Adaptive Output Feedback Based on Closed-Loop Reference Models for Hypersonic Vehicles

Transcription

1 Adaptive Output Feedback Based on Closed-Loop Reference Models for Hypersonic Veicles Daniel P. Wiese and Anurada M. Annaswamy Massacusetts Institute of Tecnology, Cambridge, MA 02139, USA Jonatan A. Muse and Micael A. Bolender U.S. Air Force Researc Laboratory, 2210 Eigt St., Wrigt-Patterson Air Force Base, Oio 45433, USA Eugene Lavretsky Te Boeing Company, Huntington Beac, CA 92647, USA Tis paper presents a new metod of syntesizing an output feedback adaptive controller for a class of uncertain, non-square, multi-input multi-output systems tat often occur in ypersonic veicle models. Te main callenge tat needs to be addressed is te determination of a corresponding square and strictly positive real transfer function. Tis paper proposes a new procedure to syntesize two gain matrices tat allows te realization of suc a transfer function, tereby allowing a globally stable adaptive output feedback law to be generated. Te unique features of tis output feedback adaptive controller are a baseline controller tat uses a Luenberger observer, a closed-loop reference model, manipulations of a bilinear matrix inequality, and te Kalman- Yakubovic Lemma. Using tese features, a simple design procedure is proposed for te adaptive controller, and te corresponding stability property is establised. Te proposed adaptive controller is compared to te classical multi-input multi-output adaptive controller. A numerical example based on a 6 degree-of-freedom nonlinear, scramjet powered, blended wing-body generic ypersonic veicle model is presented. Te adaptive output feedback controller is applied to result in stable tracking of uncertainties tat destabilize te baseline linear output feedback controller. I. Introduction A state feedback linear quadratic regulator (LQR) baseline controller wit integral action and augmented wit an adaptive component as proven to be an effective coice for accommodating te parametric uncertainties present in fligt control applications, and ensuring satisfactory reference tracking. 1 7 However, suc a controller requires tat te state is measurable, wic may not always be possible. Also, inaccuracies in te system output measurements may render state feedback controllers sensitive to measurement errors and tus not applicable. For tese reasons tere as been an increasing drive to develop an adaptive output feedback extension of te robust integral-augmented LQR baseline plus adaptive controller. Existing classical metods of multi-input multi-output (MIMO) adaptive control are applicable for plants tat are square. An m m transfer matrix is used to represent te dynamic beavior of te plant, and te existence of a stable adaptive solution depends on te available prior information about tis plant transfer matrix. 8, 9 Te solution relies on non-minimal controller representations to dynamically decouple te plant, and te controller structure consists of a feedforward gain and two filters in te feedback pat, te order of wic depends on m and an upper bound on Graduate Student, Mecanical Engineering, 77 Massacusetts Avenue Rm 3-441, Student Member AIAA Senior Researc Scientist, Mecanical Engineering, 77 Massacusetts Avenue Rm 3-348, Member AIAA Researc Aerospace Engineer, Autonomous Control Branc, Senior Member AIAA Senior Aerospace Engineer, Autonomous Control Branc, Associate Fellow AIAA Boeing Senior Tecnical Fellow, Associate Fellow AIAA 1

2 te observability index of te plant,. Te resulting classical MIMO adaptive solution will introduce 2m controller states and 2m 2 adjustable parameters. More recent metods of MIMO output feedback adaptive control ave adopted a Luenberger observer-based approac in wic a minimal observer is used to generate a state estimate to use for feedback control. 7, Tis observer also serves as te reference model wic is used by te adaptive controller, and te presence of te observer feedback gain L provides te structure known as te closed-loop reference model, or CRM Tese CRM based approaces ave relied on te so-called squaring-up procedure 17 to add fictitious inputs to a tall system (one were te dimension of te output is greater tan te dimension of te input) making it square and ensuring any transmission zeros are stable. Tese fictitious inputs are used only to syntesize a postcompensator S 1 and te CRM gain L wic ultimately render a set of underlying error dynamics strictly positive real (SPR). Tese SPR error dynamics allowed stable update laws to be cosen to guarantee system stability. We note tat systems wit transmission zeros cannot be squared up using te metod as described in Reference [17], wic as led to a recent modification to overcome tis limitation and allow te design of output feedback controllers for systems wit stable transmission zeros. 18 Te CRM based output feedback design procedure proposed in tis paper takes an alternative approac to syntesizing S 1 and L wic does not require te system first be squared-up. Instead, te postcompensator S 1 is determined as a generalized inverse of te system matrices, and a state feedback approac is used to stabilize a related lower order plant subsystem. Tis results in a feasible linear matrix inequality (LMI) wic is solved to yield L. We consider in tis work te case of tall systems, but te case of wide systems olds by duality. Furtermore, because L is a component of bot te baseline and adaptive controllers, it is crucial tat it be selected to provide good frequency domain properties for te baseline control system, as well as desirable adaptive control performance. Tis procedure is able to exploit te structure of te given system to obtain a large amount of freedom in te selection of L in order to acieve a robust baseline control design and te desired adaptive performance. In Section II we introduce te class of uncertain plants wic we wis to control, propose a control arcitecture, and formulate te control problem of interest. In Section III we provide some preliminaries wic will be used in te syntesis of a globally stable controller for te uncertain system. Section IV provides a constructive procedure for obtaining an update law for an adaptive controller wic guarantees global stability, and compares te proposed controller to te existing classical controller. Finally, a numerical example is presented in Section V to illustrate te 6, 19 efficacy of tis metod wen applied to a 6-DOF nonlinear generic ypersonic veicle (GHV) model. II. Control Problem Formulation Consider te following MIMO uncertain open-loop system ẋ p = A p x p + B p ( u + y p = C p x p z = C pz x p + D pz ( u + > p x p ) > p x p ) were A p 2 R np np, B p 2 R np m, C p 2 R` np, C pz 2 R ne np, D pz 2 R ne m are constant known matrices. Te nonsingular matrix 2 R m m and p 2 R m np, wic represents constant matced uncertainty weigts tat enter te system troug te columns of B p, are unknown. Te measured output is y p, te regulated output is z, and te number of regulated outputs cannot exceed te number of inputs, tat is n e apple m. Te goal is to design a control input u wic will make z track te reference command z cmd wit bounded errors in te presence of te uncertainties and p. We make te following assumptions about te system in (1). Assumption 1 A) (A p,b p ) is controllable. B) (A p,c p ) is observable. C) B p, C p, and C p B p are full rank. D) Any finite transmission zeros of (A p,b p,c p, 0) are strictly stable, and te rank of te following matrix is full " #! rank A p C pz B p D pz = n p + n e (1) 2

