Accurate estimation of a system's unknown parameters is critical in many engineering applications [1]. One such application is the classical model-reference adaptive control problem, which has been studied since the 1960s [2] and has experienced a recent resurgence with the advent of machine learning applications [3]. This estimation algorithm computes online an estimate of the unknown parameters by exploiting the available input signals and information about the structure of the system [4], [5]. In the case of linear time-varying systems, analyzing the convergence rate of the parameter estimate can be translated into showing exponential stability of the origin [1], [5]. A persistence of excitation condition is necessary and sufficient for exponential stability of linear time-varying systems [6].
Motivated by the recent results on finite-time parameter estimation for linear regression models [7], this paper presents a hybrid estimator for linear dynamical systems that guarantees convergence of the parameter estimate to the true value in finite time. In Section III-A, we show that, for the case of a constant unknown parameter, the parameter estimate converges to the true value after one jump. Then, in Section III-B, we generalize this result to piecewise constant unknown parameters by allowing the algorithm to jump multiple times. Robustness of the proposed algorithm to measurement noise is discussed in Section IV, and simulation results are presented in Section V.
Our approach is related to the ones in [7], [8] -both results provide a finite-time estimator using a hybrid system framework. In comparison to our work, the approach in [8] is different in two aspects. First, we use a different estimation algorithm based on the use of two coupled estimators. Second, while the results in [8] rely on a persistence of excitation condition to ensure that their hybrid system is well defined and to guarantee completeness of solutions, we impose a condition related to the invertibility of the solution components and show, using a numerical example, that our algorithm is capable of converging in finite time when the regressor is exciting over only a finite time interval. Moreover, our work differs from the one in [7] in three aspects. First, the authors of [7] deal with a linear regression model while we deal with a dynamical model. Second, the construction of the update law is different from the one in [7]. Finally, the analysis of the convergence properties of the system is more complicated for dynamical models than for linear regression models. Indeed, while the authors in [7] rely on a quadratic Lyapunov function, our analysis requires a more involved Lyapunov function so as to account for the coupling between the state and parameter estimators. Due to space constraints, the proofs of some results are sketched or omitted and will be published elsewhere.
We denote the set of real, nonnegative, positive, and natural numbers (including zero) as R, R ≥0 , R >0 , and N, respectively. The matrix I denotes the identity matrix of appropriate dimension. The set of symmetric positive definite matrices of dimension n × n is dentoed S n ++ . The Euclidean norm of vectors and the induced matrix norm is denoted | • |, and the infinity norm
α is zero at zero, strictly increasing, and unbounded.
A classical model-reference adaptive (closed-loop) control system is given by ẋ
where x ∈ R n is the known state vector, t → B(t) ∈ R n×p is the known regresson matrix, the matrix A 0 ∈ R n×n is known, and θ ∈ R p is an unknown vector of parameters.
The unknown parameter θ can be estimated using an update law of the form [1] θ = γB (t)P (x -x)
where θ ∈ R p is the estimate of the unknown parameter θ, x is the estimate of the state x, A ∈ R n×n is a user-selected Hurwitz matrix, P = P ∈ R n×n is a positive definite matrix that solves A P + P A = -Q, where Q = Q ∈ R n×n is a user-selected positive definite matrix, and γ > 0 is a design parameter that modifies the convergence rate. Denote the parameter estimation error as θ := θ -θ and the state estimation error as x := x -x. Then, the error dynamics can be written as
The convergence properties of (3) are typically analyzed using a quadratic Lyapunov function of the form V (x, θ) = x P x + γ -1 θ θ whose time derivative satisfies V (x, θ) ≤ -µ(Q)|x| 2 . It can be shown using Barbalat's Lemma [4] that t → x(t) converges to zero as time t goes to infinity. In order to show convergence of θ to θ as t goes to infinity, a persistence of excitation condition [1], [5], [9] is required (see Definition 2.2 below for a formal definition of persistence of excitation).
The objective of this paper is to estimate the unknown parameter vector θ in finite time using hybrid systems tools. For this reason, we review some preliminaries on hybrid systems in the following section.
In this paper, a hybrid system H is defined as in [10] by (C, F, D, G) as
where ξ ∈ R n is the state, u ∈ R m is the input, F : R n × R m → R n is the flow map defining a differential equation capturing the continuous dynamics, and C ⊂ R n defines the flow set on which flows are permitted. The mapping G : R n × R m → R n is the jump map defining the law resetting ξ at jumps, and D ⊂ R n is the jump set on which jumps are permitted.
