Lyapunov-Stable Discrete-Time Model Reference Adaptive Control

Transcription

1 25 merican Control Conference June 8-, 25 Portl, OR, US ThC3 Lyapunov-Stable Discrete-Time Model Reference daptive Control Suhail khtar Dennis S Bernstein Department of erospace Engineering, The University of Michigan, nn rbor, MI , akhtars@umichedu bstract Discrete-time model reference adaptive control (MRC) has been studied extensively lthough the framework of the analyses is very general, the results obtained are restricted to boundedness convergence the important question of Lyapunov stability is not addressed Lyapunov functions are an important tool for understing quantifying transient response, robustness disturbance rejection, thus merit attention In this paper we investigate the use of a logarithmic Lyapunov function to establish Lyapunov stability of MRC in the deterministic setting complete Lyapunov proof is given for stability convergence The results extend the approach of 8] to include Lyapunov stability for MRC when the normalized projection algorithm is used for parameter identification Introduction In model reference adaptive control (MRC) theory the objective is to have the plant emulate the dynamics of a specified model in response to a family of comm signals MRC has been extensively developed for continuoustime systems ] discrete-time systems 2] where the boundedness of controller parameters convergence of the tracking error are demonstrated using the Gronwall- Bellman lemma the key technical lemma respectively The objective of the present paper is to unify extend the foundation of discrete-time MRC by constructing Lyapunov functions to demonstrate Lyapunov stability as well as error convergence The results of this paper are used in a companion paper 3] to prove Lyapunov stability for a discrete-time adaptive disturbance rejection algorithm Discrete-time MRC algorithms have been based on a variety of parameter identification algorithms For example, the recursive least squares (RLS) algorithm the projection algorithm are used in 4], where convergence is based on the key technical lemma This method of proof yields convergence but does not imply Lyapunov stability of the error system MRC is considered in the presence of additive noise in 2], 5 7] In these results, convergence of the tracking error parameters is guaranteed almost surely, but stochastic Lyapunov stability is not demonstrated Lyapunov stability of discrete-time MRC convergence of the error to a finite set is demonstrated in 8], where the RLS algorithm is used for parameter identification Lyapunov cidate is applied to the timevarying error system, which requires appropriate bounds on the Lyapunov difference Stochastic Lyapunov stability of MRC is addressed ] Supported in part by the ir Force Office of Scientific Research under grant number F /5/$25 25 CC 374 The novel Lyapunov construction of 8 2] is of independent interest since it involves the logarithm of a quadratic form similar construction was used in 3] for full-statefeedback adaptive stabilization extended in 4] to a more general class of gradient based gain update laws In view of these developments, in the present paper we extend the result of 8] by constructing a Lyapunov proof of MRC for the projection algorithm These constructs remove the need for the key technical lemma used in 4] The contents of the paper are as follows In Section 2 we present the solution to the model matching control problem in the case of a known plant n adaptive control law with projection algorithm based parameter identification is presented in Section 