The necessary conditions for finite horizon time varying order optimal control of Caputo systems

The necessary conditions for finite horizon time varying order optimal control of Caputo systems Abstract This article presents a general solution scheme for finite horizon optimal control of variable order Caputo dynamic systems with specified initial and final conditions. The main approach of the article is using variable order calculus of variations. After deriving the necessary optimality conditions, as the main contribution of the article, a change of variable is used to rearrange the set of conditions as a set of coupled variable order differential equations. Then, a numerical method is introduced to solve the corresponding equations. Finally, a case study shows the effectiveness of the proposed method. 1. Introduction One of the commonly used mathematical tools for the modelling of dynamical systems is the differential equation. About 300 years ago, a generalization for differential equations was suggested having non-integer order of derivation. This topic was introduced as fractional order calculus and was mostly developed theoretically in the 19th century, however, its interesting applications in modelling, physics, and engineering belong to recent decades. Fractional operators are introduced in Podlubny (1998). Butzer & Westphal (2000) have illustrated the basic properties of fractional systems, and some other theoretical and applicational advances of fractional calculus are reported in Sabatier et al. (2007). Flexibility, generality and more degrees of freedom in non-integer order differential equations make them powerful in describing some real world events where integer order dynamics partly fail to do so. Ability of fractional dynamics in modelling of dynamic events has recently attracted lots of attentions. The main property of such dynamics is the order. The order of a non-integer order dynamic has been interpreted as the memory in electronic devices (Maundy et al., 2012), viscoelastic damping (Diaz & Coimbra, 2009; Wharmby & Bagley, 2013), human’s ability to forget and remember (Tabatabaei et al., 2012, 2013), and properties of lung tissues (Ionescu, 2009). As an extension, the definition of variable order derivation, firstly introduced in Samko & Ross (1993), was a great novelty. Using this definition, the order of a dynamic system is not limited to be even constant. It varies in time and even it can have its own dynamical evolution depending on the system’s evolution. Although there are few works on the variable order dynamics, however, it should be noted that such dynamic systems are the most general tools for modelling and they will definitely play an important role in modelling dynamic systems in future researches. Although the frame works for non-integer order optimal control (Agrawal, 2004; Agrawal & Baleanu, 2007; Frederico & Torres, 2008), robust control (Tan et al., 2009; Moornani & Haeri, 2010), and adaptive control (Ladaci & Charef, 2006) have been already developed, however, for the variable order case, no systematic controller design approach has been proposed yet. As a result, variable order systems are difficult to control. In this regard, the main purpose of the present article, as a theoretical article, is to facilitate the control process of such systems through introducing an optimal controller as the first effort to represent a systematic approach to design a controller for variable order dynamics. The problem we aim to solve is to design a control signal to make a variable order system reach from a known initial condition to a specified final condition while minimizing a cost function. The solution presented here can be used for all of the similar problems. Applying variational calculus on such dynamics leads to a set of differintegral equations. The main point of this article is to rearrange such equations to obtain a dynamic system whose response is the optimal control signal. Optimal control is an efficient control method used to minimize criterion such as the energy consumed, the time spent, etc., while the system is behaving in a desired manner. One of the common approaches for solving optimal control problems is the calculus of variations (Pinch, 1993) which is also used in this article. Non-integer constant order optimal control problem has been studied in many researches, using different approaches. Kamocki (2014) has used Pontryagin principle, Agrawal (2004) and Frederico & Torres (2008) have used calculus of variation, and Agrawal & Baleanu (2007) has used the Hamiltonian function. However, in the variable order case, to the best knowledge of the authors, this article can be considered as the first article studding the time varying order optimal control problem. The variational calculus of the variable order operator has been introduced, studied and developed recently (Odzijewicz et al., 2013a; Pooseh et al., 2013; Almeida & Torres, 2013; Odzijewicz et al., 2013b). In Odzijewicz et al. (2013a), the variation of a variable-order operator is calculated; in Odzijewicz et al. (2013b) this result is used to prove the Neother’s theorem for cost functions with variable order derivation operator, and in Tavares et al. (2015), the problem is studied for a special combined operator. However, this article, in-line with the above-mentioned ones, basically deals with the optimal control problem, i.e. the above results are used to derive the optimal control signal. In this regard, the novelty of this article is to use the variational calculus of variable-order to obtain a set of equations comparable to the traditional optimal control (with an extra equation) whose response is the optimal control signal. After defining the problem statement, and posing some related lemmas, necessary condition for optimality with respect to the first variation of cost function is obtained. Afterwards, the solution formulation is presented as a dynamical system. This system will then be solved using a numerical scheme. Hence, the rest of the article is organized as follows: Section 2 contains the definitions of non-integer order derivation and non-integer order systems. Next, in Section 3 the problem statement will be introduced. Then, the variable order optimal control (VOOC) formulation which is the main contribution of this article will be represented in Section 4. Afterwards, the numerical solution scheme and a case study will be presented in Sections 5 and 6, respectively. Finally, the article will be concluded in Section 7. 