Optimality Condition-Based Sensitivity Analysis of Optimal Control for Hybrid Systems and Its Application

Abstract

Gradient-based algorithms are efficient to compute numerical solutions of optimal control problems for hybrid systems (OCPHS), and the key point is how to get the sensitivity analysis of the optimal control problems. In this paper, optimality condition-based sensitivity analysis of optimal control for hybrid systems with mode invariants and control constraints is addressed under a priori fixed mode transition order. The decision variables are the mode transition instant sequence and admissible continuous control functions. After equivalent transformation of the original problem, the derivatives of the objective functional with respect to control variables are established based on optimal necessary conditions. By using the obtained derivatives, a control vector parametrization method is implemented to obtain the numerical solution to the OCPHS. Examples are given to illustrate the results.

1. Introduction

In many fields of applications, such as powertrain systems of automobiles and multistage chemical processes, dynamics of the systems involve a sequence of distinct modes with fixed mode transition order, forming a hybrid system characterized by the coexistence and interaction of discrete and continuous dynamics (the mode is commonly denoted by a discrete state of the systems in hybrid systems literature). To achieve some overall optimal performance for the systems, the duration and the admissible continuous control function of each mode must be determined as a whole [1–3]; thus, it necessitates the use of theories and techniques for the analysis and synthesis of hybrid dynamical systems. With the growing importance of hybrid models, various classes of hybrid systems for analysis, design, and optimization have been addressed by research communities in recent years. For more discussions on various literature results, the reader is referred to [4–8], and the references therein.

The existed results on OCPHS can be divided into the following two categories. One is about the optimal control theory on OCPHS. The theory inherits conventional optimal control theory and can be regarded as the extension of conventional optimal control theory [3, 9–14]. When control can take any value, Xu and Antsaklis [3] and Hwang et al. [9] addressed the variational method for hybrid systems. Sussmann [10], Shaikh and Caines [11], and Dmitruk and Kaganovich [12] established the Maximum Principle for hybrid systems with control constraints. Branicky et al. [14] and Bensoussan and Menaldi [13] provided the dynamic programming principle for general hybrid systems.

The other results focus on how to compute optimal control for OCPHS, which can be carried out by using a wide variety of methods (see [3, 6, 11, 15–20] and the references therein). Given a prespecified order of mode transitions, Xu and Antsaklis [3] obtained the optimal continuous control and optimal switching instants based on parameterization of the switching instant for switching hybrid systems with free control. Under a fixed switching sequence of modes, Attia et al. [19] considered an optimization problem for a class of impulsive hybrid systems where continuous control function is not involved. When switching hybrid systems with control constraints are considered, Shaikh and Caines [11] proposed two algorithms for obtaining the optimal control. As far as switching hybrid systems without external continuous control function are concerned, Egerstedt et al. [6] and Johnson and Murphey [18] derived the gradients and second-order derivatives of the cost functional, respectively, and used them to design an associated algorithm to get the mode transition instants. Based on the hybrid Maximum Principle, Taringoo and Caines [20] provided gradient geodesic and Newton geodesic algorithms for the optimization of autonomous hybrid systems, and convergence analysis for the algorithms was also provided. From the view of dynamic programming, Seatzu et al. [16] provided an optimal state feedback control law to switched piecewise affine autonomous systems. Generally, these algorithms pose the hierarchy [17, 21, 22], and the basic module of the hierarchical algorithms is how to get optimal continuous control and optimal mode transition instants, though the main challenge of OCPHS is how to get the optimal mode transition order. The basic module of the hierarchical algorithms is commonly gradient based due to that gradient information can provide a better searching direction and hence reduce computation burden and help the gradient-based algorithms converge quickly, which motivates us to pay attention to the sensitivity analysis of optimal control for hybrid systems.

Although the derivative of cost functional with respect to switching instants has been discussed in the aforementioned literature [3, 6, 18], the derivative of cost functional with respect to control function is not involved. When hybrid systems are considered, due to the coexistence and interaction of discrete and continuous dynamics, the derivative of cost functional w.r.t control functions is nontrivial and is not directly formulated by 𝜕𝐻/𝜕𝑢 as conventional optimal control indicates, where 𝐻 is the Hamiltonian function. The derivative will be a function of the derivatives of continuous states w.r.t control functions at the instants of subsequent modes. In this paper, the derivatives of cost functional w.r.t control functions are established analytically, which can facilitate the design of associated gradient-based algorithms.

Motivated by the work of Vassiliadis et al. [1, 2] and Jennings et al. [23], in this paper, optimal control problem of hybrid systems (OCPHS) with mode invariants which describe the conditions that continuous states have to satisfy at this mode are considered. Based on optimal necessary conditions, the derivatives of the objective functional w.r.t control variables, that is, the mode transition instant sequence and admissible continuous control functions, are derived analytically. As a result, a control vector parametrization method is implemented to obtain the numerical solution to optimal control of the hybrid systems with the obtained derivatives. The sensitivity analysis in Vassiliadis et al. [1, 2] is similar to the work, in which the sensitivity of states w.r.t control parameters is directly obtained from the state equations and the sensitivity of objective functional with respect to control parameters is not involved. In contrast, this paper derives the derivatives of cost functional w.r.t control variables based on the optimality conditions and gives the explicitly expression of the derivatives. Therefore, the main contributions of this paper are listed as follows. (a) Optimality conditions-based sensitivity analysis of optimal control for hybrid systems with mode invariants are given explicitly, and (b) following the given derivatives, a control vector parameterization method is designed to obtain the numerical solution. Compared with the existing results on the OCPHS with fixed mode transition order, the settings in this paper cover not only the control constraints, but also the continuous states constraints, which makes the results here more general.

The paper is organized as follows. In the next section, the hybrid system with mode invariants and its optimal control problem are formulated. In Section 3, the equivalent problem and associated optimal conditions are analyzed. The derivatives of the objective functional w.r.t control variables are established in Section 4, and a control vector parametrization approach is also proposed in this section. Some numerical examples are presented in Section 5, and Section 6 contains conclusions.

Terminology and Notation
ℕ denotes the set of positive integers. ℝ and ℝ₊ denote the set of real numbers and nonnegative real numbers, respectively. 𝐴^𝑇 denotes the transpose of a vector (or a matrix) 𝐴. 𝐶^𝑙([𝑎,𝑏],ℝ^𝑛) denotes the family of continuous functions 𝑓 from [𝑎,𝑏] to ℝ^𝑛 with up to 𝑙 order derivatives. ‖⋅‖ denotes the Euclidean norm.

2. Hybrid Systems and Its Optimal Control Problem

2.1. Hybrid Systems

Engineered systems, such as chemical engineering systems and powertrain systems of automobiles, always undergo multiple modes which are represented by a discrete state 𝑖 taking values from set 𝐼≐{1,2,…,𝑀} and pose hybrid characters. The evolution of discrete state 𝑖 is determined by mode transition sequence. A mode transition sequence schedules the sequence of active modes 𝑖_𝑗,𝑖_𝑗∈𝐼 and is a sequence of pairs of (𝑡_𝑗−1,𝑖_𝑗), which can be defined by {(𝑡₀,𝑖₁),(𝑡₁,𝑖₂),…}≐(𝜃,𝜋) where 𝜃≐{𝑡₀,𝑡₁,…} and 𝜋≐{𝑖₁,𝑖₂,…} are referred to as mode transition instants and mode transition order, respectively. A pair of (𝑡_𝑗−1,𝑖_𝑗) indicates that at instant 𝑡_𝑗−1, the hybrid system transits from mode 𝑖_𝑗−1 to mode 𝑖_𝑗. During the time interval [𝑡_𝑗−1,𝑡_𝑗), mode 𝑖_𝑗 is active and unchanged.

The mode transition order 𝜋 of the considered hybrid dynamical systems is known a priori. Without loss of generality, it is supposed that the mode transition order is {𝑖₁,𝑖₂,…,𝑖_𝐾} over the finite horizon [𝑡₀,𝑡_𝑓], 𝑖_𝑗∈𝐼, 𝑗=1,2,…,𝐾. Moreover, according to each distinct mode, the continuous states are restricted in a specified range which is referred to as mode invariants. Here, the mode invariants are formulated by a set of inequalities. Thus, for each mode 𝑖_𝑗∈𝐼 and its active horizon [𝑡_𝑗−1,𝑡_𝑗), the dynamics of the considered systems can be formulated by 𝑥̇𝑝=𝑓_𝑖𝑗(𝑥,𝑢),_𝑖𝑗𝑥𝑡(𝑥)<0,_𝑗−1g=𝜓_𝑖𝑗𝑥𝑡^−𝑗−1_𝑖𝑗,𝑥𝑡^−𝑗=0,(2.1) where 𝑥∈ℝ^𝑛, 𝑢∈𝐔_𝑖𝑗⊆ℝ^𝑚 is a piecewise continuous function, 𝑓_𝑖𝑗∶ℝ^𝑛×𝐔_𝑖𝑗→ℝ^𝑛, 𝑡_𝑗 is the mode transition instant when a particular mode transition occurs, 𝑝_𝑖𝑗, 𝜓_𝑖𝑗, and 𝑔_𝑖𝑗 are ℎ_𝑖𝑗<𝑛, 𝑛 and 𝑟_𝑖𝑗≤𝑛 dimensional vectors for 𝑖_𝑗∈𝐼, respectively. 𝑛,𝑚,ℎ_𝑖𝑗,𝑟_𝑖𝑗∈ℕ. To make the hybrid systems formulated by (2.1) well defined, the following assumption is needed.

