Dev.Chem. Eng. Mineral Process., 9(lR).pp.l43-150,200I. Hybrid Automaton Model and Control of Hybrid Systems Wei Zhang* and Youxian Sun National Laboratoly of Industrial Control Technology, Institute of Industrial Process Control, Zhejiang University, Hangzhou 310027, PR.China A hybrid automaton model is proposed in this paper for hybrid systems. The model, with particular emphasis on process control applications, is based on the dynamic features of hybrid systems. It takes into account the discrete dynamics of hybrid systems in particular and can clearly separate the controller j i o m the closed-loop system. The definition of controllability of hybrid systems with respect to the marked regions is also given. An analyzing algorithm and sufficient and necessary condition for controllability based on the hybrid automaton model are discussed. At the same time,the propem of the loops in the trajectory of the closed-loop plant is studted. Finally, the synthesis scheme for a hybrid controller is given. In this scheme, the controller consists of two parts, the discrete supervisor and the continuous regulator The closed-loopplant with the controller is state-conmilable. Introduction Hybrid systems, which are pervasively existing in production processes, are complicated with both continuous and discrete dynamics. Hybrid control systems involve both the dynarmcs and continuous and discrete controls. In recent years, with particular emphasis on process control applications, modeling, analysis and optimization of hybrid systems have given rise to much research [ 1-81. Antsaklis et al. [4] studied the hybrid control of continuous systems and proposed a DES (discrete event system) plant model. The model is a simplification of a continuous system and interface and the simplification results in a loss of information about the internal dynamics, leading to nondeterministic behavior. Lennartson et al. [ 5 ] presented a general model structure for hybrid systems with particular emphasis on process control applications. They discussed the control of hybrid systems based on this structure. Tittus and Egardt [6] defined the controllability of hybrid systems and proposed a synthesis scheme for integrator hybrid systems, which are simpler. correspondingly. In references [4-61,the control for hybrid systems has been studed through discrete control commands. The controller generates the discrete commands first,which are then imposed on the plant after being translated into a piecewise constant continuous control. This necessarily reduces the ability of the controller to regulate the system in real-time and also the nondeteministic behavior is inevitable. Branicky [7] has discussed the hybrid features in real-world systems and presented a fiamework for hybrid systems. In the light of the terms of the hnework, the hybrid *Authorfor correspondence (email: wzhang@i@c.zju.edu.cn). 143 W.&ng and Y. Sun system is named by the general hybrid dynamical system under control. In thls structure, the hybrid controller is designed and the optimal control of the system is studied. Deshpande and Varraiya [S] have proposed a hybrid automaton model and have studied the viable control of hybrid systems. There are no explicit controllers which are modeled in [7] or [El. In this paper, as a slight generalization of the previously suggested models [4-81, a hybrid automaton model is presented first. Then the controllability of hybrid systems related to the marked regions is studied. Furthermore, a controller synthesis scheme is proposed. In this scheme, the controller is composed of two parts, the discrete supervisor and the continuous regulator. The closed-loop plant controlled by the controller is state-controllable. The Hybrid Automaton Model Hybrid Features in Process Control Hybrid systems, such as variable structure control system, Bang-Bang control system, computer control system, flexible manufacturing system, and chemical batch process, are existing in production processes. In chemical industries, the material flows typically go through a number of stages of various treatments, e.g. heating, mixing, reaction, separation, cooling, etc. The occurrence of these treatments can be viewed as a time-discrete event. This process obviously possesses the hybrid features. Hybrid dynamic systems contain both continuous and discrete dynamics, the evolution of the system depends on both of the dynamics. The continuous dynamics are usually expressed by differential equations: where x ( t ) is the continuous state vector, u(t) is the vector of external inputs. In a complicated hybrid dynamic system, the continuous dynamics of the system at the different stages is often described by various differential equations. We refer to these stages as the phases of the system. When continuous states satisfy some