2. Definitions After developing the concept of non-integer order derivation, different definitions were introduced for this operator. The most prevalent definitions are derivations in the senses of Caputo and Riemann-Liouville (R-L) (Baleanu et al., 2012). Either of these definitions, has useful properties, however, Caputo’s derivation has attracted much attentions in engineering applications (Kilbas & Marzan, 2005). However, these definitions are equivalent under some conditions (Podlubny, 1998). The difference between them usually can be interpreted as a known time domain function (Podlubny, 1998; Hartley & Lorenzo, 2002). Hence, the issue of the operator type can be neglected in the control problems. Therefore, because of its engineering applications, due to directly using the initial conditions (as shown in Podlubny (1998)), the Caputo operator is chosen to define the non-integer order system. Moreover, there are some other derivation operators, i.e. left and right derivation operators or constant and variable order definitions. In this section, we will explain some of these definitions used in the article. Also, regarding to these definitions, one can describe a non-integer order system as a set of differential equations in which the derivation operator in each equation can follow one of the above mentioned ones. 2.1. Non-integer order derivation A non-integer order scalar differential equation can be described as follows. Here, scalar $$x$$ is the response of the differential equation, $$u$$ is the exogenous input and $$D^\alpha$$ is the derivation operator of order $$\alpha$$, where $$\alpha$$ can be a constant or varying function and the derivation operator can be of different types. Dαx=f(x,u) (2.1) The differential equation is supposed to be analysed in a time interval such as $$[0\ T]$$, where $$0$$ is the initial and $$T$$ is the final time. It is often constrained to a boundary condition. In most of the real world dynamics, these conditions are related to the initial or final value of $$x$$, shown here by $$x_0$$ and $$x_f$$, respectively, where $$x_0=x(0),x_f=x(T)$$. In the above-mentioned interval, operator $$D^\alpha$$ can be defined in different ways. Some of these definitions are followed (Lorenzo & Hartley, 2002):  0CDtα(t)f(t) =1Γ(1−α(t))∫0t(t−τ)−α(t)ddτf(τ)dτ0 <α(t)<1,∀t,0≤t≤T (2.2)  tCDTα(t)f(t) =−1Γ(1−α(t))∫tT(τ−t)−α(t)ddτf(τ)dτ0 <α(t)<1,∀t,0≤t≤T (2.3)  0RLDtα(t)f(t) =1Γ(1−α(t))ddτ∫0t(t−τ)−α(t)f(τ)dτ0 <α(t)<1,∀t,0≤t≤T (2.4)  tRLDTα(t)f(t) =−1Γ(1−α(t))ddτ∫tT(τ−t)−α(t)f(τ)dτ0 <α(t)<1,∀t,0≤t≤T (2.5) The definition of left varying order derivation in the sense of Caputo is shown in (2.2) (Lorenzo & Hartley, 2002). Right Caputo derivation can be defined as shown in (2.3) (Lorenzo & Hartley, 2002). Left and right Riemann–Liouville derivatives of variable order $$\alpha(t)$$ is shown in (2.4) and (2.5), respectively (Lorenzo & Hartley, 2002). Also, there are definitions for variable order integration operator as follows:  0Itα(t)f(t) =1Γ(α(t))∫0t(t−τ)α(t)−1f(τ)dτ0 <α(t)<1,∀t,0≤t≤T (2.6)  tITα(t)f(t) =1Γ(α(t))∫tT(τ−t)α(t)−1f(τ)dτ0 <α(t)<1,∀t,0≤t≤T (2.7) Here, left and right time varying order integration operators are defined in (2.6) and (2.7), respectively (Lorenzo & Hartley, 2002). One can use a constant order of derivation/integration instead of the time varying one in each of the above definitions to get a constant non-integer order derivation/integration by setting $$\alpha(t)=\alpha_0, \forall t,0\leq t \leq T$$, where $$ 0<\alpha_0<1$$ is a real number. 2.2. Non-integer order dynamic systems Generally speaking, a non-integer order dynamic system is a set of coupled non-integer order differential equations in which the derivation operators can be of different definitions. Each differential equation can have derivative operator in the sense of Caputo or Riemann–Liouville, left sided or right sided and constant or variable order. Note that the word ‘non-integer’ does not mean that the system cannot include traditional integer derivation of order one. It just implies that the order is not restricted to the integers. The general form of a non-integer system can be shown as the following equations. {Dα1x1=f1(x1,…,xk,u,t)⋮Dαkxk=fk(x1,…,xk,u,t)  (2.8) Here, $$\alpha_{i},i=1,2,\dots,k$$ are generally functions of time and $$u$$ is the exogenous input vector. In specific cases, the above definition can be reduced, for instance, to a constant order dynamic system or a Caputo dynamic system. In this article, we are focusing on the optimal control of a variable order Caputo system, i.e. a set of coupled differential equations all with Caputo left variable order operator of derivation. 3. Problem statement This article aims to develop a scheme to solve optimal control problem for a specific type of non-integer order dynamical systems. In fact, as a widely applicable type of such systems, we concentrate on commensurate variable order dynamics with left Caputo derivations with known initial conditions. Here, the word commensurate means that all of the orders have the same functionality of time, i.e. $$\alpha_i(t)=\alpha_j(t),i,j=1,2,\dots,k,\forall t$$. Since Caputo derivation operator directly uses initial conditions, such systems are proper tools for describing variable order dynamics. This system can be written in detail as: { 0CD tα(t)x1=f1(x1,…,xk,u,t),x1(0)=x1,0⋮ 0CD tα(t)xk=fk(x1,…,xk,u,t),xk(0)=xk,0 0<α(t)<1 (3.1) Or, in brief: Dα(t)x=f(x,u,t),x(0)=x0, (3.2) where $$x={[x_1\ x_2\ \dots\ x_k]}^T$$ is called as the pseudo-state vector. Since the state of the system cannot be defined using $$x$$ the word ‘state’ seems not appropriate and the phrase ‘pseudo-state’ is more precise (Hartley & Lorenzo, 2002). The goal is to design the exogenous input (as a function of time) for the above defined system to make it reach from the known initial condition to a predefined final condition in a way that a cost function is minimized. The problem can be defined as follows. Consider the system described in (3.2). This system is supposed to reach to final condition $$x(T)=x_f$$ in the finite time $$T$$, whereas the functional $$J=\int_0^T g(x,u,t)dt$$ should be minimized or, in brief: {min J=∫0Tg(x,u,t)dts.t. Dα(t)x=f(x,u,t)x(0)=x0,x(T)=xf  (3.3) 4. Optimality conditions 4.1. Preliminaries In this part, the goal is to apply calculus of variation to obtain necessary conditions pertaining to optimal control. First, Lemma 1 provides a reformation formula for dealing with some type of singular integrals (which is used in the definitions of the non-integer derivation). Then, the variable order part by part integration is represented and at last, a relation between variable order derivation and integration is proven. Consider the integral $$H(t)=\int_t^T f(\tau) d\tau$$ where the integrand $$f$$ is singular near the upper integration limit, $$T$$, i.e. $$\lim_{\tau \to T^-}f(\tau)=\infty$$. In this case, the value of $$H$$ at $$T$$ is defined as: $$H(T)=\lim_{t \to T^-}H(t)=\lim_{\epsilon\to 0^+}H(T-\epsilon)$$. Therefore, $$H(T)$$ exists if this limit converges. Note that it is not valid for singular integrands to immediately set $$H(T)=0$$ due to the fact that the lower and upper bounds of the integral are equal (Sladek et al., 2000). In special cases in which the integrand includes a factor like $$(\tau-t)^{-\alpha},0<\alpha<1$$, we can somehow cope with the singularity and rewrite $$H$$ as an integral with a non-singular integrand. Lemma 1 In the case that $$\lambda$$ is non-singular in $$[t\ \ T]$$ and $$0<\alpha<1$$ one has: ∫tT(τ−t)−αλ(τ)dτ=11−α∫0(T−t)(1−α)λ(r11−α+t)dr$$\alpha$$ can be generally a function of $$t$$. Proof. The statement can be easily proven by the change of variable: $$\tau=r^{\frac{1}{1-\alpha}}+t$$. □ Remark If we define $$\bar{H}(t)=\int_t^T (\tau-t)^{-\alpha}\lambda(\tau)d\tau$$, since $$\lambda(r^{\frac{1}{1-\alpha}}+t)$$ is non-singular, now we can write:  limt→T−H¯(t)=limϵ→0+H¯(T−ϵ) =limϵ→0+[11−q∫0ϵ(1−α)λ(r11−α+T−ϵ)dr]=0 (4.1) Hence, when the integrand singularity is because of a $$(\tau-t)^{-\alpha}$$ factor, $$lim_{t\rightarrow T^-}\bar{H}(t)$$ exists, thus $$\bar {H}(T)=0$$. Lemma 2 When $$0<\alpha(t)<1$$ is a smooth enough function so that $${d\alpha}/{dt}$$ exists everywhere on the interval $$[0\ \ T]$$, we have Variable Order Part by Part Integration in the sense of Caputo formula as shown in (4.2), where $$\psi(z)=\frac{d}{dz}ln{\it{\Gamma}}(z)=\frac{\frac{d{\it{\Gamma}}(z)}{dz}}{{\it{\Gamma}}(z)}$$.)  ∫0TgT(t) 0CD tα(t)f(t)dt=fT(t) tI T1−α(t)g(t)|0T     −∫0T(dα(t)dtψ(1−α(t))fT(t) tI T1−α(t)g(t)−fT(t) tRLD Tα(t)g(t))dt (4.2) Proof. See Section 3 of (Odzijewicz et al., 2013a). □ Lemma 3 for $$0<\alpha(t)<1;$$ ddt(Γ(1−α(t)) tI T1−α(t)λ(t))=−Γ(1−α(t)) tRLD Tα(t)λ(t) Proof. It is directly obtained from the definitions (2.5) and (2.7):  tIT1−α(t)λ(t)=1Γ(1−α(t))∫tT(τ−t)−α(t)λ(τ)dτ ⇒Γ(1−α(t)) tIT1−α(t)λ(t)=∫tT(τ−t)−α(t)λ(τ)dτ ⇒ddt(Γ(1−α(t)) tIT1−α(t)λ(t))=ddt∫tT(τ−t)−α(t)λ(τ)dτ =−Γ(1−α(t)) tRLD Tα(t)λ(t) □ 4.2. Augmented cost function and its variation One of the prevalent methods for solving optimal control problems is to derive the first variation of the augmented cost function. Augmented cost function is a functional which is constructed by using Lagrange multipliers and adjoining the constraint as a term of the integrand to the cost function (Pinch, 1993). In the problem posed in (3.3), we can define the augmented cost function as (4.3). Here, $$\lambda$$ is known as co-state, a vector with the same size as $$x$$. Ja=∫0T(g(x,u,t)+λT(f(x,u,t)− 0CD tα(t)x(t)))dt (4.3) Using common techniques in calculus of variations, the first variation of the above functional will be obtained as follows. δJa=∫0T(∂g∂xδx+∂g∂uδu+δλT(f− 0CD tα(t)X)+λT∂f∂xδx+λT∂f∂uδu−λTδ( 0CD tα(t)x))dt (4.4) According to Lemma 2;  ∫0TλTδ( 0CD tα(t)x)dt=∫0TλT 0CD tα(t)(δx)dt=λT tI T1−α(t)δx|0T     −∫0T(dα(t)dtψ(1−α(t)) tI T1−α(t)λTδx− tRLD Tα(t)λTδx)dt (4.5) Substituting (4.5) in (4.4), the first variation of $$J_a$$ is derived as: δJa=∫0T(∂g∂xδx+∂g∂uδu+δλT(f− 0CD tα(t)x) +λT∂f∂xδx+λT∂f∂uδu +(dα(t)dtψ(1−α(t)) tI T1−α(t)λT− tRLD Tα(t)λT)δx)dt − tI T1−α(T)λT(T)δx(T)+ tI T1−α(0)λT(0)δx(0) (4.6) Now, since the initial and final values of $$x$$ are both specified, their variations are equal to zero; $$\delta x(T)=\delta x(0)$$ and (4.6) can be simplified to: δJa=∫0T((∂g∂x+λT∂f∂x+dα(t)dtψ(1−α(t)) tI T1−α(t)λT− tRLD Tα(t)λT)δx    +(∂g∂u+λT∂f∂u)δu+(fT− 0CD tα(t)xT)δλ)dt (4.7) For the optimality, the first variation of the augmented cost function should be equal to zero, leading to the following set of equations: { 0CD tαx−f=0 tRLD Tαλ−dαdtψ(1−α) tI T1−αλ−(∂g∂x)T−(∂f∂x)Tλ=0∂g∂u+λT∂f∂u=0 x(0)=x0,x(T)=xf (4.8) Regarding the above equations: 1. Since the second equation contains both variable integration and derivation operators, it is a differintegral equation rather than a pure differential one. Hence, it cannot be considered as a system. As it will be shown, a variable redefinition will allow us to rewrite the set of equations as a non-integer order system similar to (2.8). 2. The third equation is an algebraic one through which the input $$u$$ can be calculated with respect to $$x$$ and $$\lambda$$. 4.3. Optimal input as the output of a non-integer order system In this part, we aim to rearrange the set of equations (4.8) to get a non-integer order system. This formation agrees with the classical optimal control problems and, it also makes the numerical scheme easier. To this purpose, we define an auxiliary variable $${\it{\Lambda}}$$ as follows. Λ= tI T1−αλ (4.9) According to Lemma 3; ddt(Γ(1−α)Λ)=−Γ(1−α) tRLD Tαλ (4.10) The left side of equation leads to: ddt(Γ(1−α)Λ)=−dαdtψ(1−α)Γ(1−α)Λ+Γ(1−α)Λ˙ ⇒Λ˙−dαdtψ(1−α)Λ+ tRLD Tαλ=0 We can replace $${_t^{RL}}D{_T^\alpha}\lambda$$ from (4.8) to obtain the dynamic of $${\it{\Lambda}}$$:  tRLD Tαλ=dαdtψ(1−α)Λ+(∂g∂x)T+(∂f∂x)Tλ⇒Λ˙+(∂g∂x)T+(∂f∂x)Tλ=0 (4.11) To avoid redundancy, a boundary condition for $${\it{\Lambda}}$$ is needed. It can be calculated from its definition (4.9) and using the remark of Lemma 1: Λ(T)=limt→T−∫tT(τ−t)−α(t)λ(τ)dτ=0 So, the dynamic of the auxiliary state variable $${\it{\Lambda}}$$ is: Λ˙+(∂g∂x)T+(∂f∂x)Tλ=0,Λ(T)=0 (4.12) It could be augmented to (4.8) to form a non-integer order system: { 0CD tα(t)x=f(x,u,t),x(0)=x0,x(T)=xf tRLD Tα(t)λ=−dαdtψ(1−α)Λ−(∂g∂x)T−(∂f∂x)TλΛ˙=−(∂g∂x)T−(∂f∂x)Tλ,Λ(T)=0(∂f∂u)Tλ+(∂g∂u)T=0  (4.13) This system includes traditional derivation (for $${\it{\Lambda}}$$), left variable order derivation in the sense of Caputo (for $$x$$) and right variable order Riemann–Liouville derivation (for $$\lambda$$). There are two boundary conditions for $$x$$ and one for $${\it{\Lambda}}$$. The next section provides a numerical scheme for solving such set of equations. 5. Numerical solution scheme As shown, the necessary conditions of optimality lead to a set of nonlinear differential equation, whose response is the optimal control signal. The existence and uniqueness of the solution are studied in Razminia et al. (2012); Xu & He (2013); Zhang (2013) where it is shown that, just like the traditional case, the solution existence of such equations is very case-dependent. However, the necessary conditions indicate that the control signal $$u$$, calculated from the above set of non-linear equations, is an extremum and its nature (whether it is a maximum, a minimum or a saddle) depends on the nature of the problem. As long as the problem is basically a minimization, one can make sure that $$u$$ is a minimum. As shown in (Kirk, 1970), as long as the Hamiltonian ($$H=g+\lambda^Tf$$) can be expressed as $$H=f+c^Tu+\frac{1}{2}u^TR(t)u$$, where $$c$$ is a vector function that does not have any terms containing $$u$$ and $$R$$ is a positive definite matrix. After calculating $$u$$ with respect to $$\lambda$$ and $$x$$ and substituting it in the first equation of (4.13), all we have is with respect to $$x,\lambda,{\it{\Lambda}}$$. Hence, we have a pure non-integer order autonomous system which is difficult to be solved analytically and the numerical methods should be used. The numerical approach used for solving the variable order differential equations is an extension for the approach used to solve constant order differential equations. Here, we have used an approach similar to the one addressed in Li & Wu (2015). The algorithm is explained in this section. More details can be found in Li & Wu (2015) or Diethelm et al. (2005). The following equation indicates the Grunwald–Letnikov (G–L) definitions of non-integer derivation.  0GLD tαy(t)=limh→01hα∑j=0[th](−1)j(αj)y(t−jh) tGLD Tαy(t)=limh→01hα∑j=0[T−th](−1)j(αj)y(t+jh), (5.1) where $$[.]$$ indicates the integer part and $$\alpha \choose j$$ is defined as: (αj)=Γ(α+1)Γ(j+1)Γ(α−j+1) There are consistencies between the definitions which have been introduced in this article (i.e., R–L, G–L and Caputo definitions) and for a wide class of functions, these three best known definitions are equivalent under some conditions (Podlubny, 1998). In fact, R–L and G–L definitions can be considered the same in most of the situations (Podlubny, 1998) and when $$n-1<\alpha<n$$ the relationship between Caputo and R–L definition is (Podlubny, 1998):  0RLD tαy(t)= 0CD tαy(t)+∑k=0n−1tk−αΓ(k−α+1)y(k)(0), (5.2) where $$y^{(k)}(0)$$ is the $$k$$th derivative of $$y$$ evaluated at $$0$$. Our approach for solving non-integer order systems is to use Grunwald–Letnikov approximation of non-integer order derivation (Ford & Simpson, 2001). It is very common in constant order non-integer order differential equation to approximate the right and left derivatives in the interval $$[0\ \ T]$$ by splitting the interval to $$N$$ sub-intervals. It leads to $$N+1$$ point, say $$t_0,t_1,\dots,t_N$$ in a way that $\begin{bmatrix}0&T\end{bmatrix}=\bigcup _{i=0}^{N-1}\begin{bmatrix}t_{i}&t_{i+1}\end{bmatrix}$ . All sub-intervals have the same length $$h$$, i.e., $$t_{i+1}-t_i=h,i=0,1,\dots,N-1$$. According to the relations between R–L, G–L and Caputo derivative operators when $$0<\alpha<1$$, left Caputo and right R–L constant order derivation operators can be approximated by the following equations, where $$y_i=y(t_i)$$ and $$t_i=ih$$ (Ford & Simpson, 2001; Diethelm et al., 2005).  