On-line Reference Management for Discrete-Time Servo systems ...

On-line Reference Management for Discrete-Time Servo systems Under State and Control Constraints Yuzo Ohta, Kohei Mori, Koji Yukimoto, and Ryoichi Mishio Kobe University, Nada, Kobe 657-8501, Japan E-mail [email protected] URL http://wwwcs23.cs.kobe-u.ac.jp/ohta/

Abstract— This paper proposes a method of discrete time servo systems design and on-line management of reference signals so as not to violate state and control constraints. The stabilizing and reference shaping controllers are designed off-line by solving LMI optimization problems that takes into account both LQR characterization and state and control constraints. Our managed reference is a piecewise constant signal which is determined online. The better performance of the proposed approach for cases that the initial state is disturbed is demonstrated through an example.

I. I NTRODUCTION For almost all practical control systems, we need to take into account the constraints on state and/or control input caused by amplitude limitation of state variables, saturation property of actuators and so on. If we ignore these constraints, then the real performance of the system degrade or, in worst cases, the control system becomes unstable. In these respect, extensive researches have been done to cope with such constraints [1] – [13]. In this paper, we study techniques that govern, manage, or shape reference signals so that servo systems depicted in Fig. 1 track the original constant reference signal successfully. The results in [3]–[13] belongs to this category. In [3] – [8], the notion of reference governor(RG) or reference management(RM) are adopted, where reference signals are managed on-line. They assumes that the closed loop system has been desined beforehand. In this approach, the on-line computing cost is very cheap. However, it is hard to expect much improvement of the control performance in many cases since governing or managing scheme is quite simple compared to the complexity of the tasks. On the contrary, in [9]–[12], predictive reference management(PRM) is used. The approach in [9]-[11] requires large amount of on-line computing, and, hence, it does not suit for systems that work rapidly. In [12], a method for improving the computational tractability was proposed where the computational effort is moved off-line. However, it seems that this method does not fit to servo systems, because it mainly focuses on regulation property. In [13], reference shaping (RS) is prox[k]

RG r

∗

RM RS

r[k]

y[k]

Servo system

Fig. 1. Constant reference command r ∗ and the governed (managed, shaped) reference signal r[k].

0-7803-9252-3/05/$20.00 ©2005 IEEE

posed. In the method, the reference signal which minimizes the quadratic cost and prevents constraint violation is computed offline. Since computed reference signals depend on initial conditions and reference command, the shape of the reference signal must be computed again when the initial state or reference input changes. In other words, this approach does not suit for the cases when the reference inputs are not known in advance. In this paper, we propose an approach which is a mixture of these approaches described above. In Sections 2 and 3, we assume that the closed loop system has already been designed. We apply the ”outer” feedback/feedforward control using the gain matrix derived from the solution of the LQR problem. Moreover, we adopt RG to manage external reference signals for the system closed by the ”outer” feedback/feedforward. The idea to prevent the violation of constraints by switching external reference signals is commonly used in the previous result [3] – [8]. The only but essential difference between our method and previous method is the existence of the ”outer” feedback/feedforward control depicted in Fig. 3 below. Since the way of managing reference signals is rather simple, the closed system must be designed so that this simple managing scheme works well to improve the control performance. In this point of view, we propose to simultaneous design of controllers of inner and outer loops in Section 4. Notation. For symmetric matrices P and Q, P  Q, P  Q, P ≺ Q, and P  Q mean that P − Q is positive definite, positive semidefinite, negative definite and negative semidefinite, respectively. For a matrixp A. [A]i denotes the i-th row vector of A. For a vector y, |y|2 = y > y. II. P ROBLEM S TATEMENT

Let us consider a discrete-time servo system given by  x[k + 1] = Ax[k] + Br[k], x[0] = x0 y[k] = Cx[k],

(1)

where k ∈ Z+ = {0, 1, · · · }, x ∈ Rn˜ is the closed loop state, y ∈ Rm is the output to be controlled, and r ∈ Rm is the reference input of the closed system. Throughout this paper, we assume the following assumptions (A-1)–(A-3). (A-1). The pair (A, B) is controllable. (A-2). A is a stable matrix. (A-3). For any rˆ ∈ Rm there exists a unique x ˆ(ˆ r ) such that x ˆ(ˆ r ) = Aˆ x(ˆ r ) + Bˆ r, C x ˆ(ˆ r ) = rˆ.

