Unit 3-State-Space Analysis_all.pptx

SVERI’s College of Engineering, Pandharpur
Department of Electrical Engineering
Advance Control Systems
Dr. Dipti A. Tamboli
HoD, Electrical Engg. Department

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 Unit 3- State-Space Analysis No of Lectures -07
 C 3.1 Syllabus
 Concept of state, state variable & state model,
 State-space representation of transfer function of electrical and
mechanical systems,
 State transition matrix & its properties,
 Solution of homogeneous and non-homogeneous state equation,
 Controllability & Observability
Content

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Text Books:
1. I. J. Nagrath, M. Gopal “Control System Engineering”, 5th
Edition. New Age International Publishers.
2. Control System Engineering by R Anandanatrajan, P Ramesh
Babu, 2nd Edition, Scitech
3. K. Ogata, “Modern Control Engineering”, Prentice Hall of India
Pvt. Ltd.

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 State Space analysis
 The "state space" is the Euclidean space [a space in any finite
number of dimensions, in which points are designated by
coordinates (one for each dimension) and the distance between two
points is given by a distance formula.] in which the variables on the
axes are the state variables.
 The process by which the state of a system is determined is called
state variable analysis.
 The unit which deals with analysis of the state space of a dynamical
system is called as state space analysis.
 The state space of a dynamical system is the set of all
possible states of the system. Each coordinate is a state variable, and
the values of all the state variables completely describes the state of
the system.
C 3.2 About title of the unit 3

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 State Space analysis (also known as state variable analysis) is a
commonly used method nowadays for analysing the control system.
 The analysis that involves providing a complete idea about the
behaviour of the system at any given time utilizing the history of the
system is known as state-space analysis.
 The analysis of the control system in this approach is based on the
state of the system.
 In this unit, we are going to study concept of state, state variable &
state model.
 Methods of representation of transfer function of electrical and
mechanical systems in state-space form.
 The State transition matrix & its properties, and the concept of
Controllability & Observability
C 3.4 Central idea of the unit 3

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 State space analysis of control system is based on the modern theory
which is applicable to all types of systems like single input single
output systems, multiple inputs and multiple outputs systems, linear
and non linear systems, time varying and time invariant systems.
 Differences between the conventional theory of control system and
modern theory of control system.
1. The conventional control theory is completely based on the
frequency domain approach while the modern control system theory
is based on time domain approach.
2. In the conventional theory of control system we have linear and time
invariant single input single output (SISO) systems only. But with the
help of theory of modern control system we can easily do the
analysis of even non linear and time variant multiple inputs multiple
outputs (MIMO) systems also.
C 3.5 Importance of the Unit 3 in the subject

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3. In the modern theory of control system the stability analysis and time
response analysis can be done by both graphical and analytically
method very easily.
4. State space is one of the key concepts of system theory. In general, a
state space is introduced into a system description without examining
its specific physical meaning.
5. It is known, however, that if we select a suitable state space
representation, it becomes easier for us to understand or to
manipulate the property of a system.
6. Thus, although it may not be meaningful to try to arrive at a physical
meaning of the state space, it is important to find a way where a
choice of a state space can be made in accordance with the system of
interest, taking into account the theoretical meaning of that choice.
C 3.5 Importance of the unit 3 in the subject

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 Find a mathematical model, called a state-space representation,
for a linear, time invariant system
 Model electrical and mechanical systems in state space
 Convert a transfer function to state space and vice versa
 Controllability and observability
C 3.6 Unit Outcome

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 Despite having a transfer function approach why do we
need state-space analysis?
 The conventional method was easy to apply in order to analyse the
system but at the same time, there were some major drawbacks
associated with it. The drawbacks associated with the conventional
method are as follows:
1. In the transfer function approach during analysis, the initial
conditions are considered to be 0. However, this is not the case with
state variable analysis because it considers all the initial conditions
associated with the elements of the control system.
2. And the consideration of initial conditions provides a more precise
response. Hence the consideration of all the initial conditions
associated with the system in the state model serves as a major
advantageous factor.
Need for State Space Analysis

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3. The transfer function approach shows inconvenience towards the
analysis of multiple input multiple output systems. As against the
state variable approach supports analysing both SISO as well as
MIMO systems.
4. Due to the effectiveness in analysing linear and non-linear as well as
time-varying or time-invariant systems, the state-space approach is
better than the transfer function approach which is applicable to
linear time-invariant systems.
5. As the transfer function approach is a frequency domain analysis
thus it offers difficulty in finding the time domain solutions in case of
a higher-order system. Whereas state space analysis is a time-domain
approach.
6. The transfer function approach uses some standard test input signals
for the analysis of the system, while this is not the case of state-space
analysis.

