Output-Feedback I&I Adaptive Control for Linear Systems with Time-Varying Parameters

This paper combines the so-called congelation of variables method with the adaptive immersion and invariance (I&I) approach to control linear single-input-single-output systems with time-varying parameters via output feedback. The system is first reparameterized using a pair of reduced-order filters and the reparameterization error dynamics show that the input is coupled with a time-varying perturbation. By exploiting the input-to-state stability (ISS) of the inverse dynamics, which is regarded as a counterpart in the time-varying setting of the classical minimum-phase property, the coupling between the input and the time-varying perturbation is transformed into a coupling between the output and another time-varying perturbation that can be dominated in the controller design stage. A pair of high-gain filters are then implemented so that the reparameterization error dynamics are ISS. Finally, output regulation is achieved by strengthened damping design, which invokes a small-gain-like argument from a Lyapunov perspective.


I. INTRODUCTION
Since the 1980s, or even earlier, adaptive control has undergone extensive research (see e.g. [1]- [5]), although only a few works have investigated systems with time-varying parameters. Some pioneering works on adaptive control for time-varying systems (see e.g. [6]) exploit persistence of excitation to guarantee stability by ensuring that parameter estimates converge to the true parameters. Subsequent works (see e.g. [7], [8]) have removed the restriction of persistence of excitation by requiring bounded and slow (in an average sense) parameter variations.
More recent works can be mainly categorized into two trends. One of them is based on the so-called robust adaptive law, see [3], which adopts a switching parameter update law called σ -modification. This approach achieves asymptotic tracking when the parameters are constant, otherwise the tracking error is nonzero and related to the rates of the parameter variations, see [9]. In [10] and [11] the parameter variations are modelled in two parts: known parameter variations and unknown variations, so that the residual tracking error only depends on the rates of the unknown parameter variations. The other trend exploits filtered transformations and projection operation, see [12], [13] and [14]. These methods can guarantee asymptotic tracking provided that the parameters are confined within a compact set, their derivatives are L 1 and the disturbance on the state evolution is additive and L 2 . Moreover, a priori knowledge on parameter variations is not needed and the residual tracking error is independent of the rates of parameter variations.
The methods mentioned above cannot guarantee zeroerror regulation when the unknown parameters are persistently varying. To achieve asymptotic state/output regulation when the time-varying parameters are neither known nor asymptotically constant, in [15] and [16] a method called the congelation of variables has been proposed and developed on the basis of the adaptive backstepping approach and the adaptive immersion and invariance (I&I) approach, respectively. In the spirit of the congelation of variables method each unknown time-varying parameter is treated as a nominal unknown constant parameter perturbed by the difference between the true parameter and the nominal parameter, which causes a time-varying perturbation term. The controller design is then divided into a classical adaptive control design, with constant unknown parameters, and a damping design via dominance to counteract the time-varying perturbation terms. This method is compatible with most adaptive control schemes using parameter estimates, as it does not change the original parameter update law for time-invariant systems.
In the output-feedback control design with the congelation of variables method, the major difficulty is the coupling between the input and the time-varying perturbation, which prevents stabilization via dominance. In [15], this is achieved by forcing the input coefficients to be constants, which eliminates the time-varying perturbation terms. Instead of decoupling the input, [17] manages to transform the coupling between the input and the time-varying perturbation into a coupling between the output and another time-varying perturbation by exploiting a modified minimum-phase property for time-varying systems, which enables the use of the dominance design again. In this paper we follow a similar idea but on the basis of the so-called adaptive I&I approach.

II. SYSTEM REPARAMETERIZATION
We consider an n-dimensional single-input-single-output linear system in observable canonical form with relative 2019 IEEE 58th Conference on Decision and Control (CDC) Palais des Congrès et des Expositions Nice Acropolis Nice, France, December 11-13, 2019 degree r, as described by the equationṡ . . .
x n = −a n (t)x 1 + b n (t)u, or, in compact form, by the equationṡ where x(t) ∈ R n is the state vector, u(t) ∈ R is the input, y(t) ∈ R is the output, S n is the n × n upper-shift matrix, and e 1 = [1, 0, . . . , 0] ∈ R n . The unknown time-varying parameters are denoted by the vector We now introduce some assumptions that are useful for the proof of the main results.

Assumption 1 (Bounded parameters):
The vector of 2n − r + 1 unknown time-varying parameters θ (t) belongs to a compact set Θ, ∀t ≥ 0. This set is also unknown, but we know its size, that is, we know where ∆ θ = θ − θ , with θ ∈ Θ a constant parameter to be selected.
Since θ is typically defined as a box and θ is typically selected as the center of the box, δ ∆ θ can be interpreted as the "radius" of the compact set Θ. The "≥" sign indicates that the knowledge of Θ can be imperfect and conservative, but in this paper we only consider the case in which max θ ∈Θ |∆ θ | = sup t≥0 |∆ θ | and always refer to the latter.
This assumption is only needed when implementing a change of coordinates (see Algorithm 1 and Algorithm 2) to guarantee boundedness of (vectors) of time-varying coefficients.
Assumption 3: |b r (t)| ≥ ε > 0 and the sign of b r (t), i.e. sgn(b r (t)) is known and does not change.
In the spirit of the congelation of variables method the system parameters θ (t) = θ + ∆ θ (t) can be regarded as the sum of a vector of constant "congealed " parameters θ of a nominal time-invariant system and a vector of differences ∆ θ (t) between the actual time-varying parameters and the "congealed" parameters. Therefore we first design a set of filters for system (1) with the nominal constant parameters θ = [ b r , . . . , b n , a 1 , . . . , a n ] . Note that its input-output relation is described by the differential equation Since the time derivatives of u and y are not available for measurement, we apply a stable filter Λ(s) = s n−1 + λ n−1 s n−2 + · · · + λ 2 s + λ 1 to both sides of (3), which yields Noting that s n−1 Consider the state-space realization of the filters described byζ where (5) can then be written aṡ where This is also the parameterization used in [5] (Section 4.4.1). For linear time-invariant systems considered in classical adaptive schemes, η 0 is an exponentially decaying error term because of the Hurwitz property of A λ and it is typically ignored in analysis and design. However, as shown in what follows, when θ (t) is time-varying the perturbation terms coupled with y and with u appear due to the substitution of θ for θ (t). Rearranging (7) yields Define η 1 = η 0 + ∆ a 1 y to separate the perturbation term −∆ a 1 y from the expression of η 1 : this allows avoiding differentiating unknown time-varying parameters when deriving the error dynamics and yieldṡ Define η 2 =η 1 + (−λ e n−1 ∆ a 1 + ∆ a 2 )y, which yieldṡ Repeating the procedures above for η 3 , . . . , η r−2 and then defining η r−1 =η r−2 + (−λ A r−3 λ e n−1 ∆ a 1 − · · · − λ e n−1 ∆ a r−2 + ∆ a r−1 )y yieldṡ Repeating again the procedures for η r . . . , η n−2 and, finally, defining Theorem 1: The reparameterization error η 0 is described by the systeṁ where η = [η 1 , . . . , η n−1 ] and Remark 1: The filters (6) are reduced-order filters (n − 1 state variables each) because we directly reparameterize the expression ofẏ instead of the expression of y.
Due to the perturbation terms coupled with y and with u we cannot directly determine the stability of the dynamics of η and ignore this term, like in classical adaptive control schemes, even if A λ is Hurwitz. The coupling with y can be dominated by a strengthened filter and controller design, as shown in [15], while the coupling with u is merely avoided by assuming a constant vector of input coefficients b, i.e. ∆ b (t) = 0, ∀t ≥ 0. This restriction can be removed by exploiting a minimum-phase property, which merges the coupling with u into the coupling with y.

