Controlling the Spectrum of Linear Completely Regular Differential-Algebraic Systems with Delays

Abstract

For linear autonomous completely regular differential-algebraic systems with commensurable delays in the state and control, we study the control problem for the spectrum, treating it as follows: each characteristic quasi-polynomial given in advance is assigned to the system by closing it by a differential-difference controller. We formulate modal controllability problems and weak controllability problems: they characterize various control possibilities for the spectrum of the original system. Reducing the investigated system to a special form, we obtain the necessary and sufficient solvability conditions for the specified problems. The proof of the main assertions is constructive: for each particular system with the given number matrices, we can construct the corresponding controller. We provide illustrative examples.

APPENDIX

It is well known that, in the case of ordinary linear autonomous differential systems

$$\dot {z}(t) = Az(t) + Bu(t),\quad A \in {{\mathbb{R}}^{{n \times n}}},\quad B \in {{\mathbb{R}}^{{n \times r}}},$$

the condition

$${\text{rank}}[p{{I}_{n}} - A,B] = n\quad \forall p \in \mathbb{C}$$

is necessary and sufficient for the existence of a controller of the type $u(t) = Qz(t)$, $Q \in {{\mathbb{R}}^{{r \times n}}}$, ensuring each predefined spectrum in a closed system (the model controllability problem).

The case where the system contains a delay is harder even if the modal controllability conditions are satisfied. First, in the general case, controllers with distributed delays (apart from controllers with concentrated delays) can be used for systems with delays. Another difficulty is as follows: it is not always possible to express the controller explicitly, i.e., in the form $u(t) = F(z(t),z(t - h),...,z(t - \rho h)),$ where $F({{\zeta }_{0}},...,{{\zeta }_{\rho }})$$(\rho \in \mathbb{N}{\text{,}}\,h = {\text{const}} > 0)$ is a linear function. The best illustration of this effect is the case where a delay is contained in the control. For example, the scalar equation $\dot {z}(t) = u(t) + u(t - h)$ appears to be simple but there is no feedback of the type

$$u(t) = \sum\limits_{i = 0}^m \,{{d}_{i}}z(t - ih)\quad ({{d}_{i}} \in \mathbb{R}),$$

reducing it to an arbitrary stable $({\text{Re}}{{p}_{i}} < 0)$ finite spectrum. In fact, if this is possible, then the transcendental equation

$$p - (1 + {{e}^{{ - ph}}})\sum\limits_{i = 0}^m \,{{d}_{i}}{{e}^{{ - iph}}} = 0$$

has a finite number of roots ${{p}_{i}}$ satisfying the condition ${\text{Re}}{{p}_{i}} < 0.$ However, this does not take place for any ${{d}_{i}} \in \mathbb{R}$ and $m \in \mathbb{N} \cup \{ 0\}$. Another example is the problem to construct an integral controller of the type

$$u(t) = \int\limits_{ - {{h}_{1}}}^0 \,[dR(s)]z(t + s)ds,$$

where R(s) is a matrix such that its elements are bounded variation functions on segment $[ - {{h}_{1}},0]$, ${{h}_{1}} = {\text{const}} > 0$. This does not appear to be a completely clear task but the formulated control problem for the spectrum of the considered scalar equation is resolved sufficiently easily if the feedback is constructed as a controller described by an integrodifferential equation. In fact, assign $u(t) = {{z}_{1}}(t)$ and ${{\dot {z}}_{1}}(t) = w(t),$ where z₁ is a new auxiliary variable, w is a controller sought in the form

$$w(t) = \int\limits_{ - {{h}_{1}}}^0 \,[d{{R}_{1}}(s)]Z(t + s)ds,$$

$Z = {\text{col}}[z,{{z}_{1}}]$, and ${{R}_{1}}(s)$ is a matrix such that its elements are bounded variation functions on segment $[ - {{h}_{1}},0].$ Thus, we arrive at the control problem for the spectrum, formulated above; however, this refers to the spectrum of the system $\dot {z}(t) = {{z}_{1}}(t) + {{z}_{1}}(t - h)$, and ${{\dot {z}}_{1}}(t) = w(t)$; this problem contains no delays in the control and has been well studied (see [7, 8, 11, 12]).

