An LMI based scheduling algorithm for constrained stabilization problems

Systems & Control Letters 57 (2008) 255 – 261 www.elsevier.com/locate/sysconle

An LMI based scheduling algorithm for constrained stabilization problems N. Wada ∗ , M. Saeki Department of Mechanical System Engineering, Hiroshima University, 1-4-1, Kagamiyama, Higashi-Hiroshima 739-8527, Japan Received 20 August 2006; received in revised form 22 June 2007; accepted 3 September 2007 Available online 5 November 2007

Abstract We propose a gain-scheduling control algorithm for locally stabilizing a discrete-time linear system with input saturation. The proposed control law has a structure that a high-gain control law and a low-gain control law are interpolated by using a single scheduling parameter. The scheduling parameter is computed on-line by solving a convex optimization problem with LMI constraints. © 2007 Elsevier B.V. All rights reserved. Keywords: Saturation; Local stabilization; Gain-scheduling; Discrete-time; LMIs

1. Introduction Recently, various attempts have been made to construct a control law for constrained control [1,4–14]. The scheduling control scheme [4,7,10,12,14] is one of the effective methods for dealing with the problems. In this scheme, a controller which interpolates a high-gain controller and a low-gain controller is scheduled so that the closed-loop system achieves high control performance near the origin and large region of attraction. The scheduling parameter is determined by solving an optimization problem on-line. By applying this approach, if the plant is asymptotically null controllable with bounded controls (see e.g., [7]), the global asymptotic stability can be achieved. The semi-global stabilization [6–9] is one of the main research topics of the constrained control problems. In [6–9], it is shown that, in the case where a plant is asymptotically null controllable with bounded controls, the region of attraction of the closed-loop system with an appropriately designed linear controller can be made arbitrary large. Moreover, in [8], it is shown that, by introducing a high-gain component to the linear controller, control performance can be significantly improved. On the other hand, when the plant has exponentially unstable poles, it is impossible to globally stabilize the plant by ∗ Corresponding author. Tel.: +81 82 424 7585; fax: +81 82 422 7193.

E-mail addresses: [email protected] (N. Wada), [email protected] (M. Saeki). 0167-6911/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2007.09.006

using any bounded control law. In such a case, local asymptotic stability in a neighborhood of the origin can be only achieved. In [5], a novel polytopic representation of a saturation function is proposed and an analysis condition of the region of attraction based on the representation is derived. Further, a synthesis condition of a controller which guarantees local asymptotic stability is derived. This representation has several remarkable features. By using the polytopic representation, a necessary and sufficient condition for estimating region of attraction in the case where the system is single input and the estimation is performed by a single quadratic Lyapunov function can be derived. Also, in multivariable case, a less conservative analysis condition as compared with the circle criterion can be derived. Further, both an analysis condition and a synthesis condition can be reduced to complete LMI conditions. In this paper, we propose a scheduling control algorithm for locally stabilizing discrete-time linear systems with input saturation. The proposed control law has a structure that a highgain control law and a low-gain control law are interpolated by a single scheduling parameter. In this paper, we propose a method of determining the scheduling parameter so that the region of attraction becomes large and the rate of convergence of the state variable becomes fast. The problem of computing the scheduling parameter is reduced to an optimization problem with LMI constraints. This property enables us to utilize the polytopic representation of a saturation function of [5] to construct the feedback control law. As a result, the proposed scheme achieves large region of attraction. Two numerical

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examples are provided to illustrate effectiveness of the proposed method. Notations: For a vector u ∈ Rm , we define the standard multivariable saturation function as (u) := ((u1 ), . . . , (um ))T , where  sgn(ui ), |ui | > 1, (ui ) := |ui |1. ui , Let P ∈ Rn×n be a positive definite matrix. Denote E(P ) := {x ∈ Rn : x T P x 1}.

(1)

For a matrix F ∈ Rm×n , denote the ith row of F as f (i) and define L(F ) := {x ∈ Rn : |f (i) x|1, i = 1, . . . , m}.

