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
∞ 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
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
where the transfer functions
G and
H are parameterized by a vector
θ ∈ ℝ
n. 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
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
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:
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]):
where
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
Φκ. The spectrum distribution is as follows:
where
Φu is the input signal spectrum and
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
Φκ, and is described as [
30]
where
and
with
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
Φu and the cross-spectrum
Φue 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]:
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
∞ 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:
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
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
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
where
n is the number of parameters of the model
G(
ω,
θ).
The spectrum
Φr of the external excitation signal
r(
t) in the closed-loop system design is related to the spectra
Φu and
Φue described by the following formulas [
13]:
In this paper, the simpler optimization problem is solved concerning the fixed operating compensator and the design variable
Φu. The optimal external excitation spectrum
can be reconstructed from the optimal
and the optimal
by inverting the following formulas [
13]:
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:
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
Φκ defined in (7). This matrix can be expressed as
where the scalar basis functions
are proper, stable, and rational with
. Using proper parameterization of the signal spectrum, it is possible to solve a finite-dimensional convex optimization problem with matrix coefficients
. We assume that
so that
ℝ. Therefore, the factors
must satisfy the below constraint:
The most common choice of the basis function is
A finite-dimensional parameterization is then acquired by substituting the spectrum
Φκ(
ω) in (19) with the following approximation:
where
Ψ(
ejω) is a finite-dimensional estimation of the positive real part of
Φκ(
ω) with
Ψ*(
ejω) =
ΨT(
e−jω). Finally,
m initial components of (19) are used to obtain the input spectrum:
where
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 , then a Q = QT like the following exists:if and only if 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
ck 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]:
Using the quality measure (24), we require that about 95% of the estimated models will fulfill the constraints imposed by the ellipsoidal region. Let
Wn,
Wd,
Xn, and
Xd be finite-dimensional transfer functions, and Y
n is as follows:
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
where
The transfer function G(ω, η) includes the parameter vector η which is not affected by the noise, and Wn, Wd, Xn, Xd, Yn, Yd, Kn, and Kd 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 Wn = Wd = 1, Xn = −G0, Yn = T, Xd = Kn = Kd = 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
using some base functions, namely,
,
k = 0,…,
K − 1, and there is a set {
Λk} expressed by
where the elements {
τk} and the ingredients of the matrix
R in (26) are linear in {
Λk}. Then, there exists a state-space realization {
Aτ,
Bτ,
Cτ,
Dτ} of the positive part of
τ(
ω), where the variables {
τk} are linear in
Cτ and
Dτ. Similarly, the same applies to the state-state representation {
AΛ,
BΛ,
CΛ,
DΛ}. Furthermore, it holds (26) if only
and
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
τk in (28) are the auto-covariance sequence
rk 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
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
Qin(
t) [m
3/s] into the first tank to the outflow of water
Qout(
t) [m
3/s] through the valve at the second tank. The dynamics of the water flow in each tank can be formulated as
where
A [m
2] 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
where
a is the cross-sectional area of the hole, and the gravitational constant value
g is equal to 9.8 m/s
2.
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:
where
i = 1, 2 represents the corresponding tank. We have performed the following substitutions
Qin(
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:
where
x1 =
x1(
t,
a1),
x2 =
x2(
t,
a1,
a2). The tanks’ water levels have obvious physical limitations:
The physical parameters and the limitations of the water tanks are shown in
Table 1.
The nonlinear model of the system (34) was linearized around the steady-state point
= 1.274 [m],
= 0.885 [m], and then discretized. The linearized model matrices, computed using a first-order Taylor expansion, are
Then, the zero-order hold discretization of (34) can be formulated as
The discrete-time state-space water tanks model obtained for the 1 Hz sampling rate is
In the experimental part of this paper, the input-output description of the process plant defined by (1) is considered.
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 ∞ 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 , 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.