Control Systems/Time Variant System Solutions
Contents
General Time Variant Solution[edit]
The statespace equations can be solved for timevariant systems, but the solution is significantly more complicated than the timeinvariant case. Our timevariant state equation is given as follows:
We can say that the general solution to timevariant stateequation is defined as:
[TimeVariant General Solution]
Matrix Dimensions:
A: p × p
B: p × q
C: r × p
D: r × q
The function φ is called the statetransition matrix, because it (like the matrix exponential from the timeinvariant case) controls the change for states in the state equation. However, unlike the timeinvariant case, we cannot define this as a simple exponential. In fact, φ can't be defined in general, because it will actually be a different function for every system. However, the statetransition matrix does follow some basic properties that we can use to determine the statetransition matrix.
In a timevariant system, the general solution is obtained when the statetransition matrix is determined. For that reason, the first thing (and the most important thing) that we need to do here is find that matrix. We will discuss the solution to that matrix below.
State Transition Matrix[edit]
The state transition matrix φ is a matrix function of two variables (we will say t and τ). Once the form of the matrix is solved, we will plug in the initial time, t_{0} in place of the variable τ. Because of the nature of this matrix, and the properties that it must satisfy, this matrix typically is composed of exponential or sinusoidal functions. The exact form of the statetransition matrix is dependent on the system itself, and the form of the system's differential equation. There is no single "template solution" for this matrix.
The state transition matrix φ is not completely unknown, it must always satisfy the following relationships:
And φ also must have the following properties:

1. 2. 3. 4.
If the system is timeinvariant, we can define φ as:
The reader can verify that this solution for a timeinvariant system satisfies all the properties listed above. However, in the timevariant case, there are many different functions that may satisfy these requirements, and the solution is dependent on the structure of the system. The statetransition matrix must be determined before analysis on the timevarying solution can continue. We will discuss some of the methods for determining this matrix below.
TimeVariant, Zero Input[edit]
As the most basic case, we will consider the case of a system with zero input. If the system has no input, then the state equation is given as:
And we are interested in the response of this system in the time interval T = (a, b). The first thing we want to do in this case is find a fundamental matrix of the above equation. The fundamental matrix is related
Fundamental Matrix[edit]
Given the equation:
The solutions to this equation form an ndimensional vector space in the interval T = (a, b). Any set of n linearlyindependent solutions {x_{1}, x_{2}, ..., x_{n}} to the equation above is called a fundamental set of solutions.
A fundamental matrix FM is formed by creating a matrix out of the n fundamental vectors. We will denote the fundamental matrix with a script capital X:
The fundamental matrix will satisfy the state equation:
Also, any matrix that solves this equation can be a fundamental matrix if and only if the determinant of the matrix is nonzero for all time t in the interval T. The determinant must be nonzero, because we are going to use the inverse of the fundamental matrix to solve for the statetransition matrix.
State Transition Matrix[edit]
Once we have the fundamental matrix of a system, we can use it to find the state transition matrix of the system:
The inverse of the fundamental matrix exists, because we specify in the definition above that it must have a nonzero determinant, and therefore must be nonsingular. The reader should note that this is only one possible method for determining the state transition matrix, and we will discuss other methods below.
Example: 2Dimensional System[edit]
Given the following fundamental matrix, Find the statetransition matrix.
the first task is to find the inverse of the fundamental matrix. Because the fundamental matrix is a 2 × 2 matrix, the inverse can be given easily through a common formula:
The statetransition matrix is given by:
Other Methods[edit]
There are other methods for finding the state transition matrix besides having to find the fundamental matrix.
 Method 1
 If A(t) is triangular (upper or lower triangular), the state transition matrix can be determined by sequentially integrating the individual rows of the state equation.
 Method 2
 If for every τ and t, the state matrix commutes as follows:
 Then the statetransition matrix can be given as:
 The state transition matrix will commute as described above if any of the following conditions are true:
 A is a constant matrix (timeinvariant)
 A is a diagonal matrix
 If , where is a constant matrix, and f(t) is a scalarvalued function (not a matrix).
 If none of the above conditions are true, then you must use method 3.
 Method 3
 If A(t) can be decomposed as the following sum:
 Where M_{i} is a constant matrix such that M_{i}M_{j} = M_{j}M_{i}, and f_{i} is a scalarvalued function. If A(t) can be decomposed in this way, then the statetransition matrix can be given as:
It will be left as an exercise for the reader to prove that if A(t) is timeinvariant, that the equation in method 2 above will reduce to the statetransition matrix .
Example: Using Method 3[edit]
Use method 3, above, to compute the statetransition matrix for the system if the system matrix A is given by:
We can decompose this matrix as follows:
Where f_{1}(t) = t, and f_{2}(t) = 1. Using the formula described above gives us:
Solving the two integrations gives us:
The first term is a diagonal matrix, and the solution to that matrix function is all the individual elements of the matrix raised as an exponent of e. The second term can be decomposed as:
The final solution is given as:
TimeVariant, Nonzero Input[edit]
If the input to the system is not zero, it turns out that all the analysis that we performed above still holds. We can still construct the fundamental matrix, and we can still represent the system solution in terms of the state transition matrix φ.
We can show that the general solution to the statespace equations is actually the solution: