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Nonlinear Discrete-time System Identification without Persistence of Excitation: A Finite-time Concurrent Learning

This paper deals with the problem of finite-time learning for unknown discrete-time nonlinear systems’ dynamics, without the requirement of the persistence of excitation. A finitetime concurrent learning approach is presented to approximate the uncertainties of the discrete-time nonlinear systems in an on-line fashion by employing current data along with recorded experienced data satisfying an easy-to-check rank condition on the richness of the recorded data which is less restrictive in comparison with persistence of excitation condition. Rigorous proofs guarantee the finite-time convergence of the estimated parameters to their optimal values based on a discrete-time Lyapunov analysis. Compared with the existing work in the literature, simulation results illustrate that the proposed method can timely and precisely approximate the uncertainties.


I. INTRODUCTION
Learning a high-fidelity model of a nonlinear system via stream of data is of vital importance in many engineering applications since such systems are highly subjected to uncertainties that can degrade the performance of the system controllers. It is well known that many learning strategies, such as least-square and gradient descent [1], depend heavily on the persistency of excitation (PE) condition that permanently requires a complete span of the space over which the learning is performed, and failure to fulfill this condition will lead to poor learning results. However, the PE condition might be hard to achieve or even might not be feasible in some scenarios, especially in the context of on-line learning. Concurrent learning [2][3][4][5][6] has emerged as a promising paradigm in the direction that guarantees the exponential convergence of the estimated parameters to their optimal values with relaxing the strict assumption of the PE condition to some easy-to-check verifiable conditions on the richness of data. Concurrent learning technique benefits from recorded experienced data along with current data to replace the PE condition on the regressor with a rank condition on the memory stack of the regressor recorded data. Based on this rank condition, the matrix of the regressor recorded data must contain the same number of linearly independent elements as the dimension of the independent basis functions in the regressor.
In many practical situations, the system dynamics are employed in on-line monitoring and control applications; therefore, learning the system's unknown dynamics over a finite-time interval is required. Finite-time learning is of more interest rather than learning with asymptotic or exponential convergence rate since it is more physically realizable than concerning infinite time. Moreover, such finite-time learning scheme is of utmost importance for learning-based controlled systems that demand fast and reliable actions. Although, finite-time control methods have been extensively employed for discrete [7][8][9] and continuous-time [10][11][12] systems, fewer attempts have been proposed by several researchers to tackle finite-time learning such as [13][14][15][16][17][18][19]. While these results have studied finite-time learning schemes, the majority of them are concerning with continuoustime systems. The results on finite-time learning of continuous-time systems are more fruitful in comparison with discrete-time ones due to the existence of abundant mathematical tools. In many practical applications, however, identifying precise discrete-time dynamics of the system is required due to the development of computer technology and the introduction of digital controllers and sensors. Hence, it is of great practical importance to investigate finite-time learning of discrete-time systems. There are only a few results discussing the identification of discrete-time systems including [19,20]. The work of [19], studied the discrete-time systems uncertainty identification in finite-time, where the proposed method required the on-line invertibility check of a regressor matrix and its inverse computation, along with interval excitation of the regressor. In [20], a concurrent learning-based method for discrete-time function approximation is presented that relaxed the PE condition and guaranteed the asymptotic convergence and boundedness of the parameters estimation errors, respectively, for function approximators with zero and non-zero residual approximation errors in case of optimal parameters.
According to the aforementioned discussions, it is favorable to employ finite-time learning schemes that can alleviate the restrictive condition of the PE. To partially bridge this gap, recently, the authors in [18] proposed a finite-time identification method for continuoustime systems based on concurrent learning. However, the counterpart results in the discrete-time domain are not available. Discrete-time systems are quite different with continuous-time systems; therefore, the tools applied to the continuous-time domain cannot be directly employed for the discrete-time domain. Moreover, different from finite-time stability analysis of continuous-time systems, which can draw support from many mathematical tools, the mathematical tools for finite-time stability analysis of discrete-time systems are not plenty. Therefore, the research on finite-time convergent concurrent learning identification method for discrete-time systems is more challenging and complex.
