Uncertain bang–bang control problem for multi-stage switched systems

Journal Pre-proof Uncertain bang-bang control problem for multi-stage switched systems Hongyan Yan, Ting Jin, Yun Sun

PII: DOI: Reference:

S0378-4371(19)32272-1 https://doi.org/10.1016/j.physa.2019.124115 PHYSA 124115

To appear in:

Physica A

Received date : 12 September 2019 Revised date : 23 December 2019 Please cite this article as: H. Yan, T. Jin and Y. Sun, Uncertain bang-bang control problem for multi-stage switched systems, Physica A (2020), doi: https://doi.org/10.1016/j.physa.2019.124115. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier B.V.

Journal Pre-proof

Uncertain Bang-Bang Control Problem for Multi-Stage Switched Systems Hongyan Yan1∗ Ting Jin1,2 1 School

of Science, Nanjing Forestry University, Nanjing 210037, China

of Science, Nanjing University of Science and Technology, Nanjing 210094, China

of Science, Nangjing University of Posts and Telecommunications, Nanjing 210023, China

Abstract

p ro

3 School

Yun Sun3

of

2 School

and

1

Introduction

Pr e-

A control problem for multi-stage switched systems with subsystems disturbed by uncertainty is presented. This problem is formulated as computing the optimal continuous input and the optimal switching strategy that jointly optimizes a linear performance index. Analytic expressions are derived for both of them. Enumeration method for solving such a control problem suffers a high computational requirement due to the fact that an exponential number of switching sequences must be explored. Genetic Algorithm is chosen to improve the computing efficiency. The examples validate the effectiveness of the method. Keywords: bang-bang control; uncertain theory; multi-stage switched system; optimal control; Genetic Algorithm

Jo

urn

al

Switched systems are essentially a particular category of hybrid systems. A finite number of subsystems and a switching law among them compose these systems. Mathematically, a collection of indexed differential or difference equations are usually used to describe such systems. Different from traditional optimal control problems, the distinctive character of optimal control problems of switched systems involves finding both the switching law and continuous inputs to jointly optimize certain performance criterion. Due to its diverse engineering applications, more and more attentions [1–5] have been attracted by such problems. The existing approaches to the problems of optimal control for switched systems can be classified into two groups: the ones pointing to the continuous-time case and others pertaining to the multi-stage case. For the continuous-time case, when the sequence of the deployed system is preassigned, many computational results are proposed to optimize on the switched times. For example, in order to seek both optimal switching times and optimal continuous inputs, a two-stage optimization strategy was presented by Xu and Antsaklis [6].A control parameterization technique and the time scaling transform method were proposed by Teo etal [7, 8] to find the approximate optimal control inputs and switching instants, which have been used extensively. Furthermore, a general continuous-time switching problem was studied in [9] based on maximum principle and an embedding method. For the multi-stage case, the linear quadratic control problems for multi-stage switched systems were investigated in [10]. Moreover, based on an efficient branch and bound algorithm, Gao et al [11] studied the problems with a constant switching cost. Motivated by the problems of viral mutation in HIV infection, [12] considered discrete-time control for switched positive systems. By far, researches on deterministic optimal control models of switched systems are the mainstream. However, indeterminacy is ubiquitous in reality. As a consequence, the controller may not be able to serve its purpose or even lead systems to be inactive due to the influence of such indeterminacy. Therefore, it’s necessary to discuss optimal control problems of uncertain switched systems. Most literatures characterized the involved uncertainty as randomness and implement optimal schemes in the stochastic environment [13–15]. As is well known, adequate historical data is required to estimate the probability distribution when applying probability distribution. However, in many cases, No data is available. Based ∗ E-mail:

[email protected]

