parameter estimator that uses the state and perturbation esti- mates to .... ified high-gain observer, and then the parameter estimator. ..... projection [2, App. A].
Observer Design and Parameter Estimation for Linear Systems with Nonlinearly Parameterized Perturbations H˚avard Fjær Grip⋆
Ali Saberi†
Abstract— We introduce a method of observer design for systems described by a linear part with a nonlinear perturbation, where the perturbation is parameterized by a vector of unknown, constant parameters. The design is modular, and consists of a modified high-gain observer that estimates the states of the system together with the full perturbation, and a parameter estimator. The parameter estimator is constructed by the designer to identify the unknown parameters, by dynamically inverting a nonlinear equation. We apply the method to observer design for a DC motor with friction modeled by the dynamic LuGre friction model, estimating the internal state in the friction model and an unknown parameter representing Coloumb friction.
I. I NTRODUCTION A common problem in model-based estimation and control is the handling of unknown perturbations to system equations. Such perturbations can be the result of external disturbances or internal plant changes, such as a configuration change, system fault, or changes in physical plant characteristics. In this article, we consider observer design for systems that can be described by a linear part with a nonlinear time-varying perturbation that is parameterized by a vector of unknown, constant parameters. Our design extends the results of [1], [2], where parameter identification for systems with full state measurement was considered. In literature, many of the results that deal with the handling of unknown perturbations are based either on high-gain designs, where the perturbations are suppressed by using a sufficiently high gain, or adaptive designs, where online estimates of unknown parameters are used to cancel the perturbations. Many of the results on high-gain observer design for nonlinear systems rely on placing the system in some variation of the normal form, where it is represented as a chain of integrators, possibly with a separate subsystem representing zero dynamics. This includes, for example, [3], [4], where the system is transformed by repeated differentiation of the output, so that all the nonlinearities occur in the highest-order derivative; and [5], where nonlinearities are allowed at each stage of the integrator chain, but with upper and lower linear bounds on growth rates. An alternative to the normal form is the special coordinate basis (SCB) introduced in [6]. The SCB is a structural transformation that can be applied to any linear multiple-input multipleoutput (MIMO) system, which divides the system into distinct This research is supported by the Research Council of Norway. of Engineering Cybernetics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway. † Department of Electrical Engineering and Computer Science, Washington State University, Pullman, WA 99163. ⋆ Department
Tor A. Johansen⋆
subsystems that reflect its internal workings. The SCB is particularly well-suited for high-gain design, as shown in [7], [8]. While designs based on the normal form are limited to square-invertible, minimum-phase systems of uniform relative degree, [8] shows that high-gain observers can be designed for more general systems that are left-invertible and minimum-phase with respect to a nonlinearity. In each of the above cases, the high-gain designs rely on Lipschitz-like conditions on the nonlinearities, which must hold at least locally in some region of interest. In the presence of unknown perturbations, high-gain observer designs can be extended to estimate the perturbation as well, by considering the perturbation as an additional state. The perturbation estimate can be used for control purposes, for example to cancel the actual perturbation. This type of approach has been adopted recently in the context of performance recovery, in [9], [10], where the perturbation appears in the first-order derivatives, and in [11], where it appears at the end of an integrator chain in a single-input single-output (SISO) system. When extending the state to include the perturbation, Lipschitz-like conditions must be imposed on the derivative of the perturbation. Such conditions may often fail to hold; in particular, the derivative may contain non-vanishing, time-varying components that cannot be removed perfectly, but must instead be attenuated by using a sufficiently high gain. A strength of the high-gain designs mentioned above is that little structural information about the perturbation is needed for implementation. Conversely, however, they do not exploit structural information, even when such information is available. Adaptive techniques are based on online estimation of unknown parameters that can be considered constant or slowly-varying. Most of the adaptive literature deals with systems that are linear with respect to the unknown parameters (see, e.g., [12], [13]). Ways of handling nonlinear parameterizations include the extended Kalman filter (see [14]), various methods for nonlinearities with convex or concave parameterizations ([15]–[18]), and methods for firstorder systems with fractional parameterizations ([19]–[21]). Adaptive techniques related to our work include [22]–[24]. In [22], perturbations with matrix fractional parameterizations are considered, by introducing an auxiliary estimate of the full perturbation that is used in estimation of the unknown parameters. In [23], adaptive update laws for a class of nonlinearly parameterized perturbations in the first derivatives are designed, by first creating virtual algorithms based on information from the derivative of the measurement, and then transforming these into implementable form without explicit
