This example shows how to design a controller and analyze the design for multiple plant models using the SISO Design Tool.
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Consider the typical feedback problem, shown in the following figure, where the controller C is designed to satisfy some performance objective.
Typically, the dynamics of the system, G, are not exactly known and can vary based on operating conditions. For example:
The variations in system dynamics may be a result of manufacturing tolerances that are typically defined as a range about the nominal value (e.g. the value of resistance for a resistor 5 ohms +/- 1%) .
The system dynamics vary with operating condition (e.g. aircraft dynamics change based on altitude and speed).
When designing controllers for these types of systems, the performance objectives for all variations of the system must be satisfied.
You can model such systems as a set of LTI models and create an LTI array to store the collection of these models. Then, use the SISO Design to design a controller for a nominal plant in the array and analyze the controller design for the set of plants.
The following list summarizes how to obtain an array of LTI models:
Control System Toolbox™
Simulink® Control Design™
Robust Control Toolbox™
System Identification Toolbox™
In this example, the plant model is a second order system defined as:
The first step is to construct the LTI array for the combinations of and .
wn = [1,1.5,2]; zeta = [.2,.5,.8]; ct = 1; for ct1 = 1:length(wn) for ct2 = 1:length(zeta) zetai = zeta(ct2); wni = wn(ct1); G(1,1,ct) = tf(wni^2,[1,2*zetai*wni,wni^2]); ct = ct+1; end end size(G) % Note that the array must be a row or column array.
9x1 array of transfer functions. Each model has 1 outputs and 1 inputs.
Next, start the SISO Design Tool.
The SISO Design Tool opens with a Bode and Root Locus open-loop editors.
By default, the nominal model used for design is the first element in the LTI array.
The root locus editor displays the root locus for the nominal model and the closed loop pole locations associated with the set of plants.
The Bode editor displays both the nominal model response and responses of the set of plants.
In these editors, you can interactively tune the gain, poles and zeros of the compensator while simultaneously visualizing the effect on the set of plants.
To change the nominal model:
1. Go to the Architecture tab of the SISO Design Task.
2. Click Multimodel Configuration.
The Multimodel Configuration Dialog window allows you to change the nominal model.
For example, selecting 5 for the nominal model results in the following changes to the Bode and Root Locus editors.
The options for plotting responses of the set of plants are accessed by right-clicking the plots. Use Multimodel Display to:
Turn off the responses.
Show them as individual responses or as an envelope encapsulating the individual responses as shown below.
The SISO Design Tool provides you design and analysis tools for multimodel systems. The tools allow you to analyze the performance and stability of a set of systems simultaneously.