Encyclopedia of Systems and Control

Living Edition
| Editors: John Baillieul, Tariq Samad

Linear Systems: Continuous-Time, Time-Varying State Variable Descriptions

  • P. J. AntsaklisEmail author
Living reference work entry
DOI: https://doi.org/10.1007/978-1-4471-5102-9_190-1


Continuous-time processes that can be modeled by linear differential equations with time-varying coefficients can be written in terms of state variable descriptions of the form \(\dot{x}(t) = A(t)x(t) + B(t)u(t),\ y(t) = C(t)x(t) + D(t)u(t)\). The response of such systems due to a given input and initial conditions is derived using the Peano-Baker series. Equivalence of state variable descriptions is also discussed.


Linear systems Continuous-time Time-varying State variable descriptions 


Dynamical processes that can be described or approximated by linear high-order ordinary differential equations with time-varying coefficients can also be described, via a change of variables, by state variable descriptions of the form
$$\begin{array}{c} \dot{x}(t) = A(t)x(t) + B(t)u(t);\quad x(t_{0}) = x_{0} \\ y(t) = C(t)x(t) + D(t)u(t),\\ \end{array}$$
where x(t) (\(t \in \mathbb{R}\), the set of reals) is a column vector of dimension n (\(x(t) \in {\mathbb{R}}^{n}\)) and A(t), B(t), C(t), D(t) are matrices with entries functions of time t. \(A(t) = [a_{ij}(t)],\ a_{ij}(t) : \mathbb{R} \rightarrow \mathbb{R}\). \(A(t) \in {\mathbb{R}}^{n\times n},B(t) \in {\mathbb{R}}^{n\times m},C(t) \in {\mathbb{R}}^{p\times n},D(t) \in {\mathbb{R}}^{p\times m}\). The input vector is \(u(t) \in {\mathbb{R}}^{m}\) and the output vector is \(y(t) \in {\mathbb{R}}^{p}\). The vector differential equation in (1) is the state equation, while the algebraic equation is the output equation.

The advantage of the state variable description (1) is that given an input \(u(t),\ t \geq 0\) and an initial condition x(t 0) = x 0, the state trajectory or motion for \(t \geq t_{0}\) can be conveniently characterized. To derive the expressions, we first consider the homogenous state equation and the corresponding initial value problem.

Solving \(\dot{x}(t) = A(t)x(t);\ x(t_{0}) = x_{0}\)

Consider the homogenous equation with the initial condition
$$x(t) = A(t)x(t);\quad x(t_{0}) = x_{0}$$
where x(t) = [x 1(t), , x n (t)] T is the state (column) vector of dimension n and A(t) is an n ×n matrix with entries functions of time that take on values from the field of reals (\(A \in {\mathbb{R}}^{n\times n}\)).

Under certain assumptions on the entries of A(t), a solution of (2) exists and it is unique. These assumptions are satisfied, and a solution exists and is unique in the case, for example, when the entries of A(t) are continuous functions of time. In the following we make this assumption.

