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Robust Simple Adaptive Control for Delayed Measurements Systems
Robust Simple Adaptive Control for Delayed Measurements Systems R. Ben Yamin ∗ I. Yaesh ∗∗ U. Shaked ∗∗∗ ∗ School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, 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: In this paper an output-feedback problem is solved for a class of linear time delayed measurements systems with polytopic-type parameter uncertainties. The objective is to make the system output follow the output of a system model. The time-delay is either constant (known) or time-varying. In both cases, the problem is tackled by applying a combination of a simple direct adaptive control scheme, a Smith-predictor, and a low-pass filter. Sufficient conditions for closed-loop stability of the proposed control scheme are given, in terms of bilinear matrix inequalities. Two numerical examples are given, which demonstrate the applicability of the proposed methods and the simplicity of their implementation. Keywords: Adaptive direct control, dead time systems, Smith-predictor, simple adaptive control, time-varying delay 1. INTRODUCTION Time-delay often appears in control systems, especially in the measurements(Zavarei & Jamshidi (1987), Kolmanovskii et al. (1999) , Meinsma & Zwart (2000)). Timedelay becomes, in many cases, a source of closed-loop instability. A popular scheme for controlling processes affected by time delay was proposed by O.J.M. Smith (Smith. (1957)). The Smith Predictor (SP)attempts to remove the effect of the time-delay from the closed-loop system, so that the controller can be designed as if there were no timedelay present. The SP works perfectly if the delay τ and the plant are known. Clearly, the delay is not eliminated there and the system still requires τ seconds to respond to an input. The SP was mainly designed to achieve good reference signal tracking and good disturbance signal rejection, but it is critical that the model parameters exactly match the plant parameters. Different modifications, based on application of the SP, have been proposed to robustify and to simplify the controllers (e.g. Matausek & Micic (1996), Normey-Rico & Camacho (1999)). In reality, neither the plant nor the time-delay are perfectly known, and the plant can never be perfectly represented by its model. A Simple Adaptive Control (SAC) system can be added to a SP to change the model parameters, so that they continuously match the changing plant parameters. The resulting system has good performance characteristics, but it tracks input signals with a time delay. SAC is a class of direct adaptive controller schemes which has received considerable attention in the literature for continuoustime systems (Sobel et al. (1982), Kaufman et al. (1998)). The stability of closed-loop SAC is related to the Almost
Strictly Positive Real (ASPR) property of the controlled plant. If a plant is ASPR, then there exists a static outputfeedback gain (possibly parameter-dependent) which stabilizes the plant and makes it Strictly Positive Real ((SPR) Kaufman et al. (1998)). For ASPR plants, SAC stabilizes the closed-loop dynamics and consequently leads to zero tracking errors. Robustness of SAC controllers for state delay systems facing polytopic uncertainties has already been established in Ben-Yamin et al. (2009). For systems with state delay, positive realness has been studied by Lu et al.(2000) and Niculescu & Lozano (2001). In Lu et al. (2000), delay-independent sufficient conditions in terms of LMIs have been derived. Fridman & Shaked (2002) introduced delay-dependent sufficient conditions for the passivity of neutral type systems by applying descriptor-type Lyapunov-Krasovskii functionals that were applied in Fridman et al. (2002) in delay-dependent analysis of stability and control synthesis. Fridman et al. (2002), also an output-feedback controller has been derived via a solution of two LMIs by applying an additional lowpass filter. Solutions are offered there for cases of online and delayed measurements. In the present paper we combine the SP scheme and the SAC method to obtain an efficient control design method for systems with measurement delay. The structure of the paper is as follows. Section 2 reviews the sufficient conditions for stability and model following of SAC for systems with state delay. In section 3 a constant (known) delay and a system with polytopic-type parameter uncertainties will be considered. The objective there is to obtain sufficient conditions for closed-loop stability of the proposed SAC and SP scheme. Section 4 brings a solution for TimeVarying Delayed Measurements (TVDM) systems which
are systems with a delayed output where the delay may be time-varying. It will be shown that SAC can stabilize and make the output of an uncertain plant follow a model without knowing the delay; only the bounds on the delay and the delay-rate should be given. Sufficient conditions are obtained for the plant to follow the model. These conditions are expressed in terms of Bilinear Matrix Inequalities (BMI), which can be solved using various methods that will be mentioned in the sequel. Finally, in Section 5 the results are demonstrated by two examples. 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. The trace of a matrix Z is denoted by tr{Z}. The convex hull defined by the polytope vertices Ωj , j = 1, ...N is denoted by Co{Ωj , j = 1, ...N } and col{a, b} for vectors a and b denotes the augmented vector [aT bT ]T . In symmetric block matrices we use ∗ as an ellipsis for terms that are induced by symmetry. 2. PRELIMINARIES In this section we briefly review relevant results that have been obtained in (Ben-Yamin et al. (2009)) for systems with state delays. These results will be used in Section 3 and Section 4 below to derive the combined control scheme of the SP and the SAC for systems with delayed output.
