Introduction

The estimation of unknown parameters in dynamical systems has been an active research area for years [36]. Parameter estimation algorithms typically rely on exploiting information about the structure of the system along with the available input and output signals to compute online an estimate of the unknown parameters. One of the most popular estimation problems is recursive linear regression, for which the estimation scheme is often based on the gradient descent algorithm [22,36]. For dynamical systems, control strategies leveraging estimation algorithms, such as model-reference adaptive control, are used in several engineering applications [10,11].

More recently, there has been a growing interest in hybrid dynamical systems. These systems are characterized by state variables that may evolve continuously (flow) and, at times, evolve discretely (jump) [27]. Hybrid systems provide new and promising modeling frameworks for a wide range of applications including robotics, aerospace, automotive, and power systems [5, 7-9, 20, 26, 32]. However, the hybrid nature of these systems limits the applicability of existing continuous-time or discrete-time estimation algorithms.

Much work has been done on parameter estimation and system identification for specific sub-classes of hybrid systems, such as switched systems [12,24,25] and piecewise-affine systems [1,3]. However, these systems exhibit nonsmooth but continuous evolution of the state variables, rather than jumps in the state variables, hence such results are not applicable to a general class of hybrid systems. Recently, the work [30] proposed a hybrid estimation algorithm for linear regression with hybrid signals, and [21] proposed an algorithm for identification of hybrid systems with linear dynamics. However, to the best of our knowledge, before our preliminary results related to this work reported in [17], an algorithm for estimating unknown parameters for a general class of hybrid systems had not been established in the literature. The goal of this paper is to fill that gap. Note that this paper focuses only on online estimation of unknown parameters in the dynamics of hybrid systems. Simultaneous estimation of the flow and jump maps and the flow and jump sets of a hybrid system, i.e., the identification of an entire hybrid system, is still an open problem.

In this paper, we propose a hybrid algorithm for estimating unknown parameters for a class of hybrid systems with nonlinear dynamics that are affine in the unknown parameters. We establish sufficient conditions that guarantee exponential convergence of the parameter estimate to the true value, and we lower bound the convergence rate of the parameter estimate. The main contributions in this paper are the following:

1. Estimation under hybrid persistence of excitation: in the main stability result (Theorem 5.1), we establish that our algorithm guarantees exponential convergence of the parameter estimate to the true value under a notion of hybrid persistence of excitation (PE) inspired by [30]. To the best of our knowledge, it is the first hybrid PE sufficient condition that ensures estimation of parameters for the considered class of hybrid systems.

2. Estimator robustness to hybrid disturbances: to prove the main robustness result (Theorem 6.2), we generalize the error dynamics of our algorithm to a class of hybrid systems, denoted by H, that includes hybrid disturbances. Lemma B.3 and Theorem 5.5 construct an input-to-state stability (ISS) Lyapunov function for H by extending ISS results for continuous-time and discrete-time systems [4,6,15,19,34,35].

In [17], we proposed a hybrid algorithm for estimating unknown parameters in a class of hybrid systems with linear dynamics. We showed that the parameter estimate converges to the true value if the hybrid regressor satisfies the classical continuous-time PE condition during flows and the classical discrete-time PE condition at jumps. In comparison to [17], this paper considers a wider class of hybrid systems with nonlinear dynamics and proposes a new hybrid algorithm to solve the estimation problem. Moreover, here we relax the classical PE conditions imposed in [17], and instead impose a hybrid PE condition inspired by [30].

The hybrid PE condition is exploited in more recent work involving authors of this paper and of [30] to establish uniform exponential stability for a general class of time-varying hybrid dynamical systems [29]. In comparison to [29], this paper focuses on deriving and analyzing the properties induced by an algorithm for estimating unknown parameters in a class of hybrid dynamical systems when the state of the hybrid plant is measured, while [29] covers the more general case of output measurements. Due to such differences and the different techniques used, we provide explicit bounds on the convergence rate of the parameter estimate, and on the estimation error when noise is present whereas [29] establishes only the existence of such bounds. Our analysis does not impose completeness of maximal solutions, while the results in [29] rely on completeness of maximal solutions to ensure well-posedness. The approach to the proof of exponential stability in this paper differs from that in [29]. Specifically, we leverage a property of input-to-state stability with respect to an exponentially convergent hybrid signal, while the analysis in [29] relies on a general, but more abstract, result involving uniform observability properties formulated to establish uniform exponential stability for hybrid systems.

The remainder of this paper is organized as follows. Preliminaries on continuous-time and discrete-time estimation algorithms are presented in Section 2. In Section 3, we present a motivational example that highlights the limitations of these algorithms, that our hybrid algorithm aims at overcoming. Our algorithm is described in Section 4, and the stability and robustness properties are analyzed in Section 5 and Section 6, respectively. Simulation results are in Section 7. Conclusions and future work are in Section 8. The proofs of Theorems 5.5 and 5.6 are in the Appendix. Due to space constraints, the proofs of Lemmas 5.7, 5.8, and 6.3 are in [18].

