Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 916040, 9 pages http://dx.doi.org/10.1155/2013/916040
Research Article Self-Triggered Model Predictive Control Using Optimization with Prediction Horizon One Koichi Kobayashi and Kunihiko Hiraishi School of Information Science, Japan Advanced Institute of Science and Technology, Ishikawa 923-1292, Japan Correspondence should be addressed to Koichi Kobayashi;
[email protected] Received 1 August 2013; Revised 12 October 2013; Accepted 13 October 2013 Academic Editor: Bo Shen Copyright Β© 2013 K. Kobayashi and K. Hiraishi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Self-triggered control is a control method that the control input and the sampling period are computed simultaneously in sampleddata control systems and is extensively studied in the field of control theory of networked systems and cyber-physical systems. In this paper, a new approach for self-triggered control is proposed from the viewpoint of model predictive control (MPC). First, the difficulty of self-triggered MPC is explained. To overcome this difficulty, two problems, that is, (i) the one-step input-constrained problem and (ii) the N-step input-constrained problem are newly formulated. By repeatedly solving either problem in each sampling period, the control input and the sampling period can be obtained, that is, self-triggered MPC can be realized. Next, an iterative solution method for the latter problem and an approximate solution method for the former problem are proposed. Finally, the effectiveness of the proposed approach is shown by numerical examples.
1. Introduction In recent years, analysis and synthesis of networked control systems (NCSs) have been extensively studied [1, 2]. An NCS is a control system in which plants, sensors, controllers, and actuators are connected through communication networks. In distributed control systems, subsystems are frequently connected via communication networks, and it is important to consider analysis and synthesis of distributed control systems from the viewpoint of NCSs. In the design of NCSs, several technical issues such as packet losses, transmission delays, and communication constraints are included. However, it is difficult to consider these issues in a unified way, and it is suitable to discuss an individual problem. From this viewpoint, several results have been obtained so far (see, e.g., [3β6]). In this paper, the periodic paradigm is focused as one of the technical issues in NCSs. The periodic paradigm is that the controller is periodically executed at a given unit of time. The period is chosen based on CPU processing time, communication bandwidth, and so on. However, in NCSs, communication should occur, when there exists important information, which must be transmitted from the controller to the actuator and/or from the sensor to the controller. In
this sense, the periodic paradigm is not necessarily suitable, and it is important to consider a new approach for the design of NCSs. One of the methods to overcome this drawback of the periodic paradigm, self-triggered control has been proposed so far (see, e.g., [7β12]). Also in the field of cyberphysical systems, this control method is focused. In selftriggered control, the next sampling time at which the control input is recomputed is computed. That is, both the sampling period and the control input are computed simultaneously. In many existing works, first, the continuous-time controller is obtained, and after that, the sampling period such that stability is preserved is computed. However, few results on optimal control have been obtained so far. From the viewpoint of optimal control, for example, a design method based onestep finite horizon boundary has been recently proposed in [13, 14]. In this method, the first sampling period such that the optimal value of the cost function is improved, is computed under the constraint that other sampling periods are given as a constant. However, a nonlinear equation must be solved. Furthermore, input constraints cannot be considered in this method. In [10], the authors have proposed self-triggered model predictive control using Taylor series expansions. In this method, the control input and the sampling period are computed by solving a quadratic programming (QP)
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Mathematical Problems in Engineering
problem, but the convexity of the obtained QP problem has not been guaranteed. In this paper, we propose two methods for self-triggered model predictive control (MPC) using optimization with horizon one. First, the optimal control problem with horizon one is formulated. However, in constrained systems, one-step prediction may be insufficient, and a longer time interval, in which the input constraint is imposed, is required. Focusing on this fact, another problem in which the time interval with input constraints is enlarged is also formulated. In the former problem, the first sampling period and the first control input are optimized. In the latter problem, the first sampling period and the control input sequence are optimized. Next, an iterative solution method for the latter problem and an approximate solution method for the former problem are proposed. In the iterative solution method, a QP problem is repeatedly solved. In the approximate solution method, the problem is approximated by one QP problem. The obtained QP problem is in general not convex, and we discuss the convexity. By solving either problem according to the receding horizon policy, self-triggered MPC can be realized. Finally, the effectiveness of the proposed approach is shown by a numerical example. The proposed approach provides us a basic result for self-triggered optimal control. Notation. Let R denote the set of real numbers. Let πΌπ , 0πΓπ denote the π Γ π identity matrix, the π Γ π zero matrix, respectively. For simplicity, we sometimes use the symbol 0 instead of 0πΓπ , and the symbol πΌ instead of πΌπ .
2. Self-Triggered Model Predictive Control Consider the following continuous-time linear system: π₯Μ (π‘) = π΄π₯ (π‘) + π΅π’ (π‘) ,
(1)
where π₯ β Rπ is the state, and π’ β [π’min , π’max ] β Rπ is the control input with the input constraint. The vectors π’min , π’max β Rπ are a given constant vector. Let π‘π , π = 0, 1, . . . denote the sampling time. The sampling period is defined by βπ := π‘π+1 β π‘π , which is a nonnegative scalar. Assume that the control input is piecewise constant, that is, the control input is given by π’ (π‘) = π’ (π‘π ) ,
π‘ β [π‘π , π‘π+1 ) .