3 E) (a) is nonsingular and diagonal wit entries of known sign (b) k pk 2 < < 1, were is known In order to facilitate command tracking, we introduce integral action, and for tis purpose an additional state x e is defined as ẋ e = z cmd z (2) Tis integral error state is appended to te plant in (1) leading to te following augmented open-loop dynamics " # " #" # " # " # ẋ p A = p 0 x p B + p > 0 ( u + p x p )+ z cmd ẋ e C pz 0 x e D pz I " # " #" # (3) y p C = p 0 x p 0 I x e Te system in (3) can be written more compactly as follows x e ẋ = Ax + B( u + y = Cx > x)+b cmd z cmd (4) were =(A, B, C, 0), and were A 2 R n n, B 2 R n m, B cmd 2 R n ne, and C 2 R p n are te known matrices given by " # " # " # " # A A = p 0 np n e B B = p 0 B cmd = np m C C = p 0` ne C pz 0 ne n e D pz I ne n e 0 ne n p I ne n e and were =[ > p 0 m ne ] > is unknown. Note tat p = ` + n e. We make te following assumptions about te system in (4), wic are equivalent to Assumption 1 related to te plant in (1): Assumption 2 a) (A, B) is controllable. b) (A, C) is observable. c) B, C, and CB are full rank. d) Any finite transmission zeros of are strictly stable. e) (a) is nonsingular and diagonal wit entries of known sign (b) k k 2 < < 1, were is known f) is tall: p>m. Remark 1 Te system in (1) satisfying Assumption 1A-D wen augmented wit te integral error state as sown in (3) also satisfies Assumption 2a-d. In oter words, under Assumption 1A-D, integral error augmentation does not destroy controllability, observability, or te rank conditions. Nor does it add any transmission zeros. 10 Remark 2 Assumptions 2a and 2b are standard. Assumption 2c implies tat inputs and outputs are not redundant, as well as a MIMO equivalent of relative degree unity. Assumption 2d is a standard requirement for output feedback adaptive control. Assumption 2f can be considered witout loss of generality as te case of wide systems p<molds by duality. Te case of square systems as been given in Reference [20] and is discussed in Section IV. A. Baseline Control Design Te underlying problem ere is to design a control input u in (4) so tat te closed-loop system as bounded solutions and z tends to z cmd wit bounded errors in te presence of te uncertainties and. In tis section, we describe te baseline control design for te nominal case wen tere are no uncertainties present, tat is wen =I and =0. 3

4 A controller along te lines of References [10 12] is proposed, as it leads to a low order robust controller. Tis controller includes a Luenberger observer togeter wit LQR feedback control gains. As our ultimate goal is to develop an adaptive controller wic in turn requires a reference model, we propose a control design were te reference model as components of an observer as well. In particular, we introduce a feedback component into te reference model, wit te corresponding feedback gain L cosen similar to a Luenberger gain, tat is, so tat it ensures adequate stability margins for te nominal closed-loop system. Te resulting reference model is referred to as a closed-loop reference model (CRM) wic as been sown recently to result in igly desirable transient properties and is described as follows: ẋ m = A m x m + B cmd z cmd + L(y m y) y m = Cx m (5) were A m = A + BK x > and K x is selected suc tat A m is Hurwitz. Furtermore, K x sould be selected to provide te desired closed-loop performance of te nominal system. Wit suc a K x, we can propose te following baseline controller tat can guarantee command tracking and a certain amount of stability margins for te nominal closed-loop system. u bl = K x > x m (6) B. Adaptive Controller Wit te baseline controller determined as above, te next step is to design an adaptive controller in te presence of 6= I and 6= 0. Suppose we augment te nominal controller in (6) as u = K x + (t) > x m (7) were (t) is to be determined by a suitable update law. Te question is if te introduction of te parameter as in (7) is sufficient to accommodate te parametric uncertainties. For tis purpose, we introduce a matcing condition as described in Remark 3 below. Remark 3 (Matcing condition) Te selection of te reference model state matrix as A m = A + BK x > guarantees te existence of a parameter tat satisfies te following matcing condition. were is given by A m = A + B > + B > + K > x > =( 1 I)K > x 1 > Given a system satisfying Assumption 2, te matcing condition in Remark 3, and te proposed control arcitecture, te reference tracking control problem is reduced to selecting te CRM gain L in (5) and a suitable adaptive law for updating (t) in (7). In summary, te problem tat is addressed in tis paper is te determination of an adaptive augmented robust baseline output feedback controller as in (7) to control te plant in (4) using te CRM/Observer as in (5). Tis in turn necessitates finding an adaptive law for adjusting in (7) and te observer gain L in (5). Te main tools used for determining te adaptive controller involve te Kalman-Yakubovic 9 and matrix elimination lemmas, 21 wic elp reduce te problem of finding L to a state feedback problem of a related lower order subsystem. Preliminaries pertaining to tese tools are described in Section III. Te complete adaptive control design and te corresponding stability result are presented in Section IV, and a numerical example is presented in Section V. III. Preliminaries Te following well-known lemma gives necessary and sufficient conditions to ensure tat te system (A, B, C, 0) is SPR. Lemma 1 (Kalman-Yakubovic) Given te strictly proper transfer matrix G(s) wit stabilizable and detectable realization (A, B, C, 0), were A 2 R n n is asymptotically stable, B 2 R n m and C 2 R m n, ten G(s) is SPR if and only if tere exists a P = P > > 0 suc tat A > P + PA < 0 (8) PB = C > (9) 4

