ContentsPreface xvAcknowledgments xxiiiI Linear Control1 Some mathematical basis for control of linear systems1.1 Linear systems vs nonlinear systems .1.1.1 Superposition principle for static functions1.1.2 Superposition principle for differential equations .1.1.3 Stability and transient response of linear systems .1.1.4 The rationale behind control design based on timeresponse1.1.5 Response to sinusoidal excitations1.1.6 The rationale behind control design based on frequencyresponse1.1.7 Frequency response in nonlinear differential equations1.1.8 Laplace transform and nonlinear systems1.2 Classical control1.2.1 Root locus method .1.2.2 Frequency response method .1.2.3 The nonminimum phase systems approach1.3 Sensitivity1.4 Controllability .1.5 Linear state feedback control1.6 LQR control1.7 Differential flatness and the state space-transfer function relationship1.8 Linear approximation of nonlinear state equationsBibliography2 Wheeled pendulum model2.1 Robot description2.1.1 The path following task2.2 Mathematical model of wheeled pendulum2.2.1 Unconstrained dynamics2.2.2 Mathematical model subject to nonholonomic constraints2.2.3 Computing torques to be applied2.3 The experimental prototype2.3.1 Robot mechanical subsystem2.3.2 Hardware used for controller implementation2.4 Linear approximate model2.4.1 Orientation control2.4.2 The undisturbed wheeled pendulum model 2.4.3 Pole(eigenvalue)-placement design criterion2.5 Differential flatness-based model2.6 Conclusions2.7 ExercisesBibliography3 Root locus based controller design3.1 Design procedure3.2 Controller gains selection3.3 Experimental results3.4 Simulation results3.5 Effect of Cz on the open-loop pole3.6 One-degree-of-freedom root-locus-based design3.7 Conclusions3.8 ExercisesBibliography4 Frequency response-based design4.1 The proposed design procedure4.2 Controller gains selection4.3 Experimental results4.4 Simulation results4.5 Conclusions4.6 Exercises Bibliography5 The nonminimum-phase systems approach 1455.1 The design procedure5.2 Controller gains selection5.3 Experimental results5.4 Simulation results5.5 Conclusions5.6 ExercisesBibliography6 LQR control 1676.1 The design procedure6.2 Controller gains selection6.3 Experimental results6.4 Simulation results6.5 Conclusions6.6 ExercisesBibliography7 Effects of diverse model parameters7.1 The disturbed wheeled pendulum model7.2 Effects of Cz, R, and Mp in closed-loop system performance7.2.1 Robot control using different values for Cz7.2.2 Robot control using different values for wheel radius 7.2.3 Increasing pendulum body mass7.3 Induced limit cycles as control gains change7.3.1 Study through experiments7.4 Coupling between θ− and (α, ν)−subsystems7.5 Discrete-time model of wheeled pendulum7.5.1 Closed-loop stability when sampling period and wheelradius change7.6 Conclusions7.7 ExercisesBibliographyII Nonlinear Control 2298 Some mathematical basis for control of nonlinear systems 2318.1 Lyapunov stability8.2 Positive and negative definiteness8.3 Stability theorems8.4 Passivity8.5 Hamiltonian formulation8.6 Linear systems and exponential stability8.7 A particular second order differential equationBibliograph9 Hamiltonian formulation 2579.1 Hamiltonian formulation of arbitrary Euler-Lagrange systems 9.2 Hamiltonian formulation of arbitrary nonholonomic Euler-Lagrangesystems9.2.1 Nonholonomic constraints9.2.2 Hamiltonian modeling (Van Der Schaft [2000])9.2.3 Obtaining the nonholonomic Hamiltonian model in(9.33)9.3 Hamiltonian representation of the error equation9.4 Hamiltonian representation of the error equation subject tononholonomic constraints9.5 ConclusionsBibliography10 Constrained Hamiltonian representation of wheeled pendulum 10.1 The constrained Hamiltonian model10.2 Obtaining Hec(˜q + q∗, ˜p1e) from ˜He(˜q + q∗, ˜pe)10.3 Simplifying the Hamiltonian model for control design10.4 Conclusions11 Path following control11.1 Hamiltonian approach11.2 Feedback linearization control11.2.1 A first proposal11.2.2 A successful feedback linearization controller11.3 Energy-shaping is equivalent to feedback linearization 11.4 Experimental results11.4.1 Linear simplification of controller in Proposition11.4.2 Linear simplification of controller in Proposition11.4.3 Linear simplification of controller in Proposition11.4.4 Controller gains selection11.4.5 Experimental tests11.5 Simulation results11.6 Exercises11.7 ConclusionsBibliographyA Program CodesA.1 Robot parametersA.2 C code for microcontroller programmingA.3 Matlab code used to draw figs. 2.6, 2.7, and 2.8A.4 Matlab code used to compute controller gains in SectionA.5 Simulation of linear control schemesA.5.1 Straight lineA.5.2 8−shaped trajectoryA.6 Matlab code used to compute controller gains in SectionA.7 Matlab code used to compute controller gains found in Section5.2A.8 Matlab code used to compute LQR controller gainsA.9 Matlab code used to compute data shown in Tables 7.1 and 7.2A.10 Matlab code used for limit cycle analysisA.11 Programs for discrete-time analysisA.11.1 Matlab code used for Z−transform computationA.11.2 Matlab-Simulink block diagram for discrete-time simulations A.12 Matlab code used to compute controller gains in Section 11.4.4A.13 Matlab-Simulink block diagram used for simulations in Chapter11BibliographyIndex