differentiation of the measurements. In [24], adaptive control for linear parameterizations is designed for the case of fullstate measurement, and then extended to the case when the full state is not measured, by using high-gain estimates of error variables. In this paper, we adopt a modular approach, consisting of two separate but connected modules. One module is a modified high-gain observer that estimates the states of the system and the unknown perturbation. The other module is a parameter estimator that uses the state and perturbation estimates to find the unknown parameters. In turn, the parameter estimates are fed back to the modified high-gain observer. This approach allows us to exploit structural information about the perturbation, and at the same time, it allows us to handle perturbations with nonlinear parameterizations, occuring in higher-order derivatives of the output. By using structural information, we can in many cases deal in an asymptotically exact fashion with perturbations containing non-vanishing, time-varying components, rather than having to suppress such disturbances using a high gain. The price we pay is that the method is restricted to a limited class of perturbations that satisfy persistency-of-excitation requirements allowing for exponentially stable estimation of the unknown parameters. We base our design on the SCB rather than the normal form, enabling us to deal comfortably with a large class of MIMO systems. A. Preliminaries For a set of vectors z1 , . . . , zn , we denote by col(z1 , . . . , zn ) the column vector obtained by stacking the elements of z1 , . . . , zn . The operator k · k denotes the Euclidian norm for vectors and the induced Euclidian norm for matrices. For a symmetric positive-semidefinite matrix P, the maximum and minimum eigenvalues are denoted λmax (P) and λmin (P). For a set E ⊂ Rn , we write (E − E) := {z1 − z2 ∈ Rn | z1 , z2 ∈ E}. When considering systems of the form z˙ = F(t, z), we assume that F : R≥0 × Rn → Rn is piecewise continuous in t and locally Lipschitz continuous in z, uniformly in t, on R≥0 × Rn . The solution of this system, initialized at time t0 with initial condition z(t0 ) is denoted z(t). We shall often simplify notation by omitting function arguments. II. P ROBLEM F ORMULATION We consider systems of the following type: x˙ = Ax + Bu + E φ , y = Cx, y ∈ Rr ,
n
x∈R ,
m
u∈R ,
this observer design are most easily overcome if the time derivative u˙ is well-defined and piecewise continuous, and x, u, u˙ and θ are known a priori to belong to compact sets. We shall therefore make this assumption throughout the paper. In most estimation problems, the restriction of the variables to compact sets is is reasonable, because the states and inputs are typically derived from physical quantities with natural bounds. When designing update laws for parameter estimates, we also assume that a parameter projection can be implemented as described in [12], restricting the parameter estimates to a compact, convex set Θ ⊂ R p , defined slightly larger than the set of possible parameter values. The parameter projection is denoted Proj( · ). We denote by X ⊂ Rn , U ⊂ Rm , and U ′ ⊂ Rm the compact sets to which x, u, and u˙ belong, and by Φ the compact image of U × X × Θ under g. For ease of notation, we define
∂g ∂g (u, x, θ )u˙ + (u, x, θ )(Ax + Bu + E φ ), ∂u ∂x representing the time-derivative of the perturbation φ . Assumption 1: The triple (A,C, E) is left-invertible and minimum-phase. Assumption 2: There exists a number β > 0 such that for ˆ θˆ , φˆ ) ∈ all (u, u, ˙ x, θ , φ ) ∈ U ×U ′ × X × Θ × Φ and for all (x, n k ˆ ˆ R × Θ × R , kd(u, u, ˙ x, θ , φ ) − d(u, u, ˙ x, ˆ θ , φ )k ≤ β kcol(x − x, ˆ θ − θˆ , φ − φˆ )k. Remark 1: In Assumption 2, we make a Lipschitz-like assumption about d, which is global in the sense that there is no upper bound on xˆ and φˆ . While this may appear restrictive, note that we are free to redefine g(u, x, θ ) as we please outside U ×X ×Θ without altering the accuracy of the system description (1). This includes introducing a smooth saturation on x outside X. With such a modification, the Lipschitz-like assumption can be satisfied by requiring that g(u, x, θ ) is sufficiently smooth. In the following sections, we shall first present the modified high-gain observer, and then the parameter estimator. While the modified high-gain observer has a fixed structure, the parameter estimator can be designed in a number of different ways to solve the equation φ = g(u, x, θ ) with respect to θ . In Section IV, we discuss several ways to design the parameter estimator. d(u, u, ˙ x, θ , φ ) :=
III. M ODIFIED H IGH -G AIN O BSERVER
φ ∈R , k
(1a) (1b)
where x is the state; y is the output; φ is a perturbation to the system equations; and u is a time-varying input that is well-defined for all t ∈ R≥0 and may include control inputs, references, measured disturbances or other known influences. The perturbation is given by the expression φ = g(u, x, θ ), where g : Rm × Rn × R p → Rk is differentiable in all the arguments, and θ ∈ R p is a vector of constant, unknown parameters. We shall design an observer to estimate both the state x and the unknown parameter vector θ . The technicalities of
Our goal is to estimate both the state x and the perturbation φ . To accomplish this goal we extend the system by introducing φ as a state. The extended system then becomes x˙ 0 A E x B u + d(u, u, ˙ x, θ , φ ), = + (2) Ik 0 0 φ 0 φ˙ with y = Cx. We shall implement an observer for this system on the following form: ˆ x˙ˆ = Axˆ + Bu + E φˆ + Kx (ε )(y −Cx), ∂g ∂ g z˙ = − (u, x, ˆ θˆ )θ˙ˆ − (u, x, ˆ ˆ θˆ )Kx (ε )(y −Cx) ∂θ ∂x + Kφ (ε )(y −Cx), ˆ
(3a)
(3b)
φˆ = g(u, x, ˆ θˆ ) + z.
(3c)
where Kx (ε ) ∈ Rn×r and Kφ (ε ) ∈ Rk×r are gain matrices parameterized by a number ε ≤ 1, to be specified later. In (3), we have made use of a parameter estimate θˆ and its time derivative θ˙ˆ . These values are produced by the parameter estimation module, which we will discuss in Section IV. The perturbation estimate is seen to consist of g(u, x, ˆ θˆ ) plus an internal variable z. The variable z can be viewed as an adjustment made to g(u, x, ˆ θˆ ) to procude a more accurate estimate of the perturbation. Taking the time-derivative of φˆ yields φ˙ˆ = d(u, u, ˙ x, ˆ θˆ , φˆ ) + Kφ (ε )(y − Cx). ˜ Defining the errors x˜ := x − xˆ and φ˜ := φ − φˆ , we may therefore write the error dynamics of the observer as x˙˜ A E x˜ 0 ˜ Kx (ε ) = + d − Cx, ˜ (4) 0 0 φ˜ Ik Kφ (ε ) φ˙˜ where d˜ := d(u, u, ˙ x, θ , φ ) − d(u, u, ˙ x, ˆ θˆ , φˆ ). In the error dy˜ namics (4), the term d acts as an unknown disturbance. Our goal is to design a family of gains K(ε ) such that, as the number ε becomes small, the modified high-gain observer produces stable estimates with a diminishing effect from the parameter error θ˜ := θ − θˆ . We write K(ε ) = [KxT (ε ), KφT (ε )]T . Lemma 1: The error dynamics (4) with the gain K(ε ) = 0 is minimum-phase and left-invertible with d˜ as the input and Cx˜ as the output. Proof: Let the transfer matrix from φ to y in system (1) be denoted G(s). Then the transfer matrix from d˜ to Cx˜ in (4) with K(ε ) = 0 is G(s)/s. The matrix G(s)/s loses rank for precisely the same values of s as G(s); hence, the zeros of (4) are the same as for (1). It follows that G(s)/s is minimum-phase. Since both G(s) and Ik /s are left-invertible, the cascade G(s)/s is also left-invertible.
