1. A Talk for AFC Summer Course, USU Numerical Issues on Fractional-Order Control
Dingyü Xue, Professor, D Phil
Northeastern University, P R China

2. Outline of the Talk Why numerical issues be addressed
Early talks of Professor Li, theoretical
Question: if a system is given in a complicated form, how can we model/analyze/design (MAD) it?
Motivations of the talk: MAD loop
Linear Fractional-Order Systems
A FOTF (transfer function) Toolbox
Motivations of FOTF Toolbox
Mathematical form and parameters
Designing a MATLAB FOTF object
Inter-connections of FOTF blocks
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

3. continued Analysis of Linear Fractional-Order Systems
Mittag-Leffler functions
Analytical solutions to FO differential equations
Grünwald-Letnikov approximations
Stability and time-, frequency-domain analysis
Block-Diagram Based Simulation of Nonlinear Fractional-Order Systems
Approximations to fractional-order differentiator
A Simulink block
How to construct block diagrams for FOC
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

4. continued Design of Controllers A IO PID controller design tool, a GUI
Optimal control, objective functions
Optimal control solutions
OCD program, a general GUI
PID optimizer, a PID design GUI
Optimal fractional-order controller design
Mixed integer programming and applications
A case study
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

5. 1 Motivations of the Talk
A Systematic View
MAD Process, the loop of modelling, analysis and design for fractional-order systems
No details in math, more on MATLAB solutions
For Linear Systems
Design a FOTF based toolbox
Enable analysis of the system
For Nonlinear FO Systems
Simulation analysis, general methods
Optimum Design of Controllers
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

6. Motivations of a Toolbox
Control System Toolbox, MATLAB
Simple way to denote transfer functions
G=tf(1,[ ])
Easy for connections
*, +, feedback()
Easy to analysis
bode(G)
step(feedback(G,1))
Similar thing should be done to FO systems
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

7. 2. Linear Fractional-Order Transfer Functions
Linear Fractional-Order Differential Equations
Fractional-Order Transfer Functions
Building a FOTF Object in MATLAB
Interconnections of FOTF Blocks
Overload functions for FOTF
Applications by examples
Simplifications of FOTF Blocks
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

8. 2.1 Linear Fractional-Order Transfer Function Models
Fractional-order differential equations
Laplace transform, zero initials, GL definitions
Mathematical form
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

9. 2.2 Creating an Object in MATLAB
Procedures
Create folder
Establish essential *.m files
fotf.m
display.m
Proposed syntaxes for the creation of object fotf
Design other files --- overload functions
mplus.m, mtimes.m, feedback.m
bode.m, step.m and others
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

10. Fotf.m Lile Listing fotf.m file
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

11. display.m file listings
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

12. An Example A give fractional-order plant model
Enter the coefficients/orders first, then call the fotf function to construct a FOTF model
MATLAB code
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

13. 2.3 Interconnections of FOTF Blocks
Typical connections
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

14. Overload Functions Redefining * Algorithms MATLAB implementation
Math form
Terms
Combine the polynomials
MATLAB implementation
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

15. A common function unique.m
Listings
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

16. Redefining + operators
Algorithms (parallel)
Math form
Terms
Implementations
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

17. Redefining feedback() function
Algorithm for feedback connections
Math form
Collecting terms
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

18. A Illustrative Example
Consider a typical feedback control system
Unity negative feedback
Plant
Controller
Find the closed-loop overall system model
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

19. 2.4 Simplifications of FOTF Blocks
For connections
uminus, inv, simple, mpower (limited use)
Example FO-PID
With the overload functions, complicated modelling of FOTF system is possible
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

20. 3. Analysis of Linear Fractional-Order Systems
Mittag-Leffler Functions and Applications
Analytical Solutions to Commensurate-Order Systems
Grünwald-Letnikov Definitions and Applications
Definitions and Computations
A closed-form Solutions of FO differential equations
Stability Analysis of FO Systems
Frequency-Domain Analysis
Time-Domain Analysis
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

21. 3.1 Mittag-Leffler Functions and Applications
Definitions of Mittag-Leffler Functions
One-parameter ML function
An extension to exponential function a=1
Special Examples
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

22. Mittag-Leffler Functions in Two Parameters
Definition
A special case of ML with one parameter
Integer-order derivative
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

23. MATLAB Implementations
Syntax
Math form
Code
Problems with the code: if the terms are too many, xn for large x may not be converge
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

24. Examples of Mittag-Leffler Functions
Draw respectively
Show that
Fail for large x, try Podlubny’s mlf.m function, however no derivatives found
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

25. Analytical Solutions to Linear Fractional-Order Differential Equations
Linear Differential Equation
u(t) is a step function
The general form of the analytical solution is quite complicated, compared with integer-order differential equations
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

26. Mittag-Leffler’s Algorithm Based Step Response Analysis
Formula, complicated
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

27. Simplified Three-Term System
When the model is
Analytical solutions of step response
where
MATLAB implementation
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

28. MATLAB Code Call internally ml_fun.m
Limitation: applies only to three-term systems
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

29. An Illustrative Example
A three-term system
The parameters
Solutions
Analytical Solution
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

30. 3.2 Analytical Solutions to Commensurate-Order Systems with Laplace Transforms
Useful Laplace Transforms
Suitable for impulse signals
Suitable for step signals
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

31. Commensurate-Order Systems
General form of a commensurate-order system
Let
Partial Fraction Expansions can be used
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

32. Impulse Responses Relevent Laplace Transform One has
Analytical Solutions
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

33. Step Response Relevant Laplace Transform One has Analytical Solutions
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

34. An Illustrative Example
System model
Commensurate-order conversion
Partial fraction expansion
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

35. MATLAB functions for ML function
Impulse response
Step response
MATLAB functions for ML function
My ml_fun.m: extremely fast but sometimes fails
Podlubny’s mlf.m: slow but works well
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

36. Draw Responses Impulse Response Step Response (ml_fun fails for )
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

37. 3.3 Grünwald-Letnikov’s Definition of Fractional-Order Differentiation
where
Equivalent to Riemann-Liouville definition
Caputo definition may not be suitable
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

38. Syntax MATLAB code Limitations Samples of y must be known
Cannot be used inside a system
Validation of results, with different h’s
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

39. Closed-Form Solutions to Fractional-Order Differential Equations
Consider the equation below first
Closed-form solution
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

40. Consider the whole equation
Closed-form solutions
Evaluate the right hand side first
Use the previous closed-form solution
Advantage over previous method
Applicable to non-commensurate-order systems
Limitations: cannot be used inside a system
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

41. 3.4 Stability Analysis Only applicable to commensurate-order systems
Characteristic equation, for
Impulse response
Stability assessment
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

42. Also works for non-commensurate-orders
Approximate order setting, by 0.01
A MATLAB function
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

43. An Example A FOTF model Approximate commensurate-order model
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

44. 3.5 Frequency-Domain Analysis
Evaluation of fractional-order transfer functions
Substituting directly into
Write an overload function folder
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

45. Other functions Syntaxes are the same with bode of LTIs
Other overload functions, folder
Nyquist plots
Nichols plots
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

46. Frequency-Domain Analysis
Example
Plant model
Draw Bode diagram
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

47. 3.6 Time-Domain Analysis Based on 3.3, Overload function lsim.m
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

48. Other Overload Functions
Step response
Impulse Response
Example
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

49. Example: A Complicated System
Fractional-order ODE
Transfer function
MATLAB code
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

50. 4 Simulation Analysis of Nonlinear Fractional-Order Systems
Approximating FO Differentiator
Oustaloup filter
Modified filter
Optimal Rational Approximations
Design of a FO Differentiator Block
Block Diagram-Based System Modelling in Simulink, with Examples
Validation of Simulation Results
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

