7'6 The solution of state equations and the transition matrix

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7.6 The solution of state equations and the
transition matrix

• The exponential function eat

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7.6 The solution of state equations and the
transition matrix

• The matrix exponential eAt is defined as

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7.6 The solution of state equations and the
transition matrix

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7.6.1 The transition matrix

• For a given continuous system,
• we can obtain the solution as the follow.
• First, we can re-write the equation as

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7.6.1 The transition matrix

• Pre-multiplying both sides of the above equation
  by e-At, we obtain
• Integrating the preceding equation between 0 and
  t gives

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7.6.1 The transition matrix

• Or
• Pre-multiplying both sides by eAt, we have

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7.6.1 The transition matrix

• That is
• Suppose the input is zero, at time t0 we have

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7.6.1 The transition matrix

• The solution of the state equation starting with
  the initial condition X(t0) is

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7.6.1 The transition matrix

• The output of the system is
• The matrix eAt?(t) is called as Transition
  Matrix.

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7.6.2 Properties of the transition matrix

• Given the transition matrix ?(t) eAt, then
• ,

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7.6.3 Calculation of the transition matrix

• As
• apply Laplace transform to the above equations,
  we have

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7.6.3 Calculation of the transition matrix

• Finally we have
• That means we can use inverse Laplace transform
  to calculate the transition matrix.

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7.6.4 Examples of the transition matrix
calculation

• Example 1 Given the transfer function of a
  system as the follow,
• find its transition matrix?(t).

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7.6.4 Examples of the transition matrix
calculation

• First, find its state equations,
• then,

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7.6.4 Examples of the transition matrix
calculation

• Next, find the inverse,

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7.6.4 Examples of the transition matrix
calculation (cont)

• Finally, calculate the transition matrix,

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7.7 State variable feedback

• For a continuous-time system given by,
• suppose that all the state variables are
  accessible, a new feedback scheme state
  variable feedback can be implemented.

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7.7 State variable feedback

u(t)
r(t)
y(t)
X(t)
C

-
KT
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7.7 State variable feedback

• How does this closed loop system
  behave?where u(t)r(t)-KTX(t), KTk1 k2...
  kn.

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7.7 State variable feedback

• Substitute u(t) into the above state equations,
  we have

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7.7 State variable feedback

• For these state equations, we can find the
  transfer function as

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7.7 State variable feedback

• Example 2 Given the system transfer function as
• suppose a state variable feedback scheme is
  implemented for this system, find its closed loop
  transfer function and draw a diagram to show the
  implementation.

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7.7 State variable feedback

• First, find its state equations A, B and C.

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7.7 State variable feedback

• According to the controllable canonical form, we
  obtain
• where

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7.7 State variable feedback

• Then the closed loop transfer function is

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7.7 State variable feedback

• Finally we have
• This means that the characteristic equation roots
  may be placed anywhere in the s-plane by choice
  of state variable feedback coefficients.

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7.7 State variable feedback

• The implementation is shown as the follow

r(t)
y(t)
X(t)
C

-
KT
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7.7 State variable feedback

• First, the original system can be implemented as
  the follow

x3
e(t)
x2
y(t)
x1

u(t)
?
?
a
?
?
-5
-4
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7.7 State variable feedback

• Then, the system with state variable feedback can
  be implemented as the follow

-k3
x2
x3
x1
y(t)

r(t)
u(t)
?
?
?
?
?
a
-5
-4
-k2
-k1
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7.8 Transition matrix for discrete-time state
equations

• For a discrete-time system,
• we will present the solution by a recursion
  procedure and then by the z-transform method.
  Finally, we discuss methods for computing
  (zI-A)-1.

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7.8.1 Solving discrete-time state equations

• The solution for any positive k may be obtained
  directly by recursion, as follows

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7.8.1 Solving discrete-time state equations

• By repeating this procedure, we obtain
• Clearly, X(k) consists of two parts, one
  representing the contribution of the initial
  state X(0) and the other the contribution of the
  input u(j), where j0,1,2,k-1. The output y(k)
  is given by y(k)CX(k).

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7.8.1 Solving discrete-time state equations

• Notice that it is possible to write the solution
  as the follow if the input is zero.
• as
• where ?(k) is a unique n?n matrix satisfying the
  condition

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7.8.1 Solving discrete-time state equations

• Therefore, we obtain the Transition Matrix for
  discrete-time systems. This transition matrix
  contains all the information about the free
  motions of the system from initial conditions.
  Based on the transition matrix, we can re-write
  X(k) as

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7.8.2 Computing of transition matrix using
z-transform

• Consider the discrete-time system described as
  the follow
• Applying the z-transform of both sides of the
  above equation, we get

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7.8.2 Computing of transition matrix using
z-transform

• Applying the inverse z-transform of both sides of
  the above equation, we get
• Comparing the above equation with
• We obtain

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7.8.2 Computing of transition matrix using
z-transform

• Example Obtain the transition matrix of the
  following discrete-time system
• Solution find the state equations,

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7.8.2 Computing of transition matrix using
z-transform

