Integration of the Stability Overlay (SO) with the Robust Multiple-Model Adaptive Control (RMMAC) Paulo Rosa, Jeff S. Shamma, Carlos Silvestre and Michael Athans
This work was partially supported by Fundac¸a˜ o para a Ciˆencia e a Tecnologia (ISR/IST pluriannual funding) through the POS Conhecimento Program that includes FEDER funds, by the PTDC/EEA-ACR/72853/2006 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. 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]
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Posterior Probability Evaluator
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I. I NTRODUCTION In some practical applications, a robust non-adaptive controller is enough to achieve the desired closed-loop performance, i.e., to guarantee a given level of attenuation from the exogenous disturbances inputs to the performance outputs. Notwithstanding, whenever there is a large model uncertainty (unknown and unmeasurable parameters and/or unmodeled dynamics), such a non-adaptive controller with maximum performance guarantees may not exist. To overcome this problem, several solutions are available in the literature of adaptive control. In this paper, we consider an important class of adaptive control, referred to as multiple-model adaptive control (MMAC). For a list of advantages of this type of control see, for instance, [10]. However, adaptive control also has certain shortcomings. The most stringent one is the lack of stability guarantees whenever the plant has unmodeled dynamics – see [16]. We stress that every non-ideal plant possesses such dynamics. Several MMAC methods are available in the literature. For instance, the authors of [18], [22], [1] evaluate the performance of the closed-loop obtained with each controller, providing them with “rewards”. Thereafter, based upon the the rewards received after its most recent utilization, each controller is disqualified or not. The SO differs from those approaches, since it is only responsible for the input/output stability of the plant, which means that another algorithm should run in parallel in order to satisfy the proposed performance requirements. An important MMAC architecture is the so-called Robust Multiple-Model Adaptive Control (RMMAC). The RMMAC
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Abstract— The Stability Overlay (SO) is a “safety net” that can be integrated with virtually any multiple-model adaptive control (MMAC) architecture, guaranteeing the stability of the closed-loop system. However, the arbitrary interconnection of the SO with a MMAC architecture can lead to severe performance deterioration. Thus, this paper proposes a systematic integration of the SO with the Robust Multiple-Model Adaptive Control (RMMAC), which provides stability guarantees, while maintaining the high levels of performance of the standard RMMAC observed in numerous simulations when the design assumptions are not violated.
Select the largest probability
Switching logic
The switching RMMAC architecture with N models.
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 identification subsystem utilizes a bank of Kalman filters (KFs), while the control subsystem consists of a set of mixed-µ controllers. Figure 1 depicts the RMMAC architecture, for the case where N regions for the uncertain real parameters are used. It is not our intention to give an in-depth explanation of the RMMAC – the interested reader is referred to [2], [6], [7] – but rather to use the Stability Overlay (SO) to endow the RMMAC with input/output stability guarantees, while keeping the high level of performance of the standard RMMAC observed in the simulations whenever the design assumptions are not violated. The SO, introduced in [17], is a “safety net” that can be integrated with virtually any MMAC architecture, guaranteeing the stability of the closed-loop system. As stressed in [17], the choice of the parameters of the algorithm may be very sensitive, depending upon the plant dynamics and the disturbance intensities. In fact, if the norm of the output of the closed-loop system grows very fast whenever a destabilizing controller is picked, and if the time required to disqualify a controller is very large, one may not get “practical stability”. This means that, although a stabilizing controller is eventually selected, the large transients may not be acceptable from a practical point of view. Nevertheless, the SO only constrains the controllers that can be selected at each sampling time. As pointed out in [17], another algorithm, such as the RMMAC, should be used to select the appropriate controller whenever the set of eligible controllers does not have a single element. However, this performance-related algorithm (the RMMAC, in the present
case) is usually more sensitive to model errors, as expected. This means that, if some of the design assumptions are violated, the controller with highest probability may not be the correct one. Therefore, in this paper, we propose a modification of the RMMAC that, integrated with the SO, guarantees stability of the closed-loop system, while preserving the high levels of performance observed in the simulation of the standard RMMAC whenever the design assumptions are not violated. We take advantage of the fact that, although the Posterior Probability Evaluator (PPE) of the RMMAC can be misled by the mismatch between the plant noise/disturbances intensity and the values used during the design of the KFs, the state estimation can sometimes be less sensitive to such errors. This paper is organized as follows: section II proposes an integration of the RMMAC with the SO; section III presents some simulation results for a mass-spring-dashpot (MSD) plant; and, finally, in section IV, some conclusions on the results obtained are provided. II. I NTEGRATING THE S TABILITY OVERLAY WITH THE RMMAC We define |x| as the euclidian norm of x ∈ Rn . We further define, for any σ > 0,
σ
z|[t1 ,t2 ] = sup e−σ(t2 −τ ) |z(τ )|, τ ∈[t1 , t2 ]
and
z|[t ,t ] = 1 2
sup
|z(τ )|.
