International Journal of Control Vol. 84, No. 1, January 2011, 180–196
Direct adaptive control using an adaptive reference model Suresh M. Joshia*, Gang Taob and Parag Patrea a
NASA-Langley Research Center, Mail Stop 308, Hampton, Virginia, USA; bUniversity of Virginia, Charlottesville, Virginia, USA
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(Received 8 September 2010; final version received 30 November 2010) Direct model reference adaptive control is considered when the plant-model matching conditions are violated due to large changes in the plant or incorrect knowledge of the plant’s mathematical structure. Because of the mismatch, the plant can no longer track the original reference model, but may be able to track a modified reference model that still provides satisfactory performance. The proposed approach uses a time-varying ‘adaptive’ reference model that reflects the achievable performance of the changed plant. The approach consists of direct adaptation of state feedback gains for state tracking and simultaneous estimation of the plant-model mismatch. The reference model adapts to the changed plant, and is redesigned if the estimated plant-model mismatch exceeds a bound determined via robust stability and/or performance criteria. The resulting controller offers asymptotic state tracking in the presence of plant-model mismatch as well as matched parameter deviations. Keywords: adaptive control; adaptive reference model; plant-model mismatch; mismatch estimation; hybrid adaptive control
1. Introduction Direct Model Reference Adaptive Control (MRAC) methods for uncertain systems aim to adjust the controller to achieve performance that is close to that of a reference model. The system uncertainties may be parametric, structural, or environmental, and the goal is to achieve system stability and close tracking of the reference model response. Direct MRAC has been known to be an effective method for state- or outputtracking (Narendra and Annaswamy 1989; Ioannou and Sun 1996). Direct MRAC schemes that use state feedback for state-tracking (SFST) have the advantage of simplicity of implementation while achieving effective state tracking in the presence of parameter uncertainties, and has been applied to important problems such as flight control (Zaid, Ioannou, Gousman, and Rooney 1991; Dydek, Annaswamy, and Lavretsky 2010; Lavretsky, Gadient, and Gregory 2010). Such schemes have also been extended to the case with actuator failures (e.g. Tao, Joshi, and Ma 2001; Tao, Chen, Tang, and Joshi 2004; Annaswamy, Jang, and Lavretsky 2008). A commonly used assumption in direct MRAC SFST schemes consists of certain plantmodel matching conditions in order to ensure stability and asymptotic state tracking. The reference model is usually designed to represent the desired (optimal) performance of the plant under ideal conditions. Therefore, it is customary to design the reference
model based on the nominal model of the plant to incorporate the desired closed-loop performance characteristics. Thus the nominal plant satisfies the matching conditions. However, the actual plant usually differs from the nominal plant because of modelling errors, uncertainties and parameter variation. If the differences between the actual plant and the nominal plant (which is the basis for the reference model design) are such that the matching conditions are still satisfied, the MRAC SFST adaptive control laws can still ensure stability (signal boundedness) and asymptotic state tracking. In practice, however, unknown modelling errors as well as parameter changes occurring during operation may alter the mathematical structure of the plant (e.g. due to ice build up on aircraft fuselage, wings and control surfaces, or due to structural damage). The changed (‘impaired’) plant may no longer meet the plant-model matching conditions. Because the proofs of signal boundedness and asymptotic tracking assume that matching conditions are satisfied, they are no longer valid in the presence of mismatch. For this reason, plant-model mismatch has been recognised as an important problem (Cao and Hovakimyan 2008; Gibson, Crespo, and Annaswamy 2009; Joshi, Tao, and Khong 2009). If the plant-reference model mismatch is relatively small (e.g. caused by small-magnitude structural changes in the plant), it may still be possible (and
*Corresponding author. Email:
[email protected] ISSN 0020–7179 print/ISSN 1366–5820 online This work was authored as part of the Contributor’s official duties as an Employee of the United States Government and is therefore a work of the United States Government. In accordance with 17 USC 105, no copyright protection is available for such works under US Law. DOI: 10.1080/00207179.2010.544756 http://www.informaworld.com
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International Journal of Control desirable) to follow a modified, slightly perturbed reference model, provided it has good stability and performance characteristics. If, however, the changes in the plant are large, the impaired plant can no longer be expected to follow the original reference model that was designed for nominal conditions, and the reference model must be redesigned – based on parameter estimates for the changed plant – to reflect the reduced capability of the plant. As a result of reference model redesign, the matching conditions would be once again met. The reference model redesign must be done online; thus this approach represents an ‘adaptive’ reference model that recognises the plant changes and adapts to suit the achievable performance of the varying plant as it undergoes large changes. To determine when reference model redesign is needed, it is necessary to estimate the plant-model mismatch. Under the direct MRAC framework, the mismatch will need to be estimated in real time while simultaneously performing control gain adaptation and maintaining stability and tracking. This article presents a direct MRAC method for simultaneously estimating the plant-model mismatch while accomplishing state tracking. The approach presented in this article uses a modified time-varying and ‘adaptive’ reference model which is updated (redesigned) whenever the estimated mismatch exceeds predetermined bounds. This approach was introduced in a conference presentation (Joshi, Tao, and Khong 2009); this article focuses on the method in greater detail, including mismatch characterisation and new, more meaningful application examples. This article is organised as follows. Section 2 develops a direct MRAC formulation for the case when unmatchable changes occur in the plant. Section 3 presents adaptive control laws and parameter estimation equations to accomplish signal boundedness and asymptotic tracking. Section 4 presents a parameter projection algorithm to ensure the stability and tracking properties. Section 5 contains example applications of the scheme and the concluding remarks are given in Section 6.
2. Adaptive control with modified time-varying reference model 2.1 Problem description Consider a linear time-invariant system described by _ ¼ ðA0 þ aÞxðtÞ þ ðB0 þ bÞuðtÞ, xðtÞ nn
nm
ð1Þ n
where A0, a 2 R , B0, b 2 R , x(t) 2 R and u(t) 2 Rm. A0, B0 represent the nominal values of the system matrices, and a, b denote the deviations caused by unknown parameter changes (assumed to be constant matrices).
The objective is to make the plant state vector track the reference model’s state vector using an adaptive state feedback control signal u(t) 2 Rm. The reference model given by x_ m ðtÞ ¼ Am xm ðtÞ þ Bm rðtÞ,
ð2Þ
n
where xm 2 R is the reference model state vector, Am 2 Rnn, Bm 2 Rnm and rðtÞ 2 Rmr (1 mr m) is a bounded reference input used in system operation (e.g. pilot input in the case of aircraft). The aim is to design an adaptive control law that will ensure closed-loop signal boundedness and asymptotic state tracking despite uncertainties, i.e. limt!1(x(t) xm(t)) ¼ 0. It is assumed that the nominal system (A0, B0) and the reference model (Am, Bm) satisfy the SFST matching conditions, i.e. there exist gains K1 2 Rnm , K2 2 Rmmr such that Am ¼ A0 þ B0 KT1 ; Bm ¼ B0 K2 :
ð3Þ
The reference model is customarily designed to capture the desired closed-loop response of the nominal plant. For example, the reference model may be designed using optimal and robust control methods such as LQR, H2 or H1 methods, based on imperfect knowledge of the plant, i.e. based on the nominal plant. The resulting gains K1, K2 can be used as initial estimates of the adaptive gains. For the adaptive control scheme, only Am and B0 need to be known.
2.2 Plant-model mismatch In many applications such as aircraft or spacecraft flight control, the actual plant differs from the nominal plant because of model errors, damage or changes such as icing. The actual plant is represented by (A0 þ a, B0 þ b), where some of the elements of a, b can be ‘matched’, while other elements cannot be matched, and thus represent the plant-model mismatch. Because of these latter elements, the matching conditions cannot be satisfied for any values of K1, K2; therefore stability (signal boundedness) and asymptotic tracking are no longer guaranteed. To illustrate this point with a simple example, suppose that 0 1 0 ; B0 ¼ ; A0 ¼ 4 5 1 ð4Þ b1 a11 a12 ; b ¼ , a ¼ a21 a22 b2 Am ¼
0 3
1 ; 8
Bm ¼
0 : 6
ð5Þ
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The deviations a21, a22 and b2 can be compensated perfectly (matched) by some state feedback gains K1, K2 (Equation (3)). However, structural perturbations a11, a12 and b1 cannot be matched by state feedback control law. That is, the ‘unmatched’ uncertainties (i.e. the plant-reference model mismatch terms) are a11 a12 b1 a ¼ ; b ¼ , ð6Þ 0 0 0
and thus, columns of b 2 R[B0]. Writing (10) as b ¼ B0L2 and substituting in (7), a ¼ B0 ðKT1 T1 L2 T1 Þ,
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For the general case, suppose the matching conditions (3) are satisfied for the nominal plant. Then the matching conditions are satisfied for the actual (perturbed) plant (1) if and only if there exist gains 1 and 2 such that B0 KT1 ¼ a þ ðB0 þ bÞT1 ¼ Am A0 ,
ð7Þ
B0 K2 ¼ ðB0 þ bÞ2 ¼ Bm :
ð8Þ
The following proposition gives necessary and sufficient conditions under which the matching conditions are satisfied for the actual plant.
œ
i.e. columns of a 2 R[B0].
From the above discussion, it is possible to decompose parameter perturbations into matched and unmatched parts, i.e.
where the same symbols a and b are reused to represent only unmatched uncertainty. 2.2.1 Matching conditions for perturbed plant
ð11Þ
a ¼ a0 þ a00 ;
b ¼ b0 þ b00 ,
where columns of a0 , b0 2 R[B0], and columns of a00 , b00 2 R[B0]?. Of course, the perturbations are not known; however, the mathematical structure of the perturbations may provide a method to obtain the structure of the unmatched perturbations, which may reduce the number of mismatch parameters that need to be estimated, while the matched parameters will be compensated for by the adaptive control gains. Now, for the general problem, define the adaptive control law as b2 r, bT x þ K u¼K 1
ð13Þ
b1 , K b2 are the adaptive gains. where K The closed-loop plant is given by
Proposition 2.1: Suppose that the matching conditions in (3) are satisfied for the nominal plant. Then the matching conditions (7), (8) are satisfied for the perturbed plant iff
bT x þ K b2 rÞ x_ ¼ ðA0 þ aÞx þ ðB0 þ bÞðK 1 T eT x þ K e2 rÞ ¼ ðA0 þ aÞx þ B0 ðK x þ K2 rÞ þ B0 ðK
rankðB0 þ bÞ ¼ rankðB0 Þ, and R½a,R½b R½B0 , ð9Þ
eT x þ K e2 rÞ ¼ ðA0 þ B0 KT1 þ aÞx þ B0 K2 r þ B0 ðK 1
where R[] denotes the range space.
