Nonlinear optimization method for stochastic predictive control

Abstract

A predictive controller controls a system under uncertainty subject to constraints on states and control variables of the system. At each control step, the predictive controller solves a nonlinear dynamic optimization problem with inequality constraints to produce control commands, and controls operation of the system using the control commands, the dynamic optimization problem including a probabilistic chance constraint representing the uncertainty. The predictive controller solves the dynamic optimization problem based on a two-level optimization that alternates propagation of a covariance matrix of the probabilistic chance constraint within a prediction horizon for fixed values of the states and control variables with optimization of the states and control variables for fixed values of the covariance matrix within the prediction horizon until a termination condition is satisfied.

Description

Technical Field

[0001] This invention relates generally to predictive control, and more specifically, to an optimization method and apparatus based on inaccurate derivatives for stochastic predictive control of nonlinear dynamic systems in the presence of uncertainty. Background Technology

[0002] Nonlinear model predictive control (NMPC) has matured and demonstrated its ability to handle relatively complex constrained processes. Predictive controllers such as NMPC are used in many applications to control complex dynamic systems described by a set of nonlinear differential equations, namely systems of ordinary differential equations (ODEs), differential algebraic equations (DAEs), or partial differential equations (PDEs). Examples of such systems include production lines, vehicles, satellites, engines, robots, generators, and other numerically controlled machines.

[0003] While NMPC exhibits inherent robustness due to feedback, this controller does not directly account for uncertainty and therefore cannot guarantee compliance with safety-critical constraints in the presence of model uncertainties or external disturbances. An alternative approach is robust NMPC, which relies on optimizing the control strategy for worst-case scenarios in the presence of bounded uncertainties. However, because the probability of the worst-case scenario occurring is extremely low, robust NMPC may result in conservative control performance.

[0004] Stochastic NMPC aims to reduce the conservatism of robust NMPC by directly incorporating a probabilistic description of uncertainty into the formulation of the optimal control problem (OCP). It requires that constraints be satisfied with a certain probability; that is, by formulating so-called chance constraints, it allows for a specific, but not zero, probability of constraint violation. Furthermore, stochastic NMPC is advantageous in achieving high-performance closed-loop operation near the limits of the plant's feasible region. In general, chance constraints are difficult to compute and often require approximate formulations.

[0005] Sampling techniques use a finite set of stochastic implementations of uncertainties to characterize stochastic system dynamics, which can lead to considerable computational costs because uncertainty propagation typically requires a large number of samples. Scenario-based methods utilize adequate representations of probability distributions, but the task of determining the number of scenarios results in a trade-off between robustness and computational efficiency. Gaussian mixture approximations can be used to describe the transitional probability distributions of states, but the adaptation of weights is often computationally expensive. Another approach relies on the use of polynomial chaos (PC), which replaces implicit mappings with expansions of orthogonal polynomial basis functions; however, for time-varying uncertainties, PC-based stochastic NMPC requires many expansion terms. Therefore, in stochastic predictive control of nonlinear system dynamics, a direct but approximate propagation of uncertainties is needed to formulate probabilistic chance constraints.

[0006] Direct optimal control (OCP) methods, based on the discretization of the control domain and the corresponding parameterization of control actions in the prediction domain, rely on the discretization of continuous-time differential equations. Furthermore, for stochastic predictive control applications, the direct OCP formula can include discrete-time or discretized equations to propagate the uncertainty of the nonlinear system dynamics based on the parameterization of control feedback within the prediction domain. The resulting large-scale nonlinear optimization problem, or nonlinear programming (NLP), can be solved by any nonlinear optimization solver. However, in real-time applications of predictive control for nonlinear systems, this nonlinear optimization problem needs to be solved under strict time constraints on embedded hardware with limited computational power and available memory.

[0007] For stochastic predictive control of systems described by nonlinear differential equations, the nonlinear stochastic optimization control problem needs to be solved at each control time step. Instead of completely solving each problem, a sequential quadratic programming (SQP) method can be used for one real-time iteration to update the guess of the solution from one time point to the next. This Newton-type SQP algorithm requires linearizing the discrete nonlinear dynamics in each iteration. This linearization can be costly, especially for the equations describing the uncertainty propagation of the nonlinear system dynamics, and requires Jacobian estimation when using explicit integration methods, and may also require iterative processes of matrix decomposition, matrix-matrix multiplication, and / or implicit integration methods to solve the nonlinear equations.

[0008] Therefore, it is necessary to reduce the computational cost of SQP solvers in real-time applications of stochastic predictive control for nonlinear dynamic systems with uncertainties. Summary of the Invention

[0009] One implementation aims to provide a system and method for controlling a system under uncertainty by solving an inequality-constrained nonlinear dynamics optimization problem, based on the discretization of the nonlinear differential equations describing the system's dynamics model and the discrete-time propagation of uncertainties in the nonlinear system dynamics. Each probabilistic constraint is designed to ensure that the probability of violating the corresponding inequality constraint is below a certain probability threshold.

[0010] Some embodiments of the present invention use a probabilistic chance constraint formula based on a contraction of each inequality constraint with terms depending on the backoff coefficient value, the constraint Jacobian matrix, and the covariance matrix for the predicted state value at a specific time step. Some embodiments of the present invention are based on the understanding that the covariance matrix can be efficiently computed for the state value at each time step within the control domain using covariance propagation based on approximate linearization. The covariance equation can be discretized based on linearized uncertainty propagation of continuous-time nonlinear system dynamics. Alternatively, discrete-time covariance propagation can be accomplished directly based on the linearization of the discretized nonlinear dynamic equations.

[0011] Some implementations are based on the understanding that discrete-time covariance propagation equations can reduce computational costs while maintaining the positive definiteness of the covariance matrix at each control time step. Some implementations may include nonlinear constraints in the linearized covariance propagation to ensure that the covariance matrix is ​​an overestimate of the exact covariance of the predicted state value at each time step, thereby ensuring that each probabilistic chance constraint is violated below a certain threshold.

[0012] Some embodiments of the present invention are based on the understanding that feedback control actions should be considered in the forward propagation of uncertainties regarding predicted state values. Some embodiments use time-invariant or time-varying sequences of affine feedback gains to pre-stabilize the dynamics of nonlinear systems, thereby generating covariance propagation equations that directly consider the impact of feedback control actions on future uncertainties. For example, an infinite-domain linear quadratic regulator of the linearized system dynamics under a reference steady state and input values ​​can be used to pre-stabilize the system dynamics in a stochastic nonlinear OCP formula.

[0013] Using approximate formulas with probabilistic chance constraints, and based on individual contraction of each inequality constraint, the resulting nonlinear dynamic optimization problem with inequality constraints can be solved using a Newton-type optimization algorithm based on continuous linearization of optimality and feasibility conditions. Examples of such Newton-type optimization algorithms include the interior-point method (IPM) and sequential quadratic programming (SQP). Some embodiments of the present invention are based on the understanding that the SQP algorithm approximates the quadratic programming (QP) of the stochastic nonlinear OCP in each iteration of the SQP optimization algorithm, based on linearized approximations of the objective function, the discrete system dynamics and discrete-time covariance propagation equations, and linearized approximations of each inequality constraint and each contracted probabilistic chance constraint.

[0014] When the initial dynamic model of a system is described by a set of continuous-time differential equations, some embodiments of the present invention use explicit or implicit numerical integration methods to discretize the system dynamics, and linearization requires corresponding Jacobi estimates to construct the discrete-time or discretized covariance propagation equations. Some embodiments are based on the understanding that, in each iteration of a Newton-type optimization algorithm, linearizing the covariance propagation equations requires evaluating higher-order derivatives of the nonlinear system dynamics, which constitutes a computationally expensive step, especially when the dynamics are high-dimensional, involve lengthy nonlinear expressions, or are described by a set of rigid or implicitly defined differential equations.

[0015] Furthermore, some embodiments of the present invention are based on the understanding that solving precise linearization-based optimization algorithms for stochastic nonlinear predictive control involves significantly greater computational complexity and memory requirements than the nominal implementation that does not directly consider uncertainties. More specifically, to solve the block-structured QP approximation in nominal NMPC, the memory requirement asymptotically expands to O(Nm). 2 ), where N represents the length of the control domain, and m represents the number of states and control variables at each time step in the control domain. Furthermore, when solving the block-structured QP in nominal NMPC, the computational complexity asymptotically expands to O(N m). 3 Therefore, due to the m×m covariance matrix of the predicted state values ​​at each time step and the corresponding equations, the memory requirement and computational complexity of the accurate linearization-based optimization algorithm for stochastic nonlinear predictive control are O(Nm). 4 ) and O(N m 6 The approach is to expand incrementally. Note that, based on the understanding that the covariance propagation dynamics are linear with respect to the covariance, and that the matrix is ​​defined by the Kronk product of the constrained Jacobian matrix, the memory requirement of stochastic NMPC can be reduced to O(N m). 3 ).

[0016] Some embodiments of the present invention propose an optimization algorithm based on inaccurate derivatives for stochastic nonlinear predictive control. For this optimization algorithm, the memory requirement and computational complexity are respectively O(Nm). 2 ) and O(N m 3The invention expands incrementally in a manner that allows for the numerical elimination of the covariance matrix of each QP subproblem while preserving the sparsity of the block-structured problem, thus significantly reducing memory requirements and computational complexity. The inexact linearization-based optimization algorithm does not compute any derivatives of the covariance propagation equations with respect to the state or control variables, therefore it does not require any higher-order derivatives of the system dynamics. Furthermore, each QP subproblem in the inexact SQP algorithm only includes the state and control variables within the control domain, and the covariance matrix can be explicitly computed in a separate propagation procedure.

[0017] In some embodiments of the present invention, the proposed inexact optimization algorithm for stochastic nonlinear predictive control comprises three main computational steps. The first step prepares a linear quadratic objective function, computes the Jacobian matrix to prepare linearized equality and inequality constraints, and computes the trajectory of the covariance matrix by evaluating the nonlinear covariance propagation equation for a given trajectory of the predicted state and control values. This covariance matrix represents the uncertainty of the predicted state values ​​in the control domain. Therefore, unlike standard optimization algorithms, the proposed algorithm maintains the positive definiteness of the covariance matrix at each time step in each iteration of the inexact SQP algorithm. The second step involves solving the resulting block-structured QP subproblems, approximating each chance constraint with one or more contracted inequality constraints. The final third step involves a Newton-type update of the trajectories of the predicted state and control values.

