# Self-Triggered Model Predictive Control Using Optimization with Prediction Horizon One

Authors: Koichi Kobayashi; Kunihiko Hiraishi
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

Self-triggered control is a control method that the control input and the sampling period are computed simultaneously in sampled-data 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) theN-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.


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## Body

## 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., [3–6]).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., [7–12]). Also in the field of cyber-physical systems, this control method is focused. In self-triggered 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 one-step 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) 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 In, 0m×n denote the n×n identity matrix, the m×n zero matrix, respectively. For simplicity, we sometimes use the symbol 0 instead of 0m×n, and the symbol I instead of In.


## 2. Self-Triggered Model Predictive Control

Consider the following continuous-time linear system:(1)x˙(t)=Ax(t)+Bu(t),
where x∈ℛn is the state, and u∈[umin,umax]⊆ℛm is the control input with the input constraint. The vectors umin,umax∈ℛm are a given constant vector. Let tk, k=0,1,… denote the sampling time. The sampling period is defined by hk:=tk+1-tk, which is a nonnegative scalar. Assume that the control input is piecewise constant, that is, the control input is given by
(2)u(t)=u(tk),t∈[tk,tk+1).
Hereafter, we denote u(tk) as uk. In addition, assume that a pair (A,B) 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 t0, the initial state x(t0)=x0, the final time tf, and the final step f are given. Then, find both a control input sequence u0,u1,…,uf-1 and a sampling period sequence h0,h1,…,hf-1 minimizing the following cost function:
(3)J=∫t0tf‍{xT(t)Qx(t)+uT(t)Ru(t)}dt,
under the following two constraints:
(4)hmin≤hk≤hmax,umin≤uk≤umax,h0+h1+⋯+hf-1=tf,
where Q is positive semidefinite, R is positive definite, and hmin,hmax≥0 are a given constant.Next, we present a procedure of MPC based on the self-triggered strategy.Procedure of Self-Triggered MPCStep  1. Set t0=0, and give the initial state x(0)=x0.Step  2. Solve Problem 1.Step  3. Apply only u(t), t∈[t0,t0+h0) to the plant.Step  4. Compute the predicated state x^(t0+h0) by using x(t0), u0, and h0.Step  5. Solve Problem 1 by using x^(t0+h0) as x0.Step  6. Wait until time t0+h0.Step  7. Update t0:=t0+h0, measure x(t0), 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, Problem1 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, onlyu(t), t∈[t0,t0+h0) is applied to the plant. Hence, it is important to consider the problem of finding suitable h0 and u0. By solving this problem repeatedly, we can continue to obtain the control input. Thus, a sampling period sequence h0,h1,…,hf-1 is approximated as follows: (i) only h0 is a decision variable (i.e., the other sampling periods are given in advance), (ii) h1=h2=⋯=hf-1 holds. In addition, the time interval [t0,tf) in Problem 1 is enlarged to [t1,∞). Here, we consider the following two optimal control problems with prediction horizon one.  (i)
One-step input-constrained problem.(ii)
N-step input-constrained problem.In the former problem, the constraint is imposed for onlyu0. In the latter problem, the constraint is imposed for u0,u1,…,uN-1, where N≤f is given in advance.Before these problems are formally given, some preparations are given. In the time interval[t1,tf), suppose that h1=h2=⋯=hf-1=h is satisfied, where h is a given constant (see also Remark 4). In addition, the input constraint umin≤uk≤umax, k=1,2,…,f-1 is ignored. Then, the optimal value of the cost function J=∫t1∞‍{xT(t)Qx(t)+uT(t)Ru(t)}dt can be derived as xT(t1)P(h)x(t1), where P(h) is a symmetric positive definite matrix, which is a solution of the following discrete-time algebraic Riccati equation:
(5)A~T(h)P(h)A~(h)-P(h)-(A~T(h)P(h)B~(h)+S~(h))×(B~T(h)P(h)B~(h)+R~(h))-1×(B~T(h)P(h)A~(h)+S~T(h))+Q~(h)=0,
where
(6)A~(h):=eAh,B~(h):=∫0h‍eAτdτB,(7)[Q~(h)S~(h)S~T(h)R~(h)]:=∫0h‍eFTt[Q00R]eFtdt,F:=[AB00].
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 t0, the initial state x(t0)=x0, and h1=h2=⋯=h are given. Then, find both a control input u0 and a sampling period h0 minimizing the following cost function:
(8)J=∫t0t1‍{xT(t)Qx(t)+uT(t)Ru(t)}dt+xT(t1)P(h)x(t1)
under the following two constraints:
(9)h≤h0≤hmax,umin≤u0≤umax,
where hmax≥0 is a given constant.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, 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, one-step 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, theN-step input-constrained problem is formulated.First, suppose that the input constraint is imposed in the time interval[t0,t0+h0+h(N-1)), where N≥1 is a given integer. In addition, suppose that no input constraint is imposed in the time interval [t0+h0+h(N-1),∞), then, consider the following cost function:
(10)J=J1+J2+J3,J1=∫t0t0+h0‍{xT(t)Qx(t)+uT(t)Ru(t)}dt,J2=∫t0+h0t0+h0+h(N-1)‍{xT(t)Qx(t)+uT(t)Ru(t)}dt,J3=∫t0+h0+h(N-1)∞‍{xT(t)Qx(t)+uT(t)Ru(t)}dt=xT(t0+h0+h(N-1))×P(h)x(t0+h0+h(N-1)).
The optimal value of J3 can be characterized by x(t0+h0+h(N-1)), because no input constraint is imposed in the time interval [t0+h0+h(N-1),∞).Under the above preparation, consider the followingN-step input-constrained problem.Problem 3 (N-step input-constrained problem).
Suppose that for the system (1), the initial time t0, the initial state x(t0)=x0, h1=h2=⋯=h, N≥1, and γ≥1 are given. Then, find a control input sequence u0,u1,…,uT-1 maximizing a sampling period h0 under the following constraints:
(11)h≤h0≤hmax,umin≤uk≤umax,k=0,1,…,N-1,(12)Jh0*≤γJh*,
where hmax≥0 is a given constant, and Jh0* is the optimal value of the cost function of (10), Jh* is the optimal value of the cost function of (10) under h0=h.In Problem3, control performance can be adjusted by suitably giving γ. We remark that in this problem, h0 is maximized under certain constraints. Furthermore, in this problem, a control input sequence is computed, but only the first one (h0) is computed on sampling periods. In this sense, this problem is regarded as a kind of the optimal control problem with prediction horizon one.Hereafter, in Section4, an iterative solution method for the N-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 parameterh 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 t0+h0∈[t0+h,t0+hmax], 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 forN-Step Input-Constrained Problem

