Nonlinear control is a branch of control engineering that deals with systems whose behavior is not linear, meaning that the output is not proportional to the input. Nonlinear systems are ubiquitous in nature and technology, posing many challenges and opportunities for analysis, design, and optimization. In this blog post, we’ll trace the history of nonlinear control from its origins in the late 19th century to its current developments in the 1990s.
The Pioneers: Poincaré, Lyapunov, and Duffing
One of the pioneers of nonlinear control was Henri Poincaré, a French mathematician and physicist who studied the n-body problem in celestial mechanics. Poincaré discovered the complexity and sensitivity to initial conditions of this problem, leading to chaotic orbits. He introduced concepts like limit cycles and bifurcation theory, which study periodic solutions and qualitative changes in system behavior.
Another important figure was Aleksandr Lyapunov, a Russian mathematician who developed a general method for analyzing the stability of dynamic systems. He defined stability as the property where small perturbations in the initial state do not cause large deviations in the final state and introduced the Lyapunov function, for proving stability or instability.
Georg Duffing, a German engineer, modeled a dynamic chaotic system using a second-order nonlinear differential equation, now known as the Duffing equation. This equation describes the motion of a mass-spring-damper system with a nonlinear restoring force and exhibits complex phenomena like bifurcations, chaos, and subharmonic oscillations.
The Interwar Period: Van der Pol, Bode, Krylov, and Bogoliubov
In the 1920s, Balthasar van der Pol, a Dutch physicist and electrical engineer, studied limit cycles in relation to ooscillator dynamics. He discovered self-sustained oscillations, now called van der Pol oscillations, and the phenomenon of entrainment or phase locking.
In the 1930s, Hendrik Bode, an American electrical engineer, developed the Bode plot, an asymptotic representation of the frequency response of linear systems, for analyzing stability and performance of feedback systems.
Also in the 1930s, Nikolay Krylov and Nikolay Bogoliubov, two Soviet mathematicians, made significant contributions to nonlinear dynamics and control. They proved the existence of the invariant measures theorem, which describes the long-term statistical behavior of bounded solutions, and developed the describing function method for approximating nonlinear systems by linear ones.
The Postwar Era: Nyquist, Lur’e, Emelyanov, and Lorenz
In 1932, Harry Nyquist, a Swedish-American electrical engineer, published his regeneration (or feedback) theory, which is a criterion for determining the stability of feedback systems based on their frequency response, known as the Nyquist criterion.
In 1944, Aleksandr Lur’e, a Soviet mathematician and engineer,
formulated the absolute stability problem, which is to find sufficient conditions for ensuring the stability of feedback systems with nonlinear elements. He proposed a class of nonlinearities and derived algebraic conditions for stability based on Lyapunov functions.
In the 1950s, Sergey Emelyanov, a Soviet mathematician and engineer, introduced the theory of variable structure systems with sliding mode control, a robust control technique that switches between linear controllers depending on the system’s state.
In the 1960s, Edward Norton Lorenz, an American mathematician and meteorologist, pioneered chaos theory, which is the study of deterministic systems that exhibit unpredictable behavior due to sensitivity to initial conditions. Lorenz discovered chaos while studying atmospheric convection and coined the term “butterfly effect.”
The Modern Era: Kalman, Popov, Yakubovich, and Zames
In 1960, Rudolf Kalman and Richard Bertram, American mathematicians and engineers, recovered Lyapunov’s work in the control context and developed the Kalman-Bucy filter, an optimal estimator for linear systems with Gaussian noise.
In 1961, V.M. Popov, a Soviet mathematician and engineer, proposed the circle criterion for asymptotic stability of feedback systems with nonlinear elements and introduced the concept of hyperstability, a stronger form of stability.
In 1962, V.A. Yakubovich, another Soviet mathematician and engineer, established connections between Lur’e and Popov’s results and generalized the circle criterion to systems with time-varying parameters and multiple nonlinearities. He also developed the frequency theorem, a necessary and sufficient condition for absolute stability based on the frequency response.
In the 1980s, George Zames, a Canadian-American electrical engineer, formulated the H∞ control problem, which is to find a controller that minimizes the worst-case gain from a disturbance input to an output of interest, generalizing classical control problems.
Recent Developments: Ortega, Spong, Sontag, Wang, and Isidori
In 1989, Romeo Ortega and Mark Spong, American electrical engineers, introduced passivity-based control (PBC), a control technique that exploits the energy properties of nonlinear systems to achieve desired behavior.
In the 1990s, Eduardo Sontag and Yorai Wardi, American mathematicians, developed the theory of input-to-state stability (ISS) for nonlinear systems, which relates the effect of an input signal on the system’s state.
In 1995, Alberto Isidori, an Italian mathematician and engineer, published his book on geometric control theory, a framework for analyzing and designing nonlinear control systems using differential geometric methods.
In this blog post, we’ve reviewed the main milestones and achievements in the history of nonlinear control, from Poincaré to Isidori. We’ve seen how nonlinear control emerged from the study of natural phenomena and developed into a rich and diverse field that encompasses various methods and applications, closely related to other branches of mathematics and engineering.
