MPC with Laguerre Functions

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MPC with Laguerre Functions
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Recall
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Discrete-time Laguerre Networks
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Where a is the pole of the discrete-time Laguerre
network, and 0 a lt 1 for stability of the
network. The free parameter, a, is required to be
selected by the user this is also called the
scaling factor. The Laguerre networks are well
known for their orthonormality.
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Discrete Laguerre network
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Inverse z-transform
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For example
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The orthonormality in time domain
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The Special Case when a 0
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Laguerre functions become a set of pulses when a
0.
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Example1
The difference equation for the first three
Laguerre functions is
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with a 0.5, the Laguerre functions decay to
zero in less than 15 samples. By contrast, with a
0.9, the Laguerre functions decay to zero at a
much slower speed (approximately 50 samples are
required). Also, the initial values for the
Laguerre functions with the smaller a value are
larger than the corresponding functions with a
larger a, particularly with the first function in
each set.
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To investigate the orthonormal property of the
Laguerre functions, we calculate the finite sums,
for a 0.5 (S1) and for a 0.9 (S2)
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We increase the number of samples from 50 to 90,
and a 0.9, we obtain that
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Use of Laguerre Networks in System Description
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Impulse response of a stable system is H(k), then
with a given number of terms N, H(k) is written
as
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MATLAB Tutorial Use of Laguerre Functions in
System Modelling
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when a 0 which was the case of the pulse
operator, there are 60 parameters required to
capture the response. However, with the Laguerre
polynomial with a 0.8, there were only 4
parameters required to perform the same task.
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Design Framework
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Within this design framework, the control horizon
Nc from the previous approach has vanished.
Instead, the number of terms N is used to
describe the complexity of the trajectory in
conjunction with the parameter a. For instance, a
larger value of a can be selected to achieve a
long control horizon with a smaller number of
parameters N required in the optimization
procedure. We note that when a 0, N Nc
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Cost Functions
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Orthonormal properties of the Laguerre functions
with a sufficiently large prediction horizon Np
so that
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Discrete-time linear quadratic regulators (DLQR)
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A- Regulator Design where the Set-point r(k) 0
if Q is chosen to be CTC both equations are the
same
The purpose of the control is to maintain
closed-loop stability and reject disturbances
occurring in the plant.
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B- Inclusion of Set-point Signal r(k) ? 0
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Cost function
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Minimization of the Objective Function
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For simplicity of the expression, we define
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The Minimum of the Cost
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To compute the prediction, the convolution sum
needs to be computed.
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Receding Horizon Control
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The Optimal Trajectory of Incremental Control
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Example 2. Suppose that a first-order system is
described by the state equation
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Examine solutions where N increases from 1 to 4
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Example 3
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