multiple delays τ1 (t)...τk (t), k > 1 (Fridman & Shaked (2002)). 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), xm (0) = 0 (3) ym (t) = Cm xm (t) + Dm um (t) 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) As shown in (Ben-Yamin et al. (2009)), there is an ideal control u∗ (t), that allows (P T ) for the system of (1). We use the SAC (Kaufman et al. (1998)) that leads, in the steady state, to the same control signal that would have been achieved by u∗ (t). 2.2 SAC law We consider the following modified SAC scheme (BenYamin et al. (2009)): ∆
u(t) = uI (t) + uP (t) = KI (t)r(t) + Kp ey (t)
(4)
2.1 Ideal control
where: KI (t) = [ Ke (t) Kx (t) Ku (t) ] , Consider the linear system with time-varying delays K˙ I (t) = T ey (t)rT (t) − βKI (t), KI (0) = 0, x(t) ˙ = A0 x(t)+A1 x(t−τ1 (t))+Bu(t), x(t) = 0, t ∈ [−h1 , 0] r(t) = col{ey (t), xm (t), um (t)}, (1) y(t) = C x(t) + D u(t) ey (t) = ym (t) − y(t) where x(t) ∈ Rn is the system state, u(t) ∈ Rm is the control input and y(t) ∈ Rm is the plant output. τ1 (t) is a differentiable uncertain delay function, satisfying for all t ≥ 0 0 ≤ τ1 (t) ≤ h1 , τ˙1 (t) ≤ d1 < 1, (2) where h1 and d1 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 = ǫI where ǫ 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): x(t)=A ˙ 0 x(t)+A1 x(t − τ1 (t)) + B u(t), x(t) = 0, t ∈ [−h1 , 0] y(t)=C x(t)+D u(t) 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. For simplicity, we consider only one uncertain delay, the results obtained can be easily extended to the case of
(5)
and where β and T are positive scalars and Kp ∈ Rm×m is a proportional gain. 2.3 Stability and model following conditions of SAC Following (Ben-Yamin et al. (2009)), we first define b e (t) = (I + Ke (t)D)−1 Ke (t) K
e ≡ (A0 − B K b e (t)C), B e ≡ B(I − K b e (t)D), A e b e b e (t)D) C ≡ (I − DKe (t))C and D ≡ D (I − K and Tb = T Kp . The proof of the following theorem can be found also in Ben-Yamin et al. (2009): Theorem 2. Given that there exist: β > 0, T > 0, n × n matrices P1 > 0, P2 , P3 , S = S T , W1 , W2 , W3 , W4 , R1 = R1T , R2 , R3 = R3T and m × m matrix Tb, and b e (t) ∈ K which satisfy a compact set K where for all K −1 b 0 < Ke (t) ≤ D , the following BMI is feasible: 0 Ψ1 Ψ2 Ψ4 h1 Φ1 −W3T A1 e h1 Φ2 −W4T A1 h1 AT1 R3 ∗ Ψ3 P3T B ∗ ∗ Ψ5 0 0 0 < 0. (6) ∗ ∗ ∗ −h1 R 0 0 ∗ ∗ ∗ ∗ −S(1−d1) 0 ∗ ∗ ∗ ∗ ∗ −h1 R3 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
e + A1 )T P3 + AT1 W4 , Ψ2 = P1 − P2T + (A Ψ3 = −P3 − P3T
e−C eT T − C eT (Tb + TbT )D, e Ψ4 = P2T B
e −D e T (Tb + TbT )D, e eT T − T T D Ψ5 = −D T Φ1 = W1 + P1 W3T + P2T , T R1 R2 T T Φ2 = W2 W4 + P3 , R = . R2T R3
3.2 Problem Solution
Then the adaptive scheme, consisting of the plant (1), the control law (4) and the gain adaptation formula (5) satisfy the following: i) They create bounded gains and states for any input um (t), ii) They attain perfect tracking for a constant um (t). If the conditions in (6) are satisfied, the controller is given by (4)-(5), where Kp = T −1 Tb. Remark 3. The second inequality in (6) is a BMI. It can be solved either by directly using the PENBMI (Kocvara & 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 (6) is of high order. Both PENBMI and SDP T 3 are conveniently accessed via the YALMIP Toolbox (Kocvara & Stingl (2005)). Remark 4. (Ben-Yamin et al. (2009)) The BMI of (6) is affine in the system matrices, therefore Theorem 2 can be used to derive a condition that will guarantee the stability in the case where the system matrices are not exactly known and they reside within a given polytope. Denoting Ω = { A0 A1 B } where Ω ∈ Co{Ωj , j = 1, ...N }, namely, Ω=
N X
f j Ωj
for some
j=1
0 ≤ fj ≤ 1,
N X
that the system (7) is observable and controllable, and that the delay h1 is a known constant. Consider the asymptotically stable model (3) and a SAC controller obtained by the measurement feedback scheme (4) with the gain adaptation formula (5). The problem is to find conditions which assure that this control law i) Creates bounded gains and states for any input um (t), ii) Attains perfect tracking for a constant um (t).
fj = 1
j=1
where the vertices of the polytope are described by n o (j) Ωj = A(j) B (j) , j = 1, 2..., N. 0 A1
The BMI of (6) is affine in A0 , A1 and B. Thus, we readily obtain, by multiplying it by fj and summing over j = 1, 2, ..., N , that the stability condition is satisfied over Ω. 3. CASE 1: CONSTANT AND KNOWN DELAY 3.1 Problem formulation
We apply the model-following SAC and the SP scheme to solve the problem. For the analysis of the above scheme we use the following ’SP system’ x˙ N (t) = AN xN (t) + BN u(t) yN (t) = CN (xN (t) − xN (t − h1 )) (8) xN (t) = 0, t ∈ [−h1 , 0] where xN (t) ∈ Rn , y(t) ∈ Rm and u(t) ∈ Rm are the state, the measured output and the control input, respectively. AN , BN and CN are the nominal matrices of (7). Remark 1. The system (8) is not applicable for analysis when the system (7) is unstable, even in part of its uncertainty region. In the latter case, the modified SP scheme should be used (Normey-Rico & Camacho (1999)) that leads to a stable SP system (8). Since the stability condition of the SAC is given in terms of state delay systems in (Ben-Yamin et al. (2009)), we use (following Fridman et al. (2002)) a low-pass filter on the output, described by y(t) ˙ = −ρy(t) + ρ(yP (t) + yN (t)) (9) with ρ ≫ 1. The state of this filter is almost identical to yP (t)+yN (t) when ρ → ∞. The idea of the SAC with SP is given in the configuration of Fig. 1. The feedforward path input 1
r(t) y(t)
r(t) y(t)
Plant
SAC
Delay Nominal Plant
where xP (t) ∈ Rn is the plant state, yP (t) ∈ Rm is the measured plant output, and u(t)∈Rm is the control input. AP , BP and CP are constant matrices of appropriate dimensions that are not exactly known and suffer from standard polytopic parameter uncertainty. It is assumed
yN(t)
Filter
y(t)
Delay
Fig. 1. The SP and SAC configurtion in Fig.1 is described by the following augmented system: ˙ ξ(t) =
k X i=0
Consider the following continuous-time linear system: x˙ P (t) = AP xP (t) + BP u(t) yP (t) = CP xP (t − h1 ) (7) xP (t) = 0, t ∈ [−h1 , 0]
Output 1
yP(t)
Ai ξ(t − τi (t)) + Bu(t)
(10)
y(t) = Cξ(t) + Du(t) where k = 1, ξ = col{xP (t), xN (t), y(t)} ,τ0 (t) = 0, τ1 (t) = h1 and where # " # " 0 0 0 AP 0 0 0 0 A0 = 0 AN 0 , A1 = 0 0 ρCN −ρI ρCP −ρCN 0 " # BP B = BN , C =[ 0 0 I ] , D = ǫI 0
and where the small D, which no physical significance, is added to assure the feasibility of the BMI condition that will be solved below. To the system (10) the sufficient condition of Theorem 2 is readily applied to assure stability and the model-following property. 4. CASE 2: TIME-VARYING UNKNOWN DELAY 4.1 Problem formulation Consider the following continuous linear TVDM system: x˙ P (t) = AP xP (t) + BP u(t) yP (t) = CP xP (t − τ2 (t)) xP (t) = 0, t ∈ [−h2 , 0]
τ˙2 (t) ≤ d2 < 1.
