Control Systems Design
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1 ELEC4410 Control Systems Design Lecture 14: Controllability Julio H. Braslavsky School of Electrical Engineering and Computer Science Lecture 14: Controllability p.1/23
2 Outline Controllability Definition and Tests Controllability and Algebraic Equivalence Controllability Gramian Controllability after Sampling Examples Lecture 14: Controllability p.2/23
3 Controllability In the next few classes we will discuss two fundamental concepts of system theory: those of controllability and observability. These two concepts describe the interaction in a system between the external world (inputs and outputs) and the internal variables (states). Lecture 14: Controllability p.3/23
4 Controllability In the next few classes we will discuss two fundamental concepts of system theory: those of controllability and observability. These two concepts describe the interaction in a system between the external world (inputs and outputs) and the internal variables (states). Loosely speaking, Lecture 14: Controllability p.3/23
5 Controllability In the next few classes we will discuss two fundamental concepts of system theory: those of controllability and observability. These two concepts describe the interaction in a system between the external world (inputs and outputs) and the internal variables (states). Loosely speaking, Controllability is the property that indicates if the behaviour of a system can be controlled by acting on its inputs. Observability is the property that indicates if the internal behaviour of a system can be detected at its outputs. Lecture 14: Controllability p.3/23
6 Controllability In the next few classes we will discuss two fundamental concepts of system theory: those of controllability and observability. These two concepts describe the interaction in a system between the external world (inputs and outputs) and the internal variables (states). Loosely speaking, Controllability is the property that indicates if the behaviour of a system can be controlled by acting on its inputs. Observability is the property that indicates if the internal behaviour of a system can be detected at its outputs. We start by considering controllability of continuous-time systems. Lecture 14: Controllability p.3/23
7 Controllability Consider the LTI system represented by the n-states, q-inputs state equation ẋ(t) = Ax(t) + Bu(t), (SE) where A R n n and B R n q. Controllability only relates inputs and states, thus the output equation y(t) = Cx(t) + Du(t) is irrelevant. Lecture 14: Controllability p.4/23
8 Controllability Consider the LTI system represented by the n-states, q-inputs state equation ẋ(t) = Ax(t) + Bu(t), (SE) where A R n n and B R n q. Controllability only relates inputs and states, thus the output equation y(t) = Cx(t) + Du(t) is irrelevant. Controllability: The state equation (SE) or the pair (A, B) is said to be controllable if for any initial state x(0) = x 0 and any final state x 1, there exists an input that transfer x 0 to x 1 in a finite time. Otherwise, (SE) or (A, B) is said to be uncontrollable. This definition requires only that the input be capable of driving the state anywhere in the state space space in a finite time; what trajectory the state takes is not specified. Lecture 14: Controllability p.4/23
9 Controllability Example (Uncontrollable systems). Consider the electric system on the left in the figure below. It is a system of first order; state variable x: voltage on the capacitor. If the capacitor has no initial charge, x(0) = 0, then x(t) = 0 for all t 0, no matter what input is applied. The input has no effect over the voltage across the capacitor. This system, or more precisely, a state equation that describes it, is not controllable. + u + 1Ω + x 1F 1Ω y 1Ω 1Ω Lecture 14: Controllability p.5/23
