<!DOCTYPE OMNotebook>
<Notebook>
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   <TextCell style="Title" >
    <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Arial; font-size:38pt; font-weight:600; color:#000000;">Sampled Systems&lt;/p>&lt;/body>&lt;/html></Text>
   </TextCell>
   <GroupCell closed="false" >
    <TextCell style="Section" >
     <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times; font-size:18pt; font-weight:600; color:#000000;">Simple Periodic Sampler&lt;/p>&lt;/body>&lt;/html></Text>
    </TextCell>
    <TextCell style="Text" >
     <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times New Roman; font-size:12pt; color:#000000;">The following model is a &lt;span style=" font-style:italic;">simple periodic sampler&lt;/span> with a sampling period &lt;span style=" font-family:Courier New;">T&lt;/span> that is constant and defined as a parameter that can be changed by the model user. This model has only one kind of event, the sampling event. We use the built-in function &lt;span style=" font-family:Courier New;">sample&lt;/span> in the &lt;span style=" font-family:Courier New;">when&lt;/span>-condition &lt;span style=" font-family:Courier New;">sample(0, T)&lt;/span> to periodically generate the sampling events with a period time &lt;span style=" font-family:Courier New;">T&lt;/span>. This is a simple model using the state space equations.&lt;br />&lt;/p>&lt;/body>&lt;/html></Text>
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     <Input>model SimplePeriodicSampler
  parameter Real T = 1        "Sample period";
  input Real u            "Input used at sample events";
  discrete output Real y      "Output computed at sample events";
  replaceable function f = Modelica.Math.atan2;
  replaceable function h = Modelica.Math.atan2;
protected
  discrete Real x;          // discrete state variable
equation
  when sample(0, T) then
    x = f(pre(x), u);        // state update expression
    y = h(pre(x), u);        // output expression
  end when;
end SimplePeriodicSampler;</Input>
     <Output></Output>
    </InputCell>
   </GroupCell>
   <GroupCell closed="false" >
    <TextCell style="Section" >
     <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times; font-size:18pt; font-weight:600; color:#000000;">Base Class for Sampling Models&lt;/p>&lt;/body>&lt;/html></Text>
    </TextCell>
    <TextCell style="Text" >
     <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times New Roman; font-size:12pt; color:#000000;">It is possible to formulate a base class for scalar sampling models using state-space equations.The class &lt;span style=" font-family:Courier New;">BaseSampler&lt;/span> below can for example be specialized to either a periodic sampler or an aperiodic sampler.&lt;br />&lt;/p>&lt;/body>&lt;/html></Text>
    </TextCell>
    <InputCell style="Input" closed="true" >
     <Input>partial class BaseSampler
  input Real u;
  discrete output Real y;
  Boolean doSample;
  replaceable function f = Modelica.Math.atan2;
  replaceable function h = Modelica.Math.atan2;
protected
  discrete Real x;
equation
  when doSample then
    x = f(pre(x), u);                  // state update expression
    x = h(pre(x), u);                  // output expression
    end when;
end BaseSampler;</Input>
     <Output></Output>
    </InputCell>
    <TextCell style="Text" >
     <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times New Roman; font-size:12pt; color:#000000;">&lt;br />Below the base model is extended to a periodic sampling model &lt;span style=" font-family:Courier New;">PeriodicSampler&lt;/span> with a fixed period time T.&lt;br />&lt;/p>&lt;/body>&lt;/html></Text>
    </TextCell>
    <InputCell style="Input" closed="true" >
     <Input>model PeriodicSampler
  extend BaseSampler;
  parameter Real T = 1        "sample period";
equation
  doSample = sample(0, T);
end PeriodicSampler;</Input>
     <Output></Output>
    </InputCell>
   </GroupCell>
   <GroupCell closed="false" >
    <TextCell style="Section" >
     <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times; font-size:18pt; font-weight:600; color:#000000;">Aperiodic Sampler&lt;/p>&lt;/body>&lt;/html></Text>
    </TextCell>
    <TextCell style="Text" >