51. 4.1 Filter Approximations to Fractional-Order Differentiators
Why filters
Essential block
Previous simulation algorithms methods not suitable for signals inside a system
Filters
Continuous filters
Carlson, Matsuda, Oustaloup, modified Oustaloup
Discrete Filters, IIR FIR, not discussed here
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

52. Oustaloup’s Filter Target, to fit a fractional-order differentiator
Straight lines
Gain, slope 20g
Phase, constant pg/2
Not possible to fit in entire range
Region of interest
Oustaloup’s algorithm
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

53. A Modified Filter Why modifying Modified filter
Cannot fit well in the selected range
Fitting quality needs improvement
Modified filter
Normally
Fit more accurately in the whole interval of interest
Higher order, limitations
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

54. MATLAB code Oustaloup’s filter Modified filter
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

55. Comparisons of the Filters
An example, 0.5th order derivative
Bode diagram comparisons
0.5th-order derivative for
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

56. Comparisons of a FOTF System model Comparisons of the two filters
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

57. 4.2 Optimal Rational Approximations
An example
High-order integer-order model obtained
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

58. An Sub-Optimal Model Reduction Algorithm
Reduced-order model
Criterion
MATLAB function opt_app()
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

59. An Example Original model Reduced-order model
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

60. Reduction Results Different order combinations
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

61. Frequency Response Fitting by Integer-Order Models
Frequency response fitting functions, invfreqs and invfreqz, can be used to get IO-models
Continuous filter
Discrete filter
Get the transfer function
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

62. A counter-example An open-loop model Frequency response fitting
Optimal reduction
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

63. 4.3 Design of a Simulink block
Why a Simulink block
No global algorithm for fractional-order system, or a FO plant inside a system
Block-diagram based simulation is feasible
Design of a block
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

64. 4.4 Block-Diagram Based Simulation with Examples
Linear system simulation
Let
Converted model
Simulink model c10mfod1.mdl
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

65. A Talk at CSOIS, Utah State University
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

66. A nonlinear system System model Solve
Construct a Simulink block diagram c10mfo2.mdl
Validation of simulation results
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

67. A Talk at CSOIS, Utah State University
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

68. 4.5 Validating Simulation Results
Changing parameters and see whether consistent results can be obtained
Changing Parameters
Simulation control parameters
RelTol
Algorithms
Filter parameters
Frequency range
Order N
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

69. 5 Design of IO/FO Controllers
A PID controller design tool, a GUI
Optimal control, objective functions
Optimal control solutions
OCD program, a general GUI
PID optimizer, a PID design GUI
Optimal fractional-order controller design
Mixed integer programming and applications
A case study
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

70. 5.1 A PID Controller Design Interface
Model type FOPDT
Other Models, by approximation
Time-domain
Frequency-domain
Sub-optimal approximation
Hundreds of tuning rules implemented
Easy to use graphical user interface (GUI)
Test plant
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

71. 5.2 Optimal Control: Optimum Criterion Selection
In control literatures LQ
Good math formula, closed-form solution
No direct relationship with responses
Optimality artificial: Q, R, S selection?
error minimization
, too conservative
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

72. What is the “best” or reasonable criterion?
Other criteria
The fastest time: bang-bang control
The smallest cost criterion
Other error-based criteria
ITAE
IAE
Finite-time ITAE
What is the “best” or reasonable criterion?
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

73. Which is the Most Suitable?
Comparison: , ITAE and finite-time ITAE
For , the following test model is used
plant
Objective: design an optimum PID controller
Not suitable for LQ design
Optimization key point: how to describe the objective function for criterion?
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

74. Finding the optimum controller
Pure unconstrained numerical optimization
Optimal PID controller
Closed-loop response
Control signal
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

