State Feedback

State Feedback

A state-feedback control law sets the input to a linear combination of the current state plus a reference:
$u(t)=-Kx(t)+Fv(t),$
where $K\in {\mathbb{R}}^{m\times n}$ is the feedback gain matrix, $F\in {\mathbb{R}}^{m\times r}$ is the feedforward gain, and $v(t)\in {\mathbb{R}}^r$ is an external reference (often a desired state or setpoint). Substituting into $\dot{x}=Ax+Bu$ gives the closed-loop dynamics
$\dot{x}=(A-BK)x+BFv=A_{c}x+B_{c}v,$
with closed-loop system matrix $A_c=A-BK$ and closed-loop input matrix $B_c=BF$.

The core design question is: choose $K$ so that $A_c=A-BK$ has a desired eigenvalue spectrum. This is the pole placement problem. Its solvability is the subject of 7.7. Controllability: if the pair $(A,B)$ is controllable, then for any desired set of $n$ eigenvalues (closed under complex conjugation), a $K$ exists that places the closed-loop eigenvalues there. For single-input systems ($m=1$), $K$ is unique; for multi-input systems, there are many $K$‘s that achieve the same spectrum, and the extra freedom is used for additional design goals (mode shapes, robustness, decoupling).

The feedforward gain $F$ does not affect stability — only $K$ does — but it sets the steady-state response. For unit DC gain from reference to state (so that $x\to v$ in steady state when $v$ is constant), choose $F$ so that $-A_c^{-1}BF=I$, i.e., $F=-(C{A}_c^{-1}B{)}^{-1}$ if tracking the output, or $F=-A_c^{-1}B$ for state tracking.

State Feedback Requires the Full State $u=-Kx$ uses every component of $x$. If sensors only measure a subset (an output $y=Cx$ with $C\ne I$), the unmeasured states must be estimated, typically by an observer (Luenberger observer or Kalman filter). Observer design is the dual of state-feedback design and closes the loop as $u=-K\hat{x}+Fv$ where $\hat{x}$ is the estimated state. The separation principle guarantees that observer and feedback designs can be done independently.

The control law

Intuition

Feedback is the move that transforms the open-loop behavior of a plant — which may be slow, oscillatory, or unstable — into the closed-loop behavior the designer wants. The key insight is that feedback can move the eigenvalues of the system: instead of living with $A$‘s natural modes, the designer chooses new modes by picking $K$.

To see why, imagine a scalar system $\dot{x}=ax+bu$ with $a>0$ (unstable). Apply $u=-kx$: the closed-loop system is $\dot{x}=(a-bk)x$. For any target rate ${\lambda }_d<0$, choose $k=(a-{\lambda }_d)/b$, and the closed-loop system decays at rate ${\lambda }_d$. The unstable plant is stabilized by proportional feedback — and in fact by choosing $k$ aggressively, it can be made arbitrarily fast. The only price is that $u$ grows with the gain, which will eventually saturate actuators.

Matrix pole placement is the same story in higher dimensions. $K$ is a row vector (or multi-row matrix) of gains, one per state component. Choosing $K$ is choosing how each state component contributes to the correction $u$. The closed-loop eigenvalues of $A-BK$ are functions of $K$, and for a controllable system, the map $K\mapsto \text{spec}(A-BK)$ is onto: every spectrum is achievable.

The reference input $v$ tells the controller where to go, separately from how to stabilize. $K$ handles regulation (suppressing disturbances and decaying initial conditions); $F$ handles tracking (making the steady state match $v$). A well-designed controller does both simultaneously.

Properties

Closed-Loop Structure

(Closed-loop matrix) The feedback $u=-Kx$ replaces $A$ by $A_c=A-BK$.

Every tool that applied to the open-loop system — $e^{At}$, phase portraits, stability criteria — applies to the closed-loop system with $A$ replaced by $A_c$. Closing the loop is an algebraic operation on the system matrix; it doesn’t change the structure of the theory, only the numbers.

(Closed-loop eigenvalues) $\det (\lambda I-A+BK)=0$ determines closed-loop poles.

For single-input systems, this is a polynomial in $\lambda$ of degree $n$ whose coefficients are affine in $K$‘s entries. Expanding, the characteristic polynomial is parameterized by $K$, and matching coefficients to a desired polynomial $\prod (\lambda -{\lambda }_i^{\text{desired}})$ gives $n$ linear equations in the $n$ entries of $K$ — solvable iff $(A,B)$ is controllable.

(Independence of $F$ from stability) $F$ scales the closed-loop input but does not affect $A_c$.

Stability depends only on $A_c=A-BK$, not on $F$. Setting $F=0$ gives a regulator (drives $x$ to $0$); setting $F\ne 0$ adds tracking. For output tracking of a constant reference $v$, choose $F$ so the closed-loop DC gain from $v$ to $y$ is the identity.

Pole Placement

(Design recipe) Pick desired eigenvalues, form the target characteristic polynomial, solve for $K$ by coefficient matching.

For single-input systems, transform $(A,B)$ to controllable canonical form, where $K$ reads off directly as the coefficient differences. Standard tools (MATLAB’s place or acker) compute $K$ from $(A,B,\text{desired spectrum})$ directly. For multi-input systems, place uses a robust algorithm that chooses among the many valid $K$‘s for numerical conditioning.

(Desired-pole trade-offs) Faster poles mean larger gains, noisier control, saturation risk.

