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An Introduction to Fractional Control [electronic resource] / Duarte Valério and José Sá da Costa

Author
Valério, Duarte
Published
Stevenage : IET, 2012.
Physical Description
1 online resource (200 pages).
Additional Creators
Costa, José Sá da.
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Contents
Machine generated contents note: 1.Preliminaries -- 1.1.The Gamma function -- 1.1.1.Definition and basic properties -- 1.1.2.Combinations -- 1.2.Calculus: integer orders -- 1.3.Laplace transforms -- 1.4.Continued fractions -- 1.4.1.Continued fraction expansions of real numbers -- 1.4.2.Evaluation of continued fractions -- 1.4.3.Continued fraction expansions of functions -- Further reading -- Appendix to Chapter 1 -- 2.Fractional calculus: real orders -- 2.1.Simple examples -- 2.1.1.Derivatives of eλt -- 2.1.2.Derivatives of tλ -- 2.1.3.Derivatives of sin(λt) and cos(λt) -- 2.2.Definitions -- 2.2.1.Non-local operator -- 2.2.2.Terminals -- 2.2.3.Linear operator -- 2.2.4.Geometrical interpretation -- 2.2.5.Alternative notations and names -- 2.2.6.Relations between the different definitions -- 2.2.7.Law of exponents -- 2.3.Laplace transforms -- 2.3.1.Riemann[—]Liouville and Grunwald[—]Letnikoff definitions -- 2.3.2.Caputo definition -- 2.3.3.Some Laplace transforms -- 2.4.Some fractional derivatives -- 2.5.Numerical evaluation -- 2.5.1.Approximations based upon the definitions -- 2.5.2.Short memory principle -- Further reading -- Appendix to Chapter 2 -- 3.Fractional transfer functions -- 3.1.Definitions -- 3.1.1.SISO transfer functions -- 3.1.2.Fractional MIMO transfer function matrixes -- 3.2.Time responses -- 3.3.Stability -- 3.4.Frequency response -- 3.4.1.Frequency response of a generic transfer function -- 3.4.2.Study of sa -- 3.4.3.Study of 1 / (s/a)a+1 -- 3.4.4.Study of 1 / (s/a)2a+2ζ(s/a)a+1 -- 3.4.5.Steady-state errors -- 3.4.6.Some irrational fractional transfer functions -- 3.5.Stability, continued -- Further reading -- Appendix to Chapter 3 -- 4.Approximations of fractional transfer functions -- 4.1.Continuous approximations -- 4.1.1.Crone approximation -- 4.1.2.Carlson approximation -- 4.1.3.Matsuda approximation -- 4.2.Discrete-time approximations of sa -- 4.2.1.Griinwald[—]Letnikoff approximation -- 4.2.2.Approximations based on truncated series -- 4.2.3.FIR approximations based on time responses -- 4.2.4.Impulse response approximation -- 4.2.5.Step response approximation -- 4.2.6.Poles and zeros -- 4.3.General considerations -- 4.3.1.Approximations of sa -- 4.3.2.Approximations of general fractional transfer functions -- 4.3.3.Approximations of irrational fractional transfer functions -- Further reading -- Appendix to Chapter 4 -- 5.Fractional identification -- 5.1.Identification from a time response -- 5.1.1.Known commensurability order -- 5.1.2.Non-commensurable model -- 5.1.3.Filtering -- 5.1.4.Order optimisation -- 5.2.Identification from a frequency response -- 5.2.1.Levy's method -- 5.2.2.Levy's method, first formulation -- 5.2.3.Levy's method, second formulation -- 5.2.4.Making use of data from many frequencies -- 5.2.5.Levy's method: summing matrixes -- 5.2.6.Levy's method: stacking matrixes -- 5.2.7.Weighted Levy's method -- 5.2.8.Iterations of Sanathanan and Koerner -- 5.3.Identification from the phase of a frequency response -- 5.3.1.Integer transfer function -- 5.3.2.Discrete-time transfer function without zeros -- 5.3.3.Discrete-time transfer function without poles -- 5.3.4.Fractional transfer function without zeros -- 5.3.5.Fractional transfer function without poles -- 5.3.6.Solving the four cases above -- 5.3.7.Fractional and discrete-time transfer functions with both poles and zeros -- 5.3.8.The effect of noise -- Further reading -- Appendix to Chapter 5 -- 6.First and second-generation Crone control -- 6.1.Robustness towards plant gain variations -- 6.2.First-generation Crone controller -- 6.3.Second-generation Crone controller -- 6.4.Filters -- Further reading -- Appendix to Chapter 6 -- 7.Fractional PIDs -- 7.1.Definition -- 7.1.1.Integer PID -- 7.1.2.Fractional PID -- 7.1.3.Frequency behaviour -- 7.1.4.Tuning methods -- 7.2.Analytical tuning: frequency response -- 7.3.Analytical tuning: internal model control -- 