1
þ
bT x bðK 1
1
b2 rÞ þK
T
c2 rÞ b xþK þ bðK 1 T
Proof: To prove sufficiency, because R[a], R[b] R[B0], there exist matrices L1 2 Rmn, L2 2 Rmm such that a ¼ B0 L1 , b ¼ B0 L2 : Therefore, a þ ðB0 þ bÞT1 ¼ B0 L1 þ B0 ðI þ L2 ÞT1 ; ðB0 þ bÞ2 ¼ B0 ðI þ L2 Þ2 : Setting T1 ¼ ðI þ L2 Þ1 ðKT1 L1 Þ and 2 ¼ (I þ L2)1K2 yields (7), (8). The rank condition in (9) ensures the existence of the inverse. To prove necessity, by hypothesis, for any K1 2 Rnm, K2 2 Rmm, there exist 1 2 Rnm, 2 2 Rmm such that (7), (8) hold. Choosing K2 to be an arbitrary non-singular matrix, from (8), the rank condition in (9) must hold, and also 2 is nonsingular. The following is obtained from (8): b ¼ B0 ðK2 2 Þ1 2
ð12Þ
ð10Þ
b1 Þx þ ðBm þ bK b2 Þr ¼ ðAm þ a þ bK f1 T x þ K f2 rÞ, þ B0 ðK
ð14Þ
e1 ¼ K b1 K1 ; K e2 ¼ K b2 K2 . where K Because of the a and b terms, the structure of the above state equation is different from that of the nominal reference model (2); therefore, it is not possible to follow the nominal reference model. In order to accommodate plant-model mismatch and parameter deviation, the reference model is modified as bK bK b þ b bT Þxm þ ðBm þ b b2 Þr, x_ m ¼ ðAm þ a 1
ð15Þ
b denote (time-varying) estimates of a b b where a, and b. Denote c ¼ a bK b þ b c1 T ; DA
c¼b b2 DB bK
ð16Þ
and c AM ¼ Am þ DA;
c BM ¼ Bm þ DB:
ð17Þ
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x_ m ¼ AM xm þ BM r:
2.3 Stability of modified reference model The time-varying matrix AM can be expressed as an affine function of a parameter vector p 2 Rnp that lies in a convex polytope S having vertices pj, j ¼ 1, . . . , nv. For example, AM can be expressed as np X
pi ðtÞAMi
ð19Þ
1
where AMi are constant matrices and pi ðtÞ 2 ½pi , pi (pi is the ith component of p). In this case S is a hyperrectangular region with nv ¼ 2np vertices. Let AM(pj) denote the value of the reference model system matrix at vertex pj. Suppose that there exist positive definite matrices P ¼ PT, Q ¼ QT 2 Rnn, such that AM ðpj ÞT P þ PAM ðpj Þ 5 Q,
j ¼ 1, . . . nv :
AM ðpðtÞÞT P þ PAM ðpðtÞÞ 5 Q
ð18Þ
As mentioned in Joshi, Tao, and Khong (2009), this representation of the reference model indicates that the changed plant can no longer follow the original reference model (2), but can possibly follow the modified time-varying reference model (18). The reference model depends on the mismatch parameter estimates, i.e. it ‘adapts’ to the changed plant, and is therefore an ‘adaptive’ reference model. However, it is necessary to first ensure the suitability of the modified reference model, i.e. that it must have acceptable stability and performance characteristics. Conventional MRAC schemes use a time-invariant reference model; therefore this time-varying reference model is a departure from standard MRAC. Timevarying reference models were considered in Annaswamy, Jang, and Lavretsky (2008), which employed a time-varying Lyapunov function weighting matrix (P(t)). In the present approach, a time-invariant Lyapunov function weighting matrix is used along with a quadratic stability argument to ensure stability of the time-varying reference model. This can be accomplished by calculating the permissible limits on the estimated perturbations within which the stability (and possibly performance) will remain satisfactory for all (time-varying) perturbations within these limits. Subsequently, it will also be necessary to derive b that c2 , a, b b) c1 , K adaptive laws (for updating K guarantee that x(t) ! xm(t), and that all signals remain bounded.
AM ðpðtÞÞ ¼ Am þ
of (20), the Lyapunov inequality
ð20Þ
For a given Q, (20) represents a set of nv linear matrix inequalities (LMIs) in the unknown variable P. In view
is satisfied 8p(t) 2 S, and the autonomous part of the reference model (i.e. Equation (18) with r ¼ 0) is exponentially stable 8p(t) 2 S with a guaranteed mini1 1 _ mum decay rate e2min ðP QÞt (since V=V 1 min ðP QÞ). Express pi and pi as pi ¼ i ;
pi ¼ i ;
i ¼ 1, . . . , np ,
ð21Þ
where i , i are known constants (chosen to normalise the parameter ranges), and i 0 i . The problem is to obtain the maximal region S for which (20) holds, i.e. find a positive definite symmetric matrix P that maximises subject to the set of LMIs in (20). The resulting estimate of the stability region S is usually conservative. One reason for the conservatism is that arbitrarily fast parameter variation is allowed. If the parameter variations (_p) can be rate-bounded, less conservative estimates can be obtained. The stability of c while DB c the reference model depends only on DA, affects the closed-loop gain; therefore, if (20) is c is reasonably small, the performance satisfied and DB of the modified reference model would be acceptable. Instead of just quadratic stability, a degreeof-stability () requirement may be imposed on the modified reference model by replacing AM(pj) by [AM(pj) þ I ] to generate a decay rate greater than et. Alternatively an H2 or H1 performance requirement can be imposed on the reference model dynamics, AM, BM. Such requirements can be formulated as linear matrix inequalities at the vertices of a polytope in the parameter space and an estimate of the maximal region S can be obtained, which provides the c If DA c permissible bounds on the elements of DA. c (and DB when performance is considered) approach their permissible bounds, the reference model will need to be updated (redesigned) and the new permissible bounds will need to be computed online.
