Clemson University College of Engineering and Science Control and Robotics (CRB) Technical Report

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1 Clemson University College of Engineering an Science Control an Robotics (CRB) Technical Report Number: CU/CRB/9/1/4/#1 Title: A Moular Controller for a Class of Uncertain MIMO Nonlinear Systems with Non-Symmetric Input Gain Matrix Authors: TZhang, DM Dawson, A Behal, an B ian

2 A Moular Controller for a Class of Uncertain MIMO Nonlinear Systems with Non-Symmetric Input Gain Matrix Zhang,DMDawson,ABehal 1,anBian Department of Electrical an Computer Engineering, Clemson University, Clemson, SC Department of Electrical an Computer Engineering, Clarkson University, Potsam, NY Department of Electrical an Computer Engineering, Duke University xzhang,awson@clemsoneu; abehal@clarksoneu; xbin@ukeeu; Abstract In this paper, we consier a general class of MIMO nonlinear systems with unstructure uncertainty in both the rift vector as the input matrix With a positive eþnite restriction on the input matrix that is require for controllability, we are able to construct a continuous state feeback control mechanism that achieves semi-global ultimate boune tracking The Lyapunov base stability result is facilitate through a ecomposition of the aforementione input matrix into a symmetric positive eþnite matrix an a unity upper triangular matrix Furthermore, a moular fee-forwar compensation scheme is introuce in the form of neural network or fuzzy logic schemes 1 Introuction We consier the control esign problem for the class of high-orer MIMO nonlinear systems that are affine in the control input an represente by (n) x t = h(x, úx,,x (), φ(t),t)+ G(x, úx,,x (), φ(t),t)u (1) where x(t) R m is the output vector, h( ) R m represents an uncertain nonlinear function vector; φ(t) R l enotes an unknown, time-varying parameter vector; G( ) R m m is an uncertain nonlinear input gain matrix, an u(t) R m enotes the control input vector Aaptive control schemes for linear time-invariant uncertain SISO systems (that are minimum phase) have been known for quite some time, eg, see [1] However, generalization to the MIMO counterpart has not been easy; a multitue of authors have ealt with this problem by various assumptions on the high-frequency gain (HFG) matrix K p In[1],itisassumethatK p is known; an upper boun on the norm of K p is assume to 1 Corresponing author be known in []; the approach of Weller an Goowin [3] partitions K p = LU while requiring apriori knowlege of the lower bouns of the iagonal entries of the upper triangular matrix U; Þnally, the metho propose in [4] assumes the existence of a matrix S p such that K p S p is positive eþnite an symmetric Most recently, Costa et al [5] solve the MIMO aaptive control problem for minimum-phase systems with relative egree 1 uner the assumption that the signs of the leaing principal minors of the HFG matrix are known A literature survey reveals that the results for uncertain nonlinear systems have been restricte to sub-classes of systems For SISO nonlinear systems in the strict feeback form that are non-affine in the unknown parameters, [6] obtaine a global uniform ultimate boune result by employing a Nussbaum gain an a smooth parameter projection algorithm In [7], Ding et al obtain an ultimately boune output feeback result for uncertain SISO systems via a moiþcation of the backstepping technique using a Nussbaum gain an a Lyapunov function that is ßat in a speciþable region aroun the origin The above results are robust to isturbances an o not require any knowlege of the sign of the highfrequency gain For multi-input nonlinear systems that are representable in the parametric strict feeback form, Krstic et al [8]were able to formalize the aaptive backstepping esign proceure; however, the gain matrix premultiplying the control is assume to be known In [9], a general proceure was presente for esigning switching aaptive controllers for multi-input nonlinear systems which inclues feeback linearizable systems, parametric-pure feeback systems, an systems with a control Lyapunov function that is linear in the parameters In [1], a neural network-base aaptive controller was formulate for the class of systems elineate in (1) with the restriction that G ( ) be uniformly positive or negative eþnite The propose controller was shown to guarantee the semi-global convergence of the tracking error to a resiual set The rawback of the control p 1

3 strategy is that the estimation strategy utilize can lea to loss of controllability in which case the control input tens to zero In [11], the following subset of MIMO nonlinear systems similar was consiere (n) x t = h(x, úx,,x (), θ 1 )+ G(x, úx,,x (n ), θ )u () with the C uncertain functions h ( ) R m an G ( ) R m m being affine in the unknown constant parameter vectors θ i < l i i =1, The propose aaptive controller was proven to ensure the global asymptotic convergence of the tracking error to zero In this paper, we exten the work in [11] for the broaer class of systems given by (1) To satisfy the controllability requirement, it is imperative that the smallest singular value of G ( ) be lowerboune by a positive constant; hence, we make the assumption that G ( ) is positive-eþnite (p) However, we rop the requirement that G ( ) be symmetric since many practical nonlinear control systems o not possess a symmetric input gain matrix [1, 13] Motivate by a matrix ecomposition introuce in [14] an subsequently utilize in [5], we ecompose G( ) into the prouct of a symmetric p matrix an a unity upper triangular matrix The symmetric p matrix is exploite in the Lyapunov base stability analysis while the unity upper triangular matrix allows for an algebraic loop free sequential synthesis of control signals u i (t) i =1,, m The unstructure uncertainty is ealt with by assuming a C smoothness property for h ( ) an G ( ) The result obtaine in this paper is a continuous state feeback control mechanism that achieves semi-global ultimate boune tracking In orer to broaen the applicability of the approach, we introuce a moular fee-forwar scheme which is shown to be achievable via neural network or fuzzy logic compensation The rest of this paper is organize as For the sake of clarity, we Þrst present the control esign for a Þrst-orer, two-input an two-output system of the form (1) We then illustrate how the control metho canbeapplietothesystemeþne by (1) The theoretical evelopment is complemente with a numerical simulation for a two egrees-of-freeom (DOF) mechanical system Simulation results prove the efficacy of the control esign an clearly illustrate the effectiveness of fee-forwar compensation First-Orer MIMO System To explain the control technique, we Þrst examine a Þrst-orer, two-input, two-output, nonlinear system having the following general form úx = h p (x)+g p (x)u (3) where x(t) = x 1 (t) x (t) T R is the system state vector, u(t) = u 1 (t) u (t) T R is the control input vector, an h p (x) R, G p (x) R are uncertain nonlinear functions We will assume that G p (x) is not symmetric but positive eþnite (ie, ν T G p (x)ν for ν(t) R ) To eal with the fact that G p (x) is not symmetric, we will employ a ecomposition tool which is etaile by the following lemma Lemma 1 Any positive-eþnite, non-symmetric matrix P R m m can be ecompose as P = RQ (4) where R R m m is symmetric an positive eþnite, an Q R m m is an unity upper triangular matrix Proof: We can use the fact that all leaing principal minors of a real, positive eþnite matrix are positive ([16], Theorem 51) along with Lemma 1 given in [5] BaseonLemma1,G p (x) can be ecompose as G p (x) =S(x)T (x) (5) where S(x) R is symmetric an positive eþnite, T (x) R is a unity upper triangular matrix explicitly eþne as 1 T1 (x) T (x) =, (6) 1 an T 1 (x) R We can now utilize (5) to rewrite the ynamic moel in (3) as M(x) úx = f(x)+t (x)u (7) where M(x), S 1 (x) R is a symmetric, positive eþnite matrix, an the auxiliary function f(x) R is eþne as f(x), S 1 (x)h p (x) = f1 (x) f (x) T To facilitate the control evelopment, we assume that M(x), f(x), an T (x) satisfy the following assumptions: Assumption F1 The matrix M( ) is assume to be boune as m kξk 6 ξ T M( )ξ 6 m( ) kξk ξ R (8) where m R enotes a positive constant, an m( ) enotes a positive, non-ecreasing function Assumption F The functions M( ), f( ) an T ( ) are assume to be secon-orer ifferentiable an also assume to satisfy the following properties M( ), M( ), ú M( ) L if x, úx, ẍ L f( ), f( ), ú L if x, úx, ẍ L (9) T ( ), T ú ( ), T ( ) L if x, úx, ẍ L p

4 Remark 1 For simplicity of presentation, we have assume that h p (x) an G p (x) o not epen explicitly on time or on unknown time-varying parameters However, it shoul be emphasize that the propose control approach can compensate for these phenomena provie the time-varying effects satisfy secon-orer ifferentiability conitions That is, the functions h p (x) an G p (x) coul be easily replace by h p (x, θ 1 (t),t) an G p (x, θ (t),t) where θ i (t), i = 1, enote unknown time-varying parameter vectors an other time-varying isturbance that may appear nonlinearly in the moel It shoul be note that θ i (t), i =1, must of course satisfy similar type of assumptions as given in Assumption F 1 Control Objective an Control Law To set up the tracking control problem, we let x (t) = x1 (t) x (t) T R enote a reference trajectory signal that must be continuous ifferentiable up to its thir erivative such that i x (t) t i L for i =, 1,, 3 (1) To quantify the control objective, the tracking error e(t) = e 1 (t) e (t) T R is eþne as e = x x (11) Our primary objective requires x(t) to practically track x (t) (ie, UUB tracking) with a continuous control law using full state feeback an norm-base, inequality bouns on the unknown functions M(x), f(x), an T (x) Base on the subsequent stability analysis, we propose the following control law 1 to achieve the state control objective u(t) = (K s + I)e(t) (K s + I)e(t )+ Z t ((K s + I)e(τ)+ ˆf(τ))τ (1) t ks1 where K s = R k is a positive eþnite, iagonal control gain matrix, I R enotes the s ientity matrix, ˆf(t) = ˆf1 (t) ˆf (t) T R enotes a user esigne fee-forwar component It shoul be note that it is assume that the control esigner must ensure that ˆf(t) L (The reaer is referre to Section 3 for speciþc etails on how this might be one) To simplify the analysis, the scalar control gains k s1 an k s are eþne in terms auxiliary control gains as k s1 = k n1 + k n + k n3 (13) k s = k n1 + k n3 1 The secon term in (1) is use to ensure that u(t )= where k n1, k n an k n3 are positive constants As illustrate in the subsequent sections, the control law of (1) ensures semi-global, uniformly ultimately boune tracking provie the control gains (ie, the gains insie the matrix K s as given by (13)) are selecte sufficiently large relative to the norm of the initial tracking error an a reference trajectory-base boun The proof of this result is presente in the following sections Error System Development We begin by eþning the following Þltere tracking error signal [17] r(t) = r 1 (t) r (t) T R as r = úe + e (14) where e(t) was introuce in (11) After ifferentiating (14) an then multiplying both sies of the resulting equation by M(x), wehave M(x) úr = 1 ú M(x)r e ú T (x)u T (x) úu+ñ 1 +N 1 (15) where the auxiliary functions Ñ 1 ( ), N 1 ( ), N 1 ( ) R are eþne as Ñ 1 = N 1 N 1 N 1 = M(x)ẍ + úm(x) úx f(x)+m(x) ú úe + 1 M(x)r ú + e N 1 = M(x )ẍ + úm(x ) úx f(x ú ) (16) The open loop error system given by (15) can be rewritten as M(x) úr = 1 ú M(x)r e T1 (x) úu ú T 1 (x)u úu + Ñ 1 + N 1 (17) where (6) an (16) have been utilize To facilitate further error system evelopment with regar to (17), we utilize (11) an (14) to rewrite Tú 1 (x) as Tú 1 (x) = T 1(x) úx x = T 1(x) (e r)+( T 1(x) x x T 1 (x ) úx x T 1(x ) x ) úx + = Ñ + T 1(x ) x úx (18) where the auxiliary function Ñ( ) R is eþne as Ñ = T 1(x) (e r)+( T 1(x) T 1(x ) ) úx (19) x x x Likewise, further error system evelopment with regar to (17) is fostere by utilizing (7) to rewrite u (t) as p 3

5 u = m 1 (x) úx 1 + m (x) úx f (x) = ũ + u () where the auxiliary functions ũ ( ), u ( ) R are e- Þne as ũ = m 1 (x) úx 1 + m (x) úx f (x) u (1) u = m 1 (x ) úx 1 + m (x ) úx f (x ) We can now use (18) an () to rewrite the Tú 1 (x)u term in (17) as Tú 1 (x)u = Ñ 3 + T 1(x ) úx u () x where the auxiliary function Ñ 3 ( ) R is eþne as Ñ 3 = Ñ (ũ + u )+ T 1(x ) úx ũ (3) x To complete the error system evelopment with regar to (17), we take the time erivative of (1) to obtain úu =(K s + I)r + ˆf (4) which can be rewritten as úu1 (ks1 +1)r = 1 ˆf1 + (5) úu (k s +1)r ˆf after utilizing (14) The expression in (5) can now be use to rewrite the T 1 (x) úu term in (17) as T 1 (x) úu = ( T 1 + T 1 (x ))((k s +1)r + ˆf ) = Ñ 4 + T 1 (x ) ˆf (6) where the auxiliary functions T 1 ( ), Ñ 4 ( ) R are e- Þne as T 1 = T 1 (x) T 1 (x ) Ñ 4 = T 1 (k s +1)r + T 1 (x )(k s +1)r + T 1 ˆf (7) After substituting (4) into (17) an employing () an (6), we can formulate the close loop error system as M(x) úr = 1 M(x)r ú e (K s + I)r Ñ3 Ñ4 + Ñ1 + N (8) Remark To facilitate the stability analysis, we Þrst note that we can use (9) an the fact that ˆf(t) L to show that N (t) L It is also not ifficult to show by the Mean Value Theorem an Assumption F that Ñ 1 ( ) eþne in (16) can be upper boune as Ñ1 6 ρ 1 (kzk) kzk (3) where z(t) R 4 is eþne as z = e T r T T (31) an k k enotes the stanar Eucliean norm, an ρ 1 (kzk) R enotes a positive, bouning function that is non-ecreasing in kzk (Note that this bouning function oes not contain any control gains) The inequality given by (3) will be utilize in the following stability analysis Likewise, the Mean Value Theorem an Assumption F can be employe to show that Ñ 3 ( ) eþne in (3) an Ñ 4 ( ) eþne in (7) can be upper boune as Ñ 3 6 ρ (kzk) kzk Ñ4 6 ρ g (kzk) kzk (3) where z(t) was eþne in (31) anρ (kzk), ρ g (kzk) R are positive, bouning functions that are non-ecreasing in kzk It is easy to show that ρ (kzk) oes not contain any control gains an that ρ g (kzk) oes not epen on the control gain k n introuce in (1) 3 Stability Analysis Before presenting the main result of this section, we state the following lemma which will be invoke later Lemma Let γ 1 (kx(t)k), γ (kx(t)k) : R + R + enote class K functions (x, t) R n R +, γ 3 (kx(t)k) : R + R + enote a function that assumes positive values, an V (x(t),t)) : R n R R + enote continuously ifferentiable functions satisfying γ 1 (kx(t)k) 6 V (x(t),t)) 6 γ (kx(t)k) (33) If there exists positive constants η 1 an η satisfying η > (γ 1 1 γ )(η 1 ) an the time erivative of V (x(t),t)) along the trajectory of the system satisþes = 1 M(x)r ú e (K s + I)r + Ñ + N úv (x(t),t) < γ 3 (kx(t)k) for η 1 < kx(t)k < η (34) where the auxiliary functions Ñ( ), N ( ) R are e- then kx(t)k, withinitialstatex(t ),hasthefollowing Þne as properties Ñ3 Ñ4 Ñ = + Ñ 1 (i) Locally uniformly boune SpeciÞcally, kx(t )k 6 s implies kx(t)k 6 (s) for all t [t T 1 (x ) N = N 1 úx u x T1 (x ) ˆf, ), where ˆf ½ (γ 1 (s) = 1 γ )(s) if η 1 6 s 6 (γ 1 γ 1 )(η ) (γ 1 1 γ )(η 1 ) if s 6 η 1 (9) (35) p 4

