AN-1489| Application Note

Techniques of State Space Modeling

Techniques of State Space Modeling
Introduction
Over the past several decades many techniques have been used to model the PWM switch. These include both analytical and circuit based models that become unwieldy for fourth, and some second order systems. The simplest approach is to use the state space analytical method. This method may be used in conjunction with computational software, such as Matlab or Maple, to quickly and easily model a given power stage. In this paper state space modeling is presented in a step-by-step manner such that one may easily implement the approach in software by following a prescribed recipe.

National Semiconductor Application Note 1489 Joel Steenis June 2006

of its two states using convenient shorthand for the internal variables as shown in Figure 2. The equations for both states may be time weighted and averaged using the following relationships:

State Space Modeling
State space modeling is a technique that describes a given system using a system of linear differential equations. These equations are easily manipulated using matrix operations and may be used to relate the internal, or state variables to the system input and output. The state equations may be expressed in matrix form as the following: = Ax + Bu y = Qx + Ru Where is the time derivative of the state variable vector, A is the state matrix, x is the state variable vector, B is a vector, u is the input, y is the output, Q is a transposed vector relating the state variables to the output, and R is a vector relating the input to the output.

Where the matrix subscript refers to the state of the network. Alternatively, these relationships may be expressed as the following: = [A1 d + A2 d'] x + [B1 d + B2 x d'] u y = [Q1d + Q2d'] x + [R1d + R2d'] u Where

The variables x, d, u, and y have both large and small signal components. Each variable in relation to its components may be expressed as: x=X+ d=D+ u = VIN + vin y = VOUT + vout Where the first and second terms on the right hand side of the equality correspond to the large and small signal components of a given variable.

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FIGURE 1. SEPIC Technology Because a given network has two states in CCM , S1 on, S2 off and S1 off, S2 on, the response of the network in each state may be time weighted and averaged. For example, the SEPIC topology shown in Figure 1 may be redrawn for each

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FIGURE 2. Network states `ton' and `toff'

2006 National Semiconductor Corporation

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