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Adding An Integral Term to State-Space Control

Recall that in classical control, adding an integral in the compensator eliminates steady-state errors, usually at the cost of worsening the stability. To minimize the distabilizing effects of adding the integral term, we often adjust the other terms (such as the proportional and derivative terms in the PID controller). We can also add an integral to the state-space controller to eliminate steady-state errors, and in the case of state-space control, we have more feedback gains to modify, so we can usually avoid a stability penalty.

State-space control without an integral term

The generic state-space equations are

\textbf{E}\frac{d}{dt}\textbf{x} = \textbf{Ax} + \textbf{B}u

y = \textbf{Cx}

u = -\textbf{Kx} + K_r y_d

where x is the state, u is the "plant" input, y_d is the primary input and is usually the desired output of the system, and y is the actual output of the system. Once one has determined \textbf{E}, \textbf{A}, \textbf{B}, and \textbf{C}, the next step is to determine a good set of values for \textbf{K} that result in good closed-loop performance.

A key point to note is that the state-feedback gains do not depend on the scaling of the primary input, that is, the K's do not depend on K_r. One can determine K to optimize dynamic performance, and then scale the primary input by K_r so as to achieve a desired steady-state without changing the dynamics! Changing K_r does not change the system poles, just the steady-state.

Once K and the K_r have been determined, the closed-loop system is given by

\textbf{E}\frac{d}{dt}\textbf{x} = \left(\textbf{A-BK}\right)\textbf{x} + \textbf{B}K_r y_d = \textbf{A}_{clp} \textbf{x} + \textbf{B}_{clp} y_d,

y = \textbf{Cx}.

where we introduced the subscripted-with-clp notation to note that the closed-loop system has the same structure as the open loop system, but with modified matrices. Note also that the output remains the same weighted combination of states, so \textbf{C}_{clp} = \textbf{C}.

We can compute K_r from the formula

K_r = -\left(\textbf{C} \left(\textbf{A-BK}\right)^{-1}\textbf{B}\right)^{-1},
which follows from insisting that if y_d is a constant with respect to time, then y \rightarrow y_d as t \rightarrow \infty. Since K_r is dependent on the state-space matrices and the state-feedback gains, any modeling errors can easily lead to an incorrect K_r, resulting in a persistent error in the steady-state.

An alternative is to create an extra state for the system, one that integrates y_d - y. If one can find a set of feedback gains for this augmented system that yields a stable closed-loop system, then one guarantees a steady-state match without K_r, as we show below.

Adding the tracking-error-integral state

Consider adding an extra state to the system, denoted by w, and defined by the differential equation,

\frac{d}{dt} w(t) = y(t) - y_d(t) = \textbf{Cx}(t) - y_d(t).
Based on the above differential equation, w(t) will be the integral of the tracking error.

We can add this extra state to a state-space model by appending w to \textbf{x}, as in

\textbf{x}_+=\begin{bmatrix} \textbf{x}\\ w \end{bmatrix}.

The extra state means that there is a row added to \textbf{B}, a column added to \textbf{C}, and a row and column added to both \textbf{E} and \textbf{A}. The expanded matrices, denoted with a subscript + are:

\textbf{A}_+ = \begin{bmatrix}\textbf{A} & 0\\ \textbf{C} & 0\end{bmatrix}
\textbf{B}_+ = \begin{bmatrix}\textbf{B} \\ 0 \end{bmatrix}
\textbf{C}_+ = \begin{bmatrix}\textbf{C} & 0\end{bmatrix}
and
\textbf{E}_+ = \begin{bmatrix}\textbf{E} & 0 \\ 0 & 1\end{bmatrix}

To include the integral term, we alter the representation of the state-space control problem to include the reference input in a different way. That is,

\textbf{E}_+\frac{d}{dt}\textbf{x}_+ = \textbf{A}_+ \textbf{x}_+ + \textbf{B}_+ u + \begin{bmatrix} \textbf{0}\\ -1 \end{bmatrix} y_d,

y = \textbf{C}_+ \textbf{x}_+

u = -\textbf{K}_+ \textbf{x}_+

Note that for the extra row in B_+, we used the value of zero, which means that the extra state has the equation

\frac{d}{dt} w = y - y_d = \textbf{C} x - y_d,
and ensures a steady-state match without requiring a scaling of the reference input.

Substituting for u gives us a closed-loop state-space system,

\textbf{E}_+\frac{d}{dt}\textbf{x}_+ = \left( \textbf{A}_+ - \textbf{B}_+ \textbf{K}_+ \right) \textbf{x}_+ + \begin{bmatrix} \textbf{0}\\ -1 \end{bmatrix} y_d = \textbf{A}_{clp} \textbf{x}_+ + \textbf{B}_{clp} y_d,
where we can used the subscript clp to denote closed-loop for the augmented system. Recall that the output remain the same weighted combination of states,
y = \textbf{C}_+ \textbf{x}_+ = \textbf{C}_{clp} \textbf{x}_+.

For the case of the three-state model of the propeller, adding the tracking-error-integral state generates a four-state model. The new state vector is then

\textbf{x}_+ =\begin{bmatrix} \theta \\ \omega \\ v_{bemf} \\ w \end{bmatrix},
and \textbf{E}_+ and \textbf{A}_+ are 4 \times 4, \textbf{B}_+ is 4 \times 1, and \textbf{C}_+ is 1 \times 4.

Determining the gains with the extra state

When we add the extra state w, we also need an extra state-feedback gain, K_w, so that

\textbf{K}_+ = \begin{bmatrix}\textbf{K}_{+_x} & K_{+_w}\end{bmatrix}
where we have used the notation \textbf{K}_{+_x} and K_{+_w} to remind us that ALL the state-feedback gains determined from analyzing the system with the integrator will be different from the gains determined from analyzing the system without the integrator.

We can determine \textbf{K}_+ using the expanded matrices, \textbf{E}_+, \textbf{A}_+ and \textbf{B}_+. For example, in matlab we could use 

Kp = place(Ep\Ap,Ep\Bp,[-1,-2,-3,-4]).
where we can used the subscript p because matlab will not let us use + in a variable label.