D
Discrete-Time Dynamic Systems
This brief review is meant as a refresher for readers who are familiar with the topic. It summarizes those concepts that are used within the textbook. It also introduces the notations adopted in this book.
D.1 Discrete-Time Dynamic Systems
A dynamic system is a system whose variables proceed in time. A concise representation of such a system is the state space model. The model consists of a state vector x(i), where i is an integer variable representing the discrete time. The dimension of x(i), called the order of the system, is M. We assume that the state vector is real-valued. The finite-state case is introduced in Chapter 4.
The process can be influenced by a control vector (input vector) u(i) with dimension L. The output of the system is given by the measurement vector (observation vector) z(i) with dimension N. The output is modelled as a memoryless vector that depends on the current values of the state vector and the control vector.
By definition, the state vector holds the minimum number of variables that completely summarize the past of the system. Therefore, the state vector at time i + 1 is derived from the state vector and the control vector, both valid at time i:
where f(.) is a possible non-linear vector function, the system function, that may depend explicitly on time and h(.) is the measurement function.
Note that if the time series starts at i = i0 and the initial condition x(i0) is given, along with the input sequence u(i0), u(i0 + 1), …, then (D.1) can be used to solve x(i) for every i > i0. Such a solution is called a trajectory.
D.2 Linear Systems
If the system is linear, then the equations evolve into matrix–vector equations according to
The matrices F(i), L(i), H(i) and D(i) must have the appropriate dimensions. They have the following names:
- F(i) system matrix
- L(i) distribution matrix (gain matrix)
- H(i) measurement matrix
- D(i) feedforward gain
Often, the feedforward gain is zero.
The solution of the linear system is found by introduction of the transition matrix Φ(i, i0), recursively defined by
Given the initial condition x(i0) at i = i0, and assuming that the feedforward gain is zero, the solution is found as
D.3 Linear Time-Invariant Systems
In the linear time-invariant case, the matrices become constants:
and the transition matrix simplifies to
Therefore, the input/output relation of the system is
The first term on the r.h.s. is the free response, the second term is the particular response and the third term is the feedforward response.
D.3.1 Diagonalization of a System
We assume that the system matrix F has M distinct eigenvectors vk with corresponding (distinct) eigenvalues λk. Thus, by definition, Fvk = λkvk. We define the eigenvalue matrix 
as the M × M diagonal matrix containing the eigenvalues at the diagonal, that is 
k,k = λk, and the eigenvector matrix V as the matrix containing the eigenvectors as its column vectors, that is V = [v0 ⋅⋅⋅ vM–1]. Consequently:
A corollary of (D.8) is that the power of F is calculated as follows:
Equation (D.9) is useful to decouple the system into a number of (scalar) first-order systems, that is to diagonalize the system. For that purpose, define the vector
Then the state equation transforms into
and the solution is found as
The kth component of the vector y(i) is
The measurement vector is obtained from y(i) by substitution of x(i) = Vy(i) in Equation (D.5).
If the system matrix does not have distinct eigenvalues, the situation is a little more involved. In that case, the matrix Λ gets the Jordan form with eigenvalues at the diagonal, but with ones at the superdiagonal. The free response becomes combinations of terms pk(i − i0)λi − i0k where pk(i) are polynomials in i with order one less than the multiplicity of λk.
D.3.2 Stability
In dynamic systems there are various definitions of the term stability. We return to the general case (D.1) first and then check to see how the definition applies to the linear time-invariant case. Let xa(i) and xb(i) be the solutions of (D.1) with a given input sequence and with initial conditions xa(i0) and xb(i0), respectively.
The solution xa(i) is stable if, for every ϵ> 0, we can find δ > 0 such that for every xb(i0) with ||xb(i0) – xa(i0)|| < δ is such that ||xb(i) – xa(i)|| < ϵ for all i ≥ i0. Loosely speaking, the system is stable if small changes in the initial condition of stable solution cannot lead to very large changes of the trajectory.
The solution is xa(i) is asymptotically stable if it is stable and if ||xb(i) – xa(i)|| → 0 as i → ∞ provided that ||xb(i0) – xa(i0)|| is not too large. Therefore, the additional requirement for a system to be asymptotically stable is that the initial condition does not influence the solution in the long range.
For linear systems, the stability of one solution assures the stability of all other solutions. Thus, stability is a property of the system and not just one of its solutions. From Equation (D.13) it can be seen that a linear time-invariant system is asymptotically stable if and only if the magnitude of the eigenvalues are less than one; that is the eigenvalues must all be within the (complex) unit circle.
A linear time-invariant system has BIBO stability (bounded input, bounded output) if a bounded input sequence always gives rise to a bounded output sequence. Note that asymptotical stability implies BIBO stability, but the reverse is not true. A system can be BIBO stable, while it is not asymptotical stable.
Selected Bibliography
- Åström, K.J. and Wittenmark, B., Computer-Controlled Systems – Theory and Design, Second Edition, Prentice Hall International, Englewood Cliffs, NJ, 1990.
- Luenberger, D.G., Introduction to Dynamic Systems, Theory, Models and Applications, John Wiley & Sons, Toronto, 1979.