A hierarchical controller synthesis method based on dynamic quantization linear sequential logic specifications
By adopting a hierarchical controller synthesis method under the linear time-sequential logic specification with dynamic quantization, the problem of high computational complexity in complex nonlinear systems is solved, achieving efficient controller synthesis and system control, and meeting the requirements of complex LTL specifications.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- DALIAN UNIV OF TECH
- Filing Date
- 2026-03-16
- Publication Date
- 2026-06-02
AI Technical Summary
Existing controller synthesis methods based on abstract models have high computational complexity in complex nonlinear systems, making it difficult to meet complex LTL specifications. Furthermore, the computational load of global control strategies is large, making them difficult to apply to practical systems.
A hierarchical controller synthesis method under the linear time-series logic specification with dynamic quantization is adopted. By constructing local and dynamic abstract models and synthesizing controllers, the computational complexity is reduced. This includes model checking, dynamic quantizer generation of quantization intervals, local abstract model establishment and local LTL specification decomposition, and finally, a global hybrid controller is combined.
It effectively reduces computational complexity, improves the overall computational performance and versatility of the controller, ensures that the system meets the given LTL specification, and reduces the computational load and state space requirements.
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Figure CN122131629A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of nonlinear system control technology, and relates to a hierarchical controller synthesis method based on dynamic quantization under the linear temporal logic (LTL) specification. Under this control technique, the system can complete a given linear temporal logic specification, such as reaching a designated region, traveling between regions, and obstacle avoidance. The hierarchical strategy reduces the computational complexity problems associated with global control strategies, thus performing better in solving large-scale problems. Background Technology
[0002] Nonlinear system control has wide applications in industry, such as robotic arms, CNC machine tools, and printing production lines in manufacturing and automation; wind turbines, photovoltaic maximum power point tracking, and nuclear reactor power control in energy and power systems; and drone swarm flight, multi-robot cooperative handling, and vehicle-road cooperation in unmanned autonomous swarms. However, the high dimensionality, strong coupling, and uncertainty of nonlinear systems also pose significant challenges to traditional control methods. To address this challenge, researchers have gradually shifted to control strategies based on abstract models—constructing abstract models (such as finite state machines) that faithfully represent the behavior of the original nonlinear system but with a simpler structure. This significantly reduces the complexity of controller design and verification while preserving key dynamic characteristics. This "abstract first, then control" paradigm not only facilitates formal analysis and synthesis but also effectively supports seamless integration between high-level task planning and low-level continuous control, providing a new path for the intelligent and autonomous control of complex nonlinear systems.
[0003] The control method based on abstract models includes three steps: (1) constructing an abstract model of the original system; (2) solving for the abstract controller through backward search; and (3) obtaining a hybrid controller by refining the abstract controller and applying it to the original system. The key to this method is the construction of the abstract model, and a definite equivalence relationship needs to be satisfied between the abstract model and the original system. Scholars have proven that abstract models can be constructed for various systems, such as nonlinear systems, time-delay systems, switching systems, networked control systems, and stochastic systems.
[0004] Despite the significant progress made in the field of abstract model-based control, the synthesis of controllers for nonlinear systems under the LTL specification remains challenging. On the one hand, for complex systems, only simple LTL specifications such as safety and reachability can be satisfied; more general LTL specifications can only be studied on systems with simple structures. Therefore, controller synthesis of complex nonlinear systems under general LTL specifications remains an unsolved problem. On the other hand, the derivation of control strategies is usually global, requiring the system's state space to be considered as a whole. While scholars have proposed different equivalence relations in this regard, abstract model construction and controller synthesis are still time-consuming and computationally complex. For example, in the paper "Tabuada P. An Approximate Simulation Approach to Symbolic Control[J]. IEEE Transactions on Automatic Control, 2008, 53(6): 1406-1418," the authors use a static quantizer to approximate the entire state space, which leads to extremely high computational complexity. For example, in the paper "Meyer PJ, Dimarogonas D V. Hierarchical Decomposition of LTLSynthesis Problem for Nonlinear Control Systems[J]. IEEE Transactions on Automatic Control, 2019, 64(11): 4676-4683.", the authors adopted a bottom-up hierarchical structure. The first step is to obtain an acceptable path for the LTL specification; the second step is to find a high-dimensional discrete program through coarse partitioning; and the last step is to refine the partitioning for controller synthesis. However, regardless of whether it is a bottom-up or top-down hierarchical structure, the state space is divided into several disjoint units, and a graph is constructed through adjacency relationships. In this way, the synthesis of the controller is still (semi-)global and cannot be solved efficiently. Therefore, how to effectively perform controller synthesis based on abstract models still needs further research.
[0005] In summary, existing control methods based on abstract models can no longer simultaneously meet the complex LTL specifications of complex systems, as well as the system's requirements for speed and accuracy. There is an urgent need for a high-performance control method. Summary of the Invention
[0006] The problem this invention aims to solve is the excessive computational burden of existing controller synthesis techniques based on abstract models. It proposes a hierarchical controller synthesis method based on a dynamic quantization linear temporal logic specification. The construction of the abstract model and the controller synthesis are both local and dynamic, thus effectively reducing computational complexity. Since each quantization interval can be considered independently, the proposed hierarchical mechanism is more effective than existing methods in solving larger and more complex problems.
[0007] To achieve the above objectives, the technical solution adopted by the present invention is as follows: A hierarchical controller synthesis method based on dynamic quantization under a linear time-series logic (LTL) specification is proposed. First, for a nonlinear system and an LTL specification, a model checking algorithm is applied to obtain an acceptable path for the LTL specification, providing a high-dimensional solution for subsequent controller synthesis. Second, a dynamic quantizer is applied to generate a quantization interval sequence to verify the implementation of the obtained acceptable path. Only when the acceptable path is implemented does subsequent controller synthesis proceed. Third, the generated quantization interval sequence is then used... - Establish a local abstract model based on approximate mutual simulation relationships. Fourth, decompose the global LTL specification into a finite number of local LTL specifications within the generated quantization interval sequence. Finally, apply the obtained local abstract model and local LTL specifications to construct an auxiliary control problem, solve this problem to obtain a local abstract controller, refine the local abstract controller to obtain a hybrid controller, and then combine the hybrid controllers to obtain a global hybrid controller. This controller acts on the original nonlinear system to satisfy the given LTL specification. Specifically: Step 1: Represent the system under consideration in the standard form of a nonlinear system and give the LTL specification that the system needs to satisfy. Next, use a model checking algorithm on the LTL specification to obtain an acceptable path. This acceptable path provides a high-dimensional solution for subsequent controller synthesis; specifically: Step 1.1, consider n-dimensional Euclidean space. m-dimensional Euclidean space The system's state space and input space Given a nonlinear system Its dynamic model is represented as: (1) in, For the sake of convenience, we will refer to the system state as 'x' from now on. ; This is system input; for ease of description, it will be referred to as 'u' from now on. ; System status Regarding time The derivative of represents the rate of change of the system state with respect to time; It is a local Lipschitz function, which describes the rate of change of state. Compared with the current state and input The relationship between them. Let... This is the initial state of the system. Representing the state space, It is the initial state set; set and set All are compact convex sets and sets The origin is included. Furthermore, the nonlinear system represented by Equation (1) is incrementally globally asymptotically stable.
[0008] In state space In the diagram, the set of obstacles is represented as... ,in , It is a set of positive integers. Indicates a certain value in a set Positive integers in Represents a set One obstacle. To represent the tasks the system needs to accomplish as an LTL specification, consider the set of propositions existing in the system. ,in For atomic propositions, that is The propositions in the equation can only be true or false. Each Both with a subset Related, if ,So , The statement is true, where , For a set of propositions cardinality and It is finite; if So, proposition It is false. (Use) To represent atomic propositions and state space The relationship. Among them, The region of interest is the area that the nonlinear system represented by (1) needs to pass through or stay in under the LTL specification.
[0009] Based on the set of regions of interest, construct the following quadruple: (2) in, It is the set of all possible initial regions; It is a transfer relationship; It is a label function, where It is a set The power set of . An infinite path can be formed by an infinitely long sequence of atomic propositions. This indicates that, for all of them... and All satisfied An infinitely long sequence of atomic propositions in An infinite word is generated above, defined as .
[0010] Step 1.2, consider the LTL task of the nonlinear system (1) in Step 1.1. The model detection algorithm is applied to obtain an acceptable path; specifically: Consider that an LTL task can be recursively expressed as follows: (3) Among them, the symbol " " means "defined as"; For atomic propositions, their values are either 'true' or 'false'; LTL specifications considered for this invention; and It is not a specific LTL specification but a placeholder that represents any valid LTL specification defined by this syntax rule; Indicates logical NOT; This indicates logical conjunction; Indicates the 'next' sequence operator. This indicates the timing operator 'until'.
[0011] Any LTL specification can be represented as nondeterministic. Automata: (4) in, Indicates nondeterminism Automata; It is a finite set of states; It is the initial state set; It is the set of input alphabets; It is a transfer relationship; This is the set of acceptable states. Based on... Define an infinite sequence For infinite words, among which, , For infinitely repeating operators. All operators that satisfy a given LTL task. The infinite set of words is represented as Consider the LTL specification. A sequence of atomic propositions ,when Sometimes, ;when Sometimes, The quadruple represented by equation (2) and the nondeterministic quadruple represented by equation (4) Automata perform Cartesian product to obtain the product. Automata: (5) in, This represents a product automaton; Represents the set of states of a product automaton; , represents the initial set of states of the product automaton; For a transfer relation, if and only if and hour, Defined as If two states do not satisfy the transition relationship, then they cannot transition between each other. , where represents the set of acceptable states for a product automaton.
[0012] The above product automaton For a finite-state system, searching using Dijkstra's algorithm yields an infinitely long sequence of atomic propositions. ,satisfy Then this sequence of atomic propositions It is called An acceptable path.
[0013] Step 2: Apply a dynamic quantizer to generate a quantization interval sequence, and verify the implementation of the acceptable path obtained in Step 1. First, the definition of a dynamic quantizer is given. Second, the constraints that the quantization interval sequence generated by the dynamic quantizer should satisfy are given. Finally, the quantization interval sequence generation algorithm is given to verify the implementation of the acceptable path; subsequent controller synthesis will only proceed if the acceptable path is implemented. Specifically: Step 2.1 first defines an acceptable path as realizable: an acceptable path is realizable only when there exists a connected set that intersects with the corresponding region of the acceptable path. Based on this explanation, a dynamic quantizer is used to generate several quantization intervals to form a connected set to verify the realization of the acceptable path. The dynamic quantizer used is as follows: (6) in, These are quantization parameters, among which It is a positive real number, initialized to ; It is a quantitative center; ,in .
[0014] Extending the one-dimensional dynamic quantizer shown in formula (6) to higher dimensions, for n-dimensional vectors... An n-dimensional dynamic quantizer is defined as: (7) The quantization interval generated by the n-dimensional dynamic quantizer shown in formula (7) is expressed as follows: (8) in, Indicates that x is the center. A hypercube with radius .
[0015] Step 2.2: The generation of the quantization interval sequence needs to meet the corresponding constraints in order to verify the implementation of the acceptable path.
[0016] That is, (a) the initial quantization interval should not intersect with the set of obstacles; (b) the quantization intervals should intersect each other so that the nonlinear system shown in formula (1) can move in different quantization intervals. In other words, the quantization interval should be a connected set or there should be a connected subset within the quantization interval; (c) the quantization interval should intersect with the region of interest; specifically: First, select the initial quantization interval. The generation should satisfy , and .
[0017] Secondly, to ensure that the quantization intervals intersect each other, it is necessary to define the set. Contract. Given a set Its - Contraction is defined as: (9) in, , and It is a set Two unequal states; A measure of state. Conversely, Called of - Expansion. Collection of obstacles. of -Inflation is defined as Based on the above explanation, given the current quantization interval... The next quantization interval The generation of should meet the following requirements: (i) ; (ii) The following connection requirements must be met: (a) If If it is connected, then no further action is needed. Apply other constraints; (b) Otherwise, there exists a connected subregion. ,satisfy and .