     <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times New Roman; font-size:12pt; color:#000000;">Besides the above periodic sampler, an aperiodic sampling model with a time-varying sample period can be obtained by specializing the base class. A variable &lt;span style=" font-family:Courier New;">nextSampling&lt;/span> is used for the next sampling time, which is current time &lt;span style=" font-family:Courier New;">+ periodTime&lt;/span> when the current time has reached the previous value of &lt;span style=" font-family:Courier New;">nextSampling&lt;/span>. In this example we increase the period time by the logarithm of the current time just to have a time varying period time.&lt;br />&lt;/p>&lt;/body>&lt;/html></Text>
    </TextCell>
    <InputCell style="Input" closed="true" >
     <Input>model APeriodicSampler
  extend BaseSampler;
  discrete Real nextSampling(start = 0);
  discrete Real periodTime(start = 1)    "time-varying sample period";
equation
  doSample = time > nextSampling;
  when doSample then
    nextSampling = pre(nextSampling) + periodTime;
    periodTime = pre(periodTime) + log(pre(periodTime));
  end when;
end APeriodicSampler;</Input>
     <Output></Output>
    </InputCell>
   </GroupCell>
   <GroupCell closed="false" >
    <TextCell style="Section" >
     <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times; font-size:18pt; font-weight:600; color:#000000;">Discrete Scalar State Space[NonBreakingSpace]Sampling Model&lt;/p>&lt;/body>&lt;/html></Text>
    </TextCell>
    <TextCell style="Text" >
     <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times New Roman; font-size:12pt; color:#000000;">A discrete first order scalar state space model with periodic sampling can be formulated as follows, essentially being a special case of the above BaseSampler model even though we do not extend that class in this particular case.&lt;br />&lt;/p>&lt;/body>&lt;/html></Text>
    </TextCell>
    <InputCell style="Input" closed="true" >
     <Input>model DiscreteScalarStateSpace
  parameter Real a, b, c, d;
  parameter Real T = 1;
  input Real u;
  discrete output Real y;
protected
  discrete Real x;
equation
  when sample(0, T) then
    x = a * pre(x) + b * u;
    y = c * pre(x) + d * u;
  end when;
end DiscreteScalarStateSpace;</Input>
     <Output></Output>
    </InputCell>
    <TextCell style="Text" >
     <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times New Roman; font-size:12pt; color:#000000;">&lt;br />Alternatively, the discrete scalar state space model can be expressed by extending the &lt;span style=" font-family:Courier New;">BaseSampler&lt;/span> model for periodic sampling and redeclaring the functions &lt;span style=" font-family:Courier New;">f&lt;/span> and &lt;span style=" font-family:Courier New;">g&lt;/span> as below. However, in this case the result is not shorter or more readable since the syntax for expressing the local functions &lt;span style=" font-family:Courier New;">f&lt;/span>1 and &lt;span style=" font-family:Courier New;">h1&lt;/span> is not very concise. On the other hand, if the functions &lt;span style=" font-family:Courier New;">f&lt;/span>1 and &lt;span style=" font-family:Courier New;">h1&lt;/span> instead would be used in several places, the modeling style based on replaceable functions would be preferable.&lt;br />&lt;/p>&lt;/body>&lt;/html></Text>
    </TextCell>
    <InputCell style="Input" closed="true" >
     <Input>model DiscreteScalarStateSpace2
  extends BaseSampler(redeclare function f = f1, redeclare function h = h1);
  parameter Real a, b, c, d;
  parameter Real T = 1        "sample period";
  function f1
    input Real x1;
    input Real x2;
    input Real y1;
  algorithm
    y1 := a * x1 + b * x2;
  end f1;
  function h1
    input Real x1;
    input Real x2;
    input Real y1;
  algorithm
    y1 := c * x1 + d * x2;
  end h1;
equation
  doSample = sample(0, T);
end DiscreteScalarStateSpace2;</Input>
     <Output></Output>
    </InputCell>
   </GroupCell>
   <GroupCell closed="false" >
    <TextCell style="Section" >
     <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times; font-size:18pt; font-weight:600; color:#000000;">Discrete Vector State Space Sampling Model&lt;/p>&lt;/body>&lt;/html></Text>
    </TextCell>
    <TextCell style="Text" >
     <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times New Roman; font-size:12pt; color:#000000;">Usually state space models are expressed using the more general vector form, as in the &lt;span style=" font-family:Courier New;">DiscreteVectorStateSpace&lt;/span> periodic sampling model below:&lt;/p>&lt;/body>&lt;/html></Text>
    </TextCell>
    <InputCell style="Input" closed="true" >
     <Input>model DiscreteVectorStateSpace
  parameter Integer n = 5, m = 4, p = 2;
  parameter Real A[n, n] = fill(1, n, n);
  parameter Real B[n, m] = fill(2, n, m);
  parameter Real C[p, n] = fill(3, p, n);
  parameter Real D[p, m] = fill(4, p, m);