τ ∈[t1 , t2 ]
Figure 2 illustrates the SO algorithm. The notation S = S\K(n) means “the exclusion of element K(n) from set S”. The reward, r(n), after using controller K(n) during the time interval tn−1 ≤ t < tn is defined as (
1, z|σ[0, tn ] ≤ γ z|σ[0,tn−1 ] + l(n) r(n) = (1) 0, otherwise, where γ is a fixed scalar with 0 < γ < 1 and l(n) is going to be specified next. z(t) is the (possibly augmented and/or filtered) output. The initial set of eligible control laws is denoted S0 , while K0 is the first control law selected, ∆T (n) is the time-period that control law K(n) is used, l0 is the initial value of l(n) in (1), and linc and ∆Tinc are the increments for l(n) and ∆T (n), respectively, whenever all the control laws have failed in their most recent utilization. Consider the integration of the SO depicted in Fig. 2 with the RMMAC. We are now deriving a procedure for controller selection, based on the posterior probabilities – computed by the PPE – and an auxiliary norm condition, that complements the SO norm condition in (1). Assume that the dynamics of the closed-loop system, for a given controller selection, is described by x(t) ˙ = Ax(t) + Lξ(t), T
(2)
where x(t) = [xp (t), xc (t)] , xp (t) and xc (t) are the states of the plant and controller, respectively, and ξ(t) comprises disturbances acting upon the plant and sensor noise. The time response of (2) is given by Z t A(t−t0 ) x(t) = e x(t0 ) + eA(t−τ ) Lξ(τ )dτ. t0
Initialize S=S0, K(1)=K0, DT(1)=DT0, l(1)=l0, n=1
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Fig. 2.
Stability Overlay (SO) algorithm.
If the controller is able to stabilize the plant, A is Hurwitz. Hence, we have that |eA(t−t0 ) | < 1 for sufficiently large t. Therefore, |x(t)| must be bounded if a stabilizing controller is used, i.e., |x(t)| < |x(t0 )| + for sufficiently large t, where t0 ≤ t and depends upon L and the bound on ξ(.). We use the state estimates of the KFs of the RMMAC to disqualify controllers that are not being able to reduce the norm of the state. As an example, suppose that we are experimenting controller number 1 among N possible controllers. We start using this controller at time t0 , and evaluate its behavior at time t1 . Let |ˆ xi (t0 )| be the norm of the state estimate of KF #i and |ˆ x(t0 )| =
max |ˆ xi (t0 )|.
i=1,··· ,N
Then, controller #1 is said to violate the norm condition at time t1 if |ˆ x1 (t1 )| ≥ |ˆ x(t0 )| + . We define the set Q(t) as the set of all the controllers that have not violated the norm condition up to time instant t, and Q(0) = S0 . Furthermore, S0 denotes the initial set of eligible controllers and S(t) the set of eligible controllers at time instant t. Since we want the proof of stability of the SO to hold, the structure of the algorithm cannot be changed. Hence, T a controller can only be selected if it belongs to S. If S Q 6= ∅, as depicted in Fig. 3, we select T the controller with the highest posterior probability in S Q. Otherwise, we select the controller with highest probability in S, as shown in Fig. 4. In reference to Fig. 2, instead of selecting any controller in S, we select T • theT controller with highest probability in Q S, if Q S 6= ∅; • the controller with highest probability in S, otherwise. The set Q(t) may be reset, i.e., Q(t) = S0 , whenever the S(t) is also reset, although this is not a requirement of the algorithm.
