Claims:The utility and scope of the invention is defined by the following claims:

Claims:
1. A method for real time identifying and modeling the time varying, time-delayed continuous time system dynamics includes the steps of:
a) Use the current data samples for estimating the discrete time-delayed model and extract the fractional time-delay parameter from discrete time-delayed model.
b) Update the integer time-delay parameter by extracting integer part from fractional time-delay parameter and evaluate the convergence condition on fractional time-delay.
c) Recover the continuous time-delayed model through exact formulation and using the identified model for analyzing and predicting the true plant’s behavior.
2. As per claim 1, a general exact formulation is done to relate continuous and discrete time-delayed systems. The time-delay parameter in the form fractional delay part is extracted in terms of discrete model parameters
3. As per claim 1, the overall time-delay parameter is updated through recovering the integer part value from fractional delay part, as mentioned in claim 3. The convergence condition is continuously validated, through which the status of other model parameter convergence are ensured.
4. As per claim 1, According to claim 5, when the convergence condition is validated, the formulated relations are utilized to recover the continuous time-delayed model parameters.
5. As per claims 3 & 6, the continuous or discrete models are directly used for analyzing and predicting the plants behavior in virtual or simulated environment, without disturbing the actual process operations. , Description:Field of Invention
It relates the invention to the field of System Identification, where the data-driven empirical statistical approaches are used for identification and modeling of unknown process dynamics. The targeted process models includes rational dynamics as well as time-delays. The estimation of time-delay is a challenging task because it does not directly appear in the model formulation. In this invention we developed a novel approach where the time-delay can be estimated simultaneously along with other model parameters.
Background of the invention
At present, most of the industrial processes use model based controlling techniques, where the designing or re-tuning of controllers essentially requires the identification of process models at present operating conditions or in real time. The model based controllers commonly use identified first or second-order plus time-delay process dynamics for its fine tuning (Ljung, L., [2010], Perspectives on system identification, Annual Reviews in Control, 34(1), pp. 1-12.). (Tangirala, A.K., [2018], Principles of system identification: theory and practice. CRC Press.). However, the problem of identifying time-delayed processes in real-time is challenging mainly because of: its non-linear optimization nature, the absence of time-delay in the parameter vector, limitation on time-delay to be an integer multiple of the sample time, the absence of an accurate convergence condition, etc. The existing methods suffer from these limitations and highly dependable on accurate parameter and time-delay initialization for achieving convergence (Yang, X., & Gao, H. [2014], Multiple model approach to linear parameter varying time-delay system identification with EM algorithm. Journal of the Franklin Institute, 351(12), pp. 5565-5581.).
The identified delayed models are used to virtually simulate the plant's behavior for analysis, testing, predictions, fault detection and most importantly for controller synthesis. The virtual simulations are also cost effecting and danger free as far as the safety and economy of the actual plant is concerned (Ren, X.M., Rad, A.B., Chan, P.T. and Lo, W.L., [2005], Online identification of continuous-time systems with unknown time delay, in IEEE Transactions on Automatic Control, 50(9), pp. 1418-1422.). The main difficulty associated with optimization problem for delayed systems is that the cost function has several local minima’s with respect to estimated delay. The contributions are reported to enhance the complexity of the cost function through single or multiple low pass filtering. In (Chen, F., Zhuan, X., Garnier, H. and Gilson, M., [2018], Issues in separable identification of continuous-time models with time-delay. Automatica, 94, pp. 258-273.) it is shown that the cost function associated with delayed systems has multiple local minima. Therefore, the convergence to the global minima can be obtained by improving the convexity through incorporating multiple low-pass filters. Two types of techniques are used for identifying continuous delayed models. First one is the fully discrete domain approach where the discrete model parameters and only integer delay are identified, the continuous delayed model is obtained by transforming the discrete model to continuous one. In contrast, this technique is computationally efficient but due to the ignorance of fractional delay the accuracy of the identified continuous delay model has to be compromised (Liu, T., Dong, S., Rong, S. and Zhong, C., [2016], Identification of discrete-time model with integer delay and control design for cooling processes with application to jacketed crystallizers. IEEE Transactions on Control Systems Technology, 25(5), pp. 1775-1789.).
