EE6B M. Lustig, EECS UC Berkeley EE6B Designing Information Devices and Systems II Lecture 6B Cont. stability of Linear State Models Controllability
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
EE6B M. Lustig, EECS UC Berkeley
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
Back to the Pendulum EE6B M. Lustig, EECS UC Berkeley
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
Back to the Pendulum EE6B M. Lustig, EECS UC Berkeley
Back to the Pendulum x x x long l x short l EE6B M. Lustig, EECS UC Berkeley
Predicting System Behavior Discrete Time 2.5 2.5 λ=.9 λ=.4 2.5 2.5.5.5 -.5 -.5 - - -.5 -.5-2 -2-2.5 -.5.5.5 2 2.5 3 3.5 4 4.5 5-2.5 -.5.5.5 2 2.5 3 3.5 4 4.5 5 2.5 2 λ= - 2.5 2 λ=. 2.5 2 λ= -.2 2.5 2 λ= -.9.5.5.5.5.5.5.5.5 -.5 -.5 -.5 -.5 - - - - -.5 -.5 -.5 -.5-2 -2-2 -2-2.5 -.5.5.5 2 2.5 3 3.5 4 4.5 5-2.5 -.5.5.5 2 2.5 3 3.5 4 4.5 5-2.5 -.5.5.5 2 2.5 3 3.5 4 4.5 5-2.5 -.5.5.5 2 2.5 3 3.5 4 4.5 5 EE6B M. Lustig, EECS UC Berkeley
If λ is complex (Euler) EE6B M. Lustig, EECS UC Berkeley
EE6B M. Lustig, EECS UC Berkeley 4 3.8 2.6.4.2 -.2 -.4 - -.6 -.8 5 5 2 25-2 4 3 2 Im(l) -3 5 5 2 25 3.5 3 2.5 2 -.5-2 -3.5-4 5 5 2 25 2 5 5 2 25 Re(l).5.5.5.5 -.5 - -.5-2 5 5 2 25 -.5 5 5 2 25 2.9.5.8.7.6.5.4.3.5.2. 5 5 2 25 -.5 - -.5-2 5 5 2 25
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
EE6B M. Lustig, EECS UC Berkeley.8.6.4.2 -.2 -.4 -.6 -.8-2 3 4 5 6 7 8 9.8.6.4.2 -.2 -.4 -.6 -.8-2 3 4 5 6 7 8 9.5.5 -.5 - -.5 2 3 4 5 6 7 8 9 Im(l) 3 2 - -2-3 2 3 4 5 6 7 8 9 F v N a u z a 3.5.8.8 2.6.4.6.4.5.2.2 -.2 -.2 -.4 -.6 -.4 -.6 -.5 - -.8 -.8-2 3 4 5 6 7 8 9-2 3 4 5 6 7 8 9 - -2 -.5 2 3 4 5 6 7 8 9-3 2 3 4 5 6 7 8 9 2 2.8.8 2.6.6 2.4.4.2 2.2 2.9.8.9.8.8.8.7.6.7.6.6.5.4.6.4.4.5.3.2.4.2.2. 2 3 4 5 6 7 8 9.3 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 Re(l)
Example EE6B M. Lustig, EECS UC Berkeley
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
Controllability Discrete-time: From last time: EE6B M. Lustig, EECS UC Berkeley
Controllability EE6B M. Lustig, EECS UC Berkeley
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
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
Controllability What about R n+? EE6B M. Lustig, EECS UC Berkeley
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
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> 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
Continuous Time (no derivation here) The continuous-time system is controllable if and only if has rank = n EE6B M. Lustig, EECS UC Berkeley
Example 3 + Quiz Write the state model: For: EE6B M. Lustig, EECS UC Berkeley
Quiz Write the state model: For: EE6B M. Lustig, EECS UC Berkeley
Example 3 Controllability: Rank =! Not controllable EE6B M. Lustig, EECS UC Berkeley
Physical explanation Why can t I drive the currents x and x2 freely using U(t)? EE6B M. Lustig, EECS UC Berkeley
Given an initial condition, I can only move along the line by changing U EE6B M. Lustig, EECS UC Berkeley
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
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