Simple Adaptive PI Control for Linear Time-Delay Systems ? R. Ben Yamin ∗ I. Yaesh ∗∗ U. Shaked ∗∗∗ ∗
Advanced Systems Division, Control Dept., I.M.I, Israel. (Tel: +972-3640-6175; e-mail:
[email protected]). ∗∗ Advanced Systems Division, Control Dept., I.M.I, Israel. (Tel: +972-3640-6351; e-mail:
[email protected]). ∗∗∗ School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel (e-mail:
[email protected]) Abstract: A simple direct adaptive proportional-plus-integral (PI) control scheme is introduced, for linear time-delay systems with polytopic uncertainties. Sufficient conditions for closed-loop stability of the proposed control scheme are given, in terms of bilinear matrix inequalities (BMIs). The addition of a proportional-gain feedback to previous results, where only integral-type gain adaptation was considered, reduces the conservatism entailed in the BMIs. Stability is analyzed using the Lyapunov-Krasovskii functional method and both delay-dependent and delay-independent results are obtain. A numerical example is given, which demonstrates applicability of the proposed method and the simplicity of its implementation.Copyright c IFAC 2009 ° Keywords: Adaptive direct control, SAC, time-delay systems,simplified adaptive control 1. INTRODUCTION A class of direct adaptive control schemes for continuoustime systems, known as Simplified Adaptive Control (SAC), has received considerable attention in the literature (Sobel et al (1982)-Ben Yamin et al (2007b)). Robustness of SAC controllers facing polytopic type uncertainties has been established (Kaufman et al (1998)-Yossef et al (1998)) allowing application to real engineering problems (see e.g. reference Yossef et al (1998)). The stability of closedloop SAC is related to the Almost Strictly Positive Real (ASPR) property of the controlled plant. Namely, there exists a static output-feedback gain which stabilizes the plant and makes it Strictly Positive Real (SPR). In such a case, SAC stabilizes the closed-loop dynamics and consequently leads to zero tracking errors. The ASPR property can be verified, under quadratic stability assumption Green and Limebeer (1995), by solving a set of LMIs or by using a parameter-dependent Lyapunov function (Yossef et al (1998),Yaesh and Shaked (2006)). For systems with state delay, positive realness has been studied by Niculescu and Lozano (2001)-Lu em et al. (2000). In Niculescu and Lozano (2001), delayindependent sufficient conditions in terms of LMIs have been derived. In Lu em et al. (2000), frequency domain approach is applied and sufficient conditions are obtained. Fridman and Shaked Fridman and Shaked (2002a) introduced delay-dependent sufficient conditions for the passivity of neutral type systems by applying a descriptortype Lyapunov-Krasovskii functionals that were recently ? This work was supported in part by C&M Maus Chair at Tel Aviv University.
introduced in Fridman and Shaked (2002b) for delaydependent analysis of stability and control synthesis. Mirkin and Gutman Mirkin and Gutman (2003) introduced a new approach for output Model Reference Adaptive Control (MRAC) of linear continuous-time plants with known, constant state time-delays, where ‘compensation’ for the delay in the system states is achieved by using delayed states of the reference model. This new approach can be used only if the delay is constant and exactly known. For this purpose, an adaptive dynamic prefilter from the delayed reference model is introduced. In the present paper, the output-feedback and the model