Assumption 2.1. For any 𝑖_𝑗∈𝐼, 𝑓_𝑖𝑗∈𝐶^𝑙(ℝ^𝑛×𝐔_𝑖𝑗;ℝ^𝑛),𝑙≥1,𝑙∈ℕ, and such that a uniform Lipschitz condition holds, that is, there exists 𝐾_𝑓<∞ such that ‖‖𝑓_𝑖𝑗(𝑥,𝑢)−𝑓_𝑖𝑗𝑥^,𝑢‖‖≤𝐾_𝑓‖𝑥−𝑥′‖,(2.2) where 𝑥,𝑥^∈ℝ^𝑛,𝑢∈𝐔_𝑖𝑗.

Remark 2.2. 𝑝_𝑖𝑗(𝑥)<0 indicates mode invariant for mode 𝑖_𝑗∈𝐼, which describes the conditions that the continuous states have to satisfy at this mode and can be referred to as the path constraints of the continuous states in Vassiliadis et al. [1, 2].

Remark 2.3. 𝑔_𝑖𝑗(𝑥(𝑡^−𝑗))=0 can be referred to as mode transition conditions which describe the conditions on the continuous states under which a particular mode transition takes place. When mode 𝑖_𝑗 is active over [𝑡_𝑗−1,𝑡_𝑗), then, at 𝑡^−𝑗, 𝑥 meets an (𝑛−𝑟_𝑖𝑗)-dimensional smooth manifold 𝑆_𝑖𝑗={𝑥∣𝑔_𝑖𝑗(𝑥)=0} and mode transition from 𝑖_𝑗 to 𝑖_𝑗+1 occurs. The mode transition conditions implicitly define the mode 𝑖_𝑗's active horizon [𝑡_𝑗−1,𝑡_𝑗). To prevent Zeno behavior from occurrence, 𝑡_𝑗−1<𝑡_𝑗 is assumed. Physically, the mode transition conditions are always the boundary of closure of the mode invariant 𝑝_𝑖𝑗<0.

Remark 2.4. 𝑥(𝑡_𝑗−1)=𝜓_𝑖𝑗(𝑥(𝑡^−𝑗−1)) is the outcome of the mode transition and describes the effect that the transition will have on the continuous states. It can be viewed as junction conditions in Vassiliadis et al. [1, 2]. It is assumed that 𝜓_𝑖𝑗∈𝐶^𝑙(ℝ^𝑛), 𝑙≥1, 𝑙∈ℕ.

Remark 2.5. Basically, for general hybrid systems, the evaluation of 𝑖 should be formulated by a function of impulsive control or a graph, which generates mode transition sequence, as formulated in Song and Li [24] and Cassandras and Lygeros [8]. However, the order of the mode transition 𝜋 is known a prior here thus, the evaluation of 𝑖 is determined only by the transition instants 𝑡_𝑗, and the evaluation function of 𝑖 is omitted here.
Besides Assumption 2.1, to make the considered systems to be well defined, there are some additional assumptions on mode invariants and mode transition conditions should be imposed. Here, it is supposed that the mode invariants and mode transition conditions meet the requirements as in Taringoo and Caines [20].

2.2. Optimal Control Problem for Hybrid Systems

Let 𝐿_𝑖∈𝐶^𝑙(ℝ^𝑛×𝐔_𝑖;ℝ) be a running cost function, 𝜑_𝑖𝑗∈𝐶^𝑙(ℝ^𝑛;ℝ₊) be a discrete state transition cost function, and 𝜙∈𝐶^𝑙(ℝ^𝑛;ℝ₊) be a terminal cost function, 𝑖,𝑗∈𝐼, 𝑙≥1, 𝑙∈ℕ, respectively. The optimal control problem for the hybrid systems (2.1) is stated as follows.

Optimal Problem A
Consider a hybrid system formulated by (2.1), given a fixed time interval [𝑡₀,𝑡_𝑓] and a prespecified mode transition order 𝜋={𝑖₁,𝑖₂,…,𝑖_𝐾}, find a continuous control 𝑢∈𝐔_𝑖𝑗 in each mode 𝑖_𝑗∈𝐼 and mode transition instants 𝜃={𝑡₁,…,𝑡_𝐾−1}, such that the corresponding continuous state trajectory 𝑥 departs from a given initial state 𝑥(𝑡₀)=𝑥₀ and meets an (𝑛−𝑙_𝑓)-dimensional smooth manifold 𝑆_𝑓={𝑥∣𝜗(𝑥)=0,𝜗∶ℝ^𝑛→ℝ^𝑙𝑓}, 𝑙_𝑓∈ℕ, at 𝑡_𝑓 and the cost functional 𝐽(𝜃,𝑢)=𝜙𝑥𝑡_𝑓+^𝑡𝑓_𝑡0𝐿_𝑖(𝑡)(𝑥(𝑡),𝑢(𝑡))𝑑𝑡+^𝐾−1_𝑗=1𝜑_{𝑖𝑗𝑖𝑗+1}𝑥𝑡^−𝑗(2.3) is minimized.

Remark 2.6. As it is well known, when 𝑡₀ and 𝑡_𝑓 are unknown points in some fixed interval 𝑇⊂ℝ₊, this problem can be transformed to one with fixed time essentially by introducing an additional state variable.

There are fruitful strategies about how to compute OCPHS (see [15] and the references therein), and the basic idea is briefly reviewed as follows for completeness.

Obtaining the optimal control for hybrid systems is very difficult due to the interactions between the continuous states and discrete states which produce a mode transition sequence that increases the feasibility range of the decision variables. One algorithm framework for dealing with this complexity is the decomposition method as follows: min_{((𝜋,𝜃),𝑢)}𝐽((𝜋,𝜃),𝑢)=min_(𝜋,𝜃)min_𝑢𝐽(𝑢∣(𝜋,𝜃))=min_𝜋min_𝜃min_𝑢𝐽((𝑢,𝜃)∣𝜋),(2.4) where 𝐽(⋅∣𝑏) means that 𝑏 is given.

According to this framework, the master problem is how to get the optimum of the inner functional, that is, minimize 𝐽(𝑢,𝜃) given 𝜋. The key point of finding the optimal solution of 𝐽(𝑢,𝜃) is how to get the sensitivity of the objective with respect to control variables, which provides a better direction for searching and hence reduces computational burden and help associated algorithms converge quickly and accelerate the primary problem convergence eventually.

In next section, the derivatives of cost functional with respect to control variables are established analytically based on optimality condition, which can facilitate the design of associated gradient-based algorithms.

3. Equivalent Problem and Its Optimal Conditions

When control vector parametrization methods are implemented to obtain numerical solution to the OCPHS, updating the parameters of control profiles should be at the same time point when iterative procedure is running. However, the fact is that the mode active horizon [𝑡_𝑗−1,𝑡_𝑗) for mode 𝑖_𝑗∈𝐼 is varying during the procedure running, so a fixed horizon should be introduced, which will guarantee the updating of parameters of control profiles is at the same time point. For this purpose, let 𝜏∈[0,𝐾] be a time independent variable, and 𝑡∈[𝑡_𝑗−1,𝑡_𝑗) can be formulated by 𝑡=𝑡_𝑗−1+(𝜏−(𝑗−1))𝑡_𝑗−𝑡_𝑗−1,𝜏∈[𝑗−1,𝑗),𝑗=1,…,𝐾.(3.1)

In addition, to deal with mode invariants constraints 𝑝_𝑖𝑗(𝑥)<0, slack algebraic variable 𝑠_𝑖𝑗=[𝑠_𝑖𝑗1,…,𝑠_{𝑖𝑗ℎ𝑖𝑗}]^𝑇∈ℝ^ℎ𝑖𝑗₊ is introduced for each mode 𝑖_𝑗∈𝐼, such that 𝑝_𝑖𝑗(𝑥)+diag[𝑠_𝑖𝑗1,…,𝑠_{𝑖𝑗ℎ𝑖𝑗}]𝑠_𝑖𝑗=0. For 𝜏∈[𝑗−1,𝑗), denote 𝐱_𝑗(𝜏)≐𝑥(𝑡_𝑗−1+(𝜏−(𝑗−1))(𝑡_𝑗−𝑡_𝑗−1)), 𝐮_𝑗(𝜏)≐𝑢(𝑡_𝑗−1+(𝜏−(𝑗−1))(𝑡_𝑗−𝑡_𝑗−1)), 𝐬_𝑗(𝜏)≐𝑠_𝑖𝑗(𝑡_𝑗−1+(𝜏−(𝑗−1))(𝑡_𝑗−𝑡_𝑗−1)), and let 𝐱=[𝐱₁,…,𝐱_𝐾]^𝑇, 𝐮=[𝐮₁,…,𝐮_𝐾]^𝑇, and 𝐬=[𝐬₁,…,𝐬_𝐾]^𝑇.