1

Journal Pre-proof

2

Uncertainty Theory

Pr e-

p ro

of

on this situation, some domain experts participate to evaluate the degree of belief that each event may happen. However, human beings usually overweight unlikely events [16] and it will result in a much wider range of values than the true frequency [17]. In order to deal with degree of belief reasonably, uncertainty theory was founded in 2007 [18] and developed by Liu [17] as a branch of axiomatic mathematic for modeling human uncertainty in 2010. After that, uncertainty theory has been widely applied to many other fields, such as portfolio selection [19–21], production-inventory [22], option pricing [23–25] and differential game [26]. Nowadays probability theory and uncertainty theory are complementary mathematical systems that provide two acceptable mathematical models to handle the indeterminate world. Probability theory is a branch of mathematics for modelling frequencies. We choose uncertainty theory to model belief degrees. Based on uncertainty theory, the expected value model of uncertain optimal control problem was introduced and dealt with by Zhu [19] in 2010. The equation of optimality for this model was presented and applied to solve a portfolio selection problem. Then, with fixed switching sequence, a bang-bang control model with optimistic value criterion for uncertain switched systems was investigated in [27]. Two-stages algorithms were used to solve this problem. The first stage was to seek the minimum value of the cost function. In the second stage, GA and PSO algorithms were used to solve optimization problems. However, many realistic switched systems are discrete-time. Sometimes, continuous-time systems even need to be discretized in order to solve the problems as in [28, 29]. Therefore, in this paper, we focus on the optimal control problems of multi-stage switched systems with subsystems perturbed by the above uncertain factors. The goal is to jointly design a deterministic switching sequence and a continuous control law to optimize the expectation of a linear cost function. The optimal control strategy is characterized analytically. However, an exponential number of switching sequences must be explored. Genetic Algorithm is chosen to improve the computing efficiency. This paper is organized as follows. In section 2, some basic concepts of uncertainty theory are reviewed. In section 3, a bang-bang control problem for uncertain multi-stage switched system is formulated. In section 4, its value function and optimal control strategy are derived analytically. Moreover, a numerical framework is proposed. In section 5, two numerical examples are given to validate the effectiveness of the method.

al

For convenience, in this section, we review some concepts such as uncertain measure, uncertain variable and expected value of an uncertain variable.

Jo

urn

Definition 1 (Liu [18]) Let Γ be a nonempty set, and L a σ -algebra over Γ. Each element Λ ∈ L is called an event. A set function M defined on the σ -algebra L is called an uncertain measure if it satisfies following axioms: (i) (Normality Axiom) M(Γ) = 1; (ii)(Self-Duality Axiom) M(Λ) + M(Λc ) = 1 for any event Λ ∈ L; S ∞ (iii)(Countable Subadditivity Axiom) M( ∞ i=1 Λi ) ≤ ∑i=1 M(Λi ) for every countable sequence of events {Λi } ∈ L. The triplet (Γ, L, M) is called an uncertainty space. Besides, an axiom called product measure axiom was given by Liu [30] for the operation of uncertain variables in 2009. (iv)(Product Measure Axiom) Let (Γk , Lk , Mk ) be uncertainty spaces for k = 1, 2, · · · . Then the product uncertain measure M is an uncertain measure satisfying ∞

M{ ∏ Λk } = k=1

∞ ^

k=1

Mk {Λk }

where Λk are arbitrarily chosen events from Lk for k = 1, 2, · · · , respectively. An uncertain variable is a measurable function from an uncertainty space (Γ, L, M) to the set R of real numbers, and an uncertain vector is a measurable function from an uncertainty space to Rn . In order to describe an uncertain variable, the concept of an uncertainty distribution is defined as follows. Definition 2 (Liu [18]) The uncertainty distribution Φ : R → [0, 1] of an uncertain variable ξ is defined by Φ(x) = M{ξ ≤ x} 2

Journal Pre-proof for any real number x. Example 1 (Liu [18]) An uncertain variable ξ is called linear if it has a linear uncertainty distribution   0, if x ≤ a,   Φ(x) = (x − a)/(b − a), if a ≤ x ≤ b,    1, if x ≥ b,

(1)

of

where a and b are real numbers with a < b. Such a linear uncertain variable is denoted by ξ ∼ L(a, b).

denoted by N(e, σ ) where e and σ are real numbers with σ > 0.

p ro

Example 2 (Liu [18]) An uncertain variable ξ is said to be normal if it has a normal uncertainty distribution −1   π(e − x) √ , x ∈ R. Φ(x) = 1 + exp 3σ Definition 3 (Liu [18]) The expected value of an uncertain variable ξ is defined by E[ξ ] =

Z +∞ 0

M{ξ ≥ r}dr −

−∞

M{ξ ≤ r}dr

Pr e-

provided that at least one of the two integrals is finite.

