Multivariable Control Systems Ali Karimpour Assistant Professor Ferdowsi University of Mashhad <<<1.1>>> ###Control System Design### {{{Control, Design}}}

Topics to be covered include: Chapter 2 Introduction to Multivariable Control Topics to be covered include: Multivariable Connections Multivariable Poles Multivariable Zeros Directions of Poles and Zeros Smith Form of a Polynomial Matrix Smith-McMillan Forms Matrix Fraction Description (MFD) Scaling Performance Specification Trade-offs in Frequency Domain Bandwidth and Crossover Frequency

Multivariable Connections Cascade (series) interconnection of transfer matrices Generally Parallel interconnection of transfer matrices

Multivariable Connections Feedback interconnection of transfer matrices A useful relation in multivariable is push-through rule Exercise 1: Proof the push-through rule

Multivariable Connections MIMO rule: To derive the output of a system, start from the output and write down the blocks as you meet them when moving backward (against the signal flow) towards the input. If you exit from a feedback loop then include a term or according to the feedback sign where L is the transfer function around that loop (evaluated against the signal flow starting at the point of exit from the loop). Parallel branches should be treated independently and their contributions added together.

Multivariable Connections Example 2-1: Derive the transfer function of the system shown in figure

Multivariable Poles ( State Space Description ) Definition 2-1: The poles of a system with state-space description are eigenvalues of the matrix A. The pole polynomial or characteristic polynomial is defined as Thus the system’s poles are the roots of the characteristic polynomial Note that if A does not correspond to a minimal realization then the poles by this definition will include the poles (eigenvalues) corresponding to uncontrollable and/or unobservable states.

Multivariable Poles ( Transfer Function Description ) Theorem 2-1 The pole polynomial corresponding to a minimal realization of a system with transfer function G(s) is the least common denominator of all non-identically-zero minors of all orders of G(s). A minor of a matrix is the determinant of the square matrix obtained by deleting certain rows and/or columns of the matrix.

Multivariable Poles ( Transfer Function Description ) Example 2-4 The non-identically-zero minors of order 1 are The non-identically-zero minor of order 2 are By considering all minors we find their least common denominator

Multivariable Poles Exercise 2: Consider the state space realization a- Find the poles of the system directly through state space form. b- Find the transfer function of system ( note that the numerator and the denominator of each element must be prime ). c- Find the poles of the system through its transfer function.

Multivariable Zeros ( State Space Description ) Laplace transform The zeros are then the values of s=z for which the polynomial system matrix P(s) loses rank resulting in zero output for some nonzero input. Let Then the zeros are found as non trivial solutions of This is solved as a generalized eigenvalue problem.

Multivariable Zeros ( State Space Description ) For square systems with m=p inputs and outputs and n states, limits on the number of transmission zeros are:

Multivariable Zeros ( State Space Description ) Example 2-5: Consider the state space realization

Multivariable Zeros ( Transfer Function Description ) Definition 2-2 is a zero of G(s) if the rank of is less than the normal rank of G(s). The zero polynomial is defined as . Where is the number of finite zeros of G(s).

Multivariable Zeros ( Transfer Function Description ) Theorem 2-2 The zero polynomial z(s) corresponding to a minimal realization of the system is the greatest common divisor of all the numerators of all order-r minors of G(s) where r is the normal rank of provided that these minors have been adjusted in such a way as to have the pole polynomial as their denominators.

Multivariable Zeros ( Transfer Function Description ) Example 2-9 according to example 2-4 the pole polynomial is: The minors of order 2 with as their denominators are The greatest common divisor of all the numerators of all order-2 minors is

Multivariable Poles Exercise 3: Consider the state space realization a- Find the zeros of the system directly through state space form. b- Find the transfer function of system ( note that the numerator and the denominator of each element must be prime ). c- Find the zeros of the system through its transfer function.

Directions of Poles and Zeros Zero directions: Let G(s) have a zero at s = z, Then G(s) losses rank at s = z and there will exist nonzero vectors and such that is input zero direction and is output zero direction We usually normalize the direction vectors to have unit length

Directions of Poles and Zeros Pole directions: Let G(s) have a pole at s = p. Then G(p) is infinite and we may somewhat crudely write is input pole direction and is output pole direction

Directions of Poles and Zeros Example: It has a zero at z = 4 and a pole at p = -2 . Output zero direction Input zero direction

Directions of Poles and Zeros Example: It has a zero at z = 4 and a pole at p = -2 . ??!! Output pole direction Input pole direction

Smith Form of a Polynomial Matrix Suppose that is a polynomial matrix. Smith form of is denoted by , and it is a pseudo diagonal in the following form is a factor of and

Smith Form of a Polynomial Matrix is a factor of and gcd {all monic minors of degree 1} gcd {all monic minors of degree 2} ……………………………………………………….. ……………………………………………………….. gcd {all monic minors of degree r}

Smith Form of a Polynomial Matrix The three elementary operations for a polynomial matrix are used to find Smith form. Multiplying a row or column by a constant; Interchanging two rows or two columns; and Adding a polynomial multiple of a row or column to another row or column.