A. SISO Systems Consider first the error dynamics (4) with K(ε ) = 0. Let na denote the number of zeros in the system, and nq = n − na the relative degree of the system. Let Λ1 , Λ2 , and Λ3 denote the nonsingular state, output, and input transformations that take the system (4) with K(ε ) = 0, and with Cx˜ considered the output, to an SCB representation. We apply these transformations to the system (4) (without setting K(ε ) = 0), by writing col(x, ˜ φ˜ ) = Λ1 χ , Cx˜ = Λ2 γ , and d˜ = Λ3 δ . Based on [8, Theorem 2.2] and Lemma 1, the transformed state vector χ is partitioned as χ = col(χa , χq ), where χa has dimension na and χq has dimension nq , and the resulting system is written as
χ˙ a = Aa χa + Laq γ − Ka γ , χ˙ q = Aq χq + Bq (δ + Da χa + Dq χq ) − Kq (ε )γ , γ = Cq χq ,
(5a) (5b) (5c)
where Bq = [0, . . . , 0, 1]T , Cq = [1, 0, . . . , 0], and 0 1 ··· 0 .. .. . . . . .. Aq = . . . 0 0 · · · 1 0 0 ··· 0
(6)
The matrices Ka and Kq (ε ) in (5) are observer gains, related to the gain K(ε ) by K(ε ) = Λ1 [KaT , KqT (ε )]T Λ−1 2 . Once we have chosen Ka and Kq (ε ), we can therefore implement the modified high-gain observer (3). 1) Discussion of SCB Structure: In (5), the χa subsystem represents the zero dynamics of (4). In particular, the eigenvalues of the matrix Aa correspond to the invariant zeros of (4). Since the system is minimum-phase, this implies that Aa is Hurwitz. The χq subsystem represents the infinite zero structure of (4), and consists of a single chain of integrators, from the input δ to the output γ , with an interconnection to the zero dynamics at the lowest level of the integrator chain. 2) Design of SISO Gains: To design the observer gains, let K¯ q := [K¯ q1 , . . . , K¯ qnq ]T be chosen such that the matrix H := Aq − K¯ qCq is Hurwitz. Because of the special structure of Aq , this is always possible by using regular poleplacement techniques. Then, let Ka = Laq and Kq (ε ) = col(K¯ q1 /ε , . . . , K¯ qnq /ε nq ).
B. MIMO Systems Consider again the error dynamics (4) with K(ε ) = 0. Let na and nq denote the number of invariant zeros and infinite zeros in the system, respectively, and define nb = n − na − nq . Let Λ1 , Λ2 , and Λ3 denote the nonsingular state, output, and input transformations that take the system (4) with K(ε ) = 0 to an SCB representation. We apply these transformations to the system (4) (without setting K(ε ) = 0), by writing col(x, ˜ φ˜ ) = Λ1 χ , Cx˜ = Λ2 γ , and d˜ = Λ3 δ . Based on [8, Th. 2.6] and Lemma 1 the transformed state χ can be partitioned as χ = col(χa , χb , χq ), where χa has dimension na , χb has dimension nb , and χq has dimension nq . The subsystem χq can be further partitioned as χq = col(χq1 , . . . , χqm ), where each χq j , j = 1, . . . , m, has dimension nq j . The transformed output is partitioned as γ = col(γq , γb ), where γq has dimension m and is further partitioned as γq = col(γq1 , . . . , γqm ), and γb has dimension rb := r − m. The resulting system can be written as
χ˙ a = Aa χa + Laq γq + Lab γb − Kaq γq − Kab γb , χ˙ b = Ab χb + Lbq γq − Kbq γq − Kbb γb , χ˙ q j = Aq j χq j + Lqq j γq − Kq j (ε )γq + Bq j (δ j + Da j χa + Db j χb + Dq j χq ),
γb = Cb χb ,
γq j = Cq j χq j ,
(7a) (7b) (7c) (7d)
where Aq j , Bq j , and Cq j have the same special structure as Aq , Bq and Cq in Section III-A and where (Ab ,Cb ) is an observable pair. The gains Kaq , Kab , Kbq , Kbb , and Kq j (ε ),
j = 1, . . . , m, are related to K(ε ) by Kaq Kab Kbb Λ−1 K(ε ) = Λ1 Kbq 2 , Kq (ε ) 0nq ×rb
(8)
where Kq (ε ) = [KqT1 (ε ), . . . , KqTm (ε )]T . Once we have chosen Kaq , Kab , Kbq , Kbb , and and Kq (ε ), we can therefore implement the modified high-gain observer (3) with the gains given by (8). 1) Discussion of SCB Structure: In (7), the χa subsystem represents the zero dynamics of (4), just as in the SISO case, with the eigenvalues of Aa corresponding to the invariant zeros of (4). Since the system is minimum-phase, this implies that Aa is Hurwitz. The system (7) has a χb subsystem that is not present in the SISO case. This subsystem represents states that are observable from the output γb , but that are not directly affected by any inputs. As in the SISO case, the χq subsystem represents the infinite zero structure of (4). In the MIMO case, it is divided into m integrator chains, from component j of the input δ to component j of the output γ j , with interconnections to other subsystems at the lowest level of each integrator chain. 2) Design of MIMO Gains: Let Kbb be chosen such that the matrix Ab − KbbCb is Hurwitz. This is always possible and can be carried out using standared pole-placement techniques, since the pair (Ab ,Cb ) is observable. For each j = 1, . . . , m, select K¯ q j := [K¯ q j 1 , . . . , K¯ q j nq j ]T such that the matrix H j := Aq j − K¯ q j Cq j is Hurwitz. Then, let Kaq = Laq , Kab = Lab , Kbq = Lbq , and let Kq j (ε ) be given by ¯ K¯ q j nq Kq 1 0nq ×( j−1) col n¯q −njq j +1 , . . . , ε n¯q j 0nq ×(m− j) +Lqq j , j
