Control and Estimation in Linear Time-Varying Systems Based on Ellipsoidal Reachability Sets

Abstract

Linear continuous or discrete time-varying systems in which the sum of a quadratic form of the initial state and the integral or sum of quadratic forms of a disturbance on a finite horizon is bounded above by a given value are considered. It is demonstrated that the reachability set of such a continuous- or discrete-time system is an evolving ellipsoid, and its ellipsoid matrix satisfies a linear matrix differential or difference equation, respectively. The optimal ellipsoidal observer and identification algorithm that yield the best ellipsoidal estimates of the system’s state and unknown parameters are designed. In addition, the optimal controllers ensuring that the system’s state will fall into a target set or that the system’s trajectory will stay within the ellipsoidal tube are designed. A connection between the optimal ellipsoidal observer and the Kalman filter is established. Some illustrative examples for the Mathieu equation, which describes the parametric oscillations of a linear oscillator, are given.

Appendix

Proof of Lemma 2.1. Using the polar factorization [32, p. 490], write the matrix S as $S={(S{S}^{{\rm{T}}})}^{1/2}U$, where UU^T = I. Then $x=Sg={(S{S}^{{\rm{T}}})}^{1/2}Ug={(S{S}^{{\rm{T}}})}^{1/2}w$, where w = Ug. Since w^Tw ⩽ 1 and for each such w there exists g = U^Tw, g^Tg = w^Tw ⩽ 1, the proof of Lemma 2.1 is complete.

Proof of Lemma 4.1. Write the equivalent conditions

$$u\in {\mathcal{E}}({Q}_{u})\quad \forall \ x\in {\mathcal{E}}(P)\iff \max {x}^{{\rm{T}}}{\Theta }^{{\rm{T}}}{Q}_{u}^{-1}\Theta x\leqslant 1,\quad x={P}^{1/2}w\quad \forall \ w:\ {w}^{{\rm{T}}}w\leqslant 1.$$

Introducing the Lagrange function

$$L={w}^{{\rm{T}}}{P}^{1/2}{\Theta }^{{\rm{T}}}{Q}_{u}^{-1}\Theta {P}^{1/2}w+\mu \left(1-{w}^{{\rm{T}}}w\right),$$

arrive at the condition

$$\mu ={\lambda }_{\max }\left({P}^{1/2}{\Theta }^{{\rm{T}}}{Q}_{u}^{-1}\Theta {P}^{1/2}\right)={\lambda }_{\max }\left({Q}_{u}^{-1/2}\Theta P{\Theta }^{{\rm{T}}}{Q}_{u}^{-1/2}\right)\leqslant 1,$$

which is equivalent to the inequality ΘPΘ^T ⩽ Q_u. In accordance with the pseudo inverse characterization, P = PP⁺P; as a result, the inequality above can be written as ΘPP⁺PΘ^T ⩽ Q_u. Then Schur’s complement lemma for singular matrices [29, p. 190] finally gives (4.2). The proof of Lemma 4.1 is complete.

Proof of Theorem 4.2. The closed loop system is described by the equation

$$x(t+1)={A}_{c}(t)x(t)+B(t)v(t),\quad {A}_{c}(t)=A(t)+{B}_{u}(t)\Theta (t).$$

The reachability sets of this system are the ellipsoids ${\mathcal{E}}({P}_{c}(t))$ with the matrices P_c(t) defined by Theorem 2.1. Introduce the matrices Y(t)Y^T(t) > 0, t = t₀, ⋯ , t_f, satisfying the inequalities

$$Y(t+1)\geqslant {A}_{c}(t)Y(t){A}_{c}^{{\rm{T}}}(t)+B(t)G(t){B}^{{\rm{T}}}(t),\quad Y({t}_{0})\geqslant R.$$

(A.1)

Using the notation Θ(t)Y(t) = Z(t) and Schur’s complement lemma, transform this inequality into the first inequality of (4.5). From (A.1) it follows that the relations

$$Y(t+1)-{P}_{c}(t+1)={A}_{c}(t)[Y(t)-{P}_{c}(t)]{A}_{c}^{{\rm{T}}}(t)+M(t),\quad Y({t}_{0})-{P}_{c}({t}_{0})\geqslant 0$$

hold for some matrices M(t) = M^T(t) ⩾ 0. Then

$$Y(t)-{P}_{c}(t)=\Phi (t,{t}_{0})[Y({t}_{0})-{P}_{c}({t}_{0})]{\Phi }^{{\rm{T}}}(t,{t}_{0})+\mathop{\sum }\limits_{i={t}_{0}}^{t-1}\Phi (i,{t}_{0})M(i){\Phi }^{{\rm{T}}}(i,{t}_{0})\geqslant 0,$$

where Φ(t, t₀) is the transition matrix of the closed loop system. Hence, P_c(t) ⩽ Y(t) and ${\mathcal{E}}({P}_{c}(t))\subseteq {\mathcal{E}}(Y(t))$, i.e., the state of the closed loop system belongs to the interior of the ellipsoid ${\mathcal{E}}(Y(t))$, and consequently the target output belongs to the interior of the ellipsoid ${{\mathcal{E}}}_{z}({Q}_{z}(t))$, where Q_z(t) = [C(t) + D(t)Θ(t)]Y(t)[C(t)+D(t)Θ(t)]^T. In this case, the inequality Q_z(t) ⩽ Q(t), which turns into the third and fourth inequalities of (4.5) with the notation Z(t) = Θ(t)Y(t) due to Schur’s complement lemma, will guarantee the corresponding goal of control. The second inequality of (4.5) (see Lemma 4.1) means that $u(t)\in {\mathcal{E}}({Q}_{u}(t))$. Thus, if inequalities (4.5) hold, the controller guarantees the corresponding goal of control. The proof of Theorem 4.2 is complete.