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Model predictive control of switching continuous‐time systems with stochastic jumps: Application to an electric current source
2022; Institution of Engineering and Technology; Volume: 16; Issue: 4 Linguagem: Inglês
10.1049/cth2.12242
ISSN1751-8652
AutoresAlessandro N. Vargas, João Y. Ishihara, Constantin F. Caruntu, Lixian Zhang, A. A. Nanha Djanan,
Tópico(s)Fault Detection and Control Systems
ResumoIET Control Theory & ApplicationsVolume 16, Issue 4 p. 454-463 ORIGINAL RESEARCH PAPEROpen Access Model predictive control of switching continuous-time systems with stochastic jumps: Application to an electric current source Alessandro N. Vargas, Alessandro N. Vargas Universidade Tecnológica Federal do Paraná, Cornelio, Procópio-PR, BrazilSearch for more papers by this authorJoão Y. Ishihara, João Y. Ishihara Universidade de Brasília, UnB, FT, Brasília-DF, BrazilSearch for more papers by this authorConstantin F. Caruntu, Constantin F. Caruntu Department of Automatic Control and Applied Informatics, Gheorghe Asachi Technical University of Iasi, Iasi, RomaniaSearch for more papers by this authorLixian Zhang, Lixian Zhang Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin, ChinaSearch for more papers by this authorArmand A. Nanha Djanan, Corresponding Author Armand A. Nanha Djanan nandjaor@yahoo.fr Laboratory of Modelling and Simulation in Engeneering, Biomimetics and Prototypes, Faculty of Sciences, University of Yaoundé I, Yaoundé, Cameroon Correspondence Armand A. Nanha Djanan, Laboratory of Modelling and Simulation in Engeneering, Biomimetics and Prototypes, Faculty of Sciences, University of Yaoundé I, P.O. Box 812 Yaoundé, Cameroon. Email: nandjaor@yahoo.frSearch for more papers by this author Alessandro N. Vargas, Alessandro N. Vargas Universidade Tecnológica Federal do Paraná, Cornelio, Procópio-PR, BrazilSearch for more papers by this authorJoão Y. Ishihara, João Y. Ishihara Universidade de Brasília, UnB, FT, Brasília-DF, BrazilSearch for more papers by this authorConstantin F. Caruntu, Constantin F. Caruntu Department of Automatic Control and Applied Informatics, Gheorghe Asachi Technical University of Iasi, Iasi, RomaniaSearch for more papers by this authorLixian Zhang, Lixian Zhang Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin, ChinaSearch for more papers by this authorArmand A. Nanha Djanan, Corresponding Author Armand A. Nanha Djanan nandjaor@yahoo.fr Laboratory of Modelling and Simulation in Engeneering, Biomimetics and Prototypes, Faculty of Sciences, University of Yaoundé I, Yaoundé, Cameroon Correspondence Armand A. Nanha Djanan, Laboratory of Modelling and Simulation in Engeneering, Biomimetics and Prototypes, Faculty of Sciences, University of Yaoundé I, P.O. Box 812 Yaoundé, Cameroon. Email: nandjaor@yahoo.frSearch for more papers by this author First published: 08 February 2022 https://doi.org/10.1049/cth2.12242AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract This paper proposes an extension of the model predictive control framework for switching continuous-time linear systems. The switching times follow a stochastic process with limited statistical information. At each switching time, the controller knows the system state, but it is blind with respect to the switching continuous-time subsystems. In this setting, the paper's main contribution is to show how to compute the model predictive control gain. The paper also illustrates the implications of our approach for applications. The approach was used in practice to control an electric current source that supplied a switching load. The experimental data support the usefulness of our approach. 