10 Controllability Example (Uncontrollable systems). Consider the electric system on the left in the figure below. It is a system of first order; state variable x: voltage on the capacitor. If the capacitor has no initial charge, x(0) = 0, then x(t) = 0 for all t 0, no matter what input is applied. The input has no effect over the voltage across the capacitor. This system, or more precisely, a state equation that describes it, is not controllable. The system on the right has two state variables. The input can transfer x 1 or x 2 to any desired value, but no matter what input is applied, x 1 (t) will always equal x 2 (t). This system is not controllable either. u + 1Ω + x 1F 1Ω + y u + 1F + + 1F x 1 x 2 1Ω 1Ω 1Ω 1Ω Lecture 14: Controllability p.5/23
11 Controllability Tests Theorem (Controllability Tests). equivalent. The following statements are 1. The n-dimensional pair (A, B) is controllable. 2. The Controllability Matrix [ ] C B AB A 2 B... A n 1 B has rank n (full row rank). 3. The n n matrix W c (t) = is nonsingular for all t > 0. t 0 e Aτ BB T e AT τ dτ Lecture 14: Controllability p.6/23
12 Controllability Tests Example. The linearised state space equation of an inverted pendulum system is given by θ m [ ẏ ] ÿ θ θ = [ ] [ ] y ẏ θ θ + [ ] u u M l mg y We compute the controllability matrix C = [ ] B AB A 2 B A 3 B = [ ] which has rank 4 (i.e., it is full rank) the system is controllable. If θ were slightly different from 0, we know then that there exists a control u that will return it to the equilibrium in finite time. Lecture 14: Controllability p.7/23
13 Controllability & Algebraic Equivalence Lecture 14: Controllability p.8/23
14 Controllability & Algebraic Equivalence Controllability is a system property that is invariant with respect to algebraic equivalence transformations (change of coordinates). Lecture 14: Controllability p.9/23
15 Controllability & Algebraic Equivalence Controllability is a system property that is invariant with respect to algebraic equivalence transformations (change of coordinates). Indeed, consider the pair (A, B) with controllability matrix [ ] C = B AB A n 1 B and an algebraic equivalent pair (Ā, B), where Ā = PAP 1 and B = PB, and P is a nonsingular matrix. Then the controllability matrix of the pair (Ā, B) is [ ] C = B Ā B Ā n 1 B [ ] = PB PAP 1 PB PA n 1 P 1 PB [ ] = P B AB A n 1 B = PC. Because P is nonsingular, rank C = rank C Lecture 14: Controllability p.9/23
16 Controllability Gramian Lecture 14: Controllability p.10/23
17 Controllability Gramian The matrix W c (t) introduced to check controllability of (A, B) can be used to construct an open loop control signal u(t) that will take the state x from any x 0 to any x 1 in finite time. W c (t) = t 0 e Aτ BB T e AT τ dτ Lecture 14: Controllability p.11/23
18 Controllability Gramian The matrix W c (t) introduced to check controllability of (A, B) can be used to construct an open loop control signal u(t) that will take the state x from any x 0 to any x 1 in finite time. Such a control law is given by u(t) = B T e AT (t 1 t) W 1 c (t 1 )(e At 1 x 0 x 1 ). Lecture 14: Controllability p.11/23
19 Controllability Gramian The matrix W c (t) introduced to check controllability of (A, B) can be used to construct an open loop control signal u(t) that will take the state x from any x 0 to any x 1 in finite time. Such a control law is given by u(t) = B T e AT (t 1 t) W 1 c (t 1 )(e At 1 x 0 x 1 ). This control law uses the least amount of energy to transfer x from x 0 to x 1 in time t 1. This means that for any other control ũ(t) performing the same transfer, t1 0 ũ(τ) 2 dτ t1 0 u(τ) 2 dτ. Lecture 14: Controllability p.11/23
20 Controllability Gramian The matrix W c (t) introduced to check controllability of (A, B) can be used to construct an open loop control signal u(t) that will take the state x from any x 0 to any x 1 in finite time. Such a control law is given by u(t) = B T e AT (t 1 t) W 1 c (t 1 )(e At 1 x 0 x 1 ). This control law uses the least amount of energy to transfer x from x 0 to x 1 in time t 1. This means that for any other control ũ(t) performing the same transfer, t1 0 ũ(τ) 2 dτ t1 0 u(τ) 2 dτ. For example, if x 0 = 0, the minimum control energy is t1 0 ( t1 u(τ) 2 dτ = x T 1 Wc 1 (t 1 ) 0 = x T 1 W 1 c (t 1 )x 1 = W 1 2 c (t 1 )x 1 2. ) e A(t1 τ) BB T e AT (t 1 τ) dτ Wc 1 (t 1 )( x 1 ) Lecture 14: Controllability p.11/23
