2. دانشگاه صنعتي اميركبير
دانشكده مهندسي پزشكي
استاد درس
دكتر فرزاد توحيدخواه
بهمن 1387
MPC with Laguerre Functions
کنترل پيش بين-دکتر توحيدخواه 2

3. Recall:

4. Discrete-time Laguerre Networks

5. 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.

6. Discrete Laguerre network

8. Inverse z-transform

9. For example

10. The orthonormality in time domain:

11. The Special Case when a = 0

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

13. Example1
The difference equation for the first three Laguerre
functions is:

14. 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.

15. 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)

16. We increase the number of samples from 50 to
90, and a = 0.9, we obtain that:

17. Use of Laguerre Networks in System Description

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

19. MATLAB Tutorial: Use of Laguerre Functions in System Modelling

21. 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.

22. Design Framework

24. 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

26. Cost Functions

28. Orthonormal properties of the Laguerre functions
with a sufficiently large prediction horizon Np so that:

29. Discrete-time linear quadratic regulators (DLQR)

30. 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.

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

32. Cost function:

33. Minimization of the Objective Function:

36. For simplicity of the expression, we define:

37. The Minimum of the Cost

39. To compute the prediction, the convolution sum
needs to be computed.

41. Receding Horizon Control

44. The Optimal Trajectory of Incremental Control

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

46. Examine solutions where N increases from 1 to 4

50. Convergence of the Incremental Control Trajectory
The optimal controller that minimizes the cost function is also called a discrete-time linear quadratic regulator (DLQR):

51. Example 3