T T Ψ2 Ψ4 h1 Φ11 h2 Φ21 −W13 A1 −W23 A2 e h1 Φ12 h2 Φ22 −W T A1 −W T A2 Ψ3 P3T B 14 24 ∗ Ψ5 0 0 0 0 <0, (13) ∗ ∗ −h1 R1 0 0 0 ∗ ∗ ∗ −h2 R2 0 0 ∗ ∗ ∗ ∗ −S1 (1−d1 ) 0 ∗ ∗ ∗ ∗ ∗ −S2 (1−d2 )
where
2 2 2 X X X T e e Ai )T P2+P2T (A+ Ai )+ (Wi3 Ai + ATi Wi3 ) Ψ1 = (A+ i=1 2 X
+
i=1
(11)
It is assumed that the system (11) is observable and controllable. The delay τ2 (t) is a differentiable function, satisfying for all t ≥ 0: 0 ≤ τ2 (t) ≤ h2 ,
Ψ1 ∗ ∗ ∗ ∗ ∗ ∗
(12)
Consider also the asymptotically stable model (3) and a SAC controller obtained by the measurement feedback scheme (4) with the gain adaptation formula (5). The problem is to find conditions which assure that this control law provides the boundedness and tracking properties of Section 3.1. 4.2 Problem solution Using again the SP system (8) and the low-pass filter of (9), with ρ ≫ 1, the system is converted into a system with state delay. The feedforward path in now described by the augmented system (10) with 2 delays (k = 2) where ξ = col{xP (t), xN (t), y(t)}, τ0 (t) = 0, τ1 (t) = h1 , τ2 (t) is a time-varying unknown delay satisfies (12) and where " # " # AP 0 0 0 0 0 0 0 , A0 = 0 AN 0 , A1 = 0 0 ρCN −ρI 0 −ρCN 0 " # " # 0 0 0 BP A2 = 0 0 0 , B1 = BN , C =[ 0 0 I ] , D = ǫI ρCP 0 0 0 The concluding arguments of Section 3.2 can now be applied. To this end, the results of Theorem 2 will now be extended to the case of multiple delays τ1 (t), τ2 (t)(Fridman & Shaked (2002)). The following corollary establishes the stability condition of SAC for time-varying multiple delays. For simplicity’s sake we define n b = 2 × n + m. Corollary 5. Consider the adaptive scheme consisting of the plant (10) with the control law (4) and the gain adaptation formula (5). Given that there exist: β > 0, T > 0, n b×n b matrices P1 > 0, P2 , P3 , Si = SiT , Wi1 , T T Wi2 , Wi3 , Wi4 , Ri1 = Ri1 , Ri2 , Ri3 = Ri3 , i=1,2 and m × m matrix , Tb, and a compact set K where for all b e (t) ∈ K which satisfy 0 < K b e (t) ≤ D−1 , the following K BMI is feasible:
i=1
e T (Tb + TbT )C, e Si − C
e+ Ψ2 = P1 − P2T + (A
Ψ3 = −P3 − P3T +
2 X
2 X
Ai )T P3 +
i=1
i=1
2 X
ATi Wi4 ,
i=1
hi ATi Ri3 Ai ,
i=1
e−C eT T − C e T (Tb + TbT )D, e Ψ4 = P2T B eT T − T T D e −D e T (Tb + TbT )D, e Ψ5 = −D T T T Φi1 = Wi1 + P1 Wi3 + P2 , T Ri1 Ri2 T T Φ2 = Wi2 Wi4 + P3 , Ri = . T Ri2 Ri3 Then, the adaptive scheme ensures the boundedness and the tracking properties of Sections 3.1 and 4.1. If the conditions in (13) are satisfied, the controller is given by (4)-(5), where Kp = T −1 Tb. Remark 6. The BMI in (13) can be transformed by the Schur complements formula (boyd et al. (1994)) so that it is affine in the system matrices Ai , i = 0, 1, 2 and B. Therefore, Corollary 5 can be used to derive robust stability criteria that guarantee stability in the face of standard parameter polytopic uncertainty. 5. NUMERICAL EXAMPLES We bring two numerical examples to demonstrate the applicability of the theory developed above, both based on the process example of Normey-Rico & Camacho (1999). The example is first modified to include polytopic type parameter uncertainty and is used to demonstrate the method of Section 3; then, on top of the parameter uncertainty, the delay is made unknown time-varying to demonstrate the method of Section 4. 5.1 Case 1: Polytopic uncertainty, known constant delay The two-vertex uncertain plant is described by x˙ P (t) = AP xP (t) + BP u(t) yP (t) = CP xP (t − τ1 ); τ1 = 9 xP (t) = 0, t ∈ [−τ1 , 0]
(14)
where onlyAP resides between the vertices (see Remark 4): −16 −6.2 −1.7 0.0 −3.9 −1.7 −1.2 0.0 0.0 0.0 0.0 (2) 8.0 0.0 0.0 0.0 (1) 8.0 , ,A = AP = 0.0 1.0 0.0 0.0 P 0.0 3.0 0.0 0.0 0.0 0.0 0.50 0.0 0.0 0.0 0.50 0.0
and where CP = BPT = [ 0 0 0 0.5 ]. The nominal plant is: −13.0 −4.0 −1.25 0 0.5 8.0 0.0 0.0 0.0 0 x˙ N (t)= x (t) + u(t) 0.0 2.0 0.0 0.0 N 0 (15) 0 0.0 0.0 0.50 0 yN (t) = [ 0 0 0 0.5 ] xN (t − 9); xN (t) = 0, t ∈ [−9, 0] The concluding arguments of Section 3 can now be applied to the above vertices. We readily find that the BMI (6) is feasible (over the whole uncertainty polytope) for d1 = 0, h1 = 9, ρ = 108 , D = 10−5 , T = 1 and Tb = 0.5.