Z 0

T

The uncertain variables ξ1 , ξ2 , · · · , ξm are said to be independent if M{ m i=1 {ξ ∈ Bi }} = min1≤i≤m M{ξ ∈ Bi } for any Borel sets B1 , B2 , · · · , Bm of real numbers. For numbers a and b, E[aξ + bη] = aE[ξ ] + bE[η] if ξ and η are independent uncertain variables. Remark 1 The difference between probability theory (Kolmogorov [31]) and uncertainty theory (Liu [17] ) does not lie in whether the measures are additive or not, but how the product measures are defined. The product probability measure is the multiplication of the probability measures of the individual events, i.e., Pr(Λ1 × Λ2 ) = Pr(Λ1 ) × Pr(Λ2 ),

al

while the product uncertain measure is the minimum of the uncertain measures of the individual events, i.e.,

3

urn

M(Λ1 × Λ2 ) = M(Λ1 ) ∧ M(Λ2 ).

Problem Statement

Jo

When switched systems are disturbed by indeterminant factors, but no samples are available to estimate distribution function, uncertainty theory is effective to solve problems. This section aims to introduce this kind of problem. Moreover, recurrence equation for the problem will be derived. Considering the following uncertain multi-stage linear switched systems described by x(k + 1) = fy(k) x(k) + gy(k) u(k) + σk+1 ξk+1 , k = 0, 1, · · · , N − 1

(2)

where (i) for each k ∈ K , {0, 1, · · · , N − 1}, x(k), u(k) are respectively the state and the continuous input at stage k and u(k) ∈ [−1, 1]; (ii) M , {1, · · · , m} is a finite set collecting the indices of the different subsystems, y(k) ∈ M refers to the value taken by the switching signal (also called the discrete input) at time k; (iii) for each i ∈ M, the pair ( fi , gi ) is associated with the subsystem i; (iv) for each k ∈ K, σk+1 6= 0, ξk is the disturbance and ξ1 , ξ2 , · · · , ξN are independent uncertain variables. Throughout the paper, we use notations of the types u(·) and y(·) to designate the control input u(k)|N−1 and the 0 switching input y(k)|N−1 . y(k) is assumed to be selected freely in K without any constraint. For system (2), we wish to 0 3

Journal Pre-proof optimize the objective ∑N−1 k=0 (ay(k) x(k) + by(k) u(k)) + sN xN . Since x(k) is an uncertain variable, which can’t be regarded as a real value to be optimized. In this paper, we introduce expected value to rank it. Therefore, the performance of u(·) and y(·) can be measured by the following expected value. " # E

N−1

∑ (ay(k) x(k) + by(k) u(k)) + sN xN

(3)

k=0

p ro

of

where, for any i ∈ M, (ai , bi ) constitutes the cost pair of the i-th subsystem. The goal of this paper is to solve the following problem. P roblem 1. Find u∗ (·) and y∗ (·) to maximize (3) subject to the difference system (2) with initial state x(0) = x0 . Let J(k, xk ) denote the performance index corresponding to the situation when at stage k we are in state x(k) = xk . Then, we have " #  N−1    max E ∑ (ay( j) x( j) + by( j) u( j)) + sN xN   J(k, xk ) = u( j)∈[−1,1],y( j)∈M,  j=k   k≤ j≤N     subject to (4)   x( j + 1) = f x( j) + g u( j) + σ ξ , j+1 j+1  y( j) y( j)     j = k, · · · , N − 1,      x(k) = xk . Theorem 1 For model (4), we have J(k, xk ) =

max

Pr e-

For P roblem 1, the following recurrence equation can be derived.

u(k)∈[−1,1] y(k)∈M

for k = 0, 1, · · · , N − 1.

J(N, xN ) = sN xN ,   E (ay(k) x(k) + by(k) u(k)) + J(k + 1, xk+1 ) ,

=

E

∑ (ay( j) x( j) + by( j) u( j)) + sN xN

j=k

(

max

u( j)∈[−1,1] y( j)∈M, k≤ j≤N

max

u( j)∈[−1,1] y( j)∈M, k≤ j≤N

E (ay(k) x(k) + by(k) u(k)) + E (

E (ay(k) x(k) + by(k) u(k)) +

=

max

u( j)∈[−1,1] y( j)∈M, k≤ j≤N

"

