Curriculum: MechEng, UC Berkele ME 132, 3-units, required, 2 nd or 3 rd ear students Prerequisites: Programming and Scientific computing (E7); Linear Algebra and ODEs (Math 54) Emphasis: mathematics of feedback sstems, at elementar level No lab No formal text Subsequent slides discuss Extensive use of computation/simulation first 8 hours of class ME 134, 3-units, elective, classical control, w/ lab Significant overlap with 132. Popular, nevertheless and repetition is OK Uses traditional text Cross-listed w/ EECS control (which has extensive signals/sstems prereqs) ME 135, 3-units, elective, microprocessors, real-time OS: hardware, software for control (project) ME 190I, 1-unit, elective, Modeling from Data ME 190L, 1-unit, elective, Loopshaping (Nquist/Bode, Glover-McFarlane, and more) ME 190M, 1-unit, elective, Model-Predictive di Control ME 190Y, 1-unit, elective, Youla-parametrization, closed-loop design optimization ME 102A, 3-units units, required, 3 rd ear, instrumentation, signals and measurement (w/ lab) ME 102B, 3-units, required, 4 th ear, mechatronics design (w/ individual project)
First few hours IEEE CSM, IEEE CSM, IEEE CSM, then Simulating feedback sstems Students come in knowing : Basic ODE theor for Runge-Kutta, ode45, function handles Simple conservation laws and constitutive eqs behavior described b ODEs Feedback examples, before an analsis results Motivate feedback architecture and strategies error, gain, speed-of-response, filters/smoothing, Dela-differential equations Look at families of (r,d,n,δ) (,u) responses d r Simulating delas with ode45 u G Δ (solve in chunks), ode45dela F K z u + + n F m m Experience/internalize common behaviors/objectives/tradeoffs in time domain Build appreciation for (and understanding of) graphical-based simulation tools (eg., Simulink) b writing general purpose simulation interconnection scripts Data structure: dnamical sstems (of a component) represented b 2 functions, (f,h) Interconnection-specific routines for composite (f,h)
Initial move to analsis (3 hours) First-order linear sstem, Response (derive as convolution integral, from integrating factor) Free response, forced response Linearit of Definition of stabilit, time-constant Stable sstems if bounded input has a limit, then response has a limit (ie., notion of stead-state gain) If input is bounded, output is bounded (deriving bound-to-bound relationship) If input is sinusoidal, output converges to sinusoidal (direct, from convolution) Frequenc-response function Dela-differential equation Observe (via ode45dela) onset (as T increases) of instabilit Make claim that at critical-value of T, a purel sinusoidal solution emerges Equation to determine T crit and ω crit
Analzing feedback sstems (3 hours) First-order closed-loop loop sstems 1 st order plant, proportional control Increasing speed of response Stabilizing an unstable G Unstable G is more problematic than stable G r m Influence induced b feedback: (d on u) and (Δ on u) to reduce influence on Influence of (n on u) and (n on ) through feedback 0 th order plant, 1 st order control Obtain 0-stead-state gain d with (and onl with) pure integral in controller K z u d u G Δ Simple studies, analtical, from the 1 st order theor established. Supplemented b Time responses: (r,d,n,δ) (,u) Gang-of-6 FRF Make informal, ad-hoc connections between time-responses and FRFs No genuine mention of Fourier transforms: observe patterns; anticipate and verif on new cases Margins (anwhere in loop) Dela margin Gain margin Saturation/Antiwindup ti ti i Discrete (time) implementation of K Effect of sample time + + n At this point, student has basic knowledge of all aspects of single-loop feedback sstems.
Remaining Topics Generalize to higher order ODEs Routh-Hurwitz (no/es: 15 minutes) Use quadratic formula to prove 2 nd order version State 3 rd and 4 th order test with examples Onl state as stable/unstable, not with number of sign changes Memorize 2 nd and 3 rd order results, use as needed Give historical account, credit to Routh and Hurwitz, and reference to work PI, PD, PID for 1 st and 2 nd order plants Elementar loopshaping, and arithmetic of feedback loops Pole-placement No Nquist (190L) No root locus (as a technique) N th order plant, (2N-1) th order desired char-pol, solve for (N-1) th order controller N th order plant, (2N) th order desired char-pol, solve for (N) th order controller w/ integral action Theoreticall interesting, relativel eas, student s like it, feels empowering GCD, coprimeness, Bezout identit, Euclid s algorithm Generalizes PI control of 1 st order sstem Less important for higher order sstems (where do ou put the poles?) Robustness margins (gain margin, time-dela margin, percentage-uncertaint margin) If robustness margins are important in chapter 8, then the are important in chapter 10 too Example, popular text: 4 th order SISO plant; pole-placement with seemingl reasonable poles; one good-looking simulation; no mention of margins (<1degree, <0.1 db) Jacobian Linearizations Linear state-space: solutions, realizations, model conversions
Heaviside Yes, Laplace No Starting with Heaviside, smbolic calculus had been shown to be an effective tool for linear time-invariant dnamical sstems. Under the influence of circuit theor, it had become evident that these methods allowed to analze complex sstems, b combining series, parallel, and feedback interconnections. The spirit of Heaviside s smbolic calculus was to be able to think of a differential Oliver Heaviside 1850 operator or a dela as a formal indeterminate for which a differential operator or a dela can be substituted. Unfortunatel, analsts had squeezed this marvelous idea in the mathematical rigor (mortis) of Laplace transforms, using complex functions, with domains of convergence and other cumbersome but largel irrelevant mathematical traps. Oliver Heaviside, 1850-1925, man remarkable accomplishments Jan Willems, In Control, Almost from the Beginning Until the Da After Tomorrow, European Journal of Control, 2007
Single-Loop, linear sstems Input/output ODEs are treated as the foundational mathematical framework Understand stabilit, homogeneous solutions Frequenc-response functions Described b Linear Differential Operators, Manipulation of interconnections v r v r v G G TF is simpl notation for ODE, suggestive of Substitution, Arithmetic manipulations, Decompositions, which are then verified as correct (b appealing back to ODEs) No mention of integral transform, ROCs, etc Lose out on one result: closed-loop time domain constraints (on various responses) due to RHP poles/zeros in plant