3. Adaptation laws 3.1 Asymptotic tracking and signal boundedness The state tracking error is given by e ¼ x xm. Equations (14) and (15) yield e þ B0 ðK e bu eT x þ K e2 rÞ, e_ ¼ AM e ax 1
ð22Þ
e ¼ b b b. For simplicity of e ¼ a b a, b where a c remains presentation, it will be first assumed that DA within the quadratic stability bounds. The next section will present a parameter projection algorithm to ensure that this condition is met. The following theorem gives
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adaptation and mismatch estimation laws that ensure asymptotic state tracking with signal boundedness. Theorem 3.1: The system given by (1), (15), the adaptive controller (13), gain adaptation laws _ T c K 1 ¼ 1 xe PB0 , _ T T c K 2 ¼ 2 B0 Per
Therefore, _ eT fPexT 1b V_ eT Qe 2Tr½a a ag _ eT fPeuT 1b 2Tr½b b bg _T eT fxeT PB0 þ 1 K b þ 2Tr½K 1 1 1g
ð23Þ
_ eT fBT PerT þ 1 K b þ 2Tr½K 2 g: 2 0 2
and parameter estimation laws
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_ b a ¼ a PexT , _ b b ¼ b PeuT ,
ð24Þ
where 1, 2, a, b are symmetric positive definite matrices and P is the (-maximising) solution of the LMI system in Section 2.2, guarantee that all closed-loop signals including adaptive gains and parameter estimates are bounded and the tracking error e(t) ¼ x(t) xm(t) goes to zero asymptotically. Proof:
Define V ¼ eT Pe þ
n X
eT 1 a ei þ a i a
1
þ
m X
m X
eT 1 b ei b i b
1
eT 1 K e1i þ K 1i 1
mr X
eT 1 K e2i , K 2i 2
ð25Þ
1
1
where the subscript i denotes the ith column of e K e b, f1 and K f2 ; a 2 Rnn, b 2 Rnn, 1 2 Rnn, a, mm are positive definite and symmetric. After 2 2 R differentiating (25) with respect to t, using (22), (20), and properties of matrix trace, the following is obtained upon simplification: fT x þ K e þ B0 ðK e bu e2 rÞ V_ ¼ 2eT P½AM ax 1 _ _ b eT 1 b eT 1 a b þ 2Tr½a þ 2Tr½b a b _ _ eT 1 K eT 1 c c þ 2Tr½K 1 þ 2Tr½K2 2 K2 : 1 1
ð26Þ
Note that the following facts are used in obtaining the above equation: _ _ c e_ c e_1 ¼ K K 1 ; K2 ¼ K2 and Tr½XRXT ¼
m X
xTi Rxi ,
1
where X 2 Rmn, R 2 Rnn and xTi denotes the ith row of X. We also have eT Pe ¼ Tr½a e ¼ xT a eT PexT , eT Pax eT Pe ¼ Tr½b e ¼ uT b eT PeuT , eT Pbu eT x ¼ Tr½K eT xeT PB0 , eT PB0 K 1 1 e2 r ¼ Tr½rT K eT BT Pe ¼ Tr½K eT BT PerT : eT PB0 K 2 0 2 0
ð27Þ
ð28Þ
b in (23), (24), b and b, The adaptive laws for K^1 , K^2 , a were chosen to make all terms except the first term in the RHS of (28) equal to zero, and V_ eT Qe: ð29Þ That is, V(T ) is bounded for all T, and b are all bounded and b b eðtÞ, xðtÞ, K^1 , K^2 , a, e(t) 2 L2. From (22) and closed-loop signal boundedness, we have e_ 2 L1, therefore limt!1 e(t) ¼ 0. T _ _ Because eðtÞ 2 L2 L1 and x(t) 2 L1, K^1j , K^2j , T _ _ b € € b a, b 2 L2 L1 . It can be verified that K^1j , K^2j , € _ € b c b a, b 2 L1 , therefore limt!1 K 1j ðtÞ ¼ 0; limt!1 _ _ _ b c2j ðtÞ ¼ 0, limt!1 aðtÞ b ¼ 0; limt!1 bðtÞ ¼ 0. That is, K all signals and estimates are bounded, and limt!1(x(t) xm(t)) ¼ 0. œ Remark 3.1: . For ensuring that the parameter estimates remain within the polytope, it would be necessary to use projection methods (e.g. Tao et al. 2004). The reference model is updated when the parameter estimates approach the polytope boundaries. The parameter projection is addressed in the next section. . When the mismatch exists only in some of the elements of A and B, the parameter estimation laws can be simplified. For the example in (4), the parameter estimation equations can be written as _ T T b a 11 ¼ "1 a Pex "1 , ð30Þ _ T T b a 12 ¼ "1 a Pex "2 , where "i 2 Rn denotes a unit vector in the i-direction. In fact, the parameter estimation equations in Theorem 3.1 can be readily modified for individual cij ’s, with individual gains ij. a . In the presence of sufficient persistent excitation, e a and e b would converge to zero. 3.2 Reference model redesign/update The adaptive reference model proposed in this article has two levels of adaption. At the first level, the timevarying reference model AM(t), BM(t) addresses relatively small-magnitude mismatch (bounded by
