Key Words : transient response, model reference adaptive control system, smooth projection, dynamic certainty equivalence

Transcription

1 SICE Journal of Control, Measurement, and System Integration, Vol. 6, No. 4, pp. 43 5, July 3 An Improvement of Transient Response of Dynamic Certainty Equivalent Model Reference Adaptive Control System Based on New Smooth Parameter Projection High Order Tuner Masataka SAWADA and Keietsu ITAMIYA Abstract : A special adaptation law which has the ability to constrain adjustable parameters of an adaptive controller to the specified convex space is often used in an adaptive control system (In the following, this is referred to as a projection adaptation law. In particular, the projection adaptation law which guarantees the existence of time derivative or high order time derivatives of adjustable parameters is referred to as a smooth projection adaptation law and it is used for an adaptive control system based on the adaptive back-stepping method or the dynamic certainty equivalent principle. In the conventional projection adaptation law, the convex set to be constrained is a hyper-rectangular or a hyper-sphere in order to accomplish a realization of an adaptive controller, an efficient estimation and robust stability of adaptation loop. This paper indicates new two smooth adaptation laws (high order tuner to improve the transient response of a control system in addition to the ability of conventional restraint. Also, the authors propose design schemes of a model reference adaptive control system using them. They can constrain adjustable parameters to an internal space of hyperparallelogram. Effectiveness of the proposed scheme is illustrated by the theoretical analysis and simple numerical simulation results. Key Words : transient response, model reference adaptive control system, smooth projection, dynamic certainty equivalence control.. Introduction In a practical adaptive control system, the adaptation law which has some projection functions is often designed to constrain the adjustable parameters into the pre-specified convex set. A projection algorithm in an adaptation law was needed (i when necessary to avoid a division by zero in an adaptive control input synthesis, or (ii when to achieve an efficient adaptive estimation based on prior information, or (iii when it is used as a robust adaptation law to uncertainties, that is, it eliminates drift phenomenon of adjustable parameters in an adaptive controller and holds these within a specified boundary despite the presence of system disturbance, measurement noise and modeling error [],[] due to incorrect estimates of the degree and relative degree of a controlled object, or (iv in the adaptive trajectory control system for a rigid link robot arm [3], the positive definiteness of the estimated inertia matrix has to be guaranteed with using the adaptation law [4], or for other reasons. In this way, an adaptive control system needs somehow a projection function to update the adjustable parameters. Several schemes have been proposed to constrain adjustable parameters into a convex set. The parameter projection algorithms can be roughly classified into two categories according to whether the time derivatives of adjustable parameters always exist or not. Graduate School of Science and Engineering, National Defense Academy, - Hashirimizu -chome, Yokosuka , Japan Department of Electrical and Electronic Engineering, National Defense Academy, - Hashirimizu -chome, Yokosuka , Japan ed5@nda.ac.jp, itamiya@nda.ac.jp (Received October, One is the switching type parameter projection methods [5] [] in which the integrator input is compulsorily switched so that adjustable parameters stay in the convex set. In the latest improved version the projection algorithm [9] can be also applied to an arbitrary convex set such as hyper-sphere, hyperellipsoid and hyper-polyhedron and so on. Moreover, this method guarantees the existence and uniqueness of the solution. However, in those methods [5] [], time derivatives of adjustable parameters do not exist at a start time of the projection or at an end time of it. Therefore, the response of the adaptive control system with such a projection algorithm may be deteriorated. Also, it cannot be applied to an adaptive control based on the dynamic certainty equivalent (DyCE principle [] or the adaptive backstepping (BS method []. On the other hand, to solve the above problems, the smooth parameter projection methods which guarantee the existence and uniqueness of the solution have been proposed in [3] [7]. The smooth parameter projection methods of Akella and Subbarao [3],[4] improve