Also, the three most common drawbacks of these techniques, first of which is the parameters and delay initialization problem, second is the presence of nonlinearities in the cost function which produces multiple local equilibrium points and third is the lack of suitable convergence condition around global optimum (Chen, F., Garnier, H., Padilla, A. and Gilson, M., [2020], Recursive IV identification of continuous-time models with time delay from sampled data. IEEE Transactions on Control Systems Technology, 28(3), pp. 1074-1082.).
An apparatus is developed for identifying sub-models and their quality testing in a multivariable process. These sub-models are then used for designing suitable controllers in a Model predictive control (MPC) application (WO2010/138452Al). In (WO2010/077038A3) a method is developed for single input single output model based predictive controller tuning. A model development procedure using historical input-output data is developed for a given process and quality prediction and fault detection is done based on the obtained model parameters mean and standard deviation values (US2015/0324329A1). Method and system for data driven empirical modeling of parameter varying systems has been reported to estimate differential and difference model equations and their parameters (US2015/0039280A1). A multi-model identification approach is reported in (US2014/0128997A1), where the best model is selected based on the minimum rate of errors between observed and estimated outputs. In (US2007/0078533A1), individual process models are identified between different control loops for a given process control system. Apart from these, some related inventions are also reported, to identify Hammerstein models with linear dynamic and static nonlinear parts (US8346712B2), to identify building system model using input-output data with filtering and Kalman gain parameter based method (US10088814B2), to develop process model based model predictive control algorithm (US7451004B2). However, by analyzing the existing prior inventions, we found that the modeling of real time, time varying time-delayed processes and corresponding convergence are not highlighted.
The aim of our invention is to develop a data-driven device, which can use input/output data information to model unknown time-varying, time-delayed processes in real-time. The estimation of time-delay and rational model parameters of process can be carried-out simultaneously in real-time with least computation efforts. We have invented a new data-driven identification approach which allows the user to track the time varying time-delayed process dynamics in real-time, where the convergence of all model parameters can be ensured by a single parameter value only. There are no such methods are proposed/invented yet, which can identify time-varying time-delays along with rational model parameters in real-time.
Summary of the invention
A generalized indirect modeling of continuous time-delay systems is developed, this method is suitable for real-time applications with time-varying parameters. The invented method is based on real-time indirect modeling of continuous time-delay processes thorough discrete sampled data, where the transfer function parameters and time-delay (integer and fractional parts) are estimated simultaneously.
The invention is based on explicit computation of time-delay parameter in real time, which is in general not present in the parameter vector. The integer part is continuously extracted from the identified fractional time-delay until it becomes “zero” or in other words till the fractional part becomes purely fractional. The identification approach can be embedded on a microcontroller based device like Arduino and can interact with MATLAB/Simulink, Scilab, Python, etc. in real time along with various sensors.
Detailed description of the invention
This invention involves the Indirect modeling of continuous time-delay systems from discrete time-delay systems through data acquisition. An iterative discretized modeling is formulated to indirectly compute the continuous time-delayed process models. The obtained generalized continuous time-delay model can be directly used for controller and process synthesis. Also, an exclusive extraction of fractional time-delay from discrete model enables the simultaneous identification of model as well as time-delay parameters. This approach is suitable for both online and offline identification with fixed and varying process parameters.
Defining the problem statement for a general SISO output-error model, which is used to relate of a continuous time-delay system with linear dynamics G(s) and time-delay 𝜏 ∈ ℝ+ as:
x(t)=L^(-1) [G(s) e^(-τs) ]*u(t) (1)
y(t)=x(t)+w(t) (2)
where, the terms are defined as input 𝑢(𝑡) ∈ ℝ, noise free output 𝑥(𝑡) ∈ ℝ and noisy output 𝑦(𝑡) ∈ ℝ with measurement noise 𝑤(𝑡) ∈ ℝ.
