Lyapunov Analysis of Least Squares Based Direct Adaptive Control
LLyapunov Analysis of Least Squares Based Direct Adaptive Control
Nursefa Zengin and Baris Fidan
Abstract — Adaptive control studies mostly utilize gradientbased parameter estimators for convenience in Lyapunov anal-ysis based constructive design. However, simulations and real-time experiments reveal that, compared to gradient based on-line parameter identifiers, least squares (LS) based parameteridentifiers, with proper selection of design parameters, exhibitbetter transient performance from the aspects of speed ofconvergence and robustness to measurement noise. The existingliterature on LS based adaptive control mostly follow theindirect adaptive control approach as opposed to the direct one,because of the difficulty in integrating an LS based adaptive lawwithin the direct approaches starting with a certain Lyapunov-like cost function to be driven to (a neighborhood) of zero.In this paper, a formal constructive analysis framework forintegration of recursive LS (RLS) based estimation to directadaptive control is proposed following the typical steps forgradient adaptive law based direct model reference adaptivecontrol, but constructing a new Lyapunov-like function forthe analysis. Adaptive cruise control simulation application isstudied in Matlab/Simulink and CarSim.
I. INTRODUCTIONStability and convergence analysis of adaptive controllerschemes has traditionally been based on Lyapunov stabilitynotions and techniques [1]–[5] . Lyapunov-like functions areselected in the design of adaptive control to penalize themagnitude of the tracking or regulation error but at the sametime useful in designing an adaptive law to generate theparameter estimates to feed the control law. Control designstargeting to drive the Lyapunov-like functions to zero leadto gradient based adaptive laws using a constant adaptivegains due to its convenient structure. On the other hand, itis well observed that least-squares (LS) algorithms have theadvantage of faster convergence; hence, LS based adaptivecontrol has potential to enhance convergence performance indirect adaptive control approaches as well [2], [6]–[9].Despite wide use of gradient on-line parameter basedidentifiers, LS adaptive algorithms with forgetting factor aredeveloped to be capable of faster settling and/or being lesssensitive to measurement noises in [6],[7]. Such propertiesbeing justified by simulation and experiment results. LSparameter estimation have been used for convergence androbustness analysis in either indirect adaptive controllers orcombination of indirect adaptive controller with direct one[8]–[18].In addition to the existing LS based adaptive control theorystudies, there are some publications in the recent literatureon real-time applications to robotic manipulators [19]–[21],unmanned aerial vehicles [22], [23], and passenger vehicles[24]–[30].
Authors are with School of Engineering, University of Waterloo, ON,Canada nyarbasi,[email protected]
Most of the existing studies on LS based adaptive controlfollow the indirect approach as opposed to direct adaptivecontrol. One reason for this is that the analysis of directadaptive control is complicated for producing an LS basedadaptive control scheme in the Lyapunov-based design. Un-like indirect ones, in direct adaptive control schemes, theestimated parameters are those directly used in the adaptivecontrol laws without any intermediate step.This paper proposes a constructive analysis frameworkfor recursive LS (RLS) on-line parameter identifier baseddirect adaptive control. In the literature, [1], [2] consideredthe possible use of LS on-line parameter identifier in directmodel reference adaptive control (MRAC); however, full de-tails of design was not provided, and Lyapunov analysis withLS parameter estimation was not mentioned. ConstructiveLyapunov analysis of RLS parameter estimation based directadaptive control, aiming to construct the adaptive and controllaws of the adaptive control scheme via this