Act-and-wait control concept for discrete-time systems with ... - CiteSeerX

IET Control Theory & Applications, 1(3) (2007) pp. 553–557.

Act-and-wait control concept for discrete-time systems with feedback delay Tam´as Insperger

∗

G´abor St´ep´an

Budapest University of Technology and Economics Department of Applied Mechanics Budapest, H-1521, Hungary e-mail: [email protected], [email protected]

Abstract Linear n-dimensional discrete-time systems are considered with feedback delay r at the m-dimensional input. These systems have n + mr poles. The act-andwait control concept is introduced: the feedback is periodically switched on and off during the control with period k. It is on for one step (act) and off for (k − 1) number of steps (wait). It is shown that if the act-and-wait period is larger than the time delay (i.e., k > r), then mr poles of the system are equal to zero, and the remaining poles can be placed arbitrarily if the system matrices satisfy certain controllability conditions.

1

Introduction

Time-delay in the feedback loop of control systems often leads to instability or poor performance of the system, therefore, pole placing of delayed systems is a crucial problem in modern control theory. Classical pole placement techniques of ordinary differential equations cannot be applied for delayed systems, since the number of poles to be controlled is much larger than the degrees of freedom in the controller (see, e.g., [1]-[3]). Although, complete pole placement is usually not possible for delayed systems, finding the optimal control parameters that results in the smallest spectral radius is still a difficult task. Periodic controllers can effectively be used in pole optimization problems. Several papers have been published on the possible stabilization effect of periodic feedback for both discrete-time systems (see, e.g., [4], [5]) and continuous-time systems (see, e.g., [6], [7]). An effective technique is the use of generalized sampled-data hold functions by sampling the output periodically and using special hold functions [8]. In this paper, a special case of periodic controllers is introduced for discrete-time systems that we call act-and-wait controller. By using this method, all the poles of the system can be assigned to zero if certain controllability conditions hold for the system and input matrices. ∗

Corresponding author.

1

2

Discrete-time systems with input delay and the autonomous controller

We consider the linear discrete-time systems with input delay in the form x(j + 1) = Ax(j) + Bu(j − r),

(1)

where x ∈ Rn is the state, u ∈ Rm is the input, A ∈ Rn×n , B ∈ Rn×m are constant and j ∈ Z. We assume that the feedback delay r ∈ Z+ is a given parameter of the system and cannot be tuned during the control design. There might be several reasons for such time delay, e.g., acquisition of response and excitation data, information transmission, on-line data processing, computation and application of control forces. Note that the general form of (1) includes the semi-discrete system ˜ ˜ ˙ x(t) = Ax(t) + Bu(j − r),

t ∈ [j, j + 1),

(2)

too. This system describes a continuous-time system under digital control like computer controlled machines (see, e.g., [9]). The solution of system (2) over the interval [j, j + 1) results in the discrete system (1) with
1 ˜ ˜ A ˜ A=e , B= eA(1−s) dsB. (3) 0

Consider the autonomous controller u(j) = Dx(j),

(4)

with D ∈ Rm×n . Equations (1) and (4) imply the discrete map zj+1 = Ψzj , with zj = (xT (j), uT (j − 1), . . . , uT (j − r))T ∈ Rn+mr . The coefficient matrix   A 0 ... 0 B D 0 . . . 0 0      Ψ = 0 I . . . 0 0  .. ..  .. . . . 0 0 ... I 0

(5)

(6)

is actually the (n + mr) × (n + mr) monodromy matrix of the system (see,e.g., [10]-[12]). The identity submatrices I below the diagonal in Ψ represent the delay effect in the feedback. Stability properties of the system are determined by the eigenvalues of matrix Ψ, which are also called characteristic multipliers or poles. The system is asymptotically stable if all the (n + mr) poles are in the open unit disc of the complex plane. The characteristic equation of Ψ can be given in the form  1 (7) det (µI − Ψ) = det µI − A − r BD = 0. µ In the rest of the paper, we make the assumption: Assumption 1. The pair (A, B) in system (1) is controllable.