Motivated by the above-mentioned discussions, this work aims to propose a finite-time concurrent learning (FTCL) scheme for discrete-time systems that guarantees finite-time parameter convergence without the restrictive PE condition by employing a memory stack of data, satisfying a rank condition. To this end, the proposed method guarantees the finite-time convergence of the parameters estimation errors to the origin for adaptive approximators with zero minimum functional approximation errors (MFAEs) where MFAE is the residual approximation error in the case of optimal parameters. And for adaptive approximators with non-zero MFAE, the presented method ensures that the parameters estimation errors are uniformly ultimately bounded. The proposed adaptive estimation method in this work is simpler in comparison with the continuous finitetime estimation laws in [16,17], containing fractional functions, and it is able to approximate the unknown dynamics of higher-order discrete-time nonlinear systems. Contrary to [14,15,19], the proposed method does not need the inverse of the recorded regressor matrix or approximating it using an auxiliary matrix.
Compared with previous results on finite-time learning and concurrent learning approaches, contributions of the current work are summarized as follows: 1) A novel FTCL scheme is presented for learning the unknown dynamics of nonlinear discrete-time systems. 2) Rigorous proofs ensure the finite-time convergence of the parameters estimation errors to the origin for adaptive approximators with zero MFAE using the proposed FTCL method and discrete-time Lyapunov analysis. It is also guaranteed that for adaptive approximators with non-zero MFAE, the parameter estimation errors are uniformly ultimately bounded.
3) The settling-time functions for the finite time of the parameters' error convergence to zero and to a bound around zero are respectively given for the adaptive approximators with zero and non-zero MFAEs. In addition, based on the finite-time analysis, a condition on the learning rate is derived for finitetime convergence. Notation: Throughout the paper, the following notation is adopted. R, Z, and N`respectively show the set of real, integer and natural numbers without zero. . is used to denote the Euclidean norm for vectors and induced 2-norm for matrices. Trace of a matrix is indicated with trp.q. The minimum and maximum eigenvalues of matrix A are respectively denoted by λ min pAq and λmaxpAq. The matrix I is the identity matrix of appropriate dimensions. t.u : R Þ Ñ Z is the floor function. In some cases, the time variable k is dropped from the state or input variables for the sake of brevity.

II. PROBLEM FORMULATION
Consider a discrete-time nonlinear system as follows, where x P Dx Ă R n is the state vector and u P Du Ă R m is the control input vector, Dx and Du are compact sets; f : Dx Þ Ñ R n , and g : Dx Þ Ñ R nˆm are respectively the unknown nonlinear drift and input terms. The integers n and m denote the dimensions of the state and input vectors, respectively. This paper aims to learn the system unknown dynamics in (1), namely to approximate the uncertain functions f pxq and gpxq in a finite time using a concurrent learning technique. Assumption 1: xpkq is a measurable state vector, and f pxpkqq and gpxpkqq are both locally Lipschitz in xpkq.
In order to learn the uncertain functions f pxq and gpxq in the system dynamics, first the unknown nonlinear functions f pxq and gpxq are formulated in regressor forms. Here, linearly parameterized adaptive approximation models are used to, respectively, represent f pxpkqq and gpxpkqq as follows, gpxpkqq "ĝpxpkq, Θg q`egpxpkqq, wheref pxpkq, Θf q " Θ˚T f ϕpxpkqq, gpxpkq, Θg q " Θ˚T g χpxpkqq.