(2)

Theorem 1. Consider system (5). For positive definite matrices Qi , (i = 0, 1), assume that there exist matrices Yi , Zi satisfying the following matrix inequalities:   ∗ Qi > 0, AQi + B(Ej Yi + Ej− Zi ) Qi ∀i ∈ [0, 1], j ∈ [1, 2m ], (7)  (l)  1 zi 0, ∗ Qi ∀i ∈ [0, 1], ∀l ∈ [1, m], (8) Q0 < Q1 , (9) (l)

where zi denotes the lth row of the matrix Zi and the symbol ∗ stands for symmetric block in matrix inequalities. Further, assume that there exists a constant  ∈ [0, 1] such that x(0) ∈ E(P ()), where P () := Q()−1 , Q() := (1 − )Q0 + Q1 . Then, by applying the feedback control law

2. Preliminary

u(t) = F ()x(t),

In this section, we introduce a polytopic model of a saturation function of [5]. Let V be the set of m × m diagonal matrices whose diagonal element are either 1 or 0. There are 2m elements in V. Suppose that each element of V is labeled as Ej , j = 1, 2, . . . , 2m , and denote Ej− := I − Ej . Clearly, Ej− is also an element of V.

where F () := Y ()Q()−1 , Y () := (1−)Y0 +Y1 to system (5), the relation x(t) → 0, (t → ∞) holds.

Lemma 1 (Hu and Lin [5]). Let u, v ∈ Rm . Suppose that |vj |1 for all j ∈ [1, m], then (u) ∈ co{Ej u + Ej− v : j ∈ [1, 2m ]},

(3)

where co denotes the convex hull. This means that, for |vj |1, we can rewrite (u) as m

(u) =

2  j =1

j (Ej u + Ej− v),

where 0 j 1,

2m

j =1 j

(4)

= 1.

3. Problem formulation Consider the following discrete-time system with a saturation nonlinearity: x(t + 1) = Ax(t) + B(u(t)),

(5)

(10)

Proof. From Lemma 1, while x ∈ L(H ()), the saturation nonlinearity (F ()x) can be represented as m

(F ()x) =

2  j =1

j {Ej F () + Ej− H ()}x,

where H () := Z()Q()−1 , Z() := (1 − )Z0 + Z1 . From (11), while x(t) ∈ L(H ()), the closed-loop system (5), (10) can be rewritten as x(t + 1) = A((t))x(t), (12) 2 m where A((t)) := j =1 j (t)Aj , Aj := A + B{Ej F () + Ej− H ()}. From (8), we have   1 z()(l) 0, ∀l ∈ [1, m], (13) ∗ Q() where z()(l) denotes the lth row of the matrix Z(). By performing a congruence transformation with block-diag [1, Q()]−1 on (13), we have   1 h()(l) 0, ∀l ∈ [1, m]. (14) ∗ P () By applying Schur complement [2] to (14), we have

where x ∈ Rn , u ∈ Rm . We consider the following problem.

P () − h()(l)T h()(l) 0,

Problem 1. Find a state feedback control law

Eq. (15)implies that E(P ()) ⊆ L(H ()).

u(t) = F (t)x(t) that stabilizes system (5). 4. Main result We initially introduce the following theorem.

(6)

(11)

∀l ∈ [1, m].

(15)

Then, we show that if x(0) ∈ E(P ()), then x(t) ∈ E(P ()), ∀t 0 and the relation x(t) → 0, (t → ∞) holds. From (7), we have   Q() ∗ > 0, AQ() + B(Ej Y () + Ej− Z()) Q() ∀j ∈ [1, 2m ]. (16)

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Then, by performing a congruence transformation with blockdiag[Q(), Q()]−1 on (16), and substituting Q()−1 = P () for the resulting inequality, we obtain   P () ∗ > 0, A + B(Ej F () + Ej− H ()) P ()−1 ∀j ∈ [1, 2m ]. (17) Then, by multiplying (17) by j (t), and summing them up for j = 1, . . . , 2m , we have   P () ∗ > 0. (18) A((t)) P ()−1 By applying Schur complement to (18), we have P () − A((t))T P ()A((t)) > 0.