183

(2)

This assumption is not unnatural when m = 1. We require constraints on state and control are satisfied. The constraints are given by z[k] = Lx[k] + Dr[k] ∈ Z ⊆ R

Nc

∀ k ∈ N,

sidered in [13]. Problem 1: Given a step reference rˆ ∈ Rm . Find a managed N reference input {r[k]}N ˆ) k=0 which minimize J({r[k]}k=0 ; x0 , r subject to (1) and (3), where J({r[k]}N ˆ) k=0 ; x0 , r N X

k=0

III. C-LQR


|y[k] − rˆ|22 + wr2 |r[k] − rˆ|22 .

AND

− A> PR B(wr I + B > PR B)−1 B > PR A,

r˜[k; rˆ] = Kr e[k; rˆ],

Theorem 1: Assume that there exist a real number ε and matrices Q and Y such that Q = Q> , Q − I  0,  Q (∗21 )> −(AQ + BY ) Q −(SQ + RY ) 0

R EFERENCE M ANAGEMENT

Given a step reference rˆ ∈ Rm , and define x ˆ(ˆ r ) by (2). Let x[k] be the solution of (1), and let e[k; rˆ] = x[k] − x ˆ(ˆ r ), e0 = x0 − xˆ(ˆ r ), y˜[k; rˆ] = y[k] − rˆ, r˜[k; rˆ] = r[k] − rˆ. Then, by (1) – (3), we have  e[k + 1; rˆ] = Ae[k; rˆ] + B˜ r [k; rˆ], e[0; rˆ] = e0 , y˜[k; rˆ] = Ce[k; rˆ], z[k; rˆ] = Le[kˆ r] + D˜ r [k; rˆ] + zˆ(ˆ r ) ∈ Z ∀ k ∈ N, zˆ(ˆ r ) = Lˆ x(ˆ r ) + Dˆ r,

(6) (7)

and J({r[k]}N ˆ) in (5) is represented by k=0 ; x0 , r ˜ r[k; rˆ]}N J({˜ ˆ) = k=0 ; x0 , r

N X

k=0

=

N X

k=0

(|˜ y [k; rˆ]|22 + wr2 |˜ r [k; rˆ]|22 ) |Se[k; rˆ] + R˜ r[k; rˆ]|22 ,



(∗31 ) 0   0. εI

(17)

P = Q−1 , Kr = Y P, G = A + BKr .

(18)

Then we have 1 G> P G − P + (S + RKr )> (S + RKr ) ≺ 0. (19) ε Moreover, if the constraint (9) holds for the solution of (15), then we have 2 ˜ r [k; rˆ]}N J({˜ ˆ) ≤ εe> k=0 ; x0 , r 0 P e0 ≤ ε|e0 |2 .

(20)

In view of (20), smaller ε means better performance. We solve the following optimization problem to determine ε, Q and Y satisfying (16), (17) and (20). ( min ε ε,Q,Y LMI 1 : subject to (16) and (17) . Let us define Kr and G by (18), and we consider the constraint (9). By (14) and (4), the constraint (9) becomes

(11)

where     C 0 S= , R= . 0 wr I

(16) >

where ∗21 = −(AQ + BY ) and ∗31 = −(SQ + RY ). Let

(8) (9) (10)

(14)

to get a ”good” approximate solution of C-LQR. Then, from (8) and (14), we have  e[k + 1; rˆ] = (A + BKr )e[k; rˆ], e[0; rˆ] = e0 , (15) y˜[k; rˆ] = Ce[k; rˆ].