1 INTRODUCTION Model predictive control (MPC) has attracted much attention from researchers because it has proved to be successful in several applications, such as in electric vehicle [1], cooling water system [2], sea-wave energy harvest [3], electronic converter [4], management of diabetes [5], among others [6]. The basic idea underlying the MPC scheme is that it requires solving a step-by-step optimization control problem [7, 8]. Regarding switching (deterministic) linear systems, MPC has been studied in the context of stability [9], saturated control [10], parameter variation [11], and estimation [12]. Stochastic MPC has been studied as well. For instance, studies have considered stochastic MPC in the context of probability constraints under state feedback [13, 14], tracking control [15], tube-based control [16], and inverse control [17] (see [18] and [19] for further details). For most investigations, the MPC setting is developed for discrete-time models. It makes sense because continuous-time systems can be readily reduced to the usual discrete-time framework when taking a constant sampling rate. However, constant sampling rate is difficult to achieve when the controller receives information through wireless communication. It is then necessary to construct an MPC strategy under non-constant sampling rate. This paper follows in this path, as detailed next. The main contribution of this paper is showing an algorithm that computes a candidate for the optimal solution of the MPC problem. The MPC problem is designed for switching linear systems. As for switching continuous-time linear systems, most results (not MPC) have been developed under the hypothesis that the controller knows exactly the mode's value for each time t ≥ 0 $t\ge 0$ , a quite usual assumption [20-22]. This assumption implies that the control does measure the instantaneous mode through a sensor, that is, the control knows exactly and instantaneously what mode is active at any time. However, this assumption is unrealistic in many real-time processes—that sensor may not exist, as exemplified in [23-26]. In this paper, we consider the controller blind with respect to the switching mode in the MPC setup. Showing an algorithm for that controller represents the main contribution of this paper. Our approach finds novelty in the fact that the switching times are random and their statistics are incomplete (see Assumption 3.1). The controller, blind with respect to the switching times, minimizes the corresponding predictive horizon. To the best of the authors' knowledge, this paper is the first to introduce such an approach. The motivation for this study stems from a real-time application. Indeed, this paper presents laboratory experiments for an electric circuit. It is shown that the circuit feeds a load that changes its value according to random switching times. The MPC approach was