digital control systems state space analysis(1). introduction
TRANSCRIPT
![Page 1: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/1.jpg)
Digital Control Systems
State Space Analysis(1)
![Page 2: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/2.jpg)
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.
![Page 3: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/3.jpg)
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:
![Page 4: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/4.jpg)
INTRODUCTION
For Linear Time-varying discrete-time systems:
![Page 5: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/5.jpg)
INTRODUCTION
For Linear Time-invariant discrete-time systems:
![Page 6: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/6.jpg)
INTRODUCTION
For Linear or Nonlinear continuous-time systems:
For Linear Time-varying continuous time systems:
![Page 7: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/7.jpg)
INTRODUCTION
For Linear Time Invariant continuous time systems:
![Page 8: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/8.jpg)
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:
![Page 9: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/9.jpg)
STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMSControllable Canonial Form:
![Page 10: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/10.jpg)
STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMSControllable Canonical Form:
If we reverse the order of the state variables:
![Page 11: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/11.jpg)
STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMSObservable Canonical Form
![Page 12: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/12.jpg)
STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMSObservable Canonical Form:
If we reverse the order of the state variables:
![Page 13: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/13.jpg)
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:
![Page 14: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/14.jpg)
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:
![Page 15: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/15.jpg)
STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMSExample:
![Page 16: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/16.jpg)
STATE SPACE REPRESENTATIONS OF DISCRETE TIME SYSTEMSExample:
![Page 17: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/17.jpg)
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.
![Page 18: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/18.jpg)
Properties of Rank of a Matrix (cntd.)
EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
![Page 19: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/19.jpg)
Eigenvalues of a Square Matrix
EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
![Page 20: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/20.jpg)
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.
![Page 21: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/21.jpg)
Eigenvectors of an n×n Matrix
Similar Matrices
EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
![Page 22: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/22.jpg)
Diagonalization of Matrices
EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
![Page 23: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/23.jpg)
Jordan Canonical Form
EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
![Page 24: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/24.jpg)
Jordan Canonical Form
EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
![Page 25: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/25.jpg)
Jordan Canonical Form
: only one eigenvector
: two linearly independent eigenvectors
: three linearly independent eignvectors
EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
![Page 26: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/26.jpg)
Jordan Canonical Form
EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
![Page 27: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/27.jpg)
Similarity Transformation When an n×n Matrix has Distinct Eigenvalues
EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
![Page 28: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/28.jpg)
Similarity Transformation When an n×n Matrix has Distinct Eigenvalues
EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
![Page 29: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/29.jpg)
Similarity Transformation When an n×n Matrix Has Multiple Eigenvalues
EIGENVALUES,EIGENVECTORSAND SIMILARITY TRANSF.
![Page 30: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/30.jpg)
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
![Page 31: Digital Control Systems State Space Analysis(1). INTRODUCTION](https://reader030.vdocuments.mx/reader030/viewer/2022012908/56649d765503460f94a57afd/html5/thumbnails/31.jpg)
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
≡