{'fraction_non_alphanumeric': 0.05784490359182735, 'fraction_numerical': 0.016636638938935102, 'mean_word_length': 4.094317516110134, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 10, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 10, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Adaptive Observers and Parameter Estimation for a Class of Systems Nonlinear in the Parameters

We consider the problem of asymptotic reconstruction of the state and parameter values in systems of ordinary differential equations. A solution to this problem is proposed for a class of systems of which the unknowns are allowed to be nonlinearly parameterized functions of state and time. Reconstruction of state and parameter values is based on the concepts of weakly attracting sets and non-uniform convergence and is subjected to persistency of excitation conditions. In absence of nonlinear parametrization the resulting observers reduce to standard estimation schemes. In this respect, the proposed method constitutes a generalization of the conventional canonical adaptive observer design.


Introduction
Often when studying and modeling real-life systems we come across a dynamical system of which we can register the input-output relations as functions of time, but are unable to fully control the input the system receives over time.
Suppose we know enough about the system to specify its equations, but to complete its description so that prediction of the system behavior over some interval of time is possible, we must recover the values of the internal system variables and parameters that are not available for direct observation. In this case, a powerful method for doing so is Adaptive Observer Design.
Observer-based methods can reconstruct both state variables and parameters, provided that the original system has, or can be transformed into, the canonical adaptive observer form [3]: Function g : R ≥0 → R n+1 represents the known input, function ϕ : R ×R ≥0 → R n+1 × R d is a known function of the observed output y(t), k = (k 1 , . . . , k n ) is a vector of known constants, F is a known n × n matrix (usually diagonal), and θ ∈ R d is a vector of unknown parameters. Algorithms are available in [3], [17] 1 that guarantee asymptotically stable (and sometimes even exponentially fast with arbitrary rate of convergence [19]) reconstruction of the unknowns, i.e. the state and parameter vectors.
However, many important systems in, for instance, chemical kinetics [7], [2] and biology [12], are not in canonical adaptive observer form. Unknown parameters in F and nonlinear parametrization characterize these systems. Absence of a canonical representation for these systems prevents straightforward application of standard observer-based techniques for state and parameter estimation.
A usual way to deal with this obstacle is to search for a bijective coordinate and parameter transformation reshaping the original system equations into the canonical form. Finding such a transformation, however, is a non-trivial problem. Available necessary and sufficient results in this direction apply to a too narrow class of systems (see e.g. [16]). Alternative, and possibly parameterdependent, transformations may not be satisfactory either.
This leaves us with the undesirable options to use heuristic methods or exhaustive search. Whereas the former are not guaranteed to work, the latter can be forbiddingly expensive in terms of computational costs.
We propose a solution that combines the advantage of exponentially fast convergence with the flexibility of explorative behavior. Fast convergence is reserved for available estimators for state and linear parameters θ of (1); exploration is reserved for the non-linear parameters an and the unknowns in F. The proposed method therefore bears an overall similarity to [25], [11], [20], [21], [27]. However, instead of sticking to the framework of Lyaunov stability we relax the problem further and invoke the concepts of weakly attracting sets and relaxation times [22], [8], [9] in our argument. The method is based on results of [28] that allow us to analyze asymptotic convergence in nonlinear systems beyond the framework of Lyapunov design. We show that, subject to a condition of persistent excitation of the parameterized uncertainty, it is possible to reconstruct both linear and nonlinear parameters of the system. The latter result is illustrated with three examples.
The paper is organized as follows. In Section 2 we define the notation used throughout the paper. Section 3 provides the formal statement of the problem, Section 4 contains the main results, Section 5 provides illustrative examples, and Section 6 concludes with a brief discussion. The proofs of all formal statements are provided in the Appendix.