The matrices Θf P D f Ă R pˆn and Θg P Dg Ă R qˆn denote the unknown optimal parameters of the adaptive approximation models, defined as follows Θg " arg min where D f and Dg are compact sets. The vectors ϕ : Dx Þ Ñ R p and χ : Dx Þ Ñ R q , denote the basis functions, whereas p and q are the number of linearly independent basis functions for approximating f pxpkqq and gpxpkqq, respectively. The quantities e f pxpkqq P R n and egpxpkqq P R nˆm are the MFAEs for f pxpkqq and gpxpkqq respectively, representing the residual approximation error in the case of optimal parameters. In the special case that the unknown functions f pxpkqq and gpxpkqq can be approximated exactly by the adaptive approximation modelsf pxpkq, Θ f q andĝpxpkq, Θgq respectively, MFAEs are zero, i.e., e f pxpkqq " egpxpkqq " 0. By using (2)-(5), the system dynamics (1) can be rewritten as where Θ˚" rΘ˚T f , Θ˚T g s T P R pp`qqˆn , zpxpkq, upkqq " rϕ T pxpkqq, u T pkqχ T pxpkqqs T P R pp`qq , and εpxpkq, upkqq " e f pxpkqq`egpxpkqqupkq. Assumption 2: For the given compact set Dx, the approximators' basis functions are bounded, and the approximation error εpxpkq, upkqq is bounded for admissible controls upkq by an upper bound bε ě 0 (i.e., sup xPDx,uPDu }εpxpkq, upkqq} ď bε).
In this paper, since xpk`1q is not available, a regressor filtering method is used. In order to proceed with regressor filtering, one can reformulate (8) as where C " cI,´1 ă c ă 1. The following lemma reveals the filtered regressor for (8).
To fulfill the finite-time learning of the uncertainties f pxq and gpxq in system (1), the paper objective is to propose a finite-time concurrent learning-based estimation method that relaxes the PE condition and satisfies the following criteria: 1) For adaptive approximators with zero MFAE, the parameters' estimation errorΘpkq converges to zero in finite time. 2) For adaptive approximators with non-zero MFAE, the parameters' estimation errorΘpkq are uniformly ultimately bounded.

III. FINITE-TIME CONCURRENT LEARNING OF THE UNKNOWN DYNAMICS
In this section, the concurrent learning-based parameter estimation method for approximating the unknown dynamics of system (1) in finite time is presented and the required preliminaries for the finitetime convergence analysis of the proposed estimation method are developed.
For using the idea of concurrent learning, which employs recorded experienced data along with current data in the identifier's parameter estimation law, the past data is collected and stored in the memory stacks M P R pp`qqˆP , L P R nˆP and X P R nˆP , at time steps τ 1 , ..., τ P as follows, M " rdpτ 1 q,dpτ 2 q, ...,dpτ P qs, L " rlpτ 1 q,lpτ 2 q, ...,lpτ P qs, X " rxpτ 1 q,xpτ 2 q, ...,xpτ P qs, where P is the number of data points stored in every stack. Note that P is chosen so that M contains at least as many linearly independent elements as the dimension of dpkq (i.e., the total number of linearly independent basis functions for approximating f pxq and gpxq), given in (10), that is called as M rank condition and requires P ě p`q. Define the error e h pkq for the h th recorded sample as wherex is the state estimation at 0 ď τ h ă k time step, h " 1, ..., P , using the current estimated parameters matrixΘpkq and the recordeddpτ h q andlpτ h q. Substitutingxpτ h q into (17), one has Now, the proposed FTCL law for estimating the parameters of the system approximator is given aŝ where all the matrices Γ, Ξ G , Ξ C P R pp`qqˆpp`qq , Γ " γI is the learning rate matrix with positive constant γ ą 0, β is a design constant parameter satisfying β ą } ř P h"1d pτ h qε T pτ h q} " P bε, Ξ C " ξ C I and Ξ G " ξ G I with positive constants ξ C ą 0 and ξ G ą 0. The above estimation law has two learning terms where the term Ξ Gd pkqe T pkq, containing the current state approximation error, is also widely used in gradient descent method and the term q, containing the past experienced data, is called the concurrent learning term. The learning weights Ξ C and Ξ G do not need to be equal and by setting appropriate ξ C and ξ G , respectively, one of the two learning terms of gradient descent and concurrent learning can be prioritized over the other. Before proceeding further, the following definitions and lemma are needed. In the following, Lemma 2, inspired by Lemma 3.1 in [21], contains a fundamental result on Lyapunov representation of finitetime stability and convergence for discrete-time systems, not being reported in the literature. It is worth mentioning that in Lemma 2, the required conditions on the difference of the Lyapunov function candidate are completely different from the ones in [21] and the settling-time function is provided which is not given in [21]. Definition 1: [22] The bounded signal dpkq is said to be persistently exciting if there exist positive scalars µ 1 , µ 2 and T P N`such that @τ P N`, µ 1 I ď ř τ`T k"τ dpkqd T pkq ď µ 2 I. Definition 2: [23] Consider the system where y P Dy and F : Dy Þ Ñ R n is a nonlinear function on the open neighborhood Dy of the origin and the equilibrium point of (21) is the origin. The system (21) is said to be finite-time stable, if it is Lyapunov stable and finite-time convergent where @yp0q P Dy any solution ypkq of (21) reaches the origin at some finite time moment, i.e., ypkq " 0, @k ą Kpyp0qq where K : Dyzt0u Þ Ñ N`is the settling-time function. Definition 3: [24] The solution of (21) is uniformly ultimately bounded with ultimate bound α 2 , if there exist positive constants α 2 and c 0 where for every 0 ă α 1 ă c 0 there exists a constant Kpα 1 , α 2 q P N`, such that }yp0q} ă α 1 ñ }ypkq} ă α 2 , @k ą Kpα 1 , α 2 q.