(2)
Hereafter, we denote π’(π‘π ) as π’π . In addition, assume that a pair (π΄, π΅) is controllable. First, for the system (1), the self-triggered optimal control problem is formulated as follows. Problem 1 (self-triggered optimal control problem). Suppose that for the system (1), the initial time π‘0 , the initial state π₯(π‘0 ) = π₯0 , the final time π‘π , and the final step π are given. Then, find both a control input sequence π’0 , π’1 , . . . , π’πβ1 and a sampling period sequence β0 , β1 , . . . , βπβ1 minimizing the following cost function: π‘π
π½ = β« {π₯π (π‘) ππ₯ (π‘) + π’π (π‘) π
π’ (π‘)} ππ‘, π‘0
under the following two constraints: βmin β€ βπ β€ βmax , π’min β€ π’π β€ π’max , β0 + β1 + β
β
β
+ βπβ1 = π‘π , where π is positive semidefinite, π
is positive definite, and βmin , βmax β₯ 0 are a given constant. Next, we present a procedure of MPC based on the selftriggered strategy. Procedure of Self-Triggered MPC Step 1. Set π‘0 = 0, and give the initial state π₯(0) = π₯0 . Step 2. Solve Problem 1. Step 3. Apply only π’(π‘), π‘ β [π‘0 , π‘0 + β0 ) to the plant. Μ 0 +β0 ) by using π₯(π‘0 ), Step 4. Compute the predicated state π₯(π‘ π’0 , and β0 . Μ 0 + β0 ) as π₯0 . Step 5. Solve Problem 1 by using π₯(π‘ Step 6. Wait until time π‘0 + β0 . Step 7. Update π‘0 := π‘0 + β0 , measure π₯(π‘0 ), and return to Step 3. Note here that in this procedure, the timing (i.e., the sampling time) to measure the state and to recompute the control input is computed. In this sense, self-triggered control is realized. In the above procedure, Problem 1 must be solved repeatedly. However, Problem 1 is in general reduced to a nonlinear programming problem, and it is difficult to solve this problem. Then, it is important to compute a suboptimal and approximate solution for Problem 1. To compute a suboptimal and approximate solution, two problems, which are solved in Steps 2 and 5 instead of Problem 1, will be formulated in the next section.
3. Problem Formulation In self-triggered MPC, only π’(π‘), π‘ β [π‘0 , π‘0 + β0 ) is applied to the plant. Hence, it is important to consider the problem of finding suitable β0 and π’0 . By solving this problem repeatedly, we can continue to obtain the control input. Thus, a sampling period sequence β0 , β1 , . . . , βπβ1 is approximated as follows: (i) only β0 is a decision variable (i.e., the other sampling periods are given in advance), (ii) β1 = β2 = β
β
β
= βπβ1 holds. In addition, the time interval [π‘0 , π‘π ) in Problem 1 is enlarged to [π‘1 , β). Here, we consider the following two optimal control problems with prediction horizon one. (i) One-step input-constrained problem.
(3)
(4)
(ii) π-step input-constrained problem.
Mathematical Problems in Engineering
3
In the former problem, the constraint is imposed for only π’0 . In the latter problem, the constraint is imposed for π’0 , π’1 , . . . , π’πβ1 , where π β€ π is given in advance. Before these problems are formally given, some preparations are given. In the time interval [π‘1 , π‘π ), suppose that β1 = β2 = β
β
β
= βπβ1 = β is satisfied, where β is a given constant (see also Remark 4). In addition, the input constraint π’min β€ π’π β€ π’max , π = 1, 2, . . . , π β 1 is ignored. Then, the optimal β value of the cost function π½ = β«π‘ {π₯π (π‘)ππ₯(π‘)+π’π (π‘)π
π’(π‘)}ππ‘ 1
can be derived as π₯π (π‘1 )π(β)π₯(π‘1 ), where π(β) is a symmetric positive definite matrix, which is a solution of the following discrete-time algebraic Riccati equation: Μ (β) β π (β) Μπ (β) π (β) π΄ π΄
the above simplified problem is considered for analyzing the convexity. In constrained systems, when control is started, the control input is frequently saturated. It is important to determine the time interval of input saturation. Then, onestep prediction may be insufficient, and a longer time interval in which the input constraint is imposed is required. From this viewpoint, another problem, that is, the π-step inputconstrained problem is formulated. First, suppose that the input constraint is imposed in the time interval [π‘0 , π‘0 + β0 + β(π β 1)), where π β₯ 1 is a given integer. In addition, suppose that no input constraint is imposed in the time interval [π‘0 + β0 + β(π β 1), β), then, consider the following cost function: π½ = π½1 + π½2 + π½3 ,
Μπ (β) π (β) π΅Μ (β) + πΜ (β)) β (π΄ Μ (β))β1 Γ (π΅Μπ (β) π (β) π΅Μ (β) + π
(5)
π½1 = β«
π‘0 +β0
π‘0
Μ (β) + πΜπ (β)) + π Μ (β) = 0, Γ (π΅Μπ (β) π (β) π΄
π½2 = β«
{π₯π (π‘) ππ₯ (π‘) + π’π (π‘) π
π’ (π‘)} ππ‘,
π‘0 +β0 +β(πβ1)
π‘0 +β0
where
π½3 = β«
Μ (β) := ππ΄β , π΄ β
π΅Μ (β) := β« ππ΄π πππ΅,
(6)
0
β Μ (β) πΜ (β) π π π 0 πΉπ‘ ππΉ π‘ [ ] := [ Μπ ] π ππ‘, β« Μ (β) 0 π
π (β) π
0
π΄ π΅ πΉ := [ ]. 0 0
π‘0 +β0 +β(πβ1)
(10)
{π₯π (π‘) ππ₯ (π‘) + π’π (π‘) π
π’ (π‘)} ππ‘
= π₯π (π‘0 + β0 + β (π β 1)) Γ π (β) π₯ (π‘0 + β0 + β (π β 1)) .