ε
j
where n¯ q := max j=1,...,m nq j . Lemma 2: If Assumptions 1 and 2 hold and the gains are chosen according to Section III-A.2 (SISO) or Section III-B.2 (MIMO), then there exists 0 < ε ∗ ≤ 1 such that for all ε ≤ ε ∗ , the error dynamics (4) is ISS with respect to θ˜ . Proof: See Appendix. IV. PARAMETER E STIMATOR As previously mentioned, the goal of the parameter estimation module is to produce an estimate of θ based on the perturbation estimate φˆ and the state estimate x. ˆ For this to work, there needs to exist an update law
θ˙ˆ = uθ (u, x, ˆ φˆ , θˆ ),
(9)
which, in the hypothetical case of perfect state and perturbation estimates (φˆ = φ and xˆ = x), would provide an unbiased asymptotic estimate of θ . This is formally stated by the following assumption on the dynamics of the error variable θ˜ = θ − θˆ . Assumption 3: There exist a continuously differentiable function Vu : R≥0 × (Θ − Θ) → R≥0 and positive constants a1 , . . . , a4 such that for all (t, θ˜ ) ∈ R≥0 × (Θ − Θ), a1 kθ˜ k2 ≤ Vu (t, θ˜ ) ≤ a2 kθ˜ k2 ,
(10)
∂ Vu ˜ ∂ Vu ˜ (t, θ )uθ (u(t), x(t), φ (t), θˆ ) ≤ −a3 kθ˜ k2 , (11) (t, θ ) − ∂t ∂ θ˜
∂ Vu
˜ ˜ (12)
∂ θ˜ (t, θ ) ≤ a4 kθ k.
Furthermore, the update law (9) ensures that if θˆ (t0 ) ∈ Θ, then for all t ≥ t0 , θˆ (t) ∈ Θ. Satisfying Assumption 3 constitutes the greatest challenge in applying the method in this paper, and this is therefore discussed in detail in the next section. A. Satisfying Assumption 3 Assumption 3 guarantees that the origin of the error dynamics θ˙˜ = −uθ (u, x, ˆ φˆ , θ − θ˜ ), (13)
is uniformly exponentially stable with (Θ − Θ) contained in the region of attraction, whenever φˆ = φ and xˆ = x. Essentially, this amounts to asymptotically solving the inversion problem of finding θ given φ = g(u, x, θ ). In the following, we shall discuss some possibilities for how to satisfy Assumption 3. As a useful reference, we point to [25], which deals with the use of state observers for inversion of nonlinear maps. The material in this section is a straightforward adaptation of [2, Sec. 3.2], which also contains examples of each of the propositions below applied to particular perturbations. The most obvious way to satisfy Assumption 3 is to invert the equality φ = g(u, x, θ ) algebraically, and to let θˆ be attracted to this solution. This may be possible the whole time (Proposition 1), or just part of the time (Proposition 2). Proposition 1: Suppose that for all (u, x, φ ) ∈ U × Rn × k R , we can find a unique solution for θ from the equation φ = g(u, x, θ ). Then Assumption 3 is satisfied with the update law uθ (u, x, ˆ φˆ ) − θˆ )), where ˆ φˆ , θˆ ) = Proj(Γ(θ ∗ (u, x, θ ∗ (u, x, ˆ φˆ ) denotes the solution of the inversion problem ˆ and Γ is a symmetric positive-definite found from φˆ and x, gain matrix. Proof: The proof follows from using the Lyapunov function Vu = 12 θ˜ T Γ−1 θ˜ when φˆ = φ and xˆ = x. The proofs of the remaining propositions in this section are found in the Appendix. Proposition 2: Suppose that there exists a known, piecewise continuous function l : U × Rn × Rk → [0, 1], and that for all (u, x, φ ) ∈ U × Rn × Rk , l(u, x, φ ) > 0 implies that we can find a unique solution for θ from the equation φ = g(u, x, θ ). Suppose furthermore R that there exist T > 0 and ε > 0 such that for all t ∈ R≥0 , tt+T l(u(τ ), x(τ ), φ (τ )) dτ ≥ ε . Then Assumption 3 is satisfied with the update ˆ φˆ ) − θˆ )), where ˆ φˆ , θˆ ) = Proj(l(u, x, ˆ φˆ )Γ(θ ∗ (u, x, law uθ (u, x, ˆ φˆ ) denotes the solution of the inversion problem θ ∗ (u, x, found from φˆ whenever l(u, x, ˆ φˆ ) > 0, and Γ is a symmetric positive-definite gain matrix. When it is not possible or desirable to solve the inversion problem explicitly, it is often possible to implement the update function as a numerical search for the solutions. Proposition 3: Suppose that there exist a positive-definite matrix P and a function M : U × Rn × Θ → R p×k such that
for all (u, x) ∈ U × Rn , and for all pairs θ1 , θ2 ∈ Θ,