the transient response of a model reference adaptive control system (MRACS based on the certainty equivalent (CE control with the projection operation. It guarantees the stability of the adaptive loop including the sigmoid function by using the adaptive immersion and manifold invariance (I&I [8],[9] technique. These methods based on prior information can constrain the adjustable parameters into the inside of a hyper-rectangular region. The methods of Tanahashi and Itamiya [5],[6] are extensions of [3],[4] to the gradient type adaptation law and applied to the MRACS based on the DyCE control [5] and the adaptive BS control with a tuning function [6]. The projection algorithm which is proposed by Z. Cai et al.[7] is useful for a robust adaptation law in an adaptive BS system based on over parameterization. It can constrain JCMSI 4/3/64 43 c SICE

2 44 SICE JCMSI, Vol. 6, No. 4, July 3 Fig. MRACS based on DyCE principle for the first-order delay system. the adjustable parameters into the inside of a hyper-sphere region. These conventional methods [3] [7] which use a priori information can accomplish a realization of an adaptive controller, an efficient estimation of adjustable parameters and robust stability of the adaptive loop. Therefore, these are useful methods for an MRACS based on the DyCE principle and an adaptive BS method. However, in a practical adaptive control system, if we use these methods for an MRACS based on the DyCE principle, the response of the adaptive control system may be deteriorated because the solution trajectory of adjustable parameters transiently passes through the unstable region. The following simple example is useful to understand a current problem. Now, we consider an MRACS based on the DyCE principle for the first-order delay plant (the sign of the high frequency gain is positive and known a priori like shown in Fig. (a >, b >. In this case, this system has two parameters (ˆθ (t andˆθ (t and the control input needs time derivatives of these adjustable parameters. Moreover, it needs the division by ˆθ (t in the control input synthesis. Figures and 3 are simulation results. Then, we use the High Order Tuner (HOT which is shown in [5] or [7]. We set ˆθ >. Figures and 3 show the response of the tracking error and the solution trajectory of ˆθ(t. These results demonstrate that the response of the adaptive control system y(t y m (t maybe deteriorated because the solution trajectory of ˆθ(t transiently passes through the unstable region. In this case, the unstable region is ˆθ (t a ˆθ b (t and the unfeasible region is ˆθ (t, since θ >. These regions are shown as Fig. 4. Thus, these regions of the adaptive control system become the outside of hyper-planes which pass through the origin. If we set a rectangular region (Fig. 5 or a circle region (Fig. 6 inside of the stable region, the optimal parameters of the adjustable parameters do not necessarily exist inside of these regions. Consequently, the response of control performance may be deteriorated (See below for further details. The convex set of the adjustable parameters which satisfies the following conditions becomes a hyper-parallelogram region. (C It avoids a division by zero in an adaptive control input synthesis. (C It achieves an efficient adaptive estimation based on prior information. (C3 It is used as a robust adaptation law to uncertainties. (C4 It prevents the deterioration of the transient response which occurs when the adjustable parameters pass through the unstable region of an adaptive controller or of an adaptive control system. This paper proposes a new design scheme of the smooth parameter projection HOT for an MRACS based on the DyCE principle to constrain the adjustable parameters into the hyperparallelogram region. This proposed method always guarantees the realization of the time derivatives of adjustable parameters, and can prevent the deterioration of the transient response which occurs when the adjustable parameters pass through the unstable region of an adaptive controller or of an adaptive control system. Moreover, we see that we can design two HOTs. This paper is organized as follows; the next section states basic configuration of an MRACS based on the DyCE principle. Section 3 presents the conventional method with using the smooth parameter projection HOT and reveals that it has some restrictions. Section 4 presents new two HOTs which can improve the transient response when the adjustable parameters are constrained into the inside of hyperparallelogram. Section 5 presents the stability analysis of the adaptive loop. Simple numerical simulation results are indicated in Section 6 