0CD tαyi ≈1hα∑j=0iωj(α)yi−j−ti−αΓ(1−α)y(0)i =1,2,…,N (5.3)  tRLD Tαyi≈1hα∑j=0N−iωj(α)yi+j,i=N−1,N−2,…,0 (5.4) Coefficient $$\omega^{(\alpha)}_{j}$$ are computed through: (Ford & Simpson, 2001). ω0(α)=1;ωj(α)=(1−α+1j)ωj−1(α),j=1,2,…,N (5.5) These formulation can be extended to variable order derivation. In this case, $$\alpha$$ varies with time, i.e. $$\alpha=\alpha(t)$$ and $$\alpha_i=\alpha(t_i)$$. Hence:             0CD tαiyi ≈1hαi∑j=0iωj(αj)yi−j−ti−αiΓ(1−αi)y(0)i =1,2,…,N (5.6)  tRLD Tαiyi≈1hαi∑j=0N−iωj(αj)yi+j,i=N−1,N−2,…,0 (5.7) The coefficients can be computed similar to (5.5): ω0(αj) =1;ωj(αj)=(1−αj+1j)ωj−1(αj)j =1,2,…,N (5.8) Equations (5.6) and (5.7) are used to approximate $${_0^C}D{_t^{\alpha(t)}}x$$ and $${_t^{RL}}D{_T^{\alpha(t)}}\lambda$$ in (4.13), respectively. The traditional derivative of $${\it{\Lambda}}$$ is also easy to be approximated using forward Euler method: Λ˙(i)≈Λ(i+1)−Λ(i)h,i=1,2,…,N (5.9) If we consider $$p$$ as the size of the pseudo-state vector $$x$$, since $$x,\lambda,{\it{\Lambda}}$$ are all $$p\times 1$$ vectors in each point, we have entirely $$3p(N+1)$$ samples of these functions during the entire interval. However, $$x_0,x_N,{\it{\Lambda}}_N$$ are known and it reduces the number of the unknown values by $$3p$$. So, there are totally $$3pN$$ unknown values. We can apply the formula (5.6) on $$x$$ in samples $$1,2,\dots,N$$ and derive $$pN$$ equations. Also, applying (5.7) on $$\lambda$$ for $$i=0,1,\dots,N-1$$ gives $$pN$$ equations. At last, we can obtain $$pN$$ equations by applying (5.9) on $${\it{\Lambda}}$$ in samples $$1,2,\dots,N$$. Therefore, there are totally $$3pN$$ coupled non-linear equations with respect to unknown variables $$x_i,i=1,2,\dots,N,\ \lambda_i,i=0,1,\dots,N,\ {\it{\Lambda}}_i,i=0,1,\dots,N-1$$. As a result, we have a set of $$3pN$$ equations and also $$3pN$$ unknown variables which can be calculated from the equations by applying a non-linear solver. In the next section, the formulation and the numerical scheme are applied on a case study. 6. Simulation study In this section, we aim to apply our proposed method on a simple case study system to show its efficiency. The system we have chosen is a simple scalar one which is a common case study for the non-integer optimal control problems, as studied in Agrawal (2008); Kumar Biswas & Sen (2011); Agrawal (2004). In fact, choosing a scalar system makes it possible to test and justify the proposed algorithm without loss of generality and the method can be applied on a multi-dimensional system in a similar way. To the best of our knowledge, so far, there has been no other systematic control approach for time varying order systems. Hence, there is no way to examine the proposed method except comparing it with itself to see how it works in different cases. Accordingly, in this section, we consider the following system with different cost functions and apply the proposed optimal control formulation on it. Comparing the results will convince us that the method works efficiently.  tCD Tα(t)x=−x+u,x0=1 (6.1) Consider the system described by (6.1). The order is chosen to be $$\alpha(t)=0.9e^{-\frac{t}{10}}$$. The goal is to make the pseudo-state $$x$$ to reach to $$x_f=0.5$$ in the finite time $$T=1$$. A cost function of the form (6.2) is also considered to be minimized. This cost function relates to the total (input and pseudo-state) energy of the system. Coefficient $$k$$ is an adjusting parameter which makes it possible to set different weights for the pseudo-state’s energy. J=∫0T12(kx2+u2)dt (6.2) After applying the formulation (4.13) on it, the following equations are obtained.  0CD tα(t)x =−x+u,x0=1,xf=0.5 tRLD Tα(t)λ =−kx−u−dαdtψ(1−α)ΛΛ˙ =−kx−u,Λ(T)=0λ+u =0 (6.3) The above equations can be simplified as follows. Now, the non-integer order system is derived and a numerical scheme should be used to solve it.  0CD tα(t)x =−x−λ,x0=1,xf=0.5 tRLD Tα(t)λ =−kx+λ−dαdtψ(1−α)ΛΛ˙ =−kx+λ,Λ(T)=0 (6.4) Since the system under consideration is linear, hence, the numerical solution leads to a set of linear equations. As long as it is non-singular, one can make sure that its response uniquely exists. Therefore, the optimal control signal $$u$$ exists and since it is unique, its nature is undoubtedly a minimizer. We set $$N=150$$ and run the simulation for different values of $$k$$, $$k=0,0.5,1$$. By increasing $$k$$, in fact, the pseudo-state’s energy is being more significant. So it is more important to keep it small. To this aim more control effort (input energy) will be needed. By setting $$k=0$$, we are neglecting the pseudo-state’s energy and it lets $$u$$ get free. Therefore, no extra control energy should be consumed for keeping $$x$$ small. Any other value of $$k$$ will lead to more input energy consumed. Hence, what we expect is that increasing $$k$$ should lessen the pseudo-state’s energy and enhance the input energy. Simulation results are shown in Figs 1 and 2. Fig. 1. View largeDownload slide The pseudo-state $$x$$ after applying the optimal control. Fig. 1. View largeDownload slide The pseudo-state $$x$$ after applying the optimal control. Fig. 2. View largeDownload slide The optimal control signal $$u.$$ Fig. 2. View largeDownload slide The optimal control signal $$u.$$ From Figs 1 and 2, it is obvious that the goal is met and the pseudo-state is reached to its predefined final value. Also, as we expected, it can be seen in Fig. 1 that energy of $$x$$ increases by increasing $$k$$ and, as it is shown in Fig. 2, increasing $$k$$ makes us to provide more control effort to make the pseudo-state reach to the specified final value. As another comparison criterion, the values of the cost functions for different values of $$k$$ are calculated in each moment $$t$$ and shown in Fig. 3. The moment values of $$J$$ can be computed through the following equation. J(t)=∫0t12(kx2(τ)+u2(τ))dτ (6.5) Fig. 3. View largeDownload slide The moment cost functions. Fig. 3. View largeDownload slide The moment cost functions. It is clear from Fig. 3 that increment of $$k$$ leads to consuming more energy and the total system’s energy in each moment (indicated by $$J(t)$$) is increased by increasing $$k$$. It is just what we expect, can be considered as another proof for justifying the proposed solution method. 7. Conclusion In this article, we presented a novel formulation for finite horizon optimal control of time varying order systems. Such systems are difficult to be analysed and so far, there has been no systematic method for controlling them. The contribution presented in this article opens the doors for other control techniques, especially optimal control methods to be developed for variable order systems. Variable order dynamics are useful tools which are currently used to model some complex physical systems, thus the method proposed in this article can be used for different applications. The numerical scheme used here, is an extension of the methods used before to solve constant order dynamic systems and here, it is shown that it can also be applied for solving all non-integer systems. Our focus in this article was mainly on the commensurate systems, i.e. the order of derivation for all of the pseudo-states was considered to be the same. Future works on this field can extend the formulation to the incommensurate ones. Also, infinite horizon optimal control problems can be discussed. Furthermore, other boundary conditions such as unspecified initial or final conditions are interesting topics for future researches. Constraints on the input and states can be also considered. Moreover, some theoretical issues on the sufficiency conditions, solvability, existence and uniqueness should be studied for the general case to obtain the conditions on which the problem is solvable in terms of $$f(x,u)$$, $$g(x,u)$$ and $$\alpha(t)$$. References Agrawal O. P. 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The necessary conditions for finite horizon time varying order optimal control of Caputo systems

