Optimal control of mechanical systems (Q2802889)

The main topic of the reviewed book is an application of the optimal control theory for an optimal synthesis of mechanical systems. The emphasis is put on practical applications, mainly on methods of reducing vibrations in various technical devices caused by external noise sources of harmonic character. From the mathematical point of view, the proposed methods are based on well-known optimality criteria, including the Pontryagin Maximum Principle in the first place. The crucial point here is a careful choice of both the cost functions and the class of systems to study, which should balance the generality of the formulation and more practical aspects such resource effectiveness, the latter understood for example as the energy consumption or the computational power needed to steer the considered device in an optimal way. I had a feeling that this balance is well-established in the presented results, which are neither to abstract nor too specific. Their practical significance is well-illustrated by a few interesting realizations of industrial character discussed in the book.NEWLINENEWLINENEWLINENEWLINEI think that this book can be particularly interesting for more applied researchers and engineers, yet the language barreer (the book was written in Polish) has to be taken into account. Most of the discussed results can, however, be found in English publications provided in the extensive bibliography -- the book is a summary several years of scientific activity of the author.NEWLINENEWLINENEWLINENEWLINEChapter 2 has an introductory character. It contains a brief resume of the optimal control theory for nonlinear control systems, including the Pontryagin Maximum Principle and Hamilton-Jacobi-Bellman equations. An emphasis is put on practical methods of finding optimal solutions, in particular methods based on the Taylor expansion, Galerkin approximation and an approach of Tang and Gao of optimization of linear systems with harmonic noise.NEWLINENEWLINENEWLINENEWLINEChapter 3 is devoted to the problem of the reduction of vibrations in mechanical controlled systems. From the practical point of view it is usually important to perform such a reduction not in the whole spectrum of frequencies, but only in a few distinguished ones. For example, in drilling devices this could be the frequency of rotation of the drill, and in suspension systems -- the resonant frequencies of the vehicle (or of the internal organs of the driver). Another practical motivation behind such a selective reduction of vibrations is that it should be much cheaper -- regarding the necessary energy -- than the reduction of vibrations in the whole spectrum.NEWLINENEWLINENEWLINENEWLINEIn the discussed chapter we consider a linear control system (with constant coefficients) of the form NEWLINE\[NEWLINE\dot x(t)=A x(t)+B u(t)+ Dw(t),\tag{1}NEWLINE\]NEWLINE with the usual conventions: \(x(t)\) being the trajectory of the system, \(u(t)\) the control, and \(w(t)\) the noise. It is additionally assumed that \(A\) is feedback-stabilizable, hence we may restrict our attention to a situation where \(A\) is stable (i.e., all its eigenvalues have negative real parts). The noise \(w(t)\) is assumed to consist of a finite number of harmonic inputs with fixed frequencies \(\omega_j\), \(j=1,\hdots,n\). The author proposed an original method of reducing the vibrations of the system at these particular frequencies. This is achieved by solving an optimal control problem with a specific choice of the cost function.NEWLINENEWLINENEWLINENEWLINEThe first step in his method is introducing a space \(\widetilde U\) consisting of bounded measurable functions \(u:[0,\infty)\rightarrow\mathbb{R}^n\) which have well-defined limits NEWLINE\[NEWLINE\lim_{T\to +\infty}\frac 1T\int_0^Tu(t)\sin(\omega t+\phi)\mathrm{d} tNEWLINE\]NEWLINE for every \(\omega\) and \(\phi\). It is proved that \(\widetilde U\) is a Banach space (Theorem 1) containing bounded functions vanishing at plus infinity, periodic functions (Theorem 3), as well as uniformly almost periodic functions (Theorem 4). The important property of functions \(u(t)\in \widetilde U\) is that they can be uniquely decomposed into harmonic components NEWLINE\[NEWLINEu(t)=\sum_{j=1}^n\left[u_\alpha^{(j)}\sin(\omega_jt)+u_\beta^{(j)}\cos(\omega_jt)\right]+\tilde{u}(t)=:\sum_{j=1}^nu^{(j)}(t)+\tilde u(t)\;,NEWLINE\]NEWLINE which are pairwise-orthogonal with respect to a natural scalar product \(\langle f,g\rangle=\lim_{T\to +\infty}\frac 1T\int_0^T f(t)g(t)\mathrm{d} t\) in \(\widetilde U\) (Theorem 2). Due to the uniqueness of this decomposition the following cost function NEWLINE\[NEWLINE J_{\omega}(x,u)=\limsup_{T\to+\infty}\frac 1T\int_0^T\left(x^T(t)Q x(t)+\sum_{j=1}^n\left(u^{(j)}(t)\right)^TR_ju^{(j)}(t)\right)\mathrm{d} t\tag{2} NEWLINE\]NEWLINE is well-defined for the class of controls \(u(t)\in \widetilde U\). Here \(Q\) and \(R_j\) are arbitrary positively-defined matrices. In Theorem 8 the author finds a solution \(u_\omega(t)\) of the optimal control problem (1)--(2) which does not depend on the position \(x(t)\) but only on the noise \(w(t)\). In fact this optimal solution is a linear combination of the harmonic noise \(w(t)\) and of the noise \(w(t)\) rotated by the angle \(\pi/2\). The main idea of the proof is to decompose the system (1) into harmonic components (here the linearity is used), solve the optimal control problem for each component separately and then prove that the sum of these partial solutions is the solution of the initial optimal control problem. In fact the partial solutions solve an optimal control problem in a bigger class of piece-wise continuous functions. It is also worth to notice that the obtained solution is not unique, as an addition of a bounded control vanishing at infinity does not affect the value of the cost functional \(J_\omega\).NEWLINENEWLINENEWLINENEWLINEThe proposed method is then tested theoretically and practically on a model of an active multi-step car suspension. Matrices (in this case numbers) \(R_j\) are chosen in such a way that the vibration transition functions achieve desired values of -30Db at the chosen frequencies \(\omega_j\) (by modifying these matrices one can improve vibration reduction performance of the system). The results of both numerical and practical experiments show that the method works as intended. The two issues of practical importance are: resonant frequencies of the system (at these frequencies the vibration reduction is usually a bit smaller than desired) and the adjustment time of the system. Regarding the latter, the harmonic components of the noise \(w(t)\) should be observed in real-time. In practice this is not possible and these components are synthesized by observing the noise \(w(t)\) through a certain peroid of time preceding the present moment. This peroid for \(\omega_j\)-component should be of order \(\frac {10}{\omega_j}\) and thus the system adjusts slower to the changes in the low-frequency noise.NEWLINENEWLINENEWLINENEWLINEIn chapter 4 we discuss a problem of an optimal control of a system linear-in-controls with a nonlinear drift NEWLINE\[NEWLINE \dot x(t)=f(x(t))+B(x(t))u(t).\tag{3} NEWLINE\]NEWLINE We assume that the components of controls \(u(t)\) belong to fixed intervals \([u^i_{\min},u^i_{\max}]\). We consider the cost function depending only on the state variables NEWLINE\[NEWLINEJ(x)=\int_0^\infty f_0(x(t))\mathrm{d} t\;.NEWLINE\]NEWLINE The data \(f\), \(B\), \(f_0\) are assumed to be \(C^1\) maps. Additionally we assume that the system is stabilizable and that the above optimal control problem has a unique solution for each initial condition. In such a setting it follows easily from the Pontryagin Maximum Principle that the optimal control has essentially bang-bang character with switching points described by the signs of the components of the vector \(\psi^T(x)B(x)\). Here \((x,\psi(x))\) is a (unique) solution of the Hamiltonian system described by the Pontryagin Maximum Principle. Since determining an analytical solution of this system is in general difficult, the author proposed a method of finding an approximate solution in a given class of functions (say, polynomials of fixed degree or splines) using the method of least squares. Actually from practical reasons, this solution is found for a problem with a finite (yet big) time horizon. In this way we obtain an approximation of an optimal synthesis for the discussed problem.NEWLINENEWLINENEWLINENEWLINEFrom a practical point of view the above approximate optimal synthesis may be applied in a semi-active damping system (non-lineraities are usually caused by the very nature of mechanical elements used in the construction). The author proposed a concrete realization of a hydraulic damper controlled by a piezoelectric stack. For such a system we were able to obtain equations of motion (3). We considered optimal synthesis for two particular problems -- minimization of the shift of the damped mass and minimization of the force acting on this mass. In the second case it was necessary to modify the method a bit, as the cost function depended additionally on the control. For these two cases the author performed numerical tests of the damping properties of the proposed regulator, by calculating the (properly defined) vibration transition function for single-frequency harmonic noises of different amplitudes. These were compared with the performance of constant-input damping schemes in the same system. The results do not favor any method, however in certain range of frequencies the proposed semi-active method was better.NEWLINENEWLINENEWLINENEWLINEWith a little effort the results from chapter 4 can be extended to synthesize an active damping system. This happens in chapter 5. We consider the following modification of system (3) NEWLINE\[NEWLINE \dot x(t)=f(x(t))+B(x(t),w(t),\dot w(t))u(t),\tag{4} NEWLINE\]NEWLINE where \(x\), \(u\) are as before and, precisely as in chapter 3, vector \(w(t)\) models an external noise consisting of a finite set of harmonic inputs of fixed frequencies \(\omega_j\), \(j=1,\hdots,n\). (Note that \(\dot w(t)\) is, up to a linear transformation, \(w(t)\) rotated by the angle \(\pi/2\).) The cost function is postulated in the form NEWLINE\[NEWLINEJ(x)=\limsup_{T\to +\infty}\frac 1T\int_0^\infty f_0(x(t))\mathrm{d} t\;,NEWLINE\]NEWLINE with the same assumptions as before, i.e. \(f\), \(B\) and \(f_0\) are of class \(C^1\), the system is stabilizable and has a unique solution. Note that if we treat vectors \(w(t)\) and \(\dot w(t)\) as additional state variables, system (4) takes form (3), and for the latter we already know how to perform an approximate optimal synthesis.NEWLINENEWLINENEWLINENEWLINEThis trick is then tested practically on the same model of hydraulic damper controlled by a piezoelectric stack discussed before and for the same two problems -- minimization of the shift of the damped mass and minimization of the force acting on this mass. Numerical tests show that the damping properties of the resulting active controller beat the performance of constant-input damping schemes at the whole spectrum of frequencies.NEWLINENEWLINENEWLINENEWLINEThe final chapter 6 discusses an original method of optimal synthesis for linear control systems with quadratic costs and with additional non-linearities caused by the properties of controlling devices (e.g. histeretic behavior of magnetic devices). The proposed approach modifies the so-called clipped-LQR (linear quadratic regulator) method. The idea is first to derive the optimal regulator for the system with unbounded controls and then to modify it so that the actual bounds are taken into account. In this way one obtains a regulator which is only sub-optimal, yet easy to derive (for non-linear system it is usually impossible to calculate the optimal synthesis). The resulting sub-optimal solution is then applied as a semi-active regulator of a damping system.NEWLINENEWLINENEWLINENEWLINEThis idea is then tested numerically for a model of a magnetorheological damper. The results show that damping properties of the semi-active controller obtained from the modified clipped-LRQ method are better then in the case of a passive controller.NEWLINENEWLINENEWLINENEWLINEThe book ends with a short summary of the presented results and a brief discussion of a possible new directions of study. It also announces industrial implementations of the discussed methods.