H 2 Adaptive Control. Tansel Yucelen, Anthony J. Calise, and Rajeev Chandramohan. WeA03.4

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1 1 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July, 1 WeA3. H Adaptive Control Tansel Yucelen, Anthony J. Calise, and Rajeev Chandramohan Abstract Model reference adaptive control laws are derived using anh optimization approach. In addition, this leads to H forms of well known σ- and e- modification based adaptive control architectures. It is shown thath adaptive control leads to smaller tracking errors without requiring high adaptation gain. Instead it provides an optimal time varying adaptation gain that results in improved performance without tuning of the adaptation gain. I. INTRODUCTION There have been a number of efforts in the recent years focused on improving model reference adaptive control (MRAC) laws [1] [9]. These approaches involve approximately enforcing linear constraints on the weights in the adaptive law, and are commonly referred as composite adaptation [1]. In general, the gradient of a norm of the error in the linear constraints produces terms that are used to modify an existing MRAC law. However, using a gradient method can result in slow parameter convergence towards a local minimum [11]. In addition, gradient based modification terms have a fixed adaptation gain, that often have to be chosen large to obtain satisfactory results, which can interact negatively with unmodeled dynamics, and amplify the effect of sensor noise. To overcome these problems, a Kalman filter (KF) based optimization approach has been proposed to approximately enforce linear constraints on the weights [1] [13]. This approach to deriving modification terms from the imposed linear constraints is shown to give better system response in tracking, when compared with gradient based modification terms. Furthermore, the KF optimization based approach results in a variable gain that does not require tuning, and gives better results than a fixed gain based adaptive control law, especially under system uncertainties and failures. H adaptive control (H -AC) proposed in this paper is an extension of the KF optimization approach. Here, we generalized the linear constraint assumption [1] [13], and approximately enforce this constraint on the weights by employing H optimization. This generalization allows one to replace a gradient based adaptive control law with its H form, rather than simply replace the modification term as was done using the KF approach. It is shown that the system response with H -AC law leads to smaller tracking errors without requiring high adaptation gains when compared with This research is supported by NASA Langley Research Center, under grant number NNX8AC61A. T. Yucelen, A. J. Calise, and R. Chandramohan are with the School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA, 333-1, USA. s: tansel@gatech.edu, anthony.calise@aerospace.gatech.edu,rajeev.chandramohan@gatech.edu. its counterpart gradient based adaptive control form. We further develop improved forms of well known σ- [1] and e- modification [] based adaptive control architectures. Both of these architectures can be interpreted as the gradient of a norm measure of the error in the linear constraint, and we use the corresponding linear constraint to develop their counterpart H -AC versions. In what follows, we will use the following notations: tr[a] is the trace of matrix A, denotes vector norm, denotes matrix Frobenius norm, λ min (A) denotes the smallest eigenvalue of matrix A, and λ max (A) denotes the largest eigenvalue of matrix A. The organization of the paper is as follows. Section II provides a brief background needed for the rest of the paper. Section III describes the main concept for H -AC. Section IV presents an illustrative example on a model of wing rock dynamics, while Section V summarizes the conclusions. II. PRELIMINARIES AND MOTIVATIONS In this section, we present necessary results for the MRAC problem. Specifically, consider