Rethinking Strict Dissipativity for Economic MPC

This paper introduces the novel concept of two-storage strict dissipativity to bridge the gap between storage functions and value functions in economic model predictive control, proving it as a necessary and sufficient condition for asymptotic stability while offering easier verification and new terminal cost designs.

The Gist

Here is an explanation of the paper "Rethinking Strict Dissipativity for Economic MPC" using simple language, analogies, and metaphors.

The Big Picture: Driving a Car to Save Money

Imagine you are driving a car (the System) and your goal is to get from point A to point B while spending as little money on gas as possible (the Economic Cost).

In a standard driving scenario (called Tracking MPC), you have a specific destination in mind (like a parking spot), and your goal is simply to get there as smoothly as possible. It's easy to prove that if you follow the rules, you will eventually stop at the parking spot without crashing.

But in Economic MPC, the goal is different. You aren't just trying to park; you are trying to make money or save fuel by driving in a specific way. Maybe the best way to save gas is to drive in a zig-zag pattern, or maybe the "best" steady state is actually a specific speed where the engine is most efficient. The problem is: How do we know that by trying to save money, we won't accidentally drive off a cliff or get stuck in a loop?

Mathematicians use a concept called Dissipativity to guarantee safety. Think of Dissipativity as a "Energy Budget."

• The Rule: You can only spend energy (or money) if you have stored some up first.
• Strict Dissipativity: This is a stricter rule. It says, "Not only must you have a budget, but every time you take a step, you must lose a tiny bit of potential energy, ensuring you eventually settle down at the best spot (the optimal steady state)."

The Problem: The "Magic" Storage Function

To prove that the car will settle down safely, mathematicians usually need a "Storage Function."

• Analogy: Imagine a magical bank account. If you deposit money (store energy) when you are far from the destination, and withdraw it as you get closer, the math proves you will arrive safely.
• The Catch: In "Economic" driving, this magical bank account is very hard to find. The paper argues that the standard way of defining this bank account is too rigid. It requires a specific mathematical trick (rotating the cost) that doesn't always work or is very hard to check in real-world, messy systems.

The Solution: The "Two-Storage" System

The author, Mario Zanon, proposes a new idea called Two-Storage Strict Dissipativity.

Instead of trying to find one perfect magical bank account, he suggests using two different accounts that talk to each other.

1. Account A (The Forward Look): This account tracks the cost of driving forward from where you are now to the destination. It asks: "How much will it cost to get there?"
2. Account B (The Backward Look): This account tracks the cost of driving backward from the destination to where you are now. It asks: "If we were at the destination, how much would it have cost to get here?"

The New Rule:
For the system to be safe, the "Forward Cost" must always be strictly higher than the "Backward Cost" (plus a little safety buffer) whenever you are not at the perfect destination.

• The Metaphor: Imagine you are hiking up a mountain (the Forward Cost) and looking down at the valley (the Backward Cost).
  • If you are at the top (the optimal state), the view up and the view down are the same.
  • If you are anywhere else, the "cost to climb up" must be strictly greater than the "cost to come down."
  • If this gap exists, it proves that you are "sliding" down the hill toward the bottom (the optimal state) and won't get stuck on a ledge.

Why is this better?

1. Easier to Check: Finding one perfect "Magic Bank Account" is like finding a unicorn. Finding two accounts that just need to be different from each other is much easier. It's like checking if two friends have different bank balances rather than checking if one friend has a specific, magical balance.
2. It Works Both Ways: The paper proves that if this "Two-Storage" rule holds, the car will eventually settle at the most efficient spot. It also proves that if the car does settle safely, this rule must be true. It's a perfect two-way street.
3. Connecting the Dots: The paper shows that this new rule is actually just a different way of looking at the old rules. It connects the "cost to travel" (how much it costs to get from A to B) directly to the stability of the system.

