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In tuning a controller for a process in a feedback control system, a method is provided for bringing the system into asymmetric self-excited oscillations for measuring the frequency of the oscillations, average over the period value of the process output signal and average over the period control si

In tuning a controller for a process in a feedback control system, a method is provided for bringing the system into asymmetric self-excited oscillations for measuring the frequency of the oscillations, average over the period value of the process output signal and average over the period control signal and tuning the controller in dependence of the measurements obtained. An element having a non-linear characteristic is introduced into the system in series with the process and set point signal is applied to excite asymmetric self-excited oscillations in the system. An algorithm and formulas are given for identification of the process model having the form of first order plus dead time transfer function. PI controller settings are given as a function of the dead time/time constant ratio. An apparatus for performing the method is disclosed.

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I claim: 1. A method of tuning a controller for a process in a feedback control system comprising the steps of: (a) bringing the system into asymmetric self-excited oscillations via the introduction into the system an element having a nonlinear characteristic in series with the process, and applyin

I claim: 1. A method of tuning a controller for a process in a feedback control system comprising the steps of: (a) bringing the system into asymmetric self-excited oscillations via the introduction into the system an element having a nonlinear characteristic in series with the process, and applying a constant set point signal f0 to the system, so that the error signal is the difference between the set point and the process output signal, the error signal is an input of the nonlinear element, and the output of the nonlinear element is a control signal for the process; (b) measuring the frequency of the oscillations Ω, the average value of the process output signal y0, and the average control signal u0, and wherein the averages are taken over the period of the oscillations; (c) choosing the process model, parameters of which have unknown values; (d) identifying of the parameters of the process model with the measurements obtained; (e) tuning the controller in dependence on the identified parameters of the process model, wherein the original process model with unknown parameters is transformed into the form of a formula of the locus of a perturbed relay system (LPRS) being a complex function, which has same unknown parameters. 2. The method as recited in claim 1, wherein the nonlinear element is the hysteresis relay characteristic with positive, zero or negative hysteresis. 3. The method as recited in claim 1, wherein with the measurement obtained, the process model parameters identification step further includes the steps of: (a) calculation of the static gain of the process as: description="In-line Formulae" end="lead"K=y 0/u0description="In-line Formulae" end="tail" (b) and calculation of one point of the LPRS J(Ω) of the process at the frequency of the oscillations Ω(which is the measured value of the LPRS at the frequency Ω) as: Im J(Ω)=-πb/(4c) where f0 if the set point, y0 is the average value of the process output signal, u0 is the average value of the control signal, b is a half of the hysteresis of the relay, c is the amplitude of the relay. 4. The method as recited in claim 1, wherein two unknown process model parameters are identified as the values that provide equality of the LPRS calculated on the basis of the chosen process model at the frequency of the oscillations to the point of the LPRS calculated through the measurements obtained via the asymmetric relay feedback test. 5. The method as recited in claim 1, wherein multiple asymmetric relay feedback tests are performed with different values of the hysteresis of the relay to measure several points of the LPRS. 6. The method as recited in claim 1, wherein in case of nonlinear character of the process, multiple asymmetric relay feedback tests are performed with decreasing values of the output amplitude of the relay to obtain a better local model of the process. 