# Free Final Time Input Design Problem for Robust Entropy-Like System Parameter Estimation

## Abstract

**:**

## 1. Introduction

## 2. Problem Statement

^{−1}) is minimized is often used. Chosen measures of optimal design performance could be found in [32]:

- A-optimality (tr(M
^{−1})) minimizes the total variance of the parameters estimates. - E-optimality (λ
_{max}(M^{−1})) minimizes the variance of the maximum eigenvalue of M^{−1}. - D-optimality minimizes the generalized variance of the parameters and minimizes the volume of the ellipsoidal confidence region of parameter estimates with respect to the input.

- D-efficiency inequality constraint can be expressed according to (4).
- Amplitude constraints on inputs, outputs or state variables in a form:$${u}_{\mathrm{min}}\le u(t)\le {u}_{\mathrm{max}}.\text{}$$
- Energy constraints imposed on inputs:$$\underset{0}{\overset{T}{\int}}{u}^{\mathrm{T}}\left(t\right)}u\left(t\right)dt\le E.\text{$$

## 3. System Identification Method

_{r}is a parameter vector to be estimated, y

_{i}is the output samples sequence, and ε

_{i}is the Gaussian white noise with restricted variance. The model prediction error estimators are obtained utilizing the regression residuals:

## 4. Optimal Input Design Problem

_{f}] and to replace free final time with fixed final time problem utilizing a scaling factor as an augmented state variable, which scales the duration of the time interval. We solve this problem using the transcription of the below optimal control formulation into a similar optimal control task represented in the Lagrange form with the set of constraints. To verify the suitability of this technique to the model parameter identification, a first-order time-invariant inertial system is considered:

_{a}= ∂x/∂a, x

_{b}= ∂x/∂b, and R is 2 × 2 matrix given by:

_{n}= 1 to obtain an optimal input signal for system parameters identification where measurements do not include additive white noise. To maximize the FIM determinant, let us define the augmented state vector given by [1]:

_{0}, t

_{f}] denotes time duration, q = {1, …, q} and l, g, and h are a priori linear or nonlinear functions. The functions g(·,·) and l(·,·,·) with indexes tc, eec, and eic are trajectory constraint, endpoint equality constraint and endpoint inequality constraint, respectively.

_{f}] and to replace the free final time problem with the fixed final time case utilizing the free final time scaling factor as an augmented state variable, which scales the duration of the time interval. For this reason, the scale factor and the scaled time are expressed by extra states which enables minimization over initial value of the further states to fit the scaling.

_{n+}

_{1}and x

_{n+}

_{2}to (30), the free terminal time problem can be modified into the similar fixed final time optimal control problem with an augmented state vector:

_{f}] and the objective function can be written as:

_{f}], x

_{n+}

_{2}is the duration scale coefficient to be minimized, x

_{n+}

_{1}= tx

_{n+}

_{2}denotes free termination time, and T

_{f}is the fixed termination time chosen arbitrarily. When considering the autonomous dynamic systems, the extra state variable x

_{n+}

_{1}is not obligatory. Therefore, the autonomous free final time problem can be solved by augmenting state equations considering only one state variable representing the free final time scaling factor.

## 5. Problem Reformulation

_{70}is an initial state condition to be optimized. It should be noted that an additional constraint was imposed on the state variable x

_{1}(t) to enable unexpected changes of the control signal u(t), which is restricted to the interval [−1, +1]. Using the proposed methodology defined by Equations (34)–(36), the optimal solution for free final time is t

_{f}= Tζ.

## 6. Numerical Results

_{1}(0) = 5, x

_{7}(0) = 1, and the initial value of the input signal was set as u(0) = 1. The free final time scaling factor ζ which scales the duration of the time is optimized from the interval 0.1 ≤ ζ ≤ 10, so the time duration could be varied from 1 to 100 s. The numerical results were computed using the fixed step-size fourth-order Runge–Kutta integration method with grid period of 0.2 s. The expression for the cost function, given by Equation (35), can be presented as:

_{1}denotes the free final time scaling factor ζ, and J

_{2}is the integral of the squared input signal.

#### 6.1. Free-Final Time Constraint Input Design

_{eff}= 90% of its optimal value. The suboptimal input signals obtained for different desired values of the input energy factor q and D-efficiency constant value D

_{eff}= 90% are displayed in Figure 1c,d.