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Assumption 1 implies that if the delay is zero (r = 0), then system (1) is controllable, consequently, its poles can be placed arbitrarily. If r ≥ 1 then pole placement is not possible, there are always at least r poles that cannot arbitrarily be placed as it is demonstrated below. Consider the single input case (m = 1, u ∈ R). Due to Assumption 1, the pair (A, B) can be written in the control canonical form     0 1 0    .  . . .. ..   .   (8) Ac =   , Bc =  .  .  0 0 1  −a1 −a2 . . . −an −1 Let D = (d1 , d2 , . . . , dn ). Equation (7) yields the characteristic equation n

µ +

n



ai + µ−r di µi−1 = 0.

(9)

i=1

This equation has r + n roots. Direct placement of n different poles is always possible using the n control parameters d1 , d2 , . . . , dn , since (9) depends linearly on the elements of D. However, the control over the location of the remaining r poles is lost and these poles may cause instability. In the multiple input case (m > 1), the characteristic equation has n + mr roots. Using the corresponding control canonical form of (A, B), it can be shown that r coefficients in the characteristic polynomial do not depend on any of the control parameters d1 , d2 , . . . , dn . Consequently, in this case, again, there are at least r eigenvalues that cannot arbitrarily be placed.

3

Act-and-wait controller

Consider the act-and-wait controller u(j) = g(j)Dx(j), where g(j) is the k-periodic switching function defined as

1 if j = hk, h ∈ Z g(j) = . 0 otherwise

(10)

(11)

In the rest of the paper, integer r will be called delay parameter and integer k will be called period parameter. While r is a given system parameter, k can be chosen during the control design. If k = 1 then g(j) ≡ 1. This case corresponds to the autonomous controller (4). If k ≥ 2 then g(j) alternates between one and zero. In the first discrete step, g(j) ≡ 1 and the control is active (act), while in the following (k − 1) number of steps, g(j) ≡ 0 and the control term is switched off (wait), then in the (k + 1)st step, the control is active again, etc. Theorem 1. If k ≥ r + 1 then system (1) with controller (10) has mr poles at zero. The remaining n poles can arbitrarily be placed if and only if the pair (Ak , Ak−r−1B) is controllable.

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Proof.

System (1) with controller (10) can be written in the discrete map form zj+1 = Gj zj ,

where, zj is the same vector as in (5), and  A g(j)D   Gj =  0  ..  . 0

the coefficient matrix is  0 ... 0 B 0 . . . 0 0  I . . . 0 0 . ..  .. . . 0 ... I 0

(12)

(13)

Due to (11), Gj is also k-periodic: Gj = Gj+k

with G0 = G1 = . . . = Gk−1 .

(14)

The coupling of the solutions over k subsequent steps (i.e., over a full act-and-wait period) results in the map j ∈ Z, (15) zj+k = Φj zj , where Φj = Gj+k−1Gj+k−2 . . . Gj

(16)

is the monodromy matrix over period k associated with the initial instant j. Stability properties of the system are determined by the eigenvalues of Φj . Note that Φj = Φj+k due to the k-periodicity of Gj . Consequently, k number of different monodromy matrices can be given, namely, Φ0 , Φ1 , . . . , Φk−1 , which are similar. Thus, consider the monodromy matrix Φ0 associated with j = 0, only. It can be seen that (17) Φ0 = Gk−1 Gk−2 . . . G0 = G1k−1 G0 due to (14). In the following, the structure of the monodromy matrix Φ0 is presented for different time period parameters k. If the period parameter is k = 1, then the monodromy matrix is Φ0 = G0 that is just identical to Ψ in (6). This case corresponds to the autonomous controller (4). If k = 2, then the monodromy matrix reads   A2 0 . . . 0 B AB  0 0 ... 0 0 0    D 0 ... 0 0 0    Φ0 = G1 G0 =  0 (18) . I . . . 0 0 0     .. .. .. ..  . . . .  0 0 ... I 0 0 Note that the second row in Φ0 is zero. As k is increased further by one, the rows of the submatrix D and the identity matrices move downwards by one, and another zero row appears instead. Increasing k step-by-step, the rank of the system and consequently the number of nonzero poles decreases this way. If k = r + 1, that is, if the time period is just larger than the time delay, then all the submatrices except the ones in the first row become zeros. For the general case k ≥ r + 1,

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the monodromy matrix reads  M 0  Φ0 = G1k−1 G0 =  ..  . 0 where

Ak−r B 0 .. .