KRMMAC
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T Fig. 3. Controller selection when S Q 6= ∅. S0 is the initial set of eligible controllers, S is the actual set of eligible controllers, Q is the set of controllers that did not violate the norm condition, KRM M AC is the controller with highest posterior probability, according to the PPE of the RMMAC, and, finally, K is the selected controller.
KRMMAC K S
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Posterior Probability
Q
T Fig. 4. Controller selection when S Q = ∅. S0 is the initial set of eligible controllers, S is the actual set of eligible controllers, Q is the set of controllers that did not violate the norm condition, KRM M AC is the controller with highest posterior probability, according to the PPE of the RMMAC, and, finally, K is the selected controller.
Remark 1 Alternatively, we can use Lyapunov-based arguments to select the controllers. Define V (x(t)) := kx(t)kP := xT (t)P x(t). If the selected controller is able to stabilize the plant, then there exists P > 0 such that AT P + P A < 0. Thus, V˙ (x(t)) = xT (P A + AT P )x + xT P Lξ + ξ T LT P x, which means that, for sufficiently large x(t), the norm kx(t)kP decreases if a stabilizing controller is selected. This way, we can evaluate the controllers based upon their behavior when connected to the loop. The problem with this approach is that, since there is parametric uncertainty in the model of the plant, we have to find a P > 0 that verifies ATi P + P Ai < 0 for every possible value of Ai . This may be a shortcoming in some cases, where such P > 0 does not exist, although every Ai is Hurwitz – see [5]. III. S IMULATIONS In this section, we present some simulation results of the RMMAC integrated with the SO for the mass-spring-dashpot (MSD) plant described in the sequel and depicted in Fig. 5. This is the same testbed used in many of our previous studies – see [2], [6], [7] –, and by [13], [14]. The unknown parameters of the plant are the spring stiffness k1 ∈ K := [0.25
1.75] N/m
and the constant input time-delay, bounded by 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 non-adaptive controller deteriorates significantly the performance of the overall system. Moreover, there are several realistic applications that share the dynamics of the
x2=z
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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. τ is an uncertain time-delay bounded by 0 < τ < 0.05s, as in [6], [7].
MSD plant, for instance in seismic and vibration systems [9], [15], [19], automotive suspension systems [20], [8], flexible space structures [12], [3], [4], [11], among others, as stressed in [13], [21]. Following the RMMAC synthesis methodology and using the same design choices as the ones described in [6], [7], 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 the value of the uncertain parameter, k1 , been known. 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 Ξ, according to α ξ(s) = Wd (s)ξ(s). d(s) = s+α The sensor noise considered is continuous-time white noise θ(t), with zero mean and intensity Θ. In the following simulations, two different values for the spring constant are considered: k1 ∈ {0.25, 1.75} N/m. For each value of the spring constant, the following cases are considered: • nominal case (Ξ = 1 and Θ = 0.001); • both disturbance and sensor noise intensities increased (Ξ = 100 and Θ = 0.01), and added sinusoidal disturbance with frequency 1 rad/s and amplitude 100 N. This represents a severe violation of the assumptions used to design the KFs. Remark 2 Notice that having a sinusoidal disturbance is a realistic assumption, since it can model, for instance, electrical and magnetic field interference in low-power circuits. For comparison purposes, we also use the control architecture introduced in [13], [14], referred to as multiple model adaptive control with mixing (MMACwM1 ). The MMACwM methodology resorts to a parameter estimator to decide which controllers signals should be selected or mixed, providing robust stability guarantees. We stress that the SO can also be used with a modified MMACwM architecture, if we switch the controllers, instead of mixing the output signals. A. Soft Spring Constant, k1 = 0.25 N/m - Nominal Case For the first simulation, we consider a soft spring constant, k1 = 0.25 N/m. The results obtained for the nominal case 1 For the simulations of the MMACwM algorithm, we use the same parameters as in [13]. However, the authors have stressed that such parameters can be further tuned, and hence the results can be improved. However, herein we are interested in showing that adaptive control laws integrated with the SO can guarantee stability, even when the parameters of the corresponding adaptive algorithm are not properly tuned.