The objective is to make the plant output follow the output of the reference model. Simulation results for a square step reference command, where β = 0.9, are shown in Figures 2-4. The reference model is x˙ m (t) = −0.1xm (t) + 0.25u(t), xm (t) = 0 (16) ym (t) = 0.4xm (t) For a convenient comparison, the model output is displayed with a 9 seconds delay. The plots depict the output of the reference model and the plant output at the two vertices (Fig. 2); the adaptive gains Ke (t), Kx (t) and Ku (t), and the control u(t) at the two vertices (Fig. 34). Evidently, the plant output tracks the reference model output by the proposed control scheme. 5.2 Case 2: Polytopic uncertainty, time-varying delay To demonstrate the method of Section 4 the above example is modified by letting the delay vary in time. Consider the following TVDM system: x˙ P (t) = AP xP (t) + BP u(t) yP (t) = CP xP (t − τ2 (t)) τ2 (t) = 9 + 4 cos(0.1t) xP (t) = 0, t ∈ [−13, 0] where AP , BP and CP are defined in the previous subsection. The concluding arguments of Section 4 applied to this TVDM system with the nominal plant (15). We readily find that the BMI (13) is feasible for h1 = 9, d1 = 0, h2 = 13, d2 = 0.4, ρ = 108 , D = 10−5 , T = 1 and Tb = 0.1. Simulation results with the reference model (16) are shown in Figures 5-7, for a square step reference command. Figure 7 depicts the output of the reference model and the plant output at the two vertices and the (satisfactory) performance at several inner points of the uncertainty polytope. The adaptive gains Ke (t), Kx (t) and Ku (t), and the control u(t) at the two vertices are shown in Figures 6-7. It is seen that the plant output successfully tracks the reference model output in presence of timevarying delay and polytopic parameter uncertainty. 6. CONCLUSIONS In this paper the existing theory of simple adaptive control for continuous-time state-delay systems is generalized to cover output-feedback measurements delay in tracking problems. Both constant (known) delay and time varying delay are considered. The results, given in terms of bilinear matrix inequalities, assure closed-loop stability and
satisfactory model-following. Similar conditions are shown to be valid also for systems with polytopic parameter uncertainties. The results are illustrated via two numerical examples, which show that the output of the plant successfully tracks the output of the reference system. The results encourage further research in related areas, such as simple adaptive control for problems with exogenous disturbances, measurement noise, and measurements time-varying delays. REFERENCES Ben Yamin R., Yaesh I. and Shaked U. (2009) Simple adaptive PI control for linear time-delay systems, 8th IFAC Workkshop on Time-Delay Systems, Sinaia, Romania, September 1 - 3. Boyd S., Ghaoui L. El, Feron E. and Balakrishnan V. (1994) Linear matrix inequality in systems and control theory,SIAM Frontier Series. Fridman, E., Shaked, U. (2002) On delay-dependent passivity,IEEE Trans. Automat. Contr, (47), pp. 664–669. Fridman, E., Shaked, U. (2002) A descriptor system approach to H∞ control of linear time-delay systems,IEEE Trans. Automat. Contr, (47), pp. 253–270. Gahinet, P. , Nemirovski, A., Laub, A.J. and Chilali, M. (1995) LMI Control Toolbox for Use with MATLAB.,The Mathworks Inc. Kaufman, H., Barkana, I. and Sobel, K.(1998) Direct Adaptive Control Algorithms - Theory and Applications, Springer, New-York, Second Edition. Kocvara, M., Stingl, M. (2005) PENBMI User’s Guide,(Version 2),www.penopt.com. Kolmanovskii, V., Niculescu, S. I. and Richard, J. P. (1999) On the LiapunovKrasovskii functionals for stability analysis of linear delay systems, International Journal of Control, 72, pp. 374384. , 1999. Niculescu, S. I. , Lozano, R.,(2006) On the passivity of linear delay systems,IEEE Trans. Automat. Contr, (46), pp. 460464. Normey-Rico J.E. and Camacho E.F. , (1996) Smith predictor and modifications, A comparactive study, Ecc 99, Karlsruhe, Germany, Aug. 31-Sept. 3. Matausek M.R., Micic A.D., (1996) A modified Smith Predictor for controlling a process with an integrator and long dead-time, ser. IEEE Trans. Automat. Contr., vol. 41, pp. 11991203. Meinsma Gjerrit , Zwart Hans. (2000) On H∞ control for dead time systems, IEEE Trans. on Automat. Contr., vol. 45, pp. 272-285. Smith, O.J.M.(1957) Closer control of loops with dead time, ser. Systems and Control. Chemical Engineering Progress, vol. 53, pp. 217219. Sobel, K., Kaufman, H. and Mabius, I.(1982) Implicit Adaptive Control for a Class of Mimo Systems, IEEE Trans. on Aerospace and Electronic Systems, (18), 576589. Toh, K-C., Todd, M. J., and Tutuncu., R. H. (2009) A MATLAB software for semidefinite-quadratic-linear programming.,http://www.math.nus.edu.sg/mattohkc/sdpt3.html. ˜ Malek-Zavarei M. and Jamshidi M. (1987) Time-Delay Systems. Analysis, Optimization and Applications, ser. Systems and Control. Amsterdam, North Holland, vol. 9.
1.5
model output plant output ver. 1 plant output ver. 2
1 0.8
model output plant output ver.1 plant output ver. 2 plant output inner points
1
0.6 0.5
p
y (t) & y (t)
0.2
P m
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y (t) & y (t)
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−0.2 −0.5
−0.4 −0.6
−1
−0.8 −1
−1.5
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Fig. 2. Constant delay case: the outputs of the reference model and the plant at the two vertices.
Adaptive gains
1
k (t)
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u
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x
ku(t)
0 −0.5
u(t)
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k (t) u
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−1 −2 0
k (t)
Fig. 6. Time-varying delay case: the adaptive gains and the control signal (vertex 1).
u(t)
Adaptive gains
k (t)
350
ke(t)
0.8
ke(t)
0.5
300
0
−1 0
1
250
−0.5
300
Fig. 3. Constant delay case: the adaptive gains and the control signal (vertex 1).
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u(t)
u(t)
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−1
50
Fig. 5. Time-varying delay case: the outputs of the reference model and the plant at the two vertices and the plant output at several inner points of the uncertainty polytope.
ke(t)
0.5
0
Time [sec]
Adaptive Gains
0
50
100
150 Time [sec]
200
250
300
Fig. 4. Constant delay case: the adaptive gains and the control signal (vertex 2).
−1 0
Fig. 7. Time-varying delay case: the adaptive gains and the control signal (vertex 2).
