Lecture abstract. EE C128 / ME C134 Feedback Control Systems. Chapter outline. Intro, [1, p. 664] Lecture Chapter 12 Design via State Space

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1 EE C18 / ME C134 Feedback Control Systems Lecture Chapter 1 Design via State Space Lecture abstract Alexandre Bayen Department of Electrical Engineering & Computer Science University of California Berkeley Topics covered in this presentation Controller design Controllability Observer design Observability September 10, 013 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, 013 / 58 Chapter outline 1 Design via state space 1.1 ntro Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 ntro, [1, p. 664] 1 Design via state space 1.1 ntro 1 Design via state space Design via state space Can be applied to a wider class of systems than transform methods Systems with nonlinearities Multiple-input, multiple output (MMO) systems Systems of higher order than We will focus on the application to linear systems Specify all CL poles Parameters for each CL pole Technique for finding these parameter values Cannot specify CL zero locations Sensitive to parameter changes Wide range of computational support Loss of graphic insight into a design problem Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58

2 1 Design via state space State-variable FB control, [1, p. 665] 1 Design via state space State-variable FB control, [1, p. 666] n th -order CL characteristic equation (CE) There are n coe det(s A) =s n + a n 1 s n a 1 s + a 0 =0 cients whose values determine the n CL poles ntroduce n adjustable parameters into the system and relate them to the n coe cients, so that we can place the n CL poles Before, output-variable FB, now, state-variable FB Each state variable is fed back to the control, u, through a gain, k i State-variable FB gain: K CL system is plant with state-variable FB ẋ = Ax + Bu = Ax + B( Kx + r) =(A BK)x + Br y = Cx Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 1 Design via state space State-variable FB control, [1, p. 666] 1 Design via state space State-variable FB control, [1, p. 666] Figure: Plant with state-variable FB Figure: State-space representation of a plant Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Methodology, [1, p. 668] 1 Design via state space 1 Design via state space State-variable FB control in PV form, [1, p. 668] Pole placement for plants in phase-variable (PV) form 1. Represent the plant in PV form. FB each PV to the input of the plant through a gain, k i 3. Find the CE for the CL system 4. Decide upon all CL pole locations and determine equivalent CE 5. Equate like coe cients of the CE and solve for k i Plant Plant CE A = ; B = ; a 0 a 1 a... a n 1 1 C = c 1 c... c n det(s A) = s n + a n 1 s n a 1 s + a 0 =0 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58

3 1 Design via state space State-variable FB control in PV form, [1, p. 668] State-variable FB CL system u = Kx; K = k 1 k... k n A BK = (a 0 + k 1 ) (a 1 + k ) (a + k 3 )... (a n 1 + K n ) CL system CE det(s (A BK)) = s n +(a n 1 + k n )s n (a 1 + k )s +(a 0 + k 1 )=0 1 Design via state space State-variable FB control in PV form, [1, p. 669] Desired CL system CE det(s (A BK)) = s n + d n 1 s n d 1 s + d 0 =0 d i = a i + k i+1 ; i =0, 1,,...,n 1 CL system TF Denominator polynomial: the CE Numerator polynomial: formed from the coe cients of the output coupling matrix, C, forsystemswritteninpvform Same for plant and CL system Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Example, [1, p. 669] 1 Design via state space 1 Design via state space Example (Controller design for PV form) Problem: Design the PV FB gains to yield %OS =9.5% Ts =0.74 seconds Solution: On the board Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Definitions, [1, p. 67] 1 Design via state space 1 Design via state space Controllability by inspection, [1, p. 673] Controllability: f an input to a system can be found that takes every state variable from a desired initial state to a desired final state, the system is said to be controllable; otherwise the system is uncontrollable. Control variable, u, can be used to control the behavior of each state variable in x Poles of the control system can be placed where we desire Determine, a priori, whether pole placement is a viable design technique for a controller When the system matrix, A, is in diagonal or parallel form, it is apparent whether or not the system is controllable A system with distinct (no repeat) eigenvalues and a diagonal system matrix, A, is controllable if the input coupling matrix, B, does not have any rows that are zero Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58

