Robust steady-state target calculation for model predictive control

A method of controlling a system in accordance with an objective function J, the operation of said system being describable by an equation Δy=GΔu, where y represents one or more system-controlled output variables, u represents one or more system-manipulated input variables, and G represents system gain parameters, said method comprising the steps of:

• (a) modeling said system according to an equation Δy={tilde over (G)}Δu, where {tilde over (G)} is a nominal estimate of G and has a known uncertainty description;
• and
• (b) extremizing said objective function J by solving the quadratic program min Δ ⁢ ⁢ u , ϵ ⁢ c T ⁢ Δ ⁢ ⁢ u + d T ⁢ G ~ ⁢ Δ ⁢ ⁢ u + e T ⁢ ϵ + Δ ⁢ ⁢ u T ⁢ Q 0 ⁢ Δ ⁢ ⁢ u + ϵ T ⁢ Q 1 ⁢ ϵ subject ⁢ ⁢ to g ~ T ⁡ ( D i ⁢ Δ ⁢ ⁢ u ) + α 1 / 2 ⁢ [LeftDoubleBracketingBar] V 1 / 2 ⁢ D i ⁢ Δ ⁢ ⁢ u [RightDoubleBracketingBar] + a i T ⁢ b ≤ b yi + ϵ i A u ⁢ Δ ⁢ ⁢ u ≤ b u ϵ ≥ 0 for ⁢ ⁢ i = 1 , … ⁢ , m
• where
• c represents weights on said input variables u in the objective function J,
• d represents such weights on said output variables y in the objective function J,
• d represents such weights on said output variables y in the objective function,
• ε=└ε1 . . . εn,┘T≧0∈ny is a slack variable that allows for violation in constraints on output variables y,
• Au∈4nuxnu and Ay∈2nyxny are matrices corresponding to desired ranges of values of said system manipulated and system-controlled variables,
• bu∈4nu and by∈2ny are vectors corresponding to desired ranges of values of said system manipulated and system-controlled variables,
• U represents said uncertainty description,
• Q0∈Rnuxnu is a positive-definite symmetric (or possibly zero) matrix that allows for quadratic tradeoffs in system manipulated variables, and
• Q1∈Rnyxny is a positive-definite symmetric (or possibly zero) matrix that allows for quadratic tradeoffs in slack variables;
• and
• (c) using a solution obtained in (b), controlling said system.

An apparatus controlling a system in accordance with an objective function J, wherein the operation of said system is describable by an equation Δy=GΔu, where y represents one or more system-controlled output variables, u represents one or more system-manipulated input variables, and G represents system gain parameters having a known uncertainty description, said apparatus comprising:

• (a) estimator circuitry for deriving a nominal estimate {tilde over (G)} of system gain parameters G;
• and
• (b) computation circuitry, responsive to at least one past value of said system-controlled output variables and of said system-manipulated input variables, and responsive to said nominal estimate {tilde over (G)}, for extremizing said objective function J by solving the equation min Δ ⁢ ⁢ u , ϵ ⁢ c T ⁢ Δ ⁢ ⁢ u + d T ⁢ G ~ ⁢ Δ ⁢ ⁢ u + e T ⁢ ϵ + Δ ⁢ ⁢ u T ⁢ Q 0 ⁢ Δ ⁢ ⁢ u + ϵ T ⁢ Q 1 ⁢ ϵ subject ⁢ ⁢ to g ~ T ⁡ ( D i ⁢ Δ ⁢ ⁢ u ) + α 1 / 2 ⁢ [LeftDoubleBracketingBar] V 1 / 2 ⁢ D i ⁢ Δ ⁢ ⁢ u [RightDoubleBracketingBar] + a i T ⁢ b ≤ b yi + ϵ i A u ⁢ Δ ⁢ ⁢ u ≤ b u ϵ ≥ 0 for ⁢ ⁢ i = 1 , … ⁢ , m
• where
• c represents weights on said input variables u in the objective function J,
• d represents such weights on said output variables y in the objective function,
• ε=└ε1 . . . εn,┘T≧0∈ny is a slack variable that allows for violation in constraints on output variables y,
• Au∈4nuxnu and Ay∈2n,xny are matrices corresponding to desired ranges of values of said system manipulated and system-controlled variables,
• bu∈4nu and by∈2ny are vectors corresponding to desired ranges of values of said system manipulated and system-controlled variables,
• U represents said uncertainty description,
• Q0∈Rnuxnu is a positive-definite symmetric (or possibly zero) matrix that allows for quadratic tradeoffs in system manipulated variables, and
• Q1∈Rnyxny is a positive-definite symmetric (or possibly zero) matrix that allows for quadratic tradeoffs in slack variables such that said computation circuitry provides target values for system manipulated input variables for controlling said system.

A computer program product comprising:

• a computer usable medium for controlling a system in accordance with an objective function J, the operation of said system being describable by an equation Δy=GΔu, where y represents one or more system-controlled output variables, u represents one or more system-manipulated input variables, G represents system gain parameters, and where J is a function of G;
• and
• a set of computer program instructions embodied on the computer usable medium, including instructions to:
• model said system according to an equation Δy={tilde over (G)}Δu, where {tilde over (G)} is a nominal estimate of G and has a known uncertainty description and where {tilde over (J)} is said objective function J applied to {tilde over (G)};
• and
• compute steady-state targets for said system-manipulated variables u including extremizing said objective function J cast as a quadratic program, such that all of said system-controlled variables will remain feasible at steady-state for all possible values of the system parameters within said known uncertainty description;
• and
• use the computed steady-state targets to control said system.