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THE PROBLEM OF OBSERVATION OF A LINEAR DYNAMIC SYSTEM AND EQUATIONS WITH RETARDED ARGUMENT
N. N. Krasovskii
The article discusses the connection between the problem of observation of a linear dynamic system [1, 2] and the problem of the canonical representation of motions described by differential equations with time delay [3, 4].
§ 1. Consider a linear dynamic system described by the vector differential equation
\[ \dot{x}(t)=Ax(t), \tag{1.1} \]
where \(x\) is an \(n\)-dimensional vector, and \(A\) is a constant \((n\times n)\)-matrix. All vectors considered below, unless otherwise stated, will be treated as column vectors, denoted by lowercase Latin letters. Matrices will be denoted by capital Latin letters. A prime as a superscript will denote transposition. Consequently, in particular, the symbol \(q'\) denotes a row vector. Scalars will, as a rule, be denoted by lowercase Greek letters (the exception being \(t\), time).
Suppose that some quantity \(\eta(t)\) is observed, related to the vector \(x(t)\) by the relation
\[ \eta(t)=p'x(t), \tag{1.2} \]
where \(p\) is a constant \(n\)-vector. For simplicity we shall restrict ourselves to the case in which \(\eta\) is a scalar. However, the arguments remain valid (with inessential changes) also in the more general case in which a vector quantity \(y=Px\) is observed, where \(y\) is an \(r\)-vector \((r>1)\), and \(P\) is a constant \((r\times n)\)-matrix.
Consider the observation problem [1] for the quantity
\[ \xi(t)=q'x(t) \tag{1.3} \]
from the quantity \(\eta(t+\vartheta)\) \((-\tau\leq \vartheta\leq 0,\ \tau>0\) constant), where \(q'\) is some constant \(n\)-vector. We shall seek the resolving operation \(\varphi[\eta]\) in the form [2]
\[ \xi(t)=\varphi[\eta]=\int_{-\tau}^{0}\eta(t+\vartheta)\,d\gamma(\vartheta), \tag{1.4} \]
where \(\gamma(\vartheta)\) is a function of bounded variation [5] on \([-\tau,0]\). Among all operations (1.4), we shall choose the optimal operation \(\varphi^{0}[\eta]\)
\[ \xi(t)=\varphi^{0}[\eta]=\int_{-\tau}^{0}\eta(t+\vartheta)\,d\gamma^{0}(\vartheta), \tag{1.5} \]
having the least possible norm
\[ \rho^*[\varphi^0]=V^0_{-\tau}[\gamma^0]=\min . \tag{1.6} \]
Here the symbol \(V^0_{-\tau}[\gamma]\) denotes the total variation of the function \(\gamma(\vartheta)\) on the interval \([-\tau,0]\).
Denote by the symbol \(\{\eta(t+\vartheta)\mid \xi(t)=1\}\) the observed signal \(\eta(t+\vartheta)\) \((-\tau\leq \vartheta\leq 0)\) corresponding to the value \(\xi(t)=1\). According to [2], the observation problem (1.1)—(1.4) has a solution if and only if
\[ \min_{\eta}\max_{\vartheta}\,|\{\eta(t+\vartheta)\mid \xi(t)=1\}|=\alpha>0, \tag{1.7} \]
\[ (-\tau\leq \vartheta\leq 0). \]
Under condition (1.7), the norm (1.6) of the optimal resolving operation \(\varphi^0[\eta]\) (1.5) is determined by the equality
\[ \rho^*[\varphi^0]=V^0_{-\tau}[\gamma^0]=\frac{1}{\alpha}, \tag{1.8} \]
and the operation \(\varphi^0[\eta]\) itself has the maximum property
\[ \varphi^0[\{\eta^0(t+\vartheta)\mid \xi(t)=1\}] = \max_{\varphi}\varphi[\{\eta^0(t+\vartheta)\mid \xi(t)=1\}] = 1 \tag{1.9} \]
for
\[ \rho^*[\varphi]=\frac{1}{\alpha}, \]
where \(\{\eta^0(t+\vartheta)\mid \xi(t)=1\}\) is the solution of problem (1.7).
