1. LOGO A Path –Following Method for solving BMI Problems in Control Author: Arash Hassibi Jonathan How Stephen Boyd Presented by: Vu Van PHong American Control Confedence San Diago, California –June 1999 Southern Taiwan University

2. www.themegallery.com Contents Click to edit text styles Edit your company slogan Introduction 1 Linearization method for solving BMIs in “Low-authority ” 2 Path-Following method for solving BMIs in control 3 Example 4 Inconclusion 5

3. www.themegallery.com Introduction  Purpose to develop a new method is to formulate the analysis or synthesis problem in term of convex and bi- convex matrix optimization problems  We have some methods: Semi-definite Progamming problem(SDP), Linear matrix inequalities( LMIs).  Use “Bilinear matrix inequalities( BMIs)” to solve some control problems such as: synthesis with structured uncertainly, fixed-order controller design, output feed- back stabilization, … Click to edit text styles Edit your company slogan

4. www.themegallery.com Introduction  This paper present a path-following method for solving BMI in control:  BMI is linearized by using a first order perturbation approximation  Perturbation is computed to improve the controller performance by using DSP.  Repeat this process until the desired performance is achieved

5. www.themegallery.com Linearization method for solving BMIs in “low-authority” control  It can predict the performance of the closed-loop system accurately.  BMIs can be solved as LMIs that can be solved very efficently.  To illustrate this method we consider the problems of linear output-feedback design with limits on the feedback gain.

6. www.themegallery.com  Consider the linear time-invariant as below:  Open-loop system has a damping rate of at least.  Design feedback gain matrix in order to control law has an additional damping of  The constraints: X: state variable, u: input, y output Linearization method for solving BMIs in “low-authority” control

7. www.themegallery.com Linearization method for solving BMIs in “low-authority” control  According to Lyapunov theory, this problem is equivalent to the existence of that full-fill BMIs:   In order for linearization of BMIs we carry following step: Are variable

8. www.themegallery.com Linearization method for solving BMIs in “low-authority” control  Step 1: Consider open-loop system that has a decay rate at least Compute P o >0 that satisfies:  Step 2: Assign (2) Rewrite (1) we gain:

9. www.themegallery.com Linearization method for solving BMIs in “low-authority” control  Step 3: Assume that are small. Ignore second order: We obtain:  (4) is an LMI with variables which can solve efficiently for desired feedback matrix  Powerful method and can be applied in many other control problems.

10. www.themegallery.com Path-Following method for solving BMIs in control  Step 1:  Carry out Linearization BMIs  Step 2:  Starting from initial system( Open-loop system)  Iterate many times until get result that satisfies condition of BMIs.  The important thing to apply this method is choice initial value.

11. www.themegallery.com Example: sparse linear constant output-feedback design  We have to design sparse linear constant output-feedback u=Ky for system  Which results in a decay rate of at least  Consider the BMIs optimization problem.

12. www.themegallery.com Example: sparse linear constant output-feedback design  Step1: Let K:=0  Step 2: Calculate Lyapunov P 0 by solving: With is the smallest negative real part of the eigenvalues of A,  Step 3: linearization (5) around P 0 and K we have:

13. www.themegallery.com Example: sparse linear constant output-feedback design Where And such that the perturbation is small and linear approximation is valid  Step 4:. Iteration will stop when exceeds the desired or if cannot improved any further is feasible for any

14. www.themegallery.com Example: sparse linear constant output-feedback design  With :  With open-loop we have:

15. www.themegallery.com Example: sparse linear constant output-feedback design  The purpose is to design a sparse K so that decay rate at least is larger that 0.35.  Iteration 6 times with we get

16. www.themegallery.com Example: simultaneous state-feedback stabilization with limits on the fedd back gains  Consider system:  Compute K that satisfies so that  The close-loop system below is stable:  It means that we have to solve BMIs as below: 

17. www.themegallery.com Example: simultaneous state-feedback stabilization with limits on the fedd back gains  Step 1: compute the minimum condition number Lyapunov matrices P k, k=1,2,3  Step 2: Linearization around K, and P k  Step 3: update K and A k as:

18. www.themegallery.com Example: simultaneous state-feedback stabilization with limits on the fedd back gains  Example:  With and iterate 15 times we have:   the three systems are simulaneously stabilizable

19. www.themegallery.com Example:H 2 /H ∞ controller design  Consider system:  Find a feedback gain matrix K such that for u=Kx the H 2 norm from w to z 2 is minimized while H ∞ norm from w to z 1 is less than some prescribed

20. www.themegallery.com Example:H 2 /H ∞ controller design  It equivalent to solve BMIs:

21. www.themegallery.com Example:H 2 /H ∞ controller design  Step 1: Compute an initial K and suppose that P 1 is Lyapunov matrix obtained.  Step 2: u=Kx, compute the H 2 norm of close-loop system and P 2 is Lyapunov matrix.  Step 3: Solve the linearized BMIs around and get perturbation  Step 4:

22. www.themegallery.com Example:H 2 /H ∞ controller design  Step 5: Solve the SDP: Get Lyapunov P which proves a level of in H ∞ norm for closed-loop system. Let P1:=P and go to step 2.  Iterate until can not improved any further.

23. www.themegallery.com Example:H 2 /H ∞ controller design  Example:  Result:

24. www.themegallery.com Conclusion  BMIs is a very powerful method to solve control problem in term of convex or bi-convex matrix optimization problems.  However its weakness is to select initial value. Because if initial value is not good, it will not convergence to an acceptable solution.

25. LOGO