Ak−r+1B 0 .. .

... ...

0

0

...

 Ar−1 B 0   , ..  .

(19)

0

M = Ak + Ak−r−1 BD.

(20)

Clearly, mr eigenvalues of Φ0 are zeros, while the remaining n eigenvalues are determined by M. Since matrix M corresponds to a system with pair (Ak , Ak−r−1B), its poles can be placed arbitrarily if and only if the pair (Ak , Ak−r−1B) is controllable. Remark 1. M can also be presented as the transition matrix of system (1) corresponding to r subsequent steps in the open loop system, followed by a close loop step using (4), then followed again by (k − r − 1) open loop steps: x(r) = Ar x(0), x(r + 1) = Ax(r) + BDx(r − r) = (Ar+1 + BD)x(0) x(k) = Ak−r−1 (Ar+1 + BD)x(0) = Mx(0).

(21) (22) (23)

Corollary 1. If the pair (Ak , Ak−r−1B) is controllable, then all the eigenvalues of M can be assigned to zero. Since in this case Mn = 0, the system is deadbeat within period of length nk. This corollary is a straightforward consequence of Theorem 1. Since the deadbeat period is proportional to the period parameter k, the optimal choice is k = r + 1 that results in the smallest deadbeat period.

4

Case study

As a case study, the semi-discrete system (2) subjected to controller (10) will be investigated with    T 0 1 0 −d1 ˜ ˜ A= , B= , D= . (24) a 0 1 −d2 This system with a > 0 corresponds to the digital control of the inverted pendulum (see [9]) where the feedback delay is r-multiple of the sampling period. If a > 0 then the open loop system is unstable, if a < 0 then it is stable. The case a = 0 gives a double integrator. Here, we will concentrate on the case a ≥ 0. Using Eqs. (3), the semi-discrete system (2) with (24) can be transformed into the discrete map (1) with the corresponding matrices  √   √ √ 1 √ 2 (1 − cosh ( a)) − √1 sinh ( a) cosh ( a) a a √ √ √ A= √ , B= (25) √1 sinh ( a) a sinh ( a) cosh ( a) a

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if a > 0 and with the continuous extension  1 1 A= , 0 1

 B=

1/2 1

(26)

if a = 0. Note that rank[B, AB] = 2, that is, the pair (A, B) is controllable. Consequently, if the delay is zero (r = 0), then arbitrary pole placement is always possible. If k = 1 in controller (10) (that corresponds to the autonomous controller (4)) and r ≥ 1, then the system is described by the monodromy matrix Ψ given in (6). Since in this case study, n = 2 and m = 1, the system has n + mr = 2 + r poles that cannot arbitrarily be placed using the control parameters (d1 , d2). Moreover, it can be shown that the system cannot even be stabilized if 2    r(r + 1) + 1 + 2r(r + 1) + 1 . (27) a > acrit = ln r(r + 1) For r = 1, this formula reduces to



acrit = ln2

√  3+ 5 2

(28)

that was derived for the digitally controlled inverted pendulum in [9]. For the further analyses, we introduce the decay index. Definition 1. The decay index is ρ = |µ1 |1/k , where µ1 is the critical eigenvalue, i.e., |µ1| ≥ |µi|, i = 2, 3, . . . , n + rm. Decay index ρ characterizes the average error decay over a single discrete step while µ1 characterizes the average error decay over the principal period k: |zj+1| ≤ ρ|zj |

and

|zj+k | ≤ |µ1 ||zj |.

(29)

The decay index is a measure that will be used to compare systems with different periods k of the act-and-wait controller for a given delay r of the system. In the next subsections, the delay parameter is fixed to r = 5, consequently the system has n + mr = 7 poles. In this case, formula (27) gives acrit = 0.0663. The act-and-wait controller (10) with k = r + 1 = 6 will be compared to the case k = 1 (to the autonomous controller (4)). Two cases will be investigated: a = 0 and a = 0.1.

