Application-Oriented Input Spectrum Design in Closed-Loop Identification

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Growing requirements for efficiency and environmental protection force the use of more and more advanced methods of industrial process control. Identification of the system in a closed loop frequently allows us to reduce the cost and duration of the experiment.

Abstract

In the paper, a problem of real-time input design, where the spectral characteristic of the external excitation signal is optimized considering the fixed bound on the ellipsoidal constraint applied to the quality measure of the estimated model, has been presented. We designed a compensator close to the complementary sensitivity function such that the resulting closed-loop system is stable and guarantees robust performance. The contribution of this work comprises a comparison of the KYP and frequency grid methods for the worst-case minimization between the identified plant and all models in the uncertainty region. It has been noted that the input design problem considering the frequency grid is far more numerically robust than the KYP method and guarantees that more than 90% of the estimated model parameters are clustered inside the assumed confidence region. The proposed approach is verified by numerical examples, and the sensitivity of the input spectrum design to the estimates of model parameters is discussed.

1. Introduction

Identification methods for control have received great interest during the last decades [1,2,3]. Many industrial processes reveal slow response which can cause long and costly identification experiments. It has been reported that three-quarters of the total costs related to real-life control projects are absorbed by modeling [4]. The costs of the identification experiment can be associated with the experiment duration and properties of the input signal. Another complex cost can be caused by the requirement of acceptable control performance when providing the model-based control design. The typical industrial practice for model identification purposes is to use the application-oriented input design [5]. The framework is based on an optimal input design, where the objective is to find a trade-off between minimal departure from nominal working conditions and the accuracy of the model parameters to be estimated [6,7]. The advantage of this model-based method relies on including the cost of the experiment in the input design framework (IDF) through the objective function to be minimized. Application-oriented input design is the cheapest identification method, ensuring the performance of the plant model [8].

In system identification, the relation between the assumed accuracy of the designated model and the experimental costs should be balanced. The optimal input design methods in the open-loop system have received considerable attention [1]. Typical optimal input design problem corresponds to non-convex solvers and, therefore, the applicability of the input signal design in the industry has been limited. A way to solve the problem of model accuracy has been applied to high-order expressions (i.e., Box-Jenkins models) which endow the identified model with a certain robustness [9]. Unfortunately, these expressions are not convenient for handling frequency-wise constraints used in many control applications. Considering these limitations, particular attention to the optimal input design has been paid to finite-order models [10,11]. The results indicate reasonable approximation accuracy also for models that do not exceed the real system order. It has been shown that convex optimization methods are efficient solutions to many open-loop input design problems [12].

In many real-world processes, the control performance degrades when the accuracy of the model of the system is getting worse. The feedback control instability often causes unexpected consequences. The industrial processes are mostly run as closed-loop control systems, and it is often not feasible to open the loop. When the prior process data are achievable, it is possible to execute a closed-loop identification experiment using the chosen controller. Such a closed-loop experiment should be stable and have a short duration [13,14]. Another important thing is to ensure that the identification experiment is plant-friendly and meets industrial demands [15,16,17]. The spectrum of the excitation signal affects the model parameters to be estimated during the identification experiment. Therefore, the input design problem could be obtained by the signal spectrum approximation using the finite impulse response (FIR) process, where the optimal filter coefficients are computed by convex optimization [18].

Some interesting approaches to the problem of optimal input signal synthesis in the frequency domain for dedicated fields of applications have been published in recent years, and some of them have been presented in [19]. In the paper [20], a heuristic solution has been suggested to tune a bounded amplitude multisine control signal in a closed-loop setup. The solution is based on the combination of two algorithms—the first one minimizes the Crest Factor of the control signal and the second one shapes (tunes) the spectrum of the control to obtain convergence to the desired reference spectrum. Some simplifications of the cost function of the identification task (with respect to unknown model parameters), which are claimed to be more suited for practical applications, have been proposed in [21]. A systematic heuristic procedure to build an input signal for the identification of the nonlinear multi-input multi-output (MIMO) dynamic model of the closed-loop rotor/radial active magnetic bearing (rotor/RAMB) system has been proposed in [22]. For the synthesis of the input signal which can reduce a potential correlation problem caused by perturbation signals, an optimal parallel amplitude-modulated pseudo-random binary sequence (PAPRBS) has been proposed. By applying the (PAPRBS) generator, the accuracy of the identified MIMO dynamics could be guaranteed to some extent.

Identification of Frequency Response Functions (FRFs) is essential in different control issues of mechatronic systems—from system design and validation to controller design and diagnostics. In the last years, optimal design of experiments for FRF identification of multivariable motion systems focused the interest of researchers. In [23], the method of FRF identification subject to element-wise power constraints has been proposed. Developed computational techniques for multivariable excitation design frameworks are fast and suitable for application to large dimensional realistic systems. For the synthesis of control algorithms for industrial robots, both the modeling and the identification problems are challenging (since the required accuracy is very high, the system complex and the real-time requirements are sometimes critical). In [24], it has been proved that by conducting experiments only in the selected robot configurations, the experiment time can be reduced and the accuracy of the parameter estimate can be increased (both in terms of bias and standard deviation). The study also indicates that conducting more experiments in the optimal configurations can be equivalent to experiments in additional configurations.