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Abstract

Abstract This article presents a general solution scheme for finite horizon optimal control of variable order Caputo dynamic systems with specified initial and final conditions. The main approach of the article is using variable order calculus of variations. After deriving the necessary optimality conditions, as the main contribution of the article, a change of variable is used to rearrange the set of conditions as a set of coupled variable order differential equations. Then, a numerical method is introduced to solve the corresponding equations. Finally, a case study shows the effectiveness of the proposed method. 1. Introduction One of the commonly used mathematical tools for the modelling of dynamical systems is the differential equation. About 300 years ago, a generalization for differential equations was suggested having non-integer order of derivation. This topic was introduced as fractional order calculus and was mostly developed theoretically in the 19th century, however, its interesting applications in modelling, physics, and engineering belong to recent decades. Fractional operators are introduced in Podlubny (1998). Butzer & Westphal (2000) have illustrated the basic properties of fractional systems, and some other theoretical and applicational advances of fractional calculus are reported in Sabatier et al. (2007). Flexibility, generality and more degrees of freedom in non-integer order differential equations make them powerful in describing some real world events where integer order dynamics partly fail to do so. Ability of fractional dynamics in modelling of dynamic events has recently attracted lots of attentions. The main property of such dynamics is the order. The order of a non-integer order dynamic has been interpreted as the memory in electronic devices (Maundy et al., 2012), viscoelastic damping (Diaz & Coimbra, 2009; Wharmby & Bagley, 2013), human’s ability to forget and remember (Tabatabaei et al., 2012, 2013), and properties of lung tissues (Ionescu, 2009). As an extension, the definition of variable order derivation, firstly introduced in Samko & Ross (1993), was a great novelty. Using this definition, the order of a dynamic system is not limited to be even constant. It varies in time and even it can have its own dynamical evolution depending on the system’s evolution. Although there are few works on the variable order dynamics, however, it should be noted that such dynamic systems are the most general tools for modelling and they will definitely play an important role in modelling dynamic systems in future researches. Although the frame works for non-integer order optimal control (Agrawal, 2004; Agrawal & Baleanu, 2007; Frederico & Torres, 2008), robust control (Tan et al., 2009; Moornani & Haeri, 2010), and adaptive control (Ladaci & Charef, 2006) have been already developed, however, for the variable order case, no systematic controller design approach has been proposed yet. As a result, variable order systems are difficult to control. In this regard, the main purpose of the present article, as a theoretical article, is to facilitate the control process of such systems through introducing an optimal controller as the first effort to represent a systematic approach to design a controller for variable order dynamics. The problem we aim to solve is to design a control signal to make a variable order system reach from a known initial condition to a specified final condition while minimizing a cost function. The solution presented here can be used for all of the similar problems. Applying variational calculus on such dynamics leads to a set of differintegral equations. The main point of this article is to rearrange such equations to obtain a dynamic system whose response is the optimal control signal. Optimal control is an efficient control method used to minimize criterion such as the energy consumed, the time spent, etc., while the system is behaving in a desired manner. One of the common approaches for solving optimal control problems is the calculus of variations (Pinch, 1993) which is also used in this article. Non-integer constant order optimal control problem has been studied in many researches, using different approaches. Kamocki (2014) has used Pontryagin principle, Agrawal (2004) and Frederico & Torres (2008) have used calculus of variation, and Agrawal & Baleanu (2007) has used the Hamiltonian function. However, in the variable order case, to the best knowledge of the authors, this article can be considered as the first article studding the time varying order optimal control problem. The variational calculus of the variable order operator has been introduced, studied and developed recently (Odzijewicz et al., 2013a; Pooseh et al., 2013; Almeida & Torres, 2013; Odzijewicz et al., 2013b). In Odzijewicz et al. (2013a), the variation of a variable-order operator is calculated; in Odzijewicz et al. (2013b) this result is used to prove the Neother’s theorem for cost functions with variable order derivation operator, and in Tavares et al. (2015), the problem is studied for a special combined operator. However, this article, in-line with the above-mentioned ones, basically deals with the optimal control problem, i.e. the above results are used to derive the optimal control signal. In this regard, the novelty of this article is to use the variational calculus of variable-order to obtain a set of equations comparable to the traditional optimal control (with an extra equation) whose response is the optimal control signal. After defining the problem statement, and posing some related lemmas, necessary condition for optimality with respect to the first variation of cost function is obtained. Afterwards, the solution formulation is presented as a dynamical system. This system will then be solved using a numerical scheme. Hence, the rest of the article is organized as follows: Section 2 contains the definitions of non-integer order derivation and non-integer order systems. Next, in Section 3 the problem statement will be introduced. Then, the variable order optimal control (VOOC) formulation which is the main contribution of this article will be represented in Section 4. Afterwards, the numerical solution scheme and a case study will be presented in Sections 5 and 6, respectively. Finally, the article will be concluded in Section 7. 