Optimal control software

Integrating with your existing POS, accounting platform, and supplier history, our software offers more than 70 reports to choose from, allowing you to pinpoint where in your business there is money to be saved.

Lower food cost and increase margins through simplified inventory management in your independent restaurant. Ensure consistency and transparency throughout your multi-unit chain with consolidated cloud-based reporting. Increase accountability by managing your establishment's multiple profit and cost centers in one unified platform.

Put an end to costly errors in your operations by utilizing targeted reporting and actionable insights. Complement and enhance your existing processes, save labor and increase employee buy-in through easy-to-use, purpose-built software.

Whether you're in the kitchen, the back office, or at HQ, there's valuable data available at your fingertips. Could not load branches. Could not load tags. Latest commit. Git stats 29 commits. Failed to load latest commit information. View code. Literature and list of software packages for optimal control Literature Lectures Books Survey papers Software High level optimal control modeling languages and optimal control software High level numerical optimization modeling languages Numerical optimization solver Non-linear programming Linear, quadratic, convex programming Integer, mixed-integer programming Automatic differentiation Other material.

Calculus of variations and optimal control theory: a concise introduction. Princeton University Press, Dynamic system is described with the following set of differential equations:. Normally, the dimension of the control vector does not exceed the dimension of the state vector.

In this statement, vector m is independent on vector x , and may be considered as control of open-loop system. The open-loop control system with input control vector m and state vector x is depicted below:.

In most of the cases, Q and Z are diagonal matrices but not necessarily! Each diagonal element of them defines relative weight of square of difference between desirable and real values of a parameter at given moment of time.

The matrices may be time dependent, e. The problem formulated here is often referred to as "tracking problem. Before we come to a solution, let's discuss a special case of dynamic systems. The most well studied class of the systems is linear dynamic systems, i. For such system, equation 1 may be rewritten as follows:. First, we will provide a solution for these kind of systems, and then spread it to nonlinear dynamic systems. Since we are going to offer a numeric solution, we will deal with the system in discrete moments of time.

Now equations 1 may be rewritten for discrete time:. Cost function 2 in discrete time will be reshaped to:. A numeric solution of the problem may be obtained based on Bellman's principle of optimality. Results are quite bulky and presented in the appendix. For general understanding, it is essential to know that m k control policy is calculated in two major runs. First, inverse calculation run is performed starting with final time step N-1 and going down to initial time 0.

The inverse calculation run using system matrices F and H , desirable state r and control u vectors and weight matrices Q , Z yields some vector c k and matrix L k. Then the direct run is carried out from time 0 till N-1 obtaining vector m k at each k time step as:. Vectors c N-k and matrices L N-k are known from the inverse run and dependent on system matrices F and H , user defined desirable vectors and weights.

They also depend on auxiliary matrices P N-k and vectors v N-k participating in inverse run and starting from zero initial conditions. On each step of direct calculation run, we obtain control vector m k based on state vector x k for this time. This procedure is repeated until the last time moment N-1 giving us both open-loop control policy m k and states of the system x k vs. Statement 7 provides us with close-loop control as depicted below:. These parameters allow designer to generate vector m based on actual state of the system, thus making the system more stable and robust.

For linear systems, control policy m k minimum of cost function 6 is achieved with just one iteration consisting of one inverse and one direct runs. As we have stated above, this policy depends on system matrices and user chosen parameters, namely desirable vectors and weight matrices.

These parameters are intuitive enough to reduce the control policy design to a play with them. The user chooses a set of parameters and calculates the control policy and appropriate state trajectories. For linear case system matrices F , H are normally time invariant. So, to attain time invariant feedback, user should choose constant weight matrix Q , i. In real life by proper choice of Q , we can achieve a desirable final state in acceptable time, even before our final time.

Although it is convenient to implement control policy using state vector coordinates, in real life usually not all those coordinates are available for measurement.

Various techniques have been developed to overcome this difficulty. Different kinds of state coordinate observers may be used to estimate unmeasured coordinates using information for the coordinates available for measurement. The nonlinear system case is much more complex.