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7. As the state variable approach is associated with matrix/vector
modelling thus is considered as an efficient computational approach.
Hence it facilitates accurate response thus is applicable to dynamic
systems.
8. Thus, we can say state-space analysis overcomes the limitations of
the transfer function approach.
 Disadvantages of State Space Analysis
1. State-space analysis involves complex technique and requirement of
high computations.

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 What is meant by ‘State Space’?
 The analysis of the system using conventional methods like
root locus or bode plot etc. utilizes the transfer function
approach where the system is analyzed on the basis of output
and input.
 In this approach, the interior conditions of the system are not
taken into consideration and are restricted to only a single input
single output.
 But in the case of a modern approach, brief information on the
internal conditions of the system is considered. These internal
conditions specify the state of the system. Thus called state
space.
Introduction

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 What is State?
 It represents every smallest past information of the system in order
to predict response. Basically a state separates the future i.e., the
response of the system from the past.
 Thus more simply we can say that to have the accurate response of
the system for any given input, the state holds the information
related to the history of the system.
 Therefore, at any instant of time, the state signifies the combined
effect of each element of the system. Hence the state is a vector in
nature that provides the values from each component associated
with initial conditions of the system.
 Due to vector nature, it is generally known as a state vector.
 As we have discussed that it is a vector term thus the various
variables that constitute state vector are known as state variables.

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 State: The state of a dynamic system is the smallest set of variables ((
𝒙𝟏, 𝒙𝟐, … … , 𝒙𝒏) called state variables or state vector such that the
knowledge of these variables at time t=to (Initial condition), together
with the knowledge of input for ≥𝑡0 , completely determines the
behaviour of the system for any time 𝑡≥𝑡0. State of a system describes
about the future behaviour of a system by using present (input
variables) and past (State variables).
 State vector: A set of state variable expressed in a matrix is called
state vector. If n state variables are needed to completely describe the
behaviour of a given system, then these n state variables can be
considered the n components of a vector X. Such a vector is called a
state vector.

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 State space: The n-dimensional space whose co-ordinate axes
consists of the x1 axis, x2 axis,.... xn axis, where x1 , x2 ,..... xn are
state variables: is called a state space.
 Any n-dimensional state vector determines a point (called the state
point) in an n-dimensional space called the state space.
 Set of possible values that the state variables can assume is called
state space.
 Now, suppose we have a multi input multi output system with m
number of inputs, n number of state variables and p number of
output variables. The systems will be represented through state
equations as follows
 Hence u is m×1 input vector, x is n×1 state variable vector and y is
p×1 output vector.

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 A, B, C, D are matrices with their dimensions as A (n×n), B (n×m), C
(p×n), D (p×m)
 Consider electrical networks, consisting any number of inductors and
capacitors. The state variables may represent the magnetic and
electric fields of the inductors and capacitors, respectively.
 Consider a spring-mass-dashpot system. The state variables may
represent the compression of the spring, or the acceleration at the
dashpot.

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 Input variables: A SISO (Single-Input Single-Output) system
will only have one input value, but a MIMO(Multiple-Input
Multiple-Output) system may have multiple inputs. We need to
define all the inputs to the system and arrange them into a vector.
 Output variables: This is the system output value, and in the
case of MIMO systems we may have several. Output variables
should be independent of one another, and only dependent on a
linear combination of the input vector and the state vector.
 We denote the input variables with u, the output variables with y,
and the state variables with x. In essence, we have the following
relationship:
 Where f(x, u) is our system. Also, the state variables can change
with respect to the current state and the system input:
Where x' is the rate of change of the state variables.

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 State Variables: The variables that represent the status of the
system at any time t, are called state variable.
 The state variables represent values from inside the system that
can change over time. In an electric circuit for instance, the node
voltages or the mesh currents can be state variables. In a
mechanical system, the forces applied by springs, gravity, and
dashpots can be state variables.
 Here x(t) denotes the state vector while x1(t), x2(t), etc. are the
state variables that are forming state vector.
 State Trajectory: The curve traced out by the state point from
𝑡 = 𝑡0 to 𝑡 = 𝑡1 in the direction of increasing time is known as
the state trajectory.

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State Model: The state vectors with input/output equations constitute
the state model of the system.
Consider a linear system with two inputs u1(t) and u2(t), while two
outputs y1(t) and y2(t).
Suppose the system is having two states x1(t) and x2(t).

