Adaptive output feedback tracking for a class of nonlinear systems

Automatica 48 (2012) 2372–2376

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Adaptive output feedback tracking for a class of nonlinear systems✩ Xu Zhang, Yan Lin 1 School of Automation, Beijing University of Aeronautics and Astronautics, Beijing 100191, China

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Article history: Received 27 August 2011 Received in revised form 23 February 2012 Accepted 19 May 2012 Available online 2 July 2012 Keywords: Nonlinear systems Output feedback Global asymptotic tracking High-gain observer Adaptive backstepping

abstract We design an adaptive output feedback controller for global asymptotic tracking of a class of nonlinear systems involving unknown parameters and unmeasured states multiplied by output nonlinearities. A modified high-gain observer is introduced, which consists of a series of high-gain K -filters with one dynamic gain and the controller design employs the adaptive backstepping approach. It is proved that by using the proposed method, global stability of the closed loop system can be guaranteed and the output tracking error converges to zero asymptotically. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction One of the important issues in control theory is global asymptotic tracking of nonlinear systems by output feedback. Considerable research efforts have been directed towards the development of schemes with less conservativeness. Early works focused on the nonlinear systems in the output-feedback canonical form (Krstic, Kanellakopoulos, & Kokotovic, 1995; Marino & Tomei, 1995), where the unknown parameters were allowed to appear in the case when the system nonlinearities only depend on the measured output. Recently, many related results (Kvaternik & Lynch, 2011; Mirkin & Gutman, 2010; Zhou & Wen, 2008) have also been worked out on such systems. When the nonlinear systems involve unmeasured states, it has been realized that global output feedback tracking is usually a very difficult task (Andrieu, Praly, & Astolfi, 2009; Freeman & Kokotovic, 1996; Gong & Qian, 2007; Krishnamurthy, Khorrami, & Chandra, 2003; Krishnamurthy, Khorrami, & Jiang, 2002; Praly & Kanellakopoulos, 2000). The problem of global asymptotic tracking was concentrated on the nonlinear systems linear in the unmeasured states (Freeman & Kokotovic, 1996; Krishnamurthy

✩ This work was supported by the NSF of China under Grant 60874044 and the Doctoral Fund of Ministry of Education under Grant 20111102110006. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Chunjiang Qian under the direction of Editor André L. Tits. E-mail addresses: [email protected] (X. Zhang), [email protected] (Y. Lin). 1 Tel.: +86 010 82338435; fax: +86 010 82338435.

0005-1098/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2012.06.002

et al., 2002; Praly & Kanellakopoulos, 2000). In Krishnamurthy et al. (2003), the systems of Freeman and Kokotovic (1996), Krishnamurthy et al. (2002) and Praly and Kanellakopoulos (2000) were extended to a wider class of nonlinear systems and by employing a global high-gain-based observer and backstepping control, global asymptotic tracking was successfully achieved. These schemes, however, are invalid to deal with the parameter uncertainty. In this paper, we consider the following nonlinear system x˙ 1 = x2 + ϕ0,1 (y) +

r 

aj ϕj,1 (y),

j =1

x˙ i = xi+1 + ϕ0,i (y) +

r 

aj ϕj,i (y)

j =1

+

i 

fj,i (y)xj + bgn−i (y)u,

2 ≤ i ≤ n,

j =2

y = x1 ,

(1)

where x ∈ Rn is the state, gn−i (y) = xn+1 ≡ 0 for 2 ≤ i ≤ ρ − 1, u ∈ R and y ∈ R are the system input and output respectively, r is a positive integer, ρ (≥ 1) denotes the relative degree of the system, b andai , 1 ≤ i ≤ r,  are unknown constants, and ϕi,j (y), fi,j (y) and gi (y) gn−ρ (y) ̸= 0 are known smooth functions. The objective of this paper is to design an output feedback controller for system (1) such that y globally asymptotically tracks a given reference signal yr which, together with its first ρ derivatives, is continuous and bounded. Note that the system not only includes the nonlinear terms linear in the unmeasured

