(12c) Data-Driven Controller Synthesis through the Learning of Integral Quadratic Constraints

Data-driven control, through the incorporation of statistical and machine learning into control theory, provides complementary strength to classical model-based control. We are especially interested in model-free control, where the identification of governing equations can be avoided and only essential control-relevant information is learned from data. In our previous works, we have developed the Dissipativity Learning Control (DLC) framework, where the dissipative property of the system is the control-relevant information to be learned from input and output trajectories [1,2]. This is motivated by the fact that process systems, obeying thermodynamic laws and transport kinetics, must be dissipative, i.e., there exists a storage function whose rate of change is bounded by an input- and output-dependent supply rate [3].

However, the existing DLC framework is restricted to quadratic supply rate functions, based on which the controller can only be obtained in P/PI/PID forms. To accommodate more complex process dynamics and also to improve control performance, methods for data-driven, model-free synthesis of more generic forms of controllers are needed. To this end, we generalize the quadratic dissipativity form to the concept of integral quadratic constraints (IQC). In IQC, dynamic multipliers are allowed to transform the input and output signals into augmented dimensions and define dissipativity on the transformed variables. IQC was first proposed as a unifying approach for uncertainty characterization in robust stability and performance analysis [4], and theoretical connections between its frequency-domain or time-domain forms have been established [5], allowing the expression of IQC conditions as linear matrix inequalities (LMIs) according to the Kalman-Yakubovich-Popov (KYP) Lemma.

In this talk, we present the framework of IQC Learning Control as follows.

1. With a given matrix of dynamic multipliers, the IQC is parameterized by a matrix of dissipativity parameters. Its range (called the dissipativity set) is the dual cone of another set (called the dual dissipativity set) that can be characterized using trajectory samples.

2. The transfer function matrix of the controller is optimized by minimizing an upper bound on the L₂-gain from disturbances to inputs and outputs. The controller synthesis problem is formulated as semidefinite programming (SDP) through the KYP Lemma.

We also perform a case study on a reactive distillation column example. By comparing the performance of the obtained controllers based on IQC learning, PID controllers based on dissipativity learning, and model identification-based LQG controller, we demonstrate the advantage of this proposed generic, input-output data-driven, model-free controller synthesis approach.

References

1. Tang, W., & Daoutidis, P. (2019). Dissipativity learning control (DLC): A framework of input–output data-driven control. Comput. Chem. Eng., 130, 106576.

2. Tang, W., & Daoutidis, P. (2021). Dissipativity learning control (DLC): Theoretical foundations of input–output data-driven model-free process control. Syst. Control Lett., 147, 104831.

3. Alonso, A. A., & Ydstie, B. E. (1996). Process systems, passivity and the second law of thermodynamics. Comput. Chem. Eng., 20, S1119–S1124.

4. Megretski, A., & Rantzer, A. (1997). System analysis via integral quadratic constraints. IEEE Trans. Autom. Control, 42, 819–830.

5. Seiler, P. (2014). Stability analysis with dissipation inequalities and integral quadratic constraints. IEEE Trans. Autom. Control, 60, 1704-1709.