On the dimension reduction of systems with ... - Semantic Scholar

IMA Journal of Mathematical Control and Information Advance Access published October 5, 2010 IMA Journal of Mathematical Control and Information Page 1 of 17 doi:10.1093/imamci/dnq020

On the dimension reduction of systems with feedback delay by act-and-wait control TAMAS I NSPERGER∗ AND G ABOR S TEPAN Department of Applied Mechanics, Budapest University of Technology and Economics, H-1521 Budapest, Hungary ∗ Corresponding author: [email protected] [Received on 11 February 2010; revised on 14 June 2010; accepted on 31 August 2010]

Keywords: feedback delay; periodic control; dimension reduction.

1. Introduction Stabilization of control systems in the presence of feedback delay is a crucial task in industrial control design. The main problem with such systems is that the corresponding characteristic equation has infinitely many roots that all should be placed to the left half of the complex plane (or, alternatively, inside the unit circle for sampled/periodic systems) in order to assure stability. The calculation of the critical (rightmost) poles for delayed systems is not an issue nowadays since there exist several powerful numerical techniques both for time-invariant (Stepan, 1989; Olgac & Sipahi, 2002; Breda et al., 2004; Yi et al., 2010) and for time-periodic systems (Butcher et al., 2004; Szalai et al., 2006; Insperger et al., 2008; Lampe & Rosenwasser, 2010; Mann & Patel, 2010). The difficulty of the problem lies rather in the fact that while the number of poles to be controlled is infinite, the number of control parameters is finite. One technique to deal with the problem is to place the rightmost poles only while monitoring the other uncontrolled poles with large real part (see, Michiels et al., 2002, 2010). This technique requires the numerical calculation of some relevant poles for different control parameters. In this case, the infinitely many poles are controlled by finite number of control parameters. An alternative approach is to increase the number of control parameters. One way to do this is the application of distributed delays in the controller, where the weight function of the distributed delay serves as a kind of infinite-dimensional vector of control gains. Another way is the application of timeperiodic controllers, where the time dependency of the gains can be considered as a set of infinitely many control parameters. A special case for distributed delay applications is when the feedback is generated based on a prediction of the state. This method is called finite spectrum assignment since the resulted closed-loop system c The author 2010. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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Dimension reduction properties of the act-and-wait controller for systems with feedback delay are analysed. The point of the act-and-wait concept is that the feedback is switched on and off periodically. Earlier works have shown that if the switch-off (waiting) period is longer than the feedback delay, then the system can be described by an n × n monodromy matrix with n being the dimension of the delay-free system. In this study, it is shown that for certain combinations of the waiting and the acting (or switch on) periods, the system can be still be transformed to a finite, kn-dimensional system with k > 2 even if the waiting period is shorter than the delay. It is shown that the control technique can be a candidate as a solution to the Brockett problem posed for systems with feedback delay.

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T. INSPERGER AND G. STEPAN

2. The Brockett problem for systems with delayed feedback Stabilization by means of time-periodic feedback gains in non-delayed systems has been presented by Brockett (1998) as one of the challenging open problems in control theory. Consider the linear system x(t) ˙ = Ax(t) + Bu(t), y(t) = C x(t),

(2.1) (2.2)