21 Controllability Gramian If the matrix A is Hurwitz (all eigenvalues with negative real part), then W c (t) converges for t, and then we denote it simply by W c Wc = 0 e Aτ BB T e AT τ dτ, and we call it the Controllability Gramian of (A, B). Lecture 14: Controllability p.12/23
22 Controllability Gramian If the matrix A is Hurwitz (all eigenvalues with negative real part), then W c (t) converges for t, and then we denote it simply by W c Wc = e Aτ BB T e AT τ dτ, and we call it the Controllability Gramian of (A, B). 0 If we desire to drive the state x from 0 to x 1 in infinite time, t 1, we find that the least required control energy would be 0 u(τ) 2 dτ = W 1 2 c x 1 2. Note that the closer to zero any eigenvalue of W c is, the closer to singular would W c be, and the larger would be the minimum energy required to drive the state to x 1. Lecture 14: Controllability p.12/23
23 Controllability Gramian Note that we do not need to slove an infinite integral to compute W c. If (A, B) is controllable (C has full row rank), W c is the unique solution of the linear Lyapunov matrix equation AW c + W c A T = BB T, which can be solved with MATLAB with Wc = lyap(a,b*b ), or by using the function Wc = gram(sys, c ). Lecture 14: Controllability p.13/23
24 Controllability after Sampling Lecture 14: Controllability p.14/23
25 Controllability after Sampling Most control systems are implemented digitally, for which we need a discrete-time model of the system. u[k] {A d, B d, C, D} y[k] ZOH {A, B, C, D} T T Lecture 14: Controllability p.15/23
26 Controllability after Sampling Most control systems are implemented digitally, for which we need a discrete-time model of the system. u[k] {A d, B d, C, D} y[k] ZOH {A, B, C, D} T T Using the Parameter Variation Formula, we have seen how to obtain a discrete-time state space model that is exact at the sampling instants. Lecture 14: Controllability p.15/23
27 Controllability after Sampling Most control systems are implemented digitally, for which we need a discrete-time model of the system. u[k] {A d, B d, C, D} y[k] ZOH {A, B, C, D} T T Using the Parameter Variation Formula, we have seen how to obtain a discrete-time state space model that is exact at the sampling instants. ẋ(t) = A x(t) + B u(t) y(t) = C x(t) + D u(t) x[k + 1] = A d x[k] + B d u[k] y[k] = C x[k] + D u[k] where A d = e AT and B d = T 0 eaτ B dτ. Lecture 14: Controllability p.15/23
28 Controllability after Sampling Most control systems are implemented digitally, for which we need a discrete-time model of the system. u[k] {A d, B d, C, D} y[k] ZOH {A, B, C, D} T T Using the Parameter Variation Formula, we have seen how to obtain a discrete-time state space model that is exact at the sampling instants. ẋ(t) = A x(t) + B u(t) y(t) = C x(t) + D u(t) x[k + 1] = A d x[k] + B d u[k] y[k] = C x[k] + D u[k] where A d = e AT and B d = T 0 eaτ B dτ. If the continuous-time system is controllable, would the discretised system be always controllable? Lecture 14: Controllability p.15/23
29 Controllability after Sampling The controllability of the discretised system depends on the sampling period T and the eigenvalues of the continuous-time plant. Controllability can be lost after sampling. Lecture 14: Controllability p.16/23
30 Controllability after Sampling The controllability of the discretised system depends on the sampling period T and the eigenvalues of the continuous-time plant. Controllability can be lost after sampling. Theorem (Nonpathological Sampling). If the pair [A, B] is controllable, then the discretised pair [A d, B d ] is controllable with sampling time T if for any two eigenvalues λ i, λ j of A such that Re[λ i λ j ] = 0, the nonpathological sampling condition is satisfied. Im[λ i λ j ] 2πm, for m = 1, 2,... T Lecture 14: Controllability p.16/23
