Stability Overlay for Linear and Nonlinear Time-Varying Plants Paulo Rosa, Jeff S. Shamma, Carlos Silvestre and Michael Athans Abstract— This paper proposes a strategy referred to as Stability Overlay (SO) for linear and nonlinear time-varying plants, that provides input/output stability guarantees for a wide set of adaptive control schemes. We use this methodology to endow Multiple-Model Adaptive Control (MMAC) architectures with robust stability properties when the plant to be controlled is uncertain and time-varying. We emphasize that adaptive control strategies should be able to handle timevarying plants, since these methodologies are usually applied to plants with drifting parameters. The results presented herein clearly indicate that multiple-model strategies augmented with the SO for time-varying plants ensure stability of the closedloop for a wide class of disturbances and model uncertainty, while providing high performance capabilities whenever the stabilizing controllers have been designed for performancerobustness.
I. I NTRODUCTION Several adaptive laws for linear and nonlinear, timeinvariant and time-varying plants are available in the literature and are required whenever a single non-adaptive controller is not able to ensure the desired performance or stability requirements. This happens because every physical system can only be known up to some finite bound on the accuracy and, in particular, when the uncertain parameters change with time. Thus, the plant uncertainties impose rigid bounds on the performance (and even stability) capabilities of non-adaptive controllers. Hence, adaptive control strategies arise naturally to tackle this kind of problems. However, most of these adaptive control laws are not robust to a wide class of disturbances and/or plant uncertainty. In fact, some of them can lead to closed-loop instability (cf. [8]) due to unmodeled high-frequency dynamics. Hence, this paper proposes a method referred to as Stability Overlay (SO), extending the ideas in [12], that can also be used with time-varying plants. The method in [12] is also extended to nonlinear plants. The strategy developed herein is based upon the algorithm originally suggested in [1], and assesses the “rewards” received by each controller after its most recent utilization. A controller is then disqualified or not, based on its rewards, in a similar way to what is done in [13], [16], [2] and in the references therein. However, in our approach, the SO is only responsible for the input/output stability of the plant, and thus another algorithm should run in parallel in order to satisfy the posed performance requirements. Notice that, for instance, in [13], the rewards are also used to achieve performance. This strategy, however, P. Rosa, C. Silvestre and M. Athans are with Institute for Systems and Robotics - Instituto Superior Tecnico, Av. Rovisco Pais, 1, 1049-001 Lisboa, Portugal
[email protected],
[email protected],
[email protected] M. Athans is also Professor of EECS (emeritus), M.I.T., USA J. S. Shamma is with Georgia Institute of Technology, School of Electrical and Computer Engineering, Atlanta, Georgia, United States of America
[email protected]
also has its shortcomings, since we do not take advantage of any knowledge regarding the model of the plant, and hence we may have a significant loss in terms of performance. This paper was primarily motivated by the lack of theoretical stability guarantees for many multiple-model adaptive control architectures, when applied to time-varying plants. For instance, control strategies such as the Robust MultipleModel Adaptive Control (RMMAC – see [3], [4], [5], [10], [9] and references therein), the Multiple Model Adaptive Control with Mixing (see [7], [6]), and the Unfalsified Control (see [13]), among others [16], [2], have successfully shown high levels of performance against complex and real uncertainties in the plant, although they can lead the closedloop to instability if the uncertain parameters of the plant vary with time. This article, thus, proposes the use of the SO to ensure input/output stability of this kind of methodologies when applied to time-varying plants. We emphasize that adaptive control strategies should be able to handle timevarying plants, since these methodologies are usually applied to plants with drifting parameters, and hence guaranteeing stability of plants with time-frozen parameters is usually not enough. The applicability of the SO turns out to be very wide, in the sense that it can be used in parallel with several adaptive control laws, as long as a few set of natural assumptions is satisfied, in particular that at least one controller in the set of eligible controllers is able to stabilize the plant. The wellknown Rohrs et al. counterexample [8], for instance, can be tackled using this kind of approach, as shown in [12]. The SO for adaptive control laws, introduced in [12], does not guarantee stability of the closed-loop if the plant is timevarying. In fact, there can happen that the parameters ∆T and l of the algorithm described in [12] increase without bounds, and hence so does the output of the system. Therefore, this article suggests two possible modifications of the algorithm to handle time-varying plants while establishing an upper bound for the rate of variation of the corresponding dynamics. This paper is organized as follows: Section II introduces the SO for time-varying plants; Section III presents an application of the SO with the RMMAC with BMI/LPV controllers [11] (rather than mixed-µ controllers) and shows how one can guarantee the input/output stability of the closed-loop even when the plant is time-varying; finally, in Section IV, some conclusions about the SO for time-varying plants are presented. II. S TABILITY OVERLAY FOR T IME -VARYING P LANTS Consider a time-varying plant described by x˙ = f (x, u, w, ρ), x(0) = x0 , y = g(x, u, w, ρ), P(ρ(t)) := (1) z = h(x, u, w, ρ),
u(t)
Plant Disturbance ξ(t)
Measurement Noise θ(t)
Plant with uncertain model
+
y(t)
K1(.)