Abstract: In this paper an output-feedback problem is solved for a class of linear time delayed measurements systems with polytopic-type parameter uncertainties. The objective is to make the system output follow the output of a system model. The time-delay is either constant (known) or time-varying. In both cases, the problem is tackled by applying a combination of a simple direct adaptive control scheme, a Smith-predictor, and a low-pass filter. Sufficient conditions for closed-loop stability of the proposed control scheme are given, in terms of bilinear matrix inequalities. Two numerical examples are given, which demonstrate the applicability of the proposed methods and the simplicity of their implementation. Keywords: Adaptive direct control, dead time systems, Smith-predictor, simple adaptive control, time-varying delay 1. INTRODUCTION Time-delay often appears in control systems, especially in the measurements(Zavarei & Jamshidi (1987), Kolmanovskii et al. (1999) , Meinsma & Zwart (2000)). Timedelay becomes, in many cases, a source of closed-loop instability. A popular scheme for controlling processes affected by time delay was proposed by O.J.M. Smith (Smith. (1957)). The Smith Predictor (SP)attempts to remove the effect of the time-delay from the closed-loop system, so that the controller can be designed as if there were no timedelay present. The SP works perfectly if the delay τ and the plant are known. Clearly, the delay is not eliminated there and the system still requires τ seconds to respond to an input. The SP was mainly designed to achieve good reference signal tracking and good disturbance signal rejection, but it is critical that the model parameters exactly match the plant parameters. Different modifications, based on application of the SP, have been proposed to robustify and to simplify the controllers (e.g. Matausek & Micic (1996), Normey-Rico & Camacho (1999)). In reality, neither the plant nor the time-delay are perfectly known, and the plant can never be perfectly represented by its model. A Simple Adaptive Control (SAC) system can be added to a SP to change the model parameters, so that they continuously match the changing plant parameters. The resulting system has good performance characteristics, but it tracks input signals with a time delay. SAC is a class of direct adaptive controller schemes which has received considerable attention in the literature for continuoustime systems (Sobel et al. (1982), Kaufman et al. (1998)). The stability of closed-loop SAC is related to the Almost
Strictly Positive Real (ASPR) property of the controlled plant. If a plant is ASPR, then there exists a static outputfeedback gain (possibly parameter-dependent) which stabilizes the plant and makes it Strictly Positive Real ((SPR) Kaufman et al. (1998)). For ASPR plants, SAC stabilizes the closed-loop dynamics and consequently leads to zero tracking errors. Robustness of SAC controllers for state delay systems facing polytopic uncertainties has already been established in Ben-Yamin et al. (2009). For systems with state delay, positive realness has been studied by Lu et al.(2000) and Niculescu & Lozano (2001). In Lu et al. (2000), delay-independent sufficient conditions in terms of LMIs have been derived. Fridman & Shaked (2002) introduced delay-dependent sufficient conditions for the passivity of neutral type systems by applying descriptor-type Lyapunov-Krasovskii functionals that were applied in Fridman et al. (2002) in delay-dependent analysis of stability and control synthesis. Fridman et al. (2002), also an output-feedback controller has been derived via a solution of two LMIs by applying an additional lowpass filter. Solutions are offered there for cases of online and delayed measurements. In the present paper we combine the SP scheme and the SAC method to obtain an efficient control design method for systems with measurement delay. The structure of the paper is as follows. Section 2 reviews the sufficient conditions for stability and model following of SAC for systems with state delay. In section 3 a constant (known) delay and a system with polytopic-type parameter uncertainties will be considered. The objective there is to obtain sufficient conditions for closed-loop stability of the proposed SAC and SP scheme. Section 4 brings a solution for TimeVarying Delayed Measurements (TVDM) systems which
are systems with a delayed output where the delay may be time-varying. It will be shown that SAC can stabilize and make the output of an uncertain plant follow a model without knowing the delay; only the bounds on the delay and the delay-rate should be given. Sufficient conditions are obtained for the plant to follow the model. These conditions are expressed in terms of Bilinear Matrix Inequalities (BMI), which can be solved using various methods that will be mentioned in the sequel. Finally, in Section 5 the results are demonstrated by two examples. 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. The trace of a matrix Z is denoted by tr{Z}. The convex hull defined by the polytope vertices Ωj , j = 1, ...N is denoted by Co{Ωj , j = 1, ...N } and col{a, b} for vectors a and b denotes the augmented vector [aT bT ]T . In symmetric block matrices we use ∗ as an ellipsis for terms that are induced by symmetry. 2. PRELIMINARIES In this section we briefly review relevant results that have been obtained in (Ben-Yamin et al. (2009)) for systems with state delays. These results will be used in Section 3 and Section 4 below to derive the combined control scheme of the SP and the SAC for systems with delayed output.
multiple delays τ1 (t)...τk (t), k > 1 (Fridman & Shaked (2002)). 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), xm (0) = 0 (3) ym (t) = Cm xm (t) + Dm um (t) 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) As shown in (Ben-Yamin et al. (2009)), there is an ideal control u∗ (t), that allows (P T ) for the system of (1). We use the SAC (Kaufman et al. (1998)) that leads, in the steady state, to the same control signal that would have been achieved by u∗ (t). 2.2 SAC law We consider the following modified SAC scheme (BenYamin et al. (2009)): ∆
u(t) = uI (t) + uP (t) = KI (t)r(t) + Kp ey (t)
(4)
2.1 Ideal control