The methodology presented in this paper is classified as an optimal average performance input design for control [25]. The $\mathscr{H}$_∞ norm of the weighted relative error and the positive real Kalman-Yakubovich-Popov (KYP) lemma are used to appraise the stability and performance indices of the closed-loop system [26,27]. The minimum FIR filter output power and the corresponding spectrum selection problem have been specified as the least-costly experiment design [28].

In the manuscript, the context of real-time closed-loop system identification is considered. The industrial processes are mostly run as closed-loop control systems, and it is often not possible to open the loop. Generally, optimal experiment design problems correspond to non-convex programming tasks, and their applicability to the industry has been restricted. The contribution of this work is to propose a framework for converting the input design problem into a convex optimization under linear matrix inequality constraints. The solution of the input spectrum design, applying minimization of the worst-case quality measure (over all models in an ellipsoidal region), has been presented in [10]. We propose a different approach where the input spectrum power minimization is performed, subject to the fixed bound on the ellipsoidal quality constraint. The results of the closed-loop system identification considering the optimization problem performed jointly, concerning the fixed compensator and the optimal spectrum of the input signal, are presented. Another objective has been to check the numerical stability issues of the methods used for evaluating the ellipsoidal quality constraints, which have a great impact on real-time system identification. The frequency grid and KYP methods have been compared for the worst-case minimization between the identified plant and models in the uncertainty region.

2. Robust Estimation of Model Parameters

The objective of the application-oriented experiment design is to deliver a model of a system using open-loop data, which guarantees an acceptable performance of a control system [5,29]. We examine the identification of a discrete-time linear time-invariant system within the prediction error framework [30]. The response of the single-input and single-output discrete true system can be expressed as

$$y\left(t\right)=G_0\left(q,{\theta }_0\right)u\left(t\right)+H_0\left(q,{\theta }_0\right)e_0\left(t\right),$$

(1)

where u(t) is the input signal, y(t) is the output sequence, e₀(t) is the white Gaussian noise with the zero mean value and the variance λ₀, q is the time-shift operator, and t represents the experiment duration. G₀ and H₀ are transfer functions, where H₀ is the stable, monic, and minimum phase, and θ₀ is a vector of the true parameters of the system. Let us assume the sufficient flexibility of a model; i.e., it is able to capture the dynamics of the true system. Then, the model output can be described as

$$\hspace{0.17em}y\left(t\right)=G\left(q,\theta \right)u\left(t\right)+H\left(q,\theta \right)e\left(t\right),$$

(2)

where the transfer functions G and H are parameterized by a vector θ ∈ ℝⁿ. The estimates of the unknown parameters of the model (2) are found by minimization of the criterion function based on the prediction error (PEM), with respect to θ. The PEM is defined as the measure of the difference between the output of the real plant (the true system) and the output signal obtained from the model [30]. The output of model (2), estimated one step ahead, becomes

$$\widehat{y}\left(t,\theta \right)=H^{-1}\left(q,\theta \right)G\left(q,\theta \right)u\left(t\right)+\left[1-H^{-1}\left(q,\theta \right)\right]y\left(t\right).$$

(3)

The closed-loop system setup, assumed for system identification purposes [27], is shown in Figure 1.

Referring to the above scheme of a control system, the input signal u(t) is given by

$$u\left(t\right)=-T\left(q\right)y\left(t\right)+r\left(t\right),$$

(4)

where r(t) is a reference input signal (external and uncorrelated with noise), e(t) is a noise sequence, and T(q) is a linear compensator. Furthermore, it has been assumed that all signals have spectral representation. It should be noted that the PEM method is not restricted only to the systems described above. The system may be either continuous or discrete, including nonlinear multivariable systems [30].

To obtain the vector θ of unknown parameters, based on N observations of input-output data and using the one-step-ahead predicted output (3), the following criterion function is minimized:

$${\widehat{\theta }}_N=\underset{\theta }{\arg \min }\frac{1}{2N}{\displaystyle \sum _{t=1}^N{\left(y\left(t\right)-\widehat{y}\left(t,\theta \right)\right)}^2.}$$

(5)

To determine the uncertainty in the parameter estimates, a study of their asymptotic properties is required. The properties of the distribution of the estimates define their precision. The true system is assumed to have the same structure as the model with the parameter vector θ₀ and the noise variance λ₀, such that: G(θ₀) = G₀, H(θ₀) = H₀. Then, using some assumptions, the parameter estimates have the following asymptotic distribution (see [30]):

$$\begin{array}{l}\sqrt{N}\left({\widehat{\theta }}_N-{\theta }_0\right)\to \mathcal{N}\left(0,P\right),N\to \infty ,\\ \underset{N\to \infty }{\lim }N\hspace{0.17em}\mathrm{E}\left[\left({\widehat{\theta }}_N-{\theta }_0\right){\left({\widehat{\theta }}_N-{\theta }_0\right)}^T\right]=P,\\ P={\lambda }_0{\left(\mathrm{E}\left[\psi \left(t,{\theta }_0\right){\psi }^T\left(t,{\theta }_0\right)\right]\right)}^{-1},\\ \psi \left(t,{\theta }_0\right)=\frac{\partial }{\partial \theta }\widehat{y}\left(t,\theta \right){|}_{\theta ={\theta }_0},\end{array}$$