(7)
Under the above preparation, we formulate the one-step input-constrained problem as follows. Problem 2 (one-step input-constrained problem). Suppose that for the system (1), the initial time π‘0 , the initial state π₯(π‘0 ) = π₯0 , and β1 = β2 = β
β
β
= β are given. Then, find both a control input π’0 and a sampling period β0 minimizing the following cost function:
The optimal value of π½3 can be characterized by π₯(π‘0 + β0 + β(π β 1)), because no input constraint is imposed in the time interval [π‘0 + β0 + β(π β 1), β). Under the above preparation, consider the following πstep input-constrained problem. Problem 3 (π-step input-constrained problem). Suppose that for the system (1), the initial time π‘0 , the initial state π₯(π‘0 ) = π₯0 , β1 = β2 = β
β
β
= β, π β₯ 1, and πΎ β₯ 1 are given. Then, find a control input sequence π’0 , π’1 , . . . , π’πβ1 maximizing a sampling period β0 under the following constraints: β β€ β0 β€ βmax ,
π‘1
π½ = β« {π₯π (π‘) ππ₯ (π‘) + π’π (π‘) π
π’ (π‘)} ππ‘ π‘0
β
{π₯π (π‘) ππ₯ (π‘) + π’π (π‘) π
π’ (π‘)} ππ‘,
(8)
+ π₯π (π‘1 ) π (β) π₯ (π‘1 )
π’min β€ π’π β€ π’max ,
π = 0, 1, . . . , π β 1,
π½ββ0 β€ πΎπ½ββ ,
(11) (12)
(9)
where βmax β₯ 0 is a given constant, and π½ββ0 is the optimal value of the cost function of (10), π½ββ is the optimal value of the cost function of (10) under β0 = β.
In [13, 14], a related problem has been discussed, but in these existing results, the above two constraints cannot be imposed. In [10], the authors have considered a more complicated problem with delay compensation. In this paper,
In Problem 3, control performance can be adjusted by suitably giving πΎ. We remark that in this problem, β0 is maximized under certain constraints. Furthermore, in this problem, a control input sequence is computed, but only the first one (β0 ) is computed on sampling periods. In this sense, this problem is regarded as a kind of the optimal control problem with prediction horizon one.
under the following two constraints: β β€ β0 β€ βmax , π’min β€ π’0 β€ π’max , where βmax β₯ 0 is a given constant.
4
Mathematical Problems in Engineering
Hereafter, in Section 4, an iterative solution method for the π-step input-constrained problem (Problem 3) will be proposed. In Section 5, an approximate solution method for the one-step input-constrained problem (Problem 2) will be proposed. Remark 4. The parameter β in Problems 2 and 3 is chosen based on the computation time for solving the problem and the dynamics of a given plant. If computation of the problem is not finished until time π‘0 + β0 β [π‘0 + β, π‘0 + βmax ], then the next control input cannot be applied to the plant. See also the procedure of self-triggered MPC in Section 2.
4. Iterative Solution Method for π-Step InputConstrained Problem First, for a fixed β0 , consider deriving π½ββ0 . The value of π½ββ can be derived by a similar method. The value of π½ββ0 is given by the optimal value of the following optimal control problem. Problem 5. Suppose that for the system (1), the initial time π‘0 , the initial state π₯(π‘0 ) = π₯0 , β0 and β1 = β2 = β
β
β
= β, π β₯ 1 are given. Then, find a control input sequence π’0 , π’1 , . . . , π’πβ1 minimizing the cost function (10) under the input constraint (11). From the conventional result on sampled-data control theory, Problem 5 can be equivalently rewritten as the following optimal control problem of time-varying discretetime linear systems with the input constraint (11). Problem 6. Suppose that the initial time π‘0 , the initial state π₯(0) = π₯0 , β0 , and β1 = β2 = β
β
β
= β, π β₯ 1 are given. Consider the following discrete-time linear system: Μ (β0 ) π₯0 + π΅Μ (β0 ) π’0 , π₯1 = π΄ Μ (β) π₯π + π΅Μ (β) π’π , π₯π+1 = π΄
π β₯ 1,
[π’0π π’1π where
β
β
β
π
π π’πβ1 ] . Then, we can obtain π₯ = π΄π₯0 + π΅π’,
πΌ ] Μ (β0 ) π΄ ] Μ Μ π΄ (β) π΄ (β0 ) ] ] Μ2 (β) π΄ Μ (β0 ) ] π΄ ], ] .. ] ] . πβ1 Μ Μ (β) π΄ (β0 )] [π΄
[ [ [ [ π΄=[ [ [ [ [
0 0 [ [ 0 π΅Μ (β0 ) [ [ [ π΄ π΅Μ (β) [ Μ (β) π΅Μ (β0 ) π΅=[ [ Μ2 Μ (β) π΅Μ (β) π΄ [ π΄ (β) π΅Μ (β0 ) [ . .. [ .. [ . Μπβ1 (β) π΅Μ (β0 ) π΄ Μπβ2 (β) π΅Μ (β) π΄ [
β
β
β
0 .. . .. . .. .