[0018] Some embodiments of the present invention are based on the understanding that inexact linearization-based optimization algorithms converge to solutions of stochastic nonlinear OCPs that are feasible for system dynamics, covariance propagation equations, inequality constraints, and probabilistic chance constraints, but may be suboptimal due to inexact derivative calculations. Conversely, some embodiments of the present invention are based on inexact linearization-based optimization algorithms that utilize adjoint gradient calculations, which converge to solutions of both feasible and optimal stochastic nonlinear OCPs. Note that the adjoint calculation of the covariance propagation equations requires estimating higher-order derivatives of the system dynamics, but corresponds to a single gradient rather than the complete Jacobian matrix. The latter adjoint gradient calculation can be efficiently performed using algorithms or a single sweep of the adjoint pattern of automatic differentiation.

[0019] In some embodiments of the present invention, the proposed inexact optimization algorithm for stochastic nonlinear predictive control, utilizing adjoint gradient computation, comprises three main computational steps. The first step prepares a linear quadratic objective function, computes the Jacobian matrix with respect to the state and control variables to prepare linearized equality and inequality constraints, computes an adjoint gradient-based evaluation, and numerically eliminates the covariance matrix from each objective and constraint function given the current trajectory of the predicted state and control values ​​and the corresponding covariance matrix. The second step involves solving the resulting block-structured QP subproblems, approximating each chance constraint with one or more contracted inequality constraints. The final third step involves performing a Newton-type update on the trajectory of the predicted state and control values, along with a corresponding update to the Lagrange multipliers, and updating the trajectory of the covariance matrix within the control domain.

[0020] Some embodiments of the present invention are based on the understanding that, in addition to the covariance matrix, state variables in stochastic optimal control problems can also be numerically eliminated in each iteration based on a compression procedure that uses discrete-time system dynamics to define the state variables for each stage in the prediction domain as a function of the initial state values ​​and all control variables of the previous stages within the prediction domain. This full or partial compression procedure results in a smaller but typically denser optimization problem with fewer or no equality constraints and the same number of inequality and probabilistic chance constraints, described by the remaining optimization variables in the OCP. Some embodiments of the present invention are based on the understanding that the same inexact linearization-based optimization algorithm can be used in conjunction with such a compression procedure. More specifically, the numerical elimination of state variables is performed additionally in the first step, the dense QP solution is obtained in the second step, and the expansion of the compressed state variables is performed additionally in the third step of the inexact optimization algorithm for stochastic nonlinear predictive control.

[0021] Some embodiments of the present invention solve the nonlinear stochastic OCP using a real-time iterative method by performing one iteration of the proposed inexact SQP method at each control step of the predictive controller. This means that at each control step, only one preparation, solution, and expansion step is required for a block-structured local QP approximation of the nonlinear stochastic optimization problem. The QP preparation includes: linearization of the nonlinear equations, which impose discrete nonlinear system dynamics; linearization of nonlinear inequality constraints; compression or elimination of the covariance matrix; and optional calculation of the adjoint gradient. Based on this preparation, the resulting block-structured QP is solved, followed by an expansion step to update all original optimization variables and Lagrange multiplier values, so that a control solution for the control system is generated at each step of the predictive controller.

[0022] Therefore, one embodiment discloses a predictive controller for controlling a system under uncertainty, the uncertainty being constrained by the system's state and control variables. The predictive controller includes: at least one processor; and a memory storing instructions that, when executed by the at least one processor, cause the predictive controller to: at each control step of the predictive controller, solve a nonlinear dynamic optimization problem with inequality constraints to generate a control command, the dynamic optimization problem including probabilistic chance constraints representing the uncertainty, wherein the predictive controller solves the dynamic optimization problem based on two-level optimization, the two-level optimization alternating between the propagation of the covariance matrix of the probabilistic chance constraints for fixed values ​​of the state and control variables within the prediction domain and the optimization of the state and control variables for fixed values ​​of the covariance matrix within the prediction domain until a termination condition is met; and use the control command to control the operation of the system.

[0023] Another embodiment discloses a predictive control method for controlling a system under uncertainty, the uncertainty being constrained by the state and control variables of the system, wherein the method uses a processor coupled with stored instructions for implementing the method, wherein the instructions, when executed by the processor, implement the steps of the method, the method comprising: at each control step of the predictive control method, solving a nonlinear dynamic optimization problem of inequality constraints to generate a control command, the dynamic optimization problem including probabilistic chance constraints representing the uncertainty, wherein the predictive controller solves the dynamic optimization problem based on two-level optimization, the two-level optimization alternating between propagation of the covariance matrix of the probabilistic chance constraints for fixed values ​​of the state and control variables within the prediction domain and optimization of the state and control variables for fixed values ​​of the covariance matrix within the prediction domain until a termination condition is met; and using the control command to control the operation of the system.

[0024] Another embodiment discloses a non-transitory computer-readable storage medium having embodied a program executable by a processor for performing a predictive control method to control a system under uncertainty, the uncertainty being constrained by the state and control variables of the system. The method includes: at each control step of the predictive control method, solving a nonlinear dynamic optimization problem with inequality constraints to generate a control command, the dynamic optimization problem including probabilistic chance constraints representing the uncertainty, wherein the predictive controller solves the dynamic optimization problem based on two-level optimization, the two-level optimization alternating between the propagation of the covariance matrix of the probabilistic chance constraints for fixed values ​​of the state and control variables within the prediction domain and the optimization of the state and control variables for fixed values ​​of the covariance matrix within the prediction domain until a termination condition is met; and using the control command to control the operation of the system. Attached Figure Description

[0025] [ Figure 1A ] Figure 1A It is a block diagram of a predictive controller and feedback loop for a system with uncertainty, according to some implementation methods;

[0026] [ Figure 1B ] Figure 1B It is a block diagram of a stochastic predictive controller and feedback loop for a system with uncertainty, according to some implementation methods.

[0027] [ Figure 2A ] Figure 2A This is a block diagram of a controller and feedback system according to some embodiments of the present invention, wherein the controller uses a CPU processor and memory;

[0028] [ Figure 2B ] Figure 2B This is a block diagram of a two-stage optimization process in a stochastic predictive controller according to some implementations, in which the propagation of the covariance matrix alternates with the optimization of the state and control variables.

[0029] [ Figure 3A ] Figure 3A It is a block diagram for implementing a stochastic nonlinear model predictive control (SNMPC) method for a controller according to some implementations;

[0030] [ Figure 3B ] Figure 3B This is a block diagram of the SNMPC method based on some implementation methods. This method solves the nonlinear programming (NLP) of direct optimal control structure based on discrete-time system dynamics and covariance propagation equations.

[0031] [ Figure 4A ] Figure 4AIt is an approximate block diagram of the probabilistic chance constraint in a stochastic predictive controller according to some implementations;

[0032] [ Figure 4B ] Figure 4B This illustrates the rationale behind the formulation and approximation of probabilistic chance constraints in stochastic predictive controllers.

[0033] [ Figure 5A ] Figure 5A This is a block diagram illustrating the propagation of state covariance based on linearization over a continuous time, according to some implementation methods.

[0034] [ Figure 5B ] Figure 5B This is a block diagram illustrating the propagation of state covariance based on linearization in discrete time according to some implementation methods;

[0035] [ Figure 5C ] Figure 5C This is a block diagram illustrating linearized state covariance propagation for the dynamics of a pre-stabilized nonlinear system in discrete time, according to some implementation methods.

[0036] [ Figure 5D ] Figure 5D According to some embodiments of the present invention, due to state feedback control actions, the formula for control limits and the approximation are block diagrams of probabilistic chance constraints.

[0037] [ Figure 6A ] Figure 6A It is a block diagram of an optimization procedure based on iterative derivatives, used to solve constrained nonlinear optimal control problems at each time step in a stochastic predictive controller;

[0038] [ Figure 6B ] Figure 6B This shows a compact formula for optimal control structured NLP that needs to be solved by a stochastic predictive controller;

[0039] [ Figure 6C ] Figure 6C This is a block diagram illustrating an accurate Jacobi-based local quadratic programming (QP) approximation of the optimal control structured NLP in a stochastic predictive controller, according to some embodiments of the present invention.

[0040] [ Figure 7A ] Figure 7A It is a block diagram of explicit and sequential calculation of the state covariance matrix value given the current state and control value in the prediction time domain, according to some implementation methods.

[0041] [ Figure 7B ] Figure 7BThis is a block diagram of an iterative inexact SQP optimization algorithm for effectively implementing stochastic predictive controllers without estimating higher-order derivatives of nonlinear system dynamics.

[0042] [ Figure 8A ] Figure 8A It is a block diagram for implementing the Jacobian matrix approximation and the corresponding adjoint gradient correction and Lagrange multiplier extension steps of the adjoint inexact SQP optimization algorithm.

[0043] [ Figure 8B ] Figure 8B It is a block diagram for improving the convergence characteristics of an iterative inexact SQP optimization algorithm for effectively implementing stochastic predictive controllers based on adjoint gradient correction.

[0044] [ Figure 8C ] Figure 8C According to some embodiments of the present invention, an algorithmic description based on a real-time variant of the accompanying inaccurate SQP optimization algorithm is provided for implementing stochastic nonlinear model predictive control.

[0045] [ Figure 9A ] Figure 9A It is a flowchart of the block structured sparsity of the constraint Jacobian matrix in forward recursion to compute compressed inequality constraint values ​​in the adjoint SQP optimization algorithm.

[0046] [ Figure 9B ] Figure 9B It is a flowchart of the process of calculating the updated Lagrange multiplier values ​​in the adjoint SQP optimization algorithm by utilizing the block-structured sparsity of the constrained Jacobian matrix in backward recursion.

[0047] [ Figure 10A ] Figure 10A It is a vehicle schematic diagram including a controller employing the principles of some implementation methods; and

[0048] [ Figure 10B ] Figure 10B This is a schematic diagram illustrating the interaction between a controller employing the principles of some implementation methods and a controller of vehicle 1001; and

[0049] [ Figure 10C ] Figure 10C This is a schematic diagram of a motion planning and / or predictive control method for a controlled vehicle that employs the principles of some implementation methods.