(6)

where $\mathcal{N}$ represents the normal distribution, and P is the asymptotic parameter covariance matrix. When the model set contains the system under consideration, then the estimate converges to the true value. However, the signal spectrum also influences estimation accuracy. Let us introduce κ(t) = [u(t) e₀(t)]^T where e₀(t) represents the true noise. We consider the information matrix, P⁻¹, as a function of the frequency distribution of the spectrum Φ_κ. The spectrum distribution is as follows:

$${\Phi }_{\kappa }=\left[\begin{array}{cc}{\Phi }_u& {\Phi }_{ue}\\ {\Phi }_{ue}^{*}& {\lambda }_0\end{array}\right],$$

(7)

where Φ_u is the input signal spectrum and ${\Phi }_{ue}^{*}$ denotes the complex conjugate of the cross-spectrum between u(t) and e₀(t). The information matrix, P⁻¹, is a measure of the errors in the parameters, is related to the spectrum distribution Φ_κ, and is described as [30]

$$P^{-1}=\frac{1}{2\pi {\lambda }_0}{\displaystyle \underset{-\pi }{\overset{\pi }{\int }}F\left(q,{\theta }_0\right){\Phi }_{\kappa }{F}^{*}\left(q,{\theta }_0\right)}d\omega ,$$

(8)

where $\psi \left(t,{\theta }_0\right)=F\left(q,{\theta }_0\right)\kappa \left(t\right)$ and $F\left(q,{\theta }_0\right)=\left[\begin{array}{cc}{F}_u\left(q,{\theta }_0\right)\hspace{0.17em}& F_e\left(q,{\theta }_0\right)\hspace{0.17em}\end{array}\right]$ with

$$F_u\left(q,\theta \right)=H^{-1}\left(q,\theta \right)\frac{dG\left(q,\theta \right)}{d\theta },$$

(9)

$$F_e\left(q,\theta \right)=H^{-1}\left(q,\theta \right)\frac{dH\left(q,\theta \right)}{d\theta }.$$

(10)

The relation between the parameters of the model and the covariance matrix P is obtained, using Parseval’s theorem. It should be noted that the input spectrum Φ_u and the cross-spectrum Φ_ue are used to create the information matrix P⁻¹. The other variables in (8) apply to the real system [27].

In control design applications, it is a common practice to have frequency-by-frequency conditions on the function estimate error. One such condition is robust stability, expressed as the weighted relative error in the following form [31]:

$$\Delta \left(\omega ,\theta \right)=T\left(\omega \right)\frac{G_0\left(\omega ,{\theta }_0\right)-G\left(\omega ,\theta \right)}{G\left(\omega ,\theta \right)},$$

(11)

where G₀(ω, θ₀) is the true system and G(ω, θ) is the model of the system. When the weight function T(ω) is designed as the sensitivity complementary function, then $\mathscr{H}$_∞ norm of (11) is used to quantify the stability and performance of the closed-loop system [31]. When the parameter vector of the model G(ω, θ) is estimated during an identification experiment, then it is included in an uncertainty set. The parameter uncertainty region is restricted by the χ constant:

$$U_{\theta }=\left\{\theta :N{\left(\theta -{\theta }_0\right)}^T{P}^{-1}\left({\Phi }_{\chi }\right)\left(\theta -{\theta }_0\right)\le \chi \right\},$$

(12)

where χ defines the size of the uncertainty region [30]. The goal of such an identification experiment is to guarantee that the weighted relative error ${\Vert \Delta \left(\omega ,\theta \right)\Vert }_{\infty }\le \gamma$ covers all models in the uncertainty set (12) and γ is the preselected constant value. The complementary sensitivity function is computed as the zero-order-hold discretization of

$$T\left(s\right)=\frac{{\omega }_0^2}{s^2+2\xi {\omega }_{0}s+{\omega }_0^2},$$

(13)

where ω₀ is used to change the bandwidth of T(s) with a damping value ξ = 0.7. In the experimental part, it was assumed that the size of the confidence region χ is determined like that of $Pr\left({\chi }^2\left(n\right)\le \chi \right)=0.95$ where n is the number of parameters of the model G(ω, θ).

The spectrum Φ_r of the external excitation signal r(t) in the closed-loop system design is related to the spectra Φ_u and Φ_ue described by the following formulas [13]:

$${\Phi }_u={\lambda }_0{\left|{\left(1+T\left(\omega \right)G_0\left(\omega ,{\theta }_0\right)\right)}^{-1}T\left(\omega \right)H_0\left(\omega ,{\theta }_0\right)\right|}^2+{\left|1+T\left(\omega \right)G_0\left(\omega ,{\theta }_0\right)\right|}^{-2}{\Phi }_r,$$

(14)

$${\Phi }_{ue}=-{\lambda }_0{\left(1+T\left(\omega \right)G_0\left(\omega ,{\theta }_0\right)\right)}^{-1}T\left(\omega \right)H_0\left(\omega ,{\theta }_0\right).$$

(15)