7. The method as recited in claim 1, wherein a combination of multiple asymmetric relay feedback tests with different values of the hysteresis of the relay and decreasing values of the output amplitude of the relay are performed to measure several points of the LPRS and obtain a better local model of the process in case of its nonlinear character. 8. The method as recited in claim 1, wherein the LPRS for a process model in the form of transfer function Wp(s) of the process is calculated with the use of the following formula: where m=0 for type 0 servo systems (non-integrating process) and m=1 for type 1 servo systems (integrating process), ω is a frequency, M1, M2 are numbers sufficiently large for the above formula being an adequate approximation of respective infinite series. 9. The method as recited in claim 1, wherein the LPRS for at process model in the form of transfer function Wp(s) of the process is calculated on the basis of expansion of the process transfer function Wp(s) into partial fractions, calculation of the partial LPRSs with the use of formulas of the following Table and summation of the partial LPRSs: Formulas of LPRS J(ω) Transfer function W(s) LPRS J(ω) K/s 0-jπ2K/(8ω) K/(Ts + 1) 0.5K(1-α cosech α)-j0.25 πK th(α/2), α = π/(Tω) K/[( T1s + 1)(T2s + 1)] 0.5K[1-T1/(T1 -T2) α1 cosech α1 -T2/(T2 -T1) α 2 cosech α2)]- j0.25 πK/(T1 -T2) [T1 th(α 1/2)-T2 th(α2/2)], α1 = π /(T1ω), α2 = π/(T2ω) K/(s2 + 2 ξs + 1) 0.5K[(1-(B + γC)/(sin2β + sh2α )]- j0.25 πK(shα-γsinβ)/(chα + cosβ) α = πξ/ω, β = π(1-ξ2) 1/2/ω, γ = α/β, B = αcosβshα + βsinβchaα, C = αsinβchα-βcosβshα Ks/(s2 + 2ξs + 1) 0.5K [ξ (B + γC)-π/ω cosβshα] /(sin2β + sh2α)]- j0.25 K π(1-ξ2)-1/2 sinβ/(chα + cosβ) α = πξ/ω, β = π(1-ξ2) 1/2/ω, γ = α/β, B = αcosβshα + βsinβchα, C = αsinβchα-βcosβshα, Ks/(s + 1)2 0.5K[α(-shα + αchα)/sh2α-j0.25 πα/(1 + chα)], α = π/ω Ks/[(T1s + 1) (T2s + 1)] 0.5K/(T2 -T1) [α2 cosech α 2 -α1 cosech α1]- j0.25 K π/(T2 -T1) [th(α1/2)-th(α2/2)] , α1 = π/(T1ω), α2 = π/(T2ω) 10. The method as recited in claim 1, wherein the chosen model of the process is given by the transfer function Wp(s) of 1st order with a dead time: description="In-line Formulae" end="lead"W p(s)=Kexp(-τs)/(Ts+1)description="In-line Formulae" end="tail" where K is a static gain, T is a time constant, τ is a dead time. 11. The controller using the method recited in claim 1, wherein the controller is realized as a processor based device and all above formulas, the nonlinear element, and the tuning rules are realized as computer programs with the use of applicable programming languages. 12. The method as recited in claim 3, wherein the LPRS for a process model in the form of state space description for type 0, or non-integrating process servo systems is calculated with the use of the following formula: where A, B and C are matrices of the following state space description of a relay system: description="In-line Formulae" end="lead"{dot over (x)}=Ax+Budescription="In-line Formulae" end="tail" description="In-line Formulae" end="lead"y=Cxdescription="In-line Formulae" end="tail" where A is an n횞n matrix, B is an n횞1 matrix, C is an 1횞n matrix, f0 is the set point, σ is the error signal, 2b is the hysteresis of the relay function, x is the state vector, y is the process output, u is the control, n is the order of the system; or the LPRS for type 1, or integration process, servo systems is calculated with the use of the following formula: where A, B and C are matrices of the following state space description of a relay system: description="In-line Formulae" end="lead"{dot over (x)}=Ax+Budescription="In-line Formulae" end="tail" description="In-line Formulae" end="lead"{dot over (y)}=Cx-f0description="In-line Formulae" end="tail" where A is an (n-1)횞(n-1) matrix, B is an (n-1)횞1 matrix, C is an 1횞(n-1) matrix, n is the order of the system. 