[0050] [ Figure 10D ] Figure 10D It is a schematic diagram of a motion planning and / or predictive control method for a controlled vehicle that employs the principles of some implementation methods. Detailed Implementation

[0051] Some embodiments of the present invention provide a system and method for controlling the operation of a system with uncertainty or a system using a stochastic predictive controller. One embodiment of the stochastic predictive controller is stochastic model predictive control (SMPC), which determines the control input based on a controlled system model and an uncertainty model.

[0052] Figure 1A An example system 120 with uncertainty 125 is illustrated according to some embodiments, connected to a predictive controller 110 via a state estimator 131. In some embodiments, the predictive controller is a model predictive controller (MPC) planned according to a dynamic model 140 of the system. This model may be a set of equations representing the state and output 103 of system 120 as a function of current and previous inputs 111 and previous output 103. The model may include constraints 142 representing physical and operational limitations of the system. During operation, the controller receives a command 101 instructing the desired behavior of the system. This command may be, for example, a motion command. In response to receiving the command 101, the controller generates a control signal 111, which serves as an input to the real system 120 with uncertainty 125. In response to this input, the system updates the system output 103. Based on measurements of the system output 103, the estimator updates the estimated state 121 of the system. The estimated state 121 of the system provides state feedback to the controller 110.

[0053] The system 120 referred to herein can be any machine or device that controls and returns controlled output signals 103 (outputs) that may be related to physical quantities (such as voltage, pressure, force, torque) via manipulating input signals 111 (inputs). These output signals may be related to physical quantities (such as current, flow rate, speed, or a position indicating the transition of the system's state from a previous state to the current state). The output value is partially related to the system's previous output value and partially to the previous and current input values. The dependence on the previous input and output is encoded in the system state. The operation of the system (e.g., the movement of system components) may include a sequence of output values ​​generated by the system after the application of certain input values.

[0054] Uncertainty 125 can be any time-varying uncertainty, including any external disturbance, force or torque acting on system 120, any unmodeled dynamics, or uncertainty in any physical quantity, such as an uncertain coefficient of friction, the mass of an object, or uncertainty factors and parameters in the dynamic model equations describing the physical behavior of the real system 120. Most implementations of MPC controllers use simplified dynamic models 140, resulting in a large amount of physical behavior in the real system remaining unmodeled to reduce the computational complexity of the controller, or because some physical behaviors are too complex to be modeled. Note that time-invariant uncertainties can be estimated or learned (whether online or offline) as part of the state and parameter estimator 131.

[0055] The system's dynamic model 140 may include a set of mathematical equations describing how the system output changes over time as a function of current and previous inputs and outputs. The system's state is any set of information, generally time-varying, such as an appropriate subset of current and previous inputs and outputs, which, together with the system model and future inputs, can uniquely define the system's future motion. The real system 120 may be subject to physical constraints and gauge constraints 142, which limit the range of possible system states that are allowed to operate.

[0056] The controller 110 may be implemented in hardware or as a software program that executes in a processor (e.g., a microprocessor) that receives estimated states and desired motion commands 101 from the system 121 at fixed or variable control cycle sampling intervals and uses this information to determine inputs (e.g., control signals 111) for the operating system.

[0057] The estimator 131 may be implemented in hardware or as a software program executed in a processor (which may be the same as or different from the controller 110), which receives the system output 103 at a fixed or variable control cycle sampling interval and uses new and previous output measurements to determine the estimated state 121 of the system 120.

[0058] Figure 1BAn example system 120 with uncertainty 125 is illustrated according to some embodiments, the system being connected to a stochastic predictive controller 150 via a state estimator 131 and an uncertainty estimator 132. In some embodiments, the stochastic predictive controller is a stochastic model predictive controller (SMPC) planned according to a dynamic model 140 of the system and the uncertainty. This dynamic model includes an uncertainty model 141 to model the uncertainty 125 and its relationship to system behavior. The uncertainty model includes a model of the linear and / or nonlinear relationship between the uncertainty and the dynamic model equations describing the system's dynamic behavior. Furthermore, the uncertainty model includes a probability distribution model for each time-varying uncertainty in the dynamic model.

[0059] In some embodiments of the invention, the dynamic model 140 of the stochastic predictive controller 150 may include one or more probabilistic chance constraints 143. Any physical limitations and specification constraints of the system may be formulated as one or more probabilistic chance constraints 143, which are designed to force the probability of violating the corresponding constraints to be below a certain probability threshold.

[0060] In some embodiments of the invention, uncertainty estimator 132 provides estimate 122, which is, for example, an estimate of the first and higher-order moments of the probability distribution of one or more uncertainties in a dynamic model 140 used by a stochastic predictive controller 150. In some embodiments of the invention, state estimator 131 and uncertainty estimator 132 are implemented together in an estimator assembly 130 that receives system output 103 at sampling intervals of a fixed or variable control period and determines uncertainty 125 and the uncertainties of estimated state 121 and estimate 122 of system 120 using new and previous output measurements.

[0061] Figure 2A A block diagram of a stochastic predictive controller 150 according to some embodiments is shown, which actuates a system such that the estimated state 121 and output 103 of the system follow a command 101 given an uncertainty of an estimate 122. The stochastic predictive controller 150 includes a computer, for example, in the form of a single central processing unit (CPU) or multiple CPU processors 201 connected to a memory 202 for storing a dynamic model 140, an uncertainty model 141, constraints 142, and probabilistic chance constraints 143 regarding the operation of a real system 120 with an uncertainty 125.

[0062] Figure 2BA block diagram of a two-stage optimization procedure is shown for solving a nonlinear dynamics optimization problem involving inequality constraints at each control step of a stochastic predictive controller 150 to generate a control command. The nonlinear dynamics optimization problem includes probabilistic chance constraints representing uncertainty. The two-stage optimization procedure alternates between propagation of the covariance matrix 250 of the probabilistic chance constraints 265 for fixed values ​​265 of the state and control variables within the prediction domain and optimization 260 of the state and control variables for fixed values ​​255 of the state covariance matrix within the prediction domain. Once the termination condition of the two-stage optimization procedure is met, the control signal 111 is computed.

[0063] In some embodiments of the invention, the two-stage optimization procedure 150 includes three stages or computational steps. The first step prepares a linear quadratic objective function, computes the Jacobian matrix to prepare linearized equality and inequality constraints, and propagates a covariance matrix trajectory 250 by estimating a nonlinear covariance propagation equation for the current trajectory of the state and control values ​​265, the covariance matrix representing the uncertainty of the predicted state values ​​in the control domain. The second step involves solving the resulting block-structured QP subproblems with one or more contracted inequality constraints to approximate each chance constraint 270. The final third step involves a Newton-type update 275 to the current trajectory of the optimal state and control values.

[0064] In some embodiments of the invention, adjoint gradient calculation is used to correct for inaccurate Jacobian information in the stochastic predictive controller, resulting in a two-stage optimization procedure comprising three main computational steps. The first step is to prepare a linear quadratic objective function, calculating the Jacobian matrix with respect to the state and control variables to prepare linearized equality and inequality constraints, the calculation of which is based on the adjoint gradient estimate, and numerically eliminating the covariance matrix from each objective and constraint function by propagating the trajectory of the covariance matrix 250 for the current trajectory of the predicted state and control values ​​265. The second step involves solving the resulting block-structured QP subproblems with one or more contracted inequality constraints to approximate each chance constraint 270. The final third step involves a Newtonian update 275 for the trajectory of the optimal state and control values, and a corresponding update to the Lagrange multiplier.

[0065] Figure 3AA block diagram of a system and method for implementing stochastic nonlinear model predictive control (SNMPC) 150 according to some embodiments of the present invention is shown. This stochastic predictive controller computes a control signal 111 given a current state estimate 121, an uncertainty estimate 122, and a control command 101 for the system. Specifically, the SNMPC computes a control solution (e.g., a solution vector 365) by solving a constrained optimization problem 350 at each control time step. This control solution includes a series of future optimal or near-optimal control inputs within the prediction time domain of the system 360. The objective function, equality and inequality constraints 345 in the optimization problem 350 depend on the dynamic model and system constraints 340, the current state estimate of system 121, the estimated uncertainty 122, and the control command 101.

[0066] Embodiments of the present invention employ a direct optimal control method to formulate the continuous-time SNMPC problem as a nonlinear dynamic optimization problem with inequality constraints. Some embodiments of the present invention use derivative-based optimization algorithms to solve the inequality-constrained optimization problem 350 precisely or approximately using an iterative process. This iterative process is based on a Newtonian-type method for the optimization problem and continuous linearization of feasibility and optimization conditions. Examples of such Newtonian-type optimization algorithms include interior-point method (IPM) and sequential quadratic programming (SQP). Some embodiments of the present invention are based on the understanding that the inequality-constrained optimization problem 350 has the form of an optimal control structure optimization problem (OCP), such that the structure implemented using derivative-based optimization algorithms can be used to compute the solution vector 365 at each control time step.

[0067] In some embodiments of the invention, the solution to the inequality-constrained optimization problem 350 uses precise or approximate state and / or control values ​​in the prediction time domain, which can be read from memory from the previous control time step 310, as a solution guess, in order to reduce the computational workload of solving the inequality-constrained optimization problem 350 in the current control time step. This concept of calculating a solution guess from the solution information of the previous control time step 310 is called a warm start or hot start of the optimization algorithm, which can reduce the required computational workload of the SNMPC controller in some embodiments of the invention. Similarly, the corresponding solution vector 365 can be used to update and store the sequence of precise or approximate state and / or control values ​​for the next control time step 360.

[0068] Figure 3BA block diagram of an SNMPC controller is shown, which solves a constrained optimal control structured nonlinear programming problem (OCP-NLP) 350 to compute a control signal 111 at each control time step, given the current state estimate of system 121, the uncertainty 122 of the estimate, and the control command 101. OCP-NLP 350 includes state variables x = [x0, x1, ..., x...] in the prediction time domain. N The state covariance matrix variable P = [P0, P1, ..., P N ] and control input variables u = [u0, u1, ..., u N-1 ], which serves as the variable in the optimization problem that needs to be solved for each control time step.