(22) Lemma 2: Consider the nonlinear system (21). Suppose there is a Lyapunov function V : Ω Þ Ñ R which is positive definite in Ω where Ω is an open neighborhood of the origin, and there exist real numbers 0 ă µ ă 1, 0 ă a ă 1, and b ą 0 such that V pk`1q " p1´aqV pkq´bpV pkqq µ , on Ω. Then, system (21) is finite-time stable and its states converge to zero for k ě KpV pyp0qqq where is the settling-time function.
Proof. Note that (23) suffices to show the Lyapunov stability of the system, because it guarantees that the difference V pk`1q´V pkq, along the discrete-time system trajectories is negative definite and it will be zero if and only if V pkq " 0.
Consider that for any finite positive initial value of the Lyapunov function V p0q, along with an arbitrary trajectory for the discrete-time system (21), one has where a unique positive scalar r 0 ą 0 exists satisfying (24).
Defining r 1 " p1´aqr 0´r µ 0 , (25) can be rewritten as Note that if r 0 ď´1 1´a¯1 1´µ , then the above implies that r 1 " 0, since V p1q has to be non-negative from the definition of the Lyapunov function. In this case, the value of the Lyapunov function already converges to zero in the first step, i.e., for K˚" 1. Now, let r 0 ą 1 1´a¯1 1´µ . Substituting (26) into (23), gives a similar expression where r 2 " p1´aqr 1´r µ 1 . In the same manner, one obtains V pk`1q as follows, with r k`1 " p1´aqr k´r µ k where r k`1 can be rewritten as Since V pk`1q ă V pkq, one has r k`1 ă r k , @k, and V pk`1q is positive definite, consequently r k`1 ě 0. Hence, from the right side of (29), one can see that there exists a k such that Therefore, for a finite integer k " K˚´1 satisfying (30), i.e., r K˚´1 ď p 1 1´a q 1 1´µ , one has r K˚" 0. Hence, it follows from (28) that Using (23), one has V p1q´V p0q "´aV p0q´bV µ p0q, V p2q´V p1q "´aV p1q´bV µ p1q, . . . V pK˚´1q´V pK˚´2q "´aV pK˚´2q´bV µ pK˚´2q, V pK˚q´V pK˚´1q "´aV pK˚´1q´bV µ pK˚´1q, which leads to Since V pk`1q ă V pkq, (32) can be rewritten as V pK˚q´V p0q ď K˚p´aV pK˚´1q´bV µ pK˚´1qq. (33) Using (31), (33) results in the following settling-time Hence, r k " 0 and V pkq " 0 for all k ě KpV p0qq where KpV p0qq " t V p0q u`1 is the settling-time function. Therefore, the system state converges to zero for all k ě KpV p0qq that assures the finite-time stability of the system. This completes the proof.
Fact 1: For every matrix A and B of the same dimensions, it is known that }A}´}B} ď }A´B} ď }A}`}B}.