5 PROOF Te proof can be found in Reference [22]. Corollary Tere exists a matrix P = P > > 0 tat satisfies (9) if and only if CB =(CB) > > 0 (10) Furtermore, wen (10) olds, all solutions of (9) are given by P = C > (CB) > C + B? XB?> (11) were X = X > > 0 is arbitrary and B? 2 R n (n B. Tat is B? must satisfy B > B? =0. PROOF Te proof can be found in Reference [20]. Lemma 2 (Matrix Elimination) Given m) is a basis for te ortogonal complement, or anniilator of G + C > L > P + PLC < 0 (12) were G 2 R n n, C 2 R p n, and P = P > 2 R n n is full rank, an L 2 R n p exists wic satisfies (12) if and only if te following inequality olds C >?> GC >? < 0 were C >? 2 R n (n p) satisfies CC >? =0. PROOF Te proof can be found in Reference [21]. IV. Adaptive Control Design In tis section we provide te process for selecting te CRM gain L in (5) and te update law for in (7). To accomplis te goal of reference tracking we take an approac wic focuses on te error between te closed-loop plant and te reference model states, as opposed to eac of tese trajectories individually. Tus, te goal of reference tracking can be ensured by appropriately selecting te update law to drive tis state error to zero. Similarly, we consider te error between te parameter in (7) and in Remark 3. Te resulting state tracking error and parameter error, respectively, can be defined as e x = x e = x m Te problem of finding an adaptive law for tat guarantees stability depends on te relationsip between te two errors above. Tis relation, denoted as error model, in turn provides cues for determining te adaptive law. In te problem under consideration, te underlying error model can be described as ė x =(A + LC + B > )e x + B e > x m (13) e y = Ce x were e y is te measured output error. As mentioned earlier, te problem of finding a stabilizing adaptive controller is equivalent to finding an L and an adaptive law for adjusting e in (13). Determining a stable adaptive law for an error model as in (13) relies on properties of an underlying transfer function tat is SPR, 9 wic in turn enables te use of Lemma 1 in Section III. However, te definition of SPR is restricted to square transfer functions. As suc, for tese properties to be applicable to te error model in (13), a suitable static postcompensator S 1 2 R m p as to be cosen suc tat S 1 C(sI A LC B > ) 1 B 2 R m m p (s) were R p (s) denotes te ring of proper rational transfer functions wit coefficients in R. Tat is te underlying transfer matrix is square, and terefore can be evaluated in terms of SPR properties. We terefore introduce a syntetic output error e s as e s = S 1 Ce x 5

6 Wit tis postcompensator, te underlying error model is modified as ė x =(A + LC + B > )e x + B e > x m (14) e s = S 1 Ce x Tus, te design of an output feedback adaptive controller is reduced to selecting matrices S 1 2 R m p and L 2 R n p suc tat te error dynamics in (14) are SPR. In Section IV-A we provide a procedure to construct S 1 and L. Tis procedure requires S 1 to be solved as a generalized inverse based on te matrices of in (4) alone. L is found by satisfying Lemma 1 (Kalman-Yakubovic), te solution of wic is reduced to a state-feedback problem of a lower-order plant subsystem wic ultimately leads to a feasible LMI wic is solved numerically to obtain L. A. Finding S 1 and L In tis section we provide a metod for selecting S 1 and L wic ensure te system in (14) is SPR. Te conditions from Lemma 1 to ensure (A + LC + B >,B,S 1 C) is SPR are given by (A + LC + B > ) > P + P (A + LC + B > ) < 0 (15) PB =(S 1 C) > (16) were, by te corollary to Lemma 1, a P exists wic satisfies (16) if and only if S 1 CB =(S 1 CB) >. 1. Finding S 1 Te matrix S 1 satisfying (16) can be computed as a generalized left inverse of CB as Note tat tis coice of S 1 is not unique. 2. Finding L S 1 = (CB) > CB 1 (CB) > (17) Te anniilator matrices B? and C >? in Section III are not unique. In te following subsection we will use te notation N and M to represent particular anniilators tat satisfy NB =0, CM =0, and a few additional desired properties. Tat is, N represents a particular B?> and M a particular C >?. Given arbitrary anniilators B? and C >? we provide a constructive process for obtaining N and M, and use tese matrices to find L. Te inequality (15) is satisfied if te following inequality is satisfied (A + LC) > P + P (A + LC)+Q<0 (18) for Using (16), te inequality (19) can be written as B > P + PB > <Q (19) S 1 C +( S 1 C) > <Q (20) Note tat Q satisfying (20) is independent of P. Using Lemma 2, an L exists satisfying (18) if and only if a P exists wic satisfies M > (A > P + PA)M < M > QM (21) Using (11), P is given by Substituting (22) into (21) we obtain P =(S 1 C) > (S 1 CB) > S 1 C + N > XN (22) (NAM) > XNM +(NM) > X(NAM) < M > QM (23) 6