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has only finite number of poles that can be assigned arbitrarily provided that there is no uncertainty in the system and control parameters (for details, see Manitius & Olbrot, 1979; Wang et al., 1998). If the open-loop system is unstable, then prediction by means of the solution of the differential equation cannot stabilize the system since it involves an unstable pole-zero cancellation even for high accuracy solutions (Engelborghs et al., 2001). The conditions for stabilizing via distributed delays that approximates the solution of the system was analysed in Mondi´e et al. (2002), and a safe implementation of finite spectrum assignment for unstable systems was provided in Mondi´e & Michiels (2003). Stabilization by means of periodic control gains is a field of intensive research even for delay-free systems (see, for instance, Brockett, 1998; Leonov, 2002; Moreau & Aeyels, 2004; Allwright et al., 2005). The combined effect of feedback delay and time periodicity may result in intricate behaviour of the system (Just, 2000). The analyses of time-periodic time-delayed systems is of great importance since these systems typically occur during the stabilization around a periodic orbit of a non-linear system with feedback delay. A paradigm for time-periodic time-delayed systems is the delayed Mathieu equation for which the stability diagram was constructed in Insperger & Stepan (2002). Sampling can be considered also as a special case of periodic controls since it corresponds to a periodic variation of the feedback delay in time (Rosenwasser & Lampe, 2000). It is known that zero order hold sampled-data systems with feedback delay can be transformed to finite-dimensional sys˚ om & Wittenmark, 1984), while other generalized sampled tems by state augmentation (Kuo, 1977; Astr¨ data hold function can also effectively be used to improve control performance (Kabamba, 1987). A special case of generalized hold discrete-time control is the intermittent predictive control, where the sequence of open-loop trajectories is punctuated by intermittent feedback. This concept was introduced in Ronco et al. (1999) and further developed in Gawthrop & Wang (2007, 2009). Intermittent predictive control is also a candidate to human motion balancing (Gawthrop et al., 2009; Loram et al., 2009; Asai et al., 2009). The current study focuses on the act-and-wait control concept that is a special case of periodic controllers: the feedback term is switched off and on periodically. The technique was introduced by Insperger (2006) and Stepan & Insperger (2006) for continuous-time systems and by Insperger & Stepan (2007) for discrete-time systems. It was shown that if the waiting period is longer than the feedback delay, then the system can be transformed to a discrete map of finite dimension, giving this way a finite spectrum assignment problem. As it was shown in Gawthrop (2010), the act-and-wait controller is related to the intermittent controller in the sense that both techniques have a generalized hold interpretation. Similar to intermittent control, the act-and-wait control may also be relevant to human control systems (Milton et al., 2009b,c; Asai et al., 2009). In this paper, the dimension reduction properties of the act-and-wait controller is analysed. As it was shown earlier, if the waiting period is longer than the feedback delay, then the infinite-dimensional system can be transformed to an n-dimensional one, with n being the order of the delay-free dynamics. Here, it is shown that finite-dimensional discrete maps can be constructed for certain acting and waiting period combinations even if the waiting period is shorter than the feedback delay.

DIMENSION REDUCTION BY THE ACT-AND-WAIT CONTROLLER

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with state x(t) ∈ Rn , input u(t) ∈ Rm and output y(t) ∈ Rl . The Brockett problem can be posed as in the following problem. P ROBLEM 2.1 For given matrices A, B and C, under what circumstances does there exist a time-varying controller u(t) = G(t)y(t),

(2.3)

such that system (2.1)–(2.2) is asymptotically stable?

u(t) = Dy(t − τ ),

(2.4)

where τ is the time delay of the feedback loop. It is assumed that the delay is a fixed parameter of the control system and cannot be eliminated or tuned during the control design. There are several sources of such time delays, e.g., acquisition of response and excitation data, information transmission, on-line data processing, computation and application of control forces. A possible adaptation of the Brockett problem to systems with feedback delay can be composed as in the following problem. P ROBLEM 2.2 For given matrices A, B and C and for given feedback delay τ , under what circumstances does there exist a time-varying controller u(t) = G(t)y(t − τ ),

(2.5)

such that the system is asymptotically stable? Here, we restrict our analysis to the class of periodic controllers, i.e., we assume that G(t + T ) = G(t) with T being the period of the controller. In this case, system (2.1)–(2.2) and controller (2.5) implies the time-periodic delay-differential equation (DDE) x(t) ˙ = Ax(t) + BG(t)C x(t − τ ).

(2.6)

The general solution of (2.6) for the initial function x0 can be formulated as xt = U (t)x0 ,

(2.7)

where U(t) is the solution operator of the system, and the function xt is defined by the shift x t (s) = x(t + s),

s ∈ [−τ, 0].

(2.8)

Stability properties are determined by the monodromy operator U (T ). The non-zero elements of the spectrum of U (T ) are called characteristic multipliers (or poles) also defined by Ker(μI − U (T )) 6= {0},

μ 6= 0,

(2.9)

with I denoting the identity operator. The system is asymptotically stable if all the infinitely many characteristic multipliers lie in the open unit disc of the complex plane. Similar to the time-independent

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Partial results on the Brockett problem have been presented by Leonov (2002) and Allwright et al. (2005) for piecewise constant control gains and by Moreau & Aeyels (2004) for sinusoidal control gains. The solution to the problem for a wide class of systems–without delay—was presented by Boikov (2005). Consider now the delayed feedback controller