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State Equation
We know that state variables show variation with time. Thus writing
the differential equation of the state variable, we will have
Thus the matrix representation will be given as:

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 Therefore, the generalized form:
 Output Equation
 The output of the system will be represented as the linear
combination of the state of the system and the applied input. Thus
is given as:
 The coefficients c and d taken over here are constants. Thus
writing the above equation in the form of a matrix, we will have:

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 Therefore, in general form, we can write it as:
 Thus the two equations combinedly form the state model of the
linear system:

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General State Space form of Physical System
x = state vector
𝑥 = derivative of the state vector with respect to time
Y output vector
u= input or control vector
A= system matrix
B= input matrix
C= output matrix
D= Feed forward matrix

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 The roots of characteristic equation that we have described above
are known as eigen values of matrix A.
 Now there are some properties related to eigen values and these
properties are written below-
1. Any square matrix A and its transpose 𝐴𝑇
have the same eigen
values.
2. Sum of eigen values of any matrix A is equal to the trace of the
matrix A.
3. Product of the eigen values of any matrix A is equal to the
determinant of the matrix A.
4. If we multiply a scalar quantity to matrix A then the eigen values
are also get multiplied by the same value of scalar.
5. If we inverse the given matrix A then its eigen values are also get
inverses.
6. If all the elements of the matrix are real then the eigen values
corresponding to that matrix are either real or exists in complex
conjugate pair.
Concept of Eigen Values and Eigen Vectors

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Eigen Vectors
 Any non zero vector 𝑚𝑖that satisfies the matrix equation
𝜆𝑖𝐼 − 𝐴 𝑋𝑖 = 0 (1)
is called the eigen vector of A associated with the eigen value 𝜆𝑖,
Where in 𝜆𝑖, i = 1, 2, 3, ……..n denotes the 𝑖𝑡ℎ
eigen values of A.
 This eigen vector may be obtained by taking cofactors of matrix
𝜆𝑖𝐼 − 𝐴 along any row & transposing that row of cofactors.
 Once we find eigen values 𝜆𝑖, then from Eq. (1), we can find eigen
vectors.
 Diagonalization
 Let 𝑋𝑖 = 𝑚𝑖 be the solution of Eq. (1). The solution of 𝑚𝑖 is called
as eigen vectors of A associated with eigen value 𝜆𝑖.
 Let 𝑚1, 𝑚2, …….. 𝑚𝑛 be the eigenvectors corresponding to the
eigen value 𝜆1, 𝜆2,…….. 𝜆𝑛 respectively.
 Then M= 𝑚1 ⋮ 𝑚2 ⋮ ……. ⋮ 𝑚𝑛 is called diagonalizing or modal
matrix of A.

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 Consider the 𝑛𝑡ℎ
order MIMO state model
𝑋 𝑡 = 𝐴 𝑋 𝑡 + 𝐵 𝑈 𝑡
𝑌 𝑡 = 𝐶 𝑋 𝑡 + 𝐷 𝑈(𝑡)
 System matrix A is non diagonal, so let us define a new state vector
𝑉(𝑡) such that 𝑋 𝑡 = 𝑀 𝑉(𝑡).
 Under this assumption original state model modifies to
𝑉 𝑡 = 𝐴 𝑋 𝑡 + 𝐵𝑈 𝑡
 Where A = M−1AM =diagonal matrix, B = M−1B, C = CM
 If the system matrix A is in companion form & if all its n eigen
values are distinct, then modal matrix will be special matrix called
the Vander Monde matrix.

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 If the system matrix A is in companion form & if all its n eigen
values are distinct, then modal matrix will be special matrix called
the Vander Monde matrix.
 A=
0
0
0
1 0 ⋯
0 1 ⋯
0 0 ⋯
0
0
0
⋮ ⋱ ⋮
−𝑎0 −𝑎1 −𝑎2 ⋯ −𝑎𝑛−1
 V=
1
𝜆1
𝜆1
2
1 1 ⋯
𝜆2 𝜆3 ⋯
𝜆2
2
𝜆3
2
⋯
1
𝜆n
𝜆𝑛
2
⋮ ⋱ ⋮
𝜆1
𝑛−1
𝜆2
𝑛−1
𝜆3
𝑛−1
⋯ 𝜆𝑛
𝑛−1

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State transition matrix & its properties
 We are here interested in deriving the expressions for the state
transition matrix and zero state response.
 Again consider the state equations that we have derived above as a
state model
 Taking their Laplace transformation we have,
𝑠 𝑋 𝑠 − 𝑋 0 = 𝐴 𝑋 𝑠 + 𝐵 𝑈(𝑠)
 Now on rewriting the above equation we have
𝑠𝐼 − 𝐴 𝑋 𝑠 = 𝑋 0 + 𝐵 𝑈(𝑠)
𝑋 𝑠 = 𝑠𝐼 − 𝐴 −1
[𝑋 0 + 𝐵 𝑈 𝑠 ]
 Put θ 𝑠 = 𝑠𝐼 − 𝐴 −1
, the equation becomes
𝑋 𝑠 = θ 𝑠 . 𝑋 0 + θ 𝑠 . 𝐵 𝑈 𝑠