4 1 Design via state space The controllability matrix, [1, p. 674] Example, [1, p. 675] 1 Design via state space n other forms, the existence of paths from the input to the state variables is not a criterion for controllability since the equations are not decoupled n th -order plant ẋ = Ax + Bu is completely controllable if the matrix C M = B AB... A n 1 B is of rank n, wherec M is called the controllability matrix Example (Controllability via the controllability matrix) Problem: Determine if the system is controllable A = ; B = Solution: On the board Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 1 Design via state space 1 Design via state space Alternative approaches to controller design, [1, p. 676] For systems not represented in PV form approaches Matching coe cients: Matching the coe cients of det(s (A BK)) with coe cients of the desired CE Same method used for systems in PV representation Transformation: Transforming the system to PV form, designing the control FB gain, & transforming the designed system back to its original state-variable representation Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, 013 / 58 1 Design via state space Approach matching coe cients, [1, p. 677] Example, [1, p. 677] 1 Design via state space Matching the coe with coe cients of cients of the desired CE det(s (A BK)) Leads to di cult calculations of the control gains, especially for higher-order systems not represented with PVs Example (Controller design by matching coe cients) Problem: Design state-variable control FB gain for the plant to yield %OS = 15% Ts =0.5 second Solution: On the board G(s) = 10 (s + 1)(s + ) Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58

5 1 Design via state space Approach transformation, [1, p. 678] 1 Design via state space Approach transformation, [1, p. 678]. Design the state-variable control FB gain 3. Transform the system in PV representation back to the original representation Plant not in PV representation ż = Az + Bu y = Cz with controllability matrix C Mz = B AB A B... A n 1 B Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 1 Design via state space Approach transformation, [1, p. 678] 1 Design via state space Approach transformation, [1, p. 678] Assume the system can be transformed into the PV representation with the transformation z = Px Transformed plant. Design the control FB nclude both state-variable control FB and input u = K x x + r with controllability matrix ẋ = P 1 AP x + P 1 Bu y = CPx Transformed plant with state-variable control FB ẋ =(P 1 AP P 1 BK x )x + P 1 Br y = CPx C Mx = P 1 C Mz Zeros of this CL system are determined from the polynomial formed from the elements of CP Solving for P P = C MzC 1 Mx Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 1 Design via state space Approach transformation, [1, p. 679] Example, [1, p. 679] 1 Design via state space 3. Transform the system in PV representation back to the original representation Plant not in PV representation with state-variable control FB ż =(A BK x P 1 )z + Br y = Cz State-variable control FB gain K z = K x P 1 Example (Controller design by transformation) Problem: Design state-variable control FB gain for the plant to yield %OS = 0.8% Ts =4seconds Solution: On the board G(s) = s +4 (s + 1)(s + )(s + 5) Zeros of the CL TF are the same as the zeros of the uncompensated plant Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58

6 1 Design via state space 1 Design via state space Observer motivation, [1, p. 68] Controller design relies upon access to the state variables for FB through adjustable gains Estimate states can be fed to the controller Observer: Estimator used to calculate state variables that are not accessible from the plant Plant Observer ẋ = Ax + Bu y = Cx ˆx = Aˆx + Bu ŷ = C ˆx Observer error e x =ẋ ˆx = A(x ˆx) y ŷ = C(x ˆx) = Ce x Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 1 Design via state space Observer motivation, [1, p. 683] 1 Design via state space Observer motivation, [1, p. 683] Figure: State-FB design using an OL observer Figure: State-FB design using a CL observer Dynamics of the di erence between the actual & estimated states is unforced, & if the plant is stable, this di erence, due to the di erences in initial state vectors,! 0 Speed of convergence between the actual state & the estimated state is the same as the TR of the plant since the CE of the observer error is the same as the plant Convergence is too slow,! speed up the observer and make its response time much faster than that of the controlled CL system,! the controller will receive the estimated states instantaneously Error between the outputs of the plant and the observer is fed back to the derivatives of the observer s states The system corrects to drive this error! 0 Design a desired TR into the observer that is much quicker than that of the plant or controlled CL system Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 1 Design via state space Observer motivation, [1, p. 683] 1 Design via state space Observer motivation, [1, p. 683] Observer canonical form yields an easy solution for the observer FB gain Observer FB gain, L: Ensures the TR of the observer is faster than the response of the controlled loop in order to yield a rapidly updated estimate of the state vector Figure: State-FB design using a CL observer exploded view showing FB arrangement to reduce state-variable estimation error Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58