(19)

By multiplying (19) from the left by x(t) and from the right by x(t) and using (12), we obtain T

V (x(t + 1)) < V (x(t)),

(20)

where V (x(t)) := x(t)T P ()x(t). From (20) and x(0) ∈ E(P ()), we have V (x(t)) < V (x(0)),

∀t 0.

(21)

Eqs. (20) and (21) imply that x(t) ∈ E(P ()), ∀t 0 and x(t) converges to zero as t → ∞. By the way, as mentioned above, the saturation nonlinearity (F ()x(t)) can be represented as (11) while x(t) ∈ L(H ()). From Eqs. (15), (21), we can conclude that the relation x(t) ∈ L(H ()), ∀t 0 holds.  In this paper, based on Theorem 1, we design a gain F (1) = Y1 Q−1 1 which enlarges region of attraction E(P (1)) and a gain F (0) = Y0 Q−1 0 which achieves fast convergence of the state variable in E(P (0)). Then we construct a control law (10) by interpolating the obtained gains. In the following, we propose a gain-scheduling algorithm for the control law (10) which achieves both large region of attraction and fast convergence of the state variable. Algorithm 1. Step 0: Set t = 0 and (−1) = 1. Step 1: Measure x(t). Step 2: If (t − 1) where  is a small positive scalar, set (t) = (t − 1) and go to Step 4. Otherwise, go to Step 3. Step 3: Solve min∈[0,1] , s.t.   1 x(t)T 0. (22) ∗ Q() Then, set (t) = . Step 4: Apply u(t) = F ((t))x(t) to the plant (5). Step 5: t ← t + 1 and go to Step 1. Remark 1. The condition (22) is an LMI with respect to a scalar . Hence, the optimization problem of Step 3 in Algorithm 1 can be solved very efficiently [3]. The following theorem can be stated.

Fig. 1. Invariant set.

Theorem 2. Consider system (5). Assume that there exist matrices Qi , Yi , Zi which satisfy the matrix inequality conditions (7)–(9). Further, assume that x(0) ∈ E(P (1)). Then by applying Algorithm 1 to system (5), x(t) converges to zero as t → ∞. Proof. In the following, we initially show that by applying Algorithm 1 (t) monotonically decreases until the condition (t)  holds. We assume that for time t, the condition (t − 1) >  holds and the optimization problem of Step 3 in Algorithm 1 is feasible. In this case, it is clear that x(t) ∈ E(P ((t))) holds. When the control signal u(t)=F ((t))x(t) is applied to system (5), x(t)T P ((t))x(t) > x(t+1)T P ((t))x(t+ 1) holds from Theorem 1. Hence, for some scalar  < 1, x(t +1) ∈ E(P ((t))/) holds (see Fig. 1). In the following, we show that the relation E(P ((t))/) ⊂ E(P ()) ⊂ E(P ((t))) holds for a scalar  such that (t) <  < (t). • E(P ()) ⊂ E(P ((t))): Since Q0 < Q1 and  < (t) hold from the assumption, we obtain 0 < ((t) − )(Q1 − Q0 ) ⇐⇒ Q() < Q((t)) ⇐⇒ E(P ()) ⊂ E(P ((t))).

(23) (24)

Therefore, E(P ()) ⊂ E(P ((t))) holds. • E(P ((t))/) ⊂ E(P ()): From the assumption, (t) <  < (t) holds. Further, since  < 1 and (t) 1, (1 − )(t) (1 − ) holds. Hence, (t) (t) + (1 − ) holds. Therefore, we obtain (t) <  < (t) + (1 − ).

(25)

From (25) and Q0 < Q1 , we have 0 < [(1 − ) − ( − (t))]Q0 + [ − (t)]Q1 .

(26)

Eq. (26)is equivalent to 1 Q((t)) < Q(). 