then applied to the circuit. The experimental data illustrate the usefulness of our approach for applications. Remark 1.1.The idea of using blind controllers for handling systems with limited statistics have been studied in the context of Markov jump linear systems [26-29]. Although blind controllers represent this paper's main research topic, the results and methods we develop here are completely detached from those found in the literature of Markov jump linear systems. Even the assumption that the system forms a Markov process is unnecessary here. The paper is organized as follows. Section 2 presents the notation, Section 3 problem formulation, and Section 4 the main result—this section presents necessary conditions for optimality and a method to compute this optimal condition. Section 5 presents data from real-time experiments for the control of an electric current source. Finally, Section 6 presents concluding remarks. 2 NOTATION AND PRELIMINARIES The standard basis for the n-dimensional Euclidean space is represented by e 1 , … , e n $e_1,\ldots ,e_n$ . The identity matrix on R n × n $\mathbb {R}^{n \times n}$ is denoted by I n $I_n$ . The symbol ⊗ represents the Kronecker product, which allows us to write [30, T.2.4] ( I s ⊗ S ) ( I s ⊗ V ) = I s ⊗ S V , \begin{equation*} (I_s \otimes S)(I_s \otimes V) = I_s \otimes SV, \end{equation*} where S and V are matrices of compatible dimensions. Taking the basis vectors e i ∈ R s $e_i\in \mathbb {R}^{s}$ and e k ∈ R p $e_k\in \mathbb {R}^{p}$ , we can construct the matrix E i , k ( s × p ) = e i e k ′ $E_{i,k}^{(s\times p)}=e_i e_k^{\prime }$ , which allows us to define the permutation matrix [30] U s × p = ▵ ∑ i = 1 s ∑ k = 1 p E i , k ( s × p ) ⊗ E k , i ( p × s ) . \begin{equation*} U_{s\times p} \overset{\triangle }{=} \sum _{i=1}^s \sum _{k=1}^p E_{i,k}^{(s\times p)} \otimes E_{k,i}^{(p\times s)}. \end{equation*} The above matrix has dimension ( s p × s p ) $(sp \times sp)$ and has a single "1" in each row and in each column. Other useful matrix is U ¯ s × p = ▵ ∑ i = 1 s ∑ k = 1 p E i , k ( s × p ) ⊗ E i , k ( s × p ) . \begin{equation*} \bar{U}_{s\times p} \overset{\triangle }{=} \sum _{i=1}^s \sum _{k=1}^p E_{i,k}^{(s\times p)} \otimes E_{i,k}^{(s\times p)}. \end{equation*} The matrix U ¯ s × p $\bar{U}_{s\times p}$ above has dimension ( s 2 × p 2 ) $(s^2 \times p^2)$ . The derivative of a matrix Y = [ y i k ] $Y=[y_{ik}]$ with respect to a scalar g is [30, Sec. IV] ∂ Y ∂ g = ∂ y i k ∂ g . \begin{equation*} \frac{\partial Y}{\partial g} = {\left[ \frac{\partial y_{ik}}{\partial g} \right]}. \end{equation*} When the derivative is taken with respect to a matrix G ∈ R s × p $G\in \mathbb {R}^{s \times p}$ , the Vetter's construction applies [30, Sec. IV]. First, it can be shown that ∂ G ∂ G = U ¯ s × p and ∂ G ′ ∂ G = U s × p . \begin{equation} \frac{\partial G}{\partial G} = \bar{U}_{s\times p} \quad \text{and} \quad \frac{\partial G^{\prime }}{\partial G} = U_{s\times p}. \end{equation} (1)Assuming that both X and Y are matrix functions with G ∈ R s × p $G\in \mathbb {R}^{s \times p}$ as input argument, then ∂ ( X Y ) ∂ G = ∂ X ∂ G ( I p ⊗ Y ) + ( I s ⊗ X ) ∂ Y ∂ G . \begin{equation} \frac{\partial (X Y)}{\partial G} = \frac{\partial X}{\partial G} (I_p \otimes Y) + (I_s \otimes