Preliminaries

Notation

We denote the set of real, nonnegative real, and positive real numbers by R, R ≥0 , and R >0 , respectively. We denote the set of natural numbers (including zero) as N.

Given sets S, U ⊂ R n , cl(S) denotes the closure of S, and S \ U denotes set subtraction. Given a measure space M and a function f

the exponential is e x or exp(x), equivalently.

Review of Parameter Estimation Algorithms

In preparation for the proposed hybrid parameter estimation algorithm, we review relevant continuous-time and discrete-time estimation algorithms.

• Consider the continuous-time system

where

is a vector of unknown constant parameters, and n, p, m ∈ N.

To estimate θ, we convert (1) into a form similar to a linear regression model by introducing [23] the state variables ψ ∈ R n×p and η ∈ R n with dynamics

where λ c > 0 is a design parameter. Defining ε := x + η -ψθ and y := x + η, it follows that ψ and ε are related via y = ψθ + ε. Since θ is constant, differentiating ε along trajectories of (1), ( 2) yields ε = -λ c ε. Thus, ε converges exponentially to zero. Moreover, we have the following equivalences:

Hence, y, ψ, and θ are related via a linear regression model plus an exponentially convergent term.

Denoting the estimate of θ as θ, the gradient algorithm for θ is [22]

where γ c > 0 is a design parameter.

• Consider the discrete-time system

where x ∈ R n is the known state vector, j → u(j) ∈ R m is the known input, j → ϕ d (j) ∈ R n×p is the known regressor, (x, u) → g d (x, u) ∈ R n is a known continuous function, θ ∈ R p is a vector of unknown constant parameters, and n, p, m ∈ N.

Using similar reasoning as in the continuous-time case, we estimate θ using a gradient algorithm as θ(j + 1) = θ(j)

+ ψ(j + 1) ⊤ γ d + |ψ(j + 1)| 2 (y(j + 1) -ψ(j + 1) θ(j)), with

where γ d > 0, λ d ∈ (0, 2) are design parameters.

To compute the update law for θ in (4), we require measurements of x for two consecutive discrete time steps. Moreover, two computational steps are required to update θ at time j ∈ N. The first step computes ψ(j + 1), η(j + 1), and y(j + 1) in (5), and the second step computes θ(j + 1) in (4). For simplicity, we omit the first computational step in (4).

It is shown in [22] that, if ε = 0, the following PE condition is necessary and sufficient for convergence of θ in (3) to θ:

(C1) The signal t → ψ(t) is uniformly bounded and there exist T > 0 and µ > 0 such that

For the discrete-time case in (4), the PE condition is [36]:

(C2) The signal j → ψ(j) is uniformly bounded and there exist J ∈ N \ {0} and µ > 0 such that

Hybrid Dynamical Systems

In this paper, a hybrid system H is defined by (C, F, D, G) as [27] H

where ξ ∈ R n is the state, F : C → R n is the flow map defining the continuous dynamics, and C ⊂ R n is the flow set on which flow is permitted. The mapping G : D → 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 [27] 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 dom ξ ∩ ([0, T ] × {0, 1, . . . , J })

A solution ξ to H is called maximal if it cannot be extended -that is, if there does not exist another solution ξ ′ to H such that dom ξ is a proper subset of dom ξ ′ and ξ(t, j) = ξ ′ (t, j) for all (t, j) ∈ dom ξ. A solution is called complete if its domain is unbounded. The operations sup t dom ξ and sup j dom ξ return the supremum of the t and j coordinates, respectively, of points in dom ξ.

The length of dom ξ is sup t dom ξ + sup j dom ξ.

We employ the following notion of stability [27]. Definition 2.1: Given a hybrid system H with data as in (8), a nonempty closed set A ⊂ R n is said to be semiglobally pre-exponentially stable 1 for H if, for each compact X 0 ⊂ R n , there exist κ, λ > 0 such that each solution ξ to H from ξ(0, 0) ∈ X 0 satisfies

If there exist κ, λ > 0 such that each solution ξ to H satisfies (9), then A is said to be globally pre-exponentially stable for H.

Given a hybrid arc (t, j) → ξ(t, j) ∈ R n , we denote the supremum norm of ξ from (0, 0) to (t, j) as

where Υ(dom ξ) := {(t, j) ∈ dom ξ : (t, j + 1) ∈ dom ξ}. (10)

Motivational Example

To motivate our parameter estimation algorithm, consider the hybrid arcs ϕ c , ϕ d :

] for all (t, j) ∈ E. For such ϕ c and ϕ d , consider a hybrid system as in (8) with an added input 2 u :

where θ = [1 1] ⊤ is a vector of unknown parameters. The flow and jump sets are

, respectively. The input u(t, j) = t -2πj for all (t, j) ∈ E is a sawtooth function that periodically ramps to a value of 2π and then resets to zero. 3 Given a solution x to (11) from x(0, 0) = (3, 6), we want to estimate θ. To do so, we first separately analyze the flows and jumps of these signals. We define the continuous-time signals t → xc (t) :=foot_0-cos(t) 6 and 1 The term "pre-exponential," as opposed to "exponential," indicates the possibility of maximal solutions that are not complete. This allows for separating the conditions for completeness from the conditions for stability and attractivity.

