# 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:

- Unstable
- A system is said to be unstable if the system response approaches infinity as time approaches infinity. If our system is G(t), then, we can say a system is unstable if:

Also, a key concept when we are talking about stability of systems is the concept of an **equilibrium point**:

- Equilibrium Point
- Given a system
*f*such that:

A particular state *x*_{e} is called an **equilibrium point** if

for all time *t* in the interval , where t_{0} is the starting time of the system.

The definitions below typically require that the equilibrium point be zero. If we have an equilibrium point *x _{e} = 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**:

- Zero State
- A state
*x*_{z}is a**zero state**if*x*_{z}= 0. A zero state may or may not be an equilibrium 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 t_{0}, there exists an associated finite constant *k*(t_{0}) 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
- The system is defined to be
**uniformly stable**if it is stable for all initial values of t_{0}:

Uniform stability is a more general, and more powerful form of stability than was previously provided.

- Asymptotic Stability
- A system is defined to be
**asymptotically stable**if:

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.

- Uniform Asymptotic Stability
- A system is defined to be
**uniformly asymptotically stable**if the system is asymptotically stable for all values of t_{0}.

- Exponential Stability
- A system is defined to be
**exponentially stable**if the system response decays exponentially towards zero as time approaches infinity.

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 x_{0} = 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 t_{0} 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**.

[Lyapunov Equation]

Where A is the system matrix, and M and N are *p* × *p* square matrices.

- Lyapunov Stability Theorem
- An LTI system is stable if there exists a matrix M that satisfies the
**Lyapunov Equation**where N is an arbitrary positive definite matrix, and M is a unique positive definite matrix.

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:

- Lyapunov Stability Theorem (alternate)
- If all the eigenvalues of the system matrix A have negative real parts, then the Lyapunov Equation has a unique solution M for every positive definite matrix N, and the solution can be calculated by:

If the matrix M can be calculated in this manner, the system is asymptotically stable.