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system (2.1)–(2.2) with (2.4), stabilization of system (2.6) requires the control over infinitely many poles, but now the control gains are function of time over the period [0, T ] that corresponds to a kind of infinite-dimensional vector of control gains. The question is how can all the infinitely many poles of the system be placed inside the unit circle by tuning the control gain function G(t) over [0, T ]. One possible approach to the problem is to decrease the dimension of the system using the act-and-wait control technique. 3. Dimension reduction by the act-and-wait controller

where Γ (t): [tw , T ] → Rm×l is an integrable matrix function. Here, ta and tw are the length of the acting and the waiting periods, respectively, and ta + tw = T is the length of one act-and-wait period. In Insperger (2006), it was shown that if tw > τ , then the system can be described by an n × n monodromy matrix, consequently, only n poles determine the stability instead of infinitely many ones. This feature is useful during the stabilization of systems since the control function Γ (t) should be chosen such that these n poles lie within the unit circle of the complex plane (see the examples in Insperger, 2006 or in Stepan & Insperger, 2006). In this paper, it is shown that the dimension of the monodromy operator is finite for certain parameter combinations even if tw < τ . The main results are composed in the following theorem. T HEOREM 3.1 The dimension of system (2.6) with (3.1) is equal to Case 1: n

if tw > τ ,

1 k+1 τ − ta , k = 1, 2, . . ., if tw < τ , ta 6 kτ and tw > 2 2 1 k 1 k+1 τ− ta and tw 6 τ − ta , k = 1, 2, . . .. Case 3: (k + 1)n if tw > k+1 k+1 k k The geometric representation of Theorem 3.1 is given in Fig. 1. The dimension of the system for the different cases are presented in the corresponding triangles. The figures denoted by NA (not analysed) are not analysed here. The proof for this theorem is based on the construction of a finite-dimensional monodromy matrix for the system. The three cases are considered separately in the subsequent sections. Case 2: (k + 1)n

4. Proof for Case 1 In this section, it is shown that if tw > τ , then system (2.6) with (3.1) can be associated with an n-dimensional monodromy matrix. Consider the general case (k −1)τ < ta 6 kτ , where k is an arbitrary positive integer. According to the method of steps for DDEs, the solution can be constructed piecewise over the succeeding intervals [0, tw ], [tw , tw +τ ], . . . , [tw +(k −1)τ, T ] as follows (see Fig. 2 for k = 3). Since the delayed term is switched off during the waiting period, the first section of the solution can be given as x (1) (t) = Φ (1) (t)x(0),

0 6 t 6 tw ,

with Φ (1) (t) = e At . Here, superscript (1) refers to the number of the segment of the solution.

(4.1)

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The act-and-wait controller is a special case of the time-varying controller (2.5) with the T -periodic matrix ( 0 if 0 6 t mod T < tw , G(t) = (3.1) Γ (t) if tw 6 t mod T < tw + ta = T,

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FIG. 2. Piecewise solution segments of (2.6) with (3.1) for tw > τ and 2τ < ta 6 3τ (k = 3).

Now, we utilize the facts that the waiting period is larger than (or equal to) the time delay, and the solution over 0 6 t 6 tw is already given by (4.1). Thus, in the interval tw < t 6 tw + τ , (2.6) with (3.1) can be written as x(t) ˙ = Ax(t) + BΓ (t)CΦ (1) (t − τ )x(0),

tw < t 6 tw + τ.

(4.2)

The solution for the initial condition x(tw ) = x (1) (tw ) = Φ (1) (tw )x(0) can be given in the form x (2) (t) = Φ (2) (t)x(0), with Φ (2) (t) = e At +

Z

t tw

tw < t 6 tw + τ,

e A(t−s) BΓ (s)CΦ (1) (s − τ )ds.

(4.3)

(4.4)

Provided that the solution in the hth interval is given as x (h) (t) = Φ (h) (t)x(0),

tw + (h − 2)τ < t 6 tw + (h − 1)τ,

(4.5)

the solution in the next interval can be given by the recursive form x (h+1) (t) = Φ (h+1) (t)x(0),

tw + (h − 1)τ < t 6 tw + hτ,

(4.6)

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FIG. 1. Chart on the dimension of system (2.6) with (3.1). Domains denoted by NA are not analysed here.

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T. INSPERGER AND G. STEPAN

with Φ (h+1) (t) = e At +

Z

t tw −(h−1)τ

e A(t−s) BΓ (s)CΦ (h) (s − τ )ds.

(4.7)

Finally, the solution at t = T is given as x(T ) = x (k+1) (T ) = Φ (k+1) (T )x(0).