6 (ii) Locally uniformly ultimately boune SpeciÞcally, given a constant η with (η 1 ) < η 6 η, kx(t )k 6 s implies kx(t)k 6 η for all t [t + T (η,s), ), where T (η,s) enotes the time in which the ultimate boun is guarantee Proof: See Theorem 15 in [18] for proof as well as an explicit formula for T (η,s) Given the above Lemma, the main result of this section can be state by the following Theorem Theorem 1 The control law of (1) ensures that all system signals are boune uner close-loop operation an ensures the tracking error variable z(t) exhibits the locally uniformly boune an the locally uniformly ultimately boune properties given by Lemma Proof: See Appenix A 3 Fee-forwar Component Design In general, one expects that the use of a fee-forwar term in control law will reuce the magnitue of the control input The aition of the fee-forwar component ˆf( ) in (1) oes not alter the type of tracking result presente in Section 3 as long as the control esigner ensures that ˆf( ) L To illustrate how one might esign ˆf( ), we now elineate three ifferent methos 31 Best-Guess Estimation In this case, the functional form of M(, θ 1 ), f(, θ ), an T (, θ 3 ) in (7) is known, but the constant parameter vectors θ 1, θ an θ 3 may not be known precisely In this case, the following certainty equivalence-type component ˆf( ) canbeaetotheproposecontrollaw u(t) in (1) ˆf( ) = N 1 (x, úx, ẍ, ˆθ 1, ˆθ,t) T 1(x, ˆθ 3 ) x úx u (x, úx, ˆθ 1, ˆθ ) (38) Remark 3 We easily illustrate that the proof of the above theorem an Lemma actually yiel a semi-global uniformly boune result for the tracking error variable z(t) by illustrating that the controller gains can be increase to cover a preetermine value of the initial conitions enote by kz(t )k First, by using (14) an (31), we can write an explicit expression for kz(t )k as q kz(t )k = e T (t )e(t )+(úe(t )) + e(t )) T ( úe(t )) + e(t )) (36) From (7), (11), an the fact that u(t )=(see (1)), we can evelop the following expression for úe(t ) úe(t )= úx (t ) M 1 (x(t ))f(x(t )) (37) From the expression given by (36) an (37), it is clear that kz(t )k oes not epen on any of the control gain parameters Secon,we can see from the eþnition of η 1 given by (66) an the inequality conition given by (61) thatη 1 can be mae arbitrarily small by increasing the control gain k n3 Lastly, we can see from the eþnition of η given by (66) an the iscussion regaring ρ (kzk) an ρ g (kzk) given in Remark that η can be mae arbitrarily large by increasing the control gains k n1 an k n Wealsonotethattheultimateboungiven in the Lemma can be arbitrarily small by increasing the control gain k n3 That is, as illustrate above, η 1 can be mae arbitrarily small by increasing the control gain k n3, an hence, ue to the structure of γ 1 ( ), γ ( ) given by (58), the lower boun for the ultimate boun given in Lemma (ie, (η 1 )) can be mae arbitrarily small where N 1 ( ) was eþnein(16),u ( ) was eþne in (1), ˆθ 1, ˆθ an ˆθ 3 enote the best-guess constant estimates of the unknown parameter vectors 3 Neural Network Base Estimation In this case, the function form of M( ), f( ), an T ( ) in (7) is unknown Accoring to the approximation properties of neural network (See [19] an []), the boune, a continuous uncertain nonlinear function f(χ) =N 1 ( ) T 1 (x ) úx u ( ) x in (9), can be approximate by a two-layer, neural network such that f(χ) = W T σ(v T χ)+ε 1 (χ) (39) for a hien-layer with p number of neurons where χ x, úx, ẍ, x R s represents the boune inputs to neural network, W R p an V R s p enote the ieal constant weights, σ( ) R p is the corresponing neuron activation function, ε 1 ( ) R enotes the boune reconstruction error (ie, kε 1 (χ)k 6 ε 1 where ε 1 is some positive constant ) We can construction a fee-forwar neural network to approximate the unknown continuous nonlinear function It contains a hien layer with p neurons an a output layer with neurons The output of neural network is esigne as ˆf( ) =Ŵ T σ( ˆV T χ) (4) The reconstruction error is boune by a constant because it epens on the esire trajectory as oppose to the actual trajectory p 5

7 where Ŵ (t) R p an ˆV (t) R s p are the weight estimates of W an V, respectively To meet the requirement that ˆf( ) L, the upate law of Ŵ(t) an ˆV (t) shoul be esigne such that Ŵ (t), ˆV (t) L (In the simulation section, we illustrate how this can be one) There are many choices for the activation function σ( ), such as raial basis function, hyperbolic tangent function, or the sigmoi function (See [19] an []) 33 Fuzzy Logic Base Estimation In this case, the functional form of M( ), f( ) an T ( ) is unknown Base on the Fuzzy Logic (FL) approximation property (See [19] an []), the boune, continuous nonlinear function f(χ) = N 1 ( ) T 1 (x ) úx u ( ) x in (9) can be approximate as f(χ) = Φ T ξ(χ)+ε (χ) (41) where Φ R p is the ieal FL constant parameters, p is the number of fuzzy logic rules, ξ( ) R p enotes the output of the fuzzy system, χ x, úx, ẍ, x R s,an ε ( ) R represents the reconstruction error which can be upper boun by some positive constant ( ie, kε (χ)k 6 ε where ε is some positive constant) By using the techniques in [], we can construct a rulebase FL term to approximate the uncertain nonlinear function as ˆf( ) =ˆΦ(t) T ξ(χ) (4) where ˆΦ(t) R p enote the weight estimates of Φ To meet the requirement that ˆf( ) L, we must ensure that the upate law for ˆΦ(t) is esigne such that ˆΦ(t) L 4 Extension to Higher-Orer Multi-Input System In this section, we iscuss the extension of the propose control law for the following system x (n) = f(x, úx,,x () )+G(x, úx,,x () )u (43) where x (i) (t) R m, i =, 1,, n 1 are the system states, ( ) (i) (t) enotes the i th erivative with respect to time, u(t) R m represents the control input, f ( ) R m an G ( ) R m m are uncertain nonlinear functions For presentation purposes, we