Wu (.)
}
z(t)
K2(.) ...
α(t)
Wy (.)
K N (.) Fig. 1. Feedback interconnection between the plant (1) and the controllers, which are selected by the signal α(t). In this case, all the controllers are running in parallel. However, the results in this paper still hold if only one or more controllers are running at a time. The exogenous plant disturbances are represented by ξ(t) and the measurement noise by θ(t).
where x0 ∈ Rn is a bounded and fixed (but unknown) initial condition, w (·) ∈ L∞ is a bounded (but unknown) exogenous disturbance (which may include measurement noise), ρ(·) is a vector of (possibly time-varying) parameters, u (·) is the control input and t ∈ R+ 0 explicitly denotes the time-dependence of the plant dynamics. We also define x(·) as the state of the plant and y(·) and z(·) as the measurement and the performance outputs, respectively. The variable z(·) can include performance outputs such as the ones obtained by filtering the actual plant output and the control input with the weights Wy (s) and Wu (s), respectively – see [17], [15] for more details on using performance weights. Let a finitely switching control input be defined as Kα(t) (y(t)), 0 ≤ t < t0 ; ufs (t) = (2) K∗ (y(t)), t ≥ t0 , where α(t) is a positive integer. N is the number of controllers, and Ki (·), for i ∈ {1, 2, ..., N }, represents a controller. Figure 1 depicts the output feedback interconnection between the plant (1) and the controllers Ki (·), selected through signal α(t)1 . We define |x| as the euclidian norm of x ∈ Rn , and kAk as the induced norm of the matrix A, i.e., kAk = sup |Ax|/|x|. x6=0
We further define, for any constant σ > 0,
σ
z|[t1 ,t2 ] = sup e−σ(t2 −τ ) |z(τ )|, τ ∈[t1 , t2 ]
and z|[t1 ,t2 ] =
sup
|z(τ )|. We are going to suppose
τ ∈[t1 , t2 ]
that the following assumptions are satisfied throughout this paper. Assumption 1: There exist continuous strictly increasing functions σ1 : R+ → R+ and σ2 : R+ → R+ and constant σ > 0, such that, for any finitely switching input and any ∆T > 0, t0 ≥ 0 and any (possibly time-varying) ρ(·),
σ
z|[0,t0 +∆T ] ≤ σ1 (∆T ) z|σ[0,t0 ] + σ2 (∆T ). 1 If all the controllers are stable, they can run in parallel, which usually leads to smaller transients during the switching. However, the results presented in the sequel are independent of this design choice.