The example above illustrates the motivation to apply controllers determined by differential, difference, or integral (or another) equations. Sometimes, such controllers are called dynamical controllers (see, e.g., [20]).

As we wrote above, integral-type controllers are used for general systems with delays. However, it is hard to implement such controllers in practice. Therefore, in practice, the integrals are changed for finite sums. For example, in [20], a stabilizing dynamical controller for the system

$$\dot {z}(t) = {{A}_{0}}z(t) + {{A}_{1}}z(t - h) + Bu(t - \tau ),\quad {{A}_{i}} \in {{\mathbb{R}}^{{n \times n}}},\quad {{B}_{i}} \in {{\mathbb{R}}^{{n \times r}}},$$

(A.1)

is constructed in the form of the integrodifferential equation

$$\begin{gathered} \frac{{du(t)}}{{dt}} = (G + {{F}_{0}}B)u(t) + {{F}_{1}}Bu(t - h) + Q(\tau )z(t) \\ \, + \int\limits_{ - \tau }^0 \,Q( - \xi )Bu(t + \xi )d\xi + \int\limits_{ - h}^0 \,Q(\tau - h - \theta ){{A}_{1}}z(t + \theta )d\theta ,\quad t \geqslant 0, \\ \end{gathered} $$

(A.2)

where G and ${{F}_{i}},i = 0,1,$ are matrices and Q(t) is a matrix function. Further, the closed system (A.1) and (A.2) is represented as the system

$$\frac{{dy(t)}}{{dt}} = \sum\limits_{j = 0}^l \,{{\mathcal{A}}_{j}}y(t - {{\tau }_{j}}) + \int\limits_{ - \tau }^0 \,\mathcal{P}(\theta )y(t + \theta )d\theta $$

with concentrated and distributed delays and is approximated by the following system with concentrated delays:

$$\frac{{dy(t)}}{{dt}} = \sum\limits_{j = 0}^l \,{{\mathcal{A}}_{j}}y(t - {{\tau }_{j}}) + \sum\limits_{k = 1}^N \,\left( {\int\limits_{ - k\delta }^{ - (k - 1)\delta } \,\mathcal{P}(\theta )d\theta } \right)y(t - {{h}_{k}}).$$

(A.3)

Here, ${{\mathcal{A}}_{j}}$ and $\mathcal{P}(\theta )$ are easily found when reducing system (A.1) and (A.2) to form (A.3) and y = ${\text{col}}[z,u]$. It is easy to see that this procedure is equivalent to the approximation of the integral controller (A.2) by a differential-difference controller of the type

$$\frac{{du(t)}}{{dt}} = \sum\limits_{j = 0}^l \,{{\mathcal{L}}_{j}}y(t - {{\tau }_{j}}) + \sum\limits_{k = 1}^N \,\left( {\int\limits_{ - k\delta }^{ - (k - 1)\delta } \,{{\mathcal{P}}_{0}}(\theta )d\theta } \right)y(t - {{h}_{k}}),$$

where the matrices ${{\mathcal{L}}_{j}}$ and ${{\mathcal{P}}_{0}}(\theta )$ consist of the last r rows of matrices ${{\mathcal{A}}_{j}}$.

In [21] the approximation controller determined by the integral equation

$$u(t) = Fz(t) + \int\limits_{ - r}^0 \,G(\theta )u(t + \theta )d\theta $$

(where $F \in {{\mathbb{R}}^{{r \times n}}}$ and $G(\theta ) \in {{\mathbb{R}}^{{r \times r}}}$ is a matrix function), used for the system

$$\dot {z}(t) = Az(t) + Bu(t - r),\quad A \in {{\mathbb{R}}^{{n \times n}}},\quad B \in {{\mathbb{R}}^{{n \times r}}},$$

by a controller of the type

$$u(t) = Fz(t) + \sum\limits_{k = 1}^K \,{{G}_{k}}u(t - {{r}_{k}}),\quad {{G}_{k}} \in {{\mathbb{R}}^{{r \times r}}}$$

is considered.

The provided examples confirm the need to obtain sufficient conditions for the existence of differential-difference controllers and methods to synthesize them, allowing us to avoid (in their practical implementation) additional problems related to the finite-sum approximation of the integrals contained in these controllers.