the controlled nonlinear uncertain dynamical system given by ẋ Ax + B[u + (x)], (1) where x R n is the state vector, u R m is the control input, A R n n and B R n m are known matrices, and : R n R m is the unknown matched uncertainty. Furthermore, we assume that the full state is available for feedback and the control input u is restricted to the class of admissible controls consisting of measurable functions. In order to achieve trajectory tracking, we construct a reference system given by ẋ m A m x m + B m r, () where x m R n is the reference state vector, r R r is a bounded piecewise continuous reference input, A m R n n is Hurwitz, and B m R n r. The following assumption is needed for the main results of this section. Assumption.1. The unknown matched uncertainty in (1) can be linearly parameterized as (x) W T β(x) + ε(x), (3) where W R s m is the unknown constant weight matrix, β : R n R s is the bounded basis function vector of the form β(x) [1, β 1 (x), β (x),..., β s 1 (x)] T R s 1, where β i (x), a 1+e a i x i >, i 1,...,s 1, and ε : R n R m is the residual error that satisfies ε(x) < ǫ on a sufficiently large bounded set in the state space /1/$6. 1 AACC 111

2 Now, consider the following control law u u n u ad, () where u n is the nominal (or baseline) control law given by u n K 1 x + K r, () with K 1 R m n and K R m r are nominal control gains, and u ad is the adaptive control law given by u ad Ŵ T β(x), (6) with Ŵ Rs m is an estimate of W obtained from the adaptive weight update law Ŵ γ[β(x)e T PB + Ŵ m ], (7) where γ is a positive fixed gain, e x x m is the error, P R n n is a positive-definite solution of the Lyapunov equation A T m P + PA m + L, (8) for any L L T >, and Ŵ m R s m is a modification term, i.e. Ŵ m σŵ for σ- modification term [1], and Ŵ m σ e Ŵ for e- modification term [], where σ is a positive fixed gain. The following matching conditions are needed for the results in this section. Assumption.. There exist K 1 R m n and K R m r such that A m A BK 1 and B m BK. The system dynamics (1) can now be written as ẋ A m x + B m r B W T β(x) + Bε(x), (9) where W Ŵ W, with the system error dynamics and error weight update given by and ė A m e B W T β(x) + Bε(x), (1) W γ[β(x)e T PB + Ŵ m ]. (11) The next theorem highlights the uniformly ultimately boundedness [1] (UUB) property of the closed-loop system error given by (1) and (11) for the σ- and e- modification cases. Theorem.1. Consider the nonlinear uncertain dynamical system given by (1), with u given by (), and reference model given by (). Then, the closed-loop system error given by (1) and (11) is UUB. Proof. Consider the Lyapunov function candidate V(e, W) 1 et Pe + 1 γ tr[ W T W], (1) where P > satisfies (8). Differentiating (1) along the trajectories of (1) and (11) yields V( ) e T P[A m e B W T β(x) Now, let V( ) +Bε(x)] + tr[ W T β(x)e T PB] +tr[ W T Ŵ m ] 1 et [A T m P + PA m]e +tr[ W T Ŵ m ] + e T PBε(x) 1 et Le + tr[ W T Ŵ m ] +e T PBε(x). (13) Ŵ m σŵ. Then, (13) becomes 1 et T Le σtr[ W Ŵ] +e T PBε(x) c 1 e + c e c 3 W + c (c 1 1) e c 3 W + c + c, (1) where c 1 λmin(l) > 1, c PB ǫ, c 3 σ, and σ W c. Either e Θ e or W Θ W renders V(e, W) <, where Θe Θ W c + c c 3 c + c c 1 1 and. Then, the closed-loop system error given by (1) and (11) is UUB, for the case Ŵ m σŵ. Similar to σ- modification case, consider e- modification case. For this purpose, let Ŵ m σ e Ŵ. Then, (13) becomes V( ) 1 et T Le σ e tr[ W Ŵ] +e T PBε(x) e (d 1 e + d W d 3 ), (1) where d 1 λmin(l), d σ, and d 3 σ W + PB ǫ, and V(e, W) is negative as long as the term in braces is positive. Consequently, either e Ω e, or W Ω W renders V(e, W) <, where Ω e d3 d 1, and Ω W d 3 d. Then, the closed-loop system error given by (1) and (11) is UUB, for the case Ŵ m σ e Ŵ. Remark.1. The UUB property of Theorem.1 is obtained for σ- and e- modification cases for illustrative purposes. In other words, Ŵ m can be selected to be any modification term such that the closed-loop system error given by (1) and (11) satisfies the UUB property. Lemma.1. The gradient based σ- modification adaptive control law given by (7) for Ŵ m σŵ can be obtained by using a gradient approach to enforcing the linear constraint W T ζ 1 ζ (x) (16) on the weights, where ζ 1 σi and ζ (x) 11