The "Terminal Cost" (The Safety Net)

In real life, we can't drive forever; we have to stop after a certain time (a Finite Horizon). To make sure we don't crash when the timer runs out, we need a "Terminal Cost"—a penalty or reward for where we end up.

The paper discusses how to design this safety net.

• Old Way: You need a very specific, complex safety net that is hard to build.
• New Way: As long as your "Two-Storage" rule is true, you can use simpler safety nets. Even if you don't have a perfect safety net, if you drive long enough (a long prediction horizon), the math guarantees you will still end up safe and stable.

Summary in One Sentence

The paper introduces a new, easier-to-check mathematical rule (using two "cost accounts" instead of one) that guarantees an economic control system will naturally settle down to its most efficient state without crashing, making it much easier to design safe and smart controllers for complex machines.

Technical Summary

Here is a detailed technical summary of the paper "Rethinking Strict Dissipativity for Economic MPC" by Mario Zanon.

1. Problem Statement

The paper addresses the challenge of guaranteeing asymptotic stability in Economic Model Predictive Control (EMPC). Unlike standard tracking MPC, where the cost function is designed to drive the system to a setpoint, EMPC uses a generic economic stage cost $\ell(x, u)$ to optimize performance (e.g., energy efficiency, profit).

• The Core Issue: Proving asymptotic stability for EMPC is difficult because the optimal steady state is not necessarily the minimum of the raw stage cost.
• Current State of the Art: Stability is typically proven under the assumption of strict dissipativity. This condition implies the existence of a "storage function" $\lambda(x)$ such that the "rotated" stage cost (modified by $\lambda$) has its minimum at the optimal steady state.
• The Limitation: While strict dissipativity is sufficient for stability, it is not immediately clear how to relate the storage function to the value function of the original economic optimal control problem (OCP). Furthermore, verifying strict dissipativity can be computationally or analytically difficult. The paper questions whether strict dissipativity is the most natural or verifiable condition for stability.

2. Methodology

The author introduces a novel framework based on two-storage strict dissipativity and utilizes value functions defined over infinite horizons (forward and backward in time).

Key Definitions and Tools

• Forward Value Function ($V^+$): The optimal cost-to-go from state $x$ to the origin (steady state) over an infinite horizon.
• Backward Value Function ($V^-$): The optimal cost-to-reach state $x$ from the origin over an infinite horizon (essentially the negative of the cost-to-travel from the origin to $x$).
• Relaxed OCPs: To handle technical issues regarding feasibility and domain definitions, the author introduces relaxed problems ($V^\oplus, V^\ominus$) that allow for fictitious control inputs $z$ with a penalty $p\|z\|_1$.
• Two-Storage Strict Dissipativity: A new condition requiring two storage functions, $\lambda_1(x)$ and $\lambda_2(x)$, such that:
  1. Both satisfy the standard dissipativity inequality (with $\rho=0$).
  2. They are separated by a positive definite function $\gamma(\|x\|)$:
     $$\lambda_1(x) \geq \lambda_2(x) + \gamma(\|x\|)$$
     The paper proposes using the negative value functions as these storage functions: $\lambda_1(x) = -V^\ominus(x)$ and $\lambda_2(x) = -V^\oplus(x)$.

Analytical Approach

1. Infinite Horizon Analysis: The author establishes the relationship between dissipativity and the boundedness of $V^+$ and $V^-$. It is proven that if dissipativity holds, $V^+$ and $V^-$ are bounded and satisfy specific inequalities relative to the storage functions.
2. Stability Proofs:
   • Sufficiency: It is proven that if two-storage strict dissipativity holds, the feedback law derived from the forward OCP ($u^+$) is asymptotically stabilizing. The proof constructs a Lyapunov function using the rotated value functions.
   • Necessity: It is proven that if the system is asymptotically stabilizing with bounded cost, two-storage strict dissipativity must hold.
3. Finite Horizon Analysis: The paper analyzes the standard finite-horizon EMPC with terminal costs. It derives conditions on the terminal cost $V_f$ (specifically, $V_f(x) \geq V^\ominus(x) + \eta(\|x\|)$) required to guarantee stability as the prediction horizon $N$ increases.

3. Key Contributions

1. Introduction of Two-Storage Strict Dissipativity:
   The paper proposes a new stability condition that relies on two storage functions separated by a positive definite function. This condition is shown to be necessary and sufficient for asymptotic stability in EMPC.

2. Direct Link to Value Functions:
   Unlike standard strict dissipativity, where the storage function is abstract, the new condition allows the storage functions to be directly identified as the negative of the forward and backward value functions ($-V^+$ and $-V^-$). This provides a clear physical and control-theoretic interpretation: stability is guaranteed if the cost to reach the steady state is strictly greater than the cost to leave it (except at the steady state itself).

3. Equivalence and Verification:

   • The paper proves that standard strict dissipativity implies two-storage strict dissipativity.
   • It is conjectured (and proven for linear-quadratic cases) that the two conditions are equivalent in general, but the two-storage formulation is argued to be easier to verify because it relies on the difference between two value functions rather than finding a single storage function that satisfies a strict inequality.

4. Terminal Cost Design:
   The paper provides rigorous conditions for terminal costs in finite-horizon EMPC. It shows that a terminal cost of the form $V_f(x) = V^\ominus(x) + \eta(\|x\|)$ guarantees asymptotic stability for sufficiently long horizons, even without terminal constraints. It also proves that $V_f(x) \geq V^-(x)$ is a necessary condition for stability.

4. Results

• Theoretical Results:

  • Theorem 3.6: Two-storage strict dissipativity is sufficient for asymptotic stability of the closed-loop system.
  • Theorem 3.13: Two-storage strict dissipativity is necessary for asymptotic stability.
  • Theorem 3.11: Standard strict dissipativity implies two-storage strict dissipativity.
  • Theorem 4.5 & 4.6: For finite-horizon MPC, if the terminal cost satisfies specific lower bounds involving the backward value function, the closed-loop system is asymptotically stable as $N \to \infty$.

• Numerical Examples:

  • Constrained Linear Quadratic (LQ): Demonstrates how constraints affect the validity of rotated costs. It shows that a storage function derived from an unconstrained problem ($V^u_+$) may fail to satisfy dissipativity when constraints are active, whereas the constrained value function ($V^+_+$) succeeds.
  • Nonlinear System: A nonlinear example illustrates the difference between $V^+$ and $V^-$. It highlights that while $V^+(x) > V^-(x)$ generally, there are cases where the optimal forward and backward controls coincide at non-zero states, adding complexity to the proofs. The example also validates the convergence of finite-horizon value functions to the infinite-horizon value function as the prediction horizon increases.

5. Significance

• Bridging Theory and Practice: By linking the abstract concept of storage functions directly to the value functions of the economic OCP, the paper makes the stability conditions more tangible and potentially easier to compute or verify using dynamic programming or approximate dynamic programming techniques.
• Unifying Framework: The work unifies the concepts of "available storage" and "required supply" (from classical dissipativity theory) with the forward and backward value functions of EMPC, providing a deeper understanding of why strict dissipativity is required for stability.
• Practical Design Guidelines: The derived conditions for terminal costs offer concrete guidelines for designing stable EMPC controllers without requiring complex terminal constraints, provided the prediction horizon is long enough.
• Foundational Shift: The paper suggests that "two-storage strict dissipativity" might be the more fundamental condition for EMPC stability, offering a potentially more robust path for analysis than the traditional single-storage strict dissipativity.

In summary, Mario Zanon's paper redefines the stability analysis of Economic MPC by introducing a two-storage framework that is mathematically equivalent to (or implied by) strict dissipativity but offers a more direct connection to the underlying optimal control value functions, thereby simplifying verification and deepening the theoretical understanding of EMPC stability.