following control problems will be extended to linear systems with time-varying delay and polytopic type parameter uncertainties. The addition of a gain adaptation type feedback to previous results Ben Yamin et al. (2009), where only integral-type gain adaptation was considered, reduces the conservatism in the BMIs. The objective is to obtain sufficient conditions for closed-loop stability of the proposed simplified adaptive control scheme. The proposed adaptive controller can stabilize the system and make it follow the output of the system model without knowing the explicit system dynamics. Sufficient conditions are derived for stability of the closed-loop dynamics of the SAC model-following scheme. These sufficient conditions are expressed in terms of BMIs, which can be solved using the YALMIP and PENBMI Toolboxes Kocvara and Stingl (2005). A numerical example is given which illustrates the application of the new results. Notation: Throughout the paper the superscript ‘T ’ stands for matrix transposition, Rn denotes the n dimensional Euclidean space, Rn×m is the set of all n × m real
matrices, and the notation P > 0, for P ∈ Rn×n , means that P is symmetric and positive definite. 2. PRELIMINARIES AND PROBLEM FORMULATION Consider the linear system with time-varying delays x(t) ˙ = A0 x(t)+A1 x(t−τ (t))+Bu(t), x(t) = 0, t ∈ [−h, 0] (1) y(t) = C x(t) + D u(t) where x(t) ∈ Rn is the system state, u(t) ∈ Rm is the control input and y(t) ∈ Rm is the plant output. τ (t) is a differentiable uncertain delay function, satisfying for all t≥0 0 ≤ τ (t) ≤ h,
τ˙ (t) ≤ d < 1,
(2)
where h and d are positive scalars. The matrices A0, A1, B, C and D > 0 are constant matrices of appropriate dimensions. Remark 1. In the general case, for strictly proper systems, we define D = ²I where ² is a small positive scalar. The small D > 0 has no physical significance and it is added only in order to assure the feasibility of the BMI conditions that appear below. For any proper system with D which is not positive definite but of full row rank, say m, ¯ we define ¯ u(t) = Hu(t) where H ∈ Rm×m and DH > 0, and obtain the following representation for (1):
Proof : Equation (4) describes n equations in n × (q + m) variables, thus the existence of F (t) and G(t) is guaranteed for all 0 ≤ t < ∞. Moreover, since the number of variables is greater than the number of equations we can impose additional constraints, which will help in the sequel to prove statements about F (t) and G(t). Define Fb(t) = ˙ b = CG(t), and require Fb CF (t) and G(t) (t) = β(Cm −Fb(t)) ˙b b and G(t) = β(Dm − G(t)), where β > 0 is a scalar. In the common case where n > m these additional requirements can be fulfilled by choosing a large enough model order q.QED Remark 3. Note that F (t) and G(t) are not required for implementation and that only their existence matters. Define Kx∗ (t) ≡ D−1 (Cm − CF (t)) Ku∗ (t) ≡ D−1 (Dm − CG(t)).
(5)
˙ ˙ b It follows from the above equations for Fb(t) and G(t) that: K˙ x∗ (t) = −βKx∗ (t)
(6)
K˙ u∗ (t)
(7)
=
−βKu∗ (t)
If the ideal control u∗ (t), defined as: u∗ (t) = Kx∗ (t)xm (t) + Ku∗ (t)um (t),
(8)
x(t)=A ˙ 0 x(t)+A1 x(t − τ (t)) + B u(t), x(t) = 0, t ∈ [−h, 0] y(t)=C x(t)+D u(t)
is substituted in (1b), we obtain that y(t)=ym (t). Therefore, the ideal control u∗ (t) allows (P F ).
where B = BH and D = DH > 0. H can be chosen to be DT or, if a less conservative result is desired, H can be solved for from the BMI that appears below. In the sequel we thus assume, without loss of generality, that D > 0.