According to the above definition, the Optimal Problem A can be transcribed into an equivalent Optimal Problem B as follows:

Optimal Problem B
Given a fixed interval [0,𝐾], find continuous inputs 𝐮∈𝐔_𝑖1×⋯×𝐔_𝑖𝐾, 𝐬∈ℝ^ℎ𝑖1₊×⋯×ℝ^ℎ𝑖𝐾₊ and 𝜃, such that the corresponding continuous state trajectory 𝐱₁ departs from a given initial state 𝐱₁(0)=𝑥₀ and 𝐱_𝐾 meets an (𝑛−𝑙_𝑓)-dimensional smooth manifold 𝑆_𝑓={𝐱_𝐾∣𝜗(𝐱_𝐾)=0,𝜗∶ℝ^𝑛→ℝ^𝑙𝑓} at 𝐾, and the cost functional 𝐽(𝜃,𝐮,𝐬)=𝜙𝐱_𝐾(𝐾)+^𝐾_𝑗=1^𝑗𝑗−1𝐿_𝑖𝑗𝐱_𝑗(𝜏),𝐮_𝑗(𝜏),𝐬_𝑗(𝜏)𝑑𝜏+^𝐾−1_𝑗=1𝜑_{𝑖𝑗𝑖𝑗+1}𝐱_𝑗(𝑗⁻)(3.2) is minimized, subject to 𝑑𝐱_𝑗(𝜏)𝑑𝜏=𝑓^𝑖𝑗𝐱_𝑗𝐱(𝜏),𝐮_𝑗(𝜏)≐𝑡_𝑗−𝑡_𝑗−1𝑓_𝑖𝑗𝐱_𝑗(𝜏),𝐮_𝑗(𝜏)_𝑗,𝑔_𝑖𝑗(𝑗−1)=𝜓_𝑖𝑗𝐱_𝑗−1((𝑗−1)⁻),𝐱_𝑗(𝑗⁻)=0,(3.3) where 𝐿_𝑖𝑗𝐱_𝑗,𝐮_𝑗,𝐬_𝑗=𝑡_𝑗−𝑡_𝑗−1𝐿_𝑖𝑗,𝐿_𝑖𝑗=𝐿_𝑖𝑗𝐱_𝑗,𝐮_𝑗+𝑀^ℎ𝑖𝑗_𝑙=1𝑝^𝑖_𝑗^𝑙𝐱_𝑗+𝑠^2𝑖_𝑗𝑙²,(3.4) and 𝑀 is a large positive constant.

According to Theorems 2 and 3 in Dmitruk and Kaganovich [12], when 𝑀 is big enough Optimal Problem B is equivalent to Optimal Problem A.

Remark 3.1. The penalty function term, say, 𝑀∑^ℎ𝑖𝑗_𝑙=1(𝑝_𝑖𝑗𝑙(𝐱_𝑗)+𝐬^2𝑖_𝑗𝑙)², cannot always guarantee the state satisfies the mode invariant conditions. However, the method works well in practice; moreover, the mode transition order is fixed in this paper which reduces the negative effect of the penalty function method for OCPHS.

For 𝜏∈[𝑗−1,𝑗), 𝑗=1,…,𝐾, let 𝜆_𝑗∈ℝ^𝑛, and define Hamiltonian function 𝐻_𝑗 by 𝐻_𝑗𝜆_𝑗,𝐱_𝑗,𝐮_𝑗,𝐬_𝑗=𝐿_𝑖𝑗𝐱_𝑗,𝐮_𝑗,𝐬_𝑗+𝜆^𝑇𝑗𝑓_𝑖𝑗𝐱_𝑗,𝐮_𝑗,(3.5) and according to Sussmann [10], Shaikh and Caines [11], and Dmitruk and Kaganovich [12], the following Theorem 3.2 holds.

Theorem 3.2. In order that 𝐮 and 𝐬 are optimal for Optimal Problem B, it is necessary that there exist vector functions 𝜆_𝑗, 𝑗=1,…,𝐾, such that the following conditions hold:(a)for almost any 𝜏∈[𝑗−1,𝑗), the following state equations hold: 𝑑𝐱_𝑗(𝜏)𝑑𝜏=𝑓^𝑖𝑗𝐱_𝑗(𝜏),𝐮_𝑗(𝜏),(3.6)(b)for almost any 𝜏∈[𝑗−1,𝑗), the following costate equations hold: 𝜆̇_𝑗=−𝜕𝐿^𝑖𝑗𝜕𝐱_𝑗^𝑇−⎛⎜⎜⎝𝜕𝑓^𝑖𝑗𝜕𝐱_𝑗⎞⎟⎟⎠^𝑇𝜆_𝑗,(3.7)(c)for a.e. 𝜏∈[𝑗−1,𝑗), 𝐻_𝑗𝜆^∗𝑗,𝐱^∗𝑗,𝐮^∗𝑗,𝐬^∗𝑗=0,(3.8)(d)minimality condition: for all 𝜏∈[𝑗−1,𝑗), min_{𝐮𝑗∈𝐔𝑖𝑗,𝐬𝑗∈ℝℎ𝑖𝑗+}𝐻_𝑗𝜆^∗𝑗,𝐱^∗𝑗,𝐮_𝑗,𝐬^∗𝑗=0,(3.9)(e)transversality conditions for 𝜆_𝑗, 𝜆𝜆_𝑗+1(𝑗)=𝛽_𝑗,𝑗=1,…,𝐾−1,_𝑗(𝑗⁻)=𝜕𝑔^𝑖𝑗𝜕𝐱_𝑗(𝑗⁻)^𝑇𝛼_𝑗−𝜕𝜓^𝑖𝑗+1𝜕𝐱_𝑗(𝑗⁻)^𝑇𝛽_𝑗+𝜕𝜑^{𝑖𝑗𝑖𝑗+1}𝜕𝐱_𝑗(𝑗⁻)^𝑇𝜆,𝑗=1,…,𝐾−1,_𝐾(𝐾)=𝜕𝜙𝜕𝐱_𝐾(𝐾)^𝑇+𝜕𝜗𝜕𝐱_𝐾(𝐾)^𝑇𝛼_𝐾,(3.10)where 𝛼_𝑗∈ℝ^ℎ𝑖, 𝛽_𝑗∈ℝ^𝑛 are Lagrangian multipliers. Based on Theorem 3.2, the sensitivity analysis is established in the next section for Optimal Problem B.

4. Sensitivity Analysis and Parametrization Method

For finding numerical solution to the OCPHS effectively, based on Theorem 3.2, the derivatives of the objective functional 𝐽(⋅) with respect to the control 𝐮, 𝐬, and the mode transition instant 𝑡_𝑗,𝑗=1,…,𝐾−1 are established in this section, and by using the obtained derivatives associated parametrization method is proposed.

4.1. Sensitivity Analysis

Lemma 4.1. The derivatives of 𝐱_𝑗(𝑗⁻),𝑗=1,…,𝐾, w.r.t 𝑡_𝑘 and 𝐮_𝑘 are given, respectively, as follows for 𝑘=1,…,𝐾−1, 𝑑𝐱_𝑗(𝑗⁻)𝑑𝑡_𝑘𝑑𝐱_𝑘=0,𝑗=1,…,𝑘−1,(𝑘⁻)𝑑𝑡_𝑘𝑑𝐱_𝑘+1=𝑓_𝑖𝑘𝐱_𝑘(𝑘⁻),𝐮_𝑘(𝑘⁻),((𝑘+1)⁻)𝑑𝑡_𝑘𝑑𝐱_𝑗=Ω_𝑘+1,(𝑗⁻)𝑑𝑡_𝑘=^𝑗_𝑙_=𝑘+2Φ_𝑙(𝑙,𝑙−1)𝑑𝜓^𝑖𝑙𝑑𝐱_𝑙−1((𝑙−1)⁻)Ω^𝑘+1𝛿𝐱_𝑗,𝑗=𝑘+2,…,𝐾,(4.1)(𝑗⁻)𝛿𝐮_𝑘𝛿𝐱=_𝑘0,𝑗=1,…,𝑘−1,(𝑘⁻)𝛿𝐮_𝑘𝛿𝐱_𝑗=Γ_𝑘(𝜏),(𝑗⁻)𝛿𝐮_𝑘=^𝑗_𝑙_=𝑘+1Φ^𝑙(𝑙,𝑙−1)𝑑𝜓^𝑖𝑙𝑑𝐱_𝑙−1((𝑙−1)⁻)Γ^𝑘(𝜏),𝑗=𝑘+1,…,𝐾,(4.2) where Ω_𝑘+1=Φ_𝑘+1(𝑘+1,𝑘)𝑑𝜓^𝑖𝑘+1𝑑𝐱_𝑘(𝑘⁻)𝑓^𝑖𝑘𝐱_𝑘Γ(𝑘⁻),𝐮_𝑘(𝑘⁻)−𝑓_𝑖𝑘+1𝐱_𝑘+1(𝑘),𝐮_𝑘+1(𝑘)_𝑘,(𝜏)=𝑡_𝑘−𝑡_𝑘−1Φ_𝑘(𝑘,𝜏)𝜕𝑓^𝑖𝑘𝜕𝐮_𝑘,Φ_𝑙(𝜏,𝑣)=exp^𝜏𝑣𝑡_𝑙−𝑡_𝑙−1𝜕𝑓^𝑖𝑙𝜕𝐱_𝑙𝑑𝑎.(4.3)