where: KI (t) = [ Ke (t) Kx (t) Ku (t) ] , Consider the linear system with time-varying delays K˙ I (t) = T ey (t)rT (t) − βKI (t), KI (0) = 0, x(t) ˙ = A0 x(t)+A1 x(t−τ1 (t))+Bu(t), x(t) = 0, t ∈ [−h1 , 0] r(t) = col{ey (t), xm (t), um (t)}, (1) y(t) = C x(t) + D u(t) ey (t) = ym (t) − y(t) where x(t) ∈ Rn is the system state, u(t) ∈ Rm is the control input and y(t) ∈ Rm is the plant output. τ1 (t) is a differentiable uncertain delay function, satisfying for all t ≥ 0 0 ≤ τ1 (t) ≤ h1 , τ˙1 (t) ≤ d1 < 1, (2) where h1 and d1 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 = ǫI where ǫ 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): x(t)=A ˙ 0 x(t)+A1 x(t − τ1 (t)) + B u(t), x(t) = 0, t ∈ [−h1 , 0] y(t)=C x(t)+D u(t) 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. For simplicity, we consider only one uncertain delay, the results obtained can be easily extended to the case of
(5)
and where β and T are positive scalars and Kp ∈ Rm×m is a proportional gain. 2.3 Stability and model following conditions of SAC Following (Ben-Yamin et al. (2009)), we first define b e (t) = (I + Ke (t)D)−1 Ke (t) K
e ≡ (A0 − B K b e (t)C), B e ≡ B(I − K b e (t)D), A e b e b e (t)D) C ≡ (I − DKe (t))C and D ≡ D (I − K and Tb = T Kp . The proof of the following theorem can be found also in Ben-Yamin et al. (2009): Theorem 2. Given that there exist: β > 0, T > 0, n × n matrices P1 > 0, P2 , P3 , S = S T , W1 , W2 , W3 , W4 , R1 = R1T , R2 , R3 = R3T and m × m matrix Tb, and b e (t) ∈ K which satisfy a compact set K where for all K −1 b 0 < Ke (t) ≤ D , the following BMI is feasible: 0 Ψ1 Ψ2 Ψ4 h1 Φ1 −W3T A1 e h1 Φ2 −W4T A1 h1 AT1 R3 ∗ Ψ3 P3T B ∗ ∗ Ψ5 0 0 0 < 0. (6) ∗ ∗ ∗ −h1 R 0 0 ∗ ∗ ∗ ∗ −S(1−d1) 0 ∗ ∗ ∗ ∗ ∗ −h1 R3 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
e + A1 )T P3 + AT1 W4 , Ψ2 = P1 − P2T + (A Ψ3 = −P3 − P3T
e−C eT T − C eT (Tb + TbT )D, e Ψ4 = P2T B
e −D e T (Tb + TbT )D, e eT T − T T D Ψ5 = −D T Φ1 = W1 + P1 W3T + P2T , T R1 R2 T T Φ2 = W2 W4 + P3 , R = . R2T R3
3.2 Problem Solution
Then the adaptive scheme, consisting of the plant (1), the control law (4) and the gain adaptation formula (5) satisfy the following: i) They create bounded gains and states for any input um (t), ii) They attain perfect tracking for a constant um (t). If the conditions in (6) are satisfied, the controller is given by (4)-(5), where Kp = T −1 Tb. Remark 3. The second inequality in (6) is a BMI. It can be solved either by directly using the PENBMI (Kocvara & 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 (6) is of high order. Both PENBMI and SDP T 3 are conveniently accessed via the YALMIP Toolbox (Kocvara & Stingl (2005)). Remark 4. (Ben-Yamin et al. (2009)) The BMI of (6) is affine in the system matrices, therefore Theorem 2 can be used to derive a condition that will guarantee the stability in the case where the system matrices are not exactly known and they reside within a given polytope. Denoting Ω = { A0 A1 B } where Ω ∈ Co{Ωj , j = 1, ...N }, namely, Ω=
N X
f j Ωj
for some
j=1
0 ≤ fj ≤ 1,
N X
that the system (7) is observable and controllable, and that the delay h1 is a known constant. Consider the asymptotically stable model (3) and a SAC controller obtained by the measurement feedback scheme (4) with the gain adaptation formula (5). The problem is to find conditions which assure that this control law i) Creates bounded gains and states for any input um (t), ii) Attains perfect tracking for a constant um (t).
fj = 1
j=1
where the vertices of the polytope are described by n o (j) Ωj = A(j) B (j) , j = 1, 2..., N. 0 A1
The BMI of (6) is affine in A0 , A1 and B. Thus, we readily obtain, by multiplying it by fj and summing over j = 1, 2, ..., N , that the stability condition is satisfied over Ω. 3. CASE 1: CONSTANT AND KNOWN DELAY 3.1 Problem formulation
We apply the model-following SAC and the SP scheme to solve the problem. For the analysis of the above scheme we use the following ’SP system’ x˙ N (t) = AN xN (t) + BN u(t) yN (t) = CN (xN (t) − xN (t − h1 )) (8) xN (t) = 0, t ∈ [−h1 , 0] where xN (t) ∈ Rn , y(t) ∈ Rm and u(t) ∈ Rm are the state, the measured output and the control input, respectively. AN , BN and CN are the nominal matrices of (7). Remark 1. The system (8) is not applicable for analysis when the system (7) is unstable, even in part of its uncertainty region. In the latter case, the modified SP scheme should be used (Normey-Rico & Camacho (1999)) that leads to a stable SP system (8). Since the stability condition of the SAC is given in terms of state delay systems in (Ben-Yamin et al. (2009)), we use (following Fridman et al. (2002)) a low-pass filter on the output, described by y(t) ˙ = −ρy(t) + ρ(yP (t) + yN (t)) (9) with ρ ≫ 1. The state of this filter is almost identical to yP (t)+yN (t) when ρ → ∞. The idea of the SAC with SP is given in the configuration of Fig. 1. The feedforward path input 1
r(t) y(t)
r(t) y(t)
Plant
SAC
Delay Nominal Plant
where xP (t) ∈ Rn is the plant state, yP (t) ∈ Rm is the measured plant output, and u(t)∈Rm is the control input. AP , BP and CP are constant matrices of appropriate dimensions that are not exactly known and suffer from standard polytopic parameter uncertainty. It is assumed
yN(t)
Filter
y(t)
Delay
Fig. 1. The SP and SAC configurtion in Fig.1 is described by the following augmented system: ˙ ξ(t) =
k X i=0
Consider the following continuous-time linear system: x˙ P (t) = AP xP (t) + BP u(t) yP (t) = CP xP (t − h1 ) (7) xP (t) = 0, t ∈ [−h1 , 0]
Output 1
yP(t)
Ai ξ(t − τi (t)) + Bu(t)
(10)
y(t) = Cξ(t) + Du(t) where k = 1, ξ = col{xP (t), xN (t), y(t)} ,τ0 (t) = 0, τ1 (t) = h1 and where # " # " 0 0 0 AP 0 0 0 0 A0 = 0 AN 0 , A1 = 0 0 ρCN −ρI ρCP −ρCN 0 " # BP B = BN , C =[ 0 0 I ] , D = ǫI 0
and where the small D, which no physical significance, is added to assure the feasibility of the BMI condition that will be solved below. To the system (10) the sufficient condition of Theorem 2 is readily applied to assure stability and the model-following property. 4. CASE 2: TIME-VARYING UNKNOWN DELAY 4.1 Problem formulation Consider the following continuous linear TVDM system: x˙ P (t) = AP xP (t) + BP u(t) yP (t) = CP xP (t − τ2 (t)) xP (t) = 0, t ∈ [−h2 , 0]
τ˙2 (t) ≤ d2 < 1.
