Causal Affine Control Policies

• Causal affine control policies are state-feedback controllers that depend affinely and causally on the current state, enabling robust and constraint-satisfying control synthesis.
• They leverage LMI-based synthesis and initial-state-to-peak gain techniques to enforce quadratic state and input constraints, ensuring performance under uncertainty.
• The approach incorporates a reciprocal change of variables to convexify the design problem, extending its applicability to uncertain, infinite, and receding-horizon formulations with reduced conservatism.

Causal affine control policies are a class of state-feedback controllers for discrete-time linear (possibly time-varying) systems, wherein the control action at each time depends affinely and causally on the current system state. These policies are formulated to enable synthesis of robust, constraint-satisfying feedback for finite, infinite, and receding-horizon control problems subject to convex quadratic state and input constraints. The synthesis leverages robust control techniques, initial-state-to-peak gain arguments for constraint satisfaction, and a convexification strategy via reciprocal change of variables, leading to a semidefinite programming framework that is computationally scalable and less conservative than standard tube-based robust model predictive control approaches (Gramlich et al., 2023).

1. Affine Feedback Policy Parameterization and Causality

Causal affine policies are defined for discrete-time systems of the form

$$x_{k+1} = f_k + A_k x_k + B^1_k u_k \quad \text{(with possible disturbance term } + B^2_k w_k \text{)},$$

where $x_k \in \mathbb{R}^n$ is the state, $u_k \in \mathbb{R}^m$ is the control input, and $w_k \in \mathbb{R}^\ell$ is a disturbance (if considered). Policies are restricted to causal affine state-feedback: $$u_k = \pi_k(x_k) = k^1_k + K^2_k x_k = K_k \begin{pmatrix} 1 \ x_k \end{pmatrix}, \quad K_k = \begin{pmatrix} k^1_k & K^2_k \end{pmatrix},$$ for each $k = 0, \ldots, N-1$. Causality is immediate: $u_k$ depends only on $x_k$ and fixed matrices precomputed at the beginning of control; no future state or disturbance terms enter into the feedback calculation at $k$.

2. LMI-Based Synthesis for Finite-Horizon, Unconstrained Problems

For finite-horizon optimal control in the absence of uncertainty, control design is posed in terms of synthesizing a sequence of quadratic value functions

$$V_k(x) = \begin{pmatrix} 1 \ x \end{pmatrix}^\top P_k \begin{pmatrix} 1 \ x \end{pmatrix}, \quad P_k \in \mathbb{S}^{1+n}_{{\succ} 0}$$

and feedback parameters $K_k$, subject to a decreasing Bellman inequality,

$$V_k(x) \geq y^\top y + V_{k+1}(x^+), \quad x^+ = f_k + B^1_k k^1_k + (A_k + B^1_k K^2_k)x,$$

and quadratic state/input constraints of the form

$$V_k(x) \leq \nu \implies v_{ki}^\top v_{ki} \leq 1 \quad \forall i = 1,\ldots,s,$$

where stage-cost and constraint outputs are affine in $x_k$ and $u_k$: $y_k = g^1_k + C^1_k x_k + D^{11}_k u_k$, $v_{ki} = g^2_{ki} + C^2_{ki} x_k + D^{21}_{ki} u_k$. Terminal and initial value function constraints $V_0(\bar{x}) \leq \nu$, $$V_N(x) \geq \begin{pmatrix} 1 \ x \end{pmatrix}^\top P_f \begin{pmatrix} 1 \ x \end{pmatrix}$$ (with $P_f \succ 0$) are similarly imposed. These synthesis conditions can be written as linear matrix inequalities (LMIs) in $(P_k, K_k, \nu)$, making the problem tractable via convex optimization.

3. Initial-State-to-Peak Gain and Constraint Satisfaction

Constraint satisfaction, especially for all quadratic state and input constraints, is enforced via an initial-state-to-peak gain (energy-to-peak) argument. Specifically, for all $x$ in the sublevel set $V_k(x) \leq \nu$, it is ensured that

$$\| v_{ki} \|^2 = \| g_{ki}^2 + C_{ki}^2 x + D_{ki}^{21} u_k \|^2 \leq 1,$$

by imposing the LMI

$$P_k \succeq \nu \begin{pmatrix} g^2_{ki} + D^{21}_{ki} k^1_k & C^2_{ki} + D^{21}_{ki} K^2_k \end{pmatrix}^\top \begin{pmatrix} g^2_{ki} + D^{21}_{ki} k^1_k & C^2_{ki} + D^{21}_{ki} K^2_k \end{pmatrix}.$$

This condition guarantees robust satisfaction of all convex quadratic constraints for every trajectory initialized within $V_0(\bar{x}) \leq \nu$.