(5)

A. A Preliminary Result

(13)

and KR = −(wr I + B > PR B)−1 B > PR A. In our case, we take the condition (9) into account and consider applying the control

(4)

where Nc = {1, 2, . . . , Nc }. The following problem was con-

=

PR = Q + A> PR A

(3)

where N = {0, 1, · · · , N } ⊆ Z+ and Z ⊆ RNc is a polyhedral set given by Z = {z = [z1 z2 · · · zNc ]> | |zi | ≤ 1, i ∈ Nc },

given by r˜[k; rˆ] = KR e[k; rˆ] and the optimal value is e> 0 PR e0 , where PR is the positive semi-definite matrix satisfying

(12)

Problem 1 is reduced to the constrained LQR problem: ( ˜ r [k; rˆ]}N ; x0 , rˆ), J({˜ min k=0 {˜ r[k;ˆ r ]}N C-LQR : k=0 sub. to (8) and (9).

|[L + DKr ]i e[k; rˆ] + [ˆ z (ˆ r )]i | ≤ 1, i ∈ Nc ,

(21)

where zˆ(ˆ r ) is defined by (10). Theorem 2: Given rˆ ∈ Rm . Assume that there exists a real number δ(ˆ r ) ∈ (0, 1) such that |[ˆ z (ˆ r )]i | ≤ δ(ˆ r ) < 1 ∀ i ∈ Nc .

(22)

Further assume that there exist a real number σ and a matrix P˜ such that I  P˜ = P˜ > G> P˜ G − P˜ ≺ 0   σ [L + DKr ]i  0, i ∈ Nc . [L + DKr ]> P˜

From a practical point of view, N need to be sufficiently large. If we ignore the condition (9), C-LQR problem is a normal LQR problem. Moreover, if N = ∞, then the optimal solution is

i

184

(23) (24) (25)

Then, we have ˜ ρ(ˆ ˜ ρ(ˆ Ge ∈ E[0; r )] ∀ e ∈ E[0; r )],

(26)

˜ ρ(ˆ (L + DKr )e + zˆ(ˆ r ) ∈ Z ∀ e ∈ E[0; r )],

(27)

and

Suppose that we have solved both LMI 1 and LMI 2, and, hence, ε, Kr , G, σ and P˜ are determined. Given rˆ. If x[k0 ] ∈ ˜ x(ˆ E[ˆ r ); ρ(ˆ r )], then Theorems 2 and 1 means that ˜ x(ˆ 1) x[k] ∈ E[ˆ r ); ρ(ˆ r )] for all k ≥ k0 , 2) x[k] converges to x ˆ(ˆ r ), 3) the constraint (3) is satisfied, and 4) J({r[k]}N ˆ) in Problem 1 satisfies k=0 ; x0 , r

where 2

ρ(ˆ r ) = [1 − δ(ˆ r )] /σ, ˜ E[˜ x; ρ(ˆ r )] = {x |(x − x ˜)> P˜ (x − x ˜) ≤ ρ(ˆ r )} 1/2 2 ˜ = {x | |P (x − x˜)| ≤ ρ(ˆ r )}. 2

(28) (29)

˜ Therefore, if e0 ∈ E[0; ρ(ˆ r )], then the solution e[k; rˆ] of (15) satisfies ˜ ρ(ˆ e[k; rˆ] ∈ E[0; r )] ∀ k ≥ 0 (L + DKr )e[k; rˆ] + zˆ(ˆ r ) ∈ Z ∀ k ≥ 0,

(30) (31)

and e[k; rˆ] → 0 as k → ∞. ˜ ρ(ˆ To obtain the ”largest” E[0; r )], we solve the following optimization problem: ( min σ σ,P˜ LMI 2 : subject to (23), (24) and (25) .