[0018] The quantization intervals generated by satisfying the above conditions (i) and (ii) are intersecting.
[0019] Finally, consider the quantization interval and region of interest. The relationship between them. For acceptable paths. Introducing a robust and acceptable path ,satisfy So, if If it is feasible, then It must be implemented. Furthermore, define two logical variables. and express Relationship: (10) in, Represents a set With sets The intersection of all subsets is not empty. To ensure that the generated quantized interval sequence intersects with all regions of interest, the following conditions must be met: .
[0020] Step 2.3, the quantization interval generation algorithm that satisfies the constraints of Step 2.2 consists of the following four steps: (1) Set the initial quantization interval according to the constraints that need to be satisfied in step 2.2. .
[0021] (2) If If it does not intersect with obstacles, then ,in The coefficient of thermal expansion; if If it intersects with an obstacle, then ,in This is the contraction coefficient. The reason for modifying the quantization parameter is that if... If it does not intersect with obstacles, then the quantization parameters of the next quantization interval should be expanded to further explore the entire state space; if If the object intersects with an obstacle, the quantization parameters of the next quantization interval should be reduced to avoid collision with the obstacle.
[0022] (3) Determine the quantization center of the next quantization interval by random sampling. ,satisfy , And the connection requirements described in step 2.2.
[0023] (4) Based on the newly generated quantization interval Update logical variables and The value of .
[0024] Iteratively run steps (2)-(4) of the algorithm until... If the generated quantization interval sequence intersects with all regions of interest, the acceptable path is feasible, and the process proceeds to subsequent controller synthesis; otherwise, if the algorithm cannot terminate within a finite number of steps, the acceptable path is not feasible, and subsequent controller synthesis is unnecessary (controller synthesis refers to controller synthesis design).
[0025] Step 3: Establish a local abstract model of the nonlinear system shown in formula (1) using the quantization interval sequence obtained by the algorithm in step 2.3. First, approximate the state space of the system. Second, approximate the input space of the system. Finally, give the system a model that satisfies the following conditions: - A local abstract model approximating the mutual simulation relationship; specifically: Step 3.1: First, discretize the nonlinear system shown in equation (1) and express it as the following migration system: (11) in, The nonlinear system represented by formula (1) is expressed in terms of time intervals. The transition system represented by discretization; the set of states Initial state set The input set is ,in This indicates that the nonlinear system shown in (1) is in state x under the action of input u. The state that is reached after a certain time; If and only if for and have The output set is ; Output mapping is ,in This represents the identity function. To establish a local abstract model, the transfer system... In the quantization range The local migration system on is represented as .
[0026] Apply the dynamic quantizer shown in formula (7) in step 2 to the local migration system. The system state can be approximated as the following embedded dot matrix: (12) in, ; yes The quantitative center.
[0027] Step 3.2, for each quantization interval Local migration system The output set is ,satisfy ,in This means that for an input , Local migration system medium state The reachable set is: (13) Given any The set of reachable approximate states is defined as follows: (14) Given function Satisfying all There is an input Make This function is used to approximate the input set: (15) Step 3.3, applying the state and input approximations from steps 3.1 and 3.2, based on... - Establishing a local migration system based on approximate mutual simulation relationships Local abstract model: (16) in, It is a set of states; It is the initial state set; It is the input set; if and only if it satisfies Sometimes, ; It is the output set; It is an identity mapping.
[0028] Step 4: In the quantization interval sequence obtained in Step 2, decompose the global LTL specification shown in Formula (2) into a finite number of local LTL specifications. If all local LTL specifications are satisfied, it means that the global LTL specification is satisfied; specifically: Consider global LTL specifications , Is it a robust and acceptable path? The corresponding LTL specification. Within each quantization interval, it is decomposed into a local LTL specification. Each local LTL specification mainly consists of three parts, and each part is represented as follows: (1) Local migration system The state remains constant And it satisfies obstacle avoidance constraints, that is ,in, and .
[0029] (2) Local migration system The state reaches the local target region .if , .Right now , .
[0030] (3) Internal target area Represented as and , , If the aforementioned internal target region exists, then the local migration system... The state requires traversing all internal target regions a finite number of times or periodically. That is, or ,in, ,and .
[0031] Local migration system The corresponding local LTL specification needs to be completed sequentially in each quantization interval. If all local LTL specifications are implemented, it means that the global LTL specification is implemented.
[0032] Step 5: First, based on the local abstract model obtained in Step 3 and the local LTL specification obtained in Step 4, solve the auxiliary control problem to obtain the local abstract controller. Then, refine the local abstract controller into a local hybrid controller. Finally, combine the local hybrid controllers to obtain a global hybrid controller, which acts on the nonlinear system shown in Equation (1) to satisfy the given LTL specification; specifically: Step 5.1, after obtaining the local LTL specification and local abstract model Then, a solution is needed. satisfy The local abstract controller. Since the local abstract model is discrete and the states and transition relationships are known, it can be represented as a directed weighted graph. Specifically, the nodes of the directed weighted graph correspond to the local abstract model. The state; if the local abstract model There exists a transition relationship between any two states. If there are control inputs, then there exists a directed edge between them, and the cost of the control input is the weight of the edge. Dijkstra's algorithm can be used to obtain a local abstract controller. This controller enables local abstraction models. Satisfying the local LTL specification .
[0033] Step 5.2, due to the local abstract model and local migration system satisfy - Approximate mutual simulation relationship, then if a local abstract controller exists Enable local migration system Satisfying the local LTL specification Therefore, there must exist a local hybrid controller. Enable local migration system Satisfying the local LTL specification Therefore, the local hybrid controller is represented as follows: (17) in, for - Approximate mutual simulation relationship.
[0034] Step 5.3, due to the local migration system The migration system shown in formula (11) The subsystem uses the LTL specification decomposition method shown in step 4. It will also certainly enable the system satisfy Therefore, by combining all the local hybrid controllers, a system can be obtained. Meets LTL Specification Global promiscuous controller: (18) The global hybrid controller shown in Equation (18) can control the nonlinear system shown in Equation (1) to meet the given LTL specification.