4. Convexification via Reciprocal Change of Variables

The coupled LMIs in $(P_k, K_k, \nu)$ are generally non-affine. A reciprocal variable transformation is introduced to render the synthesis problem fully convex. Specifically, set

$$\widetilde{P}_k = P_k^{-1}, \quad \widetilde{K}_k = K_k P_k^{-1}, \quad \tilde{\nu} = \nu^{-1}.$$

A slack variable $Z$ is incorporated to linearize the initial-state bound $\mathrm{trace}(P_0 \Sigma_0) \leq \nu$, where $\Sigma_0$ is an initial-state covariance. The transformed LMIs in these reciprocal variables are fully linear and given by:

• Linearized Bellman LMI (cost decrease)
• Linearized initial-state bound
• Linearized terminal bound
• Linearized constraint satisfaction

Once these LMIs are imposed for each $k = 0, \ldots, N$ and $\tilde{\nu}$ is maximized, the problem reduces to a semidefinite program (SDP) in $$\{\widetilde{P}_k, \widetilde{K}_k, \tilde{\nu}, Z\}$$. The original controller parameters are recovered via $K_k = \widetilde{K}_k \widetilde{P}_k^{-1}$.

5. Extensions to Uncertain, Infinite, and Receding-Horizon Formulations

Uncertainty Modeling

Robust synthesis for disturbed systems models disturbances as satisfying quadratic constraints,

$$[z_k; w_k]^\top M_k [z_k; w_k] \geq 0, \quad \forall M_k \in \mathcal{M}.$$

The Bellman decrease is replaced by a robust version using the lossless S-procedure, leading to LMIs affine in augmented decision variables, including multipliers for uncertainty.

Infinite-Horizon Control

Infinite-horizon control assumes time-invariant system data for $k \geq N$, enforcing stationarity ($P_k = P_N, K_k = K_N$ for all $k \geq N$). The terminal bound is replaced with a tail inequality at step $N$, yielding a single LMI that certifies infinite-horizon robust performance and constraint satisfaction.

Receding-Horizon (Robust Model Predictive Control)

A receding-horizon scheme solves the infinite-horizon SDP at each stage $j$ with shifted data $(G_{j+k})_{k=0}^N$, obtaining stage-specific policies $\{P_{k,j}, K_{k,j}, \nu_j\}$. After applying the first control action, the state is updated and the optimization is repeated at the next time step. This scheme achieves recursive feasibility, robust constraint satisfaction, and asymptotic performance $y_j \to 0$ as $j \to \infty$.

6. Comparison to Tube-Based Robust Model Predictive Control

Tube-based robust MPC employs a nominal trajectory and an offline-designed, time-invariant error tube (e.g., via robust invariant sets). Online optimization is limited to the nominal variables via a quadratic program (QP), with tightened constraints imposed. While recursive feasibility is maintained and computational scaling is $O(N)$ in scalar variables, conservatism is typically high. This arises due to lack of online tube optimization and limitations of uncertainty modeling (commonly norm-bounds, not general LFT).

In contrast, the LMI-based MPC synthesis with causal affine policies jointly optimizes the feedback gains online via an SDP, can handle general linear fractional transformation (LFT) uncertainties, and results in significantly less conservatism. Computationally, each step involves an SDP of block-dimension $O(N)$, heavier per iteration than QP but efficient with structure-exploiting solvers (Gramlich et al., 2023). Decision-variable scaling is notably distinct among approaches:

Approach	Decision Variables	Conservatism
Tube-MPC	$O(N)$ scalars	High
Disturbance/Min-Max MPC	$O(N^2)$	Moderate–High
LMI-based Affine Policy	$O(N)$ blocks	Significantly lower

7. Summary and Context

Causal affine control policies parameterized as state-feedback laws enable robust synthesis for constrained linear systems under both deterministic and uncertain dynamics. The key technical innovations—energy-to-peak (initial-state-to-peak) gain for robust constraint satisfaction and convexification via reciprocal transformation—yield a unified, LMI-based synthesis applicable to finite, infinite, and receding-horizon settings. The resulting controllers offer significant advantages in tractability, flexibility with respect to uncertainty descriptions, and worst-case performance, while maintaining computational scalability comparable to, but less conservative than, traditional tube-based MPC approaches (Gramlich et al., 2023).