Unmanned system trajectory tracking control method without initial stabilization learning strategy
By introducing offset factors and policy iterative learning algorithms into unmanned systems, the dependence on initial stabilization learning strategies in trajectory tracking control of unmanned systems is resolved, realizing model-free trajectory tracking control and ensuring the stability and performance of the system.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- BEIJING INST OF TECH
- Filing Date
- 2023-07-18
- Publication Date
- 2026-06-23
AI Technical Summary
Existing trajectory tracking control methods for unmanned systems rely on accurate system models and initial stabilization learning strategies, which cannot effectively handle model uncertainties, leading to difficulties in controller design and performance analysis.
A trajectory tracking control method without initial stabilization learning strategy is designed. By establishing an unmanned system model, introducing an offset factor to construct a new closed-loop system, and using a strategy iterative learning algorithm to find the optimal control strategy, a model-free trajectory tracking control that does not depend on initial stabilization is achieved.
This invention enables an autonomous vehicle system to effectively track the leader's trajectory without relying on an accurate model, ensuring the system's steady-state and transient performance and overcoming the dependence of traditional methods on initial stabilization.
Smart Images
Figure CN116909277B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of unmanned systems technology, and more specifically to a trajectory tracking control method for unmanned systems that does not require an initial stabilization learning strategy. Background Technology
[0002] Cooperative control of unmanned systems has extremely wide applications in military and civilian fields such as cooperative rescue, search and rescue, cooperative reconnaissance, and cooperative strike. In practical applications, the operation of unmanned systems is affected by the external environment, which can easily lead to inaccurate or even uncertain unmanned system models. The uncertainty of the system model poses a great challenge to controller design and performance analysis.
[0003] The design of the controller is crucial for ensuring trajectory tracking control in autonomous vehicle systems. Existing controller design techniques rely on accurate system models and assume that the initial learned control strategy must be stable. However, if the system model cannot be accurately obtained, existing initial stabilization control strategies become ineffective. Therefore, designing a reasonable and efficient controller that does not require an initial stabilization learning strategy to guarantee the steady-state and transient performance of the system, and that achieves trajectory control of the autonomous system without relying on an accurate model, is one of the urgent problems to be solved.
[0004] Therefore, it is necessary to research and provide a trajectory tracking control method for unmanned systems that does not require an initial stabilization learning strategy. Summary of the Invention
[0005] In view of this, the present invention provides a trajectory tracking control method for unmanned systems that does not require an initial stabilization learning strategy. This method can overcome the dependence of the initial stabilization control strategy on the system model and ensure that the unmanned vehicle system can track the leader's trajectory under the designed reinforcement learning controller.
[0006] To achieve the above objectives, the technical solution of the present invention includes the following steps:
[0007] Step 1: Establish an unmanned system model.
[0008] Step 2: Design the trajectory tracking controller for the follower and the corresponding cost function to achieve optimal trajectory tracking of the follower.
[0009] Step 3: Introduce an offset factor based on the augmented system of the unmanned system to construct a new closed-loop system.
[0010] Step 4: Generate system data based on the new closed-loop system and collect system data. Use the strategy iterative learning algorithm to find the optimal control strategy to achieve model-free trajectory tracking control that does not depend on the initial stabilization control strategy.
[0011] Further, step one: establish an unmanned system model; specifically:
[0012] The dynamic model of the unmanned system is established as follows:
[0013]
[0014] Where z represents the state of the unmanned system. Let z represent the first derivative of z; A represents the state coefficient matrix of the unmanned system; B represents the control input coefficient matrix of the unmanned system; C represents the output coefficient matrix of the unmanned system; y represents the output of the unmanned system; u represents the control input of the unmanned system; matrices A and B are both unknown.
[0015] If an unmanned system includes a navigator and followers, then the dynamic model of the navigator is as follows:
[0016]
[0017] Where z0 represents the state of the Navigator unmanned system. Let z0 represent the first derivative; A0 represent the state coefficient matrix of the navigator unmanned system; B0 represent the control input coefficient matrix of the navigator unmanned system; C0 represent the output coefficient matrix of the navigator unmanned system; y0 is the output of the navigator unmanned system; matrices A and B are unknown.