_{1}= 0.88, qJ

_{2}= 1.0 × 10

^{−4}and t

_{f}= 8.79 [s]) is shown in Figure 1a. The input energy factor was increased (Figure 1b) to obtain the suboptimal input signal, which corresponds to performance index components values: J

_{1}= 0.92, qJ

_{2}= 5.00 and t

_{f}= 9.23 [s]. For comparison, Figure 1c shows the suboptimal input signal, which correlates with the objective function values: J

_{1}= 0.97, qJ

_{2}= 8.70 at the final time level of t

_{f}= 9.67 [s]. Figure 1d contains the graphical display of the suboptimal excitation received for the cost function integrants values: J

_{1}= 1.01, qJ

_{2}= 33.44, where time duration was t

_{f}= 10.11 [s].

_{eff}< 100% are safer for system identification purposes until the FIM determinant is not dominated by the input energy component of the minimized performance index. The comparison of the performance index components obtained for increasing values of the input energy factor and for decreasing values of D-efficiency from the interval [100%, 80%] of its maximum value are presented in Table 1, Table 2 and Table 3.

_{1}= 1, q = 1 ×10

^{−6}and D

_{eff}= 100%) is equal to 10 s.

#### 6.2. LS and LEL Estimators for LTI Model Identification

_{f}), computed as solutions of the free final time optimization problems (35) and (36), were then utilized as excitations in the plant model parameter estimation procedure. The physical system (14), used in system identification procedure, can be described by the following single input–single output state space model:

_{f}), and we collect data on its output y(t

_{f}).

^{2}≤ 0.7 is added to the control input to the system. The model of the plant (38) depends on a vector of unknown parameters θ = [a, b]

^{T}and the aim of such an experiment is to estimate unknown model parameters values which should be the most similar to the true values of the plant parameters. The difference between the output of the plant y(t

_{f}) and the output of the model y

_{m}(t

_{f}) was minimized. The initial state condition of the inertial model was selected from the interval −5 ≤ x

_{1}(0) ≤ 5 and the experiment duration depends on chosen D-optimal signal according to Table 2. Numerical results were obtained utilizing the Nelder–Mead simplex method.

_{eff}= 90%), while Figure 3b shows the results (for the same values of initial states and noise variance) with excitation signals that were computed when an input energy coefficient increases its value and was selected as q = 0.10 (green dashed line) and 0.40 (red dash-dot line). To compare results, Figure 3 contains the graphical display of the ellipsoidal confidence region of parameter estimates, where the system (Figure 2) was perturbed utilizing a step input signal (blue solid line). The comparison of the ellipsoidal confidence regions of the plant model parameter estimates indicates some similarities. The D-optimal input signal, calculated for q ≈ 0, causes the minimal time duration of the parameter identification experiment and a minimal volume of the ellipsoidal confidence region of parameter estimates. When the value of the input energy factor increases, the area under the curve increases its size for the same initial conditions and noise variance values. Increasing the desired ratio of input energy constraint yields the increase in the input signal duration, but the excitation is safer for the plant. In such a manner, we avoid rapid switching of the excitation signal in the real identification experiments. The prediction error LS, and the relative squared error LEL estimators were compared based on maximum, average, and minimum residuals of the parameter values (Table 4 and Table 5).

_{1}(0) = 1. The reason for this obstruction is the excitation signal (i.e., unit step-like input) which is not able to unbalance the inertial system with respect to the initial condition equal to 1. Therefore, data outlier points were removed from the set of the parameter estimates. Finally, it should be noticed that there is no guarantee for relative squared residuals formulation to have a unique solution with respect to the parameters. Thus, the minimization should be performed very carefully with special attention given to the initialization of the parameters.