analysis, isapproached in this paper following the steps of the analysisfor gradient parameter identification based schemes in theliterature. The main difference is replacement of the constantadaptation gain in gradient based schemes, with a timevarying adaptive gain (covariance) matrix. Aiming to makethis replacement systematic and to formally quantify theuse of the time-varying adaptive gain, a new Lyapunov-likefunction is constructed through the provided analysis. Laterin the paper, to demonstrate the transient performance ofRLS parameter identification based direct adaptive controla simulation based case study on adaptive cruise control(ACC) is provided, where the performance is comparedin detail with that of gradient parameter estimation basedschemes. The comparative simulation testing and analysis areperformed in Matlab/Simulink and CarSim environments.The following sections of the paper are organized asfollows. Section II is dedicated to basics of direct MRACdesign. Section III provides Lyapunov-like function com-position and analysis. Comparative simulation testing andanalysis of an ACC case study for RLS based vs gradientbased MRAC is presented in Section IV. Final remarks ofthe paper are given in Section V.II. B
ACKGROUND : D
IRECT M ODEL R EFERENCE A DAPTIVE C ONTROL (MRAC) D
ESIGN
In MRAC, desired plant behaviour is described by areference model which is often formulated in the form of atransfer function driven by a reference signal. Then, a controllaw is developed via model matching so that the closed loopsystem has a transfer function equal to the reference model[1]–[4]. In MRAC, the plant has to be minimum phase, i.e, a r X i v : . [ ee ss . S Y ] J u l ll zeros have to be stable. Consider the SISO LTI plant˙ x p = A p x p + B p u p , x ( ) = x , y p = C Tp x p , (1)where x p ∈ R n , y p , u p ∈ R and A p , B p , C p have the appropriatedimensions. The transfer function of the plant is given by G p ( s ) = k p Z p ( s ) R p ( s ) , (2)where k p is the high frequency gain. The reference model isdescribed by˙ x m = A m x m + B m r , x m ( ) = x m , y m = C Tm x m (3)The transfer function of the reference model (3) is given by W m ( s ) = k m Z m ( s ) R m ( s ) , (4)with constant design parameter k m . The control task [1], [2]is to find the plant input u p so that all signals are bounded andthe plant output y p tracks the reference model output y m withgiven reference input r ( t ) , under the following assumptions: Assumption 1:
Plant Assumptions i Z p ( s ) is a monic Hurwitz polynomial.ii Upper bound n of the degree n p of R p ( s ) is known.iii Relative degree n ∗ = n p − m p of G p ( s ) is known and m p is the degree of Z p ( s ) .iv The sign of k p is known. Assumption 2:
Reference Model Assumptions i Z m ( s ) , R m ( s ) are monic Hurwitz polynomials of thedegree of q m , p m , respectively.ii Relative degree n m = p m − q m of W m ( s ) is the same asthat of G p ( s ) , i.e, n ∗ = n ∗ m .Consider the fictitious feedback control law [1], [2] u p = θ ∗ T α ( s ) Λ ( s ) u p + θ ∗ T α ( s ) Λ ( s ) y p + θ ∗ y p + c ∗ r , (5)where c ∗ = k m k p , α ( s ) (cid:44) α n − ( s ) = [ s n − , s n − , · · · , s , ] T for n ≥ , α ( s ) (cid:44) n = . (6) Λ ( s ) is an arbitrary monic Hurwitz polynomial of degree n − Z m ( s ) as a factor, i.e., Λ ( s ) = Λ ( s ) Z m ( s ) implying that Λ ( s ) is monic and Hurwitz. The fictitiousideal model reference control (MRC) parameter vector θ ∗ = (cid:2) θ ∗ T θ ∗ T θ ∗ c ∗ (cid:3) T is chosen so that the transfer functionfrom r to y p is equal to W m ( s ) . The closed-loop referenceto output relation for the MRAC scheme above is derived in[1], [2] as y p = G c ( s ) r , (7) where G c ( s ) = c ∗ k p Z p Λ Λ ( Λ − θ ∗ T ( α ) R p − k p Z p [ θ ∗ T α + θ ∗ Λ ] . The ideal MRC parameter vector θ ∗ is selected to match thecoefficients of G c ( s ) in (7) and W m ( s ) in (4).Nonzero intial conditions will affect the transient responseof y p ( t ) . A state-space realization of the control law (5) isgiven by [1], [2]˙ ω = F ω + gu p , ω ( ) = , ˙ ω = F ω + gy p , ω ( ) = , u p = θ ∗ T ω , (8)where ω , ω ∈ R n − , θ ∗ = (cid:2) θ ∗ T θ ∗ T θ ∗ c ∗ (cid:3) T , ω = (cid:2) ω ω y p r (cid:3) T , F = − λ n − − λ n − − λ n − · · · − λ · · ·
