Control Systems/State-Space Stability
State-Space Stability[edit | edit source]
If a system is represented in the state-space domain, it doesn't make sense to convert that system to a transfer function representation (or even a transfer matrix representation) in an attempt to use any of the previous stability methods. Luckily, there are other analysis methods that can be used with the state-space representation to determine if a system is stable or not. First, let us first introduce the notion of unstability:
Also, a key concept when we are talking about stability of systems is the concept of an equilibrium point:
The definitions below typically require that the equilibrium point be zero. If we have an equilibrium point xe = a, then we can use the following change of variables to make the equilibrium point zero:
We will also see below that a system's stability is defined in terms of an equilibrium point. Related to the concept of an equilibrium point is the notion of a zero point:
Stability Definitions[edit | edit source]
The equilibrium x = 0 of the system is stable if and only if the solutions of the zero-input state equation are bounded. Equivalently, x = 0 is a stable equilibrium if and only if for every initial time t0, there exists an associated finite constant k(t0) such that:
Where sup is the supremum, or "maximum" value of the equation. The maximum value of this equation must never exceed the arbitrary finite constant k (and therefore it may not be infinite at any point).
Uniform stability is a more general, and more powerful form of stability than was previously provided.
A time-invariant system is asymptotically stable if all the eigenvalues of the system matrix A have negative real parts. If a system is asymptotically stable, it is also BIBO stable. However the inverse is not true: A system that is BIBO stable might not be asymptotically stable.
For linear systems, uniform asymptotic stability is the same as exponential stability. This is not the case with non-linear systems.
Marginal Stability[edit | edit source]
Here we will discuss some rules concerning systems that are marginally stable. Because we are discussing eigenvalues and eigenvectors, these theorems only apply to time-invariant systems.
- A time-invariant system is marginally stable if and only if all the eigenvalues of the system matrix A are zero or have negative real parts, and those with zero real parts are simple roots of the minimal polynomial of A.
- The equilibrium x = 0 of the state equation is uniformly stable if all eigenvalues of A have non-positive real parts, and there is a complete set of distinct eigenvectors associated with the eigenvalues with zero real parts.
- The equilibrium x = 0 of the state equation is exponentially stable if and only if all eigenvalues of the system matrix A have negative real parts.
Eigenvalues and Poles[edit | edit source]
A Linearly Time Invariant (LTI) system is stable (asymptotically stable, see above) if all the eigenvalues of A have negative real parts. Consider the following state equation:
We can take the Laplace Transform of both sides of this equation, using initial conditions of x0 = 0:
Subtract AX(s) from both sides:
Assuming (sI - A) is nonsingular, we can multiply both sides by the inverse:
Now, if we remember our formula for finding the matrix inverse from the adjoint matrix:
We can use that definition here:
Let's look at the denominator (which we will now call D(s)) more closely. To be stable, the following condition must be true:
And if we substitute λ for s, we see that this is actually the characteristic equation of matrix A! This means that the values for s that satisfy the equation (the poles of our transfer function) are precisely the eigenvalues of matrix A. In the S domain, it is required that all the poles of the system be located in the left-half plane, and therefore all the eigenvalues of A must have negative real parts.
Impulse Response Matrix[edit | edit source]
We can define the Impulse response matrix, G(t, τ) in order to define further tests for stability:
[Impulse Response Matrix]
The system is uniformly stable if and only if there exists a finite positive constant L such that for all time t and all initial conditions t0 with the following integral is satisfied:
In other words, the above integral must have a finite value, or the system is not uniformly stable.
In the time-invariant case, the impulse response matrix reduces to:
In a time-invariant system, we can use the impulse response matrix to determine if the system is uniformly BIBO stable by taking a similar integral:
Where L is a finite constant.
Positive Definiteness[edit | edit source]
These terms are important, and will be used in further discussions on this topic.
- f(x) is positive definite if f(x) > 0 for all x.
- f(x) is positive semi-definite if for all x, and f(x) = 0 only if x = 0.
- f(x) is negative definite if f(x) < 0 for all x.
- f(x) is negative semi-definite if for all x, and f(x) = 0 only if x = 0.
A Hermitian matrix X is positive definite if all its principle minors are positive. Also, a matrix X is positive definite if all its eigenvalues have positive real parts. These two methods may be used interchangeably.
Positive definiteness is a very important concept. So much so that the Lyapunov stability test depends on it. The other categorizations are not as important, but are included here for completeness.
Lyapunov Stability[edit | edit source]
Lyapunov's Equation[edit | edit source]
For linear systems, we can use the Lyapunov Equation, below, to determine if a system is stable. We will state the Lyapunov Equation first, and then state the Lyapunov Stability Theorem.
Where A is the system matrix, and M and N are p × p square matrices.
Notice that for the Lyapunov Equation to be satisfied, the matrices must be compatible sizes. In fact, matrices A, M, and N must all be square matrices of equal size. Alternatively, we can write:
If the matrix M can be calculated in this manner, the system is asymptotically stable.