In the more general case, we seek a controller of the form Ben Yamin et al. (2009):
For simplicity, we consider only one uncertain delay, the results obtained can be easily extended to the case of multiple delays τ1 (t)...τk (t), k > 1 Fridman and Shaked (2002a). The output of the plant (1) is required to follow the output of the asymptotically stable reference model: x˙ m (t) = Am xm (t) + Bm um (t), ym (t) = Cm xm (t) + Dm um (t)
xm (0) = 0
(3)
where xm (t) ∈ Rq is the system state, ym (t) ∈ Rm is the plant output, um (t) ∈ Rm is the control input and Am , Bm , Cm and Dm are constant matrices of appropriate dimensions. The reference model (3) is used to define the desired input-output behavior of the plant. It is important to note that the dimension of the reference model state may be less than the dimension of the plant state. However, since y(t) is to track ym (t), the number of the model outputs m must be equal to number of the plant outputs. Perfect Tracking (P T ) is defined as tracking with zero tracking error y(t) = ym (t) The next lemma determines the relation that exists between the plant’s and the model’s state vectors. Lemma 2. There exist F (t) ∈ Rn×q and G(t) ∈ Rn×m such that the trajectories of (1) are of the form: x(t) = F (t)xm (t) + G(t)um (t)
(4)
u(t) = K ∗ (t)r(t) − u e(t)
(9)
K ∗ (t) = [ Ke (t) Kx∗ (t) Ku∗ (t) ] r(t) = col{ey (t), xm (t), um (t)} ey (t) = ym (t) − y(t),
(10)
where:
where Ke (t) ∈ Rm×m is a stabilizing gain which is calculated in the sequel, Kx∗ (t) ∈ Rm×q and Ku∗ (t) ∈ Rm define in (5), are stabilizing and bounded gains, and where u e(t) is an auxiliary input signal. Note that for ey (t) ≡ 0 and u e(t) ≡ 0 the controller (9)-(10) reduces to (8). The latter control law, requires the calculation of F (t) and G(t) for all 0 ≤ t < ∞ and the explicit knowledge of the system dynamics. Instead, we use the direct adaptive control scheme known as SAC Kaufman et al (1998) to calculate the gains which lead, in the steady state, to the same control signal that would have been achieved by Ke (t), Kx∗ (t) and Ku∗ (t). The application of the SAC requires neither the explicit knowledge of the gains matrix nor the exact knowledge of the system dynamics. 3. SIMPLE ADAPTIVE CONTROL LAW We consider the following SAC scheme ∆
u(t) = uI (t) + uP (t) = KI (t)r(t) + Kp ey (t) where:
(11)
KI (t) = [ Ke (t) Kx (t) Ku (t) ] ,
(12)
K˙ I (t) = T ey (t)rT (t) − βKI (t)
(13)
and where β>0 is a scalar, Kp ∈ Rm×m is a proportional gain, T is a constant weighting matrix, and the initial condition is KI (0) = 0. Remark 4. When β = 0 in (13), KI (t) steadily increases while y(t) = 6 0. With the β > 0, KI (t) is obtained from a first-order filtering of ey (t)eTy (t) and thus cannot diverge, unless y(t) diverges Kaufman et al (1998).
3.1 Lyapunov-Krasovskii functional for linear time-delay systems Following Fridman and Shaked (2002a), we represent (21a) in the following equivalent descriptor form: e˙ x (t) = µ(t) e ex (t) + A1 ex (t − τ (t)) + Be e u(t) µ(t) = A
The latter is equivalent to the following descriptor system with time-varying delay in the variable µ(t): e˙ x (t) = µ(t)
Zt
∗
We define δ(t) = K (t) − KI (t), that is, the difference between the ideal gain K ∗ (t) and the current SAC gain KI (t). The control law of (11) can now be expressed by the following choice of the auxiliary control signal u e(t) of (9): u e(t) = δ(t)r(t) − Kp ey (t)
(14)
Next, following Kaufman et al (1998), we define the state error ex (t), the output error ey (t) and the control errors eu (t): ex (t) = x∗ (t)−x(t), ey (t) = y ∗ (t)−y(t), eu (t) = u∗ (t)−u(t)
e + A1 ) ex (t) − A1 µ(t) = (A
e u(t) µ(λ)dλ + Be
(23)
t−τ (t)
A Lyapunov-Krasovskii functional for the system (23) has the form · ¸ Zt £T ¤ ex (t) T V (ex (t), µ(t)) = ex (t)µ (t) EP + eTx (λ) µ(t) t−τ (t)
Z0 Zt
(24)
µT (s)AT1 R3 A1 µ(s)dsdθ
Sex (λ)dλ +
and find that eu (t) is given by
−h t+θ
eu (t) = −Ke (t)ey (t) + u e(t)
(15)
e˙ x (t) = A0 ex + A1 ex (t − τ (t)) + Beu (t) ey (t) = C ex (t) + D eu (t)
(16)
Substituting (16b) in (15), we obtain: eu (t) = −Ke (t) (Cex (t) + Deu (t)) + u e(t)
(17)
In order to extract eu (t) from (17), we first define b e (t)=(I + Ke (t)D)−1 Ke (t) K
(18)
b e (t)D. An upperand note that (I + Ke (t)D)−1 = I − K c bound on Ke (t) is calculated as follows: ce (t) = (I +Ke (t)D) Ke (t)DD K = D−1 −(I +Ke (t)D)−1 D−1 = D−1−ν(t) −1
(19)
The algebraic loop for eu (t) in (17) thus results in b e (t)Cex (t) + (I − K b e (t)D)e eu (t) = −K u(t).