[0018] Further, step two: Design the trajectory tracking controller for the follower and the corresponding cost function to achieve optimal trajectory tracking of the follower, specifically as follows:
[0019] Based on the dynamics model of the unmanned system, the dynamics model of the follower, and the controller u, construct the augmented system:
[0020]
[0021]
[0022] Where x represents the state of the augmented system. Denotes the first derivative of x. This represents the state coefficient matrix in the augmented system, and u represents the control input of the unmanned system. This represents the control input coefficient matrix of the augmented system. Let represent the output coefficient matrix of the augmented system, and e represent the output trajectory tracking error;
[0023]
[0024] The designed follower trajectory tracking controller is specifically as follows:
[0025]
[0026] Where: u represents the controller of the unmanned system, K1 represents the feedback gain of the controller, and K2 represents the feedforward gain of the controller. Represents the augmented gain matrix.
[0027] The cost function J is
[0028]
[0029] in: This indicates that the tracking error of the unmanned system corresponds to a known matrix. Q>0 represents any chosen known matrix, R=R T >0 indicates that the control input of the unmanned system is an arbitrarily selected known matrix, s is the integration variable, and t is the time.
[0030] Furthermore, step three: Based on the augmented system of the unmanned system, an offset factor is introduced to construct a new closed-loop system, specifically as follows:
[0031] A bias factor β is introduced into the augmented system of the unmanned system to make the initial learning independent of the initial stabilization learning strategy, and a new learning matrix W is defined. [m] for:
[0032]
[0033]
[0034] Among them, W [m] The superscript [m] in the text represents the number of iterations, W [m] This represents the learning matrix corresponding to the m-th iteration. 'a' is an arbitrarily chosen variable, and λ represents the matrix. eigenvalues, Re denotes the matrix The real part of the eigenvalues, s denotes the iteration sign, m denotes the total number of iterations, and θ s I represents the accumulation factor corresponding to the iteration number s. n An identity matrix with dimension n×n;
[0035] Construct the following new closed-loop system
[0036]
[0037] Where x represents the state of the augmented system. To augment the first derivative of the system state, W [m] The superscript [m] in I represents the iteration number. n An identity matrix with dimension n×n This represents the augmented gain matrix corresponding to the m-th iteration;
[0038] Design value function for new closed-loop system
[0039] V [m] =x T P [m] x
[0040] Among them, P [m] V represents the matrix used for iterative learning, where the initial iteration values are arbitrarily chosen positive definite matrices; [m] The matrix P representing iterative learning [m] The corresponding value function.
[0041] Furthermore, system data is generated and collected based on the new closed-loop system, specifically: system data is collected at time intervals [t, t+dt], where d represents a constant greater than zero and arbitrarily set; the system data generated based on the new closed-loop system is...
[0042]
[0043] in: Represents a column vector. x1,…,x n Let x = [x1, ..., x2] represent states respectively. n ] T The element; P [m] =[p 11 ,p 12 ,…,p 22 ,p 23 ,…,p nn ] T p 11 ,p 12 ,p nn Let p represent the first element of the first row of matrix P. 11 The first and second rows contain elements p. 12 and the element p in the nth row and nth column nn ,symbol Indicates the Kronecker product; Representation matrix Row vectorization; This represents the known matrix corresponding to the tracking error of the unmanned system.
[0044] The system collects data and stores it in four data storage units, namely Λ x Δ x (t i ,t i+1 ), Γ x ,Γ u :
[0045]
[0046]
[0047]
[0048]
[0049] In the time interval [t0, t] l The system collects data generated within the interval and divides the interval into several sub-intervals with a certain sampling step size; t0 is the initial time of data collection, t1 is the time after the system has passed the sampling step size at the initial data collection time t0, and so on, t... l The termination time for system data collection;
[0050] Let a vector be a vector whose elements satisfy the following condition: in Representing column vectors respectively In t i+1 ,t i The column vector corresponding to time step 1. Indicates the interval [t] i ,t i+1 A column vector;
[0051] Don't indicate that the corresponding follower is in the subinterval [t0,t1], [t1,t2], ..., [t l-1 ,t l System data collected within the system;
[0052] These represent the followers in the subintervals [t0, t1], [t1, t2], ..., [t...]. l-1 ,t l The system collects and associates input data.