7.4.Numerical tuning of fractional PIDs -- 7.5.Tuning rules for fractional PIDs -- 7.5.1.S-shaped step response -- 7.5.2.Critical gain -- 7.5.3.Changing the time unit -- Further reading -- Appendix to Chapter 7 -- 8.Fractional reset control -- Further reading -- Appendix to Chapter 8 -- 9.Fractional H2 and Hinfinity control -- 9.1.H2 norm -- 9.1.1.Definition -- 9.1.2.H2 norm of 1/sβ -- 9.1.3.H2 norm of K/sa+a -- 9.2.Hinfinity norm -- 9.2.1.Definition -- 9.2.2.Numerical computation -- 9.2.3.Analytical computation for a SISO plant -- 9.3.H2 and Hinfinity controllers -- Further reading -- 10.Pseudo-state-space representations -- 10.1.Pseudo-state-space representations of MIMO systems -- 10.1.1.Linear system -- 10.1.2.Commensurable system -- 10.1.3.Fractional MIMO transfer function matrix -- 10.1.4.Non-linear fractional systems -- 10.2.Pseudo-state-space representations of SISO systems -- 10.3.Discretisation -- Further reading -- Appendix to Chapter 10 -- 11.Fractional sliding mode control -- 11.1.The commensurable SISO case -- 11.1.1.Sliding surface -- 11.1.2.Upper bound for the error -- 11.1.3.Following the sliding surface -- 11.1.4.Uncertainty in [ƒ]A(x(t)) -- 11.1.5.Uncertainty in [ƒ]A(x(t)) and in [ƒ]B(x(t)) -- 11.1.6.Avoiding chattering -- 11.2.The more general SISO case -- 11.3.The commensurable MIMO case -- 11.3.1.Sliding surface -- 11.3.2.Uncertainty in f(x(t)) and in B(t) -- 11.4.The more general MIMO case -- Further reading -- Appendix to Chapter 11 -- 12.Trajectory planning -- 12.1.Avoiding obstacles -- 12.2.Reaching the target -- 12.2.1.Maximum acceptable danger level -- 12.2.2.Fractional repulsive force -- Further reading -- Appendix to Chapter 12 -- 13.Fractional calculus: complex orders -- 13.1.Preliminaries -- 13.2.Definitions of complex derivatives -- Further reading -- Appendix to Chapter 13 -- 14.Complex order transfer functions -- 14.1.Complex order transfer functions -- 14.1.1.Definitions -- 14.1.2.Time and frequency responses -- 14.1.3.Frequency response of sz -- 14.2.Approximations of complex order transfer functions -- 14.2.1.Fractional approximations of complex order transfer functions -- 14.2.2.Frequency response of GR, G, GR and G, -- 14.2.3.Non-linearities in the frequency response of GR, G, GR and G -- 14.2.4.Integer approximations of complex order transfer functions -- 14.2.5.Crone approximation of a linear phase -- 14.2.6.Logarithmic phase Crone controller -- 15.Third generation Crone control -- 15.1.For SISO plants -- 15.1.1.Uncertainties in the Nichols chart -- 15.1.2.Multiple orders -- 15.2.For MIMO plants -- Further reading -- Appendix to Chapter 15 -- 16.Fractional calculus: variable orders -- 16.1.First definition -- 16.2.Second definition -- 16.3.Third definition -- 16.4.Linear operator -- 16.5.Non-linear operator -- 16.6.Distributed order derivatives -- 16.7.Memory of z -- 16.8.Evaluation of fractional derivatives -- Further reading -- Appendix to Chapter 16 -- 17.Fractional time-varying control -- 17.1.Time-varying transfer functions -- 17.2.Approximations of fractional time-varying transfer functions -- 17.2.1.Approximation for real time-varying orders -- 17.2.2.Approximation for complex time-varying orders -- 17.2.3.Approximation for time-varying linear phase plants -- 17.2.4.Approximations with memory -- 17.3.Application examples -- 17.3.1.Plant with a constant phase -- 17.3.2.Plant with a variable phase -- 17.3.3.Plant with a varying slope linear phase -- Further reading -- Appendix to Chapter 17 -- 18.General considerations -- 18.1.Types of fractional control -- 18.2.Role of fractional derivatives -- 18.3.Controllers for SISO plants -- 18.4.Controllers for MIMO plants -- 18.5.Control of non-linear plants -- 18.6.Control of time-varying plants or with time-varying specifications -- Further reading.
Summary
The book reviews developments in the following fields: fractional calculus; real orders; fractional transfer functions; fractional identification; crone control; fractional PIDs; fractional reset control; H2 control; Hinfinity control; pseudo-state-space representation; fractional sliding mode control; trajectory planning; complex orders; variable orders; and time-varying control.
Subject(s)
Other Subject(s)
ISBN
9781849195461
View MARC record | catkey: 25792635