• That is
• As
• therefore, we need to find out the inverse

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7.8.2 Computing of transition matrix using
z-transform

• That is

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7.8.2 Computing of transition matrix using
z-transform

• The transition matrix ?(k) is now obtained as the
  follows

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7.8.2 Computing of transition matrix using
z-transform

• That is

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7.8.2 Computing of transition matrix using
z-transform

• Assume that the initial state is given by
• And the input is unit step function, find the
  y(k).
• Solution Substitute transition matrix into the
  formula below, we have

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7.8.2 Computing of transition matrix using
z-transform
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7.8.3 State variable feedback control
For a discrete-time system given by, suppose
that all the state variables are accessible, a
new feedback scheme state variable feedback can
be implemented.
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7.8.3 State variable feedback
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7.8.3 State variable feedback
How does this closed loop system
behave?where u(k)r(k)-KTX(k), KTk1 k2...
kn.
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7.8.3 State variable feedback
Substitute u(k) into the above state equations,
we have
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7.8.3 State variable feedback
For these state equations, we can find the
transfer function as
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Tutorial

• Exercise1 Given system differential equation as
  below,
• Suppose the state variables are chosen as
  x1(t)?(t) and x2(t)d?(t)/dt, find out
• The state equations.
• The state equations in form of A, B C.
• The system transfer function using A, B C.
• The transition matrix ?(t)

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Tutorial

• Solution As x1(t)?(t), x2(t)d?(t)/dt and
• we have

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Tutorial
A, B and C
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Tutorial
System transfer function
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Tutorial
Transition matrix
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Tutorial
Exercise 2 Given find the transition
matrix?(t) (answer is on study book page 7.24).
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7.9 Derivation of discrete models from continuous
ones

• For a given continuous-time system,
• we can obtain the solution as the follow.

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7.9 Derivation of discrete models from continuous
ones

• For a given discrete-time system,
• we can obtain the solution as the follow.

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7.9 Derivation of discrete models from continuous
ones

• If given a continuous-time system in state
  equation form, you are required to design a
  computer controlled system. Or other way around,
  given a discrete-time system, you are looking for
  the output in continuous-time domain. In these
  cases, how can you manage that?
• Direct derivation
• Via transfer function
• Other methods

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7.9.1 Discrete-time system with ZOH

• As for the continuous-time system
• During the sample interval
• So that

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7.9.1 Discrete-time system with ZOH

• Now make a change of variable in the integral
  that is, let
• so that
• when
• and when
• This changes the solution to

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7.9.1 Discrete-time system with ZOH

• Then we have the discrete-time solution
• As we know that
• Therefore is the plant matrix,
• the driving matrix, and C is the connection
  matrix as the continuous one.

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7.9.2 Steps of discrete model derivation

• 1. Derive the continuous state equations
• 2. Compute
• then

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7.9.2 Steps of discrete model derivation

• 3. Compute
• 4. Finally we have

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7.9.3 Examples of discrete model derivation

• Example Derive the discrete-time system
  corresponding to the following continuous-time
  system when a ZOH circuit is used.
• Notice Suppose there is a ZOH if not given.

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7.9.3 Examples of discrete model derivation

• 1. Derive the continuous-time state equations
• Applying Laplace transform to the both sides of
  the differential equation.

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7.9.3 Examples of discrete model derivation

• Write the state equations in controllable
  canonical form
• That is

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7.9.3 Examples of discrete model derivation

• 2. Compute eAt and eAT

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7.9.3 Examples of discrete model derivation

• 2. Compute eAt and eAT.

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7.9.3 Examples of discrete model derivation

• 3. Compute ?.

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7.9.3 Examples of discrete model derivation

• 4. Finally we have

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7.9.3 Examples of discrete model derivation

• Example Obtain the discrete-time state and
  output equations and the transfer function of the
  following continuous-time system (T1)

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7.9.3 Examples of discrete model derivation

• Write the continuous-time state equations in
  controllable canonical form

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7.9.3 Examples of discrete model derivation

• Compute eAt and eAT

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7.9.3 Examples of discrete model derivation

• Compute ?

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7.9.3 Examples of discrete model derivation

• Finally we have
• As T1

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7.9.3 Examples of discrete model derivation

• The transfer function

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7.9.4 Other approaches to the discretization

• The same transfer function can be obtained by
  taking z-transform of G(s) when it is preceded by
  a sampler and ZOH.

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7.9.4 Other approaches to the discretization

• Matlab has a convenient command to discretize the
  continuous-time state equation
• into
• The command is
• ?,?c2d(A, B, T)

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7.9.4 Other approaches to the discretization

• Example
• A0 10 -2
• B01
• Ad,Bdc2d(A,B,1)
• Ad 1.0000 0.4323
• 0 0.1353
• Bd 0.2838
• 0.4323

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Reading

• Study book
• Module 7 State space and transition matrix (Page
  7.1 to 7.16)

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Exercise

• Exercise 1 Given a continuous-time system as
  below,
• if this system is sampled with T1 sec, represent
  the sampled system in the form as