## 7. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Kalaba, R.; Spingarn, K. Control, Identification, and Input Optimization; Plenum Press: New York, NY, USA, 1982; pp. 225–299. ISBN 978-1-4684-7662-0. [Google Scholar]
- Ljung, L. System Identification: Theory for the User; Prentice-Hall: Englewood Cliffs, NJ, USA, 1999; pp. 358–406. ISBN 0-13-881640-9. [Google Scholar]
- Pintelon, R.; Schoukens, J. System Identification: A Frequency Domain Approach, 2nd ed.; John Wiley & Sons: New York, NY, USA, 2001; pp. 151–224. ISBN 978-0-470-64037-1. [Google Scholar]
- Mehra, R. Choice of Input Signals. In Trends and Progress in Systems Identification; Eykhoff, P., Ed.; Pergamon Press: New York, NY, USA, 1981; pp. 305–366. [Google Scholar]
- Hugo, A.J. Process Controller Performance Monitoring and Assessment. Control. Arts Inc. 2001. Available online: http://www.controlarts.com/ (accessed on 28 May 2018).
- Gevers, M.; Ljung, L. Optimal experiment designs with respect to the intended model application. Automatica
**1986**, 22, 543–554. [Google Scholar] [CrossRef] - Bombois, X.; Scorletti, G.; Gevers, M.; Van den Hof, P.M.J.; Hildebrand, R. Least costly identification experiment for control. Automatica
**2006**, 42, 1651–1662. [Google Scholar] [CrossRef] - Bombois, X.; Hjalmarsson, H.; Scorletti, G. Identification for robust deconvolution filtering. Automatica
**2010**, 46, 577–584. [Google Scholar] [CrossRef] - Hussain, M. Review of the applications of neural networks in chemical process control-simulation and on-line implementation. Artif. Intell. Eng.
**1999**, 13, 55–68. [Google Scholar] [CrossRef] - Rivera, D.; Lee, H.; Braun, M.; Mittelmann, H. Plant friendly system identification: A challenge for the process industries. In Proceeding of the SYSID 2003, Rotterdam, The Netherlands, 27 August 2003; pp. 917–922. [Google Scholar]
- Narasimhan, S.; Rengaswamy, R. Plant friendly input design: Convex relaxation and quality. IEEE Trans. Automat. Control
**2011**, 56, 1467–1472. [Google Scholar] [CrossRef] - Steenis, R.; Rivera, D. Plant-friendly signal generation for system identification using a modified simultaneous perturbation stochastic approximation (SPSA) methodology. IEEE Trans. Control Syst. Technol.
**2011**, 19, 1604–1612. [Google Scholar] [CrossRef] - Rafajłowicz, E.; Rafajłowicz, W. A variational approach to optimal input signals for parameter estimation in systems with spatio–temporal dynamics. In Proceedings of the 10th International Workshop in Model-Oriented Design and Analysis, Łagów Lubuski, Poland, 10–14 June 2013; Springer: Heidelberg, Germany, 2013; pp. 219–227. [Google Scholar] [CrossRef]
- Rafajłowicz, E.; Rafajłowicz, W. More safe optimal input signals for parameter estimation of linear systems described by ODE. In Proceedings of the 26th Conference on System Modeling and Optimization (CSMO), Klagenfurt, Austria, 9–13 September 2013; Springer: Heidelberg, Germany, 2014; Volume 443, pp. 267–277. [Google Scholar] [CrossRef]
- Kumar, A.; Nabil, M.; Narasimhan, S. Economical and plant friendly input design for system identification. In Proceedings of the European Control Conference, Strasbourg, France, 24–27 June 2014; Volume 48, pp. 732–737. [Google Scholar] [CrossRef]
- Potters, M.G.; Bombois, X.; Forgione, M.; Modén, P.E.; Lundh, M.; Hjalmarsson, H.; Van den Hof, P.M.J. Optimal experiment design in closed loop with unknown, nonlinear and implicit controllers using stealth identification. In Proceedings of the European Control Conference, Strasbourg, France, 24–27 June 2014; pp. 726–731. [Google Scholar] [CrossRef]