(27)

Therefore, E(P ((t))/) ⊂ E(P ()) holds. From the above discussion, we can conclude that for a scalar  such that (t) <  < (t), the relation E(P ((t))/) ⊂ E(P ()) ⊂ E(P ((t))) holds (see Fig. 1). Then we set (t +1)=. In this case, it is clear that x(t +1) ∈ E(P ((t + 1))) holds. Namely, the optimization problem of Step 3 in Algorithm 1 is feasible at t +1, and the solution (t +1) satisfies (t + 1) < (t). The same arguments also hold for

N. Wada, M. Saeki / Systems & Control Letters 57 (2008) 255 – 261

t + 2, t + 3, . . . . Therefore, (t) decreases monotonically. Further, (t) is bounded from below by zero. Hence, there exists some time T such that the condition (T ) holds. In Algorithm 1, once the condition (T ) holds at time T, the control law u(t) = F ((T ))x(t) is applied to system (5) for t T . Hence, from Theorem 1, we can conclude that x(t) converges to zero as t → ∞. 

2.5 2 1.5 1 x1, x2

258

0.5

5. Numerical Example I

0

Consider system (5) with the coefficient matrices A = diag[1.000, 1.025], B = [0.050, 0.051]T . Clearly, this system has an exponentially unstable eigenvalue. For this system, we set E1 = 1, E2 = 0 and solve a feasibility problem with the LMI constraints in Theorem 1 and obtain     1.290 0.826 8.948 2.764 , Q1 = (28) Q0 = 0.826 0.556 2.764 2.738

-1 -1.5 0

50

100

150

200

250 t

300

350

400

450

500

350

400

450

500

350

400

450

500

Fig. 3. x1 (dashed), x2 (solid).

1 0.8 0.6 0.4 0.2 Φ(u)

and Y0 = [−4.320, −4.042], Y1 = [−54.954, −55.609], Z0 = [−0.864, −0.647], Z1 = [−0.047, −1.385]. In Fig. 2, the dash-dot line shows E(P (1)) and the dashed line shows E(P (0)). We can confirm that E(P (0)) ⊂ E(P (1)). Further, in this case, the feedback gains at  = 0 and 1 are F (0) = [27.365, −47.962], F (1) = [0.191, −20.505]. The solid line in Fig. 2 shows the state trajectory x(t) for x(0) = [2.100, 1.575]T with Algorithm 1. We set  = 0.0001. We can confirm that x(t) converges to zero as t → ∞. Further, Figs. 3–5 show responses of x(t), (u(t)) and (t) for the same conditions. We can confirm that the scheduling parameter (t) monotonically decreases until the condition (t) holds. Figs. 6 and 7 show the response of the system with u(t) = F (1)x(t) for the same initial condition. From these figures, we can see that when the fixed feedback gain F (1) is utilized the transient response is quite slow. From the above results, we can conclude that the proposed control algorithm achieves both large region of attraction and fast convergence of the state variable in this numerical example.

-0.5

0 -0.2 -0.4 -0.6 -0.8 -1 0

50

100

150

200

250 t

300

Fig. 4. Control signal (u).

2

1

1.5

0.9

1

0.8 0.7

0.5 α

x2

0.6

0

0.5

-0.5

0.4

-1

0.3 0.2

-1.5 0.1

-2 -3

-2

-1

0 x1

1

2

Fig. 2. E(P (1))(dash-dot), E(P (0))(dashed), state trajectory (solid).

3

0 0

50

100

150

200

250 t

300

Fig. 5. Scheduling parameter .

N. Wada, M. Saeki / Systems & Control Letters 57 (2008) 255 – 261

259

6

2.5 2

4

1.5

2 x1, x2, x3

x1, x2

1 0.5

0

0

-2 -0.5

-4

-1 -1.5 0

50

100

150

200

250 t

300

350

400

450

500

Fig. 6. x1 (dashed), x2 (solid).

0.6 0.4

Φ(u)

0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0

50

100

150

200

250 t

300

350

400

450

500

Fig. 7. (u).

6. Numerical Example II In this section, we compare the proposed method with the control laws proposed in [8,9]. Let us consider system (5) with the following coefficient matrices [8]: A = ⎣1 0

5

10

15

20

25 t

30

35

40

45

50

constraints in Theorem 1 and obtain  5.1249 1.6084 −2.4774 Q0 = 1.6084 1.8435 −0.4345 , −2.4774 −0.4345 1.5096  107.2712 0.7921 −4.2402 Q1 = 0.7921 102.2605 2.3333 −4.2402 2.3333 106.1782

0.8

0

0

Fig. 8. x1 (solid), x2 (dashed), x3 (dash-dot): Controller I.