13. The method as recited in claim 3, wherein if an external unknown constant or slowly changing disturbance is applied to the process, each static gain of the process is calculated on the basis of a pair of asymmetric relay feedback tests, each with different average control signal value, as a quotient of the increment of the average process output signal and increment of the average control signal. 14. The method as recited in claim 5 or 7, wherein with the measurement obtained, the process model parameters identification step further includes the steps of: (a) calculation of the static gain of the process as: (b) and calculation of N points of the LPRS J(Ω k) of the process at the frequencies of the oscillations Ω k, k=1, 2, . . . , N (which are the measured values of the LPRS at frequencies Ωk, k=1, 2, . . . , N) as: Im J(Ωk)=-πbk/(4c) where Ωk, y0k,, u0k, bk (=1, 2 . . . , N) are values of Ω, y0,, u0, and b for k-th asymmetric relay feedback test, f0 if the set point, y0 is the average value of the process output signal, u0 is the average value of the control signal, b is a half of the hysteresis of the relay, c is the amplitude of the relay. 15. The method as recited in claim 5 or 7, wherein 2N unknown parameters of the process model are found as the solution of 2N nonlinear algebraic equations, which are obtained as conditions of equality of the N measured points via the asymmetric relay feedback tests of the LPRS the process to the N points of the LPRS calculated on the basis of the chosen process model at the frequencies of the oscillations. 16. The method as recited in claim 5 or 7, wherein the unknown parameters of the process model are found through the least squares criterion or another criterion fitting of the LPRS expressed via certain process model parameters to the LPRS points obtained through the asymmetric relay feedback tests. 17. The method as recited in claim 10, wherein the corresponding LPRS of the process is given by: where α=π/ωT, γ=τ/T. 18. The method as recited in claim 10, wherein with the measurements of the frequency of the oscillations Ω, average output signal y0 and average control signal u0 obtained, calculation of the parameters K, T and τ of the process transfer function comprises the following steps: (a) calculating the static gain K as: (b) solving the following equation for α: (c) calculating the time constant T as: (d) calculating the dead time τ as: 19. The method as recited in claim 10, wherein the proportional gain Kp and the integrator gain Ki in PI control are calculated for desired overshoot being a constraint, proportional gain Kp and integrator gain Ki are sought as a solution of the parameter optimization problem with settling time being an objective function. 20. The method as recited in claim 10, wherein for the desired overshoot being a given value, the calculation of the proportional gain Kp and the integrator gain Ki comprises the following steps: (a) calculation of the normalized values of Kp and Ki, denoted as K0p and K 0i respectively, as follows: first the normalized time constants are calculated as: for desired overshoot 20% integrator time constant is calculated as: description="In-line Formulae" end="lead"T 0i=1.60τ/T;description="In-line Formulae" end="tail" for desired overshoot 10%, T0i=1.80τ/T; for desired overshoot 5%, T0i=1.95τ/T with intermediate values determined by using interpolation; after that the normalized gains K0i are to be calculated as reciprocals of the respective time constants as: description="In-line Formulae" end="lead"K 0i=1/T0idescription="In-line Formulae" end="tail" wherein the normalized proportional gain K0p is taken from the following Table with intermediate values determined by using interpolation: Normalized proportional gain settings Overshoot [%] τ/T = 0.1 τ/T = 0.2 τ/T = 0.3 τ/T = 0.4 τ/T = 0.5 τ/T = 0.6 τ/T = 0.7 τ/T = 0.8 τ/T = 0.9 τ/T = 1.0 τ/T = 1.5 20 K0p = 7.177 3.702 2.564 2.007 1.683 1.473 1.329 1.225 1.146 1.086 0.915 10 K0p = 5.957 3.058 2.120 1.673 1.419 1.258 1.148 1.068 1.008 0.963 0.833 5 K0p = 5.203 2.624 1.823 1.483 1.294 1.170 1.082 1.014 0.964 0.924 0.808 (b) The proportional gain Kp and integrator gain Ki are calculated as Kp=K0p/K and Ki=K0i/T/K, where K is the static gain of the process, T is the time constant of the process.