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MPC with Laguerre Functions

Recall:

Discrete-time Laguerre Networks

Where a is the pole of the discrete-time Laguerre network, and 0 ≤ a < 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.

Discrete Laguerre network

Inverse z-transform

For example

The orthonormality in time domain:

The Special Case when a = 0

Laguerre functions become a set of pulses when a = 0.

Example1

The difference equation for the first three Laguerrefunctions is:

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.

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)

We increase the number of samples from 50 to90, and a = 0.9, we obtain that:

Use of Laguerre Networks in System Description

Impulse response of a stable system is H(k), then with a given number of terms N, H(k) is written as:

MATLAB Tutorial: Use of Laguerre Functions in System Modelling

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.

Design Framework

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 beselected 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

Cost Functions

Orthonormal properties of the Laguerre functions

with a sufficiently large prediction horizon Np so that:

Discrete-time linear quadratic regulators (DLQR)

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.

B- Inclusion of Set-point Signal r(k) ≠ 0

Cost function:

Minimization of the Objective Function:

For simplicity of the expression, we define:

The Minimum of the Cost

To compute the prediction, the convolution sum

needs to be computed.

Receding Horizon Control

The Optimal Trajectory of Incremental Control

Example 2. Suppose that a first-order system is described by the state equation:

Examine solutions where N increases from 1 to 4

Convergence of the Incremental Control Trajectory

The optimal controller that minimizes the cost function is also called a discrete-time linear quadratic regulator (DLQR):

Example 3