∂g ∂g T (u, x, θ2 )M T (u, x, θ1 )≥2P. (14) (u, x, θ2 )+ ∂θ ∂θ Then Assumption 3 is satisfied with the update law uθ (u, x, ˆ φˆ , θˆ ) = Proj(ΓM(u, x, ˆ θˆ )(φˆ − g(u, x, ˆ θˆ ))), where Γ is a symmetric positive-definite gain matrix. Proposition 3 applies to certain monotonic perturbations for which a solution can be found arbitrarily fast by increasing the gain Γ. In many cases this is not possible, because the inversion problem is singular the whole time or part of the time. The following proposition applies to cases where a solution is only available by using data over longer periods of time, by incorporating a persistency-of-excitation condition. Proposition 4: Suppose that there exists a piecewise continuous function S : U × Rn → S+ (p), where S+ (p) is the cone of p × p positive-semidefinite matrices, and a function M : U × Rn × Θ → R p×k , both bounded for bounded x, such that for all (u, x) ∈ U × Rn and for all pairs θ1 , θ2 ∈ Θ, M(u, x, θ1 )
∂g ∂g T (u, x, θ2 )+ (u, x, θ2 )M T (u, x, θ1 )≥2S(u, x). ∂θ ∂θ (15) Suppose furthermore that there exist numbers T > 0 and R ε > 0 such that for all t ∈ R≥0 , tt+T S(u(τ ), x(τ )) dτ ≥ ε Ip , and that for all (u, x, θ , θˆ ) ∈ U × Rn × Θ × Θ, kg(u, x, θ ) − g(u, x, θˆ )k ≤ Lg (θ˜ T S(u, x)θ˜ )1/2 . Then Assumpˆ φˆ , θˆ ) = tion 3 is satisfied with the update law uθ (u, x, Proj(ΓM(u, x, ˆ θˆ )(φˆ − g(u, x, ˆ θˆ ))), where Γ is a symmetric positive-definite gain matrix. When looking for the function M, a good starting point ˆ θˆ ). This choice makes the is M(u, x, ˆ θˆ ) = [∂ g/∂ θ ]T (u, x, parameter update law into a gradient search in the direction of steepest descent for the function kg(u, x, θ ) − g(u, x, θˆ )k2 , scaled by the gain Γ. This choice of M often works even if it fails to satisfy either of Propositions 3 and 4. In the special case of a linear parameterization, this choice of M always satisfies (15), and the remaining conditions in Proposition 4 coincide with standard persistency-of-excitation conditions for parameter identification in linear adaptive theory (see, e.g., [26, Ch. 5]). M(u, x, θ1 )
V. S TABILITY OF I NTERCONNECTED S YSTEM When connecting the two modules, we need one additional assumption about the parameter update law. Assumption 4: The parameter update law uθ (u, x, ˆ φˆ , θˆ ) is ˆ ˆ Lipschitz continuous in (x, ˆ φ ), uniformly in (u, θ ), on U × Rn × Rk × Θ. Remark 2: The Lipschitz condition in Assumption 4 is a global one, in the sense that xˆ and φˆ are not presumed to be bounded. This may fail to hold in many cases. However, if a local Lipschitz condition holds, then the update law is easily modified to satisfy Assumption 4, by introducing a saturation on xˆ and φˆ outside X and Φ. We also remark that when checking Assumption 4, the projection in the update law may be disregarded, since the Lipschitz property is retained under projection [2, App. A].
Theorem 1: If Assumptions 1–4 hold and the gains are chosen according to Section III-A.2 (for SISO systems) or Section III-B.2 (for MIMO systems), then there exists 0 < ε ∗ ≤ 1 such that for all ε ≤ ε ∗ , the origin of the error dynamics (4), (13) is exponentially stable with Rn+k × (Θ − Θ) contained in the region of attraction. Proof: See Appendix. VI. R E -D ESIGN FOR I MPROVED S TATE E STIMATES The modified high-gain observer is designed to estimate not only the states x, but also the perturbation φ . This configuration may not result in optimal estimates of x. In some cases, less noisy estimates may be obtained by implementing a second observer that does not include a perturbation estimate. Instead, this observer has the form x˙ˆ = Axˆ + Bu + Eg(u, x, ˆ θˆ ) + Kx (ε )(y − Cx), ˆ with θˆ obtained from the first observer. The second observer can be designed using the same high-gain methodology as the modified highgain observer, by transforming the error dynamics x˙˜ = Ax˜ + E g˜ − Kx (ε )Cx˜ to the form (7) and designing gains. In this case, g˜ = g(u, x, θ ) − g(u, x, ˆ θˆ ), rather than d,˜ is the input that is transformed to δ , and we need to impose the same Lipschitz-like assumptions on g as we did on d. The resulting ISS property of the second observer with respect to the parameter error θ˜ , and the boundedness of θ˜ , justifies this type of cascaded design. VII. D ISCUSSION The point of the modified high-gain observer described in Section III is to estimate the states and the perturbation with sufficient accuracy to guarantee stability in closed loop with the parameter estimator. The design methodology in Section III ensures that this is possible by making the parameter ε sufficiently small. As the name implies, this may require a high gain in some cases. Such cases are of little practical interest, because it usually results in unacceptable levels of noise amplification. Often, however, only a low or moderate gain is required. Since the required gain depends on quantities that are difficult to determine, such as Lipschitz constants, the observer is typically tuned by starting with ε = 1, and then decreasing ε in small decrements until a sufficiently stable system is obtained. The high-gain design methodology works by assigning time scales in the state estimates, to make some estimates faster than others. In particular, stability is guaranteed by making the χq j subsystems faster than the χa and χb subsystems (see [8] for details). To obtain sensible gains using this methodology, it is important to start with a well-conditioned problem. In Section VIII, we discuss a problem that is poorly conditioned, due to a very fast mode in the zero dynamics, and how to resolve this issue. VIII. S IMULATION E XAMPLE We consider the example of a DC motor with friction modeled by the LuGre friction model. This example is borrowed from [27], [28]. The model is described by J ω˙ = u − F, where ω is the measured angular velocity,
u is the motor torque, F is the friction torque, and the parameter J = 0.0023 kg/m2 is the motor and load inertia. The friction torque is given by the dynamic LuGre friction model: F = σ0 η + σ1 η˙ + α2 ω , where the internal friction state η is given by η˙ = ω − σ0 η |ω |/ζ (ω ), with ζ (ω ) = α0 + α1 exp(−(ω /ω0 )2 ). The model is parameterized by σ0 = 260.0 Nm/rad, σ1 = 0.6 Nm s/rad, α0 = 0.28 Nm, α1 = 0.05 Nm, α2 = 0.176 Nm s/rad, and ω0 = 0.01 rad/s. We shall assume that these parameters are known, except for the uncertain parameter θ := α0 , which represents static Coloumb friction. To indicate that ζ depends on the unknown parameter, we shall henceforth write ζ (ω , θ ). We assume that θ is known a priori to belong to the range Θ := [0.05 Nm, 1 Nm]. Let us define the perturbation φ = g(ω , η , θ ) := σ0 η |ω |/ζ (ω , θ ). It is then straightforward to confirm that the system with input φ and output ω is minimum-phase and left-invertible, as required by Assumption 1. By extending the state space as described in Section III, we obtain the system J ω˙ = u − (α2 + σ1 )ω − σ0 η + σ1 φ , η˙ = ω − φ , φ˙ = d(u, ω , η , θ ), with y = ω . A technical problem arises due to the presence of |ω | in the perturbation, which causes the Lipschitz-like condition on d in Assumption 2 to fail. To circumvent this, we implement the function g(ωˆ , ηˆ , θˆ ) as σ0 ηˆ |y|/ζ (ωˆ ); that is, we use the absolute value of the output ω instead of the estimate ωˆ . A. Observer Design We start our observer design by creating a modified high-gain observer according to (3). We then obtain error dynamics of the form (4): ˙ 1 ω˜ 0 − J (α2 + σ1 ) − 1J σ0 1J σ1 K (ε ) η˙˜ = ω˜ , 1 0 −1 + 0 d˜− x Kφ (ε ) ˙ ˜ 1 0 0 0 φ where Kx (ε ) and Kφ (ε ) are to be designed. It can be confirmed that the transformations 0 1 0 σ1 − σJ1 0 , Λ = 1, Λ = J , Λ1 = σ0 2 3 2 σ1 J σ −σ α −σ 1 − 0 σ12 2 1 σJ1 1
takes the error dynamics to the form (5):
σ0 σ0 (−σ1 α2 + J σ0 ) χa + γ − Ka γ , σ1 σ13 χ˙ q1 = χq2 − Kq1 (ε )γ , J σ0 − σ1 α2 − σ12 σ0 (−σ1 α2 + J σ0 ) + χ˙ q2 = − χ χq 2 q 1 σ1 J J σ12 σ0 + χa + δ − Kq2 (ε )γ , γ = χq1 . J We may now proceed to design the gains according to Section III-A.2. However, the expression for χ˙ a reveals the existence of a pole in the zero dynamics at −σ0 /σ1 , which evaluates numerically to −433.3. This means that the transfer function from d˜ to ω˜ has an invariant zero located very far into the left-half plane, which makes for a poorly conditioned problem. Because the high-gain design methodology is based χ˙ a = −
on making the χq subsystem even faster than the zero dynamics, a high gain is required just to stabilize the linear part of the system. A better approach is to disregard the zero when designing the gains, by assuming that σ1 /σ0 ≈ 0. Applying this to the original error dynamics, we obtain the simplified design system ˙ 1 ω˜ 0 − J α2 − 1J σ0 0 Kx (ε ) ˜ η˙˜ = 1 ω˜ , 0 −1 + 0 d − Kφ (ε ) ˙ ˜ 1 0 0 0 φ
which is transformed to χ˙ q1 = χq2 − Kq1 (ε )γ , χ˙ q2 = χq3 − Kq2 (ε )γ , χ˙ q3 = − σJ0 χq2 − αJ2 χq3 + δ − Kq3 (ε )γ , γ = χq1 by the transformations 1 0 0 J −α J Λ1 = σ02 − σ0 0 , Λ2 = 1, Λ3 = . σ α2 0 J 1 σ0 σ0
We now design the gains according to Section III-A.2, placing the poles of H = Aq − K¯ qCq at −1, −2, and −3. To design the parameter estimator, we note that whenever φ 6= 0, we have θ = σ0 η |ω |/φ − α1 exp(−(ω /ω0 )2 ). We may therefore apply Proposition 2, by letting θ˙ˆ = Proj(l(φˆ )Γ(θ ∗ (ωˆ , ηˆ , φˆ ) − θˆ )), where θ ∗ represents the algebraic solution θ ∗ (ωˆ , ηˆ , φˆ ) := σ0 ηˆ |y|/φˆ − α1 exp(−(ωˆ /ω0 )2 ), found whenever l(φˆ ) > 0. We define l(φˆ ) as l(φˆ ) = 1 when |φˆ | ≥ 1, and l(φˆ ) = 0 otherwise, and choose the gain Γ = 1. According to Proposition 2, this approach works, assuming that the system is sufficiently excited, meaning that |φ | > 1 is guaranteed to occur some portion of the time. For simplicity, we do not use any projection or saturations in this example. B. Simulation Results We simulate with the output y corrupted by noise. The noisy output can be seen together with the actual angular rate ω in the top plot in Figure 1. Using ε = 1 gives stable estimates, which results in the gains K(ε ) ≈ [6, 0, 6]T . The resulting estimates of ω and η are somewhat noisy; not primarily due to the injection term, but due to the use of the noisy output y in constructing g(ωˆ , ηˆ , θˆ ). To improve the state estimates, we use the re-design discussed in Section VI, by creating a second observer that uses the parameter estimate θˆ from the first observer. Since in this case, Lipschitz-like conditions must only be placed on g, and not on the derivative d, we can implement g(ωˆ , ηˆ , θˆ ) in the second observer using only the state estimates, and not the noisy output y. We follow the same high-gain methodology to construct the second observer, placing the poles of the matrix H at −1 and −2 and selecting ε = 1. This gives the gains K(ε ) ≈ [3, 0]T . The resulting state estimates can be seen in Figure 1, together with the parameter estimate. A PPENDIX Proof of Lemma 2: This proof is based on the theory of [8]. The proof is stated for the MIMO case, but is valid for SISO systems as a special case. Define ξa := χa , ξb := χb , and ξq := col(ξq1 , . . . , ξqm ), where for each j = 1, . . . , m, ξq j :=
ω and y (rad/s) ω and ωˆ (rad/s) z and zˆ (rad)
θ and θˆ (Nm)
2 0 −2
0
5
10
15
20
25
0 −3 x 10
5
10
15
20
25
0
5
10
15
20
25
5 0 −5 2 0 −2 0.4 0.2 0
0
5
10
15
20
25
Time (s)
Fig. 1. From top: angular velocity ω (solid) and noisy measurement y (dashed); angular velocity ω (solid) and estimate ωˆ (dashed); internal friction state η (solid) and estimate ηˆ (dashed); unknown parameter θ (solid) and parameter estimate θˆ (dashed)
n¯ −n
S j χq j , where S j = ε q q j diag(1, . . . , ε the following equations:
nq j −1
). We then obtain
(16a) ξ˙a = Aa ξa , ξ˙b = (Ab − KbbCb )ξb n¯ q ε ˙ ε ξq j = H j ξq j + Bq j ε (Da j ξa + Db j ξb + δ j + Dq j ξq ), (16b) −1 ), Let P , P , and P , where Dεq j = Dq j diag(S1−1 , . . . , Sm a qj b j = 1, . . . , m, be symmetric, positive-definite solutions of the Lyapunov equations Aa Pa + Pa AT a = −Qa , (Ab − KbbCb )Pb + Pb (Ab −KbbCb )T = −Qb , and H j Pq j +Pq j H Tj = −Qq j , respectively, for some symmetric, positive-definite matrices Qa , Qb , and Qq j . Define W = ξaT Pa ξa + ξbT Pb ξb + ε ∑mj=1 ξqTj Pq j ξq j . We then have m
W˙ ≤ −ξaT Qa ξa − ξbT Qb ξb − ∑ ξqTj Qq j ξq j + ε n¯q j=1
m
∑ 2kPq j k
j=1
· (kDa j kkξa k + kDb j kkξa k + kδ j k + kDεq j kkξq k)kξq j k. From the Lipschitz-like condition on d from Assumption 2, we know that for each j = 1, . . . , m, there exist constants βa j , βb j , βθ j , and βq j such that kδ j k ≤ βa j kχa k + βb j kχb k + βθ j kθ˜ k + βq j kχq k, which means that kδ j k ≤ βa j kξa k + βb j kξb k + βθ j kθ˜ k + ε −(n¯q −1) βq j kξq k. We furthermore have kDεq j k ≤ ε −(n¯q −1) kDq j k. Let λq = min j=1,...,m λmin (Q j ), and let ρa = ∑mj=1 2kPq j k(kDa j k + βa j ), ρb = ∑mj=1 2kPq j k(kDb j k + βb j ), ρq = ∑mj=1 2kPq j k(kDq j k + βq j ), and ρθ = ∑mj=1 2kPq j kβθ j . Then we may write W˙ ≤ −λmin (Qa )kξa k2 − λmin (Qb )kξb k2 − (λq − ερq )kξq k2 + ε n¯q ρa kξa kkξq k + ε n¯q ρb kξb kkξq k + ε n¯q ρθ kθ˜ kkξq k.
Note that ρq is multiplied by ε relative to λq . Furthermore, note that the cross-terms between kξa k and kξq k, and between kξb k and kξq k, are multiplied by ε n¯q relative to the stabilizing quadratic terms in kξa k2 , kξb k2 and kξq k2 . It is therefore straightforward to show that, by decreasing ε , the cross-terms are dominated, and there exist positive constants ca , cb , and cq such that W˙ ≤ −ca kξa k2 − cb kξb k2 − cq kξq k2 + ε n¯q ρθ kθ˜ kkξq k. This expression shows that W˙ is negative outside a ball around the origin, the size of which is proportional to θ˜ . From [29, Th. 4.19], we conclude that (16) is ISS with respect to θ˜ . Since (16) is obtained through a nonsingular transformation of (4), the same holds for the origin of (4). Proof of Theorem 1: We first note that Assumption 3 ensures that if θˆ (t0 ) ∈ Θ, then for all t ∈ R≥0 , θˆ (t) ∈ Θ. Hence, any trajectory of (4), (13) originating in Rn+k × (Θ − Θ) remains on that set for all future time. In the remainder of this proof, we shall deal with the transformed system (16) from the proof of Lemma 2, and prove that this system, together with (13), is exponentially stable with Rn+k × (Θ − Θ) contained in the region of attraction. This implies that the theorem holds with respect to the error dynamics (4), (13). By following the same argument as for kδ j k in the proof of Lemma 2, based on Assumption (4), we know that there exist positive constants β¯a , β¯b , β¯q such that kuθ (u, x, φ , θˆ ) − ˆ φˆ , θˆ )k ≤ β¯a kξa k + β¯b kξb k + ε −(n¯q −1) β¯q kξq k. Define uθ (u, x, V := W + ε 2n¯q −1Vu , where W is from the proof of Lemma 2. We then obtain V˙ ≤ −ca kξa k2 − cb kξb k2 − cq kξq k2 − a3 ε 2n¯q −1 kθ˜ k2 + ε n¯q [(ρθ +a4 β¯q )kξq kkθ˜ k+a4 β¯a kξa kkθ˜ k+a4 β¯b kξb kkθ˜ k]. To show that we may dominate all the cross-terms in the expression above by decreasing ε , note that for arbitrary positive constants α1 , α2 , and α3 , the expression −α1 ζ12 − α2 ε 2n¯q −1 ζ22 + α3 ε n¯q ζ1 ζ2 can be made negative definite by selecting ε < 4α1 α2 /α32 . Since we may split out expressions like the one above for each of the cross-terms (by letting α1 and α2 be small fractions of the negative quadratic terms, and α3 the constant in the cross-term) and make them negative definite by decreasing ε , it is clear that there exists ε ∗ and a constant c > 0 such that for all ε < ε ∗ , V˙ ≤ −c(kξa k2 + kξb k2 + kξq k2 + kθ˜ k2 ). This holds on the set Rn+k × (Θ − Θ). By the comparison lemma [29, Lemma 3.4], we may therefore conclude exponential stability of the origin of (16), (13), and therefore also of (4), (13). Proof of Proposition 2: For ease of notation, we adopt the notation l(t) = l(u(t), x(t),φ (t)). We use the LFC Vu = R 1 ˜T Γ−1 − µ t∞ et−τ I p l(τ ) dτ θ˜ , where µ > 0 is a constant 2θ yet to be specified. We first note that 12 θ˜ T Γ−1 − µ I p θ˜ ≤ Vu ≤ 12 θ˜ T Γ−1 θ˜ . Hence, Vu is positive-definite provided µ < λmin (Γ−1 ). With φˆ = φ and xˆ = x, we get θ˙˜ = −Proj(l(t)Γθ˜ ). Using the property [12, Lemma E.1] that −θ˜ T Γ−1 Proj(τ ) ≤ −θ˜ T Γ−1 τ , we have 1 1 V˙u = µ θ˜ T I p l(t)θ˜ − µ θ˜ T 2 2
Z ∞ t
et−τ I p l(τ )dτ θ˜
Z −1 T ˜ Γ −µ −θ
t
∞
t−τ
e
I p l(τ )dτ Proj(l(t)Γθ˜ )
1 1 ≤ −(1 − µ )l(t)θ˜ T θ˜ − µε e−T θ˜ T θ˜
2
Z ∞2
t− τ
˜ e I p l(τ )dτ + µ kθ˜ k
Proj(l(t)Γθ )
t
√ 1 1 ≤ −(1 − µ − µ κ kΓk)l(t)kθ˜ k2 − µε e−T kθ˜ k2 , (17) 2 2 where κ is the ratio of the largest and smallest eigenvalue of Γ−1 . Above, we have used the property [12, Lemma T −1 T −1 implies that E.1] that Proj( √τ ) Γ Proj(τ ) ≤ τ Γ τ , which R ∞ t−τ kProj( τ )k ≤ κ k τ k. We have also used that e l(τ )dτ ≥ t R R t+T t−τ e l(τ )dτ ≥ e−T tt+T l(τ )dτ ≥ e−T ε . From the calt culation above, we see that the √ time derivative is negative definite provided µ < 1/( 21 + κ kΓk). Proof of Proposition 3: For the sake of brevity, we write M = M(u, x, θˆ ). With φˆ = φ and xˆ = x, we get θ˙˜ = −Proj(ΓM(g(u, x, θ ) − g(u, x, θˆ ))). We use the LFC Vu = 12 θ˜ T Γ−1 θ˜ . Using the property [12, Lemma E.1] that −θ˜ T Γ−1 Proj(τ ) ≤ −θ˜ T Γ−1 τ , we have V˙u ≤ − 12 θ˜ T M(g(u, x, θ ) − g(u, x, θˆ )) − 21 (g(u, x, θ ) − g(u, x, θˆ ))T M T θ˜ . Since g(u, x, θ ) is continuously differentiable with respect to θ , we may write, according to Taylor’s theorem (see, R e.g., [30, Theorem 11.1]), g(u, x, θ ) − g(u, x, θˆ ) = 01 [∂ g/∂ θ ](u, x, θˆ + pθ˜ )θ˜ dp. R Hence, we have V˙u ≤ − 12 01 θ˜ T (M[∂ g/∂ θ ](u, x, θˆ + pθ˜ ) + R [∂ g/∂ θ ](u, x, θˆ + pθ˜ )T M T )θ˜ dp ≤ − 01 θ˜ T Pθ˜ dp = −θ˜ T Pθ˜ , which proves that Assumption 3 holds. Proof of Proposition 4: We use the LFC Vu = R 1 ˜T −1 − µ ∞ et−τ S(u(τ ), x(τ ))dτ θ˜ , where µ > 0 is Γ θ t 2 a constant yet to be specified. First, we confirm that the Lyapunov function Vu is positive-definite. We have 1 1 ′ −1 −1 ˜ 2 ˜ 2 2 (λmin (Γ ) − µλS )kθ k ≤ Vu ≤ 2 λmin (Γ )kθ k , where ′ λS = sup(u,x)∈U×X λmax (S(u, x)). It follows from this that Vu is positive-definite provided λmin (Γ−1 ) − µλS′ > 0, which is guaranteed if µ < λmin (Γ−1 )/λS′ . When we insert φˆ = φ and xˆ = x, we get the same error dynamics as in the proof of Proposition 3. Following a calculation similar to the proof of Proposition 2, we√ get V˙u ≤ −(1 − 21 µ )θ˜ T S(u, x)θ˜ − 12 µε e−T kθ˜ k2 + µ κ MS kΓkMM Lg kθ˜ k(θ˜ T S(u, x)θ˜ )1/2 , where MS and MM are bounds on kS(u, x)k and kM(u, x, θˆ )k respectively, and κ is ratio of the largest and smallest eigenvalue of Γ−1 . We may write this as a quadratic expression with respect to col((θ˜ T S(u, x)θ˜ )1/2 , kθ˜ k]). It is then easily confirmed that the expression is negative definite if µ < 2/(1 + 2 L2 ε −1 eT ). κ MS2 kΓk2 MM g R EFERENCES [1] H. F. Grip, T. A. Johansen, and L. Imsland, “Estimation and control for systems with nonlinearly parameterized perturbations,” in Proc. 17th IFAC World Congress, Seoul, Korea, 2008. [2] H. F. Grip, T. A. Johansen, L. Imsland, and G.-O. Kaasa, “Parameter estimation and compensation in systems with nonlinearly parameterized perturbations,” 2009, provisionally accepted for publication in Automatica. [3] H. Khalil and F. Esfandiari, “Semiglobal stabilization of a class of nonlinear systems using output feedback,” IEEE Trans. Automat. Contr., vol. 38, pp. 1412–1415, 1993.
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