In this paper, the simpler optimization problem is solved concerning the fixed operating compensator and the design variable Φ_u. The optimal external excitation spectrum ${\Phi }_r^{opt}$ can be reconstructed from the optimal ${\Phi }_u^{opt}$ and the optimal ${\Phi }_{ue}^{opt}$ by inverting the following formulas [13]:

$$T\left(\omega \right)=-{\Phi }_{ue}^{opt}{\left({\lambda }_0{H}_0\left(\omega ,{\theta }_0\right)+G_0\left(\omega ,{\theta }_0\right){\Phi }_{ue}^{opt}\right)}^{-1},$$

(16)

$${\Phi }_r^{opt}={\left|1+T\left(\omega \right)G_0\left(\omega ,{\theta }_0\right)\right|}^2\left({\Phi }_u^{opt}-{\lambda }_0^{-1}{\left|{\Phi }_{ue}^{opt}\right|}^2\right).$$

(17)

The objective of the optimization is to obtain the minimal energy α of the spectrum of the reference signal. Based on expressions (8)–(13) and applying the frequency sampling range, it is possible to solve the input spectrum design problem expressed as the minimization task:

$$\begin{array}{c}\underset{{\Phi }_u}{minimize}\hspace{0.17em}\alpha \hfill \\ \mathit{subject}\mathit{to}\hspace{0.17em}\hspace{0.17em}{\left|T\left(\omega \right)\frac{G_0\left(\omega ,{\theta }_0\right)-G\left(\omega ,\theta \right)}{G\left(\omega ,\theta \right)}\right|}^2\le {\gamma }^2,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\forall \omega ,\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}N{\left(\theta -{\theta }_0\right)}^T{P}^{-1}\left({\Phi }_u\right)\left(\theta -{\theta }_0\right)\le \chi ,\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\frac{1}{2\pi }{\displaystyle {\int }_{-\pi }^{\pi }{\Phi }_u\left(\omega \right)d\omega \le \alpha ,}\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\Phi }_u\left(\omega \right)\ge 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\forall \omega .\hfill \end{array}$$

(18)

Based on the worst-case quality definition of Δ (i.e., the expression in the first constraint of the set (18)) over all models in the uncertainty region (12), the relative error should be smaller than the predefined value of γ for all models inside the uncertainty set [10]. Thus, the objective of such an optimization task is to obtain the input signal, which spectrum has minimal energy. As has been noted before, the spectrum of the excitation signal used in the identification experiment disrupts the parameter estimates. Thus, the input spectrum can be obtained using the frequency grid method or by implementing the KYP technique [32].

3. Spectrum Representation

The only factor affecting the covariance matrix P is the spectrum of the excitation signal. Proper parameterization of the input spectrum leads to the formulation of the finite-dimensional convex problem [33]. Consider that the closed-loop system identification is performed using the spectral matrix Φ_κ defined in (7). This matrix can be expressed as

$${\Phi }_{\kappa }\left(\omega \right)={\displaystyle \sum _{k=-\infty }^{\infty }{\tilde{c}}_k{\Re }_k\left(e^{j\omega }\right)},$$

(19)

where the scalar basis functions ${\left\{{\Re }_k\left(e^{j\omega }\right)\right\}}_{k=0}^{\infty }$ are proper, stable, and rational with ${\tilde{c}}_k={\tilde{c}}_{-k}^T$. Using proper parameterization of the signal spectrum, it is possible to solve a finite-dimensional convex optimization problem with matrix coefficients ${\tilde{c}}_k$. We assume that ${\Re }_k\left(e^{-j\omega }\right)={\Re }_k^{*}\left(e^{j\omega }\right)$ so that ${\tilde{c}}_k\in$ℝ. Therefore, the factors ${\tilde{c}}_k$ must satisfy the below constraint:

$${\Phi }_{\kappa }\left(\omega \right)\underset{\_}{\succ }0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\forall \omega .$$

(20)

The most common choice of the basis function is ${\Re }_k\left(e^{j\omega }\right)=e^{-j\omega k}.$ A finite-dimensional parameterization is then acquired by substituting the spectrum Φ_κ(ω) in (19) with the following approximation:

$${\Phi }_{\kappa }\left(\omega \right)=\Psi \left(e^{j\omega }\right)+{\Psi }^{*}\left(e^{j\omega }\right),$$

(21)

where Ψ(e^jω) is a finite-dimensional estimation of the positive real part of Φ_κ(ω) with Ψ^*(e^jω) = Ψ^T(e^−jω). Finally, m initial components of (19) are used to obtain the input spectrum:

$$\Psi \left(e^{j\omega }\right)={\displaystyle \sum _{k=0}^{m-1}{c}_k{\Re }_k\left(e^{j\omega }\right),}$$

(22)

where $c_{-k}^T=c_k$ and m is chosen large enough. Finite-dimensional parameterization can be achieved using various methods. One of them is the positive real lemma that arises from the KYP technique [27]:

Lemma 1.