in order to verify the effectiveness of the proposed scheme. We conclude the study in the last section. Notation: The symbol T expresses the transpose of a vector or a matrix. For a constant vector or vector signal x R n, x i means the i-th element of x. I means an identity matrix of appropriate size. Also, I n and n mean an identity matrix of order n and an n-dimensional zero vector respectively. For an operator matrix W(s with using s which is the Laplace operator, a vector signal w(t which is the inverse Laplace transform of W(s and a vector signal v(t of appropriate size, the following notations are applied; (W(s[v](t := w(t τv(τdτ, where the symbol := means that the left-hand is defined by the right-hand. For an integer j, a design constant λ>, D d/dt and a vector signal x(t, x ( j (tandx [ j] (t are defined as x ( j (t d j x(t, dt j x [ j] (t := ( (s + λ j I[x] (t for j (D + λ j x(t for j >.. Basic Configuration of MRACS Based on DyCE Principle In this section, we state the controlled object and the adaptive control input used in an MRACS based on the DyCE principle.. System Representation The controlled object (in the following referred to as plant considered here is a linear time invariant, SISO and minimum phase system. It has the system degree n and the relative degree n ( n. Then, the plant output y(t in time t can be parameterized by y(t = θ T ζ(t + ɛ(t, ( where θ R n is the plant parameter vector, elements of the regressor vector ζ(t R n are defined by { u [ (n ζ i (t := +i ] (t (i = n y [ (n n+i ], ( (t (i = n + n

3 SICE JCMSI, Vol. 6, No. 4, July 3 45 Fig. Simulation result the method of [5] (case. Fig. 3 Simulation result the method of [7] (case. Fig. 4 Convex set C o. Fig. 5 Convex set C r. Fig. 6 Convex set C s. where u(t is the plant input and ɛ(t is the signal which decays exponentially with order e λt and its magnitude depends upon the initial state of the plant. Assumptions : The plant is assumed that (A n and n of plant are known a priori. (A It is a controllable and observable system. (A3 The sign of the high frequency gain θ is known a priori. Hereafter, it is assumed that θ is positive without loss of generality. (A4 Available signals are only u(tandy(t.. Adaptive Control Input The adaptive control input is synthesized by ] u(t := y[n m (t n i= ˆθ i (tζ [n ] i (t f (t, (3 ˆθ (t where the signal y m (t is a desired output of y(t. It is generated by y m (t := ( N m (s D m (s [r] (t. The reference input r(t isa piece-wise continuous and bounded signal. N m (s/d m (sisthe transfer function of the stable reference model for the closed loop system and its relative degree is not less than n. Adjustable parameters ˆθ i (t(i = n of the adaptive controller mean estimates corresponding to θ i. The auxiliary input f (t is defined as f (t := n j= n C j (ˆθ ( j (t T ζ [n j] (t. (4 The right-hand of (3 is called the DyCE control input and it is derived from the surrogate model control law []; y m (t = ŷ(t. ŷ(t := ˆθ T (tζ(t is the identifier corresponding to (. The surrogate model control leads to a special control error equation which does not depend on ˆθ(tas ỹ(t := y(t y m (t (5 = θ T ζ(t + ɛ(t; θ(t := θ ˆθ(t. (6 Hence, the transient response of y(t does not deteriorate depending on the adaptation speed while ỹ(t is always influenced from ˆθ(t when f (t (certainty equivalent control. Therefore, a special adaptation law which can generate ˆθ ( j (t( j = n without using y ( j (t( j = n is needed for synthesizing (3 since the realization of f (t needs ˆθ ( j (t( j = n and high order differential values of y(t are not available from (A4. Such a special adaptation law is known as HOT and it is proposed by Morse [] and the improved versions of the feasibility have been proposed in [],[]. In the next section, we introduce the conventional methods [3] [7] and reveal their restriction. 3. Conventional Methods and Their Restriction In a practical adaptive control system, if we use conventional methods for an MRACS based on the DyCE principle, the response of the adaptive control system may be deteriorated because the solution trajectory of adjustable parameters transiently passes through the unstable region. Therefore, the adjustable parameters have to be constrained not to enter the unstable region. However, conventional methods can not always constrain the adjustable parameters not to enter the unstable region. In this section, we introduce conventional methods and reveal their restriction. The authors have already shown a conventional adaptation law with the smooth parameter projection method (or HOT. The conventional methods can be classified into two types as follows. One method can constrain the adjustable parameters into the inside of a hyper-rectangular region [3] [6], and the other method can constrain the adjustable parameters into the inside of a hyper-sphere region [7].

