Search

A basic mathematical formalism of dynamical systems is presented from the view-point of control engineering. The notion of input-output systems is given in a general logistic format in which most of the detailed system models can be embedded. Concepts of time, time system, causality, and state are introduced. The time system is of a generalized form in which the time domains of component functions of input and output may be different from each other. The causality is a term for the traditional non-anticipation. The state is introduced firstly in a pure form. Then a state representation, state mapping, state transition, and a static output system are formalized. The results of discussion and propositions show close interrelations between the causality, state mapping, and static output system.

It is shown in this paper that it is possible to derive generalized equations for computation of electrical circuits containing SCR elements. Any classical method of circuit analysis can be used applying the nonlinear mathematical model given for the SCR. In order to get this model, a binary logic state variable is doped to the equation which is obtained by a Boolean function. The derivation of this mathematical model and the application of the method to an impulse-commutated inverter with an RL load are presented.

The scope of static ac drives is increasing rapidly, and with this increase has evolved the need to devise control strategies for a variety of new applications. Purely analytical approaches to design of control systems for ac drives are hindered by the highly coupled nature of the ac induction motor equations which, in the past, necessitated lengthy manual derivations of transfer functions. This paper presents an alternative approach using a state variable formulation. The equations are arranged in a form such that the entire derivation procedure can be relegated to a digital computer. Transfer function poles, zeros, and gain for any practical input-output pair of variables can be rapidly obtained.

The problem of determining whether or not a dynamical system ℒ controllable has been an active research topic in recent years. However, the complementary problem of determining just why a system ℒ uncontrollable has received relatively little attention. In this paper the physical acid mathematical mechanism of uncontrollability is studied with particular emphasis on how uncontrollability of a general class of non-linear dynamical systems can originate with the inadvertent selection of a non-minimal dimension ‘ state vector ’. The state-apace geometry corresponding to this particular mode of uncontrollability, which we call ‘state over-description’, is examined in detail. A fundamental necessary condition for the existence of state over-description in non-linear dynamical systems is derived and its application is illustrated by five examples. Application of the present results to the special case of time-invariant linear dynamical systems will be presented in a subsequent paper.

The state variable feedback design of time invariant linear multivariable systems with piece-wise constant inputs is considered. The formulation of the sampled data problem is given, along with a study of the structural properties of the problem when decoupled by state variable feedback.

The state variable description for creeping materials is discussed. It is shown that the majority of the physical theories are of the single state variable kind and that the appropriate rate equations can be determined from a systematic series of experiments. The testing programme required to define the appropriate functions for a two state variable theory are prohibitive in a practical sense, and it is suggested that limitations should be made on the type of stress histories considered. From a practical point of view, cyclical histories of loading are important and it is suggested how the physical theories can help to provide a method of description for this situation.

In this paper it is shown that a number of ambiguous points arising in transient stability analysis of power systems can be clarified if partial or output stability concepts are used instead of stato stability. This solves the question of the number of state variables and the validity of using positive semi-definite Lyapunov functions. It is further discussed how a very general Lur'e-type Lyapunov function can be derived for the power system stability problem, which unifies and generalizes numerous Lyapunov functions available in the technical literature.

Techniques for the design of feedback controllers for systems having linear time delay models are considered and reappraised. Design methods based on the quadratic performance index are presented. These lead to a controller in the form of a linear functional operating on a segment of the past trajectory. Simplified feedback controllers involving only “current” state variables are then considered. The Nyquist technique of synthesis and analysis is generalized to cover this time delay feedback structure and a design approach based on triangularization of the forward loop transfer function by elementary transformation type compensation matrices and application of the single input-single output theory is proposed.

Several algorithms have been proposed for generating satisfactory state assignments for normal fundamental mode asynchronous sequential circuits. Programmed versions of minimal and near-minimal techniques have been incorporated in automated synthesis systems, but are known to require computational efforts which increase exponentially with flow table size. Other methods, requiring much less effort, produce assignments with far more state variables than minimal assigments.

The main purpose of this paper is to present some results concerning the stabilization of nonlinear sampled–data control systems. The stabilization is investigated by means of a particular decomposition of the state space, based on some state variable feedback, into two sub–spaces, one of them being an invariant sub– space.

