EE16B Designing Information Devices and Systems II

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1 EE6B M. Lustig, EECS UC Berkeley EE6B Designing Information Devices and Systems II Lecture 6B Cont. stability of Linear State Models Controllability

2 Today Last time: Derived stability conditions for disc. and cont. systems Easy to analyze using eigenvalues Today: Eigenvalues can predict system behaviour Envelope (decay) and Oscillation (frequency) Controllability of systems EE6B M. Lustig, EECS UC Berkeley

3 EE6B M. Lustig, EECS UC Berkeley

4 Stability -- Summary Discrete-Time Continuous-Time Stable regions Stay away from boundaries! System uncertainty can Move you over to unstable region EE6B M. Lustig, EECS UC Berkeley

5 Back to the Pendulum EE6B M. Lustig, EECS UC Berkeley

6 Back to the Pendulum If, i.e, sqrt is real, then So, λ,2 always negative -- stable! If, i.e, sqrt is imaginary, then Re{λ,2 } = So, Re{λ,2 } always negative -- stable! EE6B M. Lustig, EECS UC Berkeley

7 Back to the Pendulum EE6B M. Lustig, EECS UC Berkeley

8 Back to the Pendulum x x x long l x short l EE6B M. Lustig, EECS UC Berkeley

9 Predicting System Behavior Discrete Time λ=.9 λ= λ= λ= λ= λ= EE6B M. Lustig, EECS UC Berkeley

10 If λ is complex (Euler) EE6B M. Lustig, EECS UC Berkeley

11 EE6B M. Lustig, EECS UC Berkeley Im(l) Re(l)

12 If λ is complex (Euler) Continuous time: Q) What does e λt look like for different choices of λ? A) EE6B M. Lustig, EECS UC Berkeley

13 EE6B M. Lustig, EECS UC Berkeley Im(l) F v N a u z a Re(l)

14 Example EE6B M. Lustig, EECS UC Berkeley

15 Big Picture State space modeling is powerful Linear state-space are awesome We can say a lot about them! We can approximate non-linear as linear at equilibrium points What can we say about linear systems? We can tell if they are stable we have a test! We can can predict system behaviour for initial conditions! What about controls? Can test if the system can be controlled to reach all states (controllability) We can control a system to move to a certain state (open loop) We can control a system to stay around a state (feedback) EE6B M. Lustig, EECS UC Berkeley

16 Controllability Discrete-time: From last time: EE6B M. Lustig, EECS UC Berkeley

17 Controllability EE6B M. Lustig, EECS UC Berkeley

18 Controllability Q) Given any x(), can we find u(t) s.t. x(t) = x target for some t? A) Depends if it is in the span of R t EE6B M. Lustig, EECS UC Berkeley

19 Controllability Q) For t < n? A) No Q) At t=n, If columns are independent? A) Absolutely! Q) If not independent, does increasing t helps? A) No! Cayley-Hamilton Theorem: If A is n x n, then A n can be written as a linear combination of A n-, A, So does: EE6B M. Lustig, EECS UC Berkeley

20 Controllability What about R n+? EE6B M. Lustig, EECS UC Berkeley

21 Controllability Test If R doesn t have n independent columns at t=n, it t never will for t > n either! Therefore, we need only to examine R for n controllability: Conclusion: is controllable if and only if EE6B M. Lustig, EECS UC Berkeley

22 Example : Rank {R2} =, < n=2 Not controllable! State equations: (not stable) Can not control x 2, not with u and not with x EE6B M. Lustig, EECS UC Berkeley

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sha_base64="zhpsqmnxal6g7ybhbfmqgirh6s=">aaacoxicbzdlsgmxfiyz3q23qks3wsiogsyioaufghuxfwwrdiassc+wuxmsm4izzix8tv8abe6c+lgxkvyhpbbpvaym//n+qkx5hkydbx52z2bn5hcwl5dlk6tr6rnlzq2gsthoo8qm+jzkbqrqueebem5tdswojttdu8tb3rwhbusibrcfqhczrhkr4ayts6f+xiibfe/hk5qodoa9yucfztdxpvwpfvhzvqxxgqa9ft4cbvadtcsu9codff4q3lspkxlv2+d3vjdylqsgxzjiw56yy2htrcalfyc8mpizfss6rfqsbhpkw8wdm86hrol2i6fdoj+ppgz2jh+hnromghptgcd87+slwffurcprmc4qnbusypjnsajhaebo6ybwxjwti3ut5jmngycemcky4ykjkwfzb8fcjo88q69pktvadyroieyqxbjpphjkquskeidcpjansgzexeentfnw/kctc44y6zbzkkcr28zf6a</latexit> <latexit sha_base64="zhpsqmnxal6g7ybhbfmqgirh6s=">aaacoxicbzdlsgmxfiyz3q23qks3wsiogsyioaufghuxfwwrdiassc+wuxmsm4izzix8tv8abe6c+lgxkvyhpbbpvaym//n+qkx5hkydbx52z2bn5hcwl5dlk6tr6rnlzq2gsthoo8qm+jzkbqrqueebem5tdswojttdu8tb3rwhbusibrcfqhczrhkr4ayts6f+xiibfe/hk5qodoa9yucfztdxpvwpfvhzvqxxgqa9ft4cbvadtcsu9codff4q3lspkxlv2+d3vjdylqsgxzjiw56yy2htrcalfyc8mpizfss6rfqsbhpkw8wdm86hrol2i6fdoj+ppgz2jh+hnromghptgcd87+slwffurcprmc4qnbusypjnsajhaebo6ybwxjwti3ut5jmngycemcky4ykjkwfzb8fcjo88q69pktvadyroieyqxbjpphjkquskeidcpjansgzexeentfnw/kctc44y6zbzkkcr28zf6a</latexit> Example 2: p(t + ) = p(t)+tv(t)+ 2 T 2 u(t) v(t + ) = v(t)+tu(t) apple p(t + ) v(t + ) = Rank = 2 Controllable! EE6B M. Lustig, EECS UC Berkeley

24 Continuous Time (no derivation here) The continuous-time system is controllable if and only if has rank = n EE6B M. Lustig, EECS UC Berkeley

25 Example 3 + Quiz Write the state model: For: EE6B M. Lustig, EECS UC Berkeley

26 Quiz Write the state model: For: EE6B M. Lustig, EECS UC Berkeley

27 Example 3 Controllability: Rank =! Not controllable EE6B M. Lustig, EECS UC Berkeley

28 Physical explanation Why can t I drive the currents x and x2 freely using U(t)? EE6B M. Lustig, EECS UC Berkeley

29 Given an initial condition, I can only move along the line by changing U EE6B M. Lustig, EECS UC Berkeley

30 EE6B M. Lustig, EECS UC Berkeley Q) What if A =? Can the system be controllable? A) Only if u(t) is a vector with the same number of elements as the number of states

31 Summary Described and derived conditions for controllability of linear state models. Rank of R n for both discrete and continuouse Showed how to discretize continuous systems Showed examples of controllable and noncontrollable systems Next time: Open loop and state feedback control Controllers to make systems do what we want! EE6B M. Lustig, EECS UC Berkeley

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