4 a proof of stability is given in Section 5 Finally, Section 6 presents simulation results 2 Model Reference Control for a Known Plant Consider a SISO process described by the DRM model n m y(k) = a i y(k i)+ b j u(k j), k (2) i= j= The model (2) can be written in terms of the forward shift operator q as (q)y(k) =B(q)u(k), (22) where B are polynomials of degree n m, respectively, defined by (q) = q n + a q n + + a n (23) B(q) = b q m + b q m + + b m, (24) where b We define the delay d = n m make the following assumptions about the plant ssumption 2 The realization (22) is minimal, ie, B are coprime ssumption 22 ll roots of B(q) are inside the unit circle ssumption 23 n m are known, m<n ssumption 24 b is known To modify the dynamics (22) we consider the 2-DOF model matching control law u(k) = T(q) R(q) u c(k) S(q) y(k), (25) R(q) where u c is the comm signal We want the response from the comm signal u c to the output y to be described by the reference model

2 y m (k) = B m(q) m (q) u c(k) (26) We make the following assumptions about the reference model ssumption 25 m (q) is monic stable ssumption 26 deg m (q) deg B m (q) =d, ie, the reference model has the same delay as the plant The closed-loop system (22)-(25) the reference model (26) have the same forced response if B(q)T(q) (q)r(q)+b(q)s(q) = B m(q) m (q), (27) which is equivalent to T(q) (q)r(q)+b(q)s(q) = B m(q) m (q)b(q) (28) The roots of the closed-loop characteristic polynomial m (q)b(q) consist of the roots of m (q) as well as the roots of B(q), all of which are stable by assumption Let n m = deg m (q) define P(q) = m (q)b(q) = b q (nm+m) + p q (nm+m) + + p nm +m To satisfy (28) it suffices to choose T(q) =B m (q) (29) require that R(q) S(q) satisfy (q)r(q)+b(q)s(q) =P(q) (2) Defining n R = deg R(q), ns = deg S(q), ne = n+nr +, n u = nr + n S +2, (2) ] can be written as C(R) M = C(P), (2) C(S) where M R ne nu is the Sylvester matrix given by (n ) d (d+) (n+d ) a b b b a 2 a a b b b a 2 b 2 a n a n b m b m a n b m (n ) (n 2) (n ) (n 2) C(R) = r r r nr ] T, C(R) = s s s ns ] T, C(R) = b p p nm+m] T are vectors containing the coefficients of R(q), S(q), P(q), respectively In the remainder of the paper, we omit the explicit dependence of polynomial operators on q Proposition 2 ssume that n m =2n m n S = n Then, for each n R there exist unique polynomials R S satisfying (2) Furthermore, if n R n S then the control law (25) is causal Proof Since n S = n it follows that M R (n+n R+) (n+n R +) is square lso, since B are relatively coprime, M is nonsingular the solution to (2) is unique From (29) we have deg T = deg B m = deg m d = n (22) 375 The condition n R n S implies that deg R deg S = deg T, thus the model matching controller (25) is causal Henceforth in accordance with Proposition 2 we assume that deg S = n deg m =2n m so that deg P =2n lso, to obtain a minimum degree causal controller we assume that deg R = n so that M R 2n 2n Hence we write R = r q n + r q n r n (23) S = s q n + s q n s n, (24) where r s are nonzero In fact, it follows from (2) (23) that r = b Next to obtain a linear estimation model in terms of the controller we define the filtered output signal y f (k) = q n d+ m y(k) = q n d+ m B u(k) (25) With the model matching condition (2), y f satisfies y f (k + d) = q n+ (R + BS) u(k) = Ru(k n +)+Sy(k n +) (26) Since r = b (26) can be written as the linear identification model y f (k + d) =b u(k)+ϕ T (k)θ, (27) where the parameter vector θ R 2n the regressor ϕ(k) R 2n are defined by θ =r r n s s n ] T (28) ϕ(k) =u(k ) u(k n +)y(k) y(k n + )] T (29) Using (26) (27) the