IV. FINITE-TIME CONVERGENT ANALYSIS FOR THE PROPOSED FTCL
The following theorem demonstrates the finite-time convergence properties of the proposed learning method. It should be noted that in the proposed FTCL method, the stored data in M and other stacks is selected based on data recording algorithm in [25] to maximize λ min pSq λmaxpSq where S " ř P h"1d pτ h qd T pτ h q, and due to the satisfaction of M rank condition, S ą 0.
Hence, provided that the rank condition on M is met and γ satisfies (56), the FTCL method guarantees thatΘpkq is uniformly ultimately bounded with S ũ Θ being invariant. To obtain the finite time K2 pV p0qq that }Θpkq} reaches the invariant set S ũ Θ and remains thereafter, first using (53), (59) is written as Then, similar to the proof of Lemma 2, since, for k " 1, ..., K2´1, K2 , V pkq ă V pk´1q, (64) leads to V pK2 q´V p0q ď K2 p´aγ V pK2 q`bu }Θp0q}`cq. (65) Using V pK2 q ď γ´1pbΘq 2 , (65) leads to K2 ď K2 pV p0qq where, Therefore, for k ě K2 pV p0qq,Θpkq reaches the invariant set S ũ Θ and remains thereafter. This completes the proof. Remark 1: As (58) and (66) show, bγ and aγ dictate how fast Θpkq converges to zero and the specified bound around zero for respectively the cases withεpkq " 0 andεpkq ‰ 0. To have a faster convergence time in (58), maximizing bγ , given in (55), by maximizing λ min pSq λmaxpSq is beneficial. Maximizing λ min pSq λmaxpSq (maximizing λ min pSq and minimizing λmaxpSq) also leads to enlarging aγ which causes faster settling-time in (66). This result completely coincides with the obtained results in [20] and even supports the continuoustime framework studies in [2,4,26]. Therefore, while applying the proposed FTCL, the data recording algorithm in [25] is used, where the appropriate data is selected to maximize λ min pSq λmaxpSq . Remark 2: In approximators with non-zero MFAE (εpkq ‰ 0), maximizing λ min pSq λmaxpSq , using the data recording algorithm in [25], helps to respectively enlarge and reduce the amplitudes of a and c that leads to narrow down the error bound in (36), whereas maximizing λ min pSq λmaxpSq matches with the concepts of concurrent learning in continuous-time [2,26]. Furthermore, choosing P " p`q (satisfying P ě pp`qq) not only helps to maximize λ min pSq λmaxpSq [25], but also, keeps c small.

V. SIMULATION RESULTS AND DISCUSSION
In this section, the performance of the proposed finite-time concurrent learning for both adaptive approximators with zero and non-zero MFAEs is examined in comparison with asymptotically converging concurrent learning [20] and traditional gradient descent [1] whose estimation laws are respectively given as follows, where Γ G " γ G I, Γ C " γ C I, Σ G " σ G I and Σ C " σ C I with positive constants γ G ą 0, γ C ą 0, σ G ą 0 and σ C ą 0. For both cases, the simulation time span is rk 0 , k f s where k 0 " 0 and k f " 1000, and the x domain is defined by Dx " rx L , x H s where x L ă x H and x L , x H P R and Dx is quantized by rx L : x H´xL k f´k0 : x H s. In the proposed FTCL method, γ is chosen to satisfy (35). For concurrent learning according to [20], γ C satisfies γ C ď 2 2σ G`σC λmaxpSq where it is chosen as γ C " 1 2σ G`σC λmaxpSq . In all cases, the initial values and the controllers are all set to zero. A small exponential sum of sinusoidal input is injected to the system controller for ensuring the rank condition on the collected data and the data selection procedure in [25] is employed for both FTCL and concurrent learning methods. To fairly compare the speed and precision of the intended on-line learning methods for approximatinĝ f pxq andĝpxq on the whole domain of x as time evolves, the following learning errors are computed on-line.
In the simulations, the results of the proposed finite-time concurrent learning, the traditional concurrent learning and gradient descent methods are respectively labeled by FTCL, CL and GD.
