Digital Control Systems
State Space Analysis(1)
INTRODUCTION
State :The state of a dynamic system is the smallest set of variables (called state variables) such that
knowledge of these variables at t = t0, together with knowledge of the input for t ≥t0, completely determines
the behavior of the system for any time t ≥ t0.
State variables:The state variables of a dynamic system are the variables making up the smallest set of
variables that determines the state of the dynamic system.
If at least n variables x1,x2,… xn are needed to completely describe the behavior of a dynamic system (so that,
once the input is given for t ≥ t0. and the initial state at t=t0 is specified, the future state of the system is
completely determined), then those n variables are a set of state variables.
State vector:If n state variables are needed to completely describe the behavior of a given system, then those
state variables can be considered the n components of a vector x called a state vector. A state vector is thus a
vector that uniquely determines 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.
INTRODUCTION
State space: The n-dimensional space whose coordinate axes consist of the x1-axis, x2-axis,..xn-axis is called a
state space.
State-space equations: In state-space analysis, we are concerned with three types of variables that are
involved in the modeling of dynamic systems: input variables, output variables, and state variables.
For Linear or Nonlinear discrete-time systems:
INTRODUCTION
For Linear Time-varying discrete-time systems:
INTRODUCTION
For Linear Time-invariant discrete-time systems:
INTRODUCTION
For Linear or Nonlinear continuous-time systems:
For Linear Time-varying continuous time systems:
INTRODUCTION
For Linear Time Invariant continuous time systems:
STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMS
Canonical Forms for Discrete Time State Space Equations
or
There are many ways to realize state-space representations for the discrete time system represented by these
equations:
STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMS
Controllable Canonial Form:
STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMS
Controllable Canonical Form:
If we reverse the order of the state variables:
STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMS
Observable Canonical Form
STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMS
Observable Canonical Form:
If we reverse the order of the state variables:
STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMS
Diagonal Canonical Form:
If the poles of pulse transfer function are all distinct, then the state-space representation may be put in the
diagonal canonical form as follows:
STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMS
Jordan Canonical Form:
If the poles of pulse transfer function involves a multiple pole of orde m at z=p1 and all other poles are distinct:
STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMS
Example:
STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMS
Example:
EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Rank of a Matrix
A matrix A is called of rank m if the maximum number of linearly independent rows (or columns) is m.
Properties of Rank of a Matrix
EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Properties of Rank of a Matrix (cntd.)
EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Eigenvalues of a Square Matrix
EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Eigenvalues of a Square Matrix
The n roots of the characteristic equation are called eigenvalues of A. They are also called the characteristic
roots.
• An n×n real matrix A does not necessarily possess real eigenvalues.
• Since the characteristic equation is a polynomial with real coefficients, any compex eigenvalues must ocur
in conjugate pairs.
• If we assume the eigenvalues of A to be λi and those of to be μi then μi = (λi)-1
EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Eigenvectors of an n×n Matrix
Similar Matrices
EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Diagonalization of Matrices
EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Jordan Canonical Form
EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Jordan Canonical Form
EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Jordan Canonical Form
: only one eigenvector
: two linearly independent eigenvectors
: three linearly independent eignvectors
EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Jordan Canonical Form
EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Similarity Transformation When an n×n Matrix has Distinct Eigenvalues
EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Similarity Transformation When an n×n Matrix has Distinct Eigenvalues
EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
Similarity Transformation When an n×n Matrix Has Multiple Eigenvalues
STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMS
Nonuniqueness of State Space Representations:
For a given pulse transfer function syste the state space representation is not unique. The state equations,
however, are related to each other by the similarity transformation.
1
Let us define a new state vector
by
2
where P is a nonsingular matrix. By substituting 2 to 1
STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMS
Nonuniqueness of State Space Representations:
Let us define
then
≡
Since matrix P can be any nonsingular nn matrix, there are infinetely many state space representations for a
given system.
If we choose P properly:
(If diagonalization is not possible)