A solution ξ to H is a hybrid arc that is parameterized by (t, j) ∈ R ≥0 × N, where t is the elapsed ordinary time and j is the number of jumps that have occurred. The domain of ξ, denoted dom ξ ⊂ R ≥0 × N, is a hybrid time domain, in the sense that for every (T, J) ∈ dom ξ, there exists a nondecreasing sequence {t j } J+1 j=0 with t 0 = 0 such that
We employ the following notions of excitation for timevarying signals [4].
and ess sup
and ess sup
In this section, we present a hybrid parameter estimator for finite-time convergence with respect to constant and piecewise constant unknown parameters. We first focus on the case of constant unknown parameters.
Recall the update laws for x and θ in (2) and denote z := (x, θ) and Γ := (A, γ, Q). Then, given t → B(t), we express the dynamics of ẑ := (x, θ) in a compact form as
Next, we express the error dynamics in (3) as
where z := (x, θ) and the functional Ψ is given by
where Γ denotes explicitly the dependence of Ψ on (A, γ, Q).
The continuous evolution of z between times t 0 and t with 0 ≤ t 0 < t is given by z(t) = Ω(t, t 0 )z(t 0 ) where, given Γ and t → B(t), Ω is the state transition matrix for (8).
We extend the finite-time parameter estimation approach in [7] to classes of continuous-time systems whose solutions satisfy (1). We begin by explaining the intuition behind the algorithm before providing a formal statement of the result. The algorithm is expressed as a hybrid system, denoted H, and operates as follows. Given
++ are design parameters, denote the state of H as ξ = (x, ẑ1 , ẑ2 , Φ 1 , Φ 2 , q), where • x is the plant state, with dynamics given in (1); • ẑ1 , ẑ2 ∈ R n+p are estimates of z, with dynamics
where h is given in (7). The dynamics of the errors z1 := z -ẑ1 and z2 := z -ẑ2 are given by ż1
where Ψ is given in (9);
where Ψ is given in (9). Hence, when initialized as the identity matrix, Φ 1 and Φ 2 are equivalent to the the state transition matrices Ω 1 and Ω 2 for, respectively, the systems in (10); • q ∈ {0, 1} is a logic variable. Next, let (t, j) → ξ(t, j) be a solution to H -hence, defined on a hybrid time domain -and consider the initial interval of flow I 0 , where I j := {t : (t, j) ∈ dom ξ }, with initial conditions Φ 1 (0, 0) = Φ 2 (0, 0) = I, and ẑ1 (0, 0) = ẑ2 (0, 0) arbitrary. At any time t ∈ I 0 , the solutions components Φ 1 and Φ 2 satisfy
Then, if there exists a positive time t 1 ∈ I 0 such that the matrix Φ 1 (t 1 , 0) -Φ 2 (t 1 , 0) is invertible, resetting ẑ1 and ẑ2 to the value of the function R(ξ
where the argument of K 1 and K 2 is omitted for readability. Hence, we have finite-time convergence of ẑ1 and ẑ2 to z.
To ensure the existence of a jump time (t 1 , 0), we choose A 1 and A 2 such that ẑ1 is convergent but ẑ2 is divergent. To avoid the solution components ẑ2 and Φ 2 from growing unbounded, for all (t, j) ∈ dom ξ satisfying t ≥ t 1 and j ≥ 1, we assign ẑ2 the dynamics of ẑ1 and Φ 2 the dynamics of Φ 1 . After the jump, we have ẑ2 (t 1 , j) = ẑ1 (t 1 , j) and Φ 2 (t 1 , j) = Φ 1 (t 1 , j), and thus ẑ2 (t, j) = ẑ1 (t, j) and Φ 2 (t, j) = Φ 1 (t, j) for all t ≥ t 1 and j ≥ 1.
We implement the estimation scheme outlined above as a hybrid algorithm, denoted H, whose jump map computes ẑ1 and ẑ2 as in the first line of (12). The hybrid system H has state ξ :
where
and
The logic variable q is used to ensure that the algorithm jumps only one time, when q = 0, and to prevent the solution components ẑ2 and Φ 2 from growing unbounded.
The following proposition provides sufficient conditions for the invertibility of the matrix Φ 1 -Φ 2 .
Proposition 3.1: Given a hybrid system H with data as in (13) where the matrices A 1 and -A 2 are Hurwitz, γ 1 , γ 2 > 0, and
++ , suppose the regressor t → B(t) is persistently exciting as in Definition 2.2. Then, there exists a time T > 0 such that for each maximal solution ξ to H from ξ(0, 0) ∈ {ξ ∈ X : Φ 1 = Φ 2 = I, q = 0 }, the gains K 1 and K 2 are well defined at hybrid time {T } × {0}, and there exists ε > 0 such that | det(Φ 1 (T, 0) -Φ 2 (T, 0))| = ε.
The proof is sketched in Appendix I.