T T Ψ2 Ψ4 h1 Φ11 h2 Φ21 −W13 A1 −W23 A2 e h1 Φ12 h2 Φ22 −W T A1 −W T A2 Ψ3 P3T B 14 24 ∗ Ψ5 0 0 0 0 <0, (13) ∗ ∗ −h1 R1 0 0 0 ∗ ∗ ∗ −h2 R2 0 0 ∗ ∗ ∗ ∗ −S1 (1−d1 ) 0 ∗ ∗ ∗ ∗ ∗ −S2 (1−d2 )
where
2 2 2 X X X T e e Ai )T P2+P2T (A+ Ai )+ (Wi3 Ai + ATi Wi3 ) Ψ1 = (A+ i=1 2 X
+
i=1
(11)
It is assumed that the system (11) is observable and controllable. The delay τ2 (t) is a differentiable function, satisfying for all t ≥ 0: 0 ≤ τ2 (t) ≤ h2 ,
Ψ1 ∗ ∗ ∗ ∗ ∗ ∗
(12)
Consider also the asymptotically stable model (3) and a SAC controller obtained by the measurement feedback scheme (4) with the gain adaptation formula (5). The problem is to find conditions which assure that this control law provides the boundedness and tracking properties of Section 3.1. 4.2 Problem solution Using again the SP system (8) and the low-pass filter of (9), with ρ ≫ 1, the system is converted into a system with state delay. The feedforward path in now described by the augmented system (10) with 2 delays (k = 2) where ξ = col{xP (t), xN (t), y(t)}, τ0 (t) = 0, τ1 (t) = h1 , τ2 (t) is a time-varying unknown delay satisfies (12) and where " # " # AP 0 0 0 0 0 0 0 , A0 = 0 AN 0 , A1 = 0 0 ρCN −ρI 0 −ρCN 0 " # " # 0 0 0 BP A2 = 0 0 0 , B1 = BN , C =[ 0 0 I ] , D = ǫI ρCP 0 0 0 The concluding arguments of Section 3.2 can now be applied. To this end, the results of Theorem 2 will now be extended to the case of multiple delays τ1 (t), τ2 (t)(Fridman & Shaked (2002)). The following corollary establishes the stability condition of SAC for time-varying multiple delays. For simplicity’s sake we define n b = 2 × n + m. Corollary 5. Consider the adaptive scheme consisting of the plant (10) with the control law (4) and the gain adaptation formula (5). Given that there exist: β > 0, T > 0, n b×n b matrices P1 > 0, P2 , P3 , Si = SiT , Wi1 , T T Wi2 , Wi3 , Wi4 , Ri1 = Ri1 , Ri2 , Ri3 = Ri3 , i=1,2 and m × m matrix , Tb, and a compact set K where for all b e (t) ∈ K which satisfy 0 < K b e (t) ≤ D−1 , the following K BMI is feasible:
i=1
e T (Tb + TbT )C, e Si − C
e+ Ψ2 = P1 − P2T + (A
Ψ3 = −P3 − P3T +
2 X
2 X
Ai )T P3 +
i=1
i=1
2 X
ATi Wi4 ,
i=1
hi ATi Ri3 Ai ,
i=1
e−C eT T − C e T (Tb + TbT )D, e Ψ4 = P2T B eT T − T T D e −D e T (Tb + TbT )D, e Ψ5 = −D T T T Φi1 = Wi1 + P1 Wi3 + P2 , T Ri1 Ri2 T T Φ2 = Wi2 Wi4 + P3 , Ri = . T Ri2 Ri3 Then, the adaptive scheme ensures the boundedness and the tracking properties of Sections 3.1 and 4.1. If the conditions in (13) are satisfied, the controller is given by (4)-(5), where Kp = T −1 Tb. Remark 6. The BMI in (13) can be transformed by the Schur complements formula (boyd et al. (1994)) so that it is affine in the system matrices Ai , i = 0, 1, 2 and B. Therefore, Corollary 5 can be used to derive robust stability criteria that guarantee stability in the face of standard parameter polytopic uncertainty. 5. NUMERICAL EXAMPLES We bring two numerical examples to demonstrate the applicability of the theory developed above, both based on the process example of Normey-Rico & Camacho (1999). The example is first modified to include polytopic type parameter uncertainty and is used to demonstrate the method of Section 3; then, on top of the parameter uncertainty, the delay is made unknown time-varying to demonstrate the method of Section 4. 5.1 Case 1: Polytopic uncertainty, known constant delay The two-vertex uncertain plant is described by x˙ P (t) = AP xP (t) + BP u(t) yP (t) = CP xP (t − τ1 ); τ1 = 9 xP (t) = 0, t ∈ [−τ1 , 0]
(14)
where onlyAP resides between the vertices (see Remark 4): −16 −6.2 −1.7 0.0 −3.9 −1.7 −1.2 0.0 0.0 0.0 0.0 (2) 8.0 0.0 0.0 0.0 (1) 8.0 , ,A = AP = 0.0 1.0 0.0 0.0 P 0.0 3.0 0.0 0.0 0.0 0.0 0.50 0.0 0.0 0.0 0.50 0.0
and where CP = BPT = [ 0 0 0 0.5 ]. The nominal plant is: −13.0 −4.0 −1.25 0 0.5 8.0 0.0 0.0 0.0 0 x˙ N (t)= x (t) + u(t) 0.0 2.0 0.0 0.0 N 0 (15) 0 0.0 0.0 0.50 0 yN (t) = [ 0 0 0 0.5 ] xN (t − 9); xN (t) = 0, t ∈ [−9, 0] The concluding arguments of Section 3 can now be applied to the above vertices. We readily find that the BMI (6) is feasible (over the whole uncertainty polytope) for d1 = 0, h1 = 9, ρ = 108 , D = 10−5 , T = 1 and Tb = 0.5.