III. INVERSE DYNAMICS
In this section we adapt some results of [17], which have been developed for a more general class of nonlinear systems, to the present context. Consider the inverse dynamics of system (1), i.e. assuming the system is driven by y and its time-derivatives instead of u, which yields . . .
We now perform a change of coordinates to eliminate the time derivatives of y, which are not desirable in the design and the analysis. To this end, note that the identities 1 hold for any pair of smooth signals s 1 (t) and s 2 (t). Using (18) and defining the new coordinatē which does not contain time derivatives of y. In the spirit of the above procedure, we proceed with the change of coordinates using Algorithm 1 (see next page), which yields the inverse dynamics in the new coordinates described by the equationṡ b r , . . . , b n b r ] ∈ R n−r , and S n−r is the (n−r)×(n−r) uppershift matrix. b y (t), a y (i) (t) are unknown due to the unknown θ (t), but bounded ∀t ≥ 0 due to Assumption 2.
Assumption 4 (Strong minimum-phase property): System (1) has a strong minimum-phase property in the sense that the inverse dynamics (21) are input-to-state stable (ISS) with respect to the input y. Moreover, there exists an ISS Lyapunov function Vx(x) = γ 2 xx Pxx, with a constant Px = P x 0 and the time derivative of Vx along the trajectories of the inverse dynamics satisfies the inequalitẏ Algorithm 1 Change of coordinates x r+1 , . . . , x n . Input: x r+1 , . . . , x n ,ẋ r+1 , . . . ,ẋ n .

4:
Rewrite x i in terms ofx i in the expression ofẋ i−1 and leave the feedback term − b i b r x r+1 unchanged.
Update the old coordinates before the next iteration. 9: end while where ε b y > 0 is constant and δ b y = sup t≥0 |b y (t)|.

Remark 2:
Due to the linearity of the inverse dynamics (21), the exponential stability of the origin of the zero dynamics (which is (21) with y ≡ 0) is equivalent to the ISS of the inverse dynamics. For linear time-invariant systems the ISS property can be equivalently replaced by the condition that A b is Hurwitz, which is the typical assumption made in classical adaptive control schemes.

