Hello researchers,

I am developing a control for optimization of the trajectory to be followed by valve for extreme operation such as emergency shutdown so that pressure surges caused during this in a pipe attached to valve are within limits.

1. The system dynamics are given by transient flow model based on classical water hammer theory. The actual model consists of a pair of coupled nonlinear partial differential equations implemented after their discretization using method of characteristics and finite difference method.

2. A simple version of the system is assumed here with reservoir at one end of pipe and valve at other end. The states of the system are pressure and flow rate values at different nodes inside pipe and control signal is flow rate of the node at the valve end.

3. The optimal control is to be designed using direct method with control parametrization.

4. After control parametrization the parameters are slope of each segment (currently I have represented each segment with a line which I am planning to modify with a 2nd degree curve afterwards ) and switching times of the segments.

5. My cost function is just the total time.

6. I am using SQP for solving NLP.

My questions are –

1. How should I take the gradients of the states? (I need them for gradients of constraints) Should I differentiate the discretized system equations (iterative difference equations) and propagate the gradient information along the same grid (grid of characteristics using method of characteristics on original PDEs) and solve for gradients at the time of state calculation itself or should I differentiate original PDEs for gradients and solve them separately?

2. Should my cost function be more sophisticated? Like additional term for residual error etc.

3. How can I guarantee convergence to at least suboptimal solution?

 Thank you!