EC-3.3 Systems of Equations, and the State Model

What this is and why it exists

Everything so far has been one equation at a time. Real systems are several coupled equations at once, and writing them as a list quickly becomes unmanageable.

The state model is the answer, and it rests on one observation. Any number of coupled equations, of any order, can be rewritten as a single first-order equation in vectors and matrices. Once that is done, the eigenvalues of one matrix answer the stability question in a single step, no matter how large the system.

This is the form modern control and modern simulation both use. Meeting it here, as mathematics, means that when a control course presents it you are recognising an idea rather than learning notation.

The vocabulary

• State — the smallest set of quantities that, together with future input, determines everything about the system's future.
• State vector — those quantities collected into a column.
• State equation — the equation saying how the state vector changes, in terms of itself and the input.
• Output equation — the equation saying which combination of state quantities is actually measured.
• System matrix — the matrix multiplying the state in the state equation. It carries the system's own behaviour.
• Input matrix — the matrix saying how the input pushes each state quantity.
• Mode — one exponential behaviour contributed by one eigenvalue.
• Equilibrium — a state the system would remain in if placed there and left alone.
• Matrix exponential — the matrix-valued function that plays the role e to the power a t plays for a single equation.

The mental model

Start with the reduction. Take a second-order equation and give a name to the unknown, then a second name to its derivative. Now the derivative of the first name is the second name, which is one first-order equation. And the derivative of the second name is whatever the original equation says it is, which is another. Two first-order equations, no information lost.

The same move works at any order. An equation of order n becomes n first-order equations, at the cost of carrying n unknowns instead of one.

Collect those unknowns into a vector and the whole system reads as one line. The derivative of the state vector equals the system matrix times the state vector, plus the input matrix times the input. Alongside it sits a second statement: the output equals some matrix times the state, possibly plus a term in the input.

Separating those two statements is what makes the form general. The first says what is happening inside. The second says what you can see from outside. They are different questions and most systems answer them differently.

Now the payoff. Each eigenvalue of the system matrix contributes one mode, and that mode is an exponential in time with the eigenvalue as its exponent. So the eigenvalues, and nothing else about the matrix, decide the natural behaviour.

A negative real part means that mode decays. A positive real part means it grows without bound. An imaginary part means it oscillates. One eigenvalue with a positive real part makes the whole system unstable, however well behaved every other mode is. There is no averaging.

This is the moment eigenvalues earn their place. If you have met them only as an exercise in finding roots of a determinant, this is what they were for.

The solution itself generalises directly from the single-equation case. There the answer was the initial value multiplied by e to the power a t. Here it is the matrix exponential applied to the initial state, plus an integral carrying the input's contribution. Computing the matrix exponential by diagonalising the matrix is the case worth doing by hand. It makes plain that the matrix exponential is the ordinary exponential applied to each eigenvalue separately.

One practical note on choosing states for a circuit. Capacitor voltages and inductor currents are the natural choice, because those are the quantities that cannot change instantaneously. Choose them and the matrices fall out of the two circuit laws with almost no algebra. Choose anything else and you will fight the problem.

What you should now be able to explain or do

• Rewrite a higher-order equation as a first-order system, and say why nothing is lost.
• Write down the state and output equations and say what each matrix does.
• Predict stability and oscillation from the eigenvalues of the system matrix.
• Choose sensible state quantities for a circuit and explain the choice.
• Say what the matrix exponential is and how it relates to the scalar exponential.

Check yourself