Fixed-Structure Quadratic Controller Tuning

LQR chooses any full-state gain allowed by its mathematical model. Fixed-structure tuning solves a different problem: the controller must keep a prescribed architecture—perhaps one gain, a PID, selected feedback channels, or a fixed-order compensator—and only its parameters may vary. That restriction usually destroys the Riccati closed form and makes the optimization nonconvex.

Problem statement #

Let a controller family be

$$K(s,\theta),\qquad \theta\in\Theta\subset\mathbb R^r,$$

and let the closed-loop state, output, and control generated by an initial condition or disturbance be $x(t;\theta)$, $z(t;\theta)$, and $u(t;\theta)$. A typical objective is

$$\boxed{ J(\theta)=\int_0^\infty \left[z(t;\theta)^TQ_z z(t;\theta) +u(t;\theta)^TRu(t;\theta) \right]dt.}$$

The admissible parameter set is not merely the box of knob limits. It must include internal closed-loop stability and any required properness, algebraic-loop, actuator, or implementation constraints:

$$\Theta_{\mathrm{ad}} =\{\theta\in\Theta:\ A_{cl}(\theta)\text{ is Hurwitz and all implementation constraints hold}\}.$$

If the loop is unstable and the response is not identically zero in its unstable modes, the infinite-horizon quadratic integral diverges. Assigning a finite numerical penalty to that case is an optimizer convention, not the original cost.

Static state feedback: evaluate cost with a Lyapunov equation #

For

$$\dot x=Ax+Bu, \qquad u=-K(\theta)x,$$

define

$$A_c(\theta)=A-BK(\theta),$$

and

$$W(\theta)=Q+K(\theta)^TRK(\theta).$$

If $A_c$ is Hurwitz, the cost from $x(0)=x_0$ is

$$J(\theta)=\int_0^\infty x(t)^TW(\theta)x(t)dt.$$

Let $P(\theta)$ solve

$$\boxed{A_c^TP+PA_c+W=0.}$$

Since

$$\frac{d}{dt}(x^TPx) =x^T(A_c^TP+PA_c)x =-x^TWx,$$

integration from zero to infinity gives

$$\boxed{J(\theta)=x_0^TP(\theta)x_0.}$$

For a random initial condition with covariance

$$\Sigma_0=\mathbb E[x_0x_0^T],$$

the expected cost is

$$\boxed{\mathbb E[J]=\mathop{\mathrm{tr}}(P\Sigma_0).}$$

This evaluates a candidate architecture exactly for the linear model without time simulation. It does not make the search over $\theta$ convex.

Frequency-domain evaluation #

For a stable causal signal $z(t)$ with Fourier transform $Z(j\omega)$, Parseval's identity gives

$$\boxed{ \int_0^\infty z(t)^*z(t)dt =\frac{1}{2\pi}\int_{-\infty}^{\infty} Z(j\omega)^*Z(j\omega)d\omega.}$$

For weighted multiple outputs, factor $Q_z=C_q^TC_q$ and $R=C_r^TC_r$, stack

$$w(t)=\begin{bmatrix}C_qz(t)\\C_ru(t)\end{bmatrix},$$

and evaluate the energy of $w$. This is the reliable core behind the KPI lecture's Parseval/determinant method.

If a scalar stable strictly proper transfer response is

$$F(s)=\frac{N(s)}{D(s)},$$

with no poles on the imaginary axis, contour integration gives

$$\boxed{ \int_0^\infty f(t)^2dt =\sum_{\lambda_i\in\mathbb C_-} \mathop{\mathrm{Res}}\limits_{s=\lambda_i} \left[F(s)F(-s)\right].}$$

For simple stable poles $\lambda_i$ of $F$,

$$\mathop{\mathrm{Res}}\limits_{s=\lambda_i}[F(s)F(-s)] =\frac{N(\lambda_i)N(-\lambda_i)} {D'(\lambda_i)D(-\lambda_i)}.$$

The residue expression, numerical frequency integration, and the Lyapunov equation are three representations of the same stable-response energy. Agreement among them is a strong implementation check.