2 See [27] for details on hybrid systems with inputs. 3 With CP , DP , and u given below (11), the hybrid time domain of each maximal solution to the hybrid system in ( 11) is equal to the hybrid time domain E of ϕc and ϕ d .

t → φc (t) := sin(t) 0 0 0 , which are obtained by neglecting the resets of x and ϕ c , respectively, at jumps. The signals xc and φc are solutions to the continuous-time system ẋc (t) = φc (t)θ for all t ≥ 0. Next, we define the discretetime signals j → xd (j) := [ 3 6 ] and j → φd (j) := [ 1 2 2 4 ], which are obtained by neglecting the evolution of x and ϕ d , respectively, during flows. The signals xd and φd are solutions to the discrete-time system xd (j + 1) = φd (j)θ for all j ∈ N. Using the transformations in Section 2.2, we employ the continuous-time and discrete-time algorithms ( 3) and ( 4) to estimate θ in (11). The parameter estimation error for both algorithms fails to converge to zero, as shown in Figure 1. 4 To see why the continuous-time algorithm fails to estimate θ, note that for φc , the value of

φc (s)ds for all t ≥ 0. Since e -λct ψ(0) converges exponentially to zero and the second column of φc (t) is zero for all t ≥ 0, t → ψ(t) does not satisfy (C1) for any T > 0. Similarly for φd , the value of j → ψ(j) in ( 5) is

φd (i) for all j ∈ N. Since (1 -λ d ) j ψ(0) converges exponentially to zero and φd (j) is constant and singular for all j ∈ N, j → ψ(j) does not satisfy (C2) for any J ∈ N \ {0}. On the other hand, the hybrid algorithm proposed in this paper successfully estimates θ by leveraging the information available during both flows and jumps, as shown in Figure 1.

Fig. 1: The projection onto t of the norm of the parameter estimation error for the continuous-time and discretetime estimation algorithms, and our hybrid algorithm. The continuous-time and discrete-time algorithms produce nonzero steady-state error, whereas the error for our algorithm converges to zero.

A Hybrid Parameter Estimation Algorithm

Problem Statement

Motivated by the limitations of the continuous-time and discrete-time estimation algorithms highlighted in Section 3, we develop a hybrid algorithm for estimating pa-rameters in hybrid dynamical systems of the form

where x ∈ R n is the known state vector and (x, u) → f c (x, u) ∈ R n and (x, u) → g d (x, u) ∈ R n are known continuous functions. The regressors (t, j) → ϕ c (t, j) ∈ R n×p and (t, j) → ϕ d (t, j) ∈ R n×p and the input (t, j) → u(t, j) ∈ R m are known, and are defined on hybrid time domains as described in Section 2.3, but are not necessarily hybrid arcs. 5 The flow set is

and n, p, m ∈ N. Note that ϕ c plays no role in the dynamics of ( 12) at jumps and ϕ d plays no role in the dynamics of ( 12) during flows.

Our goal is to estimate the parameter vector θ in (12). Since ϕ c and ϕ d may exhibit both flows and jumps, it is important to update the parameter estimate θ continuously whenever ϕ c flows, and discretely each time ϕ d jumps, which is possible when jumps are detected instantaneously. Hence, we propose to estimate θ using a hybrid algorithm, denoted H g , of the form

with data designed to solve the following problem.

Problem Statement: Design the data (C g , F g , D g , G g ) of H g in ( 13) and determine conditions on ϕ c and ϕ d that ensure the parameter estimate θ converges to the unknown parameter vector θ in (12).

Next, we present our solution to this problem.

Problem Solution

Given ϕ c , ϕ d : E → R n×p and u : E → R m , where E := dom ϕ c = dom ϕ d = dom u is a hybrid time domain, we define the state ξ of H g as ξ :

where x is the state of the plant in (12), θ is the estimate of θ, and ψ, η are auxiliary state variables. The state components τ and k have dynamics such that they evolve as t and j, respectively, from the hybrid time domain E. Including τ and k in ξ allows ϕ c , ϕ d , and u to be part of the definitions of F g and G g , rather than modeled as inputs to H g . Thus, we can express H g as an autonomous hybrid system, which allows us to leverage recent results on stability and robustness properties [13,27].

During flows, we update θ with dynamics inspired by the continuous-time algorithm in ( 2) and ( 3),

where y := x+η, with ψ and η generated by ψ = -λ c ψ+ ϕ c (τ, k) and η = -λ c (x+η)-f c (x, u(τ, k)), and γ c , λ c > 0 are design parameters. Hence, for all ξ ∈ C g , the flow map for H g in ( 13) is

.

At jumps, we update θ using a reset map inspired by the discrete-time algorithm in ( 4) and ( 5),

, and y + := x + + η + , where x + gives the plant state x after a jump per (12), and γ d > 0, λ d ∈ (0, 2) are design parameters. Hence, for all ξ ∈ D g , the jump map for H g in ( 13) is

The flow and jump sets of H g are defined so that the algorithm flows when ϕ c flows, and jumps when ϕ d jumps.

Since dom

Remark 4.1: We assume for simplicity that the plant state x has the same hybrid time domain as ϕ c , ϕ d , and u. As a result, the flow set C P and jump set D P of the plant are not part of the construction of H g . Our algorithm can be extended to the case where x, ϕ c , ϕ d , and u have different hybrid time domains by considering the flow and jump sets in (12). In this case, we can reparameterize the domains of ϕ c , ϕ d , and u to express x, ϕ c , ϕ d , and u on a common hybrid time domain. See, e.g., [2].

Remark 4.2: For simplicity, the hybrid algorithm H g in ( 13) is expressed such that jumps in the parameter estimate coincide with jumps in x. This results in H g being noncausal since measurements of x + are not available until after a jump. We can remove the simplification at the price of letting the algorithm jump twice for each jump in x, as follows. Immediately before a jump in x, the algorithm jumps once to reset the values of ψ, η, and k per the jump map (14). Immediately after a jump in x, the algorithm jumps a second time to update the parameter estimate using the current value of x. A logic variable ensures that, after the second jump, the algorithm flows or jumps in accordance with the hybrid time domain E. Since θ in ( 12) is constant, the stability properties induced by H g in ( 13) are equivalent to the stability properties induced by the causal modification, after we reparameterize the domain of solutions to H g to match the domain of solutions to the causal system. Hence, for simplicity, we focus our analysis on (13).

Stability Analysis

We now establish our main stability result stating conditions that ensure the hybrid system H g in ( 13) induces semiglobal pre-exponential stability 6 of the set

where

Semiglobal pre-exponential stability of A g implies that, given any compact set of initial conditions, for each solution ξ to H g from such compact set, the distance from ξ to the set A g is bounded above by an exponentially decreasing function of the initial condition -see Definition 2.1. As a consequence, for each complete solution ξ to H g , the parameter estimate θ converges exponentially to θ, and ε converges exponentially to zero. Theorem 5.1: Given the hybrid system H g in (13), γ c , λ c , γ d > 0, and λ d ∈ (0, 2), suppose that ϕ c , ϕ d : E → R n×p satisfy the following:

6 Since each solution Hg inherits the hybrid time domain of ϕc, ϕ d , and u, the use of "pre-exponential," as opposed to "exponential," stability means that ϕc, ϕ d , and u do not need to be complete. 7 The hybrid time instants (t ′ , j ′ ) and (t * , j * ) are the beginning and the end, respectively, of a hybrid time interval with length satisfying (18), over which (19) holds.

the following hybrid PE condition holds:

max{t ′ ,tj } ψ(s, j) ⊤ ψ(s, j)ds

where {t j } J j=0 is the sequence defining E as in Section 2.3, t J+1 := T , with J := sup j E and T := sup t E, and (t, j) → ψ(t, j) is generated by (13).

Then, for each ψ 0 ≥ 0, q M ≥ q m > 0, and each ζ ∈ (0, 1), there exist κ g , λ g > 0 such that each solution ξ to H g from ξ(0, 0)

for all (t, j) ∈ dom ξ. In particular, suitable choices of κ g and λ g are given in Appendix A.

A General Class of Hybrid Gradient Algorithms

Convergence to θ for the solution component θ of H g in ( 13) is achieved when the parameter estimation error θ = θ -θ of H g converges to zero. We denote the hybrid system resulting from expressing H g in error coordinates as H g , with state ξ = (x, θ, ψ, η, τ, k) ∈ X g and dynamics

where C g := C g and D g := D g , with C g , D g in ( 15), and

with ε as in (17) and ε + := x + + η + -ψ + θ, where x + gives the plant state x after a jump per (12).

To analyze the stability properties induced by H g , we use that H g in ( 21) belongs to a class of hybrid systems, denoted by H, with state ξ = (ϑ, τ, k) ∈ X := R p × E and dynamics Remark 5.2: The hybrid system H in (22) reduces to H g in ( 21) when ϑ = θ,

for all (τ, k) ∈ E, andfoot_3

for all (τ, k) ∈ Υ(E), with Υ as in (10), where ε = x + η -ψθ is a hybrid disturbance and x, η, ψ satisfy the dynamics in (21).

We impose on A and B the following structural properties, which are similar to those imposed in the design of continuous-time and discrete-time gradient algorithms. Assumption 5.3: Given A, B : E → R p×p , where E := dom A = dom B is a hybrid time domain, 1. A(t, j) = A(t, j) ⊤ ≥ 0 for all (t, j) ∈ E; 2. B(t, j) = B(t, j) ⊤ ≥ 0 for all (t, j) ∈ Υ(E); 3. there exists a M > 0 such that ess sup {|A(t, j)| : (t, j) ∈ E} ≤ a M ; 4. |B(t, j)| < 1 for all (t, j) ∈ Υ(E).