4 46 SICE JCMSI, Vol. 6, No. 4, July 3 Fig. 7 Convex set C r. Fig. 8 Projection mechanism for ˆθ(t C r. In the adaptive regulation problem, Akella and Subbarao have proposed the smooth projection methods [3],[4] which can constrain ˆθ(t into the following convex set; C r := {ˆθ(t θ i < ˆθ i (t < θ i ; i = n}. In the case of n =, the convex set C r is indicated as the inside of the rectangular shown in Fig. 7. These methods improved the transient response of the control system which has deteriorated with using the conventional switching type projection operation. In [3],[4], the projection can be achieved by using the intermediate variables which are the outputs of the integrator designed by the adaptive I&I methods [8],[9]. The adjustable parameters of the controller are generated by using the intermediate valuables which pass through a sigmoid function (Fig. 8. The methods of Tanahashi and Itamiya [5],[6] are extensions of the adaptation law based on the smooth projection algorithms of Akella and Subbarao [3],[4] to the gradient type adaptation law and applied to the MRACS based on the DyCE control [5] and the adaptive BS control with a tuning function [6]. These methods [5],[6] can efficiently estimate the adjustable parameters and avoid a division by zero in the adaptive control input synthesis. In [5], Tanahashi and Itamiya have proposed the following HOT in order to constrain the adjustable parameters into the convex set C r. ˆθ i (t = β i + g i tanh ˆψ i (t for i = n ˆψ(t = Γ [ q(t R(tˆθ(t ], (7 where ˆψ i ( := {ln(ˆθ i ( θ i ln(θ i ˆθ i (}/, θ i < ˆθ i ( < θ i, β i := (θ i + θ i /, g i := (θ i θ i / andγ := diag{,,, n }. q(t andr(t are defined as follows; q(t := (H(s[y N ζ N ] (t, R(t := ( H(s[ζ N ζ T N ] (t, (8 where H(s := λ n /(s + λ n, y N (t := y(t/n(t, ζ N (t := ζ(t/n(t, N(t := {ρ + m(t} /, ṁ(t := σm(t + σ(u (t + y (t; m(, ρ and σ are positive design parameters. λ > is a design constant. Z. Cai et al.[7] have proposed the smooth projection adaptation law which can constrain ˆθ(t into the inside of the hypersphere region as follows; C s := {ˆθ(t ˆθ(t θ + ε} for all t, where θ is a design parameter and ε is an arbitrary positive constant. This method improves the switching type parameter projection method [8] which guarantees the existence and uniqueness of the solution. It is useful for a robust adaptation law in an adaptive BS system based on over parameterization. They have proposed the following adaptation law; η η ˆθ(t = ν p(ˆθ(t, (9 4(ε + εθ n+ θ Fig. 9 Convex set C s. Fig. Projection mechanism for ˆθ(t C s. where ˆθ(t = ν is an adaptation law which does not use parameter projection. ν generates n times differentiable ˆθ(t. We set ν := ˆψ(t. The function ˆψ(t is defined in (7. p(ˆθ(t, η and η are defined as follows; p(ˆθ(t := ˆθ T (tˆθ(t θ {, ( p η := n+ (ˆθ(t if p(ˆθ(t >, ( otherwise ( η := pt (ˆθ(tν + pt (ˆθ(tν + δ. ( δ is an arbitrary positive constant. In this method, when ˆθ(t which starts from ˆθ( <θ enters the region of θ < ˆθ(t < θ +ε, (for example, when n =, it is shown in Fig. 9 the righthand second term of (9 operates (Fig.. Therefore, ˆθ(t does not overflow C s. These conventional methods can ensure the high order derivatives of ˆθ(t, can avoid a division by zero in an adaptive control input synthesis, can adjust an efficient adaptive estimation based on prior information and can use as a robust adaptation law to uncertainties. However, in a practical adaptive control, the response of an adaptive control system may be deteriorated because the solution trajectory of ˆθ(t transiently passes through the unstable region. This region becomes the outside of hyper-planes which pass through the origin. For example, in case of the control system like shown in Fig., the stable region is the convex set C asshowninfig.4. Then, although these conventional methods can guarantee the existence of ˆθ(t and satisfy ˆθ >, ỹ(t may be deteriorated because the solution trajectory of ˆθ(t transiently goes out of the stable region. Even if we set the convex set inside of the stable region, the optimal parameters (θ ofˆθ(t is not necessarily included in the stable region like shown as Fig. or Fig.. Therefore, ỹ(t may not converge to zero when t.ifwe can roughly know about θ by using prior information, we can choose the convex set which contains θ.however,itismore difficult to only know θ than to select the convex set which contains θ. Therefore, it is important to develop a new HOT which can constrain the adjustable parameters into the convex set different from the hyper sphere or hyper rectangular region. In the next section, we propose the new HOT which can constrain the adjustable parameters into the hyper-parallelogram region. The proposed HOT can not only have the ability of conventional methods but also improve a transient response of an adaptive control system. In addition, we indicate that two HOTs can be designed. 4. New High Order Tuner Which Can Improve the Transient Response In this section, we propose HOTs which can constrain the