X. Zhang, Y. Lin / Automatica 48 (2012) 2372–2376

states but also takes the parameter uncertainty into account. In this sense, system (1) is a direct extension of the adaptive output feedback canonical form Krstic et al. (1995), represents a class of important nonlinear systems and has a wider range of applications. In Fu (2009) and Krishnamurthy and Khorrami (2001), nonlinear systems similar to (1) were also considered but the nonlinearities fi,j (y) are required to match a stringent Hurwitz-like condition so that the controller design method of Krstic et al. (1995) can still be used. The challenge here lies in the fact that, without the Hurwitz-like condition, the problem of how to eliminate the effect of the unknown parameters when the system is coupled with the unmeasured states is quite difficult in the global asymptotic tracking and, to the best of our knowledge, remains unclear and open. In this paper, to solve the challenging problem, a modified highgain observer implemented by a series of high-gain K -filters with one dynamic gain updated online is first introduced in Section 2, which well synthesizes the ideas of Kreisselmeier and high-gain observer and guarantees the convergence of the observer error if the dynamic gain is bounded. Then, following the backstepping approach, an adaptive output feedback controller with a new structure is presented. In the stability analysis, it is shown that by combining the modified high-gain observer and the adaptive backstepping controller, global boundedness of the closed loop signals and asymptotic tracking of system (1) can be achieved. At the end of this section, we point out that the following assumptions are still required for the nonlinear system. A.1. The sign of b is known. Without loss of generality, it is assumed that b > 0. A.2. Minimum-Phase Assumption. The system x˙ i = xi+1 + ϕ0,i (y) +

r 

aj ϕj,i (y) +

j =1

+

gn−i (y) gn−ρ (y)

i 

˙l = −κ l2 + κ l + lγ (y),

l(0) = 1

with κ a positive design parameter and γ (y) a nonnegative smooth function. It can be proved by contradiction that l (t )  ≥ 1 for all r t ≥ 0. Then the state estimate is formed as xˆ = ξ0 + i=1 ai ξi + bv . Define the observer error ϵ = x − xˆ , whose dynamics can be expressed as

  ϵ˙ = A − lLqc T ϵ + F (y)ϵ.

(7)



Note that the matrix A − lLqc T is similar to l A − qc



 T

, or more



specifically, A − lLqc T = lL A − qc T L−1 . Therefore, under the change of coordinates ϵ¯ = l−µ L−1 ϵ with µ a positive design parameter, (7) is transformed into

˙l   ϵ˙¯ = l A − qc T ϵ¯ + L−1 F (y)Lϵ¯ − (µI + D) ϵ¯ ,

(8)

l

where D = diag {0, 1, . . . , n − 1}. Since A − qc T is Hurwitz, then there is a symmetric positive definite matrix P satisfying

T







A − qc T P + P A − qc T = −I. Let the quadratic Lyapunov function Vϵ¯ = ϵ¯ T P ϵ¯ , whose derivative along the solution of (8) is computed as2 V˙ ϵ¯ = −l ∥¯ϵ ∥2 + 2ϵ¯ T PL−1 F (y)Lϵ¯

˙l − ϵ¯ T (DP + PD + 2µP ) ϵ¯ .

ρ + 1 ≤ i ≤ n,

(2)

(9)



2ϵ¯ T PL−1 F (y)Lϵ¯ ≤ 2 ∥P ∥ γ1 (y) ∥¯ϵ ∥2 .

(10)

Since P is a symmetric positive definite matrix, then by choosing a sufficiently large µ, we can obtain that

σ1 I ≤ DP + PD + 2µP ≤ σ2 I ,

It should be pointed out that the minimum-phase assumption A.2 is a natural extension of that of Krstic et al. (1995).

which together with (6) implies that

2. Observer design

˙l − ϵ¯ T (DP + PD + 2µP ) ϵ¯

(11)

l

To design the adaptive output feedback controller, we introduce a modified high-gain observer to estimate the unmeasured states. First of all, we rewrite (1) in the form r

x˙ = Ax + ϕ0 (y) +

(6)

Note that l≥ 1. Thenthere is a nonnegative smooth function γ1 (y) such that L−1 F (y) L ≤ γ1 (y), from which it follows that

fj,i (y)xj

with inputs y0 , y, x2 , . . . , xρ and output xρ+1 is boundedinput to bounded-output (BIBO) stable.