Remark 1 Notice that a system complying with Assumption 1 cannot have a finite escape time. Consider that ρ(t) ∈ Ω ∈ Rnρ for all t ≥ 0. The set Ω is the uncertainty region of the vector of parameters, ρ. Let Ωj denote a subset of Ω. Then, we posit the following assumption. nρ Ω⊂ S Assumption 2: Let Ωj ⊂ R , j = 1, 2, ..., ∗N satisfy ∗ Ω . There exist strictly positive constants l , ∆T , ν and j j γ, with γ < 1, such that, for any finitely switching input (2) with K∗ = Kj , if for all t ≥ to , i) ρ(t) ∈ Ωj and ii) |ρ(t)| ˙ ≤ ν, then
σ
z|[0,to +∆T ] ≤ γ z|σ[0,to ] + l∗ , for all ∆T ≥ ∆T ∗ and for σ as in Assumption 1. Remark 2 The value of ν is dictated by each controller and is referred to as the allowable time-rate of variation of the vector of parameters, ρ(·). The following definition is important when taking into account the allowable time-rate of variation of the dynamics of a plant. Definition 1: Suppose that ρ(tA ) ∈ Ωj . We say that the plant dynamics drifted at time instant t = tA if there exists δ ∗ > 0 such that, for every 0 < δ ≤ δ ∗ , we have ρ(tA + δ) ∈ / Ωj . Furthermore, t = tA is referred to as a drifting time instant. Assumption 3: If ρ(t) ∈ Ωj , then there exist t1 and t2 such that 1) |t2 − t1 | ≥ Tmin ; 2) t1 ≤ t ≤ t2 ; 3) ρ(τ ) ∈ Ωj for all τ ∈ [t1 , t2 ]. This last assumption guarantees that the plant dynamics remain in the same “region” for a time interval of at least Tmin . In other words, it means that if tA and tB are drifting time instants with tA 6= tB , then |tA − tB | ≥ Tmin . Remark 3 Assumptions 2–4 are not stringent, since they arise naturally from the control problem at hands. Namely, • Assumption 1 ensures that, even if we select the wrong controller during a finite amount of time, the norm of the output of the plant does not increase to infinity in finite time – this is valid, for instance, for every linear timeinvariant system [12], and for a wide class of nonlinear systems; • Assumption 2 simply states that, at each time instant to , at least one of the eligible controllers must be able to stabilize the plant with parameters ρ(t) ≡ ρ(to ), ∀t≥to ; • Assumption 3 will guarantee that the adaptive control algorithm has enough time to adapt itself to the changes in the plant. The reward after using controller Ki (·) is defined as
(
1, z|σ[0, tn ] ≤ γ z|σ[0,tn−1 ] + l(n) r(n) = (3) 0, otherwise, where σ > 0 is as in Assumptions 1–2, γ is a fixed positive scalar with γ < 1 and l(n) is going to be specified in the sequel.
Initialize S=S0, K(1)=K0, n=1
Use controller K(n) during time-interval D T*
receives a positive reward, we have
σ
z|[0,(i+1)∆T ∗ ] ≤ γ z|σ[0,i∆T ∗ ] + l∗ . Then, if for given integer n∗ , we have Tmin > (2N + n )∆T ∗ , we conclude that
σ
z|[0,to +Tmin ] ≤ a z|σ[0,to ] + b, ∗
r(n)=0?
n
y S=S\K(n)
n S=Æ ? y
S=S0
∗
where a ≤ Γ2N γ n and b is a function of L, l∗ , Γ, γ, N and n∗ , since in every Tmin interval, a correct controller must be used at least n∗ times. Notice that, if a < 1, then
∃z0 ≥0 : z|σ[0,to ] ≤ z0 ⇒ z|σ[0,to +Tmin ] ≤ z0 . ∗
K(n+1)=any controller in S n=n+1
Fig. 2. Stability Overlay (SO) Algorithm #1 for time-varying plants with known ∆T ∗ and l∗ . The notation S = S\K(n) means “the exclusion of element K(n) from set S”. S0 is the initial set of illegible controllers.
A. Stability Overlay for Time-Varying Plants with Parameters ∆T ∗ and l∗ Known The Stability Overlay Algorithm 1, depicted in Fig. 2, for time-varying plants, with known ∆T ∗ and l∗ , is described next. Theorem 1: Under Assumptions 1–3, for sufficiently large Tmin , the Stability Overlay Algorithm #1 for Time
Varying Plants with known ∆T ∗ and l∗ results in z|σ[0,t] bounded. The proof for sufficiently large Tmin is similar to the one in [12]. In the sequel, an upper bound for Tmin is derived. In the algorithm depicted in Fig. 2, we suppose that ∆T ∗ and l∗ are known (these can actually be upper bounds for ∆T ∗ and l∗ , respectively). This assumption is going to be relaxed in subsection II-B. The difference between the algorithms for time-invariant and for time-varying plants, is that in the later we never increase ∆T and l, since we know ∆T ∗ and l∗ , and these are the values used for every controller. Suppose that Tmin > 2N ∆T ∗ . Then, during any timeinterval ∆T such that ∆T ≥ Tmin , at least one controller receives a positive reward, even if the controllers are selected in a non-sequential fashion. To see this, consider that all the rewards we get during that time-interval are zero rewards. Then, all the N controllers have failed in a row, and, when connected again to the loop, they all failed once more. Since the plant dynamics can only drift once during the timeinterval 2N ∆T ∗ (see Definition 1), this is a contradiction. According to Assumption 1, whenever a controller receives a zero reward, we have
σ
z|[0,(i+1)∆T ∗ ] ≤ Γ z|σ[0,i∆T ∗ ] + L, where Γ = σ1 (∆T ∗ ) and L = σ2 (∆T ∗ ). On the other hand, and according to Assumption 2, whenever a controller
Thus, if γ n < Γ−2N the system is input/output stable. Therefore, a sufficient condition for Tmin is Tmin > (2N + logγ Γ−2N )∆T ∗ . This means that, the faster the stabilizing controllers are capable of reducing the norm of the output, i.e., the smaller the value of γ is, the smaller Tmin can be. However, the larger the value of Γ and the larger the number of controllers, N , the larger the interval Tmin must be. Notice that Γ > 1, otherwise the solution is trivial. B. Stability Overlay for Time-Varying Plants with ∆T ∗ and l∗ Unknown l Define z (l, ) = 1−γ
+ , for γ as in ∗Assumption 2 and
σ l , and a controller > 0. Notice that, if z|[0,t] ≥ z ∗ := 1−γ that receives a positive reward (3) is selected, then, for ∆T ≥ ∆T ∗ ,
σ
z|[0,t+∆T ] ≤ z|σ[0,t] .
