Model predictive control solver

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What is Model Predictive Control (MPC)?

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Discrete-Time Model Predictive Control

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View code. More details are available on our project website here This is still an early alpha release, be prepared for some rough spots and get in touch if you have any questions! Releases 2 tags. Packages 0 No packages published. Contributors 5.

You signed in with another tab or window. Reload to refresh your session. You signed out in another tab or window. Accept Reject.Over the years, control systems for robotics systems have drastically changed to allow for more complex actions and nonlinear dynamics. Model predictive control is one strategy to allow for these more complex behaviors.

All these applications involve either dynamic environments or dangerous inaccessible environments that do not allow for human intervention. To add, most of these robot models are highly nonlinear making control strategies more difficult.

Typical industrial control strategies such as PD control and PID control can fail in guaranteeing many kinds of features. Though there is a lot of research with different optimal control strategies for requirements such as adaptive control and imitation control, one control strategy clearly stands out in the state of the art research in the domain.

It is Model Predictive Control MPCwhich has taken years of researchers developing control strategies curated specifically for different applications. This article will establish the basic fundamentals before picking up MPC. Imagine walking in a dark room. You try to sense the surroundings, predict the best path in the direction of a goal, but take only one step at a time and repeat the cycle.

Similarly, the MPC process is like walking into a dark room. The essence of MPC is to optimize the manipulatable inputs and the forecasts of process behavior. MPC is an iterative process of optimizing the predictions of robot states in the future limited horizon while manipulating inputs for a given horizon. The forecasting is achieved using the process model. Thus, a dynamic model is essential while implementing MPC.

These process models are generally nonlinear, but for short periods of time, there are methods such as tailor expansion to linearize these models. These approximations in linearization or unpredictable changes in dynamic models might cause errors in forecasting.

Thus, MPC only takes action on first computed control input and then recalculates the optimized forecasts based on feedback. This implies MPC is an iterative, model-based, predictive, optimal, and feedback based control strategy.

MPC has three basic requirements to work. The first one is a cost function J, which describes the expected behavior of the robot. This generally involves parameters of comparison between different possibilities of actions, such as minimization of error from the reference trajectory, minimization of jerk, obstacle avoidance, etc.

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The above cost function minimizes error from a reference trajectory as well as jerk caused by drastic deviations in inputs to the robot. The second requirement is a dynamic model of the robot. This dynamic model enables MPC to simulate states of a robot in a given horizon with different possibilities of inputs.

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The third is the optimization algorithm used to solve given optimization function J. Along with these requirements, MPC provides flexibility to mention certain constraints to be taken into consideration while performing optimization.

These constraints can be the minimum and maximum value of states and inputs to the robot. In order to understand the working of MPC consider the robot is at current time k in the simulated robot movement and has a reference trajectory that needs to be followed for a given horizon p.Advances in Discrete Time Systems. More than 25 years after model predictive control MPC or receding horizon control RHC appeared in industry as an effective tool to deal with multivariable constrained control problems, a theoretical basis for this technique has started to emerge, see [ 1 - 3 ] for reviews of results in this area.

The focus of this chapter is on MPC of constrained dynamic systems, both linear and nonlinear, to illuminate the ability of MPC to handle constraints that makes it so attractive to industry. We first give an overview of the origin of MPC and introduce the definitions, characteristics, mathematical formulation and properties underlying the MPC. Furthermore, MPC methods for linear or nonlinear systems are developed by assuming that the plant under control is described by a discrete-time one.

Although continuous-time representation would be more natural, since the plant model is usually derived by resorting to first principles equations, it results in a more difficult development of the MPC control law, since it in principle calls for the solution of a functional optimization problem.

As a matter of fact, the performance index to be minimized is defined in a continuous-time setting and the overall optimization procedure is assumed to be continuously repeated after any vanishingly small sampling time, which often turns out to be a computationally intractable task. On the contrary, MPC algorithms based on discrete-time system representation are computationally simpler.

The system to be controlled which usually described, or approximated by an ordinary differential equation is usually modeled by a difference equation in the MPC literature since the control is normally piecewise constant.

Hence, we concentrate our attention from now onwards on results related to discrete-time systems. By and large, the main disadvantage of the MPC is that it cannot be able of explicitly dealing with plant model uncertainties.

For confronting such problems, several robust model predictive control RMPC techniques have been developed in recent decades. We review different RMPC methods which are employed widely and mention the advantages and disadvantages of these methods.

The basic idea of each method and some method applications are stated as well. However model and measurement uncertainties are often stochastic, and therefore RMPC can be conservative since it ignores information on the probabilistic distribution of the uncertainty. It is possible to adopt a stochastic uncertainty description instead of a set-based description and develop a stochastic MPC SMPC algorithm. Some of the recent advances in this area are reviewed.

The main advantages of NCSs are low cost, simple installation and maintenance, and potentially high reliability. However, the use of the network will lead to intermittent losses or delays of the communicated information.

These losses will tend to deteriorate the performance and may even cause the system to become unstable.Optimal control is a widespread field that involve finding an optimal sequence of future actions to take in a system or environment.

This is the most useful in domains when you can analytically model your system and can easily define a cost to optimize over your system. MPC is a powerhouse in many real-world domains ranging from short-time horizon robot control tasks to long-time horizon control of chemical processing plants. More recently, the reinforcement learning community, strife with poor sample-complexity and instability issues in model-free learninghas been actively searching for useful model-based applications and priors.

Going deeper, model predictive control MPC is the strategy of controlling a system by repeatedly solving a model-based optimization problem in a receding horizon fashion. At each time step in the environment, MPC solves the non-convex optimization problem. We focus on the box-DDP heuristic which adds control bounds to the problem. There has been an indisputable rise in control and model-based algorithms in the learning communities lately and integrating these techniques with learning-based methods is important.

PyTorch is a strong foundational Python library for implementing and coding learning systems.

model predictive control solver

Before our library, there was a significant barrier to integrating PyTorch learning systems with control methods. All of these sound like fun! We have baked in a lot of tricks to optimize the performance.

model predictive control solver

More details on this are in the box-DDP paper that we implement. Our MPC layer is also differentiable!

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You can do learning directly through it. The backwards pass is nearly free. When this happens, our solver cannot be used to differentiate through the controller, because it assumes a fixed point happens. Treating the iLQR procedure as a compute graph and differentiating through the unrolled operations is a reasonable alternative in this scenario that obtains surrogate gradients to the control problem, but this is not currently implemented as an option in this library.

GradMethods is defined in our mpc module. This example shows how our package can be used to solve a time-varying linear control LQR problem of the form.

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This code is available in a notebook here. This example shows how to do control in a simple pendulum environment that we have implemented in PyTorch here.

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The full source code for this example is available in a notebook here. Thus this optimization problem will find the control sequence that minimizes this distance.

In proper quadratic form, this becomes. If you find this repository helpful for your research please consider citing the control-limited DDP paper and our paper on differentiable MPC.

Control is important! Control in PyTorch has been painful before now There has been an indisputable rise in control and model-based algorithms in the learning communities lately and integrating these techniques with learning-based methods is important. Our library is fast We have baked in a lot of tricks to optimize the performance. Internally we solve a sequence of quadratic programs More details on this are in the box-DDP paper that we implement.Skip to Main Content.

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For IEEE to continue sending you helpful information on our products and services, please consent to our updated Privacy Policy. Email Address. Sign In. High-performance small-scale solvers for linear Model Predictive Control Abstract: In Model Predictive Control MPCan optimization problem needs to be solved at each sampling time, and this has traditionally limited use of MPC to systems with slow dynamic.

In recent years, there has been an increasing interest in the area of fast small-scale solvers for linear MPC, with the two main research areas of explicit MPC and tailored on-line MPC. State-of-the-art solvers in this second class can outperform optimized linear-algebra libraries BLAS only for very small problems, and do not explicitly exploit the hardware capabilities, relying on compilers for that.

This approach can attain only a small fraction of the peak performance on modern processors. In our paper, we combine high-performance computing techniques with tailored solvers for MPC, and use the specific instruction sets of the target architectures. The resulting software called HPMPC can solve linear MPC problems 2 to 8 times faster than the current state-of-the-art solver for this class of problems, and the high-performance is maintained for MPC problems with up to a few hundred states.

model predictive control solver

Article :. DOI: Need Help?A basic Model Predictive Control MPC tutorial demonstrates the capability of a solver to determine a dynamic move plan. In this example, a linear dynamic model is used with the Excel solver to determine a sequence of manipulated variable MV adjustments that drive the controlled variable CV along a desired reference trajectory. Download Excel Tutorial File. Model Predictive Control MPC predicts and optimizes time-varying processes over a future time horizon.

This control package accepts linear or nonlinear models. Three example files are contained in this directory that implement a controller for Linear Time Invariant LTI systems:. Continuous and discrete state space models are used in a Python script for Model Predictive Control. The model1. Other versions are model2.

Each is applied in a model predictive controller to follow a reference trajectory and reach a target value of 7. APMonitor enables the use of empirical, hybrid, and fundamental models directly in control applications. This option is set to 6 or 9 for nonlinear control. Nonlinear control adjusts variables that are declared as Manipulated Variables MVs to meet an objective. The MVs are the handles that the optimizer uses to minimize an objective function.