(20)
Substituting (20) in (16) and defining e ≡ (A0 − B K b e (t)C), B e ≡ B(I − K b e (t)D), A e ≡ (I − DK b e (t))C and D e ≡ D (I − K b e (t)D) C we obtain the closed-loop system e ex (t) + A1 ex (t − τ ) + Be e u(t) e˙ x (t) = A e ex (t) + De e u(t). ey (t) = C
·
¸ · ¸ P 0 In 0 ,P = 1 , P1 > 0, S > 0, R3 > 0. P2 P3 0 0
The first term of (24) corresponds to the descriptor system, and the other terms correspond to the delay-dependent conditions with respect to uncertain delays. For the sake of simplicity, we define Tb = T Kp , and we are now in a position to state the main result of this section. Theorem 5. For any β > 0, if there exist n × n matrices P1 > 0, P2 , P3 , S = S T , W1 , W2 , W3 , W4 , R1 = R1T , R2 , R3 = R3T and m × m matrices T , Tb, and a compact set K, b e (t) ∈ K satisfies (19), and if the BMI where K T Ψ1 Ψ2
(21)
Ψ4
hΦ1 −W3 A1
0
e hΦ2 −W4T A1 hAT1 R3 ∗ Ψ3 P3T B ∗ ∗ Ψ 0 0 0 5 < 0. ∗ ∗ ∗ −hR 0 0 ∗ ∗
where ν(t) = (D + DKe (t)D)−1 . Then the following holds: b e (t) ≤ D−1 . 0
where E=
which, after simple algebraic manipulations, leads to:
−1
(22)
∗ ∗
∗ ∗
∗ ∗
(25)
−S(1−d) 0 ∗ −hR3
where T T e T T e Ψ1 = (A+A 1 ) P2 +P2 (A + A1 ) + W3 A1 + A1 W3 + T T e (Tb + Tb )C, e S−C T e Ψ2 = P1 − P2 + (A + A1 )T P3 + AT1 W4 , Ψ3 = −P3 − P3T , e−C eT T − C e T (Tb + TbT )D, e Ψ4 = P2T B T T T T e T −T D e −D e (Tb + Tb )D, e Ψ5 = −D £ T ¤ T T Φ1 = W1 + P1 W3 + P2 , ¸ · £ ¤ R1 R2 Φ2 = W2T W4T + P3T , R = R2T R3 is feasible, then the adaptive scheme consisting of the plant (1), the control law (11) and the gain adaptation formula (13) i) Creates bounded gains and states for any input um (t),
ii) Attains perfect tracking for a constant um (t). In such a case, the controller is given by (11)-(13), where Kp = T −1 Tb.