5 SICE JCMSI, Vol. 6, No. 4, July 3 47 Fig. 4 The definition of convex set C p. Fig. Simulation result the method of [5] (case. example, if we choose n = andk =, the convex set C p is shown as Fig. 4. In this case, C p is the inside of the parallelogram region and k is equal to the number of parallel straight lines. Moreover, if we choose k =, C p is a zonal region between two straight lines. In particular, in the case of m, =, m, =, m, = andm, =, C p is equal to C r shown as Fig. 7. In addition, in the case of m, = κ / m, = / κ +, m, = κ / κ + andm, = / κ +, κ +, C p is the region shown as Fig. 4. In generalized structure, C p means the inside of the hyper-parallelogram region. Then, k is equal to the number of pairs of hyper-plains. We propose here the following HOT which achieves ˆθ(t C p for all t ; [HOT] ˆθ(t := Γ(t [ q(t R(tˆθ(t ], (5 Fig. Simulation result the method of [7] (case. where ˆθ(t C p, ˆθ( <, Γ(t := Φ(t/N (t, [ ] P Φ(t := T k (ˆθ(t O T T, O I n k (6 N (t := [ ρ + trace{φ T (tφ(t} ] /. (7 Fig. 3 The definition of convex set C p. adjustable parameters into the following convex set (C p ; C p := {ˆθ(t α i <α i (ˆθ(t < α i ; i = k, k n}, (3 α i (ˆθ(t := m T i ˆθ(t; m i := [m i,, m i,,, m i,n ] T, (4 where α i, α i and k are design parameters. k is an integer, α i, α i are real numbers which satisfy α i >α i. m i, j ( j =,,, n is a real number which satisfies ( n j= m i, j / =, m i,n > and m i κm j (κ>isareal number, i j. When k < n, α i (ˆθ(t := ˆθ i (t(i = k + n. Then, α i (ˆθ(t (i = k + n is not constrained. Remark : We choose a common area when parallel convex set regions exist in the convex set C P. For example, if the control system has two adjustable parameters (ˆθ (t, ˆθ (t, the convex set C P is selected as shown in Fig. 3. That is, the constraint region becomes the common area when the same inclination exists. C p includes the class of the hyper-rectangular regions C r.for > is a design constant and ρ is a small positive constant. q(t andr(t are defined in (8. The square matrix T R n n which depends on C p is defined as T := m, m, m,k m,k+ m,n m, m, m,k m,k+ m,n m k, m k, m k,k m k,k+ m k,n O I n k The diagonal matrix P k (ˆθ(t is defined as. (8 P k (ˆθ(t := diag{p (ˆθ(t, p (ˆθ(t,, p k (ˆθ(t}. (9 Remark : The element of m i, j (i = k, j = n of the convex set C P is chosen not to have the same inclination. Therefore, the rank of the square matrix T is n and it is a nonsingular matrix. The constraint condition for ˆθ(t C p is directly expressed by the elements of diagonal matrix P k (ˆθ(t. This matrix is not unique. Next, we propose two schemes in which different P k (ˆθ(t are used. A. Scheme The elements of P k (ˆθ(t in (6 are defined as p i (ˆθ(t := (α i α i (ˆθ(t(α i (ˆθ(t α i /Δ i ( ; Δ i := (α i α i /, i = k.

6 48 SICE JCMSI, Vol. 6, No. 4, July 3 Remark 3: When we set I to the square matrix T, the proposed scheme is the same as the HOT which is eliminating the intermediate parameters ˆψ(t(i = n in (7. Therefore, the proposed scheme is a natural extension of the conventional method [5]. In addition, this method can be used as robust adaptation law because this method can constrain the adjustable parameters into the specified bounded closed convex set []. B. Scheme The elements of P k (ˆθ(t in (6 are defined as follows p i (ˆθ(t := { α i α i (ˆθ(t } { αi (ˆθ(t α i } /Δ i. ( Remark 4: When the adjustable parameters approaches the boundary of the convex set, the eigenvalue of P k (ˆθ(t converges to asymptotically with time. Therefore, when ˆθ(t approaches the boundary, Scheme converges to asymptotically faster than Scheme. Hence, Scheme is more strongly constrained. We designed the Scheme to avoid approaching the boundary because the real control system includes the error on the numerical calculation. In this sense, we should use the Scheme. 