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quadratic stability or performance bounds). The second level of adaptation addresses large magnitude mismatch; i.e. the reference model is updated (redec DB c exceed a predetermined threshsigned) when DA, old. This is expected to occur occasionally as finite number of discrete events, and the adaptive control scheme can be implemented as follows: (1) A reference model Am, Bm is designed based on the initial approximate knowledge of the plant (A0, B0). The matching conditions (3) are satisfied. c for (2) Permissible limits on the elements of DA quadratic stability (or other additional performance criteria) are computed. (3) Adaptive control law and mismatch estimation Equations (13), (23), (24) are applied. c approach the limits (4) If any elements of DA obtained in Step (2) (e.g. within 80–90% of the bounds), the system parameter estimates are b and B0 by updated by replacing A0 by A0 þ a, b (both constant) and the reference B0 þ b model is redesigned using state feedback design methods (such as LQR), thus re-establishing the matching conditions. b are set to zero, and the procedure returns b b (5) a, to Step (2). Steps (2) and (4) (respectively) require computations for LMI solution and reference model redesign (state feedback controller design). It is assumed that these computations can be carried out rapidly in real time, which is feasible because of the availability of efficient algorithms and the fast accelerating trend in computational capability. Therefore, the computation time required is assumed to be sufficiently small so that the parameter estimates will not exceed the permissible bounds during this computation. As an additional assurance that the permissible bounds will not be exceeded, a parameter projection algorithm can be used, as described next. 4. Parameter projection The quadratic stability condition (20) requires that the reference model system matrix AM ðtÞ ¼ bK c be a stable matrix for b þ b bT ¼ Am þ DAðtÞ Am þ a 1 b and K b b b1 . The all possible parameter estimates a, c remains within the stability of AM is assured if DA bounds obtained from maximising in (21), denoted as ca , DA c b Þ. The adaptive laws (23)–(24) generating (DA ij ij these parameter estimates do not automatically guarantee such a property. The reference model update/ redesign procedure (Step 4 in Section 3.2) aims to ensure that the parameter estimates remain within the bounds. As a further assurance, it is desirable to
employ parameter projection in the adaptive gain update and mismatch estimation laws. In order to implement a parameter projection adaptive law, the knowledge of the ranges of the nominal parameters to be estimated is needed, i.e. the lower and upper bounds aaij and abij of the components aij of a ¼ {aij} 2 Rnn, baij and bbij of the components bij of b ¼ {bij} 2 Rnm, and ka1ij and kb1ij of the components k1ij of K1 ¼ {k1ij} 2 Rnn, such that for b b whose components a bij 2 ½aa , ab , any b any a ij ij bij 2 ½ba , bb and any K c1 whose whose components b ij ij a b b components k1ij 2 ½k1ij , k1ij , the matrix AM ¼ Am þ bK b þ b b T is stable. The lower and upper bounds are a 1 to be used in the construction of the parameter projection based adaptive laws. This implies that the parameter uncertainties should stay within the parameter bounds after the adaptive laws are constructed with such bounds, in order to ensure the stability of the c depends on a, b and K b b, c1 , it adaptive laws. Since DA would be necessary to determine the bounds ½aaij , abij , c remains within the ½baij , bbij , ½ka1ij , kb1ij , such that DA a cb c bounds ½DAij , DAij . Investigating appropriate methods for this mapping remains an area of future research. There are two expected desirable properties of parameter projection: one is to ensure that bij 2 ½ba , bb and k b1ij 2 ½ka , kb for bij 2 ½aa , ab , b a ij ij ij ij 1ij 1ij all t 0 so that AM is stable, and the other is to maintain the desired Lyapunov stability properties established for the case without parameter projection. For this purpose, the adaptive laws (23), (24) are modified as _ T c K 1 ¼ 1 xe PB0 þ F1 , ð31Þ _ T T c K 2 ¼ 2 B0 Per , _ b a ¼ a PexT þ Fa , _ b b ¼ b PeuT þ Fb ,
ð32Þ
where F1(t) ¼ { f1ij} 2 Rnm, Fa(t) ¼ {faij} 2 Rnn and Fb(t) ¼ {fbij} 2 Rnm are projection functions with their ijth components being f1ij, faij and fbij, respectively, and the corresponding adaptation gain matrices are chosen to be 1 ¼ diag{ 11, 12, . . . , 1n}, a ¼ diag{ a1, a2, . . . , an} and b ¼ diag{ b1, b2, . . . , bn}, with 1i 4 0, ai 4 0 and bi 4 0, for i ¼ 1, 2, . . . , n, while 2 ¼ T2 4 0 with 2 2 Rmm, which can be non-diagonal, as c2 . parameter projection is not needed for K For specifying the projection functions F1(t), Fa(t) and Fb(t), denote G1 ¼ 1 xeT PB0 , Ga ¼ a PexT , T