If A, B, C, D is a controllable state-space implementation of the positive real part of the input spectrum $\Psi \left(e^{j\omega }\right)$, then a Q = Q^T like the following exists:

$$\mathrm{K}\left(Q,\left\{A,B,C,D\right\}\right)=\left[\begin{array}{cc}Q-A^{T}QA& -A^{T}QB\\ -B^{T}QA& -B^{T}QB\end{array}\right]+\left[\begin{array}{cc}0& C^T\\ C& D+D^T\end{array}\right]\underset{\_}{\succ }0,$$

(23)

if and only if ${\Phi }_{\kappa }\left(\omega \right)={\displaystyle \sum _{k=0}^{m-1}{c}_k\left[{\Re }_k\left(e^{j\omega }\right)+{\Re }_k^{*}\left(e^{j\omega }\right)\right]}\ge 0,\hspace{0.17em}\forall \omega ,$ and K is the Kautz function.

Thus, the necessary condition considering (20) to hold for the shortened sequence is the inequality (23) if a Q matrix exists. The formulation becomes an LMI in c_k and Q if only C and D matrices are linearly dependent on the c_k coefficients [27]. Since the finite-dimensional spectrum parameterization can be approximated using the m-order FIR filter, the positive real KYP lemma is verified in the paper.

4. Finite-Dimensional Parameterization and Ellipsoidal Constraints

We introduce the ellipsoidal quality constraints, which are required to fulfill the demands for models within the system identification set (12). We consider the robust stability definition (11) for all models inside the system uncertainty set [33]:

$$\underset{\omega ,\theta \in U_D}{max}{|\Delta |}^2\le \gamma .$$

(24)

Using the quality measure (24), we require that about 95% of the estimated models will fulfill the constraints imposed by the ellipsoidal region. Let W_n, W_d, X_n, and X_d be finite-dimensional transfer functions, and Y_n is as follows:

$$Y_n=Y_n^{*}{Y}_n,$$

(25)

where Y_n is some stable finite-dimensional transfer function and Y_d, K_d, and K_n are defined in the same way. Furthermore, R is a positive definite matrix. A suggested quality measure of the model, including (24), is formulated as

$$\begin{array}{l}F\left(\omega ,\eta \right)\le \gamma ,\hspace{0.17em}\forall \omega \hspace{0.17em}\mathrm{and}\hspace{0.17em}\forall \hspace{0.17em}\eta \in \mathsf{\Gamma },\\ \mathsf{\Gamma }=\left\{\eta :{\left(\eta -{\eta }_0\right)}^{T}R\left(\eta -{\eta }_0\right)\le 1\right\},\end{array}$$

(26)

where

$$F\left(\omega ,\eta \right)=\frac{{\left[W_{n}G\left(\eta \right)+X_n\right]}^{*}{Y}_n\left[W_{n}G\left(\eta \right)+X_n\right]+{\mathrm{K}}_n}{{\left[W_{d}G\left(\eta \right)+X_d\right]}^{*}{Y}_d\left[W_{d}G\left(\eta \right)+X_d\right]+{\mathrm{K}}_d}.$$

(27)

The transfer function G(ω, η) includes the parameter vector η which is not affected by the noise, and W_n, W_d, X_n, X_d, Y_n, Y_d, K_n, and K_d are finite-dimensional stable transfer functions. The Equation (26), is a max-norm bound on F, subject to ω, and it should be fulfilled for all η which belong to the set Γ. The uncertainty set Γ is obtained using the prediction error method. In the considerations that follow it has been assumed that transfer functions are given by W_n = W_d = 1, X_n = −G₀, Y_n = T, X_d = K_n = K_d = 0. Considering the closed-loop robust stability criterion (11) and the transfer functions defined above, we need to fulfill (12) within the system identification set, using a fixed bound on the ellipsoidal quality constraint. This assumption will be implemented in the experimental part of the work.

Another performance criterion that guarantees that input design is convex in P⁻¹ arises from the KYP lemma. If the base functions are rational, the positivity of the input signal spectrum is represented by an LMI with the shortened coefficient vector. Then, τ(ω) is defined by

$$\begin{array}{l}\tau \left(\omega \right)=\Psi \left(e^{j\omega }\right)+{\Psi }^{*}\left(e^{j\omega }\right),\\ \Psi \left(e^{j\omega }\right)={\displaystyle \sum _{k=0}^{K-1}{\tau }_k{\Re }_k\left(e^{j\omega }\right),}\end{array}$$

(28)

using some base functions, namely, ${\Re }_k$, k = 0,…, K − 1, and there is a set {Λ_k} expressed by

$$\begin{array}{l}\mathsf{\Lambda }\left(\omega \right)\ge 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\forall \omega ,\\ {\displaystyle \sum _{k=0}^p{\mathsf{\Lambda }}_k\left(e^{-kj\omega }+e^{kj\omega }\right)}\ge 0,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\forall \omega ,\end{array}$$

(29)

where the elements {τ_k} and the ingredients of the matrix R in (26) are linear in {Λ_k}. Then, there exists a state-space realization {A_τ, B_τ, C_τ, D_τ} of the positive part of τ(ω), where the variables {τ_k} are linear in C_τ and D_τ. Similarly, the same applies to the state-state representation {A_Λ, B_Λ, C_Λ, D_Λ}. Furthermore, it holds (26) if only $Q_{\tau }=Q_{\tau }^T$ and $Q_{\mathsf{\Lambda }}=Q_{\mathsf{\Lambda }}^T$ exists. All the theorems and their proofs used to formulate the input design problem (30) considering the KYP lemma are available in [27].