RMMAC
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0.024
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TABLE I RMS OF THE OUTPUT FOR THE CLOSED - LOOP WITH THE DIFFERENT ADAPTIVE CONTROL METHODOLOGIES , FOR C ASE A.
TABLE II RMS OF THE OUTPUT FOR THE CLOSED - LOOP WITH THE DIFFERENT ADAPTIVE CONTROL METHODOLOGIES , FOR C ASE B.
are depicted in Fig. 6. As illustrated in Fig. 7, the posterior probabilities of each of the N models are not significantly affected by the SO. Therefore, the performance of the standard RMMAC is close to that of the RMMAC integrated with the SO, and to that of the MMACwM. For this case, the standard switching RMMAC and the RMMAC integrated with the SO are faster in terms of the identification of the uncertain parameter than the RMMACwM, as illustrated in Fig. 7. Nevertheless, the transients obtained with those two control strategies are slightly larger, as shown in Fig. 6 and Table I.
B. Soft Spring Constant, k1 = 0.25 N/m - High Disturbance and Sensor Noise Intensity The next simulation results are more interesting, since they reveal the usefulness of the SO. Suppose that the actual plant disturbances intensity is Ξ = 100 and that the sensor noise intensity is Θ = 0.01. Further suppose that we add a sinusoidal disturbance with frequency 1 rad/s and amplitude 100 N. However, assume that such information is not available, and hence we do not re-design the KFs of the RMMAC, nor the bandpass filters of the MMACwM. The results are depicted in Fig. 8 and the RMS of the output is summarized in Table II. It should be noticed that the KFs become very “confused” in this case, and, thus, the corresponding posterior probabilities are not shown. However, the SO is eventually able to decide that controllers #1, #2 and #3 should be removed from the set of eligible controllers, since they do not meet the performance requirements.
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Fig. 6. Time-evolution of the output of the MSD plant for k1 = 0.25 N/m, nominal case.
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Fig. 7. Time-evolution of the posterior probability, Pi (t), of each of the N models, controller selection for the RMMAC integrated with the SO, and controllers weights for the MMACwM, for k1 = 0.25 N/m, nominal case.
Remark 3 Note that the SO does not allow an arbitrarily fast switching of the controllers. Thus, a short and sudden change in the posterior probability of a region does not necessarily imply a switching of the controllers. This is the main reason for the differences between the standard switching RMMAC and the RMMAC integrated with the SO, for the present case.
Although it may not be clear from Fig. 8, after 375 secs, the RMMAC integrated with the SO performs better than the standard switching RMMAC, since controller #4 is considered as the only eligible controller by the SO. However, the identification subsystem of the MMACwM is faster, and hence, as shown in Table II, the transients are smaller. Remark 4 Notice that, from Table II, one can only conclude that the transients obtained with the RMMAC integrated with the SO are larger than those obtained with the standard switching RMMAC. Notwithstanding, since the KFs are not able to correctly identify the region where the uncertain parameter takes value, the performance of the RMMAC integrated with the SO is better than that of the RMMAC as time goes to infinity. C. Stiff Spring Constant, k1 = 1.75 N/m - Nominal Case For the next simulations, we use a stiffer spring constant, k1 = 1.75 N/m. The output for the nominal case is depicted
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Fig. 9. Controller selection for the RMMAC integrated with the SO, and controllers weights for the MMACwM for k1 = 0.25 N/m, with increased disturbance and sensor noise intensities. The black-dashed lines indicate the time-instants a controller has failed, and hence was removed from the set of eligible controllers. First, controller #3 fails, then controller #2 and, finally, controller #1.
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TABLE III RMS OF THE OUTPUT FOR THE CLOSED - LOOP WITH THE DIFFERENT ADAPTIVE CONTROL METHODOLOGIES , FOR C ASE C.
in Fig. 10. The posterior probabilities of each of the N models are illustrated in Fig. 11, where it can also be seen that the SO does not intervene in the selection of the controllers.