Gb ¼ b Peu ,
ð33Þ
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with their ijth components being g1ij, gaij and gbij, respectively, that is, G1(t) ¼ {g1ij} 2 Rnm, Ga(t) ¼ {gaij} 2 Rnn and Gb(t) ¼ {gbij} 2 Rnm, and similarly, b ¼ fb bij g and K b1ij g. c1 ¼ fk b ¼ fa bij g, b a Choose the initial parameter estimates to satisfy bij ð0Þ 2 ½ba , bb , b ij ij
bij ð0Þ 2 ½aa , ab , a ij ij b1ij ð0Þ 2 ½ka , kb k 1ij 1ij and set the projection 8 0 > > < f1ij ðtÞ ¼ > > : g1 ij ðtÞ
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8 > > <
ð34Þ
function components as b1ij ðtÞ 2 ðka , kb Þ, or if k 1ij 1ij b1ij ðtÞ ¼ ka , g1ij ðtÞ 0, or if k 1ij b1ij ðtÞ ¼ kb , g1ij ðtÞ 0, if k 1ij otherwise ð35Þ
bij ðtÞ 2 ðaa , ab Þ, or if a ij ij bij ðtÞ ¼ aa , gaij ðtÞ 0, or if a ij faij ðtÞ ¼ > bij ðtÞ ¼ ab , gaij ðtÞ 0, if a > ij : gaij ðtÞ otherwise ð36Þ 8 bij ðtÞ 2 ðba , bb Þ, or 0 if b > ij ij > > < bij ðtÞ ¼ ba , gbij ðtÞ 0, or if b ij fbij ðtÞ ¼ > b b > if b ðtÞ ¼ b > ij ij , gbij ðtÞ 0, : gbij ðtÞ otherwise ð37Þ 0
for the projection functions F1(t), Fa(t) and Fb(t) in the adaptive laws (31), (32). The above choice of F1(t) ¼ { f1ij}, Fa(t) ¼ { faij} and bij 2 bij 2 ½aa , ab , b Fb(t) ¼ {fbij} ensures that a ij ij a b a b b ½bij , bij , and k1ij 2 ½k1ij , k1ij for all t 0 so that AM is stable. It is verified below that the choice of F1(t), Fa(t) and Fb(t) (whose kth columns are denoted as f1k, fak and fbk, respectively) also guarantees that T 1 fk T 1 fb k 0, K fk T 1 fak 0, b g a 1k 1 f1k 0, a b ð38Þ f e f g e where ak , bk and K1k are the kth column of a, b and e1 , respectively. To see this desired parameter projecK tion property, note that n X 1 g fk T 1 fak ¼ a ð39Þ a ik ai faik , a i¼1
f g where a ik is the ith component of ak . When a b faik ¼ gaik, we have either aik ðtÞ ¼ aik and ga b b ik(t) 5 0, or aik ðtÞ ¼ aik and gaik(t) 4 0, and for both cases, it follows that 1 1 b g a ik a i fa ik ¼ a i ðaik aik Þ ga ik 0
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Using the parameter projection functions F1(t), Fa(t) and Fb(t) and the adaptive laws (31) and (32), Pn eT 1 Pn eT 1 and the i¼1 ai a fa i , i¼1 bi b fbi Pn terms eT 1 i¼1 K1i 1 f1 i (which are all non-positive as shown) appear in the time-derivative V_ in (28) of the positive definite function V defined in (25), i.e.
ð40Þ
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which indicates that the desired stability properties of the closed-loop system remain unchanged by bij 2 parameter projection which ensures that a bij 2 ½ba , bb and k b1ij 2 ½ka , kb , to make ½aaij , abij , b ij ij 1ij 1ij bK b þ b bT stable as ensured by (20). AM ðtÞ ¼ Am þ a 1
5. Application – large transport aircraft In this section, simulation studies are presented to demonstrate the performance of the proposed control method and compare it with non-adaptive and direct adaptive controllers with a fixed reference model. Two examples are shown. Example 1 considers smaller deviations in the system matrices such that all control laws yield stable operation and a comparison can be made across all control laws. Example 2 has larger deviations such that the control laws which attempt to follow the fixed original reference model result in instability, while the proposed method, which uses a time-varying reference model with updates, is able to achieve stable operation with asymptotic tracking.
5.1 Dynamic model A fourth-order longitudinal dynamics model of a large transport aircraft in cruise condition is used. The state variables are: pitch rate q(t) ( /s), true airspeed v(t) (m/s), angle of attack (t) ( ), and pitch angle (t) ( ). The actuators are elevator and engine throttle input, which produce control inputs ue(t) ( ) and ut(t), respectively. The throttle input ut(t) represents the thrust multiplied by a constant scale factor, and hence no units are used. The linear time-invariant plant is described by (1), where the nominal values of the system matrices (continuous-time) are 2 3 0:6803 0:0115 1:0490 0 6 0:0026 0:0062 0:0815 0:1709 7 6 7 A0 ¼ 6 7, 4 1:0050 0:03444 0:5717 5 0 1:000 0 3 44:5192 0:8824 6 0 1:3287 7 6 7 B0 ¼ 6 7: 4 11:4027 0:0401 5 0 0 2
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Figure 1. Example 1, Case 1: plant and reference model states.
The state isx(t) ¼ [q v ]T, and the control input is T uðtÞ ¼ ue ut .