conditions, the system makes a transition from one phase to the next. In general, the discrete dynamics of hybrid systems may contain discrete input-output, discrete external control, switching of phase, etc. We can use the automaton or Petri net to model these discrete phenomena. In the following section we will formally give a hybrid automaton model to describe the dynamics of hybrid systems. The Hybrid Automaton Model The hybrid open-loop plant is defined as a hybrid automaton. It describes the system dynamics using discrete-event system modehg and classical continuous state-space representation. In different discrete states, the continuous state evolves according to different continuous dynamic laws. Now, a hybrid automaton is formally stated as a tuple: where Q = Q p x Q, is the f ~ t set e of &Crete states and C = C, u C, is the finite set Hybrid Automaton Model and Control of Hybrid Systems of dscrete events. Q, is the set of control modes corresponding to different settings of the plant's control variables. These control variables, in turn,are manipulated via signals from the controller, the control events C, . Q p is the set of physical modes of the open-loop plant itself contains. Each physical mode corresponds to some region p i E P , where P denotes a partition of R" , induced by the plant's discrete features. Each region p i is associated with a unique event w , E Z p ,when the continuous state enters p i the event wp happens. This case is finished by an event-generator r p : P + Z p . I c Q x R " is the set of the initial states and R" is the set of continuous states. The state of hybrid systems is a pair (q,X ) E Q x R" , the discrete state and the continuous state. E c Q x A(R") x Z x {R" + R")x Q (3) is the finite set of edges, where A(*) denotes the set of all subsets of ( 0 ) . The edges model the discrete dynamics of the system. An e E E is a state-transition function and denoted as I x,, we .Ye (4) 1 we say e E E is enabled when the discrete state is q, and the continuous state is in A', . When e is enabled and the event we E C happens, the transition through e is taken. As a result, the continuous state is reset according to the map ye and the system enters the discrete state q; . If the number of the q; is more than one, we say the system is nondeteministic. When the discrete state is q E Q , the continuous state evolves according to a controlled vector field i,(t) = f,( X C ( O J 4 t ) ) fq : R" x R" + R" with (5) F is the set of f,: F = { f q ( x c ( W 4 0 ) l 4 E €?I (6) Given an open-loop plant model and a specification for the desired closed-loop behavior, OUT purpose is to design a controller to restrict the behavior of the controlled-plant to agree with the specification. In this paper, we study the controllability of hybrid systems and design a hybrid controller which generates the discrete and continuous control laws at the same time to transfer the hybrid plant between predefined subsets of the hybrid state space. The closed-loop system is given in Figurel, in which the part encircled by the dashed-lines is the open-loop plant. Controllability of Hybrid Systems Controllability means that the state of hybrid systems can be controlled to transfer from one state to another. In this section, we are concerned with the controllability of hybrid systems. At first, we give some definitions. 145 W.utang and Y.Sun ..................................................... i .................................................................. Figure 1. Closed-loopsystem. Some Generic Definitions Definition 1. A marked region A, = (q,,,,x,) is a region or singular points in the hybrid state space Q x R" with q, E Q and X, G R" . Definition 2. A path is an edges sequence e,,...,e, if forVi = 1,...,1-1 we have qe, = qq-, . Definition 3. A loop is a path with qe, = q;, . A path is A simple loop if it is a loop andsatisfiesthat V z , j = 1 , . . . , I , i # j * q , f q , . Definition 4. An acceptable trajectory of a hybrid system H is any trajectory of H generated by some path connecting two marked regions A,, and A,, ,R=A,,,,ce,,.--,e, >A,,,? , with e, E E that transfers H from some initial state (qo.x,) E &* to some f i state ( q p x , ) E ;ln2 * Definition 5. A hybrid system is said to be state-controllable with respect to the marked regions A,, and Am2 if there exist hybrid control inputs that give H an acceptable trajectory along some path IC . A hybrid system is said to be controllable if any ordered pair of marked regions is state-controllable. According to the above definitions, the necessary condition for controllable of any ordered pair of marked regions is that there exists at least one path of the hybrid system connecting the two marked regions. Denote the path as x = A,, c e, ,..