#

N−1

∑

j=k+1

max

u( j)∈[−1,1] y( j)∈M, k+1≤ j≤N

(ay( j) x( j) + by( j) u( j)) + sN xN

E



N−1

∑

j=k+1

In addition, for any u(i), y(i), k ≤ i ≤ N, we have " J(k, xk ) ≥ E

N−1

∑ (ay( j) x( j) + by( j) u( j)) + sN xN

"

j=k

= E ay(k) x(k) + by(k) u(k) +

N−1

∑

j=k+1

4

#)

(ay( j) x( j) + by( j) u( j)) + sN xN

  E (ay(k) x(k) + by(k) u(k)) + J(k + 1, xk+1 )

Jo

≤

max

u( j)∈[−1,1] y( j)∈M, k≤ j≤N

N−1

urn

J(k, xk ) =

"

al

Proof: It is obvious that J(N, xN ) = sN xN . For any k = N − 1, N − 2, · · · , 1, 0, we have

#

(ay( j) x( j) + by( j) u( j)) + sN xN

) (5)

#

Journal Pre-proof Since J(k, xk ) is independent of u(i), y(i) for k + 1 ≤ i ≤ N, we have (  J(k, xk ) ≥ E ay(k) x(k) + by(k) u(k) +

max

u( j)∈[−1,1] y( j)∈M, k+1≤ j≤N

N−1

∑

E

j=k+1

(ay( j) x( j) + by( j) u( j)) + sN xN

)

  = E (ay(k) x(k) + by(k) u(k)) + J(k + 1, xk+1 )

Taking the maximum of u(k), y(k) in the previous inequality yields max

u( j)∈[−1,1] y( j)∈M

  E (ay(k) x(k) + by(k) u(k)) + J(k + 1, xk+1 ) .

of

J(k, xk ) ≥

(6)

4

p ro

According to Eqs.(5) and (6), the recurrence equation is proved. By using the recurrence equation, we can obtain the exact solution for P roblem 1.

Bang-Bang Control

4.1

Pr e-

In this section, the optimal switching control and continue control of P roblem 1 will be derived in details. In order to implement the optimal control, we also design the specialized algorithm.

Analytical Solutions of Optimal Control

Firstly, when k = N, we have

J(N, xN ) = sN xN .

Denote pN = sN , qN = 0, rN = 0. So J(N, xN ) can be written as

J(N, xN ) = pN xN + qN + rN . For k = N − 1, the following equation u(N−1)∈[−1,1] y(N−1)∈M

max

u(N−1)∈[−1,1] y(N−1)∈M

+qN + rN ] =

  E (ay(N−1) x(N − 1) + by(N−1) u(N − 1)) + J(N, xN )

 E (ay(N−1) x(N − 1) + by(N−1) u(N − 1)) + PN ( fy(N−1) x(N − 1) + gy(N−1) u(N − 1) + σN ξN )

urn

=

max

al

J(N − 1, xN−1 ) =

max

u(N−1)∈[−1,1] y(N−1)∈M

 (ay(N−1) + fy(N−1) pN )x(N − 1) + (by(N−1) + gy(N−1) pN )u(N − 1) + qN

+pN σN E[ξN ] + rN ]

(7)

Jo

holds by Theorem 1. let {Hi }Ni=0 denote the set of ordered pairs of vectors defined recursively: H0 = {(pN , qN , rN )}, Hk+1 =

with Γk (α) =

[

i∈M

for k = 0, 1, · · · , N − 1 where

[

Γk (α)

α∈Hk

{ρi (α)}, α ∈ Hk


ρi (pk , qk , rk ) = ai + fi pk , |bi + gi pk | + qk , pk σk E[ξk ] + rk . 5

Journal Pre-proof

p ro

of

The definition of the sets {Hk } is shown clearly by Fig.1:

Figure 1: The definition of Hk

Pr e-

According to the definition of ρi (pk , qk , rk ) and Hk , Eq.(7) can be written as J(N − 1, xN−1 ) = Moreover, y∗N−1 satisfies y∗ (N − 1) = arg

max

(pN ,qN ,rN )∈H1

{pN−1 xN−1 + qN−1 + rN−1 }.

max

(pN−1 ,qN−1 ,rN−1 )∈H1

Thus,

{pN−1 xN−1 + qN−1 + rN−1 }.

is a bang bang control. Furthermore,

al

u∗ (N − 1) = sgn{by∗ (N−1) + gy∗ (N−1) pN }

where

urn

∗ J(N − 1, xN−1 ) = p∗N−1 xN + q∗N−1 + rN−1

∗ (p∗N−1 , q∗N−1 , rN−1 ) = ρy∗ (N−1) (pN , qN , rN ).