] ] ] ] ] ] ]. ] ] ] ] 0 ] π΅Μ (β)]
β
β
β
β
β
β
β
β
β
β
β
β
β
β
β
(15)
In addition, we define Μ (β0 ) , π Μ (β) , . . . , π Μ (β) , π (β)) , π := block-diag (π block-diag (πΜ (β0 ) , πΜ (β) , . . . , πΜ (β)) π := [ ], 0πΓ(πβ1)π
(16)
Μ (β0 ) , π
Μ (β) , . . . , π
Μ (β)) . π
:= block-diag (π
Then, the cost function (14) can be rewritten as follows: π½ = π₯π ππ₯ + 2π₯π ππ’ + π’π π
π’
(17)
= π’π πΏ 2 π’ + πΏ 1 π’ + πΏ 0 , where π
π
π
πΏ 2 = π
+ π΅ π + π π΅ + π΅ ππ΅, (13)
π
πΏ 1 = 2π₯0π π΄ (π + ππ΅) ,
(18)
π
where π₯π := π₯(π‘π ). Then, find a control input sequence π’0 , π’1 , . . . , π’πβ1 minimizing the cost function (10), that is,
π Μ π (β) πΜ (β) π₯π π₯ + β [ π ] [ Μπ Μ (β)] [π’π ] π’π π (β) π
π=1
π Finally, π’min := [π’min
π π’min
π
β
β
β
π
π π’min ] and π’max :=
π π π π’max β
β
β
π’max ] are also defined. [π’max Under the above preparation, Problem 6 is equivalent to the following QP problem.
π Μ π (β ) πΜ (β ) π₯ π₯ π½ = [ 0 ] [ Μπ 0 Μ 0 ] [ 0 ] π’0 π (β0 ) π
(β0 ) π’0 πβ1
πΏ 0 = π₯0ππ΄ ππ΄π₯0 .
(14)
π + π₯π π (β) π₯π
under the input constraint (11). Next, consider reducing Problem 6 to a QP probπ π ] and π’ := lem. Define π₯ := [π₯0π π₯1π β
β
β
π₯π
Problem A. Consider find
π’0 , π’1 , . . . , π’πβ1 ,
min
π’π πΏ 2 π’ + πΏ 1 π’ + πΏ 0 ,
(19)
subject to π’min β€ π’ β€ π’max . A QP problem can be solved by using a suitable solver such as MATLAB and IBM ILOG CPLEX [15]. Third, by using the obtained QP problem, we propose an algorithm for solving Problem 3.
Mathematical Problems in Engineering
5
Algorithm 7. Step 1. Derive π½ββ by solving Problem A with β0 = β.
where Ξ¦0 = [
Step 2. Set π = β and π = βmax , and give a sufficiently small positive real number π.
Ξ¦1 = [
Step 3. Set β0 = (π + π)/2. Step 4. Derive π½ββ0 by solving Problem A. Step 5. If π½ββ0 β€ πΎπ½ββ in Problem 3 is satisfied, then set π = β0 , otherwise set π = β0 . Step 6. If |πβπ| < π is satisfied, then the optimal β0 in Problem 3 is derived as π, and the optimal control input sequence is also derived. Otherwise go to Step 3. In a numerical example (Section 6.1), we will discuss the computation time of Algorithm 7. Finally, we discuss the stabilization issue. For Problem 6, consider imposing the constraint π(π₯π+1 ) β€ ππ(π₯π ), where π(π₯) is a given nonnegative function, and π β [0, 1) is a given constant. If π(π₯) is restricted to π(π₯) = βππ₯ββ , where π β RπΓπ and β β
ββ is an infinity matrix norm, then the constraint π(π₯π+1 ) β€ ππ(π₯π ) can be transformed into a set of linear inequalities (see, e.g., [16]), and can be embedded in Problem A. Then, the closed-loop system in which the control input derived by Algorithm 7 is applied is asymptotically stable. See, for example, [17, 18] for further details.