[0053] The following equivalence relation is obtained.
[0054]
[0055] in: Represents a row vector. The controller gain matrix representing the follower The numerical value learned in the (m+1)th iteration.
[0056] Furthermore, the optimal control policy is found using a policy iterative learning algorithm, specifically through the following steps:
[0057] S401: Initial iteration value m = 0 and known constants μ′ > α > 2θ 0 >0, 'a' is an arbitrarily chosen variable, and θ 0 This is the accumulation factor corresponding to iteration number 0;
[0058] Applying it to the system to collect data so that it satisfies the following rank criterion:
[0059]
[0060] Where, n z Represents the state of the augmented system The dimension of n u Represents the state of the augmented system dimensionality;
[0061] S402: Let ε ← 1, ← indicates assignment. Solve the following equation to obtain the matrix.
[0062]
[0063] in, These represent the column vectors corresponding to the initial iteration m = 0. Let P represent the matrix corresponding to the initial iteration m = 0. This represents the augmented system matrix gain corresponding to m=1 in the first iteration;
[0064] If the matrix It is a non-positive definite matrix, execute ε←2 Continue solving the above equation And so on, until the matrix is judged. It is a positive definite matrix;
[0065] S403: W [0] Here is the Herwitz matrix, and the coefficients are... The following equation is guaranteed to converge. For matrix The largest eigenvalue;
[0066]
[0067] S404: By solving the following equation
[0068]
[0069] Get P [m] and Represents the controller gain matrix The numerical value learned in the (m+1)th iteration; vecs(P) is a column vector consisting of distinct elements in matrix P;
[0070] S405: If Established, among which If the selected value is any value greater than zero, the iteration stops, and the control gain is then [value missing]. The controller is
[0071] if If this is not true, then m = m + 1, and continue executing step four until... Established.
[0072] Beneficial effects:
[0073] This invention presents a trajectory tracking control method for unmanned systems that does not require an initial stabilization learning strategy. First, addressing the issue of a completely unknown system model, this invention introduces a bias factor into traditional reinforcement learning, proposing a learning matrix with a bias factor to overcome the dependence of the initial stabilization control strategy on the system model. Second, a trajectory tracking control scheme with a bias factor based on reinforcement learning is designed, proposing a reinforcement learning algorithm that relies solely on online state and input information. Through Lyapunov stability analysis, the unmanned vehicle system is guaranteed to achieve trajectory tracking of the leader under the designed reinforcement learning controller. The novel model-free reinforcement learning method provided by this invention completely relaxes the traditional reinforcement learning algorithm's requirement for a stabilizing initial control strategy. Attached Figure Description
[0074] Figure 1 The present invention provides a design flowchart for a trajectory tracking control method for unmanned systems that does not require an initial stabilization learning strategy. Detailed Implementation
[0075] The present invention will now be described in detail with reference to the accompanying drawings and embodiments.
[0076] This invention provides a trajectory tracking control method for unmanned systems that does not require an initial stabilization learning strategy, the process of which is as follows: Figure 1 As shown, it includes the following steps:
[0077] Step 1: Establish an unmanned system model; In this embodiment of the invention, an unmanned system model is established; specifically:
[0078] The dynamic model of the unmanned system is established as follows:
[0079]
[0080] Where z represents the state of the unmanned system. Let z represent the first derivative of z; A represents the state coefficient matrix of the unmanned system; B represents the control input coefficient matrix of the unmanned system; C represents the output coefficient matrix of the unmanned system; y represents the output of the unmanned system; u represents the control input of the unmanned system; matrices A and B are both unknown.
[0081] If an unmanned system includes a navigator and followers, then the dynamic model of the navigator is as follows:
[0082]
[0083] Where z0 represents the state of the Navigator unmanned system. Let z0 represent the first derivative; A0 represent the state coefficient matrix of the navigator unmanned system; B0 represent the control input coefficient matrix of the navigator unmanned system; C0 represent the output coefficient matrix of the navigator unmanned system; y0 is the output of the navigator unmanned system; matrices A and B are unknown.
