ControlDesign[ParameterIdentify]
TimeDomain
identify parameters of a first-order with time-delay (FOTD) model using time-domain techniques
Calling Sequence
Parameters
Options
Description
Examples
References
TimeDomain(sys, opts)
sys
-
System; a DynamicSystems system object in continuous-time domain; must be single-input single-output (SISO) and stable.
opts
(optional) equation(s) of the form option = value; specify options for the TimeDomain command
method : one of moments, stepPercentage, stepIntegrate or doubletPulse
Specifies the method of parameter identification.
moments uses the values of the transfer function and its first and second derivatives at zero to estimate the parameters. This is the method of moments as described in Astrom et. al. (1995). This method is the default.
stepPercentage uses the points on the system step-response for which the response reaches 1/3 and 2/3 of its final value, as described in O'Dwyer (2006).
stepIntegrate is based on certain areas under the step response plot as described in Nishikawa et. al. (1984).
doubletPulse uses the system response to a doublet pulse input as described in Astrom et. al. (2006). The amplitude and period of the doublet pulse input signal must be specified by the user. This method only applies to systems with no symbolic parameters.
Amp = realcons
Amplitude of the doublet pulse input signal in the doubletPulse method. Must be a positive-real number.
Tp = realcons
The half period of the doublet pulse input signal in the doubletPulse method. Must be a positive-real number.
simplifyOutput = truefalse
In case that the system contains symbolic parameters, specifies whether the symbolic expressions for the identified parameters should be simplified. If true then the identification output is simplified using the simplify command. The default value is true. For details see: simplify.
The TimeDomain command estimates the parameters of an FOTD model for the given system, sys, using the time-domain method specified in the options. The command returns a list containing gain, time constant, and time delay of the identified FOTD model, respectively. In other words, TimeDomain(sys,opts) computes the parameters Kp,T,L for the model Kpⅇ−sLsT+1 that best approximates sys according to the chosen method, with Kp≠0, 0≤T, and 0<L.
When the system has symbolic parameters, the FOTD model parameters are returned as symbolic expressions of the system parameters. The computed results are theoretically valid for the parameter space only when the following conditions hold:
The system is open-loop stable. The parametric stability condition can be obtained using the DynamicSystems[RouthTable] command with the stablecondition option set to true. For details see: DynamicSystems[RouthTable].
The computed algebraic expressions for the FOTD model parameters time-constant (T) and time-delay (L) are positive-real, and that of the FOTD model gain (Kp) is either positive-real (corresponding to systems with positive DC gain) or negative-real (corresponding to systems with negative DC gain).
withControlDesignParameterIdentify:
withDynamicSystems:
Basic Usage
Use each of the methods to identify the parameters K,T,L for the following system.
sys≔NewSystem1s2+2s+3:
params≔TimeDomainsys,method=stepPercentage
params≔13,0.5461468460,0.3747961646
params≔TimeDomainsys,method=stepIntegrate
params≔13,0.6721581969,0.2695254684
params≔TimeDomainsys,method=doubletPulse,Amp=1,Tp=0.6
params≔13,0.619100864277216,0.492805324230899
params≔TimeDomainsys,method=moments
params≔13,23,13
Symbolic System
Identify the parameters for a system with symbolic coefficients.
sys≔NewSystems+as3+bs2+35s+c:
sys:-tf
s+as3+bs2+35s+c
params≔TimeDomainsys
params≔ac,2cb−1225a2+c2cc2a,−2cb−1225a2+c2c2+35a−c35c2aa2+−c2cb−1225a2+c2ca+2cb−1225a2+c2c4+2abc−1225a+35cac4a2c3a2
Use DynamicSystems[RouthTable] to find the domains of b and c that give a stable system.
stable≔RouthTabledenomsys:-tf1,1,s,stablecondition=true
stable≔0<band0<cand0<35−cb
Verify that the following values for a, b, and c are within the stable region, then substitute into params to find the corresponding FOTD parameters.
vals≔a=1,b=1,c=5
evalstable,vals
0<1and0<5and0<30
evalparams,vals
15,11905,11905−3−562500250
FOTD Approximation
Starting with the FOTD transfer function, use series to generate a second-order polynomial approximation for the delay term, create a DynamicSystems model from the rational polynomial, then use TimeDomain and simplify to extract the original parameters.
fotd≔Kpexp−sL1+sT
fotd≔Kpⅇ−sLsT+1
fotd2≔convertseriesnumerfotd,s,3,polynomdenomfotd
fotd2≔Kp−KpLs+12KpL2s2sT+1
sys≔NewSystemfotd2:
params≔Kp,KpTKp,KpL+TKp−KpT+−Kp2T+L2+2LT+2T2Kp2−Kp2T2Kp2Kp2
simplifyparamsassuming0<Kp,0≤T,0<L
Kp,T,L
Unstable System
Create a system with a zero at −1, poles at −2, 1+I, and 1−I, and a gain of 3.
sys≔NewSystem−1,−2,1+I,1−I,3:
Attempt to identify the parameters for this unstable system; doing so raises an error.
TimeDomainsys
Error, (in ControlDesign:-ParameterIdentify:-TimeDomain) model is unstable. Cannot apply parameter identification to unstable models.
[1] K. J. Astrom and T. Hagglund, Automatic Tuning of PID Controllers, ISA, 1995.
[2] K. J. Astrom and T. Hagglund, Advanced PID Control, ISA, 2006.
[3] Y. Nishikawa, N. Sannomiya, T. Ohta and H. Tanaka, A Method for Auto_Tuning of PID Control Parameters, Automatica, Vol. 20, No. 3, pp. 321-332, 1984.
[4] A. O'Dwyer, Handbook of PI and PID Controller Tuning Rules, 2nd Edition, Imperial College Press, 2006.
See Also
ControlDesign
ControlDesign[ParameterIdentify][FrequencyDomain]
DynamicSystems[RouthTable]
simplify
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