where the gain matrix q = [q1 , . . . ,qn ]T is chosen to make A − qc T Hurwitz, L = diag 1, l, . . . , ln−1 , and l is the observer gain updated by

l

j=2

(by0 − xρ+1 ),

2373



ai ϕi (y) + F (y)x + bG(y)u, (3)

where (A, c ) is the corresponding observable canonical pair, ϕ0 (y) = [ϕ0,1 , . . . , ϕ0,n ]T , ϕi (y) = [ϕi,1 , . . . , ϕi,n ]T , and 0 0 · · · 0    0ρ−1×1 . .  .. ..   gn−ρ  0 f2,2   F (y) =  G(y) =  (4) ,  ..  . . . .. .  .. . . 0 . g0 0 f2,n · · · fn,n The following modified high-gain K -filters are given

  ξ˙0 = A − lLqc T ξ0 + lLqy + ϕ0 (y) + F (y)ξ0 ,   ξ˙i = A − lLqc T ξi + ϕi (y) + F (y)ξi , 1 ≤ i ≤ r ,   v˙ = A − lLqc T v + G(y)u + F (y)v,

(12)

where σ1 , σ2 are positive constants. From (10) and (12), Eq. (9) can be rewritten as V˙ ϵ¯ ≤ −[(1 − κσ2 ) l + σ1 γ (y) − 2 ∥P ∥ γ1 (y)] ∥¯ϵ ∥2 .

i =1

y = c T x,

≤ κσ2 l ∥¯ϵ ∥2 − σ1 γ (y) ∥¯ϵ ∥2 ,

(5)

By choosing κ and γ (y) to satisfy κ 2 ∥P ∥ γ1 (y)/σ1 , we arrive at 1 V˙ ϵ¯ ≤ − l ∥¯ϵ ∥2 . 2

≤

1/2σ2 , γ (y)

(13)

≥

(14)

Remark 1. Note that, if L−1 F (y)L ≤ γ¯1 for some positive constant γ¯1 , one only needs to choose l (≥ 1) to be a sufficiently large constant without updating it online. In particular, when   L−1 F (y)L = 0, i.e., F (y) = 0, l can be explicitly chosen as 1 and the proposed observer reduces to the conventional Kreisselmeier observer.





2 Throughout the paper, we use ∥·∥ to denote the Euclidean norm of a vector or the corresponding induced norm of a matrix.

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X. Zhang, Y. Lin / Automatica 48 (2012) 2372–2376

3. Controller design

Step i (3 ≤ i ≤ ρ − 1). Similarly, define zi+1 = vi+1 − αi and αi be chosen as

In this section, the controller is constructed by employing the observer-based adaptive backstepping technique (Krstic et al., 1995). The recursive design procedure is given as follows.

αi = −zi−1 − ci zi − Bi +

Step 1. Define the tracking error z1 = y − yr , whose derivative is computed as3 z˙1 = x2 + ϕ0,1 +

r 

ai ϕi,1 − y˙ r

i=1

= bv2 + ξ0,2 + ϕ0,1 − y˙ r + θ T ω¯ + ϵ2 , (15)  T  where ω ¯ = ξ1,2 + ϕ1,1 , . . . , ξr ,2 + ϕr ,1 , 0 and θ = a1 , . . . , T ar , b . As usual in the adaptive backstepping design, let z2 be the error between v2 and α1 , i.e., z2 = v2 − α1 , and the first stabilizing function α1 = pˆ α¯ 1 , where α¯ 1 = −c1 z1 − ξ0,2 − ϕ0,1 + y˙ r − θˆ T ω¯ − l1+2µ z1

+

p˙ˆ = −κ1−1 z1 α¯ 1 ,

(18)

and using the inequality z1 ϵ2 ≤

l1+2µ z12

+ lϵ¯ /4, we have   1 ˙ V˙ 1 ≤ bz1 z2 − c1 z12 + lϵ¯22 + θ˜ T Γ τ1 − θˆ . 2 2

4

(19)

∂α1 T ∂α1 ∂α1 ˙ θ ω− ϵ2 − θˆ , ∂y ∂y ∂ θˆ

rewritten as

∂α1 T θ˜ ω − l1+2µ ∂y  ∂α1 ∂α1  τ2 − θ˙ˆ . − ϵ2 + ∂y ∂ θˆ



∂α1 ∂y

2 z2

2

(22)