(36) (37)

then we make x[k] converge to x ˆ(r ∗ ) by setting (38)

where {k` } is determined on-line by RG, kL+1 = ∞, and rˆL = r∗ (see Fig. 2). Finally, we note that we always choose {ˆ r ` }L `=1 satisfying (36) and (37): Let

(33)

rˆ` = (1 − λ` )ˆ r0 + λ` r∗ ,

(34)

If we put x[k] = xˆ(ˆ r ) in the right hand side of (34), then we have x[k + 1] = x ˆ(ˆ r ), and, hence, x ˆ(ˆ r ) is an equilibrium of the system (34). On the other hand, x ˆ(ˆ r ) must be the unique equilibrium of (34) since G is a stable matrix. Note that xˆ(ˆ r ) is defined by (2), and, hence, it is independent of K r . B. Reference Management The constraint (3) requires some additional conditions on x0 and rˆ. For example, Lˆ x(ˆ r ) + Dˆ r ∈ Z and Lx0 + D0 ∈ Z must be satisfied. From the practical point of view, we also assume the following assumptions (A-4) and (A-5). (A-4). There exists δ ∈ (0, 1) such that for any step reference input rˆ belonging to a given compact set R ⊆ Rm the following inequalty holds ∀ rˆ ∈ R, |[ˆ z (ˆ r )]i | ≤ δ < 1 ∀ i ∈ Nc

˜ x(ˆ x ˆ(ˆ r`−1 ) ∈ int E[ˆ r` ); ρ(ˆ r` )], ` = 1, 2, · · · , L ∗ ∗ ˜ x ˆ(ˆ rL ) ∈ int E[ˆ x(r ); ρ(r )],

(32)

Then, by (1) and (2), the state of the servo system satisfies x[k + 1] = Ax[k] + B{Kr [x[k] − x ˆ(ˆ r )] + rˆ} = (A + BKr )[x[k] − x ˆ(ˆ r )] + x ˆ(ˆ r) = G[x[k] − x ˆ(ˆ r )] + xˆ(ˆ r ).

Suppose that a reference command r ∗ ∈ R is given. Let k0 = 0. ˜ x(r∗ ); ρ(r∗ )], then we just set rˆ = r∗ . In this case, If x[k0 ] ∈ E[ˆ ˜ x[k] ∈ E[ˆ x(r∗ ); ρ(r∗ )] for all k ≥ k0 , and it converges to xˆ(r ∗ ). ˜ x(r∗ ); ρ(r∗ )]. As we We consider the case when x[k0 ] 6∈ E[ˆ ˜ assumed in (A-5), x[0] ∈ E[ˆ x(ˆ r0 ); ρ(ˆ r0 )] for some rˆ0 ∈ R. If ˜ x(r∗ ); ρ(r∗ )], then we set rˆ[k] = rˆ0 for k ≥ k0 . x ˆ(ˆ r0 ) ∈ int E[ˆ ˜ x(r∗ ); ρ(r∗ )] Then, there exists k1 > k0 such that x[k1 ] ∈ E[ˆ since x[k] converges x ˆ(ˆ r0 ) by Theorem 2. We now set rˆ[k] = r ∗ ˜ x(r∗ ); ρ(r∗ )] for all k ≥ k1 and for k ≥ k1 so that x[k] ∈ E[ˆ x[k] converges to x ˆ(r∗ ). This determination of k1 and switching of r[k] are done by the reference governor (RG). In general, if there exists {ˆ r` } L `=1 such that

rˆ[k] = rˆ`−1 , k ∈ [k`−1 , k` ), ` = 1, 2, · · · , L + 1,

By (7) and (14), r[k] is given by r[k] = r˜[k; rˆ] + rˆ = Kr [x[k] − x ˆ(ˆ r )] + rˆ.