π 0 ], 0 π
ππ΄ + π΄π π ππ΅ ], 0 π΅π π
1 Ξ¦2,11 π΄π ππ΅ + ππ΄π΅] [ 2 Ξ¦2 = [ ], 1 π΅π ππ΅ π΅π ππ΄ + π΅π π΄π π 2 [ ]
(23)
π 1 (ππ΄2 + (π΄2 ) π) . 2
Ξ¦2,11 = π΄π ππ΄ +
Then, the weight matrices of (7) can be rewritten as Μ (β ) π [ Μπ 0 π (β0 )
1 πΜ (β0 ) 2 Μ (β0 )] = Ξ¦0 β0 + 2 Ξ¦1 β0 + β
β
β
. π
(24)
Next, we focus on the term π₯1π π(β)π₯1 . From π₯1 of (21), π₯1π π(β)π₯1 can be rewritten as π Ξ¨ Ξ¨12 π₯0 π₯ π₯1π π (β) π₯1 = [ 0 ] [ 11 ][ ], π π’0 Ξ¨12 Ξ¨22 π’0
(25)
where
5. Approximate Solution Method for One-Step Input-Constrained Problem
π
Ξ¨11 = ππ΄ π
In this section, first we derive a solution method for the onestep input-constrained problem (Problem 2). In the proposed solution method, Problem 2 is approximately reduced to a quadratic programming (QP) problem, but the convexity is not guaranteed. Next, we discuss the convexity. 5.1. Proposed Solution Method. First, noting that the control input is piecewise constant, from (7), the cost function of (8) can be equivalently rewritten as π Μ π (β ) πΜ (β ) π₯ π₯ π½ = [ 0 ] [ Μπ 0 Μ 0 ] [ 0 ] + π₯1π π (β) π₯1 , π’0 π (β0 ) π
(β0 ) π’0 β0
π₯1 = ππ΄β0 π₯0 + β« ππ΄π πππ΅π’0 . 0
(20) (21)
We focus on the weight matrices. By using Taylor series expansions, we can obtain the following relation:
π
ππΉ π‘ [
π 0 πΉπ‘ ] π = Ξ¦0 + Ξ¦1 π‘ + Ξ¦2 π‘2 + β
β
β
, 0 π
Ξ¨12 = ππ΄ π
β0
β0
π (β) ππ΄β0 ,
π (β) (ππ΄β0 β πΌ) π΄β1 π΅, π
Ξ¨22 = π΅π (π΄β1 ) (ππ΄
β0
(26)
β πΌ) π (β) (ππ΄β0 β πΌ) π΄β1 π΅.
We use Taylor series expansions for ππ΄β0 . Then, we can obtain Ξ¨11 = π (β) + (π (β) π΄ + π΄π π (β)) β0 π 1 1 + (π΄π π (β) π΄ + π (β) π΄2 + (π΄2 ) π (β)) β02 2 2
+ β
β
β
, 1 Ξ¨12 = π (β) π΄β0 + (π΄π π (β) π΅ + π (β) π΄π΅) β02 + β
β
β
, 2 Ξ¨22 = π΅π π (β) π΅β02 + β
β
β
.
(27)
From these results, the cost function can be expressed as follows: π
(22)
π₯ π₯ π½ = [ 0 ] (Ξ0 + Ξ1 β0 + Ξ2 β02 + β
β
β
) [ 0 ] , π’0 π’0
(28)
6
Mathematical Problems in Engineering Proof. By rewriting π½π of (31), the cost function in Problem B is derived. In addition, by using V0 , β0 , the input constraint π’min β€ π’0 β€ π’max in Problem 2 is equivalent to π’min β0 β€ V0 β€ π’max β0 , which is a linear inequality constraint.
where π (β) 0 Ξ0 = [ ], 0 0 Ξ1 = [
Ξ1,11 π (β) π΅ ], π
π΅π π (β)
Ξ2 = [
Ξ2,12 Ξ2,11 ], π Ξ2,12 π΅π π (β) π΅
Ξ1,11 = π (β) π΄ + π΄π π (β) + π, Ξ2,11
(29)
π 1 = (ππ΄ + π΄π π + π (β) π΄2 + (π΄2 ) π (β)) 2
+ π΄π π (β) π΄, 1 (π (β) π΄π΅ + ππ΅) . 2 In this paper, the second-order truncated Taylor series is used. Furthermore, the term π’0π(β0 π
)π’0 appeared in (28) is approximated as follows: Ξ2,12 = π΄π π (β) π΅ +
π’0π (β0 π
) π’0 β π’0π (β02 π
σΈ ) π’0 ,
(30)
where π
σΈ := π
/((βmax +β)/2). Then, we consider the following cost function π½π :
By solving Problem B, we can obtain suboptimal π’0 (= V0 /β0 ) and β0 . Problem B is a QP problem, but the cost function is in general nonconvex. A local optimal solution can be derived by using a suitable solver, for example, MATLAB. Remark 9. In Theorem 8, the accuracy of approximations is not considered. Several existing results on analysis of the truncation error in Taylor series have been obtained so far. In addition, by suitably setting π
σΈ in (30), the approximation of (30) can be regarded as an over-approximation of π’0π(β0 π
)π’0 . Using the above discussion, the upper bound of the optimal value in Problem 2 can be evaluated by solving Problem B. 5.2. Discussion on Convexity. The cost function in Problem B is in general nonconvex. In other words, the matrix π2 is not a positive definite matrix generally. In this subsection, we clarify the reason why π2 is not positive definite. The matrix π2 can be rewritten as π2 = [
π
π₯ π₯ π½ β π½π = [ 0 ] (Ξ0 + Ξ1σΈ β0 + Ξ2σΈ β02 ) [ 0 ] , π’0 π’0
(31)
Ξ2σΈ = [
Ξ2,12 Ξ2,11 ]. π Ξ2,12 π
σΈ + π΅π π (β) π΅