[0035] Compared with the prior art, the present invention has the following beneficial effects: (1) Implementation verification of acceptable paths: For an LTL specification, the acceptable paths it generates are not unique, nor are they necessarily implementable. Traditional controller synthesis under LTL specifications often couples high-dimensional LTL acceptable path acquisition with low-dimensional controller synthesis, making these methods have poor portability. This invention adds the implementation verification step shown in step 2 before controller synthesis to ensure that the designed controller can control the system to meet the given LTL specification. If the acceptable path is not implementable, it will not proceed to subsequent controller synthesis. This method avoids useless calculations, and the acceptable path implementation verification method can be easily ported to other systems, improving the versatility of this invention.
[0036] (2) In global controller synthesis methods, the entire state space is usually considered as a whole, which results in excessive computation and limits the practical application of abstract model-based controller synthesis methods. To address this, the present invention adopts a dynamic and local controller design scheme, which constructs an abstract model through local construction and iterative optimization. This effectively reduces computational complexity while ensuring the equivalence between the abstract model and the original system.
[0037] (3) Feasibility of the solution: Currently, most methods for control based on abstract models involve abstracting the entire state space, which requires a large amount of sampling data. When the state space is large, this method is obviously difficult to implement. In this invention, the construction of the quantization interval depends on the selected parameters, and the design of the quantization interval that only covers the acceptable path further reduces the size of the state space that needs to be calculated. This ensures the superiority of the proposed technical solution in terms of computational performance. Attached Figure Description
[0038] Figure 1 This is the overall framework of the hierarchical controller synthesis technology based on dynamic quantization proposed in this invention.
[0039] Figure 2 The mechanism for generating quantization intervals; Figure 2 (a) in the diagram represents the mechanism for generating the next quantization interval when the quantization interval is a connected set; Figure 2 (b) in the diagram represents the mechanism for generating the next quantization interval when there are connected subintervals in the quantization interval.
[0040] Figure 3 This represents an acceptable path that cannot be achieved otherwise. Yellow indicates the starting region, and blue indicates the region that needs to be traversed.
[0041] Figure 4 The state trajectory (blue) generated by applying this invention and the state trajectory (green) obtained by the global method are shown when the number of quantization intervals is 13.
[0042] Figure 5 The state trajectory (blue) generated by this invention is applied when the number of quantization intervals is 14.
[0043] Figure 6 for Figure 4 The evolution diagram of the control input corresponding to the blue trajectory in the middle; Figure 6 (a) in the middle is Figure 4 The evolution of vehicle speed over time corresponding to the blue trajectory; Figure 6 (b) in the middle is Figure 4 The evolution of the vehicle steering angle corresponding to the blue trajectory over time.
[0044] Figure 7 for Figure 5 The evolution diagram of the control input corresponding to the blue trajectory in the middle; Figure 7 (a) in the middle is Figure 5 The evolution of vehicle speed over time corresponding to the blue trajectory; Figure 7 (b) in the middle is Figure 5 The evolution of the vehicle steering angle corresponding to the blue trajectory over time. Detailed Implementation
[0045] The technical solution proposed in this invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.
[0046] Example 1 like Figure 1 As shown, the specific implementation measures of the hierarchical controller synthesis method based on the linear sequential logic specification of dynamic quantization are as follows: Step 1: Represent the system under consideration in this invention in the standard form of a nonlinear system, and give the LTL specification that the system needs to satisfy. Then, apply a model checking algorithm to this LTL specification to obtain an acceptable path, which provides a high-dimensional solution for subsequent controller synthesis; specifically: Step 1.1, consider 3-dimensional Euclidean space. 2-dimensional Euclidean space The system's state space and input space Given a nonlinear system For an autonomous vehicle, its dynamics can be represented as: (1) in, Vehicle status ,in, For vehicle location, The vehicle's orientation in a 2D plane; control input ,in, Rear wheel speed, This refers to the steering angle. Assume the range of the control input is... The state space is .
[0047] In state space In the diagram, the set of obstacles is represented as... ,in , It is a set of positive integers. Indicates a certain value in a set Positive integers in Represents a set One obstacle. To represent the tasks the system needs to accomplish as an LTL specification, consider the set of propositions existing in the system. ,in For atomic propositions, that is The propositions in the equation can only be true or false. Each Both with a subset Related, if ,So , The statement is true, where , For a set of propositions cardinality and It is finite; if So, proposition It is false. (Use) To represent atomic propositions and state space The relationship. Among them, This is called the region of interest, which is the area that the system (1) needs to pass through or stay in in the LTL specification.
[0048] After describing the atomic propositions and regions of interest, the LTL specification considered in this invention can be expressed as follows: (2) in, , , .
[0049] Step 1.2, consider the LTL specification obtained in Step 1.1. The model detection algorithm is applied to obtain an acceptable path; specifically: An LTL specification can be represented as a quadruple: (3) in, It is the set of all possible initial regions; It is a transfer relationship; It is a label function, where It is a set The power set of . An infinite path can be formed by an infinitely long sequence of atomic propositions. This indicates that, for all of them... and All satisfied An infinitely long sequence of atomic propositions in An infinite word is generated above, defined as Consider the LTL specification. A sequence of atomic propositions ,when Sometimes, ;when Sometimes, Any LTL specification can be represented as nondeterministic. Automata: (4) in, Indicates nondeterminism Automata; It is a finite set of states; It is the initial state set; It is the set of input alphabets; It is a transfer relationship; This is the set of acceptable states.
[0050] The quadruple represented by equation (3) and the nondeterministic quadruple represented by equation (4) Automata perform Cartesian product to obtain the product. Automata: (5) in, This represents a product automaton; Represents the set of states of a product automaton; , represents the initial set of states of the product automaton; For a transfer relation, if and only if and hour, Defined as If two states do not satisfy the transition relationship, then they cannot transition between each other. , where represents the set of acceptable states for a product automaton.