  parameter Real T = 1;
  input Real u[m];
  discrete output Real y[p];
protected
  discrete Real x[n];// = fill(2, n);
equation
  when sample(0, T) then
    x = A * pre(x) + B * u;
    y = C * pre(x) + D * u;
  end when;
end DiscreteVectorStateSpace;</Input>
     <Output></Output>
    </InputCell>
    <TextCell style="Text" >
     <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style="-qt-paragraph-type:empty; margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times New Roman; font-size:12pt; color:#000000;">&lt;/p>&lt;/body>&lt;/html></Text>
    </TextCell>
    <InputCell style="Input" closed="true" >
     <Input>model DVSSTest
  DiscreteVectorStateSpace dvss;
equation
  dvss.u= fill(time,dvss.m);
end DVSSTest;</Input>
     <Output></Output>
    </InputCell>
    <GroupCell closed="false" >
     <TextCell style="Subsection" >
      <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times; font-size:14pt; font-weight:600; color:#000000;">Simulation of DVSSTest&lt;/p>&lt;/body>&lt;/html></Text>
     </TextCell>
     <InputCell style="Input" closed="true" >
      <Input>simulate( DVSSTest, stopTime=10 );</Input>
      <Output></Output>
     </InputCell>
     <TextCell style="Output" >
      <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style="-qt-paragraph-type:empty; margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Courier New; font-size:11pt; color:#000000;">&lt;/p>&lt;/body>&lt;/html></Text>
     </TextCell>
    </GroupCell>
   </GroupCell>
   <GroupCell closed="false" >
    <TextCell style="Section" >
     <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times; font-size:18pt; font-weight:600; color:#000000;">Two Rate Sampling Model&lt;/p>&lt;/body>&lt;/html></Text>
    </TextCell>
    <TextCell style="Text" >
     <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times New Roman; font-size:12pt; color:#000000;">In the &lt;span style=" font-family:Courier New;">TwoRateSampler&lt;/span> model below the slow sampler is five times slower than the fast sampler, i.e. the &lt;span style=" font-family:Courier New;">slowSample&lt;/span> when-equation is evaluated at every 5th activation of the &lt;span style=" font-family:Courier New;">fastSample when&lt;/span>-equation.The two samplers are synchronized via equations involving the &lt;span style=" font-family:Courier New;">cyCounter&lt;/span> variable. &lt;br />&lt;/p>&lt;/body>&lt;/html></Text>
    </TextCell>
    <InputCell style="Input" closed="true" >
     <Input>model TwoRateSampler
  discrete Real x,y;
  Boolean fastSample;
  Boolean slowSample;
  Integer cyCounter(start=0);          // Cyclic count 0,1,2,3,4, 0,1,2,3,4,... 
 equation 
  fastSample = sample(0,1);          // Define the fast clock
  when fastSample then   
    cyCounter  = if pre(cyCounter) &lt; 5 then pre(cyCounter)+1 else 0;
    slowSample = pre(cyCounter) == 0;       // Define the slow clock

  end when;
 equation
  when fastSample then              // fast sampling
    x = sin(time);               
  end when;
 equation
  when slowSample then                // slow sampling (5-times slower)
    y = log(time);               
  end when;
end TwoRateSampler;</Input>
     <Output></Output>
    </InputCell>
    <GroupCell closed="false" >
     <TextCell style="Subsection" >
      <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times; font-size:14pt; font-weight:600; color:#000000;">Simulation of TwoRateSampler&lt;/p>&lt;/body>&lt;/html></Text>
     </TextCell>
     <InputCell style="Input" closed="true" >
      <Input>simulate( TwoRateSampler, stopTime=100 );</Input>
      <Output></Output>
     </InputCell>
     <TextCell style="Output" >
      <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style="-qt-paragraph-type:empty; margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Courier New; font-size:11pt; color:#000000;">&lt;/p>&lt;/body>&lt;/html></Text>
     </TextCell>
     <InputCell style="Input" closed="true" >
      <Input>plot( { x, y }, xrange={7,100} );</Input>
      <Output>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style="-qt-paragraph-type:empty; margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Courier New; font-size:11pt; color:#000000;">&lt;/p>&lt;/body>&lt;/html></Output>
     </InputCell>
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      <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times New Roman; font-size:12pt; color:#000000;">&lt;img src="file:///C:/OpenModelica132/bin/OMNoteboook_tempfiles/7.png" width="288" height="214" />&lt;/p>&lt;/body>&lt;/html></Text>
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