X) \frac{\partial Y}{\partial G}. \end{equation} (2) In particular, using (1), we have ∂ ( A i + B i G ) ∂ G = ( I s ⊗ B i ) U ¯ s × p \begin{equation} \frac{\partial (A_i+B_iG)}{\partial G} = (I_s \otimes B_i) \bar{U}_{s\times p} \end{equation} (3)and ∂ ( A i + B i G ) ′ ∂ G = U s × p ( I p ⊗ B i ′ ) . \begin{equation} \frac{\partial (A_i+B_iG)^{\prime }}{\partial G} = U_{s\times p} (I_p \otimes B_i^{\prime }). \end{equation} (4) Also, if X, Y, and Z are functions depending on G, then ∂ ( X Y Z ) ∂ G = ∂ X ∂ G ( I p ⊗ Y Z ) + ( I s ⊗ X ) ∂ Y ∂ G ( I p ⊗ Z ) + ( I s ⊗ X Y ) ∂ Z ∂ G . \begin{eqnarray} \frac{\partial (X YZ)}{\partial G}& =& \frac{\partial X}{\partial G} (I_p \otimes YZ) \nonumber \\ &&+\, (I_s \otimes X) \frac{\partial Y}{\partial G} (I_p \otimes Z) + (I_s \otimes XY) \frac{\partial Z}{\partial G}. \end{eqnarray} (5) If X is a matrix function depending on G ∈ R s × p $G\in \mathbb {R}^{s \times p}$ , then (for all t ≥ 0 $t\ge 0$ ) ∂ exp ( X t ) ∂ G = ∫ 0 t exp ( I s ⊗ X ) ( t − τ ) ∂ X ∂ G ( I p ⊗ exp ( X τ ) ) d τ . \begin{eqnarray} && \frac{\partial \exp (X t)}{\partial G}\nonumber \\ &&\quad = \int _{0}^t \exp {\left((I_s \otimes X)(t-\tau ) \right)} \, \frac{\partial X}{\partial G} \, (I_p \otimes \exp (X \tau ) ) \, \mathrm{d}\tau .\qquad \end{eqnarray} (6) 3 MPC PROBLEM FOR SWITCHING CONTINUOUS-TIME SYSTEMS 3.1 The switched system Consider the following continuous-time switching linear system x ̇ ( t ) = A σ ( t ) x ( t ) + B σ ( t ) u ( t ) , ∀ t ≥ t 0 , x ( t 0 ) = x 0 ∈ R n , \begin{align} \dot{x}(t)&=A_{\sigma (t)}x(t) + B_{\sigma (t)}u(t), \quad \forall t\ge t_0, \, x(t_0)=x_0 \in \mathbb {R}^n, \end{align} (7)where x ( t ) $x(t)$ on R n $\mathbb {R}^n$ denotes the system state and u ( t ) $u(t)$ on R s $\mathbb {R}^s$ represents the control input. The switching signal σ ( t ) $\sigma (t)$ represents a right-continuous stochastic process taking values in a finite set, that is, σ ( t ) ∈ { 1 , … , N } $\sigma (t) \in \lbrace 1,\ldots ,N\rbrace$ , and the corresponding points of discontinuity t 0 < t 1 < t 2 < ⋯ < t k < ⋯ $t_0<t_1<t_2<\cdots <t_k<\cdots$ occur randomly ( t k → ∞ $t_k\rightarrow \infty$ as k → ∞ $k\rightarrow \infty$ with probability one). Each point t k $t_k$ is called switching time [31-35]. At the switching time t k $t_k$ , the matrix A σ ( t ) $A_{\sigma (t)}$ switches instantaneously, say, from A i $A_i$ to A j $A_j$ , for some i ≠ j $i \not= j$ ; namely, set σ ( t k − ) = lim t ↑ t k σ ( t ) $\sigma (t_k^-)=\lim _{t\uparrow t_k}\sigma (t)$ to write σ ( t k − ) = i $\sigma (t_k^-)=i$ and σ ( t k ) = j $\sigma (t_k)=j$ . Similar reasoning applies for B σ ( t ) $B_{\sigma (t)}$ . Assumption 3.1.The switching signal { σ ( t ) } $\lbrace \sigma (t)\rbrace$ forms a stochastic process for which the probability distribution is known only at the switching times, that is, Pr ( σ ( t k ) = i ) = w i , k , i = 1 , … , N $\Pr (\sigma (t_k)=i)=w_{i,k}, i=1,\ldots ,N$ , is known for each k ≥ 0 $k\ge 0$ . In applications, the probability distribution given in Assumption 3.1 could be obtained through estimation or other statistical methods [36, 37]. Remark 3.1.The system (7) can model a switching system with the mode signal σ ( t ) $\sigma (t)$ switching at all time (as in the Markov jump approach, for example, [38, 39]) but sampled only at some instants t k $t_k$ . In this case, the proposed MPC framework can be used for continuously switching systems with the probability Pr ( σ ( t ) = i ) $\Pr (\sigma (t)=i)$ unknown for t k < t < t k + 1 $t_k<t<t_{k+1}$ . To design an MPC for a system obtained by this modeling, we develop a method completely detached from the Markov jump approach. Markov chain properties are not used in this paper. 