For k = N − 2, we have

J(N − 2, xN−2 ) =

u(N−2)∈[−1,1] y(N−2)∈M

max

u(N−2)∈[−1,1] y(N−2)∈M

Jo

=

max

  E (ay(N−2) x(N − 2) + by(N−2) u(N − 2)) + J(N − 1, xN−1 )

 (ay(N−2) + fy(N−2) p∗N−1 )x(N − 2) + (by(N−2) + gy(N−2) p∗N−1 )

∗ u(N − 1) + q∗N−1 + p∗N−1 σN−1 E[ξN−1 ] + rN−1

=

max

(pN−2 ,qN−2 ,rN−2 )∈H2




{pN−2 xN−2 + qN−2 + rN−2 }.

By the similar method, we can obtain y∗ (N − 2) = arg

max

(pN−2 ,qN−2 ,rN−2 )∈H2

{pN−2 xN−2 + qN−2 + rN−2 },

u∗ (N − 2) = sgn{by∗ (N−2) + gy∗ (N−2) p∗N−1 } 6

Journal Pre-proof and ∗ J(N − 2, xN−2 ) = p∗N−2 xN−2 + q∗N−2 + rN−2 , ∗ where (p∗N−2 , q∗N−2 , rN−2 ) = ρy∗ (N−2) (pN−1 , qN−1 , rN−1 ). By induction, we can obtain the following theorem.

Theorem 2 At stage k, for given xk , the optimal switching control of P roblem 1 is max

(pk ,qk ,rk )∈HN−k

{pk xk + qk + rk },

and the optimal continuous control

(

(

(

The optimal value of P roblem 1 is

(9)

pk = ay(k) + fy(k) p∗k+1 pN = sN ,

qk = |by(k) + gy(k) p∗k+1 | + q∗k+1 qN = 0, ∗ rk = p∗k+1 σk+1 E[ξk+1 ] + rk+1 rN = 0.

Pr e-

is a bang bang control, where

(8)

p ro

u∗ (k) = sgn{by∗ (k) + gy∗ (k) p∗k+1 }

of

y∗ (k) = arg

J(0, x0 ) = p∗0 x0 + q∗0 + r0∗ .

(10)

4.2

urn

al

Theorem 2 reveals that, at iteration k, the optimal value and the optimal control law at all the future iterations depend on the current set Hk and the state x(k). Unfortunately, only the initial state x0 is available. It turns out that the optimal switching control (8) and the optimal continuous control (9) can be implemented at the price of computing Hk . However, the cardinality of Hk is mk which grows exponentially with respect to k. So the above implementation requires an exponential load in terms of both computational and storage resources. This complexity restricts the applicability of enumeration method to the controls of switched systems with small number of subsystems and small control horizons. Being presented with such difficulties, Genetic Algorithm which offers a high degree of flexibility and robustness in dynamic environment may be a good choice to solve the problem.

GA to Implement the Optimal Control

4.2.1

Jo

Genetic Algorithm(GA) is an efficient optimization method motivated by Darwin’s theory of natural selection. The algorithm operates by repeating updating a group of chromosomes which are called the population. All chromosomes from each populations are ranked by their fitness at each iteration. The fitness evaluates the quality of chromosome compared to the other chromosomes in the population. A new generation is updated by selecting specific chromosomes from the existing population. Some of these selected chromosomes are copied into the next generation population and the others are used in crossover and mutation operation to produce new offsprings. In order to use GA to implement the optimal controls (8) and (9), the chromosome representation in the population, the fitness evaluation and the genetic operations should be determined. Specific details associated with GA for P roblem 1 are described in the following. Encoding scheme

Each chromosome in the population represents a feasible solution to the problem. For P roblem 1, we want to find the optimal switching control, so a chromosome is a vector of integers of size N, which is used to present the switching law y(k)|N−1 k=0 = (y(0), · · · y(N − 1)), where y(k) ∈ M , {1, · · · , m}. Fig.2 shows the chromosome representation. It is shown that we have a set of candidates for each stage and the chromosome is represented as a vector of selected candidates. 7

Journal Pre-proof

p ro

of

The length of the vector equals N which is the number of stages in P roblem 1 as shown in the bottom of Fig.2. Every chromosome represents one of the switching control. For example, Fig.2 represents the switching control y(k)|N−1 k=0 = (2, 1, 2, · · · M − 2).

