Linear Physical Systems ELEX 7110

Learning Outcomes

Definition of Systems and Model Terminology

• Purpose of models in engineering, model suitability, example systems
• Classifications and terminology
• Memory-less versus dynamic systems
• Causal versus non-causal systems
• Time invariant versus time varying systems
• Stable versus unstable systems
• Linear versus nonlinear systems
• Distributed parameter versus lumped parameter systems
• Internal versus input/output system descriptions
• Continuous time versus discrete time systems
• Sampled data systems
• Time domain versus transform domain system models
• Order of a system, model simplification - reduction of order
• Quantization and digital systems
• Multiple input multiple output systems versus single input single output systems
• System variables versus system parameters
• Dynamic analogies
• Analogous circuits - through and across variables
• Mechanical linear systems
• Mechanical rotary systems
• Hydraulic systems
• Electric circuits
• Thermal circuits
• Economic systems
• Energy in physical systems
• Power
• Stability
• Conservation of energy
• Energy and power in mechanical, electric, thermal, and hydraulic systems

Modeling - Continuous Time

• Review of differential equation theory
• Nonlinear systems and linearization
• Saturation
• Hysterisis
• Static nonlinearities
• Dynamic nonlinearities
• Linearization
• Deviation variables and operating points
• Definition of linear system, superposition, homogeneity, convolution
• The phase plane and state space systems
• Equilibrium and types of fixed points in second order systems
• Review of the Laplace transform
• Bilateral and unilateral transform
• Region of convergence
• Properties of the Laplace transform
• Common signals and transforms
• Step
• Impulse
• Ramp
• Exponential/complex exponential
• Sine/cosine
• Time delay
• Inverse Laplace transform
• Solution of differential equations using the Laplace transform
• Convolution and applications
• Steady state error and the final value theorem
• Initial value theorem
• Block diagrams
• Generalized impedance and loading/interaction considerations
• System responses and pole zero locations
• Transfer functions
• Feedback interconnection and closed loop stability via the Laplace transform

Modeling - Discrete Time

• Examples of inherently discrete systems
• Sampled data systems and digital systems
• Difference equations and their solution
• The z-transform
• Definition of z-transform
• Region of convergence
• Common transforms
• Unit impulse, unit step, unit ramp
• Exponential
• Decaying exponential
• Sinusoid
• Damped sinusoid
• Delay
• Final value theorem
• Inverse z-transform
• Solution of difference equations using the z-transform
• Discrete time transfer functions
• Discrete convolution and applications
• System responses
• Zero order hold and discrete transfer functions of sampled analogue systems
• Sampling consideration
• Relation between the s- and z-plane
• Closed loop stability of digital feedback control systems
• Approximations of continuous systems
• Tustin approximation
• Frequency pre-warping
• Step equivalence

State Space Systems and Computer Simulation

• Review of linear algebra
• Continuous time state space descriptions
• State space feedback systems
• Similarity transformations
• Eigenvalue/eigenvector problems
• Realizations
• Relationship between state space and transfer function descriptions
• Observability, controllability, observer design
• Discrete time state space descriptions

Sinusoidal Steady State Response

• Frequency response
• Bode diagrams
• Resonance/mode shapes
• Gain and phase margins, stability, Nyquist encirclement criterion
• Unmodelled dynamics

Identification

• Model structures
• Finite impulse response
• Infinite impulse response
• Moving average models
• ARMA/ARMAX models
• Selecting model structures
• Model validation
• Lease squares estimation
• Geometric interpretation
• Curve fitting
• Estimation (BLUE property)
• Least squares system identification
• Model order selection
• Difficulties with autocorrelation in the data
• Nonsense correlations and detrending
• Noise
• Random variables and random processes
• Autocorrelation
• Cross correlation
• Heteroscedasticity
• Noise models
• PRBS signals

Effective as of Fall 2003

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