3.2 Control problem Let us consider the system (7) when the switching time t k $t_k$ occurs. At this time, the controller reads the value of x ( t k ) ∈ R n $x(t_k) \in \mathbb {R}^n$ , but it is blind with respect to σ ( t k ) $\sigma (t_k)$ , that is, the only information available to the controller is the probability distribution Pr ( σ ( t k ) = i ) = w i , k , i = 1 , … , N $\Pr (\sigma (t_k)=i)=w_{i,k}, i=1,\ldots ,N$ (see Assumption 3.1). This feature motivated us to consider the control in the linear state-feedback form u ( t ) = G k x ( t ) , ∀ t , t k ≤ t < t k + 1 . \begin{equation} u(t) = G_k x(t), \quad \forall t, t_k \le t < t_{k+1}. \end{equation} (8) Note that G k $G_k$ does not depend on σ ( t ) $\sigma (t)$ . Presenting a method for obtaining G k $G_k$ represents the main contribution of this paper. Remark 3.2.The controller knows the value of x ( t ) ∈ R n $x(t) \in \mathbb {R}^n$ for all t ≥ 0 $t\ge 0$ , in a linear state-feedback form. However, the controller is blind with respect to σ ( t ) $\sigma (t)$ for all t ≥ 0 $t\ge 0$ . At the instant t = t k $t=t_k$ , we have that σ ( t ) = i $\sigma (t)=i$ for all t ∈ [ t k , t k + T i , k ) $t\in [t_k,t_{k}+T_{i,k})$ . It follows that the predictive cost (related to the gain G k $G_k$ and mode i) of the system (7) must be calculated through the expression J G k ( i , x ( t k ) ) = ∫ t k t k + T i , k x ( t ) ′ Q i x ( t ) + u ( t ) ′ R i u ( t ) d t \begin{equation} J_{G_k}(i,x(t_k)) = \int _{t_k}^{t_k+T_{i,k}} {\left(x(t)^{\prime } \mathcal {Q}_i x(t) + u(t)^{\prime }\mathcal {R}_i u(t) \right)} \mathrm{d}t \end{equation} (9)subject to x ̇ ( t ) = ( A i + B i G k ) x ( t ) , ∀ t ≥ t k , x ( t k ) ∈ R n , \begin{equation} \dot{x}(t)=(A_i+B_i G_k) x(t), \quad \forall t\ge t_k, \quad x(t_k)\in \mathbb {R}^n, \end{equation} (10)where Q i $\mathcal {Q}_i$ and R i $\mathcal {R}_i$ denote symmetric positive definite matrices on R n × n $\mathbb {R}^{n \times n}$ and R s × s $\mathbb {R}^{s \times s}$ , respectively. Remark 3.3.Note in (9) that T i , k > 0 $T_{i,k}>0$ acts as a prediction horizon. Its value should be chosen according to the control designer's experience on the specific application to be controlled.. For instance, in the application of Section 5, we set T i , k ≡ 2 $T_{i,k} \equiv 2$ milliseconds for all i and all k ≥ 0 $k\ge 0$ . At t = t k $t=t_k$ , the controller does not know which mode i ∈ { 1 , … , N } $i\in \lbrace 1,\ldots ,N\rbrace$ is active. For this reason, it is necessary to use Pr ( σ ( t k ) = i ) = w i , k , i = 1 , … , N $\Pr (\sigma (t_k)=i)=w_{i,k}, i=1,\ldots ,N$ , to calculate all mode-dependent costs J G k ( 1 , x ( t k ) ) , … , J G k ( N , x ( t k ) ) $J_{G_k}(1,x(t_k)),\ldots ,J_{G_k}(N,x(t_k))$ at once. Namely, the control problem consists in solving min G k E J G k σ ( t k ) , x ( t k ) | x ( t k ) . \begin{align} & \min _{G_k} \, \mathrm{E}{\left[J_{G_k} {\left(\sigma (t_k),x(t_k) \right)}|x(t_k)\right]}. \end{align} (11) In words, (11) represents the optimal mean-value predictive cost of the system (7) in the scenario of limited statistical information upon σ ( t k ) $\sigma (t_k)$ . Remark 3.4.In the particular case in which σ ( t k ) $\sigma (t_k)$ is known, we have w i 0 , k = 1 $w_{i_0,k}=1$ for some i0, and w i , k = 0 $w_{i,k}=0$ for all i ≠ i 0 $i\not= i_0$ . In this case, the control problem in (11) reduces to the classical deterministic linear-quadratic regulation problem, which has solution given by the algebraic Riccati equation [40, Ch. 3.3], [41]. Thus the difficulty on solving (11) arises when 0 < w i , k < 1 $0<w_{i,k} t k $t>t_k$ reaches t = t k + 1 $t=t_{k+1}$ . At this time, set k + 1 $k+1$ and return to Step 1. Remark 3.5.Algorithm 3.1 defines the model predictive control to be used in applications (e.g., Section 5). As can be seen, each random interval [ t k , t k + 1 ) $[t_k,t_{k+1})$ yields a control gain. Since t k $t_k$ tends to infinity as k tends to infinity (with probability one), the model predictive control in Algorithm 3.1 works indefinitely, generating infinitely many gain matrices, with probability one. The main contribution of this paper is showing an algorithm that computes a candidate for the optimal solution of (11). Only a candidate can be guaranteed. Finding the optimal solution of (11) is an open problem. Remark 3.6.The expression in (11) is a nonlinear functional, possibly with many local minimizers (see Section 4.2). This feature signifies that any local optimal point, which we show how to calculate, may not coincide with the global minimum. Our contribution has practical implications as well. Experiments were carried out in a laboratory testbed, and they reveal the benefits of Algorithm 3.1—this algorithm was used to design a controller for an electric current source, a device used extensively in applications. 4 MAIN RESULTS 4.1 Gradient matrix for the i th $i^{\text{th}}$ mode Define the evolution matrix Φ i , G k ( t ) = exp ( ( A i + B i G k ) t ) , i = 1 , … , N . \begin{equation} \Phi _{i,G_k}(t)=\exp ((A_i+B_iG_k)t), \quad i=1,\ldots ,N. \end{equation} (12)The cost (9) evaluated on the system (10) with x ( t k ) = x k $x(t_k)=x_k$ can be recast as J G k ( i , x k ) = x k ′ ∫ 0 T i , k Φ i , G k ( t ) ′ ( Q i + G k ′ R i G k ) Φ i , G k ( t ) d t x k . \begin{eqnarray} && J_{G_k}(i,x_k)\nonumber \\ &&\quad = x_k^{\prime } {\left(\int _0^{T_{i,k}} \Phi _{i,{G_k}}(t)^{\prime }(\mathcal {Q}_i + G_k^{\prime }\mathcal {R}_i G_k)\Phi _{i,{G_k}}(t) \, \mathrm{d}t \!\right)} x_k.\qquad \end{eqnarray} (13) Now, we use the expression in (13) to obtain the gradient matrix of J G k ( i , x k ) $J_{G_k}(i,x_k)$ —the gradient matrix appears when the derivative of (13) is taken with respect to G k $G_k$ . With a straightforward application of the derivative rule from (5) into (13), it follows that the gradient matrix function of the cost (13) equals the expression in (14) (shown in the top of the next page). ∂ J G k ( i , x k ) ∂ G k = ( I s ⊗ x k ′ ) ∫ 0 T i , k ∂ Φ i , G k ( t ) ′ ( Q i + G k ′ R i G k ) Φ i , G k ( t ) ∂ G k d t ( I p ⊗ x k ) . \fontsize{9.5}{12}\begin{eqnarray} &&\hspace*{-6pt}\frac{\partial J_{G_k}(i,x_k)}{\partial G_k} = \nonumber\\ &&\hspace*{-6pt}\quad (I_s \otimes x_k^{\prime }) {\left(\!