·
e T (Tb + TbT )C e S−C 0 + 0 hAT1 R3 A1 and where
Proof : In order to establish the desired model following of (1), the asymptotic stability of the closed-loop system of (21) should be proven. We consider the following radially-unbounded Lyapunov function candidate Ve (ex (t), µ(t), KI (t)) = (26) = V (ex (t), µ(t)) + T r{δ(t)δ(t)T } > 0. Note that V (0, 0, K ∗ (t)) = 0 and V (ex (t), µ(t), KI (t)) > 0 for all {ex (t), µ(t), KI (t)} 6= {0, 0, K ∗ (t)}. Note also that Vk (ex (t), µ(t), KI (t)) → ∞ if kex (t)k → ∞,kµ(t)k → ∞ or kKI (t)k → ∞. The derivative of (26) is given by ˙ ˙ T} (27) Ve (t) = V˙ (ex (t), µ(t)) + 2T r{δ(t)δ(t) where the following holds. ˙ T } = T r{δ(t)K˙ ∗T (t) − δ(t)K˙ T (t)} T r{δ(t)δ(t) I
(28)
Substituting (29) in (27) and noting that Tb = T Kp , we obtain ˙ Ve (t) = V˙ (ex (t), µ(t)) + 2T r{δ(t)K˙ ∗T (t) − u e(t)eTy (t)T (30) T T −Tbey (t)ey (t) + βδ(t)KI (t)} Following Fridman and Shaked (2002a), we note that · ¸ ¤ d £ T ex (t) T { ex (t) µ (t) EP } dt · µ(t)¸ £ ¤ e˙ (t) = 2 eTx (t) µT (t) P x 0 e˙ x (t) e + A1 ) ex (t)− −µ(t) + (A £ T ¤ t T Z . = 2 ex (t) µ (t) P e u(t) A1 µ(λ)dλ + Be
η(t) = −2
· ¸ · ¸ eT T − C e T (Tb + TbT )D e 0 C ΨP − ˙e ξ e V (t) = ξ T B 0 ∗ Ψ5 T
(31)
Zt µT (λ)AT1 R3 A1 µ(λ)dλ
·
¸ 0 µ(s)ds A1
∆ λ(t) = 2T r{δ(t)K˙ ∗T (t) + βδ(t)KIT (t)} Note that λ(t) is non-definite. But, using (6), (7) and (13a) we obtain λ(t) = 2T r{−βδ(t)δ T (t) + T δ(t)χT } where £ ¤ χ = ey eTy 0 0 . Since: δ(t) = K ∗ (t) − K(t) (33) = [ 0 Kx∗ (t) − Kx (t) Ku∗ (t) − Ku (t) ] ,
λ(t) = 2T r{−βδ(t)δ T (t)} Thus, for any β > 0, we have that λ(t) is negative. Then, a sufficient condition is obtained by deleting the negative term λ(t). For any 2n × 2n-matrices R > 0 and M, the following inequality hold Park (1999) for a(s), b(s) ∈ R2n : Zt −2 bT (s)a(s)ds ≤ Zt
t−h
£
aT (s) bT (s)
¤
·
R RM M T R φ(2, 2)
¸·
¸ a(s) ds b(s)
t−h
Here φ(2, 2)= (M T R+I)R−1 (RM +I). Using this inequality for a(s) = col{0, A1 }µ(s) and b(s) = P col{ex (s), µ(s)}, we obtain · ¸ £ T T¤ T ex T −1 η(t) ≤ h ex µ P (M R + I)R (RM + · I)P¸ µ £ ¤ e +2(eTx (t) − eTx (t − h)) 0 AT1 RM P x µ · ¸ Zt £ ¤ 0 T T + µ (s) 0 A1 R µ(s)ds A1 We substitute the latter into (31) and integrate the resulting inequality in t from 0 to t1 > 0. We obtain (by Schur ˙ complements) Ve (t) < 0 if the following BMI, has a feasible solution: · ¸ · ¸ h i T T T e T 0 b P T 0 − Ce T − Ce (Tb + Tb )D Ψ hΦ −W e A1 B 0 <0 (34) ∗ Ψ5 0 0 ∗ ∗ −hR 0 ∗
t−h
where
∆
where ξ = col{ex (t), µ(t), u e(t)} and ∆ eT T − T T D e −D e T (Tb + TbT )D e Ψ5 = −D · ¸ · ¸ T eT 0 I ∆ Ψ = PT e + 0 A + A1 P A + A1 −I I −I
¤ eTx (s) µT (s) P T
t−h
−
£
t−h
t−τ (t)
−eTx (t − τ (t))S(1 − d)ex (t − τ (t)) − η(t) + λ(t)
Zt
∆
it turns out that δ(t)χT ≡ 0 and hence
Using (13) and (14) and using the fact that T r(AB) = T r(BA), we obtain ˙ T } = T r{δ(t)K˙ ∗T (t) − u T r{δ(t)δ(t) e(t)eTy (t)T (29) T −T Kp ey (t)ey (t) + βδ(t)KIT (t)}
Thus
¸
(32)
∗
∗
−S(1−d)
W = RM P, Φ = WT + PT · ¸ · ¸ 0 0 ∆ 0 AT1 b= Ψ Ψ + WT + W A1 0 0 0
BMI (25) results from the latter BMI by expanding of the block matrices.