[0069]

[0070]

[0071] Some implementations are based on initial state value constraints. The current state estimate 121 in the system leads to a system dynamics model with linear and / or nonlinear equality constraints 352. Through system dynamics, this leads to an approximation of linearized uncertainty propagation based on linearized uncertainty propagation for linear and / or nonlinear covariance propagation equations 353, control input bounds 354, and linear and / or nonlinear inequality constraints 355, as well as a linear quadratic or nonlinear objective function 351, each of which is defined in the prediction time domain within the optimal control structure optimization problem. The OCP-NLP data 345 for the objective function, equality, and inequality constraints in this optimization problem 350 depends on the dynamics model and system constraints 340, the current state of system 121, the estimated uncertainty 122, and the control command 101. Implementations of the OCP-NLP data 345 include objective functions (e.g., l(·) and m(·)) and constraint functions (e.g., f(·) and h). i (·)). Other embodiments of OCP-NLP data 345 include constraint vectors (e.g., u min and u max ) and matrices (e.g.) K and C k，i ).

[0072] In some embodiments of the present invention, nonlinear equality constraint 352

[0073] x k+1 =f(x) k u k +Kx k(0) Approximate the system dynamics with a discrete-time representation, which can be defined by a set of continuous-time differentials or a set of continuous-time differential algebraic equations. Examples of such discrete-time approximations of system dynamics include numerical simulation techniques, such as linear multistep methods, explicit or implicit Runge-Kutta methods, backward differential formulas, or finite element methods. When the original dynamic model of the system is described by a set of continuous-time differential equations, some embodiments of the invention use explicit or implicit numerical integration methods 352 to discretize the system dynamics; linearization requires a corresponding Jacobian evaluation to construct the discrete-time or discretized covariance propagation equations 353. In some embodiments of the invention, the initial state covariance matrix 356 is defined as... in This represents the state uncertainty corresponding to the current state estimate 121.

[0074] In some embodiments of the present invention, the nonlinear inequality constraint 355 can be defined by any nonlinear smooth function, including convex and / or nonconvex constraints. In embodiments of the present invention, one or more inequality constraints 355 can be defined as probabilistic chance constraints, designed to ensure that the probability of violating the corresponding inequality constraint is below a certain probability threshold, i.e., for probabilistic chance constraints, the backoff coefficient value α... i >0, for standard deterministic inequality constraints, α i =0. Note that deterministic inequality constraints are designed to ensure that the corresponding inequality constraints are satisfied for the expected values ​​of the trajectories of the state and control values.

[0075] Using an approximation of the probabilistic chance constraint 355, based on individual contraction of each inequality constraint, the resulting nonlinear dynamics optimization problem with inequality constraints can be solved using a Newton-type optimization algorithm based on continuous linearization of optimality and feasibility conditions. Examples of such a Newton-type optimization algorithm include the interior-point method (IPM) and sequential quadratic programming (SQP). Some embodiments of the invention are based on the understanding that the SQP algorithm, in each iteration of the SQP optimization algorithm, solves a quadratic programming (QP) approximation of the stochastic nonlinear OCP based on a linear quadratic equation approximation of the objective function and linearized approximations of the discrete system dynamics and discrete-time covariance propagation equations, as well as linearized approximations of each inequality constraint and each contracted probabilistic chance constraint. In some embodiments of the invention, the stage and / or terminal costs in the objective function 351 can be defined by any linear, linear quadratic, and / or nonlinear smooth function (including convex and / or nonconvex functions). The objective function 351 of the optimal control problem may include a cost term corresponding to each time point in the prediction time domain. In some implementations, the objective function includes a (non-linear) least squares-type penalty for the deviation of a certain output function of the system from a sequence of reference output values ​​at each time point in the prediction time domain, resulting in a reference-tracking-type formula for the cost function in the stochastic predictive controller 150.

[0076] Figure 4A A block diagram is shown illustrating how one or more deterministic inequality constraints 401 on one or more state and / or control input variables are formulated into probabilistic chance constraints 405 and their approximations 406 by tightening the corresponding constraint bounds 410 in the constrained OCP formula of the stochastic predictive controller. The probabilistic chance constraints aim to ensure that violations of inequality constraints h are addressed. i (x k u k The probability that ) ≤ 0 is lower than a certain probability threshold ∈ i That is, Pr(h) i (x k u k )>0)<∈ i Equivalently, the probability chance constraint aims to ensure that the probability h of satisfying the inequality constraint is... i (x k u k )≤0 is higher than a certain probability threshold 1-∈ i That is, Pr(h) i (x k u k )≤0)≥1-∈ i .

[0077] In some embodiments of the present invention, the formulation of one or more probability constraints is performed using constraint tightening procedure 410. To approximate the implementation of 406, this is based on the state covariance matrix P. k =cov(x k x k ), constrained Jacobian matrix and depends on the probability threshold ∈ i The backoff coefficient α of the probability distribution of uncertainty and the approximate probability distribution generated by the predicted state trajectory. i 420. The state covariance matrix P can be calculated using the linearized covariance propagation equation 353. k The constrained Jacobian matrix C can be efficiently estimated using symbolic differentiation or algorithmic differentiation (AD) tools. k，i In some embodiments of the invention, the backoff coefficient value α in each contracted inequality constraint 355 can be calculated using the Cantelli-Chebyshev inequality. i 420 (i.e., This holds true regardless of the underlying probability distribution, but it may lead to relatively conservative constraint tightening. Other embodiments of the invention are based on a less conservative approximation, assuming a normally distributed state trajectory, such that the backoff coefficient value can be chosen as... Among them erf -1 (·) represents the inverse Gaussian error function.

[0078] Figure 4B A sample set of trajectories 451 showing the variation of constraint function value 430 over time 435 is presented, each trajectory corresponding to a different uncertainty realization, and constraint bounds 450 are shown to illustrate the probabilistic chance constraint 405. The probabilistic chance constraint aims to ensure that the probability of violating inequality constraint 445 is below a certain probability threshold ∈ i That is, Pr(h) i (x k u k )>0)<∈ i Equivalently, the probability constraint aims to ensure that the probability of satisfying inequality constraint 440 is higher than 1-∈ i That is, Pr(h) i (x k u k )≤0)≥1-∈ i .

[0079] Sampling techniques use a finite set of randomized implementations of uncertainties to characterize the dynamics of stochastic systems, which can lead to considerable computational costs because the propagation of uncertainties typically requires a large number of samples. Scenario-based methods utilize sufficient representations of probability distributions, but the task of determining the number of scenarios results in a trade-off between robustness and computational efficiency. Therefore, embodiments of the present invention are based on the understanding that a direct but approximate propagation of uncertainties is needed to computationally represent the expression satisfying the inequality constraint h. i (x k u k The lower limit of 460 and / or the upper limit of 465 for a specific percentage of the trajectory required to ≤0 are used to formulate probabilistic chance constraints in a stochastic prediction controller. Figure 4B In the diagram, 455 represents a trajectory that violates the inequality constraint. Due to the use of probabilistic chance constraints in the stochastic predictive controller, the probability of this situation should be kept at a certain probability threshold ∈ i the following.

[0080] Figure 5A A block diagram is shown to illustrate a set of dynamics of a continuous-time nonlinear system given a time-varying disturbance w(t) 505. The propagation is based on linearized state covariance. In some embodiments of the invention, time-varying disturbances are modeled as a set of normally distributed random variables w(t) ~ N(0, ∑) over continuous time. c Therefore, the continuous-time state covariance propagation equation 510 is...

[0081]

[0082] Given a continuous-time constrained Jacobian matrix 515

[0083]

[0084] The direct optimal control method uses numerical integration to discretize the continuous-time state covariance propagation equation 511, thereby producing a numerical approximation of the continuous-time trajectory of the state covariance matrix. However, even if the initial state covariance matrix 512 is positive definite, i.e. The state covariance matrix of a series of numerical simulations may not necessarily maintain the positive definiteness of the state covariance matrix. Some embodiments of the present invention are based on the understanding that the latter leads to numerical problems in derivative-based optimization algorithms for stochastic predictive controllers. Figure 5B A block diagram is shown to illustrate the effect of a given time-varying disturbance w. k In the case of 525, a set of discrete-time nonlinear system dynamics 520x k+1 =f(x) k u k wk The linearized state covariance propagation is used. In some embodiments of the invention, time-varying disturbances are modeled as a set of discrete-time normally distributed random variables w. k ~N(0, ∑). Therefore, the discrete-time state covariance propagation equation 530 is...

[0085]

[0086] in It is the predicted state value x k The covariance matrix, and given the discrete-time constrained Jacobian matrix 535

[0087]

[0088] Unlike the continuous-time state covariance propagation equation 511, some embodiments of the present invention are based on the understanding that the discrete-time state covariance propagation equation 531 does not require numerical integration methods, provided that the initial state covariance matrix 532 is positive definite. They automatically maintain the positive definiteness of the state covariance matrix.

[0089] Figure 5C A block diagram is shown to illustrate the pre-stabilization of nonlinear system dynamics so that feedback control actions are considered as part of the state covariance propagation of a stochastic predictive controller. Some implementations are based on parameterization of the feedback control actions to formulate the pre-stabilized nonlinear system dynamics 560 given discrete-time dynamics 520 and time-varying disturbances 525.

[0090] In some embodiments of the present invention, a linear quadratic regulator is used to define the time-invariant affine feedback gain u. k =Kx k 555, Formulating the Dynamics of Prestabilized Nonlinear Systems 560

[0091] x k+1 =f(x) k u k +Kx k w k )

[0092] Among them, due to the pre-stabilized controller gain K, the overall control action is in the form of feedforward feedback u. k +Kx k For example, the reference steady state and the input value (x) ref u ref This can be used to define the reference constraint Jacobian matrix A. r 541 and B r 542

[0093]

[0094] With an infinite domain performance index of 545

[0095]

[0096] Combined to define the time-invariant affine feedback gain u k =Kx k 555. Some embodiments of the present invention solve the discrete-time algebraic Riccati equation (DARE). 550

[0097]

[0098] The time-invariant affine feedback gain is calculated as shown below 556.

[0099]

[0100] Where Q > 0 and R > 0 represent the weighting matrices in the infinite-domain performance index 545. Based on the pre-stabilized nonlinear system dynamics 560, and given the discrete-time constraint Jacobian matrix 575 of the pre-stabilized nonlinear system dynamics, the discrete-time state covariance propagation equation 570 is:

[0101]

[0102] Discrete-time constrained Jacobian matrix 575 is

[0103]

[0104] In some embodiments of the present invention, reference values The time-varying trajectory is used instead of the time-varying trajectory that defines the feedback control law, for example, the trajectory of the affine feedback gain based on the same linear-quadratic regulator design formula.