[0051] The above product automaton For a finite-state system, searching using Dijkstra's algorithm yields an infinitely long sequence of atomic propositions. ,in This indicates an infinite loop, satisfying the condition. , That is An acceptable path.
[0052] Step 2: Apply a dynamic quantizer to generate a quantization interval sequence, and verify the implementation of the acceptable path obtained in Step 1. First, the definition of a dynamic quantizer is given. Second, the constraints that the quantization interval sequence generated by the dynamic quantizer should satisfy are given. Finally, the quantization interval sequence generation algorithm is given to verify the implementation of the acceptable path; subsequent controller synthesis will only proceed if the acceptable path is implemented. Specifically: Step 2.1 first defines an acceptable path as realizable: an acceptable path is realizable only when there exists a connected set that intersects with the corresponding region of the acceptable path. Based on the above description, this invention uses a dynamic quantizer to generate several quantization intervals to form a connected set to verify the realization of the acceptable path. The dynamic quantizer used is as follows: (6) in, These are quantization parameters, among which It is a positive real number, initialized to ; It is a quantitative center; ,in .
[0053] Extending the one-dimensional dynamic quantizer shown in formula (6) to higher dimensions, for n-dimensional vectors... An n-dimensional dynamic quantizer is defined as: (7) The quantization interval generated by the n-dimensional dynamic quantizer shown in formula (7) is expressed as follows: (8) in, Indicates that x is the center. A hypercube with radius .
[0054] Step 2.2: The generation of the quantization interval sequence needs to meet the corresponding constraints in order to verify the implementation of the acceptable path.
[0055] That is, (a) the initial quantization interval should not intersect with the set of obstacles; (b) the quantization intervals should intersect each other so that the autonomous vehicle shown in formula (1) can move in different quantization intervals. In other words, the quantization interval should be a connected set or there should be a connected subset within the quantization interval; (c) the quantization interval should intersect with the region of interest; specifically: First, select the initial quantization interval. The generation should satisfy , and .
[0056] Secondly, to ensure that the quantization intervals intersect each other, it is necessary to define the set. - Shrink. Given a set Its - Contraction is defined as: (9) in, and It is a set Two unequal states; A measure of state. Conversely, Called of - Expansion. Collection of obstacles. of Expansion is defined as Based on the above explanation, given the current quantization interval... The next quantization interval The generation of should meet the following requirements: (i) ; (ii) The following connection requirements must be met: (a) If If it is connected, then no further action is needed. Apply other constraints; (b) Otherwise, there exists a connected subregion. ,satisfy and .
[0057] The quantization intervals generated by satisfying the above conditions (i) and (ii) are intersecting.
[0058] Finally, consider the quantization interval and the region of interest. shrink The relationship between them. For acceptable paths. Introducing a robust and acceptable path ,satisfy So, if If it is feasible, then It must be implemented. Furthermore, define two logical variables. and express Relationship: (10) in, Represents a set With sets The intersection of all subsets is not empty. To ensure that the generated quantized interval sequence intersects with all regions of interest, the following conditions must be met: .
[0059] Step 2.3, the quantization interval generation algorithm that satisfies the constraints of Step 2.2 consists of the following four steps: (1) Set the initial quantization interval according to the constraints that need to be satisfied in step 2.2. .
[0060] (2) If If it does not intersect with obstacles, then ,in The coefficient of thermal expansion; if If it intersects with an obstacle, then ,in This is the contraction coefficient. The reason for modifying the quantization parameter is that if... If it does not intersect with obstacles, then the quantization parameters of the next quantization interval should be expanded to further explore the entire state space; if If the object intersects with an obstacle, the quantization parameters of the next quantization interval should be reduced to avoid collision with the obstacle.
[0061] (3) Determine the quantization center of the next quantization interval by random sampling. ,satisfy , And the connection requirements described in step 2.2.
[0062] (4) Based on the newly generated quantization interval Update logical variables and The value of .
[0063] Iteratively run steps (2)-(4) of the algorithm until... If the generated quantization interval sequence intersects with all regions of interest, the acceptable path is feasible, and subsequent controller synthesis can proceed; otherwise, if the algorithm cannot terminate within a finite number of steps, the acceptable path is not feasible, and subsequent controller synthesis is unnecessary.
[0064] For the quantization interval sequence generation algorithm described above, different parameter choices will lead to different quantization interval sequence generation results. When the parameters are selected... At that time, the generated quantization interval sequence is as follows Figure 3 and Figure 4 The dashed rectangle in the image is shown. Figure 3 In this context, the entire state space is controlled by a gate ( Figure 3 The gray area is divided into two disconnected parts. First, six quantization intervals are generated and proven from... arrive It is reachable. However, this closed door divides the 7th generated quantization interval into two disconnected parts. Subsequent quantization interval generation cannot meet the connectivity requirements. Therefore, the generated quantization intervals cannot intersect with all regions of interest, making acceptable paths unrealizable and eliminating the need for subsequent controller synthesis. Figure 4 In this process, the closed door is opened, and the entire state space is now connected. Thirteen quantization intervals are generated to verify the implementation of acceptable paths. The specific parameters of the quantization interval sequence are shown in Table 1. It should be noted that since the quantization center is generated through sampling, the generated quantization interval sequence is not unique. Figure 5 Another feasible quantization interval sequence is provided, using 14 quantization intervals to verify the implementation of acceptable paths.
[0065] Table 1: Quantization Interval Parameters
[0066] Step 3: Establish a local abstract model of the autonomous vehicle shown in formula (1) based on the quantization interval sequence obtained using the algorithm in step 2.3. First, approximate the state space of the system. Second, approximate the input space of the system. Finally, give the system a model that satisfies the following conditions: - A local abstract model approximating the mutual simulation relationship; specifically: Step 3.1: First, discretize the nonlinear system shown in equation (1) and express it as the following migration system: (11) in, The nonlinear system represented by formula (1) is expressed in terms of time intervals. The transition system represented by discretization; the set of states Initial state set The input set is ,in This indicates that the autonomous vehicle shown in (1) is in state x and is affected by the input u. The state that is reached after a certain time; If and only if for and have The output set is ; Output mapping is ,in This represents the identity function. To establish a local abstract model, the transfer system... In the quantization range The local migration system on is represented as .
