STATE SPACE ANALYSIS
Introduction
In contrast to the frequency domain analysis of the classical control theory, modern control theory utilizes the time-domain state space representation, a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. To abstract from the number of inputs, outputs and states, the variables are expressed as vectors and the differential and algebraic equations are written in matrix form (the latter only being possible when the dynamical system is linear). The state space representation (also known as the "time-domain approach") provides a convenient and compact way to model and analyze systems with multiple inputs and outputs. With inputs and outputs, we would otherwise have to write down Laplace transforms to encode all the information about a system. Unlike the frequency domain approach, the use of the state space representation is not limited to systems with linear components and zero initial conditions. "State space" refers to the space whose axes are the state variables. The state of the system can be represented as a vector within that space.
For MIMO systems, pole placement can be performed mathematically using a state space representation of the open-loop system and calculating a feedback matrix assigning poles in the desired positions. In complicated systems this can require computer-assisted calculation capabilities, and cannot always ensure robustness. Furthermore, all system states are not in general measured and so observers must be included and incorporated in pole placement design.
State
a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations.
State space variable
The internal state variables are the smallest possible subset of system variables that can represent the entire state of the system at any given time.[1] The minimum number of state variables required to represent a given system, n, is usually equal to the order of the system's defining differential equation. If the system is represented in transfer function form, the minimum number of state variables is equal to the order of the transfer function's denominator after it has been reduced to a proper fraction.
Atime-invariant(TIV)systemis a system whose output does not depend explicitly on time. Such systems are regarded as a class of systems in the field ofsystem analysis. Lack of time dependence is captured in the following mathematical property of such a system:
If the input signalproduces an outputthen any time shifted input,, results in a time-shifted output
This property can be satisfied if thetransfer functionof the system is not a function of time except expressed by the input and output.
Linearity
A linear function, we have seen is a function whose graph lies on a straight line, and which can be described by giving its slope and its y intercept.
There is a special kind of linear function, which has a wonderful and important property that is often useful. These are linear functions whose y intercepts are 0 (for example functions like 3x or 5x). This means their graphs pass right through the origin, (the point with coordinates (0, 0)). Such functions are called homogeneous linear functions. They have the property that their values at any combination of two arguments is the same combination of their values at those arguments. In symbols this statement is:
f(ax + bz) = af(x) + bf(z)
Non uniqueness Of State models
State Space representation of canonical forms
Observer Canonical form
Diagonal Cannonical form
Jordan Cannonical form
Physical Systems
Examples :Electrical Systems, Mechanical Systems
Inverted pendulum
Wrt to centre of gravity
Wrt to Horizontal position
Wrt to vertical position
Horizontal position of the cart
Linearizing the equations
Final equations are
Solution of linear time Invariant Systems
Laplace Transform approach of Homogenous systems
Properties of State transition Matrix
Solution of Nonhomogenous Systems
Solution of Nonhomegenous systems using Laplace Transform approach
Cayley Hamilton Theorem
Characteristic Equation
Minimal Polynomial
Sylvester interpolation formula
Minimal polynomial which involves distinct roots
Minimal involves Sameeigen values
Responses of Systems with respect to impulse,step and ramp inputs
At t=0-
Step response
For A is Nonsingular
For Ramp input