Self-organizing probability state variable (PSV) parameter search algorithms possessing long-term memory have been formulated to cope with systems that must avoid high performance-penalty operating regions. The information gained from all previous experiments is efficiently encoded in multivariate probability distribution functions (pdf's). This long-term memory capability enables the PSV algorithms to avoid effectively future experiments in high penalty regions. The systems considered are resource-limited, and catastrophic failure may occur if parameter values lying in high penalty regions are implemented. Those cases in which the high penalty regions are not known in advance were investigated. The PSV algorithms have the capability of adaptively learning the location and hypervolume of these regions as the search proceeds. The algorithms are explicitly guided in their internal strategies as a function of the remaining system resources and the updated probability distribution functions. Clustering analysis is used both in the discovery of new operating regions and for updating the pdf's. As a by-product of this research, clustering was also investigated as a presearch scheme. It is shown that this procedure has great promise as a means of assessing the complexity of an optimization problem. Experimental results are presented to demonstrate the utility of the self-organizing PSV search algorithms.

The purpose of this paper is to look at the analysis of a nuclear reactor system as a total analysis task and to examine the assumptions which are made in the separation of the analysis task into its component disciplines. The structural analysis discipline is then examined in more detail to try to define a workable approach to an integrated structural analysis of the reactor system. We will start with a general discussion of the total analysis task, starting from the initial concept of the reactor plant. The total task will then be subdivided into the respective disciplines and an attempt will be made to rationalize or criticize the division into separate disciplines. The discipline of structural mechanics will then be examined in view of its interactions with other disciplines such as fluid flow and nuclear analysis to determine the degree of coupling which exists among these disciplines. This will be done by examining the interactions of the state variables which apply at each point of the system. The state variables considered will include fluence, temperature, displacement and pressure. The state variables defined will then be used as the basis for the definition of an overall structural model of the reactor system. Such an overall model can be conceived in terms of the present status of analytical techniques, by the use of such concepts as substructuring, constraint equations, coupled solutions for heat transfer, stress and dynamic analysis, along with fourth generation computer capabilities. A block design for an overall structural model will be discussed and also the areas which require new analysis techniques. The last section will present an outline of a mode of operation of a structural design/analysis activity which is established to implement a comprehensive integrated structural analysis of an entire reactor system. The concept of an evolving model of the system will be presented and the coordination required to successfully manage such a design/analysis approach will be discussed. A brief discussion of the effects of non-linear effects such as creep, plasticity, gaps on the overall approach will be included.

A spatial correlation method is formulated for linear dynamical problems in continuum mechanics with random boundary data. The essential feature of the method is the formulation of a nonstochastic mixed initial-boundary value problem for the (matrix) spatial correlation function of the (vector) state variable. Whenever the Green's function of the (stochastic) problem can not be obtained in terms of known functions, a numerical solution of the meansquare response and other second order response statistics by the spatial correlation method is several hundred folds more efficient than any other available method. Further improvements in the computational efficiency of the method for a steady state stationary response process are also noted.

The problem of characterizing optimal controls for a class of distributed parameter systems is considered. The system dynamics are characterized mathematically by a finite number of coupled partial differential equations involving first-order time and space derivatives of the state variables. Boundary conditions on the state are in the form of a finite number of algebraic relations between the state and boundary control variables. Multiple distributed controls extending over the entire spatial region occupied by the system are also included. The performance index is an integral over the spatial domain of penalty functions on the terminal state and on the distributed state and controls. Under certain differentiability and well-posedness assumptions, variational methods are used to derive first- and second-order necessary conditions for a control which minimizes the performance index. Of particular interest are conditions on the boundary value of the costate and on the optimal boundary controls.

The induced safety algorithm (ISA) is a method for calculating the margin of safety in the design of hydrologic structures under the uncertainty due to finite sample size. The ISA may be applied to the design of a new structure or the redesign of an existing one; the initial design criterion may be calculated by benefit-cost or be prescribed by regulation. The optimum decision is reached by maximizing the sum of three terms (in case of redesign to a larger value): the discounted economic benefit due to uncertainty reduction, the average loss averted, and the incremental construction cost. The uncertainty on the distribution function of the state variable (yearly peak flow) is encoded by the probability density function of flow pertaining to a fixed return period. Simulation is used to calculate the distribution of the state variable. Two examples of levee design in Hungary illustrate the method.

High order state variable models of chemical processes can be reduced to lower order models by matching the frequency responses for those states which it is desired to retain. The method places no restriction on the choice of outputs and all the inputs are retained.

Some subsidiary conditions for determining the superiority of the periodic control to the steady control are derived. The necessary condition for the optimality of period is established by adding a new condition to the well-known condition which can yield merely the stationary value of the objective function. Some considerations are also given for a system having a single state variable and a single control variable and for the linear control problem, which very often appears in many practical cases.

Digital filters based on state-space methods are developed whereby an "observer" is used to provide estimates of system state variables. Selected observer estimates, with appropriate gains, become the feedback control signal for the system. Procedures for independently setting the closed-loop poles of the plant and observer are outlined. Missile performance is assessed at critical flight cases. The stabilization loop is first analyzed with results presented in terms of sample calculations and time responses obtained from digital simulations for various observer/plant closed-loop pole locations. The problem of vehicle flexibility is also discussed and an observer design for bending control is outlined. The multivariable input/output observer design is treated upon closure of the steering loop around the stabilization loop.

A systematic design procedure is presented for suboptimal nonlinear controller synthesis. Such a control yields faster response characteristics than are normally possible with a linear controller. The technique is geared to single-input linear systems where excursions of one state variable are of more importance than others. Design steps are formulated in terms of readily applicable algebraic equations. The proposed method is applied to several design examples, and the advantages are shown by comparison with the results obtained from existing techniques.

An air photographic mosaic covering an area of 44.5  105 m z was subdivided into 741 rectangular cells (60  100 m). Pattern frequency, center relief, shape, and wedge image clarity were tabulated using three states for each character on a nominal scale. These state variables were converted to an htterval scale by the application of a spatial smoothing filter. The new values were subjected to a principal components analysis which indicated that a parsimonious classification of pattern spatial variation couM be constructed by equally weighting the first three nominal variables (frequency, relief, shape). The maps derived from this scheme indicate the areas on the tundra surface where polygon evolution may be occurring at the present time.

In this paper two identification algorithms for determining the parameter matrices A, B, C and D of a linear dynamical system S : [xdot] = A x + B u S : y= C x + D u are presented. It is assumed that all state variables are accessible for measurement.

We develop solutions to two fire distribution problems for a homogeneous force in Lanchester combat against heterogeneous enemy forces. The combat continues over a period of time with a choice of tactics available to the homogeneous force and subject to change with time. In these idealized combat situations the lethality of each force's fire (as expressed by the Lanchester attrition-rate coefficient) depends upon time. Optimal fire distribution rules are developed through the combination of Lanchester-type equations for combat attrition and deterministic optimal control theory (Pontryagin maximum principle). Additionally, the theory of state variable inequality constraints is used to treat the nonnegativity of force levels. The synthesis of optimal fire distribution policies was facilitated by exploiting special mathematical structures in these problems.

A design approach is presented for systems operating over a relatively short period of time about various operating points with state variable constraints. This class of problems is especially relevant to certain flight control problems. The design approach is applied to a simplified model of longitudinal dynamics of the F-4 aircraft operating in three widely separated flight conditions. A linear model is assumed about each operating point. Control is achieved via constrained state feedback. The basic problem is then to minimize a suitable integral quadratic performance measure subject to state variable constraints. The main theoretical result is theorem 1 which supplies the inequality constraints required to guarantee short-time stability. The short-time optimization problem is ultimately reduced to a nonlinear programming problem with inequality constraints.

The state variable approach to systems analysis and synthesis is used to develop a lumped mathematical model of overland flow. The model describes the unsteady nonuniform flow in terms of a set of first order ordinary differential equations and a set of arithmetic equations. Conceptually the model consists of a series of interacting reaches with unsteady uniform flow subjected to impulse inputs. The change in hydraulic roughness due to the rainfall impact effect is accounted for. The regeneration performance of the model is tested by comparison with experimental hydrographs of overland flow from a laboratory catchment. Close agreement between experimental and calculated hydrographs is obtained for both uniformly distributed as well as temporally and spatially varied rainfall inputs. The state variable approach is not limited to overland flow; it provides a comprehensive mathematical framework for a general hydrologic model.

We state and prove existence theorems for problems of optimal control which are linear in the state variables. As in previous work by L. W. Neustadt and C. Olech, no convexity condition is required. Examples are given.

The paper begins with an investigation of the scattering properties of commensurate TEM transmission line networks, and emphasis is placed upon the distinction between normal networks and general (including non-normal) networks. It is then shown that the state space concept of linear discrete systems is applicable to these networks. This provides a general and powerful systemtheoretical method for determining the network behaviour in the time and frequency domains. The waves on the transmission line elements are chosen as state variables; and for the important technical case of normal networks it is shown that each transmission line element contributes one state variable only. The entries in the state space matrices are formed by the scattering parameters of the connections and endings of the line elements; and the general method of obtaining these matrices is explained by detailed examples. An automatic network analysis program based on the state space approach needs only matrix multiplications. The program works primarily in the time domain, and the frequency behaviour is then computed by Fast Fourier Transform. The advantages of the program are demonstrated by a typical example. Finally an application to digital filters imitating TEM transmission line networks is presented.

For a class of nonlinear plants which can be decoupled by state variable feedback, the problem of decoupling by output feedback is considered. A necessary and sufficient condition for the existence of state variable feedback decoupling control law for a system composed of a nonlinear compensator in cascade with the plant is derived. The use of compensator allows decoupling by the feedback of the output and its time derivatives. Using some results on the existence and uniqueness of solutions of nonlinear equations two sufficient conditions for the decoupling by output feedback are presented.

The main objective of any time optimal design is to develop a control law to drive the state variables of the system, from a given set of initial values to a desired set of final values. Sometimes, it is required to drive the systems such that the output for a unit step input is reached in a minimum time, without overshoot [5-7]. Although the problem is normally set in the frequency domain, it is usual to convert it into the time domain for determining the optimal control sequence. It is shown in the paper that the entire time optimal design can be effectively carried out if the transfer function of the plant is given in the Z domain. ? 1974.

Conditions under which the asymptotic stabilization of uniformly decoupled time-varying multivariate systems is possible are explored. This is accomplished by developing a canonical form for integrator uniformly decoupled system in which the coefficient matrices have a simple structure. The procedures developed rely on certain conditions on the given system and yield explicit expressions for the stabilization compensators.

Description of a computer-oriented technique to generate the necessary control inputs to guide an aircraft in a given time from a given initial state to a prescribed final state subject to the constraints on airspeed, acceleration, and pitch and bank angles of the aircraft. A discrete-time mathematical model requiring five state variables and three control variables is obtained, assuming steady wind and zero sideslip. The guidance problem is posed as a discrete nonlinear optimal control problem with a cost functional of Bolza form. A solution technique for the control problem is investigated, and numerical examples are presented. It is believed that this approach should prove to be useful in automated air traffic control schemes near large terminal areas.

It is shown that if there equivalently exist the virtual resistive elements in parallel with the inductive elements of the over-normal tree of a given linear active network or in series with the capacitive elements of the corresponding co-tree, an increase in the number of state variables arises. It is also shown that when a virtual resistor equivalently appears in parallel with the distinct resistor in a tree or in series with the distinct resistor in the co-tree, a decrease in the number of state variables may arise. This is, however, a rare case in the usual types of network. Two algebraic methods for obtaining the state equation of linear active networks are presented. One is useful for the networks in which the decrease in the number of state variables does not arise. From the other, the output equation for the required variables is obtained at the same time as the state equation. Further, the initial values are simply determined without iteration in many cases.

The feasibility of simultaneous optimization of member sizing and structural configuration of truss structures is demonstrated. The structural analysis is treated by the finite element displacement method and the optimization accomplished by the steepest descent method. Inequality constraints including limitations on both state variables (stress and displacement) and design variables (element cross sectional areas and nodal point placement) are included. The computational results show that in the presence of displacement constraints, the configuration of the optimum design sometimes differs considerably from the fully stressed design. The techniques can be extended to other structures such as beams, frames, plates, etc. and to include the possibility of Euler buckling.

For a large class of nonlinear, time-variable parameter systems, a concept of weak decoupling is introduced. This concept is particularly appropriate for terminal value control problems and as an alternative to decoupling. Some of the rather restrictive assumptions needed in the theory of decoupling are not needed for weak decoupling. Also weak decoupling by state variable feedback is often possible when decoupling cannot be accomplished by state variable feedback. Conditions for the existence of weak decoupling control laws are derived and examples are presented which compare decoupling and weak decoupling.

The thermodynamics of crystalline elastic-viscoplastic materials developed earlier by the authors is extended to account for finite strains. The present theory utilizes the basic physical concepts derived from the theory of dislocations in crystals and the thermodynamics of continua with internal state variables. A crystalline simple solid is considered to be a homogeneous elastic-plastic continuum containing dislocations which constitute the intrinsic, internal mechanism of plasticity phenomena. The appropriate internal state variables and the basic constitutive relations for the description of the elastic-viscoplastic behavior are deduced from the consideration of dislocation dynamics. The theory proposed in the present paper is based on the assumption that both elastic and inelastic deformations take place at every stage of loading and unloading. This assumption has a definite physical basis and is the underlying consideration in the field of dislocation dynamics. Unlike other theories, the present one does not require the specification of a yield criterion or the prior determination of whether the material is loading or unloading. It can be shown that under appropriate assumptions on the constitutive laws the present theory reduces to the case of classical thermoelasticity or thermoplasticity. The general finite-element formulation of an initial boundary value problem is briefly presented. A specific example is carried out in detail which involves the solution of equations governing the coupled thermomechanical response of titanium. To demonstrate the effectiveness of the present formulation, a scheme for calculating the deformations, the stresses and temperature distribution is developed and applied to a three-dimensional structure subjected to high temperature, surface heat flux, volume heat supply as well as mechanical loading. The theory and the analysis should be beneficial in the study of the behavior of reactor materials at high temperatures.

In this paper a controller design incorporating a non-linear state reconstruotor is considered for a class of unstable non-linear plants. A basic assumption is that if all state variables x of the process to be controlled were available for measurement, then there is some control law Φ(x) that would result in an asymptotically stable closed-loop system. When only a limited number of measurements are available, it is shown that a non-linear state reconstructor can be used to provide estimates x of the process state. Conditions are given for the stability of the controlled system using Φ(x). Simulation results of the performance of a non-linear regulator are included.

The problem of transferring all members of a bounded initial set to a desired target set in fixed time is addressed. It is found that for (A, B) controllable, such a transfer is always possible via constant state variable feedback. Expansion of state in matrix Lagrange polynomials enables one to develop a synthesis algorithm to obtain the appropriate feedback. The feedback synthesis procedure is exemplified for a 3-dimenaional unstable plant where the problem of stabilizability in prespecified time is considered as a special case of the above.

High order state variable models of chemical processes can be reduced to lower order models by matching the moments of the impules responses for those states which it is desired to retain. The moments of the full model are calculated directly from the state equations and the parameters of the reduced model are calculated from these moments. The method places no restriction on the choice of outputs and all the inputs are retained.

The state equation of creep deformation is generalized, and evolutionary internal state variables are introduced. These concepts are then used to rationalize several creep theories currently in use. Finally, a simple recovery model of creep deformation is proposed within the framework of the general evolutionary theory. The model exhibits several important characteristics of creep in metals and also illustrates some important features of the theory.

A synthesis method for multiple-input change asynchronous sequential machines is proposed. The method is based on the self-synchronization principle. The internal states are realized with edge-sensitive flip-flops which are triggered selectively. The new concept of selective triggering or controlled excitation results in considerable saving in logic and more flexible design. The state assignment is arbitrary, and the number of state variables required can be made absolutely minimum. A detailed comparison of the speed of operation is made with one of the well-known methods of realization. It is found that our realization is, in general, as fast, and sometimes could be even faster.

Although dynamic models are now an important part of applied demand analysis, some of their fundamental concepts still suffer from a substantial lack of clarity. A particular case in point is the distinction between stock adjustment and habit formation in the Houthakker-Taylor model (1970). Empirical considerations prompted Houthakker and Taylor to specify a single state variable for each good and then to conclude that a good is habit forming or stock adjusting depending on the state variable's sign. However, besides having an obvious drawback of being ex post, this procedure is subject to the further criticism that, by treating expenditures for nondurables and services in the same way as expenditures for durables, it implicitly views habit formation and stock adjustment as arising from symmetrical processes. This is unfortunate, for stock adjustment and habit formation reflect intrinsically different phenomena, and to elaborate upon this, without the constraints imposed by thoughts of immediate empirical application, is one of the primary purposes of this paper.

The problem of designing a linear feedback when all state variables are not available is discussed. The design scheme is based on computation of a complete state feedback and a reduction to a specified structure. The reduction is made by approximation of the eigenspace corresponding to a set of dominant eigenvalues. The method consists of successive choices of weightings on this space. The method is applied to the control of a boiler and a three-machine power system. In the power system case the complete state feedback can be replaced by local output feedback without significant decrease in performance. The examples indicate that the proposed method is a realistic design method for multivariable systems.

The explicit expressions for the change in the amplitudes of one-dimensional acceleration and shock waves propagating through arbitrary homogeneous materials described by the strain and internal state variables/parameters/are derived. The existence of a critical amplitude β for the acceleration wave and a critical strain gradient λ for the shock wave is established. For an infinitesimal shock wave the general form of the solution of the governing differential equation is furnished. The differential equations for the amplitudes of these two kind of waves are applied to an elastic-viscoplastic material.

The steady two-dimensional problem of reflexion of an oblique partly dispersed plane shock wave from a plane wall is studied analytically. Viscosity, diffusion and heat conduction are neglected. The thermodynamic state of the gas is assumed to be determined by the instantaneous values of the specific entropy s, pressure p and a finite number of internal state variables. Results for the flow field behind the reflected shock are obtained by a perturbation method which is based on the assumption that the influence of relaxation is relatively weak.

A stability criterion for non-linear multivariable systems with separable linear plants is obtained. Interactions among state variables are allowed. Moreover, each of the non-linearities is allowed to be a function of several state variables.

An operating reactor has intrinsically fluctuating components in its state variables such as the thermal power, the outlet coolant temperature, the occupied region of control rods, etc. The fluctuations are generically called the reactor noise, the analysis of which, from both the theoretical and the experimental approach, is one of the problems of major concern in the field of nuclear science and engineering. The ultimate goal of the analysis lies in presenting the systematized methodology of diagnosis upon abnormality and malfunction of reactor components. The theoretical analysis of reactor noise has the most characteristic feature of integrity just as the nuclear industry has in industry. Its physical-mathematical analysis can be naturally regarded as a branch of statistical physics when one recalls the circumstances which the state of the art of the present statistical physics occupies in science. The integral nature of the reactor noise can be only clarified upon the standpoint that the theory of our reactor noise stands as a field of non-equilibrium classical statistical physics. The major effort of the present paper will be devoted to make as clear as possible the underlying physical-mathematical concepts which have been and will be used in the due course of development of the operating reactor noise theory. It is tried to make the above presentation as tasteful as possible by supplementing it with practical physical facts. Technical terms which are commonly used in modern statistical physics are explained so as to be accessible to reactor physicists and engineers.

Applying the identity that biomass equals number of individuals multiplied by average individual weight, simultaneous equations for change with respect to time in biomass, number of individuals, and average individual weight are obtained for Kostitzin's predator–prey equations and for the Lotka–Volterra competition equations. By the same procedure applied here, simultaneous equations for these three variables can be obtained for other predator–prey and competition equations. These equations can be applied to determine the biomass, number of individuals, and average individual weight of interacting fish populations under different rates of exploitation.

The purpose of clinical diagnosis is to determine the most fitting treatment for individual pathological conditions, and the process of diagnosis and treatment is one of feedback control system. In this paper, application of “optimal control theory” to diagnostic process was discussed and one example of computer program was showed. When diagnostic process is regarded as one of control system, medical treatment, pathological internal conditions and clinical findings are replaced to input variables, state variables and output variables respectively, and if the interrelationship of these three variables is quantitatively represented, it becomes to a problem of optimal control theory. As a clinical application, “insulin therapy in diabetic coma” was proposed. Injection of insulin and infusion therapy as input variables, and blood glucose level and insulin level state variables were used respectively, and the iterrelationship was mathematically represented based on the simulation study of blood glucose kinetics. This model was programmed for digital computer. When observed data of blood glucose level and details treatment were entered, the dosis of insulin to be injected now, fitting to the desired blood glucose level after adequate time, is computed.

A dynamic simulation model for use in analyzing the performance of specific designs of solar-heated-and- cooled buildings has been developed. The name of the program is SIMSHAC, an acronym for SImulation Model for Solar-Heated-And-Cooled buildings. To use the design program, one has merely to specify the components (that is, each subsystem; for example, collectors, storage units, splitting and mixing valves, house heating and cooling loads, auxiliary heating, auxiliary cooling, heat exchanger, etc.), and the manner in which they are connected and all initial conditions. Program SIMSHAC then writes the program for the specific system to be analyzed. Each subsystem is described by a set of time-dependent differential equations or, possibly, algebraic equations. System state variables include tempera ture, mass flow rate, and enthalpy. The model can handle three types of incident solar radiation data models. These are (1) deterministic (e.g., an algebraic-equation sun model), (2) random simulation (e.g., a model based upon cloud-cover statistics), and (3) actual tabulated input information based upon collected solar data for a specific site. The model has been used for the analysis of five different types of buildings in five locations within the United States. The types include single-family residences, small businesses, three-story apartments, schools, and mobile homes. The locations considered are Atlanta, Georgia; Madison, Wisconsin; Wilmington, Delaware; Mobile, Alabama; and Santa Maria, California. The program has also been used to simulate the solar- heated-and-cooled experimental laboratory presently being built by the National Science Foundation on the campus at Colorado State University.