In case of constant reference signal um (t), at the steady state the states xm (t) do not change and then K˙ ∗ (t) = 0. ˙ Since Ve (t) = 0 is achieved only when ex (t)=0, µ(t) = 0, u e(t) = 0 and δ(t) = 0, we find that Ve (t) ≥ 0 and that Ve (t) is radially unbounded and monotonically decreasing whenever ex (t) 6= 0 and µ(t) 6= 0 and u e(t) 6= 0 and δ(t) 6= 0. It therefore follows that from any ex (0) ∈ Rn , µ(0) ∈ Rn and K(0) ∈ Rm×(2m+q) the state-vectors ex (t), µ(t) and K(t) will tend, as t → ∞, to ξ(t), µ0 (t) and e˙ K 0 (t), thus satisfying V(ξ(t), µ0 (t),K 0 (t))=0 which by (25) can only happen when ex (t) = δ(t) = u e(t) = 0. P F is thus obtained if the reference signal becomes constant. QED Remark 6. BMI (25) can be either solved directly by using the PENBMI Kocvara and Stingl (2005), or by using local iterations, where at each step an LMI is solved. In the latter case, SDP T 3 Toh et al. (2009) may be used if (25) is of high order. Both PENBMI and SDP T 3 are conveniently accessed via the YALMIP Toolbox Kocvara and Stingl (2005). Remark 7. For SPR systems, if Ke (t) ≡ 0, T ≡ I and Kp ≡ 0, the BMI (25) becomes an LMI and it is similar to the passivity condition of Fridman and Shaked (2002a) for linear time-delay systems. Remark 8. From the definitions of δ(t) it follows that
for some
0 ≤ fj ≤ 1,
N X
fj = 1
(38)
where the vertices of the polytope are described by n o (j) (j) Ωj = A(j) , j = 1, 2..., N. A B 0 1
(39)
j=1
fj Ωj
j=1
The following theorem describes conditions which assure that the closed-loop system (21) is stable over Co{Ωj }. Theorem 10. For any β > 0 over Co{Ωi }, if there exist n × n matrices P1 > 0, P2 , P3 , S = S T , W1 , W2 , W3 , W4 , R1 = R1T , R2 , R3 = R3T and m × m matrices T , Tb, b e (t) ∈ K satisfies (19), and and a compact set K, where K if the BMI (25) is feasible for all the N vertices of Ω, where for each vertex the matrices A0 ; A1 ; B are replace (j) (j) by A0 ; A1 ; B (j) , then the adaptive scheme consisting of the plant (1), the control law (11) and the gain adaptation formula (13) i) Creates bounded gains and states for any input um (t), ii) Attains perfect tracking for a constant um (t). Proof : The BMI of (25) is affine in A0 , A1 and B. Thus, we readily obtain, by multiplying (25) by fj and summing over j = 1, 2, ..., N , that the stability condition is satisfied over Ω.
We bring a numerical example to demonstrate the applicability of the theory developed above.
