8. Stability, controllability and observability

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8. Stability, controllability and observability

• Stability
• Controllability
• Observability

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8.1 Stability

• For continuous-time system, the state space model
  is
• The corresponding transfer function is
• where det(sI-A) is a polynomial which forms the
  denominator of the system transfer function. That
  means the roots of det(sI-A)0 (eigenvalues of A)
  are the poles of G(s).

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8.1 Stability

• For discrete-time system, the state space model
  is
• The corresponding transfer function is
• where det(zI-?) is a polynomial which forms the
  denominator of the system transfer function. That
  means the roots of det(zI-?)0 (eigenvalues of ?
  ) are the poles of G(z).

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8.1 Stability

• Therefore, we can tell the stability of a system
  by calculating the eigenvalues of A or ?.
• We can easily tell the stability of a system by
  inspecting its plant matrix A in continuous-time
  system or ? in discrete-time system if A or ? are
  in Diagonal or Jordan canonical form.

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8.1 Stability

• Liapunov stability analysis plays an important
  role in the stability analysis of control systems
  described by state space equations. From the
  classical theory of mechanics, we know that a
  vibratory system is stable if its total energy is
  continually decreasing until an equilibrium state
  is reached. Liapunov stability is based on a
  generalization of this fact.

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8.1 Stability

• Liapunov stability If a system has an
  asymptotically stable equilibrium state, then the
  stored energy of the system displaced within a
  domain of attraction decays with increasing time
  until it finally assumes its minimum value at the
  equilibrium state (Details for Liapunov stability
  and theorems are given in section 5.6 on page
  321- 336 in our textbook).

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8.1 Stability

• Liapunov theorem on stability Suppose a system
  is described by xf(x,t) where f(0,t)0 for all
  t. If there exists a scalar function V(x,t)
  having continuous first partial derivatives and
  satisfying the conditions
• V(x,t) is positive define (V(x,t)gt0) .
• V(x,t) is negative semi-definite (V(x,t)lt0).
• Then the equilibrium state at the origin is
  stable.

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8.1 Stability

• Liapunov stability of Linear system Consider the
  system described by xAx, where x is a state
  vector and A is an nonsingular matrix. A
  necessary and sufficient condition for the
  equilibrium state x0 to be asymptotically stable
  is that, given any positive definite Hermitian
  (or any positive defined real asymmetric) matrix
  Q, there exists a positive definite Hermitian (or
  a positive definite real asymmetric) matrix P
  such that ATPPA-Q.

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8.1 Stability

• Liapunov stability of Linear system Consider the
  system described by x(k1)?x(k), where x is a
  state vector and ? is an nonsingular matrix. A
  necessary and sufficient condition for the
  equilibrium state x0 to be asymptotically stable
  is that, given any positive definite Hermitian
  (or any positive defined real asymmetric) matrix
  Q, there exists a positive definite Hermitian (or
  a positive definite real asymmetric) matrix P
  such that ?TP? -P -Q.

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8.2 Controllability

• Controllability A control system is said to be
  completely state controllable if it is possible
  to transfer the system from any arbitrary initial
  state to any desired state in a finite time
  period. That is, a control system is controllable
  if every state variable can be controlled in a
  finite time period by some unconstrained control
  signal. If any state variable is independent of
  the control signal, then it is impossible to
  control this state variable and therefore the
  system is uncontrollable.

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8.2.1 Controllability for a discrete-time control
system

• Consider the discrete-time system defined by
• We assume that u(k) is a constant for the
  sampling period (ZOH). If given the initial state
  X(0), can we find a control signal to drive the
  system to a desired state X(k)? In other words,
  is this system controllable?
• Using the definition just given, we shall now
  derive the condition for complete state
  controllability.

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8.2.1 Controllability for a discrete-time control
system

• Since the solution of the above equation is

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8.2.1 Controllability for a discrete-time control
system

• That is

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8.2.1 Controllability for a discrete-time control
system

• As ? is an k?1 matrix, we find that each term of
  the matrix is an k?1 matrix or column vector.
  That is the matrix
• is an k?k matrix. If its determinant is not
  zero, we can get the inverse of this matrix. Then
  we have

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8.2.1 Controllability for a discrete-time control
system

• That means that if we chose the input as the
  above, we can transfer the system from any
  initial state X(0) to any arbitrary state X(k) in
  at most k sampling periods.
• That is the system is completely controllable if
  and only if the inverse of controllability matrix
• is available. Or the rank of the
  controllability matrix is k.

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8.2.1 Controllability for a discrete-time control
system

• Why do we concern about the controllability of a
  system?