Note that 𝐱(𝑡_𝑗) is a functional vector of 𝐮_𝑘, and the expression 𝛿𝐱_𝑗/𝛿𝐮_𝑘 is used, where the notation 𝛿𝐱_𝑗/𝛿𝐮_𝑘 is the functional derivatives which describe the response of the functional 𝐱_𝑗 to an infinitesimal change in the function 𝐮_𝑘 at each point.

Proof. The proof of (4.1) is only going to be shown for easily reading. The proof for (4.2) can be found in Appendix.
When 𝑗=1,…,𝑘−1, 𝐱_𝑗(𝑗⁻) and 𝐱_𝑗+1(𝑗) are independent of 𝑡_𝑘, and obviously 𝑑𝐱_𝑗(𝑗⁻)/𝑑𝑡_𝑘=0 holds. In the case of 𝑗=𝑘, 𝐱_𝑘(𝑘⁻) is a function of 𝑡_𝑘 which gives rise to 𝑑𝐱_𝑘(𝑘⁻)/𝑑𝑡_𝑘=𝑓_𝑖𝑘(𝐱_𝑘(𝑘⁻),𝐮_𝑘(𝑘⁻)).
Case i. (𝑗=𝑘+1). In this case, 𝐱_𝑘+1 is a function of 𝑡_𝑘 and 𝐱_𝑘+1(𝑘), and we have 𝑑𝐱_𝑘+1(𝜏)𝑑𝑡_𝑘=𝜕𝐱^𝑘+1𝜕𝑡_𝑘+𝜕𝐱^𝑘+1𝜕𝐱_𝑘+1(𝑘)𝜕𝐱^𝑘+1(𝑘)𝜕𝑡_𝑘.(4.4)
Note that in (4.4), 𝜕𝐱_𝑘+1/𝜕𝑡_𝑘 is produced by the perturbation of 𝑡_𝑘, and (𝜕𝐱_𝑘+1/𝜕𝐱_𝑘+1(𝑘))(𝜕𝐱_𝑘+1(𝑘)/𝜕𝑡_𝑘) is produced by the perturbation of 𝐱_𝑘+1(𝑘) with respect to 𝑡_𝑘. Obviously, for 𝜏∈[𝑘,𝑘+1), 𝜕𝐱_𝑘+1(𝜏)𝜕𝑡_𝑘=−𝑓_𝑖𝑘+1𝐱_𝑘+1(𝑘),𝐮_𝑘+1(𝑘).(4.5)
The solution to 𝜕𝐱_𝑘+1(𝜏)/𝜕𝐱_𝑘+1(𝑘) is given by 𝜕𝐱_𝑘+1(𝜏)𝜕𝐱_𝑘+1𝑘⁺=𝐼+𝑡^𝑘+1−𝑡_𝑘^𝜏𝑘𝜕𝑓_𝑖𝑘+1𝜕𝐱_𝑘+1𝜕𝐱_𝑘+1(𝑣)𝜕𝐱_𝑘+1(𝑘)𝑑𝑣.(4.6)
Equation (4.6) is a linear system about 𝜕𝐱_𝑘+1/𝜕𝐱_𝑘+1(𝑘). Define the state transition matrix Φ_𝑙(𝜏,𝑣) by Φ_𝑙(𝜏,𝑣)=exp^𝜏𝑣𝑡_𝑙−𝑡_𝑙−1𝜕𝑓^𝑖𝑙(𝑎)𝜕𝐱_𝑙(𝑎)𝑑𝑎,(4.7) according to (4.6), and we have 𝜕𝐱_𝑘+1(𝜏)𝜕𝐱_𝑘+1(𝑘)=Φ^𝑘+1(𝜏,𝑘).(4.8)
Thus, 𝑑𝐱_𝑘+1(𝜏)𝑑𝑡_𝑘=Φ_𝑘+1(𝜏,𝑘)𝜕𝐱^𝑘+1(𝑘)𝜕𝑡_𝑘−𝑓_𝑖𝑘+1𝐱_𝑘+1(𝑘),𝐮_𝑘+1(𝑘).(4.9)
At transition instants 𝑡_𝑗, since 𝐱_𝑗+1(𝑗)=𝜓_𝑖𝑗+1(𝐱_𝑗(𝑗⁻)), so 𝑑𝐱_𝑗+1(𝑗)𝑑𝑡_𝑘=𝑑𝜓^𝑖𝑗+1𝑑𝐱_𝑗(𝑗⁻)𝑑𝐱^𝑗(𝑗⁻)𝑑𝑡_𝑘,(4.10) which implies 𝜕𝐱_𝑘+1(𝑘)𝜕𝑡_𝑘=𝑑𝜓^𝑖𝑘+1𝑑𝐱_𝑘(𝑘⁻)𝑑𝐱^𝑘(𝑘⁻)𝑑𝑡_𝑘=𝑑𝜓^𝑖𝑘+1𝑑𝐱_𝑘(𝑘⁻)𝑓^𝑖𝑘𝐱_𝑘(𝑘⁻),𝐮_𝑘(𝑘⁻).(4.11)
According to (4.9), and we have 𝑑𝐱_𝑘+1((𝑘+1)⁻)𝑑𝑡_𝑘=Φ_𝑘+1(𝑘+1,𝑘)𝑑𝜓^𝑖𝑘+1𝑑𝐱_𝑘(𝑘⁻)𝑓^𝑖𝑘𝐱_𝑘−𝑓_𝑖𝑘+1(𝑘⁻),𝐮_𝑘(𝑘⁻)𝐱_𝑘+1(𝑘),𝐮_𝑘+1(𝑘)≐Ω_𝑘+1.(4.12)
Case ii. (𝑗=𝑘+2,…,𝐾). When 𝑗=𝑘+2,…,𝐾, the following holds: 𝑑𝐱_𝑗(𝜏)𝑑𝑡_𝑘=𝑑𝐱^𝑗(𝑗−1)𝑑𝑡_𝑘+𝑡_𝑗−𝑡_𝑗−1^𝜏𝑗−1𝜕𝑓_𝑖𝑗𝜕𝐱_𝑗𝑑𝐱_𝑗(𝑣)𝑑𝑡_𝑘𝑑𝑣,𝜏∈[𝑗−1,𝑗).(4.13)
Then, 𝑑𝐱_𝑗(𝜏)𝑑𝑡_𝑘=Φ_𝑗(𝜏,𝑗−1)𝑑𝐱^𝑗(𝑗−1)𝑑𝑡_𝑘.(4.14)
Substituting the term 𝑑𝐱_𝑗(𝑗−1)/𝑑𝑡_𝑘 in (4.14) by (4.10), we obtain 𝑑𝐱_𝑗(𝑗⁻)𝑑𝑡_𝑘=^𝑗_𝑙_=𝑘+2Φ_𝑙(𝑙,𝑙−1)𝑑𝜓^𝑖𝑙𝑑𝐱_𝑙−1((𝑙−1)⁻)Ω^𝑘+1.(4.15)