When the input is shaped by the FIR filter sequence with m frequency points and the coefficients τ_k in (28) are the auto-covariance sequence r_k of the input Φ_u(ω), the input variance constraint can be defined as r₀ ≤ α (see Lemma 3.3 in [27]). Then, the constraint input design problem based on the KYP lemma is fulfilled for all models inside the ellipsoidal region (12). The minimum required power α of the input signal can be obtained from

$$\begin{array}{l}\underset{Q_{\tau },Q_{\mathsf{\Lambda }},{\tau }_0,\dots ,{\tau }_{K-1},r_0,\dots ,r_{m-1}}{minimize}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\alpha \\ subject\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}to\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{K}\left(Q_{\tau },\left\{A_{\tau },B_{\tau },C_{\tau },D_{\tau }\right\}\right)\ge 0,\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{K}\left(Q_{\mathsf{\Lambda }},\left\{A_{\mathsf{\Lambda }},B_{\mathsf{\Lambda }},C_{\mathsf{\Lambda }},D_{\mathsf{\Lambda }}\right\}\right)\ge 0,\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{K}\left(Q,\left\{A,B,C,D\right\}\right)\ge 0,\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{Q}_{\tau }^T=Q_{\tau },\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{Q}_{\mathsf{\Lambda }}^T=Q_{\Lambda },\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{Q}^T=Q,\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{r}_0\le \alpha .\end{array}$$

(30)

By solving optimization problems (18) or (30) we can obtain the optimal input spectrum for the closed-loop system identification. The robust performance problem consists in checking whether the feedback configuration shown in Figure 1 is stable for all admissible excitations and whether the minimum of the selected cost function can be obtained using available inputs.

5. The Nonlinear Cascaded Double Tank Model

The nonlinear model of two water tanks, presented in Figure 2, is used as an illustrative example. The system can be described by the dependence of the volumetric flow Q_in(t) [m³/s] into the first tank to the outflow of water Q_out(t) [m³/s] through the valve at the second tank. The dynamics of the water flow in each tank can be formulated as

$$A\frac{dh(t)}{dt}=Q_{in}(t)-Q_{out}(t)\hspace{0.17em},$$

(31)

where A [m²] is the cross-sectional area of the tank, and h(t) [m] denotes the height of the water level in the tank.

When the cross-sectional surface of the tank outlet hole, considering an ideal sharp-edged orifice, is known, the outflow of water of each tank can be established using Torricelli’s law and is given by

$$Q_{out}(t)=a\cdot \sqrt{2gh(t)}\hspace{0.17em},$$

(32)

where a is the cross-sectional area of the hole, and the gravitational constant value g is equal to 9.8 m/s².

Substituting expression (32) with expression (31) and assuming that the output of the first tank is interconnected with the input of the second tank, the system of two water tanks can be described by the following nonlinear differential equations:

$$\{\begin{array}{l}{A}_1\frac{d{h}_1(t)}{dt}=-a_1\cdot \sqrt{2g{h}_1(t)}+Q_{in}(t)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{h}_1\left(0\right)=h_{10},\hspace{0.17em}\\ A_2\frac{d{h}_2(t)}{dt}=a_1\cdot \sqrt{2g{h}_1(t)}-a_2\cdot \sqrt{2g{h}_2(t)}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{h}_2\left(0\right)=h_{20}.\end{array}$$

(33)

where i = 1, 2 represents the corresponding tank. We have performed the following substitutions Q_in(t) = u(t), x₁(t) = h₁(t), x₂(t) = h₂(t), y(t) = h₁(t) to obtain the ordinary form of the first-order model state-space equations:

$$\{\begin{array}{l}{\dot{x}}_1=-\frac{a_1}{A_1}\cdot \sqrt{2g{x}_1}+\frac{1}{A_1}u\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{x}_1\left(0\right)=h_{10},\hspace{0.17em}\\ {\dot{x}}_2=\frac{a_1}{A_2}\cdot \sqrt{2g{x}_1}-\frac{a_2}{A_2}\cdot \sqrt{2g{x}_2}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{x}_2\left(0\right)=h_{20}.\end{array}$$

(34)

where x₁ = x₁(t, a₁), x₂ = x₂(t, a₁, a₂). The tanks’ water levels have obvious physical limitations:

$$h_{i,\hspace{0.17em}max}\ge x_i(t)\hspace{0.17em}\ge 0\hspace{1em}\hspace{1em}i=1,\hspace{0.17em}2.$$

(35)

The physical parameters and the limitations of the water tanks are shown in

Table 1. The plant model parameters and limitations.

Parameter	Value	Unit	Description
h1,max	4.00	m	Max. water level of tank 1
h1,min = h2,min	0.00	m	Min. water level of tanks 1, 2
h2,max	2.00	m	Max. water level of tank 2
h10	0.75	m	Initial condition of tank 1
h20	0.50	m	Initial condition of tank 2
a1	0.05	m2	Area of water outlet hole
a2	0.06	m2	Area of water outlet hole
A1	1.50	m2	Cross-section of tank 1
A2	0.75	m2	Cross-section of tank 2
u0	0.25	m3/s	Initial water inflow

.