3 1 σ β(x)e T PB. Proof. Consider the cost J σ (W) 1 W T ζ 1 ζ (x), (17) where the negative gradient of J (W) with respect to W gives σ- modification adaptive control law in (7) for Ŵ m σŵ. Lemma.. The gradient based e- modification adaptive control law given by (7) for Ŵ m σ e Ŵ can be obtained by using a gradient approach to enforcing the linear constraint W T ξ 1 (x) ξ (x) (18) on the weights, where ξ 1 (x) σ e I and ξ (x) 1 σ e β(x)e T PB. Proof. Consider the cost J e (W) 1 W T ξ 1 (x) ξ (x), (19) where the negative gradient of J (W) with respect to W gives e- modification adaptive control law in (7) for Ŵ m σ e Ŵ. For motivational purposes, we show by Lemma.1 and Lemma. that gradient based σ- and e- modification adaptive control laws can be obtained by imposing the linear constraints given by (16) and (18) on the weights. In the next section, we introduce a different treatment for a general class of linear constraints to obtain H -AC laws. III. H ADAPTIVE CONTROL ARCHITECTURE The H -AC architecture is applicable to most direct adaptive control schemes available in the literature (i.e. [1] [1]), in full state feedback control form or in output feedback control form. For simplicity, and without loss in generality, we consider the state feedback formulation given in Section II. For the main results of this section, the following assumption is needed. Assumption 3.1. The constraint on the ideal weight matrix in an adaptive control design has the linear form W T φ 1 (t, x, u) φ (t, x, u), () where W R s m is an unknown weight matrix, φ 1 ( ) : [, ) R n R m R s l is a given regressor, and φ ( ) : [, ) R n R m R m l is also a given regressor. The problem of estimating W while enforcing the linear constraint in () can be viewed as a standard problem in estimation theory by defining the stochastic process Ẇ D w1 q, y φ 1 (t, x, u) T W + D w q, (1) where q is a zero-mean, Gaussian, white noise process, and it is assumed that D T w1 D w. Now, one can construct an estimator for W as Ŵ A e Ŵ + B e y. () Now, let z be the performance variable given by z E(W Ŵ) Ee w, (3) where E is a weighting matrix and e w W Ŵ. The goal is to maintain z small under q. For this purpose, the dynamics of e w can be given by ė w D w1 q (A e Ŵ + B e y) A e Ŵ B e φ 1 (t, x, u) T W + Dq A e e w + Dq. () where D (D w1 B e D w ) and we selected A e B e φ 1 (t, x, u) T. Then, define the error system in transfer function representation from q to z as G e (s) E(sI A e ) 1 D. () Now, one needs to determine Kalman gain B e by minimizing the L norm of G e (s), J(B e ) G e (s) tr[eset ] tr[sω] (6) where Ω E T E, subject to Ṡ A e S + SA T e + D D T. (7) To optimize Eq. (6) subject to the constraint in Eq. (7), form the Lagrangian [1] L(B e, µ) tr[sω] + tr[µ(a e S +SA T e + D D T Ṡ)], (8) where µ is the Lagrange multiplier. Now, the covariance update law and the Kalman gain follows from L(B e, µ) S Ṡ A es + SA T e + D D T (9) L(B e, µ) B e B e Sφ(t, x, u)r 1. (3) where R D w Dw T >. Since the objective is to approximately enforce the constraint in (), the logical choice for y when employing this estimator is y. Hence, it follows from () that H -AC can be given as Ŵ Sφ 1 (t, x, u)r 1 [ φ 1 (t, x, u) T Ŵ φ (t, x, u) T], (31) where S satisfies the differential Riccati equation given by Ṡ Sφ 1 (t, x, u)r 1 φ 1 (t, x, u) T S + Q, (3) where Q D T w1 D w1 >. Now, the following result presents the necessary condition for the solution of (3), S, to be positive definite. The stability proof of the H -AC law given by (31) and (3) depends on the constraint given by (), hence it is problem dependent. Therefore, we need to limit our presentation by giving the H -AC forms of well known σ- 113