First, for a fixedh0, consider deriving Jh0*. The value of Jh* can be derived by a similar method. The value of Jh0* is given by the optimal value of the following optimal control problem.Problem 5.
Suppose that for the system (1), the initial time t0, the initial state x(t0)=x0, h0 and h1=h2=⋯=h, N≥1 are given. Then, find a control input sequence u0,u1,…,uN-1 minimizing the cost function (10) under the input constraint (11).From the conventional result on sampled-data control theory, Problem5 can be equivalently rewritten as the following optimal control problem of time-varying discrete-time linear systems with the input constraint (11).Problem 6.
Suppose that the initial timet0, the initial state x(0)=x0, h0, and h1=h2=⋯=h, N≥1 are given. Consider the following discrete-time linear system:
(13)x1=A~(h0)x0+B~(h0)u0,xk+1=A~(h)xk+B~(h)uk,k≥1,
where xk:=x(tk). Then, find a control input sequence u0,u1,…,uN-1 minimizing the cost function (10), that is,
(14)J=[x0u0]T[Q~(h0)S~(h0)S~T(h0)R~(h0)][x0u0]+∑k=1N-1‍[xkuk]T[Q~(h)S~(h)S~T(h)R~(h)][xkuk]+xNTP(h)xN
under the input constraint (11).Next, consider reducing Problem6 to a QP problem. Define x-:=[x0Tx1T⋯xNT]T and u-:=[u0Tu1T⋯uN-1T]T. Then, we can obtain x-=A-x0+B-u-, where
(15)A-=[IA~(h0)A~(h)A~(h0)A~2(h)A~(h0)⋮A~N-1(h)A~(h0)],B-=[00⋯0B~(h0)0⋯⋮A~(h)B~(h0)B~(h)⋯⋮A~2(h)B~(h0)A~(h)B~(h)⋯⋮⋮⋮⋯0A~N-1(h)B~(h0)A~N-2(h)B~(h)⋯B~(h)].
In addition, we define
(16)Q-:=block-diag(Q~(h0),Q~(h),…,Q~(h),P(h)),S-:=[block-diag(S~(h0),S~(h),…,S~(h))0n×(N-1)m],R-:=block-diag(R~(h0),R~(h),…,R~(h)).
Then, the cost function (14) can be rewritten as follows:
(17)J=x-TQ-x-+2x-TS-u-+u-TR-u-=u-TL2u-+L1u-+L0,
where
(18)L2=R-+B-TS-+S-TB-+B-TQ-B-,L1=2x0TA-T(S-+Q-B-),L0=x0TA-TQ-A-x0.
Finally, u-min:=[uminTuminT⋯uminT]T and u-max:=[umaxTumaxT⋯umaxT]T are also defined.Under the above preparation, Problem6 is equivalent to the following QP problem.Problem A.
Consider(19)findmmllmu0,u1,…,uN-1,minmmllllu-TL2u-+L1u-+L0,subject  tomu-min≤u-≤u-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 Problem3.Algorithm 7.
Step  1. Derive Jh* by solving Problem A with h0=h.Step  2. Set a=h and b=hmax, and give a sufficiently small positive real number ε.Step  3. Set h0=(a+b)/2.Step  4. Derive Jh0* by solving Problem A.Step  5. If Jh0*≤γJh* in Problem 3 is satisfied, then set a=h0, otherwise set b=h0.Step  6. If |a-b|<ε is satisfied, then the optimal h0 in Problem 3 is derived as a, and the optimal control input sequence is also derived. Otherwise go to Step 3.In a numerical example (Section6.1), we will discuss the computation time of Algorithm 7.Finally, we discuss the stabilization issue. For Problem6, consider imposing the constraint V(xk+1)≤ρV(xk), where V(x) is a given nonnegative function, and ρ∈[0,1) is a given constant. If V(x) is restricted to V(x)=∥Px∥∞, where P∈ℛn×n and ∥·∥∞ is an infinity matrix norm, then the constraint V(xk+1)≤ρV(xk) 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.