5. Stability Analysis of the Adaptive Loop In this section, we indicate the stability analysis of the adaptive loop which uses Scheme and Scheme. Theorem The adaptive loop which is composed of the control error equation (6 and HOT (5 satisfies the following properties. (P- ˆθ(t C p for all t, (P- ˆθ(, ˆθ( L, (P-3 J( /N ( L, R T ( θ( /N (, ˆθ( L, (P-4 θ T ( ζ N (, ỹ N ( L, (P-5 ˆθ ( j ( L ( j = n, where ỹ N (t := ỹ(t/n(tand J(t := h(t τ { y N (τ ˆθ T (tζ N (τ } dτ, ( h(t := L H(s. (3 (Proof Using (4 and (8 we obtain α(ˆθ(t = T ˆθ(t, (4 where α(ˆθ(t := [α (ˆθ(t,α (ˆθ(t,,α n (ˆθ(t] T.Inthefollowing, for simplicityofnotation we use ˆα(t instead of α( ˆθ(t. Using (4, the adaptation law of (5 becomes ˆα(t = [ ] P k (T ˆα O [ q(t R(t ˆα(t ] (5 N (t O I n k ; ˆα i (t > (i = n, ˆα( <, where N (t isgivenby ˆα(t which is replaced with ˆθ(t = T ˆα(t. Let q(t := T T q(t, R(t := T T R(tT, whereq(t and R(t are defined in (8. The diagonal matrix P k (T ˆα isdefinedas P k (T ˆα = diag{p (T ˆα, p (T ˆα,, p k (T ˆα}. (6 The elements of P k (T ˆα are defined as follows : A. Scheme Fig. 5 The definition of convex set C pα. p i (T ˆα = (α i ˆα i (t( ˆα i (t α i /Δ i. (7 B. Scheme p i (T ˆα = { α i ˆα i (t } { ˆαi (t α i } /Δ i. (8 Using coordinate transformation of (4, C p which is the inside of the hyper-parallelogram on ˆθ(t-space is mapped into C pα on ˆα(t-space. C pα which has parallel straight lines for each axis is the following convex set whose adjustable parameters are constrained into the inside of the hyper-rectangular region. C pα := { ˆα(t α i < ˆα i (t < α i ; i = k, k n}. (9 For example, in the case of n = andk =, C pα is rectangular as shown in Fig. 5. Now, using ˆα(t-space, we indicate the proof of the theorem instead of directly proving from (P- to (P-5. The proof of (P- : We can indicate that ˆα(t which is adjusted by the adaptation law (5 does not overflow C pα (see Appendix A. The proof of (P- : The stability of the adaptive loop is indicated as follows. Let us introduce the positive definite functions V( ˆα fork < n, respectively. A. Scheme V( ˆα := + k i= n i=k+ [ α il ln (α i ˆα i (t + α il ˆα i (t α i + α ih ln τ t α ] ih α i ˆα i (t h(τ σɛ N(σdσdτ, (3 where α := [α,α,,α n ] T is the optimal parameter of ˆα(t, α il := α i α i and α ih := α i α i. B. Scheme V( ˆα := + n i=k+ k i= [ (ˆαi (t υ i ( ˆα i (t α i (α i ˆα i (t( ˆα i (t α i α i υ i Δ i (α i ˆα i (t + { ln ˆα i(t α i α i α i τ t ln α i ˆα i (t }] α i α i h(τ σɛ N(σdσdτ, (3 where υ i := (α i + α i /. In the case of k = n, the nd term of (3 and (3 are removed. Then, the evaluation V( ˆα along the trajectry of (5 becomes (see Appendix B V( ˆα J(t/. (3

7 SICE JCMSI, Vol. 6, No. 4, July 3 49 From V( ˆα, V( ˆα and ˆα(t <, we obtain ˆα( L. Since V( ˆα is a positive definite function and V( ˆα, we have V( ˆα L. From this fact and (5, we obtain ˆα( L. Therefore, this fact implies that (P- from (4. The proof of (P-3 : Let us now integrate the both sides of (3. We obtain the following relation; J(t dt = V( ˆα( lim V( ˆα(t. (33 t This means J( /N ( L. From this fact, (8 and (, we obtain R / ( T(α ˆα( /N / ( L. Also we obtain ˆα( L from (5. Therefore, this fact implies that (P-3 from (4. The proof of (P-4 can be shown in the same way as in [5],[]. The proof of (P-5 can be proven from (P-, (5, (4, (5 and mathematical induction. The detailed proof can be shown in the same way as [5],[]. Remark 5: The proof of the stability of MRACS and lim t ỹ(t = can be shown in the same way as in [5],[]. tings are as follows. The plant is chosen as ( y(t = s(s +.5 [u] (t (34 and the reference model is selected as ( y m (t = s +.6s + [r] (t (35 where r(t is the rectangular wave with the period [sec], varying between - and. Other parameters are set as follows. λ =, λ =., ρ =.5, m( =.5, σ =, = 35, ˆθ( = [7, 6,, ] T. Then, the unstable region of the adaptive controller parameters are shown in Fig. 6. This adaptive controller becomes always stable when the adjustable parameters are constrained into this stable region. In this simulation, ˆθ(t is set near the boundary. Then, the optimal parameters θ become [,.5,.75,.75] T. The upper and lower values are given by [ᾱ,α ] = [, ], [ᾱ,α ] = Remark 6: Scheme is composed of the square of p i (ˆθ(t (i = k in Scheme. Although we think that we can choose the third power or the fourth power of p i (ˆθ(t, we need to indicate the Lyapunov-like function V( ˆα. 6. Numerical Simulations Simple numerical simulations was performed to verify that the adjustable parameters do not overflow the specified convex set and that the transient response can be improved. In addition, we compared the conventional method [5] which achieves ˆθ (t > with the proposed methods. Simulation set- Fig. 6 The definition of convex set C p and normal vectors. Fig. 7 Simulation result the method of [5]. Fig. 8 Simulation result the method of Scheme. Fig. 9 Simulation result the method of Scheme.