The objective is formulated from Controlled Variables CVs. The CVs may be controlled to a range, a trajectory, maximized, or minimized. The CVs are an expression of the desired outcome for the controller action.

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Model Elements. APMonitor Documentation. Model Predictive Control.The Model Predictive Control technique is widely used for optimizing the performance of constrained multi-input multi-output processes. However, due to its mathematical complexity and heavy computation effort, it is mainly suitable in processes with slow dynamics. Based on the Exact Penalization Theorem, this paper presents a discrete-time state-space Model Predictive Control strategy with a relaxed performance index, where the constraints are implicitly defined in the weighting matrices, computed at each sampling time.

The performance validation for the Model Predictive Control strategy with the proposed relaxed cost function uses the simulation of a tape transport system and a jet transport aircraft during cruise flight. Without affecting the tracking performance, numerical results show that the execution time is notably decreased compared with two well-known discrete-time state-space Model Predictive Control strategies.

This makes the proposed Model Predictive Control mainly suitable for constrained multivariable processes with fast dynamics. Model Predictive Control MPC for linear systems is now a well-established discipline providing stability, feasibility, and robustness [ 1 — 6 ]. Due to its inherent ability to take into account constraints and deal with multi-input multi-output variables [ 7 — 10 ], it has been applied in a wide range of applications, including chemical processes, industrial systems, energy, health, environment, and aerospace [ 11 — 16 ].

In [ 17 ], a robust MPC strategy is presented to handle the trajectory tracking problem for an underactuated two-wheeled inverted pendulum vehicle.

Moreover, based on an MPC scheme, in [ 18 ], a control strategy is designed to an unmanned aerial vehicle for its automatic carrier landing system. Nevertheless, the computation complexity makes the multivariable MPC ineffectual for high speed applications where the controller must execute in a few milliseconds [ 19 — 23 ]. Moreover, the problem becomes much more complicated solving such an online constrained optimization problem by computing a numerical solver [ 24 — 26 ].

Several MPC techniques are used to overcome these problems. For instance, in [ 27 ], an explicit model predictive control moves major part of computation offline, which makes it enable to be implemented in real time for wide range of fast systems.

Also, in [ 28 ], to reduce the online computational time, all the state trajectories are included in the optimal control problem as the constraints in the prediction horizon, then only a quadratic programming problem is solved. In [ 29 ], based on a mixed integer quadratic programming problem, the control input is calculated at each discrete time.

In contrast to common MPC approaches, where an optimization toolbox is required, this work presents a relaxed performance index, in which the weighting matrices are computed online using the concept of Taylor series expansion and standard inverse distance weighting IDW functions. Then, tracking performance under input-output constraints is well obtained, lighter computation load is achieved, and execution time to solve a Quadratic Program QP is reduced. Thus, a computationally efficient constrained MPC for discrete-time state-space multivariable systems is obtained.

The paper is organized as follows. Section 2 gives the preliminaries of the proposed MPC strategy. Section 3 describes the proposed relaxed cost function. Section 4 presents a tape transport system and a jet transport aircraft as study cases. Simulation results show the performance of the proposed MPC strategy and the execution time improvement compared with two well-known MPC strategies.

Finally, Section 5 discusses the conclusions. Acknowledgments and the list of references finish the paper.

model predictive control solver

This section presents a brief review of MPC based on discrete-time state-space model. The original controller is proposed by Alamir in [ 7 ]. In this previous work, considering the predictions of the states, the control action is obtained through the solution of a constrained optimization problem by using a cost function with constant weighting matrices. At each sampling time, an optimal control problem is solved whose results are computationally expensive. The system dynamics is denoted by the Linear Time Invariant LTI State-Space Model taking the following structure: where is the state vector, is the controlled input vector, is the output vector, is the state matrix, stands for the input matrix, is the output matrix, and denotes the sampling instant number.

Hence, as in [ 7 ], from 1the state predictions for the consecutive sampling instants are where and are used to represent the N-step-ahead prediction map for Linear Time Invariant LTI systems in a compactness form 2. Thus, system 2 can be reformulated using the following vector-matrix notation: where and is the whole state trajectory ofconcatenates the computed sequence ofand is the output trajectory of :. Now, any candidate sequence of actions has the corresponding future behavior of the system contained in the state trajectory and consequently the output trajectory.

Defining the projection matrixwhere is the vector length, corresponds to the prediction horizon, and stands for the desired term, then the state, control, and output vectors at a specific instant can be obtained as follows:. To find the best sequence of control action, in the present work, the value of the cost function is defined over the prediction horizon as follows: where considering a past control action valuethe last two terms are added to penalize the rate excursion of the control vector by the weighting matrix.

Moreover, in the first term, the pondering matrix penalizes the error between the output trajectory and the reference of the vectorwhich is expressed as follows:.


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