[0105] Figure 5D A block diagram is shown for the following operation: According to some implementations, it will be in the form of feedforward feedback u k +Kx k One or more deterministic inequality constraints 580 of one or more control actions are formulated as probabilistic chance constraints 585, and the probabilistic chance constraints 585 are approximated 406 by the corresponding constraint bounds 590 in the constraint OCP formula of the compressed stochastic predictive controller. In some embodiments of the invention, based on the state covariance matrix P k =cov(x k x k )415, Affine feedback gain matrix K 591 and depends on probability threshold ∈ iThe backoff coefficient α is the probability distribution of uncertainty and the approximate probability distribution of the predicted state trajectory. i 420, The formula for one or more probability constraints uses the constraint tightening procedure 595

[0106] Approximately implemented as 406. State covariance matrix P k The linearized discrete-time state covariance propagation equation 571 based on the dynamics of pre-stabilized nonlinear systems can be used for calculation, given...

[0107] Notice, Figure 5C and Figure 5D The feedback control actions in the figure are predictions of future feedback control actions in the prediction time domain within a control step in a dynamic optimization problem. Therefore, these feedback control actions should not be confused with the actual feedback of the estimated state of system 121 in Figure 1 to the predictive controller 110 in Figure 1.

[0108] Figure 6A A block diagram of a derivative-based iterative optimization procedure is shown, which solves a constrained, optimally controlled structured nonlinear programming (NLP) 350 for each control time step of a stochastic predictive controller using a continuous approximation 605 based on local linearization. The solution guess of NLP 601 is used to construct this local approximation, and the solution of the local approximation of the constrained NLP 610 is used to update the current sequence of state, control, and state covariance values ​​in the prediction time domain 615, resulting in an update of the current solution guess of the constrained NLP 601 in each iteration of the algorithm. Each iteration of the optimization procedure checks whether a solution to the constrained NLP has been found and / or whether the maximum number of iterations 607 has been reached. If the termination condition 607 is met, a control solution 365 has been found; otherwise, the constrained Jacobian matrix (its approximation) 620 is estimated to construct the local linearization-based approximation 605 in the next iteration of the optimization algorithm. The state and control values ​​from the previous control time step 310 can be used to form the initial solution guess and linearization point of the constrained NLP 601.

[0109] Based on the nonlinear objective and constraint function 345, and using the current solution guess as the linearization point 601 (including the trajectory of state, control, and state covariance values ​​in the prediction time domain), a local approximation of NLP 605 is constructed in each iteration of the algorithm program. For this, the constraint Jacobian matrix 620 needs to be computed or approximated to form the linearization of the discrete system with complex nonlinear system dynamics and / or nonlinear inequality constraints. If the locally approximated solution forms a sufficiently accurate solution 607 for the NLP, then the optimal control solution 365 is obtained. When the maximum number of iterations 607 is reached, a suboptimal and / or infeasible solution 365 is obtained instead. If a sufficiently accurate solution for the NLP has not yet been found and the maximum number of iterations 607 has not been reached, the locally approximated solution 610 is used to update the trajectory 615 of state, control, and state covariance values ​​in the prediction time domain, and the solution guess 601 of the NLP is updated.

[0110] Different types of optimization algorithms can be used to solve inequality-constrained, optimally controlled, structured, nonlinear programming (NLP) problems at each control time step using successive local approximations. Some implementations are based on sequential quadratic programming (SQP), where a quadratic program (QP) is constructed and solved in each iteration as a local approximation of the initial NLP. Conversely, some implementations are based on the interior point (IP) method, where each local approximation is a linearization of the first-order necessary condition for the optimality of the NLP, where the complementary conditions corresponding to the inequality constraints are typically smoothed. In some implementations, a barrier function is used to iteratively enforce the inequality constraints, and a local approximation for the barrier reconstruction problem is constructed and solved in each iteration.

[0111] In derivative-based optimization algorithms, different Newtonian approximation techniques can be used for the constraint Jacobian and Hessian matrices when constructing the local subproblems 605 and 610 in each iteration. Some implementations are based on the exact linearization of some or all constraint functions by calculating the exact constraint Jacobian matrix 620. Other implementations use quasi-Newtonian update formulas, iteratively updating the approximation of the constraint Jacobian matrix using low-rank update techniques. Similarly, different Newtonian approximation techniques can be used for the Lagrange-Hessian matrix in NLP. Some implementations are based on the estimation of the exact Lagrange-Hessian matrix when constructing each local approximation for NLP. Other implementations use quasi-Newtonian update formulas, iteratively updating the approximation of the Hessian matrix using symmetric low-rank update techniques. When the objective function of NLP includes a (nonlinear) least squares type cost term, some implementations are based on a Gaussian-Newtonian Hessian approximation.

[0112] Figure 6BA more compact NLP formula, 630, is shown, which is equivalent to 625 for the optimal control structured optimization problem 350.

[0113]

[0114]

[0115] This needs to be solved at each control time step in the stochastic predictive controller. The compact NLP formula 630 refers to the state and control variables in the prediction time domain as y 636, and the covariance matrix variable as z 637.

[0116]

[0117]

[0118] This allows the dynamics of discrete-time nonlinear systems and the covariance propagation equation based on discrete-time linearization to be defined as: 0 = F(y)632 and 0 = E(y,z)633, respectively.

[0119]

[0120]

[0121] Some embodiments of the present invention are based on the understanding that each state covariance matrix represents a symmetric matrix, and therefore can be expressed in vectorized form. Instead, z 637 is defined to reduce the computational complexity and memory requirements of the stochastic predictive controller. Furthermore, the compact NLP formula 630 may include one or more linear and / or nonlinear inequality constraints 634 and a linear-quadratic or nonlinear objective function 631, wherein the linear and / or nonlinear inequality constraints 634 include both deterministic and approximate probabilistic chance constraints. In some embodiments of the invention, the objective function is defined as a least squares function. 631, where the linear or nonlinear function L(y) can, for example, refer to the deviation of a certain output function of the system from a series of reference output values ​​at each time point in the prediction time domain. Figure 6C A block diagram is shown of an exact Jacobi-based quadratic program (QP) 640 based on some implementations of a sequential quadratic programming method for implementing a stochastic predictive controller, which forms a local approximation 605 of an optimal control structured NLP 630. The linear equality constraints 642 in the QP subproblems correspond to the linearization of the covariance propagation equation 633, which is an estimate of the discrete-time system dynamics 632 and the complete constraint Jacobi matrix 652. Furthermore, for the inequality constraints 634 in the initial NLP formula, local linearization 643 is required, which necessitates an estimate of the exact Jacobi matrix for each nonlinear inequality constraint.

[0122] In the optimal control structured QP 640, the linear quadratic objective 641 is locally approximated as the nonlinear objective 631. As mentioned earlier, the Hessian matrix H... i 651 can be based on an exact estimate of the Lagrange Hessé, or using a quasi-Newtonian update formula or a Gauss-Newton Hessé approximation for each interval of the prediction time domain. In some embodiments of the invention, the Gauss-Newton Hessé approximation is used for the nonlinear least squares objective function 631, as shown below.

[0123]

[0124] The Lagrange definition of NLP is as follows:

[0125]

[0126] And vector g i As shown below, the gradient of the nonlinear least squares objective function 631 is defined accordingly.

[0127]

[0128] Among them, y i and z i These represent the current values ​​of the state and control variables, as well as the state covariance matrix variable in the prediction time domain during the i-th iteration of the SQP optimization algorithm. In some embodiments of the invention, the objective function 631 additionally depends on one or more elements of the state covariance matrix variable in z 637, so that the Hessian and gradient estimates simultaneously depend on y. i and z i Some embodiments of the present invention are based on the understanding that the Hessian matrix 651, the equality constraint Jacobian matrix 652, and the inequality constraint Jacobian matrix 653 exhibit block-structured sparsity due to the staged coupling between the states of subsequent stages in the prediction time domain and the covariance matrix variables in the separable objective function 351, the staged single inequality constraints 354 to 355, and the equality constraints 352 to 353 of the constrained NLP 350. Therefore, in some embodiments of the present invention, the block-sparse structure can be used in the SQP optimization algorithm to solve each local QP approximation 640 of the optimal control structured NLP 630 to implement a stochastic predictive controller. Examples of block-sparse structures using QP optimization algorithms include primal, dual, or primal-dual active set methods, interior point methods, projected gradient methods, forward-backward partitioning methods, or alternating direction multiplier methods (ADMM).

[0129] In some embodiments of the invention, one or more nonlinear inequality constraints 634 may be locally approximated by one or more nonlinear but convex inequality constraints, resulting in the need to solve a locally convex programming (CP) approximation 605 in the sequential convex programming (SCP) implementation of the stochastic predictive controller 610. For example, in some embodiments, one or more probabilistic chance constraints may be locally approximated by convex second-order cone constraints and / or convex quadratic inequality constraints. Each convex cone constraint specifies that a linear combination of state, control, and / or covariance matrix variables is restricted to the interior of the convex cone. Embodiments of the convex cone may include positive quadrants, sets of positive semidefinite matrices, and / or second-order cones. Some embodiments of the invention are based on the understanding that the locally convex programming approximation 605 of the optimal control structured constraint NLP 350 may be linear programming (LP), quadratic programming (QP), quadratic constraint quadratic programming (QCQP), second-order cone programming (SOCP), or positive semidefinite programming (SDP), and that each of these categories of problems can be solved by a structured approach using convex optimization algorithms.

[0130] Some embodiments of the present invention are based on the understanding that, in each iteration of a Newton-type SQP optimization algorithm, linearizing the covariance propagation equation 633 requires estimating the higher-order derivatives of the nonlinear system dynamics. This constitutes a computationally expensive step, especially when the dynamics are high-dimensional, involve lengthy nonlinear expressions, or are described by a set of rigidly or implicitly defined differential equations. More specifically, because the covariance propagation equation 633 requires estimation of the Jacobian matrix...

[0131] The dependency of the constraint Jacobian matrix in 652 The estimation requires estimating the higher-order derivatives of the nonlinear system dynamics f(·) in 632, thus enabling the estimation of the higher-order derivatives in 652. The estimation of the constrained Jacobian matrix constitutes a computationally expensive step in the implementation of the stochastic predictive controller. Therefore, some embodiments of the present invention are based on the Jacobian approximation technique in the inexact SQP optimization algorithm, which avoids the constraints on the Jacobian matrix in step 652. The estimation is thus obtained, thus avoiding the estimation of higher-order derivatives of the nonlinear system dynamics f(·) as in 632.