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 Taking the inverse Laplace of the above equation we have
𝑋 𝑡 = θ t . 𝑋 0 + ℒ−1[θ 𝑠 . 𝐵 𝑈 𝑠 ]
 The expression θ(t) is known as state transition matrix(STM).
 The response is given as
𝑋 𝑡 = ℒ−1[θ t . 𝑋 0 ] + ℒ−1[θ 𝑠 . 𝐵 𝑈 𝑠 ]
Zero-input response Zero-state response
To calculate the value of state transition matrix in time domain, consider
𝑋 𝑡 = 𝑏0+ 𝑏1𝑡 + 𝑏2𝑡2+….+𝑏𝑘𝑡𝑘 + 𝑏𝑘+1𝑡𝑘+1

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 Properties of the state transition matrix
𝝓 𝒕 = 𝒆𝑨𝒕 =𝓛−𝟏[ 𝐬𝐈 − 𝑨 −𝟏]
 Property 1: 𝜙 0 is an identity matrix.
𝑋 𝑡 = 𝑒𝐴𝑡
𝑋 0 = 𝜙 𝑡 𝑋 0
 At t=0
𝑋 0 = 𝜙 0 𝑋 0 = 𝑒𝐴.0 𝑋 0 =I
 𝜙 0 = 𝐼 , which simply states that the state response at time t =
0 is identical to the initial conditions.
 Property 2: 𝜙 −𝑡 = 𝜙−1 𝑡
𝜙 𝑡 = 𝑒𝐴𝑡
=(𝑒−𝐴𝑡
)−1
= 𝜙 −𝑡 −1
𝜙 𝑡 = 𝜙 −𝑡 −1
 Taking inverse to both sides
𝜙−1 𝑡 = 𝜙 −𝑡
𝜙 −𝑡 = 𝜙−1 𝑡

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 The response of an unforced system before time t = 0 may be
calculated from the initial conditions x(0),
𝑋 −𝑡 = 𝜙 −𝑡 𝑋 0 = 𝜙−1 𝑡 𝑋 0
 Property 3: 𝜙 𝑡1 . 𝜙 𝑡2 = 𝜙 𝑡1 + 𝑡2
𝜙 𝑡1 + 𝑡2 = 𝑒𝐴 𝑡1+𝑡2 = 𝑒𝐴𝑡1 𝑒𝐴𝑡2= 𝜙 𝑡1 𝜙 𝑡2
𝜙 𝑡1 + 𝑡2 = 𝜙 𝑡1 . 𝜙 𝑡2
 With this property the state response at time t may be defined
from the system state specified at some time other than t = 0, for
example at 𝑡 = 𝑡0.
 Using Property (2) i.e. 𝑋 −𝑡 = 𝜙 −𝑡 𝑋 0 , the response at
time t = 0 (𝑡 = 𝑡0) is
𝑋 0 = 𝜙 −𝑡0 𝑋 𝑡0
𝑋 𝑡 = 𝜙 𝑡 𝑋 0 = 𝜙 𝑡 𝜙 −𝑡0 𝑋 𝑡0

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 Property 4: If A is a diagonal matrix then 𝑒𝐴𝑡
is also diagonal,
and each element on the diagonal is an exponential in the
corresponding diagonal element of the A matrix, that is 𝑒𝑎𝑖𝑖𝑡.
 This property is easily shown by considering the terms 𝐴𝑛 in the
series definition and noting that any diagonal matrix raised to an
integer power is itself diagonal.
 Property 5: 𝜙 𝑡 𝑛 = 𝜙 𝑛𝑡
𝜙 𝑡 𝑛= 𝑒𝐴𝑡 𝑛 =𝑒𝐴𝑛𝑡= 𝜙 𝑛𝑡

87
 There are many tests for checking controllability and observability
and these tests are very essential during the design of a control
system using state space approach.
 Particular state variable is found uncontrollable, then it is left
untouched and any other state variable which is controllable is
selected for operations.
 Definition:
 A system is said to be completely state controllable if it is
possible to transfer the system state from any initial state
X(to) to any desired state X(t) in specified finite time by a
control/input vector u(t).
Controllability and Observability