The nonlinear model of the system (34) was linearized around the steady-state point $x_1^0$ = 1.274 [m], $x_2^0$ = 0.885 [m], and then discretized. The linearized model matrices, computed using a first-order Taylor expansion, are

$$\begin{array}{l}{A}_l=\left[\begin{array}{cc}-{\tau }_1& 0\\ {\tau }_3& -{\tau }_4\end{array}\right],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{B}_l=\left[\begin{array}{c}\frac{u_0}{A_1}\\ 0\end{array}\right],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{C}_l=\left[\begin{array}{cc}1& 0\end{array}\right],\\ {\tau }_1=\frac{a_1}{A_1}\sqrt{\frac{g}{2x_1^0}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\tau }_3=\frac{a_1}{A_2}\sqrt{\frac{g}{2x_1^0}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\tau }_4=\frac{a_2}{A_2}\sqrt{\frac{g}{2x_2^0}}.\hspace{0.17em}\end{array}$$

(36)

Then, the zero-order hold discretization of (34) can be formulated as

$$A_d=e^{A_l},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{B}_d={\displaystyle \underset{0}{\overset{1}{\int }}{e}^{A_l\left(1-t\right)}{B}_{l}dt,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{C}_d=C_l.}$$

(37)

The discrete-time state-space water tanks model obtained for the 1 Hz sampling rate is

$$A_d=\left[\begin{array}{cc}0.937& 0\\ 0.116& 0.828\end{array}\right],\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{B}_d=\left[\begin{array}{c}0.161\\ 0.010\end{array}\right].$$

(38)

In the experimental part of this paper, the input-output description of the process plant defined by (1) is considered.

6. Average Performance Input Design Problem Solution

The input design and the model parameter estimation are performed using the double tank system (34) described in the previous section. To perform the experiments, the Matlab-dedicated Moose2 package has been applied [34]. The Moose2 solver requires the YALMIP and SDPT3 toolboxes to be installed [35,36].

6.1. The Parameter Estimation of the Upper Tank

To illustrate the advantages of the input design for the closed-loop system parameter estimation, the output-error (OE) system, considering only the upper tank dynamics (34), is used:

$$y\left(t\right)=G_0\left(q,{\theta }_0\right)u\left(t\right)+e_0\left(t\right),$$

(39)

where

$$G_0\left(q,{\theta }_0\right)=\frac{0.161q^{-1}}{1-0.937q^{-1}},$$

(40)

where e₀(t) is a white Gaussian process with a zero mean, and the variance is λ₀ = 0.1. We estimate model parameters based on N = 500 input-output data with the following sampling period T_s = 1 [s]:

$$G\left(q,\theta \right)=\frac{{\theta }_1{q}^{-1}}{1+{\theta }_2{q}^{-1}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\theta ={\left[{\theta }_1,{\theta }_2\right]}^T,$$

(41)

The optimization problem is described by Formulas (18) and (30) with the arbitrarily selected value γ = 0.1 and the excitation signal is modeled by the FIR filter with the order of m = 20. The ellipsoidal constraint in (18) is defined by (12) with χ = χ²(2) = 5.99, which is related to the confidence level of 95%, and the frequency grid [0, π] with 50 frequency points is used to evaluate the ellipsoidal constraint (26). The weighting function (13) is obtained using the damping ratio ξ = 0.7 and the carefully selected bandwidth ω₀ = 0.85 [rad/s] and then is converted into the following discrete form:

$$T\left(q\right)=\frac{0.239q^{-1}+0.159q^{-2}}{1-0.906q^{-1}+0.304q^{-2}}.$$

(42)

The optimal input spectrum obtained using (18), the complementary sensitivity function T(q), and the open-loop system G₀(q, θ₀) for the upper tank process are shown in Figure 3.

The parameter estimates of the model (41), obtained using the frequency (18) and the KYP (30) methods, are displayed in Figure 4.

Solving the optimization problem (18), the optimal input spectrum for the upper tank system has been obtained (Figure 3). The objective of optimization was to design the input spectrum of minimum energy, considering constraints established by the contour ${\Vert \Delta \left(\omega ,\theta \right)\Vert }_{\infty }\le 0.1$. To reduce the required input power, most of the excitation energy is located around the first resonance peak. The water sloshing of the tank becomes weaker at higher natural frequency modes. It can be noticed that a large quantity of the excitation energy is concentrated inside the expected bandwidth of the considered system. The minimum designated input power α obtained for the upper tank system is 0.140 (Figure 3). However, the input design method that focuses the input energy around the first spectrum peak is very sensitive to inaccurate estimates of model parameters at this point. Then, the designed optimal input has been used for the estimation of the parameters when the output of the upper tank model has been disturbed by the white Gaussian process with the zero mean and the variance λ₀ = 0.1. The results of the model parameters estimation using frequency (18) and KYP (30) methods for the upper tank system are shown in Figure 4. For parameter estimation purposes, 100 Monte-Carlo independent runs have been made. From Figure 4 it can be observed that the upper tank model (41) parameter estimates, for both presented methods, are clustered inside the assumed contour (i.e., for γ = 0.1). In fact, about 95% of the estimated parameters from the frequency method and about 92% of the estimated parameters from the KYP method satisfy the ellipsoidal constraint (12). The greater dispersion of the estimated parameters observed for the KYP method (Figure 4) results from numerical issues related to the assumed frequency grid. The mean absolute percentage error (MAPE) of the estimated model parameters, considering the frequency method, does not exceed 3.12% for θ₁ and 0.27% for θ₂, respectively. The MAPE obtained for the KYP method is larger and has the following values: 8.43% for parameter θ₁ and 0.72% for parameter θ₂.