31 Controllability after Sampling The controllability of the discretised system depends on the sampling period T and the eigenvalues of the continuous-time plant. Controllability can be lost after sampling. Theorem (Nonpathological Sampling). If the pair [A, B] is controllable, then the discretised pair [A d, B d ] is controllable with sampling time T if for any two eigenvalues λ i, λ j of A such that Re[λ i λ j ] = 0, the nonpathological sampling condition is satisfied. Im[λ i λ j ] 2πm, for m = 1, 2,... T The Theorem gives a sufficient condition that preserves controllability after sampling. This condition is also necessary for systems with a single input. Lecture 14: Controllability p.16/23
32 Controllability after Sampling Example (Pathological Sampling). Consider the controllable continuous-time system ẋ(t) = 0 1 x(t) + 0 u(t) Lecture 14: Controllability p.17/23
33 Controllability after Sampling Example (Pathological Sampling). Consider the controllable continuous-time system ẋ(t) = 0 1 x(t) + 0 u(t) Its exact discretisation with sampling period T is x[k + 1] = cos T sin T x[k] + 1 cos T u[k]. sin T cos T sin T Lecture 14: Controllability p.17/23
34 Controllability after Sampling Example (Pathological Sampling). Consider the controllable continuous-time system ẋ(t) = 0 1 x(t) + 0 u(t) Its exact discretisation with sampling period T is x[k + 1] = cos T sin T x[k] + 1 cos T u[k]. sin T cos T sin T Note that if T = mπ, with m = 1, 2,..., this system becomes uncontrollable, i.e., x[k + 1] = ( 1)m 0 x[k] + 1 ( 1)m u[k]. 0 ( 1) m 0 Lecture 14: Controllability p.17/23
35 Controllability after Sampling A couple of final remarks concerning controllability and sampling: The nonpathological sampling condition only applies to systems with complex eigenvalues; a discretised system with only real eigenvalues is controllable for all T > 0 if its continuous-time counterpart is. Lecture 14: Controllability p.18/23
36 Controllability after Sampling A couple of final remarks concerning controllability and sampling: The nonpathological sampling condition only applies to systems with complex eigenvalues; a discretised system with only real eigenvalues is controllable for all T > 0 if its continuous-time counterpart is. The nonpathological sampling condition is only sufficient for a MIMO system; if sampling is pathological, controllability may be lost after sampling. Lecture 14: Controllability p.18/23
37 Controllability Examples Lecture 14: Controllability p.19/23
38 Controllability Examples Example. x 1 u y x 2 In the hydraulic system on the left it is obvious that the input cannot affect the level x 2, so it is intuitively evident that the 2-tank system is not controllable. Lecture 14: Controllability p.20/23
39 Controllability Examples Example. x 1 u y x 2 In the hydraulic system on the left it is obvious that the input cannot affect the level x 2, so it is intuitively evident that the 2-tank system is not controllable. A linearised model of this system with unitary parameters gives ẋ(t) = [ ] x(t) + [ 10 ] u(t) y(t) = [ 1 0 ] x(t) Lecture 14: Controllability p.20/23
40 Controllability Examples Example. x 1 u y x 2 In the hydraulic system on the left it is obvious that the input cannot affect the level x 2, so it is intuitively evident that the 2-tank system is not controllable. A linearised model of this system with unitary parameters gives ẋ(t) = [ ] x(t) + [ 10 ] u(t) y(t) = [ 1 0 ] x(t) The controllability matrix is C = [ B AB ] = [ which is not full rank, so the system is not controllable. ] Lecture 14: Controllability p.20/23
41 Controllability Examples Example. u x x x y The controllability of the hydraulic system on the left is not so obvious, although we can see that x 1 (t) and x 3 (t) cannot be affected independently by u(t). Lecture 14: Controllability p.21/23
42 Controllability Examples Example. u x x x y The controllability of the hydraulic system on the left is not so obvious, although we can see that x 1 (t) and x 3 (t) cannot be affected independently by u(t). The linearised model in this case is ] ẋ(t) = x(t) + [ y(t) = [ ] x(t) [ 01 0 ] u(t) Lecture 14: Controllability p.21/23