[0067] Apply the dynamic quantizer shown in formula (7) in step 2 to the local migration system. The system state can be approximated as the following embedded dot matrix: (12) in, ; yes The quantitative center.
[0068] Step 3.2, for each quantization interval Local migration system The output set is ,satisfy ,in This means that for an input , Local migration system medium state The reachable set is: (13) Given any The set of reachable approximate states is defined as follows: (14) Given function Satisfying all There is an input Make This function is used to approximate the input set: (15) Step 3.3, applying the state and input approximations from steps 3.1 and 3.2, based on... - Establishing a local migration system based on approximate mutual simulation relationships Local abstract model: (16) in, It is a set of states; It is the initial state set; It is the input set; if and only if it satisfies Sometimes, ; It is the output set; It is an identity mapping.
[0069] Select parameters , A local abstract model is established in each quantization interval. This is based on the initial quantization interval. For example, it took a total of 30.75 seconds to build the local abstract model. It contains A transfer relationship.
[0070] Step 4: In the quantization interval sequence obtained in Step 2, decompose the global LTL specification shown in Formula (2) into a finite number of local LTL specifications. If all local LTL specifications are satisfied, it means that the global LTL specification is satisfied; specifically: Consider global LTL specifications , Is it a robust and acceptable path? The corresponding LTL specification. Within each quantization interval, it is decomposed into a local LTL specification. Each local LTL specification mainly consists of three parts, and each part is represented as follows: (1) Local migration system The state remains constant And it satisfies obstacle avoidance constraints, that is ,in, and .
[0071] (2) Local migration system The state reaches the local target region .if , .Right now , .
[0072] (3) Internal target area Represented as and , , If the aforementioned internal target region exists, then the local migration system... The state requires traversing all internal target regions a finite number of times or periodically. That is, or ,in, ,and .
[0073] Local migration system The corresponding local LTL specification needs to be completed sequentially in each quantization interval. If all local LTL specifications are implemented, it means that the global LTL specification is implemented.
[0074] For example, in the initial quantization interval In the middle, local migration system The task is to keep the state within the quantization range. and reach the interval This molecular task can be represented as: ,in, and , For the timing operator 'always', For the 'final' time series operator, satisfying , .
[0075] Step 5: First, based on the local abstract model obtained in Step 3 and the local LTL specification obtained in Step 4, solve the auxiliary control problem to obtain the local abstract controller. Then, refine the local abstract controller into a local hybrid controller. Finally, combine the local hybrid controllers to obtain a global hybrid controller, which acts on the autonomous vehicle shown in Equation (1) to satisfy the given LTL specification; specifically: Step 5.1, after obtaining the local LTL specification and local abstract model Then, a solution is needed. satisfy The local abstract controller. Since the local abstract model is discrete and the states and transition relationships are known, it can be represented as a directed weighted graph. Specifically, the nodes of the directed weighted graph correspond to the local abstract model. The state; if the local abstract model There exists a transition relationship between any two states. If there are control inputs, then there exists a directed edge between them, and the cost of the control input is the weight of the edge. Dijkstra's algorithm can be used to obtain a local abstract controller. This controller enables local abstraction models. Satisfying the local LTL specification .
[0076] Step 5.2, due to the local abstract model and local migration system satisfy - Approximate mutual simulation relationship, then if a local abstract controller exists Enable local migration system Satisfying the local LTL specification Therefore, there must exist a local hybrid controller. Enable local migration system Satisfying the local LTL specification Therefore, the local hybrid controller is represented as follows: (17) in, for - Approximate mutual simulation relationship.
[0077] Step 5.3, due to the local migration system The migration system shown in formula (11) The subsystem uses the LTL specification decomposition method shown in step 4. It will also certainly enable the system satisfy Therefore, by combining all the local hybrid controllers, a system can be obtained. Meets LTL Specification Global promiscuous controller: (18) The global hybrid controller shown in formula (18) can control the autonomous vehicle shown in formula (1) to meet the LTL specification shown in formula (2).
[0078] Different quantization interval sequences will lead to different controller synthesis results. When the number of quantization intervals is 13, the controller obtained according to the above process is as follows: Figure 6 (a) and Figure 6 As shown in (b), under the action of the controller, the state trajectory of the autonomous vehicle shown in formula (1) is as follows: Figure 4 The blue trajectory is shown in the middle; when the number of quantization intervals is 14, the controller obtained according to the above process is as follows: Figure 7 (a) and Figure 7 As shown in (b), under the action of the controller, the state trajectory of the autonomous vehicle shown in formula (1) is as follows: Figure 5 As shown by the blue trajectory.
[0079] Comparative Example 1 Considering a global approach, for the autonomous vehicle shown in (1), all parameters are the same as in Example 1. Applying the global controller synthesis method, the generated trajectory is... Figure 5 As shown by the green curve.
[0080] Verification of Example 1 and Comparative Example 1: To further demonstrate the performance of the present invention, Example 1 was compared with Comparative Example 1, and the results are shown in Table 2. The results show that, in terms of the number of transition relationships, the time to establish the abstract model, and the controller synthesis time, the present invention is significantly superior to the global-based controller synthesis method.
[0081] Table 2 Comparison of the present invention with global-based methods
[0082] The above-described embodiments are merely illustrative of the implementation methods of the present invention, but should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the protection scope of the present invention.
