State Space 1

9- State space analysis
Modern control theory is contrasted with conventional control theory in that the former is applicable to multi-input-multi-output systems which may be linear or nonlinear, time-invariant or time-varying, while the later is applicable only to linear time invariant single-input-single-output systems. Also, modern control theory is essentially a time domain approach, while conventional control theory is a complex frequency-domain approach.
State : The state of a dynamic system is the smallest set of variables (called state variables) such that the knowledge of these variables at time t=to together with the input for t ? to completely determines the behavior of the system for any time t ? to.
State variables: The state variables of a dynamic system are the smallest set of variables which determine the state of the dynamic system. If at least n variables x1(t), x2(t), …, xn(t) are needed to completely describe the behavior of a dynamic system (such that once the input is given for t?to and the initial state at t=to is specified, the future state of the system is completely determined).
State vector : If n-state variables are needed to completely describe the behavior of a given system, then these n state variables can be considered to be the n components of a vector x(t).
State space : The n-dimensional space whose coordinates axis consists of the x1 axis, x2 axis, …, xn axis is called a state space. Any state can be represented by a point in the state space.
Example 9.1 Consider the RLC network shown below

Figure 9.1: RLC network
Choose i(t) and vc(t) as the state variables, then find the state-space representation for this network.
Solution
The equations describing the system dynamics are



State space representation of systems
A dynamic system consisting of a finite number of lumped elements may be described by ordinary differential equations in which time is the independent variable. By use of vector-matrix notation, an nth-order differential equation may be represented by a first-order vector-matrix differential equation.
State-Space Representation of nth-Order Systems of Linear Differential Equations in which the Forcing Function Does Not Involve Derivative Terms.

Consider the following nth-order system
(9.1)
Noting that the knowledge of y(0), y(1)(0), . . . , y(n-1) (0), together with the input u(t) for t? 0, determines completely the future behavior of the system. Let us define




Then eq. (9.1) can be written as
(9.2)
or
(9.3)
where

The output equation becomes

or
(9.4)
Where

Example 9.2
Consider the mechanical system shown in Figure 3-16. We assume that the system is linear. The external force u(t) is the input to the system, and the displacement y(t) of the mass is the output. The displacement y(t) is measured from the equilibrium position in the absence of the external force. This system is a single-input-single-output system.

Figure 9.2: mechanical system.
From the diagram, the system equation is

This system is of second order. This means that the system involves two integrators. Let us define state variables x1(t) and x2(t) as

Then we obtain

or

The output equation is

The vector-matrix form can be written as




Figure 9.3: Block diagram of mechanical system