STATE SPACE ANALYSIS

 The transfer function is the classical approach using the frequency domain
technique deals with input and output only.
 It is unable to give any information about the internal condition.
 On the modern control theory based on the state variable approach.
 It is a time-domain technique.
 It is include the initial conditions in the system.
 It is used as a mathematical tool to solve linear, non-linear, time-varying
and time-varying control problems.
 In this approach to analysis of control system with basic knowledge of
matrix algebra.

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Limitations of Classical or Transfer Function Approach

 It is based on transfer functions derived from linear differential equations

with constant coefficients.
 Analysis and design of linear time-invariant SISO control systems.
 Not applicable to nonlinear, time-varying and MIMO systems.
 The initial conditions are are set to zero.
 Initial conditions cannot be included.
 Does not give any information about the internal variables.
 The test signals like unit impulse, step, ramp and unit parabolic inputs.

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Advantages of the State Space Method

1. Applicable to be linear, nonlinear, time-invariant and time varying systems

2. MIMO systems can be represented and analyzed
3. Initial conditions are incorporated
4. It is based on certain internal variables in a system, and these variables, at
any time, can be known.
5. Discrete-time model is obtained which can be analyzed using a digital
computer.

Disadvantages of the State Space Method

1. This technique is complex over the conventional methods.

2. In this techniques many computations are required.

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Concept of State Space Response
 Consider a linear time-invariant continuous time system with an input u(t) and output response
y(t) as shown in Figure l.

Figure 1 : Input-output relationship in a LTI system

 The output response y(t) is given by the convolution integral 1
t
y (t )   h (t   )u ( ) d 1

where h(t) is the impulse response of the system.
 Consider that the input is applied at time t = 0, then the response can be written as
0 t
y (t )   h(t   )u ( ) d   h(t   )u ( ) d
 0

 The first term on the right hand side is the response of the system at t = 0, and this is the initial
condition of the system.
 Then y(t) can be written as
t
y (t )  y (0)   h (t   )u ( ) d
0
 The output y(t) can be determined completely for a given input u(t), if the initial condition y(0) is
known.
 Given the initial states at t = to and the inputs for t ≥ to, it is possible to determine the system
response completely for any time t > to.
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Basic Concept of State Space Response
 Consider the system defined by following differential equation
a 
y (t )  b y (t )  c y (t )  du (t ) (1)

where y is the output and u is input of the system.

 This system is of second order.
 This means that the system involves two integrators.
 Let us define state variables x1 and x2 as
x1  y
and x2  x1  y
 Then equation (2) can be written as
ax2  bx2  cx1  d u

Now, x1  x2 (2)

and x2   ac x1  ba x2  da u (3)

 We can write this two equations (2) and (3) in vector-matrix form as
 x1   0 1   x1   0 
 x    c b     d u (4)
 2    a  a   x2   a 
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Concept of State Space Response

 The output equation is

 x1 
y  x1   1 0   
(5)
 x2 
 Equation (4) and (5) are in the standard form:

x  Ax  B u (6)
and y  Cx  D u (7)

 0 1 0
A c b ; B  d ; C  1 0 ; and D = [ 0 ]
  a  a   a 

 Equation (6) is a state equation and equation (7) is an output equation of the system.

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State Model for Linear System

 Consider a third-order linear time-invariant system, which is described by a third-order

differential equation.
 Let the state variables are x1 (t ), x2 (t ) and x3 (t ) and two inputs r1 (t ) and r2 (t ) and
two outputs y1 (t ) and y2 (t ) .
 A set of three first-order differential equations can be written in the form

x1  a11 x1 (t )  a12 x2 (t )  a13 x3 (t )  b11 r1 (t )  b12 r2 (t ) (1)

x2  a21 x1 (t )  a22 x2 (t )  a23 x3 (t )  b21 r1 (t )  b22 r2 (t ) (2)
x3  a31 x1 (t )  a32 x2 (t )  a33 x3 (t )  b31 r1 (t )  b32 r2 (t ) (3)

 The state equations (1), (2) and (3) can be written in matrix form as

 x1   a11 a12 a13   x1 (t )   b11 b12 

 x    a  r (t ) 
 2   21 a22 a23   x2 (t )   b21 b22   1 
 r2 (t ) 
 x3   a31 a32 a33   x3 (t )  b31 b32 

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State Model for Linear System

 The output equations are

y1 (t )  c11 x1 (t )  c12 x2 (t )  c13 x3 (t )  d11 r1 (t )  d12 r2 (t ) (5)
y2 (t )  c21 x1 (t )  c22 x2 (t )  c23 x3 (t )  d 21 r1 (t )  d 22 r2 (t ) (6)

 The output equations (5) and (6) can be expressed in matrix form as
 x (t ) 
 y1 (t )   c11 c12 c13   1   d11 d12   r1 (t ) 
 y (t )    c  x2 (t )   
 2   21 c22 c23   d d 22   r2 (t ) 
 x3 (t )   21

 The two equations (4) and (7) together are said to be the state model of the system.

 In general, these are written as,

x  Ax  B u - state equation
and y  Cx  Du - output equation

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Definitions concerning state space approach
State
The state of a dynamical system is the minimal amount of information required, together
with initial conditions at t ≥ t0 and input excitation, to completely determine the behaviour
of the system for any time t ≥ t0.
State Variable
The state variables of a dynamical system are the smallest set of variables which determine
the state of the dynamic system. If at least n variables, x1, x2, x3, … , xn, are needed to
completely describe the function behaviour of the system, together with the initial state and
input excitation, then these n variables x1, x2, x3, … , xn, are a set of state variables. Note that
the state variable need not be physically measurable or observable quantities.
State Vector
The n state variables can be considered the n state variables can be considered the n
components of a state vector x(t), described in n-dimensional vector-space. Such a vector is
called a state vector. A state vector is thus a vector that determines uniquely the system
state x(t) for any time t ≥ t0, once the state at t = t0 is given and the input u(t) for t ≥ t0, is
specified.
State Space
The n-dimension space whose coordinate axes consist of the x1-axis, x2-axis, x3-axis, . . . , xn-
axis, where x1, x2, x3, … , xn, are state variables; is called a state space. Any state can be
represented by a point in the state space.
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Block Diagram Representation of State and Output Equations

 The state space representation of multiple inputs multiple outputs (MIMO) system
are given by the state and the output equations:
x  Ax  B u
and y  Cx  Du

 The block diagram of the system based on state and the output equations are shown
in Fig. 2 below:

Fig. 2 : Block diagram representation of the state space equations.

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Electromechanical System

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