When P F is attained, δ(t) = 0. Then, T r{(Kx∗ (t)−Kx (t)KxT (t))+(Ku∗ (t)−Ku (t))KuT (t)} = 0 (35) Note that P F does not require K ∗ (t) = K(t), it is suffice that the LHS of (35) vanishes. Note also that K ∗ (t) may not be unique. Remark 9. For W = −P and R = ²Ih2n , BMI (34) becomes for ² → 0+ delay-independent. Then, for the delayindependent case, the following BMI must be feasible: eT P3 Ψ8 P2T A1 Ψ7 P1 −P2T + A ∗ e P3T A1 −P3 −P3T P3T B <0 (36) ∗ ∗ Ψ5 0 ∗ ∗ ∗ −S(1−d) where eTP2 +P2T A+S e e T (Tb + TbT )C e Ψ7 = A −C e −C eT T − C e T (Tb + TbT )D e Ψ8 = P2T B 4. ROBUST SAC WITH UNCERTAINTIES FOR LINEAR TIME-DELAY SYSTEMS The BMI of (25) is affine in the system matrices, therefore Theorem 1 can be used to derive a criterion that will guarantee the stability in the case where the system matrices are not exactly known and they reside within a given polytope. We extend the results of Theorem 5 to the case where the A0 , A1 , and B of the system (1) are not exactly known. Denoting
where Ω ∈ Co{Ωj , j = 1, ...N }, namely,
N X
5. NUMERICAL EXAMPLE
T r{δ(t)K T (t)} = T r{(Kx∗ (t) − Kx (t)KxT (t))+ (Ku∗ (t) − Ku (t))KuT (t)}
Ω = { A0 A1 B }
Ω=
(37)
The unstable second-order time-delay system of Mirkin and Gutman (2003) is considered, where x(t) ˙ = A0 x(t)+A1 x(t − τ (t))+Bu(t), x(t) = 0, t ∈ [−4, 0] y(t) = C x(t) + D u(t) and where · ¸ · ¸ 0 1 −0.2 −0.1 A0 = , A1 = −1 1 0 0 ·¸ 1 , C = [1 0.5 ] and D = 10−5 . B= 0 It should be noted that the nonzero small D = 10−5 , which has no physical significance, is added only in order to assure the feasibility of (25). Note that in Mirkin and Gutman (2003) only a constant and known delay τ = 4 has been considered. Using (25) and Kp = 0, that is when the controller contains only an integral term Ben Yamin et al. (2009), we find that the BMI is feasible for d = 0, 0 < τ (t) ≤ h, h = 11.2; that is, the maximum value of the delay is 11.2. Using the method proposed here, where a proportional term is also active (Kp is not zero and it is solved for), the BMI is feasible for any τ (t) ≥ 0 for Kp = 3500. Moreover, the BMI of the delay-independent case (36) has a feasible solution for Kp = 3500. Simulation results are shown in Figures 1a-c, for a square wave reference command where h = 4, d = 0, β = 0.99, T = 1 and Kp = 5. Our aim is to make the plant output track the reference model output. The reference system is: · ¸ · ¸ −2 −1 1 A= , B= , C = [1 1] (40) 1 0 0
The plots depict the output of the reference model and the output of the plant(Fig. 1), the tracking error ey (t) (Fig. 2) and the control u(t)(Fig. 3). Evidently, the plant output tracks the reference model output extremely well by the proposed control law (11) and the gain adaptation formula (13). 6. CONCLUSIONS In this paper the existing theory of model-following simplified adaptive control has been extended to systems with state time-varying delays. Both delay-dependent and delay-independent results are obtained, using the Lyapunov-Krasovskii functional method. The results assure closed-loop stability in terms of a Bilinear Matrix Inequalitys(BMIs). A similar condition is shown to be valid also for systems with polytopic parameter uncertainties. The results attained in the example are better than those achieved (for the same system) by other approaches. The addition of a proportional-gain feedback (relative to previous methods) reduces the conservatism entailed in the BMIs, thus enabling - as shown in the example - a feasible solution to the BMIs in the delay-independent case. The results encourage further research in related areas such as simplified adaptive control for problems with exogenous disturbances, measurement noise, and state time-varying delays. 0.6
y
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Fig. 1. Time history of the plant and the reference model output. 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 0
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Fig. 2. Time history of the control signal. 0.15 0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 0
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Fig. 3. Time history of the tracking error. REFERENCES K. Sobel, H. Kaufman and I. Mabius. “Implicit adaptive control for a class of MIMO systems”, IEEE Transactions on Aerospace and Electronic Systems, Vol. AES18, No. 5, pp. 576–589, 1982.
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