Stochastic model predictive control — how does it work? Audio slides - YouTube

welcome to this five minute presentation of the  paper stochastic model predictive control how  

does it work this is published in computers and  chemical engineering and available online I'm tor  

heirung and my co-authors are John Paulson Jared  O'Leary and Ali Mesbah we're all at the department  

of chemical and biomolecular engineering at the  University of California Berkeley before MPC we  

first look at the general constrained stochastic  optimal control problem for linear systems we have  

a state-space formulation with state x control  U and measurement Y weather disturbance W and  

measurement noise in E and both are stochastic  with known distributions the goal is to minimize  

the expected value of a standard quadratic control  cost over a finite horizon with a terminal cost  

on the state we want to minimize this subject to  the model and constraints on the state X and input  

U the chance constraint specifies the permitted  probability of violating the state's constraints  

and post a key challenge in constrained stochastic  optimal control without perfect information on the  

entire state vector we have to estimate the  state here we have the patient framework in  

terms of the hyper state which is a probability  distribution for the state conditioned on the  

current information in linear systems control and  estimation are generally treated separately which  

turns out to be optimal under model assumptions  one case in which the separation principle holds  

meaning control and observer design do not  interact is linear quadratic Gaussian or  

LQG control this is an unconstrained problem and  therefore significantly less challenging in this  

problem the disturbance W and the measurement  noise V are both Gaussian sequences the optimal  

state estimator is the well-known Kalman filter  which can be derived from the general Bayesian  

recursion above the expected value cost function  has a simple deterministic form with the expected  

value of the state the minimizing control law  is linear in the state estimate with gain K this  

results in stable closed loop dynamics a minus  BK it is here clear that the separation principle  

holds when looking at the state and estimate  error dynamics together the bottom left zero in  

the highlighted matrix means the eigenvalues for  control and estimation can be placed independently  

so many of these ideas are used in stochastic  model to achieve control but the question is  

how things change from the reference we shall  win the paper that the objective function is  

not affected by the presence of constraints it  retains its deterministic quadratic form in mean  

state predictions the shown on the right in the  constrained stochastic optimal control problem  

shown two slides back the state constraints  are probabilistic we show how to arrive at  

a deterministic reformulation on the chance  constraints for the main challenge being how to  

modify the right-hand side often called the back  off this results in a standard quadratic program  

the initial condition is a state estimate here  from the Kalman filter for simplicity stochastic  

MPC is solving this deterministic QP on a receding  horizon just like in standard MPC the result is an  

implicit control law now comparing stochastic MPC  and standard MPC we have the stochastic system on  

the left and the deterministic systems which NPC  is applied on the right for s MPC we have the  

expected value pulse function as mentioned this as  a quadratic deterministic form shown here in red  

this is identical to the standard MPC cost except  for a change of variables this is also shown here  

in red similarly with the optimal control problems  the stochastic formulation has this equivalent  

deterministic form highlighted here this is  the QP and again identical in complexity to  

the QP in standard MPC also shown in red the only  significant difference is the right-hand sides in  

the state constraints highlighted in red in the  paper we discuss how to determine the modified  

right-hand side or the constraint pack of which is  generally done offline in other words stochastic  

MPC is nothing but a slight modification of  standard MPC we use a case study throughout  

the paper to illustrate the main points it is  a two-stage chemical reaction with a constraint  

on the second species here we first see many  simulations with MPC on stochastic MPC and see  

that with the stochastic MPC the constraint on  X 2 is not violated nearly as frequently or by  

as much as with MPC the second figure is a face  plot which also shows constraint modification  

using the content each abusive inequality as  well as lqg and compares the extent to which  

the controller's lead to acceptable levels of  constraint satisfaction we also look at how  

specifying the required probability of constraint  satisfaction compares with manually adjusting the  

constrained backup in terms of the effect on  the cost function the term through multiple  

simulations finally thank you for seeing this  presentation we hope you go read the paper