Claims
1. A hierarchical controller synthesis method based on a linear sequential logic specification with dynamic quantization, characterized in that, The hierarchical controller synthesis method includes the following steps: Step 1: Represent the system under consideration in the standard form of a nonlinear system and give the LTL specification that the system needs to satisfy; then, use a model checking algorithm on the LTL specification to obtain an acceptable path, which provides a high-dimensional solution for subsequent controller synthesis. Step 2: Apply a dynamic quantizer to generate a quantization interval sequence and verify the implementation of the acceptable path obtained in Step 1. First, the definition of a dynamic quantizer is given. Second, the constraints that the quantization interval sequence generated by the dynamic quantizer should satisfy are given. Finally, the algorithm for generating the quantization interval sequence is given to verify the implementation of the acceptable path. Only when the acceptable path is implemented can subsequent controller synthesis be performed. Step 3: Establish a local abstract model of the nonlinear system using the quantized interval sequence from Step 2; first, approximate the system's state space; second, approximate the system's input space; finally, give the model that satisfies the system's... - A local abstract model that approximates the mutual simulation relationship; Step 4: In the quantization interval sequence obtained in Step 2, the global LTL specification is decomposed into a finite number of local LTL specifications. If all local LTL specifications are satisfied, it means that the global LTL specification is satisfied. Step 5: First, based on the local abstract model obtained in Step 3 and the local LTL specification obtained in Step 4, solve the auxiliary control problem to obtain the local abstract controller; then, refine the local abstract controller into a local hybrid controller; finally, combine the local hybrid controllers to obtain a global hybrid controller, which acts on the nonlinear system of Step 1 to make it satisfy the given LTL specification.
2. The hierarchical controller synthesis method based on dynamic quantization under a linear sequential logic specification as described in claim 1, characterized in that, Step 1 specifically involves: Step 1.1, consider n-dimensional Euclidean space. m-dimensional Euclidean space The system's state space and input space Given a nonlinear system Its dynamic model is represented as: (1) in, For the sake of convenience, we will refer to the system state as 'x' from now on. ; This is system input; for ease of description, it will be referred to as 'u' from now on. ; System status Regarding time The derivative of represents the rate of change of the system state with respect to time; It is a local Lipschitz function, which describes the rate of change of state. Compared with the current state and input The relationship between them; let This is the initial state of the system. Representing the state space, It is the initial state set; set and set All are compact convex sets and sets It includes the origin; furthermore, the nonlinear system represented by Equation (1) is incrementally globally asymptotically stable; In state space In the diagram, the set of obstacles is represented as... ,in , It is a set of positive integers. Indicates a certain value in a set Positive integers in Represents a set One obstacle; in order to represent the tasks that the system needs to accomplish as an LTL specification, consider the set of propositions that exist in the system. ,in For atomic propositions, that is The propositions in the text can only be true or false; each Both with a subset Related, if ,So , The statement is true, where , For a set of propositions cardinality and It is finite; if So, proposition It is false; use To represent atomic propositions and state space The relationship; among them, The region of interest is the area that the nonlinear system represented by (1) needs to pass through or stay in under the LTL specification. Based on the set of regions of interest, construct the following quadruple: (2) in, It is the set of all possible initial regions; It is a transfer relationship; It is a label function, where It is a set The power set; An infinite path can be formed by an infinitely long sequence of atomic propositions. It means; among them, for all and All satisfied A sequence of atomic propositions of infinite length in An infinite word is generated above, defined as ; Step 1.2, consider the LTL task of the nonlinear system (1) in Step 1.
1. An acceptable path is obtained by applying a model detection algorithm; specifically: Consider that an LTL task can be recursively expressed as follows: (3) Among them, the symbol " "Indicates" means "defined as"; For atomic propositions, their values are either 'true' or 'false'; LTL specifications considered for this invention; and It is not a specific LTL specification but a placeholder that represents any valid LTL specification defined by this syntax rule; Indicates logical NOT; This indicates logical conjunction; Indicates the 'next' sequence operator. This indicates the timing operator 'until'; Any LTL specification can be represented as nondeterministic. Automata: (4) in, Indicates nondeterminism Automata; It is a finite set of states; It is the initial state set; It is the set of input alphabets; It is a transfer relationship; A set of acceptable states; based on Define an infinite sequence For infinite words, among which, , For infinitely repeating operators; all operators that satisfy a given LTL task The infinite set of words is represented as Considering LTL specifications A sequence of atomic propositions ,when Sometimes, ;when Sometimes, The quadruple represented by equation (2) and the nondeterministic quadruple represented by equation (4) Automata perform Cartesian product to obtain the product. Automata: (5) in, This represents a product automaton; Represents the set of states of a product automaton; , represents the initial set of states of the product automaton; For a transfer relation, if and only if and hour, Defined as If two states do not satisfy the transition relationship, then they cannot transition between each other. , representing the set of acceptable states for a product automaton; The above product automaton For a finite-state system, Dijkstra's algorithm can be used to search for an infinitely long sequence of atomic propositions. ,satisfy Then this sequence of atomic propositions It is called An acceptable path.
3. The hierarchical controller synthesis method based on dynamic quantization under a linear timing logic specification as described in claim 2, characterized in that, Step 2 specifically involves: Step 2.1: First, the condition for an acceptable path to be realized is given: an acceptable path is realized only when there is an intersection between a connected set and the corresponding region of the acceptable path; then, a dynamic quantizer is used to generate several quantization intervals to form a connected set to verify the realization of the acceptable path. Step 2.2: The generation of the quantization interval sequence needs to meet the corresponding constraints in order to verify the implementation of the acceptable path; That is, (a) the initial quantization interval should not intersect with the set of obstacles; (b) the quantization intervals should intersect each other so that the nonlinear system shown in formula (1) can move in different quantization intervals; in other words, the quantization interval should be a connected set or there should be a connected subset in the quantization interval; (c) the quantization interval should intersect with the region of interest; specifically: First, select the initial quantization interval; initial quantization interval The generation should satisfy , and ; Secondly, to ensure that the quantization intervals intersect each other, the set is defined. - Shrink; given a set Its - Contraction is defined as: (9) in, , and It is a set Two unequal states; For the measure of state; conversely, Called of - Expansion; collection of obstacles of -Inflation is defined as Based on the above explanation, given the current quantization interval Determine the next quantization interval The generation requirements; Finally, consider the quantization interval and region of interest. The relationship between them; for acceptable paths Introducing a robust and acceptable path ,satisfy ; then, if If it is feasible, then It must be implemented; further, define two logical variables. and express Relationship: (10) in, Represents a set With sets The intersection of all subsets is not empty; to ensure that the generated quantized interval sequence intersects with all regions of interest, the following conditions must be met: ; Step 2.3, the quantization interval generation algorithm that satisfies the constraints of Step 2.2 consists of the following four steps: Step 2.3.1: Set the initial quantization interval according to the constraints that need to be satisfied in Step 2.