Theorem 4.2. The derivatives of the objective functional 𝐽(⋅) w.r.t 𝑡_𝑘, 𝐮_𝑘 and 𝐬_𝑘 are given, respectively, as follows: 𝑑𝐽𝑑𝑡_𝑘=𝐿+_𝑖𝑘𝜆_𝑘𝐱_𝑘(𝑘⁻),𝐮_𝑘(𝑘⁻),𝐬_𝑘(𝑘⁻)−𝐿_𝑖𝑘+1𝐱_𝑘+1(𝑘),𝐮_𝑘+1(𝑘),𝐬_𝑘+1(𝑘)−(𝑘⁻)^𝑇𝑓_𝑖𝑘𝐱_𝑘(𝑘⁻),𝐮_𝑘(𝑘⁻)−𝜆_𝑘+1(𝑘)^𝑇𝑓_𝑖𝑘+1𝐱_𝑘+1(𝑘),𝐮_𝑘+1(𝑘)_𝐾−1_𝑗=𝑘𝛼^𝑇𝑗𝜕𝑔_𝑖𝑗𝜕𝐱_𝑗(𝑗⁻)𝑑𝐱^𝑗(𝑗⁻)𝑑𝑡_𝑘−𝛼^𝑇𝐾𝜕𝜗𝜕𝐱_𝐾(𝐾)𝑑𝐱^𝐾(𝐾)𝑑𝑡_𝑘𝛿𝐽𝛿𝐮_𝑘=𝜕𝐻^𝑘𝜕𝐮_𝑘−^𝐾−1_𝑗=k𝛼^𝑇𝑗𝜕𝑔_𝑖𝑗𝜕𝐱_𝑗(𝑗⁻)𝛿𝐱^𝑗(𝑗⁻)𝛿𝐮_𝑘−𝛼^𝑇𝐾𝜕𝜗𝜕𝐱_𝐾(𝐾)𝛿𝐱^𝐾(𝐾)𝛿𝐮_𝑘𝛿𝐽𝛿𝐬_𝑘=𝜕𝐻^𝑘𝜕𝐬_𝑘.(4.16)

Before proving Theorem 4.2, Lemma 4.3 is firstly given as follows.

Lemma 4.3. For 𝑗=𝑘+2,…,𝐾, 𝑑𝑑𝑡_𝑘^𝑗𝑗−1𝐿_𝑖𝑗𝐱_𝑗,𝐮_𝑗,𝐬_𝑗𝑑𝜏=𝜆_𝑗(𝑗−1)^𝑇𝑑𝐱_𝑗(𝑗−1)𝑑𝑡_𝑘−𝜆_𝑗(𝑗⁻)^𝑇𝑑𝐱_𝑗(𝑗⁻)𝑑𝑡_𝑘.(4.17)

Proof. For any 𝑗=𝑘+2,…,𝐾, we have 𝑑𝑑𝑡_𝑘^𝑗𝑗−1𝐿_𝑖𝑗𝐱_𝑗,𝐮_𝑗,𝐬_𝑗𝑑𝜏=^𝑗𝑗−1𝑑𝑑𝑡_𝑘𝐻^𝑗𝜆_𝑗=,𝐱_𝑗,𝐮_𝑗,𝐬_𝑗−𝜆^𝑇𝑗𝑓_𝑖𝑗𝑑𝜏_𝑗𝑗−1𝜕𝜕𝐻𝐱^𝑗_𝑗𝑑𝐱_𝑗𝑑𝑡_𝑘+𝜕𝐻^𝑗𝜕𝜆_𝑗𝑑𝜆_𝑗𝑑𝑡_𝑘−𝑑𝜆^𝑗𝑑𝑡_𝑘^𝑇𝑓_𝑖𝑗−𝜆^𝑇𝑗𝑑𝑑𝑡_𝑘𝑓_𝑖𝑗𝑑𝜏.(4.18)
Since the following holds by Theorem 3.2, 𝜕𝜕𝐻𝐱^𝑗_𝑗^𝑇=−𝜆̇_𝑗,𝜕𝐻^𝑗𝜕𝜆_𝑗^𝑇=𝑓_𝑖𝑗,(4.19) then 𝑑𝑑𝑡_𝑘^𝑗𝑗−1𝐿_𝑖𝑗𝐱_𝑗,𝐮_𝑗,𝐬_𝑗𝑑𝜏=^𝑗𝑗−1−𝜆̇^j^𝑇𝑑𝐱_𝑗𝑑𝑡_𝑘+𝑓_𝑖𝑗^𝑇𝑑𝜆_𝑗𝑑𝑡_𝑘−𝑑𝜆^𝑗𝑑𝑡_𝑘^𝑇𝑓_𝑖𝑗−𝜆^𝑇𝑗𝑑𝑑𝑡_𝑘𝑓_𝑖𝑗=𝑑𝜏_𝑗𝑗−1−𝜆̇^𝑗^𝑇𝑑𝐱_𝑗𝑑𝑡_𝑘−𝜆^𝑇𝑗𝑑𝑑𝑡_𝑘𝑓_𝑖𝑗𝑑𝜏=−^𝑗𝑗−1𝑑𝑑𝜏𝜆^𝑇𝑗𝑑𝐱_𝑗=𝜆_𝑗𝑑𝑡_𝑘𝑑𝜏(𝑗−1)^𝑇𝑑𝐱_𝑗(𝑗−1)𝑑𝑡_𝑘−𝜆_𝑗(𝑗⁻)^𝑇𝑑𝐱_𝑗(𝑗⁻)𝑑𝑡_𝑘.(4.20)

Obviously, when 𝑗=𝑘,𝑘+1, we have 𝑑𝑑𝑡_𝑘^𝑘𝑘−1𝐿_𝑖𝑘𝐱_𝑘,𝐮_𝑘,𝐬_𝑘𝑑𝜏=𝑑𝑑𝑡_𝑘^𝑡𝑘_𝑡𝑘−1𝑑𝐿_𝑖𝑘𝑥,𝑢,𝑠_𝑖𝑘𝑑𝑡=𝐿_𝑖𝑘𝐱_𝑘(𝑘⁻),𝐮_𝑘(𝑘⁻),𝐬_𝑘(𝑘⁻),(4.21)𝑑𝑡_𝑘^𝑘𝑘+1𝐿_𝑖𝑘+1𝐱_𝑘+1,𝐮_𝑘+1,𝐬_𝑘+1𝑑𝜏=𝜆_𝑘+1(𝑘)^𝑇𝑑𝐱_𝑘+1(𝑘)𝑑𝑡_𝑘−𝜆_𝑘+1((𝑘+1)⁻)^𝑇𝑑𝐱_𝑘+1((𝑘+1)⁻)𝑑𝑡_𝑘−𝐿_𝑖𝑘+1𝐱_𝑘+1(𝑘),𝐮_𝑘+1(𝑘),𝐬_𝑘+1(𝑘).(4.22)

Now we prove Theorem 4.2. We are only going to show 𝑑𝐽/𝑑𝑡_𝑘 for easily reading. The proofs for 𝛿𝐽/𝛿𝐮_𝑘 and 𝛿𝐽/𝛿𝐬_𝑘 can be found in Appendix.