4.1

Case r = 5, a = 0

If the autonomous controller (4) is used (k = 1) then the system can be stabilized , since a < acrit = 0.0663. However, arbitrary placement of the 7 poles is not possible using the two control parameters (d1 , d2 ). The optimal control parameters resulting minimal decay index can be found numerically: (d1 , d2 ) = (0.0026, 0.0840). The corresponding decay index is ρ = 0.8987. If the act-and-wait controller is used with period k = r+1 = 6 then 5 poles are directly assigned to zero as it is shown in Theorem 1. The remaining 2 poles are determined by  1 − 12 d1 6 − 12 d2 6 M = A + BD = . (30) −d1 1 − d2 6

k=1

0.01

0

1 0.9 5

0

0.02

1

x1

x1

(d1 , d2 ) = (0.0026, 0.0840) ρ = 0.8987

10

20

0.6

d1

0.5 0 0

0.4

0.2

d1 1

1.1

1

0

4

1.0

1

0.95

d2

1

1.05

1

2

1.0 5

8

0.9

1.02

0

1. 02

0.98

d2

0.1

1.1.05 1

3

1

0.2

k=6

1

0.3

4

1.04

1.02

30

40

0.5 0 0

50

t

(d1 , d2 ) = (0.1667, 1.9167) ρ=0

10

20

30

40

50

t

Figure 1: Stability diagrams (top) and optimal time histories (bottom) for system (2) with (10), (24), a = 0, r = 5. Left and right panels correspond to k = 1 (autonomous controller) and k = 6 (act-and-wait controller), respectively. Note the different scales of the top plots. It can be seen that rank[B, A6 B] = 2, that is, the pair (A6 , B) is controllable. Thus, according to Corollary 1, the eigenvalues of M can also be assigned to zero. Indeed, it can be shown that the two remaining eigenvalues are zeros if d1 = 1/6 ≈ 0.1667 and d2 = 23/12 ≈ 1.9167. Using these parameters with k = 6, all the 7 poles are placed to the origin, and the control is deadbeat in period of length nk = 12. Contour plots of the decay index ρ in the plane (d1 , d2 ) are shown in Fig. 1. Stability boundaries (ρ = 1) are denoted by thick lines. Time histories corresponding to the optimal control parameters are also presented (here, x1 denotes the first element of vector x). The instants of acting are denoted by circles, while instants of waiting steps are denoted by dots. In the case k = 1 (left panels), the control is continuously active, and the convergence is relatively slow. In the case k = 6 (right panels), the system actually converges to zero within period of length 12. The controller is switched on at instant t = 5 forcing the system to zero (we are acting), then it is switched off (we are waiting) until t = 11, when it is switched on again, and this completes the deadbeat convergence.

4.2

Case r = 5, a = 0.1

In this case, a > acrit = 0.0663, consequently, the system cannot be stabilized with the autonomous controller (4) (k = 1). The “optimal” control parameters (d1 , d2 ) = (0.0996, 0.5700) result in the decay index ρ = 1.0448 > 1 that corresponds to an unstable behavior. If the act-and-wait controller is applied with k = r+1 = 6, then, according to Theorem

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0

1 .3

0.5

0

1

1. 2

1.4

1.3

1.2 1.3

0

5 1.1

d2

d2

1.2

0.5

0

2

d1

0 0

3

10

20

30

40

x1

x1

20

4

d1

(d1 , d2 ) = (0.0996, 0.5700) ρ = 1.045

40

1

1.2

1.2

1. 3

1.4

1.3

1.3

1.5 1

1.3 1.1

k=6

1. 1.1 2

10

k=1

1

2

2 1 0 0

50

t

(d1 , d2) = (1.835, 5.796) ρ=0

10

20

30

40

50

t

Figure 2: Stability diagrams (top) and optimal time histories (bottom) for system (2) with (10), (24), a = 0.1, r = 5. Left and right panels correspond to k = 1 (autonomous controller) and k = 6 (act-and-wait controller), respectively. Note the different scales of the plots. 1, all the poles can be placed to the origin. In this case, matrix M reads  −0.5042d1 + 3.409 10.3064 − 0.5042d2 M= . 1.0306 − 1.0168d1 −1.0168d2 + 3.4091