## 5. Approximate Solution Method for One-Step Input-Constrained Problem

In this section, first we derive a solution method for the one-step input-constrained problem (Problem2). 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
(20)J=[x0u0]T[Q~(h0)S~(h0)S~T(h0)R~(h0)][x0u0]+x1TP(h)x1,(21)x1=eAh0x0+∫0h0‍eAτdτBu0.
We focus on the weight matrices. By using Taylor series expansions, we can obtain the following relation:
(22)eFTt[Q00R]eFt=Φ0+Φ1t+Φ2t2+⋯,
where
(23)Φ0=[Q00R],Φ1=[QA+ATQQBBTQ0],Φ2=[Φ2,11ATQB+12QABBTQA+12BTATQBTQB],Φ2,11=ATQA+12(QA2+(A2)TQ).
Then, the weight matrices of (7) can be rewritten as
(24)[Q~(h0)S~(h0)S~T(h0)R~(h0)]=Φ0h0+12Φ1h02+⋯.
Next, we focus on the term x1TP(h)x1. From x1 of (21), x1TP(h)x1 can be rewritten as
(25)x1TP(h)x1=[x0u0]T[Ψ11Ψ12Ψ12TΨ22][x0u0],
where
(26)Ψ11=eATh0P(h)eAh0,Ψ12=eATh0P(h)(eAh0-I)A-1B,Ψ22=BT(A-1)T(eATh0-I)P(h)(eAh0-I)A-1B.
We use Taylor series expansions for eAh0. Then, we can obtain
(27)Ψ11=P(h)+(P(h)A+ATP(h))h0+(ATP(h)A+12P(h)A2+12(A2)TP(h))h02+⋯,Ψ12=P(h)Ah0+(ATP(h)B+12P(h)AB)h02+⋯,Ψ22=BTP(h)Bh02+⋯.
From these results, the cost function can be expressed as follows:
(28)J=[x0u0]T(Γ0+Γ1h0+Γ2h02+⋯)[x0u0],
where
(29)Γ0=[P(h)000],Γ1=[Γ1,11P(h)BBTP(h)R],Γ2=[Γ2,11Γ2,12Γ2,12TBTP(h)B],Γ1,11=P(h)A+ATP(h)+Q,Γ2,11=12(QA+ATQ+P(h)A2+(A2)TP(h))+ATP(h)A,Γ2,12=ATP(h)B+12(P(h)AB+QB).
In this paper, the second-order truncated Taylor series is used. Furthermore, the term u0T(h0R)u0 appeared in (28) is approximated as follows:
(30)u0T(h0R)u0≈u0T(h02R′)u0,
where R′:=R/((hmax+h)/2). Then, we consider the following cost function Ja:
(31)J≈Ja=[x0u0]T(Γ0+Γ1′h0+Γ2′h02)[x0u0],
where
(32)Γ1′=[Γ1,11P(h)BBTP(h)0],Γ2′=[Γ2,11Γ2,12Γ2,12TR′+BTP(h)B].
Under these preparations, we can obtain the following theorem.Theorem 8.
The one-step input-constrained problem of Problem2 is approximately reduced to the following QP problem.Problem B.
Consider(33)find1111111v0,h0,min111111[v0h0]TM2[v0h0]+M1[v0h0]+M0,subject  to111.h≤h0≤hmax,11111111111uminh0≤v0≤umaxh0,
where v0:=h0u0, and
(34)M2=[R′+BTP(h)BΓ2,12Tx0x0TΓ2,12x0TΓ2,11x0],M1=[2x0TP(h)Bx0TΓ1,11x0],M0=x0TP(h)x0.Proof.
By rewritingJa of (31), the cost function in Problem B is derived. In addition, by using v0,h0, the input constraint umin≤u0≤umax in Problem 2 is equivalent to uminh0≤v0≤umaxh0, which is a linear inequality constraint.By solving Problem B, we can obtain suboptimalu0(=v0/h0) and h0. 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 Theorem8, 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 R′ in (30), the approximation of (30) can be regarded as an over-approximation of u0T(h0R)u0. 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 matrixM2 is not a positive definite matrix generally. In this subsection, we clarify the reason why M2 is not positive definite.The matrixM2 can be rewritten as
(35)M2=[I00x0]TN2[I00x0],
where
(36)N2=N2′-12CTP-1(h)C,N2′=[R′000]+12[BA]TP(h)[BA]+12DTD,C=[0ATP(h)+Q],D=[P1/2(h)BP-1/2(h)(P(h)A+ATP(h)+Q)].
Since P(h) is positive definite, N2′ is also positive definite. However, it is obvious that -(1/2)CTP-1(h)C is not positive definite. Therefore, M2 is not a positive definite matrix generally.Consider approximating the cost function in Problem B, that is,Ja in (31) by a convex function. We define the following positive definite matrix:
(37)M2′:=[I00x0]TN2′[I00x0],
and we rewrite Ja in (31). Then, we can obtain
(38)Ja=[v0h0]TM2′[v0h0]+M1[v0h0]+M0+αh02,α=-12x0T(ATP(h)+Q)T×P-1(h)(ATP(h)+Q)x0.
By approximating the negative term αh02 to a linear term, Problem B can be in general reduced to a convex QP problem. Noting that h≤h0≤hmax, we can make a two-dimensional graph on h0 and αh02. Using the two-dimensional graph obtained, we can evaluate whether the approximation is reasonable.