7 Tus, te problem of finding an SPR L wic satisfies (15) is now reduced to finding te matrix X satisfying (23). An X satisfying (23) specifies a P as in (22) tat reduces (15) to a feasible LMI in L. Tis feasible LMI can ten be easily solved using any widely available numerical LMI solver. Reference [20] gave te inequality (23) for a square system, suggesting tat X be obtained by solving tis LMI numerically. However, it was sown in Reference [23] tat for a square system, te eigenvalues of N AM are te transmission zeros of te system and te anniilators N and M can be always be selected suc tat NM = I. Given a square system wit only stable transmission zeros, tis selection reduces (23) to a Lyapunov equation were te matrix NAM is stable, and te existence of X>0 satisfying tis inequality is guaranteed. 24 Tus, wen te system in (4) is square, (23) can be solved to obtain X, and P can be computed using (22). Te inequality (15) can ten be solved for L. For a non-square systems te matrix NAM is not square, and so determining X>0 satisfying (23) requires additional steps. DETERMINING A SIMILARITY TRANSFORM We will now define a similarity transform tat will allow anniilator matrices N and M in (23) to be computed given arbitrary anniilators B? and C >?. Defining as 25 i = B F C >? (24) it is always possible to coose F 2 R n (p m) so tat is invertible and i C = C 0p (n p) i > 1 B = I m m 0 m (n m) were C 2 R p p. Define Ā = 1 A = Ā 11 Ā 12 Ā 21 Ā (25) Ā 31 Ā 32 were Ā22 2 R (p m) (n p) and Ā32 2 R (n p) (n p). Define te following transformed eliminators N 0 and M 0 wic satisfy N 0 1 B =0 (n m) m and C M 0 =0 p (n p) as i N 0 = 0 (n m) m I (n m) (n m) (26) i > M 0 = 0 (n p) p I (n p) (n p) (27) Note tat tese coices are not unique. Define N = N 0 1 (28) M = M 0 (29) Note tat wit te selection of M 0 in (27) and wit in (24) tat M = C >?. Te matrix NM is given by i > NM = 0 (n p) (p m) I (n p) (n p) (30) Te inverse of is given by 2 3 R = 4 5 (31) N 1 were R 2 R m n, N 1 2 R (p m) n and N 2 2 R (n p) n, and 1 must obviously satisfy 2 3 R i = 4 5 B F C >? = I N 1 N 2 N 2 7

8 were te matrix F, and tus 1 are yet to be determined. From tis we ave tat N 2 B =0 (n p) m, N 2 F = 0 (n p) (p m), and N 2 C >? = I (n p) (n p). Wit tis coice of te matrix NAM can be expressed as NAM = "Ā22 Ā 32 # = 2 i 6 0 (n m) m I (n m) (n m) 4 Ā 11 Ā 12 Ā 21 Ā 22 Ā 31 Ā 32 3 " p (n p) I (n p) (n p) # (32) Note tat wit te coice of NM satisfying (30), we can partition X as " # X X = 11 0 (p m) (n p) 0 (n p) (p m) X 22 (33) were X 11 2 R (p m) (p m) and X 22 2 R (n p) (n p) and X>0if X 11 > 0 and X 22 > 0. Evaluating XNM we ave " # 0 XNM = (p m) (n p) X 22 And so (23) is equivalent to te following Lyapunov equation Ā > 32X 22 + X 22 Ā 32 = M > Q M (34) for Q >Q. Te matrix F in (24) must be selected suc tat Ā 32 is Hurwitz, tus allowing X 22 to be obtained as te solution to te Lyapunov equation in (34). X 11 > 0 can ten be selected arbitrarily to specify X. Expanding (32) using (28) and (29) we ave "Ā22 Ā 32 From wic we can see 2 3 # i R 6 7 = 0 (n m) m I (n m) (n m) 4 5 A N 1 N 2 Ā 32 = N 2 AC >? i " # B F C >? 0 p (n p) I (n p) (n p) Recall tat N 2 as to satisfy N 2 B =0, and N 2 C >? = I. To satisfy tese two conditions we see tat N 2 lies in te nullspace of B > and so we ave N 2 = KB?> (35) were K 2 R (n p) (n m). Te last condition requires KB?> C >? = I. We can satisfy N 2 F =0by selecting F to lie in te nullspace of N 2, were dimension of te nullspace of N 2 is equal to p. Wit tis coice of N 2 we can express Ā32 as Ā 32 = KB?> AC >? Te requirements described above are stated as: find K 2 R (n p) (n m) suc tat KB?> C >? = I (n p) (n p) (36) Ā 32 = KB?> AC >? is Hurwitz (37) AN EQUIVALENT STATE FEEDBACK PROBLEM We continue by sowing ow te selection of F wic defines te similarity transform in (24) depends on N 2 in (31), wic depends on K in (36) and (37), can be found by solving a state feedback problem. Te requirement in (36) is tat K is a left inverse of te tall matrix B?> C >?. Tis matrix as full rank by Assumption 2c. Te generalized inverse of a tall matrix T 2 R (n m) (n p) wit full rank is given by T = T + U(I (n m) (n m) TT ) 8