Choose in (1.3) \(q'=p'A\). Then problem (1.1)—(1.6) will necessarily have a solution. Indeed, we have
\[ \eta(t+\vartheta)=p'x(t+\vartheta)=p'X(\vartheta)x(t), \tag{1.10} \]
where \(X(t)\) is the fundamental matrix of system (1.1), which becomes the identity matrix at \(t=0\). For condition (1.7), which guarantees the solvability of problem (1.1)—(1.6), to hold, it is sufficient that expression (1.10) not vanish identically for \(-\tau\leq\vartheta\leq 0\) for any value of the vector \(x(t)\) constrained by the condition
\[ \xi(t)=p'Ax(t)=1. \tag{1.11} \]
Differentiating the right-hand side of equality (1.10) with respect to \(\vartheta\), taking into account the equality \(dX(\vartheta)/d\vartheta=AX(\vartheta)\) and relation (1.11), we obtain, for \(\vartheta=0\),
\[ \left(\frac{d}{d\vartheta}[p'X(\vartheta)x(t)]\right)_{\vartheta=0} = p'Ax(t)=1. \tag{1.12} \]
It follows from (1.12) that the identity \(p'X(\vartheta)x(t)\equiv 0\) under condition (1.11) is impossible. This proves the solvability of problem (1.1)—(1.6) for \(q'=p'A\).
Thus, one can construct an operation \(\varphi^0[\eta]\) satisfying conditions (1.5), (1.6). It follows from (1.12) that the function \(|\{\eta^0(t+\vartheta)\mid \xi(t)=1\}|\) can have only isolated maxima at the points \(\vartheta_i=-\tau_i\) \((i=1,\ldots,m;\ -\tau\leq-\tau_i\leq0)\). We shall number the values \(\vartheta_i=-\tau_i\) in increasing order of the numbers \(\tau_i\). From the maximum condition (1.9) we now conclude that the operation \(\varphi^0[\eta]\) has the form
\[ \xi(t)=\varphi^0[\eta(t+\vartheta)] = \sum_{i=1}^{m}a_i\eta(t-\tau_i). \tag{1.13} \]
Here the constants \(a_i\) are determined by the equalities
\(a_i=\gamma^0(-\tau_i+0)-\gamma^0(-\tau_i-0)\), where \(\gamma^0(\vartheta)\) is a piecewise-constant function. (If \(\tau_1=0\) or \(\tau_m=\tau\), then \(a_1=\gamma^0(0)-\gamma^0(-0)\), \(a_m=\gamma^0(-\tau+0)-\gamma^0(-\tau)\).)
Differentiate equality (1.2) with respect to \(t\). Taking into account (1.1), (1.3), and (1.13), we arrive at the conclusion that the variable \(\eta(t)\) satisfies the following differential equation with delays \(\tau_i\) of the argument \(t\):
\[ \dot{\eta}(t)=\sum_{i=1}^{m} a_i \eta(t-\tau_i). \tag{1.14} \]
Thus, the following conclusion is valid.
Whatever the vector-function \(x(t)\), which is a solution of equation (1.1), may be, the function \(\eta(t)\) (1.2) is a solution of equation (1.14).
The converse conclusion is, generally speaking, false. Not for every solution \(\eta(t)\) of equation (1.14) can one choose a vector \(x(t)\) satisfying equation (1.1) and such that the functions \(\eta(t)\) and \(x(t)\) satisfy relation (1.2). This is natural, since equality (1.2) and equation (1.1) determine, according to (1.10), a finite-dimensional family of functions \(\eta(t)\). However, as is known [6], solutions \(\eta(t)\) of equation (1.14) for \(t>t_0\) may be generated, for example, by all possible continuous initial curves \(\eta(t_0+\vartheta)\) \((-\tau\leq \vartheta \leq 0)\), and, consequently, these solutions in the general case constitute an infinite-dimensional family of functions.
Thus, in this paragraph we have verified that, whatever the system (1.1) and relation (1.2) may be, one can construct equation (1.14) with delays of the argument \(t\), among whose solutions \(\eta(t)\) all functions \(\eta(t)\) determined by equality (1.2) and equation (1.1) will be contained.
§ 2. Let us discuss in more detail the connection between equation (1.14) and system (1.1), (1.2). In order to distinguish the general solution of equation (1.14) from the family of functions \(\eta(t)\) determined by the system (1.1), (1.2), let us introduce a new variable \(\xi\) and consider the equation
\[ \dot{\xi}(t)=\sum_{i=1}^{m} a_i \xi(t-\tau_i). \tag{2.1} \]
We agree on the following notation. The set of particular solutions \(\xi(t)\) of equation (2.1), generated by all possible continuously differentiable initial curves \(\xi(\vartheta)\) \((-\tau\leq \vartheta\leq 0)\), will be denoted by the symbol \(\{\xi(t)\}\) \((t>0)\). The subset of these solutions \(\xi(t)\) which coincides with the family of functions \(\eta(t)\) determined by equalities (1.1) and (1.2) will be denoted by the symbol \(\{\eta(t)\}\). Accordingly, in what follows the symbol \(\xi(t)\) will denote some particular solution of equation (2.1) contained in \(\{\xi(t)\}\), and the symbol \(\eta(t)\) will denote a particular solution of equation (2.1) contained in \(\{\eta(t)\}\).
It is known [3, 4] that in any motion \(\xi(t)\) from \(\{\xi(t)\}\) one can single out canonical coordinates \(v_i(t)\) \((i=1,2,\ldots)\), which are projections of a certain infinite-dimensional vector \(v(t)\). These coordinates \(v_i(t)\) satisfy ordinary linear differential equations and in the general case are divided into groups corresponding to the roots \(\lambda_j\) of the characteristic equation
\[ \left|\lambda-\sum_{i=1}^{m} a_i \exp -\lambda \tau_i\right|=0. \tag{2.2} \]
It is convenient to construct the system of equations describing the change of the quantities \(v_i(t)\) so that the matrix of this system has Jordan form. If we choose some \(n\) canonical coordinates \(v_i\) (for definiteness, \(i=1,\ldots,n\)), restricting ourselves, for simplicity of notation, to the case when all the corresponding eigenvalues \(\lambda_i\) are simple, then we obtain the system of equations
\[ \dot v_i(t)=\lambda_i v_i(t)\quad (i=1,\ldots,n). \tag{2.3} \]
The canonical coordinates \(v_i(t)\) are extracted from the motion \(\xi(t)\) of system (2.1) by means of linear operations \(\psi_i[\xi(t+\vartheta)]\), which are performed on segments \(\xi(t+\vartheta)\) \((-\tau\leq \vartheta\leq 0)\) of these curves. The operations \(\psi_i[\xi]\) have the form
\[ v_i(t)=\psi_i[\xi(t+\vartheta)] =\nu_i\xi(t)+\sum_{j=1}^{m}\int_{-\tau_j}^{0}\xi(t+\vartheta)\nu_{ij}(\vartheta)\,d\vartheta, \tag{2.4} \]
where the quantities \(\nu_i\) and the continuous functions \(\nu_{ij}(\vartheta)\) are expressed in a known way through the solutions \(\xi_i^*(t)\) of the equation with advanced argument
\[ \dot \xi^*(t)=-\sum_{i=1}^{m}a_i\xi^*(t+\tau_i), \tag{2.5} \]
adjoint to equation (2.1).
Thus, by means of the linear operations (2.4), from the motions \(\xi(t)\in\{\xi(t)\}\) one can extract finite-dimensional systems of canonical coordinates \(v_i(t)\). To each such system of coordinates \(v_i(t)\), satisfying equations (2.3), there will correspond a certain finite-dimensional subset \(\{\beta(t)\}\) of functions \(\beta(t)\) having, according to (2.3), the form
\[ \beta(t)=\sum_{i=1}^{n}s_i v_i(t)=\sum_{i=1}^{n}l_i\exp \lambda_i t, \tag{2.6} \]
where \(s_i\) are certain fixed nonzero constants, and \(l_i\) are arbitrary constants.
Let us now return to system (1.1) and (1.2), which generated equation (2.1). Suppose again, for simplicity, that the matrix \(A\) in equation (1.1) is simple and that all its eigenvalues \(\rho_i\) \((i=1,\ldots,n)\) are distinct. In addition, assume that system (1.1) is completely observable [1] with respect to the quantity \(\eta(t)\) in (1.2). This means that, for any coordinate \(x_i(t)\) of the vector \(x(t)\), one can construct a linear operation \(\varphi_i[\eta]\) which, on motions of system (1.1), (1.2), satisfies the equality
\[ x_i(t)=\varphi_i[\eta(t+\vartheta)]\quad (-\tau\leq \vartheta\leq 0,\ i=1,\ldots,n). \tag{2.7} \]
If system (1.1)
\[ \dot x_i=\sum_{j=1}^{n}a_{ij}x_j(t) \tag{2.8} \]
is brought by a nonsingular linear transformation \(w(t)=Tx(t)\) to canonical form
\[ \dot w_i(t)=\rho_i w_i(t)\quad (i=1,\ldots,n), \tag{2.9} \]
where the relation (1.2) takes the form
\[ \eta(t)=\sum_{i=1}^{n}s_i w_i(t), \tag{2.10} \]
then the condition of complete observability of the system (2.9) with respect to the quantity \(\eta(t)\) (2.10) is equivalent to the conditions
\[ s_i \ne 0 \quad (i=1,\ldots,n), \tag{2.11} \]
since in this and only in this case the inequality \(\{\eta(t)\}\times \{w_i(t)=1\}\ne 0\) is satisfied, whatever the vector \(w(t)\) may be. But condition (2.11) means that the family \(\{\eta(t)\}\) consists of functions \(\eta(t)\) of the form
\[ \eta(t)=\sum_{i=1}^{n} l_i \exp \rho_i t, \tag{2.12} \]
where \(l_i\) are arbitrary constants. Hence, and from the properties of the solutions \(\xi(t)\) considered above, one may conclude that among the canonical coordinates \(v_i(t)\) \((i=1,2,\ldots)\) of equation (2.1) one can choose \(n\) such coordinates \(v_i\) (for definiteness we renumber them as follows: \(i=1,\ldots,n\)) which will coincide with the coordinates \(w_i\) of the original system (1.1), (1.2) (on the functions \(\eta(t)\) from \(\{\eta(t)\}\)). Returning now to the original variables \(x_i(t)\), we arrive at the following conclusion.
Let the equation with delay (2.1) be constructed for the completely observable system (1.1), (1.2) as described in § 1. Then one can indicate \(n\) linear operations
\[ \varepsilon_i[\xi(t+\vartheta)] = \varepsilon_i\xi(t) + \sum_{j=1}^{m}\int_{-\tau_j}^{0} \xi(t+\vartheta)\,\varepsilon_{ij}(\vartheta)\,d\vartheta \tag{2.13} \]
\[ (i=1,\ldots,n), \]
which determine the variables
\[ x_i(t)=\varepsilon_i[\xi(t+\vartheta)] \quad (i=1,\ldots,n), \tag{2.14} \]
satisfying equation (1.1), and, on the functions \(\xi(t)=\eta(t)\) from \(\{\eta(t)\}\), the quantities \(x_i(t)\) coincide with the coordinates \(x_i(t)\) of the phase vector \(x(t)\) of the system (1.1), (1.2).
The operations \(\varepsilon_i[\xi]\) are constructed as linear combinations of the operations \(\psi_i[\xi]\) (2.4).