Proof. 𝐽(𝜃,𝐮,𝐬) can be formulated as 𝐽(𝜃,𝐮,𝐬)=𝜙𝐱_𝐾(𝐾)+^𝑘−1_𝑗=1^𝑗𝑗−1+𝐿_𝑖𝑗𝐱_𝑗,𝐮_𝑗,𝐬_𝑗𝑑𝜏_𝐾_𝑗=𝑘^𝑗𝑗−1𝐿_𝑖𝑗𝐱_𝑗,𝐮_𝑗,𝐬_𝑗𝑑𝜏+^𝐾−1_𝑗=1𝜑_{𝑖𝑗𝑖𝑗+1}𝐱_𝑗(𝑗⁻).(4.23)
Since 𝐿_𝑖𝑗(⋅) is independent of 𝑡_𝑘 for 𝑗=1,…,𝑘−1, then 𝑑𝐽/𝑑𝑡_𝑘 can be obtained by 𝑑𝐽𝑑𝑡_𝑘(𝜃,𝐮,𝐬)=𝜕𝜙𝐱^𝐾(𝐾)𝜕𝐱_𝐾(𝐾)𝑑𝐱^𝐾(𝐾)𝑑𝑡_𝑘+𝑑𝑑𝑡_{𝑘𝐾𝑗=𝑘}^𝑗𝑗−1𝐿_𝑖𝑗𝐱_𝑗,𝐮_𝑗,𝐬_𝑗𝑑𝜏+^𝐾−1_𝑗=1𝜕𝜑_{𝑖𝑗𝑖𝑗+1}𝜕𝐱_𝑗(𝑗⁻)𝑑𝐱^𝑗(𝑗⁻)𝑑𝑡_𝑘.(4.24)
Substituting (4.17), (4.21), and (4.22) into (4.24), we have 𝑑𝐽𝑑𝑡_𝑘(𝜃,𝐮,𝐬)=𝜕𝜙𝐱^𝐾(𝐾)𝜕𝐱_𝐾(𝐾)𝑑𝐱^𝐾(𝐾)𝑑𝑡_𝑘−+𝐿_𝑖𝑘𝐱_𝑘(𝑘⁻),𝐮_𝑘(𝑘⁻),𝐬_𝑘(𝑘⁻)𝐿_𝑖𝑘+1𝐱_𝑘+1(𝑘),𝐮_𝑘+1(𝑘),𝐬_𝑘+1(𝑘)+𝜆_𝑘+1(𝑘)^𝑇𝑑𝐱_𝑘+1(𝑘)𝑑𝑡_𝑘+𝜕𝜑^{𝑖𝑘𝑖𝑘+1}𝜕𝐱_𝑘(𝑘⁻)𝑑𝐱^𝑘(𝑘⁻)𝑑𝑡_𝑘−^𝐾−1_𝑗=_𝑘+1𝜆^𝑗(𝑗⁻)^𝑇𝑑𝐱_𝑗(𝑗⁻)𝑑𝑡_𝑘−𝜆_𝑗+1(𝑗)^𝑇𝑑𝐱_𝑗+1(𝑗)𝑑𝑡_𝑘−𝜕𝜑^{𝑖𝑗𝑖𝑗+1}𝜕𝐱_𝑗(𝑗⁻)𝑑𝐱^𝑗(𝑗⁻)−𝜆_𝐾𝑑𝑡_𝑘(𝐾)^𝑇𝑑𝐱_𝐾(𝐾)𝑑𝑡_𝑘.(4.25)
Due to Theorem 3.2 and (4.10), 𝑑𝐽/𝑑𝑡_𝑘 can be formulated by 𝑑𝐽𝑑𝑡_𝑘(𝜃,𝐮,𝐬)=𝜕𝜙𝐱^𝐾(𝐾)𝜕𝐱_𝐾(𝐾)−𝜆^𝐾(𝐾)^𝑇𝑑𝐱^𝐾(𝐾)𝑑𝑡_𝑘−+𝐿_𝑖𝑘𝐱_𝑘(𝑘⁻),𝐮_𝑘(𝑘⁻),𝐬_𝑘(𝑘⁻)𝐿_𝑖𝑘+1𝐱_𝑘+1(𝑘),𝐮_𝑘+1(𝑘),𝐬_𝑘+1(𝑘)+𝜆_𝑘(𝑘⁻)^𝑇𝑑𝐱_𝑘(𝑘⁻)𝑑𝑡_𝑘−−𝜆𝛼_{𝑘+1𝑇𝑘}(𝑘)^𝑇𝑓_𝑖𝑘+1𝐱_𝑘+1(𝑘),𝐮_𝑘+1(𝑘)𝜕𝑝_𝑖𝑘𝜕𝐱_𝑘(𝑘⁻)𝑑𝐱^𝑘(𝑘⁻)𝑑𝑡_𝑘−^𝐾−1_𝑗=𝑘𝛼^𝑇𝑗𝜕𝑔_𝑖𝑗𝜕𝐱_𝑗(𝑗⁻)𝑑𝐱^𝑗(𝑗⁻)𝑑𝑡_𝑘=𝐿+_𝑖𝑘𝜆_𝑘𝐱_𝑘(𝑘⁻),𝐮_𝑘(𝑘⁻),𝐬_𝑘(𝑘⁻)−𝐿_𝑖𝑘+1𝐱_𝑘+1(𝑘),𝐮_𝑘+1(𝑘),𝐬_𝑘+1(𝑘)−(𝑘⁻)^𝑇𝑓_𝑖𝑘𝐱_𝑘(𝑘⁻),𝐮_𝑘(𝑘⁻)−𝜆_𝑘+1(𝑘)^𝑇𝑓_𝑖𝑘+1𝐱_𝑘+1(𝑘),𝐮_𝑘+1(𝑘)_𝐾−1_𝑗=𝑘𝛼^𝑇𝑗𝜕𝑔_𝑖𝑗𝜕𝐱_𝑗(𝑗⁻)𝑑𝐱^𝑗(𝑗⁻)𝑑𝑡_𝑘−𝛼^𝑇𝐾𝜕𝜗𝜕𝐱_𝐾(𝐾)𝑑𝐱^𝐾(𝐾)𝑑𝑡_𝑘.(4.26)

Note that when second-order derivatives are needed, there is no difficulty to obtain the second-order derivatives following the above procedure.

4.2. Parametrization Method

To obtain the numerical solution to optimal control for hybrid systems, continuous control profiles are parameterized on each mode active horizon in this section. Then the numerical solution to optimal controls can be computed based on the obtained sensitivity analysis results. The basic idea behind the proposed method using finite parameterizations of the controls is to transcribe the original infinite dimensional problem, that is, 𝐶-problem, into a finite dimensional nonlinear programming problem, that is, 𝑃-problem [25]. Here, the parametrization method that the control profiles are approximated by a family of Lagrange form polynomials is implemented.

Partition each horizon [𝑗−1,𝑗) into 𝑁_𝑗 elements as 𝑗−1=𝜏_𝑗0<𝜏_𝑗1<⋯<𝜏_𝑗𝑁𝑗=𝑗 where 𝜏_𝑗𝑙 are referred to as collocation points, 𝑙=0,…,𝑁_𝑗. Let 𝐮_𝑗𝑙 denote the value of 𝐮_𝑗 at 𝜏_𝑗𝑙, 𝑙=0,…,𝑁_𝑗. Thus, the control variable 𝐮_𝑗 is represented approximately by a Lagrange interpolation profile for 𝑗=1,…,𝐾, 𝐮_𝑗(𝜏)=^𝑁𝑗_𝑙=0𝑙_𝑙̂(𝜏)𝑢_𝑗𝑙,𝜏∈[𝑗−1,𝑗),(4.27) where 𝑙_𝑙̂(𝜏)=∏^𝑁𝑗_{𝑚=0,𝑚≠𝑙}((𝜏−𝜏_𝑗𝑚)/(𝜏_𝑗𝑙−𝜏_𝑗𝑚)). 𝐬_𝑗 is also parameterized by 𝐬_𝑗(𝜏)=^𝑁𝑗_𝑙=0𝑙_𝑙̂(𝜏)𝐬_𝑗𝑙,𝜏∈[𝑗−1,𝑗),(4.28) where 𝐬_𝑗𝑙 is the value of 𝐬_𝑗 at the collocation points 𝜏_𝑗𝑙, 𝑙=0,…,𝑁_𝑗.

As a result, based on the obtained derivatives, the numerical solution of 𝑢 and 𝜃 to optimal control for the hybrid systems can be solved simultaneously and efficiently by adopting gradient-based algorithms as described in Xu and Antsaklis [3] and Egerstedt et al. [6]. Note that the derivatives are functions of costate 𝜆_𝑗 as formulated in Theorem 4.2. When control polynomial profiles are implemented, a multipoint boundary value problem about state and costate expressed by (3.6), (3.7), and (3.10) will be solved, which produces the derivatives.

Although the Lagrange interpolation profiles may cause the state or/and control trajectories violate their constraints, this parameterizations method has been proved useful in practice. Moreover, there are some techniques to decrease the defect [1, 2].

Remark 4.4. Control variable 𝐮_𝑗 can be approximated by several piecewise Lagrange interpolation profiles by further partitioning the element [𝑗−1,𝑗). More detail of the parameterizations methods can be found in Vassiliadis et al. [1, 2], Kameswaran and Biegler [26], and the references therein. Only one Lagrange interpolation profile is used here to show the process of the proposed method.

5. Some Examples

To illustrate the effectiveness of the developed method, two examples with different situations are presented in the following. Numerical examples are conducted on an ThinkPad X61 2.10-GHz PC with 2G of RAM. The program is implemented using MatLab 7. The order of Lagrange polynomials in the examples is 3.

Example 5.1. The prototype of this example comes from Vassiliadis et al. [1]. The hybrid system consists of two batch reactors as shown in Figure 1. The first reactor denoted by mode 1 is fitted with a heating coil which can be used to manipulate the reactor temperature 𝑢 over time and is initially loaded with 0.1 m³ of an aqueous solution of component 𝑥₁ of concentration 2000 mol/m³. This reacts to form components 𝑥₂ according to the consecutive reaction scheme 2𝑥₁→𝑥₂. After completion of the first reaction, an amount of dilute aqueous solution of component 𝑥₂ of concentration 600 mol/m³ is added instantaneously to the products of the first reactor, and the mixture is loaded into the second reactor denoted by mode 2 where the reaction 𝑥₂→𝑥₃ takes place under isothermal conditions at a fixed temperature. The decision variables are the temperature 𝑢 of the mode 1, and the durations of the two mode over the horizon [0,180]. The dynamics of the hybrid systems can be described by
Mode 1: 𝑥𝑥̇̇₁₂=−0.0888𝑒^{(−2500/𝑢)}𝑥²¹,𝑥̇₃=0.0444𝑒^{(−2500/𝑢)}𝑥²¹−6889.0𝑒^{(−5000/𝑢)}𝑥₂,=0.(5.1)
Mode 2: 𝑥𝑥̇̇₁₂=0,𝑥̇₃=−0.07𝑥₂−8.0×10⁻⁵𝑥²²,=0.02𝑥₂,(5.2) with 𝑥(0)=[200000]^𝑇. The system transits once at 𝑡=𝑡₁(𝑡₀<𝑡₁<𝑡_𝑓) from mode 1 to 2 with 𝑥₁(𝑡₁)=𝑥₁(𝑡⁻¹)/1.7,𝑥₂(𝑡₁)=(𝑥₂(𝑡⁻¹)+420)/1.7. The OCPHS is to find an optimal mode transition instant 𝑡₁ and an optimal input 298≤𝑢(𝑡)≤398, 𝑡∈[𝑡₀,𝑡₁], to maximize the cost functional max_𝑡1,𝑢𝑥₃𝑡_𝑓,(5.3) with 𝑥₃(𝑡_𝑓)≥150 must be satisfied.