model matching control law (25) can be written as u(k) = ϕ T (k)θ q n+ B m u c (k) ] (22) b The filtered plant model (27) the control law (22) are now in a form suitable for direct adaptive control 3 Model Matching Error Dynamics When the plant (22) is unknown we cannot solve (2) for the controller parameters R S Hence, let ˆR(k) Ŝ(k) be polynomials in q that are estimates of R S at time k Then in place of (25), the estimated model matching controller is u(k) = B m ˆR(k) u c(k) Ŝ(k) y(k) (3) ˆR(k) With (3) the closed loop system has the form BB m y(k) = ˆR(k)+BŜ(k)u c(k) (32) Next let ˆθ(k) denote an estimate of θ at time k define the parameter error θ(k) = ˆθ(k) θ (33)

3 q n d+ B m e f (k) + Filter Plant y f (k) B + u(k) B y(k) mˆr q n d+ u m c (k) Controller Estimator ŜˆR Fig Fig 2 Model Reference daptive Control Block Diagram u(k) ξ(k n) y(k) B B m m B (a) u c (k) ξ m (k n) y m (k) B (b) Fraction Forms of the Plant the Reference Model the filtered output error signal (see Figure ) e f (k) = y f (k) q n d+ B m u c (k) (34) To express e f (k) in terms of θ, note that y f (k + d) =q n+ m BB m ˆR(k)+BŜ(k)u c(k) b u(k)+ϕ T (k)θ = ]B m u c (k) q b n u(k)+ϕ T (k)ˆθ(k) (35) Combining (27) (35) yields b u(k)+ϕ T (k)ˆθ(k) =q n+ B m u c (k) (36) In a similar manner the reference model can be written in the (2n ) th order non-minimal fraction form (see Figure 2(b)) From (3) it follows that m Bξ m (k n) =B m u c (k), (3) y m (k) =Bξ m (k n) (32) q n+ B m u c (k) =q n+ m Bξ m (k n) = q 2n+ Pξ m (k) (33) Using (3) (33), the d-step ahead filtered output error can now be written as where e f (k + d) =y f (k + d) q n+ B m u c (k) = q 2n+ Pξ e (k), (34) ξ e (k) = ξ(k) ξ m (k) (35) Next define plant, reference model, model-matching error states by Then Since x(k) =ξ(k ) ξ(k 2n + )] T, (36) x m (k) =ξ m (k ) ξ m (k 2n + )] T, (37) x e (k) = x(k) x m (k) (38) x e (k) =ξ e (k ) ξ e (k 2n + )] T (39) ξ e (k) =qξ e (k ) ] = q q 2n+ P ξ e (k ) + e f (k + d) b b a state equation for x e in controllable canonical form is given by From (27), (34) (36) it follows that e f (k + d) = y f (k + d) q n+ B m u c (k) = (37) To formulate the model matching error dynamics we note that the plant (22) can be written in the n th order fraction form as 5] (see Figure 2(a)) ξ(k n) =u(k), (38) y(k) =Bξ(k n) (39) From (25) (39) it follows that y f (k + d) =q n+ m y(k) =q n+ m Bξ(k n) = q 2n+ Pξ(k) (3) 376 x e (k +)=x e (k)+ b Be f (k + d), k, (32) where = p /b p 2n /b I (2n 2) (2n 2),B = Note that is asymptotically stable lternatively, using (37) the model matching dynamics (32) can be written as x e (k +) = x e (k) b B, k (32) Next we show that the state x defined in (36) is related to ϕ through a nonsingular transformation

4 Lemma 3 The plant state x defined by (36) the regressor (36) are related by ϕ(k) =M x(k), (322) where the nonsingular matrix M R (2n ) (2n ) is given by a a 2 a n (n 2) (n 2) a a 2 a n (d ) b b b m (n ) (n+d 2) b b b m Proof It follows from (29), (38) (39) that a a n ]ξ(k ) ξ(k n )] T a ϕ(k)= a n ]ξ(k n+) ξ(k 2n+)] T b b b m ]ξ(k d) ξ(k n)] T b b b m ]ξ(k n d+) ξ(k 2n+)] T (323) From (36) (323) it follows that ϕ(k) =M x(k) It can be seen that M is the (2n ) (2n ) submatrix of M T, formed by omitting the first row first column of M T Note that det M = det M Since B are relatively co-prime by assumption, it follows that M is nonsingular, thus M is nonsingular 4 Projection daptive Control lgorithm Consider the cost function ) = J λ (ˆθ(k), ˆθ(k ),k 2 ˆθ(k) ˆθ(k ) 2 2+ ] λ y f (k) b u(k d) ϕ T (k d)ˆθ(k), (4) Define 5 Lyapunov Stability Θ(k) = θ(k) θ(k + d ) (5) Then