Outline Constrained MPC Eduardo F. Camacho Univ. of Seville. Constraints in Process Control. Constraints and MPC 3. Formulation of Constrained MPC 4. Illustrative Examples 5. Feasibility. Constraint Management Paris'9 ECCI Eduardo F. Camacho MPC Constraints Constraints in Process Control Constraints in Process Control In practice all processes are subject to constraints Actuators have a limited range of action and a limited slew rate (physical limits, as in control valves) Safety Limits: pressure or temperature limits Operating conditions: due to technological limits or economic or environmental reasons (temperature profile in a furnace) Input/Output Constraints Output constraints must be controlled in advance since output variables are affected by process dynamics Input (manipulated) variables can be kept in bound by clipping Neglecting output constraints can reduce economic profit and cause damage to people or equipment Paris'9 ECCI Eduardo F. Camacho MPC Constraints 3 Paris'9 ECCI Eduardo F. Camacho MPC Constraints 4 Optimal operation and constraints Close to the limit but without overpass In practice, the operating points are determined to satisfy economic goals The control system normally operates close to the limits: constraint violations are likely to occur For safety reasons the setpoint is changed: quantity/quality decrease and profit loss Fine 3 Ymax Paris'9 ECCI Eduardo F. Camacho MPC Constraints 5 P P Paris'9 ECCI Eduardo F. Camacho MPC Constraints 6
Conventional way of handling constraints Loss of optimality Conventional strategy: Input (manipulated variable) constraints solved by clipping Output (controlled variable) cannot be addressed. Solution: setpoint far from the constraint (in a safe zone). Heuristic solutions such as override MPC: Compute u(t). If it violates the constraint, it is saturated to its limits (by either the control program or by the saturator). The case of the future control actions violating the limits is not even considered Optimality may be lost Output constraints (related to safety and cost) cannot be addressed Saturation of output variables can produce higher values of the cost function: optimality may be lost. Even undesired behaviour or instability Even if u(t) does not violate the limit, the future sequence can do it. u(t+) Optimum Solution u(t) Saturated control signal J (cost function) Optimum u(t+) Control signal u(t) Paris'9 ECCI Eduardo F. Camacho MPC Constraints 7 Paris'9 ECCI Eduardo F. Camacho MPC Constraints 8 Constraints and MPC MPC is the only methodology that can incorporate constraints in a systematic way in the design phase: great industrial success Since a dynamic model of the process is available: use the prediction capabilities of MPC Input constraints: although they can always be fulfilled by clipping, MPC must guarantee the optimum Output constraints: the controller must compute the future output evolution and anticipate Paris'9 ECCI Eduardo F. Camacho MPC Constraints 9 How to formulate constraints? Constraints must be formulated as functions of the unknowns: control signals u Actuators: amplitude and slew-rate Outputs: amplitude. Other types to force the response of the process to have certain characteristics as: Bandconstraints Overshoot constraints Monotonic behaviour Nonminimum phase behaviour Terminal state equality constraints Terminal set constraints Paris'9 ECCI Eduardo F. Camacho MPC Constraints Types of constraints Depending on the type of variable Input or manipulated variables (MV) Output or controlled variables (CV) Depending on the way they are treated Hard Soft Penalized in the cost function Input constraints: Amplitude in u Slew-rate in u U u( t) U u u( t) u( t ) u Constraints formulation In matrix form U Tu + u( t ) U u u u For all t Paris'9 ECCI Eduardo F. Camacho MPC Constraints Paris'9 ECCI Eduardo F. Camacho MPC Constraints
Output constraints Output constraints must be expressed as functions of u using the prediction equations The prediction is computed as: y = Gu + f Free response + Forced response Constraints general form Notice that these constraints are inequalities involving vector u (increment of the manipulated variables) and can be written in compact form as Ru c with the following matrix and vector: Past Current time Future Depends on future control actions Amplitude contraints: y y( t) y For all t In matrix form y Gu + f y Paris'9 ECCI Eduardo F. Camacho MPC Constraints 3 Paris'9 ECCI Eduardo F. Camacho MPC Constraints 4 Bandcontraints The output must be kept inside a band (Temperature profiles in the food industry) Like amplitude constraints, but with variable limits Bands defined by maximum and minimum vectors: Overshoot constraints: In some processes overshoots are not desirable Every time a change is produced in the setpoint (considered to be constant), the following constraints are added: which, in matrix form, results in: That can also be expressed in terms of u Paris'9 ECCI Eduardo F. Camacho MPC Constraints 5 Paris'9 ECCI Eduardo F. Camacho MPC Constraints 6 Monotonic behaviour: To avoid kick-back (oscillations on the controlled variable before it has gone over the setpoint) Each time a setpoint changes new constraints with the following form are added: Nonminimum Phase behaviour: NMP: when the process is excited by a step, the output tends to frist move in the opposite direction (not desirable). The constraints take the form: Clipping the last n rows of G and f Paris'9 ECCI Eduardo F. Camacho MPC Constraints 7 Paris'9 ECCI Eduardo F. Camacho MPC Constraints 8