The objective is to make the plant output follow the output of the reference model. Simulation results for a square step reference command, where β = 0.9, are shown in Figures 2-4. The reference model is x˙ m (t) = −0.1xm (t) + 0.25u(t), xm (t) = 0 (16) ym (t) = 0.4xm (t) For a convenient comparison, the model output is displayed with a 9 seconds delay. The plots depict the output of the reference model and the plant output at the two vertices (Fig. 2); the adaptive gains Ke (t), Kx (t) and Ku (t), and the control u(t) at the two vertices (Fig. 34). Evidently, the plant output tracks the reference model output by the proposed control scheme. 5.2 Case 2: Polytopic uncertainty, time-varying delay To demonstrate the method of Section 4 the above example is modified by letting the delay vary in time. Consider the following TVDM system: x˙ P (t) = AP xP (t) + BP u(t) yP (t) = CP xP (t − τ2 (t)) τ2 (t) = 9 + 4 cos(0.1t) xP (t) = 0, t ∈ [−13, 0] where AP , BP and CP are defined in the previous subsection. The concluding arguments of Section 4 applied to this TVDM system with the nominal plant (15). We readily find that the BMI (13) is feasible for h1 = 9, d1 = 0, h2 = 13, d2 = 0.4, ρ = 108 , D = 10−5 , T = 1 and Tb = 0.1. Simulation results with the reference model (16) are shown in Figures 5-7, for a square step reference command. Figure 7 depicts the output of the reference model and the plant output at the two vertices and the (satisfactory) performance at several inner points of the uncertainty polytope. The adaptive gains Ke (t), Kx (t) and Ku (t), and the control u(t) at the two vertices are shown in Figures 6-7. It is seen that the plant output successfully tracks the reference model output in presence of timevarying delay and polytopic parameter uncertainty. 6. CONCLUSIONS In this paper the existing theory of simple adaptive control for continuous-time state-delay systems is generalized to cover output-feedback measurements delay in tracking problems. Both constant (known) delay and time varying delay are considered. The results, given in terms of bilinear matrix inequalities, assure closed-loop stability and
satisfactory model-following. Similar conditions are shown to be valid also for systems with polytopic parameter uncertainties. The results are illustrated via two numerical examples, which show that the output of the plant successfully tracks the output of the reference system. The results encourage further research in related areas, such as simple adaptive control for problems with exogenous disturbances, measurement noise, and measurements time-varying delays. REFERENCES Ben Yamin R., Yaesh I. and Shaked U. (2009) Simple adaptive PI control for linear time-delay systems, 8th IFAC Workkshop on Time-Delay Systems, Sinaia, Romania, September 1 - 3. Boyd S., Ghaoui L. El, Feron E. and Balakrishnan V. (1994) Linear matrix inequality in systems and control theory,SIAM Frontier Series. Fridman, E., Shaked, U. (2002) On delay-dependent passivity,IEEE Trans. Automat. Contr, (47), pp. 664–669. Fridman, E., Shaked, U. (2002) A descriptor system approach to H∞ control of linear time-delay systems,IEEE Trans. Automat. Contr, (47), pp. 253–270. Gahinet, P. , Nemirovski, A., Laub, A.J. and Chilali, M. (1995) LMI Control Toolbox for Use with MATLAB.,The Mathworks Inc. Kaufman, H., Barkana, I. and Sobel, K.(1998) Direct Adaptive Control Algorithms - Theory and Applications, Springer, New-York, Second Edition. Kocvara, M., Stingl, M. (2005) PENBMI User’s Guide,(Version 2),www.penopt.com. Kolmanovskii, V., Niculescu, S. I. and Richard, J. P. (1999) On the LiapunovKrasovskii functionals for stability analysis of linear delay systems, International Journal of Control, 72, pp. 374384. , 1999. Niculescu, S. I. , Lozano, R.,(2006) On the passivity of linear delay systems,IEEE Trans. Automat. Contr, (46), pp. 460464. Normey-Rico J.E. and Camacho E.F. , (1996) Smith predictor and modifications, A comparactive study, Ecc 99, Karlsruhe, Germany, Aug. 31-Sept. 3. Matausek M.R., Micic A.D., (1996) A modified Smith Predictor for controlling a process with an integrator and long dead-time, ser. IEEE Trans. Automat. Contr., vol. 41, pp. 11991203. Meinsma Gjerrit , Zwart Hans. (2000) On H∞ control for dead time systems, IEEE Trans. on Automat. Contr., vol. 45, pp. 272-285. Smith, O.J.M.(1957) Closer control of loops with dead time, ser. Systems and Control. Chemical Engineering Progress, vol. 53, pp. 217219. Sobel, K., Kaufman, H. and Mabius, I.(1982) Implicit Adaptive Control for a Class of Mimo Systems, IEEE Trans. on Aerospace and Electronic Systems, (18), 576589. Toh, K-C., Todd, M. J., and Tutuncu., R. H. (2009) A MATLAB software for semidefinite-quadratic-linear programming.,http://www.math.nus.edu.sg/mattohkc/sdpt3.html. ˜ Malek-Zavarei M. and Jamshidi M. (1987) Time-Delay Systems. Analysis, Optimization and Applications, ser. Systems and Control. Amsterdam, North Holland, vol. 9.
1.5
model output plant output ver. 1 plant output ver. 2
1 0.8
model output plant output ver.1 plant output ver. 2 plant output inner points
1
0.6 0.5
p
y (t) & y (t)
0.2
P m
0
0
m
y (t) & y (t)
0.4
−0.2 −0.5
−0.4 −0.6
−1
−0.8 −1
−1.5
50
100
150 Time [sec]
200
250
300
Fig. 2. Constant delay case: the outputs of the reference model and the plant at the two vertices.
Adaptive gains
1
k (t)
0.8
k (t)
0.6
x
u
0 −0.5 −1 0
50
100
150 Time [sec]
200
250
300
0
0.5
100
150 Time [sec]
200
250
Adaptive Gains
x
ku(t)
0 −0.5
u(t)
−1 0
50
100
150 Time [sec]
200
250
x
k (t) u
50
100
150 200 Time [sec]
250
300
350
50
100
150 200 Time [sec]
250
300
350
ku(t)
0.2 0
1
1
0.5
0
kx(t)
0.4
−0.2 0
300
ke(t)
0.6
2
50
100
150 200 Time [sec]
250
300
350
50
100
150 200 Time [sec]
250
300
350
0 −0.5
−1 −2 0
k (t)
Fig. 6. Time-varying delay case: the adaptive gains and the control signal (vertex 1).
u(t)
Adaptive gains
k (t)
350
ke(t)
0.8
ke(t)
0.5
300
0
−1 0
1
250
−0.5
300
Fig. 3. Constant delay case: the adaptive gains and the control signal (vertex 1).
200
0
1
50
150
0.2
1
u(t)
u(t)
2
−2 0
100
0.4
−0.2 0
−1
50
Fig. 5. Time-varying delay case: the outputs of the reference model and the plant at the two vertices and the plant output at several inner points of the uncertainty polytope.
ke(t)
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Time [sec]
Adaptive Gains
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100
150 Time [sec]
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Fig. 4. Constant delay case: the adaptive gains and the control signal (vertex 2).
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Fig. 7. Time-varying delay case: the adaptive gains and the control signal (vertex 2).