Figure 1 

Two batch reactors system.

By using the proposed method, the optimal mode transition instant is 𝑡₁=105 and the corresponding optimal cost is 𝐽^∗=150.0285. The corresponding continuous control and state trajectories are shown in Figure 2. In Vassiliadis et al. [1], the transition instants and the optimal cost are 𝑡₁=106, 𝐽^∗=150.294, respectively, which are solved by software package DAEOPT.

Figure 2 

State trajectories and control input of Example 5.1.

Example 5.2. Example 5.2 comes from Xu and Antsaklis [3] and is also reconsidered by Hwang et al. [9]. Different from the example in the two references, the control constraint is imposed. The example can be referred to as autonomous switching hybrid systems with mode invariants. Consider the hybrid system consisting of
Mode 1: 𝑥̇=⎛⎜⎜⎝10.501⎞⎟⎟⎠𝑥+⎛⎜⎜⎝11⎞⎟⎟⎠𝑢,(5.4)
Mode 2: 𝑥̇=⎛⎜⎜⎝00.8.5660−.806.56⎞⎟⎟⎠𝑥+⎛⎜⎜⎝11⎞⎟⎟⎠𝑢,(5.5) with 𝑥₀=[11]^𝑇. Assume that 𝑡₀=0, 𝑡_𝑓=2 and the system transits once at 𝑡=𝑡₁(𝑡₀<𝑡₁<𝑡_𝑓) from Mode 1 to 2 when the state trajectories intersect the linear manifold defined by 𝑚(𝑥)=𝑥₁+𝑥₂−7=0. Mode 1 is active with its mode invariant 𝑥₁+𝑥₂−7<0 and Mode 2 is active with its mode invariant 𝑥₁+𝑥₂−7>0. The OCPHS is to find an optimal mode transition instant 𝑡₁ and an optimal input 𝑢(𝑡)∈[−1,1] such that the cost functional 𝐽𝑡₁,𝑢=12𝑥¹𝑡_𝑓−10²+𝑥₂𝑡_𝑓−6²+^𝑡𝑓_𝑡0𝑢²(𝑡)𝑑𝑡(5.6) is minimized.

By using the method developed here, the optimal mode transition instant is 𝑡₁=1.1857 and the corresponding optimal cost is 𝐽^∗=0.1246. The corresponding continuous control and state trajectories are shown in Figure 3. In Xu and Antsaklis [3], the transition instants and the optimal cost are 𝑡₁=1.1624, 𝐽^∗=0.1130, respectively. The bad performance results from that the optimal control is approximated by polynomial.

Figure 3 

State trajectories and control input of Example 5.2.

6. Conclusions

The optimal control problem for hybrid systems (OCPHS) with mode invariants and control constraints is addressed under a priori fixed mode transition order. By introducing new independent variables and auxiliary algebraic variables, the original OCPHS is transformed into an equivalent optimal control problem, and the optimality conditions for the OCPHS is stated. Based on the optimality conditions, the derivatives of the objective functional w.r.t control variables, that is, mode transition instant sequence and admissible continuous control functions, are established analytically. As a result, a control vector parametrization method is implemented to obtain the numerical solution by using gradient-based algorithms with the obtained derivatives. Compared with the existing results on the OCPHS with fixed mode transition order, the settings cover not only the control constraints but also the continuous states constraints, which makes the obtained results more general. Note that when no information about the mode transition sequence is known a priori, the discrete model methods formulated in Bemporad and Morari [27], Barton et al. [15], and Song et al. [28] seem appropriate. In addition, when uncertainties are considered in the systems, the reader is referred to Hu et al. [29] and the references therein.

Appendix

For any 𝜏∈[𝑘−1,𝑘),𝑘=1,…,𝐾, let 𝐮_𝑘(𝜏)∈𝐔_𝑖𝑘 be given and let 𝛿𝐮_𝑘(𝜏)∈𝐔_𝑖𝑘 be arbitrary but fixed. Define a perturbation of 𝐮_𝑘 as 𝐮_𝑘(𝜏;𝜀)=𝐮_𝑘(𝜏)+𝜀𝛿𝐮_𝑘(𝜏),(A.1) where 𝜀∈ℝ is arbitrarily small such that 𝐮_𝑘(𝜏;𝜀)∈𝐔_𝑖𝑘. For the time being, assume that the other controls, 𝐮_𝑗,𝑗=1,…,𝐾,𝑗≠𝑘, be given and fixed. For brevity, let 𝐱_𝑗 and 𝐱_𝑗(⋅;𝜀) denote the state trajectories corresponding to 𝐮_𝑘 and 𝐮_𝑘(𝜏;𝜀), respectively. Similarly, let 𝜆_𝑗 and 𝜆_𝑗(⋅;𝜀) denote the costate trajectories corresponding to 𝐮_𝑘 and 𝐮_𝑘(𝜀), respectively, which are the solutions of the costate equations 𝜆𝐱_𝑗(⋅;𝜀)=𝐱_𝑗(⋅)+𝜀𝛿𝐱_𝑗(⋅),_𝑗(⋅;𝜀)=𝜆_𝑗(⋅)+𝜀𝛿𝜆_𝑗(⋅).(A.2)

Proof of (4.2) in Lemma 4.1. When 𝑗=1,…,𝑘−1, obviously in these cases 𝐱_𝑗 is independent of 𝐮_𝑘, that is, 𝛿𝐱_𝑗(𝑗⁻;𝜀)=0, which leads to 𝛿𝐱_𝑗(𝑗⁻)𝛿𝐮_𝑘=0,𝑗=1,…,𝑘−1.(A.3)
Case i (𝑗=𝑘). Since 𝛿𝐱̇_𝑘=𝑡_𝑘−𝑡_𝑘−1𝜕𝑓^𝑖𝑘𝜕𝐱_𝑘𝛿𝐱_𝑘+𝜕𝑓^𝑖𝑘𝜕𝐮_𝑘𝛿𝐮_𝑘,(A.4) with 𝛿𝐱_𝑘(𝑘−1)=0, thus we have 𝛿𝐱_𝑘(𝑘⁻)=^𝑘𝑘−1Φ_𝑘(𝑘,𝜏)𝑡_𝑘−𝑡_𝑘−1𝜕𝑓^𝑖𝑘𝜕𝐮_𝑘𝛿𝐮_𝑘𝑑𝜏,(A.5) where Φ_𝑘 is the state transition matrix defined in Section 3. Based on the definition of functional derivative, there exists 𝛿𝐱_𝑘(𝑘⁻)𝛿𝐮_𝑘=𝑡_𝑘−𝑡_𝑘−1Φ_𝑘(𝑘,𝜏)𝜕𝑓^𝑖𝑘𝜕𝐮_𝑘≐Γ_𝑘(𝜏).(A.6)
Case (ii) (𝑗=𝑘+1,…,𝐾). In this case, 𝛿𝐱̇_𝑗(𝜏;𝜀)=𝑡_𝑗−𝑡_𝑗−1𝜕𝑓^𝑖𝑗𝜕𝐱_𝑗𝛿𝐱_𝑗,𝜏∈[𝑗−1,𝑗),(A.7) which gives rise to 𝛿𝐱_𝑗(𝑗⁻;𝜀)=Φ_𝑗(𝑗,𝑗−1)𝛿𝐱_𝑗(𝑗−1).(A.8)
At mode transition instant 𝑡_𝑗,𝑗=1,…,𝐾−1, 𝐱_𝑗+1(𝑗)=𝜓_𝑖𝑗+1(𝐱_𝑗(𝑗⁻)) holds, which results in 𝛿𝐱_𝑗+1(𝑗)=𝑑𝜓^𝑖𝑗+1𝑑𝐱_𝑗(𝑗⁻)𝛿𝐱^𝑗(𝑗⁻).(A.9)
Substituting (A.9) into (A.8), we obtain 𝛿𝐱_𝑗(𝑗⁻)=^𝑗_𝑙_=𝑘+1Φ^𝑙(𝑙,𝑙−1)𝑑𝜓^𝑖𝑙𝑑𝐱_𝑙−1((𝑙−1)⁻)𝛿𝐱^𝑘(𝑘⁻).(A.10)
According to the definition of functional derivative, we have 𝛿𝐱_𝑗(𝑗⁻)𝛿𝐮_𝑘=^𝑗_𝑙_=𝑘+1Φ^𝑙(𝑙,𝑙−1)𝑑𝜓^𝑖𝑙𝑑𝐱_𝑙−1((𝑙−1)⁻)Γ^𝑘(𝜏).(A.11)
This completes the proof.