## 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
(20)J=[x0u0]T[Q~(h0)S~(h0)S~T(h0)R~(h0)][x0u0]+x1TP(h)x1,(21)x1=eAh0x0+∫0h0‍eAτdτBu0.
We focus on the weight matrices. By using Taylor series expansions, we can obtain the following relation:
(22)eFTt[Q00R]eFt=Φ0+Φ1t+Φ2t2+⋯,
where
(23)Φ0=[Q00R],Φ1=[QA+ATQQBBTQ0],Φ2=[Φ2,11ATQB+12QABBTQA+12BTATQBTQB],Φ2,11=ATQA+12(QA2+(A2)TQ).
Then, the weight matrices of (7) can be rewritten as
(24)[Q~(h0)S~(h0)S~T(h0)R~(h0)]=Φ0h0+12Φ1h02+⋯.
Next, we focus on the term x1TP(h)x1. From x1 of (21), x1TP(h)x1 can be rewritten as
(25)x1TP(h)x1=[x0u0]T[Ψ11Ψ12Ψ12TΨ22][x0u0],
where
(26)Ψ11=eATh0P(h)eAh0,Ψ12=eATh0P(h)(eAh0-I)A-1B,Ψ22=BT(A-1)T(eATh0-I)P(h)(eAh0-I)A-1B.
We use Taylor series expansions for eAh0. Then, we can obtain
(27)Ψ11=P(h)+(P(h)A+ATP(h))h0+(ATP(h)A+12P(h)A2+12(A2)TP(h))h02+⋯,Ψ12=P(h)Ah0+(ATP(h)B+12P(h)AB)h02+⋯,Ψ22=BTP(h)Bh02+⋯.
From these results, the cost function can be expressed as follows:
(28)J=[x0u0]T(Γ0+Γ1h0+Γ2h02+⋯)[x0u0],
where
(29)Γ0=[P(h)000],Γ1=[Γ1,11P(h)BBTP(h)R],Γ2=[Γ2,11Γ2,12Γ2,12TBTP(h)B],Γ1,11=P(h)A+ATP(h)+Q,Γ2,11=12(QA+ATQ+P(h)A2+(A2)TP(h))+ATP(h)A,Γ2,12=ATP(h)B+12(P(h)AB+QB).
In this paper, the second-order truncated Taylor series is used. Furthermore, the term u0T(h0R)u0 appeared in (28) is approximated as follows:
(30)u0T(h0R)u0≈u0T(h02R′)u0,
where R′:=R/((hmax+h)/2). Then, we consider the following cost function Ja:
(31)J≈Ja=[x0u0]T(Γ0+Γ1′h0+Γ2′h02)[x0u0],
where
(32)Γ1′=[Γ1,11P(h)BBTP(h)0],Γ2′=[Γ2,11Γ2,12Γ2,12TR′+BTP(h)B].
Under these preparations, we can obtain the following theorem.Theorem 8.
The one-step input-constrained problem of Problem2 is approximately reduced to the following QP problem.Problem B.
Consider(33)find1111111v0,h0,min111111[v0h0]TM2[v0h0]+M1[v0h0]+M0,subject  to111.h≤h0≤hmax,11111111111uminh0≤v0≤umaxh0,
where v0:=h0u0, and
(34)M2=[R′+BTP(h)BΓ2,12Tx0x0TΓ2,12x0TΓ2,11x0],M1=[2x0TP(h)Bx0TΓ1,11x0],M0=x0TP(h)x0.Proof.
By rewritingJa of (31), the cost function in Problem B is derived. In addition, by using v0,h0, the input constraint umin≤u0≤umax in Problem 2 is equivalent to uminh0≤v0≤umaxh0, which is a linear inequality constraint.By solving Problem B, we can obtain suboptimalu0(=v0/h0) and h0. 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 Theorem8, 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 R′ in (30), the approximation of (30) can be regarded as an over-approximation of u0T(h0R)u0. 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 matrixM2 is not a positive definite matrix generally. In this subsection, we clarify the reason why M2 is not positive definite.The matrixM2 can be rewritten as
(35)M2=[I00x0]TN2[I00x0],
where
(36)N2=N2′-12CTP-1(h)C,N2′=[R′000]+12[BA]TP(h)[BA]+12DTD,C=[0ATP(h)+Q],D=[P1/2(h)BP-1/2(h)(P(h)A+ATP(h)+Q)].
Since P(h) is positive definite, N2′ is also positive definite. However, it is obvious that -(1/2)CTP-1(h)C is not positive definite. Therefore, M2 is not a positive definite matrix generally.Consider approximating the cost function in Problem B, that is,Ja in (31) by a convex function. We define the following positive definite matrix:
(37)M2′:=[I00x0]TN2′[I00x0],
and we rewrite Ja in (31). Then, we can obtain
(38)Ja=[v0h0]TM2′[v0h0]+M1[v0h0]+M0+αh02,α=-12x0T(ATP(h)+Q)T×P-1(h)(ATP(h)+Q)x0.
By approximating the negative term αh02 to a linear term, Problem B can be in general reduced to a convex QP problem. Noting that h≤h0≤hmax, we can make a two-dimensional graph on h0 and αh02. Using the two-dimensional graph obtained, we can evaluate whether the approximation is reasonable.


