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. Let us define a new state vector by where P is a nonsingular matrix. By substituting to 1 2 2 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) ≡