6.2. The Parameter Estimation of the Interacting Tanks System

The results of the input spectrum design for the identification of the water tanks system, considering the dynamics of both interconnected tanks (34), are obtained using the following output-error (OE) system:

$$G_0\left(q,{\theta }_0\right)=\frac{0.010q^{-1}+0.009q^{-2}}{1-1.766q^{-1}+0.776q^{-2}}.$$

(43)

The model parameters are estimated under the same conditions as in (41), except for the KYP method which has numerical stability issues, when applied to more complex problems, and has not been presented. The choice of parameters to be estimated is as follows:

$$G\left(q,\theta \right)=\frac{{\theta }_1{q}^{-1}+{\theta }_2{q}^{-2}}{1-1.766q^{-1}+0.776q^{-2}},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\theta ={\left[{\theta }_1,{\theta }_2\right]}^T,$$

(44)

The weighting function polynomial form obtained for the same damping value ξ = 0.7 and selected bandwidth ω₀ = 2.5 [rad/s] is given by

$$T\left(q\right)=\frac{0.007q^{-1}+0.006q^{-2}}{1-1.825q^{-1}+0.839q^{-2}}.$$

(45)

The optimal input spectrum obtained using (18), the weighting function T(q), and the open-loop system G₀(q, θ₀) for the two interconnected tanks are shown in Figure 5.

The parameter estimates of the model (44), obtained using the frequency method (18) are shown in Figure 6.

Solving the minimization problem (18), the optimal input spectrum for the interacting tank system has been obtained (Figure 5). The objective was to design the spectral representation of the input signal with the minimum amount of energy that meets the constraints given by the assumed contour (i.e., for γ = 0.1). The minimum required input power α obtained for the interacting water tanks system is 0.074 and the largest part of the excitation energy is injected with the first resonance peak of the system under consideration (Figure 5). The optimal input spectrum has been used for the estimation of model parameters when the output of the model contains additive white Gaussian noise with a zero mean and a variance of λ₀ = 0.1. The result of the model parameters estimation using the frequency method (18), based on 100 Monte-Carlo independent runs, is shown in Figure 6. All estimated parameters are clustered inside the assumed contour and about 90% of them are inside the confidence region (12), the MAPE for θ₁ is 6.90%, and for θ₂ is 9.06%, respectively. The optimal input spectrum and the corresponding confidence region for the interacting water tanks system, using the KYP method (30), could not be solved due to the numerical stability problems. The linear matrix inequalities that arise during the implementation of the KYP lemma (to be solved in the Moose2 package) become too large for more complex systems [34].

7. Conclusions

The main focus of this work is on real-time optimal input spectrum design for the closed-loop system identification task. The minimization of the input spectrum power has been performed as a standard convex optimization problem, considering LMI constraints. When the compensator is close to the complementary sensitivity function, the $\mathscr{H}$_∞ norm of weighted relative error (11) has been used to ensure stability and measure the performance indices of the cascaded tanks model of the real system. It has been observed that all estimated models are clustered inside the contour ${\Vert \Delta \left(\omega ,\theta \right)\Vert }_{\infty }=0.1$, as desired. It has also been proven that the identification performed subject to the fixed bound on the ellipsoidal quality constraint yields consistent estimates of the true system. In fact, more than 90% of the models with the obtained parameter values meet the performance constraint.

The results show the benefits of conducting closed-loop system identification experiments, as discussed in the paper, where the time for acquiring measurement data is shortened and the power of the excitation signal is minimized. It has been found that the input design problem considering the frequency grid is much more numerically stable than the KYP method implemented in Moose2. It is related to linear matrix inequalities that become too large when using the KYP lemma, which is why they are the bottleneck for solving optimal input design problems. The problems encountered using the KYP lemma can probably be solved sequentially using an adaptive frequency grid instead and will be reformulated in future works. For real-time system identification purposes, the frequency grid method is preferred. The multi-objective optimization concerning both the parameters of the compensator and the external reference signal power will also be studied.

There are still several open problems in application-oriented input design for closed-loop system identification, that should be resolved. The results of the optimal input design could be extended to MIMO systems used in industry. The more general class of controllers with implicit control laws also should be analyzed. It should be investigated whether the lower order of the chosen model has an impact on the quality of closed-loop identification. The application-oriented input design for nonlinear system identification (not linearized) is another topic that should be further explored. The fractional-order models often guarantee a more precise description of system dynamics, because they are represented as the infinite distribution of time constants. Computation time has great importance in real-time applications, which is why truncated time window approximation methods (e.g., the long-memory prediction error method) for fractional-order system identification should be applied.