Rate Conversion Method Selection for Linearization

Simulink Control Design offers the ability to specify the rate conversion method for linearization of a multirate model. The choice of rate conversion methodology can affect the resulting linearized model. This demo illustrates the extraction of a discrete linear time invariant model using two different rate conversion methods. For more information about the various rate conversion methods, see the Control System Toolbox Documentation.

Example Problem

In the Simulink model scdmrate.mdl there are three different sample rates specified in five blocks. These blocks are

sysC - a continuous linear block,
Integrator - a continuous integrator,
sysTs1 - a block that has a sample time of 0.01 seconds,
sysTs2 - a block that has a sample time of 0.025 seconds, and
Zero-Order Hold - a block that samples the incoming signal at 0.01 seconds.

Open the Simulink model

scdmrate

In this example, we linearize the model between the output of the block sysTs1 and the block Zero-Order Hold. Additionally a loop opening is added at the block Zero-Order Hold to extract the plant model for the system.

model = 'scdmrate';
io(1) = linio('scdmrate/sysTs1',1,'in');
io(2) = linio('scdmrate/Zero-Order Hold',1,'out','on');

Using these linearization points the linearization effectively results in the linearization of the model scdmrate_ol.

scdmrate_ol

To get the operating point, use the command

op = operpoint('scdmrate');

When linearizing a model that contains both continuous and discrete signals, the continuous signals are first converted to discrete signals, using a rate conversion method. The default rate conversion method is zero-order hold. To view or change the rate conversion method, use the RateConversionMethod property in the linoptions function. The following command shows that RateConversionMethod is set to the default setting, zoh:

opt = linoptions

 
Options for LINEARIZE:
    LinearizationAlgorithm        : blockbyblock
    SampleTime (-1 Auto Detect)   : -1
    UseFullBlockNameLabels (on/off): off
 
  Options for 'blockbyblock' algorithm
    BlockReduction (on/off)                   : on
    IgnoreDiscreteStates (on/off)             : off
    RateConversionMethod (zoh/tustin/prewarp) : zoh
    PreWarpFreq                               : 10
 
  Options for 'numericalpert' algorithm
    NumericalPertRel : 1.000000e-005
    NumericalXPert   : []
    NumericalUPert   : []
 
Options for FINDOP:
    OptimizationOptions        : [1x1 struct]
    OptimizerType              : graddescent_elim
    DisplayReport (on/iter/off): on

The following command performs a linearization using the zero-order hold method. Since the linearization includes the Zero-Order Hold block, the sample time of the linearization is 0.01.

sys_zoh = linearize(model,op,io,opt);

The following commands change the rate conversion method to the Tustin (Bilinear transformation) method and then perform a linearization using this method. The sample time of this linearized model is also 0.01.

opt.RateConversionMethod = 'tustin';
sys_tust = linearize(model,op,io,opt);

It is also possible to create a continuous-time linearized model by specifying the sample time as 0 in the options object. The rate conversion method still creates a discrete-time linearized model but then converts the discrete-time model to a continuous-time model.

opt.SampleTime = 0;
sys_c = linearize(model,op,io,opt);

The Bode plots for the three linearizations show the effects of the two rate conversion methods. In this example the Tustin rate conversion method gives the most accurate representation of the phase response of the continuous system and the zero-order hold gives the best match to the magnitude response.

p = bodeoptions('cstprefs');
p.YLimMode = {'manual'};
p.YLim = {[-100 0];[-180 -30]};
p.Grid = 'on';
bodeplot(sys_c,sys_zoh,sys_tust,p);
h = legend('sys_c','sys_zoh','sys_tust','Location','SouthWest');
set(h,'Interpreter','none')