9 were U 2 R (n p) (n m) is arbitrary. Tis gives a form of all K satisfying (36) as Tis can be simplified as K =(B?> C >? ) + U I (n m) (n m) (B?> C >? )(B?> C >? ) K =(B?> C >? ) + U I (n m) (n m) J (38) J =(B?> C >? )(B?> C >? ) (39) were J 2 R (n m) (n m) is a rank n p matrix. Tus Ā32 is given by apple Ā 32 = (B?> C >? ) + U I (n m) (n m) J B?> AC >? wic can be written were G 2 R (n p) (n p) and H 2 R (n m) (n p) are given by Ā 32 = G + UH (40) G =(B?> C >? ) B?> AC >? H = I (n m) (n m) J B?> AC >? (42) (41) Selecting U suc tat Ā 32 is Hurwitz is possible in general if (G >,H > ) is controllable. Te uncontrollable modes of (G >,H > ) correspond to te transmission zeros of. 26 If te system as any unstable zeros, no U can be found suc tat Ā32 is Hurwitz. If te system as stable transmission zeros, (G >,H > ) is stabilizable, and U can be selected to stabilize te remaining modes. If te system as no transmission zeros, (G >,H > ) is controllable, and U can be picked to make te poles of Ā32 arbitrarily. By Assumption 2d as no unstable transmission zeros, so (G >,H > ) will be at least stabilizable. Wit U computed using te desired state-space tecnique, Ā 32 is determined as in (40). K can ten be solved for from (38) and (39) and N 2 computed using (35). F is ten selected to lie in te nullspace of N 2, and to make full rank. Wit tis coice of F, te matrix is completely specified, and N can be solved for from (28) and M given by M = C >?. Finally, (34) must be solved to obtain X 22, wic requires te specification of Q>0. Te following paragrap and teorem provide a metod to select an appropriate Q. SOLVING THE LMI TO OBTAIN L All tat remains to solve te LMI in (18) for L is to specify P as given by (22) and Q. Solving (34) for X 22 also requires Q, altoug tis equation places no restriction on ow Q>0 is selected. However, we must coose an appropriate Q wic guarantees te feasibility of te LMI in (18) by satisfying (20), as given by te following teorem. Teorem 1 If Q is cosen as were C s = S 1 C and is defined as in Assumption 1E-(b), ten (20) olds. PROOF Using C s = S 1 C te inequality (20) can be written Q = 2 kc s k 2 I n n (43) C s +( C s ) > <Q Using C s applek C s k 2 I applek k 2 kc s k 2 I< kc s k 2 I te matrix Q in (43) satisfies (20). Wit Q picked as in (43) and Ā32 made stable by selection of U in (40), te Lyapunov equation in (34) can be solved to obtain X 22. Tis procedure guarantees te feasibility of te LMI in (18) wic can be solved numerically wit any widely available solver. Tis procedure is summarized in te following subsection. B. Summary of te Design Procedure for S 1 and L Subsection IV-A provided a procedure to determine S 1 and L for te system satisfying Assumption 2 wic render (14) SPR. Tis subsection summarizes te overall procedure. Given known plant matrices A, B, B cmd, C, and uncertainties and in (4), reference model in (5), and control law in (7), te following steps provide a procedure to determine S 1 and L suc tat te underlying error dynamics in (14) are SPR: 9

10 1. Solve for S 1 as in (17). 2. Determine arbitrary anniilators B? and C >? suc tat B > B? =0and CC >? =0. 3. Calculate matrices G and H using (39), (41), and (42) and ten solve for U suc tat Ā32 in (40) is Hurwitz. 4. Compute K using (38) and define N 2 as in (35). 5. Determine F in null(n 2 ) suc tat =[B F C >? ] is full rank. 6. Define N 0 as in (26). Calculate N = N 0 1 and set M = C >?. 7. Select Q as in (43) and solve (34) to obtain X 22. Assemble X as sown in (33) were X 11 > 0 is arbitrary. 8. Solve for P as in (22). 9. Solve te LMI in (18) to obtain L Remark 4 In te case were p m n p, we are ensured tat H in (40) is a matrix of full column rank and so H H = I (n p) (n p). Tis provides te freedom in selecting U to not only make Ā32 stable, but to select it to be any stable matrix. Tis allows us to select X 22 > 0 arbitrarily, and ten solve for Ā 32 as te solution to te Lyapunov equation Ā > 32 X 22 + X 22 Ā 32 = M > Q M. Ten U can be picked in step 3 as U = Ā 32 G H (44) Remark 5 Te calculation of L sould conclude wit te verification tat A + LC + BK x > is Hurwitz. Wile tis is not a teoretical requirement, for practical implementation on systems suc as te one presented in Section V, tis requirement is enforced to ensure te reference model in (5) is stable. C. Adaptive Law and Stability Proof Using te closed-loop reference model defined in (5) wit L selected as described in Section IV-A, we ten propose te following update law: ė = x m (S 1 e y ) > sgn( ) (45) were S 1 is cosen using (17). Globally stability of te closed-loop system is guaranteed by te following teorem. Teorem 2 Given te uncertain linear system in (4) wic satisfies Assumption 2, te reference model in (5), and control law as in (7), te update law in (45) results in global stability, wit e x (t)! 0 as t!1. PROOF Propose te following radially unbounded Lyapunov function candidate V (e x, e ) = e > x Pe x + tr e > using (45) te time derivative of V along system trajectories can be evaluated as V = e > b x Qe x apple 0 were Q>0, b and indeed te proposed candidate does serve as a Lyapunov function for tis system. Furtermore, it can be sown using Barbalat s lemma tat e x (t)! 0 as t!1. Wit e x (t)! 0 as t!1, we acieve desired bounded reference tracking of z cmd by z. In te following section we demonstrate te efficacy of te proposed metod by providing a numerical example, but first we examine te applicability of te CRM based metod as compared to te classical MIMO adaptive control metod. D. Comparison Between CRM based and Classical MIMO Adaptive Control Given te classical approaces used in te literature tus far, te obvious question tat is raised is ow te proposed MIMO controller fares compared to te classical ones. Te first point to note ere is tat te classical approaces are limited to square plants wile our approac proposed ere is not. Tis is te most obvious advantage of our metod. Te next question tat arises is a comparison of te two approaces wen te underlying plant is square. Tis is addressed below. As a first step, we provide relevant definitions below: Definition 1 (Markov Parameters) [27] Given a transfer matrix G(s), te Markov Parameters are given by and so fort. H 0 = lim s!1 G(s), H 1 = lim s!1 s(g(s) H 0), H 2 = lim s!1 s2 (G(s) H 0 H 1 s 1 ) 1 e 10

