A Talk for AFC Summer Course, USU Numerical Issues on Fractional-Order Control Dingyü Xue, Professor, D Phil Northeastern University, P R China xuedingyu@mail.neu.edu.cn

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Motivations of a Toolbox Control System Toolbox, MATLAB Simple way to denote transfer functions G=tf(1,[1 2 3 4]) Easy for connections *, +, feedback() Easy to analysis bode(G) step(feedback(G,1)) Similar thing should be done to FO systems 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

2.1 Linear Fractional-Order Transfer Function Models Fractional-order differential equations Laplace transform, zero initials, GL definitions Mathematical form 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

2.2 Creating an Object in MATLAB Procedures Create an @fotf 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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Fotf.m Lile Listing fotf.m file 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

display.m file listings 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

2.3 Interconnections of FOTF Blocks Typical connections 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Overload Functions Redefining * Algorithms MATLAB implementation Math form Terms Combine the polynomials MATLAB implementation 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

A common function unique.m Listings 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Redefining + operators Algorithms (parallel) Math form Terms Implementations 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Redefining feedback() function Algorithm for feedback connections Math form Collecting terms 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

A Illustrative Example Consider a typical feedback control system Unity negative feedback Plant Controller Find the closed-loop overall system model 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Mittag-Leffler Functions in Two Parameters Definition A special case of ML with one parameter Integer-order derivative 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Mittag-Leffler’s Algorithm Based Step Response Analysis Formula, complicated 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Simplified Three-Term System When the model is Analytical solutions of step response where MATLAB implementation 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

MATLAB Code Call internally ml_fun.m Limitation: applies only to three-term systems 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

An Illustrative Example A three-term system The parameters Solutions Analytical Solution 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Commensurate-Order Systems General form of a commensurate-order system Let Partial Fraction Expansions can be used 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Impulse Responses Relevent Laplace Transform One has Analytical Solutions 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Step Response Relevant Laplace Transform One has Analytical Solutions 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

An Illustrative Example System model Commensurate-order conversion Partial fraction expansion 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Draw Responses Impulse Response Step Response (ml_fun fails for ) 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Closed-Form Solutions to Fractional-Order Differential Equations Consider the equation below first Closed-form solution 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

3.4 Stability Analysis Only applicable to commensurate-order systems Characteristic equation, for Impulse response Stability assessment 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Also works for non-commensurate-orders Approximate order setting, by 0.01 A MATLAB function 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

An Example A FOTF model Approximate commensurate-order model 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

3.5 Frequency-Domain Analysis Evaluation of fractional-order transfer functions Substituting directly into Write an overload function in @fotf folder 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Other functions Syntaxes are the same with bode of LTIs Other overload functions, in @fotf folder Nyquist plots Nichols plots 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Frequency-Domain Analysis Example Plant model Draw Bode diagram 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

3.6 Time-Domain Analysis Based on 3.3, Overload function lsim.m 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Other Overload Functions Step response Impulse Response Example 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Example: A Complicated System Fractional-order ODE Transfer function MATLAB code 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

MATLAB code Oustaloup’s filter Modified filter 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Comparisons of the Filters An example, 0.5th order derivative Bode diagram comparisons 0.5th-order derivative for 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Comparisons of a FOTF System model Comparisons of the two filters 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

4.2 Optimal Rational Approximations An example High-order integer-order model obtained 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

An Sub-Optimal Model Reduction Algorithm Reduced-order model Criterion MATLAB function opt_app() 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

An Example Original model Reduced-order model 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Reduction Results Different order combinations 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

A counter-example An open-loop model Frequency response fitting Optimal reduction 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

4.4 Block-Diagram Based Simulation with Examples Linear system simulation Let Converted model Simulink model c10mfod1.mdl 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

A Talk at CSOIS, Utah State University 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

A nonlinear system System model Solve Construct a Simulink block diagram c10mfo2.mdl Validation of simulation results 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

A Talk at CSOIS, Utah State University 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Finding the optimum controller Pure unconstrained numerical optimization Optimal PID controller Closed-loop response Control signal 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Objective function (unconstrained) Design an controller Constraint function Design 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

A Case Study: Servo Mechanism Problem model Requirements Actuator saturation: Tricky one: 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Mathematical Model State variables State space model The output equation 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

The results in ACC’06 paper Best IO-PID best FO-PID Seems not global optimal controllers found 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Robustness comparisons For load changes +50% changes -50% changes 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

Robustness for nonlinearities and parameters Coulomb friction Dead zone nonlinearity +50% in kq -50% in kq 11/13/2018Tuesday, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems

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, 2009-7- 14, 23:33:28 A Talk at CSOIS, Utah State University Numerical Issues on Fractional-Order Control Systems