eþne =[ x T úx T (x () ) T ] T R mn We assume that G( ) is not symmetric but positive eþnite Base on Lemma 1, G( ) can be ecompose as G() =S()T () (44) where S( ) R m m is symmetric, positive eþnite, an T ( ) R m m is an unity upper triangular matrix We can rewrite the system given (43) as M()x (n) = F ()+T ()u (45) where the auxiliary functions M( ) R m m an F ( ) R m are eþne as M( ), S 1 ( ) F ( ), S 1 ( )f( ) (46) It is easy to see that M( ) is symmetric an positive eþnite WeassumethatM( ) an F ( ) satisfy the following assumptions: Assumption M1 The matrix M( ) is boune by m kξk 6 ξ T M( )ξ 6 m( ) kξk ξ R m (47) where m R enotes a positive constant, an m( ) R enotes a positive, non-ecreasing function Assumption M The functions M( ), f( ), ant ( ) are secon-orer ifferentiable such that M( ), M( ), ú M( ),F( ), F ú ( ), F ( ),T( ), T ú ( ), T ( ) L if x, úx, ẍ,, x (n+1) L (48) Let x (t) R m enote the reference trajectory signal that is continuously ifferentiable such that x (i) (t) L for i =, 1,, n + (49) To simplify the analysis, we also eþne = [ x T úx T (x () ) T ] T R mn To quantify the control objective, we eþne the tracking error signal, enote by e 1 (t) R m, as e 1, x x (5) As before, the control objective is to obtain practical tracking with a continuous law with full-state feeback (ie, x (i) (t), i=, 1,, are assume measurable) an ensure that all the signals remain boune uring the close-loop operation To simplify the subsequent control esign an stability analysis, we introuce the following auxiliary error signals e i (t) R m, i =, 3,, n e, úe 1 + e 1, e 3, úe + e + e 1, e 4, úe 3 + e 3 + e, e i, úe i 1 + e i 1 + e i, e n, úe + e + e n (51) p 6

8 where e 1 (t) was eþne in (5) An expression can be erive for e i (t) for i =3, 4,, n in terms of e 1 (t) an its erivatives as following (see [15] for etails) i 1 e i = a ij e (j) 1 (5) j= where the constant coefficients a ij R are eþne as " a i = Θ(i) = 1 5 a ij = i 1 P k=1 ( 1+ 5 ) i ( 1 # 5 ) i Θ(i k j +1)a k+j 1,j 1,i=3, 4,, n an j =1,,, i a i,i 1 =1,,,, n (53) Note that e i (t) for i =1,,, n is measurable since it is a function of the system states an the reference trajectory 41 Control Law Base on subsequent stability analysis, we esign the control input u(t) as u(t) = (K s + I m )e n (t) (K s + I m )e n (t )+ Z t t [(K s + I m )e n (τ)+ ˆf(τ)]τ (54) where K s R m m is positive-eþnite, iagonal, control gain matrix, I m R m m represents the m m ientity matrix, ˆf( ) = ˆf1 ( ) ˆf ( ) ˆfm ( ) T R m enotes the user esigne fee-forwar component (It is assume that ˆf( ) L ) It is easy to see that u(t )= To simplify the analysis, the iagonal components of the control gain matrix K s are eþne as k s1 = k m + k 1 + k n k si = k m + k i + k n (55) k sm 1 = k m + k m 1 + k n k sm = k m + k n where k si enotes the i th iagonal component in K s i =1,,, m; ank m, k n, k 1,, k m 1 enotes positive control gains 4 Error System Development See Appenix B 43 Stability Analysis The main result of this section can be state by the following theorem Theorem The control law of (54) ensures that all system signals are boune uner close-loop operation an ensures the tracking error variable z(t) exhibits locally uniformly boune an the locally uniformly ultimately boune properties given by Lemma Proof: See Appenix C Remark 4 As one in Remark 3, we easily illustrate that the above the proof of Theorem an Lemma actually yiel a semi-global uniformly boune result for,i=, 3,, n the tracking error variable z(t) by illustrating that the controller gains can be increase to cover a preetermine value of the initial conitions enote by kz(t )k In aition, one can evelop fee-forwar terms for the control law as one in Section without altering the type of tracking result as long as the control esigner ensures that ˆf( ) L See Appenix D 5 Simulation 6 Conclusion In this paper, we consiere the tracking control problem for a class of MIMO nonlinear systems for which the input matrix is positive eþnite but non-symmetric By utilizing a ecomposition property of the input matrix, a continuous control strategy was propose in orer to compensate for uncertain nonlinear functions associate with the system ynamic moel an ensure semi-global uniform ultimately boune tracking uner a smoothness restriction on the uncertain system nonlinearities Best-guess, neural network an fuzzy logic base approximation techniques were employe in the fee-forwar control esign to broaen the applicability of the approach Simulation results were presente to emonstrate the efficacy of the control an fee-forwar strategies References [1] SS Sastry an M Boson, Aaptive Control: Stability, Convergence, an Robustness, Englewoo Cliffs, NJ: Prentice-Hall, 1989 [] M e Mathelin an M Boson, Multivariable Moel Reference Aaptive Control without Constraints on the High-Frequency Gain Matrix, Automatica, Vol 31, No 4, pp , 1995 [3] SR Weller an GC Goowin, Hysteresis Switching Aaptive Control of Linear Multivariable Systems, IEEE Transactions on Automatic Control, Vol 39, No 7, pp , 1994 [4] P Ioannou an K Sun, Robust Aaptive Control Englewoo Cliffs, NJ: Prentice-Hall, 1996 p 7