75. Simulation Results Analysis
Advantages of controller
Closed-form solution can be obtained
Disadvantages
Treat the error at any time equally, which results in the oscillation in the output
Control signal is extremely large at the very beginning, causes damages in the hardware
Conclusion:
Not a good choice, and cannot handle nonlinearities
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

76. ITAE and Finite-time ITAE
Pure ITAE based controller cannot be designed
And it may neglect the details at initial time
Finite-time ITAE controller design
Simulation must be used to evaluate FT-ITAE
Simulink model: c5moptpid.mdl
Design an objective function
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

77. Design a FT-ITAE-PID Controller
Running optimization procedure
The controller
Closed-loop response
The control signal, saturation can also be used
Finite-time tf selection
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

78. Comparisons among the Criteria
ISE cannot accept saturation elements
Why ISE is not as good as FT-ITAE?
ISE treat the error at any time equally
ITAE puts heavy weights on t. For large t, the optimization forces the error tend to zero much faster.
Why in literatures, ISE or is widely used?
Can easily be solve via norms
The performance is sacrificed
With MATLAB, one should consider FT-ITAE
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

79. Constrained Optimal Controller Design with MATLAB
If it is required that s≤3%
traditional algorithms cannot be used
Constrained optimization solver can be used
A plant
Simulink model c5mopta.mdl
Find an optimal controller
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

80. Objective function (unconstrained)
Design an controller
Constraint function
Design
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

81. 5.3 OCD --- an Optimal Controller Designer Interface
Run OCD interface under MATLAB
The user has to draw the Simulink model first, with unoptimized parameters
No need to write programs
Powerful and useful
An demo of OCD
Plant model: c7mopt1.mdl
Finite-time 30, Actuator saturation
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

82. 5.4 An Integer-Order PID Controller Designer: PID_Optimizer
Can be used to design PID-type controller for linear/nonlinear plant model
Allow different criteria
Enter tf model or draw plant model
Allow actuator saturation
Allow overshoot constraints
Use different optimization such as normal optimization and PSO, GA etc
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

83. 5.5 Optimum Fractional-Order PID Controller Design
For plant , with search, nothing better found than IO PID controller
Different combination of parameters tried, for (1,0.2),(1,2), no better FO PID found
Few other plants should be used to test whether consistent results be obtained
For complicated control problems, fractional-order controller may behave better
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

84. 5.6 Optimal Fractional-Order PID Controller Design using Mixed Integer Programming Approach
Why/How Mixed Integer Programming?
With a step of 0.01 for l and m, too much mesh grids to be used, say 10,000 optimizations needed
Let
Then x4,x5 are integers
Branch and bound algorithm can be used for mixed integer programming, e.g., bnb20 function
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

85. A Case Study: Servo Mechanism
Problem model
Requirements
Actuator saturation:
Tricky one:
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

86. Mathematical Model State variables State space model
The output equation
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

87. The results in ACC’06 paper
Best IO-PID best FO-PID
Seems not global optimal controllers found
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

88. Robustness comparisons
For load changes
+50% changes % changes
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

89. Robustness for nonlinearities and parameters
Coulomb friction
Dead zone nonlinearity
+50% in kq
-50% in kq
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

90. Question: Is There Exist a Plant where PIlDm is Superior to PID Controller?
Topics: Optimum Finite-time ITAE criterion
Models tested, however IO-PID performs better than FO-PID controllers
FOPDT:
for T=1, L=0.2, 1, 2
Servo Mechanism
Fractional-order plant
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems

91. Concluding Remarks A mini-toolbox is designed for FOTF object
Connections *, +, feedback
Stability analysis
Frequency-domain analysis, bode, nyquist, nichols
Time-domain analysis, step, lsim
Filter approximation to FO differentiators
Oustaloup’s filter and the modified version
Model order reduction
A Simulink block for complicated systems
Optimal design of FO controller is explored
11/13/2018Tuesday, , 23:33:28
A Talk at CSOIS, Utah State University
Numerical Issues on Fractional-Order Control Systems