(4.8)

As it can be seen, the state x(T ) depends only on the initial state x(0), and it does not depend on the initial function x0 . Matrix Φ (k+1) (T ) therefore serves as an n × n monodromy matrix for the system that proofs Theorem 3.1 for Case 1. As a consequence, the infinite-dimensional monodromy mapping (4.9)

can be written in the form 

  (k+1) Φ (T ) x(T ) = x˜ T f˜k+1

O O



 x(0) , x˜0

(4.10)

where function x˜t is defined by the shift x˜t (s) = x(t + s),

s ∈ [−τ, 0).

(4.11)

Note that s = 0 is excluded here as opposed to (2.8). In (4.10), O denotes the zero functional, O denotes the zero operator and f˜k+1 is the function ( (k) Φ (s + T ) if − τ 6 s < (k − 1)τ − ta , ˜ f k+1 (s) = (4.12) (k+1) (s + T ) if (k − 1)τ − ta 6 s < 0. Φ Equation (4.10) shows that function x T can be determined using only the initial state x(0) and does not depend on the initial function x˜0 . The stability properties are determined by the n × n matrix Φ (k+1) (T ). The system is asymptotically stable if all the eigenvalues of Φ (k+1) (T ) are in modulus less than 1. Therefore, in this case, stability analysis is equivalent to the calculation of the eigenvalues of the n × n matrix Φ (k+1) (T ). For instance, if k = 1, i.e., 0 < ta 6 τ , then Φ (2) (T ) = e AT +

Z

T

e A(T −s) BΓ (s)C e A(s−τ ) ds.

If k = 2, i.e., τ < ta 6 2τ , then Φ

(3)

(T ) = e Z

AT

+

s1 −τ tw

Z

T

e tw

A(T −s)

(4.13)

tw

BΓ (s)C e

A(s−τ )

ds +

Z

T tw +τ

e A(s1 −s2 −τ ) BΓ (s2 )C e A(s2 −τ ) ds2 ds1 .

e A(T −s1 ) BΓ (s1 )C (4.14)

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x T = U (T )x0

DIMENSION REDUCTION BY THE ACT-AND-WAIT CONTROLLER

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5. Proof for Case 2 In this case, the condition for the waiting and the acting periods is ta 6 kτ

tw < τ,

tw >

and

k+1 1 τ − ta , 2 2

(5.1)

Ik (t, ϑ) :=

Z

tw −τ ϑ

tw −τ

e A(ϑ−s) BΓ (s + kτ )C Ik−1 (t, s)ds

Jk (t) := Ik (t, 0)

for k > 2,

for k ∈ Z+ .

(5.3) (5.4)

Note that Ik (t, ϑ) ∈ Rn and Jk (t) ∈ Rn . It will be shown that the solution can be constructed piecewise over the succeeding intervals [0, tw ], [tw , τ ], [τ, tw + τ ], [tw + τ, 2τ ], . . . using the above weighted integrals. The cases k = 1, k = 2 and k > 3 are considered separately in the subsequent subsections. 5.1 Case 2 with k = 1 In this particular case, the condition for the waiting and the acting periods is ta 6 τ

tw < τ,

and

1 t w > τ − ta . 2

(5.5)

The sketch of the piecewise solution of the system is shown in Fig. 3, panel (a). Since the delayed term is switched off during the waiting period, the first section of the solution can be given as x (1) (t) = e At x(0),

0 < t 6 tw .

(5.6)

In the interval tw < t 6 τ , (2.6) with (3.1) reads

x(t) ˙ = Ax(t) + BΓ (t)C x0 (t − τ ),

tw < t 6 τ

with the initial condition x(tw ) = x (1) (tw ) = e Atw x(0). The corresponding solution segment is Z t−τ (2) At x (t) = e x(0) + e A(t−s−τ ) BΓ (s + τ )C x 0 (s)ds, tw < t 6 τ. tw −τ

The state at t = τ is given as x(τ ) = x

(2)

(τ ) = e

Aτ

x(0) +

Z |

where J1 (0) is defined according to (5.4) and (5.2).