9 [5] RR Costa, L Hsu, AK Imai, an P Kokotović, Lyapunov-Base Aaptive Control of MIMO Systems, Automatica, Vol 39, No 7, pp , July 3 [6] J Wang an Z Qu, Robust Aaptive Control of Strict-feeback Nonlinear System with Nonlinear Parameterization, Proceeings of the American Control Conference, Denver, Colorao June 4-6, pp , 3 [7] Z Ding, Aaptive Control of Nonlinear Systems with Unknown Virtual Control Coefficients, Int J Aaptive Control an Signal Processing, Vol 14, pp , [8] M Krstić, I Kanellakopoulos, an P Kokotović, Nonlinear an Aaptive Control Design, New York: John Wiley & Sons, 1995 [9] EB Kosmatopoulos an PA Ioannou, Robust Switching Aaptive Control of Multi-Input Nonlinear Systems, IEEE Trans Automatic Control, Vol 47, No 4, pp 61-64, Apr [1] H u an PA Ioannou, Robust Aaptive Control for a Class of MIMO Nonlinear Systems with Guarantee Error Bouns, IEEE Trans Automatic Control, Vol 48, No 5, pp 78-74, May 3 [11] T Zhang, DM Dawson, MS e Queiroz, an B ian, Aaptive Control for a Class of MIMO Nonlinear Systems with Non-Symmetric Input Matrix, Proceeings of the IEEE Conference on Control Applications, Taipei, Taiwan, accepte, to appear, September 4 [1] E Zergeroglu, DM Dawson, MS e Queiroz, an A Behal, Vision-Base Nonlinear Tracking Controllers with Uncertain Robot-Camera Parameters, IEEE/ASME Trans Mechatronics, Vol6,No3,pp 3-337, Sept 1 [13] P Setlur, J Wagner, D Dawson, an J Chen, Nonlinear Controller for Automotive Thermal Management Systems, Proc American Control Conf, pp , Denver, CO, June 3 [14] AS Morse, A Gain Matrix Decomposition an Some of Its Applications, Systems an Control Lett, Vol 1, pp 1-1, 1993 [15] Bin ian, Marcio S e Queiroz, an Darren M Dawson, A Continuous Control Mechanism for Uncertain Nonlinear Systems, Optimal Control, Stabilization, an Nonsmooth Analysis, Lecture Notes in Control an Information Sciences, Heielberg, Germany: Springer-Verlag, to appear, 4 [16] S Perlis, Theory of Matrices, New York: Dover Publications, 1991 [17] JJ Slotine an W Li, Applie Nonlinear Control, New York: Prentice Hall, 1991 [18] Z Qu, Robust Control of Nonlinear Uncertain System, John Willey & Sons, Inc, 1998 [19] F L Lewis, J Campos, R Selmic, Neuro-Fuzzy ControlofInustrialSystemswithActuatorNonlinearities, Siam, [] Jeffrey T Spooner, Manfrei Maggiore, Raul Oronez, an Kevin M Passino, Stable Aaptive Control an Estimation for Nonliear Systems, John Wiley & Sons, Inc, New York, [1] AP Aguiar an JP Hespanha, Position Tracking of Uneractuate Vehicles, Proc American Control Conf, pp , Denver, CO, June 3 [] LCremean,WDumbar,DvanGogh,JHickey, E Klavins, J Meltzer, an R Murray, The Caltech Multi-Vehicle Wireless Testbe, Proc IEEE Conf Decision an Control, Las Vegas, NV, pp 86-88, Dec [3] I Kanellakopoulos, P V Kokotovic, an A S Morse, Aaptive Output-Feeback Control of Systems with Output Nonlinearities, IEEE Trans Automatic Control, Vol 37, No 11, pp , Nov 199 [4] H K Khalil, Aaptive Output Feeback Control of Nonlinear System Represente by Input-Output Moels, IEEE Trans Automatic Control, Vol 41, No, pp , Feb 1996 [5] R Marino an P Tomei, Global Aaptive Output-Feeback Control of Nonlinear Systems, Part I: Linear Parameterization, IEEE Trans Automatic Control, Vol 38, No 1, pp 17-3, Jan 1993 A Proof of Theorem 1 We eþne the positive function V (z, t) :R + R 4 as V = 1 et e + 1 rt Mr (56) where z(t) was eþnein(31) Notethat(56)canbe boune as γ 1 (kzk) 6 V 6 γ (kzk) (57) where the scalar class K functions, enote by γ 1 ( ), γ ( ) R, areeþne as γ 1 (kzk) = 1 min(1, m) kzk γ (kzk) = 1 max(1, m(x)) (58) kzk where (8) has been utilize After taking the time erivative of (56) an substituting from (14) an (8), we have úv = e T e r T r r T K s r + r T Ñ + r T N = kzk + r T Ñ 1 r 1 Ñ 3 r 1 Ñ 4 k n1 krk k n r 1 k n3 krk + r T N (59) p 8

10 where the eþnition of K s in (13) has been utilize After applying (3), (3) an the fact that k n3 krk + r T N 6 1 N, we can upper boun the right-han k n3 sie of (59) as h úv 6 kzk + krk ρ(kzk) kzk k n1 krk i + h r 1 ρ g (kzk) kzk k n r 1 i + ε (6) where ρ( ) R is eþne as ρ = ρ 1 + ρ an ε R is some positive constant that satisþes 1 N 6 ε (61) k n3 (See Remark on N ( )) After completing the squares on the brackete terms in (6), we obtain úv 6 (1 1 ρ (kzk) 1 ρ 4k n1 4k g(kzk)) kzk + ε (6) n We can now use (6) to state that úv 6 λ 3 kzk + ε (63) provie that the following sufficient conitions are satisþe k n1 > 1 ρ (kzk) (or kzk < ρ 1 ( k n1 )) k n > 1 ρ g(kzk) (or kzk < ρ 1 g ( k n )) (64) where λ 3 R is some positive constant It is easy to see the gain conitions given in (64) can be satisþe given the facts that ρ(kzk) oes not contain any control gains an that ρ g (kzk) oes not epen on the control gain k n (See Remark ) From (63), we form the following inequality úv 6 γ 3 (kzk) for η 1 < kz(t)k < η (65) where the positive function γ 3 (kzk) R an the positive constants η 1, η R are eþne as γ 3 (kzk) =λ 4 kzk η 1 = ε η =min( ρ 1 ( k n1, ρ 1 g ( (66) k n )) with λ 4 R being some positive constant We can now apply Lemma to (57) an (65) to yiel the locally uniformly boune an the locally uniformly ultimately boune properties for tracking error variable z(t) given by the theorem statement; hence, if kz(t )k satisþes the requirements given in the locally uniformly boune property of Lemma, then z(t) L Stanar signal chasing arguments along with Assumption F can now