Multi-Observable Peak-Minimization Techniques

• Multi-observable peak-minimization techniques are methods that use linear programming and nonconvex optimization to reduce peak distortions and improve system robustness.
• They address quantitative criteria like peak-to-peak gain and PAR in both continuous/discrete control and large-scale MIMO-OFDM communications.
• The approaches ensure scalable, constraint-satisfying designs that handle system delays, hardware efficiency, and trade-offs between peak suppression and average power.

Multi-observable peak-minimization techniques address the problem of simultaneously suppressing peak excursions across multiple output channels or measurement signals―commonly called "observables"―to satisfy system constraints, improve robustness, or enable hardware efficiency. In control, estimation, and wireless communications, the goal is to minimize the induced peak-to-peak gain or the peak-to-average-power ratio (PAR) across all relevant outputs under input or system constraints. Recent advances have yielded tractable formulations for both linear interval observer design in control theory and joint precoding/PAR reduction in large-scale MIMO-OFDM systems, leveraging positivity theory, linear programming, and nonconvex smooth norm surrogates to obtain practical and theoretically sound solutions (Briat et al., 2015, Taner et al., 2021).

1. System Models and Problem Definition

Multi-observable peak-minimization arises in both deterministic and stochastic dynamical systems. In control-oriented interval observer design, the system is typically a continuous- or discrete-time plant subject to persistent (essentially bounded) disturbances and possibly point delays:

• Continuous-time: $\dot{x}(t) = A x(t) + A_h x(t-h) + E w(t)$, $y(t) = C x(t) + C_h x(t-h) + F w(t)$
• Discrete-time: $x(k{+}1) = A_d x(k) + E_d w(k)$, $y(k) = C_d x(k) + F_d w(k)$

Here, $w$ represents unknown but bounded disturbance inputs, and $x$, $y$ denote the system state and output. For estimation, interval observers produce lower and upper state estimates $x^-(t) \le x(t) \le x^+(t)$, maintained through observer dynamics constructed to keep error variables nonnegative.

In MIMO-OFDM communications, the observables are the time-domain waveforms of $B$ transmit antennas ($\bm t_b \in \mathbb{C}^W$) in a massive MU-MIMO setup, subject to data transmission, subcarrier energy, and PAR constraints (Taner et al., 2021).

2. Peak-to-Peak Gain and PAR: Quantitative Criteria

The induced peak-to-peak gain for a stable linear time-invariant operator $G$ mapping disturbance $w$ to output $z$ is defined as

$$\|G\|_{pp} = \sup_{\|w\|_{L_\infty}=1} \|G w\|_{L_\infty}$$

where $$\|w\|_{L_\infty} = \operatorname{ess\,sup}_{t \ge 0} \|w(t)\|_\infty$$. In discrete time, the $\ell_\infty \to \ell_\infty$ analog applies (Briat et al., 2015). This operator norm quantifies the worst-case amplification of bounded disturbances observed in any channel.

The PAR in communications expresses the ratio of the maximum to mean squared power in a transmit vector. Surrogate metrics such as

$$\mathrm{PAR}_p^q(\bm x) = \frac{N^{2/q - 2/p} \|\bm x\|_p^2}{\|\bm x\|_q^2}$$

generalize this with $q < p$ to balance focus between peak suppression ($p \to \infty$) and bulk (mean) energy ($q = 1,2$) (Taner et al., 2021).

3. Observer and Precoder Design: Optimization Formulations

For interval observer design, the estimation error system

$\dot{e}(t) = (A-LC) e(t) + (E-LF) \delta(t)$

is required to be positive (i.e., $A-LC$ Metzler, $E-LF\ge0$), where $L$ is the observer gain. The minimization of induced $L_\infty$ gain from disturbance to any weighted error output is reformulated as a finite-dimensional linear program (LP) with diagonal scaling and positivity constraints (Briat et al., 2015). The LP optimizes over $(X, U, \alpha, \gamma)$, where $X = \operatorname{diag}(\lambda) > 0$, $U = X L$, and $\gamma$ upper bounds the peak-to-peak gain.

