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Integral State Feedback (LQI / Servo)

Overview & Motivation

Plain LQR drives the system state toward the origin but leaves a persistent steady-state offset when a constant reference or disturbance is present. The integral state feedback controller (LQI, or servo LQR) removes this offset by augmenting the plant with integrators on the tracking error. A single LQR solve on the augmented system produces two gains: one that acts on the physical states and one that closes the integral loop, guaranteeing zero steady-state error to constant references with no manual trim.

Mathematical Theory

Plant Model

The discrete-time plant is

$$x_{k+1} = A x_k + B u_k, \quad y_k = C x_k$$

with state $x \in \mathbb{R}^n$, input $u \in \mathbb{R}^m$, and tracked output $y \in \mathbb{R}^p$.

Augmented Plant

Define the integral-of-error state $x_i \in \mathbb{R}^p$:

$$x_{i,k+1} = x_{i,k} + (r_k - y_k) T_s$$

Stacking $\xi = [x^\top ; x_i^\top]^\top$ gives the augmented system

$$\xi_{k+1} = A_a \xi_k + B_a u_k + E_a r_k$$

$$A_a = \begin{bmatrix} A & 0 \ -C T_s & I \end{bmatrix}, \quad B_a = \begin{bmatrix} B \ 0 \end{bmatrix}, \quad E_a = \begin{bmatrix} 0 \ T_s I \end{bmatrix}$$

LQR Design on the Augmented Plant

Minimise the infinite-horizon quadratic cost

$$J = \sum_{k=0}^{\infty} \bigl(\xi_k^\top Q \xi_k + u_k^\top R u_k\bigr)$$

by solving the Discrete Algebraic Riccati Equation (DARE) for $P$:

$$P = A_a^\top P A_a - A_a^\top P B_a (R + B_a^\top P B_a)^{-1} B_a^\top P A_a + Q$$

The optimal gain partitions as $K_a = [K_x \mid K_i]$ where $K_x \in \mathbb{R}^{m \times n}$ acts on the physical states and $K_i \in \mathbb{R}^{m \times p}$ acts on the integral states.

Control Law

$$u_k = -K_x x_k - K_i x_{i,k}$$

At equilibrium $r - y = 0$, so $x_{i}$ stops changing, and the control law holds $y = r$ exactly.

Complexity Analysis

Case Time Space Notes
Design $O((n+p)^3)$ $O((n+p)^2)$ DARE iteration on augmented system
Update $O(m(n+p))$ $O(p)$ Two matrix-vector products

Design is a one-time offline computation. The per-sample update cost is dominated by the two gain-state products and the integral accumulation.

Step-by-Step Walkthrough

Consider a scalar plant ($n=1$, $m=1$, $p=1$, $T_s = 0.01$):

  1. Form $A_a$ (2×2), $B_a$ (2×1) from plant matrices and $T_s$.
  2. Choose $Q$ (2×2 diagonal) and $R$ (scalar) to weight states and input.
  3. Solve DARE → $P$ (2×2) → $K_a = [k_x ; k_i]$ (1×2).
  4. Per step: accumulate $x_i \mathrel{+}= (r - y) T_s$, output $u = -k_x x - k_i x_i$.
  5. After ~200 steps the output converges to $r$ within numerical tolerance.

Pitfalls & Edge Cases

  • Integral windup: when the actuator saturates, the integral keeps growing because the control cannot reach the demanded value. Apply a clamp on $x_i$ or back-calculate (anti-windup) to prevent divergence after the saturation clears.
  • Marginally stable plant: a plant with an open-loop integrator combined with the error integrator yields a double-integrator augmented system. The DARE still converges if the augmented pair $(A_a, B_a)$ is stabilisable; verify that condition before deploying.
  • Slow integral weighting: under-weighting the integral state in $Q$ allows steady-state error to persist for many steps before correcting; over-weighting causes overshoot.
  • Sample-time mismatch: the discrete integral $x_i$ accumulates $T_s$-scaled errors. Using the wrong $T_s$ at run time shifts the effective integral gain and breaks zero-error convergence.
  • Slow DARE convergence: the augmented plant $(A_a, B_a)$ has eigenvalues at exactly 1, and at high sampling rates those eigenvalues approach 1 from many directions, requiring many more DARE iterations than the default 300. Raise the cap via the MaxIterations template parameter: IntegralStateFeedbackLqi<float, StateSize, InputSize, OutputSize, 30000>.

Variants & Generalizations

  • Continuous-time LQI: replace the discrete integrator with $\dot{x}_i = r - y$ and solve the continuous ARE.
  • Output-feedback LQI (LQGI): combine with a Kalman filter when only the output (not the full state) is measurable — the separation principle still holds.
  • Anti-windup: add a saturation block on $x_i$ with back-calculation to recover from actuator limits without integral drift.
  • Multiple outputs: the design extends directly to $p &gt; 1$ by stacking $p$ integral states.

Applications

  • Motor position/speed servo: eliminate gravity or friction offsets without manual trim.
  • Process control: temperature or pressure regulation with constant load disturbances.
  • Aerospace attitude control: integral action compensates for persistent aerodynamic moments.
  • Robotics: joint torque control with payload-induced constant forces.

Connections to Other Algorithms

  • Lqr: the base regulator; LQI delegates the DARE solve to Lqr on the augmented plant.
  • DiscreteAlgebraicRiccatiEquation: the inner solver used by Lqr.
  • Lqg: pairs Lqr with a Kalman filter; LQI could similarly pair with Lqg for output-feedback servo control.
  • Pid: the integral channel of a PID is the scalar, single-output analogue of $K_i x_i$.
  • SaturationRateLimiter: provides output clamping for anti-windup on the LQI integral.

References & Further Reading

  • B. D. O. Anderson, J. B. Moore, Optimal Control: Linear Quadratic Methods, Prentice Hall, 1990.
  • G. F. Franklin, J. D. Powell, A. Emami-Naeini, Feedback Control of Dynamic Systems, 8th ed., Pearson, 2019. Chapter 9.
  • K. J. Åström, B. Wittenmark, Computer-Controlled Systems: Theory and Design, 3rd ed., Prentice Hall, 1997. Chapter 5.