Romeo Ortega - PID Passivity-Based Control of Nonlinear Systems with Applications

Throughout this book, we consider nonlinear systems described by differential equations of the form

(1)

where is the state vector, , , is the control vector, is an output of the system defined via the mappings and , and is the input matrix, which is full rank. In the sequel, we will refer to this system as or system.

We also consider the case of port‐Hamiltonian systems when the vector field may be factorized as

(2)

where is the Hamiltonian, and , with and , are the interconnection and damping matrices, respectively. To simplify the notation in the sequel, we define the matrix ,

Motivated by current practice, in this book, we explore the possibility of applying the industry‐standard proportional‐integral‐derivative (PID) controllers to regulate the behavior of nonlinear systems. As is well known, PID controllers are universal, in the sense that they incorporate knowledge of the system's past, present, and future, and they are overwhelmingly dominant in engineering practice. PIDs are highly successful when the main control objective is to drive a given output signal to a constant value. PIDs, however, have two main drawbacks, first, the task of tuning the gains is far from obvious when the system's operating region is large; second, in some practical applications, the control objective cannot be captured by the behavior of output signals.

In this book we show that, for a wide class of systems, these two difficulties can be overcome by exploiting the property of passivity, which in the case of physical systems captures the universal feature of energy conservation. To achieve this end, we propose a new class of controllers called PID passivity‐based controls (PBCs), whose main construction principle is to wrap the PID around a passive output of the plant. Since PIDs define (output strictly) passive systems for all positive gains, and the feedback interconnection of passive systems is stable, the proposed architecture yields a highly robust design that preserves stability for all tuning gains – considerably simplifying the task of commissioning the controller. To enable potential designers to use PID‐PBCs, we present in the book a comprehensive coverage of this topic.

Since passivity for physical systems is simply a reformulation of energy balancing, it is possible in many practical examples to easily identify some passive outputs. However, in many examples, either these outputs are not the ones we would like to regulate, and/or their desired value is not equal to zero. To address the first problem, we explore in the book the possibility of adding an integral action to nonpassive outputs preserving some stability properties. For the second problem, we propose to wrap the PID around the error of the output signal, and then we investigate whether the system is passive with respect to this error signal – a property called shifted passivity .

Another scenario of practical interest is when the control objective is to drive the full system state to a desired constant value. A classical example is mechanical systems, whose passive outputs are the actuated velocities, but in many applications – e.g. robotics – the objective is to drive all positions to some desired constant values. To formulate mathematically this objective, we aim at achieving Lyapunov stability of the desired equilibrium, a task that entails the need to construct a Lyapunov function, i.e., a nonincreasing function of the state with a minimum at the desired equilibrium. The approach we adopt in the book to solve this new task is to identify passive outputs whose integral can be expressed as a function of the system's state. The identification of these outputs boils down to finding first integrals for the closed‐loop dynamics that, in its turn, requires the solution of partial differential equations. The design is completed by projecting the closed‐loop dynamics onto the invariant manifold defined by the first integrals and verifying that the resulting function, which depends only on the state of the system, is positive definite.