0 tw −τ

e−As BΓ (s + τ )C x 0 (s)ds , } {z = J1 (0)

(5.7)

(5.8)

(5.9)

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with k ∈ Z+ . Here, the method of steps presented in the Section 4 cannot be applied in the same way since the waiting period tw is shorter than the time delay τ . It can be shown, however, that the system can still be transformed into a finite-dimensional discrete map if the state is augmented by some weighted integrals of the initial function x0 . To start with, we define the functions Z ϑ I1 (t, ϑ) := e A(ϑ−s) BΓ (s + τ )C x t (s)ds for k = 1, (5.2)

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T. INSPERGER AND G. STEPAN

In the interval τ < t 6 T = tw + ta , (2.6) with (3.1) reads

x(t) ˙ = Ax(t) + BΓ (t)C x (1) (t − τ ),

with the initial condition x(τ ) =

x (2) (τ ) (1)

=

e Aτ x(0) +

(1)

(1) Ψ11 (t) = e At + (1)

Z

t

(5.10)

J1 (0). The third solution segment is

x (3) (t) = Ψ11 (t)x(0) + Ψ12 (t)J1 (0), with

τ < t 6 T, τ < t 6 T,

e A(t−s) BΓ (s)C e A(s−τ ) ds,

(5.11)

(5.12)

τ

Ψ12 (t) = e A(t−τ ) .

(5.13)

The state after one act-and-wait period can be given by setting t = T : (1) (1) x(T ) = Ψ11 (T )x(0) + Ψ12 (T )J1 (0).

(5.14)

Here, x(T ) is determined as a linear combination of x(0) and J1 (0). In order to obtain a discrete map, the integral J1 (T ) should also be given as a linear combination of x(0) and J1 (0). The evaluation of J1 (T ) requires the piecewise construction of the solution function x T :  (1)   x (s + T ) if − τ 6 s 6 tw − T, (5.15) x T (s) = x (2) (s + T ) if tw − T < s 6 τ − T,   (3) x (s + T ) if τ − T < s 6 0.

Note that function x (2) depends on the initial function x0 , while x (1) and x (3) depend only on the initial state x(0) and on the weighted integral J1 (0). Condition (5.5) implies that tw −τ > τ −T , thus, the domain of the integral in J1 (T ) (i.e., [tw −τ, 0], see (5.2) and (5.4)) is entirely within the domain of the solution segment x (3) . Consequently, this integral can be given as a linear combination of x(0) and J1 (0): Z 0 (1) (1) J1 (T ) = e−As BΓ (s + τ )C x (3) (s)ds = Ψ21 (T )x(0) + Ψ22 (T )J1 (0), (5.16) tw −τ

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FIG. 3. Graphs of the solution of (2.6) with (3.1) for different act-and-wait periods T . Panel (a): Case 2 with k = 1 (tw < τ , ta 6 τ and tw > τ − ta /2). Panel (b): Case 2 with k = 2 (tw < τ , ta 6 2τ and tw > (3τ − ta )/2).

DIMENSION REDUCTION BY THE ACT-AND-WAIT CONTROLLER

where (1)

Ψ21 (T ) = (1) Ψ22 (T )

=

Z Z

0 tw −τ 0 tw −τ

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e− As BΓ (s + τ )CΨ11 (s + T )ds,

(5.17)

e− As BΓ (s + τ )CΨ12 (s + T )ds.

(5.18)

(1)

(1)

(5.19)

Here, Ψ (1) (T ) serves as a 2n × 2n monodromy matrix. The stability of the system is determined by the 2n eigenvalues of Ψ (1) (T ). 5.2

Case k = 2

The condition for the waiting and the acting periods is tw < τ,

ta 6 2τ

and tw >

3 1 τ − ta . 2 2

(5.20)

The method of steps can be applied in the same way as it was done for the case k = 1. The sketch of the piecewise solution of the system is shown in Fig. 3, panel (b). In this case, the construction of the solution segments results in the 3n-dimensional monodromy map     x(0) x(T )     (2) (5.21)  J1 (T ) = Ψ (T )  J1 (0) , J2 (T ) J2 (0) where J1 (t) and J2 (t) are defined by (5.2), (5.3) and (5.4). The elements of matrix Ψ (2) (T ) (not detailed here) can be determined in a straightforward way by piecewise integration, similar to the case k = 1. Here, matrix Ψ (2) (T ) serves as a 3n × 3n monodromy matrix. The stability of the system is determined by the 3n eigenvalues of Ψ (2) (T ). 5.3

Case k > 3

The condition for the waiting and the acting periods is defined by (5.1). The same algorithm can be used as for the cases k = 1 and k = 2, and multiple application of the method of steps results in the (k + 1)n-dimensional monodromy map     x(0) x(T )  J (0)  J (T )  1    1  .  = Ψ (k) (T )  .  , (5.22)  .   .   .   .  Jk (T ) Jk (0)