J({r[k]}N ˆ) ≤ ε|x0 − x ˆ(ˆ r )|22 . k=0 ; x0 , r

(35)

where zˆ(ˆ r ) is defined by (10). (A-5). The initial state x0 locates in a neighborhood of the equilibrium point x ˆ(ˆ r0 ) corresponding to a constant reference input rˆ0 ∈ R.

(39)

then, by (2), x ˆ(ˆ r` ) − x ˆ(ˆ r`−1 ) = (λ` − λ`−1 )(I − A)−1 B(r∗ − rˆ0 ), and, hence, we have (36) and (37) if we choose λ` − λ`−1 > 0 sufficiently small. Remark 1: Block diagram of (33) is shown in Fig. 3. Note that RG generate rˆ[k] but r[k] is generated by feedback control, and, hence, there is no need to compute r[k]. Generating rˆ[k] is much easier than generating r[k] as you will see in Fig. 4 Fig. 7. Example 1: Let us consider a position servo system considered in [8]. The plant consists of a DC servo motor, a reduction gear and a load connected by a hard shaft. The state equation of the plant is given by       0 0 1 x + 2 u, y = 1 0 xP , x˙ P = ωn 0 −2ζωn P   where xP = θL θ˙L , ζ = 0.7, ωn = 7, θL [rad] denotes the positions (i.e., angles) of the load, and u [V] is the input voltage. Because of the saturation constraint of the power source, the input voltage u has to be constrained with the range: |u| ≤ Vmax , Vmax = 2.4.

185

(40)

In [8], the controller is given by x˙ C = e, u = 3(e + xC ), e = r − y.

(41)

The closed system is transformed into discrete-time by zeroorder hold assuming a sampling period of Ts = 10[ms], and it is given by  x[k + 1] = Ax[k] + Br[k], x[0] = x0 (42) y[k] = Cx[k],  > where x = x> x> . P C Let r∗ = 1.5708 [rad] (= 90 [deg]) and let x0 = 0. We examined 3 cases: Case 1). Set r[k] ≡ r ∗ . The reference signal r(t) = r∗ is supplied to the (continuous time) servo system; Case 2). Compute r[k] off-line by using the method proposed in [13]. Piecewise continuous time reference signal r(t) is given by r(t) = r[k], t ∈ [kTs , (k + 1)Ts ). This r(t) is supplied to the servo system; Case 3). Apply our method, i.e., the reference signal rˆ(t) is determined on-line by RG and it is supplied system, where  to the ”outer” feedback/feedforward  Kr = −2.0171 −0.11773 −0.72734 . As the result, r(t) is generated automatically by the systems. In Fig. 4, we show rˆ(t) and r(t), y(t) and u(t) for these 3 cases at left-upper, at right-upper, and at left-lower,respectively. Moreover, we show rˆ(t) for the Case 3) at left-upper of Fig. 4. The signals for Case 1), Case 2) and Case 3) are represented by dot broken lines, broken lines and solid lines, respectively. The constraint (40) is violated for Case 1). The output y(t) is computed by ignoring the violation of the constraint (40) for this case. The constraint (40) is satisfied for the case 2) and the Case 3). Among these transient characteristics shown in Fig. 4, the output by the method in [13] seems to attain the best performance. However, when x0 6= 0, the situation might change. For  > example, when x0 = 1.1496 0 0 , the transient characteristic by the method in [13], which is in Fig. 5, is not good since {r[k]} is computed off-line for the case when x0 = 0. On the other hand, the tangent characteristic by our method is fairly good. This advantage of our method is based on the fact that our method generates rˆ(t) on-line and r(t) is automatically generated by the feedback/feedforward control.

IV. S IMULTANEOUS D ESIGN OF I NNER AND O UTER L OOP C ONTROLLERS In the previous section, we proposed the method of managing reference signal using additional feedback control(the outer loop) under the assumption that the closed system has been designed somehow. This outer loop controller works well for reference signal shaping and contribute to the improvement of the control performance. Then it is natural to expect that it is more effective if we design the inner loop controller (the controller for the servo system in Fig. 3) and the outer loop controller simultaneously. Let us consider a linear plant given by xp [k + 1] = Ap xp [k] + Bp u[k], y[k] = Cp xp [k].