11 Teorem 3 Te set (A, B, C, D) is a realization of G(s) if and only if PROOF Te proof can be found in Reference [27]. H 0 = D H i = CA i 1 B, i =1, 2,... Definition 2 (Relative Degree One) Te MIMO system G(s) wit realization (A, B, C, D) is said to be Relative Degree One if H 0 =0and H 1 = CB is full rank. Lemma 3 Reference [9] Given a square nonsingular strictly proper transfer matrix W p (s) 2 R m m p (s), its Hermite form is diagonal if and only if te constant matrix E(W p (s)) is nonsingular, were E is calculated as follows. Calculate r i as te minimum relative degree in te i t row of W p (s) and te rows of E are were W p,i (s) corresponds to te i t row of W p (s). PROOF Te proof can be found in Reference [28] E i = lim s!1 sri W p,i (s) (46) Given W p (s) 2 R m m p (s), te assumptions tat must be satisfied for a classical adaptive control solution to exist are as follows. 9 Assumption 3 (i) Te ig frequency gain matrix K p is of te form K p = K p were K p is known and sign( ) is known. (ii) Te rigt Hermite normal form H p (s) of W p (s) is known. (iii) An upper bound on te observability index of W p (s) is known. (iv) Te zeros of W p (s) lie in C. Teorem 4 Given te square plant W p (s) 2 R m m p wit realization (A, B, C, 0), te Hermite form H p (s) of W p (s) is diagonal if CB is full rank. Furtermore, te ig frequency gain matrix is given by K p = CB. PROOF Teorem 3 connects te Markov Parameters of Relative Degree One systems to te realization of W p (s) wit H 0 =0and H 1 = CB. Wit tis and Definition 1 we ave tat lim s!1 sw p (s) =CB is full rank, and so te minimum relative degree in eac row of W p (s) is r i =1. By Lemma 3 E(W p (s)) = CB and te Hermite form H p (s) of W p (s) is diagonal. In Reference [9] it is sown tat E(W p (s)) = K p. Using Definitions 1 and 2 as well as Teorems 3 and 4, we sow in Proposition 1 tat te classical and our CRM based MIMO adaptive control solution in tis paper are equally applicable wen te system in (1) is square. Proposition 1 Consider te uncertain system in (1) were ` = m and te plant transfer matrix is given by W p (s) =C p (si A p B p > p ) 1 B p (47) if te plant in (1) satisfies Assumption 1, ten te corresponding W p (s) in (47) satisfies Assumption 3. PROOF Assumption 1E-(a) and Teorem 4 can be sown to imply tat te corresponding K p satisfies Assumption 3(i). Assumption 1C togeter wit Teorem 4 implies tat te corresponding Hermite form is diagonal wit known entries and is terefore known, wic leads to Assumption 3(ii). Assumption 3(iii) follows from te fact tat n p is known. Finally Assumption 1D is equivalent to Assumption 3(iv). In addition to te main advantage of our proposed metod of applicability to non-square plants, our proposed controller is of lower order, requiring only n controller states and nm adjustable parameters, as compared wit te classical solution wic will introduce 2m states and 2m 2 parameters. Tis reduces te number of states and parameters necessary by at least two, since n apple m. 29 Finally our solution is based on a CRM, wic as been sown to possess a superior transient performance It sould be noted tat of Assumptions 1A-E, wic are required to be satisfied for te proposed controller, te most restrictive one is Assumption 1C, wic implies tat te MIMO system must ave Relative Degree One. In most aerial platforms including ypersonic veicles, tis assumption is easy to satisfy as te relative degree of te transfer functions between te control surface deflections and aircraft angular rates is unity. Additionally, te structure of te 11

12 plant as in (1) wic as matced uncertainties is also commonly present in fligt control applications were muc of te plant uncertainty is in te aerodynamic moment coefficients and loss of control effectiveness, wic are spanned by te columns of B. It is owever required tat te uncertainty p satisfy Assumption 1E-(b), wic is not required in te classical approac. V. Numerical Example We now present a numerical example using tis output feedback design procedure applied to te control design for a 6-DOF GHV model. 6, 19 Te GHV is a small blended wing-body veicle, wit 3-D inlet and nozzle, and axisymmetric troug-flow scramjet engine. Te nonlinear equations of motion describing te GHV are linearized about a nominal fligt condition of Mac 6 at an altitude of 80,000 feet, corresponding to a dynamic pressure of 1474 psf. Modal analysis allowed te linearized equations of motion to be decoupled, and te resulting uncertain longitudinal and lateral-directional plant subsystems are represented as in Equation (1). In Reference [6] a state feedback LQR baseline controller wit integral action and augmented wit an adaptive component was applied to design two independent CRM based state feedback adaptive controllers - one for eac of te plant subsystems. Tis approac was very effective at maintaining stability and tracking performance in te presence of uncertainty, but assumed availability of angle-of-attack and sideslip angle measurements. In tis work, we no longer assume tat tese incidence angles are measurable, wic is more realistic for tis class of veicle, tus turning te problem into one of output feedback. Te adaptive control design procedure described in Section IV was used to design two independent CRM based output feedback adaptive controllers - one for eac of te two plant subsystems. A. Longitudinal Subsystem by Te state, control, output, and regulated output for te linear longitudinal subsystem as represented in (1) are given x p = qi > u = e y p = q z = q respectively, were represents te angle of attack, and q is te pitc rate. Te control input e represents te elevator deflection angle. Te pitc rate is measurable but te angle of attack is not. Te control goal is to track pitc rate commands z cmd = q cmd. Te longitudinal subsystem satisfies Assumption 1. Te state vector x p is augmented wit te integral error state as in (2) resulting in a system of te form (4) wic satisfies Assumption 2. Te augmented state and output vector are i > i > x = q x e y = q x e Te baseline control gain K x in (6) was computed resulting in te following state feedback gain and pase margin GM sf =[ 14.5, ] db PM sf = 60 Te controller was ten tuned by selecting X 11 > 0 and solving for X 22 in step 7 resulting in " # 2 0 X = Tis provided te following gain and pase margin for te resulting output feedback compensator deg GM of =[ 14.2, 33.2 ] db PM of = 59 Te numerical values for te linear system matrices and LQR weigting matrices can be found in te appendix. deg 12