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This way, we have constructed a 2n-dimensional map ! ! ! (1) (1) Ψ11 (T ) Ψ12 (T ) x(T ) x(0) = . (1) (1) J1 (T ) J1 (0) Ψ21 (T ) Ψ22 (T ) } | {z (1) =: Ψ (T )

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T. INSPERGER AND G. STEPAN

where Jk (t) is defined by (5.3) and (5.4). Here, Ψ (k) serves as an (k + 1)n × (k + 1)n monodromy matrix. Consequently, the dimension of the system is equal to (k + 1)n. 6. Proof for Case 3 In this case, the condition for the waiting and the acting periods are tw 6

1 k+1 τ− ta k k

and tw >

1 k τ− ta . k+1 k+1

(6.1)

The proof is based on the construction of the solution segments according to the method of steps (see Fig. 4). Let us first define the functions Z

T −τ tw −τ

e A(T −τ −s) BΓ (s + τ )C xt (s + (k − 1)T )ds,

k = 1, 2, . . . .

(6.2)

Note that K k (t) ∈ Rn . It will be shown that the solution can be constructed piecewise over the succeeding intervals [0, tw ] and [tw , T ] using the weighted integrals K k (t). The cases k = 1, k = 2 and k > 3 are considered separately in the subsequent subsections. 6.1

Case 3 with k = 1

The condition for the waiting and the acting periods is tw 6 τ − 2ta

and tw >

1 1 τ − ta . 2 2

(6.3)

FIG. 4. Graphs of the solution of (2.6) with (3.1) for different act-and-wait periods T . Panel (a): Case 3, k = 1 (tw 6 τ − 2ta and tw > (τ − ta )/2). Panel (b): Case 3, k = 2 (tw 6 (τ − 3ta )/2 and tw > (τ − 2ta /3).

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K k (t) :=

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DIMENSION REDUCTION BY THE ACT-AND-WAIT CONTROLLER

The sketch of the piecewise solution of the system is shown in Fig. 4, panel (a). The solution of the system can be constructed according to the method of steps as follows: x (1) (t) = e At x(0), x (2) (t) = e At x(0) +

Z

t−τ tw −τ

e A(t−s−τ ) BΓ (s + τ )C x 0 (s)ds,

The state at t = T is given as x(T ) = x

(2)

(T ) = e

AT

x(0) +

Z

tw −τ

(6.4)

tw < t 6 T.

(6.5)

e A(T −s−τ ) BΓ (s + τ )C x 0 (s)ds . } {z = K 1 (0)

(6.6)

Calculation of the weighted integral K 1 (T ) requires the construction of the solution function x T :  if − τ 6 s 6 −T,  x0 (s + T ) (6.7) x T (s) = x (1) (s + T ) if − T < s 6 tw − T,  (2) x (s + T ) if tw − T < s 6 0. Note that function x (2) depends on the initial function x0 , while x (1) depends only on the initial state x(0). Now, condition (6.3) implies that T − τ 6 tw − T and tw − τ > −T , thus, the domain of the integral in K 1 (T ) (i.e., [tw − τ, T − τ ], see (6.2)) is entirely within the domain of the solution segment x (1) . Consequently, K 1 (T ) depends only on x(0): Z T −τ K 1 (T ) = e A(T −s−τ ) BΓ (s + τ )C x (1) (s + T )ds = Θ1 x(0), (6.8) tw −τ

where Θ1 =

Z

T −τ tw −τ

e A(T −s−τ ) BΓ (s + τ )C e A(s+T ) ds.

This way, we can construct an (n + m)-dimensional map !  !  x(T ) x(0) e AT I = Θ1 0 K 1 (T ) K 1 (0) | {z } =: Θ (1) (T )

(6.9)

(6.10)

with I denoting the n × n identity matrix. Here, Θ (1) (T ) serves as a 2n × 2n monodromy matrix. The stability of the system is determined by the 2n eigenvalues of Θ (1) (T ). 6.2 Case 3 with k = 2 The condition for the waiting and the acting periods is tw 6

1 3 τ − ta 2 2

and tw >

1 2 τ − ta . 3 3

(6.11)

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|

T −τ

0 < t 6 tw ,

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The method of steps can be applied in the same way as it was done for the case k = 1. The sketch of the piecewise solution of the system is shown in Fig. 4, panel (b). In this case, the construction of the solution segments results in the (n + 2m)-dimensional monodromy map      x(0)  x(T ) e AT I 0     (6.12) 0 I   K 1 (0) ,  K 1 (T ) =  0 Θ2 0 0 (0) K 2 (T ) K 2 | {z } =: Θ (2) (T )

where

T −τ tw −τ

e A(T −s−τ ) BΓ (s + τ )C e A(s+2T ) ds.