4.2.2

Pr e-

Figure 2: Chromosome representation

Fitness evaluation

The purpose of the optimization is to search the optimal control with maximum value of Eq.(3). So the optimal value J(0, x0 ) of P roblem 1 is chosen for the fitness of a chromosome. 4.2.3

Parent selection

Genetic operations

urn

4.2.4

al

Parent selection procedure aims at selecting a set of parents to generate the offsprings. Parent selection emulates the survival of the fittest mechanism in nature. Tournament selection [32] is used to produce offsprings, which is efficient to code. It allows the selection pressure to be easily adjusted.

Jo

Genetic operations comprise crossover operators and mutation operators. Crossover is a recombination operator used to produce offsprings. Better chromosomes can be explored by exchanging information contained in two parents. There are many crossover operations. In this paper, the single-point crossover is used to generate the offsprings. A position on both parents chromosomes is picked randomly. Genes to the right of that position are swapped between the two parent chromosomes. For example: Chromosome(Parent1) : 6 5 1 1 4 3 7 Chromosome(Parent2) : 4

6

3 7

2

4 1

Number 3 is the randomly selected position. Now, the genes behind position Number 3 are cutting and vice versa to produce the following new offspring. Chromosome(O f f spring1) : 6

5

1 7

2 4

1

Chromosome(O f f spring2) : 4 6

3 1

4

7

3

This results in two offsprings each carrying some genetic information from both parents. Mutation is used to maintain genetic diversity from one generation of a population to the next generation. It allows a certain offsprings to obtain new features that are not in its parents. In this paper, the external mutation is adopted which 8

Journal Pre-proof considers the paths apart from initial population. So it can avoid the convergence of the solutions to local optimal. It is a very efficient approach for mutation that operates as follows. A random point (ie 5) is chosen to be displaced and another random point which doesn’t appear in this chromosome is inserted as following: Be f ore : 6

5 1

7 2

4 1

Take the 5 out of the sequence and reinsert the 3 which doesn’t appear in the previous chromosome at the same position: 3 1

7 2

4 1

of

A f ter : 6

The effectiveness of such implementing method will be illustrated in the following section.

Illustrative Examples

p ro

5

Pr e-

Example 3 Consider the uncertain discrete-time optimal control problem 1 with N = 3, m = 3, s3 = 1, x(0) = −1, ξi ∼ L(0, 1) and   ! ! ! ! ! ! 0.01 f1 2 f2 −1 f3 −2   = , = , = , σ = 0.03 , g1 1 g2 2 g3 2 0.02 ! ! ! ! ! ! a1 5 a2 1 a3 −2 = , = , = . b1 −2 b2 −2 b3 5 The number of elements in H3 is 33 = 27. First we list all these 27 elements in HN and substitute them into Eq.(10), the optimal switching control y∗ (·) = (y∗ (0), y∗ (1), y∗ (2)) = (3, 1, 1) and the optimal input control

u∗ (·) = (u∗ (0), u∗ (1), u∗ (2)) = (1, 1, −1)

al

are found. Then we apply the Genetic Algorithm stated in Section 4.2 to this example. The parameters are shown in Table 1. It yields the same result as above. It turns out that Genetic Algorithm is feasible to implement the optimal control of

Jo

P roblem 1.

urn

Table 1: The parameter of Example 3 The parameter Selected value population 20 crossover probability pc 0.7 mutation probability pm 0.3 iteration 100

Example 4 Consider the uncertain discrete-time optimal control problem 1 with N = 15, m = 5, s3 = 1, x(0) = 1, ξi ∼ L(0, 1) and             f1 0.2 g1 0.1 a1 5              f2  −0.1 g2   0.2  a2  −6              f3  = −0.2 , g3  = −0.2 , a3  =  2  ,                          f4   0.1  g4  −0.2 a4   3  f5 0.2 g5 0.1 a5 −3