13 B. Lateral Subsystem Te state, control, output, and regulated output for te linear lateral-directional subsystem as represented in (1) are given by x p = p r i > > u = a ri y p = p r i > z = respectively, were represents te sideslip angle, p te roll rate, r te yaw rate, and te roll angle. Te control inputs a and r represent te aileron and rudder deflection angles, respectively. All states are measurable except te sideslip angle. Te control goal is to track roll angle commands z cmd = cmd. Te lateral-directional subsystem can be sown to satisfy Assumption 1. Te state vector x p is augmented wit te integral error state as in (2) resulting in a system of te form (4) wic satisfies Assumption 2. Te augmented state and output vector are x = i > i > p r x e y = p r x e Te baseline control gain K x in (6) was computed resulting in te following state feedback gain and pase margin GM sf =[ 6.5, 161 ] db PM sf = 60 deg Since te conditions of Remark 4 are satisfied, X 22 can be selected arbitrarily. Te controller was ten tuned by selecting X 11 > 0 and X 22 > 0 resulting in X = Tis provided te following gain and pase margin for te resulting output feedback compensator GM of =[ 7.3, 14.5 ] db PM of = 48 Te numerical values for te linear system matrices and LQR weigting matrices can be found in te appendix. C. Simulations deg Te linear, adaptive augmented baseline controllers designed for te longitudinal and lateral-directional systems as described by (1) wit reference model in (5), control law as in (7), and update law in (45) were ten implemented in a simulation of te GHV model. Second order actuator dynamics were included in te simulation on te elevators, ailerons, and rudders. Te numerical property values are listed in Table 1. Table 1. Second order aerodynamic control surface actuator parameters Parameter Unit Value Surface deflection limit [deg] 30 to 30 Surface rate limit [deg/s] 100 to 100 Damping ratio 0.7 Natural frequency! n [rad/s] 150 To demonstrate te performance and robustness of te adaptive controller, uncertainties were introduced in te nonlinear model, wic manifest temselves in te uncertain linear system as given in (1). Te uncertainty is as follows: 13

14 Control effectiveness on all surfaces is reduced to 60% of te nominal value. Center of gravity is sifted 1.6 feet rearward, effectively representing an unknown center of pressure location. Te rolling moment coefficient C l is reduced to 10% of te nominal value. Te simulation block diagram is sown in Figure 1. Figure 1. Simulation block diagram. A 45 degree roll angle command was given, wile commanding zero pitc rate. Figure 2 sows te response of te nominal system, wit no uncertainty and only te baseline control law: =0. Figure 3 sows te response of te te system wit te above uncertainty introduced and only te baseline controller: =0. Figure 4 sows te response of te system wit te uncertainties wen te adaptive augmentation is used. Te baseline control law applied to te nominal linear system provides good stability margins and closed-loop performance on te 6-DOF nonlinear evaluation model as sown in Figure 2. Te system as a small rise time, minimal oversoot, and a small settling time. Wit te baseline control law only, introduction of te uncertainties causes greater initial oversoot, followed by significant ig frequency oscillations, ultimately leading to instability and loss of te aircraft, as sown in Figure 3. Te use of te adaptive component in te control law recovers te baseline control performance, as sown in Figure 4. Wen using te adaptive controller, bot te control deflections and rates are well witin acceptable limits. Figure 2. Time response of te nominal system: baseline controller wit no uncertainty. 14

15 Figure 3. Time response of baseline control applied to uncertain plant. Figure 4. Time response of baseline controller wit adaptive augmentation applied to uncertain plant. 15