To find the unique solution of (2), we use the method of successive approximations which when applied to
$$\dot{x}(t) = f(t,x(t)),\quad x(t_{0}) = x_{0}$$
is described by
$$\begin{array}{c} \phi _{0}(t) = x_{0} \\ \phi _{m}(t) = x_{0} +\displaystyle\int \limits _{ t_{0}}^{t}f(\tau ,\phi _{m-1}(\tau ))d\tau ,\quad m = 1,2,\ldots \\ \end{array}$$
As m, ϕ m converges to the unique solution of (3), assuming the f satisfies certain conditions.
Applying the method of successive approximations to (2) yields
$$\begin{array}{c} \phi _{0}(t) = x_{0} \\ \phi _{1}(t) = x_{0} +\displaystyle\int \limits _{ t_{0}}^{t}A(\tau )x_{0}d\tau \\ \phi _{2}(t) = x_{0} +\displaystyle\int \limits _{ t_{0}}^{t}A(\tau )\phi _{1}(\tau )x_{0}d\tau \\ \vdots \\ \phi _{m}(t) = x_{0} +\displaystyle\int \limits _{ t_{0}}^{t}A(\tau )\phi _{m-1}(\tau )x_{0}d\tau \\ \end{array}$$
from which
$$\displaystyle\begin{array}{rcl} \phi _{m}(t)& =& \left [I +\displaystyle\int \limits _{ t_{0}}^{t}A(\tau _{ 1})d\tau _{1}+\displaystyle\int \limits _{t_{0}}^{t}A(\tau _{ 1})\displaystyle\int \limits _{t_{0}}^{\tau _{1} }A(\tau _{2})d\tau _{2}d\tau _{1}+\ldots \right . \\ & & +\displaystyle\int \limits _{t_{0}}^{t}A(\tau _{ 1})\displaystyle\int \limits _{t_{0}}^{\tau _{1} }A(\tau _{2})\ldots \displaystyle\int \limits _{t_{0}}^{\tau _{m-1} }\left .A(\tau _{m})d\tau _{m}\ldots d\tau _{1}\right ]x_{0} \\ \end{array}$$
When m, and under the above continuity assumptions on A(t), ϕ m (t) converges to the unique solution of (2), i.e.,
$$\phi (t) = \Phi (t,t_{0})x_{0}$$
$$\Phi (t,t_{0}) = I +\displaystyle\int \limits _{ t_{0}}^{t}A(\tau _{ 1})d\tau _{1}+\displaystyle\int \limits _{t_{0}}^{t}A(\tau _{ 1})\left [\displaystyle\int \limits _{t_{0}}^{\tau _{1} }A(\tau _{2})d\tau _{2}\right ]d\tau _{1}+\ldots$$
Note that \(\Phi (t_{0},t_{0}) = I\) and by differentiation it can be seen that
$$\dot{\phi }(t,t_{0}) = A(t)\phi (t,t_{0}),$$
as expected, since (5) is the solution of (2). The n ×n matrix \(\Phi (t,t_{0})\) is called the state transition matrix of (2). The defining series (6) is called the Peano-Baker series.
Note that when A(t) = A, a constant matrix, then (6) becomes
$$\Phi (t,t_{0}) = I +\displaystyle\sum \limits _{ k=1}^{\infty }\frac{{A}^{k}{(t - t_{ 0})}^{k}} {k!}$$
which is the defining series for the matrix exponential \({e}^{A(t-t_{0})}\) (see “Linear Systems: Continuous-Time, Time-Invariant, State Variable Descriptions,” Panos J. Antsaklis).

System Response

Based on the solution (5) of \(\dot{x} = A(t)x(t)\), the solution of the non-homogenous equation
$$\dot{x}(t) = A(t)x(t) + B(t)u(t);\quad x(t_{0}) = x_{0}$$
can be shown to be
$$\phi (t) = \Phi (t,t_{0})x_{0} +\displaystyle\int _{ t_{0}}^{t}\Phi (t,\tau )B(\tau )u(\tau )d\tau .$$
Equation (10) is the variation of constants formula. This result can be shown via direct substitution of (10) into (9); note that \(\phi (t) = \Phi (t_{0},t_{0})x_{0} = x_{0}\). That (10) is a solution can also be shown using a change of variables in (9), namely,
$$z(t) = \Phi (t_{0},t)x(t).$$
Equation (10) is the sum of two parts, the state response (when u(t) = 0 and the system is driven only by the initial state conditions) and the input response (when x 0 = 0 and the system is driven only by the input u(t)); this illustrates the linear system principle of superposition.
In view of (10), the output y(t\((= C(t)x(t) + D(t)u(t))\) is
$$y(t) = C(t)\Phi (t,t_{0})x_{0} +\displaystyle\int \limits _{ t_{0}}^{t}C(t)\Phi (t,\tau )B(\tau )u(\tau )d\tau + D(t)u(t)$$
$$= C(t)\Phi (t,t_{0})x_{0} +\displaystyle\int \limits _{ t_{0}}^{t}[C(t)\Phi (t,\tau )B(\tau ) + D(t)\delta (t-\tau )]u(\tau )d\tau$$
The second expression involves the Dirac (or impulse or delta) function δ(t) and is derived based on the basic property for δ(t), namely,
$$f(t) =\displaystyle\int \limits _{ -\infty }^{+\infty }\delta (t-\tau )f(\tau )d\tau ,$$
where δ(tτ) denotes an impulse applied at time τ = t.

