Linear-Quadratic Problems in Systems and Controls via Covariance Representations and Linear-Conic Duality: Finite-Horizon Case

Abstract

Linear-Quadratic (LQ) problems that arise in systems and controls include the classical optimal control problems of the Linear Quadratic Regulator (LQR) in both its deterministic and stochastic forms, as well as $H^\infty$-analysis (the Bounded Real Lemma), the Positive Real Lemma, and general Integral Quadratic Constraints (IQCs) tests. We present a unified treatment of all of these problems using an approach which converts linear-quadratic problems to matrix-valued linear-linear problems with a positivity constraint. This is done through a system representation where the joint state/input covariance (the outer product in the deterministic case) matrix is the fundamental object. LQ problems then become infinite-dimensional semidefinite programs, and the key tool used is that of linear-conic duality. Linear Matrix Inequalities (LMIs) emerge naturally as conal constraints on dual problems. Riccati equations characterize extrema of these special LMIs, and therefore provide solutions to the dual problems. The state-feedback structure of all optimal signals in these problems emerge out of alignment (complementary slackness) conditions between primal and dual problems. Perhaps the new insight gained from this approach is that first LMIs, and then second, Riccati equations arise naturally in dual, rather than primal problems. Furthermore, while traditional LQ problems are set up in $L^2$ spaces of signals, their equivalent covariance-representation problems are most naturally set up in $L^1$ spaces of matrix-valued signals.

Figures (3)

• Figure 1: Examples of several cones ${\sf P}$ and their duals ${\sf P}'$ in ${\mathbb R}^2$. Duality is with respect to the standard Euclidean inner product. If the cone is a "right-angled wedge" (center figure), then it is equal to its dual. Otherwise the narrower the cone's angle, the larger is the dual cone's angle as the relation (\ref{['dual_cone_contain.eq']}) implies.
• Figure 2: Examples demonstrating the maximal property of the solution ${\bar{\Lambda}}$ of the Differential Riccati Equation (DRE) with final conditions (\ref{['DRE_max_lem.eq']}) over all possible solutions of the Differential Riccati Inequality (DRI) (\ref{['DRE_lem.eq']}). The equation here is scalar ($n=1$), and therefore becomes $-\dot{{\bar{\lambda}}}(t)= q+2a{\bar{\lambda}}(t) - m {\bar{\lambda}}^2(t)$, and various combinations of the signs of $q$ and $m$ are given. The gray curves represent 100 different solutions of the DRI, while the thick blue curve is the solution of the DRE. Notice how the maximality property is maintained even when the DRE and/or DRI solutions have finite escape time.
• Figure 3: Depiction of a quadratic form on input-state pairs $(v,x)$. The blue line represents the linear-affine space of all input-state pairs satisfying the dynamics $\dot{x}=Ax+Bv$, $x(0)=\rm{\rm x}_{\rm i}$. When the input $v=0$, the state trajectory is the initial condition response $\bar{x}(t) = e^{At} \rm{\rm x}_{\rm i}$, depicted here as the intersection of the linear-affine space with the "$x$-axis". The quadratic form ${\hbox{\large $\mathbf q$}}(x,v)$ over all signal pairs (not necessarily constrained by the dynamics) is depicted as the grey surface. ${\hbox{\large $\mathbf q$}}$ can have mixed signature as depicted here. The LQ Problem is to determine the infimum of the values of the quadratic form over the linear-affine constraint set (those values depicted here as the dashed red curve). Whether the infimum is finite or $-\infty$ depends on both the quadratic form and the system dynamics. In the case where the initial condition is zero (${\bar{x}}=0$), the constraint set is a subspace, and the constrained infimum can only be either $0$ or $-\infty$.

Theorems & Definitions (20)

• Definition 1
• Example 2
• Lemma 3
• Lemma 4
• proof
• Definition 5
• Example 6
• Lemma 7
• Theorem 8: Weak Linear-Conic Duality
• proof