EE 613

Final Objectives

At the completion of EE613, you should be able to perform the following objectives:

Exam I Objectives

• Find - the state transition matrix for continuous time state variable models
• Find - the state transistion matrix for discrete time state variable models
• Determine observability, detectability, stabilizability and controllability properties of continuous and discrete time state variable models
• Perform state variable regulator and estimator design via pole-placement techniques
• Find differentials, derivatives, and partial derivatives of vector-valued functions of vectors (vector calculus)
• Determine the positive definiteness of a function/matrix
• Perform free static optimization and determine the nature of critical points
• Draw contours of quadratic functions to illustrate the nature of critical points
• Use the Lagrange multiplier to solve constrained static optimization problems
• Define the Hamiltonian for constrained optimization problems
• Solve optimization problems numerically using the gradient method (i.e., the method of steepest descent)
• Perform the following tasks in discrete-time optimal control:
  • Derive the first order necessary conditions and the Hamiltonian for a general, nonlinear, time-varying
  • discrete time plant
  • Prove the matrix inversion lemma
  • Solve the Linear Quadratic Regulator Problem for both fixed and free final state x_N
  • Design discrete LQR control laws to satisfy transient and deviation specifications for continuous systems
  • Develop software to simulate your design
  • Solve the Linear Quadratic Regulator Problem in steady-state.
  • Solve the Linear Quadratic Tracking Problem (both transient and steady-state).
  • Solve the derterministic Disturbance Rejection Problem
• Perform the following discrete Kalman Filter related tasks:
  • Find the optimal static estimate, , which minimizes the mean-square error,
  • Derive E[x|z] from the regression formula
  • Derive the equations for and covariance M_k using only apriori knowledge
  • Derive the discrete Kalman Filter using the aposteriori estimate and covariance .
  • Derive the analogous LQR formula for the Kalman filter (ricatti equation, kalman gain, etc.)
  • Utilize the Discrete Kalman Filter in engineering problems
  • Solve the discrete optimal disturbance rejection problem
    

Exam II Objectives

• Perform the following tasks in continuous-time optimal control:
  • Use the calculus of variations to obtain the first order necessary conditions and the Hamiltonian
  • Use the first order necessary conditions to solve the continuous Linear Quadratic Regulator Problem
  • for both fixed and free final time, t_f
  • Use runge-kutta backward integration to solve the continuous Riccati equation
  • Design continuous LQR control laws to satisfy transient and deviation specifications for continuous systems
  • Develop software to simulate your solution
  • Solve the continuous Linear Quadratic Regulator Problem in steady-state (i.e., the sub-optimal control problem)
  • Solve the continuous Linear Quadratic Tracking Problem for both optimal and sub-optimal strategies
  • Solve the continuous optimal distrubance rejection problem
  • Solve minimum time optimal nonlinear control problems
  • State the Pontryagin Minimum Principle and use it in lieu of the stationary condition
  • Solve minimum time optimal control problems with constrained inputs
  • Solve minimum fuel optimal control problems with constrained inputs
  • Design Variable Structure Control regulators for nonlinear systems
  • Design Variable Structure Control tracking control for nonlinear systems
• Derive the solution to the continuous time Kalman Filter from the discrete solution

Post Exam II Objectives:

• Perform dynamic programming using Bellman's principle of optimality for networks and discrete systems
• Derive and apply the Hamilton-Jacobi-Bellman equation for both the continuous and discrete cases.
• Understand MIMO Robust Control and the importance of Singular Values