4

lϵ¯22

   ∂α1  + θ˜ Γ τ2 − θ˙ˆ + z2 τ2 − θ˙ˆ . ∂ θˆ T

(24)

i 

cj zj2 +

j =1

i 4

lϵ¯22 +

i   ∂αj−1  τj − θ˙ˆ zj ∂ θˆ j=2

−

j−1 i   ∂αj−1 ∂αk−1 −1 zk zj Γ ω. ∂y ∂ θˆ j=3 k=2

(25)

˙

Step ρ . In the final step, the actual control u and the adaptive law θˆ are chosen as 1 1 u = gn−−ρ αρ − gn−−ρ vρ+1 ,

∂αρ−1 ∂αρ−1 T τρ θˆ ω + αρ = −zρ−1 − cρ zρ − Bρ + ∂y ∂ θˆ  2 ρ−1  ∂αρ−1 ∂αi−1 −1 ∂αρ−1 − l1+2µ zρ − zi Γ ω, ∂y ∂y ∂ θˆ i=2 ∂αρ−1 −1 θ˙ˆ = τρ = τρ−1 − zρ Γ ω. ∂y

(26)

(27)

V˙ ρ ≤ −

ρ  i=1

ci zi2 +

ρ 4

l ∥¯ϵ ∥2 .

(28)

Remark 2. Actually, the design parameter µ can also be chosen as a small positive constant. As claimed in Krishnamurthy et al. (2003), a smaller µ leads to a better guaranteed convergence of the observer errors and allows the controller gains to be smaller. However, the price we must pay is a more restrictive choice of the observer poles. Remark 3. For physical realization, a modification for (18) and (27), such as the sigma modification (Sastry & Bodson, 1989), is necessary to counteract unmeasured bounded disturbances. 4. Stability analysis

Let the Lyapunov function V2 = V1 + z22 /2. Using (19) and (22), a simple calculation yields V˙ 2 ≤ z2 z3 − c1 z12 − c2 z22 +

zi

  + θ˜ T Γ τi − θ˙ˆ

(20)

 where B2 denotes all the known terms except v3 and ω = ξ1,2 + T ϕ1,1 , . . . , ξr ,2 + ϕr ,1 , v2 . Define z3 = v3 − α2 , where the second stabilizing function α2 is chosen as ˆ 1 − c2 z2 − B2 + ∂α1 θˆ T ω α2 = −bz ∂y 2  ∂α1 ∂α1 τ2 (21) − l1+2µ z2 + ∂y ∂ θˆ with τ2 = τ1 − z2 (∂α1 /∂ y)Γ −1 ω + Γ −1 [01×r , z1 z2 ]T . Then (20) is ˆ 1 − c2 z2 − z˙2 = z3 − bz

2

Define the Lyapunov function Vρ = Vρ−1 + zρ2 /2. Differentiating Vρ yields

Step 2. By considering (5), (6) and (16), we have z˙2 = v3 + B2 −

∂αi−1 ∂y

i−1  ∂αi−1 ∂αi−1 ∂αj−1 −1 τi − zj Γ ω ˆ ∂y ∂θ ∂ θˆ j =2

V˙ i ≤ zi zi+1 −

withc1 a positive constant, θˆ the estimate of θ , and pˆ the estimate  of p = b−1 . Then (15) is rewritten as

choosing



with τi = τi−1 − zi (∂αi−1 /∂ y)Γ −1 ω. Let the Lyapunov function Vi = Vi−1 + zi2 /2. Differentiating Vi yields

(16)

z˙1 = bz2 − c1 z1 − l1+2µ z1 + ϵ2 + θ˜ T ω ¯ − bp˜ α¯ 1 , (17) where θ˜ = θ − θˆ and p˜ = p − pˆ . Define the quadratic function V1 = z12 /2 + θ˜ T Γ θ˜ /2 + κ1 bp˜ 2 /2 with Γ a symmetric positive definite matrix and κ1 a positive constant. Let τ1 = z1 Γ −1 ω ¯ . By