In other words, if a controller that only receives positive rewards (3) is used long enough, then, for all > 0,
σ
z|[0,t+∆T ] ≤ z ∗ + . Let S0 denote the set of available controllers for the SO. The Stability Overlay Algorithm #2 for Time-Varying Plants with Unknown ∆T ∗ and l∗ is shown in Fig. 3. This algorithm has two differences when compared to the time-invariant case: 1) The sets of eligible controllers, S and Q, are “reset” (S = S0 and Q = S0 ) whenever the discounted norm of the output is below a given threshold, z (l(n), ); 2) A controller can
only
be disqualified if it fails twice
in a row and z|σ[0,t] ≥ z (l(n), ). As explained in the sequel, these modifications guarantee the boundedness of ∆T and l, while keeping the input/output stability of the closed-loop. Theorem 2: Under Assumptions 1–3, for sufficiently large Tmin , the Stability Overlay Algorithm #2 for TimeVarying Plants results in z bounded. The proof of Theorem 2 can be found on the extended version of this paper, available on the Internet.
Initialize S=S0, K(1)=K0, D T(1)=D T0, l(1)=l0, n=1 Use controller K(n) during time-interval D T(n) n
r(n)=0? y n K(n)Î Q? y Q=Q\K(n)
S=S\K(n)
||z|s z(l(n),e )? [0,t]||
n
y S=S0, Q=S0
III. U SING BMI/LPV C ONTROLLERS TO G UARANTEE T IME -VARIATIONS ROBUSTNESS
n S=Æ ? n Q=Æ ?
y
An interpretation of this result is as follows: the larger the number of controllers we have to test, N , the larger the amount of time the plant dynamics must remain in the same region; moreover, if m∗ is small, we can switch to another region sooner than if the time to recover from the use of a wrong controller is large; finally, the faster we can exclude a controller and the faster the good controllers decrease the output norm, the smaller the time the parameters must stay in the same region. This result shows the compromise between performance for time-varying and for time-invariant systems. For instance, it is well known that the performance of the multiple-model adaptive control architectures, for time-invariant plants, increases with the number of eligible non-adaptive controllers. However, this new result shows that, the more controllers we have, the larger the transients are going to be.
y S=S0, Q=S0, K(n+1)=any K(n+1)=any controller in S controller in Q D T(n+1)=D T(n)+D Tinc l(n+1)=l(n)+linc K(n+1)=K0 D T(n+1)=D T(n), l(n+1)=l(n)
n=n+1
Fig. 3. Stability Overlay (SO) Algorithm #2 for time-varying plants with unknown ∆T ∗ and l∗ .
C. Computation of an Upper Bound for Tmin This subsection suggests a method to compute an upper bound Tmin , when the SO Algorithm #3 is
used. Define zi := z|σ[0,i∆T ∗ ] . Let tA , tB and tC be drifting time instants, satisfying Assumption 3, i.e., min {|tA − tB |, |tA − tC |, |tB − tC |} ≥ Tmin . Also, consider ∆T ≥ ∆T ∗ and l ≥ l∗ . It can be shown that
σ
z|[0,t ] ≤ z(l, ), A0
for some tA0 ∈ [tA , tB ]. Hence, similarly to what was derived in subsection II-A, we have
σ
z|[0,tB ] ≤ ΓN −1 z(l, ) + ψ =: z 1 , where N is the number of controllers, ψ is a continuous function of l∗ , L, Γ, γ and m∗ , and where m∗ is the number of time-intervals ∆T ∗ needed for a controller receiving positive rewards (3) to reduce the discounted norm of the output from z 1 to z (l, ). Tmin is some time interval large enough so that the stability of the SO Algorithm #2 is guaranteed. Tmin ≥ tB0 − tB , where tB0 ∈ [tB , tC ] is
Thus,
σ
such that
z|[0,tB0 ] ≤ z(l, ). Therefore, Tmin ≥ (2N + m∗ )∆T ∗ . Notice that this result is similar to that of the SO algorithm #1, except that m∗ is in general larger than n∗ .