Before proving the 𝛿𝐽/𝛿𝐮_𝑘 in Theorem 4.2, Lemma A.1 is firstly given as follows.

Lemma A.1. For any 𝑗=𝑘+1,…,𝐾, 𝛿^𝑗𝑗−1𝐿_𝑖𝑗𝐱_𝑗,𝐮_𝑗,𝐬_𝑖𝑑𝜏=𝜆_𝑗(𝑗−1)^𝑇𝛿𝐱_𝑗(𝑗−1)−𝜆_𝑗(𝑗⁻)^𝑇𝛿𝐱_𝑗(𝑗⁻).(A.12)

Proof. Note that 𝛿^𝑗𝑗−1𝐿_𝑖𝑗𝐱_𝑗,𝐮_𝑗,𝐬_𝑗𝑑𝜏==𝛿^𝑗𝑗−1𝐻^𝑗𝜆_𝑗,𝐱_𝑗,𝐮_𝑗,𝐬_𝑗−𝜆^𝑇𝑗𝑓_𝑖𝑗𝑑𝜏_𝑗𝑗−1𝜕𝜕𝐻𝐱^𝑗_𝑗𝛿𝐱_𝑗+𝜕𝐻^𝑗𝜕𝜆_𝑗𝛿𝜆_𝑗−𝛿𝜆_𝑗^𝑇𝑓_𝑖𝑗−𝜆^𝑇𝑗𝛿𝑓_𝑖𝑗𝑑𝜏.(A.13)
Since the following holds by Theorem 3.2: 𝜕𝜕𝐻𝐱^𝑗_𝑗^𝑇=−𝜆̇_𝑗,𝜕𝐻^𝑗𝜕𝜆_𝑗^𝑇=𝑓_𝑖𝑗,(A.14) therefore, 𝛿^𝑗𝑗−1𝐿_𝑖𝑗𝐱_𝑗,𝐮_𝑗,𝐬_𝑗𝑑𝜏==−−^𝑗𝑗−1_𝑗𝑗−1𝜆̇^𝑗^𝑇𝛿𝐱_𝑗+𝜆^𝑇𝑗𝛿𝑓_𝑖𝑗𝑑𝜏=−^𝑗𝑗−1𝜆̇^𝑗^𝑇𝛿𝐱_𝑗+𝜆^𝑇𝑗𝛿𝐱̇_𝑖𝑗𝑑𝜏𝑑𝑑𝜏𝜆^𝑇𝑗𝛿𝐱_𝑗𝑑𝜏=𝜆_𝑗(𝑗−1)^𝑇𝛿𝐱_𝑗(𝑗−1)−𝜆_𝑗(𝑗⁻)^𝑇𝛿𝐱_𝑗(𝑗⁻).(A.15)

Obviously, when 𝑗=𝑘, we have 𝛿^𝑘𝑘−1𝐿_𝑖𝑘𝐱_𝑘,𝐮_𝑘,𝐬_𝑘𝑑𝜏=𝜆_𝑘(𝑘−1)^𝑇𝛿𝐱_𝑘(𝑘−1)−𝜆_𝑘(𝑘⁻)^𝑇𝛿𝐱_𝑘(𝑘⁻)+^𝑘𝑘−1𝜕𝐻_𝑘𝜕𝐮_𝑘𝛿𝐮_𝑘𝑑𝜏.(A.16)

Proof of 𝛿𝐽/𝛿𝐮_𝑘 in Theorem 4.2. 𝐽(𝜃,𝐮(𝜀),𝐬) can be rewritten by 𝐽(𝜃,𝐮(𝜀),𝐬)=𝜙𝐱_𝐾(𝐾)+^𝑘−1_𝑗=1^𝑗𝑗−1+𝐿_𝑖𝑗𝐱_𝑗,𝐮_𝑗,𝐬_𝑗𝑑𝜏+^𝑘𝑘−1𝐿_𝑖𝑘𝐱_𝑘(𝜀),𝐮_𝑘(𝜀),𝐬_𝑘𝑑𝜏_𝐾𝑗=_𝑘+1^𝑗𝑗−1𝐿_𝑖𝑗𝐱_𝑗(𝜀),𝐮_𝑗,𝐬_𝑗𝑑𝜏+^𝐾−1_𝑗=1𝜑_{𝑖𝑗𝑖𝑗+1}𝐱_𝑗(𝑗⁻).(A.17) Applying a 𝛿-operation to (A.17) leads to 𝛿𝐽=𝑑𝐽(𝜌,𝐮(𝜀),𝐬)𝑑𝜀||||_𝜀=0=𝜕𝜙𝐱^𝐾(𝐾)𝜕𝐱_𝐾(𝐾)𝛿𝐱^𝐾(𝐾)+^𝑘𝑘−1𝜕𝐻_𝑘𝜕𝐮_𝑘+𝛿𝐮_𝑘𝑑𝜏_𝐾_𝑗=𝑘𝜆_𝑗(𝑗−1)^𝑇𝛿𝐱_𝑗(𝑗−1)−𝜆_𝑗(𝑗⁻)^𝑇𝛿𝐱_𝑗(𝑗⁻)+^𝐾−1_𝑗=1𝜕𝜑_{𝑖𝑗𝑖𝑗+1}𝜕𝐱_𝑗(𝑗⁻)𝛿𝐱^𝑗=𝜕𝜙𝐱_𝐾(𝑗⁻)(𝐾)𝜕𝐱_𝐾(𝐾)𝛿𝐱^𝐾(𝐾)+^𝑘𝑘−1𝜕𝐻_𝑘𝜕𝐮_𝑘−𝛿𝐮_𝑘𝑑𝜏+𝜆_𝑘(𝑘−1)^𝑇𝛿𝐱_𝑘(𝑘−1)_𝐾−1_𝑗=𝑘𝜆^𝑗(𝑗⁻)^𝑇𝛿𝐱_𝑗(𝑗⁻)−𝜆_𝑗+1(𝑗)^𝑇𝛿𝐱_𝑗+1(𝑗)−𝜕𝜑^{𝑖𝑗𝑖𝑗+1}𝜕𝐱_𝑗(𝑗⁻)𝛿𝐱^𝑗−𝜆_𝐾(𝑗⁻)(𝐾)^𝑇𝛿𝐱_𝐾(𝐾).(A.18)
Due to Theorem 3.2 and (A.9), 𝛿𝐽 can be reformulated by 𝛿𝐽=𝜕𝜙𝐱^𝐾(𝐾)𝜕𝐱_𝐾(𝐾)−𝜆^𝐾(𝐾)^𝑇𝛿𝐱^𝐾(𝐾)+^𝑘𝑘−1𝜕𝐻_𝑘𝜕𝐮_𝑘−𝛿𝐮_𝑘𝑑𝜏_𝐾−1_𝑗=𝑘𝛼^𝑇𝑗𝜕𝑔_𝑖𝑗𝜕𝐱_𝑗(𝑗⁻)𝛿𝐱^𝑗=_𝑘𝑘−1(𝑗⁻)𝜕𝐻_𝑘𝜕𝐮_𝑘𝛿𝐮_𝑘𝑑𝜏−^𝐾−1_𝑗=𝑘𝛼^𝑇𝑗𝜕𝑔_𝑖𝑗𝜕𝐱_𝑗(𝑗⁻)𝛿𝐱^𝑗(𝑗⁻)−𝛼^𝑇𝐾𝜕𝜗𝜕𝐱_𝐾(𝐾)𝛿𝐱^𝐾(𝐾).(A.19)
Then according to the definition of functional derivative, we have 𝛿𝐽𝛿𝐮_𝑘=𝜕𝐻^𝑘𝜕𝐮_𝑘−^𝐾−1_𝑗=𝑘𝛼^𝑇𝑗𝜕𝑔_𝑖𝑗𝜕𝐱_𝑗(𝑗⁻)𝛿𝐱^𝑗(𝑗⁻)𝛿𝐮_𝑘−𝛼^𝑇𝐾𝜕𝜗𝜕𝐱_𝐾(𝐾)𝛿𝐱^𝐾(𝐾)𝛿𝐮_𝑘.(A.20)
Obviously, the functional derivative of 𝐽 with respect to 𝐬_𝑘 can be directly given by 𝛿𝐽𝛿𝐬_𝑘=𝜕𝐻^𝑘𝜕𝐬_𝑘.(A.21)
This completes the proof.

Acknowledgments

The authors would like to express their sincere thanks to Professor Michael Fu at the University of Maryland (College Park) for suggestions on this work. This work was supported by the NSF under grant 60974023 of China, the State Key Development Program for Basic Research of China (2012CB720503), and the Fundamental Research Funds for the Central Universities.