(43)

We assume that the pair (Ap , Bp ) is controllable, (Ap , Cp ) is observable, and   I − Ap −Bp 6= 0. (44) det −Cp 0 By (44), for each constant reference command rˆ, there exists a unique (ˆ xp (ˆ r ), u ˆ(ˆ r )) such that x ˆp (ˆ r ) = Ap x ˆp (ˆ r ) + Bp u ˆ(ˆ r ), Cp x ˆp (ˆ r ) = rˆ.

(45)

We consider the following controller xc [k + 1] = xc [k] + r(t) − y(t), u[k] = v[k] + xc [k] v[k] = Ku [x[k] − x ˆ(ˆ r )] + uˆ(ˆ r ), r[k] = Kr [x[k] − xˆ(ˆ r )] + rˆ,     xp [k] x ˆp (ˆ r) x[k] = , x ˆ(ˆ r) = . xc [k] 0 Then, by (45) – (50), we have    Ap Bp Bp x[k + 1] = x[k] + −Cp 1 0 | {z } | {z Ae

Be

  y[k] = Cp 0 x[k], | {z }

0 I

 }

(46) (47) (48) (49) (50)

 v[k] , r[k]

(51)

(52)

Ce



     uˆ(ˆ r) Ku v[k] . [x[k] − xˆ(ˆ r )] + = rˆ Kr r[k] | {z }

(53)

K

We note that x ˆ(ˆ r ) is the unique equilibrium of the closed system when (Ae + Be K) is stable. That is, it is independent of the stabilizing K. To design K, we show the following. r∗ RG

rˆ[k]

r[k] +

Servo system

+

y[k]

Kr Fig. 2. An example of {ˆ x(ˆ r` ) = (I − A)−1 Bˆ r` }, ˜ r` ; ρ(ˆ and {E(ˆ r` ))}, where n ˜ = 2, m = 1, xˆ(ˆ r` ) and ˜ r` ; ρ(ˆ E(ˆ r` )) are denoted by • and ellasped. In this case, R and (I − A)−1 B(R) = {ˆ x(r)| r ∈ R} are linesegments.

Kf

x ˆ(ˆ r) −

x[k] +

Fig. 3. Block diagram of (33), where Kf = (I − A)−1 B.

186

120 100

y(t)

60

rˆ(t)

40

100

140

80

120

80 60

0

20

0.5

1

t

1.5

0

2

0

0.5

1

t

1.5

−20

2

u(t)

u(t)

Our method 1 Method in [13] r(t) ≡ r∗

2 0 −2

0

0.5

1

1.5

100

rˆ(t)

100

90

90

y(t)

rˆ(t), r(t)

1

1.5

t

0

2

0

0.5

1

t

1.5

2

80 70

80 70

60 60

50 0

0.5

1

t

1.5

2

50

0

0.5

1

t

1.5

2

3 2 1

Our method 1 Method in [13] r(t) ≡ r∗

0 −1 −2 0

0.5

1

1.5

Our method 2 Method in [13] r(t) ≡ r∗

2 0

−4

2

110

u(t)

0.5

−2

t Fig. 4. Reference signals rˆ(t), r(t), outputs y(t) and input voltages u(t) for 3 cases when x0 = 0.

−3

20 0

4

4

40

60

6

6

−4

80

40

0

20 0

rˆ(t)

40

40 20

100

60

y(t)

140

80

rˆ(t), r(t)

rˆ(t), r(t)

100

0.5

1

1.5

2

t Fig. 6. Reference signals rˆ(t), r(t), outputs y(t) and input voltages u(t) for 3 cases when x0 6= 0.