9

Journal Pre-proof     −0.3 b1     b2   0.2      b3  = −0.1 , σ = 0.01 (i = 1, · · · , 15).         b4   0.2  0.1 b5

al

Pr e-

Table 2: The optimal results of Example 4 y∗ (k) optimal(pk , qk , rk ) u∗ (k) 1 (5.4751, 13.4671, -0.1321) -1 4 (2.3755, 13.4047, -0.1439) 1 2 (-6.2454, 11.9556, -0.1127) 1 4 (2.4536, 11.2649, -0.1250) 1 2 (-5.4638, 9.9721, -0.0977) -1 2 (-5.3625, 9.0996, -0.0708) -1 2 (-6.3752, 8.0246, -0.0390) 1 1 (3.7525, 7.0741, -0.0577) -1 2 (2.3764, 5.4750, -0.0384) 1 4 (-6.2356, 4.0279, -0.0072) 1 2 (2.3556, 3.3568, -0.0190) 1 4 (-6.4440, 1.8680, 0.0132) 1 2 (4.4400, 0.7800, -0.0090) 1 1 (-2.8000, 0.2000, 0.0050) -1 5 (1.0000, 0.0000, 0.0000) 1

x(k) 1.0000 0.0938 -0.1900 0.2290 -0.1760 -0.1819 -0.1786 0.2278 -0.0507 0.2140 -0.1737 0.2236 -0.1752 0.2178 -0.4990

Jo

urn

k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

p ro

of

As a matter of fact the cardinality of H15 for the current experiment is 515 ≈ 3.0518 × 1010 , which is impossible by enumeration. Applying GA to this example yields the results presented in Table 2. The parameters pc and pm are the same as Example 3, population = 80 and iteration = 800. The second column shows the optimal switching control y∗ (·) = (1, 4, 2, 4, 2, 2, 2, 1, 2, 4, 2, 4, 2, 1, 5). The optimal input control u∗ (·) = (−1, 1, 1, 1, −1, −1, 1, −1, 1, 1, 1, 1, 1, −1, 1) is listed in the forth column. The optimal switching control and input control are shown in Fig.3 and Fig.4 respectively. The optimal value is J(0, x0 ) = 18.8101.

Figure 3: The optimal switching control

6

Figure 4: The optimal input control

Conclusion

In this paper, a bang bang control problem for uncertain multi-stage switched systems is presented, together with a method to design a switching law and a continuous control strategy. The analytical solutions of the optimal strategy and the optimal 10

Journal Pre-proof objective function can be exactly characterized by Hk whose size grows exponentially with respect to the length of control horizon. Genetic Algorithm is applied to implement the optimal control. The examples validate the effectiveness of the method.

Acknowledgments

of

This work is supported by the Natural Science Foundation of Jiangsu Province (No.BK20170916 and No.BK20190723), National Natural Science Foundation of China (No.61673011), and Youth Science and Technology Innovation Fund of Nanjing Forestry University (NO.CX2016021).