1

⎡

-6

−1 0 0 0

⎤

0⎦ , 1

1 B = −2 . 1 

(29)

The eigenvalues of the matrix A are ±j and 1. In the following, we show three controllers which will be compared in this section. • Controller I (Proposed scheduling controller): We set E1 = 1, E2 = 0 and solve a feasibility problem with the LMI

and Y0 =[2.2861, 0.9224, −1.1412], Y1 =[36.4615, 16.7315, −19.2021], Z0 = [2.1295, 0.8471, −1.0634], Z1 = [5.7563, 1.3821, −3.8499]. In this case, the feedback gains at  = 0 and 1 are F (0) = [0.1751, 0.2544, −0.3954], F (1) = [0.3319, 0.1650, −0.1712]. By using this control law, the relation {x ∈ R3 : x 2 10} ⊂ E(P (1)) holds. • Controller II (Low-gain controller): Based on [9], we design a low-gain controller. Specifically, by choosing the design parameter H = 6.628 × 10−6 I, we obtain u = FL x, where FL := [0.0032, 0.0016, −0.0026]. When this control law is utilized, {x ∈ R3 : x 2 10} becomes the positive invariant set. • Controller III (Low-gain controller with a high-gain component): Based on [8], we design a low-gain controller with a high-gain component. Specifically, by choosing the design parameters H = 6.628 × 10−6 I, (x, 0) = 1,  = 1, we obtain u = FH x, where FH := [0.3037, 0.1523, −0.2403]. When this control law is utilized, {x ∈ R3 : x 2 10} becomes the positive invariant set. In Figs. 8–11, we show the responses of the system for x(0)= [6, 6, −2]T . As shown in Fig. 9, when the low-gain controller is utilized, the response of the system is quite slow. On the other hand, as shown in Figs. 8 and 10, in the cases where the lowgain controller with a high-gain component and the proposed scheduling controller are utilized, the state variables converge to zero rapidly. In the following, we would like to make several comments on the relation between the proposed method and [8]. In the

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6

1 0.8

4

0.6 0.4

2 Φ(u)

x1, x2, x3

0.2

0

0 -0.2

-2

-0.4 -0.6

-4

-0.8 -1

-6 0

5

10

15

20

25 t

30

35

40

45

50

5

10

15

20

25 t

30

35

40

45

50

Fig. 11. (u) (solid: Controller I, dash-dot: Controller II, dashed: Controller III).

Fig. 9. x1 (solid), x2 (dashed), x3 (dash-dot): Controller II.

7. Conclusions

6

In this paper, we have proposed a scheduling control algorithm for locally stabilizing discrete-time linear systems with input saturation. The scheduling parameter is determined by solving a convex optimization problem with LMI constraints with respect to a scalar parameter. Introducing the LMI based scheduling control algorithm enables us to utilize the polytopic representation of a saturation function of [5] for constructing the control law. As a result, the proposed control algorithm can achieve large region of attraction. Further, through two numerical examples, we have shown effectiveness of the proposed method.

4

2 x1, x2, x3

0

0

-2

-4

-6 0

5

10

15

20

25 t

30

35

40

45

50

Fig. 10. x1 (solid), x2 (dashed), x3 (dash-dot): Controller III.

unsaturated region near the origin, the closed-loop system behaves as a linear system. In such a region, the eigenvalues of the closed-loop system with the proposed control law are (A + BF (0)) = {0.1354 ± 0.2081j, 0}, and those with the control law of [8] are (A + BF H ) = {0.3794 ± 0.3088j, 0}. Hence, we can expect that the proposed control law achieves more aggressive performance in a neighborhood of the origin. Moreover, the proposed method has a merit that it can be applied to unstable systems. On the other hand, the control law of [8] has an advantage that it can make the positive invariant set arbitrary large in the case where the matrix A of the plant has all its eigenvalues inside or on the unit circle. Further, in the case of singe input, the control law of [8] becomes time invariant. Therefore, computational burden of the control law of [8] is light as compared with that of the proposed control law.

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