4 and e- modification based adaptive control laws. However, we first need to state the following lemmas for the main results of this section. Lemma 3.1. The solution of (3), S, exists, and is symmetric and nonnegative definite for all t. In addition, if S(τ) is positive definite for some τ, then S is positive definite for all t > τ, or if Q is positive definite, then S is positive definite for all t. Proof. See Proposition 1.1 in [16]. Since Q is defined to be a symmetric and positive definite matrix in (3), then the solution of (3), S, is positive definite for all t by Lemma 3.1. Next, we present the boundedness property for the solution of (3), S. Lemma 3.. The solution of (3), S, is uniformly bounded. Proof. See Theorem A. in [13]. Now, we are ready to present H -AC forms of well known σ- and e- modification based adaptive control laws. From the constraint (16) that satisfies Assumption 3.1, and from the equations (31) and (3), σ- modification based H -AC (H σ - AC) law can be given by Ŵ SR 1[ β(x)e T PB σŵ], (33) Ṡ σsr 1 S + Q. (3) Similarly, from the constraint (18) that satisfies Assumption 3.1, and from the equations (31) and (3), e- modification based H -AC (H e -AC) law can be given by Ŵ SR 1[ β(x)e T PB σ e Ŵ],(3) Ṡ σ e SR 1 S + Q. (36) It is important to note here that Γ SR 1 is the positive definite, variable, and uniformly bounded H gain in both H σ-ac and He -AC laws. Theorem 3.1. The nonlinear uncertain dynamical system given by (1), with the control law given by (), and with the H σ -AC law in (33), is UUB. Proof. Consider the Lyapunov function candidate V(e, W) 1 et Pe + 1 tr[ W T Γ 1 W], (37) where P > satisfies (8). The time derivative of (37) can be expressed as V( ) 1 et T Le σtr[ W Ŵ] + 1 [ W T (σi M) W] +e T PBε(x) (c 1 1) e ĉ 3 W + (c + c ) (38) where ĉ 3 c 3 η σ >, η σ λmax(σi M), and M (SR 1 ) QR 1 > by Lemma 3.1. Note that Lemma.1 in [17] is used in (38). Either e Θ e or W ˆΘ W renders V(e, W) <, where Θ e c + c c 1 1 and c ˆΘ W + c ĉ 3. Therefore, the closed-loop signals e and W are UUB. Theorem 3.. The nonlinear uncertain dynamical system given by (1), with the control law given by (), and with the H-AC e law in (3), is UUB. Proof. Consider the Lyapunov function candidate V(e, W) 1 et Pe + 1 tr[ W T Γ 1 W], (39) where P > satisfies (8). The time derivative of (39) can be expressed as V( ) 1 et T Le σ e tr[ W Ŵ] + 1 [ W T (σ e I M) W] +e T PBε(x) e (d 1 e + ˆd W d 3 ) () where ˆd d η e >, η e λmax(σ e I M), and M (SR 1 ) QR 1 > by Lemma 3.1. Note that Lemma.1 in [17] is used in (). Consequently, either e Ω e, or W ˆΩ W renders V(e, W) <, where Ω e d3 d 1, and ˆΩ W d3 ˆd. Therefore, the closed-loop signals e and W are UUB. IV. ILLUSTRATIVE EXAMPLE In this section we compare standard gradient based e- modification adaptive control with the H-AC e law, on a model of wing rock dynamics [], [18], given by ] [ ] [ ] [ ] [ẋ1 1 x1 [u ] + + Π(x), (1) ẋ x 1 where Π(x) α +α 1 +α x +α 3 x + α x x + α x 3 1 with α, α 1 1.1, α 1.98, α 3.6, α.9, and α.1. In (1), represents the roll angle, and x represents the roll rate. In this example, the control objective is to minimize the oscillations of the wing rock dynamics in order to stabilize the system at the zero trim condition. Therefore, the roll command is considered to be zero. We selected the initial state values as x() [1 o, o /s] T, and the design parameters as L I for e:6, and Q.1I and R.1I for H e -AC law given by (3) and (36). The reference system is selected to be second order with a natural frequency of 1. rad/s, and a damping of.8. Figures 1-7 present the results. Fig. 1 shows the nominal (or baseline) control response with the matched uncertainty Π(x), where u ad. Figs. - presents the standard e- modification based adaptive controller results on this system for different gains. It is obvious from all these figures that the achieved system performance is not desirable due to high frequency oscillations in the system responses. Fig. 6 shows the performance of H e-ac law on this system, where its variable H gain is further given in Fig. 7. The performance is significantly 11