## 6. Numerical Examples

### 6.1. Iterative Solution Method

First, we show an example of the iterative solution method proposed in Section4.Consider the following system:(39)x˙(t)=[01-5-8]x(t)+[01]u(t).
The input constraint is given as u(t)∈[-10,+10]. Parameters in Problem 3 are given as follows: h=0.5, γ=1.001, hmax=5, Q=103In, and R=1. In Algorithm 7, we set ε=10-4. Then, P(h) can be derived as
(40)P(h)=[10615593593575].
In addition, we consider two cases, that is, the case of N=1 and the case of N=10.We show the computational result on self-triggered MPC with theN-step input-constrained problem of Problem 3. The initial state is given as x0=[1010]T, and the case of N=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.Figure 1
State trajectory.Figure 2
Control input.Next, compare two cases. In these cases, the obtained state trajectories are almost the same. The difference between two cases is as follows. In Figures1 and 2, that is, the case of N=10, the control input at each time is shown as follows:
(41)u(t)=-10.00,t∈[0,1.83),h0=1.83,u(t)=-1.90,t∈[1.83,3.23),h0=1.41,u(t)=-0.49,t∈[3.23,4.49),h0=1.25,u(t)=-0.14,t∈[4.49,5.69),h0=1.21,u(t)=-0.04,t∈[5.69,6.89),h0=1.20,u(t)=-0.01,t∈[6.89,8.08),h0=1.19.In the case ofN=1, the control input at each time is derived as follows:
(42)u(t)=-10.00,t∈[0,0.62),h0=0.62,u(t)=-10.00,t∈[0.62,1.66),h0=1.03,u(t)=-2.62,t∈[1.66,2.88),h0=1.22,u(t)=-0.75,t∈[2.88,4.08),h0=1.20,u(t)=-0.22,t∈[4.08,5.27),h0=1.19,u(t)=-0.06,t∈[5.27,6.47),h0=1.19,u(t)=-0.02,t∈[6.47,7.66),h0=1.19,u(t)=-0.01,t∈[7.66,8.85),h0=1.19.
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 N=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 N=10, Problem 3 is solved six times. Hence, it is important to choose a suitable N. We remark that in this example, the computational result in the case of N=20 is the same as that in the case of N=10. In this sense, N=10 is one of the suitable horizons.In addition, we discuss the effect of changingγ in (12). In the case of N=10, consider the following cases: γ=1.001, 1.005, 1.010, 1.015, 1.020. For each case, the first h0 is obtained as follows:
(43)γ=1.001:h0=1.83,γ=1.005:h0=2.38,γ=1.010:h0=2.79,γ=1.015:h0=3.12,γ=1.020:h0=3.44.
From these results, we see that h0 becomes longer by setting a larger γ. Since control performance decreases for a larger γ, it is important to consider the trade-off between γ and h0.Finally, we discuss the computation time for solving theN-step input-constrained problem of Problem 3. In the case of N=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 N=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 Section5.Consider the system (39) again. The input constraint is given as u(t)∈[-10,+10]. Parameters in the one-step input-constrained problem of Problem 2 are given as follows: h=0.2, hmax=1, Q=103In, and R=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 h and hmax is large. Therefore, h and hmax must set carefully. Furthermore, in this example, Problem B is transformed into a convex QP problem. From h=0.2 and hmax=1, the term αh02 is approximated by αh0.We show the computational result on self-triggered MPC with the transformed Problem B. The initial state is given asx0=[1010]T. Figure 3 shows the obtained state trajectory, and Figure 4 shows the control input trajectory. The obtained control input is shown as follow:
(44)u(t)=-10.00,t∈[0,0.86),h0=0.86,u(t)=-10.00,t∈[0.86,1.38),h0=0.52,u(t)=-6.37,t∈[1.38,1.88),h0=0.50,u(t)=-3.57,t∈[1.88,2.40),h0=0.52,u(t)=-2.19,t∈[2.40,2.90),h0=0.50,u(t)=-1.28,t∈[3.41,3.92),h0=0.51,u(t)=-0.77,t∈[3.92,4.43),h0=0.51,u(t)=-0.45,t∈[4.43,4.94),h0=0.51.
From these results, we see that also in this example, the sampling period is nonuniform.Figure 3
State trajectory.Figure 4
Control input.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.


