Professor Walter W. Olson Department of Mechanical, Industrial and Manufacturing Engineering University of Toledo State Space Models Video: Diwheel

Outline of Today’s Lecture Review Models Dynamics States Phase Plots Example: Predator Prey Model Engineering Modeling Procedure State Space Models

Models A model is a representation of something The something can be an idea, a concrete object or an abstract object It is NOT the real thing: they are simplifications it is a fiction of our imagination Models can take many forms Solid Blocks Equations Computer programs Word descriptions Symbols

Dynamics Models of dynamics used in this course: Based on functions of time Differential equations time is considered continuous Difference equations time is considered discrete

State A state is a set of variables whose values when known completely define the dynamics (motion) State variables for nose wheel example: The parameters of the example are

State So, what about ? These are completely determined by the state variables! We can rewrite the equations as

Phase Plot A phase plot is a plot of a state variable vs. another state variable Useful in understanding how the dynamics change with changes in state

Predator Prey Model Volterra-Lotke Model The model is subject to x(t) = prey population size y(t) = predator population size a = growth rate of prey b = rate of prey predation m = death rate of predators n = rate of predator sustenance Model Solution State Variables: x, y Note the controls in this nonlinear, coupled, model

Predator Prey Model Lynx – Hare ( data from Leigh, 1968)

Discrete Predator Prey Model To discretize, choose time step, h replace The difference equations are If h = 1 year, the model is Evaluation: with x(0)=80, y(0)=30 a = 0.7, b = 0.03, m=0.99, n = 0.03 Initially h = 1 year

Discrete Predator Prey Model Unstable Model! Problem: the time step is too big! with h = 0.1 years Choice of time step is critical to discrete modeling The smaller the time step, the more accurate the model but more computations

Models REAL WORLDOBSERVATIONS MATHEMATICAL MODEL EXPLANATION/ PREDICTION SENSE FORMULATE INTERPRET TEST

Engineering Modeling Procedure Understand the problem What are the factors and relevant relationships? What assumptions can be made? What equilibrium conditions exist? What should the result look like? Draw and label an engineering sketch Free body diagram Hydraulic schematic Electrical schematic Write the equilibrium equations (usually differential or difference) Newton 2 nd Law Kirchoff Laws for current and voltages Flow continuity laws Solve the equations for the desired result Check the validity of the results

Modeling is an Iterative Process Understand the Problem Sketch Mathematical Model Solve the Model Validate the Results Use the Model Can you formulate a model? Can you solve the model? Do the results represent reality? YES NO

Modeling Terms System: a functional group of interrelated things State: A condition (which may or may not be physical) of the system regarding form, structure, location, thermodynamics or composition State vector: a collection of variables that fully describe the object over time Input: an external object provide to the system Output: a dependent variable (often a state) from within the system that can be measured or quantified Dynamics: a process of the state variables over time

Modeling Example What is the deflection of the nose in response to the runway? System: landing gear State: positions and velocities of the components State vector Input: runway roughness, z r Output: Nose deflections, z

State Space Formulation Continuous Models Let x be a vector formed of the state variables The number of components of the state vector is called the order Formulate the system as The matrices A, B, C and D have constant elements The matrix A is the called the State Dynamics Matrix The matrix B is called the Input or Control Matrix The matrix C is called the Output or Sensor Matrix The matrix D is called the Pass Through or Direct term

State Space Formulation Discrete Models Let x be a vector formed of the state variables The number of components of the state vector is called the order Formulate the system as The matrices A, B, C and D have constant elements The matrix A is the called the State Dynamics Matrix The matrix B is called the Input or Control Matrix The matrix C is called the Output or Sensor Matrix The matrix D is called the Pass Through or Direct term

State Space Formulation Advantages Time domain model Ability to determine which variables can be controlled Ability to determine which variable can be sensed Ability to design predictors for control The formulations are not unique You might have several different choices for the state variables The states might not be physical You might want different outputs (things you measure from the model)

State Space Formulation Procedure: Develop the equations of equilibrium Put the equilibrium equations in the form of the highest derivative equal the remainder of the terms Make a choice of states, the input and the outputs Write the equilibrium equations in terms of the state variables Construct the dynamics, the input, the output and the pass through matrices Write the state space formulation

State Space Formulation Example: The Nose Wheel

State Space Formulation % Nose Wheel Problem % Programmed by WW Olson 8 March 2008 % % Parameters: k = ; % Nose wheel spring stiffness N/m kt = ; % Nose wheel tire stiffness in N/m b = ; % Nose wheel shock absorber in N/m/s m = ; % Mass allocated to the nose wheel kg mu = 50; % Mass of the landing gear kg % % State Matrix A = [ ; -k/m -b/m k/m b/m; ;... k/mu b/mu -(k+kt)/mu -b/mu]; B = [0;0;0;kt/mu]; % Input Matrix C = [1,0,0,0]; % Output matrix for nose deflection D = [ 0 ]; % Pass through matrix sys = ss(A,B,C,D); % State space formulation step(0.01*sys); % Step input of 1 cm grid; title('Nose Wheel 0.01 m step input'); t = [ 0:.01:100]; % Time for runway n = length(t); % Length of time array u =0.01*exp(.1*(rand(1,n)-.5)); % Runway profile lsim(sys,u,t); % Nose response to runway grid; title('Nose Wheel response to runway'); legend('nose deflection in m');

Consider the following problem A secret chemical, we will call zonium will change state if to much force is applied to it. This chemical is badly needed in at Galena, Alaska. The only mode of transportation is by air. It has been determined that take off and landing forces by a well trained pilot can be tolereated. However, because of the mountains that need to be flown over, there are air pockets which will cause the airplane to fall or lose altitude quickly. Model the forces on the zonium and determine the state space model for this situation. Zonium Mass m z kb Aircraft floor u(t)

Derivatives of the Forcing Function Consider Problem: handling the derivatives of the forcing function Using a change of variable as indicated below: Where Note that:

Derivatives of the Forcing Function

Consider the following problem Zonium Mass m z kb Aircraft floor u(t)

Summary Modeling Definitions of terms An Engineering Procedure Understand the problem Sketch Write the equilibrium equations Solve the model Validate the model Use the model Iterative Process Choose the model to answer your question State Space Continuous Discrete Next: Lumped parameter systems

Example