[0132] Figure 7A A block diagram is shown illustrating the computation 705 of the trajectory of the state covariance matrix in the prediction time domain, which uses the covariance propagation equation 633 based on discrete-time linearization and given the current state and control value 701 in the prediction time domain with y 636. Computation 705 needs to be evaluated in the prediction time domain for the constraint Jacobian matrix 575 for each interval k = 0, ..., N-1. Initial state covariance matrix Given or estimated, P1 can be given P0, The covariance matrix is ​​evaluated in the case of Σ, and these operations are repeated sequentially for k = 0, ..., N-1 until the prediction time domain ends, while preserving the positive definiteness of the covariance matrix for each state. This explicit and sequential evaluation of the covariance matrix value z in the prediction time domain is equivalent to the compact notation z = E in the compact notation. z (y)706.

[0133] Figure 7B A block diagram of an inexact SQP optimization algorithm for an efficient implementation of a stochastic predictive controller, according to some embodiments of the present invention, is shown. The derivative-based iterative optimization procedure avoids problems with the constraint Jacobian matrix in 652. The estimation thus avoids the estimation of higher-order derivatives of the nonlinear system dynamics f(·) in 632. More specifically, some embodiments of the invention provide values ​​for the covariance matrix of a fixed sequence. The local QP approximation for the nonlinear OCP is as follows:

[0134]

[0135]

[0136] It only includes state and control deviation variables as optimization variables.

[0137]

[0138] Therefore, the computational complexity and memory requirements for implementing inaccurate SQP optimization algorithms in stochastic predictive controllers are greatly reduced.

[0139] Unlike the exact Jacobi-based QP approximation of 640, Figure 7B The inexact QP approximation 720 of the nonlinear OCP includes linearization of the nonlinear system dynamics in equality constraint 722 and covariance matrix values ​​of the fixed sequence in inequality constraint 723 of the QP approximation. Linearization of nonlinear inequality constraints. The optimal Lagrange multiplier values ​​for equality and inequality constraints are expressed as follows: 725 and 726 is used for the local QP solution in the i-th iteration of the inexact SQP optimization algorithm.

[0140] Some embodiments of the present invention are based on the understanding that if the objective function does not directly depend on the state covariance matrix variables in z 637, then the linear quadratic objective function 721 is equivalent to the objective function 641 of the exact Jacobi-based QP approximation 640. In other embodiments, the linear quadratic objective function 721 is the covariance matrix value of the fixed sequence. An inaccurate approximation.

[0141] The solution guess 601 of NLP is used to construct and solve the local QP approximation 720 in order to update the state, control, and state covariance values ​​730 of the current sequence in the prediction time domain, i.e., y. i+1 =y i +Δy i 731 and 732, where Δy i The solution 720 represents the original optimization variables in the local QP approximation. This represents the explicit and sequential estimates of the covariance matrix values ​​in the prediction time domain (705). In some embodiments of the invention, a globalization strategy is used to ensure the convergence of the Newton-type optimization algorithm, for example, by using a modified update y. i+1 =y i +α i Δy i 731, where the step size α i The selection can be based on a line search procedure combined with a specific merit function of constraint optimization. Other embodiments of the present invention use the trust region method to ensure the convergence of Newton-type optimization algorithms.

[0142] Each iteration of the optimization program checks whether a solution to the constrained NLP has been found and / or whether the maximum number of iterations (607) has been reached. If the termination condition (607) is met, a (near) optimal and / or feasible control solution (365) has been found; otherwise, the program needs to estimate the constraint Jacobian matrix. Similar to Hessian H i 740 and the target gradient g i and constraint function vector F(y) i )and 735, so as to construct a local QP approximation 720 in the next iteration of the inexact SQP optimization algorithm. The state and control values ​​from the previous control time step 310 can be used to form the initial solution guess and linearization point for the constrained nonlinear optimal control problem 601.

[0143] Some embodiments of the present invention are based on the following understanding: Figure 7B The inexact SQP optimization algorithm in the text avoids the constraints of the Jacobian matrix in 652. The high cost of estimation is avoided, thus avoiding the estimation and storage of higher-order derivatives of the nonlinear system dynamics f(·) in 632. Furthermore, some implementations are based on the understanding that the local QP approximation 720 only includes state and control deviation variables as optimization variables, unlike the accurate Jacobi-based QP approximation 640, thereby... Figure 7BThe computational complexity and memory requirements of implementing a stochastic predictive controller based on the inexact SQP optimization algorithm are significantly reduced. More specifically, if N represents the number of intervals in the prediction time domain, n x n represents the number of state variables. u To indicate the number of control input variables, then... and The structure requires incremental optimization algorithms. memory and The calculation is used to solve the QP approximation 640 in the exact SQP algorithm based on Jacobi. Conversely, the structure asymptotically requires only O(N(n)) using an optimization algorithm. x +n u ) 2 ) memory and O(N(n x +n u ) 3 The calculation is used to solve the inaccurate Jacobi-based local QP approximation 720 for implementing a stochastic predictive controller.

[0144] Figure 8A A block diagram is shown of a specific approximation of the constraint Jacobian matrix 801, which allows for numerical elimination of the trajectory of the state covariance matrix variables, resulting in an extended step of the local QP approximation of the constrained nonlinear OCP based on the adjoint gradient correction 805 and the calculation of the Lagrange multiplier of the covariance propagation equality constraint 810. More specifically, with Figure 7B Similar to the inexact SQP optimization algorithm in [the original text], the constrained Jacobian approximation 802 avoids the [problem] in 652. Estimation of the derivative

[0145]

[0146] Thus, the state covariance matrix variables can be numerically eliminated from the local QP approximation with relatively low computational cost, while maintaining the block-structured sparsity of the optimization problem.

[0147] Some embodiments of the present invention are based on the understanding that the gradient vector in the local QP approximation needs to include an adjoint gradient correction to ensure that the resulting SQP optimization algorithm converges to a feasible and optimal solution constrained to NLP 350. This gradient correction depends on the quality of the approximation of the constraint Jacobian matrix of the equation and / or inequality constraints in the local QP approximation of NLP, i.e.

[0148]

[0149] Among them, g i It is the target gradient vector, as shown in 641. Based on the estimation of the adjoint derivative The corrected gradient vector is 806. Some embodiments of the present invention are based on the understanding that, with the complete Jacobian matrix estimation... In contrast, for example, by using the backward or adjoint mode of the algorithmic differentiation (AD), the adjoint derivative vector estimate can be computed. The cost is much lower.

[0150] Based on the constrained Jacobian approximation 802, we can use Numerical elimination of the state covariance matrix variables yields a compressed evaluation of inequality constraint 807, which is:

[0151]

[0152] After solving the obtained local QP approximation, the sequence of state covariance matrix values ​​can be updated to z. i+1 =E z (y i +1 ), or use z i+1 =z i +Δz i and Lagrange multiplication numerical method based on inequality constraints from local QP solutions The Lagrange multiplier value of the covariance propagation constraint can be calculated through the following extended step 811.

[0153]

[0154] This is used to perform adjoint gradient-based correction of local QP subproblems in the next iteration of the inexact SQP optimization algorithm.

[0155] Figure 8B A block diagram of an inexact SQP optimization algorithm for efficient implementation of a stochastic predictive controller, according to some embodiments of the invention, is shown. This algorithm uses gradient-based adjoint correction to improve convergence characteristics. The derivative-based iterative optimization procedure avoids problems with the complete constraint Jacobian matrix in 652. The estimate then depends at most one adjoint derivative estimate in each SQP iteration. More specifically, some embodiments of the present invention are applicable to a given set of states and control values ​​y. i and the covariance matrix value z i The local QP approximation 820 for the nonlinear OCP is solved as follows:

[0156]

[0157]

[0158] It only includes state and control deviation variables as optimization variables.

[0159]

[0160] This significantly reduces the computational complexity and memory requirements for implementing adjoint-based inexact SQP optimization algorithms in stochastic predictive controllers.

[0161] and Figure 7B The inexact SQP optimization algorithm differs from that in other languages. Figure 8B The nonlinear OCP based on the adjoint inexact QP approximation 820 includes the compressed evaluation of the linearized inequality constraints 823 and 807 in the linear quadratic objective function 821, based on the adjoint gradient correction, and the local QP approximation. The optimal Lagrange multiplier values ​​of the equality and inequality constraints are respectively expressed as the local QP solution in the i-th iteration of the adjoint inexact SQP optimization algorithm. and

[0162] The solution guess 601 in NLP is used to construct and solve the local QP approximation 820 in order to update the current state, control, and state covariance value sequence 830 in the prediction time domain, i.e., y. i+1 =y i +Δy i 731 and z i+1 =E z (y i+1 )732, where Δy i Let z represent the solution of the original optimization variables for the local QP approximation 820. i+1 =E z (y i+1 () represents the explicit and sequential evaluation of the covariance matrix values ​​in the prediction time domain 705. The optimization algorithm updates the Lagrange multiplier values ​​of the equality and inequality constraints respectively using the optimal Lagrange multiplier values ​​from the local QP solution in the i-th SQP iteration. and Furthermore, the accompanying inexact SQP optimization algorithm is used.

[0163]

[0164] The Lagrange multiplier value of the covariance propagation equation 833 is calculated as follows:

[0165] Each iteration of the optimization procedure checks whether a solution to the constrained NLP has been found and / or whether the maximum number of iterations (607) has been reached. If the termination condition (607) is met, a (near) optimal and / or feasible control solution (365) has been found; otherwise, the procedure needs to evaluate the constraint Jacobian matrix. Approximate value H of Hessell i 740 and based on the adjoint gradient and constraint function vector F(y) i )and In order to construct a local QP approximation 820 in the next iteration based on the adjoint inexact SQP optimization algorithm. The state and control values ​​from the previous control time step 310 can be used to form the initial solution guess and linearization point of the constrained nonlinear optimal control problem 601.

[0166] Some embodiments of the present invention are based on the following understanding: Figure 8B The adjoint-based inexact SQP optimization algorithm avoids the constraints of 652 Jacobian matrices. Instead of high-cost evaluation, each SQP iteration requires at most one adjoint derivative evaluation. This significantly reduces computational and efficient storage costs. More specifically, for the accompanying derivative... An assessment only needs to be conducted as follows Figure 7A This represents a small fraction of the computation required for a single evaluation of the discrete-time covariance propagation equation as described above. Furthermore, some embodiments of the invention are based on the understanding that the adjoint gradient correction in the local QP approximation 820 causes the SQP optimization algorithm to converge to a feasible and optimal solution of the constrained NLP 350, unlike the inexact QP approximation 720 which results in a feasible but suboptimal solution to the constrained NLP 350.