In the MIMO-OFDM context, the joint precoding and PAR-minimization (JPP) is formulated as

$$\min_{\bm t_1,\dots,\bm t_B \in \mathbb{C}^W} \sum_{b=1}^B \left(W^{2/q-2/p} \|\bm t_b\|_p^2 - \|\bm t_b\|_q^2\right)$$

subject to multi-user interference (MUI) cancellation and out-of-band (OOB) spectral constraints (Taner et al., 2021). The $\ell^p$–$\ell^q$ penalty realizes a “peak-minus-bulk” minimization, and the problem is solved via forward–backward splitting (FBS), alternating between a gradient step on the smooth penalty and projection onto the affine linear constraints.

4. Uniformity and Multi-Observable Guarantees

A central theoretical result in interval observer design is the uniformity of the optimal $L^*$ computed from the “all-ones” weighting: it simultaneously minimizes the $L_\infty$ gain for all possible error-to-output mappings $M$, $N$ (nonnegative weightings) without the need to re-solve the LP (Briat et al., 2015). This implies that the solution is robust to changes in performance priorities across observables.

For MIMO-OFDM, the structured penalty sums over per-antenna time-domain waveforms, directly addressing the joint minimization of peaks across observables (antennas). The methodology embeds all system-level constraints (MUI, OOB) into the projection step, ensuring no performance trade-off in error-vector magnitude or spectral mask—only the peak suppression is at play across all observables (Taner et al., 2021).

5. Extensions to Delays and Large-Scale Systems

The LP formulation for observer design extends to systems with point delays ($A_h$, $C_h$) by adding positivity constraints for the delayed channels and to discrete-time systems by suitable augmentation of the state-transition matrix ($A_d-I$). No change of principle is required, only of matrices and constraint blocks (Briat et al., 2015).

In communications, the FBS solver naturally generalizes to multiple observables, such as additional antennas, carriers, polarizations, or other spatial dimensions. Computational scalability is ensured by per-iteration complexities of $O(BW)$ (gradient) and $O(BWU + WU^3)$ (projection), which are feasible for $B \sim 10^2$ and $W \sim 10^3$—a typical regime in massive MIMO (Taner et al., 2021).

6. Performance, Trade-Offs, and Practical Considerations

Explicit examples demonstrate the feasibility, boundaries, and trade-offs inherent to these techniques:

System Example	Result	Interpretation
Decoupling (observer, (Briat et al., 2015))	$L^*=[1;2],\gamma^*=0$	Disturbance exactly rejected
Non-decoupling (observer)	$L^*=[-1;2],\gamma^*\approx1.43$	Some disturbance leakage remains
JPP-(4,2) (precoder, (Taner et al., 2021))	PAR $\approx$ 6 dB, PINC $\approx$ 1.1 dB	Improved PAR with moderate power
JPP-(2,1) (precoder)	PAR $\approx$ 5.5 dB, PINC $\approx$ 1.8 dB	Flatter envelope, higher PINC

Selecting weighting matrices in the observer setting redistributes focus among observed error components without affecting the optimal $L^*$ due to the uniformity property. Larger weights highlight specific components of $e$, shifting minimization focus as needed (Briat et al., 2015).

Choice of $(p,q)$ in PAR minimization dictates the stringency of peak control versus bulk energy. Larger $p$ mimics $\ell^\infty$-penalties (single largest peaks), while $q$ in $\ell^1$ or $\ell^2$ moderates average power. In practice, $p \approx 2q$ offers a favorable trade-off (Taner et al., 2021).

7. Limitations, Open Problems, and Future Directions

While the observer LPs guarantee global optimality under positivity, infeasibility can occur if no $L$ satisfies both Metzlerness and nonnegativity. For the JPP-$p\!q$ formulation, nonconvexity implies only local convergence guarantees; global optimality is not in general ensured (Taner et al., 2021).

Further research directions include:

• Extensions to nonlinear or time-varying systems (e.g., time-varying channels, coordinated multi-cell scenarios)
• Incorporation of additional real-world constraints (hardware non-idealities, mixed-integer optimization)
• Automated parameter tuning for $(p, q, \tau)$ in nonconvex solvers
• Analytical characterization of the trade-off bridges between peak, average, and power-increase metrics

Multi-observable peak-minimization, through both positive-system LPs in control and smooth nonconvex surrogates in communications, provides systematic mechanisms for enforcing joint performance guarantees across high-dimensional observable sets, with provable properties and scalable algorithms (Briat et al., 2015, Taner et al., 2021).