Considering, the overall system dynamics with time-delay is expressed as:
G_C (s)=K β(s)/α(s) e^(-τs), (3)
where 𝐾 is the system's gain and 𝜷(𝑠) and 𝜶(𝑠) are the numerator and denominator terms of orders 𝑚 and 𝑛, respectively, are expressed as follows
β(s)=∏_(i=1)^m▒(β_i s+1) (4)
α(s)=∏_(j=1)^n▒(α_j s+1) (5)
if the time-delay 𝜏 consists of its integer 𝑑 and fractional 𝜆 multiples of sample time ℎ as
τ=(d-1)h+λh. (6)
Then the model of GC(s) can be presented by its equivalent discretized model as
G_D (z)=(b_0+b_1 z^(-1)+⋯+b_n z^(-n))/((1-a_1 z^(-1) )(1-a_2 z^(-1) )…(1-a_n z^(-1) ) ) z^(-d) (7)
The objective of this invention is to use the sampled input/output data to compute all the parameters of the continuous-time system GC(s) using
θ_C=[K,τ,α_1,α_2,…,α_n,β_1,β_2,…,β_m ]^T (8)
Indirectly, through estimating the parameters of the discrete-time system as
θ_D=[a_1,a_2,…,a_n,b_0,b_1,…,b_n ]^T. (9)
In following discussions the formulation for the discretization of continuous-time model and the relation between continuous and discrete model parameters has been developed.
If the discretization of GC(s) is performed using zero-order-hold, with the given transfer function as
G_D (z)=Z[├ L^(-1) {(1-e^(-hs) ) (G_C (s))/s}┤|_(t=kh) ]. (10)
Now, to solve the partial factors of above expression, the following term can be computed by
(G_C (s))/s=K(1/s+∑_(j=1)^n▒C_j/((s+1\/α_j ) )) e^(-τs), (11)
where the coefficient terms are computed by
C_j=-(α_j^(n-1))/(∏_(l=1;l≠j)^n▒(α_j-α_l ) );for  m=0 (12)
C_j=-(α_j^(n-(m+1) ∏_(r=1)^m▒(α_j-β_r ) ))/(∏_(l=1;l≠j)^n▒(α_j-α_l ) );for  m≠0 (13)
by putting the time delay expression, one can get
(G_C (s))/s=K(e^(-λhs)/s+∑_(j=1)^n▒(C_j e^(-λhs))/((s+1\/α_j ) )) e^(-(d-1)s), (14)
Now, the following proves are used to solve this expression, and to formulate the relationship between continuous and discrete model parameters.
Proposition 1: The modified discretization of the following function type can be computed by
Z[├ L^(-1) {1/((s+1\/α_j ) ) e^(-λhs) }┤|_(t=kh) ]=(a_j^((1-λ) ) z^(-1))/(1-a_j z^(-1) ), (15)
where  a_j=e^(-h/α_j )  and  0≤|λ|<1.
Proof: According to time shifting property of Laplace transform, we can have
Z[├ L^("-" 1) {1/((s+1\/α_j ) ) e^(-λhs) }┤|_(t=kh) ]=Z[├ e^((-(t"-" λh))/α_j ).1(t"-" λh)┤|_(t=kh) ], (16)
now using the definition of Z-transform, one can have
Z[├ L^("-" 1) {1/((s+1\/α_j ) ) e^(-λhs) }┤|_(t=kh) ]=e^(λ h/α_j ) ∑_(k=1)^∞▒〖e^("-" k h/α_j ) z^("-" k) 〗, (17)
by applying sum of infinite geometric progression, we have
Z[├ L^("-" 1) {1/((s+1\/α_j ) ) e^(-λhs) }┤|_(t=kh) ]=(e^(-(1-λ)h/α_j ) z^("-" 1))/(1-e^(h/α_j ) z^("-" 1) ). (18)
Corollary 1: The discretization of the following function is:
Z[├ L^("-" 1) {1/s e^("-" λhs) }┤|_(t=kh) ];0≤λ<1=z^("-" 1)/(1"-" z^("-" 1) ), (19)
Proof: In proposition 1 by substituting 1/αj = 0, we can have
Z[├ L^("-" 1) {1/s e^("-" λhs) }┤|_(t=kh) ]=z^("-" 1)/(1"-" z^("-" 1) ). (20)
Finally, the equivalent discrete model is given as
G_D (z)=K[1+∑_(j=1)^n▒〖C_j a_j^(-λ) 〗-∑_(j=1)^n▒(C_j a_j^(-λ) (1-a_j ))/((1-a_j z^(-1) ) )] z^(-d), (21)
However, to compute the above expression the model is estimated using data and expressed in the residual form as:
G_D (z)=(∑_(i=0)^n▒〖b_i z^(-i) 〗)/(∏_(j=1)^n▒(1-a_j z^(-j) ) ) z^(-d)=[K+∑_(j=1)^n▒R_j/((1-a_j z^(-j) ) )] z^(-d), (22)
where, the following terms are computed as
K=K[1+∑_(j=1)^n▒〖C_j a_j^(-λ) 〗],R_j=K[C_j a_j^(-λ) (a_j-1)]. (23)
Finally, the parameters can be recovered by
K=lim┬(z→1)⁡G (z)=(∑_(i=0)^n▒b_i )/(∏_(j=1)^n▒(1-a_j ) ); (24)
and α_j=-h/log⁡(a_j ) ;β_i=roots(∑_(i=0)^n▒〖b_i z^(-i) 〗) (25)
λ=-log⁡(R_j \/KC_j (1-a_j ))\/ log⁡(a_j ) (26)
The following parameter vector is estimated using real time recursive method:
θ ̂_D=[c ̂_1,…,c ̂_n,b ̂_0,b ̂_1,…,b ̂_n ]^T. (27)
where, the coefficients cj are computed in terms of sum and product terms of aj.
Now, consider the prediction error, which is calculated from measures output and estimated output as
ε(k)=y(k)-y ̂(k). (28)
The cost function to be minimized is:
V(θ ̂_d,k)=1/2 ∑_(i=1)^k▒Δ^(k-i) ε^2 (i), (29)
where, Δ represents the forgetting factor.
Let us now define the gradient vectors for prediction error minimization as
ϕ_F (k)=∂/(∂(θ_D ) ̂ ) (y_F ) ̂(k)=[-(y_F ) ̂(k-1),…,(-1)^n (y_F ) ̂(k-n),u_F (k-d ̂ ),…,u_F (k-n-d ̂ )]^T, (30)
where the subscript term F represents the filtered version of signals.
Finally, by introducing a covariance matrix (𝑘), the overall recursive formulation for updating the parameter vector can be summarized as
S(k)=(P(k-1) ϕ_F (k))/(Δ+ϕ_F^T (k)P(k-1) ϕ_F (k) ), (31)
ε(k)=y(k)-y ̂(k,(θ_D ) ̂(k-1)), (32)
(θ_D ) ̂(k)=(θ_D ) ̂(k-1)+S(k)ε(k), (33)
P(k)=1/Δ [P(k-1)-(P(k-1) ϕ_F (k) ϕ_F^T (k)P(k-1))/(Δ+ϕ_F^T (k)P(k-1) ϕ_F (k) )], (34)
Where, at each recursive step the continuous model parameters are computed directly from the estimated parameter vector, which are initialized to zero at 𝑘=0. Also, the integer time-delay part d (also initialized to zero) is updated by extracting integer component from fractional time-delay, while 𝜆>1, when a pre-defined number of samples (considered being 200 samples for generating results) are passed. Further the covariance matrix is initialized with 𝑃0=103𝐼.
5 claims & 3 Figures
Brief description of Drawing
Figure 1 Block Diagram of Real-time, Time-Varying Parameter Estimation Method
Figure 2 Real-time Identification of First-order Time-varying, Time-delayed Process Parameters
Figure 3 Real-time Identification of a Heating Prototype System Parameters
Detailed description of the drawing
In Figure 1, the block diagram representation of the identification and modeling scheme used in this invention is depicted. The block diagram includes the data collection, parameter initialization, auxiliary model based filtering, parameter estimation, convergence condition checking, update of integer time-delay parameter through fractional time-delay parameter, and recovery of continuous time-delay transfer function model. The Figure 2 is used to show the real-time identification of time-varying first and second order time-delayed systems using the proposed invention. From these figures one can clearly observe that the time varying parameters can be accurately tracked in online or real time mode.
A real experimental setup of heating prototype system is also utilized to validate the invention and the real-time identified parameters are plotted in Figure 3.