[0167] like Figure 7B The same as in the middle, Figure 8B The adjoint SQP-based optimization algorithm significantly reduces computational complexity and memory requirements for implementing stochastic predictive controllers. More specifically, if N represents the number of intervals in the prediction time domain, n x n represents the number of state variables. u To indicate the number of control input variables, then... and The structure requires incremental optimization algorithms. memory and The computation is used to solve the QP approximation 640 in the Jacobi-based exact SQP algorithm. Conversely, the structure asymptotically requires only O(N(n)) time using an optimization algorithm. x +n u ) 2 ) memory and O(N(n x +n u ) 3 The calculation is used to solve the inexact Jacobian local QP approximation 820 based on the adjoint to implement a stochastic predictive controller.

[0168] Figure 8CAlgorithm descriptions of real-time variants of the adjoint inexact SQP optimization algorithm according to some embodiments of the present invention are shown to implement a stochastic nonlinear model predictive controller 840. Based on solution guessing (y... i ,z i ,λ i ,μ i ,κ i ) and the affine feedback gain matrix K 841, and in real time based on the adjoint SQP optimization algorithm 840, the solution guess (y) for the updated constrained NLP 350 is calculated. i+1 ,z i+1 ,λ i+1 ,μ i+1 ,κ i+1 )856. Figure 8C The algorithm described herein is a real-time optimization algorithm that performs a finite number of iterations to update the NLP solution guess from one control time step to the next, thereby providing rapid feedback to the control system 120 under uncertainty 125, while respecting the strict timing constraints of online computation on the embedded microprocessor and allowing the optimization algorithm 840 to converge in real time to the feasible and / or optimal control solution of the constrained NLP. In some embodiments of the invention, only one real-time SQP iteration 840 is performed at each control time step to implement a stochastic predictive controller.

[0169] The real-time SQP optimization algorithm 840 includes a preparatory step of local QP approximation 845, followed by a block sparse QP solution 850 that allows for rapid feedback of control actions to the real process 855, and an extension step 860 that eliminates the primal and dual optimization variables. The preparatory step estimates the Jacobian matrix for each interval in the prediction time domain (i.e., for k = 0, ..., N in the equality and inequality constraints of the local QP approximation 820). and Some embodiments of the present invention use the forward mode of Algorithmic Differentiation (AD) to efficiently estimate these constrained Jacobian matrices 846. Furthermore, the preparation step uses the adjoint mode 847 of AD to evaluate the gradient-based adjoint correction in the linear quadratic objective function 821 for k = 0, ..., N. Finally, the preparation steps use the forward (or adjoint) mode of AD 848 to calculate the compression evaluation of inequality constraint 823.

[0170] After preparing the local QP subproblem 845, and after receiving the current state estimate After step 851, step 850 is solved by solving the block-structured QP in order to obtain Δy. i , and 852, then update the original optimization variable y. i+1 ←y i +Δy i And update the dual optimization variables. and 853. Based on the update trajectory of status and control values, the real-time SNMPC controller provides control feedback actions to the process. 855. Finally, the extension steps involve using the adjoint AD to calculate the Lagrange multiplier μ of the state covariance propagation equation 861. i+1 and z i+1 Forward propagation of the updated state covariance matrix values ​​862.

[0171] Figure 9A The constraint Jacobian matrix 900 is shown.

[0172] A block diagram of the numerical exploitation of block-structured sparsity, specifically the numerical exploitation of the block-structured sparsity of the constraint Jacobian matrix 900, is presented to compute the inequality constraints in the local QP approximation 820 of the adjoint inexact SQP optimization algorithm for the SNMPC controller. Compression assessment of 901. Constraint Jacobi. Block diagonal sparse structure of 902 and reversible constrained Jacobian The block bidiagonal sparsity structure of 903 can be directly used to compute the compression evaluation of inequality constraints 901 based on the following forward recursive formula 905.

[0173]

[0174]

[0175] Use the middle value So that the compression value can be calculated in 823.

[0176] Figure 9B The constrained Jacobian matrix 910 is shown.

[0177] The block diagram for the numerical utilization of block-structured sparsity, constraining the Jacobian matrix 910, is used to compute the state covariance propagation equation in step 861 of the extended step of the adjoint inexact SQP optimization algorithm for the SNMPC controller. The Lagrange multiplier value of 911. Constrained Jacobian. 912 block diagonal sparsity structure and reversible constrained Jacobi The block bidiagonal sparse structure of 913 can be directly used to compute the updated values ​​of the Lagrange multipliers based on the following backward recursive formula 915:

[0178]

[0179]

[0180] Use the middle value To calculate the updated value in 861

[0181] Figure 10A A schematic diagram of a vehicle 1001 is shown, incorporating a stochastic predictive controller 1002 employing principles from several implementations. As used herein, vehicle 1001 can be any type of wheeled vehicle, such as a bus, coach, or off-road vehicle. Additionally, vehicle 1001 can be autonomous or semi-autonomous. For example, some implementations control the movement of vehicle 1001. Embodiments of movement include lateral movement of the vehicle controlled by a steering system 1003 of vehicle 1001. In one implementation, the steering system 1003 is controlled by the controller 1002. Furthermore or alternatively, the steering system 1003 can be controlled by the driver of vehicle 1001.

[0182] The vehicle may also include an engine 1006, which may be controlled by the controller 1002 or other components of the vehicle 1001. The vehicle may also include one or more sensors 1004 to sense the surrounding environment. Embodiments of sensor 1004 include rangefinders, radar, lidar, and cameras. The vehicle 1001 may also include one or more sensors 1005 to sense its current motion and internal state. Embodiments of sensor 1005 include Global Positioning System (GPS), accelerometers, inertial measurement units, gyroscopes, shaft rotation sensors, torque sensors, deflection sensors, pressure sensors, and flow sensors. These sensors provide information to the controller 1002. The vehicle may be equipped with a transceiver 1006, enabling the controller 1002 to communicate via wired or wireless communication channels.

[0183] Figure 10BA schematic diagram illustrating the interaction between a stochastic predictive controller 1002 and a controller 1020 of a vehicle 1001, according to some embodiments, is shown. For example, in some embodiments, the controller 1020 of the vehicle 1001 is a steering mechanism 1025 and a brake / throttle controller 1030 that control the rotation and acceleration of the vehicle 1020. In this case, the stochastic predictive controller 1002 outputs control inputs to controllers 1025 and 1030 to control the state of the vehicle. The controller 1020 may also include a higher-level controller, such as a lane-keeping assist controller 1035 that further processes the control inputs of the stochastic predictive controller 1002. In both cases, the controller 1020 maps the output of the stochastic predictive controller 1002 to control at least one actuator of the vehicle (e.g., the vehicle's steering wheel and / or brakes) to control the movement of the vehicle.

[0184] Figure 10C A schematic diagram of an autonomous or semi-autonomous controlled vehicle 1050 is shown, for which a dynamically feasible, and typically optimal, trajectory 1055 can be calculated using embodiments of the present invention. The resulting trajectory is designed to keep the vehicle within specific road boundaries 1052 and to avoid other uncontrolled vehicles, i.e., obstacles 1051 of the controlled vehicle 1050. In some embodiments, each obstacle 1051 may be represented by one or more inequality constraints in a time or space formula of a constrained optimal control problem. For example, based on an embodiment configured to implement a stochastic model predictive controller, the autonomous or semi-autonomous controlled vehicle 1050 may make decisions in real time, such as overtaking another vehicle on the left or right, or staying behind another vehicle in the current lane of road 1052. Embodiments of the present invention are based on an SNMPC controller that directly considers uncertainties regarding the current and predicted states of the vehicle 1050, uncertainties regarding parameters in the vehicle model, and uncertainties regarding the current and predicted states of the environment (e.g., including obstacles 1051 within a certain distance of the location of the autonomous or semi-autonomous controlled vehicle 1050).

[0185] Figure 10D A schematic diagram of a vehicle 1065 controlled by an SNMPC controller is shown, which is designed to track the dynamic feasibility and optimal trajectory 1070 of a rapid lane change maneuver within the upper road boundary 1060 and the lower road boundary 1061 using an embodiment of the present invention. Figure 10DThe diagram illustrates: vehicle location 1065 at a first time point, including the propagation of uncertainty regarding the predicted state trajectory by the SNMPC controller 1071; vehicle location 1066 at a second time point and the corresponding propagation of uncertainty regarding the predicted state trajectory 1072; and vehicle location 1067 at a third time point and the corresponding propagation of uncertainty regarding the predicted state 1073. According to some embodiments of the invention, using a stochastic predictive controller with probabilistic chance constraints allows the probability of the controlled vehicle violating road boundary constraints 1060 and / or 1061 to be below a certain probability threshold. More specifically, Figure 10D The stochastic tube predicting the state trajectory 1072 reaches the upper limit of the road constraint 1060 at the second time point 1075, illustrating the behavior of a stochastic predictive controller designed to satisfy both deterministic and probabilistic constraints of a controlled system under uncertainty.

[0186] Implementations of uncertainties in the system and its environment may include any parameters related to the frictional behavior between the vehicle's tires and the road surface, such as parameters in a Pacejka tire force model that can be learned or estimated offline and / or online when controlling the vehicle. According to embodiments of the invention, the estimated parameter values ​​and the estimated uncertainties can be defined as time-varying and uncertain disturbance variables in the formula for the direct optimal control problem of a stochastic nonlinear model predictive controller.

[0187] The embodiments of the present invention described above can be implemented in any of a number of ways. For example, the embodiments can be implemented using hardware, software, or a combination thereof. When implemented in software, the software code can execute on any suitable processor or set of processors, whether provided in a single computer or distributed across multiple computers. Such a processor can be implemented as an integrated circuit, with one or more processors in a single integrated circuit assembly. However, the processor can be implemented using circuitry of any suitable format.

[0188] Furthermore, the various methods or processes outlined herein can be encoded as software that can execute on one or more processors, employing any of a variety of operating systems or platforms. Moreover, such software can be written using any of many suitable programming languages ​​and / or programming or scripting tools, and can also be compiled into executable machine language code or intermediate code that executes on a framework or virtual machine. Typically, in various implementations, the functionality of program modules can be combined or distributed as needed.