Actuators nonlinearities Dead zones and other type of nonlinearities Dead zones: impose contraints in order to generate control signals outside the zone Feasible region is nonconvex (optimization very difficult to solve) Terminal State Equality Constraints The predicted output is forced to follow the predicted reference during a number of sampling periods m after the costing horizon Ny Equality constraints for ym: Terminal Set Constraints: The final state at the end of the prediction horizon is forced to belong to a terminal set, defined by the polyhedron The vector of predicted states is expressed as: And the constraint is given by Paris'9 ECCI Eduardo F. Camacho MPC Constraints 9 Paris'9 ECCI Eduardo F. Camacho MPC Constraints 3. Formulation of constrained MPC All the constraints shown (except the dead zone) are inequalities depending on u that can be described in matrix form by Ru r+ Vz where z is a vector composed of present and past signals. It is equal to the current state if a state-space representations if used, or composed of current output and past input and outputs in CARIMA models (a way of representing the state). Therefore: Decision variable Depend on process parameters and signal bounds (not frequent changes) Ru r+ Vx(t) State that changes at every sampling time Paris'9 ECCI Eduardo F. Camacho MPC Constraints Solution The implementation fo MPC with constraints involves the minimization of a quadratic cost function subjet to linear inequalities: Quadratic Programming (QP) minimize Subject to: There are many reliable QP algorithms Active Set methods Feasible Direction methods Pivoting methods, etc. All methods use iterative algorithms (computation time) Paris'9 ECCI Eduardo F. Camacho MPC Constraints Solution Method Examples Disturbances can cause constraint violation Output constrained -.5 p= N=3 m=5 Q= R=,NOISE,RFILTER p= N=3 m=5 Q= R=,CONST,NOISE,RFILTER MODEL COST FUNCTION CONSTRAINTS SOLUTION Linear Quadratic None Explicit Linear Quadratic Linear QP Linear -Norm Linear LP Paris'9 ECCI Eduardo F. Camacho MPC Constraints 3 Ref Ref Output Output - - 4 6 8 4 6 8 4 6 8 4 6 8 Input.5 Input - 4 6 8 4 6 8 4 6 8 4 6 8. DeltaU - DeltaU - -. 4 6 8 4 6 8 4 6 8 4 6 8 Actuator slewrate No physical meaning! Actuator limit limit Paris'9 ECCI Eduardo F. Camacho MPC Constraints 4
.7.6.5 Illustrative examples.7.6.5 Overshoot constraint Nonminimum Phase Constraints N =3 Nu= Q =. R =,CONST N =3 Nu= Q =. R =. Ref Output.4.4.5.3.3.8...6...5.4 5 5.6.5.4 Monotonicity constraint 5 5 Kick-back Notice that overshoot constraints do not eliminate kick-back. -.5. - -. 5 5 5 5 Ref Output.3 Inverse peaks limited to.5.. Output never decreases nonminumum phase system Paris'9 ECCI Eduardo F. Camacho 5 MPC Constraints 5 5 Paris'9 ECCI Eduardo F. Camacho MPC Constraints 6 Feasibility Constraint Management Sometimes the region defined in the decision variables by the set of constraints is empty Unfeasibility. The optimization algorithm cannot find any solution Steady-state regime: due to unobtainable control objectives. Can be handled during the design phase Transitory regime: the constraints can become temporarily incompatible due to disturbances or setpoint changes Feasibility is closely related to stability Unfeasibility usually appears when the optimum is near the constraints The constraint manager must recover feasibility by acting on the constraints according to varying criteria This must done during the normal operation of the controller Paris'9 ECCI Eduardo F. Camacho MPC Constraints 7 Paris'9 ECCI Eduardo F. Camacho MPC Constraints 8 Techniques for improving feasibility Possible solutions to this problem Recover feasibility relaxing or eliminating certain constraints The constraint manager computes the real limits and send them to the QP solver, always avoiding violation of the physical and security limits Types of limits to be considered: Physical: can never be exceeded. Equipment construction (actuators) Security: should never be violated (danger). Associated to process controlled variables Operational: fixed by operators (can be exceede in certain circumstances) Reals: those used by the control algorithm at each instant Paris'9 ECCI Eduardo F. Camacho MPC Constraints 9 Disconnection of the controller Constraint elimination: temporary, Indiscriminate (all constraints). Fast solution, but dangerous if constraints are related to safety Hierarchical. A priority is given during the desing phase. Feasibility is checked at every sampling period to reinsert those constraints that were temporarily dropped Constraint relaxation Temporarily relaxing the bound Changing hard to soft: and adding a term to the cost function: Paris'9 ECCI Eduardo F. Camacho MPC Constraints 3
Constraints and stability Stability and constraints (example) y(t+)=. y(t)+. u(t-) con -4 < u(t) < 4, N=5 Difference in input and output constraints: manipulated variables can always be kept in bound by the controller by clipping the control action or by the actuator. Output constraints are mainly due to safety reasons, and must be controlled in advance because output variables are affected by process dynamics. Paris'9 ECCI Eduardo F. Camacho MPC Constraints 3 Paris'9 ECCI Eduardo F. Camacho MPC Constraints 3 Conclusions Constraints are present in all processes Considering is important not only because loss of optimality but because of stability, In most cases can be solved with a QP or LP Al commercial MPCs consider constraints There are very efficient QP and LPs algorithms around The solution is a PWA controller on the state Paris'9 ECCI Eduardo F. Camacho MPC Constraints 33