the error state vector consisting of the model matching error states the parameter identification error states is defined by ] T X(k) = xe (k) (52) Θ(k) Using (33),(37), (32) (44) the closed-loop error dynamics with projection identification can be represented in the state space form, for k, by x e (k +)=x e (k) B, (53) b θ(j +)= I ϕ(j d ] +)ϕt (j d +) θ(j), ϕ T (j d +)ϕ(j d +) (54) where j = k,k+ d Remark 5 The error system (53)-(54) is time varying since the regressor ϕ(k) is a function of the exogenous signal u c (k) lso, the origin is an equilibrium of the error system Remark 52 lthough the future parameter errors θ(k + ) to θ(k+d ) are not computed by the algorithm at time k, they are included in the state vector X(k) to facilitate the stability analysis To demonstrate that the origin of the system (53)-(54) is Lyapunov stable, the following results are required Fact 5 Define V θ( θ) = θ T θ (55) where λ is the Lagrange multiplier recursive expression for ˆθ(k) that minimizes (4) is given by 2, p 5] ˆθ(k) =ˆθ(k )+ ] ϕ(k d) y f (k) b u(k d) ϕ T (k d)ˆθ(k ) ϕ T (k d)ϕ(k d) (42) Therefore, the adaptive control law is u(k) = ] ϕ T (k)ˆθ(k) q n+ B m u c (k) (43) b with the parameter update ˆθ(k) = ϕ(k d)y f (k) b u(k d) ϕ {ˆθ(k )+ T (k d)ˆθ(k )],ϕ(k d), ϕ T (k d)ϕ(k d) ˆθ(k ),ϕ(k d) = (44) 377 = θ T (k +) θ(k +) θ T (k) θ(k) V θ(k) Then for k (See 2, p 5]) ϕ T (k d +) θ(k) V θ(k) ϕ T (56) (k d +)ϕ(k d +) Lemma 5 For all k>, ϕ T (k)ϕ(k) k+d ϕ T (i d +)ϕ(i d +) Proof From successive self substitutions of (54) it follows that θ(k+d )= θ(k) k+d 2 ϕ(i d+)ϕ T (i d+) θ(i) ϕ T (57) (i d+)ϕ(i d+)]

5 Dividing both sides of (57) by ϕ T (k)/ ϕ T (k)ϕ(k) yields ϕ T (k)ϕ(k)] = ϕt (k) θ(k + d ) /2 ϕ T (k)ϕ(k)] /2 k+d 2 ϕ T (k)ϕ(i d +) + ϕ T (k)ϕ(k)] /2 ϕ T (i d +)ϕ(i d + )] The triangle inequality implies ϕ T (k)ϕ(k)] /2 ϕ T (k) θ(k + d ) ϕ T (k)ϕ(k)] /2 k+d 2 ϕ T (k)ϕ(i d +) + ϕ T (k)ϕ(k)] /2 ϕ T (i d +)ϕ(i d + )] /2 ϕ T (i d +)ϕ(i d + )] /2 Now using the Cauchy-Schwarz inequality we have ϕ T (k)ϕ(k)] /2 = k+d Lemma 52 Define ϕ T (i d +)ϕ(i d + )] /2 V Θ( Θ) = Θ T Θ (58) V Θ(k) = Θ T (k +) Θ(k +) Θ T (k) Θ(k) (59) Then ϕ(k) θ(k) V Θ(k) ϕ T, (k)ϕ(k) k (5) Proof From (5) (59) it follows that V Θ(k) = k+d + k+d θ T (i) θ(i) θ T (i) θ(i) (5) Use of Lemma 5 Lemma 5 yields k+d V Θ(k) = ϕ T (i d +)ϕ(i d +) ϕ T (k)ϕ(k) Lemma 53 Let P, R R n n > be positive-definite matrices that satisfy let P = T P+ R + I, (52) σ = Furthermore let µ> define λ max ( T P) (53) 378 Then for k V xe (x e ) = ln( + µx T e Px e ) (54) V xe (k) = V xe (x e (k + )) V xe (x e (k)) V xe (k) x T e (k)rx e (k)+b 2 (σ2 +)B T PB ϕ(k) θ(k) µ +µx T e (k)px e (k) Proof Define Then, F = σ P /2, G = σp /2 B, J(x e ) = x T e Px e J xe (k) = x T e (k +)Px e (k +) x T e (k)px e (k) Omitting the explicit dependence on k we have J xe (k) =x T e T Px e x T e T PBb ϕt θ b ϕt θb T Px e + b ϕt θb T PBb ϕt θ x T e Px e dding subtracting b 2 (ϕt θ) 2 G T G yields J xe (k) =x T ( e T P P + F T F ) x e + ( B T PB + G T G ) b 2 (ϕt 2 θ) x T e b ϕt θ ] ] F T F F T G x T ] G T F G T e G b ϕt θ x T ( e T P P + F T F ) x e + ( B T PB + G T G ) b 2 (ϕt θ) 2 (55) Noting that G T G = σ 2 B T PB F T F λ ( max T P ) I n λ max ( T = I n, P) it follows from (52) (55) that J xe (k) x T e (k)rx e (k)+(σ 2 +)B T PBb 2 Now, since ln λ λ for all λ>, ϕ(k) θ(k) V xe (k) x T e (k)rx e (k)+b 2 (σ2 +)B T PB ϕ(k) θ(k) µ +µx T e (k)px e (k) We now present the main stability result Theorem 5 ssume that the reference signal u c (k) is bounded Then the origin of the error system (53)-(54) is Lyapunov stable, y(k) y m (k) as k Proof Consider the Lyapunov function cidate V (X) = av xe (x e )+V Θ( Θ) (56)