1 π2 = π2σΈ β πΆπ πβ1 (β) πΆ, 2 (32)
Theorem 8. The one-step input-constrained problem of Problem 2 is approximately reduced to the following QP problem. Problem B. Consider find V0 , β0 , π
V V V [ 0 ] π2 [ 0 ] + π1 [ 0 ] + π0 , β0 β0 β0
subject to
(36)
π· = [π1/2 (β) π΅ πβ1/2 (β) (π (β) π΄ + π΄π π (β) + π)] . Since π(β) is positive definite, π2σΈ is also positive definite. However, it is obvious that β(1/2)πΆπ πβ1 (β)πΆ is not positive definite. Therefore, π2 is not a positive definite matrix generally. Consider approximating the cost function in Problem B, that is, π½π in (31) by a convex function. We define the following positive definite matrix: π2σΈ := [
π’min β0 β€ V0 β€ π’max β0 ,
π
πΌ 0 πΌ 0 ] π2σΈ [ ], 0 π₯0 0 π₯0
(37)
and we rewrite π½π in (31). Then, we can obtain
where V0 := β0 π’0 , and
π
V V V π½π = [ 0 ] π2σΈ [ 0 ] + π1 [ 0 ] + π0 + πΌβ02 , β0 β0 β0
π π₯0 π
σΈ + π΅π π (β) π΅ Ξ2,12 ], π2 = [ π π π₯0 Ξ2,12 π₯0 Ξ2,11 π₯0
π0 = π₯0π π (β) π₯0 .
1 1 π π
σΈ 0 ] + [π΅ π΄] π (β) [π΅ π΄] + π·π π·, 0 0 2 2
(33)
β β€ β0 β€ βmax ,
π1 = [2π₯0π π (β) π΅ π₯0π Ξ1,11 π₯0 ] ,
π2σΈ = [
πΆ = [0 π΄π π (β) + π] ,
Under these preparations, we can obtain the following theorem.
min
(35)
where
where π (β) π΅ Ξ ], Ξ1σΈ = [ π 1,11 0 π΅ π (β)
π
πΌ 0 πΌ 0 ] π2 [ ], 0 π₯0 0 π₯0
(34)
π 1 πΌ = β π₯0π (π΄π π (β) + π) 2
Γ πβ1 (β) (π΄π π (β) + π) π₯0 .
(38)
Mathematical Problems in Engineering 12
0
10
β1
8
β2
6
β3 Control input
x1
4 State
7
2 0
β5 β6 β7
β2
β8
x2
β4
β9
β6 β8
β4
β10 0
1
2
3
4 Time
5
6
7
8
0
1
2
3
4 Time
5
6
7
Figure 2: Control input.
Figure 1: State trajectory.
πΌβ02
By approximating the negative term to a linear term, Problem B can be in general reduced to a convex QP problem. Noting that β β€ β0 β€ βmax , we can make a two-dimensional graph on β0 and πΌβ02 . Using the two-dimensional graph obtained, we can evaluate whether the approximation is reasonable.
cases is as follows. In Figures 1 and 2, that is, the case of π = 10, the control input at each time is shown as follows: π’ (π‘) = β10.00,
π‘ β [0, 1.83) ,
β0 = 1.83,
π’ (π‘) = β1.90,
π‘ β [1.83, 3.23) ,
β0 = 1.41,
π’ (π‘) = β0.49,
π‘ β [3.23, 4.49) ,
β0 = 1.25,
π’ (π‘) = β0.14,
π‘ β [4.49, 5.69) ,
β0 = 1.21,
6. Numerical Examples
π’ (π‘) = β0.04,
π‘ β [5.69, 6.89) ,
β0 = 1.20,
6.1. Iterative Solution Method. First, we show an example of the iterative solution method proposed in Section 4. Consider the following system:
π’ (π‘) = β0.01,
π‘ β [6.89, 8.08) ,
β0 = 1.19.
π₯Μ (π‘) = [
0 1 0 ] π₯ (π‘) + [ ] π’ (π‘) . β5 β8 1
(39)
The input constraint is given as π’(π‘) β [β10, +10]. Parameters in Problem 3 are given as follows: β = 0.5, πΎ = 1.001, βmax = 5, π = 103 πΌπ , and π
= 1. In Algorithm 7, we set π = 10β4 . Then, π(β) can be derived as π (β) = [
10615 593 ]. 593 575
8
(40)
In addition, we consider two cases, that is, the case of π = 1 and the case of π = 10. We show the computational result on self-triggered MPC with the π-step input-constrained problem of Problem 3. The initial state is given as π₯0 = [10 10]π , and the case of π = 10 is considered. Figure 1 shows the obtained state trajectory, and Figure 2 shows the control input trajectory. From these figures, we see that the sampling period is nonuniform. Next, compare two cases. In these cases, the obtained state trajectories are almost the same. The difference between two
(41)
In the case of π = 1, the control input at each time is derived as follows: π’ (π‘) = β10.00,
π‘ β [0, 0.62) ,
β0 = 0.62,
π’ (π‘) = β10.00,
π‘ β [0.62, 1.66) ,
β0 = 1.03,
π’ (π‘) = β2.62,
π‘ β [1.66, 2.88) ,
β0 = 1.22,
π’ (π‘) = β0.75,
π‘ β [2.88, 4.08) ,
β0 = 1.20,
π’ (π‘) = β0.22,
π‘ β [4.08, 5.27) ,
β0 = 1.19,
π’ (π‘) = β0.06,
π‘ β [5.27, 6.47) ,
β0 = 1.19,
π’ (π‘) = β0.02,
π‘ β [6.47, 7.66) ,
β0 = 1.19,
π’ (π‘) = β0.01,
π‘ β [7.66, 8.85) ,
β0 = 1.19.