8 5 SICE JCMSI, Vol. 6, No. 4, July 3 7. Conclusion In conventional studies [3] [7], the convex sets are the inside of the hyper-rectangular and the hyper-sphere in order to accomplish a realization of an adaptive controller, an efficient estimation of adjustable parameters and robust stability of the adaptive loop. In this paper, the authors have proposed new two smooth projection HOTs to improve the transient response of adaptive control system in addition to the ability of conventional restraint. According to these proposed methods, the adjustable parameters can be constrained into the inside of the hyper-parallelogram region. These methods are a natural extension of the conventional projection method [5]. In addition, the proposed methods do not use the intermediate variables. Therefore, these methods can be applied to the robust adaptation law without guaranteeing the boundedness of the intermediate variables. The future study is to extend the proposed method to an arbitrary convex set and apply to a robust adaptation law. Fig. Trajectory of Scheme and Scheme. [, ], [ᾱ 3,α 3 ] = [, ] and [ᾱ 4,α 4 ] = [, ]. The matrix T of (8 can be obtained as follows λ + λ T = (36 The normal vector is defined as shown in Fig. 6. Figures 7, 8 and 9 are the simulation results. These results show the response of the tracking error y(t y m (t, the control input u(t, the positive definite function V(t, the phase-plane trajectory ˆθ(t, respectively. In Fig. 7, u(t increased when the trajectory of ˆθ(t passes through the unstable region. Consequently, the response of the adaptive control system y(t y m (t deteriorated. On the other hand, the simulation results of Figs. 8 and 9 demonstrate that the new adaptation law with the smooth parameter projection function can effectively constrain the adjustable parameter without overflowing the boundary C p of C p. Consequently, the proposed control input u(t does not increase and y(t y m (t is improved. In general, although we do not make reference to the control performance, it seems that the adjustable parameters of Scheme approaches the boundary more slowly than those of Scheme near the boundary (Fig.. Therefore, if we want to avoid approaching the boundary for the adjustable parameters, we should use Scheme. References [] P.A. Ioannou and J. Sun: Robust Adaptive Control, Prentice Hall, 996. [] T. Suzuki: Adaptive Control, Corona Publishing, (in Japanese. [3] R.H. Middleton and G.C. Goodwin: Adaptive computed torque control for rigid link manipulators, Systems & Control Letters, Vol., No., pp. 9 6, 988. [4] M. Sawada and K. Itamiya: An improvment of transient response in adaptive trajectory control system based on the computed torque law for robot arm: Applying smooth projection algorithm, SICE Transactions on Industrial Application, Vol., pp , (in Japanese. [5] A. Nagurney and D. Zhang: Projected Dynamical Systems and Variational Inequalities with Applications, Kluwer Academic Publishers, 996. [6] F. Ikhouane and M. Krstic: Adaptive backstepping with parameter projection: Robustness and asymptotic performance, Automatica, Vol. 34, No. 4, pp , 998. [7] G.C. Goodwin and D.Q. Mayne: A parameter estimation perspective of continuous time model reference adaptive control, Automatica, Vol. 3, No., pp. 57 7, 987. [8] J.-B. Pomet and L. Praly: Adaptive nonlinear regulation: Estimation from the Lyapunov equation, IEEE Transactions on Automatic Control, Vol. 37, No. 6, pp , 99. [9] K. Kuhnen and P. Krejci: An adaptive gradient law with projection for non-smooth convex boundaries, European Journal of Control, Vol., No. 6, pp , 6. [] K. Kuhnen and P. Krejci: Identification of linear error-models with projected dynamical systems, Mathematical and Computer Modelling of Dynamical Systems, Vol., No., pp. 59 9, 4. [] A.S. Morse: High order parameter tuners for the adaptive control of linear and nonlinear systems, Proceedings of the US- Italy Joint Seminar on Systems, Models and Feedback: Theory and Application, pp , 99. [] M. Krstic, I. Kanellakopoulos, and P. Kokotovic: Nonlinear and Adaptive Control Design, John Wiley and Sons, 995. [3] M.R. Akella: Adaptive control: A departure from the certaintyequivalence paradigm, The Journal of the Astronautical Sciences, Vol. 5, No., pp. 75 9, 4. [4] M.R. Akella and K. Subbarao: A novel parameter projection mechanism for smooth and stable adaptive control, Systems & Control Letters, Vol. 54, No., pp. 43 5, 5. [5] S. Tanahashi and K. Itamiya: A design scheme of model reference adaptive control system to