A. Example 1: Simulations for approximators with zero MFAE (εpkq " 0) Consider the following system xpk`1q " p 1 e´x pkq`p 2 e´x pkq cospxpkqq`p 3 where the parameters rp 1 , p 2 , p 3 s are unknown and the regressors are fully known as zpxpkq, upkqq " re´x pkq , e´x pkq cospxpkqq, 1 1`xpkq upkqs, with p`q " 3. The unknown parameters are rp 1 , p 2 , p 3 s " r´1, 1.5, 1s and Dx is limited with x L " 0 and x H " 2. We set P " 3 for both FTCL and concurrent learning methods. Let γ G " 0.7 for gradient descent method, and σ G " 1, σ C " 0.4 for concurrent learning method, and ξ G " 1, ξ C " 0.3 and β " 0.3 for FTCL method. Fig. 1 depicts the true parameters and the approximated parameters for FTCL, concurrent learning and gradient descent methods. In Fig.  1, while gradient descent could not converge to the true parameters due to the lack of persistence of excitation in the learning time length, FTCL and concurrent learning succeeded in convergence. However, FTCL resulted in faster convergence to the true parameters in comparison with concurrent learning. The on-line learning errors E f pkq and Egpkq for the FTCL, concurrent learning and gradient descent are plotted in Fig. 2 where FTCL shows faster converging to the origin. Fig. 3 shows that the state estimation error of all methods has converged to zero where the results of gradient descent illustrate that the convergence of the state estimation error to zero does not necessarily imply that the parameters' estimation error converges to zero. The integral absolute errors (IAEs) of E f pkq and Egpkq for FTCL, concurrent learning and gradient descent methods are computed in Table 1 where FTCL with IAEs 36.21 and 22.08 respectively for E f pkq and Egpkq has resulted in the best precision of on-line learning in comparison with concurrent learning and gradient descent. Now, consider the following system xpk`1q " 0.5xpkq sinp0.5xpkqq`p2`cospxpkqqqupkq, (69) where the associated f pxq and gpxq are fully unknown uncertainties. In this example, radial basis function neural networks which are linearly parameterized universal approximators are used. Here, we consider 5 radial basis functions defined as e´}  upkqs T , with 10 independent basis functions which leads to setting P " 10.
Employing γ G " 0.8 for gradient descent method, σ G " 1.2 and σ C " 0.1 for concurrent learning method, and ξ G " 1, ξ C " 0.1 and β " 0.65 for FTCL method leads to the approximated parameters depicted on Fig. 4. Due to the lack of persistence of excitation in the learning time length, gradient descent parameters in Fig. 4 could not converge to the appropriate parameters, while FTCL and concurrent learning methods using a memory stack of data, satisfying the rank condition, succeeded in convergence to the suitable parameters. The steady state approximations for the uncertainties f pxq and gpxq are given in Fig. 5. As the comparison of the learning errors E f pkq and Egpkq in Fig. 6 shows, the gradient descent method did not perform well in learning the uncertainty, however both FTCL and concurrent learning errors showed bounded convergence near zero, where FTCL method is faster in converging to a smaller bound near zero in comparison with concurrent learning approach. Fig. 7 shows that the state estimation error of all methods has converged to a bound very close to zero. Furthermore, based on the results of IAE for E f pkq and Egpkq in Table 1, FTCL results in lower errors during the whole time of on-line learning in comparison with concurrent learning and gradient descent.

VI. CONCLUSION AND FUTURE WORK
This paper addressed finite-time identification of discrete-time system dynamics where finite-time learning could speed up the learning and concurrent learning technique relaxed the persistence of excitation condition on the regressor to a rank condition on the memory stack of recorded data. Learning rate condition was obtained for finite-time convergence based on discrete and finite time analysis. It was discussed that the precision and speed of the proposed finitetime learning method depend on the well-conditioning properties of the memory data. Simulation results are given where it is shown that the proposed finite-time concurrent learning has better performance in comparison with the traditional gradient descent and asymptotic convergent concurrent learning methods in both terms of precision and convergence speed. For future work, it is intended to extend the existing results for the identification of time-varying systems.