Scalar gain: full analytic solution #

Consider

$$\dot x=ax+bu, \qquad u=-kx,$$

with $b>0$ and cost

$$J(k)=\int_0^\infty(qx^2+ru^2)dt, \qquad q>0,\ r>0.$$

The closed-loop state is

$$x(t)=x_0e^{(a-bk)t}.$$

The stability domain is

$$k>\frac{a}{b}.$$

Inside it,

$$\begin{aligned} J(k) &=(q+rk^2)x_0^2 \int_0^\infty e^{2(a-bk)t}dt\\ &=\boxed{\frac{(q+rk^2)x_0^2}{2(bk-a)}}. \end{aligned}$$

Differentiate:

$$\frac{dJ}{dk} =\frac{x_0^2}{2} \frac{brk^2-2rak-bq}{(bk-a)^2}.$$

The stationary gains satisfy

$$brk^2-2rak-bq=0.$$

The stabilizing root is

$$\boxed{ k^*=\frac{a}{b} +\sqrt{\frac{a^2}{b^2}+\frac{q}{r}}.}$$

The other root lies outside the stability domain. In this one-state example the allowed static structure is already the unrestricted full-state structure, so the result agrees with scalar LQR. That coincidence disappears when feedback channels or controller order are restricted.

Parameter sensitivities #

Let $P$ satisfy

$$A_c^TP+PA_c+W=0.$$

For parameter $\theta_i$, differentiate the equation:

$$A_c^TP_i+P_iA_c +A_{c,i}^TP+PA_{c,i}+W_i=0,$$

where

$$P_i=\frac{\partial P}{\partial\theta_i}, \quad A_{c,i}=\frac{\partial A_c}{\partial\theta_i}, \quad W_i=\frac{\partial W}{\partial\theta_i}.$$

This is another Lyapunov equation. For deterministic $x_0$,

$$\boxed{ \frac{\partial J}{\partial\theta_i}=x_0^TP_ix_0,}$$

and for covariance $\Sigma_0$,

$$\boxed{ \frac{\partial\mathbb E[J]}{\partial\theta_i} =\mathop{\mathrm{tr}}(P_i\Sigma_0).}$$

Adjoint Lyapunov formulas can compute the entire gradient with fewer matrix solves when $r$ is large. Regardless of method, derivatives are meaningful only inside one connected stable region; the cost tends to infinity at a generic stability boundary.

Dynamic fixed-order controllers #

For

$$\dot x_c=A_c(\theta)x_c+B_c(\theta)y, \qquad u=C_c(\theta)x_c+D_c(\theta)y,$$

augment plant and controller states:

$$\xi=\begin{bmatrix}x\\x_c\end{bmatrix}, \qquad \dot\xi=A_{aug}(\theta)\xi.$$

The same Lyapunov construction applies to $A_{aug}$ and the augmented quadratic output. The controller order and signal connections remain fixed while the matrix entries vary.

Why deleting LQR paths is not redesign #

Suppose unrestricted LQR gives $u=-Kx$, but only selected measurements are available. Setting unavailable columns of $K$ to zero changes the closed-loop matrix and cost without re-solving the constrained problem. In general,

$$\mathop{\mathrm{project}}(K_{\mathrm{LQR}}) \ne \underset{K\in\mathcal S}{\arg\min}\ J(K),$$

where $\mathcal S$ is the allowed sparsity or architecture set.

The 1977 STOL report describes path deletion as an engineering expedient and also notes that weight selection has no foolproof recipe. The durable procedure is:

1. define the permitted controller structure explicitly;
2. search only the stable admissible parameter region;
3. evaluate the actual structured cost;
4. inspect modes, eigenvectors, control effort, and robustness;
5. verify the final controller against saturation, noise, uncertainty, and implementation dynamics.

Common failure modes #

• optimizing across unstable candidates while treating integration failure as a harmless number;
• confusing a quadratic penalty with physical energy;
• using a single initial condition that hides poorly controlled modes;
• accepting a local optimum without multiple starts or structural analysis;
• tuning continuous-time parameters and deploying them in a sampled controller without rediscretization;
• preserving the LQR label after imposing sparsity, order, or output-feedback restrictions.

Fixed-structure optimization is valuable because real controllers are structured. Its discipline is to make that restriction part of the mathematical problem rather than modifying an unrestricted solution after the fact.