Lecture 24 Introduction to state variable modeling Overall idea Example Simulating system response using MATLAB Related educational modules: –Section 2.6.1, 2.6.2

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 N th 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: 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): The previous equations can be written in matrix form:

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 computer- generated) 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 special- purpose 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 Choosing states as before results in previous state variable model Output equation:

MATLAB syntax Type at MATLAB’s command prompt (>>): >> 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 0]; >> b = [1/L1; 0; 0]; >> c = [1 0 0]; >> d = 0; >> step(A,b,c,d)

Simulated Response

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 input- output representation of a system