Placing poles far into the LHP makes the system respond quickly, but $K$ scales with pole distance — doubling the decay rate roughly doubles the gain. Large $K$ amplifies sensor noise, demands large actuator effort, and may drive actuators into saturation (after which the linear analysis no longer applies). Rule of thumb: place closed-loop poles $2$–$10\times$ faster than the dominant open-loop dynamics, not more.

(Dominant vs non-dominant poles) Long-time response is controlled by the slowest pole.

If the closed-loop spectrum contains ${\lambda }_{\text{slow}}$ and ${\lambda }_{\text{fast}}$ with $|\mathrm{Re}({\lambda }_{\text{slow}})|\ll |\mathrm{Re}({\lambda }_{\text{fast}})|$, the fast modes decay quickly and the response is dominated by the slow pole. Designers sometimes explicitly structure the spectrum as one or two dominant poles plus a cloud of much-faster ones.

Multi-Input and Robust Design

(Multi-input freedom) For $m>1$, many $K$‘s achieve the same closed-loop spectrum.

The extra degrees of freedom are used for secondary objectives: minimizing $\Vert K\Vert$ (low gain for noise rejection), shaping eigenvectors (mode decoupling), or optimizing a quadratic cost. LQR (linear-quadratic regulator) is the standard tool: given weights $Q⪰0$ on state and $R\succ 0$ on input, find $K$ minimizing ${\int }_0^{\infty }(x^{T}Qx+u^{T}Ru)dt$. Solving the algebraic Riccati equation yields an optimal $K$ that is automatically stabilizing and has strong robustness margins.

(Robustness) Pole placement alone does not guarantee robustness to model error.

A system whose closed-loop poles match the design exactly under nominal parameters can still become unstable under modest parameter variation. LQR controllers have intrinsic gain and phase margins ($\ge 6$ dB, $\pm 60°$ for single-input); general pole-placement designs do not. For robustness-critical applications, prefer LQR or $H_{\infty }$ design over arbitrary pole placement.

(Integral action for tracking) Adding an integrator state eliminates steady-state error.

Basic state feedback tracks step references only up to steady-state offset if there are any model or parameter mismatches. Augmenting the state with $\xi =\int (v-y)dt$ and applying feedback to the augmented state guarantees $y\to v$ exactly in steady state, regardless of DC plant uncertainty. The augmented system must still be controllable.

Examples

Scalar Stabilization

Stabilize $\dot{x}=2x+u$ so the closed-loop pole is at $-3$.

Solution $A=2$, $B=1$. Closed-loop: $A_c=2-k$. Want $A_c=-3$, so $k=5$.

Control law $u=-5x$. Closed-loop: $\dot{x}=-3x$, response $e^{-3t}$.

Note: gain $k=5$ to move the pole $5$ units in the $s$-plane — roughly linear in the distance moved.

Pole Placement for a 2D System

$A=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$, $B=\left[\begin{array}{c}0\\ 1\end{array}\right]$ (double integrator). Place poles at $-1,-2$.

Solution $(\lambda +1)(\lambda +2)={\lambda }^2+3\lambda +2$.

Desired characteristic polynomial:

$K=[k_1,k_2]$. $A-BK=\left[\begin{array}{cc}0& 1\\ -k_1& -k_2\end{array}\right]$.

Characteristic polynomial: ${\lambda }^2+k_2\lambda +k_1$. Matching: $k_1=2$, $k_2=3$.

Control law $u=-2x_1-3x_2$ — classical PD control for a double integrator: position gain $2$, derivative gain $3$.

Inverted Pendulum Stabilization

The inverted-pendulum linearization (see Linearization) is $A=\left[\begin{array}{cc}0& 1\\ g/\ell & 0\end{array}\right]$, $B=\left[\begin{array}{c}0\\ 1/(m{\ell }^2)\end{array}\right]$. Place closed-loop poles at $-\sqrt{g/\ell }(1\pm i/2)$ (damping ratio $\sim 0.9$, natural frequency $\sim \sqrt{g/\ell }$).

Solution ${\omega }_0=\sqrt{g/\ell }$. Desired polynomial: $(\lambda +{\omega }_0{)}^2+({\omega }_0/2{)}^2={\lambda }^2+2{\omega }_0\lambda +\frac{5}{4}{\omega }_0^2$.

Let

$A-BK=\left[\begin{array}{cc}0& 1\\ {\omega }_0^2-k_1/(m{\ell }^2)& -k_2/(m{\ell }^2)\end{array}\right]$.

Open-loop polynomial: ${\lambda }^2+(k_2/m{\ell }^2)\lambda +(k_1/m{\ell }^2-{\omega }_0^2)$.

Matching: $k_2=2m{\ell }^2{\omega }_0$; $k_1=m{\ell }^2(\frac{5}{4}{\omega }_0^2+{\omega }_0^2)=\frac{9}{4}m{\ell }^2{\omega }_0^2$.

The plant was unstable (saddle); feedback has moved its eigenvalues into the LHP, stabilizing it. The required gain $k_1$ scales with $\ell \cdot m\cdot g$, consistent with the physical intuition that larger or heavier pendulums need more torque to stabilize.

See Also

• 7.7. Controllability — the conditions under which pole placement is solvable
• 7.5. Stability — the criterion feedback is designed to satisfy
• 7.1. State-Space Representation — the form $u$, $x$, $y$ live in
• 7.3. Solving LTI Systems — the closed-loop response $x(t)=e^{A_{c}t}x(0)+\int e^{A_c(t-\tau )}{B}_{c}v(\tau )d\tau$