88
 Consider a single-input system (u∈R):
𝑋 𝑡 = 𝐴 𝑋 𝑡 + 𝐵 𝑈 𝑡 and 𝑌 𝑡 = 𝐶 𝑋 𝑡
 A linear system with state vector x is called controllable if and only
if the system states can be changed by changing the system input, u.
 In order to be able to do whatever we want with the given dynamic
system under control input, the system must be controllable.
 The Controllability Matrix is defined as
 We say that the above system is controllable if its controllability
matrix 𝐶(𝐴,𝐵) is invertible. This test for controllability is called
Kalman’s test.
 A LTI system is completely controllable if and only if its
controllability matrix or composite matrix 𝑄𝑐,
 𝑸𝒄 = 𝐁 𝐀𝐁 𝐀𝟐𝐁 … … . 𝐀𝐧−𝟏𝐁 has a full rank of n.
 For SISO system, 𝑸𝒄 ≠ 𝟎

89
 Consider a system :
𝑋 𝑡 = 𝐴 𝑋 𝑡 + 𝐵 𝑈 𝑡 and 𝑌 𝑡 = 𝐶 𝑋 𝑡 + 𝐷 𝑈(𝑡)
 Lets assume that matrix A is diagonalised by transforming the vector X
to Z, then we have
𝑍 𝑡 = 𝐴 𝑍 𝑡 + 𝐵𝑈 𝑡 , and 𝑌 𝑡 = 𝐶 𝑍 𝑡 + 𝐷 𝑈(𝑡)
 The state equation can be written as
𝑍𝑖 = 𝜆𝑖𝑍𝑖 + 𝑗=1
𝑚
𝑏𝑖𝑗𝑢𝑖 i=1, 2, …..n
 If any element of this vector is zero, then the corresponding state
variable is not controllable. In such condition state variable 𝑍𝑖 is not
connected to any of the input. 𝑍𝑖 is decoupled from the remaining (n-1)
state variables.
 Hence there is no link between the input and corresponding state
variable. Therefore system is uncontrollable.
 The necessary and sufficient condition of complete controllability is
simply that the vector B should not have any zero element.
 𝑸𝒄 = 𝐁 𝐀𝐁 𝐀𝟐𝐁 … … . 𝐀𝐧−𝟏𝐁 has a full rank of n.
 For SISO system, 𝑸𝒄 ≠ 𝟎

90
Proof of controllability matrix

93
Comment on controllability of the system described by

97
 Definition:
 A system is said to be completely state observable, if every
state X(to) can be completely identified by measurements of
the outputs y(t) over a finite time interval (𝑡0 ≤ 𝑡 ≤ 𝑡1)
OR
 A system is said to be completely state observable, if the
knowledge of output Y and input U over a finite time interval
(𝑡0 ≤ 𝑡 ≤ 𝑡f) is sufficient to determine every state 𝑋 𝑡
 The state equation can be transform to canonical form by linear
transformation
 𝑍 𝑡 = 𝐴 𝑍 𝑡 + 𝐵𝑈 𝑡 , and 𝑌 𝑡 = 𝐶 𝑍 𝑡 + 𝐷 𝑈(𝑡)
Observability

98
 For MIMO system, the output equation can be represented by
𝑦𝑖 = 𝑐𝑖1𝑧1 + 𝑐𝑖2𝑧2 + 𝑐𝑖3𝑧3 + ⋯ + 𝑐𝑖𝑛𝑧𝑛 + 𝑗=1
𝑚
𝑑𝑖𝑗𝑢𝑖 i=1, 2, …..p
 𝑐𝑖𝑛 = 0, then the state 𝑧𝑛 will 𝑛𝑜𝑡 𝑎𝑝𝑝𝑒𝑎𝑟 𝑖𝑛 the expression of
output 𝑦𝑖.
 Since diagonalization decouples the states, the state 𝑧𝑛 can’t be
observed at output if 𝑐1𝑛, 𝑐2𝑛, …. 𝑐𝑘𝑛 are all zero
 Thus, the necessary condition for complete observability is that
none of the column of matrix 𝐶 should be zero.
 The necessary and sufficient condition for the system to be
completely observable is
 𝑸𝟎 =
𝐂
𝑪𝑨
⋮
𝑪𝐀𝐧−𝟏
has a full rank of n.

99
Proof of observability matrix

109
Principle of Duality:
It gives relationship between controllability & observability.
 The Pair (AB) is controllable implies that the pair (𝐴𝑇
𝐵𝑇
) is
observable.
 The pair (AC) is observable implies that the pair (𝐴𝑇𝐶𝑇 ) is
controllable.

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Unit 3-State-Space Analysis_all.pptx