We impose the following hybrid PE condition [30]. Assumption 5.4:

where t J+1 := T , with J := sup j E and T := sup t E.

Stability Analysis for H

In this section, we establish sufficient conditions on A, B, d c , and d d that ensure the hybrid system H induces global pre-exponential stability of the set

We first establish the following ISS result for H. Theorem 5.5: Given the hybrid system H in (22), let Assumptions 5.3 and 5.4 hold. Then, for each q M ≥ q m > 0 and each ζ ∈ (0, 1), each solution ξ to H satisfies

for all (t, j) ∈ dom ξ, where

and

with Υ defined in (10) and a M , µ 0 , ∆ from Assumptions 5.3 and 5.4.

Motivated by the fact that, for each complete solution to H g in ( 13), the signal (t, j) → ε(t, j) in ( 23 for all (t, j) ∈ E. Then, for each q M ≥ q m > 0 and each ζ ∈ (0, 1), each solution ξ to H satisfies

for all (t, j) ∈ dom ξ, where

with p m , p M , ρ, and ω from Theorem 5.5.

We use Theorem 5.6 in the next section to prove the stability properties induced by our algorithm.

Stability Analysis for H g

To prove Theorem 5.1, we require the following results for the error dynamics of our algorithm.

Lemma 5.7: Given the hybrid system H g in (21), for each λ c > 0, λ d ∈ (0, 2), and each solution ξ = (x, θ, ψ, η, τ, k) to H g , (t, j) → ε(t, j) := x(t, j) + η(t, j) -ψ(t, j)θ in (17) satisfies

where b := 1 2 min {2λ c , -ln (1 -λ d (2 -λ d ))}. Lemma 5.8: Given the hybrid system H g in (21), suppose that ϕ c : E → R n×p satisfies item 1 of Theorem 5.1 and let ϕ M > 0 come from that item. Then, for each ψ 0 ≥ 0,

where

We now have all the ingredients to prove Theorem 5.1. γ d +|ψ(t,j)| 2 < 1 for all (t, j) ∈ Υ(E). Hence, items 3 and 4 of Assumption 5.3 hold with a M := γ c ψ 2 M .

Proof of Theorem

Next, using Lemma 5.8 and item 2 in Theorem 5.1, we show that Assumption 5.4 holds with A, B in (23). Substituting A, B into (24), we have that, for all (t ′ , j ′ ), (t * , j * ) ∈ E satisfying (18),

Hence, Assumption 5.4 holds with ∆ from item 2 of Theorem 5.1 and µ 0 := min γ c ,

Finally, we show that ( 28) is satisfied with d in (27) and d c , d d in (23). By item 1 of Theorem 5.1, it follows from Lemmas 5.7 and 5.8 that, for each solution ξ to H g from X 0 , |d c (t, j)| ≤ γ c |ψ(t, j)||ε(t, j)| ≤ γ c ψ M e -b(t+j) |ε(0, 0)| for all (t, j) ∈ dom ξ, with ψ M from Lemma 5.8 and b from Lemma 5.7. Furthermore, using that |ψ(t,j+1)|

√ γ d e -b(t+j) |ε(0, 0)| for all (t, j) ∈ Υ(dom ξ). Thus, we conclude that (28) holds with a := max γ c ψ M , 1 2 √ γ d , b from Lemma 5.7, and |d(0, 0)| = |ε(0, 0)|. Hence, the conditions of Theorem 5.6 hold and, from the equivalence between the data of H g in ( 21) and H in (22) with A, B, d c , d d in ( 23), 10we have from Theorem 5.6 that the θ component of each solution ξ to H g from X 0 satisfies

for all (t, j) ∈ dom ξ, with κ, λ in (30).

Using the definition of A g in ( 16), we rewrite |ξ| Ag for all (t, j) ∈ dom ξ as |ξ(t, j)

Substituting the bounds in ( 31) and ( 33) and using that κ ≥ 1 and, for any α, β ∈ R, αβ ≤ 1 2 (α 2 + β 2 ), we conclude that |ξ(t, j)| Ag ≤ √ 3κe -min{λ,b}(t+j) |ξ(0, 0)| Ag for all (t, j) ∈ dom ξ. Hence, (20) holds.

Robustness Analysis

In this section, we study the robustness properties induced by H g with respect to bounded (hybrid) noise on the state measurements.

Given ϕ c , ϕ d : E → R n×p and u : E → R m , where E := dom ϕ c = dom ϕ d = dom u is a hybrid time domain, consider additive noise ν : E → R n in the measurements of the plant state x in (12). 11 We denote the hybrid system H in (13) under the effect of the measurement noise ν as H ν , with state ξ = (x, θ, ψ, η, τ, k) ∈ X g and dynamics

where 15), and we define y ν := x+ν(τ, k)+η and y + ν := x + +ν(τ, k+1)+η + , where x + gives the plant state x after a jump per (12).

