Routh reducibility and controllability of unstable mechanical systems

Acta Mech https://doi.org/10.1007/s00707-022-03146-1 O R I G I NA L PA P E R Mate B. Vizi · Daniel M. Horvath · Gabor Stepan Routh reducibility and controllability of unstable mechanical systems Received: 30 August 2021 / Revised: 4 December 2021 / Accepted: 11 January 2022 © The Author(s) 2022 Abstract Routh reduction presents the minimum number of differential equations that uniquely describe the state of nonlinear mechanical systems where the state variables can be separated into essential ones and cyclic ones. This work extends Routh reducibility for a relevant set of controlled mechanical systems. A chain of theorems is presented for identifying the conditions when reduced order rank conditions can be applied for determining the Kalman controllability of Routh reducible mechanical systems where actuation takes place along the cyclic coordinates only, while some of the essential coordinates and their derivatives are observed. Four mechanical examples represent the advantages of using reduced rank conditions to check and/or to exclude linear controllability in such systems. 1 Introduction Routh introduced his technique [22] for conservative mechanical systems in the form of a hybrid Lagrangian and Hamiltonian description. The advantage of this Routhian formalism becomes apparent when so-called cyclic coordinates are present in the system. By decoupling the cyclic coordinates and the related hidden motion of the system, Routh’s method gives fewer number of ordinary differential equations, which are also called the equations of essential motion. The reduced model captures the essential dynamics of the given system. The so-called hidden motion, that is, the time evolution of the cyclic coordinates, can be reconstructed based on the essential motion. The model reduction also makes it easier to investigate the dynamical behavior either analytically or numerically. Moreover, the stability of certain steady-state motions can be analyzed by means of Lyapunov functions that are based on the so-called Routh potential [23]. To this day, the Routh reduction is still part of active research. For example, a new application of the Routh reducibility is presented in [20] and novel examples are revisited such as the tipple top on a cylinder’s surface [1]. The Routhian approach served as basis for several generalizations like the new model reduction techniques in [19], or for the extension of the theory like the one for discrete systems in [13]. Controllability is a crucial property of every control system; a system is controllable if any initial state can be transferred to any desired state in a finite length of time by some control action [15–17]. Main applications include redundancy/safety checking, optimal control, filter design, computer vision or stabilizing unstable states by feedback [3,14,17,24], to mention a few only. In the present paper, the Kalman controllability of cyclic mechanical systems is analyzed where external actuation is restricted to the cyclic coordinates, while the essential coordinates serve as the output states. As it is The research reported in this paper has been supported by the Hungarian National Science Foundation under Grant No. NKFI K 132477 and NKFI KKP 133846. M. B. Vizi (B) · D. M. Horvath · G. Stepan Department of Applied Mechanics, Budapest University of Technology and Economics, Muegyetem rkp. 5, 1111 Budapest, Hungary E-mail: M. B. Vizi et al. illustrated by several examples, this is a quite common and natural scenario in practice. In Sect. 2, the general methodology is presented for obtaining the reduced mechanical models, and the state-space model of the reduced system is given in closed form. A chain of theorems is presented which provide conditions of Kalman controllability based on reduced size rank conditions. By means of the kinetic energy of the Routh reducible system, one of the theorems present necessary conditions for Kalman controllability without constructing the full mechanical model or the corresponding state-space model. In Sect. 3, these results are demonstrated on the control of four nonlinear Routh reducible mechanical systems. The first two examples, the well-known Furuta pendulum and the double inverted pendulum highlight the advantages of the extended reduction methodology: compared with the literature (see [5,18]), reduced size models and reduced rank controllability conditions are obtained. The third example of the Wilson pendulum [11] demonstrates the application of the theorems when uncontrollability is proven in a Routh reducible system. The last example of a rotor model [10] represents the limitation of the reduction methodology when the Kalman controllability condition applied for the full state model cannot be simplified to reduced rank conditions. 2 Routh reducible systems and their control 2.1 Setup Consider an n ≥ 2-degree-of-freedom (DoF) holonomic mechanical system with external active forces; the equations of motion can be obtained by the Lagrangian equations of the 2nd kind in the form d ∂L ∂L − = Qk , dt ∂ ẏk ∂ yk (1) where L is the Lagrangian function, yk , k = 1, . . . , n are the generalized coordinates and Q k , k = 1, . . . , n are the generalized forces. The Lagrangian L can be expressed as L = T − V, where the kinetic energy T is a function of the generalized coordinates yk and velocities ẏk , while the potential function V depends on yk only. For scleronomic mechanical systems, where only time-independent geometric constraints are present, the general form of the kinetic energy is: T = 1 T ẏ M(y)ẏ , 2 (2)     where y = col y1 · · · yn = yk is the vector of generalized coordinates, ẏ is the vector of generalized velocities and M is the positive definite nonlinear mass matrix. At this point, the usual Einstein summation notation is introduced: a repeated a product means the  index in n summation along that index. In this particular case, the ẏk Mkl (yl ) ẏl stands for nk=1 l=1 ẏk Mkl ẏl replacing the matrix products in the vector notation (2). The Lagrangian can be independent of some of the generalized coordinates, which are called cyclic coordinates. The cyclic coordinates can be eliminated from the equations of motion resulting not only in fewer variables, but also in fewer equations [22]. Assume that the n degree of freedom mechanical system has m essential coordinates qi , i = 1, . . . , m (on which the Lagrangian depends) and n − m cyclic coordinates ϕα , α = 1, . . . , n − m. This way the generalized coordinates can be split in the form     q qi y= = , ϕ ϕα separating and distinguishing the essential and cyclic coordinates. Similarly, the nonlinear mass matrix can be partitioned as     Ai j Biα A B M= = [Mkl ] = , (3) T D BT D Bαi αβ Routh reducibility and controllability of unstable mechanical systems where A = [Ai j ] is related to the essential velocities only, D = [Dαβ ] is related to the cyclic velocities only, and B = [Biα ] refers to their mi (...truncated)