Smith Form of a Polynomial Matrix Example 2-11 Exercise 4: Derive R(s) and L(s) that convert to

Smith-McMillan Forms Theorem 2-3 (Smith-McMillan form) Let be an m × p matrix transfer function, where are rational scalar transfer functions, G(s) can be represented by: Where Π(s) is an m× p polynomial matrix of rank r and DG(s) is the least common multiple of the denominators of all elements of G(s) . Then, is Smith McMillan form of G(s) and can be derived directly by

Smith-McMillan Forms Theorem 2-3 (Smith-McMillan form)

Smith-McMillan Forms Example 2-13

Matrix Fraction Description (MFD) polynomial polynomial This is a Right Matrix Fraction Description (RMFD) polynomial polynomial This is a Left Matrix Fraction Description (LMFD)

Matrix Fraction Description (MFD) A model structure that is related to the Smith-McMillan form is matrix fraction description (MFD). Right matrix fraction description (RMFD) Left matrix fraction description (LMFD) Let is a matrix and its the Smith McMillan is Let define:

Matrix Fraction Description (MFD) We know that This is known as a right matrix fraction description (RMFD) where

Matrix Fraction Description (MFD) We know that This is known as a left matrix fraction description (LMFD) where

Matrix Fraction Description (MFD) We can show that the MFD is not unique, because, for any nonsingular matrix we can write G(s) as: Where is said to be a right common factor. When the only right common factors of and is unimodular matrix, then, we Say that the RMFD is irreducible. It is easy to see that when a RMFD is irreducible, then

Matrix Fraction Description (MFD) Example 2-16 RMFD:

Matrix Fraction Description (MFD) Example 2-16 LMFD:

Scaling

Performance Specification Nominal stability NS: The system is stable with no model uncertainty. Nominal Performance NP: The system satisfies the performance specifications with no model uncertainty. Robust stability RS: The system is stable for all perturbed plants about the nominal model up to the worst case model uncertainty. Robust performance RP: The system satisfies the performance specifications for all perturbed plants about the nominal model up to the worst case model uncertainty.

Performance Specification Time Domain Performance Frequency Domain Performance

Time Domain Performance Although closed loop stability is an important issue, the real objective of control is to improve performance, that is, to make the output y(t) behave in a more desirable manner. Actually, the possibility of inducing instability is one of the disadvantages of feedback control which has to be traded off against performance improvement. The objective of this section is to discuss ways of evaluating closed loop performance.

Time Domain Performance Step response of a system Rise time, tr , the time it takes for the output to first reach 90% of its final value, which is usually required to be small. Settling time, ts , the time after which the output remains within 5% ( or 2%) of its final value, which is usually required to be small. Overshoot, P.O, the peak value divided by the final value, which should typically be less than 20% or less.

Time Domain Performance Step response of a system Decay ratio, the ratio of the second and first peaks, which should typically be 0.3 or less. Steady state offset, ess, the difference between the final value and the desired final value, which is usually required to be small.

Time Domain Performance Step response of a system Excess variation, the total variation (TV) divided by the overall change at steady state, which should be as close to 1 as possible. The total variation is the total movement of the output as shown.

Time Domain Performance ISE : Integral squared error IAE : Integral absolute error ITSE : Integral time weighted squared error ITAE : Integral time weighted absolute error

Frequency Domain Performance Let L(s) denote the loop transfer function of a system which is closed-loop stable under negative feedback. Bode plot of

Frequency Domain Performance Let L(s) denote the loop transfer function of a system which is closed-loop stable under negative feedback. Nyquist plot of

Frequency Domain Performance Stability margins are measures of how close a stable closed-loop system is to instability. From the above arguments we see that the GM and PM provide stability margins for gain and delay uncertainty. More generally, to maintain closed-loop stability, the Nyquist stability condition tells us that the number of encirclements of the critical point -1 by must not change. Thus the actual closest distance to -1 is a measure of stability

Complementary sensitivity Frequency Domain Performance One degree-of-freedom configuration Complementary sensitivity function Sensitivity function The maximum peaks of the sensitivity and complementary sensitivity functions are defined as

Frequency Domain Performance Thus one may also view Ms as a robustness measure.

Frequency Domain Performance There is a close relationship between MS and MT and the GM and PM. For example, with MS = 2 we are guaranteed GM > 2 and PM >29˚. For example, with MT = 2 we are guaranteed GM > 1.5 and PM >29˚.

Complementary sensitivity Trade-offs in Frequency Domain One degree-of-freedom configuration Complementary sensitivity function Sensitivity function

Trade-offs in Frequency Domain Performance, good disturbance rejection Performance, good command following Mitigation of measurement noise on output Small magnitude of input signals Physical controller must be strictly proper Nominal stability (stable plant) Stabilization of unstable plant

Bandwidth and Crossover Frequency Definition 2-3 Definition 2-4 Definition 2-5

Bandwidth and Crossover Frequency Specifically, for systems with PM < 90˚ (most practical systems) we have In conclusion ωB ( which is defined in terms of S ) and also ωc ( in terms of L ) are good indicators of closed-loop performance, while ωBT ( in terms of T ) may be misleading in some cases.

Bandwidth and Crossover Frequency Example 2-17 : Comparison of and as indicators of performance.

Bandwidth and Crossover Frequency Example 2-17 : Comparison of and as indicators of performance.