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# Optimal Control Problem for Uncertain Linear Systems with Multiple Input Delays

*Journal of Uncertainty Analysis and Applications*
**volume 4**, Article number: 5 (2016)

## Abstract

Based on the concept of uncertain process, uncertain linear systems with multiple input delays are considered in this paper. An optimal problem for such systems is introduced and solved by using the equation of optimality and maximum principle. Finally, an example is given to show how to solve an uncertain optimal control model with time-delay.

## Introduction

The delay system was first proposed in the eighteenth century. Until the twentieth century, with the maximum principle and dynamic programming theory, delay system attracted significant interest from many researchers. In recent decades, the study of time-delay systems had a great development not only in theory but also in applications. Optimal control problems of linear time-delay systems were discussed in Basin [1] and Eller et al. [2]. Stochastic control problems with delay were studied by Bauer and Rieder [3] and Larssen and Risebro [4]. A set of infinite-dimensional differential equations could model the dynamic systems which have multiple input delays. Hamiltonian function can be used to solve the control problems with delay [5]. Moreover, the phenomenon of systems with multiple input delays are ubiquitous. It widely exists in communications. With the rapid development of information and technology, multi-input time-delay systems become a research hot spot.

In the real world, we will encounter a lot of indeterminacy; these phenomena cannot be explained by random events, such as high speed, about 50 kg. This fact prompted researchers to create a new mathematical tool. Liu [6] found an uncertainty theory through introducing an uncertain measure based on normality, self-duality, countable subadditivity, and product measure axioms. Based on an uncertain variable, uncertain process, and canonical process, Liu [7] presented uncertain differential equation. It aims to describe the evolution of dynamic uncertain systems and has widely applied in finance so far. Liu [8] assumed the stock price follows a time-homogenous uncertain differential equation and proposed the first uncertain stock model. Ji and Zhou [9] proved the existence and uniqueness of a multi-dimensional uncertain differential equation. Based on the uncertain differential equation, in 2010, Zhu [10] presented an equation of optimality for an uncertain optimal control problem by using dynamic programming. This equation plays a significant role in solving uncertain optimal control problems. It has been applied to uncertain bang-bang control problems by Xu and Zhu [11] and uncertain time-delay systems control problem by Chen and Zhu [12].

In this paper, we will introduce an uncertain linear system with multiple input delays and propose an optimal control problem subject to this system. The organization of the paper is as follows. In the “Preliminary” section, some basic concepts are reviewed. In the “Problem Statement” section, an uncertain linear system with multiple input delays is formulated, and the optimal control is derived in the problem by using the equation of optimality. In the “Example” section, we will give a numerical example of uncertain linear systems with multiple input delays to illustrate the result obtained in the previous section.

## Preliminary

Some concepts about uncertain measure, uncertain variable, and uncertain process can be found in Liu [6]. In convenience, we give some useful concepts. Let *Γ* be a nonempty set and *L* be a *σ*-algebra over *Γ*. The set function *M* defined on the *σ*-algebra *L* is called an uncertain measure if it satisfies the three axioms: *M*{*Γ*}=1; *M*{*Λ*}+*M*{*Λ*
^{c}}=1 for any event *Λ*∈*L*; and \(M\left \{\bigcup ^{\infty }_{i=1}\Lambda _{i}\right \}\leq \sum ^{\infty }_{i=1}M\{\Lambda _{i}\}\) for every countable sequence of events {*Λ*
_{
i
}}⊂*L*. Then, the triplet (*Γ*,*L*,*M*) is called an uncertainty space. 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 *R*
^{n}. The uncertainty distribution *Φ*: *R*→[0,1] of an uncertain variable *ξ* is defined by *Φ*(*x*)=*M*{*ξ*≤*x*} for any real number *x*. An uncertain process is a measurable function from *V*×(*Γ*,*L*,*M*) to the set of real numbers where *V* is an index set.

###
**Definition 1**

(Liu [8]) The uncertain variables *ξ*
_{1},*ξ*
_{2},⋯*ξ*
_{
m
} are said to be independent if \(M\{\bigcap ^{m}_{i=1}(\xi _{i}\in B_{i})\}=\min _{1\leq i\leq m}M\{\xi _{i}\in B_{i}\}\) for any Borel sets *B*
_{1},*B*
_{2},⋯*B*
_{
m
} of real numbers.

###
**Definition 2**

(Liu [6]) The expected value of an uncertain variable *ξ* is defined by

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

For any numbers *a* and *b*, *E*[*a*
*ξ*+*b*
*η*]=*a*
*E*[*ξ*]+*b*
*E*[*η*] if *ξ* and *η* are independent uncertain variables.

