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Multi-objective robust controller synthesis with integral quadratic constraints in discrete-time

Lukas Schwenkel, Johannes Köhler, Matthias A. Müller, Carsten W. Scherer, Frank Allgöwer

TL;DR

This work develops a comprehensive framework for robust controller design in uncertain discrete-time systems using dynamic integral quadratic constraints (IQCs). It introduces finite-horizon IQCs with terminal costs and a loop transformation to enable time-domain, LMI-based analysis of stability and multiple performance criteria, including the $\mathcal{H}_\infty$-norm, energy-to-peak, and peak-to-peak gains. The synthesis side then leverages a factorization of $\Psi^* M \Psi$ to convexify the design problem, yielding an alternating analysis-synthesis algorithm with warm starts that can optimize a mix of performance criteria across channels using a common Lyapunov matrix and a common IQC multiplier. The approach is demonstrated on numerical examples, showing competitive performance with $\mu$-synthesis and enabling multi-objective trade-offs and handling of a broad class of uncertainties. Overall, the framework extends IQC-based robust synthesis to discrete time and non-standard performance channels, with practical implications for constraint satisfaction and reachability in uncertain systems.

Abstract

This article presents a novel framework for the robust controller synthesis problem in discrete-time systems using dynamic Integral Quadratic Constraints (IQCs). We present an algorithm to minimize closed-loop performance measures such as the $\mathcal H_\infty$-norm, the energy-to-peak gain, the peak-to-peak gain, or a multi-objective mix thereof. While IQCs provide a powerful tool for modeling structured uncertainties and nonlinearities, existing synthesis methods are limited to the $\mathcal H_\infty$-norm, continuous-time systems, or special system structures. By minimizing the energy-to-peak and peak-to-peak gain, the proposed synthesis can be utilized to bound the peak of the output, which is crucial in many applications requiring robust constraint satisfaction, input-to-state stability, reachability analysis, or other pointwise-in-time bounds. Numerical examples demonstrate the robustness and performance of the controllers synthesized with the proposed algorithm.

Multi-objective robust controller synthesis with integral quadratic constraints in discrete-time

TL;DR

This work develops a comprehensive framework for robust controller design in uncertain discrete-time systems using dynamic integral quadratic constraints (IQCs). It introduces finite-horizon IQCs with terminal costs and a loop transformation to enable time-domain, LMI-based analysis of stability and multiple performance criteria, including the -norm, energy-to-peak, and peak-to-peak gains. The synthesis side then leverages a factorization of to convexify the design problem, yielding an alternating analysis-synthesis algorithm with warm starts that can optimize a mix of performance criteria across channels using a common Lyapunov matrix and a common IQC multiplier. The approach is demonstrated on numerical examples, showing competitive performance with -synthesis and enabling multi-objective trade-offs and handling of a broad class of uncertainties. Overall, the framework extends IQC-based robust synthesis to discrete time and non-standard performance channels, with practical implications for constraint satisfaction and reachability in uncertain systems.

Abstract

This article presents a novel framework for the robust controller synthesis problem in discrete-time systems using dynamic Integral Quadratic Constraints (IQCs). We present an algorithm to minimize closed-loop performance measures such as the -norm, the energy-to-peak gain, the peak-to-peak gain, or a multi-objective mix thereof. While IQCs provide a powerful tool for modeling structured uncertainties and nonlinearities, existing synthesis methods are limited to the -norm, continuous-time systems, or special system structures. By minimizing the energy-to-peak and peak-to-peak gain, the proposed synthesis can be utilized to bound the peak of the output, which is crucial in many applications requiring robust constraint satisfaction, input-to-state stability, reachability analysis, or other pointwise-in-time bounds. Numerical examples demonstrate the robustness and performance of the controllers synthesized with the proposed algorithm.

Paper Structure

This paper contains 12 sections, 10 theorems, 85 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Setup and Preliminaries
  3. Performance criteria and stability
  4. Loop transformation
  5. Discrete-time Integral Quadratic Constraints
  6. Robustness analysis
  7. Synthesis
  8. On the factorization of $\Psi ^*M\Psi$
  9. Convexifying transformation of the decision variables
  10. Algorithm to design multi-objective robust controllers
  11. Examples
  12. Conclusion

Key Result

theorem 1

Assume for some $\rho\in(0,1]$ and for all $\Delta\in\boldsymbol{\Delta}$ that $\Delta_\rho$ satisfies the finite horizon IQC with terminal cost defined by $(A_\Psi,B_\Psi,C_\Psi,D_\Psi,X,M)$. Further, assume that there exists $P\in\mathbb{S}^{n_\chi}$ and $\mu\geq 0$ such that hold. Then $\Delta\star G \star K$ is $\ell_{2,\rho}$-stable for all $\Delta \in \boldsymbol{\Delta}$.

Figures (4)

  • Figure 1: Interconnection of the system $G$, the uncertainty $\Delta$, and the controller $K$ with the performance channel from $w$ to $z$.
  • Figure 2: Interconnection after the loop transformation.
  • Figure 3: The value of $\gamma_1+\gamma_2+\gamma_3$ after each (common) analysis step of the algorithm. After each iteration, we performed an additional individual analysis step for comparison.
  • Figure 4: Step responses of the synthesized robust controller (upper plot) and the nominal controller (lower plot) for different uncertainties $\Delta\in\boldsymbol{\Delta}$.

Theorems & Definitions (36)

  • definition 1: Energy and peak induced norms
  • remark 1: (Peak- and $\ell_\infty$-norm)
  • definition 2: $\ell_{2,\rho}$-stability
  • remark 2: (Exponential and input-to-state stability)
  • definition 3: Finite horizon IQC with terminal cost
  • remark 3: (Hard and soft IQCs)
  • remark 4: ($\rho$-hard IQCs)
  • theorem 1: Stability
  • proof
  • theorem 2: $\mathcal{H}_\infty$-norm
  • ...and 26 more