State Feedback Control with Integral Action

While standard state feedback control offers precise dynamic shaping through pole placement, it has a critical weakness: it cannot guarantee zero error in the presence of constant disturbances or for step reference commands. This steady-state offset is unacceptable for high-performance systems like precision machine tools, chemical process controllers, or robotic arms. State feedback with integral action solves this by merging the stability benefits of pole placement with the inherent disturbance-rejection capability of an integrator, creating a robust controller that achieves perfect steady-state tracking.

The Limitation of Standard State Feedback

Standard state feedback control operates by feeding back a linear combination of all the system's internal states to modify its dynamic response. For a linear system represented in state-space form:

$$\dot{x}(t)=Ax(t)+Bu(t)$$ $$y(t)=Cx(t)$$

the control law is $u(t)=-Kx(t)$, where $K$ is the feedback gain matrix. By choosing $K$ appropriately, you can place the closed-loop poles (eigenvalues of $(A-BK)$) anywhere in the complex plane to achieve desired transient characteristics like rise time and overshoot.

However, consider a step reference input $r$. The standard control law has no knowledge of the persistent error $e(t)=r-y(t)$. If a constant disturbance $d$ enters the system, or if there is a slight modeling error, the output $y(t)$ will settle at a value different from $r$. This occurs because the controller lacks a mechanism to generate a control effort based on the accumulated history of the error.

Augmenting the System with Integral States

The core innovation of integral action is to introduce new states that represent the integral of the tracking error. For a single-output system, we define an integral error state $x_i(t)$ with the dynamics: $${\dot{x}}_i(t)=r(t)-y(t)=e(t)$$ This equation means the integrator state accumulates the error over time. It will continue to grow (or shrink) as long as any error exists, thereby providing an ever-increasing (or decreasing) control signal to drive that error to zero.

We then create an augmented state vector by appending this integral state to the original plant states: $$x_a(t)=\left[\begin{array}{c}x(t)\\ x_i(t)\end{array}\right]$$

The dynamics of this new augmented system are derived by combining the original plant equations with the integrator dynamics: $${\dot{x}}_a(t)=\left[\begin{array}{c}\dot{x}(t)\\ {\dot{x}}_i(t)\end{array}\right]=\left[\begin{array}{c}Ax(t)+Bu(t)\\ -Cx(t)+r(t)\end{array}\right]$$ Substituting $y=Cx$, we can write this in standard augmented state-space form: $${\dot{x}}_a(t)=\underset{A_a}{\underbrace{\left[\begin{array}{cc}A& 0\\ -C& 0\end{array}\right]}}{x}_a(t)+\underset{B_a}{\underbrace{\left[\begin{array}{c}B\\ 0\end{array}\right]}}u(t)+\left[\begin{array}{c}0\\ 1\end{array}\right]r(t)$$

Designing the Controller via Pole Placement

With the augmented system $(A_a,B_a)$, we now design a full-state feedback controller for all states, including the integral state. The new control law is: $$u(t)=-\left[\begin{array}{cc}K& k_i\end{array}\right]\left[\begin{array}{c}x(t)\\ x_i(t)\end{array}\right]=-Kx(t)-k_i{x}_i(t)$$ Here, $K$ is the feedback gain for the original plant states, and $k_i$ is the feedback gain for the integral state.

This structure reveals the controller's dual nature: the $-Kx$ term provides stabilizing state feedback for good transient response, while the $-k_i{x}_i$ term provides the integral action. The closed-loop augmented system becomes: $${\dot{x}}_a(t)=(A_a-B_a\left[\begin{array}{cc}K& k_i\end{array}\right])x_a(t)+\left[\begin{array}{c}0\\ 1\end{array}\right]r(t)$$

The designer now performs pole placement on the augmented system. You select desired closed-loop pole locations for the matrix $(A_a-B_a{K}_a)$, where $K_a=[K{k}_i]$. This places $n+1$ poles (for an $n$-state original plant), simultaneously specifying the transient performance of the original states and the convergence speed of the integrator. A properly designed set of poles ensures stability and a fast, well-damped response that settles to zero error.

How Integral Action Achieves Zero Steady-State Error

The power of this method lies in its automatic disturbance rejection. Consider a constant disturbance $w$ entering the system: $\dot{x}=Ax+Bu+w$. At steady state ($\dot{x}=0$, ${\dot{x}}_i=0$), the integrator dynamics give ${\dot{x}}_i=r-y=0$, which forces $y=r$. The original state equation becomes $0=A{x}_{ss}+B{u}_{ss}+w$. The key is that the steady-state control effort $u_{ss}$ is no longer zero; it is precisely the value generated by the integral state $x_i$ to counteract the disturbance $w$. Because the integrator has "remembered" the total error needed to overcome the disturbance, it maintains the exact control input required to force the output to match the reference, eliminating the offset.

Common Pitfalls

1. Integrator Windup: This is the most critical practical issue. If the control input $u$ saturates (e.g., a valve hits 100% open), the integrator continues to accumulate error while the plant receives only the limited, saturated signal. This causes the integral state $x_i$ to grow very large ("winds up"). When the error finally changes sign, it takes a long time to "unwind" this large value, leading to poor transient response or instability. The solution is to implement anti-windup schemes, such as clamping the integrator or using back-calculation to limit its growth during saturation.

1. Poor Pole Selection for the Integrator: Placing the pole associated with the integral state too far into the left-half plane can make the integrator action excessively fast. This often introduces significant overshoot and ringing into the response, as the aggressive integral action fights with the dominant dynamics of the system. The integrator pole should be placed to provide a slow, smooth correction—often as the slowest pole in the closed-loop set.

1. Applying to Non-Step References: Integral action is designed to eliminate steady-state error for step inputs and disturbances. It will not generally provide zero error for ramp or parabolic reference signals. For those, you would need to add additional integrators (leading to a Type 2 or higher system), which significantly complicates stability.

1. Ignoring the Effect on System Type: Adding an integrator increases the system type number by one. While this guarantees zero error for a step, it also reduces the phase margin, making the system less robust to other uncertainties. The designer must verify that the gain and phase margins of the final design are still acceptable.

Summary

• State feedback with integral action eliminates steady-state error by augmenting the plant state vector with integral error states that accumulate the tracking error over time.
• The control law $u=-Kx-k_i{x}_i$ combines the transient shaping of standard state feedback with the persistent correction of integral control.
• Design is achieved via pole placement on the augmented system $(A_a,B_a)$, allowing simultaneous tuning of transient response and integral action speed.
• The integrator automatically generates the precise steady-state control effort needed to reject constant disturbances and track step references with zero offset.
• Practical implementation must guard against integrator windup during control saturation, and careful pole placement is required to avoid introducing undesirable overshoot.