(6.13)

Matrix Θ (2) (T ) serves as a 3n × 3n monodromy matrix. 6.3

Case 3 with k > 3

The condition for the waiting and the acting periods is defined by (6.1). The same algorithm can be used as for the cases k = 1 and k = 2, and multiple application of the method of steps results in the (k + 1)n-dimensional monodromy map   AT    x(T ) x(0) e I 0 ... 0   K 1 (T )   0  0 I 0      K 1 (0)    .     . . .  .. = . ,  .. .. ... (6.14) . ..    .          K k−1 (T )  0 0 0 I   K k−1 (0) K k (T ) K k (0) Θk 0 0 ... 0 | {z } =: Θ (k) (T )

where

Θk =

Z

T −τ tw −τ

e A(T −s−τ ) BΓ (s + τ )Ce A(s+kT ) ds.

(6.15)

Here, Θ (k) serves as a (k + 1)n × (k + 1)n monodromy matrix. Consequently, the dimension of the system is equal to (k + 1)n. 7. A numerical examples In order to demonstrate the practical importance of Theorem 3.1, a first- and a second-order system is analysed for different acting and waiting periods. 7.1

First-order system

Consider system (2.1)–(2.2) and the continuous controller (2.4) with A = (a),

B = (1),

C = (1),

D = (d),

τ = 1.

(7.1)

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Θ2 =

Z

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The corresponding scalar equation reads x(t) ˙ = ax(t) + d x(t − 1).

(7.2)

This system has infinitely many poles, and it is unstable for all d if a > 1 (see, e.g., Stepan, 1989). Consider now system (2.1)–(2.2) and the act-and-wait controller (2.5) with (3.1) such that Γ (t) ≡ d. This is a special case of act-and-wait controllers: the feedback gain is switched between zero and the constant d. The corresponding scalar equation is x(t) ˙ = ax(t) + g(t)x(t − 1) with

0

if 0 6 t mod T < tw ,

d

if tw 6 t mod T < tw + ta = T.

(7.4)

The poles were determined numerically using the first-order semi-discretization method that provides a finite-dimensional approximation of the infinite-dimensional system. The delay parameter for the semi-discretization was set to r = 200, i.e., the system was approximated by an r + 1 =201-dimensional system for which the eigenvalues were computed numerically (for details on the method, see Insperger et al., 2008). The state and the control parameters were fixed to a = 1 and d = 1. The length of the the waiting and the acting periods was changed between 0.01 and 1.3 in 120 steps and 0.01 and 3.3 in 360 steps, respectively. An eigenvalue was declared to be zero if it was less than 10−15 . The resulted diagram with the number of non-zero (>10−15 ) poles is shown in Fig. 5. It can be seen that the figure is very similar to the one shown in Fig. 1, although small discrepancies occur due to numerical inaccuracies. Figure 6 shows the stability diagrams in the plane of (a, d) for the parameter points A, B and C. The stability boundaries for the autonomous equation (7.2) are also presented by dashed lines for reference. Case A has already been analysed in Stepan & Insperger (2006) in detail. In this case, the monodromy mapping can be constructed according to (4.13) in the form Φ (2) (T ) = ea(ta +tw ) (1 + be−aτ ta ).

(7.5)

It can be seen that for this case, the system can be stabilized for any a, furthermore, the control parameter b = −eaτ /ta

(7.6)

FIG. 5. Number of poles with magnitude larger than 10−15 for (7.3) with (7.4) determined numerically by the semi-discretization method.

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g(t) =

(

(7.3)

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results in deadbeat control. This case is denoted by dotted line in Fig. 6. For the cases B and C, the monodromy mappings are 2- and 3-dimensional, and the stable domains do not reach large values of a. 7.2

Second-order system

Consider system (2.1)–(2.2) and the continuous controller (2.4) with

A=



 0 1 , −a 0

B=

  0 , 1

C=



1 0

 0 , 1

D=



 −p , −d

τ = 1.