Example 2: Let us consider the position servo system considered in Example 1. We transform the plant into discrete-time by zero-order hold assuming a sampling period of T s = 10[ms]. To design K in (53), we solve the following optimization problem. ( min ε LMI 3 ε,Q,Y subject to (16) and (54) . Then, we have Ku = [−0.13979 −7.1326 · 10−3 −0.010588] and Kr = [−1.6553 −0.15141 −0.41473]. Next we solve the following optimization problem: LMI 4

2

t Fig. 5. Reference signals rˆ(t), r(t), outputs y(t) and input voltages u(t) for 3 cases when x0 6= 0.

0

(

min

σ

subject to

(23), (57) and (25) ,

σ,P˜

where ˜ ˜ G> e P Ge − P ≺ 0.

Theorem 3: Assume that there exist a real number ε and matrices Q and Y satisfying (16),   > Q •> 21 •31 −(Ae Q + Be Y ) Q 0   0, (54) −(SQ + RY ) 0 εI where •21

˜ ) and = −(Ae Q + Be Y ), •31 = −(SQ + RY     0 0 C (55) S =  0  , R = wu I 0  . 0 wr I 0

Let P = Q−1 , K = Y P , Ge = Ae + Be K. Then, we have 1 > G> e P Ge − P + (S + RK) (S + RK) ≺ 0. ε

(56)

(57)

In Figures 6 and 7, rˆ(t), r(t), y(t), and u(t) are shown. From Figures 4 – 7, we can conclude this simultaneous design method is very useful. V. C ONCLUDING R EMARKS We proposed two methods for servo systems design including on-line management of reference signals. We design controllers by solving LMI optimization problems, and we manage reference signals to prevent violation of constraints on the state and the control input. The performance of the designed servo system was demonstrated through an example. As concerns the example, the proposed method designing the inner and the outer loop simultaneously works much better than the control based on the previously proposed methods when we allow the disturbance of the initial conditions.

187

we have

100

rˆ(t)

100

90

 Q − ∗> 21

90

y(t)

rˆ(t), r(t)

110

80 70

80 70

60 60

50 40

0

0.5

1

t

1.5

2

50

0

0.5

1

t

1.5

2

3

u(t)

2 1

Our method 2 Method in [13] r(t) ≡ r∗

0 −1 −2 −3

0

0.5

1

1.5

∗> 31

  Q 0

−1   0 ∗21 εI ∗31

= Q − (AQ + BY )> Q−1 (AQ + BY ) 1 − (SQ + RY )> (SQ + RY ) ε = Q − Q(A> + Q−1 Y > B > )Q−1 (A + BY Q−1 )Q 1 − Q(S + RY Q−1 )> (S + RY Q−1 )Q  0. ε

(58)

Multiplying P = Q−1 from both side of (58), and using (12) and (18), we have P − (A + BKr )> P (A + BKr ) 1 − (S + RKr )> (S + RKr )  0 ε ε(P − G> P G)  (S + RKr )> (S + RKr ),

2

t Fig. 7. Reference signals rˆ(t), r(t), outputs y(t) and input voltages u(t) for 3 cases when x0 = 0.

(59)

which is (19). 2) Derivation of (20). From the above inequality, we compute εe> [k](P − G> P G)e[k] ≥ |(S + RKr )e[k]|22 .

R EFERENCES [1] [2] [3]

[4]

[5]

[6]

[7]

[8] [9]

[10] [11]

[12]

[13]

[14]

