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What Is Linearization?

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Linearization Background Analytic Representations of Linear Models

Linearization Background

A linearized model is an approximation to a nonlinear system, which is valid in a small region around the operating point of the system. Engineers often use linearization in the design and analysis of control systems and physical models.

The following figure shows a visual representation of a nonlinear system as a block diagram. The diagram consists of an external input signal, u(t), a measured output signal, y(t), and the nonlinear system that describes the system's states and its dynamic behavior, P.

You can also express a nonlinear system in terms of the state space equations

where x(t) represents the system's states, u(t) represents the inputs, and y(t) represents the outputs. In these equations, the variables vary continuously with time. Discrete-time and multi-rate models are discussed in Analytic Representations of Linear Models. A linear time-invariant approximation to this nonlinear system is valid in a region around the operating point at t=t₀, x(t₀)=x₀, and u(t₀)=u₀. If the values of the system's states, x(t) and inputs, u(t) are close enough to the operating point, the system will behave approximately linearly.

Simulink uses a series of connected blocks to model physical systems and control systems. Input and output signals connect the blocks, which represent mathematical operations. The nonlinear system, P, in the previous figure, represents a series of connected Simulink blocks.

The Simulink Control Design software linearizes both continuous and discrete-time nonlinear systems by computing the state-space matrices of the linearized model, A, B, C, and D, using one of the linearization algorithms described in Choosing Linearization Settings and Algorithms.

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Analytic Representations of Linear Models

Linearization of Nonlinear Models

To describe the linearized model, it helps to first define a new set of variables centered about the operating point of the states, inputs, and outputs:

The value of the outputs at the operating point is given by y(t₀)=g(x₀,u₀,t₀)=y₀.

Note   When comparing a linearized model with the original model, remember that the convention used in this book is to write the linearized model in terms of δx, δu, and δy. The value of each of these variables at the operating point is zero.

The linearized state space equations written in terms of δx(t), δu(t), and δy(t) are

where A, B, C, and D are constant coefficient matrices. These matrices are defined as the Jacobians of the system, evaluated at the operating point

The transfer function of the linearized model can be used in place of the system, P, in the previous figure. To find the transfer function, divide the Laplace transform of δy(t) by the Laplace transform of δu(t):

Linearization of Discrete-Time Models

Discrete-time models are similar to continuous models, discussed in the previous section, with the exception that the values of system variables change at discrete times, t_k, where k is an integer value. The state-space equations for a nonlinear, discrete-time system are

A linear time-invariant approximation to this system is valid in a region around the operating point

If the values of the system's states, x_k, inputs, u_k, and outputs, y_k, are close enough to the operating point, the system will behave approximately linearly. As with continuous time systems it is helpful to define variables centered about the operating point values

where the value of the outputs at the operating point are defined as:

The linearized state-space equations can then be written in terms of these new variables

where A, B, C, and D are given by

Linearization of Multirate Models

Multirate models involve states with various sampling rates. This means that the state variables change values at different times and with different frequencies, with some variables possibly changing continuously. The general state-space equations for a nonlinear, multirate system are

where k₁,..., k_m are integer values and ,..., are discrete times.

The linearized equations will approximate this system as a single-rate discrete model:

For more information, see the Simulink Control Design demo "Linearization of Multirate Models".

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Exact Linearization Using the GUI

Ways to Linearize Models

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