2. Definitions After developing the concept of non-integer order derivation, different definitions were introduced for this operator. The most prevalent definitions are derivations in the senses of Caputo and Riemann-Liouville (R-L) (Baleanu et al., 2012). Either of these definitions, has useful properties, however, Caputo’s derivation has attracted much attentions in engineering applications (Kilbas & Marzan, 2005). However, these definitions are equivalent under some conditions (Podlubny, 1998). The difference between them usually can be interpreted as a known time domain function (Podlubny, 1998; Hartley & Lorenzo, 2002). Hence, the issue of the operator type can be neglected in the control problems. Therefore, because of its engineering applications, due to directly using the initial conditions (as shown in Podlubny (1998)), the Caputo operator is chosen to define the non-integer order system. Moreover, there are some other derivation operators, i.e. left and right derivation operators or constant and variable order definitions. In this section, we will explain some of these definitions used in the article. Also, regarding to these definitions, one can describe a non-integer order system as a set of differential equations in which the derivation operator in each equation can follow one of the above mentioned ones. 2.1. Non-integer order derivation A non-integer order scalar differential equation can be described as follows. Here, scalar $$x$$ is the response of the differential equation, $$u$$ is the exogenous input and $$D^\alpha$$ is the derivation operator of order $$\alpha$$, where $$\alpha$$ can be a constant or varying function and the derivation operator can be of different types. Dαx=f(x,u) (2.1) The differential equation is supposed to be analysed in a time interval such as $$[0\ T]$$, where $$0$$ is the initial and $$T$$ is the final time. It is often constrained to a boundary condition. In most of the real world dynamics, these conditions are related to the initial or final value of $$x$$, shown here by $$x_0$$ and $$x_f$$, respectively, where $$x_0=x(0),x_f=x(T)$$. In the above-mentioned interval, operator $$D^\alpha$$ can be defined in different ways. Some of these definitions are followed (Lorenzo & Hartley, 2002):  0CDtα(t)f(t) =1Γ(1−α(t))∫0t(t−τ)−α(t)ddτf(τ)dτ0 <α(t)<1,∀t,0≤t≤T (2.2)  tCDTα(t)f(t) =−1Γ(1−α(t))∫tT(τ−t)−α(t)ddτf(τ)dτ0 <α(t)<1,∀t,0≤t≤T (2.3)  0RLDtα(t)f(t) =1Γ(1−α(t))ddτ∫0t(t−τ)−α(t)f(τ)dτ0 <α(t)<1,∀t,0≤t≤T (2.4)  tRLDTα(t)f(t) =−1Γ(1−α(t))ddτ∫tT(τ−t)−α(t)f(τ)dτ0 <α(t)<1,∀t,0≤t≤T (2.5) The definition of left varying order derivation in the sense of Caputo is shown in (2.2) (Lorenzo & Hartley, 2002). Right Caputo derivation can be defined as shown in (2.3) (Lorenzo & Hartley, 2002). Left and right Riemann–Liouville derivatives of variable order $$\alpha(t)$$ is shown in (2.4) and (2.5), respectively (Lorenzo & Hartley, 2002). Also, there are definitions for variable order integration operator as follows:  0Itα(t)f(t) =1Γ(α(t))∫0t(t−τ)α(t)−1f(τ)dτ0 <α(t)<1,∀t,0≤t≤T (2.6)  tITα(t)f(t) =1Γ(α(t))∫tT(τ−t)α(t)−1f(τ)dτ0 <α(t)<1,∀t,0≤t≤T (2.7) Here, left and right time varying order integration operators are defined in (2.6) and (2.7), respectively (Lorenzo & Hartley, 2002). One can use a constant order of derivation/integration instead of the time varying one in each of the above definitions to get a constant non-integer order derivation/integration by setting $$\alpha(t)=\alpha_0, \forall t,0\leq t \leq T$$, where $$ 0<\alpha_0<1$$ is a real number. 2.2. Non-integer order dynamic systems Generally speaking, a non-integer order dynamic system is a set of coupled non-integer order differential equations in which the derivation operators can be of different definitions. Each differential equation can have derivative operator in the sense of Caputo or Riemann–Liouville, left sided or right sided and constant or variable order. Note that the word ‘non-integer’ does not mean that the system cannot include traditional integer derivation of order one. It just implies that the order is not restricted to the integers. The general form of a non-integer system can be shown as the following equations. {Dα1x1=f1(x1,…,xk,u,t)⋮Dαkxk=fk(x1,…,xk,u,t)  (2.8) Here, $$\alpha_{i},i=1,2,\dots,k$$ are generally functions of time and $$u$$ is the exogenous input vector. In specific cases, the above definition can be reduced, for instance, to a constant order dynamic system or a Caputo dynamic system. In this article, we are focusing on the optimal control of a variable order Caputo system, i.e. a set of coupled differential equations all with Caputo left variable order operator of derivation. 3. Problem statement This article aims to develop a scheme to solve optimal control problem for a specific type of non-integer order dynamical systems. In fact, as a widely applicable type of such systems, we concentrate on commensurate variable order dynamics with left Caputo derivations with known initial conditions. Here, the word commensurate means that all of the orders have the same functionality of time, i.e. $$\alpha_i(t)=\alpha_j(t),i,j=1,2,\dots,k,\forall t$$. Since Caputo derivation operator directly uses initial conditions, such systems are proper tools for describing variable order dynamics. This system can be written in detail as: { 0CD tα(t)x1=f1(x1,…,xk,u,t),x1(0)=x1,0⋮ 0CD tα(t)xk=fk(x1,…,xk,u,t),xk(0)=xk,0 0<α(t)<1 (3.1) Or, in brief: Dα(t)x=f(x,u,t),x(0)=x0, (3.2) where $$x={[x_1\ x_2\ \dots\ x_k]}^T$$ is called as the pseudo-state vector. Since the state of the system cannot be defined using $$x$$ the word ‘state’ seems not appropriate and the phrase ‘pseudo-state’ is more precise (Hartley & Lorenzo, 2002). The goal is to design the exogenous input (as a function of time) for the above defined system to make it reach from the known initial condition to a predefined final condition in a way that a cost function is minimized. The problem can be defined as follows. Consider the system described in (3.2). This system is supposed to reach to final condition $$x(T)=x_f$$ in the finite time $$T$$, whereas the functional $$J=\int_0^T g(x,u,t)dt$$ should be minimized or, in brief: {min J=∫0Tg(x,u,t)dts.t. Dα(t)x=f(x,u,t)x(0)=x0,x(T)=xf  (3.3) 4. Optimality conditions 4.1. Preliminaries In this part, the goal is to apply calculus of variation to obtain necessary conditions pertaining to optimal control. First, Lemma 1 provides a reformation formula for dealing with some type of singular integrals (which is used in the definitions of the non-integer derivation). Then, the variable order part by part integration is represented and at last, a relation between variable order derivation and integration is proven. Consider the integral $$H(t)=\int_t^T f(\tau) d\tau$$ where the integrand $$f$$ is singular near the upper integration limit, $$T$$, i.e. $$\lim_{\tau \to T^-}f(\tau)=\infty$$. In this case, the value of $$H$$ at $$T$$ is defined as: $$H(T)=\lim_{t \to T^-}H(t)=\lim_{\epsilon\to 0^+}H(T-\epsilon)$$. Therefore, $$H(T)$$ exists if this limit converges. Note that it is not valid for singular integrands to immediately set $$H(T)=0$$ due to the fact that the lower and upper bounds of the integral are equal (Sladek et al., 2000). In special cases in which the integrand includes a factor like $$(\tau-t)^{-\alpha},0<\alpha<1$$, we can somehow cope with the singularity and rewrite $$H$$ as an integral with a non-singular integrand. Lemma 1 In the case that $$\lambda$$ is non-singular in $$[t\ \ T]$$ and $$0<\alpha<1$$ one has: ∫tT(τ−t)−αλ(τ)dτ=11−α∫0(T−t)(1−α)λ(r11−α+t)dr$$\alpha$$ can be generally a function of $$t$$. Proof. The statement can be easily proven by the change of variable: $$\tau=r^{\frac{1}{1-\alpha}}+t$$. □ Remark If we define $$\bar{H}(t)=\int_t^T (\tau-t)^{-\alpha}\lambda(\tau)d\tau$$, since $$\lambda(r^{\frac{1}{1-\alpha}}+t)$$ is non-singular, now we can write:  limt→T−H¯(t)=limϵ→0+H¯(T−ϵ) =limϵ→0+[11−q∫0ϵ(1−α)λ(r11−α+T−ϵ)dr]=0 (4.1) Hence, when the integrand singularity is because of a $$(\tau-t)^{-\alpha}$$ factor, $$lim_{t\rightarrow T^-}\bar{H}(t)$$ exists, thus $$\bar {H}(T)=0$$. Lemma 2 When $$0<\alpha(t)<1$$ is a smooth enough function so that $${d\alpha}/{dt}$$ exists everywhere on the interval $$[0\ \ T]$$, we have Variable Order Part by Part Integration in the sense of Caputo formula as shown in (4.2), where $$\psi(z)=\frac{d}{dz}ln{\it{\Gamma}}(z)=\frac{\frac{d{\it{\Gamma}}(z)}{dz}}{{\it{\Gamma}}(z)}$$.)  ∫0TgT(t) 0CD tα(t)f(t)dt=fT(t) tI T1−α(t)g(t)|0T     −∫0T(dα(t)dtψ(1−α(t))fT(t) tI T1−α(t)g(t)−fT(t) tRLD Tα(t)g(t))dt (4.2) Proof. See Section 3 of (Odzijewicz et al., 2013a). □ Lemma 3 for $$0<\alpha(t)<1;$$ ddt(Γ(1−α(t)) tI T1−α(t)λ(t))=−Γ(1−α(t)) tRLD Tα(t)λ(t) Proof. It is directly obtained from the definitions (2.5) and (2.7):  tIT1−α(t)λ(t)=1Γ(1−α(t))∫tT(τ−t)−α(t)λ(τ)dτ ⇒Γ(1−α(t)) tIT1−α(t)λ(t)=∫tT(τ−t)−α(t)λ(τ)dτ ⇒ddt(Γ(1−α(t)) tIT1−α(t)λ(t))=ddt∫tT(τ−t)−α(t)λ(τ)dτ =−Γ(1−α(t)) tRLD Tα(t)λ(t) □ 4.2. Augmented cost function and its variation One of the prevalent methods for solving optimal control problems is to derive the first variation of the augmented cost function. Augmented cost function is a functional which is constructed by using Lagrange multipliers and adjoining the constraint as a term of the integrand to the cost function (Pinch, 1993). In the problem posed in (3.3), we can define the augmented cost function as (4.3). Here, $$\lambda$$ is known as co-state, a vector with the same size as $$x$$. Ja=∫0T(g(x,u,t)+λT(f(x,u,t)− 0CD tα(t)x(t)))dt (4.3) Using common techniques in calculus of variations, the first variation of the above functional will be obtained as follows. δJa=∫0T(∂g∂xδx+∂g∂uδu+δλT(f− 0CD tα(t)X)+λT∂f∂xδx+λT∂f∂uδu−λTδ( 0CD tα(t)x))dt (4.4) According to Lemma 2;  ∫0TλTδ( 0CD tα(t)x)dt=∫0TλT 0CD tα(t)(δx)dt=λT tI T1−α(t)δx|0T     −∫0T(dα(t)dtψ(1−α(t)) tI T1−α(t)λTδx− tRLD Tα(t)λTδx)dt (4.5) Substituting (4.5) in (4.4), the first variation of $$J_a$$ is derived as: δJa=∫0T(∂g∂xδx+∂g∂uδu+δλT(f− 0CD tα(t)x) +λT∂f∂xδx+λT∂f∂uδu +(dα(t)dtψ(1−α(t)) tI T1−α(t)λT− tRLD Tα(t)λT)δx)dt − tI T1−α(T)λT(T)δx(T)+ tI T1−α(0)λT(0)δx(0) (4.6) Now, since the initial and final values of $$x$$ are both specified, their variations are equal to zero; $$\delta x(T)=\delta x(0)$$ and (4.6) can be simplified to: δJa=∫0T((∂g∂x+λT∂f∂x+dα(t)dtψ(1−α(t)) tI T1−α(t)λT− tRLD Tα(t)λT)δx    +(∂g∂u+λT∂f∂u)δu+(fT− 0CD tα(t)xT)δλ)dt (4.7) For the optimality, the first variation of the augmented cost function should be equal to zero, leading to the following set of equations: { 0CD tαx−f=0 tRLD Tαλ−dαdtψ(1−α) tI T1−αλ−(∂g∂x)T−(∂f∂x)Tλ=0∂g∂u+λT∂f∂u=0 x(0)=x0,x(T)=xf (4.8) Regarding the above equations: 1. Since the second equation contains both variable integration and derivation operators, it is a differintegral equation rather than a pure differential one. Hence, it cannot be considered as a system. As it will be shown, a variable redefinition will allow us to rewrite the set of equations as a non-integer order system similar to (2.8). 2. The third equation is an algebraic one through which the input $$u$$ can be calculated with respect to $$x$$ and $$\lambda$$. 4.3. Optimal input as the output of a non-integer order system In this part, we aim to rearrange the set of equations (4.8) to get a non-integer order system. This formation agrees with the classical optimal control problems and, it also makes the numerical scheme easier. To this purpose, we define an auxiliary variable $${\it{\Lambda}}$$ as follows. Λ= tI T1−αλ (4.9) According to Lemma 3; ddt(Γ(1−α)Λ)=−Γ(1−α) tRLD Tαλ (4.10) The left side of equation leads to: ddt(Γ(1−α)Λ)=−dαdtψ(1−α)Γ(1−α)Λ+Γ(1−α)Λ˙ ⇒Λ˙−dαdtψ(1−α)Λ+ tRLD Tαλ=0 We can replace $${_t^{RL}}D{_T^\alpha}\lambda$$ from (4.8) to obtain the dynamic of $${\it{\Lambda}}$$:  tRLD Tαλ=dαdtψ(1−α)Λ+(∂g∂x)T+(∂f∂x)Tλ⇒Λ˙+(∂g∂x)T+(∂f∂x)Tλ=0 (4.11) To avoid redundancy, a boundary condition for $${\it{\Lambda}}$$ is needed. It can be calculated from its definition (4.9) and using the remark of Lemma 1: Λ(T)=limt→T−∫tT(τ−t)−α(t)λ(τ)dτ=0 So, the dynamic of the auxiliary state variable $${\it{\Lambda}}$$ is: Λ˙+(∂g∂x)T+(∂f∂x)Tλ=0,Λ(T)=0 (4.12) It could be augmented to (4.8) to form a non-integer order system: { 0CD tα(t)x=f(x,u,t),x(0)=x0,x(T)=xf tRLD Tα(t)λ=−dαdtψ(1−α)Λ−(∂g∂x)T−(∂f∂x)TλΛ˙=−(∂g∂x)T−(∂f∂x)Tλ,Λ(T)=0(∂f∂u)Tλ+(∂g∂u)T=0  (4.13) This system includes traditional derivation (for $${\it{\Lambda}}$$), left variable order derivation in the sense of Caputo (for $$x$$) and right variable order Riemann–Liouville derivation (for $$\lambda$$). There are two boundary conditions for $$x$$ and one for $${\it{\Lambda}}$$. The next section provides a numerical scheme for solving such set of equations. 5. Numerical solution scheme As shown, the necessary conditions of optimality lead to a set of nonlinear differential equation, whose response is the optimal control signal. The existence and uniqueness of the solution are studied in Razminia et al. (2012); Xu & He (2013); Zhang (2013) where it is shown that, just like the traditional case, the solution existence of such equations is very case-dependent. However, the necessary conditions indicate that the control signal $$u$$, calculated from the above set of non-linear equations, is an extremum and its nature (whether it is a maximum, a minimum or a saddle) depends on the nature of the problem. As long as the problem is basically a minimization, one can make sure that $$u$$ is a minimum. As shown in (Kirk, 1970), as long as the Hamiltonian ($$H=g+\lambda^Tf$$) can be expressed as $$H=f+c^Tu+\frac{1}{2}u^TR(t)u$$, where $$c$$ is a vector function that does not have any terms containing $$u$$ and $$R$$ is a positive definite matrix. After calculating $$u$$ with respect to $$\lambda$$ and $$x$$ and substituting it in the first equation of (4.13), all we have is with respect to $$x,\lambda,{\it{\Lambda}}$$. Hence, we have a pure non-integer order autonomous system which is difficult to be solved analytically and the numerical methods should be used. The numerical approach used for solving the variable order differential equations is an extension for the approach used to solve constant order differential equations. Here, we have used an approach similar to the one addressed in Li & Wu (2015). The algorithm is explained in this section. More details can be found in Li & Wu (2015) or Diethelm et al. (2005). The following equation indicates the Grunwald–Letnikov (G–L) definitions of non-integer derivation.  0GLD tαy(t)=limh→01hα∑j=0[th](−1)j(αj)y(t−jh) tGLD Tαy(t)=limh→01hα∑j=0[T−th](−1)j(αj)y(t+jh), (5.1) where $$[.]$$ indicates the integer part and $$\alpha \choose j$$ is defined as: (αj)=Γ(α+1)Γ(j+1)Γ(α−j+1) There are consistencies between the definitions which have been introduced in this article (i.e., R–L, G–L and Caputo definitions) and for a wide class of functions, these three best known definitions are equivalent under some conditions (Podlubny, 1998). In fact, R–L and G–L definitions can be considered the same in most of the situations (Podlubny, 1998) and when $$n-1<\alpha<n$$ the relationship between Caputo and R–L definition is (Podlubny, 1998):  0RLD tαy(t)= 0CD tαy(t)+∑k=0n−1tk−αΓ(k−α+1)y(k)(0), (5.2) where $$y^{(k)}(0)$$ is the $$k$$th derivative of $$y$$ evaluated at $$0$$. Our approach for solving non-integer order systems is to use Grunwald–Letnikov approximation of non-integer order derivation (Ford & Simpson, 2001). It is very common in constant order non-integer order differential equation to approximate the right and left derivatives in the interval $$[0\ \ T]$$ by splitting the interval to $$N$$ sub-intervals. It leads to $$N+1$$ point, say $$t_0,t_1,\dots,t_N$$ in a way that $\begin{bmatrix}0&T\end{bmatrix}=\bigcup _{i=0}^{N-1}\begin{bmatrix}t_{i}&t_{i+1}\end{bmatrix}$ . All sub-intervals have the same length $$h$$, i.e., $$t_{i+1}-t_i=h,i=0,1,\dots,N-1$$. According to the relations between R–L, G–L and Caputo derivative operators when $$0<\alpha<1$$, left Caputo and right R–L constant order derivation operators can be approximated by the following equations, where $$y_i=y(t_i)$$ and $$t_i=ih$$ (Ford & Simpson, 2001; Diethelm et al., 2005).  