5.2 Reference model The reference model is chosen as the closed-loop system Am ¼ A0 þ B0 KT1 ,
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5.3 Simulation cases In order to evaluate the performance of the proposed method, four different control laws are implemented within each example: . Case 1: fixed-gain controller . Case 2: adaptive controller with a fixed reference model (Am, Bm) designed for the nominal plant . Case 3: adaptive controller with a time-varying reference model (AM, BM) without model updates . Case 4: adaptive controller with a time-varying reference model with model updates (as in Section 3.2)
5.4 Example 1 In this example, the following uncertain deviations (of arbitrary magnitude) in the system matrices are considered: 2 3 2 3 5:0 0 1:3 0 0 0 6 0 6 0 0 0:8 0 7 0:5 7 6 7 6 7 a ¼ 6 7 7, b ¼ 6 4 0 4 0 0 0 05 0 5 0
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No adaptation is used (i.e. 1, 2, a, b ¼ 0), and the reference model is fixed. The plant and reference model states and tracking errors are shown in Figures 1
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Figure 2. Example 1, Case 1: state tracking errors (e).
and 2, respectively. Although the closed-loop system is stable, the tracking errors approach some non-zero constant values. Case 2: Adaptive controller with a fixed reference model (Am, Bm) designed for the nominal plant The control gains are initialised to the ideal gain values for the nominal plant, and updated using the adaptive update law (23). c1 ð0Þ ¼ K1 , K
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2 ¼ diag½0:01, 0:004: The reference model is unchanged (i.e. a, b ¼ 0). Case 2 (results not shown) only showed a marginal improvement in tracking error performance over Case 1, with a somewhat smaller tracking error than for Case 1. Case 3: Adaptive controller with a time-varying reference model (AM, BM) without model updates The control gains are initialised to the ideal gain values for the nominal plant, and updated using the adaptive update law (23). c1 ð0Þ ¼ K1 , K
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2 ¼ diag½0:01, 0:004: The reference model is time-varying (17), wherein d d d d the estimates a 11 ðtÞ, a23 ðtÞ, b11 ðtÞ, b22 ðtÞ are generated by the mismatch estimation laws in (24), and the adaptation gains are chosen as a ¼ 0:1I4 ,
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c However, the maximum permissible variation DAðtÞ required for quadratic stability in the LMI formulation of Section 2 is not calculated, and the reference model is not redesigned. Case 3 (results not shown) showed improvement in tracking error performance over Case 2, with the tracking errors approaching zero, although the convergence is very slow. Case 4: Adaptive controller with a time-varying reference model with model updates (proposed method) The same adaptive gains as in Case 3 are used. c for quadratic stability are Permissible limits on DAðtÞ computed using the LMI formulation of Section 2 with a decay rate of Q ¼ 0.001I and the permissible variation threshold was arbitrarily set at 80% of the maximum permissible limits, calculated as [1.4730, 1.4730]. At approximately t ¼ 1.425 s, one of the d11 ðtÞ) exceeds this threshold and the elements (DA reference model is updated/redesigned using c b c Ad 0ðnewÞ ¼ A0 þ a, with the current values of K1 ðtÞ, c2 ðtÞ (rather than resetting at new values that K accomplish plant-model matching) to allow continuous evolution and avoid jumps in the control input. The d new permissible bounds on DA ij ðtÞ are calculated which are not exceeded for the rest of the simulation run, and no further reference model redesigning is required. As stated in Section 3.2, the reference model redesign and permissible bound computation are assumed to be carried out rapidly in real time. b are b The tracking errors, and estimates aðtÞ, bðtÞ shown in Figures 3, 4 and 5, respectively. Case 4 showed an improved tracking error performance compared to Cases 1–3. In particular, the tracking errors converge to a zero steady-state faster than
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In this example, the following deviations in the system matrix A are considered: 3 3 2 2 2 0 5 0 0 0 7 7 6 6 60 17 60 0 3 07 7 6 6 7 a ¼ 6 7: 7, b ¼ 6 60 0 0 07 60 07 5 4 5 4 0 0 0 0 0 0 (As in Example 1, these deviations are chosen arbitrarily to illustrate the method and do not necessarily represent a physical situation.) Again, it can be verified that the matching conditions are not satisfied for any values of the gains K1 and K2. The four simulation cases are repeated as in Example 1 with a reference step input of magnitude [8.5 7]T beginning at t ¼ 1 s, superposed with white noise. All adaptation/ estimation gains (0 s) are chosen to be the same as in Example 1.