- Larsson, C.A.; Annergren, M.; Hjalmarsson, H.; Rojas, C.R.; Bombois, X.; Mesbah, A.; Modén, P.E. Model predictive control with integrated experiment design for output error systems. In Proceedings of the European Control Conference, Zurich, Switzerland, 17–19 July 2013; pp. 3790–3795. [Google Scholar]
- Larsson, C.A.; Rojas, C.R.; Bombois, X.; Hjalmarsson, H. Experiment evaluation of model predictive control with excitation (MPC-X) on an industrial depropanizer. J. Process Control
**2015**, 31, 1–16. [Google Scholar] [CrossRef] - Annergren, M.; Larson, C.A.; Hjalmarsson, H.; Bombois, X.; Wahlberg, B. Application-oriented input design in system identification. Optimal input design for control. IEEE Control Syst. Mag.
**2017**, 37, 31–56. [Google Scholar] [CrossRef] - Malakara, N.K.; Knuth, K.H. Entropy-Based Search Algorithm for Experimental Design. AIP Conf. Proc.
**2011**, 1305, 157–164. [Google Scholar] - Rousseeuw, P.J.; Leroy, A.M. Robust Regression and Outlier Detection; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2003; pp. 21–145. ISBN 0271-6356. [Google Scholar]
- Huber, P.J. Robust Statistic; John Wiley & Sons, Inc.: New York, NY, USA, 1981; pp. 43–68. ISBN 0-47141805-6. [Google Scholar]
- Golan, A.; Judge, G.G.; Miller, D. Maximum Entropy Econometrics: Robust Estimation with Limited Data; John Wiley & Sons Inc.: Chichester, UK, 1996; ISBN 978-0-471-95311-1. [Google Scholar]
- Indiveri, G. An entropy-like estimator for robust parameter identification. Entropy
**2009**, 11, 560–585. [Google Scholar] [CrossRef] - Zorzi, M. Multivariate spectral estimation based on the concept of optimal prediction. IEEE Trans. Autom. Control
**2015**, 60, 1647–1652. [Google Scholar] [CrossRef] - Zorzi, M. An interpretation of the dual problem of the THREE-like approaches. Automatica
**2015**, 62, 87–92. [Google Scholar] [CrossRef] [Green Version] - Stojanovic, V.; Filipovic, V. Adaptive input design for identification of output error model with constrained output. Circuits Syst. Signal Process.
**2014**, 33, 97–113. [Google Scholar] [CrossRef] - Jakowluk, W. Plant friendly input design for parameter estimation in an inertial system with respect to D-efficiency constraints. Entropy
**2014**, 16, 5822–5837. [Google Scholar] [CrossRef] - Jakowluk, W. Optimal input signal design for a second order dynamic system identification subject to D-efficiency constraints. In Computer Information Systems and Industrial Management; Lecture Notes in Computer Science; Springer: Cham, Switzerland, 2015; pp. 351–362. [Google Scholar]
- Chianeh, H.A.; Stigter, J.D.; Keesman, K.J. Optimal input design for parameter estimation in a single and double tank system through direct control of parametric output sensitivities. J. Process Control
**2011**, 21, 111–118. [Google Scholar] [CrossRef] - Tricaud, C.; Chen, Y. An approximate method for numerically solving fractional order optimal control problems of general form. Comput. Math. Appl.
**2010**, 59, 1644–1655. [Google Scholar] [CrossRef] - Atkinson, A.; Donev, A.; Tobias, R. Optimum Experimental Design with SAS; Oxford University Press: Oxford, UK, 2007; pp. 135–147. ISBN 9780199296606. [Google Scholar]
- Schwartz, A.; Polak, E.; Chen, Y. Riots a Matlab Toolbox for Solving Optimal Control Problems. Version 1.0 for Windows, May 1997. Available online: http://www.schwartz-home.com/RIOTS/ (accessed on 28 May 2018).