Notation
The following notational conventions are used throughout the paper: • Z denotes the set of integers, and N stands for the set of positive integers.
• The Euclidian norm of x ∈ R n is denoted by x .
• By L n ∞ [t 0 , T ], t 0 ≥ 0, T ≥ t 0 we denote the space of all functions f : R ≥0 → R n such that f ∞,[t 0 ,T ] = ess sup{ f(t) , t ∈ [t 0 , T ]} < ∞; f ∞,[t 0 ,T ] stands for the L n ∞ [t 0 , T ] norm of f(t). • C r denotes the space of continuous functions that are at least r times differentiable. • Let A be a subset of R n , then for all x ∈ R n , we define dist(A, • Finally, let ǫ ∈ R >0 , then x ǫ stands for the following:

Problem Formulation
We consider the following class of single-input-single-output nonlinear systems: . . , n} are continuous and known functions, u : R ≥0 → R, u ∈ C 0 is a known function of time modelling the control input, and ξ i : R ≥0 → R, ξ i ∈ C 0 are functions that are unknown, yet bounded. The functions ξ i (t) represent unmodeled dynamics (e.g. noise).
Variable y in system (2) is the output, and the variables x i , i ≥ 1 are the components of state x, that are not available for direct observation. Vectors θ i ∈ R d i consist of linear parameters of uncertainties in the right-hand side of the i-th equation in (2). Parameters τ i ∈ R, i = {1, . . . , n} are the unknown parameters of time-varying relaxation rates, β i (x 0 , τ i , t), of the state variables x i , and vectors p i ∈ R m i , q i ∈ R r i , consist of the nonlinear parameters of the uncertainties. The functions c i (x 0 , q i , t) are supposed to be bounded.
Systems (2) model the dynamics of neural membranes [12]; they describe various chemical oscillators [2]; a number of important models in ecology, e.g. Lotka-Volterra equations, can be reduced to (2) by a, possibly parameterdependent, change of coordinates (see our Examples section for details). For these reasons alone equations (2) constitute an interesting class of systems with remarkable practical relevance. On the other hand, and from a more theoretical viewpoint, system (2) may be viewed as a natural nonlinear extension of the standard adaptive observer canonic form (1) in which the regressors are allowed to be nonlinearly parameterized and relaxation parameters, β i (·), are time-varying and uncertain.
The system state x = col(x 0 , x 1 , · · · , x n ) is not measured; only the values of the input u(t) and the output y(t) = x 0 (t), t ≥ t 0 in (2) are accessible over any time interval [t 0 , t] that belongs to the history of the system. The actual values of parameters θ, λ are assumed to be unknown a-priori. For simplicity of the proofs, we assume them to lay within a hypercube with known bounds: To reconstruct the unknown state and parameters of system from the values of y(t) = x 0 (t), u(t), we should find an auxiliary systeṁ such that for some given h : R q → R p , p = dim(col(θ T , λ T , x T )), q 0 , δ ∈ R >0 and all t 0 ∈ R ≥0 the following property holds: where w(t) = col(θ, λ, x(t, x 0 )). System (3) is usually referred to as an adaptive observer.
The goal requirement for our adaptive observer is stated in the form of inequality (4), instead of the more usual requirement lim t→∞ h(q(t, q 0 )) − w(t) = 0. This is because we allow unmodeled dynamics, ξ i (t), in the right-hand side of (2). As a result of such practically important addition, there is a possibility that (4) is satisfied not only for the original system exclusively, but it may also hold for a set of systems. The vector-fields in the right hand side of each element of this set are to be sufficiently close to that of (2), yet their parameters could be different. Instead of just one value of unknown parameter vectors θ, λ we therefore have to deal with a set of θ, λ corresponding to the solutions of (2) that over time are sufficiently close. This set of model parameters is referred to as an equivalence class of (2).
In the next section we demonstrate that despite the non-uniqueness of the solution, asymptotic reconstruction of state and parameters of (2) is achievable, subject to a standard persistency of excitation condition.