\int _0^{T_{i,k}}\!\! \frac{\partial \, \Phi _{i,{G_k}}(t) ^{\prime }(\mathcal {Q}_i + G_k^{\prime }\mathcal {R}_iG_k) \Phi _{i,G_k}(t)}{\partial G_k} \, \mathrm{d}t\! \right)} (I_p \!\otimes\! x_k).\nonumber\\ \end{eqnarray} (14)In (14), we still need to characterize the partial derivative of the term Φ i , G k ( t ) ′ ( Q i + G k ′ R i G k ) Φ i , G k ( t ) $\Phi _{i,G_k}(t)^{\prime }(\mathcal {Q}_i + G_k^{\prime }\mathcal {R}_i G_k)\Phi _{i,G_k}(t)$ with respect to G k $G_k$ . First, in order to compute the partial derivative of Φ i , G k ( t ) = exp ( ( A i + B i G k ) t ) $\Phi _{i,G_k}(t)=\exp ((A_i+B_iG_k)t)$ with respect to G k $G_k$ , substitute X ≡ A i + B i G k $X\equiv A_i+B_iG_k$ into (6) and use (3) to compute ∂ X / ∂ G k ${\partial X}/{\partial G_k}$ . Then, for all t ∈ [ 0 , T ] $t\in [0,T]$ , the equations (15) and (16) hold (equations shown in the top of the next page) ∂ Φ i , G k ( t ) ∂ G k = ∫ 0 t exp ( I s ⊗ ( A i + B i G k ) ) ( t − τ ) ( I s ⊗ B i ) × U ¯ s × p I p ⊗ exp ( ( A i + B i G k ) τ ) d τ , \begin{align} \frac{\partial \Phi _{i,G_k}(t)}{\partial G_k} = &\int _{0}^t \exp {\left[ (I_s \otimes (A_i+B_iG_k))(t-\tau ) \right]} (I_s \otimes B_i)\nonumber\\ &\times\,\bar{U}_{s \times p} {\left[ I_p \otimes \exp ((A_i+B_iG_k) \tau ) \right]} \, \mathrm{d}\tau , \end{align} (15) ∂ Φ i , G k ( t ) ′ ∂ G k = ∫ 0 t exp ( I s ⊗ ( A i + B i G k ) ′ ) ( t − τ ) U s × p × ( I p ⊗ B i ′ ) I p ⊗ exp ( ( A i + B i G k ) ′ τ ) d τ . \begin{align} \frac{\partial \Phi _{i,G_k}(t)^{\prime }}{\partial G_k} =& \int _{0}^t \exp {\left[ (I_s \otimes (A_i+B_iG_k)^{\prime })(t-\tau ) \right]} U_{s \times p}\nonumber\\ &\times\, (I_p \otimes B_i^{\prime }) {\left[ I_p \otimes \exp ((A_i+B_iG_k)^{\prime } \tau ) \right]} \, \mathrm{d}\tau . \end{align} (16) Recall that we aim to evaluate the integral term inside the parentheses of (14). So far, we have revealed the expression for the partial derivative of both Φ i , G k ( · ) $\Phi _{i,G_k}(\cdot )$ and Φ i , G k ( · ) ′ $\Phi _{i,G_k}(\cdot )^{\prime }$ ; next we characterize the partial derivative of the compounded term Φ i , G k ( t ) ′ ( Q i + G k ′ R i G k ) Φ i , G k ( t ) $\Phi _{i,G_k}(t)^{\prime }(\mathcal {Q}_i + G_k^{\prime }\mathcal {R}_i G_k) \Phi _{i,G_k}(t)$ . Setting Q i , G k = ▵ Q i + G k ′ R i G k $\mathcal {Q}_{i,G_k} \overset{\triangle }{=} \mathcal {Q}_i + G_k^{\prime }\mathcal {R}_i G_k$ and using (1) and (5) we have that ∂ Q i , G k ∂ G k = U s × p ( I p ⊗ R i G k ) + ( I s ⊗ G k ′ R i ) U ¯ s × p . \begin{equation} \frac{\partial \mathcal {Q}_{i,G_k}}{\partial G_k} = U_{s \times p} (I_p \otimes \mathcal {R}_iG_k) + (I_s \otimes G_k^{\prime }\mathcal {R}_i) \bar{U}_{s \times p}. \end{equation} (17) Substituting X ≡ Φ G k ′ $X\equiv \Phi _{G_k}^{\prime }$ , Y ≡ Q G k $Y\equiv \mathcal {Q}_{G_k}$ , and Z ≡ Φ G k $Z\equiv \Phi _{G_k}$ into (5), we have ∂ Φ G k ′ Q G k Φ G k ∂ G k = ∂ Φ G k ′ ∂ G k ( I p ⊗ Q G k Φ G k ) + I s ⊗ Φ G k ′ × ∂ Q G k ∂ G k ( I p ⊗ Φ G k ) + ( I s ⊗ Φ G k ′ Q G k ) ∂ Φ G k ∂ G k . \begin{multline} \dfrac{\partial \, \Phi _{G_k} ^{\prime }\mathcal {Q}_{G_k} \Phi _{G_k}}{\partial G_k} \, = \, \dfrac{\partial \, \Phi _{G_k}^{\prime }}{\partial G_k} (I_p \otimes \mathcal {Q}_{G_k}\Phi _{G_k} ) + {\left(I_s \otimes \Phi _{G_k}^{\prime } \right)}\\ \times\, \dfrac{\partial \, \mathcal {Q}_{G_k}}{\partial G_k}(I_p \otimes \Phi _{G_k}) \\ + (I_s \otimes \Phi _{G_k}^{\prime }\mathcal {Q}_{G_k}) \frac{\partial \, \Phi _{G_k}}{\partial G_k}.