-,en> A,, . It is noted that a path may contain a possible loop. Loops are interesting from the controllability point of view. If a path contains a loop, then 146 Hybrid Automaton Model and Control of Hybrid Systems the number of the path connecting two marked regions may be infinite. Denote a path with a loop in a generic form as: IC =A,,,,< e,,...,ei(e,,,e,,,...,elm,~i~*,ei+l,...,e, > A,,* (7) where the operator (-)* means that the sequence inside the parenthesis can be repeated an arbitrary number of times. Below we first analyze the suffkient and necessary condition for controllabilityand then study the property found in loops. Controllability Analysis To investigate the sufficient and necessary condition for controllability, some notations are needed, which will be introduced below. For f, E F , let the state evolution map be $q : R, x R" + A(R"), i.e.,4,(r,x0) is the set of continuous states that can be reached from xo at time t under the controlled vector field & 0 )= fq( x , ( 0 3 m,x , (0)= xo (8) for some u(t) .Where R, is the set of nonnegative real numbers. Define the backward projection of x E R" under bq as P,W I&{ = x E $q(xo)} (9) and for X E R" define a,(XI = u x c xa,(4 pq(X)is the set of continuous states that can reach X q .For e E E define the operator n, :R" n e ( y )= (10) Under the dynamical law of + R" by x,nu;~(P~; (y)) (11) n , ( Y ) is the set of continuous states fiom which a transition through e can be taken immediately (given the discrete state is 4,) and then Y can be reached after the passage of some time in the discrete state 4;.where y,' (.) is the reversion of the map ye(-). For the operator ne(Y) we have the following properties. Proposition 1. n,(Y) is monotone, i.e., Y, c Yz = n,(Y,) c n,(Y,) . Proposition 2. n,(Y) distributes over U, i.e., n,(Y, UY,) = n , ( Y , ) U n , ( Y , ) . Using these notations, we have the following backward algorithm to analysis controllabihtyfor two given marked regions Aml and Am2. to A,,,,: R . Algorithm Step 1. search for a path connecting A,,,, 147 W.Zhang and Y. Sun = nen-,(2) next i else goto Step4. Step4. end Utilizing the algorithm, if the n , (2) is not null then the controllabihty of the hybrid systems is guaranteed by the following theorem. Theorem 1. A hybrid system is controllable with respect to the marked regions A,,,, and Am, if and only if there exist a path a = & c e , , . * . , e > , Amz and hybrid (n, n, --.ne.(x,,,~)). controls such that x,,,, E pqml qm I To simplify the presentation, we denote ne1(riel (nen(.))...) as ncln,, - . - n e m ( - )According , to the definition of the operator n , ( Y ) and note that Proof. (.a- qL1 = qeland q;, = qm2, we have the following equalities. then, we have accordingly ne, --n,-,(ne,nrl.--n,m)*ne,ne,+l - n , ( x m 2 ). Denote the sequence I?,, as 148 Hybrid Automaton Model and Control of Hybrid Systems fin =('JN~, i = 1,2,-.. n . i=O Then, in terms of the monotonicity of the sequence, we have Theorem 2. If the ,'A is bounded and ye is continuous, then the limit of the sequence A, exists, l k f i n = N ' . n-m Once N * has been generated, we can omit the loop and substitute riel .-.ne,-l" for the ne, --.ne,-l( n , n , ,...n,m)*ne,ne,+l - - - n e m ( x m 2then, ) , from which we can continue our algorithm. Controller Synthesis Now according to the theorem 1., we give a hybrid controller for H as a map S(q,x ) = ( S , ,S, ) : Q x R" + C x R" , where S , = {we[ e E IC and x E Ze(x,,)} S, = {u I u E R"' and 3 e E IC, x E b q ( Z e ( x m 2 ) and ) x 4 Ze(x,,)} The hybrid controller consists of two parts, discrete supervisor S , and the continuous regulator S, . The dscrete supervisor S, takes the discrete actions to ensure that the discrete state transitions will take place as expected. In every discrete state the continuous regulator S, dnves the continuous state to the regions where the edges we expected will be enabled. Under the control of the hybrid controller, the controllability of the closed-loop with respect to the marked regions will be guaranteed by the theorem 1. Conclusions We have considered the hybrid automaton model of hybrid systems. Our hybrid automaton model is a slight modification of previously suggested structures. In this model the open-loop plant is clearly separated from the controlled closed-loop plant. This separation is very valuable from a control theoretical point of view. The controllability of hybrid systems related to the marked regions has also been discussed. An algorithm and sufficient and necessary condtion for controllability have been proposed. Based on the condition a controller synthesis scheme has been given. The controller issues the discrete and the continuous control commands based on the discrete and the continuous feedback information of hybrid systems. References Nerodeand, A., and Kohn, W. 1993. 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