Main Results
Traditionally, methods of observer design are based on the notion of Lyapunov stability of solutions. If a globally asymptotically Lyapunov-stable solution exists in the state space of the combined system (2), (3) such that the requirements (4) are satisfied, then (3) is the observer needed. However, as we mentioned earlier, in systems with nonlinear parameterization and unmodeled perturbations distinct parameter values in the right-hand side of (2) may correspond to indistinguishable input-output behavior. Hence multiple distinct invariant sets in the extended state space will co-exist, and only local stability would generally be guaranteed. In order to deal with these issues, novel concepts of observer design are needed.
We propose a method for finding adaptive observers (3) for (2) that rests on a two-fold idea. First, instead of the standard requirement that the dynamics of errors of state and parameter estimates is globally asymptotically stable in the sense of Lyapunov, we will require mere convergence of the estimates as specified by (4). This relaxation allows us to invoke a wider range of tools for accessing convergence such as in [28].
Second, similarly to canonical observer schemes [16], [17], [18], [19] we shall be relying on the possibility to evaluate the integrals at a given time t and for the given values of τ i , p i within a given accuracy. In classical adaptive observer schemes, the values of β i (x 0 , τ i , t) are constant. This allows us to transform the original equations by a (possibly parameterdependent) non-singular linear coordinate transformation, Φ : x → z, x 1 = z 1 , into an equivalent form in which the values of all time constants are known.
In the new coordinates the variables z 2 , . . . , z n can be estimated by integrals (5) in which the values of β i (x 0 , τ i , t) are constant and known. This is usually done by using auxiliary linear filters. In our case, the values of β i (x 0 , τ i , t) are not constant and are unknown due to the presence of τ i . Yet if the values of τ i would be known we could still estimate the values of integrals (5) as follows where T ∈ R >0 is sufficiently large and t ≥ T + t 0 .
are periodic with rationally -dependent periods and satisfy the Dini condition in t, integrals (5) can be estimated invoking a Fourier expansion. Notice that for continuous and Lipschitz in p i functions µ i (t, τ i , p i ) the coefficients of their Fourier expansion remain continuous and Lipschitz with respect to p i .
In what follows we show that the availability of a computational scheme which is capable of evaluating the integrals (5) with sufficient accuracy enables us to design an adaptive observer that, under some mild and standard persistency of excitation assumptions, ensures asymptotic estimation of the unknown parameter values of (2).

Observer definition and assumptions
Consider the following function ϕ( . . . , c n (x 0 , q n , t)µ n (t, τ n , p n )) T The function ϕ(x 0 , λ, t) is a concatenation of φ 0 (·) and integrals (5). We assume that the values of ϕ(x 0 , λ, t) can be efficiently estimated for all x 0 , λ, t ≥ 0 up to a small mismatch. In other words, we suppose that there exists a functionφ(x 0 , λ, t) such that the following property holds: where values ofφ(x 0 , λ, t) are efficiently computable for all x 0 , λ, t (see e.g. (6) for an example of such approximations), and ∆ ϕ is sufficiently small.
If parameters τ i , p i , and q i in the right-hand side of (2) would be known and where η i are the solutions of the following auxiliary system (filter)η with zero initial conditions. Systems like (9) are inherent components of standard adaptive observers [13], [17]. In our case we suppose that the values of τ i , q i , p i are not know a-priori and that c i (x 0 , q i , t), β i (x 0 , τ i , t) are not constant. Therefore, we replace η i with their approximations, e.g. as in (6): a Fourier expansion can be employed to defineφ(x 0 , λ, t). The value of ∆ ϕ in (8) stands for the accuracy of approximation, and as a rule of thumb the more computational resources are devoted to approximate ϕ(x 0 , λ, t) the smaller is the value of ∆ ϕ .
To reconstruct the nonlinear part of the uncertainties, λ, we will require that ϕ(x 0 , λ, t) is nonlinearly persistently exciting in λ. Here we adopt the definition of nonlinear persistent excitation from [6] with a minor modification. The modification is needed to account for a possibility that which is the case, for example ifφ(x 0 , λ, t) is periodic in λ. The modified notion is presented in Definition 2 below.
In our work we use condition (16) to ensure state boundedness of the observer and to enable reconstruction of linear parameters θ i in the original equations (2). Condition (17) is used to formulate the requirements for successful reconstruction of nonlinear parameters p i and unknown relaxation times τ i .