5 3 1 x x Fig. 1. Nominal control response. u x u W u W Fig.. Standard e- modification based adaptive controller, γ 1 and γ σ 1. improved relative to the results obtained using the standard e- modification term. V. CONCLUSION The intent of this paper has been to present a procedure for constructing a direct adaptive control law, based on H optimization. The result using a model of wing rock dynamics illustrates the presented theory for the case of H e- modification based adaptive control, and this controller shows significant improvement over the standard (gradient) e- modification based adaptive controller. One key difference in the H based approach to direct adaptive control laws is that the resulting gain is time varying. The proposed H approach can be used in place of all direct adaptive control Fig. 3. Standard e- modification based adaptive controller, γ and γ σ x u W Fig.. Standard e- modification based adaptive controller, γ 1 and γ σ 1. 11

6 1 1 1 x u W Fig.. Standard e- modification based adaptive controller, γ and γ σ x u W H Gain 1 S R Fig. 7. Variable H gain of H e -AC law in Fig. 6. laws that can be equivalently viewed as the gradient of a norm of the error in the linear constraint. REFERENCES [1] Ioannou, P., Kokotovic, P., Instability analysis and improvement of robustness of adaptive control, Automatica, Vol., No., pp. 83-9, 198. [] Narendra, K., Annaswamy, A., A new adaptive law for robust adaptation without persistent excitation, IEEE Trans. on Automatic Control, Vol. 3, No., pp. 13-1, [3] Calise, A. J., Rysdyk, R., Nonlinear adaptive flight control using neural networks, IEEE Control Systems Magazine, Vol. 18, No. 6, [] Johnson, E. N., Oh, S. -M., Adaptive control using combined online and background learning neural network, in AIAA Guidance, Navigation and Control Conference,. [] Volyanskyy, K. Y., Calise, A. J., Yang, B. -J., A novel Q- modification term for adaptive control, Proc. IEEE American Contr. Conf., Minneapolis, MN, 6. [6] Volyanskyy, K. Y., Calise, A. J., Yang, B. -J., Lavretsky, E., An error minimization method in adaptive control, AIAA Guidance, Navigation and Control Conference, 6. [7] Nguyen, N., Krishnakumar, K., Boskovic, J., An optimal control modification to model-reference adaptive control for fast adaptation, AIAA Guidance, Navigation and Control Conference, 8. [8] Calise, A. J., Yucelen, T., Muse, J. A., Yang, B. -J., An loop recovery method for adaptive control, AIAA Guidance, Navigation and Control Conference, 9. [9] Volyanskyy, K. Y., Haddad, W. M., Calise, A. J., A new neuroadaptive control architecture for nonlinear uncertain dynamical systems: Beyond σ- and e- modifications, accepted to IEEE Trans. on Neural Networks. [1] Slotine, J. -J. E., Li, W., Applied nonlinear control, Prentice Hall, [11] Boyd, S., Vandenberghe, L., Convex optimization, Cambridge University Press, Cambridge, NY,. [1] Yucelen, T., Calise, A. J., A Kalman filter optimization approach to direct adaptive control, AIAA Guidance, Navigation and Control Conference, 9. [13] Yucelen, T., Calise, A. J., A Kalman filter optimization approach to direct adaptive control, submitted to AIAA Journal of Guidance, Control, and Dynamics. [1] Haddad, W. M., Chellaboina, V., Nonlinear dynamical systems and control: A Lyapunov-based approach, Princeton, 8. [1] Bernstein, D. S., Matrix mathematics: Theory, facts, and formulas with applications to linear systems theory, Princeton, NJ,. [16] Dieci, L., Eirola, T., Positive definiteness in the numerical solution of Riccati differential equations, Numer. Math., Vol. 67, pp , 199. [17] Fang, Y., Loparo, K. A., Feng, X., New estimates for solutions of Lyapunov equations, IEEE Transactions on Automatic Control, Vol., 1997, pp [18] Singh, S. N., Yim, W., Wells, W. R., Direct adaptive and neural control of wing-rock motion of slender delta wings, Journal of Guidance, Control, and Dynamics, Vol. 18, 199, pp. -3. Fig. 6. H e- modification based adaptive controller (H e -AC), σ