For analyzing the effect of the noise, we make the following Lipschitz continuity assumption. Assumption 6.1: Given the hybrid plant in (12), there exist L c , L d > 0 such that, for all

We now establish our main robustness result stating conditions that ensure A g in ( 16) is ISS for H ν . Theorem 6.2: Given the hybrid system H ν in (34), γ c , λ c , γ d > 0, and λ d ∈ (0, 2), suppose that Assumption 6.1 holds and that ϕ c , ϕ d : E → R n×p satisfy items 1 and 2 of Theorem 5.1. Then, for each ψ 0 ≥ 0, q M ≥ q m > 0, and each ζ ∈ (0, 1), each solution ξ to H ν from ξ(0, 0) ∈ X 0 satisfies

for all (t, j) ∈ dom ξ, where

with ρ, ω from Theorem 5.5, p m , p M from Theorem 5.1, d ν (t, j) := ∥d∥ 2 (t,j) + ∥d ε ∥ 2 (t,j) , with d as in (27) and

where, for all (t, j) ∈ E,

and, for all (t, j) ∈ Υ(E),

with ε as in (17). Moreover, for all (t, j) ∈ E,

and for all (t, j) ∈ Υ(E),

with ψ M from Theorem 5.1, L c , L d from Assumption 6.1, and ε(0, 0) = x(0, 0) + η(0, 0) -ψ(0, 0)θ.

To prove Theorem 6.2, we require the following result. Lemma 6.3: Given the hybrid system H ν in (34), suppose that Assumption 6.1 holds. Then, for each λ c > 0, λ d ∈ (0, 2), ζ ∈ (0, 1), and each solution ξ = (x, θ, ψ, η, τ, k) to H ν , (t, j) → ε(t, j) := x(t, j) + η(t, j) -ψ(t, j)θ in (17) satisfies, for all (t, j) ∈ dom ξ,

with λ ε , ρ ε > 0 and (t, j) → d ε (t, j) from Theorem 6.2.

We now have all the ingredients to prove Theorem 6.2.

Proof of Theorem 6.2: Using the same arguments as in the proof of Theorem 5.1, we conclude that, by items 1 and 2 of Theorem 5.1, the conditions of Theorem 5.5 are satisfied with µ 0 and a M from Theorem 5.1. It can be shown that, under Assumption 6.1, the hybrid system that is obtained by expressing H ν in error coordinates is equivalent to H in (22) with A, B in ( 23) and d c , d d in (37). Hence, it follows from Theorem 5.5 that, for each solution ξ to H ν from X 0 , the parameter estimation error satisfies, for all (t, j) ∈ dom ξ,

with (t, j) → d(t, j) as in (27) and ρ, ω from Theorem 5.5, with p m , p M substituted by p m , p M from Theorem 5.1.

Using the definition of A g in ( 16), we rewrite |ξ| Ag for all (t, j) ∈ dom ξ as |ξ(t, j)

Since, by Assumption 6.1, the conditions of Lemma 6.3 are satisfied, we substitute the bounds in (38) and (39). Using that, for any α,

for all (t, j) ∈ dom ξ. Hence, (35) holds.

To conclude the proof, we upper bound d c , d d , α c , and α d for all (t, j) ∈ dom ξ. The bounds for α c and α d in Theorem 6.2 follow directly from Assumption 6.1 and the definitions of α c and α d in (37). Moreover, since, by item 1 of Theorem 5.1 and Assumption 6.1, the conditions of Lemmas 5.8 and 6.3 are satisfied, we have from (37) that, for each solution ξ to H ν from X 0 ,

for all (t, j) ∈ dom ξ, with λ ε and ρ ε from Theorem 6.2, where the last inequality follows from (38) the definition of d ε in (36). Next, using that

for all (t, j) ∈ Υ(dom ξ), we have that, for all (t, j) ∈ Υ(dom ξ),

Remark 6.4: A similar ISS result as in Theorem 6.2 can be developed without Assumption 6.1 by constraining the range of the plant state x and the input u to a compact set. Under such conditions, it follows from the continuity of f c and g d that d c , d d , α c , α d in (37) can be upper bounded by functions of only ν. Then, ISS follows from similar arguments as in the proof of Theorem 6.2.

Spacecraft Bias Torque Estimation

In this section, we present a case study that demonstrate the merits of our hybrid algorithm. Simulations are performed using the Hybrid Equations Toolbox [28].

Consider the problem of estimating a constant disturbance torque applied to a spacecraft, controlled by reaction wheels (RW) and reaction control system (RCS) thrusters. Such bias torques may arise in practice due to aerodynamic effects, gravity gradients, or solar radiation pressure differentials. For simplicity, we consider the dynamics of a spacecraft rotating about only a single principle axis of inertia, although our approach can be extended to three-axis rotation. In the following, we derive the closed-loop dynamics of the spacecraft when controlled by RW and RCS thrusters separately, and then combine the results into a single hybrid model.

