- 2. Multidisciplinary Dynamic System Design Optimization (MDSDO) • MDSDO is an emerging new area of MDO that addresses the unique properties of dynamic systems. Established MDO formulations have been applied to dynamic system design, but do not address directly or capitalize on the unique nature of dynamic systems. • Important Research Fronts for MDSDO: A. Balanced Co-Design B. Passive Dynamics C. Direct Transcription D. Dynamic System Topology Optimization E. Surrogate Modeling of Dynamic Systems zs ms /4 ks cs zus mus /4 kt v ct z0 Example Application Areas: Active automotive suspension design, wave energy converters, wind turbines Engineering System Design Lab Important Characteristics of MDSDO: • Formulations are Intrinsically Dynamic • Multidisciplinary and Integrated • Systems-Oriented • Capitalize on Passive Dynamics • Support Parallel Computation Refs: J7-J10, C12-C19, C21, C22, TH1, TR1 2
- 3. Co-Design: Integrated Physical and Control System Design Conventional methods for active engineering system design utilize sequential design processes. First the physical system (plant) is designed, and then control engineers develop the control system based on this control design: min xp s.t. gp (⇠(t), xp ) 0 ˙ ⇠(t) f (xp , ⇠(t), t) = 0 3 Mech. design Elements of Co-Design Research • Most co-design researchers approach codesign from a controls-centric perspective. At the ESDL, we take the opposite approach. We seek to learn how to solve the co-design problem when considering physical design in a more comprehensive way. • We develop of new numerical strategies, such as the extension of direct transcription to co-design, that solve co-design problems efficiently. Mech. Subspace Plant Design Optimization Sequential design does not capitalize on coupling between plant and control design. At the ESDL, we develop and investigate design methods that integrate control and mechanical design problems and produce system-optimal designs efficiently. Integrated physical and control system design is often called co-design. Engineering System Design Lab (⇠(t), xp ) x p⇤ min u(t) (⇠(t), u(t), xp⇤ ) s.t. gp (⇠(t), xp⇤ ) 0 ˙ ⇠(t) f (⇠(t), u(t), xp⇤ , t) = 0 Control Design Optimization Control design Control Subspace Above: Conventional design methods explore only one design domain at a time. While iterating this sequential approach can in some cases converge on the right solution, it is inefficient. Co-design allows designers to explore both domains simultaneously; we can trace a more direct path toward the system optimum. Refs: J7-J10, C12-C18, C21-C23
- 4. Direct Transcription (DT) • • • • • Numerical method for optimal control that has emerged as a very important strategy for solving realistic co-design problems. In DT we discretize the state and control trajectories in time: min u(t),⇠(t) Z tF L(u(t), ⇠(t), t)dt 0 ˙ subject to: ⇠(t) = fd (u(t), ⇠(t), t) (⇠(t), u(t)) ! (⌅, U) Infinite-dimensional, continuous problem and then we convert the state equations to algebraic equations using collocation. In DT we solve simultaneously for the optimal control trajectories and the corresponding state trajectories (AAO/SAND). Related to pseudospectral methods Midpoint Quadrature, Trapezoidal Collocation Finite-dimensional, discrete problem Advantages: • Most important reason for using DT with co-design: it accommodates nonlinear inequality constraints, which are present in realistic co-design problems. • While DT results in a large-dimension NLP, it is sparse and can be solved efficiently using parallel computing. • DT can solve singular optimal control problems. min U,⌅ nt 1 X L(ui , ⇠i , ti )hi i=1 subject to: ⇣(U, ⌅) = hi (fd i + fd i+1 ) 2 + ⇠i+1 ⇠i = 0 i = 1, 2, . . . , nt 1 Refs: J8, J10, C13, C16 – C19, C22, C23 Engineering System Design Lab 4
- 5. Wave Energy Converter (WEC) Design • • • Ocean waves have the highest energy density among renewable energy sources We are studying co-design (integrated optimal control and physical design) of heaving buoy WECs. Energy extraction for heaving buoy WECs is given by: E= • • • • • Z T P (t)dt = 0 Z z z0 T z(t)FP T O (t)dt ˙ 0 Active control of the power take-off force (FPTO) increases energy production. Direct Transcription used for optimal control. We have studied WEC design in the presence of irregular ocean waves. We are investigating a variety of PTO architectures, including linear and rotary electric machines as well as hydraulic systems. Future work will involve other WEC types, physical testing, and co-design using high-fidelity multiphysics modeling. ¼ FP T O PTO How do we design the FPTO trajectory? What PTOs and control systems will produce the optimal force trajectory? Refs: C16, C22 Engineering System Design Lab 5
- 6. Co-Design of an Active Suspension • • • Objective function: zs The physical design of early active suspension designs were based on previous passive designs. The physical aspects of active systems should be designed differently than passive systems. Co-design enables us to design the best overall system. ms /4 mus /4 • • Engineering System Design Lab ct Damper Design Spring and damper geometry Stress, fatigue, packaging, and thermal constraints Our study revealed the importance of incorporating nonlinear inequality constraints in co-design problems, motivating the use of direct transcription (DT) for optimal control. Simultaneous co-design with DT was the most efficient and reliable design optimization strategy. v z0 Previous co-design studies of active suspensions used simplified plant models. Here we developed a detailed plant model that includes: – – cs zus kt • Dynamic Model: ks Spring Design Rod D d Extension Chamber Piston Compression Valve Pressure tube Fs Piston Extension Valve Working Piston Foot Valve L0 Compression Chamber Foot Chamber Floating Piston p Ls Gas Chamber Refs: J10, C13 6
- 7. Active Automotive Suspension Testbed • • Suspension with reconfigurable linkage: Validating co-design methods requires a physical testbed that permits both control and physical system changes. We are developing a reconfigurable suspension testbed where the following properties can be modified: – – – – Linkage geometry Spring stiffness Damping rate (using magnetorheological damping) Feedback control design • Testbed reconfigurations are fully automated • This testbed will also be used in several courses including: – – – • GE 100 (Introduction to ISE) GE 410 (Component Design) GE 413 (Engineering Design Optimization) In GE 410 students design their own suspension system, including detailed linkage, spring, and shaft design. At the end of the course they will validate their designs on the testbed. Engineering System Design Lab Refs: J10, C13 7
- 8. Design of Genetic Regulatory Networks • Synthetic biology is an area that focuses on construction of functional biological devices and systems. • Existing design methods in synthetic biology are limited in their ability to manage large-scale and complex biological systems; We need new design principles and methods to break this complexity barrier and develop synthetic biological systems of practical importance. FA FB • • • Previous numerical design studies based on enumeration were limited to three-node networks. We are exploring several techniques for making possible the topological design of much larger networks, including Mathematical Programs with Complementarity Constraints (MPCCs), and Generative Algorithms. Our design methods manage both topological and continuous design variables. System dynamics are modeled using MichaelisMenten equations (ODEs), and are optimized using Direct Transcription (DT). A B C I2 Input • Input Example: Adaptive Network Design I1 Output O peak (Opeak O1 ) / O1 sensitivity (I2 I1 ) / I1 O2 Output O1 precision (O2 O1 ) / O1 ( I 2 I1 ) / I1 1 Refs: C19 Engineering System Design Lab 8
- 9. Simultaneous Structural and Control Design Optimization Co-Design of Wind Turbines problem considers the structural and control design simultaneously. y, a combined weighted objective function is considered as following. Simultaneous Plant and Control Design: Z tf • Considering pslant aw2 control d⌧opt (t))2 dt min w1 m (x) + nd (⌧ (t) esign x=[xp ,⌧ (t)]T Blade flap-wise bending mode together produces system-‐op7mal wind s.t. xp 0 turbine designs. ky k1 y max 0 (3.16) • Direct TranscripAon was utsed to tsolve the k⇣k1 ⇣max 0 simultaneous co-‐design problem, accounAng Pm (vrated ) Prated 0 for many plant constraints including x = Ax + Bu ˙ Elements of Wind Turbine Co-‐Design structural dynamics. , w1 > 0 and w2 > 0 are the weights on structural design and control design involves independent geometric design Plant Extension to Farm-‐Level Co-‐design n objective function respectively. variables. Control design includes open-‐loop opAmal • AccounAng for dynamic wake eﬀects allows control of rotor torque based on actual wind data. us to opAmize system-‐level performance Model Analysis instead of individual turbine performance. • Simultaneous Refs: C18, C23 notonicity Analysis opAmal placement, turbine The continuous problem 3.16, after this direct transcription becomes: design, and distributed control. 0 R Rh Ht der the simultaneous structural and control problem: Z tf min w1 ms (x) + w2 (⌧ (t) ⌧opt (t))2 dt x=[xp ,⌧ (t)]T 0 s.t. xp 0 kyt k1 yt max 0 k⇣k1 ⇣max 0 Pm (vrated ) Prated 0 x = Ax + Bu ˙ 13 Engineering System Design Lab Tower aft-tofore bending mode Tower side-toside bending mode min w1 ms (x) + w2 x=[x p , x s , x c ]T (xc (i) ⌧o p t (i))2 i =1 s.t. (4.1) nX t +1 max xs (1, i)) yt m max xs (2, i)) ⇣m a x a x xp 0 0; for i = 1 . . . nt + 1 0; for i = 1 . . . nt + 1 Pm (vr a t e d ) Pr a t e d 0 dt (x(i + 1) + x(i)) = 0; for i = 1 . . . nt ˙ ˙ 2 (5.1) tf t0 where, dt = n t is the step size of simulation. Since there are 7 states 9 in the considered system and 1 control input along with 5 structural dex(i + 1) x(i)
- 10. ˙ x been made in the area of black-box surrogate modeling, includx (t) ⇡ j (x (t), xp ) + Bu(t) ing the use of a family of surrogate models where the best (or weighted average) surrogate model is used as required [31], and x x where j (x (t), xp ) ⇡ f(x (t), xp ). The co-design problem based extension of surrogate modeling to multi-objective optimization this surrogate model is: problems where high accuracy is maintained in regions near the Z Pareto front [32]. x min J = L(x (t), xp , u(t))dt While in many cases surrogate modeling has been applied xp ,u(t) In some dynamic system design problems, the time derivative function for the to a single engineering discipline at a time [32] (e.g. structural state space equations is the computational bottleneck. x s.t. g(x (t), xp ) 0 design [33], multibody dynamic systems [34], design based on • This occurs often in hierarchical, multi-scale models, especially x h(x (t), xp ) = 0 aerodynamics and aero-acoustics [17], etc.), it can be extended models involving multiple analysis domains, such as aeroelasticity. ˙ to multidisciplinary problems [35]. Co-design problems are mulx(t) ⇡ j(x (t), xp ) + Bu(t) x New method: iteratively construct a surrogate model of the derivative tidisciplinary design optimization problems that involve the coufunction instead of the entire dynamic response. pled physical and control system design disciplines [36]. This • Conventional surrogate modeling cannot capitalize on the dynamic The design method proposed here consists of an inner loop t introducesproperties. complexity to the surrogate modeling probadditional system solves Prob. (3), and an outer loop that iteratively enhances lem, as accuracy must be provided not only in the design space surrogate model. The method consists of the following ﬁve ste • Approximating only the derivative function preserves the fundamental innature of dynamic of the optimum designis still used. in the the neighborhood system as simulation point, but also 1. Deﬁne the sampling domain in state space and design sp state space in the neighborhood of the state trajectory that corSignificant computational savings: responds to the optimum design point. The latter requirement 2. Sample test points in the combined state and design spac • in is An order of magnitude improvement in about accuracy in a more difﬁcult because we are concerned computation time observed3. Build and validate the state derivative surrogate model cy only in regions of studies. interest (e.g., near the j (·): initial case strategic region near an entire path as opposed to a single point. This arti4. [29, 30]. A signiﬁcant number of developments surrogate modeling (DFSM): Solve the co-design problem Unique challenges for derivative function have cle introduces one possible approach for tackling this challenge ˙ (Sample surrogate model x in the• areaMust ensure surrogate model accuracy across design/state/control 5. x (t) ⇡accuracyxp ) + Bu(t) ofrequirements, repeat st of black-box surrogate modeling, includCheck j (x (t), and convergence state derivative (2) associated with co-design problems. for a simple second-order dynamic system) spaces simultaneously. where the best (or 1–4 until requirements are satisﬁed e of a family Consider a general co-design optimization problem formulaof surrogate models Design of Dynamic Systems with Surrogate Models of Derivative Functions • Convergence verage) surrogateinvolvesis used ascurrently unknown. of physical sys-(x (t), x ) ⇡ f(x (t), x ). process is illustrated in Fig. 1, and describ tion that modelproperties required [31], and the simultaneous optimization x where j x p This iterative The co-design problem based on p Refs: C23, TH1 of surrogate modeling tosystem designs: optimization tem and control multi-objective in detail this surrogate model is:in the following subsections. where high accuracy is maintained in regions near the Original Formulation: With surrogate derivative function: Z tF Z t [32]. 2.1 Constructing thepSampling Plan x min J = L(x (t), x , u(t))dt x min in many cases surrogate,u(t) J = has (t), xp , u(t))dt modeling L(x been applied xp ,u(t) process starts with a deﬁnition of the modeling doma xp The 0 engineering discipline at a time [32] (e.g. structural i.e., the regions within the state and design spaces where the s x s.t. g(x (t), xp ) 0 (1) (3) x s.t. ], multibody dynamic systemsg(x (t), design0based on [34], xp ) rogate model will be constructed, and the regions from wh x h(x (t), xp ) = 0 x ics and aero-acoustics [17], etc.),(t),can = 0extended h(x it xp ) be samples will be obtained. Here the modeling domain is deﬁn ˙ ciplinary problems [35]. Co-design= f(x (t), x are mul˙ using simple bounds (t),the)state and design spaces that are e x x (t) ⇡ j (x on xp + Bu(t) x x (t) problemsp ) + Bu(t). ry design optimization problems that involve the coumates of the maximum and minimum values that the plant des and state variables will attain. Sample points are chosen fr cal and control is a cost functiondisciplines [36]. overall system design system design that This Here J The 10 design method proposed here domain using Latin Hypercube Sampl consists of an inner loop that Engineering System Design represents the Lab within the modeling additional complexity to the surrogate modeling prob-
- 11. Plant Limited Co-Design (PLCD) Many engineering systems are too complex or expensive to consider clean-sheet design of a completely new system. If an existing system no longer meets current needs, a common approach is to redesign limited elements of the system. System redesign is a commonly encountered challenge in industry. PLCD Design Strategy: Sensitivity analysis is used to identify what design changes would have the greatest impact on system performance. Co-design is then used to produce a limited redesign that meets new system requirements. Example PLCD Problem: Counterbalanced Robotic Manipulator Design. 1: IdenAfy Candidate Plant ModiﬁcaAons Identify a limited number of system elements to redesign while still meeting new system requirements. 2: Develop System Model 3: Formulate and Solve PLCD Problem 4: Verify and Repeat if Needed Clean-sheet co-design produces a manipulator that exploits passive dynamics to produce near-zero energy consumption, but is very expensive to implement. Refs: J7, J9, C14, C15 PLCD produces a limited redesign with energy consumption almost as low as the co-design result, but with much lower implementation cost. Engineering System Design Lab 11
- 12. Generative Algorithms for Design Optimization Topological design of non-continuum systems is an extremely complex problem, particularly when continuous variables must be optimized to evaluate how good each topological design really is. We use generative algorithms to abstract system topology designs for efficient design space exploration. • Instead of optimizing system elements directly, we generate system designs using iterative rules. • We then optimize with respect to generative algorithm rules, reducing design problem dimension and complexity, and supporting exploration of designs across a wide range of dimensions. Example: L-Systems for Structural Truss Design • Lindenmayer parallel rewriting system used to perform cellular-division to produce truss designs. • Continuous truss design variables are optimized in an inner loop using sequential linear programming. • Stability constraints are embedded within the generative algorithm, guaranteeing that any truss generated is structurally stable. 12 Engineering System Design Lab Modified cellular division process for truss generation: Sample generated trusses:
- 13. Origami-Based Generative Design • • • Many generative algorithms are inspired by biological processes (e.g., cellular division, venation). In this study we instead observe the creative process of an origami artist, and infer new rules for generative algorithms based on artistic exploration. These new generative algorithms can be applied to the design of folded engineering systems. Example: • Simple operations can be identified based on the folding of the four basic origami bases (shown below). These operation have been used successfully in the development of a new generative algorithm. Kite Base Engineering System Design Lab Fish Base Bird Base 13 13 Frog Base
- 14. GE 100 Trebuchet Project • • • • • Engineering System Design Lab Members of the ESDL worked together to design and build 30 reconfigurable trebuchet kits that have been used in GE 100 since Fall 2012. All ISE freshmen take GE 100. This project provides with a hands-on design experience, and helps them learn firsthand the value of modeling and simulation in design. GE 100 students learn basic Design of Experiments (DOE), and then tested their trebuchet designs on the Bardeen Engineering Quad. After learning about the physics of trebuchets and refining their designs using a SimMechanicsTM model, GE 100 students tested their trebuchets again with substantially better results. The trebuchet kits have been used in outreach efforts, including UIUC Engineering Open House and local K-12 outreach activities. 14

- JTA: I simplified and focused this a little more on what I thought was most important. I’m not really addressing formatting, since I know these slides show up differently on my Mac. I installed TeXPoint, but some of the equations still don’t show up correctly, so I can’t check all of the equations.
- Equation Notes:(\bm{\xi}(t),\mathbf{u}(t))\rightarrow(\bm{\Xi},\mathbf{U})\min_{\mathbf{u}(t),\bm{\xi}(t)}\quad\quad &\int_{0}^{t_F}L(\mathbf{u}(t),\bm{\xi}(t),t)dt \\ \mbox{subject to:}\quad &\dot{\bm{\xi}}(t) = \mathbf{f_d}(\mathbf{u}(t),\bm{\xi}(t),t)\min_{\mathbf{U},\bm{\Xi}}\quad\quad &\sum_{i=1}^{n_t-1}L(\mathbf{u}_i,\bm{\xi}_i,t_i)h_i \\ \mbox{subject to:}\quad &\bm{\zeta}(\mathbf{U},\bm{\Xi}) = -\frac{h_i}{2}\left(\mathbf{f_d}_i+\mathbf{f_d}_{i+1}\right) \\ &\quad\quad\quad\quad\quad + \bm{\xi}_{i+1}-\bm{\xi}_i=\mathbf{0}\\ &i = 1,2,\ldots ,n_t-1
- Equation Notes:
- CD used before for frame design. Here we embedded constraints into algorithm for trusses. New figure? Check WCSMO slides.