The persistence of excitation condition imposed in Proposition 3.1 is sufficient for the existence of a time T such that the matrix Φ 1 (T, 0)-Φ 2 (T, 0) is invertible, but it is not necessary. Indeed, Section V-A shows an example where finite-time convergence of the parameter estimate is achieved using the proposed hybrid algorithm when the regressor is exciting on only a finite time interval, in the sense of Definition 2.1. A formal study of the invertibility of the matrix Φ 1 -Φ 2 under relaxed excitation conditions is left as future research.
Next, we study the stability properties induced by the proposed estimator. To this end, we define the following set:
Theorem 3.3: Given a hybrid system H with data as in (13) where the matrices A 1 and -A 2 are Hurwitz, γ 1 , γ 2 > 0, and
++ , suppose the regressor t → B(t) is such that there exists a time T > 0 when | det(Ω 1 (T, 0)-Ω 2 (T, 0))| = ε, where Ω 1 and Ω 2 are the state transition matrices for, respectively, the systems in (10). Then, there exist
for all (t, j) ∈ dom ξ. Moreover, ξ(t, j) ∈ A for all (t, j) ∈ dom ξ satisfying t ≥ T and j ≥ 1.
For each solution ξ to H from X 0 , we have by assumption that the regressor t → B(t) is such that | det(Ω 1 (T, 0) -Ω 2 (T, 0))| = ε, and, since Φ 1 and Φ 2 are initialized as the identity matrix, it follows that ξ(T, 0) ∈ D. Then, at the jump, according to the jump map, we have that ẑ1 (T, 1) = z from (12). Thus, the set A is finite-time attractive from X 0 for H.
To show stability of A, consider the Lyapunov function
++ solves the equation
++ . It can be shown that ∇V (ξ), F (ξ, B(t)) ≤ 0 for all ξ ∈ C. Furthermore, for all ξ ∈ D, we have from ( 12) that V (ξ(t, j + 1)) -V (ξ(t, j)) = -V (ξ(t, j)) ≤ 0. Hence, it follows that there exist α 1 , α 2 ∈ K ∞ such that (15) is satisfied. Theorem 3.3 states that if the unknown parameter θ is constant, then the proposed estimator converges to the exact value θ after one jump. Proposition 3.1 shows that the persistence of excitation condition in Definition 2.2 is sufficient to ensure the existence of a time T such that
When the unknown parameter θ is piecewise constant, it is also possible to estimate it in finite time. However, one jump is not sufficient for the estimate to converge. Hence, we consider the following adaptation law, denoted H, with state ξ : n+p) , input B : R ≥0 → R n×p , and data
where
The new structure of the set D makes it possible to have multiple jumps, which are necessary to estimate the piecewise constant parameter.
The following theorem states the stability properties induced by H in (16) for a piecewise constant unknown parameter.
Theorem 3.4: Given a hybrid system H with data as in (16) where the matrices A 1 and -A 2 are Hurwitz, γ 1 , γ 2 > 0, and
++ , suppose that the following conditions hold:
1. The regressor t → B(t) is such that there exists a sequence of time instants {τ k } k∈N>0 such that, for all
where Ω 1 and Ω 2 are the state transition matrices for, respectively, the systems in (10).
2. The unknown parameter θ is piecewise constant, where the time instants at which the value of θ changes are defined by a sequence of times {δ i } i∈N>0 satisfying 0 ≤ δ i < δ i+1 for all i ∈ N >0 .
Then, for each solution ξ to H from ξ(0, 0) ∈ {ξ ∈ X : Φ 1 = Φ 2 = I } and each i, k ∈ N >0 such that
the following property holds: ξ(t, j) ∈ A for all (t, j) ∈ dom ξ satisfying t ∈ [τ k+2 , δ i+1 ) and j ≥ k + 2.
From items II and III of Theorem 3.4, we have that the value of the unknown parameter vector θ changes in the time interval [τ k , τ k+1 ] and is constant in the interval (τ k+1 , τ k+2 ). By assumption, we have that ξ(τ k+1 , k) ∈ D and at the jump, according to the jump map, ξ(τ k+1 , k + 1) ∈ {ξ ∈ X : ẑ1 = ẑ2 , Φ 1 = Φ 2 = I }. Finally, ξ(τ k+2 , k + 1) ∈ D by assumption, and the result follows from Theorem 3.3. Theorem 3.4 shows that, each time the value of the parameter θ changes, the proposed estimator converges to the exact value θ after no more than two jumps. Let us also mention that as in Theorem 3.3, the bound in (15) can be established.
In this section, we analyze the robustness of the proposed hybrid parameter estimator with respect to bounded timevarying measurement noise. For the sake of readability, we focus on constant unknown parameters. However, the robustness results can be generalized using the same approach to deal with piecewise constant unknown parameters.