The rate of variation of the uncertain dynamics of the plant to be controlled is constrained by 1) Tmin ; 2) the maximum rate of variation allowed by each of the non-adaptive controllers. In what follows, we assume that the plant to be controlled can be described or approximated by a (possibly timevarying) linear model. In reference to Fig. 1, if, for instance, mixed-µ controllers are used, the plant must be time-invariant. Otherwise, the stability of the closed-loop system cannot be guaranteed. Another possible approach is to use BMI/LPV controllers (see [11],[14] and references therein), which are, loosely speaking, output feedback controllers that guarantee the same stability and performance robustness properties of the mixedµ controllers, while ensuring that the closed-loop remains stable even if the rate of time-variation of the plant dynamics are not zero but instead only bounded (by a known constant). This type of controllers can be synthesized using the D-BMI iterations introduced in [11]. The SO Algorithms #1 and #2 do not describe how one should choose the controllers when the set of eligible ones has more than a single element. The appropriate choice of such controllers is going to be responsible for the performance of the closed-loop. As an example, we are going to use the identification part of the robust multiple model adaptive control (RMMAC) methodology. The RMMAC is a multiple model approach that computes and uses the posterior probabilities of the uncertain parameters of the process model being in a specific region to switch or blend the outputs of a set of controllers, each of which designed for a given uncertainty region. The estimation part is done by a bank of Kalman filters (KFs), while for the control part a set of mixed-µ controllers is used. Fig. 4 depicts the switching RMMAC architecture, for the case where N regions are used. It is not our intention to give an in-depth explanation of the RMMAC – the interested reader is referred to [3], [4], [5] – but rather to use the SO to endow the RMMAC with input/output stability capabilities.
Plant disturbances x(t)
Sensor noise q(t)
2
1.8
u(t)
y(t)
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KF #1
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r1(t)
LNARC #1 1.2
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Posterior Probability Evaluator
Switching logic
Select the largest probability
...
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Fig. 6. x2=z
x1 k1 u(t)
e
-st
m1
b1
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m2
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Fig. 5. MSD system with uncertain spring constant, k1 , and disturbances denoted by d(t). u(t) is the control input and z(t) is the system output.
50 40 30 20
We are going to use the mass-spring-dashpot (MSD) testbed example of Fig. 5 to illustrate the use of the SO for a time-varying plant. We stress that this testbed has recently been used with the so-called multiple model adaptive control with mixing (MMACwM) architecture – see [7], [6] – with uncertain time-invariant parameters. Regarding the MSD plant, the unknown parameters are the time-varying spring stiffness k1 ∈ K := [0.25 1.75] N/m and the constant input time-delay 0 < τ < 0.05 s. We point out that this is a very challenging adaptive control problem, since the control input is noncolocated with the performance variable, the position of mass m2 , and that the use of a nonadaptive controller deteriorates significantly the performance of the overall system. The disturbance force d(t) shown in Fig. 5 is a stationary first-order (colored) stochastic process generated by driving a low-pass filter, with transfer function Wd (s), with continuous-time white noise ξ(t), with zero mean and intensity β(t)Ξ, according to α d(s) = ξ(s) = Wd (s)ξ(s). s+α The sensor noise considered is continuous-time white noise θ(t), with zero mean and intensity β(t)Θ. β(·) is a timevarying unknown positive scalar, that is going to be described next. The KFs of the RMMAC/BMI were designed using β(·) = β = 1. This design assumption is going to be violated in the following simulation, in order to evaluate the behavior of the SO. Following the RMMAC synthesis methodology and using the same design choices as the ones described in [4], [5], we obtain N = 4 local non-adaptive robust controllers (LNARCs) – which are mixed-µ controllers in the original RMMAC design – in order to achieve at least 70% of the performance we would have obtained had we known the value of the uncertain parameter, k1 . Then, as explained in [11], the mixed-µ controllers are replaced by BMI/LPV controllers with similar specifications, but assuming nonzero bounds on the rate of variation of the parameter, k1 . In this design, we assume a bound of
10 0
Fig. 7.