The dynamics of a spacecraft rotating along a principle axis of inertia under the effect of RW are [33] J

where z ∈ R is the known pointing angle of the spacecraft, Ω ∈ R is the known rotational velocity of the RW, J s > 0 is the known spacecraft moment of inertia, J w > 0 is the known RW moment of inertia, and θ ∈ R is an unknown bias torque.

Suppose RW control the attitude to a pointing angle, z des ∈ R. The dynamics of the reaction wheel are [33] J w Ω = α(t),

where t → α(t) ∈ R is the RW motor torque that is designed to maintain the spacecraft pointing angle. Substituting (41) into (40), we obtain

When the bias torque is nonzero, the industry-standard proportional-derivative (PD) control scheme for the RW motor fails to yield zero pointing error in steady-state. In this case, a feedfoward term is added that compensates for the effect of the bias torque using an estimate of the bias, denoted by θ [33]. Hence, the RW torque is

where K P , K D > 0 are design parameters. From (41), (42), (43), the dynamics of the closed-loop system are

The spacecraft pointing angle can be maintained only if an equivalent RW torque is delivered to counteract the bias torque. If the bias torque is nonzero, the angular velocity of the RW constantly increases in order to counteract the disturbance and the RW motor eventually reaches its maximum angular velocity. In order to avoid the RW motor from becoming saturated, "momentum dumping" is applied to decrease the angular velocity of the RW [33]. This procedure involves firing the RCS thrusters to generate a torque that is compensated by the attitude controller by actions that cause the RW to reduce their angular momentum.

The dynamics of a spacecraft rotating along a principle axis of inertia under the effect of RCS thrusters are [33] J

where M ∈ R is the known RCS thruster torque. For simplicity, we assume that the velocity of the RW is constant for the duration of each thruster firing. As a result, the RW dynamics do not play a role in (45).

Suppose that, at time t ≥ 0, the thrusters are fired for δ > 0 seconds. Integrating (45) over the time interval [t, t + δ] yields ż(t + δ) = ż(t) + δ Js (M + θ). If the thruster firing duration δ is negligibly small compared to the other time scales of the system, which is appropriate due to the slow spacecraft attitude maneuvering, we model the thruster firing as an instantaneous jump in the angular velocity of the spacecraft, given by

To avoid chatter, a timer, denoted by τ s , is used to briefly inhibit the RCS thrusters after each thruster firing. Each time the thrusters are fired, the timer is reset to zero.

By combining the expression in (44) and ( 46), we express the closed-loop dynamics of the spacecraft as a hybrid system as in (12). Given an input u := (z des , θ), where z des ∈ R is the desired constant spacecraft pointing angle and θ ∈ R is an estimate of the unknown bias torque, the hybrid model of the spacecraft has state x = (z, ż, Ω, τ s ) ∈ R 4 and data

where α(x, u(t, j)) := -K P (z des -z)+K D ż+ θ(t, j). The flow and jump sets of the hybrid spacecraft model implement the momentum dumping procedure. The system jumps each time the angular velocity of the RW exceeds a design parameter Ω max > 0 and the timer τ s exceeds a design parameter τ * > 0, and flows otherwise, as

We employ H g to estimate the unknown bias torque. The closed-loop system is simulatedfoot_6 with initial conditions x(0, 0) = (0, 0, 0, 0), θ(0, 0) = 0, ψ(0, 0) = 0, and η(0, 0) = -x(0, 0). The hybrid spacecraft model has parameters z des = 0 rad, Ω max = 10000 RPM, J s = 5000 kg-m 2 , J w = 0. The bias torque estimation error from H g converges exponentially to zero in accordance with Theorem 5.1, as shown in Figure 2. The spacecraft pointing angle er-Fig. 2: The projection onto t of the bias torque estimation error for H g .

ror and RW angular velocity are shown in the top and bottom plots, respectively, in Figure 3, where the control performance resulting from our hybrid algorithm is compared against an industry-standard PID control scheme that is tuned to achieve a similar pointing error convergence rate during flows. For the PID controller, we inhibit accumulation of the integrator during each thruster firing, otherwise the spacecraft pointing angle fails to converge to the set point. With the exception of the transients caused by the thruster firings, the pointing error converges to zero for both controllers. However, the hybrid algorithm converges faster due to our estimator's ability to leverage information during both flows and jumps to estimate the unknown bias torque.

Fig. 3: The projection onto t of the spacecraft pointing angle error (top) and the RW angular velocity (bottom).

Conclusion

In this paper, we proposed a hybrid algorithm for estimating unknown parameters in a class of hybrid systems with nonlinear dynamics that are affine in the unknown parameter. We showed that our algorithm guarantees exponential convergences of the parameter estimate to the true value under a notion of hybrid persistence of excitation that relaxes the classical continuous-time and discrete-time persistence of excitation conditions. Moreover, we showed that the parameter estimate is ISS with respect to hybrid noise in the measurements of the plant state. where t J+1 := T , with J := sup j E and T := sup t E.