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Fig. 11. Time-evolution of the posterior probability, Pi (t), of each of the N models, controller selection for the RMMAC integrated with the SO, and controllers weights for the MMACwM, for k1 = 1.75 N/m, nominal case.
disturbance with frequency 1 rad/s and amplitude 100 N. However, the KFs for the RMMAC and the bandpass filters of the MMACwM are designed for the nominal case. This violation of the design assumptions leads to the results depicted in Fig. 12. (We omit the RMS values of the output, since we can draw all the conclusions from Fig.s 12 and 13.) The SO detects that controllers #2, #3 and #4 do not achieve the desired performance, and hence are eliminated from the set of eligible controllers, as illustrated in Fig. 13. Therefore, the only controller that can be selected is controller #1. Since this controller meets the performance specifications, it is not disqualified. MMACwM RMMAC without SO RMMAC with SO
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As expected, for the nominal case, i.e., if the design assumptions of the KFs are not violated, the results obtained for the RMMAC with and without the SO are very similar. However, the transients obtained with the MMACwM are much larger, as shown in Table III, since the identification subsystem takes about 5 secs to identify the correct region, while the KFs of the RMMAC only take about 1 sec. D. Stiff Spring Constant, k1 = 1.75 N/m - High Disturbance and Sensor Noise Intensity The last scenario considered is as follows. The actual plant disturbances intensity is Ξ = 100 and the sensor noise intensity is Θ = 0.01. Moreover, we add a sinusoidal
Fig. 12. Time-evolution of the output of the MSD plant for k1 = 1.75 N/m, with increased disturbance and sensor noise intensities. The output transients of the MMACwM exceed 107 m.
From the posterior probabilities of the regions (not depicted), we can conclude that the KFs are not able to identify the correct model, and hence the performance of the RMMAC without the SO deteriorates significantly. In fact, for this case, it can be even worse than the performance obtained with a non-adaptive controller. After a initial transient, the RMMAC with the SO is able to switch to the best LNARC, and hence achieves the highest level of performance as time goes to infinity. In comparison to the MMACwM, the transients are significantly better. In fact, the amplitude of the output for the MMACwM exceeds 107 m, and hence it is not reasonable from a practical point of view. Furthermore, the RMMAC integrated with the SO only takes about 60 secs to switch to the best controller, while keeping the output transients
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V. ACKNOWLEDGMENTS 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.
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Fig. 13. Controller selection for the RMMAC integrated with the SO, and controllers weights for the MMACwM, for k1 = 1.75 N/m, with increased disturbance and sensor noise intensities. The black-dashed lines indicate the time-instants a controller has failed, and hence was removed from the set of eligible controllers. Firstly, controller #4 fails, then controller #3 and, finally, controller #2.
below about 50 m. Although the RMMAC integrated with the SO takes longer to identify the region where the parameter takes value than the MMACwM, it disqualifies much faster the destabilizing controllers. This avoids the large transients depicted in Fig. 12. In comparison to the simulation in case B (k1 = 0.25 N/m - high disturbance and sensor noise intensity), the transients shown in Fig. 12 are much larger. This happens since, in general, using a destabilizing controller, during a given amount of time, for a plant with a stiff spring constant, increases the norm of the output faster than using a destabilizing controller, during that same amount of time, for a plant with a soft spring constant. Remark 5 Although only a small set of simulation results has been presented in this paper, the RMMAC integrated with the SO was tested for numerous different situations. The extreme values (0.25 and 1.75 N/m) of the uncertain parameter were selected, since they allows us to illustrate more clearly the differences between the experimented strategies. IV. C ONCLUSIONS This paper presented a solution to the mismatch problem between the true plant noise/disturbances intensities and the values used to design the KFs of the RMMAC. This adaptive control methodology, integrated with the SO, provides stability guarantees, while showing good performance in the simulations. We take advantage of the state estimations of the KFs to (as quickly as possible) disqualify controllers that are not being able to stabilize the plant. We stress that, in all the simulations presented, the performance of the RMMAC integrated with the SO is at least as high as that of the standard RMMAC, as time goes to infinity. In comparison to another multiple-model adaptive control (MMAC) methodology – the so-called MMAC with Mixing (MMACwM) – we conclude, resorting to numerous simulation results, that the RMMAC integrated with the SO provides smaller bounds on the transients for the general case, although in some cases there can be a small decrease in terms of performance.
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