00 1 0 · · · · , Λ ( s ) = s n − + λ n − s n − + · · · + λ s + λ = det ( sI − F ) , g = (cid:2) · · · (cid:3) T . (9)Following the certainty equivalence approach,state-space re-alization of the actual adaptive control law is given by (8):˙ ω = F ω + gu p , ω ( ) = , ˙ ω = F ω + gy p , ω ( ) = , u p = θ T ω , (10)where θ ( t ) is the online estimate of the unknown idealMRC parameter vector θ ∗ . In order to find the adaptive lawgenerating θ ( t ) , first, a composite state space representationof the plant and controller [2] is considered as follows:˙ Y c = A Y c + B c u p , y p = C Tc Y c , u p = θ T ω , (11)where Y c = [ x TP , ω T , ω T ] T , A = A p F gC Tp F , B c = B p g , C Tc = (cid:2) C Tp , , (cid:3) . (12)The system equation (11) can be further written [2] incompact form˙ Y c = A c Y c + B c c ∗ r + B c ( u p − θ ∗ T ω ) , Y c ( ) = Y y p = C Tc Y c , (13)where A c = A p + B p θ ∗ C Tp B p θ ∗ T B p θ ∗ T g θ ∗ C Tp F + g θ ∗ T g θ ∗ T gC Tp F . (14)et the state error be e = Y c − Y m , (15)and output tracking error be e = y p − y m . (16)Error equation is written using (15) and (16) as follows:˙ e = A c e + B c ( u p − θ ∗ T ω ) , e ( ) = e , e = C Tc e , (17)where A c , B c , C c represent the parameter matrices of the plantin state space realization. We have W m ( s ) = C Tc ( sI − A c ) − B c c ∗ , (18)then e becomes e = W m ( s ) ρ ∗ ( u p − θ ∗ T ω ) . (19)The estimate ˆ e of e is defined asˆ e = W m ( s ) ρ ( u p − θ T ω ) , (20)where ρ is the estimate of ρ ∗ . Since the control input is u p = θ T ( t ) ω , (21)the estimate ˆ e and the estimation error ε becomesˆ e = , ε = e − ˆ e = e . (22)Substituting (22) into (17), we obtain˙ e = A c e + ¯ B c ρ ∗ θ T ω ) , e = C Tc e , (23)where ˙ e = A c e + B c ˜ θ T ω , e = C Tc e , (24)where ˜ θ = θ ( t ) − θ ∗ . (25)III. L YAPUNOV -L IKE F UNCTION C OMPOSITION AND A NALYSIS FOR L EAST -S QUARES B ASED D IRECT
MRACIn the typical direct adaptive control designs of the literature,which are gradient adaptive law based, the Lyapunov-likefunction is chosen as V ( ˜ θ , e ) = e T P c e + ˜ θ T Γ − ˜ θ | ρ ∗ | , (26)where ˜ θ = θ − θ ∗ , θ ∗ and θ , respectively, are the idealMRC and actual MRAC parameter vectors defined in SectionII, P c = P Tc is a positive definite matrix satisfying certainconditions to be detailed in the sequel, and Γ = Γ T is a con-stant positive definite adaptive gain matrix. P c is selected tosatisfy the Meyer-Kalman-Yakubovich Lemma [2] algebraicequations P c A c + A Tc P c = − qq T − ν c L c , P c B c c ∗ = C c , (27) where q is a vector, L c = L Tc >
0, and ν c > V of V along the solution of (26) is˙ V = − e T qq T e − ν c e T L c e + e T P c B c c ∗ ρ ∗ ˜ θ T ω + ˜ θ T Γ − ˙˜ θ | ρ ∗ | . (28)Since e T P c B c c ∗ = e T C c = e and ρ ∗ = | ρ ∗ | sgn ( ρ ∗ ) , ˙ V ≤ θ = − Γ e ω sgn ( ρ ∗ ) , (29)which, noting that ˙˜ θ = ˙ θ leads to˙ V = − e T qq T e − ν c e T L c e . (30)(26) and (30) imply that V , e , ˜ θ ∈ L ∞ . Since e = Y c − Y m and Y m ∈ L ∞ , Y c ∈ L ∞ that gives use x p , y p , ω , ω ∈ L ∞ . We alsoknow that u p = θ T ω and θ , ω ∈ L ∞ ; therefore, u p ∈ L ∞ . Allthe signals in the closed-loop plant are bounded. Hence, thetracking error e = y p − y m goes to zero as time goes toinfinity.With the gradient based adaptive law (29) with constantadaptive Γ gain, fast adaptation can be achieved only byusing a large adaptive gain to reduce the tracking errorrapidly. However, introduction of a large adaptive gain Γ in many cases leads to high-frequency oscillations whichadversely affect robustness of the adaptive control law. Asadaptive gain increases, time delay for a standard MRACdecreases causing loss of robustness.Unlike the gradient based adaptive law (29) with constantadaptive gain Γ , generation of a time varying adaptive lawgain matrix P ( t ) that is