§ 3. In the preceding sections we have shown that the system (1.1) can be regarded as part of an infinite-dimensional system of differential equations describing the canonical coordinates \(v_i(t)\) of the solutions \(\xi(t)\) of equation (2.1). At the same time, the operations (2.13) can be regarded as such linear operations which, in a completely observable system (1.1), (1.2), reconstruct the coordinates \(x_i(t)\) from the quantity \(\eta(t+\vartheta)\) \((-\tau\le \vartheta\le 0)\). These operations differ in form from the operations \(\varphi\) (1.4) considered above in § 1, since there operations having the least norm of the form (1.6) were sought. However, for the system (1.1), (1.2) one can construct operations \(\varphi_i\) reconstructing the coordinates \(x_i(t)\) from the signal \(\eta(t+\vartheta)\) and having the form
\[ x_i(t)=\varphi_i[\eta(t+\vartheta)] = \varphi_i\eta(t) + \sum_{j=1}^{m}\int_{-\tau_j}^{0} \eta(t+\vartheta)\,\varphi_{ij}(\vartheta)\,d\vartheta, \tag{3.1} \]
an analogous form of the operations \(\varepsilon_i[\xi]\) (2.13), without having recourse here to equation (2.1), just as the operations \(\varepsilon_i[\xi]\) were constructed above.
The operations \(\varepsilon_i[\xi]\) (2.13) can then be regarded as an extension [5] of the operations \(\varphi_i[\eta]\) from the linear manifold \(\{\eta(t)\}\) to the set \(\{\xi(t)\}\). Operations \(\varphi_i[\eta]\) of the form (3.1) for the problem of observing the system (1.1), (1.2) can be obtained in a sufficiently natural way if this problem is solved under the additional condition
\[ \rho^*[\varphi_i] = \min \quad (i = 1,\ldots,n), \tag{3.2} \]
where the norm \(\rho^*[\varphi_i]\) of the operation \(\varphi_i\) is defined by the equality
\[ \rho^*[\varphi_i] = \left[\int_{-\tau}^{0} \chi^2(\vartheta)\,d\mu(\vartheta)\right]^{1/2}, \tag{3.3} \]
and the measure \(d\mu(\vartheta)\) on the interval \([-\tau,0]\) is chosen so that \(\mu(0)-\mu(-0)=\mu \ne 0\). The operations \(\varphi_i[\eta]\) (3.1) are then conveniently written in the form
\[ x_i(t) = \varphi_i[\eta(t+\vartheta)] = \int_{-\tau}^{0} \eta(t+\vartheta)\chi_i(\vartheta)\,d\mu(\vartheta), \tag{3.4} \]
where the function \(\chi_i(\vartheta)\) is related to the quantities \(\varphi_i\) and \(\varphi_{ij}(\vartheta)\) by known relations. (A general method for solving such problems is described, for example, in [2].) In this way, for the problem of observing the system (1.1), (1.2), one can construct optimal operations \(\varphi_i^0[\eta]\) having the least possible norms \(\rho^*[\varphi_i^0]\) (3.2), (3.3). Since the operations \(\varepsilon_i[\xi]\) are extensions of the operations \(\varphi_i^0[\eta]\) from the family of functions \(\{\eta(t)\}\) to the family of functions \(\{\xi(t)\}\), and the norm \(\rho^*[\varphi_i^0]\) is the minimum possible, the inequality
\[ \rho^*[\varphi_i^0] \leq \rho^*[\varepsilon_i]. \tag{3.5} \]
must hold.
Thus, it is now of interest to investigate the problem of the closeness of the norms \(\rho^*[\varphi_i^0]\) and \(\rho^*[\varepsilon_i]\) for sufficiently large values of \(n\), with a suitable choice of the measure \(d\mu(\vartheta)\) in (3.3) and (3.4). This investigation is intended to be the subject of a separate work.
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Received by the editors
June 15, 1965
V. A. Steklov Mathematical Institute,
Sverdlovsk Branch