p ro

References

Jo

urn

al

Pr e-

[1] M. Rinehart, M. Dahleh, D. Reed, & I. Kolmanovsky, Suboptimal control of switched systems with an application to the DISC engine, IEEE Transactions on Control Systems Technology, 16(2)(2008): 189-201. [2] K. Uthaichana, R. Decarlo, S. Bengea, S. Pekarek, & M. Zefran, Hybrid optimal theory and predictive control for power management in hybrid vehicle. Journal of Nonlinear Systems and Applications, 2(1-2)(2011): 96-110. [3] X. Li, J. Lou, Z.Wang & F. Alsaadi, Synchronization of nonlinearly coupled dynamical networks under hyrid pinning impulsive controllers, Journal of the Franklin Institute, 355(2018): 6520-6530. [4] Y. Li, J. Liu, J. Lou, J. Lu, Z.Wang & Y. Liu, Output tracking of boolean control networks driven by constant reference signal, IEEE Access, 7(2019): 112572-112577. [5] C. Liu, Z. Gong & E. Feng, Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture, Journal of Industrial and Management optimization, 5(4)(2017): 835-850. [6] X. Xu & P. Antsaklis, Optimal control of switched systems based on parameterization of the switching instants, IEEE Transactions on Automatic Control, 49(1)(2004): 2-16. [7] K. Teo, C. Goh & K. Wong, (1991). A Unified Computational Approach to Optimal Control Problems. Longman Scientific and Technical, New York. [8] H. Lee, K. Teo, V. Rehbock & L. Jennings, Control parametrization enhancing technique for optimal discrete-valued control problems. Automatica, 35(8)(1999): 1401-1407. [9] S. Benga & R. Decarlo, Optimal control of switching systems. Automatica, 41(1)(2005): 11-27. [10] W. Zhang, J. Hu & A. Abate, On the value function of the discrete-time switched lqr problem, IEEE Transactions on Automatic Control, 54(2009): 2669-2674. [11] J. Gao & L. Duan, Linear-quadratic switching control with switching cost, Automatica, 48(2012): 1138–1143. [12] H. Esteban, C. Patrizio, M. Richard, & B. Franco, Discrete-time control for switched positive systems with application to mitigating viral escape. International Journal of Robust and Nonlinear Control, 21(10)(2011): 1093-1111. [13] Z. Majid & A. Alessand, Approxminately bisimilar symbolic models for randomly switched stochastic systems, Systems & Control Letters, 69(2014): 38-46. [14] L. Wu, D. Ho & C. Li, Sliding mode control of switched hybrid systems with stochastic perturbation, Systems & Control Letters, 60(8)(2011): 531-539. [15] C. Ahmet & H. Tomohisa, Feedback control of switched stochastic systems using randomly available active mode, Automatica, 52(2015): 55-62. [16] D. Kahneman & A. Tversky, Prospect theory: An analysis of decision under risk, Econometrica, 47(1979): 263-292. [17] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Berlin: Springer-Verlag, 2010. [18] B. Liu, Uncertainty Theory, 2nd ed., Berlin: Springer-Verlag, 2007. [19] Y. Zhu, Uncertain optimal control with application to a portfolio selection model, Cybernetics and Systems: An International Journal, 41(7)(2010): 535-547. [20] K. Yao & X. Ji, Uncertain decision making and its application to portfolio selection problem, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 22(1)(2014):113-123. [21] B. Li, Y. Zhu, Y. Sun, G. Aw & K. Teo, Multi-period portfolio selection problem under uncertain environment with bankruptcy constraint, Applied Mathematical Modelling, 56(2018): 539-550. [22] L. Sheng, Y. Zhu & K. Wang, Uncertain dynamical system-based decision making with application to production-inventory problems, Applied Mathematical Modelling, 56(2018): 275-288. [23] T. Jin, Y. Sun & Y. Zhu, Extreme value for solution to uncertain fractional application to American option pricing model, Physcia A, 534(2019), 122357.

11

Journal Pre-proof

Jo

urn

al

Pr e-

p ro

of

[24] Z. Lu, H. Yan & Y. Zhu, European option pricing model based on uncertain fractional differential equation, Fuzzy Optimization and Decsion Making, 18(2)(2019): 199-217. [25] X. Chen, American option pricing formula for uncertain financial market. International Journal of Operational Research, 8(2011): 32-37. [26] Y. Sun, C. Ding & Y. Zhu, Saddle point equilibrium under uncertain environment, Mathematical Methods in the Applied Sciences, 41(5)(2018): 2063-2073. [27] H. Yan & Y. Zhu, Bang-bang control model with optimistic value criterion for uncertain switched systems, Journal of Intelligent Manufacturing, 28(2017): 527-534. [28] L. Wang, A. Beydoun, J. Sun & I. Kolmanasovsky, Optimal hybrid control with application to automotive powertrain systems, Lecture Notes in Control and Information Science, 222(1997): 190-200. [29] S. Boubakera, M. Djemaic, N. Manamannid & F. M’Sahlie, Active modes and switching instants identification for linear switched systems based on discrete particle swarm optimization, Applied Soft Computing, 14(Part C)(2014): 482-488. [30] B. Liu, Some research problems in uncertainty theory, Journal of Uncertain System, 3(2009): 3-10. [31] A. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung, Julius Springer, Berlin, 1933. [32] B. Miller & D. Goldberg, Genetic Algorithms, Tournament Selection, and the Effects of Noise, Complex Systems 9(1995): 193212.

12

Journal Pre-proof

urn

al

Pr e-

p ro

of

A control problem for uncertain discrete time switched systems is discussed. Analytic expressions of optimal control are derived. Implement by GA to overcome exponential number of switching sequences.

Jo

  

Journal Pre-proof

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Jo

urn

al

Pr e-

p ro

of

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: