Abstract

We develop computational mechanisms for in-telligently simulating nonlinear control systems.Thesemechanismsenhancenumerical simulationswith deep domain knowledge of dynamical sys-tems theory and control theory, a qualitativephase-spacerepresentation of dynamical systems,symbolic and geometric manipulation capabilities,and a high-level interface. Programs equippedwith these capabilities are able to autonomouslysimulate a dynamical system, analyze the simula-tion results, and utilize the analysis to performdesign tasks. We demonstrate the mechanismswith an implemented computational environmentcalled the Control Engineer’s Workbench.

Keywords. Qualitative reasoning, scientific com-puting, numeric/symbolic processing,control sys-tem design.

IntroductionThe purpose of computing is insight, not numbers.

— R. W. Hamming

Computationally simulating complex physical sys-temsin engineering design has becomea commonprac-tice. Yet most of today’s simulations rely entirely onextensive numerical computations and laborious hu-man analysis. Human engineershave to shoulder theburden of translating physics and constraints into mod-els, preparing numerical simulations, interpreting con-sequencesof the experiments, and performing designtasks. Moreover, nonlinear systems can exhibit ex-tremely complicated behaviors that defy human anal-ysis and pure numerical simulations. The complexitiesof thesesystemsare largely due to nonlinearities, high

‘This research was supported in part by the Na-tional ScienceFoundation grants CCR-9308639and MIP-9001651,and in part by the Advanced ResearchProjectsAgency of the Department of DefenseunderOffice of NavalResearch contract N00014-89-J-3202.An early version ofthis paper appeared in Mathematics and Computers inSimulation.

dimensionality, and uncertainties of the systems andenvironments the systemsoperate in.

The difficulties in the traditional engineering simu-lation arise from the lack of(1) parsimonious represen-tations capturing the essenceof physical systemsandamenableto efficient computations, (2) efficient mod-eling algorithms for constructing the representations,and (3) effective reasoning methods that can use therepresentationsto computeand synthesizeuseful prop-erties for the systems. The lack of computable repre-sentations for physical dynamical systemshinders theexploitation of the special properties of the systemsand the attainment of the maximum performance forthe design. The simulation and design of the dynami-cal systemsare limited by the available computationalpower and the complexities of the systems.

While the traditional numerical computing has suc-cessfully attacked many practical problems, we cangreatly enhance its effectivenessand significantly ex-pand the scopeof what can be done with this styleof engineering computing by integrating the numericalcomputation with advancedartificial intelligence tech-nology and symbolic computing methods. For exam-ple, programs equipped with Al reasoning techniquesand deep domain knowledge have already helped engi-neers solve an open problem in hydrodynamics [Yip,1991], given new insights into behaviors of a heartmodel in cardiology [Sacks & Widman, 1993], and de-signed a high-performance nonlinear controller in ma-glev engineering [Zhao & Thornton, 1992]. Abelsonet al. described a collection of computer programsthat analyze dynamical systems at the level of ex-pert dynamicists [Abelson, 1989]. Other related workis discussed in [Nishida et al., 1991; Bradley, 1992;Kant et al., 1992; Amador, Finkelstein & Weld, 1993].

Task Domain: Control SystemSimulation and Design

We study the analysis and design of nonlinear con-trol systems. A real-world control system is a com-plex closed-loop systemwith extremely rich dynamics.Computation and reasoningare pervasive in the designand operation of the controller. Sensorscollect a large

Intelligent Computing AboutComplex Dynamical Systems*

Feng ZhaoLaboratory for Artificial Intelligence ResearchandDepartment of Computer and Information Science

The Ohio State University

Columbus, OH 43210 U.S.A.E-mail: fz@cis . ohio—state . edu

amount of quantitative information. State and param-eter estimators infer hidden information about the sys-tem from the senseddata. The systemis modeled witha representation appropriate for further analysis anddesign based on available information. The model isthen analyzed to extract behaviors that are consideredsignificant for the control objective. To meet the con-trol objective, a control law is synthesized to changethe natural dynamics of the system.

The domain of automatic control brings together is-suesof sensing, estimation, control synthesis,and con-trol execution. The study of their common themes—computation and reasoning—servesas a framework forcoherently addressing theseissues and makes it possi-ble to employ advancedcomputationaltechniquestodrastically improve modern control design. We focuson the control synthesis that maps a model of a phys-ical system togetherwith somecontrol objective to asynthesizedcontrol law:

Control Design:model + control objective —* control law.

A control engineergoesthrough the following designsteps to synthesizea controller for a given dynamicalsystem:

1. Analysis: analyze the model of the dynamical sys-tem. The physics and the constraints of the systemare often modeled with a quantitative mathematicalmodel, typically a set of differential equations. Thisstep examinesthe model and analyzesthe behaviorsof the system.

2. Design: design a controller for the system. Basedon the analysis, this step arrives at a control designaccording to the prespecifled control constraints.

3. Verification: verify the control design. This step en-suresthat the designmeetsthe control specification.

The steps 1, 2, and 3 of the above design proce-dure are often iterated before a reasonablecontrol lawis synthesized. Except for very few cases in whichanalytic-form solutions are available, computer simu-lations are the main tools for analyzing nonlinear sys-tems and for designing and verifying the controllers.

Existing control simulation softwares are inadequatefor automatically designing highly complex nonlinearsystems. Commercially available programs like MAT-

LAB and SIMULAB [MathWorks, 1989j rely on numer-ical simulations. These programs are, at their verybest, semi-automatic and serve as interactive designaids to human engineers. Although they are equippedwith elaborate graphic interfaces, theseprograms pro-vide only fragmented, limited capabilities such as inte-gration and root finding for performing the simulationtask; human users need to prepare the simulation andto interpret the result. The specialized control tool-boxes embedded in these programs are “shallow” ex-pert systems; they lack deepdomain knowledgeand do

not have mechanismsfor computationally representingand manipulating a control design.

Smart Simulation of DynamicalSystems

We address the difficulties of traditional numericalcomputing by developing computer representation andsimulation technologies necessary for enabling pro-grams to autonomously perform and interpret numeri-cal simulations of control systems. The design of com-plex control systemsrequires powerful computationalmechanisms to represent, reason about, and manip-ulate the dynamics of nonlinear systems. We demon-strate that difficult control synthesis taskscan be auto-mated, using a computational workbench that activelyexploits knowledge of nonlinear dynamics and phasespace. More specifically, we develop a computer repre-sentation for dynamical systemsin terms of qualitative,geometric phase-spacefeatures and equivalenceclassesof behaviors. We employ hybrid computation integrat-ing symbolic and numerical methodswith Al reasoningand representation techniques and exploiting sophisti-cated mathematical domain knowledge. We providea high-level interface for communicating the result ofanalysis in qualitative terms and for visualizing thephase-spacedynamics.

We have constructed a computational environment,the Control Engineer’s Workbench, integrating a suiteof programs that automatically analyze and designhigh-performance, global controllers for a large classof nonlinear systemsusing the qualitative phase-spacerepresentation [Zhao, 1992]. Given a model of a phys-ical system and a control objective, the Control En-gineer’s Workbench analyzes the system anddesignsa control law achieving the control objective. A usertypically interacts with the Workbench in the followingway.

The user first tells the Workbench about the sys-tem: he inputs a systemmodel in terms of an ordinarydifferential equation, parameter values, and bounds onstatevariables for analysisin the form of a phase-spaceregion. The user also tells the Workbench about therequirements on the control design: he specifiesthe de-sired state for the systemto settle in, the initial statesof the system, the allowable control parameter values,and the constraints on the control responses.

The user then asks the Workbench to analyze thesystem within the parameter ranges of the model. TheWorkbench visualizes the totality of the behaviors ofthe systemover the parameter ranges; it represents thequalitative aspects of the system in a data structureand reports to the user a high-level, symbolic summaryof the systembehaviors and, if necessary, a graphicvisualization of the phase-spacequalitative features.

Next, the user instructs the Workbench to synthe-size a control law for the system, subject to the spec-ified design requirements. The Workbench searchesfor the global control paths that connect the initial

Figure 1: Workbench Input: the model for a bucklingsteel column and the control objective of stabilizingthe buckling motion.

statesof the system and the desired goal state, usingthe qualitative description about the system. Morespecifically, the search is conducted in a collection ofdiscrete entities representing trajectory Rowsin phasespace. After the global control paths are established,the Workbench determines the controllable region ofthe system and the switching surfaces where controlparameters should change values. A synthesized con-trol reference trajectory consists of a sequenceof tra-jectory segments,each of which is under a constantcontrol.

The following example ifiustrates how the Work-bench autonomously analyzes the buckling motion ofa steel column under compression and synthesizes acontrol law to stabilize the motion. Figure 1 showsthe input to the Workbench—the model and controlobjective—and Figure 2 reports the synthesized con-trol reference trajectory superimposed on the phasespaceof the column. The global portion of the refer-encetrajectory shown in Figure 2 consists of four seg-ments, eachof which starts at aswitching state markedas a small circle; the reference trajectory connectstheinitial state with the goal state at the origin of thephase space. The control parameter value is held con-stant for each segment denoted by Ui, U2, U3, andU4, respectively.

The analysis of the nonlinear dynamics of the buck-ling column accountsfor alarge percentageof the com-putation in arriving at the desired control design. Theequation of motion for the column is described as anonlinear ordinary differential equation. An exhaus-tive numerical simulation can be very expensive. TheWorkbench employs a geometric analysis of dynami-cal systemsthat uses a phase-spacerepresentation tocapture the qualitative features of the system, first de-veloped by Poincaré at the beginning of the century.

;; The Synthesized Control Law specifying the time;; instance, switching state, and corresponding;; control value for each switching:((time 0.)

(switching—state #(—1 —3))(control .2))

((time .284)(switching—state #(—1.82 —2.71))(control 0.))

((time 1.06)(switching—state #(—1.86 2.49))(control — .2))

((time 2.71)(switching—state #(1.35 1.82))(control 0.))

((time 6.76)(switching—state #(— .0023 — .0692))(control *local—control*))

Figure 2: Workbench Output: the control referencetrajectory and the control law for stabilizing the col-umn. In the plot, the reference trajectory is drawn insolid lines, and the phase-spacestability boundaries ofthe uncontrolled column in dashed lines.

Representation, Simulation, andInterface

The Workbench has demonstrated the following capa-bilities in designinga control law for the buckling col-umn:

• Deciding what behaviors are significant. The Work-bench looks for qualitative features like equilibriumpoints, stability regions, and trajectory flows.

• Describing the behaviors qualitatively in computa-tional terms. The Workbench models the stabilityregions and trajectory flows geometrically.

• Reasoningabout the geometry of a phase space.

• Performing the control design autonomously.

These capabilities are supported in the Work-bench by (1) qualitative representation consisting ofdimension-independent geometric constructs, (2) hier-archical extraction of the behaviors, (3) modeling and

— p2

z~— p3z2+ U

equation_of inot ion:J dz1/dt=~ dz2/dt= —p~x~

state_variable:state_variable:parameter:parameter:parameter:boundingbox:

zi

= -2.0

P2 1.0p~= 0.2x1E [—3.0,3.0],

z2E [—4.0,4.0](0.0, 0.0)(—1.0, —3.0)

goalstate:initialst ate:range_of_control:

— — — — _ - ~‘ \

I / - - _ \‘ \ \

uE [—0.2,0.2]

manipulation mechanismsfor trajectory flows, and (4)algorithms implementing the geometric, combinatorial,and numerical computations.

Our intelligent simulation environment has threegeneric layers: (1) computer representations of thephysical world, (2) simulation technology that makesfeasiblethe computation about the physical world, and(3) high-level interfaces for communications betweenhuman usersand the computer. For our intended taskdomain of control design and analysis, we present thefollowing structure for our environment:

1. Computer representation:

We develop a qualitative representation describ-ing a dynamical system in terms of its phase-space geometric features: the qualitative phase-spacestructure. This qualitative representation cap-tures asymptotic and transient behaviors of a dy-namical system and essentialconstraints of the sys-tem useful for the control synthesistask.

2. Simulation technology:

Our simulation of a control system comprises nu-merical and symbolic computation about the systemmodel and geometric modeling of phase space. TheWorkbench also embodiesdeep domain knowledgeofdynamical systems theory and control theory thatcan guide quantitative simulation and reduce theamount of computation necessaryfor analyzing thesystem. We chooseScheme,a dialect of LISP, as theimplementation languagefor the Workbench [Han-son, 1991] and use extensively the Scheme math-ematical library supporting generic numerical andsymbolic manipulations. Scheme supports proce-dural abstraction that facilitates the extraction ofcommon patterns in numerical computation and thecomposition of numerical procedures.

3. High-level interface:

The Workbench presentsahigh-level, qualitative de-scription of the result of its analysis and design tohuman designers; this level of interaction is moreintuitive and direct than that of pure quantitativepresentation. Other programs in the Workbench canefficiently accessand manipulate the result. The in-ternal data structure for the analysis ensures thatthe result is sensible to human engineersand ma-nipulable by other programs.

The rest of the paper describesour representation forcontrol systemsand an implemented simulation envi-ronment.

Representing and ManipulatingConstraints of a Control Design

The complexity of a nonlinear system necessitatestheneed for a design vocabulary capable of describingimplicit constraints of a control design and provid-ing means to manipulate and reason about these con-straints and to build abstractions. We present a corn-

putational mechanism that allows one to represent andmanipulate constraints of a control design in terms ofphase-spacegeometryand topology of a dynamical sys-tem. A control design will be specified in terms of thecomposition of geometric objects in phase space.

A qualitative representation

We have interpreted a control designas a mappingfromthe model of a physical systemto be controlled andthe control objective to the control law, under the in-fluenceof which the physical systemwill behave in thedesired way. The synthesized control law alters natu-ral dynamics of the system through the selection andcomposition of the natural behaviors.

We describe a qualitative representation for com-plex behaviors of dynamical systems in phase spaceand a design vocabulary for computationally express-ing and manipulating these behaviors [Zhao, 1991;Zhao, 1993]. A phase space of a dynamical systemis an n-dimensional geometric space, each dimensionof which represents a state variable of the system. Weare interested in representing the qualitative behav-iors of dynamical systemsfor control analysis and de-sign. One useful qualitative representation of the phasespace of a dynamical system is in terms of equilib-rium points and limit cycles, stability regions, trajec-tory flows exhibiting the samequalitative features, andthe spatial arrangement of these geometric objects inphase space. This qualitative representation capturesthe gross aspects of dynamics in a relational graphof phase-spacestructure and a set of discrete objectscalled flow pipes—theequivalenceclassesof behaviors.The design vocabulary describesa control design taskin terms of well-defined geometric, combinatorial op-erations on the flow pipes. This vocabulary formalizesaspectsof implicit expert reasoningof control engineersin solving control design problemswith the phase-planemethod. The representation and the vocabulary aredevelopedindependently of the orders of systems, i.e.,the dirnensionality of phase spaces.

Designing control by manipulatingequivalence classesof trajectories

The flow pipes group infinite numbers of distinct be-haviors into a manageablediscrete set that becomesthe basisfor establishing control referencetrajectories;each flow pipe models an equivalence class of trajecto-ries exhibiting similar qualitative features.

The geometric modeling of a phase spacewith flowpipes makes the phase-spacecontrol planning and nav-igation possible. Given adiscrete set of possiblecontrolactions, the search for a control path from an initialstate to a destination is a reachability problem, i.e.,the problem of finding a sequenceof connectedpathsegmentseachof which is under a single control action,as schematically illustrated in Figure 3. This point-to-point planning can be naturally described within the

/ goal poii~t...

~U2

//~ •:initial point

~.. phaseportrait I

phaseportrait 2

~phaseportrait 3

Figure 3: Searchfor a control path from aninitial pointto a goal point in a stack of phase spaces.

flow-pipe representation of phase space:a system un-der control jumps from one flow pipe to another uponthe switching of control, eventually arriving at the des-tination.

To make this approach computationally feasible, thephasespacesof the dynamical systemindexed by differ-ent control actions are first parsed into a discrete set oftrajectory flow pipes. Theseflow pipes are then aggre-gated to intersect each other and are pasted togetherto form a graph, the flow-pipe graph. The flow-pipegraph is adirected graph where nodesare intersectionsof flow pipes and edgesare segmentsof flow pipes. Theinitial state and the goal state are nodes in the graph.Each edge of the graph is weighed accordingto travel-ing time, smoothness,etc. With this graph represen-tation, the searchfor optimal paths is formulated as asearch for shortest paths in the directed graph.

An Environment: the ControlEngineer’s Workbench

The Control Engineer’s Workbench is an implementedsystemthat analyzesand designs control systemsandinteracts with users qualitatively. The Workbenchserves as an intelligent assistant to control engineers.The componentsof the Workbench are shown in Fig-ure 4:

• MAPS program for simulation and interpretation

• Phase SpaceNavigator for control synthesis

• A graphic program for visualizing the design

• A user interface for communication with the system.

MAPS, standing for Modeler and Analyzer for PhaseSpaces, is an autonomous phase-spaceanalysis andmodeling program that extracts and represents qual-itative phase-spacestructures of nonlinear dynamicalsystems [Zhao, 1991]. MAPS generates a high-leveldescription of the behaviors of a dynamical system,sensible to humans and manipulable by other pro-grams. Phase Space Navigator visualizes the phase-spacestructure of agiven systemcomputed by MAPS,

plans global control reference trajectories, and navi-gates the system along the planned trajectories [Zhao,1992]. The component for model building in the figurehas not yet been implemented.

The programs in the Workbench implement variousalgorithms: symbolic differentiation, numerical algo-rithms on differential equations, modeling of geomet-ric structures, clustering of equivalence classes,graphalgorithms, etc. The inference mechanismof the Work-bench uses these programs to construct a qualitativephase-spacestructure for representing a system, tocheck for the consistencyof the structure, and to rea-son about and manipulate the representation througha graph of flow pipes.

The flow of computation within the Workbench isillustrated in Figure 5. Given a model of a system, abounded phase-spaceregion of interest, allowable pa-rameter values, and control objectives and constraints,the Workbench performs stability and trajectory flowanalysis for the system in phase spaceand interpretsthe result in a phase-spacegraph. The Workbenchthen exploresthe control spaceto synthesizea desiredcontrol law subject to the design constraints. It re-ports the control law specifying reference trajectoriesand performance properties. The control design forsteering towards an equilibrium is performed in thisway: for a point-to-point control, the output is a ref-erence trajectory, whose control law consists of a se-quence of tuples of time, switching state and the cor-responding control value; if an initial operating regionis given, the output is a controllable region geometri-cally represented as a polyhedral structure.

The current implementation of the Workbench takesas input the model for a dynamical system in termsof an ordinary differential equation. Incorporatingmodel formulation capability that constructs models

Figure 4: The Control Engineer’s Workbench.

Figure 5: The flow of computation in the Workbench.

from physical principles or measurementscan broadenthe scopeof control systems the Workbench can de-sign. Although the analysis and design algorithm ofthe Workbench applies to dynamical systemsof any or-der, the computational complexity of high-dimensionalsystemsremains as a future research topic.

Conclusion

We have developed the Control Engineer’s Workbenchcomprising programs MAPS and Phase Space Navi-gator that automate a significant portion of a controlengineer’s simulation and design task. The Workbenchemployscomputational meansto represent and manip-ulate dynamics of control systems,using a qualitativerepresentation for encoding dynamics and a flow-pipebasedphase-spacemethod for designing nonlinear con-trollers; it combines numerical simulation with sym-bolic techniques and Al reasoning; it provides an in-terface for visualizing the result of control design andanalysis. The Workbench has been applied to the de-sign of a nonlinear controller for a magnetic levita-tion vehicle. We plan to extend the capabilities ofthe Workbench to support interactive editing of phase-spacegeometric representation of dynamical systemsfor the purpose of experimenting and testing new ideasin control design.

The Control Engineer’s Workbench complementsand enhances human design activities. By providingmanipulation and visualization mechanismsfor the de-sign, the Workbench relieves engineers from routine,

tedious low-level tasks of simulation and interpreta-tion, allows the engineers to focus on higher-level de-sign issues,and enlargesthe design spacethe engineerscan explore.

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