guarantee the surrogate model control law, Transactions of the SICE, Vol. 43, No., pp. 7 35, 7 (in Japanese. [6] S. Tanahashi and K. Itamiya: A design method for tuning functions scheme with parameter projection algorithm, Transactions of the ISCIE, Vol., No., pp. 3 36, 8 (in Japanese. [7] Z. Cai, M.S. de Queiroz, and D.M. Dawson: A sufficiently smooth projection operator, IEEE Transactions on Automatic Control, Vol. 5, No., pp , 6. [8] A. Astolfi and R. Ortega: Immersion and invariance: A new tool for stabilization and adaptive control of nonlinear systems, IEEE Transactions on Automatic Control, Vol. 48, No. 4, pp , 3. [9] A. Astolfi, D. Karagiannis, and R. Ortega: Nonlinear and Adaptive Control with Applications, Springer-Verlag, 7. [] K. Itamiya, F. Yamano, and T. Suzuki: A model reference adaptive control scheme based on dynamic certainty equivalence

9 SICE JCMSI, Vol. 6, No. 4, July 3 5 principle and its stability, Transactions of the SICE, Vol. 34, No. 9, pp. 5 3, 998 (in Japanese. [] K. Itamiya, M. Sawada, and T. Suzuki: Model reference adaptive control system based on surrogate model control using plant parameter estimates, Proceedings of the 38th IEEE Conference on Decision and Control, pp , 999. [] M. Sawada and K. Itamiya: A new smooth parameter projection high order tuner for robust adaptive control, Preprints of the 7th IFAC Symposium on Robust Control Design, pp ,. Appendix A The Proof of (P- C pα is composed of the hyper-plains which have the normal vectors for outside as follows; n α := [, T n ]T, n α := [,, T n ]T,, n αk := [ T k,, T n k ]T, n α := [, T n ]T, n α := [,, T n ]T,, n αk := [ T k,, T n k ]T, where i = k. Then, ˆθ(t C p is satisfied since lim ˆα(t C pα n T α i ˆα(t =, lim ˆα(t C pα n T α i ˆα(t =, (A. where C pα means the boundary of C pα. Equations (A. mean that the adaptation speed converges to zero when the adjustable parameters go to near the boundary. When i =, lim ˆα(t C pα n T α ˆα(t = lim [, T n N (t ] P k(t ˆα ν(t ˆα(t ᾱ = lim [p (T ˆα, T n N (t ] ν(t ˆα(t ᾱ [ ] (ᾱ ˆα(t( ˆα(t α = lim, T n ν(t N (t ˆα(t ᾱ Δ =, where ν(t := q(t R(t ˆα(t. In addition, lim n T ˆα(t C α ˆα(t pα = lim [, T n N (t ] P k(t ˆα ν(t ˆα(t α = lim [ p (T ˆα, T n N (t ] ν(t ˆα(t α [ = lim (ᾱ ] ˆα(t( ˆα(t α, T n ν(t N (t ˆα(t α Δ =. In the case of i = k, we can indicate them in the same way as i =. We can also indicate ˆα(t of Scheme does not overflow in the same way. Appendix B Derivation of V(ˆα (i Scheme The evaluation V( ˆα along the trajectry of (5 becomes V( ˆα = k Δ i i= (ᾱ ˆα(t( ˆα(t α α i(t ˆα(t n α i (t ˆα(t i=k+ where α i (t := α i ˆα i (t. From (5 and (B. V( ˆα = N (t αt (t [ q(t R(t ˆα(t ] h(t τɛ Ndτ, (B. h(t τɛ Ndτ. (B. In addition, from (8 and (3 V( ˆα = N (t αt (t [ h(t τy N(τ(T T ζ N (τdτ h(t τ(t ˆα(tT ζ N (τ(t T ζ N (τdτ ] t h(t τɛ Ndτ = [ h(t τ ( y N (τ (T ˆα(t T ζ N (τ ( (Tα T ζ N (τ (T ˆα(t T ζ N (τ dτ ] /N (t h(t τɛ Ndτ. (B. 3 Using y N (t = (Tα T ζ N (t + ɛ N (t, ( and (4, V( ˆα becomes [ V( ˆα = N (t J(t J / (t h(t τɛ Ndτ + t h(t τɛ Ndτ ] = J(t ( J / (t h(t τɛ Ndτ J(t. (B. 4 (ii Scheme The evaluation V( ˆα along the trajectry of (5 becomes V( ˆα = k i= (ᾱ α ˆα(t (ˆα(t α i (t ˆα(t Δ i n α i (t ˆα(t i=k+ Using (5 and (B. 5, we obtain (B.. h(t τɛ Ndτ. (B. 5 Appendix C Unstable Condition of Adaptive Controller Using (3, the control input is given by u(t = ( y [] m (t 4 i= ˆθ i (tζ [] i (t f (t /ˆθ (t. (C. Then, we fix ˆθ(t = ˆθ. The Laplace transform of (C. is obtained as follows; U(s = Nm(s(s+λ 3 sˆθ +λˆθ +ˆθ D m (s R(s (s+λˆθ 3 +ˆθ 4 s+λ Y(s F(s. (C. sˆθ +λˆθ +ˆθ sˆθ +λˆθ +ˆθ Therefore, the unstable condition is obtained as λˆθ + ˆθ. In addition, ˆθ is unfeasible region, since θ >. Masataka SAWADA (Student Member He received his B.E. and M.E. degrees from National Defense Academy, Japan, in 995 and, respectively. In 995, he joined the Japan Air Self Defense Force. His research interests include adaptive control system design. He is currently a student of the doctor s course of Equipment and Structural Engineering, Graduate School of Science and Engineering, National Defense Academy. Keietsu ITAMIYA (Member He received his M.E. and D.E degrees from the University of Tsukuba, Japan, in 99 and 993, respectively. In 993, he joined the faculty of National Defense Academy, where he is currently an Associate Professor of the Department of Electrical and Electronic Engineering. His research interests include adaptive and learning control system design, wavelet application, intelligent control of robot. He is a member of ISCIE, IEEJ and IEEE.