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Optimal Control

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Process Control

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Abstract

Theoretically, optimal control refers to any type of control that refers to optimization methods. In reality, two main domains can be distinguished, on one side open-loop control that is also called dynamic optimization for continuous nonlinear state-space systems and dynamic programming for discrete systems. This will be explained with reference to variational calculus and includes Hamilton-Jacobi theory, Pontryagin’s maximum principle and Bellman optimality principle. On the other side, these theories are applied to closed-loop control in linear quadratic control, for perfect or stochastic systems, in continuous or discrete time. Application examples for multivariable systems illustrate linear quadratic control under different forms.

The original version of this chapter has been revised: Figs. 14.12, 14.13 and 14.14 have been corrected. The erratum to this chapter is available at https://doi.org/10.1007/978-3-319-61143-3_22.

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Notes

  1. 1.

    A functional is a function of functions: the function \(F(\mathbf {x}(t),\mathbf {u}(t),t)\) depends on functions \(\mathbf {x}(t)\) and \(\mathbf {u}(t)\).

  2. 2.

    Several mathematical relations are useful:

    (a) We denote by \(y_z\) the partial derivative \(\partial y / \partial z\), where z is a scalar. If y is scalar and \(\mathbf {z}\) a vector, the notation \(y_{\mathbf {z}}\) is the gradient vector of the partial derivatives \(\partial y / \partial z_i\). If \(\mathbf {y}\) and \(\mathbf {z}\) are vectors, the notation \(\mathbf {y}_{\mathbf {z}}\) represents the Jacobian matrix of the current element \(\partial y_i / \partial z_j\).

    (b) The derivative with respect to \(\mathbf {f}\) of the integral with fixed boundaries

    $$ I = \int _{x_0}^{x_1} F(x,\mathbf {f}(x),{\dot{\mathbf {f}}}(x)) dx $$

    is equal to

    $$ \displaystyle {\frac{dI}{d\mathbf {f}}} = \int _{x_0}^{x_1} \left[ F_{\mathbf {f}} - \displaystyle {\frac{d}{dx}} F_{{\dot{\mathbf {f}}}} \right] dx $$

    (c) According to the Euler–Lagrange lemma (Cartan 1967), if \(\mathbf {C}(x)\) is a continuous function (vector) on [a, b] verifying

    $$ \int _{a}^{b} \mathbf {C}^T(x) \mathbf {v}(x) dx = 0 $$

    for all function (vector) \(\mathbf {v}(x)\) which is continuous and becomes zero at the boundaries, then \(\mathbf {C}(x)\) is zero everywhere on [a, b].

  3. 3.

    Other authors use the definition of the Hamiltonian with an opposite sign before the functional, i.e.

    $$ H(\mathbf {x}(t),\mathbf {u}(t),\mathbf {\psi }(t),t) = F(\mathbf {x}(t),\mathbf {u}(t),t) + \mathbf {\psi }^T(t) \, \mathbf {f}(\mathbf {x}(t),\mathbf {u}(t),t) $$

    which changes nothing, as long as we remain at the level of first-order conditions. However, the sign changes in condition (14.87). See also the footnote in Sect. 14.4.6.

  4. 4.

    In many articles, authors refer to the Minimum Principle, which simply results from the definition of the Hamiltonian H with an opposite sign of the functional. Comparing to definition (14.102), they define their Hamiltonian as

    figure a

    With that definition, the optimal control \(u^*\) minimizes the Hamiltonian.

  5. 5.

    This notation is that of Pontryaguine et al. (1974). The superscript corresponds to the rank i of the coordinate while the subscripts (0 and 1) or (0 and f), according to the authors, are reserved for the terminal conditions.

  6. 6.

    A matrix \(\mathbf {A}\) of dimension \((2n \times 2n)\) is called Hamiltonian if \(\mathbf {J}^{-1} \, \mathbf {A}^T \, \mathbf {J} = - \mathbf {A}\) or \(\mathbf {J} = - \mathbf {A}^{-T} \, \mathbf {J} \, \mathbf {A}\), where \(\mathbf {J}\) is equal to: \(\left[ \begin{array}{ll} \mathbf {0} &{} \mathbf {I} \\ \mathbf {I} &{} \mathbf {0} \end{array} \right] \).

    An important property (Laub 1979) of Hamiltonian matrices is that if \(\lambda \) is an eigenvalue of a Hamiltonian matrix, \(-\lambda \) is also an eigenvalue with the same multiplicity.

  7. 7.

    A matrix \(\mathbf {A}\) is symplectic, when, given the matrix \(J = \left[ \begin{array}{ll} \mathbf {0}&{} \mathbf {I} \\ -\mathbf {I} &{} \mathbf {0} \end{array} \right] \), the matrix \(\mathbf {A}\) verifies \(\mathbf {A}^T \, \mathbf {J} \, \mathbf {A} = \mathbf {J}\).

    If \(\lambda \) is an eigenvalue of a symplectic matrix \(\mathbf {A}\), \(1/\lambda \) is also an eigenvalue of \(\mathbf {A}\); \(\lambda \) is thus also an eigenvalue of \(\mathbf {A}^{-1}\) (Laub 1979).

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  1. LRGP-CNRS-ENSIC, University of Lorraine, Nancy Cedex, France

    Jean-Pierre Corriou

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Corriou, JP. (2018). Optimal Control. In: Process Control. Springer, Cham. https://doi.org/10.1007/978-3-319-61143-3_14

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