43 Controllability Examples Example. u x x x y The controllability of the hydraulic system on the left is not so obvious, although we can see that x 1 (t) and x 3 (t) cannot be affected independently by u(t). The linearised model in this case is ] ẋ(t) = x(t) + [ y(t) = [ ] x(t) [ 01 0 ] u(t) The controllability matrix is C = [ B AB A 2 B ] = [ ] which has rank 2, showing that the system is not controllable. Lecture 14: Controllability p.21/23
44 Controllability Examples Example. x 1 u x 2 x 3 y Now in the previous system suppose that the input is applied in the first tank, as shown in the figure. In this case the linearised model is the same as before, except that the matrix B is now different Lecture 14: Controllability p.22/23
45 Controllability Examples Example. x 1 u Now in the previous system suppose that the input is applied in the first tank, as shown in the figure. In this case the linearised model is the same as before, except that the y matrix B is now different [ ] [ 10 ] ẋ(t) = x(t) + u(t) y(t) = [ ] x(t) x 2 x 3 Lecture 14: Controllability p.22/23
46 Controllability Examples Example. x 1 u Now in the previous system suppose that the input is applied in the first tank, as shown in the figure. In this case the linearised model is the same as before, except that the y matrix B is now different [ ] [ 10 ] ẋ(t) = x(t) + u(t) y(t) = [ ] x(t) x 2 x 3 The controllability matrix is now C = [ B AB A 2 B ] = [ ] which has rank 3, showing that the system is controllable. Lecture 14: Controllability p.22/23
47 Summary Controllability is a fundamental property of a system which determines if it is possible to drive its state x to any desired position by an appropriate choice of the input signal u(t). Lecture 14: Controllability p.23/23
48 Summary Controllability is a fundamental property of a system which determines if it is possible to drive its state x to any desired position by an appropriate choice of the input signal u(t). If a system is controllable, we know that there exists an input capable of transferring the state to any desired position in an arbitrary, nonzero, finite time. Lecture 14: Controllability p.23/23
49 Summary Controllability is a fundamental property of a system which determines if it is possible to drive its state x to any desired position by an appropriate choice of the input signal u(t). If a system is controllable, we know that there exists an input capable of transferring the state to any desired position in an arbitrary, nonzero, finite time. Controllability only depends on the matrices A and B of the system. The pair (A, B) is controllable if and only if [ ] rank C = rank B AB A n 1 B = n Lecture 14: Controllability p.23/23
50 Summary Controllability is a fundamental property of a system which determines if it is possible to drive its state x to any desired position by an appropriate choice of the input signal u(t). If a system is controllable, we know that there exists an input capable of transferring the state to any desired position in an arbitrary, nonzero, finite time. Controllability only depends on the matrices A and B of the system. The pair (A, B) is controllable if and only if [ ] rank C = rank B AB A n 1 B = n Controllability is independent of the choice of coordinates. Lecture 14: Controllability p.23/23
51 Summary Controllability is a fundamental property of a system which determines if it is possible to drive its state x to any desired position by an appropriate choice of the input signal u(t). If a system is controllable, we know that there exists an input capable of transferring the state to any desired position in an arbitrary, nonzero, finite time. Controllability only depends on the matrices A and B of the system. The pair (A, B) is controllable if and only if [ ] rank C = rank B AB A n 1 B = n Controllability is independent of the choice of coordinates. Controllability may be lost after sampling if sampling is pathological. Lecture 14: Controllability p.23/23
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