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Constraints in MPC

بين-دکتر کنترل پيشتوحيدخواه

•Infeasibility implies that, for the current state, the constraints within the MPC algorithm cannot be satisfied.

•Without feasibility the MPC optimisation is ill posed and there is no assurance that the answer has any useful meaning.

•MPC looks at mechanisms for overcoming or avoiding this.

Feasibility in MPC

Hard constraintsHard constraints are constraints which must be satisfied.

Soft constraints

• Soft constraints are those which should be satisfied if possible. Soft constraints can be violated (ignored).

• Usually soft constraints are on outputs/states although they could also be applied to inputs. Such violations may have no effect on nominal stability results.

Terminal constraints

Summary: Constraints are a combination of hard, soft and terminal constraints.

Model uncertainty may cause infeasibility because the actual behaviour differs from the predicted behaviour. Hence, even though the nominal predictions could satisfy constraints over the entire future, a small change in the model will cause the actual behaviour to differ, and the associated predictions at the subsequent sampling instantcould then violate constraints.

Model uncertainty

One can form algorithms based on invariant sets (Section 11.11) to handle model uncertainty; however, the results are usually very conservative, as guarantees must allow for the worst case (which will arise with negligible probability).

A more pragmatic approach is to accept that guarantees cannot be given where there is significant uncertainty and make other contingencies for the rare occasions where infeasibility arises.

Model uncertainty (cont.)

•The stronger a guarantee you want, the more conservative your control law will be.

•In practise there must be a compromise between feasibility assurances and performance.

•Feasibility would usually be ensured by a systematic relaxation of soft constraints. This would be determined at a supervisory level.

•Using artificially tight constraints on future predictions automatically builds in some slack which can be used to retain feasibility in the presence of moderate uncertainty. The slack should be montonically increasing with the horizon.

Example: input limits:

Example 1. A mathematical model for an undamped oscillator is given by:

What happens if the control amplitude is limited to +/- 25 ?

closed-loop eigenvalues are at −0.1946, 0, 0.

Case A. Closed loop control without saturation

Case B- Closed-loop control with saturation

If we do not pay attention to the saturation of the control, then in the presence of constraints, the closed-loop control performance could severely deteriorate.

Example 2:

A common practice in dealing with saturation is to let the model know the difference in Δu(k) when saturation becomes effective.

over-shoot in the closed-loop response is significantly reduced.

Formulation of Constrained Control Problems

There are three major types of constraints frequently encountered in applications:

The first two types deal with constraints imposed on the control variables u(k),

and the third type of constraint deals with output y(k) or state variable x(k) constraints.

Constraints on the Amplitude of the Control Variable

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Constraints on the Control Variable Incremental Variation

Output Constraints

Output constraints are often implemented as ‘soft’ constraints in the way that a slack variable sv > 0 is added to the constraints

Constraints in a Multi-input and Multi-output Setting

Constraints as Part of the Optimal Solution

Numerical Solutions Using Quadratic Programming

E is assumed to be symmetric and positive definite.

Quadratic Programming for Equality Constraints

Example 4. Minimize

Solution. The global minimum, without constraint, is at

Illustration of constrained optimal solution

Lagrange Multipliers

Example 5. Minimize

subject to:

Solution:

Without the equality constraints, the optimal solution is:

Example 6. what happens to the constrained optimal solution when the linear constraints are dependent.

There is no feasible solution of x1 and x2

Matrix MTE−1M is not invertible

Solution:

Illustration of no feasible solution of the constrained optimization problem.Solid-line x1 + x2 = 1; darker-solid-line 2x1 +2x2 = 6

Example 7:

How the number of equality constraints is also an issue in the constrained optimal solution? (Ex. 5)

We add an extra constraint to the original constraints so that:

The only feasible solution:

In summary, the number of equality constraints is required to be less than or equal to the number of decision variables (i.e., x).

If the number of equality constraints equals the number of decision variables, the only feasible solution is the one that satisfies the constraints and there is no additional variable in x that can be used to optimize the original objective function.

Minimization with Inequality Constraints

In the minimization with inequality constraints, the number of constraints could be larger than the number of decision variables.

An inequality Mix ≤ γi is said to be active if Mix = γi and inactive if Mix < γi.

Kuhn-Tucker Conditions

Example 8.

Illustration of the constrained optimization problem with inequality constraints.Solid-line x1 + x2 = 1; darker-solid-line 3x1 +3x2 = 6

Active Set Method

Example 9

The third constraint is an inactive constraint and will be dropped from the constrained equation set.

The first two constraints as the active constraints, and solve the optimization problem as minimizing:

subject to:

From Example 5:

We drop the second constraint and solve the optimization problem as:

subject to:

1. In the case of equality constraints, the maximum number of equality constraints equals the number of decision variables. In this example, it is 3, and the only feasible solution x is to satisfy the equality constraints (see (2.50)). In contrast, in the case of inequality constraints, the number of inequality constraints is permitted to be larger than the number of decision variables, as long as they are not all active. In this example, only one constraint becomes active so it becomes an equality constraint. Once the optimal solution is found against this active constraint, the rest of the inequalities are automatically satisfied.

2. It is clear that an iterative procedure is required to solve the optimization problem with inequality constraints, because we did not know which constraints would become active constraints. If the active set could be identified in advance, then the iterative procedure would be shortened.

3. Note that the conditions for the inequality constraints are more relaxed than the case of imposing equality constraints. For instance, the number of constraints is permitted to be greater than the number of decision

variables, and the set of inequality constraints is permitted to be linearly dependent. However, these relaxations are only permitted to the point that the active constraints need to be linearly independent and the number of active constraints needs to be less than or equal to the number of decision variables.

Primal-Dual Method

Hildreth’s Quadratic Programming Procedure

Example 10. Minimize the cost function:

Example 11