K. J. Astrom & L. Runqwist, ”Integrator windup and how to avoid it,” Proc. American Control Conf. pp. 1693-1698, 1898. M. V. Kothare & M. Morari, ”Multiplier theory for stability analysis of anti-windup control systems,” Automatica, vol. 35, pp. 917-928, 1999. P. Kapasouris, M. Athans & G. Stein, ”Design of feedback control systems for stable plants with saturating actuators,” Proc. IEEE Conf. Decision and Control, pp. 469-479, 1988. E. G. Gilbert & K. T. Tan, ”Linear system with state and control constraints: the theory and application of maximal output admissible sets,” IEEE Trans. on Automatic Control, Vol. 36, pp. 1008-1020, 1991. E. G. Gilbert & I. Kolmanovsky, ”Nonlinear tracking control in the presence of state and control constraints : a generalized reference governor,” Automatica, vol. 38, pp. 2063-2073, 2002. K. Hirata & M. Fujita, ”Analysis of conditions for non-violation of constraints on linear discrete-time systems with exogenous inputs,” Proc. IEEE Conf. Decision and Control, pp. 1477-1478, 1997. K. Hirata, M. Fujita, Set of admissible reference signals and control of systems with state and control constraints, Proc. IEEE Conf. Decision and Control, pp. 1427-1432, 1999. K. Kogiso, K. Hirata, A reference governor in a piecewise state affine function, Proc. IEEE Conf. Decision and Control, pp. 1747-1752, 2003. A. Bemporad, A. Casavola & E. Mosca, ”Nonlinear control of constrained linear systems via predictive reference management,” IEEE Trans. on Automatic Control, vol. 42, pp. 340-349, 1997. A. Bemporad & E. Mosca, ”Fulfilling hard constraints in uncertain linear systems by reference managing,” Automatica, vol. 34, pp. 451-461, 1998. A. Casavola, E. Mosca, & D. Angeli, ”Robust command governors for constrained linear systems,” IEEE Trans. on Automatic Control, Vol. 45, pp. 2071-2077, 2000. A. Bemporad, M. Morari, V. Dua and E. N. Pistikopoulos, ”The explicit linear quadratic regulator for constrained systems,” Automatica, Vol. 38, pp. 3-20, 2002. T. Sugie & H. Yamamoto, ”Reference management for closed loop systems with state and control constraints,” Proc. American Control Conf., pp. 1426-1431, 2001. S. Boyd, L. I. Ghaoui, E. Feron, & V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM Publisher, Philadelphia, PA, 1994.

A PPENDIX Proof of Theorem 1. 1). Derivation of (19). By applying Schur complement to (17),

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Since e[k + 1] = Ge[k], we get ε[e> [k]P e[k] − e> [k + 1]P e[k + 1]] ≥ |(S + RKr )e[k]|22 , and, hence, ˜ r [k; rˆ]}N ˆ) = J({˜ k=0 ; x0 , r

N X

k=0

≤ε

N X

|(S + RKr )e[k]|22

[e> [k]P e[k] − e> [k + 1]P e[k + 1]]

k=0

= ε[e> (0)P e(0) − e> (N + 1)P e(N + 1)] ≤ εe> (0)P e(0) ≤ ε|e0 |22 .

To obtain the last inequality, we used the condition that 0 ≺ P = Q−1 ≺ I. Proof of Theorem 2. 1) By (24), we immediately have (26). 2) Derivation of (27). By applying Schur complement to (25), we have σ  ([L]i + [D]i Kr )P˜ −1 [([L]i + [D]i Kr )]> = |[L]i + [D]i Kr )P˜ −1/2 |22 .

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˜ Suppose that e ∈ E(0; ρ(ˆ r )). By using (61) and (35), for each i ∈ Nc , we have |([L]i + [D]i Kr )e + [ˆ z (ˆ r )]i | ≤ |([L]i + [D]i Kr ) e| + |[ˆ z (ˆ r )]i | −1/2 ˜ 1/2 ˜ ≤ |([L]i + [D]i Kr )P ||P e| + δ(ˆ r) √ p < σ ρ(ˆ r ) + δ(ˆ r ) = 1 − δ(ˆ r ) + δ(ˆ r ) = 1,

which is (21), and, hence, we have (27). From (26), we immediately have (30), and by (30) and (27), we have (31). Finally, e[k; rˆ] → 0 as k → ∞ follows from (24).

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