Properties of the State Transition Matrix \(\Phi (t,t_{0})\)

In general it is difficult to determine \(\Phi (t,t_{0})\) explicitly; however, \(\Phi (t,t_{0})\) may be readily determined in a number of special cases including the cases in which A(t) = A, A(t) is diagonal, A(t)A(τ) = A(τ)A(t).

Consider \(\dot{x} = A(t)x\). We can derive a number of important properties which are described below. It can be shown that given n linearly independent initial conditions x 0i , the corresponding n solutions ϕ i (t) are also linearly independent. Let a fundamental matrix \(\Psi (t)\) of \(\dot{x} = A(t)x\) be an n ×n matrix, the columns of which are a set of linearly independent solutions \(\phi _{1}(t),\ldots ,\phi _{n}(t)\). The state transition matrix \(\Phi \) is the fundamental matrix determined from solutions that correspond to the initial conditions \({[1,0,0,\ldots ]}^{T},\ {[0,1,0,\ldots ,0]}^{T},\ldots {[0,0,\ldots ,1]}^{T}\) (recall that \(\Phi (t_{0},t_{0}) = I\)). The following are properties of \(\Phi (t,t_{0})\):
  1. (i)

    \(\Phi (t,t_{0}) = \Psi (t){\Psi }^{-1}(t_{0})\) with \(\Psi (t)\) any fundamental matrix.

  2. (ii)

    \(\Phi (t,t_{0})\) is nonsingular for all t and t 0.

  3. (iii)

    \(\Phi (t,\tau ) = \Phi (t,\sigma )\Phi (\sigma ,\tau )\) (semigroup property).

  4. (iv)

    \({[\Phi (t,t_{0})]}^{-1} = \Phi (t_{0},t)\).

In the special case of time-invariant systems and \(\dot{x} = Ax\), the above properties can be written in terms of the matrix exponential since
$$\Phi (t,t_{0}) = {e}^{A(t-t_{0})}.$$

Equivalence of State Variable Descriptions

Given the system
$$\begin{array}{c} \dot{x} = A(t)x + B(t)u \\ y = C(t)x + D(t)u\\ \end{array}$$
consider the new state vector \(\tilde{x}\)
$$\tilde{x}(t) = P(t)x(t)$$
where P − 1(t) exists and P and P − 1 are continuous. Then the system
$$\begin{array}{c} \dot{\tilde{x}} =\tilde{ A}(t)\tilde{x} +\tilde{ B}(t)u\\ y =\tilde{ C}(t)\tilde{x} +\tilde{ D}(t)u \\ \end{array}$$
$$\begin{array}{c} \tilde{A}(t) = [P(t)A(t) +\dot{ P}(t)]{P}^{-1}(t) \\ \tilde{B}(t) = P(t)B(t) \\ \tilde{C}(t) = C(t){P}^{-1}(t) \\ \tilde{D}(t) = D(t)\\ \end{array}$$
is equivalent to (1). It can be easily shown that equivalent descriptions give rise to the same impulse responses.


State variable descriptions for continuous-time time-varying systems were introduced and the state and output responses to inputs and initial conditions were derived. The equivalence of state variable representations was also discussed.


Recommended Reading

Additional information regarding the time-varying case may be found in Brockett (1970), Rugh (1996), and Antsaklis and Michel (2006). For historical comments and extensive references on some of the early contributions, see Sontag (1990) and Kailath (1980).


  1. Antsaklis PJ, Michel AN (2006) Linear systems. Birkhauser, BostonGoogle Scholar
  2. Brockett RW (1970) Finite dimensional linear systems. Wiley, New YorkGoogle Scholar
  3. Kailath T (1980) Linear systems. Prentice-Hall, Englewood CliffsGoogle Scholar
  4. Miller RK, Michel AN (1982) Ordinary differential equations. Academic, New YorkGoogle Scholar
  5. Rugh WJ (1996) Linear system theory, 2nd edn. Prentice-Hall, Englewood CliffsGoogle Scholar
  6. Sontag ED (1990) Mathematical control theory: deterministic finite dimensional systems. Texts in applied mathematics, vol 6. Springer, New YorkGoogle Scholar

Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  1. 1.Department of Electrical EngineeringUniversity of Notre DameNotre DameUSA