Transcript Lecture 24 •Introduction to state variable modeling • Overall idea • Example •Simulating system response using MATLAB •Related educational materials: –Chapter 9.1 - 9.3

Lecture 24
•Introduction to state variable modeling
• Overall idea
• Example
•Simulating system response using MATLAB
•Related educational materials:
–Chapter 9.1 - 9.3
Overview of our approach to date
• We wrote one differential equation in one unknown
• This is called an input-output equation
• The equation relates the output directly to the input
• This approach is useful only for low-order systems
• We applied the approach to first and second order
systems
• Allows us to obtain insight into the overall system
behavior relative to input-output equation coefficients
• We want to simulate higher-order systems
Notes on generating input-output equations
• I defined unknowns as voltages across capacitors
and currents through inductors
• Used KVL and KCL to write multiple equations in
these unknowns
• Combined them into a single equation in one unknown
• Why did I recommend this approach?
State-variable models
• State-variable models consist of N first-order
differential equations in N unknowns
• Mathematically equivalent to a single Nth order equation
in one unknown
• The unknowns are the state variables
• The state variables are the minimum amount of
information necessary to completely describe the
system’s state
Definition of system state
• The state of a system is the information necessary
to completely characterize the system at some
instant in time
• Given the state at some time, and the input at all
subsequent times, we can determine the output at any
subsequent time
• Comparable to initial conditions
• The state uniquely determines the energy in all the
system’s energy storage elements
Choice of state variables
• The choice of state variables is not unique
• We will choose as our state variables:
• Voltages across capacitors
• Currents through inductors
• As we have seen, these variables will describe the
energy stored in all energy storage elements
• They are a legitimate choice for state variables
Form of the state variable model
• State equations are in the form:
x1  a11 x1  a12 x2    a1 N x N  b1u
x 2  a21 x1  a22 x2    a2 N x N  b2 u

x n  a N 1 x1  a N 2 x2    a NN x N  bN u
• x1 , x2 , , x N are the state variables and u is the input
• “dot” notation denotes differentiation with respect to time
Output equations
• Any other circuit parameter can be determined from
the states (this relation is called the output equation):
y  c1 x1  c2 x2   c N x N  du
• The previous equations can be written in matrix form:
x ( t )  A x( t )  bu( t )
y( t )  c x( t )  du( t )
Matrix form – continued
Example 1
• Determine state equations governing the circuit below
Example 1 continued
Matrix form of state equations
Example 1 (still continued)
Determine output equation
Simulating system response using MATLAB
• Analytical solution of higher-order differential
equations is generally not feasible
• Typically, we resort to numerical (or computergenerated) solutions
• Numerical simulation of the system response
• MATLAB is a common tool for solving differential
equations numerically
• Contains a number of built-in simulation capabilities
Prerequisite MATLAB background
• It is assumed that you have access to:
• MATLAB
• MATLAB’s control systems toolbox
• It is also assumed that you can use MATLAB to:
• Set up arrays
• Perform basic matrix operations
• Perform basic plotting operations
Natural and step responses using MATLAB
• MATLAB’s control systems toolbox provides specialpurpose commands to determine the natural and step
responses of a given system
• System is (for us) defined by a state variable model
{A,b,c,d}
• MATLAB commands and syntax:
• step(A,b,c,d) plots the system’s unit step response
• initial(A,b,c,d,x0) plots the system’s natural
response to the initial conditions provided in the vector x0
Example 1 – simulated step response
• Use MATLAB to simulate the current through the 0.1H
inductor
x1
+ x3 -
x2
• Choosing states as before results in previous state variable
model
• Output equation:
MATLAB syntax
• Type at MATLAB’s command prompt (>>):
>>
>>
0
>>
>>
>>
>>
R = 100; L1 = 0.1; L2 = 0.5; C = 1e-3;
A = [-R/L1 R/L1 -1/L1; R/L2 -R/L2 0; 1/C
0];
b = [1/L1; 0; 0];
c = [1 0 0];
d = 0;
step(A,b,c,d)
Simulated Response
Step Response
0.08
0.06
Amplitude
0.04
0.02
0
-0.02
-0.04
-0.06
0
0.2
0.4
0.6
0.8
Time (sec)
1
1.2
1.4
1.6
State variable models and input-output equations
• Advantages of state variable models:
• Fairly easy to write governing equations for higher-order
systems
• State variable models easy to solve numerically
• Changing output does not affect state equations!
• {A,b,c,d} can give direct information about system behavior
• Advantages of input-output equations
• Higher-order system responses are often interpreted in
terms of first and second order system responses
• Laplace transform based techniques often rely on inputoutput representation of a system