(42)
From these results, we can discuss the following topic. In this example, input saturation is needed to improve the transient behavior. However, in the case of π = 1, the time interval of input saturation was not computed suitably. As a result, to derive the state trajectory in time interval [0, 8], Problem 3 must be solved eight times. In the case of π = 10, Problem 3 is solved six times. Hence, it is important to choose a suitable π. We remark that in this example, the computational result in the case of π = 20 is the same as that in the case of π = 10. In this sense, π = 10 is one of the suitable horizons.
8
Mathematical Problems in Engineering 12
0
10
β1
8
β2
6
Control input
4 State
β3
x1
2 0
β5 β6 β7
β2 x2
β4
β8 β9
β6 β8
β4
0
1
2
3
4
5
6
7
8
β10
0
1
2
Time
4 Time
5
6
7
8
Figure 4: Control input.
Figure 3: State trajectory.
In addition, we discuss the effect of changing πΎ in (12). In the case of π = 10, consider the following cases: πΎ = 1.001, 1.005, 1.010, 1.015, 1.020. For each case, the first β0 is obtained as follows: πΎ = 1.001: β0 = 1.83, πΎ = 1.005: β0 = 2.38, πΎ = 1.010: β0 = 2.79,
3
(43)
πΎ = 1.015: β0 = 3.12, πΎ = 1.020: β0 = 3.44. From these results, we see that β0 becomes longer by setting a larger πΎ. Since control performance decreases for a larger πΎ, it is important to consider the trade-off between πΎ and β0 . Finally, we discuss the computation time for solving the π-step input-constrained problem of Problem 3. In the case of π = 10, Problem 3 with the different initial state is solved six times. Then, the mean computation time for solving Problem 3 was 6.51 [sec], where we used IBM ILOG CPLEX 11.0 [15] as the MIQP solver on the computer with the Intel Core2 Duo 3.0 GHz processor and the 2 GB memory. In the case of π = 1, Problem 3 with the different initial state is solved eight times. Then, the mean computation time was 6.22 [sec]. From these results, it is difficult at this stage to solve Problem 3 in real-time. It is significant to consider several approaches for reducing the computation time. One of the simple methods is that the number of iterations in Algorithm 7 is limited to some integer depending on the computer environment. 6.2. Approximate Solution Method. Next, we show an example of the approximate solution method proposed in Section 5. Consider the system (39) again. The input constraint is given as π’(π‘) β [β10, +10]. Parameters in the one-step input-constrained problem of Problem 2 are given as follows:
β = 0.2, βmax = 1, π = 103 πΌπ , and π
= 1. Since the approximate solution method in Section 5 is derived using an approximation via Taylor series expansions, it is not desirable that the difference between β and βmax is large. Therefore, β and βmax must set carefully. Furthermore, in this example, Problem B is transformed into a convex QP problem. From β = 0.2 and βmax = 1, the term πΌβ02 is approximated by πΌβ0 . We show the computational result on self-triggered MPC with the transformed Problem B. The initial state is given as π₯0 = [10 10]π . Figure 3 shows the obtained state trajectory, and Figure 4 shows the control input trajectory. The obtained control input is shown as follow: π’ (π‘) = β10.00,
π‘ β [0, 0.86) ,
β0 = 0.86,
π’ (π‘) = β10.00,
π‘ β [0.86, 1.38) ,
β0 = 0.52,
π’ (π‘) = β6.37,
π‘ β [1.38, 1.88) ,
β0 = 0.50,
π’ (π‘) = β3.57,
π‘ β [1.88, 2.40) ,
β0 = 0.52,
π’ (π‘) = β2.19,
π‘ β [2.40, 2.90) ,
β0 = 0.50,
π’ (π‘) = β1.28,
π‘ β [3.41, 3.92) ,
β0 = 0.51,
π’ (π‘) = β0.77,
π‘ β [3.92, 4.43) ,
β0 = 0.51,
π’ (π‘) = β0.45,
π‘ β [4.43, 4.94) ,
(44)
β0 = 0.51.
From these results, we see that also in this example, the sampling period is nonuniform. Finally, we discuss the computation time for solving the transformed Problem B. The transformed Problem B with the different initial state is solved 17 times. Then, the mean computation time was 0.01 [sec], where we used IBM ILOG CPLEX 11.0 as the QP solver. Since in this example the number of decision variables is only two, computation is very fast.