[0084] Step 2: Design the trajectory tracking controller for the follower and the corresponding cost function to achieve optimal trajectory tracking of the follower; specifically:
[0085] Based on the dynamics model of the unmanned system, the dynamics model of the follower, and the controller u, construct the augmented system:
[0086]
[0087]
[0088] Where x represents the state of the augmented system. Denotes the first derivative of x. This represents the state coefficient matrix in the augmented system, and u represents the control input of the unmanned system. This represents the control input coefficient matrix of the augmented system. Let represent the output coefficient matrix of the augmented system, and e represent the output trajectory tracking error;
[0089]
[0090] The designed follower trajectory tracking controller is specifically as follows:
[0091]
[0092] Where: u represents the controller of the unmanned system, K1 represents the feedback gain of the controller, and K2 represents the feedforward gain of the controller. Represents the augmented gain matrix.
[0093] The cost function J is
[0094]
[0095] in: This indicates that the tracking error of the unmanned system corresponds to a known matrix. Q>0 represents any chosen known matrix, R=R T >0 indicates that the control input of the unmanned system is an arbitrarily selected known matrix, s is the integration variable, and t is the time.
[0096] Step 3: Introduce an offset factor into the augmented system of the unmanned system to construct a new closed-loop system; specifically:
[0097] A bias factor β is introduced into the augmented system of the unmanned system to make the initial learning independent of the initial stabilization learning strategy, and a new learning matrix W is defined. [m] for:
[0098]
[0099]
[0100] Among them, W [m] The superscript [m] in the text represents the number of iterations, W [m] Let μ' be the learning matrix corresponding to the m-th iteration. The maximum value between the eigenvalues and zero, s represents the iteration sign, m represents the total number of iterations, and θ s I represents the accumulation factor corresponding to the iteration number s. n An identity matrix with dimension n×n;
[0101] 'a' is an arbitrarily chosen variable, and λ represents the matrix. eigenvalues, Re denotes the matrix The real part of the characteristic roots;
[0102] Construct the following new closed-loop system
[0103]
[0104] Where x represents the state of the augmented system. To augment the first derivative of the system state, W [m] The superscript [m] in I represents the iteration number. n An identity matrix with dimension n×n This represents the augmented gain matrix corresponding to the m-th iteration;
[0105] Design value function for new closed-loop system
[0106] V [m] =x T P [m] x
[0107] Among them, P [m] V represents the matrix used for iterative learning, where the initial iteration values are arbitrarily chosen positive definite matrices; [m] The matrix P representing iterative learning [m] The corresponding value function.
[0108] Step 4: Generate system data based on the new closed-loop system and collect system data. Use the strategy iterative learning algorithm to find the optimal control strategy to achieve model-free trajectory tracking control that does not depend on the initial stabilization control strategy.
[0109] The system data is generated and collected based on the new closed-loop system, specifically as follows:
[0110] System data is collected over time intervals [t, t+dt], where d represents a constant greater than zero and arbitrarily set; system data is generated based on the new closed-loop system.
[0111]
[0112] in: Represents a column vector. x1,…,x n Let x = [x1, ..., x2] represent states respectively. n ] T The element; P [m] =[p 11 ,p 12 ,…,p 22 ,p 23 ,…,p nn ] T p 11 ,p 12 ,p nn Let p represent the first element of the first row of matrix P. 11 The first and second rows contain elements p. 12 and the element p in the nth row and nth column nn ,symbol Indicates the Kronecker product; Representation matrix Row vectorization; This represents the known matrix corresponding to the tracking error of the unmanned system.
[0113] The system collects data and stores it in four data storage units, namely Λ x Δ x (t i ,t i+1 ), Γ x ,Γ u :
[0114]
[0115]
[0116]
[0117]
[0118] In the time interval [t0, t] l The system collects data generated within the interval and divides the interval into several sub-intervals with a certain sampling step size; t0 is the initial time of data collection, t1 is the time after the system has passed the sampling step size at the initial data collection time t0, and so on, t... l The termination time for system data collection;
[0119] Let a vector be a vector whose elements satisfy the following condition: in Representing column vectors respectively In t i+1 ,t i The column vector corresponding to time step 1. Indicates the interval [t] i ,t i+1 A column vector;
[0120] Don't indicate that the corresponding follower is in the subinterval [t0,t1], [t1,t2], ..., [t l-1 ,t l System data collected within the system;
[0121] These represent the followers in the subintervals [t0, t1], [t1, t2], ..., [t...]. l-1 ,t l The system collects and associates input data.