∂αi−1 T θˆ ω − l1+2µ ∂y

(23)

3 Here and throughout the paper, we use ς to denote the jth entry of a vector i ,j ςi .

In this section, we present the stability analysis of the closed loop system. The main results can be summarized as the following theorem. Theorem 1. Consider the nonlinear system (1) and the observerbased adaptive backstepping controller in Sections 2 and 3. Then, all the closed loop signals are globally bounded and the output tracking error y − yr converges to zero. Proof. The closed loop system is formed by the differential equations (1), (5), (6), (18) and   (27) and its states can be taken as

x, ξ0 , ξ1 , . . . , ξr , v, l, θˆ , pˆ . The locally Lipschitz condition

ensures the existence and uniqueness of the solutions on the right

X. Zhang, Y. Lin / Automatica 48 (2012) 2372–2376

2375

maximum time interval 0, Tf for some Tf ∈ (0, +∞]. Consider the Lyapunov function V = ρ Vϵ¯ + Vρ . From (14) and (28), we have



V˙ ≤ −

ρ 

ci zi2 −

i =1

ρ 4



l ∥¯ϵ ∥2 ,

(29)

from which it follows that ϵ¯ , z1 , . . . , zρ , θˆ and pˆ are bounded. Consider the change of coordinates in Section 3 z1 = y − yr ,





z2 = v2 − α1 y, yr , y˙ r , ξ0,2 , ξ1,2 , . . . , ξr ,2 , l, θˆ , pˆ , zi = vi − αi−1 (y, yr , y˙ r , y(ri−1) , ξ0 , ξ1 , . . . , ξr ,

ˆ pˆ ), v1 , . . . , vi−1 , l, θ,

3 ≤ i ≤ ρ.

Fig. 1. Tracking error y − yr .

(30)

Then the boundedness of z1 and yr implies that y is bounded, which follows from (6) that l is bounded. Owing to ϵ = lµ Lϵ¯ , ϵ is bounded. Using the boundedness of y and l and defining the transformation ξ¯i = l−µ L−1 ξi , 0 ≤ i ≤ r, and the corresponding Lyapunov function Vξ¯i = ξ¯iT P ξ¯i , it is not difficult to prove that ξ0 , ξ1 , . . . , ξr are bounded. Note that x = ξ0 +

r 

ai ξi + bv + ϵ.

(31)

i =1

From the boundedness of y, ξi,1 and ϵ1 , v1 is bounded. Therefore, by induction, it can be checked from (30) that v2 , . . . , vρ are bounded. Then from (31), x2 , . . . , xρ are bounded. With assumption A.2, we can obtain that xρ+1 is bounded and therefore vρ+1 is bounded. By viewing (26), u is bounded. Similar to ξi , using the boundedness of y and u and defining the transformation v¯ = l−µ L−1 v and the corresponding Lyapunov function Vv¯ = v¯ T P v¯ , the boundedness of v can be inferred. Finally, it can be proved that all the closed loop signals are bounded on 0, Tf . Hence, Tf must be +∞. Moreover, from (29), z1 , . . . , zρ and ϵ¯ converge to zero as t → +∞, i.e., limt →+∞ (y − yr ) = 0. This completes the proof. 

Fig. 2. Control input u.

Remark 4. When yr = 0 and ϕi,j (0) = 0, asymptotic regulation of the state x can be achieved via the proposed approach. Moreover, if a lower-order output feedback is required, a reduced-order modified high-gain observer and backstepping controller can also be constructed. Fig. 3. Observer gain l.

5. Illustrative example In this section, we present a typical example to illustrate the effectiveness of the proposed scheme: x˙ 1 = x2 , x˙ 2 = x3 + a sin y + y2 x2 + u, x˙ 3 = −x3 − y2 x3 − 0.1u, y = x1 ,

(32)

where a (= 1) is an unknown constant. The control objective is to design an output feedback controller such that the output y globally asymptotically tracks the reference signal yr = sin t. It is not difficult to check that (32) satisfies assumptions A.1 and A.2. Note that (32) cannot be transformed into the adaptive output feedback canonical form (Krstic et al., 1995) and the methods (Krishnamurthy et al., 2003, 2002; Praly & Kanellakopoulos, 2000) are not able to deal with the unknown parameter a. Following the approach in Sections 2 and 3, an adaptive output feedback controller can be designed for global asymptotical tracking. Firstly, design the dynamic high-gain K -filters given by (5) and (6) with q = [3, 3, 1]T , κ = 0.5 and γ (y) = 1.6y2 . According to the

backstepping design in Section 3, the control law and the adaptive law are constructed as

