poles−#finite zeros

= π/3 for k = 0

= π for k = 1

= 5π/3 for k = 2

Notice that we have one real-axis intercept but multiple angles.

There are three asymptotes—one for each infinite zero. We obtainthree unique values for θa before the angles start to repeat.

Page 134: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

We now compute the real-axis intercept and the angles of allasymptotes:

σa =

∑finite poles−

∑finite zeros

#finite poles−#finite zeros

=(0− 1− 2− 4)− (−3)

4− 1= −4/3

θa =(2k + 1)π

#finite poles−#finite zeros

= π/3 for k = 0

= π for k = 1

= 5π/3 for k = 2

Notice that we have one real-axis intercept but multiple angles.There are three asymptotes—one for each infinite zero.

We obtainthree unique values for θa before the angles start to repeat.

Page 135: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

We now compute the real-axis intercept and the angles of allasymptotes:

σa =

∑finite poles−

∑finite zeros

#finite poles−#finite zeros

=(0− 1− 2− 4)− (−3)

4− 1= −4/3

θa =(2k + 1)π

#finite poles−#finite zeros

= π/3 for k = 0

= π for k = 1

= 5π/3 for k = 2

Notice that we have one real-axis intercept but multiple angles.There are three asymptotes—one for each infinite zero. We obtainthree unique values for θa before the angles start to repeat.

Page 136: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

The following is our complete RL sketch.

Page 137: Unit 7: Part 1: Sketching the Root Locus - Computer Science · Root Locus 1 Root Locus Vector Representation of Complex Numbers De ning the Root Locus Properties of the Root Locus

The following is our complete RL sketch.