Cheng Zhao

Controlling multi‐input large‐scale systems is a challenging task, as a high‐dimensional control input space may lead to increased computational demands, difficulties in controller design, and system‐wide time delays. In this paper, we propose a novel input dimension reduction method that transforms any linear time‐invariant multi‐input system into an input‐coupled nonlinear system with only two control inputs, thereby significantly reducing the dimensionality of the input space and is scalable. We demonstrate that the nonlinear system retains global controllability and certain stability properties under certain suitable assumptions. In addition, we propose straightforward design methods for the feedback gain matrix to globally stabilize the nonlinear system. Using numerical simulations, we verify the theoretical results and test the effectiveness of the input dimension reduction method under noise and time delay.

We investigate the stabilizability of Nash equilibrium for a general class of the nonlinear game-based control system (GBCS), which was initially introduced to model the control systems with strategic behavior, such as social, economic, and “intelligent” systems. The GBCS exhibits a hierarchical structure comprising a higher-level regulator and multiple lower-level rational agents. The regulator functions as the global controller and makes decisions first, after which the agents attempt to optimize their respective objective functions. For a given control of the regulator, the lower-level agents engage in a noncooperative game. The stabilizability problem addresses whether the regulator can stabilize the system by regulating the Nash equilibrium established by the agents at the lower level. In this paper, we will first formulate the stabilizability problem of the general nonlinear GBCS. Some explicit necessary and/or sufficient algebraic conditions on the stabilizability of Nash equilibrium are given by investigating the solvability relationship between the associated Hamilton–Jacobi–Isaacs equations and the algebraic Riccati equations related to an approximated linear-quadratic GBCS.

As is well known, classical PID control is ubiquitous in industrial processes, yet a rigorous and explicit design theory for nonlinear uncertain MIMO second-order systems remains underdeveloped. In this paper we consider a class of such systems with both uncertain dynamics and an unknown but strictly positive input gain, where the nonlinear uncertainty is characterized by bounds on the Jacobian with respect to the state variables. We explicitly construct a three-dimensional region for the PID gains that is sufficient to guarantee global stability and asymptotic tracking of constant references for all nonlinearities satisfying these Jacobian bounds. We then derive a corresponding necessary region, thereby revealing the inherent conservatism required to cope with worst-case uncertainties. Moreover, under additional structural assumptions on the nonlinearities, these sufficient and necessary regions coincide, yielding a precise necessary-and-sufficient characterization of all globally stabilizing PID gains. All these regions are given in closed form and depend only on the prescribed Jacobian bounds and the known lower bound of the input gain, in contrast to many qualitative tuning methods in the literature.

This article focuses on the adaptive linear quadratic Gaussian control problem, where both the state matrix $\mathsf{A}$ and the control gain matrix $\mathsf{B}$ are unknown. We only assume that $(\mathsf{A}, \mathsf{B})$ is stabilizable and $(\mathsf{A}, \mathsf{Q}^{1/2})$ is detectable, where $\mathsf{Q}$ is the weighting matrix of the state in the quadratic cost function. This significantly weakens the classic assumptions used in the literature. To design an optimal adaptive control, a weighted least squares algorithm is modified by using random regularization method, which can ensure uniform stabilizability and uniform detectability of the family of estimated models. At the same time, a diminishing excitation is incorporated into the design of the proposed adaptive control to guarantee strong consistency of the desired components of the estimates. Finally, although some components of the estimates may not converge to the true values, it is still demonstrated that a certainty equivalence control with diminishing excitation remains optimal for an ergodic quadratic cost function.

This paper studies control problems for a class of multi-input multi-output (MIMO) strongly coupled nonlinear uncertain systems with mixed relative degrees one and two. We will design an uncoupled PID controller, where each component of the input vector is determined only by a component of the system output regulation error. Under some suitable conditions on the system nonlinear uncertain functions, we will show that the proposed controller can ensure global stability of the closed-loop system while guaranteeing that the output regulation error converges to zero at an exponential rate. We will also demonstrate that the PID parameters can be chosen from an open and unbounded set constructed based on some prior knowledge regarding the nonlinear functions of the system, and that the designed PID controller exhibits two-sided robustness in terms of the uncertain system structure and the selection of controller parameters. A simulation will be provided to verify the theoretical findings.

In this paper, we are concerned with the tracking performance and antidisturbance ability of the widely used proportional-integral-derivative (PID) controllers in practice. Towards this end, we consider a basic class of second-order nonlinear stochastic control systems subject to model uncertainties and external disturbances, and focus on the ability of the classical PID controller to track time-varying reference signals. First, under some suitable conditions on the system nonlinear functions, reference signals, and external disturbances, we show that such control systems can be stabilized in the mean square sense, provided that the three PID gains are selected from a stability region constructed in the paper. Besides, it is shown that the steady-state tracking error has an upper bound proportional to the sum of the varying rates of reference signals, the varying rates of external disturbances, and the intensity of random noises. Meanwhile, its proportional coefficient depends on the selection of PID gains, which can be made arbitrarily small by choosing suitably large PID gains. Finally, by introducing a desired transient process which is shaped from the reference signal, a new PID tuning rule is presented, which can guarantee both the expected steady state and transient tracking performance.

In this article, we focus on the global stabilizability problem for a class of uncertain stochastic control systems, where both the drift term and the diffusion term are nonlinear functions of the state variables and the control variables. We will show that the widely applied proportional-derivative (PD) control in engineering practice has the ability to globally stabilize such systems in the mean square sense, provided that the upper bounds of partial derivatives of the nonlinear functions satisfy a certain algebraic inequality. It will also be proved that the stabilizing PD parameters can only be selected from a two dimensional bounded convex set, which differs significantly from the existing results for PD controlled stochastic systems where the diffusion term does not depend on the control variables. Moreover, a particular polynomial on these bounds is introduced, which can be used to determine under what conditions the system is not stabilizable by the PD control, and thus demonstrating the fundamental limitations of PD control.

In this paper, we will consider a class of continuous-time stochastic control systems with both unknown nonlinear structure and unknown disturbances, and investigate the capability of the classical proportional-integral-derivative(PID) controller in tracking time-varying reference signals. First, under some suitable conditions on system nonlinear functions, reference signals, and unknown disturbances, we will show that PID controllers can be designed to globally stabilize such systems and ensure the boundedness of the tracking error. Analytic design formulae for PID gain matrices are also provided, which only involve some prior knowledge of the partial derivatives of system structural nonlinear functions. Besides, it will be shown that the steady-state tracking error hinges on three critical factors: i) the change rate of reference signals and external disturbances; ii) the intensity of random noises; iii) the selection of PID gains, and can be made arbitrarily small by choosing PID gains suitably large. Finally, by introducing a desired transient process which is shaped from the reference signal, we will present a new PID tuning rule, which can guarantee both nice steady-state and superior transient control performances.