2. ; Step 2.3.2, if If it does not intersect with obstacles, then ,in The coefficient of thermal expansion; if If it intersects with an obstacle, then ,in This is the shrinkage coefficient; the reason for modifying the quantization parameter is that if If it does not intersect with obstacles, then the quantization parameters of the next quantization interval should be expanded to further explore the entire state space; if If the object intersects with an obstacle, the quantization parameter of the next quantization interval should be reduced to avoid collision with the obstacle. Step 2.3.3: Determine the quantization center of the next quantization interval through random sampling. ,satisfy , and the connection requirements described in step 2.2; Step 2.3.4, based on the newly generated quantization interval Update logical variables and The value; Iteratively run steps (2)-(4) of the algorithm until... If the generated quantization interval sequence intersects with all regions of interest, the acceptable path is feasible, and subsequent controller synthesis can proceed; otherwise, if the algorithm cannot terminate within a finite number of steps, the acceptable path is not feasible, and subsequent controller synthesis is unnecessary.
4. The hierarchical controller synthesis method based on dynamic quantization under a linear timing logic specification as described in claim 3, characterized in that, In step 2.1, the dynamic quantizer is as follows: (6) in, These are quantization parameters, among which It is a positive real number, initialized to ; It is a quantitative center; ,in ; Extending the one-dimensional dynamic quantizer shown in formula (6) to higher dimensions, for n-dimensional vectors... An n-dimensional dynamic quantizer is defined as: (7) The quantization interval generated by the n-dimensional dynamic quantizer shown in formula (7) is expressed as follows: (8) in, Indicates that x is the center. A hypercube with radius .
5. The hierarchical controller synthesis method based on dynamic quantization under a linear timing logic specification as described in claim 4, characterized in that, In step 2.2, the next quantization interval The generation of should meet the following requirements: （i） ; (ii) The following connection requirements must be met: (a) If If it is connected, then no further action is needed. Apply other constraints; (b) Otherwise, there exists a connected subregion. ,satisfy and ; The quantization intervals generated by satisfying the above conditions (i) and (ii) are intersecting.
6. The hierarchical controller synthesis method based on dynamic quantization under a linear timing logic specification as described in claim 5, characterized in that, Step 3 specifically involves: Step 3.1, discretize the nonlinear system shown in equation (1) and express it as the following migration system: (11) in, The nonlinear system represented by formula (1) is expressed in terms of time intervals. The transition system represented by discretization; the set of states Initial state set The input set is ,in This indicates that the nonlinear system shown in (1) is in state x under the action of input u. The state that is reached after a certain time; If and only if for and have The output set is ; Output mapping is ,in Represents the identity function; in order to establish a local abstract model, the transfer system will be... In the quantization range The local migration system on is represented as ; Apply the dynamic quantizer shown in formula (7) in step 2 to the local migration system. The system state can be approximated as the following embedded dot matrix: (12) in, ; yes The quantitative center; Step 3.2, for each quantization interval Local migration system The output set is ,satisfy ,in This means that for an input , Local migration system medium state The reachable set is: (13) Given any The set of reachable approximate states is defined as follows: (14) Given function Satisfying all There is an input Make Apply this function to approximate the input set: (15) Step 3.3, applying the state and input approximations from steps 3.1 and 3.2, based on... - Establishing a local migration system based on approximate mutual simulation relationships Local abstract model: (16) in, It is a set of states; It is the initial state set; It is the input set; if and only if it satisfies Sometimes, ; It is the output set; It is an identity mapping.
7. The hierarchical controller synthesis method based on dynamic quantization under a linear sequential logic specification as described in claim 5, characterized in that, Step 4 specifically involves: Consider the global LTL specification shown in formula (2) , Is it a robust and acceptable path? The corresponding LTL specification; within each quantization interval, it is decomposed into a local LTL specification. Each local LTL specification mainly consists of three parts, and each part is represented as follows: (1) Local migration system The state remains constant And it satisfies obstacle avoidance constraints, that is ,in, and ; (2) Local migration system The state reaches the local target region ;if , ;Right now , ; (3) Internal target area Represented as and , , If the aforementioned internal target region exists, then the local migration system... The state requires traversing all internal target regions a finite number of times or periodically; that is, or ,in, ,and ; Local migration system The corresponding local LTL specification needs to be completed sequentially in each quantization interval. If all local LTL specifications are implemented, it means that the global LTL specification is implemented.
8. The hierarchical controller synthesis method based on dynamic quantization under a linear timing logic specification according to claim 7, characterized in that, Step 5 specifically involves: Step 5.1, after obtaining the local LTL specification and local abstract model Then, a solution is needed. satisfy The local abstract controller is represented as a directed weighted graph; Step 5.2, due to the local abstract model and local migration system satisfy If there is an approximate mutual analog relationship, then if a local abstract controller exists... Enable local migration system Satisfying the local LTL specification Therefore, there must exist a local hybrid controller. Enable local migration system Satisfying the local LTL specification Therefore, the local hybrid controller is represented as follows: (17) in, for -Approximate mutual simulation relationship; Step 5.3, due to the local migration system The migration system shown in formula (11) The subsystem uses the LTL specification decomposition method shown in step 4. It will also certainly enable the system satisfy Therefore, by combining all local hybrid controllers, the system is obtained. Meets LTL Specification Global promiscuous controller: (18) The global hybrid controller shown in Equation (18) can control the nonlinear system shown in Equation (1) to meet the given LTL specification.
9. A hierarchical controller synthesis method based on dynamic quantization and linear sequential logic specifications according to claim 8, characterized in that, In step 5.1, the nodes of the directed weighted graph correspond to the local abstract model. The state; if the local abstract model There exists a transition relationship between any two states. If there are control inputs, then there is a directed edge between them, and the cost of the control input is the weight of the edge; Dijkstra's algorithm can be used to obtain the local abstract controller. This controller enables local abstraction models. Satisfying the local LTL specification .