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8.2.2 Examples for controllability

• Example 1 Considering a system defined by
• is this system controllable?
• First, write the state equation for the above
  system

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8.2.2 Examples for controllability

• That is
• Next, find the controllability matrix

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8.2.2 Examples for controllability

• Finally, determine the rank of controllability
  matrix
• The rank is 2. Therefore, the system is
  controllable.

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8.2.2 Examples for controllability

• Exercise 1 Given the following system
• is it controllable?

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8.3 Observability

• Observability A control system is said to be
  observable if every initial state X(0) can be
  determined from the observation of Y(k) over a
  finite number of sampling periods. The system,
  therefore, is completely observable if every
  transition of the state eventually affects every
  element of the output vector.
• In the state feedback scheme, we require the
  feedback of all state variables. In practical,
  however, some of the state variables are not
  accessible for direct measurement. Then it
  becomes necessary to estimate the un-measurable
  variables in order to implement the state
  variable feedback scheme.

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8.3.1 Observability for a discrete-time control
system

• Consider the discrete-time system defined by
• we assume that u(k) is a constant for the
  sampling period (ZOH). If given the input signal
  u(k) and the output y(k), can we track back to
  the initial state X(0)? Or, is this system
  observable?

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8.3.1 Observability for a discrete-time control
system

• Since the solution of the above equation is
• And y(k) is

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8.3.1 Observability for a discrete-time control
system

• Let k0, 1, 2n-1, we have

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8.3.1 Observability for a discrete-time control
system

• Rewrite all the above equations in matrix
  equation, we have

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8.3.1 Observability for a discrete-time control
system

• As C is an 1?n matrix, we find that each term of
  the following matrix is an 1?n matrix or row
  vector. That is the matrix is an n?n matrix. If
  its determinant is not zero, we can get the
  inverse of the observability matrix.

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8.3.1 Observability for a discrete-time control
system

• Then we have

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8.3.1 Observability for a discrete-time control
system

• This means that if we knew the input and the
  output in n sampling periods, we can track back
  to the system initial state.
• That is the system is completely observable if
  and only if the inverse of observability matrix
  is available. Or the rank of the observability
  matrix is n.

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8.3.1 Observability for a discrete-time control
system

• Why do we concern about the observability of a
  system?

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8.3.2 Examples for observability

• Example 2 Given the following system
• Determine its observability?

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8.3.2 Examples for observability

• Build up the observability matrix
• Find the determinant of observability matrix

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8.3.2 Examples for observability
Exercise 2 Given the following
system determine its observability?
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8.4 Controllability and observability for
continuous system

• For a continuous system
• We can draw the similar conclusions as the
  discrete system

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8.4.1 Controllability of continuous system

• A system is completely controllable if and only
  if the inverse of controllability matrix is
  available. Or the rank of the controllability
  matrix is n.

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8.4.2 Observability of continuous system

• A system is completely observable if and only if
  the inverse of observability matrix is available.
  Or the rank of the observability matrix is n.

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8.5 Effects of discretization on controllability
and observability

• When a continuous-time system with complex poles
  is discretized, the introduction of sampling may
  impair the controllability and observability of
  the resulting discretized system. That is,
  pole-zero cancellation may take place in passing
  from the continuous-time case to the
  discrete-time case. Thus, the discretized system
  may lose controllability and observability.

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8.5 Effects of discretization on controllability
and observability

• Example 3 Consider the following continuous-time
  system
• Determine its controllability and observability.
  If the system is discretized, is the discretized
  system still controllable and observable?

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8.5 Effects of discretization on controllability
and observability

• This system is controllable and observable since
• and

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8.5 Effects of discretization on controllability
and observability

• The discrete-time system can be obtained by

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8.5 Effects of discretization on controllability
and observability

• The discrete-time system obtained by
  discretization is

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8.5 Effects of discretization on controllability
and observability

• The controllability matrix
• The observability matrix
• If , the system will lose its
  controllability and observability.

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8.5 Effects of discretization on controllability
and observability
Exercise 3 Consider the system given by
following transfer function Represent it as
controllable canonical form and observable
canonical form in state equations, and determine
its controllability and observability.
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Tutorials
Exercise 1 2 Given the following
system determine its controllability and
observability?
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Tutorials
Solution
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Tutorials
Exercise 3 Consider the system given by
following transfer function Represent it as
controllable canonical form and observable
canonical form in state equations, and determine
its controllability and observability.
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Tutorials
Solution
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Tutorials
Controllability matrix Observability matrix
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Tutorials
The apparent difference in the controllability
and observability of the same system is caused by
the fact that the original system has a pole-zero
cancellation in the transfer function. If a
pole-zero cancellation occurs in the transfer
function, then the controllability and the
observability vary, depending on how the state
variables are chosen.