[0122] The following equivalence relation is obtained.
[0123]
[0124] in: Represents a row vector. The controller gain matrix representing the follower The numerical value learned in the (m+1)th iteration.
[0125] The optimal control policy is found using a policy iterative learning algorithm, specifically through the following steps:
[0126] S401: Initial iteration value m = 0 and known constants μ′>α>2θ°>0, 'a' is an arbitrarily chosen variable, and θ° is the accumulation factor corresponding to iteration number 0.
[0127] Applying it to the system to collect data so that it satisfies the following rank criterion:
[0128]
[0129] Where, n z Represents the state of the augmented system The dimension of n u Represents the state of the augmented system dimensionality;
[0130] S402: Let ε ← 1, ← indicates assignment. Solve the following equation to obtain the matrix.
[0131]
[0132] in, These represent the column vectors corresponding to the initial iteration m = 0. Let P represent the matrix corresponding to the initial iteration m = 0. This represents the augmented system matrix gain corresponding to m=1 in the first iteration;
[0133] If the matrix It is a non-positive definite matrix, execute ε←2 Continue solving the above equation And so on, until the matrix is judged. It is a positive definite matrix;
[0134] S403: W [0] Here is the Herwitz matrix, and the coefficients are... The following equation is guaranteed to converge. For matrix The largest eigenvalue;
[0135]
[0136] S404: By solving the following equation
[0137]
[0138] Get P [m] and Represents the controller gain matrix The numerical value learned in the (m+1)th iteration; vecs(P) is a column vector consisting of distinct elements in matrix P;
[0139] S405: If Established, among which If the selected value is any value greater than zero, the iteration stops, and the control gain is then [value missing]. The controller is
[0140] if If this is not true, then m = m + 1, and continue executing step four until... Established.
[0141] In summary, the above are merely preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.
Claims
1. A trajectory tracking control method for unmanned systems without requiring an initial stabilization learning strategy, characterized in that, Includes the following steps: Step 1: Establish an unmanned system model; Step 2: Design the trajectory tracking controller for the follower and the corresponding cost function to achieve optimal trajectory tracking of the follower; Step 3: Based on the augmented system of the unmanned system, introduce an offset factor to construct a new closed-loop system, specifically as follows: Introducing an offset factor based on the augmented system of unmanned systems. This allows initial learning to be independent of the initial stabilization learning strategy, defining a new learning matrix. for: in, Top bid It is the number of iterations. This represents the learning matrix corresponding to the m-th iteration. , These are arbitrarily chosen variables. Representation matrix eigenvalues, Re denotes the matrix The real part of the eigenvalues, s denotes the iteration sign, and m denotes the total number of iterations. This represents the accumulation factor corresponding to the iteration number s. This represents the identity matrix with dimension n×n; This represents the state coefficient matrix in the augmented system. This represents the control input coefficient matrix of the augmented system. Represents the augmented gain matrix; Construct the following new closed-loop system in x To augment the state of the system, To augment the first derivative of the system state, Top bid It is the number of iterations. This represents the identity matrix with dimension n×n. This represents the augmented gain matrix corresponding to the m-th iteration; Design value function for the new closed-loop system in, The matrix represents the iterative learning process, where the initial iteration values are arbitrarily chosen positive definite matrices. Matrix representing iterative learning The corresponding value function; Step 4: Generate system data based on the new closed-loop system, collect the system data, and use the strategy iterative learning algorithm to find the optimal control strategy to achieve model-free trajectory tracking control that does not depend on the initial stabilization control strategy.
2. The trajectory tracking control method for unmanned systems without an initial stabilization learning strategy as described in claim 1, characterized in that, Step one: Establishing an unmanned system model; specifically: The dynamic model of the unmanned system is established as follows: Where z represents the state of the unmanned system. Denotes the first derivative of z. The state coefficient matrix representing the unmanned coefficients, This represents the control input coefficient matrix of the unmanned system. This represents the output coefficient matrix of the unmanned system; y is the output of the unmanned system. Represents the control input of an unmanned system; matrix and All unknown; The unmanned system includes a navigator and followers; the dynamic model of the navigator is as follows: in This indicates the status of the Navigator unmanned system. express The first derivative, Represents the state coefficient matrix of the Navigator unmanned system. This represents the control input coefficient matrix of the Navigator unmanned system. This represents the output coefficient matrix of the Navigator unmanned system; For the output of the Navigator unmanned system; matrix A and B All are unknown.