###
**Definition 3**

(Liu [8]) An uncertain process *C*
_{
t
} is said to be a canonical process if (i) *C*
_{0}=0 and almost all sample paths are Lipschitz continuous; (ii) *C*
_{
t
} has stationary and independent increments; (iii) every increment *C*
_{
s+t
}−*C*
_{
s
} is a normal uncertain variable with expected value 0 and variance *t*
^{2}, denoted by \(C_{s+t}-C_{s}\thicksim N(0,t)\), whose distribution is

###
**Definition 4**

(Liu [8]) Let *X*
_{
t
} be an uncertain process and *C*
_{
t
} be a canonical process. For any partition of closed interval [*a*,*b*] with *a*=*t*
_{1}<*t*
_{2}<⋯<*t*
_{
k+1}=*b*, the mesh is written as △=max_{1≤i≤k
}∣*t*
_{
i+1}−*t*
_{
i
}∣. Then, uncertain integral of *X*
_{
t
} with respect to *C*
_{
t
} is

provided that the limit exists almost surely and is finite.

###
**Definition 5**

(Liu [8]) Suppose *C*
_{
t
} is a canonical process, and *f* and *g* are two functions. Then,

is called an uncertain differential equation. A solution is an uncertain process *X*
_{
t
} that satisfies (1) identically in *t*.

###
**Remark**

The uncertain differential *Eq.* (1) is equivalent to the uncertain integral equation

## Problem Statement

Consider an uncertain linear system with multiple time-delays in the control input

with the initial condition *X*(*t*
_{0})=*X*
_{0}, where *t*
_{0} is the initial time. Here, *X*
_{
s
} is the state vector of *n* dimension, *u*
_{
s
} is the control vector of *m* dimension; *h*
_{
i
}>0(*i*=1,⋯,*p*) are positive time-delays; *h*= max{*h*
_{1},⋯,*h*
_{
p
}} is the maximum delay shift; and \(C_{s}=(C_{s_{1}},C_{s_{2}},\cdots,C_{s_{p}})\phantom {\dot {i}\!}\), where \(C_{s_{1}},C_{s_{2}},\cdots,C_{s_{p}}\phantom {\dot {i}\!}\) are independent canonical process. And *a*
_{0}(*s*),*a*
_{1}(*s*),*b*(*s*), and *B*
_{
i
}(*s*)(*i*=1,2,⋯,*p*) are piecewise continuous matrix functions of appropriate dimensions.

The quadratic cost function to be maximized is defined as follows:

where *R*(*s*) is positive definite, *Ψ*
_{
T
},*L*(*s*) are nonnegative definite symmetric matrices, and *T*>0 is a time moment.

The optimal control problem is to find the control *u*
^{∗}(*t*)(*t*∈[*t*
_{0},*T*]) that maximizes the criterion *J*(*t*,*x*) along with the trajectory *x*
^{∗}(*t*)(*t*∈[*t*
_{0},*T*]), generated upon substituting *u*
^{∗}(*t*) into the state Eq. (2).

###
**Theorem 1**

Let *C*
_{
t
} be a canonical process, *u*
_{1t
} be an *n*×*n* integrable uncertain process, and *u*
_{2t
} and *v*
_{2t
} be two n-dimensional integrable uncertain processes. Then, the n-dimensional linear uncertain differential equation

has a solution

where

###
*Proof*

At first, we define two uncertain processes *U*
_{
t
} and *V*
_{
t
} via uncertain differential equations,

It follows from the integration by parts that

That is, the uncertain process *X*
_{
t
}=*U*
_{
t
}
*V*
_{
t
} is a solution of the uncertain differential Eq. (4). Write *U*
_{
t
} as

Taking differentiation operations on both sides, we have

Thus, the solution of uncertain differential equation *d*
*U*
_{
t
}=*u*
_{1t
}
*U*
_{
t
}
*d*
*t* is

In addition, we have

Taking *U*
_{0}=*I* and *V*
_{0}=*X*
_{0}, we get the solution (5). The theorem is proved. □

###
**Theorem 2**

The solution to the optimal control problem for the uncertain linear system with input delay (2) and the quadratic criterion (3) are given as follows. The optimal control law for *t*≥*t*
_{0} is given by

where *P*(*t*) satisfies

and *Q*(*t*) is a solution of the following differential equation

where \(M_{i}(t)=\exp (-\int _{t-h_{i}}^{t} a_{1}(s)ds)\). The optimal value for *t*≥*t*
_{0} is given by

where

###
*Proof*

Using the equation of optimality in Zhu [10] for the optimal control problems (2) and (3), we get