(31)

It can be shown that both eigenvalues of M are zeros for the control parameters (d1 , d2 ) = (1.835, 5.796). Consequently, these parameters with k = 6 result in a deadbeat control with deadbeat period nk = 12. The corresponding contour plots of the decay index and optimal time histories are shown in Fig. 2. Left panels show that the system with k = 1 is always unstable, even in the optimal case. In the case k = 6, there is a stable parameter domain bounded by thick lines. Time history corresponding to the optimal parameters shows the deadbeat convergence of the system with deadbeat period 12. It can be seen that the system growths exponentially in the first wait period [0, 5), but after the first acting at t = 5, this tendency is reversed, and the deadbeat convergence is completed in the next act-and-wait period.

4.3

Summary of the case studies

Besides the capability of attaining deadbeat control, some other features of the act-andwait control technique should also be mentioned. Both case studies showed that the optimal control parameters for the act-and-wait system are much larger than those of the autonomous system. Note, however, that the large control gains of the act-and-wait

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controller are used every k th step only, while the small control gains of the autonomous controller are used continuously. This is the point of the method: instead of using small control parameters continuously according to the autonomous control technique, large control gains are used in the acting steps and zero gains in the waiting periods. Clearly, its application could be limited by the saturation of actuators in certain cases. Although, dead beat control can be achieved by the act-and-wait technique, it should be mentioned that this pole assignment process is very sensitive to parameter variations. In the case a = 0, k = 6, 10% perturbation of the control parameters around the optimal deadbeat values may result in an increase of the decay ratio up to ρ = 0.9073. This is a significant increase, but it is still better than the autonomous case with k = 1 where a 10% perturbation of the optimal control parameters results in ρ = 0.9570. In the case a = 0.1, the robustness of the act-and-wait controller is even weaker, but in this case the autonomous controller cannot stabilize the system at all.

5

Conclusions

Linear n-dimensional discrete-time control systems with m-dimensional delayed input and delay r were considered. The system is described by n + mr poles that cannot arbitrarily be placed using an autonomous controller with mn control parameters. As a special case of periodic controllers, the act-and-wait controller was introduced: the controller is periodically switched off and on with period k. It was shown that if the period of being switched off is just larger than the time delay (i.e., k ≥ r + 1), then rm poles of the system are assigned directly to zero, and the remaining n poles can also be placed arbitrarily if the pair (Ak , Ak−r−1B) is controllable. As a case study, a 2-dimensional semi-discrete system with delayed feedback was investigated. It was shown that all the poles can be assigned to zero using the act-and-wait method resulting deadbeat control even in the case when the original autonomous controller cannot stabilize the system. The main conclusion of this paper is that the act-and-wait concept provides an alternative for control systems with feedback delays. The traditional way is the continuous use of small control gains according to the autonomous controller (4), when a cautious, slow feedback is applied with small gains resulting slow convergence, if it can stabilize the system at all (see cases k = 1 in Figs. 1 and 2). The proposed alternative way is the act-and-wait control concept, when large control gains are used in the acting steps and zero gains are used in the waiting periods according to cases k = 6 in Figs. 1 and 2. Several (actually, infinitely many) periodic functions could be chosen as time-periodic controllers. The main idea behind choosing the one that involves waiting periods just larger than the feedback delay is that it kills the memory effect by waiting for the system answer induced by the previous action. This explains our choice in (11). Although it might seem unnatural not to actuate during the wait period, act-and-wait concept is still a natural control logic for time-delayed systems. This is the way, for example, that one would adjust the shower temperature considering the delay between the controller (tap) and the sensed output (skin).

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Acknowledgement This research was supported in part by the J´anos Bolyai Research Scholarship of the Hungarian Academy of Sciences, and by the Hungarian National Science Foundation under grant no. OTKA F047318 and OTKA T043368.

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