**Figure 1.**Free final time inputs to inertial model perturbation: (

**a**) optimal input signal for q ≈ 0 and D

_{eff}= 90%; (

**b**) suboptimal input signal for q = 0.05 and D

_{eff}= 90%; (

**c**) suboptimal input signal for q = 0.10 and D

_{eff}= 90%; and (

**d**) suboptimal input signal for q = 0.40 and D

_{eff}= 90%.

**Figure 3.**Ellipsoidal confidence regions of LS parameter estimates (

**a**); and LEL parameter estimates for different inputs: D-optimal input signal (black dotted, D

_{eff}= 90%, q = 1 × 10

^{−6}), Suboptimal input signal (green dashed, D

_{eff}= 90%, q = 0.10), Suboptimal input signal (red dash-dotted, D

_{eff}= 90%, q = 0.40), and Step input signal (blue solid line) (

**b**).

**Figure 4.**The comparison of the estimates values obtained using LEL (black solid line), and LS (red dashed line) estimators for various initial conditions, different noise variance, and D-efficiency value 90%: (

**a**) confrontation between rival estimators for parameter estimate a based on the 80 runs; and (

**b**) confrontation between rival estimators for parameter estimate b based on the 80 runs.

D_{eff}/D_{opt} | FIM | J_{1} | q | qJ_{2} | t_{f} [s] |
---|---|---|---|---|---|

100% | −43.87 | 1.00 | 1 × 10^{−6} | 1 × 10^{−4} | 10.00 |

1.06 | 0.05 | 4.79 | 10.57 | ||

1.10 | 0.10 | 8.99 | 11.00 | ||

1.13 | 0.20 | 17.42 | 11.36 | ||

1.15 | 0.30 | 25.91 | 11.53 | ||

1.16 | 0.40 | 34.44 | 11.62 | ||

1.17 | 0.50 | 42.97 | 11.70 |

D_{eff}/D_{opt} | FIM | J_{1} | q | qJ_{2} | t_{f} [s] |
---|---|---|---|---|---|

90% | −35.60 | 0.880 | 1 × 10^{−6} | 1 × 10^{−4} | 8.79 |

0.923 | 0.05 | 5.00 | 9.23 | ||

0.967 | 0.10 | 8.70 | 9.67 | ||

0.991 | 0.20 | 16.86 | 9.91 | ||

1.041 | 0.30 | 25.14 | 10.04 | ||

1.011 | 0.40 | 33.44 | 10.11 |

D_{eff}/D_{opt} | FIM | J_{1} | q | qJ_{2} | t_{f} [s] |
---|---|---|---|---|---|

80% | −28.10 | 0.763 | 1 × 10^{−6} | 1 × 10^{−4} | 7.63 |

0.807 | 0.05 | 4.35 | 8.07 | ||

0.837 | 0.10 | 8.27 | 8.37 | ||

0.858 | 0.20 | 16.25 | 8.58 | ||

0.866 | 0.30 | 24.22 | 8.66 |

Index | q = 1 × 10^{−6} | q = 0.10 | ||||||
---|---|---|---|---|---|---|---|---|

Estimator | LS | LEL | LS | LEL | ||||

Parameters | a | b | a | b | a | b | a | b |

average value [%] | 9.08 | 8.99 | 6.01 | 6.57 | 10.97 | 11.75 | 8.05 | 8.68 |

maximum value [%] | 61.27 | 63.54 | 47.54 | 40.44 | 55.69 | 57.82 | 40.74 | 42.18 |

minimum value [%] | 2.0 × 10^{−2} | 1.6 × 10^{−1} | 2.1 × 10^{−4} | 3.3 × 10^{−4} | 1.9 × 10^{−1} | 7.0 × 10^{−2} | 1.5 × 10^{−4} | 1.2 × 10^{−4} |

Index | q = 0.40 | Step Input Signal | ||||||
---|---|---|---|---|---|---|---|---|

Estimator | LS | LEL | LS | LEL | ||||

Parameters | a | b | a | b | a | b | a | b |

average value [%] | 12.29 | 12.45 | 9.02 | 9.49 | 14.35 | 14.23 | 9.83 | 10.56 |

maximum value [%] | 63.27 | 47.43 | 49.98 | 43.15 | 83.19 | 86.95 | 54.49 | 59.76 |

minimum value [%] | 4.6 × 10^{−2} | 9.6 × 10^{−1} | 1.3 × 10^{−6} | 9.5 × 10^{−5} | 1.9 × 10^{−1} | 2.3 × 10^{−2} | 4.0 × 10^{−4} | 1.5 × 10^{−4} |

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**MDPI and ACS Style**

Jakowluk, W.
Free Final Time Input Design Problem for Robust Entropy-Like System Parameter Estimation. *Entropy* **2018**, *20*, 528.
https://doi.org/10.3390/e20070528

**AMA Style**

Jakowluk W.
Free Final Time Input Design Problem for Robust Entropy-Like System Parameter Estimation. *Entropy*. 2018; 20(7):528.
https://doi.org/10.3390/e20070528

**Chicago/Turabian Style**

Jakowluk, Wiktor.
2018. "Free Final Time Input Design Problem for Robust Entropy-Like System Parameter Estimation" *Entropy* 20, no. 7: 528.
https://doi.org/10.3390/e20070528