6 Let P, R R n n > be positive definite satisfy (52), let a> Then using lemmas it follows that V (k) = V (X(k + )) V (X(k)) x T e (k)rx e (k)+(σ 2 +)B T PB aµ +µx T e (k)px e (k) ϕ T (k)ϕ(k) Now from (322) it follows that ϕ(k) θ(k) /b 2 ϕ T (k)ϕ(k) =x T M T M x x T e M T M x e + x T mm T M x m (57) Let µ > satisfy µ P>M T M (58) By assumption, the comm signal u c (k) is bounded the m is stable It thus follows that there exists β> such that x T m(k)x m (k) β (59) Using (57)-(59) we have ϕ T (k)ϕ(k) µ x T e Px e + βµ λ max (P ) (52) Defining µ = µ /( + βµ λ max (P )) a = ( b 2 /( + µ σ 2 + ) B T PB) we have x T e (k)rx e (k) V µa +µx T e (k)px e (k) (52) Since V (X) is positive definite radially unbounded it follows from (52) that the origin of the error system (53)- (54) is Lyapunov stable Furthermore, using Theorem of 2] it follows that x e (k) as k Then using (39) (32) we have that y(k) y m (k) as k 6 Example Example 6 Consider the unstable minimum phase SISO plant with relative degree d =2given by y(q) u(q) = q +5 q 3 + q 2 + q +5 (6) To track reference signals we choose an FIR filter as the reference model We require deg m =2n m =4 deg B m = deg m d =2 Let the reference model be y m (q) u c (q) = q2 q 4 (62) let u c (k) be a square wave with period of samples The plant (6) with the control law (43) the parameter update (44) is simulated in MTLB The simulation results are shown in Figure Output Time in Samples Fig 3 Tracking Performance REFERENCES ] P Ioannou J Sun, Robust daptive Control, Prentice Hall, 996 2] G C Goodwin K S Sin, daptive Filtering Prediction Control Prentice Hall, 984 3] S khtar D S Bernstein Discrete-Time daptive Stabilization Disturbance Rejection for Minimum Phase Plants, Submitted to Conf Dec Contr 25 4] G C Goodwin, P J Ramadage P E Caines, Discrete-Time Multivariable daptive Control, IEEE Trans utom Contr, vol 25, pp , 98 5] R Becker P R Kumar, daptive Control with the Stochastic pproximation lgorithm: Geometry Convergence, IEEE Trans utom Contr, vol C-3, NO 4, pp , 985 6] P R Kumar, Convergence of daptive Control Schemes Using Least Squares Parameter Estimates, IEEE Trans utom Contr, vol 35, pp , 99 7] L Guo, H Chen, The strom-wittenmark Self Tuning Regulator Revisited the ELS-Based daptive Trackers, IEEE Trans utom Contr, vol 36, pp 82-82, 99 8] R Johansson, Global Lyapunov Stability Exponential Convergence of Direct daptive Control, Int J Cont, Vol 5, No 3, pp , 989 9] R Johansson, Lyapunov Functions for daptive Systems, Proc of the 22nd IEEE Conf Dec Contr, pp , 983 ] R Johansson, Stability of Direct daptive Control with RLS Identification, Modelling, Identification Robust Control, edited by CI Byrnes Lindquist (msterdam: North Holl), pp ] R Johansson, Supermartingale nalysis of Minimum Variance daptive Control, Control-Theory dvanced Technology, Vol, No 4, Part 2, pp 993-3, 995 2] S khtar, R Venugopal D S Bernstein, Logarithmic Lyapunov Functions for Direct daptive Stabilization with Normalized daptive Laws, Int J Cont, Vol 77, No 7, pp , 24 3] W M Haddad, T Hayakawa Leonessa, Direct daptive Control for Discrete-Time Non-Linear Uncertain Dynamical Systems, Proc mer Contr Conf, pp , 22 4] S khtar, R Venugopal D S Bernstein Discrete-Time Direct daptive Control, Proc mer Contr Conf, pp 24-29, 23 5] R H Middelton G C Goodwin, Digital Control Estimation: Unified pproach Prentice Hall, 99 6] E lbert, Regression the Moore-Penrose Pseudoinverse New York: cademic Press, 972 7] H K Khalil, Nonlinear Systems, Upper Saddle River, NJ: Prentice Hall, 996