16 VI. Summary and Conclusions Tis paper as proposed a new alternative metod for syntesizing a CRM based output feedback adaptive controller for a class of uncertain MIMO systems wic do not ave any unstable transmission zeros. Te controller is composed of a baseline control gain augmented wit an adaptive component to accommodate control effectiveness uncertainty and matced plant uncertainty, and makes use of te closed-loop reference model to improve te transient properties of te overall adaptive system. Te adaptive controller requires te underlying error dynamics be made SPR troug te syntesis of te postcompensator S 1 and CRM gain L, and te SPR relationsip is enforced by reducing an underlying bilinear matrix inequality to a feasible linear matrix inequality troug appropriate selection of a tuning matrix X. Te procedure does not require te plant first be squared-up. It is computationally simple, and it requires only te calculation of some generalized inverses, te solution of te Lyapunov equation, and te solution of a reduced order state feedback problem. Tis procedure is summarized in nine straigtforward steps. Furtermore, te degrees of freedom in te tuning matrix X capture a large subset of all possible solutions wic ensure te SPR property. Using tese degrees of freedom, X can be tuned to provide te desired stability margins for te baseline system, and a globally stable update law. Te result is a baseline output feedback controller wit good stability margins and adaptive augmentation capable of accommodating matced uncertainties. Tis resulting robust baseline output feedback controller wit adaptive augmentation is sown in simulation to provide good tracking performance wen applied to a 6-DOF simulation of a ypersonic veicle wit significant uncertainty in control effectiveness, CG sift, and te rolling moment coefficient. Te simulation results sowed tat for tis uncertainty te robust baseline controller alone was unable to maintain stability, wereas te adaptive controller recovered nominal performance even wit te presence of unmodeled actuator dynamics. VII. Acknowledgment Approved for Public Release; Distribution Unlimited. Case Number 88ABW References 1 Jang, J., Annaswamy, A. M., and Lavretsky, E., Adaptive Fligt Control in te Presence of Multiple Actuator Anomalies, American Control Conference, IEEE, New York, NY, July 2007, pp Gibson, T. E. and Annaswamy, A. M., Adaptive Control of Hypersonic Veicles in te Presence of Trust and Actuator Uncertainties, AIAA Guidance, Navigation and Control Conference and Exibit, American Institute of Aeronautics and Astronautics, Honolulu, HI, August Matsutani, M., Gibson, T. E., Jang, J., Crespo, L. G., and Annaswamy, A. M., An Adaptive Control Tecnology for Safety of a GTM-like Aircraft, American Control Conference, IEEE, St. Louis, MO, June 2009, pp Crespo, L. G., Matsutani, M., Jang, J., Gibson, T. E., and Annaswamy, A. M., Design and Verification of an Adaptive Controller for te Generic Transport Model, AIAA Guidance, Navigation, and Control Conference, AIAA, Cicago, IL, August Dydek, Z. T., Annaswamy, A. M., and Lavretsky, E., Adaptive Control and te NASA X-15-3 Fligt Revisited, IEEE Control Systems Magazine, Vol. 30, No. 3, June 2010, pp Wiese, D. P., Annaswamy, A. M., Muse, J. A., and Bolender, M. A., Adaptive Control of a Generic Hypersonic Veicle, AIAA Guidance, Navigation, and Control (GNC) Conference, AIAA, Boston, MA, August Lavretsky, E. and Wise, K. A., Robust and Adaptive Control, Springer, Sing, R. and Narendra, K. S., Prior Information in te Design of Multivariable Adaptive Controllers, IEEE Transactions on Automatic Control, Vol. 29, No. 12, December 1984, pp Narendra, K. S. and Annaswamy, A. M., Stable Adaptive Systems, Dover Books on Electrical Engineering Series, Dover, Lavretsky, E., Adaptive Output Feedback Design Using Asymptotic Properties of LQG / LTR Controllers, AIAA Guidance, Navigation, and Control Conference, AIAA, Toronto, Ontario, Canada, August Wise, K. A. and Lavretsky, E., Fligt Control Design Using Observer-based Loop Transfer Recovery, AIAA Guidance, Navigation, and Control (GNC) Conference, AIAA, Boston, MA, August Qu, Z., Lavretsky, E., and Annaswamy, A. M., An Adaptive Controller for Very Flexible Aircraft, AIAA Guidance, Navigation, and Control (GNC) Conference, AIAA, Boston, MA, August Gibson, T. E., Annaswamy, A. M., and Lavretsky, E., Improved Transient Response in Adaptive Control Using Projection Algoritms and Closed Loop Reference Models, AIAA Guidance, Navigation, and Control Conference, AIAA, Minneapolis, MN, August Gibson, T. E., Annaswamy, A. M., and Lavretsky, E., Closed-loop reference models for output-feedback adaptive systems, European Control Conference (ECC), Zuric, Switzerland, July 2013, pp Gibson, T. E., Annaswamy, A. M., and Lavretsky, E., On Adaptive Control Wit Closed-Loop Reference Models: Transients, Oscillations, and Peaking, IEEE Access, Vol. 1, September 2013, pp Gibson, T. E., Annaswamy, A. M., and Lavretsky, E., Adaptive systems wit closed-loop reference-models, part I: Transient performance, American Control Conference, IEEE, Wasington DC, June 2013, pp

17 17 Misra, P., A Computational Algoritm for Squaring-up Part I: Zero Input-Output Matrix, Proceedings of te 31st IEEE Conference on Decision and Control, IEEE, Tucson, AZ, December 1992, pp Qu, Z., Wiese, D. P., Annaswamy, A. M., and Lavretsky, E., Squaring-up Metod in te Presence of Transmission Zeros, Te 19t World Congress of te International Federation of Automatic Control, IFAC, Cape Town, Sout Africa, August 2014, pp Rollins, E., Valasek, J., Muse, J. A., and Bolender, M. A., Nonlinear Adaptive Dynamic Inversion Applied to a Generic Hypersonic Veicle, AIAA Guidance, Navigation, and Control (GNC) Conference, AIAA, Boston, MA, August Huang, C., Ioannou, P. A., Maroulas, J., and Safonov, M. G., Design of Strictly Positive Real Systems Using Constant Output Feedback, IEEE Transactions on Automatic Control, Vol. 44, No. 3, Mar 1999, pp Boyd, S. P., Gaoui, L. E., Feron, E., and Balakrisnan, V., Linear Matrix Inequalities in System and Control Teory, Studies in Applied Matematics, Society for Industrial and Applied Matematics, Narendra, K. S. and Taylor, J. H., Frequency Domain Criteria for Absolute stability, Electrical science series, Elsevier Science, Kouvaritakis, B. and MacFarlane, A., Geometric approac to analysis and syntesis of system zeros Part 1. Square systems, International Journal of Control, Vol. 23, No. 2, 1976, pp Barkana, I., Comments on Design of Strictly Positive Real Systems Using Constant Output Feedback, IEEE Transactions on Automatic Control, Vol. 49, No. 11, Nov 2004, pp Owens, D., Invariant zeros of multivariable systems: A geometric analysis, International Journal of Control, Vol. 26, No. 4, 1977, pp Kouvaritakis, B. and MacFarlane, A., Geometric approac to analysis and syntesis of system zeros Part 2. Non-square systems, International Journal of Control, Vol. 23, No. 2, 1976, pp Antsaklis, P. J. and Micel, A. N., Linear systems, Springer, Cen, C., Introduction to Linear System Teory, Holt, Rineart and Winston, Cen, C., Linear System Teory and Design, Oxford University Press, Longitudinal Subsystem Numerical Data Appendix Te nominal longitudinal plant matrices for a fligt condition of Mac 6 at an altitude of 80,000 feet are: " # " # i A p = B p = C p = 0 1 C pz = i Te following weigting matrices were used to compute K x as in (6) using te MATLAB command lqr Lateral Subsystem Numerical Data Q lqr = diag [0, 0.3, 170 ] R lqr = Te nominal lateral-directional plant matrices for a fligt condition of Mac 6 at an altitude of 80,000 feet are: 2 A p = 6 4 C p = B p = C pz = Te following weigting matrices were used to compute K x as in (6) using te MATLAB command lqr Q lqr = diag [0, 0.2, 40, 0, 0.2 ] R lqr = diag [0.005, 0.005] i