(7.7)

The corresponding scalar equation reads x(t) ¨ + ax(t) = − px(t − 1) − d x(t ˙ − 1).

(7.8)

This equation describes the proportional-derivative control of a second-order system with p and d being the proportional and differential gains. It is an often referred example in dynamics and control theory and also relevant to human balancing problems (see, e.g., Stepan, 1989; Milton et al., 2009a). It is known that (7.8) is unstable for all pairs ( p, d) if a < −2. Consider now system with the act-and-wait controller in the form x(t) ¨ + ax(t) = −g(t)( px(t − 1) + d x(t ˙ − 1)),

(7.9)

where g(t) is defined in (7.4) with d = 1. The diagram presenting the number of non-zero poles in the plane (tw , ta ) for (7.9) shows similar structure as the ones shown in Fig. 1 or in Fig. 5. Stability diagrams in the plane of control parameters ( p, d) associated with the parameter points A, B and C (see Fig. 5) are presented in Fig. 7 for the system parameter a = −2.5. The diagrams also present the contour curves of the decay ratio ρ = |μ1 |1/T , where μ1 is the critical Floquet multiplier obtained by the first-order semi-discretization method. This decay ratio characterizes the decay over a unit time step, i.e., kx(t + 1)k 6 ρkx(t)k. The contour curves were determined by the analysis of the critical Floquet multipliers over a 200 × 200 grid of control parameters. The stability boundaries where ρ = 1 are denoted by thick lines. It can be seen that stable regions arise for cases A and B, while for case C,

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FIG. 6. Stability diagrams for (7.3) with (7.4). A: tw = 1.2, ta = 0.8. B: tw = 0.8, ta = 0.8. C: tw = 0.5, ta = 0.2. Stability boundaries for (7.2) are also presented by dashed lines.

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FIG. 7. Stability diagrams for (7.9) with (7.4) with a = −2.5. A: tw = 1.2, ta = 0.8. B: tw = 0.8, ta = 0.8. C: tw = 0.5, ta = 0.2.

8. Conclusion The dimension of systems with feedback delay under act-and-wait control was analysed for different values of acting and waiting times. The geometric representation of the results are presented in Fig. 1. It was shown that the system with feedback delay can be reduced to a finite-dimensional system even if the waiting period tw is shorter than the feedback delay τ . If the dimension of the delay-free system is n then a kn-dimensional monodromy map (with k > 2) can be constructed even for certain combinations of acting and waiting periods ta and tw . The main point of the analysis was the systematic exploration of the nested integrals in the method of steps. Note, however, that similar estimates can also be done using the Lyapunov spectrum of the system (Otto & Radons, 2010). The dimension of the system was found to be the smallest if the waiting period was chosen to be larger than the feedback delay. In this case, the resulting time-periodic and time-delayed system can be described by an n × n monodromy matrix, and the stability depends only on n poles. This property can be useful for control design applications when the feedback delay plays an important rule in the system’s dynamics. In this sense, the act-and-wait controller is a candidate for the Brockett problem for systems with feedback delay (see Problem 2.2). The problem for the act-and-wait control concept can be rephrased in the following problem. P ROBLEM 8.1 Consider system (2.1)–(2.2) with the periodic controller (2.5). Assume that matrices A, B and C and the feedback delay τ are given. Assume that G(t) is given as in (3.1) and tw > τ , thus an n × n monodromy matrix can be constructed. Under what circumstances does there exist a timedependent function Γ (t): [tw , T ] → Rm×l such that the system is asymptotically stable, i.e., all the n poles are in modulus less than one? In the current paper, the act-and-wait controller was implemented for a control system with a single point delay, but it can be extended to systems with multiple or even with distributed delays. The conditions for having an n-dimensional monodromy map is then tw > τmax , where τmax is the maximum delay occurring in the system. However, the conditions for having a kn-dimensional monodromy map might be more complicated than it was for the single point delay case.

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no stable region was found by the given resolution. Note that for the case a = −2.5, the autonomous system (7.8) is unstable for any ( p, d), while the act-and-wait controller with tw = 1.2, ta = 0.8 (case A) and tw = 0.8, ta = 0.8 (case B) provides a finite region of stabilizing control parameters.

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Funding The J´anos Bolyai Research Scholarship of the Hungarian Academy of Sciences and the Hungarian National Science Foundation (K72911 and K68910).

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