[0189] Alternatively, the implementation may be embodied as a method, wherein embodiments of the method have been provided. The actions performed as part of the method can be ordered in any suitable manner. Therefore, implementations can be constructed in which actions are performed in a different order than those shown, which may include performing some actions simultaneously, although shown as sequential actions in the illustrative implementation.

[0190] Although the invention has been described by way of preferred embodiments, it should be understood that various other adjustments and modifications can be made within the spirit and scope of the invention. Therefore, the appended claims are intended to cover all such variations and modifications within the true spirit and scope of the invention.

Claims

1. A stochastic model predictive controller (150) for controlling a system under uncertainty, the uncertainty being constrained by the state of the system and control variables, the stochastic model predictive controller comprising: At least one processor (201); and a memory (202) storing instructions that, when executed by the at least one processor (201), cause the stochastic model prediction controller (150) to: In each control step of the stochastic model predictive controller (150), a nonlinear dynamic optimization problem with inequality constraints is solved to generate a control command (101), the dynamic optimization problem including probabilistic chance constraints representing the uncertainty. The system is characterized in that the constraints on its state and control variables include inequality constraints, and wherein the probabilistic chance constraints are based on the prediction domain at each time step and depend on the backoff coefficient value (α). i 420) Tightening of terms of the constraint Jacobian matrix and the covariance matrix of the predicted state variables with respect to one or more inequality constraints to ensure that the probability of violating each corresponding inequality constraint is below a probability threshold (∈ i ),and, The stochastic model predictive controller (150) solves the dynamic optimization problem based on two-level optimization, which alternates between the propagation of the covariance matrix of the probability chance constraint for the fixed values ​​of the state and control variables at each time step within the prediction domain and the optimization of the state and control variables for the fixed values ​​of the covariance matrix within the prediction domain, until the termination condition (607) is met; and The system is controlled using the generated control commands (101) that include optimized control variables. The propagation of the covariance matrix at each time step within the prediction domain is performed by evaluating a nonlinear covariance propagation equation with respect to the current values ​​of the state and control variables optimized for the prediction domain. The nonlinear covariance propagation equation is based on a linearized discrete-time Lyapunov equation, which uses a pre-computed Jacobian matrix with respect to the current values ​​of the state and control variables optimized for the prediction domain, maintaining the positive definiteness of the covariance matrix at each control time step.

2. The stochastic model prediction controller (150) according to claim 1, wherein, The optimization of the state and control variables involves solving the nonlinear dynamic optimization problem using a local quadratic programming approximation of these state and control variables.

3. The stochastic model prediction controller (150) according to claim 1 or 2, wherein, For a pre-stabilized system with nonlinear system dynamics, propagation of the covariance matrix is ​​performed at each time step within the prediction domain to account for future feedback control actions in the forward propagation of the uncertainty of the predicted state variables within the prediction domain.

4. The stochastic model prediction controller (150) according to claim 3, wherein, The pre-stabilized system of the nonlinear system dynamics uses a time-invariant or time-varying sequence of affine feedback gain.

5. The stochastic model prediction controller (150) according to claim 3, wherein, The nonlinear dynamics optimization problem with inequality constraints includes one or more probabilistic chance constraints on one or more inequality constraints for subsequent feedback control actions within the prediction domain, to ensure the feasibility of a pre-stabilized system for the nonlinear system dynamics.

6. The stochastic model prediction controller (150) according to claim 1, wherein, The optimization of the state and control variables within the prediction domain with a fixed value for the covariance matrix is ​​performed at each quadratic programming (QP) subproblem using an inexact sequential quadratic programming (SQP) with the fixed value for the covariance matrix. The QP subproblems maintain block-structured sparsity without calculating the derivatives of the covariance propagation equation with respect to the state and control variables, and without calculating the derivatives of the probabilistic chance inequality constraints with respect to the covariance matrix variables. The optimization of the state and control variables within the prediction domain with a fixed value for the covariance matrix is ​​performed using adjoint gradient computation, thereby solving the block-structured QP subproblems involves an adjoint gradient-based evaluation in a linear quadratic objective function. An additional adjoint-based gradient is evaluated using the Lagrange multiplier value of the covariance propagation equation, and the gradient represents the effect of the covariance equation on the optimization of the state and control variables. After each solution of the QP subproblem in the inexact SQP optimization algorithm, the Lagrange multiplier value of the covariance propagation equation is updated using an extension step based on the Lagrange multiplier value of the inequality constraints.

7. The stochastic model prediction controller (150) according to claim 1, wherein, The stochastic model predictive controller (150) starts from the optimal or suboptimal sequence of states, control values ​​and covariance matrix values ​​in the prediction domain in the previous control step, and solves the nonlinear dynamic optimization problem of the inequality constraints in each control step using only one or a predetermined number of iterations of the two-level optimization algorithm.

8. The stochastic model prediction controller (150) according to claim 7, wherein, The two-stage optimization is based on an inaccurate SQP optimization method, which solves only one or a predetermined number of block-structured QP subproblems at each control step to update the optimal or suboptimal sequence of state and control values ​​and covariance matrix values ​​in the prediction domain.

9. The stochastic model prediction controller (150) according to claim 1, wherein, The stochastic model prediction controller (150) is configured to control the vehicle (1001) forming the controlled system.

10. The stochastic model prediction controller (150) according to claim 9, wherein, The state of the vehicle includes one or a combination of the vehicle's (1001) position, orientation, velocity, angular velocity, slip ratio, and slip angle, wherein the control inputs include one or a combination of acceleration, braking torque, steering angle, and steering ratio, and wherein the uncertainty includes time-varying disturbances, which include one or a combination of uncertainties in the mass value, inertia value, or both in the model of the vehicle (1001), uncertainties in the steering model of the vehicle (1001), and uncertainties in one or more parameter values ​​representing the friction between the tires of the vehicle (1001) and the road surface.

11. A predictive control method for controlling a system under uncertainty, wherein the uncertainty is constrained by the state of the system and control variables, wherein, The method uses a processor (201) in conjunction with stored instructions for implementing the method, wherein the instructions, when executed by the processor (201), implement the steps of the method, the method comprising the following steps: In each control step of the predictive control method, a nonlinear dynamic optimization problem with inequality constraints is solved to generate a control command (101), the dynamic optimization problem including probabilistic chance constraints representing the uncertainty. The system is characterized in that the constraints on its state and control variables include inequality constraints, and wherein the probabilistic chance constraints are based on the prediction domain at each time step and depend on the backoff coefficient value (α). i 420) Tightening of terms of the constraint Jacobian matrix and the covariance matrix of the predicted state variables with respect to one or more inequality constraints to ensure that the probability of violating each corresponding inequality constraint is below a probability threshold (∈ i ),and, The stochastic model predictive controller (150) solves the dynamic optimization problem based on two-level optimization, wherein the propagation of the covariance matrix of the probability chance constraint for fixed values ​​of the state and control variables at each time step in the prediction domain is alternated with the optimization of the state and control variables for fixed values ​​of the covariance matrix in the prediction domain until a termination condition (607) is met; and The system is controlled using the generated control commands (101) that include optimized control variables. The propagation of the covariance matrix at each time step within the prediction domain is performed by evaluating a nonlinear covariance propagation equation with respect to the current values ​​of the state and control variables optimized for the prediction domain. The nonlinear covariance propagation equation is based on a linearized discrete-time Lyapunov equation, which uses a pre-computed Jacobian matrix with respect to the current values ​​of the state and control variables optimized for the prediction domain, maintaining the positive definiteness of the covariance matrix at each control time step.

12. The predictive control method according to claim 11, wherein, The optimization of the state and control variables involves solving the nonlinear dynamic optimization problem using a local quadratic programming approximation of these state and control variables.

13. The predictive control method according to claim 11 or 12, wherein, The constraints on the state and control variables of the system include inequality constraints, wherein the probability constraint is based on a contraction of one or more inequality constraints at each time step within the prediction domain, depending on the backoff coefficient value, the constraint Jacobian matrix, and the covariance matrix of the predicted state variables, to ensure that the probability of violating each corresponding inequality constraint is below a probability threshold.

14. The predictive control method according to claim 11 or 12, wherein, The propagation of the covariance matrix at each time step within the prediction domain is performed by evaluating a nonlinear covariance propagation equation with respect to the current values ​​of the state and control variables optimized for the prediction domain.

15. The predictive control method according to claim 11 or 12, wherein, The controlled system is a vehicle (1001).

16. A non-transitory computer-readable storage medium having embodied a program executable by a processor (201), the processor being configured to perform a predictive control method for controlling a system under uncertainty, the uncertainty being constrained by the state of the system and control variables, the method comprising the steps of: In each control step of the predictive control method, a nonlinear dynamic optimization problem with inequality constraints is solved to generate a control command (101), the dynamic optimization problem including probabilistic chance constraints representing the uncertainty. The system is characterized in that the constraints on its state and control variables include inequality constraints, and wherein the probabilistic chance constraints are based on the prediction domain at each time step and depend on the backoff coefficient value (α). i 420) Tightening of terms of the constraint Jacobian matrix and the covariance matrix of the predicted state variables with respect to one or more inequality constraints to ensure that the probability of violating each corresponding inequality constraint is below a probability threshold (∈ i ),and, The stochastic model predictive controller (150) solves the dynamic optimization problem based on two-level optimization, wherein the propagation of the covariance matrix of the probability chance constraint for the fixed values ​​of the state and control variables at each time step in the prediction domain is alternated with the optimization of the state and control variables for the fixed values ​​of the covariance matrix in the prediction domain, until the termination condition (607) is met; and The system is controlled using the generated control commands (101) that include optimized control variables. The propagation of the covariance matrix at each time step within the prediction domain is performed by evaluating a nonlinear covariance propagation equation with respect to the current values ​​of the state and control variables optimized for the prediction domain. The nonlinear covariance propagation equation is based on a linearized discrete-time Lyapunov equation, which uses a pre-computed Jacobian matrix with respect to the current values ​​of the state and control variables optimized for the prediction domain, maintaining the positive definiteness of the covariance matrix at each control time step.

17. The non-transitory computer-readable storage medium according to claim 16, wherein, The optimization of the state and control variables involves solving the nonlinear dynamic optimization problem using a local quadratic programming approximation of these state and control variables.

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