Asymptotic properties of the observer
The main results of this section are provided in Theorems 3 and 5. Theorem 3 establishes conditions for state boundedness of the observer, and states its general asymptotic properties. Theorem 5 specifies a set of conditions for the possibility of asymptotic reconstruction of θ i , τ i , and p i , up to their equivalence classes and small mismatch due to errors.
Proofs of Theorems 3, 5 and other auxiliary results are provided in Appendix.

1) trajectories of the closed loop system (11) -(13) are bounded and
Remark 4 Theorem 3 assures that the estimatesθ(t),λ(t) asymptotically converge to a neighborhood of the actual values θ, λ. It does not specify, however, how close these estimates are to the true values of θ, λ.
Our next result states that if the values of ∆ ϕ and ∆ ξ : (8), (10) are small, e.g.φ(x 0 (t), λ, t) approximates ϕ(x 0 (t), λ, t) with sufficiently high accuracy and the unmodeled dynamic is negligible, the estimateŝ θ(t),λ(t) will converge to small neighborhoods of the equivalence classes of θ, λ. The sizes of these neighborhoods are shown to be bounded from above by monotone functions ∆: vanishing at zero. Formally this result is stated in Theorem 5 below is nonlinearly persistently exciting with respect to λ, then the estimatesλ(t) converge into a small vicinity of E(λ):

Examples
We illustrate our method with three examples. The first example deals with the problem of parameter reconstruction of the Duffing oscillator assuming that only one state variable is available for direct observation. This problem received considerable attention in the past and hence can serve as a benchmark. Existence of an output feedback steering the state of the Duffing system to an arbitrary small neighborhood of the origin in the presence of parametric uncertainties was demonstrated in [24], [15]. Recently, a method for asymptotic reconstruction of all parameters of the Duffing system was published in [4]. Here we show how our own method performs in this task provided that the frequency of external perturbations is known.
The second and the third examples address a more difficult class of problems in which the uncertain parameters enter the model nonlinearly and/or the functions c i (x 0 , q i , t) in (2) are not constant. In particular, we consider a twocomponent bio-reactor model [2] and a bilinear Lotka-Volterra system. We show that in both cases, subject to the chosen tolerance margin, state and parameters of these models can be successfully estimated from input-output observations with our adaptive observer.

Example 1. Parameter estimation of the Duffing oscillator.
Consider the following system of differential equationṡ where δ ∈ (0.1, 1), β, α, γ ∈ R, are parameters of which the values are supposed to be unknown, and x 0 measured. The value the frequency of periodic forcing, i.e. ω, is supposed to be known. In this example we set and the task is to reconstruct the values of δ, α, β, and γ asymptotically from the measurements of x 0 .
For convenience we rewrite (24) using the notational agreements as in (2): According to (7), we start with constructing the following regressor function where λ = τ 1 . For the given parametrization of (24) the regressor satisfies the requirement of uniform persistency of excitation (the value of µ ranges from 1.2 for τ 1 = 1 to 1000 for τ 1 = 0.2 at L = 500). Following (11) -(13), the structure of the observer is defined as follows: where σ(·) = tanh(·) and the parameters of the observer are set as follows: γ = 0.2, γ θ = 2, λ min = 0.1, λ max = 1.1, ε = 0.01. Performance of adaptive observer system (26), (27), (28) where s 0 , s 1 are products (concentrations), d is the dilution rate, r(s 0 , k) is a Monod-type nonlinearity, r max is the maximum growth rate, b > 0 is the growth yield, and k is the "half saturation" constant. The measured output is the variable s 0 . The values of parameters r max , b are supposed to be known, and precise values of parameters d, k are considered unknown.
According to the requirements of our method we need that the system equations are presented in the form (2) where β i (·, ·, ·) > 0. For the chosen model (29) this requirement may not hold for the values of d, k are unknown, and hence we cannot tell if β 1 = d − r(x 0 (t), k) remains positive at all t. Let us therefore transform (29) into (2) by the following parameter-dependent nonsingular transformation: In the new coordinates the system dynamics is described as follows: where φ 0 (x 0 , p 0 , t) = (x 0 , u(t)), φ 1 (t) = u(t).
Taking (6) into account, and that p 0 = q 0 = q 1 = k we definē According to (11) -(13), the structure of the observer is: Parameters ω 1 , ω 2 in (33) are to be rationally independent. In our example we set these parameters as follows: ω 1 = π, ω 2 = 2 √ 2. Other parameters of the observer were set as γ = 0.0001, ǫ = 0.03, γ θ = 0.001, α = 1. Parameters d and k of the reactor model were set to 0.3 and 70 respectively, and the input u(t) = 40(sin(0.2t) + 1.5). Results of computer simulations of the closed-loop system with this observer are provided in Fig. 2. As we can see from these figures, state and parameter estimates converge into a vicinity of the true values of s 1 and θ 1 , θ 2 , θ 3 , τ 1 , k as t → ∞.
Observer (33), with minor modifications, can be easily tailored to deal with other similar models of chemical oscillators, such as e.g. [5], in which the function r(s 0 , k) is replaced with an exponential nonlinearity. Furthermore, it can be used to estimate state and parameter values of various models of interacting species. We illustrate this below using the Lotka-Volterra system as an example.
Example 3. Lotka-Volterra system. Consider the following system: where x, y are the variables modeling the number of prey and predators in an isolated population. Parameters α, γ, and δ are supposed to be unknown, and x is measured. We wish to be able to estimate the values of all model parameters and also recover the values of y from the measurement of x over time.