0CD tαyi ≈1hα∑j=0iωj(α)yi−j−ti−αΓ(1−α)y(0)i =1,2,…,N (5.3)  tRLD Tαyi≈1hα∑j=0N−iωj(α)yi+j,i=N−1,N−2,…,0 (5.4) Coefficient $$\omega^{(\alpha)}_{j}$$ are computed through: (Ford & Simpson, 2001). ω0(α)=1;ωj(α)=(1−α+1j)ωj−1(α),j=1,2,…,N (5.5) These formulation can be extended to variable order derivation. In this case, $$\alpha$$ varies with time, i.e. $$\alpha=\alpha(t)$$ and $$\alpha_i=\alpha(t_i)$$. Hence:             0CD tαiyi ≈1hαi∑j=0iωj(αj)yi−j−ti−αiΓ(1−αi)y(0)i =1,2,…,N (5.6)  tRLD Tαiyi≈1hαi∑j=0N−iωj(αj)yi+j,i=N−1,N−2,…,0 (5.7) The coefficients can be computed similar to (5.5): ω0(αj) =1;ωj(αj)=(1−αj+1j)ωj−1(αj)j =1,2,…,N (5.8) Equations (5.6) and (5.7) are used to approximate $${_0^C}D{_t^{\alpha(t)}}x$$ and $${_t^{RL}}D{_T^{\alpha(t)}}\lambda$$ in (4.13), respectively. The traditional derivative of $${\it{\Lambda}}$$ is also easy to be approximated using forward Euler method: Λ˙(i)≈Λ(i+1)−Λ(i)h,i=1,2,…,N (5.9) If we consider $$p$$ as the size of the pseudo-state vector $$x$$, since $$x,\lambda,{\it{\Lambda}}$$ are all $$p\times 1$$ vectors in each point, we have entirely $$3p(N+1)$$ samples of these functions during the entire interval. However, $$x_0,x_N,{\it{\Lambda}}_N$$ are known and it reduces the number of the unknown values by $$3p$$. So, there are totally $$3pN$$ unknown values. We can apply the formula (5.6) on $$x$$ in samples $$1,2,\dots,N$$ and derive $$pN$$ equations. Also, applying (5.7) on $$\lambda$$ for $$i=0,1,\dots,N-1$$ gives $$pN$$ equations. At last, we can obtain $$pN$$ equations by applying (5.9) on $${\it{\Lambda}}$$ in samples $$1,2,\dots,N$$. Therefore, there are totally $$3pN$$ coupled non-linear equations with respect to unknown variables $$x_i,i=1,2,\dots,N,\ \lambda_i,i=0,1,\dots,N,\ {\it{\Lambda}}_i,i=0,1,\dots,N-1$$. As a result, we have a set of $$3pN$$ equations and also $$3pN$$ unknown variables which can be calculated from the equations by applying a non-linear solver. In the next section, the formulation and the numerical scheme are applied on a case study. 6. Simulation study In this section, we aim to apply our proposed method on a simple case study system to show its efficiency. The system we have chosen is a simple scalar one which is a common case study for the non-integer optimal control problems, as studied in Agrawal (2008); Kumar Biswas & Sen (2011); Agrawal (2004). In fact, choosing a scalar system makes it possible to test and justify the proposed algorithm without loss of generality and the method can be applied on a multi-dimensional system in a similar way. To the best of our knowledge, so far, there has been no other systematic control approach for time varying order systems. Hence, there is no way to examine the proposed method except comparing it with itself to see how it works in different cases. Accordingly, in this section, we consider the following system with different cost functions and apply the proposed optimal control formulation on it. Comparing the results will convince us that the method works efficiently.  tCD Tα(t)x=−x+u,x0=1 (6.1) Consider the system described by (6.1). The order is chosen to be $$\alpha(t)=0.9e^{-\frac{t}{10}}$$. The goal is to make the pseudo-state $$x$$ to reach to $$x_f=0.5$$ in the finite time $$T=1$$. A cost function of the form (6.2) is also considered to be minimized. This cost function relates to the total (input and pseudo-state) energy of the system. Coefficient $$k$$ is an adjusting parameter which makes it possible to set different weights for the pseudo-state’s energy. J=∫0T12(kx2+u2)dt (6.2) After applying the formulation (4.13) on it, the following equations are obtained.  0CD tα(t)x =−x+u,x0=1,xf=0.5 tRLD Tα(t)λ =−kx−u−dαdtψ(1−α)ΛΛ˙ =−kx−u,Λ(T)=0λ+u =0 (6.3) The above equations can be simplified as follows. Now, the non-integer order system is derived and a numerical scheme should be used to solve it.  0CD tα(t)x =−x−λ,x0=1,xf=0.5 tRLD Tα(t)λ =−kx+λ−dαdtψ(1−α)ΛΛ˙ =−kx+λ,Λ(T)=0 (6.4) Since the system under consideration is linear, hence, the numerical solution leads to a set of linear equations. As long as it is non-singular, one can make sure that its response uniquely exists. Therefore, the optimal control signal $$u$$ exists and since it is unique, its nature is undoubtedly a minimizer. We set $$N=150$$ and run the simulation for different values of $$k$$, $$k=0,0.5,1$$. By increasing $$k$$, in fact, the pseudo-state’s energy is being more significant. So it is more important to keep it small. To this aim more control effort (input energy) will be needed. By setting $$k=0$$, we are neglecting the pseudo-state’s energy and it lets $$u$$ get free. Therefore, no extra control energy should be consumed for keeping $$x$$ small. Any other value of $$k$$ will lead to more input energy consumed. Hence, what we expect is that increasing $$k$$ should lessen the pseudo-state’s energy and enhance the input energy. Simulation results are shown in Figs 1 and 2. Fig. 1. View largeDownload slide The pseudo-state $$x$$ after applying the optimal control. Fig. 1. View largeDownload slide The pseudo-state $$x$$ after applying the optimal control. Fig. 2. View largeDownload slide The optimal control signal $$u.$$ Fig. 2. View largeDownload slide The optimal control signal $$u.$$ From Figs 1 and 2, it is obvious that the goal is met and the pseudo-state is reached to its predefined final value. Also, as we expected, it can be seen in Fig. 1 that energy of $$x$$ increases by increasing $$k$$ and, as it is shown in Fig. 2, increasing $$k$$ makes us to provide more control effort to make the pseudo-state reach to the specified final value. As another comparison criterion, the values of the cost functions for different values of $$k$$ are calculated in each moment $$t$$ and shown in Fig. 3. The moment values of $$J$$ can be computed through the following equation. J(t)=∫0t12(kx2(τ)+u2(τ))dτ (6.5) Fig. 3. View largeDownload slide The moment cost functions. Fig. 3. View largeDownload slide The moment cost functions. It is clear from Fig. 3 that increment of $$k$$ leads to consuming more energy and the total system’s energy in each moment (indicated by $$J(t)$$) is increased by increasing $$k$$. It is just what we expect, can be considered as another proof for justifying the proposed solution method. 7. Conclusion In this article, we presented a novel formulation for finite horizon optimal control of time varying order systems. Such systems are difficult to be analysed and so far, there has been no systematic method for controlling them. The contribution presented in this article opens the doors for other control techniques, especially optimal control methods to be developed for variable order systems. Variable order dynamics are useful tools which are currently used to model some complex physical systems, thus the method proposed in this article can be used for different applications. The numerical scheme used here, is an extension of the methods used before to solve constant order dynamic systems and here, it is shown that it can also be applied for solving all non-integer systems. Our focus in this article was mainly on the commensurate systems, i.e. the order of derivation for all of the pseudo-states was considered to be the same. Future works on this field can extend the formulation to the incommensurate ones. Also, infinite horizon optimal control problems can be discussed. Furthermore, other boundary conditions such as unspecified initial or final conditions are interesting topics for future researches. Constraints on the input and states can be also considered. Moreover, some theoretical issues on the sufficiency conditions, solvability, existence and uniqueness should be studied for the general case to obtain the conditions on which the problem is solvable in terms of $$f(x,u)$$, $$g(x,u)$$ and $$\alpha(t)$$. References Agrawal O. P. 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