IV. FILTER DESIGN
In the previous section we have shown that the control input u can be equivalently written in terms of the inverse dynamics state variablex ρ+1 and the time derivatives of y of order up to r. Since the time derivatives of y is not desirable in the design, in this section we continue to exploit the lowpass characteristics of the dynamics of the reparameterization error η to eliminate the time derivatives of y. Substituting (22) into (13) yieldṡ , which is unknown, but bounded due to Assumption 1. It should be noted that due to the structure of B λ , the firstr − 1 elements of∆ b (t) are 0, or equivalently, u is separated from η 1 (or η 0 ) byr integrators, wherer is defined bȳ To guarantee that no time derivative of y appears in η 0 , at least r integrators between u and η 1 are required, as u Algorithm 2 Change of coordinates η 1 , . . . , η n−1 . Input: η 1 , . . . , η n−1 ,η 1 , . . . ,η n−1 . Output:η 1 , . . . ,η n−1 ,η 1 , . . . ,η n−1 . 1: while time derivatives of y appear in the expression oḟ η 1 , . . . ,η n−1 do This while-loop should only iterate for once if Assumption 5 is satisfied. 2: for i = n − 1 → 2 do 3: Updateη i andη i using (18). 4: Rewrite η i in terms ofη i in the expression oḟ η i−1 .
Remark 3: The restriction of Assumption 5 is only related to the relative degree of the original system (1) and ∆ b (t) yet independent of whether we use reduced-order filters or full-order filters (as is implemented in [17]). Using filters of different order can only provide a different non-minimal realization of the system while cannot change the properties related to the relative degree.
Similarly to what is implemented in the previous section we use a change of coordinates to eliminate the time derivatives of y. Applying Algorithm 2 yields the dynamics (24) in the new coordinateṡ where, by Assumption 2,b y (t) andā y (t) are unknown but bounded. Proposition 1: The reparameterization error dynamics (26) are ISS with respect to the inputsx ρ+1 and y if the vector of filter gains is given by λ = 1 2 e n−1 Pη , Pη = P η 0, and Pη satisfies the Riccati inequality S n−1 Pη + Pη S n−1 − Pη e n−1 e n−1 Pη + Qη 0, where Qη = 1 ε Pη∆ b > 0 and ε Pηb y > 0. Moreover, there exists an ISS Lyapunov function Vη (η) =η Pηη and the time derivative of Vη along the trajectories of the reparameterization error dynamics (26) satisfies the inequalitẏ where δ Pηb y = sup t≥0 |Pηb y (t)| and δ Pη∆ b = sup t≥0 |Pη∆ b (t)|.
V. CONTROLLER DESIGN To proceed with the controller design we first rewrite the reparameterization error η 0 in terms of the new coordinate η 1 , which yields whereā y (t) is unknown, but bounded due to Assumption 2.
As discussed in Section II, the input-output relation can be reparameterized using (7) and the filters (6). For the sake of implementing an I&I controller conveniently, we rewrite the reparameterized system in the equivalent form 2 where In the spirit of reduced-order observer design, the adaptive I&I approach adopts a dynamic (integral) parameter estimatê ϑ and a static (proportional) parameter estimate β together for parameter estimation, that is,θ +β is used for estimating ϑ . Define now the virtual control laws and the actual control law with the virtual control errorsν i = ν i − ν * i , i = 1, . . . , r − 1, and the update law for the dynamic parameter estimatė 2 We only discuss the case in which the relative degree r ≥ 2, since when r = 1 the results can be easily obtained without backstepping techniques.
where β = [β 1 (y, d), β 2 (y,θ 1 , d), β 3 (y,θ 1 ,θ 2 , d)] is the static parameter estimate, with γ 1 > 0, γ 2 > 0, γ 3 > 0 constants and σ y , σ 1 , . . . , σ m damping terms to be defined. Remark 4: β 1 is independent ofθ , β 2 is only dependent ofθ 1 , and β 3 is only dependent ofθ 1 ,θ 2 . Therefore ∂ β ∂θ is a lower triangular matrix with all-zero diagonal terms and Proposition 2: Consider the system (1), the filters specified by Proposition 1 and the adaptive controller described by (33)-(40) with the damping terms where c y > 0, c 1 > 0, . . . , c r−1 > 0, ε > 0,ε > ε Pη∆ b δ Pη∆ b . Then all signals in the overall closed-loop system are globally uniformly bounded and lim t→∞ y(t) = 0. Remark 5: In Section 4.2.4 of [5] stability properties of the same system, yet with constant parameters, are discussed. It is shown that when the parameters are constants, the stable dynamics of the reparameterization error η can be directly ignored, and the dynamics of the parameter estimation error z satisfies Φ z(t) ∈ L 2 . The controller-plant subsystem is cascaded with the parameter estimator, driven by Φ z. However, when there are time-varying parameters in the system, the y-ν-dynamics, the z-dynamics, theη-dynamics, and thex-dynamics are all coupled together with loops (instead of cascaded relations), which requires a small-gain like argument (in this paper, from a Lyapunov perspective) to establish the stability of the overall closed-loop system. This comparison reveals the challenges emerging in adaptive control problems when the parameters are time-varying.

VII. CONCLUSIONS AND FUTURE WORK
This paper discusses the combination of the outputfeedback adaptive I&I approach with the congelation of variables method to solve the adaptive output regulation problem when time-varying parameters exist. The system is reparameterized using reduced-order filters, the reparameterization error dynamics are derived and discussed. The coupling between the input and a time-varying perturbation is replaced by the coupling between the output and another time-varying perturbation by exploiting a so-called strong minimum-phase property, which is defined as ISS of the inverse dynamics of the system. This definition is an extension and modification of the classical minimum-phase property for establishing stability of time-varying systems. High-gain filters are then implemented to guarantee ISS of the reparameterization error dynamics. Finally an adaptive I&I controller is designed by synthesizing small-gain-like loops among the controllerplant dynamics, parameter estimation error dynamics, reparameterization error dynamics, and the inverse dynamics, which guarantees output regulation and boundedness of all signals. The simulations show performance improvement on the classical design under parameter variations.
Future work can be carried out to further relax Assumption 5, which is still an obstacle for the proposed method to be implemented for arbitrary linear single-input-singleoutput systems with time-varying parameters.