## 6.1. Iterative Solution Method

First, we show an example of the iterative solution method proposed in Section4.Consider the following system:(39)x˙(t)=[01-5-8]x(t)+[01]u(t).
The input constraint is given as u(t)∈[-10,+10]. Parameters in Problem 3 are given as follows: h=0.5, γ=1.001, hmax=5, Q=103In, and R=1. In Algorithm 7, we set ε=10-4. Then, P(h) can be derived as
(40)P(h)=[10615593593575].
In addition, we consider two cases, that is, the case of N=1 and the case of N=10.We show the computational result on self-triggered MPC with theN-step input-constrained problem of Problem 3. The initial state is given as x0=[1010]T, and the case of N=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.Figure 1
State trajectory.Figure 2
Control input.Next, compare two cases. In these cases, the obtained state trajectories are almost the same. The difference between two cases is as follows. In Figures1 and 2, that is, the case of N=10, the control input at each time is shown as follows:
(41)u(t)=-10.00,t∈[0,1.83),h0=1.83,u(t)=-1.90,t∈[1.83,3.23),h0=1.41,u(t)=-0.49,t∈[3.23,4.49),h0=1.25,u(t)=-0.14,t∈[4.49,5.69),h0=1.21,u(t)=-0.04,t∈[5.69,6.89),h0=1.20,u(t)=-0.01,t∈[6.89,8.08),h0=1.19.In the case ofN=1, the control input at each time is derived as follows:
(42)u(t)=-10.00,t∈[0,0.62),h0=0.62,u(t)=-10.00,t∈[0.62,1.66),h0=1.03,u(t)=-2.62,t∈[1.66,2.88),h0=1.22,u(t)=-0.75,t∈[2.88,4.08),h0=1.20,u(t)=-0.22,t∈[4.08,5.27),h0=1.19,u(t)=-0.06,t∈[5.27,6.47),h0=1.19,u(t)=-0.02,t∈[6.47,7.66),h0=1.19,u(t)=-0.01,t∈[7.66,8.85),h0=1.19.
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 N=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 N=10, Problem 3 is solved six times. Hence, it is important to choose a suitable N. We remark that in this example, the computational result in the case of N=20 is the same as that in the case of N=10. In this sense, N=10 is one of the suitable horizons.In addition, we discuss the effect of changingγ in (12). In the case of N=10, consider the following cases: γ=1.001, 1.005, 1.010, 1.015, 1.020. For each case, the first h0 is obtained as follows:
(43)γ=1.001:h0=1.83,γ=1.005:h0=2.38,γ=1.010:h0=2.79,γ=1.015:h0=3.12,γ=1.020:h0=3.44.
From these results, we see that h0 becomes longer by setting a larger γ. Since control performance decreases for a larger γ, it is important to consider the trade-off between γ and h0.Finally, we discuss the computation time for solving theN-step input-constrained problem of Problem 3. In the case of N=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 N=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 Section5.Consider the system (39) again. The input constraint is given as u(t)∈[-10,+10]. Parameters in the one-step input-constrained problem of Problem 2 are given as follows: h=0.2, hmax=1, Q=103In, and R=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 h and hmax is large. Therefore, h and hmax must set carefully. Furthermore, in this example, Problem B is transformed into a convex QP problem. From h=0.2 and hmax=1, the term αh02 is approximated by αh0.We show the computational result on self-triggered MPC with the transformed Problem B. The initial state is given asx0=[1010]T. Figure 3 shows the obtained state trajectory, and Figure 4 shows the control input trajectory. The obtained control input is shown as follow:
(44)u(t)=-10.00,t∈[0,0.86),h0=0.86,u(t)=-10.00,t∈[0.86,1.38),h0=0.52,u(t)=-6.37,t∈[1.38,1.88),h0=0.50,u(t)=-3.57,t∈[1.88,2.40),h0=0.52,u(t)=-2.19,t∈[2.40,2.90),h0=0.50,u(t)=-1.28,t∈[3.41,3.92),h0=0.51,u(t)=-0.77,t∈[3.92,4.43),h0=0.51,u(t)=-0.45,t∈[4.43,4.94),h0=0.51.
From these results, we see that also in this example, the sampling period is nonuniform.Figure 3
State trajectory.Figure 4
Control input.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.


## 7. Conclusion

In this paper, we discussed self-triggered MPC of linear systems. Since it is difficult to solve the original problem (Problem1), two control problems (the one-step input-constrained problem of Problem 2 and the N-step input-constrained 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 Problem3. 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 closed-loop system. The stabilization issue for the one-step input-constrained problem is also important as one of the future works.



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*Source: 916040-2013-11-19.xml*