Conclusion
We derived, under suitable assumptions, an observer that can reconstruct the unknown parameters of a general class of nonlinear systems, specified by equation (2). Reconstruction is subject to conditions of linear/nonlinear persistency of excitation. According to Theorem 5, the values of parameters can be reconstructed with arbitrarily small errors (up to their equivalence class). Hence there exists a time instance t ′ such that solutions of (12) will enter a sufficiently small neighborhood of x i (t) at t = t ′ and will stay there for all t ≥ t ′ provided that the values of ∆ ϕ , ∆ ξ in (8), (10) are sufficiently small. Having solved the parameter reconstruction problem, we obtain the state variables by applying the filter (12).
To ensure convergence we have gone beyond the usual requirement of Lyapunov stability. The set to which the estimates converge is not stable in the Lyapunov sense; any neighborhood of this set contains a nonempty subset of points corresponding to trajectories escaping this neighborhood in finite time.
Yet the set is attracting in a weaker, Milnor sense, cf. [22], and furthermore it is made Poisson stable by design. The amount of time required for convergence in our approach depends heavily on the subspace being searched, it does not depend on the dimension of the linearly parameterized part. This renders the method more efficient than exhaustive search; the fewer uncertainties, relatively speaking, belong to the nonlinear part, the more advantageous our method becomes.
First, we notice that trajectoriesx 1,j (t),x 2,j (t) are globally bounded. Furthermore, the right-hand side of (13) is locally Lipschitz inx 1,j ,x 2,j . Hence the following estimate holds: As follows from the assumptions of the lemma, the functionφ(x 0 (t), λ, t) is a λ-uniform PE. This implies existence of L, µ ∈ R >0 such that Consider the following matrix: Using the inequality Hz ≤ max k,l |h k,l | z , H ∈ R m×n , z ∈ R n and given that φ(x 0 (t), λ, t) ≤ B for all t ≥ 0, λ ∈ Ω λ , we can conclude that the matrix (A.13) satisfies Taking (A.11), (A.14) into account we can conclude that Then choosing γ for instance such that 16) we can ensure that In other words the functionφ(x 0 (t),λ(t), t) is persistently exciting.

A.2 Proof of Theorem 5
Similar to the proof of Theorem 3, equation (A.7), we notice that equations governing the dynamics of our observer (11), (13) relative to the dynamics of the original system can be described as followṡ is uniformly exponentially stable. Taking (A.6) into account we can express υ(λ, λ, t) as: In the equation above υ 0 (λ, λ, t) denotes the term θ T (φ(x 0 ,λ, t)−φ(x 0 , λ, t))+ c 0 (x 0 ,λ, t) −c 0 (x 0 , λ, t), and d(θ, λ, t) stands for θ T η(t, λ) −χ(t) + ǫ(t). Hence (A.25) transforms intȯ To proceed further we will need the following lemma Lemma 7 Consider the following system and each entry of A(t) is a globally bounded function of t. Suppose that the following properties hold for (A.27): 1) the origin of system (A.27) is asymptotically stable at u(t) = 0, d(t) = 0 2) the term c T b(t) is separated away from zero:

3) the time derivative of u(t) is bounded uniformly in t:
Then provided that ε is sufficiently small.
Proof. First, notice that closed form solutions of (A.27) are where Φ(t, t 0 ) is the fundamental system of solutions of (A.27) generated bẏ According to (A.31) we have: Consider y(t) − y(t − T ), T ∈ R >0 : and applying the mean-value theorem to (A.33) we obtain Hence Because (A.27) is asymptotically stable, the term |ν(t, T )| → 0 as t → ∞, and there exists t 1 ≥ R such that On the other hand, condition (A.30) assures the existence of t 0 such that Therefore there exists t ′ = max(t 1 , t 0 ) such that for all T ≥ 0 and t ≥ t ′ the following holds: According to (A.29), (A.28), inequality (A.34) implies Notice that Φ(t, t) = I ∀ t ∈ R and that Φ(t, τ ′ ) is (uniformly) continuous in τ ′ and |c T Φ(t, t)b(t)| ≥ β. Then there exists T ′ ∈ R >0 such that Optimizing the left-hand side of (A.36) for T yields i.e. when ε is sufficiently small.
As follows from Theorem 3, property (20), there exists a time instant t ′ such that In addition, vectors c, b in (A.25) satisfy c T b = 1, and d(θ, λ, t) satisfies where ǫ(t) is an exponentially decaying term and ∆ = θ ∆ ϕ + ∆ ξ . Let t ′′ : |ǫ(t)| ≤ θ ∆ for all t ≥ t ′′ , and t 0 = min{t ′ , t ′′ }. This implies that Finally, according to assumptions of the theorem there exists a constant ∂U ∈ R >0 such that Hence conditions of Lemma 7 are satisfied for system (A.26) for all t ≥ t 0 . Therefore, according to Lemma 7 there exists t 1 ≥ t 0 such that provided that ε is sufficiently small. Taking (A.38) into account, we can conclude that Given that (A.26) is exponentially stable and that (A.1), (A.40) hold, the upper bound of the difference θ (t) − θ at t → ∞ can be estimated as follows: Thus statement 1) of the theorem is proven.
To prove that statement 2) of the theorem holds we notice that, according to the definition of nonlinear persistency of excitation, constants L, β must exist such that for all t ∈ R there exists t ′ ∈ [t − L, t]: Combining this with (A.39) yields: Given thatλ(t ′ i ) are bounded we can choose t ′ j such that On the other hand, according to Theorem 3λ(t) converges to a point in Ω λ as t → ∞: lim t→∞λ (t) = λ * . Therefore λ * =λ and The latter inequality assures that Hence taking into account the bound for ε defined by (A.44), statement 2) of the theorem follows.

A.3 Lemma 8
Lemma 8 Consider an interconnection of systems that is governed by the following set of equations: where β : R → R is a strictly monotone and asymptotically decreasing to zero function. Then trajectories x(t), y(t), h(t) passing through Without loss of generality we introduce a strictly decreasing sequence such that σ 0 = 1, and σ i asymptotically converge to zero. Let be an ordered sequence of time instances such that 3 Similar to our earlier work [28] we wish to show that the amount of time needed to reach the set specified by x(t) = 0 from a given domain of initial conditions is infinite.
The time difference T i = t i − t i−1 can be estimated as Suppose that x(τ ) ∞,[t i−1 ,t i ] − ε ≤ 0. In this case the amount of time needed for the system to reach the set h = h 0 σ i is infinite. Hence h(t) ∈ [h 0 σ i , h 0 σ i−1 ] for all t ≥ t i−1 , and according to (A.42), trajectories x(t), y(t), h(t) will be bounded in forward time. Consider the case when x(τ ) ∞,[t i−1 ,t i ] − ε > 0 for all i.