Note that the term q m I in (B.10) was chosen for simplicity -any positive definite matrix would suffice.

We first show that (B.3) holds. Since, for all (t, j) ∈ E, P c (t, j) ≥ 0 and P d (t, j) ≥ 0, a lower bound on P in (B.10) is P (t, j) ≥ q m I for all (t, j) ∈ E. Next, we develop an upper bound on P . Since, by Assumptions 5.3 and 5.4, the conditions of Theorem B.1 are satisfied, it follows from (B.2) and from the equivalence between |ξ| A and |ϑ| that, for each solution ξ = (ϑ, τ, k) to H 0 and each (s, i), (t, j) ∈ dom ξ satisfying s ≥ t and i ≥ j, |ϑ(s, i)| ≤ κ 0 e -λ0(s+i-t-j) |ϑ(t, j)| with κ 0 and λ 0 from Theorem B.1. By substituting (B.9) into the expression above, we have that, for each (s, i), (t, j) ∈ dom ξ satisfying s ≥ t and i ≥ j, |Φ(s, i, t, j)ϑ(t, j)| ≤ κ 0 e -λ0(s+i-t-j) |ϑ(t, j)| which, if |ϑ(t, j)| ̸ = 0, implies that |Φ(s, i, t, j)ϑ(t, j)|/|ϑ(t, j)| ≤ κ 0 e -λ0(s+i-t-j) . Since this inequality holds for any ϑ(t, j) ∈ R p \ {0}, it follows from the equivalence between dom ξ and E that, for each (s, i), (t, j) ∈ E satisfying s ≥ t and i ≥ j, |Φ(s, i, t, j)| = sup r∈R p \{0} |Φ(s, i, t, j)r|/|r| ≤ κ 0 e -λ0(s+i-t-j) . Then, from the definitions of P c and P d below (B.10), P c (t, j) ≤ q M ∞ 0 κ 2 0 e -2λ0s dsI = q M κ 2 0 2λ0 I and P d (t, j) ≤ q M ∞ i=0 κ 2 0 e -2λ0i I = q M κ 2 0 e 2λ 0 e 2λ 0 -1 I. From 13 Since each solution ξ to H0 inherits the hybrid time domain E, it follows that dom ξ = E, and thus Φ(t, j, t ′ , j ′ ) is well defined for all (t, j), (t ′ , j ′ ) ∈ dom ξ, the bounds above and the definition of P in (B.10), we conclude that (B.3) holds with p m , p M as in Lemma B.3.

Next, we show that (B.4) holds. We differentiate P during flows and use that, for each (s, i) ∈ E and each j ∈ N and for almost all t ∈ I j , d dt Φ(s, i, t, j) = Φ(s, i, t, j)A(t, j). This property follows from (B.6) and from the definition of Φ in (B.8). Using the Leibniz integral rule, we obtain that, for each j ∈ N and for almost all t ∈ I j , d dt P c (t, j) = A(t, j) ⊤ P c (t, j)+P c (t, j)A(t, j)-Q(t, j) and d dt P d (t, j) = A(t, j) ⊤ P d (t, j)+P d (t, j)A(t, j). Combining these expressions and using (B.10) and item 1 of Assumption 5.3, we conclude that (B.4) holds.

To conclude the proof, we show that (B.5) holds. We use that, for each (t, j), (s, i) ∈ Υ(E), Φ(s, i, t, j + 1)(I -B(t, j)) = Φ(s, i, t, j). This property follows from (B.7) and the definition of Φ in (B.8). Then, for each (t, j) ∈ Υ(E), (I -B(t, j)) ⊤ P c (t, j + 1)(I -B(t, j)) = J i=j+1 ti+1 max{t,ti} Φ ⊤ (s, i, t, j)Q(s, i)Φ(s, i, t, j)ds. Since the value of ordinary time t is the same immediately before and after each jump, it follows that, for each (t, j) ∈ Υ(E), max{t, t j } = t = t j+1 . Hence, (I -B(t, j)) ⊤ P c (t, j + 1)(I -B(t, j)) = P c (t, j). Next, for each (t, j) ∈ Υ(E), since t = t j+1 at each jump, (I -B(t, j)) ⊤ P d (t, j + 1)(I -B(t, j)) -P d (t, j) = -Φ ⊤ (t, j, t, j)Q(t, j)Φ(t, j, t, j) = -Q(t, j). Combining these expressions and using (B.10) and item 2 of Assumption 5.3, we conclude that (B.5) holds.

We now have all the ingredients to prove Theorem 5.5.

Proof of Theorem 5.5: Since, by Assumptions 5.3 and 5.4, the conditions of Lemma B.3 are satisfied, given q M ≥ q m > 0 and a symmetric matrix function Q : E → R p×p satisfying q m I ≤ Q(t, j) ≤ q M I for all (t, j) ∈ E, there exists a symmetric matrix function P : E → R p×p satisfying (B. (B.12)