\qquad\qquad \end{multline} (18)Substituting (17) in the right-hand side of (18) we have ∂ ∂ G k [ Φ i , G k ( t ) ′ ( Q i + G k ′ R i G k ) Φ i , G k ( t ) ] = ∂ Φ i , G k ( t ) ′ ∂ G k ( I p ⊗ ( Q i + G k ′ R i G k ) Φ i , G k ( t ) ) + I s ⊗ Φ i , G k ( t ) ′ U s × p ( I p ⊗ R i G k ) + ( I s ⊗ G k ′ R i ) U ¯ s × p ( I p ⊗ Φ i , G k ( t ) ) + I s ⊗ Φ i , G k ( t ) ′ ( Q i + G k ′ R i G k ) ∂ Φ i , G k ( t ) ∂ G k . \begin{align*} \frac{\partial }{\partial G_k} & {[\Phi _{i,G_k}(t) ^{\prime }(\mathcal {Q}_i + G_k^{\prime }\mathcal {R}_i G_k) \Phi _{i,G_k}(t)]} \nonumber \\ &= \frac{\partial \Phi _{i,G_k}(t)^{\prime }}{\partial G_k} (I_p \otimes (\mathcal {Q}_i + G_k^{\prime }\mathcal {R}_i G_k)\Phi _{i,G_k}(t) ) \nonumber \\ &\quad + {\left(I_s \otimes \Phi _{i,G_k}(t)^{\prime } \right)} \left[U_{s \times p} (I_p \otimes \mathcal {R}_iG_k)\right. \nonumber \\ & \left.\quad + (I_s \otimes G_k^{\prime }\mathcal {R}_i)\bar{U}_{s \times p} \right] (I_p \otimes \Phi _{i,G_k}(t)) \nonumber \\ &\quad + {\left(I_s \otimes \Phi _{i,G_k}(t)^{\prime } (\mathcal {Q}_i+G_k^{\prime }\mathcal {R}_iG_k) \right)} \frac{\partial \, \Phi _{i,G_k}(t)}{\partial G_k}. \end{align*} Finally, we have that the gradient matrix function for the cost (13) is identical to (19) (shown in the top of the page). ∂ J G k ( i , x k ) ∂ G k = ( I s ⊗ x k ′ ) ∫ 0 T i , k { ∂ Φ i , G k ( t ) ′ ∂ G k I p ⊗ ( Q i + G k ′ R i G k ) Φ i , G k ( t ) + I s ⊗ Φ i , G k ( t ) ′ U s × p ( I p ⊗ R i G k ) + ( I s ⊗ G k ′ ) ( I s ⊗ R i ) U ¯ s × p ( I p ⊗ Φ i , G k ( t ) ) + I s ⊗ Φ i , G k ( t ) ′ ( Q i + G k ′ R i G k ) ∂ Φ i , G k ( t ) ∂ G k } d t ( I p ⊗ x k ) . \begin{align} &\frac{\partial J_{G_k}(i,x_k)}{\partial G_k} = (I_s \otimes x_k^{\prime }) \int _0^{T_{i,k}} \bigg \lbrace \frac{\partial \Phi _{i,G_k}(t)^{\prime }}{\partial G_k} {\left(I_p \otimes (\mathcal {Q}_i + G_k^{\prime }\mathcal {R}_i G_k)\Phi _{i,G_k}(t) \right)} + {\left(I_s \otimes \Phi _{i,G_k}(t)^{\prime } \right)} \left[U_{s \times p} (I_p \otimes \mathcal {R}_iG_k) \right.\nonumber \\ &\left.\quad + (I_s \otimes G_k^{\prime })(I_s \otimes \mathcal {R}_i) \bar{U}_{s \times p} \right] (I_p \otimes \Phi _{i,G_k}(t))+ {\left(I_s \otimes \Phi _{i,G_k}(t)^{\prime }(\mathcal {Q}_i+G_k^{\prime }\mathcal {R}_iG_k) \right)} \frac{\partial \, \Phi _{i,G_k}(t)}{\partial G_k} \bigg \rbrace \mathrm{d}t \, (I_p \otimes x_k). \end{align} (19) 4.2 Necessary optimality condition Recall the optimal control problem in (11). For the sake of simplicity, set x k = x ( t k ) $x_k=x(t_k)$ , and w i , k = Pr ( σ ( t k ) = i ) $w_{i,k}=\Pr (\sigma (t_k)=i)$ . It follows that E J G k ( σ ( t k ) , x k ) = ∑ i = 1 N w i , k J G k ( i , x k ) . \begin{equation*} \mathrm{E}{\left[J_{G_k}(\sigma (t_k),x_k)\right]} = \sum _{i=1}^N w_{i,k} J_{G_k}(i,x_k). \end{equation*} The main contribution of this paper is presented next—a necessary optimality condition for the problem (11). Propo
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