3. The trajectory tracking control method for unmanned systems without an initial stabilization learning strategy as described in claim 2, characterized in that, Step two: Design the trajectory tracking controller for the follower and the corresponding cost function to achieve optimal trajectory tracking of the follower, specifically as follows: Based on the dynamics model of the unmanned system, the dynamics model of the follower, and the controller Constructing an augmenting system: in x Indicates the state of the augmented system. express x The first derivative, Indicates the control input of the unmanned system. e This indicates the output trajectory tracking error. This represents the output coefficient matrix of the augmented system; The designed follower trajectory tracking controller is specifically as follows: in: This refers to the controller of an unmanned system. This indicates the feedback gain of the controller. This represents the feedforward gain of the controller. Represents the augmented gain matrix. ; Cost function for in: This indicates that the tracking error of the unmanned system corresponds to a known matrix. , Represents an arbitrarily chosen known matrix. This represents any known matrix corresponding to the control input of the unmanned system. s For integration variables, t For a moment.
4. The trajectory tracking control method for unmanned systems without an initial stabilization learning strategy as described in claim 1, 2, or 3, characterized in that, The process of generating system data based on the new closed-loop system and collecting the system data specifically includes: In time interval Collect system data, where d represents a constant greater than zero and arbitrarily set; generate system data based on the new closed-loop system. in: Represents a column vector. , Representing states respectively Element; , These represent the first element of the first row in matrix P. first and second row elements and the element in the nth row and nth column ,symbol Indicates the Kronecker product; Representation matrix Row vectorization; This represents the known matrix corresponding to the tracking error of the unmanned system. The system collects data and stores it in four data storage units, which are respectively... , , , : Within the time interval [ t 0 ,t l The system collects data generated by the internal collection system and divides the interval into several sub-intervals with a certain sampling step size. t 0 represents the initial time when the system starts collecting data. t 1 represents the initial data collection time of the system. t The time after the sampling step size is 0, and so on. t l The termination time for system data collection; Let a vector be a vector whose elements satisfy the following condition: ,in Representing column vectors respectively exist The column vector corresponding to time step 1. Indicates the interval Column vectors; "Don't indicate that the corresponding follower is in the sub-interval" t 0 ,t 1], [ t 1 ,t 2],...,[ t l-1 ,t l System data collected within the system; These represent the followers in the sub-intervals [ t 0 , t 1], [ t 1 ,t 2],...,[ t l-1 ,t l The system collects and associates input data. The following equivalence relation is obtained. in: Represents a row vector. , In The controller gain matrix representing the follower The numerical value learned in the (m+1)th iteration.
5. The trajectory tracking control method for unmanned systems without an initial stabilization learning strategy as described in claim 4, characterized in that, The process of finding the optimal control strategy using a policy iterative learning algorithm involves the following steps: S401: Initial iteration value m=0 and known constants , + , Since it is an arbitrarily chosen variable, This is the accumulation factor corresponding to iteration number 0; Apply it to the system to collect data so that it satisfies the following rank criterion: in, Represents the state of the augmented system dimensionality Represents the state of the augmented system dimensionality; S402: Solve the following equations to obtain the matrix. in, These represent the column vectors corresponding to the initial iteration m=0. ; Let P represent the matrix corresponding to the initial iteration m=0. This represents the augmented system matrix gain corresponding to m=1 in the first iteration. If the matrix Since it is a non-positive definite matrix, continue solving the above equation. And so on, until the matrix is judged. It is a positive definite matrix; S403: Here is the Herwitz matrix, and the coefficients are... The following equation is guaranteed to converge. For matrix The largest eigenvalue; ； S404: By solving the following equation get ; Represents the controller gain matrix The numerical value learned in the (m+1)th iteration; vec(P) is a column vector consisting of distinct elements in matrix P; S405: If Established, among which If the selected value is any value greater than zero, the iteration stops, and the control gain is then [value missing]. The controller is ; if If this is not true, then m = m + 1, and continue executing step four until... Established.