Mathematical Problems in Engineering
7. Conclusion In this paper, we discussed self-triggered MPC of linear systems. Since it is difficult to solve the original problem (Problem 1), two control problems (the one-step inputconstrained problem of Problem 2 and the π-step inputconstrained problem of Problem 3) were formulated, instead of Problem 1. For Problem 3, the iterative solution method was proposed. For Problem 2, the approximate solution method was proposed. In the latter, we also discussed the convexity. The effectiveness of these proposed method was shown by numerical examples. The proposed methods are useful as a new method of self-triggered optimal control. In the future works, first, it is important to develop a more efficient method for solving Problem 3. Then, the continuation method [19] may be useful. Next, since the proposed method for the one-step input-constrained problem (Problem 2) is an approximate method, it is difficult at the current stage to guarantee the stability of the closedloop system. The stabilization issue for the one-step inputconstrained problem is also important as one of the future works.
Conflict of Interests The authors declare that they have no conflict of interests.
Acknowledgment This work was partially supported by Grant-in-Aid for Young Scientists (B) 23760387.
References [1] C. T. Abdallah and H. G. Tanner, βComplex networked control systems: introduction to the special section,β IEEE Control Systems Magazine, vol. 27, no. 4, pp. 30β32, 2007. [2] P. Antsaklis and J. Baillieul, βSpecial issue on technology of networked control systems,β Proceedings of the IEEE, vol. 95, no. 1, pp. 5β8, 2007. [3] L.-S. Hu, T. Bai, P. Shi, and Z. Wu, βSampled-data control of networked linear control systems,β Automatica, vol. 43, no. 5, pp. 903β911, 2007. [4] H. Ishii, βStabilization under shared communication with message losses and its limitations,β in Proceedings of the 45th IEEE Conference on Decision and Control (CDC β06), pp. 4974β4979, December 2006. [5] H. Ishii, βπ»β control with limited communication and message losses,β Systems & Control Letters, vol. 57, no. 4, pp. 322β331, 2008. [6] K. Kobayashi and K. Hiraishi, βOptimal control of a class of networked systems based on MLD framework,β in Proceedings of the 18th IFAC World Congress, pp. 66β71, 2011. [7] A. Anta and P. Tabuada, βSelf-triggered stabilization of homogeneous control systems,β in Proceedings of the American Control Conference (ACC β08), pp. 4129β4134, Seattle, Wash, USA, June 2008. [8] A. Anta and P. Tabuada, βTo sample or not to sample: selftriggered control for nonlinear systems,β IEEE Transactions on Automatic Control, vol. 55, no. 9, pp. 2030β2042, 2010.
9 [9] A. Camacho, P. MartΒ΄Δ±, M. Velasco et al., βSelf-triggered networked control systems: an experimental case study,β in Proceedings of the International Conference on Industrial Technology (ICIT β10), pp. 123β128, ViΛna del Mar ValparaΒ΄Δ±so, Chile, March 2010. [10] K. Kobayashi and K. Hiraishi, βSelf-triggered model predictive control with delay compensation for networked control systems,β in Proceedings of the 38th Annual Conference of the IEEE Industrial Electronics Society, pp. 3182β3187, 2012. [11] M. Mazo Jr. and P. Tabuada, βOn event-triggered and selftriggered control over sensor/actuator networks,β in Proceedings of the 47th IEEE Conference on Decision and Control (CDC β08), pp. 435β440, CancΒ΄un, Mexico, December 2008. [12] X. Wang and M. D. Lemmon, βSelf-triggered feedback control systems with finite-gain πΏ 2 stability,β IEEE Transactions on Automatic Control, vol. 54, no. 3, pp. 452β467, 2009. [13] M. Velasco, P. Marti, J. Yepez et al., βQualitative analysis of a onestep finite horizon boundary for event-driven controllers,β in Proceedings of the 50th IEEE Conference on Decision and Control and European Control, pp. 1662β1667, 2011. [14] J. Yepez, M. Velasco, P. Marti, E. X. Martin, and J. M. Fuertes, βOne-step finite horizon boundary with varying control gain for event-driven networked control systems,β in Proceedings of the 37th Annual Conference of the IEEE Industrial Electronics Society, pp. 2531β2536, 2011. [15] IBM ILOG CPLEX Optimizer, http://www-01.ibm.com/ software/commerce/optimization/cplex-optimizer/. [16] M. Lazar, βFlexible control lyapunov functions,β in Proceedings of the American Control Conference (ACC β09), pp. 102β107, St. Louis, Mo, USA, June 2009. [17] S. Di Cairano, M. Lazar, A. Bemporad, and W. P. M. H. Heemels, βA control Lyapunov approach to predictive control of hybrid systems,β in Proceedings of the 11th International Conference on Hybrid Systems: Computation and Control, Lecture Notes in Computer Science, no. 4981, Springer, 2008. [18] K. Kobayashi, J.-i. Imura, and K. Hiraishi, βStabilization of finite automata with application to hybrid systems control,β Discrete Event Dynamic Systems, vol. 21, no. 4, pp. 519β545, 2011. [19] T. Ohtsuka, βA continuation/GMRES method for fast computation of nonlinear receding horizon control,β Automatica, vol. 40, no. 4, pp. 563β574, 2004.
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