∂α1 aˆ ξ1,2 ∂y  2 ∂α1 ∂α1 − l1+2µ z2 + τ2 , ∂y ∂ aˆ

u = −v3 − z1 − c2 z2 − B2 +

∂α1 −1 Γ ξ1,2 , aˆ (0) = 0. (33) ∂y In the simulation, the design parameters Γ , c1 , c2 and µ are chosen as Γ = 0.05, c1 = c2 = 2, and µ = 1, respectively. Figs. 1– a˙ˆ = τ2 = z1 Γ −1 ξ1,2 − z2

4 show the convergence of the output tracking error and the boundedness of partial closed loop signals under the initial conditions (x1 (0) , x2 (0) , x3 (0)) = (1, 2, 5). 6. Conclusion An adaptive output feedback control scheme has been proposed for global asymptotic tracking of a class of nonlinear systems having both unknown parameters and nonlinearities linear in

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X. Zhang, Y. Lin / Automatica 48 (2012) 2372–2376

Fig. 4. Parameter estimate aˆ .

unmeasured states. An observer-based adaptive backstepping controller has been constructed, which contains a series of modified high-gain K -filters with one dynamic gain. We have proved that, by using the proposed adaptive scheme, the global boundedness of all closed loop signals and the convergence of the tracking error can be guaranteed. References Andrieu, V., Praly, L., & Astolfi, A. (2009). Asymptotic tracking of a reference trajectory by output-feedback for a class of nonlinear systems. Systems and Control Letters, 58, 652–663.

Freeman, R. A., & Kokotovic, P. V. (1996). Tracking controllers for systems linear in the unmeasured states. Automatica, 32, 735–746. Fu, J. (2009). Extended backstepping approach for a class of nonlinear systems in generalised output feedback canonical form. IET Control Theory and Applications, 3, 1023–1032. Gong, Q., & Qian, C. J. (2007). Global practical tracking of a class of nonlinear systems by output feedback. Automatica, 43, 184–189. Krishnamurthy, P., & Khorrami, F. (2001). Global adaptive output feedback tracking for nonlinear systems linear in unmeasured states. In Proceedings of the ACC. (pp. 4814–4819). Arlington, VA. Krishnamurthy, P., Khorrami, F., & Chandra, R. S. (2003). Global high-gainbased observer and backstepping controller for generalized output-feedback canonical form. IEEE Transactions on Automatic Control, 48, 2277–2284. Krishnamurthy, P., Khorrami, F., & Jiang, Z. P. (2002). Global output feedback tracking for nonlinear systems in generalized output-feedback canonical form. IEEE Transactions on Automatic Control, 47, 814–819. Krstic, M., Kanellakopoulos, I., & Kokotovic, P. V. (1995). Nonlinear and adaptive control design. New York: Wiley. Kvaternik, K., & Lynch, A. (2011). Global tracking via output feedback for nonlinear MIMO systems. IEEE Transactions on Automatic Control, 56, 2179–2184. Marino, R., & Tomei, P. (1995). Nonlinear control design: geometric, adaptive and robust. London: Prentice-Hall. Mirkin, B., & Gutman, P. (2010). Robust adaptive output-feedback tracking for a class of nonlinear time-delayed plants. IEEE Transactions on Automatic Control, 55, 2418–2424. Praly, L., & Kanellakopoulos, I. (2000). Output feedback asymptotic stabilization for triangular systems linear in the unmeasured state components. In Proceedings of the IEEE CDC. (pp. 2466–2471). Sydney, Australia. Sastry, S., & Bodson, M. (1989). Adaptive control: stability, convergence, and robustness. NJ: Prentice-Hall. Zhou, J., & Wen, C. (2008). Decentralized backstepping adaptive output tracking of interconnected nonlinear systems. IEEE Transactions on Automatic Control, 53, 2378–2384.