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 SYSTEMSCanonical 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 SYSTEMSControllable Canonial Form:

STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMSControllable Canonical Form:

If we reverse the order of the state variables:

STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMSObservable Canonical Form

STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMSObservable Canonical Form:

If we reverse the order of the state variables:

STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMSDiagonal 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 SYSTEMSJordan 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 SYSTEMSExample:

STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMSExample:

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

EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.

STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMSNonuniqueness 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.

Let us define a new state vector by

where P is a nonsingular matrix. By substituting to

1

2

2 1

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)

STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMS

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