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Case 1: Fixed-gain controller The plant and reference model states, tracking errors and control inputs are shown in Figures 6, 7, 8, respectively. As the mismatch in the system matrices is large and the matching conditions are violated, the fixed-gain controller results in instability, and the
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The plant and reference model states, tracking errors and control inputs are shown in Figures 9, 10, 11, respectively. With large deviations in the system matrices that violate the matching conditions, the standard adaptive controller fails to achieve stable operation while attempting to track the states of the fixed reference model designed for the
The plant and reference model states, tracking errors, b are shown in b and bðtÞ control inputs, estimates aðtÞ Figures 12, 13, 14, 15 and 16, respectively. The reference model is changed continuously using the adaptive mismatch estimation. The adaptive controller is able to track the reference model states asymptotically, however, the system states and the control inputs
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become unbounded, indicating that it is necessary to update/redesign the reference model when any of the dij ðtÞ exceeds the permissible limits. elements DA Case 4: Adaptive controller with a time-varying reference model with model updates (proposed method) The plant and reference model states, tracking errors, b are shown in b and bðtÞ control inputs, estimates aðtÞ Figures 17, 18, 19, 20 and 21, respectively. The reference model is updated/redesigned twice (when c ij ðtÞ exceeds the permissible bounds) approximately DA at t ¼ 1.2 s and again at t ¼ 9.37 s. The adaptive
controller is able to track the reference model states asymptotically and all the system states, control inputs and adaptive parameters are bounded. The tracking error remains small and approaches zero (Note that the discontinuities in the parameter estimates (Figures 20 and 21) are due to resetting of the estimates to zero whenever the reference model is redesigned (Section 3.2)). In summary, based on the numerical results, the method can provide satisfactory state tracking performance in the presence of large plant-model mismatch, with the tracking error asymptotically approaching zero. For the case with relatively mild
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mismatch (Example 1), the method provided only minor improvement in the transient tracking error magnitude; however, the tracking error asymptotically approached zero, which represents significant improvement over the nominal fixed-gain controller and the conventional (fixed-reference model) adaptive controller, as well as some (although smaller) improvement over an adaptive controller with a time-varying reference model but with no model updates. For the case with more severe mismatch (Example 2), only the proposed adaptive controller using a time-varying reference model with model updates was able to provide stability and asymptotic tracking. This
controller yielded satisfactory tracking performance, with the tracking error approaching zero. The mismatch parameter estimates approach some constant values that are not necessarily their true values, which is to be expected because of lack of persistent excitation.
6. Concluding remarks This article addressed direct model reference adaptive control for state tracking using state feedback when the plant-model matching conditions do not hold.
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Matching condition violations can result from modelling errors or large changes in the plant dynamics, including changes in the plant’s mathematical structure. Because of the changes, the impaired plant may have a reduced capability and may no longer be able to track the original reference model. A new adaptive control approach was developed wherein the reference model is modified to be cognisant of the changed plant. The approach includes gain adaptation with simultaneous mismatch estimation, along with reference model redesign whenever the estimated mismatch exceeds predetermined bounds. Thus the reference model ‘adapts’ to the changing plant, in that it incorporates the altered achievable performance of the plant. Application examples were presented, which indicate that satisfactory performance and asymptotic tracking can be accomplished in the presence of large plant-model mismatch. In future work, the issue of canonical parametrisation of the plant-model mismatch needs to be addressed. For general system structures, the mismatch terms would be fully populated matrices, and such a mismatch structure can result in over-parametrisation. Some preliminary results regarding decomposition of the parameter perturbations into matched and unmatched parts were presented, but further research is needed in that area. It would also be desirable to investigate the method further by application to more complex realistic models. Another useful extension would be to address a broad class of actuator and sensor failures in addition to the plant-model mismatch, a topic that is currently being investigated by the authors.
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Acknowledgements The authors are grateful to Prof. Carsten Scherer of University of Stuttgart for his help with LMI implementation. The third author would like to acknowledge the support of the NASA Postdoctoral Program at Langley Research Center, administered by Oak Ridge Associated Universities through a contract with NASA.
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References Annaswamy, A.M., Jang, J., and Lavretsky, E. (2008), ‘Adaptive Gain-scheduled Controller in the Presence of Actuator Anomalies’, 2008 AIAA Guidance, Navigation, and Control Conference, Honolulu, HI, 18–21 August. Cao, C., and Hovakimyan, N. (2008), ‘L1 Adaptive Output Feedback Controller for Non-strictly Positive Real Reference Systems with Applications to Aerospace Examples’, 2008 AIAA Guidance, Navigation, and Control Conference, Honolulu, HI, 18–21 August. Dydek, Z.T., Annaswamy, A.M., and Lavretsky, E. (2010), ‘Adaptive Control and the NASA X-15 Program: A Concise History, Lessons Learned, and a Provably Correct Design’, IEEE Control Systems Magazine, 30, 32–48. Gibson, T.E., Crespo, L.G., and Annaswamy, A.M. (2009), ‘Adaptive Control of Hypersonic Vehicles in the Presence
of Modeling Uncertainties’, in Proceedings of 2009 American Control Conference, St. Louis, MO, 10–12 June, pp. 3178–3183. Ioannou, P.A., and Sun, J. (1996), Robust Adaptive Control, Englewood Cliffs, NJ: Prentice-Hall. Joshi, S.M., Tao, G., and Khong, T. (2009), ‘A Direct Adaptive Control Approach in the Presence of Model Mismatch’, 2009 AIAA Guidance, Navigation, and Control Conference, Chicago, IL, 10–13 August. Lavretsky, E., Gadient, R., and Gregory, I.M. (2010), ‘Predictor-based Model Reference Adaptive Control’, Journal of Guidance, Control, and Dynamics, 33, 1195–1201. Narendra, K.S., and Annaswamy, A.M. (1989), Stable Adaptive Systems, Englewood Cliffs, NJ: Prentice-Hall. Tao, G., Chen, S.H., Tang, X.D., and Joshi, S.M. (2004), Adaptive Control of Systems with Actuator Failures, London: Springer-Verlag. Tao, G., Joshi, S.M., and Ma, X. (2001), ‘Adaptive State Feedback and Tracking Control of Systems with Actuator Failures’, IEEE Transactions on Automatic Control, 46, 78–95. Zaid, A., Ioannou, P., Gousman, K., and Rooney, R. (1991), ‘Accommodation of Failures in the F-16 Aircraft using Adaptive Control’, IEEE Control Systems Magazine, 11, 73–78.