Let

Setting \(\frac {\partial g(u_{t})}{\partial u_{t}}=0\) yields

where \(M_{i}(t)=\frac {\partial u_{t-h_{i}}}{\partial u_{t}}\). Hence,

By Eq. (9), we have

Since \(J(T,X_{T})=X_{T}^{\tau }\Psi X_{T}\), we guess

Then,

and

Substituting Eqs. (10) and (13) into Eq. (11) yields

and

Since \(J(T,x)=\frac {1}{2}x^{\tau } P(T)x+Q(T)x+K(T)=x^{\tau }\Psi _{T} x\), we have *P*(*T*)=2*Ψ*
_{
T
},*Q*(*T*)=0, and *K*(*T*)=0. By Eq. (15), we obtain Eqs. (6) and (7). By Eq. (16), Eq. (8) holds. Therefore,

is the optimal value of the uncertain linear system with input delay Eq. (2) and the quadratic criterion Eq. (3), and

Let us find the value of matrices *M*
_{
i
}(*t*) for this problem. Substituting the optimal control law Eq. (17) into the Eq. (2) gives

The multi-dimensional uncertain differential Eq. (18) is equivalent to the multi-dimensional uncertain integral equation

where *t*,*r*≥*t*
_{0}, and

and we know

Since the integral terms in the right-hand side of Eq. (19) do not explicitly depend on *u*
_{
t
},

It can be converted to

Hence, the equality

holds, where *T*∈*R*
^{n×m} and *K*
_{1},*K*
_{2}∈*R*
^{n×n} can be selected the same for any *t*,*r*≥*t*
_{0}. Writing the last equality for \(x_{t+h_{i}},h_{i}>0\), we have

Thus,

which leads to

Now setting *T*=*B*
_{
i
}(*t*) and using *t*−*h*
_{
i
} instead of *t* yields

for *t*≥*t*
_{0}+*h*
_{
i
}.

So,

The theorem is proved. □

## Example

We consider the following example of uncertain linear systems with multiple time-delays in the control input

We have *a*
_{0}(*t*)=0,*a*
_{1}(*t*)=1,*B*(*t*)=1,*b*
_{0}(*t*)=0,*b*
_{1}(*t*)=1,*R*(*t*)=1,*L*(*t*)=1,and *Ψ*(*T*)=1. So, we get *M*
_{1}= exp(−0.1) and *M*
_{2}(*t*)=1. By Theorem 3.1, the function *Q*(*t*) satisfies

Thus, *Q*(*t*)=0 for *t*∈[0,2], and then *K*(*t*)=0 for *t*∈[0,2]. So, we get that the optimal control \(u^{*}_{t}\) is

and the optimal value is \(J(0,X_{0})=\frac {1}{2}P(0){X_{0}^{2}}\), and *P*(*t*) satisfies

Now, we consider the numerical solution of this model. Let *S*=*t*
_{0},*t*
_{1},⋯*t*
_{200} be an average partition of [0,2](i.e., 0=*t*
_{0}<*t*
_{1}<⋯<*t*
_{200}=2), and *Δ*
*t*=0.01. Thus,

Since *Δ*
*C*
_{
t
} is a normal uncertain variable with expected value 0 and variance *Δ*
*t*
^{2}, the distribution function is \(\Phi (x)=\left (1+\exp \left (-\frac {\pi x}{\sqrt {3}\Delta t}\right)\right)^{-1}\), *x*∈*R*. So, we maybe can get a sample point \(\tilde {c_{t}}\) of *Δ*
*C*
_{
t
} from \(\tilde {c_{t}}=\Phi ^{-1}(\text {rand}(0,1))\) that \(\tilde {c_{t}}=\frac {\sqrt {3}\Delta t}{-\pi }\ln \left (\frac {1}{\text {rand}(0,1)}-1\right)\). Thus, *x*
_{
t
} and *u*
_{
t
} may be given by the following iterative equations:

for *j*=0,1,2,⋯,200, and \(u_{t_{j}-0.1}=0\), where *t*
_{
j
}∈[0,0.1]. The numerical solution *P*(*t*
_{
j
}) of (23) is provided by

for *j*=200,199,⋯,2,1 with *P*(*t*
_{200})=2.

Therefore, the optimal value of the example is *J*(0,*X*
_{0})=1.067544, and the optimal controls and corresponding states are obtained in Table 1 and Figs. 1 and 2.

## Conclusion

In this paper, an optimal control problem for uncertain linear systems with multiple input delays was investigated. By using uncertain optimality equation and uncertain differential equation, then the optimal control of this problem was obtained. Finally, an example was used to illustrate the result of uncertain optimal control.

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- 2
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- 3
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## Acknowledgements

This work is supported by the National Natural Science Foundation of China (No. 61273009, No. 61673011).

### Authors’ contributions

YJ carried out the study in the paper and drafted the first version of the manuscript. YJ and YZ designed the framework together. HY and YZ revised the first version of the manuscript. All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

## Author information

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### Keywords

- Optimal control
- Uncertain system
- Input delay