President UniversityErwin SitompulModern Control 11/1 Dr.-Ing. Erwin Sitompul President University Lecture 11 Modern Control

President UniversityErwin SitompulModern Control 11/2 Homework 9 Chapter 10Optimal Control Consider again the control system as given before, described by Assuming the linear control law Determine the constants k 1 and k 2 so that the following performance index is minimized Consider only the case where the initial condition is x(0)=[c 0] T and the undamped natural frequency (ω n ) is chosen to be 2 rad/s. Recall again the standard form of a second order transfer function (FCS) Calculate the transfer function of the system if compensated with k Determine the value of corresponding k (k 1 or k 2 ?) to obtain ω n as requested

President UniversityErwin SitompulModern Control 11/3 Solution of Homework 9 Chapter 10Optimal Control Substituting the state feedback and finding the transfer function,

President UniversityErwin SitompulModern Control 11/4 Solution of Homework 9 Chapter 10Optimal Control

President UniversityErwin SitompulModern Control 11/5 Algebraic Riccati Equation Consider again the n-dimensional state space equations: Chapter 10Optimal Control with the following performance index to be minimized: :symmetric, positive semidefinite The control objective is to construct a stabilizing linear state feedback controller of the form u(t) = –K x(t) that at the same time minimizes the performance index J. The state feedback equation u(t) = –K x(t) is also called the “control law.”

President UniversityErwin SitompulModern Control 11/6 First, assume that there exists a linear state feedback optimal controller, such that the optimal closed-loop system: Chapter 10Optimal Control is asymptotically stable. Then, there exists a Lyapunov Function V = x T (t)P x(t) with a positive definite matrix P, so that dV/dt evaluated on the trajectories of the closed-loop system is negative definite. Algebraic Riccati Equation The synthesis of optimal control law involves the finding of an appropriate Lyapunov Function, or equivalently, the matrix P.

President UniversityErwin SitompulModern Control 11/7 Chapter 10Optimal Control The appropriate matrix P is found by minimizing: Algebraic Riccati Equation If u(t) = –K x(t) is so chosen that min{f(u(t)) = dV/dt + x T (t)Q x(t) + u T (t)R u(t)} = 0 for some V = x T (t)P x(t), Then the controller using u(t) as control law is an optimal controller. For unconstrained minimization, Optimal Solution

President UniversityErwin SitompulModern Control 11/8 Chapter 10Optimal Control Algebraic Riccati Equation The differentiation yields: if P symmetric

President UniversityErwin SitompulModern Control 11/9 Hence, incorporating the fact that P and R are symmetric, the optimal control law can be written as: or Algebraic Riccati Equation Chapter 10Optimal Control

President UniversityErwin SitompulModern Control 11/10 Algebraic Riccati Equation Chapter 10Optimal Control Performing the “Second Derivative Test”, If the weight matrix R is chosen to be a positive definite matrix, then the optimal solution u * (t) is indeed a solution that minimizes f(u(t)). We now need to perform the “Second Derivative Test” to find out whether u * (t) is a solution that minimizes f(u(t)). Second Derivative Test If f’(x) = 0 and f”(x) > 0 then f has a local minimum at x If f’(x) = 0 and f”(x) < 0 then f has a local maximum at x If f’(x) = 0 and f”(x) = 0 then no conclusion can be drawn

President UniversityErwin SitompulModern Control 11/11 Algebraic Riccati Equation Chapter 10Optimal Control Now, the appropriate matrix P must be found, in order to obtain the optimal closed-loop system in the form of: The optimal controller with matrix P minimizes the cost function f(u(t)), and will yield: After some substitutions of x(t) and later u * (t), 

President UniversityErwin SitompulModern Control 11/12 Algebraic Riccati Equation Chapter 10Optimal Control After regrouping, we will obtain: The equation above should hold for any x(t), which implies that: Algebraic Riccati Equation (ARE) After solving the ARE for P, the optimal control law given by: can be applied to the linear system of

President UniversityErwin SitompulModern Control 11/13 Example 1: Algebraic Riccati Equation Chapter 10Optimal Control Consider the following model: along with the performance index: Find the optimal control law for the system. The matrices are: The ARE is solved as:

President UniversityErwin SitompulModern Control 11/14 Example 1: Algebraic Riccati Equation Chapter 10Optimal Control The control law is: The optimal closed-loop system is described by:

President UniversityErwin SitompulModern Control 11/15 Example 2: Algebraic Riccati Equation Chapter 10Optimal Control Consider the following continuous-time system: Design an optimal controller that minimizes with give weight to x 1 (t), no restriction for x 2 (t)

President UniversityErwin SitompulModern Control 11/16 Example 2: Algebraic Riccati Equation Chapter 10Optimal Control P is found by solving the ARE:

President UniversityErwin SitompulModern Control 11/17 Example 2: Algebraic Riccati Equation Chapter 10Optimal Control Three equations can be obtained: Thus, the optimal gain is given by: The requested control law is:

President UniversityErwin SitompulModern Control 11/18 Homework 10 Chapter 10Optimal Control The regulator shown in the figure below contains a plant that is described by and has a performance index Determine a)The Riccati matrix P b)The state feedback matrix k c)The closed-loop eigenvalues