Starting from the noise-free hybrid estimator H in (13), we denote our proposed estimator under the effect of the state measurement noise t → ν(t) ∈ R n as a hybrid system, H ν , with state ξ ν := (x, ẑ1 , ẑ2 , Φ 1 , Φ 2 , q) ∈ X := R n × R n+p ×R n+p ×R (n+p)×(n+p) ×R (n+p)×(n+p) ×{0, 1}, input B : R ≥0 → R n×p , and data
where C ν := C and D ν := D given below (13), and
Next, we show that if the regressor is persistently exciting, the noisy system H ν is input-to-state stable (ISS) for any essentially bounded measurement noise t → ν(t). The question of establishing ISS properties under relaxed excitation conditions is left as future research. Proposition 4.1: Given a hybrid system H ν with data as in (17) where the matrices A 1 and -A 2 are Hurwitz, γ 1 , γ 2 > 0, and Q 1 , Q 2 ∈ S n ++ , suppose the regressor t → B(t) is persistently exciting as in Definition 2.2. Then, for any solution ξ ν to H ν from ξ ν (0, 0) ∈ X 0 := {ξ ν ∈ X : ẑ1 = ẑ2 , Φ 1 = Φ 2 = I, q = 0}, there exist constants w M , T, Φ M > 0, σ 1 , σ 2 , σ 3 > 0 with σ 1 ≤ σ 2 , and υ > 0 such that, for all (t, j) ∈ dom ξ ν ,
where ρ(0) = 1 and ρ(j) = 0 for j ∈ N >0 ,
In this section, we present simulation results that demonstrate the practicality of the proposed algorithms. Simulations are performed using the Hybrid Equations Toolbox [11].
Consider a scalar system with dynamics as in (1), where A 0 = -0.5, θ = -2, and
with measurement noise ν(t) = 4 sin(30t) added to the state. Note that B does not satisfy the persistence of excitation condition imposed in Lemma 3.1. Instead, B is exciting over only a finite time interval, in the sense of Definition 2.1, with η 1 = 1 and η 2 = 1.
The proposed finite-time parameter algorithm H in ( 13) is applied to estimate θ, with parameters
1, and ε = 1, which verifies the conditions of Theorem 3.3. The system is simulated from initial conditions x(0, 0) = 1, ẑ1 (0, 0) = ẑ2 (0, 0) = (0, 0), Φ 1 (0, 0) = Φ 2 (0, 0) = I, q(0, 0) = 0, producing the results in Figure 1 (only the trajectory of z1 is shown). 1 Due to the fact that B is zero over the interval [0, 1), the dynamics of θ1 are zero until t = 1, followed by finite-time convergence of the state and parameter estimates in accordance with Theorem 3.3. 1 Code at https://github.com/HybridSystemsLab/MRAC FT Estimator
Consider the following frequency estimation problem: given a signal t → ζ(t) = Υ sin(ωt) where Υ ∈ R >0 is the magnitude and ω ∈ R >0 is the frequency, estimate ω from measurements of ζ. To formulate this problem as the problem of estimating a parameter θ of a model as in (1), let x = (x 1 , x 2 ) be such that x 1 = ζ and ẋ1 = x 2 , and Υ is related to the initial condition. Note that a similar problem was studied in [12]. Then, we obtain the following parametric form:
where A 0 = 0 1 -1 0 , B(x) = 0 -x1 , and θ = ω 2 -1. Then, with η 1 = π ω , it can be shown that for all t ≥ 0, t+η1 t B (x(s))B(x(s))ds = πΥ 2 2ω . Hence, B is persistently exciting in the sense of Definition 2.2 with η 2 = πΥ 2 2ω . The unknown parameter θ is piecewise constant, with a value of θ = 3 if t ∈ [0, 2) and θ = 1 if t ≥ 2, and measurement noise ν(t) = sin(50t) cos(50t) is added to the state.
The proposed estimation algorithm H in ( 16) is applied to estimate θ in (19), with parameters A 1 = -A 2 = -I, γ 1 = γ 2 = 10, Q 1 = Q 2 = I, and ε = 0.5, which verifies the conditions of Theorem 3.4. The system is simulated from initial conditions x(0, 0) = (1, 0), ẑ1 (0, 0) = (0.5, 0, 0), ẑ2 (0, 0) = (0, 0, -0.2), Φ 1 (0, 0) = Φ 2 (0, 0) = I, producing the results in Figure 2. 1 The estimation errors x1 and θ1 converge to zero first at t = 1.6 and, after the value of θ changes at t = 2, converge again at t = 3.23 in accordance with Theorem 3.4. The figure also illustrates the robustness of the proposed estimator to time-varying measurement noise.