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Disturbance and noise intensities factor, β(t).
0.001 (N/m)/s for the slope (time-variation) of the parameter k1 . The state of the plant is augmented by the D-scalings obtained from the µ-analysis, increasing the state up to 16 variables. For the LPV realization, we have adopted a polytopic description for the system, since, for each region, the plant dynamics can be naturally described as the convex combination of the dynamics on the boundaries of that region. The RMMAC/BMI architecture is afterwards modified by the SO for time-varying plants, that serves as a supervisor to guarantee input/output stability of the closed-loop. The SO constrains the set of eligible controllers that the RMMAC/BMI can select, resorting to the control residuals, and should only act if the RMMAC/BMI is persistently choosing controllers that are not capable of stabilizing the plant. Figure 6 depicts the time-evolution of the spring constant considered for simulation purposes. It is important to notice that |k˙ 1 (t)| is within the predefined bounds. Hence, every time-instant, there is at least one controller that is able to stabilize the plant. The disturbance and noise intensities used during the design of the KFs were multiplied by the variable β(t), depicted in Fig. 7. The simulation results are illustrated in Figs. 8 and 9. The standard switching RMMAC is used for comparison purposes. During the first 3000 secs, the switching RMMAC/BMI with and without the SO behave approximately the same way. This happens because β(t) = 1 for t ∈ [0, 3000] secs, which means that the design assumptions of the KFs are not violated (except for the time-variations of the plant), and hence the identification subsystem of the RMMAC/BMI converges to the correct parameter region. Therefore, the SO
3
uals (typically faster than the control residuals) to enhance performance. Therefore, strategies such as the RMMAC/BMI with the SO for time-varying plants ensure stability of the closed-loop for a wide class of disturbances and model uncertainty, while providing high performance capabilities when the process model matches closely the actual plant.
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Fig. 8. Mass 2 position, x2 (t), time-evolution for the closed-loop, using the switching RMMAC/BMI integrated with the Stability Overlay for timevarying plants. Ctrl. #1
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V. ACKNOWLEDGMENTS This work was partially supported by Fundac¸a˜ o para a Ciˆencia e a Tecnologia (FCT), ISR/IST pluriannual funding, through the POS Conhecimento Program that includes FEDER funds, by the PTDC/MAR/64546/2006 OBSERVFLY project, and by the NSF project #ECS-0501394. The work of P. Rosa was supported by a PhD Student Scholarship, SFRH/BD/30470/2006, from the FCT. We wish to thank our colleagues Antonio Pascoal, Pedro Aguiar, Vahid Hassani and Jos´e Vasconcelos for the many discussions on the field of robust adaptive control.
0
Ctrl. #3
1 0.5 0
Ctrl. #4
1 0.5
Sw.RMMAC Sw.RMMAC w/SO
0 0
1000
Fig. 9. Controller selection for the switching RMMAC/BMI with and without the Stability Overlay.
does not interfere in the selection of the controllers, since none of them has failed. The same comments are valid for t ∈ [4200, 6000] secs. For t ∈ [3000, 4200] secs, the KFs do not converge. Therefore, the standard switching RMMAC/BMI selects almost arbitrarily the controllers at each sampling time, as illustrated in Fig. 9. This leads to a severe deterioration in terms of performance of the closed-loop. The SO, however, is able to disqualify controllers #1 and #2 after a few seconds, since they are not being able to stabilize the plant. Hence, we get the performance improvements depicted in Fig. 8. Remark 4 Assumptions 1–3 can easily be verified resorting to arguments similar to the ones used in [1], [12] for LTI systems. In summary, this simulation illustrated the usefulness of the SO for time-varying plants, whenever the assumptions of the high-performance adaptive algorithm (the RMMAC/BMI, in the present case) are not satisfied in practice. IV. C ONCLUSIONS This paper proposed a strategy referred to as Stability Overlay (SO) for Time-Varying (TV) Plants, that provides input/output stability guarantees for a wide set of adaptive control schemes. As an example to illustrate the applicability of the algorithm, the SO for TV plants was used to endow the “Robust Multiple-Model Adaptive Control with BMI/LPV Controllers” (RMMAC/BMI) architecture with robust stability properties when the plant to be controlled is uncertain and time-varying. We claim that adaptive control strategies must somehow resort to control residuals to guarantee stability of the closedloop, while they should also lie with the identification resid-
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