adjusted based on identificationerror during estimation process, would allow an initial largeadaptive gain to be set arbitrarily and then to be drivento lower values to adaptively achieve the desired trackingperformance.For generation of the time varying gain P ( t ) , an efficientsystematic approach is use of LS based adaptive laws, whichare observed to have the advantage of faster convergence androbustness to measurement noises [2], [6]–[9]. Next, we pro-pose a formal constructive analysis framework for integrationof recursive LS (RLS) based estimation to direct adaptivecontrol, following the typical steps above, but constructing anew Lyapunov-like function for the analysis to replace (26)and end up with a control law that is either the same as orsimilar to (21) together with and adaptive law that is the RLSbased alternative of (29).A reverse process of this analysis, starting with the fol-lowing RLS based alternative of the adaptive law will begiven in the following design. We define ˙ θ in terms of RLSalgorithm by replacing the constant adaptive gain Γ withthe time-varying gain matrix P ( t ) . In this regard, we writeLyapunov-like function in (26) as follows: V ( ˜ θ , e ) = e T P c e + ˜ θ T P − ˜ θ | ρ ∗ | , (31)he time derivative ˙ V of V along the solution of (31) is˙ V = − e T qq T e − ν c e T L c e + e T P c B c c ∗ ρ ∗ ˜ θ T ω +
12 ˜ θ T d ( P − ) dt ˜ θ | ρ ∗ | + ˜ θ T P − ˙˜ θ | ρ ∗ | , (32)where d ( P − ) dt = − P − ˙ PP − . (33)If P ( t ) is updated according to the RLS adaptive law withforgetting factor, ˙ P = β P − P ωω T P , (34)then (33) becomes d ( P − ) dt = − β P − + ωω T . (35)Substituting (35) into (32), we have˙ V = − e T qq T e − ν c e T L c e + e ρ ∗ ˜ θ T ω − β θ T P − ˜ θ | ρ ∗ | + ˜ θ T P − ˙˜ θ | ρ ∗ | + ε | ρ ∗ | , (36)where ε = ˜ θ T ω . (37)˙ V ≤ θ = − Pe ω sgn ( ρ ∗ ) + P εω , (38)and noting that ˙˜ θ = ˙ θ . The equations (38) and (34) constitutea new adaptive law based on RLS algorithm for generatingthe time-varying gain P ( t ) . Substituting these equations into(36), (32) becomes˙ V = − e T qq T e − ν c e T L c e ≤ , (39)leading to the following theorem, which summarizes thestability properties of the LS based direct MRAC scheme(10),(34),(38). Theorem 3.1:
The RLS parameter estimation basedMRAC scheme (10),(34),(38) has the following properties:i. All signals in the closed-loop are bounded and trackingerror converges to zero in time for any reference input r ∈ L ∞ .ii. If the reference input r is sufficiently rich of order 2 n ,˙ r ∈ L ∞ , and Z p ( s ) , R p ( s ) are relatively coprime, then ω is persistently exciting (PE), viz., (cid:90) t + T t ω ( τ ) ω T ( τ ) d τ ≥ α T I , α , T > , ∀ t ≥ , (40)which implies that P , P − ∈ L ∞ and θ ( t ) → θ ∗ as t → ∞ .When β >
0, the parameter error (cid:107) ˜ θ (cid:107) = (cid:107) θ − θ ∗ (cid:107) andthe tracking error e converges to zero exponentiallyfast. Proof: i. e ∈ L , θ , ω , ∈ L ∞ , and ˙ e ∈ L ∞ . Therefore, all signalsin the closed loop plant are bounded. In order tocomplete the design, we need to show tracking error e converges to the zero asymptotically with time. Using(31), (39), we know that e , e ∈ L . Using, θ , ω , e ∈ L ∞ in (24), we have ˙ e , ˙ e ∈ L ∞ . Since ˙ e , ˙ e ∈ L ∞ and e ∈ L , the tracking error e goes to zero as t goesto infinity.ii. By Theorem 3.4.3 of [2], if r is sufficiently rich of order2 n then the 2 n dimensional regressor vector ω is PE.Let Q = P − and (35) can be rewritten as˙ Q = − β Q + ωω T . (41)and the solution becomes Q ( t ) = e − β t Q + (cid:90) t e − β ( t − τ ) ω ( τ ) ω T ( τ ) d τ . (42)Since ω ( t ) is PE, Q ( t ) ≥ (cid:90) tT e − β ( t − τ ) ω ( τ ) ω T ( τ ) d τ ≥ ¯ α e − β T (cid:90) tT e − β ( t − τ ) ω ( τ ) ω T ( τ ) d τ ≥ β e − β T I , ∀ t ≥ T , (43)where β = ¯ α α T , and α , ¯ α , T > t ≤ T , Q ( t ) ≥ e − β T Q ≥ λ min ( Q ) e − β T I ≥ γ I ∀ t ≥ , (44)where γ = min { α T β , λ min ( Q ) } e − β T . Since ω is PE, Q ( t ) ≤ Q + β (cid:90) t e − β ( t − τ ) d τ I ≤ γ I , β > . (45)where γ = λ max ( Q ) + β β >
0. Using (44) and (45), weobtain γ − I ≤ P ( t ) = Q (( t ) ≤ γ − I . (46)Therefore, P ( t ) , Q ( t ) ∈ L ∞ . Exponential convergence isestablished following steps similar to those in [2].Comparing two adaptive laws in (29) and (38), we can clearlysee the effect of time varying covariance matrix reflected asan additional term to the similar part of (29).IV. C ASE S TUDY
For the application of RLS based adaptive control, ACCcase study is considered. A basic ACC scheme is given inFig. 1. ACC regulates the following vehicle’s speed v towardsthe leading vehicle’s speed v l and keeps the distance betweenvehicles x r close to desired spacing s d . The control objective 𝑣𝑣 𝑣𝑣 𝑙𝑙 𝑥𝑥 𝑟𝑟 𝑠𝑠 𝑑𝑑 𝛿𝛿 Fig. 1: Leading and following vehicles.n ACC is to make the speed error close to zero as timeincreases. This objective can be expressed as v r → , δ → , t → ∞ , (47)where v r = v l − v which is defined as the speed error orsometimes relative speed, δ = x r − s d is the spacing error.The desired spacing is proportional to the speed since thedesired spacing between vehicles is given as s d = s + hv (48)where s is the fixed spacing for safety so that the vehiclesare not touching each other at zero speed and h is constanttime headway. Control objective should also satisfies that a min ≤ ˙ v ≤ a max , and small | ¨ v | . First constraint restrictsACC vehicle generating high acceleration and the secondone is given for the driver’s comfort. For ACC system, asimple model is considered approximating the actual vehiclelongitudinal model without considering nonlinear dynamicswhich is given by ˙ v = − av + bu + d , (49)where v is the longitudinal speed, u is the throttle/brakecommand, d is the modeling uncertainty, a and b are pos-itive constant parameters. We assume that d , ˙ dv l , ˙ v l are allbounded. MRAC is considered so that the throttle command u forces the vehicle speed to follow the output of thereference model v m = a m s + a m ( v l + k δ ) , (50)where a m and k are positive design parameters.We firstassume that a , b , and d are known and consider the controllaw as follows: u = k ∗ v r + k ∗ δ + k ∗ , (51) k ∗ = a m − ab , k ∗ = a m kb , k ∗ = av l − db . (52)Since a , b , and d are unknown, we change the control lawas u = k v r + k δ + k , (53)where k i is the estimate of k ∗ i to be generated by the adaptivelaw so that the closed-loop stability is guaranteed. Thetracking error is given as e = v − v m = bs + a m ( k ∗ v r + k ∗ δ + k ∗ + u ) . (54)(54) is in the form of B-DPM. Substituting the control lawin (53) into (54), we obtain e = bs + a m ( ˜ k v r + ˜ k δ + ˜ k ) , (55)where ˜ k i = k i − k ∗ i for i = , ,
3. In order to find the adaptivelaw, consider the Lyapunov function and its time derivative[2] as V = e + ∑ i = b γ i ˜ k i γ i > , b > , (56) ˙ V = − a m e + be ( ˜ k v r + ˜ k δ + ˜ k ) + ∑ i = b γ i ˜ k i ˙˜ k i . (57)Therefore, the following gradient based adaptive laws areapplied to ACC ˙ k = Pr {− γ ev r } , ˙ k = Pr {− γ e δ } , ˙ k = Pr {− γ e } , (58)where the projection operator keeps k i within the lower andupper intervals and γ i are the positive constant adaptive gains.These adaptive laws lead to˙ V = − a m e − b γ i ˜ k i ˙ k ∗ , (59)where ˙ k ∗ = a ˙ ν l − ˙ db . By projection operator, estimated parame-ters are guaranteed to be bounded by forcing them to remaininside the bounded sets, ˙ V implies that e ∈ L ∞ , in turn allother signals in the closed loop are bounded. We apply RLSbased adaptive law to (56) and obtain following equations tobe used in simulationsFig. 2: ACC comparison results, speed tracking and separa-tion error in Matlab/Simulink.˙ˆ θ = Pr { P ii e φ } , ˙ P = β P − P φ φ T P , P ( ) = I × , (60)with e = v − v m , θ = (cid:2) k , k , k (cid:3) T , φ = (cid:104) v r s + a m , δ s + a m , s + a m (cid:105) T , (61)where P ii are the diagonal elements of P covariance matrix,i=1,2,3.For gradient based adaptive law, γ = I , γ = I , γ = I constant gains are given. For RLS based algorithm β = .
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