7 1 Design via state space Design methodology, [1, p. 684] 1 Design via state space 1. Find error system, i.e., state equations for error between actual state vector & estimated state vector, x ˆx. Find CE for error system ˆx = Aˆx + Bu + L(y ŷ) ŷ = C ˆx 3. Evaluate required observer FB gain, L, to meet rapid TR for observer 4. Select eigenvalues of observer to yield stability & desired TR that is faster than controlled CL response Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Definitions, [1, p. 690] 1 Design via state space 1 Design via state space Observability by inspection, [1, p. 690] Observability: f the initial-state vector, x(t 0 ), can be found from inputs, u(t), andmeasurements, y(t), over a finite interval of time from t 0, the system is said to be observable; otherwise the system is said to be unobservable. Knowledge of measured output variables, y, and control inputs, u(t), can be used to observe the behavior of each state variable in x Poles of the observer system can be placed where we desire Determine, a priori, whether pole placement is a viable design technique for an observer When the system matrix, A, is in diagonal or parallel form, it is apparent whether or not the system is observable A system with distinct (no repeat) eigenvalues and a diagonal system matrix, A, is observable if the output coupling matrix, C, does not have any columns that are zero Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 1 Design via state space The observability matrix, [1, p. 691] Example, [1, p. 691] 1 Design via state space n other forms, the existence of paths from the output to the state variables is not a criterion for observability since the equations are not decoupled n th -order plant ẋ = Ax + Bu y = Cx is completely observable if the matrix C 3 CA O M = CA n 1 Example (Observability via the observability matrix) Problem: Determine if the system is observable A = ; B = 405 ; C = Solution: On the board is of rank n, whereo M is called the observability matrix Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58

8 Example, [1, p. 69] 1 Design via state space Example (Unobservability via the observability matrix) Problem: Determine if the system is observable apple 0 1 A = Solution: On the board apple 0 ; B = 1 ; C = 5 4 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Alternative approaches to controller design, [1, p. 676] Approach - matching coe cients, [1, p. 693] For systems not represented in observer canonical form approaches Matching coe cients: Matching the coe cients of det(s (A LC)) with coe cients of the desired CE Same method used for systems in PV representation Transformation: Transforming the system to observer canonical form, designing the observer FB gain, & transforming the designed system back to its original state-variable representation Matching the coe with coe cients of cients of the desired CE det(s (A LC)) Leads to di cult calculations of the observer FB gain, especially for higher-order systems not in PV representation Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Example, [1, p. 698] Approach transformation, [1, p. 695] Example (Observer design by matching coe cients) Problem: Design an observer FB gain for the system in PV representation with a TR described by =0.7!n = (s ) G(s) = (s +1.7)(s +.69) Solution: On the board. Design the observer FB gain 3. Transform the system in PV representation back to the original representation Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58

9 Approach transformation, [1, p. 695] Approach transformation, [1, p. 695] Plant not in PV representation with observability matrix ż = Az + Bu y = Cz O Mz = 6 4 C CA. CA n Assume the system can be transformed into the PV, x, representation with the transformation z = Px Transformed plant with observability matrix ẋ = P 1 AP x + P 1 Bu y = CPx O Mx = O Mz P Solving for P P = O 1 Mz O Mx Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Approach transformation, [1, p. 695] Approach transformation, [1, p. 695]. Design the observer FB Transformed plant with FB gains ė x =(P 1 AP L x CP)e x y ŷ = CPe x 3. Transform the system in PV representation back to the original representation Plant not in PV representation with observer FB gain y ė z =(A ŷ = Ce z PL x C)e z Observer FB gain L z = PL x Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Example, [1, p. 695] 1 Design via state space Example (Observer design by transformation) Problem: Design an observer in cascade form Solution: On the board G(s) = 1 (s + 1)(s + )(s + 5) Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58

10 s, [1, p. 700] 1 Design via state space Design systems in state-space representation for steady-state error Error is fed forward to the controlled plant via an integrator Additional state variable Plant Control FB u = ẋ N = r Cx ẋ = Ax + Bu ẋ N = y = Cx Cx + r Kx + K e x N = K K e apple x x N s, [1, p. 701] 1 Design via state space Error is fed forward to the controlled plant via an integrator Augmented representation apple apple ẋ A BK BKe = ẋ N C 0 y = C 0 apple x x N apple apple x B + x N 0 K and K e can be selected to yield the desired TR We have an additional pole to place Keep an eye on the CL zeros and their e ect on TR u + apple 0 1 r Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Example, [1, p. 701] 1 Design via state space Bibliography 1 Design via state space Example (Design of integral control) Problem: a. Design a controller without integral control to yield %OS =10% T s =0.5 second Evaluate the steady-state error for a unit step b. Repeat the design using integral control. Evaluate the steady-state error for a unit step input. Solution: On the board Norman S. Nise. Control Systems Engineering, 011. Bayen (EECS, UCB) Feedback Control Systems September 10, / 58 Bayen (EECS, UCB) Feedback Control Systems September 10, / 58

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