be use to show that all the signals remain boune B Error System Development To begin the error system evelopment, we eþne the Þltere error signal r(t) R m as r = úe n + e n (67) where e n (t) was eþne in (51) After taking the time erivative of (67), multiplying both sies of the resulting equation by M( ), an then substituting from the erivative of (5) for i = n, wehave M úr = M a nj e (j+) 1 + M úe n (68) j= After utilizing (5) an the last expression of (53), we can rewrite (68) as n M úr = Mx (n+1) Mx (n+1) + M( a nj e (j+) 1 + úe n ) = M(x (n+1) n + j= ú F ú Tu T úu j= a nj e (j+) 1 + úe n )+ úmx (n) (69) where the time erivative of (45) has been use We now arrange (69) into the following avantageous form M úr = 1 ú Mr e n T u T úu úu + N 1 (7) where the auxiliary function N 1 ( ) R m is eþne as N 1, M(x (n+1) n + j= a nj e (j+) 1 + úe n )+ úmx (n) + 1 Mr ú + e n F ú (71) with the strict upper triangular matrix, enote by T ( ) R m m,iseþne as T = T I m Finally, after eþning the auxiliary expression N 1 (t), N(x, úx,,x (n),t) Rm, (7) can be rewritten as M úr = 1 ú Mr e n T u T úu úu + N 1 + Ñ1 (7) where the auxiliary function Ñ1( ) R m is eþne as Ñ1 = N 1 N 1 Notethatitisnotifficult to show that N 1 (t), ún 1 (t) L after utilizing (48) an (49) In aition, we note that as in Section, we can show that Ñ1 6 ρ 1 (kzk) kzk where z = e T 1 e T n r (73) T T R m(n+1) where ρ 1 (kzk) R is some positive bouning function that is non-ecreasing in kzk an oes not contain any control gains p 9

11 To facilitate the analysis, we note that the matrix T () in (7) can be expresse as h i T = T úij () for i =1,,,m 1; j = i +1,,m (74) where T ij () R is the ij th element of the unit uppertriangularmatrixt () It is obvious that T () is a strict upper triangular matrix since all its iagonal elements are zeros By utilizing (5), we can write Tú ij () of (74) as ú T ij () = = k= k= T ij () x (k) T ij () ( x (k) e (k+1) 1 ) T ij ( ) = Ñ ij + k= x (k) (75) where the auxiliary function Ñ ij ( ) R is eþne as Ñ ij = k= 1 + T ij () e (k+1) x (k)! T ij ( ) x (k) It is easy to show that Ñ ij ( ) given in (76) oes not contain any control gains Similarly, base on (45), we can explicitly express the i th component of the control vector u(t) in (7) as u i = m j=1 M ij ()x (n) j F i () m T ij ()u j (77) where i =1,,, m, M ij ( ) R represents the ij th component in matrix M( ), anf i ( ) R enotes the i th component in vector F ( ) We further rewrite (77) as u i =ũ i + u i (78) where the auxiliary functions ũ i (t), u i (t) R are e- Þne as m ũ i = ( M ij ()x (n) j j=1 j=1 F i () m T ij ()u j ) m ( M ij ( )x (n) j F i ( ) (79) m T ij ( )u j ) an u i = m j=1 M ij ( )x (n) j F i ( ) m T ij ( )u j (8) It is easy to show that ũ i (t) an u i (t) in (78) o not contain any control gains After utilizing (75) an (79), we now obtain the following expression for Tú ij ()u j Tú ij ()u j = Ñ 3ij +( k= T ij ( ) x (k) )u j (81) where the auxiliary function Ñ 3ij ( ) R is eþne as Ñ 3ij = Ñ ij ũ j + u j Ñ ij +( k= T ij ( ) x (k) )ũ j (8) for i =1,,,m 1; j = i +1,,m The expression givenby(81)cannowbeusetorewritethevector term T u R m in (7) as mp Tú µ T u = ij ()u j Tij () x (k) k= P = Ñ3i + m ( P k= (76) T ij ( ) x (k) )u j (83) where i =1,,,m 1, an the auxiliary function Ñ 3i ( ) R is eþne as Ñ 3i = m Ñ 3ij (84) From (76), (79), (8), an (84), we can utilize the Mean Value Theorem to upper boun the auxiliary function Ñ 3i ( ) as 6 ρ i (kzk) kzk (85) Ñ3i where ρ i (kzk) R is a positive bouning function that is non-ecreasing in kzk an oes not contain any control gains We also note that mp ( P T ij ( ) k= x (k) )u j in (83) an its time erivative are boune ue to the facts elineate in (48), (49) an (8) Continuing in the same manner, we can write the vector item T úu R m in (7) in the following form mp T T úu = ij () úu j for i =1,,,m 1 (86) p 1

12 where the time erivative of (54) can be use to write the term T ij () úu j R as T ij () úu j = (T ij () T ij ( )+T ij ( )) ((k sj +1)r j + ˆf j ) = Ñ 4ij + T ij ( ) ˆf j (87) with the auxiliary functions Ñ 4ij ( ) R being eþne as Ñ 4ij = T ij (k sj +1)r j + T ij ˆfj +T ij ( )(k sj +1)r j, (88) an the auxiliary functions T ij ( ) R being eþne as T ij = T ij () T ij ( ) (89) Theexpressiongivenby(87)cannowbeusetocan rewrite (86) as P T úu = Ñ4i + m T ij ( ) ˆf j for i =1,,,m 1 (9) where the auxiliary functions Ñ 4i ( ) R is eþne as Ñ 4i ( ) = m P Ñ 4ij ( ) As similarly one before, we can use the Mean Value Theorem to upper boun Ñ 4i ( ) as 6 ρ gi (kzk) kzk (91) Ñ4i where ρ gi (kzk) R is a positive bouning function that is non-ecreasing in kzk It shoul be note that ρ gi (kzk) oes contain the control gain k sj for j = i +1,,m Base on (54), (7), (83) an (9), we can now formulate the close loop system for r(t) as M úr = 1 Mr ú Ñ3i e n mp Ñ4i ( P k=1 T ij ( ) x (k) )u j T ( ) ˆf (K s + I m )r ˆf + Ñ1 + N 1 (9) where i =1,,,m 1 To facilitate the stability analysis, (9) can be rearrange into the following avantageous form M úr = 1 ú Mr e n (K s + I m )r + Ñ + N (93) where the auxiliary functions Ñ( ), N ( ) R m are e- Þne as Ñ3i Ñ4i Ñ = Ñ 1 mp N = N 1 ( P T ij ( ) k= x (k) )u j ˆf (94) for i =1,,,m 1 Itisnotifficult to show that the assumptions given earlier allow us to show that N ( ), T ( ) L C Proof of Theorem We eþne the positive function V (t, z) : R + R m(n+1) as V = 1 n e T i e i + 1 rt Mr (95) It is easy to see that γ 1 (kzk) 6 V 6 γ (kzk) (96) where γ 1 ( ), γ ( ) R are the scalar class K functions eþne as γ 1 (kzk) = 1 min(1, m) kzk γ (kzk) = 1 max(1, m(x)) (97) kzk where (8) has been utilize After taking the time erivative of (95) an substituting from (51), (55), (67) an (93), we obtain n úv = e T i e i e T e n r T r r T K s r + r T Ñ + r T N n = e T i e i e T e n k m krk k n krk m 1 m 1 k i kr i k + r T Ñ 1 ri T Ñ 3i m 1 ri T Ñ 4i + r T N (98) By employing the fact that e T e n 6 1 (ke k + ke n k ), k n krk + r T N 6 1 kn k,wecanuse(73), k n (85) an (91) to obtain an upper boun on (98) as úv 6 1 kzk k m krk + r T ρ(kzk) kzk m 1 m 1 k i krk + 6 ( 1 ρ (kzk) 4k m kr i k ρ gi (kzk) kzk + 1 k n kn k m 1 ρ gi (kzk) 4k i ) kzk + ε (99) p 11

13 where the scalar class K function ρ( ) R is eþne as m 1 ρ = ρ 1 + ρ i (1) an ε R enotes some positive constant that satisþes ε > 1 kn k (11) k n Given the fact that ρ( ) oes not contain control gain k m,anρ gi ( ) oes not contain the control gain k i,the proof of Theorem 1 can now be followe to prove the result state in Theorem provie r k m > ρ λ (kz(t )k) ( kz(t )k), λr 1 k i > (m 1)ρ gi ( kz(t )k kz(t )k) i =1,,m 1 λ 1 (1) We note that (67) an (73) can be use to obtain an explicit expression for kz(t )k which is inepenent of all control gains as one in Remark 3 D Simulation D1 Simulation Moel In orer to emonstrate the performance of the propose control law, we present the results of a numerical simulation To this en, the following two egree-offreeom (DOF) system was consiere [17] τ 1 H11 H = 1 + τ H 1 h úq H q h ( úq 1 + úq ) q1 h úq 1 where q i (t) enote the i th DOF position, úq1 úq (13) H 11 = a 1 +a 3 cos q +a 4 sin q H 1 = H 1 = a + a 3 cos q + a 4 sin q (14) H = a h = a 3 sin q a 4 sin q τ 1 τ 1 1 = α(q 1,q ) 1 u1 u, (15) a 1 =44, a =97, a 3 =14, ana 4 =6 In (15), u 1 (t), u (t) enote the control inputs, an the function α(q 1,q ) = H 11 H H 1 H 1 R The scalar the function α(q 1,q ) coul also be moiþe to represent environment relate factors as shown in [1, ] or other effects as shown in [3, 4, 5] The control objective is to ensure that q(t) = q1 (t) q (t) T track the following reference trajectory q (t) = 3 sin(t) 1 exp( 3t 3 ) 45 sin(t) 1 exp( 3t 3 ) eg; (16) hence, the tracking error variable is eþne as e 1 (t) = q (t) q (t) The initial conition was set to q i () = 6 egree for i =1, First, we carrie on the simulation without feeforwar terms in the control input u(t) in (54) (ie, ˆf(t) =) The feeback control gains were set as K s = iag {5, 3} Figure 1 shows the tracking error e 1 (t) while Figure shows the control input u(t) We note that one can ecrease the magnitue of the tracking error e 1 (t) by increasing control gains insie the matrix K s D Neural Network Base Compensation For simplicity, we employe a raial basis neural network (RBNN) [19] to approximate the unknown nonlinear function f( ) =N 1 ( ) úx u ( ) T 1 (x ) x in (9) The RBNN is comprise of a layer of raial basis activation functions with the neurons number p =1 an the output of the neural network system is esigne as ˆf( ) =Ŵ (t) T σ( V T χ 1 ) (17) where χ 1 1,x, úx, ẍ, x R 9,theÞrst component in χ 1 (t) is set to 1 to prouce a basis, V R 9 is set to constant values in orer to provie a basis [19], Ŵ (t) R 1 enotes the weight estimates of iea weight gain 1 matrix W A sigmoi function σ(z) = 1+exp( s) is utilize as the activation function The following weight tuning law for Ŵ (t) is utilize Ŵ = α 1 Ŵ + Γ 1 σ(v T χ 1 )sat(e + ζ 1 ) ζ 1 = 1 ε 1 ( η 1 + e ) úη 1 = 1 ε 1 ( η 1 + e ) (18) where α 1, ε 1 R are some small positive constant, Γ 1 R 1 1 is a iagonal, p upate gain matrix, sat( ) R is eþne as sat( ) = sat(ξ 1 ) sat(ξ ) ξ= ξ 1 ξ T (19) with sat( ) as the stanar saturation function, while ζ 1, η 1 R are auxiliary Þlter signals It is not ifficult to check that Ŵ (t) L as require The control gain matrix K s was selecte as in Section D1 while V in (17) was selecte as V = T (11) The upper an lower value for the saturation function in (18) were set as 1 an 1, respectively The p 1

14 weight tuning gains in (18) were tune by trial-anerror until goo tracking performance was achieve The turning proceure resulte in the following gain values α 1 =1 ε 1 =1 Γ 1 = iag(1,,, 1, 15, 15, 4,, 16, 8) (111) The tracking error an control input for the propose control law are shown in Figure 3 an Figure 4, respectively While the control gains were set to the same values as in Section D1, the tracking error e 1 (t) was riven to a very small value aroun zero with the inclusion of neural network base fee-forwar component in the control input Link 1 (eg) Link 1 (eg) Link (eg) Time (sec) Figure 3: Tracking error for the propose controller with neural-network base fee forwar component ˆf(t) Link (eg) Time (sec) Figure 1: Tracking error for propose controller without fee forwar component ˆf(t) 4 Link 1 (Nm) Link 1 (Nm) Link (Nm) Time (sec) Link (Nm) Time (sec) Figure : Control input for the propose controller without feeforwar component ˆf(t) Figure 4: Control input for the propose controller with neural network base fee forwar component ˆf(t) p 13